Chern-Simons Theory and S-duality

TL;DR

This work unifies analytically continued SL$(2)$ Chern-Simons theory with higher-dimensional dualities by embedding CS in the 6d $(2,0)$ framework and exploiting the Hitchin moduli space $Y=box{ \\mathcal{M}_H(G,C) }$ as a common phase space. It develops three holomorphic coordinate systems on $Y$ (loop, shear, Fenchel-Nielsen) and clarifies how mapping cylinders and tori induce Lagrangian correspondences that generate duality kernels, linking to ${ m N}=2$ gauge theories, Liouville/Teichmüller theory, and 3d domain walls via the AGT/Witten-DGG programs. A central achievement is the explicit construction of ${ m N}=4$ S-duality as a factorization of quantum Teichmüller, loop, and Fenchel-Nielsen algebras into dual pairs, with kernels that are annihilated by dual operator equations, thereby providing a concrete, checkable framework for dualities across multiple dimensions. The paper also shows how classical invariants like the A-polynomial and its quantum counterpart arise as twisted superpotentials and D-modules of dual branes, highlighting the geometric and representation-theoretic underpinnings of knot complements and mapping tori. Overall, the results illuminate how S-duality acts on flat connections and their moduli, yielding a coherent, cross-disciplinary picture that connects Chern-Simons theory, supersymmetric gauge theories, Liouville/Tek theory, and geometric representation theory.

Abstract

We study S-dualities in analytically continued SL(2) Chern-Simons theory on a 3-manifold M. By realizing Chern-Simons theory via a compactification of a 6d five-brane theory on M, various objects and symmetries in Chern-Simons theory become related to objects and operations in dual 2d, 3d, and 4d theories. For example, the space of flat SL(2,C) connections on M is identified with the space of supersymmetric vacua in a dual 3d gauge theory. The hidden symmetry "hbar -> - (4 pi^2)/hbar" of SL(2) Chern-Simons theory can be identified as the S-duality transformation of N=4 super-Yang-Mills theory (obtained by compactifying the five-brane theory on a torus); whereas the mapping class group action in Chern-Simons theory on a three-manifold M with boundary C is realized as S-duality in 4d N=2 super-Yang-Mills theory associated with the Riemann surface C. We illustrate these symmetries by considering simple examples of 3-manifolds that include knot complements and punctured torus bundles, on the one hand, and mapping cylinders associated with mapping class group transformations, on the other. A generalization of mapping class group actions further allows us to study the transformations between several distinguished coordinate systems on the phase space of Chern-Simons theory, the SL(2) Hitchin moduli space.

Figures (7)

• Figure 1: Two compactifications of the fivebrane theory. Eventually we will twist the product $C\times{\mathbb R}$ by an action "$\varphi$" of $\mathcal{N}=2$ S-duality group ( i.e. mapping class group of $C$) to form more interesting manifolds $M$.
• Figure 2: Loops on the punctured torus.
• Figure 3: Sandwiching of coordinate-transformation cylinders
• Figure 4: The basic flip for a punctured torus.
• Figure 5: Gluing a hyperbolic pair of pants to form $C=T^2\backslash\{p\}$.