Infinite number of local conservation laws for the self-dual SU(2) Yang-Mills system within 't Hooft's ansatz

Abstract

We derive infinite sets of local continuity equations for the four-dimensional classical self-dual SU(2) Yang-Mills fields subjected to 't Hooft's ansatz. In striking analogy to the two-dimensional CP( n) non-linear sigma model where local conservation laws obtain either from complex Cauchy-Riemann analyticity or from a matrix Riccati equation, our local sets derive from quaternionic Fueter analyticity or a Riccati equation associated with the geometric prolongation structure implied by the Belavin-Zakharov linear spectral problem for the self-dual Yang-Mills system. Our analysis underlines the close connection between local and non-local conservation laws and suggests that infinite sets of local continuity equations should be present in the general self-(antiself-)dual gauge field case.