A note on the secondary simplicity constraints in loop quantum gravity

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 shape-matching conditions, reducing twisted geometries to Regge geometries. We address the question using a simple model with 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 answer to the question. In our model, these extra conditions come from the different graph localizations of the Hamiltonian (faces) and primary simplicity constraints (links). 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 ξ 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 Levi-Civita and shape-matching conditions using twisted geometries and a four-dimensional version of the vertex condition appearing in ’t Hooft’s polygon model.