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

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.

91 pages including appendices; v2: corrected typos and added references, JHEP version

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