Fuzzy Sphere Dynamics and Non-Abelian DBI in Curved Backgrounds

We consider the non-Abelian action for the dynamics of $N Dp'$-branes in the background of $M Dp$-branes, which parameterises a fuzzy sphere using the SU(2) algebra. We find that the curved background leads to collapsing solutions for the fuzzy sphere except when we have $D0$ branes in the $D6$ background, which is a realisation of the gravitational Myers effect. Furthermore we find the equations of motion in the Abelian and non-Abelian theories are identical in the large $N$ limit. By picking a specific ansatz we find that we can incorporate angular momentum into the action, although this imposes restriction upon the dimensionality of the background solutions. We also consider the case of non-Abelian non-BPS branes, and examine the resultant dynamics using world-volume symmetry transformations. We find that the fuzzy sphere always collapses but the solutions are sensitive to the combination of the two conserved charges and we can find expanding solutions with turning points. We go on to consider the coincident $NS$5-brane background, and again construct the non-Abelian theory for both BPS and non-BPS branes. In the latter case we must use symmetry arguments to find additional conserved charges on the world-volumes to solve the equations of motion. We find that in the Non-BPS case there is a turning solution for specific regions of the tachyon and radion fields. Finally we investigate the more general dynamics of fuzzy $\mathbb{S}^{2k}$ in the $Dp$-brane background, and find collapsing solutions in all cases.