Lattice simulations of global ZN\mathbb{Z}_N string networks in a scalar model with vacuum manifold PSU(N)=SU(N)/ZN\mathrm{PSU}(N)=\mathrm{SU}(N)/\mathbb{Z}_N show that the networks reach a scaling regime for N≤8N\le 8, with a string-density normalization growing as N2−1N^2-1, implying a gravitational-wave amplitude ΩGW∝μ2(N2−1)2\Omega_{\rm GW}\propto\mu^2(N^2-1)^2 for the confining-gauge-theory-motivated scenario. Two source-stated questions remain open: (i) whether scaling, unit-charge dominance, and the N2−1N^2-1 normalization persist for N>8N>8, where the 3(N2−1)3(N^2-1) adjoint scalar components and the O(N2−1)\mathcal{O}(N^2-1) strings per Hubble volume make simulations rapidly more expensive; and (ii) the behavior of the local counterpart - flux tubes of the dual PSU(N)\mathrm{PSU}(N) gauge theory with baryon vertices, whose reconnection probabilities must be determined and whose gauge fields screen the angular gradient energy that drives the global-string infrared logarithm. The problem is to determine the cosmological network evolution law (scaling versus frustration, charge composition, and density normalization) of ZN\mathbb{Z}_N/baryon-vertex string networks in these two regimes.

Background

Cosmic strings are line defects formed when a symmetry breaking leaves a vacuum manifold with nontrivial first homotopy group. In 'global' string models only scalar fields are present, so the angular gradient energy extends to large distance and the string tension grows logarithmically with the ratio of the separation scale to the core width; in 'local' models a gauge field screens this gradient energy, giving finite-tension flux tubes. A string network is said to reach a scaling regime when the mean inter-string separation grows in proportion to the cosmological horizon, so that a fixed number of long strings and junctions reside per Hubble volume; scaling underpins robust predictions of the gravitational-wave (GW) background produced by the network, of current interest in the pulsar-timing-array era.

For the projective group PSU(N)=SU(N)/ZN\mathrm{PSU}(N)=\mathrm{SU}(N)/\mathbb{Z}_N one has π1(PSU(N))=ZN\pi_1(\mathrm{PSU}(N))=\mathbb{Z}_N, so symmetry breaking to PSU(N)\mathrm{PSU}(N) produces strings carrying a conserved ZN\mathbb{Z}_N charge q=1,…,N−1q=1,\dots,N-1. A junction of NN unit-charge strings is charge-neutral; for N=3N=3 this is analogous to the QCD baryon as three color flux tubes meeting at a Y-shaped junction. Such NN-string junctions are called baryon vertices (in holographic confinement - confinement modeled via the gauge/gravity duality between Yang–Mills theory and gravity in anti-de Sitter space - Witten's wrapped-brane object with NN strings attached). This structure is motivated by the dual-superconductor picture of confinement in pure SU(N)\mathrm{SU}(N) Yang–Mills theory - the idea that confinement is the electromagnetic dual of superconductivity, with electric flux squeezed into tubes as dual Abrikosov-type vortices - and by the Higgs vacua of the mass-deformed N=4\mathcal{N}=4 supersymmetric Yang–Mills theory (N=1∗\mathcal{N}=1^* theory): Yamada and Yonekura showed that cosmic strings form at the confining transition of pure Yang–Mills theory and computed their GW signatures, making the network dynamics of color flux tubes with baryon vertices a question with observational consequences.

A network with charge-conserving junctions could in principle become frustrated, with an ever-growing number of strings and junctions per horizon, as happens for non-Abelian string networks whose distinct string types cannot reconnect. Whether ZN\mathbb{Z}_N networks instead scale had been studied only at the N=3N=3 level: Vachaspati and Vilenkin (1987) and Ng, Kibble, and Vachaspati (2008) studied Z3\mathbb{Z}_3-type networks without cosmic expansion, and Copeland–Saffin (2005), Hindmarsh–Saffin (2006, in an SU(2)/Z3\mathrm{SU}(2)/\mathbb{Z}_3 model), and Urrestilla–Vilenkin (2008) simulated cosmic-superstring networks with three-string junctions.

The source paper (Yamada, 2026, arXiv:2607.21701) closes this gap for N≤8N\le 8. It simulates, in a radiation-dominated universe, a scalar-only model of three adjoint fields with vacuum manifold PSU(N)\mathrm{PSU}(N), forming global ZN\mathbb{Z}_N strings with baryon-vertex-like junctions. For N=2,3,4,5,8N=2,3,4,5,8 the winding density ζ\zeta (the string length per horizon volume reconstructed from plaquette windings) approaches a scaling plateau rather than a frustrated network; the reduced normalization ζ~=ζ/(N2−1)\tilde\zeta=\zeta/(N^2-1) is approximately NN-independent (≈0.47\approx 0.47–0.670.67), i.e. the density grows with the number of Goldstone modes N2−1N^2-1; non-minimal-charge strings stay subdominant (about 12–14% of the length for N=4,5N=4,5; a few percent for N=8N=8). This implies ΩGW∝μ2(N2−1)2\Omega_{\rm GW}\propto\mu^2(N^2-1)^2, with μ\mu the unit-string tension. The paper also reports as completed checks that the normalization depends on the initial time only logarithmically and is insensitive to the initial fluctuation amplitude and to core-growth parameters. What the paper explicitly leaves open is the extension beyond this validated window: larger NN, and the local (gauged) flux-tube theory rather than the global scalar model. In the local theory the gauge screening removes the global-string infrared logarithm, and the reconnection probability - the probability that two colliding strings intercommute, i.e. reconnect by exchanging partners, which controls loop production and hence the GW signal - is a separate nonperturbative input that the scalar-only simulations cannot determine.

Problem Statement

Determine the cosmological network evolution law of ZN\mathbb{Z}_N string networks with baryon-vertex junctions in the two regimes left open by the validated global, N≤8N\le 8 simulations of arXiv:2607.21701 - whether the network reaches a scaling regime (as opposed to frustration), with what charge composition, and with what normalization of the string density and hence of the gravitational-wave signal. Concretely:

  1. (Large NN, global) For global ZN\mathbb{Z}_N strings in the scalar PSU(N)\mathrm{PSU}(N) model, extend the formation-and-evolution analysis to N>8N>8: establish whether the winding density still approaches a scaling regime, whether non-minimal-charge strings remain subdominant or become important, and whether the normalization continues to grow as N2−1N^2-1 - thereby testing the extrapolated law ΩGW∝μ2(N2−1)2\Omega_{\rm GW}\propto\mu^2(N^2-1)^2 and its refinement ΩGW∝(N−1)2(N2−1)2\Omega_{\rm GW}\propto(N-1)^2(N^2-1)^2.

  2. (Local flux tubes) Formulate and analyze the local counterpart: networks of flux tubes of the dual PSU(N)\mathrm{PSU}(N) gauge theory with baryon vertices, in which gauge fields screen the angular gradient energy (removing the infrared logarithm of the global model), including the determination of the relevant string–string and string–junction reconnection probabilities; establish whether such local networks reach scaling and with what normalization.

These two components jointly serve one objective: the scaling law and GW prediction of ZN\mathbb{Z}_N/baryon-vertex string networks across the model class motivated by confining Yang–Mills theory, outside the regime already simulated.

References

  1. Masaki Yamada, "Formation and scaling of ZN\mathbb{Z}_N strings for global SU(N)/ZN\mathrm{SU}(N)/\mathbb{Z}_N symmetry", 2026, arXiv:2607.21701, DOI 10.48550/arXiv.2607.21701, https://arxiv.org/abs/2607.21701 (source paper)
  2. Kazuto Nakamura and Masaki Yamada, "Delayed scaling of multitype cosmic FF- and DD-strings in velocity-dependent one-scale models", Phys. Rev. D 113 (2026) 063537, arXiv:2512.16821, DOI 10.1103/4tkt-pfk2, https://arxiv.org/abs/2512.16821
  3. Mark Hindmarsh and P. M. Saffin, "Scaling in a SU(2)/Z_3 model of cosmic superstring networks", JHEP 08 (2006) 066, arXiv:hep-th/0605014, DOI 10.1088/1126-6708/2006/08/066, https://arxiv.org/abs/hep-th/0605014
  4. Yifung Ng, T. W. B. Kibble, and Tanmay Vachaspati, "Formation of Non-Abelian Monopoles Connected by Strings", Phys. Rev. D 78 (2008) 046001, arXiv:0806.0155, DOI 10.1103/PhysRevD.78.046001, https://arxiv.org/abs/0806.0155
  5. Jon Urrestilla and Alexander Vilenkin, "Evolution of cosmic superstring networks: A Numerical simulation", JHEP 02 (2008) 037, arXiv:0712.1146, DOI 10.1088/1126-6708/2008/02/037, https://arxiv.org/abs/0712.1146
  6. Edmund J. Copeland and P. M. Saffin, "On the evolution of cosmic-superstring networks", JHEP 11 (2005) 023, arXiv:hep-th/0505110, DOI 10.1088/1126-6708/2005/11/023, https://arxiv.org/abs/hep-th/0505110
  7. Masaki Yamada and Kazuya Yonekura, "Cosmic strings from pure Yang–Mills theory", Phys. Rev. D 106 (2022) 123515, arXiv:2204.13123, DOI 10.1103/PhysRevD.106.123515, https://arxiv.org/abs/2204.13123
  8. Masaki Yamada and Kazuya Yonekura, "Dark baryon from pure Yang-Mills theory and its GW signature from cosmic strings", JHEP 09 (2023) 197, arXiv:2307.06586, DOI 10.1007/JHEP09(2023)197, https://arxiv.org/abs/2307.06586
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