  Space and matter may both be manifestations of a single fundamental quantum
dynamics, as it may become evident during black-hole evaporation. Inspired by
the fact that quantum electrodynamics underlies the classical theory of
elasticity, that in turn has a natural and well-known geometric description in
terms of curvature and torsion, related to topological defects, here we move
some necessary steps to find the map from such fundamental quantum level to the
emergent level of classical space and quantum matter. We proceed by adapting
the boson transformation method of standard quantum field theory to the quantum
gravity fundamental scenario and successfully obtain the emergence of curvature
and torsion, our main focus here. In doing so, we have been able to overcome
difficult issues of interpretation, related to the Goldstone modes for
rotational symmetry. In fact, we have been able to apply the boson
transformation method to disclinations, to relate them to the spin structure
and to give an heuristic derivation of the matter field equation on curved
space. We also improve results of previous work on the emergence of geometric
tensors from elasticity theory, as the non-Abelian contributions to the torsion
and curvature tensors, postulated in those papers, here emerge naturally. More
work is necessary to identify the type of gravity theories one can obtain in
this way.
