nLab
N=4 D=4 super Yang-Mills theory

Idea

The super Yang-Mills theory in dimension 4 with the maximum number $N=4$ of supersymmetries.

Properties of 4d SYM

Conformal invariance

$N=4$ $D=4$ SYM is an SCFT.

Closed expressions for physical observables

Among all gauge theory Lagrangians that of $N=4$, $D=4$ SYM is special in several ways, in particular of course in that it is conformally invariant and in that it has maximal supersymmetry; and ultimately by the fact that it is the KK-reduction of the very special 6d (2,0)-superconformal QFT and related by AdS7-CFT6 duality to the very special theory of 11-dimensional supergravity/M-theory.

Accordingly, it is to be expected that the quantum observables of $N=4$, $D=4$ SYM satisfy special relations that make them more tractable than the observables of a generic gauge theory, in particular by having closed-form expressions. Indeed, such relations have been and are being uncovered in the last years, in particular in what is called the planar limit of the theory, where scattering amplitudes are dominated by Feynman diagrams that can be given the structure of planar graphs.

This includes notably the following phenomena:

1. The operator spectrum of the dilatation operator (the part of the stress-energy tensor which induces conformal transformations) can be expressed in closed form, indeed when regarded as a Hamiltonian it defines an integrable system equivalent to spin chain? models. This has been used in particular to explicitly check aspects of the conjectured AdS-CFT duality of $N=4$, $D=4$ SYM with type II string theory on anti de Sitter spacetimes. See the review (Beisert et al).

2. Certain scattering amplitudes called maximally helicity violating amplitudes (“MHV amplitudes”) simplify drastically as compared to the generic situation and in fact are controled by a certain twistor string theory whose target space is a twistor space. See (Monteiro) for a review.

3. Generally, the scattering amplitudes of the theory in the planar limit have certain closed-form combinatorial expressions. See (Arkani-Hamed et al).

Such “exact solutions” of the theory are of interest in that even though N=4, D=4 SYM is very different from phenomenologically viable models such that QCD in the standard model of particle physics in that it is highly (super-)symmetric and conformal, it is still similar enough (being a nonabelian gauge theory minimally coupled to fermions) that one can or can hope to deduce from these exact results approximate information about these less symmetric theories.

In other words, because understanding observables in QCD/Yang-Mills theory in general is difficult, going to special points in the space of all such theories – such as the point of N=4, D=4 SYM – may be hoped to yield a tractable approximation. For more on this way of studying QCD and other realistic theories by studying instead their highly symmetric but phenomenologically unrealistic siblings, see also at string theory results applied elsewhere.

Holography

$N=4$ $d=4$ SYM is supposed to be related under the AdS/CFT correspondence to type II superstring theory compactified on a 5-sphere to an asymptotically anti de Sitter spacetime.

Topological twists

There is a topological twist of 4d SYM to a TQFT – the Kapustin-Witten TQFT. Its S-duality is supposed to contain geometric Langlands duality as a special case.

Related concepts

• N=2 D=4 super Yang-Mills theory

• N=2 D=3 super Yang-Mills theory?

Table of branes appearing in supergravity/string theory

brane	in supergravity	charged under gauge field	has worldvolume theory
black brane	supergravity	higher gauge field	SCFT
D-brane	type II	RR-field	super Yang-Mills theory
$(D=2n)$	type IIA	$$	$$
D0-brane	$$	$$	BFSS matrix model
D2-brane	$$	$$	$$
D4-brane	$$	$$	D=5 super Yang-Mills theory with Khovanov homology observables
$(D=2n+1)$	type IIB	$$	$$
D1-brane	$$	$$	2d CFT with BH entropy
D3-brane	$$	$$	N=4 D=4 super Yang-Mills theory
D5-brane	$$	$$	$$
D7-brane	$$	$$	$$
NS-brane	type I, II, heterotic	circle n-connection	$$
string	$$	B2-field	2d SCFT
NS5-brane	$$	B6-field	little string theory
M-brane	11D SuGra/M-theory	circle n-connection	$$
M2-brane	$$	C3-field	ABJM theory, BLG model
M5-brane	$$	C6-field	6d (2,0)-superconformal QFT
topological M2-brane	topological M-theory	C3-field on G2-manifold
topological M5-brane	$$	C6-field on G2-manifold

References

General

An introduction to $d=4$ SYM is in

• Joseph A. Minahan Review of AdS/CFT Integrability, Chapter I.1: Spin Chains in $N=4$ Super Yang-Mills (arXiv:1012.3983)

More recent results are in

• Simon Caron-Huot, Superconformal symmetry and two-loop amplitudes in planar N=4 super Yang-Mills (arXiv:1105.5606)

Superconformal invariance of $N=4$, $D=4$ SYM can be shown with the result of

• Robert Leigh, Matthew Strassler, Exactly Marginal Operators and Duality in Four Dimensional N=1 Supersymmetric Gauge Theory (arXiv:hep-th/9503121)

(after regarding it as $N=1$ SYM with three adjoint chiral superfield?s).

Planar sector, integrability, MHV amplitudes

A comprehensive discussion of the integrability related to anomalous dimension in the planar sector is in

• N. Beisert et al., Review of AdS/CFT Integrability, An Overview Lett. Math. Phys. vv, pp (2011), (arXiv:1012.3982).

A review of MHV amplitudes is in

• Gustavo Machado Monteiro, MHV Tree Amplitudes in Super-Yang-Mills and in Superstring Theory (2010) (pdf) {Monteiro}

Discussion of special properties of scattering amplitudes in the planar sector is in

• Nima Arkani-Hamed, Jacob L. Bourjaily, Freddy Cachazo, Alexander B. Goncharov, Alexander Postnikov, Jaroslav Trnka, Scattering Amplitudes and the Positive Grassmannian (arXiv:1212.5605)

For mathematical background see

• Alexander Postnikov, Total positivity, Grassmannians, and networks (arXiv:math/0609764)

Scattering amplitudes

For the moment see at

• scattering amplitude

and also at

• motivic multiple zeta values.