“Geometrical web models”

Jesper Jacobsen

We introduce a family of geometrical lattice models generalising the well-known loop model on the hexagonal lattice. These models have a $U_q(sl_n)$ quantum group symmetry, the loop model being the $n=2$ case. The general models give rise to branching webs and describe, at a special point, the interfaces in $Z_n$ symmetric spin models. We mainly discuss the $n=3$ case of bipartite cubic webs, which is based on the Kuperberg $A_2$ spider. We exhibit a local vertex-model reformulation, analogous to the well-known correspondence between the loop model and the nineteen-vertex model. The local formulation allows us in particular to study the model by means of transfer matrices and conformal field theory. We find that it has a rich phase diagram, including a dense and a dilute phase that generalise those known for the loop model.

http://scgp.stonybrook.edu/video_portal/video.php?id=4889

Simons- Program: Probability, integrability, and conformal invariance