aInstitute for Theoretical Physics,

V~'e review an iterative construction of the supersymmetric non-abelian Born-Infeld action. We obtain the action through second order in the fieldstrength. Kappa-invariance fixes the ordenings which turn out to deviate from the symmetrized trace proposal. 1. In t roduct ion On a Dp-brane there are eight propagating fermionic and eight propagating bosonic world-volume degrees of freedom. In the static gauge, the bosonic ones appear as a U(1) vector field in p + 1 dimensions and 9- p scalar fields. The for-mer describe open strings attached to the brane while the latter describe the transversal positions of the brane. The effective action for slowly vary-ing fields is known through all orders in cd, it is the ten-dimensional U(1) Born-Infeld theory dimensionally reduced to p + 1 dimensions [1], [2]. Its fully covariant supersymmetric generaliza-tion is known as well [3], [4], [5]. It contains the embedding coordinates X'(a) (of which, because of the worldvolume reparametrization invariance, only the transverse coordinates are physical de-grees of freedom), a vector field V/(a), and an N = 2 spacetime fermionic field O(a) (we denote by #, ~ = 0,...,9 the spacetime indices and by i, j = 0,...,p the worldvolume coordinates a~). Because of the presence of a local fermionic sym-metry, called R-symmetry, half of the fermionic fields can be gauged away. This leaves, in the static gauge, the fieldcontent of a U(1) super-symmetric Yang-Mills theory in p + 1 dimensions which indeed describes 8 + 8 degrees of freedom. Once several, say n, Dp-branes coincide, ad-ditional massless states appear enhancing the U(1) n gauge symmetry to a full U(n) gauge sym