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

arXiv:hep-th/9708135 · Nuclear Physics B

Minimal models with integrable local defects

André LeClair, Andreas W. W. Ludwig

Published Jun 1, 1999• 28 citations
arXivHTMLPDF

Abstract

We describe a general way of constructing integrable defect theories as perturbations of conformal field theory by local defect operators. The method relies on folding the system onto a boundary field theory of twice the central charge. The classification of integrable defect theories obtained in this way parallels that of integrable bulk theories which are a perturbation of the tensor product of two conformal field theories. These include local defect perturbations of all c < 1 minimal models, as well as of the coset theories based on SO(2n), obtained in this way. We discuss in detail the former case of all the Virasoro minimal models. In the Ising case our construction corresponds to having a spin field as a defect operator; in the folded formulation this is mapped onto an orbifolding of the boundary sine-Gordon theory at β2/8π = 18, or a version of the anisotropic Kondo model.

DOI
10.1016/s0550-3213(99)00111-x

Related papers

Non-local charges from perturbed defects via SymTFT in 2d CFT

Federico Ambrosino, Ingo Runkel B, G.M.T. Watts

Abstract We investigate non-local conserved charges in perturbed two-dimensional conformal field theories (CFT) from the point of view of the 3d symmetry TFT (SymTFT) of the unperturbed theory. In the SymTFT we state a simple commutation condition which results in a pair of compatible bulk and defect perturbations, such that the perturbed line defects are conserved in the perturbed CFT. In other words, the perturbed defects are rigidly translation invariant, and such defects form a monoidal category which extends the topological symmetries. As examples we study the A-type Virasoro minimal models M ( p , q ) . Our formalism provides one-parameter families of commuting non-local conserved charges for perturbations by a primary bulk field with Kac label (1, 2), (1, 3), or (1, 5), which are the standard integrable perturbations of minimal models. We find solutions to the commutation condition also for other bulk perturbations, such as (1, 7), and we contrast this with the existence of local conserved charges. There has been recent interest in the possibility that in certain cases perturbations by fields such as (1, 7) can be integrable, and our construction provides a new way in which integrability can be found without the need for local conserved charges.

arXiv:2504.05277•Journal of Physics A Mathematical and Theoretical•Sep 24, 2025•3 citations

Entropy of conformal perturbation defects

Anatoly Konechny, Cornelius Schmidt-Colinet

We consider perturbation defects obtained by perturbing a 2D conformal field theory (CFT) by a relevant operator on a half-plane. If the perturbed bulk theory flows to an infrared fixed point described by another CFT, the defect flows to a conformal defect between the ultraviolet and infrared fixed point CFTs. For short bulk renormalization group flows connecting two fixed points which are close in theory space we find a universal perturbative formula for the boundary entropy of the corresponding conformal perturbation defect. We compare the value of the boundary entropy that our formula gives for the flows between nearby Virasoro minimal models Mm with the boundary entropy of the defect constructed by Gaiotto in [1] and find a match at the first two orders in the 1/m expansion.

arXiv:1407.6444•Jul 24, 2014•3 citations

Entropy of conformal perturbation defects

Anatoly Konechny, Cornelius Schmidt-Colinet

We consider perturbation defects obtained by perturbing a 2D conformal field theory (CFT) by a relevant operator on a half-plane.If the perturbed bulk theory flows to an infrared fixed point described by another CFT, the defect flows to a conformal defect between the ultraviolet and infrared fixed point CFTs.For short bulk renormalization group flows connecting two fixed points which are close in theory space we find a universal perturbative formula for the boundary entropy of the corresponding conformal perturbation defect.We compare the value of the boundary entropy that our formula gives for the flows between nearby Virasoro minimal models  m with the boundary entropy of the defect constructed by Gaiotto (2012 J. High Energy Phys. JHEP12(2012) 103) and find a match at the first two orders in the m 1 expansion.

Journal of Physics A Mathematical and Theoretical•Nov 11, 2014•12 citations

Conformal defects from string field theory

Kasia Budzik, Miroslav Rapčák, Jairo M. Rojas

Abstract Unlike conformal boundary conditions, conformal defects of Virasoro minimal models lack classification. Alternatively to the defect perturbation theory and the truncated conformal space approach, we employ open string field theory (OSFT) techniques to explore the space of conformal defects. We illustrate the method by an analysis of OSFT around the background associated to the (1, 2) topological defect in diagonal unitary minimal models. Numerical analysis of OSFT equations of motion leads to an identification of a nice family of solutions, recovering the picture of infrared fixed points due to Kormos, Runkel and Watts. In particular, we find a continuum of solutions in the Ising model case and 6 solutions for other minimal models. OSFT provides us with numerical estimates of the g-function and other coefficients of the boundary state.

arXiv:2009.06567•Journal of High Energy Physics•Jan 20, 2021•4 citations

Exploring defects with degrees of freedom in free scalar CFTs

Vladimir Bashmakov, Jacopo Sisti

A bstract Defects in conformal field theories are interesting objects to study from both formal and applied points of view. In this paper, we construct conformal defects in free scalar field CFTs in diverse dimensions. After discussing the possible quadratic defects, we explore interacting setups. These are realized by coupling the bulk free scalar to lower-dimensional theories, including the unitary family of minimal models $$ \mathcal{M}\left(m,m+1\right) $$ M m m + 1 . Another example involves coupling to a two-dimensional free scalar field, from which we construct surface defects for the bulk dimensions three and five. Additionally, we consider monodromy defects associated with a global U(1) flavour symmetry. In these theories, we study both self-defect interactions and couplings to Minimal Models, finding new IR defect fixed points. For all our examples, we provide results for correlation functions, such as those involving the bulk stress tensor and the displacement operator, and for the defect central charges.

arXiv:2410.01716•Journal of High Energy Physics•Mar 19, 2025•4 citations

Symmetry Breaking Boundary States and Defect Lines

Thomas Quella, Volker Schomerus

We present a large and universal class of new boundary states which break part of the chiral symmetry in the underlying bulk theory. Our formulas are based on coset constructions and they can be regarded as a non-abelian generalization of the ideas that were used by Maldacena, Moore and Seiberg to build new boundary states for SU(N). We apply our expressions to construct defect lines joining two conformal field theories with possibly different central charge. Such defects can occur e.g. in the AdS/CFT correspondence when branes extend to the boundary of the AdS-space.

•Jan 1, 2002•77 citations
We describe a general way of constructing integrable defect theories as perturbations of conformal field theory by local defect operators. The method relies on folding the system onto a boundary field theory of twice the central charge. The classification of integrable defect theories obtained in this way parallels that of integrable bulk theories which are a perturbation of the tensor product of two conformal field theories. These include local defect perturbations of all c < 1 minimal models, as well as of the coset theories based on SO(2n), obtained in this way. We discuss in detail the former case of all the Virasoro minimal models. In the Ising case our construction corresponds to having a spin field as a defect operator; in the folded formulation this is mapped onto an orbifolding of the boundary sine-Gordon theory at β2/8π = 18, or a version of the anisotropic Kondo model.
Abstract We investigate non-local conserved charges in perturbed two-dimensional conformal field theories (CFT) from the point of view of the 3d symmetry TFT (SymTFT) of the unperturbed theory. In the SymTFT we state a simple commutation condition which results in a pair of compatible bulk and defect perturbations, such that the perturbed line defects are conserved in the perturbed CFT. In other words, the perturbed defects are rigidly translation invariant, and such defects form a monoidal category which extends the topological symmetries. As examples we study the A-type Virasoro minimal models M ( p , q ) . Our formalism provides one-parameter families of commuting non-local conserved charges for perturbations by a primary bulk field with Kac label (1, 2), (1, 3), or (1, 5), which are the standard integrable perturbations of minimal models. We find solutions to the commutation condition also for other bulk perturbations, such as (1, 7), and we contrast this with the existence of local conserved charges. There has been recent interest in the possibility that in certain cases perturbations by fields such as (1, 7) can be integrable, and our construction provides a new way in which integrability can be found without the need for local conserved charges.
We consider perturbation defects obtained by perturbing a 2D conformal field theory (CFT) by a relevant operator on a half-plane. If the perturbed bulk theory flows to an infrared fixed point described by another CFT, the defect flows to a conformal defect between the ultraviolet and infrared fixed point CFTs. For short bulk renormalization group flows connecting two fixed points which are close in theory space we find a universal perturbative formula for the boundary entropy of the corresponding conformal perturbation defect. We compare the value of the boundary entropy that our formula gives for the flows between nearby Virasoro minimal models Mm with the boundary entropy of the defect constructed by Gaiotto in [1] and find a match at the first two orders in the 1/m expansion.
We consider perturbation defects obtained by perturbing a 2D conformal field theory (CFT) by a relevant operator on a half-plane.If the perturbed bulk theory flows to an infrared fixed point described by another CFT, the defect flows to a conformal defect between the ultraviolet and infrared fixed point CFTs.For short bulk renormalization group flows connecting two fixed points which are close in theory space we find a universal perturbative formula for the boundary entropy of the corresponding conformal perturbation defect.We compare the value of the boundary entropy that our formula gives for the flows between nearby Virasoro minimal models  m with the boundary entropy of the defect constructed by Gaiotto (2012 J. High Energy Phys. JHEP12(2012) 103) and find a match at the first two orders in the m 1 expansion.
Abstract Unlike conformal boundary conditions, conformal defects of Virasoro minimal models lack classification. Alternatively to the defect perturbation theory and the truncated conformal space approach, we employ open string field theory (OSFT) techniques to explore the space of conformal defects. We illustrate the method by an analysis of OSFT around the background associated to the (1, 2) topological defect in diagonal unitary minimal models. Numerical analysis of OSFT equations of motion leads to an identification of a nice family of solutions, recovering the picture of infrared fixed points due to Kormos, Runkel and Watts. In particular, we find a continuum of solutions in the Ising model case and 6 solutions for other minimal models. OSFT provides us with numerical estimates of the g-function and other coefficients of the boundary state.
A bstract Defects in conformal field theories are interesting objects to study from both formal and applied points of view. In this paper, we construct conformal defects in free scalar field CFTs in diverse dimensions. After discussing the possible quadratic defects, we explore interacting setups. These are realized by coupling the bulk free scalar to lower-dimensional theories, including the unitary family of minimal models M(m,m+1) \mathcal{M}\left(m,m+1\right) M(m,m+1) M m m + 1 . Another example involves coupling to a two-dimensional free scalar field, from which we construct surface defects for the bulk dimensions three and five. Additionally, we consider monodromy defects associated with a global U(1) flavour symmetry. In these theories, we study both self-defect interactions and couplings to Minimal Models, finding new IR defect fixed points. For all our examples, we provide results for correlation functions, such as those involving the bulk stress tensor and the displacement operator, and for the defect central charges.
We present a large and universal class of new boundary states which break part of the chiral symmetry in the underlying bulk theory. Our formulas are based on coset constructions and they can be regarded as a non-abelian generalization of the ideas that were used by Maldacena, Moore and Seiberg to build new boundary states for SU(N). We apply our expressions to construct defect lines joining two conformal field theories with possibly different central charge. Such defects can occur e.g. in the AdS/CFT correspondence when branes extend to the boundary of the AdS-space.