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

TL;DR

This work links 3d ${ m N}=2$ theories to 3-manifold topology by unifying the DGG construction with a 6d ${ m A}_1$ (2,0) compactification picture. It introduces a topological criterion for when $U(1)_{X_A}$ enhances to ${SU(2)}$ or ${SO(3)}$ in torus-boundary theories and extends the framework to closed 3-manifolds via Dehn filling, predicting infinite 3d dualities from knot-surgery calculus. A striking result is the discovery of ${SU}(3)$ symmetry enhancements for all hyperbolic twist knots, arising from nontrivial interplay between boundary data types and dualities. The approach reveals deep connections between 3d dualities and 3-manifold operations, offering a robust, topological perspective on IR symmetries and dualities in ${ m N}=2$ 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.

Figures (6)

• Figure 1: The choice of a knot $K$ inside a closed 3-manifold $M$ can be alternatively described by a choice of knot complement $N$ and a boundary cycle $A \in H_1 (\partial N, \mathbb{Z})$.
• Figure 2: Edge parameters $(z,z',z")$ of an ideal tetrahedron. Left: ideal tetrahedron in $\mathbb{H}^3=\{(y,w):y\in \mathbb{R}_+, w \in \mathbb{C}\}$ with metric $ds^2 (\mathbb{H}^3) = \frac{dy^2 +d w d \bar{w}}{y^2}$. Using the isometry of $\mathbb{H}^3$, $PSL(2,\mathbb{C})$, four asymptotic vertices can be placed at $(y,w) = (0,0),(0,1),(0,z)$ and $(\infty, \cdot)$. Right: topologically, ideal tetrahedron is a tetrahedron with truncated vertices.
• Figure 3: The simplest ideal triangulation of $m004=S^3\backslash \mathbf{4_1}$.
• Figure 4: Whitehead link ($\mathbf{5}^2_1$). It is one of the simplest (having smallest volume $\simeq 3.664$) two-component hyperbolic link.
• Figure 5: A rational surgery calculus rolfsen1984rational shows that $(S^3\backslash \mathbf{5^2_1})_{\mu_2 +k \lambda_2}$ is a twist knot $K_k$. For example, $K_{k=1}= \mathbf{4_1}, K_{k=-2}=\mathbf{5_2}$ and $K_{k=2}= \mathbf{6}_1$.