A topologically twisted index for three-dimensional supersymmetric theories

TL;DR

This paper derives a general formula for the topologically twisted index of 3d $ ext{N}=2$ gauge theories on $S^2 imes S^1$, enabling computation of chiral state counts in magnetic backgrounds and connecting to topological invariants.

Contribution

It introduces a unified residue-based formula for the index, applicable to various 3d theories and backgrounds, including dualities and higher-dimensional generalizations.

Findings

Validates the formula with examples of 3d Yang-Mills-Chern-Simons theories.

Shows the formula's consistency with known dualities like Aharony and Giveon-Kutasov.

Extends the approach to compute topological amplitudes and Gromov-Witten invariants.

Abstract

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 $\Omega$-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.