Mirror symmetry: from categories to curve counts

Abstract

We work in the setting of Calabi-Yau 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 Hodge-theoretic mirror symmetry (which relates genus-zero Gromov-Witten 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 third-named 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 Calabi-Yau structure on the Fukaya category. The crucial tool is the `cyclic open-closed map' from the cyclic homology of the Fukaya category to quantum cohomology, defined by the first-named author in [Gan]. We give precise statements of the important properties of the cyclic open-closed map: it is a homomorphism of variations of semi-infinite Hodge structures; it respects polarizations; and it is an isomorphism when the Fukaya category is non-degenerate (i.e., when the open-closed map hits the unit in quantum cohomology). The main results are contingent on works-in-preparation [PS,GPS] on the symplectic side, which establish the important properties of the cyclic open-closed 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.

37 pages; v2 updated to include arXiv identifiers of papers posted concurrently in bibliography

References in corpus (3)

• Calabi-Yau algebras
• Symplectic cohomology and duality for the wrapped Fukaya category
• Lectures on Noncommutative Geometry

Cited by in corpus (21)

• Relative Calabi-Yau structures
• Cyclic homology, $S^1$-equivariant Floer cohomology, and Calabi-Yau structures
• Calabi-Yau structures on topological Fukaya categories
• Automatically generating Fukaya categories and computing quantum cohomology
• Computing a categorical Gromov-Witten invariant
• Odd sphere bundles, symplectic manifolds, and their intersection theory
• Homological mirror symmetry at large volume
• Weil-Petersson geometry on the space of Bridgeland stability conditions
• Categorical primitive forms of Calabi-Yau $A_{\infty }$-categories with semi-simple cohomology
• Calabi-Yau structures on categories of matrix factorizations
• Categorical primitive forms and Gromov-Witten invariants of $A_n$ singularities
• Categorical Saito theory, II: Landau-Ginzburg orbifolds
• Pairings in mirror symmetry between a symplectic manifold and a Landau-Ginzburg $B$-model
• Categorical Enumerative Invariants, II: Givental formula
• Categorical Enumerative Invariants, I: String vertices
• Versality in mirror symmetry
• Fukaya categories of Lagrangian cobordisms and duality
• The wrapped Fukaya category for semi-toric SYZ fibrations
• Full exceptional collections for anticanonical log del Pezzo surfaces
• Open WDVV Equations and Frobenius Structures for Toric Calabi-Yau 3-Folds
• Ring isomorphisms of closed string via homological mirror symmetry