We offer a novel perspective on ${ \cal N}=4$ supersymmetric Yang-Mills (SYM) theory through the framework of the Nicolai map, a transformation of the bosonic fields that allows one to compute quantum correlators in terms of a free, purely bosonic functional measure. Generally, any Nicolai map is obtained through a path-ordered exponential of the so-called coupling flow operator. The latter can be canonically constructed in any gauge using an ${ \cal N}=1$ off-shell superfield formulation of ${ \cal N}=4$ SYM, or alternatively through dimensional reduction of the result from ${ \cal N}=1$ $D=10$ SYM, in which case we need to restrict to the Landau gauge. We propose a general theory of the ${ \cal N}=4$ coupling flow operator, arguing that it exhibits an ambiguity in form of an R-symmetry freedom given by the Lie algebra $\mathfrak{ su}(4)$. This theory incorporates our two construction approaches as special points in $\mathfrak{ su}(4)$ and defines a broad class of Nicolai maps for ${ \cal N}=4$ SYM.
Categories: hep-th math-ph math.MP