$\mathcal{N}{=}1$ supersymmetric indices and the four-dimensional A-model

TL;DR

This paper computes the supersymmetric index of 4D $ ext{N}=1$ theories on complex manifolds using a topological A-model approach, revealing modular properties and testing Seiberg duality.

Contribution

It introduces a novel A-model framework for calculating 4D $ ext{N}=1$ supersymmetric indices on complex fibered manifolds, connecting to surface defects and modular properties.

Findings

Derived a new formula for the three-sphere index as a sum over 2D vacua.

Demonstrated the modular behavior governed by 4D 't Hooft anomalies.

Provided new tests of Seiberg duality using the generalized indices.

Abstract

We compute the supersymmetric partition function of $\mathcal{N}{=}1$ supersymmetric gauge theories with an $R$-symmetry on $\mathcal{M}_4 \cong \mathcal{M}_{g,p}\times S^1$, a principal elliptic fiber bundle of degree $p$ over a genus-$g$ Riemann surface, $\Sigma_g$. Equivalently, we compute the generalized supersymmetric index $I_{\mathcal{M}_{g,p}}$, with the supersymmetric three-manifold ${\mathcal{M}_{g,p}}$ as the spatial slice. The ordinary $\mathcal{N}{=}1$ supersymmetric index on the round three-sphere is recovered as a special case. We approach this computation from the point of view of a topological $A$-model for the abelianized gauge fields on the base $\Sigma_g$. This $A$-model---or $A$-twisted two-dimensional $\mathcal{N}{=}(2,2)$ gauge theory---encodes all the information about the generalized indices, which are viewed as expectations values of some canonically-defined surface defects wrapped on $T^2$ inside $\Sigma_g \times T^2$. Being defined by compactification on the torus, the $A$-model also enjoys natural modular properties, governed by the four-dimensional 't Hooft anomalies. As an application of our results, we provide new tests of Seiberg duality. We also present a new evaluation formula for the three-sphere index as a sum over two-dimensional vacua.