# CP as a Symmetry of Symmetries

## TL;DR

This paper explores CP as an outer automorphism of gauge and space-time symmetries, illustrating how it can be spontaneously violated or preserved depending on the symmetry structure, with implications for understanding CP violation.

## Contribution

It introduces the concept of CP as a symmetry of symmetries via outer automorphisms and demonstrates its role in physical theories and symmetry breaking scenarios.

## Key findings

- CP corresponds to an outer automorphism of gauge and space-time symmetries.
- Outer automorphisms can be absent or present, affecting CP violation.
- A toy model shows spontaneous CP violation and topological vacuum protection.

## Abstract

It is explained that the Standard Model combined charge conjugation and parity transformation (CP) is a simultaneous complex conjugation outer automorphism transformation of gauge and space-time symmetries. Simple examples are given for the general concept of outer automorphisms ("symmetries of symmetries"), as well as for their possible actions on physical theories. It is highlighted that complex conjugation outer automorphisms do not, in general, exist for all symmetries. Examples are given for cases in which the physical CP transformation is violated as a consequence of requiring another symmetry. A toy model is illustrated in which CP is spontaneously violated in the broken phase of a continuous gauge symmetry, while an unbroken outer automorphism protects the topological vacuum angle at $\theta=0$.