Comments on twisted indices in 3d supersymmetric gauge theories

We study three-dimensional $$ mathcal{n} $$ = 2 supersymmetric gauge theories on σ g × s 1 with a topological twist along σ g , 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-simons-matter theory, in terms of the associated bethe equations for the theory on $$ {mathbb{r}}^2times {s}^1 $$ . this also provides a powerful and simple tool to study 3d $$ mathcal{n} $$ = 2 seiberg dualities. finally, we study a- and b-twisted indices for $$ mathcal{n} $$ = 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 $$ mathcal{n} $$ = 4 moduli spaces.