Is there some way to make 2+1 quantum gravity non-topological?

Question

Is there some modification that makes 2+1 dimensional quantum gravity loose its topogical nature? If yes what is this contruction, how is it motivated and what lesson are there to learn for 3+1 d?

Answer 1

Gravity as a theory of self-interacting spin-two field has zero number of physical degrees of freedom in 3d. No modification can alter this fact. Instead, you can consider, without any physical motivation though, some theories that are vaguely related to or inspired by gravity, e.g. massive gravity, new massive gravity etc.