Liouville Blocks from Spectral Networks

TL;DR

This paper explores the connection between spectral networks and quantum Liouville theory, introducing new formalisms and conjectures to better understand Liouville conformal blocks and their spectral network origins.

Contribution

It constructs a quantum transport formalism for Fenchel-Nielsen spectral networks and proposes a new approach to Liouville conformal blocks via spectral coverings and R-matrices.

Findings

Constructed q-parallel transport for Fenchel-Nielsen networks.

Reproduced Liouville conformal blocks using free-field formalism.

Conjectured a full spectrum generation via extended formalism with R-matrix.

Abstract

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.