noncommutative topology, noncommutative geometry
noncommutative stable homotopy theory
genus, orientation in generalized cohomology
Riemann integration, Lebesgue integration
line integral/contour integration
integration of differential forms
integration over supermanifolds, Berezin integral, fermionic path integral
Kontsevich integral, Selberg integral, elliptic Selberg integral
integration in ordinary differential cohomology
integration in differential K-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.
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).
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
This is such that $D_x$ graded-commutes with this $Cl_n$-action. Hence the assignment $x \mapsto \tilde D_x$ defines a map
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
This represents the push-forward class in $K^{dim(Y_x)}(X)$, and this construction gives a map
For $dim(Y_x)$ even and hence ignoring the compatibility with the $Cl_n$-action, this is discussed in (Carey-Wang 05).
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
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).
(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
in terms of which the push-forward on operator K-theory is induced by postcomposition:
where $d = dim(X) - dim(Y)$.
Now, if we could “thicken” $X$ a bit, namely to a tubular neighbourhood
of $h(X)$ in $Y$ without changing the K-theory of $X$, then the element in question will just be the KK-element
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
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
has a spin^c structure. Write $S(N_Y X)$ for the associated spinor bundle.
Then there is an invertible element in KK-theory
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
This induces a KK-equivalence
Therefore the push-forward in operator K-theory along $f \colon X \hookrightarrow Y$ is given by postcomposing in KK-theory with
(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
using that by the Whitney embedding theorem every compact $X$ may be embedded into some $\mathbb{R}^{2q}$ such as to yield an embedding
using that there is a KK-equivalence
The resulting push-forward is then given by postcomposition in KK-theory with
Specifically, for $\pi \colon X \to Z$ a smooth fibration over a closed smooth manifold whose fibers $X/Z$ are
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
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
which defines an element in KK-theory
Postcompositon with this is the push-forward map in K/KK-theory, equivalently the index map of the collection of Dirac operators.
(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 $Y$:
Hence push-forward along such a general map is postcomposition in KK-theory with
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
with notation as in this definition. By this proposition the dual morphism is of the form
If we assume that $X$ has a spin^c structure then this is
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 $X$:
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
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
is called the corresponding D-brane charge.
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:
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.