Classical geometry from the quantum Liouville theory

Details Classical geometry from the quantum Liouville theory Classical geometry from the quantum Liouville theory

2005-04-25

arxiv.org/abs/hep-th/0504204v2

Nucl.Phys. B724 (2005) 529-554

Physics

High Energy Physics

High Energy Physics - Theory

26 pages, 9 eps figures. Version 2: references added

Scientific paper

10.1016/j.nuclphysb.2005.07.003

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.

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