Alexi Morin-Duchesne

Boundary algebras and Kac modules for logarithmic minimal models

Virasoro Kac modules were originally introduced indirectly as representations whose characters arise in the continuum scaling limits of certain transfer matrices in logarithmic minimal models, described using Temperley–Lieb algebras. The lattice transfer operators include seams on the boundary that use Wenzl–Jones projectors. If the projectors are singular, the original prescription is to select a subspace of the Temperley–Lieb modules on which the action of the transfer operators is non-singular. However, this prescription does not, in general, yield representations of the Temperley–Lieb algebras and the Virasoro Kac modules have remained largely unidentified. Here, we introduce the appropriate algebraic framework for the lattice analysis as a quotient of the one-boundary Temperley–Lieb algebra. The corresponding standard modules are introduced and examined using invariant bilinear forms and their Gram determinants. The structures of the Virasoro Kac modules are inferred from these results and are found to be given by finitely generated submodules of Feigin–Fuchs modules. Additional evidence for this identification is obtained by comparing the formalism of lattice fusion with the fusion rules of the Virasoro Kac modules. These are obtained, at the character level, in complete generality by applying a Verlinde-like formula and, at the module level, in many explicit examples by applying the Nahm–Gaberdiel–Kausch fusion algorithm.

Jordan cells of periodic loop models

Jordan cells in transfer matrices of finite lattice models are a signature of the logarithmic character of the conformal field theories that appear in their thermodynamical limit. The transfer matrix of periodic loop models, TN, is an element of the periodic Temperley?Lieb algebra , where N is the number of sites on a section of the cylinder, and ? = ?q ? q?1 = 2cos?? and ? the weights of contractible and non-contractible loops. The thermodynamic limit of TN is believed to describe a conformal field theory of central charge c = 1 ? 6?2/(?(? ? ?)). The abstract element TN acts naturally on (a sum of) spaces , similar to those upon which the standard modules of the (classical) Temperley?Lieb algebra act. These spaces known as sectors are labeled by the numbers of defects d and depend on a twist parameter v that keeps track of the winding of defects around the cylinder. Criteria are given for non-trivial Jordan cells of TN both between sectors with distinct defect numbers and within a given sector.

Fusion hierarchies, T-systems and Y-systems of logarithmic minimal models

A Temperley?Lieb (TL) loop model is a Yang?Baxter integrable lattice model with nonlocal degrees of freedom. On a strip of width , the evolution operator is the double-row transfer tangle D(u), an element of the TL algebra TLN(?) with loop fugacity ? = 2cos?, . Similarly, on a cylinder, the single-row transfer tangle T(u) is an element of the so-called enlarged periodic TL algebra. The logarithmic minimal models comprise a subfamily of the TL loop models for which the crossing parameter ? = (p?? p)?/p? is a rational multiple of ? parameterized by coprime integers 1 ? p 2 takes the form of functional relations for D(u) and T(u) of polynomial degree p?. These derive from fusion hierarchies of commuting transfer tangles Dm, n(u) and Tm, n(u), where D(u) = D1,1(u) and T(u) = T1,1(u). The fused transfer tangles are constructed from (m, n)-fused face operators involving Wenzl?Jones projectors Pk on k = m or k = n nodes. Some projectors Pk are singular for k ? p?, but we argue that Dm, n(u) and Tm, n(u) are nonsingular for every in certain cabled link state representations. For generic ?, we derive the fusion hierarchies and the associated T- and Y-systems. For the logarithmic theories, the closure of the fusion hierarchies at n = p? translates into functional relations of polynomial degree p? for Dm, 1(u) and Tm, 1(u). We also derive the closure of the Y-systems for the logarithmic theories. The T- and Y-systems are the key to exact integrability, and we observe that the underlying structure of these functional equations relate to Dynkin diagrams of affine Lie algebras.

A homomorphism between link and XXZ modules over the periodic Temperley-Lieb algebra

We study finite loop models on a lattice wrapped around a cylinder. A section of the cylinder has N sites. We use a family of link modules over the periodic Temperley–Lieb algebra introduced by Martin and Saleur, and Graham and Lehrer. These are labeled by the numbers of sites N and of defects d, and extend the standard modules of the original Temperley–Lieb algebra. Besides the defining parameters β = u2 + u−2 with u = eiλ/2 (weight of contractible loops) and α (weight of non-contractible loops), this family also depends on a twist parameter v that keeps track of how the defects wind around the cylinder. The transfer matrix TN(λ, ν) depends on the anisotropy ν and the spectral parameter λ that fixes the model. (The thermodynamic limit of TN is believed to describe conformal field theory of central charge c = 1 − 6λ2/(π(λ − π)).) The family of periodic XXZ Hamiltonians is extended to depend on this new parameter v, and the relationship between this family and the loop models is established. The Gram determinant for the natural bilinear form on these link modules is shown to factorize in terms of an intertwiner between these link representations and the eigenspaces of Sz of the XXZ models. This map is shown to be an isomorphism for generic values of u and v, and the critical curves in the plane of these parameters for which fails to be an isomorphism are given.

Dimer representations of the Temperley–Lieb algebra

A new spin-chain representation of the Temperley–Lieb algebra TLn(β = 0) is introduced and related to the dimer model. Unlike the usual XXZ spin-chain representations of dimension 2^n, this dimer representation is of dimension 2^(n−1). A detailed analysis of its structure is presented and found to yield indecomposable zigzag modules.

Integrability and conformal data of the dimer model

The central charge of the dimer model on the square lattice is still being debated in the literature. In this paper, we provide evidence supporting the consistency of a

On the reality of spectra of Uq(sl2)-invariant XXZ Hamiltonians

c=-2

Symmetry classes of alternating sign matrices in a nineteen-vertex model

description. Using Liebs transfer matrix and its description in terms of the Temperley-Lieb algebra

The Gram matrix as a connection between periodic loop models and XXZ Hamiltonians

TL_n

Conformal partition functions of critical percolation from

at