One-dimensional action for simplicial gravity in three dimensions

We present a derivation of the Ponzano-Regge model from a one-dimensional spinor action. The construction starts from the first-order Palatini formalism in three dimensions. We introduce a simplicial decomposition of the three-dimensional manifold and study the discretized action in the spinorial representation of loop gravity. A one-dimensional refinement limit along the edges of the discretization brings us back to a continuum formulation. The three-dimensional action turns into a line integral over the one-skeleton of the simplicial manifold. All fields are continuous but have support only along the one-dimensional edges. We define the path integral and remove the redundant integrals over the local gauge orbits through the usual Faddeev-Popov procedure. The resulting state sum model reproduces the Ponzano-Regge amplitudes.