350 research outputs found
Half-lattice paths and Virasoro characters
We first briefly review the role of lattice paths in the derivation of
fermionic expressions for the M(p,p') minimal model characters of the Virasoro
Lie algebra. We then focus on the recently introduced half-lattice paths for
the M(p,2p+/-1) characters, reformulating them in such a way that the two cases
may be treated uniformly. That the generating functions of these half-lattice
paths are indeed M(p,2p+/-1) characters is proved by describing weight
preserving bijections between them and the corresponding RSOS lattice paths.
Here, the M(p,2p-1) case is derived for the first time. We then apply the
methods of Bressoud and Warnaar to these half-lattice paths to derive fermionic
expressions for the Virasoro characters X^{p,2p+/-1}_{1,2} that differ from
those obtained from the RSOS paths.
This work is an extension of that presented by the third author at the "7th
International Conference on Lattice Path Combinatorics and Applications",
Siena, Italy, July 2010.Comment: 22 page
A quartet of fermionic expressions for Virasoro characters via half-lattice paths
We derive new fermionic expressions for the characters of the Virasoro
minimal models by analysing the recently introduced half-lattice
paths. These fermionic expressions display a quasiparticle formulation
characteristic of the and integrable perturbations.
We find that they arise by imposing a simple restriction on the RSOS
quasiparticle states of the unitary models . In fact, four fermionic
expressions are obtained for each generating function of half-lattice paths of
finite length , and these lead to four distinct expressions for most
characters . These are direct analogues of Melzer's
expressions for , and their proof entails revisiting, reworking and
refining a proof of Melzer's identities which used combinatorial transforms on
lattice paths.
We also derive a bosonic version of the generating functions of length
half-lattice paths, this expression being notable in that it involves
-trinomial coefficients. Taking the limit shows that the
generating functions for infinite length half-lattice paths are indeed the
Virasoro characters .Comment: 29 pages. v2: minor improvements, references adde
Off-critical local height probabilities on a plane and critical partition functions on a cylinder
We compute off-critical local height probabilities in regime-III restricted
solid-on-solid models in a -quadrant spiral geometry, with periodic
boundary conditions in the angular direction, and fixed boundary conditions in
the radial direction, as a function of , the winding number of the spiral,
and , the departure from criticality of the model, and observe that the
result depends only on the product . In the limit ,
, such that is finite, we recover the
off-critical local height probability on a plane, -away from
criticality. In the limit , , such
that is finite, and following a conformal transformation,
we obtain a critical partition function on a cylinder of aspect-ratio .
We conclude that the off-critical local height probability on a plane,
-away from criticality, is equal to a critical partition function on a
cylinder of aspect-ratio , in agreement with a result of Saleur and
Bauer.Comment: 28 page
CFT, BCFT, ADE and all that
These pedagogical lectures present some material, classical or more recent,
on (Rational) Conformal Field Theories and their general setting ``in the
bulk'' or in the presence of a boundary. Two well posed problems are the
classification of modular invariant partition functions and the determination
of boundary conditions consistent with conformal invariance. It is shown why
the two problems are intimately connected and how graphs -ADE Dynkin diagrams
and their generalizations- appear in a natural way.Comment: Lectures at Bariloche, Argentina, January 2000. 36 pages, 4 figure
Logarithmic Superconformal Minimal Models
The higher fusion level logarithmic minimal models LM(P,P';n) have recently
been constructed as the diagonal GKO cosets (A_1^{(1)})_k oplus (A_1^{(1)})_n /
(A_1^{(1)})_{k+n} where n>0 is an integer fusion level and k=nP/(P'-P)-2 is a
fractional level. For n=1, these are the logarithmic minimal models LM(P,P').
For n>1, we argue that these critical theories are realized on the lattice by n
x n fusion of the n=1 models. For n=2, we call them logarithmic superconformal
minimal models LSM(p,p') where P=|2p-p'|, P'=p' and p,p' are coprime, and they
share the central charges of the rational superconformal minimal models
SM(P,P'). Their mathematical description entails the fused planar
Temperley-Lieb algebra which is a spin-1 BMW tangle algebra with loop fugacity
beta_2=x^2+1+x^{-2} and twist omega=x^4 where x=e^{i(p'-p)pi/p'}. Examples are
superconformal dense polymers LSM(2,3) with c=-5/2, beta_2=0 and superconformal
percolation LSM(3,4) with c=0, beta_2=1. We calculate the free energies
analytically. By numerically studying finite-size spectra on the strip with
appropriate boundary conditions in Neveu-Schwarz and Ramond sectors, we argue
that, in the continuum scaling limit, these lattice models are associated with
the logarithmic superconformal models LM(P,P';2). For system size N, we propose
finitized Kac character formulas whose P,P' dependence only enters in the
fractional power of q in a prefactor. These characters involve Motzkin and
Riordan polynomials defined in terms of q-trinomial coefficients. Using the
Hamiltonian limit, we argue that there exist reducible yet indecomposable
representations for which the Virasoro dilatation operator L_0 exhibits rank-2
Jordan blocks confirming that these theories are indeed logarithmic. We relate
these results to the N=1 superconformal representation theory.Comment: 55 pages, v2: comments and references adde
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
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