A one-dimensional action for simplicial gravity in three dimensions

Abstract

This article presents 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 then introduce a simplicial decomposition of the three-dimensional manifold and study the discretised action in the spinorial representation of loop gravity. A one-dimensional refinement limit along the edges of the discretisation 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.

19 pages, two figures

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Cited by in corpus (6)

• Quasi-local holographic dualities in non-perturbative 3d quantum gravity I - Convergence of multiple approaches and examples of Ponzano-Regge statistical duals
• New action for simplicial gravity in four dimensions
• Spin foam models as energetic causal sets
• Timelike twisted geometries
• Quantum gravity in three dimensions, Witten spinors and the quantisation of length
• The thermodynamics of quantum spacetime histories