The $ \mathcal{N} $ = 1 Chiral Multiplet on T 2 × S 2 and Supersymmetric Localization

Abstract

We compute the supersymmetric partition function of an $ \mathcal{N} $ = 1 chiral multiplet coupled to an external Abelian gauge field on complex manifolds with T 2 × S 2 topology. The result is locally holomorphic in the complex structure moduli of T 2 × 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 $ \mathcal{M} $ 4 preserving two supercharges of opposite chiralities, via supersymmetric localization. In the case of $ \mathcal{M} $ 4 = S 3 × 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 × S 1 complex structure moduli.