N = 1 supersymmetric indices and the four-dimensional A-model
We compute the supersymmetric partition function of N = 1 supersymmetric
gauge theories with an R-symmetry on M4 ∼= Mg,p × S1, a principal elliptic
fiber bundle of degree p over a genus-g Riemann surface, Σg. Equivalently, we compute
the generalized supersymmetric index IMg,p, with the supersymmetric three-manifold
Mg,p as the spatial slice. The ordinary 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 Σg.
This A-model—or A-twisted two-dimensional 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 T2 inside Σg × T2. 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.
