Lump dynamics in the CP^1 model on the torus

TL;DR

This paper explores the topology and geometry of the moduli space of degree 2 solutions in the CP^1 model on a torus, analyzing lump dynamics through geodesic motion and revealing its incomplete and finite diameter nature.

Contribution

It characterizes the moduli space M_2 as a coset space, analyzes lump dynamics via geodesics, and identifies a geodesic submanifold with explicit solutions, advancing understanding of CP^1 model solutions on a torus.

Findings

M_2 is homeomorphic to a coset space G/G_0.

Lump dynamics approximated by geodesic motion on M_2.

M_2 is geodesically incomplete with finite diameter.

Abstract

The topology and geometry of the moduli space, M_2, of degree 2 static solutions of the CP^1 model on a torus (spacetime T^2 x R) are studied. It is proved that M_2 is homeomorphic to the left coset space G/G_0 where G is a certain eight-dimensional noncompact Lie group and G_0 is a discrete subgroup of order 4. Low energy two-lump dynamics is approximated by geodesic motion on M_2 with respect to a metric g defined by the restriction to M_2 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 (M_2,g) is geodesically incomplete and has only finite diameter. A low dimensional geodesic submanifold is identified and a full description of its geodesics obtained.