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Non-perturbative summation over 3D discrete topologies

Laurent Freidel, David Louapre

TL;DR

This work addresses how to give a rigorous meaning to summing over all spacetime topologies in 3D quantum gravity by embedding the sum in a group field theory. It introduces modified Boulatov-type GFTs with positive potentials so that their perturbative expansions are uniquely Borel-summable, providing a non-perturbative definition of the full triangulation sum including all topologies. The approach connects familiar 3D gravity amplitudes (Ponzano-Regge) to dynamical triangulations and matrix-model ideas via non-perturbative resummation, and reorganizes generalized triangulations into a controlled real-triangulation framework using pillow interactions. The main contributions are the positivity-driven modification of the GFT potential, the proof of unique Borel summability, and the explicit reinterpretation of the perturbative series as a sum over (regular and irregular) triangulations, offering a concrete path to topology change in quantum gravity. The results open avenues for extending these non-perturbative resummation techniques to higher dimensions and more complex spin-foam models.

Abstract

We construct a group field theory which realizes the sum of gravity amplitudes over all three dimensional topologies trough a perturbative expansion. We prove this theory to be uniquely Borel summable. This shows how to define a non-perturbative summation over triangulations including all topologies in the context of three dimensional discrete gravity.

Non-perturbative summation over 3D discrete topologies

TL;DR

This work addresses how to give a rigorous meaning to summing over all spacetime topologies in 3D quantum gravity by embedding the sum in a group field theory. It introduces modified Boulatov-type GFTs with positive potentials so that their perturbative expansions are uniquely Borel-summable, providing a non-perturbative definition of the full triangulation sum including all topologies. The approach connects familiar 3D gravity amplitudes (Ponzano-Regge) to dynamical triangulations and matrix-model ideas via non-perturbative resummation, and reorganizes generalized triangulations into a controlled real-triangulation framework using pillow interactions. The main contributions are the positivity-driven modification of the GFT potential, the proof of unique Borel summability, and the explicit reinterpretation of the perturbative series as a sum over (regular and irregular) triangulations, offering a concrete path to topology change in quantum gravity. The results open avenues for extending these non-perturbative resummation techniques to higher dimensions and more complex spin-foam models.

Abstract

We construct a group field theory which realizes the sum of gravity amplitudes over all three dimensional topologies trough a perturbative expansion. We prove this theory to be uniquely Borel summable. This shows how to define a non-perturbative summation over triangulations including all topologies in the context of three dimensional discrete gravity.

Paper Structure

This paper contains 14 sections, 1 theorem, 114 equations, 7 figures.

Table of Contents

  1. Introduction
  2. 3D Group Field Theory models
  3. The Boulatov model
  4. Group field model on the sphere and 3D matrix models
  5. Other group field theories
  6. Non-perturbative resummation of an asymptotic series
  7. Modification of the group field theory model
  8. Proof of the Borel-summability of the perturbative expansion
  9. Sokal criteria for the modified models
  10. Reorganization of the perturbative expansion
  11. Conclusion
  12. Representations of $\mathrm{SO}(3)$ and Racah-Wigner symbols
  13. The non-commutative 2-sphere
  14. Note on the convergence of graphs

Key Result

Theorem 1

Let $f$ be analytic in the circle $C_R=\{z | \Re(z^{-1}) > R^{-1}\}$ (see figure fig:analyticity) and having an asymptotic expansion with then the asymptotic series is uniquely Borel summable and $f(z)$ can be uniquely reconstructed from it.

Figures (7)

  • Figure 1: Graphical representation of the propagator and potential. The black square stands for the sum over different permutations of the three lines, see (\ref{['eqn:propagator']}).
  • Figure 2: Triangulation generated by the Feynman diagrams
  • Figure 3: Contour for the analytic continuation of $Z(\lambda)$
  • Figure 4: a) Original contour $C$ for defining $D(-\omega)$; b) Deformed contour $C'$to apply the saddle point method.
  • Figure 5: Domain $C_R$ of analyticity for $f$, and the corresponding domain of analyticity for its Borel transform $B_f$, which contains an open neighbourhood of the real positive axis.
  • ...and 2 more figures

Theorems & Definitions (1)

  • Theorem 1: Sokal-Nevalinna