Closure constraints for hyperbolic tetrahedra

TL;DR

This paper extends the concept of closure constraints from flat to hyperbolic tetrahedra within loop gravity, introducing two new constraints that describe hyperbolic geometries and their duals, relevant for modeling a non-zero cosmological constant.

Contribution

It introduces two novel closure constraints for hyperbolic tetrahedra, generalizing twisted geometries to include hyperbolic curvature in loop gravity.

Findings

Two new closure constraints for hyperbolic tetrahedra are proposed.

Both constraints define a unique dual hyperbolic tetrahedron.

The constraints are shown to be equivalent, describing hyperbolic geometries.

Abstract

We investigate the generalization of loop gravity's twisted geometries to a q-deformed gauge group. In the standard undeformed case, loop gravity is a formulation of general relativity as a diffeomorphism-invariant SU(2) gauge theory. Its classical states are graphs provided with algebraic data. In particular closure constraints at every node of the graph ensure their interpretation as twisted geometries. Dual to each node, one has a polyhedron embedded in flat space R^3. One then glues them allowing for both curvature and torsion. It was recently conjectured that q-deforming the gauge group SU(2) would allow to account for a non-vanishing cosmological constant Lambda, and in particular that deforming the loop gravity phase space with real parameter q>0 would lead to a generalization of twisted geometries to a hyperbolic curvature. Following this insight, we look for generalization of the closure constraints to the hyperbolic case. In particular, we introduce two new closure constraints for hyperbolic tetrahedra. One is compact and expressed in terms of normal rotations (group elements in SU(2) associated to the triangles) and the second is non-compact and expressed in terms of triangular matrices (group elements in SB(2,C)). We show that these closure constraints both define a unique dual tetrahedron (up to global translations on the three-dimensional one-sheet hyperboloid) and are thus ultimately equivalent.