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The super Yang-Mills theory in dimension 4 with the maximum number $N = 4$ of supersymmetries.
$N=4$ $D=4$ SYM is an SCFT.
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:
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).
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.
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
M-theory perspective via AdS7-CFT6 | F-theory perspective |
---|---|
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 invariance | M5-brane worldvolume theory |
$\;\;\;\; \downarrow$ KK-compactification on Riemann surface | double dimensional reduction on M-theory/F-theory elliptic fibration |
N=2 D=4 super Yang-Mills theory with Montonen-Olive S-duality invariance; AGT correspondence | D3-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 |
$\,$
gauge theory induced via AdS5-CFT4 |
---|
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 |
There is a natural reformulation of the theory using twistor fields. See the references below. And see at twistor string theory.
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 topologically twisted D=4 super Yang-Mills theory.
Table of branes appearing in supergravity/string theory (for classification see at brane scan).
An introduction to $d=4$ SYM is in
More recent results are in
Superconformal invariance of $N=4$, $D=4$ SYM can be shown with the result of
(after regarding it as $N=1$ SYM with three adjoint chiral superfields).
A comprehensive discussion of the integrability related to anomalous dimension in the planar sector is in
A review of MHV amplitudes is in
Discussion of special properties of scattering amplitudes in the planar sector is in
For mathematical background see
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
See also (Monteiro) below.
For the moment see at
and also at