Geometric models of soliton vortex dynamics

TL;DR

This paper investigates the geometry and dynamics of soliton vortices in the gauged O(3) Sigma model, analyzing moduli space properties, vortex interactions, and effects of additional fields and terms on solutions.

Contribution

It introduces new geometric and analytical insights into vortex moduli spaces, including bounds, solution existence, and deformation effects in the gauged O(3) Sigma model.

Findings

Moduli space is incomplete on both Euclidean plane and compact surfaces.

Numerical computation shows vortex-antivortex scattering differs from Ginzburg-Landau vortices.

Bounds on Chern-Simons constant for solution existence are established.

Abstract

We focus on BPS solutions of the gauged O(3) Sigma model, due to Schroers, and use these ideas to study the geometry of the moduli space. The model has an asymmetry parameter $\tau$ breaking the symmetry of vortices and antivortices on the field equations. It is shown that the moduli space is incomplete both on the Euclidean plane and on a compact surface. On the Euclidean plane, the L2 metric on the moduli space is approximated for well separated cores and results consistent with similar approximations for the Ginzburg-Landau functional are found. The scattering angle of approaching vortex-antivortex pairs of different effective mass is computed numerically and is shown to be different from the well known scattering of approaching Ginzburg-Landau vortices. The volume of the moduli space for general $\tau$ is computed for the case of the round sphere and flat tori. The model on a compact surface is deformed introducing a neutral field and a Chern-Simons term. A lower bound for the Chern-Simons constant $\kappa$ such that the extended model admits a solution is shown to exist, and if the total number of vortices and antivortices are different, the existence of an upper bound is also shown. Existence of multiple solutions to the governing elliptic problem is established on a compact surface as well as the existence of two limiting behaviours as $\kappa \to 0$. A localization formula for the deformation is found for both Ginzburg-Landau and the O(3) Sigma model vortices and it is shown that it can be extended to the coalescense set. This rules out the possibility that this is Kim-Lee's term in the case of Ginzburg-Landau vortices, moreover, the deformation term is compared on the plane with the Ricci form of the surface and it is shown they are different, hence also discarding that this is the term proposed by Collie-Tong to model vortex dynamics with Chern-Simons interaction.