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The coupling flow of ${\cal N}=4$ super Yang-Mills theory

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.