Ruben L. Mkrtchyan

Universality in simple Lie algebras and Chern-Simons theory.

We review the notion of Vogel's universality in simple Lie algebras and their applications.

We present universal functions for: the (quantum) dimensions of adjoint and some other (series of) representations; generating function of eigenvalues of higher Casimir operators on adjoint representation; invariant volume of the compact simple Lie groups; the central charge and partition function of Chern-Simons theory on 3d sphere; colored (adjoint) knots polynomials for torus knots, etc. Universal Chern-Simons partition function is presented in terms of multiple Barnes' gamma functions and multiple sine functions with universal arguments. In particular case of SU(N) gauge group this representation establishes an exact nonperturbative Chern-Simons / topological string duality. An anomaly of Vogel's symmetry is calculated. When restricted to the SU(N) line it is shown to coincide with Kinkelin's relation on Barnes' G-function.

Universality naturally leads to classification of simple Lie algebras in terms of solutions of certain cubic Diophantine equations on three variables. One of these equations establishes the connection with McKay correspondence and equivelar maps. We discuss Deligne's hypothesis on universal characters and Vogel's conjectures on Lambda-algebra and their implications for universality.

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