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In four dimensions one can use the chiral part of the spin connection as the main object that encodes geometry. The metric is then recovered algebraically from the curvature of this connection. We address the question of how isometries can be identified in this pure connection formalism. We show that isometries are recovered from gauge transformation parameters satisfying the requirement that the Lie derivative of the connection along a vector field generating an isometry is a gauge transformation. This requirement can be rewritten as a first order differential equation involving the gauge transformation parameter only. Once a gauge transformation satisfying this equation is found, the isometry generating vector field is recovered algebraically. We work out examples of the new formalism being used to determine isometries, and also prove a general statement: a negative definite connection on a compact manifold does not have symmetries. This is the precise pure connection analog of the well-known Riemannian geometry statement that there are no Killing vector fields on compact manifolds with negative Ricci curvature.
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We state and prove a simple Theorem that allows one to generate invariant quantities in Metric-Affine Geometry, under a given transformation of the affine connection. We start by a general functional of the metric and the connection and consider transformations of the affine connection possessing a certain symmetry. We show that the initial functional is invariant under the aforementioned group of transformations iff its $Gamma$-variation produces tensor of a given symmetry. Conversely if the tensor produced by the $Gamma$-variation of the functional respects a certain symmetry then the functional is invariant under the associated transformation of the affine connection. We then apply our results in Metric-Affine Gravity and produce invariant actions under certain transformations of the affine connection. Finally, we derive the constraints put on the hypermomentum for such invariant Theories.
We provide an exact mapping between the Galilian gauge theory, recently advocated by us cite{BMM1, BMM2, BM}, and the Poincare gauge theory. Applying this correspondence we provide a vielbein approach to the geometric formulation of Newtons gravity where no ansatze or additional conditions are required.
We present a conformal isometry for static extremal black hole solutions in all four-dimensional Einstein-Maxwell-scalar theories with electromagnetic duality groups `of type $E_7$. This includes, but is not limited to, all supergravity theories with $mathcal{N}>2$ supersymmetry and all $mathcal{N}=2$ supergravity theories with symmetric scalar manifolds. The conformal isometry is valid for arbitrary electromagnetic charge configurations and relies crucially on the notion of Freudenthal duality.
The main goal of the present work is to analyze the cosmological scenario of the induced gravity theory developed in previous works. Such a theory consists on a Yang-Mills theory in a four-dimensional Euclidian spacetime with $SO(m,n)$ such that $m+n=5$ and $min{0,1,2}$ as its gauge group. This theory undergoes a dynamical gauge symmetry breaking via an Inonu-Wigner contraction in its infrared sector. As a consequence, the $SO(m,n)$ algebra is deformed into a Lorentz algebra with the emergency of the local Lorentz symmetries and the gauge fields being identified with a vierbein and a spin connection. As a result, gravity is described as an effective Einstein-Cartan-like theory with ultraviolet correction terms and a propagating torsion field. We show that the cosmological model associated with this effective theory has three different regimes. In particular, the high curvature regime presents a de Sitter phase which tends towards a $Lambda$CDM model. We argue that $SO(m,n)$ induced gravities are promising effective theories to describe the early phase of the universe.
In a spacetime divided into two regions $U_1$ and $U_2$ by a hypersurface $Sigma$, a perturbation of the field in $U_1$ is coupled to perturbations in $U_2$ by means of the holographic imprint that it leaves on $Sigma$. The linearized gluing field equation constrains perturbations on the two sides of a dividing hypersurface, and this linear operator may have a nontrivial null space. A nontrivial perturbation of the field leaving a holographic imprint on a dividing hypersurface which does not affect perturbations on the other side should be considered physically irrelevant. This consideration, together with a locality requirement, leads to the notion of gauge equivalence in Lagrangian field theory over confined spacetime domains. Physical observables in a spacetime domain $U$ can be calculated integrating (possibly non local) gauge invariant conserved currents on hypersurfaces such that $partial Sigma subset partial U$. The set of observables of this type is sufficient to distinguish gauge inequivalent solutions. The integral of a conserved current on a hypersurface is sensitive only to its homology class $[Sigma]$, and if $U$ is homeomorphic to a four ball the homology class is determined by its boundary $S = partial Sigma$. We will see that a result of Anderson and Torre implies that for a class of theories including vacuum General Relativity all local observables are holographic in the sense that they can be written as integrals of over the two dimensional surface $S$. However, non holographic observables are needed to distinguish between gauge inequivalent solutions.
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