An L^2-Index Theorem for Dirac Operators on S^1 * R^3

TL;DR

This paper derives an explicit formula for the $L^2$-index of Dirac operators on $S^1 imes R^3$, incorporating boundary conditions, and applies it to count zero modes in caloron backgrounds.

Contribution

It provides a new $L^2$-index theorem for Dirac operators on $S^1 imes R^3$ with specific boundary conditions, extending previous index results.

Findings

Derived an explicit $L^2$-index formula for Dirac operators on $S^1 imes R^3$.

Connected the index to the adiabatic limit of the $ ext{eta}$-invariant.

Applied the theorem to count zero modes in caloron backgrounds.

Abstract

An expression is found for the $L^2$-index of a Dirac operator coupled to a connection on a $U_n$ vector bundle over $S^1\times{\mathbb R}^3$. Boundary conditions for the connection are given which ensure the coupled Dirac operator is Fredholm. Callias' index theorem is used to calculate the index when the connection is independent of the coordinate on $S^1$. An excision theorem due to Gromov, Lawson, and Anghel reduces the index theorem to this special case. The index formula can be expressed using the adiabatic limit of the $\eta$-invariant of a Dirac operator canonically associated to the boundary. An application of the theorem is to count the zero modes of the Dirac operator in the background of a caloron (periodic instanton).