Hopf link volume simplicity constraints in spin foam models

TL;DR

This paper investigates volume simplicity constraints in spin foam models of quantum gravity, showing their relation to Hopf link constraints and estimating the number of geometricity conditions needed for bivector geometries.

Contribution

It demonstrates that volume simplicity constraints can be formulated through Hopf link volume simplicity constraints, providing a new perspective on geometricity conditions in spin foam models.

Findings

Volume simplicity constraints relate to Hopf link constraints.

Number of geometricity conditions equals Hopf links minus one.

Constraints can be formulated by deforming Hopf link volume simplicity constraints.

Abstract

In this article we consider specific bivector geometries which arise in the large-spin limit of the extension of the Engle-Pereira-Rovelli-Livine spin foam model for quantum gravity by Kaminski, Kisielowski and Lewandowski. We address the implementation of volume simplicity constraints, which are required to ensure that a $4d$ metric can be reconstructed from the bivector geometry. We find that the necessary conditions are closely related, but not quite equal to the Hopf link volume simplicity constraints introduced in earlier works. We estimate the number of independent geometricity conditions for arbitrary bivector geometries, and find that they always agree with the number of Hopf links on the graph minus one, suggesting that the geometricity conditions can generically be formulated by deformation of the Hopf link volume simplicity constraints.