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

Understand

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.