Gauge symmetry of the 3BF theory for a generic Lie 3-group

TL;DR

This paper analyzes the gauge symmetry of the 3BF action derived from higher category theory, performing a Hamiltonian analysis for a generic Lie 3-group, and identifies new gauge transformations relevant for quantum gravity models.

Contribution

It provides the first complete Hamiltonian analysis of the 3BF action for arbitrary semistrict Lie 3-groups, revealing new gauge symmetries not previously discussed.

Findings

Identified the full gauge symmetry group including new M- and N-transformations.

Performed the first Hamiltonian analysis of 3BF theory for arbitrary Lie 3-groups.

Established groundwork for quantizing 3BF theories in quantum gravity.

Abstract

The higher category theory can be employed to generalize the BF action to the so-called 3BF action, by passing from the notion of a gauge group to the notion of a gauge 3-group. In this work we determine the full gauge symmetry of the 3BF action. To that end, the complete Hamiltonian analysis of the 3BF action for an arbitrary semistrict Lie 3-group is performed, by using the Dirac procedure. This analysis is the first step towards a canonical quantization of a 3BF theory. This is an important stepping-stone for the quantization of the complete Standard Model of elementary particles coupled to Einstein-Cartan gravity, formulated as a 3BF action with suitable simplicity constraints. We show that the resulting gauge symmetry group consists of the already familiar G-, H-, and L-gauge transformations, as well as additional M- and N-gauge transformations, which have not been discussed in the existing literature.