Standard and geometric approaches to quantum Liouville theory on the pseudosphere

TL;DR

This work contrasts the standard (ZZ) and geometric formulations of quantum Liouville theory on the pseudosphere by performing perturbative calculations up to third order. It demonstrates that the Hadamard regulator in the geometric approach yields discrepancies in higher cumulants and two-point structures, while the ZZ regulator restores full agreement with the standard approach and the bootstrap conjectures. The key distinction lies in the regulator choice, which fixes the quantum conformal dimensions of the cosmological term and the boundary behavior of correlators. The findings clarify regulator-dependent aspects of Liouville theory on curved backgrounds and show how to align geometric and canonical formulations with conformal bootstrap data.

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.