Cartan gravity, matter fields, and the gauge principle

Abstract

Gravity is commonly thought of as one of the four force fields in nature. However, in standard formulations its mathematical structure is rather different from the Yang-Mills fields of particle physics that govern the electromagnetic, weak, and strong interactions. This paper explores this dissonance with particular focus on how gravity couples to matter from the perspective of the Cartan-geometric formulation of gravity. There the gravitational field is represented by a pair of variables: 1) a 'contact vector' V A which is geometrically visualized as the contact point between the spacetime manifold and a model spacetime being 'rolled' on top of it, and 2) a gauge connection A AB µ , here taken to be valued in the Lie algebra of SO(2, 3) or SO(1, 4), which mathematically determines how much the model spacetime is rotated when rolled. By insisting on two principles, the gauge principle and polynomial simplicity, we shall show how one can reformulate matter field actions in a way that is harmonious with Cartan's geometric construction. This yields a formulation of all matter fields in terms of first order partial differential equations. We show in detail how the standard second order formulation can be recovered. Furthermore, the energy-momentum and spin-density three-forms are naturally combined into a single object here denoted the spin-energy-momentum three-form. Finally, we highlight a peculiarity in the mathematical structure of our first-order formulation of Yang-Mills fields. This suggests a way to unify a U (1) gauge field with gravity into a SO(1, 5)-valued gauge field using a natural generalization of Cartan geometry in which the larger symmetry group is spontaneously broken down to SO(1, 3) × U (1). The coupling of this unified theory to matter fields and possible extensions to non-Abelian gauge fields are left as open questions.

Constructing an explicit AdS/CFT correspondence with Cartan geometry

“…This is the choice used for the familiar MacDowell-Mansouri action [12,22,19]. In [18] Westman and Z lośnik use this type of approach to investigate the AdS/CFT correspondence. Above, however, we have chosen to build the action using tensors of the more restricted local H-symmetry of the quotient manifold.…”

“…Investigations into SUSY gravitational theories renewed interest in these techniques in the late 70s through the 80s [12,13,14,15,16]. For a modern construction see [8,17,18,19].…”

Gravitational behavior of an effective topological field theory in a gravitational instanton background

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Holographic correspondence in topological superconductors

“…belong to the Clifford representation of Spin (3,2), which is the double covering of SO(3, 2) [10,11]. With this representation one can show that the transformations (3) are the only non-trivial orthogonal ones that leave invariant the Dirac mass in action (2).…”