aainstitutetext: School of Physics, Harbin Institute of Technology, Harbin
150001, Chinabbinstitutetext: Asia Pacific Center for Theoretical Physics,
Pohang, 37673, Koreaccinstitutetext: Department of Physics, Ben-Gurion
University of the Negev, Beer-Sheva 84105, Israel

# All order effective action for charge diffusion from Schwinger-Keldysh
holography

Yanyan Bu b Tuna Demircik c and Michael Lublinsky<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

An effective action for diffusion of a conserved $U(1)$ charge is derived to
all orders in the derivative expansion within a holographic model dual to the
Schwinger-Keldysh closed time path. A systematic approach to solution of the
5D Maxwell equations in a doubled Schwarzschild-AdS5 black brane geometry is
developed. Constitutive relation for the stochastic charge current is shown to
have a term induced by thermal fluctuations (coloured noise). All transport
coefficient functions parameterising the effective action and constitutive
relations are computed analytically in the hydrodynamic expansion, and then
numerically for finite momenta.

###### Keywords:

AdS-CFT Correspondence, Gauge-gravity correspondence, Holography and quark-
gluon plasmas, Holography and condensed matter physics (AdS/CMT)

††arxiv: 2012.08362

##  1 Introduction

Hydrodynamics landau ; forster ; 1963AnPhy..24..419K  is an effective long-
time long-distance description of many-body systems at nonzero temperature.
Within the hydrodynamic approximation, the entire dynamics of a microscopic
theory is reduced to that of conserved macroscopic currents, such as
expectation values of energy-momentum tensor or charge current operators
computed in a locally near equilibrium thermal state. An essential element of
any hydrodynamics is a constitutive relation which relates the macroscopic
currents to fluid-dynamic variables (fluid velocity, conserved charge
densities, etc), and to external forces. Derivative expansion in the fluid-
dynamic variables accounts for deviations from thermal equilibrium. At each
order, the derivative expansion is fixed by thermodynamics and symmetries, up
to a finite number of transport coefficients (TCs) such as viscosity and
diffusion coefficients. The latter are not calculable from hydrodynamics
itself, but have to be determined from underlying microscopic theory or
extracted from experiments. In general, relativistic hydrodynamics truncated
to any fixed order has a well-known major conceptual problem—it violates
causality. To restore causality one has to introduce higher order gradient
terms. Generically, causality is restored only after all (infinite) order
gradients are resummed, in a way providing a UV completion of the “old"
hydrodynamics. A compact way of organising the resummation is by introducing,
instead of order by order transport coefficients, momenta-dependent transport
coefficient functions (TCFs) Lublinsky:2009kv .

The focus of the present paper will be on $U(1)$ charge diffusion. The all
order constitutive relation for the spatial current density $J^{i}$ has the
following general form

$J^{i}=-{\mathcal{D}}\partial_{i}J^{v}+\sigma_{e}\mathcal{F}_{iv}+\sigma_{m}\partial_{k}\mathcal{F}_{ik},$
(1)

where $J^{v}$ is the charge density, and $\mathcal{F}$ is field strength of
external $U(1)$ field $\mathcal{A}_{\mu}$. The coefficients $\mathcal{D}$,
$\sigma_{e}$, and $\sigma_{m}$ are generalised diffusion constant, electric
and magnetic conductivities. These coefficients are not constants but rather
TCFs, that is, they are functions of four-momentum in Fourier space or
functionals of space-time derivatives in the real space:

$\mathcal{D}[\partial_{v},\vec{\partial}^{\,2}]\rightarrow\mathcal{D}[\omega,q^{2}],~{}~{}~{}~{}~{}~{}~{}~{}~{}\sigma_{e}[\partial_{v},\vec{\partial}^{\,2}]\rightarrow\sigma_{e}[\omega,q^{2}],~{}~{}~{}~{}~{}~{}~{}\sigma_{m}[\partial_{v},\vec{\partial}^{\,2}]\rightarrow\sigma_{m}[\omega,q^{2}].$
(2)

Generically, (1) is a non-local constitutive relation expressible in terms of
memory functions, the inverse Fourier transforms of the TCFs Bu:2015ika .

AdS/CFT correspondence Maldacena:1997re ; Gubser:1998bc ; Witten:1998qj  is
the only known framework, which provides a tractable approach to strongly
coupled regime of non-Abelian gauge theories at finite temperature and opens a
possibility to explore their transport properties exactly, at least for a
class of gauge theories for which gravity duals can be constructed. The
holographic duality maps hydrodynamic fluctuations of a boundary fluid into
gravitational perturbations of a stationary black brane in an asymptotic AdS
space Kovtun:2004de ; Policastro:2001yc ; Policastro:2002se ;
Policastro:2002tn ; Son:2002sd ; Bhattacharyya:2008jc . The original papers on
the subject focused on shear viscosity over entropy density ratio and two-
point retarded correlators. For the latter, Son and Starinets proposed a
computational prescription Son:2002sd ; Son:2007vk , to be discussed below.
Since then, the field has developed in different directions. Second and higher
order TCs were computed for various bulk models, while our team has focused on
development of all order resummation technique Bu:2014sia ; Bu:2014ena ;
Bu:2015ika . Particularly, in Bu:2015ame  we used the Maxwell theory probing
the Schwarzschild-AdS5 background to compute the TCFs introduced in (1) and
(2).

Classical hydrodynamics is dissipative with the TCs, or more generally TCFs,
parameterising the rate of dissipation in the fluids. Yet, dissipation in non-
equilibrium dynamics is tightly related to thermal fluctuations via
fluctuation-dissipation relations (FDRs). The latter originate from the energy
momentum conservation in a closed system, which includes an open subsystem and
a thermal bath. A canonical example is Brownian motion of a particle in a
thermal bath. FDR renders the diffusive motion of the particle into a
stochastic process described by Langevin equation. Similarly, proper account
of thermal fluctuations in fluids, and particularly in relativistic fluids,
should render classical hydrodynamics into stochastic one. These ideas have
sparked many interesting developments in formulating an effective field theory
(EFT) approach to dissipative hydrodynamics Dubovsky:2011sj ; Dubovsky:2011sk
; Endlich:2012vt ; Grozdanov:2013dba ; Nicolis:2013lma ; Kovtun:2014hpa ;
Harder:2015nxa ; Haehl:2015foa ; Haehl:2015uoc ; Crossley:2015evo ;
Crossley:2015tka ; deBoer:2015ija ; Glorioso:2016gsa ; Glorioso:2017fpd ;
Glorioso:2017lcn ; Jensen:2017kzi ; Haehl:2018lcu ; Jensen:2018hse , from
which constitutive relations for the currents could be straightforwardly
derived. In presence of fluctuations the conserved current is expected to take
the form111Strictly speaking, the constitutive relation (3) holds for
quadratic EFTs only. Beyond that, the constitutive relation is non-linear.

$J_{i}=-\mathcal{D}\partial_{i}J^{v}+\sigma_{e}\mathcal{F}_{iv}+\sigma_{m}\partial_{k}\mathcal{F}_{ik}+\
J_{i}^{\rm noise}.$ (3)

Here $J_{i}^{\rm noise}$ is a noise term representing thermal force.

There is also a phenomenological interest in fluctuating hydrodynamics,
largely driven by studies of quark gluon plasma (QGP), for which relativistic
hydrodynamics is instrumental. Phenomenological implications of fluctuating
hydrodynamics for realistic systems such as QGP have been discussed in, e.g.,
Kovtun:2003vj ; Kovtun:2011np ; Young:2014pka ; Akamatsu:2016llw ; Chen-
Lin:2018kfl ; Jain:2020fsm ; Bluhm:2020mpc ; Jain:2020hcu . Particularly, in a
model independent way, thermal fluctuations can be integrated out, resulting
in emergence of “effective” TCs and shifts in positions of the hydrodynamic
poles. This idea was first considered in Kovtun:2003vj ; Kovtun:2011np ;
Kovtun:2012rj  and recently revisited within an EFT framework in Chen-
Lin:2018kfl . While in most phenomenological applications the noise is assumed
to be white, it is generically non-Gaussian and momenta dependent (coloured),
see e.g Stephanov:2008qz ; Kapusta:2014dja . The discussion in Kapusta:2014dja
was based on an Israel-Stewart-type model for causal diffusion. The goal of
the present paper is to put the above mentioned ideas into a more firm ground
by learning about the noise structure to all orders in the gradient expansion
from a holographic model, in which such questions can be addressed via a first
principle calculation.

While traditional holographic approach based on black hole in AdS (BH-AdS)
captures dissipative effects of the boundary dynamics, it does not include any
fluctuation. In finite temperature QFT, the unified framework that includes
both the fluctuation and dissipation is a closed time path (CTP) integral,
also referred to as Schwinger-Keldysh (SK) formalism Chou:1984es ; Kamenev2011
; Calzetta-Hu2009 . From the holographic perspective, a dual geometry must
have SK contour at its conformal boundary Herzog:2002pc ; Skenderis:2008dg ;
Skenderis:2008dh ; Barnes:2010jp ; Nickel:2010pr ; Son:2009vu ; deBoer:2008gu
; Sonner:2012if ; deBoer:2015ija ; deBoer:2018qqm ; Crossley:2015tka ;
Glorioso:2018mmw ; CaronHuot:2011dr ; Chesler:2011ds ; Botta-Cantcheff:2018brv
. In contrast to the single BH-AdS geometry, this is achieved via patching two
Lorentzian BH-AdS geometries with an Euclidean BH-AdS geometry. Proper
matching conditions for the bulk fields should be imposed at space-like
surfaces at which the geometries are glued Skenderis:2008dh ; Skenderis:2008dg
; vanRees:2009rw ; deBoer:2018qqm .

An alternative prescription has been proposed in Glorioso:2018mmw , in which,
instead of gluing geometries, the radial (holographic) coordinate has been
complexified and analytically continued around the event horizon, forming a
geometry with two copies of BH-AdS space. This latter approach will be
referred to as SK holography. Over the last couple of years, the SK holography
was applied to open quantum systems. The questions about non-Gaussian noise
and KMS relations for fermionic degrees of freedom were addressed in
Chakrabarty:2019aeu ; Jana:2020vyx ; Loganayagam:2020eue ; Loganayagam:2020iol
; Chakrabarty:2020ohe . However, the open systems considered in
Chakrabarty:2019aeu ; Jana:2020vyx ; Loganayagam:2020eue ; Loganayagam:2020iol
; Chakrabarty:2020ohe  do not involve hydrodynamical low energy degrees of
freedom, for which an EFT formalism to be discussed below is required.

After this general introduction, we briefly review our setup. We are going to
study the $U(1)$ charge diffusion in a thermal plasma in 4d. This will be
derived from a probe Maxwell theory in the doubled Schwarzschild-AdS5
geometry. For the holographic SK formalism we will closely follow
Glorioso:2018mmw , which derived an effective action for diffusion, up to
second order in the derivative expansion. One of our results will be the
effective action computed to all orders in the derivative expansion. We will
demonstrate that, thanks to linearity of the Maxwell equations in the bulk,
the resulting effective action is quadratic in the dynamical fields and takes
the precise form proposed in Crossley:2015evo  (see (4)). The latter was
derived from general symmetry-based considerations. In the next section we
will flash the relevant results from Crossley:2015evo . The core of our
calculation is in solving the bulk equations of motion (EOMs) in the doubled
Schwarzschild-AdS5. Following the idea introduced by two of us in Bu:2014sia ;
Bu:2014ena , we will be solving the dynamical equations only, leaving the
constraint aside. This makes it possible to construct the “off-shell”
constitutive relations and “off-shell” hydrodynamic effective action. This
approach is nowadays referred to as “off-shell” holography Crossley:2015tka ;
deBoer:2015ija . At a technical level, our treatment of the bulk EOMs will be
somewhat different and more systematic compared to that of Glorioso:2018mmw :
we will first search for a complete set of independent solutions in a single
copy of the doubled Schwarzschild-AdS5, and then will carefully match the two
segments of the doubled Schwarzschild-AdS5 near the event horizon. In this
respect our formalism is more in spirit of Skenderis:2008dh ; Skenderis:2008dg
. The latter, however, glued geometries along the space-like surfaces. When
expanded to second order in the derivatives, our results could be compared
with those of Glorioso:2018mmw ; deBoer:2018qqm . While most of the
coefficients are found to match, there are also some disagreements between all
three results. The comparison and discussion are presented in subsection 5.1.

The main results of this paper are

$\bullet$ Derivation from the SK holography of the effective action
Crossley:2015evo  for the charge diffusion, from which the constitutive
relation with the noise term in the form (3) follows straightforwardly.

$\bullet$ Computation of all the TCFs parameterising the effective action.
These are computed analytically up to the second order in the derivative
expansion and then numerically for finite (large) momenta. All the TCFs are
analytically shown to satisfy the symmetry-imposed relations introduced in
Crossley:2015evo .

$\bullet$ The noise-noise correlator is computed, showing non-locality in the
space-time.

$\bullet$ Derivation of the prescription Son:2002sd  for retarded two-point
correlators, starting from the SK holography, as opposed to the original work
based on a single BH-AdS geometry 222The prescription Son:2002sd  was derived
in Herzog:2002pc ; vanRees:2009rw ; Iqbal:2009fd  for a probe scalar field.
Yet, to the best of our knowledge, it has not been derived for a bulk gauge
field.(see Appendix C).

The paper is structured as follows. In Section 2, the effective action
Crossley:2015evo  for the diffusion at quadratic order and the TCFs
parameterising it are reviewed. This Section also introduces the symmetry-
induced relations among the TCFs and a discussion of the constitutive
relations for the current with noise. The SK holography is introduced in
Section 3. Solutions to the Maxwell’s equation in the bulk are presented in
Section 4. The results for the TCFs as well as noise-noise correlator are
presented in Section 5. A brief summary and outlook is presented in Section 6.
The effective action for the charge diffusion proposed in Crossley:2015evo  is
derived in Appendix A. A subtle point regarding the near-horizon matching
condition for the time component of the bulk gauge field is further clarified
in Appendix B. In Appendix C, the prescription Son:2002sd  for retarded
current-current correlators is derived starting from the SK holography. In
Appendix D, the numerical results for independent TCFs (say, $w_{5,7,8,9}$)
parameterising the effective action are presented.

Note added: While preparing this paper for release, we got aware of the recent
work Ghosh:2020lel . Just like us, Ghosh:2020lel  considers the Maxwell’s
theory within the SK holography and constructs an EFT for stochastic
diffusion. Both papers employ the time-reversal symmetry to relate the ingoing
modes (dissipation) with the outgoing modes (fluctuation/Hawking radiation).
Our first impression is that Ghosh:2020lel  constructed EFT on-shell only,
whereas we have obtained results for both on-shell and off-shell EFTs.
Particularly, if our understanding is correct, Chapter 8 of Ghosh:2020lel  is
quite similar to our Appendix C. Admittedly, a much more careful study of
Ghosh:2020lel  would be needed to fully appreciate the degree of overlap and
agreement between the two papers.

##  2 Effective field theory for charge diffusion

In this section, we review the hydrodynamic effective action derived in
Crossley:2015evo  and the symmetry properties of the TCFs parameterising it.
We will also address the constitutive relation for the fluctuating $U(1)$
current. Finally, we present the general structure of the momenta-dependent
(coloured) noise and the noise-noise correlator.

###  2.1 Effective action

At quadratic level in the dynamical fields, the most general form of the
effective action for the $U(1)$ charge diffusion was derived in
Crossley:2015evo :

$\displaystyle S_{\rm eff}=\int d^{4}x\mathcal{L}_{\rm eff}(x),$ (4)

where the effective Lagrangian is

$\displaystyle\mathcal{L}_{\rm eff}=$
$\displaystyle\frac{i}{2}B_{av}(x)w_{1}B_{av}(x)+\frac{i}{2}B_{ak}(x)w_{2}B_{ak}(x)+\frac{i}{2}\partial_{k}B_{ak}(x)w_{3}\partial_{l}B_{al}(x)$
$\displaystyle+$ $\displaystyle
iB_{av}(x)w_{4}\partial_{k}B_{ak}(x)+B_{av}(x)w_{5}B_{rv}(x)+B_{av}(x)w_{6}\partial_{v}\partial_{k}B_{rk}(x)$
$\displaystyle+$
$\displaystyle\partial_{k}B_{ak}(x)w_{7}B_{rv}(x)+B_{ak}(x)w_{8}\partial_{v}B_{rk}(x)+\frac{1}{2}\mathcal{F}_{akl}(x)w_{9}\mathcal{F}_{rkl}(x),$
(5)

where

$\displaystyle B_{r\mu}=\frac{1}{2}(B_{1\mu}+B_{2\mu}),\qquad
B_{a\mu}=B_{1\mu}-B_{2\mu}.$ (6)

In (2.1), $\mathcal{F}_{r\mu\nu}$ and $\mathcal{F}_{a\mu\nu}$ are the field
strengths of $B_{r\mu}$ and $B_{a\mu}$, respectively. Here, $B_{1\mu}$ and
$B_{2\mu}$ live on the upper and lower branches of the SK contour, and are
defined as independent $U(1)$ gauge transforms of the background gauge fields
$\mathcal{A}_{1\mu}$ and $\mathcal{A}_{2\mu}$ Crossley:2015evo

$\displaystyle
B_{1\mu}\equiv\mathcal{A}_{1\mu}+\partial_{\mu}\varphi_{1},\qquad
B_{2\mu}\equiv\mathcal{A}_{2\mu}+\partial_{\mu}\varphi_{2},$ (7)

where the gauge transformation parameters $\varphi_{1}$ and $\varphi_{2}$ are
treated as low energy hydrodynamical modes.

The parameters $w_{1\cdots 9}$ are the TCFs: they are $SO(3)$ scalar
functionals of the space-time derivatives. As explained in section 1, in
momentum space these TCFs become functions of frequency $\omega$ and spatial
momentum $\vec{q}$.

The generating functional $W[\mathcal{A}_{a\mu},\mathcal{A}_{r\mu}]$ is
obtained by integrating over the dynamical fields $\varphi_{1}$ and
$\varphi_{2}$ or, alternatively, over $\varphi_{r}$ and $\varphi_{a}$:

$\displaystyle e^{W[\mathcal{A}_{a\mu},\mathcal{A}_{r\mu}]}\equiv\int
D\varphi_{r}D\varphi_{a}\ e^{iS_{\rm eff}[B_{a\mu},B_{r\mu}]}.$ (8)

Normalisation of $W$ is such that
$W[\mathcal{A}_{a\mu}=0,\mathcal{A}_{r\mu}]=0$.

The hydrodynamic effective action (4) could be thought of as being obtained by
integrating out the gapped modes of an underlying microscopic theory defined
on the CTP (SK contour). While the microscopic theory is formulated on the SK
contour, the low energy EFT (4) (also (8)) is defined with time running
forward only, along the real axes.

Two currents defined as

$\displaystyle J_{r}^{\mu}(x)=\frac{\delta S_{\rm
eff}}{\delta{\mathcal{A}}_{a\mu}(x)},\qquad\qquad J_{a}^{\mu}(x)=\frac{\delta
S_{\rm eff}}{\delta{\mathcal{A}}_{r\mu}(x)}$ (9)

are conserved by the EOMs for the dynamical fields $\varphi_{r}$ and
$\varphi_{a}$, which are derived from variation of the effective action (4).

###  2.2 Discrete symmetries

The effective action $S_{\rm eff}$ possesses several discrete symmetries,
including parity $\mathcal{P}$ and time reversal $\mathcal{T}$, inherited from
the underlying microscopic theory. These symmetries impose relations among the
TCFs $w_{i}$’s, which we review here, see Crossley:2015evo  for details.

$\bullet$ $Z_{2}$-reflection symmetry:

$\displaystyle S_{\rm eff}[B_{1\mu};B_{2\mu}]=-S_{\rm
eff}[B_{2\mu};B_{1\mu}],$ (10)

which implies that the coefficients of the leading terms in the derivative
expansion of $w_{i}$’s must be non-negative.

$\bullet$ $\mathcal{T}$-symmetry of (2.1) translates into local KMS
conditions,

$\displaystyle w_{1}=-\frac{i}{2}\coth\frac{\beta\omega}{2}(w_{5}-w_{5}^{*}),$
(11) $\displaystyle
w_{2}+w_{3}q^{2}=-\frac{\omega}{2}\coth\frac{\beta\omega}{2}(w_{8}+w_{8}^{*}),\quad
w_{3}=\frac{i}{2}\coth\frac{\beta\omega}{2}(w_{9}-w_{9}^{*}),$ (12)
$\displaystyle w_{4}=-\frac{1}{2}\coth\frac{\beta\omega}{2}(\omega
w_{6}-iw_{7}^{*}).$ (13)

$\bullet$ $\mathcal{PT}$-symmetry leads to Onsager relations,

$\displaystyle w_{4}=-w_{4}^{*},\qquad\omega w_{6}=-iw_{7},$ (14)

which makes it possible to rewrite the relation (13) as

$\displaystyle
w_{4}=\frac{i}{2}\coth\frac{\beta\omega}{2}(w_{7}+w_{7}^{*})=-\frac{\omega}{2}\coth\frac{\beta\omega}{2}(w_{6}-w_{6}^{*}).$
(15)

The TCFs $w_{1},w_{2},w_{3}$ are real functions of $\omega$ and $q$, while
$w_{4}$ is purely imaginary. Overall, there are four independent parameters in
(2.1), which could be taken as $w_{5}$, $w_{7}$ (or equivalently $w_{6}$),
$w_{8}$ and $w_{9}$. The non-fluctuating $U(1)$ current has three TCFs only
(see (19)). Yet, stochastic $U(1)$ current possesses an additional TCF. While
the effective action/constitutive relations are parameterised by four
(independent) coefficients, the number of independent two-point correlators is
only two, with the others being related by the FDR.

The main goal of the present paper is to derive (2.1) from a holographic model
and compute $w_{i}$’s to all orders in the derivative expansion. It will be
demonstrated analytically that all the symmetry induced relations introduced
above are automatically satisfied by the holographic construction.

###  2.3  $U(1)$ current with thermal noise

From (9) (see also equations (4.21)-(4.23) in Crossley:2015evo ),

$\displaystyle
J_{r}^{v}=w_{5}B_{rv}+w_{6}\partial_{v}\partial_{k}B_{rk}+iw_{1}B_{av}+iw_{4}\partial_{k}B_{ak}$
$\displaystyle
J_{r}^{i}=-w_{7}\partial_{i}B_{rv}+w_{8}\partial_{v}B_{ri}+w_{9}\partial_{k}\mathcal{F}_{rik}+iw_{2}B_{ai}-iw_{3}\partial_{i}\partial_{l}B_{al}-iw_{4}^{*}\partial_{i}B_{av}$
$\displaystyle J_{a}^{v}=w_{5}^{*}B_{av}+w_{7}^{*}\partial_{k}B_{ak}$
$\displaystyle
J_{a}^{i}=w_{6}^{*}\partial_{v}\partial_{i}B_{av}-w_{8}^{*}\partial_{v}B_{ai}+w_{9}^{*}\partial_{k}\mathcal{F}_{aik}$
(16)

When $B_{a}=0$, $J_{a}^{\mu}$ vanishes while $J_{r}^{\mu}$ becomes the
hydrodynamic current $J_{hydro}^{\mu}$ Crossley:2015evo :

$\displaystyle
J_{hydro}^{v}=w_{5}B_{rv}+w_{6}\partial_{v}\partial_{k}B_{rk}=(w_{5}+w_{6}\vec{\partial}^{\,2})\mu-
w_{6}\partial_{k}\mathcal{F}_{rkv},$ $\displaystyle
J_{hydro}^{i}=-w_{7}\partial_{i}B_{rv}+w_{8}\partial_{v}B_{ri}+w_{9}\partial_{k}\mathcal{F}_{rik}=(w_{8}-w_{7})\partial_{i}\mu-
w_{8}\mathcal{F}_{riv}+w_{9}\partial_{k}\mathcal{F}_{rik}$ (17)

$J_{hydro}^{v}$ is the charge density and $\mu=B_{rv}$ is identified with the
chemical potential. With $\mu$ replaced by the charge density, the current
density $\vec{J}_{hydro}$ is cast into the same form (1) as that of Bu:2015ame

$\displaystyle
J_{hydro}^{i}=-\mathcal{D}\partial_{i}J_{hydro}^{v}+\sigma_{e}\mathcal{F}_{riv}+\sigma_{m}\partial_{k}\mathcal{F}_{rik},$
(18)

where

$\displaystyle\mathcal{D}=\frac{w_{7}-w_{8}}{w_{5}+w_{6}\vec{\partial}^{\,2}},\quad\sigma_{e}=\frac{w_{8}-w_{7}}{w_{5}+w_{6}\vec{\partial}^{\,2}}w_{6}\vec{\partial}^{\,2}-w_{8},\quad\sigma_{m}=w_{9}-\frac{w_{8}-w_{7}}{w_{5}+w_{6}\vec{\partial}^{\,2}}w_{6}\partial_{v}.$
(19)

Thermal fluctuations are turned on by relaxing $B_{a}=0$ approximation. We can
still set $\mathcal{A}_{1\mu}=\mathcal{A}_{2\mu}=\mathcal{A}_{\mu}$, since
$\mathcal{A}_{\mu}$ is an external field, which is not necessarily assumed to
be fluctuating:

$B_{r\mu}\,=\,\mathcal{A}_{\mu}\,+\,\partial_{\mu}{\varphi_{r}}\,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}B_{a\mu}\,=\partial_{\mu}{\varphi_{a}}$
(20)

The $\varphi_{a}$ field acts as a source of noise both for the charge density
and hydrodynamic current $\vec{J}_{hydro}$:

$\displaystyle
J_{r}^{v}=J_{hydro}^{v}+i(w_{1}\partial_{v}+w_{4}\vec{\partial}^{\,2})\varphi_{a},\qquad\quad
J_{r}^{i}=J_{hydro}^{i}+i(w_{2}-w_{3}\vec{\partial}^{\,2}-w_{4}^{*}\partial_{v})\partial_{i}\varphi_{a},$
$\displaystyle
J_{a}^{v}=(w_{5}^{*}\partial_{v}+w_{7}^{*}\vec{\partial}^{\,2})\varphi_{a},\qquad\qquad\qquad\quad
J_{a}^{i}=(w_{6}^{*}\partial_{v}^{2}\partial_{i}-w_{8}^{*}\partial_{v}\partial_{i})\varphi_{a}$
(21)

The first line of (21) is an all order stochastic constitutive relation for
the conserved current $J_{r}^{\mu}$, which can be recast into

$J_{r}^{i}=-\mathcal{D}\partial_{i}J_{r}^{v}+\sigma_{e}\mathcal{F}_{iv}+\sigma_{m}\partial_{k}\mathcal{F}_{ik}+\Xi\partial_{i}\varphi_{a},$
(22)

where the $\Xi$-term acts as a thermal force:

$\Xi=i\mathcal{D}(w_{1}\partial_{v}+w_{4}\vec{\partial}^{\,2})+i(w_{2}-w_{3}\vec{\partial}^{\,2}-w_{4}^{*}\partial_{v})$
(23)

While $J_{r}^{\mu}$ is conserved, in presence of thermal fluctuations, the
hydrodynamical current $J_{hydro}^{\mu}$ is not:

$\partial_{\mu}J_{hydro}^{\mu}=\xi,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\xi\equiv
G_{0}\varphi_{a}$ (24)

with

$\displaystyle G_{0}=$
$\displaystyle-i\left[w_{1}\partial_{v}^{2}+w_{4}\partial_{v}\vec{\partial}^{\,2}+(w_{2}-w_{3}\vec{\partial}^{\,2})\vec{\partial}^{\,2}-w_{4}^{*}\partial_{v}\vec{\partial}^{\,2}\right]$
$\displaystyle=i\coth\frac{\beta\omega}{2}\left\\{\omega^{2}Im(w_{5})-\omega
q^{2}Re(w_{8})+2\omega q^{2}Re(w_{7})\right\\}$
$\displaystyle=i\coth\frac{\beta\omega}{2}\left\\{\omega^{2}Im(w_{5})-\omega
q^{2}Re(w_{8})-2\omega^{2}q^{2}Im(w_{6})\right\\}$ (25)

$G_{0}$ is clearly purely imaginary. We have recast the continuity equation
into the usual stochastic form. $G_{0}$ is related to the retarded current-
current correlator $G_{R}$ (173). Up to $\coth\frac{\beta\omega}{2}$ pre-
factor, $G_{0}$ is the real part of the denominator in the expression for
$G_{R}$ obtained from (19) (see (173))

$\displaystyle
G_{R}^{vv}=\frac{q^{2}\,\sigma_{e}}{-i\omega+q^{2}\mathcal{D}}=\frac{i\omega
q^{4}w_{6}^{2}-q^{2}w_{5}w_{8}}{-i\omega w_{5}+2i\omega
q^{2}w_{6}-q^{2}w_{8}}$ (26)

Hence $G_{0}$ vanishes at the poles of the $G_{R}$ correlator, which in
holography are determined by the quasi-normal modes.

It is worth noticing that the noise is a scalar, that is, only the
longitudinal sector is fluctuating. This reflects the fact that physically the
quantity that actually fluctuates is the charge density. There are no
fluctuations in the transverse sector, in which the current is induced by the
external fields, assuming that the latter are not fluctuating.

The noise is Gaussian but coloured as it depends on four-momentum. Changing
variable from $\varphi_{a}$ to $\xi$ in the action $S_{\rm eff}$ results in
the following noise-noise correlator

$\langle\xi(x)\,\xi(0)\rangle\,=\tilde{G}_{0}(x)$ (27)

where $\tilde{G}_{0}(x)$ is the inverse Fourier transformation of $-i\,G_{0}$.
Since $-i\,G_{0}$ is a real function in the momentum space, the noise-noise
correlator is symmetric as expected. Numerical results for $\tilde{G}_{0}$
will be presented in subsection 5. Contrary to the white noise behaviour
($\delta$-functional form for $\tilde{G}_{0}$) we will observe non-local
space-time effects in the noise sector.

##  3 Holographic setup

###  3.1 The geometry

The metric of Schwarzschild-AdS5 in the ingoing Eddington-Finkelstein (EF)
coordinate system $x^{M}=(r,v,x^{i})$ is given by the line element

$\displaystyle
ds^{2}=g_{MN}dx^{M}dx^{N}=-f(r)dv^{2}+2dvdr+r^{2}\delta_{ij}dx^{i}dx^{j},\qquad
i,j=1,2,3,$ (28)

where $f(r)=r^{2}-r_{h}^{4}/r^{2}$. We will also use the Schwarzschild
coordinate system $\tilde{x}^{M}=(r,t,x^{i})$, for which the metric (28) is

$\displaystyle
ds^{2}=\tilde{g}_{MN}d\tilde{x}^{M}d\tilde{x}^{N}=\frac{dr^{2}}{f(r)}-f(r)dt^{2}+r^{2}\delta_{ij}dx^{i}dx^{j},\qquad
i,j=1,2,3.$ (29)

In both (28) and (29), the curvature radius of the AdS space is set to unity.

The holographic geometry dual to thermal state with the SK contour at the
boundary is a doubled Schwarzschild-AdS5. We will closely follow the
holographic prescription of Glorioso:2018mmw , which doubled the geometry (28)
by complexifying the radial coordinate $r$ along the contour illustrated in
Figure 1.

Figure 1: The holographic SK contour of Glorioso:2018mmw : the complexified
radial coordinate analytically continued around the event horizon $r=r_{h}$.
The dashed arrows indicate that the horizontal segments are infinitesimally
close to the real axis.

The two ends of the $r$ contour are identified with the SK contour of the
boundary theory Kamenev2011 ; Calzetta-Hu2009 , the infinitesimally small
horizon circle is mapped into initial thermal state, while the horizontal
segments reflect the CTP. The holographic contour of Figure 1 is obtained by
taking two exteriors of an eternal AdS5 black hole Maldacena:2001kr  and
identifying their future horizons.

The EF time $v$ is related to the Schwarzschild time $t$ by

$\displaystyle
t=v-\zeta_{2}(r),\qquad\zeta_{2}(r)\equiv\int_{\infty_{2}}^{r}\frac{dy}{f(y)},\qquad
r\in[r_{h}-\epsilon,\infty_{2}),$ $\displaystyle
t=v-\zeta_{1}(r),\qquad\zeta_{1}(r)\equiv\int_{\infty_{1}}^{r}\frac{dy}{f(y)},\qquad
r\in[r_{h}-\epsilon,\infty_{1}),$ (30)

where the integration constants are fixed by requirement that $t$ and $v$
coincide on the AdS boundaries. An interesting observation is that viewed in
the ingoing EF coordinate (that is the EF time $v$ is identical everywhere
along the radial contour), the Schwarzschild time $t$ is discontinuous at
$r=r_{h}-\epsilon$,

$\displaystyle t^{\rm up}(r_{h}-\epsilon)-t^{\rm
dw}(r_{h}-\epsilon)=\int_{\infty_{1}}^{\infty_{2}}\frac{dy}{f(y)}=-\frac{i\pi}{2r_{h}}=-\frac{i\beta}{2},$
(31)

where $\beta$ is inverse of the black brane temperature $T$. This becomes
important when gluing bulk fields of the upper and lower segments of the
contour in Figure 1.

###  3.2 Maxwell field in the bulk

The holographic model for the $U(1)$ diffusion is a probe Maxwell field in the
above described geometry. The bulk action is

$\displaystyle S_{0}$ $\displaystyle=-\frac{1}{4}\int
d^{4}x\int_{\infty_{2}}^{\infty_{1}}dr\sqrt{-g}F_{MN}F^{MN}$ (32)
$\displaystyle=-\frac{1}{4}\int
d^{4}\tilde{x}\int_{\infty_{2}}^{\infty_{1}}dr\sqrt{-\tilde{g}}\tilde{F}_{MN}\tilde{F}^{MN},$
(33)

where $C_{N}$ and $\tilde{C}_{N}$ are the $U(1)$ bulk gauge fields in EF and
Schwarzschild coordinate systems respectively;
$F_{MN}=\nabla_{M}C_{N}-\nabla_{N}C_{M}$ and
$\tilde{F}_{MN}=\tilde{\nabla}_{M}\tilde{C}_{N}-\tilde{\nabla}_{N}\tilde{C}_{M}$.
To remove the UV divergences near the AdS boundaries $r=\infty_{1}$ and
$r=\infty_{2}$, the bulk action (32) (or (33)) should be supplemented with a
counter-term action:

$\displaystyle S_{\rm c.t.}=\frac{1}{4}\log r\int
d^{4}x\sqrt{-\gamma}F_{\mu\nu}F^{\mu\nu}\bigg{|}_{r=\infty_{1}}-\frac{1}{4}\log
r\int d^{4}x\sqrt{-\gamma}F_{\mu\nu}F^{\mu\nu}\bigg{|}_{r=\infty_{2}},$ (34)

where the indices are contracted with the induced metric $\gamma_{\mu\nu}$

$\displaystyle
ds^{2}|_{\Sigma}=\gamma_{\mu\nu}dx^{\mu}dx^{\nu}=-f(r)dv^{2}+r^{2}\delta_{ij}dx^{i}dx^{j}.$
(35)

Here $\Sigma$ denotes either the hypersurface $r=\infty_{1}$ or
$r=\infty_{2}$. The counter-term action in the Schwarzschild coordinate system
is the same as (34), since $v=t$ at the AdS boundaries. The minimal
subtraction scheme (34) differs from that used in Glorioso:2018mmw .

Transformation rule for the fields from the EF coordinate system to the
Schwarzschild can be easily derived from the coordinate-invariants

$\displaystyle\tilde{C}_{M}d\tilde{x}^{M}=C_{M}dx^{M},$ (36)

leading to

$\displaystyle\tilde{C}_{t}(r,t,\vec{x})=C_{v}(r,v,\vec{x}),\qquad\tilde{C}_{i}(r,t,\vec{x})=C_{i}(r,v,\vec{x}).$
(37)

The radial components of the bulk gauge field differ

$\displaystyle\tilde{C}_{r}(r,t,\vec{x})-\frac{\tilde{C}_{t}(r,t,\vec{x})}{f(r)}=C_{r}(r,v,\vec{x})\Longrightarrow\tilde{C}_{r}(r,t,\vec{x})=C_{r}(r,v,\vec{x})+\frac{C_{v}(r,v,\vec{x})}{f(r)}.$
(38)

Thus it is important to distinguish between two radial gauge choices:

$\displaystyle{\rm Schwarzschild~{}radial~{}gauge}:\qquad$
$\displaystyle\tilde{C}_{r}=0\Longleftrightarrow C_{r}=-\frac{C_{v}}{f(r)},$
$\displaystyle{\rm EF~{}radial~{}gauge}:\qquad$ $\displaystyle
C_{r}=0\Longleftrightarrow\tilde{C}_{r}=\frac{\tilde{C}_{t}}{f(r)}.$ (39)

The EF radial gauge is most commonly used, including in Glorioso:2018mmw .
Yet, for reasons related to time-reversal symmetry which will be explained in
the next section, we chose to perform calculations in the Schwarzschild radial
gauge.

Bulk EOMs are derived by variation of (32) and (33):

$\displaystyle\nabla_{M}F^{MN}=0\Longrightarrow\frac{1}{\sqrt{-g}}\partial_{M}(\sqrt{-g}F^{MN})=0,$
(40)
$\displaystyle\tilde{\nabla}_{M}\tilde{F}^{MN}=0\Longrightarrow\frac{1}{\sqrt{-\tilde{g}}}\tilde{\partial}_{M}(\sqrt{-\tilde{g}}\tilde{F}^{MN})=0.$
(41)

With the help of (37) and (38), the two sets of the Maxwell equations can be
related

$\displaystyle\tilde{\nabla}_{M}\tilde{F}^{Mr}=\nabla_{M}F^{Mr},\quad\tilde{\nabla}_{M}\tilde{F}^{Mt}=\nabla_{M}F^{Mv}-\frac{1}{f(r)}\nabla_{M}F^{Mr},~{}~{}~{}\tilde{\nabla}_{M}\tilde{F}^{Mi}=\nabla_{M}F^{Mi}.$
(42)

The first equation (the $r$-component) is a gauge invariant constraint, which
will play a special role in our construction. When all the bulk equations are
solved (i.e., on-shell holography), (40) and (41) are absolutely equivalent as
is obvious from (42). Yet, in Bu:2015ame  we argued that in order to compute
the TCFs parameterising the (off-shell) constitutive relations for the
current, it is sufficient to solve the bulk dynamical equations only, while
leaving the constraint aside. Within the holographic prescription the
constraint is mapped into continuity equation for the current at the boundary,
which is the dynamical equation for the low energy modes $\varphi_{1}$ and
$\varphi_{2}$. Derivation of the effective action follows the very same
strategy as introduced in Bu:2014sia ; Bu:2014ena ; Bu:2015ika ; Bu:2015ame ,
now frequently referred to as off-shell holography Crossley:2015tka ;
deBoer:2015ija .

We are going to solve the EOMs in the Schwarzschild coordinates and then re-
express the results in the EF coordinates using (37) and (38). In the spirit
of the off-shell formalism, we will not impose the constraint equation. Hence,
the dynamical equations which will be solved are

$\displaystyle\nabla_{M}F^{Mv}-\frac{1}{f(r)}\nabla_{M}F^{Mr}=0\Longleftrightarrow\tilde{\nabla}_{M}\tilde{F}^{Mt}=0,\qquad\nabla_{M}F^{Mi}=0\Longleftrightarrow\tilde{\nabla}_{M}\tilde{F}^{Mi}=0.$
(43)

Notice that the first equation differs from $\nabla_{M}F^{Mv}=0$ derived in
the EF coordinates. The seemingly freedom to modify the dynamical equation is
eliminated when the Schwarzschild radial gauge is implemented in the effective
action.

We will find the Bianchi identity Glorioso:2018mmw  is quite useful:

$\displaystyle\partial_{r}(\sqrt{-g}\nabla_{M}F^{Mr})+\partial_{v}(\sqrt{-g}\nabla_{M}F^{Mv})+\partial_{k}(\sqrt{-g}\nabla_{M}F^{Mk})=0.$
(44)

The dynamical equations (43) are instrumental in deriving a holographic RG
flow-like equation for $\nabla_{M}F^{Mr}$:

$\displaystyle\partial_{r}\left(\sqrt{-g}\nabla_{M}F^{Mr}\right)=\frac{i\omega}{f(r)}\sqrt{-g}\nabla_{M}F^{Mr},$
(45)

which is solved by

$\displaystyle\sqrt{-g}\nabla_{M}F^{Mr}=\mathcal{C}^{\rm
up}(k)e^{i\omega\zeta_{2}(r)},\qquad r\in[r_{h}-\epsilon,\infty_{2}),$
$\displaystyle\sqrt{-g}\nabla_{M}F^{Mr}=\mathcal{C}^{\rm
dw}(k)e^{i\omega\zeta_{1}(r)},\qquad r\in[r_{h}-\epsilon,\infty_{1}).$ (46)

Here, $\mathcal{C}^{\rm up,\,dw}$ are $r$-independent integration constants in
either upper or lower segments of the $r$-contour. Both vanish on-shell.

###  3.3 Boundary effective action

The basic procedure of how to derive a hydrodynamic effective action from
gravity has been realized in Crossley:2015tka  (see also Glorioso:2018mmw ),
based on early attempts of formulating a holographic Wilsonian RG flow
Heemskerk:2010hk ; Faulkner:2010jy . It amounts to identifying hydrodynamical
variables (gapless modes) of the boundary theory, as proposed in Nickel:2010pr
. The remaining degrees of freedom are then integrated out from the bulk
action, in Wilsonian sense. The procedure is outlined below for a free $U(1)$
gauge field in the bulk.

The starting point is the bulk partition function:

$\displaystyle Z=\int DC_{r}DC_{\mu}e^{iS_{0}[C_{\mu},C_{r}]+iS_{\rm c.t.}}.$
(47)

No gauge-fixing has been applied at this stage. Integrating by parts, the bulk
action $S_{0}$ (32) can be expressed as

$\displaystyle S_{0}=$ $\displaystyle-\frac{1}{2}\int
d^{4}x\int_{\infty_{2}}^{\infty_{1}}dr\sqrt{-g}\left[\nabla_{M}\left(C_{N}F^{MN}\right)-C_{N}\nabla_{M}F^{MN}\right]$
$\displaystyle=$ $\displaystyle\frac{1}{2}\int
d^{4}x\int_{\infty_{2}}^{\infty_{1}}dr\sqrt{-g}\left[C_{r}\nabla_{M}F^{Mr}+C_{v}\nabla_{M}F^{Mv}+C_{k}\nabla_{M}F^{Mk}\right]$
$\displaystyle-\frac{1}{2}\int
d^{4}x\sqrt{-\gamma}n_{M}C_{N}F^{MN}\bigg{|}_{r=\infty_{2}}^{r=\infty_{1}}$
$\displaystyle=$ $\displaystyle\frac{1}{2}\int
d^{4}x\int_{\infty_{2}}^{\infty_{1}}dr\sqrt{-g}\left\\{C_{v}\left[\nabla_{M}F^{Mv}-\frac{1}{f(r)}\nabla_{M}F^{Mr}\right]+C_{k}\nabla_{M}F^{Mk}\right.$
$\displaystyle\left.+\left(C_{r}+\frac{C_{v}}{f(r)}\right)\nabla_{M}F^{Mr}\right\\}-\frac{1}{2}\int
d^{4}x\sqrt{-\gamma}n_{M}C_{N}F^{MN}\bigg{|}_{r=\infty_{2}}^{r=\infty_{1}},$
(48)

where $n_{M}$ is a out-pointing unit vector normal to the hypersurface
$\Sigma$.

Our goal now is to derive the effective action. We will demonstrate how the
bulk gauge invariance is realised. Our derivation is an explicit holographic
construction that closely follows the original ideas of Crossley:2015evo .

The discussion below applies separately to each segment of the contour. The
asymptotic values of the gauge field are identified with the boundary external
background field $\mathcal{A}_{\mu}$,

$\displaystyle C_{\mu}(r=\infty,x^{\alpha})\equiv\mathcal{A}_{\mu}.$ (49)

Consider a generic gauge transformation parameterised by
$\Lambda(r,x^{\alpha})$ transforming the bulk gauge field $C_{M}$ as

$\displaystyle C_{\mu}\to
C_{\mu}^{\prime}=C_{\mu}+\partial_{\mu}\Lambda(r,x^{\alpha}),\qquad C_{r}\to
C_{r}^{\prime}=C_{r}+\partial_{r}\Lambda(r,x^{\alpha}).$ (50)

In the path integral (47), the gauge transformation (50) can be viewed as a
change of integration variables, say, from $C_{r}$ to $\Lambda$. In other
words, the gauge transformation defines a map from any given configuration of
$C_{r}$ into a configuration with some fixed radial gauge. For example, for
$C_{M}^{\prime}$ to be in the EF radial gauge,

$\displaystyle
C_{r}^{\prime}=0\Rightarrow\Lambda(r,x^{\alpha})=\int_{r}^{r_{c}}d\xi
C_{r}(\xi,x^{\alpha}).$ (51)

The boundary value of $C_{\mu}^{\prime}$:

$\displaystyle
C_{\mu}^{\prime}(r=\infty,x^{\alpha})=\mathcal{A}_{\mu}+\partial_{\mu}\varphi,$
(52)

where $\varphi\equiv\Lambda(r=\infty,x^{\alpha})$ is identified with the
hydrodynamic field associated with the $U(1)$ charge. $\Lambda$ can be
similarly constructed for the Schwarzschild radial gauge
$C_{r}^{\prime}=-C_{v}^{\prime}/f(r)$.

Integration over $C_{\mu}$ in (47) can be performed in the saddle point
approximation, which is exact for the Maxwell theory. That is, solutions of
the dynamical equations for $C_{\mu}$ are plugged into $S_{0}[C_{\mu},C_{r}]$.
As long as no radial gauge is imposed, $C_{\mu}$ are functionals of $C_{r}$.
Generically, the partition function (47) turns into

$\displaystyle Z=\int DC_{r}e^{iS_{0}[C_{\mu}[C_{r}],C_{r}]+iS_{\rm c.t.}},$
(53)

where $S_{0}[C_{\mu}[C_{r}],C_{r}]$ is a “partially on-shell” bulk action
deBoer:2015ija  which depends on $C_{r}$ both explicitly and through solutions
for $C_{\mu}$. The action is gauge invariant. Since in practice
$C_{\mu}[C_{r}]$ are computed in a fixed gauge, the next step is to fix the
gauge using (50), while changing the integration variable from $C_{r}$ to
$\Lambda$. The resulting partially on-shell action does not depend on the
whole $\Lambda$ but on $\varphi$ only. (53) becomes

$\displaystyle Z\sim\int D\varphi e^{iS_{0}|_{\rm p.o.s.}+iS_{\rm c.t.}}$ (54)

where $S_{0}|_{\rm p.o.s.}+S_{\rm c.t.}$ is the boundary effective action
$S_{eff}$ of the $\varphi$ field. Here, an overall coefficient due to Jacobian
of the change of variables has been absorbed into $D\varphi$ measure. Below we
will explicitly derive $S_{0}|_{\rm p.o.s.}$ for the EF radial gauge
$C_{r}^{\prime}=0$ and Schwarzschild radial gauge
$C_{r}^{\prime}=-C_{v}^{\prime}/f(r)$, and demonstrate the equivalence of the
results. In fact, it is straightforward to show that the result for
$S_{0}|_{\rm p.o.s.}$ is independent of the gauge-fixing.

In the EF radial gauge $C_{r}^{\prime}=0$, the partially on-shell bulk action
is evaluated as

$\displaystyle S_{0}|_{\rm p.o.s}$ $\displaystyle=\frac{1}{2}\int
d^{4}x\int_{\infty_{2}}^{\infty_{1}}dr\sqrt{-g}C_{r}\nabla_{M}F^{Mr}-\frac{1}{2}\int
d^{4}x\sqrt{-\gamma}n_{M}C_{N}F^{MN}\bigg{|}_{r=\infty_{2}}^{r=\infty_{1}}$
$\displaystyle=\frac{1}{2}\int
d^{4}x\int_{\infty_{2}}^{\infty_{1}}dr\sqrt{-g}(-\partial_{r}\Lambda)\nabla_{M}F^{Mr}-\frac{1}{2}\int
d^{4}x\sqrt{-\gamma}n_{M}C_{N}F^{MN}\bigg{|}_{r=\infty_{2}}^{r=\infty_{1}}$
$\displaystyle=\frac{1}{2}\int
d^{4}x\left[-\sqrt{-g}\Lambda\nabla_{M}F^{Mr}-\sqrt{-\gamma}n_{M}C_{N}F^{MN}\right]\bigg{|}_{r=\infty_{2}}^{r=\infty_{1}}$
$\displaystyle=-\frac{1}{2}\int
d^{4}x\sqrt{-g}C_{\mu}^{\prime}F^{r\mu}[C_{\mu}^{\prime},C_{r}^{\prime}=0]\bigg{|}_{r=\infty_{2}}^{r=\infty_{1}}.$
(55)

Similarly, in the Schwarzschild radial gauge
$C_{r}^{\prime}=-C_{v}^{\prime}/f(r)$, the partially on-shell bulk action is
calculated as

$\displaystyle S_{0}|_{\rm p.o.s}$ $\displaystyle=\frac{1}{2}\int
d^{4}x\int_{\infty_{2}}^{\infty_{1}}dr\sqrt{-g}\left(C_{r}+\frac{C_{v}}{f(r)}\right)\nabla_{M}F^{Mr}-\frac{1}{2}\int
d^{4}x\sqrt{-\gamma}n_{M}C_{N}F^{MN}\bigg{|}_{r=\infty_{2}}^{r=\infty_{1}}$
$\displaystyle=-\frac{1}{2}\int
d^{4}x\int_{\infty_{2}}^{\infty_{1}}dr\sqrt{-g}\left(\partial_{r}\Lambda+\frac{\partial_{v}\Lambda}{f(r)}\right)\nabla_{M}F^{Mr}-\frac{1}{2}\int
d^{4}x\sqrt{-\gamma}n_{M}C_{N}F^{MN}\bigg{|}_{r=\infty_{2}}^{r=\infty_{1}}$
$\displaystyle=-\frac{1}{2}\int
d^{4}x\int_{\infty_{2}}^{\infty_{1}}dr\left(\partial_{r}+\frac{\partial_{v}}{f(r)}\right)\left[\Lambda\sqrt{-g}\nabla_{M}F^{Mr}\right]-\frac{1}{2}\int
d^{4}x\sqrt{-\gamma}n_{M}C_{N}F^{MN}\bigg{|}_{r=\infty_{2}}^{r=\infty_{1}}$
$\displaystyle=\frac{1}{2}\int
d^{4}x\int_{\infty_{2}}^{\infty_{1}}dr\left[-\Lambda\sqrt{-g}\nabla_{M}F^{Mr}-\sqrt{-\gamma}n_{M}C_{N}F^{MN}\right]\bigg{|}_{r=\infty_{2}}^{r=\infty_{1}}$
$\displaystyle=-\frac{1}{2}\int
d^{4}x\sqrt{-g}C_{\mu}^{\prime}F^{r\mu}[C_{\mu}^{\prime},C_{r}^{\prime}=-C_{v}^{\prime}/f(r)]\bigg{|}_{r=\infty_{2}}^{r=\infty_{1}}.$
(56)

Both in (55) and (56), the terms representing the total derivatives at the AdS
boundaries were dropped. The Bianchi identity (44) was instrumental to convert
the bulk integral into the surface term. Gauge invariance of the field
strength $F^{MN}[C_{\mu},C_{r}]=F^{MN}[C_{\mu}^{\prime},C_{r}^{\prime}]$ was
used as well. (55) and (56) are identical and become a superposition of two
surface terms

$\displaystyle S_{0}|_{\rm p.o.s.}=-\frac{1}{2}\int
d^{4}x\sqrt{-\gamma}n_{M}C_{N}^{\prime}F^{MN}[C_{\mu}^{\prime},C_{r}^{\prime}]\bigg{|}_{r=\infty_{2}}^{r=\infty_{1}}.$
(57)

Notice that (57) could be obtained from (a primed version of) (48) by dropping
all the bulk terms, either because of the dynamical equations (43) or as a
result of the Schwarzschild radial gauge333In Glorioso:2018mmw , the EF radial
gauge $C_{r}=0$ was taken, combined with the dynamical equations
$\nabla_{M}F^{Mv}=0$ and $\nabla_{M}F^{Mi}=0$, which also eliminated the bulk
terms in (48) leading to the very same boundary action..

There is a subtle point in deriving (57). As will become clear in the next
section, in the off-shell formalism, the gauge potential and/or field strength
develop discontinuity near the event horizon. Hence, in principle, there might
emerge an additional boundary term at the horizon surface. This term will be
eliminated by a specific choice of the boundary conditions to be discussed
below.

Below we will drop the prime in $C_{M}^{\prime}$ and permanently stick to the
Schwarzschild radial gauge for $C_{M}$. The bulk EOMs (43) will be solved
subject to boundary conditions at two AdS boundaries $r=\infty_{1}$ and
$r=\infty_{2}$:

$\displaystyle
C_{\mu}\xrightarrow[]{r\to\infty_{1}}B_{1\mu}(x^{\alpha})\equiv\mathcal{A}_{1\mu}+\partial_{\mu}\varphi_{1},\qquad\qquad
C_{\mu}\xrightarrow[]{r\to\infty_{2}}B_{2\mu}(x^{\alpha})\equiv\mathcal{A}_{2\mu}+\partial_{\mu}\varphi_{2}.$
(58)

In order to compute the boundary action (57), near boundary asymptotic
expansion of the bulk fields is required. It has the form

$\displaystyle
C_{\mu}\xrightarrow[]{r\to\infty_{1}}B_{1\mu}(x^{\alpha})+\frac{\partial_{v}B_{1\mu}(x^{\alpha})}{r}-\frac{1}{2}\partial^{\nu}\mathcal{F}_{1\mu\nu}(x^{\alpha})\frac{\log
r}{r^{2}}+\frac{C_{1\mu}^{(2)}(x^{\alpha})}{r^{2}}+\cdots,$ (59)
$\displaystyle
C_{\mu}\xrightarrow[]{r\to\infty_{2}}B_{2\mu}(x^{\alpha})+\frac{\partial_{v}B_{2\mu}(x^{\alpha})}{r}-\frac{1}{2}\partial^{\nu}\mathcal{F}_{2\mu\nu}(x^{\alpha})\frac{\log
r}{r^{2}}+\frac{C_{2\mu}^{(2)}(x^{\alpha})}{r^{2}}+\cdots,$ (60)

where the coefficient functions $C_{1\mu}^{(2)}$ and $C_{2\mu}^{(2)}$ are
functionals of both $B_{1\mu}$ and $B_{2\mu}$. They will be determined through
the solution of the dynamical equations (43) over the entire contour in
Figure. 1. This is the subject of the next section.

Once (59) and (60) are substituted into the total action $S_{\rm
eff}=S_{0}|_{\rm p.o.s.}+S_{\rm c.t.}$, the latter takes the form (4) with the
effective Lagrangian $\mathcal{L}_{\rm eff}(x)$ being a quadratic functional
of the boundary fields $B_{1\mu}$ and $B_{2\mu}$ (in the $(r,a)$-basis)

$\displaystyle\mathcal{L}_{\rm eff}=$ $\displaystyle-
B_{rv}(C_{1v}^{(2)}-C_{2v}^{(2)})-\frac{1}{2}B_{av}(C_{1v}^{(2)}+C_{2v}^{(2)})+B_{rk}(C_{1k}^{(2)}-C_{2k}^{(2)})$
$\displaystyle+\frac{1}{2}B_{ak}(C_{1k}^{(2)}+C_{2k}^{(2)}){+\frac{1}{2}\partial_{k}B_{ak}\partial_{v}B_{rv}-\frac{1}{2}B_{av}\partial_{v}\partial_{k}B_{rk}}$
$\displaystyle-\frac{1}{2}B_{ak}\partial_{v}^{2}B_{rk}+\frac{1}{4}\mathcal{F}_{akj}\mathcal{F}_{rkj}+\frac{1}{2}B_{av}\vec{\partial}^{\,2}B_{rv}{+B_{av}\partial_{v}^{2}B_{rv}}.$
(61)

With the help of a basis decomposition procedure introduced in Bu:2015ame ,
the effective Lagrangian (61) can be recast into the form (2.1), thus
providing a holographic derivation of the latter, which is fully consistent
with the general analysis of Crossley:2015evo . This derivation is presented
in Appendix A.

##  4  Bulk dynamics: solutions and analysis

This section is devoted to solutions of the bulk EOMs (43) over the contour
displayed in Figure 1. Our strategy will be different from that of
Glorioso:2018mmw . Instead of integrating the bulk EOMs (43) along the entire
$r$-contour, we split the radial contour at $r=r_{h}-\epsilon$ into two
segments, the upper and lower one. Each segment “lives” in a single copy of
the doubled Schwarzschild-AdS5 geometry. In each segment, there is a set of
independent solutions to the dynamical EOMs (43) forming a basis. Full
solutions obeying the respective boundary conditions (58) will be constructed
as linear superpositions of the basis solutions. Thus-constructed piecewise
solutions will be carefully glued at the cutting slice $r=r_{h}-\epsilon$,
under proper matching conditions to be derived in subsection 4.2. One of the
advantages of our approach is that it avoids the subtleties related to non-
commutativity between two limits: the hydrodynamic derivative expansion and
$\epsilon\rightarrow 0$, the latter has to be taken first.

###  4.1 Discrete symmetries

Symmetries in classical theories are used in order to generate solutions, if
one is already known. Maxwell’s theory in the bulk (single copy Schwarzschild-
AdS) has a number of discrete symmetries, such as parity and time reversal,
which will be employed in our quest after a full set of independent solutions.
However, in different coordinate systems, the symmetries are represented
differently. Particularly, while in the Schwarzschild coordinates the time
reversal symmetry is realised trivially, its representation in the EF
coordinates is much less transparent. This is essentially the main reason we
have chosen to first find all the solutions in the Schwarzschild coordinates
and then translate those into the EF system.

In the Schwarzschild coordinates (without any gauge fixing yet), the Fourier
mode $\tilde{C}_{M}(r,k^{\mu})$ defined by

$\displaystyle\tilde{C}_{M}(r,t,x^{i})=\int\frac{d^{4}k}{(2\pi)^{4}}e^{ik\cdot
x}\tilde{C}_{M}(r,k^{\mu}),\qquad k^{\mu}=(\omega,\vec{q}),$ (62)

satisfies the following system of ODEs:

$\displaystyle\tilde{\nabla}_{M}\tilde{F}^{Mr}=0\Rightarrow 0=$
$\displaystyle(i\omega\partial_{r}\tilde{C}_{t}-\omega^{2}\tilde{C}_{r})+r^{-2}f(r)(q^{2}\tilde{C}_{r}+iq_{k}\partial_{r}\tilde{C}_{k}),$
(63) $\displaystyle\tilde{\nabla}_{M}\tilde{F}^{Mt}=0\Rightarrow 0=$
$\displaystyle\partial_{r}(r^{3}\partial_{r}\tilde{C}_{t})+i\omega\partial_{r}(r^{3}\tilde{C}_{r})-\frac{r}{f(r)}(q^{2}\tilde{C}_{t}+\omega
q_{k}\tilde{C}_{k}),$ (64)
$\displaystyle\tilde{\nabla}_{M}\tilde{F}^{Mi}=0\Rightarrow 0=$
$\displaystyle\partial_{r}[rf(r)\partial_{r}\tilde{C}_{i}]-iq_{i}\partial_{r}[rf(r)\tilde{C}_{r}]+\frac{r}{f(r)}(\omega^{2}\tilde{C}_{i}+\omega
q_{i}\tilde{C}_{t})$
$\displaystyle+r^{-1}(-q^{2}\tilde{C}_{i}+q_{i}q_{k}\tilde{C}_{k}).$ (65)

The full set of bulk EOMs (63)-(65) is obviously invariant under the following
time-reversal transformation:

$\displaystyle\omega\to-\omega,\qquad\tilde{C}_{t}(r,k^{\mu})\to\tilde{C}_{t}(r,\bar{k}^{\mu}),\qquad\tilde{C}_{i}(r,k^{\mu})\to-\tilde{C}_{i}(r,\bar{k}^{\mu}),$
$\displaystyle\tilde{C}_{r}(r,k^{\mu})\to-\tilde{C}_{r}(r,\bar{k}^{\mu}),\qquad\qquad{\rm
with}~{}~{}\bar{k}^{\mu}=(-\omega,\vec{q}),$ (66)

which, in the coordinate space $(r,t,\vec{x})$, turns into

$\displaystyle
t\to-t,\qquad\tilde{C}_{t}(r,t,\vec{x})\to\tilde{C}_{t}(r,-t,\vec{x}),\qquad\tilde{C}_{i}(r,t,\vec{x})\to-\tilde{C}_{i}(r,-t,\vec{x}),$
$\displaystyle\tilde{C}_{r}(r,t,\vec{x})\to-\tilde{C}_{r}(r,-t,\vec{x}).$ (67)

Furthermore, the dynamical EOMs (64) and (65) are invariant under the time
reversal independently, regardless of the constraint equation (63) being
imposed or not. Transformations (4.1) and (4.1) can be recognised as a linear
realisation of the time-reversal symmetry Chakrabarty:2019aeu .

In the ingoing EF coordinates (28), the Fourier mode $C_{M}(r,k^{\mu})$
defined by

$\displaystyle C_{M}(r,x^{\mu})=\int\frac{d^{4}k}{(2\pi)^{4}}e^{ik\cdot
x}C_{M}(r,k^{\mu}),\qquad k^{\mu}=(\omega,\vec{q}),$ (68)

obeys another system of ODEs:

$\displaystyle\nabla_{M}F^{Mr}=0~{}\Rightarrow~{}0=r^{3}(\omega^{2}C_{r}-i\omega\partial_{r}C_{v})-rf(r)(q^{2}C_{r}+iq_{k}\partial_{r}C_{k})-r(q^{2}C_{v}+\omega
q_{k}C_{k}),$ (69)
$\displaystyle\nabla_{M}F^{Mv}-\frac{1}{f(r)}\nabla_{M}F^{Mr}=0~{}\Rightarrow~{}0=\partial_{r}(r^{3}\partial_{r}C_{v}+i\omega
r^{3}C_{r})+r(q^{2}C_{r}+iq_{k}\partial_{r}C_{k})$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\frac{r^{3}}{f(r)}(\omega^{2}C_{r}-i\omega\partial_{r}C_{v})-\frac{r}{f(r)}(q^{2}C_{v}+\omega
q_{k}C_{k})-r(q^{2}C_{r}+iq_{k}\partial_{r}C_{k}),$ (70)
$\displaystyle\nabla_{M}F^{Mi}=0~{}\Rightarrow~{}0=\partial_{r}[rf(r)(\partial_{r}C_{i}-iq_{i}C_{r})]-\partial_{r}[r(i\omega
C_{i}+iq_{i}C_{v})]$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-r(i\omega\partial_{r}C_{i}+\omega
q_{i}C_{r})+r^{-1}(-q^{2}C_{i}+q_{i}q_{k}C_{k}).$ (71)

Apparently, the transformation like (4.1) is not a symmetry of EOMs (69)-(71).
This is related to the fact that $v\to-v$ is not a symmetry, because the
transformation does not leave the metric (28) invariant. This point was made
clear in Chakrabarty:2019aeu , which carried out a somewhat similar analysis
of a probe string in Schwarzschild-AdS background.

What is a nonlinear realisation of the underlying time-reversal symmetry in
the ingoing EF coordinates? This could be worked out with the help of (37) and
(38). The Fourier modes in the Schwarzschild and ingoing EF coordinates are
related,

$\displaystyle C_{\mu}(r,k)=\tilde{C}_{\mu}(r,k)e^{i\omega\zeta_{s}(r)},\qquad
C_{r}(r,k)=\left[\tilde{C}_{r}(r,k)-\frac{\tilde{C}_{t}(r,k)}{f(r)}\right]e^{i\omega\zeta_{s}(r)},$
$\displaystyle r\in[r_{h}-\epsilon,\infty_{s}),\qquad s=1~{}{\rm or}~{}2,$
(72)

where $\zeta_{1}(r)$ and $\zeta_{2}(r)$ are introduced in (30). Thus, in the
ingoing EF coordinates, the time-reversal symmetry is realized as

$\displaystyle\omega\to-\omega,\quad C_{v}(r,k)\to
C_{v}(r,\bar{k})e^{2i\omega\zeta_{s}(r)},\quad C_{i}(r,k)\to-
C_{i}(r,\bar{k})e^{2i\omega\zeta_{s}(r)},$ $\displaystyle
C_{r}(r,k)\to-\left[C_{r}(r,\bar{k})+\frac{2C_{v}(r,\bar{k})}{f(r)}\right]e^{2i\omega\zeta_{s}(r)},\qquad
s=1,2.$ (73)

So far, no gauge choice has been specified. In Glorioso:2018mmw , the EF
radial gauge $C_{r}=0$ was chosen, which is in tension with the linear
realisation of the time-reversal symmetry (4.1). In fact, a time reversed
solution is gauge transformed with respect to $C_{r}=0$. Hence, in order to
benefit from the simplicity of (4.1), we search for solutions in the
Schwarzschild radial gauge $\tilde{C}_{r}=0$.

The bulk EOMs (63)-(65) are also invariant under $\mathcal{P}$-symmetry (space
inversion):

$\displaystyle\vec{q}\to-\vec{q},\qquad\tilde{C}_{t}(r,k^{\mu})\to\tilde{C}_{t}(r,-\bar{k}^{\mu}),\qquad\tilde{C}_{i}(r,k^{\mu})\to-\tilde{C}_{i}(r,-\bar{k}^{\mu}),$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\tilde{C}_{r}(r,k^{\mu})\to\tilde{C}_{r}(r,-\bar{k}^{\mu}),$
(74)

or, alternatively, in the coordinate space $(r,t,\vec{x})$

$\displaystyle\vec{x}\to-\vec{x},\qquad\tilde{C}_{t}(r,t,\vec{x})\to\tilde{C}_{t}(r,t,-\vec{x}),\qquad\tilde{C}_{i}(r,t,\vec{x})\to-\tilde{C}_{i}(r,t,-\vec{x}),$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\tilde{C}_{r}(r,t,\vec{x})\to\tilde{C}_{r}(r,t,-\vec{x}).$
(75)

Since the coordinate transformation from (28) to (29) does not involve spatial
directions, the $\mathcal{P}$-symmetry in the EF coordinates takes exactly the
same form as (74) and (75), which can be straightforwardly checked from the
bulk EOMs (69)-(71).

The time-reversal $\mathcal{T}$ symmetry and $\mathcal{P}$-symmetry help to
examine the symmetry relations for TCFs in the effective action, as reviewed
in subsection 2.2.

###  4.2 Horizon matching conditions

As has been already mentioned, we first derive independent solutions in the
upper and lower segments, and then glue them at the surface
$r=r_{h}-\epsilon$. In this way a complete solution valid along the whole
radial contour in Figure 1 is constructed. A necessary element of this
construction is a set of matching conditions for the bulk fields to be
discussed in this subsection.

The contour in Figure 1 is cut along the surface $r=r_{\pm}\equiv
r_{h}-\epsilon$, where the subscripts + and - are used to distinguish the
upper and lower segments. In the spirit of Skenderis:2008dg , the matching
conditions at $r=r_{h}-\epsilon$ can be obtained by demanding the total bulk
action to be extremal with respect to variation of the horizon data
$C_{M}(r_{h}-\epsilon,x^{\mu})$. Consider variation of the bulk action444The
counter-term action $S_{\rm c.t.}$ does not contribute to the variational
problem near horizon.  (32)

$\displaystyle\delta S_{0}=$ $\displaystyle-\int
d^{4}x\sqrt{-\gamma}n_{M}\delta C_{N}F^{MN}\bigg{|}_{r_{-}}^{\infty_{1}}+\int
d^{4}x\sqrt{-\gamma}n_{M}\delta C_{N}F^{MN}\bigg{|}_{r_{+}}^{\infty_{2}}$
$\displaystyle+\int
d^{4}x\int_{\infty_{2}}^{\infty_{1}}dr\sqrt{-g}\left[\delta
C_{r}\nabla_{M}F^{Mr}+\delta C_{v}\frac{\nabla_{M}F^{Mr}}{f(r)}\right]$ (76)

where we imposed the dynamical equations (43). The extremum condition (at the
surface $r=r_{h}-\epsilon$) gives

$\displaystyle\frac{\delta S}{\delta C_{v}(r_{h}-\epsilon)}=0\Rightarrow
F^{rv}(r_{+})-F^{rv}(r_{-})=\lim_{\Delta\to
0}\int_{r_{+}+\Delta}^{r_{-}-\Delta}dr\frac{\nabla_{M}F^{Mr}}{f(r)},$ (77)
$\displaystyle\frac{\delta S}{\delta C_{i}(r_{h}-\epsilon)}=0\Rightarrow
F^{ri}(r_{+})-F^{ri}(r_{-})=0,$ (78)

where $\Delta$ is an infinitesimal interval along the circle in Figure 1. The
field strength components $F^{ri}$ are continuous through the cutting surface.
Yet, the $F^{rv}$ component, being continuous for on-shell theory, may develop
discontinuity if the constraint equation is relaxed, that is,
$\nabla_{M}F^{Mr}\neq 0$. When implementing the Schwarzschild radial gauge
$\tilde{C}_{r}=0$, the matching condition (78) translates into

$\displaystyle
f(r)\partial_{r}C_{i}\big{|}_{r=r_{+}}=f(r)\partial_{r}C_{i}\big{|}_{r=r_{-}}.$
(79)

Since $f(r)$ vanishes at the horizon, $\partial_{r}C_{i}$ may not be
continuous. Similarly, $C_{r}$ component is discontinuous.

The condition (77) is very non-trivial. It suggests that $\partial_{r}C_{v}$
is discontinuous too. Yet, this conclusion is based on assumption that
$C_{\mu}$ is continuous across the cutting slice:

$\displaystyle C_{\mu}(r_{-})=C_{\mu}(r_{+}).$ (80)

(80) is a natural choice that could be realised via a residual gauge
transformation implementable on each segment independently. Furthermore,
thanks to the residual gauge freedom, we could set

$\displaystyle C_{v}(r_{h}-\epsilon,x^{\mu})=0.$ (81)

This choice has been also implemented in Glorioso:2018mmw , though through a
somewhat different chain of arguments. Once the condition (81) is imposed,
(77) fixes the discontinuity of $\partial_{r}C_{v}$ uniquely. Below we will
construct the full solution of EOMs with the matching condition (81) and then
check that (77) is indeed satisfied. This consistency check will be presented
in Appendix B.

Finally, an added value of the choice (81) is that it makes the horizon
contribution to the effective action $S_{0}$ vanish. Hence, (57) is correct
555Working with another residual gauge would lead us to different matching
conditions and, as a consequence, different solutions of the EOMs, and also to
a modified expression for the effective action. The final action is however
gauge invariant and hence should not depend on a particular choice of the
residual gauge. .

###  4.3 Linearly independent solutions

In this subsection, we derive and analyse all linearly independent solutions
of the dynamical EOMs (43) in a single Schwarzschild-AdS5. The solutions are
equally valid both in the upper and lower segments in Figure 1. As argued
previously, in order to benefit from the linear realisation (4.1) of the time-
reversal symmetry, we temporarily work in the Schwarzschild coordinate system
combined with the Schwarzschild radial gauge. Eventually, linearly independent
solutions in the ingoing EF coordinates will be deduced via the transformation
rule (72). Without loss of generality, the spatial momentum $\vec{q}$ is taken
to be along the $x$-direction. Then, the bulk fields $\tilde{C}_{\mu}$
decouple between two sectors: the transverse sector
$\tilde{C}_{\perp}=\\{\tilde{C}_{y},\tilde{C}_{z}\\}$ and the longitudinal
sector $\tilde{C}_{\parallel}=\\{\tilde{C}_{t},\tilde{C}_{x}\\}$.

Transverse sector.

The transverse mode $\tilde{C}_{\perp}$ obeys a single ODE:

$\displaystyle
0=\partial_{r}[rf(r)\partial_{r}\tilde{C}_{\perp}]+\frac{\omega^{2}r}{f(r)}\tilde{C}_{\perp}-q^{2}r^{-1}\tilde{C}_{\perp},\qquad\perp=y,z.$
(82)

Eq. (82) has two independent solutions distinguishable by their near horizon
behaviour.

Near the horizon $r=r_{h}$, the ingoing solution $\tilde{C}_{\perp}^{\rm
ig}(r,k^{\mu})$ behaves as

$\displaystyle\tilde{C}_{\perp}^{\rm ig}(r,k^{\mu})\xrightarrow[]{r\to
r_{h}}(r-r_{h})^{-i\omega/(4r_{h})}\left[\tilde{C}_{\perp}^{h}+\tilde{C}_{\perp}^{1}(r-r_{h})+\tilde{C}_{\perp}^{2}(r-r_{h})^{2}+\cdots\right],$
(83)

where $\tilde{C}_{\perp}^{h}$ is an integration constant (initial condition),
which we refer to as horizon data. The remaining coefficients
$\tilde{C}_{\perp}^{1},\tilde{C}_{\perp}^{2},\cdots$ are uniquely fixed in
terms of $\tilde{C}_{\perp}^{h}$.

The outgoing solution666The outgoing solution can be identified with the
Hawking radiation. $\tilde{C}_{\perp}^{\rm og}(r,k^{\mu})$ is obtained from
the ingoing one by the time-reversal symmetry (4.1)

$\displaystyle\tilde{C}_{\perp}^{\rm og}(r,k^{\mu})=-\tilde{C}_{\perp}^{\rm
ig}(r,\bar{k}^{\mu}).$ (84)

Both solutions are functions of $q^{2}$ as is obvious from (82), and hence
they are $\mathcal{P}$-invariant. From (72), the solutions in the ingoing EF
coordinates read:

$\displaystyle C_{\perp}^{\rm ig}(r,k^{\mu})=\tilde{C}_{\perp}^{\rm
ig}(r,k^{\mu})e^{i\omega\zeta_{s}(r)},\quad C_{\perp}^{\rm
og}(r,k^{\mu})=\tilde{C}_{\perp}^{\rm
og}(r,k^{\mu})e^{i\omega\zeta_{s}(r)},\quad r\in[r_{h}-\epsilon,\infty_{s}).$
(85)

Thus, the ingoing and outgoing solutions are related to each other via

$\displaystyle C_{\perp}^{\rm og}(r,k^{\mu})=-C_{\perp}^{\rm
ig}(r,\bar{k}^{\mu})e^{2i\omega\zeta_{s}(r)},\qquad
r\in[r_{h}-\epsilon,\infty_{s}),\qquad s=1~{}{\rm or}~{}2,$ (86)

where (84) is used.

Near the AdS boundary the ingoing solution $C_{\perp}^{\rm ig}$ can be
expanded

$\displaystyle C_{\perp}^{\rm ig}(r,k^{\mu})\xrightarrow[]{r\to\infty}$
$\displaystyle C_{\perp}^{\rm ig(0)}(k^{\mu})-\frac{i\omega C_{\perp}^{\rm
ig(0)}(k^{\mu})}{r}+\frac{1}{2}(\omega^{2}-q^{2})C_{\perp}^{\rm
ig(0)}(k^{\mu})\frac{\log r}{r^{2}}+\frac{C_{\perp}^{\rm
ig(2)}(k^{\mu})}{r^{2}}+\cdots.$ (87)

In principle, one could tune $\tilde{C}_{\perp}^{h}$ so that $C_{\perp}^{\rm
ig(0)}(k^{\mu})=1$, though for a while we prefer to keep it unspecified.

Longitudinal sector.

The EOMs for the longitudinal sector
$\tilde{C}_{\parallel}=\\{\tilde{C}_{t},\tilde{C}_{x}\\}$ are

$\displaystyle
0=\partial_{r}(r^{3}\partial_{r}\tilde{C}_{t})-\frac{r}{f(r)}(q^{2}\tilde{C}_{t}+\omega
q\tilde{C}_{x}),$ $\displaystyle
0=\partial_{r}[rf(r)\partial_{r}\tilde{C}_{x}]+\frac{r}{f(r)}(\omega^{2}\tilde{C}_{x}+\omega
q\tilde{C}_{t}).$ (88)

Compared to the transverse case, the coupling between $\tilde{C}_{t}$ and
$\tilde{C}_{x}$ makes the longitudinal sector much more involved. The system
of coupled equations (88) has four linearly independent solutions, which could
be differentiated by their near horizon behaviour.

$\bullet$ The first solution is the ingoing solution $\\{\tilde{C}_{t}^{\rm
ig},\tilde{C}_{x}^{\rm ig}\\}$,

$\displaystyle\tilde{C}_{t}^{\rm ig}(r,k^{\mu})\xrightarrow[]{r\to
r_{h}}(r-r_{h})^{1-i\omega/(4r_{h})}\left[\frac{4iq\tilde{C}_{x}^{{\rm
ig}\,h}}{4r_{h}^{2}-i\omega r_{h}}+\cdots\right],$
$\displaystyle\tilde{C}_{x}^{\rm ig}(r,k^{\mu})\xrightarrow[]{r\to
r_{h}}(r-r_{h})^{-i\omega/(4r_{h})}\left[\tilde{C}_{x}^{{\rm
ig}\,h}+\frac{i\omega\tilde{C}_{x}^{{\rm ig}\,h}(8r_{h}^{2}+2i\omega
r_{h}+\omega^{2}-4q^{2})}{8r_{h}^{2}(8r_{h}^{2}-6i\omega
r_{h}-\omega^{2})}(r-r_{h})+\cdots\right],$ (89)

where $\cdots$ are higher powers of $(r-r_{h})$. Both functions
$\\{\tilde{C}_{t}^{\rm ig},\tilde{C}_{x}^{\rm ig}\\}$ are uniquely determined
in terms of single horizon data $\tilde{C}_{x}^{{\rm ig}\,h}$. It is important
to observe that $\tilde{C}_{t}^{\rm ig}$ is an odd function of $q$ while
$\tilde{C}_{x}^{\rm ig}$ is even.

$\bullet$ The second solution of (88) is the outgoing solution
$\\{\tilde{C}_{t}^{\rm og},\tilde{C}_{x}^{\rm og}\\}$, which is obtained from
the ingoing solution by the time-reversal transformation (4.1)

$\displaystyle\tilde{C}_{t}^{\rm og}(r,k^{\mu})=\tilde{C}_{t}^{\rm
ig}(r,\bar{k}^{\mu}),\qquad\tilde{C}_{x}^{\rm
og}(r,k^{\mu})=-\tilde{C}_{x}^{\rm ig}(r,\bar{k}^{\mu}).$ (90)

$\bullet$ The third solution is the pure gauge (pg) solution Policastro:2002tn
:

$\displaystyle\tilde{C}_{t}^{\rm
pg}(r,k^{\mu})=-i\omega\tilde{\Lambda}(k^{\mu}),\qquad\tilde{C}_{x}^{\rm
pg}(r,k^{\mu})=iq\tilde{\Lambda}(k^{\mu}),$ (91)

where $\tilde{\Lambda}(k^{\mu})$ is an $r$-independent gauge parameter of the
residual gauge symmetry. As we have argued above, $\tilde{C}_{t}=0$ at the
horizon can be imposed as a residual gauge fixing. This choice of the gauge is
equivalent to setting $\tilde{\Lambda}=0$.

$\bullet$ The fourth solution is the polynomial (pn) solution
$\\{\tilde{C}_{t}^{\rm pn},\tilde{C}_{x}^{\rm pn}\\}$. Near horizon it has a
Taylor expansion in powers of $(r-r_{h})$:

$\displaystyle\tilde{C}_{t}^{\rm pn}(r,k^{\mu})\xrightarrow[]{r\to
r_{h}}(r-r_{h})\left[\tilde{C}_{t}^{{\rm pn}\,h}+\frac{\tilde{C}_{t}^{{\rm
pn}\,h}(4q^{2}-48r_{h}^{2}-3\omega^{2})}{2r_{h}(16r_{h}^{2}+\omega^{2})}(r-r_{h})+\cdots\right],$
(92) $\displaystyle\tilde{C}_{x}^{\rm pn}(r,k^{\mu})\xrightarrow[]{r\to
r_{h}}(r-r_{h})\left[-\frac{\tilde{C}_{t}^{{\rm pn}\,h}\omega
q}{16r_{h}^{2}+\omega^{2}}-\frac{\tilde{C}_{t}^{{\rm pn}\,h}\omega
q(4q^{2}-32r_{h}^{2}-3\omega^{2})}{2r_{h}(16r_{h}^{2}+\omega^{2})(64r_{h}^{2}+\omega^{2})}(r-r_{h})+\cdots\right],$

where $\cdots$ refer to terms that are uniquely fixed in terms of the horizon
data $\tilde{C}_{t}^{{\rm pn}\,h}$. Thus the boundary values of
$\tilde{C}_{t}^{\rm pn}$ and $\tilde{C}_{x}^{\rm pn}$ are not independent.
$\tilde{C}_{t}^{\rm pn}$ is a function of $\omega^{2}$ and $q^{2}$, that is,
it is both $\mathcal{T}$ and $\mathcal{P}$ even. Similarly, because
$\tilde{C}_{x}^{\rm pn}$ has an overall extra factor $\omega q$, it is
$\mathcal{T}$ and $\mathcal{P}$ invariant too.

Thus we have found all four linearly independent solutions. So far, they have
been identified by their near horizon behaviours. In subsection 5.2, we will
also construct them numerically, for finite momenta. Under the rule (72), the
linearly independent solutions in the ingoing EF coordinate are

$\displaystyle C_{\parallel}^{\rm ig}(r,k^{\mu})=\tilde{C}_{\parallel}^{\rm
ig}(r,k^{\mu})e^{i\omega\zeta_{s}(r)},\qquad C_{\parallel}^{\rm
og}(r,k^{\mu})=\tilde{C}_{\parallel}^{\rm
og}(r,k^{\mu})e^{i\omega\zeta_{s}(r)},$ $\displaystyle C_{\parallel}^{\rm
pg}(r,k^{\mu})=\tilde{C}_{\parallel}^{\rm
pg}(r,k^{\mu})e^{i\omega\zeta_{s}(r)},\qquad C_{\parallel}^{\rm
pn}(r,k^{\mu})=\tilde{C}_{\parallel}^{\rm
pn}(r,k^{\mu})e^{i\omega\zeta_{s}(r)},$ (93)

where $s=2$ when $r\in[r_{h}-\epsilon,\infty_{2})$ and $s=1$ when
$r\in[r_{h}-\epsilon,\infty_{1})$. Here, the subscript ∥ collectively denotes
the time component and $x$ component of the bulk gauge field. Similarly as in
the transverse case (86), the ingoing solution $\\{C_{v}^{\rm ig},C_{x}^{\rm
ig}\\}$ and the outgoing solution $\\{C_{v}^{\rm og},C_{x}^{\rm og}\\}$ are
related:

$\displaystyle C_{v}^{\rm og}(r,k^{\mu})=C_{v}^{\rm
ig}(r,\bar{k}^{\mu})e^{2i\omega\zeta_{s}(r)},\qquad C_{x}^{\rm
og}(r,k^{\mu})=-C_{x}^{\rm ig}(r,\bar{k}^{\mu})e^{2i\omega\zeta_{s}(r)},$
$\displaystyle r\in[r_{h}-\epsilon,\infty_{s}),\qquad s=1~{}{\rm or}~{}2.$
(94)

Near the AdS boundary the linearly independent solutions can be expanded

$\displaystyle C_{v}^{\rm S}(r,k^{\mu})\xrightarrow[]{r\to\infty}$
$\displaystyle C_{v}^{\rm S(0)}(k^{\mu})-\frac{i\omega C_{v}^{\rm
S(0)}(k^{\mu})}{r}+\frac{1}{2}\partial^{\mu}{\cal F}_{v\mu}^{\rm
S(0)}(k^{\mu})\frac{\log r}{r^{2}}+\frac{C_{v}^{\rm
S(2)}(k^{\mu})}{r^{2}}+\cdots,$ $\displaystyle C_{x}^{\rm
S}(r,k^{\mu})\xrightarrow[]{r\to\infty}$ $\displaystyle C_{x}^{\rm
S(0)}(k^{\mu})-\frac{i\omega C_{x}^{\rm
S(0)}(k^{\mu})}{r}+\frac{1}{2}\partial^{\mu}{\cal F}_{x\mu}^{\rm
S(0)}(k^{\mu})\frac{\log r}{r^{2}}+\frac{C_{x}^{\rm
S(2)}(k^{\mu})}{r^{2}}+\cdots,$ (95)

where $\rm S$ stands for any of the three solutions, $\rm S=(ig,\,og,\,pn)$
and ${\cal F}^{\rm S(0)}$ is the corresponding field strength. The expansion
(4.3) is not valid for the pure gauge solution. In principle, one could tune
the horizon data for each solution independently, so that $C_{v}^{\rm
S(0)}(k^{\mu})=1$. Then, there is no freedom left to also set $C_{x}^{\rm
S(0)}(k^{\mu})$ to one: the value of $C_{x}^{\rm S(0)}(k^{\mu})$ would have to
be determined from the dynamical equations.

It is important to notice that the solutions $\\{\tilde{C}_{t}^{\rm
ig},\tilde{C}_{x}^{\rm ig}\\}$, $\\{\tilde{C}_{t}^{\rm og},\tilde{C}_{x}^{\rm
og}\\}$, and $\\{\tilde{C}_{t}^{\rm pg},\tilde{C}_{x}^{\rm pg}\\}$ satisfy the
constraint equation (63) automatically, which makes it possible to relate the
near boundary expansions of these functions. Particularly,

$\displaystyle\omega\tilde{C}_{t}^{\rm S(2)}(k^{\mu})+q\tilde{C}_{x}^{\rm
S(2)}(k^{\mu})=0,$ $\displaystyle C_{\mu}^{\rm
S(0)}(k^{\mu})=\tilde{C}_{\mu}^{\rm S(0)}(k^{\mu}),\qquad C_{\mu}^{\rm
S(2)}(k^{\mu})=\tilde{C}_{\mu}^{\rm
S(2)}(k^{\mu})-\frac{1}{2}\omega^{2}\tilde{C}_{\mu}^{\rm S(0)}(k^{\mu}),$ (96)

where the last two relations follows from (72). The polynomial solution, on
the other hand, does not satisfy the constraint (63). Consequently, having the
theory put on-shell is equivalent to setting the coefficient of the polynomial
solution to zero.

###  4.4 Solutions over the entire radial contour: gluing at the horizon

In the previous subsection, we have found the independent solutions to
dynamical EOMs (43) for a single copy of the doubled Schwarzschild-AdS5. Our
next task is to construct a full solution over the entire contour in Figure 1.
To this end, the independent solutions on the upper and lower segments will be
glued at the horizon, employing the matching conditions (79), (80) and (81)
derived in subsection 4.2.

$\bullet$ Transverse sector.

The most general solution for $C_{\perp}$ expressed in a piecewise form is:

$\displaystyle C_{\perp}^{\rm up}(r,k^{\mu})=c_{\perp}^{\rm up}C_{\perp}^{\rm
ig}(r,k^{\mu})-h_{\perp}^{\rm up}C_{\perp}^{\rm
ig}(r,\bar{k}^{\mu})e^{2i\omega\zeta_{2}(r)},\qquad
r\in[r_{h}-\epsilon,\infty_{2}),$ $\displaystyle C_{\perp}^{\rm
dw}(r,k^{\mu})=c_{\perp}^{\rm dw}C_{\perp}^{\rm ig}(r,k^{\mu})-h_{\perp}^{\rm
dw}C_{\perp}^{\rm ig}(r,\bar{k}^{\mu})e^{2i\omega\zeta_{1}(r)},\qquad
r\in[r_{h}-\epsilon,\infty_{1}).$ (97)

Here $c_{\perp}^{\rm up,dw}=c_{\perp}^{\rm up,dw}(k_{\mu})$ and
$h_{\perp}^{\rm up,dw}=h_{\perp}^{\rm up,dw}(k_{\mu})$ are linear
superposition coefficients. Near the horizon, $C_{\perp}^{\rm ig}$ is regular
while $C_{\perp}^{\rm og}$ oscillates as $e^{2i\omega\zeta_{1}(r)}$ in the
upper segment and as $e^{2i\omega\zeta_{2}(r)}$ in the lower one. The matching
condition (79) and the continuity condition (80) imply

$\displaystyle h_{\perp}^{\rm up}=h_{\perp}^{\rm
dw}e^{\beta\omega},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}c_{\perp}^{\rm
up}=c_{\perp}^{\rm dw}.$ (98)

Eventually, the solution for the transverse mode is

$\displaystyle C_{\perp}^{\rm up}(r,k^{\mu})=c_{\perp}C_{\perp}^{\rm
ig}(r,k^{\mu})-h_{\perp}C_{\perp}^{\rm
ig}(r,\bar{k}^{\mu})e^{2i\omega\zeta_{2}(r)},\qquad\qquad
r\in[r_{h}-\epsilon,\infty_{2}),$ $\displaystyle C_{\perp}^{\rm
dw}(r,k^{\mu})=c_{\perp}C_{\perp}^{\rm
ig}(r,k^{\mu})-h_{\perp}e^{-\beta\omega}C_{\perp}^{\rm
ig}(r,\bar{k}^{\mu})e^{2i\omega\zeta_{1}(r)},\qquad
r\in[r_{h}-\epsilon,\infty_{1}),$ (99)

where $c_{\perp}^{\rm up}\to c_{\perp}$ and $h_{\perp}^{\rm up}\to h_{\perp}$
relabelling was made. The piecewise solution (99) could be put into a more
compact form:

$\displaystyle C_{\perp}(r,k^{\mu})=c_{\perp}C_{\perp}^{\rm
ig}(r,k^{\mu})-h_{\perp}C_{\perp}^{\rm
ig}(r,\bar{k}^{\mu})e^{2i\omega\zeta(r)},\qquad r\in(\infty_{2},\infty_{1}),$
(100)

where $\zeta(r)$ is defined similarly to $\zeta_{2}(r)$ but with the $r$
interval extended over the whole contour:

$\displaystyle\zeta(r)\equiv\int_{\infty_{2}}^{r}\frac{dy}{f(y)},\qquad
r\in(\infty_{2},\infty_{1}).$ (101)

The decomposition coefficients $c_{\perp}$ and $h_{\perp}$ are fixed from the
boundary conditions at $r=\infty_{1}$ and $r=\infty_{2}$ (see (87)).

$\displaystyle c_{\perp}C_{\perp}^{\rm ig(0)}(k^{\mu})-h_{\perp}C_{\perp}^{\rm
ig(0)}(\bar{k}^{\mu})=B_{2\perp}(k^{\mu}),$ $\displaystyle
c_{\perp}C_{\perp}^{\rm
ig(0)}(k^{\mu})-h_{\perp}e^{-\beta\omega}C_{\perp}^{\rm
ig(0)}(\bar{k}^{\mu})=B_{1\perp}(k^{\mu}),$ $\displaystyle\Rightarrow$
$\displaystyle
c_{\perp}=\frac{1}{2}\coth\frac{\beta\omega}{2}\frac{B_{a\perp}(k^{\mu})}{C_{\perp}^{\rm
ig(0)}(k^{\mu})}+\frac{B_{r\perp}(k^{\mu})}{C_{\perp}^{\rm
ig(0)}(k^{\mu})}\qquad
h_{\perp}=\frac{B_{a\perp}(k^{\mu})}{(1-e^{-\beta\omega})C_{\perp}^{\rm
ig(0)}(\bar{k}^{\mu})}.$ (102)

As mentioned earlier, $C_{\perp}^{\rm ig(0)}$ can be set to one without loss
of generality.

$\bullet$ Longitudinal sector

With the four independent solutions presented in subsection 4.3, we construct
the most general solution for the longitudinal sector $\\{C_{v},C_{x}\\}$ in a
piecewise form:

$\displaystyle C_{v}^{\rm up}(r,k^{\mu})=$ $\displaystyle c_{\parallel}^{\rm
up}C_{v}^{\rm ig}(r,k^{\mu})+h_{\parallel}^{\rm up}C_{v}^{\rm
ig}(r,\bar{k}^{\mu})e^{2i\omega\zeta_{2}(r)}+p_{\parallel}^{\rm up}C_{v}^{\rm
pg}(r,k^{\mu})$ $\displaystyle+n_{\parallel}^{\rm up}C_{v}^{\rm
pn}(r,k^{\mu}),\qquad r\in[r_{h}-\epsilon,\infty_{2}),$ $\displaystyle
C_{x}^{\rm up}(r,k^{\mu})=$ $\displaystyle c_{\parallel}^{\rm up}C_{x}^{\rm
ig}(r,k^{\mu})-h_{\parallel}^{\rm up}C_{x}^{\rm
ig}(r,\bar{k}^{\mu})e^{2i\omega\zeta_{2}(r)}+p_{\parallel}^{\rm up}C_{x}^{\rm
pg}(r,k^{\mu})$ $\displaystyle+n_{\parallel}^{\rm up}C_{x}^{\rm
pn}(r,k^{\mu}),\qquad r\in[r_{h}-\epsilon,\infty_{2}),$ $\displaystyle
C_{v}^{\rm dw}(r,k^{\mu})=$ $\displaystyle c_{\parallel}^{\rm dw}C_{v}^{\rm
ig}(r,k^{\mu})+h_{\parallel}^{\rm dw}C_{v}^{\rm
ig}(r,\bar{k}^{\mu})e^{2i\omega\zeta_{1}(r)}+p_{\parallel}^{\rm dw}C_{v}^{\rm
pg}(r,k^{\mu})$ $\displaystyle+n_{\parallel}^{\rm dw}C_{v}^{\rm
pn}(r,k^{\mu}),\qquad r\in[r_{h}-\epsilon,\infty_{1}),$ $\displaystyle
C_{x}^{\rm dw}(r,k^{\mu})=$ $\displaystyle c_{\parallel}^{\rm dw}C_{x}^{\rm
ig}(r,k^{\mu})-h_{\parallel}^{\rm dw}C_{x}^{\rm
ig}(r,\bar{k}^{\mu})e^{2i\omega\zeta_{1}(r)}+p_{\parallel}^{\rm dw}C_{x}^{\rm
pg}(r,k^{\mu})$ $\displaystyle+n_{\parallel}^{\rm dw}C_{x}^{\rm
pn}(r,k^{\mu}),\qquad r\in[r_{h}-\epsilon,\infty_{1}),$ (103)

where $\\{C_{v}^{\rm ig},C_{x}^{\rm ig}\\}$, $\\{C_{v}^{\rm pg},C_{x}^{\rm
pg}\\}$ and $\\{C_{v}^{\rm pn},C_{x}^{\rm pn}\\}$ are related to the solutions
in the Schwarzschild coordinates by the rule (93). Next, the piecewise
solutions (103) are glued via the matching conditions (80), (79), and (81). At
the horizon surface, both $C_{v}^{\rm ig}$ and $C_{v}^{\rm pn}$ vanish, while
$C_{v}^{\rm pg}$ is generically nonzero. Thus, the condition (81)
requires777In fact, $p_{\parallel}^{\rm up,dw}$ can be absorbed into
redefinition of $\tilde{\Lambda}$ in (91).

$\displaystyle p_{\parallel}^{\rm up}=p_{\parallel}^{\rm dw}=0.$ (104)

The condition (79) implies

$\displaystyle h_{\parallel}^{\rm up}=h_{\parallel}^{\rm dw}e^{\beta\omega},$
(105)

where we have used the fact that

$\displaystyle f(r)\partial_{r}C_{x}^{\rm ig}(r,k^{\mu})\xrightarrow[]{r\to
r_{h}}0,\qquad\qquad f(r)\partial_{r}C_{x}^{\rm
pn}(r,k^{\mu})\xrightarrow[]{r\to r_{h}}0,$ $\displaystyle
f(r)\partial_{r}C_{x}^{\rm og}(r,k^{\mu})\xrightarrow[]{r\to r_{h}}-2i\omega
C_{x}^{\rm ig}(r,\bar{k}^{\mu})e^{2i\omega\zeta_{1}(r)},\qquad
r\in[r_{h}-\epsilon,\infty_{1}),$ $\displaystyle f(r)\partial_{r}C_{x}^{\rm
og}(r,k^{\mu})\xrightarrow[]{r\to r_{h}}-2i\omega C_{x}^{\rm
ig}(r,\bar{k}^{\mu})e^{2i\omega\zeta_{2}(r)},\qquad
r\in[r_{h}-\epsilon,\infty_{2}).$ (106)

Finally, the matching condition (80) implies

$\displaystyle c_{\parallel}^{\rm up}=c_{\parallel}^{\rm dw}.$ (107)

Eventually, the entire solution for the longitudinal sector is

$\displaystyle C_{v}^{\rm up}(r,k^{\mu})=$ $\displaystyle
c_{\parallel}C_{v}^{\rm ig}(r,k^{\mu})+h_{\parallel}C_{v}^{\rm
ig}(r,\bar{k}^{\mu})e^{2i\omega\zeta_{2}(r)}+n_{\parallel}^{\rm up}C_{v}^{\rm
pn}(r,k^{\mu}),$ $\displaystyle C_{x}^{\rm up}(r,k^{\mu})=$ $\displaystyle
c_{\parallel}C_{x}^{\rm ig}(r,k^{\mu})-h_{\parallel}C_{x}^{\rm
ig}(r,\bar{k}^{\mu})e^{2i\omega\zeta_{2}(r)}+n_{\parallel}^{\rm up}C_{x}^{\rm
pn}(r,k^{\mu}),$ $\displaystyle C_{v}^{\rm dw}(r,k^{\mu})=$ $\displaystyle
c_{\parallel}C_{v}^{\rm ig}(r,k^{\mu})+h_{\parallel}e^{-\beta\omega}C_{v}^{\rm
ig}(r,\bar{k}^{\mu})e^{2i\omega\zeta_{1}(r)}+n_{\parallel}^{\rm dw}C_{v}^{\rm
pn}(r,k^{\mu}),$ $\displaystyle C_{x}^{\rm dw}(r,k^{\mu})=$ $\displaystyle
c_{\parallel}C_{x}^{\rm ig}(r,k^{\mu})-h_{\parallel}e^{-\beta\omega}C_{x}^{\rm
ig}(r,\bar{k}^{\mu})e^{2i\omega\zeta_{1}(r)}+n_{\parallel}^{\rm dw}C_{x}^{\rm
pn}(r,k^{\mu}),$ (108)

where $c_{\parallel}^{\rm up}\to c_{\parallel},h_{\parallel}^{\rm up}\to
h_{\parallel}$ relabelling is made. Due to the presence of the polynomial
solution $\\{C_{v}^{\rm pn},C_{x}^{\rm pn}\\}$, it is not possible to cast the
final result (108) into a more compact form, similar to (100).

Recall that near the AdS boundary all the linearly independent solutions have
asymptotic expansions similar to the general case (59) and (60), cf. (4.3).
The coefficients $c_{\parallel},h_{\parallel},n_{\parallel}^{\rm
dw},n_{\parallel}^{\rm up}$ in (108) are fixed by the AdS boundary conditions

$\displaystyle c_{\parallel}C_{v}^{\rm ig(0)}(k^{\mu})+h_{\parallel}C_{v}^{\rm
ig(0)}(\bar{k}^{\mu})+n_{\parallel}^{\rm up}C_{v}^{\rm
pn(0)}(k^{\mu})=B_{2v}(k^{\mu}),$ $\displaystyle c_{\parallel}C_{x}^{\rm
ig(0)}(k^{\mu})-h_{\parallel}C_{x}^{\rm
ig(0)}(\bar{k}^{\mu})+n_{\parallel}^{\rm up}C_{x}^{\rm
pn(0)}(k^{\mu})=B_{2x}(k^{\mu}),$ $\displaystyle c_{\parallel}C_{v}^{\rm
ig(0)}(k^{\mu})+h_{\parallel}e^{-\beta\omega}C_{v}^{\rm
ig(0)}(\bar{k}^{\mu})+n_{\parallel}^{\rm dw}C_{v}^{\rm
pn(0)}(k^{\mu})=B_{1v}(k^{\mu}),$ $\displaystyle c_{\parallel}C_{x}^{\rm
ig(0)}(k^{\mu})-h_{\parallel}e^{-\beta\omega}C_{x}^{\rm
ig(0)}(\bar{k}^{\mu})+n_{\parallel}^{\rm dw}C_{x}^{\rm
pn(0)}(k^{\mu})=B_{1x}(k^{\mu}),$ (109)

which yield

$\displaystyle
c_{\parallel}=\frac{1}{2}G_{1}^{-1}\left\\{2B_{rx}(k^{\mu})C_{v}^{\rm
pn(0)}(k^{\mu})-2B_{rv}(k^{\mu})C_{x}^{\rm pn(0)}(k^{\mu})\right.$
$\displaystyle\qquad\qquad\left.+\coth\frac{\beta\omega}{2}\left[B_{ax}(\omega,q)C_{v}^{\rm
pn(0)}(k^{\mu})-B_{av}(k^{\mu})C_{x}^{\rm pn(0)}(k^{\mu})\right]\right\\},$
(110) $\displaystyle
h_{\parallel}=(1-e^{-\beta\omega})^{-1}G_{2}^{-1}\left[B_{ax}(k^{\mu})C_{v}^{\rm
pn(0)}(k^{\mu})-B_{av}(k^{\mu})C_{x}^{\rm pn(0)}(k^{\mu})\right],$ (111)
$\displaystyle n_{\parallel}^{\rm dw}-n_{\parallel}^{\rm
up}=G_{2}^{-1}\left[B_{ax}(k^{\mu})C_{v}^{\rm
ig(0)}(\bar{k}^{\mu})+B_{av}(k^{\mu})C_{x}^{\rm ig(0)}(\bar{k}^{\mu})\right],$
(112) $\displaystyle\frac{1}{2}(n_{\parallel}^{\rm dw}+n_{\parallel}^{\rm
up})=-G_{1}^{-1}\left[B_{rx}(k^{\mu})C_{v}^{\rm
ig(0)}(k^{\mu})-B_{rv}(k^{\mu})C_{x}^{\rm
ig(0)}(k^{\mu})\right]-\frac{1}{2}\coth\frac{\beta\omega}{2}$
$\displaystyle\qquad\qquad\qquad\quad\times
G_{1}^{-1}G_{2}^{-1}G_{3}\left[B_{ax}(k^{\mu})C_{v}^{\rm
pn(0)}(k^{\mu})-B_{av}(k^{\mu})C_{x}^{\rm pn(0)}(k^{\mu})\right],$ (113)

where

$\displaystyle G_{1}=C_{v}^{\rm pn(0)}(k^{\mu})C_{x}^{\rm
ig(0)}(k^{\mu})-C_{v}^{\rm ig(0)}(k^{\mu})C_{x}^{\rm pn(0)}(k^{\mu}),$
$\displaystyle G_{2}=C_{v}^{\rm pn(0)}(k^{\mu})C_{x}^{\rm
ig(0)}(\bar{k}^{\mu})+C_{v}^{\rm ig(0)}(\bar{k}^{\mu})C_{x}^{\rm
pn(0)}(k^{\mu}),$ $\displaystyle G_{3}=C_{v}^{\rm
ig(0)}(\bar{k}^{\mu})C_{x}^{\rm ig(0)}(k^{\mu})+C_{v}^{\rm
ig(0)}(k^{\mu})C_{x}^{\rm ig(0)}(\bar{k}^{\mu}).$ (114)

Notice that $G_{2}=G_{1}^{*}$. Without loss of generality two of the
coefficients, say, $C_{v}^{\rm pn(0)}$ and $C_{v}^{\rm ig(0)}$ could be set to
one. The remaining coefficients would have to be found from the solutions of
the EOMs. Finally, it is not difficult to verify that $G_{1}\neq 0$.

###  4.5 From the bulk solutions to the effective action

With the entire solution for $C_{M}(r,v,\vec{x})$ derived in subsection 4.4,
we are now ready to evaluate the effective Lagrangian (61), which can be split
into transverse and longitudinal parts:

$\displaystyle\mathcal{L}_{\rm eff}=\mathcal{L}_{\rm
eff}^{\perp}+\mathcal{L}_{\rm eff}^{\parallel}$ (115)

We will need near-boundary expansion coefficients $C_{M}^{(2)}$ and the
expansion of $e^{2i\omega\zeta_{s}(r)}$:

$\displaystyle
e^{2i\omega\zeta_{s}(r)}\xrightarrow[]{r\to\infty_{s}}1-\frac{2i\omega}{r}-\frac{2\omega^{2}}{r^{2}}+\cdots,\qquad
s=1,~{}{\rm or}~{}~{}2.$ (116)

$\bullet$ $\bf\mathcal{L}_{\rm eff}^{\perp}$

In the $(r,a)$-basis, the coefficients $C_{\perp}^{(2)}$ (normalizable modes)
are

$\displaystyle C_{a\perp}^{(2)}(k^{\mu})=$ $\displaystyle-\frac{C_{\perp}^{\rm
ig(2)}(\bar{k}^{\mu})}{C_{\perp}^{\rm
ig(0)}(\bar{k}^{\mu})}B_{a\perp}(k^{\mu}),$ (117) $\displaystyle
C_{r\perp}^{(2)}(k^{\mu})=$
$\displaystyle\frac{1}{2}\coth{\frac{\beta\omega}{2}}\left[\frac{C_{\perp}^{\rm
ig(2)}(k^{\mu})}{C_{\perp}^{\rm ig(0)}(k^{\mu})}-\frac{C_{\perp}^{\rm
ig(2)}(\bar{k}^{\mu})}{C_{\perp}^{\rm
ig(0)}(\bar{k}^{\mu})}\right]B_{a\perp}(k^{\mu})+\frac{C_{\perp}^{\rm
ig(2)}(k^{\mu})}{C_{\perp}^{\rm ig(0)}(k^{\mu})}B_{r\perp}(k^{\mu}).$ (118)

From (61), the transverse part of the effective Lagrangian is

$\displaystyle\mathcal{L}_{\rm
eff}^{\perp}=B_{r\perp}(x)C_{a\perp}^{(2)}(x)+B_{a\perp}(x)C_{r\perp}^{(2)}(x)-\frac{1}{2}B_{a\perp}(x)\partial_{v}^{2}B_{r\perp}(x)-\frac{1}{2}B_{a\perp}(x)\vec{\partial}^{\;2}B_{r\perp}(x),$
(119)

which in the momentum space reads

$\displaystyle\mathcal{L}_{\rm eff}^{\perp}(k)=$ $\displaystyle
B_{a\perp}(-k)\left[\frac{2C_{\perp}^{\rm ig(2)}(k)}{C_{\perp}^{\rm
ig(0)}(k)}+\frac{1}{2}\omega^{2}+\frac{1}{2}q^{2}\right]B_{r\perp}(k)$
$\displaystyle+B_{a\perp}(-k)\frac{1}{2}\coth\frac{\beta\omega}{2}\left[\frac{C_{\perp}^{\rm
ig(2)}(k)}{C_{\perp}^{\rm ig(0)}(k)}-\frac{C_{\perp}^{\rm
ig(2)}(\bar{k})}{C_{\perp}^{\rm ig(0)}(\bar{k})}\right]B_{a\perp}(k).$ (120)

Comparing (120) with (2.1), the TCFs in (2.1) are expressed in terms of the
results obtained in the bulk:

$\displaystyle-i\omega w_{8}(k)+q^{2}w_{9}(k)=\frac{2C_{\perp}^{\rm
ig(2)}(k)}{C_{\perp}^{\rm ig(0)}(k)}+\frac{1}{2}\omega^{2}+\frac{1}{2}q^{2},$
(121)
$\displaystyle\frac{i}{2}w_{2}(k)=\frac{1}{2}\coth\frac{\beta\omega}{2}\left[\frac{C_{\perp}^{\rm
ig(2)}(k)}{C_{\perp}^{\rm ig(0)}(k)}-\frac{C_{\perp}^{\rm
ig(2)}(\bar{k})}{C_{\perp}^{\rm ig(0)}(\bar{k})}\right].$ (122)

While $w_{2}$ is determined entirely by the transverse sector, the
coefficients $w_{8},w_{9}$ will be uniquely fixed only with addition of the
longitudinal sector.

$\bullet$ $\bf\mathcal{L}_{\rm eff}^{\parallel}$

In order to cast the results into a more compact form, the following ratios
are introduced

$\displaystyle R_{v}^{\rm ig}(k)=\frac{C_{v}^{\rm ig(2)}(k)}{C_{v}^{\rm
ig(0)}(k)},\qquad R_{v}^{\rm pn}(k)=\frac{C_{v}^{\rm pn(2)}(k)}{C_{v}^{\rm
pn(0)}(k)},\qquad R_{xv}^{\rm ig}(k)=\frac{C_{x}^{\rm ig(2)}(k)}{C_{v}^{\rm
ig(0)}(k)},$ $\displaystyle R_{xv}^{\rm pn}(k)=\frac{C_{x}^{\rm
pn(2)}(k)}{C_{v}^{\rm pn(0)}(k)},\qquad\bar{R}_{xv}^{\rm
ig}(k)=\frac{C_{x}^{\rm ig(0)}(k)}{C_{v}^{\rm
ig(0)}(k)},\qquad\bar{R}_{xv}^{\rm pn}(k)=\frac{C_{x}^{\rm
pn(0)}(k)}{C_{v}^{\rm pn(0)}(k)}.$ (123)

These ratios are determined by solving the dynamical EOMs (43) in a single
copy of doubled Schwarzschild-AdS5, see subsection 4.3. It is important to
recall that $C_{v}^{\rm ig(2)}$ is related to $C_{x}^{\rm ig(2)}$ via (4.3).
Furthermore, there is still the freedom to set both $C_{v}^{\rm ig(0)}$ and
$C_{v}^{\rm pn(0)}$ to one.

Near the AdS boundaries $r=\infty_{1}$ and $r=\infty_{2}$, we extract the
normalizable modes in $C_{v}$ and $C_{x}$ (in the $(r,a)$-basis):

$\displaystyle C_{rv}^{(2)}(k)=$ $\displaystyle c_{\parallel}C_{v}^{\rm
ig(2)}(k)+\frac{1}{2}h_{\parallel}(1+e^{-\beta\omega})C_{v}^{\rm
ig(2)}(\bar{k})+\frac{1}{2}(n_{\parallel}^{\rm dw}+n_{\parallel}^{\rm
up})C_{v}^{\rm pn(2)}(k)$ $\displaystyle=$ $\displaystyle\frac{R_{v}^{\rm
ig}(k)-R_{v}^{\rm pn}(k)}{\bar{R}_{xv}^{\rm ig}(k)-\bar{R}_{xv}^{\rm
pn}(k)}B_{rx}(k)+\frac{\bar{R}_{xv}^{\rm ig}(k)R_{v}^{\rm
pn}(k)-\bar{R}_{xv}^{\rm pn}(k)R_{v}^{\rm ig}(k)}{\bar{R}_{xv}^{\rm
ig}(k)-\bar{R}_{xv}^{\rm pn}(k)}B_{rv}(k)$ (124)
$\displaystyle+\frac{1}{2}\coth\frac{\beta\omega}{2}\left\\{\frac{R_{v}^{\rm
ig}(k)-R_{v}^{\rm pn}(k)}{\bar{R}_{xv}^{\rm ig}(k)-\bar{R}_{xv}^{\rm
pn}(k)}+\frac{R_{v}^{\rm ig}(\bar{k})-R_{v}^{\rm pn}(k)}{\bar{R}_{xv}^{\rm
ig}(\bar{k})+\bar{R}_{xv}^{\rm
pn}(k)}\right\\}\left[B_{ax}(k)-\bar{R}_{xv}^{\rm pn}(k)B_{av}(k)\right],$
$\displaystyle C_{av}^{(2)}(k)=$ $\displaystyle-
h_{\parallel}(1-e^{-\beta\omega})C_{v}^{\rm
ig(2)}(\bar{k})+(n_{\parallel}^{\rm dw}-n_{\parallel}^{\rm up})C_{v}^{\rm
pn(2)}(k)$ $\displaystyle=$ $\displaystyle R_{v}^{\rm
pn}(k)\frac{B_{ax}(k)+\bar{R}_{xv}^{\rm
ig}(\bar{k})B_{av}(k)}{\bar{R}_{xv}^{\rm ig}(\bar{k})+\bar{R}_{xv}^{\rm
pn}(k)}+R_{v}^{\rm ig}(\bar{k})\frac{-B_{ax}(k)+\bar{R}_{xv}^{\rm
pn}(k)B_{av}(k)}{\bar{R}_{xv}^{\rm ig}(\bar{k})+\bar{R}_{xv}^{\rm pn}(k)},$
(125) $\displaystyle C_{rx}^{(2)}(k)=$ $\displaystyle c_{\parallel}C_{x}^{\rm
ig(2)}(k)-\frac{1}{2}h_{\parallel}(1+e^{-\beta\omega})C_{x}^{\rm
ig(2)}(\bar{k})+\frac{1}{2}(n_{\parallel}^{\rm dw}+n_{\parallel}^{\rm
up})C_{x}^{\rm pn(2)}(k)$ $\displaystyle=$ $\displaystyle R_{xv}^{\rm
ig}(k)\frac{B_{rx}(k)-\bar{R}_{xv}^{\rm pn}(k)B_{rv}(k)}{\bar{R}_{xv}^{\rm
ig}(k)-\bar{R}_{xv}^{\rm pn}(k)}+R_{xv}^{\rm
pn}(k)\frac{-B_{rx}(k)+\bar{R}_{xv}^{\rm ig}(k)B_{rv}(k)}{\bar{R}_{xv}^{\rm
ig}(k)-\bar{R}_{xv}^{\rm pn}(k)}$ (126)
$\displaystyle+\frac{1}{2}\coth\frac{\beta\omega}{2}\left\\{\frac{R_{xv}^{\rm
ig}(k)-R_{xv}^{\rm pn}(k)}{\bar{R}_{xv}^{\rm ig}(k)-\bar{R}_{xv}^{\rm
pn}(k)}-\frac{R_{xv}^{\rm ig}(\bar{k})+R_{xv}^{\rm pn}(k)}{\bar{R}_{xv}^{\rm
ig}(\bar{k})+\bar{R}_{xv}^{\rm
pn}(k)}\right\\}\left[B_{ax}(k)-\bar{R}_{xv}^{\rm pn}(k)B_{av}(k)\right],$
$\displaystyle C_{ax}^{(2)}(k)=$ $\displaystyle
h_{\parallel}(1-e^{-\beta\omega})C_{x}^{\rm
ig(2)}(\bar{k})+(n_{\parallel}^{\rm dw}-n_{\parallel}^{\rm up})C_{x}^{\rm
pn(2)}(k)$ $\displaystyle=$ $\displaystyle R_{xv}^{\rm
ig}(\bar{k})\frac{B_{ax}(k)-\bar{R}_{xv}^{\rm
pn}(k)B_{av}(k)}{\bar{R}_{xv}^{\rm ig}(\bar{k})+\bar{R}_{xv}^{\rm
pn}(k)}+R_{xv}^{\rm pn}(k)\frac{B_{ax}(k)+\bar{R}_{xv}^{\rm
ig}(\bar{k})B_{av}(k)}{\bar{R}_{xv}^{\rm ig}(\bar{k})+\bar{R}_{xv}^{\rm
pn}(k)}.$ (127)

From (61), the longitudinal part of the effective Lagrangian is

$\displaystyle\mathcal{L}_{\rm eff}^{\parallel}=$ $\displaystyle-
B_{rv}(x)C_{av}^{(2)}(x)-B_{av}(x)C_{rv}^{(2)}(x)+B_{rx}(x)C_{ax}^{(2)}(x)+B_{ax}C_{rx}^{(2)}(x)$
$\displaystyle+\frac{1}{2}\partial_{x}B_{ax}(x)\partial_{v}B_{rv}(x)-\frac{1}{2}B_{av}(x)\partial_{v}\partial_{x}B_{rx}(x)-\frac{1}{2}B_{ax}(x)\partial_{v}^{2}B_{rx}(x)$
$\displaystyle+\frac{1}{2}B_{av}(x)\vec{\partial}^{\;2}B_{rv}(x)+B_{av}(x)\partial_{v}^{2}B_{rv}(x),$
(128)

which in the momentum space becomes

$\displaystyle\mathcal{L}_{\rm eff}^{\parallel}(k)=$ $\displaystyle
B_{av}(-k)\frac{i}{2}w_{1}(k)B_{av}(k)+B_{ax}(-k)\left(\frac{i}{2}w_{2}(k)+\frac{i}{2}q^{2}w_{3}(k)\right)B_{ax}(k)$
$\displaystyle+B_{av}(-k)[-qw_{4}(k)]B_{ax}(k)+B_{av}(-k)w_{5}(k)B_{rv}(k)$
$\displaystyle+B_{av}(-k)\omega
qw_{6}(k)B_{rx}(k)+B_{ax}(-k)[-iqw_{7}(k)]B_{rv}(k)$
$\displaystyle+B_{ax}(-k)[-i\omega w_{8}(k)]B_{rx}(k),$ (129)

with the TCFs given by the following expressions

$\displaystyle\frac{i}{2}w_{1}(k)=\frac{1}{2}\coth\frac{\beta\omega}{2}\left[\frac{R_{v}^{\rm
ig}(k)-R_{v}^{\rm pn}(k)}{\bar{R}_{xv}^{\rm ig}(k)-\bar{R}_{xv}^{\rm
pn}(k)}+\frac{R_{v}^{\rm ig}(\bar{k})-R_{v}^{\rm pn}(k)}{\bar{R}_{xv}^{\rm
ig}(\bar{k})+\bar{R}_{xv}^{\rm pn}(k)}\right]\bar{R}_{xv}^{\rm pn}(k),$ (130)
$\displaystyle\frac{i}{2}w_{2}(k)+\frac{i}{2}q^{2}w_{3}(k)=\frac{1}{2}\coth\frac{\beta\omega}{2}\left[\frac{R_{xv}^{\rm
ig}(k)-R_{xv}^{\rm pn}(k)}{\bar{R}_{xv}(k)-\bar{R}_{xv}^{\rm
pn}(k)}-\frac{R_{xv}^{\rm ig}(\bar{k})+R_{xv}^{\rm pn}(k)}{\bar{R}_{xv}^{\rm
ig}(\bar{k})+\bar{R}_{xv}^{\rm pn}(k)}\right],$ (131) $\displaystyle-
qw_{4}(k)=$
$\displaystyle-\frac{1}{2}\coth\frac{\beta\omega}{2}\left[\frac{R_{v}^{\rm
ig}(k)-R_{v}^{\rm pn}(k)}{\bar{R}_{xv}^{\rm ig}(k)-\bar{R}_{xv}^{\rm
pn}(k)}+\frac{R_{v}^{\rm ig}(\bar{k})-R_{v}^{\rm pn}(k)}{\bar{R}_{xv}^{\rm
ig}(\bar{k})+\bar{R}_{xv}^{\rm pn}(k)}\right]$
$\displaystyle+\frac{1}{2}\coth\frac{\beta\omega}{2}\left[\frac{R_{xv}^{\rm
ig}(\bar{k})+R_{xv}^{\rm pn}(k)}{\bar{R}_{xv}^{\rm
ig}(\bar{k})+\bar{R}_{xv}^{\rm pn}(k)}-\frac{R_{xv}^{\rm ig}(k)-R_{xv}^{\rm
pn}(k)}{\bar{R}_{xv}^{\rm ig}(\bar{k})-\bar{R}_{xv}^{\rm
pn}(k)}\right]\bar{R}_{xv}^{\rm pn}(k),$ (132) $\displaystyle
w_{5}(k)=-\omega^{2}-\frac{1}{2}q^{2}+2\frac{R_{v}^{\rm
ig}(k)\bar{R}_{xv}^{\rm pn}(k)-R_{v}^{\rm pn}(k)\bar{R}_{xv}^{\rm
ig}(k)}{\bar{R}_{xv}^{\rm ig}(k)-\bar{R}_{xv}^{\rm pn}(k)},$ (133)
$\displaystyle\omega qw_{6}(k)=-\frac{1}{2}\omega q-\frac{R_{v}^{\rm
ig}(k)-R_{v}^{\rm pn}(k)}{\bar{R}_{xv}^{\rm ig}(k)-\bar{R}_{xv}^{\rm
pn}(k)}+\frac{\bar{R}_{xv}^{\rm ig}(k)R_{xv}^{\rm pn}(k)-\bar{R}_{xv}^{\rm
pn}(k)R_{xv}^{\rm ig}(k)}{\bar{R}_{xv}^{\rm ig}(k)-\bar{R}_{xv}^{\rm pn}(k)},$
(134) $\displaystyle-iqw_{7}(k)=\omega qw_{6}(k),$ (135)
$\displaystyle-i\omega w_{8}(k)=\frac{1}{2}\omega^{2}+2\frac{R_{xv}^{\rm
ig}(k)-R_{xv}^{\rm pn}(k)}{\bar{R}_{xv}^{\rm ig}(k)-\bar{R}_{xv}^{\rm
pn}(k)}.$ (136)

We observe that all the TCFs are expressed in terms of the ratios (4.5), which
are extracted from the linearly dependent solutions to the dynamical EOMs in a
single copy of the doubled Schwarzschild-AdS5. In the next section these
results are presented explicitly. Finally, it is very important to realise and
straightforward to check that all the symmetry relations imposed by the
discrete symmetries (see subsection 2.2) are satisfied by (121), (122),
(130)-(136) automatically. Furthermore, to demonstrate that one does not
actually need to solve the bulk EOMs at all.

##  5 Results for the TCFs

In this section, all the results for the parameters in the effective
Lagrangian (2.1) are presented. For completeness and consistency check, we
first consider limits in which analytical calculations could be performed,
hereby recovering some of the results available in the literature. Next we
switch to numerical analysis.

###  5.1 Analytical results

$\bullet$ $\bf q=0$

When $q=0$, $SO(3)$ rotational symmetry is recovered,
$\tilde{C}_{\perp}=\tilde{C}_{x}=\tilde{C}_{i}$ and the dynamical EOM (88) for
this component decouples from that of $\tilde{C}_{t}$. The analytical results
for the basic set of solutions are

$\displaystyle\tilde{C}_{t}^{\rm
ig}(r,\omega,q=0)=0~{}~{}\Rightarrow~{}~{}\tilde{C}_{t}^{\rm
ig(0)}(\omega,q)=\tilde{C}_{t}^{\rm ig(2)}(\omega,q)=0,$
$\displaystyle\tilde{C}_{i}^{\rm
pn}(r,\omega,q=0)=0~{}~{}\Rightarrow~{}~{}\tilde{C}_{i}^{\rm
pn(0)}(\omega,q)=\tilde{C}_{i}^{\rm pn(2)}(\omega,q)=0.$ (137)
$\displaystyle\tilde{C}_{t}^{\rm
pn}(r,\omega,q=0)=1-\frac{r_{h}^{2}}{r^{2}}~{}~{}\Rightarrow~{}~{}\tilde{C}_{t}^{\rm
pn(0)}(\omega,q=0)=1,\qquad\tilde{C}_{t}^{\rm pn(2)}(\omega,q=0)=-r_{h}^{2}.$
(138)

The only non-trivial result is related to the spatial component
$\tilde{C}_{i}^{\rm ig}$, which is however well known Horowitz:2008bn ;
Myers:2007we :

$\displaystyle\tilde{C}_{i}^{\rm ig}(r,\omega,q=0)=$
$\displaystyle\left(1-\frac{r_{h}^{2}}{r^{2}}\right)^{-i\omega/(4r_{h})}\left(1+\frac{r_{h}^{2}}{r^{2}}\right)^{-\omega/(4r_{h})}\left(\frac{r_{h}^{2}}{r^{2}}\right)^{(1+i)\omega/(4r_{h})}$
$\displaystyle\times{{}_{2}}F_{1}\left[1-\frac{(1+i)\omega}{4r_{h}},-\frac{(1+i)\omega}{4r_{h}},1-\frac{i\omega}{2r_{h}},\frac{1}{2}\left(1-\frac{r^{2}}{r_{h}^{2}}\right)\right],$
(139)

where ${{}_{2}}F_{1}$ is a hypergeometric function. Near the AdS boundary,

$\displaystyle\frac{\tilde{C}_{i}^{\rm ig(2)}(\omega,q=0)}{\tilde{C}_{i}^{\rm
ig(0)}(\omega,q=0)}=$ $\displaystyle-
r_{h}^{2}\tilde{\omega}\left\\{i+\left[2\gamma_{e}-1+\log(2r_{h}^{2}/L^{2})\right]\tilde{\omega}+\tilde{\omega}\psi\left(-\frac{(1+i)}{2}\tilde{\omega}\right)\right.$
$\displaystyle\qquad\qquad\left.+\tilde{\omega}\psi\left(\frac{(1-i)}{2}\tilde{\omega}\right)\right\\},$
(140)

where $\tilde{\omega}=\omega/(2r_{h})=\omega\beta/(2\pi)$ is introduced for
compactness, $\psi(z)=d\Gamma(z)/dz$, and $\gamma_{e}$ is the Euler constant.
Here we have reinstalled the AdS curvature radius $L$ in the logarithmic term.

Thus, in the limit $q=0$ we can fix $w_{1}$, $w_{2}$, $w_{5}$ and $w_{8}$
while the remaining TCFs decouple:

$\displaystyle w_{1}(q=0)=0,$ $\displaystyle
w_{2}(q=0)=2\coth(\pi\tilde{\omega}){\rm Im}\left[\frac{\tilde{C}_{\perp}^{\rm
ig(2)}(\omega,q=0)}{\tilde{C}_{\perp}^{\rm ig(0)}(\omega,q=0)}\right],$
$\displaystyle w_{5}(q=0)=-2\frac{\tilde{C}_{t}^{\rm
pn(2)}(\omega,q=0)}{\tilde{C}_{t}^{\rm pn(0)}(\omega,q=0)}=2r_{h}^{2},$
$\displaystyle
w_{8}(q=0)=-\frac{1}{2}i\omega-\frac{2}{i\omega}\frac{\tilde{C}_{x}^{\rm
ig(2)}(\omega,q=0)}{\tilde{C}_{x}^{\rm ig(0)}(\omega,q=0)}.$ (141)

$\bullet$ $\bf\omega=q$

In this limit analytical results are available for the transverse mode
$\tilde{C}_{\perp}$ only CaronHuot:2006te . The ingoing solution of the
dynamical EOM (82) is

$\displaystyle\tilde{C}_{\perp}^{\rm ig}(r,\omega=q)=$
$\displaystyle\left(1-\frac{r_{h}^{2}}{r^{2}}\right)^{-i\omega/(4r_{h})}\left(1+\frac{r_{h}^{2}}{r^{2}}\right)^{\omega/(4r_{h})}$
$\displaystyle\times{{}_{2}}F_{1}\left[1-\frac{(1+i)\omega}{4r_{h}},-\frac{(1+i)\omega}{4r_{h}},1-\frac{i\omega}{2r_{h}},\frac{1}{2}\left(1-\frac{r_{h}^{2}}{r^{2}}\right)\right].$
(142)

Near the AdS boundary $r=\infty$,

$\displaystyle\frac{\tilde{C}_{\perp}^{\rm
ig(2)}(\omega=q)}{\tilde{C}_{\perp}^{\rm
ig(0)}(\omega=q)}=r_{h}^{2}\left[1-\tilde{\omega}+\left(\frac{1+i}{2}\tilde{\omega}-1\right)\frac{{{}_{2}}F_{1}(2-(1+i)\tilde{\omega}/2,-(1+i)\tilde{\omega}/2,1-i\tilde{\omega},1/2)}{{{}_{2}}F_{1}(1-(1+i)\tilde{\omega}/2,-(1+i)\tilde{\omega}/2,1-i\tilde{\omega},1/2)}\right].$
(143)

In the absence of analytical solution in the longitudinal sector, only $w_{2}$
and the combination $w_{8}+i\omega w_{9}$ can be determined:

$\displaystyle w_{2}(\omega=q)=2\coth(\pi\tilde{\omega}){\rm
Im}\left[\frac{\tilde{C}_{\perp}^{\rm ig(2)}(\omega=q)}{\tilde{C}_{\perp}^{\rm
ig(0)}(\omega=q)}\right],$ $\displaystyle w_{8}(\omega=q)+i\omega
w_{9}(\omega=q)=-\frac{2}{i\omega}\frac{\tilde{C}_{\perp}^{\rm
ig(2)}(\omega=q)}{\tilde{C}_{\perp}^{\rm ig(0)}(\omega=q)}.$ (144)

$\bullet$ The hydrodynamic limit $\bf\omega\ll T\sim r_{h},q\ll T\sim r_{h}$.

In the hydrodynamic limit, the results available in the literature (see
deBoer:2018qqm  and Glorioso:2018mmw ) pertain to the effective Lagrangian
(2.1) up to second order in the derivatives of $B_{r\mu}$ and $B_{a\mu}$.
Since our formalism is somewhat different from the others, it makes sense to
perform a comparison. Hence we have to obtain analytical results accurate up
to second order. Naively, one would expect to achieve this accuracy by solving
the bulk EOMs also up to second order in the derivatives of $B_{r\mu}$ and
$B_{a\mu}$. However, as can be seen from (122), (131), and (132), in fact one
has to solve for the ingoing solutions keeping the third order terms in the
derivative expansion.

It is convenient to introduce a new radial coordinate $u$:

$\displaystyle
u=r_{h}^{2}/r^{2}\Longrightarrow\tilde{C}_{\mu}(r,\omega,q)\to\tilde{C}_{\mu}(u,\omega,q).$
(145)

The ingoing solution for the transverse mode $\tilde{C}_{\perp}$ is well known
in the literature, see e.g. Policastro:2002se . Up to third order in momenta
the solution is

$\displaystyle\tilde{C}_{\perp}^{\rm
ig}(u,\omega,q)=(1-u^{2})^{-i\tilde{\omega}/2}\left\\{1+i\tilde{\omega}\log(1+u)+\frac{1}{24}\pi^{2}(3\tilde{\omega}^{2}-2\tilde{q}^{2})-\frac{1}{4}\tilde{\omega}^{2}\log^{2}2\right.$
$\displaystyle\left.+\frac{1}{2}\tilde{\omega}^{2}\log(1-u)\log\frac{2}{1+u}-\frac{1}{4}\log(1+u)\left[2(\tilde{\omega}^{2}-\tilde{q}^{2})\log
u+\tilde{\omega}^{2}\log(1+u)\right]\right.$
$\displaystyle\left.+\frac{1}{2}(\tilde{q}^{2}-\tilde{\omega}^{2})\left[{\rm
Li}_{2}(1-u)+{\rm Li}_{2}(-u)\right]-\frac{1}{2}\tilde{\omega}^{2}{\rm
Li}_{2}\left(\frac{1+u}{2}\right)+\tilde{C}_{\perp}^{\rm
ig[3]}(u,\omega,q)+\cdots\right\\},$ (146)

where $\rm Li_{2}$ is the Polylogarithm function, and $\tilde{C}_{\perp}^{\rm
ig[3]}(u,\omega,q)$ is the third order solution, which is too lengthy to be
shown here. Near the AdS boundary $u=0$,

$\displaystyle\tilde{C}_{\perp}^{\rm ig(0)}=1,$
$\displaystyle\tilde{C}_{\perp}^{\rm
ig(2)}=r_{h}^{2}\left[i\tilde{\omega}-\tilde{q}^{2}+\tilde{\omega}^{2}-\tilde{\omega}^{2}\log
2+(\tilde{q}^{2}-\tilde{\omega}^{2})\log(r_{h}^{2}/L^{2})+\frac{\pi^{2}}{12}i\tilde{\omega}(2\tilde{\omega}^{2}-3\tilde{q}^{2})\right].$
(147)

Most of the results about the longitudinal sector available in the literature
are based on the on-shell holography, which for this reason cannot be recycled
for our study. In the off-shell formalism similar to ours, recently the
authors of deBoer:2018qqm  worked out the hydrodynamic limit of the
independent solutions $\\{\tilde{C}_{t},\tilde{C}_{x}\\}$ (see appendix B
there), though up to second order only. We have computed the expansion up to
third order888Our results are somewhat different from deBoer:2018qqm . The
origin of the difference is in the freedom to arbitrarily select the horizon
data.:

$\displaystyle\tilde{C}_{t}^{\rm ig}(u,\omega,q)$
$\displaystyle=(1-u^{2})^{1-i\tilde{\omega}/2}\left[\frac{i\tilde{q}}{1+u}+\frac{\tilde{\omega}\tilde{q}}{1-u^{2}}\left(\log\frac{2}{1+u}+u\log
u\right)+\tilde{C}_{t}^{\rm ig[3]}(u,\omega,q)+\cdots\right],$
$\displaystyle\tilde{C}_{x}^{\rm ig}(u,\omega,q)$
$\displaystyle=(1-u^{2})^{-i\tilde{\omega}/2}\left\\{1+i\tilde{\omega}\log\frac{1+u}{2}+\frac{\pi^{2}\tilde{\omega}^{2}}{24}-\frac{1}{2}\tilde{\omega}^{2}\log\frac{1-u}{2}\log\frac{1+u}{2}\right.$
$\displaystyle\left.-\frac{1}{4}\tilde{\omega}^{2}\log^{2}\frac{1+u}{2}-\frac{1}{2}\tilde{\omega}^{2}\log
u\log(1+u)-\frac{1}{2}\tilde{\omega}^{2}{\rm
Li}_{2}(1-u)-\frac{1}{2}\tilde{\omega}^{2}{\rm Li}_{2}(-u)\right.$
$\displaystyle\left.-\frac{1}{2}\tilde{\omega}^{2}{\rm
Li}_{2}\left(\frac{1+u}{2}\right)+\tilde{C}_{x}^{\rm
ig[3]}(u,\omega,q)+\cdots\right\\},$ (148)

where the third order solutions $\tilde{C}_{t}^{\rm ig[3]}(u,\omega,q)$ and
$\tilde{C}_{x}^{\rm ig[3]}(u,\omega,q)$ have been worked out by us, but are
too lengthy to be presented here. The near AdS boundary data are

$\displaystyle\tilde{C}_{t}^{\rm ig(0)}(\omega,q)=$ $\displaystyle
i\tilde{q}+\tilde{\omega}\tilde{q}\log
2+\frac{1}{2}i\tilde{q}(-\tilde{\omega}^{2}\log^{2}2+2\tilde{q}^{2}\log
2)+\cdots,$ $\displaystyle\tilde{C}_{t}^{\rm ig(2)}(\omega,q)=$ $\displaystyle
r_{h}^{2}\left[-i\tilde{q}-\tilde{\omega}\tilde{q}+\tilde{\omega}\tilde{q}\log(r_{h}^{2}/L^{2})\right]-\frac{r_{h}^{2}}{12}i\tilde{q}\left[12\tilde{q}^{2}+\pi^{2}\tilde{\omega}^{2}-12\tilde{\omega}^{2}\log
2\right.$
$\displaystyle\left.+6\tilde{\omega}^{2}\log^{2}2+12\tilde{\omega}^{2}\log
2\log(r_{h}^{2}/L^{2})-12\tilde{q}^{2}\log(2r_{h}^{2}/L^{2})\right]+\cdots,$
$\displaystyle\tilde{C}_{x}^{\rm ig(0)}(\omega,q)=$ $\displaystyle
1-i\tilde{\omega}\log
2-\frac{1}{12}\tilde{\omega}^{2}(\pi^{2}+6\log^{2}2)+\frac{1}{12}i\omega\pi^{2}(\tilde{\omega}^{2}\log
2-\tilde{q}^{2})$
$\displaystyle+\frac{1}{6}i\tilde{\omega}^{3}\left(\log^{3}2-3\zeta(3)\right)+\cdots,$
$\displaystyle\tilde{C}_{x}^{\rm ig(2)}(\omega,q)=$ $\displaystyle
r_{h}^{2}\left[i\tilde{\omega}+\tilde{\omega}^{2}-\tilde{\omega}^{2}\log(r_{h}^{2}/L^{2})\right]+\frac{r_{h}^{2}}{12}i\tilde{\omega}\left[12\tilde{q}^{2}+\pi^{2}\tilde{\omega}^{2}-12\tilde{\omega}^{2}\log
2\right.$
$\displaystyle\left.+6\tilde{\omega}^{2}\log^{2}2+12\tilde{\omega}^{2}\log
2\log(r_{h}^{2}/L^{2})-12q^{2}\log(2r_{h}^{2}/L^{2})\right]+\cdots.$ (149)

Here we have kept terms up to third order in momenta.

Next, we present the hydrodynamic limit of the polynomial solution
$\\{\tilde{C}_{t}^{\rm pn},\tilde{C}_{x}^{\rm pn}\\}$:

$\displaystyle\tilde{C}_{t}^{\rm
pn}(u,\omega,q)=2(1-u)+2\tilde{q}^{2}\left[u\log
u+(1+u)\log\frac{2}{1+u}\right]+\cdots,$ $\displaystyle\tilde{C}_{x}^{\rm
pn}(u,\omega,q)=-\tilde{\omega}\tilde{q}\left\\{\frac{1}{4}(\pi^{2}-2\log^{2}2)+\log
u\log\frac{1+u}{1-u}+\frac{1}{2}\log(1+u)\log\frac{4}{1+u}\right.$
$\displaystyle\qquad\qquad\qquad\qquad\quad\left.-{\rm
Li}_{2}\left(\frac{1-u}{2}\right)+{\rm Li}_{2}(-u)-{\rm
Li}_{2}(u)\right\\}+\cdots.$ (150)

Near the AdS boundary,

$\displaystyle\tilde{C}_{t}^{\rm pn(0)}(\omega,q)=2+2\tilde{q}^{2}\log
2+\cdots,$ $\displaystyle\tilde{C}_{t}^{\rm
pn(2)}(\omega,q)=r_{h}^{2}\left[-2+2\tilde{q}^{2}(\log
2-1)+2\tilde{q}^{2}\log(r_{h}^{2}/L^{2})\right]+\cdots,$
$\displaystyle\tilde{C}_{x}^{\rm
pn(0)}(\omega,q)=-\frac{1}{6}\pi^{2}\tilde{\omega}\tilde{q}+\cdots,$
$\displaystyle\tilde{C}_{x}^{\rm
pn(2)}(\omega,q)=r_{h}^{2}\left[-2\tilde{\omega}\tilde{q}(\log
2-1)-2\tilde{\omega}\tilde{q}\log(r_{h}^{2}/L^{2})\right]+\cdots.$ (151)

Finally, we are ready to compute the TCFs in (2.1), based on the hydrodynamic
expansion for the bulk fields. From the solution in the transverse sector, we
obtain

$\displaystyle w_{8}=-r_{h}-\frac{1}{2}i\omega\left[\log
2+\log(r_{h}^{2}/L^{2})\right]+\cdots,$ $\displaystyle
w_{9}=\frac{1}{2}\log(r_{h}^{2}/L^{2})+\cdots,$ $\displaystyle
w_{2}=\frac{2r_{h}^{2}}{\pi}+\frac{\pi}{4}\omega^{2}-\frac{\pi}{8}q^{2}+\cdots.$
(152)

From the solutions in the longitudinal sector,

$\displaystyle w_{1}=0+\mathcal{O}(\lambda^{3}),$ $\displaystyle
w_{3}=\frac{\pi}{8}+\cdots,$ $\displaystyle
w_{4}=-\frac{\pi}{24}i\omega+\cdots,$ $\displaystyle
w_{5}=2r_{h}^{2}-\frac{1}{2}q^{2}\left[2\log
2+\log(r_{h}^{2}/L^{2})\right]+\cdots,$ $\displaystyle
w_{6}=-\frac{\log(2r_{h}^{2}/L^{2})}{2}+\cdots,$ $\displaystyle
w_{7}=-\frac{1}{2}i\omega\log(2r_{h}^{2}/L^{2})+\cdots.$ (153)

The TCFs of (22) as well as the noise-noise correlator $G_{0}$ are expanded as

$\displaystyle\mathcal{D}=\frac{1}{2r_{h}}+\mathcal{O}(\lambda^{2}),\qquad\sigma_{e}=r_{h}+\frac{1}{2}i\omega\log\frac{2r_{h}^{2}}{L^{2}}+\mathcal{O}(\lambda^{2}),\qquad\sigma_{m}=\frac{1}{2}\log\frac{r_{h}^{2}}{L^{2}}+\mathcal{O}(\lambda^{1}),$
$\displaystyle\Xi=\frac{2ir_{h}^{2}}{\pi}+\mathcal{O}(\lambda^{2}),\qquad
G_{0}=-\frac{2ir_{h}^{2}}{\pi}q^{2}+\mathcal{O}(\lambda^{4}),$ (154)

where $\lambda\sim\partial_{\mu}$ is the bookkeeping parameter for the
derivative expansion. Notice that the relaxation time (order $\omega$ term)
for the diffusion TCF vanishes. This is not in agreement with the results of
Bu:2015ame . We postpone the comparison with Bu:2015ame  to the end of this
section.

We now compare our analytical results in the hydrodynamic limit with those of
Glorioso:2018mmw ; deBoer:2018qqm . It is important to keep in mind that the
minimal subtraction counter-term $S_{\rm c.t.}$ introduced in the present work
as well as in deBoer:2018qqm  differs from the one used in Glorioso:2018mmw .
Ref. deBoer:2018qqm  claimed agreement with Glorioso:2018mmw  on the values of
the transport coefficients. Yet, after careful examination of the results of
both papers and taking into account differences originating from the different
counter-terms, we fail to see a complete agreement. Below, we detail on the
comparison.

Comparison with deBoer:2018qqm . Ref. deBoer:2018qqm  focused on the
longitudinal sector only. Hence, it does not have any results on $w_{9}$,
neither separately on $w_{2}$ and $w_{3}$ (only the combination
$w_{2}+q^{2}w_{3}$ was determined). To ease the comparison, the results of
deBoer:2018qqm  are summarised below.

$\displaystyle w_{1}^{\rm dBHPF}=0+\mathcal{O}(\lambda^{3}),$ $\displaystyle
w_{2}^{\rm dBHPF}+q^{2}w_{3}^{\rm
dBHPF}=\frac{2\pi^{2}T^{2}L}{g_{A}^{2}}\frac{1}{\pi}+\mathcal{O}(\lambda^{3}),$
$\displaystyle w_{4}^{\rm dBHPF}=0+\mathcal{O}(\lambda^{2}),$ $\displaystyle
w_{5}^{\rm
dBHPF}=\frac{2\pi^{2}T^{2}L}{g_{A}^{2}}-\frac{2\pi^{2}T^{2}L}{g_{A}^{2}}\frac{\log
2}{2\pi^{2}T^{2}}q^{2}+\mathcal{O}(\lambda^{3}),$ $\displaystyle w_{6}^{\rm
dBHPF}=-\frac{2\pi^{2}T^{2}L}{g_{A}^{2}}\frac{\log
2}{4\pi^{2}T^{2}}+\mathcal{O}(\lambda^{1}),$ $\displaystyle w_{7}^{\rm
dBHPF}=-\frac{2\pi^{2}T^{2}L}{g_{A}^{2}}\frac{\log
2}{4\pi^{2}T^{2}}i\omega+\mathcal{O}(\lambda^{2}),$ $\displaystyle w_{8}^{\rm
dBHPF}=-\frac{2\pi^{2}T^{2}L}{g_{A}^{2}}\frac{1}{2\pi
T}-\frac{2\pi^{2}T^{2}L}{g_{A}^{2}}\frac{\log
2}{4\pi^{2}T^{2}}i\omega+\mathcal{O}(\lambda^{2}),$ (155)

where $g_{A}$ is the gauge coupling constant which has been set to one in our
work. The $\log(r_{h}/L)$ terms do not appear in deBoer:2018qqm , because they
have been set to zero.

Our results are largely consistent with those of deBoer:2018qqm  except

$\bullet$ The $\omega^{2}$-term in $w_{2}+q^{2}w_{3}$;

$\bullet$ The $\omega$-term in $w_{4}$.

The differences can be attributed to the lack of the third order accuracy in
the ingoing solutions $\\{\tilde{C}_{t},\tilde{C}_{x}\\}$ in deBoer:2018qqm ,
which is necessary for correct determination of $w_{2}+q^{2}w_{3}$ (up to
second order) and $w_{4}$ (up to first order)999deBoer:2018qqm  adopted a
derivative counting scheme, in which $B_{r\mu}\sim\mathcal{O}(\partial^{0})$
while $B_{a\mu}\sim\mathcal{O}(\partial^{1})$. Consequently, the terms under
discussion appear as of higher order and hence cannot be extracted from the
effective action truncated at the second order. .

Comparison with Glorioso:2018mmw . We quote the results of Glorioso:2018mmw :

$\displaystyle w_{1}^{\rm GCL}=0+\mathcal{O}(\lambda^{3}),$ $\displaystyle
w_{2}^{\rm
GCL}=\frac{2\pi}{\beta^{2}}+\frac{\pi}{4}\omega^{2}-\frac{\pi}{8}q^{2}+\mathcal{O}(\lambda^{3}),$
$\displaystyle w_{3}^{\rm GCL}=\frac{\pi}{8}+\mathcal{O}(\lambda^{1}),$
$\displaystyle w_{4}^{\rm
GCL}=-\frac{2\pi}{\beta}-\frac{48G+7\pi^{2}}{96\pi}i\omega+\mathcal{O}(\lambda^{2}),$
$\displaystyle w_{5}^{\rm
GCL}=\frac{2\pi^{2}}{\beta^{2}}+\left(\frac{1}{2}-\log
2\right)q^{2}+\mathcal{O}(\lambda^{3}),$ $\displaystyle w_{6}^{\rm
GCL}=\frac{1-\log 2}{2}-\frac{\pi}{4}+\mathcal{O}(\lambda^{1}),$
$\displaystyle w_{7}^{\rm GCL}=\left(\frac{1-\log
2}{2}+\frac{\pi}{4}\right)i\omega+\mathcal{O}(\lambda^{2}),$ $\displaystyle
w_{8}^{\rm GCL}=-\frac{\pi}{\beta}+\frac{1-\log
2}{2}i\omega+\mathcal{O}(\lambda^{2}),$ $\displaystyle w_{9}^{\rm
GCL}=-\frac{1}{2}+\mathcal{O}(\lambda^{1}).$ (156)

In order to compare the results, we have to account for the difference between
the subtraction terms used here and in Glorioso:2018mmw . In order to
represent the results of Glorioso:2018mmw  within the minimal subtraction
scheme, some $w_{i}^{\rm GCL}$’s in (156) have to be shifted:

$\displaystyle w_{5}^{\rm GCL}\to w_{5}^{\rm
GCL}+q^{2}\left[\log(r_{h}/L)+1/2\right],\qquad w_{6}^{\rm GCL}\to w_{6}^{\rm
GCL}+\log(r_{h}/L)+1/2,$ $\displaystyle w_{7}^{\rm GCL}\to w_{7}^{\rm
GCL}+i\omega\left[\log(r_{h}/L)+1/2\right],\qquad w_{8}^{\rm GCL}\to
w_{8}^{\rm GCL}+i\omega\left[\log(r_{h}/L)+1/2\right],$ $\displaystyle
w_{9}^{\rm GCL}\to w_{9}^{\rm GCL}-\log(r_{h}/L)-1/2,\qquad\qquad{\rm
others~{}are~{}not~{}changed.}$ (157)

With the differences in the counter-terms taken into account, our results
agree with those of Glorioso:2018mmw  except

$\bullet$ $w_{4}$ is completely different. The relevant result in
Glorioso:2018mmw  appears to be also in disagreement with deBoer:2018qqm . We
also notice that $w_{4}^{\rm GCL}$ does not seem to satisfy the KMS condition
(15).

$\bullet$ $w_{6}$ and $w_{7}$. Both $w_{6}^{\rm GCL},\,w_{7}^{\rm GCL}$ do not
obey the KMS condition (14).

We have attempted to trace the origin of these differences. First of all, our
approach to solving the bulk dynamics is quite different from that of
Glorioso:2018mmw . Particularly, the horizon limit ($\epsilon\to 0$) in our
formalism is always taken before the hydrodynamic limit ($\partial_{\mu}\to
0$). This is in contrast to Glorioso:2018mmw , which performs the hydrodynamic
expansion at the level of the dynamical EOMs. These two limits do not always
commute. Particularly, it is important to keep the oscillating factors like
$e^{i\omega\zeta_{1}(r)}$ unexpanded.

###  5.2 Numerical results at finite $\omega$ and $q$

Except for the couple of special cases of vanishing three-momentum ($q=0$) and
light-like momenta ($\omega=q$), solutions of the ODEs (70), (71) at finite
frequency and momentum are not known analytically. Therefore, in order to
provide complete information about the TCFs, we resort to numerical technique.
As has been extensively explained above, we have to solve the dynamical EOMs
in a single Schwarzschild-AdS for the ingoing modes and also for the
polynomial one, in the longitudinal sector. We solve the equations in the
Schwarzschild coordinates and then transform to EF coordinates. Once the
solutions are found, we first numerically extract the coefficients of the near
boundary expansion and then compute $w_{i}$ and other TCFs according to (121),
(122), (130)-(136) and (19).

In the bulk model there are in principle two independent length parameters:
$r_{h}$ and the AdS radius $L$. In the metric (28) $L=1$. For the numerical
results to be presented next, we also set $r_{h}=1$. There are two
consequences of this choice. First, the results do not include the logarithmic
branch proportional to $\log(r_{h}/L)$, though it is not difficult to recover
it analytically. Second, $r_{h}$ becomes a unit of length for all dimension-
full quantities such as frequency and momentum. Thus all the results below
will be shown for dimensionless $\omega\rightarrow\omega/r_{h}$ and
$q\rightarrow q/r_{h}$, while we stick to the same notations to avoid
introducing new ones.

While the effective Lagrangian (2.1) is parameterised by nine TCFs, the
discrete symmetries induce relations (11)-(15) which leave only four of them
independent. Specifically, we have chosen to take $w_{5}$, $w_{7}$, $w_{8}$
and $w_{9}$ as independent for which the numerical results are presented in
Appendix D.

The TCFs that have physical interpretation and hence are more interesting are
the ones parameterising the constitutive relation for the physical current
(22). Those are the diffusion TCF $\mathcal{D}$, the electric conductivity
$\sigma_{e}$, the magnetic conductivity $\sigma_{m}$, and the thermal force
$\Xi$, whose expressions in terms of $w_{i}$ are given in (19) and (23).

The results are summarized in Figures 2, 3, 4 and 5. At vanishing frequency
and momentum both the diffusion constant and electric conductivity are well
known: $\mathcal{D}[\omega=q=0]=1/2$ Policastro:2001yc  and
$\sigma_{e}[\omega=q=0]=1$ CaronHuot:2006te . When $q=0$, $\sigma_{e}$ is
known analytically Horowitz:2008bn ; Myers:2007we . Our numerical results are
fully consistent with all known analytical results.

Beyond the hydrodynamic limit, both the diffusion TCF $\mathcal{D}$, the
magnetic conductivity $\sigma_{m}$, and $\Xi$ vanish at large frequencies
$\omega$. In contrast, $\sigma_{e}$ is monotonically increasing function of
$\omega$; particularly $Re(\sigma_{e})\sim\omega$ asymptotically. As functions
of the three-momentum $q$, we mostly notice a very mild dependence reflecting
quasi-locality. $Re(\Xi)$ scales with $q^{2}$ at not too large momentum, hence
we plot $Re(\Xi/q^{2})$ in Figure 5.

The TCFs $\mathcal{D}$, $\sigma_{e}$, $\sigma_{m}$ have been originally
computed in Bu:2015ame , using the off-shell holography in a single
Schwarzschild-AdS5 geometry. Compared with the results of Bu:2015ame , the
present results for the TCFs have different profiles. In appendix E, we
briefly review the formalism of Bu:2015ame  compared with the present one. The
discrepancy is entirely within the longitudinal sector when the currents are
taken off-shell. Despite the disagreement in the TCFs, the current-current
retarded correlators $G^{\mu\nu}_{R}$ (173) are all found to coincide. The
equivalence is proven analytically in Appendix C. We have also cross-checked
the result numerically.

(a)

(b)

Figure 2: The $\omega$-, $q$-dependence of (a)
$Re(\mathcal{D}(\omega,q^{2}))$, (b) $Im(\mathcal{D}(\omega,q^{2}))$.

(a)

(b)

Figure 3: Ther $\omega$-, $q$-dependence of (a)
$Re(\sigma_{e}(\omega,q^{2}))$, (b) $Im(\sigma_{e}(\omega,q^{2}))$.

(a)

(b)

Figure 4: The $\omega$-, $q$-dependence of (a) $Re(\sigma_{m}(\omega,q^{2}))$,
(b) $Im(\sigma_{m}(\omega,q^{2}))$.

(a)

(b)

Figure 5: The $\omega$-, $q$-dependence of the coefficients (a)
$Re(\Xi(\omega,q^{2}))$ (b) $Im(\Xi(\omega,q^{2}))$

### The noise-noise correlator

Finally, the noise-noise correlator $-iG_{0}$ is displayed in Figure 6 as a
function of $\omega$ and $q^{2}$. In Figure 7(a), we plot the same function,
but as a 2d slices at fixed representative values of $q^{2}$. Since $-iG_{0}$
is proportional to $q^{2}$ up to sufficiently large momentum, we scale this
dependence out in Figures 6 and 7(a). As is clear from the plots, $-iG_{0}$
initially oscillates as function of $\omega$ but quickly vanishes at large
frequencies.

In order to better illustrate the coloured nature of the noise-noise
correlator, we perform an inverse Fourier transform of $-iG_{0}$ with respect
to the frequency, thus obtaining the time dependence of the correlator. The
analysis is performed for fixed values of momentum $q$ and the results are
displayed in Figure 7(b). It might be interesting to additionally perform the
inverse Fourier transform in the spatial momentum, so to obtain the full
space-time dependence of the correlator. Yet, this turns out to be numerically
too expensive and we have decided not to pursue this analysis.

Figure 6: The $\omega$-, $q^{2}$-dependence of $-iG_{0}(\omega,q^{2})/q^{2}$

(a)

(b)

Figure 7:  (a) The $\omega$-dependence of $-iG_{0}(\omega)/q^{2}$ at fixed
$q$-slices; (b) The inverse Fourier transform of $-iG_{0}(\omega)/q^{2}$ at
fixed $q$-slices.

##  6 Summary and Outlook

In this work we have further developed the off-shell SK holography. While the
core element of the formalism is the geometry proposed in Glorioso:2018mmw ,
our approach to solving the bulk EOMs is different from that of
Glorioso:2018mmw . When discussing the hydrodynamic expansion, one has to be
very careful with the non-commutativity of the hydrodynamic limit vs the near
horizon limit. Particularly, in order to reach a certain accuracy in the
effective action, EOMs must be expanded to higher orders in the momenta. Our
formalism completely avoids this subtlety since at no place it relies on the
hydrodynamic expansion.

Starting from the off-shell SK holography, we have derived the effective
action Crossley:2015evo  for the charge diffusion and computed all order TCFs
($w_{1}...w_{9}$) parameterising it. These TCFs display various types of
behaviour as functions of momenta, without any clear pattern. The evolution of
$w_{1}...w_{9}$ from small momenta (the hydrodynamic limit) to very large
momenta can be thought as a flow of the effective action from IR to UV. While
we have not analysed it in any detail, the flow corresponds to integrating in
of the heavy quasi-normal modes. It would be interesting to better understand
this relation. Putting the effective action on-shell, we reproduced the
prescription Son:2002sd  for the retarded two-point current-current
correlators.

The constitutive relation (3) for the current follows from the off-shell
effective action. It is parameterised by four TCFs: the diffusion, electric
conductivity, magnetic conductivity, and a thermal force. The latter is a new
element emerging from the SK formalism. The thermal force is responsible for
fluctuations/noise in the current. Due to linearity of the Maxwell’s theory in
the bulk, the noise is Gaussian, though coloured (non-local in space-time). We
have demonstrated that by explicitly performing the inverse Fourier transform
of the noise-noise correlator to the real time.

Our new results for the diffusion TCF, electric and magnetic conductivities
are different from the ones reported earlier in Bu:2015ame , even though they
lead to identical two-point retarded correlators. The disagreement originates
from different holographic dictionaries used to define the off-shell currents.
As discussed in Appendix E, the off-shell formalism of Bu:2015ame  defines the
current entirely in terms of the normalisable modes, while from the SK
holography we now learn that there appear additional terms contributing to the
current.

The Maxwell’s theory in the bulk, even though much more complicated than a
free scalar theory, provides a simple Gaussian theory on the boundary. A much
more challenging problem would be to consider stochastic neutral flows, which
amounts to embedding the fluid/gravity correspondence Kovtun:2004de ;
Policastro:2001yc ; Policastro:2002se ; Policastro:2002tn ; Son:2002sd ;
Bhattacharyya:2008jc  (with dissipations only) into an EFT framework. This
problem would require us to further develop the SK holography, particularly
addressing the questions of non-Gaussian noise, dynamical horizon, etc.

Another very interesting direction would be to learn about hydrodynamic
fluctuations associated with transport induced by chiral anomaly. While some
relevant discussion could be found in Glorioso:2017lcn ; Haehl:2013hoa ;
Iatrakis:2015fma ; Lin:2018nxj ; Liang:2020sgr , the topic remains largely
unexplored. From the perspective of the SK holography, the problem could be
addressed within the Maxwell-Chern-Simons theory in the bulk. Stochastic
chiral hydrodynamics is expected to be rich with many new phenomena.

##  Appendix A The effective action from the basis decomposition

In this appendix we demonstrate how the effective action (2.1) is derived from
(61). Following the formalism of Bu:2015ame , the bulk gauge fields $C_{\mu}$
can be linearly decomposed in terms of the basic tensor structures built from
$B_{r\mu}$ and $B_{a\mu}$,

$\displaystyle
C_{v}=S_{1}B_{rv}+S_{2}\partial_{k}B_{rk}+S_{3}B_{av}+S_{4}\partial_{k}B_{ak},$
$\displaystyle
C_{i}=V_{1}B_{ri}+V_{2}\partial_{i}B_{rv}+V_{3}\partial_{i}\partial_{k}B_{rk}+V_{4}B_{ai}+V_{5}\partial_{i}B_{av}+V_{6}\partial_{i}\partial_{k}B_{ak},$
(158)

where the decomposition coefficients (bulk-to-boundary propagators)
$S_{i},V_{i}$ are $SO(3)$ scalar functionals of the spacetime derivative
operators $\partial_{\mu}$, and functions of the radial coordinate $r$:

$\displaystyle S_{i}=S_{i}(r,\partial_{v},\vec{\partial}),\qquad
V_{i}=V_{i}(r,\partial_{v},\vec{\partial}).$ (159)

The boundary conditions for $S_{i},V_{i}$ are translated from those of
$C_{\mu}$ 101010Our current prescription is somewhat different from that of
Bu:2015ame . :

$\displaystyle r=\infty_{1}:\quad S_{1}=V_{1}=1,\quad
S_{3}=V_{4}=\frac{1}{2},\qquad{\rm others}=0,$ $\displaystyle
r=\infty_{2}:\quad S_{1}=V_{1}=1,\quad S_{3}=V_{4}=-\frac{1}{2},\quad\;{\rm
others}=0.$ (160)

In Fourier space, $(\partial_{v},\vec{\partial})\to(-i\omega,i\vec{q})$, these
decomposition coefficients become functions of the momenta,

$\displaystyle S_{i}(r,\partial_{v},\vec{\partial})\to
S_{i}(r,\omega,q^{2}),\qquad V_{i}(r,\partial_{v},\vec{\partial})\to
V_{i}(r,\omega,q^{2}).$ (161)

With the help of (158) the original PDEs (43) reduce to a system of linear
ODEs. Near $r=\infty_{1,2}$, the decomposition coefficients $S_{i},V_{i}$ can
be expanded. Taking into account the boundary conditions (A), the expansion
takes the form

$\displaystyle S_{i}(r\to\infty_{1})=\cdots+\frac{s_{1i}}{r^{2}}+\cdots,\qquad
S_{i}(r\to\infty_{2})=\cdots+\frac{s_{2i}}{r^{2}}+\cdots,$ $\displaystyle
V_{i}(r\to\infty_{1})=\cdots+\frac{v_{1i}}{r^{2}}+\cdots,\qquad
V_{i}(r\to\infty_{2})=\cdots+\frac{v_{2i}}{r^{2}}+\cdots.$ (162)

where the coefficients $s_{1i},v_{1i}$ and $s_{2i},v_{2i}$ are respective
normalizable modes.

Substituting the decomposition (158) into (61), the effective Lagrangian in
the $(r,a)$-basis reads

$\displaystyle\mathcal{L}_{\rm eff}=$ $\displaystyle-
B_{rv}s_{a1}B_{rv}-B_{rv}s_{a2}\partial_{k}B_{rk}+B_{rk}v_{a1}B_{rk}+B_{rk}v_{a2}\partial_{k}B_{rv}+B_{rk}v_{a3}\partial_{k}\partial_{l}B_{rl}$
$\displaystyle+(-B_{rv}s_{a3}B_{av}-B_{av}s_{r1}B_{rv})+(-B_{rv}s_{a4}\partial_{k}B_{ak}+B_{ak}v_{r2}\partial_{k}B_{rv})$
$\displaystyle+(-B_{av}s_{r2}\partial_{k}B_{rk}+B_{rk}v_{a5}\partial_{k}B_{av})-B_{av}s_{r3}B_{av}+(-B_{av}s_{r4}\partial_{k}B_{ak}+B_{ak}v_{r5}\partial_{k}B_{av})$
$\displaystyle+(B_{rk}v_{a4}B_{ak}+B_{ak}v_{r1}B_{rk})+(B_{rk}v_{a6}\partial_{k}\partial_{l}B_{al}+B_{ak}v_{r3}\partial_{k}\partial_{l}B_{rl})+B_{ak}v_{r4}B_{ak}$
$\displaystyle+B_{ak}v_{r6}\partial_{k}\partial_{l}B_{al}+\frac{1}{2}\partial_{k}B_{ak}\partial_{v}B_{rv}-\frac{1}{2}B_{av}\partial_{v}\partial_{k}B_{rk}+\frac{1}{2}B_{av}{\vec{\partial}}^{\,2}B_{rv}-\frac{1}{2}B_{ak}\partial_{v}^{2}B_{rk}$
$\displaystyle+\frac{1}{4}\mathcal{F}_{ajk}\mathcal{F}_{rjk}+B_{av}\partial_{v}^{2}B_{rv}.$
(163)

Here, the coefficients are represented in the $(r,a)$-basis:

$\displaystyle s_{ai}=s_{1i}-s_{2i},\qquad
s_{ri}=\frac{1}{2}(s_{1i}+s_{2i}),\qquad i=1,2,3,4,$ $\displaystyle
v_{ai}=v_{1i}-v_{2i},\qquad v_{ri}=\frac{1}{2}(v_{1i}+v_{2i}),\qquad
i=1,2,\cdots,6.$ (164)

The first line of (A) must vanish within the usual $(r,a)$ scheme. Indeed
within the holographic representation of the SK contour:

$\displaystyle s_{a1}=s_{a2}=v_{a1}=v_{a2}=v_{a3}=0.$ (165)

To compare with Crossley:2015evo , (A) has to be rewritten so that all the
$a$-fields are placed on the left, in all terms, say

$\displaystyle\int
d^{4}xB_{rv}(x)s_{a3}(\partial_{t},\vec{\partial})B_{av}(x)=\int
d^{4}xB_{av}(x)s_{a3}(-\partial_{v},-\vec{\partial})B_{rv}(x).$ (166)

Introducing $w_{i}=w_{i}(\partial_{v},\vec{\partial})$ as

$\displaystyle\frac{i}{2}w_{1}(\partial_{v},\vec{\partial})=-s_{r3}(\partial_{v},\vec{\partial}),$
$\displaystyle\frac{i}{2}w_{2}(\partial_{v},\vec{\partial})=v_{r4}(\partial_{v},\vec{\partial}),$
$\displaystyle\frac{i}{2}w_{3}(\partial_{v},\vec{\partial})=-v_{r6}(\partial_{v},\vec{\partial}),$
$\displaystyle
iw_{4}(\partial_{v},\vec{\partial})=-s_{r4}(\partial_{v},\vec{\partial})-v_{r5}(-\partial_{v},-\vec{\partial}),$
$\displaystyle
w_{5}(\partial_{v},\vec{\partial})=\partial_{v}^{2}+\frac{1}{2}\vec{\partial}^{2}-s_{r1}(\partial_{v},\vec{\partial})-s_{a3}(-\partial_{v},-\vec{\partial})$
$\displaystyle
w_{6}(\partial_{v},\vec{\partial})\partial_{v}=-\frac{1}{2}-s_{r2}(\partial_{v},\vec{\partial})-v_{a5}(-\partial_{v},-\vec{\partial}),$
$\displaystyle
w_{7}(\partial_{v},\vec{\partial})=\frac{1}{2}\partial_{v}-s_{a4}(-\partial_{v},-\vec{\partial})-v_{r2}(\partial_{v},\vec{\partial})$
$\displaystyle
w_{8}(\partial_{v},\vec{\partial})\partial_{v}=-\frac{1}{2}\partial_{v}^{2}+v_{a4}(-\partial_{v},-\vec{\partial})+v_{r1}(\partial_{v},\vec{\partial})+\left[v_{a6}(-\partial_{v},-\vec{\partial})+v_{r3}(\partial_{v},\vec{\partial})\right]{\vec{\partial}}^{2},$
$\displaystyle
w_{9}(\partial_{v},\vec{\partial})=\frac{1}{2}+v_{a6}(-\partial_{v},-\vec{\partial})+v_{r3}(\partial_{v},\vec{\partial}),$
(167)

the effective Lagrangian (A) is cast into (2.1), consistently with
Crossley:2015evo .

##  Appendix B Validating (77)

In subsection 4.2, we derived the discontinuity relation (77). Our goal here
is to verify this condition by computing both left-hand side (LHS) and right-
hand side (RHS) of (77), starting from the solutions constructed in the
subsections 4.3 and 4.4.

The RHS of (77) is proportional to $\nabla_{M}F^{Mr}$. Recall that in general,
$\nabla_{M}F^{Mr}$ has a very simple dependence on $r$, see (46). In (46) the
values of $\mathcal{C}^{\rm up}$ and $\mathcal{C}^{\rm dw}$ should be
determined from the solutions established in subsections 4.3 and 4.4. As has
been emphasised towards the end of subsection 4.3, among the four linearly
independent solutions in the longitudinal sector, only the polynomial solution
$\\{C_{v}^{\rm pn},C_{x}^{\rm pn}\\}$ does not automatically satisfy the
constraint equation (69). Hence $\mathcal{C}^{\rm up,\,dw}$ must be
proportional to the coefficients $n_{\parallel}^{\rm up,\,dw}$ multiplying the
polynomial solutions. More precisely, by taking the near horizon limit of
(46), we have

$\displaystyle\mathcal{C}^{\rm up}(k)=i\omega r_{h}^{3}\tilde{C}_{t}^{{\rm
pn}\,h}n_{\parallel}^{\rm up},\qquad\qquad\mathcal{C}^{\rm dw}(k)=i\omega
r_{h}^{3}\tilde{C}_{t}^{{\rm pn}\,h}n_{\parallel}^{\rm dw}.$ (168)

Thus the RHS of (77) reads

$\displaystyle\lim_{\Delta\to
0}\int_{r_{-}-\Delta}^{r_{+}+\Delta}dr\frac{\nabla_{M}F^{Mr}}{f(r)}$
$\displaystyle=\frac{\mathcal{C}^{\rm dw}(k)}{i\omega
r_{h}^{3}}e^{i\omega\zeta_{1}(r_{h}-\epsilon)}-\frac{\mathcal{C}^{\rm
up}(k)}{i\omega r_{h}^{3}}e^{i\omega\zeta_{2}(r_{h}-\epsilon)}$
$\displaystyle=\tilde{C}_{t}^{{\rm pn}\,h}\left[n_{\parallel}^{\rm
dw}e^{i\omega\zeta_{1}(r_{h}-\epsilon)}-n_{\parallel}^{\rm
up}e^{i\omega\zeta_{2}(r_{h}-\epsilon)}\right].$ (169)

Based on the solutions found in 4.3 and 4.4, the discontinuity of
$\partial_{r}C_{v}$ is

$\displaystyle\partial_{r}C_{v}(r_{+})-\partial_{r}C_{v}(r_{-})=$
$\displaystyle n_{\parallel}^{\rm up}\partial_{r}C_{v}^{\rm
pn}\big{|}_{r=r_{+}}-n_{\parallel}^{\rm dw}\partial_{r}C_{v}^{\rm
pn}\big{|}_{r=r_{-}}$ $\displaystyle=$
$\displaystyle\frac{i\omega}{f(r)}n_{\parallel}^{\rm up}C_{v}^{\rm
pn}(r)\bigg{|}_{r=r_{+}}-\frac{i\omega}{f(r)}n_{\parallel}^{\rm dw}C_{v}^{\rm
pn}(r)\bigg{|}_{r=r_{-}}$ $\displaystyle+\tilde{C}_{t}^{{\rm
pn}\,h}n_{\parallel}^{\rm
up}e^{i\omega\zeta_{2}(r_{h}-\epsilon)}-\tilde{C}_{t}^{{\rm
pn}\,h}n_{\parallel}^{\rm dw}e^{i\omega\zeta_{1}(r_{h}-\epsilon)}$ (170)

While $C_{v}$ is continuous across the cutting slice, $C_{r}=-C_{v}/f(r)$ has
a jump:

$\displaystyle\frac{C_{v}(r)}{f(r)}\bigg{|}_{r=r_{+}}-\frac{C_{v}(r)}{f(r)}\bigg{|}_{r=r_{-}}=\frac{n_{\parallel}^{\rm
up}}{f(r)}C_{v}^{\rm pn}(r)\bigg{|}_{r=r_{+}}-\frac{n_{\parallel}^{\rm
dw}}{f(r)}C_{v}^{\rm pn}(r)\bigg{|}_{r=r_{-}}.$ (171)

So, the LHS of (77) is computed as

$\displaystyle F^{rv}(r_{+})-F^{rv}(r_{-})$
$\displaystyle=\left[\frac{i\omega}{f(r)}-\partial_{r}\right]C_{v}\bigg{|}_{r=r_{+}}-\left[\frac{i\omega}{f(r)}-\partial_{r}\right]C_{v}\bigg{|}_{r=r_{-}}$
$\displaystyle=\tilde{C}_{t}^{{\rm pn}\,h}n_{\parallel}^{\rm
dw}e^{i\omega\zeta_{1}(r_{h}-\epsilon)}-\tilde{C}_{t}^{{\rm
pn}\,h}n_{\parallel}^{\rm up}e^{i\omega\zeta_{2}(r_{h}-\epsilon)}$ (172)

which is exactly the same as (169).

##  Appendix C Son-Starinets prescription for retarded correlators revisited

At the early days of the fluid-gravity correspondence, Son and Starinets
Son:2002sd  proposed a prescription for computing Minkowski-space retarded
correlators. The prescription is formulated entirely within a single copy of
the doubled BH-AdS, and does not rely on SK holography. Yet, a proper
derivation of the prescription from the SK holography is missing, and in this
appendix we provide one. While there have been earlier works in this
direction, particularly Herzog:2002pc , which considered SK matrix propagator
for a scalar field starting from an eternal black hole in AdS space
Maldacena:2001kr , the SK geometry of Glorioso:2018mmw  adopted here is
different. Furthermore, we are not aware of any derivation for the $U(1)$
field available in the literature.

The prescription of Son:2002sd  relates the retarded correlators to the
ingoing solution in a single BH AdS. Starting from the SK holography, we can
reproduce the result by taking a few alternative paths. First, the correlators
could be obtained from the off-shell effective Lagrangian (2.1) by integrating
out the dynamical fields $\varphi_{r}$ and $\varphi_{a}$ (8). The result is
the boundary generating functional $W[\mathcal{A}_{a\mu},\mathcal{A}_{r\mu}]$
of the external fields only, form which the correlators could be read off
straightforwardly. For the Lagrangian quadratic in the dynamical fields, like
(2.1), integrating out $\varphi_{r}$ and $\varphi_{a}$ could be done by
imposing their classical EOMs. On the bulk side, this corresponds to imposing
the constraint equation. Putting the solutions (108) on-shell is equivalent to
setting $n_{\parallel}^{\rm up}=n_{\parallel}^{\rm dw}=0$, which via (112) and
(113) yields classical solutions for $\varphi_{r}$ and $\varphi_{a}$.

Alternatively, we could start with the constitutive relation (18), and use the
continuity equation, which leads to the retarded current-current correlators
expressed in terms of the TCFs Bu:2015ame :

$\displaystyle G_{R}^{\perp\perp}=i\omega\sigma_{e}+q^{2}\sigma_{m},\qquad
G_{R}^{vv}=\frac{q^{2}\sigma_{e}}{-i\omega+q^{2}\mathcal{D}},\qquad
G_{R}^{vx}=\frac{\omega q\sigma_{e}}{-i\omega+q^{2}\mathcal{D}},$
$\displaystyle
G_{R}^{xx}=\frac{\omega^{2}\sigma_{e}}{-i\omega+q^{2}\mathcal{D}}.$ (173)

These expressions could be algebraically traced back to the ingoing solution
in a single copy of BH-AdS.

Yet, we believe the most illuminating derivation is to reconsider the problem
from the very beginning, starting within the on-shell SK holography, which
offers a possibility to work directly with gauge invariant fields

$\displaystyle
E_{\perp}=\partial_{\perp}C_{v}-\partial_{v}C_{\perp},\qquad\qquad
E_{x}=\partial_{x}C_{v}-\partial_{v}C_{x}.$ (174)

EOMs for the bulk electric fields $E_{\perp}$ and $E_{x}$ are