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Caveats for differential geometers [0489]

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Caveats for differential geometers

However, in the differential geometric literature on the Thomas-Yau conjecture, one often attempts to go beyond all these aspects of mirror symmetry, and tries to directly compare the space of Lagrangian submanifolds inside XX, with the space of holomorphic vector bundles equipped with Hermitian metrics over X∨X^{\vee}. This comparison must be treated with caution, because there is no general bijective correspondence between these two infinite dimensional spaces, and because mirror symmetry is properly a quantum phenomenon which is not fully captured by classical considerations. In fact, mirror symmetry only relates F​u​k​(X)Fuk(X) and C​o​h​(X∨)Coh(X^{\vee}) after taking the derived category, and there is no general reason for the heart of a preferred tt-structure on the derived Fukaya category (e.g. the tt-structure defined by some almost calibrated condition) to agree with C​o​h​(X∨)Coh(X^{\vee}). Having vaccinated the reader with these precautions, we shall regard such comparisons as useful formal analogies, and we consider those aspects with quantum interpretations as more reliable.

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