Semiclassical and quantum Liouville theory on the sphere

TL;DR

This paper solves the classical and quantum Liouville theory on the sphere with singularities, computing key functions and dimensions, and demonstrating the invariance of conformal dimensions under quantum corrections.

Contribution

It provides a perturbative solution to the Riemann-Hilbert problem for Liouville theory on the sphere, extending results to multiple charges and analyzing quantum corrections with a novel regularization.

Findings

Computed the semiclassical four-point vertex function with three finite and one infinitesimal charge.

Derived the exact Green function on the sphere with three finite singularities.

Showed quantum conformal dimensions remain unchanged at two-loop order with a specific regularization.

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.