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 $G=U(1)\times SU(N)$ 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 $\mathrm{\mid }\text{Om}\mathrm{\mid }$, the existence of a unique multiple vortex solution representing $n_1,\mathrm{.}\mathrm{.}\mathrm{.},n_N$ respectively prescribed vortices arising in the $N$ species of the Higgs fields is established under the explicitly stated necessary and sufficient condition [ n_i<\frac{g^2v^2}{8\pi N}|\Om|+\frac{1}{N}(1-\frac{1}{N}[\frac{g}{e}]^2)n,\quad i=1,…,N,] where $e$ and $g$ are the U(1) electromagnetic and SU(N) chromatic coupling constants, $v$ measures the energy scale of broken symmetry, and $n={\sum }_{i=1}^N{n}_i$ 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.