Index from a point

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.

💡 Research Summary

The paper proposes a novel algebro‑geometric framework for computing the Schur and Macdonald indices of four‑dimensional 𝒩=2 superconformal field theories (SCFTs). The authors conjecture that for any such theory T there exists an affine scheme X whose reduced variety coincides with the Higgs branch M_H(T). Unlike the Higgs branch, X may carry nilpotent structure, reflecting OPE “vanishing” relations that are invisible in the reduced variety.

Given X, one considers its coordinate ring ℂ