Math Physics Seminar
Previously, it was shown that that there exists a correspondence between actions for field theories and symmetry groups. This better understood through the lens of coadjoint orbits. The coadjoint orbit action of the Kac-Moody and Virasoro semi-direct product algebra obtains the WZNW action and the Polyakov action. The pure Kac-Moody coadjoint element corresponds to the Yang-Mills gauge potential in higher dimen- sions and the pure Virasoro coadjoint element corresponds to the diffeomorphism field, a component of the Thomas-Whitehead connection (The connection on a volume bundle of the projective equivalence class of affine connections for a manifold M.) The dynamics of the diffeomorphism field is described by Thomas- Whitehead (TW) theory just as gauge potential is described by Yang-Mills theory. To start with, we will reformulate TW gravity in terms of fields that are both tensorial and projectively invariant, present their corresponding field equations, and show that the field equations obtained are equiva- lent across formulations, up to total divergences. In General Relativity, the effects of graviational radiation are understood through geodesic deviation. For now, we focus on the diffeomorphism field with a back- ground Minkowski metric and fundamental projective invariant from Levi Civita connection. The discussion will treat the diffeomorphism field in the form of the projective Schouten tensor, P. The field equations for P then support wavelike solutions that are transverse-traceless, longitudinal-traceless, and transverse scalar solutions. The field P will then be considered for supporting geodesic deviation independent of gab.