Multigluon amplitudes, $\mathcal{N}=4$ constraints and the WZW model

Classical $\mathcal{N}=4$ Yang-Mills theory is defined by the superspace constraints. We obtain a solution of a subset of these constraints and show that it leads to the maximally helicity violating (MHV) amplitudes. The action which leads to the solvable part of the constraints is a Wess-Zumino-Witten (WZW) action on a suitably extended superspace. The non-MHV tree amplitudes can also be expressed in terms of this action.