The Use of Torsion in Supergravity Uplifts and Covariant Fractons

The aim of this Thesis is twofold. On the one hand, we find the necessary and sufficient conditions for a maximally supersymmetric supergravity theory in three dimensions to be a solution of eleven-dimensional supergravity (but the result is general and also holds for ten-dimensional supergravities), with eight dimensions compactified into a coset space. The used method is based on the formalism of generalised geometry, useful for the study of dualities in string theory and supergravity. The analysis extends the known results to the case in which the duality group of the reduced theory is E8(8) whose generalised geometry is still little understood. On the other hand, we study properties of the so-called covariant fracton gauge theory, computing the BRST cohomology and consistent anomalies, and showing that its solutions describe a specific subsector of an extension of General Relativity, called Møller-Hayashi-Shirafuji theory. Covariant fracton theory is the gauge theory of a symmetric rank-two tensor, invariant under gauge transformations depending on the second derivative of a scalar parameter, and is the Lorentz-covariant extension of the continuous limit of spin-chain theories admitting excitations with reduced mobility (called “fractons”), due to the conservation of the dipole moment. In both cases, the Weitzenböck torsion plays a crucial rôle. In the usual formulation of General Relativity, the spacetime curvature is responsible for the gravitational interaction. However, an alternative formulation exists, in which spacetime is flat and gravitation is an effect of the Weitzenböck torsion. In this context, the Møller-Hayashi-Shirafuji theory is the extension of General Relativity, in parallelisable spacetime, which spoils local Lorentz invariance. At the linearised level, it corresponds to add an antisymmetric rank-two tensor to the symmetric rank-two tensor describing the perturbation of the metric. Consistent reductions, studied by Scherk and Schwarz as an extension of the Kaluza-Klein compactifications, are reductions of (d+n)-dimensional (super)gravity theories to d-dimensional theories, the remaining n dimensions being compactified into an internal space, in such a way that all solutions of the reduced theory are also solutions of the original one. Consistency requires the internal geometry to be parallelisable with constant Weitzenböck torsion. The internal space is the gauge group of the reduced theory, and the torsion corresponds to the structure constants of the group algebra. Generalised geometry allows to consider cases where the internal space is not a group, by extending the notions of parallelisability and torsion. The torsion is identified with the so-called “embedding tensor”, which captures the gauge couplings of the reduced theory. The Thesis includes a self-contained review of the main notions needed to understand the original results, comprising, in addition to the already mentioned topics, duality in supergravity theories, the generalised Lie derivative, formulation of eleven-dimensional supergravity suitable to reductions with duality groups given by exceptional Lie groups, salient aspects of three-dimensional gravity, and the BRST formalism for computing anomalies in field theories.