Bounds on the volume of A branes in Fano varieties

Question

I have been recently reading "String theory and M theory: a modern introduction" of Becker et al. They consider type IIB string theory compactified on Calabi Yau threefold $X$ with a nowhere vanishing (3,0) form $\Omega$. They derive a bound (9.155, page 407) for volume $V$ of brane wrapping a special Lagrangian submanifold $\Sigma$ $\mathit{V} \geq e^{-K}\lvert \int_{\Sigma}\Omega \rvert$ where $K$ is a certain function on the moduli space of Calabi Yau manifolds. Now my question is: can this bound be generalized for sigma models on Fano varieties (like $\mathbb{CP}^{2}$)? I am a mathematician and thus not quite interested in phenomenology (and dimensionality of spacetime). The question I have in mind is following conjecture: for any special Lagrangian torus $Y$ in $\mathbb{CP}^{2}$ we have $Vol(Y) \geq \frac{4 \pi^{2}}{3 \sqrt{3}}$; I just wanted to find out whether it can be established physically.

This post imported from StackExchange Physics at 2017-02-25 13:18 (UTC), posted by SE-user Aknazar Kazhymurat

Comments

Do you have a reference for the $\mathrm{Vol}(Y) \geq \frac{4\pi^2}{3\sqrt{3}}$ result?

This post imported from StackExchange Physics at 2017-02-25 13:18 (UTC), posted by SE-user JamalS

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@JamalS, unfortunately it is still a conjecture. You can look up motivation and history in Ma et al preprint "Energy functional for lagrangian tori in CP2".

This post imported from StackExchange Physics at 2017-02-25 13:18 (UTC), posted by SE-user Aknazar Kazhymurat