  The large $k$ asymptotics (perturbation series) for integrals of the form
$\int_{\cal F}\mu e^{i k S}$, where $\mu$ 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.
