nLab large N limit

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 05, 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).

Related concepts

• planar limit

• large 1/N limit

• 't Hooft coupling

• light-cone gauge quantization

• AdS-CFT duality

• BFSS matrix model

• volume conjecture

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:

• Gerard 't Hooft, A Planar Diagram Theory for Strong Interactions, Nucl. Phys. B72 (1974) 461 (spire:80491, doi:10.1016/0550-3213(74)90154-0)

First inkling of holographic QCD and matrix models:

• Tohru Eguchi, Hikaru Kawai, Reduction of dynamical degrees of freedom in the large-$N$ gauge theory, Phys. Rev. Lett. 48, 1063 (1982) (spire:176459, doi:10.1103/PhysRevLett.48.1063)

• A. Gonzalez-Arroyo, M. Okawa, A twisted model for large $N$ lattice gauge theory, Physics Letters B120:1–3 (1983) 174–178 (doi:10.1016/0370-2693(83)90647-0)

• A. Gonzalez-Arroyo, M. Okawa, Twisted-Eguchi-Kawai model: A reduced model for large- $N$ lattice gauge theory, Phys. Rev. D 27, 2397 (1983) (doi:10.1103/PhysRevD.27.2397)

First observation that various observables in QCD for $N=3$ are actually well-approximated by the large $N$-limit:

• B. Lucini, M. Teper, SU(N) gauge theories in four dimensions: exploring the approach to $N = \infty$, JHEP 0106 050 (2001) [doi:10.1088/1126-6708/2001/06/050]

See:

• Edward Witten, Some Milestones in the Study of Confinement, talk at Prospects in Theoretical Physics 2023 – Understanding Confinement, IAS (2023) [web, YT]

  26:26: “by now it’s clear [Lucini & Teper 2001] that lattice gauge theory, at least for the glueball sector, has made it clear that the $1/N$-expansion is a good approximation to the real world, especially if you include a leading correction to the large $N$ limit. Now unfortunately this is best established in the glueball sector, which is not very accessible experimentally.”

  28:56: “but the $1/N$ expansion doesn’t explain everything. In fact, it’s not hard to find phenomena in meson physics where the $1/N$-expansion does not work well.”

  36:15: “I suspect the $1/N$-expansion works reasonably well for many aspects of baryons. However, as for mesons, it is easy to point to things that won’t work well for baryons. In particular, among other things, I don’t think the $1/N$ expansion will be successful for nuclei as opposed to individual nucleons.”

  37:12: “I don’ t think the phenomenological models used by nuclear physicists would have any success at of if the large N limit was a good description of nuclei.”

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)

• Paul Romatschke, Quantum Field Theory in Large $N$ Wonderland: Three Lectures [arXiv:2310.00048]

• Francesco Giacosa, Introductory visual lecture on QCD at large-$N_c$: bound states, chiral models, and phase diagram [arXiv:2402.14097]

See also:

• 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

• Juan Maldacena, The Large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2:231, 1998 (hep-th/9711200)

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)

• Alexander 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)

  (for Chern-Simons theory as a topological string theory)

• 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

• Jan Albert, Johan Henriksson, Leonardo Rastelli, Alessandro Vichi, Bootstrapping mesons at large $N$: Regge trajectory from spin-two maximization [arXiv:2312.15013]

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

• Grégory Ginot, Owen Gwilliam, Alastair Hamilton, Mahmoud Zeinalian, Large $N$ phenomena and quantization of the Loday-Quillen-Tsygan theorem, Adv. Math. 409A (2022) 108631 [arXiv:2108.12109, doi:10.1016/j.aim.2022.108631]
• Owen Gwilliam, Alastair Hamilton, Mahmoud Zeinalian, A homological approach to the Gaussian Unitary Ensemble, arXiv:2206.04256

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:

• Dror Bar-Natan, Lie Algebras and the Four Color Theorem, Combinatorica 17-1(1997) 43–52 (arXiv:q-alg/9606016, doi:10.1007/BF01196130)

On the large N limit in lattice gauge theory:

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

See also:

• Marco Bochicchio, Mauro Papinutto, Francesco Scardino, On the structure of the large-$N$ expansion in $SU(N)$ Yang-Mills theory [arXiv:2401.09312]

Open/closed string duality

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

• Rajesh Gopakumar, Cumrun Vafa, On the Gauge Theory/Geometry Correspondence, Adv. Theor. Math. Phys. 3 (1999) 1415–1443 (arXiv:hep-th/9811131)

• Davide Gaiotto, Leonardo Rastelli, A paradigm of open/closed duality: Liouville D-branes and the Kontsevich model, JHEP 0507:053,2005 (hep-th/0312196)

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.

• Rajesh Gopakumar, Free Field Theory as a String Theory?, Comptes Rendus Physique 5 (2004) 1111-1119 (hep-th/0409233)

• Marcos Marino, Chern-Simons Theory and Topological Strings, Rev. Mod. Phys. 77:675-720, 2005 (arXiv:hep-th/0406005)

In relation to quantum error correction:

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

category: physics