Generalized Schur limit, modular differential equations and quantum monodromy traces

TL;DR

The paper investigates the generalized Schur limit of 4d $\mathcal{N}=2$ SCFTs, proposing that $\hat{\mathcal{Z}}(q,\alpha)$ is governed by a fixed-order MLDE with coefficients that depend polynomially on $\alpha$. It demonstrates, through extensive examples including rank-one SU(2) $N_f=4$ and higher-rank USp/SU theories, that negative integer values of $\alpha$ reproduce traces of higher powers of the quantum monodromy $M(q)$, i.e. $\mathrm{Tr}\,M(q)^{-\alpha}$, hinting at a Higgs–Coulomb correspondence. The work develops a practical MLDE-bootstrap method to determine the MLDOs from positive-$\alpha$ data and extends the analysis to higher-rank DC-like theories, where MLDE orders rise (up to 6th order) and many $q$-series coincide with known VOA characters. Overall, the results unify modular-differential constraints, wall-crossing traces, and VOA data across both Higgs and Coulomb branches, with potential implications for identifying RCFT characters from 4d SCFT structures.

Abstract

We explore some aspects of the generalized Schur limit, defined in arXiv:2506.13764. Based on several examples, we conjecture that the generalized Schur limit as a function of $α$ solves a modular linear differential equation of fixed order, with coefficients depending on $α$. We also observe in examples that for Argyres-Douglas theories of type $(A_1,G)$ with $G=A_n,D_n$, the generalized Schur limit for certain negative integer values of $α$, coincides with the trace of higher powers of the quantum monodromy operator. This hints at a more general correspondence between the wall-crossing invariant traces on the Coulomb branch and the generalized Schur limit, which is related to the Higgs branch.

Figures (5)

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