The index from a nonpoint and RG flows

Abstract

We extend the algebro-geometric construction of bifiltered affine schemes that encode both the Higgs branch and the Macdonald index of 4d $\mathcal{N}=2$ superconformal field theories to theories with non-trivial Higgs branches. For the Argyres--Douglas theories $\mathcal{D}_p(SU(N))$, we find that a universal potential, given by a function in the Casimir invariants of $\mathfrak{sl}(N)$, determines the scheme. The scheme is given by a `fattened' nilpotent orbit closure, encoding the OPE decoupling structure of the 4d theory. A procedure we call syzygy maximization fixes the precise potential in a broad range of examples, and the resulting schemes reproduce the 4d data. The underlying graded ring gives conjectural presentations of Zhu's $C_2$-algebra of the associated vertex operator algebra, while the bifiltration refines Arakawa's associated scheme by retaining the R-symmetry information. We find that Higgs branch renormalization group flows are realized by pulling back the same universal potential to the corresponding Slodowy slices, including flows for which the infrared Higgs branch is a point. In particular, our construction provides a systematic method for determining the Macdonald indices of the $\mathcal{D}_p(SU(N), O)$ theories, for which no general expressions were previously known.