Spectral Networks and Snakes

TL;DR

This paper explores spectral networks in lifted A_1 theories, connecting them with higher Teichmuller theory and Fock-Goncharov coordinates, and uses these techniques to analyze BPS spectra.

Contribution

It demonstrates how spectral networks relate to Fock-Goncharov coordinates and applies this to study BPS spectra in lifted A_1 theories.

Findings

Darboux coordinates match Fock-Goncharov coordinates

Spectral networks can determine BPS spectra

Spectrum generators are identified for specific theories

Abstract

We apply and illustrate the techniques of spectral networks in a large collection of A_{K-1} theories of class S, which we call "lifted A_1 theories." Our construction makes contact with Fock and Goncharov's work on higher Teichmuller theory. In particular we show that the Darboux coordinates on moduli spaces of flat connections which come from certain special spectral networks coincide with the Fock-Goncharov coordinates. We show, moreover, how these techniques can be used to study the BPS spectra of lifted A_1 theories. In particular, we determine the spectrum generators for all the lifts of a simple superconformal field theory.