We consider the vortex equations for a U(n) gauge field A coupled to a Higgs field phi with values on the n x n matrices. It is known that when these equations are defined on a compact Riemann surface Sigma, their moduli space of solutions is closely related to a moduli space of tau-stable holomorphic n-pairs on that surface. Using this fact and a local factorization result for the matrix phi, we show that the vortex solutions are entirely characterized by the location in Sigma of the zeros of det phi and by the choice of a vortex internal structure at each of these zeros. We describe explicitly the vortex internal spaces and show that they are compact and connected spaces.
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机译:我们考虑耦合到具有希格斯场phi的U(n)规范场A的涡旋方程,其值在n x n矩阵上。众所周知,当这些方程在紧致的Riemann表面Sigma上定义时,其解的模空间与该表面上tau稳定全纯n对的模空间紧密相关。利用这一事实和矩阵phi的局部分解结果,我们表明,涡旋解的特征完全是dit phi的零在Sigma中的位置以及在每个零处选择的涡旋内部结构。我们明确描述了涡旋内部空间,并表明它们是紧凑且连通的空间。
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