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The 1/N expansion of colored tensor models

Razvan Gurau

TL;DR

The paper proves a systematic 1/N topological expansion for the colored Boulatov tensor model, extending the 1/N paradigm from matrix models to colored GFTs. It uses bubble routing and core graphs to classify contributions, deriving a bound on Core Graph amplitudes and showing that the leading large-N behavior is governed by graphs dual to the three-sphere $S^3$. Dipole moves provide a homeomorphism-based equivalence that helps reduce graphs without changing the underlying topology. The main contribution is establishing that, at leading order, only manifolds corresponding to $S^3$ contribute, while more intricate topologies are suppressed by powers of $\delta^N(e)$, giving a controlled, topological expansion in three dimensions.

Abstract

In this paper we perform the 1/N expansion of the colored three dimensional Boulatov tensor model. As in matrix models, we obtain a systematic topological expansion, with more and more complicated topologies suppressed by higher and higher powers of N. We compute the first orders of the expansion and prove that only graphs corresponding to three spheres S^3 contribute to the leading order in the large N limit.

The 1/N expansion of colored tensor models

TL;DR

The paper proves a systematic 1/N topological expansion for the colored Boulatov tensor model, extending the 1/N paradigm from matrix models to colored GFTs. It uses bubble routing and core graphs to classify contributions, deriving a bound on Core Graph amplitudes and showing that the leading large-N behavior is governed by graphs dual to the three-sphere . Dipole moves provide a homeomorphism-based equivalence that helps reduce graphs without changing the underlying topology. The main contribution is establishing that, at leading order, only manifolds corresponding to contribute, while more intricate topologies are suppressed by powers of , giving a controlled, topological expansion in three dimensions.

Abstract

In this paper we perform the 1/N expansion of the colored three dimensional Boulatov tensor model. As in matrix models, we obtain a systematic topological expansion, with more and more complicated topologies suppressed by higher and higher powers of N. We compute the first orders of the expansion and prove that only graphs corresponding to three spheres S^3 contribute to the leading order in the large N limit.

Paper Structure

This paper contains 5 sections, 1 theorem, 35 equations, 8 figures.

Table of Contents

  1. Introduction
  2. The Colored Boulatov Model
  3. Ribbon Graphs
  4. Dipoles
  5. Bubble routing and Core Graphs

Key Result

Theorem 1

The amplitude of a Core Graph at order $p$, ${\cal G}_p$, with set of bubble ${\cal P}$ respects

Figures (8)

  • Figure 1: Colored GFT vertex and line.
  • Figure 2: Examples of Colored GFT graphs.
  • Figure 3: The bubbles of ${\cal G}_1$.
  • Figure 4: The jackets of ${\cal G}_1$.
  • Figure 5: Deletion of a ribbon line.
  • ...and 3 more figures

Theorems & Definitions (2)

  • Definition 1: Core Graph
  • Theorem 1: The Core Graph bound