Hawking fluxes, fermionic currents, $W_{1+\infty }$ algebra, and anomalies

We complete the analysis carried out in previous papers by studying the Hawking radiation for a Kerr black hole carried to infinity by fermionic currents of any spin. We find agreement with the thermal spectrum of the Hawking radiation for fermionic degrees of freedom. We start by showing that the near-horizon physics for a Kerr black hole is approximated by an effective two-dimensional field theory of fermionic fields. Then, starting from two-dimensional currents of any spin that form a $W_{1+\infty }$ algebra, we construct an infinite set of covariant currents, each of which carries the corresponding moment of the Hawking radiation. All together they agree with the thermal spectrum of the latter. We show that the predictive power of this method is based not on the anomalies of the higher-spin currents (which are trivial) but on the underlying $W_{1+\infty }$ structure. Our results point toward the existence in the near-horizon geometry of a symmetry larger than the Virasoro algebra, which very likely takes the form of a $W_{\infty }$ algebra.