aa r X i v : . [ m a t h . K T ] J a n Chern-Simons invariants in
K K theory
Omar MohsenJanuary 16, 2018
Abstract
For a unitary representation φ of the fundamental group of a com-pact smooth manifold, Atiyah, Patodi, Singer defined the so called α -invariant of φ using the Chern-Simons invariants. In this articleusing traces on C ∗ -algebras, we give an intrinsically(i.e without usingChern character) define an element in KK with real coefficients theorywhose pullback by the representation φ is the α -invariant. Introduction
Chern and Simons[6] defined invariants associated to flat vector bun-dles over a compact smooth manifold. Their invariants were originallydefined as differential forms and hence as elements in the De Rhamcohomology.Atiyah, Patodi, and Singer[3] in their celebrated article highlightedthe connection between the Chern-Simons invariants and index theory.They transported the Chern-Simons invariants to K theory. To thisend they defined the K theory with coefficients in C and C / Z , andthen using Atiyah-Hirzebruch theorem on the bijectivity of the Cherncharacter they transported the Chern-Simons invariants to K theory.The resulting element is the so-called α -invariant of a flat vector bun-dle or equivalently of the holonomy representation of the fundamentalgroup of a compact smooth manifold.The α -invariant lives in the K theory of the underlying manifoldwith coefficients in C / Z . If V is a flat vector bundle associated to arepresentation of the fundamental group of a manifold M , then theAtiyah-Hirzebruch theorem implies that the element [ V ] − [ C dim( V ) ] in K ( M ) is torsion. A property of the α -invariant is that its boundaryunder Bockstein homomorphism is equal to [ V ] − [ C dim( V ) ] .A closely related invariant is the relative Chern-Simons invariantsand relative α invariants which are defined respectively in the De Rham ohomology with coefficients in C and the K theory with coefficientsin C . These invariants are defined for flat vector bundles which areequipped with trivialisation. The relation between the two is thatwhen one takes the relative invariant modulo Z then the choice of atrivialisation disappears, and the relative invariant becomes the usualinvariant.When the holonomy representation is unitary, all the different in-variants stated above become either in R or R / Z . In this article, werestrict ourselves to the case of unitary representation.It was suggested in APS that the α -invariants should have an in-trinsic definition in terms on K theory that uses the theory of VonNeuman algebras of type II . This motivated research in this directionin particular [1, 4, 8, 14], etc ...We continue this line of research by constructing a universal clas-sifying element intrinsically in the KK -theory of the classifying spaceof flat vector bundles. The KK theory is instrumental to our under-standing of index theory. To this end an element defined in KK theorywithout adhering to the bijectivity of the Chern-character might shedsome light on the interaction between Chern-Simons invariants and KK theory.The appropriate KK theory we use is the equivariant KK theoryfor groupoids constructed by Le Gall in the general case followingKasparov’s construction in the case of group action[15, 16].As the invariants of Chern-Simons are usually real invariants, wefollow Antonini, Azzali, Skandalis definition of KK theory with realcoefficients[1].In particular we construct an element in KK U n ⋊ U δn , R ( C ( U n ) , C ( U n )) which when pulled back with the classifying map (seen as a generalisedhomomorphism in the sense of Hilsum-Skandalis[10]) of a flat vectorbundle f : M → U n ⋊ U δn gives the α -invariant.In our construction we assume that we are given a trivialisation ofthe bundle. We explain in remark 4.4 that we can avoid this assump-tion but with taking coefficients in R / Z instead of R . The Kasparov product of the α -invariant with the class of a Diracoperator [ D ] ∈ KK ( M, C ) gives the η -invariant as proved in APS.The product with the class of a leafwise Dirac operator D over a folia-tion F equipped with a transverse measure [ D ] ∈ KK ( M, N ( M, F )) is given by leafwise eta invariants as proved by Peric[18].The organisation of the article is as follows;1. In section 1, we recall Chern-Weil theory for tracial C ∗ -algebras.This section follows closely the presentation in the article byFomenko and Mishchenko[17]. . In section 2, we recall the definition of KK theory with realcoefficients, and the basic results that we need from KK theory.This section follows closely the presentation in the article by [1].3. In section 3, the definition of the Chern Simons invariants in KK theory is given.4. In section 4, we construct a more primitive element in equivariant KK theory, that is done for any compact group. Acknowledgement
I wish to thank my PhD advisor, G. Skandalis for his precious support,suggestions and remarks that were essential to this article.
In this section, we recall Chern-Weil theory for tracial C ∗ -algebras.This section follows closely the presentation in the article by Fomenkoand Mishchenko[17].Let M be a smooth manifold, A a unital C ∗ -algebra, τ : A → C afinite trace τ such that τ (1) = 1 . The trace τ extends to M n ( A ) , bythe formula τ ( M ) = P τ ( M i,i ) . The trace extends as well to End( P ) for an A -projective module P by using a complementary module Q ,as follows End( P ) ⊆ End( P ⊕ Q ) ≃ M n ( A ) τ −→ C . It is easy to verify that this extension doesn’t depend on the choice of Q . An A -smooth vector bundle V , will mean a locally finitely gener-ated projective right C ∞ ( M, A ) -module. Usually we will write V tomean the total space of the vector bundle even though the vector bun-dle is only defined by its section which are denoted by Γ( V ) . We willdenote by Ω · ( M, A ) the graded algebra of differential forms on M withvalues in A , and by Ω · ( M, V ) := Γ( V ) ˆ ⊗ C ∞ ( M,A ) Ω · ( M, A ) the gradedright C ∞ ( M, A ) module of differential forms with values in V. Let
V, W be A -vector bundles, then V ⊕ W is defined as Γ( V ) ⊕ Γ( W ) . The vector bundle
Hom A ( V, W ) is defined by the following mod-ule Hom C ∞ ( M,A ) (Γ( V ) , Γ( W )) . The vector bundle Hom A ( V, W ) is lo-cally equal to C ∞ ( U, hom( P, Q )) , where P , and Q are the local fibersof V and W , respectively. This doesn’t exactly fall under the defini-tion given above of a vector bundle but one finds that it will not causeissues in what follows. f A, B are C ∗ -algebras, and V (respectively W ) is A (and respec-tively B ) vector bundle, then the maximal tensor product V ⊗ max W ,and the minimal tensor product V ⊗ W are defined as the vector bun-dles which are locally equal to C ∞ ( U, P ⊗ max Q ) and respectively C ∞ ( U, P ⊗ min Q ) , where U is an open set, P ( Q ) is a projectivefinitely generated A ( B ) module such that V | U = C ∞ ( U, P ) , and W | U = C ∞ ( U, Q ) . Definition 1.1. An A connection on V is a C -linear map ∇ : Γ( V ) → Ω ( M, V ) which satisfies Leibniz rule ∇ ( sf ) = ∇ ( s ) f + sdf, ∀ s ∈ Γ( V ) , f ∈ C ∞ ( M, V ) . Like in the classical theory, a connection ∇ extends to a C -linearmap ∇ : Ω · ( M, V ) → Ω · ( M, V ) satisfying Leibniz rule. The map ∇ is given by a the multiplication by a form in Ω ( M, End A ( V )) calledthe curvature of ∇ .Let φ : Γ → GL ( P ) be a representation, where P is a projective A -module, then one can define a flat A -vector bundle by ˜ M × Γ P . Definition 1.2.
A flat structure on an A -vector bundle on M is thechoice of an A -vector bundle isomorphism to ˜ M × Γ P for some repre-sentation φ : Γ → GL ( P ) , and for some finitely generated projective A -module P . Furthermore if P is a C ∗ -module, then we say that V isunitary flat, if φ (Γ) ⊆ U ( P ) .A flat vector bundle is a vector bundle equipped with a flat struc-ture. Proposition 1.3.
A flat structure on an A -vector bundle can be equiv-alently given by the choice of an A -connection ∇ such that ∇ = 0 ;furthermore the bundle is unitary flat if and only if the connection canbe chosen to be unitary with respect to some C ∗ -metric on the vec-tor bundle. The representation associated to ∇ is called the holonomyrepresentation of ∇ . Definition 1.4.
A trivial A -connection is a flat A -connection ∇ whoseholonomy is trivial. Remark 1.5.
It is clear from the definition that giving a trivial con-nection on a bundle is the same as giving a trivialization of the bundle.We will use this remark through this text without mention
Let V be an A -vector bundle, and ∇ an A -connection on V , thenthe Chern character is defined by the formula Ch τ ( V, ∇ ) := exp( 12 πi ∇ ) = ∞ X k =0 k !(2 πi ) k τ ( ∇ k ) ∈ Ω even ( M ) . roposition 1.6. (see for example [7])The Chern character is a closeddifferential form whose class is independent of the choice of the con-nection ∇ . Independence from the choice of the connection follows from thefact that if ∇ t is a C -path of A -connections on V , then the so-calledChern-Simons form CS τ ( V, ∇ t ) := Z ∞ X k =0 k !(2 πi ) k τ ( ˙ ∇ t ∧ ∇ kt ) ∈ Ω odd ( M ) (1)satisfies the following identity d CS τ ( V, ∇ t ) = Ch τ ( V, ∇ ) − Ch τ ( V, ∇ ) . (2) Proposition 1.7. [19]The forms CS τ ( V, ∇ t ) ∈ Ω odd ( M ) /d Ω even ( M ) satisfy eq. (2) , and are independent of the path ∇ t connecting ∇ and ∇ . Therefore the notation CS τ ( V, ∇ , ∇ ) ∈ Ω odd ( M ) /d Ω even ( M ) isjustified.Proof. Let V × [0 , be the vector bundle on M × [0 , whose sectionsare Γ( V ) ˆ ⊗ C C ∞ ([0 , . A path of connections ∇ t defines a connectionon ∇ on V × [0 , given by ∇ | M ×{ t } = ∇ t , ∇ ddt s = ddt s, ∀ s ∈ Γ( V × [0 , . One verifies directly that i ∗ t Ch τ ( ∇ ) = Ch τ ( ∇ t ) for all t ∈ [0 , , where i t : M → M × [0 , is the map x → ( x, t ) . Hence by Poincare formula i ∗ Ch τ ( ∇ ) − i ∗ ( ∇ ) = Z L ddt ch τ ( ∇ ) = d Z i ddt Ch τ ( ∇ ) . This gives eq. (2). Independence from the choice the path ∇ t uses thesame argument applied to M × [0 , × [0 , . Given two connections ∇ , ∇ , then there is a preferred path t ∇ +(1 − t ) ∇ . In this case formula becomes CS τ ( ∇ , ∇ ) = ∞ X k =0 ( − k k !(2 πi ) k +1 (2 k + 1)! τ (( ∇ − ∇ ) k +1 ) (3)The following proposition is stated in [19]. Proposition 1.8.
Let
V, W be A -vector bundles and ∇ V , ∇ V , ∇ V , ∇ W , ∇ W be A -connections on the indicated bundles then we have . Ch τ ( V ⊕ W, ∇ V ⊕ ∇ W ) = Ch τ ( V, ∇ V ) + Ch τ ( W, ∇ W ) Ch τ ( V ⊗ W, ∇ V ⊗ ∇ W ) = Ch τ ( V, ∇ V ) ∧ Ch τ ( W, ∇ W ) CS τ ( ∇ V , ∇ V ) + CS τ ( ∇ V , ∇ V ) = CS τ ( ∇ V , ∇ V ) CS τ ( ∇ V ⊕ ∇ W , ∇ V ⊕ ∇ W ) = CS τ ( ∇ V , ∇ V ) + CS τ ( ∇ W , ∇ W ) CS τ ( ∇ V ⊗ ∇ W , ∇ V ⊗ ∇ W ) = Ch τ ( ∇ V ) CS τ ( ∇ W , ∇ W )+ Ch τ ( ∇ W ) CS τ ( ∇ V , ∇ V ) In identity (5) , the product Ch τ ( ∇ V ) CS τ ( ∇ W , ∇ W ) is well definedmodule exact forms because Ch τ ( ∇ V ) is closed. The same holds for Ch τ ( ∇ W ) CS τ ( ∇ V , ∇ V ) . The odd Chern-character (cf. [9]) is defined as follows. Let u be an A -linear automorphism of an A -vector bundle V , and ∇ a connectionon V , then the odd Chern character is defined by the formula Ch τ ( u, ∇ ) = ∞ X k =0 ( − k k !(2 k + 1)!(2 πi ) k +1 τ (( u − d ∇ u ) k +1 ) . (4)The -form u − d ∇ u ∈ Ω ( M, End A ( V )) is given by the C ∞ ( M, A ) -linear map Ω ( M, V ) → Ω ( M, V ) s → u − ∇ ( us ) − ∇ s Let u t be a contnuous path of A -linear automorphisms on V , thenthe (even)-Chern-Simons form is defined by the formula CS τ ( u t ) = ∞ X k =0 ( − k k !(2 k )!(2 πi ) k +1 τ ( u − t ˙ u t ( u − t ∇ u t ) k ) . It satisfies the identitie d CS τ ( u t ) = Ch τ ( u ) − Ch τ ( u ) . Proposition 1.9.
1. The form Ch τ ( u ) is closed, and its class de-pends only on the homotopy class of u .2. The form CS τ ( u t ) ∈ Ω even ( M ) /d Ω odd ( M ) don’t depend on thepath u t from u to u . The proof is similar to the proof of proposition 1.7. roposition 1.10. Let ∇ be an A -connection on an A -vector bundle V and T : V → V a A -linear automorphism, then we have CS τ ( T − ∇ T, ∇ ) = Ch τ ( T, ∇ ) (5) Proof.
This follows from eq. (3) and eq. (4), and the definition of T − d ∇ T given by T − ∇ ( T s ) − ∇ ( s ) = ( T − d ∇ T ) s for every section s . Remark 1.11.
The normalisation constants in the definition of Cherncharacter and hence the Chern-Simons forms are not uniform acrossthe literature. Some authors don’t divide by πi . Theorem 1.12 (Atiyah-Hirzebruch) . [12]Let M be a compact smoothmanifold, then the Chern character Ch : K i ( M ) ⊗ C → ⊕ n H n + i ( M, C ) is a ring isomorphism for i ∈ { , } . Let ∇ , ∇ be flat A -connections on an A -vector bundle V , thenit follows immediately from the definition that Ch τ ( ∇ ) = Ch τ ( ∇ ) =dim τ ( V ) . It follows from eq. (2) that CS τ ( ∇ , ∇ ) gives a cohomologyclass in H odd ( M, C ) . Definition 1.13.
The α -invariant of ( V, ∇ , ∇ ) is defined as α V, ∇ , ∇ = Ch − (CS τ ( ∇ , ∇ )) ∈ K ( M, C ) . If a vector bundle is equipped with a flat connection ∇ and a trivialconnection ∇ , then if it is clear from the context we will simply write α V, ∇ instead of α V, ∇ , ∇ From now on we restrict our selves to the case of unitary represen-tations. In this case the imaginary part of the α invariant is zero ascan be immediately seen from eq. (3). In general we have the following Proposition 1.14.
Let V be an A -flat vector bundle with fiber P ,then if there exists a non degenerate sesquilinear form Q on V suchthat φ (Γ) ⊆ U ( Q ) , then the imaginary part of α . Here U ( Q ) denotesthe group of isometries of Q . We follow the definition of nondegenerate sesquilinear forms givenby Skandalis and Hilsum[11]. A nondegenerate sesquilinear form is a C -bilinear form Q : P × P → A such that Q ( p, q ) = Q ( q, p ) ∗ , Q ( p, qa ) = Q ( p, q ) a , and that there exists a bijective linear operator T : P → P such that Q ( · , T · ) is a C ∗ -metric. roof. Let T an operator, and g a C ∗ metric such that Q ( · , · ) = g ( · , T · ) and T = 1 . (This is always possible see [11]). Let s, s ′ ∈ Γ( V ) be twosections and X ∈ Γ( T M ) a vector field Q ( ∇ X s, s ′ ) = g ( ∇ X , T s ′ ) = X · g ( s, T s ′ ) − g ( s, ∇ ∗ X s ′ )= X · Q ( s, s ′ ) − Q ( s, T ∇ ∗ X T s ′ ) It follows that ∇ = T ∇ ∗ T . It follows that the form ω = ∇ − ∇ ∗ anticommutes with T . The imaginary part of Chern-Simons forms iszero because we have τ ( ω k ) = τ ( ω k T ) = τ (( − k T ω k T ) = ( − k τ ( ω k ) . K K theory with real coefficients
Let G be a Lie groupoid. The KK ∗ G ( A, B ) group is usually definedonly for separable C ∗ -algebras only. We follow the remarks given bySkandalis[20] in order to define KK ∗ G ( A, B ) for arbitrary C ∗ -algebras A and B by KK ∗ G ( A, B ) := lim ←− D KK ∗ G ( D, B ) where the projective limit and injective limit are over all separable C ∗ -algebra with morphisms φ : C → B and ψ : B → D .When the groupoid G is not second countable (but G is alwaysassumed second countable) then KK ∗ G ( A, B ) := lim ←− H KK ∗ H ( A, B ) where the projective limit is over all second countable Lie subgroupoids. Definition 2.1.
Let C be the category whose objects are unital tracial C ∗ -algebras and whose morphisms are C ∗ -homomorphisms preservingthe trace. As assumed in this article, the traces are normalised. Definition 2.2. [2]Let G be a Lie groupoid, A and B be two G - C ∗ -algebras. Equivariant KK theory with real coefficients is defined by KK ∗ G, R ( A, B ) := lim −→ C ∈C KK ∗ G ( A, B ⊗ C ) . Here the groupoid G actson C trivially.This definition is justified by the Kunneth formula roposition 2.3. Let M be a compact smooth manifold, then KK ∗ C ( M ) , R ( C ( M ) , C ( M )) = KK ∗ R ( C , C ( M )) = K ∗ ( M, R ) . Proof.
Using the GNS construction, one sees that any traced C ∗ -algebra admits a trace preserving C ∗ -homomorphism into a Von-Neumannalgebra factor of type II . The theorem then follows from the Kunnethformula, and the fact that the natural trace map K ( N ) τ −→ R is anisomorphism, and that K ( N ) = 0 if N is a factor of type II . Theorem 2.4. (Künneth formula) (see for example [5]) Let A be aseparable C ∗ algebra in the bootstrap category and B any C ∗ -algebrathen the following sequences are exact → K ∗ ( A ) ⊗ K ∗ ( B ) → K ∗ ( A ⊗ B ) → Tor Z ( K ∗ ( A ) , K ∗ ( B )) → Where the first map is degree 0 and the second if of degree Theorems and propositions in [16] pass through the direct limit to KK G, R . In particular Kasparov product exists KK iG, R ( A, B ) × KK jG, R ( B, C ) → KK i + jG, R ( A, C ) . Functoriality and Morita equivalence remains true that is if f : G → G ′ is a generalised morphism of groupoids then f ∗ : KK ∗ G ′ , R ( A, B ) → KK ∗ G, R ( f ∗ A, f ∗ B ) is well defined. In particular if f is a Morita equivalence, then f ∗ isan isomorphism.If τ : A → C is a trace on a C ∗ -algebra, then τ defines naturallyan element in [ τ ] ∈ KK R ( A, C ) . Proposition 2.5.
Let A be a unital C ∗ -algebra, and Γ a countablediscrete group, and c : Γ → U ( A ) a group homomorphism, where U ( A ) is the group of unitaries of A . The Γ - C ∗ -algebra A with thetrivial Γ action is Γ -Morita equivalent to the Γ - C ∗ -algebra A with theinner action given by c .Proof. The module A with the action γ · a = c ( γ ) a is the Moritaequivalence. Generalised morphisms were introduced in [10] roposition 2.6. Let Γ be a countable discrete group, X a locallycompact space with a Γ right action, A , B two X ⋊ Γ - C ∗ -algebras, C a Γ - C ∗ -algebra, τ a Γ -invariant trace on C . There exists a morphismdefined in the proof KK ∗ X ⋊ Γ ( A, B ⊗ C ) → KK ∗ Γ , R ( A, B ) . (6) Proof.
Let i : C → C ⋊ Γ be the inclusion morphism. Notice that i is Γ -equivariant, when Γ acts on C ⋊ Γ by γ · z = γzγ − ∀ z ∈ C ⋊ Γ . Byproposition 2.5, the Γ - C ∗ -algebra C ⋊ Γ with the adjoint action is Γ -Morita equivalent to C ⋊ Γ with the trivial action. The composition of [ i ] with the Morita equivalence defines an element in KK ( C, C ⋊ Γ) .Composing the last element with the trace τ ⋊ Γ gives an element in z ∈ KK ∗ R Γ ( C, C ) . The map · ⊗ C τ B ( z ) : KK ∗ X ⋊ Γ ( A, B ⊗ C ) → KK ∗ , R X ⋊ Γ ( A, B ) , is the desired map.We will also need the Green-Julg theorem Theorem 2.7. [13] Let G be a proper groupoid acting on a C ∗ -algebra A , then we have a canonical isomorphism KK ∗ G ( C ( G ) , A ) ∼ −→ KK ∗ ( C , A ⋊ G ) . K K theory definition of α -invariants Let V be a C -vector bundle , ∇ a C -unitary flat connection, and ∇ triv ,a trivial connection. Proposition 3.1. [1]There exists a unital C ∗ -algebra A equipped witha trace τ such that τ (1) = 1 , and a unitary flat A -bundle ( W, ∇ W ) whose fiber is equal to A , and an isomorphism T : V ⊗ W → V ⊗ W such that T − ( ∇ triv ⊗ ∇ W ) T = ∇ V ⊗ ∇ W . (7) Proof.
We can take A = C ( U n ) ⋊ Γ , where Γ acts on U n on the rightby multiplication by φ ( γ ) , where φ : Γ → U n is the holonomy repre-sentation of ∇ . The algebra A is equipped with the trace τ ( f γ ) = δ e ( γ ) Z U n f dµ, where µ is the normalised Haar measure. We take ψ : Γ → A theinclusion map, W the associated unitary flat A -bundle, and ∇ W the ssociated flat connection. Let u ∈ M n ( C ( U n )) ⊆ M n ( A ) be theunitary defined as the ’inclusion function’ u : U n → M n ( C ) . Noticethat both V ⊗ W , and C n ⊗ W are flat with holonomy representation γ → φ ( γ ) γ ∈ M n ( A ) , and γ → γ ∈ M n ( A ) , respectively. The unitary u satisfies uφ ( γ ) γu − = γ. Therefore u defines a map T : V ⊗ W → C n ⊗ W such that T − ( d ⊗ ∇ W ) T = ∇ V ⊗ ∇ W , where d is the trivial connection on C n . Let T : C n → V be thetrivialisation given by ∇ triv . This means that d = T − ∇ triv T . Themap T = ( T ⊗ Id W ) ◦ T satisfies eq. (8). Proposition 3.2.
Let A be a tracial unital C ∗ -algebra, V an A -flatvector bundle coming whose holonomy representation is ψ : Γ → U ( A ) ,and ∇ triv a trivial connection on V . There exists a unital C ∗ -algebra C equipped with a trace τ such that τ (1) = 1 , and a unitary flat C -bundle ( W, ∇ W ) whose fiber is equal to C , and an isomorphism T : V ⊗ W → V ⊗ W such that T − ( ∇ triv ⊗ ∇ W ) T = ∇ V ⊗ ∇ W . (8) Proof.
The free product B = A ⋆ C C ( S ) can be described as theuniversal unital C ∗ -algebra that is generated by elements a ∈ A , and aunitary z , with relation coming from elements of A , and whose unit isthe same as the one from A . See [21] for more details on free product.Let u ∈ A be a unitary, then u − z is a unitary in B , hence byuniversality of B , there exists a unique C ∗ -homomorphism φ u : B → B such that φ ( a ) = a , and φ ( z ) = u − z . By uniqueness one has φ u ◦ φ v = φ uv . Since φ = Id , it follows that φ u is an automorphismfor every u .Let τ A : A → C be a finite trace. The algebra A ⋆ C C ( S ) admitsa trace τ = τ A ⋆ R S (see section 1 [21] for more details). This trace isdefined as the unique trace satisfying following two properties1. If a ∈ A, f ∈ C ( S ) , then τ ( a ) = τ A ( a ) , and τ ( f ) = R f.
2. If a , . . . a k ∈ A , and f , . . . f k ∈ C ( S ) , such that τ A ( a i ) = R S f i = 0 ∀ i, then τ ( a f a f · · · a k f k ) = 0 . Lemma 3.3.
For every unitary u ∈ A , the automorphism φ u preservesthe trace. roof. We suppose first that τ A ( u ) = 0 . It follows that τ A ( u − ) = τ A ( u ∗ ) = τ A ( u ) = 0 . In this case, we verify that two properties of τ ,are also verified by τ ◦ φ u . In both properties, by Stone–Weierstrasstheorem, we can replace all continuous functions by z k for k ∈ Z .1. If a ∈ A , then τ ( φ u ( a )) = τ ( a ) = τ A ( a ) . If k ∈ Z , then τ ( φ u ( z k )) = τ (( u − z ) k ) = τ ( u − zu − z . . . u − z ) = 0 if k>0 τ ( z − uz − u . . . z − u ) = 0 if k<0 τ (1) = 1 if k=02. To verify the second property, we first prove that τ satisfies astronger hypothesis; let A ⊆ A ⋆ C ( S ) denote the subalgebraalgebra generated by z ∈ C ( S ) and A . The subalgebra B = C z ⊕ zA z is the algebra of words containing only positive powersof z and that start and end with a strictly positive power of z .Similarly let A = A ∗ , and B = B ∗ = C z − ⊕ z − A z − be thealgebra of words containing only negative powers of z and thatstart and end with a strictly negative power of z .We claim that the trace τ satisfies the following; if a , . . . a n ∈ A such that τ A ( a i ) = 0 for ≤ i ≤ n − and such that n ≥ ,and x i ∈ B ∪ B that alternatively belong to B or B , then τ ( a x a . . . x n a n ) = 0 .To see this it is enough to see that any element in B can bewritten as the sum of elements z k h z k h . . . h l z k l with h i ∈ A elements with trace and k i > . A similar statement holds for B . The trace τ ◦ φ u satisfies the same property because φ u ( B ) = u − B and φ u ( B ) = B u , and so the element φ u ( a x a . . . x n a n ) = b y b . . . b n a n with b = a or a u − according to whether x ∈ B or B , and b = a or ua u − according to whether x ∈ B or B , similarlyfor the other terms.The case of arbitrary u is reduced to the case where τ A ( u ) = 0 byembedding A inside A ⊕ A and applying the previous argument to ( u, − u ) . It follows from lemma 3.3, that if Γ → U ( A ) is a unitary representa-tion, then ( A ⋆ C ( S )) ⋊ Γ , admits a trace defined by τ ⋊ δ e ( P b i γ i ) = τ ( b e ) , where b i ∈ A ⋆ C ( S ) . The proof of proposition 3.1 is then generalised to any representa-tion flat A -vector bundle V coming from a representation φ : Γ → ( A ) by replacing the algebra C ( U n ) ⋊ Γ with the algebra ( A ⋆ C C ( S )) ⋊ Γ and replacing the unitary u with the unitary z ∈ C ( S ) ⊆ A ⋆ C C ( S )) ⋊ Γ .The following is the key proposition of this article. Proposition 3.4.
The element [ T ] ⊗ A [ τ ] ∈ KK R ( C , C ( M )) = K ( M, R ) where T is given by proposition 3.2 and τ is the trace on A , is equal tothe α -invariant α ∇ , ∇ triv . In particular it is independent of the choiceof B and W .Proof. By propoisiton 1.10, we have Ch τ ([ T ]) =[CS τ ( T − ( ∇ triv ⊗ ∇ W ) T, ∇ triv ⊗ ∇ W ]=[CS τ ( ∇ V ⊗ ∇ W , ∇ triv ⊗ ∇ W )]=[CS τ ( ∇ V , ∇ triv )][Ch τ ( ∇ W )]= Ch( α V, ∇ ) τ (1) = Ch( α V, ∇ ) It follows that the class [ T ] in K ( M, R ) is equal to α ∇ , ∇ triv Remarks 3.5.
1. In general, it is impossible to find a commutativealgebra A satisfying proposition 3.1. Because if such an algebraexists, the Chern-Simons invariants become rational by the ra-tionality of the Chern-character on locally compact spaces whichdoesn’t hold in general.2. The proposition is in general false for non unitary flat connec-tions, because in the case were it holds, the imaginary part ofChern-Simons invariant is equal to . K K theory with realcoefficients
Definition 4.1.
Let G be a compact group. The group G δ is thegroup G with the discrete topology. The following morphism definedbelow is denoted Ψ G Ψ G : KK ∗ ( C , C ( G )) → KK ∗ G δ , R ( C , C ( G )) = KK ∗ G ⋊ G δ , R ( C ( G ) , C ( G )) . (9)The morphism is the successive composition of the following mor-phisms1. One writes ( G × G ) /G = G , using the map ( x, y ) → yx − . Here G acts on G × G by right diagonal action. So we have C ( G ) is orita equivalent to C ( G × G ) ⋊ G . Finally using theorem 2.7,we obtain a morphism KK ∗ ( C , C ( G )) → KK ∗ G ( C , C ( G ) ⊗ C ( G )) .
2. The forgetful map KK ∗ G ( C , C ( G × G )) → KK ∗ G δ ( C , C ( G × G )) .
3. One applies proposition 2.6, with C = C ( G ) , and τ the Haarmeasure to deduce the desired morphism. Remark 4.2.
The composition of the morphism (9) with the forget-ful morphism KK R , ∗ G δ ( C , C ( G )) → KK R , ∗ ( C , C ( G )) is the inclusionmorphism R . Let M be a compact manifold, then the groupoids M and ˜ M ⋊ π ( M ) are Morita equivalent. Hence a flat vector bundle V whoseholonomy representation is φ : π ( M ) → U δn defines a functor ofgroupoids ˜ M ⋊ π ( M ) → U n by sending ( x, γ ) to φ ( γ ) , and hencea generalised morphism f φ : M → U δn . One sees easily that giving a trivialisation of a ˜ M × φ C n is the samething as a map β : ˜ M → U n such that β ( xγ ) = β ( x ) φ ( γ ) for every x ∈ ˜ M and γ ∈ Γ .In particular if V is flat vector bundle equipped with a triviali-sation, then a groupoid morphism ˜ M ⋊ Γ → U n ⋊ U δn is given by ( x, γ ) → ( β ( x ) , φ (Γ) . ) Hence by Morita equivalence we get a gener-alised morphism f V : M → U n ⋊ U δ . Theorem 4.3.
Let G = U n , and [ Id ] ∈ K ( U n ) be classical iden-tity element. The image of [ Id ] by the morphism defined in 4.1 is aclassifying element for Chern-Simons invariants in KK -theory. Bythis we mean that if V is a unitary flat vector bundle equipped witha trivialization, then V defines a generalised morphism f : M → U n ⋊ U δn . The pull back of Ψ U n ([ Id ]) by f which is an element in KK C ( M ) , R ( C ( M ) , C ( M )) is equal to the α -invariant of V .Proof. We will follow the construction of Ψ U n ([ Id ]) . We will checkthat the final element obtained is the map T constructed in 3.1. Theresult then follows from proposition 3.4. The following enumerationfollows each successive composition starting with step to denote theelement [ Id ] ∈ K ( U n ) .0. The identity element [ Id ] ∈ K ( U n ) will be seen as a unitaryisomorphism from U n × C n → U n × C n sending ( x, v ) → ( x, xv ) . . The first morphism changes this element to become a a uni-tary isomorphism from U n × U n × C n → U n × U n × C n sending L (( x, y, u )) = ( x, y, yx − v ) . This will be regarded as the compo-sition of two isomorphisms L = L L L , L : U n × U n × C n → U n × U n × C n L ( x, y, v ) = ( x, y, x − v ) , L ( x, y, v ) = ( x, y, yv ) . Notice that the group U n acts trivially on C n and L is U n equiv-ariant. The group U n doesn’t act trivially on the C n appearing inthe codomain and domain of the maps L , and L respectively.It acts by z · ( x, y, v ) = ( xz − , yz − , zv ) . Both the maps L and L become equivariant for this action. For L we have L ( z · ( x, y, v )) = L ( xz − , yz − , v ) = ( xz − , yz − , zy − v )= z · L ( x, y, v ) For L , we have L ( z · ( x, y, v )) = L ( xz − , yz − , zv ) = ( xz − , yz − , yv )= z · L ( x, y, v )
2. This is the forgetful map, only changing the topology in the lastpicture of the group U n to U δn .3. One views the bundles U n × U n × C n as a bundle over the firstcopy of U n with coefficients in C ( U n ) . Applying proposition 2.6amounts to extending the coefficient algebra to C ( U n ) ⋊ U δn .4. We will use the notation of proposition 3.1. Let φ : Γ → U δn be arepresentation of Γ = π ( M ) , and P = ˜ M × Γ U n .The pull back of the element obtained in step 3 by φ , becomesthe vector bundle C n ⊗ W over P . The middle vector bundle instep 1, becomes V ⊗ W , L and L become respectively T ⊗ Id W ,and T . Remarks 4.4.
1. In this remark we outline how the α invariantsof flat vector bundle without choosing a trivialisation is definedin the picture given by theorem 4.3. Let φ : Γ → U n be theholonomy representation of a flat vector bundle, f φ : M → U δn theassociated generalised morphism. If we see the element Ψ U n ([ Id ]) as an element in KK U δn , R ( C , C ( U n )) , then its pullback by f φ isan element in KK R ( C , C ( ˜ M × φ U n )) = K ( ˜ M × φ U n , R ) . Thiselement is equal to the α -invariant defined without the choice f a trivialisation, and in fact this follows closely the originalconstruction by Chern and Simons where their invariants whereclosed differential forms on ˜ M × φ C n . To descend an elementin K theory with coefficients in R / Z , we follow the constructiongiven [3].2. In most of this paper compactness is not needed. Most notably,let φ : π ( M ) → U n is the holonomy representation of a flatvector bundle equipped with a trivialisation on a not necessar-ily compact manifold, and f : M → U n ⋊ U δn the correspond-ing generalised morphism , then f ∗ Φ U n ( Id n ) is an element in KK M, R ( C ( M ) , C ( M )) . This later group is isomorphic to K theory with real coefficients without compact support as proved in[15]. References [1] Paolo Antonini, Sara Azzali, and Georges Skandalis. Flat bundles,von Neumann algebras and K -theory with R / Z -coefficients. J. K-Theory , 13(2):275–303, 2014.[2] Paolo Antonini, Sara Azzali, and Georges Skandalis. Bivariant K -theory with R / Z -coefficients and rho classes of unitary repre-sentations. J. Funct. Anal. , 270(1):447–481, 2016.[3] M. F. Atiyah, V. K. Patodi, and I. M. Singer. Spectral asymmetryand Riemannian geometry. III.
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