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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].
We use the Beilinson $t$-structure on filtered complexes and the Hochschild-Kostant-Rosenberg theorem to construct filtrations on the negative cyclic and periodic cyclic homologies of a scheme $X$ with graded pieces given by the Hodge-completion of t
We correct a mistake regarding almost complex structures on Hilbert schemes of points in surfaces in arXiv:1510.02449. The error does not affect the main results of the paper, and only affects the proofs of invariance of equivariant symplectic Khovan
We prove $L_{infty}$-formality for the higher cyclic Hochschild complex $chH$ over free associative algebra or path algebra of a quiver. The $chH$ complex is introduced as an appropriate tool for the definition of pre-Calabi-Yau structure. We show th
We define a version of spectral invariant in the vortex Floer theory for a $G$-Hamiltonian manifold $M$. This defines potentially new (partial) symplectic quasi-morphism and quasi-states when $M//G$ is not semi-positive. We also establish a relation
Let $M$ be an exact symplectic manifold with contact type boundary such that $c_1(M)=0$. In this paper we show that the cyclic cohomology of the Fukaya category of $M$ has the structure of an involutive Lie bialgebra. Inspired by a work of Cieliebak-