3d-3d Correspondence Revisited

TL;DR

This work extends the 3d-3d correspondence by enforcing a complete accounting of all flat $G_{ ext{C}}$-connections on a 3-manifold $M_3$ in the associated 3d $ ext{N}=2$ theory $T[M_3]$. It builds explicit abelian-Chern-Simons–matter UV realizations for knot complements (on $S^3ackslash K$) whose SUSY vacua map precisely to flat connections, and presents contour-integral representations of colored knot homology that reproduce Poincaré polynomials $P_K^r(t;q)$ as partition functions of $T[M_3]$, with $x o q^r$ limits interpretably linked to Higgsing and line defects. Taking the $t o-1$ limit recovers DGG theories while preserving non-abelian content, clarifying how abelian branches decouple and how boundary/gluing data are encoded via branes in Hitchin moduli space and via matrix models. The results unify knot-homology categorification with 3d-3d physics, illuminate cutting and gluing along boundaries, and expose open questions about extending to broader manifolds and higher rank groups.

Abstract

In fivebrane compactifications on 3-manifolds, we point out the importance of all flat connections in the proper definition of the effective 3d N=2 theory. The Lagrangians of some theories with the desired properties can be constructed with the help of homological knot invariants that categorify colored Jones polynomials. Higgsing the full 3d theories constructed this way recovers theories found previously by Dimofte-Gaiotto-Gukov. We also consider the cutting and gluing of 3-manifolds along smooth boundaries and the role played by all flat connections in this operation.

Figures (6)

• Figure 1: The space of SUSY vacua (parameters) of the 3d ${\mathcal{N}}=2$ theory $T[M_3]$ has several branches, which often touch at singular points. Turning on $t\neq -1$ resolves (some of) the singularities and reconnects different branches into a single component.
• Figure 2: The index of 3d ${\mathcal{N}}=2$ theories can be generalized to include domain walls and boundary conditions GGP-walls. It is obtained from two copies of the half-index ${\mathcal{I}}_{S^1 \times_q D^{\pm}} (T^{\pm}) \simeq Z_{\text{vortex}} (T^{\pm})$ convoluted via the index (flavored elliptic genus) of the wall supported on $S^1 \times S^1_{\text{eq}}$, where $D^{\pm}$ is the disk covering right (resp. left) hemisphere of the $S^2$ and $S^1_{\text{eq}}:= \partial D^+ = - \partial D^-$ is the equator of the $S^2$.
• Figure 3: Possible integration contours for the trefoil, drawn on the cylinder parametrized by $\log s$. There are three half-lines of poles in the integrand $\Upsilon_{\mathbf{3_1}}(s,x,t;q)$, coming from $(s)^-_\infty,\, (-1/(qst))^-_\infty,\, (x/s)^-_\infty$ in the denominator; and a full line of zeroes from $\theta^-(q^{\frac{3}{2}} sxt^3)$ in the numerator. On the right, we demonstrate a pinching of contours as $x\to q^r$.
• Figure 4: $a)$ M-theory on a Seifert fibered 3-manifold $M_3$, and $b)$ its reduction to type IIA string theory with D6-branes. Upon reduction on the $S^1$ fiber the fivebrane system \ref{['surfeng']} turns into a system of D4-branes wrapped on the (orbifold) surface $\Sigma$ intersecting D6-branes at finitely many points on $\Sigma$.
• Figure 5: Plumbing graph of a Seifert fibered homology 3-sphere with $n$ exceptional fibers.