Dynamics of CP^1 lumps on a cylinder

TL;DR

This paper studies the slow dynamics of topological solitons called lumps in the CP^1 sigma-model on a cylinder, showing that their behavior can be described by geodesic flow on moduli spaces with incomplete metrics in the multilump case.

Contribution

It extends the geodesic approximation of lump dynamics to the case of a cylinder and characterizes the associated Kaehler metrics, including explicit forms for two-lump interactions.

Findings

Metrics are incomplete in multilump interactions.

Two-lump metric can be computed explicitly using elliptic integrals.

Certain geodesics are identified and analyzed for lump interactions.

Abstract

The slow dynamics of topological solitons in the CP^1 sigma-model, known as lumps, can be approximated by the geodesic flow of the L^2 metric on certain moduli spaces of holomorphic maps. In the present work, we consider the dynamics of lumps on an infinite flat cylinder, and we show that in this case the approximation can be formulated naturally in terms of regular Kaehler metrics. We prove that these metrics are incomplete exactly in the multilump (interacting) case. The metric for two-lumps can be computed in closed form on certain totally geodesic submanifolds using elliptic integrals; particular geodesics are determined and discussed in terms of the dynamics of interacting lumps.