T-duality and Differential K-Theory

T -DUALITY AND DIFFERE NTIAL K -THEOR Y ALEXANDER KAHLE AND ALESSANDR O V ALENTINO Abstract. W e give a precise formulation of T -dualit y for Ramond-Ramond fields. This give s a c a nonic al isomorphism b et w een the “geometrically in- v arian t” subgroups of the twisted diffe r ential K -theory gr oups asso ciated to certain principal t orus bundles. Our result com bines top ological T -dualit y with the Busc her rules found in physics. 1. Introduction 1.1. T -duali t y in ph ysi cs. Much interesting mathematics has co me from the in- teraction betw een geometry , top ology and supe rstring theory . In this paper we fo cus our a tten tion on a to ol which is b elieved to be o f fundamental impor tance in sup e rstring theor y: T - du ality . 1 T -duality is a physical statemen t asser ting tha t the type I IA- and type I IB - sup e rstring theorie s constructed o n pairs of spacetime manifo lds p ossess ing iso- metric torus a ctions a re in a cer ta in sense equiv alent . It fir st appe a red a s a low energy r elation b etw een t yp e I IA-s uper string theo r y on R d × S 1 r and type I IB on R d × S 1 1 /r , wher e r denotes the radius o f the circle. In this simple, or “unt wisted” case, T -duality co rresp onds to sending r to 1 /r in the str ing theory σ -mo del. 2 Int ro ducing B - fi elds allows one to consider top ologic ally non-trivial situations. In this g enerality , the spacetime ma nifolds are tota l s paces of (usually top ologic ally inequiv alen t) principal torus bundles over a common base. T -duality then asse r ts that there is a cano nic a l “eq uiv a lence” b etw een the type I IA theory in the pres ence of the B -field on the first spacetime and the type IIB theory on the second spa cetime in the pres e nc e of its B - field, as long a s a particula r conditio n on the geometry of the bundles and their B -fields is satis fie d. Such pairs of spa c e-times and their B - fie lds are sa id to b e T -dual . Even more genera l settings hav e b een co nsidered, where ele ments of the pair may be bundles of “non- commutativ e” tor i [22]. The lo cal r ules implemen ting the eq uiv a lence for the fields aris ing from the low energy limit of sup erstring theo ry are refer red to in the physics litera ture as the Buscher rules (see [11, 4, 24, 3] and re ferences therein). 1.2. T op olog ical T -duali ty. Witten r ealised that D -bra ne ch arge s are classified by K - the ory in the low energy limit of type I IA/B sup ers tr ing theor y . In the presence of a B -field, the K -theor y gr oup clas sifying the D -br ane charges b ecomes t wisted 3 [28, 7 ]. T -duality is exp ected to b e an equiv alenc e o f the low e nergy theory on T - dua l pa irs and thus, restricting a tten tion to the D -bra ne charges, predicts a canonical isomo rphism betw een the appropr iate t wisted K -theory groups The first author was supp orted b y a gr an t fr om the German Research F oundation (Deutsc he F orsch ungsgemeinscha ft (DF G)). The s econd author wa s supported by the German Researc h F oun- dation (Deutsc he F orsch ungsgemeinsc haft (DFG)) through the Institutional Strategy of the Uni- v ersity of G¨ ottingen. 1 F or a general o ve rview of T -dualit y in sup erstring theory , see [26] and [20] and references therein. 2 In this paper we set the string coupling constan t α ′ to 1. 3 See [1, 2, 17] f or details on twiste d K -theory . 1 2 ALEXANDER KAHLE AND ALE SSANDRO V ALE NTINO on principal T -bundles. This prediction ca n b e made mathematically r igoro us, and is known as top olo gic al T -duality . It ha s b een widely inv es tigated and shown to hold [5, 6, 9, 8]. In order to give a feeling for the ingredients in volv e d, we will briefly de s crib e the usual mathema tica l framework of top olo gical T -duality in the simplest situa tio n, namely when the torus bundles a re circle bundles. Let P π − → X , ˆ P ˆ π − → X b e principal U (1)-bundles, a nd and let τ , ˆ τ b e t wists of K ( P ) and K ( ˆ P ) re sp e ctively . One says ( P, τ ) is T -dual to ( ˆ P , ˆ τ ), when the condition (1.1) π ∗ [ τ ] = c 1 ( ˆ P ) , ˆ π ∗ [ ˆ τ ] = c 1 ( P ) , is satisfied, wher e [ τ ] and [ ˆ τ ] de no te the cohomolo gy classes in H 3 ( − ; Z ) classifying the twists. A consequence of the c o ndition Eq. 1.1 is tha t [ c 1 ( P )] ⌣ [ c 1 ( ˆ P )] = 0 . Suppo se now that the pairs ar e T -dual. Cons ide r the corr esp ondence dia gram (1.2) P × X ˆ P ˆ π { { x x x x x x x x x π # # F F F F F F F F F P ˆ P and define the ho momorphism T K : K • + τ ( P ) → K • + ˆ τ − 1 ( ˆ P ) by (1.3) T K := π ∗ ◦ Θ P ◦ ˆ π ∗ , where Θ P : K • + ˆ π ∗ τ ( P × X ˆ P ) → K • + π ∗ ˆ τ ( P × X ˆ P ) is a dis ting uished isomorphism betw een the twisted K -theory groups that may intuitiv ely be thought of a s b eing induced b y multiplying by the “Poincar´ e line bundle” (in the sense that on re strict- ing attention to the pre-image of any p oint in P , Θ P bec omes an automor phism of K -theory induced by tensor ing by the Poincar´ e line bundle). The statement from T -duality in this situation is that the map T K is a n atu r al isomorphi sm . Details may b e found in [5, 9]. The fo rm of the isomorphism in 1.3 allows one to think of top olog ical T -dua lity being a top olo gical version of the F ourier-Muk ai tr ansform. 1.3. Diff eren ti al K -theory and T -duality . T -duality , as understo o d by physi- cists, encompasses far more than the equiv alence of “charges” that under lies top o- logical T - duality . Our go a l in this pa per is to recover more of the physical pictur e , and in particula r, examine how T - duality acts on the s pace of fields . Just as charges in string theor y turn out to b e subtle, so to o do fields. P hysicists hav e b een accustomed to see fields a s represented by some lo ca l ob ject: a function, say , o r differential form; but in the late 1 990’s it was rea lised that certa in fields in the low e nergy limits of type I IA/B string- theory , the Ramond- Ra mond fields, have bo th lo cal a nd globa l (i.e. top o logical) as pec ts [15, 25, 16], and a new languag e was needed to describ e them. F ortunately , s uch a framework existed in mathematics: the languag e of differen tial coho mology theories. Different ial cohomo lo gy theories ar e geo metr ic refinements o f cohomolo gy theo- ries. F or example, o ne is us ed to thinking of cla ss in H 2 ( X ; Z ) as r epresenting an isomorphism clas s of a line bundle on X . By contrast, a c lass in ˇ H 2 ( X ; Z ), the second differ ential co homology group of X , represents a n isomo rphism class o f a line bundle with c onne ction . T -DUALITY AND DIFFERE NTIAL K -THEOR Y 3 F ormally , differential cohomology theories may b e thought of as co mpleting the pullback squa re (1.4) ? / /   Ω • ( X ; V )   E • ( X ) c / / H • ( X ; V ) as c ohomolo gy the ories , where E • is a g eneralised cohomolo g y theory , V = E • ( {∗} ) ⊗ Z R , and c : E • ( X ) → H • ( X ; V ) is a given map (for K -theory this is the Cher n char- acter). F or ordinary cohomolog y , such theories were first constructed by Cheeger and Simons [13], and Deligne [1 4]. The realisa tio n of the r e lev ance o f such theories in ph ysics lead to their b eing c onstructed for generalised cohomology theories [15]. The definitive g eneral cons tr uction is by Hopkins a nd Singer [1 8]. The theory of int erest for us will b e the differen tial K -theory of a spa ce, ˇ K • ( X ), and twists thereof. While Hopkins and Singer provide a genera l co nstruction for any gener alised cohomolog y theo ry that fits into the s quare (1.4), for differential K -theory ther e are much more ge o metric co nstructions [10, 27, 2 1, 12], and o ne may intuitiv ely rega r d elements of ˇ K 0 ( X ) as b eing a formal difference s of vector bundles on X with c onne ction . As hinted at ea rlier, it was r ealised that Ramond-Ramo nd fields are represented by classes 4 of the differential K -theor y of a spac e, a nd in the presence of B - fields, classes of twiste d differ ent ial K -theor y [15, 16]. T -duality would lead one to ex pe c t an equiv alence b etw een the spaces o f fields on dual torus bundles with the a ppro- priate B -fields , and this is precisely what we inv estigate. Under standing T -duality at the level o f Ramond-Ramond fields has also r ecently attracted the attention of ph ysicists, for e x ample in the work of Beck er and Bergman [3]. The main contribution of our paper is a prec is e ma thema tical for mulation of T -duality for Ramond- Ramond fields (which ex tends a formulation of top ologic a l T -duality due to Dan F ree d 5 ) and proving the existence of a c anonic al T -duality isomorphism for these. W e characterise what a T - duality pair should b e in the context of differential K - theory , and from s uch a pair, construct ca no nical t wists of the differential K -theory of the torus bundles. Having constructed these, we show that the twisted differential K -theory groups on the bundles are is o morphic, with the isomor phism furnished by a differential refinemen t o f the pus h-pull o f top ological T - duality (Eq. 1.3). In this s ense, one may see o ur work as a ge ometric generalisa tion o f top ologica l T -duality , leading to a “ differ ent ial-geo metric” F our ier- Muk a i transfo rm. F rom another p oint of view, o ur work co mb ines the glo bal a sp e cts of top olog ical T -duality with the “B uscher rules” which a re known to hold for top olo gic al ly t rivial Ramond-Ramond fields. This p oint o f view brings an ess ential new ingr e die nt of our work into fo cus. T opolo gically trivia l Ra mond-Ramond fields can be describ ed by g lobally defined different ial for ms. In the simple case descr ib ed in the previous section, a type IIA (top olo gically tr ivial) Ramond-Ramond field ha s a field-str ength given b y an element of G ∈ Ω ev ( X × S 1 ), a nd a type I IB Ra mond-Ramond field has a field-strength ˆ G ∈ Ω od d ( X × S 1 ). The Buscher rules then s tate that T -duality takes the fie ld- strength of type I IA Ramond-Ramo nd fields to that of t yp e I IB Ramond-Ramond fields via the transfor mation (1.5) G 7→ Z S 1 ˆ π ∗ G ∧ e F , 4 More precisely: equiv alence classes are represente d by element s in different ial K -theory , but the actual fields are represent ed by co-cycles. 5 Private c ommunic ati on . 4 ALEXANDER KAHLE AND ALE SSANDRO V ALE NTINO where F is the curv a tur e o f the Poincar´ e line bundle, which is usua lly referr ed to as the Hor i formula [1 9]. The crucial p oint is that the Buscher rules do not induce a n isomorphism for arbitrar y (top ologic a lly tr ivial) Ramond-Ramond fields, but only those that are invariant of the circle action. If one wan ts to under stand T - duality for Ramond- Ramond fields on general principal T -bundles, then, o ne m ust understand what the rig ht notion o f inv ariance should b e for elements of differe ntial K -theory . W e find that restricting attent ion to the fixe d cla sses in differential K -theory is insuf- ficient . The correc t subg roup, which we call the ge ometric al ly invariant s ubgroup, is essentially the subgroup of cla sses with inv ariant Chern character form. 1.4. Organis ation. This pap er is o rganised a s fo llows. Section 2 contains our main res ult (Th. 2.4). W e give a pr ecise formulation of T -dua lit y in differ ential K - theory , and pr ov e the T -dua lit y isomorphis m in this c o ntext. A new feature is the necessity of restr icting to the r ight notion of “inv ar iant” s ubgroup. Unders tanding the actio n of the torus on the v a rious cohomo logy gro ups and twists in T -duality is s ubtle, and w e discuss topics relating to this in Appendix B. In Sec. 2.5, w e relate o ur approa ch to T -dua lity to the v a rious appr oaches to t op olo gic al T -duality found in the literature, and to the the ph ysical conceptio n of T - duality captured in the Bus cher rules. The remaining a pp endix, App endix A, reviews v arious topics in differential cohomology theories. In par ticula r, we discuss the categor y str ucture induced on differential cohomolo gy by “geometric trivialisation” , which is c ent ral to our formulation of T -duality in the differ e n tial setting. W e also discuss the prop erties o f twisted differential K -theory that we require, and twists of differential K -theory . Ac knowledgemen ts . The a uthors are very grateful to Dan F reed for ex plaining his p oint of view of T -duality to them, and for man y fruitful discussio ns. They would also like to thank Thomas Schic k and Ulrich Bunke for use ful conv e r sation, and also Aaron Bergma n and Ansga r Schneider. The sec o nd a uthor would like to thank Richard Szab o, Sar a Azzali, and Alessandro F ermi for their interest in the pro ject. 2. T -duality in differential K -theor y W e give a precise for mulation of T -duality in differential K -theor y . Our p oint of v iew is inspire d by an approach to top ologica l T -duality ex plained to us by Dan F reed. F o r us, the fundament al datum is a smo oth T -duality p air (Def. 2.1). F rom this we construct cano nical twists (Sec. 2.2) of the differential K -theory of the torus bundles, and canonical homomo rphism, which we hencefo rth refer to a s the T -map, b etw ee n those twisted differential K -theory groups (Eq. 2.3). The T -map is not an isomo r phism on the entire t wisted differ e ntial K -gr oup, but ra ther on a subgroup: the ge ometric al ly invariant subgroup (Def. 2.3). The main theor e m, and in particula r the statement o f T -duality in this s etting, is Th. 2.4. 2.1. Preli minaries. Let V b e a vector space, with metric. Let Λ ⊂ V b e a full lattice. Let ˆ V = Hom ( V , R ) and ˆ Λ = Hom (Λ , Z ) ⊂ ˆ V be the duals of V and Λ resp ectively . F orm the tori T = V / Λ and ˆ T = ˆ V / ˆ Λ. T and ˆ T hav e geometry : they inherit inv ar iant metr ics fro m the inner pro ducts o n V and ˆ V , and flat connections. Let X be a smo oth manifold, a nd π : ( P , ∇ ) → X , ˆ π : ( ˆ P , ˆ ∇ ) → X be, r esp ec- tively , principal T - and ˆ T -bundles with connectio n. Let f : X → B T , ¯ f : P → E T be classifying maps adapted to the connection such that the following diagram T -DUALITY AND DIFFERE NTIAL K -THEOR Y 5 commutes: P ¯ f / / π   E T π u   X f / / B T and ˆ f : X → B ˆ T , ¯ ˆ f : ˆ P → E ˆ T class ifying maps ada pted to ( ˆ P , ˆ ∇ ) → X . W e will consider P and ˆ P w ith metrics compatible with the c o nnections as, for ins ta nce, in Sec. 5.3 o f [23]. F ro m now on, whenever we re fer to P a nd its dual, we implic- itly assume that they car ry the classifying maps with them. Principal T -bundles with co nnection determine obje cts in H 2 ( X ; Λ), the seco nd different ial co homology group oid of X with v alues in Λ. 6 W e will write P ∈ H 2 ( X ; Λ) and ˆ P ∈ H 2 ( X ; ˆ Λ) to the ob jects deter mined by ( P, ∇ ) and ( ˆ P , ˆ ∇ ). There is a pair ing · : H k ( X ; Λ) ⊗ H l ( X ; ˆ Λ) given by the sequence (2.1) H k ( X ; Λ) ⊗ H l ( X ; ˆ Λ) ⌣ / / H k + l ( X ; Λ ⊗ ˆ Λ) / / H k + l ( X ) . W e demand a (geometric) trivialis a tion σ ∈ Mor  H 4 ( X )  of P · ˆ P : σ : 0 → P · ˆ P , and in particula r that P · ˆ P be in the connected co mp one nt of 0 ∈ H 4 ( X ). This is not the sa me as demanding that the class o f P · ˆ P be zero in the gro up ˇ H 4 ( X ). T aking this together , we arrive a t the following definition. Definition 2.1 (Differen tial T -duality pair) . A differ ential T -duality p air ov er a s mo oth manifold X is g iven by the tr iple h ( P, ∇ ) , ( ˆ P , ˆ ∇ ) , σ i , where ( P, ∇ ), ( ˆ P , ∇ ) → X are r esp ectively principal T , ˆ T bundles over X with connection (and classifying maps), a nd σ : 0 → P · ˆ P is an element of Mor  H 4 ( X )  . 2.2. Canoni cal t wistings . The pullba ck ob jects π ∗ P ∈ H 2 ( P ; Λ), ˆ π ∗ ˆ P ∈ H 2 ( ˆ P ; Λ) hav e canonical geometric trivialisa tions δ P : 0 → π ∗ P , ˆ δ ˆ P : 0 → ˆ π ∗ ˆ P furnis hed by the “diag o nal sections” 7 ∆ P : p 7→ ( p, p ) , ∆ ˆ P : ˆ p 7→ ( ˆ p, ˆ p ) . W e note that δ P · π ∗ ˆ P ∈ Mo r  H 4 ( P ; Z )  . Explicitly , δ P · π ∗ ˆ P : 0 → π ∗ P · π ∗ ˆ P . Similarly ˆ π ∗ P · ˆ δ ˆ P : 0 → ˆ π ∗ P · ˆ π ∗ ˆ P . Comparing these with σ gives automorphisms of 0 ∈ H 4 ( P ; Z ) (resp. 0 ∈ H 4 ( ˆ P ; Z )): π ∗ σ − δ P · π ∗ ˆ P : 0 → 0 , ˆ π ∗ ˆ σ − ˆ π ∗ P · ˆ δ ˆ P : 0 → 0 . Automorphisms of 0 ∈ H 4 ( − ; Z ) define ob jects in H 3 ( − ; Z ), and in this way we obtain ob jects τ ∈ H 3 ( P ; Z ) a nd ˆ τ ∈ H 3 ( ˆ P ; Z ). As explained in App endix A.3, ob jects in H 3 ( − ; Z ) determine twists of differ ential K -theory . 6 The group oid structure we use throughout is the ge ometric structure described in App endix A.2. W e br iefly describ e differential cohomology with v al ues in lattices in Appendix A.1. 7 F or a careful discussion of how a section of a principal torus bundle determines a trivialis ation in differential cohomology , r efer to App endix B. 1. 6 ALEXANDER KAHLE AND ALE SSANDRO V ALE NTINO 2.3. A mo rphism of twists. Consider the fibr e pro duct P × X ˆ P ˆ π { { x x x x x x x x x π # # F F F F F F F F F P π # # G G G G G G G G G G ˆ P ˆ π { { w w w w w w w w w w X Lemma 2 .2. The pr o duct ˆ π ∗ δ P · π ∗ ˆ δ ˆ P is an element of Mor  H 3 ( X ; Z )  , and (2.2) ˆ π ∗ δ P · π ∗ ˆ δ ˆ P : ˆ π ∗ τ → π ∗ ˆ τ . Pr o of. W e will, for clarity of notation, suppress pullbac ks. As explained in App en- dix A.2, δ P ∈ C 1 (1)( P ; Λ) such that ˇ dδ P = P , with a similar statement for ˆ δ ˆ P . Then δ P · ˆ δ ˆ P ∈ C 2 (2)( P × X ˆ P ; Z ) , and ˇ d  δ P · ˆ δ ˆ P  = P · ˆ δ ˆ P − δ P · ˆ P = − ( σ − P · ˆ δ ˆ P ) + ( σ − δ P · ˆ P ) = τ − ˆ τ . This establishes the le mma.  2.4. The T -duali t y isomorphi sm. Befor e sta ting the main theorem of the paper , we need a definition. Definition 2.3 (Geometric in v ariant subgro up) . Let G be a compact and co n- nected Lie group, and X be a smo oth manifold with a s mo oth G -action. Let h ∈ Twist ˇ K ( X ) b e such that Curv h is G -inv ariant. The ge ometric al ly invariant subgroup of ˇ K h + • ( X ) is then defined to b e the subgroup ˇ K h + • ( X ) G ⊆ ˇ K h + • ( X ) of all x ∈ ˇ K h + • ( X ) such that for any g ∈ G g ∗ Curv x − Cur v x = 0 , where Curv : ˇ K τ + • ( X ) → Ω τ + • ( X ) is the map in Eq. A.6. Theorem 2.4. The se qu enc e (2.3) T ˇ K : ˇ K τ + • ( P ) T ˆ π ∗ / / ˇ K τ + • ( P × X ˆ P ) Θ P / / ˇ K ˆ τ + • ( P × X ˆ P ) π ∗ / / ˇ K ˆ τ + •− dim T ( ˆ P ) ˆ T is an isomorp hism. The midd le map, which we denote Θ P : ˇ K τ + • ( P × X ˆ P ) → ˇ K ˆ τ + • ( P × X ˆ P ) , is the isomorphism of twistings define d by the morphism in L emma 2.2. R emark 2.5 . By “for g etting the geometry” , one recovers the top o lo gical T - homo- morphism, and our statement essentially reduces to the well-known statements o f top ological T -duality found in [9, 8, 6, 5]. On the other hand, taking curv atures of the twists, T ˇ K induces a homomor phism b etw een τ -twisted differential forms o n P , and ˆ τ -twisted differential for ms o n ˆ P . One can show that the T -map induced o n t wisted differential forms is only an isomo rphism when r e stricted to the invariant differential forms. The T -maps comm ute with Curv (up to a factor of ˆ A ), and th us the mo st o ne can hope for is that the T -ma p o n differential K -theory b e a n isomorphism b etw een the g eometrically inv ariant subg r oups. This sho ws that in some sense o ur s tatement is the most g eneral statement p o s sible for the t wisted T -DUALITY AND DIFFERE NTIAL K -THEOR Y 7 K -theory gro ups in question. In Appendix B we s ee that the fixed subg roups of differential K -theory do not sur ject onto K -theory , and thus re stricting attention to these would lead to a weaker statement than the physics predicts. The details o f our pro of o f Th. 2 .4 are fairly tec hnical, but the basic idea is simple: we wish use the fact that we know that T -duality holds for K -theory (Prop. 2.18) and for inv a r iant forms (Pr o p. 2 .17). Prop. 2.12 shows precise ly how these relate to the geometrica lly in v ariant subgroup of differential K -theory , allowing us to form the commutativ e diag ram in Fig. 1. Applying the five lemma now pr ov es the theorem. Before we pro ceed in discussing the detailed pro o f o f Th. 2 .4, w e int ro duce some useful notation a nd a n easy lemma. Definition 2.6 (Averaging map) . Let G be a compact Lie gr oup with an inv aria nt volume form µ , 8 and let X b e a smooth G -space. Let m : G × X → X b e the action o f G , and π : G × X → X the canonical pro jection. Let h ∈ Twist ˇ K ( X ) with Curv h G -inv ariant. Define for any x ∈ Ω h + • ( X ), its av er age as ¯ x := 1 vol G π ∗ ( m ∗ ( x ) ∧ µ ) = 1 vol G Z g ∗ x d µ g , where d µ g is the mea sure induced by µ . Lemma 2.7. L et G b e a c omp act Lie gr oup, X a smo oth G -manifold and h ∈ Twist ˇ K ( X ) with Curv h G -invariant. Then, for any ω ∈ Ω h + • ( X ) , d h ¯ ω = d h ω . In p articular, if H h + • ( X ) is fixe d by G , and ω is d h -close d, then ω = ¯ ω + d h α for some α ∈ Ω h + •− 1 ( X ) . Pr o of. The first pa r t is direct computation. Assume now H h + • ( X ) is fixed by G , and let ω b e d h -closed. Then one may c ho ose a family η g ∈ Ω h + • ( X ) dep ending smo othly on g such that g ∗ ω = ω + d h η g , as g ∗ ω a nd ω ar e, by ass umption, d h -cohomolo gous. Averaging this equa tion ov er G gives the second part of the lemma.  Lemma 2.8. L et G b e a c onne cte d c omp act Lie gr oup, X and a smo oth G -manifold. Then G acts trivial ly on H h + • ( X ) for any h ∈ T wist ˇ K ( X ) with Curv h G -invariant. Pr o of. An y smo oth ma p f : X → X induces a homo mo rphism f ∗ : H h + • ( X ) → H f ∗ h + • ( X ). F or tw o such maps f , g : X → Y , the induced homomorphisms a re equal if ther e ex ists a homo to py from f to g holding the twist fixed (see Sec. 1.3 of [23]). Now, fo r a ny h ∈ Twist ˇ K ( X ) with Curv h G -inv ariant, i.e. g ∗ Curv h = Curv h , one has that g ∗ : H h + • ( X ) → H h + • ( X ) is the identit y morphism, as, by the co nnectedness of G , a ny g ∈ G may b e connected to the identit y by a path in G .  W e b egin with a lemma relating the “geometrically in v ar iant” differential K - groups to K -theor y . Lemma 2.9. L et G b e a c omp act c onne cte d Lie gr oup, X a c omp act smo oth G - manifold, and h ∈ Twist ˇ K ( X ) such t hat Curv h is G -invariant. Then the map δ : ˇ K h + • ( X ) G → K h + • ( X ) is a surje ction (wher e δ is the r estriction of the c orr esp onding map in Eq. A.5). 8 Henceforth we will alwa ys assume an i nv ariant measure. 8 ALEXANDER KAHLE AND ALE SSANDRO V ALE NTINO Pr o of. Let x ∈ K h + • ( X ) and c ho ose an y ˜ x ∈ δ − 1 ( x ) ⊂ ˇ K h + • ( X ). Then, by Lemma 2.7, Curv ˜ x = Curv ˜ x + d h α, for some α ∈ Ω h + •− 1 ( X ). W e then no tice tha t δ ( ˜ x − i ( α )) = δ ( ˜ x ) − δ ( i ( α )) = x, and Curv( ˜ x − i ( α )) = Curv ˜ x − d h α = Curv ˜ x, where i : Ω h + •− 1 ( X ) → ˇ K h + • ( X ) is related to the map in E q. A.5. F r om the equations ab ov e , we s ee that ˜ x − i ( α ) ∈ ˇ K h + • ( X ) G , and its image under δ is x , as r equired.  W e now wish to understand the kernel of the map δ in Lemma 2.9. Lemma 2.10. L et G b e a c omp act c onne cte d Lie gr oup, X a c omp act smo oth G - manifold, and h ∈ Twist ˇ K ( X ) such that Curv h is G -invariant. T hen any ω ∈ i − 1 ( ˇ K h + • ( X ) G ) is of the form ω = ω ′ + d h α, wher e ω ′ is invariant. Pr o of. W e notice that the sequence Ω h + •− 1 ( X ) i / / ˇ K h + • ( X ) Curv / / Ω h + • ( X ) sends ω 7→ d h ω . Thu s, ω ∈ i − 1 ( ˇ K h + • ( X ) G ) iff for a ll g ∈ G g ∗ d h ω = d h ω . Averaging this eq uation we see ¯ ω = ω + η for some d h -closed differential for m η . Av eraging ag ain s hows ¯ η = 0, s o that, by Lemma 2.7, η is exact.  One more lemma will allow us to understand i completely . Lemma 2.11. L et G b e a c omp act c onne cte d Lie gr oup, X a c omp act smo oth G - manifold, and h ∈ Twist ˇ K ( X ) such t hat Curv h is G -invariant. Then (1) i : Ω h + •− 1 ( X ) → ˇ K h + • ( X ) desc ends to a map i : Ω h + •− 1 ( X ) d h Ω h + • ( X ) → ˇ K h + • ( X ) . (2) The kernel of i : Ω h + •− 1 ( X ) d h Ω h + • ( X ) → ˇ K h + • ( K ) , is a lattic e, and invariant u nder the action of G . Explicitly, it is Ω h + •− 1 im Ch ( X ) d h Ω h + • ( X ) . Pr o of. Poin t 1 fo llows by noting that d h -exact forms are in the kernel of i by the prop erties o f twisted differential K -theory descr ibed in App endix A.3. Poin t 2 ho lds for similar re a sons – a ny tw o elements in a comp onent of ker i are cohomologo us, so differ by a d h -exact differ ent ial form. B y the naturality of i , the ker i is mapp ed to itself under the action of G .  T -DUALITY AND DIFFERE NTIAL K -THEOR Y 9 W e may gather all the ab ov e lemmas together into one prop ositio n. Prop ositio n 2.12 . L et G b e a c omp act c onne cte d Lie gr ou p, X a c omp act smo oth G -manifold, and h ∈ Twist ˇ K ( X ) such t hat Curv h is G -invariant. The se quenc e (2.4) 0 → Ω h + •− 1 im Ch ( X ) d h Ω h + • ( X ) →  Ω h + •− 1 ( X ) d h Ω h + • ( X )  G i − → ˇ K h + • ( X ) G δ − → K h + • ( X ) → 0 is exact. R emark 2.1 3 . It is not har d to see that  Ω h + • ( X ) d h Ω h + •− 1 ( X )  G Ω h + • im Ch ( X ) d h Ω h + •− 1 ( X ) ∼ = Ω h + • ( X ) Ω h + • im Ch ( X ) ! G , and that this is omorphism comm utes with all the maps of interest, allowing us, for instance, to r e-write the sequence ab ove in a more familiar form: 0 / /  Ω h + •− 1 ( X ) Ω h + •− 1 im Ch ( X )  G i / / ˇ K h + • ( X ) G δ / / K h + • ( X ) / / 0 . How ever, it seems the less familiar for m is more na tural in our pr o of, so we prefer to write things that wa y . W e now need to understand how T -duality acts on differential for ms, and on K -theory . W e b egin with differential forms. Definition 2.14. W e denote by T Ω : Ω τ + • ( P ) → Ω ˆ τ + •− dim T ( ˆ P ) the map T Ω : ω 7→ π ∗ (exp P ∧ ˆ π ∗ ω ) , where P = Cur v ˆ π ∗ δ P · π ∗ ˆ δ ˆ P . One may s ee that i ◦ T Ω = T ˇ K ◦ i, where i is as in Eq . A.5. Lemma 2. 15. The map T Ω : Ω τ + • ( P ) → Ω ˆ τ + •− dim T ( ˆ P ) sends d τ -exact forms to d ˆ τ -exact forms. Pr o of. P has the pr op erty d ˆ τ (exp P ∧ ω ) = exp P ∧ d τ ω as a c onsequence of the fact that ˆ π ∗ δ P · π ∗ δ ˆ P : τ → ˆ τ . This, alo ng with the explicit form for T Ω , now shows that T Ω (d τ ω ) = d ˆ τ T Ω ( ω ) as require d.  An immediate coro llary of the lemma is the following. Lemma 2 .16. T Ω desc ends to a map T Ω :  Ω τ + • ( P ) d τ Ω τ + •− 1 ( P )  T → Ω ˆ τ + •− dim T ( ˆ P ) d ˆ τ Ω ˆ τ + •− 1 − dim T ( ˆ P ) ! ˆ T . Now we can state the key prop osition. 10 ALEXANDER KAHLE AND ALE SSANDRO V ALE NTINO Prop ositio n 2.17. T Ω :  Ω τ + • ( P ) d τ Ω τ + •− 1 ( P )  T → Ω ˆ τ + •− dim T ( ˆ P ) d ˆ τ Ω ˆ τ + •− 1 − dim T ( ˆ P ) ! ˆ T induc e d by t he c orr esp onding map on differ ential fo rms is an isomorphism. F ur- thermor e, it induc es an isomorphi sm T Ω : Ω τ + • im Ch ( P ) d τ Ω τ + •− 1 ( P ) → Ω ˆ τ + •− dim T im Ch ( ˆ P ) d ˆ τ Ω ˆ τ + •− 1 − dim T ( ˆ P ) . Pr o of. It is well known [5, 6 , 23] that T Ω : Ω τ + • ( P ) T → Ω ˆ τ + •− dim T ( ˆ P ) ˆ T is an iso morphism. W e us e this to c o nstruct an ex plicit in verse to the map T Ω : Ω τ + • im Ch ( P ) d τ Ω τ + • ( P ) → Ω ˆ τ + •− dim T im Ch ( ˆ P ) d ˆ τ Ω ˆ τ + •− 1 − dim T ( ˆ P ) . Let y ∈  Ω ˆ τ + •− dim T ( ˆ P ) / d ˆ τ Ω ˆ τ + •− 1 − dim T ( ˆ P )  T . By Le mma 2.8, y = [ ω ], for some inv ariant ω . W e define T − 1 Ω y = [ T − 1 Ω ¯ ω ] , where we have used the fact that T Ω is an iso morphism on inv ariant forms to take the inv erse o f the representative. Lemma 2.17 shows that the inverse map is w ell defined, and it is easy to check that this map is indeed a n inv er se to T Ω . T o prove the r emaining po int, we just need to see that T Ω maps Ω τ + • im Ch ( P ) → Ω ˆ τ + •− dim T im Ch ( ˆ P ). T o this end, let ω ∈ Ω τ + • im Ch ( P ). Let x ∈ ˇ K τ + • ( P ) such that Ch τ [ x ] = [Curv x ] = [ ω ]. By the prop erties o f twisted differential K -theory (as describ ed in App. A.3 ) [Curv T ˇ K x ] = [Curv π ∗ Θ P ( ˆ π ∗ x )] = [ ˆ π ∗ ˆ A ( P / X )] ⌣ π ∗ [Curv Θ P ( ˆ π ∗ x )] , where Θ P : ˇ K ˆ π ∗ τ + • ( P × X ˆ P ) → ˇ K π ∗ ˆ τ + • ( P × X ˆ P ) is defined in Th. 2 .4. But [ ˆ A ( P / X )] = 1, as P is a principal bundle, so that [Curv T ˇ K x ] = π ∗ [Curv Θ P ( ˆ π ∗ x )] = π ∗ [exp P ∧ Curv ˆ π ∗ x ] = [ π ∗ exp P ∧ π ∗ ω ] = [ T Ω ω ] . where the second line follows from Eq. A.7. B ut then [ T Ω ω ] = Ch[ T K x ], and thus T Ω ω ∈ im Ch, as require d.  Having seen how the T -ma p acts on differential forms, we no w examine the T -map reduced to twisted K -theory . Prop ositio n 2.18. The se quenc e T K : K τ + • ( P ) ˆ π ∗ / / K ˆ π ∗ τ + • ( P × X ˆ P ) / / K π ∗ ˆ τ + • ( P × X ˆ P ) π ∗ / / K ˆ τ + •− dim T ( ˆ P ) is an isomorphism, whe r e this se quenc e (and in p articular the t wists and the iso- morphism b et we en them) is obtaine d by “for getting the ge ometry” 9 of the se quenc e Th. 2.4. 9 The passage from twiste d differen tial K -theory to K -theory i s explained in App endix A. 3. T -DUALITY AND DIFFERE NTIAL K -THEOR Y 11 As we disc us s in the next section, Prop. 2.18 is essen tially the to p o logical T - duality found in [9, 8, 6, 5]. In par ticular, (( P , τ ) , ( ˆ P , ˆ τ ) , Θ P ) is a T -duality triple in the sense of [8], and the prop ositio n follows from their theorem. W e may finally prov e Th. 2 .4. Pr o of of Th. 2.4 . W e will use the five lemma o n Fig. 1. Prop. 2.1 2 shows the top and b ottom rows ar e exact. The dia gram commutes by naturality . Th. 2 .18 and Prop. 2.1 7 then show that the requir e d maps down are isomorphisms, pr oving the theorem.  12 ALEXANDER KAHLE AND ALE SSANDRO V ALE NTINO 0 / /   Ω τ + •− 1 im Ch ( P ) d τ Ω τ + • ( P ) / / T Ω    Ω τ + •− 1 ( P ) d τ Ω τ + • ( P )  T i / / T Ω   ˇ K τ + • ( P ) T δ / / T ˇ K   K τ + • ( P ) / / T K   0   0 / / Ω ˆ τ + •− 1 − dim T im Ch ( ˆ P ) d τ Ω ˆ τ + •− dim T ( ˆ P ) / /  Ω ˆ τ + •− 1 − dim T ( ˆ P ) d ˆ τ Ω ˆ τ + •− dim T ( ˆ P )  ˆ T i / / ˇ K ˆ τ + •− dim T ( ˆ P ) ˆ T δ / / K ˆ τ + •− dim T ( ˆ P ) / / 0 Figure 1 . Diagram used to prov e T -duality for differential K -theor y T -DUALITY AND DIFFERE NTIAL K -THEOR Y 13 2.5. Re lation to other approac hes. In this section w e s how our approach to T -duality fits with the v a rious po int s of view on top olo gic al T - duality found in [5, 6, 8, 9]. W e b egin by c onsidering the very simplest situation and take the ba se X to be a p o int. In this case we have the picture T × ˆ T ˆ π } } z z z z z z z z z π ! ! D D D D D D D D D T π ! ! D D D D D D D D D ˆ T ˆ π } } z z z z z z z z z {∗} where the maps are the cano nical pro jections, and T → {∗} is obtained from the natural p ointed map {∗} → B T (and similarly for ˆ T ). W e note that Mor  H 4 ( {∗} ; Z )  is trivial, hence there is no choice o f σ . F urthermore , P and ˆ P a re b oth canonically the triv ial ob jects in H 2 ( {∗} ; Λ) and H 2 ( {∗} ; ˆ Λ) resp ectively . Pulling these o b jects back to T , ˆ T r esp ectively , we see that the morphisms δ P , δ ˆ P are automo rphisms of 0 ∈ H 2 ( T ; Λ), 0 ∈ H 2 ( ˆ T ; ˆ Λ), and are th us given by ob jects in H 1 ( T ; Λ) and H 1 ( ˆ T ; ˆ Λ). W e recall that clas s es in the g roup oid 10 ˇ H 1 ( − ; Λ ) are maps from the space to T , and w orking through the definitions, we s ee that the class of δ P rep- resents the identit y ma p id T : T → T (and simila rly , ˆ δ ˆ P is in the c lass o f the map id ˆ T : ˆ T → ˆ T ). Iden tifying the classes in the differential cohomolog y gr oups that δ P and its dual lie in suffices for our purp oses, but o ne may easily follow throug h the discussion on trivialisations in App endix B .1 to identify actual ob jects δ P , ˆ δ ˆ P . W e note that Curv δ P = θ T , the Maurer -Cartan form on T , and Curv ˆ δ ˆ P = θ ˆ T . The t wists τ = σ − δ P · ˆ P , ˆ τ = σ − P · δ ˆ P , are b oth ca nonically trivial. Nonetheless, w e will see that the morphism δ P · ˆ δ ˆ P : τ → ˆ τ . is no n-trivial. As an automorphism of 0 ∈ H 3 ( T × ˆ T ; Z ), it is canonically an ob ject in H 2 ( T × ˆ T ; Z ). Such o b jects determine line-bund les with c onne ction 11 . Chasing through the definitions, and recalling that δ P and its dual are essentially the iden tit y maps on the resp ective tor i, we see that the line bundle δ P · ˆ δ ˆ P is canonically iso morphic to the Poinc ar´ e line bund le with co nnection. F o r T = S 1 , this is the line bundle with co nnection describ ed in in App endix B.2. As the t wists τ a nd ˆ τ a re b oth trivial, w e hav e that δ P · ˆ δ ˆ P induces an automor - phism of ˇ K • ( T × ˆ T ). Concre tely , this is the automor phism induced by tensoring by the Poincar´ e line bundle, seen as a n element o f ˇ K • ( T × ˆ T ). A t the level of forms (i.e. restricting to top olo g ically trivial elements of ˇ K ), the ab ov e discussion shows that (2.5) T Ω : ω 7→ Z T exp( θ T · θ ˆ T ) ∧ ω , 10 See App endix A.2 f or a discussion of the v arious group oids asso ciated to differen tial cohomology . 11 F or a dis cussion of precisely ho w this o ccurs, see App endix A .4. 14 ALEXANDER KAHLE AND ALE SSANDRO V ALE NTINO where ω ∈ Ω • ( T ), and θ T · θ ˆ T is the differ e n tial form obtained by the s equence Ω n ( X ; V ) ⊗ Ω m ( X ; ˆ V ) V / / Ω n + m ( X ; V ⊗ ˆ V ) / / Ω n + m ( X ; R ) . with the sec ond a rrow b eing induced by the pairing V ⊗ ˆ V → R . Eq. 2.5 is expr e s sion for the T -duality is omorphism found in the physics liter ature [6, 5]. W e now return to the general case where we hav e geometric T and ˆ T bundles ( P, ∇ ), ( ˆ P , ˆ ∇ ) → X , with X a smo oth manifold, a nd a trivia lis ation σ : 0 → P · ˆ P . As expla ined in Sec. 2.2, this data a llows us to construct canonical ob jects τ ∈ H 3 ( P ; Z ), ˆ τ ∈ H 3 ( ˆ P ; Z ) explicitly given b y τ = π ∗ σ − δ P · π ∗ ˆ P , ˆ τ = ˆ π ∗ σ − ˆ π ∗ P · ˆ δ ˆ P . It is easy to see that, when dim T = 1, (2.6) π ∗ τ ∼ = ˆ P , ˆ π ∗ ˆ τ ∼ = P , where b y “ ∼ = ” we mean equality a s classes in differential coho mology (ie, iso mor- phic in the top olo gic al ca tegory). F or g etting the geometry , E q. 2.6 re duce s to the criterion of T -dua lit y describ ed in Sec. 1 .2 (sp ecifically Eq. 1 .1). The o b jects τ a nd ˆ τ in tur n give ris e to t wists of differential K -theory , a nd we construct the canonical homomo rphism defined in Th. 2.4 which, on forgetting the geometry , induces a ho momorphism T K : K τ + • ( P ) → K ˆ τ + •− dim T ( ˆ P ) . The tr iple (( P , τ ) , ( ˆ P , ˆ τ ) , Θ P ) is pre c isely a T -dua lit y triple in the sense of [8]. In particular, we note that pulling back along an y map i : {∗} → X one obtains the situatio n describ ed at the b eginning of this s ection, so that over a fibre the isomorphism of twists is induced by m ultiplying by the Poincar´ e line bundle. T -DUALITY AND DIFFERE NTIAL K -THEOR Y 15 Appendix A. Differential cohomol ogy This app endix discusses v ario us definitions and forma l prop erties in differen- tial cohomolog y theory . Section A.1 describ es a mo del fo r differential co homology with v alues in a la ttice, following Hopkins and Singer [18]. Section A.2 discusses t wo gr oup oids naturally asso ciated to differential coho mology , and in particular discusses the notion of geo metric trivialisatio n so imp ortant to our for m ulation of T -duality for differential K -theory . Section A.3 describ es the formal prop erties of t wisted differential K -theory that w e use in this paper. Finally , Section A.4 constructs a ca nonical t wist of differential K -theory from an o b ject in H 3 ( − ). A.1. Differential cohomol ogy with v alues in lattices. This section c o ntains a brief des c ription of a mo de l for different ial cohomo logy with v alues in a la ttice contained in a vector space. This is a straig ht forward g eneralisatio n of the Hopkins- Singer mo del used in [18] for o rdinary singular coho mology , and is pr esented here to settle notatio n. Let V b e a finite dimensio nal real vector space, and Λ ⊆ V a full lattice. 12 Let X be a compact manifo ld. Denote with C • ( X ; Λ) and C • ( X ; V ) the smo oth singular co chain complex e s with v a lues in Λ and V , respectively , and with H • ( X ; Λ) and H • ( X ; V ) the asso ciated co homology groups. Finally , denote with Ω • ( X ; V ) the space o f differential fo rms with v alues in V , namely Ω • ( X ; V ) := Ω • ( X ) ⊗ R V . F or any V -v a lue d fo r m, w e will denote with ˜ ω the V - v a lued singular co cyle obtained by integration, i.e. for ω = η ⊗ v we hav e ˜ ω ( σ ) :=  Z σ η  v for any σ ∈ C • ( X ). Define the bi-g raded ab elian group C p ( q )( X ; Λ) := ( C p ( X ; Λ) × C p − 1 ( X ; V ) × Ω p ( X ; V ) , p ≥ q C p ( X ; Λ) × C p − 1 ( X ; V ) , p < q and consider for any q the differential ˇ d : C p ( q )( X ) → C p +1 ( q )( X ) defined as ˇ d ( c, h, ω ) := ( δ c, c V − ˜ ω − δ h, d ω ) , p ≥ q and ˇ d ( c, h ) := ( ( δ c, c V − δ h, 0) , p = q − 1 ( δ c, c V − δ h ) , p < q − 1 where c V denotes the image o f c under the map induced by the inclusion Λ ⊆ V . W e denote the coho mology of the co mplex  C • ( q )( X ) , ˇ d  by ˇ H • ( q )( X ; Λ). Define ˇ H p ( X ; Λ) := ˇ H p ( p )( X ; Λ) , the p ’th differ ential c ohomolo gy gr oup of X with values in Λ . F or any p , the gro ups ˇ H p ( X ; Λ) fit in the following exact sequences 0 → H p − 1 ( X ; V / Λ) → ˇ H p ( X ; Λ) Curv − − − → Ω p Λ ( X ; V ) → 0 , (A.1) 0 → Ω p − 1 ( X ; V ) Ω p − 1 Λ ( X ; V ) i − → ˇ H p ( X ; Λ) δ − → H p ( X ; Λ) → 0 . (A.2) There is a cu p pr o duct ⌣ refining the cup pro duct for cohomo logy with co efficients. More precise ly , for Λ ⊆ V and Λ ′ ⊆ V ′ we ha ve a bilinear map ⌣ : C p ( q )( X ; Λ) × C p ′ ( q ′ )( X ; Λ ′ ) → C p + p ′ ( q + q ′ )( X ; Λ ⊗ Z Λ ′ ) 12 A lattice Λ ⊆ V is ful l i f Λ ⊗ Z R ≃ V . 16 ALEXANDER KAHLE AND ALE SSANDRO V ALE NTINO which is compatible with the differe ntial ˇ d , and moreover satisfies Curv( x ⌣ y ) = Curv( x ) ∧ Curv ( y ) , δ ( x ⌣ y ) = δ ( x ) ⌣ δ ( y ) . See [18] for details on the co nstruction of ⌣ . When V ′ = Hom ( V , R ) and Λ ′ = Hom (Λ , Z ) there is a pairing · : C p ( q )( X ; Λ) × C p ′ ( q ′ )( X ; Λ ′ ) → C p + p ′ ( q + q ′ )( X ; Z ) defined by the comp osition o f the cup-pro duct with the natura l pairing Λ ⊗ Λ ′ → Z . The gr oup of smo oth maps from X to V / Λ is isomo rphic to the g roup ˇ H 1 ( X ; Λ). The gr oup of iso morphism clas s es of pr incipal V / Λ-bundles ov e r X with connectio n is isomorphic to ˇ H 2 ( X ; Λ). In fact, one may a sso ciate to a c o cycle in C 2 (2)( X ; Λ) a canonical principal V / Λ -bundle with connection ov er X , in a manner ana logous to the c o nstruction of a principal U (1)-bundle with connection a sso ciated to a co cycle in C 2 (2)( X ; Z ) describ ed in Sec. A.4. A.2. Geometri c triviali sation. In this se c tion we describ e t wo gr o up o ids asso ci- ated with the differential cohomolo gy group ˇ H p ( X ): the fir st has as its co nnected comp onents the differential cohomolo gy group, and the second, which we call the ge ometric group oid, is the one we use extensively in Sec tio n 2. Both of thes e group oids ar e discussed in [15, 1 8]. In fact, b oth o f these group oids ass o ciated to ˇ H p ( X ) may b e extended to be p - group oids: we will not for malise higher structure, but it should b e clear how to extend the co ns truction recursively . Let X be a compact manifold, and denote with Z p ( X ) the kernel of the differ- ent ial ˇ d : C p ( p )( X ) → C p +1 ( p )( X ). W e define the catego ry ˇ H p ( X ) as having the following ob jects and morphisms Ob j  ˇ H p ( X )  := Z p ( X ) , Mor( x, y ) :=  α ∈ C p − 1 ( p )( X ) : x = y + ˇ dα  . The category ˇ H p ( X ) is a group oid, since all the mor phisms are in vertible, and its connected compo nent s are pr e c isely the elements of the differential co homology group ˇ H p ( X ). The o b jects of ˇ H p ( X ) ar e endowed with a s tructure of a n ab elian group, in particular it has a distinguishe d 0 ob ject. W e will say that an o b ject x of ˇ H p ( X ) is triviali sable if Mor( x, 0) is not empty , and refer to elements α in Mor( x, 0) as trivialisations . In other words, w e have x = ˇ dα if x is triv ia lisable. One can see that the cla ss of a trivialis able element x in the gro up ˇ H p ( X ) satisfies Curv( x ) = 0 , δ ( x ) = 0 , where Curv and δ a re the maps fr o m the sequences in Eqs A.1 and A.2. W e now define the ge ometric group oid a sso ciated to ˇ H p ( X ), which we denote by H p ( X ). The resultant notion o f trivialisa tion is the o ne w e use in Sec. 2. The ob jects of H p ( X ) a re again the differen tial co-cycles Z p ( X ). A geometric morphism betw een tw o co-cycles x and y is an α ∈ C p − 1 ( p − 1)( X ) such tha t x = y + ˇ dα , where we r egard ˇ dα ∈ C p ( p )( X ) via the inclusion C p +1 ( p )( X ) ⊂ C p +1 ( p + 1)( X ). A ge ometric trivialisation of an ob ject x is a geometric mor phism b etw een 0 and x . An y ob ject for whic h such a geometric trivialisa tion e x ists is ca lled ge ometric al ly trivialisabl e . Notice that the ima ge of a geo metrically tr iv ialisable ob ject x in ˇ H p ( X ) is in genera l non-zer o , sinc e α is not in gener al an element in C p − 1 ( p )( X ). The geometric group oid is the gr oup oid of relev a nce in this pap er. The cup-pro duct extends to mor phisms (for b oth gr oup oids): ⌣ : Ob j ( H p ( X )) × Mor ( H q ( X )) → Mor  H p + q ( X )  . T -DUALITY AND DIFFERE NTIAL K -THEOR Y 17 W e give a short motiv ating example for the adjective “g eometric”. Let L → X b e a U (1)-pr incipal bundle with connection ∇ , and supp ose that L → X is top ologically trivial. In pa rticular it repre s ent s the zero class in H 2 ( X ; Z ). Let c be a co cycle representativ e o f L : then c is e xact. Any sectio n s : X → L a llows to construct an explicit trivialisatio n of the cohomo logy class , na mely a co cycle α such that c = δ α . In particular , any section s g ive an isomorphism L ≃ X × C . Nevertheless, the se ction s may not induce a connection preserving isomor phism with ( X × C , d) → X , and thus no t trivialis e ( L , ∇ ) in ˇ H 2 ( X ). In the langua ge of co chains, the cocy le ˇ c ∈ Z 2 (2)( X ) repr esenting ( L , ∇ ) will in general not be exact, hence it will not r epresent the zer o class in ˇ H 2 ( X ). Ho wev er, we explain in Appendix B.1 how the sectio n s induces a geometric tr ivialisation, i.e. a mor phism 0 → ( L , ∇ ) in H 2 ( X ). A.3. Twisted di fferen tial K - theory. In this section we desc rib e the formal prop- erties of twiste d differential K -theory we r equire for our work. The study of twisted differential K -theory is still young, and some of these prop er ties have the character of “folk theorems” . How ever, Carey e t al. [12] cons truct a mo del o f twisted differ- ent ial K -theory with many o f the pro pe r ties we require. W e will in future work give a construction of twisted differential K -theory adapted to our lang uage. Our basic mo del for t wisted K -theory is that des crib ed in [17]. Let X b e a compa ct manifold. W e may form the trivial gr oup oid 13 X = ( X , X ). The group oid of twists of ˇ K • ( X ), Twist ˇ K ( X ), 14 is the g roup oid of ge ometric c entr al extensions ( P , ( L, ∇ ) , ω ) with P lo ca lly equiv alent to X . 15 By a geometric ce ntral extension of a group oid, we mean the following. Definition A.1 (Geometric cen tr al ex tens ion) . A geometr ic central extension of a group oid P = ( P 0 , P 1 ) is a tuple ( P , ( L, ∇ ) , ω ), where L → P 1 is a principal U (1)- bundle with connec tio n ∇ , and ω ∈ Ω 2 ( P 0 ), satisfying the following conditions : (1) L → P 1 is a central extension of gr oup oids, (2) ( L, ∇ ) sa tisfies the commutativ e diagrams in Def. 2.4 of [17] a s a line bundle with connection, (3) p ∗ 1 ω − p ∗ 0 ω = i 2 π Ω ∇ . W e no te tha t bundle ger b es with curving and co nnec tive structure, and P U ( H )- bundles with co nnection and B -field are b oth essentially ob jects in Twist ˇ K ( X ). In Sec. A.4 we describ e a functor fro m H 3 ( X ) to Twist ˇ K ( X ). There are functors F : Twist ˇ K ( X ) → T wist K ( X ) , (A.3) Curv : Twist ˇ K ( X ) → Ω 3 d=0 ( X ) . (A.4) 13 F or us, group oi ds are s m ooth, and assumed to be lo c al quotient gr oup oids : that is, they admit a coun table op en cov er by sub-gr oupoids, eac h of which is weakly equiv alent to a compact Lie gr oup acting on a Hausdorff space. W e denote a groupoid by a tuple ( X 0 , X 1 ), where X 0 is the ob ject space, and X 1 is the morphism space. Notational con ven tions are as in [17], whic h also con tains an excellent discussion of the basic notions surrounding groupoids. 14 As explained in [17], Twist ˇ K ( X ) is, in f act, a 2-groupoi d. 15 As of writing, no construction of t w i sted differen tial K -theory has been done us- ing these t wists, but it is reasonable to expect that this groupoid twists differential K - theory . Indeed, one m ay see precisely these ob j ects twisting differential K - theory in slides b y Dan F r eed “Dirac Charge Quan tization, K - Theory , and Orientifolds” found at http://w ww.ma.utex as.edu/users/dafr/paris_nt.pdf . 18 ALEXANDER KAHLE AND ALE SSANDRO V ALE NTINO The fir st, “forgetting the geometry ”, cons tructs a t wist o f K -theor y in the se ns e o f [17] 16 by ( P , ( L, ∇ ) , ω ) 7→ ( P , L ) . The second functor , “cur v a ture”, gives an ob ject in the categ o ry 17 Ω 3 d=0 ( X ). These are twists of the c omplex of differential for ms: from ω ∈ Ω 3 d=0 ( X ) one forms the Z / 2 Z graded co mplex (Ω ω + • ( X ) , d ω ) := (Ω • ( X ) , d + ω ) . The t wisted coho mology gr oup H ω + • ( X ) is defined to b e the cohomolog y of the complex (Ω ω + • ( X ) , d ω ). When the twist is induced b y a n ob ject τ ∈ H 3 ( X ), the functor Curv is induced by the “Curv ” morphism in Eq. A.1. An o b ject τ ∈ Twist ˇ K allows one to define a Chern character Ch τ : K τ + • ( X ) → H τ + • ( X ). T o an obje ct τ ∈ Twist ˇ K ( − ), one may asso ciate an ab elia n gr oup K τ + • ( − ) satisfying the following prop erties: - fun ctoriality : for any smo oth map f : X → Y , we hav e a pullba ck mo r- phism f ∗ : ˇ K τ + • ( Y ) → K f ∗ τ + • ( X ) , - ex act se quenc es : the following natur a l exact sequence holds 0 / /  Ω τ + •− 1 ( X ) Ω τ + •− 1 im Ch ( X )  i / / ˇ K τ + • ( X ) δ / / K τ + • ( X ) / / 0 , (A.5) 0 / / K τ + •− 1 ( X, R / Z ) / / ˇ K τ + • ( X ) Curv / / Ω τ + • K ( X ) / / 0 , (A.6) where the t wist of K -theory a nd of the differential forms are obta ined fr o m the functor s F and Curv in Eqs A.3 and A.4. The compositio n of Curv and the map Ω τ d τ =0 ( X ) → H τ + • ( X ) is the twisted Cher n character map Ch τ men tioned ab ov e. - pus hforwar d : for any differen tia l K -oriented Riemannian family f : X → Y there exists a “wr ong wa y” map f ∗ : ˇ K f ∗ τ ( X ) → ˇ K τ ( Y ) . In particular, we assume the following prop e rty of the pushforward: f ∗ ( i ([ ω ])) = i " Z X/ Y Td( ∇ X/ Y ) ∧ ω #! , for all ω ∈ Ω f ∗ τ + •− 1 ( X ), where Td( ∇ X/ Y ) denotes the T o dd for m of the Levi-Civita connection ∇ X/ Y asso ciated to the Riemannian map f . - n atur ality of twists : for an y mo rphism α ∈ Mor( τ , τ ′ ) there is a natural isomorphism φ α : K τ + • ( X ) ≃ − → K τ ′ + • ( X ) . The isomorphism φ α satisfies φ α ( i ( [ ω ])) = i  e Curv α ∧ ω  , (A.7) Curv ( φ α x ) = e Curv α ∧ Curv x, (A.8) for all ω ∈ Ω τ + •− 1 ( X ), and x ∈ ˇ K τ + • ( X ). Moreov er, if α ′ ∈ Mor( τ , τ ′ ) is such that ther e exists β ∈ Mor( α, α ′ ), then φ α ′ = φ α . 16 The t wi sts in [ 17] ha ve an additional grading. A l l our twists are taken to b e eve n in their grading. 17 Ob j ects i n Ω 3 d=0 ( X ) are closed 3-forms on X , and a morphism α : ω → ω ′ is an α ∈ Ω 2 ( X ) / dΩ 1 ( X ) s uch that ω ′ = ω + d α . T -DUALITY AND DIFFERE NTIAL K -THEOR Y 19 A.4. Twists of differen tial K -theory from Ho pkins-Singer co cycles. In this section we sketc h the construction o f a functor H 3 ( X ) → Twist ˇ K ( X ). Before con- structing the functor, we briefly recall a co nstruction “one deg ree low er”, fo und in Hopkins a nd Singer [1 8], which ass o ciates canonical cir cle bundles with connections to ob jects in H 2 ( X ). One presentation of a circle bundle P → X with connection ∇ is as an ass ignment to each o pen set U ⊂ X a (p ossibly empty) principal homogene o us space Γ( U ) for the group C ∞ ( U, U (1)). Intuitiv ely , Γ( U ) is the space of lo cal sec tio ns o f P over U . A connection is then g iven by an assig nmen t ∇ : Γ( U ) → Ω 1 ( U ) with the “eq uiv a riance” condition ∇ ( g · s ) = ∇ ( s ) + g − 1 dg for any s ∈ Γ( U ) and g ∈ C ∞ ( U, U (1)): ∇ ( s ) gives the 1-for m repr esentativ e of the connection ∇ in the trivialis ation induced by s . Such an assig nment Γ s hould “ glue” prop erly: it should b e a sheaf o f torso rs ov er the sheaf o f circle v a lued functions ov er X . W e now present a line bundle from a g iven x ∈ H 2 ( X ). F or each op en set U ⊂ X , define (A.9) Γ( U ) :=  s ∈ C 1 (1)( X ) : x | U = ˇ ds  / ∼ where s ′ ∼ s iff there exis ts t ∈ C 0 (1)( X ) suc h that s ′ = s + ˇ dt . In other words, w e assign to U the space of geometric trivialisa tions of x | U up to bo undaries. F or any s, s ′ ∈ Γ( U ), α := s − s ′ is an element in Z 1 (1)( X ) up to b o undaries. It is thus a class in ˇ H 1 ( X ), i.e. an R / Z v alued function on U . Identifying R / Z with U (1) g ives the requir e d tor sor ov er the sheaf of U (1)-v alued functions. The connection ∇ can be defined by (A.10) ∇ ( s ) := Curv( s ) ∀ s ∈ Γ ( U ), a nd the equiv a riance condition follows. W e no w e xtend this co nstruction to pro duce the desired functor H 3 ( X ) → Twist ˇ K ( X ). Let x ∈ H 3 ( X ). Defin e (A.11) Γ B ( U ) :=  s ∈ C 2 (2)( X ) : ds = x | U  By the co mment ab ov e, fo r ea ch U we hav e that Γ B ( U ) is a torso r ov e r H 2 ( U ). Define P 0 = a U op e n in X s ∈ Γ( U ) U s . In other words, a po in t o f P 0 is a p oint in an op en set U ⊂ X , “coloured” by an s ∈ Γ( U ). The spa c e P 1 ≡ P 0 × X P 0 may b e identified with P [2] = a U,V op en in X s ∈ Γ( U ) , t ∈ Γ( V ) U s ∩ V t , where her e the intersection is or der e d . It is ea s y to see there is a lo cal equiv a lence of g roup oids b etw een P = ( P 0 , P 1 ) and canonica l group o id X = ( X , X ). W e now build a canonical central extensio n of P . W e do more : w e construct a principal U (1)-bundle with connection ( L , ∇ ) → P 1 : to e a ch U s ∩ U ′ s ′ we assign the the U (1)- bundle with co nnection ( L s,s ′ , ∇ s − s ′ ) asso cia ted to the Ho pkins-Singer 2- co cyle s − s ′ , as describ ed ab ov e. T aken to gether these give the desired U (1) bundle. On P [3] = U s ∩ U ′ s ′ ∩ U ′′ s ′′ , we hav e that the eq uation (A.12) ( s − s ′ ) + ( s ′ − s ′′ ) = ( s − s ′′ ) 20 ALEXANDER KAHLE AND ALE SSANDRO V ALE NTINO induces the c a nonical isomorphism L ss ′ ⊗ L s ′ s ′′ ≃ L ss ′′ of U (1)-bundles with connectio ns. Similar co nsiderations show that all the r e quired commutativ e diagr ams are satisfied (as U (1)- bundles with c onne ction ), and th us ( P , L ) is a central extens ion of the g roup oid P . W e hav e alrea dy s een that the line bundle L has a connection, which als o satisfies the correct compatibility on triple ov erlaps. W e now cons truct the t wo-form ω ∈ Ω 2 ( P 0 ) required to make ( P , ( L, ∇ )) into a ge o metric central extension of P . T he “ B -field” is defined by ass igning to each U s the 2- form Curv( s ) ∈ Ω 2 ( U s ). The assignment is easily seen to satisfy the r e quired compatibility c ondition: Curv( s ) − Curv ( s ′ ) = i 2 π Ω ∇ ss ′ . It remains to show how geometric morphisms σ : x → y (ob jects in α ∈ C 2 (2)( X ) such that x = y + dα ) induce morphis ms of twists. The element α naturally induces a map on torso rs a sso ciated to x and y , α : Γ x ( U ) → Γ y ( U ) by sending s to s + α . This in turn, induces the r equired morphism. Appendix B. Notes on the T -a ction The action of the tor us on the twisted K -theory and differ ent ial twisted K - theory of the bundles appe a ring in T -duality is in itse lf an interesting topic. In this a ppendix, w e e x plain some basic results – in par ticula r, we sho w how the t wists aris ing in T -duality are acted on by the tor us. W e also exhibit an explicit class in K 0 ( T 2 ) that has no fixed pre-ima g e in ˇ K 0 ( T 2 ) (although it do es have a “geometr ic a lly in v aria nt ” pre-image). This sugges ts that r e s tricting attention to the action o f T -dua lit y on the fixed s ubg roup of differential K - theory is far to o restrictive, and further motiv ates our choice of the “ge o metrically inv ar iant” subgroup as the co rrect subgroup suitable fo r T -duality . B.1. The action of T and ˆ T on the twistings. The goal of this section is to examine the action of the torus a nd its dual o n the t wistings involv ed in T -duality . T o this end, w e reca ll that the bundles ( P , ∇ ) and ( ˆ P , ˆ ∇ ) come with classifying maps adapted to the connections f : X → B T , ¯ f : X → E T , a nd ˆ f : X → B ˆ T , ¯ ˆ f : X → E ˆ T so tha t the diagr am below (and its dua l) ( P, ∇ ) ¯ f / / π   ( E T , ∇ u ) π u   X f / / B T commutes. W e now fix 18 c ∈ H 2 ( B T ; Λ) r epresenting the universal U (1)-bundle with co n- nection π u : ( E T , ∇ u ) → B T . The element ( π u ) ∗ c ∈ H 2 ( E T ; Λ) is trivia l, and we fix a trivia lisation in Mor  H 2 ( E T ; Λ)  α : 0 → ( π u ) ∗ c with Curv α T -inv a riant. With this understo o d, we see that the element P ∈ H 2 ( X ; Λ) is given by P = f ∗ c. and thu s ¯ f ∗ α : 0 → π ∗ P . 18 Ev erything done henceforth foll o ws through word-for-word f or the bundle ˆ P . T -DUALITY AND DIFFERE NTIAL K -THEOR Y 21 W e note that any global sec tio n s : X → P induces a geometric triv ialisation t s : 0 → P defined by t s = s ∗ ¯ f ∗ α. W e now consider the pullback bundle p : π ∗ P → P equipp ed with the pullback connection. The map π ∗ f : P → B T cla ssifies π ∗ P , and there is a natural map p ∗ ¯ f : π ∗ P → E T such tha t π u ◦ p ∗ ¯ f = π ∗ f ◦ p . W e have, following our discuss io n in the par agra ph ab ov e, that ( p ∗ ¯ f ) ∗ α : 0 → p ∗ π ∗ P . How ever, the pullback bundle has a natural triv ia lisation – the diag onal section ∆ : P → π ∗ P . Using this, a nd that ∆ ∗ p ∗ = id, we see that ∆ ∗ ( p ∗ ¯ f ) ∗ α : 0 → π ∗ P. This is precis ely the trivia lisation δ P we encountered in Sec. 2.2. In other words δ P = ∆ ∗ ( p ∗ ¯ f ) ∗ α. The following dia gram should ho pefully make everything clear: π ∗ P p ∗ ¯ f / / p   E T π u   P π ∗ f / / ¯ f < < x x x x x x x x x ∆ H H )   π   B T X f < < x x x x x x x x There is a second piece o f natural structure on π ∗ P – it is a T -e quivariant bundle with equiv ariant connectio n with r esp ect to the action o f T on P . There are t wo comm uting a ctions of T on π ∗ P , co ming fr o m T acting on each facto r of π ∗ P = P × X P . The action that makes π ∗ P equiv ar iant with re s pe c t to the T -a c tio n on P is the actio n o n the first facto r; the action on the second factor is the action of T as the structure group of the bundle. W e w ill deno te the equiv ariant action by putting bars on group elements, a nd the structure gr oup action b y omitting the bars. The s tatement , thus, that π ∗ P → P is an equiv ar iant bundle r eads p ◦ ¯ t = t ◦ p. It is also e asy to see that ∆ ◦ t = t ◦ ¯ t ◦ ∆ , t ∗ ( p ∗ ¯ f ) ∗ = ( p ∗ ¯ f ) ∗ t ∗ , ¯ t ∗ ( p ∗ ¯ f ) ∗ = p ∗ ¯ f . W e now wis h to compar e t ∗ ∆ P and ∆ P . W e ca lc ula te fr om the a bove t ∗ δ P − δ P = ∆ ∗ ( p ∗ ¯ f ) ∗ ( t ∗ α − α ) . But, t ∗ α − α : 0 → ( t ∗ c − c ) = 0 → 0 and th us t ∗ α − α ∈ H 1 ( E T ; Λ). In fact, E T is contractible, so [ t ∗ α − α ] ∈ H 1 ( E T ) is zero . This implies that t ∗ α − α is geometrically trivialis a ble, and, once and for all, we fix a family of trivialisations η t : 0 → t ∗ α − α. W e note that bo th π ∗ σ and π ∗ ˆ P ar e inv a riant o f the action of T on P (where we recall σ ∈ Mor  H 4 ( X ; Z )  was used to build the t wist), and th us, recalling τ = π ∗ σ − δ P · π ∗ ˆ P , 22 ALEXANDER KAHLE AND ALE SSANDRO V ALE NTINO we see that t ∗ τ − τ = t ∗ ( π ∗ σ − δ P · π ∗ ˆ P ) − ( π ∗ σ − δ P · π ∗ ˆ P ) = − ( t ∗ δ P − δ P ) · π ∗ ˆ P = − ∆ ∗ ( p ∗ ¯ f ) ∗ ( t ∗ α − α ) · π ∗ ˆ P . Therefore we obtain morphisms − ∆ ∗ ( p ∗ ¯ f ) ∗ η t · π ∗ ˆ P : 0 → ( t ∗ τ − τ ) and which, in turn, allow us to define mo r phisms θ t : t ∗ τ → τ . Using the morphisms θ t , one may define the fixe d s ubgroup of ˇ K τ ( X ) as x ∈ ˇ K τ ( X ) suc h that x = θ t ( t ∗ x ) for all t ∈ T . B.2. T -duali t y and in v arian t elements. In this sectio n we argue that the ele- men t x P ∈ K 0 ( T 2 ), where T 2 = R 2 / Z 2 , with Ch x P =  1 − i 2 π V o l T 2  has no T 2 -fixed pre-image in ˇ K 0 ( T 2 ). How ever, we no te that it do es have a rep- resentativ e whose curv ature character form is in v ariant. This shows that the g eo- metrically inv aria n t subgr oup of ˇ K 0 ( T 2 ) is larger tha n the fixed subgro up, and in particular, that any statement of T -duality restricting attention only to the fixed subgroup of ˇ K 0 ( T 2 ) would b e to o weak. The c la ss x P is explicitly represented by a line bundle L P → T 2 which we now construct. Let ˜ L P → R × R b e the trivia l line bundle. Let Z × Z act on ˜ L as follows: ( n, m ) : ( θ 1 , θ 2 , ξ ) 7→ ( θ 1 + n, θ 2 + m, e 2 π i ( nθ 2 + mθ 1 ) ξ ) , where ( n, m ) ∈ Z × Z , θ 1 , θ 2 ∈ R , a nd ξ ∈ C . This makes ˜ L P int o a Z × Z - equiv ariant line bundle. W e may endow ˜ L P with a n equiv a riant connection given by ∇ ˜ L P = d + π i ( θ 1 d θ 2 − θ 2 d θ 1 ) . W e chec k that Curv ∇ ˜ L P = 2 π i d θ 1 d θ 2 . W e now define L P as ˜ L P / Z × Z . It inherits a co nnection ∇ L P from ∇ ˜ L P , and, explicitly calc ula ting the Chern character form shows tha t indeed [ L P ] = x P . W e no w consider the cla ss 19 ˇ x P = [ L P , ∇ L P , 0] ∈ ˇ K 0 ( T 2 ). A small computation shows that θ ∗ 1 ˇ x P − ˇ x P = i (CS  θ ∗ 1 ∇ ˜ L P , ∇ ˜ L P  ) = i ( π i θ 1 d θ 2 ) . F o r gener ic θ 1 , CS  θ ∗ 1 ∇ ˜ L P , ∇ ˜ L P  is no t exact, so its image by i is cer ta inly no n-zero in ˇ K 0 ( T 2 ). An y e le men t of ˇ K 0 ( T 2 ) that maps to x P ∈ K 0 ( T 2 ) m ust b e of the form ˇ x P + [ α ], where [ α ] ∈ ker δ (and δ is as in Eq. A.5 ). W e thus see a generic element in the pre-image of x P is given by ˇ x P + i ( α ), for α ∈ Ω 1 ( T 2 ). W e compute θ ∗ 1 ( ˇ x P + i ( α )) − ( ˇ x P + i ( α )) = i ( π i ( θ 1 d θ 2 ) + θ ∗ 1 α − α ) . But there is no differential for m α ∈ Ω 1 ( T 2 ) such that π i ( θ 1 d θ 2 ) = θ ∗ 1 α − α + im Ch K − 1 ( T 2 ) 19 F or the purposes of this app endix we will use the mo del of ˇ K 0 to b e found in Klonoff ’s thesis [21]. In his mo del classes in differential K -theory are form al differences of tri ples [ V , ∇ , ω ], where V is a vector bundle, ∇ a connection on i t, and ω an o dd degree differen tial form. T -DUALITY AND DIFFERE NTIAL K -THEOR Y 23 for all θ 1 . Thus there is no fixed elemen t o f ˇ K 0 ( T 2 ) in the pre-image o f x P ∈ K 0 ( T 2 ). In fact, by noticing that a principa l U (1) bundle w ith connection is determined up to equiv alence b y its holo nomy aro und every lo op, w e see that for any principal T -bundle P → X , the only line bundles that give rise to fixed classes in ˇ K 0 ( P ) are those pulled back from X . Similar reas oning shows tha t the fixed classes in ˇ K 0 ( P ) are those with basic curv ature . References 1. M. F. Atiya h and G. 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Witten, D - br anes and K-The ory , JHEP 98 12 (1998), 019. Ma themati sches In stitut, Georg-August Universit ¨ at G ¨ ottingen E-mail addr ess : kahle@uni-m ath.gwdg. de Courant Research Centre “ Higher Order Structures”, Ma thema tisches Institut, Georg-A ugust Universit ¨ at G ¨ ottingen E-mail addr ess : sandro@uni- math.gwdg .de