Integrating over Higgs Branches

We develop some useful techniques for integrating over Higgs branches in supersymmetric theories with 4 and 8 supercharges. In particular, we define a regularized volume for hyperkähler quotients. We evaluate this volume for certain ALE and ALF spaces in terms of the hyperkähler periods. We also reduce these volumes for a large class of hyperkähler quotients to simpler integrals. These quotients include complex coadjoint orbits, instanton moduli spaces on 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 Schrödinger system. We discuss some applications of our results.