Gauge Symmetries in Spin-Foam Gravity: The Case for “Cellular Quantization”

The spin-foam approach to quantum gravity rests on a quantization of $BF$ theory using 2-complexes and group representations. We explain why, in dimension three and higher, this spin-foam quantization must be amended to be made consistent with the gauge symmetries of discrete $BF$ theory. We discuss a suitable generalization, called “cellular quantization,” which (1) is finite, (2) produces a topological invariant, (3) matches with the properties of the continuum $BF$ theory, and (4) corresponds to its loop quantization. These results significantly clarify the foundations—and limitations—of the spin-foam formalism and open the path to understanding, in a discrete setting, the symmetry-breaking which reduces $BF$ theory to gravity.