Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 354 (Gauge/Gravity Duality)

Let $|\mathcal{O}⟩$ be the state associated with a scalar conformal primary operator of scaling dimension $\Delta$. In radial quantization,and $P_a^{†}=-K_a$. The norm of a level-one conformal descendant isUnitarity first gives $\Delta \ge 0$. If $\Delta =0$, every $P_a|\mathcal{O}⟩$ is null, so the local operator is translation invariant and belongs to the identity conformal family. Excluding the identity therefore gives $\Delta >0$.

Now consider the scalar level-two descendant $P^2|\mathcal{O}⟩$. The conformal algebra and the scalar-primary conditions giveApplying the second $K_a$ and summing over $a$ yieldsPositivity of this norm, together with $\Delta >0$, proves the scalar conformal unitarity boundAt equality the level-two descendant is null; in position space this is the free scalar equation of motion.

For the Poincaré metric on ${\mathrm{AdS}}_6$,The massless Klein-Gordon equation, evaluated with the Laplace-Beltrami operator, isorA boundary power $\varphi \sim z^{\alpha }$ obeys the indicial equationThusThe constant branch is the nonnormalizable source and the $z^5$ branch is the normalizable response. Since $m^2=\Delta (\Delta -d)=0$ with $d=5$, the roots are ${\Delta }_{-}=0$ and ${\Delta }_{+}=5$. Only standard quantization is unitary here: the putative alternative operator of dimension zero would be the identity, whereas the field is a nontrivial fluctuating operator. The permissible conformally invariant source-free boundary condition is therefore Dirichlet,More generally one may prescribe $J$ as an external source. The holographic dictionary gives

For a boundary-independent source, the AdS scalar-field boundary asymptotics reduce toOne may therefore extract the source and response directly:The operator expectation value is proportional to $A$; choosing the boundary-operator normalization $\mathcal{O}=A$ gives the requested expressions. A conventional action normalization can instead multiply this relation by the fixed factor $2\Delta -d=5$.

When $J(x)$ varies, the radial wave equation recursively generates local derivative terms before the normalizable response. SubstitutingintogivesBecause the boundary dimension is odd, no logarithmic term is required. Holographic renormalization subtracts these source-dependent pieces, leavingin the normalization $\mathcal{O}=A$. Equivalently,

The state–operator correspondence maps $\mathcal{O}$ to the lowest one-particle state on $S^4\times \mathbb{R}$. On a sphere of radius $r$, cylinder energy equals scaling dimension divided by $r$. The primary has $\Delta =5$.

A translation generator raises the dimension by one and transforms as a vector of $SO(5)$. States with no angular momentum arise from scalar descendant pairs $P^2$, so the allowed descendants areTheir dimensions are $5+2n$, and hence the one-quantum, zero-angular-momentum energies areThese are the $l=0$ normal-mode energies of a massless scalar in global Anti-de Sitter spacetime.

The Ryu–Takayanagi formula gives the leading entropy of a static boundary region:On the $t=0$ slice of Poincaré ${\mathrm{AdS}}_4$,The minimal surface anchored on a boundary circle of radius $R_1$ is the hemisphereCutting it off at $z=ϵ$, its area isThereforeThe universal requirement was the perimeter-law divergence $\pi R_1/(2Gϵ)$; the finite constant depends on the stated pure ${\mathrm{AdS}}_4$ vacuum geometry and equals $-\pi /(2G)$ here. Since $G\sim N^{-3/2}$, both terms are of order $N^{3/2}$.

Let ${\gamma }_1$ and ${\gamma }_2$ be the individual Ryu–Takayanagi surfaces for $D_1$ and $D_2$. Their disconnected union ${\gamma }_1\cup {\gamma }_2$ is homologous to $D_1\cup D_2$ and is therefore an admissible competitor in the minimization that defines ${\gamma }_{12}$. Minimality givesDividing by $4G$ provesand hence the leading holographic mutual information obeysThis is the geometric realization of the nonnegativity of quantum mutual information.

There are two competing extremal-surface topologies. The disconnected candidate ${\gamma }_1\cup {\gamma }_2$ has area independent of the separation $L$ and gives $I_c=0$. A connected surface joining the two boundary circles can have smaller area when the gap is small; as $L\to 0$, short-distance entanglement across the nearby boundaries makes $I_c$ positive and divergent.

Increasing $L$ makes the connected candidate less favorable. At a critical separation $L=C(R_1,R_2)$ its area equals the disconnected area, and beyond that point the disconnected candidate is minimal. This entanglement-wedge phase transition gives

Write $S_{\mathrm{conn}}(L)$ for the connected candidate andfor the $L$-independent disconnected candidate. The physical entropy is the lower envelopeNear $L=C$, the connected branch rises to meet the horizontal disconnected branch. The entropy is continuous but generically has a kink there: its derivative jumps from the positive slope of $S_{\mathrm{conn}}$ to zero. Correspondingly, $I_c=S_{\mathrm{disc}}-S(D_1\cup D_2)$ falls to zero and remains zero.

A pair of boundary circles has one dimensionless invariant under the conformal field theory group. If their radii are $R_1,R_2$ and their center separation isa convenient invariant is the inversive distanceThe choice between connected and disconnected bulk surfaces can depend only on $\chi$. Let the transition occur at the single theory-independent numerical value $\chi =X$, with $X>1$. ThensoAll dependence on the two radii is fixed by conformal symmetry; only the pure number $X$ requires the explicit minimal-surface calculation.