  Integration of Kirillov form on a coadjoint orbit of Virasoro algebra yields
the coupling of a background field to Polyakov's two dimensional quantum
gravity. This background field is used to be called the diffeomorphism field.
Einstein's gravity is dynamically trivial in two dimensions. Moreover, the
diffeomorphism field can be interpreted as the gravitational analog of a
Yang-Mills field. This raises the question of whether the diffeomorphism field
exists in higher dimensions, and plays an essential role in gravity. With this
motivation, several dynamical theories for the diffeomorphism field have been
constructed by mimicking construction of Yang-Mills theory from Kac-Moody
algebra.
  This thesis constitutes a further development for obtaining a consistent
dynamical theory of the diffeomorphism field. The previously proposed theories
are thoroughly examined, certain subtleties and problems in them have been
discovered and made apparent. Some of these problems have been solved, and for
others possible routes to follow have been laid down. Finally, alternative
geometric approaches are investigated.
