Formation and scaling of $\mathbb{Z}_N$ strings for global $\mathrm{SU}(N)/\mathbb{Z}_N$ symmetry

We numerically investigate networks of global $\mathbb{Z}_N$ strings with multi-string junctions in a scalar field model whose vacuum manifold is $\mathrm{PSU}(N)=\mathrm{SU}(N)/\mathbb{Z}_N$. The construction of the model is motivated by the Higgs vacua of mass-deformed $\mathcal{N}=4$ supersymmetric Yang-Mills theory, commonly known as $\mathcal{N}=1^*$ theory, and provides a tractable effective description of string networks with baryon-vertex-like junctions. We perform classical lattice simulations of the formation and evolution of these networks in a radiation-dominated universe. For $N=2,3,4,5,$ and $8$, we find that the networks approach a scaling regime rather than becoming frustrated. The normalization of the string density grows in proportion to the dimension of the adjoint representation, $N^2-1$, while non-minimal-charge components remain subdominant. These results imply that, at least for $N\lesssim 8$, the amplitude of the gravitational-wave energy density generated by the cosmic-string network scales as $\Omega_{\rm GW}\propto \mu^2 (N^2-1)^2$, where $\mu$ is the tension of a unit-charge string.