Black Flowers and Real Forms of Higher Spin Symmetries

Using Chern-Simons formulation, we investigate higher spin (HS) black holes in AdS$_{3}$ with soft Heisenberg hair and establish linkage with the real forms of the underlying complexified gauge symmetries taken here as $SL(N,C)_{L}\times SL(N,C)_{R}.$ We study the various conserved currents characterizing the HS black flowers (HS-BF) and show that they can be formed of layers indexed by the elements of the centre of the gauge symmetry. This feature follows from requiring the holonomy of the asymptotic gauge connection around the thermal cycle to sit in the centre $\mathbb{Z}_{N}$ of the symmetry group. With regard to the compact subgroups of the real forms of the complexified gauge symmetry, we calculate the entropies of the HS-BF and verify that, unless we are considering trivial holonomies, there are no continuous paths joining the HS-BF to the core spin 2 black holes. As explicit illustrations, we give quantum field realisations of the soft Heisenberg hair in terms of bosonic and fermionic primary conformal fields and compute the HS-BF entropy as a function of the number of fermions occupying the ground state of the Heisenberg soft hair.