Representations of the Chekanov-Eliashberg algebra from closed exact Lagrangians I

This is the first of a series of two articles aiming at relating the compact Fukaya category of a Weinstein manifold to the derived category of finite dimensional representations of the Chekanov-Eliashberg differential graded algebra of the attaching spheres of the critical handles. In this first article we associate a finite dimensional representation $V_L$ to any compact exact Lagrangian submanifold $L$ and prove that for two any such Lagrangian submanifolds $L_0$ and $L_1$ the isomorphism $$HF(L_0,L_1)\cong H^{\ast }R{\mathrm{hom}}_{\mathcal{A}}(V_{L_0},V_{L_1})$$ holds. This generalises a previous result of Ekholm and Lekili, but out techniques are different since we use an extension of the Floer theory for Lagrangian cobordisms with negative ends that we developed in collaboration with Roman Golovko.