Generalized Schur limit, modular differential equations and quantum monodromy traces

TL;DR

This paper investigates the generalized Schur limit in certain quantum field theories, proposing it satisfies modular differential equations and relates to quantum monodromy traces, revealing new links between wall-crossing invariants and Higgs branch structures.

Contribution

It conjectures that the generalized Schur limit solves modular linear differential equations and connects it to quantum monodromy traces in Argyres-Douglas theories, suggesting a broader correspondence.

Findings

Generalized Schur limit likely satisfies modular differential equations.

For specific theories, the limit matches traces of quantum monodromy powers.

Indicates a link between wall-crossing invariants and Higgs branch geometry.

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 $\alpha$ solves a modular linear differential equation of fixed order, with coefficients depending on $\alpha$. 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 $\alpha$, 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.