Entropy of quantum black holes

In the Loop Quantum Gravity, black holes (or even more general Isolated Horizons) are described by a $SU(2)$ Chern–Simons theory. There is an equivalent formulation of the horizon degrees of freedom in terms of a $U(1)$ gauge theory which is just a gauged fixed version of the $SU(2)$ theory. These developments will be surveyed here. Quantum theory based on either formulation can be used to count the horizon micro-states associated with quantum geometry fluctuations and from this the micro-canonical entropy can be obtained. We shall review the computation in $SU(2)$ formulation. Leading term in the entropy is proportional to horizon area with a coefficient depending on the Barbero–Immirzi parameter which is fixed by matching this result with the Bekenstein–Hawking formula. Remarkably there are corrections beyond the area term, the leading one is logarithm of the horizon area with a definite coefficient $-3/2$, a result which is more than a decade old now. How the same results are obtained in the equivalent $U(1)$ framework will also be indicated. Over years, this entropy formula has also been arrived at from a variety of other perspectives. In particular, entropy of BTZ black holes in three dimensional gravity exhibits the same logarithmic correction. Even in the String Theory, many black hole models are known to possess such properties. This suggests a possible universal nature of this logarithmic correction.