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

Edoardo Livolsi

Published Jan 1, 2026• 0 citations
Publisher

Abstract

The manuscript presents a detailed construction of boundary Liouville field theory based onconformal bootstrap, analytic continuation, and duality arguments. The results are technicallysophisticated and of potential interest.However, several steps that are essential for establishing the mathematical completeness ofthe construction are currently implicit. Clarifying these points would be important in orderto assess whether the framework defines a fully consistent quantum field theory, rather than aformally constrained set of correlation functions.Below I list a number of technical questions whose resolution appears necessary.

DOI
10.6084/m9.figshare.31032742

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The manuscript presents a detailed construction of boundary Liouville field theory based onconformal bootstrap, analytic continuation, and duality arguments. The results are technicallysophisticated and of potential interest.However, several steps that are essential for establishing the mathematical completeness ofthe construction are currently implicit. Clarifying these points would be important in orderto assess whether the framework defines a fully consistent quantum field theory, rather than aformally constrained set of correlation functions.Below I list a number of technical questions whose resolution appears necessary.
This thesis explores the application of numerical bootstrap techniques to two-point functions of bulk insertions in the presence of a boundary in two-dimensional conformal field theories. The first part of this thesis is devoted to presenting a comprehensive introduction to essential concepts in the numerical bootstrap of two-dimensional CFTs. Namely, an extensive introduction to the basics of conformal field theories (CFTs), their properties arising from the algebraic structure of conformal transformations, consitency conditions on the observables in a CFT, as well as the effects of the presence of defects in two-dimensional CFTs, is given. This is followed by a quick recount of the numerical techniques employed in the literature to solve for the aforementioned consistency conditions. These chapters form an overview of the seminal works that set the backdrop of the contributions made in the second part of the thesis. In the contributions of the thesis, we discuss the progress made in the investigations in (Meineri and Radhakrishnan 2025) where two-point functions in BCFTs are bootstrapped by treating them as a part of a larger system of consistency conditions involving the partition function on an annulus, the two-point function in the presence of a boundary, and four-point functions on the infinite plane. This perspective of treating the two-point function as a part of a larger system makes accessible parts of the BCFT data that have so-far remained inaccessible, and we present results on the boundary gap in the two-point function, and improvements in existing numerical bounds on the boundary entropy.
We review conformal field theory on the plane in the conformal bootstrap approach. We introduce the main ideas of the bootstrap approach to quantum field theory, and how they apply to two-dimensional theories with local conformal symmetry. We describe the mathematical structures that appear in such theories, from the Virasoro algebra and its representations, to BPZ equations and conformal blocks. Examples include Liouville theory, (generalized) minimal models, free bosonic theories, the H3+H_3^+H3+​ model, and the SU2SU_2SU2​ and SL~2(R)\widetilde{SL}_2(\mathbb{R})SL2​(R) WZW models. We also discuss relations between some of these models, and limits of these models when the central charge and/or conformal dimensions tend to particular values.
The study of conformal symmetry is motivated through an example in statistical mechanics and then rigorously developed in quantum field theories in general spatial dimensions. In particular, primary fields are introduced as the fundamental objects of such theories and then studied in the formalism of radial quantization. The implications of conformal invariance on the functional form of correlation functions are studied in detail. Conformal blocks are defined and various approaches to their analytical and numerical calculation are presented with a special emphasis on the one-dimensional case. Building on these preliminaries, a modern formulation of the conformal bootstrap program and its various extensions are discussed. Examples are given in which bounds on the scaling dimensions in a one-dimensional theory are derived numerically. Using these results I motivate the technique of using the extremal functional bootstrap which I then develop in more detail. Many technical details are discussed and examples shown. After a brief discussion of conformal field theories with a boundary I apply numerical methods to find constraints on the spectrum of the 3D Ising model. Another application is presented in which I study the 4-point function on the boundary of a particular theory in Anti-de-Sitter space in order to approximate the mass spectrum of the theory.
A bstract For QFTs in AdS the boundary correlation functions remain conformal even if the bulk theory has a scale. This allows one to constrain RG flows with numerical conformal bootstrap methods. We apply this idea to flows between two-dimensional CFTs, focusing on deformations of the tricritical and ordinary Ising model. We provide non-perturbative constraints for the boundary correlation functions of these flows and compare them with conformal perturbation theory in the vicinity of the fixed points. We also reproduce a completely general constraint on the sign of the TT‾T\overline{T }TT deformation in two dimensions.
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.
Abstract Quantum field theory with boundaries is a very rich subject both from purely theoretical point of view or applicative aims. Chapter 21 presents the basis results relative to critical systems and massive integrable field theories. It covers stress-energy tensor in boundary conformal field theories and integrable deformations of these theories (including the single Virasoro algebra), reflection amplitudes, conformal boundary operators and states (including the Hilbert space and Ishibashi state), non-renormalizable states of the Hilbert space, operator product expansion involving a boundary operator, massive integrable boundary field theory, the massive boundary Ising model, boundary magnetic fields and correlation functions.