Standard and geometric approaches to quantum Liouville theory on the pseudosphere

TL;DR

This paper compares standard and geometric methods in quantum Liouville theory on the pseudosphere, revealing discrepancies due to regularization choices but confirming agreement with bootstrap results for key functions.

Contribution

It provides a detailed perturbative comparison of the two approaches and clarifies the impact of regularization on their agreement.

Findings

Discrepancy caused by Hadamard regularization in geometric approach

Agreement between standard approach and bootstrap conjectures for certain functions

Perturbative calculations up to third order in coupling constant

Abstract

We compare the standard and geometric approaches to quantum Liouville theory on the pseudosphere by performing perturbative calculations of the one and two point functions up to the third order in the coupling constant. The choice of the Hadamard regularization within the geometric approach leads to a discrepancy with the standard approach. On the other hand, we find complete agreement between the results of the standard approach and the bootstrap conjectures for the one point function and the auxiliary two point function.