Residual group-like symmetries in selection rules without group actions

TL;DR

This paper investigates how residual group-like symmetries persist at loop levels in theories with fusion algebra structures, revealing approximate symmetries that influence coupling magnitudes and selection rules.

Contribution

It introduces the concept of 'groupification' to preserve residual symmetries at loop level and analyzes their implications in heterotic string theories on non-Abelian orbifolds.

Findings

Residual symmetries remain exact through groupification.

Approximate discrete symmetries control loop-induced couplings.

Non-invertible selection rules are naturally explained.

Abstract

We analyze loop-induced group-like symmetries in theories where fields are labeled by basis elements of a fusion algebra constructed from the conjugacy classes of finite groups. Although the fusion rules for conjugacy classes are in general violated at loop level, residual group-like symmetries, including both Abelian and non-Abelian ones, remain exact through a procedure referred to as ``groupification''. By examining various conjugacy classes of finite groups realized in heterotic string theory on non-Abelian orbifolds, we identify an approximate discrete symmetry that controls the magnitude of loop-induced couplings. As a result, most parameters appearing in non-invertible selection rules are natural in the sense of 't Hooft. Furthermore, we discuss anomalies of the groupification symmetry, which can impose additional constraints on models with non-invertible fusion rules.