Integrability of Chern-Simons-Higgs Vortex Equations and a Reduction of the Self-Dual Yang-Mills Equations to Three Dimensions

Abstract

I would like to make brief presentations on two topics, both of which focus on an issue of ‘integrability’ in equations of interest in high energy physics. In my first talk, I would like to introduce the Chern-Simons-Higgs vortex equations, which describe classical solutions of a certain (2 + 1)-dimensional field theory. In flat space-time these equations are non-integrable, but in curved spacetime the ODE describing cylindrically symmetric vortices can, by correct choice of the metric, be made to be a degenerate case of the third Painleve equation, possessing rational solutions which can be written down. Remarkably, the overall features of the solutions (found numerically) for the flat spacetime case, are very similar to those found in the integrable case, suggesting that maybe the non-integrable case should be looked on as a ‘perturbation’ of the integrable case. My second talk is on the topic of a reduction of the self-dual Yang-Mills equations from four to three dimensions: there has been considerable interest recently in the self-dual Yang-Mills equations as a ‘master equation’, from which many integrable systems can be obtained by suitable reductions. Here I focus on a method to reduce to three dimensions, but the systems that emerge are really trivial generalizations of two dimensional integrable systems.