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

<p>We compute the supersymmetric partition function of $\mathcal{N}$ = 1 supersymmetric gauge theories with an R-symmetry on $\mathcal{M}$₄ ≅ $\mathcal{M}$_<em>g,p</em> × <em>S</em>¹, a principal elliptic fiber bundle of degree <em>p</em> over a genus-<em>g</em> Riemann surface, Σ_<em>g</em> . Equivalently, we compute the generalized supersymmetric index <em>I</em>_{$\mathcal{M}$_<em>g,p</em>} , with the supersymmetric three-manifold $\mathcal{M}$_<em>g,p</em> 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 <em>A</em>-model for the abelianized gauge fields on the base Σ_<em>g</em>. This <em>A</em>-model — or <em>A</em>-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 <em>T</em>² inside Σ_<em>g</em> × <em>T</em>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.</p>