Joaquín Sánchez Guillén

The main concepts of zero curvature integrability are reviewed and a generalization to higher dimensions is proposed, based on non-abelian connections in loop spaces. Our results for their holonomies are explained, relating reparametrization invariance necessarily to abelianization. These are "no-go" theorems which restrict local formulations, but they can be evaded with sufficient conditions (constantcovariance of 2-forms), which have many useful applications. Main source: Integrable theories and loops spaces: Fundamentals, applications and new developments. (Review). O. Alvarez, L.A. Ferreira and J. S-G. Int. J. Modern Physics A 24, 10 (2009), 1825-1888. arXiv: 0901.1654 [hep-th]