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 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 L , 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 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 ≥ 1 is formulated. The Chern–Simons term is replaced by the integral over spacetime of A ∧ F ∧ 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 L is obtained. [ABSTRACT FROM AUTHOR]