# BPS Graphs: From Spectral Networks to BPS Quivers

## TL;DR

BPS graphs are a new geometric tool on punctured Riemann surfaces that connect spectral networks and BPS quivers, capturing elementary BPS states and encoding quiver topology for complex theories.

## Contribution

Introduction of BPS graphs as a bridge between spectral networks and BPS quivers, applicable to higher-rank theories and partial punctures.

## Key findings

- BPS graphs encode BPS quivers for complex theories.
- They provide a geometric realization of Fock-Goncharov triangulations.
- BPS graphs capture elementary BPS states at spectral network degenerations.

## Abstract

We define "BPS graphs" on punctured Riemann surfaces associated with $A_{N-1}$ theories of class $\mathcal{S}$. BPS graphs provide a bridge between two powerful frameworks for studying the spectrum of BPS states: spectral networks and BPS quivers. They arise from degenerate spectral networks at maximal intersections of walls of marginal stability on the Coulomb branch. While the BPS spectrum is ill-defined at such intersections, a BPS graph captures a useful basis of elementary BPS states. The topology of a BPS graph encodes a BPS quiver, even for higher-rank theories and for theories with certain partial punctures. BPS graphs lead to a geometric realization of the combinatorics of Fock-Goncharov $N$-triangulations and generalize them in several ways.