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

+ 8 like - 0 dislike
260 views

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

+ 3 like - 0 dislike

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 (5,120 points)

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysicsOver$\varnothing$lowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.