Partition function of Chern-Simons theory as renormalized q-dimension

TL;DR

This paper demonstrates that the renormalized q-dimension of certain affine Kac-Moody algebra representations converges to the universal partition function of Chern-Simons theory on a 3-sphere as q approaches 1.

Contribution

It establishes a direct link between the q-dimension of affine Kac-Moody algebra representations and the Chern-Simons partition function, providing a new perspective on their relationship.

Findings

q-dimension of affine algebra representations matches Chern-Simons partition function in the limit

Renormalization is necessary for the q-dimension to converge to the partition function

Provides a mathematical bridge between algebraic and topological quantum field theories

Abstract

We calculate $q$-dimension of $k$-th Cartan power of fundamental representation $\Lambda_0$, corresponding to affine root of affine simply laced Kac-Moody algebras, and show that in the limit $q\rightarrow 1 $, and with natural renormalization, it is equal to universal partition function of Chern-Simons theory on three-dimensional sphere.