Iterated binary fusion versus single m-fold fusion for families of modules over the enlarged periodic Temperley–Lieb algebra

Ikhlef and Morin-Duchesne (arXiv:2509.11756; published as SciPost Phys. Core 9, 053 (2026)) constructed a fusion product $\times_f$ on the category of families of modules over the enlarged periodic Temperley–Lieb algebra $\mathcal{E}PTL_N(\beta)$ — the algebra of connectivity diagrams on an annulus that governs two-dimensional loop models — and proved it is a bifunctor that is distributive, commutative and associative, with associativity established for three families through an explicit triple-fusion presentation (their Theorem 4, the $m=3$ case). They leave open, as an explicit conjecture, the generalization to any number $m \ge 4$ of input families: that the iterated two-fold fusion $(((\mathsf{M}_1 \times_f \mathsf{M}_2) \times_f \mathsf{M}_3) \times_f \cdots) \times_f \mathsf{M}_m$ is isomorphic to the single $m$-fold fusion family whose modules are $\sum \mathcal{L}(N,\widehat{N}_m)\cdot(\mathsf{M}_1(N_1)\otimes\cdots\otimes\mathsf{M}_m(N_m))$ subject to the generator-entry relations (5.2a)–(5.2c) of their Section 5. The problem is the lattice analogue of order-independence of the operator product expansion for the nonlocal connectivity operators of loop models. The full task is to establish the well-definedness of the $m$-fold presentation and its isomorphism with the iterated two-fold fusion; as one route, the authors suggest extending the diagrammatic framework to connectivity diagrams on discs with an arbitrary number of holes.

Background

The Temperley–Lieb (TL) algebra $\mathrm{TL}_N(\beta)$ is the associative algebra generated by $e_1,\dots,e_{N-1}$ with relations $e_i^2=\beta e_i$, $e_i e_{i+1} e_i=e_i$ and $e_i e_j=e_j e_i$ for $|i-j|>1$; its basis elements can be drawn as non-crossing pairings of $N$ boundary points, where $\beta$ is the weight of a closed loop. It is the common symmetry algebra of the dense $O(n)$ loop model, the XXZ spin-$\frac{1}{2}$ chain (an anisotropic Heisenberg quantum spin chain) and the RSOS (restricted solid-on-solid) lattice models, and its representation theory controls the spectra of their transfer matrices. On a periodic lattice one needs the periodic TL algebra, whose diagrams live on an annulus and include the rotation $\Omega$ of the annulus by one site. Ikhlef and Morin-Duchesne work with the enlarged periodic Temperley–Lieb algebra $\mathcal{E}PTL_N(\beta)$, generated by $\Omega^{\pm 1}$ and $e_0,\dots,e_{N-1}$ (indices modulo $N$) with the TL relations plus $\Omega e_j \Omega^{-1}=e_{j-1}$ and $\Omega^2 e_1=e_{N-1}e_{N-2}\cdots e_1$. Its elements are connectivity diagrams on an annulus, and for each pair of sizes there is a diagram space $\mathcal{L}(N,N')$ of connectivities between $N$ nodes on the outer boundary and $N'$ nodes on the inner boundary: algebraically the words in generators $c_j$ and $c^{\dagger}_j$ that respectively remove and add pairs of adjacent boundary nodes, subject to TL-type relations, and diagrammatically the annular connectivities; Theorem 1 of the source paper proves these two definitions equivalent.

A family of modules is a sequence $\mathsf{M}=\{\mathsf{M}(N)\mid N=N_0,N_0+2,\dots\}$ of modules over $\mathcal{E}PTL_N(\beta)$ for all sizes $N$ of a fixed parity $N_0\in\{0,1\}$, together with compatible actions of the diagram spaces $\mathcal{L}(N',N)$ on the modules. The paradigmatic examples are the standard modules $\mathsf{W}_{k,x}(N)$ of Graham and Lehrer: finite-dimensional modules spanned by link states, drawn on a disc with a marked point at its center, consisting of $N/2-k$ arches pairing boundary nodes and $2k$ defects (lines connecting the marked point to the boundary), with a twist parameter $x\in\mathbb{C}^{\times}$ that weights the winding of the defects around the marked point (for $k=0$, each non-contractible loop around the marked point is replaced by the factor $x+x^{-1}$). In the dense $O(n)$ loop model, the vectors of configuration probabilities on a closed contour around an inserted operator live in such a module, and every connectivity operator is naturally associated with one of them. Families also arise from the XXZ spin chain and the RSOS models, and from quotient families such as quotients of the standard modules.

In loop models, the observables of interest are connectivity operators — nonlocal operators measuring whether given points of the lattice lie on the same loop. Their correlation functions are governed by an algebraic fusion of the associated modules that is the lattice analogue of the operator product expansion (OPE) of conformal field theory. Ikhlef and Morin-Duchesne introduced this fusion in two earlier papers: first for standard modules, using link states with two marked points, which produced the OPE structure of connectivity operators of the dense $O(n)$ model; then for irreducible and quotient modules — determining the fusion rules of $\mathsf{W}\times\mathsf{W}$ and constructing the products $\mathsf{W}\times\mathsf{R}$, $\mathsf{W}\times\mathsf{Q}$ and $\mathsf{Q}\times\mathsf{Q}$, where $\mathsf{R}_{\ell,y}$ and $\mathsf{Q}_{\ell,y}$ are the radical submodule and the quotient of a reducible standard module — the fusions involving the quotient $\mathsf{Q}$ turning out to obey very simple rules reminiscent of the fusion rules of $\mathfrak{sl}(2)$ (the Lie algebra of traceless $2\times2$ matrices). Both constructions were tied to specific module classes.

The source paper defines, for the first time, a fusion product on the category of all families of $\mathcal{E}PTL_N(\beta)$-modules. For two families $\mathsf{M}_a,\mathsf{M}_b$ it sets

$$(\mathsf{M}_a \times_f \mathsf{M}_b)(N) = \sum_{N_a,N_b} \mathcal{L}(N,N_{ab})\cdot\big(\mathsf{M}_a(N_a)\otimes\mathsf{M}_b(N_b)\big), \qquad N_{ab}=N_a+N_b,$$

a quotient of the direct sum by local relations that prescribe how each generator $c_j$, $c^{\dagger}_j$ and $\Omega$ enters the tensor factor on which it naturally acts. Diagrammatically, fused states are connectivity diagrams drawn on a disc with two holes, one hole carrying each input family — exactly the intermediate states that arise in lattice correlation functions. The authors prove that $\times_f$ is a bifunctor (a construction defined both on pairs of objects and on pairs of homomorphisms, respecting composition) that is distributive over direct sums, commutative, and associative, with the vacuum family $\mathsf{V}$ — whose modules $\mathsf{V}(N)$ are spanned by link states with no marked point — as neutral element. The associativity proof goes through a triple fusion product with the explicit presentation (4.12)–(4.13): for $N_{abc}=N_a+N_b+N_c$,

$$(\mathsf{M}_a \times_f \mathsf{M}_b \times_f \mathsf{M}_c)(N) = \sum_{N_a,N_b,N_c} \mathcal{L}(N,N_{abc})\cdot\big(\mathsf{M}_a(N_a)\otimes\mathsf{M}_b(N_b)\otimes\mathsf{M}_c(N_c)\big)$$

with seven types of generator-entry relations, and Theorem 4 establishes $(\mathsf{M}_a\times_f\mathsf{M}_b)\times_f\mathsf{M}_c \simeq \mathsf{M}_a\times_f\mathsf{M}_b\times_f\mathsf{M}_c \simeq \mathsf{M}_a\times_f(\mathsf{M}_b\times_f\mathsf{M}_c)$, where two families are isomorphic when their modules are isomorphic at every admissible size. A noteworthy subtlety recorded in the paper: the presentation induced by iterating the two-fold relations involves ten types of relations, while the direct three-fold presentation needs only seven, so the isomorphism identifies genuinely different presentations and is not a formal triviality.

The paper closes by conjecturing the analogous statement for an arbitrary number of input families — the open problem stated below — and by announcing, as the authors' own work in preparation (not part of the conjecture), applications of the fusion functor to explicit decompositions of the families $\mathsf{W}_{k,x}$ and their quotients, and to lattice connectivity correlation functions.

Problem Statement

Prove or refute the following conjecture of Ikhlef and Morin-Duchesne (arXiv:2509.11756, Section 5). Let $m\ge 4$ and let $\mathsf{M}_1,\dots,\mathsf{M}_m$ be arbitrary families of modules over the enlarged periodic Temperley–Lieb algebra $\mathcal{E}PTL_N(\beta)$, for any $\beta\in\mathbb{C}$. For admissible sizes $N_i$ write $\widehat{N}_i=\sum_{k=1}^{i}N_k$ for the partial sums ($\widehat{N}_0=0$). Define the $m$-fold fusion family by

$$(\mathsf{M}_1 \times_f \mathsf{M}_2 \times_f \cdots \times_f \mathsf{M}_m)(N) = \sum_{N_1,\dots,N_m} \mathcal{L}(N,\widehat{N}_m)\cdot\big(\mathsf{M}_1(N_1)\otimes\mathsf{M}_2(N_2)\otimes\cdots\otimes\mathsf{M}_m(N_m)\big),$$

where the sum runs over all admissible $N_1,\dots,N_m$, the action of $\mathcal{L}(N',N)$ is by left multiplication, and the quotient is taken subject to the generator-entry relations, for all $u_i\in\mathsf{M}_i(N_i)$:

$$c_{j+\widehat{N}_{i-1}}\cdot(u_1\otimes\cdots\otimes u_m) = u_1\otimes\cdots\otimes c_j u_i\otimes\cdots\otimes u_m, \qquad 1\le j\le N_i-1,\; 1\le i\le m,$$ $$c^{\dagger}_{j+\widehat{N}_{i-1}}\cdot(u_1\otimes\cdots\otimes u_m) = u_1\otimes\cdots\otimes c^{\dagger}_j u_i\otimes\cdots\otimes u_m, \qquad 1\le j\le N_i+1,\; 1\le i\le m,$$ $$\Omega\cdot(u_1\otimes\cdots\otimes u_m) = c_{\widehat{N}_{m-1}}\cdots c_{\widehat{N}_2}c_{\widehat{N}_1}\cdot\big(\Omega u_1\otimes c^{\dagger}_0 u_2\otimes c^{\dagger}_0 u_3\otimes\cdots\otimes c^{\dagger}_0 u_m\big), \qquad N_1>0.$$

The conjecture asserts that (i) this presentation is well defined — the relations are consistent and the resulting quotients form a family of modules over the diagram spaces — and (ii) the iterated two-fold fusion

$$\big(\big(\big((\mathsf{M}_1 \times_f \mathsf{M}_2) \times_f \mathsf{M}_3\big) \times_f \mathsf{M}_4\big) \times_f \cdots\big) \times_f \mathsf{M}_m$$

is isomorphic to it as a family of modules, i.e. the modules are isomorphic at every admissible size $N$. (The source states the conjecture as the isomorphism (ii), taking the $m$-fold presentation as given; part (i) makes that presentation explicit and is the $m$-fold analogue of Proposition 4.1, which establishes the corresponding trifunctor for $m=3$. This task requires establishing both (i) and (ii).) The case $m=3$ is Theorem 4 of the source paper and is not part of the open problem. The authors' suggested route is to extend the diagrammatic framework to connectivity diagrams on discs with an arbitrary number of holes, including the proof that the algebraic and diagrammatic definitions of these diagram spaces are equivalent (generalizing their Theorems 1–3), but a proof by any rigorous route — diagrammatic, inductive, or categorical — is acceptable, as is a refutation by explicit counterexample.

References

1. Yacine Ikhlef and Alexi Morin-Duchesne, Fusion in the periodic Temperley–Lieb algebra: general definition of a bifunctor, SciPost Phys. Core 9, 053 (2026), doi:10.21468/SciPostPhysCore.9.3.053; preprint arXiv:2509.11756, https://arxiv.org/abs/2509.11756
2. Yacine Ikhlef and Alexi Morin-Duchesne, Fusion in the periodic Temperley–Lieb algebra and connectivity operators of loop models, SciPost Phys. 12, 030 (2022), doi:10.21468/SciPostPhys.12.1.030; preprint arXiv:2105.10240, https://arxiv.org/abs/2105.10240
3. Yacine Ikhlef and Alexi Morin-Duchesne, Fusion of irreducible modules in the periodic Temperley–Lieb algebra, SciPost Phys. 17, 132 (2024), doi:10.21468/SciPostPhys.17.5.132; preprint arXiv:2312.14837, https://arxiv.org/abs/2312.14837
4. Yacine Ikhlef, Non-invertible symmetries and modular invariance in lattice models, arXiv:2606.31313 (2026), https://arxiv.org/abs/2606.31313
5. Yacine Ikhlef and Alexi Morin-Duchesne, Temperley–Lieb modules and local operators for critical ADE models, arXiv:2602.15742 (2026), https://arxiv.org/abs/2602.15742
6. Jesper Lykke Jacobsen, Rongvoram Nivesvivat, Sylvain Ribault and Paul Roux, Three-point functions in critical loop models, SciPost Phys. 20, 125 (2026), doi:10.21468/SciPostPhys.20.5.125; preprint arXiv:2510.04701, https://arxiv.org/abs/2510.04701