  In this paper we link the physics of topological nonlinear {\sigma} models
with that of Chern-Simons insulators. We show that corresponding to every
2n-dimensional Chern-Simons insulator there is a (n-1)-dimensional topological
nonlinear {\sigma} model with the Wess-Zumino-Witten term. Breaking internal
symmetry in these nonlinear {\sigma} models leads to nonlinear {\sigma} models
with the {\theta} term. [This is analogous to the dimension reduction leading
from 2n-dimensional Chern-Simons insulators to (2n-1) and (2n-2)-dimensional
topological insulators protected by discrete symmetries.] The correspondence
described in this paper allows one to derive the topological term in a theory
involving fermions and order parameters (we shall referred to them as
"fermion-{\sigma} models") when the conventional gradient-expansion method
fails. We also discuss the quantum number of solitons in topological nonlinear
{\sigma} model and the electromagnetic action of the (2n-1)-dimensional
topological insulators. Throughout the paper we use a simple model to
illustrate how things work.
