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Chern-Simons theory and S-duality

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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, \( \mathbb{C} \)) connections on M is identified with the space of supersymmetric vacua in a dual 3d gauge theory. The hidden symmetry of SL(2) Chern-Simons theory can be identified as the S-duality transformation of \( \mathcal{N}=4 \) 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 \( \mathcal{N}=2 \) 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.

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Authors and Affiliations

  1. Institute for Advanced Study, Einstein Dr., Princeton, NJ, 08540, U.S.A.

    Tudor Dimofte

  2. Trinity College, Cambridge, CB2 1TQ, U.K.

    Tudor Dimofte

  3. California Institute of Technology, 1200 E California Blvd., Pasadena, CA, 91125, U.S.A.

    Sergei Gukov

  4. Max-Planck-Institut für Mathematik, Vivatsgasse 7, D-53111, Bonn, Germany

    Sergei Gukov

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  1. Tudor Dimofte
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Dimofte, T., Gukov, S. Chern-Simons theory and S-duality. J. High Energ. Phys. 2013, 109 (2013). https://doi.org/10.1007/JHEP05(2013)109

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