A Ponzano-Regge model of Lorentzian 3-Dimensional gravity

TL;DR

This work extends the Ponzano–Regge state-sum construction to Lorentzian 3D gravity by replacing the compact gauge group with $SL(2,\mathbb{R})$ and introducing a causal structure on triangulations. It expresses the partition function as a sum over unitary SL(2,R) representations (principal and discrete series) with a Plancherel weight and SL(2,R) $6$-$j$ symbols, distinguishing spacelike and timelike edges via representation type. Regularization ensures triangulation-independence of the bulk Lorentzian amplitude, while the causal-structure dependent pieces capture possible Lorentzian geometries; special all-$+$ or all-$-$ structures yield triangulation-invariant results. The work thus links Witten's connection-based quantization with a Lorentzian spin-network/state-sum framework, providing a computable model for Lorentzian 3D gravity and clarifying gauge fixing and boundary coupling.

Abstract

We present the construction of the partition function of 3-dimensional gravity in the Lorentzian regime as a state sum model over a triangulation. This generalize the work of Ponzano and Regge to the case of Lorentzian signature.