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

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.