Chern-Simons Theory and S-duality

TL;DR

This paper explores S-duality in SL(2) Chern-Simons theory on 3-manifolds, connecting it to dualities in higher-dimensional theories and revealing symmetries and transformations in the theory's phase space.

Contribution

It establishes a link between Chern-Simons theory dualities and S-duality in super-Yang-Mills theories, providing new insights into the symmetries and coordinate transformations of the theory.

Findings

Identification of flat SL(2,C) connections with supersymmetric vacua

Realization of S-duality as a symmetry in super-Yang-Mills theories

Analysis of mapping class group actions on the phase space

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.