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?



Mirror symmetries for generalized geometries ?
Mirror symmetry is at the most fundamental level an isomorphism of N=(2,2)-supersymmetric conformal field theories attached to different geometric data, which acts on the supersymmetries as a prescribed outer automorphism (switching A- and B-twists). Calabi-Yaus give rise to such SCFTs, hence one can ask for two CYs to be mirror -- though this is not a map in general, rather a correspondence that can then be pinned down more precisely in terms of large volume limits and SYZ fibrations etc. Mirror symmetry also has many rougher manifestations, such as an isomorphism of topological field theories of A- and B-type attached to geometric data, which now no longer need to be CY. In any case the point is that the generalized geometries you mention are precisely appropriate backgrounds to define such (2,2) SCFTs (see eg articles of Kapustin from around 04), so from the beginning they have a mirror symmetry question (look for pairs giving rise to the same SCFT up to this outer automorphism).


