Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 354 (Gauge/Gravity Duality)

Four distinct points in a conformal field theory with $d>2$ possess two independent conformal cross-ratios, conventionallyTranslations, rotations, dilations, and special conformal transformations can map three points to canonical positions $0$, a point on one chosen axis, and the point at infinity in a conformal completion. The fourth point retains one radial coordinate and one angle relative to that axis. Those two quantities are equivalent to $u$ and $v$; the stabilizer rotations remove all its remaining angular coordinates.

For identical scalar primary operators of dimension $\Delta$, conformal covariance fixes a kinematic prefactor and leaves an arbitrary function $\mathcal{G}(u,v)$:for one common convention. Since a generic conformal field theory has unrestricted dynamical dependence on both cross-ratios, no lower-dimensional set determines the function. Therefore

For a scalar operator of dimension $\Delta =2$ in a boundary theory of dimension $d=3$, the AdS scalar mass--dimension relation givesin unit-radius ${\mathrm{AdS}}_4$. In Poincare coordinates,so the free Klein-Gordon equation isThis mass is exactly that of a conformally coupled scalar field in four-dimensional anti-de Sitter spacetime. The Weyl transformationtherefore maps the equation to the flat half-space wave equation

The AdS scalar-field boundary asymptotics areThus the holographic dictionary becomes especially simple for the flat field $\psi =\varphi /z$:up to the overall normalization chosen for the bulk action and possible local counterterms. The leading mode is the source and the normalizable subleading mode is the response.

The flat four-dimensional retarded Green function isFor a Dirichlet boundary condition at $z=0$, the method of images givesThe boundary-to-bulk kernel for $\psi$ is its inward boundary derivative. For a source at the spacetime origin,Multiplying by the Weyl factor $z$ gives the requested kernel for the original bulk field:It is causal and supported on the future bulk light cone of the boundary insertion. Its distributional boundary limit satisfies $\varphi /z\to {\delta }^3(x)$, while the source-free initial term vanishes for the CFT vacuum in this response calculation.

On the $t=0$ slice, rotational symmetry lets the surface be written $z=z(r)$. The inversion symmetry fixing the boundary circle extends to an isometry of the Poincare half-space whose fixed set isThe unique extremal surface anchored at $r=R,z=0$ must be invariant under that isometry. Hence the Ryu–Takayanagi surface for a disk is the hemisphereOne may verify directly that it extremizes the area functional $2\pi \int r{z}^{-2}\sqrt{1+z^{\prime 2}}dr$.

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.

Translation invariance in $y$ lets the surface be written $z=z(x)$. Per unit $y$ length, its area and entropy functionals areThe boundary conditions are $z(\pm R)=0$, understood with an ultraviolet cutoff, and reflection symmetry gives $z^{\prime }(0)=0$ at $z(0)=z_{\max }$.

The Lagrangian $\mathcal{L}=z^{-2}\sqrt{1+z^{\prime 2}}$ has no explicit $x$ dependence, so its Beltrami identity is conserved:At the turning point $z=z_{\max }$ and $z^{\prime }=0$, the constant is $z_{\max }^{-2}$. Rearrangement gives

For the disk, the hemisphere reachesFor half of the strip surface, invert the first-order equation and integrate from the midpoint to the boundary:The beta function integral isThereforeThe strip's holographic entropy surface lies deeper in the bulk than that of a disk with the same half-width or radius.