Deformation quantization of Poisson manifolds, I

I prove that every finite-dimensional Poisson manifold X admits a canonical deformation quantization. Informally, it means that the set of equivalence classes of associative algebras close to the algebra of functions on X is in one-to-one correspondence with the set of equivalence classes of Poisson structures on X modulo diffeomorphisms. In fact, a more general statement is proven ("Formality conjecture"), relating the Lie superalgebra of polyvector fields on X and the Hochschild complex of the algebra of functions on X. Coefficients in explicit formulas for the deformed product can be interpreted as correlators in a topological open string theory, although I do not use explicitly the language of functional integrals. One of corollaries is a justification of the orbit method in the representation theory.Comment: plain TeX and epsf.tex, 46 pages, 24 figure

Frobenius Manifolds and Formality of Lie Algebras of Polyvector Fields

We construct a generalization of the variations of Hodge structures on Calabi-Yau manifolds. It gives a Mirror partner for the theory of genus=0 Gromov-Witten invariantsComment: 12 pages, AMS-TeX; typos and a sign corrected, appendix added. Submitted to IMR