Contents

# Contents

## Idea

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

$d$$N$superconformal super Lie algebraR-symmetryblack brane worldvolume
superconformal field theory
$\phantom{A}3\phantom{A}$$\phantom{A}2k+1\phantom{A}$$\phantom{A}B(k,2) \simeq$ osp$(2k+1 \vert 4)\phantom{A}$$\phantom{A}SO(2k+1)\phantom{A}$
$\phantom{A}3\phantom{A}$$\phantom{A}2k\phantom{A}$$\phantom{A}D(k,2)\simeq$ osp$(2k \vert 4)\phantom{A}$$\phantom{A}SO(2k)\phantom{A}$M2-brane
D=3 SYM
BLG model
ABJM model
$\phantom{A}4\phantom{A}$$\phantom{A}k+1\phantom{A}$$\phantom{A}A(3,k)\simeq \mathfrak{sl}(4 \vert k+1)\phantom{A}$$\phantom{A}U(k+1)\phantom{A}$D3-brane
D=4 N=4 SYM
D=4 N=2 SYM
D=4 N=1 SYM
$\phantom{A}5\phantom{A}$$\phantom{A}1\phantom{A}$$\phantom{A}F(4)\phantom{A}$$\phantom{A}SO(3)\phantom{A}$D4-brane
D=5 SYM
$\phantom{A}6\phantom{A}$$\phantom{A}k\phantom{A}$$\phantom{A}D(4,k) \simeq$ osp$(8 \vert 2k)\phantom{A}$$\phantom{A}Sp(k)\phantom{A}$M5-brane
D=6 N=(2,0) SCFT
D=6 N=(1,0) SCFT

(Shnider 88, also Nahm 78, see Minwalla 98, section 4.2)

## 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.

$N=4$ $d=4$ 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.

gauge theory induced via AdS-CFT correspondence

11d supergravity/M-theory
$\;\;\;\;\downarrow$ Kaluza-Klein compactification on $S^4$compactificationon elliptic fibration followed by T-duality
7-dimensional supergravity
$\;\;\;\;\downarrow$ topological sector
7-dimensional Chern-Simons theory
$\;\;\;\;\downarrow$ AdS7-CFT6 holographic duality
6d (2,0)-superconformal QFT on the M5-brane with conformal invarianceM5-brane worldvolume theory
$\;\;\;\; \downarrow$ KK-compactification on Riemann surfacedouble dimensional reduction on M-theory/F-theory elliptic fibration
N=2 D=4 super Yang-Mills theory with Montonen-Olive S-duality invariance; AGT correspondenceD3-brane worldvolume theory with type IIB S-duality
$\;\;\;\;\; \downarrow$ topological twist
topologically twisted N=2 D=4 super Yang-Mills theory
$\;\;\;\; \downarrow$ KK-compactification on Riemann surface
A-model on $Bun_G$, Donaldson theory

$\,$

type II string theory
$\;\;\;\;\downarrow$ Kaluza-Klein compactification on $S^5$
$\;\;\;\; \downarrow$ topological sector
5-dimensional Chern-Simons theory
$\;\;\;\;\downarrow$ AdS5-CFT4 holographic duality
N=4 D=4 super Yang-Mills theory
$\;\;\;\;\; \downarrow$ topological twist
topologically twisted N=4 D=4 super Yang-Mills theory
$\;\;\;\; \downarrow$ KK-compactification on Riemann surface
A-model on $Bun_G$ and B-model on $Loc_G$, geometric Langlands correspondence

### Twistor space formulation

There is a natural reformulation of the theory using twistor fields. See the references below. And see at twistor string theory.

### 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.

Table of branes appearing in supergravity/string theory (for classification see at brane scan).

branein supergravitycharged under gauge fieldhas worldvolume theory
black branesupergravityhigher gauge fieldSCFT
D-branetype IIRR-fieldsuper Yang-Mills theory
$(D = 2n)$type IIA$\,$$\,$
D(-2)-brane$\,$$\,$
D0-brane$\,$$\,$BFSS matrix model
D2-brane$\,$$\,$$\,$
D4-brane$\,$$\,$D=5 super Yang-Mills theory with Khovanov homology observables
D6-brane$\,$$\,$D=7 super Yang-Mills theory
D8-brane$\,$$\,$
$(D = 2n+1)$type IIB$\,$$\,$
D(-1)-brane$\,$$\,$$\,$
D1-brane$\,$$\,$2d CFT with BH entropy
D3-brane$\,$$\,$N=4 D=4 super Yang-Mills theory
D5-brane$\,$$\,$$\,$
D7-brane$\,$$\,$$\,$
D9-brane$\,$$\,$$\,$
(p,q)-string$\,$$\,$$\,$
(D25-brane)(bosonic string theory)
NS-branetype I, II, heteroticcircle n-connection$\,$
string$\,$B2-field2d SCFT
NS5-brane$\,$B6-fieldlittle string theory
D-brane for topological string$\,$
A-brane$\,$
B-brane$\,$
M-brane11D SuGra/M-theorycircle n-connection$\,$
M2-brane$\,$C3-fieldABJM theory, BLG model
M5-brane$\,$C6-field6d (2,0)-superconformal QFT
M9-brane/O9-planeheterotic string theory
M-wave
topological M2-branetopological M-theoryC3-field on G2-manifold
topological M5-brane$\,$C6-field on G2-manifold
S-brane
SM2-brane,
membrane instanton
M5-brane instanton
D3-brane instanton
solitons on M5-brane6d (2,0)-superconformal QFT
self-dual stringself-dual B-field
3-brane in 6d

## References

### General

Discussion of D=4 N=4 SYM with emphasis on multi-trace operators:

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

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

(after regarding it as $\mathcal{N}=1$ SYM with three adjoint chiral superfields).

Introductions in view of single trace operators as spin chain integrable systems in the AdS/CFT correspondence:

• Joseph A. Minahan Review of AdS/CFT Integrability, Chapter I.1: Spin Chains in $\mathcal{N}=4$ 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)

### Matrix model description

On the BMN matrix model as a KK-compactification of D=4 N=4 super Yang-Mills theory on the 3-sphere:

Using the KK-compactification of D=4 N=4 super Yang-Mills theory to the BMN matrix model for lattice gauge theory-computations in D=4 N=4 SYM and for numerical checks of the AdS-CFT correspondence:

• Masazumi Honda, Goro Ishiki, Sang-Woo Kim, Jun Nishimura, Asato Tsuchiya, Direct test of the AdS/CFT correspondence by Monte Carlo studies of N=4 super Yang-Mills theory, JHEP 1311 (2013) 200 (arXiv:1308.3525)

### Planar sector, integrability, MHV amplitudes

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

A review of MHV amplitudes is in

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

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

### Twistor space formulation

The twistor space formulation of $N=4$ $D = 4$ SYM was originally found from the B-model string theory in

A comprehensive discussion is in

• Rutger Boels, Lionel Mason, David Skinner, Supersymmetric Gauge Theories in Twistor Space, JHEP 0702:014,2007 (arXiv:hep-th/0604040)