Transiting topological sectors with the overlap

TL;DR

This paper explores how the eigenvalues of the overlap operator evolve when transitioning between different topological sectors in lattice gauge theories, providing insights into the topological structure of gauge fields.

Contribution

It demonstrates a simple low-dimensional example illustrating the flow of eigenvalues of the truncated overlap operator across topological sectors.

Findings

Eigenvalues flow between sectors as fields change

The space of gauge fields is simply connected without restrictions

Overlap operator defines winding number except on measure-zero sets

Abstract

The overlap operator provides an elegant definition for the winding number of lattice gauge field configurations. Only for a set of configurations of measure zero is this procedure undefined. Without restrictions on the lattice fields, however, the space of gauge fields is simply connected. I present a simple low dimensional illustration of how the eigenvalues of a truncated overlap operator flow as one travels between different topological sectors.