Spectral Networks and Snakes

We apply and illustrate the techniques of spectral networks in a large collection of AK-1 theories of class S, which we call “lifted A1 theories.” Our construction makes contact with Fock and Goncharov’s work on higher Teichmüller 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 A1 theories. In particular, we determine the spectrum generators for all the lifts of a simple superconformal field theory.