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Bounding bubbles: the vertex representation of 3d Group Field Theory and the suppression of pseudo-manifolds

Sylvain Carrozza, Daniele Oriti

TL;DR

This work introduces a vertex-variable reformulation of the colored Boulatov GFT to make simplicial diffeomorphisms and bubble topology manifest. By recasting edge data as vertex positions and exploiting a non-commutative metric representation, the authors derive explicit bubble amplitudes and two key scaling bounds that depend on the genus of bubbles, showing pseudo-manifolds are suppressed relative to manifolds in the large cut-off limit. They prove these bounds are optimal and demonstrate that manifold configurations dominate, providing a robust topological control essential for GFT renormalization and continuum-limit studies. The results pave the way for extending vertex-based analyses to higher dimensions and for exploring the interplay between diffeomorphism symmetries, topology, and dynamics in GFT-based quantum gravity models.

Abstract

Based on recent work on simplicial diffeomorphisms in colored group field theories, we develop a representation of the colored Boulatov model, in which the GFT fields depend on variables associated to vertices of the associated simplicial complex, as opposed to edges. On top of simplifying the action of diffeomorphisms, the main advantage of this representation is that the GFT Feynman graphs have a different stranded structure, which allows a direct identification of subgraphs associated to bubbles, and their evaluation is simplified drastically. As a first important application of this formulation, we derive new scaling bounds for the regularized amplitudes, organized in terms of the genera of the bubbles, and show how the pseudo-manifolds configurations appearing in the perturbative expansion are suppressed as compared to manifolds. Moreover, these bounds are proved to be optimal.

Bounding bubbles: the vertex representation of 3d Group Field Theory and the suppression of pseudo-manifolds

TL;DR

This work introduces a vertex-variable reformulation of the colored Boulatov GFT to make simplicial diffeomorphisms and bubble topology manifest. By recasting edge data as vertex positions and exploiting a non-commutative metric representation, the authors derive explicit bubble amplitudes and two key scaling bounds that depend on the genus of bubbles, showing pseudo-manifolds are suppressed relative to manifolds in the large cut-off limit. They prove these bounds are optimal and demonstrate that manifold configurations dominate, providing a robust topological control essential for GFT renormalization and continuum-limit studies. The results pave the way for extending vertex-based analyses to higher dimensions and for exploring the interplay between diffeomorphism symmetries, topology, and dynamics in GFT-based quantum gravity models.

Abstract

Based on recent work on simplicial diffeomorphisms in colored group field theories, we develop a representation of the colored Boulatov model, in which the GFT fields depend on variables associated to vertices of the associated simplicial complex, as opposed to edges. On top of simplifying the action of diffeomorphisms, the main advantage of this representation is that the GFT Feynman graphs have a different stranded structure, which allows a direct identification of subgraphs associated to bubbles, and their evaluation is simplified drastically. As a first important application of this formulation, we derive new scaling bounds for the regularized amplitudes, organized in terms of the genera of the bubbles, and show how the pseudo-manifolds configurations appearing in the perturbative expansion are suppressed as compared to manifolds. Moreover, these bounds are proved to be optimal.

Paper Structure

This paper contains 25 sections, 6 theorems, 65 equations, 17 figures.

Table of Contents

  1. Introduction and background
  2. Model and translation symmetry
  3. Definition of the model
  4. Translation symmetries
  5. Transformation to variables associated to vertices
  6. From edge to vertex variables
  7. GFT Action in vertex variables
  8. Action in terms of the constrained fields
  9. Action in terms of the unconstrained fields
  10. Translation symmetries in vertex variables
  11. Quantum theory
  12. Cut-off and rescaling of the fields
  13. Partition function and Feynman rules
  14. Quantum amplitudes
  15. Explicit bubble structure
  16. ...and 10 more sections

Key Result

Lemma 1

Let ${\mathcal{G}}$ be a colored graph, with a subgraph ${\mathcal{C}}_{g}$ for some $g \in \mathbb{N}$. Call ${\mathcal{G}}^{{\mathcal{C}}_{g} \rightarrow {\mathcal{C}}_{0}}$ the graph obtained after replacement of ${\mathcal{C}}_{g}$ by the graph ${\mathcal{C}}_{0}$. Then:

Figures (17)

  • Figure 1: Graphical representation of a field, and the interaction vertex in usual edge variables.
  • Figure 2: Combinatorial structure of the sunshine graph in edge variables.
  • Figure 3: Action of ${\mathcal{T}}^{3}_\epsilon$ on the interaction term, and resulting transformation of ${\tilde{\varphi}}_1$.
  • Figure 4: Map from edge to vertex variables.
  • Figure 5: Combinatorics of the interaction function in vertex variables. One of the four three valent interactions is redundant. The arrow indicates the ordering of variables.
  • ...and 12 more figures

Theorems & Definitions (12)

  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Lemma 2
  • proof
  • Proposition 1
  • proof
  • Lemma 3
  • proof
  • ...and 2 more