Advertisement

Communications in Mathematical Physics

, Volume 267, Issue 1, pp 227–263 | Cite as

Equivariant Asymptotics for Bohr-Sommerfeld Lagrangian Submanifolds

  • Marco DebernardiEmail author
  • Roberto Paoletti
Article

Abstract

Suppose given a complex projective manifold M with a fixed Hodge form Ω. The Bohr-Sommerfeld Lagrangian submanifolds of (M,Ω) are the geometric counterpart to semi-classical physical states, and their geometric quantization has been extensively studied. Here we revisit this theory in the equivariant context, in the presence of a compatible (Hamiltonian) action of a connected compact Lie group.

Keywords

Asymptotic Expansion Lagrangian Submanifolds Geometric Quantization Ample Line Bundle Fourier Integral Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. BW.
    Bates S., Weinstein A. (1997). Lectures on the geometry of quantization. Berkeley Mathematics Lecture Notes 8, AMS, Providence, RIzbMATHGoogle Scholar
  2. BSZ.
    Bleher P., Shiffman B., Zelditch S. (2000). Universality and scaling of correlations between zeros on complex manifolds. Invent. Math. 142:351–395CrossRefADSMathSciNetzbMATHGoogle Scholar
  3. BPU.
    Borthwick D., Paul T., Uribe A. (1995). Legendrian distributions with applications to relative Poincaré series. Invent. Math. 122(2):359–402CrossRefADSMathSciNetzbMATHGoogle Scholar
  4. BS.
    Boutet de Monvel L., Sjöstrand J. (1976). Sur la singularité des noyaux de Bergman et de Szegö. Astérisque 34–35:123–164Google Scholar
  5. BG.
    Burns D., Guillemin V. (2004). Potential functions and actions of tori on Kähler manifolds. Comm. Anal. Geom. 12 (1–2):281–303MathSciNetzbMATHGoogle Scholar
  6. DI.
    Dixmier J. (1964). Les C *-algebras et leurs réprésentations. Gauthier-Villars, ParisGoogle Scholar
  7. DU.
    Duistermaat J.J. (1996). Fourier integral operators. Birkhäuser, BostonzbMATHGoogle Scholar
  8. GE.
    Geiges H. (2006). Contact Geometry. In: Dillen F.J.E., Verstraelen L.C.A. (eds). Handbook of Differential Geometry 2. North Holland, Amsterdam, pp. 325–382Google Scholar
  9. GT.
    Gorodentsev, A.L., Tyurin, A.N.: Abelian Lagrangian algebraic geometry. Izv. Ross. Akad. Nauk Ser. Mat. 65:3, 15–50 (2001); English transl., Izv. Math. 65, 437–467 (2001)Google Scholar
  10. GGK.
    Guillemin V., Ginzburg V., Karshon Y. (2002). Moment maps, cobordism, and Hamiltonian group actions. Mathematics Surveys and Monographs 98, A.M.S., Providence, RIGoogle Scholar
  11. GP.
    Guillemin V., Pollack A. (1974). Differential topology. Prentice-Hall Inc., Englewood Cliffs, N.J.zbMATHGoogle Scholar
  12. GS1.
    Guillemin V., Sternberg S. (1982). Geometric quantization and multiplicities of group representations. Inv. Math. 67:515–538CrossRefADSMathSciNetzbMATHGoogle Scholar
  13. GS2.
    Guillemin V., Sternberg S. (1982). Homogeneous quantization and multiplicities of group representations. J. Func. Anal. 47:344–380CrossRefMathSciNetzbMATHGoogle Scholar
  14. GS3.
    Guillemin V., Sternberg S. (1983). The Gelfand-Cetlin system and quantization of the complex flag manifold. J. Func. Anal. 52:106–128CrossRefMathSciNetzbMATHGoogle Scholar
  15. H.
    Hörmander L. (1990). The analysis of partial differential operators I. Springer-Verlag, Berlin-Heidelberg-New YorkzbMATHGoogle Scholar
  16. K.
    Kostant, B.: Quantization and unitary representations. I. Prequantization. Lectures in modern analysis and applications, III (1965), Lecture Notes in Math., Vol. 170, Springer, Berlin, 1970, pp. 87–208Google Scholar
  17. P1.
    Paoletti R. (2003). Moment maps and equivariant Szegö kernels. J. Symplectic Geom. 2(1):133–175MathSciNetzbMATHGoogle Scholar
  18. P2.
    Paoletti R. (2005). The Szegö kernel of a symplectic quotient. Adv. Math. 197:523–553CrossRefMathSciNetzbMATHGoogle Scholar
  19. STZ.
    Shiffman B., Tate T., Zelditch S. (2004). Distribution laws for integrable eigenfunctions. Ann. Inst. Fourier (Grenoble) 54(5):1497–1546MathSciNetzbMATHGoogle Scholar
  20. SZ.
    Shiffman B., Zelditch S. (2002). Asymptotics of almost holomorphic sections of ample line bundles on symplectic manifolds. J. Reine Angew. Math. 544:181–222MathSciNetzbMATHGoogle Scholar
  21. W.
    Weinstein A. (1990). Connections of Berry and Hannay type for moving Lagrangian submanifolds. Adv. Math. 82:133–159CrossRefMathSciNetzbMATHGoogle Scholar
  22. Z.
    Zelditch S. (1998). Szegö kernels and a theorem of Tian. Int. Math. Res. Not. 6:317–331CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Dipartimento di Matematica F. CasoratiUniversità di PaviaPaviaItaly
  2. 2.Dipartimento di Matematica e ApplicazioniUniversità degli Studi di Milano BicoccaMilanoItaly

Personalised recommendations