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

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

2007-03-06

arxiv.org/abs/math/0703164v1

Mathematics

Quantum Algebra

28 pages

Scientific paper

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 $\R^2$, ie., we define also {\em non-transversal} $A_\infty$-products. This construction is motivated by homological mirror symmetry of (two-)tori, where $\R^2$ 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.

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