The N=1 Chiral Multiplet on T2 x S2 and Supersymmetric Localization

We compute the supersymmetric partition function of an N = 1 chiral multiplet coupled to an external Abelian gauge field on complex manifolds with T-2 x S-2 topology. The result is locally holomorphic in the complex structure moduli of T-2 x S-2. This computation illustrates in a simple example some recently obtained constraints on the parameter dependence of supersymmetric partition functions. We also devise a simple method to compute the chiral multiplet partition function on any four-manifold M-4 preserving two supercharges of opposite chiralities, via supersymmetric localization. In the case of M-4 = S-3 x S-1, we provide a path integral derivation of the previously known result, the elliptic gamma function, which emphasizes its dependence on the S-3 x S-1 complex structure moduli.