Integrable theories in any dimension: a perspective

TL;DR

This paper reviews a generalized zero curvature approach to studying integrable theories across any dimension, extending classical methods to higher-dimensional space-times using d-form connections.

Contribution

It introduces a new framework for integrability in arbitrary dimensions via d-form connections, broadening the scope of classical zero curvature methods.

Findings

Application to self-dual Yang-Mills and sigma models

Conditions for infinite-dimensional representations of zero curvature

Loop space formulation of integrability

Abstract

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.