Chern–Simons deformation of vortices on compact domains

Abstract

Existence of Maxwell-Chern-Simons-Higgs (MCSH) vortices in a Hermitian line bundle $\L$ over a general compact Riemann surface $\Sigma$ is proved by a continuation method. The solutions are proved to be smooth both spatially and as functions of the Chern-Simons deformation parameter $\kappa$, and exist for all $|\kappa|<\kappa_*$, where $\kappa_*$ depends, in principle, on the geometry of $\Sigma$, the degree $n$ of $\L$, which may be interpreted as the vortex number, and the vortex positions. A simple upper bound on $\kappa_*$, depending only on $n$ and the volume of $\Sigma$, is found. Further, it is proved that a positive {\em lower} bound on $\kappa_*$, depending on $\Sigma$ and $n$, but independent of vortex positions, exists. A detailed numerical study of rotationally equivariant vortices on round two-spheres is performed. We find that $\kappa_*$ in general does depend on vortex positions, and, for fixed $n$ and radius, tends to be larger the more evenly vortices are distributed between the North and South poles. A generalization of the MCSH model to compact K\"ahler domains $\Sigma$ of complex dimension $k\geq 1$ is formulated. The Chern-Simons term is replaced by the integral over spacetime of $A\wedge F\wedge \omega^{k-1}$, where $\omega$ is the K\"ahler form on $\Sigma$. A topological lower bound on energy is found, attained by solutions of a deformed version of the usual vortex equations on $\Sigma$. Existence, uniqueness and smoothness of vortex solutions of these generalized equations is proved, for $|\kappa|<\kappa_*$, and an upper bound on $\kappa_*$ depending only on the K\"ahler class of $\Sigma$ and the first Chern class of $\L$ is obtained.