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

TL;DR

The paper addresses computing multigluon amplitudes in $\mathcal{N}=4$ super Yang–Mills by solving a subset of the superspace constraints using an enlarged, holomorphically extended space and a Wess-Zumino-Witten (WZW) action. This approach yields the maximally helicity violating (MHV) amplitudes as current correlators of a WZW theory on $\mathbf{CP}^1$, connecting to twistor-space formulations and Witten's string-theoretic perspective. The authors derive a holomorphic extension of the WZW action that reproduces tree-level MHV amplitudes and outline how non-MHV amplitudes can be assembled from MHV vertices via off-shell continuation and a Wick-contraction framework. They illustrate a concrete non-MHV construction and discuss the relation to other twistor-string approaches, outlining a path toward a unified, field-theoretic description of gauge-theory amplitudes. The work provides a explicit bridge between superspace constraints, holomorphic Chern–Simons/WZW structures, and modern twistor-inspired amplitude techniques.

Abstract

Classical ${\cal N}=4$ 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.