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

Ganatra, Sheel¹

¹ Department of Mathematics, University of Southern California, Los Angeles, CA, United States

Geometry & topology, Tome 27 (2023) no. 9, pp. 3461-3584.

Résumé

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.

Détail

DOI : 10.2140/gt.2023.27.3461

Classification : 53D12, 53D37, 19D55
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

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