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

Sheel Ganatra

Geometry & Topology 27 (2023) 3461–3584

We construct geometric maps from the cyclic homology groups of the (compact or wrapped) Fukaya category to the corresponding S1–equivariant (Floer/quantum or symplectic) cohomology groups, which are natural with respect to all Gysin and periodicity exact sequences and are isomorphisms whenever the (nonequivariant) open–closed map is. These cyclic open–closed maps give 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 model in characteristic 0, and also give a purely symplectic proof of the noncommutative 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 and Sheridan, and with Cohen.

Fukaya category, Calabi–Yau structures, cyclic homology, Hochschild homology, $S^1$–equivariant homology, Floer homology, symplectic cohomology, open–closed maps, Hodge–de Rham degeneration
Mathematical Subject Classification 2010
Primary: 53D12, 53D37
Secondary: 19D55
Received: 12 January 2020
Revised: 25 September 2021
Accepted: 27 December 2021
Published: 5 December 2023
Proposed: Paul Seidel
Seconded: Ciprian Manolescu, Yakov Eliashberg
Sheel Ganatra
Department of Mathematics
University of Southern California
Los Angeles, CA
United States

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