Spectral networks and Fenchel-Nielsen coordinates

TL;DR

This work unifies two major coordinate systems on moduli spaces of flat SL(2,C) connections—Fock-Goncharov shear coordinates and Fenchel-Nielsen length-twist coordinates—via spectral networks. By developing abelianization and nonabelianization along FG, FN, contracted FN, and mixed networks, the authors show that flat SL(2) connections can be encoded by rank-1 data on a double cover, yielding complexified spectral coordinates X_γ that act as Darboux coordinates. FG networks recover traditional FG coordinates; FN networks produce complexified Fenchel-Nielsen coordinates, with twist flows realized as GL(1) holonomies under abelianization. The integrated, boundary-aware, and nonabelian constructions provide explicit monodromy representations for key surfaces and reveal how spectral coordinates capture asymptotics and line-defect data, linking to Hitchin moduli, BPS spectra, and AGT-type correspondences. Overall, spectral networks offer a versatile framework that simultaneously encompasses shear, length-twist, and mixed coordinate systems in a unified nonabelianization paradigm.

Abstract

We explain that spectral networks are a unifying framework that incorporates both shear (Fock-Goncharov) and length-twist (Fenchel-Nielsen) coordinate systems on moduli spaces of flat SL(2,C) connections, in the following sense. Given a spectral network W on a punctured Riemann surface C, we explain the process of "abelianization" which relates flat SL(2)-connections (with an additional structure called "W-framing") to flat C*-connections on a covering. For any W, abelianization gives a construction of a local Darboux coordinate system on the moduli space of W-framed flat connections. There are two special types of spectral network, combinatorially dual to ideal triangulations and pants decompositions; these two types of network lead to Fock-Goncharov and Fenchel-Nielsen coordinates respectively.

Figures (38)

• Figure 4: A Fock-Goncharov network $\mathcal{W}(\mathcal{T})$, restricted to a single triangle of $\mathcal{T}$. Walls are drawn as oriented black paths; the orange cross is a branch point; wavy orange lines are branch cuts; dashed green paths are edges of $\mathcal{T}$.
• Figure 5: Two examples of Fock-Goncharov networks on the four-punctured sphere (shown here as the plane, with the point at infinity omitted.) The triangulation on the right includes two degenerate triangles. To simplify the figures, here we chose to present the coverings with the minimal possible number of branch cuts; nevertheless one can check directly that the networks shown here are equivalent to the ones we described in the construction of the Fock-Goncharov network.
• Figure 6: Two examples of Fenchel-Nielsen spectral networks on pairs of pants. The network on the left is "molecule I" and on the right is "molecule II." Each network carries the "British resolution"; to get the "American resolution" one would reverse the orientation of every wall in the network.
• Figure 7: A Fenchel-Nielsen network on the four-holed sphere. This network is built by gluing together two pairs of pants, each containing a copy of molecule II from Figure \ref{['fig:molecules']}.
• Figure 8: Two spectral networks on the three-punctured sphere (with one puncture at infinity). Each wall spirals many times around and asymptotically approaches some puncture; in the top network, two walls approach each puncture, while in the bottom network, four walls approach the puncture at infinity, and one wall approaches each of the other two punctures. To reduce clutter we show only truncated versions of the walls, and to help distinguish the different walls, they are shown in different colors. These networks are Fock-Goncharov networks, but in the limit where the separation between walls goes to zero, they approach the British resolution of Fenchel-Nielsen molecule I (top) and molecule II (bottom).