Ginsparg-Wilson Dirac operator in the monopole backgrounds on the fuzzy 2-sphere

TL;DR

This paper explores the Ginsparg-Wilson Dirac operator in monopole backgrounds on the fuzzy 2-sphere, establishing an index theorem, classifying topological sectors, and analyzing the spectrum in these configurations.

Contribution

It introduces an index theorem in the TP monopole background on the fuzzy 2-sphere and extends it to general configurations, clarifying the topological classification.

Findings

Established an index theorem in the projected space for TP monopoles.

Extended the index theorem to non-eom configurations.

Calculated the spectrum of the GW Dirac operator in monopole backgrounds.

Abstract

In the previous papers, we studied the 't Hooft-Polyakov (TP) monopole configurations in the U(2) gauge theory on the fuzzy 2-sphere,and showed that they have nonzero topological charge in the formalism based on the Ginsparg-Wilson (GW) relation. In this paper, we will show an index theorem in the TP monopole background, which is defined in the projected space, and provide a meaning of the projection operator. We also extend the index theorem to general configurations which do not satisfy the equation of motion, and show that the configuration space can be classified into the topological sectors. We further calculate the spectrum of the GW Dirac operator in the TP monopole backgrounds, and consider the index theorem in these cases.