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

Boundary States of c=1 and 3/2 Rational Conformal Field Theories

Andrea Cappelli A, Giuseppe D’Appollonio

Published Jan 1, 2002• 7 citations
OpenAlex

Abstract

We study the boundary states for the rational points in the moduli spaces of c=1 conformal and c=3/2 superconformal field theories, including the isolated Ginsparg points. We use the orbifold and simple-current techniques to relate the boundary states of different theories and to obtain symmetry-breaking, non-Cardy boundary states. We show some interesting examples of fractional and twisted branes on orbifold spaces.

Related papers

Conformal Boundary States in

Atsushi Yamaguchi

We construct boundary states in a particular c = 1 conformal field theory, the orbifold with G a binary finite subgroup of SU(2). These states preserve the conformal symmetry, at least, but break rational symmetries of the orbifold in general.

arXiv:hep-th/0309102•Progress of Theoretical Physics•Jul 1, 2004•2 citations

Conformal and fractional boundary states in the symmetric orbifold of the level one SU(2) WZW model

Atsushi Yamaguchi

We study the space of the allowed boundary states in a particular c=1 conformal field theory, the symmetric orbifold of the level one SU(2) WZW model, where the orbifolding group G is a generic binary finite subgroup of SU(2). We especially construct generic boundary states in the orbifold, which can be extended from the Cardy states decomposed into Virasoro Ishibashi states. Our result involves conformal boundary states parameterized by the quotient SU(2)/G, and the fractional boundary states sitting at every fixed-point in the quotient. We find also that a conformal boundary state at a fixed-point in the quotient becomes a linear combination of the fractional boundary states sitting at the fixed-point. To be complete, we check the mutual consistency among all the boundary states we construct.

•Sep 10, 2003•0 citations

Two-dimensional Anomaly, Orbifolding, and Boundary States

Ken Kikuchi, Yang Zhou

We study anomalies of discrete internal global symmetry $G$ in two-dimensional rational conformal field theories based on twisted torus partition functions. The anomaly of $G$ can be seen from the noncommutativity of two symmetry lines inserted along the nontrivial cycles of two-torus and we propose a criterion to detect the anomaly, which agrees with the truncated modular $S$-matrix approach. The obstruction for orbifolding has been recently interpreted as a mixed anomaly between $G$ and large diffeomorphisms. We clarify the relations among anomaly-free conditions, orbifoldable conditions, and invariant boundary state condition, focusing on Wess-Zumino-Witten models.

arXiv:1908.02918•Aug 8, 2019•7 citations

$2d$ 't Hooft Anomaly, Orbifolding, and Boundary States

Ken Kikuchi, Yang Zhou

We study anomalies of discrete internal global symmetry $G$ in two-dimensional rational conformal field theories based on twisted torus partition functions. The anomaly of $G$ can be seen from the noncommutativity of two symmetry lines inserted along the nontrivial cycles of two-torus and we propose a criterion to detect the anomaly, which agrees with the truncated modular $S$-matrix approach. The obstruction for orbifolding has been recently interpreted as a mixed anomaly between $G$ and large diffeomorphisms. We clarify the relations among anomaly-free conditions, orbifoldable conditions, and invariant boundary state condition, focusing on Wess-Zumino-Witten models.

•Aug 8, 2019•1 citation

Perturbations of boundary conformal field theories

A. Recknagel, Volker Schomerus

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.

Cambridge University Press eBooks•Nov 5, 2013•0 citations

The logarithmic Cardy case: Boundary states and annuli

Jürgen Fuchs, Terry J. Gannon, Gregor Schaumann +1 more

We present a model-independent study of boundary states in the Cardy case that covers all conformal field theories for which the representation category of the chiral algebra is a – not necessarily semisimple – modular tensor category. This class, which we call finite CFTs, includes all rational theories, but goes much beyond these, and in particular comprises many logarithmic conformal field theories. We show that the following two postulates for a Cardy case are compatible beyond rational CFT and lead to a universal description of boundary states that realizes a standard mathematical setup: First, for bulk fields, the pairing of left and right movers is given by (a coend involving) charge conjugation; and second, the boundary conditions are given by the objects of the category of chiral data. For rational theories our proposal reproduces the familiar result for the boundary states of the Cardy case. Further, with the help of sewing we compute annulus amplitudes. Our results show in particular that these possess an interpretation as partition functions, a constraint that for generic finite CFTs is much more restrictive than for rational ones.

arXiv:1712.01922•Nuclear Physics B•Mar 28, 2018•9 citations
We study the boundary states for the rational points in the moduli spaces of c=1 conformal and c=3/2 superconformal field theories, including the isolated Ginsparg points. We use the orbifold and simple-current techniques to relate the boundary states of different theories and to obtain symmetry-breaking, non-Cardy boundary states. We show some interesting examples of fractional and twisted branes on orbifold spaces.
We study the space of the allowed boundary states in a particular c=1 conformal field theory, the symmetric orbifold of the level one SU(2) WZW model, where the orbifolding group G is a generic binary finite subgroup of SU(2). We especially construct generic boundary states in the orbifold, which can be extended from the Cardy states decomposed into Virasoro Ishibashi states. Our result involves conformal boundary states parameterized by the quotient SU(2)/G, and the fractional boundary states sitting at every fixed-point in the quotient. We find also that a conformal boundary state at a fixed-point in the quotient becomes a linear combination of the fractional boundary states sitting at the fixed-point. To be complete, we check the mutual consistency among all the boundary states we construct.
We study anomalies of discrete internal global symmetry GGG in two-dimensional rational conformal field theories based on twisted torus partition functions. The anomaly of GGG can be seen from the noncommutativity of two symmetry lines inserted along the nontrivial cycles of two-torus and we propose a criterion to detect the anomaly, which agrees with the truncated modular SSS-matrix approach. The obstruction for orbifolding has been recently interpreted as a mixed anomaly between GGG and large diffeomorphisms. We clarify the relations among anomaly-free conditions, orbifoldable conditions, and invariant boundary state condition, focusing on Wess-Zumino-Witten models.
2d2d2d 't Hooft Anomaly, Orbifolding, and Boundary States
We study anomalies of discrete internal global symmetry GGG in two-dimensional rational conformal field theories based on twisted torus partition functions. The anomaly of GGG can be seen from the noncommutativity of two symmetry lines inserted along the nontrivial cycles of two-torus and we propose a criterion to detect the anomaly, which agrees with the truncated modular SSS-matrix approach. The obstruction for orbifolding has been recently interpreted as a mixed anomaly between GGG and large diffeomorphisms. We clarify the relations among anomaly-free conditions, orbifoldable conditions, and invariant boundary state condition, focusing on Wess-Zumino-Witten models.
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 present a model-independent study of boundary states in the Cardy case that covers all conformal field theories for which the representation category of the chiral algebra is a – not necessarily semisimple – modular tensor category. This class, which we call finite CFTs, includes all rational theories, but goes much beyond these, and in particular comprises many logarithmic conformal field theories. We show that the following two postulates for a Cardy case are compatible beyond rational CFT and lead to a universal description of boundary states that realizes a standard mathematical setup: First, for bulk fields, the pairing of left and right movers is given by (a coend involving) charge conjugation; and second, the boundary conditions are given by the objects of the category of chiral data. For rational theories our proposal reproduces the familiar result for the boundary states of the Cardy case. Further, with the help of sewing we compute annulus amplitudes. Our results show in particular that these possess an interpretation as partition functions, a constraint that for generic finite CFTs is much more restrictive than for rational ones.