Deformation quantization of Poisson manifolds, I

Details Deformation quantization of Poisson manifolds, I Deformation quantization of Poisson manifolds, I

1997-09-29

arxiv.org/abs/q-alg/9709040v1

Lett.Math.Phys.66:157-216,2003

Physics

High Energy Physics

High Energy Physics - Theory

plain TeX and epsf.tex, 46 pages, 24 figures

Scientific paper

10.1023/B:MATH.0000027508.00421.

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.

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