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

## TL;DR

This paper establishes a new link between the Fukaya category of Weinstein manifolds and the Chekanov-Eliashberg algebra by associating finite-dimensional representations to compact exact Lagrangians and relating Floer homology to derived hom spaces.

## Contribution

It introduces a novel method to relate Lagrangian Floer homology to algebraic representations, extending previous results with new techniques involving Lagrangian cobordisms.

## Key findings

- Constructs finite dimensional representations for compact exact Lagrangians.
- Proves isomorphism between Floer homology and derived hom spaces of representations.
- Extends Floer theory for Lagrangian cobordisms with negative ends.

## Abstract

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 $$HF(L_0, L_1) \cong H^*R\hom_{\mathcal A}(V_{L_0}, V_{L_1})$$ 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.