Chain level loop bracket and pseudo-holomorphic disks

TL;DR

This paper details how virtual fundamental chains of pseudo-holomorphic disks induce a Maurer-Cartan element in the string topology loop bracket, linking symplectic topology with algebraic structures.

Contribution

It provides a detailed construction of the Maurer-Cartan element in the string topology loop bracket using Kuranishi structures and a specific chain model.

Findings

Establishment of a Maurer-Cartan element in the loop bracket

Connection between pseudo-holomorphic disks and string topology algebraic structures

Detailed analysis of moduli spaces using Kuranishi structures

Abstract

Let $L$ be a Lagrangian submanifold in a symplectic vector space which is closed, oriented and spin. Using virtual fundamental chains of moduli spaces of nonconstant pseudo-holomorphic disks with boundaries on $L$, one can define a Maurer-Cartan element of a Lie bracket operation in string topology (the loop bracket) defined at chain level. This observation is due to Fukaya, who also pointed out its important consequences in symplectic topology. The goal of this paper is to work out details of this observation. Our argument is based on a string topology chain model previously introduced by the author, and the theory of Kuranishi structures on moduli spaces of pseudo-holomorphic disks, which has been developed by Fukaya-Oh-Ohta-Ono.