Black Hole Entropy in Loop Quantum Gravity, Analytic Continuation, and Dual Holography

Abstract

A new approach to black hole thermodynamics is proposed in Loop Quantum Gravity (LQG), by defining a new black hole partition function, followed by analytic continuations of Barbero-Immirzi parameter to $\gamma\in i\mathbb{R}$ and Chern-Simons level to $k\in i\mathbb{R}$. The analytic continued partition function has remarkable features: The black hole entropy $S=A/4\ell_P^2$ is reproduced correctly for infinitely many $\gamma= i\eta$, at least for $\eta\in\mathbb{Z}\setminus\{0\}$. The near-horizon Unruh temperature emerges as the pole of partition function. Interestingly, by analytic continuation the partition function can have a dual statistical interpretation corresponding to a dual quantum theory of $\gamma\in i\mathbb{Z}$. The dual quantum theory implies a semiclassical area spectrum for $\gamma\in i\mathbb{Z}$. It also implies that at a given near horizon (quantum) geometry, the number of quantum states inside horizon is bounded by a holographic degeneracy $d= e^{A/4\ell_P}$, which produces the Bekenstein bound from LQG. On the other hand, the result in arXiv:1212.4060 receives a justification here.