Ramond-Ramond Fields, Fractional Branes and Orbifold Differential K-Theory

HWM–07–41 EMPG–07 –19 RAMOND-RAMOND FIELDS, FRA CTIONAL BRANES AND ORBIFOLD DIFFERENTIAL K-THEO R Y RICHARD J. SZABO A ND ALESSANDRO V ALENTINO Abstra ct. W e study D-branes and Ramond-Ramond ﬁelds on global orbifolds of T yp e II string theory with v anishing H -ﬂux using methods of equ iv ari ant K-theory and K-homology . W e illustrate ho w Bredon equiv ari ant cohomology natu rally realizes stringy orbif old cohomology . W e emphasize its role as the correct cohomological t ool which ca ptures kno wn fea tures of the lo w-energy eﬀective ﬁeld theory , and whic h provides n ew consistency conditions for fractional D-branes a nd Ramo nd-Ramond ﬁelds on orbifolds. W e use an equiv arian t Chern character from equiva rian t K-theory to Bredon cohomo logy to deﬁne new Ramond-Ramond coup lings of D-branes whic h generalize previous examples. W e prop ose a deﬁn ition for groups of d iﬀerenti al chara cters associated to equiv arian t K-theory . W e d eriv e a Dirac quantization rule for Ramond- Ramond ﬂuxes, and s tudy ﬂat Ramond -Ramond potentials on orbifolds. Introduction The stud y of ﬂ uxes and D-branes h as b een of fun damen tal imp ortance in un derstanding the nonp ertu r bativ e structures of string theory and M-theory . It has also established a common ground on whic h a fruitful interac tion b et w ee n physic s and mathematics tak es place. F or ex- ample, the seminal p ap ers [49, 62] demonstrated that D-brane charge s in T yp e I I su p erstring theory are classiﬁed by the K-theory of the sp acetime manifold, and that ordinary cohomology alone cannot accoun t for certain physical features induced by th e dynamics of D-branes. As emphasized b y refs. [2, 55, 36, 60], and analyzed in great deta il in refs. [56, 57], another descrip- tion of D-branes is provided by K-homology wh ic h sh eds ligh t on their geometrical nature and suggests that the standard picture of a D-brane as a sub manifold of s p acetime equip ed with a v ector bun dle (and connection) should b e m o diﬁed. Ramond-Ramond ﬁelds are du al ob jects to D-branes and ha ve also b een extensiv ely in ves- tigated, but until recen tly their geometric n ature has remained somewh at obscure. In ref. [52] it was pr op osed that Ramond -Ramond ﬁ elds are also classiﬁed by K-theory , a nd that their total ﬁeld s tr engths lie in the image of th e Chern c haracter homomorphism from K-theory to ordinary cohomology . This resu lt led to the un derstanding that the Ramond-Ramond ﬁeld is correctly un dersto o d as a self-dual ﬁeld qu an tized b y K-theory , and it exp lains v arious subtle issues surr ounding the p artition functions of these ﬁ elds. In refs. [28, 29] it was prop osed that these prop erties are most n aturally form ulated b y rega rding Ramond-Ramond ﬁelds as co cycles for the diﬀerential K-theory of spacetime , an elegan t description th at allo ws one to stud y the gauge theory of Ramond-Ramond ﬁelds in top ological ly n on-trivial bac kgroun ds w hic h naturally incorp orates consistency conditions such as anomaly cancellati on on b ranes in string theory and M-theory . These issues were among the motiv ations that led to the foundational pap er [37], in whic h a detailed, elab orate construction for generalized diﬀeren tial cohomolog y theories is given. The imp ortance of these mathematical theories has b een great ly emphasized in refs. [32, 33], where th ey are u sed to deﬁne and und erstand certain n o ve l prop erties of quant um Hilb ert sp aces of abelian gauge ﬁeld ﬂuxes. A t wisted ve rsion of diﬀeren tial K -theory h as b een pr op osed in 1 2 RICHARD J. SZABO AND ALESSANDRO V ALENTINO refs. [33, 11] and applied to the quantiz ation of Ramond-Ramond ﬁelds in an H -ﬂux bac kgroun d, while a r igorous geometrical deﬁnition of th is theory h as b een d ev elop ed recen tly in ref. [18]. The goal of this pap er is to extend these lines of dev elopments to study prop erties of Ramond- Ramond ﬁ elds and D-bran es in orbifolds of T yp e I I sup erstring theory with v anishing H -ﬂux. W e limit ou r study to the cases of goo d (or global) orbifolds [ X/G ], wh ere X is a manifold and G is a ﬁnite group acting via diﬀeomorphisms of X . It is p ossible to resolv e singularities in the orbifold wh ere it fails to b e a manifold, and r eplace the quotien t space b y a n on-compact manifold with appropriate asym p totic b eha v iour . Ho wev er, orbifold singularities do not p ose a p roblem and one can still ha ve consisten t su p erstrings p ropagating on orbifolds [24, 25]. It w as prop osed in ref. [62] th at D-branes on the orbifold spacetime [ X/G ] are classiﬁed b y the G -equiv ariant K-theory of the cov erin g space X , as deﬁned in ref. [58]. A recent ov erview of related deve lopmen ts in the case of ab elian orbifolds can b e foun d in ref. [41]. One of the main new ingredient s that w e int ro duce into the description of D-branes and ﬂ u xes on orbifolds is the us e of Bredon cohomology [16, 23]. T his is a p ow erful equiv ariant cohomology theory that has b oth adv anta ges and pitfalls. In con trast to th e more commonly u s ed Borel equiv ariant cohomolog y , Bred on cohomology is a go o d “appro ximation” to the classiﬁcation of D-brane charge s. W e will supp ort th is statemen t by sho wing that it correctly captures the prop erties of Ramond-Ramond ﬁelds on an orbifold, in particular it n aturally tak es in to accoun t the twisted sectors of the string theory . It thereby giv es a precise, rigorous realization of stringy orbifold cohomology . W e will also see that it n aturally arises in the Atiy ah-Hirzebru c h sp ectral sequence for equ iv ariant K-theory , a fact that w e shall exploit to d escrib e n ew consistency conditions for D-branes and ﬂuxes on orb ifolds in terms of classes in the Bredon cohomology of the co v erin g space X . Related to this feature is th e f act that this equiv arian t cohomology th eory is the target for a Ch ern c haracter homomorph ism on equ iv ariant K-theory , deﬁn ed in ref. [46], whic h in d uces an isomorphism wh en tensored o ver R . By means of this technolog y , we p resen t new compact and elegan t expr essions f or the W ess-Zumin o couplings of Ramond -Ramond ﬁelds to D-br anes on [ X/G ]. This generalizes the usual Ramond-Ramond couplings [49 ] to orbifolds, and yields app ropriate correction terms to previous ﬂat sp ace form ulas. The ma j or dra wbac k of Bredon cohomology is th at it is a rather diﬃcult, abstract theory to deﬁn e, and is ev en m ore diﬃcult to exp licitly calculate than other equiv arian t cohomology theories. Another m ain ac hieveme n t of this p ap er is a prop osed deﬁnition of diﬀerenti al K-theory s uit- able for orbifolds. T h ough extremely p o w erf ul and general, the mac hin ery deve lop ed in ref. [37] cannot b e imm ed iately applied to an equiv arian t cohomology fun ctor on the category of G - manifolds. By using Bredon cohomology and the equiv arian t Chern characte r, w e deﬁn e ab elian groups that b ehav e as natural generaliza tions of the ord in ary diﬀerential K-theory groups, in the sense that they agree in the case of a trivial group and they satisfy analogous exact sequences. Although far f rom ha ving the generalit y of the work of ref. [37], our construction giv es a system- atic framewo rk in whic h to stud y Ramond-Ramond ﬁ elds on orbifolds with a Dirac quanti zation condition, includin g non-trivial contributions from ﬂ at p oten tials, and it represen ts a ﬁ r st step in the dev elopment of generalized diﬀerenti al cohomology theories in the equiv ariant setting. It is her e that the use of Bredon cohomology is p articularly imp ortan t, b oth b ecause of the equiv ariant Chern c haracter isomorphism an d b ecause the framew ork requ ir es explicit u se of diﬀeren tial forms , neither of whic h can b e accomo dated directly b y the Borel construction. The outline of the remainder of this p ap er is as f ollo ws. In S ection 1 we sum m arize some basic notions ab out the cohomology theories of sp aces with group actions. In Section 2 we present a detailed deﬁnition of Bredon co homology and the construction of the equiv ariant Ch ern characte r of ref. [46], as these ha ve not m ade app earences b efore in th e physics literature. These ﬁr st t wo sections give the main mathematical b ac kground for the rest of the p ap er. In Section 3 we mak e a br ief excursion in to the descrip tion of D-branes using geometric equiv ariant K -homology , sho w in g that the us e of K-cycles is very w ell-suited to the description of fr actional D-branes and their top olog ical c harges compu ted using equiv arian t Dirac op erator theory . In Section 4 RAMOND-RAM OND FIELDS, FRACTIONAL BRA NES AND ORBIFOLD DIFFERENTIAL K- THEOR Y 3 w e use Bredon cohomology and the equiv arian t Ch ern c haracter to deﬁne Ramond -Ramond couplings to D-branes on orbifolds and compare it w ith p r evious examp les in the literature. Our formulas in clude the appropr iate gra v itational con tributions w hic h are derived from an equiv ariant v ers ion of the Riemann-Ro c h theorem and equiv arian t index theory . I n Section 5 we giv e a detailed mathematical construction of the orbifold diﬀeren tial K-theory groups, and prov e that they ﬁt into appropr iate exact sequences which are useful in applications. In Section 6 we use the orbif old diﬀerenti al K-theory to d escrib e the ﬂux quan tization of Ramond-Ramond ﬁelds on orbifolds by w riting an equiv ariant v ersion of the Ramond-Ramond curr en t in terms of the equiv ariant Chern c haracter. W e also study the group of ﬂ at p oten tials in detail, and illustrate ho w the sp ectral sequence for equ iv ariant K-theory can b e used to determine obstru ction classes in Bredon cohomology w hic h yield stabilit y conditions for D-br anes and ﬂuxes on orbifolds. App end ix A con tains some bac kgroun d material on fun ctor categories used in the main text, App end ix B r ecords the deﬁn itions of equ iv ariant K-homology , wh ile Ap p endix C demons trates the us e of geometric equiv ariant K-cycles in the classiﬁcation of D-brane c h arges on orb ifolds. Ac kno wlegmen t s. W e are grateful to J. Figueroa-O’F arrill, D. F reed, J . Greenlees, J. Ho wie, A. Konechn y , W. L ¨ uc k, M. La w son, R. Reis, P . T ur n er and S. Willerton for h elpful discuss ions and corr esp ondence. This w ork was supp orted in part by the Marie Cu rie Researc h T rain- ing Net work Gran t F orcesUniv erse (con tract no. MR TN-CT-2004-00 5104) from the Europ ean Comm u nit y’s Sixth F ramew ork P rogramme. 1. Cohomology of sp aces with s ymmetries In this section we will recall some basic notions ab out (generalized) equiv arian t cohomology theories th at we will n eed throughout this p ap er. In the follo wing, X denotes a top ological space and G a ﬁ nite group, un less otherwise stated. Throughout a (left) action G × X → X of G on X w ill b e d enoted ( g , x ) 7→ g · x . Th e stabilizer or isotrop y group of a p oint x ∈ X is denoted G x = { g ∈ G | g · x = x } . Recall that a conti n 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 . 1.1. G -complexes. 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 fr om X n − 1 b y “attac hing” equiv ariant cells by the follo wing pr o cedure. Deﬁne X 0 = a j ∈ J 0 G/K j , with K j a collect ion of subgrou p s of G and the stand ard (left) G -action on an y coset sp ace G/K j . F or n ≥ 1 set X n =  X n − 1 ∐ a j ∈ J n  B n j × G/K j   . ∼ (1.1) where the equiv alence relation ∼ is generated by G -equiv arian t “attac hing maps” φ n j : S n − 1 j × G/K j − → X n − 1 . (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 B n j × G/ K j (resp. ˚ B n j × G/ K j ) a c lose d (resp. op en ) n -cell of orbit t y p e G/K j . As usual, we call the sub space X n the n -sk eleton 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 C W-decomp ositio n is called a G -c omplex . 4 RICHARD J. SZABO AND ALESSANDRO V ALENTINO When G = e is the trivial group, a G -complex is ju st an ordinary CW-complex. In general, if X is a G -complex then the orb it sp ace X/G is an ordinary CW-complex. Con v ersely , there is an in timate relation b et wee n G -complexes and ordinary CW-complexes wheneve r G is a discrete group. Let X b e a G -space which is an ordin ary CW-complex. W e say that G acts c el lularly on X if 1) F or eac h g ∈ G and eac h 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 id entit y . Then we hav e the follo wing Prop osition 1.1. 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 th at X is a smo oth manifold, we requir e the G -ac tion on X to b e smo oth and there is an analogous result. Recall that the app licabilit y of algebraic top ology to manifolds relies on th e fact th at any m an if old comes equip ed with a canonical CW-decomp osition. In the case in wh ic h a group acts on the manifold one has the follo wing r esult due to Illman [38, 39]. Theorem 1.2. 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. The collection of G -complexes with G -maps as morph isms form a category . W e are in terested in equiv arian t cohomolo gy theories deﬁn ed on this catego ry (or on sub cate gories th ereof ). 1.2. Equiv ariant cohomology theories. W e will now brieﬂy sp ell out the main ingredients in volv ed in bu ilding an equiv ariant cohomology theory on th e category of ﬁnite G -co mplexes, lea ving the details to the comprehensive treatmen ts of refs . [23] and [45], and fo cusing ins tead on some explicit examples. Fix a group G and a commutat iv e r ing R . A G -c ohomolo gy the ory E • G with v alues in R -mo dules is a collection of con tr a v arian t functors E n G from the catego ry of G - CW pairs to the category of R -mo dules ind exed b y n ∈ Z together with n atural 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 inv ariance, long exact sequence of a pair, exci- sion, and disjoin t u nion. The theory is called or dinary if for an y orb it G/H one has E q G ( G/H ) = 0 for all q 6 = 0. Th ese axioms are formulated in an analogous w ay to that of ordinary cohomology . The new ingredien t in an equiv ariant cohomology theory (whic h we ha ve not yet deﬁned) are the induction structur es , whic h we sh all now describ e. Let α : H → G b e a group homomorph ism, and let X b e an H -sp ace. Deﬁne the induction of X with r esp e ct to α to b e the G -space in d α X giv en b y ind α X := G × α X . This is the quotien t of the pro d uct G × X b y 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 ′ · [ g , x ] = [ g ′ g , x ]. If H < G and α is the su bgroup inclusion, the ind uced G -space is d enoted G × H X . An e quivariant c ohomolo gy the ory E • ( − ) with values in R -mo dules consists of a collect ion of G -cohomolo gy th eories E • G with v alues in R -mo dules for eac h group G suc h that for any group homomorphism α : H → G and any H -CW pair ( X, A ) with k er ( α ) acting freely on X , th ere are for eac h n ∈ Z n atur al isomorp hisms ind α : E n G  ind α ( X, A )  ≈ − → E n H ( X, A ) (1.3) satisfying RAMOND-RAM OND FIELDS, FRACTIONAL BRA NES AND ORBIFOLD DIFFERENTIAL K- THEOR Y 5 (a) Comp atibilit y with the cob oundary homomorph isms: δ n H ◦ in d α = ind α ◦ δ n G ; (b) F unctorialit y: If β : G → K is another group homomorphism suc h that k er ( β ◦ α ) acts freely on X , then for ev ery n ∈ Z one has ind β ◦ α = ind α ◦ in d 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 -homeomorph ism and E n K ( f 1 ) is the morphism on K -cohomology ind uced by f 1 ; and (c) Comp atibilit y with conjugation: F or g , g ′ ∈ G d eﬁne Ad g ( g ′ ) = g g ′ g − 1 . T hen the homomorphism ind Ad g coincides with E n G ( f 2 ), wher e f 2 : ( X , A ) ≈ − → ind Ad g ( X, A ) x 7− →  e , g − 1 · x  is a G -homeomorphism, where throughout e den otes the identit y element in the group G . Th us the induction stru ctures connect the v arious G -cohomologies and k eep trac k of the equiv- ariance. They will b e very imp ortant in the construction of the equiv arian t Ch ern charac ter for equiv ariant K-theory in the n ext section, even if we are only in terested in a ﬁxed group G . Example 1.3 ( Bor el c ohomolo gy ) . Let H • b e a cohomology theory for C W-pairs (for example, singular cohomology). Deﬁne H n G ( X, A ) := H n  E G × G ( X, A )  where E G is the total space of th e cla ssifying pr in cipal G -bun dle E G → B G whic h is con tractible and carries a free G -ac tion. This is calle d ( e qui v ariant ) Bor e l c ohomo lo gy , and it is the most commonly used f orm of equiv arian t cohomolog y in the ph ysics literature. No te that H • G is w ell-deﬁn ed b ecause th e quotien t E G × G X is uniqu e u p to the h omotop y t yp e of X/G . Th e ordinary G -cohomolog y structures on H • G are inh er ited from the cohomology structur es on H • . The induction structures for H • G are constructed as follo w s. Let α : H → G b e a group homomorphism and X an H -space. Deﬁne b : E H × H X − → E G × G G × α X ( ε, x ) 7− →  E α ( ε ) , e , x  where ε ∈ E H , x ∈ X and E α : E H → E G is the α -equiv arian t map indu ced by α . The induction map ind α is then giv en by pu llbac k ind α := b ∗ : H n G (ind α X ) = H n ( E G × G G × α X ) − → H n ( E H × H X ) = H n H ( X ) . If k er( α ) acts freely on X , th en the map b is a homotopy equiv alence and hen ce the map in d α is an isomorphism. Example 1.4 ( Eq uivariant K-the ory ) . In ref. [58], equ iv ariant top ological K-theory is deﬁned for an y G -complex X as the ab elian group completion of the semigroup V ect C G ( X ) of complex G -v ector bu ndles o ver X , i.e. , bu ndles E → X together with a lift of the G -action on X to the ﬁbres. The higher groups are deﬁned v ia iterated su sp ension. T o deﬁne th e induction structures, recall that if X is an H -sp ace and α : H → G is a group homomorph ism, then the m ap ϕ : X − → G × α X x 7− → ( e, x ) 6 RICHARD J. SZABO AND ALESSANDRO V ALENTINO is an α -equiv arian t map w hic h em b eds X as th e subsp ace H × α X of G × α X , and whic h induces via pu llb ac k of v ector bu ndles the h omomorphism ϕ ∗ : K • G ( G × α X ) − → K • H ( X ) . This map deﬁnes the indu ction structure. It is in vertible when k er ( α ) acts freely on X , with in verse the “extension” map E 7→ G × H E for any H -v ector bun dle E o v er X . The induction structure can b e used to prov e the we ll-kno wn e quivariant exc i sion the or em K • G/ N ( X/ N ) ∼ = K • G ( X ) (1.4) where N is a normal s u bgroup of G acting freely on X . In deed, one has X/ N ∼ = ( G/ N ) × G X and if we deﬁne α : G → G/ N to b e the quotien t map, then K • G/ N  ( G/ N ) × α X  ∼ = K • G ( X ) since ker( α ) = N acts freely on X . 2. The equiv ariant Chern character In this section we will describ e the equiv ariant Chern c haracter for the K • G functor and its target cohomology th eory , Bredon cohomology . T o this end , w e will introdu ce some tec h nology related to m o dules o ver fun ctor categ ories, giving the n ecessary deﬁnitions and directing the reader to th e relev an t literature for further details. Some p ertinen t asp ects of fu nctor categories are sum marized in App endix A. 2.1. Chern character in top ological K-theory. Let us b egin b y r ecalling some b asic notions ab out the ordinary Ch ern c h aracter. Deﬁne π −• K to b e the complex K-theory rin g of th e p oin t. It is the Z -graded ring Z [[ u, u − 1 ]] of Laur en t p olynomials freely generated by an elemen t u of degree deg( u ) = 2, where u − 1 ∈ K − 2 (pt) is called the Bott elemen t and is represented by the Hopf bun dle ov er S 2 . O ne then h as a homomorph ism c h : K • ( X ) − → H( X ; R ⊗ π −• K) • whic h indu ces th e natural Z -graded ring isomorph ism K • ( X ) ⊗ R ≈ − → H( X ; R ⊗ π −• K) • for any ﬁ nite CW-complex X . Th is statement is true even if w e tens or o ver Q . The u se h ere of the K-theory of the p oint as the co eﬃcient r ing serv es ju st as a re-grading of the cohomology ring H • ( X ; R ). F or example, it is easy to chec k th at H( X ; R ⊗ π −• K) 0 ∼ = H ev en ( X ; R ) . In particular, the Chern c h aracter tells us that K-theory and cohomology are the same thing up to torsion. It is natural now to ask if th er e exists such a morph ism f or equiv ariant K-theory . One m ight naiv ely thin k that the correct target th eory for the equiv arian t Chern charact er would naturally b e Borel cohomology . But the prob lem is muc h more su btle than it ﬁ rst may seem. The crucial p oin t is that while in the ordinary cohomolo gy of (ﬁnite) CW-complexes the building blo c ks are the cohomology groups of a p oin t, in the equiv ariant case they are th e cohomology groups of the orbits G/H for all subgroups H of G , as w e saw in Section 1.1. Any equiv arian t cohomology th eory E • G on th e category of ﬁ nite G -complexes is completely sp eciﬁed by its v alue on the orbit spaces G/H . A lo c alization the or em d ue to A tiya h and Segal [4] tells us that the Borel cohomology of a G -space X is isomorphic to its equiv ariant K -theory lo calized at the augmen tation id eal in the representa tion ring R ( G ) consisting of all elements w hose charac ters v anish at the iden tity e in G (regarding K • G ( X ) as a mo d ule o ver R ( G )). Lo calizing at a prime ideal of R ( G ) corresp ond s to restricting X to the set of ﬁxed p oint s of an asso ciated conjugacy RAMOND-RAM OND FIELDS, FRACTIONAL BRA NES AND ORBIFOLD DIFFERENTIAL K- THEOR Y 7 class of cyclic sub groups of G . In this sense, Borel cohomology do es not tak e in to accoun t the “con tributions” of th e non-trivial elemen ts in G , and hence of th e ﬁxed p oin ts of the G -action. There are seve ral approac h es to the equiv ariant C hern charac ter (see refs. [5, 59, 15, 31, 1], for example) which strongly d ep end on the types of groups inv olve d (discrete, contin uous, etc. ) and on the rin g one tensors with ( R , C , etc. ). As w e are in terested in ﬁnite groups and real co eﬃcien ts for our p h ys ical applications later on, we will us e the Chern c haracter constru cted in refs. [45] and [46]. Thus we pro ceed to th e more abstract, but p o werful and compact, deﬁnition of Bredon cohomology , whic h will turn out to b e the b est suited equiv ariant cohomology theory for all of our p urp oses. 2.2. Bredon cohomology. Let G b e a discrete grou p . The orbit c ate gory Or ( G ) of G is deﬁned as the category w hose ob jects are h omogeneous spaces G/H , w ith H < G , and wh ose morphisms are G -maps b etw een them. F rom general considerations [23] it follo ws that a G -map b et ween t wo 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  (2.1) for some a ∈ G suc h that a − 1 H a < K . If F is an y family of s u bgroups of G then there is a sub category Or ( G, F ) with ob jects G/H for H ∈ F . A simple examp le is pr o vided b y the cyclic groups G = Z p with p prime, for whic h th e orb it category h as just t wo ob jects, G/e = G and G/G = pt. If Ab denotes the category of ab elian groups , then a c o eﬃcient system is a functor F : Or ( G ) op − → Ab where Or ( G ) op denotes the dual category to Or ( G ). With such a fun ctor and an y G -complex X , 1 one can deﬁne for eac h n ∈ Z the group C n G ( X, F ) := Hom Or ( G )  C n ( X ) , F  (2.2) where C n ( X ) : Or ( G ) op → Ab is the pro jectiv e fu nctor deﬁned by C n ( X )( G/H ) := C n  X H  , the cellular homology of th e ﬁxed p oin t complex X H :=  x ∈ X   h · x = x ∀ h ∈ H  . (2.3) In eq. (2.2), Hom Or ( G ) ( − , − ) denotes the group of natural transform ations b et w een t w o con- tra v arian t functors, with the group structur e inherited b y the images of the functors in Ab . The f unctorialit y prop ert y of C n ( X )( G/H ) is the natur al one induced b y the ident iﬁcation X H ∼ = Map G ( G/H, X ). Indeed, th e 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 s een to b e in verse to eac h other, and the desired homeomorphism is obtained by giving the sp ace Map G ( G/H, X ) the c omp act-op en top ology . In particular, a G -map (2.1) induces a cellular map X K → X H , x 7→ a · x . These group s can b e expressed in terms of th e G -complex stru ctur e of X . If the n -skele ton X n is obtained b y attac hing equiv arian t cells as in eq. (1.1) with K j the stabilizer of an n -cell of X , then the cellular c hain complex C • ( X ) consists of G -mod ules 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 )  . 1 When G is an inﬁn ite 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. 8 RICHARD J. SZABO AND ALESSANDRO V ALENTINO F or eac h n ≥ 0, the group C n G ( X , F ) is the direct limit functor o ver all n -cells of orbit t yp e G/K j in X of th e groups F ( G/K j ). Th is f ollo ws b y restricting eq. (2.2) to the f ull sub categ ory Or ( G, F ( X )), with F ( X ) the family of su bgroups of G which o ccur as stabilizers of the G -action on X [50]. The Z -graded group C • G ( X, F ) = L n ∈ Z C n G ( X, F ) inherits a cobou n dary oper ator δ , and hence the structur e of a co c hain complex, from the b ou n dary op erator on cellular c hains. T o a natural transformation f : C n ( X ) → F , one asso ciates the natural transformation δf deﬁn ed b y 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 b oundary op erator ∂ . Then the Br e don c ohomolo gy of X with co eﬃcien t sy s tem F is d eﬁ ned as H • G ( X ; F ) := H  C • G ( X, F ) , δ  . This d eﬁnes a G -cohomolog y theory . See ref. [44] for the p ro of that H • G ( X ; F ) is an equiv ariant cohomology theory , i.e. , for the deﬁnition of the ind uction str ucture. One can also deﬁn e cohomology groups b y r estricting the functors in eq. (2.2) to a sub categ ory Or ( G, F ). The deﬁnition of Bredon cohomology is indep endent of F as long as F conta ins the family F ( X ) of stabilizers [50]. T h is fact is useful in exp licit calculations. In particular, b y taking F = H to consist of a single su bgroup, one sh o ws that the Bred on cohomolog y of G -homogeneous spaces is giv en by H • G ( G/H ; F ) = H 0 G ( G/H ; F ) = F ( G/H ) . (2.4) Example 2.1 ( T rivial gr oup ) . When G = e is th e trivial grou p , i.e . , in the non-equiv arian t case, the functors C n ( X ) and F can b e identiﬁed with the ab elian grou p s 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 ordin ary n -th cohomology group of X w ith co eﬃcien ts in F . Example 2.2 ( F r e e action ) . If the G -action on X is fr e e , then all stabilizers K j are tr ivial and X H = ∅ for eve ry H ≤ G , H 6 = e . In this case one may tak e F = e to compute th e co chain complex C • G ( X, F ) ∼ = Hom G  C • ( X ) , F ( G/e )  and so the Bredon cohomology H • G ( X ; F ) coincides with the equiv ariant cohomology H • G  X ; F ( G/e )  of X with coeﬃcients 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 every H ≤ G and the v alue on morphisms in Or ( G ) op giv en by the ident it y h omomorp hism of Z , this group redu ces to the ord in ary cohomology H • ( X/G ; Z ). Example 2.3 ( T rivial action ) . If the G -act ion on X is trivial , then th e collection of isotrop y groups K j for the G -action is the s et of all sub groups of G and X H = X for all H ≤ G . In this case the fun ctor C n ( X ) can b e decomp osed int o a sum ov er n -cells of pro jectiv e functors P K j with K j = G [50], and so one has Hom Or ( G )  C n ( X ) , F  ∼ = Hom  C n ( X ) , lim ← − Or ( G ) op F ( G/H )  where the in v erse limit fun ctor is tak en ov er the opp osite category Or ( G ) op . It follo ws that the Bredon cohomology H • G ( X ; F ) = H •  X ; F ( G/G )  is the ordinary cohomology of X with co eﬃcien ts in the ab elian group F ( G/G ) = F (pt). RAMOND-RAM OND FIELDS, FRACTIONAL BRA NES AND ORBIFOLD DIFFERENTIAL K- THEOR Y 9 2.3. Represen ta t ion ring functors. In what follo ws we will sp ecializ e the co eﬃcien t system for Bredon cohomolog y to the r epr esentation ring functor F = R ( − ) deﬁn ed on th e orb it catego ry Or ( G ) b y sending the left coset G/H to R ( H ), the complex representa tion ring of the group H . A morph ism (2.1) is sen t to the homomorph ism R ( K ) → R ( H ) giv en b y ﬁ rst restricting the r epresen tation from K to the subgroup conjugate to H , and then conjugating b y a . S ince R ( − ) is a fu nctor to r ings, the Bredon cohomology H • G ( X ; R ( − )) naturally has a ring structure. Note that R ( H ) ∼ = K 0 G ( G/H ) = K • G ( G/H ) , (2.5) whic h follo w s from the ind uction structure of Example 1.4 with X = pt and α the subgroup inclusion H ֒ → G . By eq. (2.4) the group (2.5) also coincides w ith the Bredon cohomology group H • G ( G/H ; R ( − )), which is already an ind ication that Br ed on cohomology is a b etter relativ e of equ iv ariant K-theory than Borel cohomology . Ind eed, using the induction structure of Example 1.3 one sh o ws that the Borel cohomology H • G ( G/H ) = H • ( B H ) coincides with th e cohomology of the classifying space B H = E H /H , whic h computes the group cohomology of H and is t yp ically inﬁ nite-dimensional (ev en for ﬁnite group s H ). In this pap er w e w ill sh o w th at the Bred on cohomolog y H • G ( X ; R ( − )) giv es a m ore p recise r ealizat ion of th e stringy orbifold cohomolog y of X in the con text of op en string theory . In the construction of the equiv arian t Chern c h aracter in Section 2.4 b elo w , it will b e im- p ortant to represen t the rational Bredon cohomology H • G ( X ; Q ⊗ R ( − )) as a certain group of homomorphisms of fu nctors, similarly to the co chain groups (2.2). F or this, we in tro duce an- other category Sub ( G ). The ob jects of S ub ( G ) are the sub groups of G , 2 and the m orp hisms are giv en by Mor Sub ( H,K ) :=  f : H → K   ∃ g ∈ G , g H g − 1 ≤ K , f = Ad g   Inn( K ) . In particular, there is a fu nctor Or ( G ) → Sub ( G ) whic h s ends the ob ject G/H to H and the morphism (2.1) in Or ( G ) to the homomorphism ( g 7→ a − 1 g a ) in Sub ( G ). If a lies in th e cen tralizer Z G ( H ) :=  g ∈ G   g − 1 H g = H  (2.6) of H in G , then the morph ism (2.1) is sen t to the id en tit y map. Any functor F : Sub ( G ) op → Ab can b e naturally regarded as a fu n ctor on Or ( G ) op . Deﬁne the quotien t functors C qt • ( X ) , H qt • ( X ) : S ub ( G ) op → Ab by C qt • ( X )( H ) := C •  X H / Z G ( H )  and H qt • ( X )( H ) := H •  X H / Z G ( H )  . F or an y fu nctor 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 (2.6) is precisely th e group of automorp h isms of G/H in the orbit category Or ( G ) sent to the ident it y m ap in the subgroup category S ub ( G ) , we ﬁn ally ha ve C • G ( X, F ) = Hom Or ( G )  C • ( X ) , F  ∼ = Hom Sub ( G )  C qt • ( X ) , F  . (2.7) A t this p oin t one can apply eq. (2.7) to the r ational repr esentati on ring fun ctor F = Q ⊗ R ( − ), whic h by construction can b e r egarded as an injectiv e fu n ctor Sub ( G ) op → Ab , to prov e the Lemma 2.4 ([46]) . 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 ( − )  . 2 If G is inﬁ nite then one should restrict to ﬁnite subgroups of G . 10 RICHARD J. SZABO AND ALESSANDRO V ALENTINO 2.4. Chern c haracter in equiv a ria n t K-theory. Before sp elling out th e deﬁnition of the equiv ariant C hern charact er, w e recall some basic pr op erties of the equ iv ariant K-theory of a G -complex X . Let H b e a sub group of G , and consid er the ﬁxed p oint subspace of X deﬁned b y eq. (2.3). Th e action of G do es not preserve X H , bu t the action of the normalizer N G ( H ) of H in G do es. If we denote with i : X H ֒ → X the inclusion of X H as a sub space of X , and with α : N G ( H ) ֒ → G the inclusion of N G ( H ) as a sub group of G , th en we n aturally h a ve the equalit y i ( n · x ) = α ( n ) · i ( x ) for all n ∈ N G ( H ) and x ∈ X H . It follo ws that the ind u ced homomorph ism on equ iv ariant K-theory is a map [58] 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 kn own p rop erty [46]. Let N ✁ G be a ﬁnite normal sub group, and let Rep( N ) b e the catego ry of (isomorphism classes of ) irreducible complex r ep resen tations of N . Let X b e a (prop er) G/ N - complex, and let G act on X via th e pro jection map G → G/ N . Then for any complex G -ve ctor bundle E → X and an y represent ation V ∈ Rep( N ), deﬁn e Hom N ( V , E ) as the vec tor bund le 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 v ia the pr o jectio n map. No w if H ≤ G is a subgroup whic h comm utes with N , [ H , N ] = e , then one can induce an H -v ector bund le from Hom N ( V , E ) by deﬁnin g ( h · f )( v ) = h · f ( v ) , v ∈ V for any h ∈ H and any f ∈ Hom N ( V , E ) (remem b ering that G acts on E ). Hence there is a h omomorp hism of rin gs Ψ : K • G ( X ) − → K • H ( X ) ⊗ R ( N ) deﬁned on G -ve ctor bun dles by Ψ  [ E ]  := X V ∈ Rep( N )  Hom N ( V , E )  ⊗ [ V ] . (2.8) This homomorph ism satisﬁes some n aturalit y prop erties wh ich are describ ed in detail in r ef. [46]. Note that the sum (2.8) is ﬁnite , since N is a ﬁn ite s ubgroup. W e are now ready to constru ct th e equiv arian t Chern c haracter as a homomorp hism c h X : K 0 , 1 G ( X ) − → H ev en , o dd G  X ; Q ⊗ R ( − )  for any ﬁnite prop er G -co mplex X . The strategy u sed in ref. [46] is to construct Z 2 -graded homomorphisms c h H X : K • G ( X ) − → Hom  H • ( X H / Z G ( H )) , Q ⊗ R ( H )  (2.9) for an y ﬁnite subgroup H , and then glue them together as H v aries through the ﬁnite subgroup s of G . T o deﬁ n e the homomorphism (2.9), w e ﬁr st comp ose the ring homomorph isms 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 )  E G × X H  ⊗ R ( H ) where π 2 : E G × X H → X H is the pro jection on to the second factor. By using the induction structure of E x amp le 1.4, one then has K • Z G ( H )  E G × X H  ⊗ R ( H ) ≈ − → K •  E G × Z G ( H ) X H  ⊗ R ( H ) c h ⊗ id − − − → H  E G × Z G ( H ) X H ; Q ⊗ π −• K  • ⊗ R ( H ) where c h is the ordinary Chern charact er. One ﬁnally has H •  E G × 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 )  , RAMOND-RAM OND FIELDS, FRACTIONAL BRA NES AND ORBIFOLD DIFFERENTIAL K- THEOR Y 11 where the ﬁr st isomorphism follo ws from the Ler ay sp ectral sequence by observing that the ﬁ bres of the p ro jection E G × Z G ( H ) X H − → X H  Z G ( H ) are all classifying spaces of ﬁ nite groups, h a ving triv al r educed cohomology with Q -co eﬃcien ts and are therefore Q -acyclic. The equiv ariant Ch er n c haracter is now deﬁned as 3 c h X = M H ≤ G c h H X . (2.10) By using the v arious naturalit y prop erties of the homomorph ism (2.8) [46 ], one sees that ch X tak es v alues in Hom Sub ( G )  H qt • ( X ) , Q ⊗ R ( − )  , and by Lemma 2.4 it is th u s a Z 2 -graded map c h X : K • G ( X ) − → Hom Sub ( G )  H qt • ( X ) , Q ⊗ R ( − )  ∼ = H • G  X ; Q ⊗ R ( − )  . This map is w ell-deﬁned as a ring homomorphism b ecause all maps inv olv ed ab o ve are homomor- phisms of r ings. As with the deﬁnition of Br ed on cohomology , the sum (2.10) ma y b e restricted to any family of sub groups of G con taining the set of stabilizers F ( X ). T o conclude, w e ha ve to p ro ve that this map b ecomes an isomorph ism up on tensoring o ver Q . F or this, one p ro ves that the morph ism c h X in eq. (2.10) is an isomorph ism on h omogeneous spaces G/H , with H a ﬁ nite su b group of G , and then u ses ind uction on the num b er of orbit t yp es of cells in X along with the Ma y er-Vietoris sequen ces for the push out squares indu ced by the attac hing G -maps (1.2). Th e isomorphism on G/H is a consequence of the isomorphisms (2.4) and (2.5). The d etails ma y b e found in ref. [46]. Let π −• K G ( − ) b e the fu nctor on Or ( G ) deﬁned by G/H 7→ K • G ( G/H ). T h en on e has th e follo win g Theorem 2.5. F or any ﬁnite pr op er G -c omplex X , the Chern char acter ch 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 ( − )  • . 3. 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 h arges on global orbifolds of T yp e I I sup ers tr ing theory with v anishing H - ﬂux. Let X b e a s m o oth manifold and G a (ﬁ nite) group acting b y diﬀeomorphisms of X . Ramond-Ramond c harges on th e global orbifold [ X/G ] are classiﬁed by the equiv ariant K-theory K • G ( X ) as d eﬁned in Ex amp le 1.4 [34, 54, 62 ]. Dually , the equ iv ariant K-homology K G • ( X ) leads to an elegan t description of fr actional D-br anes pinn ed at the orbifold singularities in terms of equiv arian t K-cycles. In the f ollo wing w e will frequently refer to App endix B for detailed deﬁnitions and tec hnical asp ects of equiv ariant K-homology , fo cu sing instead here on s ome of the more qualitativ e asp ects of D-branes on orb ifolds in this language. In the remainder of this pap er we will assume for deﬁniteness th at G is a ﬁn ite group . 3.1. F ractional D-branes. As in the non-equiv arian t case G = e [56, 57, 60], a ve ry natural description of D-branes in th e orbifold space, which captures the inherent geometrica l picture of D-brane states inv olving wrapp ed cycles in spacetime, is pro vided b y the top ological realization of the equiv ariant K-homology groups K G • ( X ). Th e cycles for this homology theory , called G - equiv ariant K-cycles, liv e in an add itive categ ory 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 -v ector bundle o ver W , and f : W − → X (3.1) 3 If G is inﬁ nite then the direct sum in eq. (2.10 ) is understo o d as the inv erse limit functor ov er the dual subgroup category S ub ( G ) op . 12 RICHARD J. SZABO AND ALESSANDRO V ALENTINO is a G -map. The group K G • ( X ) is the quotient of this category by the equiv alence r elation generated by b ord ism, d irect sum, and vect or bundle m o diﬁcation, as detailed in App endix B. Note that W need n ot b e a sub manifold of spacetime. Ho wev er , since X is a manifold, we can restrict the b ordism equiv alence relation to diﬀer ential b or dism [56] and assum e that the map (3.1) 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 in clud e branes supp orted on n on-represen table cycles in spacetime. This deﬁ nition of equiv arian t K-h omology thus giv es a concrete geo metric mo del for the top ologic al classiﬁcation of D-branes ( W, E , f ) in a global orbifold [ X/G ] wh ic h captures the physical constructions of orb ifold D-branes as G -in v arian t states of br anes on th e co vering space X . In the su bsequent sections w e will stud y the p airing of Ramond-Ramond ﬁ elds with these D-branes. Consider a D-brane lo calized at a generic p oint in the orbif old [ X/G ] with the action of the r e gular r epresen tation of G on the ﬁbr es of its Ch an-P aton gauge b u ndle, i. e. , the natural action of the group algebra C [ G ] as b ounded linear op erators ℓ 2 ( G ) → ℓ 2 ( G ). It corresp ond s to a G -orbit of suc h branes on the lea v es X g = { x ∈ X | g · x = x } , g ∈ G of the co vering space X . A t a G -ﬁxed p oint, this b rane splits u p into a set of fr actional br anes according to the decomp osition of the represent ation of G on the ﬁb res of the Chan-Pato n bund le into irredu cible G -mo d ules. S table fractional D-br an es corresp ond to b ound states of branes wr apping v arious collapsed cycles at th e ﬁxed p oints. Th ey are th u s stuc k at the orb ifold p oin ts and provide th e op en strin g an alogs of “t wisted sectors”. T o f ormulate this physic al construction in the language of equiv arian t K-cycles ( W, E , f ), let G ∨ denote the set of conjugacy classes [ g ] of elemen ts g ∈ G . It can b e regarded as a set of represent ativ es for the isomorphism classes π 0 Rep( G ), where Rep( G ) is the additiv e category of irreducible complex r ep resen tations of G whic h coincides with the category of D-brane b oundary conditions at the orbifold p oints. There is a natural sub categ ory D G frac ( X ) of D G ( X ) consisting of triples ( W , E , f ) f or whic h W is a G -ﬁxed space, i.e. , for which W g = W (3.2) for all g ∈ G . By G -equiv ariance this imp lies f ( W ) g = f ( W ) for all g ∈ G , and so the image of the br ane worldv olume lies in the sub space f ( W ) ⊂ \ g ∈ G X g . This is precisely the set of G -ﬁxed p oint s of X , and so the ob jects ( W , E , f ) of the cate gory D G frac ( X ) are naturally p in ned to the orbifold p oin ts. W e ca ll D G frac ( X ) the categ ory of “maximall y fractional D-branes”. In th is case, an application of Sc hur’s lemma sh o ws that the Chan-P aton b undle admits an isotopical d ecomp osition and there is a canonical isomorphism of G -bund les E ∼ = M [ g ] ∈ G ∨ E [ g ] ⊗ 1 1 [ g ] with E [ g ] = Hom G  1 1 [ g ] , E  , (3.3) where E [ g ] is a complex v ector bundle with trivial G -action and 1 1 [ g ] is the G -bu n dle W × V [ g ] with γ : G → En d( V [ g ] ) the irreducible rep resen tation corresp ond ing to the conjugacy class [ g ] ∈ G ∨ . This isotopical d ecomp osition deﬁnes the action of G on the Ch an-P aton factors of the D-brane, an d it im p lies the well-kno w n isomorp hism K • G ( W ) ∼ = R ( G ) ⊗ K • ( W ) (3.4) for G -ﬁxed spaces W [58 ]. T his is a sp ecial case of the homomorphism Ψ d eﬁ ned in eq. (2.8). F rom the d irect su m r elation in equ iv ariant K-homology it follo ws that a D-brane, represente d b y a K -cycle ( W, E , f ), sp lits at an orbifold p oint in to a sum o ver irreducible fr actional b ranes represent ed by the K-cycles ( W , E [ g ] ⊗ 1 1 [ g ] , f ), [ g ] ∈ G ∨ . RAMOND-RAM OND FIELDS, FRACTIONAL BRA NES AND ORBIFOLD DIFFERENTIAL K- THEOR Y 13 It is imp ortant to realize that the full category D G ( X ) con tains m uc h m ore information, and in particular the fr actional D-branes w ill not generically form a spann ing set of K-cycles for th e group K G • ( X ) (except in some sp eciﬁc examples). Ho wev er , it follo w s fr om the b ordism relation in equiv arian t K-homology that an y tw o G -equiv arian t K-cycles ( W i , E i , f i ), i = 0 , 1 whic h are b ordant along the s ame G -orbit determine the same elemen t in K G • ( X ). Th is is exp ected since a pu rely top ological classiﬁcation su c h as equiv arian t K-homology cannot capture th e p ositio nal mo duli asso ciated with the regular D-br anes in X/G . A related w ay to und erstand the r ole of fractional b ranes is through the connection b et we en geometric K -homology and b ord ism theory [56]. Let MSp in c • ( X ) b e the or dinary s p in c b ordism group of X , which forgets ab out the G -action and consists of spin c b ordism classes of pairs ( W , f ). Th en there is a m ap MSpin c • ( X ) ⊗ MSpin c • (pt) Rep( G ) − → D G frac ( X ) , ( W , f ) ⊗ V 7− → ( W , W × V , f ) (3.5) whic h descends to giv e a homomorphism MSpin c • ( X ) ⊗ MSpin c • (pt) K G • (pt) − → K G • ( X ) . When G = e this is the isomorphism of K • (pt)-mo dules indu ced by the At iy ah-Bott-Shapiro orien tation, and th e map (3.5) determines K-cycle generators in terms of s p in c b ordism genera- tors [56]. The equiv arian t extension of the A tiyah-Bo tt-Shapiro constru ction is give n in ref. [42] in terms of ﬁnite-dimensional Z 2 -graded G -Cliﬀord mo du les. Since an y G -Cliﬀord mo du le can b e built as a direct sum of tensor pro ducts of G -mo dules and ordinary Cliﬀord mod ules (see App end ix B), there is an isomorphism of R ( G )-mo dules K • G (pt) ∼ = R ( G ) ⊗ K • (pt) and so these represent ation mo dules conta in n o new information ab out the orbifold group. This seems to suggest th at, at least in certain cases, span n ing sets of equiv ariant K-cycles can b e tak en to lie in the sub category D G frac ( X ). 3.2. T op ological c harges. The top ologic al charge of a fractional D-brane, in a giv en closed string t w isted secto r of the orbifold strin g theory on a G -spin c manifold X , can b e computed b y using the equiv ariant Dirac op erator theory d evelo p ed in App endix B. The equiv arian t in- dex of the G -in v ariant spin c Dirac op erator D / X E coupled to a G -v ector bundle E → X tak es v alues in K • G (pt) ∼ = R ( G ). W e can turn this into a homomorphism on K G • ( X ) with v alues in Z b y comp osing with the p ro jection R ( G ) → Z deﬁned by taking the m ultiplicit y of a giv en represent ation γ : G − → End( V γ ) (3.6) of G on a ﬁnite-dimensional complex vec tor space V γ . There is a corresp onding class in the KK-theory group [ γ ] ∈ K K •  C [ G ] , End( V γ )  whic h is represent ed b y the Kasp aro v mo d ule ( V γ , γ , 0) asso ciated with the extension of the represent ation (3.6) to a complex r epresen tation of group rin g C [ G ]. By Morita equiv alence, the Kasparo v pr o duct with [ γ ] is the homomorphism on K-theory K 0  C [ G ]  − → K 0  End( V γ )  ∼ = K 0 ( C ) ∼ = Z induced by γ : C [ G ] → En d( V γ ). W e ma y then deﬁne a homomorphism µ γ : K G 0 ( X ) − → Z of ab elia n groups by µ γ  [ W , E , f ]  = Ind ex γ  f ∗ [ D / W E ]  := ass  f ∗ [ D / W E ]  ⊗ C [ G ] [ γ ] (3.7) on equiv arian t K -cycles ( W , E , f ) ∈ D G ( X ) (and extended linearly), wh ere ass : K G • ( X ) − → K •  C [ G ]  14 RICHARD J. SZABO AND ALESSANDRO V ALENTINO is the analytic assembly map constructed in App endix B. 3.3. Linear orbifolds. Let u s now consider a simple class of examples wherein ev eryth ing can b e made v ery explicit. Let V b e a complex ve ctor space of dimension dim C ( V ) = d ≥ 1, and let G b e a ﬁ nite sub group of SL( V ). Our s p acetime X is the G -space identiﬁed with the p ro duct X = R p, 1 × V , where G acts trivially on the Mink o wski space R p, 1 . F ractional D-br an es carrying themselves a complex linear represent ation of G , wh ic h is a submo d ule of R p, 1 × V , ha ve worldv olumes W linearly emb edded in the subspace R p, 1 and ha ve trans v erse sp ace giv en b y the orthogonal complemen t f ( W ) ⊥ ∼ = V with r esp ect to a c h osen inner pro d uct. S ince the sp ace of hermitean metrics is contract ible, all top ological qu an tities b elo w are indep end en t of this c hoice. As a G -space, V is equiv ariantly cont ractible to a p oin t and hence its compactly supp orted equiv ariant K-theory is giv en by [4 ] K • G, cpt ( V ) ∼ = K • G (pt) ∼ = R ( G ) = Z | G ∨ | . This group coincides w ith the Bredon cohomolo gy H • G, cpt ( V ; R ( − )), owing to the fact that th e equiv ariant Chern charac ter c h G/H of Section 2.4 is the identit y map (since the non-equiv arian t Chern charac ter c h = c 0 : K 0 (pt) → H 0 (pt; Z ) is the iden tit y map ). It follo ws that the fractional D-branes, as deﬁn ed by element s of equiv arian t K-theory , can b e identiﬁed with representat ions of the orb ifold group 4 γ = | G ∨ | M a =1 N a γ a consisting of N a ≥ 0 copies of the a -th irreducible r epresen tation γ a : G − → End( V a ) , a = 1 , . . . ,   G ∨   , whic h deﬁn es the actio n of G on the ﬁbres of the Chan-Pa ton bu ndle. More precisely , eac h irreducible fr actional brane is asso ciated to the G -bu ndle V × V a o ve r V . By P oincar ´ e dualit y , it follo ws f r om Prop osition 2.1 of ref. [56 ] that a basis for the equ iv ariant K-homology group K G • ( V ) is pro vid ed by the geometric equiv arian t K-cycles ( V , V × V a , id V ), a = 1 , . . . , | G ∨ | . By G -homoto p y inv ariance [56, Lemma 1.4] these cycles can b e con tracted to [pt , V a , i ], where i is the in clusion of a p oin t pt ⊂ V whose induced homomorp h ism i ∗ : K G • (pt) − → K G • ( V ) can b e take n to b e the ident it y map R ( G ) → R ( G ). This is simply the physica l statemen t that the stable fr actional b ranes in this case are D0-branes in Typ e I IA string theory (the Typ e I I B theory con taining n o suc h states). Th e G -in v arian t Dirac op erator D / pt V a is just Cliﬀord multipli- cation t wisted b y th e G -mo d ule V a , and thus the top olog ical c harges (3.7) of the corresp onding fractional br anes in the twisted sector lab elled by b are giv en by µ b  [pt , V a , i ]  = Ind ex γ b  [ D / pt V a ]  =  V a ⊗ (∆ + − ∆ − )  ⊗ C [ G ] [ γ b ] , where ∆ ± are the half-spin represent ations of SO(2 d ) on V ∼ = C d regarded as C [ G ]-modu les. Acting on th e c haracter rin g th e pro jection giv es [ W ] ⊗ C [ G ] [ γ b ] = χ W ( g b ), w here χ W : G → C is the c haracter of the G -mo dule W and [ g b ] ∈ G ∨ is the conjugacy class corresp ondin g to the irreducible rep r esen tation γ b . 4 If the transverse space V is instead a r e al linear G -mo dule, t hen throughout one should restrict to th e subring of R ( G ) consisting of representations associated t o conjugacy classes [ g ] ∈ G ∨ for which the centralizer Z G ( g ) acts on the ﬁx ed p oin t subspace V g by oriented automorphisms [40]. This will follo w immediately from the isomorphism (4.7) b elow with X = V . RAMOND-RAM OND FIELDS, FRACTIONAL BRA NES AND ORBIFOLD DIFFERENTIAL K- THEOR Y 15 4. Delocaliza tion a nd Ramond-Ramo nd coupl ings The purp ose of this section is to d escrib e the delo calizatio n of Bredon cohomology and the equiv ariant Chern characte r , int ro duced in Section 2, and to apply it to the study of the coupling b et ween Ramond-Ramond p oten tials and the D-branes of the previous section. In the follo wing w e will require s ome standard facts concerning strin g theory on global orbifolds, particularly its lo w-energy ﬁeld theory con tent. F or more details see refs. [24, 25]. 4.1. Closed string sp ectrum. The b oundary states corresp ond ing to the fractional b ranes constructed in Section 3 ha ve comp on ents in the t wisted sectors of th e closed strin g Hilb ert space H of orb ifold string theory on X . The closed string is an em b edding x : S 1 × R → X of the w orldsh eet cylinder, with lo cal coordin ates ( σ, τ ) ∈ S 1 × R , in to the G -spin c spacetime manifold X . The Hilb ert space H of physica l string states decomp oses into a direct sum o ve r t wisted sectors, eac h c h aracterize d b y a conjugacy class, as H = M [ g ] ∈ G ∨ H [ g ] (4.1) with only G -in v ariant states surviving in eac h sup erselect ion sector H [ g ] . Actually , the Hilb ert space factorizes into one sector f or eac h elemen t of the group G , bu t the action of G mixes the sectors within a giv en conju gacy class. The subspaces in eq. (4.1) are th u s deﬁn ed as H [ g ] := M h ∈ [ g ] H h , where H h is the subspace of states indu ced by the t wisted string ﬁ eld b oun dary condition x ( σ + 2 π , τ ) = h · x ( σ, τ ) (4.2) with an analogous condition on the w orld s heet fermion ﬁ elds (using a lift ˆ G of the orbifold group). Then G acts on th e subspace H [ g ] , and pro jecting on to G -in v arian t states in H [ g ] is equiv alent to p r o jecting onto Z G ( h )-in v arian t states in H h for any h in [ g ]. The lo w-energy limit of Typ e I I orbifold sup er s tring theory on X con tains Ramond-Ramond ﬁelds C [ g ] coming from the v arious t w isted sectors. The t wisted b ou n dary conditions (4.2) on the string em b edding map imp oses constraints on the lo w-energy sp ectrum. F or example, th e unt wisted sector giv en b y g = e con tains Ramond-Ramond ﬁelds d eﬁned on the entire spacetime manifold X , while the twisted sector represen ted b y g 6 = e giv es rise to ﬁelds deﬁned on ly on the ﬁxed p oin t sub manifold X g . The GSO pro jection then enforces the prop erties that the Ramond-Ramond form p oten tials C [ g ] determine self-dual ﬁelds in eac h t w isted sector, and that they b e of o dd degree in Type I IA theory and of ev en d egree in Typ e I IB theory . The Ramond-Ramond ﬁ elds can thus b e “organised” in to the d iﬀeren tial complex Ω • G ( X ; R ) := M [ g ] ∈ G ∨ Ω •  X g ; R  Z G ( g ) . (4.3) Here we consider only ﬁelds coming from in equiv alent t w isted sectors and mak e a c hoice of submanifold X g , since for any conju gate elemen t h ∈ [ g ] there is a d iﬀeomorphism X g ∼ = X h . (No c h oice is needed in the case in whic h G is an ab elian group.) As d ◦ g ∗ = g ∗ ◦ d for all g ∈ G , the der iv ation is giv en by d G := M [ g ] ∈ G ∨ d g where d g = d : Ω • ( X g ; R ) → Ω • ( X g ; R ) is the usual de Rh am exterior deriv ativ e. Note that only the cen tralizer subgroup of g in G acts (prop erly) on X g . 16 RICHARD J. SZABO AND ALESSANDRO V ALENTINO 4.2. Delo calization of Bredon cohomology. W e will no w show ho w Bredon cohomology can b e used to compu te the cohomology of the complex (4.3) of orbifold Ramond -Ramond ﬁ elds by giving a delo calized description of Bredon cohomology with r e al coeﬃcients, follo wing refs. [50] and [46] where further deta ils can b e found. This is the s tr ingy o rbifold cohomology of X , deﬁned as the ordinary (real) cohomology of the orbifold r esolution e X = ` [ g ] ∈ G ∨ X g / Z G ( g ). Note that there is a natural surjectiv e map π : e X → X deﬁned by ( x, [ g ]) 7→ x , and a natural injection X ֒ → e X in to the conn ected comp onent of e X corresp onding to th e unt wisted sector [ g ] = [ e ]. Denote with R ( − ) the real representa tion ring fu nctor R ⊗ R ( − ) on the orbit category Or ( G ). Let h G i denote the set of conju gacy classes [ C ] of cyclic s u bgroups C of G . Let R C ( − ) b e the cont ra v ariant fun ctor on Or ( G ) d eﬁned by R C ( G/H ) = 0 if [ C ] con tains no representa tiv e g C g − 1 < H , and otherwise R C ( G/H ) is isomorp h ic to the cyclotomic ﬁeld R ( ζ | C | ) ov er R gen- erated b y the p rimitiv e ro ot of u nit y ζ | C | of order | C | . A standard resu lt from the repr esen tation theory of ﬁ nite groups th en giv es a natural splitting R ( − ) = M [ C ] ∈h G i R C ( − ) . By deﬁn ition, for any 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 )  ∼ = M ( G/C ) ⊗ N G ( C ) R C ( G/C ) where the normalizer subgroup N G ( C ) acts on R C ( G/C ) ∼ = R ( ζ | C | ) via id en tiﬁcation of a gen- erator of C w ith ζ | C | . These facts toge ther imply that the co chain groups (2.2) with F = R ( − ) admit a sp litting giv en by 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 whic h acts by translation on X C / Z G ( C ). Since R C ( G/C ) is a p ro jectiv e R [ W G ( C )]-mo du le, it follo ws that for an y prop er G -complex X the Bredon cohomology of X with coeﬃcient 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 ) . (4.4) A t this p oin t, we 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 ﬁn ite group G a sum o ver conjugacy classes of cyclic s u bgroups is equiv alen t to a su m o ve r conjugacy classes of elements in G , and that X h g i = X g and Z G ( h g i ) = Z G ( g ). One ﬁn ally obtains a sp litting of r eal Bredon cohomology groups 5 H • G  X ; R ⊗ R ( − )  ∼ = M [ g ] ∈ G ∨ H •  X g ; R  Z G ( g ) (4.5) in to ordinary cohomology groups H •  X g ; R  Z G ( g ) ∼ = H •  X g / Z G ( g ) ; R  ∼ = H  Ω • ( X g ; R ) Z G ( g ) , d  5 This splitting in fact holds ov er Q [50]. RAMOND-RAM OND FIELDS, FRACTIONAL BRA NES AND ORBIFOLD DIFFERENTIAL K- THEOR Y 17 with constant co eﬃcien ts R . The group on the righ t-hand side of eq. (4.5) corresp onds to the (real) “delo calized equiv ariant cohomol ogy” H • ( ` g ∈ G X g ) G ⊗ R d eﬁned b y Baum and Connes [6, 9]. Note that this group is (non-canonically) isomorp hic to R ( G ) ⊗ H • ( X ; R ) when the G -action on X is trivial. F urthermore, by usin g T h eorem 2.5 one also has a decomp osition for equiv arian t K-theory with real co eﬃcien ts giv en by K • G ( X ) ⊗ R ∼ = M [ g ] ∈ G ∨  K • ( X g ) ⊗ R  Z G ( g ) . Ho we v er, this decomp osition captures only the torsion-free part of the group K • G ( X ). 4.3. Delo calization of t he equiv ariant C he rn c haracter. The complex (4.3) of orbifold Ramond-Ramond ﬁelds can also b e used to p ro vide an explicit geometric description of the (complex) equiv ariant Chern c h aracter deﬁned in Section 2.4. W e w ill no w explain this con- struction, refering the reader to ref. [17] for the tec hnical details. Consider a complex G -bund le E o ver X equ ip ed with a G -inv arian t hermitean metric and a G -in v ariant metric connection ∇ E . One can then deﬁn e a closed G -in v ariant diﬀeren tial form c h ( E ) ∈ Ω • ( X ; C ) G in the usual wa y by the Chern -W eil constru ction 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 p oint subr ing of the action of G as automorphisms of H • ( X ; C ). By u sing the deﬁnition of the homomorphisms (2.9), 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 su bgroup, and d eﬁne 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  represent ed by c h( g , E ) := T r  γ ( g ) exp( − F E C / 2 π i )  where g is a generator of C , F E C is the restriction of th e in v ariant curv ature t wo-form F E to the ﬁxed p oint su bspace X C , and γ is a rep r esen tation of C on the ﬁb res of the restriction bu ndle E | X C whic h is an N G ( C )-bun dle o ver X C . The c haracter χ C naturally identi ﬁes R ( C ) ⊗ C with the C -v ector space of class fun ctions C → C . By using the s plitting (4.4) for complex Bredon cohomology , one can then s ho 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 kernel the ideal of elemen ts wh ose characte r s v anish on all generators of C . Using eq. (2.10) we can th en deﬁne th e map c h C : V ect C G ( X ) − → M [ g ] ∈ G ∨ Ω ev en  X g ; C  Z G ( g ) from complex G -bundles E → X giv en b y c h C ( E ) = M [ g ] ∈ G ∨ T r  γ ( g ) exp ( − F E g / 2 π i )  . (4.6) 18 RICHARD J. SZABO AND ALESSANDRO V ALENTINO A t the lev el of equiv ariant K-theory , fr om Th eorem 2.5 it follo ws that th is map induces an isomorphism c h C : K • G ( X ) ⊗ C ≈ − → H G  X ; C ⊗ π −• K G ( − )  • (4.7) where we ha v e used th e splitting (4.5). Th e map (4.6) coincides with the equiv ariant Chern c haracter deﬁn ed in ref. [5]. 4.4. W ess-Zumino pairings. W e now hav e all the n ecessary ingredients to deﬁne a coupling of the Ramond-Ramond ﬁelds to a D-brane in the orb ifold [ X/G ]. In this s ection w e will only consider R amond -Ramond ﬁelds whic h are top ologically trivial, i.e. , elemen ts of the diﬀerentia l complex (4.3), and use the delo calized cohomology th eory ab ov e b y working throughout with complex co eﬃcients. Under these conditions w e can straigh tforw ard ly mak e con tact with existing examples in the physics literature and write d o wn their appropr iate generalizations. T o th is aim, w e introdu ce th e bilinear p ro duct ∧ G : Ω • G ( X ; R ) ⊗ Ω • G ( X ; R ) − → Ω • G ( X ; R ) 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 ] (4.8) where ∧ g = ∧ is the usu al exterior pro d uct on Ω • ( X g ; R ). Th ere is also an inte gration 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 / Z G ( g ) ω [ g ] . The normalization ensures that R G X ω = R X ω when G acts tr ivially on X and ω ∈ Ω • ( X ; R ) is “diagonal” in R ( G ) ⊗ Ω • ( X ; R ). Supp ose no w that f : W → X is the smo oth immersed w orld v 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 -bu ndle E → W and an inv arian t connection ∇ E on E . W e deﬁn e the Wess-Zumino p airing WZ : D G ( X ) × Ω • G ( X ; C ) − → C b et ween such D-branes and Ramond-Ramond ﬁelds as (4.9) WZ  ( W , E , f ) , C  = Z G W ˜ C ∧ G c h C ( E ) ∧ G R ( W , f ) , where ˜ C = f ∗ C is th e pu llb ac k along f : W → X of the total Ramond -Ramond ﬁeld C = M [ g ] ∈ G ∨ C [ g ] and the equiv arian t Chern c haracter is giv en by eq. (4.6) with γ giving the action of G on the Chan-P aton factors of th e D-br an e. Th e closed worldv olume form R ( W, f ) ∈ Ω ev en G, cl ( W ; C ) represent s a complex Bredon cohomology class w hic h accoun ts for gravita tional corrections du e to curv ature in the spacetime X and d ep ends only on the b ordism class of ( W, f ). It will b e constructed explicitly in Section 4.7 b elo w in terms of the geometry of the immersed cycle f : W → X and of the G -bun dle ν → W give n by ν = ν ( W ; f ) = f ∗ ( T X ) ⊕ T W . (4.10) RAMOND-RAM OND FIELDS, FRACTIONAL BRA NES AND ORBIFOLD DIFFERENTIAL K- THEOR Y 19 It is easily seen that, mo dulo the cu r v atur e con tribu tion R ( W, f ), the v ery natural exp res- sion (4.9) reduces to the usual W ess-Zu mino coupling of top ologica lly trivial Ramond-Ramond ﬁelds to D-branes in the case G = e . But ev en if a group G 6 = e acts trivially on the brane w orld v olume W (or on the spacetime X ), there can still b e additional con tribu tions to the us u al Ramond-Ramond coupling if E is a non-trivial G -bund le . This is the situ ation, for ins tance, for fractional D-branes ( W , E , f ) ∈ D G frac ( X ) placed at orb ifold singularities. In this case, w e ma y use the isotopical decomp osition (3.3) of the Ch an -Pato n b undle along w ith eq. (3.2). Then the W ess-Zu mino pairin g (4.9) descends to a pairing WZ frac : D G frac ( X ) × Ω • G ( X ; C ) − → C with the additiv e sub categ ory of fractional br anes at orbifold singularities give n b y WZ frac  ( W , E , f ) , C  = Z W  1 | G ∨ | X [ g ] ∈ G ∨ ˜ C [ g ] ∧ ch  E [ g ]  χ γ ( g )  ∧ R ( W, f ) (4.11) where χ γ : G → C is the c haracter of the representa tion γ and R ( W , f ) ∈ Ω ev en cl ( W ; C ). One can immediately read oﬀ from the W ess-Zumin o action (4.11) the Ramond-Ramond c h arges of D0-branes, and the state corresp onding to the repr esen tation γ has (fr actional) charge Q [ g ] γ = χ γ ( g ) | G ∨ | with resp ect to the t wisted Ramond -Ramond one-form ﬁ eld C (1) [ g ] . Th ese charge s agree with b oth those of an op en string disk amplitud e compu tation an d a b oundary state analysis for fractional D0-branes [21]. O u r general formula (4.9) in cludes also th e natural extension to the Ramond-Ramond couplings of r e gular D-branes wh ic h mov e freely in the bulk of X under the action of G , as well as to other non -BPS D-br an e states suc h as tru ncated bran es. 4.5. Linear orbifolds. W e w ill no w “test” our d eﬁnition (4.9) on the class of examples con- sidered in Section 3.3. Th ese are ﬂ at orbifolds f or wh ic h there are n o non-trivial curv ature con tribu tions, i.e. , R ( W, f ) = 1. Let u s sp ecializ e to the case of cyclic orbifolds ha ving twist group G = Z n with n ≥ d . I n 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 by k = 1 , . . . , n − 1. The unt w isted sector is lab elled by k = 0. W e take a generator g of Z n whose action on V ∼ = C d is giv en by 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 inte gers satisfying a 1 + · · · + a d ≡ 0 mo d n . 6 In this case the action of an y elemen t in Z n has only one ﬁxed p oint, an orb ifold singularit y at the origin (0 , . . . , 0). Hence for any g 6 = e one h as X g ∼ = R p, 1 and the d iﬀeren tial complex (4.3) of orb ifold Ramond-Ramond ﬁ elds is give n b y Ω • Z n ( X ; R ) = Ω • ( X ; R ) ⊕  n − 1 M k =1 Ω •  R p, 1 ; R   . Consider no w a D-brane ( W, E , f ) ∈ D Z n frac ( X ) with worldv olume cycle f ( W ) ⊂ R p, 1 placed at the orbifold singularit y , i.e. , f : W → R p, 1 × (0 , . . . , 0) ⊂ X . Let the generator g act on the ﬁ bres of the Chan-Pat on bundle E → W in the n -dimensional regular representati on γ ( g ) ij = ζ i δ ij . The action on worldv olume ferm ion ﬁ elds is d etermined by a lift ˆ Z n acting on the spinor bun dle 6 Both the requirement that th e representation V b e complex and the form of the G -action are physical input s ensuring that the closed string background X p reserv es a suﬃcient amount of sup ersymmetry after orbifolding. 20 RICHARD J. SZABO AND ALESSANDRO V ALENTINO S → W . T hen the p airing (4.9) cont ains the follo win g terms . First of all, we ha ve the coup lin g of the u n t wisted Ramond-Ramond ﬁelds to W give n b y Z W ˜ C ∧ T r  exp( − F E / 2 π i )  , whic h is just the usual W ess-Z u mino coupling and h ence the Ramond-Ramond c harge of a regular (bulk) b rane is 1 as exp ected. Th en there are the con tributions fr om the twiste d sectors, whic h by recalling eq. (3.2) are giv en by the expr ession Z W 1 n n − 1 X k =1 ˜ C k ∧ T r  γ ( g k ) exp( − F E / 2 π i )  where g k is an element of Z n of order k . Since γ ( g k ) ii = ζ ik , the coupling in this case is determined by a discrete F our ier transform of the ﬁelds ˜ C k o ve r the group Z n . The brane asso ciated with the i -th irr educible representat ion of Z n has charge ζ ik /n with resp ect to the Ramond-Ramond ﬁeld in the k -th t wisted sector. F or d = 2 and d = 3 this pairing agrees with and un iformizes the gauge ﬁ eld couplings compu ted in refs. [26] and [27], resp ectiv ely . 4.6. An equiv arian t Riemann-Roch formula. L et X , W b e smo oth compact G -manifolds, and f : W → X a smo oth prop er G -map. If we w an t to make sense of the equations of motion for the R amond -Ramond ﬁeld C , whic h is a quantit y d eﬁ ned on the spacetime X , th en w e need to pu shforwa rd classes deﬁn ed on the b rane worldv olume W to classes d eﬁned on the spacetime. This will enable the construction of Ramond -Ramond cur ren ts in Section 6 indu ced b y the bac kground and D-branes which app ear as source terms in the Ramond-Ramond ﬁeld equations. Some tec h nical d etails of th e constructions b elo w are pr ovided in App endix C. Consider ﬁr st the non-equiv arian t case G = e . Let ν → W b e the Z 2 -graded bundle (4.10), i.e. , the K O-theory class of ν is the virtual bun dle [ ν ] = f ∗ [ T X ] − [ T W ] ∈ KO 0 ( W ). W e assume that ν is even-dimensional and end o wed with a spin c structure (this is automatic if b oth X and W are spin c ). Then, as r eview ed in Ap p endix C, one can deﬁn e the Gysin homomorphism in ordinary K-theory f K ! : K • ( W ) − → K • ( X ) . Using the orien tations on X and W one has Poinca r ´ e du alit y in ord inary cohomology , in ducing a Gysin homomorphism f H ! : H • ( W ; Q ) − → H • ( X ; Q ) , where here w e consider the Z 2 -grading giv en by ev en and o dd degree. The pushforward homomorphisms in K-theory and in cohomolo gy , u nder the conditions stated ab o ve, are r elated by the Riemann-Ro c h theorem w hic h states th at c h  f K ! ( ξ )  = f H !  c h( ξ ) ⌣ T o dd( ν ) − 1  (4.12) for an y class ξ in K • ( W ). Here T o d d( E ) ∈ Ω ev en cl ( W ; C ) denotes the T o d d gen u s charact eristic class of a hermitean vecto r bun dle E o ve r W , wh ose Chern -W eil r epresen tativ e is T o d d( E ) = s det  F E / 2 π i tanh  F E / 2 π i   where F E is the cur v ature of a hermitean connection ∇ E on E . The T o dd class of th e Z 2 - graded b undle (4.10) can b e computed by using m u ltiplicativit y , n aturalit y and inv ertibilit y to get T o d d( ν ) = f ∗ T o d d( T X ) / T o dd ( T W ). Thus the Chern c haracter d o es n ot commute with the Gysin p ushforward maps , an d the d efect in th e comm u tation relation is pr ecisely the T o d d gen us of the bun dle ν . Th is “t wisting” b y the bun dle ν o v er the D-br an e con tribu tes in a crucial wa y to the Ramond-Ramond current in the n on -equiv ariant case [19, 49, 54]. Let us n ow attempt to ﬁ nd an equiv ariant v ersion of the Riemann-Ro ch theorem. As the morphism f : W → X is G -equiv ariant , the Z 2 -graded bund le ν is itself a G -bundle with ev en RAMOND-RAM OND FIELDS, FRACTIONAL BRA NES AND ORBIFOLD DIFFERENTIAL K- THEOR Y 21 G -actio n . W e assume th at ν is K G -orien ted. Th is requiremen t is just th e F reed-Witten anomaly cancellat ion condition [30] in this case, generalized to global wo rldsheet anomalies for D-branes represent ed by generic G -equiv arian t K -cycles. I t enables, analogo usly to th e non-equiv arian t case, the constru ction of an equiv arian t Gysin homomorphism f K G ! : K • G ( W ) − → K • G ( X ) . (4.13) W e will demonstr ate that, u nder some v ery sp ecial conditions, one can construct a complex Bredon cohomology class wh ic h is analogous to the T o d d genus and whic h pla ys the r ole of th e equiv ariant comm utativit y defect as ab ov e. Let us sup p ose that the G -action has the prop er ty that for any element g ∈ G , the N G ( g )-bundle ν g = ν ( W ; f ) g → W g is the Z 2 -graded bun d le ν g = f ∗ | W g ( T X g ) ⊕ T W g o ve r the im m ersion f | W g : W g → X g with a Z G ( g )-in v arian t sp in c structure. Note that these are highly n on-trivial conditions, b ecause for an arb itrary G -bun dle E → X one is not ev en guaran teed in general that E g → X g is a vect or bun dle, as the dimension of the ﬁbre may ju mp from p oin t to p oin t. As a simple example of what can happ en, 7 let X = R and G = R + the group of p ositiv e reals un der m ultiplication. Consider the G -bund le X × V → X giv en by p ro jection ont o the ﬁr st factor, where V is a ﬁn ite-dimensional real v ector space and the G -action is g · ( x, v ) =  x , g x v  for all g ∈ G . F or any g 6 = 1, ( X × V ) g is not a ﬁbr e b undle ov er X g = X , as the G -inv arian t ﬁbre space ov er x = 0 is V wh ile it is the null ve ctor o ve r an y other p oint . When G is a ﬁ nite group, one can apply a construction d ue to A tiy ah and Segal [5]. If E is a complex G -v ector b undle ov er X , its restriction to the ﬁ x ed p oin t subs p ace X g for any g ∈ G carries a represen tation of the normalizer sub group N G ( g ) ﬁbrewise. W e can th u s decomp ose E | X g in to a Whitney sum of sub-b u ndles E α = Hom g (1 1 α , E | X g ) ⊗ 1 1 α o ve r the eigen v alues α ∈ sp ec( g ) ⊂ C for the action of g on the ﬁbres of E | X g , wh ere Hom g (1 1 α , E | X g ) is a Z G ( g )- bund le ov er X g and 1 1 α is the N G ( g )-bundle X g × V α with V α the corresp ond ing eigenspace. W e deﬁne the class φ g ( E ) = X α ∈ spec( g ) α [ E α ] (4.14) in the ord in ary K-theory of X g with complex coeﬃcients. By Sc hur’s lemma, ev ery element h ∈ Z G ( g ) commuting w ith g acts as a multiple of the iden tit y on the total space of th e bund le E α , and so the class obtained in this wa y is Z G ( g )-in v arian t. It follo ws th at the m ap (4.14) on V ect C G ( X ) induces a homomorphism φ g : K • G ( X ) ⊗ C − →  K • ( X g ) ⊗ C  Z G ( g ) . By setting φ = M [ g ] ∈ G ∨ φ g w e obtain a natural isomorp h ism leading to the splitting [5] K • G ( X ) ⊗ C ∼ = M [ g ] ∈ G ∨  K • ( X g ) ⊗ C  Z G ( g ) . (4.15) The equiv arian t Chern characte r (4.7) pr o vides an isomorphism comp onen t wise b et w een the equiv ariant K-theory group (4.15) and the complex Bredon cohomology of X . Supp ose no w that the equiv ariant Thom class Th om G ( ν ) ∈ K • G, cpt ( ν ) can b e decomp osed according to the splitting (4.15) in suc h a w ay that the comp onen t in any su bgroup Thom  ν g  ∈  K • cpt ( ν g ) ⊗ C  Z G ( g ) 7 W e are grateful to J. Figueroa-O’F arrill for suggesting this ex ample to us. 22 RICHARD J. SZABO AND ALESSANDRO V ALENTINO coincides with the (ordinary) Thom class of the Z 2 -graded bund le ν g → W g . Un der th ese conditions, the equiv ariant Gysin homomorph ism (4.13) constructed in App endix C decomp oses according to the splitting f K G ! = M [ g ] ∈ G ∨ f K g where f K g is the K-theoretic Gysin homomorph ism asso ciated to th e smo oth p rop er map f   W g : W g − → X g . Deﬁne the c haracteristic class T o dd G b y T o d d G ( ν ) := M [ g ] ∈ G ∨ T o d d  ν g  (4.16) where T o d d is the ordinary T o dd gen u s. This class deﬁn es an ele men t of the ev en d egree complex Bredon cohomology of the brane wo rldvo lume W . Under the conditions sp elled out ab ov e, we can no w u se the equiv ariant Chern c haracter (4.7) and the usual Riemann-Ro c h th eorem for eac h pair ( W g , X g ) to pr o v e the identi t y f H G !  c h C ( ξ ) ⌣ G T o d d G ( ν ) − 1  = c h C  f K G ! ( ξ )  (4.17) for any class ξ ∈ K • G ( W ) ⊗ C , as all quantitie s inv olve d in the formula (4.17) are compatible with the G -equiv ariant decomp ositions giv en ab o v e. When the geometric conditions assumed ab o ve are not m et, the equiv ariant T o d d class in the formula (4.17) should b e m o diﬁed by multiplying it with another equiv arian t characte r istic class Λ G ( W ) w hic h r eﬂ ects non-trivial geometry of the norm al bun dles N W g to th e em b eddings W g ⊂ W . This should come from app lying a suitable ﬁxed p oin t th eorem to the ord in ary Riemann-Ro c h formula (4 .12), but w e ha ve not found a ve rsion which is su itable to o ur particular equiv ariant Chern c haracter in the general case on the catego ry of G -spaces. When f is th e collapsing map X → p t, this is the cont en t of the index theorem used in S ection 4.7 b elo w. Th e form u la (4.17) is, how ev er, directly app licable on the category D G frac ( X ) of fractional D-branes. When G is the cyclic group Z n as in S ection 4.5 ab o ve, one can apply the Thomasson-Nori ﬁ xed p oint theorem [53, 61] to get Λ Z n ( W ) = n − 1 M k =0 ζ k c h  V − 1 N ∨ W g k  ∈ H • Z n  W ; C ⊗ R ( − )  (4.18) where V − 1 N W g = codim( W g ) X l =0 ( − 1) l  V l N W g  ∈ K 0  W g  . 4.7. Gra vit a tional pairings. W e will no w explain how to compute the curv ature con tribu tions R ( W , f ) ∈ Ω ev en G, cl ( W ; C ) to the W ess-Zumino f u nctional (4.9) f or the brane geometries describ ed in S ection 4.6 ab o v e and for v anishing B -ﬁeld. W e deriv e the cancelling form for the Ramond- Ramond gauge anomaly inﬂo w due to c h iral fermions on the in tersection wo rldvo lume for famili es of branes usin g the usual descent pro cedure [35], which is due to cur v ature of the spacetime manifold X itself. F or this, w e must exp licitly u se the G -spin c structure on X . T h e standard mathematical int uition b ehin d this correction is to mo d ify the equiv arian t Chern c haracter to an isometry with resp ect to the n atural b ilinear pairings on equiv arian t K-theory and Bredon cohomology with complex co eﬃcient s [49, 54]. The n atural sesquilinear pairing b et we en t wo cla sses of complemen tary d egrees in complex Bredon cohomology , represente d b y closed diﬀerential forms ω , η ∈ Ω • G, cl ( X ; C ), is giv en by ([ ω ] , [ η ]) H G := R G X ω ∧ G η . O n th e other hand , the natural qu adratic form on f r actional b ranes RAMOND-RAM OND FIELDS, FRACTIONAL BRA NES AND ORBIFOLD DIFFERENTIAL K- THEOR Y 23 deﬁned by classes in equiv ariant K-theory , repr esen ted by complex G -v ector bund les E , F → X , is the top ologica l c harge  [ E ] , [ F ]  K G := µ 1  [ X, E ∨ ⊗ F , id X ]  of eq. (3.7 ) in the unt w isted secto r corresp on d ing to t he represen tation γ = 1 : C [ G ] → C induced from the trivial representat ion of G . This quant it y agrees with the natural inte rsection form on b ound ary states computed as the G -in v ariant Witten in d ex o v er op en string states susp ended b et ween D-branes [51], whic h coun ts the diﬀerence b et we en the num b er of p ositiv e and negativ e c hiralit y Ramond groun d states and hence computes the required c h iral fermion anomaly . The tw o bilinear forms are related thr ou gh the lo cal index theorem wh ic h pr o vides a formula for I n dex 1  [ D / X E ]  in terms of integ rals of c haracteristic forms o ver the v arious singular strata of the orbif old [ X/G ]. It reads [17] Index 1  [ D / X E ]  = X [ g ] ∈ G ∨ 2 d g | g | Z X g / Z G ( g ) c h( g , E ) ∧ T o dd  T X g  ∧ Z V • N X g ST r S  ∆( g )  q det  1 − N ( g )  det  1 − N ( g ) exp ( − F N X g / 2 π i )  , where d g = d im( X ) − dim( X g ), | g | is the order of the element g ∈ G , and N ( g ) denotes the action of g ∈ G on the ﬁ bres of the normal bund le N X g to X g in X . The determinan t is tak en o ve r the normal bun dle N X g for eac h g ∈ G , implemen ted by a Berezin-Grassmann in tegration o ve r the exte rior algebra bu ndle V • N X g . The form ∆( g ), r egarded as an element of V • N X g under the sym b ol map, is the action of g on the ﬁbres of the spinor bun dle S | X g , and ST r S is the su p ertrace ov er the endomorp hism bund le of S . This form ula can b e thought of as an equiv arian t lo caliza tion of the us ual A tiy ah-Singer index densit y ont o the submanifolds X g ⊂ X of ﬁxed p oin ts of the G -acti on on X , with the determinan ts reﬂecting the “Euler class” con tribu tions f rom the non-trivial normal bun dles to X g . This is the antic ipated physica l resu lt arising from the closed strin g t wisted Witten ind ex, computed as the p artition f u nction on the cylinder w ith the t wisted b oundary conditions (4.2). Sup ersymmetry localizes the compu tation on to zero mo des of the string ﬁelds whic h are constan t maps to the s u bmanifolds X h ⊂ X , w hile mo d ular inv ariance requir es a (w eigh ted) su m o ver all t wisted sectors. The index can b e rewritten using the equiv arian t c haracteristic classes d eﬁned as in eqs. (4.6) and (4.16) to get Index 1  [ D / X E ]  = Z G X c h C ( E ) ∧ G T o d d G ( T X ) ∧ G Λ G ( X ) , (4.19) where we h a ve us ed ( T X ) g = T X g and the elemen t of Ω • G, cl ( X ; C ) giv en by Λ G ( X ) := M [ g ] ∈ G ∨ | g | 2 d g | G ∨ | Z V • N X g ST r S  ∆( g )  q det  1 − N ( g )  det  1 − N ( g ) exp ( − F N X g / 2 π i )  (4.20) deﬁnes a c haracteristic class in the complex Bredon cohomology of X . Note that the inte grands of eq. (4.20) are f ormally similar to the Ch ern c haracters of th e virtual bund les (4.18) ab o v e. By using multiplicat ivit y of the equiv arian t Chern charact er (4.6) to write c h C  E ∨ ⊗ F  = c h C ( E ) ∧ G c h C ( F ) , it f ollo ws from eq. (4.19) that the map (4.7) can b e turned in to an isometry by “t wisting” it with the closed d iﬀeren tial f orm p T o d d G ( T X ) ∧ G Λ G ( X ), whic h when pulled bac k along f : W → X giv es the required anomaly cancelling form on the brane w orldvolume. This should th en b e com bined with the correcti on T o dd G ( ν ) − 1 con tribu ted by the Z 2 -graded bundle (4.10) to the Riemann-Ro c h formula (4.17). Then under th e v arious conditions sp elled out in Section 4.6 24 RICHARD J. SZABO AND ALESSANDRO V ALENTINO ab o ve, th e r equired map R in eq. (4.9) fr om G -spin c b ordism classes [( W, f )] to Ω ev en G, cl ( W ; C ) is giv en by R ( W , f ) = f ∗ p T o d d G ( T X ) ∧ G Λ G ( X ) T o d d G ( ν ) = T o dd G ( T W ) ∧ G f ∗ s Λ G ( X ) T o d d G ( T X ) . (4.21) The main new ingredient in th is form u la is th e con tribu tion f rom the ﬁxed p oint sub manifolds X g ⊂ X , particularly their normal bund le c haracteristic cla sses (4.20). This corrects previous top ologica lly trivial, ﬂat space formulas, even for G -ﬁxed wo rldvo lumes W (see ref. [34] for example). Note that wh en the G -action on X is trivial, one has T o d d G ( T X ) = T o d d( T X ) an d Λ G ( X ) is constant. 5. Orbifold differ ential K-theor y The main drawbac k of the d elo calized th eory of the previous section is that it cann ot incor- p orate the interesting eﬀects of torsion, whic h ha v e b een one of the drivin g forces b ehind the K-theory descrip tion of D-branes and Ramond -Ramond ﬂuxes, and as suc h it is desirable to ha ve a description whic h utilizes the full R ( G )-mo dule K • G ( X ). In this section we w ill dev elop an extension of diﬀerentia l K-theory as deﬁn ed in ref. [37] to incorp orate th e case of a G -manifold. These are the groups needed to extend the analysis of the p revious section to top ologically n on- trivial, real-v alued Ramond -Ramond ﬁ elds. While w e d o not hav e a formal pr o of that this is a prop er deﬁn ition of an equiv arian t diﬀeren tial cohomolo gy theory , we will see that it matc hes exactly with exp ectations from str in g theory on orbifolds and also has the correct limiting prop- erties. F or this reason we dub the theory that w e deﬁne ‘orbifold’ diﬀerentia l K-theory , deferin g the terminology ‘equiv arian t’ to a more thorough treatmen t of our mo del (we discuss this in more detail in Section 5.4 b elo w). In the follo wing w e will sp ell out the deﬁnition of diﬀeren- tial K-theory group s. T he crux of the extensions of these deﬁnitions to the equiv ariant s etting will b e explicit constructions of the exact sequences they are describ ed by , which are imp ortan t for p h ys ical considerations. W e will determine concrete r ealizat ions of the v arious morphisms in volv ed, whic h are giv en in a general b u t abstract framew ork in r ef. [37]. 8 See refs. [28, 32] for an int ro duction to diﬀerential cohomology theories and their ap p lications in physic s. 5.1. Diﬀeren t ial cohomology theories. Diﬀeren tial K-theory of a manifold is an en r ic hment of its K-theory , which enco des global top ological in formation, with lo cal geometric information con tained in the d e Rham complex. Consider a (generalized) cohomology theory E • deﬁned on the category of smo oth manifolds X along w ith a canonical map ϕ : E • ( X ) − → H( X ; R ⊗ π −• E) • (5.1) whic h indu ces an isomorphism E • ( X ) ⊗ R ∼ = H( X ; R ⊗ π −• E) • , i.e. , the image of ϕ is a full lattice and its kernel is th e torsion su b group of E • ( X ). Then one can deﬁne diﬀer ential E - the ory as the cohomology theory ˇ E • whic h lifts E • via the pullbac k squ are (5.2) ˇ E • ( − ) / /   Ω cl ( − ; R ⊗ π −• E) •   E • ( − ) ϕ / / H( − ; R ⊗ π −• E) • , where Ω cl ( X ; R ⊗ π −• E) q denotes the r eal vec tor space of closed E • (pt; R )-v alued diﬀeren tial forms ω on X of total degree q , and the righ t v er tical map in th e comm utativ e diagram (5.2) is 8 An explicit p roof of these exact seq uences has b een given recently in ref. [18] using the geometric d escription of diﬀerential K- cocyles in terms of bundles with connection. Our proof is more general, but less geometric, as it exploits the realization in terms of maps t o classifying spaces. RAMOND-RAM OND FIELDS, FRACTIONAL BRA NES AND ORBIFOLD DIFFERENTIAL K- THEOR Y 25 giv en by sending ω to its d e Rham cohomology class [ ω ] dR . A class in ˇ E q ( X ) is give n b y a p air ( ξ , ω ), w ith ξ ∈ E q ( X ) suc h that ϕ ( ξ ) = [ ω ] dR , (5.3) together with an isomorphism that r ealizes the equalit y (5.3) exp licitly in H( X ; R ⊗ π −• E) q . In their found ational pap er [37] Hopkins and S in ger deﬁn e the diﬀerenti al E-theory asso ciated to an y generalized cohomolog y theory E • , and p ro ve its natur alit y and homotopy prop erties. This is done b y generalizing the concept of function sp ac e in algebraic top ology , whic h can b e used to deﬁne the cohomology of a space, to that of diﬀe r ential function sp ac e , where here the term “diﬀeren tial” t ypically means something diﬀerent from diﬀeren tiable or smo oth. Because of this, the diﬀeren tial E-groups are deﬁn ed in an abstract w ay and are diﬃ cu lt to realize explicitly . An explicit construction for diﬀerential K -theory is giv en in r ef. [37]. In the follo w ing we will go through this construction in some detail. This will b e our starting p oin t to giv e a d eﬁnition of the diﬀerent ial cohomology theory asso ciated to equiv ariant K-theory K • G , whic h will r ed uce to ordinary diﬀerential K-theory in the case where the group G is th e trivial group. The v alidit y of our d eﬁnition will b e conﬁrmed by explicit constru ction of the exact sequences, that will also b e imp ortan t in our later physica l app licatio ns. 5.2. Diﬀeren t ial K-theory. Th roughout X will denote a smo oth manifold. Let F red b e the algebra of F redholm op erators on a separable Hilb ert space. Recall that F red is a classifying space for complex K-theory through the isomorphism [ X, F red ] Index( − ) − − − − − → K 0 ( X ) whic h asso ciates to an y map f : X → F red the index b undle of f in K 0 ( X ). L et u ∈ Z ev en (F red ; R ) b e a co cycle of eve n degree wh ic h r epresen ts the Chern c haracter of th e un iv ersal bun dle. Then for any map f : X → F r ed r epresen tin g a complex v ector b undle E → X , the pullback f ∗ u is a represent ativ e of c h( E ) in H ev en ( X ; R ). The diﬀerenti al K -theory group ˇ K 0 ( X ) is deﬁned to b e the set of triples ( c, h, ω ), where c : X → F red, ω is a closed diﬀerential form in Ω ev en cl ( X ; R ), and h is a co c h ain in C ev en − 1 ( X ; R ) satisfying δ h = ω − c ∗ u . (5.4) The co c hain h in eq. (5.4) is precisely the isomorphism refered to in S ection 5.1 ab ov e, which is in visib le in the cohomology groups, and in th is equation the closed diﬀeren tial form ω is r egarded as a co c h ain und er the de Rham map ω 7→ R ( − ) ω . Two triples ( c 0 , h 0 , ω 0 ) an d ( c 1 , h 1 , ω 1 ) are equiv alent if there exists a triple ( c, h, ω ) on X × [0 , 1], w ith ω = ω ( t ) constant along t ∈ [0 , 1], suc h that (5.5) ( c, h, ω )   t =0 = ( c 0 , h 0 , ω 0 ) and ( c, h, ω )   t =1 = ( c 1 , h 1 , ω 1 ) . The equiv alence (5.5) can b e rep hrased [28] by r equiring that there exists a map F : X × [0 , 1] − → F red and a diﬀeren tial form σ ∈ Ω ev en − 2 ( X ; R ) such th at F   t =0 = c 0 , F   t =1 = c 1 , ω 1 = ω 0 , h 1 = h 0 + π ∗ F ∗ u + d σ (5.6) 26 RICHARD J. SZABO AND ALESSANDRO V ALENTINO where π : X × [0 , 1] → X is the natur al pro jection. Th e relations (5.6) sa y that c 0 and c 1 are homotopic maps, hence they represent the same class in K 0 ( X ), and th at the co chains h 0 and h 1 are related by the homotop y that connects the maps c 0 and c 1 . W e also see that the closed form ω completely charac terizes the triple ( c, h, ω ). Borro wing terminology used in representing classes in the d iﬀeren tial cohomology ˇ H 2 ( X ) as principal U(1)-bundles with connection, th e class [ c ] ∈ K 0 ( X ) is called the char acteristic class , the closed diﬀerenti al form ω is called the c u rvatur e , wh ile the co c hain h is called th e holonomy of the tr ip le. F rom the deﬁn in g pr op ert y of the un iv ersal co cycle u and eq. (5.4) it follo ws that c h  [ c ]  = [ ω ] dR . Th us the cohomology class represented b y the curv ature ω lies in th e image of th e (real) Chern c haracter, which is a lattice of maximal r ank inside the cohomology group w ith r eal co eﬃcient s. Let us now deﬁne the diﬀerentia l K-theory group ˇ K − 1 ( X ). Recall that the classiﬁyin g s p ace for K − 1 is the based lo op space ΩF red. Thus we need a cocycle u − 1 ∈ Z odd (ΩF red; R ) whic h represents the universal o dd C hern c h aracter. Consider the ev aluation map ev : ΩF r ed × S 1 − → F red . Then the co cycle u − 1 is deﬁned by u − 1 = Π ∗ ev ∗ u where Π : ΩF r ed × S 1 → ΩF red is the natural pro jection. In fact, u − 1 can b e deﬁn ed as the slant pr o duct of ev ∗ u with the fu ndamen tal class of the circle S 1 , i.e. , by integrat ing the co cycle ev ∗ u along S 1 . As ab o v e, a class in ˇ K − 1 ( X ) is r epresen ted by a triple ( c, h, ω ), where c : X → ΩF red, ω is a closed diﬀerentia l form in Ω ev en − 1 cl ( X ; R ), an d h is a co cycle in C ev en − 2 ( X ; R ) satisfying δ h = ω − c ∗ u − 1 . Tw o triples ( c 0 , h 0 , ω 0 ) and ( c 1 , h 1 , ω 1 ) are equiv alen t if a r elation like that in eq. (5.5) holds. In an analogous w ay one can deﬁne the higher diﬀerential K-theory group s ˇ K − n ( X ) for an y p ositiv e intege r n . One can pro v e that Bott p erio dicit y in co m plex K-theory induces a p eriod icity in diﬀerentia l K-theory giv en b y ˇ K − n ( X ) ∼ = ˇ K − n − 2 ( X ) . This p erio dicit y enables one to deﬁn e the higher diﬀerentia l K-theory groups in p ositiv e degrees. The group comp osition la w on ˇ K − n ( X ) is give n b y ( c 1 , h 1 , ω 1 ) + ( c 2 , h 2 , ω 2 ) := ( c 1 · c 2 , h 1 + h 2 , ω 1 + ω 2 ) where the dot denotes p oin t wise multiplicat ion. The iden tity elemen t is giv en by th e triple ( c , 0 , 0), w here throughout c denotes an y map whic h is homotopic to the (co nstan t) identit y map. T o allo w for the pr esence of the characte ristic class ω in the d eﬁnition, the ab elian groups ˇ K − n ( X ) are generally inﬁnite-dimensional. The deﬁ n ition of ˇ K − n ( X ) dep ends, u p to homotopy t yp e and cohomology class, on the c h oice of classifying sp ace and of u niv er s al co cycle u [37]. A key p rop erty is the exact sequences which c h aracterize the diﬀerentia l K-theory groups ˇ K − n ( X ) for an y n ∈ Z as extensions of top olog ical K-theory b y certain groups of diﬀerent ial forms. In eac h case th e diﬀeren tial K-theory group ˇ K • ( X ) is an extension of th e set wise ﬁbre pro du ct A • K ( X ) =  ( ξ , ω ) ∈ K • ( X ) × Ω cl ( X ; R ⊗ π −• K) •   c h( ξ ) = [ ω ] dR  b y the toru s of top ologically trivial ﬂ at ﬁelds giv en b y 0 − → K •− 1 ( X ) ⊗ R / Z − → ˇ K • ( X ) − → A • K ( X ) − → 0 . (5.7) This will b e useful b elo w wh en we deﬁne equiv ariant diﬀerenti al K-theory . RAMOND-RAM OND FIELDS, FRACTIONAL BRA NES AND ORBIFOLD DIFFERENTIAL K- THEOR Y 27 As in the case of top ological K-theory , there are geometrical realizatio ns of the groups ˇ K − n ( X ) [28]. In p articular, a class in ˇ K 0 ( X ) can b e represen ted b y a complex v ector bun dle E → X equip ed with a connection ∇ E . T o the p air ( E , ∇ E ) w e can asso ciate the triple ( f , η , ω ), where f : X → B U is a map w h ic h classiﬁes th e bu n dle E , ω = ch( ∇ E ) is a Ch ern-W eil repre- sen tativ e of the Chern c haracter of [ E ], and η is a Chern -Simons form such that d η = f ∗ ω B U − ω with ω B U = c h ( ∇ B U ) the Chern c haracter form of the u niv ers al bun d le E B U → B U w ith resp ect to the unive rsal connection ∇ B U on E B U . In the follo w ing w e will deﬁne ab elian groups t hat can b e though t of as a natural generalization of the d iﬀeren tial K-theory of a manifold X acted u p on by a (ﬁnite) group G . In this case one cannot emplo y the p o werful mac hinery dev elop ed in ref. [37], as the equiv ariant K-theory K • G ( X ) is not a cohomology theory deﬁned on the category of manifolds. Instead, we will tak e as our starting p oint the explicit deﬁn ition of the grou p s ˇ K − n ( X ) give n ab o ve, and naturally generalize it to groups ˇ K − n G ( X ) whic h acco mo date the action of the group G in such a wa y that when G = e is trivial, one has ˇ K − n G ( X ) ∼ = ˇ K − n ( X ) . 5.3. Orbifold diﬀeren tial forms. W e wan t to generalize the co m m u tativ e diagram (5. 2) to the case in which our u nderlying c ohomolo gy theory E • ( X ) is the equiv ariant K-theory K • G ( X ). W e ﬁrst need a homomorph ism ϕ from equiv arian t K-theory to a target cohomology theory whic h induces an isomorphism when tensored o ver the reals. F or this, w e will u se the Chern charac ter constructed in Section 2.4 with the target cohomology theory give n by Bredon cohomology . Then we need a reﬁn emen t of this cohomology wh ic h reduces to the de Rham complex when the group G is trivial. Th is complex may b e thought of as the complex of diﬀer ential forms on the orbifold X/G . F or this p urp ose, w e will use th e diﬀerentia l complex (Ω • G ( X ; R ) , d G ) deﬁn ed in Section 4.1. Using the d elocalization formula (4.5) one sho w s that this complex is a r eﬁnemen t for Bredon cohomology with real coeﬃcient s, in the case wh en G is a ﬁn ite group. It comes equip ed w ith a n atur al pro d uct deﬁned in eq. (4.8). As a reﬁnemen t for Bredon co homology , the complex Ω • G ( X ; R ) giv es a well -deﬁned map ω 7− → [ ω ] G − dR ∈ H  Ω • G ( X ; R ) , d G  ∼ = H • G  X ; R ⊗ R ( − )  and redu ces to the us u al de Rh am complex of d iﬀeren tial forms in the case G = e . There is an alternativ e complex one could constru ct whic h is “manifestly” equiv arian t, in the sense that its f u nctorialit y p rop erty ov er the category of groups is transparent . It can also b e generalized to th e case in whic h G is an inﬁnite discrete group. Ho we v er, it is not evident how to d eﬁ ne a ring structure on this complex, and its physic al relation to Ramond-Ramond ﬁelds is not clear. W e include its d eﬁnition h ere for completeness. 9 See App end ix A and ref. [23] for the relev an t deﬁ n itions concerning m o dules o v er a functor category and their tensor pr o ducts. Starting from the real repr esen tation rin g f unctor R ( − ) = R ⊗ R ( − ) o ver the orbit category Or ( G ), there is a n atur al map of real ve ctor s paces R ( − ) ⊗ R Or ( G ) C • ( X ; R ) − → Hom R Or ( G )  C • ( X ; R ) , R ( − )  (5.8) where C • ( X ; R ) is the left R Or ( G )-mod ule obtained by du alizing the fu nctor C • ( X ; R ) := R ⊗ C • ( X ) deﬁned in S ection 2.2. Note th at b oth C • ( X ; R ) and R ( − ), b eing con trav arian t fun ctors, are righ t R Or ( G )-mo dules. Th e map (5.8) is giv en on decomp osable elements as λ ⊗ f 7− →  σ 7→ f ( σ ) ∗ ( λ )  and it is an isomorph ism of real v ector spaces. 10 9 W e are grateful to W. L ¨ uck for suggesting t his construction to u s. 10 In general, to h a ve an isomorphism one has to require th e G -manifold X to b e co compact and prop er. 28 RICHARD J. SZABO AND ALESSANDRO V ALENTINO Deﬁne the diﬀeren tial complex Ω • G  X ; R ⊗ R ( − )  := R ( − ) ⊗ R Or ( G ) Ω • ( X ; R ) where Ω • ( X ; R ) is the fu nctor Or ( G ) → A b giv en by Ω • ( X ; R ) : G/H 7→ Ω • ( X H ; R ), and with deriv ation d orb induced by the exterior deriv ativ e d. Since the de R h am map induces a chain homotop y equ iv alence of left R Or ( G )-complexes C • ( X ; R ) → Ω • ( X ; R ), there is a G -equiv ariant c hain homotop y equiv alence R ( − ) ⊗ R Or ( G ) C • ( X ; R ) − → R ( − ) ⊗ R Or ( G ) Ω • ( X ; R ) . Com b ined with th e isomorphism (5.8) w e can th us conclude H • G  X ; R ⊗ R ( − )  ∼ = H  Ω • G ( X ; R ⊗ R ( − )) , d orb  . If one c h o oses to work with this complex, then the construction of orbifold diﬀerent ial K-theory groups give n in Section 5.4 b elo w can b e carr ied thr ough in exactly the same w a y . But since the t wo complexes Ω • G ( X ; R ) and Ω • G ( X ; R ⊗ R ( − )) are in ge neral not isomorphic, th e t w o diﬀerent ial cohomology theories obtained will b e generically distinct. 5.4. Orbifold diﬀerential K-groups. Ha vin g sorted out all the ingredients n ecessary to mak e sense of a generalizatio n of the diagram (5.2), we will no w deﬁne the diﬀeren tial equiv arian t K-theory groups ˇ K − n G ( X ). First, let us recall some fu rther basic facts ab out equiv arian t K- theory . S imilarly to ordinary K-theory , a mo del for the classifying space of the fun ctor K 0 G is giv en by the G -alge bra of F redh olm op erators F red G acting on a separable Hilb ert space whic h is a repr esen tation space for G in whic h eac h irreducible representa tion o ccurs w ith inﬁ nite m u ltiplicit y [3]. Then there is an isomorphism K 0 G ( X ) ∼ = [ X , F red G ] G where [ − , − ] G denotes the set of equiv alence classes of G -homotopic maps, and the isomorphism is giv en by taking the index b undle. There is also a u niv er s al space V ect n G , equip ed with a unive rsal G -bun d le e E n G , su c h that [ X, V ect n G ] G corresp onds to the set of isomorphism classes of n -dimens ional G -v ector bundles o ve r X [46]. Th ese sp aces are constructed as follo ws. Let E G b e the category whose ob jects are the elemen ts of G and with exactly one m orp hism b et ween eac h pair of ob jects. The geometric realization (or nerv e) of the set of isomorphism classes in E G is, as a simplicial sp ace, the total space of the classifying p rincipal G -bundle E G . With V ect n (pt) the catego r y of n - dimensional complex vec tor spaces V ∼ = C n , th e universal space V ect n G is deﬁ n ed to b e the geometric realization of the functor category [ E G, V ect n (pt)] (see Ap p endix A). The univ ersal n -dimensional G -v ector bundle e E n G is then d eﬁned as e E n G = g V ect n G × GL( n, C ) C n − → V ect n G , (5.9) where g V ect n G is the geometric realization of the fun ctor category deﬁned as ab o v 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 suﬃcient regularit y conditions on the inﬁnite-dimensional spaces F red G and e E n G . Since F red G and the group completion Ω B V ect G are b oth classifying s paces for equiv ariant K-theory , they are G -homotopic and we can thereb y choose a co cycle u G ∈ Z ev en G (F red G ; R ) represent ing the equiv ariant Chern characte r of th e unive rsal G -bun d le (5.9). Generally , the group Z ev en G ( X ; R ) is the sub group of closed co cycles in the complex C ev en G ( X ; R ) := M [ g ] ∈ G ∨ C ev en  X g ; R  Z G ( g ) (5.10) RAMOND-RAM OND FIELDS, FRACTIONAL BRA NES AND ORBIFOLD DIFFERENTIAL K- THEOR Y 29 whic h , by the results of S ection 4.2, is a cochain mo del for the Bredon cohomology group H ev en − 1 G ( X ; R ⊗ R ( − )). Th e equiv ariant Chern characte r is u ndersto o d to b e comp osed with the delo calizing isomorphism of Section 4.2. S ince it is a natural homomorph ism, for any G -bun dle E → X classiﬁed by a G -map f : X → F red G one has c h X  [ E ]  =  f ∗ u G  . Deﬁnition 5.1. The orbifold diﬀer ential K-the ory ˇ K 0 G ( X ) of the (global) orbifold [ X/G ] is the group of trip les ( c, h, ω ), where c : X → F red G is a G -map, ω is an elemen t in Ω ev en G, cl ( X ; R ), an d h is an elemen t in C ev en − 1 G ( X ; R ) satisfying δ h = ω − c ∗ u G . (5.11) Tw o triples ( c 0 , h 0 , ω 0 ) and ( c 1 , h 1 , ω 1 ) are s aid 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], suc h 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 eq. (5.11) the closed orbifold diﬀerential form ω is regarded as an orbifold co c h ain in the complex (5.10) b y app lying the de Rh am map comp onent wise on the ﬁxed p oin t su bmanifolds X g , g ∈ G . The h igher orbifold d iﬀeren tial K-theory groups ˇ K − n G ( X ) are deﬁn ed analogo usly to those of Section 5.2 ab ov e. T o conﬁ rm that th is is an appropr iate extension of the ord inary diﬀeren tial K-theory of X , w e should sho w that the orbifold diﬀeren tial K -theory groups ﬁ t int o exact sequences wh ic h redu ce to those giv en by eq. (5.7) when G is tak en to b e the trivial group. F or this, we d eﬁne the group A 0 K G ( X ) :=  ( ξ , ω ) ∈ K 0 G ( X ) × Ω ev en G, cl ( X ; R )   c h X ( ξ ) = [ ω ] G − dR  . Theorem 5.2. The orbifold diﬀer ential K -the ory gr oup ˇ K 0 G ( X ) satisﬁes the exact se qu enc e 0 − → H ev en − 1 G  X ; R ⊗ R ( − )  c h X  K − 1 G ( X )  − → ˇ K 0 G ( X ) − → A 0 K G ( X ) − → 0 . (5.12) Pr o of. Consider the su bgroup of H ev en − 1 G ( X ; R ⊗ R ( − )) deﬁn ed as the image of th e equiv arian t K-theory group K − 1 G ( X ) und er the Chern c h aracter ch X . It consists of Bredon cohomology classes of the form [ ˜ c ∗ u − 1 G ], wh er e ˜ c : X → ΩF red G . T here 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 h omomorphism, i.e. , it do es not d ep end on the c hosen representati v e of the orbifold diﬀeren tial K -theory class. By deﬁnition, the k ernel of f consists of triples of the form ( c , h, 0). W e also deﬁ ne the map g : H ev en − 1 G  X ; R ⊗ R ( − )  − → ˇ K 0 G ( X ) [ h ] 7− →  ( c , h, 0)  , whic h is a well-deﬁned homomorph ism b ecause the class [( c , h, 0)] dep ends only on the Bredon cohomology class [ h ] ∈ H ev en − 1 G ( X ; R ⊗ R ( − )). Then by constru ction one has im( g ) = ker( f ). The homomorphism g is n ot injectiv e. T o d etermine the ke rnel of g , w e u se the fact th at th e zero elemen t in ˇ K 0 G ( X ) can b e r epresen ted as  ( c , 0 , 0)  =  ( c , π ∗ F ∗ u G + d G σ, 0)  30 RICHARD J. SZABO AND ALESSANDRO V ALENTINO with F : X × S 1 → F red G and σ ∈ Ω ev en − 2 G ( X ; R ) (see eq. (5.6)). T o the map F w e can asso ciate a map ˜ c : X → Ω F red G suc h that F = ev ◦ (˜ c × id S 1 ). T his follo ws fr om the isomorp h ism K − 1 G ( X ) ∼ = k er  i ∗ : K 0 G ( X × S 1 ) → K 0 G ( X )  where i is the inclus ion i : X ֒ → X × pt ⊂ X × S 1 . Now use the fact that at th e level of (real) Bredon cohomology one has an equalit y π ∗  ˜ c × id S 1  ∗ = ˜ c ∗ Π ∗ since the p ro jection homomorph isms π ∗ and Π ∗ b oth corresp ond to in tegration (slan t pro du ct) along the S 1 ﬁbre. Then one has the iden tit y  π ∗ F ∗ u G  =  π ∗ (˜ c × id S 1 ) ∗ ev ∗ u G  =  ˜ c ∗ Π ∗ ev ∗ u G  =  ˜ c ∗ u − 1 G  . It follo w s that k er( g ) is exactly the group ch X (K − 1 G ( X )), and putting ev erything together w e arriv e at eq. (5.12).  The torus H ev en − 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 (or of “orbifold Wilson lines”). W e can rewrite the s equ ence (5.12) in v arious illuminating wa ys . Consider the char acteristic class map f cc : ˇ K 0 G ( X ) − → K 0 G ( X )  ( c, h, ω )  7− → [ c ] and the map g cc : Ω ev en − 1 G ( X ; R ) − → ˇ K 0 G ( X ) h 7− →  ( c , h, d G h )  . Let Ω ev en − 1 K G ( X ; R ) b e the su b group of element s in Ω ev en − 1 G, cl ( X ; R ) w hose Bredon cohomology class lies in c h X (K − 1 G ( X )). Then by using arguments similar to those u sed in arriving at the sequ en ce (5.12) , one ﬁnd s the Corollary 5.3 ( Characteristic class exact sequence). The orbifold diﬀer ential K-the ory gr oup ˇ K 0 G ( X ) satisﬁes the exact se qu enc e 0 − → Ω ev en − 1 G ( X ; R ) Ω ev en − 1 K G ( X ; R ) − → ˇ K 0 G ( X ) − → K 0 G ( X ) − → 0 . (5.13) The quotien t space of orbifold diﬀeren tial forms in the e xact sequence (5.1 3) is calle d the group of top olo gic al ly trivial ﬁelds . An elemen t of this group is a globally d eﬁned (and hen ce top ologi- cally trivial) gauge p oten tial on the orb ifold X/G up to large (quantized) gauge transformations, with ω the corresp onding ﬁ eld strength. Finally , consid er the ﬁeld str ength m ap f fs : ˇ K 0 G ( X ) − → Ω ev en G, cl ( X ; R )  ( c, h, ω )  7− → ω . (5.14) The ke rnel of th e h omomorphism f fs is the group wh ic h classiﬁes the ﬂat ﬁelds (whic h are n ot necessarily top olog ically trivial) and is denoted K − 1 G ( X ; R / Z ). T his group will b e describ ed in more detail in the next section, wh er e w e sh all also conjecture an essen tially purely algebraic deﬁnition of K − 1 G ( X ; R / Z ) wh ich explains the n otatio n. I n any case, w e ha v e the Corollary 5.4 ( Field strength exact sequence). The orbifold diﬀer ential K- the ory gr oup ˇ K 0 G ( X ) satisﬁes the exact se qu enc e 0 − → K − 1 G ( X ; R / Z ) − → ˇ K 0 G ( X ) − → Ω ev en K G ( X ; R ) − → 0 . (5.15) RAMOND-RAM OND FIELDS, FRACTIONAL BRA NES AND ORBIFOLD DIFFERENTIAL K- THEOR Y 31 Higher orbifold diﬀeren tial K-theory group s satisfy analogous exact sequences, with the ap- propriate d egree sh ifts th roughout. It is clear fr om our deﬁn ition that one r eco v ers the ordin ary diﬀeren tial K-theory groups in th e case of the trivial group G = e , and in this sense our orbifold diﬀeren tial K-theory may b e regarded as its equiv ariant generalization. At this p oin t we hasten to add that, although ou r groups are w ell-deﬁn ed and satisfy d esired prop erties whic h are useful for physical applications suc h as fun ctorialit y and the v arious exact sequ ences ab o ve, we hav e not prov en that our orbif old theory is a diﬀerenti al cohomolog y th eory . W e h a ve also not give n a d eﬁnition of w hat a generic orbifold (or equiv arian t) diﬀeren tial cohomology theory is. F or instance, it w ould b e interesting to deﬁne a rin g structure and an in tegration on ˇ K • G ( X ). In par- ticular, th e in tegration requires kno w ledge of a relativ e ve rsion of orbifold diﬀerenti al K-theory , whic h we hav e not dev elop ed in this p ap er. W e hav e also inv estigated the p ossibilit y that the group ˇ K • G ( X ) redu ces to the ord inary diﬀeren tial K-theory ˇ K • ( X/G ) in th e case of a f ree G -action on X , and to ˇ K • ( X ) ⊗ R ( G ) in the case of a trivial grou p action, as one might naiv ely exp ect from the analogo u s results for equiv ariant top ological K-theory (the equiv ariant excision theorem (1.4) with N = G and eq. (3.4), resp ectiv ely) and for Bredon cohomology (Examples 2.2 and 2.3, resp ectiv ely). On the con trary , these d ecomp ositio ns do not o ccur, b ecause the corresp ond ing isomorphisms in equiv ariant K -theory are estabilished via the in duction maps and these usu ally d o n ot lift at the “co c h ain level” as isomorp hisms. Prop erties suc h as in duction structures reﬂect homotopy in v ariance of top ological cohomology groups, w hic h is n ot p ossessed by diﬀerentia l cohomolo gy groups due to their “local” dep en dence on the complex of diﬀeren tial forms. W e will see an explicit example of this in the next section. With this in mind , it would b e int eresting then to deﬁne a su itable analog of the indu ction structures in an equiv ariant cohomology theory . These and v arious other in teresting mathematical issu es su rround in g th e orbifold diﬀerenti al K-theory groups that w e ha ve deﬁn ed w ill not b e purs u ed in this p ap er. 6. Flux quantiza t ion of orb if old Ramond-Ramond fields In this ﬁnal section w e will argue that th e orbifold diﬀerentia l K-theory deﬁned in the p revious section can b e used to d escrib e Ramond-Ramond ﬁelds and their ﬂ ux quan tization condition in orbifolds of T yp e I I sup erstring theory with v anishing H -ﬂux. T o formulate the self-dualit y prop erty of orbifold Ramond-Ramond ﬁelds in equiv ariant K-theory , one n eeds an approp r iate equiv ariant ve rsion of P ontrjagin dualit y [32]. T his app ears to b e a very deep and complicated problem, and is b ey ond the scop e of the p resen t pap er. F u rthermore, to generalize the pairing of Section 4.4 to top ologicall y n on-trivial Ramond-Ramond ﬁelds, on e needs to deﬁn e an in- tegration on the orbifold d iﬀeren tial cohomology theory deﬁn ed in Section 5.4, and r egard the Ramond-Ramond ﬁelds pr op erly as cocycles f or it. I n add ition, one needs a graded ring stru c- ture and an appropr iate group oid repr esenting the orbifold d iﬀeren tial K-theory , whose ob jects are the Ramond-Ramond form gauge p oten tials C and wh ose isomorphism classes are the gauge equiv alence classes in ˇ K • G ( X ). Lac king these ingredients, most of our analysis in this section will b e essentia lly p u rely “top ological” . W e shall s tudy the somewhat simpler problem of the p rop er K-theory qu an tization of orbifold Ramond-Ramond ﬁelds, in particular du e to their sourcing b y fractional D-branes, in terms of the form ulation pr o vided by orbifold diﬀerential K-theory . 6.1. Ramond-Ramond currents. W e will b egin by rephr asing the relatio n b et ween the D- brane c harge group and the group of Ramond-Ramond ﬂu xes “measured at inﬁnity” in the equiv ariant case, wh ich is a statemen t ab out the K -theoretic classiﬁcation of Ramond-Ramond ﬁelds on a global orbifold [ X/G ]. F or this, we inv ok e an argument du e to Mo ore and Witten [52] whic h will suggest that the equiv arian t Chern c h aracter c h X constructed in Section 2 giv es the r igh t quantiz ation r u le for orbifold Ramond-Ramond ﬁelds. Sup p ose that our spacetime X is a non-compact G -manifold. S upp ose fur ther that ther e are D-br anes pr esen t in T y p e I I sup ers tr ing theory on X/G . Their Ramond-Ramond c harges are classiﬁed by the equiv ariant 32 RICHARD J. SZABO AND ALESSANDRO V ALENTINO K-theory K i G, cpt ( X ) with compact sup p ort, wh ere i = 0 in Typ e I IB theory and i = − 1 in T yp e I IA theory . W e r equire that the brane b e a source for the equation of motion f or the total Ramond- Ramond ﬁ eld strength ω . This means that it creates a Ramond-Ramond current J . If we require th at the wo rldvo lu me W b e compact in equiv ariant K-cycles ( W , E , f ) ∈ D G ( X ), th en J is supp orted in the interio r ˚ X of X . Let X ∞ b e the “b oundary of X at inﬁnity” , whic h we assume is pr eserv ed b y 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 c harge liv es in the ke rnel of the n atural forgetful homomorphism f • : K • G, cpt ( X ) − → K • G ( X ) (6.1) induced by the inclusion ( X, ∅ ) ֒ → ( X , X ∞ ). W e d enote by i : X ∞ ֒ → X the canonical inclusion. The long exact sequ en ce for th e 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 th at the c harge groups are giv en by 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 th e group of T yp e I I B (resp. Typ e I IA) brane c harges is measured b y the group K − 1 G ( X ∞ ) (resp. K 0 G ( X ∞ )) of “orbifold Ramond-Ramond ﬂ uxes at inﬁn it y” whic h cannot b e extended to all of sp acetime X . W e ma y th en int erpret, for arbitrary spacetimes X , the group K − 1 G ( X ) (resp. K 0 G ( X )) as the grou p classifying Ramond-Ramond ﬁelds in the orbifold X/G whic h are not sour ced by branes in Type I I B (resp. Typ e I IA) s tr ing theory . The Ramond-Ramond cu rren t can b e describ ed explicitly in the d elocalized theory of Sec- tion 4. The W ess-Zumino p airin g (4.9) b et we en a top ologically tr ivial, complex Ramond- Ramond p oten tial and a D-brane represente d by an equiv ariant K -cycle ( W , E , f ) ∈ D G ( X ) con tribu tes a source term to the Ramond-Ramond equations of motion, wh ic h is the class  Q ( W , E , f )  ∈ H ev en G  X ; C ⊗ R ( − )  represent ed by the push forw 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 (4.17) and the f act that f ∗ is r ight adjoin t to f H G ! , i.e. , f H G ! ◦ f ∗ = id H • G ( X ; C ( − )) . Using the explicit expression for the cur v ature f orm in eq. (4.21), w e can then rewrite this class as Q ( W , E , f ) = c h C  f K G ! ( E )  ∧ G p T o d d G ( T X ) ∧ G Λ G ( X ) . (6.2) This is the complex Bredon cohomology class of the Ramond -Ramond cur ren t J created b y the D-brane ( W , E , f ). In the case G = e , the expr ession (6.2) redu ces to the standard class of the current for Ramond -Ramond ﬁelds in Type I I su p erstring theory on X [19, 49, 52, 54]. There is a natural extension of the current (6.2) w hic h allo ws us to f orm ally conclude, in analogy with the non-equiv arian t case, that the co mplex Bredon cohomolo gy class asso ciated to a class ξ ∈ K • G ( X ) ⊗ C representing a Ramond -Ramond ﬁ eld is assigned b y the equi- v ariant C hern c haracter. If the Ramond -Ramond ﬁeld is determined by a diﬀerential form RAMOND-RAM OND FIELDS, FRACTIONAL BRA NES AND ORBIFOLD DIFFERENTIAL K- THEOR Y 33 C / 2 π p T o d d G ( T X ) ∧ G Λ G ( X ) with C ∈ Ω • G ( X ; C ) and d G C = ω , then this is the class [ ω ( ξ )] in H • G ( X ; C ⊗ R ( − )) repr esen ted by the closed diﬀerential form ω ( ξ ) 2 π p T o d d G ( T X ) ∧ G Λ G ( X ) = c h C ( ξ ) . (6.3) This is jus t the ant icipated ﬂ ux quantiza tion condition from orbifold d iﬀeren tial K -theory . Th e app earance of the additional gra vitational terms in eq. (6.3) is inconsequentia l to this identiﬁ- cation. Given th e canonical m ap (5.1) in a generalized cohomology theory E • , an y other m ap E • ( X ) → H( X ; R ⊗ π −• E) • with the same pr op erties describ ed in Section 5.1 is obtained by m u ltiplying ϕ with an inv ertible elemen t in H( X ; R ⊗ π −• E) 0 . In the case at hand, the charac- teristic class p T o d d G ( T X ) ∧ G Λ G ( X ) is an inv ertible closed d iﬀerential form wh ich r ep resen ts this element in H ev en G ( X ; C ⊗ R ( − )). This class reduces to the usual gra vitational correction p T o d d( T X ) when G acts trivially on X . W e should stress that this analysis of th e delo calized theory assum es the strong conditions sp elled out in Section 4.7, w hic h require a deep geometrical compatibilit y of the equiv arian t K-cycle ( W, E , f ) with the orbifold structure of [ X/G ] (or else an explicit determination of the unknown c haracteristic class Λ G ( W ) correcting the Riemann-Ro ch formula as explained in Section 4.6). The example of the linear orbifolds considered in Sections 3.3 and 4.5, and in Section 6.2 b elo w , is simple enough to s atisfy these cond itions. It w ould b e very interesting to ﬁnd a geometrically non-trivial explicit example to test these requ iremen ts on. In an y case, the results ab o v e suggest that the orbifold d iﬀerential K-theory (or more precisely a complex v ersion of it) deﬁ ned in the p revious section is the natural framework in wh ic h to d escrib e top ologicall y non-trivial Ramond -Ramond ﬁelds on orbifolds. It w ould b e h ighly desirable to determine the correct generaliza tion of eq. (6.2) to the orbifold d iﬀeren tial K-theory group ˇ K • G ( X ) of the previous section, an d thereby extendin g the delocalized Ramond-Ramond cur ren ts to include eﬀects su ch as torsion. 6.2. Linear orbifolds. T o und er s tand certain asp ects of the orbifold diﬀerentia l K-theory groups, it is instructiv e to stud y the K-theory classiﬁcation of Ramond-Ramond ﬁ elds on the linear orbifolds considered in Sections 3.3 and 4.5. S ince the C -linear G -mo du le V is equiv ari- an tly con tr actible, one has H odd G ( V ; R ⊗ R ( − )) = 0 and K 0 G ( V ) = R ( G ). F rom Theorem 5.2 it then follo ws that ˇ K 0 G ( V ) ∼ = A 0 K G ( X ) ∼ =  ( γ , ω ) ∈ R ( G ) × Ω ev en G, cl ( V ; R )   c h G/G ( γ ) = [ ω ] G − dR  . Since the equ iv ariant Ch ern charact er ch G/H : R ( H ) → R ( H ) for H ≤ G is the iden tity map, the setwise ﬁb r e pro d u ct tr u ncates to the lattice of q u an tized orb ifold d iﬀeren tial forms and one has ˇ K 0 G ( V ) = Ω ev en K G ( V ; R ) . (6.4) This is the group of T yp e I I A Ramond-Ramond form p otenti als on V . It naturally con tains those ﬁelds wh ic h trivialize the Ramond -Ramond currents sourced by the stable fractional D0- branes of the Typ e I IA theory , corresp ondin g to c h aracteristic classes [ c ] in the represen tation ring R ( G ) as explained in S ection 3.3. This can b e explicitly describ ed as an extension of th e group of top ologically trivial Ramond- Ramond ﬁelds C of o d d d egree b y the equiv ariant K-theory of V , as implied by Corollary 5.3. Since V is connected and G -con tractible, one has Ω 0 G, cl ( V ; R ) = R ⊗ R ( G ) and the group (6.4) has a n atural splitting ˇ K 0 G ( V ) = R ( G ) ⊕  d M k =1 Ω 2 k G, cl ( V ; R )  . (6.5) 34 RICHARD J. SZABO AND ALESSANDRO V ALENTINO An y closed orbifold form ω on V of p ositiv e d egree is exact, ω = d G C , with the gauge inv 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 ( V ; R ) − → Ω odd G ( V ; R ) Ω odd K G ( V ; R ) whic h asso ciates to the ﬁeld strength ω the corresp onding glo bally well- deﬁned Ramond-Ramond p oten tial C . On the other hand, the orbifold d iﬀeren tial K-theory group ˇ K − 1 G ( V ) of T yp e I IB Ramond- Ramond ﬁelds on V can b e computed b y u sing the c haracteristic class exact sequence (5.13) with degree shifted by − 1. Using K − 1 G ( V ) = 0, one ﬁ nds ˇ K − 1 G ( V ) = Ω ev en G ( V ; R ) Ω ev en K G ( V ; R ) . (6.6) This result reﬂects the fact that the Typ e I IB theory has no stable fractional D0-branes. Hence there is no extension and the Ramond-Ramond ﬁelds are induced solely b y the closed string bac kground . Th eir ﬁeld strengths ω = d G C are determined en tirely by the p oten tials C , wh ich are globally deﬁned diﬀerential forms of ev en degree. Note that for an y G -homogeneous space G/H one has K − 1 G ( G/H ) = 0 and Ω odd G ( G/H ; R ) = 0 . ¿F rom th e c haracteristic class exact sequence (5.13) one th u s compu tes that the orb ifold d iﬀer- en tial K-theory group ˇ K 0 G ( G/H ) ∼ = K 0 G ( G/H ) ∼ = R ( H ) (6.7) is giv en by th e c haracteristic classes (of f ractional D0-branes), while Theorem 5.2 (with degree shifted by − 1) implies that the orb ifold diﬀerent ial K -theory group ˇ K − 1 G ( G/H ) ∼ = H ev en G  G/H ; R ⊗ R ( − )  c h G/H  K 0 G ( G/H )  ∼ = R ( H ) ⊗ R / Z (6 .8) is giv en b y the top ologically trivial ﬂat ﬁelds. Setting H = G in eqs. (6.7) and (6.8) sh o ws that the diﬀeren tial K G -theory groups of a p oin t generically diﬀer from the groups (6.4) and (6.6) , ev en thou gh V is G -con tractible. This exempliﬁes the G -homotop y non-inv ariance of the orbifold diﬀeren tial K-theory groups, required to capture the non-v anish ing (but top olog ically tr ivial) gauge p oten tials on V . 6.3. Flat p ot e ntials. In Section 6.2 ab o ve we encoun tered some examples of top olog ically trivial R amond -Ramond ﬁelds, corresp onding to gauge equiv alence classes with trivial K-theory ﬂux [ c ] = 0. They are the globally d eﬁned orbifold diﬀerential forms C ∈ Ω • G ( X ; R ) with the gauge symmetry C → C + ξ , wh ere d G ξ = 0 and ξ ∈ Ω • K G ( X ; R ), an d ﬁeld strength ω = d G C . The ﬂat Ramond-Ramond ﬁ elds are instead classiﬁed b y the ab elian group K i G ( X ; R / Z ), where i = 0 for Type I IB theory and i = − 1 for Type I IA theory . I n the p revious section this group w as deﬁned to b e the su bgroup of orbifold d iﬀeren tial K-theory with v anishin g curv ature. In the follo w in g w e will conjecture a v ery natural algebraic deﬁnition of these groups wh ic h ties them somewhat more dir ectly to equiv arian t K-theory groups. T o motiv ate this conjecture, we ﬁ rst compute t he groups K • G ( V ; R / Z ) for the linear o rbifolds of Section 6.2 a b o v e, wherein the asso ciated diﬀerentia l K-theory groups were determined explicitly . Using the ﬁ eld strength exact sequence (5.15) , by d eﬁnition one h as K − 1 G ( V ; R / Z ) ∼ = k er  f fs : ˇ K 0 G ( V ) → Ω ev en K G ( V ; R )  RAMOND-RAM OND FIELDS, FRACTIONAL BRA NES AND ORBIFOLD DIFFERENTIAL K- THEOR Y 35 whic h from th e natural isomorp h ism (6.4) tr ivially giv es K − 1 G ( V ; R / Z ) = 0 . (6.9) Similarly , using K − 1 G ( V ) = 0 one has K 0 G ( V ; R / Z ) ∼ = k er  f fs : ˇ K − 1 G ( V ) → Ω odd G, cl ( V ; R )  . Using the n atural isomorph ism (6.6), the ﬁ eld strength map is f fs  [ C ]  = d G C for C ∈ Ω ev en G ( V ; R ) , giving K 0 G ( V ; R / Z ) ∼ = Ω ev en G, cl ( V ; R ) Ω ev en K G ( V ; R ) . Similarly to eq. (6.5), th ere is a natural splitting of the v ector space of closed orbifold diﬀeren tial forms giv en by Ω ev en G, cl ( V ; R ) =  R ( G ) ⊗ R  ⊕  d M k =1 Ω 2 k G, cl ( V ; R )  and we arr iv e ﬁnally at K 0 G ( V ; R / Z ) = R ( G ) ⊗ R / Z . (6.10) These results of course simp ly follo w from the fact that V is G -co n tr actible, so that ev ery d G -closed Ramond-Ramond ﬁ eld is trivial, except in d egree zero where the gauge equiv alence classes are naturally parametrized by the t w isted sectors of the str in g theory in eq. (6.10). Note that b oth group s of ﬂat ﬁelds (6.9) and (6.10) are unchanged b y (equiv arian t) contrac tion of th e G -mo d ule V to a p oin t, as an analogous (but simp ler) calculation sho w s. T his suggests that th e groups K • G ( X ; R / Z ) ha ve at least some G -homotop y inv ariance pr op erties, unlike the diﬀeren tial K G -theory groups . This motiv ates th e follo win g conjectural algebraic fr amew ork for describing these groups . W e w ill prop ose that th e group K • G ( X ; R / Z ) is an extension of th e torus of top ologically trivial ﬂ at orb ifold Ramond-Ramond ﬁelds b y the torsion elemen ts in K • +1 G ( X ), as they h a ve v anish ing image un der the equ iv ariant C hern c h aracter c h X . T he r esulting group ma y b e called the “equiv ariant K-theory with coeﬃcients in R / Z ”. The s hort exact sequence of co eﬃcient groups 0 − → Z − → R − → R / Z − → 0 induces a long exact sequence of equiv arian t K-theory group s wh ic h, by Bott p er io d icit y , trun- cates 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 (6.11) The connecting homomorphism β is a s uitable v ariant of the usual Bo c kstein h omomorphism. W e assume that the equ iv ariant K-theory with real coeﬃcien ts is deﬁned simp ly b y the Z 2 -graded ring K • G ( X ; R ) = K • G ( X ) ⊗ R ∼ = H • G  X ; R ⊗ R ( − )  , where we ha ve used Theorem 2.5. Th e maps to real K -theory in eq. (6.11) ma y then b e iden tiﬁed with the equiv ariant Chern c haracter c h X , wh ose image is a full lattice in the Bredon cohomolo gy group H • G ( X ; R ⊗ R ( − )). Then the ab elian 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 . (6.12) 36 RICHARD J. SZABO AND ALESSANDRO V ALENTINO When G = e , eq. (6.11) is the us ual Bockstein exact sequence for K-theory . In this case, an explicit geometric realization of the groups K • ( X ; R / Z ) in terms of bund les with connec- tion has b een giv en b y Lott [43]. Mo reo ver, in ref. [37] a geometric construction of the map K − 1 ( X ; R / Z ) → ˇ K 0 ( X ) in the ﬁeld strength exact sequence is giv en. Unfortun ately , no such geometrica l description is immediately a v ailable for our equiv ariant diﬀeren tial K-theory , d ue to the lac k of a Ch ern-W eil th eory for the homotopy theoretic equiv arian t Chern c haracter of Section 2.4. Our conjectural deﬁnition (6.12) is satisﬁed by the linear orb ifold groups (6.9) and (6.10). In ref. [14] a v ery diﬀerent d eﬁ nition of the groups K • G ( X ; R / Z ) is giv en, by deﬁning b oth equiv ariant K-theory and cohomology us in g the Borel construction of E x amp le 1.3. Th en the Bo c kstein exact sequence (6.1 1) is written for the ordinary K-theory groups of the homotop y quotien t X G = E G × 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 (6.12). Th e r eason is that the equiv ariant C hern c h aracter used is not an isomorphism o ver the r eals, as explained in Section 2.1 (see also ref. [47] for a description of K • ( X G ) as the completion of K • G ( X ) w ith resp ect to a certain ideal). Moreo ver, an asso ciated diﬀeren tial K -theory construction would d ir ectly inv olv e diﬀeren tial forms on the in ﬁnite-dimensional space X G whic h is only homotopic to the ﬁn ite- dimensional C W-complex X/G . The physic al inte rpretation of such ﬁelds is not clear. Ev en in the simple case of the linear orbifolds V studied ab ov e, this d escrip tion predicts an inﬁnite set of equiv ariant ﬂuxes of arb itrarily h igh dimen sion on the in ﬁnite-dimensional classifying space B G , and one must p erform some n on-canonical quotien ts in ord er to try to isolate the physical ﬂuxes. The diﬀerences b et we en the equiv arian t K-theory and Borel cohomology grou p s of V also require p ostulating certain eﬀects of fractional branes on the orbifold, as in ref. [12]. I n con trast, with our constru ctions the relation b et ween orbifold ﬂu x groups and Bredon cohomology is muc h more natur al, and it inv olv es only ﬁn itely-man y orbifold Ramond-Ramond ﬁelds. 6.4. Consistency conditions. As we h av e stressed through ou t this p ap er, the u sage of Borel cohomology as a companion to equiv arian t K-theory in the top ological classiﬁcation of D-br anes and Ramond-Ramond ﬂuxes on orbifolds has v arious und esirable features, most notably the fact that it inv olves torsion classes sub s tan tially , esp ecially when ﬁnite group cohomology is inv olv ed. In our applications to string geometry , it is more con v enien t to use an equiv arian t cohomology theory with sub stan tial torsion-fr e e information. Bredon cohomology natur ally accomplishes this, as instead of group cohomology the b asic ob ject is a r epresen tation rin g. In fact, as we n ow demonstrate, th e form u lation of top ological consistency conditions f or orbifold Ramond-Ramond ﬁelds and D-branes within the f r amew ork of equiv ariant K -theory naturally necessitates the u se of classes in Bredon cohomology . Giv en a Bredon cohomology class λ ∈ H • G ( X ; R ( − )), let us ask if ther e exists a Ramond- Ramond ﬁeld f or whic h ω = λ in the sense of eq. (6.3). F or this, w e m ust ﬁnd an equiv ariant K-theory lift ξ ∈ K • G ( X ) of λ . As in th e non-equiv ariant case [22, 48], th e obstructions to suc h a lift can b e determined via a suitable sp ectral sequence. F or equiv arian t K-theory the appropriate sp ectral sequence is describ ed in refs. [20, 50] (see also ref. [58]) using the skel etal ﬁltration ( X n ) of Section 1.1. W e w ill n o w brieﬂy explain the construction of this sp ectral sequence and its natural relationship with the obstruction theory for Ramond -Ramond ﬂux es in Bredon cohomology . The E 1 -term of th e sp ectral sequence is the r elativ e G -equiv arian t K-theory group E p,q 1 = K p + q G ( X p , X p − 1 ) with diﬀerentia l d p,q 1 : E p,q 1 − → E p +1 ,q 1 induced by the long exact sequ en ce of the triple ( X p +1 , X p , X p − 1 ) in equiv ariant K-theory , i.e . , d p,q 1 is the comp osition of the map i ∗ induced by th e inclusion i : ( X p , ∅ ) ֒ → ( X p , X p − 1 ) with RAMOND-RAM OND FIELDS, FRACTIONAL BRA NES AND ORBIFOLD DIFFERENTIAL K- THEOR Y 37 the cellular cob oundary op erator of the pair ( X p +1 , X p ). F r om eq. (1.1) it follo ws th at there is a homeomorphism ` j ∈ J p  ˚ B p j × G/K j  → X p \ X p − 1 , and h ence E p,q 1 ∼ = M j ∈ J p K p + q G  ˚ B p j × G/K j  ∼ = M j ∈ J p K q G  G/K j  . Th us E p,q 1 = 0 for q o d d, while for q ev en th e group E p,q 1 is a direct sum of repr esen tation rings R ( K j ) o ve r all isotropy subgroups of p -cells of orbit t yp e G/K j . It parametrizes equiv ariant K-theory classes deﬁned on the p -sk eleton of X w hic h are trivial on the ( p − 1)-ske leton, and giv es the sup p orts of p -form ﬁelds an d c h arges on the orbifold wh ich carry no low er or higher degree ﬂu x es. The E 2 -term of the sp ectral sequence is the cohomology of the diﬀeren tial d 1 . The cohomolog y of the co chain complex assem bled fr om suc h terms is the equ iv ariant cohomology w ith co eﬃcien t system R ( − ) on Or ( G, F ( X p )) for q = 0, and thus a necessary condition for a p -form Ramond- Ramond ﬁeld to lift to K • G ( X ) is that it d eﬁne a non-trivial co cycle in Bredon cohomology . T his is consistent with Deﬁnition 5.1. The resulting Ati y ah-Hirzebru c h sp ectral sequence ma y then b e wr itten E p,q 2 = H p G  X ; π − q K G ( − )  = ⇒ K p + q G ( X ) and it lives in the ﬁrst and fou r th qu adran ts of the ( p, q )-plane. On the r -th terms E p,q r , the diﬀeren tial d p,q r has b idegree ( r, − r + 1), and E p,q r +1 is the corresp onding cohomolog y group . Note that d p,q r = 0 for all r ev en , since then either its source or its target v anishes (as K q ( C [ H ]) = 0 for all q o d d and H ≤ G ). The E ∞ -term is th e inductiv e limit E p,q ∞ = lim − → r E p,q r . F or a ﬁnite-dimensional manifold X , one h as E p,q r = E p,q ∞ for all r > dim ( X ) and the sp ectral sequence conv erges to K p + q G ( X ). T his means that the E ∞ -term is the asso ciated graded group of a decreasing ﬁnite ﬁltration ﬁlt p,q K p + q G ( X ) ⊂ ﬁlt p − 1 ,q +1 K p + q G ( X ), 0 ≤ p ≤ d im( X ) with K q G ( X ) = ﬁ lt 0 ,q K q G ( X ) and ﬁlt p,q K p + q G ( X ) ﬁlt p +1 ,q − 1 K p + q G ( X ) ∼ = E p,q ∞ . (6.13) Explicitly , if ι : X p − 1 ֒ → X denotes the inclusion of the ( p − 1)-sk eleton in X , then the ﬁ ltration groups ﬁlt p,q K p + q G ( X ) := k er  ι ∗ : K p + q G ( X ) → K p + q G ( X p − 1 )  consist of Ramond -Ramond ﬂuxes wh er e the ﬁeld stren gth ω is a form of d egree ≥ p , wh ile th e extension grou p s (6.13) consist of p -form ﬂuxes with v anish ing higher and lo w er d egree ﬂuxes. By Theorem 2.5, the equiv ariant Chern c haracter c h X determines an isomorp h ism from the limit of the sp ectral sequence to its E 2 -term. T h u s the sp ectral s equ ence collapses rationally , and s o the images of all higher diﬀeren tials d p,q r , r > 2 in the sp ectral sequence consist of torsion classes. It follo ws that the next non-trivial ob s truction to extending a Ramond-Ramond ﬁ eld is giv en b y a “cohomology op eration” d p, 0 3 : H p G  X ; R ( − )  − → H p +3 G  X ; R ( − )  . (6.14) Th us a necessary condition for a Bredon cohomology class λ ∈ H p G ( X ; R ( − )) to su rviv e to E p, 0 ∞ is giv en by d p, 0 3 ( λ ) = 0 . (6.15) W e interpret th e condition (6.15) as a (partial) requ iremen t of global worldsheet anomaly can- cellatio n for Ramond -Ramond ﬂuxes and, d ually , the w orldvol ume homology cycles that they pair with. This is the orbifold generalizat ion of the F r eed-Witten condition [22, 30, 48] for- m u lated in terms of obstr uction classes in Bredon cohomology . It is a necessary condition for 38 RICHARD J. SZABO AND ALESSANDRO V ALENTINO the existence of a fractional D-brane whose low est non-v anishing Ramond-Ramond charge is λ ∈ H p G ( X ; R ( − )). On the other h and, in computing the E 3 -term as th e cohomology of the diﬀeren tial (6.1 4), we m u s t also tak e the quotient by the image of d p − 3 , 0 3 . This means that a class λ satisfying eq. (6.15) must b e fu rther sub jected to the identiﬁcatio n s λ ∼ λ + d p − 3 , 0 3 ( λ ′ ) ( 6.16) in E p, 0 3 , f or an y class λ ′ ∈ H p − 3 G ( X ; R ( − )). W e interpret the condition (6.16) as accoun ting for Ramond-Ramond charge violation du e to D-instan ton eﬀects in th e orbifold bac kgroun d , as explained in ref. [48] for the non-equiv arian t case. It means that while there exists a fr actional brane wh ose low est Ramond-Ramond charge is d p − 3 , 0 3 ( λ ′ ), this D-br ane is u nstable. The p assage fr om the limit (6.13) with q = 0 to the actual equiv ariant K-theory group K p G ( X ) requires s olving a t ypically n on-trivial extension p roblem. Eve n when the sp ectral se- quence collapses at the E 2 -term, the extension can lead to imp ortant torsion corrections whic h distinguish the classiﬁcations of Ramond-Ramond ﬁelds based on Bredon cohomolo gy and on equiv ariant K -theory . The exte nsion problem c hanges the additiv e structure on the K-theory group of ﬂ uxes f rom that of the equiv arian t cohomology classes. This corresp ond s physicall y to non-trivial correlations b et ween Ramond-Ramond ﬁelds of d iﬀeren t degrees, w h en represented b y orbifold diﬀeren tial f orms. This torsion enh ancemen t in equiv ariant K-theory compared to Bredon cohomology can shift th e Dirac c h arge qu an tization condition on the Ramond-Ramond ﬁelds by fr actional units and can play an imp ortant role near th e orbifold p oin ts [12, 13]. In the n on-equiv ariant case G = e , th e diﬀerentia l d p, 0 3 is kn o wn to b e giv en by th e Steenro d square cohomology op eration Sq 3 . The v anishing condition Sq 3 ( λ ) = 0 imp lies the v anishing of the thir d inte ger S tieﬀel-Whitney class of the Poi ncar ´ e dual cycle to λ ∈ H p ( X ; Z ), wh ich is just the condition guarant eeing that the corresp onding brane worldv olume is a spin c submanifold of X . Unfortunately , for G 6 = e th e diﬀeren tial d p, 0 3 is not kn own an d th e geometrical meanin g of the condition (6.15) is un clear. It wo uld b e in teresting to un derstand th is requ iremen t in terms of an obstru ction theory for Bredon cohomology , analogously to the non-equiv arian t case, as this would op en up in teresting new consistency cond itions f or D-branes and Ramond-Ramond ﬁelds on global orbifolds [ X/G ]. Ho wev er, w e are not a ware of any c haracteristic class th eory underlying the Bredon cohomology grou p s H p G ( X ; R ( − )). Appendix A. Linear algebra in functor ca tegories In this app end ix w e will su mmarize some notions ab out algebra in fu nctor cate gories that w ere used in the main text of the p ap er. T h ey generalize the more commonly u sed concepts for mo dules ov er a ring. F or fu r ther details see r ef. [23]. Let R b e a comm utativ e ring, and denote th e catego r y of (left) R -mo dules b y R − Mod. Let Γ b e a smal l cate gory , 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 g ory of (co v arian t) fu nctors Γ → C . The ob jects of [Γ , C ] are (co v ariant) functors φ : Γ → C and a morphism from φ 1 to φ 2 is a natural transformation α : φ 1 → φ 2 b et ween functors. In particular, in the m ain text w e u sed the fu nctor category R Γ − Mo d := [Γ , R − Mo d] whose ob jects are called left R Γ -mo dules . If one denotes with Γ op the du al category to Γ, th en there is also the functor category Mo d − R Γ := [Γ op , R − Mo d] RAMOND-RAM OND FIELDS, FRACTIONAL BRA NES AND ORBIFOLD DIFFERENTIAL K- THEOR Y 39 of cont ra v ariant functors Γ → R − Mod, whose ob jects are called rig ht R Γ -mo dules . As an example, let G b e a discrete group regarded as a category w ith a sin gle ob ject and a morphism for eac h elemen t of G . A co v ariant functor G → R − Mo d is then the same th in g as a left mo dule o ve r the group ring R [ G ] of G o ver R . As the n ame itself su ggests, all standard deﬁn itions from the linear algebra of mo du les hav e extensions to this m ore general setting. F or ins tance, the notions of submo dule, kernel, c okernel, dir e ct sum, c opr o duct, etc. can b e n aturally deﬁn ed ob ject wise. If M and N are R Γ-mo dules, then Hom R Γ ( M , N ) is the R -mo dule of all natural transformations M → N . This notation should not b e confused with the one u sed for the set of all morph ism s b et w een t wo ob jects in Γ, and u sually it is clear fr om the con text. If M is a righ t R Γ-mo d u le and N is a left R Γ-mo du le, then one can deﬁn e th eir catego rical tensor pr o duct M ⊗ R Γ N in the follo wing w a y . I t is the R -mo du le given b y ﬁrst forming the direct su m F = M λ ∈ Ob j(Γ) M ( λ ) ⊗ R N ( λ ) and then quotien ting F by the R -su b mo dule generated by 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 pr o duct comm u tes w ith copro d ucts. If M and N are functors from Γ to the category of vect or sp aces o ver a ﬁeld K , then their tensor p ro duct is n aturally equip ed with the structure of a v ector space o ver K . When Γ is the orbit category Or ( G ) and R = Z , the tensor p ro duct has precise limiting cases. F or an arbitrary contra v ariant mo dule M and th e constant co v ariant mo dule N , the categorica l pro duct M ⊗ Z Or ( G ) N is th e tensor pro duct of the right Z [ G ]-mo dule M ( G/e ) with the constan t left Z [ G ]-mo dule N ( G/e ), M ( G/e ) ⊗ Z [ G ] N ( G/e ). On th e other hand, if the contra v ariant mo du le M is constan t and the co v arian t mo dule N is arb itrary , then M ⊗ Z Or ( G ) N is just N ( G/G ). Appendix B. Equiv ariant K-hom ology This app endix is dev oted to explaining in more detail some of th e d eﬁnitions and tec h n ical constructions in equiv arian t K-homology theories that w ere used in the m ain text to describ e states of D-br anes in orbifolds. B.1. Sp ectral deﬁnit ion. A natural wa y to deﬁne the equiv ariant homology theory K G • is b y means of a sp e ctrum for equiv arian t top ologica l K -theory K • G , whic h w ithin the con text of Section 2 is a particular co v arian t fu n ctor V ect G ( − ) fr om the orbit category Or ( G ) to the tensor catego ry Sp ec of sp ectra [20]. Giv en an y G -co m plex X , the corr esp onding p oint ed G -space is X + = X ∐ pt and one deﬁnes the lo op sp ectrum X + ⊗ G V ect G ( − ) by X + ⊗ G V ect G ( − ) = a G/H ∈ Or ( G )  X H + ∧ V ect G ( G/H )   ∼ , (B.1) where the equiv alence relation ∼ is generated b y the identi ﬁcations 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 ( − )  . (B.2) By using v arious G -homotop y equiv alences of the lo op sp ectra (B.1), one sho ws that this deﬁnition of equiv arian t K-homology comes with a natural induction structure in the sense of Section 1.2. 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 ﬁ nite group, any ﬁ nite-dimensional represent ation of G n aturally 40 RICHARD J. SZABO AND ALESSANDRO V ALENTINO extends to a complex representa tion of th e group rin g C [ G ]. Then th er e is an analytic assem bly map ass : K G • ( X ) − → K •  C [ G ]  to th e K-theory of the ring C [ G ], indu ced by th e collapsing map X → pt and the isomorphisms K •  C [ H ]  ∼ = π •  V ect G ( G/H )  ∼ = K G • ( G/H ) ∼ = R ( H ) for an y su bgroup H ≤ G . In the follo wing we will give tw o concrete realizati ons of the homotop y groups (B.2). B.2. Analytic deﬁnition. Th e simplest realiz ation of the equiv arian t K-homolo gy group K G • ( X ) is within the framew ork of an equiv arian t v ersion of Kasparo v’s KK-theory KK G • . Let A b e a G -alge bra, i.e. , a C ∗ -algebra A together with a group homomorphism λ : G − → Aut( A ) . By a Hilb ert ( G, A )-mo dule w e mean a Hilb ert A -mo dule E together w ith a G -act ion giv en by a homomorph ism Λ : G → GL( E ) such that Λ g ( ε · a ) = Λ g  ε · λ g ( a )  (B.3) for all g ∈ G , ε ∈ E and a ∈ A . Let L ( E ) d enote the ∗ -algebra of A -linear maps T : E → E admitting an adjoin t with resp ect to th e A -v alued inner pro du ct on E . The indu ced G -action on L ( E ) is given by g · T := Λ g ◦ T ◦ Λ g − 1 . Let K ( E ) b e th e sub algebra of L ( E ) consisting of generalized compact op erators. Giv en a pair ( A , B ) of G -alg eb ras, let D G ( A , B ) b e the set of trip les ( E , φ, T ) where E is a coun tably generated Hilb ert ( G, B )-mo dule, φ : A → L ( E ) is a ∗ -homomorphism which com- m u tes with the G -actio n, φ  λ g ( a )  = Λ g ◦ φ ( a ) ◦ Λ g − 1 (B.4) for all g ∈ G and a ∈ A , and T ∈ L ( E ) su c h that 1) [ T , φ ( a )] ∈ K ( E ) f or 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 n o w an alogously deﬁ n ed. The set of equiv- alence classes in D G ( A , B ) deﬁnes the equiv arian t KK-theory group s KK G • ( A , B ). If X is a smo oth prop er G -manifold without b ound ary , and G acts on X b y diﬀeomorph isms, then the algebra A = C 0 ( X ) of cont in u ous functions on X v anishin g at inﬁnit y is a G -alge bra with automorphism λ g on A giv en by λ g ( f )( x ) :=  g ∗ f  ( x ) = f  g − 1 · x  , where g ∗ denotes the pullback of the G -action on X by left tr an s lation by g − 1 ∈ G . W e deﬁn e K G • ( X ) := KK G •  C 0 ( X ) , C  (B.5) with G acting trivially on C . The conditions (B.3) and (B.4) naturally capture the p h ysical requirement s th at p h y s ical orbifold string states are G -in v ariant and also that the wo r ldv olum e ﬁelds on a fractional D-brane carry a “co v arian t represent ation” of the orbifold group [26 ]. B.3. The equiv a ria n t Dirac class. W e can determine a canonical class in the ab elian group (B.5) as f ollo ws. Let dim( X ) = 2 n , and let G be a ﬁ nite subgroup of the rotati on group SO(2 n ). 11 Let Cliﬀ (2 n ) = C liﬀ + (2 n ) ⊕ Cliﬀ − (2 n ) 11 Throughout the ex tension to K G 1 or K − 1 G and dim( X ) o dd can b e describ ed in t he same wa y as in degree zero by replacing X with X × S 1 . RAMOND-RAM OND FIELDS, FRACTIONAL BRA NES AND ORBIFOLD DIFFERENTIAL K- THEOR Y 41 denote the complex Z 2 -graded euclidean Cliﬀord algebra on n generators e 1 , . . . , e n with the relations e i e j + e j e i = − 2 δ ij . A c hoice of a complete G -in v arian t riemannian metric on X deﬁnes a G -bundle of Cliﬀord algebras Cliﬀ = Cliﬀ  T ∗ X  := F r ∗ × SO(2 n ) Cliﬀ (2 n ) whic h is an asso ciated bu ndle to the metric coframe bund le o ver X , the principal SO(2 n )-bundle F r ∗ = F r( T ∗ X ) of oriente d orthonormal frames on th e cotangen t bun dle T ∗ X = F r ∗ × SO(2 n ) R 2 n . The action of SO(2 n ) on the Cliﬀord alge b ra is through the sp in group Spin(2 n ) ⊂ Cliﬀ (2 n ). The Lie group Spin c (2 n ) ⊂ Cliﬀ (2 n ) is a cen tral extension of S O(2 n ) b y the circle group U(1), 1 − → U(1) − → S pin c (2 n ) − → SO(2 n ) − → 1 , (B.6) where the quotient map in eq. (B.6) is consistent with the dou b le co v ering of SO(2 n ) by Spin(2 n ) so that Spin c (2 n ) = Spin(2 n ) × Z 2 U(1) . 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 coframe b undle to a pr incipal Spin c (2 n )-bundle F r ∗ L o ve r X wh ic h is compatible with the G -actio n. T he extension F r ∗ L ma y b e regarded as a prin cipal circle bund le o v er F r ∗ , U(1) y y r r r r r r r r r r   ˆ G / /   Spin c (2 n ) / /   F r ∗ L / /   X , G / / SO(2 n ) / / F r ∗ < < x x x x x x x x x where the pullb ac k square on the b ottom left d eﬁ nes th e required co v ering of the orbifold group G < SO(2 n ) b y a sub group of the spin c group ˆ G < Sp in c (2 n ). T his lift is also necessary in order to accoun t for th e spacetime f ermions present in string theory . The kernel of the homomorphism ˆ G → G is identiﬁed with the circle group U(1) < Spin c (2 n ) in the Cliﬀord algebra Cliﬀ (2 n ). W e ﬁx a c h oice of lift and h ence assum e that G is a discrete su bgroup of th e spin c group. Z 2 -graded Cliﬀord mo dules are likewise extended to r epresen tations of C [ G ] ⊗ Cliﬀ (2 n ), with C [ G ] the group ring of G , calle d G -Cliﬀord mo dules. The top ological obstruction to th e existence of a G -spin c structure on X is the equiv arian t third in tegral Stiefel- Whitney class ( W 3 ) G ( T ∗ X ) ∈ H 3 G ( X ; Z ) of the cotangen t bundle T ∗ X in Borel cohomology . The asso ciated b undles of half-spinors on X are deﬁned as S ± = S  T ∗ X  ± := F r ∗ L × Spin c (2 n ) ∆ ± , (B.7) where ∆ ± are the irreducible half-spin representa tions of S O(2 n ). Sin ce G lifts to ˆ G in the spin c group, the half-spin represen tations ∆ ± restrict to represent ations of G and the half-spinor bund les (B.7) are G -bundles. The G -in v ariant Levi-Civita connection determines a connection one-form on F r ∗ , and together with a c h oice of G -in v ariant connection one-form on the pr incipal U(1)-bundle F r ∗ L → F r ∗ , they d etermin e a connection one-form on the principal Spin c (2 n )-bundle F r ∗ L → X whic h is G -in v ariant. This determines an inv arian t connection ∇ S ⊗ E := ∇ S ⊗ 1 + 1 ⊗ ∇ E : C ∞  X , S + ⊗ E  − → C ∞  X , T ∗ X ⊗ S + ⊗ E  where ∇ E is a G -in v ariant connection on a G -bund le E → X . T h e con traction giv en b y C liﬀord m u ltiplicatio n deﬁn es a map C ℓ : C ∞  X , T ∗ X ⊗ S + ⊗ E  − → C ∞  X , S − ⊗ E  42 RICHARD J. SZABO AND ALESSANDRO V ALENTINO whic h graded commutes with th e G -actio n, and the G -in v ariant spin c Dirac op erator on X with co eﬃcien ts in E is deﬁned as the comp osition D / X E = C ℓ ◦ ∇ S ⊗ E . (B.8) W e will view the op erator (B.8) 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 ) act s on the Z 2 -graded G - Hilb ert s p ace E := L 2 ( X, S ⊗ E ) by m ultiplication. Deﬁne th e b ound ed G -inv arian t op erator T := D / X E  ( D / X E ) 2 + 1  − 1 / 2 ∈ F red G . Then  D / X E  is repr esen ted by th e G -equiv ariant F r edholm mo dule ( E , T ). B.4. Geometric deﬁnition. The natural geometric description of D-branes in an orbifold space is provi ded b y the top ological ve rsion of the group s K G • ( X ) d ue to Baum, Con n es and Douglas [8, 7]. Th is can b e deﬁn ed f or an arbitrary discrete, count able group G on the category of prop er, ﬁnite G -complexes X and pr ov en to b e isomorph ic to an alytic equiv ariant K-homology [10]. Recall th at the top ological equiv arian t K-theory K • G ( X ) is deﬁned b y applyin g the Grothendiec k functor K • to th e additiv e category V ect C G ( X ) wh ose ob jects are complex G -v ector b undles ov er X , i.e. , K • G ( X ) := K •  V ect C G ( X )  . In the h omologic al setting, the r elev an t category is instead the a dditiv e catego ry of G -e quivariant K-cycles D G ( X ), w hose o b jects are tr ip les ( W, E , f ) where (a) W is a man if old without b oundary with a smo oth prop er co compact 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 arian t K-cycles ( W, E , f ) and ( W ′ , E ′ , f ′ ) are said to b e isomorp hic if there is a G -equiv ariant diﬀeomorphism h : W → W ′ preserving the G -spin c structures on W , W ′ suc h that h ∗ ( E ′ ) ∼ = E and f ′ ◦ h = f . Deﬁne an equiv alence relation ∼ on the category D G ( X ) generated by the op erations of • Bordism: ( W i , E i , f i ) ∈ D G ( X ), i = 0 , 1 are b or dant if th ere is a triple ( M , E , f ) where M is a manifold w ith b oundary ∂ M , with a smo oth prop er co compact G -act ion and G -spin c structure, E → M is a complex G -v ector b undle, and f : M → X is a G -map suc h that ( ∂ M , E | ∂ M , f | ∂ M ) ∼ = ( W 0 , E 0 , f 0 ) ∐ ( − W 1 , E 1 , f 1 ). Here − W 1 denotes W 1 with the rev ersed G -spin c structure; • Direct sum: If ( W , E , f ) ∈ D G ( X ) and E = E 0 ⊕ E 1 , then ( W , E , f ) ∼ = ( W , E 0 , f ) ∐ ( W , E 1 , f ) ; and • V ector b undle mo d iﬁcation: Let ( W, E , f ) ∈ D G ( X ) and H an even-dimensional G - spin c v ector bun dle o v er W . L et c W = S ( H ⊕ 1 1) denote th e sph ere bu ndle of H ⊕ 1 1 , whic h is canonically a G -spin c manifold, w ith G -bundle p ro jection π : c W → W . Let S ( H ) = S ( H ) + ⊕ S ( H ) − denote the Z 2 -graded G -bun dle o v er W of spinors on H . Set b E = π ∗  ( S ( H ) + ) ∨ ⊗ E  and b f = f ◦ π . Then  c W , b E , b f  ∈ D G ( X ) is the ve ctor bund le mo diﬁc ation of ( W, E , f ) b y H . W e set K G 0 , 1 ( X ) = D G ev en , o dd ( X )  ∼ where the p arit y refers to the d imension of the K-cycle, w h ic h is preserved by ∼ . RAMOND-RAM OND FIELDS, FRACTIONAL BRA NES AND ORBIFOLD DIFFERENTIAL K- THEOR Y 43 Using the equiv arian t Dirac class, one can construct a h omomorphism from the geometric to the analytic K-homology group . On K-cycles we deﬁne ( W , E , f ) 7→ f ∗  D / W E  and extend linearly . This map can b e u s ed to express G -index theorems within this homologic al fr amew ork and it extends to giv e an isomorph ism b et ween th e t wo equiv arian t K-homology group s [10]. (See also ref. [57] for a r elated construction in the non-equiv ariant case.) Appendix C. D-brane charges of eq uiv ariant K-cycle s In this app endix w e will review the construction of the equiv ariant Gysin h omomorp hism and ho w it shows that D-brane c h arges on the orbifold [ X/G ] tak e v alues in th e equ iv ariant K-theory K • G ( X ). L et X and W b e smo oth compact G -manifolds, and f : W → X a smo oth p rop er G - map. W e b egin by dealing with the non-equiv ariant setting G = e . Assu me that the Z 2 -graded bund le ν of eq. (4.10) is of ev en rank r = 2 n and endow ed with a spin c structure. W e will generalize th e construction [49 , 54, 62], establishing that the c h arge of a D-brane supp orted on W with Ch an-P aton gauge bu ndle E → W in T yp e I I sup ers tring theory without H -ﬂux tak es v alues in the complex K-theory of spacetime X , to D-branes represented b y generic top ologi- cal K-cycles ( W , E , f ), i.e . , including those D-branes whic h are not representable as w r apping em b edded cycles in X . It is based on the diagram ν ∼ = U π    % % J J J J J J J J J J W κ   f / / X X × R 2 q π 1 : : t t t t t t t t t o ve r the brane imm er s ion f , which we explain m omen tarily . The sp in c condition on th e bund le ν is the appropriate generalizati on of the F reed-Witten anomaly cancellati on condition [30] to this situation. It amounts to a c h oice of line bun dle L → W whose ﬁ rst C hern class c 1 ( L ) ∈ H 2 ( W ; Z ) ob eys c 1 ( L ) ≡ f ∗ w 2 ( T X ) − w 2 ( T W ) mo d 2, wh ere w 2 ( T X ) and w 2 ( T W ) are the second Stiefel- Whitney classes of the tangent bundles of X and W . The set of all suc h K-orienta tions is an aﬃne space mo delled on 2 H 2 ( W ; Z ). Consider ﬁrst the usual case where f : W ֒ → X is a smo othly em b edded cycle. Then th e virtual b undle ν can b e iden tiﬁ ed (in KO-theory) with the normal b undle to W with resp ect to f , whic h is the qu otien t bund le π : f ∗ ( T X ) /T W → W . Up on choosing a riemannian metric on X , we can iden tify ν with a tu b ular neighbourh o o d U of f ( W ) via a diﬀeomorphism fr om the op en em b edding  : U ֒ → X onto a n eigh b ourho o d of th e zero section em b edding W ֒ → ν . Let [ π ∗ S ( ν ) + , π ∗ S ( ν ) − ; c ( v )] b e the Ati y ah-Bott-Shapiro r ep resen tativ e of the Thom class Th om( ν ), in the K-theory with compact ve rtical su pp ort K r cpt ( ν ) := K r ( ν, ν \ W ) , whic h restricts to the Bott class u − n ∈ K − r (pt) on eac h ﬁb r e of ν . Here S ( ν ) ± − → W are th e half-spinor bun dles associated to ν and the morphism c ( v ) : π ∗ S ( ν ) + → π ∗ S ( ν ) − is giv en b y Cliﬀord m ultiplication b y the tautological section v of the bun dle π ∗ ν → ν whic h assigns to a ve ctor in ν the same v ector in π ∗ ν . Then one can deﬁne the Gysin h omomorphism in ordin ary K-theory f K ! : K • ( W ) − → K • ( X ) . 44 RICHARD J. SZABO AND ALESSANDRO V ALENTINO It is deﬁned as the comp osition of the Th om isomorphism K • ( W ) ≈ − → K • cpt ( ν ) ξ 7− → π ∗ ( ξ ) ⊗ Thom( ν ) with th e natural “extension by zero” homomorp h ism  : K • cpt ( ν ) → K • ( X ) given b y comp osin g K • ( U, U \ W ) → K • ( X, X \ W ) → K • ( X ), where the ﬁrst map is the excision isomorphism and the second map is induced by th e inclusion ( X , pt) ֒ → ( X , W ). F or a general smo oth p rop er map f : W → X , w e use the fact that ev ery smo oth compact manifold W can b e sm o othly em b edded in R 2 q for q suﬃciently large to deﬁn e a parametrized v ers ion that yields an embed ding κ : W − → X × R 2 q , whose normal bund le is sp in c . T he corresp ond ing Gysin map is a h omomorphism κ K ! : K • ( W ) − → K • cpt  X × R 2 q  . The Gysin homomorphism f K ! : K • ( W ) → K • ( X ) is th en deﬁned as the comp osition of κ K ! with the inv erse T hom isomorphism K • cpt ( X × R 2 q ) ∼ = K • ( X ) for the trivial spin c bund le π 1 : X × R 2 q − → X . By homotop y in v ariance of K-theory and functorialit y for p ushforward maps, the map f K ! is indep end en t of the c hoice of identi ﬁcation of th e normal b undle with a tubular neigh b ourho o d and of Whitney em b edd in g W ֒ → R 2 q . Let us no w consider the G -actions on W and on X . In a s im ilar w a y as in ordin ary K -theory , if ν is K G -orien ted then one has the equiv arian t Thom isomorphism K • G ( W ) ≈ − → K • G, cpt ( ν ) ξ 7− → π ∗ ( ξ ) ⊗ Th om G ( ν ) , where the equiv ariant Th om class Thom G ( ν ) ∈ K r G, cpt ( ν ) is deﬁned in th e same wa y as ab ov e using the G -spin c structure on ν and the equiv ariant v ers ion of the A tiy ah-Bott-Shapiro con- struction [42]. The asso ciated Gysin homomorphism, constru cted as ab o v e via a c h oice of G -in v arian t r iemannian metric on X and of G -inv arian t Whitney em b edding W ֒ → R 2 q with G acting trivially on R 2 q , is the p ushforward m ap f K G ! : K • G ( W ) → K • G ( X ). This establishes that the charge of a f ractional D-brane in the Typ e I I s p acetime orbifold [ X/G ], asso ciated to a generic G -equiv arian t K -cycle ( W , E , f ) ∈ D G ( X ) on the co v ering sp ace X , tak es v alues f K G !  [ E ]  ∈ K • G ( X ) in th e equiv ariant K-theory of X . Referen ces [1] A. A dem and Y. R uan. Twisted orbifold K- theory . Comm u n. Math. Phys. 237 , 533–556 (2003) [arXiv:math/010716 8]. [2] T. Asak aw a, S. Sugimoto and S . T erashima. D -branes, Matrix th eory and K -homology . J. High Energy Phys. 0203 , 034 (2002) [arXiv:hep- th/0108085 ]. [3] M.F. Atiyah. K-The ory ( Benjamin, 1967). [4] M.F. Atiyah and G.B. Segal. Equiva riant K- theory and completion. J. Diﬀ. Geom. 3 , 1–18 (1969). [5] M.F. Atiyah and G.B. Segal. On eq uiv ariant Euler chara cteristics. J. Geom. Phys. 6 , 399–405 (1989). [6] P . Baum and A. Connes. Chern character for discrete groups. in: A Fˆ ete of T op olo gy , pp . 163–232 (N orth Holland, Amsterdam, 1987). [7] P . Baum and A . Connes. Geometric K-theory for Lie groups and foliations. Enseign. Math. 46 , 3–42 (2000). [8] P . Baum and R.G. Douglas. K-homology and index th eory . Pro c. Symp. Pure Math. 38 , 117–173 (1982). [9] P . Baum, J.-L. Brylinksi and R. MacPherson. Cohomologie equiv arian te d´ elocalis´ ee. C. R. Acad. Sci. Pari s Ser. I Math. 300 , 605–608 (1985). [10] P . Baum, N. Higson and T. Schic k . On th e equival ence of geometric and analytic K- homology . Pure App l. Math. Quart. 3 , 1–24 (2007) [arXiv:math/0701484]. [11] D.M. Belo v and G.W. Mo ore. Typ e I I actions from 11-dimensional Chern-Simons theories. Preprint arXiv:hep-th/0611020 (2006). RAMOND-RAM OND FIELDS, FRACTIONAL BRA NES AND ORBIFOLD DIFFERENTIAL K- THEOR Y 45 [12] O. Bergman, E.G. Gimon and B. Kol. Strings on orbifold lines. J. High Energy Phys. 0105 , 019 (2001) [arXiv:hep-th/0102095]. [13] O. Bergman, E.G. Gimon and S. Sugimoto. O rien tifolds, RR torsion and K-theory . J. High Energy Phys. 0105 , 047 (2001) [arXiv:hep- th/0103183 ]. [14] J. de Bo er, R. Dijkgraaf, K. Hori, A. Keurentjes, J. Morgan, D .R. Morrison and S. Seth i. T riples, ﬂux es and strings. Adv . Theor. Math. Phys. 4 , 995–118 6 (2002) [arXiv:hep-th/0103170]. [15] J. Blo ck and E. Getzler. Equ iv ariant cyclic cohomolog y an d equiv arian t d iﬀerenti al forms. A n n. Sci. ENS 27 , 493–527 (1994). [16] G.E. Bredon. Equivariant C ohomolo gy The ories (Springer, 1967). [17] U. Bunke. Orbifold ind ex and equiva rian t K-homology . Math. Ann. 339 , 175–194 (2007) [arXiv:math/070176 8]. [18] A.L. Carey , J. Mick elsson and B.-L. W ang. Diﬀerential twisted K-th eory and applications. Preprint arXiv:0708.31 14 [math.KT] (2007). [19] Y.-K.E. Cheun g and Z. Yin. Anomalies, branes and currents. Nucl. Phys. B 517 , 69–91 (1998) [arXiv:h ep- th/9710206] . [20] J.F. Davis and W. L ¨ uc k. Spaces ov er a category and assem b ly maps in isomorphism conjectures in K- and L-theory . K-Theory 15 , 201–252 (1998). [21] D.-E. Diaconescu and J. Gomis. F ractional b ranes and b ound ary states in orbifold theories. J. High Energy Phys. 0010 , 001 (2000) [arXiv:hep-th /990624 2]. [22] D.-E. Diaconescu, G.W. Moore and E. Witten . E 8 gauge theory and a deriv ation of K-t heory from M-theory . Adv. Theor. Math. Phys. 6 , 1031–113 4 (2003) [arXiv:hep-th/0005090]. [23] T. tom Dieck. T r ansformation Gr oups (W alter de Gruyter, 1987). [24] L.J. Dix on, J.A. Harvey , C. V afa and E. W itten. Strings on orbifolds. Nucl. Phys. B 261 , 678–686 (1985). [25] L.J. Dixon, J.A. Harvey , C. V afa and E. Witten. Strings on orbifolds 2. Nu cl. Phys. B 274 , 285–314 (1986). [26] M.R. Douglas and G.W. Mo ore. D-branes, quivers and ALE instantons. Preprint arXiv:hep-th/9603167 (1996). [27] M.R. D ouglas, B.R. Greene and D.R. Morrison. Orbifold resolution by D- branes. Nu cl. Phys. B 506 , 84–106 (1997) [arXiv:hep-th /97041 51]. [28] D.S. F reed. Dirac charg e q uantiza tion and generalized diﬀerential cohomology . Surv. Diﬀ. Geom. V II , 129– 194 (2000) [arXiv:hep- th/0011220] . [29] D.S. F reed and M.J. H opkins. On Ramond-R amond ﬁ elds and K-theory . J. High Energy Phys. 0005 , 044 (2000) [arXiv:hep-th /00020 27]. [30] D.S. F reed and E. Witten. An omalies in string theory with D-branes. A sian J. Math. 3 , 819 –851 (1999) [arXiv:hep-th/9907189]. [31] D.S. F reed , M.J. H opkins an d C. T eleman. Twisted eq uiv ariant K-t heory with comp lex co eﬃcients. Preprint arXiv:math/0206257 (2002). [32] D.S. F reed, G.W. Moore and G.B. Segal. The un certain ty of ﬂ uxes. Commun. Math. Ph ys. 271 , 247–274 (2007) [arXiv:hep-th /06051 98]. [33] D.S. F reed, G.W. Mo ore and G.B. S egal. Heisenberg groups and noncommutativ e ﬂuxes. Ann . Phys. 322 , 236–285 (2007) [arXiv:hep- th/0605200 ]. [34] H. Garc ´ ıa-Co mp e´ an. D-branes in orbifo ld singularities and equiva rian t K-theory . Nucl. Ph y s. B 557 , 480–504 (1999) [arXiv:hep-th /98122 26]. [35] M.B. Green, J.A. H arv ey and G.W. Moore. I- brane inﬂow and anomalous coup lings on D-branes. Class . Quant. Grav. 14 , 47–52 (1997) [arXiv:hep-th/9605033]. [36] J.A. Harvey and G.W. Moore. Noncommutativ e tac hy ons and K- theory . J. Math. Ph ys. 42 , 27 65–2780 (2001) [arXiv:hep-th/0009030]. [37] M.J. Hopkins and I.M. Singer. Quadratic functions in geometry , top ology and M-theory . J. D iﬀ. Geom. 70 , 329–452 (2005) [arXiv:math/0211216]. [38] S. I llman. Smo oth equ iv ariant triangulations of G -man ifolds for G a ﬁn ite group. Math. Ann. 233 , 199–220 (1978). [39] S. Illman. T h e equiv ariant triangulation theorem for actions of compact Lie groups. Math. An n . 262 , 487–501 (1983). [40] M. Karoubi. Equiv ariant K-theory of real vector spaces and real pro jective spaces. T op ol. Appl. 12 2 , 531–546 (2002) [arXiv:math/0509497] . [41] I. Kriz, L.A. P ando Zay as and N. Quiroz. Commen ts on D-branes on orbifolds and K-theory . Preprin t arXiv:hep-th/0703122 (2007). [42] G.D. Landweber. Representation rin gs of Lie sup eralgebras. K-Theory 36 , 115 –168 (2005) [arXiv:math/040320 3]. [43] J. Lott. R / Z index theory . Comm. Anal. Geom. 2 , 279–311 (1994). [44] W. L¨ uc k. Chern characters for p roper equiv arian t homology th eories and app lications to K- and L-theory . J. Reine Angew. Math. 543 , 193–234 (2002). [45] W. L ¨ uc k. Equ iv ariant cohomological Chern chara cters. Int. J. Alg. Comp. 15 , 1025–1052 (2006). 46 RICHARD J. SZABO AND ALESSANDRO V ALENTINO [46] W. L ¨ uck and B. Oliver. Chern characters for the equiv ariant K-th eory of prop er G - CW-complexes. Progr. Math. 196 , 217–248 ( 2001). [47] W. L ¨ uck and B. Oliver. The completion theorem in K-theory for prop er actions of a d iscrete group. T op ology 40 , 585–616 (2001). [48] J.M. Maldacena, G.W. Mo ore and N. Seib erg. D-brane instantons and K-theory charge s. J. High Energy Phys. 0111 , 062 (2001) [arXiv:hep-th /010810 0]. [49] R. Minasian and G.W. Moore. K-th eory and Ramond-Ramond charge. J. High Energy Phys. 9711 , 002 (1997) [arXiv:hep-th /97102 30]. [50] G. Mislin and A. V alette. Pr op er Gr oup A ctions and the Baum-Connes Conje ctur e (Birkh¨ auser V erlag, 2003). [51] G.W. Moore and A. Par nachev. Localized tac hy ons and the quan t um McKa y correspond ence. J. H igh Energy Phys. 0411 , 086 (2004) [arXiv:hep-th /040301 6]. [52] G.W. Mo ore and E. Witten. Self-du alit y , R amond-Ramond ﬁ elds and K-theory . J. High Energy Phys. 0005 , 032 (2000) [arXiv:hep- th/9912279] . [53] M.V. Nori. The Hirzebruch-Riemann-Ro ch theorem. Michigan Math. J. 48 , 473–482 (2000). [54] K. Olsen and R.J. Szabo. Constructing D-branes from K-theory . Adv. Theor. Math. Phys. 4 , 889–1025 (2000) [arXiv:hep-th/9907140]. [55] V. Peri w al. D-brane charge s and K-homology . J . High Energy Phys. 0007 , 041 (2000) [arXiv:hep-th/0006223] . [56] R.M.G. R eis and R .J. Szab o. Geometric K- homology of ﬂat D-branes. Commun. Math. Phys. 266 , 71–122 (2006) [arXiv:hep-th /05070 43]. [57] R.M.G. Reis, R.J. S zabo and A. V alentino. KO-homolog y and T yp e I string th eory . Preprint arXiv:hep- th/0610177 (2006). [58] G.B. Segal. Equiv ariant K- theory . Publ. Math. IHES 34 , 129–151 (1968). [59] J. S lomi´ nsk a. On th e equiva rian t Chern chara cter homomorphism. Bull. Acad. Pol. Sci. 24 , 909–913 (1976). [60] R.J. Szabo. D-branes, tac hyons and K -homology . Mo d. Phys. Lett. A 17 , 2297–2316 (2002) [arXiv:hep- th/0209210] . [61] R.W. Thomasson. U ne formule de Lefsc h etz en K-theories eq uiv ariante algebraique. Duke Math. J. 68 , 447– 462 (1992). [62] E. Witten. D-branes and K-th eory . J. High Energy Phys. 9812 , 019 (1998) [arXiv:hep- th/9810188 ]. Dep ar tment of Ma the ma tics and Maxwell Institute f or Ma thema tical S ciences, Heriot-W a tt University, Colin Macla urin Buildin g, Riccar ton, Edinburgh EH14 4AS, U.K. E-mail addr ess : R.J.Szabo@ma.h w.ac.uk E-mail addr ess : A.Valentino@ma .hw.ac.uk

Ramond-Ramond Fields, Fractional Branes and Orbifold Differential K-Theory

Original Paper

Comments & Academic Discussion

Leave a Comment

Original Paper

Related Papers

Comments & Academic Discussion

Leave a Comment