Giant gravitons, Bogomol’nyi-Prasad-Sommerfield bounds, and noncommutativity

It has been recently suggested that gravitons moving in ${\mathrm{AdS}}_m\times {\mathrm{S}}^n$ spacetimes along the ${\mathrm{S}}^n$ blow up into spherical (n-2)-branes whose radius increases with increasing angular momentum. This leads to an upper bound on the angular momentum, thus “explaining” the stringy exclusion principle. We show that this bound is present only for states which saturate a BPS-like condition involving the energy E and angular momentum $J,E>~J/R,$ where R is the radius of ${\mathrm{S}}^n.$ Restriction of motion to such states lead to a noncommutativity of the coordinates on ${\mathrm{S}}^n.$ As an example of motions which do not obey the exclusion principle bound, we show that there are finite action instanton configurations interpolating between two possible BPS states. We suggest that this is consistent with the proposal that there is an effective description in terms of supergravity defined on noncommutative spaces and noncommutativity arises here because of imposing supersymmetry.