A New Approach to Integrable Theories in any Dimension

TL;DR

This paper generalizes the zero curvature representation to higher dimensions using a d-form connection, enabling new methods to construct conserved currents and solutions for various integrable field theories.

Contribution

It introduces a novel higher-dimensional zero curvature framework applicable to several physical theories, expanding the scope of integrable models and conserved quantities.

Findings

Constructed infinite conserved currents in the 2+1 dimensional CP^1 model.

Developed new methods for solutions and conserved currents in higher-dimensional theories.

Identified explicit conserved charges for each spin representation of sl(2).

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.