# An infrared bootstrap of the Schur index with surface defects

## TL;DR

This paper develops a bootstrap method for the Schur index of 4d $ ext{N}=2$ theories with surface defects, using infrared properties and $q$-difference operators to connect BPS monodromies and UV data.

## Contribution

It introduces a novel bootstrap approach for BPS monodromies with surface defects, linking infrared invariants to ultraviolet Schur index data.

## Key findings

- Trace of defect BPS monodromies matches $q$-difference operator actions.
- Bootstrap reproduces the ultraviolet characterization of the Schur index.
- Method applies to class $ ext{S}$ theories with arbitrary vortex surface defects.

## Abstract

The infrared formula relates the Schur index of a 4d $\mathcal{N}=2$ theory to its wall-crossing invariant, a.k.a. BPS monodromy. A further extension of this formula, proposed by C\'ordova, Gaiotto and Shao, includes contributions by various types of line and surface defects. We study BPS monodromies in the presence of vortex surface defects of arbitrary vorticity for general class $\mathcal{S}$ theories of type $A_1$ engineered by UV curves with at least one regular puncture. The trace of these defect BPS monodromies is shown to coincide with the action of certain $q$-difference operators acting on the trace of the (pure) 4d BPS monodromy. We use these operators to develop a "bootstrap" (of traces) of BPS monodromies, relying only on their infrared properties, thereby reproducing the very general ultraviolet characterization of the Schur index.