As noted above, quantum gravity in three spacetime dimensions has no propagating degrees of freedom. This makes it a powerful model for exploring some kinds of conceptual issues, but many other important questions can't be addressed.
One step more complicated is a model called "topologically massive gravity," proposed by Deser, Jackiw, and Templeton in 1982. This model modifies the field equations of general relativity by adding a new term with three derivatives. This is normally a dangerous thing to do -- "higher derivative" theories in physics usually have negative energies and no stable solutions -- but in this special case it is consistent. In a different context, the extra term is "topological", that is, it depends only on the topology of spacetime and not the particular geometry; hence the somewhat confusing name.
The addition of a higher-derivative term in the field equations changes the counting of degrees of freedom of a theory. For topologically massive gravity, the effect is to add a new, propagating degree of freedom, a sort of massive gravitational wave, or, in the quantum theory, a massive graviton. Recently, the model has been a subject of renewed attention because of its interesting properties in anti-de Sitter space, where it has become a testing ground for the AdS/CFT correspondence of string theory.
- Steve Carlip (Physics, UC Davis)