On the wrapped Fukaya category and the loop space of Lagrangians in a Liouville manifold

We introduce an $A_{\mathrm{\infty }}$ map from the cubical chain complex of the based loop space of Lagrangian submanifolds with Legendrian boundary in a Liouville Manifold $C_{\ast }({\mathrm{\Omega }}_L\mathcal{L}ag)$ to wrapped Floer cohomology of Lagrangian submanifold ${\mathcal{C}\mathcal{W}}^{-\ast }(L,L)$. In the case of a cotangent bundle and a Lagrangian co-fiber, the composition of our map with a previously constructed map from ${\mathcal{C}\mathcal{W}}^{\ast }(L,L)\to C_{\ast }({\mathrm{\Omega }}_{q}Q)$ shows that this map is split surjective.