Cyclic homology, S1–equivariant Floer cohomology and Calabi–Yau structures

Ganatra, Sheel

doi:10.2140/gt.2023.27.3461

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

Sheel Ganatra

Geometry & Topology 27 (2023) 3461–3584

DOI: 10.2140/gt.2023.27.3461

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 (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 $A_{\infty}$ 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.

Keywords

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

References

Publication

Received: 12 January 2020

Revised: 25 September 2021

Accepted: 27 December 2021

Published: 5 December 2023

Proposed: Paul Seidel

Seconded: Ciprian Manolescu, Yakov Eliashberg

Authors

Sheel Ganatra
Department of Mathematics University of Southern California Los Angeles, CA United States

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Volume 27, issue 9 (2023)