Compactification Without Orientation, or a Topological Scenario for $CP$ Violation

In higher dimensional theories, we often assume that the extra dimensions form an orientable space, perhaps with singularities. However, many physical theories are well-defined on non-orientable spaces, and many spaces are not orientable, so it is reasonable to explore what happens if the assumption of orientability is relaxed. Here we consider the simplest example of free 6D theories compactified on a flat Klein bottle. We focus on a Dirac fermion in 6D, with boundary conditions that define ${\rm pin}^+$ and ${\rm pin}^-$ structures. Translation invariance is broken by the boundary conditions, which leads to sharp features localized near the parity walls (fixed points of the reflection used to construct the Klein bottle). For a scalar field, there is a position-dependent energy density, peaked near the parity walls. A Dirac fermion can lead to breaking of parity, charge conjugation and $CP$ in 3+1 dimensions. Order parameters for this breaking are provided by the vevs of certain fermion bilinears, again peaked near the parity walls. As one potential application, these results suggest mechanisms for $CP$ violation and baryogenesis, enabled by compactification on a Klein bottle.