Free field representation of the Zamolodchikov-Faddeev algebra and form factors of the $SU(N) \times SU(N)$ Principal Chiral Field model

A free field representation of the Zamolodchikov-Faddeev algebra of the $SU(N) \times SU(N)$ Principal Chiral Field model is constructed, and used to derive an integral representation for form factors of a multi-parameter family of exponential fields.