nLab Poincaré duality algebra

Idea

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).

Definition

For graded-commutative algebras

Definition

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

$$A_k \otimes A_{n-k} \stackrel{\otimes}{\to} A_n \stackrel{\epsilon}{\to} k$$

are non-degenerate.

e.g. (Lambrechst-Stanley 07)

For $C^\ast$-algebras

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.

Definition

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).

Remark

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

$$\Delta^\vee \otimes_{A^{op}} \Delta = id_A \in KK(A,A)$$

and

$$\Delta^\vee \otimes_A \Delta = id_{A^{op}} \in KK(A^{op}, A^{op}) \,.$$

Definition/Proposition

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.

Remark

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

Definition

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.

Properties

For dg-Algebras

For $C^\ast$-algebras

Duals and twists

Proposition

Let $X$ be a closed manifold with spin^c-structure. Then there is a Poincaré duality isomorphism

$$K^\bullet(X) \simeq K_\bullet(X) \,.$$

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 $X$.

Definition/Notation

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

$$C_c(X) \in C^\ast Alg \to KK$$

for the corresponding twisted groupoid convolution algebra, the one whose operator K-theory is the $c$-twisted K-theory of $X$:

$$KK_\bullet(\mathbb{C}, C_c(X)) \simeq K_{\bullet + c}(X) \,.$$

Proposition

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

$$(C_c(X))^\vee \simeq C_{\frac{1}{c\otimes W_3(\tau_X)}}(X) \,,$$

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).

K-Orientation and Umkehr maps

We discuss Umkehr maps/fiber integration in generalized cohomology in K-theory using Poincaré duality algebras / dual objects in KK-theory.

Proposition

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

$$f! \coloneqq \Delta^\vee_A \otimes_{A^{op}} f^{op} \otimes_{B^{op}} \Delta_B \,.$$

(BMRS 07, 3.3)

More generally we have the following.

Example

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 def. . By prop. the dual morphism is of the form

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

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

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

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 $X$:

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

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

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

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.

See (Nuiten 13).

Examples

Example

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 $X \times X$.

(BMRS 07, example 3.2)

Example

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)

Related concepts

• fundamental class

• virtual fundamental class

• Poincaré duality complex

References

For graded associative algebras

• Pascal Lambrechts, Don Stanley, Poincaré duality and commutative differential graded algebras (arXiv:math/0701309)

For $C^\ast$-algebras

For C*-algebras/in noncommutative topology:

• Henri Moscovici, Eigenvalue inequalities and Poincaré duality in noncommutative geometry, Commun. Math. Phys. 184 , 3 (1997) 619

Chapter 6.4 $\beta$ (starting p. 601) in

• Alain Connes, Noncommutative Geometry

Def. 2.1 in

• 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)

Duality including the twisted K-theory induced by twisted spin^c structure over manifolds is discussed in section 7 of

• Jacek Brodzki, Varghese Mathai, Jonathan Rosenberg, Richard Szabo, D-Branes, RR-Fields and Duality on Noncommutative Manifolds, Commun. Math. Phys. 277:643-706,2008 (arXiv:hep-th/0607020)

and generalized to equivariant KK-theory in

• Jean-Louis Tu, Twisted K-theory and Poincaré duality (arXiv:math/0609556)

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

• Joost Nuiten, Cohomological quantization of local prequantum boundary field theory, master thesis, August 2013