On Reconstructing Finite Gauge Group from Fusion Rules

TL;DR

This paper investigates how fusion rules of surface operators from higher-gauging Wilson lines can uniquely determine finite groups, surpassing the limitations of Wilson line fusion rules alone.

Contribution

It demonstrates that surface operator fusion rules can distinguish certain groups that Wilson line fusion rules cannot, providing new insights into group reconstruction from quantum field theory data.

Findings

Fusion rules of surface operators can distinguish infinite pairs of groups.

Necessary conditions are derived for groups to have identical surface operator fusion.

A specific pair of non-isomorphic groups with identical surface operator fusion is identified.

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.