Chern - Simons Theory and S - duality

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 ~ → −4π2 ~ 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. CALT-68-2841 ar X iv :1 10 6. 45 50 v1 [ he pth ] 2 2 Ju n 20 11