Towards the Algebro-Geometric Integration of the Bogomolny Equations

Finding explicit solutions of the Bogomolny equations for (the BPS limit) of Yang-Mills-Higgs systems represents an important and challenging problem in mathematical physics because of its links to gauge theory and the standard model of elementary particles. Although the complete integrability of these equations was proven a quarter of a century ago and the relation of this integrability to algebraic curves was also elucidated, far less is known about the algebro-geometric (theta-function) solutions. In our investigation we develop the approach by Ercolani-Sinha (1989) based on the famous Atyah-Drinfeld-Hitchin-Manin-Nahm construction.

In the talk we first explain the whole construction in the case when Bogomolny equations are solvable in terms of elliptic functions where all answers are explicit. Then we discus the generalization to higher genera when Hitchin constraints appear. This leads to complications which make the problem both mathematically interesting and difficult.

Taking into account the subject of the meeting we will then concentrate on the computer algebra problems relevant to solving Hitchin's constraints and the calculation of objects of physical interest - gauge and Higgs field. These include the implementing Weierstrass-Poincaré reduction of Riemann period matrix to standard form, and calculating theta-functions whose period matrices yield poor convergence. Both problem are good candidates for incorporation within Maple packages.

More details on the work and results will be done in the next lecture of H.W.Braden.