Duality between Lagrangian and Legend

Consider a pair (X, L) of a Weinstein manifold X with an exact Lagrangian submani-fold L, with ideal contact boundary (Y, A), where Y is a contact manifold and A c Y is a Legendrian submanifold. We introduce the Chekanov-Eliashberg DG-algebra, CE*(A), with coefficients in chains of the based loop space of A, and study its relation to the Floer cohomology CF*(L) of L. Using the augmentation induced by L, CE*(A) can be expressed as the Adams cobar construction 2 applied to a Legendrian coalgebra, LC*(A). We define a twisting cochain t: LC*(A)-* 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*(A) and CF*(L) are Koszul dual. In particular, t induces a quasi-isomorphism between CE*(A) and 2CF*(L), the cobar of the Floer homology of L.This generalizes the classical Koszul duality result between C*(L) and C -*(2L) for L a simply connected manifold, where 2L is the based loop space of L, and provides the geometric ingredient explaining the computations given by Etgu and Lekili (2017) in the case when X is a plumbing of cotangent bundles of 2-spheres (where an additional weight grading ensured Koszulity of t). We use the duality result to show that under certain connectivity and local-finiteness assumptions, CE*(A) is quasi-isomorphic to C -*(2L) for any Lagrangian filling L of A.Our constructions have interpretations in terms of wrapped Floer cohomology after versions of Lagrangian handle attachments. In particular, we outline a proof that CE*(A) is quasi-isomorphic to the wrapped Floer cohomology of a fiber disk C in the Weinstein domain obtained by attaching T*(A x [0, oo)) to X along A (or, in the terminology of Sylvan (2019), the wrapped Floer cohomology of C in X with wrapping stopped by A). 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.