Northern California Symplectic Geometry Seminar: Chain level loop bracket and pseudo-holomorphic disks

Seminar | February 5 | 2:30-3:30 p.m. | 891 Evans Hall

 Kei Irie, Kyoto

 Department of Mathematics

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. In this talk I'll explain how to rigorously carry out this idea. Our argument is based on a string topology chain model previously introduced by the speaker, and the theory of Kuranishi structures on moduli spaces of pseudo-holomorphic disks, which has been developed by Fukaya-Oh-Ohta-Ono.