Logarithmic Conformal Field Theory Research Articles

AdS3/LCFT2 – Correlators in new massive gravity

We calculate 2-point correlators for New Massive Gravity at the chiral point and find that they behave precisely as those of a logarithmic conformal field theory, which is characterized in addition to the central charges cL=cR=0 by ‘new anomalies’bL=bR=−σ12ℓGN, where σ is the sign of the Einstein–Hilbert term, ℓ the AdS radius and GN Newton's constant.

A Heisenberg Double Addition to the Logarithmic Kazhdan–Lusztig Duality

For a Hopf algebra B, we endow the Heisenberg double H(B^*) with the structure of a module algebra over the Drinfeld double D(B). Based on this property, we propose that H(B^*) is to be the counterpart of the algebra of fields on the quantum-group side of the Kazhdan--Lusztig duality between logarithmic conformal field theories and quantum groups. As an example, we work out the case where B is the Taft Hopf algebra related to the U_qsl(2) quantum group that is Kazhdan--Lusztig-dual to (p,1) logarithmic conformal models. The corresponding pair (D(B),H(B^*)) is "truncated" to (U_qsl(2),H_qsl(2)), where H_qsl(2) is a U_qsl(2) module algebra that turns out to have the form H_qsl(2)=\oC_q[z,d]\tensor C[\lambda]/(\lambda^{2p}-1), where C_q[z,d] is the U_qsl(2)-module algebra with the relations z^p=0, d^p=0, and d z = q-q^{-1} + q^{-2} zd.

On staggered indecomposable Virasoro modules

In this article, certain indecomposable Virasoro modules are studied. Specifically, the Virasoro mode L0 is assumed to be nondiagonalizable, possessing Jordan blocks of rank 2. Moreover, the module is further assumed to have a highest weight submodule, the “left module,” and that the quotient by this submodule yields another highest weight module, the “right module.” Such modules, which have been called staggered, have appeared repeatedly in the logarithmic conformal field theory literature, but their theory has not been explored in full generality. Here, such a theory is developed for the Virasoro algebra using rather elementary techniques. The focus centers on two different but related questions typically encountered in practical studies: How can one identify a given staggered module, and how can one demonstrate the existence of a proposed staggered module. Given just the values of the highest weights of the left and right modules, themselves subject to simple necessary conditions, invariants are defined which together with the knowledge of the left and right modules uniquely identify a staggered module. The possible values of these invariants form a vector space of dimension 0, 1, or 2, and the structures of the left and right modules limit the isomorphism classes of the corresponding staggered modules to an affine subspace (possibly empty). The number of invariants and affine restrictions is purely determined by the structures of the left and right modules. Moreover, in order to facilitate applications, the expressions for the invariants and restrictions are given by formulas as explicit as possible (they generally rely on expressions for Virasoro singular vectors). Finally, the text is liberally peppered throughout with examples illustrating the general concepts. These have been carefully chosen for their physical relevance or for the novel features they exhibit.

Decoherence and CPT Violation in a Stringy Model of Space-Time Foam

I discuss a model inspired from the string/brane framework, in which our Universe is represented as a three brane, propagating in a bulk space time punctured by D0-brane (D-particle) defects. As the D3-brane world moves in the bulk, the D-particles cross it, and from an effective observer on D3 the situation looks like a ``space-time foam'' with the defects ``flashing'' on and off (``D-particle foam''). The open strings, with their ends attached on the brane, which represent matter in this scenario, can interact with the D-particles on the D3-brane universe in a topologically non-trivial manner, involving splitting and capture of the strings by the D0-brane defects. Such processes are described by logarithmic conformal field theories on the world-sheet. Physically, they result in effective decoherence of the string matter on the D3 brane, and as a result, of CPT Violation, but of a type that implies an ill-defined nature of the effective CPT operator. Due to electric charge conservation, only electrically neutral (string) matter can exhibit such interactions with the D-particle foam. This may have unique, experimentally detectable, consequences for electrically-neutral entangled quantum matter states on the brane world, in particular the modification of the pertinent EPR Correlation of neutral mesons in a meson factory.

On a logarithmic deformation of the supersymmetric bc-system on curved manifolds

Frenkel et al. (“Instantons beyond topological theory. I,” e-print hep-th/0610149; “Instantons beyond topological theory. II,” e-print hep-th/0803.3302) claimed that a certain class of theories on compact Kähler manifolds and, in particular, the “gauged” supersymmetric bc-system on CP1 are modeled by logarithmic conformal field theories. We discuss that proposition for the bc-system on CP1 and thereby confirm their conjecture. Consequently, the property of being a logarithmic conformal field theory can be interpreted as an effect of gravity.

The Baxter Q operator of critical dense polymers

We consider critical dense polymers , corresponding to a logarithmic conformal field theory with central chargec = −2. An elegant decompositionof the Baxter Q operator is obtained in terms of a finite number of lattice integrals of motion.All local, nonlocal and dual nonlocal involutive charges are introduceddirectly on the lattice and their continuum limit is found to agree with theexpressions predicted by conformal field theory. A highly nontrivial operatorΨ(ν) is introduced on the lattice taking values in the Temperley–Lieb algebra. ThisΨ function provides a lattice discretization of the analogous function introduced by Bazhanov,Lukyanov and Zamolodchikov. It is also observed how the eigenvalues of theQ operator reproduce the well known spectral determinant for the harmonic oscillator in thecontinuum scaling limit.

Twist operator correlation functions in O(n) loop models

Using conformal field theoretic methods we calculate correlation functions of geometric observables in the loop representation of the O(n) model at the critical point. We focus on correlation functions containing twist operators, combining these with anchored loops, boundaries with SLE processes and with double SLE processes. We focus further upon n = 0, representing self-avoiding loops, which corresponds to a logarithmic conformal field theory (LCFT) with c = 0. In this limit the twist operator plays the role of a 0-weight indicator operator, which we verify by comparison with known examples. Using the additional conditions imposed by the twist operator null states, we derive a new explicit result for the probabilities that an SLE8/3 winds in various ways about two points in the upper half-plane, e.g. that the SLE passes to the left of both points. The collection of c = 0 logarithmic CFT operators that we use deriving the winding probabilities is novel, highlighting a potential incompatibility caused by the presence of two distinct logarithmic partners to the stress tensor within the theory. We argue that both partners do appear in the theory, one in the bulk and one on the boundary and that the incompatibility is resolved by restrictive bulk–boundary fusion rules.

A generalization of the Verlinde formula in logarithmic conformal field theory

We propose a generalized Verlinde formula associated with (1, p) logarithmic models of two-dimensional conformal field theories, which have applications in statistical physics problems such as the sand-pile model and phase transitions in polymers. This formula gives the integer structure constants in the whole (3p-1)-dimensional space of vacuum torus amplitudes in which the fusion algebra is a 2p-dimensional subalgebra.

The N = 1 Triplet Vertex Operator Superalgebras

We introduce a newfamily of C 2-cofinite N = 1 vertex operator superalgebras \({\mathcal{SW}(m)}\), m ≥ 1, which are natural super analogs of the triplet vertex algebra family \({\mathcal{W}(p)}\), p ≥ 2, important in logarithmic conformal field theory. We classify irreducible \({\mathcal{SW}(m)}\)-modules and discuss logarithmic modules. We also compute bosonic and fermionic formulas of irreducible \({\mathcal{SW}(m)}\) characters. Finally, we contemplate possible connections between the category of \({\mathcal{SW}(m)}\)-modules and the category of modules for the quantum group \({U^{small}_q(sl_2)}\), \({q = e^{\frac{2 \pi i}{2m+1}}}\), by focusing primarily on properties of characters and the Zhu’s algebra \({A(\mathcal{SW}(m))}\). This paper is a continuation of our paper Adv. Math. 217, no.6, 2664–2699 (2008).

Logarithmic conformal field theory approach to topologically massive gravity

We study the topologically massive gravity at the chiral point (chiral gravity) by using the logarithmic conformal field theory. Two new tensor fields of ψnew and X are introduced for a candidate of propagating physical field at the chiral point. However, we show that (ψnew,ψL) form a dipole ghost pair of unphysical fields and X is not a primary. This implies that there is no physically propagating degrees of freedom at the chiral point.

LOGARITHMIC KNOT INVARIANTS ARISING FROM RESTRICTED QUANTUM GROUPS

We construct knot invariants from the radical part of projective modules of the restricted quantum group [Formula: see text] at [Formula: see text], and we also show a relation between these invariants and the colored Alexander invariants. These projective modules are related to logarithmic conformal field theories.

On the percolation BCFT and the crossing probability of Watts

The logarithmic conformal field theory describing critical percolation is further explored using Watts' determination of the probability that there exists a cluster connecting both horizontal and vertical edges. The boundary condition changing operator which governs Watts' computation is identified with a primary field which does not fit naturally within the extended Kac table. Instead a “shifted” extended Kac table is shown to be relevant. Augmenting the previously known logarithmic theory based on Cardy's crossing probability by this field, a larger theory is obtained, in which new classes of indecomposable rank-2 modules are present. No rank-3 Jordan cells are yet observed. A highly non-trivial check of the identification of Watts' field is that no Gurarie–Ludwig-type inconsistencies are observed in this augmentation. The article concludes with an extended discussion of various topics related to extending these results including projectivity, boundary sectors and inconsistency loopholes.

Quasi-normal modes for logarithmic conformal field theory

We generalize the one-to one correspondence between quasi normal modes in 3- dimensional anti deSitter black holes and the poles of the retarded correlators in the boundary conformal field theory to include logarithmic operators in the latter. This analysis is carried out explicitly for the logarithmic mode in chiral topologically massive gravity.

Formula omitted]-extended logarithmic minimal models

We consider the continuum scaling limit of the infinite series of Yang–Baxter integrable logarithmic minimal models LM ( p , p ′ ) as ‘rational’ logarithmic conformal field theories with extended W symmetry. The representation content is found to consist of 6 p p ′ − 2 p − 2 p ′ W -indecomposable representations of which 2 p + 2 p ′ − 2 are of rank 1, 4 p p ′ − 2 p − 2 p ′ are of rank 2, while the remaining 2 ( p − 1 ) ( p ′ − 1 ) are of rank 3. We identify these representations with suitable limits of Yang–Baxter integrable boundary conditions on the lattice. The W -indecomposable rank-1 representations are all W -irreducible while we present a conjecture for the embedding patterns of the W -indecomposable rank-2 and -3 representations. The associated W -extended characters are all given explicitly and decompose as finite non-negative sums of W -irreducible characters. The latter correspond to W -irreducible subfactors and we find that there are 2 p p ′ + ( p − 1 ) ( p ′ − 1 ) / 2 of them. We present fermionic character expressions for some of the rank-2 and all of the rank-3 W -indecomposable representations. To distinguish between inequivalent W -indecomposable representations of identical characters, we introduce ‘refined’ characters carrying information also about the Jordan-cell content of a representation. Using a lattice implementation of fusion on a strip, we study the fusion rules for the W -indecomposable representations and find that they generate a closed fusion algebra, albeit one without identity for p > 1 . We present the complete set of fusion rules and interpret the closure of this fusion algebra as confirmation of the proposed extended symmetry. Finally, 2 p p ′ of the W -indecomposable representations are in fact W -projective representations and they generate a closed fusion subalgebra.

-extended fusion algebra of critical percolation

Two-dimensional critical percolation is the member of the infinite series of Yang–Baxter integrable logarithmic minimal models . We consider the continuum scaling limit of this lattice model as a ‘rational’ logarithmic conformal field theory with extended symmetry and use a lattice approach on a strip to study the fundamental fusion rules in this extended picture. We find that the representation content of the ensuing closed fusion algebra contains 26 -indecomposable representations with eight rank-1 representations, 14 rank-2 representations and four rank-3 representations. We identify these representations with suitable limits of Yang–Baxter integrable boundary conditions on the lattice and obtain their associated -extended characters. The latter decompose as finite non-negative sums of -irreducible characters of which 13 are required. Implementation of fusion on the lattice allows us to read off the fusion rules governing the fusion algebra of the 26 representations and to construct an explicit Cayley table. The closure of these representations among themselves under fusion is remarkable confirmation of the proposed extended symmetry.

Integrable boundary conditions and -extended fusion in the logarithmic minimal models

We consider the logarithmic minimal models as ‘rational’ logarithmic conformal field theories with extended symmetry. To make contact with the extended picture starting from the lattice, we identify 4p − 2 boundary conditions as specific limits of integrable boundary conditions of the underlying Yang–Baxter integrable lattice models. Specifically, we identify 2p integrable boundary conditions to match the 2p known irreducible -representations. These 2p extended representations naturally decompose into infinite sums of the irreducible Virasoro representations (r, s). A further 2p − 2 reducible yet indecomposable -representations of rank 2 are generated by fusion and these decompose as infinite sums of indecomposable rank-2 Virasoro representations. The fusion rules in the extended picture are deduced from the known fusion rules for the Virasoro representations of and are found to be in agreement with previous works. The closure of the fusion algebra on a finite number of representations in the extended picture is remarkable confirmation of the consistency of the lattice approach.

Towards the construction of local logarithmic conformal field theories

Although logarithmic conformal field theories (LCFTs) are known not to factorise many previous findings have only been formulated on their chiral halves. Making only mild and rather general assumptions on the structure of an chiral LCFT we deduce statements about its local non-chiral equivalent. Two methods are presented how to construct local representations as subrepresentations of the tensor product of chiral and anti-chiral Jordan cells. Furthermore we explore the assembly of generic non-chiral correlation functions from generic chiral and anti-chiral correlators. The constraint of locality is studied and the generality of our method is discussed.

Factorizable ribbon quantum groups in logarithmic conformal field theories

We review the properties of quantum groups occurring as the Kazhdan-Lusztig dual to logarithmic conformal field theory models. These quantum groups at even roots of unity are not quasitriangular but are factorizable and have a ribbon structure; the modular group representation on their center coincides with the representation on generalized characters of the chiral algebra in logarithmic conformal field models.

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 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.

Pair correlations in sandpile model: A check of logarithmic conformal field theory

We compute the correlations of two height variables in the two-dimensional Abelian sandpile model. We extend the known result for two minimal heights to the case when one of the heights is bigger than one. We find that the most dominant correlation logr/r4 exactly fits the prediction obtained within the logarithmic conformal approach.