ABSTRACT
The large [k] asymptotics (perturbation series) for integrals of the form https://www.w3.org/1998/Math/MathML"> ∫ f μ e i k S https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072393/02402e9d-0cfd-4d5e-bccf-46dd4134d466/content/eq2084.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , where μ is a smooth top form and S is a smooth function on a manifold F, both of which are invariant under the action of a symmetry group G, may be computed using the stationary phase approximation. This perturbation series can be expressed as the integral of a top form on the space ℳ of critical points of S mod the action of G. In this paper we overview a formulation of the “Feynman rules” computing this top form and a proof that the perturbation series one obtains is independent of the choice of metric on F needed to define it. We also overview how this definition can be adapted to the context of 3-dimensional Chern-Simons quantum field theory, where F is infinite dimensional. This results in a construction of new differential invariants depending on a closed, oriented 3-manifold M together with a choice of smooth component of the moduli space of flat connections on M with compact structure group G. To make this paper more accessible we warm up with a trivial example and only give an outline of the proof that one obtains invariants in the Chern-Simons case. Full details will appear elsewhere.
