Gravity as a deformed topological gauge theory

Abstract

We describe gauge theories which allow to retrieve a large class of gravitational theories, including, MacDowell-Mansouri gravity and its topological extension to Loop Quantum Gravity via the Pontrjagin characteristic class involving the Nieh-Yan term. Considering symmetric spaces parametrized by mutations allows to naturally obtain a bare cosmological constant which in particular cases gives rise to a positive effective cosmological constant while having an AdS spacetime. Two examples are studied, Lorentzian geometry (including dS and AdS spacetimes) and Lorentz$\times$Weyl geometry. In the latter case we prove the equations of motion exhibit a secondary source for curvature in addition to the usual energy-momentum tensor. This additional source is expressed in terms of the spin density of matter, torsion and their variations. Finally, we show that the gauge + matter actions constructed from invariant polynomials are asymptotically topological if one assumes a vanishing bare cosmological constant together with gauge and matter fields having compact support.