BPS spectra from BPS graphs

TL;DR

This paper introduces a graphical method using BPS graphs to determine the BPS spectra of $A_1$ theories of class S, connecting spectral networks and BPS quivers for simplified spectrum computation.

Contribution

It presents a novel graphical approach to extract the quantum spectrum generator from BPS graphs, linking different frameworks for BPS state analysis.

Findings

BPS spectra can be read directly from BPS graphs.

The method simplifies spectrum calculations via quantum parallel transport.

Provides a unified framework connecting spectral networks and BPS quivers.

Abstract

I present a simple graphical method to find the BPS spectra of $A_1$ theories of class S. BPS graphs provide a bridge between spectral networks and BPS quivers, the two main frameworks for the study of BPS states. Here I show how to essentially read off from a BPS graph the quantum spectrum generator (or BPS monodromy), expressed as a product of quantum dilogarithms. Thanks to the framed wall-crossing phenomenon for line defects, the determination of the BPS spectrum reduces to the computation of quantum parallel transport across the edges of the BPS graph.