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arXiv:2509.11756 · SciPost Physics Core

Fusion in the periodic Temperley–Lieb algebra: General definition of a bifunctor

Yacine Ikhlef, Alexi Morin-Duchesne

Published Aug 27, 2026• 0 citations
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Abstract

The periodic Temperley–Lieb category consists of connectivity diagrams drawn on a ring with N N and N' N ′ nodes on the outer and inner boundary, respectively. We consider families of modules, namely sequences of modules \mathsf{M}(N) 𝖬 ( N ) over the enlarged periodic Temperley–Lieb algebra for varying values of N N , endowed with an action \mathsf{M}(N') \to \mathsf{M}(N) 𝖬 ( N ′ ) → 𝖬 ( N ) of the diagrams. Examples of modules that can be organised into families are those arising in the RSOS model and in the XXZ spin- \frac12 1 2 chain, as well as several others constructed from link states. We construct a fusion product which outputs a family of modules from any pair of families. Its definition is inspired from connectivity diagrams drawn on a disc with two holes. It is thus defined in a way to describe intermediate states in lattice correlation functions. We prove that this fusion product is a bifunctor, and that it is distributive, commutative, and associative.

DOI
10.21468/scipostphyscore.9.3.053

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The periodic Temperley–Lieb category consists of connectivity diagrams drawn on a ring with N N and N' N ′ nodes on the outer and inner boundary, respectively. We consider families of modules, namely sequences of modules \mathsf{M}(N) 𝖬 ( N ) over the enlarged periodic Temperley–Lieb algebra for varying values of N N , endowed with an action \mathsf{M}(N') \to \mathsf{M}(N) 𝖬 ( N ′ ) → 𝖬 ( N ) of the diagrams. Examples of modules that can be organised into families are those arising in the RSOS model and in the XXZ spin- \frac12 1 2 chain, as well as several others constructed from link states. We construct a fusion product which outputs a family of modules from any pair of families. Its definition is inspired from connectivity diagrams drawn on a disc with two holes. It is thus defined in a way to describe intermediate states in lattice correlation functions. We prove that this fusion product is a bifunctor, and that it is distributive, commutative, and associative.
In two-dimensional loop models, the scaling properties of critical random curves are encoded in the correlators of connectivity operators. In the dense O(n) loop model, any such operator is naturally associated to a standard module of the periodic Temperley-Lieb algebra. We introduce a new family of representations of this algebra, with connectivity states that have two marked points, and argue that they define the fusion of two standard modules. We obtain their decomposition on the standard modules for generic values of the parameters, which in turn yields the structure of the operator product expansion of connectivity operators.
Abstract The equivalent of fusion in boundary conformal field theory (CFT) can be realized quite simply in the context of lattice models by essentially glueing two open spin chains. This has led to many developments, in particular in the context of chiral logarithmic CFT. We consider in this paper a possible generalization of the idea to the case of bulk conformal field theory. This is of course considerably more difficult, since there is no obvious way of merging two closed spin chains into a big one. In an earlier paper, two of us had proposed a “topological” way of performing this operation in the case of models based on the affine Temperley-Lieb (ATL) algebra, by exploiting the associated braid group representation and skein relations. In the present work, we establish — using, in particular, Frobenius reciprocity — the resulting fusion rules for standard modules of ATL in the generic as well as partially degenerate cases. These fusion rules have a simple interpretation in the continuum limit. However, unlike in the chiral case this interpretation does not match the usual fusion in non-chiral CFTs. Rather, it corresponds to the glueing of the right moving component of one conformal field with the left moving component of the other.
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 EPTL_N(β, α) 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. Beside the defining parameters β=u^2+u^{-2} with u=e^{iλ/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 T_N(λ, ν) depends on the anisotropy νand the spectral parameter λthat fixes the model. (The thermodynamic limit of T_N is believed to describe a conformal field theory of central charge c=1-6λ^2/(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 i_N^d between these link representations and the eigenspaces of S^z 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 i_N^d fails to be an isomorphism are given.
A bstract We show that every fusion category containing a non-invertible, self-dual object a gives rise to an integrable anyonic chain whose Hamiltonian density satisfies the Temperley-Lieb algebra. This spin chain arises by considering the projection onto the identity channel in the fusion process a ⨂ a . We relate these models to Pasquier’s construction of ADE lattice models. We then exploit the underlying Temperley-Lieb structure to discuss the spectrum of these models and argue that these models are gapped when the quantum dimension of a is greater than 2. We show that for fusion categories where the dimension is close to 2, such as the Fib×Fib and Haagerup fusion categories, the finite size effects are large and they can obscure the numerical analysis of the gap.
We introduce a class of 2d2d2d quantum spin systems based on a unitary spherical fusion category with a chosen object. These models have frustration-free, reflection positive, but generally non-commuting local interactions, generalizing the Levin-Wen string-net models. Based on the theory developed in arXiv:2510.20662, we explicitly realize the Temperley-Lieb-Jones algebra, and more generally fusion spin chains as their boundary algebras.