In superstring theory, one usually considers compactifications on Calabi-Yau 3-manifolds. These manifolds are in particular compact Kähler, hence possess a Kähler class which gives rise to nontrivial cohomology classes in every even degree. To see this, note that the Kähler class ω on M is closed by definition, hence if ωk=dα for some (2k−1)-form α, we find that ωk+1=ω∧dα=d(ω∧α), therefore ωk+1 would be exact as well. But ωn, where n=dimCM, is a volume form, hence by Stokes' theorem it cannot be exact. Thus, ωk is not exact for any k≤n. Equivalent you, we have the following condition on the Hodge numbers: hp,p(X)≥1. Now, for my actual question:
Since the powers of the Kähler class always generate nontrivial cohomology classes, these can in some sense be called universal. I was wondering if there is a nice interpretation of these classes in string theory.
I vaguely recall that one can interpret the cohomology classes of the Calabi-Yau manifold that one compactifies on in terms of the multiplets (under supersymmetry) of the resulting effective four-dimensional description, and in particular I'm hoping that the Kähler class gives rise to some kind of universal multiplets.
This post imported from StackExchange Physics at 2016-11-02 09:06 (UTC), posted by SE-user Danu