large N limit (Rev #13) in nLab

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×NN \times N square matrices for any natural number NN, notably in the special unitary group SU(N)SU(N), the special orthogonal group SO(N)SO(N) or the quaternionic unitary group (“symplecticc group”) Sp(N)Sp(N), one may consider the limit of the theory’s scattering amplitudes and other quantum observables as N→∞N\to \infty (“large number of colours-limit”). In good cases the values close to but away from this large NN limit scale with 1/N1/N and allow a perturbation series around the large NN limit called the 1/N1/N expansion.

This large NN limit often has remarkable properties, often revealing an otherwise hidden relation to perturbative string theories with the parameter 1/N1/N proportional to the string coupling constant.

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

At least for the case of super Yang-Mills theories the full statement of the relation of large-NN 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 NN-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×NN\times N matrices as N→∞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/N1/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).

• planar limit

• 't Hooft coupling

• AdS-CFT duality

• BFSS matrix model

• volume conjecture

References

General

The original article observing the large NN-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)

Lecture notes:

• Sidney Coleman, 1/N1/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 NN, workshop lecture (hep-th/0204069)

• A. V. Manohar, Large NN QCD, L:es Houches Lecture 2004, pdf

• Markus Gross, Large NN, 2006 (pdf)

• McGreevy, Swingle, Large NN counting, 2008 (pdf)

See also:

• Wikipedia, 1/N expansion

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

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

• Juan Maldacena, The large NN 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:

• A. Jevicki, Instantons and the 1/N1/N expansion in nonlinear σ\sigma models, Phys. Rev. D 20, 3331–3335 (1979) pdf

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

• A. A. Migdal, Loop equations and 1/N1/N expansion, Physics Reports, 102 (4), 199-290 (1983) (doi)

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

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

• Hirosi Ooguri, Cumrun Vafa, Worldsheet Derivation of a Large NN Duality, Nucl. Phys. B641:3-34, 2002 (arXiv:hep-th/0205297)

• Semyon Klevtsov, Random normal matrices, Bergman kernel and projective embeddings, arxiv/1309.7333

Unoriented case

• S. Sinha, Cumrun Vafa, SOSO and SpSp Chern-Simons at Large NN (arXiv:hep-th/0012136)

  (for Chern-Simons theory and topological string theory)

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

• Harald Ita, Harald Nieder, Yaron Oz, Perturbative Computation of Glueball Superpotentials for SO(N)SO(N) and USp(N)USp(N), JHEP 0301:018, 2003 (arXiv:hep-th/0211261)

• Jaume Gomis, Anton Kapustin, Two-Dimensional Unoriented Strings And Matrix Models, JHEP 0406 (2004) 002 (https://arxiv.org/abs/hep-th/0310195)