Gravity as a gauge theory for the conformal group and the cosmological constant.

1. Gravity as a gauge theory for the conformal group and the cosmological constant. A. Anabalon Centro de Estudios Científicos, CECS. Universidad de Concepción. S. Willison, J. Zanelli Centro de Estudios Científicos, CECS. hep-th/0610136 hep-th/0702192

2. The objective Generalize four dimensional Einstein gravity to the status of a gauge theory, with a principal bundle structure (a la Yang-Mills).

3. Wedge products are omitted throughout All the lie Algebras considered are of the so(p,q) kind

4. Three dimensional gravity Thus, three dimensional gravity can be writen as a gauge theory for the SO(2,2) group.

5. When formulated as a gauge theory, three dimensional gravity has in general no metric interpretation, however is power counting renormalizable. In four dimensions the same construction can not be done. Instead, as will be shown, the theory must be completed with some other fields to acquire the same kind of structure that takes place in three dimensions. In order to generalize the phenomena note that

6. The transgression form reduce to the CS theory when one of the connections is set to zero. If instead we fix

8. A four dimensional field theory, gauge invariant under the SO(p,q) group, with p+q=6 has been introduced. The theory is gauge invariant, no extra constraints are required to ensure the closure of the symmetry algebra, it has only one, dimensionless, free parameter in front of the action and non geometric configuration makes sense in this description. In what follows, it will be shown how a subsector of this theory is Einstein’s gravity.

9. The Einstein dynamical sector The conformal group is particularly interesting, since contains as a subgroup, AdS, dS and Poincare, this motivates to pick G=SO(4,2). Assigning the indexes a,b=0…3 to the tangent space of the manifold, the connection is descomposed in Lorentz irreducible parts

10. It is possible to show that the group element, h, are pure Goldstone fields and, as in the Higgs mechanism, by a redefinition of the gauge field it can be reduced toThis simplifies the field equations enough to be written by components, in a Lorentz covariant way.

11. The Unitary gauge has reduced the gauge freedom to Lorentz time dilatations, furthermore, fixing b (c) to zero the dilatation invariance is broken by the background configuration and Einstein gravity with a possitive (negative) cosmological constant is recovered: In the same anzats the gauged WZW lagrangian becomes the Einstein-Hilbert one