Contents

# Contents

## Idea

For classes of gauge theories, such as (super) Yang-Mills theory or Chern-Simons theory or various matrix models, whose gauge groups may be $N \times N$ square matrices for any natural number $N$, notably in the special unitary group $SU(N)$, the special orthogonal group $SO(N)$ or the quaternionic unitary group (“symplectic group”) $Sp(N)$, one may consider the limit of the theory’s scattering amplitudes and other quantum observables as $N\to \infty$ (“large number of colours-limit”). In good cases the values close to but away from this large $N$ limit scale with $1/N$ and allow a perturbation series around the large $N$ limit called the $1/N$ expansion.

This large $N$ limit often has remarkable properties, often revealing an otherwise hidden relation to perturbative string theories with the parameter $1/N$ proportional to the string coupling constant.

Notably for Yang-Mills theory and in particular for QCD, the large $N$-behaviour is exhibited by rewriting the Feynman amplitudes in 't Hooft double line notation. If the 't Hooft coupling $g^2 N$ is held fixed as $N\to \infty$, this turns out to organize the gauge theory’s Feynman perturbation series by the Euler characteristic/genus of emerging string worldsheet surfaces, with genus 0 (planar graphs) dominating in the large $N$ limit, whence also called the planar limit.

(Open/closed string duality plays a subtle role in interpreting the 't Hooft double line notation of gauge theory Feynman diagrams in the large N limit alternatively as open string or as closed string worldsheets, see Gopakumar-Vafa 98, Gaiotto-Rastelli 03, Gopakumar 04 and notably Marino 04, Section III, p. 14).

At least for the case of super Yang-Mills theories the full statement of the relation of large-$N$ gauge theory to a perturbative string theory is the content of the AdS/CFT correspondence, which explains that the effective string worldsheets emerging from the gauge theory propagate in a higher-dimensional asymptotically anti-de Sitter spacetime (the near horizon geometry of a black brane) whose asymptotic boundary (the worldvolume of the black brane itself) is identified with the spacetime of the original gauge theory.

An extreme case of this large $N$-limit is that of the BFSS matrix model in AdS2/CFT1 duality where all spatial dependence of fields in the higher dimensional spacetime is supposedly encoded in the quantum mechanics of $N\times N$ matrices as $N\to \infty$. And for the IKKT matrix model this includes also the temporal dependence.

For non-supersymmetric gauge theories such as QCD this duality still holds in suitably adjusted form such as in the AdS/QCD correspondence. Here the $1/N$-expansion serves to provide a computational tool for describing confined hadron states (mesons and baryons, hence in particular nucleons and hence ordinary room-temperature matter) which are not seen by ordinary perturbation theory in the gauge theory coupling constant (the confinement/mass gap problem).

## References

### General

The original article observing the large $N$-behaviour and the planar limit of Yang-Mills theory in 't Hooft double line notation is:

First inkling of holographic QCD and matrix models:

Lecture notes:

• Sidney Coleman, $1/N$,

in: A. Zichichi (ed.) Pointlike Structures Inside and Outside Hadrons. The Subnuclear Series, vol 17. Springer 1982 (doi:10.1007/978-1-4684-1065-5_2)

and Chapter 8 in: Sidney Coleman, Aspects of Symmetry, Cambridge University Press 1985 (doi:10.1017/CBO9780511565045.009)

• Gerard 't Hooft, Large $N$, workshop lecture (hep-th/0204069)

• A. V. Manohar, Large $N$ QCD, Les Houches Lecture 2004, (arXiv:hep-ph/9802419, pdf)

• Markus Gross, Large $N$, 2006 (pdf)

• McGreevy, Swingle, Large $N$ counting, 2008 (pdf)

• Wikipedia, 1/N expansion

• E. Brézin, S.R. Wadia, eds. The Large $N$ Expansion in Quantum Field Theory and Statistical Physics, a book collection of reprinted historical articles, gBooks

Application of WKB method:

• V. P. Maslov, O. Yu. Shvedov, Large-N expansion as a semiclassical approximation to the third-quantized theory, Physical Review D60(10) 105012 doi

The refinement for super Yang-Mills theory to the AdS/CFT correspondence (see there for more) originates with

• Juan Maldacena, The large $N$ limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys.2:231-252, 1998 hep-th/9711200; Wilson loops in large N field theories, hep-th/9803002.

reviewed for instance in

But see at AdS/CFT correspondence for a more comprehensive list of references.

Further discussion:

• Edward Witten, Baryons in the $1/n$ Expansion, Nucl. Phys. B160 (1979) 57–115 (spire:140391, doi:10.1016/0550-3213(79)90232-3)

(on mesons and baryons in the large N limit)

• A. Jevicki, Instantons and the $1/N$ expansion in nonlinear $\sigma$ models, Phys. Rev. D 20, 3331–3335 (1979) pdf

• Laurence G. Yaffe, Large $N$ limits as classical mechanics, Rev. Mod. Phys. 54, 407–435 (1982) (pdf)

• A. A. Migdal, Loop equations and $1/N$ expansion, Physics Reports, 102 (4), 199–290 (1983) (doi)

• M. Bershadsky, Z. Kakushadze, Cumrun Vafa, String expansion as large $N$ expansion of gauge theories, Nucl.Phys. B523 (1998) 59–72 (hep-th/9803076, doi)

• Gary Horowitz, Hirosi Ooguri, Spectrum of large $N$ gauge theory from supergravity, hep-th/9802116

• S. Sinha, Cumrun Vafa, $SO$ and $Sp$ Chern-Simons at Large $N$ (arXiv:hep-th/0012136)

• Hiroyuki Fuji, Yutaka Ookouchi, Confining Phase Superpotentials for $SO/Sp$ Gauge Theories via Geometric Transition, JHEP 0302:028, 2003 (arXiv:hep-th/0205301)

• Hirosi Ooguri, Cumrun Vafa, Worldsheet derivation of a large $N$ duality, Nucl. Phys. B641:3–34, 2002 (arXiv:hep-th/0205297)

• Semyon Klevtsov, Random normal matrices, Bergman kernel and projective embeddings, J. High Energ. Phys. 2014, 133 (2014) doi arXiv:1309.7333

On a kind of BV-quantization of the Loday-Quillen-Tsygan theorem and relating to the large $N$-limit of Chern-Simons theory:

On the logical equivalence between the four-colour theorem and a statement about transition from the small N limit to the large N limit for Lie algebra weight systems on Jacobi diagrams via the 't Hooft double line construction:

On the large N limit in lattice gauge theory:

• Margarita Garcia Perez, Prospects for large $N$ gauge theories on the lattice (arXiv:2001.10859)

### Open/closed string duality

On the role of open/closed string duality in interpreting the large N limit:

Nowadays we interpret $[$ the 't Hooft double line notation $]$ quite literally as the perturbative expansion of an open string theory, either because the full open string theory is just equal to the gauge theory (as e.g. for Chern-Simons theory [27]), or because we take an appropriate low-energy limit (as e.g. for N = 4 SYM [31]).

The general speculation [1] is that upon summing over the number of holes, (1.1) can be recast as the genus expansion for some closed string theory of coupling $g_s = g_{YM}^2$. This speculation is sometimes justified by appealing to the intuition that diagrams with a larger and larger number of holes look more and more like smooth closed Riemann surfaces. This intuition is perfectly appropriate for the double-scaled matrix models, where the finite N theory is interpreted as a discretization of the closed Riemann surface; to recover the continuum limit, one must send $N\to \infty$ and tune $t$ to the critical point $t_c$ where diagrams with a diverging number of holes dominate.

However, in AdS/CFT, or in the Gopakumar-Vafa duality [2], $t$ is a free parameter, corresponding on the closed string theory side to a geometric modulus. The intuition described above clearly goes wrong here.

A much more fitting way in which the open/closed duality may come about in these cases is for each fatgraph of genus g and with h holes to be replaced by a closed Riemann surface of the same genus g and with h punctures: each hole is filled and replaced by a single closed string insertion.

In relation to quantum error correction:

• Alexey Milekhin, Quantum error correction and large $N$ (arXiv:2008.12869)
category: physics

Last revised on March 16, 2023 at 10:43:40. See the history of this page for a list of all contributions to it.