Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 354 / 2 / a / ii

Regulate the surface at $z=ϵ$, so $r_{\max }=\sqrt{R^2-{ϵ}^2}$. On the hemisphere, $\sqrt{1+z^{\prime 2}}=R/z$, and its area isThe Ryu–Takayanagi formula therefore givesThe first term is the ultraviolet area-law divergence proportional to the boundary circle's length; the finite constant is universal for the vacuum of a three-dimensional conformal field theory.