On Time-Dependent Collapsing Branes and Fuzzy Odd-Dimensional Spheres

We study the time-dependent dynamics of a collection of N collapsing/expanding D0-branes in type IIA string theory. We show that the fuzzy-S3 and S5 provide time-dependent solutions to the matrix model of D0-branes and its DBI generalisation. Some intriguing cancellations in the calculation of the non-Abelian DBI matrix actions result in the fuzzy-S3 and S5 having the same dynamics at large-N. For the matrix model, we find analytic solutions describing the time-dependent radius, in terms of Jacobi elliptic functions. Investigation of the physical properties of these configurations shows that there are no bounces for the trajectory of the collapse at large-N. We also write down a set of useful identities for fuzzy-S3, fuzzy-S5 and general fuzzy odd-spheres.