$U_q(\mathfrak{sl}_n)$ web models and $\mathbb{Z}_n$ spin interfaces

TL;DR

This paper introduces a lattice model of $U_q( ext{sl}_n)$ webs on the honeycomb lattice, revealing a connection to $ ext{Z}_n$ spin models and extending known loop models to higher $n$ with non-local weights.

Contribution

It defines a new $U_q( ext{sl}_n)$ web model on the honeycomb lattice and establishes its relation to $ ext{Z}_n$-symmetric spin models at a specific quantum parameter.

Findings

Model reduces to O(N) loop model for n=2.

Partition function relates to $ ext{Z}_n$-spin models at special q.

Web configurations correspond to domain walls in spin models.

Abstract

This is the first in a series of papers devoted to generalisations of statistical loop models. We define a lattice model of $U_q(\mathfrak{sl}_n)$ webs on the honeycomb lattice, for $n \ge 2$. It is a statistical model of closed, cubic graphs with certain non-local Boltzmann weights that can be computed from spider relations. For $n=2$, the model has no branchings and reduces to the well-known O($N$) loop model introduced by Nienhuis. In the general case, we show that the web model possesses a particular point, at $q=e^{i \pi/(n+1)}$, where the partition function is proportional to that of a $\mathbb{Z}_n$-symmetric chiral spin model on the dual lattice. Moreover, under this equivalence, the graphs given by the configurations of the web model are in bijection with the domain walls of the spin model. For $n=2$, this equivalence reduces to the well-known relation between the Ising and O($1$) models. We define as well an open $U_q(\mathfrak{sl}_n)$ web model on a simply connected domain with a boundary, and discuss in particular the role of defects on the boundary.