Mathematical Sciences: Dynamical Behaviour of Vortices and Monopoles in Classical Gauge Theories

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This project is on the time-dependent Yang-Mills-Higgs equations. These are nonlinear hyperbolic partial differential equations which are invariant under the Lorentz group and the infinite dimensional gauge group. The investigator has proved that under certain situations it is possible to construct solutions to the equations which correspond to motion of the vortices or monopoles according to a finite dimensional Hamiltonian system on the configuration space. These approximations are known to be valid on long but finite time intervals. The present aim of this investigation is to extend this understanding to more general situations, in particular to monopole scattering which is described by geodesics with respect to the Atiyah-Hitchin metric on the moduli space. A longer term aim is to extend the understanding to infinite time -- it is conceivable that asymptotically the solution approaches a combination of vortices and monopoles moving according to the Hamiltonian system on the configuration space in the maximum norm. The reason for this conjecture is the dispersion of radiation. Another long term aim is to extend the understanding to the equations of general relativity in which black holes take the place of vortices and monopoles. The work has possible applications to superconductivity and particle physics in addition to being of mathematical interest.