Index from a point

Monica Jinwoo Kang; Craig Lawrie; Jaewon Song

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arXiv · 2507.12510

Index from a point

Monica Jinwoo Kang·Craig Lawrie·Jaewon Song

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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.

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Monica Jinwoo Kang, Craig Lawrie, Jaewon Song. 2025-07-16. Index from a point. https://arxiv.org/abs/2507.12510

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