Cyclic homology, $S^1$-equivariant Floer cohomology, and Calabi-Yau structures

Abstract

We construct geometric maps from the cyclic homology groups of the (compact or wrapped) Fukaya category to the corresponding $S^1$-equivariant (Floer/quantum or symplectic) cohomology groups, which are natural with respect to all Gysin and periodicity exact sequences and are isomorphisms whenever the (non-equivariant) open-closed map is. These {\em cyclic open-closed maps} give (a) constructions of geometric smooth and/or proper Calabi-Yau structures on Fukaya categories (which in the proper case implies the Fukaya category has a cyclic A-infinity model in characteristic 0) and (b) a purely symplectic proof of the non-commutative Hodge-de Rham degeneration conjecture for smooth and proper subcategories of Fukaya categories of compact symplectic manifolds. Further applications of cyclic open-closed maps, to counting curves in mirror symmetry and to comparing topological field theories, are the subject of joint projects with Perutz-Sheridan [GPS1, GPS2] and Cohen [CG].

84 pages, 18 figures. Final version to appear in Geometry and Topology

References in corpus (2)

• Symplectic cohomology and duality for the wrapped Fukaya category
• Lectures on Noncommutative Geometry

Cited by in corpus (8)

• Categorical primitive forms of Calabi-Yau $A_{\infty }$-categories with semi-simple cohomology
• On the embedding complexity of Liouville manifolds
• Intrinsic mirror symmetry and categorical crepant resolutions
• Singular Legendrian unknot links and relative Ginzburg algebras
• Open Gromov-Witten invariants from the Fukaya category
• Categorical non-properness in wrapped Floer theory
• Open WDVV Equations and Frobenius Structures for Toric Calabi-Yau 3-Folds
• Persistence of unknottedness of clean Lagrangian intersections