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 $\mathcal{D}_p(ADE)$, with $\gcd(p,h^\vee)=1$. 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.