On refined Vogel's universality

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 analyzing the Macdonald dimensions, i.e. values of Macdonald polynomials at qρ, 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.