Kac-Moody Symmetries of Critical Ground States

TL;DR

This paper explores the Kac-Moody symmetries of critical ground states in 2D lattice models, linking lattice models to conformal field theories via vertex operator constructions, revealing specific algebraic structures.

Contribution

It demonstrates how to identify Kac-Moody symmetry algebras of critical models using a mapping to interface models and vertex operator methods, providing explicit examples.

Findings

Critical models are described by specific Kac-Moody algebras.

The approach applies to six-vertex, three-coloring, and four-coloring models.

Symmetry algebras identified are $su(2)_{k=1}$, $su(3)_{k=1}$, and $su(4)_{k=1}$.

Abstract

The symmetries of critical ground states of two-dimensional lattice models are investigated. We show how mapping a critical ground state to a model of a rough interface can be used to identify the chiral symmetry algebra of the conformal field theory that describes its scaling limit. This is demonstrated in the case of the six-vertex model, the three-coloring model on the honeycomb lattice, and the four-coloring model on the square lattice. These models are critical and they are described in the continuum by conformal field theories whose symmetry algebras are the $su(2)_{k=1}$, $su(3)_{k=1}$, and the $su(4)_{k=1}$ Kac-Moody algebra, respectively. Our approach is based on the Frenkel--Kac--Segal vertex operator construction of level one Kac--Moody algebras.