Showing papers on "Logarithmic conformal field theory published in 2007"
TL;DR: Read and Saleur as mentioned in this paper derived the boundary CFT of spin-1/2 chains with supersymmetry algebras with open (or free) boundary conditions in all cases.
173 citations
TL;DR: In this paper, the smallest deformation of the minimal model M ( 2, 3 ) that can accommodate Cardy's derivation of the percolation crossing probability is presented, which leads to a consistent logarithmic conformal field theory at c = 0.
128 citations
TL;DR: In this paper, an explicit construction of a family of W(2,2p−1) modules, which decompose as direct sum of simple Virasoro algebra modules, was obtained for every p⩾2.
Abstract: For every p⩾2, we obtained an explicit construction of a family of W(2,2p−1) modules, which decompose as direct sum of simple Virasoro algebra modules. Furthermore, we classified all irreducible self-dual W(2,2p−1) modules, we described their internal structure, and computed their graded dimensions. In addition, we constructed certain hidden logarithmic intertwining operators among two ordinary and one logarithmic W(2,2p−1) modules. This work, in particular, gives a mathematically precise formulation and interpretation of what physicists have been referring to as “logarithmic conformal field theory” of central charge cp,1=1−[6(p−1)2∕p], p⩾2. Our explicit construction can be easily applied for computations of correlation functions. Techniques from this paper can be used to study the triplet vertex operator algebra W(2,(2p−1)3) and other logarithmic models.
111 citations
TL;DR: In this paper, the solution of WZNW models on general type I supergroups is reduced to those defined on the bosonic subgroup, and the latter is shown to be non-diagonalizable.
Abstract: Extending our earlier work on PSL(2|2), we explain how to reduce the solution of WZNW models on general type I supergroups to those defined on the bosonic subgroup. The new analysis covers in particular the supergroups GL(M|N) along with several close relatives such as PSL(N|N), certain Poincare supergroups and the series OSP(2|2N). The technical foundation for this remarkable progress is a special Feigin-Fuchs type representation which allows to keep the bosonic symmetry manifest instead of reducing it to free fields. In preparation for the field theory analysis, we shall exploit a minisuperspace analogue of the resulting free fermion construction to deduce the spectrum of the Laplacian on type I supergroups. The latter is shown to be non-diagonalizable. After lifting these results to the full WZNW model, we address various issues of the field theory, including its modular invariance and the computation of correlation functions. In agreement with previous findings, supergroup WZNW models allow to study chiral and non-chiral aspects of logarithmic conformal field theory within a geometric framework. We shall briefly indicate how insights from WZNW models carry over to non-geometric examples, such as e.g. the (p) triplet models.
102 citations
TL;DR: In this paper, a new family of C_2-cofinite N = 1 vertex operator superalgebras SW(m), which are natural super analogs of the triplet vertex algebra family W(p), are introduced.
Abstract: We introduce a new family of C_2-cofinite N=1 vertex operator superalgebras SW(m), $m \geq 1$, which are natural super analogs of the triplet vertex algebra family W(p), $p \geq 2$, important in logarithmic conformal field theory. We classify irreducible SW(m)-modules and discuss logarithmic modules. We also compute bosonic and fermionic formulas of irreducible SW(m) characters. Finally, we contemplate possible connections between the category of 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(SW(m)). This paper is a continuation of arXiv:0707.1857.
43 citations
TL;DR: In this article, the authors presented fermionic quasi-particle sum representations for all characters of the logarithmic conformal field theory models with central charge c p, 1, p ⩾ 2, and suggested a physical interpretation.
32 citations
TL;DR: In this article, the authors review the properties of quantum groups occurring as Kazhdan-Lusztig dual to logarithmic conformal field theory models and show that these groups at even roots of unity are not quasitriangular but are factorizable and have a ribbon structure.
Abstract: We review the properties of quantum groups occurring as 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.
31 citations
TL;DR: In this paper, the authors investigated the two-dimensional Abelian sandpile model in terms of a logarithmic conformal field theory with central charge c = −2, by introducing two new boundary conditions.
Abstract: We continue our investigation of the two-dimensional Abelian sandpile model in terms of a logarithmic conformal field theory with central charge c = −2, by introducing two new boundary conditions. These have two unusual features: they carry an intrinsic orientation, and, more strangely, they cannot be imposed uniformly on a whole boundary (like the edge of a cylinder). They lead to seven new boundary condition changing fields, some of them being in highest weight representations (weights −1/8, 0 and 3/8), some others belonging to indecomposable representations with rank 2 Jordan cells (lowest weights 0 and 1). Their fusion algebra appears to be in full agreement with the fusion rules conjectured by Gaberdiel and Kausch.
27 citations
TL;DR: In this article, it was shown how a limiting procedure of conformal field theories may result in logarithmic conformal fields with Jordan cells of arbitrary rank, and the general construction of minimal models in conformal Field Theory with rank-two Jordan cells was discussed.
Abstract: It is discussed how a limiting procedure of conformal field theories may result in logarithmic conformal field theories with Jordan cells of arbitrary rank. This extends our work on rank-two Jordan cells. We also consider the limits of certain three-point functions and find that they are compatible with known results. The general construction is illustrated by logarithmic limits of minimal models in conformal field theory. Characters of quasirational representations are found to emerge as the limits of the associated irreducible Virasoro characters.
26 citations
TL;DR: In this paper, the finite-size corrections of the dimer model on infinity x N square lattice with two different boundary conditions: free and periodic, were studied and explained in the framework of the c = -2 logarithmic conformal field theory.
Abstract: We study the finite-size corrections of the dimer model on infinity x N square lattice with two different boundary conditions: free and periodic. We find that the finite-size corrections depend in a crucial way on the parity of N, and show that, because of certain non-local features present in the model, a change of parity of N induces a change of boundary condition. Taking a careful account of this, these unusual finite-size behaviours can be fully explained in the framework of the c = -2 logarithmic conformal field theory.
17 citations
TL;DR: For positive integers p = k + 2, a logarithmic extension of the conformal field theory of integrable representations by taking the kernel of two fermionic screening operators in a butterfly resolution of a three-boson realization of.
Abstract: For positive integers p = k + 2, we construct a logarithmic extension of the
$$\widehat{s\ell }(2)_k $$
conformal field theory of integrable representations by taking the kernel of two fermionic screening operators in a butterfly resolution of a three-boson realization of
$$\widehat{s\ell }(2)_k $$
. The currents W−(z) and W+(z) of a W-algebra acting in the kernel are determined by a highest-weight state of dimension 4p − 2 and charge 2p − 1 and by a (θ=1)-twisted highest-weight state of the same dimension 4p − 2 and opposite charge −2p+1. We construct 2p W-algebra representations, evaluate their characters, and show that together with the p−1 integrable representation characters, they generate a modular group representation whose structure is described as a deformation of the (9p−3)-dimensional representation R
p+1⊕ℂ2⊗R
p+1ʕR
p−1⊕ℂ2
R
p−1⊕ℂ3
R
p−1, where R
p−1 is the SL(2, ℤ)-representation on
$$\widehat{s\ell }(2)_k $$
integrable-representation characters and R
p+1 is a (p+1)-dimensional SL(2, ℤ)-representation known from the logarithmic (p, 1) model. The dimension 9p − 3 is conjecturally the dimension of the space of torus amplitudes, and the ℂn with n = 2 and 3 suggest the Jordan cell sizes in indecomposable W-algebra modules. We show that under Hamiltonian reduction, the W-algebra currents map into the currents of the triplet W-algebra of the logarithmic (p, 1) model.
TL;DR: In this article, the characters of the logarithmic conformal field theory ck,1 admit fermionic representations labelled by the Lie algebra Dk. In this paper, we provide a simple analytic proof of this conjecture.
Abstract: In a recent paper, Flohr, Grabow and Koehn conjectured that the characters of the logarithmic conformal field theory ck,1 admit fermionic representations labelled by the Lie algebra Dk. In this paper, we provide a simple analytic proof of this conjecture.
TL;DR: In this article, the spectrum of the Laplacian on type I supergroups has been shown to be non-diagonalizable, and a special Feigin-Fuchs type representation has been proposed for the solution of WZNW models.
Abstract: Extending our earlier work on PSL(2|2), we explain how to reduce the solution of WZNW models on general type I supergroups to those defined on the bosonic subgroup. The new analysis covers in particular the supergroups GL(M|N) along with several close relatives such as PSL(N|N), certain Poincare supergroups and the series OSP(2|2N). This remarkable progress relies on the use of a special Feigin-Fuchs type representation. In preparation for the field theory analysis, we shall exploit a minisuperspace analogue of a free fermion construction to deduce the spectrum of the Laplacian on type I supergroups. The latter is shown to be non-diagonalizable. After lifting these results to the full WZNW model, we address various issues of the field theory, including its modular invariance and the computation of correlation functions. In agreement with previous findings, supergroup WZNW models allow to study chiral and non-chiral aspects of logarithmic conformal field theory within a geometric framework. We shall briefly indicate how insights from WZNW models carry over to non-geometric examples, such as e.g. the W(p) triplet models.
TL;DR: In this paper, the authors give a mathematically precise interpretation of the logarithmic conformal field theory of central charge, which can be easily applied for computations of correlation functions.
Abstract: For every $p \geq 2$, we obtained an explicit construction of a family of $\mathcal{W}(2,2p-1)$-modules, which decompose as direct sum of simple Virasoro algebra modules. Furthermore, we classified all irreducible self-dual $\mathcal{W}(2,2p-1)$-modules, we described their internal structure, and computed their graded dimensions. In addition, we constructed certain hidden logarithmic intertwining operators among two ordinary and one logarithmic $\mathcal{W}(2,2p-1)$-modules. This work, in particular, gives a mathematically precise formulation and interpretation of what physicists have been referring to as "logarithmic conformal field theory" of central charge $c_{p,1}=1-\frac{6(p-1)^2}{p}, p \geq 2$. Our explicit construction can be easily applied for computations of correlation functions. Techniques from this paper can be used to study the triplet vertex operator algebra $\mathcal{W}(2,(2p-1)^3)$ and other logarithmic models.
TL;DR: In this article, the authors provided a simple analytic proof of this conjecture and showed that the characters of the logarithmic conformal field theory c = 1-6(k-1)^2/k admit fermionic representations labelled by the Lie algebra D_k.
Abstract: In a recent paper Flohr, Grabow and Koehn conjectured that the characters of the logarithmic conformal field theory c_{k,1}, of central charge c=1-6(k-1)^2/k, admit fermionic representations labelled by the Lie algebra D_k. In this note we provide a simple analytic proof of this conjecture.
TL;DR: In this paper, the authors investigated some aspects of the c=-2 logarithmic conformal field theory, including the various representations related to this theory, the structures which come out of the Zhu algebra and the W algebra related to the theory, and the important role of zero modes in this model.
Abstract: We investigate some aspects of the c=-2 logarithmic conformal field theory. These include the various representations related to this theory, the structures which come out of the Zhu algebra and the W algebra related to this theory. We try to find the fermionic representations of all of the fields in the extended Kac table especially for the untwisted sector case. In addition, we calculate the various OPEs of the fields, especially the energymomentum tensor. Moreover, we investigate the important role of the zero modes in this model. We close the paper by considering the perturbations of this theory and their relationship to integrable models and generalization of Zamolodchikov’s c−theorem.
TL;DR: In this paper, the authors investigated the two-dimensional Abelian sandpile model in terms of a logarithmic conformal field theory with central charge c=-2, by introducing two new boundary conditions.
Abstract: We continue our investigation of the two-dimensional Abelian sandpile model in terms of a logarithmic conformal field theory with central charge c=-2, by introducing two new boundary conditions. These have two unusual features: they carry an intrinsic orientation, and, more strangely, they cannot be imposed uniformly on a whole boundary (like the edge of a cylinder). They lead to seven new boundary condition changing fields, some of them being in highest weight representations (weights -1/8, 0 and 3/8), some others belonging to indecomposable representations with rank 2 Jordan cells (lowest weights 0 and 1). Their fusion algebra appears to be in full agreement with the fusion rules conjectured by Gaberdiel and Kausch.
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TL;DR: In this paper, the authors construct the representation of double affine hecke algebra whose symmetrization gives the center of the quantum group U_q(sl(2)) and by Kazhdan-Lusztig duality the Verlinde algebra of (1,p) models of logarithmic conformal field theory.
Abstract: We construct the representation of Double Affine Hecke Algebra whose symmetrization gives the center of the quantum group U_q(sl(2)) and by Kazhdan--Lusztig duality the Verlinde algebra of (1,p) models of logarithmic conformal field theory.