Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 354 (Gauge/Gravity Duality)

Under the state–operator correspondence, let $|O_{ij}⟩$ be the state of the antisymmetric conformal primary operator $O_{ij}$. In radial quantization $P_i^{†}=K_i$, while primarity gives $K_i|O_{jk}⟩=0$. The norm of the level-one descendant obtained by taking a divergence is therefore determined by the conformal algebra commutatorUsing the two-form action of $M_{ab}$ gives, up to a positive normalization,Positivity of norm yields the two-form conformal unitarity boundAt saturation the descendant is null, and the operator obeys the conservation equation ${\partial }^i{O}_{ij}=0$.

In $d=3$, Hodge duality turns the two-form into the vector primary $V_k={ϵ}_{kij}{O}^{ij}/2$. The vector divergence descendant has norm proportional to $\Delta -d+1=\Delta -2$. Consequently the stronger bound israther than the formal two-form value $d-2=1$.

Varying the two-form action gives the gauge-invariant equationChoose radial gauge $A_{zi}=0$ and boundary-transverse gauge ${\partial }^i{A}_{ij}=0$. In the Poincare patch of ${\mathrm{AdS}}_6$, $\sqrt{-g}=z^{-6}$, while raising the three indices of $F$ contributes $z^6$. The boundary components consequently obeyThe near-boundary indicial equation is $\alpha (\alpha -1)=0$, so

Under the bulk dilation $(z,x)\mapsto (\lambda z,\lambda x)$, a two-form component carries two powers of inverse length. Thus the leading coefficient has dimension $2$, while the coefficient of $z$ has dimension $3$:The leading freely specifiable, nonnormalizable coefficient is the CFT source $J_{ij}$. The subleading normalizable coefficient is its canonical response and determines $⟨O_{ij}⟩$. This agrees with the saturated two-form unitarity bound ${\Delta }_O=d-2=3$ in the five-dimensional boundary CFT and with the holographic dictionary for a massless bulk gauge field.

Translation invariance supplies the momentum-conserving delta function. Lorentz invariance, symmetry of $T_{ij}$, stress-tensor conservation, and tracelessness then fix the remaining tensor structure to the transverse traceless spin-two projector shown in the question, up to an overall theory-dependent coefficient. Since the stress-energy tensor has scaling dimension $d$, its momentum transform has dimension zero. The delta function has momentum dimension $-d$, so scale invariance requires

For the Wightman function, the spectral condition restricts support to the appropriate future-directed timelike momenta, with convention-dependent distributions on the null boundary. It vanishes for spacelike momentum. A time-ordered or Euclidean correlator is obtained by analytic continuation and exists more broadly, but polynomial contact terms are renormalization-scheme dependent; in even dimensions the nonlocal power is accompanied by a logarithm.

In a large-$N$ holographic CFT, the single-trace state $T_{ij}(p)|0⟩$ is dual at leading order to a one-graviton bulk state with the matching boundary momentum and polarization. Multiparticle intermediate states and graviton interactions enter at subleading orders in $1/N$.

Choose a constant-time boundary interval whose endpoints differ by $\pi$ in $\varphi$. Its Ryu–Takayanagi formula surface is the diameter through $r=0$. Cutting it off at $r_c=\pi /2-ϵ$, its length isThe holographic entanglement entropy is thereforewhere the Brown--Henneaux central charge $c=3R_{\mathrm{AdS}}/(2G)$ was used.

Let $R=R_{\mathrm{AdS}}$ and $r_c=\pi /2-ϵ$. For ${\mathrm{AdS}}_3$, $\mathcal{R}=-6/R^2$ and $\Lambda =-1/R^2$, whileThe bulk term isWith the inward normal $n^r=-\cos r/R$, the induced metric has $\sqrt{-h}=R^2\sin r/{\cos }^{2}r$ andThe Gibbons–Hawking–York boundary term is consequentlyThus, with the orientation and Lorentzian signs displayed in the question,

The divergence is local in the induced boundary metric and is cancelled by the leading holographic renormalization countertermOn the regulated cylinder this iswhich cancels the ${ϵ}^{-2}$ divergence of $I_{\mathrm{GR}}$. An overall sign changes if one uses the oppositely signed Euclidean generating functional; the local counterterm changes with it.

The stated finite valueis the Casimir energy of a two-dimensional conformal field theory on the unit spatial circle. The plane-to-cylinder conformal map shifts the Hamiltonian by $-c/12$, so the global ${\mathrm{AdS}}_3$ vacuum is dual to the CFT cylinder vacuum rather than to a state of zero cylinder energy. The sign assigned directly to the Lorentzian on-shell action depends on whether it is identified with the action density or with the vacuum-energy generating functional; the universal boundary datum is the vacuum energy $E_0=-c/12$ supplied in the question.