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arXiv · 1912.13510

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

Sheel Ganatra
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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].

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Sheel Ganatra. 2019-12-31. Cyclic homology, $S^1$-equivariant Floer cohomology, and Calabi-Yau structures. https://doi.org/10.2140/gt.2023.27.3461

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