since a constant scalar $\phi$ is not normalizable in noncompact spacetimes. This is the source of quantum inequivalence from our point of view.

[*A priori*]{} any field with gauge symmetry might possess one or more boundary modes. For example, in global AdS$_2$ all fields with a gauge symmetry have them [@Larsen:2014bqa; @Camporesi:1994ga; @Sen:2011ba]. On the other hand, in AdS$_{d+1}$ with higher $d$ it was found by explicit construction in global AdS$_{d+1}$ that boundary modes exist only for $\frac{d+1}{2}$-forms [@Camporesi:0000; @Bhattacharyya:2012ye]. In AdS$_4$ those are precisely the $2$-forms that we are interested in.

To make the discussion explicit we write the background AdS$_4$ metric $$\label{eq:globads}
ds^2_4 = \ell^2_A\left(d\rho^2 + \sinh^2\rho d\Omega^2_3\right)~.$$ We take the AdS$_4$ radius $\ell_A=1$ in the remainder of this section to avoid cluttered formulae. The normalized boundary modes in this background are $$\begin{aligned}
\label{eq:boundarybmodes}
B_{\rho i} &= \sqrt{\frac{k+1}{2}}\frac{1}{\sinh\rho}\tanh^{k+1}(\rho/2)\Theta^{(k,\sigma)}_i(\Omega_3)~, \\ \nonumber
B_{i j } & = \sqrt{\frac{1}{2(k+1)}}\tanh^{k+1}(\rho/2) [\tilde{\nabla}_ i \Theta^{(k,\sigma)}_j(\Omega_3) - \tilde{\nabla}_ j \Theta^{(k,\sigma)}_i(\Omega_3)]~.\end{aligned}$$ for $k=1,2,...$. The covariant derivative $\tilde{\nabla}$ refers to components along $S^3$ and latin indices represent these angular components. The $1$-form field $\Theta^{(k,\sigma)}_i(\Omega_3)$ is a vector spherical harmonic with eigenvalue of the Hodge de Rham operator $=(k+1)^2$. The quantum numbers $k,\sigma$ are analogous to the numbers 