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The AdS scalar-field boundary asymptotics areThus the holographic dictionary becomes especially simple for the flat field $\psi =\varphi /z$:up to the overall normalization chosen for the bulk action and possible local counterterms. The leading mode is the source and the normalizable subleading mode is the response.

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.

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

Choose the Fefferman--Graham coordinates radial variable fromNear the boundary,Consequently, through $O(Z^2)$,By the holographic dictionary, the leading boundary value of $A_{\tau }$ is the source for charge density. Thus $\mu$ is the dimensionless holographic chemical potential; the coefficient of the normalizable linear term is proportional to the CFT charge density, $\rho =\mu L/(4\pi G)$ with the action normalization and outward-orientation convention used here.