An Affine Semigroup from Orbifold Boundary Conditions: cut, phylogenetic and hierarchical models in the unit-weight sector, and weighted configurations beyond them

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 $T^2\mathrm{/}{\mathbb{Z}}_2$, the wheel $W_4$ for $S^1\mathrm{/}{\mathbb{Z}}_2\times S^1\mathrm{/}{\mathbb{Z}}_2$ — 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.