Letters in Mathematical Physics

, Volume 66, Issue 3, pp 157–216

Deformation Quantization of Poisson Manifolds

  • Maxim Kontsevich

DOI: 10.1023/B:MATH.0000027508.00421.bf

Cite this article as:
Kontsevich, M. Letters in Mathematical Physics (2003) 66: 157. doi:10.1023/B:MATH.0000027508.00421.bf


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.

deformation quantizationhomotopy Lie algebras

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© Kluwer Academic Publishers 2003

Authors and Affiliations

  • Maxim Kontsevich
    • 1
  1. 1.I.H.E.S.Bures-sur-YvetteFrance