  We construct the first explicit, smooth, horizonless black-hole microstate
geometry whose moduli space is described by an arbitrary function of one
variable and is thus infinite-dimensional. This is achieved by constructing the
scalar Green function on a simple D6 anti-D6 background, and using this Green
function to obtain the fully back-reacted solution for a supertube with varying
charge density in this background. We show that this supertube can store
parametrically more entropy than in flat space, confirming the entropy
enhancement mechanism that was predicted using brane probes. We also show that
all the local properties of the fully back-reacted solution can, in fact, be
obtained using the DBI action of an appropriate brane probe. In particular, the
supergravity and the DBI analysis yield identical functional bubble equations
that govern the relative locations of the centers. This indicates that there is
a non-renormalization theorem that protects these functional equations as one
moves in moduli space. Our construction creates configurations that are beyond
the scope of recent arguments that appear to put strong limits on the entropy
that can be found in smooth supergravity solutions.
