Entanglement entropy, conformal invariance and extrinsic geometry

TL;DR

The paper develops a conformal-invariance framework, complemented by holographic methods, to express 4d entanglement entropy as a universal function of the entangling surface's intrinsic and extrinsic geometry. It shows that the logarithmic term splits into a topology-driven type-A part and an extrinsic-curvature–driven type-B part, with holography fixing the relevant coefficients (μ = -1). The authors derive closed-form expressions for EE when Σ is a cylinder or a sphere in flat space and demonstrate how extrinsic geometry enters EE through the anomaly structure. These results connect holographic entanglement entropy with conformal anomaly data and extend the formalism to generic 4d CFTs. They clarify the distinct roles of the A and B anomalies in shaping entanglement across geometries.

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 $Σ$ 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 $Σ$ is sphere $S_2$ and when $Σ$ is two-dimensional cylinder. The contribution of the type A conformal anomaly to entanglement entropy is always determined by topology of surface $Σ$ while the dependence of the entropy on the extrinsic geometry of $Σ$ is due to the type B conformal anomaly.