Chern-Simons perturbative series revisited

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.

10 pages

References in corpus (3)

• Wilson loops in terms of color invariants
• On R-matrix approaches to knot invariants
• Difference of mutant knot invariants and their differential expansion

Cited by in corpus (5)

• Differential Expansion for antiparallel triple pretzels: the way the factorization is deformed
• Implications for colored HOMFLY polynomials from explicit formulas for group-theoretical structure
• Evolution properties of the knot's defect
• Tug-the-hook symmetry for quantum 6j-symbols
• Multistrand Eigenvalue conjecture and Racah symmetries