Yang-Mills instantons sitting on a Ricci-flat worldspace of double D4-branes

Thus far, there seem to be no complete criteria that can settle the issue as to what the correct generalization of the Dirac-Born-Infeld (DBI) action, describing the low-energy dynamics of the D-branes, to the non-Abelian case would be. According to recent suggestions, one might pass the issue of worldvolume solitons from an Abelian to non-Abelian setting by considering the stack of multiple, coincident D-branes and use it as a guideline to construct or censor the relevant non-Abelian version of the DBI action. In this spirit, here we are interested in the explicit construction of SU(2) Yang-Mills (YM) instanton solutions in the background geometry of two coincident probe D4-brane worldspaces, particularly when the metric of the target spacetime in which the probe branes are embedded is given by the Ricci-Aat, magnetic extremal 4-brane solution in type IIA supergravity theory with its worldspace metric being given by that of Taub-Newman-Unti-Tamburino (NUT) and Eguchi-Hanson solutions, the mio best-known gravitational instantons. Then we demonstrate that, with this YM instanton-gravitational instanton configuration on the probe D4-brane worldvolume, the energy of the probe branes attains its minimum value and hence enjoys a stable state provided one employs Tseytlin's non-Abelian DBI action for the description of multiple probe D-branes. In this way, we support the arguments in the literature in favor of Tseytlin's proposal for the non-Abelian DBI action.