Classical geometry from the quantum Liouville theory
Leszek Hadasz, Zbigniew Jaskolski, Marcin Piatek
TL;DR
This work numerically confirms the semiclassical limit of quantum Liouville theory for a 4-punctured sphere with four parabolic weights, showing that saddle-point momenta correspond to hyperbolic geodesic lengths and that the classical Liouville action factors consistently across channels. By employing Zamolodchikov’s recursion and its $q$-expansion, the authors compute the classical conformal block $f_{\delta}$ to high order and extract saddle-point data from a saddle-point equation, aligning with known geodesic lengths. They validate the classical bootstrap and demonstrate that the 4-point Liouville action and the accessory parameters follow from the saddle-point data, the classical block, and the 3-point action via the Polyakov conjecture. Overall, the results provide efficient numerical tools for hyperbolic geometry on the 4-punctured sphere and reinforce the deep link between Liouville theory, Teichmüller theory, and uniformization, with potential routes to rigorous proofs and extensions to more punctures.
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.