Entanglement entropy, conformal invariance and extrinsic geometry

TL;DR

This paper investigates how entanglement entropy in strongly coupled conformal field theories depends on the extrinsic geometry of the dividing surface, using conformal invariance and holography, and derives explicit formulas for specific geometries.

Contribution

It provides a complete characterization of entanglement entropy dependence on extrinsic geometry for 2d surfaces and extends the analysis to 4d conformal field theories, including explicit formulas for spheres and cylinders.

Findings

Entanglement entropy depends on extrinsic geometry via type B conformal anomaly.

Explicit formulas for entanglement entropy in flat space with spherical and cylindrical surfaces.

Topology influences the contribution of the type A conformal anomaly.

Abstract

We use the conformal invariance and the holographic correspondence to fully specify the dependence of entanglement entropy on the extrinsic geometry of the 2d surface $\Sigma$ that separates two subsystems of quantum strongly coupled ${\mathcal{N}}=4$ SU(N) superconformal gauge theory. We extend this result and calculate entanglement entropy of a generic 4d conformal field theory. As a byproduct, we obtain a closed-form expression for the entanglement entropy in flat space-time when $\Sigma$ is sphere $S_2$ and when $\Sigma$ is two-dimensional cylinder. The contribution of the type A conformal anomaly to entanglement entropy is always determined by topology of surface $\Sigma$ while the dependence of the entropy on the extrinsic geometry of $\Sigma$ is due to the type B conformal anomaly.