  Three-dimensional gravity is a topological field theory, which can be
quantized as the Ponzano-Regge state-sum model built from the $\{3nj\}$-symbols
of the recoupling of the $\SU(2)$ representations, in which spins are
interpreted as quantized edge lengths in Planck units. It describes the flat
spacetime as gluing of three-dimensional cells with a fixed boundary metric
encoding length scale. In this paper, we revisit the Ponzano-Regge model
formulated in terms of spinors and rewrite the quantum geometry of 3D cells
with holomorphic recoupling symbols. These symbols, known as Schwinger's
generating function for the $\{6j\}$-symbols, are simply the squared inverse of
the partition function of the 2D Ising model living on the boundary of the 3D
cells. They can furthermore be interpreted, in their critical regime, as
scale-invariant basic elements of geometry. We show how to glue them together
into a discrete topological quantum field theory. This reformulation of the
path integral for 3D quantum gravity, with a rich pole structure of the
elementary building blocks, opens a new door toward the study of phase
transitions and continuum limits in 3D quantum gravity, and offers a new twist
on the construction of a duality between 3D quantum gravity and a 2d conformal
theory.

High Energy Physics - Theory;General Relativity and Quantum Cosmology