Advertisement

Algebras and Representation Theory

, Volume 21, Issue 5, pp 1151–1164 | Cite as

Formal Geometric Quantization III: Functoriality in the Spinc Setting

  • Paul-Emile Paradan
Article
  • 7 Downloads

Abstract

In this paper, we prove a functorial aspect of the formal geometric quantization procedure of non-compact spin-c manifolds.

Keywords

Geometric quantization Dirac operator Equivariant index 

Mathematics Subject Classification (2010)

57S15 53C27 58J20 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Atiyah, M.F.: Elliptic Operators and Compact Groups. Lecture Notes in Mathematics, vol. 401. Springer-Verlag, Berlin (1974)CrossRefGoogle Scholar
  2. 2.
    Atiyah, M.F., Singer, I.M.: The index of elliptic operators III. Ann. Math. 87, 546–604 (1968)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Atiyah, M.F., Hirzebruch, F.: Spin Manifold and Group Actions, Essays on Topology and Related Topics (Geneva), vol. 1969. Springer-Verlag, Berlin (1970)Google Scholar
  4. 4.
    Berline, N., Getzler, E., Vergne, M.: Heat Kernels and Dirac Operators, Grundlehren, vol. 298. Springer, Berlin (1991)zbMATHGoogle Scholar
  5. 5.
    Braverman, M.: Index theorem for equivariant Dirac operators on noncompact manifolds. K-Theory 27, 61–101 (2002)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Duistermaat, J.J.: The Heat Equation and the Lefschetz Fixed Point Formula for the Spinc-Dirac operator, Progress in Nonlinear Differential Equation and Their Applications, vol. 18. Birkhauser, Boston (1996)Google Scholar
  7. 7.
    Hattori, A.: spinc-structures and S 1-actions. Invent. Math. 48, 7–31 (1978)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Hochs, P., Song, Y.: Equivariant indices of spinc-Dirac operators for proper moment maps. Duke Math. J. 166, 1125–1178 (2017)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Kawasaki, T.: The index of elliptic operators over V-manifolds. Nagoya Math. J. 84, 135–157 (1981)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Ma, X., Zhang, W.: Geometric quantization for proper moment maps: the Vergne conjecture. Acta Math. 212, 11–57 (2014)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Meinrenken, E.: Symplectic surgery and the Spinc-Dirac operator. Adv. Math. 134, 240–277 (1998)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Meinrenken, E., Sjamaar, R.: Singular reduction and quantization. Topology 38, 699–762 (1999)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Paradan, P.-E.: Localization of the Riemann-Roch character. J. Funct. Anal. 187, 442–509 (2001)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Paradan, P.-E.: Spinc-quantization and the K-multiplicities of the discrete series. Annales scientifiques de l’E.N.S. 36, 805–845 (2003)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Paradan, P.-E.: Formal geometric quantization. Ann. Inst. Fourier 59, 199–238 (2009)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Paradan, P.-E.: Formal geometric quantization II. Pac. J. Math. 253, 169–211 (2011)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Paradan, P.-E., Vergne, M.: Witten non abelian localization for equivariant K-theory and the [Q, R] = 0 Theorem, arXiv:http://arXiv.org/abs/1504.07502 (2015), accepted in Memoirs of the A.M.S.
  18. 18.
    Paradan, P.-E., Vergne, M.: Admissible coadjoint orbits for compact Lie groups. Transformation Groups (2017).  https://doi.org/10.1007/s00031-017-9457-2 MathSciNetCrossRefGoogle Scholar
  19. 19.
    Paradan, P.-E., Vergne, M.: Equivariant Dirac operators and differential geometric invariant theory. Acta Math. 218, 137–199 (2017)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Tian, Y., Zhang, W.: An analytic proof of the geometric quantization conjecture of Guillemin-Sternberg. Invent. Math. 132, 229–259 (1998)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institut Montpelliérain Alexander GrothendieckUniversité de Montpellier, CNRSMontpellierFrance

Personalised recommendations