Complex Chern-Simons theory at level k via the 3d-3d correspondence

TL;DR

This work builds a non-perturbative state-integral model for SL$(n,\mathbb C)$ Chern-Simons theory at arbitrary level $k$ using the 3d-3d correspondence and the DGG construction of $T_n[M]$, based on ideal triangulations of a 3-manifold $M$. The approach yields partition functions on squashed lens spaces $L(k,1)$ (and extensions to $L(k,p)$) with holomorphic-antiholomorphic factorization into holomorphic blocks, and it provides a precise quantization of a model phase space, operator algebras, and a functional-analytic framework controlled by angle polytopes to ensure convergence. The theory reproduces known examples (e.g., figure-eight and $5_2$ knots) and supplies a general program for extending to arbitrary triangulated manifolds, plus a detailed extension to general lens spaces via a modified holomorphic-variable formalism. The angle-polytope structure ties convergence and RG-flow properties to the IR behavior of the UV gauge theories $T_n[M]$, linking mathematical state-integrals with the existence of IR SCFTs and holomorphic-block factorization that matches the physics of boundary dualities. Overall, the paper provides a unifying, non-perturbative framework for complex Chern-Simons partition functions at general levels, with concrete computational tools and deep connections to 3d dualities and boundary indices.

Abstract

We use the 3d-3d correspondence together with the DGG construction of theories $T_n[M]$ labelled by 3-manifolds M to define a non-perturbative state-integral model for SL(n,C) Chern-Simons theory at any level k, based on ideal triangulations. The resulting partition functions generalize a widely studied k=1 state-integral as well as the 3d index, which is k=0. The Chern-Simons partition functions correspond to partition functions of $T_n[M]$ on squashed lens spaces L(k,1). At any k, they admit a holomorphic-antiholomorphic factorization, corresponding to the decomposition of L(k,1) into two solid tori, and the associated holomorphic block decomposition of the partition functions of T_n[M]. A generalization to L(k,p) is also presented. Convergence of the state integrals, for any k, requires triangulations to admit a positive angle structure; we propose that this is also necessary for the DGG gauge theory T_n[M] to flow to a desired IR SCFT.

Figures (4)

• Figure 1: Rescaled angle structure on an ideal tetrahedron: $\alpha,\alpha',\alpha">0$, subject to $\alpha+\alpha'+\alpha"=\mathfrak Q/2$. The angles $\alpha,\alpha"$ are identified \ref{['anglemu']} as the imaginary parts of $\mu,\nu$, where $x=\exp \frac{2\pi i}{k}(-ib\mu-m)$ and $y=\exp\frac{2\pi i}{k}(-ib\nu-n)$ parametrize the tetrahedron phase space.
• Figure 2: Assignment of classical edge parameters to an ideal tetrahedron
• Figure 3: Labeling of external edges of the bipyramid, and labeling of tetrahedra that compose it.
• Figure 4: Angle polytopes ${\mathfrak p}[\mathbf{4_1},\mathbf t]={\mathfrak p}[\mathbf{4_1}]$ (left) and ${\mathfrak p}[\mathbf{5_2},\mathbf t]$ (right), with axes in units of $\mathfrak Q/2$.