Advertisement

Journal of Geometric Analysis

, Volume 18, Issue 2, pp 565–611 | Cite as

Toeplitz Operators on Symplectic Manifolds

  • Xiaonan Ma
  • George Marinescu
Article

Abstract

We study the Berezin-Toeplitz quantization on symplectic manifolds making use of the full off-diagonal asymptotic expansion of the Bergman kernel. We give also a characterization of Toeplitz operators in terms of their asymptotic expansion. The semi-classical limit properties of the Berezin-Toeplitz quantization for non-compact manifolds and orbifolds are also established.

Keywords

Toeplitz operator Berezin-Toeplitz quantization Bergman kernel spinc Dirac operator 

Mathematics Subject Classification (2000)

58F06 81S10 32A25 47B35 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Adem, A., Leida, J., Ruan, Y.: Orbifolds and Stringy Topology. Cambridge Tracts in Mathematics, vol. 171. Cambridge University Press, Cambridge (2007) zbMATHGoogle Scholar
  2. 2.
    Bayen, F., Flato, M., Fronsdal, C., Lichnerowicz, A., Sternheimer, D.: Deformation theory and quantization, part I. Lett. Math. Phys. 1, 521–530 (1977) CrossRefMathSciNetGoogle Scholar
  3. 3.
    Bayen, F., Flato, M., Fronsdal, C., Lichnerowicz, A., Sternheimer, D.: Deformation theory and quantization, part II. Ann. Phys. 111, 61–110 (1978) CrossRefMathSciNetzbMATHGoogle Scholar
  4. 4.
    Bayen, F., Flato, M., Fronsdal, C., Lichnerowicz, A., Sternheimer, D.: Deformation theory and quantization, part III. Ann. Phys. 111, 111–151 (1978) CrossRefMathSciNetzbMATHGoogle Scholar
  5. 5.
    Berezin, F.A.: Quantization. Izv. Akad. Nauk SSSR Ser. Mat. 38, 1116–1175 (1974) MathSciNetGoogle Scholar
  6. 6.
    Berline, N., Getzler, E., Vergne, M.: Heat Kernels and Dirac Operators. Grundl. Math. Wiss. Band, vol. 298. Springer, Berlin (1992) zbMATHGoogle Scholar
  7. 7.
    Bismut, J.-M., Vasserot, E.: The asymptotics of the Ray–Singer analytic torsion associated with high powers of a positive line bundle. Commun. Math. Phys. 125, 355–367 (1989) CrossRefMathSciNetzbMATHGoogle Scholar
  8. 8.
    Bordemann, M., Meinrenken, E., Schlichenmaier, M.: Toeplitz quantization of Kähler manifolds and gl(N), N→∞ limits. Commun. Math. Phys. 165, 281–296 (1994) CrossRefMathSciNetzbMATHGoogle Scholar
  9. 9.
    Borthwick, D., Lesniewski, A., Upmeier, H.: Nonperturbative deformation quantization of Cartan domains. J. Funct. Anal. 113(1), 153–176 (1993) CrossRefMathSciNetzbMATHGoogle Scholar
  10. 10.
    Borthwick, D., Uribe, A.: Almost complex structures and geometric quantization. Math. Res. Lett. 3, 845–861 (1996); Erratum: Math. Res. Lett. 5, 211–212 (1998) MathSciNetzbMATHGoogle Scholar
  11. 11.
    Boutet de Monvel, L., Guillemin, V.: The Spectral Theory of Toeplitz Operators. Annals of Math. Studies, vol. 99. Princeton University Press, Princeton (1981) zbMATHGoogle Scholar
  12. 12.
    Boutet de Monvel, L., Sjöstrand, J.: Sur la singularité des noyaux de Bergman et de Szegö. In: Journées: Équations aux Dérivées Partielles de Rennes (1975). Astérisque, vol. 34–35, pp. 123–164. Soc. Math. France, Paris (1976) Google Scholar
  13. 13.
    Cahen, M., Gutt, S., Rawnsley, J.: Quantization of Kähler manifolds, part I. J. Geom. Phys. 7(1), 45–62 (1990) CrossRefMathSciNetzbMATHGoogle Scholar
  14. 14.
    Cahen, M., Gutt, S., Rawnsley, J.: Quantization of Kähler manifolds, part II. Trans. Am. Math. Soc. 337(1), 73–98 (1993) CrossRefMathSciNetzbMATHGoogle Scholar
  15. 15.
    Cahen, M., Gutt, S., Rawnsley, J.: Quantization of Kähler manifolds, part III. Lett. Math. Phys. 30, 291–305 (1994) CrossRefMathSciNetzbMATHGoogle Scholar
  16. 16.
    Cahen, M., Gutt, S., Rawnsley, J.: Quantization of Kähler manifolds, part IV. Lett. Math. Phys. 34(2), 159–168 (1995) CrossRefMathSciNetzbMATHGoogle Scholar
  17. 17.
    Charles, L.: Berezin-Toeplitz operators, a semi-classical approach. Commun. Math. Phys. 239, 1–28 (2003) CrossRefMathSciNetzbMATHGoogle Scholar
  18. 18.
    Charles, L.: Toeplitz operators and Hamiltonian torus actions. J. Funct. Anal. 236(1), 299–350 (2006) CrossRefMathSciNetzbMATHGoogle Scholar
  19. 19.
    Coburn, L.A.: Deformation estimates for the Berezin-Toeplitz quantization. Commun. Math. Phys. 149(2), 415–424 (1992) CrossRefMathSciNetzbMATHGoogle Scholar
  20. 20.
    Dai, X., Liu, K., Ma, X.: On the asymptotic expansion of Bergman kernel. J. Differ. Geom. 72(1), 1–41 (2006); announced in C.R. Math. Acad. Sci. Paris 339(3), 193–198 (2004) MathSciNetzbMATHGoogle Scholar
  21. 21.
    Fedosov, B.V.: Deformation Quantization and Index Theory. Mathematical Topics, vol. 9. Akademie, Berlin (1996) zbMATHGoogle Scholar
  22. 22.
    Guillemin, V.: Star products on compact pre-quantizable symplectic manifolds. Lett. Math. Phys. 35(1), 85–89 (1995) CrossRefMathSciNetzbMATHGoogle Scholar
  23. 23.
    Karabegov, A.V., Schlichenmaier, M.: Identification of Berezin-Toeplitz deformation quantization. J. Reine Angew. Math. 540, 49–76 (2001) MathSciNetzbMATHGoogle Scholar
  24. 24.
    Klimek, S., Lesniewski, A.: Quantum Riemann surfaces, I: the unit disc. Commun. Math. Phys. 146(1), 103–122 (1992) CrossRefMathSciNetzbMATHGoogle Scholar
  25. 25.
    Kostant, B.: Quantization and unitary representations, I: prequantization. In: Lectures in Modern Analysis and Applications, III. Lecture Notes in Math., vol. 170, pp. 87–208. Springer, Berlin (1970) CrossRefGoogle Scholar
  26. 26.
    Ma, X.: Orbifolds and analytic torsions. Trans. Am. Math. Soc. 357(6), 2205–2233 (2005) CrossRefzbMATHGoogle Scholar
  27. 27.
    Ma, X., Marinescu, G.: The Spinc Dirac operator on high tensor powers of a line bundle. Math. Z. 240(3), 651–664 (2002) CrossRefMathSciNetzbMATHGoogle Scholar
  28. 28.
    Ma, X., Marinescu, G.: Generalized Bergman kernels on symplectic manifolds. Adv. in Math. 217(4), 1756–1815 (2008); announced in: C.R. Acad. Sci. Paris 339(7), 493–498 (2004) CrossRefMathSciNetzbMATHGoogle Scholar
  29. 29.
    Ma, X., Marinescu, G.: The first coefficients of the asymptotic expansion of the Bergman kernel of the spinc Dirac operator. Int. J. Math. 17(6), 737–759 (2006) CrossRefMathSciNetzbMATHGoogle Scholar
  30. 30.
    Ma, X., Marinescu, G.: Holomorphic Morse Inequalities and Bergman Kernels. Progress in Math., vol. 254. Birkhäuser, Basel (2007) zbMATHGoogle Scholar
  31. 31.
    Moreno, C., Ortega-Navarro, P.: Deformations of the algebra of functions on Hermitian symmetric spaces resulting from quantization. Ann. Inst. H. Poincaré Sect. A (N.S.) 38(3), 215–241 (1983) MathSciNetzbMATHGoogle Scholar
  32. 32.
    Pflaum, M.J.: On the deformation quantization of symplectic orbispaces. Differ. Geom. Appl. 19(3), 343–368 (2003) CrossRefMathSciNetzbMATHGoogle Scholar
  33. 33.
    Pflaum, M.J., Posthuma, H.B., Tang, X.: An algebraic index theorem for orbifolds. Adv. Math. 210(1), 83–121 (2007) CrossRefMathSciNetzbMATHGoogle Scholar
  34. 34.
    Schlichenmaier, M.: Deformation quantization of compact Kähler manifolds by Berezin-Toeplitz quantization. In: Conférence Moshé Flato 1999, vol. II. Math. Phys. Stud., vol. 22, pp. 289–306. Kluwer Academic, Dordrecht (2000) Google Scholar
  35. 35.
    Souriau, J.-M.: Structure des Systèmes Dynamiques. Mâtrises de Mathématiques. Dunod, Paris (1970) zbMATHGoogle Scholar
  36. 36.
    Taylor, M.E.: Partial Differential Equations, 1: Basic Theory. Applied Mathematical Sciences, vol. 115. Springer, Berlin (1996) zbMATHGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2008

Authors and Affiliations

  1. 1.UFR de MathématiquesUniversité Denis Diderot—Paris 7Paris Cedex 13France
  2. 2.Mathematisches InstitutUniversität zu KölnKölnGermany
  3. 3.Institute of Mathematics ‘Simion Stoilow’Romanian AcademyBucharestRomania

Personalised recommendations