Consistent 3D quantum gravity on lens spaces

We study nonperturbative quantization of the first order formulation of three-dimensional gravity with positive cosmological constant (de Sitter space being the prototype vacuum solution, whose Euclideanization gives the three sphere) on the background topology of lens space, which is a three sphere modulo a discrete group. Instead of the strategy followed by a recent work [A. Castro, N. Lashkari, and A. Maloney, Phys. Rev. D 83, 124027 (2011); A. Castro, N. Lashkari, and A. Maloney Phys. Rev. D 84, 089904(E) (2011)], which compares results in the second and first order formulations of gravity, we concentrate solely on the latter. We note, as a striking feature, that the quantization which relies heavily on the axiomatic of topological quantum field theory, can only be consistently carried by augmenting the conventional theory by an additional topological term coupled through a dimensionless parameter. More importantly the introduction of this additional parameter renders the theory finite.