Higher-Form Symmetries, Bethe Vacua, and the 3d-3d Correspondence

TL;DR

This paper refines the understanding of 3d theories from 6d $(2,0)$ theory compactified on three-manifolds by incorporating higher-form symmetries, linking Bethe vacua, Witten index, and flat connections in the 3d-3d correspondence.

Contribution

It introduces a refined framework for 3d theories with higher-form symmetries, connecting Bethe equations, Witten index, and flat connections for Seifert manifolds.

Findings

Computed a refined Witten index for Seifert manifolds.

Established a match between the refined Witten index and flat connection counting.

Extended the 3d-3d correspondence to include higher-form symmetries.

Abstract

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 $\mathcal{N}=2$ and $\mathcal{N}=1$ 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 higher-form symmetries into account.