For the Poincaré metric on $\operatorname{AdS}_6$,
$$
\sqrt{|g|}=z^{-6},\qquad g^{ab}=z^2\eta^{ab}.
$$
The massless <Klein-Gordon equation>, evaluated with the <Laplace-Beltrami operator>, is
$$
\nabla^2\phi
=z^6\partial_z(z^{-4}\partial_z\phi)+z^2\Box_5\phi=0,
$$
or
$$
\boxed{z^2\phi_{zz}-4z\phi_z+z^2\Box_5\phi=0}.
$$
A boundary power $\phi\sim z^\alpha$ obeys the indicial equation
$$
\alpha(\alpha-5)=0.
$$
Thus
$$
\phi(z,x)=J(x)+z^5A(x)+\cdots.
$$
The 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,
$$
\boxed{J=0,\qquad \phi=O(z^5)}.
$$
More generally one may prescribe $J$ as an external source. The <holographic dictionary> gives
$$
\boxed{[\mathcal O]=5,\qquad [J]=d-\Delta=0}.
$$

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-354/1/b/i/solution