Duality between Lagrangian and Legendrian invariants

Abstract

Consider a pair $(X,L)$, of a Weinstein manifold $X$ with an exact Lagrangian submanifold $L$, with ideal contact boundary $(Y,Λ)$, where $Y$ is a contact manifold and $Λ\subset Y$ is a Legendrian submanifold. We introduce the Chekanov-Eliashberg DG-algebra, $CE^{\ast}(Λ)$, with coefficients in chains of the based loop space of $Λ$ and study its relation to the Floer cohomology $CF^{\ast}(L)$ of $L$. Using the augmentation induced by $L$, $CE^{\ast}(Λ)$ can be expressed as the Adams cobar construction $Ω$ applied to a Legendrian coalgebra, $LC_{\ast}(Λ)$. We define a twisting cochain:\[\mathfrak{t} \colon LC_{\ast}(Λ) \to \mathrm{B} (CF^*(L))^\#\]via holomorphic curve counts, where $\mathrm{B}$ denotes the bar construction and $\#$ the graded linear dual. We show under simply-connectedness assumptions that the corresponding Koszul complex is acyclic which then implies that $CE^*(Λ)$ and $CF^{\ast}(L)$ are Koszul dual. In particular, $\mathfrak{t}$ induces a quasi-isomorphism between $CE^*(Λ)$ and the cobar of the Floer homology of $L$, $ΩCF_*(L)$. We use the duality result to show that under certain connectivity and locally finiteness assumptions, $CE^*(Λ)$ is quasi-isomorphic to $C_{-*}(ΩL)$ for any Lagrangian filling $L$ of $Λ$. Our constructions have interpretations in terms of wrapped Floer cohomology after versions of Lagrangian handle attachments. In particular, we outline a proof that $CE^{\ast}(Λ)$ is quasi-isomorphic to the wrapped Floer cohomology of a fiber disk $C$ in the Weinstein domain obtained by attaching $T^{\ast}(Λ\times[0,\infty))$ to $X$ along $Λ$ (or, in the terminology of arXiv:1604.02540 the wrapped Floer cohomology of $C$ in $X$ with wrapping stopped by $Λ$). Along the way, we give a definition of wrapped Floer cohomology without Hamiltonian perturbations.