Semiclassical and quantum Liouville theory

TL;DR

This paper develops a novel functional integral approach to quantum Liouville theory, solving the Riemann-Hilbert problem for singularities, computing Green functions, and analyzing quantum determinants, with results matching conformal bootstrap predictions.

Contribution

It introduces a new functional integral method for Liouville theory, independently of the Hamiltonian approach, and computes exact quantum determinants and conformal dimensions.

Findings

Derived the semiclassical four-point vertex function with singularities.

Computed the exact Green function on the sphere with three singularities.

Obtained the quantum determinant on the pseudosphere matching bootstrap results.

Abstract

We develop a functional integral approach to quantum Liouville field theory completely independent of the hamiltonian approach. To this end on the sphere topology we solve the Riemann-Hilbert problem for three singularities of finite strength and a fourth one infinitesimal, by determining perturbatively the Poincare' accessory parameters. This provides 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 on the background of 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 of the further perturbative corrections. The zeta function regularization 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 then apply the method to the case of the pseudosphere with one finite singularity and compute the exact value for the quantum determinant. Such results are compared to those of the conformal bootstrap approach finding complete agreement.