Orbifolds from $\boldsymbol{\mathrm{Sp}(4,\mathbb Z)}$ and their modular symmetries

TL;DR

This paper explores the extension of modular symmetries in string compactifications to the Siegel modular group $ ext{Sp}(4, ext{Z})$, identifying orbifolds and their flavor symmetries relevant for Standard Model physics.

Contribution

It classifies 13 orbifolds of $ ext{Sp}(4, ext{Z})$ and determines their modular flavor symmetries, including symmetric and asymmetric cases with potential applications in flavor physics.

Findings

Identified 13 possible orbifolds of $ ext{Sp}(4, ext{Z})$.

Determined modular flavor symmetries for each orbifold.

Found connections between symmetric and asymmetric orbifolds with implications for flavor model building.

Abstract

The incorporation of Wilson lines leads to an extension of the modular symmetries of string compactification beyond $\mathrm{SL}(2,\mathbb Z)$. In the simplest case with one Wilson line $Z$, K\"ahler modulus $T$ and complex structure modulus $U$, we are led to the Siegel modular group $\mathrm{Sp}(4,\mathbb Z)$. It includes $\mathrm{SL}(2,\mathbb Z)_T\times\mathrm{SL}(2,\mathbb Z)_U$ as well as $\mathbb Z_2$ mirror symmetry, which interchanges $T$ and $U$. Possible applications to flavor physics of the Standard Model require the study of orbifolds of $\mathrm{Sp}(4,\mathbb Z)$ to obtain chiral fermions. We identify the 13 possible orbifolds and determine their modular flavor symmetries as subgroups of $\mathrm{Sp}(4,\mathbb Z)$. Some cases correspond to symmetric orbifolds that extend previously discussed cases of $\mathrm{SL}(2,\mathbb Z)$. Others are based on asymmetric orbifold twists (including mirror symmetry) that do no longer allow for a simple intuitive geometrical interpretation and require further study. Sometimes they can be mapped back to symmetric orbifolds with quantized Wilson lines. The symmetries of $\mathrm{Sp}(4,\mathbb Z)$ reveal exciting new aspects of modular symmetries with promising applications to flavor model building.