Peer review Boundary Liouville Field TheoryI. Boundary State and Boundary Two-point Function Alexander Zamolodchikov

This document provides a technical referee-style analysis of Boundary Liouville Field Theory, focusing on the mathematical conditions required for a quantum field theory with boundaries to be considered functionally complete. Rather than addressing phenomenological results, the review concentrates on structural aspects such as the existence of a generating functional, control of quadratic fluctuations, stability of boundary modes, and infrared–ultraviolet matching. To facilitate a transparent assessment, a model-independent consistency benchmark is included as an appendix. The benchmark formulates minimal functional criteria—variational closure, emergence of operator structure, spectral control, and ultraviolet behavior—that any closed theoretical framework is expected to satisfy. The purpose of this analysis is not to invalidate specific constructions, but to clarify the distinction between formally consistent sets of correlation functions and theories admitting a complete functional and spectral interpretation. The benchmark is intended as a reusable diagnostic tool applicable to a broad class of field-theoretic models with boundaries.