Generalized Indices for $\mathcal{N}=1$ Theories in Four-Dimensions

TL;DR

This paper develops generalized indices for four-dimensional $ =1$ supersymmetric theories on manifolds of the form $S^1 imes M_3$, extending previous indices and exploring their properties as holomorphic functions.

Contribution

It introduces a new class of generalized indices for $ =1$ theories on complex manifolds, including background flat connections, expanding the understanding of supersymmetric partition functions.

Findings

Generalized indices are holomorphic functions of complex structure moduli.

Indices include and extend the lens space index.

Deformation by background flat connections is demonstrated.

Abstract

We use localization techniques to calculate the Euclidean partition functions for $\mathcal{N}=1$ theories on four-dimensional manifolds $M$ of the form $S^1 \times M_3$, where $M_3$ is a circle bundle over a Riemann surface. These are generalizations of the $\mathcal{N}=1$ indices in four-dimensions including the lens space index. We show that these generalized indices are holomorphic functions of the complex structure moduli on $M$. We exhibit the deformation by background flat connections.