The closure constraint for the hyperbolic tetrahedron as a Bianchi identity

TL;DR

This paper reinterprets the closure constraint in loop quantum gravity as a Bianchi identity, extending it to hyperbolic tetrahedra and linking it to $q$-deformed geometries for positive cosmological constant.

Contribution

It develops a framework connecting closure constraints with Bianchi identities for hyperbolic tetrahedra using holonomies, advancing the understanding of quantum geometry with a cosmological constant.

Findings

Closure constraints for hyperbolic tetrahedra are defined via $SU(2)$ and $SB(2,\mathbb{C})$ holonomies.

The framework links classical phase space with $q$-deformed loop quantum gravity.

First step towards interpreting $q$-deformed twisted geometries as hyperbolic triangulations.

Abstract

The closure constraint is a central piece of the mathematics of loop quantum gravity. It encodes the gauge invariance of the spin network states of quantum geometry and provides them with a geometrical interpretation: each decorated vertex of a spin network is dual to a quantized polyhedron in $\mathbb{R}^{3}$. For instance, a 4-valent vertex is interpreted as a tetrahedron determined by the four normal vectors of its faces. We develop a framework where the closure constraint is re-interpreted as a Bianchi identity, with the normals defined as holonomies around the polyhedron faces of a connection (constructed from the spinning geometry interpretation of twisted geometries). This allows us to define closure constraints for hyperbolic tetrahedra (living in the 3-hyperboloid of unit future-oriented spacelike vectors in $\mathbb{R}^{3,1}$) in terms of normals living all in $SU(2)$ or in $SB(2,\mathbb{C})$. The latter fits perfectly with the classical phase space developed for $q$-deformed loop quantum gravity supposed to account for a non-vanishing cosmological constant $\Lambda>0$. This is the first step towards interpreting $q$-deformed twisted geometries as actual discrete hyperbolic triangulations.