A new approach to integrable theories in any dimension

Abstract

The zero curvature representation for two dimensional integrable models is generalized to spacetimes of dimension d + 1 by the introduction of a d-form connection. The new generalized zero curvature conditions can be used to represent the equations of motion of some relativistic invariant field theories of physical interest in 2 + 1 dimensions (BF theories, Chern-Simons, 2 + 1 gravity and the CP 1 model) and 3 + 1 dimensions (self-dual Yang-Mills theory and the Bogomolny equations). Our approach leads to new methods of constructing conserved currents and solutions. In a submodel of the 2+1 dimensional CP 1 model, we explicitly construct an infinite number of previously unknown nontrivial conserved currents. For each positive integer spin representation of sl(2) we construct 2j +1 conserved currents leading to 2j + 1 Lorentz scalar charges.

Some aspects of self-duality and generalised BPS theories

“…These theories are, in fact, integrable in the sense of generalised integrability [35,36] and their conservation laws may be expressed as a generalised zero curvature condition in an appropriate higher loop space. The fact that, in many cases, the conservation laws of generalised integrability are related to target space SDiff symmetries was first pointed out in [37], for a detailed discussion see [38].…”

Conservation laws in Skyrme-type models

“…This set of vector fields does not form a subalgebra, however, for general H (1) and H (2) (i.e. ∂ 3Ỹ 3 = 0 does not hold in general).…”

“…∂ 3Ỹ 3 = 0 does not hold in general). It does form a subalgebra for special choices of H (1) , H (2) , like, e.g., H (1) = H (1) (X 2 ) and H (2) = H (2) (X 1 ), or for H (1) = ∂ 2 H and H (2) = −∂ 1 H. But in these special cases Y 3 = 0 and, therefore, they are included in the subalgebra discussed above. Finally, we give the general expression for Noether currents in a relativistic field theory which correspond to the vector fields v (Y ) that generate volume preserving diffeomorphisms on target space.…”

A generalization of the submodel of nonlinear models

“…Instead of higher dimensional models themselves , they considered their submodels to construct integrable models in the sense of possessing an infinite number of conserved currents. They applied their theories to nonlinear CP 1 model in (1 + 2)-dimensions in [1].…”