A note on the secondary simplicity constraints in loop quantum gravity

TL;DR

This paper investigates the role of secondary simplicity constraints in loop quantum gravity, demonstrating their connection to shape matching conditions and discrete extrinsic geometry through a simplified flat model.

Contribution

It provides evidence that secondary simplicity constraints imply shape matching conditions, extending previous results to Lorentzian signature and arbitrary cellular decompositions.

Findings

Shape matching conditions arise from secondary simplicity constraints.

The twist angle equals the Regge dihedral angle times the Immirzi parameter.

Flatness conditions imply Levi-Civita and shape-matching conditions.

Abstract

A debate has appeared in the literature on loop quantum gravity and spin foams, over whether the secondary simplicity constraints, reducing the connection to be Levi-Civita, should imply the shape matching conditions, reducing twisted geometries to Regge geometries. We address the question using a simple model with a flat dynamics, in which secondary simplicity constraints arise from a dynamical preservation of the primary ones. We find that shape matching conditions arise, thus providing support to an affirmative question. The origin of these extra conditions is to be found in the different graph localisation of the Hamiltonian and primary simplicity constraints. Our results are consistent with previous claims by Dittrich and Ryan, and extend their validity to Lorentzian signature and arbitrary cellular decompositions. We show in particular how the (gauge-invariant version of the) twist angle xi featuring in twisted geometries equals on-shell the Regge dihedral angle multiplied by the Immirzi parameter, thus recovering the discrete extrinsic geometry from the Ashtekar-Barbero holonomy. Finally, we confirm that flatness implies both the Levi-Civita and the shape-matching conditions using twisted geometries and a 4-dimensional version of the vertex condition appearing in 't Hooft's polygon model.