Chern-Simons perturbative series revisited

TL;DR

This paper explores the algebraic structure underlying the perturbative expansion of Wilson loops in 3d Chern-Simons theory, introducing a new basis that simplifies calculations and reveals symmetries in knot invariants.

Contribution

It introduces a special basis in the center of the universal enveloping algebra, enabling representation of group factors in any finite-dimensional irreducible representation, and proves a symmetry of colored HOMFLY polynomials.

Findings

Computed higher-order Vassiliev invariants.

Proved the tug-the-hook symmetry of colored HOMFLY polynomial.

Established a group-theoretical framework for Wilson loop expansions.

Abstract

A group-theoretical structure in a perturbative expansion of the Wilson loops in the 3d Chern-Simons theory with $SU(N)$ gauge group is studied in symmetric approach. A special basis in the center of the universal enveloping algebra $ZU(\mathfrak{sl}_N)$ is introduced. This basis allows one to present group factors in an arbitrary irreducible finite-dimensional representation. Developed methods have wide applications, the most straightforward and evident ones are mentioned. Namely, Vassiliev invariants of higher orders are computed, a conjecture about existence of new symmetries of the colored HOMFLY polynomials is stated, and the recently discovered tug-the-hook symmetry of the colored HOMFLY polynomial is proved.