A new approach to integrable theories in any dimension

Orlando Alvarez, Luiz A. Ferreira, J. Sánchez Guillén

Research output: Contribution to journalArticlepeer-review

92 Scopus citations


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 ℂP1 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 ℂP1 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 languageEnglish (US)
Pages (from-to)689-736
Number of pages48
JournalNuclear Physics B
Issue number3
StatePublished - Oct 5 1998
Externally publishedYes


  • General dimensions
  • Integrability
  • New solutions
  • Zero curvature

ASJC Scopus subject areas

  • Nuclear and High Energy Physics


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