Global Minimality of Matrix-Valued Heteroclinic Connections between Vacuum Orbits

Abstract: 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 $4\sqrt2\,μ<sup>3/(3λ)$ and $9\sqrt2\,μ<sup>3/(10h)$, 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.