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

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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 you, 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.

This post imported from StackExchange Physics at 2016-11-02 09:06 (UTC), posted by SE-user Danu

edited Nov 2, 2016
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.

This post imported from StackExchange Physics at 2016-11-02 09:07 (UTC), posted by SE-user Physics Guy
@PhysicsGuy The answer is not (clearly?) stated in those notes, as far as I can tell.

This post imported from StackExchange Physics at 2016-11-02 09:07 (UTC), posted by SE-user Danu

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I dont't think that the question really makes sense because when one compactifies string theory on a Calabi-Yau 3-fold, one usually can choose any possible Kähler class: the Kähler class is a continuous parameter and there is no natural/universal Kähler class on a Calabi-Yau 3-fold (which, by the way, is one of the reason of the mathematical difficulties to understand Calabi-Yau varieties).

Consider for example type IIA string theory on a Calabi-Yau 3-fold $X$. The theory in the non-compact four dimensions has $\mathcal{N}=2$ supersymmetry. This theory contains $h^{1,1}(X)$ abelian ($U(1)$) vector multiplets (obtained by reduction of the RR 3-form of 10d IIA string on harmonic (1,1)-forms on $X$). But again, there is no natural/universal choice among these $h^{1,1}(X)$  vector multiplets (any linear combination  is as good as any of them). Classes of type $(p,p)$ are also particularly relevant in this context because RR-charges of supersymmetric D-branes are cohomology classes of type $(p,p)$ (essentially because they correspond to holomorphic submanifolds of $X$).

One could argue that what I have written above does not apply to $\mathcal{N}=1$ supersymmetric systems, like Heterotic string compactified on $X$ or II string with fluxes on $X$, where one can have a potential fixing the Kähler class at some specific value of $X$. But such potential will in general be the result of strong coupled dynamics and its geometric interpretation is probably unclear.

answered Nov 4, 2016 by (5,140 points)

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