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# Uq(sl3) web models: Locality, phase diagram and geometrical defects

1 Champs Aléatoires et Systèmes hors d'Équilibre
LPENS - Laboratoire de physique de l'ENS - ENS Paris
Abstract : We continue investigating the generalisations of geometrical statistical models introduced in [13], in the form of models of webs on the hexagonal lattice $H$ having a $Uq(sln)$ quantum group symmetry. We focus here on the $n=3$ case of cubic webs, based on the Kuperberg $A2$ spider, and illustrate its properties by comparisons with the well-known dilute loop model (the $n=2$ case) throughout. A local vertex-model reformulation is exhibited, analogous to the correspondence between the loop model and a three-state vertex model. The $n=3$ representation uses seven states per link of $H$, displays explicitly the geometrical content of the webs and their $Uq(sl3)$ symmetry, and permits us to study the model on a cylinder via a local transfer matrix. A numerical study of the effective central charge reveals that for $q=−eiγ$, in the range $γ∈[0,π/3]$, the web model possesses a dense and a dilute critical point, just like its loop model counterpart. In the dense $q=−eiπ/4$ case, the $n=3$ webs can be identified with spin interfaces of the critical three-state Potts model defined on the triangular lattice dual to $H$. We also provide another mapping to a $Z3$ spin model on $H$ itself, using a high-temperature expansion. We then discuss the sector structure of the transfer matrix, for generic $q$, and its relation to defect configurations in both the strip and the cylinder geometries. These defects define the finite-size precursors of electromagnetic operators. This discussion paves the road for a Coulomb gas description of the conformal properties of defect webs, which will form the object of a subsequent paper. Finally, we identify the fractal dimension of critical webs in the $q=−eiπ/3$ case, which is the $n=3$ analogue of the polymer limit in the loop model.
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https://hal.archives-ouvertes.fr/hal-03315296
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Submitted on : Thursday, August 5, 2021 - 3:30:23 PM
Last modification on : Tuesday, December 6, 2022 - 4:05:13 AM

### Citation

Augustin Lafay, Azat M. Gainutdinov, Jesper Lykke Jacobsen. Uq(sl3) web models: Locality, phase diagram and geometrical defects. Nucl.Phys.B, 2022, 979, pp.115789. ⟨10.1016/j.nuclphysb.2022.115789⟩. ⟨hal-03315296⟩

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