Spectral Networks and Snakes

TL;DR

The paper develops the framework of spectral networks for lifted A1 theories of class S, linking higher-rank A_{K-1} systems to the A1 parent via a lift locus on the Coulomb branch. It shows that Darboux coordinates on moduli spaces of flat connections, arising from special spectral networks, coincide with Fock–Goncharov higher Teichmüller coordinates, establishing a concrete bridge between Hitchin moduli, FG coordinates, and spectral-network data. A central contribution is an algorithmic approach to determine BPS spectra through the spectrum generator, with explicit results for K=3,4,5, and a detailed treatment of minimal Yin/Yang networks and their relation to FG triangles, snakes, and amalgamation. The work provides a toolkit for constructing and gluing coordinate systems across triangles, deriving edge coordinates, and understanding wall-crossing in lifted A1 theories, with broader implications for cluster algebras and higher-rank 4d N=2 dynamics. It also outlines open problems and future directions, including systematic minimal-network realizations for all K and deeper connections to quivers and wall-crossing identities.

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.

Figures (31)

• Figure 1: Minimal spectral networks of Yin type, for $K=4$ (left) and general $K$ (right).
• Figure 2: Minimal spectral networks of Yang type, for $K=4$ (left) and general $K$ (right).
• Figure 3: Two essentially minimal WKB spectral networks. Left: the WKB spectral network for the $K=4$ lift of $AD_1$ theory, with SW curve $\lambda^4 - 10 z ({\mathrm{d}} z)^2 \lambda^2 + 4 ({\mathrm{d}} z)^3 \lambda + 9z^2 ({\mathrm{d}} z)^4 = 0$, and $\vartheta = 0.2$. Right: the WKB spectral network for the $K=5$ lift of $AD_1$ theory, with SW curve $\lambda^5 - 20 z ({\mathrm{d}} z)^2 \lambda^3 + ({\mathrm{d}} z)^3 \lambda^2 + 64z^2 ({\mathrm{d}} z)^4 \lambda - 2z ({\mathrm{d}} z)^5 = 0$, and $\vartheta = 0.1$.
• Figure 4: Regions defined by the $z\to \infty$ asymptotics of the Higgs field, in the level $K$ lift of the $AD_1$ Hitchin system. In this figure $\zeta$ has argument $- \frac{\pi}{4}$. If we vary the argument of $\zeta$ by $\alpha$ then the regions rotate by an angle $\frac{2}{3} \alpha$. The $a$, $b$, $c$ cables of a minimal WKB spectral network are also depicted. The position of the cables depends on $\vartheta$; the situation shown here is $\vartheta = \arg \zeta$. If we vary $\vartheta$ by $\alpha$ then the cables rotate by $\frac{2}{3}\alpha$.
• Figure 5: Left: Associated to each branch point in any spectral network there is a distinguished plane in the vector space of local flat sections, containing several distinguished lines. The relation between these lines is as indicated here. Right: A $1$-triangle representing the plane determined by the branch point $\mathfrak{b}$, with the three distinguished lines at its vertices.