Overview and Warmup Example for Perturbation Theory with Instantons

TL;DR

This paper reviews a method for computing asymptotic expansions of integrals with symmetry, introduces Feynman rules for these calculations, and discusses their application to 3D Chern--Simons theory to produce new topological invariants.

Contribution

It presents a formulation of Feynman rules for perturbation series in symmetry-invariant integrals and extends this approach to infinite-dimensional Chern--Simons theory for new 3-manifold invariants.

Findings

Feynman rules for top form computation are established

Perturbation series are shown to be metric-independent

Application to 3D Chern--Simons theory yields new invariants

Abstract

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.