Consistent 3D Quantum Gravity on Lens Spaces

TL;DR

This paper investigates the non-perturbative quantization of 3D de Sitter quantum gravity on lens spaces, revealing that an extra topological term is necessary for consistency and finiteness of the theory.

Contribution

It demonstrates that adding a topological term with a dimensionless parameter is essential for consistent and finite non-perturbative 3D quantum gravity on lens spaces.

Findings

Quantization requires an additional topological term.

The extra parameter makes the theory finite.

Focus on second order formulation of gravity.

Abstract

We study non-perturbative quantization of 3d gravity with positive cosmological constant (de Sitter space being the prototype vacuum solution, whose Euclideanization of course gives the three sphere) on the background topology of lens space, which is a three spheres modulo a discrete group. Instead of the strategy followed by a recent work \cite{Castro:2011xb}, which compares results in the second and first order formulations of gravity, we concentrate on the later solely. We note, as a striking feature, that the quantization, that relies heavily on the axiomatics of topological quantum field theory (TQFT) 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.