Authors’ abstract: Consider a pair \((X,L)\) of a Weinstein manifold \(X\) with an exact Lagrangian submanifold \(L\), with ideal contact boundary \((Y,\Lambda)\), where \(Y\) is a contact manifold and \(\Lambda\subseteq Y\) is a Legendrian submanifold. We introduce the Chekanov-Eliashberg DG-algebra, \(CE^*(\Lambda)\), with coefficients in chains of the based loop space of \(\Lambda\), and study its relation to the Floer cohomology \(CF^*(L)\) of \(L\). Using the augmentation induced by \(L\), \(CE^*(\Lambda)\) can be expressed as the Adams cobar construction \(\Omega\) applied to a Legendrian coalgebra, \(LC_*(\Lambda)\). We define a twisting cochain \(\mathfrak{t}:LC_*(\Lambda)\to B(CF^*(L))^\#\) via holomorphic curve counts, where \(B\) denotes the bar construction and \(\#\) the graded linear dual. We show under simple-connectedness assumptions that the corresponding Koszul complex is acyclic, which then implies that \(CE^*(\Lambda)\) and \(CF^*(L)\) are Koszul dual. In particular, \(\mathfrak{t}\) induces a quasi-isomorphism between \(CE^*(\Lambda)\) and \(\Omega CF_*(L)\), the cobar of the Floer homology of \(L\).
This generalizes the classical Koszul duality result between \(C^*(L)\) and \(C_{-*}(\Omega L)\) for \(L\) a simply connected manifold, where \(\Omega L\) is the based loop space of \(L\), and provides the geometric ingredient explaining the computations given by T. Etgü and Y. Lekili [Geom. Topol. 21, No. 6, 3313–3389 (2017; Zbl 1378.57041)] in the case when \(X\) is a plumbing of cotangent bundles of 2-spheres (where an additional weight grading ensured Koszulity of \(\mathfrak{t}\)).
We use the duality result to show that under certain connectivity and local-finiteness assumptions, \(CE^*(\Lambda)\) is quasi-isomorphic to \(C_{-*}(\Omega L)\) for any Lagrangian filling \(L\) of \(\Lambda\).
Our constructions have interpretations in terms of wrapped Floer cohomology after versions of Lagrangian handle attachments. In particular, we outline a proof that \(CE^*(\Lambda)\) is quasi-isomorphic to the wrapped Floer cohomology of a fiber disk \(C\) in the Weinstein domain obtained by attaching \(T^*(\Lambda\times [0,+\infty))\) to \(X\) along \(\Lambda\) (or, in the terminology of Z. Sylvan [J. Topol. 12, No. 2, 372–441 (2019; Zbl 1430.53097)], the wrapped Floer cohomology of \(C\) in \(X\) with wrapping stopped by \(\Lambda\)). Along the way, we give a definition of wrapped Floer cohomology via holomorphic buildings that avoids the use of Hamiltonian perturbations, which might be of independent interest.