[Question suggested by @Lawrence B. Crowell's answer to the question on T-duality]
There are three parts to my question:
A. What is the action for ${\cal N}=4$ SUSY Yang-Mills and what is the physics of the various terms in the action?
B. Give a simple explanation for the origin of Montonen-Olive duality in this theory. This duality maps the physics at gauge coupling $g$ to the physics at $1/g$, i.e. this is a duality between the strong and weak coupling regimes of the theory.
C. Auxiliary Question: Explain how one can decompose a SUSY gauge field into its bosonic and fermionic parts.
[Background: I am familiar with non-abelian gauge theory and have had a class on SUSY some years back. So I pick up on the jargon but am missing a fuller picture.]
Best Answer
A. The action of $N=4$ SYM (Super Yang-Mills theory) in $d=4$ is the simple dimensional reduction of the 9+1-dimensional SYM, the maximal dimensional SYM that exists. The latter is $$S = \int d^{10} x\mbox{ Tr } \left( -\frac{1}{4} F_{\mu\nu}F^{\mu \nu} + \overline{\psi}D_\mu \gamma^\mu \psi \right)$$ where $D$ is the covariant derivative and $\psi$ is a real chiral spinor in 9+1 dimensions which has 16 real components, leading to 8 fermionic on-shell degrees of freedom. The dimensional reduction reduces $d^{10}x$ to $d^4 x$ but it also renames 6 "compactified" spatial components $A_\mu$ as six scalars $\Phi_I$ in $d=4$. The derivatives in the corresponding 6 directions are set to zero.
If one looks what fields and interactions we get in $d=4$ - it's straightforward to write the action - it's one gauge field; four Weyl fermions; six real scalars. All those fields transform as adjoint of the gauge group - the most popular ones are $SU(N)$. They have the standard kinetic terms; standard cubic couplings of the gauge field to all the other fields; the usual quartic Yang-Mills self-interaction of the gauge field; quartic interaction of the gauge field and the scalars; Yukawa cubic couplings for the 6 scalars that arise from the gauge interactions of the fermions in 9+1 dimensions; quartic potential for the scalars which is equal to the squared commutator. Everything has to be traced over the gauge group. All these interactions are related and determined by supersymmetry - by 16 real supercharges. The individual vertices of Feynman diagrams are self-evident but the true physics behind all of them is related by symmetries.
B. The simplest explanation of the S-duality is to represent the gauge theory as the low-energy limit of the dynamics of a stack of D3-branes in type IIB string theory. The S-duality group - actually it's an $SL(2,Z)$ group because it may also act on the $\theta$-angle (RR-axion) - is directly inherited from the same S-duality group of type IIB string theory. In particular, the $g\to 1/g$ may be interpreted in the F-theory description of type IIB as the exchange of the 11th and 12th (infinitesimal) dimension of F-theory. One may also get the gauge theory as the compactification of the $d=6$ (2,0) superconformal field theory on a tiny two-torus, and $SL(2,Z)$ acts in the obvious way - again, the exchange of the two radii is the $g\to 1/g$ map. It's a non-perturbative duality so there's no simple perturbative "field redefinition" that would prove it at the classical level. However, one may make many consistency checks that the duality seems viable - e.g. one may found the magnetic monopole solutions that turn to light elementary excitations at the strong coupling.
C. The $N=4$ $d=4$ vector multiplet contains all the physical fields in the theory and I have already written what they are: a vector field with 2 physical polarizations, 6 real scalars, and 8 fermionic degrees of freedom from 4 Weyl fermions, carrying the $SU(4) \approx SO(6)$ R-symmetry group. It's not too helpful to use superspace for $N=4$ theories, unless one wants to break it to $N=1$ or $N=2$. Too big supersymmetry.
I agree with Jeff that those things can be found in first (SUSY) chapters of any modern introductory textbook or other literature on advanced quantum field theory or string theory so in this sense, this question is a theft of the time of other users of this server.
By the way, I also want to mention that the $N=4$ theory is arguably the "most important" or "simplest" $d=4$ non-gravitational theory, by the modern criteria, and the action above is far from the only one - and maybe even from the most elementary - way to describe this theory. This theory, by the AdS/CFT correspondence, is also dual i.e. exactly equivalent to type IIB string theory on $AdS_5\times S^5$. The $N=4$ SYM also has the "dual superconformal symmetry" that, together with the original superconformal symmetry, generates an infinite-dimensional "Yangian symmetry". Twistor techniques are particularly useful for the computation of scattering amplitudes in this $N=4$ SYM and many of the twistor researchers think that the twistor formulae are more fundamental and elementary ways to describe physics of the SYM than the perturbative action above.