  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.

Condensed Matter;High Energy Physics - Theory