Generalized Schur limit, modular differential equations and quantum monodromy traces

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 $\hat{I}\pm$ solves a modular linear differential equation of fixed order, with coefficients depending on $\hat{I}\pm$. 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 $\hat{I}\pm$, 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.

14 pages + appendix, 9 tables, 1 figure. v2: references added, minor typos in text, Eq. (2.8), Eq. (2.10) and Table 1 corrected