Nahm’s equation and the search for classical solutions in Yang-Mills theory

Abstract

The history of the theory of magnetic monopoles in classical electrodynamics and unified gauge theories is reviewed, and the Atiyah-Ward and Atiyah-Drinfe1d-Hitchin-Man in constructions of exact classical solutions to the self-dual Yang-Mills equations are described. It is shown that the one-dimensional self-dual equation introduced by Nahm can be reformulated as a Rieraann-Hi1bert problem through the twister transform previously used by Ward for monopole and instanton fields, and a general formula for the patching matrix is derived. This is evaluated in some special cases, and a few simple examples are given where Nahm's equation can be solved by this method. An attempt is made to generalize the ADHM construction to treat non self dual Yang-Mills fields, with only partial success. The one-dimensional analogue of the second-order Yang-Mills equation, the so-called non self dual Nahm equation, is investigated, paying particular attention to a simple ansatz in which translation of the fields is equivalent to a mere scale transformation of the matrices T(_i)(Z). For these 'separable solutions' the matrices satisfy certain cubic equations, whose solution space depends critically on the nature of the Lie algebra under consideration. It is shown that corresponding to certain Riemannian symmetric pairs there are one-parameter families of 'interpolating solutions' to the cubic equations, which join oppositely oriented bases of a Lie subalgebra. The associated matrix-valued functions T(_i)(z) therefore interpolate between solutions of 'selfdual' and 'antiselfdual' Nahm equations.