Microlocal Euler classes and Hochschild homology

TL;DR

This paper introduces microlocal Euler classes for trace kernels on manifolds, linking Hochschild homology with index theorems, and demonstrating functoriality under kernel composition.

Contribution

It defines a new microlocal Euler class for trace kernels, unifying and simplifying existing index theorem results across sheaves, D-modules, and elliptic pairs.

Findings

Defines trace kernels and associates microlocal Euler classes

Proves functoriality of the Euler class under kernel composition

Unifies various index theorems in a common framework

Abstract

We define the notion of a trace kernel on a manifold M. Roughly speaking, it is a sheaf on M x M for which the formalism of Hochschild homology applies. We associate a microlocal Euler class to such a kernel, a cohomology class with values in the relative dualizing complex of the cotangent bundle over M and we prove that this class is functorial with respect to the composition of kernels. This generalizes, unifies and simplifies various results of (relative) index theorems for constructible sheaves, D-modules and elliptic pairs.