One-form symmetries and the 3d N=2 A-model: Topologically twisted indices for any G

We study three-dimensional N=2 supersymmetric Chern-Simons-matter gauge theories with a one-form symmetry in the A-model formalism on g S¹. We explicitly compute expectation values of topological line operators that implement the one-form symmetry. This allows us to compute the topologically twisted index on the closed Riemann surface g for any real compact gauge group G. All computations are carried out in the effective A-model on g, whose ground states are the so-called Bethe vacua. We discuss how the 3d one-form symmetry acts on the Bethe vacua, and how its 't Hooft anomaly constrains the vacuum structure. In the special case of the SU (N) K N=2 Chern-Simons theory, we obtain results for the (SU (N) /Zᵣ) ^K N=2 Chern-Simons theories, for all non-anomalous Zᵣ ZN subgroups of the center of the gauge group, and with the associated Zᵣ -angle turned on, reproducing and extending various results in the literature. In particular, we find an interesting mixed 't Hooft anomaly between gravity and the Zᵣ one-form symmetry of the SU (N) K theory (for N even, Nr odd and Kr even). This plays a key role in our derivation of the Witten index, which we explicitly compute for any N, K and r in terms of refinements of Jordan's totient function. Our results lead to precise conjectures about integrality of indices, which appear to have a strong number-theoretic flavour. Note: this paper directly builds upon unpublished notes by Brian Willett from 2020.