Effective gravity from a quantum gauge theory in Euclidean spacetime

Abstract

We consider a SO(d) gauge theory in a Euclidean d-dimensional spacetime, which is known to be renormalizable to all orders in perturbation theory for 2 ⩽ d ⩽ 4. Then, with the help of a spacetime representation of the gauge group, the gauge theory is mapped into a curved spacetime with a linear connection. Furthermore, in this mapping the gauge field plays the role of the linear connection of the curved spacetime and an effective metric tensor arises naturally from the mapping. The obtained action, being quadratic in the Riemann–Christoffel tensor, at first sight, spoils a gravity interpretation of the model. Thus, we provide a sketch of a mechanism that breaks the SO(d) color invariance and generates the Einstein–Hilbert term, as well as a cosmological constant term, allowing an interpretation of the model as a modified gravity in the Palatini formalism. In that sense, gravity can be visualized as an effective classical theory, originated from a well-defined quantum gauge theory. We also show that, in the four-dimensional case, two possibilities for particular solutions of the field equations are the de Sitter and anti-de Sitter spacetimes.