10 problems

Large-$N$ commutation-relation conjecture. In the large $N$ limit, these operators satisfy

Let $p\geq 2$, set $\epsilon_1=1$, $\epsilon_2=k+p$, and define the rescaled operators … Let $\mathcal B$ be an appropriate energy eigenbasis for the Hilbert space. Non-Abelian lar…

Let $\mathscr O^{(p)}_\infty$ be the large $N$ limit algebra of quantum observables, and let $\mathcal H_\infty$ be the large $N$ limit Hilbert space of the Chern–Simons matrix mod…

Let $(M,a)$ be a closed, area-weighted topological map, embedded in a closed, connected, orientable surface, and let $G_N=\mathrm{U}(N)$. A loop is included in a disc if its drawin…

Let $\Sigma$ be a closed, connected, orientable, two dimensional, Riemannian manifold, the Euclidean plane $\mathbb{R}^2$, or a disc in $\mathbb{R}^2$. Let $G_N$ be a classical gro…

All-orders large-$N$ conjecture. In the large-$N$ limit, the function $H_N(h,|q|)$ is exactly represented to all orders in $h$ by

Let $A$ and $B$ be arbitrary matrices, and consider the large-$N$ limits of the corresponding integral over $U(N)$ and over $SU(N)$. Large-$N$ distinction conjecture. Based on the…

Let $\Sigma$, $U$, $C$, $a$, and $c=\operatorname{area}(\Sigma)-a$ be as in the preceding conjecture, with $C$ a simple closed curve in the topological disk $U$. For…

Let $\Sigma$ be a surface of area $\operatorname{area}(\Sigma)$, let $U\subset\Sigma$ be a topological disk, and let $C$ be a simple closed curve in $U$. Assign area $a$ to the int…

Large-$n$ spectral conjecture. The spectrum remains purely discrete with a finite spectral gap and a divergent vacuum energy; specifically,