We study integrable realizations of conformal twisted boundary conditions for s ℓ(2) unitary minimal models on a torus. These conformal field theories are realized as the continuum scaling limit of critical G = A , D , E lattice models with positive spectral parameter u > 0 and Coxeter number g . Integrable seams are constructed by fusing blocks of elementary local face weights. The usual A -type fusions are labelled by the Kac labels ( r , s ) and are associated with the Verlinde fusion algebra. We introduce a new type of fusion in the two braid limits u → ±i ∞ associated with the graph fusion algebra, and labelled by nodes a , b ∊ G respectively. When combined with automorphisms, they lead to general integrable seams labelled by x = ( r , a , b , κ) ∊ ( A g −2 , H , H , 2 ) where H is the graph G for type I theories and its parent for type II theories. Identifying our construction labels with the conformal labels of Petkova and Zuber, we find that the integrable seams are in one-to-one correspondence with the conformal seams. The distinct seams are thus associated with the nodes of the Ocneanu quantum graph. The quantum symmetries and twisted partition functions are checked numerically for | G | ≤ 6. We also show, in the case of D 2ℓ , that the non-commutativity of the Ocneanu algebra of seams arises because the automorphisms do not commute with the fusions.
We study integrable realizations of conformal twisted boundary conditions for s ℓ(2) unitary minimal models on a torus. These conformal field theories are realized as the continuum scaling limit of critical G = A , D , E lattice models with positive spectral parameter u > 0 and Coxeter number g . Integrable seams are constructed by fusing blocks of elementary local face weights. The usual A -type fusions are labelled by the Kac labels ( r , s ) and are associated with the Verlinde fusion algebra. We introduce a new type of fusion in the two braid limits u → ±i ∞ associated with the graph fusion algebra, and labelled by nodes a , b ∊ G respectively. When combined with automorphisms, they lead to general integrable seams labelled by x = ( r , a , b , κ) ∊ ( A g −2 , H , H , 2 ) where H is the graph G for type I theories and its parent for type II theories. Identifying our construction labels with the conformal labels of Petkova and Zuber, we find that the integrable seams are in one-to-one correspondence with the conformal seams. The distinct seams are thus associated with the nodes of the Ocneanu quantum graph. The quantum symmetries and twisted partition functions are checked numerically for | G | ≤ 6. We also show, in the case of D 2ℓ , that the non-commutativity of the Ocneanu algebra of seams arises because the automorphisms do not commute with the fusions.
Abstract We consider unitary and nonunitary ( A , G ) coset minimal models on the cylinder with G = A , D , E . Elementary topological defects are implemented as non-contractible Verlinde, Pasquier and Ocneanu lines around the cylinder. The decomposition of compound defects, formed by fusing elementary defects together, is described by the Verlinde, Pasquier and Ocneanu graph fusion algebras. The action of the compound defects on the left- or right-vacuum boundary state builds the known conformal cylinder boundary conditions. Fusing these defects, in the presence of vacuum boundary conditions, reproduces the known general formulas for the conformal cylinder partition functions. We define the coset graph A ⊗ G / Z 2 , argue that it is a universal object and show that it encodes (i) the coset graph fusion algebra, (ii) the Affleck–Ludwig boundary g -factors, (iii) the defect g -factors given by quantum dimensions and (iv) the relative symmetry resolved Affleck–Ludwig entropies. Additionally, it is shown that the boundary and defect g -factors are related to the asymptotic counting of fusion paths on the coset graph. On the lattice, the defects are constructed as Yang–Baxter integrable seams including special braid and graph automorphism transfer matrices. Remarkably, many of the boundary conformal field theory (CFT) discrete structures, such as fusion matrices, modular matrices, quantum dimensions and defects, appear at the level of lattice Yang–Baxter integrable A - D - E restricted solid-on-solid models and these structures and their properties carry over to the CFT in the continuum scaling limit. Importantly, in the continuum scaling limit, the lattice transfer matrix T - and Y -system functional equations carry over to produce the coset graph fusion algebra for the defect lines. Moreover, the effective central charges and conformal weights are expressed in terms of dilogarithms of the braid and bulk asymptotics of the Y -system expressed in terms of the quantum dimensions.
We develop further the implementation and analysis of Kac boundary conditions in the general logarithmic minimal models with 1 ⩽ p < p ′ and p , p ′ coprime. Specifically, working in a strip geometry, we consider the ( r , s ) Kac boundary conditions. These boundary conditions are organized into infinitely extended Kac tables labeled by the Kac labels r , s = 1, 2, 3, …. They are conjugate to Virasoro Kac representations with conformal dimensions Δ r , s given by the usual Kac formula. On a finite strip of width N , built from a square lattice, the associated integrable boundary conditions are constructed by acting on the vacuum (1, 1) boundary with an s -type seam of width s − 1 columns and an r -type seam of width ρ − 1 columns. The r -type seam contains an arbitrary boundary field ξ . While the usual fusion construction of the r -type seam relies on the existence of Wenzl–Jones projectors restricting its application to r ⩽ ρ < p ′, this limitation was recently removed by Pearce et al who further conjectured that the conformal boundary conditions labeled by r are realized, in particular, for . In this paper, we confirm this conjecture by performing extensive numerics on the commuting double row transfer matrices and their associated quantum Hamiltonian chains. Letting [ x ] denote the fractional part, we fix the boundary field to the specialized values if and otherwise. For these boundary conditions, we obtain the Kac conformal weights Δ r , s by numerically extrapolating the finite-size corrections to the lowest eigenvalue of the quantum Hamiltonians out to sizes N ⩽ 32 − ρ − s . Additionally, by solving local inversion relations, we obtain general analytic expressions for the boundary free energies allowing for more accurate estimates of the conformal data.
We define generalised chiral vertex operators covariant under the Ocneanu “double triangle algebra” A, a novel quantum symmetry intrinsic to a given rational 2-d conformal field theory. This provides a chiral approach, which, unlike the conventional one, makes explicit various algebraic structures encountered previously in the study of these theories and of the associated critical lattice models, and thus allows their unified treatment. The triangular Ocneanu cells, the 3j-symbols of the weak Hopf algebra A, reappear in several guises. With A and its dual algebra  one associates a pair of graphs, G and ˜G. While G are known to encode complete sets of conformal boundary states, the Ocneanu graphs ˜G classify twisted torus partition functions. The fusion algebra of the twist operators provides the data determining Â. The study of bulk field correlators in the presence of twists reveals that the Ocneanu graph quantum symmetry gives also an information on the field operator algebra.
We derive the fusion hierarchy of functional equations for critical A-D-E lattice models related to the sℓ ( 2 ) unitary minimal models, the parafermionic models and the supersymmetric models of conformal field theory and deduce the related TBA functional equations. The derivation uses fusion projectors and applies in the presence of all known integrable boundary conditions on the torus and cylinder. The resulting TBA functional equations are universal in the sense that they depend only on the Coxeter number of the A-D-E graph and are independent of the particular integrable boundary conditions. We conjecture generally that TBA functional equations are universal for all integrable lattice models associated with rational CFTs and their integrable perturbations. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Abstract We consider general logarithmic minimal models L M ( p , p ′ ) , with p , p ′ coprime, on a strip of N columns with the ( r , s ) Robin boundary conditions introduced by Pearce, Rasmussen and Tipunin. On the lattice, these models are Yang–Baxter integrable loop models that are described algebraically by the one-boundary Temperley–Lieb algebra. The ( r , s ) Robin boundary conditions are a class of integrable boundary conditions satisfying the boundary Yang–Baxter equations which allow loop segments to either reflect or terminate on the boundary. The associated conformal boundary conditions are organized into infinitely extended Kac tables labelled by the Kac labels r ∈ Z and s ∈ N . The Robin vacuum boundary condition, labelled by ( r , s − 1 2 ) = ( 0 , 1 2 ) , is given as a linear combination of Neumann and Dirichlet boundary conditions. The general ( r , s ) Robin boundary conditions are constructed, using fusion, by acting on the Robin vacuum boundary with an ( r , s )-type seam consisting of an r -type seam of width w columns and an s -type seam of width d = s − 1 columns. The r -type seam admits an arbitrary boundary field which we fix to the special value ξ = − λ 2 where λ = ( p ′ − p ) π <m
The fused six-vertex models with open boundary conditions are studied. The Bethe ansatz solution given by Sklyanin has been generalized to the transfer matrices of the fused models. We have shown that the eigenvalues of transfer matrices satisfy a group of functional relations, which are the su(2) fusion rule held by the transfer matrices of the fused models. The fused transfer matrices form a commuting family and also commute with the quantum group Uq[sl(2)]. In the case of the parameter qh = −1 (h = 4,5, · · ·) the functional relations in the limit of spectral parameter u → i ∞ are truncated. This shows that the su(2) fusion rule with finite level appears for the six vertex model with the open boundary conditions. We have solved the functional relations to obtain the finite-size corrections of the fused transfer matrices for low-lying excitations. From the corrections the central charges and conformal weights of underlying conformal field theory are extracted. To see different boundary conditions we also have studied the six-vertex model with a twisted boundary condition. hep-th/9502053; to appear in NPB. 1