fiber integration in K-theory


Index theory

Integration theory



The special case of fiber integration in generalized cohomology/twisted Umkehr maps for KU-cohomology.


There are various different models for describing and constructing fiber integration in K-theory.

  1. In terms of bundles of Fredholm operators

  2. In operator KK-theory

In terms of bundles of Fredholm operators

We discuss here fiber integration in the model of twisted K-theory by bundles of spaces of Fredholm operators. Related literature includes (Carey-Wang 05).

  1. Along a fibration of closed spin^c manifolds

Along a fibration of closed Spin cSpin^c-manifolds

Let f:YXf\colon Y \longrightarrow X be a fiber bundle of compact smooth manifolds carrying fiberwise a spin^c structure.

For xXx\in X write

Here D xD_x depends smoothly on xx while D˜ x\tilde D_x still depends continuously on xx. Equip CL(Y) yCL(Y)_y with the Cl nCl_n-action given on elements v nCl nv \in \mathbb{R}^n \hookrightarrow Cl_n joint right Clifford product by vv and left Clifford product by the volume element

vol()v. vol \cdot (-) \cdot v \,.

This is such that D xD_x graded-commutes with this Cl nCl_n-action. Hence the assignment xD˜ xx \mapsto \tilde D_x defines a map

D˜ ():XFred (n) \tilde D_{(-)} \colon X \longrightarrow Fred^{(n)}

from the base to the space Fredholm operators graded-commuting? with Cl nCl_n, as defined here at twisted K-theory.

More generally, let then VYV \to Y be a vector bundle representing a class in K 0(Y)K^0(Y). With a choice of connection this twists the above constrction to yield VV-twisted Dirac operator D˜ x V\tilde D^V_x and hence a map

D˜ () V:XFred (n). \tilde D_{(-)}^V \colon X \longrightarrow Fred^{(n)} \,.

This represents the push-forward class in K dim(Y x)(X)K^{dim(Y_x)}(X), and this construction gives a map

f:K 0(Y)K dim(Y)dim()X(X). \int f \colon K^0(Y) \longrightarrow K^{dim(Y)-dim()X}(X) \,.

For dim(Y x)dim(Y_x) even and hence ignoring the compatibility with the Cl nCl_n-action, this is discussed in (Carey-Wang 05).

In operator KK-theory

We discuss fiber integration /push-forward/Umkehr maps/Gysin maps in operator K-theory, hence in KK-theory (Connes-Skandalis 85, BMRS 07, section 3).

The following discusses KK-pushforward

  1. Along an embedding

  2. Along a submersion

  3. Along a fibration of closed spin^c manifolds

  4. Along a general K-oriented map

  5. In twisted K-theory

The construction goes back to (Connes 82), where it is given over smooth manifolds. Then (Connes-Skandalis 84, Hilsum-Skandalis 87) generalize this to maps between foliations by KK-elements betwen the groupoid convolution algebras of the coresponding holonomy groupoids and (Rouse-Wang 10) further generalize to the case where a circle 2-bundle twist is present over these foliations. A purely algebraic generalization to (K-oriented) maps between otherwise arbitrary noncommutative spaces/C*-algebras is in (BMRS 07).

Along an embedding

(Connes-Skandalis 84, above prop. 2.8)

Let h:XYh \colon X \hookrightarrow Y be an embedding of compact smooth manifolds.

The push-forward constructed from this is supposed to be an element in KK-theory

h!:KK d(C(X),C(Y)) h! \colon KK_d(C(X), C(Y))

in terms of which the push-forward on operator K-theory is induced by postcomposition:

h !:K (X)KK (,X)h!()KK +d(,Y)KK +d(Y), h_! \;\colon\; K^\bullet(X) \simeq KK_\bullet(\mathbb{C}, X) \stackrel{h!\circ (-)}{\to} KK_{\bullet+d}(\mathbb{C},Y) \simeq KK^{\bullet+d}(Y) \,,

where d=dim(X)dim(Y)d = dim(X) - dim(Y).

Now, if we could “thicken” XX a bit, namely to a tubular neighbourhood

h:XUjY h \;\colon\; X \hookrightarrow U \stackrel{j}{\hookrightarrow} Y

of h(X)h(X) in YY without changing the K-theory of XX, then the element in question will just be the KK-element

j!KK(C 0(U),C(Y)) j! \in KK(C_0(U), C(Y))

induced directly from the C*-algebra homomorphism C 0(U)C(Y)C_0(U) \to C(Y) from the algebra of functions vanishing at infinity of UU to functions on YY, given by extending these functions by 0 to functions on YY. Or rather, it will be that element composed with the assumed KK-equivalence

ψ:C(X) KKC 0(U). \psi \colon C(X) \stackrel{\simeq_{KK}}{\to} C_0(U) \,.

The bulk of the technical work in constructing the push-forward is in constructing this equivalence. (BMRS 07, example 3.3)

In order for it to exist at all, assume that the normal bundle

N YXh *(TY)/TX N_Y X \coloneqq h^\ast(T Y)/ T X

has a spin^c structure. Write S(N YX)S(N_Y X) for the associated spinor bundle.

Then there is an invertible element in KK-theory

ι X!KK n(C(X),C 0(N YX)) \iota^X! \in KK_n(C(X), C_0(N_Y X))

hence a KK-equivalence ι X!:C(X)C 0(N YX)\iota^X! \colon C(X) \stackrel{\simeq}{\to} C_0(N_Y X), where C 0()C_0(-) denotes the algebra of functions vanishing at infinity.

This is defined as follows. Consider the pullback π n *S(N YX)N YX\pi_n^\ast S(N_Y X) \to N_Y X of this spinor to the normal bundle itself along the projection π N:N YXX\pi_N \colon N_Y X \to X. Then…

Moreover, a choice of a Riemannian metric on XX allows to find a diffeomorphism between the tubular neighbourhood U h(X)U_{h(X)} of h(X)h(X) and a neighbourhood of the zero-section of of the normal bundle

Φ:U h(X)N YX. \Phi \colon U_{h(X)} \hookrightarrow N_Y X \,.

This induces a KK-equivalence

[Φ]:C 0(N YX) KKC 0(U). [\Phi] \colon C_0(N_Y X) \stackrel{\simeq_{KK}}{\to} C_0(U) \,.

Therefore the push-forward in operator K-theory along f:XYf \colon X \hookrightarrow Y is given by postcomposing in KK-theory with

h!:C(X) KKi X!C 0(N YX) KKΦC 0(U)j!C(Y). h! \colon C(X) \underoverset{\simeq_{KK}}{i^X!}{\to} C_0(N_Y X) \underoverset{\simeq_{KK}}{\Phi}{\to} C_0(U) \stackrel{j!}{\to} C(Y) \,.

Along a proper submersion

(Connes-Skandalis 84, above prop. 2.9)

For π:XZ\pi \colon X \to Z a K-oriented proper submersion of compact smooth manifolds, the push-forward map along it is reduced to the above case of an embedding by

  1. using that by the Whitney embedding theorem every compact XX may be embedded into some 2q\mathbb{R}^{2q} such as to yield an embedding

    h:XZ× 2q h \colon X \to Z \times \mathbb{R}^{2 q}
  2. using that there is a KK-equivalence

    ι Z!:C(Z) KKC 0(Z× 2q). \iota^Z! \colon C(Z) \stackrel{\simeq_{KK}}{\to} C_0(Z \times \mathbb{R}^{2q}) \,.

The resulting push-forward is then given by postcomposition in KK-theory with

π!:C(X)h!C 0(Z× 2q) KK(ι Z!) 1C(Z). \pi! \colon C(X) \stackrel{h!}{\to} C_0(Z \times \mathbb{R}^{2}q) \underoverset{\simeq_{KK}}{(\iota^Z!)^{-1}}{\to} C(Z) \,.

(BMRS 07, example 3.4)

Along a smooth fibration of closed Spin cSpin^c-manifolds

Specifically, for π:XZ\pi \colon X \to Z a smooth fibration over a closed smooth manifold whose fibers X/ZX/Z are

the push-forward element π!KK(C 0(X),C 0(Z))\pi! \in KK(C_0(X), C_0(Z)) is given by the Fredholm-Hilbert module obatined from the fiberwise spin^c Dirac operator acting on the fiberwise spinors. (Connes-Skandalis 84, proof of lemma 4.7, BMRS 07, example 3.9).

In detail, write

T(X/Z)TX T(X/Z) \hookrightarrow T X

for the sub-bundle of the total tangent bundle on the vertical vectors and choose a Riemannian metric g X/Zg^{X/Z} on this bundle (hence a collection of Riemannian metric on the fibers X/ZX/Z smoothly varying along ZZ). Write S X/ZS_{X/Z} for the corresponding spinor bundle.

A choice of horizontal complenet TXT HXT(X/Z)T X \simeq T^H X \oplus T(X/Z) induces an affine connection X/Z\nabla^{X/Z}. This combined with the symbol map/Clifford multiplication of T *(X/Z)T^\ast (X/Z) on S X/ZS_{X/Z} induces a fiberwise spin^c Dirac operator, acting in each fiber on the Hilbert space L 2(X/Z,S X/Z)L^2(X/Z, S_{X/Z}).

This yields a Fredholm-Hilbert bimodule

(D X/Z,L 2(X/Z,S X/Z)) (D_{X/Z}, L^2(X/Z, S_{X/Z}))

which defines an element in KK-theory

π!KK(C 0(X),C 0(Z)). \pi ! \in KK(C_0(X), C_0(Z)) \,.

Postcompositon with this is the push-forward map in K/KK-theory, equivalently the index map of the collection of Dirac operators.

Along a general K-oriented map

(Connes-Skandalis 84, def. 2.1)

Now for f:XYf \colon X \to Y an arbitray K-oriented smooth proper map, we may reduce push-forward along it to the above two cases by factoring it through its graph map, followed by projection to YY:

f:Xgraph(f)X×Yp YY. f \;\colon\; X \stackrel{graph(f)}{\to} X \times Y \stackrel{p_Y}{\to} Y \,.

Hence push-forward along such a general map is postcomposition in KK-theory with

f!p Y!graph(f)!. f! \coloneqq p_Y !\circ graph(f)! \,.

(BMRS 07, example 3.5)

In twisted K-theory

We discuss push forward in K-theory more generally by Poincaré duality C*-algebras hence dual objects in KK-theory.

Let i:QXi \colon Q \to X be a map of compact manifolds and let χ:XB 2U(1)\chi \colon X \to B^2 U(1) modulate a circle 2-bundle regarded as a twist for K-theory. Then forming twisted groupoid convolution algebras yields a KK-theory morphism of the form

C i *χ(Q)i *C χ(X), C_{i^\ast \chi}(Q) \stackrel{i^\ast}{\longleftarrow} C_{\chi}(X) \,,

with notation as in this definition. By this proposition the dual morphism is of the form

C W 3(τ Q)i *χ(Q)i !C W 3(τ X)χ(X). C_{\frac{W_3(\tau_Q)}{i^\ast \chi}}(Q) \stackrel{i_!}{\longrightarrow} C_{\frac{W_3(\tau_X)}{\chi}}(X) \,.

If we assume that XX has a spin^c structure then this is

C W 3(τ Q)i *χ(Q)i !C 1χ(X). C_{\frac{W_3(\tau_Q)}{i^\ast \chi}}(Q) \stackrel{i_!}{\longrightarrow} C_{\frac{1}{\chi}}(X) \,.

Postcomposition with this map in KK-theory now yields a map from the W 3(τ Q)i *χ\frac{W_3(\tau_Q)}{i^\ast \chi}-twisted K-theory of QQ to the χ 1\chi^{-1}-twisted K-theory of XX:

i !:K +W 3(τ Q)i *χ(Q)K χ. i_! \colon K_{\bullet + W_3(\tau_Q) - i^\ast \chi}(Q) \to K_{\bullet -\chi} \,.

If we here think of i:QXi \colon Q \hookrightarrow X as being the inclusion of a D-brane worldvolume, then χ\chi would be the class of the background B-field and an element

[ξ]K +W 3(τ Q)i *χ(Q) [\xi] \in K_{\bullet + W_3(\tau_Q) - i^\ast \chi}(Q)

is called (the K-class of) a Chan-Paton gauge field on the D-brane satisfying the Freed-Witten-Kapustin anomaly cancellation mechanism. (The orginal Freed-Witten anomaly cancellation assumes ξ\xi given by a twisted line bundle in which case it exhibits a twisted spin^c structure on QQ.) Finally its push-forward

[i !ξ]K χ(X) [i_! \xi] \in K_{\bullet- \chi}(X)

is called the corresponding D-brane charge.


Discussion in KK-theory

Discussion for integration twisted K-theory over manifolds:

Discussion for integration of twisted K-theory along representable morphisms of local quotient stacks:

Review in the context of geometric quantization with KU-coefficients is in

Last revised on November 24, 2014 at 19:42:39. See the history of this page for a list of all contributions to it.