Volume of Moduli Space of Vortex Equations and Localization

TL;DR

This paper computes the volume of the moduli space of BPS vortices on compact Riemann surfaces using topological field theory and localization, confirming previous results for Abelian vortices and extending to non-Abelian cases.

Contribution

It introduces a localization-based method to evaluate vortex moduli space volumes, including non-Abelian and multi-flavor cases, linking to Kahler quotient spaces.

Findings

Volume matches previous Abelian vortex results

Extended evaluation to non-Abelian gauge groups

Compared with Kahler quotient space volume

Abstract

We evaluate volume of moduli space of BPS vortices on a compact Riemann surface by using topological field theory and localization technique developed by Moore, Nekrasov and Shatashvili. We apply this technique to Abelian (ANO) vortex and show that the volume of moduli space agrees with the previous results obtained by integrating over the moduli space metric. We extend the evaluation to non-Abelian gauge groups and multi-flavors. We also compare our results with the volume of the Kahler quotient space inspired by the brane configuration.