A Semiclassical Heat Kernel Proof of the Poincar\'e-Hopf Theorem

TL;DR

This paper uses semiclassical heat kernel techniques applied to the Witten operator to derive the Poincaré-Hopf theorem and its generalizations, connecting heat kernel asymptotics with Thom form approaches.

Contribution

It introduces a semiclassical heat kernel proof of the Poincaré-Hopf theorem and extends it to degenerate cases, linking different mathematical frameworks.

Findings

Semiclassical heat kernel methods derive the Poincaré-Hopf theorem.

Connection established between heat kernel asymptotics and Thom form approaches.

Generalizations of the theorem for degenerate cases are provided.

Abstract

We treat the Witten operator on the de Rham complex with semiclassical heat kernel methods to derive the Poincar\'e-Hopf theorem and degenerate generalizations of it. Thereby, we see how the semiclassical asymptotics of the Witten heat kernel are related to approaches using the Thom form of Mathai and Quillen.