Abstract / Description of output
We work in the setting of CalabiYau mirror symmetry. We establish conditions under which Kontsevich's homological mirror symmetry (which relates the derived Fukaya category to the derived category of coherent sheaves on the mirror) implies Hodgetheoretic mirror symmetry (which relates genuszero GromovWitten invariants to period integrals on the mirror), following the work of Barannikov, Kontsevich and others. As an application, we explain in detail how to prove the classical mirror symmetry prediction for the number of rational curves in each degree on the quintic threefold, via the thirdnamed author's proof of homological mirror symmetry in that case; we also explain how to determine the mirror map in that result, and also how to determine the holomorphic volume form on the mirror that corresponds to the canonical CalabiYau structure on the Fukaya category. The crucial tool is the `cyclic openclosed map' from the cyclic homology of the Fukaya category to quantum cohomology, defined by the firstnamed author in [Gan]. We give precise statements of the important properties of the cyclic openclosed map: it is a homomorphism of variations of semiinfinite Hodge structures; it respects polarizations; and it is an isomorphism when the Fukaya category is nondegenerate (i.e., when the openclosed map hits the unit in quantum cohomology). The main results are contingent on worksinpreparation [PS,GPS] on the symplectic side, which establish the important properties of the cyclic openclosed map in the setting of the `relative Fukaya category'; and they are also contingent on a conjecture on the algebraic geometry side, which says that the cyclic formality map respects certain algebraic structures.
Original language  English 

Publisher  ArXiv 
Number of pages  37 
Publication status  Published  15 Oct 2015 
Keywords / Materials (for Nontextual outputs)
 math.SG
 math.AG
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Nick Sheridan
 School of Mathematics  Personal Chair of Mirror Symmetry
Person: Academic: Research Active