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\begin{document}

\begin{titlepage}
\begin{flushright}
{\small $\,$}
\end{flushright}
\vskip 1cm
\centerline{\huge{\bf{ A perturbative expansion scheme}}}
\vskip 0.4cm
\centerline{\huge{\bf{for supermembrane and matrix theory}}}
\vskip 1.5cm
\begin{center}
\begingroup\scshape\Large
Olaf Lechtenfeld$^{\,\diamond}$ and
Hermann Nicolai$^{\,\star}$
\endgroup
\end{center}
\centerline{{\Large ${\,}^\diamond$}\it {Institut f\"ur Theoretische Physik and Riemann Center for Geometry and Physics}}
\centerline {\it {Leibniz Universit\"at Hannover, Appelstrasse 2, 30167 Hannover, Germany}}
\vskip 0.5cm
\centerline{{\Large ${\,}^\star$}\it {Max-Planck-Institut f\"ur Gravitationsphysik (Albert-Einstein-Institut)}}
\centerline {\it {Am M\"{u}hlenberg 1, 14476 Potsdam, Germany}}
\vskip 1.5cm
\centerline{\bf {Abstract}}
\vskip .5cm
\noindent
We reconsider the supermembrane in a Minkowski background and in the light-cone gauge
as a one-dimensional gauge theory of area preserving diffeomorphisms (APDs).
Keeping the membrane tension~$T$ as an independent parameter we show
that $T$ is proportional to the gauge coupling~$g$ of this gauge theory,
such that the small (large) tension limit of the supermembrane
corresponds to the weak (strong) coupling limit of the APD gauge theory
and its SU$(N)$ matrix model approximation. A perturbative linearization
of the supersymmetric theory suitable for a quantum mechanical path-integral treatment
can be achieved by formulating a Nicolai map for the matrix model,
which we work out explicitly to~${\cal O}(g^4)$. The corresponding formul\ae\ remain
well-defined in the limit $N\rightarrow\infty$, {\em i.e.}~for the supermembrane theory itself.
Furthermore we show that the map has improved convergence properties in
comparison with the usual perturbative expansions because its Jacobian
admits an expansion in~$g$ with a non-zero radius of convergence. Possible implications
for unsolved issues with the matrix model of M~theory are also mentioned.
\vfill
\end{titlepage}

\section{Introduction and summary}

The maximally supersymmetric supermembrane theory in spacetime
dimension $D=11$~\cite{BST,BST1} is a model `beyond' string theory that also
incorporates $D=11$ supergravity~\cite{CJS}, and is thus a candidate theory for a
non-perturbative formulation of superstring theory (besides there are three
more classically consistent supermembrane theories for target-space dimensions
$D=4,5,7$ \cite{BST1}). As shown long ago~\cite{dWHN} the supermembrane in
a flat (Minkowski) background and in the light-cone gauge can be reformulated
as a one-dimensional maximally supersymmetric gauge theory of area preserving
diffeomorphisms (APDs). Building on earlier results of~\cite{G,Hoppe} it has been shown
that this model can be equivalently obtained as the $N\rightarrow\infty$ limit of a maximally
supersymmetric SU$(N)$ matrix model~\cite{dWHN}. Much later the very same model
was re-interpreted in terms of D0 particle quantum mechanics~\cite{Witten}, and proposed
as a model of M theory in~\cite{BFSS}. For reviews of supermembrane theory
with many further references, see {\em e.g.}~\cite{Duff,HN,deWit}.

The main unsolved problem of (super-)membrane theory is its quantization.
Unlike for string theory there exists no gauge which linearizes the equations
of motion such that the determination of quantum correlators can be effectively
reduced to free field theory computations. Likewise, in view of the non-linearities
a covariant path-integral approach {\em \`a la\/} Polyakov appears hopeless for either
the bosonic or the supersymmetric membrane. A more realistically feasible
approach is based on (target-space) light-cone gauge quantization. Nevertheless,
even with this preferred gauge choice
the solution to the problem of quantization has so far remained elusive not only
because one has to deal with a fully interacting theory on the world volume,
but also because it is not obvious how to set up a perturbative expansion for the
quantized supermembrane. These difficulties are mirrored by corresponding difficulties
of the supersymmetric SU$(N)$ matrix model for $N < \infty$, as a consequence of which
key issues remain unresolved to this day.  Apart from questions regarding the
existence and  properties of the $N\rightarrow\infty$ limit for the quantized theory, there are
two main issues. One concerns the target-space Lorentz invariance of the
quantized supermembrane (or the matrix model in the $N\rightarrow\infty$ limit).
For the classical theory and for finite~$N$, Lorentz invariance is in fact violated,
but  can be recovered in the $N\rightarrow\infty$ limit \cite{dWMN,EMM}. However, there
has been almost no progress on the quantized theory, which would first of
all require a proper definition of the quantized Lorentz generators and ensuring
their quantum consistency, before actually checking the Lorentz algebra.
Amongst other things this involves the correct definition of the light-cone
target-space coordinate~$X^-$ (which matrix theory by itself `does not know about')
as a quantum operator. Consequently, it also remains an open question whether
$D_{\rm crit}=11$ is indeed the critical dimension for the supermembrane, eliminating the
other classically consistent theories (see however~\cite{MKS} for some early results
in this direction), unlike for the superstring where the well-known result $D_{\rm crit} = 10$
can be  established in more than one way.

A second key issue arises in connection with correlators and scattering amplitudes for
the putative massless supermembrane excitations corresponding to the graviton,
the gravitino and the 3-form field of $D=11$ supergravity. In particular, there is
no (super-)membrane analog known of the Veneziano and Virasoro--Shapiro
amplitudes, as this would almost certainly involve higher order and non-perturbative
contributions beyond the reach of conventional string technology. Remarkably, there do exist
classical candidate expressions for vertex operators associated to these states~\cite{DNP},
but like for the Lorentz generators, it has not been possible so far to turn
them into well-defined quantum objects.

In this paper we wish to tackle the quantization of the supermembrane
from a new and different perspective. A main ingredient here is
the fact that the supersymmetric APD gauge theory ({\em alias\/} the
supersymmetric SU$(\infty)$ matrix model) {\em is\/} the supermembrane.
Our analysis leads us to the conclusion that the membrane tension~$T$, made dimensionless,
must be identified with the gauge coupling~$g$ of the APD gauge theory.
This insight allows us to set up a systematic expansion scheme
in terms of a path-integral formulation  where the small (large) tension limit
of the membrane theory corresponds to the weak (strong) coupling limit
of the APD gauge theory -- a result to be contrasted with the
somewhat murky state of affairs with the zero-tension limit of string theory.
This expansion is introduced by means of a Nicolai map \cite{Nic0,Nic1}
(designated by ${\cal T}_g$)  which we first construct for the finite-$N$ theory
up to and including quartic order ${\cal O}(g^4)$.\footnote{
In the temporal gauge and actually including $O(g^5)$
since odd orders vanish at least up to this point.}
Our derivation is based on a
systematic procedure that relies on very recent progress in perturbatively
evaluating this map for supersymmetric Yang--Mills theories in higher dimensions,
see especially \cite{MN,LR2}. This prescription in principle allows for the determination
of the map to {\em any\/} desired order.  It then turns out
that the pertinent formul\ae\ all remain well-defined in the limit $N\rightarrow\infty$,
via the straightforward replacement of SU($N$) commutators by APD brackets,
see especially (\ref{TXAPD}).\footnote{
An accompanying $N$-dependent rescaling cancels only for the supermembrane,
indicating that the $N\rightarrow\infty$ limit does not exist for the ordinary membrane.}
These are two main results of this paper,
which should eventually permit setting up an approximation scheme also for
correlators and other quantities of physical interest. This might enable one to sidestep
the finite-$N$ approximation altogether and to deal directly with the limiting theory
for $N=\infty$. Finally, we demonstrate that the expansion of the Jacobian in
powers of~$g$ has a {\em non-zero\/} radius of convergence. This does not
yet mean that the map itself has this property, but it represents strong evidence
that the expansion is indeed better behaved than the usual perturbation
expansions, and that it may likewise have a non-zero radius of convergence.

The structure of this paper is as follows. In Section~2 we review basic results
on the light-cone gauge formulation of the supermembrane, mostly following
the exposition in~\cite{dWHN}, and set up the path integral in Subsection~2.3.
Section~3 explains the construction of the Nicolai map both for the matrix theory and
the APD gauge theory. This section contains our main result, namely an explicit
form of the map~${\cal T}_g$ in an expansion to quartic order in the coupling~$g$.
The question of the behavior of the map in the complex $g$~plane (and thus
the convergence properties of the expansion) is addressed in Section~4.
Finally an Appendix provides details of two consistency checks on the main
result derived in this paper.

\section{Supermembrane basics}

In the section we closely follow~\cite{dWHN} to which we refer for further details of
the derivation. The main difference is that we here keep the membrane tension~$T$
as an independent parameter, in order to expose the link with weakly and strongly
coupled Yang--Mills theory.

\subsection{Supermembrane in the light-cone gauge with variable tension}

Classically consistent supermembranes exist for target-space dimensions
$D=4,5,7$ and 11, but in the remainder we will restrict attention
mostly to the maximally supersymmetric case, for which $D=11$.
The target superspace coordinates
$\big\{X^\mu, \theta} \def\tb{\bar\theta\big\}  \equiv \big\{ X^\mu (\xi^i)\,,\, \theta} \def\tb{\bar\theta(\xi^i) \big\}$
with the range $\mu,\nu,\ldots=0,1,\ldots,D-1 = 10$
are then functions of the membrane  world-volume coordinates
\begin{equation}
(\xi^i) \equiv (\tau,\boldsymbol{\sigma}) \equiv (\tau,\sigma^r)
\qquad\textrm{where}\quad i,j,\ldots=0,1,2 \quad\textrm{and}\quad r,s,\ldots=1,2\ .
\end{equation}
$\theta} \def\tb{\bar\theta(\tau,\boldsymbol{\sigma})$ is a real 32-component Majorana spinor of SO(1,10) (we usually do
not write out spinor indices). The target-space vielbein is
\begin{equation}
E_i{}^\mu\ =\ \partial_i X^\mu + \bar \theta} \def\tb{\bar\theta\,\Gamma^\mu \partial_i \theta} \def\tb{\bar\theta \ .
\end{equation}
For $D=11$, the real 32-by-32 $\Gamma$-matrices generate the SO(1,10) Clifford algebra,
$\{\Gamma^\mu , \Gamma^\nu \} = 2\eta^{\mu\nu}$. The world-volume metric is
\begin{equation}
{\tt g}_{ij}\ \equiv\ E_i{}^\mu E_j{}^\nu \eta_{\mu\nu}\ .
\end{equation}
For the light-cone gauge we split the target-space coordinates as
\begin{equation}
\{ X^\mu \} \equiv \{ X^+,X^-,X^a \}  \qquad\quad \textrm{with}\qquad
X^\pm = \tfrac1{\sqrt{2}} (X^{10} \pm X^0) \qquad\textrm{and}\qquad
\{ X^a\} \equiv {\bf{X}}
\end{equation}
being the transverse components $( a,b,\ldots=1,\ldots,9)$.
We adopt the target-space light-cone gauge
\begin{equation}\label{LCG}
X^+(\tau,\boldsymbol{\sigma}) \ =\  X_0^+ \,+\, \tau \ , \qquad \Gamma_+ \theta} \def\tb{\bar\theta(\tau,\boldsymbol{\sigma})\ =\ 0
\end{equation}
thus identifying the target-space light-cone coordinate $X^+$ with the world-volume time
coordinate $\tau$. With these gauge choices the induced metric on the three-dimensional world
volume is
\begin{equation}\label{3metric}
\begin{aligned}
{\tt g}_{rs} &\ \equiv\ {\bar\gg}_{rs}\ =\ \partial_r {\bf{X}} \cdot \partial_s {\bf{X}} \ ,\\
{\tt g}_{0r} &\ \equiv\ u_r\ =\ \partial_r X^- + \partial_0 {\bf{X}}\cdot\partial_r {\bf{X}} + \bar\theta} \def\tb{\bar\theta\,\Gamma_- \partial_r \theta} \def\tb{\bar\theta \ ,\\
{\tt g}_{00} &\ =\ 2\partial_0 X^- + (\partial_0{\bf{X}})^2 + 2\bar\theta} \def\tb{\bar\theta\,\Gamma_- \partial_0 \theta} \def\tb{\bar\theta\ .
\end{aligned}
\end{equation}
The metric determinant is
\begin{equation}
{\tt g} \ \equiv\, {\rm det\,} {\tt g}_{ij} \ =\  - \Delta\,{\bar\gg}
\end{equation}
with
\begin{equation}
{\bar\gg}\ \equiv\ {\rm det\,} {\bar\gg}_{rs} \qquad\textrm{and}\qquad
\Delta\ =\ - {\tt g}_{00} + u_r {\bar\gg}^{rs} u_s  \ ,\qquad {\bar\gg}^{rs} {\bar\gg}_{st} = \delta} \def\dd{\dot{\d}^r_t\ .
\end{equation}
The supermembrane Lagrangian then becomes
\begin{equation}
{\cal L} \ =\  T\,\Bigl(- \sqrt{{\bar\gg}\,\Delta} \,+\, \epsilon} \def\he{\hat{\epsilon}^{rs} \partial_r X^a \bar\theta} \def\tb{\bar\theta\,\Gamma_- \Gamma_a \partial_s \theta} \def\tb{\bar\theta\Bigr)\ ,
\end{equation}
where we now include the membrane tension $T$ as an independent parameter.
In principle the membrane tension is of dimension [mass]$^3$, but we
here find it convenient to render all variables dimensionless by rescaling them
with appropriate powers of some reference mass scale (as was already
implicitly assumed in (\ref{LCG})). This reference scale has no physical
meaning in and by  itself, as a proper identification of the gravitational
coupling (Newton constant or Planck mass) and evaluating its relation to $T$
will require the evaluation of a graviton scattering amplitude,
as is the case in string theory. However, for the doubly
dimensionally reduced supermembrane~\cite{Duff1} such a
relation can indeed be established by noting that $T R_{10} = (\alpha')^{-1}$, where
$R_{10}$ is the radius of the compactified 11th dimension. Because the latter is related
to the string coupling by $R_{10} = g_s^{2/3}$~\cite{Witten1} (see also~\cite{T}),
we see that
\begin{equation}
T\ =\ g_s^{-2/3}\,(\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A}')^{-1} \ .
\end{equation}
In this way the parameter $T$ ties together the two key parameters of string theory,
and thus also with the APD gauge coupling  via (\ref{T=g}) below.

With these conventions the (dimensionless) canonical momenta are
\begin{equation}\label{momenta}
\begin{aligned}
P^+ &\ =\ T \sqrt{\frac{{\bar\gg}}{\Delta}}\ ,    \\
{\bf{P}} &\ =\ \frac{\delta} \def\dd{\dot{\d}{\cal L}}{\delta} \def\dd{\dot{\d} \partial_0{\bf{X}}}\ =\ T \sqrt{\frac{{\bar\gg}}{\Delta}}
\big( \partial_0 {\bf{X}} - u_r {\tt g}^{rs} \partial_s {\bf{X}} \big) \ \equiv\
P^+ \big( \partial_0 {\bf{X}} - u_r {\tt g}^{rs} \partial_s {\bf{X}}\big)\ ,  \\
S &\ =\  \frac{\delta} \def\dd{\dot{\d}{\cal L}}{\delta} \def\dd{\dot{\d} \partial_0\bar\theta} \def\tb{\bar\theta} \ =\
- T \sqrt{\frac{{\bar\gg}}{\Delta}}  \Gamma_- \theta} \def\tb{\bar\theta\ \equiv\ - P^+ \Gamma_- \theta} \def\tb{\bar\theta\ .
\end{aligned}
\end{equation}
The last formula implies a second-class constraint (entailing the replacement of Poisson
brackets by Dirac brackets). The formul\ae~(\ref{momenta}) imply the first-class constraint
\begin{equation}
\phi_r \ =\  {\bf{P}}\cdot \partial_r {\bf{X}} \,+\, P^+ \partial_r X^- \,+\, \bar{S} \partial_r \theta} \def\tb{\bar\theta \ \approx\ 0\ ,
\end{equation}
which generates spatial diffeomorphisms on the membrane. This gauge freedom can
be exploited to set $u^r = 0$ in (\ref{3metric}), which in turn implies
\begin{equation}
\partial_r X^- \ =\ - \partial_0 {\bf{X}}\cdot\partial_r {\bf{X}} \,-\, \bar\theta} \def\tb{\bar\theta\,\Gamma_- \partial_r \theta} \def\tb{\bar\theta  \ .
\end{equation}
To be able to solve this equation for $X^-$ we must impose the integrability constraint
\begin{equation}\label{APD0}
\phi \ \equiv \  \epsilon} \def\he{\hat{\epsilon}^{rs} \big( \partial_r \partial_0 {\bf{X}}\cdot \partial_s {\bf{X}} \,+\, \partial_r \bar\theta} \def\tb{\bar\theta\,\Gamma_- \partial_s\theta} \def\tb{\bar\theta\big)
\ \approx \  0\ .
\end{equation}
This constraint generates APDs on the membrane: while general (spatial) diffeomorphisms
on the membrane are generated by vector fields $\delta} \def\dd{\dot{\d}\xi^r(\boldsymbol{\sigma})$, APDs are generated by
divergence-free vector fields obeying $\partial_r (\sqrt{w}\delta} \def\dd{\dot{\d}\xi^r) = 0$ (where the reference
density ${\sqrt{w(\boldsymbol{\sigma})}}$ coincides with the one introduced in (\ref{P+}) below). The latter are
locally of the form $\sqrt{w} \delta} \def\dd{\dot{\d}\xi^r = \epsilon} \def\he{\hat{\epsilon}^{rs}\partial_s \delta} \def\dd{\dot{\d}\xi$ with a scalar
parameter $\delta} \def\dd{\dot{\d}\xi(\boldsymbol{\sigma})$. On higher-genus
membranes there are in addition topologically non-trivial diffeomeophisms formally generated
by harmonic vector fields~\cite{dWMN}, and related to global diffeomeorphisms not contractible
to the identity, which we will, however, disregard here.

With these gauge choices the (dimensionless) Hamiltonian density becomes (see also~\cite{Tanii})
\begin{equation}\label{M2}
\begin{aligned}
{\cal H}(\boldsymbol{\sigma}) \ &\equiv \  - P^-(\boldsymbol{\sigma}) \ =\ {\bf{P}}\cdot \partial_0 {\bf{X}} \,+\, P^+ \partial_0 X^-\, +\, \bar{S}\partial_0 \theta} \def\tb{\bar\theta
\,-\, {\cal L} \\
&=\ \frac{{\bf{P}}^2 \,+\, T^2 {\bar\gg}}{2P^+} \ -\  T\epsilon} \def\he{\hat{\epsilon}^{rs} \partial_r X^a \bar\theta} \def\tb{\bar\theta\,\Gamma_- \Gamma_a \partial_s \theta} \def\tb{\bar\theta
\end{aligned}
\end{equation}
whose bosonic part was already derived long ago in~\cite{G,Hoppe} (for $T=1$).
Here we see why we must choose the membrane tension  to be positive;
flipping the sign of~$T$ will change
the sign of the kinetic part of the Hamiltonian by~(\ref{momenta}),
hence result in an instability. This is, of course, in accord with expectations.

Because $P^+(\tau,\boldsymbol{\sigma})$ obeys the Hamiltonian equation of motion ${\partial}_\tau P^+ (\tau,\boldsymbol{\sigma}) = 0$
and transforms as a density we can set~\cite{dWHN}
\begin{equation}\label{P+}
P^+(\tau, \boldsymbol{\sigma})\ =\  P^+_0 {\sqrt{w(\boldsymbol{\sigma})}}
\end{equation}
where $P^+_0>0 $ is constant, and ${\sqrt{w(\boldsymbol{\sigma})}} >0$ is a reference density
normalized to $\int\!\mathrm{d}^2 \sigma \, {\sqrt{w(\boldsymbol{\sigma})}} = 1$ (with an associated
reference metric $w_{rs}(\boldsymbol{\sigma})$ on the membrane, which is however only needed when
discussing target-space Lorentz invariance~\cite{dWMN}).
This leads to the (dimensionless) mass operator
\begin{equation}\label{M1}
{\cal M}^2 \ =\  - 2 P^+_0 P^-_0  - {\bf{P}}^2_0 \ = \
\int\!\mathrm{d}^2\sigma \ \left( [{\bf{P}}^2]' \,+\, T^2 {\bar\gg}  \,-\,
2T\epsilon} \def\he{\hat{\epsilon}^{rs} \partial_r X^a \bar\Theta\,\Gamma_- \Gamma_a \partial_s \Theta \right)
\end{equation}
with $P^-_0 = \int\!\mathrm{d}^2 \sigma\,P^-(\boldsymbol{\sigma})$ and rescaled fermionic variables~\footnote{
Which obey the canonical (Dirac)  brackets
$\{ \Theta(\boldsymbol{\sigma}) \,,\, \bar\Theta(\boldsymbol{\sigma}')\}_{\rm DB} = (4{\sqrt{w(\boldsymbol{\sigma})}})^{-1} \Gamma_+ \delta} \def\dd{\dot{\d}^{(2)}(\boldsymbol{\sigma}{,}\boldsymbol{\sigma}')$ \cite{dWHN}.}
\begin{equation}
\Theta (\boldsymbol{\sigma})\ \equiv \  \sqrt{P^+_0}  \, \theta} \def\tb{\bar\theta(\boldsymbol{\sigma})\ .
\end{equation}
The prime in (\ref{M1}) indicates that zero modes have been removed
from $\int\!\mathrm{d}^2\sigma \, {\bf{P}}^2(\boldsymbol{\sigma})$.

Finally we note that the fulfilment of the constraint (\ref{APD0}) allows us
to solve for the target-space coordinate $X^-$: we have
\begin{equation}
X^-(\tau, \boldsymbol{\sigma}) \ =\ - \int\!\mathrm{d}^2\sigma' \
G^r(\boldsymbol{\sigma},\boldsymbol{\sigma}') \Big( \partial_0 {\bf{X}}\cdot\partial_r {\bf{X}}(\tau,\boldsymbol{\sigma}')  \,-\, \bar\theta} \def\tb{\bar\theta\,\Gamma_- \partial_r \theta} \def\tb{\bar\theta(\tau,\boldsymbol{\sigma}')\Big)
\end{equation}
with a suitable Green's function obeying ${\partial}_r G^r(\boldsymbol{\sigma},\boldsymbol{\sigma}') = \delta} \def\dd{\dot{\d}(\boldsymbol{\sigma}{,}\boldsymbol{\sigma}')$
\cite{G,dWMN}. This formula is needed for the target-space boost
generators and for the verification of target-space Lorentz invariance
in the classical limit~\cite{dWMN,EMM}. It is worth  pointing out that this
information is {\em not} available in the matrix model as such, where the Lorentz
boost generators must either be ``guessed" or deduced from the supermembrane
matrix-model correspondence, as in~\cite{dWMN}.

\subsection{APD gauge theory and matrix model}

With the above formula for the mass operator the supermembrane theory can be
reformulated as a one-dimensional supersymmetric gauge theory of area preserving diffeomorphisms~\cite{dWHN}.
This can be seen by exploiting the algebraic identity
\begin{equation}
{\bar\gg} \ =\  {\rm det\,} \big( \partial_r{\bf{X}}\cdot \partial_s{\bf{X}} \big)\ =\
  \big\{ X^a, X^b\big\}  \big\{ X^a, X^b\big\}\ ,
\end{equation}
where the APD bracket of any two functions $A(\boldsymbol{\sigma})$ and $B(\boldsymbol{\sigma})$ on the membrane is defined by
\begin{equation}\label{APDbracket}
\big\{ A(\boldsymbol{\sigma})\, , \, B(\boldsymbol{\sigma}) \big\} \ :=\ \frac1{{\sqrt{w(\boldsymbol{\sigma})}}}\,\epsilon} \def\he{\hat{\epsilon}^{rs} \partial_r A(\boldsymbol{\sigma})  \partial_s B(\boldsymbol{\sigma})\ .
\end{equation}
This is indeed a Lie bracket (obeying antisymmetry and the Jacobi identity)~\cite{G,Hoppe}.

Then (\ref{M2}) can be equivalently obtained from the supersymmetric
Lagrangian~\footnote{
While $\tau$ is the time coordinate on the membrane world-volume,
we denote the Yang--Mills time coordinate by $t$, but keep the erstwhile membrane
coordinates $\sigma^r$ as labels for the APD gauge group.}
\begin{equation}\label{LYM}
\tfrac1{\sqrt{w}} \,{\cal L} \ =\
\tfrac12 (D_t {\bf{X}})^2 \ +\ \bar\Theta\,\Gamma_- D_t\Theta \ -\
\tfrac14 g^2 \{ X^a,X^b\}^2\ +\  g\,\bar\Theta\,\Gamma_- \Gamma_a \{X^a , \Theta\}
\end{equation}
if we identify
\begin{equation}\label{T=g}
T \ =\ g \ .
\end{equation}
In view of our comments after (\ref{M2}) we must, however, restrict this identification
to positive values of $T$ and $g$, even though there appear to be no obstructions
to continuing the APD gauge theory to negative couplings.
Hence the small (large) tension limit of the supermembrane corresponds
to the weak (strong) coupling limit of the supersymmetric APD gauge theory.
The APD covariant derivative is given by
\begin{equation}
D_t f (t,\boldsymbol{\sigma}) \ := \ \partial_t f (t,\boldsymbol{\sigma})  \,+\, g\,\big\{ \omega (t,\boldsymbol{\sigma}) \, ,\,  f(t,\boldsymbol{\sigma}) \big\}
\end{equation}
with the APD gauge field $\omega(t,\boldsymbol{\sigma})$ which is here introduced {\em ad hoc}, as it is
absent from the supermembrane action. The Lagrangian (\ref{LYM}) is nothing but the
dimensional reduction of maximally extended super-Yang--Mills theory~\cite{SYM1} to one (time)
dimension, with the identifications $\omega \equiv A_0$ and $X_a \equiv A_a$, and $g$
the usual Yang--Mills  coupling, but now with the infinite-dimensional APD gauge group.
This works precisely in the dimensions where pure supersymmetric Yang--Mills theories
exist, namely $D=3,4,6,10$ \cite{SYM1}, in agreement with the admissible target-space
dimensions 4,5,7 and 11 for supermembranes.

The group of APDs on the membrane can be approximated by the finite-dimensional
unitary groups SU($N$), such that the full group of APDs is recovered in the limit
$N\rightarrow\infty$ \cite{G,Hoppe}. Replacing APDs by SU($N$) gives the matrix model of M theory.
For this approximation one expands all functions on the membrane into a complete
orthonormal set of functions $Y^A(\boldsymbol{\sigma})$,
\begin{equation}
 \int\!\mathrm{d}^2\sigma \ {\sqrt{w(\boldsymbol{\sigma})}}\,Y^A(\boldsymbol{\sigma})\,Y^B(\boldsymbol{\sigma}) \ =\ \delta} \def\dd{\dot{\d}^{AB} \ ,
\end{equation}
where we separate off the zero modes,
\begin{equation}
\begin{aligned}\label{ExpandAPD}
X_a(t,\boldsymbol{\sigma}) \ &=\  X_a^{(0)}(t) \ +\  \sum_{A=1}^\infty X_a^A(t) Y^A(\boldsymbol{\sigma}) \ ,\\
\omega(t,\boldsymbol{\sigma}) \ &=\  \omega^{(0)}(t) \ +\  \sum_{A=1}^\infty \omega^A(t) Y^A(\boldsymbol{\sigma}) \ ,\quad \\
\Theta(t,\boldsymbol{\sigma}) \ &=\  \Theta^{(0)}(t) \ +\  \sum_{A=1}^\infty \Theta^A(t) Y^A(\boldsymbol{\sigma}) \ .
\end{aligned}
\end{equation}
The zero modes $X_a^{(0)}(t)$ and $\Theta^{(0)}(t)$ decouple in (\ref{LYM}), where
$X_a^{(0)}(t)$ describes the center of mass motion of the membrane as a whole.
Likewise, the gauge zero mode $\omega^{(0)}(t)$ drops out in the Lagrangian (as it acts
effectively like a  U(1) gauge field, which cannot couple because both
$X_a^{(0)}$ and $\Theta^{(0)}$ are real). The remaining non-zero modes
describe the `internal' degrees of freedom of the supermembrane. The APD gauge group
can thus be approximated by  SU$(N)$, as is most easily and explicitly
done for $S^2$ \cite{G,Hoppe} and  $T^2$ \cite{Hoppe1,FFZ,dWMN}, by cutting off
the mode expansions at $N^2{-}1$ (ignoring topological modes) and replacing
the APD-brackets by SU($N$) commutators. In fact, as shown in~\cite{B} the SU($N$)
approximation works for {\em any} genus of the membrane.  Consequently, we have
\begin{equation}\label{fABC}
f^{ABC}_{\rm APD} \ \equiv \
\int d^2\sigma\, {\sqrt{w(\boldsymbol{\sigma})}} \, Y^A(\boldsymbol{\sigma}) \big\{ Y^B(\boldsymbol{\sigma}) , Y^C(\boldsymbol{\sigma})\big\}\ =\
\lim_{N\rightarrow\infty} f^{ABC}(N)
\end{equation}
with SU($N$) structure constants $f^{ABC}(N)$. Hence the expansion labels
$A,B,\ldots=1,\ldots,N^2{-}1$ are thus turned into Yang--Mills indices, while $a,b,\ldots$ are
transverse (for membrane) and space-like (for supersymmetric Yang--Mills) indices.

After these preparations,  the matrix-model Lagrangian assumes the standard
form~\footnote{
For finite-dimensional gauge groups these supersymmetric matrix
models were first obtained in~\cite{CH,BRR,Flume}.}
\begin{equation}\label{MMLag}
{\cal L} \ =\  \tfrac12 (D_t X^A_a)^2 \,-\, \mathrm{i}\,\theta} \def\tb{\bar\theta^A_\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A} D_t \theta} \def\tb{\bar\theta^A_\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A} \,- \,
 \tfrac14 g^2 (f^{ABC} X^B_b X^C_c)^2 \,-\,
 \tfrac{\mathrm{i}}{2}\,g\,f^{ABC} \theta} \def\tb{\bar\theta^A_\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A} \gamma} \def\cd{\dot{\c}^a_{\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A}\beta} \def\bd{\dot{\b}} \def{\bar u}{{\underline \b}} X^B_a \theta} \def\tb{\bar\theta^C_\beta} \def\bd{\dot{\b}} \def{\bar u}{{\underline \b}\ ,
\end{equation}
where we have now switched to SO(9) spinors $\theta} \def\tb{\bar\theta^A_\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A}$ with 16 real components
and where
\begin{equation}
D_t \theta} \def\tb{\bar\theta^a\ =\ \partial_t \theta} \def\tb{\bar\theta^A + g\, f^{ABC} \omega^B \theta} \def\tb{\bar\theta^C
\end{equation}
is the SU($N$) covariant derivative. The real symmetric 16-by-16 matrices
$\gamma} \def\cd{\dot{\c}_a$ generate the SO(9) Clifford algebra, $\{\gamma} \def\cd{\dot{\c}^a , \gamma} \def\cd{\dot{\c}^b \} = 2\delta} \def\dd{\dot{\d}^{ab}$.
Variation w.r.t.~$\omega^A$ yields the constraint
\begin{equation}
f^{ABC} \big( X^B_a D_t X^C_a + \theta} \def\tb{\bar\theta^B_\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A} \theta} \def\tb{\bar\theta^C_\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A} \big) \ \approx \  0
\end{equation}
which is equivalent to the canonical generator of SU$(N)$ gauge transformations
(after performing this variation we can put $\omega^A =0$ everywhere).
The Lagrangian (\ref{MMLag}) is the one that underlies the M theory conjecture
of \cite{BFSS}, see also \cite{Bigatti,Banks,Taylor}.

\subsection{Setting up the path integral}

Our goal is now to set up a path-integral formulation that should eventually
permit the computation of correlators of physically  relevant quantities, and
complement the canonical quantization methods underlying many treatments
of the matrix model. We shall thus be interested in evaluating correlation functions
of the type
\begin{equation}\label{Corr1}
\begin{aligned}
\Big\langle {\cal O}_1[{\bf{X}},\theta} \def\tb{\bar\theta] \cdots {\cal O}_n[{\bf{X}},\theta} \def\tb{\bar\theta] \Big\rangle_g  \ &=\
\int\!\prod  {\cal D} X_a(t,\boldsymbol{\sigma})\,{\cal D}\theta} \def\tb{\bar\theta_\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A}(t,\boldsymbol{\sigma})\,{\cal D}\omega(t,\boldsymbol{\sigma})\,{\cal D} C(t,\boldsymbol{\sigma})\,{\cal D}\bar{C}(t,\boldsymbol{\sigma}) \,\times \\
&\qquad\qquad\times \ {\cal O}_1[{\bf{X}},\theta} \def\tb{\bar\theta] \cdots {\cal O}_n[{\bf{X}},\theta} \def\tb{\bar\theta]\ \exp\bigl( \mathrm{i}\,{\rm S}_{\rm tot} \bigr)
\end{aligned}
\end{equation}
where the precise form of the functionals ${\cal O}_i[{\bf{X}},\theta} \def\tb{\bar\theta]$ need not be specified at
this point. Because this is a gauge theory, the full action
\begin{equation}
{\rm S}_{\rm tot}\ =\  {\rm S} \,+\, {\rm S}'
\end{equation}
with ${\rm S} = \int\!\mathrm{d} t\,{\cal L}$  must comprise a gauge-fixing part ${\rm S}' = \int\!\mathrm{d} t\,{\cal L}'$.
For higher-dimensional Yang--Mills theories there are two preferred choices,
namely the Lorenz gauge ${\partial}^\mu A_\mu = 0$, and the axial gauge
$n^\mu A_\mu = 0$ (which includes the light-cone gauge for null vectors $n^\mu$).
In the reduction to one time dimension the axial gauge is necessarily identical
with the temporal gauge. Consequently, we have two preferred choices for the
gauge-fixing part, namely
\begin{equation}
\begin{aligned}
{\cal L}'\ &= \ -\tfrac{1}{2\xi}(\partial_t\omega)^2\,+\,\bar{C}\,\partial_t D_t C
\quad\qquad\, \textrm{(Lorenz gauge)} \ ,\\[4pt]
{\cal L}'\ &= \ -\tfrac{1}{2\xi}\omega^2\,+\,\bar{C} D_t C
\quad\qquad\qquad  \textrm{(temporal gauge)} \ .
\end{aligned}
\end{equation}
A further peculiarity of one dimension is that the temporal gauge implies the Lorenz gauge
\begin{equation}\label{axialLorenz}
\omega(t,\boldsymbol{\sigma})  = 0 \qquad {\Rightarrow} \qquad {\partial}_t \omega(t,\boldsymbol{\sigma}) \equiv \dot{\omega}(t,\boldsymbol{\sigma}) = 0 \ .
\end{equation}
$C(t,\boldsymbol{\sigma})$ and $\bar{C}(t,\boldsymbol{\sigma})$ are the usual Faddeev--Popov ghosts~\cite{FP,tH},
and $\xi$ is a real parameter which will be eventually sent to zero to put the theory
on the gauge hypersurface.  After trading the $\boldsymbol{\sigma}$~dependence for SU$(N)$ indices,
we are left with a quantum mechanical path integral describing {\em finitely\/} many
degrees of freedom. Because of the supersymmetry there is no need for a
normalization factor in (\ref{Corr1}) (as can be easily checked for $g=0$ with both
gauge choices). In passing we note that we can of course equivalently switch to a
Euclidean formulation by flipping  the sign in the kinetic terms $(\dot X_a)^2$, $\dot\omega^2$
and for the ghosts, and by replacing the oscillatory exponent by $\exp(-{\rm S}_{\rm tot})$;
the factor $\mathrm{i}$ is then absent in the spinor kinetic term.

An important part of our construction is that we consider the path
integral in a form where the fermions (and also the ghosts) are integrated out.
For the temporal gauge and for finite~$N$ the integration over $\theta} \def\tb{\bar\theta^A_\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A}(t)$ results in the
Matthews--Salam--Seiler~(MSS) determinant~\cite{MS,S}
\begin{equation}\label{MSS}
\Delta_{\rm MSS}\big[\omega{=}0, {\bf{X}}\big]  \ =\  \Bigl[
{\rm det\,} \bigl( \delta} \def\dd{\dot{\d}^{AB} \delta} \def\dd{\dot{\d}_{\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A}\beta} \def\bd{\dot{\b}} \def{\bar u}{{\underline \b}}\,\delta} \def\dd{\dot{\d}(t_1{-}t_2)\ +\ g\,K^{AB}_{\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A}\beta} \def\bd{\dot{\b}} \def{\bar u}{{\underline \b}}(t_1.t_2)\bigr) \Bigr]^{1/2}
\end{equation}
which is actually a Pfaffian because we are integrating over {\em real\/} fermions.
The integral kernel appearing in this expression is
\begin{equation} \label{Kkernel}
K^{AB}_{\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A}\beta} \def\bd{\dot{\b}} \def{\bar u}{{\underline \b}}(t_1,t_2)\ :=\
\varepsilon(t_1{-}t_2)\,f^{ACB}\gamma} \def\cd{\dot{\c}^a_{\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A}\beta} \def\bd{\dot{\b}} \def{\bar u}{{\underline \b}} X_a^C(t_2) \ .
\end{equation}
This is a real operator which is however not symmetric because hermitean
conjugation also exchanges the arguments $t_1$ and $t_2$.
Furthermore, we have taken out trivial factors of ${\rm det\,} ({\partial}_t)$ (which
anyway cancel in the supersymmetric path integral).
The free fermion propagator~$\varepsilon$ is just the Green's function for~$\partial_t$,
\begin{equation} \label{freeprop}
\varepsilon(t-t')\ =\ \bigl[\partial_t^{-1}\bigr](t,t') \ =\
\int\!\frac{\mathrm{d} p}{2\pi}\ \frac{\mathrm{i} p}{p^2 - \mathrm{i}\epsilon} \def\he{\hat{\epsilon}} \mathrm{e}^{-\mathrm{i} p(t-t')} \ =\
\Theta(t{-}t')-\tfrac12 \ =\ -\varepsilon(t'{-}t)\ .
\end{equation}
This choice of integration constant implies $\varepsilon(0) = 0$ as well as
\begin{equation}
\int\!\mathrm{d} t \  \varepsilon(t{-}t') \ =\  0\ .
\end{equation}
Our particular choice is important for the tests in the Appendix which
otherwise cannot be satisfied (it is also consistent with the dimensional
reduction of the usual Dirac propagator). In Section~4 we will study some properties
of this determinant in more detail and prove in particular that the expansion of
$\log(\Delta_{\rm MSS})$ in powers of~$g$ has a {\em non-zero radius of convergence\/}
with suitable technical assumptions on the behavior of~$X_a^A(t)$. We also note that
we have no positivity statement about~$\Delta_{\rm MSS}$ (though the fermion determinant is
non-negative for complex fermions!). Similarly, the determinant cannot be shown to be
an even function of~$g$ because of the non-vanishing trace
${\rm tr}\,(\gamma} \def\cd{\dot{\c}^{a_1}\cdots\gamma} \def\cd{\dot{\c}^{a_9})=16\,\epsilon} \def\he{\hat{\epsilon}^{a_1\cdots a_9}$.

For the infinite-dimensional APD gauge group we must be a little more careful:
while the kinetic term of (\ref{LYM}) is local in $\boldsymbol{\sigma}$, the interaction term is not
because it contains derivatives in $\boldsymbol{\sigma}$. To take into account this non-locality
we can formally replace the integral kernel (\ref{Kkernel}) by
\begin{equation} \label{KkernelAPD}
K^{\rm APD}_{\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A}\beta} \def\bd{\dot{\b}} \def{\bar u}{{\underline \b}}(t_1,t_2\,;\,\boldsymbol{\sigma}_1,\boldsymbol{\sigma}_2)\ :=\
\varepsilon(t_1{-}t_2)\,\gamma} \def\cd{\dot{\c}^a_{\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A}\beta} \def\bd{\dot{\b}} \def{\bar u}{{\underline \b}} \, \frac{1}{\sqrt{w(\boldsymbol{\sigma_1})}}\,
\epsilon} \def\he{\hat{\epsilon}^{rs}\,\frac{{\partial} X_a(t_2,\boldsymbol{\sigma}_1)}{{\partial} \sigma^r_1}\,
\delta} \def\dd{\dot{\d}(\boldsymbol{\sigma}_1,\boldsymbol{\sigma}_2)\,\frac{{\partial}}{{\partial} \sigma^s_2}
\end{equation}
and the identity operator by $\delta} \def\dd{\dot{\d}_{\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A}\beta} \def\bd{\dot{\b}} \def{\bar u}{{\underline \b}}\delta} \def\dd{\dot{\d}(t_1{-}t_2)\,\delta} \def\dd{\dot{\d}(\boldsymbol{\sigma}_1,\boldsymbol{\sigma}_2)$, with the
proviso that folding with this kernel now also contains an integral over~$\boldsymbol{\sigma}$.
When expanding the logarithm of the MSS~determinant using log det = Tr log,
we encounter for each trace a divergent factor $\delta} \def\dd{\dot{\d}(\boldsymbol{\sigma},\boldsymbol{\sigma})$, which can thus
be uniformly factored out of the sum. To arrive at a sensible result in the $N\rightarrow\infty$ limit
we must therefore remove this factor. This `renormalization' corresponds to the replacement
of the Cartan-Killing form $f^{ACD} f^{BCD} = N\delta} \def\dd{\dot{\d}^{AB}$ (whose limit $N\rightarrow\infty$
also diverges) in the matrix action by the finite action obtained from (\ref{LYM}) by
integrating $\frac1{\sqrt{w}}{\cal L}$ over the world volume with the measure $\mathrm{d} t \,\mathrm{d}^2 \sigma {\sqrt{w(\boldsymbol{\sigma})}}$.

\subsection{Physical correlators}

With the path integral formalism at hand we can now in principle proceed to calculate gauge-variant
and gauge-invariant correlators of suitable objects. But what are the physically
relevant operators ${\cal O}[{\bf{X}},\theta} \def\tb{\bar\theta]\,$? As in string theory, for the membrane the latter
should be associated to vertex operators describing the emission or absorption of certain
one-particle excitations from the membrane. As first shown in~\cite{DNP} there
indeed exist the classical analogs of supermembrane light-cone vertex operators
exciting the massless states of the supermembrane, which comprise
the massless supermultiplet of maximal $D=11$ supergravity~\cite{CJS}.
The related expressions must satisfy various consistency
constraints (target-space and world-volume gauge invariance, linear and non-linear
supersymmetry) which are explained at length in~\cite{DNP}, corresponding to (but
more complicated than) the ones known from type~II superstring theory. In particular,
in analogy with closed-string vertex operators they are to be integrated over
the membrane world volume.
For instance, for the transverse graviton components we have \cite{DNP}
\begin{equation}
{\cal O}[{\bf{X}},\theta} \def\tb{\bar\theta] \ =\ \int\!\mathrm{d} t\,\mathrm{d} \boldsymbol{\sigma} \ V_h[{\bf{X}},\theta} \def\tb{\bar\theta]
\end{equation}
with
\begin{equation}
\begin{aligned}
V_h[{\bf{X}},\theta} \def\tb{\bar\theta] \ &=\ h_{ab} \Bigl[ D_tX^a D_t X^b\ -\ \{X^a,X^c\}\{X^b, X^c\}
\ -\ \mathrm{i}\bar\theta} \def\tb{\bar\theta\,\gamma} \def\cd{\dot{\c}^a \{ X^b , \theta} \def\tb{\bar\theta\} \\
&\quad
-\ \tfrac12 D_t X^a \,\bar\theta} \def\tb{\bar\theta \gamma} \def\cd{\dot{\c}^{bc}\theta} \def\tb{\bar\theta \, k_c\ -\ \tfrac12 \{X^a , X^c\} \, \bar\theta} \def\tb{\bar\theta \gamma} \def\cd{\dot{\c}^{bcd}\theta} \def\tb{\bar\theta \, k_c
\ +\ \tfrac12 \bar\theta} \def\tb{\bar\theta\gamma} \def\cd{\dot{\c}^{ac}\theta} \def\tb{\bar\theta \, \bar\theta} \def\tb{\bar\theta \gamma} \def\cd{\dot{\c}^{bd}\theta} \def\tb{\bar\theta \, k_c k_d \Bigr]
\ \mathrm{e}^{-\mathrm{i}\boldsymbol{k}\cdot{\bf{X}}+\mathrm{i} k^- t}
\end{aligned}
\end{equation}
where $h_{ab}$ is the transverse graviton polarization tensor,
and $\{k_a\}=\boldsymbol{k}$ denotes the transverse components of the target-space momentum.
For the light-cone gauge target-space
momentum $k^\mu$ one must furthermore assume $k^+ = 0$ in order to avoid having to
deal with the light-cone coordinate $X^-(\tau,\boldsymbol{\sigma})$ in the exponential (as is also customary
in string theory~\cite{GS}). Remarkably, and unlike for superstring theory, there do not
appear to exist analogs of the string vertex operators for massive string states.
This would be in accord with the fact that the supermembrane is not a first quantizable
({\em i.e.}~one-particle) theory~\cite{HN} and for finite~$N$ consistent with the
D0-multiparticle interpretation of~\cite{BFSS}.

Because the light-cone vertex operators are given by complicated expressions,
and because the measure in (\ref{Corr1}) is not Gaussian, no sustained attempt has been
made, as far as we are aware, to evaluate their correlators. Neither has it been
possible so far to set up a perturbative expansion, as this will also require
understanding the quantum corrections (renormalizations) that are necessary for
the vertex operators to remain well-defined in the quantized interacting theory. Our line
of attack will therefore be a different one, in that we will reformulate the above path integral
in terms of a Nicolai map. A main advantage of such an approach is that the formul\ae\
to be presented below remain perfectly well-defined in the  limit $N\rightarrow\infty$ and can
thus be consistently implemented also in the APD path integral. Consequently,  it may be possible
in this way to sidestep the  detour via the finite-$N$ matrix model, and to directly
tackle the $N=\infty$ theory  right away. Possible applications of this technology
to supermembrane vertices will, however, be left to future work.

\section{The map to fourth order}

The method that we propose here to tackle expressions like (\ref{Corr1}) is based on the
Nicolai map ${\cal T}_g$ \cite{Nic0,Nic1,Nic2,FL,DL1,DL2,L1}, exploiting recent progress
in determining this map to higher orders in $g$ \cite{ANPP,ALMNPP,MN,LR2}. This map is
a non-local and non-linear field transformation, which
maps the theory to a free theory in such a way that after integrating out the fermions
(gaugini and ghosts) the product of the resulting fermionic determinants
equals the Jacobian of the map ${\cal T}_g$ at least locally in field space.
For operators ${\cal O}_k(t_k)$ built from $X_a$ (and~$\omega$) alone,
this enables us to re-express the expectation value (\ref{Corr1}) in the matrix theory as
a free-field correlator of transformed bosonic fields, {\em viz.}
\begin{equation}\label{Corr2}
\Big\langle {\cal O}_1(t_1) \cdots {\cal O}_n(t_n) \Big\rangle_g \ =\
\Big\langle {\cal T}_g^{-1}({\cal O}_1(t_1)) \cdots {\cal T}_g^{-1}({\cal O}_n(t_n)) \Big\rangle_0
\end{equation}
where integrating out the gaugini and ghosts is trivial on the right-hand side
because the transformed operators are purely bosonic ones.
We can therefore read this relation as one in the integrated-out theory
as well as in the original one including the fermions. A key property
of the map ${\cal T}_g$ is the equality of its functional Jacobian with the
product of the fermionic determinants obtained by integrating out
all anticommuting variables, to wit,
\begin{equation} \label{detrelation}
{\rm det\,} \left( \frac{\delta} \def\dd{\dot{\d} {\cal T}_g X}{\delta} \def\dd{\dot{\d} X} \right) \ =\
\Delta_{\rm MSS}[\omega,{\bf{X}}]\ \Delta_{\rm FP} [\omega,{\bf{X}}]
\end{equation}
where $\Delta_{\rm FP}$ and $\Delta_{\rm MSS}$ are, respectively,
the Faddeev--Popov determinant~\cite{FP,tH} and the
MSS~determinant~(\ref{MSS})~\cite{MS,S}.
We refer readers to \cite{ANPP,ALMNPP,LR1,MN,LR2}
for recent progress in constructing the map ${\cal T}_g$ for pure supersymmetric
Yang--Mills theories in all relevant dimensions. A crucial simplification follows from
(\ref{axialLorenz}), since it allows us to largely ignore the distinction between
`on-shell' and `off-shell' $R$-prescriptions in \cite{MN,LR2} that must be taken into
account in more than one dimension.

\subsection{Construction by dimensional reduction}

The goal of this section is the construction of ${\cal T}_g$~\cite{Nic1,Nic2,FL,DL1,DL2,L1}
for the APD and SU($N$) supersymmetric matrix models~(\ref{LYM}) and (\ref{MMLag}).
This can be done either by repeating the construction procedure described
in~\cite{L1,Nic2} for this particular theory, or by dimensionally reducing the map
for ten-dimensional super Yang--Mills theory to one-dimensional matrix mechanics.
Let us first choose the second path.

Since we only have an on-shell formulation of supersymmetry in ten dimensions,
we cannot employ the general scheme~\cite{MN,LR2} for arbitrary gauge fixing
but have to stick to the Lorenz gauge, for which the map was presented
on the gauge hypersurface in the critical spacetime dimensions $D=3, 4, 6$ and~$10$,
to $O(g^3)$ in~\cite{ALMNPP} and to $O(g^4)$ in~\cite{MN}.
In the dimensional reduction all quantities loose their coordinate dependence
except for a dependence on time~$t$, and the $D$ components of the gauge potential
become $(D{-}1)$ dynamical matrices $X_a(t)$ and one non-dynamical matrix~$\omega(t)$.
The Lorenz gauge reduces to $\partial_t\omega\equiv\dot\omega=0$,
hence the matrix~$\omega$ is a constant on the gauge hypersurface.
It will turn out that it is invariant under the map ${\cal T}_g$.

Let us recall the salient facts of the construction,
keeping $D$ arbitrary and denoting by~$r$ the dimension
of the corresponding Majorana spinor representation.
The map ${\cal T}_g$ is a nonlinear and nonlocal field transformation
\begin{equation}
{\cal T}_g: \bigl(X_a(t),\omega\bigr)\ \mapsto\ \bigl(X'_a(t),\omega'\bigr)\ .
\end{equation}
It affords to express the quantum correlator $\langle F\rangle_g$ of an arbitrary
bosonic functional~$F$ at gauge coupling~$g$ in terms of a free correlator ($g{=}0$)
of the same functional, but with its arguments transformed by the inverse map,
\begin{equation}
\Bigl\langle F[X,\omega] \Bigr\rangle_g \ =\
\Bigl\langle F[{\cal T}_g^{-1}X,{\cal T}_g^{-1}\omega] \Bigr\rangle_0\ .
\end{equation}
An infinitesimal (in~$g$) version reads
\begin{equation} \label{infflow}
{\partial}_g \Bigl\langle F[X,\omega] \Bigr\rangle_g \ =\
\Bigl\langle \bigl( {\partial}_g + R_g[X,\omega] \bigr) F[X,\omega] \Bigr\rangle_g\ ,
\end{equation}
where the ``coupling flow operator''~$R_g$ is a linear functional
integro-differential operator
with a nonlinear and nonlocal dependence on $X$ and~$\omega$.
As the construction is perturbative in the coupling~$g$,\footnote{
There exists, however, a universal nonperturbative formula for the map, see~\cite{LR1}.}
we expand (note the index shift)
\begin{equation}\label{Rng}
R_g[X,\omega] \ =\ \sum_{k=1}^\infty g^{k-1} \mathrm{R}_k[X,\omega] \ =\
\mathrm{R}_1[X,\omega] + g\,\mathrm{R}_2[X,\omega] + g^2 \mathrm{R}_3[X,\omega] + g^3 \mathrm{R}_4[X,\omega] + \ldots\ .
\end{equation}
Integrating the infinitesimal flow equation (\ref{infflow}) yields ${\cal T}_g^{-1}$ and finally
\begin{equation} \label{TfromR}
\begin{aligned}
{\cal T}_gX_a \ &=\ X_a\ -\ g\,\mathrm{R}_1 X_a \ -\ \tfrac12g^2\bigl(\mathrm{R}_2-\mathrm{R}_1^2\bigr)X_a\ -\
\tfrac16g^3\bigl(2\mathrm{R}_3-\mathrm{R}_1\mathrm{R}_2-2\mathrm{R}_2\mathrm{R}_1+\mathrm{R}_1^3\bigr)X_a \\
&\quad -\ \tfrac{1}{24}\bigl(6\mathrm{R}^4-2\mathrm{R}_1\mathrm{R}_3-3\mathrm{R}_2^2+\mathrm{R}_1^2\mathrm{R}_2-6\mathrm{R}_3\mathrm{R}_1
+2\mathrm{R}_1\mathrm{R}_2\mathrm{R}_1+3\mathrm{R}_2\mathrm{R}_1^2-\mathrm{R}_1^4\bigr)\ +\ \ldots
\end{aligned}
\end{equation}
in terms of the flow operator's expansion coefficients.
We have displayed the result to $O(g^4)$ since we shall evaluate the map to this order,
and we omitted the analogous formula for~$\omega$ because it reduces to ${\cal T}_g\omega=\omega$.

In order to avoid cluttering the equations with indices, we mostly suppress spinor
and color indices as well as time dependence and employ the DeWitt summation
convention (suppressing also time integrals) in the remainder of this section.
We find it convenient to let the flow operator act (by functional differentiation) to the left.
It is then given by a variation $\overleftarrow{\frac{\delta} \def\dd{\dot{\d}}{\delta} \def\dd{\dot{\d} X^A_a(t)}}$ followed by a string
of matrices in color, spinor and coordinate space, such as
\begin{equation}
(X_a{\times})^{AB}(t,t') = f^{AMB} X_a^M(t)\delta(t{-}t') \qquad\Rightarrow\qquad
(X_c{\times}X_d)^A(t) = f^{AMN} X_c^M(t) X_d^N(t)
\end{equation}
and propagators $G$ and $S$ defined by
\begin{equation} \label{props}
\bigl[D_t\,G\bigr]^{AB}_{\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A}\beta} \def\bd{\dot{\b}} \def{\bar u}{{\underline \b}}(t,t') = \delta} \def\dd{\dot{\d}^{AB}\delta} \def\dd{\dot{\d}_{\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A}\beta} \def\bd{\dot{\b}} \def{\bar u}{{\underline \b}}\delta} \def\dd{\dot{\d}(t{-}t')\ ,\qquad
\bigl[(D_t+g\,\gamma} \def\cd{\dot{\c}^aX_a{\times})S\bigr]^{AB}_{\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A}\beta} \def\bd{\dot{\b}} \def{\bar u}{{\underline \b}}(t,t') = \delta} \def\dd{\dot{\d}^{AB}\delta} \def\dd{\dot{\d}_{\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A}\beta} \def\bd{\dot{\b}} \def{\bar u}{{\underline \b}}\delta} \def\dd{\dot{\d}(t{-}t')
\end{equation}
with $D_t\equiv D_0=\partial_t+g\,\omega{\times}$.
We note that $\omega{\times}$ and $X_a{\times}$ are to be considered as matrices in color space.
The product of all these objects is to be executed in canonical fashion,
where we suppress obvious unit factors in the formul\ae. Observe also
that $G$ is {\em not\/} the ghost propagator whose defining equation
contains another derivative ${\partial}_t$. This is because in all relevant
expressions the ghost propagator appears with a derivative ${\partial}_t$.

A careful dimensional reduction of the coupling flow operator
eq.(1.19) of~\cite{ALMNPP} then yields~\footnote{
Note that we have split ${\cal R}=\partial_g+R$.}
\begin{equation} \label{Rright}
 \overleftarrow{R} \ =\
 -\tfrac{1}{r} \overleftarrow{\tfrac{\delta} \def\dd{\dot{\d}}{\delta} \def\dd{\dot{\d} X_a}}\,{\rm tr}\Bigl[
\bigl(\gamma} \def\cd{\dot{\c}_a-gX_a{\times}G\bigr)\,S\,
\bigl(\tfrac12\gamma} \def\cd{\dot{\c}^{cd}X_c{\times}X_d + \gamma} \def\cd{\dot{\c}^d\omega{\times}X_d\bigr)\Bigr]
\end{equation}
where the explicit trace refers to the spinor space,
and we have dropped a term proportional to~$\tfrac1g\dot{\omega}$.
Here, the first round bracket arises from a non-abelian projector~\cite{ALMNPP}
which in the Lorenz gauge reads ($\mu=(t,a)$)
\begin{equation}
P_\mu^{\ \nu} \ =\ \delta} \def\dd{\dot{\d}_\mu^{\ \nu} - D_\mu (\partial{\cdot}D)^{-1} \partial^\nu
\qquad\stackrel{\textrm{reduction}}{\longrightarrow}\qquad
\delta} \def\dd{\dot{\d}_\mu^{\ \nu} - D_\mu D_t^{-1}\partial_t^{-1} \partial^\nu
\end{equation}
which obeys $\partial^\mu P_\mu^{\ \nu} =0= P_\mu^{\ \nu} D_\nu$  and yields
\begin{equation} \label{Pred}
P_t^{\ \nu}=0\ ,\qquad P_a^{\ b}=\delta} \def\dd{\dot{\d}_a^{\ b}
\qquad\textrm{and}\qquad
P_a^{\ t} = g\,X_a{\times}D_t^{-1}\ .
\end{equation}
This shows that $R$ does not contain a variation $\frac{\delta} \def\dd{\dot{\d}}{\delta} \def\dd{\dot{\d}\omega}$,
{\em cf.}~formula (1.19) in \cite{ALMNPP} (with $\mu = t$).
The second round bracket is just the decomposition of
$\tfrac12 A_\rho{\times}A_\lambda$ in the reduction.

\subsection{Construction in the matrix model}

Alternatively, we may take the first path and construct the map ${\cal T}_g$
directly for the matrix model, following the strategy of~\cite{L1,Nic2}.
To this end, we implement the Lorenz gauge constraint $\dot\omega=0$
by adding to the matrix model Lagrangian~(\ref{MMLag})~\footnote{
Here and below, the $\cdot$ denotes a contraction in color space, i.e.~$P{\cdot}Q:=\delta} \def\dd{\dot{\d}^{AB}P^A Q^B$.}
\begin{equation}
{\cal L}\ =\ \tfrac12(D_t X_a)^2\ -\ \tfrac14g^2(X_c{\times}X_d)^2
\ -\ \tfrac{\mathrm{i}}{2}\theta} \def\tb{\bar\theta\cdot(D_t+g\,\hat{X}{\times})\theta} \def\tb{\bar\theta
\end{equation}
with $\hat{X}:=\gamma} \def\cd{\dot{\c}^aX_a$ a ``gauge-fixing term''
\begin{equation}
{\cal L}'\ =\ -\tfrac{1}{2\xi}\dot\omega^2\ +\ \bar{C}\cdot\partial_t D_t C
\end{equation}
with a real parameter~$\xi$ and ghost matrices $C$ and~$\bar{C}$.
Taking the limit $\xi\rightarrow 0$ puts the theory on the gauge hypersurface.

We aim to directly derive a coupling flow operator~$R$ as in~(\ref{infflow})
for the matrix model, which will govern the infinitesimal change in the coupling~$g$
for the quantum correlator of an arbitrary bosonic matrix functional~$F[X,\omega]$.
Keeping in mind the $g$-dependence of the functional integral weight $\mathrm{e}^{\mathrm{i}\int({\cal L}+{\cal L}')}$,
we compute (suppressing the subscript in $\langle\cdots\rangle_g$)
\begin{equation} \label{delF}
\begin{aligned}
\partial_g\,\Bigl\langle \,F\,\Bigl\rangle \ &=\
\Bigl\langle \partial_g F + F\,\partial_g\int\!\mathrm{i}({\cal L}{+}{\cal L}') \Bigl\rangle \\
&=\ \Bigl\langle \partial_g F + F\ \mathrm{i}\int\!\Bigl[
D_t X_a\cdot\omega{\times}X_a - \tfrac12g(X_c{\times}X_d)^2
- \tfrac{\mathrm{i}}{2}\,\theta} \def\tb{\bar\theta\cdot(\omega+\hat{X}){\times}\theta} \def\tb{\bar\theta
+ \bar{C}\cdot\partial_t(\omega{\times}C) \Bigr] \\
&=\ \Bigl\langle \partial_g F + F\ \mathrm{i}\Bigl[\delta_\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A}\Delta_\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A}
+ \mathrm{i} q\int\!\theta} \def\tb{\bar\theta{\cdot}(\omega+\hat{X}){\times}\theta} \def\tb{\bar\theta
+ \int\!\bar{C}\cdot\partial_t(\omega{\times}C) \Bigr] \Bigl\rangle
\end{aligned}
\end{equation}
where
\begin{equation} \label{Delta}
\Delta_\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A}\ =\ -\tfrac{1}{r}\int\!\mathrm{d} t\;(\gamma} \def\cd{\dot{\c}^d\theta} \def\tb{\bar\theta)_\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A}{\cdot}\omega{\times}X_d
\ +\ \tfrac{1}{2r}\int\!\mathrm{d} t\;(\gamma} \def\cd{\dot{\c}^{cd}\theta} \def\tb{\bar\theta)_\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A}{\cdot}X_c{\times}X_d\ .
\end{equation}
With the supersymmetry transformations
\begin{equation}
\delta} \def\dd{\dot{\d}_\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A}\omega = -\mathrm{i}\,\theta} \def\tb{\bar\theta_\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A}\ ,\qquad \delta} \def\dd{\dot{\d}_\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A} X_a = -\mathrm{i}\,(\theta} \def\tb{\bar\theta\gamma} \def\cd{\dot{\c}_a)_\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A}\ ,\qquad
\delta} \def\dd{\dot{\d}_\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A} \theta} \def\tb{\bar\theta_\beta} \def\bd{\dot{\b}} \def{\bar u}{{\underline \b} = -\gamma} \def\cd{\dot{\c}^d_{\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A}\beta} \def\bd{\dot{\b}} \def{\bar u}{{\underline \b}} D_t X_d -\tfrac{g}{2} \gamma} \def\cd{\dot{\c}^{cd}_{\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A}\beta} \def\bd{\dot{\b}} \def{\bar u}{{\underline \b}} X_c{\times} X_d
\end{equation}
one confirms that indeed
\begin{equation}
\delta} \def\dd{\dot{\d}_\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A}\Delta_\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A}\ =\ \int\!\mathrm{d} t\;\Bigl[
D_t X_a\cdot\omega{\times}X_a\ -\ \tfrac12g(X_c{\times}X_d)^2
\ -\ \mathrm{i}\,\tfrac{D-1}{r}\,\theta} \def\tb{\bar\theta\cdot(\omega+\hat{X}){\times}\theta} \def\tb{\bar\theta \Bigr]\ .
\end{equation}
Therefore, $\delta} \def\dd{\dot{\d}_\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A}\Delta_\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A}$ in (\ref{delF}) reproduces $\partial_g\int\!{\cal L}$
but with a mismatch in the coefficient of the Majorana term,
which thus still appears there but with a coefficient
\begin{equation}
q\ =\ \tfrac{D-1}{r}-\tfrac12 \ =\ \tfrac1r \qquad\textrm{for}\quad D=3,4,6,10\ .
\end{equation}
It is noteworthy that for a temporal gauge this mismatch is absent,
\begin{equation}
A_0 = 0 \qquad\Rightarrow\qquad \omega=0 \quad\textrm{and}\quad D_t = \partial_t
\qquad\Rightarrow\qquad \delta} \def\dd{\dot{\d}_\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A} \bigl( \Delta_\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A}|_{\omega=0} \bigr)\ =\ \partial_g\int{\cal L}\ ,
\end{equation}
since effectively $D\to D{-}1$ and the ghosts decouple.

Next, we employ the broken supersymmetric Ward identity
$\langle\delta} \def\dd{\dot{\d}_\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A} Y\rangle=-\mathrm{i}\langle(\delta} \def\dd{\dot{\d}_\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A}\int{\cal L}')\,Y\rangle$
together with
\begin{equation}
\delta} \def\dd{\dot{\d}_\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A}\,\int\!{\cal L}' \ =\ -s\,\delta} \def\dd{\dot{\d}_\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A} \Delta_{\textrm{gh}}
\qquad\textrm{for}\quad \Delta_{\textrm{gh}} \ =\ \int\!\bar{C}\,\dot{\omega}
\end{equation}
and the Slavnov variations
\begin{equation}
s\,\omega = D_t C\ ,\qquad s\,X_a = g\,X_a{\times}C\ ,\qquad s\,\theta} \def\tb{\bar\theta = g\,\theta} \def\tb{\bar\theta{\times}C\ ,\qquad
s\,C = -\tfrac{g}{2}C{\times}C\ ,\qquad s\,\bar{C} = \tfrac{1}{\xi}\dot\omega
\end{equation}
to rewrite
\begin{equation} \label{dF}
\begin{aligned}
\partial_g\,\Bigl\langle \,F\,\Bigl\rangle \ &=\
\Bigl\langle \partial_g F + \mathrm{i}\,\Delta_\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A} \delta} \def\dd{\dot{\d}_\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A} F \Bigl\rangle \ +\
\Bigl\langle F\ \Bigl[ \Delta_\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A}\,s\,\delta} \def\dd{\dot{\d}_\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A} \Delta_{\textrm{gh}}
- q\int\!\theta} \def\tb{\bar\theta{\cdot}(\omega+\hat{X}){\times}\theta} \def\tb{\bar\theta
+ \mathrm{i}\int\!\bar{C}\cdot\partial_t(\omega{\times}C) \Bigr] \Bigl\rangle\\
&=\ \Bigl\langle \partial_g F + \mathrm{i}\,\Delta_\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A} \delta} \def\dd{\dot{\d}_\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A} F
- \Delta_\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A} (\delta} \def\dd{\dot{\d}_\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A} \Delta_{\textrm{gh}})\,s\,F \Bigl\rangle \\
&\ +\ \Bigl\langle F\ \Bigl[ \bigl(s\,\Delta_\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A}\bigr)\bigl(\delta} \def\dd{\dot{\d}_\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A}\Delta_{\textrm{gh}}\bigr)
- q\int\!\theta} \def\tb{\bar\theta{\cdot}(\omega+\hat{X}){\times}\theta} \def\tb{\bar\theta
- \mathrm{i}\int\!\dot{\bar{C}}\cdot(\omega{\times}C) \Bigr] \Bigl\rangle\ ,
\end{aligned}
\end{equation}
where in the last step we used the BRST Ward identity $\langle s\,Y\rangle=0$.

For the flow equation~(\ref{infflow}) to hold, the last correlator has to vanish
for any bosonic functional~$F$. Writing out
\begin{equation}
s\,\Delta_\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A}\ =\ \tfrac1r\int (\hat{X}{\times}\theta} \def\tb{\bar\theta)_\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A}\cdot\dot{C}
\qquad\textrm{and}\qquad
\delta} \def\dd{\dot{\d}_\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A} \Delta_{\textrm{gh}} \ =\ -\mathrm{i}\int\dot{\bar{C}}\cdot\theta} \def\tb{\bar\theta_\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A}
\end{equation}
and performing the functional integrations over the fermions and the ghosts,
this requirement becomes
\begin{equation} \label{Z}
0\ \stackrel{!}{=}\
-\tfrac{\mathrm{i}}{r}\int(\hat{X}_{\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A}\beta} \def\bd{\dot{\b}} \def{\bar u}{{\underline \b}}{\times}
\bcontraction{\theta} \def\tb{\bar\theta_b)\cdot}{\dot{C}}{\ \int}{\dot{\bar{C}}}
\bcontraction[2ex]{}{\theta} \def\tb{\bar\theta}{_\beta} \def\bd{\dot{\b}} \def{\bar u}{{\underline \b})\cdot\dot{C}\ \int\dot{\bar{C}}\cdot}{\theta} \def\tb{\bar\theta}
\theta} \def\tb{\bar\theta_\beta} \def\bd{\dot{\b}} \def{\bar u}{{\underline \b})\cdot\dot{C}\ \int\dot{\bar{C}}\cdot\theta} \def\tb{\bar\theta_\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A}
\ -\ q\int(\hat{X}_{\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A}\beta} \def\bd{\dot{\b}} \def{\bar u}{{\underline \b}}{\times}
\bcontraction{}{\theta} \def\tb{\bar\theta}{_\beta} \def\bd{\dot{\b}} \def{\bar u}{{\underline \b})\cdot}{\theta} \def\tb{\bar\theta}
\theta} \def\tb{\bar\theta_\beta} \def\bd{\dot{\b}} \def{\bar u}{{\underline \b})\cdot\theta} \def\tb{\bar\theta_\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A}
\ -\ q\int(\omega{\times}
\bcontraction{}{\theta} \def\tb{\bar\theta}{_\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A})\cdot}{\theta} \def\tb{\bar\theta}
\theta} \def\tb{\bar\theta_\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A})\cdot\theta} \def\tb{\bar\theta_\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A}
\ +\ \mathrm{i}\int\!
\bcontraction{(\omega{\times}}{C}{)\cdot}{\dot{\bar{C}}}
(\omega{\times}C)\cdot\dot{\bar{C}}\ ,
\end{equation}
where the contractions stand for the fermionic and ghost propagators
\begin{equation} \label{contractions}
\bcontraction{}{\theta} \def\tb{\bar\theta}{^B_\beta} \def\bd{\dot{\b}} \def{\bar u}{{\underline \b}(t)\ }{\theta} \def\tb{\bar\theta}
\theta} \def\tb{\bar\theta^B_\beta} \def\bd{\dot{\b}} \def{\bar u}{{\underline \b}(t)\ \theta} \def\tb{\bar\theta^A_\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A}(t')\ =\ -S^{BA}_{\beta} \def\bd{\dot{\b}} \def{\bar u}{{\underline \b}\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A}}(t,t')
\qquad\textrm{and}\qquad
\bcontraction{}{C}{^B(t)\ }{\bar{C}}
C^B(t)\ \dot{\bar{C}}^A(t')\ =\ \mathrm{i}\,G^{BA}(t,t')\ ,
\end{equation}
respectively (note the time derivative on $\bar{C}^A$). With
$\partial_t G(t,t')^{BA}=\partial_{t'}G^{AB}(t',t)=:\partial G^{AB}(t',t)$
the condition~(\ref{Z}) reads
\begin{equation}
0\ \stackrel{!}{=}\
-\tfrac1r\,{\rm Tr}\,\bigl[(\hat{X}{\times}S)\,\partial G\bigr]
+q\,{\rm Tr}\,\bigl[ \hat{X}{\times}S \bigr]
+q\,{\rm Tr}\,\bigl[ \omega{\times}S \bigr]
-\tfrac1r\,{\rm Tr}\,\bigl[\omega{\times}G\bigr]\ ,
\end{equation}
where the trace here refers to spin, color and time altogether.
Abbreviating the unit operator by the symbol $\mathbbm{1}$, and
inserting the useful identities
\begin{equation}
\partial G\ =\ \mathbbm{1} - g\,\omega{\times}G
\qquad\textrm{and}\qquad
S\ =\ G - g\,G\,(\hat{X}{\times}S)\ ,
\end{equation}
into the first and third term, respectively, we cancel the second and fourth terms
(provided $q=\tfrac1r$) and remain with
\begin{equation}
0\ \stackrel{!}{=}\
\tfrac{1}{r}\,g\,{\rm Tr}\,\bigl[ (\hat{X}{\times}S)\,(\omega{\times}G)\bigr]\ -\
q\,g\,{\rm Tr}\,\bigl[ (\omega{\times}G)\,(\hat{X}{\times}S)\bigr]\ ,
\end{equation}
which indeed holds in the critical dimensions.

We return to (\ref{dF}) and integrate out the fermions and ghosts
to read off the flow operator
\begin{equation}
\begin{aligned}
R_g\ &=\ \mathrm{i}
\bcontraction{}{\Delta}{_\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A}}{\delta} \def\dd{\dot{\d}}
\Delta_\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A} \delta} \def\dd{\dot{\d}_\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A} -
\bcontraction{}{\Delta}{_\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A}(}{\delta} \def\dd{\dot{\d}}
\Delta_\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A} (\delta} \def\dd{\dot{\d}_\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A}
\bcontraction{}{\Delta}{_{\textrm{gh}})\,}{s}
\Delta_{\textrm{gh}})\,s \\
&=\
\bcontraction{}{\Delta}{_\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A} \int\!}{\theta} \def\tb{\bar\theta}
\Delta_\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A} \int\!\theta} \def\tb{\bar\theta_\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A}\cdot\frac{\delta} \def\dd{\dot{\d}}{\delta} \def\dd{\dot{\d}\omega}
\ +\
\bcontraction{}{\Delta}{_\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A} \int\!(}{\theta} \def\tb{\bar\theta}
\Delta_\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A} \int\!(\theta} \def\tb{\bar\theta\gamma} \def\cd{\dot{\c}_a)_\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A}\cdot\frac{\delta} \def\dd{\dot{\d}}{\delta} \def\dd{\dot{\d} X_a} \\
&\ -\ \mathrm{i}
\bcontraction{}{\Delta}{_\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A} \int\!(}{\theta} \def\tb{\bar\theta}
\Delta_\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A} \int\!\theta} \def\tb{\bar\theta_\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A}\cdot
\bcontraction{}{\dot{\bar{C}}}{\int\!D_t}{C}
\dot{\bar{C}} \int\!D_t C\cdot\frac{\delta} \def\dd{\dot{\d}}{\delta} \def\dd{\dot{\d}\omega}
\ -\ \mathrm{i}
\bcontraction{}{\Delta}{_\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A} \int\!(}{\theta} \def\tb{\bar\theta}
\Delta_\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A} \int\!\theta} \def\tb{\bar\theta_\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A}\cdot
\bcontraction{}{\dot{\bar{C}}}{\int\!(g\,X_a{\times}}{C}
\dot{\bar{C}} \int\!(g\,X_a{\times}C)\cdot\frac{\delta} \def\dd{\dot{\d}}{\delta} \def\dd{\dot{\d} X_a}\ .
\end{aligned}
\end{equation}
Since $D_t G=\mathbbm{1}$, the two variations w.r.t.~$\omega$
(first and third terms) cancel, and we are left with
\begin{equation}
R_g\ =\
\bcontraction{}{\Delta}{_\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A} \int}{\theta} \def\tb{\bar\theta}
\Delta_\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A} \int \theta} \def\tb{\bar\theta_\beta} \def\bd{\dot{\b}} \def{\bar u}{{\underline \b} \cdot \bigl[ (\gamma} \def\cd{\dot{\c}_a)_{\beta} \def\bd{\dot{\b}} \def{\bar u}{{\underline \b}\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A}}\,\mathbbm{1}
\ +\ g\,\delta} \def\dd{\dot{\d}_{\beta} \def\bd{\dot{\b}} \def{\bar u}{{\underline \b}\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A}} G\times X_a \bigr] \cdot\frac{\delta} \def\dd{\dot{\d}}{\delta} \def\dd{\dot{\d} X_a}\ .
\end{equation}
In the curly brackets we recognize the
(dimensionally reduced) non-abelian projector~(\ref{Pred}).
Recalling $\Delta_\alpha} \def\ad{\dot{\a}} \def\ua{{\underline \a}}\def\hA{{\widehat A}$ from (\ref{Delta}),
inserting the fermion propagator~(\ref{contractions})
and reversing the multiplication order,
one again arrives at the flow operator presented in~(\ref{Rright}).

\subsection{The map to third and fourth order}

For the perturbative power series we need the expansion of the propagators,
\begin{equation}
\begin{aligned}
G\ &=\ \varepsilon - g\,\varepsilon\,\omega{\times}\varepsilon + g^2\varepsilon\,\omega{\times}\varepsilon\,\omega{\times}\varepsilon -
g^3\varepsilon\,\omega{\times}\varepsilon\,\omega{\times}\varepsilon\,\omega{\times}\varepsilon \pm\ldots\ ,\\
S\ &=\ \varepsilon - g\,\varepsilon\,(\omega+\hat{X}){\times}\varepsilon +
g^2\varepsilon\,(\omega+\hat{X}){\times}\varepsilon\,(\omega+\hat{X}){\times}\varepsilon \mp\ldots\ ,
\end{aligned}
\end{equation}
with the free fermion propagator (\ref{freeprop}).
Because spin traces vanish for an odd product of gamma matrices
(for less than nine factors), the expansion coefficients $\mathrm{R}_k$ displayed here carry
only even/odd powers of~$\omega$ for $k$ being even/odd.
This parity extends to the map ${\cal T}_g$ itself.
Carrying out the spin traces, one gets
\begin{equation}
\begin{aligned}
\overleftarrow{\mathrm{R}_1}\ &=\  -\overleftarrow{\tfrac{\delta} \def\dd{\dot{\d}}{\delta} \def\dd{\dot{\d} X_a}}\,\varepsilon\,\omega{\times}X_a \ ,\\
\overleftarrow{\mathrm{R}_2}\ &=\
\overleftarrow{\tfrac{\delta} \def\dd{\dot{\d}}{\delta} \def\dd{\dot{\d} X_a}}\,\varepsilon\,\omega{\times}\,\varepsilon\,\omega{\times}X_a +
\overleftarrow{\tfrac{\delta} \def\dd{\dot{\d}}{\delta} \def\dd{\dot{\d} X_a}}\,\varepsilon\,X_b{\times}\,\varepsilon\,X_b{\times}X_a\ , \\
\overleftarrow{\mathrm{R}_3}\ &=\
- \overleftarrow{\tfrac{\delta} \def\dd{\dot{\d}}{\delta} \def\dd{\dot{\d} X_a}}\,\varepsilon\,\omega{\times}\,\varepsilon\,\omega{\times}\,\varepsilon\,\omega{\times}X_a
- \overleftarrow{\tfrac{\delta} \def\dd{\dot{\d}}{\delta} \def\dd{\dot{\d} X_a}}\,\varepsilon\,\omega{\times}\,\varepsilon\,X_b{\times}\,\varepsilon\,X_b{\times}X_a
-\overleftarrow{\tfrac{\delta} \def\dd{\dot{\d}}{\delta} \def\dd{\dot{\d} X_a}}\,\varepsilon\,X_b{\times}\,\varepsilon\,\omega{\times}\,\varepsilon\,X_b{\times}X_a \\
&\ \quad
-\overleftarrow{\tfrac{\delta} \def\dd{\dot{\d}}{\delta} \def\dd{\dot{\d} X_a}}\,\varepsilon\,X_a{\times}\,\varepsilon\,X_b{\times}\,\varepsilon\,\omega{\times}X_b
+\overleftarrow{\tfrac{\delta} \def\dd{\dot{\d}}{\delta} \def\dd{\dot{\d} X_a}}\,\varepsilon\,X_b{\times}\,\varepsilon\,X_a{\times}\,\varepsilon\,\omega{\times}X_b
-\overleftarrow{\tfrac{\delta} \def\dd{\dot{\d}}{\delta} \def\dd{\dot{\d} X_a}}\,\varepsilon\,X_b{\times}\,\varepsilon\,X_b{\times}\,\varepsilon\,\omega{\times}X_a \\
&\ \quad
+\overleftarrow{\tfrac{\delta} \def\dd{\dot{\d}}{\delta} \def\dd{\dot{\d} X_a}}\,X_a{\times}\,\varepsilon\,\varepsilon\,X_b{\times}\,\varepsilon\,\omega{\times}X_b\ ,
\end{aligned}
\end{equation}
and so on. Inserting all these into (\ref{TfromR}), performing the functional derivatives and
observing various cancellations, we arrive at
\begin{equation} \label{TX}
\begin{aligned}
{\cal T}_gX_a \ &=\ X_a\ +\ g\,\varepsilon\,\omega{\times}X_a \ -\
\tfrac12g^2 \varepsilon\,X_b{\times}\,\varepsilon\,X_b{\times}X_a \\[2pt]
&\ \ \ + \tfrac16g^3 \Bigl[
2\,\varepsilon\,X_b{\times}\,\varepsilon\,\omega{\times}\,\varepsilon\,X_b{\times}X_a +
2\,\varepsilon\,X_a{\times}\,\varepsilon\,X_b{\times}\,\varepsilon\,\omega{\times}X_b -
2\,X_a{\times}\,\varepsilon\,\varepsilon\,X_b{\times}\,\varepsilon\,\omega{\times}X_b \\
&\qquad\quad\ \ -
\varepsilon\,X_b{\times}\,\varepsilon\,X_a{\times}\,\varepsilon\,\omega{\times}X_b +
\varepsilon\,X_b{\times}\,\varepsilon\,X_b{\times}\,\varepsilon\,\omega{\times}X_a  +
\varepsilon\,(\varepsilon\,X_b{\times}X_a)\times(\varepsilon\,\omega{\times}X_b) \Bigr] \\
&\ \ \ +\ O(g^4)\ .
\end{aligned}
\end{equation}
For the practitioner's convenience we spell this out with our shorthand notation fully
expanded,
{\small
\begin{equation} \label{TXlong}
\begin{aligned}
{\cal T}_g X^A_a(t)\ &=\ X^A_a(t)\ +\ g\,f^{ABC}\!\int\!\!\mathrm{d} s\; \varepsilon(t{-}s)\,\omega^B X^C_a(s)\\
&\ -\ \tfrac12g^2 f^{ABC}f^{CDE}\!\int\!\!\mathrm{d} s\,\mathrm{d} u\; \varepsilon(t{-}s)\,X^B_b(s)\,\varepsilon(s{-}u)\,X^D_b X^E_a(u)\\
&\ +\ \tfrac16g^3 f^{ABC}f^{CDE}f^{EMN}\!\int\!\!\mathrm{d} s\,\mathrm{d} u\,\mathrm{d} v\; \varepsilon(t{-}s)\,X^B_b(s)\,\varepsilon(s{-}u)\,\Bigl[\\
&\quad 2\,\omega^D(u)\,\varepsilon(u{-}v)\,X^M_b X^N_a(v) - X^D_a(u)\,\varepsilon(u{-}v)\,\omega^M X^N_b(v) + X^D_b(u)\,\varepsilon(u{-}v)\,\omega^M X^N_a(v)
\Bigr]\\
&\ +\ \tfrac13g^3 f^{ABC}f^{CDE}f^{EMN}\!\int\!\!\mathrm{d} s\,\mathrm{d} u\,\mathrm{d} v\;
\varepsilon(t{-}s)\,\bigl[ X^A_a(s) - X^A_a(t)\bigr] \,\varepsilon(s{-}u)\,X^D_b(u)\,\varepsilon(u{-}v)\,\omega^M X^N_b(v)\\
&\ +\ \tfrac16g^3 f^{ABC}f^{BDE}f^{CMN}\!\int\!\!\mathrm{d} s\,\mathrm{d} u\,\mathrm{d} v\; \varepsilon(t{-}s)\,
\bigl[\varepsilon(s{-}u)\,X^D_b X^E_a(u)\bigr]\bigl[\varepsilon(s{-}v)\,\omega^M X^N_b(v)\bigr] \\
&\ +\ {\cal O}(g^4)\ .
\end{aligned}
\end{equation}
}
As a check, beyond ${\cal O}(g)$ all terms linear in~$X_a^A$ (and thus of maximal
power in~$\omega^A$) cancel out, a feature that can be proven to hold in general.
Also, in the temporal gauge $\omega^A{=}0$ the map drastically simplifies and admits only
even powers in~$g$ at least up to the order considered. Moreover,
to the order displayed here all terms share the ``linear tree"
topology of the flow operator, except for the last term in $O(g^3)$, which is
the first ``branched tree''. We have checked that this result is consistent with
the final result of~\cite{MN}. It is straightforward though tedious to extend the
above computation to higher orders. Equivalently, the ${\cal O}(g^4)$ result can be read off
by dimensionally reducing the result of~\cite{MN}, but we refrain here from spelling
out this formula with non-vanishing $\omega$ because it is rather lengthy and not
very illuminating.

However, the result simplifies greatly in the temporal gauge $\omega{=}0$.
Furthermore, given our transcription rule~(\ref{fABC}), it is straightforward
to write it down right away with the APD brackets (\ref{APDbracket}).
Suppressing the common argument~$\boldsymbol{\sigma}$, we arrive at
\begin{equation}\label{TXAPD}
\begin{aligned}
{\cal T}_g X_a (t) \ &=\ X_a(t) \ -\  \tfrac12 g^2 \int\!\!\mathrm{d} s\,\mathrm{d} u\; \varepsilon(t{-}s)\,\varepsilon(s{-}u)\,
\Big\{ X_b(s) \,,\,\bigl\{ X_b(u) , X_a(u) \bigl\} \Big\} \\
&\quad +\ \tfrac18 g^4 \int\!\!\mathrm{d} s\,\mathrm{d} u\,\mathrm{d} v\,\mathrm{d} w\; \varepsilon(t{-}s)\,\varepsilon(s{-}u)\,\varepsilon(u{-}v)\,\varepsilon(v{-}w)\,\Biggl[ \\
&\qquad\qquad 6\ \biggl\{ X_b(s)\,,\,\Bigl\{ X_c(u)\,,\,\bigl\{ X_{[a}(v)\,,\,\{ X_b(w),X_{c]}(w)\}\bigr\}\Bigr\}\biggr\} \\
&\qquad\ \  +\ 2\ \biggl\{ X_b(s)\,,\,\Bigl\{ X_{[b}(u)\,,\,\bigl\{ X_{|c|}(v)\,,\,\{ X_{a]}(w),X_c(w)\}\bigr\}\Bigr\}\biggr\} \\
&\qquad\ \  +\ 2\ \biggl\{ X_a(s)-X_a(t)\,,\,\Bigl\{ X_b(u)\,,\,\bigl\{ X_c(v)\,,\,\{ X_b(w),X_c(w)\}\bigr\}\Bigr\}\biggr\}\,
\Biggr] \\
&\quad +\ \tfrac18 g^4 \int\!\!\mathrm{d} s\,\mathrm{d} u\,\mathrm{d} v\,\mathrm{d} w\; \varepsilon(t{-}s)\,\varepsilon(s{-}u)\,\varepsilon(s{-}v)\,\varepsilon(v{-}w)\,\times \\
&\qquad\qquad\ \ \biggl\{ \bigl\{ X_a(u) , X_b(u) \bigr\}\,,\,\Bigl\{ X_c(v)\,,\,\bigl\{ X_b(w) , X_c(w) \bigr\} \Bigr\}\biggr\}
\ \ +\ {\cal O}(g^6)\ .
\end{aligned}
\end{equation}
This expression is perfectly well-defined for well-behaved functions
$X_a(t,\boldsymbol{\sigma})$, whence the $N\rightarrow\infty$ limit of (\ref{TXlong}) is equally well-defined.
At higher orders we will encounter more nested APD brackets, but the expansion stays
well-defined to arbitrary order. It is noteworthy that, while (\ref{TXlong}) contains both
even and odd powers in~$g$, the expansion (\ref{TXAPD}) with the temporal gauge $\omega = 0$
contains only even powers in $g$. This can only change in higher orders (starting
with R$_7$ in (\ref{Rng}), to be completely precise) when we encounter
$\gamma} \def\cd{\dot{\c}$-traces such as ${\rm tr}\, (\gamma} \def\cd{\dot{\c}^{a_1} \cdots \gamma} \def\cd{\dot{\c}^{a_9}) = 16\,\epsilon} \def\he{\hat{\epsilon}^{a_1\cdots a_9}$.

Finally, as an independent check,
in the Appendix we also demonstrate that the ``free-action condition"
\begin{equation} \label{freeaction}
\tfrac12(\partial_t {\cal T}_gX_a)^2 \ \stackrel{!}{=}\ \tfrac12(D_t X_a)^2 - \tfrac14g^2(X_b{\times}X_c)^2
+\ \textrm{total derivative}
\end{equation}
as well as the ``determinant matching condition''~\footnote{
For the temporal gauge the last two terms on the r.h.s.~are replaced by
$\big[ {\rm Tr}\,\log D_t \,-\, \tfrac{D-1}{2}{\rm Tr}\,\log\partial_t^2 \big]$,
but the cancellations remain the same, of course.}
\begin{equation} \label{detmatching}
{\rm Tr}\,\log\Bigl(\frac{\delta} \def\dd{\dot{\d} {\cal T}_gX}{\delta} \def\dd{\dot{\d} X}\Bigr)
\ \stackrel{!}{=}\
\tfrac12{\rm Tr}\,\log\bigl(D_t+g\hat{X}{\times}\bigr)\ +\ {\rm Tr}\,\log\partial_tD_t \,-\, \tfrac{D}{2}{\rm Tr}\,\log\partial_t^2
\end{equation}
for the Jacobian, Matthews--Salam--Seiler~\cite{MS,S} and Faddeev--Popov~\cite{FP} determinants
are both fulfilled up to and including $O(g^3)$ by the maps~(\ref{TX}) and
(\ref{TXAPD}), provided
\begin{equation}
D-2 \ \stackrel{!}{=}\ \frac{r}{2}\ ,
\end{equation}
as happens to be the case for the critical dimensions $D=3,4,6$ and~$10$.
As already mentioned, the determinants are more subtle in the APD gauge theory directly:
like for the APD integral kernel (\ref{KkernelAPD}) we can in each APD bracket~(\ref{APDbracket})
separate the two $\boldsymbol{\sigma}$~arguments by inserting $\delta} \def\dd{\dot{\d}$-functions together with integrals over $\boldsymbol{\sigma}$~variables.
When expanded, the Jacobian of the map ${\cal T}_g$ then contains exactly the same divergent factor
$\delta} \def\dd{\dot{\d}(\boldsymbol{\sigma},\boldsymbol{\sigma})$ that we encountered in the expansion of $\Delta_{\rm MSS}$
and which can thus be dropped for the same reason.

The results of this section should be considered as a generalization of the polynomial map
that obtains in supersymmetric quantum mechanics (see {\em e.g.}~\cite{Nic2}), where the
perturbative expansion terminates after the first step and gives rise to a closed expression
(see also \cite{L2,AFF} for attempts to find polynomial maps in higher dimensions).
Such a feature cannot be expected for the  APD gauge theory or matrix model.
However, our expressions (\ref{TXlong}) and~(\ref{TXAPD}) are almost as good,
because they can be obtained from a universal formula for ${\cal T}_g$ in terms of
a path-ordered exponential~\cite{LR1}. This formula furnishes an algorithmic
procedure to work out the expansion of ${\cal T}_g$ systematically to any given order
in~$g$, a calculation that can be automated and implemented on a  computer.
Again the result will be much simpler with the temporal gauge~$\omega=0$.
On the technical side it is worth emphasizing that because of (\ref{axialLorenz})
the differences between the axial and the Lorenz gauge choices almost
disappear in one dimension, together with the considerable complications
accompanying gauge choices different from the Lorenz gauge in higher dimensions.

\section{The Jacobian has a non-zero radius of convergence}

One main difference between the present approach and more conventional
perturbative expansions of the path integral is that the series expansion
for ${\cal T}_g$ has better convergence properties (here we are not referring to
UV divergences, but to the non-summability of what would be the renormalized perturbation
expansion in higher dimensions).  That the convergence properties should be better
was already anticipated in~\cite{Nic2} but never actually proven. Here we present further
evidence for this conjecture by showing that with suitable technical assumptions
the Jacobian of the map admits a non-zero radius of convergence when expanded
around $g=0$ in the complex $g$~plane. This we can do by exploiting the equality
(\ref{detrelation}) of the Jacobian with (the product of) the fermionic determinants.
Since we are actually only interested in the statement for the remporal gauge let us
therefore set $\omega = 0$ for which $\Delta_{\rm FP}$ is trivial, and consider the
MSS determinant~(\ref{MSS}). To this aim we expand the logarithm of $\Delta_{\rm MSS}$
and make use of the triangle inequality,
\begin{equation}\label{MSS1}
\Big| \log {\rm det\,}^{1/2}(\mathbbm{1} + gK) \Big| \,=\, \tfrac12\,
\Bigl|  \, \sum_{n=1}^\infty \tfrac{(-1)^{n-1}}{n}\, g^n\,{\rm Tr}\,\, K^n \Bigr|\ \leq \
\tfrac12 \, \sum_{n=1}^\infty \frac{|g|^n}{n} \, \bigl| {\rm Tr}\,\, K^n \bigr| \ ,
\end{equation}
where the kernel $K$ is defined in (\ref{Kkernel}).
Let us have a look at the individual terms: we have
\begin{equation}
\begin{aligned}
{\rm Tr} \, K^n \ =\ \int\!\mathrm{d} t_1 \cdots \int\!\mathrm{d} t_n \ &\varepsilon(t_1{-}t_2) \cdots \varepsilon(t_n{-}t_1) \
 {\rm tr} \bigl( \gamma} \def\cd{\dot{\c}^{a_1}\cdots \gamma} \def\cd{\dot{\c}^{a_n}\bigr) \, \times \\
 \times \  &{\rm tr} \bigl( T^{A_1} \cdots T^{A_n}\bigr) \
X_{a_1}^{A_1}(t_1) \cdots X_{a_n}^{A_n}(t_n)
\end{aligned}
\end{equation}
where $T^A$ are the SU($N$) generators in the adjoint representation. We can now
derive an upper bound on the absolute value of this expression by
using $\bigl|\varepsilon(t)\bigr| \leq \frac12$, together with
\begin{equation}
\bigl|\,{\rm tr} \bigl( \gamma} \def\cd{\dot{\c}^{a_1}\cdots \gamma} \def\cd{\dot{\c}^{a_n}\bigr) \bigr| \ \leq \ r
\qquad \textrm{and} \qquad
\bigl|\,{\rm tr} \bigl( T^{A_1} \cdots T^{A_n}\bigr) \bigr| \ \leq \  c^n
\end{equation}
where $c\equiv c_N$ is an $N$-dependent positive constant. Furthermore
introducing the L$^1$-norm
\begin{equation}
\big|\!\big| {\bf{X}} \big|\!\big|_1 \ :=\ \sum_{a,A} \int\!\mathrm{d} t \ \bigl| X_a^A(t) \bigr|
\end{equation}
we can majorize the individual terms to obtain
\begin{equation}
 (\ref{MSS1}) \ \leq \ \frac{r}2 \, \sum_{n=1}^\infty \, \Bigl(\frac{c}2 \Bigr)^n
 \frac{|g|^n}{n}\, \big|\!\big| {\bf{X}} \big|\!\big|_1^n\ .
\end{equation}
This series converges for $|g| < 2\,c^{-1} \big|\!\big| {\bf{X}} \big|\!\big|_1^{-1}$.
Consequently if we constrain the functions $X_a^A(t)$ to belong to the
Lebesgue space L$^1({\mathbb R}^k)$ (for $k=9(N^2{-}1)$) the series always has
an ($X$-dependent)  non-zero radius of convergence.

While the fact that the Jacobian has a non-zero radius of convergence as
a function of~$g$ does not imply that the map itself has this property, it strongly
constrains the series expansion for ${\cal T}_g$, regardless of the precise
form of the functions $X_a^A(t)$.
The main reason that makes the argument work is that, unlike for higher-dimensional
Yang--Mills theories, the supersymmetric matrix model has no UV divergences
which would necessitate infinite subtractions (as in~\cite{S}).
With appropriate UV and IR regularizations the above statements remain
valid for supersymmetric Yang--Mills  theories in higher dimensions, at least
with the axial gauge choice (for which, however, ${\cal T}_g$ is considerably more
complicated than for the Lorenz gauge \cite{MN,LR2}). So in that case
{\em both\/} regulators are necessary  for the MSS~determinant to make
sense in a more rigorous context.

\section{Outlook}

We hope that the present investigations will open some new and so far unexplored
avenues for addressing several outstanding key problems of supermembrane and
matrix theory. Among the topics for future investigation we have already
highlighted two of these, namely the question of quantum target-space
Lorentz invariance, and the problem of computing physically relevant
correlations functions. Here our approach provides a perturbative expansion
scheme of a type that has not been available in the literature so far.
Finally we note that our methods may also turn out to be applicable
to matrix string theory \cite{M,DVV}, which corresponds to the reduction
of maximally extended super-Yang--Mills theory to two spacetime dimensions.

\bigskip

\noindent
{\bf Acknowledgments.\ }
We thank Hannes Malcha for help in matching (\ref{TXAPD}) with the dimensional
reduction of the  ${\cal O}(g^4)$ result of \cite{MN}, and Daniele Dorigoni for discussion.

\newpage

redpajama_set_name: RedPajamaArXiv


The information provided on this site will get you started, but there is so much more than we could possibly write here! We STRONGLY recommend that you attend the Coaches Training that we hold in the fall. More information about this and other events can be found under "Important Events" on the right. Additional coaching tips and the Virginia Coaches Handbook are all available under the "Resources" menu at the top of this page.
Odyssey of the Mind is a creative problem-solving competition for children from preschool through college. Odyssey emphasizes creativity, encouraging students to "think outside the box." One unique aspect of Odyssey is that the students are expected to do everything, from brainstorming ideas to building props and machines. The role of the coach is to supervise the students and provide encouragement and guidance when needed, but never to do something specifically for the team's solution. For additional information about the program, please visit the national organization's web site.
An Odyssey of the Mind team typically consists of five to seven students with no more than seven students allowed on one team. Everyone on the team is expected to contribute to the team's solution. Teams can be put together in many ways, but keep in mind that, especially with a new team, you never know who might be a natural with electronics, a talented writer, or an amazing actor. Chances are, at some point during the year, every one of your kids will surprise you with what they can do!
*Note: If all team members are in Grade 2 or below, then the team may choose to participate in either the Primary problem or in a Division I problem. The Primary problem is non-competitive and is a great introduction to Odyssey of the Mind for younger children. If you have questions about which division would be most appropriate for your young team, please contact your school coordinator for additional information and help. Your coordinator can also assist if you have additional questions about the divisions or are unsure in which division your team should be placed.
There are three scored components at an Odyssey of the Mind competition: Long-Term, Style, and Spontaneous.
Long-Term: The long-term solution is likely what your team while spend the most time on. Each year, teams choose which of five problems they would like to solve (or the Primary problem for primary teams only). Problems fall into different categories, and they each have different scoring components. Generally, a long-term problem is solved with the presentation of an eight-minute skit addressing the scoring components in your chosen problem. You can read brief synopses of this year's problems on our web site, but you will need to contact your coordinator to receive the full problem statement and rules.
Style: Style is incorporated into the long-term solution. It is not a separate part of the competition, but consists of elements that are presented within the long-term solution. Each problem usually has two required style elements, two elements that teams are free to choose, and one overall effect score for those style elements. Style elements contribute to the overall long-term solution, but are elements that are not already scored in the long-term solution.
The key to spontaneous is PRACTICE! Practice often and practice as many different types of problems as you can. The team members won't know anything about the problem before walking into the room, so learning strategies for each of the different types of problems, having the team plan in advance which members will participate in each type of problem, etc., is invaluable. You can come up with your own spontaneous problems for your team to practice or use the many online resources available, such as the sample problems on the international site.
For more information about the Spontaneous portion of Odyssey of the Mind, see the "Introduction to Spontaneous" video under the "Resources" menu at the top of the page.
It is vital that you read through the Odyssey of the Mind Program Guide; a link to it is available on the Resources page. All of it. Every page. Make sure your students read through it as well. While you can remind your students of the rules, especially as they relate to safety, it is ultimately the responsibility of the students to be sure that their solution conforms to Odyssey of the Mind program guidelines.
The students are responsible for all aspects of their solution. They are the ones who decide which problem they want to solve; brainstorm ideas for their solution; come up with characters and a script; build props, cars, structures, and other parts of the solution; fill out paperwork…you get the idea. You can help guide them and keep them focused in practices, but you CANNOT tell them what to do or how to do it. Please remind your team's parents, brothers, sisters, cousins, neighbors, best friends, or anyone else who might be tempted to help the team members of this fact. Help is fine and encouraged, just not in a way that tells the team how to solve a problem. Following the Socratic method generally works (e.g., "Why do you think the structure broke in that place?" instead of "That joint needs to be redesigned" or "You should use a triangle instead of a square"). If something is too dangerous for the team members to do themselves, such as using certain power tools, then either a team member needs to be trained in how to use that equipment safely or the team needs to come up with a different solution.
For additional information about Outside Assistance, please see the "Outside Assistance" section on the main "Coaches" page. In that section, we also have contracts for team members and their parents to sign. We recommend bringing these to your first team meeting and having all team members and parents sign them so that everyone knows what is expected of them from the beginning.
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We recommend that you identify your judge as early as possible. Teams' time slots for volunteer duty will be determined after the competition schedule is finalized and will be posted on the tournament schedule.
Additional information for volunteers and judges can be found under the relevant menus at the top of the page.


text:
Interpreting Isolation - Creative responses to COVID-19
Here at Sound and Music and particularly on the British Music Collection, we feel it is important to capture this moment in our history and to create a space to reflect on COVID-19 and individual responses in a creative way.
New Voices 2018
New Voices is an annual composer support scheme raising the profile of some of the most talented and exciting new voices in the UK today.
#BMC50 Things
Introducing a new series of articles mapping the last fifty years of new music in the UK
British Music Collection: 50 Things
Composers or sound artists based in or born in the UK are invited to join the British Music Collection. Scroll down to find out more...
Photo: Na'ama Zisser, London-based composer featured in the British Music Collection.
Susanna Eastburn reflects on the Collection
"Where would you go to find an eclectic and fascinating collection of 20th and 21st Century British art?"
Hugh Davies - Shozyg 1 & 2 (1969)
Video of Hugh Davies - Shozyg 1 & 2 (1969)
Artist-researcher Tim Shaw on Hugh Davies
"Davies reveals the minute musical potential of all sorts of items including screwdrivers, egg slicers and scraps of metal..."
Solo Soprano - a portrait of Lol Coxhill (extract)
Video of Solo Soprano - a portrait of Lol Coxhill (extract)
Film-maker Helen Petts on Lol Coxhill
"Lol's playing was totally unique. When Solo Soprano was showed in Cafe Oto project space, people walked in off the street saying 'I can hear Lol Coxhill'."
Sebastien Lavoie on Digitising the Collection
"The work I'm doing could be compared to monks copying manuscripts by hand. I feel I could be considered a monk of the 21st Century..."
Derriere Le Miroir
Video of Derriere Le Miroir
Sohrab Uduman
Sohrab Uduman began his musical life as a clarinettist before going on to study composition in Birmingham with Jonty Harrison and Vic Hoyland.
Getting the Most from Your Profile
We are starting to see the beginning of the transformation of the Collection into a world leading resource for new music discovery.
Have Your Say About the Collection
New Voices is a newly re-launched, annual composer support scheme that exists to raise the profile of the most talented and exciting new voices emerging in Britain today.
Solipsism Cinema
Video of Solipsism Cinema
Shaun Blezard on Stephen Cornford
"The audio starts with the romantic sound of film going through projectors, the rhythm familiar to those of us that remember pre digital..."
Intern Tihomir Ranogajec on Jacob Thompson-Bell
by previous Sound and Music Intern and composer, Tihomir Ranogajec

redpajama_set_name: RedPajamaCommonCrawl


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Houston, Texas' high poverty rates alone are a good reason to teach kids lessons about sustainability. Did you know that 1 in 5 Houston kids are at or below poverty level? Using community gardens to teach children about long-term food sources could help reduce those numbers and children will hone life skills, including knowledge on being healthy and taking care of the environment.
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Can My Houston Community Afford This?
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