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Symplectic Connections of Ricci Type and Star Products

  • Michel Cahen
  • Simone GuttEmail author
  • Stefan Waldmann
Chapter
Part of the Progress in Mathematics book series (PM, volume 287)

Abstract

In this article we relate the construction of Ricci-type symplectic connections by reduction to the construction of star products by reduction yielding rather explicit descriptions for the star product on the reduced space.

AMS Classification (2010): 53D55, 53C07, 53D20

Key words

Deformation quantization Symplectic connections Reduction Ricci-type 

References

  1. 1.
    Abraham, R., Marsden, J.E.: Foundations of Mechanics, 2nd edn. Addison-Wesley, MA (1985)Google Scholar
  2. 2.
    Baguis, P., Cahen, M.: A construction of symplectic connections through reduction. Lett. Math. Phys. 57, 149–160 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Baguis, P., Cahen, M.: Marsden-Weinstein reduction for symplectic connections. Bull. Belg. Math. Soc. Simon Stevin 10(1), 91–100 (2003)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Bayen, F., Flato, M., Frønsdal, C., Lichnerowicz, A., Sternheimer, D.: Deformation theory and quantization. Ann. Phys. 111, 61–151 (1978)CrossRefzbMATHGoogle Scholar
  5. 5.
    Bieliavsky, P., Cahen, M., Gutt, S., Rawnsley, J., Schwachhöfer, L.: Symplectic connections. Int. J. Geomet. Meth. Mod. Phys. 3(3), 375–420 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bordemann, M.: (Bi)Modules, morphismes et réduction des star-produits: le cas symplectique, feuilletages et obstructions. Preprint math.QA/0403334, 135 (2004)Google Scholar
  7. 7.
    Bordemann, M.: (Bi)Modules, morphisms, and reduction of star-products: the symplectic case, foliations, and obstructions. Trav. Math. 16, 9–40 (2005)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Bordemann, M., Neumaier, N., Waldmann, S.: Homogeneous Fedosov star products on cotangent bundles I: Weyl and standard ordering with differential operator representation. Comm. Math. Phys. 198, 363–396 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bordemann, M., Herbig, H.-C., Waldmann, S.: BRST cohomology and phase space reduction in deformation quantization. Comm. Math. Phys. 210, 107–144 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bordemann, M., Neumaier, N., Pflaum, M.J., Waldmann, S.: On representations of star product algebras over cotangent spaces on Hermitian line bundles. J. Funct. Anal. 199, 1–47 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Cahen, M., Schwachhöfer, L.J.: Special symplectic connections and Poisson geometry. Lett. Math. Phys. 69, 115–137 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Cahen, M., Gutt, S., Schwachhöfer, L.: Construction of Ricci-type connections by reduction and induction. In: Marsden, J.E., Ratiu, T.S. (eds.) The Breadth of Symplectic and Poisson Geometry. Progress in Mathematics, vol. 232, pp. 41–57. Birkhäuser Boston, MA (2005); Festschrift in honor of Alan WeinsteinGoogle Scholar
  13. 13.
    Cattaneo, A.S., Felder, G.: Coisotropic submanifolds in Poisson geometry and Branes in the Poisson sigma model. Lett. Math. Phys. 69, 157–175 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Cattaneo, A.S., Felder, G.: Relative formality theorem and quantisation of coisotropic submanifolds. Adv. Math. 208(2), 521–548 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Fedosov, B.V.: Reduction and eigenstates in deformation quantization. In: Demuth, M., Schrohe, E., Schulze, B.-W. (eds.) Pseudo-Differential Calculus and Mathematical Physics. Advances in Partial Differential Equations, vol. 5, pp. 277–297. Akademie, Berlin (1994)Google Scholar
  16. 16.
    Fedosov, B.V.: Non-abelian reduction in deformation quantization. Lett. Math. Phys. 43, 137–154 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Fisch, J., Henneaux, M., Stasheff, J., Teitelboim, C.: Existence, uniqueness and cohomology of the classical BRST charge with ghosts of ghosts. Comm. Math. Phys. 120, 379–407 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Gerstenhaber, M.: On the deformation of rings and algebras. Ann. Math. 79, 59–103 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Glößner, P.: Phasenraumreduktion von Sternprodukten für superauflösbare Constraintalgebren. PhD thesis, Fakultät für Physik, Albert-Ludwigs-Universität, Freiburg (1998)Google Scholar
  20. 20.
    Glößner, P.: Star product reduction for coisotropic submanifolds of codimension 1. Preprint Freiburg FR-THEP-98/10 math.QA/9805049 (1998)Google Scholar
  21. 21.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Gutt, S., Rawnsley, J.: Natural star products on symplectic manifolds and quantum moment maps. Lett. Math. Phys. 66, 123–139 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Vaisman, I.: Symplectic curvature tensors. Monatsh. Math. 100(4), 299–327 (1985)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Département de MathématiqueUniversité Libre de BruxellesBruxellesBelgium
  2. 2.Département de MathématiqueLMAM, Université Paul Verlaine-MetzMetz Cedex 01France
  3. 3.Fakultät für Mathematik und Physik, Physikalisches InstitutAlbert-Ludwigs-Universität FreiburgFreiburgGermany

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