An Affine Semigroup from Orbifold Boundary Conditions: cut, phylogenetic and hierarchical models in the unit-weight sector, and weighted configurations beyond them
Abstract
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 ℤ2 it is the cut configuration of an explicit graph in the sense of Sturmfels-Sullivant --- the four-cycle for T2/ℤ2, the wheel W4 for S1/ℤ2× S1/ℤ2 --- verified as an equality of configurations; over ℤm 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 ℤ3 separates local from global: the ℤ4 tripod is a complete intersection on the Zariski-open set the phylogenetics literature works in, and not globally.
Related
BibTeX
@misc{marin2026affine,
title = {{An Affine Semigroup from Orbifold Boundary Conditions: cut, phylogenetic and hierarchical models in the unit-weight sector, and weighted configurations beyond them}},
author = {Carles Marín},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.02630},
eprint = {2609.02630},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02630},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - An Affine Semigroup from Orbifold Boundary Conditions: cut, phylogenetic and hierarchical models in the unit-weight sector, and weighted configurations beyond them
AU - Carles Marín
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.02630
UR - https://arxiv.org/abs/2609.02630
AB - 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 ℤ_2 it is the cut configuration of an explicit graph in the sense of Sturmfels-Sullivant --- the four-cycle for T^2/ℤ_2, the wheel W_4 for S^1/ℤ_2× S^1/ℤ_2 --- verified as an equality of configurations; over 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 ℤ_3 separates local from global: the ℤ_4 tripod is a complete intersection on the Zariski-open set the phylogenetics literature works in, and not globally.
ER -