Existence of multiple vortices in supersymmetric gauge field theory

Two sharp existence and uniqueness theorems are presented for solutions of multiple vortices arising in a six-dimensional brane-world supersymmetric gauge field theory under the general gauge symmetry group G = U (1)× SU ( N ) and with N Higgs scalar fields in the fundamental representation of G . Specifically, when the space of extra dimension is compact so that vortices are hosted in a 2-torus of volume | Ω |, the existence of a unique multiple vortex solution representing n 1 ,…, n N , respectively, prescribed vortices arising in the N species of the Higgs fields is established under the explicitly stated necessary and sufficient condition where e and g are the U (1) electromagnetic and SU ( N ) chromatic coupling constants, v measures the energy scale of broken symmetry and is the total vortex number; when the space of extra dimension is the full plane, the existence and uniqueness of an arbitrarily prescribed n -vortex solution of finite energy is always ensured. These vortices are governed by a system of nonlinear elliptic equations, which may be reformulated to allow a variational structure. Proofs of existence are then developed using the methods of calculus of variations.