# Mirror symmetries for generalized geometries ?

For Calabi-Yau three-folds we have $\mathcal{mirror \ symmetry}$: a map that associates most Calabi-Yau three-folds $M$ another Calabi-Yau three-fold $W$ such that $h^{1,1}(M) = h^{2,1}(W)$ and $h^{1,1}(W) = h^{2,1}(M)$ where $h^{i,j}$ are the Hodge numbers of the Calabi-Yau. In string theory such a duality leads to the conjecture that the type IIA superstring theory compactified on $M$ is equvilalent to the type IIB compactified on $W$.

$\textbf{Question} :$ Are there extensions of mirror symmetry applied to generalized geometries (in the sense of Hitchin, Cavalcanti, and Gualtieri)? If so, what is the state of the art of this topic/question?

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–  Kevin H. Lin Nov 5 '10 at 6:52
As far as I know, we still don't exactly know the "mirror map" explicitly even in the (classical?) case of Calab-Yau 3-folds. Much is known about mirror symmetry at the level of topology, but much remains to be understood at the level of geometry. –  Spiro Karigiannis Nov 5 '10 at 14:46
I was under the impression that we don't know enough about Calabi-Yau 3-folds to talk about "most" of them. –  S. Carnahan Nov 6 '10 at 7:48