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 andthe volume of Σ, is found. Further, it is proved that a positive 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 andradius, 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≥1 is formulated. The Chern–Simons term is replaced by the integral over spacetime of A∧F∧ωk⁻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 andthe first Chern class of Ł is obtained.