Scattering of vortices in Abelian Higgs models on compact Riemann surfaces

Abelian Higgs models on Riemann surfaces are natural analogues of the $({2+1})$-dimensional Abelian Higgs model on the plane. The last model arises in theory of superconductivity. For this model the following result was previously obtained: if two vortices (zeros of the Higgs field) move slowly, then after the head-on collision they scatter under the right angle, and if $N$ vortices collide, then after the symmetric head-on collision they scatter on the angle $\pi/N$. In the critical case (when the parameter of the model is equal to 1) these results can be obtained with the help of so-called adiabatic principle. This principle allows to consider geodesics of so-called kinetic metric (metric that is given by kinetic energy functional) on the moduli space of static solutions as approximations to dynamical solutions of the model with small kinetic energy. Recently, the adiabatic principle was rigorously justified in the $(2+1)$-dimensional Abelian Higgs model on the plane in the critical case. Although the metric can not be written in explicit form, one can prove that required geodesics (describing the $\pi/N$ scattering) exist, using smoothness of the metric in coordinates that are given by symmetric functions on positions of vortices and symmetry properties of the kinetic metric. A local analogue of the result on $\pi/N$ scattering in $(2+1)$-dimensional Abelian Higgs model on the plane can be deduced only from smoothness property of the kinetic metric. One can suppose that this local version of the result on $\pi/N$ scattering can be generalized to Abelian Higgs models on Riemann surfaces. It is proved in this paper that one can find geodesics of the kinetic metric that describe local $\pi/N$ scattering after the symmetric collision in models on Riemann surfaces, using the fact that the kinetic metric is smooth in symmetric coordinates in the neihbourhood of a point of vortex collision. This smoothness property is established in the case of compact Riemann surfaces. With the help of adiabatic principle one could obtain local $\pi/N$ scattering after the symmetric collision for dynamical models on compact Riemann surfaces. Unfortunately, the adiabatic principle in models on compact Riemann surfaces needs the proof yet, until now it is only a heuristic statement.