Siegel modular flavor group and CP from string theory

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