The Diffeomorphism Field

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.