Localization for Yang-Mills Theory on the Fuzzy Sphere

TL;DR

This paper introduces a new model for Yang-Mills theory on the fuzzy sphere, employing localization techniques to compute the partition function explicitly, bridging fuzzy and classical gauge theories.

Contribution

It develops a novel coadjoint orbit-based model for fuzzy sphere Yang-Mills theory and applies localization methods to derive explicit partition function formulas.

Findings

Classical solutions of the fuzzy sphere gauge theory identified

Partition function expressed as sum over critical points

Classical limit recovers ordinary Yang-Mills instantons

Abstract

We present a new model for Yang-Mills theory on the fuzzy sphere in which the configuration space of gauge fields is given by a coadjoint orbit. In the classical limit it reduces to ordinary Yang-Mills theory on the sphere. We find all classical solutions of the gauge theory and use nonabelian localization techniques to write the partition function entirely as a sum over local contributions from critical points of the action, which are evaluated explicitly. The partition function of ordinary Yang-Mills theory on the sphere is recovered in the classical limit as a sum over instantons. We also apply abelian localization techniques and the geometry of symmetric spaces to derive an explicit combinatorial expression for the partition function, and compare the two approaches. These extend the standard techniques for solving gauge theory on the sphere to the fuzzy case in a rigorous framework.