Non invariant zeta-function regularization in quantum Liouville theory

TL;DR

This paper compares two zeta-function regularization schemes in quantum Liouville theory, highlighting their invariance properties and implications for conformal symmetry.

Contribution

It introduces and analyzes a non-invariant regularization scheme, demonstrating its equivalence to Zamolodchikov's approach and its full conformal invariance.

Findings

Invariant regularization does not preserve full conformal symmetry.

Naive non-invariant regularization aligns with Zamolodchikov's method.

Full conformal invariance achieved with the non-invariant scheme.

Abstract

We consider two possible zeta-function regularization schemes of quantum Liouville theory. One refers to the Laplace-Beltrami operator covariant under conformal transformations, the other to the naive non invariant operator. The first produces an invariant regularization which however does not give rise to a theory invariant under the full conformal group. The other is equivalent to the regularization proposed by Zamolodchikov and Zamolodchikov and gives rise to a theory invariant under the full conformal group.