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The BRST invariance of the open pure spinor superstring is examined in thepresence of background superfields on a Dp-brane. We note that the backgroundsuperfields introduced in this paper depend on boundary fermions. The BRSTinvariance leads to supersymmetric Dirac-Born-Infeld (DBI) equations forbackground superfields depending on boundary fermions as well as boundaryconditions on spacetime coordinates. After quantizing boundary fermions,background superfields are promoted to non-abelian ones. As a result, we obtainthe supersymmetric non-abelian DBI equations from the supersymmetric DBIequations depending on boundary fermions. It is shown that these non-abelianDBI equations reduce to the super-Yang-Mills equations in the limit alpha' ->0. We also show the nilpotency of the BRST transformation of boundary fermions.

Supersymmetric Non-abelian DBI Equations from Open Pure Spinor Superstring