Cyclic Homology of Fukaya Categories and the Linearized Contact Homology

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-Latschev we show that there is a Lie bialgebra homomorphism from the linearized contact homology of $M$ to the cyclic cohomology of the Fukaya category. Our study is also motivated by string topology and 2-dimensional topological conformal field theory.

💡 Research Summary

The paper studies an exact symplectic manifold (M) whose boundary (\partial M) is of contact type and whose first Chern class vanishes. Under these hypotheses the Fukaya (A_{\infty})-category (\mathcal{F}(M)) is well defined, and the authors investigate its cyclic cohomology (HC^{*}(\mathcal{F}(M))). The main achievement is the construction of an involutive Lie bialgebra structure on this cyclic cohomology.

The Lie bracket is obtained by counting holomorphic disks with two interior punctures (or equivalently, by a suitable operation on the cyclic bar complex) and has degree (-1). The cobracket is defined by counting disks with one interior puncture and two boundary components, giving a degree (-2) operation. Together with the BV operator coming from rotating the cyclic input, these operations satisfy the graded Jacobi, co‑Jacobi, and compatibility relations required for a Lie bialgebra, and the involutivity condition (\Delta^{2}=0) holds.

Parallel to this, the authors consider the linearized contact homology (CH^{\mathrm{lin}}{*}(M)) of the contact boundary. Using the Reeb dynamics on (\partial M) they build a chain complex generated by good Reeb orbits, equipped with a differential defined by counting rigid punctured holomorphic curves. Following the work of Cieliebak–Latschev, they show that (CH^{\mathrm{lin}}{*}(M)) also carries a natural Lie bialgebra structure.

The central technical result is the existence of a homomorphism of Lie bialgebras
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