Integrable theories in any dimension: a perspective

Details Integrable theories in any dimension: a perspective Integrable theories in any dimension: a perspective

1999-03-19

arxiv.org/abs/hep-th/9903168v1

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.

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