Modular bootstrap in Liouville field theory

TL;DR

This paper proves the modular invariance of toric 1-point functions in Liouville field theory and expresses the modular matrix in terms of fusion matrices, advancing understanding of conformal blocks and modular properties.

Contribution

It establishes the modular invariance of Liouville 1-point functions and relates the toric modular matrix to sphere fusion matrices, providing new insights into conformal blocks.

Findings

Modular invariance of Liouville 1-point functions proven

Modular matrix expressed via sphere fusion matrix

Enhanced understanding of conformal block relations

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.