Fermionic wigs for BTZ black holes

Abstract

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 AdS 3 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.

Fermionic corrections to fluid dynamics from BTZ black hole

“…the complete solution in the fermionic zero modes) associated with a boosted and dilatate BTZ black hole in Poincaré patch. 8 As explained in the previous section, the boost and the dilatation shift the mass and angular momentum of the black hole. Therefore, to get the complete solution, we first compute the wig for the BH and then we perform the shift to the mass and of the angular momentum.…”

“…In order to complete the procedure outlined in [1,2,4,8] we perform a finite boost on the BTZ solution in the t-x plane, namely…”

“…Thus, we proceed in the usual way by constructing the wig for the black hole metric (2.9) and then replacing M 0 and J 0 with M and J as defined in (2.22). This procedure is iterative and can be found in [8]. The superpartner of a generic field Φ is constructed by acting with a finite supersymmetry transformation on the original field [19]:…”

“…By construction they will indeed satisfy the Killing vectors equation. 8 Note that our Killing spinors (or anti-Killing spinors as defined in [18]) are not time independent but the fermionic black hole superpartner does not depend on t.…”

“…We consider N = 2, D = 3 supergravity with cosmological term which has two solutions, the AdS 3 space and the BTZ black hole [6,7]. In a previous paper [8], we computed the full supermultiplet of the BTZ black hole by performing a finite susy transformation. Notice that only by finite susy transformations, we are able to compute the complete orbit (wig) starting from the black hole solutions [9,10].…”