Supersymmetric partition functions on Riemann surfaces

Source abstract

We present a compact formula for the supersymmetric partition function of 2d N = ( 2 , 2 ) \mathcal {N}=(2,2) , 3d N = 2 \mathcal {N}=2 and 4d N = 1 \mathcal {N}=1 gauge theories on Σ g × T n \Sigma _g \times T^n with partial topological twist on Σ g \Sigma _g , where Σ g \Sigma _g is a Riemann surface of arbitrary genus and T n T^n is a torus with n = 0 , 1 , 2 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 S^1 . For genus g = 1 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 N partition function of ABJM theory on Σ g × S 1 \Sigma _g \times S^1 reproduces the Bekenstein-Hawking entropy of BPS black holes in AdS 4 _4 whose horizon has Σ g \Sigma _g topology.