An $A_{\infty }$-structure for lines in a plane

Abstract

As an explicit example of an $A_{\infty }$-structure associated to geometry, we construct an $A_{\infty }$-structure for a Fukaya category of finitely many lines (Lagrangians) in ${\mathbb{R}}^2$, ie., we define also {\em non-transversal} $A_{\infty }$-products. This construction is motivated by homological mirror symmetry of (two-)tori, where ${\mathbb{R}}^2$ is the covering space of a two-torus. The strategy is based on an algebraic reformulation of Morse homotopy theory through homological perturbation theory (HPT) as discussed by Kontsevich and Soibelman in math.SG/0011041, where we introduce a special DG category which is a key idea of our construction.

28 pages

References in corpus (1)

• Homological mirror symmetry on noncommutative two-tori