超对称规范域论中多重涡旋的存在性
Existence of Multiple Vortices in Supersymmetric Gauge Field Theory
摘要：
在一般规范对称群$G=U(1)\times SU(N)$下和在$G=U(1)\times SU(N)$的基本表象中含有$N$ Higgs标量场的六维膜世界超对称规范域论中，给出了多重涡旋解的两个尖锐的存在唯一性定理.具体地说，当额外维空间是紧凑的，使得涡旋被容纳在体积为$|\Om|$的2-环面中时，在显式陈述的充分必要条件\[ n_i< \frac{g^2v^2}{8πN}]下，建立了Higgs场的$N$类中产生的分别代表$n_1,...,n_N$和指定涡旋的唯一多涡旋解的存在性|\Om| + \frac{1}{N}（1- \frac{1}{N} [ \frac{g}{e} ]^2）n，\quad i=1，.，其中$e$和$g$分别是U（1）电磁耦合常数和SU（N）色耦合常数，$v$度量对称性破缺的能量尺度，$n=\sum_{i=1}^N n_i$是总涡旋数;当额外维空间是全平面时，总是保证了一个任意给定的有限能量$n$涡旋解的存在性和唯一性。这些旋涡是由一个系统的非线性椭圆方程，这可能是重新制定，以允许变分结构。存在的证明，然后开发使用变分法的方法。
英文摘要：
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 $|\Om|$, 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 \[ n_i<\frac{g^2v^2}{8πN}|\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.
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