Resumen de Physics and geometry of gravity at high energies
In this thesis we study physical and geometric aspects of gravity at high energies. On the one hand, we carry out a detailed investigation of gravitational physics in this regime with the aid of higher-order gravities. These are extensions of General Relativity including terms in higher derivatives of the metric and other fields which, in addition to an effective field theory interpretation, possess as well an intrinsic interest by themselves. More concretely, we focus on higher-order gravities of the (Generalized) Quasitopological class, defined as those admitting static and spherically symmetric solutions characterized by a single function satisfying an equation of, at most, second order. The motivation for this is twofold: they are amenable to computations and, at the same time, are generic enough so as to capture effects and phenomena introduced by higher-order corrections, which one may use to learn properties of a putative theory of Quantum Gravity.
First, we restrict ourselves to theories of pure gravity and show that all higher-order gravities can be mapped, via (perturbative) field redefinitions, to a Generalized Quasitopological Gravity. Secondly, we consider the addition of a U(1) gauge vector field and identify infinite families of Electromagnetic (Generalized) Quasitopological Gravities (E(G)QGs). We establish several intriguing properties of these theories and explore their charged, static and spherically symmetric solutions. In particular, we prove that a subset of EQGs allows for completely regular electrically-charged black holes for arbitrary mass and non-vanishing charge. Next, we move to the analysis of higher-derivative extensions of Einstein-Maxwell theory which are duality-invariant. We classify all such theories up to eight derivatives and find that, up to the six-derivative level, they all can be mapped via field redefinitions to a higher-curvature gravity with a minimally coupled vector. Also, we are able to classify all exactly duality-invariant theories which are quadratic in the Maxwell field strength. Afterwards, we revisit EQGs and examine some of their holographic aspects. We manage to obtain fully analytic and non-perturbative results that motivate us to discover a new universal result valid for all d(>=3)-dimensional CFTs, which we rigorously prove.
On the other hand, we study geometric properties of gravity at high energies. We choose Supergravity and String Theory as the scenarios in which to test such properties and we inspect distinct topics on the subject, in an attempt to form a global picture of the type of geometric structures we may encounter in this context.
We start by exploring real parallel spinors on globally hyperbolic four-manifolds. We are able to reformulate the problem in terms of a system of differential equations for a family of functions and coframes on a Cauchy surface that we call the parallel spinor flow. Remarkably, we find that the parallel spinor and the Einstein flows coincide on common initial data, thus providing an initial data characterization of a real parallel spinor on a Ricci flat globally hyperbolic four-manifold. Then, we investigate self-dual Einstein four-manifolds admitting a principal and isometric action of the three-dimensional Heisenberg group with non-degenerate orbits and manage to classify all of them. Finally, we introduce ε-contact structures, which encompass the usual notions of (three-dimensional) contact Riemannian, contact Lorentzian and para-contact metric structures, but also allow for a lightlike Reeb vector field. We show explicitly how they can be used for the construction of solutions of six-dimensional Supergravity.
