Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Cambridge University Press eBooks

Perturbations of boundary conformal field theories

A. Recknagel, Volker Schomerus

Published Nov 5, 2013• 0 citations
Publisher

Abstract

We have gathered quite a large amount of technology to construct conformal boundary conditions and study their properties. So far, most of the tools relied on detailed knowledge of the underlying algebraic structure of conformal field theories (CFTs), such as representations of symmetry algebras and modular transformations. In almost all cases, this route will lead to a restricted class of boundary conditions only, favouring those that preserve a lot of symmetry. In “real life applications” of boundary CFT, arising in string theory, in condensed matter physics, or in the study of integrable models, we are often faced with situations where we start from a simple given boundary condition and then “deform” by “turning on perturbations”. In this chapter, we will discuss methods aimed at making this procedure well defined and at determining the perturbed boundary theory. Most of the time, we will restrict ourselves to pure boundary perturbations, which leave the bulk CFT untouched. We will sometimes be led to new boundary conditions that would have been difficult to find directly. The total space of boundary CFTs takes the form of a fibration over the moduli space of bulk theories, with the fibre over each point made up from all boundary conditions (branes) that exist for that bulk theory. Marginal boundary deformations, if they exist, will generate, starting from a single boundary condition, whole continuous families within this boundary moduli space.

DOI
10.1017/cbo9780511806476.006

Related papers

Higher structures on boundary conformal manifolds: Higher Berry phase and boundary conformal field theory

Yichul Choi, Hyunsoo Ha, Dongyeob Kim +3 more

We introduce the notion of higher Berry connection and curvature in the space of conformal boundary conditions in ( 1 + 1 ) d conformal field theories, related to each other by exactly marginal boundary deformations, forming a “boundary conformal manifold.” Our definition builds upon previous works on tensor networks, such as matrix product states, where the triple inner product or multiwavefunction overlap plays a key geometric role. On the one hand, our boundary conformal field theory formulation of higher Berry phase provides a new analytic tool to study families of invertible phases in condensed matter systems. On the other hand, it uncovers a new geometric structure on the moduli space of conformal boundary conditions, beyond the usual Riemannian structure defined through the Zamolodchikov metric. When the boundary conformal manifold has an interpretation as the position moduli space of a D-brane, our higher Berry connection coincides with the Neveu-Schwarz-Neveu-Schwarz (NS-NS) -field in string theory. The general definition does not require such an interpretation and is formulated purely field theoretically, in terms of correlation functions of boundary-condition-changing operators. We also explore a connection between higher Berry connections and functional Berry connections in the loop spaces of boundary conformal manifolds.

arXiv:2507.12525•Physical review. D/Physical review. D.•Apr 10, 2026•2 citations

Numerical Bootstrap Program for Boundary Conformal Field Theories

Bharathkumar Radhakrishnan

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.

•Jan 1, 2025•0 citations

Perturbative and non-perturbative studies in low dimensional quantum field theory

Anna Lishman

A relevant perturbation of a conformal field theory (CFT) on the half-plane, by both a bulk and boundary operator, often leads to a massive theory with a particle description in terms of the bulk S-matrix and boundary reflection factor R. The link between the particle basis and the CFT in the bulk is usually made with the thermodynamic Bethe ansatz effective central charge C(_eff). This allows a conjectured S-matrix to be identified with a specific perturbed CFT. Less is known about the links between the reflection factors and conformal boundary conditions, but it has been proposed that an exact, off-critical version of Affleck and Ludwig's g-function could be used, analogously to C(_eff), to identify the physically realised reflection factors and to match them with the corresponding boundary conditions. In the first part of this thesis, this exact g-function is tested for the purely elastic scattering theories related to the ADET Lie algebras. Minimal reflection factors are given, and a method to incorporate a boundary parameter is proposed. This enables the prediction of several new flows between conformal boundary conditions to be made. The second part of this thesis concerns the three-parameter family of PT-symmetric Hamiltonians H(M,o,1) = p(^2) – (ix) (^2M) – α(ix) The positions where the eigenvalues merge and become complex correspond to quadratic and cubic exceptional points. The quasi-exact solvability of the models for M = 3 is exploited to exploreaway from M = 3 is investigated using both numerical and perturbative approaches.

•Jan 1, 2007•0 citations

Lattice regularisation of a non-compact boundary conformal field theory

Niall Robertson, Jesper Lykke Jacobsen, Hubert Saleur

Abstract Non-compact Conformal Field Theories (CFTs) are central to several aspects of string theory and condensed matter physics. They are characterised, in particular, by the appearance of a continuum of conformal dimensions. Surprisingly, such CFTs have been identified as the continuum limits of lattice models with a finite number of degrees of freedom per site. However, results have so far been restricted to the case of periodic boundary conditions, precluding the exploration via lattice models of aspects of non-compact boundary CFTs and the corresponding D-brane constructions. The present paper follows a series of previous works on a ℤ2-staggered XXZ spin chain, whose continuum limit is known to be a non-compact CFT related with the Euclidian black hole sigma model. By using the relationship of this spin chain with an integrable $$ {D}_2^2 $$ D 2 2 vertex model, we here identify integrable boundary conditions that lead to a continuous spectrum of boundary exponents, and thus correspond to non-compact branes. In the context of the Potts model on a square lattice, they correspond to wired boundary conditions at the physical antiferromagnetic critical point. The relations with the boundary parafermion theories are discussed as well. We are also able to identify a boundary renormalisation group flow from the non-compact boundary conditions to the previously studied compact ones.

arXiv:2012.07757•Journal of High Energy Physics•Feb 1, 2021•20 citations

Boundary Conformal Field Theory

John Cardy

Boundary conformal field theory (BCFT) is simply the study of conformal field theory (CFT) in domains with a boundary. It gains its significance because, in some ways, it is mathematically simpler: the algebraic and geometric structures of CFT appear in a more straightforward manner; and because it has important applications: in string theory in the physics of open strings and D-branes, and in condensed matter physics in boundary critical behavior and quantum impurity models. In this article, however, I describe the basic ideas from the point of view of quantum field theory, without regard to particular applications nor to any deeper mathematical formulations.

arXiv:hep-th/0411189•Nov 21, 2004•19 citations

Boundary Field Theory

Giuseppe Mussardo

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.

•Mar 26, 2020•1 citation
We have gathered quite a large amount of technology to construct conformal boundary conditions and study their properties. So far, most of the tools relied on detailed knowledge of the underlying algebraic structure of conformal field theories (CFTs), such as representations of symmetry algebras and modular transformations. In almost all cases, this route will lead to a restricted class of boundary conditions only, favouring those that preserve a lot of symmetry. In “real life applications” of boundary CFT, arising in string theory, in condensed matter physics, or in the study of integrable models, we are often faced with situations where we start from a simple given boundary condition and then “deform” by “turning on perturbations”. In this chapter, we will discuss methods aimed at making this procedure well defined and at determining the perturbed boundary theory. Most of the time, we will restrict ourselves to pure boundary perturbations, which leave the bulk CFT untouched. We will sometimes be led to new boundary conditions that would have been difficult to find directly. The total space of boundary CFTs takes the form of a fibration over the moduli space of bulk theories, with the fibre over each point made up from all boundary conditions (branes) that exist for that bulk theory. Marginal boundary deformations, if they exist, will generate, starting from a single boundary condition, whole continuous families within this boundary moduli space.
We introduce the notion of higher Berry connection and curvature in the space of conformal boundary conditions in ( 1 + 1 ) d conformal field theories, related to each other by exactly marginal boundary deformations, forming a “boundary conformal manifold.” Our definition builds upon previous works on tensor networks, such as matrix product states, where the triple inner product or multiwavefunction overlap plays a key geometric role. On the one hand, our boundary conformal field theory formulation of higher Berry phase provides a new analytic tool to study families of invertible phases in condensed matter systems. On the other hand, it uncovers a new geometric structure on the moduli space of conformal boundary conditions, beyond the usual Riemannian structure defined through the Zamolodchikov metric. When the boundary conformal manifold has an interpretation as the position moduli space of a D-brane, our higher Berry connection coincides with the Neveu-Schwarz-Neveu-Schwarz (NS-NS) -field in string theory. The general definition does not require such an interpretation and is formulated purely field theoretically, in terms of correlation functions of boundary-condition-changing operators. We also explore a connection between higher Berry connections and functional Berry connections in the loop spaces of boundary conformal manifolds.
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.
A relevant perturbation of a conformal field theory (CFT) on the half-plane, by both a bulk and boundary operator, often leads to a massive theory with a particle description in terms of the bulk S-matrix and boundary reflection factor R. The link between the particle basis and the CFT in the bulk is usually made with the thermodynamic Bethe ansatz effective central charge C(_eff). This allows a conjectured S-matrix to be identified with a specific perturbed CFT. Less is known about the links between the reflection factors and conformal boundary conditions, but it has been proposed that an exact, off-critical version of Affleck and Ludwig's g-function could be used, analogously to C(_eff), to identify the physically realised reflection factors and to match them with the corresponding boundary conditions. In the first part of this thesis, this exact g-function is tested for the purely elastic scattering theories related to the ADET Lie algebras. Minimal reflection factors are given, and a method to incorporate a boundary parameter is proposed. This enables the prediction of several new flows between conformal boundary conditions to be made. The second part of this thesis concerns the three-parameter family of PT-symmetric Hamiltonians H(M,o,1) = p(^2) – (ix) (^2M) – α(ix) The positions where the eigenvalues merge and become complex correspond to quadratic and cubic exceptional points. The quasi-exact solvability of the models for M = 3 is exploited to exploreaway from M = 3 is investigated using both numerical and perturbative approaches.
Abstract Non-compact Conformal Field Theories (CFTs) are central to several aspects of string theory and condensed matter physics. They are characterised, in particular, by the appearance of a continuum of conformal dimensions. Surprisingly, such CFTs have been identified as the continuum limits of lattice models with a finite number of degrees of freedom per site. However, results have so far been restricted to the case of periodic boundary conditions, precluding the exploration via lattice models of aspects of non-compact boundary CFTs and the corresponding D-brane constructions. The present paper follows a series of previous works on a ℤ2-staggered XXZ spin chain, whose continuum limit is known to be a non-compact CFT related with the Euclidian black hole sigma model. By using the relationship of this spin chain with an integrable D22 {D}_2^2 D22​ D 2 2 vertex model, we here identify integrable boundary conditions that lead to a continuous spectrum of boundary exponents, and thus correspond to non-compact branes. In the context of the Potts model on a square lattice, they correspond to wired boundary conditions at the physical antiferromagnetic critical point. The relations with the boundary parafermion theories are discussed as well. We are also able to identify a boundary renormalisation group flow from the non-compact boundary conditions to the previously studied compact ones.
Boundary conformal field theory (BCFT) is simply the study of conformal field theory (CFT) in domains with a boundary. It gains its significance because, in some ways, it is mathematically simpler: the algebraic and geometric structures of CFT appear in a more straightforward manner; and because it has important applications: in string theory in the physics of open strings and D-branes, and in condensed matter physics in boundary critical behavior and quantum impurity models. In this article, however, I describe the basic ideas from the point of view of quantum field theory, without regard to particular applications nor to any deeper mathematical formulations.
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.