Index from a point

TL;DR

This work proposes that the Schur and Macdonald indices of 4d $\mathcal{N}=2$ SCFTs admit a unified algebro-geometric interpretation via an affine scheme $X$ whose arc space $J_\infty(\mathbb{C}[X])$ encodes the index. The Schur index is recovered as $I_S(q)=\operatorname{HS}_{p,q}(J_\infty(\mathbb{C}[X]))|_{p\to q}$ and the Macdonald index via the associated graded $\mathrm{gr}(J_\infty(R))$ with $I_{Mac}(q,T)=\operatorname{HS}_{p,q,T}(\mathrm{gr}(J_\infty(R)))|_{p\to q}$, with the reduced scheme $X_{\rm red}$ equal to the Higgs branch. The authors test this framework on a wide array of Argyres--Douglas theories, showing that simple nilpotent relations (e.g., fat points $x^n=0$ or small polynomial relations) at a point can reproduce the full Schur and Macdonald indices, while more elaborate schemes capture theories with nontrivial Higgs branches. Beyond points, they demonstrate Higgsable examples (e.g., $\mathbb{C}^2/\mathbb{Z}_2$ and its $\mathbb{Z}_{n+1}$ generalizations) where multiple distinct affine schemes yield the same Higgs-branch geometry yet correspond to different SCFTs, with indices again reproduced by arc-space calculations. The work also discusses connections to Zhu’s $C_2$-algebra, associated varieties of VOAs, and potential reconstruction of VOAs from $X$, outlining future directions for geometric criteria and broader classes of theories.

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.