(geometry $\leftarrow$ Isbell duality $\to$ algebra)
algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)
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
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. 2 is called a fundamental class of $A$.
This appears as (BMRS 07, def. 2.1, following Connes, p. 601) following (Connes).
Explicitly def. 2 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. \ref{PDAlgebra}, 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. 4 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 2 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. 4. By prop. 3 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