The Supersymmetric CP problem in Orbifold Compactifications

The possibility of spontaneous breaking of CP symmetry by the expectation values of orbifold moduli is investigated with particular reference to $CP$ violating phases in soft supersymmetry-breaking terms. The effect of different mechanisms for stabilizing the dilaton and the form of the non-perturbative superpotential on the existence and size of these phases is studied. Models with modular symmetries which are subgroups of $PSL(2,Z)$, as well as the single overall modulus $T$ case with the full $PSL(2,Z)$ modular symmetry, are discussed. Non-perturbative superpotentials involving the absolute modular invariant $j(T)$, such as may arise from F-theory compactifications, are considered.