Heat equation approach to index theorems on odd dimensional manifolds

TL;DR

This paper presents a heat kernel method to prove Freed's index theorem on odd-dimensional spin manifolds with boundary, including new boundary conditions and implications for isospectral invariants.

Contribution

It offers a heat kernel proof of Freed's index theorem, extending analysis to spectral boundary conditions and providing insights into isospectral invariants without relying on cobordism invariance.

Findings

Heat kernel proof of Freed's index theorem

Extension to Atiyah-Patodi-Singer boundary conditions

Insights into isospectral invariants of boundary conditions

Abstract

D.Freed has formulated and proved an index theorem on odd dimensional spin manifolds with boundary. The proof is based on analysis by Calderon and Seeley. In this note we are going to give a proof of this theorem using the heat kernels methods for boundary conditions of Dirichlet and Von Neumann type. Moreover we consider also the Atiyah-Patodi-Singer spectral boundary condition which is not considered in Freed's paper. As a direct consequence of the method, we will obtain some information about isospectral invariants of the boundary conditions. This proof does not uses the cobordism invariance of index and are easily generalized to family case.