On Universal Quantum Dimensions

TL;DR

This paper derives universal formulas for quantum dimensions of Lie algebra representations, enabling a unified approach to supersymmetric Yang-Mills theory, knot polynomials, and testing Deligne's hypothesis.

Contribution

It provides new universal expressions for quantum dimensions of Lie algebra representations, supporting calculations in supersymmetric theories and knot invariants.

Findings

Derived universal formulas for quantum dimensions.

Validated formulas through numerical proof of Deligne's hypothesis.

Connected universal characters to knot polynomial computations.

Abstract

We represent in the universal form restricted one-instanton partition function of supersymmetric Yang-Mills theory. It is based on the derivation of universal expressions for quantum dimensions (universal characters) of Cartan powers of adjoint and some other series of irreps of simple Lie algebras. These formulae also provide a proof of formulae for universal quantum dimensions for low-dimensional representations, needed in derivation of universal knot polynomials (i.e. colored Wilson averages of Chern-Simons theory on 3d sphere). As a check of the (complicated) formulae for universal quantum dimensions we prove numerically Deligne's hypothesis on universal characters for symmetric cube of adjoint representation.