Integrable Lattice Realizations of N=1 Superconformal Boundary Conditions

We construct integrable boundary conditions for sl(2) coset models with central charges c=3/2-12/(m(m+2)) and m=3,4,... The associated cylinder partition functions are generating functions for the branching functions but these boundary conditions manifestly break the superconformal symmetry. We show that there are additional integrable boundary conditions, satisfying the boundary Yang-Baxter equation, which respect the superconformal symmetry and lead to generating functions for the superconformal characters in both Ramond and Neveu-Schwarz sectors. We also present general formulas for the cylinder partition functions. This involves an alternative derivation of the superconformal Verlinde formula recently proposed by Nepomechie.Comment: 22 pages, 12 figures; section 2 rewritten; journal-ref. adde

Fusion Algebras of Logarithmic Minimal Models

We present explicit conjectures for the chiral fusion algebras of the logarithmic minimal models LM(p,p') considering Virasoro representations with no enlarged or extended symmetry algebra. The generators of fusion are countably infinite in number but the ensuing fusion rules are quasi-rational in the sense that the fusion of a finite number of representations decomposes into a finite direct sum of representations. The fusion rules are commutative, associative and exhibit an sl(2) structure but require so-called Kac representations which are reducible yet indecomposable representations of rank 1. In particular, the identity of the fundamental fusion algebra is in general a reducible yet indecomposable Kac representation of rank 1. We make detailed comparisons of our fusion rules with the results of Gaberdiel and Kausch for p=1 and with Eberle and Flohr for (p,p')=(2,5) corresponding to the logarithmic Yang-Lee model. In the latter case, we confirm the appearance of indecomposable representations of rank 3. We also find that closure of a fundamental fusion algebra is achieved without the introduction of indecomposable representations of rank higher than 3. The conjectured fusion rules are supported, within our lattice approach, by extensive numerical studies of the associated integrable lattice models. Details of our lattice findings and numerical results will be presented elsewhere. The agreement of our fusion rules with the previous fusion rules lends considerable support for the identification of the logarithmic minimal models LM(p,p') with the augmented c_{p,p'} (minimal) models defined algebraically.Comment: 22 pages, v2: comments adde

Fusion hierarchies, TT-systems and YY-systems for the dilute A2(2)A_2^{(2)} loop models

The fusion hierarchy, $T$-system and $Y$-system of functional equations are the key to integrability for 2d lattice models. We derive these equations for the generic dilute $A_2^{(2)}$ loop models. The fused transfer matrices are associated with nodes of the infinite dominant integral weight lattice of $s\ell(3)$. For generic values of the crossing parameter $\lambda$, the $T$- and $Y$-systems do not truncate. For the case $\frac{\lambda}{\pi}=\frac{(2p'-p)}{4p'}$ rational so that $x=\mathrm{e}^{\mathrm{i}\lambda}$ is a root of unity, we find explicit closure relations and derive closed finite $T$- and $Y$-systems. The TBA diagrams of the $Y$-systems and associated Thermodynamic Bethe Ansatz (TBA) integral equations are not of simple Dynkin type. They involve $p'+2$ nodes if $p$ is even and $2p'+2$ nodes if $p$ is odd and are related to the TBA diagrams of $A_2^{(1)}$ models at roots of unity by a ${\Bbb Z}_2$ folding which originates from the addition of crossing symmetry. In an appropriate regime, the known central charges are $c=1-\frac{6(p-p')^2}{pp'}$. Prototypical examples of the $A_2^{(2)}$ loop models, at roots of unity, include critical dense polymers ${\cal DLM}(1,2)$ with central charge $c=-2$, $\lambda=\frac{3\pi}{8}$ and loop fugacity $\beta=0$ and critical site percolation on the triangular lattice ${\cal DLM}(2,3)$ with $c=0$, $\lambda=\frac{\pi}{3}$ and $\beta=1$. Solving the TBA equations for the conformal data will determine whether these models lie in the same universality classes as their $A_1^{(1)}$ counterparts. More specifically, it will confirm the extent to which bond and site percolation lie in the same universality class as logarithmic conformal field theories.Comment: 34 page

Fusion of \ade Lattice Models

Fusion hierarchies of \ade face models are constructed. The fused critical $D$, $E$ and elliptic $D$ models yield new solutions of the Yang-Baxter equations with bond variables on the edges of faces in addition to the spin variables on the corners. It is shown directly that the row transfer matrices of the fused models satisfy special functional equations. Intertwiners between the fused \ade models are constructed by fusing the cells that intertwine the elementary face weights. As an example, we calculate explicitly the fused $2\times 2$ face weights of the 3-state Potts model associated with the $D_4$ diagram as well as the fused intertwiner cells for the $A_5$--$D_4$ intertwiner. Remarkably, this $2\times 2$ fusion yields the face weights of both the Ising model and 3-state CSOS models.Comment: 41 page

Intertwiners and \ade Lattice Models

Intertwiners between \ade lattice models are presented and the general theory developed. The intertwiners are discussed at three levels: at the level of the adjacency matrices, at the level of the cell calculus intertwining the face algebras and at the level of the row transfer matrices. A convenient graphical representation of the intertwining cells is introduced. The utility of the intertwining relations in studying the spectra of the \ade models is emphasized. In particular, it is shown that the existence of an intertwiner implies that many eigenvalues of the \ade row transfer matrices are exactly in common for a finite system and, consequently, that the corresponding central charges and scaling dimensions can be identified.Comment: 48 pages, Two postscript files included

Surface Free Energies, Interfacial Tensions and Correlation Lengths of the ABF Models

The surface free energies, interfacial tensions and correlation lengths of the Andrews-Baxter-Forrester models in regimes III and IV are calculated with fixed boundary conditions. The interfacial tensions are calculated between arbitrary phases and are shown to be additive. The associated critical exponents are given by $2-\alpha_s=\mu=\nu$ with $\nu=(L+1)/4$ in regime III and $4-2\alpha_s=\mu=\nu$ with $\nu=(L+1)/2$ in regime IV. Our results are obtained using general commuting transfer matrix and inversion relation methods that may be applied to other solvable lattice models.Comment: 21 pages, LaTeX 2e, requires the amsmath packag

Solutions of the boundary Yang-Baxter equation for ADE models

We present the general diagonal and, in some cases, non-diagonal solutions of the boundary Yang-Baxter equation for a number of related interaction-round-a-face models, including the standard and dilute A_L, D_L and E_{6,7,8} models.Comment: 32 pages. Sections 7.2 and 9.2 revise