The kappa Supersymmetric Non Abelian Born Infeld Action for a D3 brane

TL;DR

This paper revisits the challenge of constructing a fully kappa supersymmetric non-abelian Born-Infeld action for D3 branes, acknowledging previous inconsistencies and clarifying that the earlier claimed solution is invalid.

Contribution

The authors critically reassess their prior attempt at formulating a kappa supersymmetric non-abelian Born-Infeld action, identifying key issues and clarifying the limitations of their previous approach.

Findings

Identification of gauge invariance issues in the non-abelian extension

Recognition of associativity problems in the symmetrized trace

Disclaiming of the previously claimed fully supersymmetric action

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.