AI 中文总结
本文确定二维orbifold边界条件对应的仿射半群，复现相关环不变量，提出加权字母表胶合树等贡献，分析不同阶群对应的系统发育等模型性质。
AI 中文摘要
二维orbifold上规范理论的边界条件等价类是由仿射半群索引的边际映射的纤维：每个字母标签对应一个生成元，按权重分级，通过其在不动点处的局部数据嵌入。本文确定了该半群。无权重时，该构型有特定名称和相关文献，本文引用了这些文献中对应我们研究案例的结果：在$\boldsymbol{\text{Z}_2}$上，它是Sturmfels-Sullivant意义下显式图的割构型——$T^2/\text{Z}_2$对应四阶环图，$S^1/\text{Z}_2\times S^1/\text{Z}_2$对应轮图$W_4$——已验证为构型相等；在具有相等锥阶的$\boldsymbol{\text{Z}_m}$上，它是爪树的群基系统发育模型；在锥阶不等时，它是一种混合阶变体，本文未在文献中发现；对于更高维乘积，它是交叉多面体边界复形的二元分层模型。乘积orbifold的环是2008年一张表格中的一行——包含余维数、次数、极小生成元、正规性等属性，本文的方法在未预先知晓该表格的情况下复现了所有这些不变量。三个覆盖都未涉及的是带权重的字母表，这些权重源于向非阿贝尔空间群的更高维不可约表示的诱导，以及在实或四元数基上的共轭对重组。该扇区与Sturmfels和Sullivant在2005年提出的非阿贝尔方向相邻但不等同，也是本文贡献所在：加权字母表的胶合树、正交与辛列，以及三脚架上的群基模型，当有限阿贝尔群的阶至多为3时，该模型是完全交。首个超出$\boldsymbol{\text{Z}_3}$的群将局部与全局分离：$\boldsymbol{\text{Z}_4}$三脚架在系统发育文献研究的Zariski开集上是完全交，但在全局上不是。
英文摘要
The equivalence classes of boundary conditions of a gauge theory on a two-dimensional orbifold are the fibres of a marginal map, indexed by an affine semigroup: one generator per alphabet label, graded by weight, embedded by its local data at the fixed points. This note identifies that semigroup. Without weights the configuration has a name and a literature, whose results about our cases are attributed here: over $\mathbb{Z}_2$ it is the cut configuration of an explicit graph in the sense of Sturmfels-Sullivant --- the four-cycle for $T^2/\mathbb{Z}_2$, the wheel $W_4$ for $S^1/\mathbb{Z}_2\times S^1/\mathbb{Z}_2$ --- verified as an equality of configurations; over $\mathbb{Z}_m$ with equal cone orders, the group-based phylogenetic model on a claw tree; with unequal orders, a mixed-order variant we do not find in the literature; for higher products, the binary hierarchical model of a cross-polytope boundary complex. The product orbifold's ring is a row of a 2008 table --- codimension, degree, minimal generators, normality --- every invariant of which our machinery reproduced without knowing of it. What none of the three covers is the alphabet with weights, which arise from induction to higher-dimensional irreducibles of a non-abelian space group and from conjugate-pair recombination over real or quaternionic ground. That sector is adjacent to, but not identified with, the non-abelian direction Sturmfels and Sullivant raised in 2005, and is where our contributions sit: gluing trees for the weighted alphabets and the orthogonal and symplectic columns, and the group-based model on the tripod, a complete intersection exactly when the finite abelian group has order at most three. The first group beyond $\mathbb{Z}_3$ separates local from global: the $\mathbb{Z}_4$ tripod is a complete intersection on the Zariski-open set the phylogenetics literature works in, and not globally.
Comments16 pages, 3 figures. Companion to Part IX-A, "The Alphabet of Orbifold Boundary Conditions", which determines the alphabet whose relations are computed here: https://doi.org/10.5281/zenodo.22254861 . This note's record, with every ancillary script and its archived receipt: https://doi.org/10.5281/zenodo.22254863
