Non-Abelian large-$N$ convergence conjecture for the Chern–Simons matrix model

Let $p\geq 2$, set $\epsilon_1=1$, $\epsilon_2=k+p$, and define the rescaled operators

$$\tilde{J}^a_{b;m,n}=\left(\frac{k+p}{p}N\right)^{-\frac{m+n}{2}}J^a_{b;m,n},\qquad \tilde{t}_{m,n}=\left(\frac{k+p}{p}N\right)^{-\frac{m+n+2\delta_{m,n}}{2}}t_{m,n}.$$

Let $\mathcal B$ be an appropriate energy eigenbasis for the Hilbert space. Non-Abelian large-$N$ convergence conjecture. The operators $\tilde{J}^a_{b;m,n}$ and $\tilde{t}_{m,n}$ converge as operators on the large $N$ limit of the Hilbert space. In the basis $\mathcal B$,

$$\tilde{t}_{n,n}=\frac{1}{n+1}\frac{p}{k+p}+O(N^{-1}),$$

and the leading terms of $\{\tilde{J}^a_{b;m,n}\}$ and $\{\tilde{t}_{m,n}:m\ne n\}$ depend only on $m-n$. This conjecture describes the still-unproved non-Abelian large-$N$ representation and convergence statement; it implies the subsequent commutation-relation conjecture.