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Lost in Translation: Topological Singularities in Group Field Theory

Razvan Gurau

TL;DR

The paper tackles the problem that GFT graphs can glue simplices into non-manifold spaces, compromising their interpretation as quantum gravity amplitudes. It analyzes wrapping singularities via dual gluings and link graphs, showing that standard GFTs produce non-pseudo-manifold structures and nontrivial topological pathologies that dominate perturbative contributions. It then proves that colored GFT models impose strand-color conservation that enforces coherent identifications of all subsimplices, guaranteeing dual gluings are normal simplicial pseudo manifolds in any dimension. This result positions colored GFT as a topology-safe framework for quantum gravity, while signaling the need to explore scaling regimes where manifold configurations dominate.

Abstract

Random matrix models generalize to Group Field Theories (GFT) whose Feynman graphs are dual to gluings of higher dimensional simplices. It is generally assumed that GFT graphs are always dual to pseudo manifolds. In this paper we prove that already in dimension three (and in all higher dimensions), this is not true due to subtle differences between simplicial complexes and gluings dual to GFT graphs. We prove however that, fortunately, the recently introduced "colored" GFT models [1] do not suffer from this problem and only generate graphs dual to pseudo manifolds in any dimension.

Lost in Translation: Topological Singularities in Group Field Theory

TL;DR

The paper tackles the problem that GFT graphs can glue simplices into non-manifold spaces, compromising their interpretation as quantum gravity amplitudes. It analyzes wrapping singularities via dual gluings and link graphs, showing that standard GFTs produce non-pseudo-manifold structures and nontrivial topological pathologies that dominate perturbative contributions. It then proves that colored GFT models impose strand-color conservation that enforces coherent identifications of all subsimplices, guaranteeing dual gluings are normal simplicial pseudo manifolds in any dimension. This result positions colored GFT as a topology-safe framework for quantum gravity, while signaling the need to explore scaling regimes where manifold configurations dominate.

Abstract

Random matrix models generalize to Group Field Theories (GFT) whose Feynman graphs are dual to gluings of higher dimensional simplices. It is generally assumed that GFT graphs are always dual to pseudo manifolds. In this paper we prove that already in dimension three (and in all higher dimensions), this is not true due to subtle differences between simplicial complexes and gluings dual to GFT graphs. We prove however that, fortunately, the recently introduced "colored" GFT models [1] do not suffer from this problem and only generate graphs dual to pseudo manifolds in any dimension.

Paper Structure

This paper contains 8 sections, 4 theorems, 42 equations, 17 figures.

Table of Contents

  1. Introduction: Group Field Theory
  2. Simplicial Pseudo Manifolds
  3. GFT Graphs
  4. Link graphs
  5. Wrapping Singularities in GFT Graphs
  6. The Colored GFT Graphs and Pseudo Ma-nifolds
  7. Conclusion
  8. Remarks on Simplicial Complexes

Key Result

Proposition 1

The Euler character of a three dimensional normal pseudo manifold $\Delta$ respects and $\chi(\text{lk}_{\Delta}(v_i)) \le 2$, thus $\chi(\Delta)\ge 0$.

Figures (17)

  • Figure 1: A graph not dual to a pseudo manifold.
  • Figure 2: Colored GFT line.
  • Figure 3: GFT vertex in $n$ dimensions.
  • Figure 4: GFT lines in n dimensions.
  • Figure 5: GFT vertex in 3 dimensions.
  • ...and 12 more figures

Theorems & Definitions (7)

  • Proposition 1
  • Lemma 1
  • Theorem 1
  • Theorem 2
  • Remark 1
  • Remark 2
  • Remark 3