Localization on Hopf surfaces

We discuss localization of the path integral for supersymmetric gauge theories with an R-symmetry on Hermitian four-manifolds. After presenting the localization locus equations for the general case, we focus on backgrounds with S^1 x S^3 topology, admitting two supercharges of opposite R-charge. These are Hopf surfaces, with two complex structure moduli p,q. We compute the localized partition function on such Hopf surfaces, allowing for a very large class of Hermitian metrics, and prove that this is proportional to the supersymmetric index with fugacities p,q. Using zeta function regularisation, we determine the exact proportionality factor, finding that it depends only on p,q, and on the anomaly coefficients a, c of the field theory. This may be interpreted as a supersymmetric Casimir energy, and provides the leading order contribution to the partition function in a large N expansion.