Deformation Quantization of Poisson Manifolds
Abstract
I prove that every finitedimensional 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 onetoone 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.
 Publication:

Letters in Mathematical Physics
 Pub Date:
 December 2003
 DOI:
 10.1023/B:MATH.0000027508.00421.bf
 arXiv:
 arXiv:qalg/9709040
 Bibcode:
 2003LMaPh..66..157K
 Keywords:

 Mathematics  Quantum Algebra;
 High Energy Physics  Theory;
 Mathematics  Algebraic Geometry
 EPrint:
 plain TeX and epsf.tex, 46 pages, 24 figures