Hidden-symmetry algebra for the self-dual Yang-Mills equation

Abstract

We find that the infinite-parameter hidden-symmetry algebra for the SL(N, C ) self-dual Yang-Mills field is the Kac-Moody algebra sl(N, C ) ⊗ C(λ,λ −1) . There is no center term for the algebra. The corresponding hidden-symmetry algebra for the real self-dual SU( N) Yang-Mills field is also derived. It is related to an infinite-dimensional symmetric space over a subalgebra su( N) ⊗ R ( λ), or can be viewed as a real form su( N) ⊗ C R( λ, λ −1) of the complex algebra sl(N, C ) ⊗ C(λ, λ −1) . A precise definition is given in the text.