From boundary to bulk in logarithmic CFT

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.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:0707.0388Journal of Physics A Mathematical and TheoreticalFeb 5, 200866 citations

A modular invariant bulk theory for the \boldsymbol{c=0} triplet model

Matthias R. Gaberdiel B, Ingo Runkel B, Simon Wood

A proposal for the bulk space of the logarithmic -triplet model at central charge zero is made. The construction is based on the idea that one may reconstruct the bulk theory of a rational conformal field theory from its boundary theory. The resulting bulk space is a quotient of the direct sum of projective representations, which is isomorphic, as a vector space, to the direct sum of tensor products of the irreducible representations with their projective covers. As a consistency check of our analysis, we show that the partition function of the bulk theory is modular invariant, and that the boundary state analysis is compatible with the proposed annulus partition functions of this model.A proposal for the bulk space of the logarithmic -triplet model at central charge zero is made. The construction is based on the idea that one may reconstruct the bulk theory of a rational conformal field theory from its boundary theory. The resulting bulk space is a quotient of the direct sum of projective representations, which is isomorphic, as a vector space, to the direct sum of tensor products of the irreducible representations with their projective covers. As a consistency check of our analysis, we show that the partition function of the bulk theory is modular invariant, and that the boundary state analysis is compatible with the proposed annulus partition functions of this model.

arXiv:1008.0082Journal of Physics A Mathematical and TheoreticalNov 30, 201045 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.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:1712.01922Nuclear Physics BMar 28, 20189 citations

BOUNDARIES, CROSSCAPS AND SIMPLE CURRENTS

Jürgen Fuchs, L.R. Huiszoon, A.N. Schellekens +2 more

Universal formulas for the boundary and crosscap coefficients are presented, which are valid for all symmetric simple current modifications of the charge conjugation invariant of any rational conformal field theory. 1 1. Boundaries and crosscaps In this letter we report progress on the problem of finding boundaries and crosscaps for all conformal field theories that can be obtained as simple current invariants of a given rational conformal field theory (RCFT). This line of research was initiated by Cardy [1], who obtained the (bulk symmetry preserving) boundaries in the case where the torus partition function is the charge conjugation invariant. Later, in [2] the corresponding crosscap coefficients were obtained. In a series of subsequent papers [3,4,5,6,7,8,9,10,11,12,13] various more general situations were considered. In particular one may choose a different Klein bottle projection and a different modular invariant partition function (MIPF) χiZij ¯χ j for the bulk theory. The basic data one would like to determine are the set {m} of Ishibashi labels, the set {a} of boundary labels, a matrix Bm,a of boundary coefficients and a vector Γm of crosscap coefficients.Universal formulas for the boundary and crosscap coefficients are presented, which are valid for all symmetric simple current modifications of the charge conjugation invariant of any rational conformal field theory. 1 1. Boundaries and crosscaps In this letter we report progress on the problem of finding boundaries and crosscaps for all conformal field theories that can be obtained as simple current invariants of a given rational conformal field theory (RCFT). This line of research was initiated by Cardy [1], who obtained the (bulk symmetry preserving) boundaries in the case where the torus partition function is the charge conjugation invariant. Later, in [2] the corresponding crosscap coefficients were obtained. In a series of subsequent papers [3,4,5,6,7,8,9,10,11,12,13] various more general situations were considered. In particular one may choose a different Klein bottle projection and a different modular invariant partition function (MIPF) χiZij ¯χ j for the bulk theory. The basic data one would like to determine are the set {m} of Ishibashi labels, the set {a} of boundary labels, a matrix Bm,a of boundary coefficients and a vector Γm of crosscap coefficients.

Jan 1, 200085 citations

BOUNDARY STATES AND BROKEN BULK SYMMETRIES IN WAr MINIMAL MODELS

ALEXANDRE F. CALDEIRA, J. F. WHEATER

We study the boundary states of (p', p) rational conformal field theories having a W symmetry of the type A(r) using the multi-component free-field formalism. The classification of primary fields for these models given in the literature is shown to be incomplete; we give the correct classification by demanding modular covariance and show that the resulting modular S matrix satisfies all the necessary conditions. Basis states satisfying the boundary conditions are found in the form of coherent states and as expected we find that W violating states can be found for all these models. We construct consistent physical boundary for all the rank 2 (p+1, p) models (of which the already known case of the 3-state Potts model is the simplest example) and find that the W violating sector possesses a direct analogue of the Verlinde formula.We study the boundary states of (p', p) rational conformal field theories having a W symmetry of the type A(r) using the multi-component free-field formalism. The classification of primary fields for these models given in the literature is shown to be incomplete; we give the correct classification by demanding modular covariance and show that the resulting modular S matrix satisfies all the necessary conditions. Basis states satisfying the boundary conditions are found in the form of coherent states and as expected we find that W violating states can be found for all these models. We construct consistent physical boundary for all the rank 2 (p+1, p) models (of which the already known case of the 3-state Potts model is the simplest example) and find that the W violating sector possesses a direct analogue of the Verlinde formula.

arXiv:hep-th/0404052WORLD SCIENTIFIC eBooksFeb 1, 20055 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.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, 200136 citations

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.