Abelian 3D TQFT gravity, ensemble holography and stabilizer states

We construct a model of 3D quantum gravity based on abelian topological quantum field theory (TQFT), by defining the gravitational path-integral as a sum over all 3D topologies with genus-$g$ boundary $\Sigma_g$. The path-integral of an abelian TQFT $\mathcal T$ on any single topology with boundary $\Sigma_g$ prepares a stabilizer state. This way, $\mathcal T$ partitions all these topologies into finitely many equivalence classes, where each topology within a class is associated with the same stabilizer state. The gravitational path-integral can thus be rephrased as a weighted sum over representative topologies, which are further organized into orbits under the mapping class group of $\Sigma_g$. One orbit is represented by handlebodies, whose average reproduces the ``Poincar\'e series of the vacuum", while additional orbits describe non-handlebody topologies. The resulting quantum gravity state is $Sp(2g,\mathbb Z)$-invariant and can be expressed as a weighted average of 2D CFT partition functions on $\Sigma_g$. This establishes a duality between a weighted sum over bulk topologies and a weighted sum over boundary CFTs. We introduce the ``$\lambda$-matrix", which relates bulk and boundary weights. The $\lambda$-matrix can be fully determined by the set of topological boundary conditions that the TQFT admits, and we present a systematic procedure to construct this set. Using this framework, we evaluate the $\lambda$-matrix and the TQFT gravity state in several tractable examples.