Abstract
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 $\mathcal D_{ab}$.
Highlights
The geometric classification of manifolds via their geodesics as opposed to distances between points is an old notion. In his inaugural professorial lecture at Cambridge University in 1863, Cayley remarked that “descriptive geometry includes metrical geometry” and “descriptive geometry is all geometry” [1]
We will conclude with remarks on geodesic deviations as it is there that contributions through gravitational radiation may become manifest
String theory may be thought of as originating from regulating Feynman diagrams in gravitational theories, by adding a tiny dimension to the point particle as initial data. This regulator quickly takes on a life of its own through the Virasoro algebra, which maintains the reparametrization invariance
Summary
The geometric classification of manifolds via their geodesics as opposed to distances between points (metrical) is an old notion In his inaugural professorial lecture at Cambridge University in 1863, Cayley remarked that “descriptive geometry includes metrical geometry” and “descriptive geometry is all geometry” [1]. The relationship between string theory and the Virasoro algebra has an even more primitive origin through its identity as a one-dimensional vector space [13] and projective structure [2,14,15]. It was suggested in [19] that the coadjoint elements of the Virasoro sector could be put on an equivalent footing with the Kac-Moody sector if the coadjoint elements of the Virasoro algebra could have an associated “gauge” field in higher dimensions. We will conclude with remarks on geodesic deviations as it is there that contributions through gravitational radiation may become manifest
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
