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.