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.
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References
Atiyah, M. F., Convexity and commuting Hamiltonians,Bull. London Math. Soc. 14, 1–15 (1982).
Atiyah, M. F. and Bott, R., The moment map and equivariant cohomology,Topology 23, 1–28 (1984).
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.
Berline, N. and Vergne, M., Zéros d'un champ de vecteurs et classes caractéristiques équivariantes,Duke Math. J. 50, 539–549 (1983).
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).
Guillemin, V., Lerman, E., and Sternberg, S., On the Kostant multiplicity formula,J. Geom. Phys 5, 721–750 (1988).
Guillemin, V. and Sternberg, S., Convexity properties of the moment mapping,Invent. Math. 67, 491–513 (1982).
Mathai, V. and Quillen, D., Superconnections, Thom classes, and equivariant differential forms,Topology 25, 85–110 (1986).
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).
Weinstein, A., SymplecticV-manifolds, periodic orbits of Hamiltonian systems, and the volume of certain Riemannian manifolds,Comm. Pure Appl. Math. 30, 265–271 (1977).
Witten, E., On quantum gauge theories in two dimensions,Comm. Math. Phys. 141, 153–209 (1991).
Witten, E., Two dimensional gauge theories revisited,J. Geom. Phys. 9, 303–368 (1992).