Quadratic gravity theories in 2+1 dimensions and the topological Chern-Simons term

Abstract

It is astonishing that a locally trivial and globally nontrivial Einstein-type gravitational theory can be built only by lowering the dimension of our space-time—which is supposed to be four dimensional—by one unit. In fact, threedimensional general relativity is dynamically trivial: outside sources, spacetime is flat—all effects of the localized sources are on the global geometry, which is fixed by the singularities of the worldlines of the particles @1#. The quantum mechanical analogue of this triviality emerges when the HilbertEinstein action related to this theory is quantized. It is easy to establish that the theory does not possess any propagating degrees of freedom. In other words, there are no gravitons. The preceding considerations lead us to raise the following important question: Is it possible to build a nontrivial generally covariant three-dimensional gravity theory having propagating degrees of freedom? The answer is affirmative. Actually, this can be done at least in two different ways: ~i! Adding a topological nontrivial term to general relativity in ~211!D @2#. Ordinary Einstein gravity which is trivial acquires now a propagating, massive, spin-2 mode. This theory is ghost-free and causal, although of the thirdderivative order. ~ii! Including the four-derivatives terms *R 2 Agd 3 x and