Gauge symmetries in spinfoam gravity: the case for "cellular quantization"

TL;DR

This work shows that standard spinfoam quantization on $2$-complexes fails to account for the discrete shift symmetry of BF theory in dimensions $d≥3$, leading to divergences and loss of topological invariance. It introduces cellular quantization, which includes higher-dimensional cells and applies a resolvent gauge-fixing to produce a finite, topological partition function equal to the twisted Reidemeister (analytic) torsion, thereby matching the continuum theory and connecting to loop quantization. The approach yields a loop-like canonical quantization with torsion-weighted transition amplitudes over the moduli space of flat connections, clarifying the relationship between spinfoam and loop pictures and making amplitudes triangulation-independent. By clarifying diffeomorphism-invariance in BF theory and framing gravity as symmetry breaking within a controlled discrete setting, the paper provides a solid foundation for understanding gravity's emergence from gauge symmetries in quantum gravity.

Abstract

The spinfoam 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 "spinfoam 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, (4) corresponds to its loop quantization. These results significantly clarify the foundations - and limitations - of the spinfoam formalism, and open the path to understanding, in a discrete setting, the symmetry-breaking which reduces BF theory to gravity.