The 3BF theory as a TQFT

TL;DR

This paper develops a topological quantum field theory (TQFT) based on 3BF theory with a 3-group gauge symmetry, providing a new framework for quantum gravity coupled with matter, and rigorously proves its mathematical consistency.

Contribution

It constructs a state sum for 3BF theory as a TQFT, satisfying all Atiyah's axioms, advancing the spinfoam approach to quantum gravity with matter.

Findings

Constructed a state sum as a topological invariant of 4-manifolds

Proved the state sum satisfies Atiyah's axioms for TQFT

Provides a foundation for quantizing gravity with matter using spinfoam models

Abstract

We study the path integral quantization of the topological 3BF theory, whose gauge symmetry is described by a 3-group. This theory is relevant for the quantization of general relativity coupled to Standard Model of elementary particles. We explicitly construct a state sum corresponding to the discretized path integral of a 3BF action. Being a topological invariant of 4-dimensional manifolds with boundary, this state sum gives rise to a topological quantum field theory (TQFT), realized as a functor between the category of cobordisms and the category of Hilbert spaces. After an introduction to appropriate category theory concepts and the construction of the state sum, we provide an explicit proof that it satisfies all Atiyah's axioms, and thus represents a genuine TQFT. The formulation of this TQFT represents a major step in the spinfoam quantization programme for a realistic theory of quantum gravity with matter.