36 research outputs found
Two-dimensional spanning webs as (1,2) logarithmic minimal model
A lattice model of critical spanning webs is considered for the finite
cylinder geometry. Due to the presence of cycles, the model is a generalization
of the known spanning tree model which belongs to the class of logarithmic
theories with central charge . We show that in the scaling limit the
universal part of the partition function for closed boundary conditions at both
edges of the cylinder coincides with the character of symplectic fermions with
periodic boundary conditions and for open boundary at one edge and closed at
the other coincides with the character of symplectic fermions with antiperiodic
boundary conditions.Comment: 21 pages, 3 figure
Kazhdan--Lusztig-dual quantum group for logarithmic extensions of Virasoro minimal models
We derive and study a quantum group g(p,q) that is Kazhdan--Lusztig-dual to
the W-algebra W(p,q) of the logarithmic (p,q) conformal field theory model. The
algebra W(p,q) is generated by two currents and of dimension
(2p-1)(2q-1), and the energy--momentum tensor T(z). The two currents generate a
vertex-operator ideal with the property that the quotient W(p,q)/R is the
vertex-operator algebra of the (p,q) Virasoro minimal model. The number (2 p q)
of irreducible g(p,q)-representations is the same as the number of irreducible
W(p,q)-representations on which acts nontrivially. We find the center of
g(p,q) and show that the modular group representation on it is equivalent to
the modular group representation on the W(p,q) characters and
``pseudocharacters.'' The factorization of the g(p,q) ribbon element leads to a
factorization of the modular group representation on the center. We also find
the g(p,q) Grothendieck ring, which is presumably the ``logarithmic'' fusion of
the (p,q) model.Comment: 52pp., AMSLaTeX++. half a dozen minor inaccuracies (cross-refs etc)
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Integrals of Motion for Critical Dense Polymers and Symplectic Fermions
We consider critical dense polymers . We obtain for this model
the eigenvalues of the local integrals of motion of the underlying Conformal
Field Theory by means of Thermodynamic Bethe Ansatz. We give a detailed
description of the relation between this model and Symplectic Fermions
including the indecomposable structure of the transfer matrix. Integrals of
motion are defined directly on the lattice in terms of the Temperley Lieb
Algebra and their eigenvalues are obtained and expressed as an infinite sum of
the eigenvalues of the continuum integrals of motion. An elegant decomposition
of the transfer matrix in terms of a finite number of lattice integrals of
motion is obtained thus providing a reason for their introduction.Comment: 53 pages, version accepted for publishing on JSTA
The Baxter Q Operator of Critical Dense Polymers
We consider critical dense polymers , corresponding to a
logarithmic conformal field theory with central charge . An elegant
decomposition of the Baxter operator is obtained in terms of a finite
number of lattice integrals of motion. All local, non local and dual non local
involutive charges are introduced directly on the lattice and their continuum
limit is found to agree with the expressions predicted by conformal field
theory. A highly non trivial operator is introduced on the lattice
taking values in the Temperley Lieb Algebra. This function provides a
lattice discretization of the analogous function introduced by Bazhanov,
Lukyanov and Zamolodchikov. It is also observed how the eigenvalues of the
operator reproduce the well known spectral determinant for the harmonic
oscillator in the continuum scaling limit.Comment: improved version, accepted for publishing on JSTA
Solvable Critical Dense Polymers
A lattice model of critical dense polymers is solved exactly for finite
strips. The model is the first member of the principal series of the recently
introduced logarithmic minimal models. The key to the solution is a functional
equation in the form of an inversion identity satisfied by the commuting
double-row transfer matrices. This is established directly in the planar
Temperley-Lieb algebra and holds independently of the space of link states on
which the transfer matrices act. Different sectors are obtained by acting on
link states with s-1 defects where s=1,2,3,... is an extended Kac label. The
bulk and boundary free energies and finite-size corrections are obtained from
the Euler-Maclaurin formula. The eigenvalues of the transfer matrix are
classified by the physical combinatorics of the patterns of zeros in the
complex spectral-parameter plane. This yields a selection rule for the
physically relevant solutions to the inversion identity and explicit finitized
characters for the associated quasi-rational representations. In particular, in
the scaling limit, we confirm the central charge c=-2 and conformal weights
Delta_s=((2-s)^2-1)/8 for s=1,2,3,.... We also discuss a diagrammatic
implementation of fusion and show with examples how indecomposable
representations arise. We examine the structure of these representations and
present a conjecture for the general fusion rules within our framework.Comment: 35 pages, v2: comments and references adde
Gepner-like models and Landau-Ginzburg/sigma-model correspondence
The Gepner-like models of -type is considered. When is multiple
of the elliptic genus and the Euler characteristic is calculated. Using
free-field representation we relate these models with -models on
hypersurfaces in the total space of anticanonical bundle over the projective
space
The logarithmic triplet theory with boundary
The boundary theory for the c=-2 triplet model is investigated in detail. In
particular, we show that there are four different boundary conditions that
preserve the triplet algebra, and check the consistency of the corresponding
boundary operators by constructing their OPE coefficients explicitly. We also
compute the correlation functions of two bulk fields in the presence of a
boundary, and verify that they are consistent with factorisation.Comment: 43 pages, LaTeX; v2: references added, typos corrected, footnote 4
adde