quantum field theory - when is the stationary phase approximation exact?

I am thinking about some topological field theories, and I am wondering when one can say that the stationary phase approximation (ie. a sum of the first-order variations about each vacuum) is exact.

I am looking perhaps for conditions on how the space of vacua is embedded into the space of all field configurations. I suspect that when the action is a Morse function (and I suppose the space of field configurations is finite dimensional) that the exactness of the stationary phase approximation implies some very strict topological constraints on the configuration space... torsion-free and so on.

Anyone have a good reference or some wisdom?

Also I'd like to dedicate this question to the memory of theoreticalphysics.stackexchange

Answer

In general, the situation where the stationary phase approximation is exact is described by the Duistermaat Heckman theorem, which states (not in its most general form) that if $M$ is a compact symplectic manifold and $H$ is a Hamiltonian generationg a torus action on $M$, then for the "partition" function

$Z = \int_M e^{it H} d_L(M)$

the stationary phase approximation is exact ($d_L(M)$ is the Liouville measure) and the integral can be computed by summing the contributions from the extrema of $H$ (fixed points of the torus action).

An equivalent characterization of the hamiltonian $H$ is that it is a perfect Morse function.

Two very known examples are the Gaussian integral and the spin partition function in a magnetic field where the classical and the quantum partition functions are exactly the same.

This theorem was applied and generalized to more complicated situations (e.g., when the fixed points are not isolated), to path integrals of certain theories (coherent state path integrals), loop spaces and to topological field theories.

Further reserach of the Duistermaat-Heckman theorem and its generalizations led to the discovery of a general phenomenon leading to this type of exactness, now called "equivariant localization".

Please see the following review article"Equivariant Localization of Path Integrals" by Richard J. Szabo, where numerous applications are described.