Exact vortex solutions in a CP^N Skyrme-Faddeev type model

We consider a four dimensional field theory with target space being CP^N which constitutes a generalization of the usual Skyrme-Faddeev model defined on CP^1. We show that it possesses an integrable sector presenting an infinite number of local conservation laws, which are associated to the hidden symmetries of the zero curvature representation of the theory in loop space. We construct an infinite class of exact solutions for that integrable submodel where the fields are meromorphic functions of the combinations (x^1+i x^2) and (x^3+x^0) of the Cartesian coordinates of four dimensional Minkowski space-time. Among those solutions we have static vortices and also vortices with waves traveling along them with the speed of light. The energy per unity of length of the vortices show an interesting and intricate interaction among the vortices and waves.

💡 Research Summary

The paper introduces a four‑dimensional field theory whose target space is the complex projective space CPN, thereby extending the familiar Skyrme‑Faddeev model that is based on CP¹ (the two‑sphere). The authors start by writing down a Lagrangian that contains three pieces: a standard kinetic term proportional to the Kähler metric of CPN, a quartic Skyrme‑type term that stabilises solitonic configurations, and a topological term analogous to the Hopf term in the original model. While the full theory is non‑integrable, the authors identify a remarkable subsector that becomes integrable once a specific ansatz is imposed.

The key observation is that if the fields depend only on the complex combinations
 z = x¹ + i x² and w = x³ + x⁰,
the equations of motion can be recast as a zero‑curvature condition in loop space. This allows the construction of a Lax pair (A_z, A_w) whose flatness condition is equivalent to the field equations. Consequently an infinite tower of local conserved currents emerges, reflecting hidden symmetries of the loop‑space formulation.

To obtain explicit solutions, the authors adopt a meromorphic ansatz: each component u_i of the CPN field is written as a product of a holomorphic function of z and a holomorphic function of w,
 u_i(z,w) = f_i(z) · h(w), i = 1,…,N‑1.
The functions f_i(z) are arbitrary meromorphic maps on the transverse plane; their poles correspond to vortex cores, and the winding numbers around these poles give the topological charge of each vortex. The function h(w) encodes dependence along the light‑like direction w; choosing h(w)=const yields static vortices, while h(w)=e^{ik w} produces a wave travelling at the speed of light along the vortex line.

For static configurations (h = const) the energy per unit length reduces to a sum of contributions from each vortex core, proportional to the square of its winding number, exactly as in the original Skyrme‑Faddeev model. When a travelling wave is present, cross‑terms appear in the energy density because derivatives with respect to w mix with those in the transverse plane. The total energy per unit length takes the schematic form

 E/L = ∑_i