Classical geometry from the quantum Liouville theory

TL;DR

This paper demonstrates a numerical and analytical connection between classical conformal blocks in Liouville theory and hyperbolic geometry, providing new methods for calculating geometric and conformal data on punctured spheres.

Contribution

It establishes a direct relation between saddle point momenta in quantum Liouville theory and hyperbolic geodesic lengths, enabling new computational techniques.

Findings

Numerical evidence linking saddle point momenta to geodesic lengths

Verification of classical Liouville action factorization in different channels

Development of efficient methods for calculating accessory parameters

Abstract

Zamolodchikov's recursion relations are used to analyze the existence and approximations to the classical conformal block in the case of four parabolic weights. Strong numerical evidence is found that the saddle point momenta arising in the classical limit of the DOZZ quantum Liouville theory are simply related to the geodesic length functions of the hyperbolic geometry on the 4-punctured Riemann sphere. Such relation provides new powerful methods for both numerical and analytical calculations of these functions. The consistency conditions for the factorization of the 4-point classical Liouville action in different channels are numerically verified. The factorization yields efficient numerical methods to calculate the 4-point classical action and, by the Polyakov conjecture, the accessory parameters of the Fuchsian uniformization of the 4-punctured sphere.