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Title:
From categories to curve counts II: variations of semi-infinite Hodge structures
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Abstract:
We will explain how to recover from homological mirror symmetry (HMS) a version of enumerative mirror symmetry, and in particular how to use Sheridan’s proof of HMS for the quintic threefold to give a new proof of enumerative predictions made by Candelas, de la Ossa, Green and Parkes in 1991. The main intermediate step, following ideas of Kontsevich and Barannikov, is extracting from HMS a form of Hodge-theoretic mirror symmetry. The crucial new ingredient on the A side is showing that an open-closed map from the cyclic homology of the Fukaya category to quantum cohomology intertwines certain Hodge-theoretic structures. On the B side, our result relies on a conjecture (some of which is in the literature) that a cyclic HKR map does the same.
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