Quantum Hall Effect and Black Hole Entropy in Loop Quantum Gravity

Abstract

In LQG, black hole horizons are described by 2+1 dimensional boundaries of a bulk 3+1 dimensional spacetime. The horizon is endowed with area by lines of gravitational flux which pierce the surface. As is well known, counting of the possible states associated with a given set of punctures allows us to recover the famous Bekenstein-Hawking area law according to which the entropy of a black hole is proportional to the area of the associated horizon $ S_{BH} \propto A_{Hor} $. It is also known that the dynamics of the horizon degrees of freedom is described by the Chern-Simons action of a $\mathfrak{su(2)}$ (or $\mathfrak{u(1)}$ after a certain gauge fixing) valued gauge field $A_{\mu}^i$. Recent numerical work which performs the state-counting for punctures, from first-principles, reveals a step-like structure in the entropy-area relation. We argue that both the presence of the Chern-Simons action and the step-like structure in the entropy-area curve are indicative of the fact that the effective theory which describes the dynamics of punctures on the horizon is that of the Quantum Hall Effect.