Integrating Over Higgs Branches

TL;DR

This paper introduces techniques for calculating volumes of Higgs branches in supersymmetric theories, simplifying complex integrals and applying them to various moduli spaces, with connections to Bethe Ansatz solutions.

Contribution

It defines a regularized volume for hyperkahler quotients and reduces complex integrals to simpler forms, enabling new calculations in supersymmetric gauge theories.

Findings

Computed volumes for ALE and ALF spaces using hyperkahler periods

Reduced complex integrals to simpler forms for various moduli spaces

Connected volume calculations to Bethe Ansatz solutions in Hitchin spaces

Abstract

We develop some useful techinques for integrating over Higgs branches in supersymmetric theories with 4 and 8 supercharges. In particular, we define a regularized volume for hyperkahler quotients. We evaluate this volume for certain ALE and ALF spaces in terms of the hyperkahler periods. We also reduce these volumes for a large class of hyperkahler quotients to simpler integrals. These quotients include complex coadjoint orbits, instanton moduli spaces on R^4 and ALE manifolds, Hitchin spaces, and moduli spaces of parabolic Higgs bundles on Riemann surfaces. In the case of Hitchin spaces the evaluation of the volume reduces to a summation over solutions of Bethe Ansatz equations for the non-linear Schroedinger system. We discuss some applications of our results.