Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 354 / 1 / b

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,