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}=left(2,0right) theory of A1-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.
Highlights
R-symmetry and allows a 1/2 BPS regular co-dimension two defect
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 N = (2, 0) theory of A1-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
A hint comes from so called 3d/3d relations [6,7,8,9,10] which says that the partition functions of the T 6d[M, K] theory on supersymmetric curved backgrounds are equal to the partition functions of purely bosonic SL(2, C) Chern-Simons (CS) theories on M with a monodromy defect along K
Summary
In [19], a combinatorial way of constructing a 3d SCFT, which we denote T DGG[N, XA], for given choice of (N, A) is proposed. In the construction of T DGG theory, we need to choose conjugate variables {PB, ΓI } These choices only affect the background Chern-Simons coupling coupled to flavor symmetries. To specify the background Chern-Simons coupling of the U(1)XA flavor symmetry associated to the knot, we sometimes specify the choice of boundary cycle B and denote the theory by. The theory constructed by six tetrahedra has a hidden additional u(1) symmetry in the low energy limit which corresponds to the hard edge in the triangulation with two tetrahedra. A possibility is that T DGG might be identified with Ti6rrded in (2.7) Both of them are genuine 3d theories and they are associated to the hyperbolic connection Ahyp which can be realized in ideal triangulation.
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