# Orientability of compactified manifolds in string theory

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?

• I don't know about string theory, but field theories on non-orientable manifolds run into trouble if there is since parity may be locally conserved but globally broken (Visser, Lorenzian Wormholes p.285-290). Dec 30, 2019 at 8:25

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.