Floer theory on based loops and fundamental groups of Lagrangians in Cⁿ

We obtain new constraints on the topology of Lagrangian submanifolds in the standard symplectic vector space Cⁿ using Floer theory on the based loop space, generalizing a theorem of Fukaya-Irie. In particular, we show that non-trivial connected sums of closed, spin, aspherical manifolds of even dimensions do not admit Lagrangian embeddings into Cⁿ. A feature of the proof is that the obstructedness of the Floer theory is used to constrain the fundamental group of the Lagrangian.