  This paper describes a semiclassical field-theory approach to the topological
properties of spatially featureless Affleck-Kennedy-Lieb-Tasaki type valence
bond solid ground states of antiferromagnets in spatial dimensions one to
three. Using nonlinear sigma models set in the appropriate target manifold and
augmented with topological terms, we argue that the path integral
representation of the ground-state wave functional can correctly distinguish
symmetry-protected topological ground states from topologically trivial ones.
The symmetry-protection feature is demonstrated explicitly in terms of a dual
field theory, where we take into account the nontrivial spatial structure of
topological excitations, which are caused by competition among the relevant
ordering tendencies. A temporal surface contribution to the action originating
from the bulk topological term plays a central role in our study. We discuss
how the same term governs the behavior of the so-called strange correlator. In
particular, we find that the path integral expression for the strange
correlator in two dimensions reduces to the well-known Haldane expression for
the two point spin correlator of antiferromagnetic spin chains.
