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\( {\text{SL}}\left( {2,\mathbb{R}} \right) \) Chern-Simons, Liouville, and gauge theory on duality walls

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Abstract

We propose an equivalence of the partition functions of two different 3d gauge theories. On one side of the correspondence we consider the partition function of 3d \( {\text{SL}}\left( {2,\mathbb{R}} \right) \) Chern-Simons theory on a 3-manifold, obtained as a punctured Riemann surface times an interval. On the other side we have a partition function of a 3d \( \mathcal{N} = 2 \) superconformal field theory on S 3, which is realized as a duality domain wall in a 4d gauge theory on S 4. We sketch the proof of this conjecture using connections with quantum Liouville theory and quantum Teichmüller theory, and study in detail the example of the once-punctured torus. Motivated by these results we advocate a direct Chern-Simons interpretation of the ingredients of (a generalization of) the Alday-Gaiotto-Tachikawa relation. We also comment on M5-brane realizations as well as on possible generalizations of our proposals.

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  1. Department of Mathematics, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo, 152-8551, Japan

    Yuji Terashima

  2. Princeton Center for Theoretical Science, Princeton University, Princeton, NJ, 08544, U.S.A.

    Masahito Yamazaki

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Terashima, Y., Yamazaki, M. \( {\text{SL}}\left( {2,\mathbb{R}} \right) \) Chern-Simons, Liouville, and gauge theory on duality walls. J. High Energ. Phys. 2011, 135 (2011). https://doi.org/10.1007/JHEP08(2011)135

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