Fusion hierarchies, T-systems, and Y-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 , the evolution operator is the double-row transfer tangle D(u), an element of the TL algebra TLN(β) with loop fugacity β = 2cosλ, . 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 λ = (p′− p)π/p′ is a rational multiple of π parameterized 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 , 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 generalization 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 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 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 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.