MnLargeSymbols’164 MnLargeSymbols’171

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

.
Maximilian<EMAIL_ADDRESS>
Institut für Theoretische Physik
and
Riemann Center for Geometry and Physics
Leibniz Universität Hannover
Appelstraße 2, 30167 Hannover, Germany

( )

###### 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.

###### Contents

  1. 1 Introduction
    1. 1.1 History and overview
    2. 1.2 Basics of the Nicolai map
    3. 1.3 Conventions and notation
  2. 2 $\mathcal{N}=\,4$ SYM action
    1. 2.1 $\mathcal{N}=\,1$ superfield formalism
    2. 2.2 Dimensional reduction
  3. 3 Coupling flow operator
    1. 3.1 From the canonical construction
    2. 3.2 From dimensional reduction
    3. 3.3 Unified R-symmetric framework
  4. 4 Nicolai maps
  5. 5 Conclusions and outlook
  6. A Details on the $\mathcal{N}=\,1$ superfield formalism
  7. B Details on the canonical construction
  8. C Infinitesimal free action condition
  9. D Explicit computation of the Nicolai maps
  10. E Testing the conditions of the Nicolai maps

##  1 Introduction

###  1.1 History and overview

Gauge theories are our best candidates for describing Nature at elementary
scales. Out of all such theories, maximally supersymmetric Yang–Mills theory
in four dimensions ($\mathcal{N}=\,4$ SYM) takes on a special role. In a sense
it is the most simple (possibly integrable) gauge theory that can be
formulated. Indeed it has been a long standing goal of theoretical physicists
to solve this particular theory analytically. While being a toy model, it is
expected that progress on this matter would have a drastic impact on our
understanding of more complex models of nature. It is therefore important to
continue this pursuit and investigate $\mathcal{N}=\,4$ SYM from every
possible angle.

A so far relatively unknown approach to supersymmetric field theories is
referred to as the Nicolai map. It is based on works from Nicolai, Dietz,
Flume and Lechtenfeld [1, 2, 3, 4, 5, 6, 7] from the 1980s. The Nicolai map is
a transformation of the bosonic Yang–Mills fields that relates the interacting
theory at some coupling $g$ to the free theory at zero coupling. It allows one
to compute quantum correlators in terms of a free, purely bosonic functional
measure, entirely bypassing the use of any anticommuting (Grassmann)
variables. After a pause of roughly 35 years there has recently been renewed
interest on this matter starting with the papers [8, 9, 10, 11]. Shortly
after, a general formula for the Nicolai map in terms of the so-called
coupling flow operator $R_{g}$ was found in [12]. For theories that have an
off-shell superfield formalism the coupling flow operator can be constructed
canonically. In these cases, a general construction method for the Nicolai map
in arbitrary gauges was developed in [13] and independently in [14]. This
framework has potential applications in all kinds of supersymmetric theories,
e.g. as recently investigated [15] in supermembrane and matrix theory.

We should highlight that the Nicolai map and its coupling flow operator are
(depending on the theory) not necessarily unique. The original proof [1, 2] by
Nicolai only shows that there exists such a map. The non-uniqueness was seen
most strikingly in [11], where it was shown that there exist two distinct
Nicolai maps in ${\cal N}=\,1$ $D=6$ SYM (at least to third order in the
coupling). In fact, one of the main results of this work is that in case of
${\cal N}=4$ SYM, there is a 15-dimensional ambiguity in the coupling flow
operator. However, by definition of the map, correlators are independent of
the particular choice of the Nicolai map (or coupling flow operator).

To this date, most works on Nicolai maps were restricted to $\mathcal{N}=\,1$
supersymmetry. Only in [9] a Nicolai map for $\mathcal{N}=\,4$ SYM was deduced
by dimensional reduction from the map for $\mathcal{N}=\,1$ $D=10$ SYM. In our
work, we develop a more extensive framework for dealing with $\mathcal{N}=\,4$
supersymmetry222other theories with extended supersymmetry such as
$\mathcal{N}=\,2$ $D=6$ SYM could be described in an analogous fashion..
Generally, we distinguish two possibilities for obtaining Nicolai maps in
$\mathcal{N}=\,4$ SYM. The first one is dimensional reduction from ten to four
dimensions, while the second one makes use of an $\mathcal{N}=\,1$ off-shell
superfield formalism for $\mathcal{N}=\,4$ SYM. The former approach treats all
four supersymmetries on an equal footing, whereas the latter singles out one
of the supersymmetries. This leads to an ambiguity in the coupling flow
operator and the corresponding Nicolai map. We explain this by an analysis of
the R-symmetry of $R_{g}$. Essentially, the operator is subject to
$\mathrm{SU}(4)$ R-symmetry transformations and to a principle of
superpositions with weight one. This suggests a general description of the
coupling flow operator in terms of an $\mathfrak{su}(4)$ R-symmetry freedom
that incorporates the two before-mentioned results as special points in the
Lie algebra.

In Section 2 we compare two formulations of the $\mathcal{N}=\,4$ SYM action
and how they are actually equivalent. We start with the $\mathcal{N}=1$
superfield formalism and then consider dimensional reduction from
$\mathcal{N}=\,1$ $D=10$ SYM to $\mathcal{N}=\,4$ $D=4$ SYM. In Section 3 we
compare the two corresponding formulations of the coupling flow operator and
place them as special cases within a general R-symmetric understanding of the
operator. In Section 4 we investigate the resulting Nicolai maps. Finally, in
Section 5 we give our conclusions and an outlook to possible future directions
that this work may point at.

Due to the technical nature of this paper, we present many of the calculations
in detailed appendices A-E in order not to disrupt the common theme of our
arguments.

###  1.2 Basics of the Nicolai map

To begin with, we recall the essentials of the Nicolai map, without
specializing to a particular theory. Any supersymmetric theory can be
expressed in terms of bosonic and fermionic (potentially including ghost)
fields. Usually the latter appear quadratically so that they can be integrated
out, giving a nonlocal functional determinant. The resulting action can be
written as

$S_{g}[\phi]\ =\ S_{g}^{\mathrm{b}}[\phi]+\hbar\ S_{g}^{\mathrm{f}}[\phi]\ ,$
(1.1)

with coupling constant $g$ and local, nonlocal parts of the action
$S_{g}^{\mathrm{b}}$, $S_{g}^{\mathrm{f}}$ respectively. Here, $\phi$ stands
for the bosonic field content of the theory, i.e. for $\mathcal{N}=\,4$ SYM we
have $\phi=(A_{\mu},\ \varphi_{i})$ with the gauge field $A_{\mu}$ and six
real scalars $\varphi_{i}$. Expectation values333by the vanishing of the
vacuum energy in supersymmetric theories, we have the normalization $\langle
1\rangle_{g}=1$. of bosonic observables $X[\phi]$ in the theory (1.1) are
given by

$\langle X[\phi]\rangle_{g}\ =\ \int\mathcal{D}\phi\
\exp\bigl{\\{}{\textstyle\frac{\mathrm{i}}{\hbar}}S_{g}[\phi]\bigr{\\}}\
X[\phi]\ .$ (1.2)

The Nicolai map is a (nonlinear and nonlocal) field transformation

$T_{g}:\ \phi(x)\ \mapsto\ \phi^{\prime}(x;\ g,\phi)\ ,$ (1.3)

invertible at least as a formal power series in $g$, with the defining
property

$\langle X[\phi]\rangle_{g}\ =\ \langle
X[T_{g}^{-1}\phi]\rangle_{0}\qquad\forall X\ ,$ (1.4)

that connects the interacting theory at coupling $g$ with the free theory
($g=0$). We stress again that the map $T_{g}$ is not necessarily unique.
However, since we construct all maps from the defining relation (1.4),
correlators do not depend on the choice of the particular map. Taking the
derivative of (1.4) with respect to the coupling gives

$\partial_{g}\langle X[\phi]\rangle_{g}\ =\
\langle(\partial_{g}+R_{g}[\phi])X[\phi]\rangle_{g}\ ,$ (1.5)

which defines the infinitesimal version of the Nicolai map, the so-called
coupling flow operator

$R_{g}[\phi]\ =\ \int\\!\mathrm{d}x\ \bigl{(}\partial_{g}T_{g}^{-1}\circ
T_{g}\bigr{)}\phi(x)\,\frac{\delta}{\delta\phi(x)}\ =:\ \int\\!\mathrm{d}x\
K[\phi;\;x]\,\frac{\delta}{\delta\phi(x)}\ ,$ (1.6)

with kernel $K$. By setting $X[\phi]=T_{g}\phi$ in (1.4), one can quickly
derive [16] the relation

$(\partial_{g}+R_{g}[\phi])T_{g}\phi\ =\ 0\ .$ (1.7)

This is a well-known differential equation solved by the path-ordered
exponential

$T_{g}\,\phi\ =\ \overrightarrow{\cal
P}\exp\Bigl{\\{}-\\!\int_{0}^{g}\\!\mathrm{d}h\ R_{h}[\phi]\Bigr{\\}}\ \phi\
,$ (1.8)

which was first found in [12]. This shows that the knowledge of $R_{g}$
completely captures the analytic $g$-dependence of the Nicolai map, allowing
its perturbative construction. It is the main objective of this work to find
the explicit and most general form of $R_{g}$ for $\mathcal{N}=\,4$ SYM.

Next, we note the characteristic properties444which were originally used as
the defining conditions of the Nicolai map, but can be traded for the single
relation (1.4). of the Nicolai map and the corresponding infinitesimal
properties for the coupling flow operator. Writing (1.4) in terms of path
integrals and collecting powers of $\hbar$, one finds

$S^{\mathrm{b}}_{0}[T_{g}\phi]\ =\
S^{\mathrm{b}}_{g}[\phi]\quad\quad\textrm{and}\quad\quad
S^{\mathrm{f}}_{0}[T_{g}\phi]-\mathrm{i}\,\mathrm{tr}\ln{\textstyle\frac{\delta
T_{g}\phi}{\delta\phi}}\ =\ S^{\mathrm{f}}_{g}[\phi]\ ,$ (1.9)

the ‘free-action’ and ‘determinant-matching’ conditions respectively. For
gauge theories, we have the additional property that the chosen gauge fixing
function $\mathcal{G}(\phi)$ is a fixed point of the Nicolai map. From (1.5),
it is straightforward to deduce [16] the corresponding infinitesimal
conditions

$(\partial_{g}+R_{g}[\phi])S^{\mathrm{b}}_{g}[\phi]\ =\
0\quad\quad\textrm{and}\quad\quad(\partial_{g}+R_{g}[\phi])S^{\mathrm{f}}_{g}[\phi]\
=\ \int\\!\mathrm{d}x\ \frac{\delta K[\phi;\;x]}{\delta\phi(x)}\ ,$ (1.10)

as well as the gauge condition

$(\partial_{g}+R_{g}[\phi])\mathcal{G}(\phi)\ =\ 0\ .$ (1.11)

For completeness, although it will not be relevant to the rest of this work,
we include here a few general remarks on regularization and renormalization.
Since the Nicolai map itself only consists of tree graphs, regularization is
not required at this stage. Only in the end, when computing correlators in the
free theory with (1.4), one has to contract trees with each other. This
generates loops (but interestingly, none of them purely fermionic) that have
to be regularized. In the case of ${\cal N}=\,1$ SYM this technique and the
subsequent renormalization are successfully carried out in the paper [7] from
1985, for example rederiving the universality of the gauge coupling to 1-loop
order. The computational effort of this method as opposed to the traditional
Feynman diagram approach is practically comparable.

###  1.3 Conventions and notation

In this paper, we work in four- and sometimes ten-dimensional Minkowski space
equipped with the mostly plus metric

$\eta^{\mu\nu}\ =\ \operatorname{diag}(-1,\ +1,\ +1,\ +1)\
,\qquad\eta^{\Sigma\Theta}\ =\ \operatorname{diag}(-1,\ +1,\ ...\ ,\ +1)$
(1.12)

respectively, where lowercase Greek indices run from 0 to 3 and uppercase
Greek indices run from 0 to 9. For the four-dimensional spinor algebra, we
adopt the conventions from [17], including the chiral basis for the gamma
matrices

$\gamma^{\mu}\ =\ \begin{pmatrix}0&\sigma^{\mu}\\\
{\bar{\sigma}}^{\mu}&0\end{pmatrix}\ ,\qquad\gamma^{5}\ =\
\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}\ =\ \begin{pmatrix}-\mathrm{i}&0\\\
0&\mathrm{i}\end{pmatrix}\ ,$ (1.13)

with sigma matrices

$\sigma^{0}\ =\ \begin{pmatrix}-1&0\\\ 0&-1\end{pmatrix}\ ,\qquad\sigma^{1}\
=\ \begin{pmatrix}0&1\\\ 1&0\end{pmatrix}\ ,\qquad\sigma^{2}\ =\
\begin{pmatrix}0&-\mathrm{i}\\\ \mathrm{i}&0\end{pmatrix},\qquad\sigma^{3}\ =\
\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}\ ,$ (1.14)

and ${\bar{\sigma}}^{0}=\sigma^{0}$, ${\bar{\sigma}}^{1,2,3}=-\sigma^{1,2,3}$.
The gamma matrices satisfy the Clifford algebra $\\{\gamma^{\mu},\
\gamma^{\nu}\\}=-2\eta^{\mu\nu}$. In this basis, the chiral projectors
$\mathrm{P}^{\pm}$ take the form

$\mathrm{P}^{+}\ =\ {\textstyle\frac{1}{2}}(1+\mathrm{i}\gamma^{5})\ =\
\begin{pmatrix}1&0\\\ 0&0\end{pmatrix}\ ,\qquad\mathrm{P}^{-}\ =\
{\textstyle\frac{1}{2}}(1-\mathrm{i}\gamma^{5})\ =\ \begin{pmatrix}0&0\\\
0&1\end{pmatrix}\ .$ (1.15)

We often use the standard Feynman slash notation

$\not{a}\ =\ \gamma^{\mu}a_{\mu}\ ,$ (1.16)

with the exception that the slashed script letters $\not{\mathscr{A}}$ and
$\not{\mathscr{D}}$ have a related but distinct meaning that is defined in the
main text. We additionally define the antisymmetric

$\gamma^{\mu\nu}\ =\
{\textstyle\frac{1}{2}}(\gamma^{\mu}\gamma^{\nu}-\gamma^{\nu}\gamma^{\mu})$
(1.17)

and generally antisymmetrize indices with weight one, indicated by square
brackets, e.g.

$a^{[\mu}b^{\nu]}\ =\ {\textstyle\frac{1}{2}}(a^{\mu}b^{\nu}-a^{\nu}b^{\mu})\
.$ (1.18)

All of our fields are in the adjoint representation of the gauge group which
we take to be $\mathrm{SU}(n_{\mathrm{c}})$ with real antisymmetric structure
constants $f^{abc}$ such that

$f^{abc}f^{abd}\ =\ n_{\mathrm{c}}\delta^{cd}\ ,$ (1.19)

where color indices run from 1 to $n_{\mathrm{c}}^{2}-1$, which we often leave
implicit. For example, we write the non-abelian field-strength tensor
$F_{\mu\nu}$ in two equivalent notations

$F_{\mu\nu}\ =\ \partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}+gA_{\mu}\times
A_{\nu}\qquad\iff\qquad F^{a}_{\mu\nu}\ =\
\partial_{\mu}A^{a}_{\nu}-\partial_{\nu}A^{a}_{\mu}+gf^{abc}A^{b}_{\mu}A^{c}_{\nu}\
,$ (1.20)

and the covariant derivative

$\mathrm{D}_{\mu}\ =\ \partial_{\mu}+gA_{\mu}\times\
\qquad\iff\qquad(\mathrm{D}_{\mu}...)^{a}\ =\
\partial_{\mu}(...)^{a}+gf^{abc}A^{b}_{\mu}(...)^{c}\ .$ (1.21)

We sum over implicit color indices of products, e.g.

$F_{\mu\nu}F^{\mu\nu}\ =\ F^{a}_{\mu\nu}F^{a\,\mu\nu}\ ,$ (1.22)

except when we write an explicit cross product, e.g.

$\varphi\ \psi{\times}\lambda\ =\ f^{abc}\varphi^{a}\psi^{b}\lambda^{c}\ .$
(1.23)

Further, e.g. when writing down Nicolai maps to second order, we often adopt
from section 4 of [10] the shorthand notations for multiplying quantities in
color and position space. This means that all objects are multiplied as color
matrices or vectors, and integration kernels are convoluted with insertions of
bosonic fields $A_{\mu}$ or $\varphi_{i}$ for $\mu=0,1,2,3$ and $i=1,...,6$.
For example, we would write in two equivalent notations the expression

$\partial^{\rho}C\varphi_{i}\partial_{\mu}CA_{\rho}{\times}\varphi_{i}\qquad\iff\qquad\int\mathrm{d}^{4}y\;\mathrm{d}^{4}z\
\partial^{\rho}C(x-y)\;(f^{abc}\varphi^{b}_{i})(y)\;\partial_{\mu}C(y-z)\;(f^{cde}A^{d}_{\rho})(z)\varphi^{e}_{i}(z)\
,$ (1.24)

with the scalar propagator $C=\Box^{-1}$. We often summarize the bosonic
fields in the symbol

$\mathscr{A}_{\Gamma}\ =\ (A_{\mu},\ \varphi_{i})\ .$ (1.25)

An overview over the various types of indices to be used in the following can
be found in Table 1.

Table 1: Types of indices used in this paper. Color and spinor indices are often left implicit. Name | Representation | Range | Alphabet
---|---|---|---
R-symmetry |  $\mathbf{4}$ of $\mathrm{SU}(4)$ | 1 to 4 | 1st half of uppercase Latin ($A,B,C,...$)
R-symmetry (broken) |  $\mathbf{3}$ of $\mathrm{SU}(3)$ | 1 to 3 | 2nd half of uppercase Latin ($I,J,K,...$)
R-symmetry |  $\mathbf{6}$ of $\mathrm{SU}(4)\ \cong\ \operatorname{SO}(6)$ | 1 to 6 | 2nd half of lowercase Latin ($i,j,k,...$)
Color | Adjoint of $\mathrm{SU}(n_{\mathrm{c}})$ | 1 to $n_{\mathrm{c}}^{2}-1$ | 1st half of lowercase Latin ($a,b,c,...$)
Lorentz (4-dim.) | Spin 1 of $\operatorname{SO}(1,3)$ | 0 to 3 | 2nd half of lowercase Greek ($\mu,\nu,\rho,...$)
Lorentz (10-dim.) | Spin 1 of $\operatorname{SO}(1,9)$ | 0 to 9 | uppercase Greek ($\Sigma,\Theta,\Gamma,...$)
Spinor | Spin ${\textstyle\frac{1}{2}}$ | 1 to 4 | 1st half of lowercase Greek ($\alpha,\beta,\gamma,...$)

Lastly, we recall the definitions of the basic building blocks of the coupling
flow operator in $\mathcal{N}=\,1$ $D=4$ SYM [13], since these also appear in
the more complicated $\mathcal{N}=\,4$ case. The free gaugino and ghost
propagators $S_{0}$ and $G_{0}$ which will be used for the perturbative
expansion of their full versions are given by

$S_{0}\ =\ \not{\partial}^{\;-1}\ =\ -\not{\partial}C\ ,\qquad G_{0}\ =\
({\textstyle\frac{\partial\mathcal{G}(\mathscr{A})}{\partial
A_{\mu}}}\partial_{\mu})^{-1}\ ,$ (1.26)

respectively with the gauge fixing function $\mathcal{G}(\mathscr{A})$. An
object that appears often in the context of the Nicolai map is the free
projector

$\Pi{\indices{{}_{\mu}^{\nu}}}\ =\
\delta\indices{{}_{\mu}^{\nu}}-\partial_{\mu}G_{0}{\textstyle\frac{\partial\mathcal{G}(\mathscr{A})}{\partial\mathscr{A}_{\nu}}}\
,$ (1.27)

which we often extend to capital greek indices

$\Pi{\indices{{}_{\Gamma}^{\Sigma}}}\ =\
\delta\indices{{}_{\Gamma}^{\Sigma}}-\partial_{\Gamma}G_{0}{\textstyle\frac{\partial\mathcal{G}(\mathscr{A})}{\partial\mathscr{A}_{\Sigma}}}\
,$ (1.28)

with the understanding that in the reduced four-dimensional theory
$\partial_{3+i}=0$ for $i=1,...,6$. Among all possible gauges, the Landau
gauge $\mathcal{G}(\mathscr{A})=\partial^{\mu}A_{\mu}$ takes on a special
role, since in that case $G_{0}\equiv C$ so that $\Pi\indices{{}_{\mu}^{\nu}}$
equals the standard transversal projector

$\amalg\indices{{}_{\mu}^{\nu}}\ =\
\delta\indices{{}_{\mu}^{\nu}}-\partial_{\mu}C\partial^{\nu}\ .$ (1.29)

The latter splits the Yang–Mills fields into transversal and longitudinal
components

$A_{\mu}\ =\ A^{\mathrm{T}}_{\mu}+A^{\mathrm{L}}_{\mu}\ ,\qquad
A^{\mathrm{T}}_{\mu}\ =\ \amalg\indices{{}_{\mu}^{\nu}}A_{\nu}\ ,\qquad
A^{\mathrm{L}}_{\mu}\ =\
(\delta\indices{{}_{\mu}^{\nu}}-\amalg\indices{{}_{\mu}^{\nu}})A_{\nu}\ =\
\partial_{\mu}C\ \partial\cdot A\ ,$ (1.30)

where we abbreviate $\partial\cdot A=\partial^{\mu}A_{\mu}$. In arbitrary
gauges (or outside of the gauge hypersurface of the Landau gauge), it is
helpful to define the ‘conjugate’ Yang–Mills field

$A^{*}_{\mu}\ :=\ A^{\mathrm{T}}_{\mu}-A^{\mathrm{L}}_{\mu}\ =\
A_{\mu}-2\partial_{\mu}C\partial\cdot A\ ,$ (1.31)

although for the most part in this work, we restrict ourselves to the Landau
gauge hypersurface, where $A=A^{*}$.

##  2 $\mathcal{N}=\,4$ SYM action

A common formulation of the $\mathcal{N}=\,4$ SYM invariant action (without a
topological term555which we neglect here for simplicity but could be included
in a future analysis. It is expected to lead to an additional chiral freedom
in our theory.) is [18]

$\displaystyle S_{\mathrm{inv}}\ =\ \int\mathrm{d}^{4}x\;\Bigl{\\{}$
$\displaystyle-{\textstyle\frac{1}{4}}F^{\mu\nu}F_{\mu\nu}-{\textstyle\frac{1}{2}}\mathrm{D}_{\mu}\varphi_{i}\mathrm{D}^{\mu}\varphi_{i}-{\textstyle\frac{\mathrm{i}}{2}}\bar{\chi}_{A}\not{\mathrm{D}}\mathrm{P}^{+}\chi^{A}-{\textstyle\frac{\mathrm{i}}{2}}\bar{\tilde{\chi}}^{A}\not{\mathrm{D}}\mathrm{P}^{-}\tilde{\chi}_{A}$
(2.1) $\displaystyle-\mathrm{i}g\
t\indices{{}^{i}_{AB}}\bar{\tilde{\chi}}^{A}\mathrm{P}^{+}\varphi_{i}\times\chi^{B}+\mathrm{i}g\
t\indices{{}^{iAB}}\bar{\chi}_{A}\mathrm{P}^{-}\varphi_{i}\times\tilde{\chi}_{B}-{\textstyle\frac{g^{2}}{4}}(\varphi_{i}\times\varphi_{j})^{2}\Bigr{\\}}\
,$

in terms of Weyl spinors $\chi^{A}$, $\tilde{\chi}_{A}$ where
$\tilde{\chi}_{A}=\mathrm{C}(\bar{\chi}^{A})^{\mathrm{T}}$ with the charge
conjugation operator $\mathrm{C}$ in four dimensions. All fields are in the
adjoint representation of the gauge group, with color indices left implicit.
Here, $\chi^{A}$ transforms as a $\mathbf{4}$ under the global
$\mathrm{SU}(4)\ \cong\ \operatorname{SO}(6)$ R-symmetry, while
$\tilde{\chi}_{A}$ transforms as a $\bar{\mathbf{4}}$. The indices
$i=1,2,...,6$ label the six bosonic fields $\varphi_{i}$ that transform as a
$\mathbf{6}$. Furthermore, the coefficients
$t\indices{{}^{i}_{AB}}=(t\indices{{}^{iAB}})^{*}$ are the structure constants
of the R-symmetry, or in other words Clebsch-Gordon coefficients that couple
two $\mathbf{4}$’s to a $\mathbf{6}$ [19]. They allow us to define anti-
symmetric complex scalars

$\varphi_{AB}\ =\ t\indices{{}^{i}_{AB}}\varphi_{i}\ ,\qquad\varphi^{AB}\ =\
t\indices{{}^{iAB}}\varphi_{i}\ =\ (\varphi_{AB})^{*}$ (2.2)

and will be specified explicitly when we construct the action below. In this
work, we find it advantageous to work with Majorana spinors instead of Weyl
spinors. To that aim, we define

$\psi^{A}\ =\ \mathrm{P}^{+}\chi^{A}\ +\ \mathrm{P}^{-}\tilde{\chi}_{A}\
,\qquad{\bar{\psi}}_{A}\ =\ \bar{\chi}_{A}\mathrm{P}^{-}\ +\
\bar{\tilde{\chi}}^{A}\mathrm{P}^{+}\ .$ (2.3)

With $\mathrm{C}\gamma_{5}=\gamma_{5}\mathrm{C}$, it is straightforward to
check that $\psi^{A}=\mathrm{C}(\bar{\psi}_{A})^{\mathrm{T}}$, which shows
that $\psi^{A}$ are indeed Majorana spinors. A slight complication with this
definition is that we need to be careful with R-symmetry transformations,
since $\psi^{A}$ transforms neither as a $\mathbf{4}$ nor a
$\bar{\mathbf{4}}$. We emphasize this point, because in this Majorana
formulation, the position of the R-symmetry indices does not indicate the
transformation properties of the corresponding quantities. When translating
objects from the Weyl formulation to the Majorana formulation, index positions
on the two sides of the equation do not match up. Hence, one generally has to
remember the R transformation properties from the Weyl formulation. However,
the Majorana formulation allows us to write the action in the more compact
form

$S_{\mathrm{inv}}\ =\
\int\mathrm{d}^{4}x\;\Bigl{\\{}-{\textstyle\frac{1}{4}}F^{\mu\nu}F_{\mu\nu}-{\textstyle\frac{1}{2}}\mathrm{D}_{\mu}\varphi_{i}\mathrm{D}^{\mu}\varphi_{i}-{\textstyle\frac{\mathrm{i}}{2}}{\bar{\psi}}_{A}(\not{\mathrm{D}}\delta\indices{{}^{A}_{B}}+g\Phi\indices{{}^{A}_{B}}\times)\psi^{B}-{\textstyle\frac{g^{2}}{4}}(\varphi_{i}\times\varphi_{j})^{2}\Bigr{\\}}\
,$ (2.4)

with

$\Phi\indices{{}^{A}_{B}}\ :=\
2\bigl{[}t\indices{{}^{i}_{AB}}\mathrm{P}^{+}-t\indices{{}^{iAB}}\mathrm{P}^{-}\bigr{]}\varphi_{i}\
\equiv\ (c^{i})\indices{{}^{A}_{B}}\varphi_{i}\ ,$ (2.5)

where we have defined a matrix-valued field $\Phi\indices{{}^{A}_{B}}$ that is
obtained from the scalars $\varphi_{i}$ through contraction with matrix-valued
coefficients $(c^{i})\indices{{}^{A}_{B}}$. We further often use the shorthand

$\not{\mathscr{D}}{\indices{{}^{A}_{B}}}\ :=\
\not{\mathrm{D}}\delta\indices{{}^{A}_{B}}+g\Phi\indices{{}^{A}_{B}}\times\ .$
(2.6)

In the following two subsections, we show how the action (2.4) is obtained
from an $\mathcal{N}=\,1$ superfield formalism and from dimensional reduction,
respectively.

###  2.1 $\mathcal{N}=\,1$ superfield formalism

It is well established that $\mathcal{N}=\,4$ SYM does not have a formulation
in which all four supersymmetries are realized off-shell. However, it is
possible to single out one of the supersymmetries to construct an
$\mathcal{N}=\,4$ action using an $\mathcal{N}=\,1$ superfield formalism [19].
The field content resides in one vector superfield $V$ and three chiral
superfields $\Phi_{I}$

$V\ =\ (A_{\mu},\ \lambda,\ \mathcal{D})\ ,\qquad\Phi_{I}\ =\ (\phi_{I},\
\psi_{I},\ F_{I})\quad\textrm{with}\quad I\ =\ 1,2,3\ .$ (2.7)

All fields are in the adjoint representation of the gauge group. The
propagating degrees of freedom are the vector field $A_{\mu}$, four Weyl- (or
equivalently Majorana-) spinors $\psi^{A}$ ($A=1,2,3,4$, with
$\lambda=\psi^{4}$) and three complex scalars $\phi_{I}$. Further, there is
one real scalar auxiliary field $\mathcal{D}$ and three complex scalar
auxiliary fields $F_{I}$. In terms of superfields, the $\mathcal{N}=\,4$
Lagrangian density in Weyl notation is the last component666recall that
$\theta^{2}\theta^{\alpha}=0$ and
${\bar{\theta}}^{2}{\bar{\theta}}_{\dot{\alpha}}=0$. of a superfield:

$\mathcal{L}\ =\
{\textstyle\frac{1}{g^{2}n_{\mathrm{c}}}}\mathrm{tr}\Bigl{[}{\textstyle\frac{1}{16}}\bigl{(}W^{\alpha}W_{\alpha}\bigr{|}_{\theta\theta}+\mathrm{h.c.}\bigr{)}\
+\
\mathrm{e}^{-2V}\Phi^{\dagger}_{I}\mathrm{e}^{2V}\Phi_{I}\bigr{|}_{\theta\theta{\bar{\theta}}{\bar{\theta}}}\
+\
\mathrm{i}{\textstyle\frac{\sqrt{2}}{3!}}\bigl{(}\epsilon_{IJK}\Phi_{I}[\Phi_{J},\Phi_{K}]\bigr{|}_{\theta\theta}+\mathrm{h.c.}\bigr{)}\Bigr{]}\
,$ (2.8)

where $...|_{\theta\theta}$ denotes the $\theta\theta$-component of a given
superfield and so on. The trace is over color space. We have also introduced
the non-abelian supersymmetric field strength $W_{\alpha}$ and its conjugate

$W_{\alpha}\ =\
-{\textstyle\frac{1}{4}}\bar{\mathrm{D}}\bar{\mathrm{D}}\mathrm{e}^{-2V}\mathrm{D}_{\alpha}\mathrm{e}^{2V}\
,\qquad\bar{W}^{\dot{\alpha}}\ =\
-{\textstyle\frac{1}{4}}\mathrm{D}\mathrm{D}\mathrm{e}^{-2V}\bar{\mathrm{D}}^{\dot{\alpha}}\mathrm{e}^{2V}\
,$ (2.9)

in chiral superspace, with the superspace covariant derivatives
$\mathrm{D}_{\alpha}$, $\bar{\mathrm{D}}^{\dot{\alpha}}$. Note that the
coupling only appears as an overall factor $1/g^{2}$ in front of (2.8). One
recovers the usual dependence on the coupling by rescaling $V\ \rightarrow\
gV$ and $\Phi_{I}\ \rightarrow\ g\Phi_{I}$. Given the various superspace
expansions, it is straightforward to obtain the Lagrangian explicitly in terms
of components. The details on the computations that lead to the following
results can be found in Appendix A. We find for the Lagrangian in the Majorana
basis

$\displaystyle g^{2}\mathcal{L}\ =$
$\displaystyle-{\textstyle\frac{1}{4}}F_{\mu\nu}F^{\mu\nu}-{\textstyle\frac{\mathrm{i}}{2}}\bar{\lambda}\gamma^{\mu}\mathrm{D}_{\mu}\lambda+{\textstyle\frac{1}{2}}\mathcal{D}^{2}-{\textstyle\frac{1}{\sqrt{2}}}\epsilon_{IJK}\bigl{(}F_{I}\phi_{J}{\times}\phi_{K}+F^{\dagger}_{I}\phi_{J}^{\dagger}{\times}\phi_{K}^{\dagger}\bigr{)}$
(2.10)
$\displaystyle-\mathrm{D}_{\mu}\phi^{\dagger}_{I}\mathrm{D}^{\mu}\phi_{I}-{\textstyle\frac{\mathrm{i}}{2}}{\bar{\psi}}_{I}\gamma^{\mu}\mathrm{D}_{\mu}\psi_{I}+F^{\dagger}_{I}F_{I}+{\textstyle\frac{1}{\sqrt{2}}}\epsilon_{IJK}\bigl{(}\phi_{I}{\bar{\psi}}_{J}\mathrm{P}^{+}{\times}\psi_{K}+\phi^{\dagger}_{I}{\bar{\psi}}_{J}\mathrm{P}^{-}{\times}\psi_{K}\bigr{)}$
$\displaystyle-\sqrt{2}\bigl{(}{\bar{\psi}}_{I}\mathrm{P}^{-}\lambda{\times}\phi_{I}+{\bar{\psi}}_{I}\mathrm{P}^{+}\lambda{\times}\phi^{\dagger}_{I}\bigr{)}-\mathrm{i}\phi_{I}^{\dagger}\mathcal{D}{\times}\phi_{I}\
,$

and for the supersymmetry transformations

$\displaystyle\delta_{\alpha}\phi_{I}\ =\
\sqrt{2}({\bar{\psi}}_{I}\mathrm{P}^{+})_{\alpha}\ ,$ (2.11)
$\displaystyle\delta_{\alpha}\phi^{\dagger}_{I}\ =\
\sqrt{2}({\bar{\psi}}_{I}\mathrm{P}^{-})_{\alpha}\ ,$
$\displaystyle\delta_{\alpha}(\mathrm{P}^{+}\psi_{I})_{\beta}\ =\
-\mathrm{i}\sqrt{2}(\mathrm{P}^{+}\gamma^{\mu})_{\beta\alpha}(\mathrm{D}_{\mu}\phi_{I})-\sqrt{2}(\mathrm{P}^{+})_{\beta\alpha}F_{I}\
,$ $\displaystyle\delta_{\alpha}(\mathrm{P}^{-}\psi_{I})_{\beta}\ =\
-\mathrm{i}\sqrt{2}(\mathrm{P}^{-}\gamma^{\mu})_{\beta\alpha}(\mathrm{D}_{\mu}\phi^{\dagger}_{I})-\sqrt{2}(\mathrm{P}^{-})_{\beta\alpha}F^{\dagger}_{I}\
,$ $\displaystyle\delta_{\alpha}F_{I}\ =\
-\mathrm{i}\sqrt{2}(\mathrm{D}_{\mu}{\bar{\psi}}_{I\beta})(\gamma^{\mu}\mathrm{P}^{-})_{\beta\alpha}-2\phi_{I}{\times}(\bar{\lambda}\mathrm{P}^{-})_{\alpha}\
,$ $\displaystyle\delta_{\alpha}F^{\dagger}_{I}\ =\
-\mathrm{i}\sqrt{2}(\mathrm{D}_{\mu}{\bar{\psi}}_{I\beta})(\gamma^{\mu}\mathrm{P}^{+})_{\beta\alpha}-2\phi_{I}^{\dagger}{\times}(\bar{\lambda}\mathrm{P}^{+})_{\alpha}\
,$ $\displaystyle\delta_{\alpha}A_{\nu}\ =\
-\mathrm{i}(\bar{\lambda}\gamma_{\nu})_{\alpha}\ ,$
$\displaystyle\delta_{\alpha}\mathcal{D}\ =\
-\mathrm{i}(\mathrm{D}_{\mu}\bar{\lambda}_{\beta})(\gamma_{5}\gamma^{\mu})_{\beta\alpha}\
,$ $\displaystyle\delta_{\alpha}\lambda_{\beta}\ =\
-{\textstyle\frac{1}{2}}(\gamma^{\mu\nu})_{\beta\alpha}F_{\mu\nu}+\mathcal{D}(\gamma_{5})_{\beta\alpha}\
.$

The decisive advantage of the superfield formalism is that we can deduce the
penultimate component777that is, the components with one less power of
$\theta$ and ${\bar{\theta}}$ than maximal for the respective contributions.
of the superfield in (2.8). It reads

$\displaystyle\mathring{\Delta}_{\alpha}\ =$
$\displaystyle{\textstyle\frac{1}{4}}\int\mathrm{d}^{4}x\;\Bigl{\\{}-\mathcal{D}\gamma_{5}\lambda-{\textstyle\frac{1}{2}}F_{\mu\nu}\gamma^{\mu\nu}\lambda+2\epsilon_{IJK}\bigl{[}\mathrm{P}^{+}\psi_{I}\phi_{J}{\times}\phi_{K}+\mathrm{P}^{-}\psi_{I}\phi^{\dagger}_{J}{\times}\phi^{\dagger}_{K}\bigr{]}+2\mathrm{i}\gamma_{5}\phi_{I}^{\dagger}\lambda{\times}\phi_{I}$
(2.12)
$\displaystyle+\mathrm{i}\sqrt{2}\bigl{[}\gamma^{\mu}\mathrm{P}^{-}\psi_{I}\mathrm{D}_{\mu}\phi_{I}+\gamma^{\mu}\mathrm{P}^{+}\psi_{I}\mathrm{D}_{\mu}\phi^{\dagger}_{I}\bigr{]}-\sqrt{2}\bigl{[}\mathrm{P}^{+}\psi_{I}F^{\dagger}_{I}+\mathrm{P}^{-}\psi_{I}F_{I}\bigr{]}\Bigr{\\}}_{\alpha}\
.$

The superfield structure now enables us to write the invariant action as a
supervariation

$S_{\mathrm{inv}}\ =\ \int\mathrm{d}^{4}x\;\ \mathcal{L}\ =\
{\textstyle\frac{1}{2g^{2}}}\ \delta_{\alpha}\mathring{\Delta}_{\alpha}\ ,$
(2.13)

which will be the central ingredient in the canonical construction of the
coupling flow operator later on. To find the on-shell invariant action, we
first need to insert the equations of motion for the auxiliary fields

$\mathcal{D}\ =\ -\mathrm{i}\phi^{\dagger}_{I}{\times}\phi_{I}\ ,\qquad F_{I}\
=\
{\textstyle\frac{1}{\sqrt{2}}}\epsilon_{IJK}\phi^{\dagger}_{J}{\times}\phi^{\dagger}_{K}\
,$ (2.14)

resulting in

$\displaystyle S_{\mathrm{inv}}\ =\
{\textstyle\frac{1}{g^{2}}}\int\mathrm{d}^{4}x\;\;\Bigl{\\{}-{\textstyle\frac{1}{4}}F^{\mu\nu}F_{\mu\nu}-\mathrm{D}_{\mu}\phi^{\dagger}_{I}\mathrm{D}^{\mu}\phi_{I}-{\textstyle\frac{\mathrm{i}}{2}}{\bar{\psi}}_{A}\not{\mathrm{D}}\psi^{A}+{\textstyle\frac{1}{\sqrt{2}}}\epsilon_{IJK}\bigl{(}\phi_{I}{\bar{\psi}}_{J}\mathrm{P}^{+}{\times}\psi_{K}+\phi^{\dagger}_{I}{\bar{\psi}}_{J}\mathrm{P}^{-}{\times}\psi_{K}\bigr{)}$
(2.15)
$\displaystyle-\sqrt{2}\bigl{(}{\bar{\psi}}_{I}\mathrm{P}^{-}\lambda{\times}\phi_{I}+{\bar{\psi}}_{I}\mathrm{P}^{+}\lambda{\times}\phi^{\dagger}_{I}\bigr{)}+{\textstyle\frac{1}{2}}(\phi^{\dagger}_{I}\times\phi_{I})^{2}-{\textstyle\frac{1}{2}}\epsilon_{IJK}\epsilon_{ILM}(\phi_{J}\times\phi_{K})(\phi^{\dagger}_{L}\times\phi^{\dagger}_{M})\Bigr{\\}}\
.$

Note that in this expression, the scalars are represented by three complex
fields $\phi_{I}$. In order to get to the formulation (2.4), we need to
replace these by six real fields $\varphi_{i}$ by a suitable identification

$\phi_{I}\ =\
{\textstyle\frac{1}{\sqrt{2}}}(\varphi_{I+3}+\mathrm{i}\varphi_{I})\
,\qquad\phi_{I}^{\dagger}\ =\
{\textstyle\frac{1}{\sqrt{2}}}(\varphi_{I+3}-\mathrm{i}\varphi_{I})\ ,$ (2.16)

giving

$\displaystyle S_{\mathrm{inv}}\ =\
{\textstyle\frac{1}{g^{2}}}\int\mathrm{d}^{4}x\;\;\Bigl{\\{}$
$\displaystyle-{\textstyle\frac{1}{4}}F^{\mu\nu}F_{\mu\nu}-{\textstyle\frac{1}{2}}\mathrm{D}_{\mu}\varphi_{i}\mathrm{D}^{\mu}\varphi_{i}-{\textstyle\frac{\mathrm{i}}{2}}{\bar{\psi}}_{A}\not{\mathrm{D}}\psi^{A}$
(2.17) $\displaystyle+{\textstyle\frac{1}{2}}$
$\displaystyle\epsilon_{IJK}\bigl{(}{\bar{\psi}}_{I}\varphi_{J+3}{\times}\psi_{K}-{\bar{\psi}}_{I}\varphi_{J}\gamma_{5}{\times}\psi_{K}\bigr{)}+{\bar{\psi}}_{I}\varphi_{I+3}{\times}\lambda+{\bar{\psi}}_{I}\varphi_{I}\gamma_{5}{\times}\lambda-{\textstyle\frac{1}{4}}(\varphi_{i}\times\varphi_{j})^{2}\Bigr{\\}}\
,$

where for the potential term the Jacobi identity in color space was used. From
this expression we can read off the coefficients $(c^{i})\indices{{}^{A}_{B}}$
from (2.5)

$(c^{I})\indices{{}^{J}_{4}}\ =\ \mathrm{i}\delta_{IJ}\gamma_{5}\
,\qquad(c^{I+3})\indices{{}^{J}_{4}}\ =\ \mathrm{i}\delta_{IJ}\mathbbm{1}_{4}\
,\qquad(c^{I})\indices{{}^{J}_{K}}\ =\ \mathrm{i}\epsilon_{IJK}\gamma_{5}\
,\qquad(c^{I+3})\indices{{}^{J}_{K}}\ =\
-\mathrm{i}\epsilon_{IJK}\mathbbm{1}_{4}\ ,$ (2.18)

wich are anti-symmetric under exchange of $A$ and $B$ and all others are zero.
We will show in Section 2.2 that these exactly match the coefficients obtained
from dimensional reduction.

To conclude this subsection, we note two results that will be used for the
canonical construction of the coupling flow operator in Section 3.1. The susy-
transformations of the six real scalars are

$\delta_{\alpha}\varphi_{i}\ =\
-\mathrm{i}{\bar{\psi}}_{J}(c^{i})\indices{{}^{J}_{4}}\ ,$ (2.19)

and the penultimate superfield component in terms of the six real scalars and
with the auxiliary fields integrated out can be brought to the compact form

$\Delta_{\alpha}\ =\
{\textstyle\frac{1}{4}}\int\mathrm{d}^{4}x\;\Bigl{\\{}-{\textstyle\frac{1}{2}}F_{\mu\nu}\gamma^{\mu\nu}\lambda-(\Phi\indices{{}^{4}_{A}})^{\dagger}\not{\mathscr{D}}\indices{{}^{A}_{B}}\psi^{B}+{\textstyle\frac{1}{2}}(\Phi\indices{{}^{4}_{A}})^{\dagger}\Phi\indices{{}^{A}_{B}}{\times}\psi^{B}\Bigr{\\}}\
.$ (2.20)

###  2.2 Dimensional reduction

As an alternative to the construction via the $\mathcal{N}=\,1$ superfield
formalism, an on-shell action of $\mathcal{N}=\,4$ $D=4$ SYM can be obtained
by dimensional reduction from $\mathcal{N}=\,1$ $D=10$ SYM [18]

$S^{(10)}\ =\
{\textstyle\frac{1}{g^{2}}}\int\mathrm{d}^{10}x\Bigl{\\{}-{\textstyle\frac{1}{4}}F^{\Sigma\Theta}F_{\Sigma\Theta}-{\textstyle\frac{\mathrm{i}}{2}}\bar{\lambda}\
\Gamma^{\Sigma}\ \mathrm{D}_{\Sigma}\ \lambda\Bigr{\\}}\ ,$ (2.21)

where capital greek indices label the ten-dimensional representation of the
Lorentz group and $\Gamma^{\Sigma}$ are the ten-dimensional gamma matrices. We
reduce the gauge field as

$A_{\Sigma}\ =\ (A_{\mu},\ \varphi_{i})\ ,$ (2.22)

which leads to the reduction of the Yang–Mills term

$-{\textstyle\frac{1}{4}}F^{\Sigma\Theta}F_{\Sigma\Theta}\quad\longrightarrow\quad-{\textstyle\frac{1}{4}}F_{\mu\nu}F^{\mu\nu}-{\textstyle\frac{1}{2}}\mathrm{D}_{\mu}\varphi_{i}\mathrm{D}^{\mu}\varphi_{i}-{\textstyle\frac{1}{4}}(\varphi_{i}\times\varphi_{j})^{2}\
.$ (2.23)

For the Dirac term, we write the gamma matrices as

$\Gamma^{\mu}\ =\ \mathbbm{1}_{8}\otimes\gamma^{\mu}\ ,\qquad\Gamma^{AB}\ =\
\begin{pmatrix}0&\rho^{AB}\\\
\rho_{AB}&0\end{pmatrix}\otimes\mathrm{i}\gamma_{5}\ ,\qquad A,B\ =\ 1,2,3,4\
,$ (2.24)

with antisymmetric $4\times 4$-matrices

$(\rho^{AB})_{CD}\ =\ \delta_{AC}\delta_{BD}-\delta_{AD}\delta_{BC}\
,\qquad(\rho_{AB})_{CD}\ =\
{\textstyle\frac{1}{2}}\epsilon_{ABFG}(\rho^{FG})_{CD}\ =\ \epsilon_{ABCD}\ .$
(2.25)

It is convenient to define antisymmetric

$\varphi_{I4}\ =\
{\textstyle\frac{1}{2}}(\varphi_{I}+\mathrm{i}\varphi_{I+3})\
,\qquad\varphi^{AB}\ =\ {\textstyle\frac{1}{2}}\epsilon^{ABCD}\varphi_{CD}\ =\
(\varphi_{AB})^{*}\ .$ (2.26)

In the literature one often finds a factor ${\textstyle\frac{1}{\sqrt{2}}}$ in
front of the first equation in (2.26). In order to match the coefficients from
our previous analysis, we prefer to chose a normalization
${\textstyle\frac{1}{2}}$ instead. Note that this explicitly determines the
Clebsch-Gordon coefficients $t\indices{{}^{i}_{AB}}$ from (2.2) to be

$\displaystyle(t^{I})_{J4}\ =\ {\textstyle\frac{1}{2}}\delta_{IJ}\ =\
(t^{I})^{J4}\ ,$ $\displaystyle(t^{I{+}3})_{J4}\ =\
{\textstyle\frac{\mathrm{i}}{2}}\delta_{IJ}\ =\ -(t^{I{+}3})^{J4}\ ,$ (2.27)
$\displaystyle(t^{I})_{JK}\ =\ {\textstyle\frac{1}{2}}\epsilon_{IJK}\ =\
(t^{I})^{JK}\ ,$ $\displaystyle(t^{I{+}3})_{JK}\ =\
-{\textstyle\frac{\mathrm{i}}{2}}\epsilon_{IJK}\ =\ -(t^{I{+}3})^{JK}\ .$

For a matching of the bosonic and fermionic degrees of freedom, the spinor
$\lambda$ has to be a Majorana-Weyl-spinor, which can be realized in the
structure

$\lambda\ =\
(\mathrm{P}^{+}\chi^{1},\ldots,\mathrm{P}^{+}\chi^{4},\mathrm{P}^{-}\tilde{\chi}_{1},\ldots,\mathrm{P}^{-}\tilde{\chi}_{4})^{\mathrm{T}}\
,\quad\text{with}\quad\tilde{\chi}_{A}\ =\
\mathrm{C}\bar{\chi}^{A\;\mathrm{T}}\ ,$ (2.28)

where $\mathrm{C}$ is the charge conjugation operator in four dimensions. We
find that the Dirac term becomes

$-{\textstyle\frac{\mathrm{i}}{2}}\bar{\lambda}\ \Gamma^{\Sigma}\
\mathrm{D}_{\Sigma}\
\lambda\quad\longrightarrow\quad-{\textstyle\frac{\mathrm{i}}{2}}{\bar{\psi}}_{A}\
\not{\mathscr{D}}\indices{{}^{A}_{B}}\ \psi^{B}\ ,$ (2.29)

with the shorthand (2.6), the Majorana-spinors (2.3) and

$\Phi\indices{{}^{A}_{B}}\ =\ (c^{i})\indices{{}^{A}_{B}}\varphi_{i}\ =\
\bigl{[}(\rho^{CD})_{AB}\mathrm{P}^{+}-(\rho_{CD})_{AB}\mathrm{P}^{-}\bigr{]}\varphi_{CD}\
=\
2\bigl{[}t\indices{{}^{i}_{AB}}\mathrm{P}^{+}-t\indices{{}^{iAB}}\mathrm{P}^{-}\bigr{]}\varphi_{i}\
.$ (2.30)

It is easy to verify that the coefficients $(c^{i})\indices{{}^{A}_{B}}$
defined this way are equivalent to those found in the superfield formalism
(2.18).

##  3 Coupling flow operator

In this section we construct the coupling flow operator (1.5) first via the
canonical construction 3.1 and then via dimensional reduction 3.2. Reconciling
the two approaches, we propose a unified framework for the operator in 3.3.

###  3.1 From the canonical construction

The basic procedure of the following construction is exactly the same as in
the $\mathcal{N}=\,1$ case (see e.g. [13]). We merely have more fields to take
into account. In order to fix the redundant degrees of freedom of the gauge
theory, following the Faddeev-Popov procedure we add a gauge fixing term
$S_{\mathrm{gf}}$ with gauge fixing function
$\mathcal{G}(\tilde{\mathscr{A}})$ and ghost fields $\widetilde{\bar{C}},\
\widetilde{C}$ to the full action $S_{\textsc{SUSY}}$. We use the on-shell
invariant action (2.17), so that

$\displaystyle
S_{\textsc{SUSY}}[\widetilde{A},\widetilde{\varphi},\widetilde{\psi},\widetilde{{\bar{\psi}}},\widetilde{C},\widetilde{\bar{C}}]\
=\
S_{\mathrm{inv}}[\widetilde{A},\widetilde{\varphi},\widetilde{\psi},\widetilde{{\bar{\psi}}}]\
+\
S_{\mathrm{gf}}[\widetilde{A},\widetilde{\varphi},\widetilde{C},\widetilde{\bar{C}}]\
,$ (3.1) $\displaystyle S_{\mathrm{inv}}\ =\
{\textstyle\frac{1}{g^{2}}}\int\mathrm{d}^{4}x\;\;\Bigl{\\{}-{\textstyle\frac{1}{4}}\widetilde{F}^{\mu\nu}\widetilde{F}_{\mu\nu}-{\textstyle\frac{1}{2}}\widetilde{\mathrm{D}}_{\mu}\widetilde{\varphi}_{i}\widetilde{\mathrm{D}}^{\mu}\widetilde{\varphi}_{i}-{\textstyle\frac{\mathrm{i}}{2}}\widetilde{{\bar{\psi}}}_{A}\widetilde{\not{\mathscr{D}}}{\indices{{}^{A}_{B}}}\widetilde{\psi}^{B}-{\textstyle\frac{1}{4}}(\widetilde{\varphi}_{i}\times\widetilde{\varphi}_{j})^{2}\Bigr{\\}}\
,$ $\displaystyle S_{\mathrm{gf}}\ =\
{\textstyle\frac{1}{g^{2}}}\int\mathrm{d}^{4}x\;\;\Bigl{\\{}-{\textstyle\frac{1}{2\xi}}\mathcal{G}(\widetilde{A},\widetilde{\varphi})^{2}+g\,\widetilde{\bar{C}}{\textstyle\frac{\partial\mathcal{G}(\widetilde{A},\widetilde{\varphi})}{\partial\widetilde{A}_{\mu}}}\widetilde{\mathrm{D}}_{\mu}\widetilde{C}+g\,\widetilde{\bar{C}}{\textstyle\frac{\partial\mathcal{G}(\widetilde{A},\widetilde{\varphi})}{\partial\widetilde{\varphi}_{i}}}\widetilde{\varphi}_{i}\times\widetilde{C}\Bigr{\\}}\
,$

where we emphasize that the fields are in the (canonical or geometric) scaling
where the coupling only occurs as an overall factor $1/g^{2}$ (and one factor
of $g$ multiplying two ghost terms) by explicitly putting tildes on all scaled
quantities. The usual dependence on the coupling is recovered after rescaling
all fields with an appropriate power of $g$ (i.e. $\widetilde{A}=gA$,
$\widetilde{\varphi}=g\varphi$). In this scaling we can write the $g$
derivative of the action as a supervariation up to a Slavnov
variation888strictly, this should be written with the auxiliary fields still
present, but since we can integrate them out after the construction, we leave
them implicit here.,

$\partial_{g}S_{\textsc{SUSY}}\ =\
-{\textstyle\frac{1}{g^{3}}}\bigr{\\{}\delta_{\alpha}\Delta_{\alpha}-\sqrt{g}\,s\,\Delta_{\mathrm{gh}}\bigr{\\}}\
,$ (3.2)

with the superfield component

$\Delta_{\alpha}\ =\
{\textstyle\frac{1}{4}}\int\mathrm{d}^{4}x\;\Bigl{\\{}-{\textstyle\frac{1}{2}}\widetilde{F}_{\mu\nu}\gamma^{\mu\nu}\widetilde{\lambda}-(\widetilde{\Phi}\indices{{}^{4}_{A}})^{\dagger}\not{\widetilde{\mathscr{D}}}\indices{{}^{A}_{B}}\widetilde{\psi}^{B}+{\textstyle\frac{1}{2}}(\widetilde{\Phi}\indices{{}^{4}_{A}})^{\dagger}\widetilde{\Phi}\indices{{}^{A}_{B}}{\times}\widetilde{\psi}^{B}\Bigr{\\}}\
,$ (3.3)

the ghost contribution

$\Delta_{\mathrm{gh}}\ =\
\int\mathrm{d}^{4}x\;\;\Bigl{\\{}\widetilde{\bar{C}}\
\mathcal{G}(\widetilde{A},\widetilde{\varphi})\Bigr{\\}}\ ,$ (3.4)

the supervariations (2.11) and the BRST (or Slavnov) variations

$\displaystyle s\widetilde{A}_{\mu}\ =\
\sqrt{g}\;\widetilde{\mathrm{D}}_{\mu}\widetilde{C}\ ,\qquad$ $\displaystyle
s\widetilde{\lambda}\ =\ \sqrt{g}\;\widetilde{\lambda}\times\widetilde{C}\
,\qquad$ $\displaystyle s\widetilde{\bar{\lambda}}\ =\
\sqrt{g}\;\widetilde{\bar{\lambda}}\times\widetilde{C}\ ,\qquad$ (3.5)
$\displaystyle s\widetilde{\mathcal{D}}\ =\
\sqrt{g}\;\widetilde{\mathcal{D}}\times\widetilde{C}\ ,$ $\displaystyle
s\widetilde{C}\ =\
-{\textstyle\frac{\sqrt{g}}{2}}\;\widetilde{C}\times\widetilde{C}\ ,$
$\displaystyle s\widetilde{\bar{C}}\ =\
{\textstyle\frac{1}{\sqrt{g}}}{\textstyle\frac{1}{\xi}}\;\mathcal{G}(\widetilde{A},\widetilde{\varphi})\
,$ $\displaystyle s\widetilde{\varphi}_{i}\ =\
\sqrt{g}\;\widetilde{\varphi}_{i}\times\widetilde{C}\ ,$ $\displaystyle
s\widetilde{\psi}_{I}\ =\ \sqrt{g}\;\widetilde{\psi}_{I}\times\widetilde{C}\
,$ $\displaystyle s\widetilde{F}\ =\
\sqrt{g}\;\widetilde{F}_{I}\times\widetilde{C}\ .$

An intermediate scaled coupling flow operator is given by [4, 6]

$\widetilde{R}[\tilde{\mathscr{A}}]\ =\
-\mathrm{i}\,\bcontraction{}{\Delta}{{}_{\alpha}[\tilde{\mathscr{A}}]\
}{\delta}\Delta_{\alpha}[\tilde{\mathscr{A}}]\
\delta_{\alpha}+{\textstyle\frac{\mathrm{i}}{\sqrt{g}}}\,\bcontraction{}{\Delta}{{}_{\textrm{gh}}[\tilde{\mathscr{A}}]\
}{s}\Delta_{\textrm{gh}}[\tilde{\mathscr{A}}]\
s-{\textstyle\frac{1}{\sqrt{g}}}\,\bcontraction{}{\Delta}{{}_{\alpha}[\tilde{\mathscr{A}}]\
\bigl{(}}{\delta}\Delta_{\alpha}[\tilde{\mathscr{A}}]\
\bigl{(}\delta_{\alpha}\bcontraction{}{\Delta}{{}_{\textrm{gh}}[\tilde{\mathscr{A}}]\bigr{)}\
}{s}\Delta_{\textrm{gh}}[\tilde{\mathscr{A}}]\bigr{)}\ s\ ,$ (3.6)

where we introduced the shorthand $\tilde{\mathscr{A}}=(\widetilde{A}_{\mu},\
\widetilde{\varphi}_{i})$ and the contractions indicate either gaugino or
ghost propagators.999Note that the coupling flow acts on observables
$X[\tilde{\mathscr{A}}]$ and $\delta_{\alpha}\tilde{\mathscr{A}}_{\Sigma}$
contains a gaugino field, whereas $s\;\tilde{\mathscr{A}}_{\Sigma}$ contains a
ghost field. The calculation can be found in Appendix B. After rescaling the
fields $\tilde{\mathscr{A}}=g\mathscr{A}$ according to the scheme developed in
[12], the final expression for the coupling flow operator is

$\stackrel{{\scriptstyle\leftarrow}}{{R_{g}}}[\mathscr{A}]\ =\
{\textstyle\frac{1}{8}}\stackrel{{\scriptstyle\longleftarrow}}{{{\textstyle\frac{\delta}{\delta\mathscr{A}_{\Gamma}}}}}P\indices{{}_{\Gamma}^{\Sigma}}\
\mathrm{tr}\bigl{\\{}(\mathscr{C}_{\Sigma})\indices{{}^{4}_{A}}S\indices{{}^{A}_{B}}\not{\mathscr{A}}{\indices{{}^{B}_{C}}}\times\not{\mathscr{A}}{\indices{{}^{*C}_{4}}}\bigr{\\}}+\stackrel{{\scriptstyle\longleftarrow}}{{{\textstyle\frac{\delta}{\delta\mathscr{A}_{\Gamma}}}}}\Pi\indices{{}_{\Gamma}^{\Sigma}}\mathscr{A}_{\Sigma}G{\textstyle\frac{\partial\mathcal{G}(\mathscr{A})}{\partial
A_{\nu}}}A^{\mathrm{L}}_{\nu}\ ,$ (3.7)

acting to the left to comply with the implicit color index and position
argument structure (adopting the notation in section 4 of [10]). Note that
this has the exact same structure as the result for $\mathcal{N}=\,1$ $D=4$
SYM [13], only with the additional R-symmetry indices. We now give a detailed
account of the various quantities involved. First of all, an object that
appears very frequently is

$(\mathscr{C}_{\Sigma})\indices{{}^{A}_{B}}\ =\
\left\\{\begin{array}[]{ll}\delta\indices{{}^{A}_{B}}\gamma_{\mu}&\text{for}\quad\Sigma\
=\ \mu\ =\ 0,1,2,3\\\ (c^{i})\indices{{}^{A}_{B}}&\text{for}\quad\Sigma\ =\
3+i\ =\ 4,5,...,9\end{array}\right.\ .$ (3.8)

With the shorthands

$\mathscr{D}_{\Gamma}\ =\ (\mathrm{D}_{\mu}\;,\ g\varphi_{i}\times\ )\
,\qquad\not{\mathscr{D}}{\indices{{}^{A}_{B}}}\ =\
\mathscr{D}^{\Sigma}(\mathscr{C}_{\Sigma})\indices{{}^{A}_{B}}\ =\
\not{\mathrm{D}}\delta{\indices{{}^{A}_{B}}}+g\Phi{\indices{{}^{A}_{B}}}\times\
\ ,$ (3.9)

we can compactly express the gaugino and ghost propagators
$S\indices{{}^{A}_{B}}$, $G$, defined by

$\bcontraction{}{\psi}{{}^{A}(x)}{{\bar{\psi}}}\psi^{A}(x){\bar{\psi}}_{B}(y)\
=\ -S\indices{{}^{A}_{B}}(x,y;\mathscr{A})\
,\qquad\not{\mathscr{D}}{\indices{{}^{A}_{C}}}\;S\indices{{}^{C}_{B}}(x,y;\mathscr{A})\
=\ \delta\indices{{}^{A}_{B}}\delta(x-y)\ ,$ (3.10)

and

$\mathrm{i}\bcontraction{}{C}{(x)}{\bar{C}}C(x)\bar{C}(y)\ =\
G(x,y;\mathscr{A})\
,\qquad{\textstyle\frac{\partial\mathcal{G}(\mathscr{A})}{\partial\mathscr{A}_{\Gamma}}}\mathscr{D}_{\Gamma}\;G(x,y;\mathscr{A})\
=\ \delta(x-y)\ ,$ (3.11)

respectively. Further we have

$\not{\mathscr{A}}{\indices{{}^{A}_{B}}}\ =\
\mathscr{A}^{\Sigma}(\mathscr{C}_{\Sigma})\indices{{}^{A}_{B}}\ =\
\not{A}\delta{\indices{{}^{A}_{B}}}+\Phi{\indices{{}^{A}_{B}}}\
,\qquad\not{\mathscr{A}}^{*}{\indices{{}^{A}_{B}}}\ :=\
\not{A}^{*}\delta{\indices{{}^{A}_{B}}}+(\Phi{\indices{{}^{A}_{B}}})^{\dagger}\
,$ (3.12)

where $A^{*}$ is the conjugate gauge field (1.31). The second term in (3.7)
also explicitly contains the longitudinal part of the gauge field
$A_{\mu}^{\mathrm{L}}$ (1.30). Lastly, we have the natural generalization
(c.f. [13]) of the covariant projector

$P\indices{{}_{\Gamma}^{\Sigma}}\ =\
\delta\indices{{}_{\Gamma}^{\Sigma}}-\mathscr{D}_{\Gamma}G{\textstyle\frac{\partial\mathcal{G}(\mathscr{A})}{\partial\mathscr{A}_{\Sigma}}}\
,$ (3.13)

and its free version

$\Pi\indices{{}_{\Gamma}^{\Sigma}}\ =\
P\indices{{}_{\Gamma}^{\Sigma}}\bigr{|}_{g=0}\ .$ (3.14)

###  3.2 From dimensional reduction

Since the ten-dimensional theory does not have an off-shell formalism, the
operator cannot be constructed canonically in any gauge. However, for the
Landau gauge, an expression for $R_{g}$ was derived and shown to satisfy all
necessary conditions in all the critical [18] dimensions $D=3,4,6,10$ [10].
Hence, in the following we restrict to the Landau gauge hypersurface, so there
will be no distinction between $A$ and $A^{*}$ since $A^{\mathrm{L}}=0$. We
start with the formula for the coupling flow operator in $\mathcal{N}=\,1$
$D=10$ SYM:

$R_{g}[\mathscr{A}]\ =\
{\textstyle\frac{1}{32}}\stackrel{{\scriptstyle\longleftarrow}}{{{\textstyle\frac{\delta}{\delta\mathscr{A}_{\Gamma}}}}}{P^{(10)}}\indices{{}_{\Gamma}^{\Sigma}}\
\mathrm{tr}^{(32)}\Bigl{\\{}\Gamma_{\Sigma}S^{\mathrm{(10)}}\not{\mathscr{A}}\times\not{\mathscr{A}}\Bigl{\\}}\
,$ (3.15)

where the trace is over $32\times 32$ spinor space and

$\not{\mathscr{A}}\ =\ \Gamma^{\Sigma}\mathscr{A}_{\Sigma}\ .$ (3.16)

We now apply the same scheme for dimensional reduction that we have used to
reduce the action in Section 2.2. It is easy to establish that
${P^{(10)}}\indices{{}_{\Gamma}^{\Sigma}}\longrightarrow{P^{(4)}}\indices{{}_{\Gamma}^{\Sigma}}$
under dimensional reduction from ten to four dimensions. We can relate
$S^{\mathrm{(10)}}$ and $S^{\mathrm{(4)}}\equiv S$ by dimensional reduction.
When leaving out the superscript indicating the number of dimensions, we
always mean the four-dimensional object. In ten dimensions, the $32\times 32$
matrix $S^{\mathrm{(10)}}$ is given by the contraction

$S^{\mathrm{(10)}}\ =\
-\bcontraction{}{\lambda}{}{\bar{\lambda}}\lambda\bar{\lambda}\ .$ (3.17)

With the dimensional reduction of the spinor $\lambda$ (2.28) and our
definition of the four-dimensional Majorana spinors (2.3), we can decompose
the $32\times 32$ matrix as

$S^{\mathrm{(10)}}\ =\
\left(\begin{array}[]{c|c}(\mathrm{P}^{+}S{\indices{{}^{A}_{B}}}\mathrm{P}^{-})_{\alpha\beta}&(\mathrm{P}^{+}S{\indices{{}^{AB}}}\mathrm{P}^{+})_{\alpha\beta}\\\
\hline\cr(\mathrm{P}^{-}S{\indices{{}_{AB}}}\mathrm{P}^{-})_{\alpha\beta}&(\mathrm{P}^{-}S{\indices{{}_{A}^{B}}}\mathrm{P}^{+})_{\alpha\beta}\end{array}\right)\
,$ (3.18)

where each block is a $16\times 16$ matrix with ‘inner’ indices $A,\ B$ and
‘outer’ indices $\alpha,\ \beta$, all ranging from one to four. Due to the
chiral projectors, the position of the indices $A,\ B$ matches the R-symmetry
transformation properties, i.e. upper indices transform as a $\mathbf{4}$ and
lower indices as a $\bar{\mathbf{4}}$. We can interpret $S^{\mathrm{(10)}}$ as
an $8\times 8$ matrix with $4\times 4$ matrix-valued entries and take a
partial trace in the $8\times 8$ matrix space so that we are left with a trace
over $4\times 4$ matrices (over the outer indices). Using the representation
of Gamma matrices (2.24) and bosonic fields (2.26), we further write
$\not{\mathscr{A}}$ in the same block notation as

$\not{\mathscr{A}}\ =\ \Gamma^{\Sigma}\mathscr{A}_{\Sigma}\ =\
\left(\begin{array}[]{c|c}\not{A}_{\alpha\beta}\delta\indices{{}^{A}_{B}}&(\mathrm{i}\gamma_{5})_{\alpha\beta}\varphi^{AB}\\\
\hline\cr(\mathrm{i}\gamma_{5})_{\alpha\beta}\varphi_{AB}&\not{A}_{\alpha\beta}\delta\indices{{}_{A}^{B}}\end{array}\right)\
.$ (3.19)

When multiplying two block matrices, we simply have to contract inner with
inner indices and outer with outer indices. This leads to

$\not{\mathscr{A}}\times\not{\mathscr{A}}\ =\
\left(\begin{array}[]{c|c}(\not{A}\times\not{A})_{\alpha\beta}\delta\indices{{}^{A}_{B}}+(\mathbbm{1}_{4})_{\alpha\beta}\varphi^{AC}{\times}\varphi_{CB}&2(\not{A}\mathrm{i}\gamma_{5})_{\alpha\beta}\times\varphi^{AB}\\\
\hline\cr
2(\mathrm{i}\gamma_{5}\not{A})_{\alpha\beta}\times\varphi_{AB}&(\not{A}\times\not{A})_{\alpha\beta}\delta\indices{{}_{A}^{B}}+(\mathbbm{1}_{4})_{\alpha\beta}\varphi_{AC}{\times}\varphi^{CB}\end{array}\right)\
.$ (3.20)

In order to perform the partial trace, one multiplies the block matrices in
(3.15) and then takes the trace over the blocks. For example, a simple
contribution would be

$\displaystyle\mathrm{tr}^{\mathrm{(32)}}\ \Bigl{\\{}\Gamma_{\mu}\
S^{\mathrm{(10)}}\ \mathbbm{1}_{8}\otimes(\not{A}\times\not{A})\Bigr{\\}}$
(3.21) $\displaystyle\ =\ \mathrm{tr}^{\mathrm{(32)}}\
\left(\begin{array}[]{c|c}(\gamma_{\mu})_{\alpha\gamma}(\mathrm{P}^{+}S\indices{{}^{A}_{B}}\mathrm{P}^{-})_{\gamma\delta}(\not{A}\times\not{A})_{\delta\beta}&(\gamma_{\mu})_{\alpha\gamma}(\mathrm{P}^{+}S^{AB}\mathrm{P}^{+})_{\gamma\delta}(\not{A}\times\not{A})_{\delta\beta}\\\
\hline\cr(\gamma_{\mu})_{\alpha\gamma}(\mathrm{P}^{-}S_{AB}\mathrm{P}^{-})_{\gamma\delta}(\not{A}\times\not{A})_{\delta\beta}&(\gamma_{\mu})_{\alpha\gamma}(\mathrm{P}^{-}S\indices{{}_{A}^{B}}\mathrm{P}^{+})_{\gamma\delta}(\not{A}\times\not{A})_{\delta\beta}\end{array}\right)$
$\displaystyle\ =\
\mathrm{tr}^{\mathrm{(4)}}\bigl{\\{}\gamma_{\mu}\mathrm{P}^{+}S\indices{{}^{A}_{A}}\mathrm{P}^{-}\not{A}\times\not{A}\bigr{\\}}\
+\
\mathrm{tr}^{\mathrm{(4)}}\bigl{\\{}\gamma_{\mu}\mathrm{P}^{-}S\indices{{}_{A}^{A}}\mathrm{P}^{+}\not{A}\times\not{A}\bigr{\\}}\
=\
\mathrm{tr}^{\mathrm{(4)}}\bigl{\\{}\gamma_{\mu}S\indices{{}^{A}_{A}}\not{A}\times\not{A}\bigr{\\}}\
,$

where in the last step we used the cyclicity of the trace to commute the
chiral projectors. In the last step, since the positions of the indices of the
gaugino propagator in the first vs. second term do not match up, the
R-symmetry transformation properties become slightly nontransparent. This is a
consequence of the way we defined our Majorana spinors (2.3). Step by step,
one establishes the relation

$\mathrm{tr}^{(32)}\Bigl{\\{}\Gamma_{\Sigma}S^{\mathrm{(10)}}\not{\mathscr{A}}\times\not{\mathscr{A}}\Bigl{\\}}\
=\
\mathrm{tr}^{(4)}\Bigl{\\{}({\mathscr{C}_{\Sigma}})\indices{{}^{A}_{B}}S{\indices{{}^{B}_{C}}}\not{\mathscr{A}}{\indices{{}^{C}_{D}}}\times\not{\mathscr{A}}^{*}{\indices{{}^{D}_{A}}}\Bigl{\\}}\
,$ (3.22)

with the same definitions (3.8), (3.9), (3.12) as in the canonical
construction. Although the R-symmetry indices in (3.22) cannot be strictly
assigned to $\mathbf{4}$’s or $\bar{\mathbf{4}}$’s, by construction through
the above dimensional reduction, this object is invariant under R-symmetry
transformations for $\Sigma=\mu$ and transforms as a $\mathbf{6}$ of
$\mathrm{SU}(4)$ for $\Sigma=3+i$.101010This can also be verified through
explicit calculation by splitting the quantities in (3.22) into their chiral
contributions that have fixed transformation properties. Due to the ‘hidden’
chiral projectors, all the contributions that do not transform appropriately
are projected out of the trace.

In total, we have shown that the coupling flow operator obtained from
dimensional reduction on the Landau gauge hypersurface is

$\stackrel{{\scriptstyle\leftarrow}}{{R_{g}}}[\mathscr{A}]\ =\
{\textstyle\frac{1}{32}}\stackrel{{\scriptstyle\longleftarrow}}{{{\textstyle\frac{\delta}{\delta\mathscr{A}_{\Gamma}}}}}{P}\indices{{}_{\Gamma}^{\Sigma}}\
\mathrm{tr}\Bigl{\\{}(\mathscr{C}_{\Sigma})\indices{{}^{A}_{B}}S\indices{{}^{B}_{C}}\not{\mathscr{A}}{\indices{{}^{C}_{D}}}\times\not{\mathscr{A}}^{*}{\indices{{}^{D}_{A}}}\Bigl{\\}}\
.$ (3.23)

###  3.3 Unified R-symmetric framework

The goal of this subsection is to reconcile the two results (3.7) and (3.23),
since they are clearly not identical. For simplicity, we for now restrict to
the Landau gauge, for which the two results read

$\displaystyle\stackrel{{\scriptstyle\leftarrow}}{{R_{g}}}[\mathscr{A}]\ =\
{\textstyle\frac{1}{8}}\stackrel{{\scriptstyle\longleftarrow}}{{{\textstyle\frac{\delta}{\delta\mathscr{A}_{\Gamma}}}}}{P}\indices{{}_{\Gamma}^{\Sigma}}\
\mathrm{tr}\Bigl{\\{}(\mathscr{C}_{\Sigma})\indices{{}^{4}_{B}}S\indices{{}^{B}_{C}}\not{\mathscr{A}}{\indices{{}^{C}_{D}}}\times\not{\mathscr{A}}^{*}{\indices{{}^{D}_{4}}}\Bigl{\\}}$
$\displaystyle\text{from canonical construction}\ ,$ (3.24)
$\displaystyle\stackrel{{\scriptstyle\leftarrow}}{{R_{g}}}[\mathscr{A}]\ =\
{\textstyle\frac{1}{32}}\stackrel{{\scriptstyle\longleftarrow}}{{{\textstyle\frac{\delta}{\delta\mathscr{A}_{\Gamma}}}}}{P}\indices{{}_{\Gamma}^{\Sigma}}\
\mathrm{tr}\Bigl{\\{}(\mathscr{C}_{\Sigma})\indices{{}^{A}_{B}}S\indices{{}^{B}_{C}}\not{\mathscr{A}}{\indices{{}^{C}_{D}}}\times\not{\mathscr{A}}^{*}{\indices{{}^{D}_{A}}}\Bigl{\\}}$
$\displaystyle\text{from dimensional reduction}\ .$ (3.25)

We begin this discussion with two universal observations. Firstly, the general
definition of the coupling flow operator $R_{g}[\phi]$ (1.5) shows that it
maps real observables to other real observables.111111That is, modulo
imaginary terms that vanish when taking the expectation value. For simplicity,
we ignore the possibility of such extra imaginary terms. Hence, we require the
kernel $K$ of the coupling flow operator

$R_{g}[\phi]\ =\ \int\\!\mathrm{d}x\
K[\phi;\;x]\,\frac{\delta}{\delta\phi(x)}$ (3.26)

to be real. Note that for our cases and notation above, the kernel carries a
ten-dimensional index $\Gamma$, so that

$\stackrel{{\scriptstyle\leftarrow}}{{R_{g}}}[\mathscr{A}]\ =\
\stackrel{{\scriptstyle\longleftarrow}}{{{\textstyle\frac{\delta}{\delta\mathscr{A}_{\Gamma}}}}}K_{\Gamma}\
,$ (3.27)

with implicit integration. Secondly, we recall the three conditions (1.10),
(1.11) for a coupling flow operator in a general gauge theory

$(\partial_{g}+R_{g}[\phi])S^{\mathrm{b}}_{g}[\phi]\ =\ 0\
,\qquad(\partial_{g}+R_{g}[\phi])S^{\mathrm{f}}_{g}[\phi]\ =\
\int\\!\mathrm{d}x\ \frac{\delta K[\phi;\;x]}{\delta\phi(x)}\
,\qquad(\partial_{g}+R_{g}[\phi])\mathcal{G}(\phi)\ =\ 0\ .$ (3.28)

It is easy to see that given two coupling flow operators $R_{g}^{(1)}$ and
$R_{g}^{(2)}$, the linear combination

$R^{\prime}_{g}\ :=\ pR_{g}^{(1)}+qR_{g}^{(2)}\quad\text{with}\quad p,q\ \in\
\mathbb{R}\quad\text{and}\quad p+q\ =\ 1$ (3.29)

is again a coupling flow operator (i.e. satisfies (3.28)). Thus, the coupling
flow operator obeys a principle of superposition with real coefficients that
add up to one.

The next ingredient is the fact that in the canonical construction we had to
single out one of the supersymmetries to obtain a working superfield
formalism. In this work, we chose the ‘fourth’ one, resulting in the index 4
at the beginning and the end of the trace in (3.24). This choice is of course
arbitrary. We could have equivalently chosen any of the three other
supersymmetries. This, together with the principle of superposition allows us
(in the Landau gauge) to build a more general coupling flow operator by
inserting a diagonal matrix with trace four (or as we prefer, the identity
plus a traceless matrix $\mathrm{L}$) in R-space in the trace in the kernel.
In anticipation of the rest of this discussion, we write down the general
ansatz

$\stackrel{{\scriptstyle\leftarrow}}{{R_{g}}}[\mathscr{A}]\ =\
{\textstyle\frac{1}{32}}\stackrel{{\scriptstyle\longleftarrow}}{{{\textstyle\frac{\delta}{\delta\mathscr{A}_{\Gamma}}}}}{P}\indices{{}_{\Gamma}^{\Sigma}}\
\mathrm{tr}\Bigl{\\{}(\mathscr{C}_{\Sigma})\indices{{}^{A}_{B}}S\indices{{}^{B}_{C}}\not{\mathscr{A}}{\indices{{}^{C}_{D}}}\times\not{\mathscr{A}}^{*}{\indices{{}^{D}_{E}}}(\delta\indices{{}^{E}_{A}}+\mathrm{L}\indices{{}^{E}_{A}})\Bigl{\\}}\
.$ (3.30)

The two cases (3.24), (3.25) correspond to

$\mathrm{L}\ =\
\operatorname{diag}(-1,-1,-1,+3)\quad\text{and}\quad\mathrm{L}\ =\ 0\ ,$
(3.31)

respectively. From the discussion so far we know that we can reach any element
of

$\Bigl{\\{}\mathrm{L}\ =\
\operatorname{diag}(q_{1},q_{2},q_{3},q_{4})\quad\text{with}\quad\sum_{i}q_{i}\
=\ 0\ \Bigr{\\}}\ =:\ \mathfrak{h}\ ,$ (3.32)

the Cartan subalgebra of the Lie algebra $\mathfrak{su}(4)$. By applying an
$\mathrm{SU}(4)$ R transformation on the various factors in (3.30), we can
deduce how $\mathrm{L}$ effectively transforms under R transformations. One
finds that the underlying structure of the chiral projectors requires that
just like the Majorana spinors (2.3), $\mathrm{L}$ splits up into two chiral
parts

$\mathrm{L}\ =\ L\ \mathrm{P}^{-}+L^{*}\ \mathrm{P}^{+}$ (3.33)

that are complex conjugate to each other. The matrix $L$ transforms in the
adjoint $\mathbf{15}$ of $\mathrm{SU}(4)$

$L\quad\longrightarrow\quad ULU^{\dagger}\ ,\quad\quad\textrm{with}\quad U\
\in\ \mathrm{SU}(4)\ .$ (3.34)

We notice that (3.34) preserves zero trace and hermiticity $L^{\dagger}=L$ so
that the group action of $\mathrm{SU}(4)$ on $\mathfrak{h}$ (3.32) generates
the entire Lie algebra $\mathfrak{su}(4)$. Any $L\ \in\ \mathfrak{su}(4)$ can
be unitarily diagonalized such that we can classify it in terms of its four
eigenvalues $q_{i}$. Invariants under the adjoint action are $\mathrm{tr}\
L^{m}$ for any integer $m\geq 1$, but with only the first four

$\mathrm{tr}\ L\ =\ \sum q_{i}\ =\ 0\ ,\quad\mathrm{tr}\ L^{2}\ =\ \sum
q_{i}^{2}\ ,\quad\mathrm{tr}\ L^{3}\ =\ \sum q_{i}^{3}\ ,\quad\mathrm{tr}\
L^{4}\ =\ \sum q_{i}^{4}$ (3.35)

functionally independent. We can equivalently characterize a generic orbit by
the eigenvalues $(q_{1},\ q_{2},\ q_{3},\ q_{4})$ with $\sum\ q_{i}=0$ or by
$(\mathrm{tr}\ L^{2},\ \mathrm{tr}\ L^{3},\ \mathrm{tr}\ L^{4})$. This gives
us three real parameters matching the dimension of the Cartan subalgebra. For
a generic $L\ \in\ \mathfrak{su}(4)$ with all eigenvalues $q_{i}$ distinct,
the stabilizer of the adjoint action is the maximal torus
$\mathrm{S}(\mathrm{U}(1)^{4})\ \cong\ \mathrm{U}(1)^{3}$ and its orbit under
the action is the 12-dimensional flag manifold

$\faktor{\mathrm{SU}(4)}{\mathrm{U}(1)^{3}}\ .$ (3.36)

For singular $L$, i.e. some eigenvalues $q_{i}$ coinciding, the stabilizer is
larger and the orbit smaller. Table 2 summarizes all cases.

Table 2: Stabilizer subgroups $X$ of $\mathrm{SU}(4)$ acting on $L\ \in\ \mathfrak{su}(4)$, depending on the degeneracy of the eigenvalues $q_{i}$. The last column indicates the number of free parameters for the coupling flow operator. It is computed by adding the number of degrees of freedom (dofs) in the choice of the $q_{i}$’s to the dimension of the orbit $\mathrm{SU}(4)/X$. Degeneracy | dofs | Stabilizer $X$ | $\operatorname{dim}(X)$ |  $\\#$ free param.
---|---|---|---|---
all $q_{i}$ distinct | 3 | $\mathrm{S}(\operatorname{U}(1)^{4})$ | 3 | 15
two $q_{i}$ equal | 2 | $\mathrm{S}(\operatorname{U}(2)\times\operatorname{U}(1)^{2})$ | 5 | 12
two equal pairs | 1 | $\mathrm{S}(\operatorname{U}(2)\times\operatorname{U}(2))$ | 7 | 9
three $q_{i}$ equal | 1 | $\mathrm{S}(\operatorname{U}(3)\times\operatorname{U}(1))$ | 9 | 7
all $q_{i}\ =\ 0$ | 0 | $\mathrm{SU}(4)$ | 15 | 0

The fully degenerate case corresponds to $L=0$ (3.25). It is a fixed point
under all $\mathrm{SU}(4)$ transformations. The canonical construction on the
other hand led to a configuration where three of the $q_{i}$’s were equal, so
that only an $\mathrm{S}(\operatorname{U}(3)\times\operatorname{U}(1))$
subgroup leaves the configuration invariant. This suggests that for $L$ with
three degenerate $q_{i}$’s, the points in its orbit under the adjoint action
of $\mathrm{SU}(4)$ are those that originate from an off-shell formalism and
hence allow for an arbitrary choice of the gauge fixing function. For these
cases, when choosing gauges other than the Landau gauge, the second term in
(3.7) has to be added to the general formula (3.30).

In Appendix C, we directly check that (3.30) indeed satisfies the
infinitesimal conditions of a coupling flow operator.

##  4 Nicolai maps

In this section we use the notation of [13] to write down Nicolai maps to
second order. We briefly recall the most important aspects of this compact
notation. Derivatives of the scalar propagator $C$ are simply written as
indices, so that e.g. $\partial_{\mu}\partial_{\nu}C\equiv C_{\mu\nu}$.
Further, all objects are implicitly matrices in color space with the exception
that the last quantity in each term is a vector in color space. Moreover,
implicit integration kernels are convoluted with insertions of bosonic fields
$A$ or $\varphi$.

A Nicolai map for $\mathcal{N}=\,4$ SYM to $\mathcal{O}(g^{2})$ in the Landau
gauge was already found in [9] by directly reducing the result from
$\mathcal{N}=\,1$ $D=10$

$T_{g}\mathscr{A}_{\Sigma}\ =\
\mathscr{A}_{\Sigma}-gC^{\Theta}\mathscr{A}_{\Sigma}\mathscr{A}_{\Theta}+{\textstyle\frac{3}{2}}g^{2}C^{\Theta}\mathscr{A}^{\Gamma}C_{[\Sigma}\mathscr{A}_{\Theta}\mathscr{A}_{\Gamma]}+\mathcal{O}(g^{3})\
,$ (4.1)

down to four dimensions with the simple prescription $\mathscr{A}\ =\
(A_{\mu},\ \varphi_{i})$ and $\partial_{3+i}\equiv 0$, which leads to

$\displaystyle T_{g}A_{\mu}\ =\
A_{\mu}-gC^{\rho}A_{\mu}A_{\rho}+{\textstyle\frac{3}{2}}g^{2}C^{\rho}A^{\lambda}C_{[\mu}A_{\rho}A_{\lambda]}+g^{2}C^{\rho}\varphi_{i}C_{[\mu}A_{\rho]}\varphi_{i}+\mathcal{O}(g^{3})\
,$ (4.2) $\displaystyle T_{g}\varphi_{i}\ =\
\varphi_{i}-gC^{\rho}\varphi_{i}A_{\rho}+g^{2}C^{[\rho}A^{\lambda]}C_{\lambda}\varphi_{i}A_{\rho}+{\textstyle\frac{1}{2}}g^{2}C^{\rho}\varphi_{j}C_{\rho}\varphi_{j}\varphi_{i}+\mathcal{O}(g^{3})\
.$ (4.3)

In this section, we want to explicitly show the ambiguity of the
$\mathcal{N}=\,4$ map by computing (Appendix D) and testing (Appendix E) four
different maps to second order in the Landau gauge corresponding to the points

$L\ =\ \operatorname{diag}(+3,-1,-1,-1)\ ,\ ...\ ,\
\operatorname{diag}(-1,-1,-1,+3)\ ,$ (4.4)

in $\mathfrak{su}(4)$ and denote the respective coupling flow operators as

$\stackrel{{\scriptstyle\leftarrow}}{{R_{g}}}{{}^{(A)}}[\mathscr{A}]\ =\
{\textstyle\frac{1}{8}}\stackrel{{\scriptstyle\longleftarrow}}{{{\textstyle\frac{\delta}{\delta\mathscr{A}_{\Gamma}}}}}{P}\indices{{}_{\Gamma}^{\Sigma}}\
\mathrm{tr}\Bigl{\\{}(\mathscr{C}_{\Sigma})\indices{{}^{A}_{B}}S\indices{{}^{B}_{C}}\not{\mathscr{A}}{\indices{{}^{C}_{D}}}\times\not{\mathscr{A}}^{*}{\indices{{}^{D}_{A}}}\Bigl{\\}}\
.$ (4.5)

with no sum over $A=1,2,3,4$. In general [12], the Nicolai map can be obtained
from the perturbative expansion

$R_{g}[\mathscr{A}]\ =\ \sum_{k=1}^{\infty}g^{k-1}\mathrm{R}_{k}[\mathscr{A}]\
=\
\mathrm{R}_{1}[\mathscr{A}]+g\,\mathrm{R}_{2}[\mathscr{A}]+g^{2}\mathrm{R}_{3}[\mathscr{A}]+\ldots\
,$ (4.6)

via

$\displaystyle T_{g}\mathscr{A}$ $\displaystyle\ =\ \mathscr{A}\ -\
g\,\mathrm{R}_{1}\mathscr{A}\ -\
{\textstyle\frac{1}{2}}g^{2}\bigl{(}\mathrm{R}_{2}-\mathrm{R}_{1}^{2}\bigr{)}\mathscr{A}\
+\ {\cal O}(g^{3})\ .$ (4.7)

When putting all the contributions together, we find the four distinct Nicolai
maps

$\displaystyle T_{g}^{(4)}A_{\mu}\ =$ $\displaystyle
A_{\mu}-gC^{\rho}A_{\mu}A_{\rho}+{\textstyle\frac{3}{2}}g^{2}C^{\rho}A^{\lambda}C_{[\mu}A_{\rho}A_{\lambda]}+g^{2}C^{\rho}\varphi_{i}C_{[\mu}A_{\rho]}\varphi_{i}$
(4.8)
$\displaystyle{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}-{\textstyle\frac{1}{2}}g^{2}\Pi\indices{{}_{\mu}^{\nu}}\epsilon_{\nu\lambda\rho\sigma}\sum_{J=1}^{3}[C^{\lambda}\varphi_{J}C^{\rho}\varphi_{J+3}A^{\sigma}-C^{\lambda}\varphi_{J+3}C^{\rho}\varphi_{J}A^{\sigma}+C^{\lambda}A^{\rho}C^{\sigma}\varphi_{J+3}\varphi_{J}]}+\mathcal{O}(g^{3})\
,$ $\displaystyle T_{g}^{(K)}A_{\mu}\ =$ $\displaystyle
A_{\mu}-gC^{\rho}A_{\mu}A_{\rho}+{\textstyle\frac{3}{2}}g^{2}C^{\rho}A^{\lambda}C_{[\mu}A_{\rho}A_{\lambda]}+g^{2}C^{\rho}\varphi_{i}C_{[\mu}A_{\rho]}\varphi_{i}$
(4.9)
$\displaystyle{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}+{\textstyle\frac{1}{2}}g^{2}\Pi\indices{{}_{\mu}^{\nu}}\epsilon_{\nu\lambda\rho\sigma}\sum_{J=1}^{3}(-)^{\delta_{KJ}}[C^{\lambda}\varphi_{J}C^{\rho}\varphi_{J+3}A^{\sigma}-C^{\lambda}\varphi_{J+3}C^{\rho}\varphi_{J}A^{\sigma}+C^{\lambda}A^{\rho}C^{\sigma}\varphi_{J+3}\varphi_{J}]}+\mathcal{O}(g^{3})\
,$

and

$\displaystyle T_{g}^{(4)}\varphi_{I}\ =$
$\displaystyle\varphi_{I}-gC^{\rho}\varphi_{I}A_{\rho}+g^{2}C^{[\rho}A^{\lambda]}C_{\lambda}\varphi_{I}A_{\rho}+{\textstyle\frac{1}{2}}g^{2}C^{\rho}\varphi_{j}C_{\rho}\varphi_{j}\varphi_{I}$
(4.10)
$\displaystyle{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}-{\textstyle\frac{1}{4}}g^{2}\epsilon_{\mu\nu\rho\lambda}[C^{\mu}\varphi_{I+3}C^{\nu}A^{\rho}A^{\lambda}+2C^{\mu}A^{\nu}C^{\rho}\varphi_{I+3}A^{\lambda}]}$
$\displaystyle{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}-{\textstyle\frac{1}{2}}g^{2}C^{\rho}\sum_{J=1}^{3}[\varphi_{I+3}C_{\rho}\varphi_{J+3}\varphi_{J}+\varphi_{J}C_{\rho}\varphi_{I+3}\varphi_{J+3}-\varphi_{J+3}C_{\rho}\varphi_{I+3}\varphi_{J}]}+\mathcal{O}(g^{3})\
,$ $\displaystyle T_{g}^{(K)}\varphi_{I}\ =$
$\displaystyle\varphi_{I}-gC^{\rho}\varphi_{I}A_{\rho}+g^{2}C^{[\rho}A^{\lambda]}C_{\lambda}\varphi_{I}A_{\rho}+{\textstyle\frac{1}{2}}g^{2}C^{\rho}\varphi_{j}C_{\rho}\varphi_{j}\varphi_{I}$
(4.11)
$\displaystyle{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}+{\textstyle\frac{1}{4}}g^{2}\epsilon_{\mu\nu\rho\lambda}(-)^{\delta_{IK}}[C^{\mu}\varphi_{I+3}C^{\nu}A^{\rho}A^{\lambda}+2C^{\mu}A^{\nu}C^{\rho}\varphi_{I+3}A^{\lambda}]}$
$\displaystyle{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}-{\textstyle\frac{1}{2}}g^{2}C^{\rho}(-)^{\delta_{IK}}\sum_{J=1}^{3}[\varphi_{I+3}C_{\rho}\varphi_{J+3}\varphi_{J}+\varphi_{J}C_{\rho}\varphi_{I+3}\varphi_{J+3}-\varphi_{J+3}C_{\rho}\varphi_{I+3}\varphi_{J}]}$
$\displaystyle{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}+g^{2}C^{\rho}[\varphi_{I+3}C_{\rho}\varphi_{K+3}\varphi_{K}+\varphi_{K}C_{\rho}\varphi_{I+3}\varphi_{K+3}-\varphi_{K+3}C_{\rho}\varphi_{I+3}\varphi_{K}]}+\mathcal{O}(g^{3})\
,$

and the blue parts of $T_{g}^{(4)}\varphi_{I+3}$, $T_{g}^{(K)}\varphi_{I+3}$
are given by those of (4.10), (4.11) with $I$ and $I{+}3$ exchanged and an
overall minus sign on the r.h.s., while the black terms are obtained by simply
replacing $I$ with $I{+}3$. Note that the black parts of all four maps exactly
equal each other and the result from dimensional reduction (4.2), (4.3),
whereas the blue parts differ in signs for the four choices $A=1,2,3,4$. By
investigating the explicit contributions (Appendix D) to the coupling flow
operators $R_{g}^{(A)}$, it is easy to check that the symmetric superposition

$R_{g}\ :=\
{\textstyle\frac{1}{4}}(R_{g}^{(1)}+R_{g}^{(2)}+R_{g}^{(3)}+R_{g}^{(4)})\ ,$
(4.12)

exactly yields the result from dimensional reduction, as expected. Note that
while we can superimpose coupling flow operators, this is not the case for the
Nicolai map (1.8), since it is not linear in $R_{g}$. In Appendix E, we show
through explicit computations that all four maps that we have found in this
section indeed satisfy the necessary conditions for a Nicolai map to second
order.

##  5 Conclusions and outlook

In this work we have initiated a systematic study of the Nicolai map in ${\cal
N}=\,4$ supersymmetric Yang–Mills theory in four dimensions. Our main result
is the explicit form of the coupling flow operator in the Landau gauge

$\stackrel{{\scriptstyle\leftarrow}}{{R_{g}}}[\mathscr{A}]\ =\
{\textstyle\frac{1}{32}}\stackrel{{\scriptstyle\longleftarrow}}{{{\textstyle\frac{\delta}{\delta\mathscr{A}_{\Gamma}}}}}{P}\indices{{}_{\Gamma}^{\Sigma}}\
\mathrm{tr}\Bigl{\\{}(\mathscr{C}_{\Sigma})\indices{{}^{A}_{B}}S\indices{{}^{B}_{C}}\not{\mathscr{A}}{\indices{{}^{C}_{D}}}\times\not{\mathscr{A}}^{*}{\indices{{}^{D}_{E}}}(\delta\indices{{}^{E}_{A}}+\mathrm{L}\indices{{}^{E}_{A}})\Bigl{\\}}\
,$

with

$\mathrm{L}\ =\ L\ \mathrm{P}^{-}+L^{*}\ \mathrm{P}^{+}\ ,$

and $L$ any element in the Lie algebra $\mathfrak{su}(4)$. Most importantly,
it follows that the operator is subject to a 15-dimensional ambiguity. This
can be traced back to the theory-intrinsic $\mathrm{SU}(4)$ R-symmetry.
However, by construction, correlators do not depend on the choice of $L$.
Building up on previous results in ${\cal N}=\,1$ SYM and with the help of
very compact notation, the knowledge of $R_{g}$ allows for an analogous
perturbative construction of the Nicolai map of ${\cal N}=\,4$ SYM via the
universal formula $T_{g}=\overrightarrow{\cal
P}\exp\bigl{\\{}-\int_{0}^{g}\mathrm{d}h\,R_{h}[\mathscr{A}]\bigr{\\}}$.

These first steps suggest many future analyses. Natural open questions relate
to the effect of including a topological term in the theory, a better
understanding of general gauges and a graphical representation for the
perturbative expansion. As a potential application of this work, one could
compute explicit ${\cal N}=\,4$ quantum correlators
$\bigl{\langle}X[\mathscr{A}]\bigr{\rangle}_{g}=\bigl{\langle}X[T_{g}^{-1}\mathscr{A}]\bigr{\rangle}_{0}$
with the inverse Nicolai map. Critical future investigations concern how
exactly the ambiguity in the coupling flow operator translates to the Nicolai
map and, more distantly related to that, whether the framework of the Nicolai
map might hint at an integrable structure of ${\cal N}=\,4$ SYM.

Acknowledgments.  It is a pleasure to thank Olaf Lechtenfeld for many helpful
discussions. We thank the referee for their comments on the initial
submission, which have helped to further clarify several important aspects of
this paper. This work is supported by a PhD grant of the German Academic
Scholarship Foundation.

##  Appendix A Details on the $\mathcal{N}=\,1$ superfield formalism

In the conventions from [17]121212up to a global sign to recover a plus sign
in the field strength and covariant derivative instead of a minus sign. and in
the Wess-Zumino (WZ) gauge, the vector superfield ($V^{\dagger}=V$) takes the
form

$\displaystyle V$ $\displaystyle\ =\
\theta\sigma^{\mu}{\bar{\theta}}A_{\mu}(x)-\mathrm{i}\theta^{2}{\bar{\theta}}\bar{\lambda}(x)+\mathrm{i}{\bar{\theta}}^{2}\theta\lambda(x)-{\textstyle\frac{1}{2}}\theta^{2}{\bar{\theta}}^{2}\mathcal{D}(x)$
(A.1) $\displaystyle\ =\
\theta\sigma^{\mu}{\bar{\theta}}A_{\mu}(y\hphantom{{}^{\dagger}})-\mathrm{i}\theta^{2}{\bar{\theta}}\bar{\lambda}(y\hphantom{{}^{\dagger}})+\mathrm{i}{\bar{\theta}}^{2}\theta\lambda(y\hphantom{{}^{\dagger}})-{\textstyle\frac{1}{2}}\theta^{2}{\bar{\theta}}^{2}[\mathcal{D}(y\hphantom{{}^{\dagger}})-\mathrm{i}\mathrm{D}^{\mu}A_{\mu}(y\hphantom{{}^{\dagger}})]$
$\displaystyle\ =\
\theta\sigma^{\mu}{\bar{\theta}}A_{\mu}(y^{\dagger})-\mathrm{i}\theta^{2}{\bar{\theta}}\bar{\lambda}(y^{\dagger})+\mathrm{i}{\bar{\theta}}^{2}\theta\lambda(y^{\dagger})-{\textstyle\frac{1}{2}}\theta^{2}{\bar{\theta}}^{2}[\mathcal{D}(y^{\dagger})+\mathrm{i}\mathrm{D}^{\mu}A_{\mu}(y^{\dagger})]$

where $y=x+\mathrm{i}\theta\sigma{\bar{\theta}}$ and
$y^{\dagger}=x-\mathrm{i}\theta\sigma{\bar{\theta}}$ parameterize
(anti-)chiral superspace. The advantage of the WZ gauge is that

$V^{2}=-{\textstyle\frac{1}{2}}\theta^{2}{\bar{\theta}}^{2}A_{\mu}A^{\mu}\ ,$
(A.2)

whereas all higher powers vanish such that the power series

$\mathrm{e}^{2V}=1+2V+2V^{2}$ (A.3)

truncates at the second order. The non-abelian supersymmetric field strength
$W_{\alpha}$ and its conjugate are given by

$\displaystyle W_{\alpha}\ =\
-{\textstyle\frac{1}{4}}\bar{\mathrm{D}}\bar{\mathrm{D}}\mathrm{e}^{-2V}\mathrm{D}_{\alpha}\mathrm{e}^{2V}\
=\
+2\mathrm{i}\lambda_{\alpha}(y\hphantom{{}^{\dagger}})-2\bigl{[}\delta\indices{{}_{\alpha}^{\beta}}\mathcal{D}(y\hphantom{{}^{\dagger}})-\mathrm{i}\sigma\indices{{}^{\mu\nu}_{\alpha}{}^{\beta}}F_{\mu\nu}(y\hphantom{{}^{\dagger}})\bigr{]}\theta_{\beta}-2\theta^{2}\not{\mathrm{D}}_{\alpha{\dot{\alpha}}}\bar{\lambda}^{\dot{\alpha}}(y\hphantom{{}^{\dagger}})\
,$ (A.4) $\displaystyle\bar{W}^{\dot{\alpha}}\ =\
-{\textstyle\frac{1}{4}}\mathrm{D}\mathrm{D}\mathrm{e}^{-2V}\bar{\mathrm{D}}^{\dot{\alpha}}\mathrm{e}^{2V}\
=\
-2\mathrm{i}\bar{\lambda}^{\dot{\alpha}}(y^{\dagger})-2\bigl{[}\delta\indices{{}^{\dot{\alpha}}_{\dot{\beta}}}\mathcal{D}(y^{\dagger})+\mathrm{i}{\bar{\sigma}}\indices{{}^{\mu\nu}{}^{\dot{\alpha}}_{\dot{\beta}}}F_{\mu\nu}(y^{\dagger})\bigr{]}{\bar{\theta}}^{\dot{\beta}}+2{\bar{\theta}}^{2}\not{\bar{\mathrm{D}}}^{{\dot{\alpha}}\alpha}\lambda_{\alpha}(y^{\dagger})\
,$

in chiral superspace, with the superspace covariant derivatives
$\mathrm{D}_{\alpha},\bar{\mathrm{D}}_{\dot{\alpha}}$. The chiral superfields
($\bar{\mathrm{D}}_{\dot{\alpha}}\Phi_{I}=0$,
$\mathrm{D}_{\alpha}\Phi^{\dagger}_{I}=0$) have the simple expansions

$\Phi_{I}\ =\ \phi_{I}(y)+\sqrt{2}\theta\psi_{I}(y)+\theta^{2}F_{I}(y)\
,\qquad\Phi^{\dagger}_{I}\ =\
\phi^{\dagger}_{I}(y^{\dagger})+\sqrt{2}{\bar{\theta}}{\bar{\psi}}_{I}(y^{\dagger})+{\bar{\theta}}^{2}F^{\dagger}_{I}(y^{\dagger})\
,$ (A.5)

in chiral superspace and the full superspace expansions

$\displaystyle\Phi_{I}$ $\displaystyle\ =\
\phi_{I}(x)+\mathrm{i}\theta\sigma^{\mu}{\bar{\theta}}\partial_{\mu}\phi_{I}(x)+{\textstyle\frac{1}{4}}\theta^{2}{\bar{\theta}}^{2}\Box\phi_{I}(x)+\sqrt{2}\theta\psi_{I}(x)-{\textstyle\frac{\mathrm{i}}{\sqrt{2}}}\theta^{2}\partial_{\mu}\psi_{I}(x)\sigma^{\mu}{\bar{\theta}}+\theta^{2}F_{I}(x)\
,$ (A.6) $\displaystyle\Phi^{\dagger}_{I}$ $\displaystyle\ =\
\phi^{\dagger}_{I}(x)-\mathrm{i}\theta\sigma^{\mu}{\bar{\theta}}\partial_{\mu}\phi^{\dagger}_{I}(x)+{\textstyle\frac{1}{4}}\theta^{2}{\bar{\theta}}^{2}\Box\phi^{\dagger}_{I}(x)+\sqrt{2}{\bar{\theta}}{\bar{\psi}}_{I}(x)+{\textstyle\frac{\mathrm{i}}{\sqrt{2}}}{\bar{\theta}}^{2}\theta\sigma^{\mu}\partial_{\mu}{\bar{\psi}}_{I}(x)+{\bar{\theta}}^{2}F^{\dagger}_{I}(x)\
.$

For the construction of the coupling flow operator we also need the
penultimate components of the various contributions to (2.8). For the first
part, we find

$\displaystyle{\textstyle\frac{1}{4}}W^{\alpha}W_{\alpha}$
$\displaystyle=-\lambda^{2}+\bigl{[}-2\mathrm{i}\mathcal{D}\lambda-2F_{\mu\nu}\lambda\sigma^{\mu\nu}\bigr{]}\theta+\bigl{[}-2\mathrm{i}\lambda\sigma^{\mu}\mathrm{D}_{\mu}\bar{\lambda}-{\textstyle\frac{1}{2}}F^{\mu\nu}F_{\mu\nu}+\mathcal{D}^{2}+{\textstyle\frac{\mathrm{i}}{4}}F^{\mu\nu}F^{\rho\lambda}\epsilon_{\mu\nu\rho\lambda}\bigr{]}\theta^{2}\
,$ (A.7)
$\displaystyle{\textstyle\frac{1}{4}}\bar{W}_{\dot{\alpha}}\bar{W}^{\dot{\alpha}}$
$\displaystyle=-\bar{\lambda}^{2}+\bigl{[}+2\mathrm{i}\mathcal{D}\bar{\lambda}-2F_{\mu\nu}\bar{\lambda}{\bar{\sigma}}^{\mu\nu}\bigr{]}{\bar{\theta}}+\bigl{[}+2\mathrm{i}\mathrm{D}_{\mu}\lambda\sigma^{\mu}\bar{\lambda}-{\textstyle\frac{1}{2}}F^{\mu\nu}F_{\mu\nu}+\mathcal{D}^{2}-{\textstyle\frac{\mathrm{i}}{4}}F^{\mu\nu}F^{\rho\lambda}\epsilon_{\mu\nu\rho\lambda}\bigr{]}{\bar{\theta}}^{2}\
.$

Next, we evaluate

$\epsilon_{IJK}\ \mathrm{tr}\ \Phi_{I}[\Phi_{J},\Phi_{K}]\ =\
\mathrm{i}\epsilon_{IJK}f^{abc}\Bigl{[}\phi^{a}_{I}\phi^{b}_{J}\phi^{c}_{K}+3\sqrt{2}\;\theta\
\psi^{a}_{I}\phi^{b}_{J}\phi^{c}_{K}+3\theta^{2}\bigl{(}F^{a}_{I}\phi^{b}_{J}\phi^{c}_{K}-\phi^{a}_{I}\psi^{b}_{J}\psi^{c}_{K}\bigr{)}\Bigr{]}\
,$ (A.8)

and the hermitian conjugate analogously. Lastly, we find

$\displaystyle{\textstyle\frac{1}{n_{\mathrm{c}}}}\mathrm{tr}\
\mathrm{e}^{-2V}\Phi^{\dagger}_{I}\mathrm{e}^{2V}\Phi_{I}\ =$
$\displaystyle\Phi^{a\dagger}_{I}\Phi^{a}_{I}+{\textstyle\frac{2}{n_{\mathrm{c}}}}\mathrm{tr}\
[T^{a},T^{b}]T^{c}\
\Phi^{a\dagger}_{I}V^{b}\Phi^{c}_{I}+{\textstyle\frac{2}{n_{\mathrm{c}}}}\mathrm{tr}\
[T^{a},T^{b}][T^{c},T^{d}]\ \Phi^{a\dagger}_{I}V^{b}V^{c}\Phi^{d}_{I}$ (A.9)
$\displaystyle\ =\ ...\ +$
$\displaystyle\theta^{2}{\bar{\theta}}\bigl{[}-\mathrm{i}\sqrt{2}{\bar{\sigma}}^{\mu}\psi^{a}_{I}\mathrm{D}_{\mu}\phi^{a\dagger}_{I}+\sqrt{2}F^{a}_{I}{\bar{\psi}}^{a}_{I}+2f^{abc}\phi^{a\dagger}_{I}\bar{\lambda}^{b}\phi^{c}_{I}\bigr{]}$
$\displaystyle+$
$\displaystyle{\bar{\theta}}^{2}\theta\bigl{[}-\mathrm{i}\sqrt{2}\sigma^{\mu}{\bar{\psi}}^{a}_{I}\mathrm{D}_{\mu}\phi^{a}_{I}+\sqrt{2}F^{a\dagger}_{I}\psi^{a}_{I}-2f^{abc}\phi^{a\dagger}_{I}\lambda^{b}\phi^{c}_{I}\bigr{]}$
$\displaystyle+$
$\displaystyle\theta^{2}{\bar{\theta}}^{2}\bigl{[}-\mathrm{D}_{\mu}\phi^{a\dagger}_{I}\mathrm{D}^{\mu}\phi^{a}_{I}+F^{a\dagger}_{I}F^{a}_{I}+\mathrm{i}\mathrm{D}_{\mu}{\bar{\psi}}_{I}^{a}{\bar{\sigma}}^{\mu}\psi_{I}$
$\displaystyle\qquad\qquad-f^{abc}\bigl{(}\mathrm{i}\phi_{I}^{a\dagger}\mathcal{D}^{b}\phi_{I}^{c}-\sqrt{2}\phi^{a\dagger}_{I}\lambda^{b}\psi_{I}^{c}+\sqrt{2}{\bar{\psi}}^{a}_{I}\bar{\lambda}^{b}\phi^{c}_{I}\bigr{)}\bigr{]}$
$\displaystyle\ +\ \textrm{total derivatives}\ ,$

where we have left out terms of power 2 or less in $\theta$ and the traces
over the $\operatorname{SU}(n_{\mathrm{c}})$ generators were evaluated with

$[T^{a},T^{b}]\ =\ \mathrm{i}f^{abc}T^{c}\ ,\qquad\mathrm{tr}\ T^{a}T^{b}\ =\
n_{\mathrm{c}}\delta^{ab}\ .$ (A.10)

From (2.8) and (A.7), (A.8), (A.9), we deduce that in Weyl notation the
Lagrangian can be written as

$\displaystyle g^{2}\mathcal{L}\ =$
$\displaystyle-{\textstyle\frac{1}{4}}F^{a}_{\mu\nu}F^{a\mu\nu}-\mathrm{i}\lambda^{a}\sigma^{\mu}\mathrm{D}_{\mu}\bar{\lambda}^{a}+{\textstyle\frac{1}{2}}\mathcal{D}^{2}-{\textstyle\frac{1}{\sqrt{2}}}\epsilon_{IJK}f^{abc}\bigl{(}F^{a}_{I}\phi_{J}^{b}\phi_{K}^{c}+F^{a\dagger}_{I}\phi_{J}^{b\dagger}\phi_{K}^{c\dagger}\bigr{)}$
(A.11)
$\displaystyle-\mathrm{D}_{\mu}\phi^{a\dagger}_{I}\mathrm{D}^{\mu}\phi^{a}_{I}-\mathrm{i}\psi_{I}^{a}\sigma^{\mu}\mathrm{D}_{\mu}{\bar{\psi}}_{I}+F^{a\dagger}_{I}F^{a}_{I}+{\textstyle\frac{1}{\sqrt{2}}}\epsilon_{IJK}f^{abc}\bigl{(}\phi^{a}_{I}\psi^{b}_{J}\psi^{c}_{K}+\phi^{a\dagger}_{I}{\bar{\psi}}^{b}_{J}{\bar{\psi}}^{c}_{K}\bigr{)}$
$\displaystyle-\sqrt{2}f^{abc}\bigl{(}\psi_{I}^{a}\lambda^{b}\phi^{c\dagger}_{I}+{\bar{\psi}}^{a}_{I}\bar{\lambda}^{b}\phi^{c}_{I}\bigr{)}-\mathrm{i}f^{abc}\phi_{I}^{a\dagger}\mathcal{D}^{b}\phi_{I}^{c}\
,$

up to total derivatives. From the superspace expansions we read off the
supersymmetry transformations

$\displaystyle\delta\phi_{I}\ =\ \sqrt{2}\theta\psi_{I}\
,\qquad\delta\psi_{I}\ =\
\mathrm{i}\sqrt{2}\sigma^{\mu}{\bar{\theta}}\mathrm{D}_{\mu}\phi_{I}+\sqrt{2}\theta
F_{I}\ ,\qquad\delta F_{I}\ =\
\mathrm{i}\sqrt{2}{\bar{\theta}}{\bar{\sigma}}^{\mu}\mathrm{D}_{\mu}\psi_{I}-2\phi_{I}{\times}\bar{\lambda}{\bar{\theta}}\
,$ (A.12) $\displaystyle\delta A^{\mu}\ =\
-\mathrm{i}\bar{\lambda}{\bar{\sigma}}^{\mu}\theta+\mathrm{i}{\bar{\theta}}{\bar{\sigma}}^{\mu}\lambda\
,\qquad\delta\lambda\ =\ \sigma^{\mu\nu}\theta
F_{\mu\nu}+\mathrm{i}\theta\mathcal{D}\ ,\qquad\delta\mathcal{D}\ =\
-\theta\sigma^{\mu}\mathrm{D}_{\mu}\bar{\lambda}-\mathrm{D}_{\mu}\lambda\sigma^{\mu}{\bar{\theta}}\
.$

For convenience, we translate the superfield formalism to a four-component
Majorana basis using

$\displaystyle\lambda^{(\mathrm{M})}\ =\ \begin{pmatrix}\lambda_{\alpha}\\\
\bar{\lambda}^{\dot{\alpha}}\end{pmatrix}\
,\qquad\bar{\lambda}^{(\mathrm{M})}\ =\ (\lambda^{\alpha},\
\bar{\lambda}_{\dot{\alpha}})\ ,\qquad\alpha\ =\
\begin{pmatrix}\theta_{\alpha}\\\ {\bar{\theta}}^{\dot{\alpha}}\end{pmatrix}\
,\qquad\bar{\alpha}\ =\ (\theta^{\alpha},\ {\bar{\theta}}_{\dot{\alpha}})\ $
(A.13) $\displaystyle\gamma_{\mu}\ =\ \begin{pmatrix}0&\sigma_{\mu}\\\
{\bar{\sigma}}_{\mu}&0\end{pmatrix}\ ,\qquad\gamma_{5}\ =\
\gamma_{0}\gamma_{1}\gamma_{2}\gamma_{3}\ =\ \begin{pmatrix}-i&0\\\
0&i\end{pmatrix}\ ,\qquad\mathrm{etc.}\ ,$

so that

$\displaystyle\bar{\lambda}^{(\mathrm{M})}\lambda^{(\mathrm{M})}\ =\
\lambda\lambda+\bar{\lambda}\bar{\lambda}\
,\qquad\bar{\lambda}^{(\mathrm{M})}\mathrm{i}\gamma_{5}\lambda^{(\mathrm{M})}\
=\ \lambda\lambda-\bar{\lambda}\bar{\lambda}\ ,$ (A.14)
$\displaystyle\bar{\lambda}^{(\mathrm{M})}\gamma^{\mu}\lambda^{(\mathrm{M})}\
=\ \lambda\sigma^{\mu}\bar{\lambda}+\bar{\lambda}{\bar{\sigma}}^{\mu}\lambda\
=\ 2\lambda\sigma^{\mu}\bar{\lambda}\
,\qquad\frac{1}{2}\bar{\lambda}^{(\mathrm{M})}\gamma^{\mu\nu}\alpha\ =\
\lambda\sigma^{\mu\nu}\theta+\bar{\lambda}{\bar{\sigma}}^{\mu\nu}{\bar{\theta}}\
,\qquad\text{etc.}\ ,$

where the l.h.s. are in the four-component Majorana basis and the r.h.s. are
in the two-component Weyl basis. Additionally, we need the chiral projectors

$\mathrm{P}^{\pm}\ =\ {\textstyle\frac{1}{2}}(1\pm\mathrm{i}\gamma_{5})\
,\qquad\bar{\lambda}^{(\mathrm{M})}\mathrm{P}^{+}\lambda^{(\mathrm{M})}\ =\
\lambda\lambda\
,\quad\bar{\lambda}^{(\mathrm{M})}\mathrm{P}^{-}\lambda^{(\mathrm{M})}\ =\
\bar{\lambda}\bar{\lambda}\ .$ (A.15)

This leads to the Lagrangian in Majorana notation (2.10) (leaving the
superscript (M) implicit from now on) and to the penultimate component

$\displaystyle\mathring{\Delta}\ =$
$\displaystyle\bar{\alpha}\Bigl{\\{}-\mathcal{D}\gamma_{5}\lambda-{\textstyle\frac{1}{2}}F_{\mu\nu}\gamma^{\mu\nu}\lambda+2\epsilon_{IJK}f^{abc}\bigl{[}\mathrm{P}^{+}\psi^{a}_{I}\phi^{b}_{J}\phi^{c}_{K}+\mathrm{P}^{-}\psi^{a}_{I}\phi^{b\dagger}_{J}\phi^{c\dagger}_{K}\bigr{]}+2\mathrm{i}f^{abc}\gamma_{5}\phi_{I}^{a\dagger}\lambda^{b}\phi^{c}_{I}$
(A.16)
$\displaystyle+\mathrm{i}\sqrt{2}\bigl{[}\gamma^{\mu}\mathrm{P}^{-}\psi^{a}_{I}\mathrm{D}_{\mu}\phi^{a}_{I}+\gamma^{\mu}\mathrm{P}^{+}\psi^{a}_{I}\mathrm{D}_{\mu}\phi^{a\dagger}_{I}\bigr{]}-\sqrt{2}\bigl{[}\mathrm{P}^{+}\psi^{a}_{I}F^{a\dagger}_{I}+\mathrm{P}^{-}\psi^{a}_{I}F^{a}_{I}\bigr{]}\Bigr{\\}}\
.$

For completeness, we note the following hermiticity properties for Majorana
spinors $\chi,\xi$:

$\displaystyle\bar{\chi}\xi\ =\ \bar{\xi}\chi\
,\qquad\bar{\chi}\gamma^{\mu}\xi\ =\ -\bar{\xi}\gamma^{\mu}\chi\
,\qquad\bar{\chi}\gamma_{5}\xi\ =\ \bar{\xi}\gamma_{5}\chi\
,\qquad\bar{\chi}\gamma^{\mu}\gamma_{5}\xi\ =\
\bar{\xi}\gamma^{\mu}\gamma_{5}\chi\ ,$ (A.17)
$\displaystyle\bar{\chi}\gamma^{\mu\nu}\xi\ =\ -\bar{\xi}\gamma^{\mu\nu}\chi\
,\qquad\bar{\chi}\gamma^{\mu\nu}\gamma_{5}\xi\ =\
-\bar{\xi}\gamma^{\mu\nu}\gamma_{5}\chi\
,\qquad\bar{\chi}\gamma^{\rho\lambda}\gamma_{\mu}\xi\ =\
\bar{\xi}\gamma_{\mu}\gamma^{\rho\lambda}\chi\ .$

Consistency checks. In order to cross-check the expressions above (in the
Majorana basis), we performed three consistency checks. Firstly, due to the
superfield structure, the penultimate component $\mathring{\Delta}$ has to
generate the Lagrangian via its supervariation up to total derivatives

${\textstyle\frac{1}{4}}\delta\mathring{\Delta}\bigr{|}_{\bar{\alpha}\alpha}\
=\ g^{2}\mathcal{L}\ ,$ (A.18)

with
$\bar{\alpha}(\ldots)\alpha\bigr{|}_{\bar{\alpha}\alpha}=-{\textstyle\frac{1}{2}}\mathrm{tr}(\ldots)$.
In practice, this requires the Fierz identity for Majorana spinors

$4\xi\bar{\chi}=-(\bar{\chi}\xi)+\gamma_{\mu}(\bar{\chi}\gamma^{\mu}\xi)+{\textstyle\frac{1}{2}}\gamma_{\mu\nu}(\bar{\chi}\gamma^{\mu\nu}\xi)+\gamma_{5}\gamma_{\mu}(\bar{\chi}\gamma_{5}\gamma^{\mu}\xi)+\gamma_{5}(\bar{\chi}\gamma_{5}\xi)\
.$ (A.19)

A second check is making sure that the Lagrangian transforms as a divergence,
i.e.

$\delta\mathcal{L}\ =\ \text{divergence}\ .$ (A.20)

A third consistency check is verifying the generation of the susy-algebra

$\\{Q_{\alpha},\bar{Q}_{\beta}\\}\ =\ 2{(\gamma^{\mu})}_{\alpha\beta}P_{\mu}\
=\ -2\mathrm{i}{(\gamma^{\mu})}_{\alpha\beta}\partial_{\mu}\ ,$ (A.21)

up to a gauge transformation. With the supercharges
$\delta(...)=\bar{\alpha}_{\alpha}Q_{\alpha}(...)$ this can be evaluated by
computing the commutator of two supervariations

$[\delta^{(1)},\delta^{(2)}]=[\bar{\alpha}_{1\alpha}Q_{\alpha},\bar{Q}_{\beta}\alpha_{2\beta}]=\bar{\alpha}_{1\alpha}\\{Q_{\alpha},\bar{Q}_{\beta}\\}\alpha_{2\beta}\
,$ (A.22)

acting on each field. One finds that the susy-algebra reads

$\\{Q_{\alpha},\bar{Q}_{\beta}\\}=-2\mathrm{i}{(\gamma^{\mu})}_{\alpha\beta}\partial_{\mu}-[\omega,\cdot]_{\alpha\beta}+G_{\alpha\beta}(A)\;,$
(A.23)

where $\omega=2\mathrm{i}\not{A}$,
$[\omega,\cdot]^{a}=f^{abc}\omega^{b}(\cdot)^{c}$ and $G_{\alpha\beta}$ is a
gauge transformation $A_{\mu}\rightarrow A_{\mu}+\partial_{\mu}\omega$ as
required in the WZ gauge.

Stripping-off the susy parameter. Lastly, we find it convenient to strip-off
the susy parameter by setting $\delta\equiv\delta_{\alpha}\alpha_{\alpha}$ and
$\mathring{\Delta}\equiv\bar{\alpha}_{\alpha}\mathring{\Delta}_{\alpha}$. This
yields the supervariations (2.11) and the penultimate component (2.12) (up to
an overall normalization). We note that the fermionic supervariations have
gained an extra minus sign, since

$\bar{\chi}\delta\lambda\ =\ \bar{\chi}_{\beta}M_{\beta\alpha}\alpha_{\alpha}\
=\ \bar{\chi}_{\beta}\delta_{\alpha}\alpha_{\alpha}\lambda_{\beta}\ =\
-\bar{\chi}_{\beta}\delta_{\alpha}\lambda_{\beta}\alpha_{\alpha}\quad\Rightarrow\quad\delta_{\alpha}\lambda_{\beta}\
=\ -M_{\beta\alpha}\ ,$ (A.24)

with some arbitrary spinor $\bar{\chi}$. In general, one has to be careful
with sign-flips, because fermionic quantities anti-commute with each other.

##  Appendix B Details on the canonical construction

In this appendix we give the detailed calculation for how to get from (3.6) to
(3.7). We need

$\displaystyle\delta_{\alpha}X[\tilde{\mathscr{A}}]$ $\displaystyle\ =\
-\mathrm{i}\int\mathrm{d}^{4}x\;\Bigl{(}\widetilde{{\bar{\psi}}}_{4}\gamma_{\mu}{\textstyle\frac{\delta}{\delta\widetilde{A}_{\mu}}}+\widetilde{{\bar{\psi}}}_{J}(c^{i})\indices{{}^{J}_{4}}{\textstyle\frac{\delta}{\delta\widetilde{\varphi}_{i}}}\Bigr{)}_{\alpha}X[\tilde{\mathscr{A}}]$
(B.1) $\displaystyle\ =\ -\mathrm{i}\int\mathrm{d}^{4}x\;\
\Bigl{(}\widetilde{{\bar{\psi}}}_{A}(\widehat{\mathscr{C}}_{\Sigma})\indices{{}^{A}_{4}}{\textstyle\frac{\delta}{\delta\tilde{\mathscr{A}}_{\Sigma}}}\Bigr{)}_{\alpha}X[\tilde{\mathscr{A}}]\
,$

where we introduced the object

$(\widehat{\mathscr{C}}_{\Sigma})\indices{{}^{A}_{4}}\ =\
\left\\{\begin{array}[]{ll}\delta\indices{{}^{A}_{4}}\gamma_{\mu}&\text{for}\quad\Sigma\
=\ \mu\ =\ 0,1,2,3\\\ (c^{i})\indices{{}^{A}_{4}}&\text{for}\quad\Sigma\ =\
3+i\ =\ 4,5,...,9\end{array}\right.\ ,$ (B.2)

with matrix-valued entries. It is defined via

$\delta_{\alpha}^{(4)}\tilde{\mathscr{A}}_{\Sigma}\ =\
-\mathrm{i}(\widetilde{{\bar{\psi}}}_{A}(\widehat{\mathscr{C}}_{\Sigma})\indices{{}^{A}_{4}})_{\alpha}\
,$ (B.3)

where the $(4)$ indicates that we have singled out one of the four
supersymmetries (the ‘fourth’ one). Further we have

$sX[\tilde{\mathscr{A}}]\ =\ \sqrt{g}\int\mathrm{d}^{4}x\;\
\widetilde{\mathscr{D}}_{\Gamma}\widetilde{C}{\textstyle\frac{\delta}{\delta\tilde{\mathscr{A}}_{\Gamma}}}X[\tilde{\mathscr{A}}]\
,$ (B.4)

as well as the gaugino and ghost propagators, given by

$\bcontraction{}{\widetilde{\psi}}{{}^{A}(x)}{\widetilde{{\bar{\psi}}}}\widetilde{\psi}^{A}(x)\widetilde{{\bar{\psi}}}_{B}(y)\
=\ -\widetilde{S}\indices{{}^{A}_{B}}(x,y;\tilde{\mathscr{A}})\
,\qquad\widetilde{\not{\mathscr{D}}}{\indices{{}^{A}_{C}}}\;\widetilde{S}\indices{{}^{C}_{B}}(x,y;\tilde{\mathscr{A}})\
=\ \delta\indices{{}^{A}_{B}}\delta(x-y)\ ,$ (B.5)

and

$\mathrm{i}\bcontraction{}{\widetilde{C}}{(x)}{\widetilde{\bar{C}}}\widetilde{C}(x)\widetilde{\bar{C}}(y)\
=\ \widetilde{G}(x,y;\tilde{\mathscr{A}})\
,\qquad{\textstyle\frac{\partial\mathcal{G}(\tilde{\mathscr{A}})}{\partial\tilde{\mathscr{A}}_{\Gamma}}}\widetilde{\mathscr{D}}_{\Gamma}\;\widetilde{G}(x,y;\tilde{\mathscr{A}})\
=\ \delta(x-y)\ ,$ (B.6)

respectively. The rescaled coupling flow operator then reads

$\stackrel{{\scriptstyle\leftarrow}}{{\widetilde{R}}}[\tilde{\mathscr{A}}]\ =\
\stackrel{{\scriptstyle\longleftarrow}}{{{\textstyle\frac{\delta}{\delta\tilde{\mathscr{A}}_{\Gamma}}}}}\widetilde{P}\indices{{}_{\Gamma}^{\Sigma}}\widetilde{R}_{\Sigma}\
+\
\stackrel{{\scriptstyle\longleftarrow}}{{{\textstyle\frac{\delta}{\delta\tilde{\mathscr{A}}_{\Gamma}}}}}\widetilde{\mathscr{D}}_{\Gamma}\widetilde{G}\
\mathcal{G}(\tilde{\mathscr{A}})\ ,$ (B.7)

where we introduced the covariant projector

$\widetilde{P}\indices{{}_{\Gamma}^{\Sigma}}\ =\
\delta\indices{{}_{\Gamma}^{\Sigma}}-\widetilde{\mathscr{D}}_{\Gamma}\widetilde{G}{\textstyle\frac{\partial\mathcal{G}(\tilde{\mathscr{A}})}{\partial\tilde{\mathscr{A}}_{\Sigma}}}\
,$ (B.8)

and

$\widetilde{R}_{\Sigma}\ =\
-{\textstyle\frac{1}{4}}\mathrm{tr}\Bigl{\\{}\bigl{[}{\textstyle\frac{1}{2}}\widetilde{F}_{\mu\nu}\gamma^{\mu\nu}\widetilde{S}\indices{{}^{4}_{C}}+(\widetilde{\Phi}\indices{{}^{4}_{A}})^{\dagger}\not{\widetilde{\mathscr{D}}}{\indices{{}^{A}_{B}}}\widetilde{S}\indices{{}^{B}_{C}}-{\textstyle\frac{1}{2}}(\widetilde{\Phi}\indices{{}^{4}_{A}})^{\dagger}\widetilde{\Phi}\indices{{}^{A}_{B}}{\times}\widetilde{S}\indices{{}^{B}_{C}}\bigr{]}(\widehat{\mathscr{C}}_{\Sigma})\indices{{}^{C}_{4}}\Bigl{\\}}\
,$ (B.9)

where the trace is over Majorana spinor space.

The original (unrescaled) coupling flow operator is given by [12]

$R_{g}[\mathscr{A}]\ =\
{\textstyle\frac{1}{g}}\,\bigl{(}\widetilde{R}[\tilde{\mathscr{A}}]-E\bigr{)}\qquad\textrm{with}\qquad
E\ =\
\tilde{\mathscr{A}}_{\Gamma}\,{\textstyle\frac{\delta}{\delta\tilde{\mathscr{A}}_{\Gamma}}}\
.$ (B.10)

To isolate the Euler operator $E$, we need the identities

$\displaystyle\gamma^{\rho\lambda}\widetilde{F}_{\rho\lambda}\ =\
2\not{\widetilde{\mathrm{D}}}\widetilde{\not{A}}+2\partial\cdot\widetilde{A}-\widetilde{\not{A}}\times\widetilde{\not{A}}\
,$ (B.11)
$\displaystyle\not{\widetilde{\mathrm{D}}}\widetilde{S}\indices{{}^{4}_{C}}\
=\
\delta\indices{{}^{4}_{C}}-\widetilde{\Phi}\indices{{}^{4}_{B}}\times\widetilde{S}\indices{{}^{B}_{C}}\
,$ (B.12)

which (next to other contributions) generate the
$\widetilde{A}_{\mu}{\textstyle\frac{\delta}{\delta\widetilde{A}_{\mu}}}$ part
of $E$. Further we use
$\widetilde{\not{\mathscr{D}}}{\indices{{}^{A}_{B}}}\;\widetilde{S}\indices{{}^{B}_{C}}=\delta\indices{{}^{A}_{C}}$
in the second term of (B.9) and

$(\widetilde{\Phi}{\indices{{}^{4}_{A}}})^{\dagger}(\widehat{\mathscr{C}}_{\Sigma})\indices{{}^{A}_{4}}\
=\ \left\\{\begin{array}[]{cl}0&\text{for}\quad\Sigma\ =\ \mu\\\
-\mathbbm{1}_{4}\widetilde{\varphi}_{I}-\gamma_{5}\widetilde{\varphi}_{I+3}&\text{for}\quad\Sigma\
=\ 3+I\\\
+\gamma_{5}\widetilde{\varphi}_{I}-\mathbbm{1}_{4}\widetilde{\varphi}_{I+3}&\text{for}\quad\Sigma\
=\ 6+I\\\ \end{array}\right.\ .$ (B.13)

With $\mathrm{tr}\ \gamma_{5}=0$, this gives the second part of the Euler
operator. Straightforward calculations lead to

$\displaystyle\widetilde{R}_{\Sigma}\ =\
\tilde{\mathscr{A}}_{\Sigma}-{\textstyle\frac{1}{4}}\mathrm{tr}\Bigl{\\{}(\mathscr{C}_{\Sigma})\indices{{}^{4}_{A}}\bigl{[}{\textstyle\frac{1}{2}}\widetilde{S}\indices{{}^{A}_{4}}(2\partial\cdot\widetilde{A}-\widetilde{\not{A}}\times\widetilde{\not{A}})-\widetilde{S}\indices{{}^{A}_{B}}\widetilde{\Phi}\indices{{}^{B}_{4}}\times\widetilde{\not{A}}-{\textstyle\frac{1}{2}}\widetilde{S}\indices{{}^{A}_{B}}\widetilde{\Phi}\indices{{}^{B}_{C}}\times(\widetilde{\Phi}\indices{{}^{C}_{4}})^{\dagger}\bigr{]}\Bigl{\\}}\
,$ (B.14)

where we flipped the order of the quantities in the trace for a more natural
implicit color structure. To do so, we have used that $\widetilde{R}_{\Sigma}$
is real and identities such as

$\displaystyle{\bar{\psi}}\ =\ \psi^{\dagger}\;\gamma_{0}\
,\qquad(\gamma_{0})^{2}\ =\ \mathbbm{1}_{4}\
,\qquad(\widetilde{S}\indices{{}^{A}_{B}})^{\dagger}\ =\
\gamma_{0}\;\widetilde{S}\indices{{}^{B}_{A}}\;\gamma_{0}\
,\qquad(\mathscr{C}_{\Sigma})\indices{{}^{4}_{A}}\ :=\ \gamma_{0}\
((\widehat{\mathscr{C}}_{\Sigma})\indices{{}^{A}_{4}})^{\dagger}\ \gamma_{0}\
,$ (B.15) $\displaystyle\gamma_{\mu}^{\dagger}\ =\
\gamma_{0}\;\gamma_{\mu}\;\gamma_{0}\ ,\qquad\gamma_{5}^{\dagger}\ =\
\gamma_{0}\;\gamma_{5}\gamma_{0}\ =\ -\gamma_{5}\;\
,\qquad\gamma_{0}\;\widetilde{\Phi}\indices{{}^{A}_{B}}\;\gamma_{0}\ =\
(\widetilde{\Phi}\indices{{}^{B}_{A}})^{\dagger}\ .$

This leads to

$(\mathscr{C}_{\Sigma})\indices{{}^{4}_{A}}\ =\
\left\\{\begin{array}[]{ll}\delta\indices{{}^{4}_{A}}\gamma_{\mu}&\text{for}\quad\Sigma\
=\ \mu\ =\ 0,1,2,3\\\ (c^{i})\indices{{}^{4}_{A}}&\text{for}\quad\Sigma\ =\
3+i\ =\ 4,5,...,9\end{array}\right.\ .$ (B.16)

Since $\widetilde{R}_{\Gamma}=\tilde{\mathscr{A}}_{\Gamma}+...$, the Euler
operator conveniently cancels and we find (after inserting
$\tilde{\mathscr{A}}=g\mathscr{A}$) for any linear gauge

$\stackrel{{\scriptstyle\leftarrow}}{{R_{g}}}[\mathscr{A}]\ =\
-{\textstyle\frac{1}{4}}\stackrel{{\scriptstyle\longleftarrow}}{{{\textstyle\frac{\delta}{\delta\mathscr{A}_{\Gamma}}}}}P\indices{{}_{\Gamma}^{\Sigma}}\
\mathrm{tr}\Bigl{\\{}(\mathscr{C}_{\Sigma})\indices{{}^{4}_{A}}\bigl{[}{\textstyle\frac{1}{2}}S\indices{{}^{A}_{4}}({\textstyle\frac{2}{g}}\partial{\cdot}A-\not{A}\times\not{A})-S\indices{{}^{A}_{B}}\Phi\indices{{}^{B}_{4}}\times\not{A}-{\textstyle\frac{1}{2}}S\indices{{}^{A}_{B}}\Phi\indices{{}^{B}_{C}}\times(\Phi\indices{{}^{C}_{4}})^{\dagger}\bigr{]}\Bigl{\\}}\
.$ (B.17)

Now that the coupling is restored, with

$\not{\mathscr{D}}{\indices{{}^{A}_{B}}}\ =\
\not{\mathrm{D}}\delta{\indices{{}^{A}_{B}}}+g\Phi{\indices{{}^{A}_{B}}}\times\
,\qquad\text{with}\qquad\mathrm{D}_{\mu}\ =\ \partial_{\mu}+gA_{\mu}\times\ \
,$ (B.18)

and $\not{\mathscr{D}}{\indices{{}^{A}_{C}}}S\indices{{}^{C}_{B}}\ =\
\delta\indices{{}^{A}_{B}}$, the fermion propagators can be expanded
perturbatively:

$S{\indices{{}^{A}_{B}}}\ =\
S_{0}\delta{\indices{{}^{A}_{B}}}-gS_{0}\not{\mathscr{A}}{\indices{{}^{A}_{C}}}S{\indices{{}^{C}_{B}}}\
=\
\sum_{l=0}^{\infty}\bigl{(}-gS_{0}\not{\mathscr{A}}\bigr{)}^{l}{\indices{{}^{A}_{B}}}S_{0}\
,$ (B.19)

with $S_{0}=\not{\partial}^{\;-1}=-\not{\partial}C$ and

$\not{\mathscr{A}}{\indices{{}^{A}_{B}}}\ =\
\not{A}\delta\indices{{}^{A}_{B}}+\Phi\indices{{}^{A}_{B}}\ .$ (B.20)

In particular $S{\indices{{}^{A}_{4}}}|_{g=0}=0$, so that the coupling flow
operator $R_{g}$ contains no term of order ${\textstyle\frac{1}{g}}$. We use
the same procedure to get rid of the
$S_{0}{\textstyle\frac{2}{g}}\partial\cdot A|_{g=0}$ contribution as in the
$\mathcal{N}=\,1$ case [13] with

$2S_{0}\partial\cdot A\ =\ -2\not{A^{\mathrm{L}}}\ =\ \not{A}^{*}-\not{A}$
(B.21)

to rewrite the first term:

$\displaystyle-{\textstyle\frac{1}{4}}\stackrel{{\scriptstyle\longleftarrow}}{{{\textstyle\frac{\delta}{\delta\mathscr{A}_{\Gamma}}}}}P\indices{{}_{\Gamma}^{\Sigma}}\
\mathrm{tr}\bigl{\\{}(\mathscr{C}_{\Sigma})\indices{{}^{4}_{A}}\bigl{[}{\textstyle\frac{1}{2}}S\indices{{}^{A}_{4}}({\textstyle\frac{2}{g}}\partial{\cdot}A-\not{A}\times\not{A})\bigr{]}\bigr{\\}}$
(B.22) $\displaystyle\ =$
$\displaystyle-{\textstyle\frac{1}{8}}\stackrel{{\scriptstyle\longleftarrow}}{{{\textstyle\frac{\delta}{\delta\mathscr{A}_{\Gamma}}}}}P\indices{{}_{\Gamma}^{\Sigma}}\
\mathrm{tr}\Bigl{\\{}(\mathscr{C}_{\Sigma})\indices{{}^{4}_{A}}\Bigl{[}\sum_{l=0}^{\infty}\bigl{(}-gS_{0}\not{\mathscr{A}}\bigr{)}^{l}{\indices{{}^{A}_{4}}}S_{0}({\textstyle\frac{2}{g}}\partial\cdot
A-\not{A}\times\not{A})\Bigr{]}\Bigr{\\}}$ $\displaystyle\ =$
$\displaystyle-{\textstyle\frac{1}{8}}\stackrel{{\scriptstyle\longleftarrow}}{{{\textstyle\frac{\delta}{\delta\mathscr{A}_{\Gamma}}}}}P\indices{{}_{\Gamma}^{\Sigma}}\
\mathrm{tr}\Bigl{\\{}(\mathscr{C}_{\Sigma})\indices{{}^{4}_{A}}\Bigl{[}{\textstyle\frac{1}{g}}\sum_{l=0}^{\infty}\bigl{(}-gS_{0}\not{\mathscr{A}}\bigr{)}^{l}{\indices{{}^{A}_{4}}}S_{0}(\not{A}^{*}-\not{A})-\sum_{l=0}^{\infty}\bigl{(}-gS_{0}\not{\mathscr{A}}\bigr{)}^{l}_{A4}S_{0}\not{A}\times\not{A}\Bigr{]}\Bigr{\\}}$
$\displaystyle\ =$
$\displaystyle-{\textstyle\frac{1}{8}}\stackrel{{\scriptstyle\longleftarrow}}{{{\textstyle\frac{\delta}{\delta\mathscr{A}_{\Gamma}}}}}P\indices{{}_{\Gamma}^{\Sigma}}\
\mathrm{tr}\Bigl{\\{}(\mathscr{C}_{\Sigma})\indices{{}^{4}_{A}}\Bigl{[}{\textstyle\frac{1}{g}}\sum_{l=0}^{\infty}\bigl{(}-gS_{0}\not{\mathscr{A}}\bigr{)}^{l}{\indices{{}^{A}_{4}}}S_{0}(\not{A}^{*}-\not{A})-\sum_{l=0}^{\infty}\bigl{(}-gS_{0}\not{\mathscr{A}}\bigr{)}^{l}{\indices{{}^{A}_{B}}}S_{0}(\not{\mathscr{A}}{\indices{{}^{B}_{4}}}-\Phi{\indices{{}^{B}_{4}}})\times\not{A}\Bigr{]}\Bigr{\\}}$
$\displaystyle\ =$
$\displaystyle-{\textstyle\frac{1}{8}}\stackrel{{\scriptstyle\longleftarrow}}{{{\textstyle\frac{\delta}{\delta\mathscr{A}_{\Gamma}}}}}P\indices{{}_{\Gamma}^{\Sigma}}\
\mathrm{tr}\Bigl{\\{}(\mathscr{C}_{\Sigma})\indices{{}^{4}_{A}}\Bigl{[}{\textstyle\frac{1}{g}}\sum_{l=0}^{\infty}\bigl{(}-gS_{0}\not{\mathscr{A}}\bigr{)}^{l}{\indices{{}^{A}_{4}}}S_{0}(\not{A}^{*}-\not{A})+{\textstyle\frac{1}{g}}\sum_{l=1}^{\infty}\bigl{(}-gS_{0}\not{\mathscr{A}}\bigr{)}^{l}{\indices{{}^{A}_{4}}}\times\not{A}+S{\indices{{}^{A}_{B}}}\Phi{\indices{{}^{B}_{4}}}\times\not{A}\Bigr{]}\Bigr{\\}}$
$\displaystyle\ =$
$\displaystyle-{\textstyle\frac{1}{8}}\stackrel{{\scriptstyle\longleftarrow}}{{{\textstyle\frac{\delta}{\delta\mathscr{A}_{\Gamma}}}}}P\indices{{}_{\Gamma}^{\Sigma}}\
\mathrm{tr}\Bigl{\\{}(\mathscr{C}_{\Sigma})\indices{{}^{4}_{A}}\Bigl{[}{\textstyle\frac{1}{g}}\delta\indices{{}^{A}_{4}}S_{0}(\not{A}^{*}-\not{A})+{\textstyle\frac{1}{g}}\sum_{l=1}^{\infty}\bigl{(}-gS_{0}\not{\mathscr{A}}\bigr{)}^{l}{\indices{{}^{A}_{4}}}\times\not{A}^{*}+S{\indices{{}^{A}_{B}}}\Phi{\indices{{}^{B}_{4}}}\times\not{A}\Bigr{]}\Bigr{\\}}$
$\displaystyle\ =$
$\displaystyle-{\textstyle\frac{1}{8}}\stackrel{{\scriptstyle\longleftarrow}}{{{\textstyle\frac{\delta}{\delta\mathscr{A}_{\Gamma}}}}}P\indices{{}_{\Gamma}^{\Sigma}}\
\mathrm{tr}\Bigl{\\{}(\mathscr{C}_{\Sigma})\indices{{}^{4}_{A}}S{\indices{{}^{A}_{B}}}\Bigl{[}-\not{\mathscr{A}}{\indices{{}^{B}_{4}}}\times\not{A}^{*}+\Phi{\indices{{}^{B}_{4}}}\times\not{A}\Bigr{]}\Bigr{\\}}-{\textstyle\frac{1}{g}}\stackrel{{\scriptstyle\longleftarrow}}{{{\textstyle\frac{\delta}{\delta\mathscr{A}_{\Gamma}}}}}P\indices{{}_{\Gamma}^{\nu}}A^{\mathrm{L}}_{\nu}$
$\displaystyle\ =$
$\displaystyle-{\textstyle\frac{1}{4}}\stackrel{{\scriptstyle\longleftarrow}}{{{\textstyle\frac{\delta}{\delta\mathscr{A}_{\Gamma}}}}}P\indices{{}_{\Gamma}^{\Sigma}}\
\mathrm{tr}\Bigl{\\{}(\mathscr{C}_{\Sigma})\indices{{}^{4}_{A}}S{\indices{{}^{A}_{B}}}\Bigl{[}-{\textstyle\frac{1}{2}}\not{\mathscr{A}}{\indices{{}^{B}_{4}}}\times\not{A}^{*}+{\textstyle\frac{1}{2}}\Phi{\indices{{}^{B}_{4}}}\times\not{A}\Bigr{]}\Bigr{\\}}+\stackrel{{\scriptstyle\longleftarrow}}{{{\textstyle\frac{\delta}{\delta\mathscr{A}_{\Gamma}}}}}\Pi\indices{{}_{\Gamma}^{\Sigma}}\mathscr{A}_{\Sigma}G{\textstyle\frac{\partial\mathcal{G}(\mathscr{A})}{\partial
A_{\nu}}}A^{\mathrm{L}}_{\nu}\ ,$

containing no term of order $1/g$. Putting everything together we find

$\stackrel{{\scriptstyle\leftarrow}}{{R_{g}}}[\mathscr{A}]\ =\
{\textstyle\frac{1}{8}}\stackrel{{\scriptstyle\longleftarrow}}{{{\textstyle\frac{\delta}{\delta\mathscr{A}_{\Gamma}}}}}P\indices{{}_{\Gamma}^{\Sigma}}\
\mathrm{tr}\Bigl{\\{}(\mathscr{C}_{\Sigma})\indices{{}^{4}_{A}}S{\indices{{}^{A}_{B}}}\bigl{[}\not{\mathscr{A}}{\indices{{}^{B}_{4}}}\times\not{A}^{*}+\Phi{\indices{{}^{B}_{4}}}\times\not{A}+\Phi{\indices{{}^{B}_{C}}}\times\Phi^{\dagger}{\indices{{}^{C}_{4}}}\bigr{]}\Bigl{\\}}+\stackrel{{\scriptstyle\longleftarrow}}{{{\textstyle\frac{\delta}{\delta\mathscr{A}_{\Gamma}}}}}\Pi\indices{{}_{\Gamma}^{\Sigma}}\mathscr{A}_{\Sigma}G{\textstyle\frac{\partial\mathcal{G}(\mathscr{A})}{\partial
A_{\nu}}}A^{\mathrm{L}}_{\nu}\ ,$ (B.23)

which after defining

$\not{\mathscr{A}}^{*}{\indices{{}^{A}_{B}}}\ =\
\not{A}^{*}\delta{\indices{{}^{A}_{B}}}+(\Phi{\indices{{}^{A}_{B}}})^{\dagger}\
,$ (B.24)

takes the simple form (3.7).

##  Appendix C Infinitesimal free action condition

In this appendix we present a direct proof that the coupling flow operator in
the Landau gauge (3.30) satisfies the three infinitesimal conditions (1.10)
(1.11). The determinant matching condition follows from the other two
conditions and the defining relation (1.4). The gauge condition (1.11) follows
automatically from the form of the covariant projector (3.13). Thus, we have
only left to show the infinitesimal free action condition

$(\partial_{g}+R_{g})S_{g}^{\mathrm{b}}[\mathscr{A}]\ =\ 0\ .$ (C.1)

The basic procedure of the proof is equivalent to the one in A.3 of [10] for
${\cal N}=\,1$ SYM in $D=3,4,6,10$, but we have to take into account
subtleties coming from the additional degrees of freedom in the ${\cal N}=\,4$
case. We can write the bosonic action as

$S_{g}^{\mathrm{b}}[\mathscr{A}]\ =\
\int\mathrm{d}^{4}x\;\bigl{\\{}-{\textstyle\frac{1}{4}}\mathcal{F}^{\Sigma\Theta}\mathcal{F}_{\Sigma\Theta}\bigr{\\}}\
,$ (C.2)

with

$\mathcal{F}_{\Sigma\Theta}\ =\
\partial_{\Sigma}\mathscr{A}_{\Theta}-\partial_{\Theta}\mathscr{A}_{\Sigma}+g\mathscr{A}_{\Sigma}\times\mathscr{A}_{\Theta}\
,\qquad\partial_{3+i}\ =\ 0\ ,\qquad\mathscr{A}_{\mu}\ =\ A_{\mu}\
,\qquad\mathscr{A}_{3+i}\ =\ \varphi_{i}\ .$ (C.3)

From these expressions, it is easy to find

$\partial_{g}S_{g}^{\mathrm{b}}\ =\
-{\textstyle\frac{1}{2}}\mathcal{F}^{\Sigma\Theta}\mathscr{A}_{\Sigma}{\times}\mathscr{A}_{\Theta}\quad\textrm{and}\quad{\textstyle\frac{\delta
S_{g}^{\mathrm{b}}}{\mathscr{A}_{\Sigma}}}\ =\
\mathscr{D}_{\Theta}\mathcal{F}^{\Theta\Sigma}\ ,$ (C.4)

with implicit integration. We first show the statement for the particular
choice of the coupling flow operator (3.24) and afterwards generalize the
result to the full Lie algebra $\mathfrak{su}(4)$. Concretely, we first prove
that

$(\partial_{g}+R_{g})S_{g}^{\mathrm{b}}[\mathscr{A}]\ =\
-{\textstyle\frac{1}{2}}\mathcal{F}^{\Sigma\Theta}\mathscr{A}_{\Sigma}{\times}\mathscr{A}_{\Theta}+{\textstyle\frac{1}{8}}\mathscr{D}_{\Theta}\mathcal{F}^{\Theta\Sigma}\
\mathrm{tr}\bigl{\\{}(\mathscr{C}_{\Sigma})\indices{{}^{4}_{B}}S\indices{{}^{B}_{C}}\not{\mathscr{A}}{\indices{{}^{C}_{D}}}\times\not{\mathscr{A}}^{*}{\indices{{}^{D}_{4}}}\bigr{\\}}$
(C.5)

vanishes. To do so, we use the identities

$\displaystyle{\textstyle\frac{1}{4}}\mathrm{tr}\bigl{\\{}(\mathscr{C}_{\Sigma})\indices{{}^{4}_{B}}(\bar{\mathscr{C}}_{\Theta})\indices{{}^{B}_{C}}(\mathscr{C}_{\Gamma})\indices{{}^{C}_{D}}(\bar{\mathscr{C}}_{\Psi})\indices{{}^{D}_{4}}\bigr{\\}}\
=\
\eta_{\Sigma\Psi}\eta_{\Theta\Gamma}-\eta_{\Sigma\Gamma}\eta_{\Theta\Psi}+\eta_{\Sigma\Theta}\eta_{\Gamma\Psi}\
,$ (C.6)
$\displaystyle(\mathscr{C}^{\Gamma})\indices{{}^{A}_{B}}\mathscr{D}_{\Gamma}\
S\indices{{}^{B}_{C}}\ =\ \not{\mathscr{D}}\indices{{}^{A}_{B}}\
S\indices{{}^{B}_{C}}\ =\ \delta\indices{{}^{A}_{C}}\ ,$ (C.7)
$\displaystyle(\mathscr{C}_{[\Sigma})\indices{{}^{A}_{B}}(\bar{\mathscr{C}}_{\Theta]})\indices{{}^{B}_{C}}(\mathscr{C}_{\Gamma})\indices{{}^{C}_{D}}\
=\ -2(\mathscr{C}_{[\Sigma})\indices{{}^{A}_{D}}\
\eta_{\Theta]\Gamma}+(\mathscr{C}_{[\Sigma})\indices{{}^{A}_{B}}(\bar{\mathscr{C}}_{\Theta})\indices{{}^{B}_{C}}(\mathscr{C}_{\Gamma]})\indices{{}^{C}_{D}}\
,$ (C.8)

that are similar to the ones used in [10]. Here we have introduced a
‘conjugate’ $\bar{\mathscr{C}}$ (in the Landau gauge), so that

$\displaystyle\mathscr{C}_{\mu}\ =\ \mathbbm{1}_{4}\gamma_{\mu}\
,\qquad\mathscr{C}_{3+i}\ =\ 2[(t^{i})^{*}\mathrm{P}^{+}-t^{i}\mathrm{P}^{-}]\
,$ $\displaystyle\not{\mathscr{A}}{\indices{{}^{A}_{B}}}\ =\
\mathscr{A}^{\Gamma}(\mathscr{C}_{\Gamma})\indices{{}^{A}_{B}}\ =\
\not{A}+\Phi\indices{{}^{A}_{B}}$ (C.9) $\displaystyle\bar{\mathscr{C}}_{\mu}\
=\ \mathbbm{1}_{4}\gamma_{\mu}\ ,\qquad\bar{\mathscr{C}}_{3+i}\ =\
2[t^{i}\mathrm{P}^{+}-(t^{i})^{*}\mathrm{P}^{-}]\ ,$
$\displaystyle\not{\mathscr{A}}^{*}{\indices{{}^{A}_{B}}}\ =\
\mathscr{A}^{\Gamma}(\bar{\mathscr{C}}_{\Gamma})\indices{{}^{A}_{B}}\ =\
\not{A}+(\Phi\indices{{}^{A}_{B}})^{\dagger}\ ,$

with the Clebsch-Gordon coefficients $t^{i}_{AB}$ as matrices in R-space. It
should be noted that (C.6) is only valid up to terms that vanish when
contracted with fields in the adjoint representation of the gauge group due to
the Jacobi identity in color space. We explicitly check (C.6) at the end of
this appendix. The identity (C.8) follows from the analogous identity for the
10d gamma matrices

$\Gamma_{\mu}\ =\
\mathbbm{1}_{8}\otimes\gamma_{\mu}\quad\textrm{and}\quad\Gamma_{3+i}\ =\
2\begin{pmatrix}0&t^{i}\\\
(t^{i})^{*}&0\end{pmatrix}\otimes(\mathrm{P}^{+}-\mathrm{P}^{-})\ ,$ (C.10)

as well as the anti-commutation relation for the Clebsch-Gordon matrices

$\\{t^{i},(t^{j})^{*}\\}\ =\
-{\textstyle\frac{1}{2}}\delta^{ij}\mathbbm{1}_{4}\ .$ (C.11)

With these identities at hand, we can rewrite the first term in (C.5) as

$\displaystyle-{\textstyle\frac{1}{2}}\mathcal{F}^{\Sigma\Theta}\mathscr{A}_{\Sigma}{\times}\mathscr{A}_{\Theta}\
\stackrel{{\scriptstyle\eqref{eq:id_fa_1}}}{{=}}$
$\displaystyle{\textstyle\frac{1}{16}}\mathcal{F}^{\Sigma\Theta}\mathrm{tr}\bigl{\\{}(\mathscr{C}_{\Sigma})\indices{{}^{4}_{B}}(\bar{\mathscr{C}}_{\Theta})\indices{{}^{B}_{C}}(\mathscr{C}_{\Gamma})\indices{{}^{C}_{D}}(\bar{\mathscr{C}}_{\Psi})\indices{{}^{D}_{4}}\bigr{\\}}\mathscr{A}^{\Gamma}{\times}\mathscr{A}^{\Psi}$
(C.12) $\displaystyle\ \stackrel{{\scriptstyle\eqref{eq:id_fa_2}}}{{=}}$
$\displaystyle{\textstyle\frac{1}{16}}\mathcal{F}^{\Sigma\Theta}\mathrm{tr}\bigl{\\{}(\mathscr{C}_{\Sigma})\indices{{}^{4}_{B}}(\bar{\mathscr{C}}_{\Theta})\indices{{}^{B}_{C}}(\mathscr{C}_{\Gamma})\indices{{}^{C}_{D}}\mathscr{D}^{\Gamma}\
S\indices{{}^{D}_{E}}\not{\mathscr{A}}{\indices{{}^{E}_{F}}}\times\not{\mathscr{A}}^{*}{\indices{{}^{F}_{4}}}\bigr{\\}}$
$\displaystyle\ \stackrel{{\scriptstyle\mathrm{ibp}}}{{=}}$
$\displaystyle-{\textstyle\frac{1}{16}}\mathscr{D}^{\Gamma}\mathcal{F}^{\Sigma\Theta}\mathrm{tr}\bigl{\\{}(\mathscr{C}_{\Sigma})\indices{{}^{4}_{B}}(\bar{\mathscr{C}}_{\Theta})\indices{{}^{B}_{C}}(\mathscr{C}_{\Gamma})\indices{{}^{C}_{D}}S\indices{{}^{D}_{E}}\not{\mathscr{A}}{\indices{{}^{E}_{F}}}\times\not{\mathscr{A}}^{*}{\indices{{}^{F}_{4}}}\bigr{\\}}$
$\displaystyle\ \stackrel{{\scriptstyle\eqref{eq:id_fa_3}}}{{=}}$
$\displaystyle-{\textstyle\frac{1}{16}}\mathscr{D}^{\Gamma}\mathcal{F}^{\Sigma\Theta}\mathrm{tr}\bigl{\\{}\bigl{[}-2(\mathscr{C}_{\Sigma})\indices{{}^{4}_{D}}\
\eta_{\Theta\Gamma}+(\mathscr{C}_{[\Sigma})\indices{{}^{4}_{B}}(\bar{\mathscr{C}}_{\Theta})\indices{{}^{B}_{C}}(\mathscr{C}_{\Gamma]})\indices{{}^{C}_{D}}\bigr{]}S\indices{{}^{D}_{E}}\not{\mathscr{A}}{\indices{{}^{E}_{F}}}\times\not{\mathscr{A}}^{*}{\indices{{}^{F}_{4}}}\bigr{\\}}$
$\displaystyle\ =$
$\displaystyle-{\textstyle\frac{1}{8}}\mathscr{D}_{\Theta}\mathcal{F}^{\Theta\Sigma}\mathrm{tr}\bigl{\\{}(\mathscr{C}_{\Sigma})\indices{{}^{4}_{D}}S\indices{{}^{D}_{E}}\not{\mathscr{A}}{\indices{{}^{E}_{F}}}\times\not{\mathscr{A}}^{*}{\indices{{}^{F}_{4}}}\bigr{\\}}\
,$

where in the last step we used the Bianchi identity
$\mathscr{D}^{[\Gamma}\mathcal{F}^{\Sigma\Theta]}=0$. This concludes the proof
for the special case $L=\operatorname{diag}(-1,-1,-1,+3)$ (and permutations
thereof). To reach the full Lie algebra we make use of the fact that we can
superimpose coupling flow operators with weight one, giving the Cartan
subalgebra and that $S_{g}^{\mathrm{b}}[\mathscr{A}]$ is invariant under
R-symmetry transformations $\mathscr{A}\rightarrow\mathscr{A}^{\prime}$. From

$0\ =\
(\partial_{g}+R_{g}[\mathscr{A}^{\prime}])S_{g}^{\mathrm{b}}[\mathscr{A}^{\prime}]\
=\ (\partial_{g}+R_{g}[\mathscr{A}^{\prime}])S_{g}^{\mathrm{b}}[\mathscr{A}]\
,$ (C.13)

we observe the transformed $R_{g}[\mathscr{A}^{\prime}]$ also satisfies the
infinitesimal free action condition, reaching all $L\ \in\ \mathfrak{su}(4)$.

Lastly, we prove (C.6) by explicitly checking the various possibilities of the
open indices. The easiest case is the one with only gamma matrices

${\textstyle\frac{1}{4}}\mathrm{tr}\bigl{\\{}(\mathscr{C}_{\mu})\indices{{}^{4}_{B}}(\bar{\mathscr{C}}_{\nu})\indices{{}^{B}_{C}}(\mathscr{C}_{\rho})\indices{{}^{C}_{D}}(\bar{\mathscr{C}}_{\sigma})\indices{{}^{D}_{4}}\bigr{\\}}\
=\
{\textstyle\frac{1}{4}}\mathrm{tr}\bigl{\\{}\gamma_{\mu}\gamma_{\nu}\gamma_{\rho}\gamma_{\sigma}\bigr{\\}}\
=\
\eta_{\mu\sigma}\eta_{\nu\rho}-\eta_{\mu\rho}\eta_{\nu\sigma}+\eta_{\mu\nu}\eta_{\rho\sigma}\
.$ (C.14)

Next, we consider the case when there are three gamma matrices (modulo chiral
projectors) in the trace, i.e. one of the four indices in the range 4 to 9 and
the three others in the range 0 to 3. In that case, the trace vanishes since
any trace over an odd number of gamma matrices vanishes and the r.h.s. of
(C.6) also clearly vanishes because in each term there is a Kronecker delta
that is zero. The next case is the one where two indices are in the range 0 to
3 and the other two indices are in the range 4 to 9. We have to distinguish
three arrangements of indices

$\displaystyle{\textstyle\frac{1}{4}}\mathrm{tr}\bigl{\\{}(\mathscr{C}_{3{+}i})\indices{{}^{4}_{B}}(\bar{\mathscr{C}}_{3{+}j})\indices{{}^{B}_{C}}(\mathscr{C}_{\mu})\indices{{}^{C}_{D}}(\bar{\mathscr{C}}_{\nu})\indices{{}^{D}_{4}}\bigr{\\}}$
$\displaystyle\ =\
{\textstyle\frac{1}{4}}\mathrm{tr}\bigl{\\{}(\mathscr{C}_{3{+}i})\indices{{}^{4}_{B}}(\bar{\mathscr{C}}_{3{+}j})\indices{{}^{B}_{4}}\gamma_{\mu}\gamma_{\nu}\bigr{\\}}$
(C.15) $\displaystyle\ =\ (t^{i})_{4J}(t^{j})^{J4}\
\mathrm{tr}\bigl{\\{}\mathrm{P}^{+}\gamma_{\mu}\gamma_{\nu}\bigr{\\}}+(t^{i})^{4J}(t^{j})_{J4}\
\mathrm{tr}\bigl{\\{}\mathrm{P}^{-}\gamma_{\mu}\gamma_{\nu}\bigr{\\}}$
$\displaystyle\ =\ -2[(t^{i})_{4J}(t^{j})^{J4}\eta_{\mu\nu}\ +\
\mathrm{c.c.}]\ =\ \delta_{ij}\eta_{\mu\nu}\ ,$
$\displaystyle{\textstyle\frac{1}{4}}\mathrm{tr}\bigl{\\{}(\mathscr{C}_{3{+}i})\indices{{}^{4}_{B}}(\bar{\mathscr{C}}_{\mu})\indices{{}^{B}_{C}}(\mathscr{C}_{3{+}j})\indices{{}^{C}_{D}}(\bar{\mathscr{C}}_{\nu})\indices{{}^{D}_{4}}\bigr{\\}}$
$\displaystyle\ =\
\mathrm{tr}\bigl{\\{}[(t^{i})_{4J}\mathrm{P}^{+}-(t^{i})^{4J}\mathrm{P}^{-}]\gamma_{\mu}[(t^{j})_{J4}\mathrm{P}^{+}-(t^{j})^{J4}\mathrm{P}^{-}]\gamma_{\nu}\bigr{\\}}$
(C.16) $\displaystyle\ =\
-(t^{i})_{4J}(t^{j})^{J4}\mathrm{tr}\bigl{\\{}\mathrm{P}^{+}\gamma_{\mu}\gamma_{\nu}\bigr{\\}}-(t^{i})^{4J}(t^{j})_{J4}\mathrm{tr}\bigl{\\{}\mathrm{P}^{-}\gamma_{\mu}\gamma_{\nu}\bigr{\\}}$
$\displaystyle\ =\ 2(t^{i})_{4J}(t^{j})^{J4}\eta_{\mu\nu}\ +\ \mathrm{c.c.}\
=\ -\delta_{ij}\eta_{\mu\nu}\ ,$
$\displaystyle{\textstyle\frac{1}{4}}\mathrm{tr}\bigl{\\{}(\mathscr{C}_{3{+}i})\indices{{}^{4}_{B}}(\bar{\mathscr{C}}_{\mu})\indices{{}^{B}_{C}}(\mathscr{C}_{\nu})\indices{{}^{C}_{D}}(\bar{\mathscr{C}}_{3{+}j})\indices{{}^{D}_{4}}\bigr{\\}}$
$\displaystyle\ =\
\mathrm{tr}\bigl{\\{}[(t^{i})_{4J}\mathrm{P}^{+}-(t^{i})^{4J}\mathrm{P}^{-}]\gamma_{\mu}\gamma_{\nu}[(t^{j})^{J4}\mathrm{P}^{+}-(t^{j})_{J4}\mathrm{P}^{-}]\bigr{\\}}$
(C.17) $\displaystyle\ =\ (t^{i})_{4J}(t^{j})^{J4}\
\mathrm{tr}\bigl{\\{}\mathrm{P}^{+}\gamma_{\mu}\gamma_{\nu}\bigr{\\}}+(t^{i})^{4J}(t^{j})_{J4}\
\mathrm{tr}\bigl{\\{}\mathrm{P}^{-}\gamma_{\mu}\gamma_{\nu}\bigr{\\}}$
$\displaystyle\ =\ -2[(t^{i})_{4J}(t^{j})^{J4}\eta_{\mu\nu}\ +\
\mathrm{c.c.}]\ =\ \delta_{ij}\eta_{\mu\nu}\ ,$

with all the other index configurations related to the three above by the
cyclicity of the trace. The trace with only one gamma matrix vanishes due to
the same reason as for three gamma matrices. We are left with the case

$\displaystyle{\textstyle\frac{1}{4}}\mathrm{tr}\bigl{\\{}(\mathscr{C}_{3{+}i})\indices{{}^{4}_{B}}(\bar{\mathscr{C}}_{3{+}j})\indices{{}^{B}_{C}}(\mathscr{C}_{3{+}k})\indices{{}^{C}_{D}}(\bar{\mathscr{C}}_{3{+}l})\indices{{}^{D}_{4}}\bigr{\\}}\
=$ $\displaystyle
4(t^{i}){\indices{{}_{4I}}}(t^{j}){\indices{{}^{IC}}}(t^{k}){\indices{{}_{CK}}}(t^{l}){\indices{{}^{K4}}}\
\mathrm{tr}\ \mathrm{P}^{+}$ (C.18) $\displaystyle+$ $\displaystyle
4(t^{i}){\indices{{}^{4I}}}(t^{j}){\indices{{}_{IC}}}(t^{k}){\indices{{}^{CK}}}(t^{l}){\indices{{}_{K4}}}\
\mathrm{tr}\ \mathrm{P}^{-}$ $\displaystyle\ =$ $\displaystyle 8\
(t^{i}){\indices{{}_{4I}}}(t^{j}){\indices{{}^{IC}}}(t^{k}){\indices{{}_{CK}}}(t^{l}){\indices{{}^{K4}}}\
+\ \mathrm{c.c.}$ $\displaystyle\ =$ $\displaystyle 8\
[(t^{i}){\indices{{}_{4I}}}(t^{j}){\indices{{}^{I4}}}(t^{k}){\indices{{}_{4K}}}(t^{l}){\indices{{}^{K4}}}$
$\displaystyle+(t^{i}){\indices{{}_{4I}}}(t^{j}){\indices{{}^{IJ}}}(t^{k}){\indices{{}_{JK}}}(t^{l}){\indices{{}^{K4}}}]\
+\ \mathrm{c.c.}$

The last expression can be evaluated with the explicit form of the Clebsch-
Gordon coefficients (2.27) and the identity

$\epsilon_{IJM}\epsilon^{MKL}\ =\
\delta\indices{{}_{I}^{K}}\delta\indices{{}_{J}^{L}}-\delta\indices{{}_{I}^{L}}\delta\indices{{}_{J}^{K}}\
.$ (C.19)

We do not quite find the desired result, because we obtain additional terms
when two of the indices $i,j,k,l$ are in the range 1 to 3 and the other two
are in the range 4 to 6. For example

${\textstyle\frac{1}{4}}\mathrm{tr}\bigl{\\{}(\mathscr{C}_{3{+}I})\indices{{}^{4}_{B}}(\bar{\mathscr{C}}_{6{+}J})\indices{{}^{B}_{C}}(\mathscr{C}_{3{+}K})\indices{{}^{C}_{D}}(\bar{\mathscr{C}}_{6{+}L})\indices{{}^{D}_{4}}\bigr{\\}}\
=\ \delta_{IL}\delta_{JK}-\delta_{IK}\delta_{JL}-\delta_{IJ}\delta_{KL}\ ,$
(C.20)

where only the second term on the r.h.s. would appear in the r.h.s. of (C.6).
However, we contract (C.6) with the $\varphi$’s in the adjoint representation
of the gauge group. It turns out that the additional terms are proportional to

$(\varphi_{I}{\times}\varphi_{J})\
(\varphi_{I+3}{\times}\varphi_{J+3})+(\varphi_{I}{\times}\varphi_{J+3})\
(\varphi_{J}{\times}\varphi_{I+3})+(\varphi_{I}{\times}\varphi_{I+3})\
(\varphi_{J+3}{\times}\varphi_{J})\ =\ 0\ ,$ (C.21)

i.e. vanish by the Jacobi identity in color space.

##  Appendix D Explicit computation of the Nicolai maps

We first collect a number of useful identities

$\displaystyle\sum_{B}\Phi\indices{{}^{A}_{B}}\times(\Phi\indices{{}^{B}_{A}})^{\dagger}\
=\
\left\\{\begin{array}[]{cl}-2\gamma_{5}\sum_{J}\varphi_{J}\times\varphi_{J+3}&\text{for}\quad
A\ =\ 4\\\
2\gamma_{5}\sum_{J}(-)^{\delta_{KJ}}\varphi_{J}\times\varphi_{J+3}&\text{for}\quad
A\ =\ K\\\ \end{array}\right.\ ,$ (D.1)
$\displaystyle\sum_{B}\Phi\indices{{}^{A}_{B}}S_{0}\Phi\indices{{}^{B}_{A}}\
=\
\left\\{\begin{array}[]{cl}\sum_{j}\varphi_{j}S_{0}\varphi_{j}-\gamma_{5}\sum_{J}(\varphi_{J+3}S_{0}\varphi_{J}-\varphi_{J}S_{0}\varphi_{J+3})&\text{for}\quad
A\ =\ 4\\\
\sum_{j}\varphi_{j}S_{0}\varphi_{j}+\gamma_{5}\sum_{J}(-)^{\delta_{KJ}}(\varphi_{J+3}S_{0}\varphi_{J}-\varphi_{J}S_{0}\varphi_{J+3})&\text{for}\quad
A\ =\ K\\\ \end{array}\right.\ ,$
$\displaystyle\sum_{B}(c^{I})\indices{{}^{A}_{B}}S_{0}\Phi\indices{{}^{B}_{A}}\
=\
\left\\{\begin{array}[]{cl}S_{0}\varphi_{I}+\gamma_{5}S_{0}\varphi_{I+3}&\text{for}\quad
A\ =\ 4\\\
S_{0}\varphi_{I}-\gamma_{5}S_{0}\varphi_{I+3}(-)^{\delta_{IK}}&\text{for}\quad
A\ =\ K\\\ \end{array}\right.\ ,$
$\displaystyle\sum_{B}(c^{I{+}3})\indices{{}^{A}_{B}}S_{0}\Phi\indices{{}^{B}_{A}}\
=\
\left\\{\begin{array}[]{cl}-\gamma_{5}S_{0}\varphi_{I}+S_{0}\varphi_{I+3}&\text{for}\quad
A\ =\ 4\\\
\gamma_{5}S_{0}\varphi_{I}(-)^{\delta_{IK}}+S_{0}\varphi_{I+3}&\text{for}\quad
A\ =\ K\\\ \end{array}\right.\ ,$

and

$\displaystyle\sum_{B,C,D}\mathrm{tr}\bigl{\\{}(c^{I})\indices{{}^{4}_{B}}S_{0}\Phi\indices{{}^{B}_{C}}S_{0}\Phi\indices{{}^{C}_{D}}\times(\Phi\indices{{}^{D}_{4}})^{\dagger}\bigr{\\}}$
$\displaystyle\ =\
\mathrm{tr}\Bigl{\\{}-2S_{0}\sum_{j}\varphi_{j}S_{0}\varphi_{j}\varphi_{I}$
(D.2) $\displaystyle-2S_{0}\sum_{J}$
$\displaystyle\Bigl{[}\varphi_{I+3}S_{0}\varphi_{J}\varphi_{J+3}-\varphi_{J}S_{0}\varphi_{I+3}\varphi_{J+3}+\varphi_{J+3}S_{0}\varphi_{I+3}\varphi_{J}\Bigr{]}\Bigr{\\}}\
,$
$\displaystyle\sum_{B,C,D}\mathrm{tr}\bigl{\\{}(c^{I})\indices{{}^{K}_{B}}S_{0}\Phi\indices{{}^{B}_{C}}S_{0}\Phi\indices{{}^{C}_{D}}\times(\Phi\indices{{}^{D}_{K}})^{\dagger}\bigr{\\}}$
$\displaystyle\ =\
\mathrm{tr}\Bigl{\\{}-2S_{0}\sum_{j}\varphi_{j}S_{0}\varphi_{j}\varphi_{I}$
(D.3) $\displaystyle-2S_{0}\sum_{J}$
$\displaystyle\Bigl{[}\varphi_{I+3}S_{0}\varphi_{J}\varphi_{J+3}-\varphi_{J}S_{0}\varphi_{I+3}\varphi_{J+3}+\varphi_{J+3}S_{0}\varphi_{I+3}\varphi_{J}\Bigr{]}(-)^{\delta_{IK}}$
$\displaystyle+4S_{0}$
$\displaystyle\Bigl{[}\varphi_{I+3}S_{0}\varphi_{K}\varphi_{K+3}-\varphi_{K}S_{0}\varphi_{I+3}\varphi_{K+3}+\varphi_{K+3}S_{0}\varphi_{I+3}\varphi_{K}\Bigr{]}\Bigr{\\}}\
,$

where in (D.2) and (D.3) half of the terms dropped out due to $\mathrm{tr}\
\gamma^{\mu}\gamma^{\nu}\gamma_{5}=0$. The analogous formulae with
$(c^{I{+}3})\indices{{}^{A}_{B}}$ are given by (D.2) and (D.3) with all
indices $I,J,K,...$ replaced by $I{+}3,J{+}3,K{+}3,...$ and vice versa so that
e.g.

$\displaystyle\sum_{B,C,D}\mathrm{tr}\bigl{\\{}(c^{I{+}3})\indices{{}^{4}_{B}}S_{0}\Phi\indices{{}^{B}_{C}}S_{0}\Phi\indices{{}^{C}_{D}}\times(\Phi\indices{{}^{D}_{4}})^{\dagger}\bigr{\\}}$
$\displaystyle\ =\
\mathrm{tr}\Bigl{\\{}-2S_{0}\sum_{j}\varphi_{j}S_{0}\varphi_{j}\varphi_{I+3}$
(D.4) $\displaystyle-2S_{0}\sum_{J}$
$\displaystyle\Bigl{[}\varphi_{I}S_{0}\varphi_{J+3}\varphi_{J}-\varphi_{J+3}S_{0}\varphi_{I}\varphi_{J}+\varphi_{J}S_{0}\varphi_{I}\varphi_{J+3}\Bigr{]}\Bigr{\\}}\
.$

To evaluate the traces, we need

$\displaystyle\mathrm{tr}\
\gamma_{5}\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}\gamma^{\sigma}$
$\displaystyle\ =\ -4\epsilon^{\mu\nu\rho\sigma}\ ,$ (D.5)
$\displaystyle\mathrm{tr}\
\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}\gamma^{\sigma}$ $\displaystyle\ =\
4(\eta^{\mu\nu}\eta^{\rho\sigma}-\eta^{\mu\rho}\eta^{\nu\sigma}+\eta^{\mu\sigma}\eta^{\nu\rho})\
,$ $\displaystyle\mathrm{tr}\
\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}\gamma^{\sigma}\gamma^{\lambda}\gamma^{\eta}$
$\displaystyle\ =\ -\eta^{\mu\nu}\ \mathrm{tr}\
\gamma^{\rho}\gamma^{\sigma}\gamma^{\lambda}\gamma^{\eta}+\eta^{\mu\rho}\
\mathrm{tr}\ \gamma^{\nu}\gamma^{\sigma}\gamma^{\lambda}\gamma^{\eta}\mp...\
.$

We now give some intermediate results for the various contributions to the
Nicolai maps.

First order: We have

$\stackrel{{\scriptstyle\leftarrow}}{{R_{1}}}{{}^{(A)}}\ =\
{\textstyle\frac{1}{8}}\stackrel{{\scriptstyle\longleftarrow}}{{{\textstyle\frac{\delta}{\delta\mathscr{A}_{\Gamma}}}}}\Pi\indices{{}_{\Gamma}^{\Sigma}}\
\mathrm{tr}\Bigl{\\{}(\mathscr{C}_{\Sigma})\indices{{}^{A}_{B}}S_{0}\not{\mathscr{A}}{\indices{{}^{B}_{C}}}\times\not{\mathscr{A}}^{*}{\indices{{}^{C}_{A}}}\Bigl{\\}}\
,$ (D.6)

where $\Pi\indices{{}_{\Gamma}^{\Sigma}}\ =\
\delta\indices{{}_{\Gamma}^{\Sigma}}-C\indices{{}_{\Gamma}^{\Sigma}}$ and
$\partial_{3+i}\equiv 0$. Further expanding

$\stackrel{{\scriptstyle\leftarrow}}{{R_{1}}}{{}^{(A)}}\ =\
{\textstyle\frac{1}{8}}\stackrel{{\scriptstyle\longleftarrow}}{{{\textstyle\frac{\delta}{\delta\mathscr{A}_{\Gamma}}}}}\Pi\indices{{}_{\Gamma}^{\Sigma}}\
\mathrm{tr}\Bigl{\\{}(\mathscr{C}_{\Sigma})\indices{{}^{A}_{B}}S_{0}\Bigl{[}\not{A}\times\not{A}\delta\indices{{}^{B}_{A}}+2\Phi\indices{{}^{B}_{A}}\times\not{A}+\Phi\indices{{}^{B}_{C}}\times(\Phi\indices{{}^{C}_{A}})^{\dagger}\Bigr{]}\Bigl{\\}}\
,$ (D.7)

we find

$R_{1}{{}^{(A)}}A_{\mu}\ =\
{\textstyle\frac{1}{8}}\Pi\indices{{}_{\mu}^{\nu}}\
\mathrm{tr}\Bigl{\\{}\gamma_{\nu}S_{0}\not{A}\times\not{A}\Bigr{\\}}\ =\
C^{\rho}A_{\mu}A_{\rho}\ ,\quad\forall\ A\ =\ 1,2,3,4\ ,$ (D.8)

where we used $(\mathscr{C}_{\nu})\indices{{}^{A}_{J}}\ =\ 0$, as well as
$\Phi\indices{{}^{A}_{J}}\times(\Phi\indices{{}^{J}_{A}})^{\dagger}\ \sim\
\gamma_{5}$ and $\mathrm{tr}\ {\gamma_{\nu}\gamma_{\rho}\gamma_{5}}\ =\ 0$.
The remaining part is

$R_{1}{{}^{(A)}}\varphi_{i}\ =\ {\textstyle\frac{1}{4}}\
\mathrm{tr}\Bigl{\\{}(c^{i})\indices{{}^{A}_{B}}S_{0}\Phi\indices{{}^{B}_{A}}\times\not{A}\Bigr{\\}}\
=\ C^{\rho}\varphi_{i}A_{\rho}\ ,\quad\forall\ A\ =\ 1,2,3,4\ ,$ (D.9)

where we used that a trace over an odd number of gamma matrices vanishes.

Second order: First, it is straightforward to obtain

$(R_{1}{{}^{(A)}})^{2}A_{\mu}\ =\
2C^{\rho}A_{[\mu}C^{\lambda}A_{\rho]}A_{\lambda}\ ,\quad\forall\ A\ =\
1,2,3,4\ ,$ (D.10) $(R_{1}{{}^{(A)}})^{2}\varphi_{i}\ =\
C_{\rho}\varphi_{i}C_{\lambda}A^{\rho}A^{\lambda}-C_{\rho}A^{\rho}C_{\lambda}\varphi_{i}A^{\lambda}\
\quad\forall\ A\ =\ 1,2,3,4\ .$ (D.11)

With the perturbative expansion of the covariant projector

$P\indices{{}_{\Gamma}^{\Sigma}}\ =\
\delta\indices{{}_{\Gamma}^{\Sigma}}-\mathscr{D}_{\Gamma}G\partial^{\Sigma}\
=\
\Pi\indices{{}_{\Gamma}^{\Theta}}\bigl{\\{}\delta\indices{{}_{\Theta}^{\Sigma}}-g\mathscr{A}_{\Theta}\sum_{k=0}^{\infty}(-g\partial{\cdot}A\;\;C)^{k}C\partial^{\Sigma}\bigr{\\}}\
,$ (D.12)

and the gaugino propagator (B.19), we further find

$\stackrel{{\scriptstyle\leftarrow}}{{R_{2}}}{{}^{(A)}}\ =\
-{\textstyle\frac{1}{8}}\stackrel{{\scriptstyle\longleftarrow}}{{{\textstyle\frac{\delta}{\delta\mathscr{A}_{\Gamma}}}}}\Pi\indices{{}_{\Gamma}^{\Sigma}}\
\mathrm{tr}\Bigl{\\{}(\mathscr{C}_{\Sigma}){\indices{{}^{A}_{B}}}S_{0}\not{\mathscr{A}}{\indices{{}^{B}_{C}}}S_{0}\not{\mathscr{A}}{\indices{{}^{C}_{D}}}\times\not{\mathscr{A}}^{*}{\indices{{}^{D}_{A}}}\Bigl{\\}}-{\textstyle\frac{1}{8}}\stackrel{{\scriptstyle\longleftarrow}}{{{\textstyle\frac{\delta}{\delta\mathscr{A}_{\Gamma}}}}}\Pi\indices{{}_{\Gamma}^{\Theta}}\mathscr{A}_{\Theta}S_{0}\partial^{\sigma}\mathrm{tr}\Bigl{\\{}\gamma_{\sigma}S_{0}\not{\mathscr{A}}{\indices{{}^{A}_{B}}}\times\not{\mathscr{A}}^{*}{\indices{{}^{B}_{A}}}\Bigl{\\}}\
.$ (D.13)

It is easy to see that the second term gives no contribution using
$\Phi{\indices{{}^{A}_{B}}}\times(\Phi{\indices{{}^{B}_{A}}})^{\dagger}\ \sim\
\gamma_{5}$ and symmetry. Evaluating the traces, one finds

$\displaystyle R_{2}{{}^{(4)}}A_{\mu}$ $\displaystyle\ =\
3C^{\rho}A^{\lambda}C_{[\mu}A_{\lambda}A_{\rho]}+2C^{\rho}A_{[\mu}C^{\lambda}A_{\rho]}A_{\lambda}+2C^{\rho}\varphi_{i}C_{[\rho}A_{\mu]}\varphi_{i}$
(D.14)
$\displaystyle\qquad+\Pi\indices{{}_{\mu}^{\nu}}\epsilon_{\nu\lambda\rho\sigma}\sum_{J=1}^{3}[C^{\lambda}\varphi_{J}C^{\rho}\varphi_{J+3}A^{\sigma}-C^{\lambda}\varphi_{J+3}C^{\rho}\varphi_{J}A^{\sigma}+C^{\lambda}A^{\rho}C^{\sigma}\varphi_{J+3}\varphi_{J}]\
,$ $\displaystyle R_{2}{{}^{(K)}}A_{\mu}$ $\displaystyle\ =\
3C^{\rho}A^{\lambda}C_{[\mu}A_{\lambda}A_{\rho]}+2C^{\rho}A_{[\mu}C^{\lambda}A_{\rho]}A_{\lambda}+2C^{\rho}\varphi_{i}C_{[\rho}A_{\mu]}\varphi_{i}$
(D.15)
$\displaystyle\qquad-\Pi\indices{{}_{\mu}^{\nu}}\epsilon_{\nu\lambda\rho\sigma}\sum_{J=1}^{3}(-)^{\delta_{KJ}}[C^{\lambda}\varphi_{J}C^{\rho}\varphi_{J+3}A^{\sigma}-C^{\lambda}\varphi_{J+3}C^{\rho}\varphi_{J}A^{\sigma}+C^{\lambda}A^{\rho}C^{\sigma}\varphi_{J+3}\varphi_{J}]\
,$

and

$\displaystyle R_{2}{{}^{(4)}}\varphi_{I}\ =$ $\displaystyle
C^{\rho}\varphi_{I}C^{\lambda}A_{\rho}A_{\lambda}-C_{\rho}A^{\rho}C_{\lambda}\varphi_{I}A^{\lambda}+2C^{[\rho}A^{\lambda]}C_{\rho}\varphi_{I}A_{\lambda}+C^{\rho}\varphi_{j}C_{\rho}\varphi_{I}\varphi_{j}$
(D.16) $\displaystyle+$
$\displaystyle{\textstyle\frac{1}{2}}\epsilon_{\mu\nu\rho\lambda}[C^{\mu}\varphi_{I+3}C^{\nu}A^{\rho}A^{\lambda}+2C^{\mu}A^{\nu}C^{\rho}\varphi_{I+3}A^{\lambda}]$
$\displaystyle+$ $\displaystyle
C_{\rho}\sum_{J=1}^{3}\bigl{[}\varphi_{I+3}C_{\rho}\varphi_{J+3}\varphi_{J}+\varphi_{J}C_{\rho}\varphi_{I+3}\varphi_{J+3}-\varphi_{J+3}\varphi_{I+3}\varphi_{J}\bigr{]}\
,$ $\displaystyle R_{2}{{}^{(K)}}\varphi_{I}\ =$ $\displaystyle
C^{\rho}\varphi_{I}C^{\lambda}A_{\rho}A_{\lambda}-C_{\rho}A^{\rho}C_{\lambda}\varphi_{I}A^{\lambda}+2C^{[\rho}A^{\lambda]}C_{\rho}\varphi_{I}A_{\lambda}+C^{\rho}\varphi_{j}C_{\rho}\varphi_{I}\varphi_{j}$
(D.17) $\displaystyle-$
$\displaystyle{\textstyle\frac{1}{2}}\epsilon_{\mu\nu\rho\lambda}(-)^{\delta_{IK}}[C^{\mu}\varphi_{I+3}C^{\nu}A^{\rho}A^{\lambda}+2C^{\mu}A^{\nu}C^{\rho}\varphi_{I+3}A^{\lambda}]$
$\displaystyle+$ $\displaystyle
C^{\rho}(-)^{\delta_{IK}}\sum_{J=1}^{3}\bigl{[}\varphi_{I+3}C_{\rho}\varphi_{J+3}\varphi_{J}+\varphi_{J}C_{\rho}\varphi_{I+3}\varphi_{J+3}-\varphi_{J+3}C_{\rho}\varphi_{I+3}\varphi_{J}\bigr{]}$
$\displaystyle-$ $\displaystyle
2C^{\rho}\bigl{[}\varphi_{I+3}C_{\rho}\varphi_{K+3}\varphi_{K}+\varphi_{K}C_{\rho}\varphi_{I+3}\varphi_{K+3}-\varphi_{K+3}C_{\rho}\varphi_{I+3}\varphi_{K}\bigr{]}\
,$ $\displaystyle R_{2}{{}^{(4)}}\varphi_{I+3}\ =$ $\displaystyle
C^{\rho}\varphi_{I+3}C^{\lambda}A_{\rho}A_{\lambda}-C_{\rho}A^{\rho}C_{\lambda}\varphi_{I+3}A^{\lambda}+2C^{[\rho}A^{\lambda]}C_{\rho}\varphi_{I+3}A_{\lambda}+C^{\rho}\varphi_{j}C_{\rho}\varphi_{I+3}\varphi_{j}$
(D.18) $\displaystyle-$
$\displaystyle{\textstyle\frac{1}{2}}\epsilon_{\mu\nu\rho\lambda}[C^{\mu}\varphi_{I}C^{\nu}A^{\rho}A^{\lambda}+2C^{\mu}A^{\nu}C^{\rho}\varphi_{I}A^{\lambda}]$
$\displaystyle-$ $\displaystyle
C_{\rho}\sum_{J=1}^{3}\bigl{[}\varphi_{I}C_{\rho}\varphi_{J+3}\varphi_{J}+\varphi_{J}C_{\rho}\varphi_{I}\varphi_{J+3}-\varphi_{J+3}\varphi_{I}\varphi_{J}\bigr{]}\
,$ $\displaystyle R_{2}{{}^{(K)}}\varphi_{I+3}\ =$ $\displaystyle
C^{\rho}\varphi_{I+3}C^{\lambda}A_{\rho}A_{\lambda}-C_{\rho}A^{\rho}C_{\lambda}\varphi_{I+3}A^{\lambda}+2C^{[\rho}A^{\lambda]}C_{\rho}\varphi_{I+3}A_{\lambda}+C^{\rho}\varphi_{j}C_{\rho}\varphi_{I+3}\varphi_{j}$
(D.19) $\displaystyle+$
$\displaystyle{\textstyle\frac{1}{2}}\epsilon_{\mu\nu\rho\lambda}(-)^{\delta_{IK}}[C^{\mu}\varphi_{I}C^{\nu}A^{\rho}A^{\lambda}+2C^{\mu}A^{\nu}C^{\rho}\varphi_{I}A^{\lambda}]$
$\displaystyle-$ $\displaystyle
C_{\rho}(-)^{\delta_{IK}}\sum_{J=1}^{3}\bigl{[}\varphi_{I}C_{\rho}\varphi_{J+3}\varphi_{J}+\varphi_{J}C_{\rho}\varphi_{I}\varphi_{J+3}-C^{\rho}\varphi_{J+3}\varphi_{I}\varphi_{J}\bigr{]}$
$\displaystyle+$ $\displaystyle
2\bigl{[}\varphi_{I}C_{\rho}\varphi_{K+3}\varphi_{K}+\varphi_{K}C_{\rho}\varphi_{I}\varphi_{K+3}-C^{\rho}\varphi_{K+3}\varphi_{I}\varphi_{K}\bigr{]}\
,$

for $K=1,2,3$.

##  Appendix E Testing the conditions of the Nicolai maps

We explicitly check the three conditions for the four distinct Nicolai maps
(4.8)-(4.11).

Check of the gauge condition: The (Landau) gauge condition
$\partial^{\mu}T_{g}A_{\mu}=\partial^{\mu}A_{\mu}+\mathcal{O}(g^{3})$ follows
from symmetry and $\partial^{\mu}\Pi\indices{{}_{\mu}^{\nu}}=0$.

Check of the free-action condition: The maps have to satisfy

$S_{0}[A^{\prime}_{\mu},\varphi^{\prime}_{I},\varphi^{\prime}_{I+3}]\ =\
S^{\mathrm{b}}_{g}[A_{\mu},\varphi_{I},\varphi_{I+3}]\ ,$ (E.1)

with the bosonic action

$\displaystyle S^{\mathrm{b}}_{g}\ =\
\int\mathrm{d}^{4}x\;\bigl{\\{}-{\textstyle\frac{1}{4}}F_{\mu\nu}F^{\mu\nu}-{\textstyle\frac{1}{2}}\mathrm{D}_{\mu}\varphi_{i}\mathrm{D}^{\mu}\varphi_{i}-{\textstyle\frac{g^{2}}{4}}(\varphi_{i}\times\varphi_{j})^{2}\bigr{\\}}\
,$ (E.2) $\displaystyle F_{\mu\nu}\ =\
\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}+gA_{\mu}\times A_{\nu}\ ,$