发表机构
School of Mathematical Sciences, Zhejiang University（浙江大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究矩阵值异宿连接在真空轨道间的全局能量极小化，通过构造辅助势能和校准不等式证明两个非阿贝尔扭结为全局极小元，并给出精确壁张力。
AI 中文摘要
我们研究真空集为连续共轭轨道而非孤立点的矩阵值异宿连接的全局能量极小化。对于一维 $\mathrm{SU}(5)\times\mathbb Z_2$ 矩阵场模型的两个特殊参数区域，我们首先将四次势能的完整零集识别为两个紧致齐性真空轨道的不相交并集。然后我们构造由到这些轨道的 Frobenius 距离控制的辅助势能。与原势能的逐点比较，结合校准不等式和一维余面积公式，为连接两个真空分量的每个有限能量连接给出一个尖锐下界；对竞争者不施加对角性、交换性或对称性约化的假设。两个显式的非阿贝尔扭结达到该下界，因此是完整轨道到轨道异宿类中的全局极小元。它们的精确壁张力分别为 $4\sqrt2\\,\mu^3/(3\lambda)$ 和 $9\sqrt2\\,\mu^3/(10h)$。等价地，相应的线段路径实现了真空流形之间的退化加权距离。我们还解决了每个固定端点扇区：对于两个真空轨道上任意给定的代表元，能量下确界是相同的壁张力，并且当且仅当给定的代表元构成两个轨道的最接近对时才达到。因此，非最接近端点扇区通过沿真空流形的越来越慢的切向运动表现出不可达性。证明将模型无关的距离-余面积准则与模型特定的轨道几何和多项式正性估计分开，为具有对称性产生的真空的矩阵值多阱势能提供了一个可复用的机制。
英文摘要
We study global energy minimization for matrix-valued heteroclinic connections whose vacuum sets are continuous conjugacy orbits rather than isolated points. For two distinguished parameter regimes of a one-dimensional $\mathrm{SU}(5)\times\mathbb Z_2$ matrix field model, we first identify the full zero set of the quartic potential as the disjoint union of two compact homogeneous vacuum orbits. We then construct auxiliary potentials controlled by the Frobenius distance to these orbits. Pointwise comparison with the original potential, combined with a calibration inequality and the one-dimensional coarea formula, gives a sharp lower bound for every finite-energy connection joining the two vacuum components; no diagonality, commutativity, or symmetry-reduced ansatz is imposed on the competitors. Two explicit non-Abelian kinks attain the bound and are therefore global minimizers in the full orbit-to-orbit heteroclinic classes. Their exact wall tensions are $4\sqrt2\,μ^3/(3λ)$ and $9\sqrt2\,μ^3/(10h)$, respectively. Equivalently, the corresponding line-segment paths realize the degenerate weighted distance between the vacuum manifolds. We also resolve every fixed-endpoint sector: for arbitrary prescribed representatives on the two vacuum orbits, the energy infimum is the same wall tension, and it is attained if and only if the prescribed representatives form a closest pair of the two orbits. Nonclosest endpoint sectors therefore exhibit non-attainment through increasingly slow tangential motion along the vacuum manifolds. The proof separates a model-independent distance--coarea criterion from the model-specific orbit geometry and polynomial positivity estimates, providing a reusable mechanism for matrix-valued multiwell potentials with symmetry-generated vacua.
Comments28 pages
