Quasi-local holography in 3d quantum gravity

Since the idea appeared in black hole physics, the concept of holography has become a guiding principle for quantum gravity. It is the notion that the dynamics of the geometry of a region of space-time can be entirely encoded in a theory living on its boundary. Although such holographic dualities have been well-developed in an asymptotical context, it remains a challenge to realize it exactly at finite distances. I will draw a possible route in 3d quantum gravity, by showing a duality between the Ponzano-Regge path integral for 3d quantum gravity as a topological field theory and the 2d (inhomogeneous) Ising model. This leads to an intriguing geometrical interpretation of the Ising critical couplings and opens the door to a possibly rich interplay between 3d quantum gravity and 2d condensed matter built out of holographic dualities.