Supersymmetric charge constraints on AdS black holes from free fields

Supersymmetric AdS black hole solutions exist only when their angular momenta and charges satisfy a certain constraint that depends on the dimension. We show that these nonlinear relations on the conserved charges agree with a computation in the dual supersymmetric CFT in its free limit, with interactions entering only through a uniform rescaling of all charges. Our computations apply to the highly non-trivial charge constraints for AdS₄, AdS₅ and AdS₇ black holes, and generalize an earlier one for the analogous constraint in AdS₃. Our results suggest a microscopic understanding of AdS black holes beyond the scope of supersymmetric indices.