Title:

Nahm's equation and the search for classical solutions in YangMills theory

The history of the theory of magnetic monopoles in classical electrodynamics and unified gauge theories is reviewed, and the AtiyahWard and AtiyahDrinfe1dHitchinMan in constructions of exact classical solutions to the selfdual YangMills equations are described. It is shown that the onedimensional selfdual equation introduced by Nahm can be reformulated as a RieraannHi1bert 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 YangMills fields, with only partial success. The onedimensional analogue of the secondorder YangMills equation, the socalled 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 oneparameter families of 'interpolating solutions' to the cubic equations, which join oppositely oriented bases of a Lie subalgebra. The associated matrixvalued functions T(_i)(z) therefore interpolate between solutions of 'selfdual' and 'antiselfdual' Nahm equations.
