Modular bootstrap in Liouville field theory

TL;DR

This work establishes a concrete modular bootstrap framework for Liouville field theory by deriving an explicit modular matrix for torus 1-point blocks from the sphere fusion kernel via FLNO-type relations and analytic continuation. It proves modular invariance of Liouville torus one-point functions with DOZZ structure constants for the continuous Liouville spectrum ($\lambda \in i\mathbb{R}$), tying torus and sphere blocks together through precise relations between $b$-parameters. Key contributions include the explicit expression ${\sf S}_{\lambda_s\lambda_t}^{c,\lambda}=2^{2(\lambda_s^2-\lambda_t^2)+1/2}{\sf F}^{c'}_{\sqrt{2}\lambda_s\,\sqrt{2}\lambda_t}[^{1/(2b')\;\lambda/\sqrt{2}}_{1/(2b')\;1/(2b')}]$ and a contour-regularity analysis for the fusion kernel, supporting crossing symmetry and modular invariance. The results provide a path to further understand the Liouville modular grupoid and extend to related models, while offering a practical route to verify modular consistency of Liouville correlation functions.

Abstract

The modular matrix for the generic 1-point conformal blocks on the torus is expressed in terms of the fusion matrix for the 4-point blocks on the sphere. The modular invariance of the toric 1-point functions in the Liouville field theory with DOZZ structure constants is proved.