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 Σg× 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 Σg for any real compact gauge group G. All computations are carried out in the effective A-model on Σg, 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)θK N=2 Chern-Simons theories, for all non-anomalous Zᵣ ⊆ ZN subgroups of the center of the gauge group, and with the associated Zᵣ θ-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, N/r odd and K/r 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 Note: this paper directly builds upon unpublished notes by Brian Willett from 2020.