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

We compute the supersymmetric partition function of $ \mathcal{N} $ = 1 supersymmetric gauge theories with an R-symmetry on $ {\mathrm{\mathcal{M}}}_4\cong {\mathrm{\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_{\mathrm{\mathcal{M}}}}_{{g,p}} $ , with the supersymmetric three-manifold $ {\mathrm{\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$_{g}$ × 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.