Recently the Navier-Stokes equations have been derived from the duality with the black branes in AdS 5 . The zero modes of black branes are reinterpreted as dynamical degrees of freedom of a conformal fluid on the boundary of AdS 5 . Here, we derive the corrections to the Navier-Stokes equations due to fermionic zero modes of the black branes. We study only the contributions due to bilinears in the fermionic zero modes in the first order of the parameter expansion. The need of a superextension of the fluid dynamics is a consequence of the full AdS/CF T correspondence and yet to be investigated.
We provide the metric, the gravitino fields and the gauge fields to all orders in the fermionic zero modes for D = 5 and D = 4, N = 2 gauged supergravity solutions starting from non-extremal AdS-Schwarzschild black holes. We compute the Brown-York stress-energy tensor on the boundary of AdS 5/AdS4 spaces and we discuss some implications of the fermionic corrections to perfect fluid interpretation of the boundary theory. The complete non-linear solution, which we denote as fermionic wig, is achieved by acting with supersymmetry transformations upon the supergravity fields and that expansion naturally truncates at some order in the fermionic zero modes
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.
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