Topology of Lagrangian submanifolds via open-closed string topology

We study Lagrangian submanifolds in standard symplectic vector spaces R^{2n} using ideas from open-closed string topology. Specifically, for a closed, oriented, spin Lagrangian L, we construct a (possibly curved) deformation of the dg associative algebra of chains on the based loop space of L (“open string”), using moduli spaces of pseudo-holomorphic discs with boundaries on L viewed as chains in the free loop space (“closed string”). As an application, we prove that if the second homotopy group of L is zero, then L has non-vanishing Maslov class, generalizing previous results due to Viterbo, Cieliebak-Mohnke, Fukaya and Irie. If time permits, we will speculate on the possible classification of the topology of such Lagrangians in dimension 3.