$\displaystyle\partial_{r}^{2}\Phi^{a}\,+\,(D-1)\partial_{r}{A}\partial_{r}\Phi^{a}\,+\,{\cal
G}^{a}_{\,\,\,\,bc}\partial_{r}\Phi^{b}\partial_{r}\Phi^{c}\,-\,\mathcal{V}^{a}$
$\displaystyle=$ $\displaystyle 0\,,$ (311)

where the sigma-model derivatives are given by $\mathcal{V}^{a}\equiv
G^{ab}\partial_{b}\mathcal{V}$, and
$\partial_{b}\mathcal{V}\equiv\frac{\partial\mathcal{V}}{\partial\Phi^{b}}$.
The Einstein equations reduce to

$\displaystyle(D-1)(\partial_{r}{A})^{2}\,+\,\partial_{r}^{2}{A}\,+\,\frac{4}{D-2}\mathcal{V}$
$\displaystyle=$ $\displaystyle 0\,,$ (312)
$\displaystyle(D-1)(D-2)(\partial_{r}{A})^{2}\,-\,2G_{ab}\partial_{r}\Phi^{a}\partial_{r}\Phi^{b}\,+\,4\mathcal{V}$
$\displaystyle=$ $\displaystyle 0\,.$ (313)

####  B.1.1 Tensor and scalar fluctuations

Following Refs. Bianchi:2003ug ; Berg:2006xy ; Berg:2005pd ; Elander:2009bm ;
Elander:2010wd , the scalar fields can be written as

$\Phi^{a}(x^{\mu},r)=\Phi^{a}(r)+\varphi^{a}(x^{\mu},r)\,,$ (314)

where $\varphi^{a}(x^{\mu},r)$ are small fluctuations around the background
solutions, $\Phi^{a}(r)$. The metric fluctuations are decomposed with the ADM
formalism Arnowitt:1962hi :

$\displaystyle\mbox{d}s_{D}^{2}$ $\displaystyle=$
$\displaystyle\left((1+\nu)^{2}+\nu_{\sigma}\nu^{\sigma}\right)\mbox{d}r^{2}+2\nu_{\mu}\mbox{d}x^{\mu}\mbox{d}r+e^{2{A}(r)}\left(\eta_{\mu\nu}+h_{\mu\nu}\right)\mbox{d}x^{\mu}\mbox{d}x^{\nu}\,,$
(315)

where

$\displaystyle h^{\mu}{}_{\nu}$ $\displaystyle=$
$\displaystyle\mathfrak{e}^{\mu}{}_{\nu}+iq^{\mu}\epsilon_{\nu}+iq_{\nu}\epsilon^{\mu}+\frac{q^{\mu}q_{\nu}}{q^{2}}H+\frac{1}{D-2}\delta^{\mu}{}_{\nu}h.$
(316)

Here, $\nu(x^{\mu},r)$, $\nu^{\mu}(x^{\mu},r)$,
$\mathfrak{e}^{\mu}{}_{\nu}(x^{\mu},r)$, $\epsilon^{\mu}(x^{\mu},r)$,
$H(x^{\mu},r)$, and $h(x^{\mu},r)$ are small fluctuations around the
background metric, of which $\mathfrak{e}^{\mu}{}_{\nu}$ is transverse and
traceless (while $\epsilon^{\mu}$ is transverse), and gauge invariant.

The other (diffeomorphism) gauge-invariant combinations are

$\displaystyle\mathfrak{a}^{a}$ $\displaystyle=$
$\displaystyle\varphi^{a}-\frac{\partial_{r}\Phi^{a}}{2(D-2)\partial_{r}{A}}h\,,$
(317) $\displaystyle\mathfrak{b}$ $\displaystyle=$
$\displaystyle\nu-\partial_{r}\left(\frac{h}{2(D-2)\partial_{r}{A}}\right)\,,$
(318) $\displaystyle\mathfrak{c}$ $\displaystyle=$ $\displaystyle
e^{-2{A}}\partial_{\mu}\nu^{\mu}-\frac{e^{-2{A}}q^{2}h}{2(D-2)\partial_{r}A}-\frac{1}{2}\partial_{r}H\,,$
(319) $\displaystyle\mathfrak{d}^{\mu}$ $\displaystyle=$ $\displaystyle
e^{-2{A}}P^{\mu}{}_{\nu}\nu^{\nu}-\partial_{r}\epsilon^{\mu}\,.$ (320)

The algebraic nature of the equations for $\mathfrak{b}$ and $\mathfrak{c}$
allows us to decouple the equations and solve them.

The tensor fluctuations $\mathfrak{e}^{\mu}{}_{\nu}$ obey the equation of
motion

$\left[\partial_{r}^{2}+(D-1)\partial_{r}{A}\partial_{r}-e^{-2{A}(r)}q^{2}\right]\mathfrak{e}^{\mu}_{\,\,\,\nu}=0\,,$
(321)

with boundary conditions given by

$\left.\frac{}{}\partial_{r}\mathfrak{e}^{\mu}_{\,\,\,\nu}\right|_{r=r_{i}}=0\,.$
(322)

The equation of motion for $\mathfrak{d}^{\mu}$ is algebraic and does not lead
to a spectrum of states. The equations of motion for $\mathfrak{a}^{a}$ obey
the following equations of motion:

$\displaystyle 0$ $\displaystyle=$ $\displaystyle\Big{[}{\cal
D}_{r}^{2}+(D-1)\partial_{r}{A}{\cal
D}_{r}-e^{-2{A}}q^{2}\Big{]}\mathfrak{a}^{a}\,\,$
$\displaystyle-\Big{[}{\mathcal{V}}^{\,\,a}{}_{\,|c}-\mathcal{R}^{a}{}_{bcd}\partial_{r}\Phi^{b}\partial_{r}\Phi^{d}+\frac{4(\partial_{r}\Phi^{a}{\mathcal{V}}^{\,b}+{\mathcal{V}}^{\,a}\partial_{r}\Phi^{b})G_{bc}}{(D-2)\partial_{r}{A}}+\frac{16{\mathcal{V}}\partial_{r}\Phi^{a}\partial_{r}\Phi^{b}G_{bc}}{(D-2)^{2}(\partial_{r}{A})^{2}}\Big{]}\mathfrak{a}^{c}\,,$

while the boundary conditions are given by

$\displaystyle\frac{2e^{2A}\partial_{r}\Phi^{a}}{(D-2)q^{2}\partial_{r}{A}}\left[\partial_{r}\Phi^{b}{\cal
D}_{r}-\frac{4{\cal V}\partial_{r}\Phi^{b}}{(D-2)\partial_{r}{A}}-{\cal
V}^{b}\right]\mathfrak{a}_{b}-\mathfrak{a}^{a}\Big{|}_{r_{i}}=0\,.$ (324)

The background covariant derivative is
$\mathcal{D}_{r}\mathfrak{a}^{a}\equiv\partial_{r}\mathfrak{a}^{a}+\mathcal{G}^{a}_{\
bc}\partial_{r}\Phi^{b}\mathfrak{a}^{c}$, and
${\mathcal{V}}^{a}{}_{|b}\equiv\frac{\partial{\mathcal{V}}^{a}}{\partial\Phi^{b}}+\mathcal{G}^{a}_{\
bc}{\mathcal{V}}^{c}$.

###  B.2 Vectors, pseudoscalars, and spurions

In this appendix, we provide the gauge fixing terms and equations of motion
for the spin-1 and spin-0 fluctuations introduced in Sect. IV. The resulting
equations for the fields defined in Eq. (130) (except for $\chi_{M}$) and
other fields interacting with them are distributed in the following
subsections. Section B.2.1 reports the equations of motion and boundary
conditions for $\mathcal{B}_{M}{}^{\tilde{\mathcal{A}}}$ and
$\mathcal{B}_{M}{}^{\hat{\mathcal{A}}}$, and for the associated spin-0 states.
Section B.2.2 discusses $\mathcal{A}_{M}{}^{4}$ and Sect. B.2.3 focuses on
$\mathcal{A}_{M}{}^{\bar{\mathcal{A}}}$, respectively.

####  B.2.1 The $\mathcal{B}_{M}{}^{\hat{\mathcal{A}}}$ and
$\mathcal{B}_{M}{}^{\tilde{\mathcal{A}}}$ sectors

Following the procedure in Ref. Elander:2018aub , we choose the gauge fixing
terms for $\mathcal{B}_{M}{}^{\hat{\mathcal{A}}}$ and
$\mathcal{B}_{M}{}^{\tilde{\mathcal{A}}}$ to be

$\displaystyle\mathcal{S}_{\hat{\xi}}^{(1)}$ $\displaystyle=$
$\displaystyle\int\mathrm{d}^{4}q\mathrm{~{}d}r\left\\{-\frac{H^{(1)}_{\hat{\mathcal{A}}\hat{\mathcal{A}}}}{2\hat{\xi}}\left[iq^{\mu}\mathcal{B}_{\mu}{}^{\hat{\mathcal{A}}}(-q)-\frac{g\sin(v)}{2v}\frac{\hat{\xi}}{H^{(1)}_{\hat{\mathcal{A}}\hat{\mathcal{A}}}}G^{(1)}_{\hat{\mathcal{A}}\hat{\mathcal{A}}}e^{2A}\pi^{\hat{\mathcal{A}}}(-q)-\frac{\hat{\xi}}{H^{(1)}_{\hat{\mathcal{A}}\hat{\mathcal{A}}}}\partial_{r}\left(H^{(1)}_{\hat{\mathcal{A}}\hat{\mathcal{A}}}e^{2A}\mathcal{B}_{5}{}^{\hat{\mathcal{A}}}(-q)\right)\right]\times\right.$
(325)
$\displaystyle\times{\left.\left[-iq^{\nu}\mathcal{B}_{\nu}{}^{\hat{\mathcal{A}}}(q)-\frac{g\sin(v)}{2v}\frac{\hat{\xi}}{H^{(1)}_{\hat{\mathcal{A}}\hat{\mathcal{A}}}}G^{(1)}_{\hat{\mathcal{A}}\hat{\mathcal{A}}}e^{2A}\pi^{\hat{\mathcal{A}}}(q)-\frac{\hat{\xi}}{H^{(1)}_{\hat{\mathcal{A}}\hat{\mathcal{A}}}}\partial_{r}\left(H^{(1)}_{\hat{\mathcal{A}}\hat{\mathcal{A}}}e^{2A}\mathcal{B}_{5}{}^{\hat{\mathcal{A}}}(q)\right)\right]\right\\}\,,}$

and

$\displaystyle\mathcal{S}_{\tilde{\xi}}^{(1)}$ $\displaystyle=$
$\displaystyle\int\mathrm{d}^{4}q\mathrm{~{}d}r\left\\{-\frac{H^{(1)}_{\tilde{\mathcal{A}}\tilde{\mathcal{A}}}}{2\tilde{\xi}}\left[iq^{\mu}\mathcal{B}_{\mu}{}^{\tilde{\mathcal{A}}}(-q)-\frac{\tilde{\xi}}{H^{(1)}_{\tilde{\mathcal{A}}\tilde{\mathcal{A}}}}\partial_{r}\left(H^{(1)}_{\tilde{\mathcal{A}}\tilde{\mathcal{A}}}e^{2A}\mathcal{B}_{5}{}^{\tilde{\mathcal{A}}}(-q)\right)\right]\times\right.$
(326)
$\displaystyle\times{\left.\left[-iq^{\nu}\mathcal{B}_{\nu}{}^{\tilde{\mathcal{A}}}(q)-\frac{\tilde{\xi}}{H^{(1)}_{\tilde{\mathcal{A}}\tilde{\mathcal{A}}}}\partial_{r}\left(H^{(1)}_{\tilde{\mathcal{A}}\tilde{\mathcal{A}}}e^{2A}\mathcal{B}_{5}{}^{\tilde{\mathcal{A}}}(q)\right)\right]\right\\}\,,}$

where $\hat{\xi}$ and $\tilde{\xi}$ are gauge-fixing parameters. The boundary-
localised gauge fixing terms at $r=r_{2}$ are

$\displaystyle\mathcal{S}_{\hat{M}}^{(1)}=$
$\displaystyle\int\mathrm{d}^{4}q\mathrm{~{}d}r\delta\left(r-r_{2}\right)\left\\{-\frac{1}{2\hat{M}_{2}}\left[iq^{\mu}\mathcal{B}_{\mu}{}^{\hat{\mathcal{A}}}(-q)+\hat{M}_{2}H^{(1)}_{\hat{\mathcal{A}}\hat{\mathcal{A}}}e^{2A}\mathcal{B}_{5}{}^{\hat{\mathcal{A}}}(-q)-\hat{M}_{2}K_{5}\frac{gv_{5}}{2}e^{2A}\cos(v)P^{\hat{\mathcal{A}}}_{5}(-q)\right.\right.$
(327)
$\displaystyle\left.\left.-\frac{g\sin(v)}{2v}\hat{M}_{2}C_{2\,\hat{\mathcal{A}}\hat{\mathcal{A}}}^{(1)}e^{2A}\pi^{\hat{\mathcal{A}}}(-q)\right]\times\left[(q\rightarrow-q)\right]\right\\}\,,$

and

$\displaystyle\mathcal{S}_{\tilde{M}}^{(1)}=$
$\displaystyle\int\mathrm{d}^{4}q\mathrm{~{}d}r\delta\left(r-r_{2}\right)\left\\{-\frac{1}{2\tilde{M}_{2}}\left[iq^{\mu}\mathcal{B}_{\mu}{}^{\tilde{\mathcal{A}}}(-q)+\tilde{M}_{2}H^{(1)}_{\tilde{\mathcal{A}}\tilde{\mathcal{A}}}e^{2A}\mathcal{B}_{5}{}^{\tilde{\mathcal{A}}}(-q)+\tilde{M}_{2}K_{5}\frac{gv_{5}}{2}e^{2A}\sin(v)P^{\hat{\mathcal{A}}}_{5}(-q)\right]\right.\times$
(328) $\displaystyle\left.\times\left[(q\rightarrow-q)\right]\right\\}\,.$

The boundary-localised gauge fixing parameters, $\hat{M}_{2}$ and
$\tilde{M}_{2}$, are independent of the bulk dynamics. Gauge fixing at
$r=r_{1}$ can be done in a similar manner. Gathering terms from the action Eq.
(123) and gauge fixing terms in Eqs. (325), (326), (327), and (328), the
equations of motion and boundary conditions for
$\mathcal{B}_{\mu}{}^{\hat{\mathcal{A}}}$ and
$\mathcal{B}_{\mu}{}^{\tilde{\mathcal{A}}}$ read

$\displaystyle 0=$
$\displaystyle{\left[q^{2}H^{(1)}_{\hat{\mathcal{A}}\hat{\mathcal{A}}}-\partial_{r}\left(H^{(1)}_{\hat{\mathcal{A}}\hat{\mathcal{A}}}e^{2A}\partial_{r}\right)+\left(\frac{g}{2}\right)^{2}G^{(1)}_{\hat{\mathcal{A}}\hat{\mathcal{A}}}e^{2A}\right]P^{\mu\nu}\mathcal{B}_{\mu}{}^{\hat{\mathcal{A}}}(q,r)},$
(329) $\displaystyle 0=$
$\displaystyle{\left.\left[q^{2}D_{2\,\hat{\mathcal{A}}\hat{\mathcal{A}}}^{(1)}+H^{(1)}_{\hat{\mathcal{A}}\hat{\mathcal{A}}}e^{2A}\partial_{r}+e^{2A}\left(\frac{g}{2}\right)^{2}C_{2\,\hat{\mathcal{A}}\hat{\mathcal{A}}}^{(1)}+e^{2A}(\frac{gv_{5}}{2})^{2}K_{5}\cos(v)^{2}\right]P^{\mu\nu}\mathcal{B}_{\nu}{}^{\hat{\mathcal{A}}}(q,r)\right|_{r=r_{2}}}$
$\displaystyle+\left.\left[q^{2}D_{2\,\hat{\mathcal{A}}\tilde{\mathcal{A}}}^{(1)}-e^{2A}(\frac{gv_{5}}{2})^{2}K_{5}\cos(v)\sin(v)\right]P^{\mu\nu}\mathcal{B}_{\nu}{}^{\tilde{\mathcal{A}}}(q,r)\right|_{r=r_{2}}\,,$
(330) $\displaystyle 0=$
$\displaystyle{\left[\frac{q^{2}}{\hat{\xi}}H^{(1)}_{\hat{\mathcal{A}}\hat{\mathcal{A}}}-\partial_{r}\left(H^{(1)}_{\hat{\mathcal{A}}\hat{\mathcal{A}}}e^{2A}\partial_{r}\right)+\left(\frac{g}{2}\right)^{2}G^{(1)}_{\hat{\mathcal{A}}\hat{\mathcal{A}}}e^{2A}\right]\frac{q^{\mu}q^{\nu}}{q^{2}}\mathcal{B}_{\mu}{}^{\hat{\mathcal{A}}}(q,r)},$
(331) $\displaystyle 0=$
$\displaystyle{\left.\left[\frac{q^{2}}{\hat{M}_{2}}+H^{(1)}_{\hat{\mathcal{A}}\hat{\mathcal{A}}}e^{2A}\partial_{r}+e^{2A}\left(\frac{g}{2}\right)^{2}C_{2\,\hat{\mathcal{A}}\hat{\mathcal{A}}}^{(1)}+e^{2A}(\frac{gv_{5}}{2})^{2}K_{5}\cos(v)^{2}\right]\frac{q^{\mu}q^{\nu}}{q^{2}}\mathcal{B}_{\nu}{}^{\hat{\mathcal{A}}}(q,r)\right|_{r=r_{2}}}$
$\displaystyle{\left.-\left[e^{2A}(\frac{gv_{5}}{2})^{2}K_{5}\cos(v)\sin(v)\right]\frac{q^{\mu}q^{\nu}}{q^{2}}\mathcal{B}_{\nu}{}^{\tilde{\mathcal{A}}}(q,r)\right|_{r=r_{2}}}\,,$
(332) $\displaystyle 0=$
$\displaystyle{\left[q^{2}H^{(1)}_{\tilde{\mathcal{A}}\tilde{\mathcal{A}}}-\partial_{r}\left(H^{(1)}_{\tilde{\mathcal{A}}\tilde{\mathcal{A}}}e^{2A}\partial_{r}\right)\right]P^{\mu\nu}\mathcal{B}_{\mu}{}^{\tilde{\mathcal{A}}}(q,r)}\,,$
(333) $\displaystyle 0=$
$\displaystyle{\left.\left[q^{2}D_{2\,\tilde{\mathcal{A}}\tilde{\mathcal{A}}}^{(1)}+H^{(1)}_{\tilde{\mathcal{A}}\tilde{\mathcal{A}}}e^{2A}\partial_{r}+e^{2A}(\frac{gv_{5}}{2})^{2}K_{5}\sin(v)^{2}\right]P^{\mu\nu}\mathcal{B}_{\nu}{}^{\tilde{\mathcal{A}}}(q,r)\right|_{r=r_{2}}}$
$\displaystyle{\left.+\left[q^{2}D_{2\,\tilde{\mathcal{A}}\hat{\mathcal{A}}}^{(1)}-e^{2A}(\frac{gv_{5}}{2})^{2}K_{5}\cos(v)\sin(v)\right]P^{\mu\nu}\mathcal{B}_{\nu}{}^{\hat{\mathcal{A}}}(q,r)\right|_{r=r_{2}}}\,,$
(334) $\displaystyle 0=$
$\displaystyle{\left[\frac{q^{2}}{\tilde{\xi}}H^{(1)}_{\tilde{\mathcal{A}}\tilde{\mathcal{A}}}-\partial_{r}\left(H^{(1)}_{\tilde{\mathcal{A}}\tilde{\mathcal{A}}}e^{2A}\partial_{r}\right)\right]\frac{q^{\mu}q^{\nu}}{q^{2}}\mathcal{B}_{\mu}{}^{\tilde{\mathcal{A}}}(q,r)}\,,$
(335) $\displaystyle 0=$
$\displaystyle{\left.\left[\frac{q^{2}}{\tilde{M}_{2}}+H^{(1)}_{\tilde{\mathcal{A}}\tilde{\mathcal{A}}}e^{2A}\partial_{r}+e^{2A}(\frac{gv_{5}}{2})^{2}K_{5}\sin(v)^{2}\right]\frac{q^{\mu}q^{\nu}}{q^{2}}\mathcal{B}_{\nu}{}^{\tilde{\mathcal{A}}}(q,r)\right|_{r=r_{2}}}$
$\displaystyle-{\left.\left[e^{2A}(\frac{gv_{5}}{2})^{2}K_{5}\cos(v)\sin(v)\right]\frac{q^{\mu}q^{\nu}}{q^{2}}\mathcal{B}_{\nu}{}^{\hat{\mathcal{A}}}(q,r)\right|_{r=r_{2}}}\,.$
(336)

Equations (329), (331), (333), and (334) can be compared to Eqs. (154), (155),
and (157) in the body of the paper.

For pseudoscalar and spurion fields, the equations are obtained from the
variation with respect to $\mathcal{B}_{5}{}^{\tilde{\mathcal{A}}}$,
$\mathcal{B}_{5}{}^{\hat{\mathcal{A}}}$, $\pi^{\hat{\mathcal{A}}}$ in the bulk
and boundary and $P_{5}^{\hat{\mathcal{A}}}$ in the boundary, respectively:

$\displaystyle 0=$
$\displaystyle{\left[q^{2}-\partial_{r}\left(\frac{\tilde{\xi}}{H^{(1)}_{\tilde{\mathcal{A}}\tilde{\mathcal{A}}}}\partial_{r}\left(H^{(1)}_{\tilde{\mathcal{A}}\tilde{\mathcal{A}}}e^{2A}\right)\right)\right]\mathcal{B}_{5}{}^{\tilde{\mathcal{A}}}(q,r)}\,,$
(337) $\displaystyle 0=$
$\displaystyle{\left.\left[\tilde{\xi}\frac{e^{-2A}}{H^{(1)}_{\tilde{\mathcal{A}}\tilde{\mathcal{A}}}}\partial_{r}\left(H^{(1)}_{\tilde{\mathcal{A}}\tilde{\mathcal{A}}}e^{2A}\right)+\tilde{M}_{2}H^{(1)}_{\tilde{\mathcal{A}}\tilde{\mathcal{A}}}\right]\mathcal{B}_{5}{}^{\tilde{\mathcal{A}}}(q,r)\right|_{r=r_{2}}+\left.\left[\frac{gv_{5}}{2}\tilde{M}_{2}K_{5}\sin(v)\right]P_{5}^{\hat{\mathcal{A}}}(q)\right|_{r=r_{2}}}\,,$
(338) $\displaystyle 0=$
$\displaystyle{\left[q^{2}H^{(1)}_{\hat{\mathcal{A}}\hat{\mathcal{A}}}-H^{(1)}_{\hat{\mathcal{A}}\hat{\mathcal{A}}}\partial_{r}\left(\frac{\hat{\xi}}{H^{(1)}_{\hat{\mathcal{A}}\hat{\mathcal{A}}}}\partial_{r}\left(H^{(1)}_{\hat{\mathcal{A}}\hat{\mathcal{A}}}e^{2A}\right)\right)+\frac{g}{2}^{2}G^{(1)}_{\hat{\mathcal{A}}\hat{\mathcal{A}}}e^{2A}\right]\mathcal{B}_{5}{}^{\hat{\mathcal{A}}}(q,r)}$
$\displaystyle+{\frac{g\sin(v)}{2v}\left[G^{(1)}_{\hat{\mathcal{A}}\hat{\mathcal{A}}}e^{2A}\partial_{r}-H^{(1)}_{\hat{\mathcal{A}}\hat{\mathcal{A}}}\partial_{r}\left(\hat{\xi}\frac{G^{(1)}_{\hat{\mathcal{A}}\hat{\mathcal{A}}}e^{2A}}{H^{(1)}_{\hat{\mathcal{A}}\hat{\mathcal{A}}}}\right)\right]\pi^{\hat{\mathcal{A}}}(q,r)}\,,$
(339) $\displaystyle 0=$
$\displaystyle{\left.\left[\frac{e^{-2A}}{H^{(1)}_{\tilde{\mathcal{A}}\tilde{\mathcal{A}}}}\partial_{r}\left(\hat{\xi}H^{(1)}_{\hat{\mathcal{A}}\hat{\mathcal{A}}}e^{2A}\right)+\hat{M}_{2}H^{(1)}_{\hat{\mathcal{A}}\hat{\mathcal{A}}}\right]\mathcal{B}_{5}{}^{\hat{\mathcal{A}}}(q,r)-\left[\frac{gv_{5}}{2}\hat{M}_{2}\cos(v)K_{5}\right]P_{5}^{\hat{\mathcal{A}}}(q)\right|_{r=r_{2}}}$
$\displaystyle+{\left.\frac{g\sin(v)}{2v}\left[\hat{\xi}\frac{G^{(1)}_{\hat{\mathcal{A}}\hat{\mathcal{A}}}}{H^{(1)}_{\tilde{\mathcal{A}}\tilde{\mathcal{A}}}}-\hat{M}_{2}C_{2\,\hat{\mathcal{A}}\hat{\mathcal{A}}}^{(1)}\right]\pi^{\hat{\mathcal{A}}}(q,r)\right|_{r=r_{2}}}\,,$
(340) $\displaystyle 0=$
$\displaystyle{\frac{\sin(v)}{v}\left[\partial_{r}\left(G^{(1)}_{\hat{\mathcal{A}}\hat{\mathcal{A}}}e^{4A}\partial_{r}\right)-G^{(1)}_{\hat{\mathcal{A}}\hat{\mathcal{A}}}e^{2A}q^{2}-\left(\frac{g}{2}\right)^{2}\frac{\hat{\xi}}{H^{(1)}_{\hat{\mathcal{A}}\hat{\mathcal{A}}}}e^{4A}(G^{(1)}_{\hat{\mathcal{A}}\hat{\mathcal{A}}})^{2}\right]\pi^{\hat{\mathcal{A}}}(q,r)}$
$\displaystyle+{\frac{g}{2}\left[\partial_{r}\left(G^{(1)}_{\hat{\mathcal{A}}\hat{\mathcal{A}}}e^{4A}\right)-\frac{\hat{\xi}}{H^{(1)}_{\hat{\mathcal{A}}\hat{\mathcal{A}}}}e^{2A}G^{(1)}_{\hat{\mathcal{A}}\hat{\mathcal{A}}}\partial_{r}\left(H^{(1)}_{\hat{\mathcal{A}}\hat{\mathcal{A}}}e^{2A}\right)\right]\mathcal{B}_{5}{}^{\hat{\mathcal{A}}}(q,r)}\,,$
(341) $\displaystyle 0=$
$\displaystyle{\left.\frac{\sin(v)}{v}\left[C_{2\,\hat{\mathcal{A}}\hat{\mathcal{A}}}^{(1)}e^{-2A}q^{2}+\hat{M}_{2}\left(\frac{g}{2}\right)^{2}(C_{2\,\hat{\mathcal{A}}\hat{\mathcal{A}}}^{(1)})^{2}+G^{(1)}_{\hat{\mathcal{A}}\hat{\mathcal{A}}}\partial_{r}\right]\pi^{\hat{\mathcal{A}}}(q,r)\right|_{r=r_{2}}}$
$\displaystyle+{\left.\left[\hat{M}_{2}v_{5}\left(\frac{g}{2}\right)^{2}\cos(v)K_{5}C_{2\,\hat{\mathcal{A}}\hat{\mathcal{A}}}^{(1)}\right]P_{5}^{\hat{\mathcal{A}}}(q)\right|_{r=r_{2}}-\left.\left[\frac{g}{2}\hat{M}_{2}(C_{2\,\hat{\mathcal{A}}\hat{\mathcal{A}}}^{(1)})H^{(1)}_{\hat{\mathcal{A}}\hat{\mathcal{A}}}-\frac{g}{2}G^{(1)}_{\hat{\mathcal{A}}\hat{\mathcal{A}}}\right]\mathcal{B}_{5}{}^{\hat{\mathcal{A}}}(q,r)\right|_{r=r_{2}}}\,,$
(342) $\displaystyle 0=$
$\displaystyle{\left.\left[K_{5}e^{-2A}q^{2}+\left(\frac{gv_{5}}{2}\right)^{2}K_{5}^{2}\left(\cos(v)^{2}\hat{M}_{2}+\sin(v)^{2}\tilde{M}_{2}\right)\right]P_{5}^{\hat{\mathcal{A}}}(q)\right|_{r=r_{2}}}$
$\displaystyle+\left.\left[\hat{M}_{2}K_{5}\left(\frac{gv_{5}}{2}\right)^{2}\frac{\cos(v)\sin(v)}{v}C_{2\,\hat{\mathcal{A}}\hat{\mathcal{A}}}^{(1)}\right]\pi^{\hat{\mathcal{A}}}(q,r)\right|_{r=r_{2}}$
$\displaystyle-{\left.\left[K_{5}\cos(v)\frac{gv_{5}}{2}\hat{M}_{2}H^{(1)}_{\hat{\mathcal{A}}\hat{\mathcal{A}}}\right]\mathcal{B}_{5}{}^{\hat{\mathcal{A}}}(q,r)\right|_{r=r_{2}}+\left.\left[K_{5}\sin(v)\frac{gv_{5}}{2}\tilde{M}_{2}H^{(1)}_{\tilde{\mathcal{A}}\tilde{\mathcal{A}}}\right]\mathcal{B}_{5}{}^{\tilde{\mathcal{A}}}(q,r)\right|_{r=r_{2}}}\,.$
(343)

We introduce convenient redefinitions

$\displaystyle\mathcal{B}_{5}{}^{\hat{\mathcal{A}}}$ $\displaystyle\equiv$
$\displaystyle\frac{X^{\hat{\mathcal{A}}}}{e^{4A}G^{(1)}_{\hat{\mathcal{A}}\hat{\mathcal{A}}}}-\frac{2}{g}\frac{\sin(v)}{v}\partial_{r}\pi^{\hat{\mathcal{A}}}\,,$
(344) $\displaystyle\pi^{\hat{\mathcal{A}}}$ $\displaystyle\equiv$
$\displaystyle\frac{v}{\sin(v)}\left(Y^{\hat{\mathcal{A}}}+\frac{(g/2)\partial_{r}X^{\hat{\mathcal{A}}}}{q^{2}e^{2A}G^{(1)}_{\hat{\mathcal{A}}\hat{\mathcal{A}}}}\right)\,,$
(345)

to separate the physical states from the gauge-dependent ones. The equations
for the gauge-independent scalar fields, $X^{\hat{\mathcal{A}}}$, and gauge
dependent, non physical $Y^{\hat{\mathcal{A}}}$ are

$\displaystyle 0=$
$\displaystyle{\left[\partial^{2}_{r}+\left(-2\partial_{r}A(r)-\frac{\partial_{r}G^{(1)}_{\hat{\mathcal{A}}\hat{\mathcal{A}}}}{G^{(1)}_{\hat{\mathcal{A}}\hat{\mathcal{A}}}}\right)\partial_{r}+\left(-q^{2}e^{-2A(r)}-\frac{g^{2}G^{(1)}_{\hat{\mathcal{A}}\hat{\mathcal{A}}}}{4H^{(1)}_{\hat{\mathcal{A}}\hat{\mathcal{A}}}}\right)\right]X^{\hat{\mathcal{A}}}(q,r)}\,,$
(346) $\displaystyle 0=$
$\displaystyle{\left[\partial^{2}_{r}+\left(2\partial_{r}A(r)+\frac{\partial_{r}H^{(1)}_{\hat{\mathcal{A}}\hat{\mathcal{A}}}}{H^{(1)}_{\hat{\mathcal{A}}\hat{\mathcal{A}}}}\right)\partial_{r}+\left(-\frac{q^{2}e^{-2A(r)}}{\hat{\xi}}-\frac{g^{2}G^{(1)}_{\hat{\mathcal{A}}\hat{\mathcal{A}}}}{4H^{(1)}_{\hat{\mathcal{A}}\hat{\mathcal{A}}}}\right)\right]Y^{\hat{\mathcal{A}}}(q,r)}\,.$
(347)

For the boundary terms we only mention the physical $X^{\hat{\mathcal{A}}}$
boundary condition:

$\displaystyle 0=$
$\displaystyle{\left.\left[\partial_{r}+\frac{G^{(1)}_{\hat{\mathcal{A}}\hat{\mathcal{A}}}}{C_{2\,\hat{\mathcal{A}}\hat{\mathcal{A}}}^{(1)}}\right]X^{\hat{\mathcal{A}}}(q,r)\right|_{r=r_{2}}}.$
(348)

Equations (346) and (348) can be compared to Eqs. (156) and (161).

####  B.2.2 The $\mathcal{A}_{M}{}^{4}$ sector

Appropriate gauge fixing terms for $\mathcal{A}_{M}{}^{4}$ in the bulk are the
following:

$\displaystyle\mathcal{S}_{\xi}^{(1)}=$
$\displaystyle\int\mathrm{d}^{4}q\mathrm{~{}d}r\left\\{-\frac{H^{(1)}_{44}}{2\xi}\left[iq^{\mu}\mathcal{A}_{\mu}{}^{4}(-q)-\frac{g}{2}\frac{\xi}{H^{(1)}_{44}}G^{(1)}_{44}e^{2A}\Pi^{4}(-q)-\frac{\xi}{H^{(1)}_{44}}\partial_{r}\left(H^{(1)}_{44}e^{2A}\mathcal{A}_{5}{}^{4}(-q)\right)\right]\times\right.$
$\displaystyle\times{\left.\left[-iq^{\nu}\mathcal{A}_{\nu}{}^{4}(q)-\frac{g}{2}\frac{\xi}{H^{(1)}_{44}}G^{(1)}_{44}e^{2A}\Pi^{4}(q)-\frac{\xi}{H^{(1)}_{44}}\partial_{r}\left(H^{(1)}_{44}e^{2A}\mathcal{A}_{5}{}^{4}(q)\right)\right]\right\\},}$
(349)

where $\xi$ is the gauge-fixing parameter. The boundary-localised, gauge-
fixing terms at $r=r_{2}$ are

$\displaystyle\mathcal{S}_{M}^{(1)}=$
$\displaystyle\int\mathrm{d}^{4}q\mathrm{~{}d}r\delta\left(r-r_{2}\right)\left\\{-\frac{1}{2M_{2}}\left[iq^{\mu}\mathcal{A}_{\mu}{}^{4}(-q)+M_{2}H^{(1)}_{44}e^{2A}\mathcal{A}_{5}{}^{4}(-q)-M_{2}K_{5}\frac{gv_{5}}{2}e^{2A}P^{4}_{5}(-q)\right.\right.$
$\displaystyle\left.\left.-\frac{g}{2}M_{2}C_{2\,44}^{(1)}e^{2A}\Pi^{4}(-q)\right]\times\left[(q\rightarrow-q)\right]\right\\}\,,$
(350)

with $M_{2}$ a free parameter. Thus, the equations of motion and boundary
conditions for $\mathcal{A}_{\mu}{}^{4}$ read as

$\displaystyle 0=$
$\displaystyle{\left[q^{2}H^{(1)}_{44}-\partial_{r}\left(H^{(1)}_{44}e^{2A}\partial_{r}\right)+\left(\frac{g}{2}\right)^{2}G^{(1)}_{44}e^{2A}\right]P^{\mu\nu}\mathcal{A}_{\mu}{}^{4}(q,r)},$
(351) $\displaystyle 0=$
$\displaystyle{\left.\left[H^{(1)}_{44}e^{2A}\partial_{r}+q^{2}D_{2\,44}^{(1)}+e^{2A}\left(\frac{g}{2}\right)^{2}C_{2\,44}^{(1)}+(\frac{gv_{5}}{2})^{2}K_{5}e^{2A}\right]P^{\mu\nu}\mathcal{A}_{\nu}{}^{4}(q,r)\right|_{r=r_{2}}},$
(352) $\displaystyle 0=$
$\displaystyle{\left[\frac{q^{2}}{\xi}H^{(1)}_{44}-\partial_{r}\left(H^{(1)}_{44}e^{2A}\partial_{r}\right)+\left(\frac{g}{2}\right)^{2}G^{(1)}_{44}e^{2A}\right]\frac{q^{\mu}q^{\nu}}{q^{2}}\mathcal{A}_{\mu}{}^{4}(q,r)},$
(353) $\displaystyle 0=$
$\displaystyle{\left.\left[H^{(1)}_{44}e^{2A}\partial_{r}+\frac{q^{2}}{M_{2}}+e^{2A}\left(\frac{g}{2}\right)^{2}C_{2\,44}^{(1)}+(\frac{gv_{5}}{2})^{2}K_{5}e^{2A}\right]\frac{q^{\mu}q^{\nu}}{q^{2}}\mathcal{A}_{\nu}{}^{4}(q,r)\right|_{r=r_{2}}}.$
(354)

Equations (351) and (352) are restated in Eqs. (155) and (160).

For the pseudoscalars and the spurion, the equations are obtained from the
variation with respect to $\mathcal{A}_{5}{}^{4}$ and $\Pi^{4}$ in bulk and
boundary and $P_{5}^{4}$ in the boundary, respectively. They are the
following:

$\displaystyle 0=$
$\displaystyle{\left[q^{2}H^{(1)}_{44}-H^{(1)}_{44}\partial_{r}\left(\frac{\xi}{H^{(1)}_{44}}\partial_{r}\left(H^{(1)}_{44}e^{2A}\right)\right)+\left(\frac{g}{2}\right)^{2}G^{(1)}_{44}e^{2A}\right]\mathcal{A}_{5}{}^{4}(q,r)}$
$\displaystyle+\frac{g}{2}{\left[G^{(1)}_{44}e^{2A}\partial_{r}-H^{(1)}_{44}\partial_{r}\left(\frac{\xi
G^{(1)}_{44}e^{2A}}{H^{(1)}_{44}}\right)\right]\Pi^{4}(q,r)}\,,$ (355)
$\displaystyle 0=$
$\displaystyle{\left.\left[\xi\frac{e^{-2A}}{H^{(1)}_{44}}\partial_{r}\left(H^{(1)}_{44}e^{2A}\right)+M_{2}H^{(1)}_{44}\right]\mathcal{A}_{5}{}^{4}(q,r)\right|_{r=r_{2}}}$
$\displaystyle+{\left.\frac{g}{2}\left[\frac{\xi}{H^{(1)}_{44}}G^{(1)}_{44}-M_{2}C_{2\,44}^{(1)}\right]\Pi^{4}(q,r)-\left[M_{2}\frac{g}{2}K_{5}v_{5}\right]P^{4}_{5}(q)\right|_{r=r_{2}}}\,,$
(356) $\displaystyle 0=$
$\displaystyle{\left[\partial_{r}\left(G^{(1)}_{44}e^{4A}\partial_{r}\right)-G^{(1)}_{44}e^{2A}q^{2}-\left(\frac{g}{2}\right)^{2}\frac{\xi}{H^{(1)}_{44}}e^{4A}(G^{(1)}_{44})^{2}\right]\Pi^{4}(q,r)}$
$\displaystyle+{\frac{g}{2}\left[\partial_{r}\left(G^{(1)}_{44}e^{4A}\right)-\frac{\xi}{H^{(1)}_{44}}e^{2A}G^{(1)}_{44}\partial_{r}\left(H^{(1)}_{44}e^{2A}\right)\right]\mathcal{A}_{5}{}^{4}(q,r)}\,,$
(357) $\displaystyle 0=$
$\displaystyle{\left.\left[C_{2\,44}^{(1)}e^{-2A}q^{2}+M_{2}\left(\frac{g}{2}\right)^{2}(C_{2\,44}^{(1)})^{2}+\partial_{v}^{2}\mathcal{V}_{4}+G^{(1)}_{44}\partial_{r}\right]\Pi^{4}(q,r)\right|_{r=r_{2}}}$
$\displaystyle+{\left.\frac{g}{2}\left[G^{(1)}_{44}-M_{2}C_{2\,44}^{(1)}H^{(1)}_{44}\right]\mathcal{A}_{5}{}^{4}(q,r)+\left[\frac{g^{2}v_{5}}{4}M_{2}K_{5}C_{2\,44}^{(1)}-\frac{1}{v_{5}}\partial_{v}^{2}\mathcal{V}_{4}\right]P^{4}_{5}(q)\right|_{r=r_{2}}}\,,$
(358) $\displaystyle 0=$
$\displaystyle{\left.\left[K_{5}e^{-2A}q^{2}+M_{2}\left(\frac{gv_{5}}{2}\right)^{2}K_{5}^{2}+\frac{1}{v_{5}^{2}}\partial_{v}^{2}\mathcal{V}_{4}\right]P^{4}_{5}(q)\right|_{r=r_{2}}}$
$\displaystyle{\left.-\left[M_{2}\frac{gv_{5}}{2}K_{5}H^{(1)}_{44}\right]\mathcal{A}_{5}{}^{4}(q,r)+\left[\frac{g^{2}v_{5}}{4}M_{2}K_{5}C_{2\,44}^{(1)}-\frac{1}{v_{5}}\partial_{v}^{2}\mathcal{V}_{4}\right]\Pi^{4}(q,r)\right|_{r=r_{2}}}\,.$
(359)

By applying the convenient definition

$\displaystyle\mathcal{A}_{5}{}^{4}$ $\displaystyle\equiv$
$\displaystyle\frac{X^{4}}{e^{4A}G^{(1)}_{44}}-\frac{2}{g}\partial_{r}\Pi^{4}\,,$
(360)

one can derive the equations for gauge invariant field $X^{4}$:

$\displaystyle 0=$
$\displaystyle{\left[\partial^{2}_{r}+\left(-2\partial_{r}A-\frac{\partial_{r}G^{(1)}_{44}}{G^{(1)}_{44}}\right)\partial_{r}+\left(-q^{2}e^{-2A}-\frac{g^{2}G^{(1)}_{44}}{4H^{(1)}_{44}}\right)\right]X^{4}(q,r)},$
(361) $\displaystyle 0=$
$\displaystyle{\left.\left[\partial_{r}+G^{(1)}_{44}\left(\frac{\partial_{v}^{2}\mathcal{V}_{4}e^{2A(r)}+K_{5}v_{5}^{2}q^{2}}{\partial_{v}^{2}\mathcal{V}_{4}K_{5}v_{5}^{2}e^{2A(r)}+C_{2\,44}^{(1)}\partial_{v}^{2}\mathcal{V}_{4}e^{2A(r)}+K_{5}C_{2\,44}^{(1)}v_{5}^{2}q^{2}}\right)\right]X^{4}(q,r)\right|_{r=r_{2}}}.$
(362)

Equations (361) and (362) are restated in Eqs. (156) and (162).

####  B.2.3 The $\mathcal{A}_{M}{}^{\bar{\mathcal{A}}}$ sector

The gauge fixing terms for $\mathcal{A}_{M}{}^{\bar{\mathcal{A}}}$ in the bulk
are chosen to be

$\displaystyle\mathcal{S}_{\bar{\xi}}^{(1)}=$
$\displaystyle\int\mathrm{d}^{4}q\mathrm{~{}d}r\left\\{-\frac{H^{(1)}_{\bar{\mathcal{A}}\bar{\mathcal{A}}}}{2\bar{\xi}}\left[iq^{\mu}\mathcal{A}_{\mu}{}^{\bar{\mathcal{A}}}(-q)-\frac{\bar{\xi}}{H^{(1)}_{\bar{\mathcal{A}}\bar{\mathcal{A}}}}\partial_{r}\left(H^{(1)}_{\bar{\mathcal{A}}\bar{\mathcal{A}}}e^{2A}\mathcal{A}_{5}{}^{\bar{\mathcal{A}}}(-q)\right)\right]\times\right.$
$\displaystyle{\left.\left[-iq^{\nu}\mathcal{A}_{\nu}{}^{\bar{\mathcal{A}}}(q)-\frac{\bar{\xi}}{H^{(1)}_{\bar{\mathcal{A}}\bar{\mathcal{A}}}}\partial_{r}\left(H^{(1)}_{\bar{\mathcal{A}}\bar{\mathcal{A}}}e^{2A}\mathcal{A}_{5}{}^{\bar{\mathcal{A}}}(q)\right)\right]\right\\},}$
(363)

where $\bar{\xi}$ is the gauge fixing parameter. The boundary-localised gauge
fixing terms at $r=r_{2}$ reads

$\displaystyle\mathcal{S}_{\bar{M}}^{(1)}=$
$\displaystyle\int\mathrm{d}^{4}q\mathrm{~{}d}r\delta\left(r-r_{2}\right)\left\\{-\frac{1}{2\bar{M}_{2}}\left[iq^{\mu}\mathcal{A}_{\mu}{}^{\bar{\mathcal{A}}}(-q)+\bar{M}_{2}H^{(1)}_{\bar{\mathcal{A}}\bar{\mathcal{A}}}e^{2A}\mathcal{A}_{5}{}^{\bar{\mathcal{A}}}(-q)\right]\times\left[-iq^{\nu}\mathcal{A}_{\nu}{}^{\bar{\mathcal{A}}}(q)+\bar{M}_{2}H^{(1)}_{\bar{\mathcal{A}}\bar{\mathcal{A}}}e^{2A}\mathcal{A}_{5}{}^{\bar{\mathcal{A}}}(q))\right]\right\\}\,,$
(364)

with $\bar{M}_{2}$, the boundary gauge fixing parameter.

Similar to the previous sections, the equations of motion and boundary
conditions for $\mathcal{A}_{\mu}{}^{\bar{\mathcal{A}}}$ are

$\displaystyle 0=$
$\displaystyle{\left[q^{2}H^{(1)}_{\bar{\mathcal{A}}\bar{\mathcal{A}}}-\partial_{r}\left(H^{(1)}_{\bar{\mathcal{A}}\bar{\mathcal{A}}}e^{2A}\partial_{r}\right)\right]P^{\mu\nu}\mathcal{A}_{\mu}{}^{\bar{\mathcal{A}}}(q,r)},$
(365) $\displaystyle 0=$
$\displaystyle{\left.\left[H^{(1)}_{\bar{\mathcal{A}}\bar{\mathcal{A}}}e^{2A}\partial_{r}+q^{2}D_{2\,\bar{\mathcal{A}}\bar{\mathcal{A}}}^{(1)}\right]P^{\mu\nu}\mathcal{A}_{\nu}{}^{\bar{\mathcal{A}}}(q,r)\right|_{r=r_{2}}},$
(366) $\displaystyle 0=$
$\displaystyle{\left[\frac{q^{2}}{\bar{\xi}}H^{(1)}_{\bar{\mathcal{A}}\bar{\mathcal{A}}}-\partial_{r}\left(H^{(1)}_{\bar{\mathcal{A}}\bar{\mathcal{A}}}e^{2A}\partial_{r}\right)\right]\frac{q^{\mu}q^{\nu}}{q^{2}}\mathcal{A}_{\mu}{}^{\bar{\mathcal{A}}}(q,r)},$
(367) $\displaystyle 0=$
$\displaystyle{\left.\left[\frac{q^{2}}{\bar{M}_{2}}+H^{(1)}_{\bar{\mathcal{A}}\bar{\mathcal{A}}}e^{2A}\partial_{r}\right]\frac{q^{\mu}q^{\nu}}{q^{2}}\mathcal{A}_{\nu}{}^{\bar{\mathcal{A}}}(q,r)\right|_{r=r_{2}}}.$
(368)

Equations (365) and (366) are restated in Eqs. (154), and (159). The fifth
component of the gauge field is pure gauge in this case.

###  B.3 Asymptotic expansions of the fluctuations

We present here some of the asymptotic expansions for the fluctuations, see
also Ref. Elander:2023aow .

####  B.3.1 IR expansions

For convenience, we put $\rho_{o}=0$ and $A_{I}=0$ in this
subsection,131313The dependence on $\rho_{o}$ and $A_{I}$ can be reinstated by
making the substitutions $\rho\rightarrow\rho-\rho_{o}$ and $q^{2}\rightarrow
e^{-2A_{I}}q^{2}$ in the expressions for the IR expansions. while $\chi_{I}=0$
in order to avoid a conical singularity.

For the fluctuations of the scalars, we have

$\displaystyle\mathfrak{a}^{\phi}=$ $\displaystyle\
\mathfrak{a}^{\phi}_{I,0}+\mathfrak{a}^{\phi}_{I,l}\log(\rho)+\frac{1}{4}\rho^{2}\bigg{[}-\frac{1}{4}\Delta\left(\mathfrak{a}^{\phi}_{I,0}\left(\Delta\left(15\phi_{I}^{2}-4\right)+20\right)+6\phi_{I}(\mathfrak{a}^{\chi}_{I,0}-\mathfrak{a}^{\chi}_{I,l})\left(\Delta\left(5\phi_{I}^{2}-4\right)+20\right)\right)$
$\displaystyle+q^{2}(\mathfrak{a}^{\phi}_{I,0}-\mathfrak{a}^{\phi}_{I,l})-\frac{1}{48}\mathfrak{a}^{\phi}_{I,l}\left(\Delta\left(25\Delta\phi_{I}^{4}+20(10-11\Delta)\phi_{I}^{2}+48(\Delta-5)\right)+400\right)$
$\displaystyle+\log(\rho)\left(\mathfrak{a}^{\phi}_{I,l}\left(-\frac{15\Delta^{2}\phi_{I}^{2}}{4}+(\Delta-5)\Delta+q^{2}\right)-\frac{3}{2}\mathfrak{a}^{\chi}_{I,l}\Delta\phi_{I}\left(\Delta\left(5\phi_{I}^{2}-4\right)+20\right)\right)\bigg{]}+\mathcal{O}\left(\rho^{4}\right)\,,$
(369) $\displaystyle\mathfrak{a}^{\chi}=$ $\displaystyle\
\mathfrak{a}^{\chi}_{I,0}+\mathfrak{a}^{\chi}_{I,l}\log(\rho)+\frac{1}{4}\rho^{2}\bigg{[}-\frac{1}{4}\Delta\phi_{I}(\mathfrak{a}^{\phi}_{I,0}-\mathfrak{a}^{\phi}_{I,l})\left(\Delta\left(5\phi_{I}^{2}-4\right)+20\right)+q^{2}(\mathfrak{a}^{\chi}_{I,0}-\mathfrak{a}^{\chi}_{I,l})$
$\displaystyle-\frac{3}{8}\mathfrak{a}^{\chi}_{I,0}\left(\Delta\phi_{I}^{2}\left(\Delta\left(5\phi_{I}^{2}-8\right)+40\right)+80\right)+\frac{13}{48}\mathfrak{a}^{\chi}_{I,l}\left(\Delta\phi_{I}^{2}\left(\Delta\left(5\phi_{I}^{2}-8\right)+40\right)+80\right)$
$\displaystyle+\log(\rho)\left(-\frac{5}{4}\mathfrak{a}^{\phi}_{I,l}\Delta^{2}\phi_{I}^{3}+\mathfrak{a}^{\phi}_{I,l}(\Delta-5)\Delta\phi_{I}+\mathfrak{a}^{\chi}_{I,l}\left(-\frac{15}{8}\Delta^{2}\phi_{I}^{4}+3(\Delta-5)\Delta\phi_{I}^{2}+q^{2}-30\right)\right)\bigg{]}+\mathcal{O}\left(\rho^{4}\right)\,,$
(370) $\displaystyle\mathfrak{a}^{\bar{A}}=$ $\displaystyle\
\mathfrak{a}^{\bar{A}}_{I,0}+\rho^{2}\left(\frac{1}{2}\mathfrak{a}^{\bar{A}}_{I,0}q^{2}\log(\rho)+\mathfrak{a}^{\bar{A}}_{I,2}\right)+\mathcal{O}\left(\rho^{4}\right)\,,$
(371) $\displaystyle\mathfrak{a}^{\hat{A}}=$ $\displaystyle\
\mathfrak{a}^{\hat{A}}_{I,0}+\rho^{2}\left(\frac{1}{2}\mathfrak{a}^{\hat{A}}_{I,0}\left(q^{2}+\frac{g^{2}\phi_{I}^{2}}{4}\right)\log(\rho)+\mathfrak{a}^{\hat{A}}_{I,2}\right)+\mathcal{O}\left(\rho^{4}\right)\,.$
(372)

For the pseudoscalar fluctuations, we have

$\mathfrak{p}^{\hat{A}}=\mathfrak{p}^{\hat{A}}_{I,0}+\rho^{2}\left[\mathfrak{p}^{\hat{A}}_{I,2}+\frac{1}{2}\mathfrak{p}^{\hat{A}}_{I,0}\left(q^{2}+\frac{g^{2}}{4}\phi_{I}^{2}\right)\log(\rho)\right]+\mathcal{O}\left(\rho^{4}\right)\,.$
(373)

For the vector fluctuations, we have

$\displaystyle\mathfrak{v}=$ $\displaystyle\
\mathfrak{v}_{I,-2}\rho^{-2}+\frac{1}{2}q^{2}\mathfrak{v}_{I,-2}\log(\rho)+\mathfrak{v}_{I,0}+\frac{1}{12288}\rho^{2}\Big{[}1536q^{2}\mathfrak{v}_{I,0}+80\Delta^{2}\mathfrak{v}_{I,-2}\phi_{I}^{4}\left(2\left(8\Delta^{2}-50\Delta+75\right)-3q^{2}\right)$
$\displaystyle+128(\Delta-5)\Delta\mathfrak{v}_{I,-2}\phi_{I}^{2}\left(-3(\Delta-5)\Delta+3q^{2}-50\right)-64\left(9q^{4}+60q^{2}-500\right)\mathfrak{v}_{I,-2}+125\Delta^{4}\mathfrak{v}_{I,-2}\phi_{I}^{8}$
$\displaystyle-1000(\Delta-2)\Delta^{3}\mathfrak{v}_{I,-2}\phi_{I}^{6}+768q^{4}\mathfrak{v}_{I,-2}\log(\rho)\Big{]}+\mathcal{O}\left(\rho^{4}\right)\,,$
(374) $\displaystyle\mathfrak{v}^{\bar{A}}=$ $\displaystyle\
\mathfrak{v}^{\bar{A}}_{I,0}+\mathfrak{v}^{\bar{A}}_{I,l}\log(\rho)+\frac{1}{96}\rho^{2}\Big{[}24q^{2}(\mathfrak{v}^{\bar{A}}_{I,0}-\mathfrak{v}^{\bar{A}}_{I,l})+\mathfrak{v}^{\bar{A}}_{I,l}\left(-5\Delta^{2}\phi_{I}^{4}+8(\Delta-5)\Delta\phi_{I}^{2}-80\right)$
$\displaystyle+24q^{2}\mathfrak{v}^{\bar{A}}_{I,l}\log(\rho)\Big{]}+\mathcal{O}\left(\rho^{4}\right)\,,$
(375) $\displaystyle\mathfrak{v}^{\hat{A}}=$ $\displaystyle\
\mathfrak{v}^{\hat{A}}_{I,0}+\mathfrak{v}^{\hat{A}}_{I,l}\log(\rho)+\frac{1}{96}\rho^{2}\Big{[}\left(24q^{2}+6g^{2}\phi_{I}^{2}\right)\mathfrak{v}^{\hat{A}}_{I,0}+\left(-80-24q^{2}-6g^{2}\phi_{I}^{2}-40\Delta\phi_{I}^{2}+\Delta^{2}\left(8\phi_{I}^{2}-5\phi_{I}^{4}\right)\right)\mathfrak{v}^{\hat{A}}_{I,l}$
$\displaystyle+\left(24q^{2}+6g^{2}\phi_{I}^{2}\right)\log(\rho)\mathfrak{v}^{\hat{A}}_{I,l}\Big{]}+\mathcal{O}\left(\rho^{4}\right)\,.$
(376)

For the tensor fluctuations, we have

$\mathfrak{e}=\mathfrak{e}_{I,0}+\mathfrak{e}_{I,l}\log(\rho)+\frac{1}{192}\rho^{2}\Big{[}48q^{2}(\mathfrak{e}_{I,0}-\mathfrak{e}_{I,l})-25\Delta^{2}\mathfrak{e}_{I,l}\phi_{I}^{4}+40(\Delta-5)\Delta\mathfrak{e}_{I,l}\phi_{I}^{2}-400\mathfrak{e}_{I,l}+48\mathfrak{e}_{I,l}q^{2}\log(\rho)\Big{]}+\mathcal{O}\left(\rho^{4}\right)\,.$
(377)

####  B.3.2 UV expansions

In this subsection, we put $\Delta=2$, and $A_{U}=0=\chi_{U}$.141414The
dependence on $\chi_{U}$ and $A_{U}$ can be reinstated by making the
substitution $q^{2}\rightarrow e^{2\chi_{U}-2A_{U}}q^{2}$ in the expressions
for the UV expansions. We write the expansions in terms of $z\equiv
e^{-\rho}$.

For the fluctuations of the scalars, we have

$\displaystyle\mathfrak{a}^{\phi}=$ $\displaystyle\
\mathfrak{a}^{\phi}_{2}z^{2}+\mathfrak{a}^{\phi}_{3}z^{3}+\frac{1}{2}\mathfrak{a}^{\phi}_{2}q^{2}z^{4}+\frac{1}{6}\mathfrak{a}^{\phi}_{3}q^{2}z^{5}+\frac{1}{24}\mathfrak{a}^{\phi}_{2}\left(q^{4}-12\phi_{V}^{2}\right)z^{6}+\mathcal{O}\left(z^{7}\right)\,,$
(378) $\displaystyle\mathfrak{a}^{\chi}=$ $\displaystyle\
\mathfrak{a}^{\chi}_{0}-\frac{1}{6}\mathfrak{a}^{\chi}_{0}q^{2}z^{2}+\frac{1}{24}\mathfrak{a}^{\chi}_{0}q^{4}z^{4}+\mathfrak{a}^{\chi}_{5}z^{5}+\frac{1}{144}\mathfrak{a}^{\chi}_{0}q^{2}\left(q^{4}-14\phi_{V}^{2}\right)z^{6}+\mathcal{O}\left(z^{7}\right)\,,$
(379) $\displaystyle\mathfrak{a}^{\bar{A}}=$ $\displaystyle\
\mathfrak{a}^{\bar{A}}_{0}-\frac{1}{2}\mathfrak{a}^{\bar{A}}_{0}q^{2}z^{2}+\mathfrak{a}^{\bar{A}}_{3}z^{3}-\frac{1}{8}\mathfrak{a}^{\bar{A}}_{0}q^{4}z^{4}+\frac{1}{10}\mathfrak{a}^{\bar{A}}_{3}q^{2}z^{5}-\frac{1}{144}\mathfrak{a}^{\bar{A}}_{0}q^{2}\left(q^{4}+10\phi_{V}^{2}\right)z^{6}+\mathcal{O}\left(z^{7}\right)\,,$
(380) $\displaystyle\mathfrak{a}^{\hat{A}}=$ $\displaystyle\
\mathfrak{a}^{\hat{A}}_{0}-\frac{1}{2}\mathfrak{a}^{\hat{A}}_{0}q^{2}z^{2}+\mathfrak{a}^{\hat{A}}_{3}z^{3}+\frac{1}{16}\mathfrak{a}^{\hat{A}}_{0}\left(g^{2}\phi_{J}^{2}-2q^{4}\right)z^{4}+\frac{1}{20}\left(\mathfrak{a}^{\hat{A}}_{0}g^{2}\phi_{J}\phi_{V}+2\mathfrak{a}^{\hat{A}}_{3}q^{2}\right)z^{5}$
$\displaystyle-\frac{1}{288}\mathfrak{a}^{\hat{A}}_{0}\left(2q^{6}+(20+g^{2})q^{2}\phi_{J}^{2}-4g^{2}\phi_{V}^{2}\right)z^{6}+\mathcal{O}\left(z^{7}\right)\,.$
(381)

For the pseudo-scalar fluctuations, we have

$\mathfrak{p}^{\hat{A}}=\mathfrak{p}^{\hat{A}}_{0}+\mathfrak{p}^{\hat{A}}_{1}z+\left(\frac{\mathfrak{p}^{\hat{A}}_{0}q^{2}}{2}+\frac{\mathfrak{p}^{\hat{A}}_{1}\phi_{V}}{\phi_{J}}\right)z^{2}+\frac{2\mathfrak{p}^{\hat{A}}_{0}q^{2}\phi_{J}\phi_{V}+\mathfrak{p}^{\hat{A}}_{1}q^{2}\phi_{J}^{2}+2\mathfrak{p}^{\hat{A}}_{1}\phi_{V}^{2}}{6\phi_{J}^{2}}z^{3}+\mathcal{O}\left(z^{4}\right)\,.$
(382)

For the vector fluctuations, we have

$\displaystyle\mathfrak{v}=$ $\displaystyle\
\mathfrak{v}_{0}-\frac{1}{6}q^{2}\mathfrak{v}_{0}z^{2}+\frac{1}{24}q^{4}\mathfrak{v}_{0}z^{4}+\mathfrak{v}_{5}z^{5}+\frac{1}{144}q^{2}\mathfrak{v}_{0}\left(q^{4}-14\phi_{V}^{2}\right)z^{6}$
$\displaystyle+\frac{1}{350}q^{2}(350\mathfrak{v}_{0}\chi_{5}-38\mathfrak{v}_{0}\phi_{V}\phi_{J}+25\mathfrak{v}_{5})z^{7}+\mathcal{O}\left(z^{8}\right)\,,$
(383) $\displaystyle\mathfrak{v}^{\bar{A}}=$ $\displaystyle\
\mathfrak{v}^{\bar{A}}_{0}-\frac{1}{2}q^{2}\mathfrak{v}^{\bar{A}}_{0}z^{2}+\mathfrak{v}^{\bar{A}}_{3}z^{3}-\frac{1}{8}q^{4}\mathfrak{v}^{\bar{A}}_{0}z^{4}+\frac{1}{10}q^{2}\mathfrak{v}^{\bar{A}}_{3}z^{5}-\frac{1}{144}q^{2}\mathfrak{v}^{\bar{A}}_{0}\left(q^{4}+10\phi_{V}^{2}\right)z^{6}$
$\displaystyle+\frac{1}{280}\left(q^{4}\mathfrak{v}^{\bar{A}}_{3}+6q^{2}\mathfrak{v}^{\bar{A}}_{0}(15\chi_{5}-\frac{26}{5}\phi_{V}\phi_{J})+45\mathfrak{v}^{\bar{A}}_{3}\phi_{V}^{2}\right)z^{7}+\mathcal{O}\left(z^{8}\right)\,,$
(384) $\displaystyle\mathfrak{v}^{\hat{A}}=$ $\displaystyle\
\mathfrak{v}^{\hat{A}}_{0}-\frac{1}{2}q^{2}\mathfrak{v}^{\hat{A}}_{0}z^{2}+\mathfrak{v}^{\hat{A}}_{3}z^{3}-\frac{1}{8}\mathfrak{v}^{\hat{A}}_{0}\left(q^{4}-\frac{g^{2}}{2}\phi_{J}^{2}\right)z^{4}+\frac{1}{10}\left(q^{2}\mathfrak{v}^{\hat{A}}_{3}+\frac{g^{2}}{2}\mathfrak{v}^{\hat{A}}_{0}\phi_{J}\phi_{V}\right)z^{5}+\mathcal{O}\left(z^{6}\right)\,.$
(385)

For the tensor fluctuations, we have

$\mathfrak{e}=\mathfrak{e}_{0}-\frac{1}{6}\mathfrak{e}_{0}q^{2}z^{2}+\frac{1}{24}\mathfrak{e}_{0}q^{4}z^{4}+\mathfrak{e}_{5}z^{5}+\mathcal{O}\left(z^{6}\right)\,.$
(386)

##  Appendix C Of two-point functions

In this short appendix, we perform an exercise to demonstrate the relation
between mass spectra of vector fluctuations and holographically renormalised
two-point functions. To this purpose, we digress and analyse a simplified
system, in which analytical calculations can be carried out, and symmetries
are manifest. We consider a new $U(1)$ gauge theory living in a new six-
dimensional spacetime, yet adopt the same ansatz as in Eqs. (8) and (9). At
variance with the body of the paper, we consider solutions for which
$A=4\chi=4\rho/3$, resulting in an AdS6 geometry. To introduce a scale, we
hold the IR cutoff, $\rho_{1}=0$, fixed, and interpret it as a confinement
scale, as in hard-wall models in the AdS/QCD literature Erlich:2005qh ;
DaRold:2005mxj . The (reduced) five-dimensional action for the new $U(1)$
gauge field, $\mathcal{A}_{M}$, is151515This action closely resembles the one
for $\mathcal{A}_{M}^{\bar{\mathcal{A}}}$, in the body of the paper.

$\mathcal{S}=\int\mbox{d}\rho\int\mbox{d}^{4}x\sqrt{-g}\,\bigg{\\{}-\frac{1}{4}\,e^{2\chi}\,g^{MP}g^{NQ}\mathcal{F}_{MN}\mathcal{F}_{PQ}\bigg{\\}}+\int\mbox{d}^{4}x\sqrt{-\tilde{g}}\,\bigg{\\{}-\frac{1}{4}D_{2}\,\tilde{g}^{\mu\rho}\tilde{g}^{\nu\sigma}\mathcal{F}_{\mu\nu}\mathcal{F}_{\rho\sigma}\bigg{\\}}\bigg{|}_{\rho=\rho_{2}}\,,$
(387)

where $\mathcal{F}_{MN}=\partial_{M}{\cal A}_{N}-\partial_{N}{\cal A}_{M}$,
and ${\cal A}_{M}$ has trivial background profile. After Fourier transforming
in the Minkowski directions, the transverse part of the gauge field satisfies
an equation of motion that can be written as follows:

$\bigg{[}\partial_{\rho}^{2}+3\partial_{\rho}-e^{-2\rho}q^{2}\bigg{]}P_{\mu\nu}\mathcal{A}^{\nu}(q,\rho)=0\,.$
(388)

In the following, we call $v(q,\rho)$ the solutions to this linear equation,
which depend on the four-momentum, $q_{\mu}$, and on the radial direction,
$\rho$. In the IR, we impose Neumann boundary conditions,
$\partial_{\rho}v(q,\rho)\big{|}_{\rho=\rho_{1}}=0$.

The presence of the boundary-localised term, proportional to $D_{2}$,
introduces (weak) gauging of the global $U(1)$ symmetry of the dual field
theory. The transverse part of the two-point function (propagator) is

$\displaystyle P^{\mu\sigma}P^{\nu\gamma}\langle{\cal A}_{\mu}(q){\cal
A}_{\nu}(-q)\rangle$ $\displaystyle=$
$\displaystyle\lim_{\rho_{2}\rightarrow+\infty}\frac{-i\,P^{\sigma\gamma}}{q^{2}}\,\frac{1}{\Pi(q^{2},\rho_{2})}\,,$
(389)

where we anticipated that we will take the limit $\rho_{2}\rightarrow+\infty$,
and where

$\displaystyle\Pi(q^{2},\rho)$ $\displaystyle\equiv$ $\displaystyle
D_{2}+\frac{1}{q^{2}}e^{3\rho}\frac{\partial_{\rho}v(q,\rho)}{v(q,\rho)}\,.$
(390)

To remove a divergence, that appears when $\rho_{2}\rightarrow+\infty$, we
require that

$\displaystyle D_{2}$ $\displaystyle\equiv$
$\displaystyle-e^{\rho_{2}}+\frac{1}{\varepsilon^{2}}\,,$ (391)

where $\varepsilon$ is a renormalisation constant. The transverse part of the
propagator in the background has the closed form:

$\displaystyle P^{\mu\sigma}P^{\nu\gamma}\langle{\cal A}_{\mu}(q){\cal
A}_{\nu}(-q)\rangle$ $\displaystyle=$
$\displaystyle\frac{-i\,P^{\sigma\gamma}}{q^{2}}\,\frac{\varepsilon^{2}}{1+\varepsilon^{2}\Pi_{o}(q^{2})}\,,$
(392)

and the physical $q^{2}$-dependence is encoded in

$\displaystyle\Pi_{o}(q^{2})$ $\displaystyle=$
$\displaystyle-\frac{\sqrt{-q^{2}}}{\tan(\sqrt{-q^{2}})}\,.$ (393)

The poles in Eq. (392) identify the mass spectrum. Equivalently, one can solve
Eq. (388), subject to Neumann boundary condition in the IR,
$\partial_{\rho}v(q,\rho)\big{|}_{\rho=\rho_{1}}=0$, impose in the UV the
constraint
$\partial_{\rho}v(q,\rho)+q^{2}D_{2}e^{-3\rho}v(q,\rho)\big{|}_{\rho=\rho_{2}}=0$,
hence identifying the discrete spectrum in $q^{2}$, and afterwards take the
limit $\rho_{2}\rightarrow+\infty$ to recover the physical spectrum.

As long as $\varepsilon^{2}$ and $|q^{2}|$ are small, the process discussed
after Eq. (392) yields sensible physical results. However, if one considers
large values of $|q^{2}|$, an unphysical tachyonic mode appears, with
$q^{2}\simeq(\varepsilon^{2})^{-2}$, which is due to having taken the limit
$\rho_{2}\rightarrow 0$.

If one, conversely, chooses $D_{2}$ to be non-negative, and retains a large,
but finite, value for $\rho_{2}$, there are no tachyons while the rest of the
spectrum is shifted only by small effects. The appearance of the tachyon is,
hence, truly unphysical, and it should be ignored. Is there a way to write a
two-point function that does not depend explicitly on the (unphysical) cutoff,
$\rho_{2}$, nor leads to the appearance of an (unphysical) tachyon? We devote
these final paragraphs to propose a simple and elegant solution to this
puzzle.

As in ordinary perturbation theory, the presence of large hierarchies between
the renormalisation scale and physical scales of interest may lead to
difficulties, which can be overcome by applying the renormalisation group to
improve perturbation theory. Inspired by the work on the holographic Wilsonian
renormalisation group Heemskerk:2010hk ; Faulkner:2010jy  (see also Refs.
Laia:2011wf ; Elander:2011vh ; Elander:2021bmt  and Ref. Papadimitriou:2010as
), we proceed as follows.

Suppose one computes the spectrum with a finite cutoff,
$\rho_{2}=\rho_{\Lambda}$, for a given choice of $D_{2}$. One can ask how to
change $D_{2}$ into a function that depends on $q^{2}$ and $\rho_{2}$ so that
the same spectrum is reproduced for any choice of the finite cutoff,
$\rho_{2}$. This can be achieved by requiring that $D_{2}(q^{2},\rho_{2})$
satisfies the first-order differential equation

$\partial_{\rho_{2}}D_{2}-e^{-3\rho_{2}}q^{2}D_{2}^{2}+e^{\rho_{2}}=0\,,$
(394)

with boundary condition $D_{2}(q^{2},\rho_{\Lambda})=D_{2,\,\Lambda}$.

For concreteness, we set $D_{2,\Lambda}=0$. From Eq. (391), we see that if one
identifies $\varepsilon^{2}=e^{-\rho_{\Lambda}}$, then the new $D_{2}$ matches
the previous expression at $\rho_{2}=\rho_{\Lambda}$. The solution is given by

$D_{2}(q^{2},\rho_{2})=-\frac{e^{\rho_{2}}}{1+e^{-\rho_{2}}\sqrt{-q^{2}}\cot\left(\sqrt{-q^{2}}\left(\varepsilon^{2}-e^{-\rho_{2}}\right)\right)}\,,$
(395)

and we can replace it into Eq. (390). One may verify that, for any $q^{2}$
such that the resulting $\Pi(q^{2},\rho_{2})=0$, one also has
$\partial_{\rho_{2}}\Pi(q^{2},\rho_{2})=0$. The zeroes of the new
$\Pi(q^{2},\rho_{2})$ do not change as a function of $\rho_{2}$, leaving the
spectrum invariant.

We may now take the limit $\rho_{2}\rightarrow\infty$, evolving
$D_{2}(q^{2},\rho_{2})$ towards the UV by using Eq. (395), after which the
renormalised two-point function is given by

$P^{\mu\sigma}P^{\nu\gamma}\langle\mathcal{A}_{\mu}(q)\mathcal{A}_{\nu}(-q)\rangle=(-i)\,\left(\frac{1}{-\frac{1}{\varepsilon^{2}}\Pi_{o}(q^{2}(\varepsilon^{2})^{2})+\Pi_{o}(q^{2})}\right)\,\frac{1}{q^{2}}\,P^{\sigma\gamma}\,.$
(396)

Equation (396) is approximately equal to Eq. (392) for small $\varepsilon^{2}$
and $q^{2}$, which validates the approximation used in the body of the paper.
Furthermore, the (unphysical) tachyon is no longer present, while the physical
states are retained. Finally, by expanding
$\langle\mathcal{A}_{\mu}(q)\mathcal{A}_{\nu}(-q)\rangle$ in powers of small
$q^{2}$, one finds a normalisation factor for the gauge fields, so that the
four-dimensional gauge coupling is

${g_{4}^{2}}=\frac{\varepsilon^{2}}{1-\varepsilon^{2}}\,{g^{2}}\approx\varepsilon^{2}{g^{2}}\,.$
(397)

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