  We identify two-dimensional three-state Potts paramagnets with gapless edge
modes on a triangular lattice protected by $(\times Z_3)^3\equiv Z_3\times
Z_3\times Z_3$ symmetry and smaller $Z_3$ symmetry. We derive microscopic
models for the gapless edge, uncover their symmetries, and analyze the
conformal properties. We study the properties of the gapless edge by employing
the numerical density-matrix renormalization group (DMRG) simulation and exact
diagonalization. We discuss the corresponding conformal field theory, its
central charge, and the scaling dimension of the corresponding primary field.
We argue that the low energy limit of our edge modes is defined by the
$SU_k(3)/SU_k(2)$ coset conformal field theory with the level $k=2$. The
discussed two-dimensional models realize a variety of symmetry-protected
topological phases, opening a window for studies of the unconventional quantum
criticalities between them.
