Perturbation Theory with Instantons

The large k asymptotics (perturbation series) for integrals of the form ∫ F μe ikS , 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. This work was supported in part by the Divisions of Applied Mathematics of the U. S. Department of Energy under contract DE-FG02-88ER25066 as well as by an Alfred P. Sloan Research Fellowship.