低秩中微子物质流的幺正不变量代数：Krylov-Plücker不变量、$3+1$环面环与受控形变
Invariant Algebras of Low-Rank Neutrino Matter Flows: Krylov-Plücker Invariants, the $3+1$ Toric Ring, and Controlled Deformations
- National University of Singapore（新加坡国立大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出统一代数框架描述中微子传播中物质形变不变量，给出维度公式及$3+1$系统环面环结构，区分精确不变量与谱分解性质。
AI中文摘要：
物质会改变中微子传播哈密顿量的本征值和混合矩阵，同时通过低维加性形变进入味基哈密顿量。我们发展了对此形变不敏感量的统一代数描述。不变量代数被表述为味重相位不变环内交换的物质平移导数的联合核。对于对角物质势，这给出了不受影响的对称方向与非对角循环代数的一般分解，以及任意味数和独立物质参数下的精确维度公式。对于具有共同低秩支撑的物质spurion，块Krylov行生成外幂协变量，其最大子式与物质无关。它们的质量基形式是Plücker展开：秩一产生熟悉的Vandermonde分解，而更高秩通常产生独立Plücker项之和。标准$3+1$系统提供了我们主要的非平凡例子。其全局非对角代数是完整四顶点味图的环面循环环，由边模量、三角形循环和四边形循环生成；常用的十一个不变量构成一个一般的局部坐标系，而非全局多项式生成集。我们进一步将独立变化的物质组成和非对角非标准相互作用分别表述为不变量代数的交换精确流和受控形变。该框架将精确不变量的存在性与更强的、特殊的单单项式谱分解性质区分开来。
英文摘要：
Matter changes the eigenvalues and mixing matrix of the neutrino propagation Hamiltonian while entering the flavor-basis Hamiltonian through a low-dimensional additive deformation. We develop a unified algebraic description of the quantities that are insensitive to this deformation. The invariant algebra is formulated as the joint kernel of commuting matter-translation derivations inside the flavor-rephasing invariant ring. For diagonal matter potentials, this gives a general decomposition into unaffected diagonal directions and an off-diagonal cycle algebra, together with an exact dimension formula for an arbitrary number of flavors and independent matter parameters. For matter spurions with common low-rank support, block-Krylov rows generate exterior-power covariants whose maximal minors are matter independent. Their mass-basis form is a Plücker expansion: rank one produces the familiar Vandermonde factorization, whereas higher rank generically produces a sum of independent Plücker terms. The standard $3+1$ system provides our principal nontrivial example. Its global off-diagonal algebra is the toric cycle ring of the complete four-vertex flavor graph, generated by edge moduli, triangle cycles, and quadrangle cycles; the commonly used eleven invariants constitute a generic local coordinate system rather than a global polynomial generating set. We further formulate independently varying matter compositions and off-diagonal nonstandard interactions as, respectively, commuting exact flows and controlled deformations of the invariant algebra. This framework separates the existence of exact invariants from the stronger and exceptional property of single-monomial spectral factorization.
