(This talk is based on joint works with Oh, Ohta, Ono)

Relation of rational homotopy type of a manifold M to its De Rham complex (differential graded algebra = DGA) Ω(M) is classical after the work by Quillen and Sullivan. In case when M is a Kahler manifold, the Kodaira-Spencer deformation theory of complex structure of M is controlled by the homotopical algebra of the DGA Ω(M; TM), the Dolbault complex of M with coefficient in the tangent bundle. Especially, the infinitesimal deformation theory of complex structure of M is controlled by the higher massey product which is nothing but the (rational) homotopy type of the DGA Ω(M; TM). The purpose of this talk is to explain its ”quantization”. Namely first an noncommutative generalization of the notion of DGA. (Note the DGA’s appeared above are graded commutative.) We develop a rational homotopy theory of (non commutative) DGA. (Actually the homotopy category of non commutative DGA is equivalent to the (rational) homotopy category of A∞ algebra. In case when L is a Lagrangian submanifold of a complex manifold, we can associate a (rational) homotopy type of an A∞ algebra. The deformation theory of it is closely related to the Floer homology (its family version also) and to the deformation theory of the Lagrangian submanifold itself. This deformation theory together with Floer cohomology is expected to be the mirror of the deformation theory and cohomology theory of (objects of the derived category of) the coherent sheaves of the mirror. The contents of the talk is based on our recent preprint, Lagrangian intersection Floer theory - Anomaly and obstruction, (the 2000 Dec version). However there are several points we made progress after that which will also be included in the talk.