Quantum gravity in anti de Sitter space in three spacetime dimensions is conjectured to be dual to two dimensional conformal field theories (CFTs) with sparse spectra. In this paper, we study "pure" quantum gravity in three dimensions, which is in some sense the simplest place to study quantum gravity. These are theories with no light states, having only graviton degrees of freedom, and black holes states.
In a 2007 paper by Witten, two dimensional purely chiral CFTs with as sparse of a spectrum as possible from modular invariance were considered as duals to pure 3d gravity. We analyze these theories, and argue that they need to be slightly modified, by adding small black holes whose entropy come solely from quantum corrections to the Bekenstein-Hawking entropy formula. In analogous systems with supersymmetry, these correction pass nontrivial consistency checks by considering a more refined count. These theories (should they exist) can also provide a very clean system in which to study small black holes.