K-Theory, D-Branes and Ramond-Ramond Fields

Heriot-W a tt University K-Theory , D-Branes and Ramond-Ramond Fields Alessandro V alen tino Submitted f or the degree of Doctor of Philosophy in Ma thema tics on completion of research in the Dep ar tment of Ma thema tics, School of Ma thema tical and Computing Sciences. This cop y of the thesis has b een supplied on the condition that an y one who consults it is understo o d to recognise that the copyrigh t rests with the author and that no quotation from the thesis and no information deriv ed from it ma y b e published without the written consen t of the author or the Univ ersit y (as ma y b e appropriate). Abstract This thesis is dedicated to the study of K-theoretical prop erties of D-branes and Ramond-Ramond ﬁelds. W e construct ab elian groups whic h deﬁne a homology theory on the category of CW-complexes, and pro ve that this homology theory is equiv alen t to the b ordism represen tation of KO-homology , the dual theory to K O-theory . W e construct an iso- morphism b et w een our geometric representation and the analytic representation of K O-homology , whic h induces a natural equiv alence of homology functors. W e apply this framew ork to describe mathematical properties of D-branes in type I String the- ory . W e in vestigate the gauge theory of Ramond-Ramond ﬁelds arising from t yp e I I String theory deﬁned on global orbifolds. W e use the machinery of Bredon cohomology and the equiv arian t Chern character to construct ab elian groups which generalize the prop erties of diﬀeren tial K-theory deﬁned b y Hopkins and Singer to the equiv ariant setting, and can b e considered as a diﬀeren tial extension of equiv ariant K-theory for ﬁnite groups. W e show that the Dirac quantization condition for Ramond-Ramond ﬁeldstrengths on a go o d orbifold is dictated b y equiv arian t K-theory and the equiv- arian t Chern character, and study the group of ﬂat Ramond-Ramond ﬁelds in the particular case of linear orbifolds in terms of our orbifold diﬀeren tial K-theory . Ac knowlegdemen ts I wan t to thank my sup ervisor Prof. Ric hard Szab o for the supp ort and patience during these y ears, for the many discussions w e had, and for allo wing me to alw a ys express and dev elop m y o wn p oin t of view in an y asp ect of m y researc h. I wan t to thank m y colleague and friend Rui Reis for the uncountable discussions on maths and science in general, whic h made me better appreciate man y aspects of these sub jects. A particular thanks go es to my examiners Jacek Bro dzki, Jos´ e Figueroa-O’F arrill, and Des Johnston, whose questions and comments help ed improv e the qualit y of the presen t w ork. I w an t to thank U.Bunke, J.Figueroa-O’F arrill, D.F reed, J.Greenlees, J.Howie, A. Konec hn y , W.L ¨ uc k, T.Sc hic k, P .T urner, and S.Willerton for helpful suggestions and corresp ondence. I w an t to thank Dr. Mark La wson for the discussions on maths, physics, and British culture. I w an t to thank Mic hele Ciraﬁci and Mauro Riccardi for the man y con versations we had in fron t of a pin t of ale. I w an t to thank F edele Lizzi for the supp ort during diﬃcult momen ts, and P atrizia Vitale for the long time in terest in m y scien tiﬁc dev elopmen ts. I w an t to thank Giorgos, Henry , and Y orgos for b eing suc h great mates, and for the man y hours of fun w e had. A thank y ou go es to my oﬃcemates Emma, Kenny , Sally , and Singy ee, for b eing the b est oﬃcemates I could ha v e ev er hop ed for. I wan t to thank Christine, Claire and Pat for helping with bureo cratic matters of any sort. A special thanks go es to m y paren ts, for letting me alwa ys pick any decision by my o wn, whic h highly con tributed to the p erson I am no w. Last, but in no wa y least, I w an t to thank Antonella. Unfortunately , w ords are useless to describ e my deep gratitude, and her relev ance to the very existence of this w ork. They say that “behind every great man there is alw ays a great woman”: m y case trivially suggests that a feminine “version” of the previous statement could not p ossibly hold. The presen t w ork of thesis is based on the follo wing articles - R.J.Szab o and A.V alentino, “Ramond-Ramond Fields, F ractional Branes and Orbifold Diﬀeren tial K-theory”, - R.M.G.Reis, R.J.Szab o, and A.V alentino, “KO-Homology and Type I String Theory”, arXiv:hep-th/0610177 Con ten ts In tro duction 1 1 Generalities on String Theory 8 1.1 The Bosonic String . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.2 The Sup ersymmetric String . . . . . . . . . . . . . . . . . . . . . . . 10 1.3 Quan tum asp ects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.4 Bac kground ﬁelds and lo w energy limit . . . . . . . . . . . . . . . . . 15 1.5 Some commen ts on the B-ﬁeld . . . . . . . . . . . . . . . . . . . . . . 17 2 D-branes and Ramond-Ramond ﬁelds 19 2.1 D-branes as b oundary conditions . . . . . . . . . . . . . . . . . . . . 19 2.2 The sp ectrum of op en strings on Dp-branes . . . . . . . . . . . . . . 21 2.3 Chan-P aton factors and Adjoin t bundles . . . . . . . . . . . . . . . . 22 2.4 D-branes and Sup ersymmetry . . . . . . . . . . . . . . . . . . . . . . 25 2.5 D-branes and Ramond-Ramond c harges . . . . . . . . . . . . . . . . . 27 2.5.1 Generalized electromagnetism and sources . . . . . . . . . . . 27 2.5.2 Ramond-Ramond c harges and anomalies . . . . . . . . . . . . 30 2.6 D-brane deca y and Sen’s conjectures . . . . . . . . . . . . . . . . . . 33 3 K-Theory , an in tro duction 36 3.1 The group K 0 (X) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.2 Relativ e K-theory and higher K-groups . . . . . . . . . . . . . . . . . 40 3.3 Multiplicativ e structures on K-theory . . . . . . . . . . . . . . . . . . 42 3.4 K-theory and classifying spaces . . . . . . . . . . . . . . . . . . . . . 45 3.4.1 Examples: K-theory of spheres and tori . . . . . . . . . . . . . 47 3.5 The A tiy ah-Bott-Shapiro isomorphism . . . . . . . . . . . . . . . . . 49 3.6 K-theory and Spin c manifolds . . . . . . . . . . . . . . . . . . . . . . 54 3.6.1 K-orien tation and Thom isomorphism . . . . . . . . . . . . . . 55 i 3.6.2 Chern Character and Gysin homomorphism . . . . . . . . . . 58 3.7 K-theory and t yp e I IA/B D-branes . . . . . . . . . . . . . . . . . . . 61 3.8 K O-theory and T yp e I D-branes: torsion eﬀects . . . . . . . . . . . . 65 4 K O-homology and Type I D-branes 67 4.1 Dual theories and sp ectral K O-homology . . . . . . . . . . . . . . . . 68 4.2 KK O-theory and analytic K O-homology . . . . . . . . . . . . . . . . 69 4.2.1 Real C ∗ -algebras . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.2.2 Hilb ert Mo dules . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.2.3 Kasparo v’s formalism for KK O-theory . . . . . . . . . . . . . 76 4.2.4 Analytic K O-homology . . . . . . . . . . . . . . . . . . . . . . 78 4.3 Geometric K O-homology . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.3.1 Homological prop erties of K O t ] . . . . . . . . . . . . . . . . . . 85 4.3.2 Pro ducts and P oincar ´ e Duality . . . . . . . . . . . . . . . . . 88 4.4 K-homology and Index Theorems . . . . . . . . . . . . . . . . . . . . 89 4.5 The equiv alence b et ween KO t ] and K O a ] . . . . . . . . . . . . . . . . . 92 4.5.1 The natural transformation µ a . . . . . . . . . . . . . . . . . . 93 4.5.2 The analytic index map ind a n . . . . . . . . . . . . . . . . . . . 97 4.5.3 The top ological index map ind t n . . . . . . . . . . . . . . . . . 99 4.5.4 The Isomorphism Theorem . . . . . . . . . . . . . . . . . . . . 102 4.6 The Real Chern Character . . . . . . . . . . . . . . . . . . . . . . . . 103 4.7 Cohomological Index form ulas . . . . . . . . . . . . . . . . . . . . . . 105 4.8 D-branes and K-homology . . . . . . . . . . . . . . . . . . . . . . . . 107 4.9 The group K O t ] (pt) and torsion branes . . . . . . . . . . . . . . . . . 109 5 Ab elian Gauge Theories and Diﬀeren tial Cohomology 115 5.1 An example: the electromagnetic case . . . . . . . . . . . . . . . . . . 115 5.2 Diﬀeren tial Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . 118 5.2.1 Cheeger-Simons c haracters . . . . . . . . . . . . . . . . . . . . 118 5.2.2 Deligne cohomology . . . . . . . . . . . . . . . . . . . . . . . . 125 5.3 Ramond-Ramond ﬁelds and c harge quan tization . . . . . . . . . . . . 132 5.4 Generalized diﬀeren tial cohomology . . . . . . . . . . . . . . . . . . . 135 6 Ramond-Ramond ﬁelds and Orbifold diﬀeren tial K-theory 142 6.1 G-CW complexes and equiv arian t cohomology theories . . . . . . . . 143 6.2 The equiv arian t Chern c haracter . . . . . . . . . . . . . . . . . . . . . 148 6.2.1 Bredon cohomology . . . . . . . . . . . . . . . . . . . . . . . . 149 ii 6.2.2 Chern c haracter in equiv arian t K-theory . . . . . . . . . . . . 154 6.3 String theory on orbifolds . . . . . . . . . . . . . . . . . . . . . . . . 156 6.3.1 D-branes and equiv arian t K-cycles . . . . . . . . . . . . . . . . 158 6.3.2 Delo calization and Ramond-Ramond ﬁelds . . . . . . . . . . . 162 6.3.3 Delo calization of the equiv arian t Chern character . . . . . . . 165 6.3.4 Ramond-Ramond couplings with D-branes . . . . . . . . . . . 166 6.4 An equiv arian t Riemann-Ro c h formula . . . . . . . . . . . . . . . . . 169 6.5 Orbifold diﬀeren tial K-theory and ﬂux quan tization . . . . . . . . . . 173 6.5.1 Orbifold diﬀeren tial K-groups . . . . . . . . . . . . . . . . . . 173 6.5.2 Flux quan tization of orbifold Ramond-Ramond ﬁelds . . . . . 178 6.6 K-homology and ﬂat ﬁelds in T yp e I I String theory . . . . . . . . . . 183 App endix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 App endix B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 App endix C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 App endix D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 Bibliograph y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 iii In tro duction “The most p owerful metho d of advanc e that c an b e suggeste d at pr esent is to employ al l the r esour c e of pur e mathematics in attempts to p erfe ct and gener alise the mathematic al formalism that forms the existing b asis of the or etic al physics, and after e ach suc c ess in this dir e ction, to try to interpr et the new mathematic al fe atur es in terms of physic al entities” P .A.M. Dirac, 1931 Since the da wn of science, the interaction betw een ph ysics and mathematics has alw a ys b een an in teresting and debated one, given the apparent diﬀerence b etw een these tw o disciplines. Indeed, on one hand, ph ysics deals with natural phenomena as they happen in an “ob jective realit y”, external to the observ er, and the aim of ph ysics is then to understand and form ulate the la ws they obey . On the other hand, mathe- matics app ears to deal with a realit y whic h is in ternal to the human b eing, p opulated b y ob jects which need only ob ey the laws of logic and consistency . F urthermore, their metho dology seems rather diﬀeren t: physics pro ceeds by exp erimen ts and particular cases, while mathematics pro ceeds by deduction and chains of logical statements. As it alwa ys happ ens when diﬀerent disciplines come in con tact, the interaction of these t w o h uman activities is b ound to enlarge our kno wledge of the realit y w e liv e in, and of the nature of the human b eing himself. In this resp ect, the 20th cen tury has b een of a crucial imp ortance. In particular, most part of it has b een dominated by the inﬂuence of the adv ances in modern mathematics, such as diﬀerential geometry , functional analysis, and algebra in the form ulation and understanding of fundamen- tal physical theories, suc h as General Relativity , Quan tum Mechanics, and Quan tum Field Theory . This inﬂuence was so prominent and surprising that Eugene Wigner w as lead to celebrate it in the now classic pap er “The Unreasonable Eﬀectiv eness of Mathematics in the Natural Sciences” [93]. This fruitful in teraction has b ecome ev en stronger in the late part of the last cen tury . In terestingly , though, we hav e somehow witnessed the shift of inﬂuence from mathematics to ph ysics, in suc h a w ay that we are lead to w onder ab out the “unreasonable eﬀectiveness of physics on mathematics”. 1 The description of Jones’ p olynomials via quan tum ﬁeld theoretical techniques, whic h earned Edward Witten a Fields Medal in 1990, is only the most eviden t result of the ab o v e in teraction. This is not something new: after all, diﬀerential calculus was b orn mainly out of the necessity to solve problems in mechanics. How ever, mathematics in the 20th century , in particular in its second half, has ev olv ed indep enden tly of any application, following more abstract and formal principles. This makes the p ossibilit y of its relation with ph ysics ev en more exciting and surprising. Despite the diﬃculties and con tro v ersies surrounding it, it is undeniable that String theory has play ed a ma jor role in these dev elopmen ts, generating an incredible num- b er of interesting problems in diﬀeren t ﬁelds of mathematics, in particular algebra, geometry and top ology . The ric h structure of String theory has lead mathematical ph ysicists and pure mathematicians to w ork on the same problems, alb eit with dif- feren t attitudes and motiv ations, allowing b oth comm unities to b etter appreciate the exten t of their o wn sub ject of study . The present w ork of thesis b elongs to the ab o ve trend, ﬁnding its place at the in ter- face of geometry , top ology , and String theory . Lo osely sp eaking, the “middle p oin t” is represented b y the circle of ideas surrounding K-theory , a generalized cohomology theory developed b y Atiy ah, Hirzebruch, and Grothendiec k, among others, which can b e deﬁned in terms of complex v ector bundles on top ological spaces. The relev ance of K-theory in String theory rests on the fact that D-branes, extended ob jects presen t in the theory , hav e charges whic h are not classiﬁed by the homology cycle of their w orldv olumes, as exp ected, but rather by the K-theory of their normal bundles in the spacetime manifold. In the ﬁrst part of this thesis, w e will exploit the fact that a more natural description of D-branes can b e giv en in terms of K-homology , whic h is the generalized homology theory dual to K-theory . The interesting mathematical asp ect concerning K-homology is the fact that it can v ery naturally b e constructed b y using b oth a geometric and an analytic representation, resp ectiv ely in terms of Spin c -manifolds and vector bundles, and of F redholm mo dules. In particular, Baum and Douglas [14] show ed that an isomorphism can b e constructed at the lev el of the represen tativ e cycles, and that suc h an isomorphism induces a natural equiv alence b et w een the geometric and analytic K-homology functors. The authors then show ed that the existence of such an isomorphism is equiv alent to the Index Theorem of A tiy ah and Singer for the canonical Dirac op erator, and conjectured that this is the case for an y ﬂa v our of K-theory . In this thesis w e will supp ort this conjecture, b y constructing a natural equiv alence 2 b et w een geometrical and analytical KO-homology , the dual theory to the K-theory of r e al vector bundles, elucidating its relation with a suitable Index Theorem. The diﬃcult aspect of the ab ov e analysis lies in the fact that the index homomorphism tak es v alues in ab elian groups with torsion, which do es not allo w a lo cal description in terms of characteristic classes. Since K O-homology describ es D-branes in type I sup erstring theory , we will b e able to construct D-branes with torsion charges which are not asso ciated to an y spacetime form ﬁeld. Indeed, D-branes can also b e realised as curren ts for Ramond-Ramond ﬁelds. These “dual” ob jects are gauge ﬁelds locally described b y diﬀerential forms of diﬀeren t de- grees, dep ending on the t yp e of String theory considered. Their ﬁeldstrengths satisfy generalized Maxw ell equations, and they interact with D-branes via the usual in tegral coupling. Since D-brane c harges are classiﬁed b y K-theory/K-homology , it is cer- tainly exp ected that K-theory pla ys some role in the Ramond-Ramond gauge theory b eha viour. This is indeed the case: the suitable mathematical formalism to describ e these ﬁelds in non trivial backgrounds is kno wn as diﬀerential K-theory , which is an example of generalized diﬀerential cohomology theories recen tly developed in [54]. In particular, the main ingredient is the Chern c haracter homomorphism from K-theory to the ev en part of the deRham cohomology ring of the spacetime manifolds, whic h roughly realizes the Ramond-Ramond curren t generated b y D-branes. In the second part of this thesis, w e will generalize the arguments leading to the ab o v e conclusions to the case of t yp e I I String theory on orbifolds. An orbifold can b e loosely describ ed as a singular ob ject whic h is locally isomorphic to the quotient of an Euclidean space by a ﬁnite subgroup of the group of linear transformations. De- spite the presence of the singularities, String theory b eha v es w ell on suc h ob jects: in particular, D-branes can b e introduced, and their c harges are classiﬁed b y equiv ariant (or orbifold) K-theory , as prop osed by [94]. W e will analyse the Ramond-Ramond gauge theory arising from the closed type I I String theory on global orbifolds, and demonstrate the role play ed by equiv ariant K-theory . In particular, w e will construct some abelian groups satisfying all the exp ected prop erties to be considered a general- ization of diﬀeren tial K-theory for global orbifolds. This will require the introduction of an equiv arian t cohomology theory not v ery p opular in the ph ysical literature, dev el- op ed by Bredon in [22], which nevertheless, as w e will argue, captures all the relev ant ph ysical prop erties of Ramond-Ramond ﬁelds on orbifolds. 3 Plan of the W ork W e will kno w give a brief description of the con ten ts of this thesis, trying to highligh t the main new results in mathematics and String theory . In Chapter 1 we introduce some generalities ab out String theory , fo cusing on the structure of its quan tum sp ectrum and on the low energy limit. W e ha v e tried throughout to express all the relev ant notions in a precise mathematical framework whenev er p ossible, and w e ha v e a v oided details of the constructions in volv ed, referring the reader to the extensive literature on the sub ject. In this w a y , the basic concepts relev an t to this work of thesis should b e accessible to a mathematically minded audi- ence acquain ted with the basics of ﬁeld theory and quan tum mec hanics. In Chapter 2 w e introduce D-branes as b oundary conditions for open String theory , emphasising the geometric and top ological prop erties of suc h ob jects. Indeed, we will fo cus mainly on properties of D-branes that are exp ected in an y top ological non trivial bac kground, suc h as the b ehaviour of the Chan-P aton v ector bundle. The main prop erties of supersymmetric D-branes are brieﬂy review ed via the analysis of the spinor bundle on the D-brane w orldvolume, which is a recurrent ingredien t throughout this thesis. W e will in tro duce the gauge theory of Ramond-Ramond ﬁelds, and discuss Ramond-Ramond c harges and the anomalous couplings with D-branes. Also in this case, we will av oid the rather lengthy computations regarding the inﬂow mechanism, since these tec hniques will not pla y an y relev ant role in this w ork of thesis. Finally , we state Sen’s conjectures on D-brane deca y and men tion Witten prop osal on D-brane c harge classiﬁcation. Chapter 3 consists of a quick introduction to top ological K-theory . F or a given CW-complex X, w e deﬁne the group K 0 (X) as the Grothediec k group asso ciated to the monoid of vector bundles o v er X, and deﬁne the higher K-groups via susp ension. W e describ e the multiplicativ e structure p ossessed b y K-theory , and discuss the A tiyah- Bott-Shapiro isomorphism in some details. W e then restrict ourselv es to the category of Spin c -manifolds, discussing the concept of K-orien tation, Thom isomorphisms, and the Chern character. Finally , we illustrate ho w the K-theoretical machinery is used in the classiﬁcation of D-brane charges in t yp e II and t yp e I String theory . This chapter is of great imp ortance, since the concepts therein will b e tacitly assumed throughout the rest of the thesis. Chapter 4 consists mostly of original material regarding KO-homology . After a brief introduction on dual theories and sp ectral K O-homology , w e will recall the basic notions ab out the theory of r e al C ∗ -algebras, emphasising the diﬀerences with the 4 analogous results in the theory of ordinary complex C ∗ -algebras. W e will then intro- duce Kasparov’s formalism for KKO-theory , whic h is based on the notions of Hilb ert mo dules and generalized F redholm mo dules, and deﬁne analytic KO-homology for a top ological space X via the C ∗ -algebra of r e al functions on X. W e pro ceed to con- struct geometric K O-homology in terms of Spin manifolds and real v ector bundles. By comparison with the b ordism description of sp ectral K-homology dev elop ed in [58], w e prov e that our construction is a generalized homology theory dual to K O-theory , and discuss the v arious relev an t homological prop erties. W e then introduce the main mathematical result of this c hapter, given b y the construction of an isomorphism µ b et w een the geometric and analytic represen tation of K O-homology which induces a natural equiv alence b etw een the geometric and analytic KO-homology functors. T o this aim w e introduce some index homomorphism on geometric and analytic K O- homology , and pro ve that the Index theorem for a suitable Dirac op erator implies that µ is indeed an isomorphism. W e point out that the pro of of the ab ov e theorem app ears also in [15], albeit it is completely diﬀeren t from the one we presen t here, whic h is more suitable for the applications w e presen t later. W e also construct a homological real Chern c haracter, and use it to giv e an alternativ e deriv ation of coho- mological index form ulas for the canonical C  n -linear A tiyah-Singer operator. F rom the physical p oin t of view, we in tro duce the concept of wr apping char ge of a wr app e d D-br ane , sho wing that in t yp e I String theory it is a gen uinely diﬀeren t notion from that of an ordinary D-brane. Finally , we construct nontrivial generators for the KO- homology of a p oin t, and in terpret these in terms of wrapp ed D-branes. In Chapter 5 we give a detailed accoun t on (generalized) diﬀeren tial cohomology theories. W e motiv ate this mathematical formalism by discussing the prop erties of ordinary electromagnetism with Dirac quan tization of c harges in top ologically non- trivial bac kgrounds. W e giv e a rather extensive treatmen t of b oth Cheeger-Simons groups and Deligne, in order to build a solid intuition for these mathematical ob jects. W e then discuss the Moore and Witten argumen t regarding the c harge quan tization of Ramond-Ramond ﬁelds, as proposed in [74]. W e conclude the chapter b y explain- ing the construction of diﬀeren tial K-theory of Hopkins and Singer, whic h will b e generalized later in the thesis. In Chapter 6 we presen t new ph ysical and mathematical results regarding String theory on global orbifolds. W e ﬁrst giv e a brief in tro duction to equiv arian t cohomol- ogy theories on the category of G-CW complexes, and consider equiv ariant K-theory as an example. W e then pro ceed to deﬁne Bredon cohomology in terms of natural transformations b et w een functors on the or bit category of a ﬁnite group. This is the 5 main ingredient used to constructed the equiv arian t Chern character deﬁned in [67], whic h has the unique prop ert y of inducing an isomorphism on rational equiv arian t K-theory . One of the main new physical result of this chapter is to demonstrate that the Dirac quantization of Ramond-Ramond ﬁelds on orbifolds is dictated b y the ab ov e Chern c haracter, which, after a pro cess of delo c alization , can be used to describe the couplings of Ramond-Ramond ﬁeld on global orbifold with fr actional D-branes, whic h w e will describ e in terms of equiv arian t K-homology . W e chec k this statement on lin- ear orbifolds, wh ic h are the usual cases studied in the physics literature. Our approac h has the main adv an tage of b eing applicable to the case of nonab elian orbifolds and for quotien ts of general Spin manifolds by ﬁnite groups. F rom the mathematical point of view, w e construct abelian groups whic h ha ve all the desired prop erties for a gener- alization of diﬀeren tial K-theory to global orbifolds. This is mathematically needed, since the general results of Hopkins and Singer in [54] hold only for generalized co- homology theories on the category of manifolds. F ar from reaching the generalit y of [54], we ﬁnd this an imp ortant step tow ards the general construction of diﬀerential extensions of equiv ariant generalized cohomology theories. W e will use our equiv ari- an t (or orbifold) diﬀeren tial cohomology theory to describ e Ramond-Ramond ﬁelds on orbifolds, and study in particular ﬂat Ramond-Ramond p oten tials. W e conclude the work with App endices whic h aim to settle the notations for some of the standard notions used throughout the thesis. 6 “F r e e dom is the fr e e dom to say that two plus two make four. If that is gr ante d, al l else fol lows.” G.Orw ell, 1984 Chapter 1 Generalities on String Theory In this chapter we will collect some basic results in String theory that will b e used as the starting p oin t for the rest of the thesis.W e will review some w ell kno wn asp ects of the p erturbative formulation of b osonic and sup ersymmetric string theories. W e direct the reader to [49],[79],[33] for more information ab out these constructions. 1.1 The Bosonic String The action functional for a string propagating in spacetime is a direct generalization of the functional describing the motion of relativistic point particle. In the case of a p oin t particle, the action functional for a given curve γ : [0 , 1] → M, where M is a d-dimensional Loren tzian manifold, is giv en b y S[ γ ] := Z γ µ g | γ (1.1.1) where µ g | γ is the in v arian t v olume form for the spacetime metric restricted to the curv e γ . The action S computes the length of the curve γ , the worldline of the p oin t particle, and its stationary p oin ts are the geo desics for the metric g on M. It is natural to generalize this action for an extend p-dimensional ob ject propa- gating in M as the “v olume” of the surface sw ept b y the ob ject while propagating. In other w ords, given f : Σ → M, where Σ is a p-dimensional manifold, called the worldshe et , and f is a smo oth immersion, w e ha v e that S[ f ] := Z Σ µ f ∗ g (1.1.2) The case of a free string propagating in M is given by p=2, and b y Σ ' S 1 × R for closed string, and Σ ' [0 , 1] × R for op en strings. 8 The Bosonic String The functional S is called the Nambu-Goto action and in a lo cal system of coordinates { σ i } o v er Σ, the functional S can b e represen ted as S[ f ] = Z Σ d p σ r − det( g µν ∂ x µ ∂ σ i ∂ x ν ∂ σ j ) (1.1.3) where x µ ( σ ) are lo cal represen tativ es for the function f . Unfortunately , this action is non-p olynomial in the x µ and its deriv atives, making its quan tization diﬃcult to deﬁne unam biguously , even in ﬂat spacetime. F or this reason, it is generally preferred to use the classically equiv alen t Polyakov action given by S[ f , γ ] := k Z Σ µ γ < γ , f ∗ g > (1.1.4) In the abov e expression, γ is a Loren tzian metric on Σ, <, > denotes || d f || 2 , deﬁned b y considering d f as an elemen t in Ω 1 (Σ; f ∗ TM), and using b oth γ on T ∗ Σ and g on TM, and k is called the tension . Notice that now the intrinsic metric γ is a dynamical v ariable, while the metric tensor g is considered as a “background”, i.e. it is a nondynamical quantit y . When also γ is held ﬁxed, the action S describ es a nonline ar sigma mo del , and its stationary p oin ts when p=2 are called harmonic maps . In the lo cal system of coordinates as ab o ve the P olyak ov action can b e represen ted as S[ f , γ ] = k Z Σ d p σ p − det( γ ) γ ij ( g µν ∂ x µ ∂ σ i ∂ x ν ∂ σ j ) (1.1.5) By construction, the P olyak ov action is inv ariant under Dif f + (Σ), the group of ori- en tation preserving diﬀeomorphisms, and under I S O ( g ), the group of isometries of g . Moreo v er, only in the p eculiar case p=2 the functional S is inv ariant under C ∞ + (Σ), the group of smo oth p ositiv e functions on Σ, acting as Weyl r esc aling of the metric γ , i.e. transformations of the type γ → ρ · γ . Hence the full group of symmetries of the string action is given by D if f + (Σ) n C ∞ + (Σ) × I S O ( g ), where n means semidirect pro duct. The in v ariance under diﬀeomorphims and W eyl scalings is crucial in giving the P oly ak o v action a more tractable form, allo wing to use canonical quan tization tec hniques, and as this is possible only for p=2, this is seen as a reason to rule out higher dimensional extended ob jects other than strings. Indeed, a 2-dimensional manifold is alw ays lo cally c onformal ly ﬂat , i.e. there alw ays exist lo cal co ordinates ( u, v ) in which the metric γ can be expressed as ρ · η ab , where η ab is the usual Minko wski metric in 2 dimensions. Diﬀeomorphisms and W eyl in v ariance then allo w to “pic k a gauge”, called the c onformal gauge , in whic h the functional S is the action functional for a vibrating string in a curv ed spacetime. 9 The Sup ersymmetric String When M is the d-dimensional Minko wski spacetime, the equations of motion of the string in the conformal gauge can b e completely solved, and its canonical quan tization carried on thanks to an additional symmetry enjo y ed b y the string functional. Indeed, conformal transformations of the Minko wski metric in t w o dimensions do not sp oil the (lo cal) co nformal gauge, implying that String theory (at an y ﬁxed γ ) is a t wo dimensional conformal ﬁeld theory , hence completely integrable b y symmetry consid- erations alone, as the conformal algebra in t w o dimensions is inﬁnite dimensional. Conformal in v ariance is so imp ortant that it is required to hold also at the quantum lev el, where an anomaly could p ossibly sp oil it: the cancellation of such a conformal anomaly , indeed, ﬁxes the spacetime dimensionality to d=26, called the critic al di- mension . Remark During this thesis, we will mainly consider b oth the w orldsheet and space- time manifolds as being equipp ed with a Riemannian as opp osed to a Loren tzian metric tensor. Ev en if this is not p er se ph ysically realistic, in most of the cases one can p erform a Wick r otation on the spacetime manifold and obtain the relativistic description w e ha v e in tro duced so far. Moreo v er, we will also consider diﬀeren t top ologies for the w orldsheet, and in partic- ular w e will regard Σ as a Riemann surface of genus g : this is essentially due to how inter actions are introduced in String theory at the quan tum lev el. This p erturbativ e form ulation of String theory also forces the use of Riemannian worldsheets, as opp osed to Lorentzian, since a compact manifold admits a Loren tzian metric if and only if the Euler n um b er v anishes. F or example, the transition amplitude for the propagation of a quan tum string will b e giv en b y A ∼ X topologies of Σ Z Met(Σ) D γ Z Map(Σ , M) D f e − S[ f ,γ ] where D γ and D f are “path integral measures” o v er a space of metrics and maps, resp ectiv ely . More precisely , the ab ov e expression should b e “gauge ﬁxed”: indeed, the path in tegral ov er the space of metrics reduces to an in tegral ov er a ﬁnite dimen- sional mo duli space. 1.2 The Sup ersymmetric String Despite the ric h structure of its symmetries, the b osonic string has some fundamental ﬂa ws. The most relev an t ones, apart from the high dimensionality of the spacetimes 10 The Sup ersymmetric String allo w ed, are the presence of a tach yonic state, which is a strong signal tow ards insta- bilit y of the quan tum theory , and the absence of fermions, which is not a feature of an inconsisten t theory , but nevertheless fermions are required for physical reasons. A wa y out of these t w o problems is the so called Neveu-Schwarz-R amond (NSR) for- mulation of String theory , whic h consists in in tro ducing additional fermionic degree on the w oldsheet. More precisely , the Poly ako v action can b e at ﬁrst generalized as S[ f , γ , ψ ] := k Z Σ µ γ  < γ , f ∗ g > + ¯ ψ / D ψ  (1.2.1) where ψ is a section of S ⊗ f ∗ TM, with S the spin bundle ov er Σ for a giv en spin structure, and / D is the Dirac op erator asso ciated to γ , coupled to f ∗ TM. Notice ﬁrst that Σ is a spin manifold, b eing 2-dimensional, and it admits, at an y gen us g , 2 2 g inequiv alen t spin structures whic h can b e distinguished by a ± sign along the homology cycles of Σ. Moreo v er in even dimensions the spin bundle S decomp oses according to the c hiralit y op erator as S + ⊕ S − . One also requires ψ to b e a Ma jorana spinor ﬁeld to ensure that the full action is real. Ma jorana spinors exist on worldsheets with Lorentzian signature, but not on those with Euclidean signature. Anyw ay , in the Euclidean case one can use an ordinary c hiral spinor ﬁeld ψ + , and choose ψ − to b e its complex conjugate, in order to preserve the degrees of freedom. More concretely , ψ + will b e a section of K 1 / 2 , the square ro ot of the canonical bundle on Σ, and ψ − a section of ¯ K 1 / 2 for the same spin structure. In the free (closed) superstring case, the topology of Σ admits a single homology cycle, and hence there are tw o spin structures, which are conv entionally referred to as R amond (R) and Neveu-Schwarz (NS), and can b e characterized by ψ ± ( τ , σ + 2 π ) = + ψ ± ( τ , σ ) R: p erio dic conditions ψ ± ( τ , σ + 2 π ) = − ψ ± ( τ , σ ) NS: anti-perio dic conditions On a ﬂat spacetime, the free sup erstring can b e quantized using canonical quan tiza- tion: the main diﬀerence with the b osonic string is giv en b y the app earance of new sectors, NS and R, giv en b y the b oundary conditions for the spinor ﬁelds ψ . In particular, the states in the F o ck space F NS for Neveu-Sc hw arz degrees of freedom can b e sho wn to b e spacetime bosons, while the states in the F o ck space F R for Ra- mond degrees of freedom can b e sho wn to b e spacetime fermions. Unfortunately , b oth the ab ov e spaces con tain ne gative norm states that are not elim- inated by an y kind of symmetry , in contrast to what happ ens in the b osonic string 11 Quan tum asp ects case thanks to the Virasoro algebra constrain ts. T o ensure this, one mo diﬁes the action (1.2.1) b y in tro ducting a gr avitino , a spin 3/2 ﬁeld, and an interaction term in order for the action to b e in v arian t under lo c al worldshe et sup ersymmetry . In analogy with the b osonic case, the mo diﬁed sup erstring action will deﬁne a sup er c onformal ﬁeld theory: the preserv ation of the sup er conformal symmetry at the quantum level requires the spacetime dimensionalit y d to b e 10. Ev en with these mo diﬁcations, the theory is still inconsisten t, as there is still a tach y- onic state in the NS sector, with no sup ersymmetric partner state, making it impos- sible to ha v e spacetime sup ersymmetry . T o ov ercome this ﬁnal problem, Gliozzi-Sc herk and Oliv e prop osed a pro cedure for a truncation of the RNS String theory that pro duces a sp ectrum with spacetime sup er- symmetry . This truncation is called the GSO pr oje ction , whic h can b e though t as a pro jection on the space of inv ariant states for the op erator ( − 1) F , which assigns to eac h state the n um b er of fermions presen t mo dulo 2. This op erator can be carefully deﬁned in b oth the Ramond and Nev eu-Sc h w arz sec tor, obtaining that the free op en GSO pro jected string is N=1 sup ersymmetric, while the free closed GSO pro jected string is N=2 sup ersymmetric. Remark In the functional integration form ulation, the GSO pro jection corresp onds to a weigh ted sum o v er the spin structures of the worldsheet, with the weigh t c ho osen in suc h a w ay that the resulting amplitudes are inv ariant under the action of the mo dular group of Σ. 1.3 Quan tum asp ects In this section we will brieﬂy recall the (massless) con ten t of the sup ersymmetric string space of states, in ﬂat 10-dimensional Mink o wski spacetime [49, 33]. Denote b y F k and ˜ F k the F o ck spaces for the b osonic degrees of freedom at momen tum k for left and right mov ers, obtained up on a holomorphic decomp osition of the ﬁelds in lo cal complex w orldsheet co ordinates. Then the full RNS F o ck space for op en and closed strings is given by F open := M k F RN S k F closed := M k F RN S k ⊗ ˜ F RN S k k ∈ R 10 (1.3.1) 12 Quan tum asp ects where F RN S k := F k ⊗ ( F R ⊕ F N S ) and the same for ˜ F RN S k . After imp osing the ph ysical constaints required b y the sup erconformal symmetry , we will obtain tw o p ositiv e-deﬁnite Hilb ert spaces for op en and closed strings, denoted b y F phys open and F phys closed , resp ectiv ely . By (1.3.1), the op en string F o c k space con tains t w o sectors, Ramond and Neveu- Sc h w arz. The ground state in the Ramond sector satisﬁes a massless Dirac equation, suggesting that it is single particle state for a spacetime fermion ﬁeld, while the ground state in the Neveu-Sc hw arz sector is a b osonic tac hy on, obtained b y a spacetime scalar ﬁeld. Moreo v er, the ﬁrst excited state is a massless ve ctor state , with its degree of freedom suggesting it is a one particle state for a Y ang-Mills ﬁeld [79, 49]. These, and an inﬁnite to w er of massiv e states, are contained in the left-mo ving sector, but b y (1.3.1) this suﬃces to construct the Hilb ert space of the op en string. F or the closed string, instead, we need to consider a tensor pro duct for left and righ t mo v ers Hilb ert spaces: w e will ha v e four sectors, c haracterized by the spin structure c hoice for left and righ t-mo ving degrees of freedom. The ground state in the NS-NS sector is a tac h y on, as for the open string, while the massless states con tains, in the same sense as b efore, a gr aviton , asso ciated to a symmetric spacetime tensor of t yp e (2,0), the B-ﬁeld , coming from a spacetime t w o form, and a dilaton , a spacetime scalar ﬁeld. In particular, the graviton state satisﬁes linearized Einstein equations, hence its name. The R-NS (and NS-R) ground state is a massless state, whic h is reducible in to a spinor state coming from a spinor ﬁeld λ , called the dilatino , and a gr avitino state, coming from a spinor-v ector ﬁeld. These are the sup erpartners of the dilaton and graviton, resp ectiv ely . Finally , the ground state in the R-R sector is massless, and can b e reduced in states that are one particle states for spacetime diﬀerential forms of degree 0,. . . ,10, called R amond-R amond ﬁelds . In a certain sense, this is the “most imp ortan t” sector for the conten t of this thesis: indeed, most of the next chapters will b e devoted to ex- plore the rich mathematical prop erties of these ob jects, and the interaction with their “sources”, called D-br anes . Of course, all the sectors describ ed ab o v e contain also a (inﬁnite) n um b er of massive excited states. 13 Quan tum asp ects As was p oin ted out in the previous section, the theory is still inconsisten t, due to the presence of the tach yon state in the NS sector, and do es not present spacetime sup ersymmetry , required b y the consistency of the coupling of the massless gra vitino app earing in the sp ectrum. This enforces the use of the GSO pro jection, b oth for the open and closed string. W e are left then with three diﬀeren t spacetime sup ersymmetric string theories, called typ e IIA , typ e IIB , and typ e I , which uses unoriented w orldsheets. Moreov er, the “I” and “I I” refers to the fact that the theory is spacetime sup ersymmetric with one and t w o sup erc harge generators, resp ectiv ely . T yp e I IA and type I IB are sup erstring theories constructed from GSO pro jecting the F o ck space of closed strings: in this case, the GSO pro jection requires a choice of c hiralit y for the R ground state in the left and right mo ving sector, but b y spacetime parit y symmetry the theories obtained b y the same c hoice of c hiralit y coincide. In- deed, under exc hange of left and right mo v ers, type IIA is a non-c hiral theory , while t yp e I IB is chiral. Moreov er, it’s a theory of oriented closed strings. The GSO pro jection mo diﬁes, among other things, the con ten t of the massless R-R sector: in t yp e IIA there will b e one particle states coming from diﬀerential form of o dd degree, while in type I IB there are states asso ciated to diﬀerential forms of even degree. Moreo ver, in t yp e I IB the 5-form ﬁeldstrengh t is required to b e selfdual. T yp e I is a sup erstring theory constructed from op en and closed strings, and will b e discussed further on, when w e will in tro duce additional degrees of freedom for op en strings, called “Chan-P aton” factors, which are essential in obtaining massless Y ang- Mills states, i.e the standar d mo del gauge inter actions . T o end, we should mention the Heterotic String theory , which is a hybrid theory ob- tained b y combining right mo v ers of t yp e I I String theory with b osonic left mo vers. W e will not discuss this theory in this thesis. Remark As p oin ted out at the b eginning of this section, the discussion ab o ve refers to the sp ectrum of a string propagating in 10-dimensional Mink owski spacetime. This is one of the v ery few cases in which the quan tization of the theory can b e done “ac- curately”, ev en if in particular gauges and with a particular choices of spacetime co ordinates, and the sp ectrum can b e found explicitly 1 . In particular, the asso ciated classic al ﬁelds for the v arious particle states can b e inferred thanks to the P oincar´ e symmetry of Minko wski spacetime. It is generally “assumed” that the ﬁeld conten t 1 In case of interests, the choice of a gauge do es no aﬀect the quantum theory , as gauge inv ariance is restored at the quantum level. 14 Bac kground ﬁelds and lo w energy limit of the eﬀectiv e theory do es not change for a more generic c hoice of spacetime mani- fold: this is a p oint of view w e will adopt in the dev elopment of the follo wing c hapters. 1.4 Bac kground ﬁelds and lo w energy limit As w e mentioned in the previous section, the closed string F o ck space con tains states in its NS-NS sector that can b e asso ciated to a symmetric, an an tisymmetric and scalar massless ﬁeld. These ﬁelds can arise as a mo diﬁcation of the Poly ako v action, describing a string propagating in b ackgr ound ﬁelds . By bac kground ﬁeld we mean a (spacetime) ﬁeld whic h is not aﬀected by the presence of the propagating string, and that do es not represent a dynamical v ariable; moreov er, background ﬁelds are not in tegrated o v er in the path in tegral. The b osonic part of the mo diﬁed action is giv en b y 1 8 π l s 2  Z Σ µ γ < γ , f ∗ g > + Z Σ f ∗ B + Z Σ µ γ R γ f ∗ Φ  (1.4.1) where g is the metric tensor on the spacetime M, B is lo cally an element in Ω 2 (M; R ), Φ ∈ C ∞ (M; R ), and R γ is the Ricci scalar for the worldsheet metric. Moreo ver we ha v e expressed the tension k in term of the worldsheet length scale l s . Notice that the ﬁrst term in the mo diﬁed string action “coincides” with the deﬁnition (1.1.4), in the sense that in the form ulation of the string dynamics as a nonlinear sigma mo del w e ha v e tacitly assumed the presence of a gra vitational bac kground. The second term con tains the B-ﬁeld, and for closed strings, i.e in the case ∂ Σ = Ø, it is in v arian t under the gauge transformation B → B + d λ , with λ ∈ Ω 1 (M; R ). The last term pla ys an important role, as for an y Riemann surface Σ of genus g one has χ (Σ) = Z Σ µ γ R γ where χ (Σ) = 2(1 − g ) is the Euler numb er of Σ, a top ological inv ariant. This mak es the expansion in p o wers of l , called the low ener gy exp ansion , somehow complicated, as the term containing the dilaton is directly dep enden t on the lo op order. Moreo v er, the action (1.4.1) is in v arian t under D if f + (Σ) and D if f (M), but not under W eyl rescaling. Indeed, enforcing these (super) symmetries constrains the c hoice for the ab o ve bac kground ﬁelds, providing equations they hav e to ob ey at the lo west order in l , whic h are essen tially giv en by the v anishing of β -functions go v erning the 15 Bac kground ﬁelds and lo w energy limit W eyl scaling of the mo del describ ed by (1.4.1). These equations can b e obtained as Euler-Lagrange equations for the follo wing action [79, 49] 2 Z M µ g e − 2Φ (R g + 4 h DΦ , DΦ i − 1 2 H ∧  H) (1.4.2) where D the usual co v arian t deriv ativ e, H is the 3-form ﬁeld strength for the B-ﬁeld, and  denotes the Hodge op erator on M. Moreov er, the critical dimension d of M is again 10. The important asp ect about the ab ov e action is that it corresponds to (part of ) the b osonic part of N=2 sup ergravit y . More precisely , the ab o v e action has to b e com- plemen ted with an additional set of ﬁelds coming from the lo w energy appro ximation of String theory , but that cannot b e incorp orated as background ﬁelds. These are essen tially the ﬁeld strengths for the Ramond-Ramond ﬁelds, giving a type I IA or t yp e I IB sup ergra vity theory . W e will discuss these additional contributions in the next chapters, as they play a prominent role in the study of Ramond-Ramond ﬁelds as a (generalized) gauge theory . W e conclude this section b y brieﬂy men tioning the main diﬀerences b et w een the p er- turbativ e analysis around general ﬁeld conﬁgurations in String theory and quan tum ﬁeld theory . In quantum ﬁeld theory , one usually starts with a set of canonical ﬁelds φ and a classical action S[ φ ] whic h is indep enden t of any quan tum parameter, suc h as } . In the functional integral formalism, p erturbation theory in the lo op expansion is carried out by c ho osing a classical ﬁeld conﬁguration φ 0 , and expanding the ﬁelds around this conﬁguration as φ = φ 0 + √ } ϕ . The F eynman rules are then obtained b y expanding the action S [ φ ] in p o w ers of √ } : the propagator for the ﬁeld ϕ and the interaction vertices will then dep end up on the choice of the background ﬁeld φ 0 . As φ 0 is a classical solution, all the F eynman graphs usually referred to as tadp oles , corresp onding to the linear term in √ } , v anish. This w orks also in the other w ay , in the sense that the v anishing of all the tadp oles graphs allows one to infer that φ 0 is a classical solution. As stated, this pro cedure has no analogue in String theory , simply b ecause there is no equiv alent for the canonical ﬁelds φ : indeed, the S-matrix in String theory is not obtained by an action go v erning string interactions, but it is deﬁned p erturbatively through a generalization of F eynman diagrams. Hence it is not p ossible to c heck di- rectly that a given conﬁguration for the ﬁelds g , B and Φ (and p ossibly other ﬁelds) is a bac kground conﬁguration, as the string action do es not include their equations of motion. One can then think of using the v anishing of tadp ole graphs as a criterion 16 Some commen ts on the B-ﬁeld to decide when a set of ﬁelds deﬁnes a bac kground solution. It is remark able, then, that asking for the action (1.4.1) to deﬁne a conformal ﬁeld theory of a given central charge, which is the case in which one is assured that there exists a Hilb ert space of states (no nonnegativ e and zero length v ectors), forces the tadp ole graphs (of certain ﬁelds) to v anish, giving us the equations of motion that the bac kground ﬁelds ha v e to satisy . 1.5 Some commen ts on the B-ﬁeld As w e hav e seen in the previous section, an additional term con taining a tw o form B can b e consistently added to the (b osonic) string action. W e ha ve also men tioned that the action so extended enjo ys a “gauge symmetry” with resp ect to the ﬁeld B. This gauge simmetry giv es the B-ﬁeld a v ery rich mathematical in terpretation: indeed, ev en if the B-ﬁeld “plays no activ e role” for the extent of this thesis, we ﬁnd it nev erthless imp ortan t to giv e some informations ab out its mathematical nature. W e start b y noticing that the description of the B ﬁeld as a t wo form on the spacetime M need not hold globally , as the B ﬁeld is deﬁned up to the transformation B → B + dλ , since the action (1.4.1) is inv ariant under this pro cess. A lo cal description, then, can b e obtained as follo ws. Let { U α } b e a op en co v er of M enjoying the prop ert y that U α and all its intersections are con tractible sets. Suc h a co v er is called a go o d c over , and an y manifold can b e equipp ed with one. 2 Moreo v er, denote with U αβ ...τ the intersection U α ∩ U β ∩ · · · ∩ U τ . As in an y op en set U α the form B α is only deﬁned up to the transfomation ab o v e, we ha v e that on an y U αβ the follo wing equation holds B β − B α = dλ αβ λ αβ ∈ Ω 1 ( U αβ ; R ) (1.5.1) These equations imply that on a triple o v erlap U αβ γ one has d ( λ αβ + λ β γ + λ γ α ) = 0 (1.5.2) Then b y P oincar ´ e’s lemma one has that λ αβ + λ β γ + λ γ α = d f αβ γ f αβ γ ∈ C ∞ ( M ; R ) (1.5.3) for some smo oth real function f αβ γ . Finally , on a quadruple intersections U αβ γ σ f β γ σ − f αγ σ + f αβ σ − f β γ α = 2 π ω αβ γ σ (1.5.4) 2 Just consider a cov er made of geo desically complete op en sets for some Riemannian metric. 17 Some commen ts on the B-ﬁeld for some real n um b ers ω αβ γ σ . Notice, at this p oin t, that the lo cal ﬁeld strengths deﬁned as H α := dB α can b e “glued” together to form a globally deﬁned three form on M. If we require H to ha v e in tegral p erio ds, then one can c ho ose ω αβ γ σ to b e integer n um b ers. According to this description, then, a B-ﬁeld conﬁguration is deﬁned as a triple (B α , λ αβ , f αβ γ ) satisying the ab o v e equations. In a mathematical language, the triple ab ov e deﬁnes a (ab elian) gerb e with c onne ction , whic h is a generalization of the idea of principal bundle with connection. As in the case of a connection on a principal bundle, the ﬁeld strength H is related to the cohomology of the manifold M: indeed, the class [H] represen ted by H in H 3 (M; R ) is the image of the class [ ω αβ γ σ ] ∈ ˇ H 3 (M; R ) under the isomorphism b et w een DeRham cohomology and C ˇ ech cohomology with co eﬃcien ts in the sheaf of lo cally constan t real functions. F or the rest of this thesis, w e will alwa ys assume that the B-ﬁeld is “turned oﬀ ”, meaning that w e will alw a ys work under the h yp othesis that B = 0. W e will mention, though, the ma jor mo diﬁcations that the presence of a non trivial B-ﬁeld induces. Notice, at this point, that the abov e deﬁnition of a gerb e has been giv en in a particular c hoice of a go o d co v er: this raises the question of the indep endence of this deﬁnition from an y particular c hoice. A more elegant form ulation, which manifestly do es not rely on any particular lo cal c hoice, will b e the sub ject of the next c hapters. 18 Chapter 2 D-branes and Ramond-Ramond ﬁelds In this c hapter we will in tro duce D-br anes and analyze the main features of open strings in the presence of such ob jects. W e will review the generalized electromag- netic theory of Ramond-Ramond ﬁelds, and discuss their anomalous coupling with D-branes. Finally , w e will discuss Sen’s conjectures, and motiv ate the relev ance of K-theory in the description of D-brane c harges. 2.1 D-branes as b oundary conditions As we hav e men tioned in the previous c hapter, the b osonic string propagating in a ﬂat 26-dimensional Minko wski spacetime describ es a tw o dimensional conformal ﬁeld theory , in which the spacetime cartesian coordinates are ﬁelds deﬁned o ver a surface of a giv en gen us. W e hav e also seen that String theory can b e deﬁned, apart from instabilities and other unpleasant eﬀects, b oth for closed and open strings: in the latter case, the worldsheet Σ has a nonempty b oundary ∂ Σ. Since Σ has a b oundary , w e are faced with the task of sp ecifying b oundary conditions for the conformal ﬁelds deﬁned on Σ. Moreo ver, one has to require that these b oundary conditions preserve the conformal in v ariance of the w orldsheet theory . F or a general conformal ﬁeld theory , the classiﬁcation of lo cal b oundary conditions preserving conformal inv ariance is very complicated to obtain; moreov er, v ery often the b oundary conditions hav e no geometrical interpretation, making it more diﬃcult to understand their relation with the fundamental strings describ ed in the previous c hapter. F or this reason, we will for the momen t fo cus on those b oundary conditions that ha v e 19 D-branes as b oundary conditions a clear geometrical interpretation, p ostp oning the discussion of more general cases to the next c hapters. Consider the P oly ak o v action (1.1.4) in the case in whic h Σ is a Riemann surface of gen us g with b oundary ∂ Σ. Recall that the dynamical v ariables are given b y maps from the w orldsheet to the spacetime M, and the w orldsheet metric γ . A t this p oin t we can consider the action restricted to a subspace of Maps(Σ , M) sp eciﬁed b y the condition f ( ∂ Σ) ⊂ Q (2.1.1) where Q ⊂ M is a submanifold of the target spacetime. W e hav e then for the moment the following geometrical working deﬁnition Deﬁnition A D-br ane is a (physic al) obje ct whose dynamic al evolution is describ e d by a submanifold Q of the sp ac etime c ontaining the wold lines of the end p oints of op en strings. In the ab ov e deﬁnition we are also supp osing that the manifold Q is c hosen in such a w a y that the worldsheet ﬁeld theory is still conformal, and are usually referred to as D-submanifold . The “D” in D-brane refers to the fact that some of the co ordinates representativ es for the map f are lo cally sub ject to Dirichlet b oundary conditions. More precisely , consider a set of co ordinates { x µ } in M in such a w ay that Q can lo cally be represen ted b y the conditio n x α = 0 , α = p + 1 , . . . , 26, with p = dim Q, and c ho ose coordinates { σ, τ } on Σ such that τ is the co ordinate along the b oundary , for σ = 0 , π . Then condition (2.1.1) implies that x µ ( τ , σ ) | σ =0 = x µ ( τ , σ ) | σ =0 = 0 , µ = p + 1 , . . . , 26 (2.1.2) whic h are referred to as Diric hlet b oundary conditions. Usually , (2.1.2) are supplemen ted with lo cal Neumann conditions for the ﬁelds repre- sen ting the co ordinates on Q . Brieﬂy , these can b e enforced by requiring that d f | ( ∂ Σ) ⊥ p = 0 , p ∈ ∂ Σ (2.1.3) where ( ∂ Σ) ⊥ p denotes the space of normal v ectors to the b oundary in p . In other w ords, the end p oin ts are free to mo v e on the submanifold Q. It is common use to refer to a D-brane represented b y a p+1-dimensional manifold Q as Dp-br ane . More precisely , in the follo wing w e will refer to Q as the worldvolume of the Dp-brane, suggesting a conceptual diﬀerence b et ween a D-brane and the submanifold it “wraps”, i.e. it is represen ted with. This diﬀerence will play an important role in 20 The sp ectrum of op en strings on Dp-branes the next c hapters of this thesis. W e conclude this section by stressing out that the ab ov e b oundary conditions do not exhaust all the p ossible conditions for a b oundary conformal ﬁeld theory , and they represen t a particular well-behav ed subset of b oundary conditions. Nev erthless, their presence in tro duces remark able additional features to String theory , as we will see in the next sections. Also, notice that D-branes can be formally added also to the theory of closed strings, since the condition (2 . 1 . 1) do es not apply . In particular, D-branes do not in teract if no op en strings are presen t. 2.2 The sp ectrum of op en strings on Dp-branes As usual in String theory , op en strings in the presence of a D p -brane can be quantized only in very sp eciﬁc settings. Indeed, one usually considers op en strings propagating in ﬂat d -dimensional Minko wski spacetime in the presence of hyp erplanar D-branes, i.e. D-branes whose worldv olume is sp eciﬁed b y linear conditions in (spatial) cartesian co ordinates. Then, the sp ectrum con tent of the eﬀectiv e theory and its prop erties are assumed to b e the same for more general D-brane conﬁgurations. Notice, at this p oin t, that in tro ducing a D-brane generally reduces the spacetime sym- metries of the conformal ﬁeld theory . F or instance, in the ﬂat case ab ov e, introducing a ﬁxed p +1-dimensional h yp erplane in Mink owski spacetime destroys the translational symmetry , and reduces the SO(1 , d ) action to that of SO(1 , p ) × SO( d − p − 1). This is analogous to the fact that introducing an electron in spacetime breaks its transla- tional symmetry , and is not surprising. The analogy is ev en deep er, in view of the fact that D-branes can arise as solitonic solutions of a lo w energy eﬀectiv e theory . Consider then a D p -brane in d -dimensional Mink o wski spacetime represen ted b y the h yp erplane x µ = ¯ x µ , µ = p + 1 , . . . , d , and where x 0 is the time co ordinate. In this situation, the F o c k space of the quan tized op en string contains, as in the case with- out an y D-brane, states that can be recognised as one particle states for a quan tized classical ﬁeld. Moreo v er, these classical ﬁelds are constrained to “live” on the w orld- v olume represen ting the D p -brane, in the sense that they can b e naturally in terpreted as ob jects deﬁned only on the brane itself. 1 Notice, ho wev er, that statemen ts about the supp ort of these ﬁelds are usually dep endent on the quantization pro cedure, and diﬀeren t pro cedures can yield in principle diﬀeren t answ ers. 1 This is due essen tially to the fact that they are states generated from the ground state by the action of creation op erators asso ciated to the momenta corresp onding to the co ordinate describing p oin ts on the brane. 21 Chan-P aton factors and Adjoin t bundles In the follo wing, we will just list the ﬁrst few states arising from the ground state, referring the reader to [96] for more details. As in the case of a b osonic op en string without the presence of a D-brane, the ground state con tains tac h y onic states, whic h arise b y a scalar ﬁeld on the brane, and consti- tute a section of the normal bundle. The next states are massless states, arising b y a v ector ﬁeld on the D-brane, with the degrees of freedom of a gauge ﬁeld, i.e. they can b e describ ed b y a 1-form A, whic h can b e shifted by an exact 1-form. This is usually in terpreted as a Maxwell ﬁeld deﬁned on the worldv olume of the D p -brane. Actually , this ﬁeld has a deep er geometrical and top ological description, as w e will see in the next c hapters. Finally , there are massless states for eac h normal direction to the D p -brane, coming from scalar ﬁelds deﬁned on the brane itself. These are usually interpreted as the ﬁelds generating the excitations of the D p -brane, describing ﬂuctuations in the D p - brane p osition in spacetime. In the case of general spacetimes and brane conﬁgurations, w e will assume that the picture describ ed ab ov e is still v alid lo cally for any given neigh b orho o d of a p oint in the D p -brane w orldv olume. 2.3 Chan-P aton factors and Adjoin t bundles In this section w e will consider how the description in the previous section is mo diﬁed when w e allo w for the presence of n D p -branes represented by the same worldv olume Q. This is indeed p ossible, due to the fact D-branes represent b oundary conditions for the worldsheet conformal ﬁeld theory , and hence they are distinguished b y the op en string b oundary . In this case, the n D-branes are said to b e c oincident . Then, an y quan tum state of an op en string constrained on the worldv olume of a set of n coincident D p -branes can b e lab elled by | k ; ij > , where k is a given collection of quan tum num b ers, and i, j = 1 , . . . , n lab el the branes whic h the end p oin ts of the strings are constrained to. Allo wing a general sup erp osition of this states, one has that a general state assumes the form | k ; a > = P i,j λ a ij | k ; ij > : the co eﬃcien ts λ a ij are called the Chan-Paton factors of the op en strings. F or ph ysical reasons ([79],[33]), the Chan-Paton factors λ a = ( λ a ij ) are constrained to b e an tihermitian, i.e. λ a † = − λ a ; moreov er, they are conserved in interactions. Hence, the F o c k space for an op en string in the presence of n coinciden t D p -branes is 22 Chan-P aton factors and Adjoin t bundles giv en b y F D open ⊗ u ( n ) where F D is the space of an op en string in the presence of a single D p -brane, and u ( n ) is the Lie algebra of U( n ). Consequen tly , all the massless ﬁelds describ ed in the previous section will tak e v alues, at least lo cally on the brane, in u ( n ). As the Chan-P aton factors are conserv ed in interactions, their contribution in the in teraction amplitude at any level is of the form of a trace of matrix pro ducts. As the trace of a matrix is preserved under the adjoin t action of the group of inv ertible matrices, the transformation λ a → g λ a g † , with g ∈ U( n ) is a symmetry of the system. 2 Hence the Lie algebra u ( n ) can b e seen as the adjoint represen tation for the matrix group U( n ). Consider now a set of n D p -branes wrapping a submanifold Q of the spacetime, and let { U α } be a goo d co v er for Q. Recall that at low energy the spectrum pro duces a u ( n )-v alued 1-form A α , for any α . Consisten tly with the symmetry of Chan-Paton factors and the prop erties of A α , on a double in tersection U αβ w e c an hav e A β = g αβ ( p )A α g − 1 αβ ( p ) + g − 1 αβ ( p )d g αβ ( p ) (2.3.1) where g αβ : U αβ → U( n ) is a smo oth function. A t the same time, other ﬁelds will ob ey similar prop erties. F or instance, the u ( n )-v alued scalar ﬁeld T describing the tac h y on will satisfy on U αβ T β = g αβ ( p )T α g − 1 αβ ( p ) (2.3.2) The function g αβ m ust b e the same as it arises from a symmetry of the transition amplitude. Consistency on triple o v erlaps requires that Ad( g αβ )Ad( g β γ )Ad( g γ α ) = 1 (2.3.3) The ab o v e condition implies that g αβ g β γ g γ α = ω αβ γ 1 (2.3.4) where ω αβ γ ∈ U(1), i.e. in the kernel of the adjoin t representation. A t this p oin t one can ask if it is p ossible to redeﬁne the functions g αβ in such a w a y that ω αβ γ = 1, i.e. if the vector bundle E ad j , of which the tac h y on ﬁeld is a section, is the adjoin t bundle asso ciated to a principle bundle P o ver Q deﬁned b y the transition functions 2 U( n ) is singled out b ecause it preserv es the antihermitian prop erty of λ a . 23 Chan-P aton factors and Adjoin t bundles g αβ . I n this case, the u ( n )-v alued forms A α can b e seen as the lo cal represen tatives of a connection deﬁned on the principal bundle P , hence as a Y ang-Mills ﬁeld o v er Q. This “lifting” pro cess is not alw a ys p ossible for general v ector bundles: an yw a y , we will alw a ys assume that this is possible. With this we mean that choosing functions g αβ b eha ving as transition functions of a principal bundle is c onsistent with the path in tegral for op en strings in the presence of D-branes being w ell deﬁned and gauge in v arian t. In this case, the path integral is supplemented with a factor hol ∂ Σ ( f ∗ A) (2.3.5) whic h only mak es sense when A is a connection on some principal bundle. In the case in whic h a top ologically non trivial B-ﬁeld is presen t, though, the equation (2.3.1) has to b e mo diﬁed in order to tak e in to account the gauge transformations for the B-ﬁeld discussed in section 1.5: this, in turn, forces a redeﬁnition of the function g αβ , whic h will now ob ey condition (2.3.4), with the co cycle ω αβ γ directly determined b y the top ological prop erties of the B-ﬁeld. See [60] for more details. Hence, in absence of a B-ﬁeld, the bundle E ad j o v er Q is c hosen to b e decomposable as E ad j ' E ⊗ ¯ E where E is a U( n )-v ector bundle, and ¯ E is its complex conjugate. It is customary to refer to the vector bundle E as the Chan-Paton bund le , as from this last we can obtain the bundle of Chan-P aton factors E ad j . W e can then summarize the crucial asp ects of this section by the following F act In the absenc e of a nontrivial B-ﬁeld and in the low ener gy appr oximation, a set of n c oincident D-br anes with worldvolume Q gives rise to a U( n )- ve ctor bund le E → Q e quipp e d with a line ar c onne ction. Notice that the top ology of the vector bundle E is not determined by the string dynamics, and has to b e speciﬁed a priori when in tro ducing a set of D-branes. Also for a single D-brane, we ha v e to assign a line bundle o v er its worldv olume: in the case of hyperplanar D-branes discussed in the previuos section, this is hidden b y the fact that an y line bundle o v er suc h a h yp ersurface is top ologically trivial. 24 D-branes and Sup ersymmetry 2.4 D-branes and Sup ersymmetry In this section w e will consider the eﬀect of the presence of a D-brane in superstring theories; we will fo cus, in particular, on the case of t yp e IIA and IIB sup erstring theories. Recall that sup estring theories, in the RNS formalism, are obtained by in tro ducing degrees of freedom on the w orldsheet Σ which are sections of S(TΣ) ⊗ f ∗ TM, where S(TΣ) is the spinor bundle asso ciated to TΣ, and that the GSO pro jection ensures spacetime supersymmetry . F or this reason, we will require M to b e a spin manifold, where M is a 10-dimensional euclidean spacetime. W e ha ve seen in the previous sections that the deﬁning prop ert y of a D p -brane with w orldv olume Q is that of constraining the maps f to satisfy f ( ∂ Σ) ⊂ Q, and w e ha v e argued that this requirement mo diﬁes the sp ectrum of the open b osonic string. Indeed, the presence of a D p -brane mo diﬁes also the fermionic sector for the op en sup ersymmetric string [90, 79]. Notice, ﬁrst, that for op en strings whose end p oints are constrained to the submanifold Q, one has f ∗ TM ' f ∗ TM | Q , where TM | Q is the restriction of TM to Q. Moreov er, as Q is a submanifold of M, the follo wing exact sequence holds 0 → TQ → TM | Q → ν Q → 0 (2.4.1) and with a choice of a metric on TM it splits orthogonally as TM | Q ' TQ ⊕ ν Q , where ν Q is the normal bundle to Q in M. Hence, the ground state in the Ramond sector will giv e rise to a section of S(TM | Q ) ' S(TQ) ⊗ S( ν Q ) (2.4.2) i.e. a w orldvolume spinor charged under an in ternal SO(9 − p ) symmetry , the struc- ture group of ν Q 3 . T o b e more precise, notice that in general neither S(TQ) nor S( ν Q ) are wel l deﬁne d as vector bundles, even if their tensor pro duct is. An yw a y , w e will for the moment require that the normal bundle ν Q admits a spin structure: together with the require- men t that M is a spin manifold, one has that TQ also admits a spin structure, i.e. Q is a spin manifold. W e refer the reader to App endix B for more details on spin manifolds and Cliﬀord algebras. 3 Actually , the decomp osition (2.4.2) dep ends on the parit y of the codimension of Q in M, since in general C ` (V ⊕ W) ' C ` (V) ˆ ⊗ C ` (W), where ˆ ⊗ denotes the Z 2 -graded pro duct. 25 D-branes and Sup ersymmetry Because of the prop erties of a system of n coinciden t D-branes discussed in section 2.3, the w orldv olume spinors obtained in the low energy approximation are sections of S(TQ) ⊗ S( ν Q ) ⊗ E ad j , and hence they are charged under the gauge ﬁeld lo calized on the w orldv olume. Then, it is natural to think that, in a certain sense, D p -branes in the lo w energy approximation are described by the p +1-dimensional gauge theories lo calized on their w orldv olume. As we ha v e mentioned in section 2.2, the introduction of a D-brane can reduce the spacetime symmetries of the v acuum conﬁguration, and hence aﬀect the prop erties of the op en and closed string sp ectrum. As should b e exp ected, a D-brane aﬀects also the amoun t of sup ersymmetry present in the theory . Indeed, the introduction of a D-brane in an interacting String theory requires the presence of op en string states in the spectrum: this is due to the fact that a closed string, though not aﬀected b y the b oundary conditions, can br e ak on the w orldv olume of the D-brane into an op en string, and op en strings ha v e at most N=1 sup ersymmetry [79, 16, 90]. A D-brane is said to b e sup ersymmetric if the sp ectrum of open and closed sup erstrings in its presence is sup ersymmetric. Moreov er, a D-brane is said to be stable if the spec- trum do es not contain a tach yonic state. In this sense, all D-branes in b osonic String theory are unstable. Sup ersymmetry and stability of a D-brane are very diﬃcult fea- tures to determine for general brane conﬁgurations and in non trivial backgrounds, but are well understo o d in the case of h yp erplanar D-branes in Minko wski spacetime. Indeed, one can show that ﬂat D p -branes in t yp e I IA on a Mink o wski spacetime are sup ersymmetric for p ev en, while p must b e o dd in type I IB. This can b e inferred by an- alyzing the conserv ed sup ersymmetry in eac h case, and by using T-duality [79, 90, 16]. As usual, we will assume these prop erties of D-branes to hold for a more general type I I conﬁgurations. As suggested b y the terminology used abov e, a D-brane app ears not only to be a geometrical region enforcing the b oundary conditions for the worldsheet conformal ﬁeld theory , but also to enjo y particle-like prop erties, lik e mass, c harge, etc., detected through the b eha vior of the strings propagating around it. Ev en if a full theory of quan tum D-branes, whic h w ould b e the righ t framew ork to describ e quan tum prop- erties like deca y , etc., has not y et b een dev elop ed, the sup erstrings in the presence of a D-brane giv e useful information ab out the dynamics of these extended ob jects. Remark The deﬁnition of sup ersymmetric and stable D-brane giv en ab ov e are sen- sible in the low energy approximation, and in the case in which we are considering a D-brane that is ﬁxe d , i.e. we are neglecting its dynamics: indeed, to take in to accoun t 26 D-branes and Ramond-Ramond c harges that a D-brane is a dynamical ob ject, and to describ e some asp ects of its quantum b eha vior, a formalism kno w as the b oundary state formalism is more suitable. In this formalism, the b oundary conditions characterizing the D-brane are imp osed after the theory of closed sup erstrings has b een quantized: this suggests to identify states of a quantum D-brane as coherent states in the F o c k space of the closed sup erstring, making it more precise the inv estigation of stabilit y , sup ersymmetry , etc. of these states. Moreov er, it mak es it p ossible to in tro duce D-branes ev en when the b oundary condition it represen ts has no clear geometrical description. As in this thesis w e will b e concerned only with a semiclassical and geometrical descrip- tion of D-branes, w e will not in vok e this formalism, directing the reader to [59, 79], and references therein, for a complete review of these topics. 2.5 D-branes and Ramond-Ramond c harges As we hav e men tioned in section 1.3, the lo w energy approximation of t yp e IIA and t yp e I IB sup erstring theory con tains particle states describ ed by a gauge theory of diﬀeren tial forms C ( p ) deﬁned on a d -dimensional spacetime manifold M, with p o dd in t yp e I IA and p even in type I IB. Moreo v er, if we denote b y F p = dC ( p ) the Ramond-Ramond ﬁeld strength, the GSO pro jection imp oses the constraint  F p = F d − p , represen ting the fact that not all the forms C ( p ) are indep endent degrees of freedom [79]. It follo ws that the Ramond-Ramond ﬁeld strengths satisfy the linearized equations d  F ( p ) = 0 dF ( p ) = 0 These equations are a generalization of the Maxw ell equations in 4 dimensions. As the ﬁeld theory they describ e pla ys a prominent role in this thesis, w e will digress sligh tly to review some of its asp ects, follo wing [29, 42, 41]. 2.5.1 Generalized electromagnetism and sources In analogy with Maxw ell electromagnetism in 4 dimensions, consider a theory of diﬀeren tial forms A ∈ Ω n (M; R ), where M = R × Y is a d -dimensional orien ted Loren tzian manifold with Y the spatial slice, and whose equations of motion are 27 D-branes and Ramond-Ramond c harges giv en b y d  G = 0 dG = 0 with G = dA. Suc h a theory is usually referred to as gener alize d ele ctr omagnetism , as the Maxwell equations (in v acuum) are obtained for n =1 and d =4. The equations of motion are in v arian t under the transformation A → A + ω , with ω a closed n -form, describing an ab elian gauge theory of n -forms. It is natural, then, to introduce an ele ctric source for the ﬁeld A in complete analogy with the electromagnetic case, i.e. by mo difying the equations of motion as d  G = j e dG = 0 (2.5.1) where j e ∈ Ω d − n s (M; R ) is a ( d − n )-form with compact supp ort on the spatial slice, called the ele ctric curr ent distribution . The equations of motion imply that d j e = 0, hence j e represen ts a class [ j e ] ∈ H d − n s (M; R ) in the cohomology of M with real co eﬃcien ts, and compact supp ort on Y. Notice that the equations of motion do not imply that the class [ j e ] is v anishing, as  G is not required to ha v e compact supp ort on the spatial slice. More precisely , if we denote with i t : Y → M the map Y → { t } × Y ⊂ M, the class Q e := [ i ∗ t j e ] ∈ H d − n cpt (Y; R ) (2.5.2) is in general non v anishing, and it is called the total char ge of the electric current distribution. Indeed, the fact that j e is a closed form implies 4 that the class Q e do es not dep end upon the choice of i t , hence it is a conserv ed quantit y for the equations of motion. This cohomological interpretation of the electric c harge ma y sound “exotic” at ﬁrst: but notice that in the ordinary electromagnetism case, with n =1, d =4, and Y = R 3 , one has H 3 cpt ( R 3 ; R ) ' R , and the total electric c harge is giv en b y a real num b er, as usual. Equations (2.5.1) can b e obtained from the functional S[A] := − 1 2 Z M G ∧  G − 1 2 Z M j e ∧ A (2.5.3) 4 This is also due to the particular choice made for M. 28 D-branes and Ramond-Ramond c harges via a v ariational principle. This functional is not gauge in v ariant, but one can sho w that the equations of motion obtained do not dep end on the particular c hoice of a gauge. Moreo v er, also the coupling term in (2.5.3) can b e given a cohomological in terpretation. As j e is a closed form with compact supp ort on M, it will induce a homomorphism ψ j e : H n (M; R ) → R deﬁned as ψ j e ([ a ]) := < [ j e ] ∪ [ a ] , [M] > = Z M j e ∧ a for an y close d n -form a . As in cohomology with real co eﬃcien ts we hav e that Hom(H n (M; R )) ' H n (M; R ), one has that there exists a class in [Q] ∈ H n (M; R ) represen ted by a compact oriented submanifold Q suc h that ψ j e ([ a ]) = < [ a ] , [Q] > = Z Q a The class [Q] is called the Poinc ar ´ e dual of [ j e ]. Indeed, using this class w e could rewrite the coupling term in (2.5.3) as Z M j e ∧ A = Z Q A | Q (2.5.4) whic h gives rise to the straigh tforward interpretation of Q as the w orldv olume of an extended ob ject acting as a source for the ﬁeld A, with coupling giv en by (2.5.4). Moreo v er, in the usual physical case in which Q ' R × e Q, the total charge of the distribution is represen ted b y [ e Q]. Unfortunately , as it is stated, equation (2.5.4) is not quite true, as A is in general not closed. The mathematical solution to this problem is to regard j e as a curr ent [29, 50], i.e. as a distribution v alued diﬀerential form, with supp ort on Q. This indeed allows equa- tion (2.5.4) to hold for a non-closed diﬀeren tial form A; notice that this is perfectly analogous to the electromagnetic case, where an electron curren t is described with a Dirac delta distribution ha ving supp ort on the w orldline of the electron itself. T o summarize, in this section we ha ve argued that a generalized ab elian gauge theory of n -forms admits extended ob jects as electric sources, whose worldv olume Q are sub- manifolds of the spacetime, and whose c harges can b e identiﬁed with the homology classes represen ted b y Q in de Rham homology . 29 D-branes and Ramond-Ramond c harges 2.5.2 Ramond-Ramond c harges and anomalies As w e hav e seen in the previous section, a generalized electromagnetic theory natu- rally admits extended ob jects as its electric sources. Since the dynamics of Ramond- Ramond ﬁelds are gov erned by such a gauge theory , it is natural to ask for their sources. Recall that Ramond-Ramond ﬁelds are given by p -forms, with p o dd in t yp e I IA String theory , and p even in t yp e I IB: sources for Ramond-Ramond ﬁelds in type I IA are necessarily o dd-dimensional submanifolds, while in type I IB they need to b e ev en-dimensional. Hence, the natural candidates for the role of sources are the super- symmetric stable D-branes of t yp e I I String theory: indeed, w e ha ve seen in section 2.4 that they o ccur with the right dimension, i.e. as o dd-dimensional in type I IA, and as even-dimensional in type I IB. This w ould indeed tak e into accoun t the stability of D-branes of certain dimensions. As it is stated here, this could b e merely a coincidence, in the sense that the only prop ert y of having the right dimension do es not necessarily iden tify D-branes with the sources of Ramond-Ramond ﬁelds. On the con trary , in the seminal pap er [78] strong supp ort was giv en to the fact that D-branes do interact with Ramond-Ramond ﬁelds: this w as ac hiev ed by scattering closed strings with D-branes, and sho wing that the interaction amplitude contains a term that can b e manipulated to a coupling of the form (2.5.4). Moreo v er, it was sho wn that D-branes coupling to Ramond-Ramond ﬁelds enjo y the BPS property: their mass, that in the case of D-branes is called the tension , coincides with their c harge. D-branes in type I I with the “wrong” dimension do not couple to Ramond- Ramond ﬁelds, and they are unstable: they are called non-BPS D-branes. Hence, let Q ⊂ M b e a Dp-brane: the term Z Q C ( p +1) | Q (2.5.5) is called the R amond c oupling , and the class [Q] is called the R amond char ge of the Dp-brane Q. It turns out [71, 48], though, that (2.5.5) is not yet the righ t coupling term, and has to b e mo diﬁed in order to take into account some p eculiar prop erties of D-branes. Before explaining the reason for this mo diﬁcation, w e add a comment on the coupling (2.5.5). Indeed, in the ab ov e discussion w e ha v e willingly neglected the fact that the diﬀerent Ramond-Ramond ﬁelds are dep endent up on the relation  F ( p ) = F ( d − p ) . This means that any electric source for the ﬁeld C ( p ) is a magnetic source for C ( d − p ) , whic h requires a shift in the meaning of C ( p ) themselv es, for the appropriate 30 D-branes and Ramond-Ramond c harges v alues of p . More precisely , in the presence of a Dp-brane, C ( d − p − 1) cannot b e a globally deﬁned diﬀeren tial d − p − 1-form, as the Bianc hi iden tities for its ﬁeld strength do not hold any longer. Moreov er, this poses a serious threat to the quantization of the Ramond-Ramond ab elian gauge theory . F or the moment, we will con tin ue to neglect the problems p osed b y the dualit y constrain ts. Recall from section 2.4 that in the low energy approximation the worldv olume of a set of n D p -branes carries a dimensionally reduced Y ang-Mills theory coupled with a spinor ﬁeld c harged under some additional internal symmetries; moreo v er, if the D p - brane is sup ersymmetric, the w orldv olume gauge theory is an N=1 sup er Y ang-Mills theory , and the tach yon ﬁeld is absent. A consistency requiremen t on the gauge theory supp orted b y the D-brane is that the theory does not present any “quan tum” anomaly , which w ould sp oil its gauge in v ariance after quan tization. Recall, indeed, that this gauge theory is obtained b y considering only the massless states in the op en string sp ectrum in the presence of a D-brane, whic h should represen t one particle states for the full quan tized gauge theory: an anomaly would render this an inconsisten t pro cedure. It turns out that the gauge theory on the D-brane suﬀers from t w o anomalies. The ab elian anomaly is related to the fact that the spinor ﬁelds on the w orldv olume are c harged under an in ternal symmetry , represented by the tensor pro duct S( ν Q ) ⊗ E ad j , where Q is the D-brane worldv olume. More precisely , in the t yp e IIB case, where Q is an ev en-dimensional (spin) manifold, the dynamics for the spinor ﬁelds is go v erned b y the Dirac op erator / D : S + (TM | Q ) ⊗ E ad j → S − (TM | Q ) ⊗ E ad j (2.5.6) where the ± splitting is giv en by the c hiral decomp osition, since M ev en-dimensional. F rom equation (2.4.2) one can see that S ± (TM | Q ) '  S + (TQ) ⊗ S ± ( ν Q )  ⊕  S − (TQ) ⊗ S ∓ ( ν Q  Hence, from the worldv olume p ersp ectiv e the spinor ﬁelds do not ha v e a deﬁnite c hiralit y: this, in general, leads to the so called “c hiral” anomaly , and is measured b y the index of the Dirac op erator in (2.5.6). The other p ossible source of anomaly is given b y the in tersection of t w o D-branes, usually referred to as I-br ane . The sp ectrum of the op en string in the presence of an I-brane is diﬀeren t from the one obtained in the presence of a D-brane wrapping the same w orldv olume [79]. In particular, the fermion sector on the I-brane develops itself an anomaly , dep ending on the dimension of the tw o intersecting branes. 31 D-branes and Ramond-Ramond c harges Both the anomalies need to b e cancelled: the mechanism used is kno wn as the inﬂow me chanism , and essentially consists in mo difying the coupling (2.5.5) in suc h a w a y that its v ariation under gauge transformations cancel the quantum anomalies ab o v e. W e refer the reader to [48, 29, 90] for a detailed exp osition, as the computations in v olv ed are rather length y , and the tec hniques used therein will not play an y relev ant role for the rest of this thesis. The coupling (2.5.5) is mo diﬁed as Z Q i ∗ C ∧ ch(E) i ∗ q ˆ A (TM) 1 ˆ A (TQ) (2.5.7) where i : Q → M is the embedding map, ch denotes the (total) Chern c haracter, and ˆ A denotes the A-ro of genus 5 . See App endix C for details on c haracteristic classes of v ector bundles. Moreo v er, in the coupling (2.5.7), C is the total Ramond-Ramond ﬁeld, deﬁned as the formal sum C := C ( i ) + C ( i +2) + · · · , with i = 0 , 1 in type I IB, type I IA, resp ectiv ely , and the in tegration is understo o d to b e 0 when the degrees do not match. The coupling term (2.5.5) is referred to as the W ess-Zumino, or Chern-Simons term for D-branes: w e will refer to it as the anomalous c oupling . The anomalous coupling forces us to rethink the c harge classiﬁcation of D-branes: indeed, the current generated b y such extended ob jects is no longer given b y the P oincar ´ e class dual to the homology cycle of their w orldv olume, but needs to b e corrected according to the equations of motion for the (total) Ramond-Ramond ﬁeld induced b y (2.5.7). W e will refer to the ch arge of this curren t as the D-br ane char ge . The mo dern in terpretation for the charge of a D-brane w as ﬁrst exploited in [71], where it was prop osed that the correct mathematical to ol to represent D-brane charges is not de Rham or singular cohomology , but K-theory , whic h represents, together with some of its “ﬂa v ours”, the main mathematical sub ject of this thesis. W e conclude this section b y noticing that the anomalous coupling (2.5.7) suﬀers from the same, and actually w orse, problems of the coupling (2.5.5). Indeed, the anomalous coupling requires that al l the Ramond-Ramond diﬀeren tial forms should b e deﬁned on the worldv olume Q of the D-brane. This is not the case when w e tak e into accoun t the fact that D-branes are electric and magnetic sources: indeed, the usual description for a gauge ﬁeld A in the presence of a magnetic distribution prescrib es that A should b e deﬁned on the c omplement of the distribution’s supp ort, which renders the expression (2.5.7) ill deﬁned [42]. Again, its correct description requires a 5 More precisely , these are the Chern-W eil forms represen ting such characteristic classes. 32 D-brane deca y and Sen’s conjectures more pow erful formalism: we will address a p ossible solution for this problem in later c hapters. 2.6 D-brane deca y and Sen’s conjectures The fact that Dp-branes are c harged under Ramond-Ramond ﬁelds allo ws the in tro- duction of anti-D-br anes . Of course, such ob jects should arise in a prop er quan tum description of D-branes, whic h w e currently lac k. In any case, one can give the fol- lo wing semiclassical deﬁnition: Deﬁnition A n anti-D-brane for a char ge d Dp-br ane Q supp orting a Chan-Paton line bund le E is given by a submanifold Q 0 c arrying the opp osite Dp-br ane char ge, and supp orting a line bund le E 0 which is top olo gic al ly e quivalent to E. As D-brane c harges are conserv ed during the dynamics, c harged D p -branes are stable, and hence cannot deca y . Again, the description of the decay pro cess w ould require a prop er quantum theory of D-branes: in this context, we will sa y that a D p -brane has de c aye d if the theory of open and closed strings in the presence of the D p -brane is e quivalent , in some sense, to a theory of only closed strings in the absence of the D p -brane itself. This is b eliev ed to b e describ ed at low energy b y the dynamics of the tac hy on ﬁeld living on the worldv olume of the D p -brane, which mimics the “slo w rolling” dynamics of the Higgs ﬁeld in the standard mo del of elemen tary par- ticles. In particular, the deca y pro cess of the D p -brane ends when the tac h y on ﬁeld reac hes a stable minim um of the p oten tial mo delling its dynamics [96]. Ev en if at the presen t stage such a pro cess cannot b e describ ed in a satisfactory wa y , it has greatly con tributed to a more basic understanding of D-branes and their charges. More pre- cisely , it is at the base of the so called Sen ’s c onje ctur es . Recall that in the b osonic String theory all D-branes are unstable, as the op en string sp ectrum con tains a tach yon. In particular, one has a spacetime ﬁlling D25-brane whic h is unstable. In [85, 84] Sen stated the following Conjectures(Sen) The op en b osonic String the ory in the pr esenc e of a sp ac etime ﬁl ling D25-br ane is such that a) The tachyon ﬁeld p otential has a stable lo c al minimum. Mor e over, the ener gy density of this minimum as me asur e d with r esp e ct to that of the initial unstable p oint is e qual with opp osite sign to the tension of the D25-br ane; b) L ower dimensional Dp-br anes c an b e obtaine d as solitonic solutions of the ﬁeld 33 D-brane deca y and Sen’s conjectures the ory living on the D25-br ane worldvolume; c) The stable minimum of the tachyon p otential c orr esp onds to the close d string vacuum with no op en string excitations The ab ov e conjecture can b e stated also in the case of a low er dimensional unsta- ble D p -brane: the adv antage of using a spacetime ﬁlling D-brane consists in the fact that the spacetime symmetries are not brok en. Moreo v er, the conjecture can b e adapted to sup erstring theory on a 10-dimensional spacetime, but some modiﬁcations are needed. F or instance, in t yp e I IB String theory the spacetime ﬁlling D9-brane is stable, and hence do es not ob ey Sen’s conjecture. In an y case, an instability app ears in a system of a coinciden t D p -brane and its anti-D- brane, denoted as D ¯ p -brane, wrapped on a submanifold Q: at an in tuitiv e lev el, the system has no conserved D-brane c harge, and hence it should b e able to annihilate b y Sen’s conjectures. This is supp orted by the fact that the sp ectrum of the op en strings “stretc hing” b et ween the D p -brane and the D ¯ p -brane con tains a tac hy onic state, whic h is not pro jected out b y the GSO pro jection. This extends to a system of n D p -branes and n D ¯ p -branes wrapping the same submanifold of the spacetime. Notice that a system of n D p -branes with Chan-P aton bundle E and m an ti-D-branes with Chan-Paton bundle F wrapping the submanifold Q has Chan-P aton bundle E ⊕ F, as the D-branes and the an ti-D-branes are distinguished b y the open string endpoints. In particular, at low energy the surviving op en string tac h y on ﬁeld is describ ed b y a section of E ⊗ F ∗ , where F ∗ denotes the dual bundle. Despite the resemblances, there is a ma jor diﬀerence b et w een an unstable b osonic D p -brane and a brane-antibrane system: indeed, while by Sen’s conjectures al l the b osonic D p -branes will even tually decay to the v acuum state, a generic brane-antibrane system can deca y to a stable D p -brane, if the starting conﬁguration has a net D-brane c harge diﬀerent from “zero”. In particular, in [83] Sen was able to construct a D p - brane as a deca y pro duct of a system of a D p +2-brane and a coincinden t D p + 2-brane, iden tifying the worldv olume wrapped by the D p -brane as a “vortex” for the tach yon ﬁeld living on the brane-antibrane w orldvolume. Indeed, recall that in this case the tac h y on ﬁeld T is a section of the complex line bundle E ⊗ F ∗ deﬁned on the brane- an tibrane worldv olume: in Sen’s construction, the D p -brane is iden tify ed with the submanifold representing the Poincar ´ e dual class to the (ﬁrst) Chern class of E ⊗ F ∗ , the zero lo ci for the section T. 6 A somewhat simple observ ation w as used by Witten in the seminal pap er [94] to giv e 6 W e are supposing that the Chern class of E ⊗ F ∗ is nontorsion and that T is a tran sversal section. 34 D-brane deca y and Sen’s conjectures an elegan t mathematical description of the v arious D-brane conﬁgurations. More pre- cisely , consider in t yp e I IB String theory a system of n spacetime ﬁlling D9-branes with Chan-P aton bundle E, and a system of n D ¯ 9-branes carrying a Chan-P aton bun- dle F; 7 lab el suc h a conﬁguration by the pair (E,F). As the pro cess of brane-an tibrane creation and annihilation do es not c hange the D-brane charge, we can add an y col- lection of m D9-branes and m D ¯ 9-branes with Chan-P aton bundle H. 8 Hence, the conﬁguration (E,F) should b e considered to hav e the same D-brane c harge as the conﬁguration (E ⊕ H,F ⊕ H): pairs of vector bundles with such an equiv alence relation constitute, in a n utshell, the basic ingredien ts for the K-theory group K(M) of the spacetime. In particular, Witten show ed that the construction used by Sen can b e generalized to arbitrary spacetime and D-brane conﬁgurations, and that suc h a generalization p erfectly corresp onds to mathematical prop erties of K-theory . More imp ortantly , Witten’s classiﬁcation of D-branes via K-theory applies not only to t yp e I IB/A su- p erstring theories, but also to t yp e I and to String theory on orbifolds : in these latter cases, indeed, the K-theory description mak es new and unexp ected predictions, as we will see in the follo wing c hapters. 7 The num b er of branes and antibranes needs to b e the same in type I IB to cancel the Ramond- Ramond tadp ole. 8 W e use the same notation H for b oth the gauge bundle on the D-brane and that on the anti-D- brane, as they are by deﬁnition top ologically equiv alent. 35 Chapter 3 K-Theory , an in tro duction 3.1 The group K 0 (X) In this section w e in tro duce the basic deﬁnitions used to construct the group K 0 (X) of a top ological space X. In particular, w e will restrict ourselv es to compact Haus- dorﬀ top ological spaces which can carry a structure of a ﬁnite CW-complex. Even if most of the constructions can be extended to more general top ological spaces, the c hoice of CW-complexes is not restrictiv e for the aim of this thesis, as we will mainly b e interested in the K-theory groups of ﬁnite dimensional manifolds, which naturally carry a canonical CW-complex structure. In the following exp osition w e tacitily refer to [4, 61], unless otherwise stated. Let V ect F (X) denote the set of isomorphism classes of topological F-v ector bundles o v er X, where F = C , R . The direct (Whitney) sum of vector bundles gives V ect F (X) the structure of an ab elian monoid: in a n utshell, the group K 0 (X) is constructed via a procedure that consists in tutiv ely in adding “in verses” to V ect F (X). In particular, the pro cedure can b e generalized to any ab elian monoid, which is the case w e will illustrate in the follo wing. Let A b e an ab elian monoid. W e can then asso ciate to A an abelian group K 0 ( A ) and a monoid homomorphism α : A → K 0 ( A ) with the the following universal prop- ert y . F or an y group G, and any morphism of the underlying monoids f : A → G, there is a unique group homomorpshim ˜ f such ˜ f α = f . Because of the uniqueness prop ert y , it is an immediate consequence that if suc h a K 0 ( A ) exists, then it is unique up to isomorphism. The group K 0 ( A ) is kno wn usually as the Gr othendie ck gr oup of A . A p ossible construction for the group K 0 ( A ) is the following. Denote with F( A ) the 36 The group K 0 (X) free ab elian group generated b y the elements of A , and let E( A ) be the subgroup of F( A ) generated by those elemen ts of the form a + b − ( a ⊕ b ), with a, b ∈ A , and ⊕ denoting the addition in A . Then for K 0 ( A ) := F( A ) / E( A ), the universalit y prop ert y ab o v e holds, with α : A → K 0 ( A ) the ob vious map. A diﬀeren t and sometimes con venien t construction of K 0 ( A ) is the follo wing. Consider the diagonal homomorphism of monoids ∆ : A → A × A , i.e the map ∆( a ) = ( a, a ), and denote with K 0 ( A ) the set of cosets of ∆( A ) in A × A . It is clearly a quotient monoid, but the interc hange of factors induces inv erses in K 0 ( A ), giving it a group structure. If w e deﬁne α : A → K 0 ( A ) to b e the comp osition of a → ( a, 0) with the natural pro jection A × A → K 0 ( A ), then the univ ersalit y prop ert y ab o ve holds. The asso ciation to a monoid A of its Grothendieck group as deﬁned ab ov e induces in an obvious w ay a unique co v ariant functor K 0 from the category A of ab elian monoids with monoid morphisms to the category G of ab elian groups with group morphisms. Indeed, to an y morphisim f : A → B , the functor K 0 asso ciates the mor- phism K 0 ( f ) : K 0 ( A ) → K 0 ( B ) sat ysfying α B ◦ f = K 0 ( f ) ◦ α A , and such a morphism is unique b y the univ ersalit y prop ert y . Example 3.1. Consider A := ( N 0 , +). Then K 0 ( A ) ' Z Example 3.2. Consider A := ( Z − { 0 } , · ). Then K 0 ( A ) ' Q − { 0 } . A fundamental example of the ab ov e construction that allo ws a higher degree of generality is the following. Let C an additiv e category , and denote with ˙ E the isomorphism class of the ob ject E. Then the set Φ( C ) of such classes can b e provided with a structure of an ab elian monoid if one deﬁnes ˙ E + ˙ F to b e ˙ E ⊕ F: the w ell- deﬁnedness of the + op eration and the algebraic iden tities derive from the additivit y of the category C . In this case one denotes with K 0 ( C ) the group K 0 (Φ( C )), called the Gr othendie ck gr oup of the c ate gory C . Moreov er, if ϕ : C → C 0 is an additiv e functor, then ϕ naturally induces a monoid homomorphism Φ( C ) → Φ( C 0 ), hence a group homomorphism K 0 (Φ( C )) → K 0 (Φ( C 0 )), denoted with ϕ ∗ . The usual comp osition rule follo ws. W e will no w sp ecialize to the case A = V ect F (X), for X a top ological space. W e denote with K 0 (X) the group K 0 (V ect F (X)), or equiv alen tly the Grothendiec k group of the additive category of top ological F-v ector bundles on X, that we will denote with V F . T o follow the notations usually found in literature, we will use K 0 (X) for F = C , and KO 0 (X) for F = R . When the results do not dep end on the c hoice of F, w e will use the notation for F = C . 37 The group K 0 (X) The basic constructions for the K 0 group describ ed ab ov e allow to pro v e the following basic, but v ery useful results. Prop osition 3.3. L et X b e a c omp act sp ac e. Then every element x ∈ K 0 (X) c an b e written in the form [E] − [F] , with E , F ve ctor bund les on X . Mor e over, [E] − [F] = [E 0 ] − [F 0 ] in K 0 (X) if and only if ther e exists a ve ctor bund le G on X such that E ⊕ F 0 ⊕ G ' E 0 ⊕ F ⊕ G Pr o of. By deﬁnition, x = [( ˙ E , ˙ F)], for some vector bundles E , F. Then w e ha v e that [( ˙ E , ˙ F)] = [( ˙ E , 0) + (0 , ˙ F)] = [( ˙ E , 0)] + [(0 , ˙ F)] = [E] − [F], where with [E] , [F] we de- note the comp osition of E → ˙ E with ˙ E → [ ˙ E]. Let [E] − [F] = [E 0 ] − [F 0 ]. Then one has that [E ⊕ F 0 ] = [E 0 ⊕ F], which implies ˙ E + ˙ F 0 + ˙ G = ˙ E 0 + ˙ F + ˙ G, for some G. Hence E ⊕ F 0 ⊕ G ' E 0 ⊕ F ⊕ G. The ab o v e prop osition allows almost immediately to prov e the following Prop osition 3.4. L et E and F b e ve ctor bund les over X . Then [E] = [F] if and only if E ⊕ θ n ' F ⊕ θ n , for some θ n a trivial bund le of r ank n. Pr o of. Recall that for an y v ector bundle G on a compact space X there exists a v ector bundle G 0 suc h that G ⊕ G 0 ' θ n , for some n , i.e. an y vector bundle is a pr oje ctive ob ject in V F . By Prop osition 3.3, if [E] = [F], then E ⊕ G ' F ⊕ G, for some vector bundle G. By adding G 0 c hosen as ab ov e, one gets that E ⊕ θ n ' F ⊕ θ n , whic h yelds the desired result. An easy consequence of Prop osition 3.3 is that any element x ∈ K 0 (X) can b e written as [H] − [ θ n ], for some v ector bundle H and some θ n . As we hav e seen, the group K 0 ( A ) “dep ends” co v arian tly on the monoid A . Ho wev er, K 0 (X) dep ends contra v ariantly on the topological space X, i.e. K 0 is a contra v ariant functor from the category T of top ological spaces with contin uous maps to G . More precisely , if w e let f : X → Y b e a contin uous map, then f induces a monoid morphism f ∗ : V ect F (Y) → V ect F (X) via the pullbac k of v ector bundles, hence a map K 0 (Y) → K 0 (X), whic h w e still denote with f ∗ . By the homotop y theory of v ector bundles one has immediately the follo wing Theorem 3.5. L et X and Y b e c omp act top olo gic al sp ac es, and let f 0 , f 1 : X → Y b e c ontinuous and homotopic maps. Then f 0 and f 1 induc es the same homomorphism K 0 (Y) → K 0 (X) . 38 The group K 0 (X) Hence, the group K 0 (X) is a top ological in v ariant, in the sense that t wo isomor- phic spaces ha ve isomorphic K 0 groups. More importantly , if the t w o spaces can b e “deformed” in to each other, they hav e isomorphic K 0 : this prop ert y is typical of a cohomology theory , and it is in tuitiv ely a ﬁrst indication that the group K 0 (X) is a building blo ck for a cohomology theory . In this sense, it is imp ortant to deﬁne the r e duc e d gr oup e K 0 (X). First notice that K 0 ( { pt } ) ' Z , as a vector bundle on a poin t is uniquely character- ized b y the dimension of its typical ﬁb er. Then, the inclusion i : x 0 → X induces the homomorphism i ∗ : K 0 (X) → K 0 ( { x 0 } ) ' Z The reduced group e K 0 (X) is deﬁned as the k ernel of i ∗ . Moreov er, the following exact sequence 0 → e K 0 (X) → K 0 (X) i ∗ − → K 0 ( { x 0 } ) → 0 canonically splits, i.e. K 0 (X) ' Z ⊕ e K 0 (X). Giv en a vector bundle E → X with E x the ﬁber of E ov er x , w e can deﬁne the r ank function of E rnk(E) : X → N 0 . As E is lo cally trivial as a vector bundle ov er X, its rank function is a lo cally constant function o ver X with v alues in N 0 , i.e. an elemen t of the ab elian monoid H 0 (X; N 0 ). Hence, the rank map extends naturally as the homomorphism rnk : K 0 (X) → H 0 (X; Z ) [E] − [F] → rnk(E) − rnk(F) (3.1.1) In the case in whic h X is a connected space, the integer (3.1.1) is called the virtual dimension of the class [(E , F)] in K 0 (X). Moreo ver, in the same case, the group e K 0 (X) is isomorphic to the kernel of the rank homomorphism, hence it consists of the subgroup of elements whose virtual dimension is zero, classes [(E , F)] with E and F of equal rank. W e conclude this section with a prop osition whic h states how the K-theory group b eha v es under disjoin t unions. Prop osition 3.6. L et X = ` n i =1 X i . Then the inclusions of the X i in X induc es the de c omp osition K 0 (X) ' K 0 (X 1 ) ⊕ K 0 (X 1 ) ⊕ · · · ⊕ K 0 (X n ) . Pr o of. Use the fact that any v ector bundle on X is c haracterized by its restrictions on the X i . 39 Relativ e K-theory and higher K-groups Remark Notice that this last prop osition is not true for the functor e K 0 (X). In- deed, if X is the disjoin t union of t w o p oin ts { x 0 } and { x 1 } , then e K 0 (X) ' Z , but e K 0 ( { x i } ) = 0, for i = 0 , 1. 3.2 Relativ e K-theory and higher K-groups Starting with the group K 0 (X) one can naturally deﬁne the r elative K-theory and the higher K-theory groups. Let T P denote the category of c omp act p airs (X , Y), where X , Y are top ological spaces, and X is suc h that it can b e equipp ed with a structure of a CW-complex such that Y is a CW-sub complex. The morphisms (X , Y) → (X 0 , Y 0 ) are given by r elative maps, i.e. con tinuous functions f : X → X 0 suc h that f (Y) ⊂ Y 0 . W e deﬁne the r elative K-the ory gr oup as K 0 (X , Y) := e K 0 (X / Y) (3.2.1) where X / Y is obtained from X b y shrinking Y to a p oint, with resp ect to which the reduced K-group is deﬁned. In the case when Y = { Ø } , w e deﬁne X / Y as X + := X ` { pt } . Hence, for a base-p oin ted space X one has K 0 (X , Ø) := e K 0 (X + ) ' K 0 (X) (3.2.2) Moreo v er, for { x 0 } ⊂ X we hav e K 0 (X , { x 0 } ) = e K 0 (X). Let us denote with π : X → X / Y . The map π induces the obvious relative map π : (X , Y) → (X / Y , { pt } ) hence the map π ∗ : K 0 (X / Y , { pt } ) → K 0 (X , Y) W e hav e the following excision the or em Theorem 3.7. The map π ∗ is an isomorphism. It is instructiv e, at this point, to explain a useful w a y to describ e vector bundles on the space X / Y , given a vector bundle E on X with typical ﬁb er F. Supp ose E 40 Relativ e K-theory and higher K-groups is trivializable on Y , i.e. there exists a homeomorphism α : E | Y ' − → Y × F which is linear on the ﬁbres. Consider the equiv alence relation on E | Y giv en b y v ∼ v 0 ⇔ p ◦ α ( v ) = p ◦ α ( v 0 ) where p : Y × F → Y is the canonical pro jection. The relation is then extended to the whole E. The corresp onding set of equiv alence classes can b e sho wn to b e the total space of a v ector bundle o ver X / Y ; moreov er, every vector bundle on X which is trivial when restricted to Y is isomorphic to the pullbac k of a vector bundle ov er X / Y . This construction, in particular, allo ws to pro v e that the follo wing sequence K 0 (X , Y) ρ ∗ − → K 0 (X) i ∗ − → K 0 (Y) (3.2.3) is exact, where the homomorphism ρ ∗ is induced b y the map ρ : X → X / Y , and i : Y  → X realizes Y as a subspace of X [61, 56]. T o deﬁne the higher K-groups, w e need to in tro duce some well known op erations on top ological spaces. Let X and Y be compact spaces with base p oin ts { x 0 } and { y 0 } , resp ectiv ely . The we dge pr o duct of X and Y is deﬁned as X ∨ Y := (X q Y) / { x 0 ∼ y 0 } (3.2.4) while the smash pr o duct of X and Y is deﬁned as X ∧ Y := X × Y / { X × { y 0 } ∪ { x 0 } × Y } (3.2.5) The t w o op erations ab o ve are related. Indeed, one has natural maps X ∨ Y → X × Y → X ∧ Y whic h allo w to write X ∧ Y = X × Y / X ∨ Y Moreo v er, the op erations ∨ and ∧ are asso ciative and commutativ e, and ∧ is dis- tributiv e ov er ∨ . This means, for example, that there is a canonical homeomorphism b et w een X ∧ Y and Y ∧ X. An imp ortan t prop ert y of the smash pro duct is that S n ' S 1 ∧ S 1 ∧ · · · ∧ S 1 n -times (3.2.6) where S n is the standard n -sphere with base p oin t. F or a given space X with base p oin t, w e deﬁne the n-th r e duc e d susp ension Σ n (X) as Σ n (X) := S n ∧ X (3.2.7) 41 Multiplicativ e structures on K-theory Because of property (3.2.6), w e ha v e that the n -th reduced susp ension Σ n (X) is the n -th iterated reduced susp ension of X. W e hav e then the following Deﬁnition 3.8. F or (X , Y) a compact pair, and for n ≥ 0 we deﬁne K − n (X , Y) := e K 0 (Σ n (X / Y)) F or X a compact space we put K − n (X) := K − n (X , Ø) := e K 0 (Σ n (X + )) and for X a compact space with base p oin t { x 0 } w e put e K − n (X) := K − n (X , { x 0 } ) := e K 0 (Σ n (X)) The K − n are con trav ariant functors, as the reduced suspension induces a co v ariant functor on T . The relation b etw een higher and reduced K-theory groups is giv en canonically b y K − n (X) ' e K − n (X) ⊕ e K − n ( { x 0 } ) with { x 0 } b eing the base p oin t. By the deﬁnitions ab ov e K − n ( { x 0 } ) ' e K 0 (S n ): we will compute these groups in section 3.4. The v arious higher K-groups are link ed together by the following semi-inﬁnite long exact sequence . . . K − ( n +1) (X) → K − ( n +1) (Y) δ − → K − n (X , Y) → K − n (X) → K − n (Y) δ − → . . . (3.2.8) where δ is the b oundary homomorphism. F or a deﬁnition of δ see [4, 61]. The se- quence ab o v e is another t ypical feature of a cohomology theory . 3.3 Multiplicativ e structures on K-theory As v ector bundles on a space X can b e “multiplied” together via tensor pro duct, it is natural to lo ok for m ultiplicativ e structures on the K-groups. The tensor pro duct of v ector bundles on X induces a m ultiplication K 0 (X) ⊗ Z K 0 (X) → K 0 (X) deﬁned as [(E , F)] ⊗ [(E 0 , F 0 )] := [(E ⊗ E 0 ⊕ F ⊗ F 0 , E ⊗ F 0 ⊕ F ⊗ E 0 )] (3.3.1) 42 Multiplicativ e structures on K-theory The expression ab ov e comes from writing [(E , F)] as [E] − [F], and formally imp osing distributivit y of the tensor pro duct on virtual bundles. Hence, K 0 (X) is actually a ring. There is another product, usually called the external tensor pro duct or cup pr o duct , whic h is a homomorphism ∪ : K 0 (X) ⊗ Z K 0 (Y) → K 0 (X × Y) (3.3.2) deﬁned as follo ws. Denote with π X : X × Y → X and π Y : X × Y → X the canonical pro jections. Hence, ∪ ([E] , [F]) ∈ K 0 (X) ⊗ Z K 0 (Y) is the class in K 0 (X × Y) deﬁned by ∪ ([E] , [F]) := π ∗ X ([E]) ⊗ π ∗ Y ([F]) and extended b y (bi)linearit y . Notice that when Y = X, we hav e that ∆ ∗ ∪ ([(E , F)] , [(E 0 , F 0 )]) = [(E , F)] ⊗ [(E 0 , F 0 )] where ∆ : X → X × X is the diagonal map. The ab ov e exterior pro duct is crucial when dealing with higher K-groups. In partic- ular, when restricted to reduced K-theory , the cup pro duct induces a homomorphism [4] e K 0 (X) ⊗ Z e K 0 (Y) → e K 0 (X ∧ Y) whic h immediately induces the homomorphism e K 0 (Σ n (X)) ⊗ Z e K 0 (Σ m (Y)) → e K 0 (Σ n + m (X ∧ Y)) (3.3.3) By substituting in the ab ov e expression X + and Y + for X and Y , resp ectively , one obtains the homomorphism K − n (X) ⊗ Z K − m (Y) → K − ( n + m ) (X × Y) (3.3.4) If we denote with K −∗ (X) := L n ≥ 0 K − n (X), the cup pro duct induces, via (3.3.4), the structure of a graded ring on K −∗ (X). Moreov er, for any space X with base p oin t { pt } , the cup pro duct mak es K −∗ (X) in to a graded mo dule o v er K −∗ (pt). Notice, at this p oin t, that a priori the ring K −∗ (X) could b e v ery complicated, ev en for the case X = { pt } . It’s an extremely remark able prop ert y of K-theory that this is not the case, as the follo wing results sho w. 43 Multiplicativ e structures on K-theory F or the ﬁrst time in this introduction, we hav e to make a diﬀerence b et w een complex and real K-theory to in tro duce a fundamen tal and deep result. Theorem 3.9. (Bott P erio dicit y) The ring K −∗ ( { pt } ) is a p olynomial algebr a gener ate d by an element u ∈ K − 2 ( { pt } ) ' e K 0 (S 2 ) , i.e. ther e is a ring isomorphism K −∗ ( { pt } ) ' Z [ u ] (3.3.5) The elemen t u can b e represented as u = [H] − [ θ 1 ], where H denotes the tautolo g- ic al complex line bundle ov er S 2 ' CP 1 . F or a pro of see [4]. The ab o ve theorem, in particular, sa ys that the map µ u : K − n ( { pt } ) → K − n − 2 ( { pt } ) induced b y m ultiplication b y u , is an isomorphism for all n . The previous theorem generalizes as Theorem 3.10. L et X b e a c omp act sp ac e. Then the map µ u : K − n (X) ' − → K − n − 2 (X) given by mo dule multiplic ation by u , is an isomorphism for al l n ≥ 0 The ab ov e theorem is usually referred to as the gener al Bott p erio dicity the or em , and essen tially states that there are only t w o “independent” K functors, namely K 0 and K − 1 . Moreov er, theorem (3.10) can b e extended to relative and reduced K-theory . Finally , the original simplest form ulation of the perio dicity theorem states that for an y compact space X, there is an isomorphism b etw een K 0 (X) ⊗ K 0 (S 2 ) and K 0 (X × S 2 ). In the real case things are a bit diﬀeren t, and sligh tly more complicated. Indeed, one has the follo wing real v ersion of the Bott p erio dicit y theorems. Theorem 3.11. The ring KO −∗ ( { pt } ) is gener ate d by elements η ∈ KO − 1 ( { pt } ) , y ∈ KO − 4 ( { pt } ) , x ∈ K O − 8 ( { pt } ) subje ct to the r elations 2 η = 0 , η 3 = 0 , η y = 0 , y 2 = 4 x i.e. ther e is a ring isomorphism K O −∗ ( { pt } ) ' Z [ η , y , x ] / < 2 η , η 3 , η y , y 2 − 4 x > (3.3.6) Theorem 3.12. L et X b e a c omp act sp ac e. Then the map µ x : K O − n (X) ' − → K O − n − 8 (X) given by mo dule multiplic ation by x , is an isomorphism for al l n ≥ 0 . 44 K-theory and classifying spaces The ﬁrst consequence of theorem (3.11) is that the ring KO −∗ ( { pt } ) is not freely generated, i.e. it contains torsion subgroups. Apart from limiting the n umber of K-groups to compute, the Bott p eriodicity theorem allo ws to deﬁne K-theory for p ositiv e degrees, whic h is the ﬁnal imp ortan t step needed to construct a cohomology theory asso ciated to the K-groups. Indeed, for n > 0 one deﬁnes K n (X , Y) as K n (X , Y) := K 0 (X , Y) , for n ev en K n (X , Y) := K − 1 (X , Y) , for n odd and analogous deﬁnitions are given for K O n (X , Y). In particular, the Bott isomor- phism is compatible with the long exact sequence (3.2.8), and hence it can b e extended to p ositiv e degrees. Because of Bott p erio dicit y , it is conv enient to deﬁne for any pair (X , Y) K ∗ (X , Y) := K 0 (X , Y) ⊕ K − 1 (X , Y) K O ∗ (X , Y) := L 7 i =0 K O − i (X , Y) The cohomology theory constructed in this wa y is referred to as K-theory , and satisﬁes all the usual axioms of a cohomology theory E on T P , but the dimension axiom , which requires the cohomology E to satisfy E i (pt) = 0, for i 6 = 0, and E 0 (pt) = Z . Finally , b ecause of Bott p erio dicity , the long exact sequence (3.2.8) for c omplex K- theory can b e truncated to the six-term exact sequence K − 1 (X) / / K − 1 (Y) δ / / K 0 (X , Y)   K − 1 (X , Y) O O K 0 (Y) δ o o K 0 (X) o o 3.4 K-theory and classifying spaces As for ordinary cohomology , there is a description of K-theory in terms of classifying spaces. As this approac h will play a very imp ortant role in later chapters, w e will review it in some detail. Let X a compact CW-complex of ﬁnite type. It is a v ery imp ortan t result in geometry that F-v ector bundles of rank k o v er X can b e classiﬁed up to isomorphism. Namely , there exists a connected top ological space BF k , equipp ed with a rank k F- v ector bundle E k , called the universal F-ve ctor bund le , such that V ect F k (X) ' [X , BF k ] 45 K-theory and classifying spaces where the isomorphism is induced by assigning to the homotopy class [ f ] the isomor- phism class [ f ∗ E k ] [56]. A mo del for BF k can b e deﬁned as follo ws. Denote with Gr( k , m ; F) the Gr assmannian manifold of F-linear subspaces in F m of dimension k . F or F = C one has Gr( k , m ; C ) ' U( m ) U( k ) × U( m − k ) while for F = R one has Gr( k , m ; R ) ' O( m ) O( k ) × O( m − k ) The inclusion F m ⊂ F m +1 induces the natural inclusion Gr( k, m ; F) ⊂ Gr( k , m + 1; F). T aking the inductiv e limit o v er suc h inclusions allo ws to deﬁne the inﬁnite Grassmannian Gr( k , ∞ ; F) := lim m →∞ Gr( k , m ; F) whic h can b e used as a mo del for our classifying space [56]. W e will use the notation BU( k ) for Gr( k , ∞ ; C ), and BO( k ) for Gr( k , ∞ ; R ). The univ ersal F-v ector bundle E k → Gr( k , ∞ ; F) is giv en b y assigning to a p oin t x ∈ Gr( k , ∞ ; F) the vector space represented by x itself. T o represen t K-theory we need a limit of the abov e construction ov er the vector bundle rank k . More precisely , denote with K 0 0 (X) the kernel of the rank homomorphism (3.1.1). Recall that when X is connected, K 0 0 (X) is isomorphic to e K 0 (X). In general one has the decomp osition K 0 (X) ' K 0 0 (X) ⊕ H 0 (X; Z ) (3.4.1) with H 0 (X; Z ) = [X , Z ]. Notice, at this point, that for [E k ] ∈ V ect F k (X), w e ha ve that [E k ] − [ θ k ] ∈ K 0 0 (X). If w e deﬁne V ect F ∞ (X) := lim k →∞ V ect F k (X) as the inductiv e limit o v er the inclusions V ect F k (X) ⊂ V ect F k +1 (X) induced b y the maps E k → E k ⊕ θ 1 , one has that the map [E k ] → [E k ] − [ θ k ] is a monoid morphism V ect F ∞ (X) → K 0 0 (X), hence V ect F ∞ (X) can b e given the structure of an ab elian group [61]. Finally , as V ect F ∞ (X) = [X , BF ∞ ] one has K 0 (X) ' V ect F ∞ (X) ⊕ [X , Z ] = [X , Z × BF ∞ ] 46 K-theory and classifying spaces where BF ∞ := S k BF k . Moreo v er one can obtain reduced K-theory for a base p oin ted space X as e K 0 (X) = [X , Z × BF ∞ ] ∗ where the base p oin t in Z × BF ∞ has b een c hosen to lie in 0 × BF ∞ . Consequently , higher relativ e K-theory can b e obtained as K − n (X , Y) = [Σ n (X / Y) , Z × BF ∞ ] ∗ ' [X / Y , Ω n ( Z × BF ∞ )] ∗ where Ω n is the iterated lo op space functor, and we hav e used the fact that for CW- complexes X , Y one has [Σ(X) , Y] ∗ = [X , Ω(Y)] ∗ . In particular, if X is a connected space, then e K 0 (X) = [X , BF ∞ ] ∗ In later sections we will see that another classifying space for K-theory is given by the space of F redholm op erators on a inﬁnite dimensional Hilb ert space. 3.4.1 Examples: K-theory of spheres and tori The reduced K-theory of spheres can b e deduced immediately b y the Bott p erio dicit y theorems (3.9) and (3.11): in this section we w an t to give a geometric view on the Bott p erio dicity theorems, at least in the complex case. Indeed, this is actually ho w the Bott theorem originated. Consider the standard n -dimensional sphere S n . W e will illustrate a pro cedure, called the clutching c onstruction , to construct v ector bundles E → S n . W rite the n -sphere S n as the union of the upp er and low er hemispheres D n + and D n − , such that D n + ∩ D n − ' S n − 1 . Given a con tin uous map f : S n − 1 → GL k ( C ), we can deﬁne the total space of a complex v ector bundle E f of rank k as E f := D n + × C k a D n − × C k / ∼ where the iden tiﬁcation ∼ is b et w een ( x, v ) ∈ ∂ D n + × C k and ( x, f ( x ) v ) ∈ ∂ D n − × C k , and the pro jection E f → S n is the ob vious one. Essen tially , the clutc hing construction is a sp ecial case of the gluing construction for lo cally trivial vector bundles, which one can see by considering a “small” strip S n × {− ,  } , and using the function f on eac h slice S n × { t } . Moreo ver, as the structure group for a complex vector bundle of rank k can alwa ys b e reduced to the unitary group U( k ), up on the introduction 47 K-theory and classifying spaces of a Hermitian metric, the function f can alw a ys be c hosen to b e a con tinuous map f : S n − 1 → U( k ). A basic prop erty of the clutc hing construction is that E f ' E g if f and g are homotopic maps. This allows to deﬁne a map Φ : [S n − 1 , U( k )] → V ect k (S n ) A fundamen tal result is the follo wing [61] Theorem 3.13. The map Φ is a bije ction. In other w ords, al l complex v ector bundles on spheres are obtained up to isomor- phism b y the clutc hing construction. By taking inductive limits, w e ﬁnally obtain V ect ∞ (S n ) = [S n − 1 , U( ∞ )] where U( ∞ ) denotes the inﬁnite unitary group. Hence w e ha v e e K 0 (S n ) = π n − 1 (U( ∞ )) (3.4.2) Because K − n (pt) ' e K 0 (S n ), w e see that the Bott p erio dicity theorem is a statement ab out the homotop y of the inﬁnite unitary group: indeed, this is ho w it w as originally stated 1 in [20]. The real case requires some modiﬁcations in the clutc hing construction, and, as it is exp ected from the structure of KO ∗ (pt), the relation with the orthogonal groups is more complicated. Notice, also, that as a top ological in v arian t for spheres, K-theory is a very coarse one, as it cannot detect the sphere dimensionalit y , in con trast to ordinary cohomology . The case for tori, instead, is quite diﬀeren t. T o compute the complex K-theory groups for tori, w e will use the follo wing result [4]: giv en t wo base p oin ted spaces X and Y w e ha v e the follo wing isomorphism e K − n (X × Y) ' e K − n (X ∧ Y) ⊕ e K − n (X) ⊕ e K − n (Y) (3.4.3) W e no w sp ecialize (3.4.3) to the case in whic h Y = S 1 , with the usual base p oint. In this case, w e ha v e that e K − n (X × S 1 ) ' e K − n (X ∧ S 1 ) ⊕ e K − n (X) ⊕ e K − n (S 1 ) ' e K − ( n +1) (X) ⊕ e K − n (X) ⊕ e K − ( n +1) ( { pt } ) 1 T o be precise, the statemen t w as on the homotop y groups of the unitary groups U( k ) in the stable r ange . 48 The A tiy ah-Bott-Shapiro isomorphism where w e ha v e used Bott p erio dicit y . Recalling that for o dd n the reduced and unreduced K-theory groups coincide, w e ha v e that K − 2 n (X × S 1 ) ' K − 1 (X) ⊕ K 0 (X) K − (2 n +1) (X × S 1 ) ' K 0 (X) ⊕ K − 1 (X) (3.4.4) Let T n ' S 1 × · · · × S 1 b e the standard n-dimensional torus. The K-theory for T n can b e computed using isomorphism (3.4.4) b y induction with base n = 2. Indeed, w e ha v e that K 0 (T 2 ) ' K − 1 (T 2 ) ' K 0 (S 1 × S 1 ) ' Z ⊕ Z As T n ' T n − 1 × S 1 , w e ha v e K 0 (T n ) ' K − 1 (T n ) ' K − 1 (T n − 1 ) ⊕ K 0 (T n − 1 ) Then, by induction w e ﬁnd that there is the following non-canonical isomorphism for n ≥ 2 K 0 (T n ) ' K − 1 (T n ) ' Z 2 ( n − 1) Hence, for tori the K-theory groups can indeed detect their dimension. 3.5 The A tiy ah-Bott-Shapiro isomorphism In this section we will introduce the Atiyah-Bott-Shapir o (ABS) isomorphism, whic h will giv e explicit representativ es for the generators of the rings K ∗ ( { pt } ) and K O ∗ ( { pt } ) via the so called “diﬀerence bundle construction”. More imp ortan tly , the ABS iso- morphism relates complex and real Cliﬀord algebras to K-theory: such a relation is someho w exp ected, giv en that the p erio dicity of K-theory is similar to the p erio dicit y of Cliﬀord algebras. The main reference is the seminal pap er [5]; moreov er, we will refer to App endix B for the basic notions of Cliﬀord algebras. T o construct the ABS isomorphism, w e need a reformulation of the relativ e K- theory group K 0 (X , Y), which will also prov e useful in relation to elliptic op erators, and in the description of D-branes, in particular in type I IA String theory . Again, this reform ulation is due to [5], and there is no diﬀerence b et w een the complex and real case. Deﬁnition 3.14. Let X and Y b e CW-complexes of ﬁnite type. F or n ≥ 1, denote with L n (X , Y) the set of elemen ts E = (E 0 , E 1 , · · · , E n ; σ 1 , σ 2 , · · · , σ n ), where E i is a 49 The A tiy ah-Bott-Shapiro isomorphism v ector bundle ov er X, σ i : E i − 1 | Y → E i | Y is a bundle morphism deﬁned on Y , suc h that 0 → E 0 | Y σ 1 − → E 1 | Y σ 2 − → . . . σ n − → E n | Y → 0 is an exact sequence of v ector bundles. W e will sa y that tw o such elemen ts E and E 0 are isomorphic if there are bundle isomorphisms ϕ i : E i → E 0 i o v er X suc h that the diagram E i − 1 | Y σ i / / ϕ i − 1   E i | Y ϕ i   E 0 i − 1 | Y σ 0 i / / E 0 i | Y comm utes for ev ery i . Finally , an elemen t E = (E 0 , E 1 , · · · , E n ; σ 1 , σ 2 , · · · , σ n ) is said to b e elementary if there is an i suc h that a) E i = E i − 1 and σ i = id b) E j = { 0 } , for j 6 = i or i − 1 The Whitney sum ⊕ of v ector bundles induces naturally an op eration on L n (X , Y). W e deﬁne the equiv alence relation ∼ on L n (X , Y) generated by isomorphisms and and addition of elementary elemen ts. Namely , we will sa y that tw o elements E , E 0 are equiv alen t if there are elemen tary elemen ts P 1 , P 2 , . . . , P k , Q 1 , Q 2 , . . . , Q l in L n (X , Y) suc h that E ⊕ P 1 ⊕ · · · ⊕ P k ' E 0 ⊕ Q 1 ⊕ · · · ⊕ Q l The set of all equiv alences classes in L n (X , Y) under ∼ is denoted b y L n (X , Y), and is an ab elian group under the op eration ⊕ . Moreov er, if Y = Ø, we will use the notation L n (X). Consider the natural map L n (X , Y) → L n +1 (X , Y) whic h asso ciates to the element (E 0 , E 1 , · · · , E n ; σ 1 , σ 2 , · · · , σ n ) the elemen t (E 0 , E 1 , · · · , E n , 0; σ 1 , σ 2 , · · · , σ n , 0). W e refer to [5] for the pro of of the following fundamental result Prop osition 3.15. F or e ach n ≥ 1 , the induc e d map L n (X , Y) → L n +1 (X , Y) is an isomorphism. Hence w e can fo cus on the group L 1 (X , Y), whose elemen ts are given by triples E = [E 0 , E 1 ; σ ]. The diﬀer enc e bund le c onstruction allows to associate to an y triple E an elemen t χ ( E ) in K 0 (X , Y) in the follo wing w a y . 50 The A tiy ah-Bott-Shapiro isomorphism First, set X k = X × { k } , for k = 0 , 1, and consider the space A = X 0 ∪ Y X 1 , obtained from the disjoin t union X 0 ` X 1 b y iden tifying y × 0 with y × 1 for an y y ∈ Y . Notice that the map ρ : A → X 1 is a retraction, i.e i ◦ ρ = id , i : X 1  → A. In this case, the exact sequence (3.2.3) b ecomes the split short exact sequence [4] 0 → K 0 (A , X 1 ) − → K 0 (A) i ∗ − → K 0 (X 1 ) → 0 (3.5.1) Moreo v er, the relativ e map (X , Y) → (A , X 1 ) whic h identiﬁes X with X 0 induces an isomorphism ϕ : K 0 (A , X 1 ) ' − → K 0 (X , Y). No w, from the element E = [E 0 , E 1 ; σ ] we construct (up to isomorphism) a vector bundle o v er A by setting F | X k := E k , and using σ to iden tify o ver Y . Recall, at this p oin t, that the map i ∗ is giv en essen tially by restricting vector bundles from A to X 1 . Hence, setting F 1 := ρ ∗ E 1 , the class [F] − [F 1 ] is in ker( i ∗ ) ⊂ K 0 (A). By (3.5.1), there exists a unique elemen t χ ( E ) ∈ K 0 (X , Y) such that π ∗ ϕ − 1 χ ( E ) = [F] − [F 1 ] In this w a y w e ha v e deﬁned a homomorphism χ : L 1 (X , Y) → K 0 (X , Y). The follo wing result allo ws the desired reform ulation of relativ e K-theory [5] Prop osition 3.16. The map χ is an isomorphism. The ab ov e proposition is easy to pro v e in the case in which Y = Ø. In this case, indeed, the map χ satisﬁes χ ([E 0 , E 1 ]) = [E 0 ] − [E 1 ] The surjectivit y of χ is ob vious. Supp ose that χ ([E 0 , E 1 ]) = 0: then there exists a vector bundle G such that E 0 ⊕ G ' E 1 ⊕ G. Hence the element E ⊕ G , where G is the elementary sequence deﬁned b y G, is isomorphic to the elemen tary sequence deﬁned b y E 1 ⊕ G, and hence it represen ts 0 in L 1 (X). With this reform ulation of relativ e K-theory , we can describe the ABS isomor- phism. Again, we refer the reader to App endix B for the relev an t notions of mo dules of Cliﬀord algebras. Let D n denote the unit disk in R n , and S n − 1 the b oundary ∂ D n . 51 The A tiy ah-Bott-Shapiro isomorphism Let W = W 0 ⊕ W 1 b e a Z 2 -graded mo dule o ver the Cliﬀord algebra C  n := C  ( R n ). T o the mo dule W w e can asso ciate the element ϕ (W) = [E 0 , E 1 ; µ ] ∈ K 0 (D n , S n − 1 ) (3.5.2) where E k := D n × W k , and µ is the bundle isomorphism E 0 → E 1 deﬁned ov er S n − 1 b y Cliﬀord m ultiplication µ ( x, v ) := ( x, x · v ) As the elemen t ϕ (W) depends only on the isomorphism class of W, and the map W → ϕ (W) is an additiv e homomorphism, it follo ws that (3.5.2) induces a homomorphism ϕ : c M C n → K 0 (D n , S n − 1 ) (3.5.3) where c M C n is the ab elian group freely generated by the irreducible complex graded C  n -mo dules. Consider no w the homomorphism i ∗ : c M C n +1 → c M C n induced b y restricting the action from C  n +1 to C  n . No w, let W b e a graded C  n -mo dule obtained from a C  n +1 - mo dule b y restriction. Then, the isomorphism µ deﬁned on S n − 1 can b e extended to all of D n b y setting ˜ µ ( x, v ) := ( x, ( x + p 1 − || x || 2 e n +1 ) · v ) where e n +1 ∈ R n +1 is a unit v ector orthogonal to R n . As E 0 and E 1 are isomorphic bundles o v er D n , the elemen t [E 0 , E 1 ; µ ] is 0 in K 0 (D n , S n − 1 ). Hence, the map (3.5.3) descends to the homomorphism ϕ n : c M C n /i ∗ c M C n +1 → K 0 (D n , S n − 1 ) (3.5.4) In complete analogy , in the real case w e ha v e the homomorphism ϕ R n : c M n /i ∗ c M n +1 → K O 0 (D n , S n − 1 ) (3.5.5) Recalling that K 0 (D n , S n − 1 ) = e K 0 (D n / S n − 1 ) ' e K 0 (S n ) Denote with c M C ∗ /i ∗ c M C ∗ +1 := M n ≥ 0 c M C n /i ∗ c M C n +1 the graded ring induced by the graded tensor pro duct of Cliﬀord mo dules, and the same for the real case. Finally , we can state the following fundamental result in K-theory 52 The A tiy ah-Bott-Shapiro isomorphism Theorem 3.17. Atiyah-Bott-Shapir o Isomorphisms. The maps ϕ n and ϕ R n induc e gr ade d ring isomorphisms ϕ ∗ : c M C ∗ /i ∗ c M C ∗ +1 → K −∗ ( { pt } ) ϕ R ∗ : c M ∗ /i ∗ c M ∗ +1 → K O −∗ ( { pt } ) As the perio dicity of the quotien ts c M C ∗ /i ∗ c M C ∗ +1 and c M ∗ /i ∗ c M ∗ +1 can be deduced b y the algebraic properties of Cliﬀord algebras, it may seem that the ab ov e theorem giv es an algebraic pro of of the Bott p erio dicity theorems. This is not the case, as the pro of of the ab o ve theorem in [5] actually inv okes the Bott result. The ABS isomorphisms can now be used to obtain explicit representativ es for K −∗ ( { pt } ) and K O −∗ ( { pt } ). Namely , consider S C = S + C ⊕ S − C , the fundamental Z 2 - graded complex represen tation of C  2 n . Then, by fundamen tal results in the theory of Cliﬀord algebras, the group c M C 2 n ' Z ⊕ Z is generated b y S C and ˜ S C , the graded mo dule obtained b y in terc hanging the factors in S C . Moreo v er, the homomorphism i ∗ maps the generator of the group c M C 2 n +1 ' Z to (S C , ˜ S C ) ∈ Z ⊕ Z . Hence, the generator for K − 2 n ( { pt } ) ' K 0 (D 2 n , S 2 n − 1 ) is given b y the elemen t σ C 2 n := [S + C , S − C ; µ ] In the case n = 1, σ C 2 is mapp ed b y the isomorphism K 0 (D 2 , S 1 ) ' e K 0 (S 2 ) to the class [H] − [ θ 1 ], where H is the tautological complex line bundle o v er S 2 ' CP 1 . Due to the presence of torsion, the generators for the ring K O −∗ ( { pt } ) are more complicated to describ e. As an example, we will consider the case n = 1, where we ha v e K O − 1 ( { pt } ) ' Z 2 . First, recall that to any graded mo dule W = W 0 ⊕ W 1 for C  n w e can assign the ungr ade d mo dule W 0 for the Cliﬀord algebra C  0 n ' C  n − 1 , where C  0 n denotes the ev en part of C  n . The con v erse is also true: giv en an ungraded mo dule W 0 for the Cliﬀord algebra C  n − 1 , the mo dule W := C  n ⊗ C ` 0 n W 0 is naturally a graded mo dule for the Cliﬀord algebra C  n . This induces the isomorphism c M n ' M n − 1 , where M n − 1 denotes the ungraded v ersion of c M n − 1 , and consequen tly the isomorphism c M n /i ∗ c M n +1 ' M n − 1 /i ∗ M n In the case n = 1, C  0 ' R and C  1 ' C : taking the real and complex dimension of the ungraded mo dules giv es the isomorphisms M 0 ' Z and M 1 ' Z , resp ectiv ely . 53 K-theory and Spin c manifolds The map i ∗ : M 1 → M 0 is giv en b y considering a complex v ector space to b e a real v ector space under restriction of scalars. It induces a homomorphism Z → Z given b y m ultiplication b y 2, hence M 0 /i ∗ M 1 ' Z 2 . Then, the generator for KO − 1 ( { pt } ) is giv en b y the elemen t in K O 0 (D 1 , S 0 ) σ 1 := [D 1 × R , D 1 × R ; µ ] where µ (1 , v ) = v µ ( − 1 , v ) = − v Moreo v er, σ 1 = [H 1 ] − [ θ 1 ] ∈ g K O 0 (S 1 ), where H 1 is the inﬁnite M¨ obius bund le ov er the circle, i.e. the tautological real line bundle ov er RP 1 ' S 1 . 3.6 K-theory and Spin c manifolds In this section, we will sp ecialize to the case in which the CW-complex X is a smo oth ﬁnite dimensional manifold. Being a top ological in v arian t, K-theory do es not dep end on the presence of an y smo oth structure: nev erthless, restricting to manifolds pro- vides a strong connection b etw een K-theory and the theory of elliptic op erators, and, moreo v er, it is the right framework for the description of D-branes in String theory . Again, all the following results for whic h a smo oth structure is not explicitly required, are to b e considered v alid in a more general con text. Before introducing the notions of K -orien tation and the Thom isomorphism, w e need some basic preliminary results. In the following w e will consider the K-theory of total spaces of vector bundles, which are lo cally compact spaces. F or a lo cally compact space X we deﬁne K 0 cpt (X) := e K 0 (X + ) and K − n cpt (X) := K 0 cpt (X × R n ) The functors K − n cpt are deﬁned on the category of locally compact spaces and prop er maps, and they constitute the K-the ory with c omp act supp ort . Moreo v er, w e ha v e that K − n cpt (X , Y) := K 0 cpt ((X − Y) × R n ) 54 K-theory and Spin c manifolds Analogous deﬁnitions can b e given for KO-theory , and the Bott p erio dicit y theorems assume the form K 0 cpt (X) ' K 0 cpt (X × C ) K O 0 cpt (X) ' KO 0 cpt (X × R 8 ) No w, let X b e a compact space, and let E → X b e a complex vector bundle. The one p oin t compactiﬁcation E + is called the Thom sp ac e , or Thom c omplex of E. By deﬁnition, e K 0 (E + ) = K 0 cpt (E). Consider the (unit) b al l and spher e bundle of E, denoted as B(E) and S(E), resp ec- tiv ely , and deﬁned as B(E) := { e ∈ E : ϕ ( e , e ) ≤ 1 } S(E) := { e ∈ E : ϕ ( e , e ) = 1 } for ϕ a Hermitian metric on E, and the pro jection is the ob vious one. By noticing that B(E) − S(E) ' E, we hav e B(E) / S(E) ' E + and consequen tly K 0 cpt (E) := e K 0 (E + ) ' e K 0 (B(E) / S(E)) := K 0 (B(E) , S(E)) The isomorphism ab o v e do es not dep end on the choice of the Hermitian metric ϕ , and a similar construction follo ws for K O-theory . In the follo wing, we will often interc hange K 0 cpt (E) and K 0 (B(E) , S(E)) freely using the isomorphism ab o v e. 3.6.1 K-orien tation and Thom isomorphism In the same wa y that an orientation of a v ector bundle E → X induces a class τ ∈ H ∗ cpt (E) and a ring isomorphism H ∗ cpt ( E ) ' H ∗ (X) via cup pro duct with τ , the existence of certain structures on v ector bundles will induce analogous results for K- and K O-theory . Let π : E → X be a complex or real v ector bundle, and denote with k ∗ the functor K ∗ cpt or K O ∗ cpt , or, more generally , any multiplicativ e generalized cohomology theory . Deﬁne the map j ∗ : k ∗ (X × E) → k ∗ (E) induced b y the map j ( e ) = ( π ( e ) , e ). Then, the comp osition k ∗ (X) × k ∗ (E) ∪ − → k ∗ (X × E) j ∗ − → k ∗ (E) 55 K-theory and Spin c manifolds canonically equips k ∗ (E) with a structure of left mo dule o v er k ∗ (X). In the follo wing we will give a somewhat general deﬁnition of orientation for k ∗ , as it will b ecome imp ortan t later in the context of K-homology . Deﬁnition 3.18. An n -dimensional v ector bundle π : E → X is said to b e k ∗ - orientable if there exists a class τ ∈ k n (E) such that i ∗ x τ is a generator of k n (E x ) for eac h ﬁb er inclusion i x : E x  → E. A choice of suc h a class τ is called a k ∗ -orientation for π , and π is said to b e k ∗ -oriente d by τ . F or a k ∗ -orien ted v ector bundle one has the follo wing fundamen tal general result [38, 88] Theorem 3.19. L et π : E → X b e an n-dimensional ve ctor bund le which is k ∗ - oriente d by the class τ ∈ k n (E) . Then the homomorphism T X , E : k i (X) → k i + n (E) deﬁne d by T X , E ( ξ ) := π ∗ ( ξ ) ∪ τ is an isomorphism. The isomorphism T X , E is called the Thom isomorphism . As a corollary of theorem (3.19), we ha ve that k ∗ (E) is a free k ∗ -mo dule with generator τ , and hence is com- pletely kno wn once k ∗ (X) is. Giv en a vector bundle, one is in terested in ﬁnding suﬃcient , if not necessary con- ditions for orien tation classes to exist, and, more generally , to inv estigate whic h are the p ossible obstructions to suc h an existence. F or this, consider the trivial bundle E = X × C m π − → X and consider the class τ (E) := [Λ ev en C C m , Λ odd C C m ; σ ] ∈ K 0 (B(E) , S(E)) where C m = π ∗ E , and where σ ( x,v ) ( ϕ ) := v ∧ ϕ − i v ∗ ϕ with i v ∗ giv en b y con traction. See App endix B for details. No w, the restriction of τ (E) to the ﬁb er is giv en b y the class [Λ ev en C C m , Λ odd C C m ; σ ] ∈ K 0 cpt ( C m ) = K 0 (D 2 m , S 2 m − 1 ) 56 K-theory and Spin c manifolds where w e ha v e iden tiﬁed C m ' R 2 m . Up on the iden tiﬁcation Λ ∗ C m ' C  2 m , and noticing that σ is giv en b y Cliﬀord m ul- tiplication b y the elemen t v , the class ab o v e can b e rewritten as [S + C , S − C ; µ ] whic h b y the ABS isomorphism is a generator of K 0 (D 2 m , S 2 m − 1 ). As any complex v ector bundle E is lo cally trivializable, and there is no obstruction to construct Λ ∗ C E globally , we hav e the following Theorem 3.20. L et π : E → X b e a c omplex hermitian ve ctor bund le over a c omp act sp ac e X . Then the class τ K (E) := [Λ ev en C π ∗ E , Λ odd C π ∗ E; σ ] ∈ K 0 cpt (E) wher e σ e ( ϕ ) := e ∧ ϕ − i e ∗ ϕ , is a K ∗ -orientation class for π . Hence, any complex vector bundle is K ∗ -orien table, and there is no obstruction presen t. The case for real vector bundles is instead diﬀerent. Indeed, consider the 8 n -dimensional real trivial v ector bundle E = X × R 8 n π − → X and consider the class τ K O ( E ) := [ π ∗ S + (E) , π ∗ S − (E); µ ] ∈ K O 0 cpt (E) where S + (E) ⊕ S − (E) = S(E) = X × W, with W the irreducible real graded C  8 n - mo dule, and µ is Cliﬀord multiplication. Again, b y Bott p erio dicit y , the class τ K O ( E ) giv es the generator of K O 0 cpt ( R 8 n ) = KO 0 (D 8 n , S 8 n − 1 ), when restricted to the ﬁb er. Giv en an arbitrary real vector bundle E, the bundle S(E) will exist if and only if E admits a spin structure. Hence, w e ha v e the follo wing Theorem 3.21. L et π : E → X b e a r e al 8n-dimensional ve ctor bund le with a spin structur e over a c omp act sp ac e X . Then the class τ K O (E) := [ π ∗ S + (E) , π ∗ S − (E); µ ] ∈ K O 0 cpt (E) wher e µ e ( ϕ ) = e · ϕ is Cliﬀor d multiplic ation, is a K O ∗ -orientation for π . 57 K-theory and Spin c manifolds In the same w a y , the follo wing theorem giv es suﬃcien t conditions for a real vector bundle to b e K ∗ -orien table. Theorem 3.22. L et π : E → X a r e al 2n-dimensional ve ctor bund le with a spin c structur e over a c omp act sp ac e X . Then the class τ K (E) := [ π ∗ S + C (E) , π ∗ S − C (E); µ ] ∈ K 0 cpt (E) wher e S C = S + C ⊕ S − C is the irr e ducible spinor bund le asso ciate d to the spin c structur e on E , and µ e ( ϕ ) = e · ϕ is Cliﬀor d multiplic ation. Then τ K (E) is a K ∗ -orientation for π . Thanks to the results abov e, we can construct an orientation class for a spin c v ector bundle of arbitrary rank in the follo wing w ay , with the spin case requiring minor mo diﬁcations. Let π : E → X b e a spin c v ector bundle of rank n , and consider the Whitney sum E ⊕ θ p , with p suc h that n + p = 2 m . If we equip θ p with its canonical spin c structure, then the sum E ⊕ θ p is a spin c v ector bundle of rank 2 m . Hence, b y Theorem 3.22 there exists a Thom class τ K (E ⊕ θ p ) ∈ K 0 (B(E ⊕ θ p ) , S(E ⊕ θ p )) By using the follo wing isomorphisms [38, 88] K 0 (B(E ⊕ θ p ) , S(E ⊕ θ p )) := e K 0 (B(E ⊕ θ p ) / S(E ⊕ θ p )) ' e K 0 (Σ p (B(E) / S(E)) ' K − p (B(E) , S(E)) ' K n (B(E) , S(E)) it follo ws that the class τ K (E ⊕ θ p ) is an orien tation class for the v ector bundle π . It is imp ortant to notice that, in con trast to the results in Theorem 3.22, the con- struction of the Thom class ab ov e is not natural, as we hav e to make several choices in the pro cess, starting from that of a spin c structure on E ⊕ θ p . Neverthless, the induced Thom isomorphism will b e an essential ingredient in the next c hapter, where w e will discuss a more natural description of D-branes in terms of K-homology . 3.6.2 Chern Character and Gysin homomorphism In this section w e will brieﬂy introduce the homomorphisms induced b y the Chern c haracter and the Thom isomorphism. W e refer to App endix C for the basic notions 58 K-theory and Spin c manifolds on c haracteristic classes for v ector bundles. Let E → X b e a complex v ector bundle. The Chern character ch(E) is an element in the rational cohomology H ev (X; Q ) constructed from the total Chern class of E. The imp ortance of the Chern c haracter lies in the fact that it resp ects the (semi)additive and m ultiplicative structure on V ect(X). Namely , for E and E 0 v ector bundles on X w e ha v e c h(E ⊕ E 0 ) = ch(E) + ch(E 0 ) c h(E ⊗ E 0 ) = ch(E) ∪ ch(E 0 ) Hence, w e can deﬁne the follo wing homomorphism c h : K 0 (X) → H ev (X; Q ) [E] − [F] → ch(E) − ch(F) (3.6.1) It is easy to sho w that the Chern character homomorphism (3.6.1) for a class x do es not dep end on its represen tation in terms of vector bundles, hence it is a w ell deﬁned ring homomorphism. The Chern c haracter homomorphism can b e extended to a group homomorphism c h : K − n (X , Y) → H ∗ (X , Y; Q ) b y deﬁning c h( x ) := α (( σ n ) − 1 c h( x 0 )) x ∈ K − n (X , Y) (3.6.2) where x 0 is the class corresp onding to x in e K 0 (Σ n (X / Y)), σ n is the susp ension iso- morphism in cohomology , and α is the canonical isomorphism α : e H ∗ (X / Y; Q ) ' − → H ∗ (X , Y; Q ) Moreo v er, one can prov e that the homomorphism (3.6.2) is compatible with Bott p erio dicit y: this allo ws to deﬁne c h( x ) for x ∈ K n (X , Y), where n is an y in teger. Finally , by using a sp ectral sequence argumen t, the follo wing fundamental result w as pro v en in [7] Theorem 3.23. L et X b e a ﬁnite CW-c omplex. Then the homomorphism c h ⊗ id : K ∗ (X) ⊗ Q → H ∗ (X; Q ) is an isomorphism and maps K 0 (X) ⊗ Q onto H ev (X; Q ) and K 1 (X) ⊗ Q onto H odd (X; Q ) . As a corollary of Theorem 3.23, when X is a ﬁnite CW-complex, K ∗ (X) is a ﬁnitely generated abelian group. This tec hnical simpliﬁcation in a sense justiﬁes the 59 K-theory and Spin c manifolds restriction to CW-complexes. In the case in which X is a smo oth manifold, the Chern character homomorphism is closely related to the Gysin homomorphism, whic h we will illustrate in the following. Consider tw o lo cally compact manifolds X and Y, such that dimY − dimX = 0 mo d 2, and let f : X → Y b e a prop er embedding 2 . Denote with ν X the normal bundle of X in Y , and supp ose it is equipp ed with a spin c structure, whic h is a restriction on the map f . Moreo v er, identify this normal bundle with a tubular neigh b ourho o d N of X in Y . Then, we can deﬁne the Gysin homomorphism f ∗ : K 0 cpt (X) → K 0 cpt (Y) as the comp osition K 0 cpt (X) T X ,ν X − − − → K 0 cpt ( ν X ) ' K 0 cpt (N) j ∗ − → K 0 cpt (Y) where the map j ∗ is induced b y the map j : Y + → N + deﬁned as j ( z ) = z , ∀ z ∈ N j ( z ) = ∞ , ∀ z / ∈ N with ∞ denoting the p oint that compactiﬁes N. Intuitiv ely , j ∗ extends a class in K 0 cpt (N) via trivial bundles, and is usually referred to as “extension b y zero”. The Gysin homomorphism is also called the “wrong wa y” morphism, as it “goes” in the direction opp osite to the con tra v ariance of the K-functo r. The Gysin homomorphism do es not dep end on the tubular neighbourho o d N, and only dep ends on the homotop y class of f , as the v ector bundle ν X is deﬁned as f ∗ (TY ) / TX, and the K-functor is homotop y in v arian t. Moreo v er, the Gysin homomorphism enjo ys the following functorialit y prop erty , whic h will b e v ery useful in later c hapters. Let f : X → Y and g : Y → Z b e prop er em b eddings satisying the hypothesis abov e. Then ( g ◦ f ) ∗ = g ∗ f ∗ (3.6.3) As the Thom isomorphism can b e extended to higher K-groups, the Gysin homomor- phism can b e extended to a homomorphism f ∗ : K − 1 cpt (X) → K − 1 cpt (Y) 2 The embedding condition can b e relaxed. See [61]. 60 K-theory and t yp e I IA/B D-branes Moreo v er, by recalling the discussion on orien tation classes in section 3.6.1, we can relax the condition that dimY − dimX = 0 mo d 2. Indeed, in general, w e will ha ve a Gysin homomorphism f ∗ : K i cpt (X) → K i + n cpt (Y) where n is the rank of the normal bundle. Again, the price we pa y is that the homomorphism f ∗ so obtained do es not induce a natural transformation. Finally , along the same lines, one can deﬁne a Gysin homomorphism in ordinary cohomology f H ∗ : H ∗ (X; Q ) → H ∗ (Y; Q ) As men tioned b efore, the Chern character homomorphism and the Gysin homo- morphism are closely related via the Atiy ah-Hirzebruch v ersion of the Riemann-Ro c h theorem [6]. Consider tw o lo cally compact manifolds X and Y and a prop er embedding f : X → Y as ab ov e, and denote with d ( ν X ) the cohomology class deﬁning the spin c structure on the normal bundle ν X . Then we ha v e the follo wing Theorem 3.24. ( Riemann-R o ch ) F or e ach class x ∈ K 0 (X) we have the r elation c h( f ∗ ( x )) = f H ∗ ( e d ( ν X ) / 2 ˆ A ( ν X ) c h( x )) This form of the Riemann-Ro ch theorem will b e used in the next section, where w e will see how the K-theoretic mac hinery developed so far is related to D-branes in String theory . 3.7 K-theory and t yp e I IA/B D-branes Ha ving developed the necessary notions in the previous sections, w e will describ e how K-theory is related to D-branes in sup erstring theory . In particular, in this section w e will consider t yp e I I String theory on a 10-dimensional spin manifold M, while in the next section w e will in v estigate t yp e I String theory . W e will presen t t w o supp orting argumen ts for the K-theoretic description of D-branes, based on the fact that D-branes are curren ts for Ramond-Ramond ﬁelds, and on Sen’s conjectures, resp ectiv ely . W e ﬁrst need some comments on the anomalous coupling (2.5.7) b etw een the total Ramond-Ramond p oten tial and a D-brane. Recall that in section 2.4 w e made the assumption that the w orldv olume Q wrapp ed by the D-brane is a spin manifold, p oin ting out that this condition is not necessary from a string theoretic p oint of view. 61 K-theory and t yp e I IA/B D-branes In presence of a D-brane the partition function of the sup erstring dev elops an anomaly , called the F r e e d-Witten anomaly [45]: in a top ologically trivial B-ﬁeld setting, this anomaly cancels exactly when the normal bundle of Q in M admits a spin c structure. Since M is a spin manifold, the manifold Q necessarily admits a spin c structure. Moreo v er, the anomalous coupling is mo diﬁed as Z Q i ∗ C ∧ e d ( ν Q ) / 2 c h(E) i ∗ q ˆ A (TM) 1 ˆ A (TQ) (3.7.1) whic h coincides with the coupling (2.5.7) when Q is spin manifold, as in this case the class d ( ν Q ) v anishes. Consider the cohomology class j Q = e d ( ν Q ) / 2 c h(E) i ∗ q ˆ A (TM) 1 ˆ A (TQ) As w e ha ve discussed in section 2.5, the class j Q represen ts the charge asso ciated to the Ramond-Ramond current generated b y the D-brane wrapping Q. More precisely , for such an in terpretation to b e v alid, the class j Q needs to b e “pushed” in to the spacetime manifold M as i H ∗ ( j Q ) (3.7.2) where w e recall that i : Q → M is the embedding map. By noticing that ν Q ' i ∗ TM / TQ and using the fact that the ro of gen us is a c haracteristic class, w e ha v e the iden tit y ˆ A ( ν Q ) = i ∗ ˆ A (TM) / ˆ A (TQ) Hence, w e can rewrite (3.7.2) as i H ∗ ( e d ( ν Q ) / 2 ˆ A ( ν Q )c h(E) 1 i ∗ q ˆ A (TM) ) By using the follo wing prop ert y of the cohomological Gysin homomorphism i H ∗ ( α ∪ i ∗ β ) = i H ∗ ( α ) ∪ β , ∀ α ∈ H ∗ (Q; Q ) , ∀ β ∈ H ∗ (M; Q ) w e ha v e that i H ∗ ( e d ( ν Q ) / 2 ˆ A ( ν Q )c h(E) 1 i ∗ q ˆ A (TM) ) = i H ∗ ( e d ( ν Q ) / 2 ˆ A ( ν Q )c h(E)) ∪ 1 q ˆ A (TM) 62 K-theory and t yp e I IA/B D-branes Finally , by using the Riemann-Ro ch theorem we hav e i H ∗ ( j Q ) = ch( i ∗ [E]) ∪ 1 q ˆ A (TM) (3.7.3) The iden tity (3.7.3) strongly suggests that the charge for a D-brane wrapping a w orld- v olume Q equipp ed with a Chan-Paton bundle E is naturally given b y the element i ∗ [E] ∈ K 0 (M). This argumen t was dev elop ed in [71], and constitutes the ﬁrst evidence for the imp or- tance of K-theory in String theory . Ho w ev er, the argumen t is strictly sp eaking only v alid o v er Q , where the map c h ∪ 1 q ˆ A (TM) : K ∗ (M) ⊗ Q → H ∗ (M; Q ) is a group isomorphism. In this case, then, the cohomological description of D-branes “coincides” with the K-theoretical one, with the latter ha ving the adv an tage of b eing more natural. An argumen t whic h relates the full K-theory groups to D-brane charges is based on Sen’s conjectures, as discussed in section 2.6: of course, the price w e pa y consists in the fact that w e hav e to in v ok e (op en) string tac hy on condensation, which is not y et fully understo o d, b oth from the physical and the mathematical p oint of view. Recall from section 2.6 that Witten made the observ ation that in type IIB String theory , a conﬁguration of n D9-branes and n D9-branes (E , F) has to be considered equiv alen t to the conﬁguration (E ⊕ H , F ⊕ H), b ecause the brane-an tibrane system (H , H) is able to deca y in the string v acuum, as conjectured b y Sen. As we ha ve seen in this chapter, the equiv alence classes of conﬁgurations (E , F) under brane-an tibrane annihilation are elements in K 0 (M), or more precisely e K 0 (M), as the bundles E and F ha v e the same rank. Moreo v er, in [94] Witten was able to K-theoretically interpret Sen’s construction, whic h allo ws to obtain D p -branes with p < 9 as the decay pro duct of a system of brane-an tibranes of higher dimension. According to this in terpretation, the group of p ossible c harges for a D p -brane wrapping a spin c submanifold Q ⊂ M is given by K 0 cpt (N) ' K 0 (B(N) , S(N)) with N denoting the tubular neighbourho o d identifying the normal bundle to Q in M. In particular, given a single D p -brane wrapping Q with trivial Chan-P aton bundle, the system of D9-D9-brane deca ying to Q is giv en b y i ∗ (1) ∈ K 0 (M) 63 K-theory and t yp e I IA/B D-branes where 1 denotes the class of the trivial line bundle o v er Q. Applying the ab ov e machinery to a ﬂat D p -brane wrapping R 1 ,p ⊂ M 10 , with M 10 the ten dimensional Minko wski spacetime, we obtain that the its c harge group is giv en b y K 0 cpt (B(N) , S(N)) ' K 0 cpt ( R 1 ,p × D 9 − p , R 1 ,p × S 9 − p ) ' e K 0 (S 9 − p ) where w e ha v e used the compact supp ort relativ e K-theory , as R 1 ,p is not compact. As e K 0 (S 9 − p ) = 0 for p even, w e ﬁnd that the K-theoretical description agrees with, and in a certain sense justiﬁes the statement that o dd-dimensional (ﬂat) D-branes in t yp e I IB String theory are necessarily unstable, as they cannot carry any D-brane c harge. The discussion for t yp e I IA is less straightforw ard. Indeed, the fact that Sen’s con- struction w orks in even co dimension let Witten prop ose to relate branes not to bun- dles on M, but to bundles on S 1 × M. Namely , given a D p -brane wrapping an o dd- dimensional submanifold Q ⊂ M, one identiﬁes Q with pt × Q in S 1 × M, where pt is an y point in S 1 . Applying Sen’s construction, the D p -brane wrapping Q determines an elemen t in K 0 (S 1 × M), whic h turns out to b e trivial when restricted to pt × M. By using the isomorphism K − 1 (M) ' ker  K 0 (S 1 × M) → K 0 (pt × M)  it follows that type I IA brane-antibrane conﬁgurations hav e to b e considered equiv a- len t if they determine the same elemen t in K − 1 (M). Of course, the construction abov e is not natural, as w e ha v e to mak e a c hoice of ho w to em b ed Q in S 1 × M. Similarly , the group of charges for a D p -brane wrapping Q is given by K − 1 cpt (B(N) , S(N)) whic h agrees with the kno wn results for ﬂat D p -branes in Mink o wski spacetime. F rom our p oin t of view, the app earence of the functor K − 1 can be motiv ated in the follo wing wa y . As w e hav e seen in section 3.6.2, a Gysin homomorphism for a map i : X → Y such that the normal bundle admits a spin c structure do es alwa ys exist. Hence, giv en a D-brane wrapping an o dd co dimension manifold Q equipp ed with a trivial Chan-P aton bundle, its D-brane c harge is giv en b y the elemen t i ∗ (1) ∈ K − 1 cpt (M) The classes in K − 1 cpt (M) do not represent a brane-an tibrane system: instead, they can can b e conv eniently represen ted as a system of D9-branes equipp ed with the tach yon ﬁeld causing the instabilit y of D9-branes in T yp e I IA String theory . W e refer the reader to [55] for details on this construction. 64 K O-theory and T yp e I D-branes: torsion eﬀects 3.8 K O-theory and T yp e I D-branes: torsion ef- fects As we men tioned in chapter 1, Type I String theory is a theory of op en and closed strings, including oriented as well as unoriented worldsheets. The F o ck space for a String theory of unorien ted worldsheets is obtained by considering states in the Type I I F o c k space which are in v arian t under the worldshe et p arity op er ator Ω, induced by rev ersing the orien tation on the w orldsheet [79, 33]. The op erator Ω induces an action also on the Chan-Paton bundle to a D-brane, forc- ing it to b e an O( n )-bundle. Moreo v er, the Ramond-Ramond ﬁeld con ten t in Type I diﬀers from that of T yp e I I: indeed, the Ramond-Ramond gauge theory in T yp e I String theory consists of p -forms, for p =2,6. Finally , F reed-Witten anomaly cancellation imposes that a D-brane can only wrap a spin manifold. The right framew ork to describe D-brane charges is then K O-theory . Namely , a conﬁg- uration of T yp e I D9-branes (E , F) mo dulo brane-antibrane annihilation is represen ted b y a class x in K O 0 (M). Moreo v er, as in Type I I String theory , given a D p -brane wrapping a spin manifold Q ⊂ M, the group of its admissable charges is given by K O 0 (B(N) , S(N)) Applying this classiﬁcation to a D p -brane wrapping R 1 ,p ⊂ M 10 , w e obtain K O 0 cpt (B(N) , S(N)) ' K O 0 cpt ( R 1 ,p × D 9 − p , R 1 ,p × S 9 − p ) ' g K O 0 (S 9 − p ) where g K O 0 (S n ) '        Z , n = 0 , 4 mo d 8 Z 2 , n = 1 , 2 mo d 8 0 , otherwise W e ha v e g K O 0 (S 9 − p ) ' Z , for p = 1 , 5 , 9: for p = 1 , 5, this agrees with the fact that the corresp onding sup ersymmetric D-branes couple to Ramond-Ramond ﬁelds, and hence are stable. Moreov er, w e would exp ect these to b e the only stable D-branes in T yp e I String theory . A new prediction of K-theory is that this is not the case. Indeed, for p = 7 , 8, w e ha v e g K O 0 (S 9 − p ) ' Z 2 : hence, D7-branes and D8-branes can carry a c harge that w ould protect them from decaying. Morever, since the group Z 2 is a pure torsion group, this c harge is not associated to an y spacetime ﬁeld coupling 65 K O-theory and T yp e I D-branes: torsion eﬀects to the D-brane, and has to b e considered as a purely top ological eﬀect. The K-theoretic description also naturally describ es stable ob jects for p = − 1 and p = 0, which are called the D-instanton and the D-p article . In contrast to Type I I String theory , constructing a system of spacetime ﬁlling brane- an tibranes which has a given D p -brane as its decay product is not v ery systematic, in the sense that there is no uniﬁed pro cedure, or homomorphism, realizing this construction. Indeed, in [94] the diﬀeren t stable D p -branes are treated with diﬀerent metho ds. This is due to the fact that the Thom isomorphism in K O-theory implies that K O 0 cpt (B(N) , S(N)) ' K O − p cpt (Q) (3.8.1) where Q is the wrapp ed manifold, and p is the rank of the normal bundle of Q in M. As we ha v e seen, elements of KO − p cpt (Q) are not represented by “diﬀerences” of v ector bundles on Q, hence there is no natural w a y to iden tify them as a system of D-branes. Indeed, the higher KO-groups only pla y a role through the isomorphism (3.8.1), whic h is one of the limitations of the K-theorical description of D-branes. W e will address a p ossible solution to this problem in the next c hapter, where w e will dev elop another description of D-branes, based on KO-homology , the dual theory to KO-theory . W e refer to [76] for an alternativ e interpretation of the higher K O-groups, and to [75] for extensiv e review on the application of K-theory to String theory . 66 Chapter 4 K O-homology and T yp e I D-branes As it was emphasized in the last sections of chapter 3, the K-theorical description of D-branes in String theory is based on the Sen-Witten mec hanism of brane-antribrane annhilation. Ho wev er, we ha v e seen that in an ordinary Maxw ell theory of p-forms the sources are extended ob jects, and their charges are related to the homology cycles they represent. W e ha v e argued that for the Ramond-Ramond gauge theory this is not the case: an ywa y , in tuitiv ely w e still exp ect the righ t mathematical framew ork to b e given by some sort of homology theory that would take in to accoun t the relation b et w een K-theory and D-branes as exp osed in the previous c hapter. In Type II String theory , this simple observ ation suggests that a m uc h more natural description of D-branes can b e given in terms of K-homology , the homological theory asso ciated to K-theory , as thouroghly emphasized in [77, 52, 69, 89]. More precisely , K-homology has t wo equiv alent representations: an analytic repre- sen tation, in terms of C ∗ -algebras and F redholm modules, and a geometric one, con- structed by Baum and Douglas in [14, 13]. In particular, the Baum-Douglas con- struction w as extensiv ely used in [80] to provide a rigorous geometric description of D-branes in T yp e I I String theory in v arious top ologically non trivial backgrounds. In this c hapter, we will presen t new results concerning KO-homology , the homology theory asso ciated to K O-theory . F rom the mathematical p ersp ective, we construct a geometric realization of KO- homology , and we argue that it is indeed isomorphic to the homology theory deﬁned via the lo op spectrum of K O-theory , which w e refer to as sp e ctr al K O-homolo gy . W e will also develop the analytic description of K O-homology , using Kasparo v’s formal- ism for real C ∗ -algebras, which represen ts a uniﬁed description of b oth K-theory and K-homology . W e then construct an homomorphism b et w een geometric and analytic K O-homology , 67 Dual theories and sp ectral K O-homology and give a detailed and explicit pro of that suc h homomorphism induces a natural equiv alence b et w een the tw o representations. W e w ant to comment, at this p oin t, that the equiv alence b et ween geometric and analytic K-homology has b een a sort of a “folklore theorem” un til recently [15], as the original work of Baum and Douglas did not contain any accurate pro of of such equiv alence. In particular, the w ork [15] also con tains a proof of the equiv alence betw een geometric and analytic K O-homology: nev erthless, the pro of presen ted in this chapter is fundamentally diﬀeren t, and the whole construction is more apt for ph ysical applications, as w e will see later on. In- deed, the approac h follo w ed here has strongly been inspired by the p oint of view in [14] that index theory is based on the equiv alence b etw een geometric and analytic K-homology: this p oin t of view is reinforced by the in tro duction of certain geometric in v arian ts that will help us to deriv e some cohomological index form ulas in the real case. F rom the ph ysical p ersp ective, we in tro duce the notion of wr app e d D-br ane and of wr apping char ge of a D-brane. In particular, we will illustate ho w the higher K- homology groups can naturally b e interpreted in terms of wrapp ed D-branes, and we will argue that the wrapping c harge of a D-brane is a genuinely diﬀerent concept in T yp e I String theory .. W e hav e men tioned in c hapter 2 that the in terpretation of a D-brane as a submanifold of the spacetime is not v ery accurate, and that a distinction should b e made some- ho w b etw een the D-brane itself and the worldv olume it wraps. This p oint of view emerges here, as we will construct stable torsion D-branes wrapping a single p oint in the spacetime. 4.1 Dual theories and sp ectral K O-homology In this section we will explain in which sense KO -homology is “asso ciated” to KO- theory , and we will deﬁne sp ectral KO-homology . First w e recall some prop erties of cohomology theories in the category of CW-complexes, referring to [88] for a detailed exp osition. An Ω -sp e ctrum , or lo op sp e ctrum for a generalized cohomology theory k ∗ is given b y a sequence of CW-complexes { K n } n ∈ Z together with homotop y equiv alences K n → ΩK n +1 (4.1.1) where Ω denotes the lo op space functor, such that the functor k n can b e represented k n (X) = [X , K n ] ∀ n ∈ Z 68 KK O-theory and analytic K O-homology for an y CW-complex X. By considering the maps σ n : Σ(K n ) → K n +1 adjoin t to the maps in (4.1.1), w e can deﬁne the unreduced generalized homology theory asso ciated to k ∗ b y setting k i (X) := ˜ k i (X + ) := lim n π n + i (X + ∧ K n ) where the inductiv e limit is tak en using the maps σ i π n + i (X + ∧ K n ) = [S n + i , X + ∧ K n ] ∗ susp − − → [Σ(S n + i ) , Σ(X + ∧ K n )] ∗ ' [S n + i +1 , X + ∧ Σ(K n )] ∗ σ i − → [S n + i +1 , X + ∧ K n +1 ] ∗ = π n + i +1 (X + ∧ K n +1 ) The relative homology theory can be deﬁned as usual, and we will refer to the homol- ogy theory k ∗ as the dual theory to k ∗ . In [10] it w as shown that a suitable sp ectrum for K O-theory can b e deﬁned in the follo wing w ay . F or n ≥ 1, let H R b e a real Z 2 -graded separable Hilb ert space whic h is a ∗ -mo dule for the real Cliﬀord algebra C  n − 1 = C  ( R n − 1 ). Let F red n b e the space of all F redholm op erators on H R whic h are o dd, C  n − 1 -linear and self-adjoint. Then F red n is the classifying space for KO − n , and there are homotop y equiv alences F red n → ΩF red n − 1 . F or n ≤ 0, we choose k ∈ N such that 8 k + n ≥ 1 and deﬁne F red n := F red 8 k + n . Then, sp e ctr al K O-homolo gy can b e deﬁned by setting K O s i (X , Y) := lim n π n + i ((X / Y) ∧ F red n ) (4.1.2) F rom a general result for sp ectrally deﬁned generalized homology theories, we hav e that K O s i (pt) = g K O 0 (S i ) The imp ortance of the functors K O s n in our context consists in the fact that any set of groups whic h are isomorphic to spectral K O-homology , in a suitable sense, for an y space X, deﬁnes necessarily an homology theory . Indeed, this will b e the case for geometric K O-homology deﬁned later on, whose homological properties will b e deduced b y “comparison” with sp ectral K O-homology . 4.2 KK O-theory and analytic K O-homology W e will giv e no w a detailed ov erview of the deﬁnition of KO-homology in terms of Kasparo v’s KK-theory for real C ∗ -algebras [62], and describ e v arious prop erties that w e will need later on. Incidentally , we will ev en tually giv e an interpretation of the KK-groups for complex C ∗ -algebras in terms of T yp e I I D-branes. 69 KK O-theory and analytic K O-homology 4.2.1 Real C ∗ -algebras A r e al algebr a is a ring A which is also an R -v ector space suc h that λ ( x y ) = ( λ x ) y = x ( λ y ) for all λ ∈ R and all x, y ∈ A . A r e al ∗ - algebr a is a real algebra A equipped with a linear inv olution ∗ : A → A suc h that ( x y ) ∗ = y ∗ x ∗ for all x, y ∈ A . A r e al Banach algebr a is a real algebra A equipp ed with a norm k−k : A → R such that k x y k ≤ k x k k y k and suc h that A is complete in the norm top ology . If A is a unital algebra then w e assume k 1 k = 1. A r e al Banach ∗ - algebr a is a real Banac h algebra whic h is also a real ∗ -algebra. A r e al C ∗ - algebr a is a real Banach ∗ -algebra such that (i) k x ∗ x k = k x k 2 for all x ∈ A ; and (ii) 1 + x ∗ x is in v ertible in ˜ A for all x ∈ A . where ˜ A denotes the unitalization of the algebra A . Remark 4.1. Although in the complex case in v ertibilit y of 1 + x ∗ x for all x ∈ A would follo w immediately from the C ∗ -algebra structure, in the real case this is no longer true. F or example, consider the real Banach ∗ -algebra C with in volution given by the iden tit y map. Then 1 + i ∗ i is not in v ertible, where i := √ − 1. This in v ertibilit y condition is fundamen tal to obtaining the usual representation theorem b elow for C ∗ - algebras in terms of b ounded self-adjoin t op erators on a real Hilb ert space. Ho wev er, C with in volution giv en by complex conjugation is a real C ∗ -algebra. Since the only R -linear inv olutions of C are the iden tity and complex conjugation, when we consider C as a real C ∗ -algebra the in volution will alwa ys b e implicitly assumed to b e complex conjugation. More generally an y complex C ∗ -algebra, regarded as a real vector space and with the same op erations, is a real C ∗ -algebra. Let us now give a num b er of examples of real C ∗ -algebras, some of which we will use later on in represen tation theorems. Example 4.2. Let H R b e a real Hilb ert space. Then the set of b ounded linear op erators B ( H R ) with the usual operations is a real C ∗ -algebra. Any closed self- adjoin t subalgebra of B ( H R ) is also a real C ∗ -algebra. More generally , an y closed self-adjoin t subalgebra of a real C ∗ -algebra is alw a ys a real C ∗ -algebra. Example 4.3. Let X b e a lo cally compact Hausdorﬀ space and C 0 (X , R ) the space of real-v alued con tin uous functions v anishing at inﬁnit y . Then C 0 (X , R ) with p oint wise op erations, the suprem um norm and inv olution given by the iden tit y map is a real C ∗ -algebra. As in the complex case, C 0 (X , R ) is unital if and only if X is compact. 70 KK O-theory and analytic K O-homology Example 4.4. With X as in Example 4.3 ab o v e, let Y b e a closed subspace of X and C 0 (X , Y; R ) the subspace of C 0 (X , C ) consisting of maps f : X → C such that f (Y) ⊂ R . Then with the op erations inherited from C 0 (X , C ), the subspace C 0 (X , Y; R ) is a real C ∗ -algebra. Example 4.5. Let X be a lo cally compact Hausdorﬀ space with in volution τ : X → X, i.e. a homeomorphism such that τ ◦ τ = id X , and consider the subset C 0 (X , τ ) of C 0 (X , C ) consisting of maps f suc h that f ◦ τ = f ∗ = f . Then C 0 (X , τ ), with the op erations inherited from C 0 (X , C ), is a real C ∗ -algebra. If τ = id X then C 0 (X , τ ) = C 0 (X , R ). If X is compact and Y is a closed subspace of X, then there is a compact Hausdorﬀ space Z with an in v olution τ such that C(X , Y; R ) ' C( Z, τ ). Ho wev er, the con v erse do es not hold in general. Example 4.6. Let V b e a real v ector space equipped with a quadratic form q , and consider the asso ciated real Cliﬀord algebra C  ( V , q ). Assume, without loss of gen- eralit y , that q ( v ) = h v , φ ( v ) i for all v ∈ V with resp ect to an inner pro duct on V , where the linear op erator φ ∈ L ( V ) is symmetric and orthogonal. W e can then de- ﬁne an in v olution on C  ( V , q ) by ( v 1 · · · v k ) ∗ = φ ( v k ) · · · φ ( v 1 ), i.e. if v ∈ V then v ∗ = φ ( v ). The isomorphism Φ : C  ( V ⊕ V , q ⊕ − q ) → L (Λ ∗ V ) induces a norm on C  ( V ⊕ V , q ⊕ − q ) by pullback of the op erator norm on L (Λ ∗ V ), and the inclusion C  ( V , q )  → C  ( V , q ) ˆ ⊗ C  ( V , − q ) ' C  ( V ⊕ V , q ⊕ − q ) giv en b y x 7→ x ˆ ⊗ 1 thereb y induces a norm on C  ( V , q ). Then C  ( V , q ) with its algebra structure, this in v olution and norm is a real C ∗ -algebra. If A , B are real ∗ -algebras then a r e al ∗ - algebr a homomorphism is a real algebra map φ : A → B , i.e. an R -linear ring homomorphism, such that φ ( x ∗ ) = φ ( x ) ∗ for all x ∈ A . The homomorphism is assumed to b e unital if b oth algebras are unital. If A is an algebra, w e denote b y M n ( A ) the algebra of n × n matrices with entries in A . Then, we ha v e the follo wing general represen tation theorem [47] Theorem 4.7. L et A b e a ﬁnite-dimensional r e al C ∗ -algebr a. Then ther e exist n 1 , . . . , n k ∈ N such that A ' M n 1 ( A 1 ) × · · · × M n k ( A k ) as r e al C ∗ -algebr as with A 1 , . . . , A k ∈ { R , C , H } . Analogously to the complex case, real C ∗ -algebras are alw ays algebras of op erators on some Hilb ert space. Theorem 4.8. ( Ingelstam ) L et A b e any r e al C ∗ -algebr a. Then ther e exists a r e al Hilb ert sp ac e H R such that A is isomorphic as a r e al C ∗ -algebr a to a close d self-adjoint sub algebr a of B ( H R ) . 71 KK O-theory and analytic K O-homology Consider no w a real C ∗ -algebra A . W e denote b y A C := A ⊗ C the complexiﬁcation of A , whic h is a complex algebra con taining A as a real algebra. Theorem 4.2.1 assures that A C can b e giv en a unique C ∗ -algebra norm suc h that the natural em b edding θ : A → A C of A on to its complexiﬁcation is an isometry . W e can deﬁne a map J A : A C → A C b y J A ( x + i y ) = x − i y for all x, y ∈ A . The map J A is a conjugate linear ∗ -isomorphism of the complex C ∗ -algebra A C . If φ : A → A is a contin uous ∗ -homomorphism, then the map J A ( φ ) : A C → A C deﬁned by J A ( φ )( x + i y ) = φ ( x ) + i φ ( y ) is a con tin uous ∗ -homomorphism such that J A ◦ J A ( φ ) = J A ( φ ) ◦ J A . Conv ersely , if J is a conjugate linear ∗ -isomorphism of a complex C ∗ -algebra B , then A = { x ∈ B | J ( x ) = x } is a real C ∗ -algebra. This implies the follo wing result. Prop osition 4.9. L et C ∗ R b e the c ate gory of r e al C ∗ -algebr as and c ontinuous ∗ - algebr a homomorphisms. L et C ∗ C , cl b e the c ate gory of p airs ( A, J ) , wher e A is a c omplex C ∗ -algebr a and J is a c onjugate line ar ∗ -isomorphism of A , and c ontinu- ous ∗ -homomorphisms c ommuting with J . Then the assignments A 7→ ( A C , J A ) φ 7→ J A ( φ ) deﬁne a functor J : C ∗ R − → C ∗ C , cl which is an e quivalenc e of c ate gories. The complexiﬁcation of a real C ∗ -algebra A is crucial in generalizing the notion of sp ectrum of an elemen t. Indeed, we deﬁne the c omplexiﬁe d sp e ctrum Sp C ( x ) of an elemen t x ∈ A as the sp ectrum of the elemen t θ ( x ) in A C , i.e. the set of λ ∈ C suc h that λ − θ ( x ) is not in v ertible in A C . This deﬁnition of the sp ectrum assures that the functional calculus in A is well behav ed. In the following, w e will use the notion of p ositiv e elemen t. An element x in a real C ∗ -algebra A is said to b e p ositive if x = x ∗ , and Sp C ( x ) ⊆ R + . As we are in terested in the applications of real C ∗ -algebras to algebraic top ology of CW-complexes, w e will no w sp ecialize to the case of comm utativ e algebras. As with complex Banac h algebras, a maximal t wo-sided ideal in a real Banac h algebra A is closed in A . If M is a maximal tw o-sided ideal of a real Banac h algebra A , then A/ M is isomorphic to one of R or C as real algebras. A char acter on a real algebra A is a non-zero real algebra map χ : A → C , assumed unital if A is unital. Let Ω A b e the space of c haracters of A . This can b e given, as in the complex case, a lo cally compact 72 KK O-theory and analytic K O-homology Hausdorﬀ space top ology suc h that Ω A is homeomorphic to Ω A C . F urthermore, A is unital if and only if Ω A is compact. Giv en x ∈ A , ev aluation at x gives a con tinuous map Γ( x ) : Ω A → C called the Gel’fand tr ansform of x . F rom this w e obtain the Gel’fand tr ansform of A , Γ : A → C 0 (Ω A , C ), which is a contin uous real algebra homomorphism of unit norm. If A is a real ∗ -algebra, then Γ is a ∗ -algebra homomorphism. The following imp ortan t results on the represen tation of commutativ e real C ∗ -algebras allo w to unify the treatment of real commutativ e C ∗ -algebras and top ological spaces with in v olution. Theorem 4.10. L et A b e a c ommutative r e al C ∗ -algebr a. Then: (i) The map τ : Ω A → Ω A deﬁne d by τ ( χ ) = χ is an involution; and (ii) The Gel’fand tr ansform Γ : A → C 0 (Ω A , τ ) is a r e al C ∗ -algebr a isomorphism. Pr o of. (i) The map τ is a bijection. The collection of sets U x,V =  χ ∈ Ω A | χ ( x ) ∈ V  for every x ∈ A and V op en in C is a sub-basis for the top ology of Ω A . The complex conjugate V of V is an op en set and τ − 1 ( U x,V ) = U x,V . Th us τ is contin uous. (ii) The map Γ is a real ∗ -algebra map with k Γ( x ) k = k x k . One also has Γ( x ) ◦ τ ( χ ) = Γ( x )( χ ) = χ ( x ) = Γ( x ) ∗ ( χ ) , and so Γ( x ) ◦ τ = Γ( x ) ∗ and Γ( A ) ⊂ C 0 (Ω A , τ ). Let θ : A → A C b e the C ∗ -algebra em b edding of A into its complexiﬁcation. The map ϑ : Ω A C → Ω A giv en b y ϑ ( f ) = f ◦ θ is a homeomorphism and there is a comm utativ e diagram A Γ / / θ   C 0 (Ω A , C ) ϑ ∗   A C Γ / / C 0 (Ω A C , C ) . Using this one then sho ws that Γ( A ) = C 0 (Ω A , τ ); see [47]. Corollary 4.11. L et A b e a c ommutative r e al C ∗ -algebr a with trivial involution. Then A is ∗ -isomorphic to C 0 (Ω A , R ) . 73 KK O-theory and analytic K O-homology 4.2.2 Hilb ert Mo dules In this section we will presen t a generalization of the notion of Hilb ert space, in which the scalar pro duct tak es v alue in a general real C ∗ -algebra. Let A b e a (not necessarily comm utativ e) real C ∗ -algebra. A pr e-Hilb ert mo dule over A is a (righ t) A -mo dule E equipp ed with an A - value d inner pr o duct , i.e. a bilinear map ( − , − ) : E × E → A such that (i) ( x, x ) ≥ 0 for all x ∈ E and ( x, x ) = 0 if and only if x = 0; (ii) ( x, y ) = ( y , x ) ∗ for all x, y ∈ E ; and (iii) ( x, y a ) = ( x, y ) a for all x, y ∈ E , a ∈ A . F or x ∈ E we deﬁne k x k E := k ( x, x ) k 1 / 2 . This deﬁnes a norm on E satisfying the Cauc h y-Sc h w artz inequalit y . If E is complete under this norm, then it is called a Hilb ert mo dule over A . As a straigh t generalization from the ordinary Hilb ert space case, w e hav e the following examples of Hilb ert mo dules. Example 4.12. A real C ∗ -algebra A can b e giv en the structure of a Hilb ert mo dule o v er itself by deﬁning ( a, b ) = a ∗ b , for any a, b ∈ A . More generally , an y closed right ideal of A is a Hilb ert mo dule o v er A . Example 4.13. Let E consists of all sequences ( a n ) n ∈ N , a n ∈ A , suc h that X n k a n k 2 < ∞ with inner pro duct (( a n ) , ( b n )) = P n a ∗ n b n . E is called the Hilb ert sp ac e over A , and is often denoted with A ∞ . Let E , F b e Hilb ert A -mo dules and T : E → F an A -linear map. W e call a map T ∗ : F → E such that (T x, y ) F = ( x, T ∗ y ) E for all x ∈ E , y ∈ F the adjoint of T. If it exists the adjoin t is unique. In con trast to ordinary b ounded op erators on separable Hilb ert spaces, not every A -linear map b etw een Hilb ert A -mo dules has an adjoint. W e denote the set of all A -linear maps T : E → F admitting an adjoint b y L ( E , F ). Elemen ts of L ( E , F ) are b ounded A -linear maps and L ( E ) := L ( E , E ) is a C ∗ -algebra with the op erator norm and in v olution giv en b y the adjoin t. Notice that a submo dule of a Hilbert A -module E in general need not b e comple- men ted, i.e. there is generally no pro jection in L ( E ) onto the given submo dule. 74 KK O-theory and analytic K O-homology Ho w ev er, one can deﬁne some sp ecial “rank 1” op erators as follows. Given x ∈ F , y ∈ E w e deﬁne an operator θ x,y ∈ L ( E , F ) b y θ x,y ( z ) = x ( y , z ) E . These operators generate an L ( E ) − L ( F )-bimo dule whose norm closure in L ( E , F ) is denoted K ( E , F ). Ele- men ts of K ( E , F ) are called gener alize d c omp act op er ators . W e will use the notation K ( E ) for K ( E , E ). Example 4.14. K ( A ) = A for any real C ∗ -algebra A . If A is unital, w e hav e also that L ( A ) = A . Example 4.15. If E is a Hilb ert A -mo dule, then L ( E n ) ' M n ( R ) ⊗ L ( E ), and K ( E n ) ' M n ( R ) ⊗ K ( E ) Example 4.16. K ( A ∞ ) ' A ⊗ K R , where K R := K ( H R ). F or a real C ∗ -algebra A , the multiplier algebr a of A , M ( A ), is the maximal C ∗ - algebra containing A as an essential ideal. Equiv alen tly , b y representing A ⊂ L ( H R ) one has M ( A ) = { T ∈ L ( H R ) | T S , S T ∈ A for all S ∈ A } . The multiplier algebra M ( A ) is a C ∗ -algebra which is ∗ -isomorphic to the C ∗ -algebra of double cen tralizers, i.e. pairs (T 1 , T 2 ) ∈ L ( A ) × L ( A ) suc h that a T 1 ( b ) = T 2 ( a ) b , T 1 ( a b ) = T 1 ( a ) b and T 2 ( a b ) = a T 2 ( b ) for all a, b ∈ A . If A is unital, then M ( A ) = A . F urthermore, M ( K R ) = L ( H R ), and M (C 0 (X , R )) = C b (X , R ) is the C ∗ -algebra of real-v alued b ounded con tinuous functions on a lo cally compact Hausdorﬀ space X. W e hav e then the follo wing prop osition, whose proof follo ws from the analogous result for complex Hilb ert mo dules [18] Prop osition 4.17. L et E b e a Hilb ert A -mo dule. Then ther e is an isomorphism L  E  ' M  K ( E )  . As with ordinary Hilb ert spaces, one can deﬁne tensor pro ducts of Hilbert mo dules in the follo wing w a y [18]. Let E i b e a Hilb ert B i -mo dule, for i=1,2, and let φ : B 1 → L ( E 2 ) b e a *-homomorphism. If w e regard E 2 as a left B 1 -mo dule via φ , w e can form the algebraic tensor pro duct E 1 ⊗ B 1 E 2 , whic h is a righ t B 2 -mo dule. Finally , w e deﬁne the B 2 -v alued pre-inner pro duct on the algebraic tensor pro duct by ( x 1 ⊗ x 2 , y 1 ⊗ y 2 ) := ( x 2 , φ (( x 1 , y 1 ) E 1 ) y 2 ) E 2 75 KK O-theory and analytic K O-homology The completion of the algebraic tensor pro duct with resp ect to this inner pro duct, with vectors of length 0 divided out, is called the tensor pr o duct of E 1 and E 2 , and is denoted with E 1 ⊗ φ E 2 . As for ordinary Hilb ert spaces, there is a natural homomorphism from L ( E 1 ) to L ( E 1 ⊗ φ E 2 ), and we will denote the image of T ∈ L ( E 1 ) as T ⊗ 1, or φ ∗ (T) when we w an t to emphatize the homomorphism φ . How ever, there is no homomorphism from L ( E 2 ) to L ( E 1 ⊗ φ E 2 ) in general. Finally , if E is a pre-Hilb ert module o ver the real C ∗ -algebra A , we assume that the complexiﬁcation E ⊗ C is a pre-Hilb ert mo dule o v er A C . This means that the A -v alued inner product extends to a sesquilinear map. W e assume that sesquilinear maps are linear in the second v ariable. 4.2.3 Kasparo v’s formalism for KK O-theory W e are no w ready to deﬁne KK O-theory b y using Kasparov’s approach, developed in [62]. In the follo wing w e will assume that a real C ∗ -algebra A is separable and a real C ∗ -algebra B is σ -unital, i.e. B contains an element h such that φ ( h ) > 0, for ev ery c haracter φ of B . This technical requirements will b e useful in the following. Deﬁnition 4.18. A (Kasp ar ov) ( A, B ) -mo dule is a triple ( E , ρ, T), where E is a coun tably generated Z 2 -graded Hilb ert B -mo dule, ρ : A → L ( E ) is an ev en ∗ - homomorphism and T ∈ L ( E ) such that (T − T ∗ ) ρ ( a ) ,  T 2 − 1  ρ ( a ) ,  T , ρ ( a )  ∈ K ( E ) (4.2.1) for all a ∈ A , and T is o dd with resp ect to the grading on E . A Kasparov mo dule ( E , ρ, T) is called de gener ate if all op erators in (4.2.1) are zero. Two Kasparov mo dules ( E i , ρ i , T i ), i = 1 , 2 are said to b e ortho gonal ly e quivalent if there is an isometric isomorphism U ∈ L ( E 1 , E 2 ) such that T 1 = U ∗ T 2 U and ρ 1 ( a ) = U ∗ ρ 2 ( a ) U for all a ∈ A . Orthogonal equiv alence is an equiv alence relation on the set of Kasparo v mo dules. W e denote the set of Kasparo v mo dules b y E ( A, B ). The subset con taining degenerate mo dules is denoted D ( A, B ). Direct sum mak es E ( A, B ) and D ( A, B ) into monoids. Deﬁnition 4.19. Let ( E i , ρ i , T i ) ∈ E ( A, B ) for i = 0 , 1, ( E , ρ, T) ∈ E ( A,B ⊗ C([0 , 1] , R )), and let f t : B ⊗ C([0 , 1] , R ) → B b e the ev aluation map f t ( g ) = g ( t ). Then 76 KK O-theory and analytic K O-homology ( E 0 , ρ 0 , T 0 ) and ( E 1 , ρ 1 , T 1 ) are said to b e homotopic and ( E , ρ, T) is called a ho- motopy if ( E ⊗ f i B , f i ◦ ρ, f i ∗ (T)) is orthogonally equiv alent to ( E i , ρ i , T i ) for i = 0 , 1. Homotop y is an equiv alence relation on E ( A, B ) and we denote the equiv alence classes by [ E , ρ, T]. It is useful to consider sp ecial kinds of homotop y . If E = C([0 , 1] , E 0 ), E 0 = E 1 and the induced maps t 7→ T t , t 7→ ρ t ( a ) for all a ∈ A are strongly ∗ -contin uous, then we call ( E , ρ, T) a standar d homotopy . If in addition ρ t = ρ is constant and T t is norm con tin uous, then ( E , ρ, T) is called an op er ator homotopy . The following result holds, whose pro of can b e obtained b y the analogous result in complex case [18]. Prop osition 4.20. L et ( E , φ, T) b e an element in D ( A, B ) . Then ( E , φ, T) is homo- topic to the zer o mo dule. W e can now give the deﬁnition of the Kasparov’s KKO-groups. Deﬁnition 4.21. The set of equiv alence classes in E ( A, B ) with resp ect to homotopy of ( A, B )-mo dules is denoted KKO( A, B ) or KKO 0 ( A, B ). F or p, q ≥ 0 we deﬁne KK O p,q ( A, B ) := KK O( A, B ⊗ C  p,q ) , where C  p,q := C  ( R p,q ) is the real Cliﬀord algebra of the vector space R p + q with quadratic form of signature ( p, q ). The equiv alence relation allo ws us to simplify the ( A, B )-mo dules required to de- ﬁne KK O( A, B ) [18]. Indeed, w e need only consider modules of the form ( B ∞ , ρ, T) with T = T ∗ . If A is unital, w e can further assume that k T k ≤ 1 and T 2 − 1 ∈ K ( B ∞ ). There is another equiv alence relation that we can deﬁne on E ( A, B ). W e sa y that t w o ( A, B )-mo dules ( E i , ρ i , T i ), i = 0 , 1 are stably op er ator homotopic , ( E 0 , ρ 0 , T 0 ) ' oh ( E 1 , ρ 1 , T 1 ), if there exist ( E 0 i , ρ 0 i , T 0 i ) ∈ D ( A, B ) suc h that ( E 0 ⊕ E 0 0 , ρ 0 ⊕ ρ 0 0 , T 0 ⊕ T 0 0 ) and ( E 1 ⊕ E 0 1 , ρ 1 ⊕ ρ 0 1 , T 1 ⊕ T 0 1 ) are op erator homotopic up to orthogonal equiv alence.One can pro ov e that the set of equiv alence classes with resp ect to ' oh coincides with the set KK O( A, B ) deﬁned abov e. F or this result to hold the h yp othesis that B is σ -unital is of particular imp ortance. The set KKO( A, B ) is an ab elian group, for an y separable real C ∗ -algebra A and an y real σ -unital C ∗ -algebra B . Moreov er, KK O( − , − ) is a cov ariant bifunctor from 77 KK O-theory and analytic K O-homology the category of separable C ∗ -algebras into the category of ab elian groups which is additiv e, i.e. KK O( A 1 ⊕ A 2 , B ) = KKO( A 1 , B ) ⊕ KKO( A 2 , B ) , KK O( A, B 1 ⊕ B 2 ) = KK O( A, B 1 ) ⊕ KKO( A, B 2 ) . Namely , any tw o ∗ -homomorphisms f : A 2 → A 1 and g : B 1 → B 2 induce group homomorphisms f ∗ : KK O( A 1 , B ) − → KKO( A 2 , B ) , g ∗ : KK O( A, B 1 ) − → KK O( A, B 2 ) deﬁned b y f ∗ [ E , ρ, T ] = [ E , ρ ◦ f , T ] , g ∗ [ E , ρ, T ] = [ E ⊗ g B 2 , ρ ⊗ 1 , T ⊗ 1] . Finally , an y tw o homotopies f t : A 2 → A 1 and g t : B 1 → B 2 induce the same homomorphism for all t ∈ [0 , 1], i.e. f ∗ t = f ∗ 0 and g t ∗ = g 0 ∗ . 4.2.4 Analytic K O-homology As men tioned in the in tro duction to this chapter, Kasparo v’s formalism represents a uniﬁed description of b oth K O-theory and K O-homology . Indeed, if w e denote with K O p ( B ) the algebraic K O-theory groups of a unital real C ∗ -algebra B , we ha ve the follo wing result [18, 62]. Theorem 4.22. L et B b e a unital r e al C ∗ -algebr a. Then, KKO( R , B ) ' KO 0 ( B ) and KK O p,q ( R , B ) ' K O p − q ( B ) Recall that for a real unital C ∗ -algebra A , the algebraic K-theory group KO 0 ( A ) is deﬁned as the Grothendiec k group of the monoid of unitarily equiv alent pro jectors in M ∞ ( A ), whic h is deﬁned as the direct limit of A -v alued matrix algebras M n ( A ) under the em b edding a → diag ( a, 0). The higher algebraic K-theory group can b e deﬁned b y K O p ( A ) := KO 0 ( C 0 ( R p ) ⊗ A ). If X is a compact Hausdorﬀ space, KKO p ( R , C(X , R )) ' KO p (C(X , R )) ' K O p (X). On the other hand, using the Gel’fand transform the con tra v arian t functor (X , τ ) 7→ C(X , τ ) induces an equiv alence of categories b etw een the category of compact Haus- dorﬀ spaces with inv olution and the category of comm utativ e real C ∗ -algebras. Since 78 KK O-theory and analytic K O-homology KK O ] ( − , R ) is also a con trav ariant functor, it follo ws that their composition (X , τ ) 7→ KK O ] (C(X , τ ) , R ) is a co v arian t functor. This motiv ates the follo wing Deﬁnition 4.23. Let (X , τ ) b e a compact Hausdorﬀ space with in v olution. The analytic K O-homolo gy gr oups of (X , τ ) are deﬁned by K O a n  X , τ  := KK O n, 0  C(X , τ ) , R  = KK O  C(X , τ ) , C  n  . In the following, we will illustrate an alternative description of Kasparov’s KO- homology groups for a real C ∗ -algebra A , referring to [53] for more details. This description is based on triples ( H R , ρ, T) which are deﬁned by the data: (i) H R is a separable real Hilb ert space; (ii) ρ : A → L ( H R ) is a *-represen tation of A ; and (iii) T is a b ounded linear op erator on H R . These are assumed to satisfy the follo wing conditions: (i) H R is equipp ed with a Z 2 -grading suc h that ρ ( a ) is ev en for all a ∈ A and T is o dd; (ii) F or all a ∈ A one has  T 2 − 1  ρ ( a ) , (T − T ∗ ) ρ ( a ) , T ρ ( a ) − ρ ( a ) T ∈ K R ; (4.2.2) and (iii) There are o dd R -linear op erators ε 1 , . . . , ε n on H R with the C  n algebra relations ε i = − ε ∗ i , ε 2 i = − 1 , ε i ε j + ε j ε i = 0 (4.2.3) for i 6 = j suc h that T and ρ ( a ) comm ute with eac h ε i . W e shall refer to the triple ( H R , ρ, T) as an n -gr ade d F r e dholm mo dule . Let us denote by ΓO n ( A ) the set of all n -graded F redholm mo dules ov er A . Consider the equiv alence relation ∼ on ΓO n ( A ) generated b y the relations: Ortho gonal e quivalenc e : ( H R , ρ, T) ∼ ( H 0 R , ρ 0 , T 0 ) if and only if there exists an isometric degree-preserving linear op erator U : H R → H 0 R suc h that U ρ ( a ) = ρ 0 ( a ) U for all a ∈ A , U T = T 0 U , and U ε i = ε 0 i U ; and 79 KK O-theory and analytic K O-homology Homotopy e quivalenc e : ( H R , ρ, T) ∼ ( H R , ρ, T 0 ) if and only if there exists a norm contin uous function t 7→ T t suc h that ( H R , ρ, T t ) is a F redholm mo dule for all t ∈ [0 , 1] with T 0 = T, T 1 = T 0 . W e deﬁne the dir e ct sum of tw o F redholm modules ( H R , ρ, T) and ( H 0 R , ρ 0 , T 0 ) to b e the F redholm mo dule ( H R ⊕ H 0 R , ρ ⊕ ρ 0 , T ⊕ T 0 ). W e ma y no w deﬁne KO n ( A ) as the free ab elian group generated by elemen ts in ΓO n ( A ) / ∼ and quotien ted by the ideal generated by the set { [ x 0 ⊕ x 1 ] − [ x 0 ] − [ x 1 ] | [ x 0 ] , [ x 1 ] ∈ ΓO n ( A ) / ∼} . In KO n ( A ) the inverse of a class represented by the mo dule ( H R , ρ, T) is giv en b y ( H o R , ρ, T), where H o R is the Hilb ert space H R with the opp osite Z 2 -grading and where the op erators ε i rev erse their signs. Moreov er, the op erator T for a triple ( H R , ρ, T) representing an elemen t in KO n ( A ) can b e tak en to b e a F redholm op erator without loss of generality [18]. F or a compact Hausdorﬀ space X we deﬁne K O a n  X  := K O n  C(X , R )  = KK O  C(X , R ) , C  n  . Moreo v er, for a compact pair (X , Y), w e can deﬁne the higher relative KO-homology groups as K O a n  X , Y  := K O n  C(X / Y , R )  = KK O  C(X / Y , R ) , C  n  . The groups K O a n  X , Y  enjo y Bott p erio dicit y K O a n  X , Y  ' K O a n +8  X , Y  , ∀ n ≥ 0 whic h can b e prov en by using the p erio dicit y of real Cliﬀord algebras, and the iso- morphisms KK O( A ⊗ C  p,q , B ⊗ C  r,s ) ' KKO( A ⊗ C  p,q ⊗ C  r,s , B ) ' KK O( A ⊗ C  p − q + s − r , 0 , B ) induced b y the interse ction pr o duct , which we will not attempt to deﬁne here. The term “KO-homology” for the groups K O a n  X , Y  is justiﬁed by the follo wing theorem [15, 53]. Theorem 4.24. (Kasp ar ov) Ther e ar e c onne cting homomorphisms ∂ : KO a n  X , Y  → K O a n − 1  X , Y  which ar e c omp atible with Bott p erio dicity, and give Kasp ar ov KO-homolo gy the struc- tur e of a Z 8 -gr ade d homolo gy the ory on the c ate gory of CW-c omplex p airs (X , Y) . On the sub c ate gory of ﬁnite CW-c omplex p airs Kasp ar ov’s K O-homolo gy is isomorphic to sp e ctr al K O-homolo gy. 80 Geometric K O-homology In particular, Theorem 4.24 implies that K O a n (pt) ' KO − n (pt) This can indeed b e pro v en b y directly computing the groups KK O( R , C  n ); see [18] for suc h a computation in the complex case. 4.3 Geometric K O-homology As w e hav e seen in the previous section, analytic KO-homology giv es a represen tation of sp ectral K O-homology based on C ∗ -algebras and linear op erators on Hilb ert spaces. In this section w e will develop a geometric version of K O-homology , whic h is analogous to the Baum-Douglas construction of K-homology [14, 13, 80]. Indeed, we will pro ve directly its homological prop erties by comparing it with other formulations of K O- homology . Deﬁnition 4.25. Let X b e a ﬁnite CW-complex. A K O-cycle on X is a triple (M , E , φ ) where (i) M is a compact spin manifold without b oundary; (ii) E is a real vector bundle ov er M; and (iii) φ : M → X is a contin uous map. There are no connectedness requirements made up on M, and hence the bundle E can ha v e diﬀeren t ﬁbre dimensions on the diﬀeren t connected components of M. It follo ws that disjoin t union (M 1 , E 1 , φ 1 ) q (M 2 , E 2 , φ 2 ) := (M 1 q M 2 , E 1 q E 2 , φ 1 q φ 2 ) is a w ell-deﬁned op eration on the set of KO-cycles on X, whic h we will denote with ΓO(X). In the follo wing w e will consider some equiv alence relations on the set ΓO(X). Deﬁnition 4.26. Tw o K O-cycles (M 1 , E 1 , φ 1 ) and (M 2 , E 2 , φ 2 ) on X are isomorphic if there exists a diﬀeomorphism h : M 1 → M 2 suc h that (i) h preserves the spin structures; (ii) h ∗ (E 2 ) ' E 1 as real v ector bundles; and 81 Geometric K O-homology (iii) The diagram M 1 h / / φ 1 ! ! D D D D D D D D M 2 φ 2   X comm utes. Deﬁnition 4.27. Two K O-cycles (M 1 , E 1 , φ 1 ) and (M 2 , E 2 , φ 2 ) on X are spin b or dant if there exists a compact spin manifold W with b oundary , a real vector bundle E → W, and a con tin uous map φ : W → X such that the t w o K O-cycles  ∂ W , E | ∂ W , φ | ∂ W  ,  M 1 q ( − M 2 ) , E 1 q E 2 , φ 1 q φ 2  are isomorphic, where − M 2 denotes M 2 with the opp osite class in H 1 (M; Z 2 ) repre- sen ting the spin structure on its tangent bundle TM 2 . The triple (W , E , φ ) is called a spin b or dism of KO-cycles. W e ﬁnally in tro duce the last equiv alence relation we will need to deﬁne geometric K O-homology . Let M be a spin manifold and F → M a C ∞ real spin v ector bundle with ﬁbres of dimension n := dim R F p ≡ 0 mod 8 for p ∈ M. Let 1 1 R M := M × R denote the trivial real line bundle o v er M. Then F ⊕ 1 1 R M is a real v ector bundle o v er M with ﬁbres of dimension n + 1 and pro jection map λ . By c ho osing a C ∞ metric on it, w e ma y deﬁne b M = S  F ⊕ 1 1 R M  (4.3.1) where S  F ⊕ 1 1 R M  denotes the spere bundle of F ⊕ 1 1 R M . The tangent bundle of F ⊕ 1 1 R M ﬁts in to an exact sequence of bundles giv en b y 0 − → λ ∗  F ⊕ 1 1 R M  − → T  F ⊕ 1 1 R M  − → λ ∗  TM  − → 0 , as for a v ector bundle E π − → M the vertical tangent bundle to the ﬁbration π is isomorphic to the v ector bundle π ∗ E → E. Upon c ho osing a splitting, it follo ws that T  F ⊕ 1 1 R M  ' λ ∗  TM  ⊕ λ ∗  F ⊕ 1 1 R M  and hence the spin structures on TM and F ⊕ 1 1 R M determine a spin structure on T  F ⊕ 1 1 R M  . Since the sphere bundle S  F ⊕ 1 1 R M  is the b oundary of the disk bundle B  F ⊕ 1 1 R M  , and using the fact that w e can equip B  F ⊕ 1 1 R M  with the spin structure induced b y that on the total space of T  F ⊕ 1 1 R M  , it follows that b M is a compact spin 82 Geometric K O-homology manifold. By construction, b M is a sphere bundle ov er M with n -dimensional spheres S n as ﬁbres. W e denote the bundle pro jection b y π : b M − → M . (4.3.2) W e may regard the total space b M as consisting of tw o copies B ± (F), with opp osite spin structures, of the unit ball bundle B (F) of F glued together b y the identit y map id S (F) on its b oundary so that b M = B + (F) ∪ S (F) B − (F) . (4.3.3) Since n ≡ 0 mod 8, the group Spin( n ) has t w o irreducible real half-spin represen ta- tions. The spin structure on F asso ciates to these representations real v ector bundles S 0 (F) and S 1 (F) of equal rank 2 n/ 2 o v er M. Their Whitney sum S(F) = S 0 (F) ⊕ S 1 (F) is a bundle of real Cliﬀord mo dules o v er TM suc h that C  (F) ' End S(F), where C  (F) is the real Cliﬀord algebra bundle of F. Let / S + (F) and / S − (F) b e the real spinor bun- dles o ver F obtained from pullbac ks to F b y the bundle pro jection F → M of S 0 (F) and S 1 (F), resp ectiv ely . Cliﬀord multiplication induces a bundle map F ⊗ S 0 (F) → S 1 (F) that deﬁnes a vector bundle map σ : / S + (F) → / S − (F) co vering id F whic h is an isomorphism outside the zero section of F. Since the ball bundle B (F) is a sub- bundle of F, we ma y form real spinor bundles o v er B ± (F) as the restriction bundles ∆ ± (F) = / S ± (F) | B ± (F) . W e can then glue ∆ + (F) and ∆ − (F) along S (F) = ∂ B (F) by the Cliﬀord m ultiplication map σ giving a real v ector bundle o v er b M deﬁned b y H(F) = ∆ + (F) ∪ σ ∆ − (F) . (4.3.4) F or eac h p ∈ M, the bundle H(F) | π − 1 ( p ) is the real Bott generator vector bundle ov er the n -dimensional sphere π − 1 ( p ). In particular, the class [H(F)] ∈ KO 0 ( b M) is the image of the Thom class of F τ F ∈ K O 0 ( B + (F) , S (F)) under the comp osition of the homomorphisms K O 0 ( B + (F) , S (F)) → K O 0 ( b M , B − (F)) → KO 0 ( b M) where the ﬁrst map is giv en b y excision, and the second map is giv en b y restriction. Deﬁnition 4.28. Let (M , E , φ ) be a KO-cycle on X and F a C ∞ real spin v ector bundle ov er M with ﬁbres of dimension dim R F p ≡ 0 mo d 8 for p ∈ M. Then the pro cess of obtaining the KO-cycle ( b M , H(F) ⊗ π ∗ (E) , φ ◦ π ) from (M , E , φ ) is called r e al ve ctor bund le mo diﬁc ation. 83 Geometric K O-homology W e are now ready to deﬁne the geometric KO-homology groups of the space X. Deﬁnition 4.29. The ge ometric KO-homolo gy gr oup of X is the ab elian group ob- tained from quotien ting ΓO(X) b y the equiv alence relation ∼ generated by the rela- tions of (i) isomorphism; (ii) spin b ordism; (iii) direct sum: if E = E 1 ⊕ E 2 , then (M , E , φ ) ∼ (M , E 1 , φ ) q (M , E 2 , φ ); and (iv) real vector bundle mo diﬁcation. The group op eration is induced b y disjoint union of KO-cycles. W e denote this group b y K O t ] (X) := ΓO(X) / ∼ , and the homology class of the KO-cycle (M , E , φ ) by [M , E , φ ] ∈ K O t ] (X). Since the equiv alence relation on ΓO(X) preserv es the dimension of M mo d 8 in K O-cycles (M , E , φ ), one can deﬁne the subgroups KO t n (X) consisting of classes of K O-cycles (M , E , φ ) for which all connected comp onents M i of M are of dimension dim M i ≡ n mo d 8. Then K O t ] (X) = 7 M n =0 K O t n (X) (4.3.5) has a natural Z 8 -grading. The geometric construction of K O-homology is functorial. If f : X → Y is a con tin u- ous map, then the induced homomorphism f ∗ : K O t ] (X) − → K O t ] (Y) of Z 8 -graded ab elian groups is giv en on classes of K O-cycles [M , E , φ ] ∈ K O t ] (X) b y f ∗ [M , E , φ ] := [M , E , f ◦ φ ] . One has (id X ) ∗ = id KO t ] ( X ) and ( f ◦ g ) ∗ = f ∗ ◦ g ∗ . Since real vector bundles o v er M extend to real vector bundles ov er M × [0 , 1], it follows b y spin b ordism that induced homomorphisms dep end only on their homotopy classes. If pt denotes a one-p oin t top ological space, then the collapsing map ζ : X → pt induces an epimorphism ζ ∗ : K O t ] (X) − → K O t ] (pt) . (4.3.6) 84 Geometric K O-homology The r e duc e d geometric KO-homology group of X is g K O t ] (X) := ker ζ ∗ . (4.3.7) Since the map (4.3.6) is an epimorphism with left in verse induced by the inclusion of a p oin t ι : pt  → X, one has KO t ] (X) ' KO t ] (pt) ⊕ g K O t ] (X) for an y space X. As for the complex case [80], the ab elian group K O t ] (X) is generated by classes [M , E , φ ] where M is connected, and if { X j } j ∈ J is the set of connected comp onents of X then K O t ] (X) = M j ∈ J K O t ] (X j ) . Moreo v er, the homology class of a cycle (M , E , φ ) on X depends only on the K O-theory class of E in K O 0 (M), and on the homotop y class of φ in [M , X]. 4.3.1 Homological prop erties of K O t ] In the previous section w e ha v e constructed a co v ariant functor K O t ] from the category of ﬁnite CW-complexes to the category of ab elian groups, whic h is homotop y in v arian t. W e will no w establish that this construction actually yields a (generalized) homology theory , and in particular is the dual homology to KO-theory . The main strategy consists in “comparing” the functors K O t n with a realization of sp ectral K O-homology dev elop ed in [58], which w e will denote with KO 0 ] . Namely , for each pair (X , Y) w e will construct a map µ s : KO t n (X , Y) → KO 0 n (X , Y) for each n ∈ Z which deﬁnes a natural equiv alence b etw een functors on the category of top ological spaces ha ving the homotop y t yp e of ﬁnite CW-pairs. The set of cycles for KO 0 ] (X) is given by triples (M , x, φ ) as in Deﬁnition 4.25, but with x ∈ KO i (M) b eing a K O-theory class ov er M such that dim M ≡ i + n mo d 8. The equiv alence relations are as in the previous section, apart from real vector bundle mo diﬁcation, whic h is modiﬁed from Deﬁnition 4.28 as follo ws. The no where zero section Σ F : M − → F ⊕ 1 1 R M (4.3.8) deﬁned b y Σ F ( p ) = 0 p ⊕ 1 for p ∈ M induces an embedding Σ F : M  → b M . (4.3.9) 85 Geometric K O-homology Then real v ector bundle mo diﬁcation is replaced b y the relation  M , x , φ  ∼  b M , Σ F ! ( x ) , φ ◦ π  , where the functorial homomorphism Σ F ! : K O i (M) → KO i + r ( b M ) is the Gysin map induced b y the embedding (4.3.9), with r = rank(F). By construction, the normal bundle to M in b M can b e iden tiﬁed with F. Since H(F) is the image of the Thom class of F, by the deﬁnition of Gysin morphism in section 3.6 it follo ws that on stable isomorphism classes of real v ector bundles [E] ∈ KO 0 (M) one has Σ F !  E  =  H(F) ⊗ π ∗ (E)  . (4.3.10) T o compare K O t ] with K O 0 ] , w e deﬁne K O t n +8 k (X) := KO t n (X) for all k ∈ Z , 0 ≤ n ≤ 7. Moreo v er, we giv e a spin b ordism description of the relativ e geometric K O-homology groups K O t n (X , Y) as follows. W e consider the set ΓO(X , Y) of isomorphism classes of triples (M , E , φ ) where (i) M is a compact spin manifold with (p ossibly empty) b oundary; (ii) E is a real vector bundle ov er M; and (iii) φ : M → X is a contin uous map with φ ( ∂ M) ⊂ Y . The set ΓO(X , Y) is then quotiented b y relations of relativ e spin b ordism, which is mo diﬁed from Deﬁnition 4.27 by the requiremen t that M 1 q ( − M 2 ) ⊂ ∂ W is a regularly embedded submanifold of co dimension 0 with φ ( ∂ W \ M 1 q ( − M 2 )) ⊂ Y , direct sum, and real v ector bundle mo diﬁcation, which is applicable in this case since S (F ⊕ 1 1 R M ) is a compact spin manifold with b oundary S (F ⊕ 1 1 R M ) | ∂ M . The collection of equiv alence classes is a Z 8 -graded ab elian group with op eration induced by disjoin t union of relativ e K O-cycles. One has KO t i (X , ∅ ) = K O t i (X) . W e can ﬁnally prov e the following Theorem 4.30. The map µ s : K O t n (X , Y) − → KO 0 n (X , Y) deﬁne d on classes of KO-cycles by µ s  M , E , φ  t =  M , [E] , φ  0 is an isomorphism of ab elian gr oups which is natur al with r esp e ct to c ontinuous maps of p airs. 86 Geometric K O-homology Pr o of. T aking in to accoun t the equiv alence relations on ΓO(X , Y) used to deﬁne b oth K O-homology groups, the map µ s is w ell-deﬁned and a group homomorphism. Let [M , x, φ ] 0 ∈ K O 0 n (X , Y) with m := dim M. W e may assume that M is connected and x is non-zero in KO i (M). Then m − i ≡ n mo d 8. Consider the trivial spin v ector bundle F = M × R n +7 m o v er M. In this case the sphere bundle (4.3.1) is b M = M × S n +7 m and the asso ciated Gysin homomorphism in K O-theory is a map Σ F ! : K O i  M  − → K O i +7 m + n  b M  . Since i + 7 m + n ≡ ( i + 7 m + m − i ) mo d 8 ≡ 0 mod 8, one has K O i +7 m + n ( b M ) ' K O 0 ( b M ). It follows that there are real vector bundles E , H → b M such that Σ F ! ( x ) = [E] − [H], and so by real v ector bundle mo diﬁcation one has [M , x, φ ] 0 = [ b M , [E] , φ ◦ π ] 0 − [ b M , [H] , φ ◦ π ] 0 in KO 0 n (X , Y). Therefore µ 0 ( [ b M , E , φ ◦ π ] t − [ b M , H , φ ◦ π ] t ) = [M , x, φ ] 0 , and we conclude that µ s is an epimorphism. No w supp ose that µ s [M 1 , E 1 , φ 1 ] t = µ s [M 2 , E 2 , φ 2 ] t are identiﬁed in KO 0 n (X , Y) through real vector bundle mo diﬁcation. Then, for instance, there is a real spin v ector bundle F → M 1 suc h that M 2 = c M 1 and [E 2 ] = Σ F ! [E 1 ]. Since Σ F !  E 1  =  H(F) ⊗ π ∗ (E 1 )  and since the class [M 2 , E 2 , φ 2 ] t dep ends only on the KO-theory class [E 2 ], it fol- lo ws that the homology classes [M 1 , E 1 , φ 1 ] t and [M 2 , E 2 , φ 2 ] t are also iden tiﬁed in K O t n (X , Y) through real v ector bundle mo diﬁcation. As this is the only relation in K O 0 n (X , Y) that might iden tify these classes without iden tifying them as KO-cycles, w e conclude that µ s is a monomorphism and therefore an isomorphism. Since K O 0 ] is a homological realization of the homology theory asso ciated with K O-theory , w e hav e thus established that geometric KO-homology is a generalized homology theory whic h is equiv alent to sp ectral K O-homology . In particular, it enjoys the standard homological prop erties, such as the existence of a long exact sequence for an y pair (X , Y). Moreo ver, the connecting homomorphism ∂ : K O t n (X , Y) − → KO t n − 1 (Y) is giv en b y the b oundary map ∂ [M , E , φ ] := [ ∂ M , E | ∂ M , φ | ∂ M ] (4.3.11) on classes of K O-cycles and extended b y linearit y . Other homological prop erties are direct translations of those of the complex case pro vided b y [80], to whic h w e refer for details. 87 Geometric K O-homology 4.3.2 Pro ducts and P oincar´ e Dualit y Since w e hav e established that geometric K O-homology is a represen tation of sp ectral K O-homology , we can deﬁne pro ducts and dualities with KO-theory . The c ap pr o duct is deﬁned as the Z 8 -degree preserving bilinear pairing ∩ : K O 0 (X) ⊗ KO t ] (X) − → K O t ] (X) giv en for an y real v ector bundle F → X and KO-cycle class [M , E , φ ] ∈ KO t ] (X) b y [F] ∩ [M , E , φ ] := [M , φ ∗ F ⊗ E , φ ] (4.3.12) and extended linearly . In particular, it mak es K O t ] (X) in to a mo dule ov er the ring K O 0 (X). As in the complex case, this pro duct can b e extended to a bilinear form ∩ : K O i (X) ⊗ KO t j (X , A) − → KO t j − i (X , A) . deﬁned as x ∩ [M , E , φ ] := ( µ s ) − 1 ([M , x ∪ φ ∗ [E] , φ ] 0 ) (4.3.13) whic h coincides with (4.3.12) when x = [F]. If X and Y are spaces, then the exterior pr o duct × : K O t i (X) ⊗ KO t j (Y) − → K O t i + j (X × Y) is giv en for classes of K O-cycles [M , E , φ ] ∈ KO t i (X) and [N , F , ψ ] ∈ KO t j (Y) b y  M , E , φ  ×  N , F , ψ  :=  M × N , E  F , ( φ, ψ )  , where M × N has the pro duct spin structure uniquely induced b y the spin structures on M and N, and E  F is the real vector bundle o v er M × N with ﬁbres (E  F) ( p,q ) = E p ⊗ F q for ( p, q ) ∈ M × N. This pro duct is natural with resp ect to con tin uous maps. Unfortunately , in contrast to the complex case, w e don’t hav e a version of the K¨ unneth theorem for K O-homology . Indeed, should such a form ula exist, one could use it to sho w that K O ] (pt) ⊗ KO ] (pt) has to b e a tensor pro duct as mo dules ov er the ring K O ] (pt). But this do es not w ork correctly as p oin ted out b y A tiy ah [3]. Let M b e a spin manifold of dimension n . The class [M] := [M , 1 1 R M , id M ] ∈ KO t n (M , ∂ M) 88 K-homology and Index Theorems is called the fundamental class , and it induces the follo wing Poinc ar ´ e Duality isomor- phism [58] Φ M : K O i (M) ' − − → K O t n − i (M , ∂ M) ξ − → ξ ∩ [M] The fundamen tal class is uniquely determined. Finally , let V π − → X b e a KO-orien ted v ector bundle of rank r o ver a space X with Thom class τ V . In analogy to K-theory , we hav e the homological v ersion of Thom isomorphism T ∗ X , V : K O t i + r  B (V) , S (V)  ≈ − → K O t i  X  . (4.3.14) deﬁned as T ∗ X , V ([M , E , φ ]) := ˜ π ∗ ( τ V ∩ [M , E , φ ]) (4.3.15) where ˜ π : B (V) → X is the bundle pro jection induced by π . W e conclude this section b y noticing that all the abov e constructions hav e an equiv- alen t description in analytic K O-homology . 4.4 K-homology and Index Theorems W e ha ve seen in the previous sections that b oth analytic KO-homology and geomet- ric KO-homology are represen tations of sp ectral KO-homology . It follows straigh t- forw ardly from comp ositions of natural equiv alences that analytic KO-homology and geometric KO-homology are naturally equiv alent. It is in teresting, at this p oin t, to ask if suc h natural isomorphism can b e induced by a map deﬁned at the level of the cycles . That this is indeed the case can be view ed as the primordial form ulation of the index theorem, and it is the philosoph y prop osed in [14, 13]. Since this p oin t of view will b e crucial in b oth the construction and the pro of of the equiv alence carried in the next section, we will brieﬂy make the ab ov e statement more precise by illustrating the case of even complex K-homology , directing the reader to [14, 13, 53] for more information. Let M b e an even dimensional compact spin c manifold, and let E b e a complex v ector bundle on M. The canonical Dirac operator on M induces the elliptic diﬀeren tial op erator / D M ⊗ I E : Γ( / S ⊗ E) → Γ( / S ⊗ E) where / S denotes the spinor bundle asso ciated to TM. After a c hoice of a smo oth metric g on M, we can construct the Hilb ert space H M E := L 2 (Γ( / S ⊗ E); dg M ) 89 K-homology and Index Theorems and deﬁne the b ounded F redholm op erator T E M as the partial isometry in the p olar decomp osition for / D M ⊗ I E . Since M is ev en-dimensional, H M E is Z 2 -graded, and there is a *-homomorphism ρ M E : C(M; C ) → L ( H M E ) as the space of section of a v ector bundle on a manifold is equipp ed with a mod- ule structure o v er the algebra of functions of the manifold itself. Hence we hav e a corresp ondence (M , E , φ ) → ( H M E , ρ E M ◦ φ ∗ , T E M ) b et w een cycles for geometric K-homology and cycles for analytic K-homology . Theorem 4.31. ([14]) L et X b e a ﬁnite CW-c omplex. Then the c orr esp ondenc e (M , E , φ ) → ( H M E , ρ E M ◦ φ ∗ , T E M ) induc es a natur al isomorphism µ a : K t 0 (X) → K a 0 (X) c ommuting with the c ap pr o duct b etwe en K-the ory and K-homolo gy. The isomorphism µ can b e now used to give an elegan t formulation of the Atiy ah- Singer index theorem. Namely , consider a closed even-dimensional smo oth manifold M, and let T ∗ M denote its cotangent bundle. By the results in [11], an y elliptic pseudo-diﬀeren tial op erator D b et w een the sections of v ector bundles on M can b e assigned to a class in K 0 cpt (T ∗ M). More precisely , let D : Γ(E) → Γ(F) b e an elliptic diﬀeren tial op erator of order m , where the v ector bundles E and F ha v e rank p and q resp ectiv ely . The diﬀeren tial op erator D can b e expressed in lo cal co ordinates ( x 1 , . . . , x n ) on M as D = X | α |≤ m A α ( x ) ∂ | α | ∂ x α where | α | := P k α k for a n -tuple on nonnegative in tegers α = ( α 1 , . . . , α n ), and where for each α A α ( x ) is a q × p matrix of smooth complex-v alued functions on M with A α ( x ) 6 = 0 for some α suc h that | α | = m . The princip al symb ol of D is deﬁned to b e the section σ (D) of the bundle (  m TM) ⊗ Hom(E , F) represented by the co eﬃcients { A α } | α | = m , and where  denotes the symmetric tensor pro duct. Since 90 K-homology and Index Theorems  m V is canonically isomorphic to the space of homogeneous p olynomial functions of degree m on V ∗ , for an y vector space V , it follows that for each cotangent v ector ξ ∈ T ∗ x (M), the principal sym b ol giv es a homomorphism σ ξ (D) : E x → F x F or elliptic op erators, b y deﬁnition, suc h a homomorphism is in v ertible for each non- zero cotangent vector ξ and any p oint x ∈ M. The principal symbol σ (D) can b e considered to liv e on the cotangen t bundle, i.e. it deﬁnes a bundle map σ (D) : π ∗ E → π ∗ F where π : T ∗ M → M. In particular, if D is elliptic, σ (D) is an isomorphism aw ay from the zero section, hence w e can assign to D the class i (D) := [ π ∗ E , π ∗ F; σ (D)] ∈ K 0 ( B (T ∗ M) , S (T ∗ M)) Con v ersely , to an y class [E , F; µ ] ∈ K 0 ( B (T ∗ M) , S (T ∗ M)) one can assign a pseudo- diﬀeren tial op erator on M with total sym b ol µ . See [63] for details on this construction. By using the pseudo-diﬀeren tial op erator asso ciated to any class in K 0 cpt (T ∗ M), one can deﬁne isomorphisms ind t : K 0 cpt (T ∗ M) → K t 0 (M) ind a : K 0 cpt (T ∗ M) → K a 0 (M) whic h in the case that M is a spin c manifold are giv en b y the composition of Thom isomorphism and Poincar ´ e dualit y . The ab o ve isomorphisms can b e used to state the follo wing elegan t v ersion of the index theorem. Theorem 4.32. (Atiyah-Singer) L et M b e a close d smo oth manifold. Then the fol lowing diagr am c ommutes K 0 cpt (T ∗ M) ind t & & L L L L L L L L L L ind a x x r r r r r r r r r r K t 0 (M) µ a / / K a 0 (M) T o reco v er the usual form of the Atiy ah-Singer index theorem, we consider the follo wing comp osition of comm utativ e diagrams K 0 cpt (T ∗ M) ind a & & M M M M M M M M M M ind t x x q q q q q q q q q q K t 0 (M) ζ ∗ & & M M M M M M M M M M µ a / / K a 0 (M) ζ ∗ x x q q q q q q q q q q q K 0 (pt) ' Z (4.4.1) 91 The equiv alence b et ween KO t ] and K O a ] where ζ is the collapsing map, and the comm utativit y of the b ottom diagram is gran ted b y the fact that the isomorphism µ is natural. The homomorphisms ζ ∗ ◦ ind t , ζ ∗ ◦ ind a coincide with the top ological and analytical index homomorphism, resp ectiv ely , deﬁned b y A tiyah and Singer, and the comm utativity of the t w o triangles implies that ζ ∗ ◦ ind t = ζ ∗ ◦ ind a whic h is the original form ulation of the index theorem. Theorem 4.32 motiv ated the authors in [14] to state that for an y ﬂa vour k of K- theory , the equiv alence b etw een geometric and analytic k -homology is related to an index theorem for the giv en theory k . In the next section w e will reinforce the ab o v e statemen t: w e will construct a natural morphism µ a b et w een K O t ] and K O a ] deﬁned at the level of the cycles, and use a suitable index theorem to prov e that µ a is indeed a natural equiv alence. 4.5 The equiv alence b et w een K O t  and K O a  As illustrated in the previous section, our primary goal is to pro v e the following result. Theorem 4.33. Ther e is a natur al e quivalenc e µ a : K O t ≈ − → KO a b etwe en the top olo gic al and analytic KO-homolo gy functors. As w e ha ve seen, geometric and analytic K O-homology are generalized cohomology theories deﬁned on the category of ﬁnite CW-pairs ( X , Y ). F or such theories, the follo wing general result holds [30]. Theorem 4.34. L et h ] and k ] b e gener alize d homolo gy the ories deﬁne d on the c ate gory of ﬁnite CW-p airs, and let φ : h ] → k ] a natur al tr ansformation such that φ : h n (pt) → k n (pt) is an isomorphism for any n ∈ Z . Then φ is a natur al e quivalenc e. W e hav e explained in the previous section ho w the fact that the natural isomor- phism µ a b et w een geometric and analytic K-homology induces the comm utativity of 92 The equiv alence b et ween KO t ] and K O a ] the b ottom triangle in (4.4.1), and consequently the Atiy ah-Singer index theorem. Our strategy to prov e Theorem 4.33 will b e instead opp osite: namely , we will con- struct surjectiv e “index” homomorphisms ind t n and ind a n suc h that the diagram K O t n (pt) µ a / / ind t n & & M M M M M M M M M M K O a n (pt) ind a n   K O − n (pt) (4.5.1) comm utes for ev ery n , thanks to a suitable index theorem. Recall that K O t,a n (pt) ' KO − n (pt) and since the groups K O − n (pt) are equal to either 0, Z or Z 2 dep ending on the particular v alue of n , the commutativit y of the diagram (4.5.1) along with surjectivit y of the index maps are suﬃcient to prov e that µ a is an isomorphism 1 . Moreov er, the index homomorphism will play a crucial role in the applications exploited in later sections, in particular in the construction of cycles representing the generators for K O t ] (pt). 4.5.1 The natural transformation µ a Let (M , E , φ ) b e a top ological K O-cycle on X with dim M = n . Recall that M is a compact spin manifold. W e construct a corresp onding class in KO a n (X) as follo ws. Consider the asso ciated v ector bundle / S (M) := P Spin (M) × λ n C  n where C  n = C  ( R n ), λ n : Spin( n ) → End(C  n ) is given b y left multiplication with Spin( n ) ⊂ C  0 n ⊂ C  n , and P Spin (M) is the principal Spin( n )-bundle ov er M asso ciated to the spin structure on the tangen t bundle TM. Since C  n = C  0 n ⊕ C  1 n is a Z 2 -graded algebra, it follo ws that / S (M) = / S 0 (M) ⊕ / S 1 (M) (4.5.2) is a Z 2 -graded real vector bundle ov er M with resp ect to the C  (TM)-action. The Cliﬀord algebra C  n acts by righ t m ultiplication on the ﬁbres whilst preserving the bundle grading (4.5.2). Since / S (M) is a v ector bundle asso ciated to P Spin (M), it carries the canonical Riemannian connection asso ciated to the spin connection, and 1 Recall that surjective group homom orphisms of Z or any ﬁnite group are isomorphisms 93 The equiv alence b et ween KO t ] and K O a ] hence it is a Dirac bundle, i.e. it admits a canonical Dirac op erator whic h is selfadjoint on the space of L 2 -sections of / S (M) with respect to the given metric, and which has ﬁnite dimensional k ernel. Indeed, as a v ector bundle, / S (M) is isomorphic to the direct sum of irreducible real spinor bundles on M. Hence, after choosing a C ∞ Riemannian metric g M on TM, w e consider the canonical Dirac op erator / D M : C ∞ (M , / S (M)) → C ∞ (M , / S (M)) whic h w e will refer to as the A tiyah-Singer operator [10] deﬁned lo cally by / D M = n X i =1 e i · ∇ M e i , (4.5.3) where { e i } 1 ≤ i ≤ n is a lo cal orthonormal basis of sections of the tangen t bundle TM, ∇ M e i are the corresp onding comp onen ts of the spin connection ∇ M , and the dot denotes Cliﬀord m ultiplication. The op erator / D M is a C  n -op erator [63], i.e. one has / D M (Ψ · ϕ ) = / D M (Ψ) · ϕ for all Ψ ∈ C ∞ (M , / S (M)) and all ϕ ∈ C  n , where · ϕ denotes right m ultiplication b y ϕ . In particular, with resp ect to decomp osition (4.5.2), the op erator / D M is of the form / D M = 0 / D M 1 / D M 0 0 ! where / D M 0 : Γ( / S 0 (M)) → Γ( / S 1 (M)) is a real, elliptic ﬁrst-order diﬀeren tial op erator which comm utes with the action of C  0 n ' C  n − 1 on / S (M). Since / D M comm utes with the C  n -action, the vector space k er / D M is a ﬁnite dimensional graded C  n -mo dule. W e can now construct a triple ( H M E , ρ M E , T M E ) comprising the follo wing data: (i) The separable real Hilb ert space H M E := L 2 R (M , / S (M) ⊗ E; d g M ); (ii) The ∗ -homomorphism ρ M E : C(M , R ) → L ( H M E ) deﬁned b y  ρ M E ( f )(Ψ)  ( p ) = f ( p ) Ψ( p ) for f ∈ C(M , R ), Ψ ∈ C ∞ (M , / S (M) ⊗ E) and p ∈ M; and (iii) The b ounded F redholm op erator T M E := / D M E q 1 +  / D M E  2 (4.5.4) 94 The equiv alence b et ween KO t ] and K O a ] acting on H M E , where / D M E is the A tiy ah-Singer op erator (4.5.3) twisted b y the real v ector bundle E → M. This triple satisﬁes the follo wing prop erties [53]: (i) H M E is Z 2 -graded according to the splitting (4.5.2) of the Cliﬀord bundle; (ii) ρ M E ( f ) is an ev en op erator on H M E for all f ∈ C(M , R ); (iii) Since M is compact, T M E is an o dd F redholm op erator whic h ob eys the compact- ness conditions (4.2.2) with ρ M E ( f ); and (iv) There are o dd op erators ε i , i = 1 , . . . , n commuting with b oth ρ M E ( f ) and T M E whic h generate a C  n -action on H M E as in (4.2.3), and which are giv en explicitly as righ t m ultiplication b y elemen ts e i of a basis of the v ector space R n . It follo ws that ( H M E , ρ M E , T M E ) is a w ell-deﬁned n -graded F redholm module ov er the real C ∗ -algebra C(M , R ). W e now deﬁne the map µ a in (4.5.1) b y µ a  M , E , φ  := φ ∗  H M E , ρ M E , T M E  =  H M E , ρ M E ◦ φ ∗ , T M E  , (4.5.5) where φ ∗ : C(X , R ) → C(M , R ) is the real C ∗ -algebra homomorphism induced by the map φ . At this stage the map µ a is only deﬁned on KO-cycles. More precisely , w e can consider the map µ a : ΓO n (X) → K O a n (X) induced by the equiv alence relations on the set of F redholm mo dules. W e need to pro v e, at this p oin t, that the map µ a giv es a w ell deﬁned homomorphism µ a : K O t n (X) → KO a n (X) W e will ﬁrst recall some useful results concerning the ab o v e construction. Let us denote with [ / D M E ] the class corresp onding to the elemen t µ a (M , E , id M ) ∈ K O a n (M). W e can then state the following result [53] Theorem 4.35. L et M − ∂ M b e the interior of a spin manifold M of dimension n with b oundary ∂ M , and let E b e a r e al ve ctor bund le on M . Equip the b oundary ∂ M with the spin structur e induc e d by that on M . Then ∂ [ / D M − ∂ M E | M − ∂ M ] = [ / D ∂ M E | ∂ M ] wher e ∂ : KO a n (M − ∂ M) → KO a n − 1 ( ∂ M) is the b oundary homomorphism. 95 The equiv alence b et ween KO t ] and K O a ] Notice that in the theorem ab ov e w e ha ve used that K O a n (M − ∂ M) ' KO n (M , ∂ M) via excision. Moreo ver, one can prov e that the class [ / D M ] := [ / D M 1 1 ] represents the fundamen tal class of M in K O a n (M), and that [ / D M E ] = [E] ∩ [ / D M ] See [24] for details. W e are now ready to prov e the following Theorem 4.36. The map µ a : ΓO n (X) → K O a n (X) induc es a wel l deﬁne d homomor- phism µ a : K O t n (X) → KO a n (X) for any CW-c omplex X , and any n ∈ Z . Pr o of. That the map µ a resp ects the algebraic sum and indep endence of the direct sum relation follo ws straigh tforw ardly from the fact that Γ( / S (M t N) ⊗ (E t F)) = Γ( / S (M) ⊗ E) ⊕ Γ( / S (N) ⊗ F) for an y compact spin manifolds M, N, with v ector bundles E, F ov er M, N resp ectively , and b y the deﬁnition of direct sum of F redholm mo dules. T o prov e that the map homomorphism µ a do es not depend on the b ordism relation is equiv alent to proving that µ a (M , E , φ ) = 0 for ev ery b ord n -cycle (M , E , φ ), i.e. a cycle for whic h there exists an n + 1-cycle (W , F , ψ ) in the appropriate relativ e group suc h that (M , E , φ ) = ( ∂ W , F | ∂ W , ψ | ∂ W ) First, w e notice that the map ψ ∂ W factors as ψ | ∂ W = ψ ◦ i , where i : ∂ W  → W denotes the inclusion of the b oundary . W e hav e that µ a (M , E , φ ) = ( ψ | ∂ W ) ∗ ([ / D ∂ W F | ∂ W ]) = ( ψ ) ∗ ◦ i ∗ ([ / D ∂ W F | ∂ W ]) = 0 whic h follo ws b y Theorem 4.35, and the long exact sequence for the pair (W , ∂ W) · · · → K O a n +1 (W − ∂ W) ∂ − → K O a n ( ∂ W) i ∗ − → K O a n (W) → · · · 96 The equiv alence b et ween KO t ] and K O a ] T o conclude the pro of, w e need to sho w that for any KO-cycle (M , E , φ ) and any real spin v ector bundle F of rank 8 r o ver M we hav e µ a (M , E , φ ) = µ a ( b M , H(F) ⊗ π ∗ E , φ ◦ π ) By deﬁnition, w e ha v e µ a ( b M , H(F) ⊗ π ∗ E , φ ◦ π ) := φ ∗ π ∗  [ / D b M H(F) ⊗ π ∗ E ]  By using that [ / D b M H(F) ⊗ π ∗ E ] = [H(F) ⊗ π ∗ E] ∩ [ / D b M ] = ([H(F)] ∪ [ π ∗ E]) ∩ [ / D b M ] = π ∗ [E] ∩  [H(F)] ∩ [ / D b M ]  w e ha v e µ a ( b M , H(F) ⊗ π ∗ E , φ ◦ π ) = φ ∗ π ∗  π ∗ [E] ∩  [H(F)] ∩ [ / D b M ]  = φ ∗  [E] ∩ π ∗  [H(F)] ∩ [ / D b M ]  By recalling that [H(F)] is the image of the Thom class of F, and by the equation (4.3.15), the class π ∗  [H(F)] ∩ [ / D b M ]  is the image of [ / D b M ] under the isomorphism K O n +8 r ( b M) → KO n (M) for the spherical ﬁbration b M π − → M induced b y the Thom isomorphism. Since the ab o v e isomorphism maps the fundamen tal class of b M to that of M, w e ha v e π ∗  [H(F)] ∩ [ / D b M ]  = [ / D M ] The follo wing equalities conclude the pro of µ a ( b M , H(F) ⊗ π ∗ E , φ ◦ π ) = φ ∗  [E] ∩ [ / D M ]  = φ ∗  [ / D M E ]  = µ a (M , E , φ ) W e refer the reader to [15] for an alternativ e pro of of Theorem 4.36. 4.5.2 The analytic index map ind a n Let ( H R , ρ, T) b e an n -graded F redholm mo dule o ver the real C ∗ -algebra C(X , R ) suc h that T is a F redholm op erator. Since T commutes with ε i for i = 1 , . . . , n , the 97 The equiv alence b et ween KO t ] and K O a ] k ernel ker T ⊂ H R is a real C  n -mo dule with Z 2 -grading induced by the grading of H R . Th us we can deﬁne ind a n (T) := [k er T] ∈ c M n /ı ∗ c M n +1 (4.5.6) where c M n is the Grothendiec k group of irreducible Cliﬀord modules deﬁned in c hapter 3. By the A tiy ah-Bott-Shapiro isomorphism, w e ha v e c M n /ı ∗ c M n +1 ' K O − n (pt) W e will call (4.5.6) the analytic or Cliﬀor d index of the F redholm op erator T. An imp ortan t prop ert y of this deﬁnition is the following result [63]. Theorem 4.37. The analytic index ind a n : F red n − → KO − n (pt) is surje ctive and c onstant on the c onne cte d c omp onents of F red n . Giv en t wo F redholm mo dules ( H R , ρ, T) and ( H R , ρ, T 0 ) o ver a real C ∗ -algebra A , w e will sa y that T is a c omp act p erturb ation of T 0 if (T − T 0 ) ρ ( a ) ∈ K R for all a ∈ A . W e then hav e the following elementary result. Lemma 4.38. If T is a c omp act p erturb ation of T 0 , then the F r e dholm mo dules ( H R , ρ, T) and ( H R , ρ, T 0 ) ar e op er ator homotopic over A . Pr o of. Consider the path T t = (1 − t ) T + t T 0 for t ∈ [0 , 1]. Then the map t 7→ T t is norm contin uous. W e will show that for an y t ∈ [0 , 1], the triple ( H R , ρ, T t ) is a F redholm mo dule ov er A , i.e. that the op erator T t satisﬁes  T 2 t − 1  ρ ( a ) , (T t − T ∗ t ) ρ ( a ) , T t ρ ( a ) − ρ ( a ) T t ∈ K R (4.5.7) for all a ∈ A . The last tw o inclusions in (4.5.7) are easily pro v en b ecause the path T t is “linear” in the op erators T and T 0 . T o establish the ﬁrst one, for an y t ∈ [0 , 1] and a ∈ A we compute (T 2 t − 1) ρ ( a ) = h (T 2 − 1) + t 2 (T − T 0 ) 2 − t (T 2 − 1) − t (T − T 0 ) 2 + t (T 0 2 − 1) i ρ ( a ) . (4.5.8) By using the fact that ( H R , ρ, T) and ( H R , ρ, T 0 ) are F redholm mo dules, that T is a compact p erturbation of T 0 , and that K R is an ideal in L ( H R ), one easily veriﬁes that the right-hand side of (4.5.8) is a compact op erator. This implies that ( H R , ρ, T t ) is a w ell-deﬁned family of F redholm mo dules ov er A . 98 The equiv alence b et ween KO t ] and K O a ] By Lemma 4.38, w e can c ho ose the op erator T in the class [ H R , ρ, T] to b e selfad- join t without loss of generalit y . Indeed, giv en a F redholm module w e simply replace the op erator T with ˜ T := 1 2 (T + T ∗ ). Moreov er, suc h a choice of ˜ T is compatible with op erator h omotop y . Hence in the follo wing w e will alwa ys assume that the op erator T is selfadjoin t and F redholm. Prop osition 4.39. The induc e d map ind a n : K O a n (X) − → K O − n (pt) given on classes of n -gr ade d F r e dholm mo dules by ind a n [ H R , ρ, T] = [k er T] is a wel l-deﬁne d surje ctive homomorphism for any n ∈ N . Pr o of. W e ﬁrst show that to the direct sum of tw o F redholm mo dules ( H R , ρ, T) and ( H 0 R , ρ 0 , T 0 ) o ver A = C(X , R ), the map ind a n asso ciates the class [k er T] + [ker T 0 ] ∈ c M n /ı ∗ c M n +1 ' K O − n (pt). The kernel k er(T ⊕ T 0 ) = ker(T) ⊕ ker(T 0 ) is a real graded C  n -mo dule. By the deﬁnition of the group c M n and of its quotien t b y ı ∗ c M n +1 , one thus has ind a n (T ⊕ T 0 ) = [ker T] + [k er T 0 ] and so the map ind a n resp ects the algebraic structure on ΓO n ( A ). Consider no w t w o F redholm mo dules ( H R , ρ, T) and ( H 0 R , ρ 0 , T 0 ) whic h are orthog- onally equiv alen t. Then there exists an even isometry U : H R → H 0 R suc h that T 0 = U T U ∗ , ε 0 i = U ε i U ∗ . This implies that k er T 0 = U (ker T), and that the graded C  n represen tations given resp ectiv ely by ε 0 i and ε i are equiv alent. In particular, they represen t the same class in c M n /ı ∗ c M n +1 . Finally , consider tw o homotopic n -graded F redholm mo dules ( H R , ρ, T), ( H R , ρ, T 0 ) o v er A . There exists b y deﬁnition a contin uous path t → T t connecting T and T 0 in F red n . Hence, by Theorem 4.37 ind a n (T) = ind a n (T 0 ). 4.5.3 The top ological index map ind t n Giv en a KO-cycle (M , E , φ ) on X with M an n -dimensional compact spin manifold, w e can assign to it the asso ciate d Atiyah-Milnor-Singer (AMS) invariant [63] deﬁned b y b A E (M) = β ◦ f ! ([E]) ∈ KO − n (pt) (4.5.9) 99 The equiv alence b et ween KO t ] and K O a ] where f ! is the Gysin homomorphism for the smo oth em b edding f : M  → R n +8 k (4.5.10) for some k ∈ N suﬃciently large, and β is induced by the Bott p erio dicit y isomorphism β : KO cpt ( R n +8 k ) → KO − n (pt) This deﬁnition do es not dep end on the embedding (4.5.10) nor on the in teger k . W e deﬁne ind t n (M , E , φ ) := b A E (M) . (4.5.11) Prop osition 4.40. The map ind t n : K O t n (X) − → K O − n (pt) induc e d by (4.5.11) is a wel l-deﬁne d surje ctive homomorphism for any n ∈ N . Pr o of. W e ﬁrst prov e that the map ind t n resp ects the algebraic structure on the ab elian group KO t n (X). Given tw o n -dimensional compact spin manifolds M 1 and M 2 , let M = M 1 q M 2 . Em b ed M in the Euclidean space R n +8 k for some k suﬃciently large as in (4.5.10). Recall no w that the Gysin homomorphism of f is the composition of Thom isomorphism with resp ect to the normal bundle of M in R n +8 k with the map “extending b y zero”, and that the Thom isomorphism is the isomorphism induced b y the Thom class. Moreov er, notice that the normal bundle ν to the em b edding of M is giv en by ν 1 q ν 2 , where ν 1 and ν 2 are resp ectively the normal bundles to the em b eddings of M 1 and M 2 induced b y the embedding of M. The Thom class of ν is giv en b y τ ν :=   ∗ / S + ( ν ) ,  ∗ / S − ( ν ) ; σ  =   ∗ 1 / S + ( ν 1 ) q  ∗ 2 / S + ( ν 2 ) ,  ∗ 1 / S − ( ν 1 ) q  ∗ 2 / S − ( ν 2 ) ; σ 1 q σ 2  = τ ν 1 + τ ν 2 ∈ K O 0 (B( ν ) , S( ν )) ' K O 0 (B( ν 1 ) , S( ν 1 )) ⊕ KO 0 (B( ν 1 ) , S( ν 2 )) . where  =  1 q  2 : ν 1 q ν 2 → M 1 q M 2 is the normal bundle pro jection. Let E 1 and E 2 b e real v ector bundles ov er M 1 and M 2 , respectively , and let E = E 1 q E 2 . Then in K O 0 (B( ν ) , S( ν )) one has τ ν (E) = τ ν  [  ∗ E] = τ ν  [  ∗ 1 E 1 q  ∗ 2 E 2 ] = τ ν 1  [  ∗ 1 E 1 ] + τ ν 2  [  ∗ 2 E 2 ] = τ ν 1 (E 1 ) + τ ν 2 (E 2 ) . 100 The equiv alence b et ween KO t ] and K O a ] Since the map extending b y zero is a homomorphism, one then ﬁnds ind t n  (M 1 , E 1 , φ 1 ) q (M 2 , E 2 , φ 2 )  := b A E (M) = b A E 1 (M 1 ) + b A E 2 (M 2 ) = ind t n (M 1 , E 1 , φ 1 ) + ind t n (M 2 , E 2 , φ 2 ) ∈ KO − n (pt) , sho wing that ind t n resp ects the algebraic sum of cycles. Next w e ha v e to chec k that the map ind t n is indep enden t of the c hoice of represen tativ e of a homology class in K O t n (X). The indep endence of the direct sum relation follo ws from the discussion abov e, while spin bordism indep endence is guaranteed by the prop ert y that the AMS inv ariant b A E (M) is a spin cob ordism in v arian t [63]. Finally , w e ha v e to verify that the map ind t n do es not dep end on real v ector bundle mo diﬁcation. Let M b e a smo oth n -dimensional compact spin manifold and let E → M b e a smo oth real v ector bundle. Let F b e a real spin vector bundle o ver M with ﬁbres of real dimension 8 l for some l ∈ N . Consider the corresp onding sphere bundle (4.3.1) with pro jection (4.3.2). Real vector bundle modiﬁcation of a KO-cycle (M , E , φ ) on X induced by F pro duces the KO-cycle ( b M , b E , φ ◦ π ), where b E = H(F) ⊗ π ∗ (E) is the real v ector bundle o v er b M suc h that  b E  = Σ F !  E  with [E] ∈ K O 0 (M), [ b E ] ∈ K O 0 ( b M ), and Σ F deﬁned as in (4.3.9). W e may compute the AMS in v arian t for the pair ( b M , b E ) b y c ho osing an em b edding b f : b M  → R n +8 k +8 l so that b A b E  b M  = β ◦ b f !  [ b E ]  = β ◦ b f ! ◦ Σ F ! [E] = β ◦  b f ◦ Σ F  ! [E] , where in the last equalit y w e ha v e used functorialit y of the Gysin map. Notice that b f ◦ Σ F : M  → R n +8 k +8 l =: R n +8 m is an em b edding of M in to a “large enough” Euclidean space. Since b A E (M) is inde- p enden t of the em b edding and the integer m , we hav e b A b E  b M  = b A E  M  as required. 101 The equiv alence b et ween KO t ] and K O a ] 4.5.4 The Isomorphism Theorem W e can now assemble the constructions of the previous sections to ﬁnally establish our main result. Notice ﬁrst of all that since ker / D M E ' k er T M E , one has ind a n ◦ µ a  M , E , φ  = ind a n  / D M E  (4.5.12) for any K O-cycle (M , E , φ ) on X with dim M = n . A t this p oint w e can use an imp ortan t result from spin geometry called the C  n -index the or em [63]. Theorem 4.41. L et M b e a c omp act spin manifold of dimension n and let E b e a r e al ve ctor bund le over M . L et / D M E : C ∞  M , / S (M) ⊗ E  − → C ∞  M , / S (M) ⊗ E  b e the C  n -line ar A tiyah-Singer op er ator with c o eﬃcients in E . Then ind a n  / D M E  = b A E  M  . The pro of of Theorem 4.33 is no w completed once w e establish the follo wing result. Prop osition 4.42. The map µ a : K O t n (pt) − → K O a n (pt) is an isomorphism for any n ∈ N . Pr o of. As noticed at the b eginning of this section, it suﬃces to establish the comm u- tativit y of the diagram (4.5.1), i.e. that ind t n = ind a n ◦ µ a . Let [M , E , φ ] b e the class of a KO-cycle ov er pt with dim M = n . Using Theorem 4.41 and (4.5.12) w e ha v e ind t n  M , E , φ  := b A E  M  = ind a n  / D M E  = ind a n ◦ µ a  M , E , φ  as required. 102 The Real Chern Character 4.6 The Real Chern Character In this section we will describ e the natural complexiﬁcation map from geometric K O-homology to geometric K-homology and use it to deﬁne the Chern c haracter homomorphism in top ological KO-homology . W e describ e v arious prop erties of this homomorphism, most notably its in timate connection with the AMS in v arian t which w as the crux of the isomorphism of the previous section. Let X b e a compact top ological space. Consider the top ological, generalized homology groups K t ] (X) and K O t ] (X), along with the corresp onding K-theory and K O-theory groups. The complexiﬁcation of a real vector bundle ov er X is a complex v ector bundle o v er X which is isomorphic to its own conjugate vector bundle. The complexiﬁcation map is compatible with stable isomorphism of real and complex v ector bundles, and th us deﬁnes a homomorphism from stable equiv alence classes of real vector bundles to stable equiv alence classes of complex v ector bundles. It thereby induces a natural transformation of cohomology theories ( ⊗ C ) ∗ : K O ∗ (X) − → K ∗ (X) induced b y [E] − [F] 7− → [E C ] − [F C ] where E C := E ⊗ C is the complexiﬁcation of the real vector bundle E → X. W e can also deﬁne a complexiﬁcation morphism relating the homology theories ( ⊗ C ) ∗ : K O t ] (X) − → K t ] (X) (4.6.1) b y [M , E , φ ] ⊗ C := [M , E C , φ ] and extended b y linearit y , where on the righ t-hand side w e regard M endo wed with the spin c structure induced b y its spin structure as a K O-cycle. One can easily see that [M , E , φ ] ⊗ C = φ ∗  [E C ]  [M] K  (4.6.2) where [M] K ∈ K ] (M) denotes the K-theory fundamental class of M. Th us in the case when X is a compact spin manifold, the homomorphism ( ⊗ C ) ∗ is just the P oincar´ e dual of ( ⊗ C ) ∗ . This is clearly a natural transformation of homology theories. A related natural transformation b et w een cohomology theories is the realiﬁcation morphism ( R ) ∗ : K ] (X) − → K O ] (X) 103 The Real Chern Character induced b y assigning to a complex vector bundle o v er X the underlying real vector bundle o v er X. Because a spin c manifold is not necessarily spin, w e cannot implemen t this transformation in the homological setting in general. Rather, w e m ust assume that X is a compact spin manifold. In this case the K-homology group K t ] (X) has generators [X × S n , E i , pr 1 ] − [X × S n , F i , pr 1 ], 0 ≤ n ≤ 7, where pr 1 : X × S n → X is the pro jection on to the ﬁrst factor [80]. W e can then deﬁne the morphism ( R ) ∗ : K t ] (X) − → K O t ] (X) b y  [X × S n , E i , pr 1 ] − [X × S n , F i , pr 1 ]  R := [X × S n , E i R , pr 1 ] − [X × S n , F i R , pr 1 ] and extending by linearity . Since this deﬁnition dep ends on a c hoice of generators for K t ] (X), the transformation is not natural. As for the complexiﬁcation morphism, the morphism ( R ) ∗ th us deﬁned is Poincar ´ e dual to ( R ) ∗ . It follo ws that the comp osition ( R ) ∗ ◦ ( ⊗ C ) ∗ is m ultiplication b y 2. W e can use the natural transformation provided b y the complexiﬁcation homo- morphism (4.6.1) to deﬁne a real homological Chern c haracter c h R • : K O t ] (X) − → H ] (X , Q ) (4.6.3) b y c h R • ( ξ ) = c h • ( ξ ⊗ C ) for ξ ∈ K O t ] (X), where on the righ t-hand side w e use the K-homology Chern c haracter c h • : K t ] (X) → H ] (X , Q ), which can b e deﬁned as c h • (M , E , φ ) = φ ∗  (c h(E) ∪ e d (TM) / 2 ˆ A(TM)) ∩ [M]  where the cohomology class d (TM) is deﬁned in App endix C. The real Chern c haracter (4.6.3) is a natural transformation of homology theories, and it preserv es the cap pro duct, i.e. the diagram K O ∗ t (X) ⊗ KO t ] (X) ∩ / / ch • ⊗ ch •   K O t ] (X) ch •   H ∗ t (X; Q ) ⊗ H t ] (X; Q ) ∩ / / H t ] (X; Q ) comm utes for an y space X. This prop ert y is indeed guaran teed b y the fact that ch R • is the natural transformation in KO-homology induced b y the real Chern c haracter in K O-theory . See [58] for details. Finally , tensoring with Q giv es a map c h R • ⊗ id Q =  c h • ⊗ id Q  ◦  ( ⊗ C ) ∗ ⊗ id Q  : K O t ] (X) ⊗ Z Q − → H ] (X , Q ) In constrast with the complex case, this map is not an isomorphism. 104 Cohomological Index form ulas 4.7 Cohomological Index form ulas W e will now explore the relation b etw een the homological real Chern c haracter and the top ological index deﬁned in (4.5.11). In particular, w e will be able to give cohomo- logical C  n -index form ulas. W e ﬁrst sho w that up to Poincar ´ e duality the top ological index is the homological morphism induced b y the collapsing map. Recall that up to isomorphism, the AMS in v arian t is giv en b y b A E (M) = ˜ ζ KO ! [E] where M is a compact spin manifold of dimension n , E is a real vector bundle ov er M, ˜ ζ : M → pt is the collapsing map on M, and ˜ ζ KO ! is the corresponding Gysin homomorphism 2 on K O-theory . In this case w e ha v e ˜ ζ KO ∗ = Φ pt ◦ ˜ ζ KO ! ◦ Φ − 1 M where ˜ ζ KO ∗ is the induced morphism on KO t ] (M), and Φ pt and Φ M are the P oincar ´ e dualit y isomorphisms on pt and M, resp ectiv ely . It then follo ws that Φ pt ◦ ind t n (M , E , φ ) = Φ pt ◦ ˜ ζ KO ! [E] = Φ pt ◦ ˜ ζ KO ! ◦ Φ − 1 M (M , E , id M ) = ˜ ζ KO ∗ [M , E , id M ] = [M , E , ˜ ζ ] = ζ KO ∗ [M , E , φ ] (4.7.1) where ζ : X → pt is the collapsing map on X with ˜ ζ = ζ ◦ φ . W e will next describ e how the real Chern character can be used to give a c haracteristic class description of the map ind t n in the torsion-free cases. Consider ﬁrst the case n ≡ 4 mo d 8. W e b egin b y sho wing that there is a commutativ e diagram K O t 4 (X) ζ KO ∗ / / ζ H ∗ ◦ ch R • & & M M M M M M M M M M K O t 4 (pt) ch R •   H 0 (pt , Q ) (4.7.2) where ζ H ∗ is the induced morphism on homology . Recall that ch R • = c h • ◦ ( ⊗ C ) ∗ , where ( ⊗ C ) ∗ is the complexiﬁcation map (4.6.1), hence it can b e expressed as c h R • (M , E , φ ) = φ ∗  (c h(E C ) ∪ ˆ A(TM)) ∩ [M]  2 W e are using the Gysin homomorphism deﬁnition for contin uous and proper maps, whic h are not necessarily embeddings. See [61] for details. 105 Cohomological Index form ulas Then one has ζ H ∗ ◦ c h R • (M , E , φ ) = ζ H ∗ ◦ φ ∗  (c h • (E C ) ∪ b A(TM)) ∩ [M]  = ( ζ ◦ φ ) ∗  (c h • (E C ) ∪ b A(TM)) ∩ [M]  = c h R • (M , E , ˜ ζ ) = ch R • ◦ ζ KO ∗ [M , E , φ ] . No w we use the fact that the map ch R • : K O t 4 (pt) → H 0 (pt , Q ) sends Z → 2 Z ⊂ Q [61]. On its image, the homomorphism c h R • is thus inv ertible and its in verse is giv en as division b y 2. This can b e pro v en as follo ws. W e hav e ζ H ∗ ◦ c h R • (M , E , φ ) = ζ H ∗ ◦ Φ M  c h • (E C ) ∪ b A (TM)  = Φ pt ◦ ζ H !  c h • (E C ) ∪ b A (TM)  =  c h • (E C ) ∪ b A (TM) , [M]  , (4.7.3) where h− , −i : H ] (M , Q ) × H ] (M , Q ) → Q is the canonical dual pairing b et w een cohomology and homology . In (4.7.3) we ha v e used the fact that Φ pt is the iden tity on H 0 (pt , Q ) ' Q , and the pro of of the last equality uses the Atiy ah-Hirzebruch version of the Grothendiec k-Riemann-Ro c h theorem [61]. Recall that for a spin manifold M of dimension 4 k + 8, one has  c h • (E C ) ∪ b A (TM) , [M]  ∈ 2 Z After using the isomorphism K O 4 (pt) ' Z , we thus deduce that ζ KO ∗ [M , E , φ ] = 1 2  c h • (E C ) ∪ b A (TM) , [M]  and from (4.7.1) w e arriv e ﬁnally at ind t n (M , E , φ ) = 1 2  c h • (E C ) ∪ b A (TM) , [M]  When n ≡ 0 mod 8, one obtains a similar result but no w without the factor 1 2 , since in that case c h R • : K O t 0 (pt) → H 0 (pt , Q ) is the inclusion Z  → Q [61]. In the remaining non-trivial cases n ≡ 1 , 2 mod 8 the homological Chern character is of no use, as K O − n (pt) is the pure torsion group Z 2 , and there is no cohomological formula for the AMS in v arian t in these instances. Ho wev er, by using Theorem 4.41 one still has an in teresting mo d 2 index form ula for the top ological index in these cases as well [63]. W e can summarize our homological deriv ations of these index formulas as follows. Theorem 4.43. L et [M , E , φ ] ∈ KO t n (X) , and let / D M E b e the A tiyah-Singer op er ator on M with c o eﬃcients in E . L et / H M E := k er / D M E denote the ve ctor sp ac e of r e al harmonic 106 D-branes and K-homology E -value d spinors on M , wher e / D M E is the Atiyah-Singer op er ator for the irr e ducible r e al spinor bund le on M . Then one has the C  n -index formulas ind t n (M , E , φ ) =                   c h • (E C ) ∪ b A (TM) , [M]  , n ≡ 0 mo d 8 , dim C / H M E mo d 2 , n ≡ 1 mo d 8 , dim H / H M E mo d 2 , n ≡ 2 mo d 8 , 1 2  c h • (E C ) ∪ b A (TM) , [M]  , n ≡ 4 mo d 8 , 0 , otherwise . 4.8 D-branes and K-homology In this section w e will show ho w geometric K-homology can b e used to describ e D- branes in t yp e II and type I String theory in a top ological nontrivial spacetime. W e will introduce the notion of wr app e d D-br ane on a given submanifold of spacetime, w e will deﬁne the group of charges of wrapp ed D-branes, and construct explicit examples of wrapp ed D-branes whic h ha v e torsion c harge. Recall by section 3.7 that the group of top ological c harges of a D p -brane realized as a spin c submanifold W ⊂ X is giv en b y K 0 cpt ( ν W ) ' K 0 (B(N(W)) , S(N(W))) as proposed by Witten. According to the Sen-Witten construction, the classes in K 0 cpt ( ν W ) are in terpreted as systems of D9 − D ¯ 9 branes whic h are unstable, and will deca y on to the worldv olume W , which corresp ond to the zero lo ci of the appropriate tac h y on ﬁeld. In particular, this pro cess happ ens in spacetime, and it depends on ho w the worldv olume is embedded in it. On the other hand, the role pla y ed b y the Chan-Paton vector bundle on the D p -brane is not manifest in this classiﬁcation. Ho w ev er, there is a natural w a y of classifying the D p -branes on W b y means whic h manifestly takes into accoun t the Chan-Paton bundle con tribution. Indeed, from the D p -brane data, w e can naturally construct the Baum-Douglas cycle (W , E , id), where E denotes the Chan-Paton bundle, and declare that its charge is giv en by the class [W , E , id] ∈ K p +1 (W). As the group K p +1 (W) contains no information ab out the em b edding of the w orldv olume W in X, w e can intuitiv ely think the charge [W , E , id] takes into account how the D-brane wraps the submanifold W . Notice that this analogous to the charge classiﬁcation of an extended ob ject in an abelian gauge theory via the homology cycle of its worldv olume. Finally , the equiv alence relation 107 D-branes and K-homology that deﬁnes the group K p +1 (W) are very natural from the ph ysical point of view. See [80] for details. A t this p oint, w e notice that by deﬁnition the elemen ts of K t p +1 (W) are giv en by (diﬀerences of ) classes [M , E , φ ] where M is a p + 1-dimensional manifold. Ho w ev er, it is not alw ays p ossible to c ho ose the map φ in [M , W] in such a w ay that φ is a diﬀeomorphism. This motiv ates the follo wing Deﬁnition 4.44. Let X b e a t yp e II String theory spacetime, described b y a 10- dimensional spin manifold, and let W ⊂ X b e a spin c submanifold. A D p -brane wr apping the w orldvolume W is deﬁned as the K-cycle (M , E , φ ), where dim M = p + 1, and φ (M) ⊂ W . W e will call E the Chan-Paton bundle asso ciated to the wrapp ed D p -brane, and we will sa y that the D p -branes ﬁl ls W if φ (M) = W . The charge of the wrapp ed D p -brane is giv en b y the class [M , E , φ ] in the group K t p +1 (W). Notice that in the ab o v e deﬁnition w e hav e relaxed the condition that dimW = p + 1, as w e are not requiring that the wrapping preserves the dimension of the D- brane. This is an attempt to tak e in to accoun t, at least at the top ological lev el, the w ell-kno wn fact that D-branes are not alwa ys representable as submanifolds equipped with vector bundles, since they are b oundary conditions for a sup erconformal ﬁeld theory , and that a distinction should b e made betw een the wrapping D-brane, in this case identiﬁed with a K-cycle representing a particular t yp e of b oundary conditions, and the worldv olume it wraps. Notice also that the group of c harges of wrapped D p -branes do es not dep end on ho w the manifold W is em b edded into the spacetime, and hence it seems to represen t a gen uine w orldvolume concept. In particular, as men tioned ab ov e, the wrapp ed D-brane deﬁnition is very natural in the ordinary case of a D-brane realized as a submanifold W of spacetime equipp ed with a Chan-Paton bundle E, as it only dep ends on how the vector bundle is deﬁned on the submanifold, and not on the pro cedure used to “extend” it to the spacetime. Finally , in the case of ordinary D-branes wrapping W with dimW = p + 1, the group K p +1 (W) coincides with the group of charges of type I IB D p -branes that can b e obtained via the Sen-Witten construction, i.e. via brane-antibrane deca y . This can b e shown as follows. Since the normal bundle ν W → W is a spin c v ector bundle, w e can use the Thom isomorphism in K-theory to establish that K 0 cpt ( ν W ) ' K 0 (W) As W is a spin c submanifold of the spacetime, w e can use P oincar ´ e Duality to get K 0 (W) ' K t p +1 (W) 108 The group K O t ] (pt) and torsion branes where p + 1 = dim W. This suggests that for ordinary type I I D p -branes the wrapping c harge is completely determined by the decay of the tac h y on ﬁeld. It is natural at this p oin t to extend the notion of wrapp ed D-brane and of wrapping charge to type I String theory . In this case, though, the tw o notions of charge do not coincide, as w e will sho w in the follo wing. Recall that in t yp e I String theory the group of top ological charges of a D p -brane realized as a spin submanifold W ⊂ X is giv en b y K O 0 cpt ( ν W ) ' KO 0 (B(N(W)) , S(N(W))) (4.8.1) By using the Thom isomorphism in K O-theory , we hav e that K O 0 cpt ( ν W ) ' KO p − 9 (W) Finally , by Poincar ´ e Dualit y , we get K O p − 9 (W) ' KO 10 (W) The group KO 10 (W) is in general not isomorphic to the group KO p +1 (W), and ex- plicitly depends on the dimension of the spacetime. W e can physically interpret the elemen ts of K O 10 (W) as equally charged systems of wrapping D9 − D ¯ 9-branes deca y- ing on the submanifold W, and via the inclusion i : W  → X they can be related to the D9-branes used in the Sen-Witten construction. This is not surprising, as the decay mec hanism is somehow at the heart of the spacetime D-brane charge classiﬁcation, and it reinforces the statement that the group (4.8.1) enco des spacetime prop erties of the D p -brane. After in tro ducing the K-theoretical description of Ramond-Ramond ﬁelds in the next c hapter, w e will argue that wrapp ed D-branes can in principle couple to Ramond-Ramond ﬁelds. W e commen t at this point that the deﬁnition of wrapped branes and of wrapp ed c harge as presen ted in this section is v ery natural from the mathematical point of view, but is still heuristic in nature. Indeed, stronger evidences for wrapped D-branes and their coupling to Ramond-Ramond ﬁelds should come from the b oundary conformal ﬁeld theory describing type I String theory in top ologically nontrivial settings, whic h has not y et b een in v estigated in full generalit y . 4.9 The group K O t  (pt) and torsion branes In this section we will ﬁnd explicit generators for the group K O t ] (pt), whic h, as ex- plained in the previous section, we can interpret as stable D-branes wrapping a p oin t 109 The group K O t ] (pt) and torsion branes in spacetime. The stabilit y of suc h D-branes is related to the fact that their charge is the “lightest” in the group of p ossible c harges, hence the conserv ation of c harge do es not allow these D-branes to deca y . T o this end, we ﬁrst give suﬃcient com bina- torial criteria on the rational homology of X which ensure that torsion-free D-branes can wrap non-trivial spin cycles of the spacetime X, whic h is an adaptation of an analogous result in [80] Theorem 4.45. L et X b e a c omp act c onne cte d ﬁnite CW-c omplex of dimension n whose r ational homolo gy c an b e pr esente d as H ] (X , Q ) = n M p =0 m p M i =1  M p i  Q , wher e M p i is a p -dimensional c omp act c onne cte d spin submanifold of X without b ound- ary and with orientation cycle [M p i ] given by the spin structur e. Then the K O-homolo gy lattic e Λ KO t ] (X) := KO t ] (X) / tor KO t ] (X) c ontains a set of line arly indep endent elements given by the classes of K O-cycles  M p i , 1 1 R M p i , ι p i  , 0 ≤ p ≤ n , 1 ≤ i ≤ m p . Pr o of. By [80] the cycles  M p i , 1 1 C M p i , ι p i  form a rational basis for the lattice Λ K t ] (X) := K t ] (X) / tor K t ] (X) in K-homology . The conclusion follo ws from the fact that  M p i , 1 1 R M p i , ι p i  ⊗ C =  M p i , 1 1 C M p i , ι p i  , i.e. that the elements c h R •  M p i , 1 1 R M p i , ι p i  form a set of generators of H ] (X , Q ). Recall no w, that we ha v e K O t n (pt) ' Z for n = 0 , 4, KO t n (pt) ' Z 2 for n = 1 , 2, and 0 otherwise. F or n = 0 , 4 Theorem 4.45 and the Chern c haracter assure that the classes [pt , 1 1 R pt , id pt ] and [S 4 , 1 1 R S 4 , ζ ] are generators of the groups KO t 0 (pt) ' Z and K O t 4 (pt) ' Z , respectively . Let us no w consider the group KO t 1 (pt). Consider the circle S 1 and assign to it a Riemannian metric. Since there is only one unit tangent v ector at an y p oint of S 1 , one has P S O (S 1 ) ' S 1 , where P S O (S 1 ) is the orthonormal frame bundle of S 1 . A spin structure on S 1 is th us giv en b y a double co v ering P Spin  S 1  − → S 1 and b y the ﬁbration Z 2 / / P Spin  S 1  .   S 1 110 The group K O t ] (pt) and torsion branes There are only tw o double cov erings of the circle, one disconnected and the other connected, giv en resp ectiv ely b y S 1 × Z 2 − → S 1 , S 1 M − → S 1 where S 1 M is the total space of the principal Z 2 -bundle asso ciated to the M¨ obius strip. W e will call these t w o spin structures the “in teresting” and the “uninteresting” spin structures, resp ectiv ely . Corresp onding to these t w o spin structures (lab elled ‘i’ and ‘u’, respectively), w e construct classes in KO t 1 (pt) giv en by [S 1 i , 1 1 R S 1 , ζ ] and [S 1 u , 1 1 R S 1 , ζ ] where ζ : S 1 → pt is as usual the collapsing map. W e will no w compute the topological indices in detail, ﬁnding the AMS in v arian ts [63] b A 1 1 R S 1  S 1 i  = 1 , b A 1 1 R S 1  S 1 u  = 0 in KO − 1 (pt) ' Z 2 . Hence the tw o classes ab ov e represen t the elemen ts of K O t 1 (pt) ' Z 2 . In particular, [S 1 i , 1 1 R S 1 , ζ ] is a generator, analogous to the non-BPS Type I D- particle that arises from tac h y on condensation. Let us ﬁrst consider the circle with the interesting spin structure. Since C  1 ' C , one has / S (S 1 ) := P Spin (S 1 ) × Z 2 C  1 ' S 1 × C . By decomp osing C = R ⊕ i R , one has the identiﬁcations / S 0 (S 1 ) = S 1 × R and / S 1 (S 1 ) = S 1 × i R . As the Cliﬀord bundle is trivial, its space of sections is given by C ∞ (S 1 , / S (S 1 )) = C ∞ (S 1 , C ). By co ordinatizing the circle S 1 with arc length s , the A tiy ah-Singer op erator can b e expressed as / D S 1 = i d d s (4.9.1) where e 1 = i is a generator of the Cliﬀord algebra C  1 . T o compute the top ological index b A 1 1 R S 1 (S 1 i ), w e use the C  1 -index Theorem 4.41 and hence determine the vector space ker / D S 1 , or equiv alently the chiral subspace ker( / D S 1 ) 0 . Since C ∞ (S 1 , / S 0 ) = C ∞ (S 1 , R ), the k ernel of the c hiral Atiy ah-Singer op erator ( / D S 1 ) 0 : C ∞ (S 1 , / S 0 ) → C ∞ (S 1 , / S 1 ) is given by the space of real-v alued constan t functions on S 1 . The dimen- sion of this v ector space, as a mo dule o v er C  0 1 ' R , is 1 and hence ind t 1  S 1 i , 1 1 R S 1 , ζ  =  k er( / D S 1 ) 0  = 1 in M 0 /ı ∗ M 1 ' K O − 1 (pt) ' Z 2 . (Note that here we are using ungr ade d Cliﬀord mo dules.) W e no w turn to the uninteresting spin structure on S 1 . This time the bundle / S (S 1 ) is the (inﬁnite complex) M¨ obius bundle. It can b e describ ed by a trivialization made 111 The group K O t ] (pt) and torsion branes of three charts U 1 , U 2 and U 3 with Z 2 -v alued transition functions g 12 = 1, g 23 = 1 and g 31 = − 1. In this case, the v ector space ker( / D S 1 ) 0 consists of lo cally constant real-v alued functions ψ i deﬁned on U i whic h satisfy ψ j = g j i ψ i on the in tersections U i ∩ U j 6 = ∅ . Because of the non-trivial transition function g 31 , there are no non-zero solutions ψ to the equation ( / D S 1 ) 0 ψ = 0. The k ernel k er( / D S 1 ) 0 is th us trivial, and so ind t 1  S 1 u , 1 1 R S 1 , ζ  = 0 . Let us no w consider the structure of the group K O t 2 (pt). Analogously to the construc- tion ab o ve, one can equip the torus T 2 = S 1 × S 1 with an “in teresting” spin structure and sho w that b A 1 1 R T 2  T 2  = 1 , and also that b A 1 1 R S 2  S 2  = 0 in K O − 2 (pt) ' Z 2 . It follo ws that the classes [T 2 , 1 1 R T 2 , ζ ] and [S 2 , 1 1 R S 2 , ζ ] represent the elemen ts of the group K O t 2 (pt) ' Z 2 . In particular, [T 2 , 1 1 R T 2 , ζ ] is a generator, and it is analogous to the T yp e I non-BPS D-instan ton which is usually constructed as the Ω-pro jection of the Type I IB D( − 1) brane-antibrane system. W e will no w giv e some details of these results. Equip T 2 with the ﬂat metric d θ 1 ⊗ d θ 1 + d θ 2 ⊗ d θ 2 , where ( θ 1 , θ 2 ) are angular co ordinates on S 1 × S 1 . Since T 2 is a Lie group, its tangen t bundle is trivializable, and hence the oriented orthonormal frame bundle is canonically given by P SO (T 2 ) = T 2 × S 1 . Consider the spin structure on T 2 giv en b y P Spin  T 2  = T 2 × S 1 id T 2 × z 2 − − − − → T 2 × S 1 . Since C  2 ' H and C  0 2 ' C , the corresp onding Cliﬀord bundles are / S (T 2 ) = T 2 × H and / S 0 (T 2 ) = T 2 × C . In the riemannian co ordinates ( θ 1 , θ 2 ), the Atiy ah-Singer op erator can b e expressed as / D T 2 = σ 1 ∂ ∂ θ 1 + σ 2 ∂ ∂ θ 2 where the P auli spin matrices σ 1 = 0 1 1 0 ! , σ 2 = 0 − i i 0 ! represen t the generators e 1 , e 2 of C  2 , acting by left m ultiplication. The c hiral op- erator ( / D T 2 ) 0 is lo cally the Cauc hy-Riemann op erator, and hence its k ernel consists 112 The group K O t ] (pt) and torsion branes of holomorphic sections of the c hiral Cliﬀord bundle / S 0 (T 2 ). These are simply the complex-v alued constan t functions on T 2 , as the torus is a compact complex manifold. As a mo dule o v er C  0 2 , this v ector space is one-dimensional and so ind t 2  T 2 , 1 1 R T 2 , ζ  =  k er( / D T 2 ) 0  = 1 in M 1 /ı ∗ M 2 ' K O − 2 (pt) ' Z 2 . Consider no w the t w o-sphere S 2 as a riemannian manifold. It is not diﬃcult to see that P SO  S 2  = SO(3) − → SO(3) / SO(2) ' S 2 is the oriented orthonormal frame bundle ov er S 2 . The (unique) spin structure on S 2 is th us giv en b y P Spin  S 2  ' SU(2) h / / U(1) * * U U U U U U U U U U U U U U U U U U U P SO  S 2  ' SO(3) SO(2)   S 2 with h : SU(2) → SO(3) the usual double cov ering, and by U(1) / / P Spin  S 2    S 2 whic h is the Hopf ﬁbration of S 2 . Recall that the group Spin(2) ' U(1) ' SO(2) acts on C  2 ' H as m ultiplication b y e i θ 0 0 e − i θ ! , θ ∈ [0 , 2 π ) . If one giv es the sphere S 2 the structure of the complex pro jective line CP 1 , then there are isomorphims / S 0 (S 2 ) = P Spin (S 2 ) × U(1) C ' T 1 , 0 CP 1 since the bundle / S 0 (S 2 ) has the same transition functions as the Hopf ﬁbration. In other words, / S 0 (S 2 ) is isomorphic to the canonical line bundle L C o v er CP 1 . The v ector space k er( / D S 2 ) 0 th us consists of the holomorphic sections of L C . The only such section on CP 1 is the zero section [72], and w e ﬁnally ﬁnd ind t 2  S 2 , 1 1 R S 2 , ζ  =  k er( / D S 2 ) 0  = 0 in M 1 /ı ∗ M 2 ' Z 2 . Remark As w e hav e seen ab o v e, the problem of ﬁnding generators of the geometric 113 The group K O t ] (pt) and torsion branes K O-homology groups of a space X becomes increasingly inv olv ed at a v ery rapid rate. Ev en in the case of spherical D-branes, we ha v e not b een able to ﬁnd a nice explicit solution. Nevertheless, at least in these cases w e can ﬁnd a formal solution as follows, whic h also illustrates the generic problems at hand. Supp ose that w e wan t to construct generating branes for the group KO t k (S n ) for some n > 0. P oincar´ e dualit y giv es the map K O n − k  S n  − → K O t k  S n  , ξ 7− → ξ ∩  S n , 1 1 R S n , id S n  . (4.9.2) As Poincar ´ e duality is a group isomorphism, pic king a generator in KO n − k (S n ) will giv e a generator in K O t k (S n ). But the problem is that the class ξ is not a (virtual or stable) v ector bundle ov er S n in the cases of interest k < n . T o this end, w e rewrite the cap pro duct in (4.9.2) b y using the susp ension isomorphism Σ and the desusp ension Σ − 1 to get ξ ∩  S n , 1 1 R S n , id S n  = Σ − 1  Σ  ξ  ∩ Σ  S n , 1 1 R S n , id S n   . As w e are in terested only in generators, we can substitute Σ( ξ ) with the generators of the K O-theory group K O 0 (Σ n − k S n ) = g K O 0 (S 2 n − k ). The generators of the latter groups are given by [61] the canonical line bundle L F o v er the pro jective line FP 1 , with F the reals R for k = 2 n − 1, the complex num b ers C for k = 2 n − 2, the quaternions H for k = 2 n − 4 and the o ctonions O for k = 2 n − 8. 114 Chapter 5 Ab elian Gauge Theories and Diﬀeren tial Cohomology W e ha ve seen in the previous chapters that D-brane charges require the use of K-theory and K-homology to b e prop erly described. Morev er, since D-branes are electric and magnetic sources for Ramond-Ramond ﬁelds, w e expect that some form of K-theory will play a relev an t role also in the description of these generalized gauge ﬁelds. In this c hapter we introduce some basic notions of gener alize d diﬀer ential c ohomolo gy the ories , which we will see constitute a p ow erful mathematical machinery to describ e ab elian gauge theories of diﬀeren tial forms, and in particular the theory of Ramond- Ramond ﬁelds. W e ﬁrst motiv ate the use of this formalism in the case of ordinary electromagnetism, follo wing [40, 44]. 5.1 An example: the electromagnetic case In ordinary electromagnetism form ulated on the four dimensional Mink o wski space- time M 4 = R t × R 3 , the Maxw ell equations are giv en b y dF = 0 d  F = j e (5.1.1) where F ∈ Ω 2 (M; R ), and where j e ∈ Ω 3 (M; R ) is the electric current distribution. Since M 4 is a con tractible space 1 , equations (5.1.1) and P oincar ´ e’s Lemma imply that there exists a form A ∈ Ω 1 (M; R ), called the ve ctor p otential , such that F = dA (5.1.2) 1 More precisely , we only use that every 2-sphere is the b oundary of a 3-ball. 115 An example: the electromagnetic case As we hav e seen in section 2.5, the total electric c harge of the distribution can b e iden tiﬁed with the class [ j e | R 3 ] in H 3 cpt ( R 3 ; R ) ' R . W e can mo dify the equations (5.1.1) by introducing a magnetic current j m ∈ Ω 3 (M; R ), and allo wing the equations dF = j m d  F = j e (5.1.3) The magnetic c harge of the system Q m is giv en b y the class [ j m | R 3 ] in H 3 cpt ( R 3 ; R ). The equations (5.1.3) ha ve changed the “global prop erties” of F: indeed, it is no longer a closed form, and equation (5.1.2) no longer holds. If we denote with W e and W m the support of the forms j e and j m resp ectiv ely , and supp osing W e ∩ W m = Ø, w e can consider the follo wing equations deﬁned on M 4 − W m dF = 0 d  F = j e (5.1.4) Notice at this p oint that equations (5.1.4) do not imply equation (5.1.2), since the space M 4 − W m is in general not con tractible. How ever, for an y con tractible op en set U α ⊂ M 4 − W m w e ha v e F | U α = dA α and on o v erlaps U α ∩ U β w e ha v e A α − A β = dg αβ g αβ ∈ C ∞ (M; R ) Finally , on triple ov erlaps U α ∩ U β ∩ U γ g αβ + g β γ + g γ α = c αβ γ where c αβ γ is a real constant o ver the en tire triple ov erlap. The fact that in the presence of magnetic c harges the v ector p oten tial is not globally deﬁned requires that the coupling term Z M 4 A ∧ j e = Z W e A b e carefully deﬁned, as W e could in tersect more patches U α . Ho wev er, one can sho w that the classical action ev aluated ov er a giv en worldline W e is ambiguous up to a constan t c αβ γ [2], and this do es not aﬀect the classical equations of motion. Indeed, equations (5.1.4) only depend on the ﬁeldstrength F, whic h is a globally deﬁned tw o form. At the quantum level, instead, this classical am biguit y leads to inconsistencies, unless some restrictions are imp osed on the collection of all { c αβ γ } . F or example, in 116 An example: the electromagnetic case the path in tegral quan tization, an am biguous phase factor of exp { i c αβ γ } is p oten- tially present at each non-empt y triple in tersection of patc hes: the only w a y to a v oid this am biguit y is to require that c αβ γ = 2 π n αβ γ , where n αβ γ are inte ger num b ers. Moreo v er, the constan ts c αβ γ can be directly related to the total magnetic ﬂux, hence the total magnetic c harge, implying that the t w o form F restricted to M 4 − W m has inte gr al p erio ds . This is the so called Dir ac quantization c ondition , and w as prop osed b y Dirac in [34], alb eit in a diﬀerent w a y . Mathematically , the Dirac quantization condition states that the class [F] in H 2 (M 4 − W m ; R ) lies in the image of the map H 2 (M 4 − W m ; Z ) → H 2 (M 4 − W m ; R ) induced in cohomology b y the inclusion Z  → R . Notice at this p oin t that the ab o v e argument applies even to the case of Maxwell equations on a spacetime M = R t × N in then absence of an y magnetic curren t, pro- vided that H 2 (N; R ) 6 = 0. Also in this case, the Dirac quantization condition requires that the ﬁeldstrength F has in teger perio ds. Diﬀeren tial geometry pro vides a b eautiful solution to the necessit y of having a lo cal vector p oten tial A in the quantum theory , without forgetting the obstructions that preven t its global existence, and including the Dirac quan tization condition. Indeed, one regards the ﬁeldstrength F as the cur- v ature of a connection A deﬁned on a principal U(1)-bundle π : L → M with ﬁrst Chern class c 1 ( L ) = [ c αβ γ ] ∈ H 2 (M; Z ). The relev ant space 2 of ﬁelds for the quan tum theory is then giv en b y the space of all principal U(1)-bundles with connection o ver M. Notice that this space extends the space of classical solutions of Maxw ell equations b y the ﬂat connections, whic h ma y con tribute non trivially to the quan tum theory . As w e hav e seen through the argumen t ab ov e, the Dirac quantization of charges is a required condition in order to hav e a w ell deﬁned quan tum theory . W e ha v e also seen that the theory of principal bundles can b e used to geometrically enco de this condition. Ho w ev er, this framework cannot b e applied to the case of generalized electromagnetic theories in tro duced in section 2.5: indeed, since the ﬁeldstrength F is giv en by a p -form, it cannot b e realized as the curv ature of some connection. One can then resort to a lo cal description of these gauge ﬁelds, as done for the B-ﬁeld in section 1.5. This approac h has several disadv an tages, though: it is usually diﬃcult in this framew ork to determine the space of gauge equiv alent ﬁeld conﬁgurations, o v er which the path in tegral should be performed, and in particular it is diﬃcult to introduce a 2 It is actually a gr oup oid , where the morphisms are given by connection preserving gauge trans- formations. 117 Diﬀeren tial Cohomology coupling of these ﬁelds with their sources, as one needs a proper notion of pullbac k, in tegration, etc. In the follo wing section, w e will in tro duce the prop er mathematical formalism to treat these ﬁelds, whose recen t developmen t has b een greatly motiv ated b y the v ery problems men tioned ab o v e. 5.2 Diﬀeren tial Cohomology In the past 30 y ears, Diﬀeren tial Cohomology has appeared in the mathematical lit- erature as the theory of diﬀerential characters, Deligne co cycles, sparks, and more recen tly diﬀerential functions [28, 23, 51, 54]. F rom the mathematical point of view, it has pro vided a reﬁnement of the theory of c haracteristic classes and characteris- tic forms, which, in the appropriate contexts, giv es rise to obstruction to conformal immersion of Riemannian manifolds in euclidean spaces; more recen tly , in com bina- tion with diﬀeren tial K-theory , it has b een used to construct an index theorem for ﬂat bundles. As mentioned ab ov e, from the physical p oin t of view it constitutes a p o w erful formalism to describ e gauge theories of p -forms whose Dirac quan tization condition is dictated b y in teger cohomology . In the follo wing w e will focus on t wo par- ticular descriptions of Diﬀerential Cohomology , whic h use Cheeger-Simons characters and Deligne co cycles, resp ectiv ely . W e will illustrate the relation b et w een diﬀerential cohomology and electromagnetism with Dirac quan tization of charges, and we will giv e a deﬁnition of gener alize d ab elian gauge the ories in terms of these ob jects. 5.2.1 Cheeger-Simons c haracters Let M denote a smo oth manifold, and let C k ⊃ Z k ⊃ B k denote the groups of smo oth singular c hains, cycles, and b oundaries with ∂ : C k → C k − 1 and δ : C k → C k +1 the usual b oundary and cob oundary op erators, resp ectiv ely . Let Ω k Z (M) ⊂ Ω k (M) denote the lattice of closed k -forms with integral perio ds. Notice that if w e denote with r the map induced on cohomology by the inclusion Z  → R , then a k -form ω has in tegral p erio ds if and only if [ ω ] = r ( u ), for some u ∈ H k (M; Z ). Given ω ∈ Ω k (M), we hav e the map Ω k (M) → C k (M; R / Z ) (5.2.1) whic h assigns to ω the R / Z -v alued co chain deﬁned as ˜ ω ( σ ) := Z σ ω mo d Z , ∀ σ ∈ C k (M) 118 Diﬀeren tial Cohomology As the integral of a generic diﬀerential form o v er the set of all cycles never takes v alues only in Z , the map (5.2.1) is an injection, and w e denote the image of Ω k (M) in C k (M; R / Z ) with e Ω k (M). W e ha v e then the follo wing Deﬁnition 5.1. The n-th Che e ger-Simons gr oup of a smo oth ma nifold M is the group ˇ H n (M) := n f ∈ Hom ( Z n − 1 , R / Z ) : f ◦ ∂ ∈ e Ω n (M) o W e set ˇ H 0 (M) = Z . The elemen ts of ˇ H ∗ (M) := ⊕ n ˇ H n (M) are called diﬀer ential char acters . A smo oth map φ : M → N naturally induces a homomorphism φ ∗ : ˇ H n (N) → ˇ H n (M) Hence, ˇ H n is a contra v ariant functor from the category of smo oth manifolds to the category of ab elian groups. A main result is the follo wing [28] Prop osition 5.2. Ther e exist surje ctive maps ˇ H n (M) F − → Ω n Z (M) ˇ H n (M) c − → H n (M; Z ) for any smo oth manifold M and any n ∈ N . The map F is c al le d the ﬁeldstrength map , and the map c is c al le d the characteristic class map . Pr o of. Let f ∈ ˇ H n (M). Consider an element T 0 ∈ Hom ( Z n − 1 , R ) such that ˜ T 0 = f where the tilde means mo d Z . Consider now the following (split) exact sequence 0 → Z n − 1 i − → C n − 1 ∂ − → B n − 2 → 0 Since Hom( · , R ) is an exact functor, the following exact sequence holds 0 → Hom( B n − 2 , R ) δ − → Hom( C n − 1 , R ) i ∗ − → Hom( Z n − 1 , R ) → 0 (5.2.2) where the map i ∗ is giv en b y restriction. Hence, there exist T ∈ Hom( C n − 1 , R ) such that T | Z n − 1 = T 0 , and consequen tly ˜ T | Z n − 1 = f 119 Diﬀeren tial Cohomology Since b y construction δ T := T ◦ ∂ , and f δ T = δ ˜ T, w e ha v e that δ ˜ T = f ◦ ∂ By assumption δ ˜ T is an elemen t of e Ω n (M). Any suc h elemen t can b e written as ω − c , where ω is a n -form regarded as a real co chain b y integration, and c ∈ C n (M , Z ). Since δ T = ω − c , we ha v e 0 = δ 2 T = δ ω − δ c = d ω − δ c (5.2.3) where d denotes the deRham diﬀeren tial. Since the map (5.2.1) is an injection, equa- tion (5.2.3) implies that d ω = 0 and δ c = 0. Finally , since δ T = ω − c we hav e that [ ω ] = r ([ c ]), with [ c ] ∈ H n (M; Z ), whic h implies that ω ∈ Ω n Z (M). The elements ω and [ c ] do not dep end on the lift T. Indeed, let T 0 b e another real co chain suc h that ˜ T 0 | Z n − 1 = f . Then ^ T − T 0 | Z n − 1 = 0, and the sequence (5.2.2) implies that T 0 = T + δ α + β (5.2.4) with α ∈ C n − 2 (M; R ) and β ∈ C n − 1 (M; Z ). By the argument ab o ve there exist ω 0 and c 0 suc h that δ T 0 = ω 0 − c 0 . Hence we ha v e ω 0 − c 0 = δ T 0 = δ T + δ β = ω − c + δ β whic h implies ω 0 − ω = c 0 − c + δ β ∈ C  (V; q ) By the same reason as b efore, w e ha v e ω 0 = ω and [ c 0 ] = [ c ]. W e deﬁne F(f ) = ω and c( f ) = [ c ], where ω and c are obtained as ab ov e. Moreo v er, w e will refer to the form ω as the ﬁeldstr ength and to the class [c] as the char acteristic class ; the use of this terminology will b e clear later. T o pro v e that the maps F and c are surjectiv e, notice that an y ω ∈ Ω n Z (M) de- termines a u ∈ H n (M; Z ) suc h that [ ω ] = r ( u ), and conv ersely for any u w e can ﬁnd suc h a ω . Let [ c ] = u . Then ω − c is exact as a real cochain, and there exists T suc h that δ T = ω − c . Then ˜ T | Z n − 1 deﬁnes a homomorphism f : Z n − 1 → R / Z with the prop ert y that f ◦ ∂ ∈ e Ω n (M). W e will now inv estigate the kernels of the ﬁeldstrength and c haracteristic class maps. Let f ∈ ˇ H n (M) satisfy F( f ) = 0. Then the real cochain T satisﬁes δ T = − c . Hence 120 Diﬀeren tial Cohomology δ ˜ T = 0, and th us ˜ T is an R / Z -v alued cocycle. If we consider another lift T 0 , equation (5.2.4) implies that ˜ T 0 = ˜ T + δ ˜ α , and hence [T 0 ] = [T] ∈ H n − 1 (M; R / Z ). So an elemen t f with F( f ) = 0 determines an R / Z cohomology class. Con v ersely , given an R / Z class represen ted by a cocyle s , w e ha ve that s | Z n − 1 deﬁnes a diﬀeren tial character f , and suc h a deﬁnition is indep enden t of the c hoice of the represen ting co cycle s . Finally , let f ∈ ˇ H n (M) satisfy c( f ) = 0. Then δ T = ω − c , with c = δ e , for some e ∈ C n − 1 (M; Z ). Hence, δ (T − e ) = ω . Since [ ω ] = r ([ c ]), by the deRham theorem w e ha ve that ω = d θ , for some θ ∈ Ω n − 1 (M). Since δ (T − e − θ ) = 0, w e hav e that T − e − θ = z , for some co cycle z ∈ Z n − 1 (M; R ). Again b y the deRham theorem, there exists a closed form φ ∈ Ω n − 1 (M) such that φ | Z n − 1 = z | Z n − 1 , which implies that f = ˜ T | Z n − 1 = ^ θ + φ + e = ] θ + φ Hence f is in the image of the map ω → ˜ ω | Z n − 1 , with ω ∈ Ω n − 1 (M), whose kernel is giv en b y Ω n − 1 Z (M). W e hav e then prov ed the following [28] Prop osition 5.3. Ther e ar e natur al exact se quenc es 0 → H n − 1 (M; R / Z ) → ˇ H n (M) F − → Ω n Z (M) → 0 0 → Ω k − 1 (M) / Ω k − 1 Z (M) → ˇ H n (M) c − → H n (M; Z ) → 0 Another useful exact sequence can b e obtained as follo ws. Consider the group A n (M) := { ( ω , u ) ∈ Ω n Z (M) × H n (M; Z ) : [ ω ] = r ( u ) } W e hav e then a surjective map ˇ H n (M) (F , c) − − → A n (M) The k ernel of this map is given by elements f ∈ ˇ H n (M) suc h that F( f ) = 0 and c( f ) = 0. By the same argument as b efore, we hav e that δ (T − e ) = 0, hence the elemen t f determines a class in H n − 1 (M; R ) represented by the closed form φ that satisﬁes T | Z n − 1 = φ + e . As φ is a closed form, ˜ T | Z n − 1 is determined only b y the class [ φ ] ∈ H n − 1 (M; R ). Conv ersely , any class [ φ ] determines a diﬀeren tial character f = ˜ φ | Z n − 1 , and the k ernel of such an assignment is giv en precisely b y r (H n − 1 (M; Z )). W e hav e then show ed that the following exact sequence holds 0 → H n − 1 (M; R ) /r (H n − 1 (M; Z )) → ˇ H n (M) (F , c) − − → A n (M) → 0 (5.2.5) The ab o ve sequences are very imp ortan t, as they are usually the only computational tec hnique a v ailable since the groups ˇ H n (M) are usually inﬁnite dimensional. 121 Diﬀeren tial Cohomology Notice that the group A ∗ (M) := ⊕ n A n (M) carries an ob vious ring structure given b y ( ω , u ) · ( φ, v ) := ( ω ∧ φ, u ∪ v ) W e exp ect then that the group ˇ H ∗ (M) carries an analogous pro duct. This is indeed the case. Ho wev er, the ring pro duct of diﬀerential c haracters is more subtle to deﬁne: this is due to the fact that given tw o forms ω 1 and ω 2 of degrees l and p , resp ectively , in general w e ha v e ω 1 ∧ ω 2 6 = ω 1 ∪ ω 2 where on the righ t hand side ω 1 and ω 2 are realized as real co c hains. Ho wev er, one has that δ E ( ω 1 , ω 2 ) = ω 1 ∧ ω 2 − ω 1 ∪ ω 2 for some E ( ω 1 , ω 2 ) ∈ C l + p − 1 (M; R ). The technical diﬃcult y consists exactly in con- structing the co c hain E ( ω 1 , ω 2 ) in a canonical w a y , and w e refer the reader to [28] for details. A deﬁnition of the ring product can then be giv en in the following wa y . Let f ∈ ˇ H l (M) and g ∈ ˇ H p (M), and choose the lifts T f ∈ C l − 1 (M; R ) and T g ∈ C p − 1 (M; R ) for f and g resp ectively . Then we can deﬁne [28] f  g := ^ T f ∪ ω f − ( − 1) l − 1 ^ ω f ∪ T g − ^ T f ∪ δ T g + E ( ω f , ω g ) | Z l + p − 1 and it can b e shown that f  g is indep enden t of the c hoice of T f and T g . Moreo ver, the pro duct  is asso ciative, graded commutativ e, and it is such that the ﬁeldstrength and c haracteristic class maps are ring homomorphisms. Ev en if the pro duct  is fairly complicated in general, there are sp ecial cases in whic h it greatly simpliﬁes. Indeed, if g ∈ k er F ⊂ ˇ H p (M), then f  g is the image of ( − 1) l c( f ) ∪ g , for f ∈ ˇ H l (M), while if g ∈ ker c ⊂ ˇ H p (M), f  g is the image of ( − 1) l F( f ) ∧ g . In the follo wing w e will giv e some examples. Example 5.4. Let M = S 1 and consider the group ˇ H 1 (S 1 ). Since Z 0 (S 1 ) is freely generated b y the p oints of S 1 , we hav e that ˇ H 1 (S 1 ) ' C ∞ (S 1 , S 1 ), where w e ha v e realized R / Z as S 1 . Consider then a smo oth function f : S 1 → S 1 An y suc h function f can b e expressed as f ( p ) = e 2 π i Θ( f ( p )) 122 Diﬀeren tial Cohomology where Θ = θ ◦ f , with θ the usual lo cal angular co ordinate on the circle. Then a lift T of f is giv en b y T = 1 2 π i log f = Θ( p ) + η ( p ) where η is a lo cally constant Z -v alued function. By assumption, we kno w that on an y curv e γ on S 1 δ ˜ T( γ ) = e 2 π i (Θ( b ) − Θ( a )) where a and b are the endp oints of the curv e. The righ t hand side of the ab ov e equation can b e written as exp  2 π i Z γ dΘ  Notice that dΘ is not an exact 1-form, but it is integral, as dΘ = f ∗ d θ . Hence we ha v e sho wn that d  1 2 π i log f  is the ﬁeldstrength of f . Since the group H 1 (S 1 ; Z ) con tains no torsion, the ﬁeld- strength determines the c haracteristic class: hence, we ha v e [ c ] = f ∗ [d θ ] The same argumen t can b e applied to the group ˇ H 1 (M) for an y smo oth manifold M. Example 5.5. F or any smo oth manifold M with n = dim M we hav e ˇ H n +1 (M) ' H n (M; R / Z ) ˇ H k (M) = 0 , k > n + 1 This can b e sho wn b y using the exact sequences in tro duced ab o ve. Example 5.6. Let M = pt. Then ˇ H n (M) =        Z , n = 0 R / Z , n = 1 0 , otherwise Example 5.7. Finally , w e presen t an example coming from diﬀerential geometry , whic h is in a certain sense the “canonical” one. Consider a complex line bundle L → M with connection ∇ , and denote with F ∇ its curv ature form. Since 1 2 π F ∇ represen ts the real ﬁrst Chern class of L , w e ha ve that F ∇ ∈ Ω 2 Z (M). Let γ b e a closed curve, and deﬁne f ( γ ) := hol ∇ ( γ ) ∈ R / Z 123 Diﬀeren tial Cohomology where hol ∇ denotes the holonomy of the connection ∇ . W e can extend the homo- morphism f to the whole of Z 1 (M) as follows. Let z ∈ Z 1 (M). Represen t z as z = P i n i γ i + ∂ y , where γ i is a closed curv e, and y ∈ C 2 (M). Then we can deﬁne f ( z ) := Y i f ( γ i ) n i + 1 2 π ˜ F ∇ ( y ) The homomorphism f abov e is w ell deﬁned, and independent of the presen tation of z , since F ∇ has integral p erio ds. Since f ◦ ∂ = 1 2 π F ∇ , w e hav e that f ∈ ˇ H 2 (M). Moreo v er, one can c hec k that F( f ) = 1 2 π ˜ F ∇ c( f ) = c 1 ( L ) Notice that the character f con tains more information than F ∇ and c 1 ( L ) together, since b oth ma y v anish when f do es not, e.g. when M = S 1 . It is immediate to realize in the ab o v e example that if we p erform a connection preserving gauge transformation on L , the homomorphism f do es not c hange, hence it is a gauge in v ariant quan tit y . Con v ersely , given an elemen t f ∈ ˇ H 2 (M) we can construct up to isomorphism a line bundle L classiﬁed b y c( f ), equipp ed with a connection ∇ deﬁned b y requiring that hol ∇ ( γ ) := f ( γ ) for any closed curve γ . Hence, for any smo oth manifold M the group ˇ H 2 (M) is equiv alen t to the space of all gauge equiv alen t complex line bundles with connection o v er M: the group structure on the latter space is induced b y tensor pro duct of line bundles. W e ha v e then realized that the group ˇ H 2 (M) is the set of orbits of the group of gauge transformations acting on the space of connections on al l line bundles o v er M: noncanonically , this space can b e expressed as [ c 1 ∈ H 2 (M; Z ) A ( L c 1 ) / G where A ( L c 1 ) is the aﬃne space of connections on the line bundle L c 1 classiﬁed b y the class c 1 , and G is the group of gauge transformations. As seen in section 5.1, this is precisely the space of all p ossible gauge inequiv alent solutions to Maxw ell equations on M, taking into accoun t the Dirac quantization condition. Realizing this space as the group ˇ H 2 (M) naturally suggests the follo wing Deﬁnition 5.8. A gener alize d ab elian gauge the ory on a smooth manifold M is a ﬁeld theory whose space of gauge inequiv alen t conﬁgurations is giv en b y ˇ H n (M), for some n ∈ N . 124 Diﬀeren tial Cohomology The generalized electromagnetism discussed in section 2.5.1, once pro vided with a suitable Dirac quantization condition, is a generalized ab elian gauge theory . Be- cause of the relation b et w een the group ˇ H n (M) and gauge theories of n -forms, one usually refers to H n − 1 (M; R / Z ) as the group of ﬂat ﬁelds , while Ω n − 1 (M) / Ω n − 1 Z (M) is called the group of top olo gic al ly trivial ﬁelds . Consequently , we can see that the group H n − 1 (M; R ) /r (H n − 1 (M; Z )) classiﬁes the (gauge equiv alence classes of ) ﬂat and top olo gic al ly trivial ﬁelds . Notice that if we classify inequiv alent ﬁeld conﬁgurations only b y curv ature and c haracteristic class, we are not taking in to accoun t the eﬀect of ﬂat and top ologically trivial ﬁelds, which ma y b e nonv anishing, according to the top ology of M. Since the electromagnetic ﬁeld can b e describ ed with complex line bundles and connections thereon, this suggets that elemen ts of ˇ H n (M) should represen t isomor- phism classes of “higher” line bundles with “higher” connections. This is indeed the case: for example, the group ˇ H 3 (M) is given by equiv alence classes of bundle gerb es with connections. Ho w ev er, to be able to talk ab out the gauge ﬁelds and higher line bundles whose equiv alen t classes are the elemen ts in ˇ H n (M), we need a mo del repre- sen ting such a group. Recall indeed, that the group ˇ H 2 (M) can b e obtained as the set of isomorphism classes for the group oid of connections on M. T o giv e such a mo del is the aim of the next section. 5.2.2 Deligne cohomology In this section we will describ e a co c hain mo del for the Cheeger-Simons groups. This is done b y introducing Deligne c ohomolo gy , which is deﬁned via the cohomology of a certain complex. The Deligne cohomology groups are isomorphic to the Cheeger- Simons groups: ho w ev er, we will not prov e this result, and will fo cus instead on a detailed construction of the co chain mo del, showing ho w it can naturally describ e the gauge ﬁelds in a generalized abelian gauge theory in topologically nontrivial bac k- grounds. In the following w e will describ e the smo oth Deligne cohomology 3 . In a nutshell, the diﬀeren tial complex used to deﬁne Deligne cohomology is a “mo di- ﬁcation” of the ˇ Cec h-de Rham complex: we will recall some basics ab out the ˇ Cec h-de Rham complex in order to clearly sho w the nature of suc h a mo diﬁcation. Let M b e a para compact smo oth manifold with n = dim M, and consider a goo d co v er 3 Deligne cohomology originated in the con text of algebraic geometry . 125 Diﬀeren tial Cohomology U = { U α } α ∈ I , where the index set I ma y b e inﬁnite. Denote U α 0 α 1 ··· α p := U α 0 ∩ U α 1 ∩ · · · ∩ U α p Let Ω p (M) b e the set of real v alued p -forms, and denote with Ω 0 (M) the space of smo oth real functions C ∞ (M; R ). Since each U α 0 α 1 ··· α p is con tractible, Poincar ´ e’s Lemma implies that for an y sequence α 0 , α 1 , · · · , α p w e ha v e an exact sequence 0 → C ∞ lc ( U α 0 α 1 ··· α p ; R ) → Ω 0 ( U α 0 α 1 ··· α p ) d − → Ω 1 ( U α 0 α 1 ··· α p ) d − → . . . d − → Ω n ( U α 0 α 1 ··· α p ) → 0 where C ∞ lc (M; R ) denotes the space of smo oth lo cally constan t real v alued functions. Deﬁne the group of p -co c hains of the co v er U with v alues in the q -forms as ˇ C p ( U ; Ω q ) := Y α 0 <α 1 < ··· <α p Ω q ( U α 0 α 1 ··· α p ) Notice that the inclusion U α 0 ··· α p  → U α 0 ··· ˆ α i ··· α p induces a map φ α i : Ω q ( U α 0 ··· ˆ α i ··· α p ) → Ω q ( U α 0 ··· α p ) whic h is giv en b y restriction. W e can then deﬁne the homomorphism δ : ˇ C p ( U ; Ω q ) → ˇ C p +1 ( U ; Ω q ) whic h on ω ∈ ˇ C p ( U ; Ω q ) is deﬁned as δ ω := p +1 X i =0 ( − 1) i φ α i ( ω ) The homomorphism δ satisﬁes δ 2 = 0, hence it can b e used to the deﬁne a cohomology theory . Ho wev er, we ha v e the following Generalized Ma y er-Vietoris exact sequence [21] 0 → Ω q (M) → Y Ω q ( U α 0 α 1 ) δ − → Y Ω q ( U α 0 α 1 α 3 ) δ − → · · · for an y q ≥ 0, whic h tells us that the δ -cohomology of the complex ˇ C ∗ ( U ; Ω q ) v anishes iden tically . Notice that for q = 0 this is related to the fact that H p (M; R ), the p -th cohomology group of M v alued in the sheaf of smo oth real v alued functions, is 0 for all p ≥ 0, since R is a ﬁne sheaf 4 . Since the de Rham diﬀeren tial d : ˇ C p (U; Ω q ) → ˇ C p ( U ; Ω q +1 ) 4 A ﬁne sheaf is lo osely sp eaking a sheaf with a “partition of unity”. 126 Diﬀeren tial Cohomology comm utes with the cob oundary homomorphism δ , w e can form the follo wing double complex, called the ˇ Ce ch-de R ham complex ˇ C ∗ (M; Ω ∗ ) := M p,q ≥ 0 ˇ C p ( U ; Ω q ) with cob oundary op erator D = δ + ( − 1) deg ( · ) d where for ω ∈ ˇ C p ( U ; Ω q ), deg ( ω ) := p . As usual for diﬀeren tial double complexes, w e can consider the diagonal sub complex K n := M p + q = n ˇ C p ( U ; Ω q ) By “tic-tac-to eing”, i.e b y c hasing diagrams, one can pro v e that H ∗ dR (M) ' H ∗ ( K ∗ ; D ) ' ˇ H ∗ Ch (M; R ) where ˇ H ∗ Ch (M; R ) denote ˇ Cec h cohomology . The ab ov e isomorphism can b e seen as a consequence of the fact that al l the “ro ws” and “columns” of the ˇ Cec h-de Rham complex are exact 5 . The basic idea b ehind Deligne cohomology consists in constructing a complex analog to the ˇ Cec h-de Rham one, but based on the follo wing exact sequence Ω 0 U ( U α 0 α 1 ··· α p ) d log − − → Ω 1 ( U α 0 α 1 ··· α p ) d − → . . . d − → Ω l ( U α 0 α 1 ··· α p ) where Ω 0 U ( V ) := C ∞ ( V ; U(1)) for an y op en set V , and 0 ≤ l ≤ n . In other words, w e truncate the complex of forms on the right for some l , and we substitute in the 0 degree real v alued functions with circle v alued ones. W e can then deﬁne a mo diﬁed ˇ Cec h-de Rham complex as ˇ C ∗ [ l ](M; Ω ∗ ) := M p ≥ 0 , 0 ≤ q ≤ l ˇ C p ( U ; Ω q ) and the asso ciated single complex K [ l ] n := M p + q = n ˇ C p ( U ; Ω q ) The ˇ Cec h-de Rham cob oundary D is unc hanged, and we will refer to an elemen t ω ∈ K [ l ] ∗ with D ω = 0 as a Deligne c o chain . F or a giv en l , the Deligne cohomology is giv en b y H ∗ (M; D l ) := H ∗ ( K [ l ] ∗ ; D ) 5 Indeed, H ∗ dR (M) and ˇ H ∗ Ch (M; R ) are the cohomology groups of the augmented column and ro w, resp ectiv ely . 127 Diﬀeren tial Cohomology As for ˇ Cec h cohomology , the result do es not dep end on the particular choice of go o d co v er. Notice that it is no longer true that all the rows of the double complex ˇ C ∗ [ l ](M; Ω ∗ ) are exact. Indeed, for q = 0 the sequence Y Ω 0 U ( U α 0 ) δ − → Y Ω 0 U ( U α 0 α 1 ) δ − → Y Ω 0 U ( U α 0 α 1 α 2 ) δ − → · · · is in general not exact, and the δ -cohomology is given b y ˇ H ∗ Ch (M; U(1)), the sheaf cohomology with v alue in the circle functions. Mo erov er, one can prov e that ˇ H ∗ Ch (M; U(1)) ' ˇ H ∗ +1 Ch ( U ; Z ) for any go o d co v er U . This inno cuous modiﬁcation implies interesting prop erties for the groups H n (M; D l ). I ndeed, consider the case n < l . Then a Deligne class is determined b y an n -tuple ( ω n 0 , ω n − 1 1 , · · · , ω 0 n ) , ω i j ∈ Y Ω i ( U α 0 α 1 ...α j ) satisfying the equations d ω n 0 = 0 δ ω n 0 − d ω n − 1 1 = 0 . . . δ ω 1 n − 1 + ( − 1) n d log ω 0 n = 0 δ ω 0 n = 0 Since the ﬁrst equation implies that ω n 0 = d α n − 1 0 , for some α n − 1 0 , one can “descend” through the equations, showing that ω 0 n = f + δ α 0 , where f ∈ Q C ∞ lc ( U α 0 α 1 ...α n ; U(1)). Moreo v er, b y the last equation δ f = 0, hence f represen ts a class in ˇ H n C h (M; R / Z ). Con v ersely , any class [ f ] in ˇ H n Ch (M; R / Z ) determines a Deligne class up to D -exact terms. W e ha v e then H n (M; D l ) ' ˇ H n Ch (M; R / Z ) , n < l Consider now the case n > l . In this case a Deligne class is represented b y an l -tuple ( ω l n − l , ω l − 1 n − ( l − 1) , · · · , ω 0 n ) satisfying the equations δ ω l n − l + ( − 1) n − ( l − 1) d ω l − 1 n − ( l − 1 = 0 δ ω l − 1 n − l − 1 + ( − 1) n − ( l − 2) d ω l − 2 n − ( l − 2) = 0 . . . δ ω 1 n − 1 + ( − 1) n d log ω 0 n = 0 δ ω 0 n = 0 128 Diﬀeren tial Cohomology Notice that in this case the element ω l n − l is not in general d-closed. Then ω 0 n represen ts a class in ˇ H n Ch (M; U(1)) ' ˇ H n +1 Ch (M; Z ). Con v ersely , given a class [ f ] in ˇ H n Ch (M; U(1)) one can reconstruct the Deligne class up to D -exact terms. This is p ossible thanks to the fact that any element ω i j for j ≥ 1 which is δ -closed, is also δ -exact. Hence we ha v e that H n (M; D l ) ' ˇ H n +1 Ch (M; Z ) , n > l The case n = l is the most in teresting one. Indeed, in this case the arguments ab ov e cannot b e applied. Instead, we hav e the following [23] Theorem 5.9. L et M b e a smo oth manifold. Then H n (M; D n ) ' ˇ H n +1 (M) for any n ≥ 0 . The complex ab o ve deﬁned to compute Deligne cohomology constitutes then a lo cal description of diﬀeren tial characters, which instead are deﬁned in an in trinsic and global wa y . One can easily deﬁne the ﬁeldstrength and the c haracteristic class for the groups H n (M; D n ) in the following wa y . A Deligne class ξ ∈ H n (M; D n ) is represen ted as ξ = [( ω n 0 , ω n − 1 1 , · · · , ω 0 n )] W e can then deﬁne the ﬁeldstrength of ξ as the n + 1-form F( ξ ) := d ω n 0 This is a globally deﬁned n + 1-form, since δ dω n 0 = d δ ω n 0 = d 2 ω n − 1 1 = 0 The c haracteristic class of ξ is instead deﬁned as c( ξ ) := [ ω 0 n ] ∈ H n (M; Z ) By c hasing diagrams one can pro v e that [F( ξ )] = r (c( ξ )) The groups H n (M; D n ) satisfy the same exact sequences as the Cheeger-Simons groups. Moreo v er, the relation betw een the Cheeger-Simons groups and Deligne cohomology clariﬁes the v ery use of the term “cohomology”. In other words, the groups ˇ H n +1 (M) 129 Diﬀeren tial Cohomology can be realized as the cohomology of a diﬀerential complex, but this do es not mean that the collection of functors ˇ H ∗ ( · ) constitute a cohomology theory on the category of manifolds in the sense of the Eilen b erg-Steenro d axioms. First, notice that the v arious groups ˇ H n +1 (M) are the cohomologies of diﬀer ent complexes, for diﬀerent n . Morever, they do not satisfy imp ortan t cohomological prop erties lik e homotopy in v ariance. This is clear, since the groups ˇ H n +1 (M) are extensions of the groups of closed forms, whic h are not homotop y in v arian t. As w e ha v e seen in section 5.2.1, the ﬁrst few Cheeger-Simons groups hav e a geometric in terpretation. Indeed, for n = 0 we hav e shown that ˇ H 1 (M) ' C ∞ (M , S 1 ) A Deligne class for H 0 (M; D 0 ) is determined b y an elemen t f ∈ Y C ∞ ( U α 0 ; U(1)) satisfying δ f = 0, whic h implies that f is a globally deﬁned function from M to U(1). F or n = 1, the group H 1 (M; D 1 ) is giv en b y isomorphism classes of line bundles with connection. An element in H 1 (M; D 1 ) can b e represen ted as a pair ( ω 1 0 , ω 0 1 ) satisying the equations δ ω 1 0 − d log ω 0 1 = 0 δ ω 0 1 = 0 W e can then see that ω 1 0 giv es lo cal representativ es { A α } of a connection on the line bundle with transition functions g αβ giv en by ω 0 1 . The representativ es in the same Deligne class diﬀer b y D -exact terms ( ˜ ω 1 0 , ˜ ω 0 1 ) = ( ω 1 0 , ω 0 1 ) + D ξ 0 0 = ( ω 1 0 + d log ξ 0 0 , ω 0 1 + δ ξ 0 0 ) whic h corresp ond to the gauge transformations ˜ A α = A α + d log f α ˜ g αβ = f − 1 β g αβ f α for some circle v alued function f α . Notice that the gauge transformations ab o v e correctly include a “c hange” in the transition functions of the line bundle. The case n = 2 is in a sense the ﬁrst in teresting case for the application to ab elian 130 Diﬀeren tial Cohomology gauge theories of p -forms. Indeed, a Deligne class in H 2 (M; D 2 ) can b e represented b y a triple ( ω 2 0 , ω 1 1 , ω 0 2 ) satisfying the equations δ ω 2 0 − d ω 1 1 = 0 δ ω 1 1 + d log ω 0 2 = 0 δ ω 0 2 = 0 The triple ( ω 2 0 , ω 1 1 , ω 0 2 ) satisﬁes the equations (1.5.2)-(1.5.4) we used in section 1.5 to deﬁne the B-ﬁeld, apart from the fact that the functions ω 0 2 are circle v alued, rather than real v alued. This is due to the fact that Deligne classes in H 2 (M; D 2 ) represen t gauge equiv alence classes of B-ﬁelds with a Dirac quan tization condition as dictated b y the group H 3 (M; Z ). Ha ving realized the B-ﬁeld as a Deligne co c hain, w e also ha v e determined the correct notion of gauge transformation, whic h consists simply in adding terms of the form D α , with α = ( α 1 0 , α 0 1 ). Using the Deligne complex as a model for the groups ˇ H n +1 (M) clariﬁes the sense of Deﬁnition 5.8, since, as we hav e sho wn ab ov e, the groups H p (M; D p ) ha ve a natural description in terms of gauge equiv alence classes of p -form gauge ﬁelds. F rom the mathematical p oin t of view, it is v ery natural to interpret the elemen ts of H p (M; D p ) as higher cir cle bund les with c onne ctions [39]. In con trast to ordinary circle bun- dles, these ob jects are not ﬁb er bundles o ver a base, since there is no total space whic h makes sense as a manifold. Neverthless, usual operations lik e pullbac k can b e p erformed. More importantly , a theory of integration can be constructed for suc h ob jects, i.e one can deﬁne the notion of the holonomy of a Deligne class around a suitable cycle. This is of ma jor imp ortance for physical applications, as the holonom y usually describes the coupling term of the gauge ﬁeld to the relativ e curren t. In String theory , in particular, this is needed to mak e sense of the term Z Σ f ∗ B describing the coupling of a fundamen tal string with the B-ﬁeld. In this sense, the holonom y of a Deligne class ξ is exactly the diﬀeren tial c haracter assigned to ξ b y the isomorphism in Theorem 5.9. F or more details on Deligne cohomology we refer the reader to [23, 27, 39]. 131 Ramond-Ramond ﬁelds and c harge quan tization 5.3 Ramond-Ramond ﬁelds and c harge quan tiza- tion As we ha v e seen in the previous sections, diﬀerential cohomology , in the Cheeger- Simons or the Deligne approach, represents a suitable formalism to describ e ﬁelds in generalized ab elian gauge theories, and we ha v e also seen how the formalism auto- matically incorporates the Dirac quan tization condition. It is natural then to try to describ e within this formalism the gauge theory of Ramond-Ramond ﬁelds. Ho wev er, b ecause of the fact that D-branes, which are sources for Ramond-Ramond ﬁelds, hav e c harges taking v alues in K-theory , it turns out that Ramond-Ramond ﬁelds them- selv es, in the absence of branes, are classiﬁed b y K-theory . In the following w e will presen t the main argumen ts developed in [74] which supp ort this statement, and the fact that the Dirac quan tization condition for Ramond-Ramond ﬁelds should be ex- pressed in a K-theoretic language. Some of the arguments w e will present here are of a heuristic nature, and should be understo o d as an educated guess for the use of diﬀeren tial K-theory , which will b e introduced in the next section. Let us ﬁrst consider the case of ordinary generalized electromagnetism on a d -dimensional spacetime M = R × Y , where Y is noncompact. As discussed in section 2.5.1, in pres- ence of a magnetic source j m w e ha v e the equation dG = j m (5.3.1) where the ﬁeldstrength G is an n -form. Recall that the (total) magnetic charge is giv en b y the class [ i ∗ t j m ] ∈ H n +1 cpt (Y; R ). Ho w ev er, to enforce equation (5.3.1), the class [ i ∗ t j m ] must b e in the kernel of the natural map i : H n +1 cpt (Y; R ) → H n +1 (Y; R ) whic h “forgets” the compact support condition. Let us deﬁne N as the b oundary of Y : the term b oundary is used in a loose w a y , e.g. when Y = R d − 1 , the boundary is considered to b e the sphere S d − 2 “at inﬁnit y”. W e ha v e then that H n +1 cpt (Y; R ) ' H n +1 (Y , N; R ) By using the long exact cohomology sequence for the pair (Y , N) we hav e · · · → H n (Y; R ) j − → H n (N; R ) → H n +1 (Y , N; R ) i − → H n +1 (Y; R ) → · · · where j is given by restriction. The sequence ab ov e can b e “broken”, and we hav e k er i ' H n (N; R ) /j (H n (Y; R )) 132 Ramond-Ramond ﬁelds and c harge quan tization The ab o v e isomorphism leads to the following in terpretation: the total c harge of the source j m can b e detected b y classes of ﬁelds on the b oundary N, i.e. at inﬁnity , that are not restrictions of ﬁelds deﬁned on Y . Moreo v er, the group H n (Y; R ) classiﬁes ﬁeld conﬁgurations G whic h do not contribute to the total charge. In other words, it classiﬁes gauge ﬁelds in the absence of sources. The same reasoning can no w b e applied to t yp e I I String theory , where the sources are classiﬁed by the in tegral K-theory group K(Y). Indeed, in type I IB String theory , for D-branes with compact supp ort in space, the brane charge can be realized as an elemen t in K 0 cpt (Y). T o ensure that the equations of motion for the Ramond-Ramond ﬁeld ha v e a solution, the brane c harge should tak e v alues in the k ernel of the map i : K 0 cpt (Y) → K 0 (Y) As b efore, the long exact sequence · · · → K − 1 (Y) j − → K − 1 (N) → K 0 (Y , N) i − → K 0 (Y) → · · · implies that k er i ' K − 1 (N) /j (K − 1 (Y)) In analogy with the cohomological case, w e can in terpret the ab o ve isomorphism as the fact that in type I IB, gauge equiv alence classes of Ramond-Ramond ﬁelds in the absence of D-branes are classiﬁed top ologically by the group K − 1 (Y). By the same argumen t, w e ﬁnd that in type IIA, Ramond-Ramond ﬁelds are top ologically classiﬁed b y the group K 0 (Y). Ha ving extablished a relation b etw een Ramond-Ramond ﬁelds and K-theory , we still m ust determine the relation b etw een Ramond-Ramond ﬁelds and cohomology . Indeed, to be able to write equations of motion for suc h ﬁelds, w e need to specify the de Rham cohomology class asso ciated to an elemen t x ∈ K(Y) that determines (the class of ) a Ramond-Ramond ﬁeldstrength G. Let us then ﬁrst consider the t yp e I IA case. Recall that in t yp e I IA the total Ramond-Ramond p otential is of the form C = C 1 + C 3 + · · · where C i is a lo cally deﬁned i -form on M = R × Y . Let us in tro duce a collection of 8-branes and 8-branes with worldw olume p × Y , with p a p oin t in R , and with arbitrary Chan-P aton bundles (E , F) ov er Y . These D-branes are in a sense instantonic conﬁgurations. The (electric) coupling term is given b y Z p × Y C ∧ 1 q ˆ A(TY ) (c h(E) − ch(F)) (5.3.2) 133 Ramond-Ramond ﬁelds and c harge quan tization where w e ha v e used q ˆ A(TM) = q ˆ A(TY ). The Ramond-Ramond equations of mo- tion can b e formally written as d  G = δ (p) 1 q ˆ A(TY ) (c h(E) − ch(F)) where δ (p) is a Dirac delta distribution supp orted on p × Y . By integrating b oth sides, w e ha v e the relation  G R −  G L = 1 q ˆ A(TY ) (c h(E) − ch(F)) (5.3.3) where G R and G L denote the v alue of the ﬁeld G on spatial slices on the “right” and on the “left” of the D-brane, where the notion of left and right is giv en by the orien tation of M and Y . Now, by imp osing the selfduality constraint on the total Ramond-Ramond ﬁeldstrength G, w e ha v e the follo wing relation G R − G L = 1 q ˆ A(TY ) (c h(E) − ch(F)) (5.3.4) Notice that this at b est a heuristic conclusion, since requiring the Ramond-Ramond ﬁeld to be selfdual implies that the coupling (5.3.2) is not deﬁned. Moreov er, notice that the right hand side of (5.3.3) and (5.3.4) are not selfdual, while the left hand side is supposed to b e. A wa y of circumna vigating the problem is to use selfdualit y to eliminate half the ﬁelds in G. F or G 0 , G 2 , G 4 one w ould introduce the magnetic coupling as arising from the equation dG = δ (p) 1 q ˆ A(TY ) (c h(E) − ch(F)) (5.3.5) and in tro duce the magnetic coupling for G 6 , G 8 , G 10 via the electric coupling (5.3.2) [74]. Apart from these diﬃculties, equation (5.3.4) gives a goo d educated guess for the cohomology class of a Ramond-Ramond ﬁeld. Indeed, let b, a ∈ K 0 (Y) classify the Ramond-Ramond ﬁeld on the righ t and on the left of the collection of D-branes, resp ectiv ely . Then w e ha v e the follo wing equation in de Rham cohomology [G R ( b )] − [G L ( a )] =  q ˆ A(TY )  − 1 ∪ [c h( x )] (5.3.6) where x = [E] − [F]. Now, by considering the limit p → −∞ , and requiring that for a = 0, the associated de Rham cohomology class v anishes, equation (5.3.6) determines the class of G on all of the spacetime M as [G( x )] =  q ˆ A(TY )  − 1 ∪ [c h( x )] (5.3.7) 134 Generalized diﬀeren tial cohomology where w e ha v e used that the fact that the Chern character is an isomorphism o ver the reals. In other w ords, in this particular conﬁguration, the elemen t x ∈ K 0 (Y) classiﬁes the class of Ramond-Ramond ﬁelds whose ﬁeldstrength is a solution of the equation (5.3.5). Morov er, the ﬁeldstrength form G asso ciated to x m ust satisfy the condi- tion (5.3.7). An analogous argumen t can b e form ulated for type I IB String theory , b y using the Chern character deﬁnition as in section 3.6.2. The condition (5.3.7) is the Dirac quantization condition for Ramond-Ramond ﬁeldstrengths in t yp e I I String theory . It is a “quan tization” condition since the Chern c haracter c h : K 0 , − 1 (Y) → H ev , o dd (Y; R ) maps the K-theory group to a lattice in real cohomology , since the group K 0 , − 1 (Y) is a Z -mo dule. A t this p oin t, one assumes that the K-theoretical classiﬁcation of Ramond-Ramond ﬁelds in the absence of branes, and the condition (5.3.7), are v alid for general space- times M not of the form R × Y , and not only in the situation used to motiv ate it. W e are then left with the following mathematical problem: assign to any manifold M ab elian groups that can naturally b e interpreted as classes of gauge inequiv alent ﬁelds carrying a top ological c harge with v alues in the group K 0 , − 1 (M), and suc h that the associated ﬁeldstrength satisﬁes the condition (5 . 3 . 7). W e see that the diﬀer- en tial cohomology deﬁned in the previous section is not the suitable formalism to solv e this problem, since ob jects in ˇ H p (M) or H p (M; D p ) carry a top ological c harge in in teger cohomology . W e then intuitiv ely need a “generalized” v ersion of diﬀeren tial cohomology , which is the sub ject of the next section. 5.4 Generalized diﬀeren tial cohomology As we hav e seen in the previous section, the K-theoretical description of Ramond- Ramond ﬁelds requires a new framew ork whic h generalizes the diﬀerential cohomology formalism. W e need indeed a theory of some sort which is able to describ e ob jects with local degrees of freedom, and at same time takes into accoun t global properties of suc h ob jects. A hin t on ho w to deﬁne such a theory is given b y a closer insp ection of the Dirac quantization condition, as app eared in the examples b efore, along the lines of the argumen ts prop osed in [40, 41]. Essen tially , given a gauge theory of ﬁelds ˇ A with ﬁeldstrength ω ∈ Ω ∗ (M), to imp ose a Dirac quan tization condition is tan tamount to requiring that [ ω ] ∈ Λ ⊂ H ∗ dR (M; R ) 135 Generalized diﬀeren tial cohomology where Λ is a lattice in H ∗ dR (M; R ). F or instance, in the case of ordinary electromag- netism, the lattice Λ H is giv en b y the image of the map i : H ∗ (M; Z )  → H ∗ dR (M; R ) while for Ramond-Ramond ﬁelds in t yp e II String theory , the lattice Λ K is giv en b y the image of the map c h : K 0 , 1 (M) → H ev , o dd dR (M; R ) up to a “scale” factor. Notice that b oth the ab o v e maps induce an isomorphism when tensored o v er the real ﬁeld. It is clear that the lattice Λ is greatly aﬀected b y the chosen (generalized) cohomology theory whic h top ologically classiﬁes the ﬁelds in the given gauge theory , and b y the map realizing the free part of the relev ant cohomology groups in real de Rham cohomology . In general, as in the ab ov e cases, the c hoice of the cohomology theory for the given gauge ﬁeld is suggested b y physical prop erties of the system. In principle, though, there is no argument to exclude a given generalized cohomology theory Γ ∗ . The ab o v e arguments suggest the following mathematical idea, which constitute the starting p oin t in [54] for the construction of generalized diﬀerential cohomology theories. Let Γ ∗ b e an arbitrary m ultiplicative 6 generalized cohomology theory on the category of smo oth manifolds. Denote with π −∗ Γ := Γ ∗ (pt) the co eﬃcien t ring of the p oin t. F or example, for Γ ∗ = H ∗ , w e ha v e π −∗ H = Z while for Γ ∗ = K ∗ w e ha v e π −∗ K = Z [[ u − 1 , u ]] where u − 1 is an elemen t of degree -2, and corresp onds to the Bott generator. F or an y generalized cohomology theory Γ ∗ there exists a c anonic al map ϕ : Γ ∗ (X) → H(X; R ⊗ π −∗ Γ) ∗ whic h induces an isomorphism when tensored o v er the reals for an y top ological space X [54]. In the ab o v e expression, the grading on the cohomology groups is such that H(X; R ⊗ π −∗ Γ) n := M p + q = n H p (X; R ⊗ π − q Γ) 6 This condition can b e relaxed. 136 Generalized diﬀeren tial cohomology In the case Γ ∗ = H ∗ , the map ϕ coincides with i , while for Γ ∗ = K ∗ , the map ϕ is giv en b y the Chern c haracter. Notice indeed that H(X; R ⊗ π −∗ K) 0 ' H ev (X; R ) , H(X; R ⊗ π −∗ K) 1 ' H odd (X; R ) W e are then in terested in assigning to an y smooth manifold M an abelian group whic h “completes the square” ? / /   Ω cl (M; R ⊗ π −∗ Γ) ∗   Γ ∗ (M) ϕ / / H(M; R ⊗ π −∗ Γ) ∗ (5.4.1) where the map Ω cl (M; R ⊗ π −∗ Γ) ∗ → H(M; R ⊗ π −∗ Γ) ∗ is giv en b y assigning to an elemen t ω ∈ Ω cl (M; R ⊗ π −∗ Γ) ∗ its de Rham class in H(M; R ⊗ π −∗ Γ) ∗ . A ﬁrst guess to complete the square w ould b e to consider the ﬁb ered pro duct of Γ ∗ (M) and Ω cl (M; R ⊗ π −∗ Γ) ∗ o v er H(M; R ⊗ π −∗ Γ) ∗ , i.e. the group A ∗ Γ := { ( ω , u ) ∈ Ω cl (M; R ⊗ π −∗ Γ) ∗ × Γ ∗ (M) : [ ω ] = ϕ ( u ) } Ho w ev er, we know that these groups, for eac h degree, are only a ﬁrst approximation to our desired generalized diﬀerential cohomology theory: indeed, we kno w that for Γ ∗ = H ∗ , the Cheeger-Simons groups are an extension of A ∗ Γ . In other w ords, by considering only the ﬁb ered pro duct we are losing information ab out the group of ﬂat and top ologically trivial ﬁelds, whic h in general ma y be non v anishing. Mathematically this can b e understo od as follows: for the tw o cohomology classes [ ω ] and [ v ] to b e equal, the co cyle represen tativ es m ust satisfy the equation ω − δ h = v for some cycle h . The cycle h realizes the homotopy b et ween the t w o represen tativ es ω and v , and the information ab out it is lost if we only consider cohomology classes. In the case in whic h Γ ∗ is obtained as the cohomology of a diﬀeren tial complex, the square (5.4.1) could b e completed by a homotopy r eﬁnement . The diﬃcult y in doing this is in the fact that the functor Ω cl ( · ; R ⊗ π −∗ Γ) ∗ is not a cohomology functor on the category of manifolds. In the case in which Γ ∗ = H ∗ the problem can b e solved b y regarding for eac h q ≥ 0 the space Ω q cl (M; R ) as the 0-th cohomology of the complex Ω q (M; R ) d − → Ω q +1 (M; R ) d − → · · · d − → Ω n (M; R ) → 0 137 Generalized diﬀeren tial cohomology where n = dim M. By using standard results, then, one can deﬁne for a giv en q the complex [54] ˇ C p ( q ) := ( C p (M; Z ) × C p − 1 (M; R ) × Ω p (M; R ) n ≥ q C p (M; Z ) × C p − 1 (M; R ) n < q (5.4.2) with diﬀeren tial d( c, h, ω ) := ( δ c, ω − c − δ h, dω ) and d( c, h ) := ( ( δ c, − c − δ h, 0) n = q − 1 ( δ c, − c − δ h ) otherwise Notice that a triple ( c, h, ω ) is a co cycle if it satisﬁes the equations δ c = 0 d ω = 0 ω − c − δ h = 0 (5.4.3) The last of the ab o ve equations implies that [ ω ] = i ([ c ]): this means that the coho- mology groups ˇ H( q ) ∗ (M) := H ∗ ( ˇ C ∗ ( q ); d) ﬁt the square (5.4.1). The groups ˇ H( q ) ∗ (M) are called the Che e ger-Simons c ohomolo gy gr oups and as one can exp ect we hav e ˇ H( q ) p (M) ' H p (M; D q ) The ﬁeldstrength map assigns to the class [( c, h, ω )] ∈ ˇ H( q ) q (M) the closed form ω , while the characteristic class map assigns the class [ c ]. Morev er, the groups ˇ H( q ) q (M) satisﬁes the same exact sequences as the Cheeger-Simons groups. Unfortunately , the approac h follo w ed in the ab o ve paragraph to construct the the- ory completing the square (5.4.1) for ordinary cohomology cannot b e used in general for generalized cohomology theories as K-theory , since these are not obtained as the cohomology of a certain diﬀerential complex. Neverthless, a homotopy reﬁnement for a giv en theory Γ can b e obtained b y substituting co cycles with maps to the classifying space BΓ, and deﬁning an analog of conditions (5.4.3). By using this strate gy , Hopkins and Singer show ed in [54] that to any arbitrary generalized cohomology theory one can assign a theory , that they call a gener alize d diﬀer ential c ohomolo gy the ory , which ﬁts the square (5.4.1) and satisﬁes analogous prop erties to the Cheeger-Simons or Deligne cohomology groups. W e will fo cus in the following on a particular deﬁnition 138 Generalized diﬀeren tial cohomology of diﬀeren tial K-theory , which will b e the basis in the follo wing chapter for a suitable generalization to the equiv arian t setting; w e refer the reader to [54] for an extensive and detailed in tro duction to this b eautiful and exciting part of mo dern mathematics. Let F red denote the space of F redholm op erators on an inﬁnite dimensional Hilb ert space: recall that F red is a classifying space for the group K 0 (X) for a giv en top ological space, i.e. K 0 (X) ' [X , F red] where the isomorphism ab o v e is giv en b y considering the index bundle [Ker F f ] − [Coker F f ] (5.4.4) where Ker F f := S x Ker f ( x ) Cok er F f := S x Cok er f ( x ) for a given map f : X → F red. Of course, the expression (5.4.4) is naiv e, since the dimension of Ker f ( x ) and Coker f ( x ) can c hange while x v aries, and needs to b e stabilized, as sho wn in [4], for instance. Let u denote a co cycle in Z (F red; R ⊗ π −∗ K) 0 = M n Z 2 n (F red; R ) represen ting the universal Chern c haracter, i.e. such that if f : X → F red classiﬁes a v ector bundle E, f ∗ u represents ch(E). F or a manifold M, an element in the diﬀer ential K-the ory group ˇ K 0 (M) is represen ted b y a triple ( c, h, ω ) where c : M → F red, h ∈ C ev − 1 (M; R ), and ω ∈ Ω ev − 1 (M; R ) such that δ h = ω − c ∗ u (5.4.5) Moreo v er, the triples ab o v e deﬁned must satisfy the following equiv alence relation. Tw o triples ( c 0 , h 0 , ω 0 ) and ( c 1 , h 1 , ω 1 ) are said to b e equiv alent if there exists a triple ( c, h, ω ) on M × [0 , 1], with ω constan t along [0 , 1], suc h that ( c, h, ω ) | 0 = ( c 0 , h 0 , ω 0 ) ( c, h, ω ) | 1 = ( c 1 , h 1 , ω 1 ) (5.4.6) Notice that equation (5.4.5) enforces the condition [ ω ] = ch([ c ]) 139 Generalized diﬀeren tial cohomology The relations (5.4.6) can b e rephrased [40] in a wa y that will b e useful later on. Indeed, t w o triple ( c 0 , h 0 , ω 0 ) and ( c 1 , h 1 , ω 1 ) are equiv alen t if there exists a map f : M × [0 , 1] → F red and an elemen t σ ∈ Ω ev − 2 (M; R ) such that f | 0 = c 0 f | 1 = c 1 ω 1 = ω 0 h 1 = h 0 + π ∗ f ∗ u + d σ where π : M × [0 , 1] → M is the pro jection onto the ﬁrst factor, and π ∗ denotes in tegration along the ﬁbre, i.e. pairing with the fundamental class of [0 , 1]. The last of the ab o v e equations is obtained b y imp osing the condition (5.4.5). T o deﬁne the higher diﬀerential K-groups, recall that the iterated lo op spaces Ω i F red classify the functors K − i . Let u − i b e the co cycle u − i ∈ Z 2 n − i (Ω i F red; R ) deﬁned as u − i := Π ∗ ev ∗ u where ev : S i × Ω i F red → F red denotes the ev aluation map, and Π ∗ denotes the integration o v er the ﬁb er of the pro jection map Π : S i × Ω i F red → Ω i F red. Hence, elements of the higher diﬀerential groups ˇ K − i (M) are represen ted b y triples ( c, h, ω ) where c : M → Ω i F red, h ∈ C ev − i (M; R ), and ω ∈ Ω ev − i (M; R ) such that δ h = ω − c ∗ u − i and satisfying the equiv alence relations (5.4.6). The diﬀeren tial K-groups satisfy the follo wing exact sequences 0 → K − i − 1 (M; R / Z ) → ˇ K − i (M) → Ω ev − i K (M) → 0 0 → Ω ev − i − 1 (M) / Ω ev − i − 1 K (M) → ˇ K − i (M) → K − i (M) → 0 0 → H ev − i − 1 (M) / c h(K − i − 1 (M)) → ˇ K − i (M) → A − i K (M) → 0 (5.4.7) 140 Generalized diﬀeren tial cohomology where Ω ev − i K (M) denotes the elements in Ω ev − i (M) whose cohomology class is in the image of the Chern c haracter. W e will giv e a pro of of the existence of the ab o v e exact sequences in the context of equiv ariant K-theory in the next chapter. Finally , the homotop y equiv alence Ω n F red ' Ω n +2 F red induces Bott perio dity on the diﬀeren tial K-groups, and in particular it allo ws to deﬁne them for p ositiv e degrees as ˇ K n (M) := ˇ K n − 2 N (M) n − 2 N < 0 This deﬁnition of diﬀeren tial K-theory will b e at the core of a generaliztion we will prop ose in the next chapter. How ever, we should p oint out that there are diﬀeren t mo dels for diﬀeren tial K-theory , suc h as the one in tro duced in [26], where axioms for generalized diﬀerential cohomology theories are giv en. W e ha v e seen that ordinary diﬀeren tial cohomology admits diﬀerent mo dels, which lead to isomorphic theories, c haracterized by the fact that they satisfy the same exact sequences. As shown in [86], these exact sequences in a sense completely c haracterize ordinary diﬀeren tial cohomology . As far as the author kno ws, a similar result do es not exist for generalized diﬀeren tial cohomology theories. 141 Chapter 6 Ramond-Ramond ﬁelds and Orbifold diﬀeren tial K-theory As we ha v e seen in the previous c hapter, Ramond-Ramond ﬁelds in t yp e I I String theory require the use of diﬀeren tial K-theory to b e prop erly describ ed. W e ha ve also explained that this is due to the fact that the Dirac quantization condition for Ramond-Ramond ﬁeldstrength is dictated by K-theory , rather than integral cohomol- ogy . Moreo ver, the lattice in whic h charges tak e their v alues has b een suggested b y the form of the coupling b et w een D-branes and Ramond-Ramond ﬁeld. In this chapter w e will generalize the ab o ve argumen ts to the case of sup erstring theory deﬁned on an orbifold. T o b e precise, we will only consider the case of pre- sen table orbifolds of the form [M / G], where M is a spin manifold, and G is a ﬁnite group acting b y spin structure preserving diﬀeomorphisms. In this case, as prop osed b y Witten in [94], D-branes are classiﬁed by the equiv ariant K-theory of the “co ver- ing” space M. It is natural, then, to ask that the Dirac quan tization condition for the total ﬁeldstrength of Ramond-Ramond ﬁelds on the orbifold [M / G] is given by a Chern c haracter homomorphism on the equiv ariant K-theory K ∗ G (M), whic h induces an isomorphism when tensored o v er R . This is supported by the fact that we exp ect to obtain t yp e I I String theory features when G = { e } . A Chern c haracter with the ab o v e prop erties has b een constructed in [67], and mak es use of Bredon cohomology , an equiv arian t cohomology theory deﬁned on the category of G-CW complexes. The imp ortan t feature of this equiv ariant cohomology theory is that it naturally tak es into accoun t some important features of String theory on orbifolds, namely the presence of t wisted sectors. W e then pass to prop ose a deﬁnition of diﬀeren tial K-theory suitable for go o d orbifolds. Indeed, the existence theorem for generalized diﬀerential cohomol- ogy theories dev elop ed in [54] cannot be applied to equiv arian t cohomology functors 142 G-CW complexes and equiv arian t cohomology theories on the category of G-manifolds. More precisely , w e deﬁne ab elian groups that b eha v e as a natural generalization of the ordinary diﬀeren tial K-theory groups, in the sense that they agree in the case of a trivial group and they satisfy analogous exact se- quences. W e will “test” our deﬁnition in the case of linear ab elian orbifolds, and give a prop osal for the group of (equiv alence classes of ) ﬂat ﬁelds. W e will also make a brief digression to describ e D-branes using geometric equiv ariant K-homology , sho w- ing that the use of K-cycle is well-suited to the description of fr actional D-branes and their top ological charges computed using equiv arian t Dirac op erator theory . By using the Chern c haracter, we will deﬁne electric Ramond-Ramond couplings to D-branes on go o d orbifolds, and compare with previous examples in the literature. Finally , w e will supp ort our deﬁnition of orbifold diﬀeren tial K-theory , or to b e precise a com- plex v ersion of it, b y generalizing Moore and Witten argumen t as in tro duced in the previous chapter to the equiv arian t setting, and by expressing the Ramond-Ramond equations of motion by writing an equiv ariant v ersion of the Ramond-Ramond cur- ren t in term of the equiv ariant Chern c haracter and an equiv arian t v ersion of the Riemann-Ro c h theorem. 6.1 G-CW complexes and equiv arian t cohomology theories In this section w e will recall some basic notions ab out (generalized) equiv ariant coho- mology theories. In the following, X denotes a top ological space and G a ﬁnite group, unless otherwise stated. In the following, a (left) action G × X → X of G on X will b e denoted ( g , x ) 7→ g · x , and we will call X a G-sp ac e . The stabilizer or isotrop y group of a p oint x ∈ X is denoted G x = { g ∈ G | g · x = x } . Recall that a contin uous map f : X → Y of G-spaces is a G -map if f ( g · x ) = g · f ( x ) for all g ∈ G and x ∈ X. Deﬁnition 6.1. A G -e quivariant CW-de c omp osition of a G-space X consists of a ﬁltration X n , n ∈ N 0 suc h that X = [ n ∈ N 0 X n and X n is obtained from X n − 1 b y “attac hing” equiv arian t cells via the follo wing pro- cedure. Deﬁne X 0 = a j ∈ J 0 G / K j , with K j a collection of subgroups of G and the standard (left) G-action on an y coset 143 G-CW complexes and equiv arian t cohomology theories space G / K j . F or n ≥ 1 set X n =  X n − 1 q a j ∈ J n  D n j × G / K j   . ∼ (6.1.1) where the equiv alence relation ∼ is generated b y G-equiv arian t “attaching maps” φ n j : S n − 1 j × G / K j − → X n − 1 . (6.1.2) One requires that X carries the colimit top ology with resp ect to (X n ), i.e., B ⊂ X is closed if and only if B ∩ X n is closed in X n for all n ∈ N 0 . W e call the image of D n j × G / K j (resp. ˚ D n j × G / K j ) a close d (resp. op en ) n -cell of orbit t yp e G / K j . As usual, we call the subspace X n the n -skeleton of X. If X = X n and X 6 = X n − 1 , then n is called the ( c el lular ) dimension of X and X is said to b e of ﬁnite typ e . A G-space with a G-equiv arian t CW-decomp osition is called a G -c omplex . When G = e is the trivial group, a G-complex is just an ordinary CW-complex. In general, if X is a G-complex then the orbit space X / G is an ordinary CW-complex. Con v ersely , there is an in timate relation b etw een G-complexes and ordinary CW- complexes whenever G is a discrete group. Let X be a G-space which is an ordinary CW-complex. W e sa y that G acts c el lularly on X if 1) F or each g ∈ G and each op en cell E of X, the left translation g · E is again an op en cell of X; and 2) If g · E = E, then the induced map E → E, x 7→ g · x is the identit y . Then w e ha v e the follo wing [92] Prop osition 6.2. L et X b e a CW-c omplex with a c el lular action of a discr ete gr oup G . Then X is a G -c omplex with n -skeleton X n . In the case that X is a smo oth manifold, w e require the G-action on X to b e smo oth and there is an analogous result. Recall that the applicability of algebraic top ology to manifolds relies on the fact that an y manifold comes equiped with a canonical CW-decomp osition. In the case in which a group acts on the manifold one has the follo wing result due to Illman [57, 57]. Theorem 6.3. If G is a c omp act Lie gr oup or a ﬁnite gr oup acting on a smo oth c omp act manifold X , then X is triangulable as a ﬁnite G -c omplex. 144 G-CW complexes and equiv arian t cohomology theories The collection of G-complexes with G-maps as morphisms form a category . W e are interested in equiv ariant cohomology theories deﬁned on this category (or on sub- categories thereof ). W e will no w brieﬂy sp ell out the main ingredien ts in volv ed in building an equiv- arian t cohomology theory on the category of ﬁnite G-complexes, lea ving the details to the comprehensiv e treatments of [92] and [66], and fo cusing instead on some explicit examples. Fix a group G and a comm utativ e ring R. A G -c ohomolo gy the ory E ∗ G with values in R -mo dules is a collection of con tra v arian t functors E n G from the category of G-CW pairs to the category of R-mo dules indexed b y n ∈ Z together with natural transformations δ n G (X , A) : E n G (X , A) − → E n +1 G (X) := E n +1 G (X , ∅ ) for all n ∈ Z satisfying the axioms of G-homotop y in v ariance, long exact sequence of a pair, excision, and disjoint union. The theory is called or dinary if for an y orbit G / H one has E q G (G / H) = 0 for all q 6 = 0. These axioms are form ulated in an analogous w ay to that of ordinary cohomology . The new ingredients in an equiv arian t cohomology theory (whic h w e hav e not yet deﬁned) are the induction structur es , whic h w e shall no w describ e. Let α : H → G b e a group homomorphism, and let X b e an H-space. Deﬁne the induction of X with r esp e ct to α to b e the G-space ind α X giv en b y ind α X := G × α X . This is the quotient of the pro duct G × X by the H-action h · ( g , x ) := ( g α ( h − 1 ) , h · x ), with the G-action on ind α X giv en b y g 0 · [ g , x ] = [ g 0 g , x ]. If H < G is a subgroup, and α is the subgroup inclusion, the induced G-space is denoted G × H X. An e quivariant c ohomolo gy the ory E ∗ ( − ) with values in R -mo dules consists of a collection of G-cohomology theories E ∗ G with v alues in R-mo dules for each group G suc h that for an y group homomorphism α : H → G and an y H-CW pair (X , A) with k er( α ) acting freely on X, there are for each n ∈ Z natural isomorphisms ind α : E n G  ind α (X , A)  ≈ − → E n H (X , A) (6.1.3) satisfying (a) Compatibility with the cob oundary homomorphisms: δ n H ◦ ind α = ind α ◦ δ n G ; 145 G-CW complexes and equiv arian t cohomology theories (b) F unctoriality: If β : G → K is another group homomorphism such that k er( β ◦ α ) acts freely on X, then for ev ery n ∈ Z one has ind β ◦ α = ind α ◦ ind β ◦ E n K ( f 1 ) where f 1 : ind β  ind α (X , A)  ≈ − → ind β ◦ α (X , A) ( k , g , x ) 7− →  k β ( g ) , x  is a K-homeomorphism and E n K ( f 1 ) is the morphism on K-cohomology induced b y f 1 ; and (c) Compatibility with conjugation: F or g , g 0 ∈ G deﬁne Ad g ( g 0 ) = g g 0 g − 1 . Then the homomorphism ind Ad g coincides with E n G ( f 2 ), where f 2 : (X , A) ≈ − → ind Ad g (X , A) x 7− →  e , g − 1 · x  is a G-homeomorphism, where throughout e denotes the identit y elemen t in the group G. Th us the induction structures connect the v arious G-cohomologies and keep track of the equiv ariance. They will b e v ery imp ortant in the construction of the equiv ari- an t Chern c haracter for equiv arian t K-theory in a later section, ev en if w e are only in terested in a ﬁxed group G. Example 6.4 ( Bor el c ohomolo gy ) . Let H ∗ b e a cohomology theory for CW-pairs (for example, singular cohomology). Deﬁne H n G (X , A ) := H n  EG × G (X , A )  where EG is the total space of the classifying principal G-bundle EG → BG which is con tractible and carries a free G-action. This is called ( e quivariant ) Bor el c ohomol- o gy , and is the most commonly used form of equiv arian t cohomology in the physics literature. Note that H ∗ G is w ell-deﬁned because the quotient EG × G X is unique up to the homotop y t yp e of X / G. The ordinary G-cohomology structures on H ∗ G are inherited from the cohomology structures on H ∗ . The induction structures for H ∗ G are constructed as follows. Let α : H → G b e a group homomorphism and X an H-space. Deﬁne b : EH × H X − → EG × G G × α X ( ε, x ) 7− →  E α ( ε ) , e , x  146 G-CW complexes and equiv arian t cohomology theories where ε ∈ EH, x ∈ X and E α : EH → EG is the α -equiv ariant map induced by α . The induction map ind α is then giv en b y pullbac k ind α := b ∗ : H n G (ind α X) = H n (EG × G G × α X) − → H n (EH × H X) = H n H (X) . If k er( α ) acts freely on X, then the map b is a homotopy equiv alence and hence the map ind α is an isomorphism. Example 6.5 ( Equivariant K-the ory ) . In [82], equiv arian t top ological K-theory is de- ﬁned for an y G-complex X as the abelian group completion of the semigroup V ect C G (X) of complex G-vector bundles ov er X, with G a compact Lie group. Recall that for a G-space X, a complex G-vector bundle is giv en by a G-space E, and a G-map π : E → X such that E is the total space of a complex v ector bundle, and suc h that for any g ∈ G and any x ∈ X, the map g : E x → E g x is an homomorphism. The compactness prop erty of G assures that the Grothendieck functor K ∗ G satisﬁes the G-homotopy inv ariance 1 . The higher groups are deﬁned via iterated susp ension, similarly to ordinary K-theory , and Bott p erio dicity holds. T o deﬁne the induction structures, recall that if X is an H-space and α : H → G is a group homomorphism, then the map ϕ : X − → G × α X x 7− → ( e, x ) is an α -equiv ariant map whic h embeds X as the subspace H × α X of G × α X, and whic h induces via pullbac k of v ector bundles the homomorphism ϕ ∗ : K ∗ G (G × α X) − → K ∗ H (X) . This map deﬁnes the induction structure. It is in v ertible when ker( α ) acts freely on X, with inv erse the “extension” map E 7→ G × H E for any H-v ector bundle E ov er X. This can b e pro v en b y using the follo wing [82] Theorem 6.6. L et G b e a c omp act gr oup, and let N b e a normal sub gr oup acting fr e ely on X . Then pr ∗ : K ∗ G / N (X / N) ' − → K ∗ G (X) (6.1.4) wher e pr : X → X / N is the usual pr oje ction. 1 This is due to the fact that pullbacks of a G-bundle via G-homotopic maps are isomorphic only if G is compact. 147 The equiv arian t Chern c haracter By noticing that X / N ' (G / N) × G X and if w e deﬁne denote with N the k ernel of α : H → G, then K ∗ G  G × α X  ' K ∗ H / N  (H / N) × α X  ' K ∗ H / N (X / N) ' K ∗ H (X) since N acts freely on X b y h yp othesis. In the case in whic h X = pt, a G-vector bundle is just a G-mo dule, hence we hav e K 0 G (pt) ' R(G) , K − 1 G (pt) ' 0 (6.1.5) where R(G) is the r espr esentation ring of G, i.e. the ring generated ov er Z by the irreducible represen tations of G. W e can use the ab ov e results to show that K 0 G (G / H) ' K 0 G (G × H pt) ' K 0 H (pt) ' R(H) , K − 1 G (G / H) ' 0 for H < G. In the case in whic h the group G acts freely on the space X, theorem 6.6 implies K ∗ (X / G) ' K ∗ G (X) On the other extreme, when G acts trivially on X, the follo wing isomorphism holds [82] K ∗ G (X) ' K ∗ (X) ⊗ R(G) (6.1.6) 6.2 The equiv arian t Chern c haracter As we mentioned in the b eginning of this chapter, we exp ect that the Dirac quan- tization condition for the total Ramond-Ramond ﬁeldstrength on a go o d orbifold is dictated by some homomorphism on equiv arian t K-theory which is a generalization of the ordinary Chern c haracter. One might naiv ely think that the correct target theory for the equiv ariant Chern c haracter w ould naturally b e Borel cohomology , as deﬁned in example 6.4, with real coeﬃcients. This is not the case, as emphasised in particular b y a c ompletion the or em of A tiyah and Segal, whic h w e brieﬂy summarize. An y conjugacy classes of an elemen t γ ∈ G induces a homomorphism ν γ : R(G) → C giv en by ν γ ( ρ ) := χ ρ ( γ ), where χ ρ is the character asso ciated to the representation ρ , whic h is constan t on conjugacy classes. The k ernel of suc h a homomorphism is a 148 The equiv arian t Chern c haracter prime ideal 2 in the ring R(G). Let us denote with I G the prime ideal asso ciated to elemen t e ∈ G. Then we hav e [8] Theorem 6.7. F or a G-sp ac e X , with G a c omp act Lie gr oup, the Bor el c ohomolo gy H ∗ G (X; Q ) is isomorphic to the c ompletion of the R(G) -mo dule K ∗ G (X) ⊗ Q with r esp e ct to the ide al I G . The completion of K ∗ G (X; Q ) := K ∗ G (X) ⊗ Q is deﬁned as the tensor pro duct K ∗ G (X; Q ) ⊗ R(G) b R(G), where b R(G) is given by the limit of the quotients R(G) / I n G · R(G) for n going to inﬁnit y . The ab ov e theorem suggests then that Borel cohomology is not the correct target theory for a Chern character inducing an isomorphism o v er R . If w e think of R(G) as the ring of functions o v er G, the prime ideal I G corresp onds to the unit element in G. Theorem 6.7 then states that Borel cohomology do es not take in to account “con tributions” of the non-trivial elements in G, and hence, in a sense, it is lo c alise d around the unit elemen t. There are several approac hes to the equiv ariant Chern c haracter (see refs. [9, 87, 19, 43, 1], for example) which strongly dep end on the t yp es of groups inv olved (discrete, con tin uous, etc. ) and on the ring one tensors with ( R , C , etc. ). As we are in terested in ﬁnite groups and real co eﬃcien ts, we will use the Chern c haracter constructed in [66] and [67]. In the follo wing section we will brieﬂy recall the basic constructions in Bredon coho- mology [22, 66, 73], which will turn out to b e the b est suited equiv ariant cohomology theory for all of our purposes. W e will refer to App endix A for some pertinent aspects of functor categories. 6.2.1 Bredon cohomology In the following, G will denote a discrete group. The orbit c ate gory Or (G) of G is deﬁned as the category whose ob jects are homogeneous spaces G / H, with H < G, and whose morphisms are G-maps b etw een them. F rom general considerations [92] it follo ws that a G-map b etw een t w o homogeneous spaces G / H and G / K exists if and only if H is conjugate to a subgroup of K, and hence an y suc h map is of the form  g H 7− → g a K  (6.2.1) 2 A prime ide al P in a commutativ e ring R is an ideal such that whenever the pro duct ab of tw o elemen ts in R lies in P, then a or b lies in P. 149 The equiv arian t Chern c haracter for some a ∈ G such that a − 1 H a < K. If F is any family of subgroups of G then there is a sub category Or (G , F ) with ob jects G / H for H ∈ F . A simple example is pro vided b y the cyclic groups G = Z p with p prime, for which the orbit category has just t wo ob jects, G /e = G and G / G = pt. If Ab denotes the category of abelian groups, then a c o eﬃcient system is a functor F : Or (G) op − → Ab where Or (G) op denotes the opp osite category to Or (G). With such a functor and any G-complex X, 3 one can deﬁne for eac h n ∈ Z the group C n G (X , F ) := Hom Or (G)  C n (X) , F  (6.2.2) where C n (X) : Or (G) op → Ab is the pro jective functor deﬁned b y C n (X)(G / H) := C n  X H  , the cellular homology of the ﬁxed p oin t complex X H :=  x ∈ X   h · x = x ∀ h ∈ H  . (6.2.3) In equation (6.2.2), Hom Or (G) ( − , − ) denotes the group of natural transformations b et w een tw o con tra v arian t functors, with the group structure inherited by the images of the functors in Ab . The functorialit y prop erty of C n (X) is the natural one induced b y the iden tiﬁcation X H ' Map G (G / H , X). Indeed, the t w o maps X H − → Map G (G / H , X) , x 7− → f x  [ g H]  = g · x , Map G (G / H , X) − → X H , f 7− → f (H) are easily seen to b e in v erse to each other, and the desired homeomorphism is obtained b y giving the space Map G (G / H , X) the c omp act-op en top ology . In particular, a G-map (6.2.1) induces a cellular map X K → X H , x 7→ a · x . These groups can be expressed in terms of the G-complex structure of X. If the n -sk eleton X n is obtained b y attac hing equiv arian t cells as in equation (6.1.1) with K j the stabilizer of an n -cell of X, then the cellular chain complex C ∗ (X) consists of G-mo dules C n (X) = L j ∈ J n Z [G / K j ] and hence C n (X)(G / H) ' M j ∈ J n Z  Mor Or (G) (G / H , G / K j )  . 3 When G is an inﬁnite discrete group, one should restrict to pr op er G-complexes, i.e., with ﬁnite stabilizer for any p oint of X. Some further minor assumptions are needed when G is a Lie group. 150 The equiv arian t Chern c haracter F or eac h n ≥ 0, the group C n G (X , F ) is the direct limit functor o v er all n -cells of orbit t yp e G / K j in X of the groups F (G / K j ). This follo ws b y restricting equation (6.2.2) to the full sub category Or (G , F (X)), with F (X) the family of subgroups of G which o ccur as stabilizers of the G-action on X [73]. The Z -graded group C ∗ G (X , F ) = L n ∈ Z C n G (X , F ) inherits a cob oundary operator δ , and hence the structure of a co c hain complex, from the b oundary op erator on cellular c hains. T o a natural transformation f : C n (X) → F , one asso ciates the natural transformation δ f deﬁned by δ f (G / H) : C n  X H  − → F (G / H) σ 7− → f (G / H)( ∂ σ ) for σ ∈ C n − 1 (X H ), with naturalit y induced from that of the cellular boundary op erator ∂ . Then the Br e don c ohomolo gy of X with co eﬃcien t system F is deﬁned as H ∗ G (X; F ) := H  C ∗ G (X , F ) , δ  . This deﬁnes a G-cohomology theory . See [65] for the pro of that H ∗ G (X; F ) is an equiv arian t cohomology theory , i.e., for the deﬁnition of the induction structure. One can also deﬁne cohomology groups by restricting the functors in equation (6.2.2) to a sub category Or (G , F ). The deﬁnition of Bredon cohomology is indep enden t of F as long as F contains the family F (X) of stabilizers [73]. This fact is useful in explicit calculations. In particular, b y taking F = H to consist of a single subgroup, one sho ws that the Bredon cohomology of G-homogeneous spaces is giv en b y H ∗ G (G / H; F ) = H 0 G (G / H; F ) = F (G / H) . (6.2.4) Example 6.8 ( T rivial gr oup ) . When G = e is the trivial group, i.e., in the non- equiv arian t case, the functors C n (X) and F can b e iden tiﬁed with the ab elian groups C n (X) = C n (X)( e ) and F = F ( e ). Then C n e (X , F) = C n (X , F) and one has H n e (X; F ) = H ( C n (X , F) , δ ), i.e., the ordinary n -th cohomology group of X with co eﬃcien ts in F. Example 6.9 ( F r e e action ) . If the G-action on X is fr e e , then all stabilizers K j are trivial and X H = ∅ for every H ≤ G, H 6 = e . In this case one may tak e F = e to compute the co c hain complex C ∗ G (X , F ) ' Hom G  C ∗ (X) , F (G /e )  151 The equiv arian t Chern c haracter and so the Bredon cohomology H ∗ G (X; F ) coincides with the equiv arian t cohomology H ∗ G  X ; F (G /e )  of X with co eﬃcien ts in the G-mo dule F (G /e ) = F (G). In the case of the constan t functor F = Z , with Z (G / H) = Z for ev ery H ≤ G and the v alue on morphisms in Or (G) op giv en b y the identit y homomorphism of Z , this group reduces to the ordinary cohomology H ∗ (X / G; Z ). Example 6.10 ( T rivial action ) . If the G-action on X is trivial , then the collection of isotrop y groups K j for the G-action is the set of all subgroups of G and X H = X for all H ≤ G. In this case the functor C n (X) can b e decomp osed in to a sum o ver n -cells of pro jectiv e functors P K j with K j = G [73], and so one has Hom Or (G)  C n (X) , F  ' Hom  C n (X) , lim ← − Or (G) op F (G / H)  where the inv erse limit functor is tak en o v er the opp osite category Or (G) op . It follows that the Bredon cohomology H ∗ G (X; F ) = H ∗  X ; F (G / G)  is the ordinary cohomology of X with coeﬃcients in the abelian group F (G / G) = F (pt). W e will no w sp ecialize the co eﬃcien t system for Bredon cohomology to the r epr e- sentation ring functor R( − ) deﬁned on the orbit category Or (G) b y sending the left coset G / H to R(H), the representation ring of the group H. A morphism (6.2.1) is sen t to the homomorphism R(K) → R(H) giv en b y ﬁrst restricting the represen tation from K to the subgroup conjugate to H, and then conjugating b y a . Since R ( − ) is a functor to rings, the Bredon cohomology H ∗ G (X; R( − )) naturally has a ring structure. By equation (6.2.4), w e ha v e H ∗ G (G / H; R( − ) ) = R(G / H) = R(H) = K ∗ G (G / H) whic h is already an indication that Bredon cohomology is a b etter relative of equiv- arian t K-theory than Borel cohomology . Indeed, using the induction structure of Example 6.4 one sho ws that the Borel cohomology H ∗ G (G / H) = H ∗ (BH) coincides with the cohomology of the classifying space BH = EH / H, which computes the group cohomology of H and is t ypically inﬁnite-dimensional (ev en for ﬁnite groups H). 152 The equiv arian t Chern c haracter In the construction of the equiv arian t Chern c haracter in the next section, it will b e imp ortan t to represen t the rational Bredon cohomology H ∗ G (X; Q ⊗ R( − )) as a certain group of homomorphisms of functors, similarly to the cochain groups (6.2.2). F or this, w e in tro duce another category Sub (G). The ob jects of Sub (G) are the subgroups of G, 4 and the morphisms are giv en b y Mor Sub (H , K) :=  f : H → K   ∃ g ∈ G , g H g − 1 ≤ K , f = Ad g   Inn(K) . In particular, there is a functor Or (G) → Sub (G) whic h sends the ob ject G / H to H and the morphism (6.2.1) in Or (G) to the homomorphism ( g 7→ a − 1 g a ) in Sub (G). If a lies in the cen tralizer Z G (H) :=  g ∈ G   g − 1 h g = h, ∀ h ∈ H  (6.2.5) of H in G, then the morphism (6.2.1) is sent to the iden tit y map. Any functor F : Sub (G) op → Ab can b e naturally regarded as a functor on Or (G) op . Deﬁne the quotien t functors C qt ∗ (X) , H qt ∗ (X) : Sub (G) op → Ab b y C qt ∗ (X)(H) := C ∗  X H / Z G (H)  and H qt ∗ (X)(H) := H ∗  X H / Z G (H)  . F or any functor F : Sub (G) op → Ab one has Hom  C ∗ (X H / Z G (H)) , F (H)  ' Hom Z G (H)  C ∗ (X H ) , F (H)  . By observing that the cen tralizer (6.2.5) is precisely the group of automorphisms of G / H in the orbit category Or (G) sen t to the identit y map in the subgroup category Sub (G), w e ﬁnally ha v e C ∗ G (X , F ) = Hom Or (G)  C ∗ (X) , F  ' Hom Sub (G)  C qt ∗ (X) , F  . (6.2.6) A t this p oin t one can apply equation (6.2.6) to the rational representation ring func- tor F = Q ⊗ R( − ), whic h by construction can b e regarded as an injective functor Sub (G) op → Ab , to pro v e the Lemma 6.11 ([67]) . F or any ﬁnite gr oup G and any G -c omplex X , ther e exists an isomorphism of rings Φ X : H ∗ G  X ; Q ⊗ R( − )  ≈ − → Hom Sub (G)  H qt ∗ (X) , Q ⊗ R( − )  . 4 If G is inﬁnite then one should restrict to ﬁnite subgroups of G. 153 The equiv arian t Chern c haracter 6.2.2 Chern c haracter in equiv arian t K-theory Before sp elling out the deﬁnition of the equiv ariant Chern c haracter given in [67], we recall some basic prop erties of the equiv arian t K-theory of a G-complex X. Let H b e a subgroup of G, and consider the ﬁxed p oin t subspace of X deﬁned in (6.2.3). The action of G does not preserv e X H , but the action of the normalizer N G (H) of H in G do es. If w e denote with i : X H  → X the inclusion of X H as a subspace of X, and with α : N G (H)  → G the inclusion of N G (H) as a subgroup of G, then we naturally ha v e the equalit y i ( n · x ) = α ( n ) · i ( x ) for all n ∈ N G (H) and x ∈ X H . It follows that the induced homomorphism on equiv- arian t K-theory is a map [82] i ∗ : K ∗ G (X) − → K ∗ N G (H)  X H  whic h is called a r estriction morphism . W e also need a somewhat less known prop erty [67]. Let N < G b e a ﬁnite normal subgroup, and let Rep(N) b e the category of (isomorphism classes of ) irreducible complex represen tations of N. Let X b e a (prop er) G / N-complex, and let G act on X via the pro jection map G → G / N. Then for any complex G-v ector bundle E → X and an y representation V ∈ Rep(N), deﬁne Hom N (V , E) as the vector bundle o v er X with total space Hom N (V , E) := [ x ∈ X Hom N (V , E x ) where N acts on the ﬁbres of E b ecause of the action of G via the pro jection map. No w if H ≤ G is a subgroup which comm utes with N, [H , N] = e , then one can induce an H-v ector bundle from Hom N (V , E) by deﬁning ( h · f )( v ) = h · f ( v ), v ∈ V for an y h ∈ H and any f ∈ Hom N (V , E) (remembering that G acts on E). Hence there is a homomorphism of rings Ψ : K ∗ G (X) − → K ∗ H (X) ⊗ R(N) deﬁned on G-v ector bundles b y Ψ  [E]  := X V ∈ Rep(N)  Hom N (V , E)  ⊗ [V] . (6.2.7) This homomorphism satisﬁes some naturality prop erties; see [67]. Note that the sum (6.2.7) is ﬁnite , since N is a ﬁnite subgroup. 154 The equiv arian t Chern c haracter W e are now ready to construct the equiv arian t Chern character as a homomor- phism c h X : K 0 , 1 G (X) − → H even , odd G  X ; Q ⊗ R( − )  for an y ﬁnite prop er G-complex X. The strategy used in [67] is to construct Z 2 -graded homomorphisms c h H X : K ∗ G (X) − → Hom  H ∗ (X H / Z G (H)) , Q ⊗ R(H)  (6.2.8) for an y ﬁnite subgroup H, and then glue them together as H v aries through the ﬁnite subgroups of G. T o deﬁne the homomorphism (6.2.8), we ﬁrst comp ose the ring homomorphisms K ∗ G (X) i ∗ − → K ∗ N G (H)  X H  Ψ − → K ∗ Z G (H)  X H  ⊗ R(H) π ∗ 2 ⊗ id − − − → K ∗ Z G (H)  EG × X H  ⊗ R(H) where π 2 : EG × X H → X H is the pro jection onto the second factor. By using the induction structure of Example 6.5, one then has K ∗ Z G (H)  EG × X H  ⊗ R(H) ≈ − → K ∗  EG × Z G (H) X H  ⊗ R(H) ch ⊗ id − − − → H ∗  EG × Z G (H) X H ; Q  ⊗ R(H) where c h is the ordinary Chern c haracter. One ﬁnally has H ∗  EG × Z G (H) X H ; Q  ⊗ R(H) ≈ − → H ∗  X H / Z G (H) ; Q  ⊗ R(H) ' Hom  H ∗ (X H / Z G (H)) , Q ⊗ R(H)  , where the ﬁrst isomorphism follows from the Leray sp ectral sequence by observing that the ﬁbres of the pro jection EG × Z G (H) X H − → X H  Z G (H) are all classifying spaces of ﬁnite groups, having triv al reduced cohomology with Q - co eﬃcien ts and are therefore Q -acyclic. The equiv arian t Chern c haracter is no w deﬁned as 5 c h X = M H ≤ G c h H X . (6.2.9) By using the v arious naturalit y properties of the homomorphism (6.2.7) [67], one sees that c h X tak es v alues in Hom Sub (G)  H qt ∗ (X) , Q ⊗ R( − )  , and by Lemma 6.11 it is th us a Z 2 -graded map c h X : K ∗ G (X) − → Hom Sub (G)  H qt ∗ (X) , Q ⊗ R( − )  ' H ∗ G  X ; Q ⊗ R( − )  . 5 If G is inﬁnite then the direct sum in equation (6.2.9) is understo od as the in verse limit functor o ver the dual subgroup category Sub (G) op . 155 String theory on orbifolds This map is well-deﬁned as a ring homomorphism b ecause all maps inv olv ed ab o ve are homomorphisms of rings. As with the deﬁnition of Bredon cohomology , the sum (6.2.9) may b e restricted to any family of subgroups of G con taining the set of stabi- lizers F (X). T o conclude, w e hav e to prov e that this map becomes an isomorphism upon ten- soring o ver Q . F or this, one prov es that the morphism c h X in equation (6.2.9) is an isomorphism on homogeneous spaces G / H, with H a ﬁnite subgroup of G, and then uses induction on the n um b er of orbit t yp es of cells in X along with the May er- Vietoris sequences for the pushout squares induced b y the attac hing G-maps (6.1.2). The isomorphism on G / H is a consequence of the isomorphisms (6.2.4) and (6.1.5). The details may b e found in [67]. Let π −∗ K G ( − ) b e the functor on Or (G) deﬁned by G / H 7→ K ∗ G (G / H). Then one has the following Theorem 6.12. ([67]) F or any ﬁnite pr op er G -c omplex X , the Chern char acter c h X extends to a natur al Z -gr ade d isomorphism of rings c h X ⊗ Q : K ∗ G (X) ⊗ Q ≈ − → H G  X ; Q ⊗ π −∗ K G ( − )  ∗ . 6.3 String theory on orbifolds The techniques of equiv arian t cohomology and K-theory illustrated in the previous section pla y an imp ortant role in understanding the behaviour of String theory deﬁned on orbifolds. As mentioned in the in tro duction to this c hapter, w e will b e in terested in go o d orbifolds, whic h are obtained as the orbit space of the action of a ﬁnite group G on a smo oth manifold X. In particular, the action of G will b e isometric, prop er, and co compact 6 . It is kno w that when G acts on X with nontrivial stabilizers, the orbit space X / G cannot b e giv en a diﬀeren tial structure suc h that the usual pro jection π : X → X / G is a smo oth map. In the case in whic h all the stabilizers are trivial, i.e. the group G acts freely , the orbit space naturally carries a manifold structure. W e will not attempt to give a deﬁnition of nonglobal orbifolds, since we will w ork in the equiv arian t “regime”. W e direct instead the reader to the seminal pap er [81] for a lo cal description of orbifolds, and to [68] for a mo dern description in terms of group oids. The quan tum b ehaviour of String theory on an orbifold [X / G] is diﬀeren t from that of 6 An action of a group G on a space X is said to b e c o c omp act if the orbit space X / G is compact 156 String theory on orbifolds a quan tum particle, as ﬁrst realized in [35, 36]. Indeed, supp ose that G acts freely on X. T o describ e the quan tum mechanics of a p oint particle propagating on the smo oth manifold X / G, one could think of ﬁrst contructing the Hilb ert space of states for a particle on the manifold X, and then restrict to the Hilbert subspace of G in v arian t states 7 . F ollowing the same logic, the ﬁrst step to the quantum string propagation on X / G consists in constructing the Hilb ert space H 0 for a string propagating on X, and restrict to the G-inv ariant states. In contrast to the particle case, this is not y et a complete Hilb ert space of states. Indeed, H 0 do es not contain states of strings which are closed on the quotien t manifold X / G, but are only mo dulo a G transformation. More precisely , consider an em b edding f : [0 , 1] × R → X of the w orldsheet strip, with lo cal co ordinates ( σ, τ ) ∈ [0 , 1] × R . The op en strings ob eying f ( σ + 2 π , τ ) = h · f ( σ, τ ) (6.3.1) for some h ∈ G are closed on the quotien t X / G, since the p oint x and h · x are iden tiﬁed 8 . Hence, the Hilb ert space H for a quan tum closed string propagating on X / G is giv en b y H := M h ∈ G H h ! G (6.3.2) where the sector H h is giv en b y the space of states of an op en string satisfying con- dition (6.3.1). A t this p oint, notice that the action of G p ermutes the sectors in the conjugacy classes of the asso ciated element h . Indeed, for any f satisfying condition (6.3.1), for any g ∈ G we hav e g · f ( σ + 2 π , τ ) = g h · f ( σ, τ ) = ( g hg − 1 ) g · f ( σ, τ ) (6.3.3) W e can then deﬁne H [ h ] := M l ∈ [ h ] H l = n h M i =1 H p i hp − 1 i where n h is the num b er of element in the conjugacy class [ h ], and { p i } is an appropriate set of elemen ts of G. An y element ξ [ h ] can b e expressed as ξ [ h ] = ( ξ h , ξ p 1 hp − 1 1 , · · · ) 7 A choice of a “lift” of the action of G on the internal degrees of freedom, should b e made, if possible. In other words, the vector bundle whose the wa ve function is a section of must b e G-equiv ariant. 8 In sup erstring theory X is a G − spin c manifold, and an analogous condition to (6.3.1) should b e imp osed on the w orldsheet fermion ﬁelds. 157 String theory on orbifolds and clearly the action of G preserv es the vector space H [ h ] . Since w e are in terested in G-in v arian t states, we hav e only to consider the action of elements g ∈ Z G ( h ), since these are the only elements of G whic h do not p ermute the sectors. More precisely , if w e deﬁne the action of g ∈ Z G ( h ) on H [ h ] as g · ξ [ h ] := ( g ξ h , p 1 g p − 1 1 ξ p 1 hp − 1 1 , · · · ) w e ha v e H ' M [ h ] H Z G ( h ) [ h ] The subspaces H [ h ] asso ciated to a nontrivial conjugacy class are called twiste d se ctors . The ab ov e construction of the Hilb ert space of closed strings also ensures that the theory is modular in v arian t [35, 36]. One expects that these t wistor sectors will app ear also in the case of a G-action with ﬁxed p oints, and that the Hilbert space of states can b e constructed as ab o v e. Moreo v er, in con trast with ordinary quantum ﬁeld theory , String theory is usually w ell deﬁned on the singular orbifold p oin ts. As it is expected, the t wisted sectors will pla y a role in the behaviour of the low-energy limit of type I I orbifold String theory and of D-branes. This will b e illustrated in the follo wing sections. W e conclude this section with a basic result that will b e constan tly used later on. Theorem 6.13. L et G b e a ﬁnite gr oup acting via isometries on a smo oth Riemannian manifold X . Then the set X g := { x ∈ X : g · x = x } is natur al ly a (p ossibly disc onne cte d) submanifold of X , for any g ∈ G . Pr o of. Let ¯ x b e a ﬁxed p oint for the action of g . The pushforward g ∗ acts linearly on the vector space T ¯ x X: denote with K ⊂ T ¯ x X the space whic h is left ﬁxed b y g ∗ . Then the exp onen tial map exp : T ¯ x X → X maps diﬀeomorphically the subspace K on the ﬁxed p oin ts of g , since g acts by isometries, and hence we can use this co ordinate system to deﬁne lo cal c harts for X g . 6.3.1 D-branes and equiv ariant K-cycles In this section we will make some remarks concerning the top ological classiﬁcation of D-branes and their c harges on global orbifolds of Type II superstring theory with v anishing H -ﬂux. As prop osed b y Witten in [94] and emphasised in [75, 46], Ramond- Ramond charges in t yp e I I String theory on a global orbifold [X / G] are classiﬁed b y 158 String theory on orbifolds the equiv arian t K-theory K ∗ G (X) of spacetime. The argumen ts are essen tially the same as those presen ted in c hapter 3, hence we will a void their restatemen t. Instead, w e will sho w ho w equiv arian t K-homology K G ∗ (X), dual to equiv arian t K-theory , leads to a description of fr actional D-br anes in terms of equiv ariant K-cycles. In the follo wing w e will refer to App endix D for the deﬁnition of equiv arian t K-homology , b oth geometric and equiv arian t. Similarly to K-homology , the cycles for equiv arian t K-homology , called G-equiv ariant K-cycles, live in an additiv e category D G (X) whose ob jects are triples (W , E , f ) where W is a G-spin c manifold without b oundary , E is a G-vector bundle ov er W , and f : W − → X (6.3.4) is a G-map. The group K G ∗ (X) is the quotient of this category by the equiv alence relation generated by b ordism, direct sum, and v ector bundle mo diﬁcation, as detailed in App endix B. Note that W need not b e a submanifold of spacetime. Ho w ev er, since X is a manifold, we can restrict the b ordism equiv alence relation to diﬀer ential b or dism and assume that the map (6.3.4) is a diﬀeren tiable G-map in equiv ariant K-cycles (W , E , f ) ∈ D G (X). In this wa y the category D G (X) extends the standard K-theory classiﬁcation to include branes supp orted on non-represen table cycles in spacetime. This deﬁnition of equiv arian t K-homology thus giv es a concrete geometric mo del for the top ological classiﬁcation of D-branes (W , E , f ) in a global orbifold [X / G] whic h captures the ph ysical constructions of orbifold D-branes as G-in v arian t states of branes on the cov ering space X. In the subsequen t sections we will study the pairing of Ramond-Ramond ﬁelds with these D-branes. Consider a D-brane lo calized on the submanifold X g of the cov ering space X. Since the Chan-P aton bundle E is G-equiv arian t, the ﬁb er of the restriction E to X g at eac h p oin t carries a representation of the cyclic group < g > . In this case the D-brane is said to b e fr actional . F ractional D-branes are stuck at the ﬁxed p oints and they couple to Ramond-Ramond ﬁelds coming for the t wisted sector labelled b y [ g ]. The term fractional is used since fractional D-branes, in the simple examples kno wn, carry a fr action of the corresp onding Ramond-Ramond charge. W e can use equiv arian t K-homology to geometrically describ e a particular class of fractional D-branes. Indeed, let G ∨ denote the set of conjugacy classes [ g ] of elements g ∈ G. There is a natural subcategory D G frac (X) of D G (X) consisting of triples (W , E , f ) for whic h W is a G-ﬁxed space, i.e., for which W g = W (6.3.5) 159 String theory on orbifolds for all g ∈ G. By G-equiv ariance this implies f (W) g = f (W) for all g ∈ G, and so the image of the brane w orldv olume lies in the subspace f (W) ⊂ \ g ∈ G X g . This is the set of G-ﬁxed p oints of X, and so the ob jects (W , E , f ) of the category D G frac (X) can naturally b e intepreted in terms of fractional branes. More precisely , w e call D G frac (X) the category of “maximally fractional D-branes”. In this case, an application of Sc h ur’s lemma sho ws that the Chan-P aton bundle admits an isotopical decomp osition and there is a canonical isomorphism of G-bundles E ' M [ g ] ∈ G ∨ E [ g ] ⊗ 1 1 [ g ] with E [ g ] = Hom G  1 1 [ g ] , E  , (6.3.6) where E [ g ] is a complex v ector bundle with trivial G-action and 1 1 [ g ] is the G-bundle W × V [ g ] with γ : G → End(V [ g ] ) the irreducible representation corresp onding to the conjugacy class [ g ] ∈ G ∨ . F rom the direct sum relation in equiv ariant K-homology it follo ws that a fractional D- brane, represen ted b y a K-cycle (W , E , f ) in D G frac (X), splits into a sum ov er irreducible fractional branes represen ted b y the K-cycles (W , E [ g ] ⊗ 1 1 [ g ] , f ), [ g ] ∈ G ∨ , which can then b e considered stable. W e then prop ose that in the framework of equiv arian t K-homology , the top ological c harge of a fractional D-brane, in a given closed string twisted sector of the orbifold String theory on a G-spin c manifold X, can b e computed b y using the equiv ariant Dirac op erator theory in tro duced in App endix D. The equiv ariant index of the G- in v arian t spin c Dirac op erator D / X E coupled to a G-vector bundle E → X takes v alues in K ∗ G (pt) ' R(G). W e can turn this in to a homomorphism on K G ∗ (X) with v alues in Z by comp osing with the pro jection R(G) → Z deﬁned b y taking the multiplicit y of a giv en represen tation γ : G − → End(V γ ) (6.3.7) of G on a ﬁnite-dimensional complex v ector space V γ . There is a corresp onding class in the KK-theory group [ γ ] ∈ KK ∗  C [G] , End(V γ )  whic h is represented b y the Kasparov mo dule (V γ , γ , 0) asso ciated with the extension of the representation (6.3.7) to a complex representation of group ring C [G]. By Morita inv ariance, the Kasparov pro duct with [ γ ] is the homomorphism on K-theory K 0  C [G]  − → K 0  End(V γ )  ' K 0 ( C ) ' Z 160 String theory on orbifolds induced b y γ : C [G] → End(V γ ) [25]. W e ma y then deﬁne a homomorphism µ γ : K G 0 (X) − → Z of ab elian groups b y µ γ  [W , E , f ]  = Index γ  f ∗ [ D / W E ]  := ass  f ∗ [ D / W E ]  ⊗ C [G] [ γ ] (6.3.8) on equiv arian t K-cycles (W , E , f ) ∈ D G (X) (and extended linearly), where ass : K G ∗ (X) − → K ∗  C [G]  is the analytic assembly map mentioned in App endix D. W e then naturally interpret 6.3.8 as the topological c harge of the D-brane represen ted b y (W , E , f ). Notice that for G = e , (6.3.8) reduces to the ordinary expression for the charge of a D-brane. W e ma y no w consider a simple class of examples. Let V be a complex vector space of dimension dim C (V) = d ≥ 1, and let G be a ﬁnite subgroup of SL(V). Our spacetime X is the G-space iden tiﬁed with the pro duct X = R p, 1 × V , where G acts trivially on the Minko wski space R p, 1 , and p is o dd. W e will consider fractional D-branes with worldv olume R p, 1  → R p, 1 × v , where v ∈ V is a ﬁxed vector under the linear action of G. In analogy with the nonequiv ariant case, the group of charge of these fractional D-branes is giv en b y the compact supp ort equiv arian t K-theory of the normal bundle, whic h in this case is giv en b y [8] K ∗ G , cpt (V) ' K ∗ G (pt) ' R(G) = Z | G ∨ | . since V is G-contractible. The same result can b e obtained by considering the equiv ariant K-theory of the w orld v olume R p, 1 , whic h is G-con tactible. It follo ws that the fractional D-branes, as deﬁned b y elemen ts of equiv arian t K-theory , can b e identiﬁed with represen tations of the orbifold group γ = | G ∨ | M a =1 N a γ a consisting of N a ≥ 0 copies of the a -th irreducible representation γ a : G − → End(V a ) , a = 1 , . . . ,   G ∨   , 161 String theory on orbifolds whic h deﬁnes the action of G on the ﬁbres of the Chan-P aton bundle. More precisely , eac h irreducible fractional brane is asso ciated to the G-bundle R p, 1 × V a o v er R p, 1 . W e can then consider the K-cycles ( R p, 1 , R p, 1 × V a , i v ), where i v : R p, 1 → R p, 1 × v ⊂ X. In equiv ariant K-homology , these cycles can b e contracted to [pt , V a , i ], where i is the inclusion of a p oin t pt ⊂ V whose induced homomorphism i ∗ : K G ∗ (pt) − → K G ∗ (X) can b e taken to b e the identit y map R(G) → R(G). The G-in v arian t Dirac operator D / pt V a is just Cliﬀord m ultiplication t wisted b y the G-module V a , and th us the top ologi- cal c harges (6.3.8) of the corresp onding fractional branes in the t wisted sector labelled b y b are giv en b y µ b  [pt , V a , i ]  = Index γ b  [ D / pt V a ]  =  V a ⊗ (∆ + ⊕ ∆ − )  ⊗ C [G] [ γ b ] , where ∆ ± are the half-spin represen tations of SO( p + 1) on C p +1 2 . Acting on the c haracter ring the pro jection giv es [W] ⊗ C [G] [ γ b ] = γ ∗ ([W]), where γ ∗ : K 0  C [G]  − → K 0  End(V γ )  is the map induced b y γ . 6.3.2 Delo calization and Ramond-Ramond ﬁelds As discussed in chapter 1, the gauge theory of Ramond-Ramond ﬁelds arises as a low- energy limit of type I I sup erstring theory from the Hilb ert space of states of closed sup erstrings. In tuitiv ely , the lo w-energy limit is the limit in which the string b ecomes p oin tlik e, i.e. the lengh t of the string go es to zero. As w e ha v e seen in the previous section, the Hilb ert space for type I I sup erstring theory deﬁned on the go o d orbifold [X / G] is giv en b y H := M h ∈ G H h ! G where the subspaces H h are spaces of states of open strings satisfying the b oundary condition (6.3.3); w e ha v e also noticed that such open strings “look lik e” closed strings on the submanifolds X h . W e then exp ect massless ﬁelds, in particular Ramond- Ramond ﬁelds, arising from eac h of these sectors, deﬁned on X h . This is due to the fact that the cen ter of mass of an op en string satisfying condition (6.3.3) is constrained to b e a p oint of X h . W e can mathematically “organize” the information about these 162 String theory on orbifolds Ramond-Ramond ﬁelds in the following wa y . W e can asso ciate to eac h G-manifold the space ˆ X := a h ∈ G X h (6.3.9) Notice that we ha v e an action of G on ˆ X, with g ∈ G inducing the diﬀeomorphism X h → X g hg − 1 . W e can then consider Ramond-Ramond ﬁelds as elements of the diﬀeren tial complex Ω ∗ G (X; R ) := Ω ∗ ( ˆ X; R ) G = M h ∈ G Ω ∗ (X h ; R ) ! G (6.3.10) equip ed with the diﬀeren tial d G := M h ∈ G d h where d h : Ω ∗ (X h ; R ) → Ω ∗ (X h ; R ) is the usual deRham exterior deriv ative. Since X h is diﬀeomorphic to X g hg − 1 for any g ∈ G, b y making a c hoice of submanifolds X g w e ha v e Ω ∗ G (X; R ) ' M [ h ] ∈ G ∨ Ω ∗ (X h ; R ) Z G ( h ) (6.3.11) The cohomology of the complex (6.3.10) with resp ect to the diﬀeren tial d G is giv en b y H ∗ (Ω ∗ G (X; R ); d G ) = M h ∈ G H ∗ (X h ; R ) ! G ' M [ h ] ∈ G ∨ H ∗ (X h ; R ) Z G ( h ) where w e ha v e used H ∗ (X h / Z G ( h ); R ) ' H ∗ (X h ; R ) Z G ( h ) The cohomology groups ab o v e corresp ond to the delo c alize d e quivariant c ohomolo gy theory deﬁned by Baum and Connes [12]. Notice that the group H ∗ (Ω ∗ G (X; R ); d G ) is non-canonically isomorphic to H ∗ (X; R ) ⊗ R(G) when the G-action on X is trivial. W e will no w show how Bredon cohomology can b e used to compute the cohomology of the complex (6.3.10) of orbifold Ramond-Ramond ﬁelds b y giving a delocalized description of Bredon cohomology with r e al co eﬃcients, following [73] and [67] where further details can b e found. Denote with R ( − ) the real represen tation ring functor R ⊗ R ( − ) on the orbit category Or (G). Let h G i denote the set of conjugacy classes [C] of cyclic subgroups C of G. Let R C ( − ) b e the con tra v arian t functor on Or (G) deﬁned b y R C (G /H ) = 0 if [C] con tains no representativ e g C g − 1 < H , and otherwise R C (G /H ) is isomorphic to the 163 String theory on orbifolds cyclotomic ﬁeld R ( ζ | C | ) o ver R generated b y the primitiv e root of unit y ζ | C | of order | C | . A standard result from the representation theory of ﬁnite groups then gives a natural splitting R ( − ) = M [C] ∈h G i R C ( − ) . By deﬁnition, for an y mo dule M ( − ) ov er the orbit category one has Hom Or (G)  M ( − ) , R C ( − )  ' Hom N G (C)  M (G / C) , R C (G / C)  ' Hom ( M (G / C); Z ) ⊗ N G (C) R C (G / C) where the normalizer subgroup N G (C) acts on R C (G / C) ' R ( ζ | C | ) via iden tiﬁcation of a generator of C with ζ | C | . These facts together imply that the co c hain groups (6.2.2) with F = R ( − ) admit a splitting giv en b y C ∗ G  X , R ( − )  ' M [C] ∈h G i C ∗  X C  ⊗ N G (C) R C (G / C) . As the cen tralizer Z G (C) acts prop erly on X C , the natural map M [C] ∈h G i C ∗  X C  ⊗ N G (C) R C (G / C) − → M [C] ∈h G i C ∗  X C / Z G (C)  ⊗ W G (C) R C (G / C) is a cohomology isomorphism, where W G (C) := N G (C) / Z G (C) is the W eyl group of C < G which acts by translation on X C / Z G (C). Since R C (G / C) is a pro jectiv e R [ W G (C)]-mo dule, it follows that for any prop er G-complex X the Bredon cohomology of X with co eﬃcien t system R ⊗ R( − ) has a splitting H ∗ G  X ; R ⊗ R( − )  ' M [C] ∈h G i H ∗  X C / Z G (C) ; R  ⊗ W G (C) R C (G / C) . (6.3.12) A t this p oin t, w e note that the dimension of the R -v ector space R C (G / C) W G (C) ' R ⊗ W G (C) R C (G / C) is equal to the num b er of G-conjugacy classes of generators for C. W e also use the fact that for a ﬁnite group G a sum o ver conjugacy classes of cyclic subgroups is equiv alen t to a sum o v er conjugacy classes of elements in G, and that X h g i = X g and Z G ( h g i ) = Z G ( g ). One ﬁnally obtains a splitting of real Bredon cohomology groups 9 H ∗ G  X ; R ⊗ R( − )  ' M [ g ] ∈ G ∨ H ∗  X g ; R  Z G ( g ) (6.3.13) 9 This splitting in fact holds o v er Q [73]. 164 String theory on orbifolds whic h is the cohomology of the diﬀeren tial complex (6.3.10). By using Theorem 6.12, one also has a decomp osition for equiv arian t K-theory with real co eﬃcien ts giv en b y K ∗ G (X) ⊗ R ' M [ g ] ∈ G ∨  K ∗ (X g ) ⊗ R  Z G ( g ) . 6.3.3 Delo calization of the equiv arian t Chern c haracter It is w ell known that the ordinary Chern character, when tensored o v er R , admits a Chern-W eyl reﬁnemen t, expressed in terms of the curv ature of an arbitrary connection on the giv en v ector bundle. This is not the case for the equiv arian t Chern c haracter deﬁned in section 6.2.2. This is exp ected, since the equiv ariant Chern c haracter w as constructed in terms of homomorphisms b et w een K-theory and cohomology , without an y reference to an y geometric description. Ho w ev er, when tensored ov er C , the equiv arian t Chern character admits a more geometric description. W e will now explain this construction, referring the reader to [25] for the tec hnical details. Consider a complex G-bundle E ov er X equip ed with a G-in v arian t hermitean metric and a G- in v arian t metric connection ∇ E . One can then deﬁne a closed G-in v arian t diﬀeren tial form c h(E) ∈ Ω ∗ (X; C ) G in the usual w a y b y the Chern-W eil construction c h(E) := T r  exp( − F E / 2 π i )  where F E is the curv ature of the connection ∇ E . It represents a cohomology class  c h(E)  ∈ H ∗ (X; C ) G in the ﬁxed point subring of the action of G as automorphisms of H ∗ (X; C ). By using the deﬁnition of the homomorphisms (6.2.8), with Q substituted b y C and H = e , one can establish the equalit y  c h(E)  = c h e X  [E]  . Let C < G b e a cyclic subgroup, and deﬁne the cohomology class  c h( g , E)  ∈ H ∗  X C ; C  Z G ( C ) ' H ∗  X C / Z G (C) ; C  ' H  Ω ∗ (X C ; C ) Z G (C) , d  represen ted b y c h( g , E) := T r  γ ( g ) exp( − F E C / 2 π i )  165 String theory on orbifolds where g is a generator of C, F E C is the restriction of the in v arian t curv ature t w o-form F E to the ﬁxed p oin t subspace X C , and γ is a represen tation of C on the ﬁbres of the restriction bundle E | X C whic h is an N G (C)-bundle o v er X C . The character χ C naturally iden tiﬁes R(C) ⊗ C with the C -v ector space of class functions C → C . By using the splitting (6.3.12) for complex Bredon cohomology , one can then sho w that c h C X  [E]  ( g ) =  c h( g , E)  up to the restriction homomorphism R(C) ⊗ C → C C (G / C) of rings with k ernel the ideal of elemen ts whose c haracters v anish on all generators of C. Using (6.2.9) w e can then deﬁne the map c h C : V ect C G (X) − → Ω even G  X ; C  from complex G-bundles E → X given by c h C (E) = M [ g ] ∈ G ∨ T r  γ ( g ) exp( − F E g / 2 π i )  . (6.3.14) A t the level of equiv arian t K-theory , from Theorem 6.12 it follo ws that this map induces an isomorphism c h C : K ∗ G (X) ⊗ C ≈ − → H G  X ; C ⊗ π −∗ K G ( − )  ∗ (6.3.15) where we ha v e used the splitting (6.3.13). The map (6.3.14) coincides with the equiv- arian t Chern c haracter deﬁned in [9]. 6.3.4 Ramond-Ramond couplings with D-branes W e no w hav e all the necessary ingredients to deﬁne a coupling of the Ramond-Ramond ﬁelds to a D-brane in the orbifold [X / G]. In this section w e will only consider Ramond- Ramond ﬁelds whic h are top ologically trivial, i.e., elemen ts of the diﬀeren tial complex (6.3.10), and use the delo calized cohomology theory ab ov e b y w orking throughout with complex coeﬃcien ts. Moreo v er, w e will only consider electric couplings to D-branes, i.e. w e will not impose selfduality . Under these conditions we can straightforw ardly mak e contact with existing examples in the physics literature and write down their appropriate generalizations. T o this aim, we introduce the bilinear pro duct ∧ G : Ω ∗ G (X; R ) ⊗ Ω ∗ G (X; R ) − → Ω ∗ G (X; R ) 166 String theory on orbifolds deﬁned on ω = L g ∈ G ω g and η = L g ∈ G η g b y ω ∧ G η := M g ∈ G ω g ∧ g η g (6.3.16) where ∧ g = ∧ is the usual exterior pro duct on Ω ∗ (X g ; R ). There is also an integration Z G X : Ω ∗ G (X; R ) − → R . If ω ∈ Ω ∗ G (X; R ) then w e set Z G X ω := 1 | G ∨ | X [ g ] ∈ G ∨ Z X g ω [ g ] . where w e hav e used in the ab ov e construction that for a G-manifold X admitting a G-equiv arian t spin structure, the ﬁxed p oint manifold X g is naturally oriented, for an y g ∈ G [17]. The normalization ensures that R G X ω = R X ω when G acts trivially on X, and ω is “diagonal” in Ω ∗ (X) ⊗ R(G). Supp ose no w that f : W → X is the smo oth immersed worldv olume cycle of a wrapp ed D-brane state (W , E , f ) ∈ D G (X) in the orbifold [X / G], i.e., W is a G- spin c manifold equip ed with a G-bundle E → W and an inv ariant connection ∇ E on E. W e deﬁne the Wess-Zumino p airing WZ : D G (X) × Ω ∗ G (X; C ) − → C b et w een suc h D-branes and Ramond-Ramond ﬁelds as WZ  (W , E , f ) , C  = Z G W ˜ C ∧ G c h C (E) ∧ G R (W , f ) , (6.3.17) where ˜ C = f ∗ C is the pullback along f : W → X of the total Ramond-Ramond ﬁeld C = M [ g ] ∈ G ∨ C [ g ] and the equiv arian t Chern character is given by (6.3.14) with γ giving the action of G on the Chan-Paton factors of the D-brane. The closed w orldv olume form R (W , f ) ∈ Ω even G , cl (W; C ) represen ts a complex Bredon cohomology class which ac- coun ts for gravitational corrections due to curv ature in the spacetime X and dep ends only on the b ordism class of (W , f ). W e refer the reader to [91], where a construction of R (W , f ) can b e found. 167 String theory on orbifolds Mo dulo the curv ature con tribution R (W , f ), the very natural expression (6.3.17) reduces to the usual W ess-Zumino coupling of top ologically trivial Ramond-Ramond ﬁelds to D-branes in the case G = e . But even if a group G 6 = e acts trivially on the brane w orldv olume W (or on the spacetime X), there can still b e additional con tributions to the usual Ramond-Ramond coupling if E is a non-trivial G -bund le . This is the situation, for instance, for fractional D-branes (W , E , f ) ∈ D G frac (X) placed at orbifold singularities. In this case, w e ma y use the isotopical decomp osition (6.3.6) of the Chan-P aton bundle along with (6.3.5). Then the W ess-Zumino pairing (6.3.17) descends to a pairing WZ frac : D G frac (X) × Ω ∗ G (X; C ) − → C with the additiv e sub category of fractional branes at orbifold singularities. Example 6.14. W e will no w “test” our deﬁnition (6.3.17) on the class of examples considered in section 6.3.1. These are ﬂat orbifolds for which there are no non-trivial curv ature con tributions, i.e., R (W , f ) = 1. Let us specialize to the case of cyclic orbifolds having twist group G = Z n with n ≥ d . In this case, as Z n is an ab elian group, one has Z ∨ n = Z n (set wise) and w e can lab el the non-trivial twisted sectors of the orbifold String theory on X b y k = 1 , . . . , n − 1. The un t wisted sector is lab elled b y k = 0. W e take a generator g of Z n whose action on V ' C d is giv en b y g ·  z 1 , . . . , z d  :=  ζ a 1 z 1 , . . . , ζ a d z d  , where ζ = exp(2 π i /n ) and a 1 , . . . , a d are in tegers satisfying a 1 + · · · + a d ≡ 0 mo d n 10 . In this case the action of an y elemen t in Z n has only one ﬁxed point, an orbifold singularit y at the origin (0 , . . . , 0). Hence for an y g 6 = e one has X g ' R p, 1 and the diﬀeren tial complex (6.3.10) of orbifold Ramond-Ramond ﬁelds is giv en b y Ω ∗ Z n (X; R ) = Ω ∗ (X; R ) ⊕  n − 1 M k =1 Ω ∗  R p, 1 ; R   . 10 Both the requirement that the representation V b e complex and the form of the G-action are ph ysical inputs ensuring that the closed string bac kground X preserves a suﬃcien t amount of sup er- symmetry after orbifolding. 168 An equiv arian t Riemann-Ro c h formula Consider no w a fractional D-brane (W , E , f ) ∈ D Z n frac (X) with w orldv olume cycle f (W) ⊂ R p, 1 placed at the orbifold singularity , i.e., f : W → R p, 1 × (0 , . . . , 0) ⊂ X. Let the generator g act on the ﬁbres of the Chan-P aton bundle E → W in the n - dimensional regular representation γ ( g ) ij = ζ i δ ij . The action on w orldvolume fermion ﬁelds is determined b y a lift ˆ Z n acting on the spinor bundle S → W . Then the pairing (6.3.17) con tains the follo wing terms. First of all, w e ha ve the coupling of the un t wisted Ramond-Ramond ﬁelds to W given by 1 n Z W ˜ C ∧ T r  exp( − F E / 2 π i )  , whic h is just the usual W ess-Zumino coupling and hence the unt wisted Ramond- Ramond charge of the brane is 1. Then there are the con tributions from the twisted sectors, whic h b y recalling (6.3.5) are giv en b y the expression 1 n Z W n − 1 X k =1 ˜ C k ∧ T r  γ ( g k ) exp( − F E / 2 π i )  where g k is an elemen t of Z n of order k . Since γ ( g k ) ii = ζ ik , the brane asso ciated with the i -th irreducible represen tation of Z n has c harge ζ ik /n with resp ect to the top Ramond-Ramond ﬁeld in the k -th t wisted sector. F or d = 2 and d = 3, the form of these couplings agrees with those computed in [37]. 6.4 An equiv arian t Riemann-Ro c h form ula Let X , W b e smo oth compact G-manifolds, and f : W → X a smo oth prop er G-map. If we w ant to make sense of the equations of motion for the Ramond-Ramond ﬁeld C , whic h is a quan tit y deﬁned on the spacetime X, then w e need to pushforward classes deﬁned on the brane w orldv olume W to classes deﬁned on the spacetime. This will enable the construction of Ramond-Ramond currents in a later section induced by the background and D-branes whic h app ear as source terms in the Ramond-Ramond ﬁeld equations. As we hav e seen in chapter 3, given a smo oth embedding f : W → X with normal bundle ν → W equiped with a spin c structure, w e ha v e c h  f K ! ( ξ )  = f H !  c h( ξ ) ∪ T o dd( ν ) − 1  (6.4.1) for any class ξ ∈ K ∗ (W), where f ! denote the Gysin homomorphism deﬁned in c hap- ter 3, and T o dd(E) ∈ Ω even cl (W; C ) denotes the T o dd gen us characteristic class of a 169 An equiv arian t Riemann-Ro c h formula hermitean v ector bundle E o v er W, whose Chern-W eil representativ e is T o dd(E) = s det  F E / 2 π i tanh  F E / 2 π i   where F E is the curv ature of a hermitean connection ∇ E on E. The imp ortant asp ect is that the Chern ch aracter do es not commute with the Gysin pushforw ard maps, and the defect in the comm utation relation is precisely the T o dd genus of the bundle ν . This “twisting” b y the bundle ν ov er the D-brane con tributes in a crucial wa y to the Ramond-Ramond curren t in the non-equiv arian t case [29, 71, 75]. W e will now attempt to ﬁnd an equiv ariant v ersion of the Riemann-Ro c h theorem. W e will consider G-equiv ariant em b eddings f : W → X: in this case normal bundle ν is itself a G-bundle. W e assume that ν is K G -orien ted. This requirement is just the F reed-Witten anomaly cancellation condition [45] in this case, generalized to global w orldsheet anomalies for D-branes represen ted b y generic G-equiv ariant K-cycles. It enables, analogously to the non-equiv arian t case, the construction of an equiv arian t Gysin homomorphism f K G ! : K ∗ G ( W ) − → K ∗ G (X) . (6.4.2) W e will show that, under some very sp ecial conditions, one can construct a complex Bredon cohomology class which is analogous to the T o dd gen us and whic h pla ys the role of the equiv arian t comm utativit y defect as ab o ve. F or this we will need the following 11 Lemma 6.15. L et π : E → X b e an e quivariant G -ve ctor bund le, wher e G is a ﬁnite gr oup acting pr op erly. Then for any g ∈ G the map π | E g : E g → X g is ve ctor bund le pr oje ction. Pr o of. Consider a p oint w ∈ E. Then dπ : T w E → T π ( w ) X is surjective, since it is the pro jection map of a ﬁb er bundle. If w ∈ E g , then π ( w ) b elongs to X g , and the elemen t g acts linearly on T π ( w ) X. If v ∈ T π ( w ) X is a g -ﬁxed v ector, one can then ﬁnd a g -ﬁxed preimage in T w E by taking an y preimage and av eraging with resp ect to the action of g . By using that (TX) g = TX g , w e ha ve that the map T w E g → T π ( w ) X g is surjectiv e, hence the restricted map π E g is a prop er surjection. An y prop er submersion is automatically a ﬁbre bundle, and the linear structure on the ﬁb ers of E g → X g is inherited b y that on the ﬁb ers of E → X. 11 W e are grateful to U.Bunk e for suggesting this result to us. 170 An equiv arian t Riemann-Ro c h formula Notice that the ab o v e lemma strongly use the prop erties that the group G is ﬁnite and acting prop erly . T o see what can happ en when these conditions are not satisﬁed 12 , let X = R and G = R + the group of p ositive reals under m ultiplication. Consider the G-bundle X × V → X given by pro jection onto the ﬁrst factor, where V is a ﬁnite-dimensional real v ector space and the G-action is g · ( x, v ) =  x , g x v  for all g ∈ G. F or an y g 6 = 1, (X × V) g is not a ﬁbre bundle ov er X g = X, as the G-in v arian t ﬁbre space o ver x = 0 is V while it is the n ull v ector ov er any other p oint. In particular, (X × V) g is not ev en a manifold. Consider no w tw o G-manifolds W and X equip ed with G-inv ariant spin c structure, and a prop er G-equiv arian t embedding f : W → X. F or any g ∈ G, the vector bundle ν g = ν (W; f ) g → W g is the normal bundle ν g = f ∗ | W g (T X g ) ⊕ T W g o v er the immersion f | W g : W g → X g . Recall that this is a Z G ( g )-bundle. W e will supp ose that ν g is equip ed with a Z G ( g )-in v arian t spin c structure. As w e ha v e seen in section 6.3.2, equiv arian t K-theory “delo calizes” when tensored o v er R , thanks to the equiv arian t Chern character. W e will sho w in the following that when working ov er C the delo calization of equiv arian t K-theory is compatible, in a certain sense, with the complex equiv arian t Chern c haracter deﬁned in (6.3.14). Let E be a G-vector bundle ov er a X. F or any g ∈ G, the restriction E g of E to the ﬁxed p oin t subspace X g giv es a Z G ( g )-in v arian t vector bundle carrying a represen tation γ of < g > on the ﬁb ers, where < g > is the cyclic group generated b y g . The decomp osition of the representation γ in terms of irreducible representations γ α of < g > induces the homomorphism K ∗ G (X) → K ∗ (X g ) ⊗ R( < g > ) giv en on v ector bundles b y E → E g ' M γ α E α ⊗ γ α where E α := Hom < g > (E , 1 1 α ), where 1 1 α = X g × V α , with V α carrying the irreducible represen tation γ α . W e can then deﬁne the morphism K ∗ (X g ) ⊗ R( < g > ) → K ∗ (X g ) ⊗ C b y tracing o v er the second factor, i.e. [E g ] → X α [E α ]T r( γ α ( g )) 12 W e are grateful to J.Figueroa-O’F arrill for suggesting this example to us 171 An equiv arian t Riemann-Ro c h formula Denoting φ g the comp osition of the ab o ve morphism, we can deﬁne φ := ⊕ [ g ] ∈ G ∨ φ g Recalling that E g is a Z G ( g )-in v arian t bundle, w e ha v e the isomorphism [9] φ : K ∗ G (X) ⊗ C ' M [ g ] ∈ G ∨ (K ∗ (X g ) ⊗ C ) Z G ( g ) (6.4.3) By using that [25] T r  γ ( g ) exp( − F E g / 2 π i )  = X α T r  exp( − F E α / 2 π i )  T r ( γ α ( g )) where F E α is the restriction of the curv ature form F E g to the subbundle E α , we ha ve that the complex equiv ariant Chern c haracter (6.3.14) coincides on the comp onen ts of the decomp osition (6.4.3) with the ordinary Chern c haracter. Supp ose no w that the equiv ariant Thom class Thom G ( ν ) ∈ K ∗ G , cpt ( ν ) can b e de- comp osed according to the splitting (6.4.3) in suc h a wa y that the comp onen t in an y subgroup Thom  ν g  ∈  K ∗ cpt ( ν g ) ⊗ C  Z G ( g ) coincides with the (ordinary) Thom class of the vector bundle ν g → W g . Under these conditions, the equiv ariant Gysin homomorphism (6.4.2) decomp oses according to the splitting f K G ! = M [ g ] ∈ G ∨ f K g where f K g is the K-theoretic Gysin homomorphism asso ciated to the smo oth map f   W g : W g − → X g . Deﬁne the c haracteristic class T o dd G b y T o dd G ( ν ) := M [ g ] ∈ G ∨ T o dd  ν g  (6.4.4) This class deﬁnes an elemen t of the even degree complex Bredon cohomology of the brane w orldvolume W. Under the conditions spelled out ab ov e, w e can no w use the equiv arian t Chern c haracter (6.3.15) and the usual Riemann-Roch theorem for each pair (W g , X g ) to pro v e the iden tit y f H G !  c h C ( ξ ) ∪ G T o dd G ( ν ) − 1  = c h C  f K G ! ( ξ )  (6.4.5) 172 Orbifold diﬀeren tial K-theory and ﬂux quan tization for an y class ξ ∈ K ∗ G (W) ⊗ C , as all quan tities in v olv ed in the expression (6.4.5) are compatible with the G-equiv arian t decomp ositions given ab ov e. This equiv arian t Riemann-Ro c h form ula will b e imp ortan t when we will argue the ﬂux quan tization for Ramond-Ramond ﬁelds on orbifolds. 6.5 Orbifold diﬀeren tial K-theory and ﬂux quan ti- zation In this section w e will prop ose an extension of diﬀerential K-theory as deﬁned in sec- tion 5.4 to incorp orate the case of a G-manifold. These are the groups needed to extend the analysis of the previous section to topologically non-trivial, real-v alued Ramond-Ramond ﬁelds. While we do not ha ve formal argumen ts that this is a prop er deﬁnition of an equiv arian t diﬀeren tial cohomology theory , we will see that it matc hes exactly with exp ectations from String theory on orbifolds and also has the correct limiting prop erties. F or this reason w e dub the theory that w e deﬁne ‘orbifold’ diﬀer- en tial K-theory , defering the terminology ‘equiv ariant’ to a more thorough treatment of our mo del. W e will use the Riemann-Ro c h formula dev elop ed in the last section to argue that the Dirac quantization condition for Ramond-Ramond ﬁeldstrengths on a go o d orbifold is dictated b y equiv ariant K-theory via the equiv ariant Chern c harac- ter, suggesting that our orbifold diﬀeren tial K-theory is the correct guess for the space gauge equiv alen t classes of Ramond-Ramond ﬁelds. Finally , we will study the classi- ﬁcation of Ramond-Ramond ﬁelds on the linear orbifolds considered in section 6.3.1, and giv e a deﬁnition for the group of ﬂat Ramond-Ramond ﬁelds on more general go o d orbifolds. 6.5.1 Orbifold diﬀeren tial K-groups As mentioned ab ov e, w e will generalize the deﬁnition of diﬀerential K-theory giv en in section 5.4 to accomo date the action of a ﬁnite group. First, let us recall some further basic facts about equiv arian t K-theory . Similarly to ordinary K-theory , a mo del for the classifying space of the functor K 0 G is giv en b y the G-algebra of F redholm op erators F red G acting on a separable Hilb ert space which is a represen tation space for G in which eac h irreducible represen tation o ccurs with inﬁnite m ultiplicit y [4]. Then there is an isomorphism K 0 G (X) ' [X , F red G ] G 173 Orbifold diﬀeren tial K-theory and ﬂux quan tization where [ − , − ] G denotes the set of equiv alence classes of G-homotopic maps, and the isomorphism is giv en b y taking the index bundle. There is also a universal space V ect n G , equip ed with a universal G-bundle e E n G , such that [X , V ect n G ] G corresp onds to the set of isomorphism classes of n -dimensional G- v ector bundles o ver X [67]. These spaces are constructed as follows. Let E G be the category whose ob jects are the elemen ts of G and with exactly one morphism b et ween eac h pair of ob jects. The geometric realization (or nerve) of the set of isomorphism classes in E G is, as a simplicial space, the total space of the classifying principal G-bundle EG. With V ect n (pt) the category of n -dimensional complex v ector spaces V ' C n , the univ ersal space V ect n G is deﬁned to b e the geometric realization of the functor category [ E G , V ect n (pt)]. The univ ersal n -dimensional G-vector bundle e E n G is then deﬁned as e E n G = g V ect n G × GL( n, C ) C n − → V ect n G , (6.5.1) where g V ect n G is the geometric realization of the functor category deﬁned as abov e but with V ect n (pt) replaced with the category consisting of ob jects V in V ect n (pt) together with an orien ted basis of V. W e assume suﬃcien t regularity conditions on the inﬁnite-dimensional spaces F red G and e E n G . Since F red G and the group completion V ect G are b oth classifying spaces for equiv arian t K-theory , they are G-homotopic and we can thereby choose a co cycle u G ∈ Z even G (F red G ; R ) represen ting the equiv ariant Chern character of the universal G-bundle (6.5.1). Gen- erally , the group Z even G (X; R ) is the subgroup of closed co cycles in the complex C even G (X; R ) := M [ g ] ∈ G ∨ C even  X g ; R  Z G ( g ) (6.5.2) whic h, by the results of section 6.3.2, is a co chain mo del for the Bredon cohomology group H even − 1 G (X; R ⊗ R( − )). The equiv ariant Chern character is understo o d to b e comp osed with the delo calizing isomorphism of section 6.3.2. Since it is a natural homomorphism, for an y G-bundle E → X classiﬁed b y a G-map f : X → F red G one has c h X  [E]  =  f ∗ u G  . Deﬁnition 6.16. The orbifold diﬀer ential K-the ory ˇ K 0 G (X) of the (global) orbifold [X / G] is the group of triples ( c, h, ω ), where c : X → F red G is a G-map, ω is an elemen t in Ω even G , cl (X; R ), and h is an elemen t in C even − 1 G (X; R ) satisfying δ h = ω − c ∗ u G . (6.5.3) 174 Orbifold diﬀeren tial K-theory and ﬂux quan tization Tw o triples ( c 0 , h 0 , ω 0 ) and ( c 1 , h 1 , ω 1 ) are said to b e e quivalent if there exists a triple ( c, h, ω ) on X × [0 , 1], with the group G acting trivially on the in terv al [0 , 1] and with ω constan t along [0 , 1], such that ( c, h, ω )   t =0 = ( c 0 , h 0 , ω 0 ) and ( c, h, ω )   t =1 = ( c 1 , h 1 , ω 1 ) . In (6.5.3) the closed orbifold diﬀerential form ω is regarded as an orbifold co c hain in the complex (6.5.2) by applying the de Rham map comp onen t wise on the ﬁxed p oint submanifolds X g , g ∈ G. The higher orbifold diﬀeren tial K-theory groups ˇ K − n G (X) are deﬁned analogously to those of section 5.4. T o conﬁrm that this is a suitable extension of the ordinary diﬀerential K-theory of X, w e should show that the orbifold diﬀeren tial K-theory groups ﬁt into exact sequences whic h reduce to those giv en b y (5.4.7) when G is tak en to b e the trivial group. F or this, w e deﬁne the group A 0 K G (X) :=  ( ξ , ω ) ∈ K 0 G (X) × Ω even G , cl (X; R )   c h X ( ξ ) = [ ω ] G − dR  . Theorem 6.17. The orbifold diﬀer ential K-the ory gr oup ˇ K 0 G (X) satisﬁes the exact se quenc e 0 − → H even − 1 G  X ; R ⊗ R ( − )  c h X  K − 1 G (X)  − → ˇ K 0 G (X) − → A 0 K G (X) − → 0 (6.5.4) Pr o of. Consider the subgroup of H even − 1 G (X; R ⊗ R( − )) deﬁned as the image of the equiv arian t K-theory group K − 1 G (X) under the Chern c haracter c h X . It consists of Bredon cohomology classes of the form [ ˜ c ∗ u − 1 G ], where ˜ c : X → ΩF red G . There is a surjectiv e map f : ˇ K 0 G (X) − → A 0 K G (X)  ( c, h, ω )  7− →  [ c ] , ω  whic h is a well-deﬁned homomorphism, i.e., it do es not dep end on the chosen rep- resen tativ e of the orbifold diﬀeren tial K-theory class. By deﬁnition, the kernel of f consists of triples of the form ( c , h, 0), where c is G-homotopic to the constant (iden tit y) map. W e also deﬁne the map g : H even − 1 G  X ; R ⊗ R( − )  − → ˇ K 0 G (X) [ h ] 7− →  ( c , h, 0)  , 175 Orbifold diﬀeren tial K-theory and ﬂux quan tization whic h is a well-deﬁned homomorphism because the class [( c , h, 0)] dep ends only on the Bredon cohomology class [ h ] ∈ H even − 1 G (X; R ⊗ R( − )). Then b y construction one has im( g ) = ker( f ). The homomorphism g is not injective. T o determine the k ernel of g , we use the fact that the zero elemen t in ˇ K 0 G (X) can b e represen ted as  ( c , 0 , 0)  =  ( c , π ∗ F ∗ u G + d G σ, 0)  with F : X × S 1 → F red G and σ ∈ Ω even − 2 G (X; R ). T o the map F w e can associate a map ˜ c : X → ΩF red G suc h that F = ev ◦ ( ˜ c × id S 1 ). This follo ws from the isomorphism K − 1 G (X) ' ker  i ∗ : K 0 G (X × S 1 ) → K 0 G (X)  where i is the inclusion i : X  → X × pt ⊂ X × S 1 . No w use the fact that at the lev el of (real) Bredon cohomology one has an equalit y π ∗  ˜ c × id S 1  ∗ = ˜ c ∗ Π ∗ since the pro jection homomorphisms π ∗ and Π ∗ b oth correspond to integration (slant pro duct) along the S 1 ﬁbre. Then one has the identit y  π ∗ F ∗ u G  =  π ∗ (˜ c × id S 1 ) ∗ ev ∗ u G  =  ˜ c ∗ Π ∗ ev ∗ u G  =  ˜ c ∗ u − 1 G  . It follows that ker( g ) is exactly the group c h X (K − 1 G (X)), and putting ev erything to- gether w e arriv e at (6.5.4). The torus H even − 1 G  X; R ⊗ R( − )  / c h X  K − 1 G (X)  ' K − 1 G (X) ⊗ R / Z is called the group of top olo gic al ly trivial ﬂat ﬁelds . Consider no w the c haracteristic class map f cc : ˇ K 0 G (X) − → K 0 G (X)  ( c, h, ω )  7− → [ c ] and the map g cc : Ω even − 1 G (X; R ) − → ˇ K 0 G (X) h 7− →  ( c , h, d G h )  . Let Ω even − 1 K G (X; R ) b e the subgroup of elements in Ω even − 1 G , cl (X; R ) whose Bredon coho- mology class lies in c h X (K − 1 G (X)). Then b y using argumen ts similar to those used in arriving at the sequence (6.5.4), one ﬁnds the 176 Orbifold diﬀeren tial K-theory and ﬂux quan tization Corollary 6.18 ( Characteristic class exact sequence). In analo gy with the or di- nary c ase, the orbifold diﬀer ential K-the ory gr oup ˇ K 0 G (X) satisﬁes the exact se quenc e 0 − → Ω even − 1 G (X; R ) Ω even − 1 K G (X; R ) − → ˇ K 0 G (X) − → K 0 G (X) − → 0 . (6.5.5) The quotient space of orbifold diﬀeren tial forms in the exact sequence (6.5.5) is called the group of top ologically trivial ﬁelds. Finally , consider the ﬁeld strength map f fs : ˇ K 0 G (X) − → Ω even G , cl (X; R )  ( c, h, ω )  7− → ω . (6.5.6) The k ernel of the homomorphism f fs is the group which classiﬁes the ﬂat ﬁelds (which are not necessarily top ologically trivial) and is denoted K − 1 G (X; R / Z ). This group will b e describ ed in more detail in the next section, where w e shall also conjecture an essen tially purely algebraic deﬁnition of K − 1 G (X; R / Z ) whic h explains the notation. In an y case, w e ha v e the Corollary 6.19 ( Field strength exact sequence). The orbifold diﬀer ential K- the ory gr oup ˇ K 0 G (X) satisﬁes the exact se quenc e 0 − → K − 1 G (X; R / Z ) − → ˇ K 0 G (X) − → Ω even K G (X; R ) − → 0 (6.5.7) Higher orbifold diﬀeren tial K-theory groups satisfy analogous exact sequences, with the appropriate degree shifts throughout. It is clear from our deﬁnition that one recov ers the ordinary diﬀerential K-theory groups in the case of the trivial group G = e , and in this sense our orbifold diﬀerential K-theory may b e regarded as its equiv arian t generalization. A t this p oint w e hasten to add that, although our groups are w ell-deﬁned and satisfy desired prop erties whic h are useful for physical applica- tions such as the v arious exact sequences ab ov e, w e hav e not show ed that our orbifold theory generalizes al l the prop erties of an ordinary diﬀerential cohomology theory . F or instance, it w ould b e interesting to deﬁne a ring structure and an integration on ˇ K ∗ G (X). W e ha v e not dev elop ed these constructions in this thesis. 177 Orbifold diﬀeren tial K-theory and ﬂux quan tization 6.5.2 Flux quan tization of orbifold Ramond-Ramond ﬁelds In this section we will argue the ﬂux quantization condition for Ramond-Ramond ﬁeldstrengths on orbifolds of Type I I sup erstring theory with v anishing H -ﬂux, show- ing that it is dictated b y equiv ariant K-theory via the complex equiv ariant Chern c haracter. Essentially , w e will closely follo w the Mo ore-Witten argumen t discussed in section 5.3. Supp ose that our spacetime X is a non-compact G-manifold. Suppose further that there are D-branes present in Type I I sup erstring theory on X / G. Their Ramond- Ramond c harges are classiﬁed b y the equiv arian t K-theory K i G , cpt (X) with compact supp ort, where i = 0 in Type I IB theory and i = − 1 in Type I IA theory . W e require that the brane b e a source for the equation of motion for the total Ramond-Ramond ﬁeld strength ω . This means that it creates a Ramond-Ramond curren t J . If w e require that the w orldv olume W b e compact in equiv arian t K-cycles (W , E , f ) ∈ D G (X), then J is supp orted in the interior ˚ X of X. Let X ∞ b e the “b oundary of X at inﬁnity”, which we assume is preserved by the action of G. Then K ∗ G , cpt (X) ' K ∗ G (X , X ∞ ). Since J is trivialized b y ω in ˚ X, the D-brane charge liv es in the k ernel of the natural forgetful homomorphism f ∗ : K ∗ G , cpt (X) − → K ∗ G (X) (6.5.8) induced b y the inclusion (X , ∅ )  → (X , X ∞ ). W e denote b y i : X ∞  → X the canonical inclusion. The long exact sequence for the pair (X , X ∞ ) in equiv ariant K-theory truncates, b y Bott p erio dicit y , to the six-term exact sequence K − 1 G (X ∞ ) / / K 0 G (X , X ∞ ) f 0 / / K 0 G (X) i ∗   K − 1 G (X) i ∗ O O K − 1 G (X , X ∞ ) f − 1 o o K 0 G (X ∞ ) . o o It follo ws that the c harge groups are giv en b y k er  f 0  ' K − 1 G (X ∞ ) i ∗  K − 1 G (X)  and k er  f − 1  ' K 0 G (X ∞ ) i ∗  K 0 G (X)  . This form ula means that the group of Type IIB (resp. Type IIA) brane charges is mea- sured b y the group K − 1 G (X ∞ ) (resp. K 0 G (X ∞ )) of “orbifold Ramond-Ramond ﬂuxes at inﬁnit y” which cannot b e extended to all of spacetime X. W e may then interpret, for arbitrary spacetimes X, the group K − 1 G (X) (resp. K 0 G (X)) as the group classify- ing Ramond-Ramond ﬁelds in the orbifold X / G in absence of branes in T yp e I IB (resp. T yp e I IA) String theory . 178 Orbifold diﬀeren tial K-theory and ﬂux quan tization The Ramond-Ramond current can b e describ ed explicitly by using the delo caliza- tion of Bredon cohomology o ver C . The W ess-Zumino pairing (6.3.17) b etw een a top o- logically trivial, complex Ramond-Ramond p otential and a D-brane represented by an equiv arian t K-cycle (W , E , f ) ∈ D G (X) con tributes a source term to the Ramond- Ramond equations of motion, whic h is the class  Q (W , E , f )  ∈ H even G  X ; C ⊗ R( − )  represen ted b y the pushforw ard Q (W , E , f ) = f H G !  c h C (E) ∧ G R (W , f )  . W e now use the Riemann-Ro ch form ula (6.4.5) and the fact that f H G ! ◦ f ∗ = id H ∗ G (X; C ( − )) . Using the explicit expression for the curv ature form and the deﬁnition for Λ G (X) giv en in [91], w e can then rewrite this class as Q (W , E , f ) = c h C  f K G ! (E)  ∧ G p T o dd G (T X ) ∧ G Λ G (X) . (6.5.9) This is the complex Bredon cohomology class of the Ramond-Ramond current J created b y the D-brane (W , E , f ). In the case G = e , the expression (6.5.9) reduces to the standard class of the curren t for Ramond-Ramond ﬁelds in T yp e I I sup erstring theory on X [29, 71, 74, 75]. W e can then formally conclude, in analogy with the non-equiv arian t case, that the complex Bredon cohomology class asso ciated to a class ξ ∈ K ∗ G (X) ⊗ C represen ting a Ramond-Ramond ﬁeld is assigned b y the equiv ariant Chern character, and that the total Ramond-Ramond ﬁeldstrength ω asso ciated to ξ satisﬁes [ ω ( ξ )] 2 π [ p T o dd G (T X ) ∧ G Λ G (X)] = [c h C ( ξ )] . (6.5.10) The ab ov e expression the suggests that the orbifold diﬀerential K-theory developed in the previous section is a suitable candidate to describ e Ramond-Ramond ﬁelds with Dirac quan tization condition, at least in the complex case. Moreo ver, we should stress that this analysis of the delocalized theory assumes the strong conditions spelled out in section 6.4, which require a deep geometrical compatibilit y of the equiv ariant K-cycle (W , E , f ) with the orbifold structure of [X / G]. The example of the linear orbifolds considered in section 6.3.1 is simple enough to satisfy these conditions. It w ould b e v ery in teresting to ﬁnd a geometrically non-trivial explicit example to test these requirements on. How ev er, the linear orbifolds case is v ery useful to understand 179 Orbifold diﬀeren tial K-theory and ﬂux quan tization certain asp ects of the orbifold diﬀeren tial K-theory groups. Since the C -linear G-mo dule X is equiv arian tly contractible, one has H odd G (X; R ⊗ R( − )) = 0 and K 0 G (X) = R(G). F rom Theorem 6.17 it then follows that ˇ K 0 G (X) ' A 0 K G (X) '  ( γ , ω ) ∈ R(G) × Ω even G , cl (X; R )   c h pt ( γ ) = [ ω ] G − dR  . Since the equiv ariant Chern c haracter c h G /H : R(H) → R(H) for H ≤ G is the iden tit y map, the set wise ﬁbre pro duct truncates to the lattice of quantized orbifold diﬀeren tial forms and one has ˇ K 0 G (V) = Ω even K G (V; R ) . (6.5.11) This is the group of Type IIA Ramond-Ramond ﬁeldstrengths on X. It naturally con tains those ﬁelds whic h trivialize the Ramond-Ramond currents sourced by the stable fractional D0-branes of the Type I IA theory , corresponding to characteristic classes [ c ] in the represen tation ring R(G). This can b e explicitly described as an extension of the equiv ariant K-theory of X b y the group of top ologically trivial Ramond-Ramond ﬁelds C of o dd degree, as implied b y Corollary 6.18. Since X is connected and G-contractible, one has Ω 0 G , cl (X; R ) = R ⊗ R(G) and the group (6.5.11) has a natural splitting ˇ K 0 G (X) = R(G) ⊕  d M k =1 Ω 2 k G , cl (X; R )  . (6.5.12) An y closed orbifold form ω on X of p ositiv e degree is exact, ω = d G C , with the gauge in v ariance C 7→ C + d G ξ . It follo ws that there is a natural map d M k =1 Ω 2 k G , cl (X; R ) − → Ω even − 1 G (X; R ) Ω even − 1 K G (X; R ) whic h asso ciates to the ﬁeld strength ω the corresp onding globally well-deﬁned Ramond- Ramond p oten tial C . On the other hand, the orbifold diﬀeren tial K-theory group ˇ K − 1 G (X) of T yp e I IB Ramond-Ramond ﬁelds on X can b e computed by using the c haracteristic class exact sequence (6.5.5) with degree shifted b y − 1. Using K − 1 G (X) = 0, one ﬁnds ˇ K − 1 G (X) = Ω even G (X; R ) Ω even K G (X; R ) . (6.5.13) 180 Orbifold diﬀeren tial K-theory and ﬂux quan tization This result reﬂects the fact that the T yp e I IB theory has no stable fractional D0- branes. Hence there is no extension and the Ramond-Ramond ﬁelds are induced solely by the closed string backgroun d. Their ﬁeld strengths ω = d G C are determined en tirely by the p oten tials C , whic h are globally deﬁned diﬀeren tial forms of even degree. Setting X = pt, we obtain the orbifold diﬀeren tial K-theory of the p oint, whic h are giv en b y ˇ K 0 G (pt) ' R(G) and ˇ K − 1 G (pt) ' R(G) ⊗ R / Z F or G = e , these groups reduce to the usual diﬀeren tial K-teory groups of the p oin t. In the previous section w e deﬁned the group of ﬂat Ramond-Ramond ﬁelds on a general orbifold [X / G] as the subgroup of orbifold diﬀeren tial K-theory with v anishing curv ature. In the following we will conjecture a very natural algebraic deﬁnition of these groups whic h ties them somewhat more directly to equiv ariant K-theory groups. T o motiv ate this conjecture, w e ﬁrst compute the groups K ∗ G (X; R / Z ) for the linear orbifolds ab ov e, wherein the asso ciated diﬀerential K-theory groups were determined explicitly . Using the ﬁeld strength exact sequence (6.5.7), b y deﬁnition one has K − 1 G (X; R / Z ) ' k er  f fs : ˇ K 0 G (X) → Ω even K G (X; R )  whic h from the natural isomorphism (6.5.11) trivially giv es K − 1 G (X; R / Z ) = 0 . (6.5.14) Similarly , using K − 1 G (X) = 0 one has K 0 G (X; R / Z ) ' k er  f fs : ˇ K − 1 G (X) → Ω odd G , cl (X; R )  . Using the natural isomorphism (6.5.13), the ﬁeld strength map is f fs  [ C ]  = d G C for C ∈ Ω even G (X; R ) , giving K 0 G (X; R / Z ) ' Ω even G , cl (X; R ) Ω even K G (X; R ) . Similarly to (6.5.12), there is a natural splitting of the vector space of closed orbifold diﬀeren tial forms giv en b y Ω even G , cl (X; R ) =  R(G) ⊗ R  ⊕  d M k =1 Ω 2 k G , cl (X; R )  181 Orbifold diﬀeren tial K-theory and ﬂux quan tization and w e arriv e ﬁnally at K 0 G (X; R / Z ) = R(G) ⊗ R / Z . (6.5.15) These results of course simply follow from the fact that X is G-con tractible, so that every d G -closed Ramond-Ramond ﬁeld is trivial, except in degree zero where the gauge equiv alence classes are naturally parametrized by the twisted sectors of the String theory in (6.5.15). Note that b oth groups of ﬂat ﬁelds (6.5.14) and (6.5.15) are unc hanged b y (equiv arian t) con traction of the G-mo dule X to a p oin t, as an analogous (but simpler) calculation shows. This suggests that the groups K ∗ G (X; R / Z ) hav e at least some G-homotop y inv ariance prop erties, unlik e the diﬀerential K G -theory groups. This motiv ates the following conjectural algebraic framework for describing these groups. W e will prop ose that the group K ∗ G (X; R / Z ) is an extension of the torus of top olog- ically trivial ﬂat orbifold Ramond-Ramond ﬁelds b y the torsion elemen ts in K ∗ +1 G (X), as they ha ve v anishing image under the equiv ariant Chern c haracter c h X . The result- ing group ma y b e called the “equiv ariant K-theory with co eﬃcien ts in R / Z ”. The short exact sequence of co eﬃcien t groups 0 − → Z − → R − → R / Z − → 0 induces a long exact sequence of equiv arian t K-theory groups whic h, by Bott p erio d- icit y , truncates to the six-term exact sequence K 0 G (X) / / K 0 G (X; R ) / / K 0 G (X; R / Z ) β   K − 1 G (X; R / Z ) β O O K − 1 G (X; R ) o o K − 1 G (X) . o o (6.5.16) The connecting homomorphism β is a suitable v ariant of the usual Bo ckstein homo- morphism. W e assume that the equiv arian t K-theory with real co eﬃcients is deﬁned simply b y the Z 2 -graded ring K ∗ G (X; R ) = K ∗ G (X) ⊗ R ' H ∗ G  X ; R ⊗ R( − )  , where w e ha v e used Theorem 6.12. The maps to real K-theory in (6.5.16) may then b e iden tiﬁed with the equiv arian t Chern c haracter c h X , whose image is a full lattice in the Bredon cohomology group H ∗ G (X; R ⊗ R( − )). Then the abelian group K ∗ G (X; R / Z ) sits in the exact sequence 0 − → K ∗ G (X) ⊗ R / Z − → K ∗ G (X; R / Z ) β − → T or  K ∗ +1 G (X)  − → 0 . (6.5.17) 182 K-homology and ﬂat ﬁelds in T yp e I I String theory When G = e , (6.5.16) is the usual Bockstein exact sequence for K-theory . In this case, an explicit geometric realization of the groups K ∗ (X; R / Z ) in terms of bundles with connection has b een given by Lott [64]. Moreo ver, in [54] a geometric construc- tion of the map K − 1 (X; R / Z ) → ˇ K 0 (X) in the ﬁeld strength exact sequence is given. Unfortunately , no suc h geometrical description is immediately av ailable for our equiv- arian t diﬀerential K-theory , due to the lack of a Chern-W eil theory for the homotopy theoretic equiv arian t Chern character. Our conjectural deﬁnition (6.5.17) is satisﬁed b y the linear orbifold groups (6.5.14) and (6.5.15). In [32] a v ery diﬀerent deﬁnition of the groups K ∗ G (X; R / Z ) is given, b y deﬁning b oth equiv ariant K-theory and cohomology using the Borel construction of Exam- ple 6.4. Then the Bo ckstein exact sequence (6.5.16) is written for the ordinary K- theory groups of the homotopy quotien t X G = EG × G X. While these groups reduce, lik e ours, to the usual K-theory groups of ﬂat ﬁelds when G = e , they do not ob ey the exact sequence (6.5.17). The reason is that the equiv ariant Chern character used is not an isomorphism ov er the reals, as explained in section 6.2. Moreov er, an asso- ciated diﬀeren tial K-theory construction w ould directly inv olve diﬀeren tial forms on the inﬁnite-dimensional space X G whic h is only homotopic to the ﬁnite-dimensional CW-complex X / G. The physical in terpretation of suc h ﬁelds is not clear. Ev en in the simple case of the linear orbifolds X studied abov e, this description predicts an inﬁ- nite set of equiv arian t ﬂuxes of arbitrarily high dimension on the inﬁnite-dimensional classifying space B G, and one must p erform some non-canonical quotients in order to try to isolate the physical ﬂuxes. In con trast, with our constructions the relation b et w een orbifold ﬂux groups and Bredon cohomology is muc h more natural, and it in v olv es Ramond-Ramond ﬁelds deﬁned on submanifolds of the co v ering space X. 6.6 K-homology and ﬂat ﬁelds in T yp e I I String theory In this ﬁnal section, w e will brieﬂy commen t on ho w the ﬂat Ramond-Ramond ﬁelds can in principle couple to the wrapp ed D-branes deﬁned in section 4.8. The analysis will b e restricted to t yp e I IA String theory . As w e ha ve seen in the previous sections, the Ramond-Ramond ﬁelds in ordinary t yp e I IA String theory are describ ed b y diﬀerential K theory . Hence, the group of ﬂat Ramond-Ramond ﬁelds is giv en by K − 1 (X; R / Z ). Consider the short exact sequence 183 K-homology and ﬂat ﬁelds in T yp e I I String theory of co eﬃcien t groups giv en b y 1 − → Z − → R − → R / Z − → 1 , It induces the corresp onding long exact sequence · · · − → K i (X) − → K i (X) ⊗ R − → K i (X; R / Z ) − → K i +1 (X) − → · · · . By truncating the ab o v e long exact sequence, w e ha v e 0 → K − 1 (X) ⊗ R / Z − → K − 1 (X; R / Z ) β − → T or(K 0 (X)) − → 0 (6.6.1) where β is the Bockstein homomorphism. Thus the identit y component of the circle co eﬃcien t K-theory group is the torus K − 1 (X; R / Z ). Supp ose now that K − 1 (X) is pure torsion. In this case, K − 1 (X; R / Z ) ' T or(K 0 (X)) , and the corresp onding ﬂat Ramond-Ramond ﬁelds can b e represented b y virtual v ector bundles o v er X. A torsion Ramond-Ramond ﬂat ﬁeld ξ ∈ K 0 (X) giv es an additional phase factor to a D-brane in the String theory path integral [32]. Generally , the origin of these phases can b e understo o d from the topological classiﬁcation of the ph ysical coupling b et ween T yp e I D-branes and Ramond-Ramond ﬁelds. Let W b e a compact spin c submanifold of X of dimension p + 1, and let the spacetime manifold X b e a spherical spin c ﬁbration π : X → W such that X / W ' S 9 − p . The group of ﬂat Ramond-Ramond ﬁelds is giv en b y K − 1  X; R / Z  = Hom  K t − 1 (X) , R / Z  ' Hom  K t p − 10 (W) , R / Z  where w e ha v e use the follo wing exact sequence [95, 80] 0 → Ext  K t i − 1 (X) , G  → K i  X; G  → Hom  K t i (X) , G  → 0 for G = R / Z , and the Thom isomorphism. Using Bott p erio dicit y , we hav e ﬁnally K − 1  X; R / Z  ' Hom  K t p +2 (W) , R / Z  . (6.6.2) The K-homology group K t p +2 (W) consists of wrapp ed D-branes [ M , E , φ ] with the prop erties dim M = p + 2 and φ ( M ) ⊂ W. The dimension shift is related to the top ological anomaly in the w orldvolume fermion path integral [74], as the following argumen t seems to suggest. Consider a one-parameter family of p + 1-dimensional brane w orldvolumes speciﬁed b y a circle bundle U → W whose total space U is a p + 2-dimensional submanifold 184 K-homology and ﬂat ﬁelds in T yp e I I String theory of spacetime X with the top ology of W × S 1 . Complex v ector bundles E g of rank n o v er generic ﬁbres U / W ' S 1 are determined b y elements g ∈ U( n ) by the clutching construction. Thus the family of self-adjoin t Atiy ah-Singer op erators / D S 1 E g determined b y (4.9.1) is parametrized by the group U( n ). The anomaly [74] arises as the deter- minan t line bundle of this family , whic h is essen tially deﬁned as the highest exterior p o w er of the kernel of the family . This deﬁnes a non-trivial line bundle on the group U( n ) called the Pfaﬃan line bund le , which has the prop ert y that its lift to Spin( n ) is the trivial line bundle. One can also construct a connection and holonomy of the Pfaﬃan line bundle [45]. The manifold U is wrapp ed b y D-branes in K t p +2 ( U ). One can no w restrict to the subgroup K t p +2 (W) ⊂ K t p +2 ( U ) b y keeping only those D-branes whic h are wrapped on the embedding W  → U by the zero section of U → W . The isomorphism (6.6.2) reﬂects the fact that the topological anomaly could in principle b e cancelled b y coupling D-branes to the Ramond-Ramond ﬁelds through a phase factor. 185 App endix A Linear algebra in F unctor categories In this app endix w e will summarize some notions ab out algebra in functor categories that were used in the main text of the pap er. They generalize the more commonly used concepts for mo dules o v er a ring. F or further details see [92]. Let R b e a comm utative ring, and denote the category of (left) R-mo dules b y R − Mo d. Let Γ be a smal l category , i.e. , its class of ob jects Ob j(Γ) is a set. If C is another category , then one denotes by [Γ , C ] the functor c ate gory of (co v arian t) functors Γ → C . The ob jects of [Γ , C ] are (cov ari- an t) functors φ : Γ → C and a morphism from φ 1 to φ 2 is a natural transformation α : φ 1 → φ 2 b et w een functors. In particular, in the main text w e used the functor category RΓ − Mo d := [Γ , R − Mo d] whose ob jects are called left RΓ -mo dules . If one denotes with Γ op the dual category to Γ, then there is also the functor category Mo d − RΓ := [Γ op , R − Mo d] of contra v arian t functors Γ → R − Mo d, whose ob jects are called right RΓ -mo dules . As an example, let G b e a discrete group regarded as a category with a single ob ject and a morphism for eac h elemen t of G. A cov ariant functor G → R − Mo d is then the same thing as a left mo dule o v er the group ring R[G] of G o v er R. As the name itself suggests, all standard deﬁnitions from the linear algebra of mo dules hav e extensions to this more general setting. F or instance, the notions of submo dule, kernel, c okernel, dir e ct sum, c opr o duct, etc. can b e naturally deﬁned ob ject wise. If M and N are RΓ-mo dules, then Hom RΓ (M , N) is the R-mo dule of all natural transformations M → N. This notation should not be confused with the one used for the set of all morphisms b etw een t w o ob jects in Γ, and usually it is clear from the con text. If M is a righ t RΓ-mo dule and N is a left RΓ-mo dule, then one can deﬁne their categorical tensor pr o duct M ⊗ RΓ N 186 in the follo wing w a y . It is the R-mo dule giv en b y ﬁrst forming the direct sum F = M λ ∈ Ob j(Γ) M( λ ) ⊗ R N( λ ) and then quotien ting F b y the R-submo dule generated b y all relations of the form f ∗ ( m ) ⊗ n − m ⊗ f ∗ ( n ) = 0 , where ( f : λ → ρ ) ∈ Mor(Γ), m ∈ M( ρ ) , n ∈ N( λ ) and f ∗ ( m ) = M( f )( m ) , f ∗ ( n ) = N( f )( n ). This tensor pro duct commutes with copro ducts. If M and N are functors from Γ to the category of v ector spaces o v er a ﬁeld K , then their tensor pro duct is naturally equip ed with the structure of a v ector space ov er K . When Γ is the orbit category Or (G) and R = Z , the tensor pro duct has precise limiting cases. F or an ar- bitrary contra v arian t mo dule M and the constan t cov ariant module N, the categorical pro duct M ⊗ Z Or (G) N is the tensor pro duct of the righ t Z [G]-mo dule M(G /e ) with the constan t left Z [G]-module N(G /e ), M(G /e ) ⊗ Z [G] N(G /e ). On the other hand, if the con tra v arian t mo dule M is constan t and the cov ariant mo dule N is arbitrary , then M ⊗ Z Or (G) N is just N(G / G). 187 App endix B Cliﬀord algebras and Spin manifolds In this app endix w e will brieﬂy recall some basic prop erties of Cliﬀord algebras and Spin manifolds used throughout this thesis. W e direct the reader to [5, 63] for an extensiv e treatmen t of these topics. Let V b e a real v ector space, and let q : V → R b e a quadratic form. Consider the tensor algebra of V F (V) := ∞ X r =0 r O V and denote with I q (V) the t wo-sided ideal in F (V) generated by all elements of the form v ⊗ v + q ( v )1 for v ∈ V . Then the Cliﬀor d algebr a C  (V; q ) associated to V and q is the asso ciativ e algebra with unit deﬁned as C  (V; q ) := F (V) / I q (V) The algebra C  (V; q ) is generated by the vector space V ⊂ C  (V; q ) sub ject to the relations v · v = − q ( v )1 whic h giv e a “univ ersal” c haracterization of the algebra. Giv en tw o v ector spaces V , V 0 , equip ed with quadratic forms q , q 0 , resp ectiv ely , an y linear map f : V → V 0 preserving the quadratic forms induces a homomorphism ˜ f : C  (V; q ) → C  (V 0 ; q 0 ). An imp ortant example is giv en by the homomorphism ˜ α induced b y the map α ( v ) = − v on V . Since α 2 = id, there is a decomp osition C  (V; q ) = C  0 (V; q ) ⊕ C  1 (V; q ) where C  i (V; q ) := { ϕ ∈ C  (V; q ) : α ( ϕ ) = ( − 1) i ϕ } . The v ector space C  0 (V; q ) is a subalgebra of C  (V; q ), and it is called the even p art , while the subspace C  1 (V; q ) is called the o dd p art of C  (V; q ). Giv en a Cliﬀord algebra C  (V; q ), w e can deﬁne the multiplic ative gr oup of units as C  ∗ (V; q ) :=  ϕ ∈ C  (V; q ) : ∃ ϕ − 1 ∈ C  (V; q ) such that ϕ − 1 · ϕ = ϕ · ϕ − 1 = 1  When dimV = n < ∞ , C  ∗ (V; q ) is a Lie group of dimension 2 n . Consider the subgroup P(V; q ) ⊂ C  ∗ (V; q ) generated b y the elements v ∈ V with q ( v ) 6 = 0. 188 Then the Pin gr oup asso ciated to the pair (V; q ) is the subgroup Pin(V; q ) of P(V; q ) generated by elements v ∈ V with q ( v ) = ± 1. The associated Spin gr oup of (V; q ) is deﬁned as Spin(V; q ) := Pin(V; q ) ∩ C  0 (V; q ) W e will now restrict to the case in which V = R n , and q ( x ) := x 2 1 + x 2 2 + · · · + x 2 n is the quadratic form induced by the usual scalar pro duct. Denote with C  n the Cliﬀord algebra C  ( R n ; q ). If e 1 , e 2 , · · · , e n is any orthonormal basis of R n , then C  n is generated as an algebra b y e 1 , e 2 , · · · , e n and 1 sub ject to the relations e i · e j + e j · e j = − 2 δ ij 1 It is straightforw ard to see that C  1 ' C , and C  2 ' H . Moreov er, one can pro v e that dim R (C  n ) = 2 n . The decomp osition of C  n in even and o dd part induces the follo wing isomorphism C  n ' C  0 n +1 An important result concerning the Cliﬀord algebra C  n is the follo wing. There is a canonical isomorphism C  n ' Λ ∗ R n , according to which the Cliﬀord m ultiplication b et w een v ∈ R n and an y ϕ ∈ C  n can b e written as v · ϕ ' v ∧ ϕ − v b ϕ where w e hav e identiﬁed R n with its dual via the scalar pro duct, and b denotes the con traction of the v ector v with an element of Λ ∗ R n . The Cliﬀord algebras can b e describ ed as matrix algebras ov er R , C , or H . Indeed, denote with C  n the complexi- ﬁcation C  n ⊗ R C of C  n . Then for all n ≤ 0 we hav e the p erio dicity isomorphisms C  n +8 ' C  n ⊗ C  8 C  n +2 ' C  n ⊗ C C  2 The ab ov e isomorphisms allo w to deduce all the algebras C  n and C  n from the follo wing table 1 2 3 4 5 6 7 8 C  n C H H ⊕ H H (2) C (4) R (8) R (8) ⊕ R (8) R (16) C  n C ⊕ C C (2) C (2) ⊕ C (2) C (4) C (4) ⊕ C (4) C (8) C (8) ⊕ C (8) C (16) where K( n ) denotes the algebra of n × n matrices ov er the ﬁeld K, with K = R , C , H . 189 The ab o v e table also dictates the theory of representations of Cliﬀord algebras. In- deed, if w e see K( n ) as an algebra o ver R , the natural represen tation of K( n ) on the v ector space K n is the only irreducible real representation of K( n ) up to isomorphism, while the algebras K( n ) ⊕ K( n ) ha v e exactly t w o equiv alence classes of irreducible real represen tations [63]. Giv en a mo dule An imp ortan t role is play ed by the Spin groups asso ciated to R n with the Euclidean quadratic form. Indeed, if w e denote with Spin n the group Spin( R n ; q ), w e ha v e the follo wing exact sequence 0 → Z 2 → Spin n ξ 0 − → SO n → 0 for all n ≥ 3. In particular, the map ξ 0 denotes the universal co vering homomorphism of SO n . The representation theory of Cliﬀord algebras can b e used to construct represen tations of Spin n ⊂ C  0 n whic h are not trivial on the element -1, hence they do not arise from represen tation of the orthogonal group SO n . Indeed, we can deﬁne the r e al spinor r epr esentation of Spin n as the homomorphism ∆ n : Spin n → GL(S) induced b y retricting an irreducible real represen tation C  n → Hom R (S , S) to Spin n . W e can also deﬁne the c omplex spinor r epr esentation of Spin n as the homomorphism ∆ C n : Spin n → GL C (S) induced b y restricing an irreducible complex represen tation C  n → Hom C (S , S) to Spin n ⊂ C  0 n ⊂ C  n . In particular, for n o dd, the complex spinor representation ∆ C n is irreducible, and indep enden t of which irreducible represention of C  n is used. When n = 2 m , there is a decomp osition ∆ C 2 m = ∆ C + 2 m ⊕ ∆ C − 2 m in to a direct sum of irreducible complex represen tations of Spin n . The representation ∆ C ± 2 m is giv en b y the comp osition of ∆ C 2 m with the pro jection 1 ± ω C , where ω C = i m e 1 · e 2 · · · e 2 m is the complex volume elemen t. Notice that ∆ C ± 2 m is not a represen tation of C  n . Similar results hold for the real spinor represen tation. Finally , a Z 2 -gr ade d mo dule for C  n is a mo dule W with a decomp osition W = W 0 ⊕ W 1 suc h that C  i n W j ⊆ W ( i + j )(mo d2) 190 Giv en a Z 2 -graded mo dule W ov er C  n , the even part W 0 is a mo dule o ver C  0 n . Con v ersely , given a mo dule W 0 o v er C  0 n , w e can form the Z 2 -graded mo dule W := C  n ⊗ C ` 0 n W 0 Hence there is an equiv alence b et w een the category of Z 2 -graded mo dules o v er C  n and the category of ungraded mo dules o v er C  n − 1 . Let E b e an oriented n -dimensional Riemannian vector bundle ov er a manifold X, and supp ose n ≥ 3. Let P SO (E) denote the orthonormal frame bundle of E. Then a spin structur e on E is a principal Spin n -bundle P Spin (E) together with a double co v ering ξ : P Spin (E) → P SO (E) suc h that ξ ( p · g ) = ξ ( p ) · ξ 0 ( g ) for all p ∈ P Spin (E) and all g ∈ Spin n . F or n = 2, a Spin structure on E is deﬁned analogously , with Spin 2 replaced by SO 2 and ξ 0 : SO 2 → SO 2 the connected double co v ering. F or n = 1, P SO (E) ' X and a spin structure is simply deﬁned to b e a double co vering of X. The existence of a spin structure on a v ector bundle E is dictated by the v anishing of the ﬁrst and second Stiefel-Whitney class w 1 (E) and w 2 (E), resp ectiv ely . A spin manifold is an oriented Riemannian manifold X with a spin structure on its tangen t bundle. Moreov er, the inequiv alen t spin structures on X are in one-to-one corresp ondence with elemen ts of H 1 (X; Z 2 ). Giv en an orien ted Riemannian vector bundle, w e can construct the asso ciated Cliﬀor d bund le deﬁned as C  (E) := P SO (E) × c ` ( ρ ) C  n where c  ( ρ ) : SO n → Aut(C  n ) is induced b y lifting orthogonal transformations of R n to the Cliﬀord algebra C  n . The Cliﬀord bundle C  (E) is a bundle of Cliﬀord algebras o v er X, and the ﬁbrewise multiplication in C  (E) gives an algebra structure to the space of sections of C  (E). Hence, all the notions regarding Cliﬀord algebras carry o v er to Cliﬀord bundles. W e can then lo ok for bundles of irreducible modules ov er the Cliﬀord bundle C  (E). These bundles can b e constructed if E has a spin structure. Indeed, a r e al spinor bund le of E is a bundle of the form S(E) := P Spin (E) × µ W where W is a left mo dule for C  n , and where Spin n acts on W by left multiplication b y elemen ts of Spin n ⊂ C  n . Analogously , a c omplex spinor bund le of E is given by S C (E) := P Spin (E) × µ W C 191 where W C is a complex left mo dule for C  n . One can easily pro v e that the sec- tions of the spinor bundle are a mo dule o v er the sections of the Cliﬀord bundle. A spinor bundle S(E) is called irr e ducible if the left mo dule W(W C ) is the irreducible real(complex) spinor represen tation of Spin n . In particular, when the vector bundle E has rank n = 2 m , the complex spinor bundle S C (E) asso ciated to the irreducible represen tation of C  2 m decomp oses as S C (E) = S + C (E) ⊕ S − C (E) where S ± C (E) is the ± 1 eigen bundle for Cliﬀord multiplication by the complex v olume elemen t ω C , and can b e represen ted as S + C (E) ' P Spin (E) × ∆ C ± 2 m C 2 m − 1 Let E b e no w equip ed with a cov ariant deriv ative ∇ : Γ(E) → Γ(E ⊗ T ∗ X) induced b y a connection τ on P SO (E). Since C  (E) is an associated vector bundle to P SO (E), τ induces a cov ariant deriv ativ e ∇ C ` on C  (E) with the prop ert y that ∇ C ` ( ϕ · ψ ) = ( ∇ C ` ϕ ) · ψ + ϕ · ( ∇ C ` ψ ) Moreo v er, the cov ariant deriv ativ e ∇ C ` preserv es the subbundles C  0 (E) and C  1 (E), and the v olume elemen t ω = e 1 e 2 · · · e n is globally parallel, i.e. ∇ C ` ω = 0. Supp ose E is also equip ed with a spin structure ξ : P Spin (E) → P SO (E). W e can then lift the connection τ on P SO (E) to a connection ˜ τ on P Spin (E). Since a spinor bundle S(E) is an associated v ector bundle to P SO (E), the connection ˜ τ induces a cov ariant deriv ativ e ∇ S on S(E). In particular, the cov ariant deriv ativ e ∇ S is compatible with ∇ C ` , in the sense that ∇ S ( ϕ · σ ) = ( ∇ C ` ϕ ) · σ + ϕ · ( ∇ S σ ) for an y ϕ ∈ Γ(C  (E)) and any σ ∈ Γ(S(E)). Giv en an n -dimensional orien ted Riemannian manifold X, w e denote with C  (X) the Cliﬀord bundle asso ciated to TX. The Cliﬀord bundle carries a canonical cov ariant deriv ativ e, whic h is induced b y the Levi-Civita connection on P SO (TX). Consider no w an y bundle S of left mo dules o v er C  ( X ), not necessarily a spinor bundle, and suppose S is Riemannian and equiped with a Riemannian connection. W e can then deﬁne a 192 ﬁrst-order elliptic diﬀerential op erator D : Γ(S) → Γ(S) called the Dir ac op er ator of S, deﬁned as D σ | x := n X j =1 e j · ( ∇ e j σ ) | x at x ∈ X, where e 1 , e 2 , . . . , e n is an orthonormal basis of T x X. If we let S = C  (X) with its canonical Riemannian connection, and view C  (X) as a bundle of left mo dules o ver itself b y Cliﬀord multiplication, then the Dirac operator is a square ro ot of the classical Ho dge laplacian. Another case of ma jor imp ortance is the following. Let X b e a spin manifold, and let S b e any spinor bundle. The vector bundle S is Riemannian and carries a canon- ical co v ariant deriv ative ∇ Spin , called the spin c onne ction , induced b y the lift of the Levi-Civita connection on P SO (TX). The Dirac op erator in this case is called the A tiyah-Singer op er ator , and pla ys a fundamen tal role in the Index theorem. Finally , a Dirac op erator associated to a bundle S of left mo dules o ver C  (X) can b e twiste d b y a Riemannian vector bundle with connection E b y considering the Cliﬀord multi- plication on Γ(S ⊗ E) induced by ϕ · ( σ ⊗ e ) := ( ϕ · σ ) ⊗ e and equiping S ⊗ E with the tensor pro duct connection ∇ deﬁned on sections of the form σ ⊗ e by ∇ ( σ ⊗ e ) := ( ∇ S σ ) ⊗ e + σ ⊗ ( ∇ E e ) where ∇ S and ∇ E are the co v arian t deriv ativ es on S and E, resp ectively . Finally , we conclude this app endix by giving the deﬁnition of a Spin c manifold. This inv olves ﬁrst constructing the group Spin c . Consider the complex spinor repre- sen tation ∆ C : Spin n → GL C (W C ) and let z : U(1) → GL C (W C ) denote the m ultiplication by scalar. W e then get the homomorphism ∆ C × z : Spin n × U(1) → Hom C (W C , W C ), whic h has the element ( − 1 , − 1) in its k ernel. Dividing by this element gives the group Spin c n := Spin n × Z 2 U(1) whic h satisﬁes the short exact sequence 0 → Z 2 → Spin c n ξ 0 − → SO n × U(1) → 1 193 Let E be an oriented Riemannian v ector bundle of rank n o v er a manifold X . A S pin c - structur e on E consists of a principal Spin c n -bundle P Spin c n (E), and also a principal U(1)-bundle P U(1) (E) o v er X with a bundle map P Spin c n (E) ξ − → P SO n (E) × P U(1) (E) suc h that ξ satisﬁes ξ ( p · g ) = ξ ( p ) ξ 0 ( g ) for all p ∈ P Spin c n (E) and all g ∈ Spin c n . The ﬁrst Chern class d ( E ) of the U(1)-bundle P U(1) (E) is called the c anonic al class of the Spin c -structure. A S pin c -manifold is an orien ted Riemannian manifold X with a Spin c -structure on the tangen t bundle TX. 194 App endix C Characteristic classes for v ector bundles In this app endix we will brieﬂy recall some basic facts from the theory of c haracter- istic classes, as dev elop ed in [70]. All the spaces are assumed to b e of the homotopy type of countable CW-complexes. Let G b e a Lie group. A classifying sp ac e for G is a connected top ological space BG, together with a principal G-bundle EG → BG such that for an y compact Hausdorﬀ space X the set of homotopy classes of maps from X to BG is in bijective corresp on- dence with the set of equiv alence classes of principal G-bundles ov er X. In particular, the ab ov e corresp ondence is induced by asso ciating to eac h map f : X → BG the pullbac k bundle f ∗ EG o v er X. The principal bundle EG → BG is called the universal princip al G-bund le . It can b e pro v en b y direct construction that for any Lie group a classifying space do es exist. Moreo v er, b y the v ery properties of a classifying space, it is unique up to homotop y t yp e. Consider the singular cohomology H ∗ (BG; Λ) with co eﬃcien ts in a ring Λ. Each non- zero class in H ∗ (BG; Λ) is a universal char acteristic class for principal G-bundles. Fix a class c ∈ H ∗ (BG; Λ). F or eac h principal G-bundle P → X we deﬁne the c- char acteristic class c (P) ∈ H ∗ (X; Λ) as c (P) := f ∗ P (c) where f P : X → P is a classifying map for P, and is well deﬁned, since f P is uniquely deﬁned up to homotopy . Moreo v er, an y suc h c haracteristic class is “natural”, in the sense that given a principal G-bundle P o v er X and a contin uous map ϕ : Y → X we ha v e c ( ϕ ∗ P) = ϕ ∗ c(P) W e will no w specialize to the case G = BO n , BU n . In these particular cases, the univ ersal bundle EG can b e obtained as the appropriate bundle of frames of a uni- versal r e al (or c omplex) ve ctor bund le E n o v er BO n (or BU n ), whic h classiﬁes real (or complex) v ector bundles. Moreov er, despite still diﬃcult to compute, the cohomology rings of the classifying spaces BO n and BU n are quite manageable. The cohomology ring H ∗ (BO n ; Z 2 ) is a Z 2 -p olynomial ring Z 2 [ w 1 , w 2 , . . . , w n ] 195 where w k is a canonical generators of H k (BO n ; Z 2 ), and is called the universal k-th Stiefel-Whitney class . T o any n -dimensional real vector bundle E → X classiﬁed b y a map f E : X → BO n , w e can asso ciate the k -th Stiefel-Whitney class of E, w k (E) := f ∗ E (w k ). In particular, the total Stiefel-Whitney class w := 1 + w 1 + · · · + w n satisﬁes w (E ⊕ E 0 ) = w(E) ∪ w(E 0 ) If X is a smo oth manifold, we can deﬁne the k -th Stiefel-Whitney class w k ( X ) of X as w k (TX). An imp ortan t prop ert y of the Stiefel-Whitney classes is that for a compact smo oth manifold they are in v arian ts of the homotopy type of the manifold. Giv en an n -dimensional real vector bundle E, the ﬁrst and second Stiefel-Whitney v anish exactly when E is orien table and admits a Spin structure, resp ectively . This can b e seen b y using the follo wing equiv alen t deﬁnition of w 1 (E) and w 2 (E). Recall that an y isomorphism class of principal G-bundles on a space X can b e rep- resen ted as a class in H 1 (X; G), the cohomology of X with co eﬃcien ts in the sheaf of G-v alued functions, via the transition functions. When G is not ab elian, H 1 (X; G) is not a group, but rather a set with a distinguished element giv en b y the trivial G- bundle. How ev er, it still preserves some cohomological prop erties. Indeed, consider the short exact sequence 0 → SO n i − → O n ρ − → Z 2 → 0 , whic h induces the exact sequence H 1 (X; SO n ) i ∗ − → H 1 (X; O n ) ρ ∗ − → H 1 (X; Z 2 ) Giv en a rank n v ector bundle, w e can deﬁne w 1 (E) = ρ ∗ ([P O (E)]), where [ P O ( E )] denotes the class of the orthonormal frame bundle of E. Hence, when w 1 (E) = 0 w e ha v e that the class [ P O ( E )] is the image of the class of an SO n -principal bundle, whic h is p ossible if and only if E is orien tabale. Similarly , the short exact sequence 0 → Z 2 → Spin n ξ 0 − → SO n → 0 induces the exact sequence H 0 (X; SO n ) δ 0 − → H 1 (X; Z 2 ) → H 1 (X; Spin n ) ξ 0 ∗ − → H 1 (X; SO n ) δ − → H 2 (X; Z 2 ) 196 W e can deﬁne the second Stiefel-Whitney class b y w 2 (E) = δ ([P SO (E)]). Hence, w 2 (E) = 0 if and only if P SO (E) is equiv alen t to the Z 2 -quotien t of a principal Spin n - bundle on X. F or the classifying space BU n , w e hav e that the cohomology ring H ∗ (BU n ; Z ) is a Z -p olynomial ring Z [ c 1 , c 2 , · · · , c n ] where c k ∈ H k (BU n ; Z ) is a canonical generator, and is called the universal k-th Chern class . Thus, to an y n -dimensional complex vector E → X classiﬁed by a map f E : X → BU n w e can asso ciated the k-the Chern class c k (E) = f ∗ E (c k ). W e can deﬁne the total Chern class c = 1 + c 1 + · · · + c n , whic h satisﬁes c (E ⊕ E 0 ) = c(E) ∪ c(E 0 ) (C.1) F or a giv en complex n -dimensional v ector bundle E → X, w e can compute the Chern classes which are nontorsion in the follo wing geometric w ay . Let F be the curv ature of an arbitrary co v ariant deriv ativ e on E. Recall that F is a diﬀerential form v alued in the adjoint represen tation of U(n), hence in n × n an tihermitian matrices. Then the k -th Chern class of E is giv en b y the deRham class of α k (E), where det(I + F t 2 π ) = n X k =1 α k (E)t k The Chern classes deﬁned ab ov e can be used as the basic ingredien t to deﬁne other imp ortan t c haracteristic class of v ector bundle ov er a manifold. This is due to the follo wing Theorem C.1 L et E b e a n -dimensional c omplex ve ctor bund le over a manifold X . Then ther e exists a manifold M E and a smo oth and pr op er ﬁbr ation π : M E → X such that i) the homomorphism π ∗ : H ∗ (X) → H ∗ ( M E ) is inje ctive ii) the bund le π ∗ E splits into the dir e ct sum of c omplex line bund les π ∗ E ' L 1 ⊕ L 2 ⊕ · · · ⊕ L n The ab o v e theorem “induces” the following splitting principle : all p olynomial iden tities in the Chern classes of complex v ector bundles can b e prov en under the assumption that all v ector bundles are direct sums of line bundles. 197 F or a real vector bundle E of dimension 2 n one can pro ve that the complexiﬁcation E ⊗ C splits on M E as π ∗ (E ⊗ C ) ' L 1 ⊕ L 1 ⊕ · · · ⊕ L n ⊕ L n where L 1 denotes the complex conjugate of the L 1 . Notice that b y naturalit y property , any c-c haracteristic class of π ∗ E is in the image of the homomorphism π ∗ . W e can then construct c haracteristic classes as the unique preimage via the splitting homomorphism π ∗ of functions of the Chern classes of the splitting line bundles. F or instance, b y using the prop erty (C.1) of the total Chern class, w e ha v e c (E) = n Y k =1 (1 + x k ) where x k = c 1 ( L 1 ). In this w ay , we can associate to formal p o w er series rational c haracteristic classes. W e can deﬁne the total T o dd class of a complex vector bundle E by Td(E) := n Y k =1 x k 1 − e − x k Giv en a real v ector bundle E of dimension 2 n , the total ˆ A -class can b e deﬁned by ˆ A(E) := n Y k =1 x k / 2 sinh( x k / 2) Finally , we can deﬁne the total Chern char acter b y c h(E) := n X k =1 e x k The Chern c harater can b e represen ted in the deRham cohomology of X as c h(E) = [T r( e F / 2 π )] ∈ H ev (X; R ) where F is the curv ature of a cov ariant deriv ativ e on E, and the trace T r is in the adjoin t represen tation of U(n). 198 App endix D Equiv arian t K-homology Sp ectral deﬁnition A natural wa y to deﬁne the equiv ariant homology theory K G • is b y means of a sp e c- trum for equiv arian t top ological K-theory K • G , whic h is a particular cov ariant functor V ect G ( − ) from the orbit category Or (G) to the tensor category Spec of sp ectra [31, 73]. Giv en any G-complex X, the corresp onding p oin ted G-space is X + = X q pt and one deﬁnes the lo op sp ectrum X + ⊗ G V ect G ( − ) b y X + ⊗ G V ect G ( − ) = a G / H ∈ Or (G)  X H + ∧ V ect G (G / H)   ∼ , (D.1) where the equiv alence relation ∼ is generated by the iden tiﬁcations f ∗ ( x ) ∧ s ∼ x ∧ f ∗ ( s ) with ( f : G / K → G / H) ∈ Mor( Or (G)), x ∈ X H + , and s ∈ V ect G (G / K) ∗ . One then puts K G ∗ (X) := π ∗  X + ⊗ G V ect G ( − )  . (D.2) By using v arious G-homotopy equiv alences of the lo op sp ectra (D.1), one shows that this deﬁnition of equiv ariant K-homology comes with a natural induction struc- ture. F or the trivial group it reduces to the ordinary K-homology K e ∗ = K ∗ giv en by the Bott sp ectrum B U . If G is a ﬁnite group, any ﬁnite-dimensional represen tation of G naturally extends to a complex represen tation of the group ring C [G]. Then there is an analytic assem bly map ass : K G ∗ (X) − → K ∗  C [G]  to the K-theory of the ring C [G], induced by the collapsing map X → pt and the isomorphisms K ∗  C [H]  ∼ = π ∗  V ect G (G / H)  ∼ = K G ∗ (G / H) ∼ = R(H) for any subgroup H ≤ G [73]. In the follo wing we will giv e t wo concrete realizations of the homotop y groups (D.2). 199 Analytic deﬁnition The simplest realization of the equiv arian t K-homology group K G ∗ (X) is within the framew ork of an equiv arian t version of Kasparov’s KK-theory KK G ∗ . Let A b e a G-algebra, i.e. , a C ∗ -algebra A together with a group homomorphism λ : G − → Aut( A ) . By a Hilb ert (G , A )-mo dule we mean a Hilb ert A -mo dule E together with a G-action giv en b y a homomorphism Λ : G → GL( E ) such that Λ g ( ε · a ) = Λ g ( ε ) · λ g ( a ) (D.3) for all g ∈ G, ε ∈ E and a ∈ A . Let L ( E ) denote the ∗ -algebra of A -linear maps T : E → E admitting an adjoint with resp ect to the A -v alued inner pro duct on E . The induced G-action on L ( E ) is given b y g · T := Λ g ◦ T ◦ Λ g − 1 . Let K ( E ) b e the subalgebra of L ( E ) consisting of generalized compact op erators. Giv en a pair ( A , B ) of G-algebras, let D G ( A , B ) b e the set of triples ( E , φ, T) where E is a coun tably generated Hilb ert (G , B )-mo dule, φ : A → L ( E ) is a ∗ -homomorphism whic h comm utes with the G-action, φ  λ g ( a )  = Λ g ◦ φ ( a ) ◦ Λ g − 1 (D.4) for all g ∈ G and a ∈ A , and T ∈ L ( E ) such that 1) [T , φ ( a )] ∈ K ( E ) for all a ∈ A ; and 2) φ ( a ) (T − T ∗ ), φ ( a ) (T 2 − 1), φ ( a ) ( g · T − T) ∈ K ( E ) for all a ∈ A and g ∈ G. The standard equiv alence relations of KK-theory are now analogously deﬁned. The set of equiv alence classes in D G ( A , B ) deﬁnes the equiv ariant KK-theory groups KK G ∗ ( A , B ). If X is a smo oth prop er G-manifold without b oundary , and G acts on X by dif- feomorphisms, then the algebra A = C 0 (X) of contin uous functions on X v anishing at inﬁnit y is a G-algebra with automorphism λ g on A giv en b y λ g ( f )( x ) :=  g ∗ f  ( x ) = f  g − 1 · x  , where g ∗ denotes the pullbac k of the G-action on X b y left translation by g − 1 ∈ G. W e deﬁne K G ∗ (X) := KK G ∗  C 0 (X) , C  (D.5) with G acting trivially on C . The conditions (D.3) and (D.4) naturally capture the ph ysical requiremen ts that physical orbifold string states are G-inv ariant and also that the w orldv olume ﬁelds on a fractional D-brane carry a “cov ariant represen tation” of the orbifold group [37]. 200 The equiv arian t Dirac class W e can determine a canonical class in the ab elian group (D.5) as follows. W e refer to App endix B for basic notions of Cliﬀord algebras. Let d im(X) = 2 n , and let G b e a ﬁnite subgroup of the rotation group SO(2 n ). 13 A c hoice of a complete G-in v arian t riemannian metric on X naturally lieft the Cliﬀord bundle C  (X). The G-manifold X is said to ha ve a G -spin c structur e or to b e K G -oriente d if there is an extension of the orthonormal frame bundle P SO 2 n (X) to a principal Spin c (2 n )- bundle P Spin c 2 n (X) ov er X which is compatible with the G-action. The extension P Spin c 2 n (X) ma y b e regarded as a principal circle bundle o v er P SO 2 n (X), U(1) w w n n n n n n n n n n n n n   ˆ G / /   Spin c (2 n ) / /   P Spin c 2 n (X) / /   X , G / / SO(2 n ) / / P SO 2 n (X) 9 9 s s s s s s s s s s where the pullbac k square on the b ottom left deﬁnes the required co v ering of the orb- ifold group G < SO(2 n ) b y a subgroup of the spin c group ˆ G < Spin c (2 n ). The kernel of the homomorphism ˆ G → G is identiﬁed with the circle group U(1) < Spin c (2 n ). W e ﬁx a c hoice of lift and hence assume that G is a discrete subgroup of the spin c group. Z 2 - graded Cliﬀord mo dules are likewise extended to representations of C [G] ⊗ Cliﬀ (2 n ), with C [G] the group ring of G, called G-Cliﬀord mo dules. Since G lifts to ˆ G in the spin c group, the the spinor bundles bundles S ± asso ciated to P Spin c 2 n (X) are naturally G-bundles. The G-inv ariant Levi-Civita connection deter- mines a connection on P SO 2 n (X), and together with a c hoice of G-in v arian t connection form on the principal U(1)-bundle P Spin c 2 n (X) → P SO 2 n (X), they determine a connec- tion one-form on the principal bundle P Spin c 2 n (X) → X which is G-in v arian t. This determines an in v arian t co v arian t deriv ative ∇ S ⊗ E : Γ  S + ⊗ E  → Γ  T ∗ X ⊗ S + ⊗ E  where ∇ E is a G-in v ariant connection on a G-bundle E → X. The contraction giv en b y Cliﬀord m ultiplication deﬁnes a map C  : Γ  T ∗ X ⊗ S + ⊗ E  → Γ  S − ⊗ E  13 Throughout the extension to K G 1 or K − 1 G and dim(X) o dd can be describ ed in the same w a y as in degree zero by replacing X with X × S 1 . 201 whic h graded commutes with the G-action, and the G-inv ariant spin c Dirac op erator on X with co eﬃcien ts in E is deﬁned as the comp osition D / X E = C  ◦ ∇ S ⊗ E . (D.6) W e will view the op erator (D.6) as an op erator on L 2 -spaces D / X E : L 2  X , S + ⊗ E  − → L 2  X , S − ⊗ E  . It induces a class  D / X E  ∈ K G 0 (X) as follo ws. The G-algebra C 0 (X) acts on the Z 2 - graded G-Hilbert space E := L 2 (X , S ⊗ E) b y m ultiplication. Deﬁne the bounded G-in v arian t op erator T := D / X E  ( D / X E ) 2 + 1  − 1 / 2 ∈ F red G . Then  D / X E  is represented by the G-equiv arian t F redholm mo dule ( E , T). Geometric deﬁnition Geometric equiv ariant K-homology can b e deﬁned for an arbitrary discrete, coun table group G on the category of prop er, ﬁnite G-complexes X and prov en to b e isomor- phic to analytic equiv arian t K-homology [15]. Recall that the top ological equiv arian t K-theory K ∗ G (X) is deﬁned b y applying the Grothendiec k functor K ∗ to the addi- tiv e category V ect C G (X) whose ob jects are complex G-vector bundles o v er X, i.e. , K ∗ G (X) := K ∗  V ect C G (X)  . In the homological setting, the relev ant category is instead the additive category of G -e quivariant K-cycles D G (X), whose ob jects are triples (W , E , f ) where (a) W is a manifold without b oundary with a smooth prop er cocompact G-action and G-spin c structure; (b) E is an ob ject in V ect C G (W); and (c) f : W → X is a G-map. Tw o G-equiv ariant K-cycles (W , E , f ) and (W 0 , E 0 , f 0 ) are said to b e isomorphic if there is a G-equiv ariant diﬀeomorphism h : W → W 0 preserving the G-spin c structures on W , W 0 suc h that h ∗ (E 0 ) ∼ = E and f 0 ◦ h = f . Deﬁne an equiv alence relation ∼ on the category D G (X) generated b y the op era- tions of i) Bordism: (W i , E i , f i ) ∈ D G (X), i = 0 , 1 are b or dant if there is a triple (M , E , f ) where M is a manifold with b oundary ∂ M, with a smo oth prop er co compact G-action and G-spin c structure, E → M is a complex G-vector bundle, and f : 202 M → X is a G-map suc h that ( ∂ M , E | ∂ M , f | ∂ M ) ∼ = (W 0 , E 0 , f 0 ) q ( − W 1 , E 1 , f 1 ). Here − W 1 denotes W 1 with the rev ersed G-spin c structure; ii) Direct sum: If (W , E , f ) ∈ D G (X) and E = E 0 ⊕ E 1 , then (W , E , f ) ∼ = (W , E 0 , f ) q (W , E 1 , f ) ; and iii) V ector bundle mo diﬁcation: Let (W , E , f ) ∈ D G (X) and H an even-dimensional G-spin c v ector bundle ov er W . 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K-Theory, D-Branes and Ramond-Ramond Fields

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