A summary of multimonopoles

• Main Author: Houghton, C. J.
• Published: University of Cambridge 1998
• Subjects: 510
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.604260

For the most part, the thesis is about how finite symmetry groups are used to study Bogomolny-Prasad-Sommerfield multimonopoles. The Nahm equations corresponding to 7-monopoles and to 5-monopoles are generally intractable. However, the Nahm equations corresponding to an icosahedrally invariant 7-monopole and to an octahedrally invariant 5-monopole are tractable and these equations are calculated and solved. From a solution of the Nahm equations the monopole fields are numerically attainable. The Donaldson and Jarvis rational maps describe monopole moduli spaces and are used here to study symmetric multimonopoles. Geodesics of symmetric monopoles are found using the Donaldson rational maps. The twisted line scattering geodesics of monopoles with rotary-reflection symmetries are studied in this way. Using the Jarvis rational map it is possible to tell precisely which symmetric monopoles there are. A symmetric monopole corresponds to a symmetric rational map and the question of which symmetric monopoles exist is reduced to the question of which symmetric rational maps there are, a question answered using elementary group representation theory. It is found that there is a geodesic of tetrahedral 4-monopoles. The Nahm equations are solved for these monopoles. There is a two-dimensional space of <I>D</I><SUB>2</SUB> 3-monopoles. The <I>D</I><SUB>2</SUB> 3-monopole Nahm equations are complex Euler-Poinsot equations and their solutions are known. There are geodesics in this space that are identical to the 2-monopole right-angle scattering geodesics. The 3-monopole twisted line scattering geodesics also lie in this space. Knowing the Nahm data for these monopoles allows the fields to be computed numerically. It is discovered that there are monopoles along the twisted line scattering geodesics with anti-zeros of the Higgs field. Many different hyperKähler manifolds are monopole moduli spaces. These hyperKähler manifolds always have an isometric SO<SUB>3</SUB> action. In the thesis new hyperKähler manifolds are derived from monopole moduli spaces by fixing monopoles. These fixed monopole spaces do not have an SO<SUB>3</SUB> action.