From pure Yang-Mills action for the SL(5, R) group in four Euclidean dimensions we obtain a gravity theory in the first order formalism. Besides the Einstein-Hilbert term, the effective gravity has a cosmological constant term, a curvature squared term, a torsion squared term and a matter sector. To obtain such geometrodynamical theory, asymptotic freedom and the Gribov parameter (soft BRST symmetry breaking) are crucial. Particularly, Newton and cosmological constant are related to these parameters and they also run as functions of the energy scale. One-loop computations are performed and the results are interpreted.

The equivalence between Chern-Simons and Einstein-Hilbert actions in three dimensions established by A. Achúcarro and P. K. Townsend (1986) and E. Witten (1988) is generalized to the off-shell case. The technique is also generalized to the Yang-Mills action in four dimensions displaying de Sitter gauge symmetry. It is shown that, in both cases, we can directly identify a gravity action while the gauge symmetry can generate spacetime local isometries as well as diffeomorphisms. The price we pay for working in an off-shell scenario is that specific geometric constraints are needed. These constraints can be identified with foliations of spacetime. The special case of spacelike leafs evolving in time is studied. Finally, the whole set up is analyzed under fiber bundle theory. In this analysis we show that a traditional gauge theory, where the gauge field does not influence in spacetime dynamics, can be (for specific cases) consistently mapped into a gravity theory in the first order formalism.

In the works of A. Achúcarro and P. K. Townsend and also by E. Witten, a duality between threedimensional Chern-Simons gauge theories and gravity was established. First (Achúcarro and Townsend), by considering an Inönü-Wigner contraction from a superconformal gauge theory to an Anti-de Sitter supergravity. Then, Witten was able to obtain, from Chern-Simons theory (in two cases: Poincaré and de Sitter gauge theories), an Einstein-Hilbert gravity by mapping the gauge symmetry in local isometries and diffeomorphisms. In all cases, the results made use of the field equations. Latter, we were capable to generalize Witten's work (in Euclidean spacetime) to the off-shell cases, as well as to four dimensional Yang-Mills theory with de Sitter gauge symmetry. The price we paid is that curvature and torsion must obey some constraints under the action of the interior derivative. These constraints implied on the partial breaking of diffeomorphism invariance. In the present work, we, first, formalize our early results in terms of fiber bundle theory by establishing the formal aspects of the map between a principal bundle (gauge theory) and a coframe bundle (gravity) with partial breaking of diffeomorphism invariance. Then, we study the effect of the constraints on the homology defined by the interior derivative. The result being the emergence of a nontrivial homology in Riemann-Cartan manifolds.