Siegel modular flavor group and CP from string theory

TL;DR

This paper explores the modular symmetries in heterotic string theory compactifications, identifying the Siegel modular group and its extension with a CP-like symmetry, which has implications for flavor physics and CP violation.

Contribution

It derives the potential modular symmetries, including the Siegel modular group and its extension, from heterotic string theory compactifications with Wilson lines.

Findings

Identifies the Siegel modular group $ ext{Sp}(4, ext{Z})$ as a symmetry in heterotic string compactification.

Shows the inclusion of $ ext{SL}(2, ext{Z})$ symmetries for geometric moduli $T$ and $U$.

Proposes a CP-like symmetry extending the modular group to $ ext{GSp}(4, ext{Z})$.

Abstract

We derive the potential modular symmetries of heterotic string theory. For a toroidal compactification with Wilson line modulus, we obtain the Siegel modular group $\mathrm{Sp}(4,\mathbb{Z})$ that includes the modular symmetries $\mathrm{SL}(2,\mathbb{Z})_T$ and $\mathrm{SL}(2,\mathbb{Z})_U$ (of the "geometric" moduli $T$ and $U$) as well as mirror symmetry. In addition, string theory provides a candidate for a CP-like symmetry that enhances the Siegel modular group to $\mathrm{GSp}(4,\mathbb{Z})$.