FiniteNeffects on the collapse of fuzzy spheres

Finite N effects on the time evolution of fuzzy 2-spheres moving in flatspacetime are studied using the non-Abelian DBI action for N D0-branes.Constancy of the speed of light leads to a definition of the physical radius interms of symmetrised traces of large powers of Lie algebra generators. Thesetraces, which determine the dynamics at finite N, have a surprisingly simpleform. The energy function is given by a quotient of a free multi-particlesystem, where the dynamics of the individual particles are related by a simplescaling of space and time. We show that exotic bounces of the kind seen in the1/N expansion do not exist at finite N. The dependence of the time of collapseon N is not monotonic. The time-dependent brane acts as a source for gravitywhich, in a region of parameter space, violates the dominant energy condition.We find regimes, involving both slowly collapsing and rapidly collapsingbranes, where higher derivative corrections to the DBI action can be neglected.We propose some generalised symmetrised trace formulae for higher dimensionalfuzzy spheres and observe an application to D-brane charge calculations.Comment: 32 pages, Late