乘积交换场论中的拓扑分层能量极小元
Topologically Stratified Energy Minimizers in a Product Abelian Field Theory
摘要：
本文研究了Tong和Wong最近发展的含磁性杂质的乘积阿贝尔规范场论。我们首先得到了一个必要和充分条件，存在一个唯一的解决方案，实现这样的杂质的形式，多个漩涡。接下来，我们重新制定的理论到一个扩展的模型，允许涡和反涡共存。模型中的两个阿贝尔规范场诱导了两种磁涡旋线，它们分别由$N_s$涡旋和$P_s$反涡旋（$s=1,2$）实现为两个复值Higgs场的零点和极点.建立了紧致Riemann曲面$S$上的控制方程的一个存在定理，该定理指出具有指定的$N_1, N_2$涡和两个指定物种的$P_1,P_2$反涡的解存在当且仅当不等式\[ \left|N_1+N_2-(P_1+P_2)\right|<\frac{|S|}π,\quad \left|N_1+2N_2-(P_1+2P_2)\right|<\frac{|S|}π, \]同时成立，该不等式给出了涡数和反涡数之差以$S$的总表面积表示的界.这些解决方案的最小能量示出承担明确的值\[ E= 4π(N_1+N_2+P_1+P_2), \]给出的几个拓扑不变量，测量的总张力的涡线。
英文摘要：
We study a recently developed product Abelian gauge field theory by Tong and Wong hosting magnetic impurities. We first obtain a necessary and sufficient condition for the existence of a unique solution realizing such impurities in the form of multiple vortices. We next reformulate the theory into an extended model that allows the coexistence of vortices and anti-vortices. The two Abelian gauge fields in the model induce two species of magnetic vortex-lines resulting from $N_s$ vortices and $P_s$ anti-vortices ($s=1,2$) realized as the zeros and poles of two complex-valued Higgs fields, respectively. An existence theorem is established for the governing equations over a compact Riemann surface $S$ which states that a solution with prescribed $N_1, N_2$ vortices and $P_1,P_2$ anti-vortices of two designated species exists if and only if the inequalities \[ \left|N_1+N_2-(P_1+P_2)\right|<\frac{|S|}π,\quad \left|N_1+2N_2-(P_1+2P_2)\right|<\frac{|S|}π, \] hold simultaneously, which give bounds for the `differences' of the vortex and anti-vortex numbers in terms of the total surface area of $S$. The minimum energy of these solutions is shown to assume the explicit value \[ E= 4π(N_1+N_2+P_1+P_2), \] given in terms of several topological invariants, measuring the total tension of the vortex-lines.
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