Aspects of Defects in 3d-3d Correspondence

TL;DR

The work extends the 3d--3d correspondence to include supersymmetric co-dimension 2 and 4 defects arising from M5-branes on 3-manifolds, encoding knot data via monodromy and Wilson-line insertions in complex CS theory. It develops and cross-checks multiple computational frameworks—state-integral models, cluster partition functions, non-Abelian $T_N[M]$ descriptions, 5d ${ m N}=2$ SYM localization, and holographic duals—providing a robust dictionary for defects and deriving general expressions for partition functions, including non-maximal punctures. Key contributions include a general cluster-partition-function formula with loop-operator insertions, a Higgsing-based refinement of CS theory, and large-$N$ holographic predictions for simple and maximal defects. The results offer new computational tools for 3d--3d studies, clarify the role of non-Abelian structures in $T_N[M]$, and yield testable predictions for partition functions and defect observables across CS, 5d gauge theories, and holography, with concrete examples such as figure-eight knot complements and mapping-torus manifolds.

Abstract

In this paper we study supersymmetric co-dimension 2 and 4 defects in the compactification of the 6d $(2,0)$ theory of type $A_{N-1}$ on a 3-manifold $M$. The so-called 3d-3d correspondence is a relation between complexified Chern-Simons theory (with gauge group $SL(N, \mathbb{C})$) on $M$ and a 3d $\mathcal{N}=2$ theory $T_{N}[M]$. We establish a dictionary for this correspondence in the presence of supersymmetric defects, which are knots/links inside the 3-manifold. Our study employs a number of different methods: state-integral models for complex Chern-Simons theory, cluster algebra techniques, domain wall theory $T[SU(N)]$, 5d $\mathcal{N}=2$ SYM, and also supergravity analysis through holography. These methods are complementary and we find agreement between them. In some cases the results lead to highly non-trivial predictions on the partition function. Our discussion includes a general expression for the cluster partition function, in particular for non-maximal punctures and $N>2$. We also highlight the non-Abelian description of the 3d $\mathcal{N}=2$ $T_N[M]$ theory with defect included, as well as its Higgsing prescription and the resulting `refinement' in complex CS theory. This paper is a companion to our shorter paper arXiv:1510.03884, which summarizes our main results.

Figures (22)

• Figure 1: Inside a closed 3-manifold $\hat{M}$, we in general simultaneously include a co-dimension $2$ defect along $K$, and then a co-dimension 4 defect along ${\cal K}$. The two knots, $K$ and ${\cal K}$, can be mutually knotted inside $\hat{M}$.
• Figure 2: The co-dimension 2 defect passes through the small neighborhood of a vertex of an ideal tetrahedron. In general the co-dimension 2 defects passing through four vertices are labeled by different $\rho$s. The octahedron decomposition should be determined for a given choice of $\rho_{1,\ldots, 4}$. There is no general known rule in the literature except when all $\rho$s are maximal, however we will discuss the non-maximal cases in the next section, where we identify octahedron structures (Fig. \ref{['quiver[21]-network']})
• Figure 3: For a single octahedron, we have the wavefunction $\psi_{\hbar, \tilde{\hbar}}(Z, \bar{Z})$ inside the phase space $P(\partial \Diamond)$. The phase space is constructed from three variables $Z, Z', Z"$, satisfying the constraint in eq. \ref{['P_Diamond']}.
• Figure 4: $N$-decomposition of a single tetrahedron. The $m$-th layer has $m(m+1)/2$ octahedra. Octahedrons are labelled by four non-negative integers $({a,b,c,d})$ satisfying $a+b+c+d=N-2$.
• Figure 5: The figure-eight knot $\bm{4}_1$ (left). We consider its knot complement in $S^3$, namely $S^3\backslash \bm{4}_1$. An ideal triangulation for the knot complement is drawn (right).