Cyclic cocycles on deformation quantizations and higher index theorems

Abstract

We construct a nontrivial cyclic cocycle on the Weyl algebra of a symplectic vector space. Using this cyclic cocycle we construct an explicit, local, quasi-isomorphism from the complex of differential forms on a symplectic manifold to the complex of cyclic cochains of any formal deformation quantization thereof. We give a new proof of Nest-Tsygan's algebraic higher index theorem by computing the pairing between such cyclic cocycles and the K-theory of the formal deformation quantization. Furthermore, we extend this approach to derive an algebraic higher index theorem on a symplectic orbifold. As an application, we obtain the analytic higher index theorem of Connes-Moscovici and its extension to orbifolds.

On the Algebraic Index for Riemannian Étale Groupoids

“…In the work [18], we obtained a complete answer to this question in the case of propeŕ etale groupoids. To answer this question for generalétale groupoids, we need more tools in Lie algebra cohomology and homological algebras.…”

“…When there is a G-invariant symplectic form on G 0 , deformation quantization of the groupoid algebra C ∞ G was constructed by the last author [22]. As a first step toward the algebraic index theory, we construct explicit cyclic (co)cycles on the deformation quantization A G of C ∞ G. More precisely, we construct a quasi-isomorphism from the cyclic chain complex of the algebra A G to the de Rham complex of compactly supported simplicial differential forms on the inertia groupoid G of G, and a quasi-isomorphism from the de Rham complex on the inertia groupoid G to the cyclic cochain complex of A G. These quasi-isomorphisms generalize our constructions in [18,Sec. 5].…”

“…The cocycle τ 2m is the untwisted cocycle of [18] on W(T s(g 0 ) B p ) which extends the Hochschild cocycle of [10], and tr θ g 0 ...g p is the twisted trace of [9] on W(T s(g 0 ) σ p ). The definition of this trace uses an auxiliary invariant almost complex structure on G 0 that we fix using the invariant Riemannian metric and symplectic form.…”

“…With this paper we continue our study of algebraic index theory over singular spaces modeled by groupoids. More precisely, omitting the assumption of properness from [17] and [18] we study here the algebraic index theory of a singular space which can be obtained as the quotient of anétale groupoid equipped with an invariant Riemannian metric on the unit space. Such groupoids are called Riemannianétale groupoids and appear naturally in the study of Riemannian foliations.…”

“…In [17] and [18], the authors of this paper studied the algebraic index theory over orbifolds using noncommutative geometry and deformation quantization. In [17], we obtained an explicit topological formula for the Chern character of an elliptic operator on a compact Riemannian orbifold.…”