Canonical models of filtered A_∞-algebras and Morse complexes

2008/12/10 by Fukaya, K., OH, Y. -G., Ohta, H. +1
#14J32 #14J45 #53D12 #53D40 #Algebraic Geometry (math.AG) #FOS: Mathematics #Symplectic Geometry (math.SG)

paper · doi:10.48550/arxiv.0812.1963

Abstract

The purpose of this paper is two-fold. First we explain the construction of the canonical model of filtered A_∞-algebras given in the authors' book [FOOO]. The canonical model plays a crucial role in the study of Lagrangian Floer theory on toric manifolds in our recent papers, arXiv:0802.1703 and arXiv:0810.5654. Then using a variation of the arguments used in that construction, we define a natural filtered A_∞-structure on the Morse complex of a Morse function and its A_∞ homotopy to the A_∞-algebras on a Lagrangian submanifold constructed in [FOOO]. The corresponding graphical moduli spaces `summing over trees' involve holomorphic discs connected by the gradient flow lines.

Related

• Perturbed gradient flow trees and A_∞-algebra structures on Morse cochain complexes
• A Weak ∞-Functor in Morse Theory
• Higher algebra of A_∞ and ΩB As-algebras in Morse theory I
• Morse complexes and multiplicative structures
• The moduli space of gradient flow lines and Morse homology