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

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.

arXiv comment31 pages

References in corpus (26)

• Fault-tolerant quantum computation by anyons
• String-net condensation: A physical mechanism for topological phases
• Spiders for rank 2 Lie algebras
• SLE for theoretical physicists
• Exact exponents for the spin quantum Hall transition
• Webs and quantum skew Howe duality
• Operator Spectrum and Exact Exponents of the Fully Packed Loop Model
• Field theory of compact polymers on the square lattice
• Topological order from quantum loops and nets
• A colored sl(N)-homology for links in S^3
• A Diagrammatic Category for the Representation Theory of U_q(sl_n)
• Conformal boundary conditions in the critical O(n) model and dilute loop models
• Multi-Colour Braid-Monoid Algebras
• SLE in the three-state Potts model - a numerical study
• Stochastic Loewner evolution for conformal field theories with Lie-group symmetries
• Critical exponents of domain walls in the two-dimensional Potts model
• Conformational Entropy of Compact Polymers
• SLE in self-dual critical Z(N) spin systems: CFT predictions
• Bulk and boundary critical behaviour of thin and thick domain walls in the two-dimensional Potts model
• Exact solution of the anisotropic special transition in the O(n) model in 2D
• The fully packed loop model as a non-rational $W_3$ conformal field theory
• Critical interfaces and duality in the Ashkin Teller model
• The Bubble Algebra: Structure of a Two-Colour Temperley-Lieb Algebra
• Phase diagram of the triangular-lattice Potts antiferromagnet
• Dilute Algebras and Solvable Lattice Models
• The quantum $\mathfrak{sl}(n,\mathbb{C})$ representation theory and its applications

Cited by in corpus (4)

• Geometric Algebra and Algebraic Geometry of Loop and Potts Models
• Quantum Groups as Global Symmetries
• On the CFT describing the spin clusters in 2d Potts model
• $U_{\mathfrak{q}}({\mathfrak{sl}}_3)$ web models: Locality, phase diagram and geometrical defects