Dangerous Liouville Wave -- exactly marginal but non-conformal deformation

TL;DR

The paper constructs a two‑dimensional quantum field theory that is scale‑invariant but non‑conformal by coupling Liouville theory to free scalars with an exactly marginal perturbation. It shows that quantum Schwinger‑Dyson analysis yields nonzero contributions to the trace of the energy‑momentum tensor, violating conformal invariance, while preserving scale invariance to all orders. The authors leverage Zamolodchikov’s higher equations of motion and careful limits to establish the presence of contact terms that break holomorphicity of the energy‑momentum tensor. They discuss implications for string worldsheet conformal anomaly and the potential hazards of such deformations in super‑critical string backgrounds.

Abstract

We give a non-trivially interacting field theory example of scale invariant but non-conformal field theory. The model is based on the exactly solvable Liouville field theory coupled with free scalars deformed by an exactly marginal operator. We show non-vanishing of the trace of the energy-momentum tensor by using the quantum Schwinger-Dyson equation for the Liouville field theory, which is a sophistication of the quantum higher equations of motion for the Liouville field theory introduced by Alyosha Zamolodchikov. Possibly dangerous implications for the super-critical string theory will be discussed.