Chain level loop bracket and pseudo-holomorphic disks

Princeton/IAS Symplectic Geometry Seminar
Topic:Chain level loop bracket and pseudo-holomorphic disks
Speaker:Kei Irie
Affiliation:Research Institute for Mathematical Sciences , Kyoto University
Date:Monday, February 26
Time/Room:4:00pm - 5:00pm/Simonyi Hall 101

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 idea is due to Fukaya, who also pointed out its important consequences in symplectic topology. In this talk I will explain how to rigorously carry out this idea. Our argument is based on a string topology chain model previously introduced by the speaker, and theory of Kuranishi structures on moduli spaces of pseudo-holomorphic disks, which has been developed by Fukaya-Oh-Ohta-Ono.