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Colored Tensor Models - a Review

Razvan Gurau, James P. Ryan

TL;DR

Colored tensor models extend random matrix models to higher dimensions by using rank-$D$ tensors with $D{+}1$ colors, so that Feynman graphs encode $D$-dimensional cellular pseudo-manifolds. A $1/N$ expansion governed by the degree $\omega(\mathcal{G})$, defined via jackets, organizes leading melonic (sphere) graphs and permits resummation to a continuum limit with a universal $\gamma_{\rm melonic}=\tfrac{1}{2}$. The framework yields embedded matrix models, boundary observables, and a rich set of combinatorial/topological tools (core graphs, dipole moves, and PJ-factorizations), while bubble equations form a Lie algebra indexed by colored $D$-ary trees. The review also develops nontrivial classical backgrounds and connects to spin foams and dynamical triangulations, highlighting universality, renormalization prospects, and potential quantum-gravity applications. Together these results position colored tensor models as a powerful, multi-faceted approach to higher-dimensional random geometry and quantum gravity regimes.

Abstract

Colored tensor models have recently burst onto the scene as a promising conceptual and computational tool in the investigation of problems of random geometry in dimension three and higher. We present a snapshot of the cutting edge in this rapidly expanding research field. Colored tensor models have been shown to share many of the properties of their direct ancestor, matrix models, which encode a theory of fluctuating two-dimensional surfaces. These features include the possession of Feynman graphs encoding topological spaces, a 1/N expansion of graph amplitudes, embedded matrix models inside the tensor structure, a resumable leading order with critical behavior and a continuum large volume limit, Schwinger-Dyson equations satisfying a Lie algebra (akin to the Virasoro algebra in two dimensions), non-trivial classical solutions and so on. In this review, we give a detailed introduction of colored tensor models and pointers to current and future research directions.

Colored Tensor Models - a Review

TL;DR

Colored tensor models extend random matrix models to higher dimensions by using rank- tensors with colors, so that Feynman graphs encode -dimensional cellular pseudo-manifolds. A expansion governed by the degree , defined via jackets, organizes leading melonic (sphere) graphs and permits resummation to a continuum limit with a universal . The framework yields embedded matrix models, boundary observables, and a rich set of combinatorial/topological tools (core graphs, dipole moves, and PJ-factorizations), while bubble equations form a Lie algebra indexed by colored -ary trees. The review also develops nontrivial classical backgrounds and connects to spin foams and dynamical triangulations, highlighting universality, renormalization prospects, and potential quantum-gravity applications. Together these results position colored tensor models as a powerful, multi-faceted approach to higher-dimensional random geometry and quantum gravity regimes.

Abstract

Colored tensor models have recently burst onto the scene as a promising conceptual and computational tool in the investigation of problems of random geometry in dimension three and higher. We present a snapshot of the cutting edge in this rapidly expanding research field. Colored tensor models have been shown to share many of the properties of their direct ancestor, matrix models, which encode a theory of fluctuating two-dimensional surfaces. These features include the possession of Feynman graphs encoding topological spaces, a 1/N expansion of graph amplitudes, embedded matrix models inside the tensor structure, a resumable leading order with critical behavior and a continuum large volume limit, Schwinger-Dyson equations satisfying a Lie algebra (akin to the Virasoro algebra in two dimensions), non-trivial classical solutions and so on. In this review, we give a detailed introduction of colored tensor models and pointers to current and future research directions.

Paper Structure

This paper contains 39 sections, 23 theorems, 158 equations, 18 figures.

Table of Contents

  1. Introduction
  2. Gaussian measures and Feynman graphs
  3. The $(D+1)$-colored models
  4. A brief review of (colored) matrix models
  5. Topology of colored graphs
  6. Cellular structure and pseudo-manifolds
  7. Homology and homotopy
  8. Boundary graphs
  9. Combinatorial moves: $k$-dipole reductions
  10. Jackets and degree
  11. Topological equivalence
  12. Graph factorization
  13. Tensor models
  14. Introducing colored tensor models
  15. Boundary graphs as observables
  16. ...and 24 more sections

Key Result

Lemma 1

Denoting the total number of $d$-bubbles by ${\cal B}^{[d]}$$($hence ${\cal B}^{[0]}$ its the number of vertices, ${\cal B}^{[1]}$ is the number of lines, etc.$)$:

Figures (18)

  • Figure 1: Colored ribbon graphs.
  • Figure 2: $3$-bubbles of a graph in $D=3$.
  • Figure 3: The dual complex in $D=3$.
  • Figure 4: Stranded representation.
  • Figure 5: Examples of 3-dimensional graphs.
  • ...and 13 more figures

Theorems & Definitions (66)

  • Definition 1
  • Example 1
  • Example 2
  • Example 3
  • Definition 2
  • Example 4
  • Example 5
  • Example 6
  • Example 7
  • Definition 3
  • ...and 56 more