Volume of moduli space of vortex equations and localization

Abstract

We evaluate volume of moduli space of BPS vortices on a compact genus h Riemann surface σh 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 multiflavors. We also compare our results with the volume of the Kähler quotient space inspired by the brane configuration.