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.
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.
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 The fusion hierarchy, T -system and Y -system of functional equations are the key to exact solvability for 2d lattice models. We derive these equations for the generic dilute loop models. The fused transfer matrices are associated with nodes of the infinite dominant integral weight lattice of . For generic values of the crossing parameter , the T - and Y -systems do not truncate. For the case rational so that is a root of unity, we find explicit closure relations and derive closed finite T - and Y -systems. The thermodynamics Bethe ansatz (TBA) integral equations and the associated TBA diagrams of the Y-systems are not of simple Dynkin type. They involve nodes if p is even and nodes if p is odd and are related to the TBA diagrams of models at roots of unity by a folding which originates from the addition of crossing symmetry. In an appropriate regime, the known central charges are . Prototypical examples of the loop models, at roots of unity, include critical dense polymers with central charge c = −2, and loop fugacity and critical site percolation on the triangular lattice with c = 0, and . Solving the TBA equations for the conformal data will determine whether these models lie in the same universality classes as their 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.
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.
I choose the Lee-Yang model and go through different approaches to analyze it using the form factor approach and the bootstrap program, the lattice description and the lattice TBA equations for a full understanding of the model. The bootstrap program aims to explicitly solve 1+1 dimensional integrable quantum field theories. In the first step, called the S-matrix bootstrap, the scattering matrix, is determined from its properties. In developing a defect form factor program the first step is the T-matrix bootstrap. Interacting integrable defect theories are purely transmitting and topological. We analyze both operators localized in the bulk and also on the defect. By finding their solutions, the spectral representation of any correlator can be determined. Next we consider the model on the lattice. The lattice approach allows to obtain both massive and massless excited TBA equations by studying the continuum scaling limit of the associated integrable lattice models. The most important input from the lattice approach is an insight into the analytic structure of the excited state solutions of the TBA equations. Previously this structure had to be guessed. In this thesis we turn our attention to the simplest example of a non-unitary minimal theory, namely, the Lee-Yang minimal model M(2,5). We study the Lee-Yang model on the lattice. We analyze the periodic, boundary and the seam cases in both massive and massless regimes. We derive their ground state TBA equations and analyze the flows between different boundary and defect conditions. Finally, in the large volume limit, we calculate the Lüscher correction terms of the TBA energy of a one-particle state. We display those ideas in the end of the thesis as this is an important direction of work in string theory, particularly in the AdS/CFT correspondence.
Abstract Using the planar Temperley–Lieb algebra, critical bond percolation on the square lattice can be reformulated as a loop model. In this form, it is incorporated as L M ( 2 , 3 ) in the Yang–Baxter integrable family of logarithmic minimal models L M ( p , p ′ ) . We consider this model of percolation in the presence of boundaries and with periodic boundary conditions. Inspired by Kuniba, Sakai and Suzuki, we rewrite the recently obtained infinite Y -system of functional equations. In this way, we obtain nonlinear integral equations in the form of a closed finite set of TBA equations described by a D 3 Dynkin diagram. Following the methods of Klümper and Pearce, we solve the TBA equations for the conformal finite-size corrections. For the ground states of the standard modules on the strip, these agree with the known central charge c = 0 and conformal weights Δ 1 , s for s ∈ Z ⩾ 1 with Δ r , s = ( ( 3 r − 2 s ) 2 − 1 ) / 24 . For the periodic case, the finite-size corrections agree with the conformal weights Δ 0 , s , Δ 1 , s with s
Determinantal functional equations satisfied by the row transfer matrix eigenvalues of critical A–D–E lattice spin models are presented. These are obtained for models associated with the Lie algebras [Formula: see text], [Formula: see text], AL, DL and E6,7,8 by exploiting connections with functional equations satisfied by the row transfer matrix eigenvalues of the six-vertex model at rational values of the crossing parameter λ=sπ/h where h is the Coxeter number. In addition, fusion is used to derive special functional equations, called inversion identity hierarchies, which provide the key to the direct calculation of finite-size corrections, central charges and conformal weights for the critical A–D–E lattice models.