Coarse-graining of bubbling geometries and the fuzzball conjecture

In the LLM bubbling geometries, we compute the entropies of black holes and estimate their "horizon" sizes from the fuzzball conjecture, based on coarse-graining on the gravity side. The differences of black hole microstates cannot be seen by classical observations. Conversely, by counting the possible deformations of the geometry which are not classically detectable, we can calculate the entropy. We carry out this method on the black holes of the LLM bubbling geometries, such as the superstar, and obtain the same result as was derived by coarse-graining directly on the CFT (fermion) side. Second, by application of this method, we can estimate the "horizon" sizes of those black holes, based on the fuzzball conjecture. The Bekenstein-Hawking entropy computed from this "horizon" agrees with that microscopic entropy above. This result supports the fuzzball conjecture.