Higher-form symmetries, Bethe vacua, and the 3d-3d correspondence

By incorporating higher-form symmetries, we propose a refined definition of the theories obtained by compactification of the 6d (2, 0) theory on a three-manifold M-3. This generalization is applicable to both the 3d N = 2 and N = 1 supersymmetric reductions. An observable that is sensitive to the higher-form symmetries is the Witten index, which can be computed by counting solutions to a set of Bethe equations that are determined by M-3. This is carried out in detail for M-3 a Seifert manifold, where we compute a refined version of the Witten index. In the context of the 3d-3d correspondence, we complement this analysis in the dual topological theory, and determine the refined counting of flat connections on M-3, which matches the Witten index computation that takes the higherform symmetries into account.