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

TL;DR

The work develops a refined framework for 3d theories T[M_3] from the 6d (2,0) theory by incorporating higher-form symmetries, necessitating a polarization data H to define T[M_3,g,H]. Using graph-manifold decompositions, it constructs widehat{T}[{fOmega},G] quivers and then decouples topological sectors to obtain physical theories with well-defined 0- and 1-form symmetries. The refined Witten index, computed via Bethe-vacua and twisted superpotentials, is matched with refined counts of flat G^C connections on M_3 through the 3d-3d correspondence, with extensive explicit results for Seifert, lens space, and Brieskorn-type manifolds. The paper also extends the analysis to 3d N=1 twists, outlining how higher-form symmetries organize the Bethe-vacua structure in these cases and providing several concrete examples. Overall, it clarifies how global structures and higher-form symmetries influence dual observables and the flat-connection dictionary in the 3d-3d correspondence, offering a detailed toolkit for refined topological and gauge-theoretic data in this compactification program.

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.

Figures (10)

• Figure 1: Gauged Trinion: this is the quiver for $T[M_3,\frak{su}(2)]$, where $M_3$ is a Seifert manifold over $S^2$ with special fibers $(k_i,1)$, $i=1,...,3$. Here the circles denote $SU(2)$ gauge nodes, and the labels are the Chern-Simons levels.
• Figure 2: Plumbing graph for Seifert manifolds with $r$ special fibers. For $r=3$ this graph corresponds to the Seifert quivers in the text. We consider here $g=0$ for the central node. This also gives the quiver for the theory $\widehat{T}[{\Omega}_{[d; 0, [k^i_1,\cdots,k^i_{n_i}]]},G]$.
• Figure 3: The relations $TST$, $S^2$ and $(TS)^{-3}$ acting on the plumbing data associated to the three-manifold.
• Figure 4: Quiver description of Lens space quiver $\widehat{T}[{\Omega}_{[k_1, k_2]},G]$, where $p/q= k_1 - 1/k_2$ and the dictionary in (\ref{['QuivBuild']}) is used. On the left we use the $T(G)$ theory, and on the right the $FT(G)$ theory, which is related by flipping an adjoint field. This flipping operation can be represented pictorially by contracting an upward arc on a node and an adjacent downward arc on an edge to leave an upward arc on the edge. Further equivalent descriptions of this theory are shown in figure \ref{['fig:tststduality']} below.
• Figure 6: Heegaard decomposition of $L(p,1)$, where the two solid tori are glued with a $T^p$ transformation. In red is shown the 2 cycle with ${\mathbb Z}_N$ coefficients. For this to be consistently extend to the right side, we must have $p=0 \mod N$, which reflects the fact that the homology group is non-trivial only in this case.