On Universal Eigenvalues of Casimir Operator

TL;DR

This paper derives a universal formula for the second Casimir operator's eigenvalues on certain representations of simple Lie algebras, revealing dualities and peculiar values in gauge theories.

Contribution

It presents a universal expression for the second Casimir eigenvalues applicable to all simple Lie algebras, linking to knot polynomials and gauge dualities.

Findings

Universal form for second Casimir eigenvalues derived

Duality between $SO(2n)$ and $Sp(2n)$ demonstrated

Non-zero Casimir values on zero representations discussed

Abstract

Motivated by the universal knot polynomials in the gauge Chern-Simons theory, we show that the values of the second Casimir operator on an arbitrary power of Cartan product of $X_2$ and adjoint representations of simple Lie algebras can be represented in a universal form. We show that it complies with $N\longrightarrow -N$ duality of the same operator for $SO(2n)$ and $Sp(2n)$ algebras (the part of $N\leftrightarrow-N$ duality of gauge $SO(2n)$ and $Sp(2n)$ theories). We discuss the phenomena of non-zero universal values of Casimir operator on zero representations.