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

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 $Z$, Kähler 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.

27 pages + appendices, 2 tables

References in corpus (4)

• Unification of Flavor, CP, and Modular Symmetries
• Eclectic flavor scheme from ten-dimensional string theory -- I. Basic results
• K3 surfaces, modular forms, and non-geometric heterotic compactifications
• Siegel modular flavor group and CP from string theory