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integration in differential K-theory
Generally, a Poincaré duality dg-algebra is a dg-algebra with structure mimicking Poincaré duality in ordinary cohomology.
On the other hand a Poincaré duality $C^\ast$-algebra is a C*-algebra which represents a space in noncommutative topology for which there is a sensible notion of Poincaré duality in K-theory (operator K-theory/K-homology).
The structure of a Poincaré duality algebra in dimension $n$ on a graded-commutative graded associative algebra $A$ is a linear function $\epsilon \colon A_n \to k$ to the ground field such that all the induced bilinear forms
are non-degenerate.
e.g. (Lambrechst-Stanley 07)
For C*-algebras hence in noncommutative topology there is the following notion of Poincaré duality, which is really Poincaré with respect not to ordinary cohomology but K-theory (operator K-theory).
We start with the definition of Poincaré self-duality and then generalize to Poincaré dual pairs.
A separable C*-algebra $A \in$ C*Alg is a Poincaré duality algebra (or PD algebra, for short ) if it is dualizable object when regarded as an object of the KK-theory-category, with dual object its opposite algebra.
The element $\Delta$ in def. is called a fundamental class of $A$.
This appears as (BMRS 07, def. 2.1, following Connes, p. 601) following (Connes).
Explicitly def. says that $A$ is a PD algebra if there exists $\Delta \in KK(A \otimes A^{op}, \mathbb{C})$ and $\Delta^\vee \in KK(\mathbb{C}, A \otimes A^{op})$ such that
and
For $A$ $B$ two Poincaré duality algebras, def. , and for $f \colon A \to B$ a homomorphism between them, regarded as a morphism $f^\ast \colon B \to A$ in KK-theory, the correspondung dual morphism $f! \colon A \to B$ is the one such that postcomposition in $KK$ with this corresponds to the Umkehr map/push forward in generalized cohomology? in KK-theory.
For more on this see below at Properties – K-Orientation and Umkehr mpas.
For $C^\ast$-algebras which are groupoid convolution algebras $C^\ast(\mathcal{G})$ the opposite algebra is Morita equivlant (since a groupoid $\mathcal{G}$ is equivalent to its opposite groupoid $\mathcal{G}^{op}$, the equivalence being induced by the functor which sends a morphism to its inverse). But given a circle 2-bundle $\chi \colon \mathcal{G} \to \mathbf{B}^2 U(1)$ the corresponding twisted groupoid convolution algebra is such that passing to the opposite corresponds to passing to the inverse twist $-\chi$.
Therefore it makes sense to consider more generally
For $A$ a C*-algebra a Poincaré dual for $A$ is a dual object $B \in C^\ast Alg \to KK$ in KK-theory.
Below in the Proposition-Section is discussed how under Poincaré-duality the twist changes.
Let $X$ be a closed manifold with spin^c-structure. Then there is a Poincaré duality isomorphism
For instance (Connes, chapter 2.7, prop. 5).
(…) The relaton between Poincaré duality on algebras of functions and spin^c-structure is discussed in (Connes, around p. 603). (…)
Notice that the obstruction to spin^c structure is the third integral Stiefel-Whitney class $W_3 \colon B SO \to B^2 U(1)$. If this does not vanish on a manifold, then a Poincaré dual/dual object in KK-theory still exists, but is the same manifold equipped with a twist shifted by $W_3(\tau_X)$, where $\tau_X$ denotes the (co)tangent bundle of $X$.
For $X$ a (compact) manifold and $c \in H^3(X,\mathbb{Z})$ the class of a circle 2-bundle/bundle gerbe $\mathcal{G}$ on $X$, write
for the corresponding twisted groupoid convolution algebra, the one whose operator K-theory is the $c$-twisted K-theory of $X$:
Let $X$ be a compact manifold with tangent bundle $\tau_X$ and let $c \in H^3(X,\mathbb{Z})$ be a twist. Then the C*-algebra $C_{c}(X)$ of def. has a dual object in the full subcategory of KK-theory on separable C*-algebras, given by
hence by the same manifold but with twist the inverse of the third integral Stiefel-Whitney class and the original twist.
The same remains true in $G$-equivariant KK-theory, for $G$ a locally compact topological group.
The non-equivariant case is in (Brodzki-Mathai-Rosenberg-Szabo 06, section 7.3) and the generalization to the equivariant case in (Tu 06, theorem 3.1) (where we use remark in order to interpret the opposite twisted convolution algebra up to equivalence as inducing the inverse twist).
We discuss Umkehr maps/fiber integration in generalized cohomology in K-theory using Poincaré duality algebras / dual objects in KK-theory.
Every homomorphism $f \colon A \to B$ between PD $C^\ast$-algebras is K-orientable in KK-theory. The K-orientation is given by the corresponding dual morphism, hence the element $f! \colon B \to A$ given as the composite
More generally we have the following.
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 def. . By prop. the dual morphism is of the form
If we redefine the twist on $X$ to absorb this “quantum correction” as $\chi \mapsto \frac{1}{\chi \otimes W_3(T X)}$ then this is
Postcomposition with this map in KK-theory now yields a map from the $i^\ast \chi \otimes W_3(N Q)$-twisted K-theory of $Q$ to the $\chi$-twisted K-theory of $X$:
This is the twisted Umkehr map in this context.
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.
See (Nuiten 13).
For $A = C_0(X)$ the algebra of functions on a compact complex manifold $X$, then $A$ is a PD algebra with fundamental class $\Delta$ in K-homology given by the Dolbeault operator on $X \times X$.
For $A = C_0(X)$ the algebra of functions vanishing at infinity of a manifold $X$ with spin^c structure. Take $B = C_0(T^\ast X) \simeq_{KK} A^{op} \simeq A$. Then $\Delta$ constructed from the Dirac operator on the Clifford algebra bundle over $T^\ast X$ is a fundamental class.
(BMRS 07, proof of theorem 2.9)
For C*-algebras/in noncommutative topology:
Chapter 6.4 $\beta$ (starting p. 601) in
Def. 2.1 in
Duality including the twisted K-theory induced by twisted spin^c structure over manifolds is discussed in section 7 of
and generalized to equivariant KK-theory in
More on dual objects in KK is in
Heath Emerson, Ralf Meyer, Bivariant K-theory via correspondences, Adv. Math. 225 (2010), 2883-2919 (arXiv:0812.4949)
Heath Emerson, Ralf Meyer, Dualities in equivariant Kasparov theory (arXiv:0711.0025)
Discussion of the twisted Umkehr map and the Freed-Witten-Kapustin anomaly in this context is in
Last revised on May 19, 2023 at 20:41:01. See the history of this page for a list of all contributions to it.