The closure constraint for the hyperbolic tetrahedron as a Bianchi identity

Understand

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).