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In the context of canonical quantum gravity in 3+1 dimensions, we introduce anew notion of bubble network that represents discrete 3d space geometries.These are natural extensions of twisted geometries, which represent thegeometrical data underlying loop quantum geometry and are defined as networksof SU(2) holonomies. In addition to the SU(2) representations encoding thegeometrical flux, the bubble network links carry a compatible SL(2,R)representation encoding the discretized frame field which composes the flux. Incontrast with twisted geometries, this extra data allows to reconstruct theframe compatible with the flux unambiguously. At the classical level this datarepresents a network of 3d geometrical cells glued together. The SL(2,R) datacontains information about the discretized 2d metrics of the interfaces between3d cells and SL(2,R) local transformations are understood as the group ofarea-preserving diffeomorphisms. We further show that the natural gluingcondition with respect to this extended group structure ensures that theintrinsic 2d geometry of a boundary surface is the same from the viewpoint ofthe two cells sharing it. At the quantum level this gluing corresponds to amaximal entanglement along the network edges. We emphasize that the nature ofthis extension of twisted geometries is compatible with the general analysis ofgauge theories that predicts edge mode degrees of freedom at the interface ofsubsystems.

Bubble Networks: Framed Discrete Geometry for Quantum Gravity