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

TL;DR

The paper develops a comprehensive framework to compute generalized N=1 supersymmetric indices on four-dimensional backgrounds M_{g,p}×S^1 by recasting the problem as a four-dimensional A-model on Σ_g×T^2. Central ingredients are the twisted superpotential and effective dilaton, whose derivatives generate flux, fibering, and handle-gluing operators, enabling a Bethe-vacua sum that produces the partition function across arbitrary g and p. The authors derive explicit formulas for chiral and vector contributions, establish modular and anomaly-induced transformation properties, and connect to familiar cases such as the S^3 index through a new Bethe-vacua evaluation. They demonstrate powerful nontrivial checks of Seiberg dualities (USp and SU) by matching Bethe vacua and all operator insertions across dual theories, and they discuss dimensional reduction to 3d and Cardy-like small-circle limits. The work provides a unifying, anomaly-aware, localization-based approach to 4d indices that integrates 2d A-model techniques, elliptic function identities, and duality tests, with broad implications for understanding IR dynamics of N=1 gauge theories.

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, $Σ_g$. 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 $Σ_g$. 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 $Σ_g \times T^2$. 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.

Figures (2)

• Figure 1: Integration contour $\mathcal{C}_R^\eta$ for $p>0$ in the rank-one case. The contour in the bulk is determined by the JK residue. Two vertical segments represent the potential singularities at $u = \rightarrow \pm \tau \infty$.
• Figure 2: Decomposition of the boundary integral $\mathcal{C}^\eta_R = \sum_{k\in \mathbb{Z}}\mathcal{C}^\eta_k$. By an appropriate choice of the $R$-charges and of $\eta$, the contour in the bulk cancel each other and only the unit circle contour $a_1=0$ remains.