nLab
Poincaré duality algebra

Context

Noncommutative geometry

Operator algebra

Integration theory

Contents

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 *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 nn on a graded-commutative graded associative algebra AA is a linear function ϵ:A nk\epsilon \colon A_n \to k to the ground field such that all the induced bilinear forms

A kA nkA nϵk 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 *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 AA \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 Δ\Delta in def. 2 is called a fundamental class of AA.

This appears as (BMRS 07, def. 2.1, following Connes, p. 601) following (Connes).

Remark

Explicitly def. 2 says that AA is a PD algebra if there exists ΔKK(AA op,)\Delta \in KK(A \otimes A^{op}, \mathbb{C}) and Δ KK(,AA op)\Delta^\vee \in KK(\mathbb{C}, A \otimes A^{op}) such that

Δ A opΔ=id AKK(A,A) \Delta^\vee \otimes_{A^{op}} \Delta = id_A \in KK(A,A)

and

Δ AΔ=id A opKK(A op,A op). \Delta^\vee \otimes_A \Delta = id_{A^{op}} \in KK(A^{op}, A^{op}) \,.
Definition/Proposition

For AA BB two Poincaré duality algebras, def. \ref{PDAlgebra}, and for f:ABf \colon A \to B a homomorphism between them, regarded as a morphism f *:BAf^\ast \colon B \to A in KK-theory, the correspondung dual morphism f!:ABf! \colon A \to B is the one such that postcomposition in KKKK 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 *C^\ast-algebras which are groupoid convolution algebras C *(𝒢)C^\ast(\mathcal{G}) the opposite algebra is Morita equivlant (since a groupoid 𝒢\mathcal{G} is equivalent to its opposite groupoid 𝒢 op\mathcal{G}^{op}, the equivalence being induced by the functor which sends a morphism to its inverse). But given a circle 2-bundle χ:𝒢B 2U(1)\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 χ-\chi.

Therefore it makes sense to consider more generally

Definition

For AA a C*-algebra a Poincaré dual for AA is a dual object BC *AlgKKB \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 *C^\ast-algebras

Duals and twists

Proposition

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

K (X)K (X). 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:BSOB 2U(1)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(τ X)W_3(\tau_X), where τ X\tau_X denotes the (co)tangent bundle of XX.

Definition/Notation

For XX a (compact) manifold and cH 3(X,)c \in H^3(X,\mathbb{Z}) the class of a circle 2-bundle/bundle gerbe 𝒢\mathcal{G} on XX, write

C c(X)C *AlgKK C_c(X) \in C^\ast Alg \to KK

for the corresponding twisted groupoid convolution algebra, the one whose operator K-theory is the cc-twisted K-theory of XX:

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

Let XX be a compact manifold with tangent bundle τ X\tau_X and let cH 3(X,)c \in H^3(X,\mathbb{Z}) be a twist. Then the C*-algebra C c(X)C_{c}(X) of def. 4 has a dual object in the full subcategory of KK-theory on separable C*-algebras, given by

(C c(X)) C 1cW 3(τ X)(X), (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 GG-equivariant KK-theory, for GG 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).

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:ABf \colon A \to B between PD C *C^\ast-algebras is K-orientable in KK-theory. The K-orientation is given by the corresponding dual morphism, hence the element f!:BAf! \colon B \to A given as the composite

f!Δ A A opf op B opΔ B. 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:QXi \colon Q \to X be a map of compact manifolds and let χ:XB 2U(1)\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 *χ(Q)i *C χ(X), C_{i^\ast \chi}(Q) \stackrel{i^\ast}{\longleftarrow} C_{\chi}(X) \,,

with notation as in def. 4. By prop. 3 the dual morphism is of the form

C 1i *χW 3(TQ)(Q)i !C 1χW 3(TX)(X). 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 XX to absorb this “quantum correction” as χ1χW 3(TX)\chi \mapsto \frac{1}{\chi \otimes W_3(T X)} then this is

C i *χW 3(i *TX)W 3(TQ)(Q)i !C χ(X), 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 *χW 3(NQ)i^\ast \chi \otimes W_3(N Q)-twisted K-theory of QQ to the χ\chi-twisted K-theory of XX:

i !:K +W 3(NQ)+i *χ(Q)K +χ. 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:QXi \colon Q \hookrightarrow X as being the inclusion of a D-brane worldvolume, then χ\chi would be the class of the background B-field and an element

[ξ]K +W 3(NQ)+i *χ(Q) [\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 ξ\xi given by a twisted line bundle in which case it exhibits a twisted spin^c structure on QQ.) Finally its push-forward

[i !ξ]K χ(X) [i_! \xi] \in K_{\bullet- \chi}(X)

is called the corresponding D-brane charge.

See (Nuiten 13).

Examples

Example

For A=C 0(X)A = C_0(X) the algebra of functions on a compact complex manifold XX, then AA is a PD algebra with fundamental class Δ\Delta in K-homology given by the Dolbeault operator on X×XX \times X.

(BMRS 07, example 3.2)

Example

For A=C 0(X)A = C_0(X) the algebra of functions vanishing at infinity of a manifold XX with spin^c structure. Take B=C 0(T *X) KKA opAB = C_0(T^\ast X) \simeq_{KK} A^{op} \simeq A. Then Δ\Delta constructed from the Dirac operator on the Clifford algebra bundle? over T *XT^\ast X is a fundamental class.

(BMRS 07, proof of theorem 2.9)

References

For graded associative algebras

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

For C *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 β\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

Discussion of the twisted Umkehr map and the Freed-Witten-Kapustin anomaly in this context is in

Revised on August 19, 2013 16:30:08 by Urs Schreiber (24.131.18.91)