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Diffeomorphisms in group field theories

Aristide Baratin, Florian Girelli, Daniele Oriti

TL;DR

The paper identifies a discrete diffeomorphism symmetry in colored 3d GFTs by employing a noncommutative metric representation, where vertex translations are generated by the quantum group \\mathcal{D}SO(3) and implemented through a braided quantum field theory framework. It shows that this symmetry leaves the GFT action invariant across the metric, group, and spin representations, and that the invariance of vertex amplitudes encodes discrete Bianchi identities and flat boundary connections, linking Regge, spin-foam, and canonical gravity constraints. The results extend to higher-dimensional BF theories and offer insights into how diffeomorphism invariance can guide GFT renormalization, model-building constraints (notably the necessity of coloring), and potential braided generalizations that preserve Ward identities. Overall, the work provides a unified field-theoretic realization of discrete diffeomorphisms in simplicial quantum gravity, clarifying how continuum diffeomorphism symmetry emerges in a GFT framework and informing both mathematical structure and physical interpretation of spin foam amplitudes.

Abstract

We study the issue of diffeomorphism symmetry in group field theories (GFT), using the recently introduced noncommutative metric representation. In the colored Boulatov model for 3d gravity, we identify a field (quantum) symmetry which ties together the vertex translation invariance of discrete gravity, the flatness constraint of canonical quantum gravity, and the topological (coarse-graining) identities for the 6j-symbols. We also show how, for the GFT graphs dual to manifolds, the invariance of the Feynman amplitudes encodes the discrete residual action of diffeomorphisms in simplicial gravity path integrals. We extend the results to GFT models for higher dimensional BF theories and discuss various insights that they provide on the GFT formalism itself.

Diffeomorphisms in group field theories

TL;DR

The paper identifies a discrete diffeomorphism symmetry in colored 3d GFTs by employing a noncommutative metric representation, where vertex translations are generated by the quantum group \\mathcal{D}SO(3) and implemented through a braided quantum field theory framework. It shows that this symmetry leaves the GFT action invariant across the metric, group, and spin representations, and that the invariance of vertex amplitudes encodes discrete Bianchi identities and flat boundary connections, linking Regge, spin-foam, and canonical gravity constraints. The results extend to higher-dimensional BF theories and offer insights into how diffeomorphism invariance can guide GFT renormalization, model-building constraints (notably the necessity of coloring), and potential braided generalizations that preserve Ward identities. Overall, the work provides a unified field-theoretic realization of discrete diffeomorphisms in simplicial quantum gravity, clarifying how continuum diffeomorphism symmetry emerges in a GFT framework and informing both mathematical structure and physical interpretation of spin foam amplitudes.

Abstract

We study the issue of diffeomorphism symmetry in group field theories (GFT), using the recently introduced noncommutative metric representation. In the colored Boulatov model for 3d gravity, we identify a field (quantum) symmetry which ties together the vertex translation invariance of discrete gravity, the flatness constraint of canonical quantum gravity, and the topological (coarse-graining) identities for the 6j-symbols. We also show how, for the GFT graphs dual to manifolds, the invariance of the Feynman amplitudes encodes the discrete residual action of diffeomorphisms in simplicial gravity path integrals. We extend the results to GFT models for higher dimensional BF theories and discuss various insights that they provide on the GFT formalism itself.

Paper Structure

This paper contains 16 sections, 95 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Colored GFT's and metric representation
  3. GFT (discrete) diffeomorphism symmetry
  4. Action of ${\mathcal{D}} \mathrm{SO}(3)$ on fields on $\mathrm{SO}(3)$
  5. GFT as a braided quantum field theory
  6. Symmetries of the GFT action
  7. Invariance of the vertex and diffeomorphisms
  8. GFT with vertex variables
  9. From GFT to simplicial gravity symmetries
  10. Diffeomorphisms in simplicial path integrals
  11. Simplicial diffeomorphisms as quantum group symmetries
  12. Non-commutative translations, invariance of the GFT amplitudes and Bianchi identities
  13. Open diagrams and n-point functions
  14. Diffeomorphisms in topological models in higher dimensions
  15. Additional insights
  16. ...and 1 more sections

Figures (10)

  • Figure 1: Geometric interpretation of the GFT field.
  • Figure 2: 3D GFT propagator and vertex.
  • Figure 3: The 'boundary' holonomies $g^\ell_i$ are in red, while the 'bulk' holonomies $h_\ell$ are in blue.
  • Figure 4: Vertex graph for the vertex $v_3$
  • Figure 5: Vertex translation and trivial vertex holonomy
  • ...and 5 more figures