Symmetries of stationary points of the $G$-invariant potential and the framework of the auxiliary group

TL;DR

This paper classifies constraints on stationary points of G-invariant potentials, showing how intrinsic and extrinsic constraints relate to symmetry groups, and demonstrates that effective theories can generate intrinsic constraints that are extrinsic in elementary theories.

Contribution

It introduces a classification of constraints on stationary points into intrinsic and extrinsic, and applies this to the auxiliary group framework to unify how constraints are generated.

Findings

Symmetry group of stationary points can exceed that of the potential.

Stabilizer under the symmetry group generates intrinsic constraints.

Effective theories can intrinsically generate constraints that are extrinsic in elementary theories.

Abstract

We classify the constraints on a stationary point of the potential invariant under a finite group into intrinsic and extrinsic based on whether they are independent of the coefficients in the potential or not. We find that the symmetry group of a set of stationary points can be larger than that of the potential and the stabilizer under this group generates intrinsic constraints. By applying these findings in the framework of the auxiliary group, we show that the constraints that can only be obtained extrinsically in an elementary theory can be generated intrinsically in an effective theory.