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

TL;DR

The paper develops a concrete framework to compute the supersymmetric partition function of a four-dimensional $\mathcal{N}=1$ chiral multiplet on $T^2\times S^2$ and more generally on any $\mathcal{M}_4$ preserving two supercharges. It provides two complementary derivations: a canonical-quantization index computation that reveals holomorphic dependence on the complex-structure moduli $(\tau,\sigma)$ and a localization-based path integral approach that yields a universal determinant structure in terms of unpaired eigenmodes. The results reproduce known special-function structures, such as theta- and elliptic gamma functions, and show how the partition function behaves under modular-like transformations, highlighting gravitational anomalies. The work also offers a practical localization method to compute $Z^{\Phi}_{\mathcal{M}_4}$ for arbitrary Hermitian backgrounds with two supercharges, with explicit applications to $T^2\times S^2$ and $S^3\times S^1$ and a unifying perspective on their holomorphic moduli dependence and spectral data.

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 $T^2 \times S^2$ topology. The result is locally holomorphic in the complex structure moduli of $T^2\times S^2$. 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 $\mathcal{M}_4=S^3\times S^1$, we provide a path integral derivation of the previously known result, the elliptic gamma function, which emphasizes its dependence on the $S^3 \times S^1$ complex structure moduli.