Black hole as topological insulator (I): the BTZ black hole case

Black holes are extraordinary massive objects which can be described classically by general relativity, and topological insulators are new orders of matter that could be use to built a topological quantum computer. They seem to be different objects, but in this paper, we claim that the black hole can be considered as kind of topological insulator. For BTZ black hole in three dimensional $AdS_3$ spacetime we give two evidences to support this claim: the first evidence comes from the black hole "membrane paradigm", which says that the horizon of black hole behaves like an electrical conductor. On the other hand, the vacuum can be considered as an insulator. 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. We also consider the coupling with the electromagnetic field to show that the boundary modes can conduct the electricity. For higher dimensional black hole the first evidence is still valid. So we conjecture that the higher dimensional black hole can also be considered as higher dimensional topological insulators. This conjecture will have far-reaching influences on our understanding of quantum black hole and the nature of gravity.