The diffeomorphism field is introduced to the physics literature in [1] where it arises as a background field coupled to Polyakov’s quantum gravity in two dimensions, where Einstein’s gravity is trivial. Moreover, it is seen in many ways as the gravitational analog of the Yang-Mills field. This raises the question of whether the diffeomorphism field exists in higher dimensions, playing an essential role in gravity either by supplementing Einstein’s theory or by modifying it. With this motivation, several distinct theories governing the dynamics of the diffeomorphism field have been constructed and developed by mimicking the construction of the Yang-Mills theory from the Kac-Moody algebra. This analogy, however, is not perfect and there are many subtleties and difficulties encountered. This thesis constitutes a further development. The previously proposed theories are carefully 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, other geometric approaches than the ones followed before are investigated.

Coadjoint orbits of Lie algebras come naturally imbued with a symplectic two-form allowing for the construction of dynamical actions. Consideration of the coadjoint orbit action for the Kac-Moody algebra leads to the Wess-Zumino-Witten model with a gauge-field coupling. Likewise, the same type of coadjoint orbit construction for the Virasoro algebra gives Polyakov’s 2D quantum gravity action with a coupling to a coadjoint element, D, interpreted as a component of a field named the diffeomorphism field. Gauge fields are commonly given dynamics through the Yang-Mills action and, since the diffeomorphism field appears analogously through the coadjoint orbit construction, it is interesting to pursue a dynamical action for D. This thesis reviews the motivation for the diffeomorphism field as a dynamical field and presents results on its dynamics obtained through projective connections. Through the use of the projective connection of Thomas and Whitehead, it will be shown that the diffeomorphism field naturally gains dynamics. Results on the analysis of this dynamical theory in two-dimensional Minkowski background will be presented.