On the Non-abelian Born-infeld Action

We discuss some aspects of the generalization of the Born-Infeld action to non-abelian gauge groups and show how the discrepancy between Tseytlin’s symmetrized trace proposal and string theory can be corrected at order F . We also comment on the possible quadratic order fermionic terms. While the talk I gave at the Gong Show session of Cargèse’ 99 Summer School was devoted to an outline of the main results of the paper “Curvature terms in D-brane actions and their M-theory origin” [1], this note will be focused on another very interesting but not yet settled aspect of the D-brane effective action : the generalization of the Dirac-Born-Infeld (DBI) action to non-abelian gauge groups. The massless degrees of freedom of an isolated D-brane are its transverse coordinates, a world-volume U(1) gauge field, and the associated fermions [2]. Its effective action contains a Dirac-Born-Infeld (DBI) (see [3] for a recent review of the DBI action) and a Wess-Zumino (WZ) piece. While the WZ part, describing the interactions of the D-brane with Ramond-Ramond field backgrounds, can be determined by anomaly cancellation and is therefore believed to be exact, the DBI piece is only reliable for small space-time curvatures, and in the low-acceleration regime. Indeed, for non constant gauge field strength, the DBI action is corrected by derivative terms [3, 4]. When N identical D-branes coincide, the transverse coordinates and world-volume gauge fields become N × N unitary matrices. The nonabelian generalization of the DBI action describing this dynamics has LPTENS-99/34 UMR 8549 du Centre National de la Recherche Scientifique et de l’École Normale Supérieure. 1