  It is shown that Wess-Zumino-Witten (WZW) type actions can be constructed in
odd dimensional space-times using Wilson line or Wilson loop. WZW action
constructed using Wilson line gives anomalous gauge variations and the WZW
action constructed using Wilson loop gives anomalous chiral transformation. We
show that pure gauge theory including Yang-Mills action, Chern-Simons action
and the WZW action can be defined in odd dimensional space-times with even
dimensional boundaries. Examples in 3D and 5D are given. We emphasize that this
offers a way to generalize gauge theory in odd dimensions. The WZW action
constructed using Wilson line can not be considered as action localized on
boundary space-times since it can give anomalous gauge transformations on
separated boundaries. We try to show that such WZW action can be obtained in
the effective theory when making localized chiral fermions decouple.
