Work

Chain level loop bracket and pseudo‐holomorphic disks

Year: 2020

Type: article

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.

Source: Journal of Topology

DOI: 10.1112/topo.12140

Author: Kei Irie

Institution: The University of Tokyo

Language: English

Topic: Geometric and Algebraic Topology

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funder: Japan Society for the Promotion of Science

Award: Symplectic Geometry and Geometry of Loop Spaces