Localization, Conformal Motions and the Duistermaat-heckman Theorem 1

Here we develop an explicitly covariant expression for the stationary phase approximation of a classical partition function based on geometric properties of the phase space. As an example of the utility of such an evaluation we show that in the case of the Hamiltonian ows generating conformal rescalings of the underlying geometry, the classical partition function is given exactly by the leading term of the stationary phase approximation. We give an explicit example of such an extension of the Duistermaat-Heckman theorem. Understanding the stationary phase approximation in general and speciically the circumstances under which a partition function is given exactly by its saddle-point approximation reveals connections between classical mechanics and the geometry and topology of the associated phase space. The object of interest is the phase space partition function d 2n x q det !(x) e iTH(x) (1) which describes the statistical dynamics (with imaginary temperature) of a classical Hamil-tonian system. Here M is a 2n-dimensional phase space and ! = 1 2 ! (x)dx ^ dx is the symplectic 2-form on M which is closed, d! = 0, and non-degenerate, det !(x) 6 = 0 8x 2 M. The matrix inverse ! of ! deenes the Poisson brackets fx ; x g = ! (x) of the dynamical system. H is a smooth Hamiltonian function on M which for simplicity we assume has a nite set of critical points I(H) = fp 2 M : dH(p) = 0g each of which is non-degenerate. In many cases we can obtain useful information about a dynamical system by considering an asymptotic expansion for large-T of (1) with coeecients determined by the method of stationary-phase approximation. Most interesting and potentially useful are the cases where the resulting series in 1=T truncates at the rst term. Documenting the conditions under which such simpliication occurs has been the basis of so-called localization theory in both mathematics and physics 4]{{10] (see 11] for a recent review). An example of such a classiication is the Duistermaat-Heckman theorem 1]{{3] which states that if the phase