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

TL;DR

This paper constructs geometric maps linking cyclic homology of Fukaya categories to $S^1$-equivariant Floer cohomology, enabling new insights into Calabi-Yau structures and the Hodge-de Rham degeneration conjecture in symplectic geometry.

Contribution

It introduces cyclic open-closed maps that establish isomorphisms and geometric structures, advancing the understanding of Fukaya categories and symplectic invariants.

Findings

Constructed geometric maps from cyclic homology to $S^1$-equivariant Floer cohomology.

Provided a symplectic proof of the Hodge-de Rham degeneration conjecture.

Enabled new applications in mirror symmetry and topological field theories.

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].