Dynamics of monopole walls

The moduli space of centred Bogomolny–Prasad–Sommerfield 2-monopole fields is a 4-dimensional manifold <math altimg="si1.gif" xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">M</mi></math> with a natural metric, and the geodesics on <math altimg="si1.gif" xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">M</mi></math> correspond to slow-motion monopole dynamics. The best-known case is that of monopoles on <math altimg="si2.gif" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><mrow><mn>3</mn></mrow></msup></math> , where <math altimg="si1.gif" xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">M</mi></math> is the Atiyah–Hitchin space. More recently, the case of monopoles periodic in one direction (monopole chains) was studied a few years ago. Our aim in this note is to investigate <math altimg="si1.gif" xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">M</mi></math> for doubly-periodic fields, which may be visualized as monopole walls. We identify some of the geodesics on <math altimg="si1.gif" xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">M</mi></math> as fixed-point sets of discrete symmetries, and interpret these in terms of monopole scattering and bound orbits, concentrating on novel features that arise as a consequence of the periodicity