Entanglement entropy, conformal invariance and extrinsic geometry

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.