Liouville Blocks from Spectral Networks

In this paper, we investigate the role of spectral networks in quantum Liouville theory, with particular emphasis on spectral networks of Fenchel-Nielsen-type. In the first part, we construct q-parallel transport for Fenchel-Nielsen networks through q-nonabelianisation, and compare with quantum parallel transport computed using the Moore-Seiberg formalism. This motivates a proposal for a quantum version of the NRS proposal. In the second part, we reproduce Liouville conformal blocks through the standard free-field formalism with Fenchel-Nielsen-type integration contours. However, we observe that this approach is not complete with respect to wall-crossing. We therefore develop an extension of the free-field formalism to smooth spectral coverings, with the Maulik-Okounkov R-matrix playing a central role. We conjecture that this new formalism generates the full spectrum of Liouville conformal blocks, and provides a first-principle definition for Goncharov-Shen conformal blocks.