Bubble networks: framed discrete geometry for quantum gravity

Abstract

In the context of canonical quantum gravity in 3+1 dimensions, we introduce a new notion of bubble network that represents discrete 3d space geometries. These are natural extensions of twisted geometries, which represent the geometrical data underlying loop quantum geometry and are defined as networks of SU(2) holonomies. In addition to the SU(2) representations encoding the geometrical flux, the bubble network links carry a compatible SL(2, R) representation encoding the discretized frame field which composes the flux. In contrast with twisted geometries, this extra data allows to reconstruct the frame compatible with the flux unambiguously. At the classical level this data represents a network of 3d geometrical cells glued together. The SL(2, R) data contains information about the discretized 2d metrics of the interfaces between 3d cells and SL(2, R) local transformations are understood as the group of area-preserving diffeomorphisms. We further show that the natural gluing condition with respect to this extended group structure ensures that the intrinsic 2d geometry of a boundary surface is the same from the viewpoint of the two cells sharing it. At the quantum level this gluing corresponds to a maximal entanglement along the network edges. We emphasize that the nature of this extension of twisted geometries is compatible with the general analysis of gauge theories that predicts edge mode degrees of freedom at the interface of subsystems.

Gravitational edge modes: from Kac–Moody charges to Poincaré networks

“…Such a su(2) × sl 2 (R) algebraic structure with the same balance equation already appeared in [14,39], where it related the su(2) geometrical flux to the 2d surface metric.…”

“…One can then envision gravity to be written as a theory of edge modes living on the boundary surfaces of these patches of 3D geometry [14,15]. This translates into a picture of space as a network of "bubbles" as in [39], which should eventually lead to quantum gravity as dynamical networks of quantum edge modes. This picture is naturally compatible with local-holography and it is designed to offer a perfect setting to study the coarse-graining of gravity both at the classical and the quantum levels [40][41][42][43].…”

Dual2+1Dloop quantum gravity on the edge

“…Furthermore, we will use 1 The cells in this decomposition can take any shape. 2 See also [28] for a more intuitive discussion and [29,30,31] for the case of 3+1-dimensional gravity. 3 The notation T * G signifies the cotangent bundle of G. 4 They satisfy anti-symmetry f ij k = − f ji k and the Jacobi identity f [ij l f k]l m = 0.…”

Generating functional for gravitational null initial data

“…If the torsionless conditions (24) and (23) are satisfied, we can use the decomposition of the difference tensor (22) to immediately see that C 3 vanishes by itself, 16 ℓ A ℓ a D a ℓ A = 0, while the three remaining constraints impose the matching conditions…”

“…The matching of ϑ (k) will then follow from the matching of the initial conditions ϑ + (k) |u o = ϑ − (k) |u o and the evolution equation (33d) 16. The null hypersurface is ruled by lightlike geodesics, the null generators ℓ a are tangent to them, hence ℓAℓ a Daℓ A = 0.…”