A path-integral approach to polynomial invariants of links

TL;DR

This paper presents a three-dimensional, non-perturbative path-integral approach to polynomial invariants of knots and links, utilizing a monodromy matrix within Chern-Simons theory and advanced quantization methods.

Contribution

It introduces a novel covariant path-integral framework using monodromy matrices and an auxiliary topological quantum mechanics model for knot invariants.

Findings

Derived monodromy matrix for arbitrary Lie groups and representations

Extended the framework to higher-dimensional Chern-Simons actions

Provided a quantization scheme using Batalin-Vilkovisky formalism

Abstract

A brief review of a self-contained genuinely three-dimensional monodromy-matrix based non-perturbative covariant path-integral approach to {\it polynomial invariants} of knots and links in the framework of (topological) quantum Chern-Simons field theory is given. An idea of ``physical'' observables represented by an auxiliary topological quantum-mechanics model in an external gauge field is introduced substituting rather a limited notion of the Wilson loop. Thus, the possibility of using various generalizations of the Chern-Simons action (also higher-dimensional ones) as well as a purely functional language becomes open. The theory is quantized in the framework of the best suited in this case {\it antibracket-antifield} formalism of Batalin and Vilkovisky. Using the Stokes theorem and formal translational invariance of the path-integral measure a {\it monodromy matrix} corresponding to an arbitrary pair of irreducible representations of an arbitrary semi-simple Lie group is derived.