Supersymmetric partition functions on Riemann surfaces

We present a compact formula for the supersymmetric partition function of 2d N = (2, 2), 3d N = 2 and 4d AT = 1 gauge theories on Sigma(g) x T-n with partial topological twist on Sigma(g), where Sigma(g) is a Riemann surface of arbitrary genus and T-n is a torus with n = 0, 1, 2, respectively. In 2d we also include certain local operator insertions, and in 3d we include Wilson line operator insertions along S-1. For genus g = 1, the formula computes the Witten index. We present a few simple Abelian and non-Abelian examples, including new tests of non-perturbative dualities. We also show that the large N partition function of ABJM theory on Sigma(g) x S-1 reproduces the Bekenstein-Hawking entropy of BPS black holes in AdS4 whose horizon has Sigma(g) topology.