Large N

G. 't Hooft

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Large N

G. 't Hooft

High Energy Physics - Theory · Submitted 8 Apr 2002

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim In large-N gauge theories, the 1/N expansion is tantamount to sorting the Feynman diagrams according to their degree of planarity, that is, the minimal genus of the plane onto which the diagram can be mapped without any crossings.

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What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.25 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.07 · compares with prior results 0.09 · mentions code or data availability 0.02.

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arXiv listing abstract v1. Results and the main claim are underlined; show every role.

In the first part of this lecture, the 1/N expansion technique is illustrated for the case of the large-N sigma model. In large-N gauge theories, the 1/N expansion is tantamount to sorting the Feynman diagrams according to their degree of planarity, that is, the minimal genus of the plane onto which the diagram can be mapped without any crossings. This holds both for the usual perturbative expansion with respect to powers of {tilde g}^2=g^2 N, as well as for the expansion of lattice theories in positive powers of 1/{tilde g}^2. If there were no renormalization effects, the tilde g expansion would have a finite radius of convergence. The zero-dimensional theory can be used for counting planar diagrams. It can be solved explicitly, so that the generating function for the number of diagrams with given 3-vertices and 4-vertices, can be derived exactly. This can be done for various kinds of Feynman diagrams. We end with some remarks about planar renormalization.

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What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	In the first part of this lecture, the 1/N expansion technique is illustrated for the case of the large-N sigma model.	contribution 0.53	0.50	0.04	asserted 0.99
s1	In large-N gauge theories, the 1/N expansion is tantamount to sorting the Feynman diagrams according to their degree of planarity, that is, the minimal genus of the plane onto which the diagram can be mapped without any crossings.	background 0.85	0.11	0.21	asserted 1.00
s2	This holds both for the usual perturbative expansion with respect to powers of {tilde g}^2=g^2 N, as well as for the expansion of lattice theories in positive powers of 1/{tilde g}^2.	background 0.56	0.13	0.10	asserted 1.00
s3	If there were no renormalization effects, the tilde g expansion would have a finite radius of convergence.	background 0.61	0.12	0.06	hedged possibility 0.49
s4	The zero-dimensional theory can be used for counting planar diagrams.	method 0.88	0.29	0.05	asserted 0.99
s5	It can be solved explicitly, so that the generating function for the number of diagrams with given 3-vertices and 4-vertices, can be derived exactly.	result 0.55	0.57	0.06	asserted 0.93
s6	This can be done for various kinds of Feynman diagrams.	contribution 0.48	0.46	0.03	asserted 0.73
s7	We end with some remarks about planar renormalization.	future work 0.54	0.47	0.06	unclear 0.90

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The paper

Original abstract

In the first part of this lecture, the 1/N expansion technique is illustrated for the case of the large-N sigma model. In large-N gauge theories, the 1/N expansion is tantamount to sorting the Feynman diagrams according to their degree of planarity, that is, the minimal genus of the plane onto which the diagram can be mapped without any crossings. This holds both for the usual perturbative expansion with respect to powers of {tilde g}^2=g^2 N, as well as for the expansion of lattice theories in positive powers of 1/{tilde g}^2. If there were no renormalization effects, the tilde g expansion would have a finite radius of convergence. The zero-dimensional theory can be used for counting planar diagrams. It can be solved explicitly, so that the generating function for the number of diagrams with given 3-vertices and 4-vertices, can be derived exactly. This can be done for various kinds of Feynman diagrams. We end with some remarks about planar renormalization.

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How the literature uses it

19 arXiv papers cite this in 22 sentences

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Compares with it 1 paper

Nonetheless, the theory exhibits a remarkable similarity with the well-known $O(N)$ vector model [ref] that we elaborate on in more detail below, and offers a new perspective on the nascent NN-QFT correspondence that may be fruitfully developed further.

The edge of chaos: quantum field theory and deep neural networks · Introduction · 0.89

Cites it as support 1 paper

This is because the perturbative proof of factorization holds in the lattice regularization [ref] at finite $a$.

Large $N$ scaling and factorization in SU($N$) Yang-Mills gauge theory · Large $$ limit at fixed lattice spacing · 0.97

Cites it as background 8 papers

Yet, here we are not interested in this type of questions, but rather in the study of $N_c$ different from $3,$ and in particular the study of the limit in which $N_c$ is a `large' number, in order to understand better our word with $N_c=3.$ This is indeed the so-called QCD in the large-$N_c$ limit, initiated by 't Hooft [ref] and further investigated in several review papers and lectures [ref] (and refs. therein).

Introductory visual lecture on QCD at large-$N_{c}$: bound states, chiral models, and phase diagram · Introduction · 0.98
These techniques have now been used in many fields of physics from atomic and molecular physics [ref] , and condensed matter [ref] , to quantum field theory [ref] , among others [ref] .

The Pauli principle in collective motion: Reimagining and reinterpreting Cooper pairs, the Fermi sea, Pauli blocking and superfluidity · Overview · 0.94
See [ref] for a review.

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