Spectral networks

TL;DR

Spectral networks provide a geometric framework to compute 4d BPS spectra and 2d–4d BPS data in class S theories via wall-like networks on a punctured curve C. They yield a nonabelianization map that translates flat GL(1) data on the Seiberg–Witten cover Σ into flat GL(K) data on C, endowing moduli spaces of flat connections with cluster-coordinate-like charts. By analyzing wall-crossing (K-walls) and S-wall dynamics, the authors derive systematic rules to extract 4d BPS degeneracies Ω(γ) and 2d–4d enhancements ω(γ,ā), and demonstrate the consistency of these structures with known wall-crossing formulas. The formalism unifies BPS counting with coordinate systems on flat-connection moduli spaces, offering a bridge to higher Teichmüller theory through canonical nonabelianization and suggesting a cluster-algebraic structure underlying BPS data. The work also extends the spectral-network framework beyond physically realized networks to a general mathematical setting, enabling generalized path lifting and moduli-space coordinates with applications to Hitchin systems and WKB analysis.

Abstract

We introduce new geometric objects called spectral networks. Spectral networks are networks of trajectories on Riemann surfaces obeying certain local rules. Spectral networks arise naturally in four-dimensional N=2 theories coupled to surface defects, particularly the theories of class S. In these theories spectral networks provide a useful tool for the computation of BPS degeneracies: the network directly determines the degeneracies of solitons living on the surface defect, which in turn determine the degeneracies for particles living in the 4d bulk. Spectral networks also lead to a new map between flat GL(K,C) connections on a two-dimensional surface C and flat abelian connections on an appropriate branched cover Sigma of C. This construction produces natural coordinate systems on moduli spaces of flat GL(K,C) connections on C, which we conjecture are cluster coordinate systems.

Figures (42)

• Figure 1: A spectral network, drawn on the stereographic projection of $C = S^2$, with a single puncture at infinity. All of the walls eventually asymptote to this puncture. The walls are labeled by pairs $ij$, where $i$, $j$ are sheets of a 3-fold covering $\Sigma \to C$. The branch points of the covering are shown as orange crosses. We have trivialized the covering over the complement of some branch cuts, shown as wavy orange lines.
• Figure 2: Some possible topologies for finite webs of BPS strings. An orange cross with label $(ij)$ denotes an $(ij)$-branch point. Wherever a string with label $ij$ appears, it could equally well have been represented by a string with label $ji$ and the opposite orientation.
• Figure 3: Some possible topologies for finite open webs of BPS strings, representing BPS solitons on the surface defect $\mathbb S_z$. Each finite open web includes one string that ends on the point $z \in C$. An orange cross with label $(ij)$ denotes an $(ij)$-branch point.
• Figure 4: Two surface defects connected by an interface. (This picture lives in the three-dimensional space where the field theory $S[\mathfrak{g},C,D]$ is defined; we have factored out the time direction.)
• Figure 5: Two tangent directions at $z$ determined by a finite web $N$: ${\tilde{z}}_1$ points "into" the finite web while ${\tilde{z}}_2$ points "away."