Cyclic cocycles on deformation quantizations and higher index theorems

TL;DR

This paper constructs cyclic cocycles on deformation quantizations of symplectic manifolds, providing a new proof of the algebraic higher index theorem and extending it to orbifolds, with applications to analytic index theorems.

Contribution

It introduces a new cyclic cocycle on the Weyl algebra and establishes an explicit local quasi-isomorphism, leading to a novel proof and extension of higher index theorems.

Findings

Constructed a nontrivial cyclic cocycle on the Weyl algebra.

Provided a new proof of the algebraic higher index theorem.

Extended the higher index theorem to symplectic orbifolds.

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.