Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 354 / 1

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 $1/N$.