I am preparing a presentation on Calabi-Yau manifolds where I do not want to use advanced mathematics, but I would like to show that under some minimal assumptions in one (complex) dimension, the compactification of spacetime in string theory would have to be by "folding the extra dimensions into a Calabi-Yau manifold", i.e. a torus.
First, we want it to allow for a Ricci-flat metric, i.e. a flat metric. Assuming Gauss-Bonnet you see that the Euler characteristic must be 0.
For supersymmetry to survive in the compact manifold, we want a Killing spinor, or (which is a bit stronger) a covariantly constant spinor. I thought to relax that to the presence of a nonvanishing vector field, so that Poincaré-Hopf gives us an Euler characteristic of 0 again.
In any case, we see that the Euler characteristic must be 0, that a torus works, but we could still have a Klein bottle.
Can anyone think of a way to make it plausible that the manifold making up the compact dimensions in string theory must be must be orientable? Or another argument why it must admit a complex structure?