Topology and the Dirac Operator Spectrum in Finite-Volume Gauge Theories

TL;DR

This paper investigates how topological features of gauge fields influence the Dirac operator spectrum in finite-volume gauge theories, providing a detailed analysis of the interplay between topology, the spectrum, and physical observables.

Contribution

It introduces a method to compute the Dirac spectrum in finite-volume gauge theories by summing over topological sectors with proper weights, enhancing understanding of topological effects.

Findings

Derived the microscopic Dirac spectrum by summing over topological sectors.

Compared mass-dependent chiral condensate in fixed topological sectors with the full sum.

Clarified the role of gauge-field winding numbers and theta-vacua in spectral properties.

Abstract

The interplay between between gauge-field winding numbers, theta-vacua, and the Dirac operator spectrum in finite-volume gauge theories is reconsidered. To assess the weight of each topological sector, we compare the mass-dependent chiral condensate in gauge field sectors of fixed topological index with the answer obtained by summing over the topological charge. Also the microscopic Dirac operator spectrum in the full finite-volume Yang-Mills theory is obtained in this way, by summing over all topological sectors with the appropriate weight.