  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.
