On the wrapped Fukaya category and based loops

TL;DR

This paper constructs a functor linking the wrapped Fukaya category of a symplectic manifold to modules over the based loop space of a Lagrangian, extending known results to the A-infinity setting.

Contribution

It introduces an A-infinity restriction functor from the wrapped Fukaya category to modules over the chain algebra of the based loop space, generalizing previous homology-level results.

Findings

Constructed an A-infinity restriction functor for Lagrangian embeddings.

Established an A-infinity equivalence in cotangent bundle cases.

Extended previous homology results to the A-infinity framework.

Abstract

Given an exact relatively Pin Lagrangian embedding Q in a symplectic manifold M, we construct an A-infinity restriction functor from the wrapped Fukaya category of M to the category of modules on the differential graded algebra of chains over the based loop space of Q. If M is the cotangent bundle of Q, this functor induces an A-infinity equivalence between the wrapped Floer cohomology of a cotangent fibre and the chains over the based loop space of Q, extending a result proved by Abbondandolo and Schwarz at the level of homology.