Algebro-geometric bootstrapping from OPE decoupling

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.

💡 Research Summary

The paper proposes a novel algebro‑geometric framework for bootstrapping four‑dimensional 𝒩=2 superconformal field theories (SCFTs) based on operator‑product‑expansion (OPE) decoupling relations. The authors observe that in many non‑Lagrangian SCFTs certain products of protected operators vanish at a finite order—an OPE “truncation” that does not rely on perturbation theory. They reinterpret such truncations as defining a scheme‑theoretic object: a bifiltered affine scheme X = Spec R, where the coordinate ring R carries two filtrations F_{p,q} reflecting the two independent grading structures of the 𝒩=2 superconformal algebra (the scaling dimension and the U(1)_r charge).

From the bifiltered ring they construct the infinite jet (or arc) space J^∞R. By expanding each generator x_i as a formal power series in an auxiliary variable t, one obtains a trigraded polynomial ring S^∞ whose variables carry a triple degree (α, deg_q, deg_T). The ideal generated by the t‑expanded relations defines J^∞R = S^∞/I^∞. Taking the associated graded with respect to the bifiltration yields a bigraded ring gr(R) and, similarly, a trigraded graded version of the jet space. The Hilbert series of this graded jet ring, denoted HS_{p,q,T}(gr J^∞R), depends on three fugacities (p,q,T). The central claim (Proposition) is that, after setting p = q, this Hilbert series reproduces the Macdonald index of the underlying SCFT: \