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 be a fiber bundle of compact smooth manifolds carrying fiberwise a spin^c structure.
For write
for the fiber over ;
for the fiberwise spin^c-principal bundle;
for the fiberwise Clifford bundle;
for the spin^c spin bundle on ;
for the Spin^c Dirac operator on the space of sections of spinors;
for the bounded version;
Here depends smoothly on while still depends continuously on . Equip with the -action given on elements joint right Clifford product by and left Clifford product by the volume element
This is such that graded-commutes with this -action. Hence the assignment defines a map
from the base to the space Fredholm operators graded-commuting? with , as defined here at twisted K-theory.
More generally, let then be a vector bundle representing a class in . With a choice of connection this twists the above constrction to yield -twisted Dirac operator and hence a map
This represents the push-forward class in , and this construction gives a map
For even and hence ignoring the compatibility with the -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 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 .
Now, if we could “thicken” a bit, namely to a tubular neighbourhood
of in without changing the K-theory of , then the element in question will just be the KK-element
induced directly from the C*-algebra homomorphism from the algebra of functions vanishing at infinity of to functions on , given by extending these functions by 0 to functions on . 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 for the associated spinor bundle.
Then there is an invertible element in KK-theory
hence a KK-equivalence , where denotes the algebra of functions vanishing at infinity.
This is defined as follows. Consider the pullback of this spinor to the normal bundle itself along the projection . Then…
Moreover, a choice of a Riemannian metric on allows to find a diffeomorphism between the tubular neighbourhood of 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 is given by postcomposing in KK-theory with
(Connes-Skandalis 84, above prop. 2.9)
For 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 may be embedded into some 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 a smooth fibration over a closed smooth manifold whose fibers are
the push-forward element 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 on this bundle (hence a collection of Riemannian metric on the fibers smoothly varying along ). Write for the corresponding spinor bundle.
A choice of horizontal complenet induces an affine connection . This combined with the symbol map/Clifford multiplication of on induces a fiberwise spin^c Dirac operator, acting in each fiber on the Hilbert space .
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 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 :
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 be a map of compact manifolds and let 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 has a spin^c structure then this is
Postcomposition with this map in KK-theory now yields a map from the -twisted K-theory of to the -twisted K-theory of :
If we here think of as being the inclusion of a D-brane worldvolume, then 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 given by a twisted line bundle in which case it exhibits a twisted spin^c structure on .) 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.