Spontaneous CP violation and symplectic modular symmetry in Calabi-Yau compactifications

TL;DR

This paper investigates the geometric origins of CP symmetry and its spontaneous violation in Calabi-Yau compactifications, linking it to symplectic modular groups and flux configurations.

Contribution

It identifies CP with an outer automorphism of the symplectic modular group and demonstrates spontaneous CP violation through fluxes in Calabi-Yau threefolds.

Findings

CP symmetry corresponds to an outer automorphism of the symplectic modular group.

Spontaneous CP violation occurs via fluxes that preserve certain discrete symmetries.

Explicit example of CP violation demonstrated on a specific Calabi-Yau threefold.

Abstract

We explore the geometrical origin of CP and the spontaneous CP violation in Calabi-Yau compactifications. We find that the CP symmetry is identified with an outer automorphism of the symplectic modular group in the large complex structure regime of Calabi-Yau threefolds, thereby enlarging the symplectic modular group to their semidirect product group. The spontaneous CP violation is realized by the introduction of fluxes, whose effective action is invariant under CP as well as the discrete $\mathbb{Z}_2$ symmetry or $\mathbb{Z}_4$ R-symmetry. We explicitly demonstrate the spontaneous CP violation on a specific Calabi-Yau threefold.