Lately, I have been interested in understanding the role of spectral networks in 2-dimensional conformal field theory. Spectral networks were originally introduced by [Gaiotto-Moore-Neitzke] to enumerate the various BPS spectra of 4-dimensional N=2 supersymmetric gauge theories. Via the AGT correspondence, initially proposed by [Alday-Gaiotto-Tachikawa], such theories are dual to 2-dimensional conformal field theories. It has long been suspected that spectral networks must play a crucial role in this correspondence. 

With Lotte Hollands, we work this out in detail for Fenchel-Nielsen-type spectral networks in Liouville field theory at arbitrary central charge, while laying the groundwork for more general spectral networks and higher-rank Toda theories. This has interesting implications for matrix model partition functions, (refined) topological recursion, the 3d-2d correspondence, quantum Teichmüller theory, and more.