Dualities in Field Theories and the Role of K-Theory

Dualities in Field Theories a nd the Role of K -Theory Jonathan Rosenberg ∗ Department of Mathematics, Universit y of Maryland, Co llege P ark, MD 20742–4 015, USA, jmr@math.u md.edu Summary . It is now kn o wn (or in some cases just b elieved) t h at many q uantum field t h eories exh ibit dualities , equiv alences with t he same or a d ifferent theory in whic h things app ear v ery differen t, but the o v erall ph ysical implicati ons are the same. W e will discuss some of these dualities from the p oint of view of a mathe- matician, fo cusing on “charge conserv ation” and the role play ed by K -theory and noncommutativ e geometry . Some of the w ork describ ed here is joint with Mathai V arghese and St efan Mendez-D iez; the last section is new. Key wor ds: duality , S-duality , T-duality , K -theory , AdS/CFT correspond ence, D-brane. Mathematics Sub ject Cl assification (2010): Primary 81T30; Secondary 81T75, 81T13, 19L50, 19L64. 1 Ov erview with So me Class i cal Examples . . . . . . . . . . . . . . . . . . 1 2 T opol ogical T-Dualit y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3 Problems Prese n ted b y S-Duality and Other Dualities . . . . 14 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1 Overview with Some Cla ssical Examples 1.1 Structure of Phy sical Theories Most ph ysical theories descr ibe fields , e.g ., the gr avitational field , ele ctric field , magnetic field , etc. Fields ca n b e ∗ This research wa s partially supp orted by NSF gran t DMS-0805003. I would like to thank the organizers and participants of the Closing Meeting on Perspectives in Deformation Quantizatio n and Noncommutativ e Geometry in Kyoto, F eb ru- ary 2011, as well as the staff of the RIMS, for a very stimulati ng and well-run conference. 2 Jonathan R osen b erg • scalar- v a lued functions ( sc alars ), • sections o f vector bundles ( ve ctors ), • connections on principal bundles ( sp e cial c ases of gauge fi elds ), • sections o f spinor bundles ( spinors ). In c la ssical ph ysic s , the fields satisfy a v ar iational principle — they are critical p oints of the action S , which in turn is the integral of a lo c al functiona l L calle d the L agr angian . This is called the principle of le ast action , a nd can be traced bac k to F ermat’s theory o f optics in 16 62. The Euler-Lag range equations fo r critica l p oints o f the action are the e qu ations of m otion . Example 1 (Y ang-Mil ls The ory). Let M b e a 4-dimensional Riemannia n or Lorentzian manifold, s ay compact. W e fix a co mpa ct Lie gro up G a nd a prin- cipal G -bundle over M . A Y ang-Mills field is a connection A on this bundle. The “field strength” F is the cur v a ture, a g -v alued 2-fo rm. The action is S = R M T r F ∧ ∗ F (up to a constant inv olving the coupling c o nstant g YM measuring the streng th of the interactions). Note that the metric o n M is needed to define the Ho dg e ∗ -o per ator F 7→ ∗ F . Example 2 (Gener al R elativity in Empty Sp ac e). F or conv enience, we consider the “Wick rota tion” of the theor y to E uclidean sig nature. Let M be a 4- manifold, say compac t. A field is a Riemannian metric g on M . The (Einstein- Hilber t) action is S = R M R d vol, R = scalar curv ature. (Strictly sp eaking one should insert a coupling consta n t in front, c 4 16 πG , where G is Newton’s gravitational constant and c is the s pee d of light, which we usually set equal to 1 in suitable units.) The asso ciated field e q uation is Einstein’s equation. Unlik e classic a l mechanics, qua ntum mechanics is not deterministic, only probabilistic. The key pro pe r ty of quantum mec ha nics is the Heisenb er g u nc er- tainty principle , that obser v a ble quantities are r epresented by nonc ommuting op er ators A represe n ted on a Hilb ert space H . In the qua n tum world, every particle has a w ave-lik e asp ect to it, and is repres ent ed by a wav e function ψ , a unit vector in H . The phase of ψ is not directly o bserv able, only its amplitude, or mo r e precis ely , the state ϕ ψ defined by ψ : ϕ ψ ( A ) = h Aψ , ψ i . But the phase is still imp ortant since interfer enc e dep ends on it. The q ua nt ization o f clas sical field theories is ba sed on p ath inte gr als . The idea (not 10 0% rig o rous in this formulation) is that al l fields c ontribut e , not just those that are cr itical p oints of the action (i.e., solutions o f the cla ssical field eq ua tions). Instead, one lo oks at the p artition fun ction Z = Z e i S ( ϕ ) / ~ d ϕ or Z e − S ( ϕ ) / ~ d ϕ , depe nding on whether one is working in Lore ntz or Euclidean sig nature. By the principle of stationary phase , o nly fields close to the classical s olutions Dualities and K -Theory 3 should contribute v ery muc h. Exp e ctation values of physical quantities are given by h A i =  Z A ( ϕ ) e i S ( ϕ ) / ~ d ϕ  / Z . 1.2 Dualiti es A duali ty is a transfor mation b etw een different-looking physical theories that, rather magically , hav e the same obs e r v able physics. O ften, such dualities ar e part of a discrete g roup, s uc h as Z / 2 o r Z / 4 or S L (2 , Z ). Example 3 (Ele ctric-magnetic duality). Let E and B b e the electric and mag- netic fields, resp ectively . Ther e is a sy mmetry of Max well’s equatio ns in free space ∇ · E = 0 , ∇ · B = 0 , ∂ E ∂ t = c ∇ × B , ∂ B ∂ t = − c ∇ × E , (1) given by E 7→ − B , B 7→ E . This is a duality of o rder 4. Example 4 (Configur ation sp ac e-momentu m sp ac e duality). Another example from standard q uantum mec hanics concerns the quantum harmonic oscillator (say in one dimension). F or an ob ject with mass m and a restor ing force with “spring cons tant” k , the Hamiltonian is H = k 2 x 2 + 1 2 m p 2 , (2) where p is the mo men tum. In classica l mechanics, p = m ˙ x . But in qua nt um mechanics (with ~ set to 1 ), [ x, p ] = i . (3) W e obtain a duality of (2) and (3 ) via m 7→ 1 k , k 7→ 1 m , x 7→ p , p 7→ − x . This is a gain a duality of or der 4, and is closely related to the F ourier tr ansform . • A big puzzle in classica l electricity and magnetism is that while there are plent y of charged particles (electrons , etc.), no mag netically charged particles ( magnetic monop oles ) hav e ever b een obse r ved, ev en tho ugh their existence would not contradict Ma xwell’s eq ua tions. • Another problem with classical electricity and magnetism is that it doe s n’t explain why charges a pp ea r to be qua n tized, i.e., o nly o ccur in units that are integral multiples o f the charge of the electron (or of the ch arges of [down-t yp e] qua rks). Dirac [21] pro po sed to solve b oth problems at once with a quantum theor y of electr icit y a nd magnetism that in mo dern terms we would call a U (1) gauge the ory . In Dira c’s theo ry , we assume spacetime is a 4-manifo ld M , say R 4 \ R ∼ = R 2 × S 2 (Mink owski spa ce with the time tra jectory of one particle taken out). 4 Jonathan R osen b erg The (magnetic) vector po tential ( A 1 , A 2 , A 3 ) a nd electric potential A 0 = φ of classica l elec tr icity a nd magnetism are combined int o a single en tit y A , a (unitary) connection on a co mplex line bundle L ov er M . Thus i A is lo c al ly a real-v alued 1-for m, a nd F = i µ d A , µ a cons tant , is a 2-fo rm enco ding b oth of the fields E (via the (0 , j ) comp onents) and B (via the ( j, k ) comp onents, 0 < j < k ). The Chern class c 1 ( L ) ∈ H 2 ( M , Z ) ∼ = Z is an inv ariant o f the top ology of the situation. Of course, F should really be i µ times the curv atur e of A , and Chern-W eil theory says that the de Rham clas s [ F ] is 2 π µ times the image of c 1 ( L ) in H 2 ( M , Z ) ∼ = Z . L is a sso ciated to a principal U (1 )-bundle P → M , a nd Dirac iden tifies a s ection of this bundle with the phase of a wav e function o f a charged particle in M . In the ab ov e setup, if w e integrate F ov er the S 2 that links the worldline we remov ed, we ge t 2 π µc 1 ( L ), and this is the flux of the magnetic field thro ugh S 2 . So the deleted worldline ca n b e identified with that of a magnetic monop ole of charge g = µc 1 ( L ) in suitable units. Suppo s e we consider the motio n of a test charge o f electric charge q ar ound a closed lo op γ in M . In qua nt um electricity and magnetism, by the Aharonov-Bohm effect [2], the exterior deriv ative is replaced by the cov ariant deriv ative (in volving the v ector potential A ). So the phase change in the wa ve function is ba s ically the holonomy of ( P → M , A ) around γ , or (taking ~ = 1 ) exp  q µ H γ A  . Since M is simply c onnected, γ bo unds a disk D and the integral is (b y Stokes’ Theorem) exp  − i q R D F  . T aking D in turn to b e the tw o hemispher e s in S 2 , w e get tw o answers which differ by a factor o f exp  i q Z S 2 F  = e 2 π i q µ c 1 ( L ) . Since this must be 1 , we get D ir ac’s quant ization c ondition qg ∈ Z . The upshot of this analysis is that we exp ect b oth electrical and magnetic charges to b e quantized, but that the b asic quanta of ele ctric al and magnetic char ge should b e inversely pr op ortional in size . In other words, the smallness of the fundamental electrica l charge mea ns that the charge of any ma gnetic monop ole has to b e larg e. In an y even t, we ex p ect the electrical and mag netic charges ( q , g ) to take v alue s in an ab elian char ge gr oup C , in this c a se Z 2 . It is also re a sonable to exp ect there to be par ticle s , usua lly called dyons , with bo th charges q and g non-zer o . Now think a bo ut the class ic al electric- magnetic duality (Example 3) that switches E and B . The Montonen-Olive Conje cture [36], for which there is now some tantalizing evidence, is that in a wide v ariety of cases this s hould extend to a duality o f quantum theories , which would necessar ily give an isomorphism of charge gro ups b etw e en a theory and its dual. In Dirac’s theory , the quantization o f magnetic charge and of electrical charge a rise from differen t origins. The former is a purely top olo gical phe- nomenon; it comes from the fact that the Chern classes liv e in inte gr al co- homology . Quantization o f electrical charge comes from the r equirement that Dualities and K -Theory 5 the action (for the field asso cia ted to a charged par ticle moving in the back- ground electro magnetic field o f a monopo le) b e w e ll- defined and not m ulti- v a lued, so this can b e viewed as a version o f anomaly c anc el lation . How e ver, since Maxwell’s e q uations a re in v a riant under electr o-magnetic duality , we can imagine an equiv a le n t dual theor y in which electr ic charge is top ologica l and magnetic charge is quantized to achieve anomaly cancellatio n. 1.3 A General F ramew ork and the Rol e of K -Theory Extrap ola ting from case ab ov e , we will b e lo oking a t the following set-up: 1. W e have a co llection C of “physical theories” on which a discrete duality gr oup G ope r ates by “equiv alences.” (Mor e generally , G migh t b e replace d by a gr oup oid.) 2. Each theory in C has an asso cia ted char ge gr oup C . If g ∈ G gives an equiv alence betw een tw o theor ies in C , it must g ive a n iso morphism be- t ween the ass o ciated charge groups . In pa rticular, the stabilizer of a fixed theory o pe r ates by automor phisms on C . 3. In many cas es, the charge groups aris e as top olog ic a l inv aria nts. W e have already seen how Pic X = H 2 ( X, Z ) a rises. (The notation Pic X deno tes the set of isomor phism cla sses of complex line bundles ov er X , which is a gr o up under tenso r pro duct.) W e will see how K -theory arises in so me cases. Many of the most interesting examples o f duality (and of to p o lo gical charge groups) arise in (sup ersymmetr ic ) string the ories . These ar e q uantum field theories based on the idea o f r e pla cing p oint particles b y strings or 1-manifolds (alwa ys compact, but mayb e with b ounda ry — contrary to mathematica l usage, physicists ca ll these “op en strings”). F or anomaly cancellation reasons, the spacetime manifold has to b e 10- dimensional. The worldsheet traced out by a string in the spa cetime X is a compact 2-manifold Σ (ag ain, p ossibly with b oundary), so we obtain fields that ar e maps f : Σ → X , with the sigma-mo del action of the form Z Σ k∇ f k 2 + Z Σ f ∗ ( B ) + (terms inv olving other fields) . (4) Here B is a 2-form on X ca lled the B-field ( not the magnetic field). The term R Σ f ∗ ( B ) is called the Wess-Zumino term . The terms in volving the “other fields” dep end o n which of the five sup erstring theories (type I, which allows unoriented strings, t yp es I IA and I IB , and the tw o t yp es of heterotic theories) one is dea ling with. They differ with rega rd to such issues as chirality a nd orientation co nditions, and whether or no t op en strings are a llow ed. In string theor ies, b oundary conditions (o f Dirichlet or Neumann type) m ust b e imp osed on the op en string s tates. These are g iven by D-br anes (D for “Dirichlet”), s ubmanifolds of the spacetime X o n which string s a re allow ed 6 Jonathan R osen b erg to “end.” If we forget cer tain complications and lo ok at type I I string theo ry , then X is a 1 0-dimensional spin manifold a nd the stable D-branes ar e spin c submanifolds, of even dimensio n for type I IB and of o dd dimensional for type IIA. (At least in the absence o f t wisting, X is generally R 4 times a Calabi- Y au 3-fold, a nd in the type I IB case , the stable D-branes a re complex submanifolds, whereas in the I IA c a se, they a re typically is o tropic subma nifolds for the symplectic structure.) There is a nother piece of str uc tur e; each D-brane carr ies a Chan-Paton ve c- tor bund le that reflects a U ( N ) gauge symmetry allowing for lo cal exchanges betw een coincident D-bra nes. The D-branes carry char ges which are not just num b ers but elements o f the K -group K ( X ) (in the type I IB theory), K − 1 ( X ) (in the type I IA theor y), or K O ( X ) (in the type I theory). The idea that the D-br ane charges should ta ke v a lues in K -the ory comes from Minasian-Mo ore [35] and Witten [46 ], around 1997 –1998 , with further elab oration by other authors later. Mo tiv atio n comes from several sour ces: • compatibility with anoma ly cancellation formulas; • better functor iality; • compatibility with analy sis of decay of unstable branes; • compatibility with what is k nown ab out string dua lit y . W e will not attempt to go through these arguments (which the reader can find in [46, 47, 26] and [7, § 6.2]) but will discuss s ome cons e q uences. F or a D-brane W   ι / / X with Chan- Paton bundle E → W , the K - theory charge is ι ! ([ E ]), where [ E ] is the class o f E in K ( W ), and ι ! is the Gysin map in K - theo ry (defined using the s pin c structures). While string dualities do no t hav e to preser ve the diffeomorphism t yp e , or even the dimension, of D-branes, they do hav e to giv e rise to an isomorphism of the K - groups in which the D-bra ne charges lie . The most imp orta n t kinds of string theo ry dualities are T-duality , a n outgrowth of c la ssical F our ier duality (“T” orig ina lly standing for “targ et space”), and S-du ality , a n outgrowth of classical electro -magnetic dualit y. The big difference b etw een them is that T-dua lit y preserves c o upling strength and changes geometry , wherea s S-duality (“S” sta nding fo r “strong - weak”) inter- changes strong and weak coupling and pres erves the geometry of spacetime, just as electro- magnetic duality in verts the mag nitude of charges. Much of the interest of these dualities is that they are non-p ertu rb ative , in o ther w o rds, don’t depend on p erturbation expansions. In some cases, a quantit y which is difficult to compute in one theory ca n b e computed by passage to a dual theor y in which the quantit y is easier to compute directly , or ca n b e computed v ia a p erturbation expansion. T-duality replac e s tori (of a fixed dimensio n k ) in the spacetime manifold X by their dual tori (quotients of the dual sp ac e by the dual lattic e ) in the dual spa c etime X ♯ , inv er ting the radii. If k is o dd, T-duality in terchanges the theories of type s IIA and I IB, so one g ets an isomorphism K ( X ) ∼ = K − 1 ( X ♯ ) Dualities and K -Theory 7 or K − 1 ( X ) ∼ = K ( X ♯ ). S-duality interc hange s type I string theory with the S O (32) heter otic string theory , and also ma ps type I IB string theor y to itself. In Sections 2 and 3 I will discuss T- duality and S-duality in more de ta il, and the wa y charge conserv ation in K -theory sheds more lig h t on them. 2 T opological T-Dualit y 2.1 The H-flux and Twisted K -Theory It’s now time to corre ct a s lig ht ov ersimplification in Section 1: the “B-field” in the sigma-mo del actio n (4) is not necessar ily globally well-defined, though its field str en gth H = d B do es make sense glo ba lly . P rop erly normalize d, one can show that H defines an integral de Rham class in H 3 . This can b e r efined to an actual c lass in [ H ] ∈ H 3 ( X, Z ). Th us the W ess-Zumino term in the path int egral should really be defined using a gerb e , fo r example a bund le gerb e in the sense of Murray [37] with cur ving B and Dixmier-Douady class [ H ]. W e usually r efer to H (or to the asso cia ted class [ H ] ∈ H 3 ( X, Z )) as the H-flux . (F or an expo sition o f ho w ger bes can b e used to make sense of the W ess-Zumino ter m, see for e xample [25] or [40, § 4.3 ].) The a sso ciation of H with a Dixmier-Doua dy c la ss is not an accident, and indee d indicates a deep er co nnection with nonco mmutative geometr y . T o set this up in the s implest wa y , choo se a stable c ontinuous-tra c e algebr a A = C T ( X , [ H ]) with b A = X a nd with Dixmier - Douady class [ H ]. Thus A is the algebra of contin uous sections v anishing at ∞ o f a bundle over X with fiber s K (the c o mpact op era tors on a separ able ∞ -dimensional Hilb e rt s pace H ) and struc tur e gro up Aut K = P U ( H ) ≃ K ( Z , 2 ). There ar e several p ossible definitions of twiste d K -the ory (see [22, 39, 5, 6, 2 9]), but for our purp os es we ca n define it as K − i ( M , [ H ]) = K i ( A ) with A = C T ( M , [ H ]) as a b ove. Up to isomorphism, this only dep ends on X and the co homology class [ H ] ∈ H 3 ( X, Z ). In the presence of a top olog ically nontrivial H-flux, the K -theoretic clas- sification of D-brane charges has to b e mo dified. A D-brane W   ι / / X in t yp e I I string theory is no long a Spin c manifold; instea d it is Spin c “up to a twist,” acc o rding to the F r e e d-Witten anomaly c anc el lation c ondition [2 4] W 3 ( W ) = ι ∗ ([ H ]). Here W 3 is the canonic a l integral lift o f the third Stiefel- Whitney class, which is the obstruction to a Spin c structure. Accor dingly , the D-brane charge will live in the twiste d K -gr oup K ( X , [ H ]) (in type IIB) or in K − 1 ( X, [ H ]) (in type IIA). Accordingly , if we hav e a T-duality b etw een string theories on ( X , H ) and ( X ♯ , H ♯ ), c onservation of char ge (for D-branes) requires an isomorphism of twisted K -groups of ( X , [ H ]) and ( X ♯ , [ H ♯ ]), with no degree s hift if we dualize with r e sp e c t to even-degree tor i, and with a degree shift if we dualize with re s pec t to o dd-degree tor i. One might wonder w ha t happ ened to the K -gro ups of o ppo site parity , viz., K − 1 ( X, [ H ]) (in t ype I IB) and K ( X , [ H ]) (in type I I A). These still hav e 8 Jonathan R osen b erg a physical significa nce in ter ms of Ra mond-R amond fi elds [47], so wan t these to ma tch up under T-dua lit y also . 2.2 T op ologi cal T-Duality and the Bunk e-Sc hi ck Construction T op olo gic al T-duality fo cuse s on the top olog ical asp ects of T-dua lit y . The first example of this phenomenon was studied by Alv arez , Alv a rez-Gaum´ e, Barb´ on, and Lozano in 19 93 [4], and gener alized 1 0 years later by Bouwknegt, E vslin, and Mathai [11, 12]. Let’s star t with the simplest nontrivial example of a circ le fibration, where X = S 3 , identified with S U (2), T is a maximal tor us. Then T acts freely o n X (say by r ight tr anslation) and the q uo tient X/T is CP 1 ∼ = S 2 , with quotient map p : X → S 2 the Hopf fi br ation . Assume for simplicit y that the B -field v anishes. W e hav e X = S 3 fiber ing ov er Z = X/T = S 2 . Think o f Z as the unio n of the tw o hemispheres Z ± ∼ = D 2 int ersecting in the equator Z 0 ∼ = S 1 . The fibratio n is trivial over each hemisphere , so we have p − 1 ( Z ± ) ∼ = D 2 × S 1 , with p − 1 ( Z 0 ) ∼ = S 1 × S 1 . So the T-dual also lo oks like the union of tw o co pies of D 2 × S 1 , joined alo ng S 1 × S 1 . How ever, we have to b e ca r eful ab out the clu tching tha t identifies the t wo copies of S 1 × S 1 . In the orig inal Ho pf fibration, the clutching function S 1 → S 1 winds once ar ound, with the r esult tha t the fundamental gro up Z of the fib er T dies in the total space X . But T-duality is supp osed to int erchange “winding” a nd “ momentum” q uantu m num ber s. So the T-dual X ♯ has no winding and is just S 2 × S 1 , while the winding of the origina l clutch ing function shows up in the H -flux of the dual. In fact, following Buscher’s metho d [19] for dualizing a sigma- mo del, we find that the B -field B ♯ on the dual side is different on the t wo copies of D 2 × S 1 ; they differ by a closed 2-form, and so H ♯ = d B ♯ , the H-flux of the dual, is nontrivial in de Rham cohomolog y (for simplicity of notation we delete the bra ck ets from now on) but well defined. Let’s chec k the principle of K -theory ma tc hing in the cas e we’ve b e e n considering, X = S 3 fiber ed by the Hopf fibr ation ov er Z = S 2 . The H - flux o n X is trivial, so D-brane charges lie in K ∗ ( S 3 ), with no twisting. And K 0 ( S 3 ) ∼ = K 1 ( S 3 ) ∼ = Z . On the T-dual side, we exp ect to find X ♯ = S 2 × S 1 , also fib ered ov e r S 2 , but simply by pro jection onto the first factor. If the H -flux on X were trivial, D-brane changes would lie in K 0 ( S 2 × S 1 ) and K 1 ( S 2 × S 1 ), b o th of which are is omorphic to Z 2 , which is to o big . On the o ther hand, we can compute K ∗ ( S 2 × S 1 , H ♯ ) for the class H ♯ which is k times a generato r of H 3 ∼ = Z , using the A tiyah-Hirzebruch Sp ectral Sequence. The differential is H 0 ( S 2 × S 1 ) k − → H 3 ( S 2 × S 1 ) , so when k = 1, K ∗ ( S 2 × S 1 , H ♯ ) ∼ = K ∗ ( S 3 ) ∼ = Z for b oth ∗ = 0 and ∗ = 1. Dualities and K -Theory 9 Axiomatics for n = 1 This discussio n sugges ts we sho uld tr y to develop an ax io matic treatment of the top olo gic al asp ects of T-duality (for cir cle bundles). No te that we ar e ignoring man y things, suc h a s the under lying metric on spacetime and the auxiliary fields. Here is a fir st attempt. Axioms 1. 1. We have a suitable class of sp ac etimes X e ach e qu ipp e d with a princi- p al S 1 -bund le X → Z . ( X might b e r e quir e d to b e a smo oth c onne cte d manifold. ) 2. F or e ach X , we assume we ar e fr e e to cho ose any H -flux H ∈ H 3 ( X, Z ) . 3. Ther e is an involution ( map of p erio d 2) ( X , H ) 7→ ( X ♯ , H ♯ ) ke eping the b ase Z fix e d. 4. K ∗ ( X, H ) ∼ = K ∗ +1 ( X ♯ , H ♯ ) . The Bunke-Sc hi c k Co ns truction Bunke and Schic k [16] sugge s ted constructing a theory satisfying these a xioms by means of a universal example . It is known that (for reaso na ble s paces X , say CW complexe s ) all principal S 1 -bundles X → Z co me by pul l-b ack from a diag ram X   / / E S 1 ≃ ∗   Z / / B S 1 ≃ K ( Z , 2 ) Here the map Z / / K ( Z , 2) is unique up to homotopy , a nd pulls the canonical clas s in H 2 ( K ( Z , 2) , Z ) back to c 1 of the bundle. Similarly , every class H ∈ H 3 ( X, Z ) co mes by pull- back from a ca nonical class v ia a map X / / K ( Z , 3) uniq ue up to homotopy . Theorem 2 (Bunk e-Schic k [ 1 6]). Ther e is a classifying sp ac e R , unique up to homotopy e quivalenc e, with a fibr ation K ( Z , 3) / / R   K ( Z , 2) × K ( Z , 2 ) , (5) and any ( X , H ) → Z as in t he axioms c omes by a pul l-b ack X   / / E p   Z / / R, 10 Jonathan R osen b erg with the hori zontal maps unique up to homotopy and H pul le d b ack fr om a c anonic al class h ∈ H 3 ( E , Z ) . Theorem 3 (Bunk e-Schic k [16]). F urthermor e, t he k -invariant of t he Post- nikov t ower (5 ) char acterizing R is t he cup-pr o duct in H 4 ( K ( Z , 2) × K ( Z , 2) , Z ) of t he two c anonic al classes in H 2 . The sp ac e E in t he fibr ation S 1 / / E p   R has t he homotopy t yp e of K ( Z , 3 ) × K ( Z , 2) . Corollary 1 . If ( X p − → Z , H ) and ( X ♯ p ♯ − → Z , H ♯ ) ar e a T-dual p air of cir cle bund les over a b ase sp ac e Z , then the bund les and fluxes ar e r elate d by the formula p ! ( H ) = [ p ♯ ] , ( p ♯ ) ! ( H ♯ ) = [ p ] . Her e [ p ] , [ p ♯ ] ar e the Euler classes of the bund les, and p ! , ( p ♯ ) ! ar e the “inte- gr ation over t he fib er” maps in t he Gysin s e quenc es. F urthermor e, ther e is a pul lb ack diagr am of cir cle bund les Y ( p ♯ ) ∗ ( p ) / / p ∗ ( p ♯ )   X p   X ♯ p ♯ / / Z. in which H and H ♯ pul l b ack to the same class on Y . The Case n > 1 W e now wan t to gener alize T-duality to the cas e of spacetimes X “co mpa cti- fied on a hig he r -dimensional torus,” or in other words, equipp ed with a princi- pal T n -bundle p : X → Z . In the s implest case, X = Z × T n = Z × n z }| { S 1 × · · · S 1 . W e can then p erfor m a string of n T-dualities, one circ le factor at a time. A single T- duality interc ha ng es type I IA and type IIB string theor ies, so this n -dimensional T- duality “preser ves t yp e” when n is even and switc hes it when n is o dd. In terms o f our Axioms 1 for top olo gical T-duality , we would there- fore exp ect an isomorphism K ∗ ( X, H ) ∼ = K ∗ ( X ♯ , H ♯ ) when n is e ven and K ∗ ( X, H ) ∼ = K ∗ +1 ( X ♯ , H ♯ ) when n is o dd. Dualities and K -Theory 11 In the hig her-dimensional case, a new problem prese nts itself: it is no longer cle ar t hat the T-dual should b e unique . In fact, if w e p erform a string of n T-dualities, one circle factor at a time, it is not clear that the result should be indep endent of the order in which these op e rations are do ne . F urthermore, a higher-dimensiona l torus do es no t split a s a pro duct in o nly one wa y , so in principle there ca n b e a lot of no n-uniqueness. The w ay out of this difficult y has therefore been to try to organiz e the information in terms of a T-duality gr oup , a discrete group of T-duality iso- morphisms p otentially inv olving a larg e n um ber o f spacetimes a nd H -fluxes. W e ca n think of this group a s o pe rating on some big metaspace of p o ssible spacetimes. Another difficulty is that there are some spacetimes with H -flux that would app ear to have no highe r -dimensional T-duals a t a ll, a t least in the s e nse we hav e de fined them so far, e.g., X = T 3 , vie w ed a s a principa l T 3 -bundle over a p oint, with H the genera tor o f H 3 ( X, Z ) ∼ = Z . 2.3 The Use of Noncommutativ e Geome try Here is the strategy of the Ma thai-Rosenberg approach [31, 32, 3 3]. Star t with a principal T n -bundle p : X → Z and an “ H -flux” H ∈ H 3 ( X, Z ). W e assume that H is trivial when r estricted to each T n -fib er of p . This of co urse is no restriction if n = 2, but it rules o ut cases with no T-dual in any se ns e. W e wan t to lift the fr e e action o f T n on X to an action o n the co nt in uous- trace alg ebra A = C T ( X , H ). Usually ther e is no hop e to get such a lifting for T n itself, so we go to the universal cov ering group R n . If R n acts on A so that the induced actio n o n b A is trivial on Z n and factors to the given action of T n = R n / Z n on b A , then we can take the crossed pro duct A ⋊ R n and use Connes’ Thom Iso morphism Theorem to get a n isomo rphism b etw e e n K −∗− n ( X, H ) = K ∗ + n ( A ) and K ∗ ( A ⋊ R n ). Under fav or able cir cumstances, we can hop e that the cro ssed pro duct A ⋊ R n will again b e a contin uous-tr ace algebra C T ( X ♯ , H ♯ ), with p ♯ : X ♯ → Z a new pr incipa l T n -bundle and with H ♯ ∈ H 3 ( X ♯ , Z ). If we then act on C T ( X ♯ , H ♯ ) with the dual action of b R n , then by T ak ai Dualit y a nd s ta bilit y , we come back to wher e we started. So we have a top olo gical T-duality b etw een ( X, H ) and ( X ♯ , H ♯ ). F urthermo re, we hav e an iso mo rphism K ∗ + n ( X, H ) ∼ = K ∗ ( X ♯ , H ♯ ) , as require d for matching o f D-br ane charges under T-dua lit y in Axio ms 1 (as mo dified for n ≥ 1). Now what ab out the pr oblems we identifie d b efor e, ab out p otential non- uniqueness of the T-dual and “missing” T-duals? Thes e can b e explained either by non-uniqueness of the lift to an action of R n on A = C T ( X , H ), or else by failure of the cro ssed pro duct to be a contin uous - trace algebra. 12 Jonathan R osen b erg A Crucial Example Let’s now ex amine what happ ens when w e try to car r y o ut this pr ogram in one of our “pro blem case s,” n = 2, Z = S 1 , X = T 3 (a trivial T 2 -bundle ov er S 1 ), and H the usual generator of H 3 ( T 3 ). First w e show that ther e is an action of R 2 on C T ( X , H ) compatible with the fr ee action of T 2 on X with quotient S 1 . W e will need the notion of a n induc e d action . W e sta rt w ith an action α of Z 2 on C ( S 1 , K ) which is trivial on the sp ectrum. This is given by a map Z 2 → C ( S 1 , Aut K ) = C ( S 1 , P U ( L 2 ( T ))) sending the t wo gener ators of Z 2 to the maps w 7→ m ultiplica tion by z , w 7→ translation by w , where w is the co or dinate on S 1 and z is the co ordinate on T . (Of cour se S 1 and T ar e ho meomorphic, but we us e different letters in or de r to distinguish them, since they play slightly different roles. These tw o unitaries commute in P U , not in U .) Now form A = Ind R 2 Z 2 C ( S 1 , K ). This is a C ∗ -algebra with R 2 -action Ind α whose sp ectrum (as an R 2 -space) is Ind R 2 Z 2 S 1 = S 1 × T 2 = X . W e can see that A ∼ = C T ( X , H ) via “inducing in s tages”. L e t B = Ind R Z C ( S 1 , K ( L 2 ( T ))) be the result of inducing o ver the first copy o f R . It’s clear that B ∼ = C ( S 1 × T , K ). W e still have another action of Z on B coming from the second ge nerator of Z 2 , and A = Ind R Z B . The action of Z on B is b y means of a ma p σ : S 1 × T → P U ( L 2 ( T )) = K ( Z , 2), whos e v alue a t ( w, z ) is the pro duct o f mult iplication by z with trans lation by w . Thus A is a CT-algebra with Dixmier-Douady inv ariant [ σ ] × c = H , wher e [ σ ] ∈ H 2 ( S 1 × T , Z ) is the homotopy class of σ and c is the usual ge ne r ator of H 1 ( S 1 , Z ). Now that we hav e an a ction of R 2 on A = C T ( X , H ) inducing the free T 2 -action on the sp ectrum X , we can compute the cros s ed product to see what the asso ciated “T-dua l” is. Since A = Ind R 2 Z 2 C ( S 1 , K ), w e can use the Green Imprimitivity Theo r em to s ee tha t A ⋊ Ind α R 2 ∼ =  C ( S 1 , K ) ⋊ α Z 2  ⊗ K . Recall that A θ is the universal C ∗ -algebra gener a ted by unitaries U and V with U V = e 2 π iθ V U . So if we lo ok at the definition of α , we s ee tha t A ⋊ Ind α R 2 is the algebra o f sections of a bundle of algebra s over S 1 , whos e fiber ov er e 2 π iθ is A θ ⊗ K . Alternatively , it is Morita equiv alent to C ∗ ( Γ ), where Γ is the discr ete Heisenb er g gr oup of str ic tly upper - triangular 3 × 3 int egral matric e s. Put another wa y , w e could argue that we’v e shown that C ∗ ( Γ ) is a nonc om- mutative T-dual to ( T 3 , H ), bo th viewed as fiber ing ov er S 1 . So we have an ex - planation for the miss ing T-dual: we c ouldn ’t find it just in the world of top ol- o gy alone b e c ause it’s nonc ommu tative . W e w ill wan t to s ee how widely this Dualities and K -Theory 13 phenomenon o ccur s, and also will wan t to res olve the ques tio n o f nonunique- ness of T-duals when n > 1. F urther analy sis of this e x ample leads to the following classification theo - rem: Theorem 4 (Mathai-Rosenberg [ 31]). L et T 2 act fr e ely on X = T 3 with quotient Z = S 1 . Consider the set of al l actions of R 2 on algebr as C T ( X , H ) inducing this action on X , with H al lowe d to vary over H 3 ( X, Z ) ∼ = Z . Then the set of ex terior e quivalenc e classes of su ch actions is p ar ametrize d by Maps( Z, T ) . The winding nu mb er of a map f : Z ∼ = T → T c an b e identi- fie d with t he Dixmier-Douady invariant H . Al l these actions ar e given by the c onstruction ab ove, with f as t he “Mackey obstruction m ap.” Consider a general T 2 -bundle X p − → Z . W e have an edge homomorphism p ! : H 3 ( X, Z ) → E 1 , 2 ∞ ⊆ H 1 ( Z, H 2 ( T 2 , Z )) = H 1 ( Z, Z ) which turns out to play a ma jor role. Theorem 5 (Mathai-Rosenberg[ 31]). L et p : X → Z b e a princip al T 2 - bund le as ab ove, H ∈ H 3 ( X, Z ) . Then we c an always fi nd a “gener alize d T-dual” by lifting the action of T 2 on X to an action of R 2 on C T ( X , H ) and forming the cr osse d pr o duct . When p ! H = 0 , we c an alway s do this in such a way as t o get a cr osse d pr o duct of t he form C T ( X ♯ , H ♯ ) , wher e ( X ♯ , H ♯ ) is a classic al T-dual ( e.g., as found though the pur ely top olo gic al the ory ) . When p ! H 6 = 0 , t he cr osse d pr o duct C T ( X, H ) ⋊ R 2 is never lo cally stably commutativ e and should b e viewe d as a noncommutativ e T - dual . 2.4 Current Di rections in T opo logical T-Dualit y Here we just summarize some o f the current tr ends in top ologica l T-duality: 1. the ab ove appr oach with actio ns o f R n on contin uous-tra ce algebra s. F or n ≥ 2, the lift of even a free a ction of T n on X to an action of R n on C T ( X , H ) is usually not ess en tially unique, and a more detailed study of non-uniqueness is required. One would also like to extend the study of top ologica l T- duality to cases where the a c tion of T n has isotr opy , as progre s s on the famous “ SYZ co njecture” [43] will req uire study of to rus bundles with so me deg eneration. These is s ues have b een studied in [33], [17], a nd [38], for exa mple. 2. the homotopy-theoretic approach of Bunke-Sc hick, ex tended to the higher - dimensional ca se. This has b een studied by B unke-Rumpf -Schic k [15], by Mathai-Rosenber g [33], and by Sch neider [41]. 3. a fancier appr oach using duality of sheaves over the Grothendieck site of (suitable) to po logical spac e s (Bunke-Sc hick-Spitzwec k-Tho m [18]). 4. a gener alization of the nonco mm uta tive g eometry approach using group oids (Daenzer [2 0]). 14 Jonathan R osen b erg 5. alge br aic ana logues, in the world of complex manifolds, schemes, etc., using Muk ai duality with g erb es (Ben-Basset, Blo ck, Pan tev [8] and Block and Daenzer [9]). 6. work of Bo uwknegt and Pande [14] relating the noncommutativ e geometry approach to Hull’s notio n of T-folds [28], which a r e certain no ngeometric backgrounds well-kno wn in string theory . 7. an approach of Bouwknegt and Mathai using duality for lo op gro up bun- dles [13]. As one can see, this is a very ac tiv e sub ject going off in many different directions, a nd it would take a muc h longer sur vey to go in to these matters in detail. 3 Pr oblems Presen ted b y S-Dualit y and Other Dualities 3.1 Ty p e I/T yp e I IA Dualit y on T 4 /K3 In this subs e ction I wan t to des c r ib e s ome joint work with Stefan Mendez- Diez [34]. As w e men tioned befo r e, there is b elieved to b e an S-duality relating type I str ing theory to one of the hetero tic str ing theories . Ther e a r e also v ar ious other dualities relating these tw o theo ries to type IIA theory . Putting these together, we exp ect a (non-p erturbative) duality b etw een typ e I st ring the ory on T 4 × R 6 and typ e IIA string the ory on K 3 × R 6 , at lea st at certa in p oints in the mo duli space. (Here K 3 denotes a K3 surface, a simply connected closed complex surface with trivia l canonical bundle. The name K3 stands for “Kummer, K¨ ahler , Ko da ira.” As a manifold, it has Betti num b ers 1 , 0, 22, 0, 1, and s ignature − 16 .) This duality is discuss e d in detail in [42]. How ca n we r econcile this with the pr inciple that brane charges in type I should take their v alues in K O , while brane charges in type I IA should take their v alues in K − 1 ? On the face o f it, this app ears ridiculous: K O ( T 4 × R 6 ) = K O − 6 ( T 4 ) has lots of 2-tor sion, while K ∗ ( K 3 ) is all torsion-free and concentrated in even degree. One side is easy compute. Recall that for any space X , K O − j ( X × S 1 ) ∼ = K O − j ( X ) ⊕ K O − j − 1 ( X ) . Iterating, we get K O − 6 ( T 4 ) ∼ = K O − 6 ⊕ 4 K O − 7 ⊕ 6 K O − 8 ⊕ 4 K O − 9 ⊕ K O − 10 ∼ = Z 6 ⊕ ( Z / 2) 4 ⊕ ( Z / 2) ∼ = Z 6 ⊕ ( Z / 2) 5 . The way we dea l with the oppos ite side of the duality is to r ecall that a K3 surface can b e obtained by blowing up the p oint singularities in T 4 /G , where G = Z / 2 acting b y multiplication by − 1 on R 4 / Z 4 . This action is semi-free Dualities and K -Theory 15 with 1 6 fixed p oints, the p oints with all four co ordinates equa l to 0 or 1 2 mo d Z . If fact one wa y of deriving the (type I on T 4 ) ↔ (type I IA on K 3) duality explicitly uses the orbifold T 4 /G . But what group should or bifold bra ne charges live in? K ∗ ( T 4 /G ) is not quite right, as this ignore s the orbifold structure. O ne solution that has b een prop osed is K ∗ G ( T 4 ), which Mendez-Diez a nd I co mputed. How ever, as we’ll see, there app ear s to b e a b etter candidate. Let M b e the res ult of removing a G -inv ar ia nt o pe n ball around each G - fixed po int in T 4 . This is a co mpa ct manifold with b oundary on which G acts fr e ely ; let N = M /G . W e get a K 3 surface back from N by gluing in 16 copies of the unit disk bundle of the tangent bundle of S 2 (known to physicists a s the Eguchi-Hanson s pace), one along ea ch RP 3 bo undary comp onent in ∂ N . Theorem 6 ([34]). H i ( N , ∂ N ) ∼ = H 4 − i ( N ) ∼ =                0 , i = 0 Z 15 , i = 1 Z 6 , i = 2 ( Z / 2) 5 , i = 3 Z , i = 4 0 , otherwise . Recall N is the manifold with b oundar y obtained from T 4 /G b y r emoving an op en cone neighborho o d of each singular po int . Theorem 7 ([34]). K 0 ( N , ∂ N ) ∼ = K 0 ( N ) ∼ = Z 7 and K − 1 ( N , ∂ N ) ∼ = K 1 ( N ) ∼ = Z 15 ⊕ ( Z / 2) 5 . Note that the r educed K -theory o f ( T 4 /G ) mod (singular p oints) is the sa me as K ∗ ( N , ∂ N ). Note the rese mblance of K − 1 ( N , ∂ N ) to K O − 6 ( T 4 ) ∼ = Z 6 ⊕ ( Z / 2) 5 . While they a re not the same, the c alculation sugge sts that the brane charges in type I string theory on T 4 × R 6 do indeed show up some w ay in t yp e I IA string theory on the orbifold limit o f K 3. Again let G = Z / 2. Equiv ariant K -theory K ∗ G is a mo dule ov er the r epr e- sentation ring R = R ( G ) = Z [ t ] / ( t 2 − 1). This ring ha s tw o imp ortant prime ideals, I = ( t − 1) and J = ( t + 1). W e hav e R/I ∼ = R/J ∼ = Z , I · J = 0, I + J = ( I , 2) = ( J, 2 ), R/ ( I + J ) = Z / 2. Theorem 8 ([34]). K 0 G ( T 4 ) ∼ = R 8 ⊕ ( R /J ) 8 , and K − 1 G ( T 4 ) = 0 . Also , K 0 G ( M , ∂ M ) ∼ = ( R/I ) 7 , K − 1 G ( M , ∂ M ) ∼ = ( R/I ) 10 ⊕ ( R/ 2 I ) 5 . Note that the equiv ar iant K -theo ry calculation is a refinemen t of the ordi- nary K -theory calculatio n (since G a cts fr eely on M and ∂ M with quotients N and ∂ N , s o that K ∗ G ( M ) and K ∗ G ( ∂ M ) are the sa me as K ∗ ( N ) and K ∗ ( ∂ N ) as ab elian gr oups , though with the addition of more structure). While we do n’t immediately need the ex tr a s tructure, it may pr ove useful later in matching brane charges from K O ( T 4 × R 6 ) on sp ecific classes o f bra nes. 16 Jonathan R osen b erg 3.2 Other Cases o f T yp e I/Type I I Charge Matc hing More gener ally , one could a s k if there a re circumstances where understanding of K -theory lea ds us to exp ect the p ossibility o f a string duality betw een type I string theory on a spacetime Y and type I I string theor y on a spacetime Y ′ . F or definiteness, we will assume we a re dealing with type IIB o n Y ′ . (This is no g reat lo s s o f gener ality since as we have seen in Section 2, types I IA a nd IIB a re r elated via T-duality .) Matching of stable D-brane charges then leads us to lo ok for a n isomor phism of the form K O ∗ ( Y ) ∼ = K ∗ ( Y ′ ) . In genera l, such isomorphis ms are quite rar e, in part b ecause of 2-to rsion in K O − 1 and K O − 2 , and in part b ecause K O -theory is usually 8-p erio dic rather than 2 -p erio dic. But there is o ne no table exception: one knows [1, p. 206] that K O ∧ ( S 0 ∪ η e 2 ) ≃ K , where S 0 ∪ η e 2 is the stable ce ll complex obtained by attaching a stable 2- cell v ia the stable 1-stem η . This is stably the same (up to a deg ree shift) a s CP 2 , since the attaching map S 3 → S 2 ∼ = CP 1 for the top cell of CP 2 is the Hopf map, whose stable homotopy class is η . Thus one might expect a duality betw een t yp e I string theory on X 6 ×  CP 2 r { pt }  and t yp e I IB string theory on X 6 × R 4 . W e pla n to lo ok for evidenc e for this. 3.3 The AdS/CFT Corres p ondence The A dS/CFT c orr esp ondenc e or holo gr aphic duality is a conjectured ph ysical duality , prop osed by J uan Maldace na [30], of a different sort, relating I IB string theory on a 10 - dimensional spa cetime manifold to a gauge theory on another space. In the o r iginal version of this dua lit y , the s tr ing theory lives on AdS 5 × S 5 , a nd the g auge theory is N = 4 sup er-Y ang-Mills theory on Minko wski space R 1 , 3 . O ther versions inv olve slig ht ly different spa ces and gauge theor ies. A go o d s urvey may b e found in [3]. Notation: • N is the sta ndard notation for the sup ersymmetry multiplicity . In other words, N = 4 means there are 4 sets of sup ercharges, a nd there is a U (4) R-symmetry group acting on them. • AdS 5 , 5-dimensio nal ant i-de Sitter sp ac e is (up to coverings) the homo- geneous spa c e S O (4 , 2) /S O (4 , 1 ). T op ologic a lly , this homogeneo us spa ce is R 4 × S 1 . It’s b e tter to pass to the universal cov e r R 5 , howev er, so tha t time isn’t p erio dic. Dualities and K -Theory 17 Nature of the Co rresp ondence W e hav e already explained that D-bra nes car ry Chan-Paton bundles . In type IIB string theor y , a co llection of N coincident D3 bra nes have 3 + 1 = 4 dimensions a nd carry a U ( N ) gaug e theory living on the Chan-Paton bundle. This gauge theor y is the holo graphic dual of the string theory , and the num b er N can be recov er ed as the flux of the Ramond-Ramo nd (RR) field s tr ength 5- form G 5 through an S 5 linking the D3 brane [3, equa tion (3.7 )]. The rotatio n group S O (6) o f R 5 is ident ified (up to cov er ing s) with the S U (4) R symmetry group of the N = 4 gaug e theory . The AdS/CFT corr esp ondence lo oks like holo graphy in that physics in the bulk of AdS space is describ ed by a theor y of o ne les s dimension “on the bo undary .” This can be expla ined b y the famous Bec kenstein-Hawking b o und for the entropy o f a bla ck hole in terms of the ar ea of its b oundary , which in turn fo r ces quantum gravity theories to ob ey a holo gr aphic principle . Recall tha t the Mont onen-Olive Conjecture (Section 1.2) asserts that clas- sical elec tr o-magnetic duality should e x tend to an exact s ymmetry of certain quantum field theor ies. 4 -dimensional sup er- Y ang-Mills (SYM) with N = 4 sup e rsymmetry is b elieved to be a case fo r which this conjecture applies. The Lagra ng ian inv olves the usual Y ang -Mills term − 1 4 g 2 YM Z T r( F ∧ ∗ F ) and the theta angle term (related to the Pontrjagin num b er or inst anton numb er ) θ 32 π 2 Z T r( F ∧ F ) . W e combine these by intro ducing the tau p ar ameter τ = 4 π i g 2 YM + θ 2 π . The ta u par ameter measur e s the relative size o f “ma gnetic” a nd “e le ctric charges.” Dyons in SYM have charges ( m, n ) living in the gr oup Z 2 ; the as- so ciated c omplex char ge is q + ig = q 0 ( m + nτ ). As in the theory of the Dirac monop ole, q uantization of magnetic charge is related to integralit y of charac- teristic clas s es in top ology , i.e., to the fact that the Pon trjagin num b er must be a n integer. The ele ctr o-magnetic duality gr oup S L (2 , Z ) acts on τ by linear frac- tional tr ansformations . Mo re precise ly , it is genera ted by t wo tr ansformations : T : τ 7→ τ + 1, which just incr eases the θ -angle b y 2 π , and has no effect on magnetic charges, a nd by S : τ 7→ − 1 τ , whic h effectiv ely interc hanges elec- tric a nd magnetic charge. By the Montonen-Olive Conjecture [36], the sa me group S L (2 , Z ) should op e r ate o n type I IB string theory in a similar wa y , and θ sho uld cor resp ond in the string theory to the exp ectatio n v alue of the RR scalar field χ . (See for example [27, 44, 4 5, 3].) 18 Jonathan R osen b erg Puzzles Ab o ut Charge Groups An imp ortant constr a int on v a riants of the AdS/CFT co rresp ondence should come from the action o f the S L (2 , Z ) S-duality gro up on the v ar ious charges. F or exa mple, this g roup is expected to act on the pair ( H, G 3 ) of type I IB string theory field strengths in H 3 ( X, Z ) × H 3 ( X, Z ) by linear fractiona l tra ns- formations. Here G 3 denotes the RR 3-form field streng th, or more precisely , its co homology class . But now we hav e so me puzzles: • The class es of RR fields are really supp ose d to live in K − 1 , no t cohomolo gy , whereas the NS class [ H ] is really exp ected to live in ordinary co ho mology . (F ortunately , since the first differential in the Atiy ah- Hirzebruch s pec tral sequence is Sq 3 , ther e is no difference when it comes to cla sses in H 3 , except when H 3 has 2- torsion. See [23, 1 0] for related discussions.) • Since the S-duality gro up mixes the NS-NS and RR sectors, it is not clea r how it should act on D-bra ne and RR field charges. • It’s also not so clea r what conditions to imp ose a t infinit y when spacetime is not compact. F or example, it would app ear that the H- flux a nd RR fields do not have to have compact supp or t, so pe rhaps K -theory with compact suppo rt is not the r ight home for the RR field charges. This p oint s eems unclear in the liter ature. Example 5. Let’s lo ok aga in at the exa mple o f t yp e I IB str ing theory on AdS 5 × S 5 , compare d with N = 4 SYM o n 4- space. How do the K -theoretic charge gro ups match up? Our spa cetime is top o logically X = R 5 × S 5 , where R 5 is the univ er sal cover o f AdS 5 . W e think of R 5 more exactly as R 4 × R + , so that R 4 × { 0 } , Minkowski spa ce, is “ at the b oundar y .” The RR field charges should live in K − 1 ( X ), according to [47], but we see this r equires clarifica- tion: the RR field strength G 5 should represent the num b er N in H 5 ( S 5 ) (since as we mentioned, N is computed by pa iring the class of G 5 with the fundamen tal class of S 5 ), s o we need to use homotopy the or etic K -theory K h here instead of K -theory w ith compac t supp or t, which we’v e implicitly b een using b efore. Indeed, note that K − 1 ( X ) ∼ = K − 1 ( R 5 ) ⊗ K 0 ( S 5 ) ∼ = H 0 ( S 5 ), while K − 1 h ( X ) ∼ = K 0 h ( R 5 ) ⊗ K − 1 ( S 5 ) ∼ = H 5 ( S 5 ), which is what we wan t. Now what ab out the D-br ane charge group for the string theo ry? This should b e Z ∼ = K 0 ( X ) ∼ = K 0 ( R 4 × Y ) ∼ = K 0 ( R 4 ) ⊗ K 0 ( Y ), where Y is the D5- brane R × S 5 , which has K 0 ( Y ) ∼ = Z . Note that this is naturally isomor phic to K 0 ( R 4 ) = e K 0 ( S 4 ), which is where the instanton nu mb er lives in the dual gauge theory . But what charge gr o up on X corr esp onds to the group o f electr ic and magnetic charges in the gaug e theory ? (This should be a gro up isomor phic to Z 2 containing the gro up Z c la ssifying the instanton num b er.) It is b elieved that the str ing /gauge co rresp ondence sho uld apply muc h more generally , to man y type I IB string theories on spaces other than AdS 5 × S 5 , and to g auge theor ies with less sup ersymmetry than the N = 4 theory that we’v e b een consider ing. Analysis of the re le v a nt charge g r oups on b o th Dualities and K -Theory 19 the string and gauge sides of the corre s po ndence should g ive us a guide as to what to exp ect. Study of these constra in ts is s till in a very ear ly stag e. References 1. J. F. Adams, Stable homotopy and gener alise d homolo gy , Chicago Lectures in Mathematics, Universit y of Chicago Press, Chicago, IL, 1995 , Reprint of the 1974 original. MR1324104 (96a:55002) 2. Y . Aharono v and D. Bohm, Signific anc e of ele ctr omagnetic p otentials in the quantum the ory , Phys. Rev . 115 (1959), n o. 3, 485–491. 3. O fer A h aron y , Steven S. Gub ser, Juan Maldacena, Hirosi Ooguri, and Y aron Oz, L ar ge N field the ories, string the ory and gr avity , Phys. Rep. 323 (2000), no. 3-4, 183–386, arXiv.org: hep-th/990511 1 . MR1743597 (2001c:81134 ) 4. E. Alva rez, L. Alv arez-Gaum´ e, J. L. F. Barb´ on, and Y . Lozano, Some glob al asp e cts of duality in string the ory , Nuclear Phys. B 415 (1994), no. 1, 71–100, arXiv.org: hep-th/ 9309039 . MR1265457 (95a:81197) 5. Michael Atiya h and Graeme Segal, Twiste d K -the ory , Ukr. Mat. Visn. 1 ( 2004), no. 3, 287–330, arXiv.org: math/0407054 . MR2172633 (2006m:550 17) 6. , Twiste d K -the ory and c ohomolo gy , In spired b y S. S. Chern, Nank ai T racts Math., vol. 11, W orld Sci. Publ., Hack ensack, NJ, 2006, math/05106 74 , pp. 5–43. MR2307274 (2008j:5 5003) 7. K atrin Beck er, Melanie Bec ker, and John H. Sch warz, String the ory and M - the ory: A mo dern intr o duction , Cambridge Universit y Press, Cambridge, 2007. MR2285203 ( 2008g:81185) 8. O . Ben-Bassat, J. Blo ck, and T. Pan tev, Non-c ommutative tori and Fourier- Mukai duality , Comp os. M ath. 143 (200 7), no. 2, 423–475, math/05091 61 . MR2309993 (2008h:53157) 9. Jonathan Blo ck and Calder Daenzer, Mukai duali ty for gerb es with c onne ction , J. Reine Angew. Math. 639 (2010), 131–171, arXiv.org: 0803.15 29 . MR260819 4 (2011e:53 025) 10. Peter Bouw knegt, Jarah Evslin, Branislav Jurˇ co, V arghese Mathai, and Hisham Sati, Fl ux c omp actific ations on pr oj e ctive sp ac es and the S -duality puzzle , Ad v. Theor. Math. Phys. 10 (2006), no. 3, 345–394. MR2250274 (2007i:81183) 11. Peter Bou wknegt, Jarah Evslin, and V arghese M athai, T -duality: to p ol- o gy change fr om H -flux , Comm. Math. Phys. 249 (2004), no. 2, 383–415 , arXiv.org: hep-th/ 0306062 . MR2080959 (2005m:81235) 12. , T op olo gy and H -flux of T -dual m anifolds , Phys. Rev . Lett. 92 (2004), no. 18, 181601, 3, arXiv.org: hep-th/031205 2 . MR2116165 (2006b:812 15) 13. Peter Bouwknegt and V arghese Mathai, T -duality as a duality of lo op gr oup bund les , J. Phys. A 42 (2009), no. 16, 162001, 8, arXiv.org: 0902.4341 . MR2539268 ( 2010j:53035) 14. Peter Bouwknegt and Ashw in S. P and e, T op olo gic al T -duality and T -folds , Adv. Theor. Math. Phys. 13 (2009), no. 5, 1519–1539, arXiv.org: 0810.4374 . MR2672468 ( 2011g:81256) 15. Ulrich Bunke, Philipp R umpf, and Thomas Schic k, The top olo gy of T -duality for T n -bund les , Rev. Math. Phys. 18 (20 06), no. 10, 1103–1 154, arXiv.o rg: math/05014 87 . MR2287642 (2007k:55023) 20 Jonathan R osen b erg 16. Ulrich Bun ke and Thomas Schic k , On the top olo gy of T -duality , Rev. Math. Phys. 17 (2005), no. 1, 77–112, arXiv.org: math/0405132 . MR2130624 (2006b:5501 3) 17. , T -duality for non-fr e e cir cle actions , An alysis, geometry and top ol- ogy of elliptic op erators, W orld S ci. Publ., Hack ensack, NJ, 2006, math/05085 50 , pp. 429–466. MR2246781 (2008f:8118 1) 18. Ulrich Bun ke, Thomas Schic k , Markus Spitzwec k, and Andreas Thom, Duality for top olo gic al ab elian gr oup stacks and T -duality , K -theory and noncommuta- tive geometry , EMS Ser. Congr. Rep., Eur. Math. So c., Z¨ urich, 2008, arXiv.o rg: math/07014 28 , pp. 227–347. MR2482327 (2010j:1802 6) 19. T. H. Busc her, A symmetry of the string b ackgr ound field e quations , Phys. Let t . B 194 (1987), no. 1, 59–62. MR901282 (88g:81057) 20. Calder Daenzer, A gr oup oid appr o ach to nonc ommutative T -duality , Comm. Math. Phys. 288 (2009), no. 1, 55–96, arXiv.org: 0704.2592 . MR 2491618 (2011a:5 8039) 21. P . A. M. Dirac, Quantise d singularities in the ele ctr omagnetic field , Pro c. Roy al Soc. London , Ser. A 133 (1931), no. 821, 60–72. Zbl0002.3050 2 22. P . Donov an and M. Karoubi, Gr ade d Br auer gr oups and K -the ory with lo c al c o- efficients , Inst. Hautes ´ Etudes Sci. Pu b l. Math. ( 1970), no. 38, 5–25. MR0282363 (43 #8075) 23. Jarah Evslin and Uday V aradara jan, K -the ory and S -duality: starting over fr om squar e 3 , J. High Energy Phys. (2003), no. 3, 026, 26, hep-th/011 2084 . MR1975994 (2004c:81202) 24. Daniel S . F reed and Edward Witten, Anomalies in string the ory with D- br anes , A sian J. Math. 3 (1999), no. 4, 819–851 , arXiv.org: hep-th/99 07189 . MR1797580 ( 2002e:58065) 25. Krzy sztof Ga w ‘ ed zki and Nun o Reis, WZW br anes and gerb es , Rev . Math. Phys. 14 (2002), n o. 12, 1281–13 34, arXiv.org: hep-th/020 5233 . MR1945806 (2003m:812 22) 26. Petr Ho ˇ ra va, T yp e IIA D-br anes, K -the ory and M atrix the ory , Adv . Theor. Math. Phys. 2 (1998), no. 6, 1373–1404 (1999), arXiv.org: hep-th/981213 5 . MR1693636 ( 2001e:81109) 27. C. M. Hull and P . K. T ownsend, Unity of sup erstring dualities , Nuclear Phys. B 438 (1995), no. 1-2, 109–137 , arXiv.or g: hep-th/9410167 . MR1321523 (96e:8306 9) 28. Christofer M. Hull, A ge ometry f or non-ge ometric string b ackgr ounds , J. High Energy Phys. (2005), no. 10, 065, 30 p p. (electronic), arXiv.org : hep-th/040 6102 . MR2180623 (2007d:81188) 29. Max Karoubi, Twiste d K -the ory—old and new , K -theory and noncommutativ e geometry , EMS Ser. Congr. Rep., Eur. Math. So c., Z ¨ uric h, 2008, math/07017 89 , pp. 117–149. MR2513335 (2010h:19010) 30. Juan Maldacena, The lar ge N lim i t of sup er c onformal field the ories and su- p er gr avity , A dv. Theor. Math. Phys. 2 (1998), n o. 2, 231–252, hep-th/971 1200 . MR1633016 (99e:81204a) 31. V arghese Mathai and Jonathan Rosenberg, T -duality for torus bund l es with H - fluxes via nonc ommutative top ol o gy , Comm. Math. Phys. 253 (2005), no. 3, 705–721 , arXiv.org: hep-th/0401168 . MR2116734 (2006b:58008) 32. , On m ysteriously missing T -duals, H -flux and the T -duality gr oup , Dif- feren t ial geometry and physics, Nank ai T racts Math., vol. 10, W orld Sci. Pub l., Dualities and K -Theory 21 Hack ensack, NJ, 2006, arXiv.org: hep-th/040 9073 , pp. 350–358. MR2327179 (2008m:580 11) 33. , T -duality f or torus bund les with H -fluxes via nonc ommutative top ol- o gy. II. The high-dimensional c ase and the T -duality gr oup , A dv. Theor. Math. Phys. 10 ( 2006), n o. 1, 123–158, arXiv.org: hep-th/0508084 . MR2222224 (2007m:580 09) 34. St efan Mendez-Diez and Jonathan R osen b erg, K-the or etic matching of br ane char ges in S- and U-duality , arXiv.org: 1007.1202 , 2010. 35. Ru b en Minasian and Gregory Moore, K -th e ory and Ramond-Ramond char ge , J. H igh Energy Phys. (1997), no. 11, Paper 2, 7 pp. (electronic), see up dated versi on at arXiv.org: hep-th/9710230 . MR1606278 (2000a:81 190) 36. C. Mon tonen an d D. Olive, Magnetic monop oles as gauge p articles? , Phys. Lett. B 72 (1977), 117–120. 37. M. K. Murray , Bund le gerb es , J. London Math. So c. (2) 54 (1996), no. 2, 403– 416, arXiv.org: dg-ga/9407015 . MR1405064 (98a:55016 ) 38. Ashwin S. P and e, T op olo gic al T -duality and Kaluza-Klein monop oles , Adv. Theor. Math. Phys. 12 (2008), no. 1, 185–215 , arXiv.org : math-ph/0612034 . MR2369414 ( 2008j:53040) 39. Jonathan Rosenberg, Continuous-tr ac e algebr as fr om the bund le the or etic p oint of view , J. A ustral. Math. So c. S er. A 47 (1989), no. 3, 368–381. MR1018964 (91d:46090) 40. , T op olo gy, C ∗ -algebr as, and string duality , CBMS Regional Conference Series in Mathematics, vol. 111 , Published for the Conference Board of the Mathematical S ciences, W ashington, DC, 2009. MR2560910 (2011c:46153) 41. An sgar Schneider, Die lokale Struktur von T-Dualit¨ ats trip el n ( the lo c al str uctur e of T-duality triples ), Ph.D. thesis, Universit¨ at G¨ ottingen, 2007, 0712.0260 . 42. A. Sen, An i ntr o duction to duali ty symmetries in string the ory , U nity from du - alit y : gravit y , gauge theory and strings (Les Houches, 2001), NA TO Ad v. Stud y Inst., EDP Sci., Les Ulis, 2003, pp. 241–322. MR2010972 (2004i:81208) 43. An drew Strominger, Shing-T ung Y au, and Eric Zaslow, Mirr or symmetry is T -duality , Nuclear Phys. B 479 (1996), no. 1-2, 243–259 , hep-th/960 6040 . MR1429831 (97j:32022) 44. A. A. Tseytlin, Self-duali ty of Born-Infeld action and Diri chlet 3 -br ane of typ e IIB sup erstring the ory , Nuclear Ph y s. B 469 (1996), no. 1-2, 51–67 , hep-th/960 2064 . MR1394827 (97e:81128) 45. Edward Witten, Bound states of strings and p -br anes , Nuclear Phys. B 460 (1996), n o. 2, 335–350, arXiv.org: hep-th/9510 135 . MR1377168 (97c:81162 ) 46. , D-br anes and K -the ory , J. High Energy Phys. (1998), n o. 12, P ap er 19, 41 pp . (electronic), arXiv.org: hep-th/9810188 . MR1674715 (2000e:811 51) 47. , Overview of K -the ory applie d to strings , Internat. J. Mod ern Phys. A 16 (2001), no. 5, 693–706, Strings 2000. Proceedings of th e International Sup er- strings Conference (Ann Arb or, MI), arXiv.or g: hep-th/0007175 . MR1827946 (2002c:81 175) Index action 2 least 2 sigma-model 5 AdS/CFT correspondence 16 Aharonov-Bohm effect 4 anomaly cancellation 5, 7 anti-de Sitter space 16 B-field 5 Chan-Pato n bundle 6, 17 charg e group 5 conti nuous-trace algebra 7 D-brane 5, 7, 16–18 Dirac q uantiza tion condition 4 Dirac monop ole 3, 17 Dixmier-Douady class 7 dual torus 6 duality 3 electric-magnetic 3, 6, 17 F ourier 3, 6 S- 6, 14 T- 6 noncommutativ e 12 top ological 8, 13 dyon 4, 17 Eguc hi-Hanson space 15 exp ectation v alue 3 field gauge 2 Ramond-Ramond 8, 17 field (in physics) 1 field strength 2, 7 F ourier transform 3 gerbe 7 Gysin map 6 H-flux 7 Heisenberg u ncertaint y principle 2 holograph y 17 instanton num b er 17, 18 K3 su rface 14, 15 magnetic monop ole 3 Maxw ell’s equations 3 Mon tonen-Olive Conjecture 4, 17 partition fun ction 2 path integral 2 quantization charg e 3 Dirac cond ition 4 R-symmetry 16 S-duality 6, 14 sigma mo del 5 stationary p hase 2 string th eory 5 sup er-Y ang-Mills theory 17 T-duality 6 noncommutativ e 12 top ological 8, 13 24 Index T-duality group 11 theta angle 17 tw isted K -theory 7 w ave function 2 W ess-Zumino term 5, 7