Skip to main content
SpringerLink
Log in
Book cover

Geometric Aspects of Analysis and Mechanics pp 257–293Cite as

Quantization of q-Hamiltonian SU(2)-spaces

  • Chapter
  • First Online:

Part of the book series: Progress in Mathematics ((PM,volume 292))

Abstract

We will explain how to define the quantization of q-Hamiltonian SU(2)- spaces as push-forwards in twisted equivariant K-homology, and prove the “quantization commutes with reduction” theorem for this setting. As applications, we show how the Verlinde formulas for flat SU(2)- or SO(3)-bundles are obtained via localization in twisted K-homology.

Mathematics Subject Classification (2010): 53D30, 19L50

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD   169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. A. Alekseev, A. Malkin, and E. Meinrenken, Lie group valued moment maps, J. Differential Geom. 48 (1998), no. 3, 445–495.

    MATH  MathSciNet  Google Scholar 

  2. A. Alekseev, E. Meinrenken, and C. Woodward, Formulas of Verlinde type for non simply-connected groups, 2000, Unfinished manuscript.

    Google Scholar 

  3. A. Alekseev, Group-valued equivariant localization, Invent. Math. 140 (2000), no. 2, 327–350.

    Google Scholar 

  4. A. Alekseev, E. Meinrenken, and C. Woodward, The Verlinde formulas as fixed point formulas, J. Symplectic Geom. 1 (2001), no. 1, 1–46.

    MATH  MathSciNet  Google Scholar 

  5. A. Alekseev, Duistermaat–Heckman measures and moduli spaces of flat bundles over surfaces, Geom. and Funct. Anal. 12 (2002), 1–31.

    Google Scholar 

  6. M. Atiyah and G. Segal, Twisted K -theory, Ukr. Mat. Visn. 1 (2004), no. 3, 287–330.

    MATH  MathSciNet  Google Scholar 

  7. M. F. Atiyah and G. B. Segal, The index of elliptic operators. II, Ann. of Math. (2) 87 (1968), 531–545.

    Google Scholar 

  8. M. F. Atiyah and I. M. Singer, The index of elliptic operators I, Ann. of Math. (2) 87 (1968), 484–530.

    Google Scholar 

  9. M. F. Atiyah and I. M. Singer, The index of elliptic operators III, Ann. of Math. (2) 87 (1968), 546–604.

    Google Scholar 

  10. A. Beauville, The Verlinde formula for PGL(p)., The mathematical beauty of physics (Saclay, 1996), Adv. Ser. Math. Phys., vol. 24, World Sci. Publishing, 1997, pp. 141–151.

    Google Scholar 

  11. N. Berline, E. Getzler, and M. Vergne, Heat kernels and Dirac operators, Grundlehren der mathematischen Wissenschaften, vol. 298, Springer-Verlag, Berlin–Heidelberg–New York, 1992.

    Google Scholar 

  12. D. Chatterjee, On the construction of Abelian gerbs, Ph.D. thesis, University of Cambridge, 1998. (1985), no. 62, 257–360.

    Google Scholar 

  13. A. Connes, Noncommutative differential geometry, Inst. Hautes Études Sci. Publ. Math.

    Google Scholar 

  14. J. Dixmier and A. Douady, Champs continus d’espaces hilbertiens et de C ∗ -algèbres, Bull. Soc. Math. France 91 (1963), 227–284.

    MATH  MathSciNet  Google Scholar 

  15. J. J. Duistermaat, Convexity and tightness for restrictions of Hamiltonian functions to fixed point sets of an antisymplectic involution,Trans.Amer.Math.Soc. 275 (1983), 412–429.

    Article  MathSciNet  Google Scholar 

  16. J. J. Duistermaat, The heat kernel Lefschetz fixed point formula for the Spin-c Dirac operator, Progress in Nonlinear Differential Equations and Their Applications, vol. 18, Birkh¨auser, Boston, 1996.

    Google Scholar 

  17. J. J. Duistermaat, V. Guillemin, E. Meinrenken, and S. Wu, Symplectic reduction and Riemann–Roch for circle actions, Math. Res. Letters 2 (1995), 259–266.

    MATH  MathSciNet  Google Scholar 

  18. J. J. Duistermaat and G. J. Heckman, On the variation in the cohomology of the symplectic form of the reduced phase space, Invent. Math. 69 (1982), 259–268.

    Article  MATH  MathSciNet  Google Scholar 

  19. D. Freed, The Verlinde algebra is twisted equivariant K -theory, Turkish J. Math. 25 (2001), no. 1, 159–167.

    MATH  MathSciNet  Google Scholar 

  20. D. Freed, M. Hopkins, and C. Teleman, Loop Groups and Twisted K-Theory I,arXiv: 0711.1906.

    Google Scholar 

  21. V. Guillemin, Reduced phase spaces and Riemann–Roch, Lie Groups and Geometry in Honor of B. Kostant (Massachusetts Institute of Technology, 1994) (R. Brylinski et al., eds.), Progress in Mathematics, vol. 123, Birkh¨auser, Boston, 1995, pp. 305–334.

    Google Scholar 

  22. V. Guillemin and S. Sternberg, Geometric quantization and multiplicities of group representations, Invent. Math. 67 (1982), 515–538.

    Article  MATH  MathSciNet  Google Scholar 

  23. N. Higson and J. Roe, Analytic K -homology, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2000, Oxford Science Publications.

    Google Scholar 

  24. N. Hitchin, Lectures on special Lagrangian submanifolds, Winter School on Mirror Symmetry, Vector Bundles and Lagrangian Submanifolds (Cambridge, MA, 1999), Amer. Math. Soc., Providence, RI, 2001, pp. 151–182.

    Google Scholar 

  25. J. Hurtubise, L. Jeffrey, and R. Sjamaar, Group-valued implosion and parabolic structures, Amer.J.Math. 128 (2006), no. 1, 167–214.

    Article  MATH  MathSciNet  Google Scholar 

  26. J. C. Hurtubise and L. C. Jeffrey, Representations with weighted frames and framed parabolic bundles, Canad. J. Math. 52 (2000), no. 6, 1235–1268.

    Article  MATH  MathSciNet  Google Scholar 

  27. L. Jeffrey and J. Weitsman, Bohr–Sommerfeld orbits in the moduli space of flat connections and the Verlinde dimension formula,Comm. Math.Phys. 150 (1992), no. 3, 593–630.

    Article  MATH  MathSciNet  Google Scholar 

  28. G. G. Kasparov, K -theory, group C ∗ -algebras, and higher signatures (conspectus), Novikov conjectures, index theorems and rigidity, Vol. 1 (Oberwolfach, 1993), London Math. Soc. Lecture Note Ser., vol. 226, Cambridge Univ. Press, Cambridge, 1995, pp. 101–146.

    Google Scholar 

  29. D. Krepski, Pre-quantization of the moduli space of flat G-bundles over a surface,J. Geom. Phys. 58 (2008), no. 11, 1624–1637.

    Article  MATH  MathSciNet  Google Scholar 

  30. E. Meinrenken, Symplectic surgery and the Spin c -Dirac operator, Adv. in Math. 134 (1998), 240–277.

    Article  MATH  MathSciNet  Google Scholar 

  31. E. Meinrenken, Twisted K -homology and group-valued moment maps, Preprint, August 2010, arXiv:1008.1261.

    Google Scholar 

  32. E. Meinrenken and R. Sjamaar, Singular reduction and quantization, Topology 38 (1999), 699–763.

    Article  MATH  MathSciNet  Google Scholar 

  33. L. O’Shea and R. Sjamaar, Moment maps and Riemannian symmetric pairs, Math. Ann. 317 (2000), no. 3, 415–457.

    Article  MATH  MathSciNet  Google Scholar 

  34. T. Pantev, Comparison of generalized theta functions, Duke Math. J. 76 (1994), no. 2, 509–539.

    Article  MATH  MathSciNet  Google Scholar 

  35. R. J. Plymen, Strong Morita equivalence, spinors and symplectic spinors, J. Operator Theory 16 (1986), no. 2, 305–324.

    MATH  MathSciNet  Google Scholar 

  36. A. Pressley and G. Segal, Loop groups, Oxford University Press, Oxford, 1988.

    Google Scholar 

  37. J. Rosenberg, Continuous-trace algebras from the bundle theoretic point of view,J. Austral. Math. Soc. Ser. A 47 (1989), no. 3, 368–381.

    Article  MATH  MathSciNet  Google Scholar 

  38. F. Schaffhauser, A real convexity theorem for quasi-Hamiltonian actions, arXiv:0705.0858.

    Google Scholar 

  39. R. Sjamaar and E. Lerman, Stratified symplectic spaces and reduction, Ann. of Math. (2) 134 (1991), 375–422.

    Google Scholar 

  40. A. Szenes, Verification of Verlinde’s formulas for SU(2), Internat. Math. Res. Notices (1991), no. 7, 93–98.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, Ontario, M5S2E4, Canada

    Eckhard Meinrenken

Authors
  1. Eckhard Meinrenken
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Eckhard Meinrenken .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Utrecht University, P.O. Box 80.010, Utrecht, 3508 TA, Netherlands

    Johan A.C. Kolk

  2. Mathematical Institute, Utrecht University, P.O. Box 80.010, Utrecht, 3508 TA, Netherlands

    Erik P. van den Ban

Additional information

Dedicated to Hans Duistermaat on the occasion of his 65th birthday

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer Science+Buisness Media, LLC

About this chapter

Cite this chapter

Meinrenken, E. (2011). Quantization of q-Hamiltonian SU(2)-spaces. In: Kolk, J., van den Ban, E. (eds) Geometric Aspects of Analysis and Mechanics. Progress in Mathematics, vol 292. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-8244-6_10

Download citation

Publish with us

Policies and ethics

Search

Navigation