Chern-Simons deformation of vortices on compact domains
Existence of Maxwell-Chern-Simons-Higgs (MCSH) vortices in a Hermitian line bundle $Ł$ over a general compact Riemann surface $Σ$ is proved by a continuation method. The solutions are proved to be smooth both spatially and as functions of the Chern-Simons deformation parameter $κ$, and exist for all $|κ|<κ_*$, where $κ_*$ depends, in principle, on the geometry of $Σ$, the degree $n$ of $Ł$, which may be interpreted as the vortex number, and the vortex positions. A simple upper bound on $κ_*$, depending only on $n$ and the volume of $Σ$, is found. Further, it is proved that a positive {\em lower} bound on $κ_*$, depending on $Σ$ 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 $κ_*$ 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 Kähler domains $Σ$ of complex dimension $k\geq 1$ is formulated. The Chern-Simons term is replaced by the integral over spacetime of $A\wedge F\wedge ω^{k-1}$, where $ω$ is the Kähler form on $Σ$. A topological lower bound on energy is found, attained by solutions of a deformed version of the usual vortex equations on $Σ$. Existence, uniqueness and smoothness of vortex solutions of these generalized equations is proved, for $|κ|<κ_*$, and an upper bound on $κ_*$ depending only on the Kähler class of $Σ$ and the first Chern class of $Ł$ is obtained.