$\mathcal{N}=1$ 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 $$\mathcal{M}$4 \cong $\mathcal{M}${g,p}\times S^1$,aprincipalellipticfiberbundleofdegree$p$overagenus-$g$Riemannsurface,$\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 ${\mathrm{\Sigma }}_g$. 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 $\Sigma_g \times T^2$\mathrm{.}Beingdefinedbycompactificationonthetorus,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.