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Quantum Field Theory

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)

Introduction

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field theory:

Lagrangian field theory

quantization

quantum mechanical system, quantum probability

free field quantization

gauge theories

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Operator algebra

Local QFT

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Phenomenology

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 (“symplectic 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 Euler characteristic/genus of emerging string worldsheet surfaces, with genus 0 (planar graphs) dominating in the large NN 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-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).

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:

First inkling of holographic QCD and matrix models:

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

See:

Lecture notes and other reviews:

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

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 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

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

Further discussion:

On a kind of BV-quantization of the Loday-Quillen-Tsygan theorem and relating to the large N 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 NN gauge theories on the lattice (arXiv:2001.10859)

See also:

  • Marco Bochicchio, Mauro Papinutto, Francesco Scardino, On the structure of the large-NN expansion in SU(N)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:

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 2g_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→∞N\to \infty and tune tt to the critical point t ct_c where diagrams with a diverging number of holes dominate.

However, in AdS/CFT, or in the Gopakumar-Vafa duality [2], tt 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:

Last revised on March 5, 2025 at 10:14:46. See the history of this page for a list of all contributions to it.