k-string tensions inSU(N)gauge theories

In the context of four-dimensional $\mathrm{SU}\mathrm{}\left(N\right)\mathrm{}$ gauge theories, we study the spectrum of the confining strings. We compute, for the SU(6) gauge theory formulated on a lattice, the three independent string tensions ${\sigma }_k$ related to sources with $Z_N$ charge $k=1,2,3,$ using Monte Carlo simulations. Our results, whose uncertainty is approximately 2% for $k=2\mathrm{}$ and 4% for $k=3,$ are consistent with the sine formula ${\sigma }_k/\sigma =\sin (k\pi /N)/\sin (\pi /N)$ for the ratio between ${\sigma }_k$ and the standard string tension σ, and show deviations from the Casimir scaling. The sine formula is known to emerge in supersymmetric $\mathrm{SU}\mathrm{}\left(N\right)\mathrm{}$ gauge theories and in M theory. We comment on an analogous behavior exhibited by two-dimensional $\mathrm{SU}\mathrm{}\left(N\right)\mathrm{}\times \mathrm{SU}\mathrm{}\left(N\right)\mathrm{}$ chiral models.