Perturbative and nonperturbative aspects of complex Chern-Simons Theory

TL;DR

The article surveys perturbative and nonperturbative aspects of complex Chern-Simons theory with gauge group $SL(N,{ m\textstyle\relax\text} C)$, emphasizing the core obstacle of defining a full TQFT due to infinite-dimensional Hilbert spaces, and highlighting the 3d-3d correspondence as a practical bridge to calculable observables via localization in the dual theory $T[M]$. It develops a concrete picture starting from quantization on the torus, including equivariant and holomorphic polarizations, to motivate how holomorphic blocks emerge and how boundary CFT data play a role in the complex setting. The paper then connects these constructions to three-manifold topology through hyperbolic-volume asymptotics, state-integral models built from angle-structure triangulations, and the $k=0$ index that counts surfaces via M2/M5 brane pictures, all enriched by the 3d-3d viewpoint and string/M-theory intuition. A final focus is the Quantum Modularity Conjecture, which ties the asymptotics of knot invariants to hyperbolic geometry and modular-type structures in the complex CS framework, with lens-space realizations offering a concrete physical realization. Together, these elements yield new 3-manifold invariants, regularization mechanisms via dualities, and deep links to geometry, topology, and number theory.

Abstract

We present an elementary review of some aspects of Chern-Simons theory with complex gauge group SL(N,C). We discuss some of the challenges in defining the theory as a full-fledged TQFT, as well as some successes inspired by the 3d-3d correspondence. The 3d-3d correspondence relates partition functions (and other aspects) of complex Chern-Simons theory on a 3-manifold M to supersymmetric partition functions (and other observables) in an associated 3d theory T[M]. Many of these observables may be computed by supersymmetric localization. We present several prominent applications to 3-manifold topology and number theory in light of the 3d-3d correspondence.

Figures (4)

• Figure 4.1: Ideal (i.e. truncated) tetrahedra, glued together along their large hexagonal faces to form an ideal triangulation of a knot complement $M$. The small triangular faces at truncated vertices become part of the boundary of $M$.
• Figure 4.2: Left: dihedral angles assigned to edges of an oriented ideal tetrahedron, obeying $\alpha+\alpha'+\alpha"=\pi$. Right: operators of the tetrahedron theory $T[\Delta]$ assigned to the same edges; these operators have R-charges $\rho,\rho',\rho"$ that obey $\rho+\rho'+\rho"=1$.
• Figure 4.3: Three incompressible surfaces inside an ideal tetrahedron, corresponding to three unconstrained operators in the theory $T[\Delta]$.
• Figure 5.1: The graph of $\mathcal{J}_K(\alpha)/\mathcal{J}_K(1/\alpha)$ when $K$ is the figure-eight knot, from DIQMF.