# Fuzzy sphere of polarized D0 branes, non-abelian worldvolume action regime validity

I am studying Myer's paper "Dielectric Branes" (https://arxiv.org/abs/hep-th/9910053v2), where it is shown that a stack of D0 branes polarizes into a spherical D0-D2 bound state. The stack of $N$ D0 branes pictures is modeled via a non-abelian worldvolume action; this is due to the fact that, when branes are together, more string modes become massless and the spectrum upgrades its symmetry as follows, $$U(1)^N\rightarrow U(N).$$ This is valid only when the branes are at a distance smaller than the string scale; if this is not the case, some string modes become massive (the string length connecting two branes adds up mass) and the non-abelian picture is not valid anymore.

That said, when the D0 stack polarizes into a spherical D0-D2 bound state, you would (naively, says Myers) like to maintain the radius of this sphere below the string scale: then you can assure that the D0-branes are within a distance that makes the premise and result self-consistent. But actually, you want to work in the regime $$R\ll\sqrt{N}\ell_s,$$ where $N$ is the number of branes and $\ell_s$ the string scale. This maintains the typical area size $4\pi R^2/N$ of a D0-brane within the two-sphere below the string scale, and this means that the typical separation between branes is less than the string scale, as required.

Question: Why is the last true? I might understand that if the surface of a brane is small enough, the next brane is to be found at a distance below the string scale, but what about the antipodal surface element in the same sphere, which also represents another brane? It seems to me that they are quite far away from each other if we decide to not set the radius of the sphere below the string scale.

Myer's elaborates on this in the same paper below equation (101).