Lump Dynamics in the ℂP<SUP>1</SUP> Model on the Torus

Abstract:The topology and geometry of the moduli space, M₂, of degree 2 static solutions of the ℂP¹ model on a torus (spacetime T₂ ×ℝ) are studied. It is proved that M₂ is homeomorphic to the left coset space G/G₀, where G is a certain eight-dimensional noncompact Lie group and G₀ is a discrete subgroup of order 4. Low energy two-lump dynamics is approximated by geodesic motion on M₂ with respect to a metric g defined by the restriction to M₂ of the kinetic energy functional of the model. This lump dynamics decouples into a trivial "centre of mass" motion and nontrivial relative motion on a reduced moduli space. It is proved that (M₂,g) is geodesically incomplete and has only finite diameter. A low dimensional geodesic submanifold is identified and a full description of its geodesics obtained.