# 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

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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.

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+1, thanks for the wonderful review (and for preventing me from posting a totally wrong answer). –  Ron Maimon Jul 23 '12 at 7:54
The Duistermaat Heckman theorem is restricted to finitley many degrees of freedom. How would it apply to a topological field theory? –  Arnold Neumaier Jul 23 '12 at 13:40
@Arnold. The generalization of the Duistermaat-Heckman theorem to field theories is only to the physics level of rigor. Actually, the field theory examples for which this formalism was applied constitute of finite dimensional reduced phase spaces. For example the Chern-Simons theory whose reduced phase space is the moduli space of flat connections, or the coherent state path integral which is reduced to the lowest landau level by supersymmetry. I think that the first one who in roduced this generalization is Antti Niemi, please see for example Arxiv: hep-th/9301059 (He has also earlier works). –  David Bar Moshe Jul 23 '12 at 14:22
Thanks for your answer, David! That review article looks like a good reference for me. –  Ryan Thorngren Jul 23 '12 at 16:53