Comments on twisted indices in 3d supersymmetric gauge theories

We study three-dimensional N = 2 supersymmetric gauge theories on Σg × S 1 with a topological twist along Σg, a genus-g Riemann surface. The twisted supersymmetric index at genus g and the correlation functions of half-BPS loop operators on S 1 can be computed exactly by supersymmetric localization. For g = 1, this gives a simple UV computation of the 3d Witten index. Twisted indices provide us with a clean derivation of the quantum algebra of supersymmetric Wilson loops, for any Yang-Mills-Chern-Simonsmatter theory, in terms of the associated Bethe equations for the theory on R 2 × S 1 . This also provides a powerful and simple tool to study 3d N = 2 Seiberg dualities. Finally, we study A- and B-twisted indices for N = 4 supersymmetric gauge theories, which turns out to be very useful for quantitative studies of three-dimensional mirror symmetry. We also briefly comment on a relation between the S 2 × S 1 twisted indices and the Hilbert series of N = 4 moduli spaces.