Letters in Mathematical Physics

, Volume 29, Issue 4, pp 311–328 | Cite as

An integration formula for the square of moment maps of circle actions

Article

Abstract

The integration of the exponential of the square of the moment map of the circle action is studied by a direct stationary phase computation and by applying the Duistermaat-Heckman formula. Both methods yield two distinct formulas expressing the integral in terms of contributions from the critical set of the square of the moment map. Certain cohomological pairings on the symplectic quotient are computed explicitly using the asymptotic behavior of the two formulas.

Mathematics Subject Classifications (1991)

53C15 57R70 41A60 55N91 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Atiyah, M. F., Convexity and commuting Hamiltonians,Bull. London Math. Soc. 14, 1–15 (1982).Google Scholar
  2. 2.
    Atiyah, M. F. and Bott, R., The moment map and equivariant cohomology,Topology 23, 1–28 (1984).Google Scholar
  3. 3.
    Audin, M., Hamiltoniens périodiques sur les variété symplectiques compactes de dimension 4, in C. Albert (ed),Géométrie symplectique et méchanique, Proceedings 1988, Lecture Notes in Mathematics 1416, Springer, Berlin, Heidelberg, New York, 1990, pp. 1–25;The Topology of Torus Action on Symplectic Manifolds, Progress in Mathematics, Vol. 93, Birkhäuser, Basel, Boston, Berlin, 1991, Ch. IV.Google Scholar
  4. 4.
    Berline, N. and Vergne, M., Zéros d'un champ de vecteurs et classes caractéristiques équivariantes,Duke Math. J. 50, 539–549 (1983).Google Scholar
  5. 5.
    Duistermaat, J. J. and Heckman, G. J., On the variation in the cohomology of the symplectic form of the reduced phase space,Invent. Math. 69, 259–268 (1982); Addendum,ibid. 72, 153–158 (1983).Google Scholar
  6. 6.
    Guillemin, V., Lerman, E., and Sternberg, S., On the Kostant multiplicity formula,J. Geom. Phys 5, 721–750 (1988).Google Scholar
  7. 7.
    Guillemin, V. and Sternberg, S., Convexity properties of the moment mapping,Invent. Math. 67, 491–513 (1982).Google Scholar
  8. 8.
    Mathai, V. and Quillen, D., Superconnections, Thom classes, and equivariant differential forms,Topology 25, 85–110 (1986).Google Scholar
  9. 9.
    Satake, I., On a generalization of the notion of manifold,Proc. Nat. Acad. Sci. U.S.A. 42, 359–363 (1956); The Gauss-Bonnet theorem forV-manifolds,J. Math. Soc. Japan 9, 464–492 (1957).Google Scholar
  10. 10.
    Weinstein, A., SymplecticV-manifolds, periodic orbits of Hamiltonian systems, and the volume of certain Riemannian manifolds,Comm. Pure Appl. Math. 30, 265–271 (1977).Google Scholar
  11. 11.
    Witten, E., On quantum gauge theories in two dimensions,Comm. Math. Phys. 141, 153–209 (1991).Google Scholar
  12. 12.
    Witten, E., Two dimensional gauge theories revisited,J. Geom. Phys. 9, 303–368 (1992).Google Scholar

Copyright information

© Kluwer Academic Publishers 1993

Authors and Affiliations

  • Siye Wu
    • 1
  1. 1.Department of MathematicsColumbia UniversityNew YorkUSA

Personalised recommendations