Because 4-dimensional $CP$ is a good symmetry of many higher-dimensional theories, this suggests the possible existence of an universal $CP$-violating phase originating from the process of compactification. Such a phase, if it existed, would not be easy to uncover since the phases in Yukawa matrices are not simply related to the observed Cabibbo-Kobayashi-Maskawa phase $\delta$. Hierarchical Yukawa models, of the type arising in $F$-theory GUT models provide an interesting exception. Recently, Heckman and Vafa studied a particular $F$-theory GUT model with hierarchical Yukawa matrices with complex phases of $O(1)$ and showed, by examining the Jarlskog invariant, that this model leads to $\sin \delta \sim O(1)$. A more detailed examination of the model, although confirming their results, is seen to be also compatible with having a phase ${\delta }_o=\pi /3$ imprinted on the Hermitian Yukawa matrices, leading to $\sin \delta \simeq \sin {\delta }_o$.

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