Hair on non-extremal D1-D5 bound states

We consider a truncation of type HB supergravity on four-torus where in addition to the Ramond-Ramond 2-form field, the Ramond-Ramond axion (w) and the NS-NS 2-form field (B) are also retained. In the (w, B) sector we construct a linearised perturbation carrying only left moving momentum on two-charge non-extremal D1-D5 geometries of Jejjala, Madden, Ross and Titchener. The perturbation is found to be smooth everywhere and normalisable. It is constructed by matching to leading order solutions of the perturbation equations in the inner and outer regions of the geometry.