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MiS Preprint
13/2004
Dynamics of $\mathbb{CP}^{1}$ lumps on a cylinder
Nuno Miguel Romao
Abstract
The slow dynamics of topological solitons in the $\mathbb{CP}^{1}$ $\sigma$-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 Kähler 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 in terms of elliptic integrals; particular geodesics are determined and discussed in terms of the dynamics of interacting lumps.
