BPSZNstring tensions, sine law and Casimir scaling, and integrable field theories

Abstract

We consider a Yang-Mills-Higgs theory with spontaneous symmetry breaking of the gauge group $G\ensuremath{\rightarrow}U(1{)}^{r}\ensuremath{\rightarrow}{C}_{G}$, with ${C}_{G}$ being the center of $G$. We study two vacua solutions of the theory which produce this symmetry breaking. We show that for one of these vacua, the theory in the Coulomb phase has the mass spectrum of particles and monopoles which is exactly the same as the mass spectrum of particles and solitons of two-dimensional affine Toda field theory, for suitable coupling constants. That result holds also for $\mathcal{N}=4$ super Yang-Mills theories. On the other hand, in the Higgs phase, we show that for each of the two vacua the ratio of the tensions of the BPS ${Z}_{N}$ strings satisfy either the Casimir scaling or the sine law scaling for $G=SU(N)$. These results are extended to other gauge groups: for the Casimir scaling, the ratios of the tensions are equal to the ratios of the quadratic Casimir constant of specific representations; for the sine law scaling, the tensions are proportional to the components of the left Perron-Frobenius eigenvector of Cartan matrix ${K}_{ij}$ and the ratios of tensions are equal to the ratios of the soliton masses of affine Toda field theories.