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arXiv · 2302.08419

On Reconstructing Finite Gauge Group from Fusion Rules

Rajath Radhakrishnan
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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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Rajath Radhakrishnan. 2023-02-16. On Reconstructing Finite Gauge Group from Fusion Rules. https://arxiv.org/abs/2302.08419

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