Vassiliev Invariants for Links from Chern-Simons Perturbation Theory

TL;DR

This paper explores the algebraic structure of Wilson loops in Chern-Simons theory, introduces new link invariants of finite type, and provides explicit calculations for low-degree invariants of small links.

Contribution

It develops a factorization theorem for Wilson lines, defines new finite type link invariants, and computes explicit numerical results for simple two-component links.

Findings

Factorization theorem for Wilson lines

New finite type link invariants introduced

Numerical results for two-component links up to six crossings

Abstract

The general structure of the perturbative expansion of the vacuum expectation value of a product of Wilson-loop operators is analyzed in the context of Chern-Simons gauge theory. Wilson loops are opened into Wilson lines in order to unravel the algebraic structure encoded in the group factors of the perturbative series expansion. In the process a factorization theorem is proved for Wilson lines. Wilson lines are then closed back into Wilson loops and new link invariants of finite type are defined. Integral expressions for these invariants are presented for the first three primitive ones of lower degree in the case of two-component links. In addition, explicit numerical results are obtained for all two-component links of no more than six crossings up to degree four.