A topologically twisted index for three-dimensional supersymmetric theories

We provide a general formula for the partition function of three-dimensional $\mathcal{N}=2$ gauge theories placed on $S^2\times S^1$ with a topological twist along $S^2$, which can be interpreted as an index for chiral states of the theories immersed in background magnetic fields. The result is expressed as a sum over magnetic fluxes of the residues of a meromorphic form which is a function of the scalar zero-modes. The partition function depends on a collection of background magnetic fluxes and fugacities for the global symmetries. We illustrate our formula in many examples of 3d Yang-Mills-Chern-Simons theories with matter, including Aharony and Giveon-Kutasov dualities. Finally, our formula generalizes to $\mathrm{\Omega }$-backgrounds, as well as two-dimensional theories on $S^2$ and four-dimensional theories on $S^2\times T^2$. In particular this provides an alternative way to compute genus-zero A-model topological amplitudes and Gromov-Witten invariants.