780 research outputs found
Integrable anyon chains: from fusion rules to face models to effective field theories
Starting from the fusion rules for the algebra we construct
one-dimensional lattice models of interacting anyons with commuting transfer
matrices of `interactions round the face' (IRF) type. The conserved topological
charges of the anyon chain are recovered from the transfer matrices in the
limit of large spectral parameter. The properties of the models in the
thermodynamic limit and the low energy excitations are studied using Bethe
ansatz methods. Two of the anyon models are critical at zero temperature. From
the analysis of the finite size spectrum we find that they are effectively
described by rational conformal field theories invariant under extensions of
the Virasoro algebra, namely and ,
respectively. The latter contains primaries with half and quarter spin. The
modular partition function and fusion rules are derived and found to be
consistent with the results for the lattice model.Comment: 43 pages, published versio
Boundary States, Extended Symmetry Algebra and Module Structure for certain Rational Torus Models
The massless bosonic field compactified on the circle of rational is
reexamined in the presense of boundaries. A particular class of models
corresponding to is distinguished by demanding the existence
of a consistent set of Newmann boundary states. The boundary states are
constructed explicitly for these models and the fusion rules are derived from
them. These are the ones prescribed by the Verlinde formula from the S-matrix
of the theory. In addition, the extended symmetry algebra of these theories is
constructed which is responsible for the rationality of these theories.
Finally, the chiral space of these models is shown to split into a direct sum
of irreducible modules of the extended symmetry algebra.Comment: 12 page
Three-leg correlations in the two component spanning tree on the upper half-plane
We present a detailed asymptotic analysis of correlation functions for the
two component spanning tree on the two-dimensional lattice when one component
contains three paths connecting vicinities of two fixed lattice sites at large
distance apart. We extend the known result for correlations on the plane to
the case of the upper half-plane with closed and open boundary conditions. We
found asymptotics of correlations for distance from the boundary to one of
the fixed lattice sites for the cases and .Comment: 16 pages, 5 figure
Higher string functions, higher-level Appell functions, and the logarithmic ^sl(2)_k/u(1) CFT model
We generalize the string functions C_{n,r}(tau) associated with the coset
^sl(2)_k/u(1) to higher string functions A_{n,r}(tau) and B_{n,r}(tau)
associated with the coset W(k)/u(1) of the W-algebra of the logarithmically
extended ^sl(2)_k conformal field model with positive integer k. The higher
string functions occur in decomposing W(k) characters with respect to level-k
theta and Appell functions and their derivatives (the characters are neither
quasiperiodic nor holomorphic, and therefore cannot decompose with respect to
only theta-functions). The decomposition coefficients, to be considered
``logarithmic parafermionic characters,'' are given by A_{n,r}(tau),
B_{n,r}(tau), C_{n,r}(tau), and by the triplet \mathscr{W}(p)-algebra
characters of the (p=k+2,1) logarithmic model. We study the properties of
A_{n,r} and B_{n,r}, which nontrivially generalize those of the classic string
functions C_{n,r}, and evaluate the modular group representation generated from
A_{n,r}(tau) and B_{n,r}(tau); its structure inherits some features of modular
transformations of the higher-level Appell functions and the associated
transcendental function Phi.Comment: 34 pages, amsart++, times. V2: references added; minor changes; some
nonsense in B.3.3. correcte
Wind on the boundary for the Abelian sandpile model
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.Comment: 26 pages, 4 figure
Infinite Symmetry in the Fractional Quantum Hall Effect
We have generalized recent results of Cappelli, Trugenberger and Zemba on the
integer quantum Hall effect constructing explicitly a for
the fractional quantum Hall effect such that the negative modes annihilate the
Laughlin wave functions. This generalization has a nice interpretation in
Jain's composite fermion theory. Furthermore, for these models we have
calculated the wave functions of the edge excitations viewing them as area
preserving deformations of an incompressible quantum droplet, and have shown
that the is the underlying symmetry of the edge
excitations in the fractional quantum Hall effect. Finally, we have applied
this method to more general wave functions.Comment: 15pp. LaTeX, BONN-HE-93-2
Logarithmic two-point correlators in the Abelian sandpile model
We present the detailed calculations of the asymptotics of two-site
correlation functions for height variables in the two-dimensional Abelian
sandpile model. By using combinatorial methods for the enumeration of spanning
trees, we extend the well-known result for the correlation of minimal heights to for
height values . These results confirm the dominant logarithmic
behaviour for
large , predicted by logarithmic conformal field theory based on field
identifications obtained previously. We obtain, from our lattice calculations,
the explicit values for the coefficients and (the latter are new).Comment: 28 page
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.Comment: 35 pages, 2 figures; v2: minor corrections, version to appear in
J.Phys.
Bits and Pieces in Logarithmic Conformal Field Theory
These are notes of my lectures held at the first School & Workshop on
Logarithmic Conformal Field Theory and its Applications, September 2001 in
Tehran, Iran.
These notes cover only selected parts of the by now quite extensive knowledge
on logarithmic conformal field theories. In particular, I discuss the proper
generalization of null vectors towards the logarithmic case, and how these can
be used to compute correlation functions. My other main topic is modular
invariance, where I discuss the problem of the generalization of characters in
the case of indecomposable representations, a proposal for a Verlinde formula
for fusion rules and identities relating the partition functions of logarithmic
conformal field theories to such of well known ordinary conformal field
theories.
These two main topics are complemented by some remarks on ghost systems, the
Haldane-Rezayi fractional quantum Hall state, and the relation of these two to
the logarithmic c=-2 theory.Comment: 91 pages, notes of lectures delivered at the first School and
Workshop on Logarithmic Conformal Field Theory and its Applications, Tehran,
September 2001. Amendments in Introductio
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