Semiclassical and quantum Liouville theory on the sphere

Abstract

We solve the Riemann-Hilbert problem on the sphere topology for three singularities of finite strength and a fourth one infinitesimal, by determining perturbatively the Poincare' accessory parameters. In this way we compute the semiclassical four point vertex function with three finite charges and a fourth infinitesimal. Some of the results are extended to the case of n finite charges and m infinitesimal. With the same technique we compute the exact Green function on the sphere with three finite singularities. Turning to the full quantum problem we address the calculation of the quantum determinant on the background of three finite charges and the further perturbative corrections. The zeta function technique provides a theory which is not invariant under local conformal transformations. Instead by employing a regularization suggested in the case of the pseudosphere by Zamolodchikov and Zamolodchikov we obtain the correct quantum conformal dimensions from the one loop calculation and we show explicitly that the two loop corrections do not change such dimensions. We expect such a result to hold to all order perturbation theory.

26 pages, latex, 1 figure, references added, 2 typos corrected

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Cited by in corpus (18)

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• Classical geometry from the quantum Liouville theory
• Liouville Quantum Gravity
• Heavy-Heavy-Light-Light correlators in Liouville theory
• Riemann-Hilbert treatment of Liouville theory on the torus
• Footballs, Conical Singularities and the Liouville Equation
• Accessory parameters for Liouville theory on the torus
• Liouville field theory with heavy charges. I. The pseudosphere
• Vacuumless kinks systems from vacuum ones, an example
• Quantum Liouville theory on the pseudosphere with heavy charges
• Liouville field theory with heavy charges. II. The conformal boundary case
• Polyakov relation for the sphere and higher genus surfaces
• Semiclassical limit of the FZZT Liouville theory
• The two-sphere partition function in two-dimensional quantum gravity
• Riemann-Hilbert treatment of Liouville theory on the torus: The general case
• Semiclassical and quantum Liouville theory
• Classical conformal blocks, Coulomb gas integrals and Richardson-Gaudin models
• Non invariant zeta-function regularization in quantum Liouville theory