Algebro-geometric bootstrapping from OPE decoupling

TL;DR

This paper proposes that decoupling relations in 4d $\ ext{N}=2$ superconformal field theories can be represented by an algebro-geometric object, offering a geometric approach to understanding and classifying these theories.

Contribution

It introduces a novel algebro-geometric framework encoding OPE decoupling relations in 4d $\ ext{N}=2$ SCFTs, linking them to bifiltered affine schemes and a geometric extremization principle.

Findings

Reproduces the Macdonald and Schur indices from the scheme

Models the Higgs branch within the algebro-geometric framework

Provides a geometric extremization principle to fix moduli

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.