Showing papers on "Logarithmic conformal field theory published in 2006"
TL;DR: The SL(2, ℤ)-representation π on the center of the restricted quantum group at the primitive 2pth root of unity is shown to be equivalent to the SL( 2, ↦)-representations on the extended characters of the logarithmic (1, p) conformal field theory model in this article.
Abstract: The SL(2, ℤ)-representation π on the center of the restricted quantum group at the primitive 2pth root of unity is shown to be equivalent to the SL(2, ℤ)-representation on the extended characters of the logarithmic (1, p) conformal field theory model. The multiplicative Jordan decomposition of the ribbon element determines the decomposition of π into a ``pointwise'' product of two commuting SL(2, ℤ)-representations, one of which restricts to the Grothendieck ring; this restriction is equivalent to the SL(2, ℤ)-representation on the (1, p)-characters, related to the fusion algebra via a nonsemisimple Verlinde formula. The Grothendieck ring of at the primitive 2pth root of unity is shown to coincide with the fusion algebra of the (1, p) logarithmic conformal field theory model. As a by-product, we derive q-binomial identities implied by the fusion algebra realized in the center of .
268 citations
TL;DR: In this paper, a logarithmic minimal model of the planar Temperley?Lieb algebra is constructed on the strip acting on link states and its associated Hamiltonian limits.
Abstract: Working in the dense loop representation, we use the planar Temperley?Lieb algebra to build integrable lattice models called logarithmic minimal models . Specifically, we construct Yang?Baxter integrable Temperley?Lieb models on the strip acting on link states and consider their associated Hamiltonian limits. These models and their associated representations of the Temperley?Lieb algebra are inherently non-local and not (time-reversal) symmetric. We argue that, in the continuum scaling limit, they yield logarithmic conformal field theories with central charges c = 1?(6(p?p')2/pp'), where p, p' = 1, 2, ... are coprime. The first few members of the principal series are critical dense polymers (m = 1, c = ?2), critical percolation (m = 2, c = 0) and the logarithmic Ising model (m = 3, c = 1/2). For the principal series, we find an infinite family of integrable and conformal boundary conditions organized in an extended Kac table with conformal weights ?r,s = (((m+1)r?ms)2?1)/4m(m+1), r, s = 1, 2, .... The associated conformal partition functions are given in terms of Virasoro characters of highest-weight representations. Individually, these characters decompose into a finite number of characters of irreducible representations. We show with examples how indecomposable representations arise from fusion.
261 citations
TL;DR: In this article, a logarithmic conformal field model that extends the (p, q ) Virasoro minimal models is defined as the kernel of the two minimal-model screening operators.
218 citations
TL;DR: In this article, the authors studied the representation category of the triplet W-algebra that is the symmetry of the (1, p) logarithmic conformal field theory model and proposed the equivalent category Cp of finite-dimensional representations of the restricted quantum group.
Abstract: To study the representation category of the triplet W-algebra \(\mathcal{W}\left( p \right)\) that is the symmetry of the (1, p) logarithmic conformal field theory model, we propose the equivalent category Cp of finite-dimensional representations of the restricted quantum group Ūqsl(2) at \(\mathfrak{q} = e^{{{i\pi } \mathord{\left/ {\vphantom {{i\pi } p}} \right \kern-
ulldelimiterspace} p}} \) We fully describe the category Cp by classifying all indecomposable representations These are exhausted by projective modules and three series of representations that are essentially described by indecomposable representations of the Kronecker quiver The equivalence of the \(\mathcal{W}\left( p \right)\)-and Ūqsl(2)-representation categories is conjectured for all p = 2 and proved for p = 2 The implications include identifying the quantum group center with the logarithmic conformal field theory center and the universal R-matrix with the braiding matrix
183 citations
TL;DR: In this paper, Coulomb gas methods were used to derive an explicit form for the scaling limit of the partition function of the critical O(n) model on an annulus, with free boundary conditions, as a function of its modulus.
Abstract: We use Coulomb gas methods to derive an explicit form for the scaling limit of the partition function of the critical O(n) model on an annulus, with free boundary conditions, as a function of its modulus. This correctly takes into account the magnetic charge asymmetry and the decoupling of the null states. It agrees with an earlier conjecture based on Bethe ansatz and quantum group symmetry, and with all known results for special values of n. It gives new formulae for percolation (the probability that a cluster connects the two opposite boundaries) and for self-avoiding loops (the partition function for a single loop wrapping non-trivially around the annulus.) The limit n→0 also gives explicit examples of partition functions in logarithmic conformal field theory.
40 citations
TL;DR: In this article, Coulomb gas methods were used to obtain an explicit form for the scaling limit of the partition function of the critical O(n) model on an annulus, with free boundary conditions, as a function of its modulus.
Abstract: We use Coulomb gas methods to propose an explicit form for the scaling limit of the partition function of the critical O(n) model on an annulus, with free boundary conditions, as a function of its modulus. This correctly takes into account the magnetic charge asymmetry and the decoupling of the null states. It agrees with an earlier conjecture based on Bethe ansatz and quantum group symmetry, and with all known results for special values of n. It gives new formulae for percolation (the probability that a cluster connects the two opposite boundaries) and for self-avoiding loops (the partition function for a single loop wrapping non-trivially around the annulus.) The limit n->0 also gives explicit examples of partition functions in logarithmic conformal field theory.
27 citations
TL;DR: In this paper, a particular type of logarithmic extension of SL ( 2, R ) Wess-Zumino-Witten models based on affine Jordan cells constructed as multiplets of quasi-primary fields organized in indecomposable representations of the Lie algebra is studied.
21 citations
TL;DR: In this paper, Zamolodchikov's method is used to derive conditions for the perturbing operator such that the perturbed model be integrable, and the integrability of the perturbation model arises out of perturbations of the c=−2 logarithmic conformal field theory.
3 citations
TL;DR: In this article, one of the most profound and foundational aspects of the subject is reviewed in detail, namely that of an operator algebra of the theory, which is then demonstrated in some modern applications, first stochastic Loewner evolution and then logarithmic conformal field theory.
Abstract: Conformal field theory in two dimensions has, over the years, been an extremely useful tool for a variety of physical and mathematical problems. In this paper, perhaps one of the most profound and foundational aspects of the subject is reviewed in detail, namely that of an operator algebra of the theory. This aspect is then demonstrated in some modern applications, first stochastic Loewner evolution and then logarithmic conformal field theory.
2 citations
TL;DR: In this article, universal bracket relations for logarithmic mode algebras are given for a vertex operator algebra and a vertex algebra for a conformal field theory at the vacuum sector.
Abstract: It is an open question whether or not it is possible to generalize the definition of a vertex operator algebra to treat logarithmic conformal field theory already at the vacuum sector With this task in mind, universal bracket relations for logarithmic mode algebras are given
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 it, 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 energy-momentum 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.