# What is the stringy interpretation of the cohomology classes arising from the Kähler class?

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 $$\omega$$ on $$M$$ is closed by definition, hence if $$\omega^k=d\alpha$$ for some $$(2k-1)$$-form $$\alpha$$, we find that $$\omega^{k+1}=\omega\wedge d\alpha=d(\omega\wedge \alpha)$$, therefore $$\omega^{k+1}$$ would be exact as well. But $$\omega^n$$, where $$n=\operatorname{dim}_{\Bbb C}M$$, is a volume form, hence by Stokes' theorem it cannot be exact. Thus, $$\omega^k$$ is not exact for any $$k\leq n$$. Equivalent, we have the following condition on the Hodge numbers: $$h^{p,p}(X)\geq 1$$. 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.

• aei.mpg.de/~theisen/lectures.pdf Look at Chapter 3.7 It's about supergravity compactified on a CY-three-fold. It's not superstring theory but a related concept. Maybe you'll find your answer there. May the force be with you. – Physics Guy Oct 30 '16 at 16:23
• @PhysicsGuy The answer is not (clearly?) stated in those notes, as far as I can tell. – Danu Oct 30 '16 at 16:27
• There is an answer at physicsoverflow.org/37611 – Arnold Neumaier Nov 5 '16 at 15:09