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

TL;DR

This paper introduces a new perspective on ${\cal N}=4$ super Yang-Mills theory using the Nicolai map, which transforms the theory into a free bosonic form, revealing a broad class of such maps with an R-symmetry ambiguity.

Contribution

It develops a general theory of the ${\cal N}=4$ coupling flow operator, unifying different construction methods and highlighting an R-symmetry freedom in the Nicolai map.

Findings

Constructed the coupling flow operator in various gauges.

Identified an R-symmetry ambiguity in the Nicolai map.

Unified different approaches within a broad theoretical framework.

Abstract

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.