  We show that all $so(N)_1$ universality class quantum criticalities emerge
when one-dimensional generalized cluster models are perturbed with Ising or
Zeeman terms. Each critical point is described by a low-energy theory of $N$
linearly dispersing fermions, whose spectrum we show to precisely match the
prediction by $so(N)_1$ conformal field theory. Furthermore, by an explicit
construction we show that all the cluster models are dual to non-locally
coupled transverse field Ising chains, with the universality of the $so(N)_1$
criticality manifesting itself as $N$ of these chains becoming critical. This
duality also reveals that the symmetry protection of cluster models arises from
the underlying Ising symmetries and it enables the identification of local
representations for the primary fields of the $so(N)_1$ conformal field
theories. For the simplest and experimentally most realistic case that
corresponds to the original one-dimensional cluster model with local three-spin
interactions, our results show that the $su(2)_2 \simeq so(3)_1$
Wess-Zumino-Witten model can emerge in a local, translationally invariant and
Jordan-Wigner solvable spin-1/2 model.
