Noncommutative geometry, quantum effects and DBI-scaling in the collapse of D0-D2 bound states

We study fluctuations of time-dependent fuzzy two-sphere solutions of thenon-abelian DBI action of D0-branes, describing a bound state of a sphericalD2-brane with N D0-branes. The quadratic action for small fluctuations is shownto be identical to that obtained from the dual abelian D2-brane DBI action,using the non-commutative geometry of the fuzzy two-sphere. For some of thefields, the linearized equations take the form of solvable Lam\'e equations. Wedefine a large-N DBI-scaling limit, with vanishing string coupling and stringlength, and where the gauge theory coupling remains finite. In this limit, thenon-linearities of the DBI action survive in both the classical and the quantumcontext, while massive open string modes and closed strings decouple. Wedescribe a critical radius where strong gauge coupling effects becomeimportant. The size of the bound quantum ground state of multiple D0-branesmakes an intriguing appearance as the radius of the fuzzy sphere, where themaximal angular momentum quanta become strongly coupled.Comment: 34 pages, Latex; v2: Minor correction in conformal transformation of couplings, references adde