  Thomas-Whitehead (TW) gravity is a projectively invariant model of gravity
over a d-dimensional manifold that is intimately related to string theory
through reparameterization invariance. Unparameterized geodesics are the
ubiquitous structure that ties together string theory and higher dimensional
gravitation. This is realized through the projective geometry of Tracy Thomas.
The projective connection, due to Thomas and later Whitehead, admits a
component that in one dimension is in one-to-one correspondence with the
coadjoint elements of the Virasoro algebra. This component is called the
diffeomorphism field $\mathcal{D}_{ab }$ in the literature. It also has been
shown that in four dimensions, the TW\ action collapses to the Einstein-Hilbert
action with cosmological constant when $\mathcal{D}_{ab}$ is proportional to
the Einstein metric. These previous results have been restricted to either
particular metrics, such as the Polyakov 2D\ metric, or were restricted to
coordinates that were volume preserving. In this paper, we review TW gravity
and derive the gauge invariant TW action that is explicitly projectively
invariant and general coordinate invariant. We derive the covariant field
equations for the TW action and show how fermionic fields couple to the gauge
invariant theory. The independent fields are the metric tensor $g_{ab}$, the
fundamental projective invariant $\Pi^{a}_{\,\,\,bc}$, and the diffeomorphism
field $\mathcal D_{ab}$.
