A Deformed Poincare Invariance for Group Field Theories

TL;DR

The paper identifies a quantum-deformed Poincaré invariance for 3d group field theories, showing that Boulatov's GFT is invariant under the DSU(2) symmetry when expressed in gauge-invariant variables and that 2d perturbations around classical backgrounds yield effective NCQFTs preserving this symmetry. By performing a gauge reduction and a Fourier transform, the authors recast Boulatov GFT as a non-commutative field theory on $\mathbb R^3$ with group-valued momenta, enabling a space-time interpretation of GFTs as momentum-space theories. They classify classical GFT solutions into flat ones, which preserve the symmetry and define a Poincaré-invariant vacuum for matter perturbations, and doped solutions, which break translational invariance and correspond to excited geometries with momentum transfer. The work extends these ideas to 4d BF-type GFTs and outlines future directions on braiding, symmetry breaking, and extensions to more general GFTs, with potential implications for the understanding of spacetime in quantum gravity and the role of geometry–matter coupling.

Abstract

In the context of quantum gravity, group field theories are field theories that generate spinfoam amplitudes as Feynman diagrams. They can be understood as generalizations of the matrix models used for 2d quantum gravity. In particular Boulatov's theory reproduces the amplitudes of the Ponzano-Regge spinfoam model for 3d quantum gravity. Motivated by recent works on field theories on non-commutative flat spaces, we show that Boulatov's theory (and its colored version) is actually invariant under a global deformed Poincare symmetry. This allows to define a notion of flat/excited geometry states when considering scalar perturbations around classical solutions of the group field equations of motion. As a side-result, our analysis seems to point out that the notion of braiding of group field theories should be a key feature to study further in this context.