{In the previous submission of this paper we claimed to have solved the problem of constructing a consistent non-abelian Born Infeld action which is $\kappa$--supersymmetric to all orders. Our method was based on an extension of target superspace to $\mathrm{N} \times \mathrm{N}$ matrix--valued objects for all the items appearing in the geometric construction. Then we applied the double first order formalism introduced by us in the previous construction of abelian brane actions to the non--abelian case and we claimed that the Bosonic action given by the prescription of the symmetrized trace could be promoted to a fully kappa supersymmetric one. Unfortunately there is an internal inconsistency, relative to $U(N)$ gauge invariance in the matricisation of target superspace geometry and there is a subtle inconsistency in the variation of the symmetric trace action due to the loss of associativity of the underlying symmetrized product. Because of that the entire construction previously submitted does not stand on its feet. Consequently the present resubmission is done in order to disclaim our previously claimed result and also to express our sincere gratitude to our friends Paul Howe and Ulf Lindstrom who, with their constructive criticism and private correspondence, have helped us to understand where the bugs in our construction were.}
hep-th0310217.tex
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\begin{document}
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DFTT- ../2003\\
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\begin{center}
{\LARGE \bf The $\kappa$-Supersymmetric Non Abelian Born Infeld Action for a $D3$-brane}\\ \vfill {\large
 Pietro Fr\'e and  Leonardo Modesto$^1$} \\
\vfill {
$^1$ Dipartimento di Fisica Teorica, Universit\'a di Torino, \\
$\&$ INFN -
Sezione di Torino\\
via P. Giuria 1, I-10125 Torino, Italy  }
\end{center}
\vfill
\begin{abstract}
{In the previous submission of this paper we claimed to have solved the problem of
constructing a consistent non-abelian Born Infeld action which is
$\kappa$--supersymmetric to all orders. Our method was based on an
extension of target superspace to $\mathrm{N} \times \mathrm{N}$ matrix--valued objects
for all the items appearing in the geometric construction. Then we
applied the \textit{double first order formalism} introduced by us in
the previous construction of abelian brane actions to the non--abelian
case and we claimed that the Bosonic action  given by the
prescription of the \textit{symmetrized trace} could be promoted to a
fully kappa supersymmetric one. Unfortunately there is   an internal
inconsistency, relative to $U(N)$ gauge invariance in the matricisation   of target superspace geometry
and there is a subtle inconsistency in the variation of the symmetric
trace action due to the loss of associativity of the underlying symmetrized
product. Because of that the entire construction previously submitted
does not stand on its feet. Consequently the present resubmission is
done in order to disclaim our previously claimed result and also to
express our sincere gratitude to our friends Paul Howe and Ulf
Lindstrom who, with their constructive criticism and private correspondence, have helped us to
understand where the bugs in our construction were.}
\end{abstract}
%\vspace{2mm} \vfill \hrule width 3.cm {\footnotesize $^ \dagger $
%This work is supported in part by the European Union RTN contracts
%HPRN-CT-2000-00122 and HPRN-CT-2000-00131.}
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\section{Introduction}
As stated in the abstract this resubmission of the paper which
appeared in the Archive with the same title is done in order to disclaim the
result we had previously claimed, due to the discovery of ill-fated
inconsistencies that make the entire construction not stand on its
feet.
\par
The previously presented non--abelian extension of the rheonomic construction of
$D$--brane actions based on the double first order formalism is
wrong.
\par
The rheonomic construction and the double first order formalism work
perfectly well and are valuable tools to establish kappa
supersymmetry in the abelian case.
\par
What is wrong and inconsistent is the matrix extension of superspace
and the consequent application of the geometric constructions to this
matrix extended superspace.
\par
As also stated in the abstract this resubmission is also done in
order to express our gratitude to our friends Paul Howe and Ulf
Lindstrom who have helped us to understand the internal subtle bugs of our
construction with their constructive criticism expressed through
private correspondence.
\newpage
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