Deformation Quantization of Poisson Manifolds
- Cite this article as:
- Kontsevich, M. Letters in Mathematical Physics (2003) 66: 157. doi:10.1023/B:MATH.0000027508.00421.bf
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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 (the ‘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 explicitly use the language of functional integrals.