Counting the Microstates of a Kerr Black Hole in M Theory

We show that an extremal Kerr black hole, appropriately lifted to M theory, can be transformed to a Kaluza-Klein black hole in M theory, or a D0–D6 charged black hole in string theory. Since all the microstates of the latter have recently been identified, one can exactly reproduce the entropy of an extremal Kerr black hole. We also show that the topology of the event horizon is not well defined in M theory.

Figure 1

The branes wrap a rational direction $k/l$ of the torus (here, $k=3$, $l=1$), so there are $2kl$ intersection points on each $T^2$.

Figure 2

By taking $R$ large, the geometry becomes a Myers-Perry black hole at the tip of Taub-NUT. A simple reflection now changes configuration (a) with $N_0=0$ and $J\ne 0$ into (b) with $N_0\ne 0$ and $J=0$. Although it appears that the black hole has been rotated by 90°, this is just an artifact of the projection down to two dimensions.