Compactification without orientation, or a topological scenario for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>C</mml:mi> <mml:mi>P</mml:mi> </mml:math> 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 nonorientable 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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:msup> <a:mi>pin</a:mi> <a:mo>+</a:mo> </a:msup> </a:math> and <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:msup> <c:mi>pin</c:mi> <c:mo>−</c:mo> </c:msup> </c:math> 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 <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mi>C</e:mi> <e:mi>P</e:mi> </e:math> in <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:mn>3</g:mn> <g:mo>+</g:mo> <g:mn>1</g:mn> </g:math> dimensions. Order parameters for this breaking are provided by the vacuum expectation values of certain fermion bilinears, again peaked near the parity walls. As one potential application, these results suggest mechanisms for <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:mi>C</i:mi> <i:mi>P</i:mi> </i:math> violation and baryogenesis, enabled by compactification on a Klein bottle.