  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 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 target spacetime in which the probe
branes are embedded is given by the Ricci-flat, magnetic extremal 4-brane
solution in type IIA supergravity theory with its worldspace metric being given
by that of Taub-NUT and Eguchi-Hanson solutions, the two best-known
gravitational instantons. And 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 stable
state provided one employs the 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.
