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:0711.3430 · Journal of Physics A Mathematical and Theoretical

Radford, Drinfeld and Cardy boundary states in the (1,p) logarithmic conformal field models

Azat M. Gainutdinov, I. Yu. Tipunin

Published Jul 14, 2009• 48 citations
arXivHTMLPDF

Abstract

We introduce ( p − 1) pseudocharacters in the space of (1, p ) model vacuum torus amplitudes to complete the distinguished basis in the 2 p -dimensional fusion algebra to a basis in the whole (3 p − 1)-dimensional space of torus amplitudes, and the structure constants in this basis are (not necessarily non-negative) integer numbers. We obtain a generalized Verlinde formula that gives these structure constants. In the context of theories with boundaries, we identify the space of vacuum torus amplitudes with the space of Ishibashi states. Then, we propose (3 p − 1) boundary states satisfying the Cardy condition.

DOI
10.1088/1751-8113/42/31/315207

Related papers

Boundary States, Extended Symmetry Algebra and Module Structure for certain Rational Torus Models

Ioannis Smyrnakis

The massless bosonic field compactified on the circle of rational R 2 is reexamined in the presense of boundaries. A particular class of models corresponding to R 2 = 1 2k is distinguished by demanding the existence of a consistent set of Newmann boundary states. The boundary states are constructed explicitly for these models and the fusion rules are derived from them. These are the ones prescribed by the Verlinde formula from the S-matrix of the theory. In addition, the extended symmetry algebra of these theories is constructed which is responsible for the rationality of these theories. Finally, the chiral space of these models is shown to split into a direct sum of irreducible modules of the extended symmetry algebra. 1

•Jan 1, 2002•1 citation

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

From boundary to bulk in logarithmic CFT

Matthias R. Gaberdiel B, Ingo Runkel B

The analogue of the charge-conjugation modular invariant for rational logarithmic conformal field theories is constructed. This is done by reconstructing the bulk spectrum from a simple boundary condition (the analogue of the Cardy ‘identity brane’). We apply the general method to the c1,p triplet models and reproduce the previously known bulk theory for p = 2 at c = −2. For general p we verify that the resulting partition functions are modular invariant. We also construct the complete set of 2p boundary states, and confirm that the identity brane from which we started indeed exists. As a by-product we obtain a logarithmic version of the Verlinde formula for the c1,p triplet models.

•Jan 1, 2007•67 citations

From boundary to bulk in logarithmic CFT

Matthias R. Gaberdiel B, Ingo Runkel B

The analogue of the charge-conjugation modular invariant for rational logarithmic conformal field theories is constructed. This is done by reconstructing the bulk spectrum from a simple boundary condition (the analogue of the Cardy `identity brane'). We apply the general method to the c_1,p triplet models and reproduce the previously known bulk theory for p=2 at c=-2. For general p we verify that the resulting partition functions are modular invariant. We also construct the complete set of 2p boundary states, and confirm that the identity brane from which we started indeed exists. As a by-product we obtain a logarithmic version of the Verlinde formula for the c_1,p triplet models.

arXiv:0707.0388•Journal of Physics A Mathematical and Theoretical•Feb 5, 2008•66 citations

Integrable and conformal twisted boundary conditions forsl(2)A-D-Elattice models

C H O Chui, Christian Mercat, Paul A Pearce

We study integrable realizations of conformal twisted boundary conditions for s ℓ(2) unitary minimal models on a torus. These conformal field theories are realized as the continuum scaling limit of critical G = A , D , E lattice models with positive spectral parameter u > 0 and Coxeter number g . Integrable seams are constructed by fusing blocks of elementary local face weights. The usual A -type fusions are labelled by the Kac labels ( r , s ) and are associated with the Verlinde fusion algebra. We introduce a new type of fusion in the two braid limits u → ±i ∞ associated with the graph fusion algebra, and labelled by nodes a , b ∊ G respectively. When combined with automorphisms, they lead to general integrable seams labelled by x = ( r , a , b , κ) ∊ ( A g −2 , H , H , 2 ) where H is the graph G for type I theories and its parent for type II theories. Identifying our construction labels with the conformal labels of Petkova and Zuber, we find that the integrable seams are in one-to-one correspondence with the conformal seams. The distinct seams are thus associated with the nodes of the Ocneanu quantum graph. The quantum symmetries and twisted partition functions are checked numerically for | G | ≤ 6. We also show, in the case of D 2ℓ , that the non-commutativity of the Ocneanu algebra of seams arises because the automorphisms do not commute with the fusions.

arXiv:hep-th/0210301•Journal of Physics A Mathematical and General•Mar 6, 2003•19 citations

Modular transformation and boundary states in logarithmic conformal field theory

Kawai, S, JOHN F. WHEATER

We study the $c=-2$ model of logarithmic conformal field theory in the presence of a boundary using symplectic fermions. We find boundary states with consistent modular properties. A peculiar feature of this model is that the vacuum representation corresponding to the identity operator is a sub-representation of a ``reducible but indecomposable'' larger representation. This leads to unusual properties, such as the failure of the Verlinde formula. Despite such complexities in the structure of modules, our results suggest that logarithmic conformal field theories admit bona fide boundary states.

•Mar 22, 2001•36 citations
We introduce ( p − 1) pseudocharacters in the space of (1, p ) model vacuum torus amplitudes to complete the distinguished basis in the 2 p -dimensional fusion algebra to a basis in the whole (3 p − 1)-dimensional space of torus amplitudes, and the structure constants in this basis are (not necessarily non-negative) integer numbers. We obtain a generalized Verlinde formula that gives these structure constants. In the context of theories with boundaries, we identify the space of vacuum torus amplitudes with the space of Ishibashi states. Then, we propose (3 p − 1) boundary states satisfying the Cardy condition.
The massless bosonic field compactified on the circle of rational R 2 is reexamined in the presense of boundaries. A particular class of models corresponding to R 2 = 1 2k is distinguished by demanding the existence of a consistent set of Newmann boundary states. The boundary states are constructed explicitly for these models and the fusion rules are derived from them. These are the ones prescribed by the Verlinde formula from the S-matrix of the theory. In addition, the extended symmetry algebra of these theories is constructed which is responsible for the rationality of these theories. Finally, the chiral space of these models is shown to split into a direct sum of irreducible modules of the extended symmetry algebra. 1
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.
The analogue of the charge-conjugation modular invariant for rational logarithmic conformal field theories is constructed. This is done by reconstructing the bulk spectrum from a simple boundary condition (the analogue of the Cardy ‘identity brane’). We apply the general method to the c1,p triplet models and reproduce the previously known bulk theory for p = 2 at c = −2. For general p we verify that the resulting partition functions are modular invariant. We also construct the complete set of 2p boundary states, and confirm that the identity brane from which we started indeed exists. As a by-product we obtain a logarithmic version of the Verlinde formula for the c1,p triplet models.
The analogue of the charge-conjugation modular invariant for rational logarithmic conformal field theories is constructed. This is done by reconstructing the bulk spectrum from a simple boundary condition (the analogue of the Cardy `identity brane'). We apply the general method to the c_1,p triplet models and reproduce the previously known bulk theory for p=2 at c=-2. For general p we verify that the resulting partition functions are modular invariant. We also construct the complete set of 2p boundary states, and confirm that the identity brane from which we started indeed exists. As a by-product we obtain a logarithmic version of the Verlinde formula for the c_1,p triplet models.
We study integrable realizations of conformal twisted boundary conditions for s ℓ(2) unitary minimal models on a torus. These conformal field theories are realized as the continuum scaling limit of critical G = A , D , E lattice models with positive spectral parameter u > 0 and Coxeter number g . Integrable seams are constructed by fusing blocks of elementary local face weights. The usual A -type fusions are labelled by the Kac labels ( r , s ) and are associated with the Verlinde fusion algebra. We introduce a new type of fusion in the two braid limits u → ±i ∞ associated with the graph fusion algebra, and labelled by nodes a , b ∊ G respectively. When combined with automorphisms, they lead to general integrable seams labelled by x = ( r , a , b , κ) ∊ ( A g −2 , H , H , 2 ) where H is the graph G for type I theories and its parent for type II theories. Identifying our construction labels with the conformal labels of Petkova and Zuber, we find that the integrable seams are in one-to-one correspondence with the conformal seams. The distinct seams are thus associated with the nodes of the Ocneanu quantum graph. The quantum symmetries and twisted partition functions are checked numerically for | G | ≤ 6. We also show, in the case of D 2ℓ , that the non-commutativity of the Ocneanu algebra of seams arises because the automorphisms do not commute with the fusions.
We study the c=−2c=-2c=−2 model of logarithmic conformal field theory in the presence of a boundary using symplectic fermions. We find boundary states with consistent modular properties. A peculiar feature of this model is that the vacuum representation corresponding to the identity operator is a sub-representation of a ``reducible but indecomposable'' larger representation. This leads to unusual properties, such as the failure of the Verlinde formula. Despite such complexities in the structure of modules, our results suggest that logarithmic conformal field theories admit bona fide boundary states.