Noncommutativity and tachyon condensation

We study the fuzzy or noncommutative Dp-branes in terms of infinitely manyunstable D0-branes, from which we can construct any Dp-branes. We show that thetachyon condensation of the unstable D0-branes induces the noncommutativity. Inthe infinite tachyon condensation limit, most of the unstable D0-branesdisappear and remaining D0-branes are actually the BPS D0-branes with thecorrect noncommutative coordinates. For the fuzzy S^2 case, we explicitly showonly the D0-branes corresponding to the lowest Landau level survive in thelimit. We also show that a boundary state for a Dp-brane satisfying theDirichlet boundary condition on a curved submanifold embedded in the flat spaceis not localized on the submanifold. This implies that the Dp-brane on it isambiguous at the string scale and solves the problem for a spherical D2-branewith a unit flux on the world volume which should be equivalent to oneD0-brane. We also discuss the diffeomorphism in the D0-brane picture.Comment: 30 pages, references added, minor corrections and clarifications, version to appear in JHE