On non-abelian generalisation of the Born-Infeld action in string theory

We point out that the part of the tree-level open string effective action for the non-abelian vector field which depends on the field strength but not on its covariant derivatives is not defined unambiguously. We show that if one adopts the definition under which one drops all terms depending on all possible products (symmetrised or antisymmetrised) of covariant derivatives then this part of the action is given by the symmetrised trace of the direct non-abelian generalisation of the Born-Infeld action, We discuss some possible applications to D-brane dynamics. (C) 1997 Elsevier Science B.V.