Integrating over Higgs branches

We develop some useful techniques for integrating over Higgs branches in su persymmetric 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 red uce these volumes for a large class of hyperkahler quotients to simpler int egrals. These quotients include complex coadjoint orbits, instanton moduli spaces on R-4 and ALE manifolds, Hitchin spaces, and moduli spaces of (para bolic) Higgs bundles on Riemann surfaces. In the case of Hit chin spaces th e evaluation of the volume reduces to a summation over solutions of Bethe A nsatz equations for the non-linear Schrodinger system. We discuss some appl ications of our results.