Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
1999-03-19
Physics
High Energy Physics
High Energy Physics - Theory
Invited talk at the Meeting ''Trends in Theoretical Physics II'', Buenos Aires, Argentina, December 1998
Scientific paper
10.1063/1.59672
We review the developments of a recently proposed approach to study integrable theories in any dimension. The basic idea consists in generalizing the zero curvature representation for two-dimensional integrable models to space-times of dimension $d+1$ by the introduction of a $d$-form connection. The method has been used to study several theories of physical interest, like self-dual Yang-Mills theories, Bogomolny equations, non-linear sigma models and Skyrme-type models. The local version of the generalized zero curvature involves a Lie algebra and a representation of it, leading to a number of conservation laws equal to the dimension of that representation. We discuss the conditions a given theory has to satisfy in order for its associated zero curvature to admit an infinite dimensional (reducible) representation. We also present the theory in the more abstract setting of the space of loops, which gives a deeper understanding and a more simple formulation of integrability in any dimension.
Alvarez Orlando
Ferreira Antonio Luis
Guillen Joaquin Sanchez
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