Chern-Simons Theory and S-duality

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.

64 pages, 18 figures

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• 4d Index to 3d Index and 2d TQFT
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• 3-Manifolds and 3d Indices
• Deformed N=2 theories, generalized recursion relations and S-duality
• Gauge Theories Labelled by Three-Manifolds
• Large N Free Energy of 3d N=4 SCFTs and AdS/CFT
• Superconformal Index with Duality Domain Wall
• Higher laminations, webs and N=2 line operators
• Superconformal Index and 3d-3d Correspondence for Mapping Cylinder/Torus
• A 5d/3d duality from relativistic integrable system
• The universal Racah-Wigner symbol for Uq(osp(1|2))
• Complex Chern-Simons from M5-branes on the Squashed Three-Sphere
• Network, Cluster coordinates and N=2 theory I
• Localization principle in SUSY gauge theories
• Static Charges in the Low-Energy Theory of the S-Duality Twist
• 6j symbols for the modular double, quantum hyperbolic geometry, and supersymmetric gauge theories
• Open Verlinde line operators