Given an exact relatively Pin Lagrangian embedding $Q \subset M$, we construct an $A^∞$ 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 $A^∞$ 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.
"On the wrapped Fukaya category and based loops." J. Symplectic Geom. 10 (1) 27 - 79, March 2012.