Black hole as topological insulator (II): the boundary modes

In the previous paper Ref.[1], it was claimed that the black hole can be considered as a kind of topological insulator. For BTZ black hole in three dimensional $AdS_3$ spacetime two evidences were given to support this claim: the first evidence comes from the black hole "membrane paradigm", and the second evidence comes from the fact that the horizon of BTZ black hole can support two chiral massless scalar field with opposite chirality. Those are two key properties of 2D topological insulator. For higher dimensional black hole the first evidence is still valid but the second fails. In this paper, starting from the boundary BF theory, which can be used to describe the boundary degrees of freedom of black hole in arbitrary dimension, we shown that the isolated horizon of $3+1-$D black hole can support massless scalar field and vector field. Those two fields can be used to construct a massless Dirac field through the $2+1-$dimensional bosonization, which also appears on the boundary of $3+1-$D topological insulators.