Quantum Holographic Entanglement Entropy to All Orders in $1/N$ Expansion

TL;DR

This paper calculates quantum corrections to holographic entanglement entropy in four-dimensional AdS gravity, revealing an exact all-order $1/N$ expansion that matches the Airy function results in ABJM theory.

Contribution

It provides the first explicit computation of all-order quantum corrections to holographic entanglement entropy using the replica trick and minisuperspace approximation.

Findings

The entanglement entropy matches the logarithm of the Airy partition function.

The entropy splits into a corrected minimal surface area and bulk entanglement entropy.

The results hold despite the absence of supersymmetry.

Abstract

We study holographic entanglement entropy in four-dimensional quantum gravity with negative cosmological constant. By using the replica trick and evaluating path integrals in the minisuperspace approximation, in conjunction with the Wheeler-DeWitt equation, we compute quantum corrections to the holographic entanglement entropy for a circular entangling surface on the boundary three sphere. Similarly to our previous work on the sphere partition function, the path integrals are dominated by a replica version of asymptotically AdS conic geometries at saddle points. As expected from a general CFT argument, the final result is minus the free energy on the three sphere which agrees with the logarithm of the Airy partition function for the ABJM theory that sums up all perturbative $1/N$ corrections despite the absence of supersymmetries. The all-order holographic entanglement entropy cleanly splits into two parts, (1) the $1/N$-corrected Ryu-Takayanagi minimal surface area and (2) the bulk entanglement entropy across the minimal surface, as suggested in the earlier literature. It is explicitly shown that the former comes from the localized conical singularity of the replica geometries and the latter from the replication of the bulk volume.