Topological Quantum Theories and Representation Theory

  • Peter Woit
Part of the NATO ASI Series book series (NSSB, volume 245)


We discuss the relationship between path integrals, geometric quantization and representation theory for a simple quantum theory whose Hilbert space is a group representation. The path integrals involved have interesting cohomological significance and can be evaluated in terms of fixed point formulas to give the Kirillov and Weyl character formulas. The relation to recent work of Witten on Chern-Simons gauge theory is also discussed.


Line Bundle Dirac Operator Loop Space Holomorphic Section Geometric Quantization 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    E. Witten, Topological quantum field theory, Comm. Math. Phys. 117 (1988), 353–386.MathSciNetADSzbMATHCrossRefGoogle Scholar
  2. [2]
    E. Witten, Quantum field theory and the Jones polynomial, Princeton IAS preprint IASSNS-HEP- 88/33 (Sept. 1988).Google Scholar
  3. [3]
    J. Klauder and B.-S. Skagerstam, “Coherent States:Applications in Physics and Mathematical Physics,” World Scientific, Singapore, Philadelphia, 1985.zbMATHGoogle Scholar
  4. [4]
    H. B. Nielsen and D. Rohrlich, A path integral to quantize spin, Nucl. Phys. B299 (1988), 471–483.MathSciNetADSCrossRefGoogle Scholar
  5. [5]
    M. Stone, Supersymmetry and the quantum mechanics of spin, Nucl. Phys. B314 (1989), 557–586.ADSCrossRefGoogle Scholar
  6. [6]
    A. Alekseev, L. Fadeev and S. Shatashvili, Quantization of the symplectic orbits of the compact Lie group by means of the functional integral, Preprint (1988).Google Scholar
  7. [7]
    B. Kostant, Quantization and unitary representations, in “Lectures in Modern Analysis III,” Lecture Notes in Mathematics vol. 170, 1970, pp. 86–208.Google Scholar
  8. [8]
    R. Bott, Homogeneous vector bundles, Annals of Mathematics 66 (1957), 203–248.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    R. Bott, On induced representations, in “The Mathematical Heritage of Hermann Weyl,” Vol. 48 Proceedings of Symposia in Pure Mathematics, AMS, 1988, pp. 1–13.CrossRefGoogle Scholar
  10. [10]
    R. Bott, Homogeneous differential operators, in “Differential and Combinatorial Topology,” S.S. Cairns, ed., 1965, pp. 167–186.Google Scholar
  11. [11]
    G.B. Segal, The representation ring of a compact group, Publ. Math. IHES 34 (1968), 113–128.zbMATHGoogle Scholar
  12. [12]
    G.B. Segal, Equivariant K-theory, Publ. Math. IHES 34 (1968), 129–151.zbMATHGoogle Scholar
  13. [13]
    J. Marsden and A. Weinstein, Reduction of symplectic manifolds with symmetry, Rep. on Math. Phys. 5 (1974), 121–130.MathSciNetADSzbMATHCrossRefGoogle Scholar
  14. [14]
    L. Alvarez-Gaumé, Supersymmetry and the Atiyah-Singer Index Theorem, Comm. Math. Phys. 90 (1983), 161–173.MathSciNetADSzbMATHCrossRefGoogle Scholar
  15. [15]
    M.F. Atiyah, Circular symmetry and the stationary phase approximation, Astérisque 131 (1985), 43–59.MathSciNetGoogle Scholar
  16. [16]
    M.F. Atiyah and R. Bott, The moment map and equivariant cohomology, Topology 23 (1984), 1–28.MathSciNetzbMATHCrossRefGoogle Scholar
  17. [17]
    N. Berline and M. Vergne, Zeros d’un champ de vecteurs et classes characteristiques équivariantes, Duke Math. J. 90 (1983), 539–549.MathSciNetCrossRefGoogle Scholar
  18. [18a]
    J.J. Duistermaat and G. J. Heckman, On the variation in the cohomology in the symplectic form of the reduced phase space, Invent. Math. 69 (1982), 259–268.MathSciNetADSzbMATHCrossRefGoogle Scholar
  19. [18b]
    J.J. Duistermaat and G. J. Heckman, On the variation in the cohomology in the symplectic form of the reduced phase space, Invent. Math. 72 (1983), 153–158.MathSciNetADSzbMATHCrossRefGoogle Scholar
  20. [19]
    N. Berline and M. Vergne, The equivariant index and Kirillov’s character formula, Am. J. of Math. 107 (1985), 1159–1190.MathSciNetzbMATHCrossRefGoogle Scholar
  21. [20]
    A. Pressley and G. Segal, “Loop Groups,” Oxford University Press, New York, 1986.zbMATHGoogle Scholar
  22. [21]
    E. Witten, Non-abelian bosonization in two dimensions, Comm. Math. Phys. 92 (1984), 455–472.MathSciNetADSzbMATHCrossRefGoogle Scholar
  23. [22]
    M.F. Atiyah and R. Bott, The Yang-Mills equations over Riemann surfaces, Phil. Trans. Roy. Soc. A308 (1982), 523–615.MathSciNetADSGoogle Scholar
  24. [23]
    T.R. Ramadas, I.M. Singer and J. Weitsman, Some comments on Chern-Simons gauge theory, MIT preprint (1989).Google Scholar

Copyright information

© Springer Science+Business Media New York 1990

Authors and Affiliations

  • Peter Woit
    • 1
  1. 1.Mathematical Sciences Research InstituteBerkeleyUSA

Personalised recommendations