An algebraic index theorem for Poisson manifolds

TL;DR

This paper proposes an algebraic index theorem for Poisson manifolds utilizing a trace density map derived from the formality theorem for Hochschild chains, linking deformation quantization to Poisson homology.

Contribution

It introduces a novel algebraic index theorem for Poisson manifolds based on the trace density map from Hochschild chains, extending deformation quantization techniques.

Findings

Establishes a trace density map from Hochschild homology to Poisson homology.

Provides a new algebraic index theorem for Poisson manifolds.

Connects deformation quantization with Poisson geometry.

Abstract

The formality theorem for Hochschild chains of the algebra of functions on a smooth manifold gives us a version of the trace density map from the zeroth Hochschild homology of a deformation quantization algebra to the zeroth Poisson homology. We propose a version of the algebraic index theorem for a Poisson manifold which is based on this trace density map.