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The Multi-Orientable Random Tensor Model, a Review

Adrian Tanasa

TL;DR

The paper reviews the multi-orientable (MO) random tensor model as a broader alternative to colored tensor models, detailing the $1/N$ expansion, large-$N$ limit, and the double scaling limit. It identifies melonic graphs as the leading sector and infinity graphs as the next-to-leading sector, and develops a thorough combinatorial framework (dipoles, chains, schemes) leading to dominant configurations described by rooted binary trees. The double scaling analysis combines $N\to\infty$ with critical coupling $\lambda\to\lambda_c$, yielding explicit scaling forms for two-, four-, and higher-point functions and revealing controlled, nonperturbative behavior through the parameter $\kappa$. The work also discusses probabilistic interpretations, renormalizability avenues, and future directions toward a continuum limit and connections with quantum gravity in higher dimensions.

Abstract

After its introduction (initially within a group field theory framework) in [Tanasa A., J. Phys. A: Math. Theor. 45 (2012), 165401, 19 pages, arXiv:1109.0694], the multi-orientable (MO) tensor model grew over the last years into a solid alternative of the celebrated colored (and colored-like) random tensor model. In this paper we review the most important results of the study of this MO model: the implementation of the $1/N$ expansion and of the large $N$ limit ($N$ being the size of the tensor), the combinatorial analysis of the various terms of this expansion and finally, the recent implementation of a double scaling limit.

The Multi-Orientable Random Tensor Model, a Review

TL;DR

The paper reviews the multi-orientable (MO) random tensor model as a broader alternative to colored tensor models, detailing the expansion, large- limit, and the double scaling limit. It identifies melonic graphs as the leading sector and infinity graphs as the next-to-leading sector, and develops a thorough combinatorial framework (dipoles, chains, schemes) leading to dominant configurations described by rooted binary trees. The double scaling analysis combines with critical coupling , yielding explicit scaling forms for two-, four-, and higher-point functions and revealing controlled, nonperturbative behavior through the parameter . The work also discusses probabilistic interpretations, renormalizability avenues, and future directions toward a continuum limit and connections with quantum gravity in higher dimensions.

Abstract

After its introduction (initially within a group field theory framework) in [Tanasa A., J. Phys. A: Math. Theor. 45 (2012), 165401, 19 pages, arXiv:1109.0694], the multi-orientable (MO) tensor model grew over the last years into a solid alternative of the celebrated colored (and colored-like) random tensor model. In this paper we review the most important results of the study of this MO model: the implementation of the expansion and of the large limit ( being the size of the tensor), the combinatorial analysis of the various terms of this expansion and finally, the recent implementation of a double scaling limit.

Paper Structure

This paper contains 17 sections, 16 theorems, 44 equations, 23 figures.

Table of Contents

  1. Introduction
  2. Definition of the model
  3. The $\boldsymbol{1/N}$ expansion and the large $\boldsymbol{N}$ limit
  4. Feynman amplitudes; the $\boldsymbol{1/N}$ expansion
  5. The large $\boldsymbol{N}$ limit -- the leading order (melonic graphs)
  6. The large $\boldsymbol{N}$ limit -- the next-to-leading order
  7. Leading and next-to-leading order series
  8. Leading order series
  9. Next to leading order series
  10. Some combinatorial developments
  11. Dipoles, chains, schemes and all that
  12. Generating functions, asymptotic enumeration and dominant schemes
  13. The double scaling limit
  14. The two-point function
  15. The four-point function
  16. ...and 2 more sections

Key Result

Proposition 3.1

A non-bipartite MO graph has at least one non-orientable jacket and thus its degree satisfies the inequality: $\delta \ge \frac{1}{2}$.

Figures (23)

  • Figure 1: The MO vertex and an MO vacuum Feynman graph.
  • Figure 2: A different representation (without strands) of the MO vertex.
  • Figure 3: Examples of MO tensor graphs which do not occur in the colorable framework.
  • Figure 4: The three jackets associated to the graph of Fig. \ref{['graf']}.
  • Figure 5: Deletion a pair of opposite corner strands in a non-MO graph.
  • ...and 18 more figures

Theorems & Definitions (25)

  • Definition 2.1
  • Proposition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Theorem 3.4
  • Theorem 3.5
  • Lemma 4.1
  • Definition 4.2
  • Definition 4.3
  • Definition 4.4
  • ...and 15 more