Letters in Mathematical Physics

, Volume 66, Issue 3, pp 157–216 | Cite as

Deformation Quantization of Poisson Manifolds

  • Maxim Kontsevich


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 quantization homotopy Lie algebras 


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  1. 1.
    Alexandrov, M., Kontsevich, M., Schwarz, A. and Zaboronsky, O.: The geometry of the master equation and topological quantum field theory, Internat. J. Modern Phys. A 12(7) (1997), 1405–1429.Google Scholar
  2. 2.
    Arnal, D., Manchon, D. and Masmoudi, M.: Choix des signes pour la formalité de M. Kontsevich. Pacific J. Math. 203(2002), 23–66.Google Scholar
  3. 3.
    Arnold, V.I., Gusein-Zade, S.M. and Varchenko, A.N.: Singularities of Differentiable Maps, Vol. I: The Classi cation of Critical Points, Caustics and Wave Fronts, Birkhäuser, Boston, 1985.Google Scholar
  4. 4.
    Bar-Natan, D., Garoufalidis, S., Rozansky, L. and Thurston, D.: Wheels, wheeling, and the Kontsevich integral of the unknot, Israel J. Math. 119 (2000), 217–237.Google Scholar
  5. 5.
    Barannikov, S. and Kontsevich, M.: Frobenius manifolds and formality of Lie algebras of polyvector fields, Internat. Math. Res. Notices 1998(4) (1998), 201–215.Google Scholar
  6. 6.
    Bayen, F., Flato, M., Frønsdal, C., Lichnerowicz, A. and Sternheimer, D.: Deformation theory and quantization. I. Deformations of symplectic structures, Ann. Phys. 111(1) (1978), 61–110.Google Scholar
  7. 7.
    Cahen, M., Gutt, S. and De Wilde, M.: Local cohomology of the algebra of C functions on a connected manifold, Lett. Math. Phys. 4 (1980), 157–167.Google Scholar
  8. 8.
    Cattaneo, A. and Felder, G.: On the AKSZ formulation of the Poisson sigma model, Lett. Math. Phys. 56(2) (2001), 163–179.Google Scholar
  9. 9.
    Cattaneo, A. and Felder, G.: A path integral approach to the Kontsevich quantization formula, Comm. Math. Phys. 212(3) (2000), 591–611.Google Scholar
  10. 10.
    Cattaneo, A., Felder, G. and Tomassini, L.: From local to global deformation quantization of Poisson manifolds, Duke Math. J. 115(2) (2002), 329–352.Google Scholar
  11. 11.
    Deligne, P.: Catégories tannakiennes, In: The Grothendieck Festschrift, Vol. II, Progr. in Math. 87, Birkhäuser, Boston, 1990, pp. 111–195.Google Scholar
  12. 12.
    De Wilde, M. and Lecomte, P.B.A.: Existence of star-products and of formal deformations in Poisson Lie algebra of arbitrary symplectic manifolds, Lett. Math. Phys. 7 (1983), 487–496.Google Scholar
  13. 13.
    Duflo, M.: Caractères des algèbres de Lie résolubles, C.R. Acad. Sci. Sér. A 269 (1969), 437–438.Google Scholar
  14. 14.
    Etingof, P. and Kazhdan, D.: Quantization of Lie Bialgebras, I, Selecta Math. (New Ser.) 2(1) (1996), 1–41.Google Scholar
  15. 15.
    Fedosov, B.: A simple geometric construction of deformation quantization, J. Differential Geom. 40(2) (1994), 213–238.Google Scholar
  16. 16.
    Felder, G. and Shoikhet, B.: Deformation quantization with traces, Lett. Math. Phys. 53(1) (2000), 75–86.Google Scholar
  17. 17.
    Fulton, W. and MacPherson, R.: Compactification of configuration spaces, Ann. of Math. (2) 139(1) (1994), 183–225.Google Scholar
  18. 18.
    Gelfand, I. M. and Kazhdan D. A.: Some problems of differential geometry and the calculation of cohomologies of Lie algebras of vector fields, Soviet Math. Dokl. 12(5) (1971), 1367–1370.Google Scholar
  19. 19.
    Gerstenhaber, M. and Voronov, A.: Homotopy G-algebras and moduli space operad, Internat. Math. Res. Notices 1995(3) (1995), 141–153.Google Scholar
  20. 20.
    Getzler, E. and Jones, J. D. S.: Operads, homotopy algebra and iterated integrals for double loop spaces, 1994, hep-th/9403055.Google Scholar
  21. 21.
    Ginzburg, V.: Method of orbits in the representation theory of complex Lie groups, Funct. Anal. Appl. 15(1) (1981), 18–28.Google Scholar
  22. 22.
    Goldman, W. and Millson, J.: The homotopy invariance of the Kuranishi space, Illinois. J. Math. 34(2) (1990), 337–367.Google Scholar
  23. 23.
    Hinich, V.: Tamarkin's proof of Kontsevich formality theorem, Forum Math. 15(4) (2003), 591–614.Google Scholar
  24. 24.
    Hinich, V. and Schechtman, V.: Deformation theory and Lie algebra homology, I. II., Algebra Colloq. 4(2) (1997), 213–240, and 4(3) (1997), 291–316.Google Scholar
  25. 25.
    Hinich, V. and Schechtman, V.: Homotopy Lie algebras, In: I. M. Gelfand Seminar, Adv. Soviet Math. 16(2), Amer. Math. Soc., Providence, RI, 1993, pp. 1–28.Google Scholar
  26. 26.
    Hochschild, G., Kostant, B. and Rosenberg, A.: Differential forms on regular affine algebras, Trans. Amer. Math. Soc. 102(1962), 383–408.Google Scholar
  27. 27.
    Hu, P., Kriz, I. and Voronov, A.: On Kontsevich's Hochschild cohomology conjecture, 2003, math. AT/0309369.Google Scholar
  28. 28.
    Kashiwara, M. and Vergne, M.: The Campbell-Hausdor. formula and invariant hyperfunctions, Invent. Math. 47 (1978), 249–272.Google Scholar
  29. 29.
    Kirillov, A.: Elements of the Theory of Representations, Springer-Verlag, Berlin, 1976.Google Scholar
  30. 30.
    Kontsevich, M.: Feynman diagrams and low-dimensional topology, In: First European Congress of Mathematics (Paris, 1992), Vol. II, Progr. in Math. 120, Birkhäuser, Basel, 1994, pp. 97–121.Google Scholar
  31. 31.
    Kontsevich, M.: Formality conjecture, In: D. Sternheimer et al.(eds), Deformation Theory and Symplectic Geometry, Kluwer, Dordrecht, 1997, pp. 139–156.Google Scholar
  32. 32.
    Kontsevich, M.: Rozansky–Witten invariants via formal geometry, Compositio Math. 115(1) (1999), 115–127.Google Scholar
  33. 33.
    Kontsevich, M.: Homological algebra of mirror symmetry, In: Proceedings of ICM, (Zürich 1994) Vol. I, Birkhäuser, Basel, 1995, pp. 120–139.Google Scholar
  34. 34.
    Kontsevich, M.: Deformation quantization of Poisson manifolds, I., 1997, q-alg/9709040.Google Scholar
  35. 35.
    Kontsevich, M.: Operads and motives in deformation quantization, Lett. Math. Phys. 48(1) (1999), 35–72.Google Scholar
  36. 36.
    Kontsevich, M.: Deformation quantization of algebaric varieties, Lett. Math. Phys. 56(3) (2001), 271–294.Google Scholar
  37. 37.
    Kontsevich, M. and Soibelman, Y.: Deformations of algebras over operads and the Deligne conjecture, In: G. Dito and D. Sternheimer(eds), Conférence Moshé Flato 1999, Vol. I (Dijon 1999), Kluwer Acad. Publ., Dordrecht, 2000, pp. 255–307.Google Scholar
  38. 38.
    McClure, M. and Smith, J.: A solution of Deligne's Hochschild cohomology conjecture, In: Recent Progress in Homotopy Theory (Baltimore, MD, 2000), Contemp. Math. 293, Amer. Math. Soc., Providence, RI, 2002, pp. 153–193Google Scholar
  39. 39.
    Manin, Y.I.: Gauge Field Theory and Complex Geometry, Springer-Verlag, Berlin, 1988.Google Scholar
  40. 40.
    Markl, M. and Voronov, A.: PROPped up graph cohomology, 2000, math. QA/0307081.Google Scholar
  41. 41.
    Quillen, D.: Superconnections and the Chern character, Topology 24(1985), 89–95.Google Scholar
  42. 42.
    Schlessinger, M. and Stashe., J.: The Lie algebra structure on tangent cohomology and deformation theory, J. Pure Appl. Algebra 38(1985), 313–322.Google Scholar
  43. 43.
    Sullivan, D.: Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math. (1977) No. 47, (1978), 269–331.Google Scholar
  44. 44.
    Voronov, A.: Quantizing Poisson manifolds, In: Perspectives on Quantization (South Hadley, MA, 1996), Contemp. Math. 214, Amer. Math. Soc., Providence, RI, 1998, pp. 189–195.Google Scholar
  45. 45.
    Tamarkin, D.: Another proof of M. Kontsevich formality theorem, 1998, math. QA/9803025.Google Scholar
  46. 46.
    Tamarkin, D.: Quantization of Lie bialgebras via the formality of the operad of little disks, in: G. Halbout(ed. ), Deformation Quantization(Strasbourg 2001), IRMA Lectures in Math. Theoret. Phys. Vol. I, Walter de Gruyter, Berlin, 2002, pp. 203–236.Google Scholar
  47. 47.
    Tamarkin, D. and Tsygan, B.: Noncommutative differential calculus, homotopy BV algebras and formality conjectures, Methods Funct. Anal. Topology 6(2) (2000), 85–100.Google Scholar

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© Kluwer Academic Publishers 2003

Authors and Affiliations

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

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