Symmetry enhancement and closing of knots in 3d/3d correspondence

TL;DR

This paper revisits the construction of 3d gauge theories from 3-manifolds, clarifies symmetry enhancements, and predicts new dualities and symmetry phenomena in theories related to knot complements and Dehn fillings.

Contribution

It introduces a topological criterion for symmetry enhancement, generalizes the construction to closed 3-manifolds, and predicts novel dualities and symmetry enhancements in 3d theories.

Findings

Symmetry enhancement criterion for $SU(2)/SO(3)$ in 3d theories.

Generalization of 3d gauge theories to closed 3-manifolds via Dehn filling.

Prediction of $SU(3)$ symmetry enhancement in hyperbolic twist knot theories.

Abstract

We revisit Dimofte-Gaiotto-Gukov's construction of 3d gauge theories associated to 3-manifolds with a torus boundary. After clarifying their construction from a viewpoint of compactification of a 6d $\mathcal{N}=(2,0)$ theory of $A_1$-type on a 3-manifold, we propose a topological criterion for $SU(2)/SO(3)$ flavor symmetry enhancement for the $u(1)$ symmetry in the theory associated to a torus boundary, which is expected from the 6d viewpoint. Base on the understanding of symmetry enhancement, we generalize the construction to closed 3-manifolds by identifying the gauge theory counterpart of Dehn filling operation. The generalized construction predicts infinitely many 3d dualities from surgery calculus in knot theory. Moreover, by using the symmetry enhancement criterion, we show that theories associated to all hyperboilc twist knots have surprising $SU(3)$ symmetry enhancement which is unexpected from the 6d viewpoint.