Connection among Equations for the Classical Heisenberg Ferromagnet, the Ernst Equation for the Axisymmetric Gravitational Field and the Yang Equations for the Self-Dual Gauge Field

Abstract

A study is made on the connection among stationary equations for the classical continuous isotropic Heisenberg model (CCIHM), the Ernst equation for the axisymmetric gravitational field (ASGF) and the Yang equations for the self-dual SU(2) gauge field (SDGF). This is done by reducing the Yang equations to a four-dimensional (4d) version of the Ernst equation and by parametrizing the Ernst potential in terms of variables θ and φ analogous to two angles of rotation of spins in the CCIHM. Both of the Ernst equation and the Yang equations are thereby reduced to a pair of equaions for θ and φ quite similar to those for the CCIHM, the latter reducing to the former by taking θ → iθ. This shows one-to-one correspondence of solutions to the field equations between the CCIHM and the SDGF or the ASGF, though spatial dimensionality, symmetry and boundary condition of physical significance may be different for these different cases. By paying attention to particular solutions for which φ satisfies the Laplace equation, three types of solutions are obtained for the SDGF by using analogy with hydrodynamics, one of which is also applicable to the ASGF. For the first type new multi-vortex or multi-instanton solutions similar to vortex solutions for the two-dimensional (2d) CCIHM are obtained for the 4d SDGF. The second type is axially symmetric solutions for the three-dimensional (3d) SDGF and the ASGF written in terms of the Painlevé transcendents of the third kind. The third type is uiform-flow solutions for the 3d SDGF given as solutions to the 2d static sinh-Gordon equation.