  We examine the effective field theory of the Bethe ansatz integrable
Heisenberg antiferromagnetic spin chains. It shows that the quantum critical
theories for the integer spin-S chains should be characterized by the
SO(3)level-S Wess-Zumino-Witten model, and classified by the third cohomology
group $H^{3}(SO(3),Z)=Z$. Depending on the parity of spin S, this integer
classification is further divided into two distinct universality classes, which
are associated with two completely different conformal field theories: the
even-S chains have gapless bosonic excitations and the odd-S chains have both
bosonic and fermionic excitations. We further show that these two classes of
critical states correspond to the boundary states of two distinct topological
phases in two dimension, which can be described by two-dimensional doubled
SO(3) topological Chern-Simons theory and topological spin theory,
respectively.
