Paul A. Pearce

Boundary conditions in rational conformal field theories

We develop further the theory of RationalConformalFieldTheories (RCFTs) on a cylinder with specified boundaryconditions emphasizing the role of a triplet of algebras: the Verlinde, graph fusion and Pasquier algebras. We show that solving Cardys equation, expressing consistency of a RCFT on a cylinder, is equivalent to finding integer valued matrix representations of the Verlinde algebra. These matrices allow us to naturally associate a graph G to each RCFT such that the conformalboundaryconditions are labelled by the nodes of G . This approach is carried to completion for sl(2) theories leading to complete sets of conformalboundaryconditions, their associated cylinder partition functions and the A -D -E classification. We also review the current status for WZW sl(3) theories. Finally, a systematic generalization of the formalism of Cardy–Lewellen is developed to allow for multiplicities arising from more general representations of the Verlinde algebra. We obtain information on the bulk-boundary coefficients and reproduce the relevant algebraic structures from the sewing constraints.

Conformal weights of RSOS lattice models and their fusion hierarchies

The finite-size corrections, central charges c and conformal weights Δ of L-state restricted solid-on-solid lattice models and their fusion hierarchies are calculated analytically. This is achieved by solving special functional equations, in the form of inversion identity hierarchies, satisfied by the commuting row transfer matrices at critically. The results are all obtained in terms of Rogers dilogarithms. The RSOS models exhibit two distinct critical regimes. For the regime III/IV critical line, we find c = [3p/(p + 2)][1 − 2(p + 2)/r(r − p)] where L = r − 1 is the number of heights and p = 1, 2, … is the fusion level. The conformal weights are given by a generalized Kac formula Δ = {[rt − (r − p)s]2 − p2}/ 4pr(r − p) + (s0 − 1)(p − s0 + 1)/ 2p(p + 2) where s = 1, 2, …, r − 1; t = 1, 2, …, r − p − 1; 1 ⩽ s0 ⩽ p + 1 and s0 − 1 = ±(t − s) mod 2p. For p = 1, 2, these models are described by the unitary minimal conformal series and the discrete superconformal series, respectively. For the regime I/II critical line, we obtain c = 2(N − 1)/(N + 2) and Δ = l(l + 2)/4(N + 2) − m2/4N for the conformal weights, independent of the fusion level p, where N = L − 1, l = 0, 1, …, N and m = −l, −l + 2, …, l − 2, l. In this critical regime the models are described by ZN parafermion theories.

Central charges of the 6- and 19-vertex models with twisted boundary conditions

A new and general analytic method for calculating finite-size corrections and central charges is applied to the 6- and 19-vertex models and their related spin-1/2 and spin-1 XXZ chains with twisted boundary conditions. Nonlinear integral equations are derived from which the central charge c can be extracted in terms of Rogers dilogarithms. For twist angle phi , the central charge is c=3S/S+1 (1-4(S-1) phi 2/ pi ( pi -2S gamma )) where gamma is the crossing parameter or chain anisotropy and spin S=1/2 or 1. For periodic boundary conditions ( phi =0) this reduces to the known results c=1 and c=3/2, respectively.

Interaction-Round-a-Face Models with Fixed Boundary Conditions: The ABF Fusion Hierarchy

We use boundary weights and reflection equations to obtain families of commuting double-row transfer matrices for interaction-round-a-face models with fixed boundary conditions. In particular, we consider the fusion hierarchy of the Andrews-Baxter-Forrester (ABF) models, for which we obtain diagonal, elliptic solutions to the reflection equations, and find that the double-row transfer matrices satisfy functional equations with the same form as in the case of periodic boundary conditions.

Analytic Calculation of Scaling Dimensions: Tricritical Hard Squares and Critical Hard Hexagons

The finite-size corrections, central chargesc, and scaling dimensionsx of tricritical hard squares and critical hard hexagons are calculated analytically. This is achieved by solving the special functional equation or inversion identity satisfied by the commuting row transfer matrices of these lattice models at criticality. The results are expressed in terms of Rogers dilogarithms. For tricritical hard squares we obtainc=7/10,x=3/40, 1/5, 7/8, 6/5 and for hard hexagons we obtainc=4/5,x=2/15, 4/5, 17/15, 4/3, 9/5, in accord with the predictions of conformal and modular invariance.

On the classification of bulk and boundary conformal field theories

The classification of rational conformal field theories is reconsidered from the standpoint of boundary conditions. Solving Cardys equation expressing the consistency condition on a cylinder is equivalent to finding integer valued representations of the fusion algebra. A complete solution not only yields the admissible boundary conditions but also gives valuable information on the bulk properties.

Solvable critical dense polymers

A lattice model of critical dense polymers is solved exactly for finite strips. The model is the first member of the principal series of the recently introduced logarithmic minimal models. The key to the solution is a functional equation in the form of an inversion identity satisfied by the commuting double-row transfer matrices. This is established directly in the planar Temperley–Lieb algebra and holds independently of the space of link states on which the transfer matrices act. Different sectors are obtained by acting on link states with s−1 defects where s = 1,2,3,... is an extended Kac label. The bulk and boundary free energies and finite-size corrections are obtained from the Euler–Maclaurin formula. The eigenvalues of the transfer matrix are classified by the physical combinatorics of the patterns of zeros in the complex spectral-parameter plane. This yields a selection rule for the physically relevant solutions to the inversion identity and explicit finitized characters for the associated quasi-rational representations. In particular, in the scaling limit, we confirm the central charge c = −2 and conformal weights Δs = ((2−s)2−1)/8 for s = 1,2,3,.... We also discuss a diagrammatic implementation of fusion and show with examples how indecomposable representations arise. We examine the structure of these representations and present a conjecture for the general fusion rules within our framework.

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 s�( 2) structure but require so-called Kac representations which are typically reducible yet indecomposable representations of rank 1. In particular, the identity of the fundamental fusion algebra p � 1 is 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 cp,p� (minimal) models defined algebraically.

Fusion algebra of critical percolation

We present an explicit conjecture for the chiral fusion algebra of critical percolation considering Virasoro representations with no enlarged or extended symmetry algebra. The representations that we take to generate fusion are countably infinite in number. The ensuing fusion rules are quasi-rational in the sense that the fusion of a finite number of these representations decomposes into a finite direct sum of these representations. The fusion rules are commutative, associative and exhibit an structure. They involve representations which we call Kac representations of which some are reducible yet indecomposable representations of rank 1. In particular, the identity of the fusion algebra is a reducible yet indecomposable Kac representation of rank 1. We make detailed comparisons of our fusion rules with the recent results of Eberle–Flohr and Read–Saleur. Notably, in agreement with Eberle–Flohr, we find the appearance of indecomposable representations of rank 3. Our fusion rules are supported by extensive numerical studies of an integrable lattice model of critical percolation. Details of our lattice findings and numerical results will be presented elsewhere.

Temperley-Lieb stochastic processes

We discuss one-dimensional stochastic processes defined through the Temperley–Lieb algebra related to the Q = 1 Potts model. For various boundary conditions, we formulate a conjecture relating the probability distribution which describes the stationary state, to the enumeration of a symmetry class of alternating sign matrices, objects that have received much attention in combinatorics.