General structure of Thomas-Whitehead gravity

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