Spectral Flow, and the Spectrum of Multi-Center Solutions

We discuss "spectral flow" coordinate transformations that take asymptotically four-dimensional solutions into other asymptotically four-dimensional solutions. We find that spectral flow can relate smooth three-charge solutions with a multi-center Taub-NUT base to solutions where one or several Taub-NUT centers are replaced by two-charge supertubes, and vice versa. We further show that multi-parameter spectral flows can map such Taub-NUT centers to more singular centers that are either D2-D0 or pure D0-brane sources. Since supertubes can depend on arbitrary functions, we establish that the moduli space of smooth horizonless black hole microstate solutions is classically of infinite dimension. We also use the physics of supertubes to argue that some multi-center solutions that appear to be bound states from a four-dimensional perspective are in fact not bound states when considered from a five- or six-dimensional perspective.