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The complete 1/N expansion of colored tensor models in arbitrary dimension

Razvan Gurau

TL;DR

This work constructs the complete $1/N$ expansion for colored tensor models in arbitrary dimensions, focusing on the i.i.d. and Boulatov–Ooguri models. It develops two complementary expansion schemes: a combinatorial amplitude-based expansion and a topology-aware (topological) expansion, each with rigorous graph-based accounting via bubbles, jackets, and dipole moves. The authors derive explicit amplitude bounds in terms of a graph degree ω and establish foundational core-graph classifications, enabling term-by-term evaluation of the free energy. The results generalize known $1/N$ results from low dimensions to arbitrary $D$, clarifying the role of topology and offering a path toward UV-complete gravitational tensor models.

Abstract

In this paper we generalize the results of [1,2] and derive the full 1/N expansion of colored tensor models in arbitrary dimensions. We detail the expansion for the independent identically distributed model and the topological Boulatov Ooguri model.

The complete 1/N expansion of colored tensor models in arbitrary dimension

TL;DR

This work constructs the complete expansion for colored tensor models in arbitrary dimensions, focusing on the i.i.d. and Boulatov–Ooguri models. It develops two complementary expansion schemes: a combinatorial amplitude-based expansion and a topology-aware (topological) expansion, each with rigorous graph-based accounting via bubbles, jackets, and dipole moves. The authors derive explicit amplitude bounds in terms of a graph degree ω and establish foundational core-graph classifications, enabling term-by-term evaluation of the free energy. The results generalize known results from low dimensions to arbitrary , clarifying the role of topology and offering a path toward UV-complete gravitational tensor models.

Abstract

In this paper we generalize the results of [1,2] and derive the full 1/N expansion of colored tensor models in arbitrary dimensions. We detail the expansion for the independent identically distributed model and the topological Boulatov Ooguri model.

Paper Structure

This paper contains 13 sections, 7 theorems, 45 equations, 9 figures.

Table of Contents

  1. Introduction
  2. The $1/N$ expansion of random matrix models
  3. Graphs, Jackets and Bubbles
  4. Colored GFT Models
  5. The Independent Identically Distributed Model
  6. Amplitude
  7. The Combinatorial $1/N$ Expansion
  8. Topological $1/N$ expansion
  9. The Boulatov Ooguri Model
  10. Power counting
  11. $k$-Dipoles of degree $0$
  12. Topological $1/N$ expansion
  13. The first terms of the $1/N$ expansion

Key Result

Lemma 1

The degrees of a graph $\omega({\cal G})$ and its $D$-bubbles $\omega({\cal B}^{\widehat{i}}_{(\rho)})$ respect In particular, by eq. eq:smeche, $p+D-{\cal B}^{[D]} \ge 0$, thus $\omega({\cal G})=0 \Rightarrow \omega({\cal B}^{\widehat{i}}_{(\rho)}) = 0 \; \forall i,\rho$ and $p+D-{\cal B}^{[D]} =0$.

Figures (9)

  • Figure 1: Contraction of a ribbon line in ${\cal G}$.
  • Figure 2: Deletion of a ribbon line in ${\cal G}$.
  • Figure 3: Line and vertex of the Colored GFT graphs.
  • Figure 4: Bubbles of a graph in $D=3$.
  • Figure 5: Contraction of a k-Dipole.
  • ...and 4 more figures

Theorems & Definitions (11)

  • Definition 1
  • Definition 2
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Definition 3
  • Definition 4
  • Lemma 5
  • Lemma 6
  • ...and 1 more