  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.
