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

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.

101 pages, 11 figures

References in corpus (10)

• Generalized Global Symmetries
• Supersymmetric Boundary Conditions in N=4 Super Yang-Mills Theory
• Seiberg Duality in Chern-Simons Theory
• Four-Dimensional SCFTs from M5-Branes
• Comments on One-Form Global Symmetries and Their Gauging in 3d and 4d
• Mirrors of 3d Sicilian theories
• The Uncertainty of Fluxes
• Down the rabbit hole with theories of class S
• Bethe/Gauge correspondence on curved spaces
• Flipping the head of T[SU(N)]: mirror symmetry, spectral duality and monopoles

Cited by in corpus (27)

• The Branes Behind Generalized Symmetry Operators
• Non-invertible Symmetries of Class $\mathcal{S}$ Theories
• 2-Group Symmetries in Class S
• Symmetry TFTs for 3d QFTs from M-theory
• Higher Symmetries of 5d Orbifold SCFTs
• Mixed Anomalies, Two-groups, Non-Invertible Symmetries, and 3d Superconformal Indices
• Relative Defects in Relative Theories: Trapped Higher-Form Symmetries and Irregular Punctures in Class S
• Liberating Confinement from Lagrangians: 1-form Symmetries and Lines in 4d N=1 from 6d N=(2,0)
• Generalized Symmetries in F-theory and the Topology of Elliptic Fibrations
• Anomalies of Generalized Symmetries from Solitonic Defects
• Zero-form and one-form symmetries of the ABJ and related theories
• Reflections on the Matter of 3d $\mathcal{N}=1$ Vacua and Local $Spin(7)$ Compactifications
• Magnetic quivers and line defects -- On a duality between 3d N=4 unitary and orthosymplectic quivers
• Generalized Symmetries and Anomalies of 3d N=4 SCFTs
• 3d $\mathcal{N}=4$ mirror symmetry with 1-form symmetry
• On $\mathcal{N}=4$ supersymmetry enhancements in three dimensions
• Twisted indices, Bethe ideals and 3d $\mathcal{N}=2$ infrared dualities
• BPS Indices, Modularity and Perturbations in Quantum K-theory
• Infrared phases of 3D Class R theories
• Comments on Global Symmetries and Anomalies of $5d$ SCFTs
• Free field realisation of boundary vertex algebras for Abelian gauge theories in three dimensions
• On the Witten index of 3d $\mathcal{N}=2$ unitary SQCD with general CS levels
• The 3d $A$-model and generalised symmetries, Part I: bosonic Chern-Simons theories
• 1-form Symmetries of 4d N=2 Class S Theories
• One-form symmetries and the 3d $\mathcal{N}=2$ $A$-model: Topologically twisted indices and CS theories
• Non-hyperbolic 3-manifolds and 3D field theories for 2D Virasoro minimal models
• Wrapped $M5$-branes and complex saddle points