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\section*{N=4 D=4 super Yang-Mills theory}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{physics}{}\paragraph*{{Physics}}\label{physics}

[[!include physicscontents]]

\hypertarget{quantum_field_theory}{}\paragraph*{{Quantum field theory}}\label{quantum_field_theory}

[[!include functorial quantum field theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{properties_of_4d_sym}{Properties of 4d SYM}\dotfill \pageref*{properties_of_4d_sym} \linebreak
\noindent\hyperlink{conformal_invariance}{Conformal invariance}\dotfill \pageref*{conformal_invariance} \linebreak
\noindent\hyperlink{ClosedExpressionsForPhysicalObservables}{Closed expressions for physical observables}\dotfill \pageref*{ClosedExpressionsForPhysicalObservables} \linebreak
\noindent\hyperlink{holography_adscft}{Holography, AdS-CFT}\dotfill \pageref*{holography_adscft} \linebreak
\noindent\hyperlink{twistor_space_formulation}{Twistor space formulation}\dotfill \pageref*{twistor_space_formulation} \linebreak
\noindent\hyperlink{topological_twists}{Topological twists}\dotfill \pageref*{topological_twists} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{planar_sector_integrability_mhv_amplitudes}{Planar sector, integrability, MHV amplitudes}\dotfill \pageref*{planar_sector_integrability_mhv_amplitudes} \linebreak
\noindent\hyperlink{ReferencesTwistorSpace}{Twistor space formulation}\dotfill \pageref*{ReferencesTwistorSpace} \linebreak
\noindent\hyperlink{scattering_amplitudes}{Scattering amplitudes}\dotfill \pageref*{scattering_amplitudes} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

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

[[!include superconformal symmetry -- table]]

\hypertarget{properties_of_4d_sym}{}\subsection*{{Properties of 4d SYM}}\label{properties_of_4d_sym}

\hypertarget{conformal_invariance}{}\subsubsection*{{Conformal invariance}}\label{conformal_invariance}

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

\hypertarget{ClosedExpressionsForPhysicalObservables}{}\subsubsection*{{Closed expressions for physical observables}}\label{ClosedExpressionsForPhysicalObservables}

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 [[Kaluza-Klein mechanism|KK-reduction]] of the very special [[6d (2,0)-superconformal QFT]] and related by [[AdS-CFT|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 \emph{[[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:

\begin{enumerate}%
\item 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 (\hyperlink{Beisert}{Beisert et al}).


\item Certain [[scattering amplitudes]] called \emph{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 (\hyperlink{Monteiro}{Monteiro}) for a review.


\item Generally, the [[scattering amplitudes]] of the theory in the planar limit have certain closed-form combinatorial expressions. See (\hyperlink{Arkani-Hamed}{Arkani-Hamed et al}).



\end{enumerate}
Such ``exact solutions'' of the theory are of interest in that even though N=4, D=4 SYM is very different from [[phenomenology|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]] [[minimal coupling|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 \emph{[[string theory results applied elsewhere]]}.

\hypertarget{holography_adscft}{}\subsubsection*{{Holography, AdS-CFT}}\label{holography_adscft}

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

[[!include gauge theory from AdS-CFT -- table]]

\hypertarget{twistor_space_formulation}{}\subsubsection*{{Twistor space formulation}}\label{twistor_space_formulation}

There is a natural reformulation of the theory using [[twistor]] [[field (physics)|fields]]. See the references \hyperlink{ReferencesTwistorSpace}{below}. And see at \emph{[[twistor string theory]]}.

\hypertarget{topological_twists}{}\subsubsection*{{Topological twists}}\label{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.

See at \emph{[[topologically twisted D=4 super Yang-Mills theory]]}.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[super Yang-Mills theory]]


\item [[N=1 D=4 super Yang-Mills theory]]


\item [[N=2 D=4 super Yang-Mills theory]]


\item [[N=4 D=3 super Yang-Mills theory]]


\item [[D=5 super Yang-Mills theory]]


\item [[topologically twisted D=4 super Yang-Mills theory]]



\end{itemize}
[[!include table of branes]]

\hypertarget{references}{}\subsection*{{References}}\label{references}

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

An introduction to $d=4$ SYM is in

\begin{itemize}%
\item Joseph A. Minahan \emph{Review of AdS/CFT Integrability, Chapter I.1: Spin Chains in $N=4$ Super Yang-Mills} (\href{http://arxiv.org/abs/1012.3983}{arXiv:1012.3983})

\end{itemize}
More recent results are in

\begin{itemize}%
\item Simon Caron-Huot, \emph{Superconformal symmetry and two-loop amplitudes in planar N=4 super Yang-Mills} (\href{http://arxiv.org/abs/1105.5606}{arXiv:1105.5606})

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

\begin{itemize}%
\item Robert Leigh, [[Matthew Strassler]], \emph{Exactly Marginal Operators and Duality in Four Dimensional N=1 Supersymmetric Gauge Theory} (\href{http://arxiv.org/abs/hep-th/9503121}{arXiv:hep-th/9503121})

\end{itemize}
(after regarding it as $N=1$ SYM with three adjoint chiral [[superfield]]s).

\hypertarget{planar_sector_integrability_mhv_amplitudes}{}\subsubsection*{{Planar sector, integrability, MHV amplitudes}}\label{planar_sector_integrability_mhv_amplitudes}

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

\begin{itemize}%
\item N. Beisert et al., \emph{Review of AdS/CFT Integrability, An Overview} Lett. Math. Phys. vv, pp (2011), (\href{http://arxiv.org/abs/1012.3982}{arXiv:1012.3982}).

\end{itemize}
A review of [[MHV amplitudes]] is in

\begin{itemize}%
\item Gustavo Machado Monteiro, \emph{MHV Tree Amplitudes in Super-Yang-Mills and in Superstring Theory} (2010) (\href{http://www.ift.unesp.br/posgrad/gustavo.pdf}{pdf})

\end{itemize}
Discussion of special properties of [[scattering amplitudes]] in the planar sector is in

\begin{itemize}%
\item [[Nima Arkani-Hamed]], Jacob L. Bourjaily, Freddy Cachazo, Alexander B. Goncharov, Alexander Postnikov, Jaroslav Trnka, \emph{Scattering Amplitudes and the Positive Grassmannian} (\href{http://arxiv.org/abs/1212.5605}{arXiv:1212.5605})

\end{itemize}
For mathematical background see

\begin{itemize}%
\item Alexander Postnikov, \emph{Total positivity, Grassmannians, and networks} (\href{http://arxiv.org/abs/math/0609764}{arXiv:math/0609764})

\end{itemize}
\hypertarget{ReferencesTwistorSpace}{}\subsection*{{Twistor space formulation}}\label{ReferencesTwistorSpace}

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

\begin{itemize}%
\item [[Edward Witten]], \emph{Perturbative Gauge Theory As A String Theory In Twistor Space}, Commun. Math. Phys. 252:189-258,2004 (\href{http://arxiv.org/abs/hep-th/0312171/}{arXiv:hep-th/0312171/})

\end{itemize}
A comprehensive discussion is in

\begin{itemize}%
\item Rutger Boels, Lionel Mason, David Skinner, \emph{Supersymmetric Gauge Theories in Twistor Space}, JHEP 0702:014,2007 (\href{http://arxiv.org/abs/hep-th/0604040}{arXiv:hep-th/0604040})

\end{itemize}
See also (\hyperlink{Monteiro}{Monteiro}) below.

\hypertarget{scattering_amplitudes}{}\subsubsection*{{Scattering amplitudes}}\label{scattering_amplitudes}

For the moment see at

\begin{itemize}%
\item \emph{[[scattering amplitude]]}

\end{itemize}
and also at

\begin{itemize}%
\item \emph{[[motivic multiple zeta values]]}.

\end{itemize}
[[!redirects N=4 D=4 SYM]]

[[!redirects N=4 D=4 sYM]]

[[!redirects D=4 N=4 SYM]] [[!redirects D=4 N=4 super Yang-Mills]]



\end{document}