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Multi-orientable Group Field Theory

Adrian Tanasa

TL;DR

The paper introduces multi-orientable GFT (MO-GFT) as a new simplification of Group Field Theory tensor graphs, distinct from the colored GFT approach. It defines MO-GFT in three dimensions via a Boulatov-like model with a vertex oriented by ± corners, and analyzes how MO graphs relate to colorable graphs, proving that all colorable graphs are MO while providing counterexamples to strict equivalence. The work shows MO excludes tadfaces and some generalized tadpoles, studies explicit Feynman amplitudes (including UV/IR-like mixing in group space), and discusses renormalizability with a favorable structure where quantum corrections align with bare-action terms. It then generalizes MO to four dimensions, proposing a two-vertex, two-coupling framework, and outlines future directions for leveraging MO as a tractable laboratory for renormalization and topological analysis in GFT. Overall, MO-GFT provides a practical avenue to explore the topology, amplitudes, and renormalization properties of GFTs, with potential to adapt insights from colored models to broader tensor-graph studies.

Abstract

Group Field Theories (GFT) are quantum field theories over group manifolds; they can be seen as a generalization of matrix models. GFT Feynman graphs are tensor graphs generalizing ribbon graphs (or combinatorial maps); these graphs are dual not only to manifolds. In order to simplify the topological structure of these various singularities, colored GFT was recently introduced and intensively studied since. We propose here a different simplification of GFT, which we call multi-orientable GFT. We study the relation between multi-orientable GFT Feynman graphs and colorable graphs. We prove that tadfaces and some generalized tadpoles are absent. Some Feynman amplitude computations are performed. A few remarks on the renormalizability of both multi-orientable and colorable GFT are made. A generalization from three-dimensional to four-dimensional theories is also proposed.

Multi-orientable Group Field Theory

TL;DR

The paper introduces multi-orientable GFT (MO-GFT) as a new simplification of Group Field Theory tensor graphs, distinct from the colored GFT approach. It defines MO-GFT in three dimensions via a Boulatov-like model with a vertex oriented by ± corners, and analyzes how MO graphs relate to colorable graphs, proving that all colorable graphs are MO while providing counterexamples to strict equivalence. The work shows MO excludes tadfaces and some generalized tadpoles, studies explicit Feynman amplitudes (including UV/IR-like mixing in group space), and discusses renormalizability with a favorable structure where quantum corrections align with bare-action terms. It then generalizes MO to four dimensions, proposing a two-vertex, two-coupling framework, and outlines future directions for leveraging MO as a tractable laboratory for renormalization and topological analysis in GFT. Overall, MO-GFT provides a practical avenue to explore the topology, amplitudes, and renormalization properties of GFTs, with potential to adapt insights from colored models to broader tensor-graph studies.

Abstract

Group Field Theories (GFT) are quantum field theories over group manifolds; they can be seen as a generalization of matrix models. GFT Feynman graphs are tensor graphs generalizing ribbon graphs (or combinatorial maps); these graphs are dual not only to manifolds. In order to simplify the topological structure of these various singularities, colored GFT was recently introduced and intensively studied since. We propose here a different simplification of GFT, which we call multi-orientable GFT. We study the relation between multi-orientable GFT Feynman graphs and colorable graphs. We prove that tadfaces and some generalized tadpoles are absent. Some Feynman amplitude computations are performed. A few remarks on the renormalizability of both multi-orientable and colorable GFT are made. A generalization from three-dimensional to four-dimensional theories is also proposed.

Paper Structure

This paper contains 12 sections, 7 theorems, 18 equations, 15 figures.

Table of Contents

  1. Introduction and motivation
  2. Multi-orientable GFT in three dimensions
  3. Multi-orientable graphs; relation with the colorable ones
  4. Tadpoles and generalized tadpoles
  5. Tadfaces
  6. Feynman amplitude computations
  7. A non-colorable, multi-orientable amplitude
  8. A colorable, multi-orientable amplitude
  9. A non-colorable, non-multi-orientable amplitude
  10. Remark on multi-orientable GFT and renormalizability; comparison with the colorable GFT
  11. Generalization to four dimensional GFT models
  12. Conclusions and perspectives

Key Result

Proposition 3.1

Every GFT graph which is colorable is also multi-orientable.

Figures (15)

  • Figure 1: The edge of the orientable GFT tensor graphs.
  • Figure 2: The vertex of the Boulatov model considered here.
  • Figure 3: The vertex of the proposed multi-orientable GFT model.
  • Figure 4: A tadpole graph which is multi-orientable. This graph is not allowed by the colorable GFT. The group elements of the external edges are $g_1$, $g_4$, $g_6$ , $g_2$ and $g_5$ while the group element $h$ is associated to the single internal edge.
  • Figure 5: A tadpole graph which is not multi-orientable. This type of tadpole is thus discarded within the multi-orientable GFT framework. The group elements of the external edges are $g_1$, $g_4$, $g_6$ , $g_3$ and $g_5$ while the group element $h$ is associated to the single internal edge.
  • ...and 10 more figures

Theorems & Definitions (9)

  • Proposition 3.1
  • Definition 3.1
  • Definition 3.2
  • Theorem 3.1
  • Lemma 3.1
  • Corollary 3.1
  • Corollary 3.2
  • Corollary 3.3
  • Theorem 5.1