large N limit (Rev #11) in nLab

Contents

Idea

In theoretical physics, one often considers gauge theory models whose symmetries are groups of large matrices, notably in the special unitary group SU(N)SU(N). The limit of such theories for N→∞N\to \infty (“large number of colours-limit”) has often remarkable properties and string-like features.

This limit sometimes appears in its own right, but sometimes it is just considered as an approximation for a system with fixed finite NN. One of the features in the large NN limit is that non-planar Feynman diagrams lose their importance and that the correlation functions satisfy certain decoupling/factorization rule. The behaviour is studied in terms of expansion in 1/N1/N whose square has a similar role to Planck's constant in the semiclassical approximation limit of quantum mechanics.

Properties

SO(N)SO(N), Sp(N)Sp(N) gauge theory and unoriented strings

(…)

Gaumis-Kapustin 04

‘t Hooft double-line diagrams for SO(N)SO(N) gauge theory and unoriented worldsheets:

Ita-Nieder-Oz 02, Figure 3

McGreevySwingle 08, Section 4

(…)

• planar limit

• AdS-CFT duality

• BFSS matrix model

• volume conjecture

• 't Hooft coupling

References

General

On the large N limit of QCD at fixed 't Hooft coupling in terms of planar Feynman diagrams.

The original article:

• 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:

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

• Markus Gross, Large NN, 2006 (pdf)

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

See also:

• Wikipedia, 1/N expansion

Further discussion:

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

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

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

• Sidney Coleman, 1/N, in Aspects of Symmetry, Cambridge University Press 1985.

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

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

• 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.

• 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

• Yuri Makeenko, Methods of contemporary gauge theory, Cambridge Monographs on Math. Physics, gBooks

• 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

• 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)