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Published December 7, 2021 | Submitted
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Symmetries of 2d TQFTs and Equivariant Verlinde Formulae for General Groups


We study (generalized) discrete symmetries of 2d semisimple TQFTs. These are 2d TQFTs whose fusion rules can be diagonalized. We show that, in this special basis, the 0-form symmetries always act as permutations while 1-form symmetries act by phases. This leads to an explicit description of the gauging of these symmetries. One application of our results is a generalization of the equivariant Verlinde formula to the case of general Lie groups. The generalized formula leads to many predictions for the geometry of Hitchin moduli spaces, which we explicitly check in several cases with low genus and SO(3) gauge group.

Additional Information

We would like to thank Dan Freed, Po-Shen Hsin, Andrew Neitzke, and Sebastian Schulz for useful discussions. We would also like to thank Simons Center for Geometry and Physics for generous hospitality during Graduate Summer School on the Mathematics and Physics of Hitchin Systems and the Simons Summer Workshop 2021. The work of SG is supported by the U.S. Department of Energy, Office of Science, Office of High Energy Physics, under Award No. DE-SC0011632, and by the National Science Foundation under Grant No. NSF DMS 1664227. The work of AS was supported by NSF grants DMS-2005312 and DMS1711692. The work of DP was supported by the Center for Mathematical Sciences and Application. This paper is partly a result of the ERC-SyG project, Recursive and Exact New Quantum Theory (ReNewQuantum) which received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme under grant agreement No 810573.

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