Chern-Simons Gauge Theory: Ten Years After

TL;DR

This paper reviews a decade of progress in Chern-Simons gauge theory, focusing on its connection to knot invariants and the development of combinatorial formulas for Vassiliev invariants through different gauge fixings.

Contribution

It introduces a new combinatorial approach in the non-covariant temporal gauge for deriving Vassiliev invariants, complementing existing integral representations.

Findings

Different gauge fixings yield distinct representations of Vassiliev invariants.

The non-covariant temporal gauge provides combinatorial expressions for invariants.

Collected formulas for invariants up to order four using this approach.

Abstract

A brief review on the progress made in the study of Chern-Simons gauge theory since its relation to knot theory was discovered ten years ago is presented. Emphasis is made on the analysis of the perturbative study of the theory and its connection to the theory of Vassiliev invariants. It is described how the study of the quantum field theory for three different gauge fixings leads to three different representations for Vassiliev invariants. Two of these gauge fixings lead to well known representations: the covariant Landau gauge corresponds to the configuration space integrals while the non-covariant light-cone gauge to the Kontsevich integral. The progress made in the analysis of the third gauge fixing, the non-covariant temporal gauge, is described in detail. In this case one obtains combinatorial expressions, instead of integral ones, for Vassiliev invariants. The approach based on this last gauge fixing seems very promising to obtain a full combinatorial formula. We collect the combinatorial expressions for all the Vassiliev invariants up to order four which have been obtained in this approach.