On the General Structure of the Non–Abelian Born–Infeld Action

We discuss the general structure of the non–abelian Born–Infeld action, together with all of the α derivative corrections, in flat D–dimensional space– time. More specifically, we show how the connection between open strings propagating in background magnetic fields and gauge theories on non–commutative spaces can be used to constrain the form of the effective action for the massless modes of open strings at weak coupling. In particular, we exploit the invariance in form of the effective action under a change of non– commutativity scale of space–time to derive algebraic equations relating the various terms in the α expansion. Moreover, we explicitly solve these equations in the simple case D = 2, and we show, in particular, how to construct the minimal invariant derivative extension of the NBI action.