Deformation Quantization of Poisson Manifolds
Abstract
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.
- Publication:
-
Letters in Mathematical Physics
- Pub Date:
- December 2003
- DOI:
- 10.1023/B:MATH.0000027508.00421.bf
- arXiv:
- arXiv:q-alg/9709040
- Bibcode:
- 2003LMaPh..66..157K
- Keywords:
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- Mathematics - Quantum Algebra;
- High Energy Physics - Theory;
- Mathematics - Algebraic Geometry
- E-Print:
- plain TeX and epsf.tex, 46 pages, 24 figures