Bubbling supertubes and foaming black holes

We construct smooth BPS three-charge geometries that resolve the zero-entropy singularity of the $U(1)\times U(1)$ invariant black ring. This singularity is resolved by a geometric transition that results in geometries without any branes sources or singularities but with nontrivial topology. These geometries are both ground states of the black ring, and nontrivial microstates of the D1-D5-P system. We also find the form of the geometries that result from the geometric transition of $N$ zero-entropy black rings, and argue that, in general, such geometries give a very large number of smooth bound-state three-charge solutions, parametrized by $6N$ functions. The generic microstate solution is specified by a four-dimensional hyper-Kähler geometry of a certain signature, and contains a “foam” of nontrivial two-spheres. We conjecture that these geometries will account for a significant part of the entropy of the D1-D5-P black hole, and that Mathur’s conjecture might reduce to counting certain hyper-Kähler manifolds.