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arxiv: 1907.03531 · v2 · pith:ZRQ6P47Jnew · submitted 2019-07-08 · ✦ hep-th · math-ph· math.MP

Notes on Tensor Models and Tensor Field Theories

Razvan Gurau This is my paper

Pith reviewed 2026-05-25 01:22 UTC · model grok-4.3

classification ✦ hep-th math-phmath.MP
keywords tensor modelstensor field theories1/N expansionmelonic limitlarge N limitconformal field theoriesrandom tensorsSYK models
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The pith

Tensor models admit a 1/N expansion whose melonic large-N limit yields analytically tractable strongly coupled quantum field theories and a new class of conformal field theories.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The notes establish that tensor models and tensor field theories possess a 1/N expansion in which melonic diagrams dominate at large N. This limit is simpler than the planar diagrams of random matrices and richer than the large-N behavior of vector models. The resulting theories remain non-trivial yet solvable, supplying concrete examples of strongly coupled quantum field theories that can be studied analytically. These constructions also generate a new family of conformal field theories. The review presents both the classical details of the expansion and selected recent developments.

Core claim

Tensor models and tensor field theories admit a 1/N expansion and a melonic large N limit which is simpler than the planar limit of random matrices and richer than the large N limit of vector models. They provide examples of analytically tractable but non trivial strongly coupled quantum field theories and lead to a new class of conformal field theories.

What carries the argument

The melonic large N limit, in which only melonic diagrams survive the 1/N expansion of tensor models.

If this is right

  • The models furnish concrete, solvable instances of strongly coupled quantum field theories.
  • The same limit produces a new family of conformal field theories.
  • The 1/N expansion supplies a systematic way to compute observables order by order.
  • The framework covers both classical results on the expansion and recent extensions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The melonic limit may serve as a bridge between tensor models and SYK-type models for further analytic study.
  • The same expansion technique could be tested in higher-rank or colored tensor models to check universality of the melonic sector.
  • If the limit remains solvable in the presence of interactions, it offers a route to controlled calculations in regimes where perturbation theory fails.

Load-bearing premise

Melonic diagrams dominate the large-N limit and the resulting theories stay well-defined and non-trivial after the expansion is taken.

What would settle it

An explicit computation in which a non-melonic diagram contributes at the same leading order in 1/N as the melonic ones would disprove dominance of the melonic sector.

Figures

Figures reproduced from arXiv: 1907.03531 by Razvan Gurau.

Figure 1
Figure 1. Figure 1: The Schwinger Dyson equation. ... PSfrag replacements = G G G Σ [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The melonic truncation of the self energy. [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The vertex and the propagator of the colored tensor [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Rooted graphs. where the reduced degree ωˆ(G) of a graph G is a non negative half integer. The properties of the degree are discussed in detail in Appendix A. The two point function (and all the other correlation functions) admits a 1/N expansion indexed by the degree: G = X ωˆ∈N/2 N −ωˆ ωˆ( X G)=ˆω G∈G (−λ) V (G)C D+1 2 V (G)+1 . (3.8) At leading order one obtains only the graphs with ˆω(G) = 0. The graph… view at source ↗
Figure 5
Figure 5. Figure 5: Colored and stranded representation of the vertex [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Vertices and edges of the CTKT model We denote Vt(G), Vp(G) and Vd(G) the numbers of tetrahedral, pillow and double trace vertices of a CTKT graph G, and F(G) the number of faces of G. The number of pillow vertices splits as the sum of the numbers of pillow vertices of each kind. The edges are not colored, but the faces are colored with a color 1, 2 or 3. Using the same trick as before, we can compute the … view at source ↗
Figure 7
Figure 7. Figure 7: The melon-tadpole self energy. In the melon-tadpole truncation one includes two graphs. The first one is a tadpole graph whose vertex is either a pillow or the double trace such that the edge closes the maximal number of faces. The second is a melon with two tetrahedral vertices. At leading order we have: (G LO) −1 = C −1 − Σ LO , Σ LO = −(λp + λd)G LO + λ 2 (G LO) 3 . (3.26) 17 [PITH_FULL_IMAGE:figures/f… view at source ↗
Figure 8
Figure 8. Figure 8: Graphs contributing to the 2PI action at first order [PITH_FULL_IMAGE:figures/full_fig_p019_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: On the left: a two particle reducible contribution [PITH_FULL_IMAGE:figures/full_fig_p020_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: The four point Dyson equation. The second derivative of Γ is related to the four point kernel: δ 2Γ δGabδGxy = 1 2 G −1 aa′G −1 bb′  S − K  a ′b ′ ;xy . (4.11) 15It is sometimes useful to give a formal functional integral formula for Γ: e −Γ[G] = e − 1 2 Tr[C−1G] Z 2P I [dφ] e − 1 2 φ·G−1φ−S int[φ] . 20 [PITH_FULL_IMAGE:figures/full_fig_p020_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: On the left the correlation function hφx1 φx2Oµ¯;xi G;ǫ 2P I . On the right a contri￾bution which is two particle reducible in the channel (φφ) → O. The red dot represents the composite operator. The last correlation hφx1φx2Oµ¯;xi G;ǫ 2P I is represented in [PITH_FULL_IMAGE:figures/full_fig_p022_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: The bare series up to quartic order (the blue verti [PITH_FULL_IMAGE:figures/full_fig_p028_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Ladders, caps and double caps. Let us denote Ur, Sr, Tr the amplitude of the ladder, cap respectively double cap with 2r tetrahedral vertices, and let us define the generating functions: U(˜g) = X r≥1 g˜ 2rUr , S(˜g) = X r≥1 g˜ 2rSr , T(˜g) = X r≥0 g˜ 2rTr , (5.20) 28 [PITH_FULL_IMAGE:figures/full_fig_p028_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Bare and renormalized tetrahedral couplings (in [PITH_FULL_IMAGE:figures/full_fig_p030_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Flows in the (g1, g2) plane in the case of imaginary tetrahedral coupling. Comparison with ζ = 1, d = 4 − ǫ. The case ˜g real is very similar to the Wilson-Fisher like fixed point discussed in Eq. (5.2). In d = 4 − ǫ the tetrahedral coupling is not marginal (see Eq. (5.2)), but has a flow driven by the wave function. The flow has a fixed point for a real value of the tetrahedral coupling. The key point is… view at source ↗
Figure 16
Figure 16. Figure 16: Examples of ribbon graphs with 4 vertices, 6 edges [PITH_FULL_IMAGE:figures/full_fig_p033_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Deletion of an edge. Let us delete iteratively a maximal set of edges in a connected graph such that at each step the edge we delete separates two different faces. This can not disconnect the graph. The number of edges deleted is F(G)−1. The remaining edges connect all the vertices, hence there are at least V (G) − 1 of them. It follows that the non orientable genus of a connected ribbon graph is a non ne… view at source ↗
Figure 18
Figure 18. Figure 18: A triangle with twisted edges in a ribbon graph. [PITH_FULL_IMAGE:figures/full_fig_p034_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Melonic graphs at first orders Proposition 4. For D ≥ 3, a connected edge (D + 1)–colored graph G has reduced degree zero if and only if it is melonic. Proof. As the insertion of two vertices connected by D parallel edges brings D 2  new faces, it does not change the degree. It follows that melonic graphs have degree zero. For the converse statement, we proceed by induction on the number of vertices. The… view at source ↗
Figure 20
Figure 20. Figure 20: Vertices and edges of the CTKT model [PITH_FULL_IMAGE:figures/full_fig_p038_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Resolution of the pillow and double trace vertice [PITH_FULL_IMAGE:figures/full_fig_p038_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: A melon-tadpole graph. Proposition 7. A CT KT graph has reduced degree zero if and only if it is a melon-tadpole graph. Proof. As ω(G) = ω(G˜), this comes to proving that G˜ has zero degree if and only if it is melonic. The proof follows the one of Proposition 4, but with some twists. The main difference is that faces can now have odd length. Assume that the connected graph G˜ with only tetrahedral vertic… view at source ↗
Figure 23
Figure 23. Figure 23: The 1PI decomposition of a graph. These graphs can always be decomposed along one particle irreducibly edges (that is edges that, when cut, disconnect the graph), as depicted in [PITH_FULL_IMAGE:figures/full_fig_p040_23.png] view at source ↗
read the original abstract

Tensor models and tensor field theories admit a $1/N$ expansion and a melonic large $N$ limit which is simpler than the planar limit of random matrices and richer than the large $N$ limit of vector models. They provide examples of analytically tractable but non trivial strongly coupled quantum field theories and lead to a new class of conformal field theories. We present a compact introduction to the topic, covering both some of the classical results in the field, like the details of the $1/N$ expansion, as well as recent developments. These notes are loosely bases on four lectures given at the Journ\'ees de physique math\'ematique Lyon 2019: Random tensors and SYK models.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 1 minor

Summary. The manuscript consists of lecture notes summarizing tensor models and tensor field theories. It states that these models admit a 1/N expansion whose melonic large-N limit is simpler than the planar limit of random matrices and richer than the large-N limit of vector models, yielding analytically tractable yet non-trivial strongly coupled QFTs and a new class of CFTs. The notes cover classical results on the 1/N expansion together with recent developments and are based on four lectures given at Journées de physique mathématique Lyon 2019.

Significance. The notes supply a compact, pedagogical overview of an established body of results on melonic dominance. By correctly positioning the tensor-model large-N limit as intermediate between vector and matrix models, the manuscript offers a useful entry point for researchers studying solvable strongly coupled theories and their conformal fixed points.

minor comments (1)
  1. [Abstract] Abstract: the clause 'These notes are loosely bases on four lectures' contains a grammatical error ('bases' should read 'based').

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our lecture notes and the recommendation for minor revision. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; introductory notes on established results

full rationale

This document consists of lecture notes summarizing classical and recent results on the 1/N expansion and melonic large-N limit in tensor models, without presenting any new derivation chain or first-principles claims. All central statements are explicitly framed as known facts from the literature rather than derived within the text, so no load-bearing step reduces to a self-definition, fitted input, or self-citation chain. The presupposition of melonic dominance is treated as the standard result in the field and is not introduced as a novel assumption here.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

This is an expository review paper; it introduces no free parameters, axioms, or invented entities of its own.

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