aainstitutetext: Institute for Theoretical Physics, University of Amsterdam,
1090 GL Amsterdam, The Netherlandsbbinstitutetext: Max Planck Institute for
Gravitational Physics, Potsdam-Golm, D-14476, Germanyccinstitutetext:
Institute for Theoretical Physics, KU Leuven, Celestijnenlaan 200D, Leuven
B-3001, Belgium

# Holographic Schwinger-Keldysh effective field theories

Jan de Boer b* Michal P. Heller,***On leave from: _National Centre for Nuclear
Research, 00-681 Warsaw, Poland_. c and Natalia Pinzani-Fokeeva
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We construct a holographic dual of the Schwinger-Keldysh effective action for
the dissipative low-energy dynamics of relativistic charged matter at strong
coupling in a fixed thermal background. To do so, we use a mixed signature
bulk spacetime whereby an eternal asymptotically anti-de Sitter black hole is
glued to its Euclidean counterpart along an initial time slice in a way to
match the desired double-time contour of the dual field theory. Our results
are consistent with existing literature and can be regarded as a fully-ab
initio derivation of a Schwinger-Keldysh effective action. In addition, we
provide a simple infrared effective action for the near horizon region that
drives all the dissipation and can be viewed as an alternative to the membrane
paradigm approximation.

##  1 Introduction

The path integral formulation of quantum mechanics is central to many among
the most important developments in modern theoretical physics. For example, it
played a pivotal role in the quantization of gauge theories. The latter are
the corner stone of the Standard Model of particle physics tHooft:2005hbu ,
provide, through holography (AdS/CFT) Maldacena:1997re ; Witten:1998qj ;
Gubser:1998bc , our current best shot at understanding quantum gravity and
find numerous applications in the condensed matter physics shankar_2017 .

Path integrals are also widely used to describe non-equilibrium phenomena. In
these cases they are defined on nontrivial time contours in which time can
evolve forward and backwards in various sequences. A special case of this type
is the Schwinger-Keldysh partition function Schwinger:1960qe ; Keldysh:1964ud
. This object incorporates a particular class of correlators which involve
time-ordering (i.e. retarded, advanced, symmetric, etc.) and has been
extensively used in numerous applications to non-equilibrium collective states
of matter both in high energy and condensed matter physics, see, e.g.,
2004cond.mat.12296K  and references therein.

One of the most active lines of research on non-equilibrium phenomena of the
past two decades concerns relativistic hydrodynamics. The key experimental
impetus on this front was the discovery that droplets of quark-gluon plasma
created in ultrarelativistic heavy-ion collisions at RHIC (and later also at
LHC) behave as the most perfect fluid known in the universe. It was also
understood relatively early on that deviations from perfect fluidity provide
an invaluable window on the microscopics of the quark-gluon plasma. See, e.g.,
ref. Busza:2018rrf  for a recent broad overview of this research field.
Altogether, this has led to a large number of developments concerning
dissipative relativistic hydrodynamics that have been occurring both on the
phenomenological and theoretical front, see, e.g., refs. Romatschke:2009im ;
Hubeny:2011hd ; Florkowski:2017olj ; Romatschke:2017ejr  for a review.

The vast majority of the aforementioned progress occurred at the level of
equations of motion, which has limited our ability to understand the
microscopic origin of dissipation and the entropy current, as well as to
systematically incorporate noise. It was only very recently that the
Schwinger-Keldysh formulation has been used to bring us closer to recasting
the relativistic fluid dynamics as a proper effective field theory with a
Lagrangian formulation, see refs. Haehl:2015foa ; Crossley:2015evo ;
Haehl:2015uoc ; Haehl:2016pec ; Haehl:2016uah ; Glorioso:2017lcn ;
Jensen:2017kzi ; Gao:2017bqf ; Glorioso:2017fpd ; Haehl:2018lcu ;
Jensen:2018hse , the precursor works Grozdanov:2013dba ; Kovtun:2014hpa ;
Haehl:2014zda ; Haehl:2015pja ; Harder:2015nxa  as well as ref.
Glorioso:2018wxw  for a review and a more comprehensive list of references.

The symmetries of the microscopic Schwinger-Keldysh path integral, such as
unitarity and CPT invariance, have been used to constrain the form of the low-
energy Schwinger-Keldysh effective action and consequently to re-derive the
conventional phenomenological formulation of hydrodynamics in some limiting
regimes. For example, it was shown in ref. Glorioso:2016gsa , see also refs.
Jensen:2018hhx ; Haehl:2018uqv , how the local second law of thermodynamics
can be obtained as the Noether current of a symmetry of the effective action.
Moreover, additional constraints on transport, invisible to the
phenomenological treatment based on equations of motion, have been found in
ref. Jensen:2018hse . As alluded earlier, the power of this approach is that
both dissipative and fluctuation effects are naturally incorporated allowing
for a systematic treatment of stochastic noises, see the recent ref. Chen-
Lin:2018kfl  which evaluates corrections to the heat diffusion coefficient due
to thermal fluctuations finding novel results with respect to previous, more
phenomenological, derivations, see, e.g., ref. Kovtun:2014nsa .

The key motivation for our present work is that the above considerations,
while agreeing with all previously known results that stood the test of
scrutiny and addressing some of their shortcomings, have not yet been derived
from any microscopic model when dissipation needed to be included. Our aim,
therefore, is to provide the first microscopic derivation of the Schwinger-
Keldysh effective actions for what seems to be the simplest hydrodynamic
setup, i.e. the charge diffusion in the limit in which the energy and momentum
carried by charge carriers is negligible and the charge current amplitude is
small.

The tool that we will use to achieve this goal is holography viewed here as an
_ab initio_ formulation of a large class of strongly-interacting quantum field
theories with a large number of microscopic constituents.

Indeed, holography provides a very natural arena in which all these ideas can
be explicitly tested. Following ref. Herzog:2002pc , the double-time contour
of the Schwinger-Keldysh formalism can be represented by the two boundaries of
the maximally-extended AdS-Schwarzschild black brane. Subsequent developments
in refs. Skenderis:2008dh ; Skenderis:2008dg  interpreted the Euclidean part
of the thermal Schwinger-Keldysh contour as a Euclidean black brane glued to a
finite time slice of the Lorentzian solution. The contour can be then closed
by gluing the future horizons of the Lorentzian solution as in ref.
vanRees:2009rw , see also ref. Botta-Cantcheff:2018brv . This simple picture,
of course, applies only if the sources and their effects are a tiny distortion
of the global thermal equilibrium. Otherwise, according to the fully
holographic prescription of refs. Skenderis:2008dh ; Skenderis:2008dg , one
would need to solve non-linear Einstein’s equations (possibly with matter) and
find a mixed signature holographic geometry specified by the asymptotic
boundary condition given by the desired Schwinger-Keldysh contour. In the
present work we circumnavigate this otherwise fascinating problem by
considering the dynamics of a probe bulk gauge field in the aforementioned
mixed-signature static background spacetime. This scenario gives then rise to
the quadratic Schwinger-Keldysh effective action for the charge diffusion in
holographic conformal field theories, which we derive to the second order in
derivatives.

Our work builds on several earlier developments. In particular, in the
previous paper deBoer:2015ija , see also ref. Crossley:2015tka , we considered
the preliminary case of dissipationless fluids, i.e. those which do not
exhibit entropy production, and derived their leading order effective action
from holography. Our approach then focused on single-sided, asymptotically AdS
black branes dual to a strongly-coupled conformal plasma around thermal
equilibrium. To remove dissipative effects, we introduced a fictitious
intermediate timelike hypersurface at some fixed radial position very close to
the horizon and considered only the remaining spacetime between this radial
cut-off and the conformal boundary. We identified the holographic dual of the
degrees of freedom relevant for the fluid dynamic behavior with Wilson lines
extending between the conformal boundary and the cutoff hypersurface, much in
the spirit of ref. Nickel:2010pr , see also refs. Heemskerk:2010hk ;
Faulkner:2010jy . After solving the linearized Einstein equations with double-
Dirichlet boundary conditions between the two boundaries at the leading order
in the hydrodynamic gradient expansion, we computed the (partially) on-shell
gravitational action which we then interpreted as the effective action for
ideal fluids. It was, of course, already clear back then that this procedure
was not enough had one wished to go beyond the leading order hydrodynamic
expansion. Not only the dissipation was absent by construction, but also non-
dissipative second order transport was suffering from divergences. The latter
could only be cured by including dissipative effects, as we showed there using
a simple membrane paradigm approximation, see in this context also our earlier
work deBoer:2014xja .

The plan of the present article is the following. In sec. 2 we summarize the
key ingredients of the Schwinger-Keldysh formalism which are necessary for the
present work. The reader familiar with the literature of the subject may skip
this part. In sec. 3 we provide a general discussion on how to parallel the
same construction in holography using the real-time formulation of refs.
Skenderis:2008dh ; Skenderis:2008dg . In sec. 4 we consider the specific
example of a probe gauge field and show how the low-energy effective action
for charge dynamics at strong coupling can be _derived_ from holography. In
sec. 5 we show how the general properties of the Schwinger-Keldysh effective
action are captured by a simple infrared (IR) piece which resides in the near
horizon region of the spacetime and how the remaining geometry between the
horizon and the boundaries only serves as a map of this information to where
the dual field theory is defined. Finally, in sec. 6 we discuss some natural
extensions of our results.

Note added: We learnt that ref. Glorioso:2018mmw , which will appear soon,
also addresses the problem of deriving Schwinger-Keldysh effective actions
using holography.

Note added (v2): The results of ref. Glorioso:2018mmw  are in agreement with
ours.

##  2 The Schwinger-Keldysh effective action

Let us briefly review the main ingredients that enter the construction of
Schwinger-Keldysh effective actions. We will mainly focus on the simplest case
of the low-energy behavior of charged matter in a fixed thermal background
which will be used in this work. This section is admittedly brief, we refer
the reader to the review Glorioso:2018wxw  and references therein for more
details.

The Schwinger-Keldysh partition function is defined as

$Z[A_{R},A_{L}]={\rm Tr}\Big{(}{\cal U}[A_{R}]\,\rho_{0}\,{\cal
U}^{\dagger}[A_{L}]\Big{)}\,,$ (1)

where $\rho_{0}$ is the density matrix of the initial state given at some time
$t=0$, ${\cal U}$ is the evolution operator from $t=0$ to the infinite future
$t=+\infty$, and ${\cal U}^{\dagger}$ is the anti-evolution operator from the
infinite future back to $t=0$ where the state is defined. The evolution
operators depend generically on external sources, collectively denoted by
$A_{R}$ and $A_{L}$, which are taken to be independent from one another. In a
path integral representation, eq. (1) takes the form

$Z[A_{R},A_{L}]=\int_{\rho_{0}}{\cal D}\psi_{R}\,{\cal
D}\psi_{L}\,e^{iS[\psi_{R};\,A_{R}]-iS[\psi_{L};\,A_{L}]}\,,$ (2)

where $S$ is the microscopic action, functional of the microscopic fields
$\psi$ and the sources $A$. Appropriate boundary conditions need to be
supplemented in the infinite future $t=+\infty$, i.e.
$\psi_{L}(t=+\infty)=\psi_{R}(t=+\infty)$, and at $t=0$. The relative minus
sign between the $R$ and $L$-type actions in (2) reflects the backward flow of
time of the anti-evolution operator in eq. (1). With these definitions, the
Schwinger-Keldysh path integral (2) can be viewed as an integral over a
double-time contour where time flows forward in the upper part of the contour
with time $t_{R}$ and then back with time $t_{L}$ to where the state is
defined. If the initial state is thermal, i.e. $\rho_{0}\sim e^{-\beta H}$
with the inverse temperature $\beta=1/T$, then the contour becomes complex, as
depicted in Fig. 1, and the imaginary time segment is also periodically
identified with period $\beta$.

Figure 1: The Schwinger-Keldysh contour at finite temperature. Time flows
forward from $t=0$ to $t=+\infty$ and back to where the initial state is
defined. The latter is represented by an imaginary time segment identified
along the circles with period $\beta=1/T$.

The main virtue of the Schwinger-Keldysh partition function (2) is that it
incorporates a large class of Minkowski signature correlators for generic
states. Defining the Schwinger-Keldysh average and difference basis

$A_{r}=\frac{1}{2}\left(A_{R}+A_{L}\right)\,,\qquad A_{a}=A_{R}-A_{L}\,,$ (3)

the retarded, advanced and symmetric connected two-point functions can be
shown to be related to _very simple_ (which is part of the formalism’s allure)
variations of the generating functional $W=-i\ln Z$ as follows

$\displaystyle\begin{split}G_{ret}(t_{1},t_{2})&=i\,G_{ra}(t_{1},t_{2})=i\frac{\delta^{2}W}{\delta
A_{a}(t_{1})\delta A_{r}(t_{2})}\bigg{|}_{A=0}\,,\\\
G_{adv}(t_{1},t_{2})&=i\,G_{ar}(t_{1},t_{2})=i\frac{\delta^{2}W}{\delta
A_{r}(t_{1})\delta A_{a}(t_{2})}\bigg{|}_{A=0}\,,\\\
G_{sym}(t_{1},t_{2})&=G_{rr}(t_{1},t_{2})=\frac{\delta^{2}W}{\delta
A_{a}(t_{1})\delta A_{a}(t_{2})}\bigg{|}_{A=0}\,,\end{split}$ (4)

see Glorioso:2018wxw  for more details. These functions are of obvious
interest in many-body physics111In particular, the structure of singularities
in the complex frequency and momentum plane of retarded two-point functions
has been the driving force behind recent progress on understanding non-
equilibrium properties of states of matter evading quasiparticle description,
see, e.g., refs. Hartnoll:2016apf ; Florkowski:2017olj ; Liu:2018crr  for
contemporary reviews..

Let us now separate in eq. (2) between the high-energy (UV) and low-energy
(IR) degrees of freedom. Assuming that the UV degrees of freedom have been
integrated out, the Schwinger-Keldysh path integral (2) can be formally
rewritten in terms of appropriately defined IR dynamical fields $\xi_{R}$ and
$\xi_{L}$ as

$Z[A_{R},A_{L}]=\int{\cal D}\xi_{R}\,{\cal
D}\xi_{L}\,e^{i\,S_{eff}[\xi_{R},\,\xi_{L};\,A_{R},\,A_{L};\,\rho_{0}]}\,,$
(5)

where $S_{eff}$ is the Schwinger-Keldysh effective action that incorporates
the low-energy dynamics around the state $\rho_{0}$. With respect to the
conventional Wilsonian effective action there are a few differences. For
example, the Schwinger-Keldysh effective action depends on two copies of the
infrared degrees of freedom which may interact with one another. It is those
interactions that allow to write dissipative terms in the Lagrangian,
something that is impossible to achieve with conventional effective actions.

In this work we are interested in the low-energy effective action $S_{eff}$
for charged matter at finite temperature in a fixed $d$-dimensional
background. In this case, the external sources are taken to be a pair of
$U(1)$ flavor fields $A_{R\,\mu}$ and $A_{L\,\mu}$ which live on two different
spacetimes, dubbed as $R$ and $L$, and transform as connections under two
independent gauge transformations parameterized by $\Lambda_{R}$ and
$\Lambda_{L}$,

$A_{R\,\mu}\rightarrow A_{R\,\mu}+\partial_{\mu}\Lambda_{R}\,,\qquad
A_{L\,\mu}\rightarrow A_{L\,\mu}+\partial_{\mu}\Lambda_{L}\,,$ (6)

where $\mu=t\,,x_{1}\,,\dots\,,x_{d}$ are spacetime indices. Notice that the
coordinates defined on the $R$ and $L$ spacetimes, $x^{\mu}=(t,\vec{x})$, can
in principle be independent as the fields in $R$ transform under independent
diffeomorphisms than those in $L$. However, as we already alluded many times,
we are only interested in the charge dynamics, thus in a probe limit where the
background is fixed to the Minkowski metric $\eta_{\mu\nu}$ and the spacetime
coordinates on which the two type of fields $R$ and $L$ depend can be taken to
be identical.

The relevant low-energy degrees of freedom are two scalar fields $\phi_{L}$
and $\phi_{R}$ that appear in the effective action in combination with the
sources as follows

$B_{R\,i}(\sigma)=\partial_{i}x^{\mu}(\sigma)A_{R\,\mu}(x(\sigma))+\partial_{i}\phi_{R}(\sigma)\,,\qquad
B_{L\,i}(\sigma)=\partial_{i}x^{\mu}(\sigma)A_{L\,\mu}(x(\sigma))+\partial_{i}\phi_{L}(\sigma)\,.$
(7)

Expressions (7) are pullbacks of the sources $A_{R\,\mu}$ and $A_{L\,\mu}$
from their respective $R$ and $L$ target spaces to a common worldvolume
spacetime parameterized by the coordinates $\sigma^{i}=(\bar{t},\vec{x})$. The
maps $x^{\mu}(\sigma)$, $\phi_{R}(\sigma)$ and $\phi_{L}(\sigma)$ define the
pullback and here we have promoted the scalar fields to be the dynamical low-
energy fields that capture the hydrodynamic behavior of the charge current.
The remaining maps $x^{\mu}(\sigma)$ are not dynamical and are taken to only
include a reparameterization of time between the target spaces and the
worldvolume. With a little bit of prescience we write them as
$x^{\mu}(\sigma)=\left(\bar{t}/\sqrt{f_{\delta}},\vec{x}\right)$ where
$\sqrt{f_{\delta}}$ is a redshift factor. As we will see again later on, this
picture is very much reminiscent of black holes where the target spaces can be
identified with the boundaries of an eternal AdS black hole and the
worldvolume with the near horizon region where the redshift can be naturally
defined through the emblackening factor. In fact, the metric on the
worldvolume is defined as

$g_{ij}(\sigma)=\partial_{i}x^{\mu}(\sigma)\partial_{j}x^{\nu}(\sigma)\eta_{\mu\nu}(x(\sigma))\,,$
(8)

which is the pullback of the boundary metric to the near horizon region.

The scalar fields $\phi_{R}$ and $\phi_{L}$ are respectively bifundamental
maps between the tangent bundles over the $R/L$ target spaces and the tangent
bundle over the worldvolume. They transform thus as follows

$\phi_{R}\rightarrow\phi_{R}-\Lambda_{R}+\Lambda\,,\qquad\phi_{L}\rightarrow\phi_{L}-\Lambda_{L}+\Lambda\,,$
(9)

where $\Lambda$ is the $U(1)$ gauge parameter on the worldvolume. With these
definitions, the combinations (7) are invariant under the $R$ and $L$ gauge
transformations, and transform as connections under the worldvolume one

$B_{R\,i}\rightarrow B_{R\,i}+\partial_{i}\Lambda\,,\qquad B_{L\,i}\rightarrow
B_{L\,i}+\partial_{i}\Lambda\,.$ (10)

In what follows, it is convenient to introduce $r$ and $a$-type combinations
of the fields and sources

$\phi_{r}=\frac{1}{2}\left(\phi_{R}+\phi_{L}\right)\,,\quad\phi_{a}=\phi_{R}-\phi_{L}\,,\quad
B_{r\,i}=\frac{1}{2}\left(B_{R\,i}+B_{L\,i}\right)\,,\quad
B_{a\,i}=B_{R\,i}-B_{L\,i}\,.$ (11)

Having defined the low-energy degrees of freedom, the most general Schwinger-
Keldysh effective action $S_{eff}$ can now be constructed requiring
compatibility with the symmetries of the Schwinger-Keldysh path integral (1),

Schwinger-Keldysh symmetry: $\displaystyle\quad Z[A_{R}=A_{L}]=1\,,$ (12)
Reality condition: $\displaystyle\quad
Z[A_{R},A_{L}]^{*}=Z[A_{L}^{*},A_{R}^{*}]\,,$ (13) KMS symmetry:
$\displaystyle\quad Z[A_{R}(t),A_{L}(t)]=Z[\eta A_{R}(-t),\eta
A_{L}(-t-i\beta)]\,,\qquad$ (14)

and with the constraint

$\text{Im}\,S_{eff}\geq 0\,.$ (15)

Expressions (12) and (13) are valid for any initial state $\rho_{0}$ while the
KMS symmetry (14) is only applicable for thermal states $\rho_{0}\sim
e^{-\beta{\cal H}}$ where $\beta=1/T$ is the inverse of the temperature, see,
again, ref. Glorioso:2018wxw  for more details. In writing eq. (14) we have
also assumed CPT invariance222The condition on CPT invariance of the
microscopic theory can be changed to any combination of the discrete
symmetries as long as it contains time invariance, see ref. Crossley:2015evo .
of the underlying microscopic theory, where $\eta$ is the CPT eigenvalue
associated to the operator dual to the corresponding source $A$. For example,
in the case $A$ is an external $U(1)$ flavor field $A_{\mu}$ depending on $t$
and $\vec{x}$ coordinates of the Minkowski metric, the corresponding CPT
eigenvalue $\eta_{\mu}$ is $\eta_{t}=\eta_{x}=-1$ and the remaining components
are $+1$.

Notice that upon reinstating $\hbar$ in the expressions above we have
$\beta\rightarrow\hbar\,\beta$, and so the KMS symmetry (14) is a nonlocal
$Z_{2}$ symmetry at the full quantum level. Although it is possible to take a
classical limit $\hbar\rightarrow 0$ where quantum fluctuations are neglected
and the condition (14) becomes local, for the probe limit case at hand this
restriction is not necessary and we may work in the full quantum regime, see,
e.g., ref. Jensen:2017kzi . Let us also notice that when
$A_{R}(t)=A_{L}(t-i\beta)$, we have from (14) that
$Z[A_{R}(t)=A_{L}(t-i\beta)]=1$, meaning that the Schwinger-Keldysh partition
function for thermal states has an additional limit where it becomes trivial,
other than the one given in eq. (12).

We cannot help but mention that the Schwinger-Keldysh symmetry (12) can be
implemented at tree level by requiring the effective action to be at least
linear in the $a$-type combinations of the fields and sources $B_{a}$. At the
full quantum level however this condition is not sufficient. The same property
can be accounted for using a BRST-type symmetry introducing additional ghost-
like degrees of freedom Haehl:2015foa ; Crossley:2015evo ; Haehl:2015uoc ;
Haehl:2016pec ; Haehl:2016uah , see also refs. Haehl:2015uoc ; Haehl:2016pec ;
Haehl:2016uah ; Jensen:2017kzi ; Gao:2017bqf ; Haehl:2018lcu ; Jensen:2018hse
for a supersymmetric implementation of this symmetry333A second BRST charge
arises when imposing the KMS symmetry (14), see refs. Crossley:2015evo ;
Jensen:2017kzi ; Gao:2017bqf ; Jensen:2018hse . The authors of refs.
Haehl:2015foa ; Haehl:2015uoc ; Haehl:2016pec ; Haehl:2016uah  have a more
general structure of an ${\cal N}=2$ equivariant cohomology.. While such a
construction is convenient as an organizational principle, it has been shown
in ref. Gao:2018bxz  that ghost degrees of freedom effectively decouple to all
loop orders in the effective action and, for the purposes of this work, we
will neglect them completely.

Another set of symmetries of the Schwinger-Keldysh effective action comes from
the doubled structure of the path integral (2): every continuous symmetry of
the microscopic action is doubled. In our case, this amounts to have a doubled
$U(1)$ flavor symmetry which can be implemented at the level of the Schwinger-
Keldysh effective action by allowing the low-energy degrees of freedom and the
external sources to appear only in the invariant pullback combinations (7),

$S_{eff}=S_{eff}[B_{r\,i},B_{a\,i}].$ (16)

The effective action (32) not only needs to be invariant under the doubled
symmetries of the $R/L$ target spaces but also under the $U(1)$ symmetry
implied on the common worldvolume spacetime by the transformation properties
of the bifundamental fields (9). This transformation is however restricted to
preserve the properties of the initial state. For example, in the coordinates
where the chemical potential is given by the time component of the average
pullback source $\mu=B_{r\,\bar{t}}$, only time-independent gauge
transformations with parameter $\Lambda(\vec{x})$ are allowed to keep $\mu$
invariant. Thus, under the “gauge-fixed” worldvolume $U(1)$ gauge
transformations, the pullback sources transform as444 The transformations (17)
are given in the so-called static gauge. See, e.g., refs. Haehl:2015pja ;
Jensen:2017kzi  for a full covariant implementation of the worldvolume gauge
invariance.

$B_{r\,i}\rightarrow B_{r\,i}+\partial_{i}\Lambda(\vec{x})\,,\qquad
B_{a\,i}\rightarrow B_{a\,i}\,,$ (17)

and the effective action $S_{eff}$ in (32) must be invariant under these
transformations.

Having introduced all the symmetries, a local low-energy Schwinger-Keldysh
effective action can be written in a hydrodynamic expansion

$S_{eff}=S^{(0)}_{eff}+S^{(1)}_{eff}+\dots$ (18)

where the superscript counts the amount of derivatives. The superficial555One
of the most interesting recent lessons about hydrodynamics is that
constitutive relations for the energy-momentum tensor truncated at low orders
of the derivative expansion can work remarkably well even if the relevant
gradient terms are of the order of the perfect fluid contributions. The naive
criterion (19) is then strongly violated. This, however, does not lead to a
contradiction since it has been explicitly shown in several cases that the
gradient expansion of the energy-momentum tensor is a divergent series for
which convergence conditions appropriate for series with a finite radius of
convergence do not apply. See ref. Florkowski:2017olj  for a broad overview of
the relevant developments. dimensionless expansion parameter is

$\frac{l_{mfp}}{l}\ll 1,$ (19)

where $l$ is the size of the gradients and, in weakly-coupled theories,
$l_{mfp}$ is the typical scale for consequent collisions of the microscopic
particles. For conformal field theories, $l_{mfp}\sim\hbar/T$ with the
proportionality constant in general depending on the interaction
strength666For holographic theories, for which the coupling constant is
infinite, the proportionality constant turns out to be just some finite
number., see, e.g., ref. Rangamani:2009xk .

To count derivatives, we choose $B_{r}\sim{\cal O}(1)$ and $B_{a}\sim{\cal
O}(\partial)$, since only in this way terms related by the KMS symmetry (14)
appear on the same footing. This can be seen by realizing that
$A_{L}(t-i\beta)\sim A_{L}(t)-i\,\beta\,\partial_{t}A_{L}(t)+{\cal
O}(\partial^{2})$ and that in the same limit the average and difference
combinations transform as $A_{r}\rightarrow A_{r}$ and $A_{a}\rightarrow
A_{a}+i\beta\partial_{t}A_{r}$ under the KMS symmetry (14). Thus, for example,
a term in the effective action of the form $B_{a}\,\partial B_{r}$ is related
to a term proportional to $B_{a}B_{a}$.

Finally, the physical equation of motion for the dynamical field $\phi_{r}$ is
associated to the variation of the effective action with respect to the field
$\phi_{a}$777There is an equivalent equation when varying the effective action
with respect to the $\phi_{r}$ fields. In the absence of difference sources
$A_{a}=0$ it is solved by setting $\phi_{a}=0$. and it can be recast into a
conservation equation

$\frac{\delta
S_{eff}}{\delta\phi_{a}}\bigg{|}_{B_{a}=0}=-\partial_{i}J^{i}=0\,,$ (20)

where the charge current is defined as

$J^{i}=\frac{\delta S_{eff}}{\delta B_{a\,i}}\bigg{|}_{B_{a}=0}\,.$ (21)

In this way it is possible to extract the off-shell ($\phi_{r}$-dependent,
i.e. obtained without solving the equations of motion for $\phi_{r}$)
constitutive relations for the charge current $J^{i}$ from an effective action
$S_{eff}$ order by order in a derivative expansion. Having $J^{i}$, the
constitutive relations in the target space(s) can be simply obtained via a
pushforward

$J^{\mu}=\partial_{i}x^{\mu}J^{i}\,.$ (22)

Given that we will be almost exclusively interested in quantities defined on
the target spaces and that the pullback maps for the $U(1)$ flavor field in
the probe limit are rather simple, in what follows we will construct the
effective action directly in the target space(s).

Combining all the ingredients together, the most general local Schwinger-
Keldysh effective action describing the dynamics of a conserved charge current
can be constructed using $B_{r}$ and $B_{a}$ fields satisfying the conditions
(12), (13), (14) and (17). To third order in a derivative expansion,888 Note
that we assigned $B_{a}\sim{\cal O}(\partial)$ and since the effective action
is at least linear in $B_{a}$, keeping terms up to third order in derivatives
in the effective action amounts to having the constitutive relations up to
second order in derivatives. to quadratic order in the fields and restricting
to the longitudinal sector where the wave vector points along the $x$
direction $\vec{k}=(k,0,\dots)$ and the only non-vanishing space component of
$B_{i}$ is $B_{x}$, the most general effective action (see Appendix A for
details) is given by

$\displaystyle\begin{split}S_{eff}&=\int
d^{d}x\Big{(}\chi\,B_{r\,t}B_{a\,t}-\sigma\,B_{a\,x}\partial_{t}B_{r\,x}+i\,\sigma\,T\,B_{a\,x}B_{a\,x}-s\,B_{a\,t}\partial_{t}B_{r\,t}+i\,s\,T\,B_{a\,t}B_{a\,t}\\\
&\qquad+k_{0}B_{a\,t}\partial_{t}^{2}B_{r\,t}+k_{1}\,B_{a\,t}\partial_{x}^{2}B_{r\,t}+k_{2}\,(\partial_{x}B_{a\,t}\partial_{t}B_{r\,x}+\partial_{t}B_{a\,x}\partial_{x}B_{r\,t})+k_{3}\,B_{a\,x}\partial^{2}_{t}B_{r\,x}\Big{)}\,.\end{split}$
(23)

Here $\chi,\,\sigma,\,\dots$ are generic functions of the temperature $T$
which would need to be determined from the microscopic theory. Notice that the
effective action (23) is linear in $B_{a}$ such that the Schwinger-Keldysh
symmetry (12) is satisfied at least at tree level and the quadratic term in
the $B_{a}$ fields is imaginary so that the reality condition (13) holds.
Moreover, the second and the third term (as well as the fourth and the fifth,
and the one proportional to $k_{2}$) in (23) are not independent since they
are related to one another by the KMS symmetry (14). Finally, for (17) to be
satisfied, the spatial components of $B_{r}$ fields only appear with time
derivatives $\partial_{t}B_{r\,x}$ .

Finally, integrating out the dynamical degrees of freedom $\phi_{a}$ and
$\phi_{r}$ in eq. (23) leads to the tree level generating functional $W=-i\ln
Z$ from which one can extract the two-point functions (4). The diffusive pole
can be computed from the retarded correlator and it is given by

$\omega=-iDk^{2}-iDk^{4}\left(k_{1}-k_{2}+k_{3}-sD\right)/\chi\,,$ (24)

where we can readily interpret

$D=\frac{\sigma}{\chi}$ (25)

as the diffusion constant, $\sigma$ as the conductivity and $\chi$ as the
susceptibility. Note that it is a linear combination of several coefficients
from the action that contributes to the second order (quartic in momentum)
correction to the dispersion relation (24). To close this part of the
analysis, let us also comment that eq. (15) constrains the first order
coefficients to be non-negative

$\sigma\geq 0\,,\qquad s\geq 0\,,$ (26)

see, e.g., refs. Glorioso:2016gsa ; Jensen:2017kzi .

##  3 Towards the holographic Schwinger-Keldysh effective action

In this section we build a gravitational parallel with the various field
theory notions introduced so far. The main goal is to give a prescription for
constructing and deriving the holographic dual of the low-energy Schwinger-
Keldysh effective action $S_{eff}$ for strongly-coupled conformal field
theories at finite temperature and a large rank of the gauge group $N$ (large
central charge).

###  3.1 The mixed signature bulk spacetime

A simple prescription to compute the Schwinger-Keldysh partition function (2)
in holography appeared first in ref. Herzog:2002pc  where the double-time
contour of the dual field theory was mimicked in the gravity side by an
eternal black hole in AdS. The key ingredient of this construction is the
presence of an independent dual conformal field theory at each of the (left
and right) boundaries, see, e.g., refs. 1976PhLA…57..107I ;
Balasubramanian:1998de ; Horowitz:1998xk ; CarneirodaCunha:2001jf ;
Maldacena:2001kr . In this way, a map between the field theory Schwinger-
Keldysh path integral (2) and the gravitational one arises quite naturally by
associating the two boundary field theories to the doubled structure in eq.
(2).

The prescription of ref. Herzog:2002pc  was subsequently refined in refs.
Skenderis:2008dh ; Skenderis:2008dg  which provided a systematic approach to
real-time holography, previous works in this context include also refs.
Balasubramanian:1998sn ; Son:2002sd ; Satoh:2002bc ; Kraus:2002iv ;
Marolf:2004fy ; Lawrence:2006ze . For example, while ref. Herzog:2002pc
relied on certain natural boundary conditions to be imposed in the interior of
the spacetime, the authors of refs. Skenderis:2008dh ; Skenderis:2008dg
adopted a fully holographic point of view where bulk dynamics depends only on
boundary data and at most on regularity conditions in the interior. Moreover,
the procedure outlined in refs. Skenderis:2008dh ; Skenderis:2008dg  is not
restricted to Schwinger-Keldysh contours but encompasses multiple forward and
backward evolutions starting from a general, not necessarily thermal, initial
state. This powerful procedure can be summarized as follows. First, to each
real-time segment of the contour associate a Lorentzian bulk spacetime and to
each imaginary time segment assign a Euclidean one. All parts of the spacetime
should then be glued smoothly together along fixed time slices and any other
field living in such mixed signature spacetime must also be smooth across the
separation surfaces. Notice that the resulting mixed signature bulk spacetimes
are not exotic and that similar constructions have been considered previously
in the context of cosmology, see ref. PhysRevD.28.2960  and a discussion of
several related developments in footnote 10.

(a)  (b)

Figure 2: On the left-hand side (a): an eternal AdS black hole where the
arrows indicate the flow of time. We cut along the an initial time slice at
$t=0$ and a late time slice at $t=\hat{t}$ (dashed lines), and we keep the
region in between them. On the right-hand side (b): a Euclidean black hole in
AdS with period $\beta$. We cut along finite time slices at $\tau=0$ and
$\tau=\beta-\sigma$ (dashed lines). For simplicity we have depicted the
spacetimes in $2+1$ dimensions in the global time and radial coordinate.

The bulk gravity solution dual to the field theory Schwinger-Keldysh contour
of Fig. 1 for a thermal state can be realized as follows. The real-time field
theory segments are given by the boundaries of a Lorentzian eternal black hole
in AdS cut along some finite time slices999We could as well have chosen to
place the initial state at some time in the infinite past $t=-\infty$. This
would correspond in the field theory side to the contour in Fig. 1 extended
all the way to $t=-\infty$.  between, say, $t=0$ and $t=\hat{t}$. The two
imaginary time segments in Fig. 1 are given by the boundary of the
analytically-continued101010For simple spacetimes, as the one considered in
this work, the analytic continuation to Euclidean time gives real spacetime
metrics. The prescription necessitates, however, complex spacetimes when
considering, say, rotating black holes, see ref. Skenderis:2008dg . Recent
works Feldbrugge:2017kzv ; Feldbrugge:2017fcc , building on Vilenkin:1982de ;
Vilenkin:1983xq ; Hartle:1983ai  revived interests in constructing complex
spacetimes also outside the realm of holography. Of course, the challenge is
to find fully non-linear solutions of Einstein equations with matter without
too restrictive symmetry assumptions on the boundary sources. One simple way
to circumnavigate this demanding problem is by constructing the bulk by gluing
pieces of time-translationally-invariant geometries and only in this
analytically-given background explicitly solve for the actual dynamics of
probe fields (gauge fields, metric perturbations, etc). Euclidean AdS black
hole with period $\beta$. Cutting along two finite time slices, at $\tau=0$
and $\tau=\beta-\sigma$ as depicted in Fig. 2, we glue $t_{L}=0$ to
$\tau_{0}=0$ and $\tau_{0}=\beta-\sigma$ to $t_{R}=0$. Then we glue
$t_{R}=\hat{t}$ to $\tau_{1}=0$ and $t_{L}=\hat{t}$ to $\tau_{1}=\sigma$. By
doing so, we are ensuring that the total periodicity of the Euclidean time is
$\beta$. The Schwinger-Keldysh contour of Fig. 1 is then realized by sending
$\hat{t}\rightarrow+\infty$ such that the late time slice corresponds to the
future horizons. Moreover, we can also choose to set $\sigma=0$ so that the
future horizons in the right and left region of the eternal black hole are
effectively identified with one another. In this way, the bulk spacetime
precisely realizes the contour in Fig. 1 where the two real time segments are
taken to be lying on top of each other. The resulting mixed signature bulk
spacetime is summarized in Fig. 3. Two Lorentzian geometries, ${\cal M}_{R}$
and ${\cal M}_{L}$, are glued together along $t=+\infty$ and to a Euclidean
cap ${\cal M}_{0}$ along $t=0$. The gluing is smooth by construction since the
metric and the extrinsic curvature are identified across these finite time
slices. A bulk spacetime of this sort appeared before in ref. vanRees:2009rw .
What is interesting is that the black hole interior is not included in this
geometrization of the thermal Schwinger-Keldysh profile in the probe
approximation and will play no role in the calculation we are about to
perform.

Figure 3: The mixed signature bulk spacetime containing two Lorentzian
regions, ${\cal M}_{R}$ and ${\cal M}_{L}$, with future horizons identified
and one Euclidean black hole ${\cal M}_{E}$ glued smoothly together along
finite time slices. Notice that in order to simplify the drawing we flipped
the left segment with respect to the way it appears in fig. 2, so that the
left time increases now upwards.

A different choice of $\sigma$ would lead to another parameterization of the
Schwinger-Keldysh path integral and the resulting correlators. For example,
$\sigma=\beta/2$ would correspond to the thermofield double partition function

$Z[A_{1},A_{2}]=\text{Tr}\left({\cal U}[A_{1}]\rho^{\beta/2}{\cal
U}[A_{2}]\rho^{\beta/2}\right)$ (27)

where $\rho^{\beta/2}=e^{-\frac{1}{2}\beta H}$. A parameterization of this
type was effectively considered in ref. Herzog:2002pc  and more recently in
ref. Botta-Cantcheff:2018brv  precisely to build the holographic dual of eq.
(27) through the real-time holography techniques of refs. Skenderis:2008dh ;
Skenderis:2008dg .

###  3.2 The degrees of freedom

Now that we have build the mixed signature bulk spacetime necessary to derive
the holographic dual of the Schwinger-Keldysh effective action, we need to
identify the bulk dual of the relevant low-energy degrees of freedom
responsible for hydrodynamic phenomena.

In the simple case of charge diffusion in a fixed thermal background, the
field theory degrees of freedom are two scalar fields $\phi_{R}$ and
$\phi_{L}$ as previously discussed around eq. (7). The holographic dual of one
of these quantities was first introduced in ref. Nickel:2010pr  as a Wilson
line of a probe bulk gauge field $A_{M}$ and subsequently covariantly
generalized in ref. deBoer:2015ija  to correspond to spacelike geodesics
extending between two boundaries of the bulk spacetime

$\phi=\int_{\gamma(\lambda_{1})}^{\gamma(\lambda_{2})}A_{M}V^{M}d\lambda\,.$
(28)

Here $V^{M}=dx^{M}/d\lambda$ is the tangent vector to spacelike geodesics
$\gamma$ with affine parameter $\lambda$ extending between two timelike
boundaries at $(x^{\mu},\gamma(\lambda_{1}))$ and
$(x^{\mu},\gamma(\lambda_{2}))$, while $M$ labels coordinates in the bulk. In
particular, the boundaries are taken to be the conformal boundary at infinity
(which can be interpreted in light of the discussion in sec. 2 as one of the
target spaces) and an intermediate timelike slice at finite radius, very close
to the horizon of a black brane in AdS (interpreted as the worldvolume).

Here we are going to consider two of these Wilson lines precisely to retain
the doubling structure of the fields and sources of the Schwinger-Keldysh
effective action. Starting from the interior of the spacetime close to the
bifurcation horizon we extend a Wilson line towards each of the left and right
conformal boundaries of the Lorentzian regions ${\cal M}_{L}$ and ${\cal
M}_{R}$ in Fig. 3. These Wilson lines will correspond to the low-energy
degrees of freedom $\phi_{R}$ and $\phi_{L}$ and in fact we will use the same
symbols when dealing with these objects.

Let us elaborate on why eq. (28) is a valid choice for the low-energy degree
of freedom describing charge diffusion. Consider bulk gauge transformations
$\delta_{\Lambda}A_{M}=\partial_{M}\Lambda$ with parameter $\Lambda$. The
transformations which are equal at the two boundaries with
$\Lambda(\lambda_{1})=\Lambda(\lambda_{2})$ leave $\phi$ in (28)
invariant111111Given the freedom in redefining the gauge parameter by a
constant, we can equivalently set the values of these gauge transformations at
the boundary to vanish.. Under a generic bulk gauge transformations with
$\Lambda(\lambda_{1})\neq\Lambda(\lambda_{2})$, the field $\phi$ transforms
instead as a bifundamental field with respect to what can be interpreted as
independent gauge transformations on the two boundaries

$\phi\rightarrow\phi+\Lambda(\lambda_{2})-\Lambda(\lambda_{1})\,,$ (29)

similarly to what happens in eq. (9). This property is almost all we want. We
still need the field $\phi$ to transform under a restricted version of the
worldvolume gauge transformations as discussed around eq. (17). This can be
achieved when one of the boundaries is pushed towards the horizon. Here, the
time component of the gauge field needs to be set to zero given that otherwise
the gauge connection on the corresponding Euclidean cap would be singular,
see, e.g., ref. Hartnoll:2011fn . Thus, to avoid the time component of the
gauge field $A_{t}$ to acquire any non-zero value at the horizon, only time-
independent gauge transformations are allowed there121212More precisely, the
covariant requirement is $\oint A=0$ at the horizon. A pure gauge boundary
condition $A_{M}dx^{M}=dx^{M}\partial_{M}f$ at the horizon with $f$ a generic
function would therefore satisfy the above constraint. In this way, any gauge
transformation is allowed at the horizon as it would preserve the covariant
condition above., precisely as in eq. (17).

These properties make it manifest that the aforementioned Wilson lines are
continuously connected to the Goldstone modes of spontaneously broken global
symmetries, see the original ref. Nickel:2010pr  and subsequent developments
in refs. Faulkner:2010jy ; deBoer:2015ija ; Crossley:2015tka . The main idea
in these works is the following. It is well known that in holography global
symmetry transformations of a dual field theory are represented by gauge
transformations of the bulk fields that act in a desired way (i.e. as a
desired global transformation) at the boundary. However, if the bulk contains
two boundaries (here, as in refs. Nickel:2010pr ; Faulkner:2010jy ;
deBoer:2015ija ; Crossley:2015tka , the UV boundary and a surface right
outside the horizon), then bulk gauge transformations induce two global
symmetries – one for each boundary. However, the connectedness of the bulk
makes independent global transformations impossible without preserving the
solution, which is a clear manifestation of a spontaneous symmetry breaking.
In the present case, the relevant symmetry breaking pattern is $U(1)\times
U(1)$ into the diagonal $U(1)$, which gives rise to a single Goldstone boson
(28).

To make sure that the quantity in (28) is well defined, we require $\phi$ to
be invariant under infinitesimal variations of the bulk gauge field $\delta
A_{M}$. The latter can be separated into longitudinal and perpendicular
contributions to the tangent vector $V^{M}$,

$\delta A_{M}=V_{M}V^{N}\delta A_{N}+P_{M}^{N}\delta A_{N}\,,$ (30)

where $P_{MN}=g_{MN}-V_{M}V_{N}/V^{2}$ is the projector onto the orthogonal
directions, satisfying $P_{MN}V^{N}=0$. It is immediate to see that orthogonal
variations do not modify $\phi$ in eq. (28) while longitudinal ones do, unless
they are of a pure gauge form $V^{M}\delta A_{M}=V^{M}\partial_{M}h$ with $h$
a generic function. In what follows we will thus work with a constrained
variational principle where longitudinal variations of the bulk gauge field
are set to zero

$V^{M}\delta A_{M}=0\,.$ (31)

Given the action for a probe gauge field

$S=-\frac{1}{4g_{A}^{2}}\int d^{d+1}x\sqrt{-g}F_{MN}F^{MN}\,,$ (32)

where $g_{A}$ is the effective gauge coupling and $F_{MN}$ is the bulk
electromagnetic field strength, the resulting equations of motion can also be
separated into longitudinal and perpendicular contributions,

$\delta S=\frac{\delta S}{\delta A_{M}}\delta
A_{M}\sim\left(V_{M}\nabla_{N}F^{NM}\right)V^{Q}\delta
A_{Q}+\left(P^{Q}_{M}\nabla_{N}F^{NM}\right)\delta A_{Q}\,.$ (33)

In practice, to implement the constrained variational principle and to have a
well defined dynamical field $\phi$ in eq. (28), we shall not solve the
equation of motion along the tangent vector $V_{M}\nabla_{N}F^{NM}=0$ which is
dubbed as the constraint equation.

A similar analysis to the one considered in this section goes through for the
case of linear fluid dynamic equations, thus, holographically, for bulk metric
perturbations, see, e.g., refs. deBoer:2015ija ; Crossley:2015tka  for a
definition of the corresponding low-energy fluid degrees of freedom in
holography. This setup, however, lies outside of the scope of the present
work.

##  4 The effective action for diffusion from holography

In this section we are going to consider in more detail the example of a
conserved charge current in (1+3)-dimensional strongly coupled conformal field
theories at large $N$. Through holography, this system corresponds on the
gravity side to the low-energy dynamics of a probe $U(1)$ gauge field in a
fixed spacetime at finite temperature given by an AdS5 black brane metric.

In Lorentzian signature the background metric can be written using
Schwarzschild-like coordinates

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle\frac{(\pi\,TL)^{2}}{u}\left(-f(u)\,dt^{2}+dx^{2}+dy^{2}+dz^{2}\right)+\frac{L^{2}du^{2}}{4\,u^{2}f(u)},$
(34)

where

$\displaystyle f(u)=1-u^{2}$ (35)

is the emblackening factor, $T$ is the Hawking temperature and $L$ is the AdS
curvature radius. The radial coordinate $u$ stretches between the bifurcation
surface at $u=1$ and the conformal boundary located at $u=0$. The future
horizon corresponds to $u=1$ and $t\rightarrow+\infty$, see, e.g., ref.
Bhattacharyya:2008mz . The Euclidean black brane metric can be obtained from
eq. (34) by a Wick rotation of the time coordinate $t=-i\,\tau$,

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle\frac{(\pi\,TL)^{2}}{u}\left(f(u)\,d\tau^{2}+dx^{2}+dy^{2}+dz^{2}\right)+\frac{L^{2}du^{2}}{4\,u^{2}f(u)}\,.$
(36)

As described in the previous section we will make use of both of these
metrics. Two Lorentzian spacetimes of the form (34), corresponding to the left
and right parts of the eternal black hole ${\cal{M}}_{L}$ and ${\cal{M}}_{R}$,
are glued to the Euclidean spacetime (36) parameterizing the Euclidean cap
${\cal{M}}_{E}$, see Fig. 3. To do so, one needs to ensure that the continuity
conditions are satisfied across the gluing surfaces. This is easily achieved
in our fixed background since the gluings are performed along fixed time
slices, where the induced metric and the extrinsic curvature of the metrics
(36) and (34) coincide. Notice that in building the mixed signature background
there are three sets of coordinates which we will parameterize with subscripts
$R$, $L$ and $E$ when referring to ${\cal M}_{R}$, ${\cal M}_{L}$ and ${\cal
M}_{E}$ respectively. However, to avoid cluttering of notation, we will also
often omit these subscripts when their meaning is clear from the context.

The action of the probe $U(1)$ gauge field $A_{M}$ in Lorentzian signature is
given by eq. (32) with $d=1+3$. It is convenient to work in Fourier space
where

$A_{M}(t,x,u)=\int\frac{d\omega\,d\vec{k}}{(2\pi)^{4}}\,e^{-i\,\omega\,t+i\,k\,x}A_{M}(u)\,,$
(37)

and, without loss of generality, we have chosen to align the wave vector along
the $x$ direction $\vec{k}=(k,0,0)$. In the following we will suppress, as was
the case in eq. (37), the dependence of functions on frequency and momentum,
when it follows from the context. The components of the gauge vector $A_{M}$
can be organized according to their transformation properties with respect to
the residual symmetry group $O(2)$ of rotations in the plane transverse to
momentum as follows

longitudinal channel: $\displaystyle\qquad A_{t},\quad A_{x},\quad A_{u}\,,$
(38a) transverse channel: $\displaystyle\qquad
A_{\alpha}\qquad\text{with}\qquad\alpha=y,z\,.$ (38b)

As a result, the equations of motion,

$\nabla_{M}F^{MN}=0,$ (39)

separate between the two channels. In the reminder of this work we will
consider the longitudinal channel (38a) as it is the only one which exhibits a
non-trivial hydrodynamic behavior.

We find it convenient to re-parameterize the fields as follows

$A_{u}(u)=-\phi^{\prime}(u)\,,\qquad
B_{t}(u)=A_{t}(u)-i\,\omega\,\phi(u)\,,\qquad
B_{x}(u)=A_{x}(u)+i\,k\,\phi(u)\,,$ (40)

where ′ denotes the derivative with respect to the radial coordinate $u$. The
field $\phi$ is a non-local Wilson line

$\phi(u)=-\int^{u}_{\delta}A_{u}(\tilde{u})\,d\tilde{u}$ (41)

extending along geodesics at fixed $(t,\vec{x})$ between two hypersurfaces
located at fixed radii $\delta$ and $u$. When these boundaries are stretched
between the horizon $\delta=1$ and the conformal boundary $u=0$, as it has
been anticipated in the previous section around eq. (28), the field (41) can
be interpreted as the holographic dual of the low-energy degree of freedom
capturing the long-lived excitations of charged matter of the dual field
theory side, see ref. Nickel:2010pr  and subsequent refs. deBoer:2015ija ;
Crossley:2015tka . The only difference between eq. (41) and the field
appearing in the definition (7) is that the former is defined in Schwarzschild
coordinates, thus boundary (target space) coordinates, while the latter is
defined on the worldvolume. We could work with combinations in (40) pulled
back to the horizon (i.e. the worldvolume) but we choose not to. Such
combinations would differ only by an overall multiplicative factor which will
be cancelled anyway given that we are interested in the physical quantities
defined on the target spaces via pushforwards, as in eq. (22).

Under bulk $U(1)$ gauge transformations $A_{M}\rightarrow
A_{M}+\partial_{M}\Lambda$, the field $\phi$ defined in eq. (41) transforms as
a bifundamental field under what can be interpreted as independent gauge
transformations on the two fixed radius hypersurfaces

$\phi(u)\rightarrow\phi(u)-\Lambda(u)+\Lambda(\delta)\,,$ (42)

similarly to eq. (9). The Wilson line (41) is clearly invariant under diagonal
combinations of the boundary gauge transformations, those for which
$\Lambda(u)=\Lambda(\delta)$, and transforms non trivially under the remaining
transformations for which $\Lambda(u)-\Lambda(\delta)\neq 0$. As discussed
shortly before, it is this reason why the field in eq. (41) is sometimes
referred to as the Goldstone mode of the broken $U(1)\times U(1)$ global
symmetry subgroup of the gauge invariances of the two boundaries down to the
diagonal combination thereof.

Finally, the equations of motion (39) for the newly redefined variables (40)
in the Lorentzian background (34) are given by

$\displaystyle
B_{t}^{\prime\prime}-\frac{\tilde{k}^{2}}{u\,f}B_{t}-\frac{\tilde{k}\,\tilde{\omega}}{u\,f}B_{x}=0\,,$
(43a) $\displaystyle
B_{x}^{\prime\prime}+\frac{f^{\prime}}{f}B_{x}^{\prime}+\frac{\tilde{\omega}^{2}}{u\,f^{2}}B_{x}+\frac{\tilde{k}\,\tilde{\omega}}{u\,f^{2}}B_{t}=0\,,$
(43b)
$\displaystyle\tilde{\omega}\,B_{t}^{\prime}+\tilde{k}\,f\,B_{x}^{\prime}=0\,,$
(43c)

where the dependence on $u$ is implied and we have introduced the rescaled
frequency and momentum

$\tilde{\omega}=\frac{\omega}{2\pi
T}\qquad\text{and}\qquad\tilde{k}=\frac{k}{2\pi T}\,.$ (44)

The first two equations (43a) and (43b) are the dynamical equations for
$B_{t}$ and $B_{x}$ while the last equation (43c) is a constraint, i.e. the
field equation projected along the geodesic on which the Wilson line (41) is
defined. Our primary goal is to solve the dynamical equations while leaving
the constraint unsolved. This will allow us to compute a partially on-shell
bulk action which will be interpreted as the local effective action for the
low-energy dynamics of charged matter in the dual field theory.

###  4.1 The bulk piecewise solution

Let us now derive the bulk solution by solving the field equations in the
various parts of the spacetime and patching them together along the gluing
surfaces at constant time. Performing the near horizon limit first
$u\rightarrow 1$ and then the hydrodynamic limit

$\tilde{\omega},\,\tilde{k}\ll 1\,,$ (45)

the equations (43a) and (43b) admit a simple solution of the form

$B_{t}=a_{c}+a_{l}(1-u)\qquad\text{and}\qquad
B_{x}=a_{+}(1-u)^{+i\tilde{\omega}/2}+a_{-}(1-u)^{-i\tilde{\omega}/2}\,,$ (46)

where $a_{c}$, $a_{l}$ and $a_{\pm}$ are integration constants. The time
component of $B_{t}$ has a constant and linear behavior at the horizon, while
$B_{x}$ is a superposition of an ingoing and outgoing mode.

At higher orders in derivatives, the equations (43a) and (43b) are coupled and
it is plausible to work with the following ansatz

$\displaystyle
B_{t}=a_{c}\,m+a_{l}\,n+a_{+}g_{+}+a_{-}g_{-}\qquad\text{and}\qquad
B_{x}=a_{c}\,p+a_{l}\,q+a_{+}h_{+}+a_{-}\,h_{-}\,,\quad$ (47)

where $m=m(\tilde{\omega},\tilde{k},u)$,
$n=n(\tilde{\omega},\tilde{k},u),\dots$ are linearly dependent mode solutions.
Only 4 solutions are independent, in fact by inserting expressions (47) into
eq. (43a) we can immediately show that one can express $p$, $q$, $h_{\pm}$ in
terms of $m$, $n$, $g_{\pm}$. We require that near the horizon the solution
(47) reproduces the behavior seen in eq. (46), thus

$\displaystyle m=1+\lambda\,m^{(1)}+\dots\,,$ (48a) $\displaystyle
n=(1-u)\left(1+\lambda\,n^{(1)}+\dots\right)\,,$ (48b) $\displaystyle
g_{\pm}=\lambda\,g_{\pm}^{(1)}+\dots\,,$ (48c) $\displaystyle
p=\lambda\,p^{(1)}+\dots\,,$ (48d) $\displaystyle
q=(1-u)\left(\lambda\,q^{(1)}+\dots\right)\,,$ (48e) $\displaystyle
h_{\pm}=(1-u)^{\pm\tilde{\omega}/2}\left(1+\lambda\,h_{\pm}^{(1)}+\dots\right)\,,$
(48f)

where $\lambda$ is a bookkeeping parameter counting the number of derivatives
and dots denote higher order terms in the hydrodynamic expansion. Thus, the
general bulk solution in the various parts of the spacetime can be
parameterized by 12 independent coefficients, 8 in the Lorentzian parts given
by $a^{R/L}_{c}$, $a_{l}^{R/L}$ and $a_{\pm}^{R/L}$,

$\displaystyle\begin{split}&B_{s\,t}(t_{s},x,u)=\int\frac{d\omega\,d\vec{k}}{(2\pi)^{4}}e^{-i\,\omega\,t_{s}+i\,k\,x}\left(a^{s}_{c}\,m+a^{s}_{l}\,n+a^{s}_{+}g_{+}+a^{s}_{-}g_{-}\right)\,,\\\
&B_{s\,x}(t_{s},x,u)=\int\frac{d\omega\,d\vec{k}}{(2\pi)^{4}}e^{-i\,\omega\,t_{s}+i\,k\,x}\left(a^{s}_{c}\,p+a^{s}_{l}\,q+a^{s}_{+}h_{+}+a^{s}_{-}\,h_{-}\right)\,,\end{split}$
(49)

where $s=R,\,L$ and 4 more in the Euclidean manifold given by $a_{c}^{E}$,
$a_{l}^{E}$ and $a_{\pm}^{E}$,

$\displaystyle\begin{split}&B_{E\,\tau}(\tau,x,u)=\int\frac{d\omega\,d\vec{k}}{(2\pi)^{4}}e^{-\omega\,\tau+i\,k\,x}\left(a^{E}_{c}\,m+a^{E}_{l}\,n+a^{E}_{+}g_{+}+a^{E}_{-}g_{-}\right)\,,\\\
&B_{E\,x}(\tau,x,u)=\int\frac{d\omega\,d\vec{k}}{(2\pi)^{4}}e^{-\omega\,\tau+i\,k\,x}\left(a^{E}_{c}\,p+a^{E}_{l}\,q+a^{E}_{+}h_{+}+a^{E}_{-}h_{-}\right)\,.\end{split}$
(50)

We now impose Dirichlet boundary conditions on the conformal boundaries of the
two Lorentzian parts of the spacetime and trivial (i.e. vanishing) boundary
conditions on the Euclidean cap,

$\displaystyle\begin{split}&B_{s\,t}(t_{s},x,u=0)=B_{s\,t}(t_{s},x)\,,\qquad
B_{s\,x}(t_{s},x,u=0)=B_{s\,x}(t_{s},x)\,,\\\
&B_{E\,\tau}(\tau,x,u=0)=0\,,\qquad B_{E\,x}(\tau,x,u=0)=0\,\end{split}$ (51)

where, again, $s=R,\,L$. We refrained from using another symbol for the
sources, the distinction between the bulk gauge field and its boundary value
should be clear from the context. Non-vanishing non-normalizable modes in the
Euclidean part of the spacetime can be interpreted as considering excited
initial states, see, e.g., ref. Botta-Cantcheff:2015sav ;
Christodoulou:2016nej . Here, instead, we have restricted ourselves to the
case where the initial state is simply thermal. The conditions
(LABEL:eq:Dirichlet) imply relations among the integration constants

$\displaystyle\begin{split}&a_{c}^{s}m^{0}+a_{l}^{s}n^{0}+a^{s}_{+}g_{+}^{0}+a^{s}_{-}g^{0}=B_{s\,t}\,,\qquad
a_{c}^{s}p^{0}+a_{l}^{s}q^{0}+a^{s}_{+}h_{+}^{0}+a^{s}_{-}h^{0}=B_{s\,x}\,,\\\
&a_{c}^{E}m^{0}+a_{l}^{E}n^{0}+a^{E}_{+}g_{+}^{0}+a^{E}_{-}g^{0}=0\,,\qquad
a_{c}^{E}p^{0}+a_{l}^{E}q^{0}+a^{E}_{+}h_{+}^{0}+a^{E}_{-}h^{0}=0\,,\end{split}$
(52)

and we have introduced the shortcut notation where, for example,
$n^{0}=n(\tilde{\omega},\tilde{k},u=0)$. Using these expressions it is
possible to fix 6 of the integrations constants, say $a_{l}^{R}$, $a_{l}^{L}$,
$a_{l}^{E}$, $a_{-}^{R}$, $a_{-}^{L}$ and $a_{-}^{E}$, in terms of the other
ones.

Another condition on the coefficients arises requiring the Euclidean cap not
to have a singular gauge connection at the horizon, see, e.g., ref.
Hartnoll:2011fn . This can be achieved by demanding that the time component of
the gauge field vanishes at the horizon as we have discussed in the previous
section. We thus impose

$a_{c}^{E}=0\,.$ (53)

Continuity conditions across the gluing surfaces imply that the same
constraints should hold in the Lorentzian parts of the bulk spacetime given
that these modes all meet at the bifurcation point,

$a_{c}^{R}=0\,,\qquad a_{c}^{L}=0\,.$ (54)

The next step in constructing the full solution in our spacetime depicted in
Fig. 3 is to impose continuity of the gauge field along the surfaces at which
the Lorentzian parts of the spacetime are glued to the Euclidean black hole.
These conditions are simply given by

$\displaystyle\begin{split}&B_{L\,t}(t_{L}=0,x,u)=B_{E\,\tau}(\tau=0,x,u)\,,\qquad
B_{R\,t}(t_{R}=0,x,u)=B_{E\,\tau}(\tau=\beta,x,u)\,,\\\
&B_{L\,x}(t_{L}=0,x,u)=B_{E\,x}(\tau=0,x,u)\,,\qquad
B_{R\,x}(t_{R}=0,x,u)=B_{E\,x}(\tau=\beta,x,u).\end{split}$ (55)

and imply the relations

$\displaystyle\begin{split}a_{+}^{E}=a_{+}^{L}\,,\qquad
a_{+}^{R}=e^{-\omega/T}a_{+}^{L}\,,\end{split}$ (56)

where $\beta=1/T$ is the period of the Euclidean cap. The requirement that
also the time derivative of the fields across the gluing surfaces is
continuous does not impose additional constraints.

To derive eq. (56) we used the fact that source insertions $B_{s\,t}$ and
$B_{s\,x}$ are vanishing at $t_{L}=0$ and $t_{R}=0$ to match the boundary
condition on the Euclidean cap and have support only at later times. After
imposing the relations (LABEL:eq:conddir), (53), (54) and (56), the
combinations in (LABEL:eq:Btmatch) depend only on the external sources, as
they should. For example, we have

$\displaystyle\begin{split}&B_{R\,x}(t_{L}=0,x,u)-B_{E\,x}(\tau=1/T,x,u)=\int\frac{d\omega\,d\vec{k}}{(2\pi)^{4}}e^{ikx}\left(\Delta_{-}^{xt}B_{R\,t}(\omega,k)+\Delta_{-}^{xx}B_{R\,x}(\omega,k)\right)=\\\
&\int
dt^{\prime}\,dx^{\prime}\left(B_{R\,t}(t^{\prime},x^{\prime})\int\frac{d\omega\,d\vec{k}}{(2\pi)^{4}}e^{+i\omega
t^{\prime}}e^{-ik(x^{\prime}-x)}\Delta_{-}^{xt}+B_{R\,x}(t^{\prime},x^{\prime})\int\frac{d\omega\,d\vec{k}}{(2\pi)^{4}}e^{+i\omega
t^{\prime}}e^{-ik(x^{\prime}-x)}\Delta_{-}^{xx}\right)\\\ \end{split}$ (57)

with $\Delta_{-}^{xt}$ and $\Delta_{-}^{xx}$ being functions of the modes
$m,h_{+}\dots$ which we will define shortly. The combination in eq.
(LABEL:eq:convergence) is vanishing for $t^{\prime}>0$ as long as
$\Delta_{-}^{xt}$ and $\Delta_{-}^{xx}$ have no poles in the upper half of the
complex $\omega$-plane where the integral is convergent. We have verified up
to second order in the derivative expansion that $\Delta_{-}^{xt}$ and
$\Delta_{-}^{xx}$ indeed satisfy the above requirement. It is also natural to
expect that this feature holds also at higher orders in derivatives given that
the propagators $\Delta_{-}$ have essentially an ingoing behavior near the
horizon, although we have not verified this statement explicitly.

Finally, let us impose a gluing condition on the late time surface $\hat{t}$
which is taken to $\hat{t}\rightarrow+\infty$ in such a way that it coincides
with the future horizons. The precise position of this late time hypersurface
is not so important as long as it is later than any source insertion. Thus, we
require

$B_{L\,t}(t_{L}=\hat{t},x,u)=B_{R\,t}(t_{R}=\hat{t},x,u)\,,\qquad
B_{L\,x}(t_{L}=\hat{t},x,u)=B_{R\,x}(t_{R}=\hat{t},x,u)\,,$ (58)

which give

$a_{+}^{L}=(1+\bar{n}(\omega))\frac{(B_{L\,x}-B_{R\,x})n^{0}-(B_{L\,t}-B_{R\,t})q^{0}}{h_{+}^{0}n^{0}-g_{+}^{0}q^{0}}\,,$
(59)

where we have defined the Boltzmann distribution

$\bar{n}(\omega)=\frac{1}{e^{\omega/T}-1}\,.$ (60)

In obtaining eq. (59) we used similar arguments as the ones appearing around
eq. (LABEL:eq:convergence). The late time surface is located at times
$\hat{t}$ bigger than any possible other time where the sources might have
been inserted making the domain of convergence of the integrals involved in
eq. (58) being the lower half of the complex $\omega$-plane. We will have to
make sure that after imposing the condition on the coefficients (59) the
argument of the integrals has poles at most in the upper half complex
$\omega$-plane.

Having fixed all the integration constants in terms of the boundary sources,
we have succeeded in finding a regular formal bulk solution interpolating
between the Lorentzian and Euclidean manifolds as a function of the sources
$B_{L\,t}$, $B_{R\,t}$, $B_{L\,x}$ and $B_{R\,x}$ inserted on the left and
right conformal boundaries. To summarize, the solution can be parameterized as
follows

$\displaystyle\begin{split}B_{s\,\mu}(t_{s},x,u)=&\int\frac{d\omega\,d\vec{k}}{(2\pi)^{4}}\,e^{-i\omega
t_{s}+ikx}\sum_{m=R,L}\sum_{\nu=t,x}\Delta^{\mu\nu}_{sm}(\tilde{\omega},\tilde{k},u)B_{m\,\nu}(\tilde{\omega},\tilde{k})\,,\\\
B_{E\,\mu}(\tau,x,u)=&\int\frac{d\omega\,d\vec{k}}{(2\pi)^{4}}\,e^{-\omega\tau+ikx}\sum_{m=R,L}\sum_{\nu=t,x}\Delta^{\mu\nu}_{Em}(\tilde{\omega},\tilde{k},u)B_{m\,\nu}(\tilde{\omega},\tilde{k})\,,\end{split}$
(61)

where $s=R,L$, $\mu=t,x$ (or $\mu=\tau,x$) and $\Delta_{sm}^{\mu\nu}$ and
$\Delta_{Em}^{\mu\nu}$ are the bulk to boundary propagators. For example,
$\Delta_{LL}^{tt}$ is the propagator which determines the bulk gauge field
$B_{L\,t}$ in ${\cal M}_{L}$ under the influence of the source $B_{L\,t}$
inserted at the conformal boundary of the same manifold. On the other hand,
$\Delta_{LR}^{tt}$ is the bulk to boundary propagator for the same bulk gauge
field $B_{L\,t}$ in ${\cal M}_{L}$, depending now on the source $B_{R\,t}$
inserted on the conformal boundary of ${\cal M}_{R}$ instead. The bulk to
boundary propagators are explicitly given by

$\displaystyle\begin{split}&\Delta_{LL}^{\mu\nu}=-\bar{n}(\omega)\Delta_{-}^{\mu\nu}+(1+\bar{n}(\omega))\Delta_{+}^{\mu\nu}\,,\qquad\Delta_{LR}^{\mu\nu}=(1+\bar{n}(\omega))\left(\Delta_{-}^{\mu\nu}-\Delta_{+}^{\mu\nu}\right)\,,\\\
&\Delta_{RL}^{\mu\nu}=-\bar{n}(\omega)\left(\Delta_{-}^{\mu\nu}-\Delta_{+}^{\mu\nu}\right)\,,\qquad\Delta_{RR}^{\mu\nu}=(1+\bar{n}(\omega))\Delta_{-}^{\mu\nu}-\bar{n}(\omega)\Delta_{+}^{\mu\nu}\,,\\\
&\Delta_{EL}^{\mu\nu}=-(1+\bar{n}(\omega))\left(\Delta_{-}^{ij}-\Delta_{+}^{\mu\nu}\right)\,,\qquad\Delta_{ER}^{\mu\nu}=(1+\bar{n}(\omega))\left(\Delta_{-}^{\mu\nu}-\Delta_{+}^{\mu\nu}\right)\,,\end{split}$
(62)

where we have defined the combinations

$\displaystyle\begin{split}&\Delta^{tt}_{\pm}=\frac{h_{\pm}^{0}n-q^{0}g_{\pm}}{h_{\pm}^{0}n^{0}-g_{\pm}^{0}q^{0}}\,,\qquad\Delta^{tx}_{\pm}=\frac{n^{0}g_{\pm}-g_{\pm}^{0}n}{h_{\pm}^{0}n^{0}-g_{\pm}^{0}q^{0}}\,,\\\
&\Delta^{xt}_{\pm}=\frac{h_{\pm}^{0}q-q^{0}h_{\pm}}{h_{\pm}^{0}n^{0}-g_{\pm}^{0}q^{0}}\,,\qquad\Delta^{xx}_{\pm}=\frac{n^{0}h_{\pm}-g_{\pm}^{0}q}{h_{\pm}^{0}n^{0}-g_{\pm}^{0}q^{0}}\,.\end{split}$
(63)

Before proceeding let us first observe a simple fact. When the external
sources are set to be equal to one another on the two sides of the Lorentzian
spacetime $B_{L\,x}=B_{R\,x}=B_{x}$ and $B_{L\,t}=B_{R\,t}=B_{t}$, the bulk
gauge field in ${\cal M}_{R}$ depends only on the combination
$\Delta^{\mu\nu}_{-}$,

$\displaystyle\begin{split}B_{R\,\mu}(t_{s},x,u)\bigg{|}_{B_{R}=B_{L}=B}=&\int\frac{d\omega\,d\vec{k}}{(2\pi)^{4}}\,e^{-i\omega
t_{R}+ikx}\sum_{\nu=t,x}(\Delta^{\mu\nu}_{RL}+\Delta^{\mu\nu}_{RR})B_{\nu}\\\
=&\int\frac{d\omega\,d\vec{k}}{(2\pi)^{4}}\,e^{-i\omega
t_{R}+ikx}\sum_{\nu=t,x}\Delta^{\mu\nu}_{-}B_{\nu}\,.\end{split}$ (64)

We will see in what follows that $\Delta^{\mu\nu}_{-}$ is related to ingoing
modes on the right part of the bulk spacetime ${\cal M}_{R}$. Thus, as it has
been previously observed in ref. vanRees:2009rw , selecting the sector with
equal sources on the boundary is equivalent to selecting ingoing mode
solutions and therefore a retarded propagation. This is not surprising and
reflects the well known prescription to compute retarded correlators in
holography by imposing ingoing boundary conditions at the horizon Son:2002sd .

Conversely, when the sources are set to be equal to one another up to a shift
by $\beta$ in imaginary time, $B_{R\,t}=e^{-\omega/T}B_{L\,t}={B}_{t}$ and
$B_{R\,x}=e^{-\omega/T}B_{L\,x}={B}_{x}$, we have

$B_{R\,{\mu}}(t_{R},x,u)\bigg{|}_{B_{R}=e^{-\omega/T}B_{L}={B}}=\int\frac{d\omega\,d\vec{k}}{(2\pi)^{4}}\,e^{-i\omega
t_{R}+ikx}\sum_{\nu=t,x}\Delta^{\mu\nu}_{+}B_{\nu}\,,$ (65)

where the combination $\Delta^{\mu\nu}_{+}$ will be related to the outgoing
mode in the bulk. Thus, this subsector is equivalent to selecting outgoing
mode solutions. We observe, therefore, a parallel between the two subsectors
for which the Schwinger-Keldysh path integral for thermal states becomes
trivial as discussed in sec. 2 and the distinction between ingoing and
outgoing modes in the bulk.

Let us conclude this section by commenting on a couple of important points.
Notice that we have been cavalier in performing Fourier transforms. In fact,
given that the sources have support only on a half time line between $t=0$ and
$\hat{t}\rightarrow\infty$, the appropriate transformation would have been
instead the Laplace transform. By working with Fourier transforms, we have
abandoned the idea of finding a unique solution. We will be content here with
determining $a$ solution and not to uniquely determine it. The reason is that
we could have as well extended the Lorentzian manifold from $t=0$ all the way
to $t=-\infty$ and worked with a spacetime where time runs from $t=-\infty$ to
$t=+\infty$. The initial state, i.e. the Euclidean cap, would have needed to
be glued to the slice $t=-\infty$. We have verified that the resulting
solution is equivalent to the one presented in the main text and that in this
case it is unique as there are no normalizable modes compatible with the
gluing conditions.

###  4.2 The holographic Schwinger-Keldysh effective action

Now that we have the formal bulk piecewise solution (61) at our disposal, we
are ready to derive the Schwinger-Keldysh effective action describing
diffusion of a $U(1)$ charge current in the dual conformal field theory in a
saddle point approximation and up to the quadratic order in the fields and
sources.

We first compute the bulk action for the Maxwell field (32) partially on-
shell, that is using only the dynamical equations of motion given by eqs.
(43a) and (43b). The resulting action is simply given by a term of the form

$\displaystyle S_{\rm
onshell}=\frac{\pi^{2}T^{2}L}{g_{A}^{2}}\int_{\partial{\cal
M}}\frac{d\omega\,d\vec{k}}{(2\pi)^{4}}\,\left(B_{t}(-\tilde{\omega},-\tilde{k},u)\,\partial_{u}B_{t}(\tilde{\omega},\tilde{k},u)-f(u)B_{x}(-\tilde{\omega},-\tilde{k},u)\,\partial_{u}B_{x}(\tilde{\omega},\tilde{k},u)\right),$

where $\partial{\cal M}$ is the boundary of the manifold ${\cal M}$. Secondly,
we evaluate the on-shell action on the boundary of the spacetime by taking the
following combination

$i\,S_{eff}=i\,S_{\rm onshell}\bigg{|}_{u_{R}=0}-i\,S_{\rm
onshell}\bigg{|}_{u_{L}=0}-S_{\rm onshell}\bigg{|}_{u_{E}=0}\,,$ (67)

where the minus sign in front of the contribution of the left-hand side of the
Lorentzian spacetime comes from the fact that we take time to run backwards in
${\cal M}_{L}$ to match the field theory contour in Fig. 1. Given that there
are no sources inserted at the conformal boundary of the Euclidean cap, the
on-shell action corresponding to the Euclidean region will be actually
vanishing.

Inserting the solution (61) we found in the previous Section into the
expression (67) leads to

$\displaystyle\begin{split}S_{eff}=&\int\frac{d\omega\,d\vec{k}}{(2\pi)^{4}}\sum_{\mu=t,x}\sum_{\nu=t,x}\bigg{(}B_{a\,\mu}(-\tilde{\omega},-\tilde{k})\Pi_{-}^{\mu\nu}(\tilde{\omega},\tilde{k})B_{r\,\nu}(\tilde{\omega},\tilde{k})+B_{r\,\mu}(-\tilde{\omega},-\tilde{k})\Pi_{+}^{\mu\nu}(\tilde{\omega},\tilde{k})B_{a\,\nu}(\tilde{\omega},\tilde{k})\\\
&\qquad+\frac{1}{2}B_{a\,\mu}(-\tilde{\omega},-\tilde{k})\coth(\pi\tilde{\omega})\left(\Pi_{-}^{\mu\nu}(\tilde{\omega},\tilde{k})-\Pi_{+}^{\mu\nu}(\tilde{\omega},\tilde{k})\right)B_{a\,\nu}(\tilde{\omega},\tilde{k})\bigg{)}\,,\end{split}$
(68)

where the summation only over $t$ and $x$ indices comes from restricting to
the longitudinal channel (the only one with the gapless excitations), see eq.
(38a). Note that given eq. (4.2), the terms mixing $t$ and $x$ come into play
from subleading behavior of $B$’s at the UV boundary $\partial{\cal M}$. In
eq. (68) we have also defined the average and difference combinations of the
pullback sources, see also eq. (7),

$B_{r}=\frac{1}{2}\left(B_{R}+B_{L}\right)\,,\qquad B_{a}=B_{R}-B_{L}\,,$ (69)

and, on top, we also defined the following quantities

$\Pi^{tt}_{\pm}=-G^{tt}_{\pm}\,,\qquad\Pi^{tx}_{\pm}=-G^{tx}_{\pm}\,,\qquad\Pi^{xt}_{\pm}=G^{xt}_{\pm}\,,\qquad\Pi^{xx}_{\pm}=G^{xx}_{\pm}\,,$
(70)

with

$G_{\pm}^{\mu\nu}=\frac{\pi^{2}T^{2}L}{g_{A}^{2}}\partial_{u}\Delta^{\mu\nu}_{\pm}\big{|}_{u=0}\,.$
(71)

Given the $\Delta^{\mu\nu}_{\pm}$’s defined in (LABEL:eq:deltas), we
explicitly have

$\displaystyle\begin{split}&G^{tt}_{\pm}=\frac{\pi^{2}T^{2}L}{g_{A}^{2}}\frac{h_{\pm}^{0}n^{\prime}-q^{0}g_{\pm}^{\prime}}{h_{\pm}^{0}n^{0}-g_{\pm}^{0}q^{0}}\bigg{|}_{u=0}\,,\qquad
G^{tx}_{\pm}=\frac{\pi^{2}T^{2}L}{g_{A}^{2}}\frac{n^{0}g_{\pm}^{\prime}-g_{\pm}^{0}n^{\prime}}{h_{\pm}^{0}n^{0}-g_{\pm}^{0}q^{0}}\bigg{|}_{u=0}\,,\\\
&G^{xt}_{\pm}=\frac{\pi^{2}T^{2}L}{g_{A}^{2}}\frac{h_{\pm}^{0}q^{\prime}-q^{0}h_{\pm}^{\prime}}{h_{\pm}^{0}n^{0}-g_{\pm}^{0}q^{0}}\bigg{|}_{u=0}\,,\qquad
G^{xx}_{\pm}=\frac{\pi^{2}T^{2}L}{g_{A}^{2}}\frac{n^{0}h_{\pm}^{\prime}-g_{\pm}^{0}q^{\prime}}{h_{\pm}^{0}n^{0}-g_{\pm}^{0}q^{0}}\bigg{|}_{u=0}\,,\end{split}$
(72)

where ′ denotes a derivative with respect to the radial coordinate $u$. Note
that the only dependence on the sources and gapless modes comes into eq. (68)
obviously through $B$’s and the functions (LABEL:eq:Gpm) depend only on
$\tilde{\omega}$, $\tilde{k}$ and otherwise, trivially, on $T$. One important
thing that we want to stress here is that the coefficients in front of various
$B$-terms in eq. (68) are not Green’s functions given by eq. (4), since the
Goldstone field has not been yet integrated out (i.e. put on-shell). In
particular, these coefficients, given by simple derivative expansions, do not
have visible singularities as opposed to Green’s functions, which we would
expect to have a diffusive pole.

Let us now integrate out the dynamical degrees of freedom by imposing their
equations of motion

$\frac{\delta S_{eff}}{\delta\phi_{r}}=0\qquad\text{and}\qquad\frac{\delta
S_{eff}}{\delta\phi_{a}}=0\,.$ (73)

As anticipated above, the holographic Schwinger-Keldysh effective action (68)
put on shell becomes the generating functional for connected correlators

$\displaystyle\begin{split}W=&\int\frac{d\omega\,d\vec{k}}{(2\pi)^{4}}\bigg{(}Z_{a}(-\tilde{\omega},-\tilde{k})\,\Pi_{-}(\tilde{\omega},\tilde{k})\,Z_{r}(\tilde{\omega},\tilde{k})+Z_{r}(-\tilde{\omega},-\tilde{k})\,\Pi_{+}(\tilde{\omega},\tilde{k})\,Z_{a}(\tilde{\omega},\tilde{k})\\\
&\qquad+\frac{1}{2}Z_{a}(-\tilde{\omega},-\tilde{k})\,\coth(\pi\tilde{\omega})\left(\Pi_{-}(\tilde{\omega},\tilde{k})-\Pi_{+}(\tilde{\omega},\tilde{k})\right)\,Z_{a}(\tilde{\omega},\tilde{k})\bigg{)}\,,\end{split}$
(74)

with

$\Pi_{\pm}=\frac{G_{\pm}^{tt}\,G_{\pm}^{xx}-G_{\pm}^{tx}\,G_{\pm}^{xt}}{\tilde{\omega}^{2}\,G_{\pm}^{tt}-\tilde{k}\,\tilde{\omega}\,\left(G_{\pm}^{tx}-G_{\pm}^{xt}\right)-\tilde{k}^{2}\,G_{\pm}^{xx}}\,,$
(75)

and we have defined the fully gauge-invariant variable for sources

$Z=-i\,\tilde{k}\,A_{t}-i\,\tilde{\omega}\,A_{x}\,.$ (76)

By varying (74) with respect to the external gauge invariant sources $Z$, the
formal expressions for the corresponding retarded, advanced and symmetric two-
point functions in momentum space are given by

$\displaystyle\begin{split}&G_{ret}(\tilde{\omega},\tilde{k})=i\left(\Pi_{-}(\tilde{\omega},\tilde{k})+\Pi_{+}(-\tilde{\omega},-\tilde{k})\right)\,,\\\
&G_{adv}(\tilde{\omega},\tilde{k})=i\left(\Pi_{+}(\tilde{\omega},\tilde{k})+\Pi_{-}(-\tilde{\omega},-\tilde{k})\right)\,,\\\
&G_{sym}(\tilde{\omega},\tilde{k})=\frac{1}{2}\coth(\pi\tilde{\omega})\left(\Pi_{-}(\tilde{\omega},\tilde{k})-\Pi_{+}(\tilde{\omega},\tilde{k})+\Pi_{+}(-\tilde{\omega},-\tilde{k})-\Pi_{-}(-\tilde{\omega},-\tilde{k})\right)\,,\\\
\end{split}$ (77)

c.f. eq. (4). It is straightforward to verify that the usual fluctuation-
dissipation theorem for two-point thermal correlators is satisfied

$i\,G_{sym}(\tilde{\omega},\tilde{k})=\frac{1}{2}\coth(\pi\tilde{\omega})\left(G_{ret}(\tilde{\omega},\tilde{k})-G_{adv}(\tilde{\omega},\tilde{k})\right)\,,$
(78)

and that

$G_{ret}(\tilde{\omega},\tilde{k})=G_{adv}(-\tilde{\omega},-\tilde{k})\,.$
(79)

The usual current-current two-point functions can now be easily recovered from
the expressions (LABEL:eq:greenfull) by multiplying with appropriate
$\tilde{\omega}$ and $\tilde{k}$ factors.

###  4.3 The Schwinger-Keldysh effective action up to second order

While the discussion in previous section was formal, it is certainly of
interest to have a look at the explicit form of the effective action up to
third order in derivatives which gives rise to constitutive relations at
second order131313Remember $B_{a}\sim{\partial}$.. Consider the modes (48a)
and their solutions. To first order in a hydrodynamic expansion, the solutions
take the form

$\displaystyle\begin{split}h_{\pm}=(1-u)^{\pm\frac{i\tilde{\omega}}{2}}\bigg{(}1\pm\frac{i\tilde{\omega}}{2}{\rm
ln}\left(\frac{2}{1+u}\right)\bigg{)}\,,\qquad n=1\,,\qquad g_{\pm}=0\,,\qquad
q=0\,.\end{split}$ (80)

We refer interested reader to appendix B for an analytic form of second order
formulae. Inserting these expressions into eqs. (LABEL:eq:Gpm) and (70) we
find

$\displaystyle\begin{split}&\Pi^{tt}_{\pm}=\frac{\pi^{2}T^{2}L}{g_{A}^{2}}\left(1-2\tilde{k}^{2}\ln(2)\right)\,,\qquad\Pi^{tx}_{\pm}=\Pi^{xt}_{\pm}=-\frac{\pi^{2}T^{2}L}{g_{A}^{2}}\tilde{k}\tilde{\omega}\ln(2)\,,\\\
&\Pi^{xx}_{\pm}=\frac{\pi^{2}T^{2}L}{g_{A}^{2}}\left(\pm
i\,\tilde{\omega}-\tilde{\omega}^{2}\ln(2)\right)\,,\end{split}$ (81)

which, as anticipated in the previous section, are just given by truncated
Taylor series in frequency and momentum, i.e. do not have any singularities at
the level of their expansions. Substituting these expressions into the
effective action (68) and taking the hydrodynamic limit to third order gives
eq. (112) from the appendix. Here we just quote its real space form, which is

$\displaystyle\begin{split}S_{eff}=&\frac{2\pi^{2}T^{2}L}{g_{A}^{2}}\int
d^{4}x\,\bigg{(}B_{r\,t}B_{a\,t}-\frac{1}{2\pi
T}B_{a\,x}\,\partial_{t}B_{r\,x}+\frac{i}{2\pi}B_{a\,x}B_{a\,x}\\\
&+\frac{\ln(2)}{2\pi^{2}T^{2}}B_{a\,t}\,\partial_{x}^{2}B_{r\,t}+\frac{\ln(2)}{4\pi^{2}T^{2}}\partial_{t}B_{a\,x}\,\partial_{x}B_{r\,t}+\frac{\ln(2)}{4\pi^{2}T^{2}}\partial_{x}B_{a\,t}\,\partial_{t}B_{r\,x}+\frac{\ln(2)}{4\pi^{2}T^{2}}B_{a\,x}\,\partial_{t}^{2}B_{r\,x}\bigg{)}\,.\end{split}$
(82)

Eq. (82) is our final result for the effective action for diffusion to third
order (second order constitutive relations) for holographic field theories and
to quadratic order in amplitudes. It obviously satisfies all the symmetries
presented in sec. 2 at tree level, i.e. the Schwinger-Keldysh symmetry (12),
the KMS symmetry (14), etc., as it should.

The constitutive relations that can be extracted from the effective action
(82) are

$\displaystyle J^{t}$
$\displaystyle=\frac{2\pi^{2}T^{2}L}{g_{A}^{2}}\mu+\frac{\ln(2)L}{g_{A}^{2}}\partial_{x}^{2}\mu+\frac{\ln(2)L}{2g_{A}^{2}}\partial_{x}V_{x}\,,$
(83) $\displaystyle J^{x}$ $\displaystyle=\frac{\pi
TL}{g_{A}^{2}}V_{x}-\frac{\ln(2)L}{2g_{A}^{2}}\partial_{t}E_{x}\,,$ (84)

where we have identified the chemical potential $\mu=B_{r\,t}$, the electric
field $E_{x}=\partial_{x}B_{r\,t}-\partial_{t}B_{r\,x}$ and the vector
$V_{x}=E_{x}-\partial_{x}\mu=-\partial_{t}B_{r\,x}$. Although it is
tantalizing to directly compare the resulting effective action and
constitutive relations with the general expressions at the end of sec. 2, one
should bear in mind that the expressions derived here are given in a
particular reference frame. Hydrodynamic variables, like the chemical
potential, can always be redefined order by order by performing a
transformation $\mu\rightarrow\mu+\delta\mu$ where $\delta\mu$ is of higher
order in derivatives. In this way it is always possible to reabsorb certain
transport coefficients into redefinitions of lower order ones. One possible
way to extract physically independent transport coefficients is to consider
frame invariant combinations of the constitutive relations, as in ref.
Bhattacharya:2011tra .

The alternative is to instead inspect the poles of the field theory
correlators and this is the route that we will take. Up to fist order in the
hydrodynamic expansion, the relevant correlators (LABEL:eq:greenfull) are

$G_{ret}=i\,\frac{\sigma}{D\,k^{2}-i\,\omega}\,,\qquad
G_{adv}=i\,\frac{\sigma}{D\,k^{2}+i\,\omega}\,,\qquad
G_{sym}=i\,\frac{T\sigma}{D^{2}\,k^{4}+\omega^{2}}\,,$ (85)

where we have identified the susceptibility and the diffusion constant as

$\chi=\frac{2\pi^{2}T^{2}L}{g_{A}^{2}}\,,\qquad
D=\frac{\sigma}{\chi}=\frac{1}{2\pi T}\,.$ (86)

These values match known results derived from holography by other means, see,
e.g., Policastro:2002se ; Kovtun:2008kx .

From the third order effective action (82) we can compute the corresponding
generating functional and extract, say, the diffusive pole to second order in
the hydrodynamic expansion

$\omega=-\frac{i}{2\pi T}k^{2}\left(1+\frac{\ln(2)}{(2\pi
T)^{2}}k^{2}\right)\,,$ (87)

which agrees with known results in literature, see, e.g., Policastro:2002se .
A comparison with the field theory result (24) gives us141414We thank P.
Kovtun for discussions on this point.

$\frac{1}{\chi}\left(k_{1}-2k_{2}+k_{3}-sD\right)=\frac{\ln(2)}{(2\pi
T)^{2}}\,.$ (88)

To close this section, let us re-stress that the key novelty of the presented
approach are not things like eq. (87), which has been already known for a long
time, but rather the ability to reproduce from an ab initio calculation
structures postulated earlier using an effective field theory reasoning.

##  5 The near-horizon region and the Schwinger-Keldysh effective action

In this section we derive the Schwinger-Keldysh effective action describing
diffusion on an intermediate radial cut-off of the bulk spacetime, which
interpolates between the horizons and the conformal boundaries. Fluid dynamics
on finite cutoff hypersurfaces for black holes in holography and beyond was
considered before in refs. Bredberg:2010ky ; Cai:2011xv ; Brattan:2011my ;
Bai:2012ci ; Emparan:2013ila ; Pinzani-Fokeeva:2014cka  and here we focus on
the charge dynamics to understand its role, via semi-holography
Faulkner:2010tq ; Faulkner:2010jy ; Nickel:2010pr , in calculations of
Schwinger-Keldysh effective actions.

###  5.1 The infrared Schwinger-Keldysh effective action

The ansatz for the gauge field in the bulk is similar to the one discussed
around eqs. (47) and (48a). The Dirichlet boundary conditions are now fixed on
timelike hypersurfaces at finite radius $u=\delta$ as follows

$\displaystyle\begin{split}B^{\delta}_{s\,\mu}(t_{s},x)=B_{s\,\mu}(t_{s},x,u=\delta)\,,\qquad
B^{\delta}_{E\,\mu}(\tau,x)=B_{E\,\mu}(\tau,x,u=\delta)\,,\end{split}$ (89)

where $s=R,L$ and $\mu=t,x$ as usual. This time we have allowed non-vanishing
sources on the (finite cutoff) boundary of the Euclidean manifold as well. As
previously, we impose that the gauge field at the horizon is vanishing, namely
we require eqs. (53) and (54), to secure regularity of the gauge field at the
tip of the Euclidean cigar. Gluing conditions along the fixed time slices at
$t=0$ and at $t=+\infty$, similarly to eqs. (56) and (59), imply

$\displaystyle\begin{split}&a_{+}^{E}=a_{+}^{L}\,,\qquad
a_{+}^{R}=e^{-\omega/T}a_{+}^{L}\,,\\\
&a_{+}^{L}=(1+\bar{n}(\omega))\frac{(B_{L\,x}-B_{R\,x})n^{\delta}-(B_{L\,t}-B_{R\,t})q^{\delta}}{h_{+}^{\delta}n^{\delta}-g_{+}^{\delta}q^{\delta}}\,,\end{split}$
(90)

where we have defined the shortcut notation for, e.g.,
$h_{+}^{\delta}=h_{+}(\tilde{\omega},\tilde{k},u=\delta)$.

The solution in the Lorentzian parts of the bulk spacetime can be written
similarly to eq. (61) as

$\displaystyle\begin{split}B_{s\,\mu}(t,x,u)=\int\frac{d\omega\,d\vec{k}}{(2\pi)^{4}}\,e^{-i\,\omega\,t_{s}+i\,k\,x}\sum_{m=R,L}\sum_{\nu=t,x}\Delta_{sm}^{\delta\,\,\mu\nu}(\tilde{\omega},\tilde{k},u)B_{m\,\nu}^{\delta}(\tilde{\omega},\tilde{k})\,,\end{split}$
(91)

where the bulk to boundary propagators now depend on the position of the
radial cutoff $\delta$,

$\displaystyle\begin{split}&\Delta_{LL}^{\delta\,\,\mu\nu}=-\bar{n}(\omega)\Delta_{-}^{\delta\,\,\mu\nu}+(1+\bar{n}(\omega)){\Delta}_{+}^{\delta\,\,\mu\nu}\,,\qquad\Delta_{LR}^{\delta\,\,\mu\nu}=(1+\bar{n}(\omega))\left(\Delta_{-}^{\delta\,\,\mu\nu}-\Delta_{+}^{\delta\,\,\mu\nu}\right)\,,\\\
&\Delta_{RL}^{\delta\,\,\mu\nu}=-\bar{n}(\omega)\left(\Delta_{-}^{\delta\,\,\mu\nu}-\Delta_{+}^{\delta\,\,\mu\nu}\right)\,,\qquad\Delta_{RR}^{\delta\,\,\mu\nu}=(1+\bar{n}(\omega))\Delta_{-}^{\delta\,\,\mu\nu}-\bar{n}(\omega)\Delta_{+}^{\delta\,\,\mu\nu}\,,\end{split}$
(92)

and we have defined the combinations

$\displaystyle\begin{split}&\Delta^{\delta\,\,tt}_{\pm}=\frac{h_{\pm}^{\delta}n-q^{\delta}g_{\pm}}{h_{\pm}^{\delta}n^{\delta}-g_{\pm}^{\delta}q^{\delta}}\,,\qquad\Delta^{\delta\,\,tx}_{\pm}=\frac{n^{\delta}g_{\pm}-g_{\pm}^{\delta}n}{h_{\pm}^{\delta}n^{\delta}-g_{\pm}^{\delta}q^{\delta}}\,,\\\
&\Delta^{\delta\,\,xt}_{\pm}=\frac{h_{\pm}^{\delta}q-q^{\delta}h_{\pm}}{h_{\pm}^{\delta}n^{\delta}-g_{\pm}^{\delta}q^{\delta}}\,,\qquad\Delta^{\delta\,\,xx}_{\pm}=\frac{n^{\delta}h_{\pm}-g_{\pm}^{\delta}q}{h_{\pm}^{\delta}n^{\delta}-g_{\pm}^{\delta}q^{\delta}}\,.\end{split}$
(93)

The solution in the Euclidean manifold can be written in an analogous fashion.

The infrared Schwinger-Keldysh effective action can be now derived evaluating
the partially on-shell effective action in the various parts of the spacetime,
as we have done around eq. (67). The result is

$\displaystyle\begin{split}S_{eff}^{\delta}=&\int\frac{d\omega\,d\vec{k}}{(2\pi)^{4}}\sum_{\mu=t,x}\sum_{\nu=t,x}\bigg{(}B^{\delta}_{a\,\mu}(-\tilde{\omega},-\tilde{k})\Pi_{-}^{\delta\,\,\mu\nu}(\tilde{\omega},\tilde{k})B^{\delta}_{r\,\nu}(\tilde{\omega},\tilde{k})+B^{\delta}_{r\,\mu}(-\tilde{\omega},-\tilde{k})\Pi_{+}^{\delta\,\,\mu\nu}(\tilde{\omega},\tilde{k})B^{\delta}_{a\,\nu}(\tilde{\omega},\tilde{k})\\\
&\qquad+\frac{1}{2}B^{\delta}_{a\,\mu}(-\tilde{\omega},-\tilde{k})\coth(\pi\tilde{\omega})\left(\Pi_{-}^{\delta\,\,\mu\nu}(\tilde{\omega},\tilde{k})-\Pi_{+}^{\delta\,\,\mu\nu}(\tilde{\omega},\tilde{k})\right)B^{\delta}_{a\,\nu}(\tilde{\omega},\tilde{k})\bigg{)}\,,\end{split}$
(94)

where we have defined

$\Pi^{\delta\,\,tt}_{\pm}=-G^{\delta\,\,tt}_{\pm}\,,\qquad\Pi^{\delta\,\,tx}_{\pm}=-G^{\delta\,\,tx}_{\pm}\,,\qquad\Pi^{\delta\,\,xt}_{\pm}=(1-\delta^{2})G^{\delta\,\,xt}_{\pm}\,,\qquad\Pi^{\delta\,\,xx}_{\pm}=(1-\delta^{2})G^{\delta\,\,xx}_{\pm}\,,$
(95)

and

$G_{\pm}^{\delta\,\,\mu\nu}=\frac{\pi^{2}T^{2}L}{g_{A}^{2}}\partial_{u}\Delta^{\delta\,\,\mu\nu}_{\pm}\big{|}_{u=\delta}\,.$
(96)

In writing eq. (94) we have omitted the contribution of the Euclidean cap for
simplicity which is not vanishing in this case. Notice that by sending the
intermediate cut-off to the conformal boundary $\delta\rightarrow 0$ the
original Schwinger-Keldysh effective action (68) is recovered. Moreover, we
can clearly see that the same characteristics of the full effective action of
the dual field theory are retained in the infrared. For instance, the
effective action is at least linear in the $a$-type sources to satisfy the
Schwinger-Keldysh symmetry of the underlying path integral.

Let us now consider the limit where the intermediate radial cutoff is taken to
be very close to the horizon $\delta\rightarrow 1$. The Schwinger-Keldysh
effective action (94) becomes

$\displaystyle\begin{split}S_{eff}^{IR}=&\frac{\pi^{2}T^{2}L}{g_{A}^{2}}\int\frac{d\omega\,d\vec{k}}{(2\pi)^{4}}\,\bigg{(}\frac{1}{(1-\delta)}\left(B^{\delta}_{r\,t}(-\tilde{\omega},-\tilde{k})B^{\delta}_{a\,t}(\tilde{\omega},\tilde{k})+B^{\delta}_{r\,t}(\tilde{\omega},\tilde{k})B^{\delta}_{a\,t}(-\tilde{\omega},-\tilde{k})\right)\\\
&\qquad-i\,\tilde{\omega}B^{\delta}_{r\,x}(-\tilde{\omega},-\tilde{k})B^{\delta}_{a\,x}(\tilde{\omega},\tilde{k})+i\,\tilde{\omega}B^{\delta}_{r\,x}(\tilde{\omega},\tilde{k})B^{\delta}_{a\,x}(-\tilde{\omega},-\tilde{k})\\\
&\qquad+i\,\tilde{\omega}\coth(\pi\tilde{\omega})B^{\delta}_{a\,x}(-\tilde{\omega},-\tilde{k})B^{\delta}_{a\,x}(\tilde{\omega},\tilde{k})\bigg{)}\,.\end{split}$
(97)

This effective action is universal and simple in the sense that it is given to
all orders in a hydrodynamic expansion since in deriving it we have not used
the explicit mode solutions but only their near horizon behaviors. In this way
this object can be thought of as a simple effective action incorporating
already all the symmetries of the full Schwinger-Keldysh effective action even
if it is restricted to some near horizon region.

To make comparison with known results we need to rescale the coordinates in a
way that the induced metric on the intermediate radial cutoff $\delta$ is
(conformally) flat

$ds^{2}=\frac{(\pi
TL)^{2}}{u}\left(-f(u)dt^{2}+dx^{2}+dy^{2}+dz^{2}\right)=\frac{(\pi
TL)^{2}}{u}\left(-d\bar{t}^{2}+dx^{2}+dy^{2}+dz^{2}\right)\,,$ (98)

with $\bar{t}=\sqrt{f_{\delta}}\,t$ and $f_{\delta}=(1-\delta^{2})$. With this
redefinition the various field theory quantities scale as

$\bar{\omega}=\omega/\sqrt{f_{\delta}}\,,\qquad\bar{T}=T/\sqrt{f_{\delta}}\,,\qquad
B_{\bar{t}}=B_{t}/\sqrt{f_{\delta}}\,,$ (99)

and the near horizon effective action becomes

$\displaystyle\begin{split}\bar{S}_{eff}^{IR}=&\frac{\pi^{2}\,f_{\delta}\,\bar{T}^{2}L}{g_{A}^{2}}\int\frac{d\bar{\omega}\,d\vec{k}}{(2\pi)^{4}}\sqrt{f_{\delta}}\,\bigg{(}2\left(B^{\delta}_{r\,\bar{t}}(-\tilde{\omega},-\tilde{k})B^{\delta}_{a\,\bar{t}}(\tilde{\omega},\tilde{k})+B^{\delta}_{r\,\bar{t}}(\tilde{\omega},\tilde{k})B^{\delta}_{a\,\bar{t}}(-\tilde{\omega},-\tilde{k})\right)\\\
&-i\,\frac{\bar{\omega}}{2\pi\bar{T}}B^{\delta}_{r\,x}(-\tilde{\omega},-\tilde{k})B^{\delta}_{a\,x}(\tilde{\omega},\tilde{k})+i\,\frac{\bar{\omega}}{2\pi\bar{T}}B^{\delta}_{r\,x}(\tilde{\omega},\tilde{k})B^{\delta}_{a\,x}(-\tilde{\omega},-\tilde{k})\\\
&+i\,\frac{\bar{\omega}}{2\pi\bar{T}}\coth\left(\frac{\bar{\omega}}{2\bar{T}}\right)B^{\delta}_{a\,x}(-\tilde{\omega},-\tilde{k})B^{\delta}_{a\,x}(\tilde{\omega},\tilde{k})\bigg{)}\,,\end{split}$
(100)

which is effectively living on the worldvolume. The diffusion constant in
these coordinates is given by

$D=\frac{\sigma}{\chi}=\frac{1}{4\pi\bar{T}}$ (101)

and matches the well-known result of ref. Bredberg:2010ky  with an _exact_
dispersion relation $\bar{\omega}=-i\,D\,k^{2}$. The diffusion constant on a
finite radial cutoff can also be easily extracted $D=1/2\pi\bar{T}(1+\delta)$
which interpolates between the field theory result defined on the conformal
boundary and the one on the horizon, see, again, ref. Bredberg:2010ky .

###  5.2 The infrared/ultraviolet coupling

Having the infrared Schwinger-Keldysh effective action (97) as a simple object
incorporating the near horizon physics we would like to show that the
remaining part of the spacetime between the horizon and the conformal boundary
acts as a mere transfer of this information to the dual conformal field
theory.

To see this, let us use the finite radial cutoff to separate the bulk
spacetime into an IR region ranging between the cutoff and the horizon and a
UV region extending from the cutoff to the conformal boundary, see, e.g.,
Nickel:2010pr ; Heemskerk:2010hk ; Faulkner:2010jy . The names associated to
the different regions in spacetime parallel the field theory interpretation
where, through the UV/IR relation in holography, short/large scale physics in
the bulk corresponds to the IR/UV regime in the dual field theory. With this
division of the bulk spacetime, also the bulk action separates artificially
into an IR and an UV contribution

$S=S^{IR}+S^{UV}\,.$ (102)

We have to imagine that the intermediate timelike cutoff hypersurface acts as
a boundary to the IR spacetime and as a second boundary to the UV spacetime on
which we impose Dirichlet boundary conditions $B^{\delta}$. The full action is
then recovered when imposing continuity conditions along the cutoff
hypersurface

$\frac{\delta S^{IR}}{\delta B^{\delta}}=\frac{\delta S^{UV}}{\delta
B^{\delta}}\,.$ (103)

The IR effective action where the cutoff is taken very close to the horizon is
given by eq. (97). The question is whether it is indeed sufficient to recover
the full Schwinger-Keldysh effective action (68) once also the UV part of the
effective action is known. The UV effective action can be computed considering
a double-Dirichlet problem along the lines of what we did in ref.
deBoer:2015ija  for linearized gravity perturbations, see also ref.
Crossley:2015tka . For instance, we first solve the dynamical equations of
motion (43a) and (43b) for the bulk gauge field subject to two Dirichlet
boundary conditions, $B^{\delta}$ at the intermediate cutoff $u=\delta$ and
$B$ at the conformal boundary $u=0$. Up to leading order in a hydrodynamic
expansion the solution is simply given by

$\displaystyle\begin{split}&B_{t}(\tilde{\omega},\tilde{k},u)=\left(1-\frac{u}{\delta}\right)B_{t}+\frac{u}{\delta}B_{t}^{\delta}\,,\\\
&B_{x}(\tilde{\omega},\tilde{k},u)=\left(1-\frac{\text{ArcTanh}(u)}{\text{ArcTanh}(\delta)}\right)B_{x}+\frac{\text{ArcTanh}(u)}{\text{ArcTanh}(\delta)}B_{x}^{\delta}\,.\end{split}$
(104)

Subsequently, we evaluate the partially on-shell bulk action (4.2) on the
solution (LABEL:eq:solDD) where, this time, the boundary $\partial{\cal M}$ is
taken to be the intermediate cutoff and the conformal boundary. Thus, taking
three such combinations

$iS_{eff}^{UV}=iS_{\rm onshell}\bigg{|}_{u_{R}=0}^{u_{R}=\delta}-iS_{\rm
onshell}\bigg{|}_{u_{L}=0}^{u_{L}=\delta}-S_{\rm
onshell}\bigg{|}_{u_{E}=0}^{u_{E}=\delta}$ (105)

gives us the total effective action in the UV region of our mixed signature
spacetime

$\displaystyle\begin{split}&S_{eff}^{UV}=\frac{\pi^{2}T^{2}L}{g_{A}^{2}}\int\frac{d\omega\,d\vec{k}}{(2\pi)^{4}}\,\bigg{(}\frac{1}{\delta}\left(B_{R\,t}(-\tilde{\omega},-\tilde{k})-B_{R\,t}^{\delta}(-\tilde{\omega},-\tilde{k})\right)\left(B_{R\,t}(\tilde{\omega},\tilde{k})-B_{R\,t}^{\delta}(\tilde{\omega},\tilde{k})\right)\\\
&-\frac{1}{\text{ArcTanh}(\delta)}\left(B_{R\,x}(-\tilde{\omega},-\tilde{k})-B_{R\,x}^{\delta}(-\tilde{\omega},-\tilde{k})\right)\left(B_{R\,x}(\tilde{\omega},\tilde{k})-B_{R\,x}^{\delta}(\tilde{\omega},\tilde{k})\right)-(R\rightarrow
L)\bigg{)}\,,\end{split}$ (106)

where we have neglected the Euclidean contribution for simplicity. Gluing
conditions along fixed time slices can be easily satisfied requiring that the
various Dirichlet values are equal at the crossing points.

Notice that taking strictly a near horizon limit of $S_{eff}^{UV}$ in eq.
(LABEL:eq:effUV) gives a very simple effective action which only depends on
the time component of the gauge field $B_{t}$. However, given that the IR
effective action $S_{eff}^{IR}$ in eq. (97) is divergent in this limit, we
will posit to stay slightly away from the horizon. In this way we can safely
impose the continuity condition (103) among the IR and UV effective actions.
Integrating out the Dirichlet values at the finite cutoff, taking the
hydrodynamic limit first and the near horizon limit second, we recover the
Schwinger-Keldysh effective action up to second order in a derivative
expansion (82). We have also verified that the same holds to third order when
the UV effective action $S_{eff}^{UV}$ is given to second order while keeping
the universal IR effective action $S_{eff}^{IR}$ in eq. (97). This strongly
suggest the picture advocated at the beginning of this subsection, that is the
near horizon part of the spacetime can be approximated by a simple term (97)
which alone includes the dissipative effects, KMS conditions etc, while the UV
part of the spacetime pushes that information to the conformal boundary where
the dual field theory is defined.

###  5.3 Interpretation of the infrared theory

As we just stated, the IR effective action in eq. (97) is believed to be
solely responsible for most of the interesting physics that underlies the full
effective action. It is therefore worthwhile to try to provide a different
perspective on this IR theory. In general, whenever we have a black hole or
black brane with a horizon at some value $r=r_{S}$ of a radial coordinate $r$,
we write $r=r_{S}+\epsilon\,\rho$ and consider the limit $\epsilon\rightarrow
0$ in order to take a near horizon limit. In this limit the metric behaves
schematically as

$ds^{2}\sim\epsilon\,(d\rho^{2}-\rho^{2}d{\cal T}^{2})+d\Omega_{H}+\ldots$
(107)

where $\rho$ and ${\cal T}$ are two-dimensional Rindler coordinates and
$\Omega_{H}$ is the metric on the horizon.

We first observe that time derivatives with respect to ${\cal T}$ will come
with an extra factor of $1/\sqrt{\epsilon}$ compared to spatial derivatives
along the horizon, and as a result the former will dominate. This is why in
the IR effective action the momenta $k$ do not appear explicitly and the
action is ultra-local along the horizon. It also explains why the IR effective
action can be obtained from a Kaluza-Klein reduction of the full theory along
the horizon directions. For the case studied in this paper, a higher
dimensional gauge field gives rise to some scalar fields and one gauge field
in two dimensions. More precisely, $B_{t}$ corresponds to a two-dimensional
gauge field, and $B_{x}$ to a two-dimensional scalar. In two dimensions these
fields do not mix and indeed, in the IR effective action they do not mix.

The two-dimensional action for scalars is conformally invariant and does
therefore not explicitly depend on the parameter $\epsilon$. Since
$\epsilon\sim(1-\delta)$, this explains the absence of a factor of $1-\delta$
in the part of the effective action that depends on $B_{x}$. The two-
dimensional action for gauge fields on the other hand will scale as
$1/\epsilon\sim 1/(1-\delta)$, which explains the corresponding factor in the
terms of the IR effective action that contains $B_{t}$.

Besides all this, there is also an emergent conformal symmetry in the near-
horizon limit. Rindler spacetime is just a wedge of two-dimensional Minkowski
spacetime and inherits its conformal symmetries. We are considering the
effective action with some cutoff $\rho_{c}={\rm const}$, but, as is familiar
from boundary states in conformal field theory, this boundary will preserve
half of the conformal symmetries. In terms of light-cone coordinates
$x^{\pm}=\rho\,e^{\pm{\cal T}}$, if we map $x^{+}\rightarrow g(x^{+})$ and
$x^{-}\rightarrow\rho_{c}^{2}/g(\rho_{c}^{2}/x_{-})$ for any function $g$ then
the boundary $x^{+}x^{-}=\rho_{c}^{2}$ will be mapped to itself.

If we interpret the IR effective action as a quantum mechanical effective
action for each value of the momenta $k$, with only $\omega$-dependence, then
this effective action will inherit the above conformal invariance (at least
for the scalar degrees of freedom). In other words, we can reinterpret part of
eq. (97) as the effective action of finite temperature conformally invariant
quantum mechanics.

Indeed, if we look at $B_{x}$, this corresponds to a massless scalar in two-
dimensions which has scaling dimension zero. At zero temperature, the retarded
and advanced two-point functions would presumably just be theta-functions of
the time difference, and the Wightman two-point function would most likely be
as in two-dimensions, i.e. be of the form $\sim\log|{\cal T}_{1}-{\cal
T}_{2}|$. We can then obtain the finite temperature two-point functions in
conformal quantum mechanics by a replacement of the form ${\cal T}_{1}-{\cal
T}_{2}\rightarrow a\sinh\left[({\cal T}_{1}-{\cal T}_{2})/a\right]$. Indeed,
taking such two-point functions and performing a Fourier transformation leads
to an effective action which is of the same form as that given for $B_{x}$ in
eq. (97).

The action for $B_{t}$ is given by two-dimensional Yang-Mills theory which is
not conformally invariant. The above consideration therefore does not
immediately yield a useful constraint on the terms in the effective action
that contain $B_{t}$. The Yang-Mills action on two-dimensional Rindler
spacetime does however have a symmetry under ${\cal
T}\rightarrow\lambda\,{\cal T}$, $A_{\rho}\rightarrow\sqrt{\lambda}\,A_{\rho}$
and $A_{\cal T}\rightarrow A_{\cal T}/\sqrt{\lambda}$ which implies that the
effective action for $B_{t}$ should behave under scale transformation as if
$B_{t}$ was an operator with scaling dimension $1/2$. This is indeed
consistent with the IR effective action.

We postpone a detailed study of this IR theory and the origin of the emergent
conformally invariant quantum mechanical system to future work.

##  6 Discussion

Motivated by recent developments in Schwinger-Keldysh effective field theories
for hydrodynamics we have explicitly derived the low-energy local effective
action capturing the dynamics of a conserved $U(1)$ charge current in strongly
coupled conformal quantum field theories at large-$N$ using holographic
techniques. Our results match existing literature both in field theory as far
as the formal structure of the effective action is concerned and in holography
for the actual values of the transport coefficients.

One main virtue of our approach, which utilizes real-time holography
techniques developed in refs. Skenderis:2008dh ; Skenderis:2008dg , is that it
allows to obtain analyticity conditions on the thermal correlators (KMS
conditions) without relying on particular analytic continuations of fields
across horizons. The KMS conditions arise naturally in this holographic setup
as a result of the gluing conditions of the Lorentzian spacetimes with the
Euclidean cap along fixed initial and final time slices, see also ref. Botta-
Cantcheff:2018brv .

Our derivation so far has been limited to the simplest case of a probe gauge
field without backreaction on the bulk metric and, as such, it is not
capturing the dynamics of a conserved stress-energy tensor. One obvious
obstruction to achieve a fully-fledged fluid dynamical behavior is the need
for a more careful treatment of the matching conditions at the fixed time
slices which now would be fluctuating. We expect our construction to
generalize at least to linearized order in the gravitational field
perturbations, capturing the linearized fluid dynamic effective action for the
dual charged conformal fluids at strong coupling. Such construction would need
a full understanding of the gauge field dynamics on top of a Reisner-Nordstrom
background with backreaction. Looking ahead, deriving the nonlinear Schwinger-
Keldysh effective action for fluid dynamics from gravity would require further
refinements of the procedure presented in this paper. Such a result could be
viewed as an alternative derivation of the fluid/gravity correspondence
Bhattacharyya:2008jc .

One feature that we have not seen in our construction is a geometrical
interpretation of the ghost fields which we have completely neglected in this
work. This is not so surprising since we have only restricted ourselves to the
tree level effective actions in the $1/N$ expansion. The ghost fields appear
in order to interpret the vanishing of the effective action for particular
source configurations as a consequence of certain BRST symmetries, see refs.
Haehl:2015foa ; Crossley:2015evo ; Haehl:2015uoc ; Haehl:2016pec ;
Haehl:2016uah , see also refs. Haehl:2015uoc ; Haehl:2016pec ; Haehl:2016uah ;
Jensen:2017kzi ; Gao:2017bqf ; Haehl:2018lcu ; Jensen:2018hse  for a
supersymmetric implementation of these symmetries. As we saw, the relevant
source configurations have a very simple IR interpretation, as they correspond
to purely left-moving and purely right-moving field configurations in the
near-horizon Rindler region. This suggests that, by gauge fixing the
2-dimensional diffeomorphisms of the IR Rindler geometry, the relevant ghosts
may naturally appear. This is quite familiar from the standard treatment of
the string world-sheet, where one also gauge fixes the 2-dimensional
diffeomorphisms and as a result the left- and right-moving stress tensor
become BRST exact. We hope to report on this direction in more detail in the
future and to see how the results parallel the discussion in ref. Gao:2018bxz
.

Another important aspect that we have not addressed in this work is the
entropy current. One of the major results in Schwinger-Keldysh effective field
theories for hydrodynamics is the possibility to define the entropy current as
a Noether current of a symmetry of the Schwinger-Keldysh effective action
Glorioso:2016gsa , see also Jensen:2018hhx ; Haehl:2018uqv ; Jensen:2018hse
for a supersymmetric implementation. Such a symmetry should have a dual
version in the bulk and we postpone this derivation to future work.

It is important to notice that our results obtained from holography are given
at finite $\hbar$. An effective action and an entropy current for diffusion
using Schiwnger-Keldysh techniques at finite $\hbar$ have been constructed in
ref. Jensen:2017kzi ; Jensen:2018hhx . Extending these results to the full
fluid dynamics case is problematic and one needs to rely on an approximation
where quantum fluctuations are neglected. This can be achieved by going to the
so-called statistical mechanics limit where $\hbar\rightarrow 0$, see refs.
Crossley:2015evo ; Glorioso:2017fpd ; Jensen:2018hse , or, equivalently, to a
high-temperature regime, see refs. Haehl:2015foa ; Haehl:2015uoc ;
Haehl:2018lcu . In holographic theories, however, such a limit cannot be
performed while keeping a well defined hydrodynamic expansion. The mean free
path in holographic field theories is $l_{mfp}\sim\hbar/T$, therefore there is
no separation of scales between the gradient expansion and quantum
fluctuations. The current work and future developments to include full fluid
dynamics might provide an invaluable testing ground to understand how to go
beyond the statistical limit in constructing Schwinger-Keldysh effective field
theories for hydrodynamics which is currently an open problem.

In this work, we have also been able to show that all the nice properties of
the field theory Schwinger-Keldysh effective action reside in the near horizon
region of the dual black hole spacetime. We have done so by defining a simple,
universal, near horizon Schwinger-Keldysh-like effective action valid to all
orders in a hydrodynamic expansion which has the same symmetry structure as
the original one (Schwinger-Keldysh symmetry, KMS symmetry, etc.). We have
then coupled this object to the remaining bulk spacetime extending between the
horizon and the conformal boundaries and showed that the full Schwinger-
Keldysh effective action for the probe gauge field is recovered to second
order in a hydrodynamic expansion. This procedure can be thought of as a
generalization of the membrane paradigm viewed as a simple boundary condition,
see, e.g., ref. Iqbal:2008by ; deBoer:2014xja , to a membrane paradigm as an
action principle given in the IR part of the spacetime. A natural
generalization of this framework is to revisit the IR effective action and its
relation to a membrane paradigm in the presence of dynamical gravity. The
general scaling arguments given in subsection 5.3 can perhaps be applied to
gravity as well. Since a higher dimensional metric yields some scalars, gauge
fields, and a 2-dimensional metric in two-dimensions, the main new ingredient
is to understand the appropriate boundary conditions for the fluctuations of
the 2-dimensional metric.

The division of the bulk spacetime between a simple infrared near horizon
region and the remaining ultraviolet spacetime is a precise implementation of
the idea of semi-holography Faulkner:2010tq . It is then natural to ask what
are the different possible near horizon Schwinger-Keldysh-like actions when
the near horizon region itself is different from the simple case analyzed in
this work. For instance, what would it be for near-extremal black holes? Near
extremal black holes develop a long AdS2 throat region, and one might try to
separate the bulk spacetime in an IR region which contains the AdS2 throat and
the rest. In this case, the IR region can perhaps be well approximated by the
Schwarzian theory describing nearly AdS2 physics, see, e.g., ref.
Maldacena:2016upp . We cannot, however, go arbitrarily close to the horizon,
as this would not preserve the nearly AdS2 property of the metric. We
therefore do not expect a precise membrane paradigm-like description similar
to the one for Rindler space, and that in order to reproduce the complete
physical picture higher order corrections to the IR theory need to be kept.
This is in line with the observations in, e.g., refs. Moitra:2018jqs  and
Castro:2018ffi .

Another natural extension of our work is to dual field theory contours defined
on multiple time segments. The effective field theories arising in these
scenarios would include out-of time ordered correlators, observables which
acquired much recent attention as probes of early-time chaotic behavior
leading to the celebrated result that black holes are the maximally chaotic
systems in nature Maldacena:2015waa . More recently, much effort has been put
into framing quantum chaos into an effective field theory language
Blake:2017ris , see also refs. Haehl:2018izb ; Cotler:2018zff  for the case of
underlying two-dimensional conformal field theories. Our techniques can be
used to test these developments via simple holographic derivations and
possibly guide to a better understanding of universal features of quantum
chaos.

Finally, it would be very interesting to use our construction to understand
the large-order behaviour of hydrodynamic gradient expansion of the Schwinger-
Keldysh effective action. What is well-established by now is that for two
classes of non-linear solutions of relativistic hydrodynamics the energy-
momentum tensor expanded in gradients receives numerical contributions scaling
like $n!$ from the combined terms having $n$ derivatives of fluid variables,
see ref. Florkowski:2017olj  for a review. What was also seen, see refs.
Heller:2016gbp  and Withers:2018srf , is that hydrodynamic dispersion
relations like eq. (87) exhibit finite radius of convergence of the small-$k$
expansion. In both cases, one can link the zero and finite radia of
convergence to the presence of fast-decaying excitations not captured directly
by hydrodynamics in the gradient expansion. In holography, these short-lived
excitations are transient quasinormal modes of black branes. Exploring what
happens at the level of an effective action is an interesting open problem
that might lead to a new chapter in the studies of hydrodynamics at large
orders, as well as understanding effective actions beyond the derivative
expansion.

## Acknowledgements

We would like to thank N. Iqbal, K. Jensen, P. Kovtun, S. Minwalla, B. van
Rees, K. Skenderis and A. Yarom for discussions. The work of NPF is supported
in part by the National Science Foundation of Belgium (FWO) grant G.001.12
Odysseus and by the European Research Council grant no. ERC-2013-CoG 616732
HoloQosmos. NPF would also like to acknowledge Technion and University of
Haifa where part of this work was done. This group was supported in part by
the Israel Science Foundation under grant 504/13 and in part at the Technion
by a fellowship from the Lady Davis Foundation. MPH and the Gravity, Quantum
Fields and Information group at AEI are supported by the Alexander von
Humboldt Foundation and the Federal Ministry for Education and Research
through the Sofja Kovalevskaja Award. NPF would also like to aknowledge KITP
for hospitality during the final stage of this project.

##  Appendix A Quadratic effective action for diffusion

Let us here justify in more detail the expression (23) from the main text for
the effective action for diffusion up to third order151515The resulting
constitutive relations are given up to second order. in the derivative
expansion.

The most general effective action for diffusion to all orders in a
hydrodynamic expansion has been given in Crossley:2015evo . The expression at
finite $\hbar$ and to quadratic order in the fields was given in eq. (4.53) of
ref. Jensen:2017kzi . Expanding that expression to third order in gradients,
going back to coordinate space, and keeping only the longitudinal sector,
where $\vec{k}=(k,0,0)$, gives

$\displaystyle\begin{split}S_{eff}&=\int
d^{d}x\Big{(}\chi\,B_{r\,t}B_{a\,t}-\sigma\,B_{a\,x}\partial_{t}B_{r\,x}+i\,\sigma\,T\,B_{a\,x}B_{a\,x}-s\,B_{a\,t}\partial_{t}B_{r\,t}+i\,s\,T\,B_{a\,t}B_{a\,t}\\\
&+k_{0}\,B_{a\,t}\partial_{t}^{2}B_{r\,t}+k_{1}\,B_{a\,t}\partial_{x}^{2}B_{r\,t}+k_{2}\,\left(\partial_{x}B_{a\,t}\partial_{t}B_{r\,x}+\partial_{t}B_{a\,x}\partial_{x}B_{r\,t}\right)+k_{3}\,B_{a\,x}\partial^{2}_{t}B_{r\,x}\Big{)}\,.\end{split}$
(108)

The parameters here are related to the ones in ref. Jensen:2017kzi  via

$\displaystyle\begin{split}&\chi=2\,F_{00}^{tt}\,,\qquad\sigma=-\frac{1}{T}\sigma_{00}^{xx}\,,\qquad
s=-\frac{1}{T}\sigma_{00}^{tt}\,,\\\
&k_{0}=2\,F_{20}^{tt}-\frac{1}{12\,T^{2}}F_{00}^{tt}+\frac{1}{T}\sigma_{10}^{tt}\,,\qquad
k_{1}=2\,F_{02}^{tt}\,,\\\ &k_{2}=F_{11}^{tx}+F_{11}^{xt}\,,\qquad
k_{3}=2\,F_{20}^{xx}+\frac{1}{T}\sigma_{10}^{xx}\,,\end{split}$ (109)

where we have set ${\cal R}=1$ (see the original reference) and we have used
the expansions

$\displaystyle\begin{split}F^{\mu\nu}(i\,\omega,i\,k)&=F^{\mu\nu}_{00}+k\omega
F_{11}^{\mu\nu}-\omega^{2}F^{\mu\nu}_{20}-k^{2}F^{\mu\nu}_{02}+{\cal
O}(\omega^{3},k^{3})\,,\\\
\sigma^{\mu\nu}(i\,\omega,i\,k)&=\sigma^{\mu\nu}_{00}+i\omega\sigma^{\mu\nu}_{10}+{\cal
O}(\omega^{2},k^{2})\,,\\\ {\cal
A}(b\,\omega)&=1+\frac{\omega^{2}}{12T^{2}}+{\cal
O}(\omega^{4})\,,\end{split}$ (110)

Here we used $b=1/T$ and the spacetime indices $\mu,\,\nu$ stand for $t,\,x$.
Notice that, for example, there is no contribution of the form $F_{00}^{xx}$
which would give rise to a term proportional to $B_{r\,x}B_{a\,x}$ in the
effective action. Such a term would not be invariant under eq. (17). Notice
also that there are terms appearing at third order which come from lower order
terms due to finite $\hbar$ effects. An example of such a term is the
contribution $F_{00}^{tt}$ to $k_{0}$.

##  Appendix B Details of the holographic results

The mode functions at second order in the hydrodynamic expansion are

$\displaystyle\begin{split}n=&1+\frac{\tilde{k}^{2}}{1-u}\left(u\,{\rm
ln}(u)+(1+u)\,{\rm ln}\left(\frac{2}{1+u}\right)\right)\,,\\\
g_{\pm}=&+\frac{\tilde{k}\,\tilde{\omega}}{2}\left((1-u)+u\,{\rm
ln}\left(\frac{2}{1-u^{2}}\right)+{\rm
ln}\left(\frac{2(1-u)}{1+u}\right)+2u\,{\rm ln}(u)\right)\,,\\\
h_{\pm}=&(1-u)^{\pm\frac{i\tilde{\omega}}{2}}\bigg{(}1\pm\frac{i\tilde{\omega}}{2}{\rm
ln}\left(\frac{2}{1+u}\right)+\frac{\tilde{\omega}^{2}}{24}\bigg{(}\pi^{2}-9{\rm
ln}^{2}(2)+12{\rm ln}(2){\rm ln}(1-u)\\\ &-6{\rm ln}(2){\rm ln}(1+u)-12{\rm
ln}\left(\frac{1-u}{2}\right){\rm ln}(1+u)-12{\rm ln}(u){\rm ln}(1+u)+3{\rm
ln}^{2}(1+u)\\\ &-12{\rm Li}_{2}(1-u)-12{\rm Li}_{2}(-u)-12{\rm
Li}_{2}\left(\frac{1+u}{2}\right)\bigg{)}\bigg{)}\,,\\\
q=&\frac{\tilde{k}\,\tilde{\omega}}{24}\bigg{(}\pi^{2}-6{\rm ln}^{2}(2)+12{\rm
ln}(2){\rm ln}(1-u)-12{\rm ln}(1-u){\rm ln}(1+u)-12{\rm ln}(u){\rm ln}(1+u)\\\
&+6{\rm ln}^{2}(1+u)-12{\rm Li}_{2}(1-u)-12{\rm Li}_{2}(-u)-12{\rm
Li}_{2}\left(\frac{1+u}{2}\right)\bigg{)}\,,\end{split}$

where ${\rm Li}_{2}$ is the Polylogarithm function. Inserting these
expressions into eq. (LABEL:eq:Gpm) and then into eq. (70) we get

$\displaystyle\begin{split}&\Pi^{tt}_{\pm}=\frac{\pi^{2}T^{2}L}{g_{A}^{2}}\left(1-2\tilde{k}^{2}\ln(2)\right)\,,\qquad\Pi^{tx}_{\pm}=\Pi^{xt}_{\pm}=-\frac{\pi^{2}T^{2}L}{g_{A}^{2}}\tilde{k}\tilde{\omega}\ln(2)\,,\\\
&\Pi^{xx}_{\pm}=\frac{\pi^{2}T^{2}L}{g_{A}^{2}}\left(\pm
i\,\tilde{\omega}-\tilde{\omega}^{2}\ln(2)\right)\,,\end{split}$ (111)

and, consequently, the Schwinger-Keldysh effective action up to second order
in the Fourier space takes the form

$\displaystyle\begin{split}S_{eff}=&\frac{\pi^{2}T^{2}L}{g_{A}^{2}}\int\frac{d\omega\,d\vec{k}}{(2\pi)^{4}}\,\bigg{(}B_{r\,t}(-\tilde{\omega},-\tilde{k})B_{a\,t}(\tilde{\omega},\tilde{k})+B_{r\,t}(\tilde{\omega},\tilde{k})B_{a\,t}(-\tilde{\omega},-\tilde{k})\\\
&-i\,\tilde{\omega}B_{r\,x}(-\tilde{\omega},-\tilde{k})B_{a\,x}(\tilde{\omega},\tilde{k})+i\,\tilde{\omega}B_{r\,x}(\tilde{\omega},\tilde{k})B_{a\,x}(-\tilde{\omega},-\tilde{k})\\\
&+\frac{i}{\pi}B_{a\,x}(-\tilde{\omega},-\tilde{k})B_{a\,x}(\tilde{\omega},\tilde{k})\\\
&-2\tilde{k}^{2}\,{\rm
ln}(2)\left(B_{a\,t}(\tilde{\omega},\tilde{k})B_{r\,t}(-\tilde{\omega},-\tilde{k})+B_{a\,t}(-\tilde{\omega},-\tilde{k})B_{r\,t}(\tilde{\omega},\tilde{k})\right)\\\
&-\tilde{k}\,\tilde{\omega}\,{\rm
ln}(2)\left(B_{a\,x}(\tilde{\omega},\tilde{k})B_{r\,t}(-\tilde{\omega},-\tilde{k})+B_{a\,x}(-\tilde{\omega},-\tilde{k})B_{r\,t}(\tilde{\omega},\tilde{k})\right)\\\
&-\tilde{k}\,\tilde{\omega}\,{\rm
ln}(2)\left(B_{a\,t}(\tilde{\omega},\tilde{k})B_{r\,x}(-\tilde{\omega},-\tilde{k})+B_{a\,t}(-\tilde{\omega},-\tilde{k})B_{r\,x}(\tilde{\omega},\tilde{k})\right)\\\
&-\tilde{\omega}^{2}\,{\rm
ln}(2)\left(B_{a\,x}(\tilde{\omega},\tilde{k})B_{r\,x}(-\tilde{\omega},-\tilde{k})+B_{a\,x}(-\tilde{\omega},-\tilde{k})B_{r\,x}(\tilde{\omega},\tilde{k})\right)\bigg{)}\,,\end{split}$
(112)

where we have used a counterterm of the form for curing the divergences on the
right boundary

$S_{ct}=\frac{L\pi^{2}T^{2}}{g_{A}^{2}}\log(u)\int\frac{d\omega\,d\vec{k}}{(2\pi)^{4}}\left(\tilde{k}\,B_{R\,t}(\tilde{\omega},\tilde{k})+\tilde{\omega}\,B_{R\,x}(\tilde{\omega},\tilde{k})\right)\left(\tilde{k}\,B_{R\,t}(-\tilde{\omega},-\tilde{k})+\tilde{\omega}\,B_{R\,x}(-\tilde{\omega},-\tilde{k})\right)\,,$
(113)

and an analogous term for the left boundary.

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