Article

# Nonlinear Grassmann Sigma Models in Any Dimension and An Infinite Number of Conserved Currents

Physics Letters B (Impact Factor: 4.57). 06/1998; DOI: 10.1016/S0370-2693(98)00981-2

Source: arXiv

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**ABSTRACT:**We review our proposal to generalize the standard two-dimensional flatness construction of Lax–Zakharov–Shabat to relativistic field theories in d+1 dimensions. The fundamentals from the theory of connections on loop spaces are presented and clarified. These ideas are exposed using mathematical tools familiar to physicists. We exhibit recent and new results that relate the locality of the loop space curvature to the diffeomorphism invariance of the loop space holonomy. These result are used to show that the holonomy is Abelian if the holonomy is diffeomorphism invariant. These results justify in part and set the limitations of the local implementations of the approach which has been worked out in the last decade. We highlight very interesting applications like the construction and the solution of an integrable four-dimensional field theory with Hopf solitons, and new integrability conditions which generalize BPS equations to systems such as Skyrme theories. Applications of these ideas leading to new constructions are implemented in theories that admit volume-preserving diffeomorphisms of the target space as symmetries. Applications to physically relevant systems like Yang–Mills theories are summarized. We also discuss other possibilities that have not yet been explored.International Journal of Modern Physics A 01/2012; 24(10). · 1.13 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**In the preceding paper,1 we constructed submodels of nonlinear Grassmann sigma models in any dimensions and, moreover, an infinite number of conserved currents and a wide class of exact solutions. In this letter, we first construct almost all conserved currents for the submodels and all those for CP1-model. We next review the Smirnov and Sobolev construction for the equations of CP1-submodel and extend the equations, the S-S construction and conserved currents to higher order ones.Modern Physics Letters A 11/2011; 14(14). · 1.11 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**A new class of non-linear O(3)O(3) models is introduced. It is shown that these systems lead to integrable submodels if an additional integrability condition (the generalized eikonal equation) is imposed. In the case of particular members of the family of the models the exact solutions describing toroidal solitons with a nontrivial value of the Hopf index are obtained. Moreover, the generalized eikonal equation is analyzed in detail. Topological solutions describing torus knots are presented. Multi-knot configurations are found as well.Physics Letters B 08/2005; 621(s 1–2):201–207. · 4.57 Impact Factor

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