Orbifolds from and their modular symmetries

The incorporation of Wilson lines leads to an extension of the modular symmetries of string compactification beyond $\mathrm{S}\mathrm{L}(2,\mathbb{Z})$. 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 $\mathrm{S}\mathrm{p}(4,\mathbb{Z})$. It includes $$\mathrm{S}\mathrm{L}$(2,\mathbb Z)_T\times$\mathrm{S}\mathrm{L}$(2,\mathbb Z)_U$aswellas$\mathbb Z_2$ mirror symmetry, which interchanges $T$ and $U$. Possible applications to flavor physics of the Standard Model require the study of orbifolds of $\mathrm{S}\mathrm{p}(4,\mathbb{Z})$ to obtain chiral fermions. We identify the 13 possible orbifolds and determine their modular flavor symmetries as subgroups of $\mathrm{S}\mathrm{p}(4,\mathbb{Z})$. Some cases correspond to symmetric orbifolds that extend previously discussed cases of $\mathrm{S}\mathrm{L}(2,\mathbb{Z})$. 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 $\mathrm{S}\mathrm{p}(4,\mathbb{Z})$ reveal exciting new aspects of modular symmetries with promising applications to flavor model building.