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?

This post imported from StackExchange MathOverflow at 2015-04-15 10:46 (UTC), posted by SE-user Yaniel Cabrera