On Refined Vogel's universality

TL;DR

This work extends Vogel's universal framework from unrefined to refined Chern-Simons theory, showing that universality persists for simply laced root systems through Macdonald dimensions evaluated at the refined Weyl vector. The key result is an explicit universal formula Md$_{Adj}(\mathfrak{a},\mathfrak{b},\mathfrak{c})$ in terms of the Vogel parameters and refined variables $q,t$, alongside analogous factorization at $x=q^{2r_k}$. The non-simply-laced cases do not admit a full universal description due to extra deformation parameters, though certain limits reveal partial coincidences. The findings connect universal adjoint invariants to knot-theoretic hyperpolynomials, particularly the unknot, and point to further directions including universal formulas for more knots/links and potential extensions of the Vogel plane.

Abstract

In accordance with P. Vogel, a set of algebra structures in Chern-Simons theory can be made universal, independent of a particular family of simple Lie algebras. In particular, this means that various quantities in the adjoint representations of these simple Lie algebras such as dimensions and quantum dimensions, Racah coefficients, etc. are simple rational functions of two parameters on Vogel's plane, giving three lines associated with $sl$, $so/sp$ and exceptional algebras correspondingly. By analyzing the partition function of refined of Chern-Simons theory, it was suggested earlier that the refinement may preserve the universality for simply laced algebras. Here we support this conjecture by analysing the Macdonald dimensions, i.e. values of Macdonald polynomials at $q^ρ$, where $ρ$ is the Weyl vector: there is a universality formula that describes these dimensions for the simply laced algebras as a function on the Vogel's plane.