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

TL;DR

This paper constructs canonical models of filtered $A_ infty$-algebras, applies them to Morse complexes, and relates these structures to Lagrangian Floer theory on toric manifolds.

Contribution

It explains the canonical model construction for filtered $A_ infty$-algebras and introduces a natural filtered $A_ infty$-structure on Morse complexes, linking to Lagrangian Floer theory.

Findings

Canonical models are crucial in Lagrangian Floer theory.

A natural filtered $A_ infty$-structure on Morse complexes is defined.

The work connects Morse theory with holomorphic disc moduli spaces.

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.