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According to the proponents of homological mirror symmetry, when a complex and symplectic manifold are mirror symmetric, we can take difficult questions about the symplectic space and transfer them across the symmetry into more tractable complex geometry questions. Can anyone provide an example of an actual problem that was solved in this way.

More specifically, one of the better understood examples of mirror symmetry is between the flag manifolds and certain Landau--Ginzburg models, see the papers of Rietsch. Can anyone provide an example of a difficult question about Landau-Ginzburg spaces that was answered in this way using mirror symmetry.

Finally, can anyone point me to the papers where the equivalence between the flag manifolds and Landau--Ginzburg models was originally established?

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  • $\begingroup$ By the way, the mirror symmetry of flag varieties is far from being understood. The best understood examples are semi-flat Calabi-Yaus and local Calabi-Yaus. Local CYs are somehow like semi-flat in some directions, see the work of Abouzaid-Auroux-Katzarkov. $\endgroup$
    – YHBKJ
    Jan 29, 2015 at 6:58
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    $\begingroup$ Another thing is I think it's not easy to establish HMS for flag varieties unless you consider sheaves on flag varieties and Fukaya-Seidel category on the mirror LG model. Even for that, you need to assume the flag variety is full to ensure that your superpotential is a Lefschetz fibration on the mirror side. Once that is true, the proof follows similarly as in the work of Auroux-Katzarkov-Orlov on del-Pezzo surfaces or weighted projective spaces. $\endgroup$
    – YHBKJ
    Jan 29, 2015 at 7:03

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A perfect example is Abouzaid-Smith's classification of genus 2 Lagrangian surfaces in $(T^4,\omega_{std})$. In this paper (http://arxiv.org/pdf/0903.3065v2.pdf), they proved that any Lagrangian genus 2 surface is Floer cohomologically indistinguishable from the Lagrangian surgery of two linear Lagrangian tori meeting transversely.

The method is roughly to prove homological mirror symmetry for $T^4$ first, then establish the corresponding result on the B-side, finally translate it in the A-side.

Recently, Abouzaid developed Fukaya's family Floer theory to produce a faithful mirror functor. The motivation for developing this is to study the symplectic topology of the Thurston manifold, which is the first example of a non-Kahler symplectic manifold. The difficulty is that the usual method of resolving the diagonal is not available here to prove that a finite set of Lagrangians split-generate the Fukaya category. However, since this space admits a Lagrangian fibration, family Floer theory can be used here to produce a mirror functor. Showing this functor is an equivalence will yield homological mirror symmetry in this case. The mirror is a rigid analytic Kodaira surface, therefore one can speculate from looking at the mirror that there are not so many non-trivial Lagrangians in the Fukaya category. This is very different from the case of $T^4$.

Currently, there is no well-established mirror transformation in the general case, which makes it difficult to explicitly transform something to the mirror side, therefore mirror symmetry, in many cases, just helps in the philosophical level. For example, a Landau-Ginzburg model is in many cases mirror to a Fano. For many Fano manifolds, a full excpetional collection of the derived category of coherent sheaves can be found by the work of Beilinson and others. The analogous theory on the A-side is Seidel's work on Lefschetz fibrations. In the converse direction, one has Dehn twist along a Lagrangian sphere on the A-side, then it's natural to expect similar algebraic twist on the B-side, this is established by Seidel and Thomas. These works are not aimed to solve explicit problems, but they really comes from the philosophy provided by mirror symmetry, and they are of course helpful in solving classical problems in symplectic topology. For example, Arnold's nearby Lagrangian conjecture, see Fukaya-Seidel-Smith's very beautiful paper: http://arxiv.org/pdf/math/0701783v2.pdf.

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