Orientability of compactified manifolds in string theory

Question

Calabi-Yau manifolds in string theory are orientable (topologically you can make "handed" structures in them). Non-orientable manifolds are perfectly respectable (the projective plane is a basic example), so have they ever been considered as candidate compactified manifolds in string theories?

Answer 1

An M-theory compactification on a Mobius strip is perfectly possible, it gives you a description of the CHL string. Now M theory compactified on a 2-torus is equivalent to the type IIA theory and by performing a T-duality along one $S^{1}$ you obtain a description of the type IIB theory. After acting with the usual orientifold projection you discover that M-theory compactified on a Klein-Bottle (the orientifold image of the compactification torus) is equivalent to type I superstrings (see Strings on Orientifolds). Then compactifications over one and two non-orientable real dimensional manifolds are possible.

Things become complex for higher (complex d=3,4) dimensional compactification manifolds were non-orientable compactification spaces are ruled out. You could see what the problem is by recalling that the Calabi-Yau condition is equivalent to the statement that the holonomy of the internal manifold is SU($3$). The lack of an orientation reduces the SU($3$) holonomy to SO($6$), then the Calabi-Yau condition is violated and supersymmetry is lost.

A curiosity: Notice that not having orientability can be seen as an obstruction to have a complex structure because complex manifolds are always orientable. Nothing like mirror symmetry can be produced in the world of non-orientable manifolds because they have Kahler moduli (associated to local volumes of orientable cycles of the manifold) and nothing to match them under the mirror map (remmember that Kahler moduli on a threefold are exchanged by complex moduli on the mirror threefold under the mirror map). So one of the most important stringy symmetries (mirror symmetry) is lost.