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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 N, the evolution operator is the double-row transfer tangle D(u), an element of the TL algebra TL_N(beta) with loop fugacity beta=2cos(lambda). Similarly on a cylinder, the single-row transfer tangle T(u) is an element of the enlarged periodic TL algebra. The logarithmic minimal models LM(p,p') comprise a subfamily of the TL loop models for which the crossing parameter lambda=(p'-p)pi/p' is parameterised by coprime integers 0<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), D(u) and T(u) are known to satisfy inversion identities that allow us to obtain exact eigenvalues in any representation and for all system sizes 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'-1, but we argue that D^{m,n}(u) and T^{m,n}(u) are well defined for all m,n. For generic lambda, 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.