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

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

2008-12-10

arxiv.org/abs/0812.1963v1

Mathematics

Symplectic Geometry

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

Scientific paper

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.

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