Pullback of the Volume Form, Integrable Models in Higher Dimensions and Exotic Textures

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A procedure allowing for the construction of Lorentz invariant integrable models living in d+1 dimensional space-time and with an n dimensional target space is provided. Here, integrability is understood as the existence of the generalized zero-curvature formulation and infinitely many conserved quantities. A close relation between the Lagrange density of the integrable models and the pullback of the pertinent volume form on target space is established. Moreover, we show that the conserved currents are Noether currents generated by the volume preserving diffeomorphisms. Further, we show how such models may emerge via abelian projection of some gauge theories. Then we apply this framework to the construction of integrable models with exotic textures. Particularly, we consider integrable models providing exact suspended Hopf maps i.e., solitons with a nontrivial topological charge of pi₄(S³). Finally, some families of integrable models with solitons of piₙ(Sⁿ) type are constructed. Infinitely many exact solutions with arbitrary value of the topological index are found. In addition, we demonstrate that they saturate a Bogomolny bound.