Selfduality and Chern-Simons Theory

TL;DR

This work proposes that the operator implementing S-duality in 4D ${ m N}=4$ SYM can be realized, after twisting and circle compactification, as a three-dimensional topological theory, conjectured to be pure Chern-Simons theory for sufficiently small gauge rank. By constructing the S-duality kernel and analyzing metric independence, the authors connect duality to a local 3D action and derive Wilson-loop correlation structures that reflect Chern-Simons-like linking phases. They develop a parallel detour through T-duality twists in 2D sigma-models, showing geometric quantization of target spaces (e.g., Hitchin moduli) as the 0+1D low-energy limit in simple cases, and then extend to the nonabelian 4D case by relating Chern-Simons levels to the S-duality twists. The paper tests the conjecture via Witten-index computations in various limits, discusses the Hitchin fibration and its cohomology, and outlines a six-dimensional perspective from the $(2,0)$ theory; while sign ambiguities remain, the framework provides a bridge between S-duality, geometric quantization, and topological field theory with potential applications to knot invariants and holography.

Abstract

We propose a relation between the operator of S-duality (of N=4 super Yang-Mills theory in 3+1D) and a topological theory in one dimension lower. We construct the topological theory by compactifying N=4 super Yang-Mills on a circle with an S-duality and R-symmetry twist. The S-duality twist requires a selfdual coupling constant. We argue that for a sufficiently low rank of the gauge group the three-dimensional low-energy description is a topological theory, which we conjecture to be a pure Chern-Simons theory. This conjecture implies a connection between the action of mirror symmetry on the sigma-model with Hitchin's moduli space as target space and geometric quantization of the moduli space of flat connections on a Riemann surface.