Title:

Conformal holonomy and theoretical gravitational physics

Conformal holonomy theory is the holonomy theory of the tractor connection on a conformal manifold. In this thesis, we present the first application of conformal holonomy theory to theoretical physics and determine the conformal holonomy groups/algebras of physically relevant spaces. After recalling some necessary background on conformal structures, tractor bundles and conformal holonomy theory in chapter 1, we begin in chapter 2 by discussing the role of conformal holonomy in the gaugetheoretic MacDowellMansouri formulation of general relativity. We show that the gauge algebra of this formulation is uniquely determined by the conformal structure of spacetime itself, in both Lorentzian and Riemannian metric signatures, through the conformal holonomy algebra. We then show that one may construct a MacDowellMansouri action functional for scaleinvariant gravity, and we discuss a geometric interpretation for the scalar field therein. In chapter 3 we study a class of spacetimes relevant to Maldacena's AdS5=CFT4 correspondence in quantum gravity. It is well known that a Lie group coincidence lies at the heart of this correspondence: the proper isometry group of the bulk precisely matches the conformal group of the boundary. It has previously been proposed that the AdS5=CFT4 correspondence be extended to socalled Poincar eEinstein spacetimes, which need not be as symmetric as antide Sitter space. We show that the conformal holonomy groups of the boundary and bulk furnish such a Lie group coincidence for 5dimensional Poincar eEinstein spacetimes in general. We completely characterise this boundarybulk conformal holonomy matching for the Riemannian theory and present partial results for the Lorentzian theory. In chapter 4 we use the tools developed in the preceding chapters to further the classiification of the conformal holonomy groups of conformally Einstein spaces. Specifically, we determine the conformal holonomy groups of generic neutral signature conformally Einstein 4manifolds subject to a condition on the conformal holonomy representation. Lastly, in chapter 5, we investigate the conformal holonomy reduction of the Fefferman conformal structures of residual twistor CR manifolds. A sufficient condition for reducible conformal holonomy is that the (Fefferman conformal structure of a) residual twistor CR manifold admit a parallel tractor. We show that this occurs if and only if the residual twistor CR manifold admits a Sasakian structure.
