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

TL;DR

The paper develops a localization framework for 4d ${\cal N}=1$ gauge theories on manifolds $M\simeq S^{1}\times M_{3}$, where $M_{3}$ is a circle bundle over a Riemann surface, thereby generalizing the lens-space and related indices. By exploiting two supercharges and a carefully chosen localizing term, the path integral localizes to flat $G$-connections on $M$ and their fermionic partners, reducing the problem to a moduli-space integral over ${\cal M}_{G}^{0}(M)$ with determinants computed via an ${\cal R}$-equivariant index theorem. The authors derive explicit one-loop determinants for matter and gauge sectors across several geometries, including lens spaces, $T^{2}\times S^{2}$, and elliptic fibrations, expressing results in terms of elliptic gamma, theta, and eta functions and clarifying holomorphic dependence on complex-structure moduli. The final generalized index takes the schematic form $Z_{G,r,M,g,d}=\int_{\mathcal{M}_{G}^{0}(g,d)} e^{-S_{\rm class}} Z_{\rm gauge}^{g,d} Z_{\rm matter}^{g,d,r}$, encoding both topological data and holomorphic moduli, and offering a powerful tool for exploring dualities and operator mappings in 4d $\mathcal{N}=1$ theories. The work also discusses limitations, zero modes, and extensions to broader manifolds, with potential applications to dualities and high-temperature limits tied to conformal anomalies.

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.