Letters in Mathematical Physics

, Volume 106, Issue 5, pp 675–692 | Cite as

Classification of Equivariant Star Products on Symplectic Manifolds



In this note, we classify invariant star products with quantum momentum maps on symplectic manifolds by means of an equivariant characteristic class taking values in the equivariant cohomology. We establish a bijection between the equivalence classes and the formal series in the second equivariant cohomology, thereby giving a refined classification which takes into account the quantum momentum map as well.


quantum momentum map equivariant cohomology star products Fedosov construction 

Mathematics Subject Classification



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  1. 1.
    Arnal D., Cortet J.C., Molin P., Pinczon G.: Covariance and geometrical invariance in ∗-quantization. J. Math. Phys. 24(2), 276–283 (1983)ADSMathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bayen F., Flato M., Frønsdal C., Lichnerowicz A., Sternheimer D.: Deformation theory and quantization. Ann. Phys. 111, 61–151 (1978)ADSCrossRefMATHGoogle Scholar
  3. 3.
    Bertelson M., Bieliavsky P., Gutt S.: Parametrizing equivalence classes of invariant star products. Lett. Math. Phys. 46, 339–345 (1998)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bertelson M., Cahen M., Gutt S.: Equivalence of star products. Class. Quantum Gravity 14, A93–A107 (1997)ADSMathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bordemann M., Brischle M., Emmrich C., Waldmann S.: Phase space reduction for star products: an explicit construction for \({{\mathbb{C} P^{n}}}\). Lett. Math. Phys. 36, 357–371 (1996)ADSMathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Deligne P.: Déformations de l’Algèbre des Fonctions d’une Variété Symplectique: Comparaison entre Fedosov et DeWilde. Lecomte. Sel. Math. New Ser. 1(4), 667–697 (1995)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Dolgushev V.A.: Covariant and equivariant formality theorems. Adv. Math. 191, 147–177 (2005)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Fedosov B.V.: A simple geometrical construction of deformation quantization. J. Differ. Geom. 40, 213–238 (1994)MathSciNetMATHGoogle Scholar
  9. 9.
    Fedosov, B.V.: Deformation Quantization and Index Theory. Akademie, Berlin (1996)Google Scholar
  10. 10.
    Guillemin, V.W., Sternberg, S.: Supersymmetry and Equivariant de Rham Theory. Springer, Berlin (1999)Google Scholar
  11. 11.
    Gutt S., Rawnsley J.: Equivalence of star products on a symplectic manifold; an introduction to Deligne’s Čech cohomology classes. J. Geom. Phys. 29, 347–392 (1999)ADSMathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Gutt S., Rawnsley J.: Natural star products on symplectic manifolds and quantum moment maps. Lett. Math. Phys. 66, 123–139 (2003)ADSMathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Hamachi K.: Quantum moment maps and invariants for G-invariant star products. Rev. Math. Phys. 14(6), 601–621 (2002)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Jansen S., Neumaier N., Schaumann G., Waldmann S.: Classification of invariant star products up to equivariant morita equivalence on symplectic manifolds. Lett. Math. Phys. 100, 203–236 (2012)ADSMathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Kontsevich M.: Deformation quantization of Poisson manifolds. Lett. Math. Phys. 66, 157–216 (2003)ADSMathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Müller-Bahns M.F., Neumaier N.: Invariant star products of wick type: classification and quantum momentum mappings. Lett. Math. Phys. 70, 1–15 (2004)ADSMathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Müller-Bahns M.F., Neumaier N.: Some remarks on \({{\mathfrak{g}}}\)-invariant Fedosov star products and quantum momentum mappings. J. Geom. Phys. 50, 257–272 (2004)ADSMathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Nest, R.: On some conjectures related to [Q, R] for Hamiltonian actions on Poisson manifolds. In: Conference Talk at the Workshop on Quantization and Reduction 2013 in Erlangen (2013)Google Scholar
  19. 19.
    Nest R., Tsygan B.: Algebraic index theorem. Commun. Math. Phys. 172, 223–262 (1995)ADSMathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Nest R., Tsygan B.: Algebraic index theorem for families. Adv. Math. 113, 151–205 (1995)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Neumaier, N.: Klassifikationsergebnisse in der Deformationsquantisierung. PhD thesis, Fakultät für Physik, Albert-Ludwigs-Universität, Freiburg (2001). https://www.freidok.uni-freiburg.de/data/2100. Accessed 12 Mar 2016
  22. 22.
    Neumaier N.: Local \({\nu}\)-Euler derivations and Deligne’s characteristic class of Fedosov star products and star products of special type. Commun. Math. Phys. 230, 271–288 (2002)ADSMathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Tsygan, B.: Equivariant deformations, equivariant algebraic index theorems, and a Poisson version of [Q, R] = 0 (2010). (Unpublished notes)Google Scholar
  24. 24.
    Waldmann S.: Poisson-Geometrie und Deformationsquantisierung. Eine Einführung. Springer, Heidelberg (2007)Google Scholar
  25. 25.
    Weinstein, A., Xu, P.: Hochschild cohomology and characteristic classes for star-products. In: Khovanskij, A., Varchenko, A., Vassiliev, V. (eds.) Geometry of Differential Equations. Dedicated to V. I. Arnold on the Occasion of his 60th Birthday, pp. 177–194. American Mathematical Society, Providence (1998)Google Scholar
  26. 26.
    Xu P.: Fedosov ∗-products and quantum momentum maps. Commun. Math. Phys. 197, 167–197 (1998)ADSMathSciNetCrossRefMATHGoogle Scholar

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© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Institut für Mathematik, Lehrstuhl für Mathematik XUniversität WürzburgWürzburgGermany

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