nLab fiber integration in K-theory

Idea

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

Models

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 $Spin^c$-manifolds

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

For $x\in X$ write

• $Y_x$ for the fiber over $x$;

• $P_x \to Y_x$ for the fiberwise spin^c-principal bundle;

• $Cl(Y)_x \coloneqq P_x \underset{Spin^c}{\times} Cl_n$ for the fiberwise Clifford bundle;

• $S_x$ for the spin^c spin bundle on $Y_x$;

• $D_x\colon \Gamma(S_x)\to \Gamma(S_x)$ for the Spin^c Dirac operator on the space of sections of spinors;

• $\tilde D_x \coloneqq \frac{D_x}{\sqrt{1+ D_x^\ast D_x}} \colon L^2(S_x)\to L^2(S_x)$ for the bounded version;

Here $D_x$ depends smoothly on $x$ while $\tilde D_x$ still depends continuously on $x$. Equip $CL(Y)_y$ with the $Cl_n$-action given on elements $v \in \mathbb{R}^n \hookrightarrow Cl_n$ joint right Clifford product by $v$ and left Clifford product by the volume element

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

This is such that $D_x$ graded-commutes with this $Cl_n$-action. Hence the assignment $x \mapsto \tilde D_x$ defines a map

$$\tilde D_{(-)} \colon X \longrightarrow Fred^{(n)}$$

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

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

$$\tilde D_{(-)}^V \colon X \longrightarrow Fred^{(n)} \,.$$

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

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

For $dim(Y_x)$ even and hence ignoring the compatibility with the $Cl_n$-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 $h \colon X \hookrightarrow Y$ be an embedding of compact smooth manifolds.

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

$$h! \colon KK_d(C(X), C(Y))$$

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

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

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

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

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

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

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

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

$$\psi \colon C(X) \stackrel{\simeq_{KK}}{\to} C_0(U) \,.$$

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

$$N_Y X \coloneqq h^\ast(T Y)/ T X$$

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

Then there is an invertible element in KK-theory

$$\iota^X! \in KK_n(C(X), C_0(N_Y X))$$

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

This is defined as follows. Consider the pullback $\pi_n^\ast S(N_Y X) \to N_Y X$ of this spinor to the normal bundle itself along the projection $\pi_N \colon N_Y X \to X$. Then…

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

$$\Phi \colon U_{h(X)} \hookrightarrow N_Y X \,.$$

This induces a KK-equivalence

$$[\Phi] \colon C_0(N_Y X) \stackrel{\simeq_{KK}}{\to} C_0(U) \,.$$

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

$$h! \colon C(X) \underoverset{\simeq_{KK}}{i^X!}{\to} C_0(N_Y X) \underoverset{\simeq_{KK}}{\Phi}{\to} C_0(U) \stackrel{j!}{\to} C(Y) \,.$$

Along a proper submersion

(Connes-Skandalis 84, above prop. 2.9)

For $\pi \colon X \to Z$ 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 $X$ may be embedded into some $\mathbb{R}^{2q}$ such as to yield an embedding

   $$h \colon X \to Z \times \mathbb{R}^{2 q}$$

2. using that there is a KK-equivalence

   $$\iota^Z! \colon C(Z) \stackrel{\simeq_{KK}}{\to} C_0(Z \times \mathbb{R}^{2q}) \,.$$

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

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

(BMRS 07, example 3.4)

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

Specifically, for $\pi \colon X \to Z$ a smooth fibration over a closed smooth manifold whose fibers $X/Z$ are

• closedsmooth spin^c manifolds of even dimension

the push-forward element $\pi! \in KK(C_0(X), C_0(Z))$ 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

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

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

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

This yields a Fredholm-Hilbert bimodule

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

which defines an element in KK-theory

$$\pi ! \in KK(C_0(X), C_0(Z)) \,.$$

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 $f \colon X \to Y$ 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 $Y$:

$$f \;\colon\; X \stackrel{graph(f)}{\to} X \times Y \stackrel{p_Y}{\to} Y \,.$$

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

$$f! \coloneqq p_Y !\circ graph(f)! \,.$$

(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 $i \colon Q \to X$ be a map of compact manifolds and let $\chi \colon X \to B^2 U(1)$ 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

$$C_{i^\ast \chi}(Q) \stackrel{i^\ast}{\longleftarrow} C_{\chi}(X) \,,$$

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

$$C_{\frac{W_3(\tau_Q)}{i^\ast \chi}}(Q) \stackrel{i_!}{\longrightarrow} C_{\frac{W_3(\tau_X)}{\chi}}(X) \,.$$

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

$$C_{\frac{W_3(\tau_Q)}{i^\ast \chi}}(Q) \stackrel{i_!}{\longrightarrow} C_{\frac{1}{\chi}}(X) \,.$$

Postcomposition with this map in KK-theory now yields a map from the $\frac{W_3(\tau_Q)}{i^\ast \chi}$-twisted K-theory of $Q$ to the $\chi^{-1}$-twisted K-theory of $X$:

$$i_! \colon K_{\bullet + W_3(\tau_Q) - i^\ast \chi}(Q) \to K_{\bullet -\chi} \,.$$

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

$$[\xi] \in K_{\bullet + W_3(\tau_Q) - i^\ast \chi}(Q)$$

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 $\xi$ given by a twisted line bundle in which case it exhibits a twisted spin^c structure on $Q$.) Finally its push-forward

$$[i_! \xi] \in K_{\bullet- \chi}(X)$$

is called the corresponding D-brane charge.

Related concepts

• geometric quantization by push-forward

• geometric quantization with KU-coefficients

References

Discussion in KK-theory

• Alain Connes, A survey of foliations and operator algebras, Proceedings of the A.M.S., 38, 521-628 (1982) (pdf)

• Alain Connes, Georges Skandalis, The longitudinal index theorem for foliations. Publ. Res. Inst. Math. Sci. 20,

  no. 6, 1139–1183 (1984) (web)

• Michel Hilsum, Georges Skandalis, Morphismes K-orienté d’espace de feuille et fonctoralité en théorie de Kasparov, Annales scientifiques de l’École Normale Supérieure, Sér. 4, 20 no. 3 (1987), p. 325-390 (numdam)

• Jacek Brodzki, Varghese Mathai, Jonathan Rosenberg, Richard Szabo, Noncommutative correspondences, duality and D-branes in bivariant K-theory, Adv. Theor. Math. Phys.13:497-552,2009 (arXiv:0708.2648)

• Paulo Carrillo Rouse, Bai-Ling Wang, Twisted longitudinal index theorem for foliations and wrong way functoriality (arXiv:1005.3842)

Discussion for integration twisted K-theory over manifolds:

• Alan Carey, Bai-Ling Wang, Thom isomorphism and Push-forward map in twisted K-theory (arXiv:0507414)

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

• Daniel Freed, Mike Hopkins, Constantin Teleman, Loop Groups and Twisted K-Theory I (2011) (arXiv:0711.1906),

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

• Joost Nuiten, section 4 of Cohomological quantization of local prequantum boundary field theory, 2013