Quantization usually means finding the energy spectrum — the discrete eigenvalues of a Hamiltonian. You solve the Schrödinger equation, impose boundary conditions, and the allowed energies fall out. This is quantization as a spectral property of local operators.
Gamboa and Tapia-Arellano show that in the infrared sector of asymptotically flat quantum gravity, quantization arises from a completely different mechanism. Starting from the Regge-Teitelboim Hamiltonian, they perform a Born-Oppenheimer separation: slow asymptotic data (what distant observers measure) separated from fast bulk gravitational fluctuations (quantum gravity in the interior). Integrating out the fast sector induces a functional Berry connection over the space of asymptotic charges.
The effective infrared dynamics is then governed by parallel transport on this charge space. Carry a gravitational state around a closed loop in the space of asymptotic configurations, and it picks up a geometric phase. The requirement that this transport be globally consistent — that holonomies around contractible loops be trivial — imposes quantization conditions. Infrared gravitational states organize into superselection sectors labeled by holonomy.
No local operator spectrum is involved. The quantization is purely geometric: a consistency condition on how states transform when transported around loops in charge space, not a property of any operator's eigenvalues.
The structural point: at large scales, quantum gravity may not be about energy levels at all. It may be about what happens when you carry information around a loop and demand that the story stays coherent.