### Abstract

The zero curvature representation for two-dimensional integrable models is generalized to space-times 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 non-trivial conserved currents. For each positive integer spin representation of sl(2) we construct 2j + 1 conserved currents leading to 2j + 1 Lorentz scalar charges.

Original language | English (US) |
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Pages (from-to) | 689-736 |

Number of pages | 48 |

Journal | Nuclear Physics B |

Volume | 529 |

Issue number | 3 |

DOIs | |

State | Published - Oct 5 1998 |

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### Keywords

- General dimensions
- Integrability
- New solutions
- Zero curvature

### ASJC Scopus subject areas

- Nuclear and High Energy Physics

### Cite this

*Nuclear Physics B*,

*529*(3), 689-736. https://doi.org/10.1016/S0550-3213(98)00400-3