Black Hole Entropy and $SU(2)$ Chern-Simons Theory

Black holes (BH’s) in equilibrium can be defined locally in terms of the so-called isolated horizon boundary condition given on a null surface representing the event horizon. We show that this boundary condition can be treated in a manifestly $SU(2)$ invariant manner. Upon quantization, state counting is expressed in terms of the dimension of Chern-Simons Hilbert spaces on a sphere with punctures. Remarkably, when considering an ensemble of fixed horizon area $a_H$, the counting can be mapped to simply counting the number of $SU(2)$ intertwiners compatible with the spins labeling the punctures. The resulting BH entropy is proportional to $a_H$ with logarithmic corrections $\Delta S=-\frac{3}{2}\log a_H$. Our treatment from first principles settles previous controversies concerning the counting of states.

Figure 1

The characteristic data for a (vacuum) spherically symmetric isolated horizon corresponds to Schwarzschild data on $\Delta$, and free radiation data on the transversal null surface.