Abstract
Gauging a finite group 0-form symmetry G of a quantum field theory (QFT) results in a QFT with a Rep(G) symmetry implemented by Wilson lines. The group G determines the fusion of Wilson lines. However, in general, the fusion rules of Wilson lines do not determine G. In this paper, we study the properties of G that can be determined from the fusion rules of Wilson lines and surface operators obtained from higher-gauging Wilson lines. This is in the spirit of Richard Brauer who asked what information in addition to the character table of a finite group needs to be known to determine the group. We show that fusion rules of surface operators obtained from higher-gauging Wilson lines can be used to distinguish infinite pairs of groups which cannot be distinguished using the fusion of Wilson lines. We derive necessary conditions for two non-isomorphic groups to have the same surface operator fusion and find a pair of such groups.
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Acknowledgments
The author thanks ICTP for support. We thank Matthew Buican and Adrian Padellaro for numerous discussions, collaborations on related projects and detailed comments on a draft of this article. We thank Sanjaye Ramgoolam for collaboration on a related project.
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Radhakrishnan, R. On reconstructing finite gauge group from fusion rules. J. High Energ. Phys. 2024, 43 (2024). https://doi.org/10.1007/JHEP02(2024)043
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DOI: https://doi.org/10.1007/JHEP02(2024)043