Structure of six-dimensional microstate geometries

We investigate the structure of smooth and horizonless microstate geometries in six dimensions, in the spirit of the five-dimensional analysis of Gibbons and Warner [arXiv: 1305.0957]. In six dimensions, which is the natural setting for horizonless geometries with the charges of the D1-D5-P black hole, the natural black objects are strings and there are no Chern-Simons terms for the tensor gauge fields. However, we still find that the same reasoning applies: in absence of horizons, there can be no smooth stationary solutions without non-trivial topology. We use topological arguments to describe the Smarr formula in various examples: the uplift of the five-dimensional minimal supergravity microstates to six dimensions, the two-charge D1-D5 microstates, and the non-extremal JMaRT solution. We also discuss D1-D5-P superstrata and confirm that the Smarr formula gives the same result as for the D1-D5 supertubes which are topologically equivalent.