Fusion hierarchies,T-systems, andY-systems of logarithmic minimal models

Abstract

A Temperley-Lieb (TL) loop model is a Yang-Baxter integrable lattice model with nonlocal degrees of freedom. On a strip of width N ∈ N, the evolution operator is the double-row transfer tangle D(u), an element of the TL algebra TL N (β) with loop fugacity β = 2 cos λ, λ ∈ R. 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 LM(p, p ′ ) comprise a subfamily of the TL loop models for which the crossing parameter λ = (p ′ − p)π/p ′ is a rational multiple of π parameterised by coprime integers 1 ≤ p < p ′ . For these special values, additional symmetries allow for particular degeneracies in the spectra that account for the logarithmic nature of these theories. For critical dense polymers LM(1, 2), with β = 0, D(u) and T (u) satisfy inversion identities that have led to the exact determination of the eigenvalues in any representation and for arbitrary finite system size N . The generalisation for 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 D m,n (u) and T m,n (u) where D(u) = D 1,1 (u) and T (u) = T 1,1 (u). The fused transfer tangles are constructed from (m, n)-fused face operators involving Wenzl-Jones projectors P k on k = m or k = n nodes. Some projectors P k are singular for k ≥ p ′ , but we argue that D m,n (u) and T m,n (u) are nonsingular for every m, n ∈ N in certain cabled link state representations. For generic λ, 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 D m,1 (u) and T m,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.

Critical dense polymers with Robin boundary conditions, half-integer Kac labels andZ4fermions

“…This paper opens several avenues for further work. It is clearly of interest to study, either numerically or analytically through more general functional equations [75], the conformal spectra of the other logarithmic minimal models LM(p, p ) to confirm more generally that Robin boundary conditions lead to conformal weights with non-integer Kac labels. It would also be interesting to continue our analysis of fusion.…”

Conformal partition functions of critical percolation fromD3thermodynamic Bethe Ansatz equations

“…The value d SLE path = 7 4 was proved by Beffara [55]. The incorporation of critical dense polymers LM (1,2) [38,[56][57][58][59][60] and critical percolation LM (2,3) into the framework of the family of logarithmic minimal models LM(p, p ′ ) [37,39,41,[61][62][63][64] establishes that these models are Yang-Baxter integrable. The transfer matrices of the logarithmic minimal models are built from so called transfer tangles of the planar Temperley-Lieb algebra [14,15], which we respectively denote by D(u) and T (u) for the boundary and the periodic cases.…”

“…These transfer tangles were constructed in terms of fused face operators in [62]. They commute as elements of TL N (β): [D m (u), D n (v)] = 0.…”

Fusion hierarchies, T-systems and Y-systems for the dilute $\boldsymbol{A_2^{(2)}}$ loop models

“…1 loop models in [18] and the A (1) 2 loop models in [31]. For the A (2) 2 models, we have not succeeded to construct a full set of (m, n) projectors.…”

“…At roots of unity, there exist extra relations, the closure relations, which take the form of linear relations between the various fused transfer tangles. Closure relations have been previously derived for the A (loop models[18] and A…”