Index from a point

TL;DR

This paper introduces an algebro-geometric framework linking the Schur and Macdonald indices of 4D $ abla=2$ SCFTs to the Hilbert series of arc spaces of affine schemes, providing a new perspective on their structure.

Contribution

It proposes a novel geometric interpretation of superconformal indices via arc spaces and affine schemes, connecting local geometric descriptions to operator relations in SCFTs.

Findings

The conjecture holds for various Argyres--Douglas theories.

Distinct local geometric descriptions correspond to different SCFTs without Higgs branches.

The approach encodes operator product relations through nilpotency conditions.

Abstract

We propose an algebro-geometric interpretation of the Schur and Macdonald indices of four-dimensional $\mathcal{N}=2$ superconformal field theories (SCFTs). We conjecture that there exists an affine scheme $X$, which reduces to the Higgs branch as a variety, such that the Hilbert series of the (appropriately-graded) arc space of its polynomial ring $J_\infty(\mathbb{C}[X])$ encodes the indices. Distinct local descriptions of a (singular) point correspond to distinct choices of $X$, giving rise to families of $\mathcal{N}=2$ SCFTs each without a Higgs branch. These local descriptions directly translate into nilpotency relations in the operator product expansions. We test our conjecture across a variety of (generalized) Argyres--Douglas theories.