  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.
