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Scattering of vortices in the Abelian Higgs model. (English) Zbl 1152.81042
Introduction: We study the scattering of vortices in the Abelian $$(2+1)$$-dimensional Higgs model. The vortices, we are considering, are solutions of the vortex equations, arising in the superconductivity theory. They are given by smooth pairs $$(A,\Phi)$$, consisting of the (electromagnetic) gauge potential $$A$$ and the (scalar) Higgs field $$\Phi$$ on $$\mathbb C$$. Such solutions are parameterized (up to gauge equivalence) by the zeros of the Higgs field $$\Phi$$, so the moduli space of $$N$$ vortices can be identified with $$\mathbb C^N$$. The dynamics of vortices in $$\mathbb C$$ is governed by the hyperbolic Ginzburg-Landau action functional. The dynamics of $$N$$ vortices may be described approximately by geodesics of $$\mathbb C^N$$ in the metric, determined by the kinetic energy of the model. Unfortunately, this metric cannot be computed explicitly. But in a special case of the symmetric scattering of $$N$$ vortices we can show, without using the explicit form of the metric, that after their head-on collision the configuration of vortices looks the same, only rotated by the angle $$\pi/N$$. In particular, in the case of two vortices, their trajectories are rotated by the angle $$\pi/2$$ after the head-on collision, so we have the right-angle scattering. This result was already obtained earlier in a number of papers.
##### MSC:
 81V22 Unified quantum theories 81T13 Yang-Mills and other gauge theories in quantum field theory 81U99 Quantum scattering theory