Universality in Chern-Simons theory

Abstract

We show that the perturbative part of the partition function in the Chern-Simons theory on a 3-sphere as well as the central charge and expectation value of the unknotted Wilson loop in the adjoint representation can be expressed in terms of the universal Vogel's parameters $\alpha, \beta, \gamma.$ The derivation is based on certain generalisations of the Freudenthal-de Vries strange formula.Comment: Slightly revised version with minor correction

Refined Chern–Simons versus Vogel universality

“…If we expect that there exist knot invariants depending on Vogel parameters, we should expect that the partition functions of ChernSimons theory for all simple Lie algebras can be expressed in an universal fashion in terms of these parameters. This has been proved for Chern-Simons theory on S 3 in [13,14], where a closed integral expression in terms of the Vogel parameters for the partition function has been derived.…”

“…The partition function of universal Chern-Simons on S 3 has been shown in [13,14] to be factorizable as…”

“…Since we have Ξ(x, 1) = 1, the combined summations over the roots can be performed in the β = 1 case using the results of [13]. However, we do not know at present how to perform the individual summations in general, hence we have to evaluate the integrand (3.8) in a case by case basis, as we will do for A N in section 3.2 and for D N in section 3.3.…”

“…Rather unrelated to these developments, very recently there has been a proposal for a three parameter generalization of Chern-Simons theory, dubbed universal Chern- Simons [13,14]. The original motivation for such a universal Chern-Simons theory goes back to the work of Vogel [15,16].…”

On universal knot polynomials

“…As shown in [9,22], in the Chern-Simons gauge theory [23] on the 3d sphere universal are such quantities as central charge, perturbative and non-perturbative partition functions, etc., and particularly unknotted Wilson average in adjoint representation. In the present paper we discuss the problem of universality of adjoint Wilson averages for some knotted curves.…”