Canonical models of filtered $A_{\infty }$-algebras and Morse complexes

Abstract

The purpose of this paper is two-fold. First we explain the construction of the canonical model of filtered $A_{\infty }$-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 $A_{\infty }$-structure on the Morse complex of a Morse function and its $A_{\infty }$ homotopy to the $A_{\infty }$-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.

28 pages, 6 figures ; to appear in the proceedings of Eliashberg's 60th birthday conference (Yashafest)

References in corpus (1)

• Quantum Structures for Lagrangian Submanifolds