A Local Index Theorem of Transversal Type on Manifolds with Locally Free $\mathbb{S}^1$-action

TL;DR

This paper develops a probabilistic approach to analyze the index of a transversal Dirac operator on manifolds with locally free circle actions, linking it to orbifold heat kernels and simplifying classical index calculations.

Contribution

It introduces a probabilistic method using the Feynman-Kac formula to study transversal heat kernels on orbifolds, enabling new insights into index theory for manifolds with circle actions.

Findings

Probabilistic approach links transversal heat kernel to orbifold heat kernel.

Uniform bounds established for small-time asymptotics.

Lower-dimensional strata contributions vanish for certain spin orbifolds.

Abstract

We study an index of a transversal Dirac operator on an odd-dimensional manifold $X$ with locally free $\mathbb{S}^1$-action. One difficulty of using heat kernel method lies in the understanding of the asymptotic expansion as $t\to 0^+$. By a probabilistic approach via the Feynman-Kac formula, the transversal heat kernel on $X$ can be linked to the ordinary heat kernel for functions on the orbifold $M=X/\mathbb{S}^1$ which is more tractable. After some technical results for a uniform bound estimate as $t\to 0^+$, we are reduced from the transversal, orbifold situation to the classical situation particularly at points of the principal stratum. One application asserts that for a certain class of spin orbifolds $M$, to the classical index problem of Kawasaki in the Riemannian setting the net contributions arising from the lower-dimensional strata beyond the principal one vanish identically.