aa r X i v : . [ m a t h . K T ] D ec EXOTIC TWISTED EQUIVARIANT K-THEORY
FEI HAN AND VARGHESE MATHAI
Abstract.
In this paper we introduce exotic twisted T -equivariant K-theory of loop space LZ depending on the (typically non-flat) holonomy line bundle L B on LZ induced from agerbe on Z . We also define exotic twisted T -equivariant Chern character that maps the exotictwisted T -equivariant K-theory of LZ into the exotic twisted T -equivariant cohomology asdefined earlier in [9], and which localises to twisted cohomology of Z . Contents
Introduction 11. Coupling of T -equivariant line bundles and weak T -invariant gerbes 32. Exotic twisted equivariant cohomology and U (1)-bundles 73. Exotic twisted equivariant K -theory and the Chern character 103.1. Gerbe modules and twisted K-theories 103.2. Exotic twisted equivariant K -theory 113.3. Exotic twisted equivariant Chern Character 12References 15 Introduction
In [9], we introduced the exotic twisted T -equivariant cohomology for the loop space LZ of a smooth manifold Z via the invariant differential forms on LZ with coefficients in the(typically non-flat) holonomy line bundle L B of a gerbe, with differential an equivariantlyflat superconnection ∇ L B − ι K + ¯ H in the sense of [13], where K is the rotation vector fieldand ¯ H is a degree 3 circle-invariant form on LZ that is completely determined by H , thecurvature of the gerbe.This exotic twisted T -equivariant cohomology theory has two applications.First we introduced in [9] the twisted Bismut-Chern character form, generalising [2], whichis a loop space refinement of the twisted Chern character form in [4] and represents classesin the completed periodic exotic twisted T -equivariant cohomology h • T ( LZ, ∇ L B : ¯ H ) of LZ . Mathematics Subject Classification.
Primary 55N91, Secondary 58D15, 58A12, 81T30, 55N20. ore precisely, we define these in such a way that the following diagram commutes,(0.1) K • ( Z, H ) Ch H ) ) ❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚ BCh H / / h • T ( LZ, ∇ L B : ¯ H ) res t t ✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐ H • (Ω( Z )[[ u, u − ]] , d + u − H )where res is the localisation map, degree( u ) = 2.Secondly, in [9] we establish a localisation theorem (about the map res ) for the completedperiodic exotic twisted T -equivariant cohomology for loop spaces and apply it to establishT-duality in a background flux in type II String Theory from a loop space perspective. Con-tinuing along this clue, we recently used in [10] the exotic twisted T -equivariant cohomologyto enhance T-duality on twisted differential forms on circle bundles, where we also showedthe exchange of winding and momentum for the first time in the model of [5, 6]. For analternate approach to T-duality on loop space using twisted chiral de Rham cohomologyinstead, see [12].In this paper, we introduce exotic twisted T -equivariant K-theory , K T ( LZ, ∇ L B : G ), forthe loop space LZ , where G is the weak T -invariant gerbe on LZ whose Dixmier-Douadyclass is ¯ H . We also define an exotic twisted T -equivariant Chern character , Ch ∇ L B : G : K T ( LZ, ∇ L B : G ) −→ h even T ( LZ, ∇ L B : ¯ H )that make the below diagram commutative :(0.2) K T ( LZ, ∇ L B : G ) res u u ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ Ch ∇L B : G * * ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ K • ( Z, H ) Ch H ) ) ❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚ h • T ( LZ, ∇ L B : ¯ H ) res t t ✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐ H • (Ω( Z )[[ u, u − ]] , d + u − H )It follows that the exotic twisted T -equivariant K-theory is the correct version of K-theorythat corresponds via a Chern character map to the exotic twisted T -equivariant cohomologyas defined in [9]. However we would like to point out that the map BCh H in figure (0.1)does not make the upper triangle of figure (0.2) commutative.Our construction of the exotic twisted T -equivariant K-theory can be done on more generalspaces rather than loop spaces, namely on the good T -manifolds, which apply to the circlebundles in the T-duality setting. Actually this paper lays the foundation for work in progress,[11], where we will use the exotic twisted T -equivariant K-theory on LZ to enhance T-dualityon objects in (twisted) K-theory on circle bundles, similarly in spirit to what we did in [10].The plan of this paper is as follows.In Section 1, we introduce the concept of weak T -invariant gerbes and study the couplingof them to T -equivariant line bundles on possibly infinite dimensional good T -manifolds. A air of coupled weak T -invariant gerbe and T -equivariant line bundles will be the initiallyinput data for an exotic twisted T -equivariant K-theory (see Section 3).In Section 2, we establish the correspondence of the exotic twisted T -equivariant cohomol-ogy about differential forms on M with coefficients in a line bundle ξ to certain cohomologytheory about differential forms on Sξ , the circle bundle of ξ (see Theorem 2.3). Such apassage from M to Sξ is needed to be established because when we attempt to developthe exotic twisted T -equivariant K-theory, we realize that it is difficult to be done on M itself, instead one needs to pass to the circle bundle of ξ . This space has more room to de-velop the correct K-theory, who possesses a Chern character landing into the exotic twisted T -equivariant cohomology.In Section 3, we introduce exotic twisted T -equivariant K-theory for possibly infinite di-mensional T -manifolds, and the exotic twisted T -equivariant Chern character that lands intoexotic twisted T -equivariant cohomology. We also establish the transgression formulae in thiscontext, using a new version of Chern-Simons forms. Acknowledgements.
The first author was partially supported by the grant AcRF R-146-000-218-112 from National University of Singapore. He would also like to thank Qin Lifor helpful discussion. The second author was partially supported by funding from theAustralian Research Council, through the Discovery Project grant DP150100008 and theAustralian Laureate Fellowship FL170100020.1.
Coupling of T -equivariant line bundles and weak T -invariant gerbes Let M be a (possibly infinite dimensional) T -manifold. We call M a good T -manifold if M has an open cover { U α } such that all finite intersections U α α ··· α p = U α ∩ U α · · · U α p are T -invariant. Let K be the Killing vector field of the T -action. Definition 1.1.
The system ( { U α } , H, B α , A αβ ) is called a gerbe on M , if H ∈ Ω ( M ) , B α ∈ Ω ( U α ) , A αβ ∈ Ω ( U αβ ) , such that πi H has integral period, H = dB α on U α ,B α − B β = dA αβ on U αβ , (1.1) and there exist C αβγ ∈ C ∞ ( U αβγ , U (1)) such that A αβ + A βγ − A αγ = d ln C αβγ . It is easy to see that different choices of C αβγ differ by a U (1) -valued constant scalar on eachconnected component of U αβγ . Remark 1.2.
Our definition of a gerbe here is slightly more general than the gerbe in theusual sense. We don’t require C βγδ C − αγδ C αβδ C − αβγ = 1 on each nonempty intersection U α ∩ U β ∩ U γ ∩ U δ . efinition 1.3. A gerbe ( { U α } , H, B α , A αβ ) is called a weak T -invariant gerbe on M if(i) H, B α , A αβ are all T -invariant;(ii) ι K A αβ + ι K A αβ − ι K A αγ takes values in πi · Z on each connected component of U αβγ . Remark 1.4.
The second condition is equivalent to L K C αβγ = 2 πinC αβγ for some n ∈ Z on each connected component of U αβγ . Actually we have ι K A αβ + ι K A αβ − ι K A αγ = ι K (cid:0) C − αβγ dC αβγ (cid:1) = C − αβγ ι K dC αβγ = C − αβγ L K C αβγ . If all the n is equal to , i.e. C αβγ ’s are T -invariant, we call it a T -invariant gerbe. Let ξ be a T -equivariant complex line bundle over M equipped with a T -invariant connec-tion ∇ ξ . Definition 1.5.
The T -equivariant line bundle ( ξ, ∇ ξ ) and the weak T -invariant gerbe ( { U α } , H, B α , A αβ ) are called coupled on M if under some T -invariant local basis { s α } ,(i) − ι K B α is the connection 1-form of ∇ ξ on U α for each α ;(ii) e − ι K A αβ is the transition function of ξ on U αβ for each α, β . Lemma 1.6.
If the T -equivariant line bundle ( ξ, ∇ ξ ) and the weak T -invariant gerbe ( { U α } , H, B α , A αβ ) are coupled on M , then the equivariant super connection ∇ ξ − uι K + u − H on ξ is equivari-antly flat, i.e. (1.2) ( ∇ ξ − uι K + u − H ) + uL ξK = 0 . Proof.
The proof is similar to the proof of Lemma 1 in [9]. (cid:3)
We provide some examples of coupled T -equivariant line bundles and weak T -invariantgerbes. Example 1.
Let M be a smooth manifold. Let { U α } be a Brylinski open cover of M , i.e. { U α } is a maximal open cover of M with the property that H i ( U α I ) = 0 for i = 2 , U α I = T i ∈ I U α i , | I | < ∞ . Then the free loop space LM is good T -manifold with the opencover { LU α } , where the T -action is the loop rotating action.Let τ be the transgression(1.3) τ : Ω • ( U α I ) −→ Ω •− ( LU α I )is the transgression map defined as(1.4) τ ( ξ I ) = Z T ev ∗ ( ξ I ) , ξ I ∈ Ω • ( U α I ) . Here ev is the evaluation map(1.5) ev : T × LM → M : ( t, γ ) → γ ( t ) . et ω ∈ Ω i ( M ). Define ˆ ω s ∈ Ω i ( LM ) for s ∈ [0 ,
1] by(1.6) ˆ ω s ( X , . . . , X i )( γ ) = ω ( X (cid:12)(cid:12) γ ( s ) , . . . , X i (cid:12)(cid:12) γ ( s ) )for γ ∈ LM and X , . . . , X i are vector fields on LM defined near γ . Then one checks that d ˆ ω s = c dω s . The i -form(1.7) ¯ ω = Z ˆ ω s ds ∈ Ω i ( LM )is T -invariant, that is, L K (¯ ω ) = 0. Moreover τ ( ω ) = ι K ¯ ω . Here K is the vector field on LM generating rotation of loops.Let ( { U α } , H, B α , A αβ ) be a gerbe on M . Associated to this gerbe, there exists a pair ofcoupled T -equivariant line bundle and weak T -invariant gerbe on LM .The holonomy of this gerbe is a T -equivariant line bundle L B → LM over the loop space LM . L B has Brylinski local sections { σ α } with respect to { LU α } such that the transitionfunctions are { e − R ι K A αβ = e − τ ( A αβ ) } , i.e. σ α = e − R ι K A αβ σ β . The Brylinski sections are T -invariant. L B comes with a natural connection, whose definition with respect to the basis { σ α } is(1.8) ∇ L B = d − ι K ¯ B α = d − τ ( B α ) . For more details, cf. [8].On the other hand, averaging the gerbe ( { U α } , H, B α , A αβ ) gives rise to a gerbe( { LU α } , ¯ H, ¯ B α , ¯ A αβ )on LM . First it is not hard to see that πi ¯ H still has integral period. It is evident that¯ H = d ¯ B α on LU α , ¯ B α − ¯ B β = d ¯ A αβ on LU αβ . (1.9)If on U αβγ ,(1.10) A αβ + A βγ − A αγ = d ln C αβγ , then(1.11) ι K ¯ A αβ + ι K ¯ A βγ − ι K ¯ A αγ = τ ( d ln C αβγ ) ∈ πi Z on each connected component of LU αβγ . By (1.11), if x is a fixed loop in U αβγ and x is anyloop in U αβγ , then(1.12) e R xx ( ¯ A αβ + ¯ A βγ − ¯ A αγ ) does not depend on the choice of paths from x to x in LU αβγ . By (1.10), it is not hard tosee that R xx ( ¯ A αβ + ¯ A βγ − ¯ A αγ ) is pure imaginary. Then we further have(1.13) ¯ A αβ + ¯ A βγ − ¯ A αγ = d ln e R xx ( ¯ A αβ + ¯ A βγ − ¯ A αγ ) , where e R xx ( ¯ A αβ + ¯ A βγ − ¯ A αγ ) is an U (1)-valued function on LU αβγ . Therefore ( { LU α } , ¯ H, ¯ B α , ¯ A αβ )is a gerbe on LM . t is obvious that ¯ H, ¯ B α , ¯ A αβ are all T -invariant. Combining (1.11), we see that the gerbe( { LU α } , ¯ H, ¯ B α , ¯ A αβ ) is a weak T -invariant gerbe on LM .As under the Brylinski sections, the local connection 1-form of (cid:16) L B , ∇ L B (cid:17) is − τ ( B α ) = − ι K ¯ B α , and the transition function of L B is e − R ι K A αβ = e − ι K ¯ A αβ , we see that (cid:16) L B , ∇ L B (cid:17) and ( { LU α } , ¯ H, ¯ B α , ¯ A αβ ) are coupled on LM . Example 2.
In [5, 6], T-duality in a background flux has the following settings. There isa principal circle bundle T → Z π → X with a T -invariant connection Θ and a background T -invariant flux H , which is a T -invariant closed 3-form on Z . Let { U α } be a good cover of X . The cover { π − ( U α ) } makes Z a good T -manifold.The T-dual circle bundle ˆ T → ˆ Z ˆ π → X with a T -invariant connection ˆΘ and a backgroundˆ T -invariant flux ˆ H . The cover { ˆ π − ( U α ) } makes ˆ Z a good T -manifold.Denote v, ˆ v the Killing vector field on Z, ˆ Z respectively. The gerbe ( { π − ( U α ) } , H, B α , A αβ )on Z and the gerbe ( { ˆ π − ( U α ) } , ˆ H, ˆ B α , ˆ A αβ ) be the gerbe on ˆ Z satisfy the following relations(1.14) e − ι v A αβ = ˆ g αβ , − ι v B α = ˆ η α , ι v H = F ˆΘ and(1.15) e − ι v ˆ A αβ = g αβ , − ι ˆ v ˆ B α = η α , ι ˆ v ˆ H = F Θ , where ˆ g αβ is the transition functions of the bundle ˆ Z , ˆ η α is the local connection 1-form ofˆΘ on U α , F ˆΘ is the curvature 2-form of ˆΘ on X and the similar meaning for the notationswithout hats on the dual side.In the setting, B α , A αβ are all chosen to be T -invariant. Moreover as e − ι v A αβ = ˆ g αβ ,we conclude that ι v A αβ + ι v A αβ − ι v A αγ takes values in 2 πi · Z on each U αβγ . Therefore( { π − ( U α ) } , H, B α , A αβ ) is a weak T -invariant gerbe on Z . Similarly ( { ˆ π − ( U α ) } , ˆ H, ˆ B α , ˆ A αβ )is a weak ˆ T -invariant gerbe on ˆ Z .( ˆ Z, ˆΘ) and the standard representation of circle on complex plane give rise to a complexline bundle with connection ( ˆ ξ, ∇ ˆ ξ ) on X . Dually, there is a similar ( ξ, ∇ ξ ) on X comingfrom ( Z, Θ). As e − ι v A αβ = ˆ g αβ , − ι v B α = ˆ η α , the T -equivariant line bundle ( π ∗ ˆ ξ, π ∗ ∇ ˆ ξ ) and the weal T -invariant gerbe ( { π − ( U α ) } , H, B α , A αβ )are coupled on Z . Dually, the ˆ T -equivariant line bundle (ˆ π ∗ ξ, ˆ π ∗ ∇ ξ ) and the ˆ T -invariant gerbe( { ˆ π − ( U α ) } , ˆ H, ˆ B α , ˆ A αβ ) are coupled on ˆ Z . . Exotic twisted equivariant cohomology and U (1) -bundles Let M be a good T -manifold, i.e. M has an open cover { U α } such that all finite intersec-tions U α α ··· α p = U α ∩ U α · · · U α p are T -invariant.Let K be the Killing vector field of the T -action. Denote by L ξK , ι K the Lie derivative andcontraction along the direction K respectively.Let ξ → M be a T -equivariant Hermitian line bundle over M equipped with a T -invariantHermitian connection ∇ ξ . Let H ∈ Ω cl ( M ) be a T -invariant closed 3-form (see [3] as a generalreference for differential forms) such that the equivariant super connection ∇ ξ − uι K + u − H is equivariantly flat, i.e.(2.1) ( ∇ ξ − uι K + u − H ) + uL ξK = 0 , where u is a degree 2 indeterminate. For relevant references to equivariant differential forms,see [13, 1].In the previous section, we have seen examples that satisfy these settings.Let π : Sξ → M be the principal U (1)-bundle of ξ . Let v be the vertical tangent vectorfield on Sξ , i.e. the Killing vector field of the U (1)-action.It is clear that Sξ also admits the induced T -action. As the action of T on the fibers of ξ is linear, i.e. g ( λ · v ) = λ · g ( v ) , ∀ g ∈ T , λ ∈ U (1), one deduces that the T -action and the U (1)-action commute. Therefore we have(2.2) [ K, v ] = 0 . The condition ( ∇ ξ − uι K + u − H ) + uL ξK = 0 is equivalent to the following three equalities,(2.3) µ ξK = L ξK − [ ∇ ξ , ι K ] = L ξK − ∇ ξK = 0( ∇ ξ ) − ι K H = 0 dH = 0Let Θ be the connection 1-form on Sξ for ( ξ, ∇ ξ ). Lemma 2.1. (2.4) ι K Θ = 0 , L K Θ = 0 and (2.5) d Θ = ι K π ∗ H. Proof.
Let { U α } be a T -cover of M . Choose a T -invariant local basis s α of ξ on U α . Let η α be the connection 1-form corresponding to s α . By the first relation in (2.3), we have0 = µ ξK ( s α ) = ( L ξK − [ ∇ ξ , ι K ])( s α ) = ( ι K η α ) ⊗ s α , and therefore we have(2.6) ι K η α = 0 . s s α is T -invariant, we get a local T -equivariant diffeomorphism φ α : U α × S → π − ( U α )such that on the left hand side, T only acts on U α . Then as φ ∗ α (Θ) | U α × S = η α + dθ , wededuce that ι K Θ = 0 , L K Θ = 0 . By the second relation in (2.3), we get d Θ + 12 Θ − ι K π ∗ H = 0or d Θ = ι K π ∗ H. (cid:3) Consider the C ∞ ( M )-module(2.7) e Ω ∗ ( Sξ ) := { ω ∈ Ω ∗ ( Sξ ) | ι v ω = 0 , L v ω = − ω } . Theorem 2.2. (cid:16)e Ω ∗ ( Sξ ) T [[ u, u − ]] , d − ι V − uι K + Θ + u − π ∗ H (cid:17) is a chain complex.Proof. We need to show that:(i) if ω ∈ e Ω ∗ ( Sξ ) T , then ( d − ι v − uι K + Θ + u − π ∗ H ) ω ∈ e Ω ∗ ( Sξ ) T ;(ii) ( d − ι v − uι K + Θ + u − π ∗ H ) + uL K = 0 . (i) holds as we have following three equalities,[ d − ι v − uι K + Θ + u − π ∗ H, ι v ]= L v − [ ι v , ι v ] − uι [ K,v ] + ι v Θ + u − ι v ( π ∗ H )= L v + ι v Θ=0 on e Ω ∗ ( Sξ );(2.8) [ d − ι v − uι K + Θ + u − π ∗ H, L v ]=[ d, L v ] − ι [ v,v ] − uι [ K,v ] + L v Θ + u − L v ( π ∗ H )=0;(2.9)and [ d − ι v − uι K + Θ + u − π ∗ H, L K ]=[ d, L K ] − ι [ v,K ] − uι [ K,K ] + L K Θ + u − L K ( π ∗ H )=0 . (2.10) o show (ii), we have( d − ι v − uι K + Θ + u − π ∗ H ) =( d − ι v − uι K ) + ( d − ι v − uι K )(Θ + u − π ∗ H ) + (Θ + u − π ∗ H ) = − L v − uL K + d Θ − ι v Θ − π ∗ ι K H =( − L v − ι v Θ) + ( d Θ − ι K π ∗ H ) − uL K = − uL K on e Ω ∗ ( Sξ ) . (2.11) (cid:3) Let π ∗ ξ be the pull back bundle of ξ on Sξ . Clearly this is a trivial bundle which has acanonical global nowhere vanishing section γ : ( x, y ) → y, x ∈ M, y ∈ π − ( x ) . Consider the map(2.12) f : Ω ∗ ( M, ξ ) → Ω ∗ ( Sξ ) , ω γ − · π ∗ ω. It is not hard to see that Im( f ) = e Ω ∗ ( Sξ ) , ker( f ) = { } . We therefore get an isomorphism of C ∞ ( M )-modules:(2.13) f : Ω ∗ ( M, ξ ) → e Ω ∗ ( Sξ ) . Since γ is a T -invariant global section of π ∗ ξ , we see that f sends T -invariant invariantparts to T -invariant invariant parts. Hence we get an isomorphism of C ∞ ( M )-modules,which we still denote by f :(2.14) f : Ω ∗ ( M, ξ ) T → e Ω ∗ ( Sξ ) T . Theorem 2.3. (2.15) f : (cid:0) Ω ∗ ( M, ξ ) T [[ u, u − ]] , ∇ ξ − uι K + u − H (cid:1) → (cid:16)e Ω ∗ ( Sξ ) T [[ u, u − ]] , d − ι v − uι K + Θ + u − π ∗ H (cid:17) is a chain map and therefore induces an isomorphism on cohomology (2.16) f ∗ : h ∗ T ( M, ∇ ξ : H ) → H ∗ (cid:16)e Ω ∗ ( Sξ ) T [[ u, u − ]] , d − ι v − uι K + Θ + u − π ∗ H (cid:17) , where h ∗ T ( M, ∇ ξ : H ) is the completed periodic exotic twisted T -equivariant coho-mology [9] .Proof. Let ω ∈ Ω ∗ ( M, ξ ) T [[ u, u − ]]. We have( d − ι v − uι K + Θ + u − π ∗ H )( f ( ω ))=( d − ι v − uι K + Θ + u − π ∗ H )( γ − · π ∗ ω )=( d − uι K + Θ)( γ − · π ∗ ω ) + u − π ∗ H ( γ − · π ∗ ω ) . (2.17)Let { U α } be an T -cover of M . Let s α be T -invariant local basis of the ξ on U α . Suppose ω | Uα = ω α ⊗ s α . hen d ( γ − · π ∗ ω )= d ( π ∗ ω α · ( γ − · π ∗ s α ))= π ∗ ( dω α )( γ − · π ∗ s α ) − π ∗ ( ω α ) d ( γ − · π ∗ s α ) . (2.18)Therefore locally, we have d ( γ − · π ∗ ω ) + Θ( γ − · π ∗ ω )= π ∗ ( dω α )( γ − · π ∗ s α ) − π ∗ ( ω α ) d ( γ − · π ∗ s α ) + Θ( π ∗ ω α )( γ − · π ∗ s α )= π ∗ ( dω α )( γ − · π ∗ s α ) − π ∗ ( ω α )( γ − · π ∗ ω )[Θ − ( γ − · π ∗ ω ) − d ( γ − · π ∗ s α )]=[ π ∗ ( dω α ) − π ∗ ( ω α ) η α ]( γ − · π ∗ s α ) , (2.19)where η α = Θ − ( γ − · π ∗ ω ) − d ( γ − · π ∗ s α ) is connection one form for the basis s α of theconnection ∇ ξ on U α . Moreover, we have ι K ( γ − · π ∗ ω )= ι K ( π ∗ ( ω α )( γ − · π ∗ s α ))= ι K ( π ∗ ( ω α ))( γ − · π ∗ s α ) . (2.20)Therefore, [ d ( γ − · π ∗ ω ) + Θ( γ − · π ∗ ω ) + ι K ( γ − · π ∗ ω )] | U α = π ∗ ( dω α + ω α η α − uι K ω α )( γ − · π ∗ s α )= γ − π ∗ [( dω α + ω α η α − uι K ω α ) ⊗ s α ]= γ − π ∗ [( ∇ ξ − uι K ) ω ] | U α = f (( ∇ ξ − uι K ) ω ) | U α . (2.21)And so we have(2.22) ( d − ι v − uι K + Θ + u − π ∗ H )( f ( ω )) = f (( ∇ ξ − uι K + u − H ) ω ) . (cid:3) Exotic twisted equivariant K -theory and the Chern character Gerbe modules and twisted K-theories.
A geometric realization of the gerbe G =( { U α } , H, B α , A αβ ) is { ( L αβ , ∇ Lαβ ) } , a collection of trivial line bundles L αβ → U αβ such thatthere are isomorphisms L αβ ⊗ L βγ ∼ = L αγ on U αβγ and collection of connections {∇ Lαβ } suchthat ∇ Lαβ = d + A αβ . Note that as here we are using slightly more general version of gerbe(see Definition 1.1 and Remark 1.2), isomorphisms L αβ ⊗ L βγ ∼ = L αγ are not uniquely fixed,but may differ by a multiplication by a constant U (1)-valued scalar. Then we have(3.1) ( ∇ Lαβ ) = F Lαβ = B β − B α . et E = { E α } be a collection of (infinite dimensional) Hilbert bundles E α → U α whosestructure group is reduced to U I , which are unitary operators on the model Hilbert space H of the form identity + trace class operator. Here I denotes the Lie algebra of trace classoperators on H . In addition, assume that on the overlaps U αβ that there are isomorphisms(3.2) φ αβ : L αβ ⊗ E β ∼ = E α , which are consistently defined on triple overlaps because of the gerbe property. Then { E α } issaid to be a gerbe module for the gerbe { L αβ } . A gerbe module connection ∇ E is a collectionof connections {∇ Eα } is of the form ∇ Eα = d + A Eα where A Eα ∈ Ω ( U α ) ⊗ I whose curvature F Eα on the overlaps U αβ satisfies(3.3) φ − αβ ( F E α ) φ αβ = F L αβ I + F E β Using equation (3.1), this becomes(3.4) φ − αβ ( B α I + F Eα ) φ αβ = B β I + F Eβ . It follows that exp( B ) Tr (cid:0) exp( F E ) − I (cid:1) is a globally well defined differential form on Z ofeven degree. Notice that Tr( I ) = ∞ which is why we need to consider the subtraction.Let E = { E α } and E ′ = { E ′ α } be a gerbe modules for the gerbe { L αβ } . Then an elementof twisted K-theory K ( Z, G ) is represented by the pair ( E, E ′ ), see [4]. Two such pairs( E, E ′ ) and ( G, G ′ ) are equivalent if E ⊕ G ′ ⊕ K ∼ = E ′ ⊕ G ⊕ K as gerbe modules for somegerbe module K for the gerbe { L αβ } . We can assume without loss of generality that thesegerbe modules E, E ′ are modeled on the same Hilbert space H , after a choice of isomorphismif necessary.Suppose that ∇ E , ∇ E ′ are gerbe module connections on the gerbe modules E, E ′ respec-tively. Then we can define the twisted Chern character as Ch H : K ( Z, G ) → H even ( Z, H ) Ch H ( E, E ′ ) = exp( − B ) Tr (cid:16) exp( − F E ) − exp( − F E ′ ) (cid:17) That this is a well defined homomorphism is explained in [4, 14]. To define the twisted Cherncharacter landing in (Ω • ( Z )[[ u, u − ]]) ( d + u − H ) − cl , simply replace the above formula by Ch H ( E, E ′ ) = exp( − u − B ) Tr (cid:16) exp( − u − F E ) − exp( − u − F E ′ ) (cid:17) . The above theory can be extended to equivariant setting with a compact group action onall the data [14].3.2.
Exotic twisted equivariant K -theory. Let M be a good T -manifold with an T -invariant cover { U α } .Let ξ → M be a T -equivariant Hermitian line bundle over M equipped with a T -invariantHermitian connection ∇ ξ . Let π : Sξ → M be the principal U (1)-bundle of ξ .Let G = ( { U α } , H, B α , A αβ ) be a weak T -invariant gerbe on M .Assume that ( ξ, ∇ ξ ) and ( { U α } , H, B α , A αβ ) are coupled on M . ssociated to these data (( ξ, ∇ ξ ); G ), we will introduce a version of twisted K -theory andtwisted Chern character in this section.It is clear that the open cover { π − ( U α ) } makes Sξ a good ( T × U (1))-manifold. Here todistinguish the two circle actions, we denote by T the circle acting on the base M and by U (1) the circle acting on the fibers.Denote G ξ := ( { π − ( U α ) } , π ∗ H, π ∗ B α , π ∗ A αβ ), which is a ( T × U (1))-invariant gerbe on Sξ .Let { ( ˆ L αβ , ∇ ˆ L αβ = d + π ∗ A αβ ) } be the system of ( T × U (1))-line bundles with ( T × U (1))-invariant connections on U αβ × U (1) be the geometrization of the gerbe G ξ .Let v be the vertical tangent vector field on Sξ , i.e. the Killing vector field of the U (1)-action. Let K be the Killing vector field of the T -action. Let u be a degree 2 indeterminate. Definition 3.1. E = { E α , ∇ E α } is called a ( T × U (1)) -equivariant gerbe module withhorizontal connection for the gerbe { ˆ L αβ } if(a) the ( T × U (1)) -invariant connections ∇ E α ’s vanish on the vertical direction, i.e. ∇ E α v ≡ ;(b) there are ( T × U (1)) -equivariant isomorphisms φ αβ : ˆ L αβ ⊗ E β ∼ = E α , which respect the connections and are consistently defined on triple overlaps because of thegerbe property. Let (
E, E ′ ) and ( G, G ′ ) be two pairs of ( T × U (1))-equivariant gerbe modules with hori-zontal connections for the gerbe { ˆ L αβ } . We say they are equivalent, denoted by( E, E ′ ) ∼ ( G, G ′ )if there exists some K , a ( T × U (1))-equivariant gerbe modules with horizontal connection,such that E ⊕ G ′ ⊕ K ∼ = E ′ ⊕ G ⊕ K as ( T × U (1))-equivariant gerbe modules with horizontal connections. Clearly this is anequivalence relation. As usual, we define(3.5) ˆ K T ( M, ∇ ξ : G ) := { ( E, ∇ E , E ′ , ∇ E ′ ) } / {∼} . If the horizontal gerbe module connections are forgotten, one defines the exotic twisted T -equivariant K -theory of the coupled pair (( ξ, ∇ ξ ) , G ), denoted as K T ( M, ∇ ξ : G ), by(3.6) K T ( M, ∇ ξ : G ) := { ( E, E ′ ) } / {∼} . Exotic twisted equivariant Chern Character.
Let E = { E α , ∇ E α } be a ( T × U (1))-equivariant gerbe module with horizontal connection for the gerbe { ˆ L αβ } . For the equivariantcurvatures along the direction v + uK , we have(3.7) φ − αβ ( F E α + µ E α v + uK ) φ αβ = ( F ˆ L αβ + µ ˆ L αβ v + uK ) I + ( F E β + µ E β v + uK ) , where µ stands for the moment. However(3.8) F ˆ L αβ = π ∗ B β − π ∗ B α , µ ˆ L αβ v + uK = ( ι v + uι K ) π ∗ A αβ = uι K π ∗ A αβ = 2 πiuθ β − πiuθ α , where θ α (resp. θ β ) is the vertical coordinates of π − ( U α ) (resp. π − ( U β )). So we have(3.10) φ − αβ ( F E α + µ E α v + uK + π ∗ B α + 2 πiuθ α ) φ αβ = F E β + µ E β v + uK + π ∗ B β + 2 πiuθ β . Let E ′ = { E ′ α } be another ( T × U (1))-equivariant gerbe module for the gerbe { ˆ L αβ } . Thenexp( − u − π ∗ B α − πiθ α ) Tr (cid:16) exp( − u − ( F E α + µ E α v + uK )) − exp( − u − ( F E ′ α + µ E ′ α v + uK )) (cid:17) can beglued together as a global differential form in Ω ∗ ( Sξ )[[ u, u − ]]. Simply denote this form by(3.11) ch ∇ ξ : G ( ∇ E , ∇ E ′ ) = exp( − u − π ∗ B − πiθ ) Tr (cid:16) − exp( u − ( F E + µ Ev + uK )) − exp( − u − ( F E ′ + µ E ′ v + uK )) (cid:17) . Theorem 3.2. (i) The following equalities hold, (3.12) ι v ch ∇ ξ : G ( ∇ E , ∇ E ′ ) = 0 , L v ch ∇ ξ : G ( ∇ E , ∇ E ′ ) = − ch ∇ ξ : G ( ∇ E , ∇ E ′ ) , (3.13) ( d − ι v − uι K + Θ + u − π ∗ H ) ch ∇ ξ : G ( ∇ E , ∇ E ′ ) = 0 . (ii) If ( ∇ E , ∇ E ′ ) , ( ∇ E , ∇ E ′ ) are two horizontal gerbe module connections, then there exists cs ( ∇ E , ∇ E ′ ; ∇ E , ∇ E ′ ) ∈ e Ω ∗ ( Sξ )[[ u, u − ]] such that (3.14) ch ∇ ξ : G ( ∇ E , ∇ E ′ ) − ch ∇ ξ : G ( ∇ E , ∇ E ′ ) = ( d − ι v − uι K + Θ + u − π ∗ H ) cs ( ∇ E , ∇ E ′ ; ∇ E , ∇ E ′ ) . Proof. (i) Consider the local expression ch ∇ ξ : G ( ∇ E , ∇ E ′ ) | π − ( U α ) = exp( − u − π ∗ B α − πiθ α ) Tr (cid:16) exp( − u − ( F E α + µ E α v + uK )) − exp( − u − ( F E ′ α + µ E ′ α v + uK )) (cid:17) . Obviously, ι v π ∗ B α = 0 . On the other hand, as ∇ E α is horizontal connection, we have ∇ E α v = 0, but this equivalent to [ ∇ E α , ι v ] = L v . Therefore ι v ( F E α ) = [ ι v , ( ∇ E α ) ] = ( L v − ∇ E α ι v ) ∇ E α − ∇ E α ( L v − ι v ∇ E α ) = [ ∇ E α , L v ] = 0 , as ∇ E α is T × U (1)-invariant. Similarly, ι v ( F E ′ α ) = 0. We therefore have ι v ch ∇ ξ : G ( ∇ E , ∇ E ′ ) | π − ( U α ) = 0 . As ∇ E α is T × U (1)-invariant, clearly L v ( F E α ) = 0. The moment µ E α v + uK = L v + uK − [ ι v + uK , ∇ E α ] . Since [ v, K ] = 0, it is easy to see that L v µ E α v + uK = 0 . Now L v π ∗ B α = 0 and L v e − πiθ α = − e πiθ α , we have L v ch ∇ ξ : G ( ∇ E , ∇ E ′ ) | π − ( U α ) = − ch ∇ ξ : G ( ∇ E , ∇ E ′ ) | π − ( U α ) . t last, as ( ξ, ∇ ξ ) and ( { U α } , H, B α , A αβ ) are coupled on M , one has2 πiθ α − π ∗ ι K B α = Θ | π − ( U α ) , where Θ is the connection 1-form on Sξ . Hence( d − ι v − uι K ) ch ∇ ξ : G ( ∇ E , ∇ E ′ ) | π − ( U α ) ==( d − ι v − uι K ) h exp( − u − π ∗ B α − πiθ α ) Tr (cid:16) exp( − u − ( F E α + µ E α v + uK )) − exp( − u − ( F E ′ α + µ E ′ α v + uK )) (cid:17)i = (cid:2) ( d − ι v − uι K ) exp( − u − π ∗ B α − πiθ α ) (cid:3) Tr (cid:16) − exp( u − ( F E α + µ E α v + uK )) − exp( − u − ( F E ′ α + µ E ′ α v + uK )) (cid:17) = (cid:2) exp( − u − π ∗ B α − πiθ α )( − u − π ∗ dB α − πidθ α + ι K π ∗ B α ) (cid:3) · Tr (cid:16) − exp( u − ( F E α + µ E α v + uK )) − exp( − u − ( F E ′ α + µ E ′ α v + uK )) (cid:17) = (cid:2) − u − π ∗ H − (2 πiθ α − π ∗ ι K B α )) (cid:3) · h exp( − u − π ∗ B α − πidθ α ) Tr (cid:16) − exp( u − ( F E α + µ E α v + uK )) − exp( − u − ( F E ′ α + µ E ′ α v + uK )) (cid:17)i = (cid:0) − u − π ∗ H − Θ (cid:1) ch ∇ ξ : G ( ∇ E , ∇ E ′ ) | π − ( U α ) , (3.15)and therefore ( d − ι v − uι K + Θ + u − π ∗ H ) ch ∇ ξ : G ( ∇ E , ∇ E ′ ) | π − ( U α ) = 0 . (ii) Let ∇ Et = (1 − t ) ∇ E + t ∇ E , ∇ E ′ t = (1 − t ) ∇ E ′ + t ∇ E ′ and F Et , F E ′ t , µ Et , µ E ′ t be the corresponding curvatures and momentums.Let A E α = ∇ E α − ∇ E α , A E ′ α = ∇ E ′ α − ∇ E ′ α . We have φ − αβ ( − u − ( F E α t + µ E α v + uK,t ) − u − π ∗ B α − πiθ α ) φ αβ = − u − ( F E β t + µ E β v + uK,t ) − u − π ∗ B β − πiθ β and φ − αβ ( − u − A E α ) φ αβ = − u − A E β . Similarly equalities hold for E ′ .Therefore we haveexp( − u − π ∗ B α − πiθ α ) · Z Tr (cid:16) − u − A E α exp( − u − ( F E α t + µ E α v + uK,t )) + u − A E ′ α exp( − u − ( F E ′ α t + µ E ′ α v + uK,t )) (cid:17) dt (3.16)can be glued together as a global differential form in Ω ∗ ( Sξ )[[ u, u − ]]. Denote this form by cs ( ∇ E , ∇ E ′ ; ∇ E , ∇ E ′ ). Since ι v A E α = 0 , L v A E α = 0, similar to proof of (i), we have ι v cs ( ∇ E , ∇ E ′ ; ∇ E , ∇ E ′ ) = 0 , L v cs ( ∇ E , ∇ E ′ ; ∇ E , ∇ E ′ ) = − cs ( ∇ E , ∇ E ′ ; ∇ E , ∇ E ′ ) nd therefore cs ( ∇ E , ∇ E ′ ; ∇ E , ∇ E ′ ) ∈ e Ω ∗ ( Sξ )[[ u, u − ]] . Moreover, by the standard Chern-Simons transgression, we have( d − ι v − uι K ) Z Tr (cid:16) − u − A E α exp( − u − ( F E α t + µ E α v + uK,t )) + u − A E ′ α exp( − u − ( F E ′ α t + µ E ′ α v + uK,t )) (cid:17) dt = Tr (cid:16) exp( − u − ( F E α + µ E α v + uK, )) − exp( − u − ( F E ′ α + µ E ′ α v + uK, )) (cid:17) − Tr (cid:16) exp( − u − ( F E α + µ E α v + uK, )) − exp( − u − ( F E ′ α + µ E ′ α v + uK, )) (cid:17) . (3.17)Then similar to (3.15), we see that( d − ι v − uι K + Θ + u − π ∗ H ) cs ( ∇ E , ∇ E ′ ; ∇ E , ∇ E ′ ) = ch ∇ ξ : G ( ∇ E , ∇ E ′ ) − ch ∇ ξ : G ( ∇ E , ∇ E ′ ) . (cid:3) This theorem shows that ch ∇ ξ : G ( ∇ E , ∇ E ′ ) is ( d − ι v − uι K + Θ + u − π ∗ H )-closed in e Ω ∗ ( Sξ ) T [[ u, u − ]]. Theorem 2.3 then tells us that f − (cid:0) ch ∇ ξ : G ( ∇ E , ∇ E ′ ) (cid:1) is ( ∇ ξ − uι K + u − H )-closed in Ω ∗ ( M, ξ ) T [[ u, u − ]].We call CS ( ∇ E , ∇ E ′ ; ∇ E , ∇ E ′ ) := f − (cid:16) cs ( ∇ E , ∇ E ′ ; ∇ E , ∇ E ′ ) (cid:17) ∈ Ω ∗ ( M, ξ )[[ u, u − ]]the exotic twisted equivariant Chern-Simons transgression term . By (3.14) andTheorem 2.3 (formula (2.22)), one has(3.18) Ch ∇ ξ : G ( ∇ E , ∇ E ′ ) − Ch ∇ ξ : G ( ∇ E , ∇ E ′ ) = ( ∇ ξ − uι K + u − H ) CS ( ∇ E , ∇ E ′ ; ∇ E , ∇ E ′ ) . We therefore can define the exotic twisted equivariant Chern character : Ch ∇ ξ : G : K T ( M, ∇ ξ : G ) → h ∗ T ( M, ∇ ξ : H ) ,Ch ∇ ξ : G ( E, E ′ ) := h f − (cid:16) ch ∇ ξ : G ( ∇ E , ∇ E ′ ) (cid:17)i . References [1] Heat kernels and Dirac operators, by N. Berline, E. Getzler and M. Vergne, Grundlehren Math. Wiss.,vol. 298, Springer-Verlag, New York, 1992. 2[2] J-M. Bismut, Index theorem and equivariant cohomology on the loop space. Comm. Math. Phys. (1985), no. 2, 213-237. 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