The planar limit, self-duality and their relation to two dimensions In the lecture notes by Beisert on integrability, it is stated that integrability is a property mainly in two-dimensional field theories, with some higher-dimensional examples. As higher-dimensional examples he explicitly quotes the following two theories:
$$\begin{align}
&\bullet ~ ~ \mathcal{N}=4 ~ ~ \rm{super-Yang ~ Mills ~ theory} \\
&\bullet ~ ~ \rm{the ~ self-dual ~ Yang ~ Mills ~ theory ~ in ~ four ~ dimensions}
\end{align}$$
and he goes on to make the following statement:
most of them have some implicit two-dimensionality (self-duality, planar limit).
I have tried to understand and figure out how self-duality and the planar limit are related and/or implied to two dimensions, but I have not made any progress. Any ideas?
 A: Not sure this is a complete answer, all I did was basically some data mining + use some sparse domain knowledge + tried to be mature enough to interpretate and correlate information. Anyways, here are my 5 cents...
Let us first set the ground here.
a) Self-duality and conformal symmetry are closely related in SDYM.
b) I'll take "planar limit" as the consequence of 't Hooft expansion over large N (colour number). This makes a spectrum composed of single trace operators only. Also, see my key point 2.
Now, the key points are

*

*The implicit two-dimensionality is related to how the S-matrix factors out. Self-duality and its relation to conformal symmetry (or the self-dual "conformality"), plus the underlying symmetries that allows the S-matrix factorization over 2-body processes, might be what the author meant as implicit two-dimensionality regarding self-duality.

*The planar limit greatly simplifies, since (and I quote 2):
"The gauge theory simplifies and exhibits string-like behaviour.The Feynman diagrams organise themselves into an expansion in topologies of the two-dimensional surfaces on which the diagrams can be written". So, here the implicit two-dimensionality might be related to the perturbative-sum-over-genus-like tractable approach that emerges.

Hope you find it useful.
