Fermionic wigs for BTZ black holes

We compute the wig for the BTZ black hole, namely the complete non-linear solution of supergravity equations with all fermionic zero modes. We use a "gauge completion" method starting from AdS₃ Killing spinors to generate the gravitinos fields associated to the BH and we compute the back-reaction on the metric. Due to the anticommutative properties of the fermionic hairs the resummation of these effects truncates at some order. We illustrate the technique proposed in a precedent paper in a very explicit and analytical form. We also compute the mass, the angular momentum and other charges with their corrections.