The topology of Lagrangian submanifolds via open-closed string topology

TL;DR

This paper explores the topology of Lagrangian submanifolds in symplectic vector spaces using open-closed string topology, constructing algebraic structures from moduli spaces of pseudo-holomorphic discs.

Contribution

It introduces a deformation of the chain algebra of the loop space of Lagrangians via string topology methods, extending previous results on Maslov class non-vanishing.

Findings

If a2(L)=0, then L has non-zero Maslov class.

Constructs a (possibly curved) dg associative algebra from moduli spaces.

Generalizes earlier results by Viterbo, Cieliebak-Mohnke, Fukaya, and Irie.

Abstract

We study the topology of Lagrangian submanifolds in standard symplectic vector spaces $\mathbb{C}^n$ 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 $L$. This is done via pushing forward moduli spaces of pseudo-holomorphic discs with boundaries on $L$, viewed as chains in the free loop space, along a string topology closed-open map. As an application, we prove that if $\pi_2(L)=0$, then $L$ has non-vanishing Maslov class, generalizing previous results due to Viterbo, Cieliebak-Mohnke, Fukaya, and Irie.