ADE Spectral Networks

TL;DR

This paper generalizes spectral networks for certain 4d $ ext{N}=2$ theories, providing a Lie algebraic framework to compute BPS spectra and study wall-crossing phenomena, including new surface defects.

Contribution

It introduces a new perspective on spectral networks for class $ ext{S}$ theories, linking them to Lie algebraic structures and surface defects, enhancing the analysis of BPS spectra.

Findings

Provides a Lie algebraic interpretation of BPS spectra.

Develops an efficient framework for studying 4d BPS spectra.

Introduces novel surface defects related to minuscule representations.

Abstract

We introduce a new perspective and a generalization of spectral networks for 4d $\mathcal{N}=2$ theories of class $\mathcal{S}$ associated to Lie algebras $\mathfrak{g} = \textrm{A}_n$, $\textrm{D}_n$, $\textrm{E}_{6}$, and $\textrm{E}_{7}$. Spectral networks directly compute the BPS spectra of 2d theories on surface defects coupled to the 4d theories. A Lie algebraic interpretation of these spectra emerges naturally from our construction, leading to a new description of 2d-4d wall-crossing phenomena. Our construction also provides an efficient framework for the study of BPS spectra of the 4d theories. In addition, we consider novel types of surface defects associated with minuscule representations of $\mathfrak{g}$.