Integrable Submodels of Nonlinear σ-models and Their Generalization

Preface The nonlinear σ-model is a field theory whose action is a energy functional of maps from a space-time to a Riemannian manifold. This theory is used in particle physics and condensed-matter physics as low-energy effective theories. The solutions of its equation of motion are called harmonic maps, which are important object in geometry. If the dimension of the space-time is 1 + n (this means one time-variable and n space-variables) and the Riemann man-ifold is a Grassmann manifold, it is called the nonlinear Grassmann model in (1 + n) dimensions. The nonlinear Grassmann model in (1+1) dimensions is a very interesting theory and is investigated in huge amount of papers. In particular, it has integrable structures such as a Lax-pair, an infinite number of conserved currents, a wide class of exact solutions and so on [Zak]. Moreover, this model carries instantons which exhibit many similarities to instantons of the four-dimensional Yang-Mills theory [Zak], [Raj]. In view of these rich structures, it may be natural to investigate the structures of the nonlinear Grassmann models in higher dimensions. We are interested in integrable structures of the nonlinear Grassmann models in higher dimensions. However, we find that it is hard to study the higher-dimensional models in a similar way as (1+1)-dimensional one because it is difficult to extend the concepts of integrability such as zero-curvature i conditions to higher dimensions.posed a new approach to higher-dimensional integrable theories [AFG1] (see also [AFG2]) in 1997. They defined a local integrability condition in higher dimensions as the vanishing of a curvature of a trivial bundle over a path-space. They investigated the condition for the nonlinear CP 1-model in (1+2) dimensions. Then they found an integrable submodel of the nonlinear CP 1-model in (1 + 2) dimensions. We call it the CP 1-submodel for short. The CP 1-submodel possesses an infinite number of conserved currents and a wide class of exact solutions. Moreover, solutions of the CP 1-submodel are also those of the nonlinear CP 1-model. Thus, by analysing the submodel, we can investigate hidden structures of the original model. After AFG's proposal, the theory of integrable submodels has been generalized and applied to some interesting models in physics to find new non-Therefore it is expected that the theory of submodels open a new way to develop exact methods in higher-dimensional field theories. In this thesis, we investigate various integral submodels and generalize them. We use …