Complete integrability in classical gauge theories

Abstract

We consider completely integrable classical field theory models with a view to identifying the properties which characterize their integrability. In particulars, we study the infinite sets of ‘hidden' symmetries, and the corresponding transformations carrying representations of infinite dimensional loop algebras, of the following models; the chiral-field equations in two dimensions, the self-dual sector of pure gauge theories in 4 dimensions, the functional (loop-space) formulation of 3-dimensional gauge theories, and some sectors of the extended superaymmetric gauge theories. We also construct an infinite number of conserved spinor currents for the latter theories. The (non-) integrability of the full four dimensional Yang-Mills equations is studied? and a local approximation for the non-integrable phase factor of gauge theories on an arbitrary, infinitesimally small, straight-line path is presented. Finally, we study classical gauge theories in dimensions greater than four5 and obtain, in analogy to the self-duality equations, algebraic equations for the field-strength which automatically imply the higher dimensional Yang-Mills equations as a consequence of the Bianchi identities. The most interesting sets of equations found are those in eight dimensions which have a structure related to the algebra of the octonions.