Time reversal and $\boldsymbol{CP}$ invariance in Calabi-Yau compactifications

We revisit the question of time reversal and $CP$ invariance in Calabi-Yau compactifications. We show that time reversal invariance is respected by quantum corrections to the prepotential. In particular, field independent $\theta$ angles whose presence is dictated by requiring integrality of relevant monodromy transformations can take precisely the quantized values compatible with time reversal invariance. Furthermore, monodromy symmetry enlarges the region on moduli space on which time reversal is not spontaneously broken. We define the action of the $CP$ transformation for multi-parameter models and argue that on the slice of moduli space where it is defined, $CP$ is trivially a symmetry of the theory. For supersymmetric vacua that lie in this slice, we derive a condition on the third cohomology of the compactification manifold which determines whether $CP$ preserving fluxes exist that stabilize the moduli to such points. In the case of one-parameter models, the condition is always satisfied.