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.
Highlights
In the last few years there has been a huge development in the study of supersymmetric quantum field theories on compact manifolds in various dimensions
We provide a general formula for the partition function of three-dimensional N = 2 gauge theories placed on S2 × S1 with a topological twist along S2, which can be interpreted as an index for chiral states of the theories immersed in background magnetic fields
For an N = 2 gauge theory on S2 × S1 with gauge group G of rank r, the topologically twisted path-integral localizes on a set of BPS configurations specified by a gauge magnetic flux on
Summary
In the last few years there has been a huge development in the study of supersymmetric quantum field theories on compact manifolds in various dimensions. We use localization to compute the partition function, with a method similar to that recently used to evaluate the elliptic genus of two-dimensional gauge theories [4, 5] and the Witten index of quantum mechanical sigma models [6,7,8]. Index by the angular momentum on S2: in the path-integral formulation this corresponds to turning on an Ω-background on S2 This more general formulation makes contact with the “factorization” of 3d partition functions [14,15,16,17] (as well as with a similar factorization in two [18, 19] and four [20, 21] dimensions). Note added. while we were finishing our work, we became aware that some overlapping results have been obtained by Closset, Cremonesi and Park, and will appear in [27]
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