1995/11/27 by Scott Axelrod, Axelrod, Scott
Mathematics · Physics and Astronomy · #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #dg-ga #hep-th #math.DG
paper · pdf · doi:10.48550/arxiv.hep-th/9511196
25 pages, latex, amssymb, epsfig(11 figures), epic & eepic(1 figure), uuencoded gz-compressed tar-file. Minor modifications, 1 figure, and notation index added. This is final version to appear in "Proceedings on Geometry & Physics" (proceedings are from workshop in Aarhus, Denmark 1995)
The large k asymptotics (perturbation series) for integrals of the form ∫\cal Fμei k S, where μ is a smooth top form and S is a smooth function on a manifold \cal F, both of which are invariant under the action of a symmetry group \cal 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 \cM of critical points of S mod the action of \cal 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 \cal 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 \cal 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.