Algebro-geometric bootstrapping from OPE decoupling

TL;DR

This work posits that OPE decouplings in 4d N=2 SCFTs can be encoded by a bifiltered affine scheme X=Spec R, with the Macdonald index realized as the arc-space Hilbert series HS_{q,q,T}(gr(J_infty R)) and the Higgs branch given by the reduced scheme. An extremality principle on jet-scheme topology selects unique geometric data, fixing moduli and yielding candidate descriptions for Argyres-Douglas theories via complete-intersection Jacobian ideals I_P whose associated arc-space Hilbert series match known Macdonald and Schur indices. The authors provide explicit constructions for (A_{k-1},A_{N-1}) theories, including detailed results for (A_1,A_{N-1}), (A_2,A_{N-1}), and (A_3,A_{N-1}) cases, with VOAs and null relations offering a physical interpretation of the geometry. If valid universally, this framework offers a geometric route to classify certain 4d N=2 SCFTs purely from algebro-geometric data, connecting jet-scheme topology, Betti-number constraints, and VOA null vectors. The approach suggests deep links between OPE decouplings, Higgs branch geometry, and the full operator algebra structure, with open questions about universality and extension to more general Higgs branches.

Abstract

We conjecture that decoupling relations in the operator product expansion of a 4d $\mathcal{N}=2$ superconformal field theory (SCFT) are encoded by an algebro-geometric object: a bifiltered affine scheme. We demonstrate how this scheme reproduces the Macdonald index (thus the Schur index) as well as the Higgs branch. Although the associated scheme typically admits continuous deformations, we find that a geometric extremization principle uniquely fixes these moduli, thereby providing a possible geometric route toward a classification of 4d $\mathcal{N}=2$ SCFTs.