350 research outputs found

    Half-lattice paths and Virasoro characters

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    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 M(k,2k±1)M(k,2k\pm1) Virasoro characters via half-lattice paths

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    We derive new fermionic expressions for the characters of the Virasoro minimal models M(k,2k±1)M(k,2k\pm1) by analysing the recently introduced half-lattice paths. These fermionic expressions display a quasiparticle formulation characteristic of the ϕ2,1\phi_{2,1} and ϕ1,5\phi_{1,5} integrable perturbations. We find that they arise by imposing a simple restriction on the RSOS quasiparticle states of the unitary models M(p,p+1)M(p,p+1). In fact, four fermionic expressions are obtained for each generating function of half-lattice paths of finite length LL, and these lead to four distinct expressions for most characters χr,sk,2k±1\chi^{k,2k\pm1}_{r,s}. These are direct analogues of Melzer's expressions for M(p,p+1)M(p,p+1), 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 LL half-lattice paths, this expression being notable in that it involves qq-trinomial coefficients. Taking the L→∞L\to\infty limit shows that the generating functions for infinite length half-lattice paths are indeed the Virasoro characters χr,sk,2k±1\chi^{k,2k\pm1}_{r,s}.Comment: 29 pages. v2: minor improvements, references adde

    Off-critical local height probabilities on a plane and critical partition functions on a cylinder

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    We compute off-critical local height probabilities in regime-III restricted solid-on-solid models in a 4N4 N-quadrant spiral geometry, with periodic boundary conditions in the angular direction, and fixed boundary conditions in the radial direction, as a function of NN, the winding number of the spiral, and τ\tau, the departure from criticality of the model, and observe that the result depends only on the product N τN \, \tau. In the limit N→1N \rightarrow 1, τ→τ0\tau \rightarrow \tau_0, such that τ0\tau_0 is finite, we recover the off-critical local height probability on a plane, τ0\tau_0-away from criticality. In the limit N→∞N \rightarrow \infty, τ→0\tau \rightarrow 0, such that N τ=τ0N \, \tau = \tau_0 is finite, and following a conformal transformation, we obtain a critical partition function on a cylinder of aspect-ratio τ0\tau_0. We conclude that the off-critical local height probability on a plane, τ0\tau_0-away from criticality, is equal to a critical partition function on a cylinder of aspect-ratio τ0\tau_0, in agreement with a result of Saleur and Bauer.Comment: 28 page

    CFT, BCFT, ADE and all that

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    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

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    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

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    Starting from the fusion rules for the algebra SO(5)2SO(5)_2 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 WB2\mathcal{W}B_2 and WD5\mathcal{W}D_5, 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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