Ratio of k-string tensions in large-N Yang-Mills theory
In plain words
Heavy test charges can be grouped by how many units k of a color charge they carry that cannot be cancelled by gluons, and each k is joined by a flux string of different tension. The exact rule giving these tensions for many colors is unknown.
Precise statement
In $4\mathrm{D}$ $\mathrm{SU}(N)$ Yang-Mills let $\sigma_k$ be the asymptotic string tension between static sources of N-ality k. Determine $\sigma_k/\sigma_1$ at large N, in particular whether corrections to the free-string value k are of order $1/N$ (Casimir scaling, $k(N-k)/(N-1)$) or of order $1/N^{2}$ (sine law, $\operatorname{sin}(\pi k/N)/\operatorname{sin}(\pi/N)$), or follow neither form.
What would settle it
Large-$N$ extrapolation of continuum lattice $\sigma_{k}$ for $N$ up to 8 or more, or an analytic large-$N$ derivation of the ratio.
Status in the literature
Lattice data in $3+1$ dimensions for $N$ up to about 8 lie between Casimir scaling and the sine law without separating $1/N$ from $1/N^{2}$ corrections; in $2+1$ dimensions the leading correction was found to be of order $1/N$ (Athenodorou and Teper 2016).