One ring to rule them all: syzygy maximization for Argyres-Douglas
Abstract
We propose a uniform construction of affine schemes encoding the Higgs branches and Macdonald indices of the Argyres--Douglas theories 𝒟p(ADE), with gcd(p,h∨)=1. The construction captures conjectural OPE decoupling relations via the Jacobian ring of a single invariant polynomial potential, obtained from a polynomial twisted period of the corresponding Coxeter orbit space. We show that these potentials are ``syzygy-maximizing'' in the first jet schemes and recover the Higgs branches as their reduced critical loci. We obtain closed determinant expressions for the classical types and explicit potentials for E6, E7, and E8. Powers of the nilradical define an intrinsic filtration on the resulting Jacobian rings, from which we compute candidate Macdonald indices with the correct Schur and Hall--Littlewood limits, in the examples studied. Restricting the potentials to Slodowy slices extends the prescription to nilpotently Higgsed theories. We further conjecture that these Jacobian rings are isomorphic to Zhu's C2-algebras of simple affine VOAs at boundary admissible levels and their quantum Drinfeld--Sokolov reductions.
Related
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BibTeX
@misc{kang2026ring,
title = {{One ring to rule them all: syzygy maximization for Argyres-Douglas}},
author = {Monica Jinwoo Kang and Craig Lawrie and Jaewon Song},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33151},
eprint = {2610.33151},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33151},
}
RIS
TY - UNPB
TI - One ring to rule them all: syzygy maximization for Argyres-Douglas
AU - Monica Jinwoo Kang
AU - Craig Lawrie
AU - Jaewon Song
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.33151
UR - https://arxiv.org/abs/2610.33151
AB - We propose a uniform construction of affine schemes encoding the Higgs branches and Macdonald indices of the Argyres--Douglas theories 𝒟_p(ADE), with gcd(p,h^∨)=1. The construction captures conjectural OPE decoupling relations via the Jacobian ring of a single invariant polynomial potential, obtained from a polynomial twisted period of the corresponding Coxeter orbit space. We show that these potentials are ``syzygy-maximizing'' in the first jet schemes and recover the Higgs branches as their reduced critical loci. We obtain closed determinant expressions for the classical types and explicit potentials for E_6, E_7, and E_8. Powers of the nilradical define an intrinsic filtration on the resulting Jacobian rings, from which we compute candidate Macdonald indices with the correct Schur and Hall--Littlewood limits, in the examples studied. Restricting the potentials to Slodowy slices extends the prescription to nilpotently Higgsed theories. We further conjecture that these Jacobian rings are isomorphic to Zhu's C_2-algebras of simple affine VOAs at boundary admissible levels and their quantum Drinfeld--Sokolov reductions.
ER -