# Holographic Schwinger-Keldysh field theory of SU(2) diffusion

Yanyan Bu<EMAIL_ADDRESS>Xiyang Sun<EMAIL_ADDRESS>(correspondence
author)  Biye Zhang<EMAIL_ADDRESS>
###### Abstract

We construct effective field theory for SU(2) isospin charge diffusion, based
on holographic Schwinger-Keldysh contour [1]. The holographic model consists
of a probe SU(2) gauge field in a doubled Schwarzschild-AdS5 geometry.
Accurate to first order in derivative expansion, we analytically compute the
effective action up to quartic order in dynamical variables. The effective
theory contains both non-Gaussianity for noises and nonlinear interactions
between noises and dynamical variables. Moreover, the effective theory
captures both thermal and quantum fluctuations, which perfectly satisfy
dynamical Kubo-Martin-Schwinger (KMS) symmetry at quantum level.
Interestingly, the dynamical KMS symmetry, which is crucial in formulating
non-equilibrium effective field theory for a quantum many-body system, is
found to have a nice holographic interpretation.

###### Contents

  1. 1 Introduction
  2. 2 Holographic setup
  3. 3 From bulk dynamics to boundary effective action
    1. 3.1 General consideration: linearize Yang-Mills system
    2. 3.2 Partially on-shell solution: generic structure
    3. 3.3 Review of leading-order solution of [43]
    4. 3.4 Boundary effective action: contour integrals
    5. 3.5 Dynamical KMS symmetry: from boundary to bulk
  4. 4 Summary and discussion
  5. A Yang-Mills equation in different coordinate systems
  6. B More on partially on-shell solution: beyond leading order

##  1 Introduction

Hydrodynamics is believed to be an effective description of a quantum many-
body system in the long distance long time regime. Classical hydrodynamics is
formulated as a set of deterministic partial differential equations (PDEs),
which stands for conservation laws of stress-energy tensor and internal
currents. However, classical hydrodynamics does not capture effects of
statistical and/or quantum fluctuations. In stochastic formulation of
hydrodynamics, fluctuations are modelled by random dissipative fluxes (with
Gaussian distributions), added to the stress-energy tensor and internal
currents. Resultantly, the deterministic conservation laws become stochastic
Langevin-type PDEs.

Recently, hydrodynamics has been reformulated as a non-equilibrium effective
field theory (EFT) [2, 3, 4, 5, 6, 7]111 See [8, 9, 10, 11, 12] for early
attempts on this subject. (see [13] for a pedagogical review) by utilizing
Schwinger-Keldysh (SK) formalism, entirely based on symmetry principles. In
contrast to stochastic formulation, hydrodynamic EFT incorporates dissipations
and fluctuations in a systematic manner. Particularly, fluctuation-dissipation
theorem (FDT) is implemented as a dynamical KMS symmetry [3, 4] satisfied by
the hydrodynamic effective action. The dynamical KMS symmetry acts on
dynamical variables of the hydrodynamic EFT. Thus, hydrodynamic EFT provides
an ideal framework for investigating fluctuation effects in a variety of
physical problems, see e.g. [14, 15, 16, 17, 18, 19, 20] for recent progress.

The development of hydrodynamic EFT has been largely inspired by the AdS-CFT
correspondence [21, 22, 23], which conjectures that strongly coupled large
$N_{c}$ (the number of colors) gauge theory is equivalent to a weakly coupled
gravitational theory in higher dimensional asymptotic AdS space. Essentially,
deriving hydrodynamic EFT from gravity amounts to implementing holographic
Wilsonian renormalization group [24, 25, 26, 27, 28] in dual gravity. A first
successful derivation was presented for a U(1) charge diffusion [29, 1]. While
[29] adopted real-time formalism for AdS-CFT correspondence [30, 31,
32]222This real-time prescription has been used to compute higher-point
correlation functions [33, 34, 35, 36]., the work [1] achieved the goal by
proposing a holographic prescription for Schwinger-Keldysh (SK) closed time
contour. The results of [29, 1] are in perfect agreement with that constructed
within the non-equilibrium EFT framework [2]. Recently, the holographic SK
contour [1] attracted a lot of attention in various holographic settings [37,
38, 39, 40, 41, 42, 43, 44, 45, 46].

In this work we extend studies of [1, 43] and present a holographic derivation
of hydrodynamic EFT for a SU(2) isospin charge diffusion333Classical
hydrodynamics with an internal SU(2) symmetry has been considered, for
instance, in [47, 48, 49, 50, 51, 52]. , using the holographic SK contour [1].
This extension is highly nontrivial, mainly due to nonlinearity of the theory.
Indeed, for a linear bulk theory, in order to consistently cover both ingoing
mode (dual to dissipation) and outgoing mode (dual to fluctuation), the bulk
fields inevitably exhibit logarithmic divergences near the event horizon [1,
43]. When nonlinearities are present, this logarithmic behavior near the
horizon will bring in technical complications in the derivation of boundary
action, which is one of the main problems we will address in this study. Apart
of this, there is a subtle issue [1] regarding the non-commutativity between
hydrodynamic limit and near horizon limit. In this work, we would like to
further explore this point and develop a more solid way to extract
hydrodynamic EFT action on the boundary.

Our study is partly motivated by a recent publication [53], which developed an
EFT for hydrodynamics with global non-Abelian symmetries. Ref. [53] assumes
two main approximations: truncating the effective action to quadratic order in
noise variables, and imposing dynamical KMS symmetry in the classical
statistical limit (i.e., $\hbar\to 0$)444Implementation of dynamical KMS
symmetry at quantum level has been considered for U(1) charge diffusion [54,
55] and in formulating an EFT for maximally quantum chaotic system [56, 57,
58].. While the former approximation does not completely kill nonlinear
interactions between noises and dynamical variables, it does ignore non-
Gaussianity in noise. Recently, Jain and Kovtun clarified importance of non-
Gaussian noises in quantifying (non-)universality of hydrodynamics [17]: non-
Gaussian noise does have non-negligible signals in hydrodynamic correlation
functions.

The second approximation made in [53] corresponds to ignoring quantum
fluctuations. While this works well from hydrodynamic EFT perspective [2], it
becomes unsatisfactory on holographic side: a clear splitting between quantum
fluctuations and statistical fluctuations would be impossible for a
holographic field theory [29]. More precisely, for a holographic theory, the
mean free path is $\sim\hbar/T$, which implies that derivative expansion would
unavoidably bear quantum fluctuations [29]. In other words, the large $N_{c}$
limit of holography does not necessarily mean classicalization of the boundary
system, which is another conceptual issue we would like to clarify through
present study. Hopefully, elaboration on this matter will shed light on the
imposition of dynamical KMS symmetry at quantum level for the construction of
hydrodynamic EFT for a dissipative fluid, which is still an open question [3,
4]. In addition, we are interested in understanding to what the dynamical KMS
symmetry proposal [3, 4] corresponds in the dual gravity theory.

Therefore, via holographic technique, we aim at deriving a more comprehensive
EFT for SU(2) isospin charge diffusion, in which the two approximations
undertaken in [53] will be relaxed. Phenomenologically, the present study will
be of potential relevance to the study of QCD plasma in the chiral limit [47,
Grossi:2020ezz]: on the one hand, the global SU(2) symmetry can mimic flavor
symmetry of QCD in two-flavor approximation in a late stage of RHIC plasma; on
the other hand, this study would be insightful in understanding fluctuations
in pion fluid at finite temperature.

Before diving into detailed calculations, we summarize our main results:

$\bullet$ Limited to first order in derivative expansion, we derive the
hydrodynamic effective action for SU(2) diffusion, valid to quartic order in
hydrodynamic fields. We analytically compute all the parameters in the
effective action. See equations (3.39), (3.49), (3.54), and (3.57) (or the
more compact ones (3.66)-(3.68)) for detailed results.

$\bullet$ We reveal that our effective action satisfies dynamical KMS symmetry
at the quantum level, say, both thermal and quantum fluctuations are accounted
for in the effective action, see subsection 3.5.

$\bullet$ We present a holographic interpretation for dynamical KMS symmetry
proposed in [3, 4]: under bulk KMS transformation, the bulk fields living on
the lower (upper) branch of the holographic contour will transform in an
analogous way as those living on the forward (backward) branch of the SK
contour in boundary theory. See subsection 3.5 for more discussions.

The rest of this paper will be organized as follows. In section 2 we present
the holographic setup: a probe SU(2) gauge field in doubled Schwarzschild-AdS5
geometry. In addition, we outline basic strategy of deriving boundary
effective action from bulk dynamics. Section 3 is the main part of this work.
First, we solve classical bulk dynamics in the partially on-shell sense; then,
we compute the partially on-shell bulk action by implementing radial contour
integrals, which gives the boundary effective action; finally, we check
dynamical KMS symmetry of our effective action, and explore a holographic
interpretation for it. In section 4, we make a summary and discussions.
Appendices A and B supplement further calculational details.

##  2 Holographic setup

Notation convention: throughout this paper, we use the upper-case Latin
letters “$M,N,\cdots$” to denote bulk spacetime indices, the Greek letters
“$\mu,\nu,\cdots$” for boundary indices, the Italic lower-case Latin letters
“$i,j,\cdots$” for spatial coordinates, and the upright lower-case Latin
letters “$\rm a,b,\cdots$” for SU(2) flavor indices.

We consider a non-Abelian SU(2) gauge theory in Schwarzschild-AdS5 geometry,
whose dynamics is described by Yang-Mils action:

$\displaystyle S_{0}=-\frac{1}{2}\int d^{5}x\sqrt{-g}\,{\rm
Tr}(F^{2})=-\frac{1}{4}\int d^{5}x\sqrt{-g}\,F_{MN}^{\rm a}F^{{\rm a}MN},$
(2.1)

where Yang-Mills field and its field strength are matrix-valued

$\displaystyle C=C_{M}dx^{M}=C_{M}^{\rm a}t^{\rm a}dx^{M},\quad
F=dC=F_{MN}dx^{M}\wedge dx^{N}=F_{MN}^{\rm a}t^{\rm a}dx^{M}\wedge dx^{N},$
$\displaystyle F_{MN}^{\rm a}=\nabla_{M}C_{N}^{\rm a}-\nabla_{N}C_{M}^{\rm
a}+\epsilon^{\rm abc}C_{M}^{\rm b}C_{N}^{\rm c},\qquad t^{\rm
a}=\frac{1}{2}\tau^{\rm a},$ (2.2)

where $\tau^{\rm a}$’s are the Pauli matrices, and repeated indices mean
summation. Here, we take the convention for normalization ${\rm Tr}(t^{\rm
a}t^{\rm b})=\frac{1}{2}\delta^{\rm ab}$. In ingoing Eddington-Finkelstein
(EF) coordinate system $x^{M}=(r,v,x^{i})$, the metric of Schwarzschild-AdS5
geometry is given by

$\displaystyle
ds^{2}=g_{MN}dx^{M}dx^{N}=2dvdr-r^{2}f(r)dv^{2}+r^{2}\delta_{ij}dx^{i}dx^{j},\qquad
i,j=1,2,3,$ (2.3)

where $f(r)=1-r_{h}^{4}/r^{4}$ with $r_{h}$ the horizon radius (the AdS radius
is set to unity). The Schwarzschild-AdS5 has Hawking temperature
$T=r_{h}/\pi$, which is identified as the temperature for boundary theory. In
asymptotic AdS space, we also need a counter-term action [59]

$\displaystyle S_{\rm ct}=\frac{1}{4}\log r\,\int
d^{4}x\sqrt{-\gamma}F_{\mu\nu}^{\rm a}F^{{\rm a}\mu\nu}\bigg{|}_{r=\infty},$
(2.4)

which is written down based on minimal subtraction scheme. Here, $\gamma$ is
the determinant of induced metric $\gamma_{\mu\nu}$ on the boundary
$r=\infty$.

For the purpose of incorporating fluctuation and dissipation in an action
principle, the boundary system shall be placed on the SK time contour [60]. A
gravity dual of the SK time contour is proposed in [1], which complexifies the
radial coordinate $r$ of (2.3) and analytically continues it around the event
horizon $r=r_{h}$, see Figure 1.

Figure 1: Gravity dual of the SK time contour [1]: complexified radial
coordinate and analytical continuation around the horizon $r_{h}$.

Here, we explain the basic strategy of deriving hydrodynamic effective action
from AdS gravity. This has been originally formulated in [24] for the problem
of stress tensor (see also [1, 43] for U(1) charge diffusion problem), based
on early attempts [26, 27, 61]. The starting point is the holographic
dictionary [22, 23]:

$\displaystyle Z_{\rm CFT}=Z_{\rm AdS}.$ (2.5)

The CFT partition function $Z_{\rm CFT}$ may be presented as a path integral
over gapless modes (collectively denoted by $X$) in the low energy EFT:

$\displaystyle Z_{\rm CFT}=\int[DX]e^{iS_{eff}[X]},$ (2.6)

where $S_{eff}$ is the desired effective action. The AdS partition function
$Z_{\rm AdS}$ is a path integral over bulk fields555In the probe limit, it is
valid to ignore dynamics of bulk spacetime.,

$\displaystyle Z_{\rm
AdS}=\int[DC_{M}^{\prime}]e^{iS_{0}[C_{M}^{\prime}]+iS_{\rm ct}},$ (2.7)

which will be computed in the saddle point approximation, i.e., based on
solving classical dynamics of the bulk Yang-Mills theory (2.1). In (2.7) the
primed configuration $C_{M}^{\prime}$ corresponds to no gauge-fixing for the
bulk gauge theory.

Thus, in order to obtain $S_{eff}$ from bulk theory we shall identity gapless
mode of low energy EFT, whose gravity dual shall not be integrated out in the
gravity partition function (2.7). This can be achieved by carefully examining
gauge symmetry in the bulk, as illustrated for U(1) case by Nickel and Son
[61] (see also [1, 43, 44]). Through a bulk gauge transformation

$\displaystyle C_{M}^{\prime}\to
C_{M}=U(\Lambda)(C_{M}^{\prime}+i\nabla_{M})U^{\dagger}(\Lambda),\qquad
U(\Lambda)=e^{i\Lambda^{\rm a}t^{\rm a}},$ (2.8)

a generic configuration $C_{M}^{\prime}$ can be brought into any gauge-fixed
form $C_{M}$, for example, the one with radial component fixed as $C_{r}=0$.
The bulk gauge transformation parameter $\Lambda^{\rm a}(x^{M})$ is determined
by solving gauge-fixing condition. At the AdS boundary, (2.8) takes the form:

$\displaystyle
A_{\mu}^{\prime}(x^{\alpha})\to\mathcal{U}(\varphi)\left[A_{\mu}^{\prime}(x^{\alpha})+i\partial_{\mu}\right]\mathcal{U}^{\dagger}(\varphi),\qquad\mathcal{U}(\varphi)=e^{i\varphi^{\rm
a}t^{\rm a}},\qquad\varphi^{\rm a}(x^{\mu})\equiv\Lambda^{\rm
a}(r=\infty,x^{\mu}),$ (2.9)

where $A_{\mu}^{\prime}$ is boundary value of $C_{\mu}^{\prime}$. Thus, the
gravity partition function (2.7) can be equivalently expressed as a path
integral over $C_{M}$ (gauge-fixed configuration) and the gauge transformation
parameter $\Lambda$:

$\displaystyle Z_{\rm AdS}$
$\displaystyle=\int[D\Lambda][DC_{\mu}]e^{iS_{0}[C_{M},\Lambda]+iS_{\rm ct}}$
$\displaystyle\simeq\int[D\varphi][DC_{\mu}]e^{iS_{0}[C_{\mu},\varphi]+iS_{\rm
ct}},$ (2.10)

where in the last equality we ignored an overall normalization constant given
that this will not affect results of physical quantities. Once $C_{\mu}$ is
integrated out, (2.10) is put into the desired form of (2.6),

$\displaystyle Z_{\rm
AdS}=\int[D\varphi]e^{iS_{0}[C_{\mu}[\varphi],\varphi]|_{\rm p.o.s}+iS_{\rm
ct}},$ (2.11)

so that the boundary effective action is read off from bulk action

$\displaystyle S_{eff}=S_{0}[C_{\mu}[\varphi],\varphi]|_{\rm p.o.s}+S_{\rm
ct}.$ (2.12)

Here, under the philosophy of [61, 1, 2], we identify $\varphi$ as the gapless
mode of low energy EFT. Note $S_{0}[C_{\mu}[\varphi],\varphi]|_{\rm p.o.s}$ is
partially on-shell bulk action to be obtained by plugging classical solution
for $C_{\mu}$ into Yang-Mills action (2.1). It is important to stress that by
partially on-shell, we determine profile for $C_{\mu}$ by solving dynamical
equations of motion (EOMs) only. Particularly, we will leave aside the
constraint equation. Such a partially on-shell formalism was ever invented in
[62, 63, 64, 65, 66] to resum (linear) all-order derivatives in off-shell
hydrodynamic constitutive relations. Below we elaborate on this point.

Under the variation

$\displaystyle C_{M}^{\rm a}\to C_{M}^{\rm a}+\delta C_{M}^{\rm
a}\Longrightarrow\delta F_{MN}^{\rm a}=\nabla_{M}\delta C_{N}^{\rm
a}-\nabla_{N}\delta C_{N}^{\rm a}+\epsilon^{\rm abc}\delta C_{M}^{\rm
b}C_{N}^{\rm c}+\epsilon^{\rm abc}C_{M}^{\rm b}\delta C_{N}^{\rm c},$ (2.13)

the Yang-Mills action (2.1) varies as

$\displaystyle\delta S_{0}=-\int d^{5}x\sqrt{-g}\left[\nabla_{M}\left(\delta
C_{N}^{\rm a}F^{{\rm a}MN}\right)-\delta C_{N}^{\rm a}\nabla_{M}F^{{\rm
a}MN}+\delta C_{N}^{\rm a}\epsilon^{\rm abc}C_{M}^{\rm c}F^{{\rm
b}NM}\right].$ (2.14)

Immediately, we read off the bulk EOMs:

$\displaystyle\nabla_{M}F^{{\rm a}MN}+\epsilon^{\rm abc}C_{M}^{\rm b}F^{{\rm
c}MN}=0.$ (2.15)

Throughout this work, we will take the gauge convention

$\displaystyle C_{r}^{\rm a}=-\frac{C_{v}^{\rm a}}{r^{2}f(r)},$ (2.16)

which means in Schwarzschild coordinate system the radial component of bulk
gauge field is fixed to zero [43, 44]. The motivation for taking such a gauge
choice is to realize time-reversal symmetry in a simple way, see [43].
Accordingly, the dynamical components of bulk EOMs are [43, 24]

$\displaystyle\nabla_{M}F^{{\rm a}Mi}+\epsilon^{\rm abc}C_{M}^{\rm b}F^{{\rm
c}Mi}=0,$ $\displaystyle\nabla_{M}F^{{\rm a}Mv}+\epsilon^{\rm abc}C_{M}^{\rm
b}F^{{\rm c}Mv}-\frac{1}{r^{2}f(r)}\left(\nabla_{M}F^{{\rm a}Mr}+\epsilon^{\rm
abc}C_{M}^{\rm b}F^{{\rm c}Mr}\right)=0.$ (2.17)

The constraint component of bulk EOMs is

$\displaystyle\nabla_{M}F^{{\rm a}Mr}+\epsilon^{\rm abc}C_{M}^{\rm b}F^{{\rm
c}Mr}=0.$ (2.18)

As emphasized below (2.12), we will solve the bulk dynamics in a partially on-
shell approach: the profile of bulk gauge potential $C_{\mu}^{\rm a}$ will be
fixed by solving dynamical components of bulk EOMs, say (2.17). The constraint
component of bulk EOMs, say (2.18), will be left aside.

Finally, the AdS boundary conditions are

$\displaystyle
C_{\mu}(r=\infty_{s},x^{\mu})=\mathcal{U}(\varphi_{s})(A_{s\mu}+i\partial_{\mu})\mathcal{U}^{\dagger}(\varphi_{s})\equiv
B_{s\mu}(x^{\alpha}),\quad\mathcal{U}(\varphi_{s})=e^{i\varphi_{s}^{\rm
a}(x)t^{\rm a}},\quad s=1,2,$ (2.19)

where $\varphi_{s}^{\rm a}$ is the hydrodynamic field associated with SU(2)
isospin charge, and $A_{s\mu}$ is the external (background) SU(2) gauge field
in the boundary theory. For brevity, in (2.19) we dropped the prime in
$A_{s\mu}^{\prime}$.

However, as explained in [1] (see also [43]), the AdS boundary conditions
(2.19) are insufficient to fully determine $C_{\mu}$. Resultantly, we have
freedom to impose

$\displaystyle C_{v}^{\rm a}(r=r_{h}-\epsilon,x^{\mu})=0.$ (2.20)

For a bulk U(1) field, the condition (2.20) has been shown to be equivalent to
discontinuity of $\partial_{r}C_{v}$ at the horizon [43]. We will not
elaborate on this point since linearized solution for $C_{M}^{\rm a}$ (which
is identical to U(1) case) is sufficient for our purpose.

##  3 From bulk dynamics to boundary effective action

In this section we solve classical dynamics of the bulk theory, and then
compute the partially on-shell bulk action
$S_{0}[C_{\mu}[\varphi],\varphi]|_{\rm p.o.s}$ of (2.12).

###  3.1 General consideration: linearize Yang-Mills system

The dynamical EOMs (2.17) are nonlinear PDEs and are thus hard to solve
analytically. Instead of solving (2.17) order-by-order in the boundary
derivative expansion, we linearize the bulk Yang-Mills theory:

$\displaystyle C_{M}^{\rm a}=\xi^{1}C_{M}^{{\rm a}(1)}+\xi^{2}C_{M}^{{\rm
a}(2)}+\xi^{3}C_{M}^{{\rm a}(3)}+\cdots,$ $\displaystyle F_{MN}^{\rm
a}=\xi^{1}F_{MN}^{{\rm a}(1)}+\xi^{2}F_{MN}^{{\rm a}(2)}+\xi^{3}F_{MN}^{{\rm
a}(3)}+\cdots,$

where

$\displaystyle F_{MN}^{{\rm a}(1)}=\nabla_{M}C_{N}^{{\rm
a}(1)}-\nabla_{N}C_{M}^{{\rm a}(1)},$ $\displaystyle F_{MN}^{{\rm
a}(2)}=\nabla_{M}C_{N}^{{\rm a}(2)}-\nabla_{N}C_{M}^{{\rm a}(2)}+\epsilon^{\rm
abc}C_{M}^{{\rm b}(1)}C_{N}^{{\rm c}(1)},$ $\displaystyle F_{MN}^{{\rm
a}(3)}=\nabla_{M}C_{N}^{{\rm a}(3)}-\nabla_{N}C_{M}^{{\rm a}(3)}+\epsilon^{\rm
abc}\left(C_{M}^{{\rm b}(2)}C_{N}^{{\rm c}(1)}+C_{M}^{{\rm b}(1)}C_{N}^{{\rm
c}(2)}\right).$ (3.1)

At each order in $\xi$-expansion, the dynamical EOMs (2.17) become a system of
linear PDEs. For instance, at $\mathcal{O}(\xi^{1})$ and
$\mathcal{O}(\xi^{2})$, we have

$\displaystyle\nabla_{M}F^{{\rm a}Mi(1)}=0,\qquad\qquad\nabla_{M}F^{{\rm
a}Mv(1)}-\frac{1}{r^{2}f(r)}\nabla_{M}F^{{\rm a}Mr(1)}=0,$ (3.2)
$\displaystyle\nabla_{M}F^{{\rm a}Mi(2)}+\epsilon^{\rm abc}C_{M}^{{\rm
b}(1)}F^{{\rm c}Mi(1)}=0,$ $\displaystyle\nabla_{M}F^{{\rm
a}Mv(2)}+\epsilon^{\rm abc}C_{M}^{\rm b(1)}F^{{\rm
c}Mv(1)}-\frac{1}{r^{2}f(r)}\left(\nabla_{M}F^{{\rm a}Mr(2)}+\epsilon^{\rm
abc}C_{M}^{\rm b(1)}F^{{\rm c}Mr(1)}\right)=0.$ (3.3)

The detailed forms of (3.2) and (3.3) can be found in appendix A. The AdS
boundary conditions (2.19) will be imposed as follows:

$\displaystyle C_{s\mu}^{{\rm a}(1)}(r=\infty_{s},x^{\alpha})=B_{s\mu}^{\rm
a}(x^{\alpha}),\qquad C_{\mu}^{{\rm a}(n)}(r=\infty_{s},x^{\alpha})=0,\quad
n=2,3,\cdots,\quad s=1,2.$ (3.4)

Meanwhile, the vanishing horizon condition (2.20) is also imposed
perturbatively

$\displaystyle C_{v}^{{\rm a}(n)}(r_{h}-\epsilon,x^{\mu})=0,\qquad
n=1,2,3,\cdots.$ (3.5)

The leading order EOMs (3.2) are homogeneous PDEs, which are identical to
those of a free Maxwell field in Schwarzschild-AdS5 geometry. Thus, the
leading order solution $C_{M}^{{\rm a}(1)}$ is essentially the same as that
constructed in [43], which will be reviewed in subsection 3.3.

The next-to-leading order EOMs (3.3) form a system of linear inhomogeneous
PDEs, which differ from leading order ones (3.2) by source terms, see appendix
A for more details. Moreover, the source terms are constructed from leading
order solution $C_{M}^{{\rm a}(1)}$. Given that linearly independent solutions
for homogeneous parts of (3.3) have been worked out in [43], the task of
solving (3.3) boils down to looking for a particular solution associated with
its source term. This can be implemented via Green’s function approach [67,
44], which will be discussed in detail in appendix B.

We turn to the Yang-Mills action (2.1), which is expanded as:

$\displaystyle
S_{0}=\xi^{2}S_{0}^{(2)}+\xi^{3}S_{0}^{(3)}+\xi^{4}S_{0}^{(4)}+\cdots,$ (3.6)

where

$\displaystyle S_{0}^{(2)}=-\frac{1}{4}\int d^{5}x\sqrt{-g}F_{MN}^{{\rm
a}(1)}F^{{\rm a}MN(1)},$ $\displaystyle S_{0}^{(3)}=-\frac{1}{4}\int
d^{5}x\sqrt{-g}\,2F_{MN}^{{\rm a}(1)}F^{{\rm a}MN(2)},$ $\displaystyle
S_{0}^{(4)}=-\frac{1}{4}\int d^{5}x\sqrt{-g}\left[2F_{MN}^{{\rm a}(1)}F^{{\rm
a}MN(3)}+F_{MN}^{{\rm a}(2)}F^{{\rm a}MN(2)}\right].$ (3.7)

The quadratic action $S_{0}^{(2)}$ will be similar to that of [43]. The cubic
action $S_{0}^{(3)}$ could be further simplified via integration by part:

$\displaystyle S_{0}^{(3)}$ $\displaystyle=-\frac{1}{2}\int
d^{5}x\sqrt{-g}\left[(\nabla_{M}C_{N}^{{\rm a}(2)}-\nabla_{N}C_{M}^{{\rm
a}(2)})F^{{\rm a}MN(1)}+\epsilon^{\rm abc}F_{MN}^{{\rm a}(1)}C^{{\rm
b}M(1)}C^{{\rm c}N(1)}\right]$ $\displaystyle=-\frac{1}{2}\int
d^{5}x\sqrt{-g}\left[2\nabla_{M}C_{N}^{{\rm a}(2)}F^{{\rm
a}MN(1)}+\epsilon^{\rm abc}F_{MN}^{{\rm a}(1)}C^{{\rm b}M(1)}C^{{\rm
c}N(1)}\right]$ $\displaystyle=-\frac{1}{2}\int
d^{5}x\sqrt{-g}\left[2\nabla_{M}(C_{N}^{{\rm a}(2)}F^{{\rm
a}MN(1)})-C_{N}^{{\rm a}(2)}\nabla_{M}F^{{\rm a}MN(1)}+\epsilon^{\rm
abc}F_{MN}^{{\rm a}(1)}C^{{\rm b}M(1)}C^{{\rm c}N(1)}\right]$
$\displaystyle=-\frac{1}{2}\int d^{5}x\sqrt{-g}\epsilon^{\rm abc}F_{MN}^{{\rm
a}(1)}C^{{\rm b}M(1)}C^{{\rm c}N(1)}.$ (3.8)

Explicitly, using the gauge-fixing (2.16), the cubic order action becomes

$\displaystyle S_{0}^{(3)}=-\int d^{5}x\sqrt{-g}\epsilon^{\rm
abc}\left\\{(\partial_{i}C_{v}^{{\rm a}(1)}-\partial_{v}C_{i}^{{\rm
a}(1)})\frac{C_{v}^{{\rm b}(1)}}{r^{4}f(r)}C_{i}^{{\rm
c}(1)}+\frac{1}{r^{4}}\partial_{i}C_{j}^{{\rm a}(1)}C_{i}^{{\rm
b}(1)}C_{j}^{{\rm c}(1)}\right\\}.$ (3.9)

The quartic order action $S_{0}^{(4)}$ of (3.7) could be simplified in the
same fashion (3.8). Eventually, the quartic order action is cast into

$\displaystyle S_{0}^{(4)}$ $\displaystyle=-\frac{1}{4}\int
d^{5}x\sqrt{-g}\left[2\epsilon^{\rm abc}F_{MN}^{{\rm a}(1)}C^{{\rm
b}M(2)}C^{{\rm c}N(1)}+\epsilon^{\rm abc}(\nabla_{M}C_{N}^{{\rm
a}(2)}-\nabla_{N}C_{M}^{{\rm a}(2)})C^{{\rm b}M(1)}C^{{\rm c}N(1)}\right.$
$\displaystyle\qquad\qquad\qquad\qquad\left.+C_{M}^{{\rm b}(1)}C^{{\rm
b}M(1)}C_{N}^{{\rm c}(1)}C^{{\rm c}N(1)}-C_{M}^{{\rm b}(1)}C^{{\rm
b}N(1)}C_{N}^{{\rm c}(1)}C^{{\rm c}M(1)}\right],$ (3.10)

where the first two terms involve the next-to-leading order solution
$C_{M}^{{\rm a}(2)}$, and the last two terms do not. Explicitly, the quartic
order action is

$\displaystyle S_{0}^{(4)}=$ $\displaystyle-\frac{1}{4}\int
d^{5}x\sqrt{-g}\left\\{-\frac{2}{r^{4}f(r)}C_{v}^{{\rm a}(1)}C_{k}^{{\rm
b}(1)}\left(C_{v}^{{\rm a}(1)}C_{k}^{{\rm b}(1)}-C_{k}^{{\rm a}(1)}C_{v}^{{\rm
b}(1)}\right)\right.$
$\displaystyle\qquad\qquad\qquad\qquad\left.+\frac{1}{r^{4}}C_{k}^{{\rm
a}(1)}C_{l}^{{\rm b}(1)}\left(C_{k}^{{\rm a}(1)}C_{l}^{{\rm b}(1)}-C_{l}^{{\rm
a}(1)}C_{k}^{{\rm b}(1)}\right)\right\\}$ $\displaystyle-\int
d^{5}x\sqrt{-g}\epsilon^{\rm abc}\left\\{(\partial_{i}C_{v}^{{\rm
a}(1)}-\partial_{v}C_{i}^{{\rm a}(1)})\frac{C_{v}^{{\rm
b}(2)}}{r^{4}f(r)}C_{i}^{{\rm c}(1)}+\frac{1}{r^{4}}\partial_{i}C_{j}^{{\rm
a}(1)}C_{i}^{{\rm b}(2)}C_{j}^{{\rm c}(1)}\right\\}$
$\displaystyle-\frac{1}{2}\int d^{5}x\sqrt{-g}\epsilon^{\rm
abc}\left\\{(\partial_{i}C_{v}^{{\rm a}(2)}-\partial_{v}C_{i}^{{\rm
a}(2)})\frac{C_{v}^{{\rm b}(1)}}{r^{4}f(r)}C_{i}^{{\rm
c}(1)}+\frac{1}{r^{4}}\partial_{i}C_{j}^{{\rm a}(2)}C_{i}^{{\rm
b}(2)}C_{j}^{{\rm c}(1)}\right\\}.$ (3.11)

In obtaining (3.9) and (3.11), we have imposed the leading-order dynamical
EOMs (3.2), but we did not impose the constraint equation (2.18). Moreover, we
have utilized the following two facts: $C_{\mu}^{{\rm a}(2)}$ vanishes at the
AdS boundaries, see (3.4); leading order counterpart of the radial gauge
choice (2.16).

While the quadratic action $S_{0}^{(2)}$ and cubic action $S_{0}^{(3)}$
involve only the leading-order solution $C_{\mu}^{{\rm a}(1)}$, the
computation of $S_{0}^{(4)}$ generally requires the next-to-leading order
solution $C_{\mu}^{{\rm a}(2)}$. In subsection 3.2, we will see that the
leading-order solution $C_{\mu}^{(1)}$ is essentially the bulk-to-boundary
propagator. In appendix B, $C_{\mu}^{{\rm a}(2)}$ is constructed via Green’s
function method and would be schematically written as

$\displaystyle C_{\mu}^{{\rm
a}(2)}\simeq\int_{\infty_{2}}^{\infty_{1}}dr^{\prime}G_{\perp,\parallel}(r,r^{\prime},k^{\alpha})\mathbb{S}_{\mu}^{{\rm
a}(2)}(r^{\prime},k^{\alpha}),$ (3.12)

where $G_{\perp,\parallel}(r,r^{\prime},k^{\alpha})$ represents the bulk-to-
bulk propagator, and the source term $\mathbb{S}_{\mu}^{{\rm
a}(2)}(r^{\prime},k^{\alpha})$ are quadratic in $C_{\mu}^{{\rm a}(1)}$, see
appendix B. Diagrammatically, the perturbative expansion of bulk action can be
drawn as tree-level Witten diagrams, see Figure 2. More precisely, the
quadratic action $S_{0}^{(2)}$ and cubic action $S_{0}^{(3)}$ correspond to
contact Witten diagrams, while the quartic action $S_{0}^{(4)}$ contains both
contact Witten diagram (the first two lines of (3.11)) and exchange Witten
diagram (the last two lines of (3.11)).

Figure 2: Witten diagrams for $S_{0}^{(2)}$ (first), $S_{0}^{(3)}$ (second)
and $S_{0}^{(4)}$ (last two). Here, the circle represents the AdS boundaries,
and the region inside the circle is for AdS interior. The straight lines stand
for bulk-to-bulk propagator (connecting two points inside the circle) and
bulk-to-boundary propagator (connecting an interior point and a point on the
circle) for bulk SU(2) gauge field propagating in doubled Schwarzschild-AdS.

From subsection 3.2, it will be clear that $C_{M}^{{\rm a}(2)}$ starts from
first order in boundary derivative expansion. This implies, through (3.11),
that in practice we do not need to solve for $C_{M}^{{\rm a}(2)}$, since in
present work we will truncate quartic order action $S_{0}^{(4)}$ at first
order in the hydrodynamic derivative expansion. Equivalently, we will only
compute contact-type Witten diagrams (the first three of Figure 2).

###  3.2 Partially on-shell solution: generic structure

Turning to Fourier space:

$\displaystyle C_{M}^{{\rm a}(n)}(r,x^{\mu})=\int\frac{d\omega
dq}{(2\pi)^{2}}e^{ik^{\alpha}x_{\alpha}}C_{M}^{{\rm a}(n)}(r,k^{\mu}),\quad
n=1,2,\cdots,\quad k^{\mu}=(\omega,q,0,0),$ (3.13)

we further reduce the system of linear PDEs, say (3.2) or (3.3), into a system
of linear ordinary differential equations (ODEs). Here, by spatially
rotational invariance, we have chosen the spatial momentum for Fourier modes
to be along $x$-direction, without losing generality. Then, the bulk gauge
field $C_{\mu}^{{\rm a}(n)}$ can be classified into two decoupled sub-sectors:
the transverse sector $C_{\perp}^{{\rm a}(n)}$ with $\perp=y,z$ versus the
longitudinal sector $C_{\parallel}^{{\rm a}(n)}=\left\\{C_{v}^{{\rm
a}(n)},~{}C_{x}^{{\rm a}(n)}\right\\}$.

The transverse mode obeys a closed second order ODE ($n=1,2,\cdots$):

$\displaystyle\partial_{r}\left[r^{3}f(r)\partial_{r}C_{\perp}^{{\rm
a}(n)}\right]+\Box_{\perp}\left(\partial_{r};\omega,q^{2}\right)C_{\perp}^{{\rm
a}(n)}=\mathbb{S}_{\perp}^{{\rm a}(n)},$ (3.14)

where the source $\mathbb{S}_{\perp}^{{\rm a}(n)}$ is built from lower order
solutions, with $\mathbb{S}_{\perp}^{{\rm a}(1)}=0$, see (A.7). From (A.6),
the symbol $\Box_{\perp}\left(\partial_{r};\omega,q^{2}\right)C_{\perp}^{{\rm
a}(n)}$ stands for terms containing boundary derivatives of $C_{\perp}^{{\rm
a}(n)}$, and contains at most one first order radial derivative of
$C_{\perp}^{{\rm a}(n)}$.

The dynamics for the longitudinal sector is more involved, see (A.6) and
(A.7). Schematically, the dynamical EOMs are ($n=1,2,\cdots$)

$\displaystyle\partial_{r}\left[r^{3}\partial_{r}C_{v}^{{\rm
a}(n)}\right]+\Box_{v}\left(\partial_{r};\omega,q\right)C_{\parallel}^{{\rm
a}(n)}=\mathbb{S}_{v}^{{\rm a}(n)},$
$\displaystyle\partial_{r}\left[r^{3}f(r)\partial_{r}C_{x}^{{\rm
a}(n)}\right]+\Box_{x}\left(\partial_{r};\omega,q\right)C_{\parallel}^{{\rm
a}(n)}=\mathbb{S}_{x}^{{\rm a}(n)},$ (3.15)

where, similar to transverse sector, the symbols
$\Box_{v}\left(\partial_{r};\omega,q\right)C_{\parallel}^{{\rm a}(n)}$ and
$\Box_{x}\left(\partial_{r};\omega,q\right)C_{\parallel}^{{\rm a}(n)}$ denote
terms having boundary derivatives of $C_{v,x}^{{\rm a}(n)}$, and contain at
most one first order radial derivative of $C_{v,x}^{{\rm a}(n)}$. Finally,
$\mathbb{S}_{v,x}^{{\rm a}(n)}$ are the source terms to be constructed from
lower order solutions, with the leading order $\mathbb{S}_{v,x}^{{\rm
a}(1)}=0$.

In general, solution for (3.14) or (3.15) consists of two parts: generic
solution for homogeneous parts of (3.14) and (3.15), which will be referred to
as homogeneous generic solution (HGS), and particular solution for the whole
inhomogeneous systems (3.14) and (3.15). First, we consider HGS by ignoring
source terms, which has been worked out in Section 4 of [43]. We cut the
radial contour of Figure 1 at the leftmost point $r=r_{h}-\epsilon$ so that
the contour is split into upper branch and lower branch. Then, we search for
linearly independent solutions on single copy AdS, i.e., either on the upper
branch or on the lower branch of the contour. For generic values of $\omega$
and $q$, these linearly independent solutions are not known analytically, and
may be classified according to their near-horizon behavior. Resultantly, on
upper (lower) branch of the radial contour, the HGS is simply linear
superposition of those linearly independent solutions. With HGS at hand, it is
straightforward to obtain particular solution for the whole inhomogeneous
system, for instance, by Green’s function approach. On upper (lower) branch,
we sum the HGS and particular solution, forming inhomogeneous generic solution
(IHGS). At the cutting slice $r=r_{h}-\epsilon$, the IHGS on the upper branch
will be properly glued to the IHGS on the lower branch. Eventually, we impose
AdS boundary conditions. This completes the solving of (3.14) and (3.15).

Finally, we briefly comment on the next-to-leading order solution
$C_{\mu}^{{\rm a}(2)}$. From (3.3), it is direct to check that the source
terms $\mathbb{S}_{\perp,v,x}^{{\rm a}(2)}$ appearing in (3.14) and (3.15) are
at least of first order in boundary derivative expansion. Given homogenous AdS
boundary conditions for $C_{\mu}^{{\rm a}(2)}$, see (3.4), one can immediately
conclude that $C_{\mu}^{{\rm a}(2)}$ will be at least of first order in
boundary derivative expansion, just as their source terms. Consequently,
combined with (3.11), this observation implies that the contribution from
$C_{\mu}^{{\rm a}(2)}$ will start from second order in boundary derivative
expansion, which will be beyond focus of present work. Since the next-to-
leading solution is irrelevant in subsequent calculations, we postpone the
detailed construction of it in appendix B.

###  3.3 Review of leading-order solution of [43]

In this subsection, we review the leading-order solution comprehensively
constructed in [43]. The strategy of solving leading-order counterparts (i.e.,
with $n=1$) of (3.14) and (3.15) will go under three steps.

First, with the radial contour of Figure 1 cut at $r=_{h}-\epsilon$, we search
for linearly independent solutions when $r$ varies on the upper branch or
lower branch. While this topic has been widely explored in the literature, a
major difference arises due to partially on-shell approach adopted in this
work. Particularly, for the longitudinal modes, one has to include a new
linearly independent solution, referred to as “polynomial solution” in [43],
which does not obey the constraint equation and is thus off-shell. Then, the
generic solution on the upper (lower) branch is simply superposition of the
linearly independent solutions.

Second, we will perform a gluing procedure so that the generic solution on the
upper branch will be properly glued with that on the lower branch. Here, the
gluing condition is derived from the requirement that variational problem for
$S_{0}^{(2)}$ of (3.7) is well-defined at the cutting slice
$r=r_{h}-\epsilon$, say $\delta S_{0}^{(2)}/\delta C_{\mu}^{{\rm
a}(1)}\big{|}_{r=r_{h}-\epsilon}=0$. Essentially, the gluing condition tells
how the radial derivatives of bulk fields will jump across the cutting slice
$r=r_{h}-\epsilon$. Here, it is natural to assume that bulk fields
$C_{\mu}^{{\rm a}(1)}$ are continuous across $r=r_{h}-\epsilon$. We refer the
reader into [43] for more details.

Finally, superposition constants in the generic solutions will be completely
fixed by the doubled AdS boundary conditions.

Without repeating the details, here we simply write down final results for
leading order solution $C_{\mu}^{{\rm a}(1)}$ [43]. For the transverse mode
$C_{\perp}^{{\rm a}(1)}$,

$\displaystyle C_{\perp}^{{\rm a}(1)~{}{\rm
up}}(r,k^{\mu})=c_{\perp}(k^{\mu})C_{\perp}^{\rm
ig}(r,k^{\mu})-h_{\perp}(k^{\mu})C_{\perp}^{\rm
ig}(r,\bar{k}^{\mu})e^{2i\omega\zeta_{2}(r)},~{}\qquad\quad
r\in[r_{h}-\epsilon,\infty_{2}),$ $\displaystyle C_{\perp}^{{\rm a}(1)~{}{\rm
dw}}(r,k^{\mu})=c_{\perp}(k^{\mu})C_{\perp}^{\rm
ig}(r,k^{\mu})-h_{\perp}(k^{\mu})e^{-\beta\omega}C_{\perp}^{\rm
ig}(r,\bar{k}^{\mu})e^{2i\omega\zeta_{1}(r)},\quad
r\in[r_{h}-\epsilon,\infty_{1}),$ (3.16)

where $\bar{k}^{\mu}=(-\omega,q,0,0)$. $C_{\perp}^{\rm ig}(r,k^{\mu})$ and
$C_{\perp}^{\rm ig}(r,\bar{k}^{\mu})e^{2i\omega\zeta_{s}(r)}$ represent the
ingoing solution and outgoing solution666Here, as in [37, 38, 42, 43], the
ingoing and outgoing modes are related via time-reversal symmetry. for
homogeneous part of (3.14). The functions $\zeta_{1,2}(r)$ are

$\displaystyle\zeta_{s}(r)=\int_{\infty_{s}}^{r}\frac{dy}{y^{2}f(y)},\qquad
r\in[r_{h}-\epsilon,\infty_{s}),\quad s=1~{}{\rm or}~{}2,$ (3.17)

which has an explicit form:

$\displaystyle\zeta_{s}(r)=-\frac{1}{4r_{h}}\left[\pi-2\arctan\left(\frac{r}{r_{h}}\right)+\log\left(1+\frac{r_{h}}{r}\right)-\log\left(1-\frac{r_{h}}{r}\right)\right].$
(3.18)

The superposition coefficients $c_{\perp},h_{\perp}$ are determined by the AdS
boundary conditions (see (3.4)):

$\displaystyle
c_{\perp}=\frac{1}{2}\coth\frac{\beta\omega}{2}\frac{B_{a\perp}^{\rm
a}(k^{\mu})}{C_{\perp}^{\rm ig(0)}(k^{\mu})}+\frac{B_{r\perp}^{\rm
a}(k^{\mu})}{C_{\perp}^{\rm ig(0)}(k^{\mu})},\qquad
h_{\perp}=\frac{B_{a\perp}(k^{\mu})}{(1-e^{-\beta\omega})C_{\perp}^{\rm
ig(0)}(\bar{k}^{\mu})},$ (3.19)

where $C_{\perp}^{\rm ig(0)}(k^{\mu})$ is the boundary value of ingoing mode
$C_{\perp}^{\rm ig}(r,k^{\mu})$. In ingoing EF coordinate system, the ingoing
solution $C_{\perp}^{\rm ig}(r,k^{\mu})$ is regular over the entire contour,
particularly near the horizon. Here, we introduced the $(r,a)$-basis:

$\displaystyle B_{r\mu}^{\rm a}=\frac{1}{2}\left(B_{1\mu}^{\rm
a}+B_{2\mu}^{\rm a}\right),\qquad\qquad B_{a\mu}^{\rm a}=B_{1\mu}^{\rm
a}-B_{2\mu}^{\rm a},\qquad\mu=v,x,\perp.$ (3.20)

For the longitudinal modes, we have

$\displaystyle C_{v}^{{\rm a}(1)~{}{\rm up}}(r,k^{\mu})=$ $\displaystyle
c_{\parallel}(k^{\mu})C_{v}^{\rm
ig}(r,k^{\mu})+h_{\parallel}(k^{\mu})C_{v}^{\rm
ig}(r,\bar{k}^{\mu})e^{2i\omega\zeta_{2}(r)}+p_{\parallel}^{\rm up}C_{v}^{\rm
pg}(r,k^{\mu})$ $\displaystyle+n_{\parallel}^{\rm up}(k^{\mu})C_{v}^{\rm
pn}(r,k^{\mu}),$ $\displaystyle C_{x}^{{\rm a}(1)~{}{\rm up}}(r,k^{\mu})=$
$\displaystyle c_{\parallel}(k^{\mu})C_{x}^{\rm
ig}(r,k^{\mu})-h_{\parallel}(k^{\mu})C_{x}^{\rm
ig}(r,\bar{k}^{\mu})e^{2i\omega\zeta_{2}(r)}+p_{\parallel}^{\rm up}C_{v}^{\rm
pg}(r,k^{\mu})$ $\displaystyle+n_{\parallel}^{\rm up}(k^{\mu})C_{x}^{\rm
pn}(r,k^{\mu}),$ $\displaystyle C_{v}^{{\rm a}(1)~{}{\rm dw}}(r,k^{\mu})=$
$\displaystyle c_{\parallel}(k^{\mu})C_{v}^{\rm
ig}(r,k^{\mu})+h_{\parallel}(k^{\mu})e^{-\beta\omega}C_{v}^{\rm
ig}(r,\bar{k}^{\mu})e^{2i\omega\zeta_{1}(r)}+p_{\parallel}^{\rm dw}C_{x}^{\rm
pg}(r,k^{\mu})$ $\displaystyle+n_{\parallel}^{\rm up}(k^{\mu})C_{v}^{\rm
pn}(r,k^{\mu}),$ $\displaystyle C_{x}^{{\rm a}(1)~{}{\rm dw}}(r,k^{\mu})=$
$\displaystyle c_{\parallel}(k^{\mu})C_{x}^{\rm
ig}(r,k^{\mu})-h_{\parallel}(k^{\mu})e^{-\beta\omega}C_{x}^{\rm
ig}(r,\bar{k}^{\mu})e^{2i\omega\zeta_{1}(r)}+p_{\parallel}^{\rm dw}C_{x}^{\rm
pg}(r,k^{\mu})$ $\displaystyle+n_{\parallel}^{\rm dw}(k^{\mu})C_{x}^{\rm
pn}(r,k^{\mu}),$ (3.21)

where, as for the transverse mode, $C_{v,x}^{\rm ig}$ and $C_{v,x}^{\rm
ig}e^{2i\omega\zeta_{s}(r)}$ are the ingoing and outgoing solutions,
respectively. $C_{v,x}^{\rm pg}$ is the pure gauge solutions, which is
obtained from a zero solution by a gauge transformation preserving the gauge
condition (2.16). Lastly, $C_{v,x}^{\rm pn}$ represents the off-shell solution
which is crucial in exhausting the full basic solutions. Among the four
linearly independent solutions, only $C_{v}^{\rm pg}$ does not vanish at the
horizon. Consequently, the condition (3.5) requires to set

$\displaystyle p_{\parallel}^{\rm up}=p_{\parallel}^{\rm dw}=0.$ (3.22)

The rest superposition coefficients
$c_{\parallel},h_{\parallel},n_{\parallel}^{\rm up},n_{\parallel}^{\rm dw}$
are determined by the AdS boundary conditions (3.4):

$\displaystyle c_{\parallel}=\frac{1}{2}G_{1}^{-1}\left\\{2B_{rx}^{\rm
a}(k^{\mu})C_{v}^{\rm pn(0)}(k^{\mu})-2B_{rv}^{\rm a}(k^{\mu})C_{x}^{\rm
pn(0)}(k^{\mu})\right.$
$\displaystyle\qquad\qquad\left.+\coth\frac{\beta\omega}{2}\left[B_{ax}^{\rm
a}(\omega,q)C_{v}^{\rm pn(0)}(k^{\mu})-B_{av}^{\rm a}(k^{\mu})C_{x}^{\rm
pn(0)}(k^{\mu})\right]\right\\},$ (3.23) $\displaystyle
h_{\parallel}=(1-e^{-\beta\omega})^{-1}G_{2}^{-1}\left[B_{ax}^{\rm
a}(k^{\mu})C_{v}^{\rm pn(0)}(k^{\mu})-B_{av}^{\rm a}(k^{\mu})C_{x}^{\rm
pn(0)}(k^{\mu})\right],$ (3.24) $\displaystyle n_{\parallel}^{\rm
dw}-n_{\parallel}^{\rm up}=G_{2}^{-1}\left[B_{ax}^{\rm a}(k^{\mu})C_{v}^{\rm
ig(0)}(\bar{k}^{\mu})+B_{av}^{\rm a}(k^{\mu})C_{x}^{\rm
ig(0)}(\bar{k}^{\mu})\right],$ (3.25)
$\displaystyle\frac{1}{2}(n_{\parallel}^{\rm dw}+n_{\parallel}^{\rm
up})=-G_{1}^{-1}\left[B_{rx}^{\rm a}(k^{\mu})C_{v}^{\rm
ig(0)}(k^{\mu})-B_{rv}^{\rm a}(k^{\mu})C_{x}^{\rm
ig(0)}(k^{\mu})\right]-\frac{1}{2}\coth\frac{\beta\omega}{2}$
$\displaystyle\qquad\qquad\qquad\quad\times
G_{1}^{-1}G_{2}^{-1}G_{3}\left[B_{ax}^{\rm a}(k^{\mu})C_{v}^{\rm
pn(0)}(k^{\mu})-B_{av}^{\rm a}(k^{\mu})C_{x}^{\rm pn(0)}(k^{\mu})\right],$
(3.26)

where

$\displaystyle G_{1}=C_{v}^{\rm pn(0)}(k^{\mu})C_{x}^{\rm
ig(0)}(k^{\mu})-C_{v}^{\rm ig(0)}(k^{\mu})C_{x}^{\rm pn(0)}(k^{\mu}),$
$\displaystyle G_{2}=C_{v}^{\rm pn(0)}(k^{\mu})C_{x}^{\rm
ig(0)}(\bar{k}^{\mu})+C_{v}^{\rm ig(0)}(\bar{k}^{\mu})C_{x}^{\rm
pn(0)}(k^{\mu}),$ $\displaystyle G_{3}=C_{v}^{\rm
ig(0)}(\bar{k}^{\mu})C_{x}^{\rm ig(0)}(k^{\mu})+C_{v}^{\rm
ig(0)}(k^{\mu})C_{x}^{\rm ig(0)}(\bar{k}^{\mu}).$ (3.27)

Here, we use the superscript (0) to denote boundary values of various linearly
independent solutions. It is not difficult to verify that $G_{1}\neq 0$.

Practically, these linearly independent solutions were computed [43] in
Schwarzschild coordinate system, which can be converted to EF coordinate
system through:

$\displaystyle C_{\perp}^{\rm ig}(r,k^{\mu})=\tilde{C}_{\perp}^{\rm
ig}(r,k^{\mu})e^{i\omega\zeta_{s}(r)},~{}\qquad C_{\parallel}^{\rm
ig}(r,k^{\mu})=\tilde{C}_{\parallel}^{\rm
ig}(r,k^{\mu})e^{i\omega\zeta_{s}(r)},$ $\displaystyle C_{\parallel}^{\rm
pg}(r,k^{\mu})=\tilde{C}_{\parallel}^{\rm
pg}(r,k^{\mu})e^{i\omega\zeta_{s}(r)},\qquad C_{\parallel}^{\rm
pn}(r,k^{\mu})=\tilde{C}_{\parallel}^{\rm
pn}(r,k^{\mu})e^{i\omega\zeta_{s}(r)},$ (3.28)

where $s=1$ when $r\in[r_{h}-\epsilon,\infty_{1})$ and $s=2$ when
$r\in[r_{h}-\epsilon,\infty_{2})$. Here, the tilded functions denote linearly
independent solutions in the Schwarzschild coordinate system.

For later calculations, we summarize the hydrodynamic expansion for all
linearly independent solutions when $\omega,q\ll T$. It is convenient to
introduce a new radial coordinate $u$:

$\displaystyle
u=r_{h}^{2}/r^{2}\Longrightarrow\tilde{C}_{\mu}(r,\omega,q)\to\tilde{C}_{\mu}(u,\omega,q).$
(3.29)

For the transverse mode, we just need the ingoing solution

$\displaystyle\tilde{C}_{\perp}^{\rm
ig}(u,\omega,q)=(1-u^{2})^{-i\tilde{\omega}/2}\left\\{1+i\tilde{\omega}\log(1+u)+\frac{1}{24}\pi^{2}(3\tilde{\omega}^{2}-2\tilde{q}^{2})-\frac{1}{4}\tilde{\omega}^{2}\log^{2}2\right.$
$\displaystyle\left.+\frac{1}{2}\tilde{\omega}^{2}\log(1-u)\log\frac{2}{1+u}-\frac{1}{4}\log(1+u)\left[2(\tilde{\omega}^{2}-\tilde{q}^{2})\log
u+\tilde{\omega}^{2}\log(1+u)\right]\right.$
$\displaystyle\left.+\frac{1}{2}(\tilde{q}^{2}-\tilde{\omega}^{2})\left[{\rm
Li}_{2}(1-u)+{\rm Li}_{2}(-u)\right]-\frac{1}{2}\tilde{\omega}^{2}{\rm
Li}_{2}\left(\frac{1+u}{2}\right)+\cdots\right\\},$ (3.30)

where $\rm Li_{2}$ is the Polylogarithm function, and the tilded frequency and
momentum are dimensionless

$\displaystyle\tilde{\omega}\equiv\frac{\omega}{2r_{h}}=\frac{\omega}{2\pi
T},\qquad\tilde{q}\equiv\frac{q}{2r_{h}}=\frac{q}{2\pi T}.$ (3.31)

For the longitudinal sector, the ingoing solution and polynomial solution are

$\displaystyle\tilde{C}_{t}^{\rm ig}(u,\omega,q)$
$\displaystyle=(1-u^{2})^{1-i\tilde{\omega}/2}\left[\frac{i\tilde{q}}{1+u}+\frac{\tilde{\omega}\tilde{q}}{1-u^{2}}\left(\log\frac{2}{1+u}+u\log
u\right)+\cdots\right],$ $\displaystyle\tilde{C}_{x}^{\rm ig}(u,\omega,q)$
$\displaystyle=(1-u^{2})^{-i\tilde{\omega}/2}\left\\{1+i\tilde{\omega}\log\frac{1+u}{2}+\frac{\pi^{2}\tilde{\omega}^{2}}{24}-\frac{1}{2}\tilde{\omega}^{2}\log\frac{1-u}{2}\log\frac{1+u}{2}\right.$
$\displaystyle\left.-\frac{1}{4}\tilde{\omega}^{2}\log^{2}\frac{1+u}{2}-\frac{1}{2}\tilde{\omega}^{2}\log
u\log(1+u)-\frac{1}{2}\tilde{\omega}^{2}{\rm
Li}_{2}(1-u)-\frac{1}{2}\tilde{\omega}^{2}{\rm Li}_{2}(-u)\right.$
$\displaystyle\left.-\frac{1}{2}\tilde{\omega}^{2}{\rm
Li}_{2}\left(\frac{1+u}{2}\right)+\cdots\right\\}.$ (3.32)
$\displaystyle\tilde{C}_{t}^{\rm
pn}(u,\omega,q)=2(1-u)+2\tilde{q}^{2}\left[u\log
u+(1+u)\log\frac{2}{1+u}\right]+\cdots,$ $\displaystyle\tilde{C}_{x}^{\rm
pn}(u,\omega,q)=-\tilde{\omega}\tilde{q}\left\\{\frac{1}{4}(\pi^{2}-2\log^{2}2)+\log
u\log\frac{1+u}{1-u}+\frac{1}{2}\log(1+u)\log\frac{4}{1+u}\right.$
$\displaystyle\qquad\qquad\qquad\qquad\quad\left.-{\rm
Li}_{2}\left(\frac{1-u}{2}\right)+{\rm Li}_{2}(-u)-{\rm
Li}_{2}(u)\right\\}+\cdots.$ (3.33)

Given that the two limits $\epsilon\to 0$ and $\tilde{\omega}\to 0$ do not
commute [1, 43]:

$\displaystyle\lim_{\epsilon\to 0}\lim_{\tilde{\omega}\to
0}(1-u^{2})^{-i\tilde{\omega}/2}\neq\lim_{\tilde{\omega}\to
0}\lim_{\epsilon\to 0}(1-u^{2})^{-i\tilde{\omega}/2},\qquad{\rm as}~{}~{}u\to
1-\epsilon,$ (3.34)

in (3.30), (3.32) and (3.33), we have kept the overall oscillating factor like
$(1-u^{2})^{-i\tilde{\omega}/2}$ (if present) unexpanded in small
$\tilde{\omega}$. Under this treatment, the leading order solutions (3.16) and
(3.21) would be schematically written as

$\displaystyle C_{\perp}^{{\rm
a}(1)}=\mathcal{C}_{\perp}(r,\omega,q)+[f(r)]^{i\tilde{\omega}}\mathcal{H}_{\perp}(r,\omega,q),$
$\displaystyle C_{\parallel}^{{\rm
a}(1)}=\mathcal{C}_{\parallel}(r,\omega,q)+[f(r)]^{i\tilde{\omega}}\mathcal{H}_{\parallel}(r,\omega,q)+[f(r)]^{i\tilde{\omega}/2}\mathcal{G}_{\parallel}(r,\omega,q),$
(3.35)

where $\mathcal{C}_{\perp,\parallel}$, $\mathcal{H}_{\perp,\parallel}$, and
$\mathcal{G}_{\parallel}$ represent regular parts of the leading order
solutions, which are valid to be expanded in terms of
$\tilde{\omega},\tilde{q}$ even near the horizon. Essentially, the results
presented in (3.30), (3.32) and (3.33) correspond to hydrodynamic expansion of
$\mathcal{C}_{\perp,\parallel}$, $\mathcal{H}_{\perp,\parallel}$, and
$\mathcal{G}_{\parallel}$.

###  3.4 Boundary effective action: contour integrals

In this subsection, we compute the radial contour integrals in the bulk action
(3.6), producing the boundary effective action (2.12), which would be expanded
in amplitude of $B_{r\mu}$ and $B_{a\mu}$,

$\displaystyle S_{eff}=S_{eff}^{(2)}+S_{eff}^{(3)}+S_{eff}^{(4)}+\cdots=\int
d^{4}x\left[\mathcal{L}_{eff}^{(2)}+\mathcal{L}_{eff}^{(3)}+\mathcal{L}_{eff}^{(4)}+\cdots\right].$
(3.36)

At each order in amplitude expansion (3.36), we will truncate the action at
first order in hydrodynamic derivative expansion.

Near the two AdS boundaries $r=\infty_{1}$ and $r=\infty_{2}$, the contour
integrals in $S_{0}^{(2)}$, $S_{0}^{(3)}$ and $S_{0}^{(4)}$ suffer from UV
divergences, which are exactly cancelled by the counter-term action like
(2.4), added on each AdS boundary:

$\displaystyle S_{\rm ct}=\frac{1}{4}\log r\,\int
d^{4}x\sqrt{-\gamma}F_{\mu\nu}^{\rm a}F^{{\rm
a}\mu\nu}\bigg{|}_{r=\infty_{1}}-\frac{1}{4}\log r\,\int
d^{4}x\sqrt{-\gamma}F_{\mu\nu}^{\rm a}F^{{\rm
a}\mu\nu}\bigg{|}_{r=\infty_{2}}$ (3.37)

where the extra minus sign for the second term is due to flipping of
$r$-orientation.

$\bullet$ Quadratic Lagrangian $\mathcal{L}_{eff}^{(2)}$

Through integration by part, the $S_{0}^{(2)}$ of (3.7) is reduced into a
surface term [43],

$\displaystyle S_{0}^{(2)}=-\frac{1}{2}\int
d^{4}x\sqrt{-\gamma}n_{M}C_{N}^{\rm a}F^{{\rm
a}MN}\bigg{|}_{r=\infty_{2}}^{r=\infty_{1}}.$ (3.38)

which helps to avoid contour integrals. Up to first order in boundary
derivative expansion, the quadratic action $S_{eff}^{(2)}$ is [43, 1, 29]

$\displaystyle\mathcal{L}_{eff}^{(2)}=$ $\displaystyle\frac{i}{2}B_{av}^{\rm
a}w_{1}B_{av}^{\rm a}+\frac{i}{2}B_{ak}^{\rm a}w_{2}B_{ak}^{\rm
a}+iB_{av}^{\rm a}w_{4}\partial_{k}B_{ak}^{\rm a}+B_{av}^{\rm
a}w_{5}B_{rv}^{\rm a}$ $\displaystyle+\partial_{k}B_{ak}^{\rm
a}w_{7}B_{rv}^{\rm a}+B_{ak}^{\rm a}w_{8}\partial_{v}B_{rk}^{\rm a},$ (3.39)

where

$\displaystyle w_{1}=0+\mathcal{O}(\partial^{2}),~{}~{}~{}\qquad
w_{2}=\frac{2r_{h}^{2}}{\pi}+\mathcal{O}(\partial^{2}),\qquad
w_{4}=0+\mathcal{O}(\partial^{1}),$ $\displaystyle
w_{5}=2r_{h}^{2}+\mathcal{O}(\partial^{2}),\qquad
w_{7}=0+\mathcal{O}(\partial^{1}),\qquad\quad
w_{8}=-r_{h}+\mathcal{O}(\partial^{1}).$ (3.40)

$\bullet$ Strategy of computing $\mathcal{L}_{eff}^{(3)}$ and
$\mathcal{L}_{eff}^{(4)}$

The computation of $S_{0}^{(3)}$ and $S_{0}^{(4)}$ inevitably involves contour
integrals. Due to presence of branch cuts in the integrands, we advance by
splitting the radial contour as

$\displaystyle\int_{\infty_{2}}^{\infty_{1}}dr=\int_{\infty_{2}}^{r_{h}+\epsilon}dr+\int_{\mathcal{C}}dr+\int_{r_{h}+\epsilon}^{\infty_{1}}dr,$
(3.41)

where $\mathcal{C}$ denotes the infinitesimal circle. First, let us examine
the near-horizon behavior for $S_{0}^{(3)}$ and $S_{0}^{(4)}$. Recall that the
leading order solution $C_{v}^{{\rm a}(1)}$ vanishes at the horizon, and the
regular part of the spatial component $C_{i}^{{\rm a}(1)}$ is finite near the
horizon. Thus, it is obvious that the integrands in $S_{0}^{(3)}$ (3.9) and
$S_{0}^{(4)}$ (3.11) are finite when $r$ varies on the infinitesimal circle.
Therefore, in both (3.9) and (3.11) contributions from integrals along the
infinitesimal circle will vanish as $\epsilon\to 0$ is taken in the end.
Eventually, the contour integrals in (3.9) and (3.11) reduce into real-
variable integrals on the interval $[r_{h}+\epsilon,\infty)$, which we
schematically write as,

$\displaystyle S_{0}^{3}$
$\displaystyle=\int\frac{d^{2}k_{1}d^{2}k_{2}d^{2}k_{3}}{(2\pi)^{6}}\delta^{(2)}(k_{1}+k_{2}+k_{3})\mathcal{I}(k_{1},k_{2},k_{3})$
$\displaystyle S_{0}^{4}$
$\displaystyle=\int\frac{d^{2}k_{1}d^{2}k_{2}d^{2}k_{3}d^{2}k_{4}}{(2\pi)^{8}}\delta^{(2)}(k_{1}+k_{2}+k_{3}+k_{4})\mathcal{J}(k_{1},k_{2},k_{3},k_{4})$
(3.42)

with (for convenience we convert to $u$-variable by (3.29))

$\displaystyle\mathcal{I}$
$\displaystyle=\int_{r_{h}+\epsilon}^{\infty}dr\sum_{m}(r-r_{h})^{i\lambda_{m}}\mathcal{I}_{m}(r,k_{1},k_{2},k_{3}),$
$\displaystyle\mathcal{J}$
$\displaystyle=\int_{r_{h}+\epsilon}^{\infty}dr\sum_{m}(r-r_{h})^{i\delta_{m}}\mathcal{J}_{m}(r,k_{1},k_{2},k_{3},k_{4}).$
(3.43)

Here, $\mathcal{I}_{m},\mathcal{J}_{m}$ are constructed from (essentially
products of) regular parts of leading order solution, cf. (3.35). Importantly,
$\mathcal{I}_{m}$ and $\mathcal{J}_{m}$ are regular functions, i.e., they does
not contain singularity over the interval $[r_{h},\infty)$. In (3.43),
$\lambda_{m}$ is certain linear combination of $k_{1}^{0},k_{2}^{0},k_{3}^{0}$
while $\delta_{m}$ is certain linear combination of
$k_{1}^{0},k_{2}^{0},k_{3}^{0},k_{4}^{0}$, whose exact forms will be
irrelevant in general analysis below.

In general, we have two different treatments in extracting hydrodynamic limits
of (3.43). In Scheme I, we will expand regular functions
$\mathcal{I}_{m},\mathcal{J}_{m}$ in terms of four-momentum, but will keep the
oscillating factors $(r-r_{h})^{i\lambda_{m},\,i\delta_{m}}$ unexpanded; then,
once the radial integrals in (3.43) are done, we need another hydrodynamic
expansion. In Scheme II, we expand both oscillating factors
$(1-u)^{i\lambda_{m},\,i\delta_{m}}$ and regular parts
$\mathcal{I}_{m},\mathcal{J}_{m}$ in the hydrodynamic limit, and then perform
the radial integrals (3.43). Generically, thanks to the subtlety (3.34), the
results obtained within these two schemes would not match. Nevertheless, we
observe that up to first order in the boundary derivative expansion, these two
schemes accidentally yield the same results. This observation relies on the
fact that both $\mathcal{I}_{m}$ and $\mathcal{J}_{M}$ can be represented by
Taylor series near the horizon

$\displaystyle\mathcal{I}_{m}=\sum_{n=0}^{\infty}\mathcal{I}_{m}^{(n)}(k_{1},k_{2},k_{3})(r-r_{h})^{n},\qquad\mathcal{J}_{m}=\sum_{n=0}^{\infty}\mathcal{J}_{m}^{(n)}(k_{1},k_{2},k_{3},k_{4})(r-r_{h})^{n},$
(3.44)

which is convergent over the whole interval $r\in[r_{h},\infty]$. Then, the
task of extracting hydrodynamic limits of $S_{0}^{(3)}$ and $S_{0}^{(4)}$
boils down to evaluating the following type integrals in the hydrodynamic
limit

$\displaystyle\mathfrak{C}_{n}=\int dr(r-r_{h})^{i\lambda_{m}}(r-r_{h})^{n}$
(3.45)

Direct calculation gives

$\displaystyle\mathfrak{C}_{n}\big{|}_{\rm
Scheme~{}I}=\sum_{l=0}^{\infty}\frac{(n+1)^{-l-1}(-i\lambda_{m})^{l}\Gamma\left[l+1,-(n+1)\log(r-r_{h})\right]}{\Gamma(l+1)},$
$\displaystyle\mathfrak{C}_{n}\big{|}_{\rm
Scheme~{}II}=\frac{(r-r_{h})^{i\lambda_{m}+n+1}}{i\lambda_{m}+n+1}$ (3.46)

which can be shown to be equivalent in hydrodynamic limit. Here, $\Gamma[s]$
and $\Gamma[s,x]$ are the Euler gamma function and incomplete gamma function,
respectively.

Therefore, in subsequent calculations, it is valid to simply expand
$C_{\mu}^{{\rm a}(1)}$ in the hydrodynamic limit (including the oscillating
factors).

With (3.30), (3.32), and (3.33), we truncate the leading order solution (3.16)
and (3.21) to first order in $\tilde{\omega}$ and $\tilde{q}$:

$\displaystyle C_{v}^{{\rm a}(1)\,{\rm
up}}(r,k^{\mu})=\left[1+i\omega\zeta_{2}(r)\right]\left[2B_{rv}^{\rm
a}F_{2}(r)-B_{av}^{\rm a}F_{2}(r)\right],$ $\displaystyle C_{v}^{{\rm
a}(1)\,{\rm
dw}}(r,k^{\mu})=\left[1+i\omega\zeta_{1}(r)\right]\left[2B_{rv}^{\rm
a}F_{2}(r)+B_{av}^{\rm a}F_{2}(r)\right],$ $\displaystyle C_{i}^{{\rm
a}(1)\,{\rm
up}}(r,k^{\mu})=\left[1+i\omega\zeta_{2}(r)\right]\left[B_{ri}^{\rm
a}+B_{ai}^{\rm
a}\left(F_{1}(r)-\frac{1}{2}\right)\right]+\frac{1}{2}\pi\tilde{\omega}(2B_{ri}^{\rm
a}+B_{ai}^{\rm a})F_{1}(r),$ $\displaystyle C_{i}^{{\rm a}(1)\,{\rm
dw}}(r,k^{\mu})=\left[1+i\omega\zeta_{1}(r)\right]\left[B_{ri}^{\rm
a}+B_{ai}^{\rm
a}\left(F_{1}(r)+\frac{1}{2}\right)\right]+\frac{1}{2}\pi\tilde{\omega}(2B_{ri}^{\rm
a}-B_{ai}^{\rm a})F_{1}(r),$ (3.47)

where $r\in[r_{h}+\epsilon,\infty_{2})$ for solutions on the upper branch and
$r\in[r_{h}+\epsilon,\infty_{1})$ for solutions on the lower branch. The
functions $F_{1,2}(r)$ are

$\displaystyle
F_{1}(r)=\frac{i}{2\pi}\left[\log\left(r^{2}+r_{h}^{2}\right)-\log\left(r^{2}-r_{h}^{2}\right)\right],\qquad\qquad
F_{2}(r)=\frac{r^{2}-r_{h}^{2}}{2r^{2}}.$ (3.48)

From (3.9) and (3.11), the leading order solution $C_{\mu}^{{\rm a}(1)}$
contributes to $S_{0}^{(3)}$ and $S_{0}^{(4)}$ through itself or its boundary
derivatives. Thus, to first order in boundary derivative, the contributions
from $i\omega\zeta_{s}(r)$-terms in (3.47) to $S_{0}^{(3)}$ and $S_{0}^{(4)}$
will vanish, due to the delta-functions in (3.42).

Plugging (3.47) into (3.9) and (3.11), we analytically implement the radial
integrals. Below, we present the final results for $S_{eff}^{(3)}$ and
$S_{eff}^{(4)}$.

$\bullet$ Leading order result of $\mathcal{L}_{eff}^{(3)}$

The cubic effective Lagrangian $\mathcal{L}_{eff}^{(3)}$ starts from first
order in boundary derivative expansion:

$\displaystyle\mathcal{L}_{eff}^{(3)}=$ $\displaystyle-4\lambda_{1}{\rm
tr}\left[B_{\\{rv}B_{ri}\partial_{i}B_{a\\}v}\right]-4\lambda_{2}{\rm
tr}\left[B_{ai}\partial_{i}B_{(av}B_{r)v}\right]-4\lambda_{3}{\rm
tr}[B_{av}B_{ai}\partial_{i}B_{av}]$ $\displaystyle-4\lambda_{4}{\rm
tr}\left[B_{\\{rv}B_{ri}\partial_{v}B_{a\\}i}\right]-4\lambda_{5}{\rm
tr}\left[B_{rv}B_{ai}\partial_{v}B_{ai}+\frac{1}{2}B_{av}B_{(ri}\partial_{v}B_{a)i}\right]$
$\displaystyle-4\lambda_{6}{\rm
tr}(B_{av}B_{ai}\partial_{v}B_{ai})-4\lambda_{7}{\rm
tr}\left[B_{\\{rk}B_{rl}\partial_{k}B_{a\\}l}\right]-4\lambda_{8}{\rm
tr}\left[B_{\\{rk}B_{al}\partial_{k}B_{a\\}l}\right]$
$\displaystyle-4\lambda_{9}{\rm
tr}\left(B_{ak}B_{al}\partial_{k}B_{al}\right).$ (3.49)

Here, $r$-,$a$-indices inside $\\{\cdots\\}$ and $(\cdots)$ shall be
understood as all possible permutations, and symmetrization,
respectively777Space-time indices $v,i$ inside $\\{\cdots\\}$ and $(\cdots)$
are not affected., e.g.

$\displaystyle A_{\\{rra\\}}\equiv A_{rra}+A_{arr}+A_{rar},\qquad\qquad
A_{(ra)}=A_{ra}+A_{ar}.$ (3.50)

The overall factor $4$ in (3.49) comes from the fact that $4\,{\rm tr}(t^{\rm
a}t^{\rm b}t^{\rm c})=i\epsilon^{\rm abc}$. Various coefficients in (3.49) are

$\displaystyle\lambda_{1}=i\log(2r_{h}),\qquad\qquad\,\lambda_{2}=\frac{\pi}{48},\qquad\qquad\,\lambda_{3}=\frac{i}{4}\log(2r_{h}),$
$\displaystyle\lambda_{4}=-\frac{i}{2}\log(2r_{h}^{2}),\qquad\quad\lambda_{5}=-\frac{\pi}{12},\qquad\quad\,\,\lambda_{6}=-\frac{i\zeta(3)}{2\pi^{2}}-\frac{i}{8}\log(2r_{h}^{2}),$
$\displaystyle\lambda_{7}=i\log(r_{h}),\qquad\qquad\,\,\,\,\lambda_{8}=\frac{\pi}{8},\qquad\qquad\,\,\,\,\lambda_{9}=\frac{21i\zeta(3)}{16\pi^{2}}+\frac{i}{4}\log(r_{h}),$
(3.51)

where $\zeta(x)$ is Riemann zeta function. By dimensional analysis, pieces
such as $\log(r_{h})$ etc. shall be understood as $\log(r_{h}/L)$ with $L$ the
AdS radius which we set to unity.

$\bullet$ Derivative expansion of $\mathcal{L}_{eff}^{(4)}$

The quartic effective Lagrangian $\mathcal{L}_{eff}^{(4)}$ starts from zeroth
order in boundary derivative expansion:

$\displaystyle\mathcal{L}_{eff}^{(4)}=\mathcal{L}_{eff}^{(4),\;\rm
LO}+\mathcal{L}_{eff}^{(4),\;\rm NLO}+\cdots.$ (3.52)

For convenience of presentation, $\mathcal{L}_{eff}^{(4),\;\rm LO}$ may be
split into four pieces:

$\displaystyle\mathcal{L}_{eff}^{(4),\;\rm
LO}=\mathscr{L}_{1}+\mathscr{L}_{2}+\mathscr{L}_{3}+\mathscr{L}_{4},$ (3.53)

where

$\displaystyle\mathscr{L}_{1}=$ $\displaystyle\chi_{1,1}\left(B_{rv}^{\rm
a}B_{rv}^{\rm a}B_{ri}^{\rm b}B_{ai}^{\rm b}+B_{rv}^{\rm a}B_{av}^{\rm
a}B_{ri}^{\rm b}B_{ri}^{\rm b}\right)+\chi_{1,2}\left(B_{rv}^{\rm
a}B_{rv}^{\rm a}B_{ai}^{\rm b}B_{ai}^{\rm b}+B_{rv}^{\rm a}B_{av}^{\rm
a}B_{ri}^{\rm b}B_{ai}^{\rm b}\right.$ $\displaystyle\left.+B_{av}^{\rm
a}B_{rv}^{\rm a}B_{ri}^{\rm b}B_{ai}^{\rm
b}\right)+\chi_{1,3}\left(B_{rv}^{\rm a}B_{av}^{\rm a}B_{ai}^{\rm
b}B_{ai}^{\rm b}+B_{av}^{\rm a}B_{av}^{\rm a}B_{ri}^{\rm b}B_{ai}^{\rm
b}\right)+\chi_{1,4}B_{rv}^{\rm a}B_{av}^{\rm a}B_{ai}^{\rm b}B_{ai}^{\rm b}$
$\displaystyle+\chi_{1,5}B_{av}^{\rm a}B_{av}^{\rm a}B_{ai}^{\rm b}B_{ai}^{\rm
b},$ $\displaystyle\mathscr{L}_{2}=$ $\displaystyle\chi_{2,1}\left(B_{rv}^{\rm
a}B_{rv}^{\rm b}B_{ri}^{\rm a}B_{ai}^{\rm b}+B_{rv}^{\rm a}B_{av}^{\rm
b}B_{ri}^{\rm a}B_{ri}^{\rm b}\right)+\chi_{2,2}\left(B_{rv}^{\rm
a}B_{rv}^{\rm b}B_{ai}^{\rm a}B_{ai}^{\rm b}+B_{rv}^{\rm a}B_{av}^{\rm
b}B_{ri}^{\rm a}B_{ai}^{\rm b}\right.$ $\displaystyle\left.+B_{av}^{\rm
a}B_{rv}^{\rm b}B_{ri}^{\rm a}B_{ai}^{\rm
b}\right)+\chi_{2,3}\left(B_{rv}^{\rm a}B_{av}^{\rm b}B_{ai}^{\rm
a}B_{ai}^{\rm b}+B_{av}^{\rm a}B_{av}^{\rm b}B_{ri}^{\rm a}B_{ai}^{\rm
b}\right)+\chi_{2,4}B_{rv}^{\rm a}B_{av}^{\rm b}B_{ai}^{\rm a}B_{ai}^{\rm b}$
$\displaystyle+\chi_{2,5}B_{av}^{\rm a}B_{av}^{\rm b}B_{ai}^{\rm a}B_{ai}^{\rm
b},$ $\displaystyle\mathscr{L}_{3}=$ $\displaystyle\chi_{3,1}B_{ri}^{\rm
a}B_{ri}^{\rm a}B_{rj}^{\rm b}B_{aj}^{\rm b}+\chi_{3,2}\left(B_{ri}^{\rm
a}B_{ri}^{\rm a}B_{aj}^{\rm b}B_{aj}^{\rm b}+B_{ri}^{\rm a}B_{ai}^{\rm
a}B_{rj}^{\rm b}B_{aj}^{\rm b}+B_{ai}^{\rm a}B_{ri}^{\rm a}B_{rj}^{\rm
b}B_{aj}^{\rm b}\right)$ $\displaystyle+\chi_{3,3}B_{ri}^{\rm a}B_{ai}^{\rm
a}B_{aj}^{\rm b}B_{aj}^{\rm b}+\chi_{3,4}B_{ai}^{\rm a}B_{ai}^{\rm
a}B_{aj}^{\rm b}B_{aj}^{\rm b},$ $\displaystyle\mathscr{L}_{4}=$
$\displaystyle\chi_{4,1}B_{ri}^{\rm a}B_{ri}^{\rm b}B_{rj}^{\rm a}B_{aj}^{\rm
b}+\chi_{4,2}\left(B_{ri}^{\rm a}B_{ri}^{\rm b}B_{aj}^{\rm a}B_{aj}^{\rm
b}+B_{ri}^{\rm a}B_{ai}^{\rm b}B_{rj}^{\rm a}B_{aj}^{\rm b}+B_{ai}^{\rm
a}B_{ri}^{\rm b}B_{rj}^{\rm a}B_{aj}^{\rm b}\right)$
$\displaystyle+\chi_{4,3}B_{ri}^{\rm a}B_{ai}^{\rm b}B_{aj}^{\rm a}B_{aj}^{\rm
b}+\chi_{4,4}B_{ai}^{\rm a}B_{ai}^{\rm b}B_{aj}^{\rm a}B_{aj}^{\rm b},$ (3.54)

where various coefficients are

$\displaystyle\chi_{1,1}=-\chi_{2,1}=-\log(2r_{h}),\qquad\chi_{1,2}=-\chi_{2,2}=\frac{i\pi}{48},\qquad\chi_{1,3}=-\chi_{2,3}=-\frac{1}{4}\log(2r_{h}),$
$\displaystyle\chi_{1,4}=-\chi_{2,4}=-\frac{\zeta(3)}{16\pi^{2}},\qquad\quad\chi_{1,5}=-\chi_{2,5}=\frac{i\pi}{192},\qquad\chi_{3,1}=-\chi_{4,1}=\log(r_{h}),$
$\displaystyle\chi_{3,2}=-\chi_{4,2}=-\frac{i\pi}{16},\qquad\chi_{3,3}=-\chi_{4,3}=\frac{1}{4}\log(r_{h})+\frac{21\zeta(3)}{16\pi^{2}},\qquad\chi_{3,4}=-\chi_{4,4}=-\frac{i\pi}{128}.$
(3.55)

Similarly, the next-to-leading order result $\mathcal{L}_{eff}^{(4),\;\rm
NLO}$ is also split into four pieces

$\displaystyle\mathcal{L}_{eff}^{(4),\;\rm
NLO}=\tilde{\mathscr{L}}_{1}+\tilde{\mathscr{L}}_{2}+\tilde{\mathscr{L}}_{3}+\tilde{\mathscr{L}}_{4},$
(3.56)

where

$\displaystyle\tilde{\mathscr{L}}_{1}=$
$\displaystyle\tilde{\chi}_{1,1}\left(B_{rv}^{\rm a}B_{rv}^{\rm a}B_{ai}^{\rm
b}\partial_{v}B_{ri}^{\rm b}-B_{rv}^{\rm a}B_{rv}^{\rm a}B_{ri}^{\rm
b}\partial_{v}B_{ai}^{\rm b}+2B_{rv}^{\rm a}B_{av}^{\rm a}B_{ri}^{\rm
b}\partial_{v}B_{ri}^{\rm b}\right)$
$\displaystyle+\tilde{\chi}_{1,2}\left(-B_{rv}^{\rm a}B_{rv}^{\rm
a}B_{ai}^{\rm b}\partial_{v}B_{ai}^{\rm b}+2B_{rv}^{\rm a}B_{av}^{\rm
a}B_{ai}^{\rm b}\partial_{v}B_{ri}^{\rm
b}\right)+\tilde{\chi}_{1,3}\left(-\frac{1}{2}B_{rv}^{\rm a}B_{av}^{\rm
a}B_{ai}^{\rm b}\partial_{v}B_{ai}^{\rm b}\right.$
$\displaystyle\left.+\frac{1}{4}B_{av}^{\rm a}B_{av}^{\rm a}B_{ai}^{\rm
b}\partial_{v}B_{ri}^{\rm b}-\frac{1}{4}B_{av}^{\rm a}B_{av}^{\rm
a}B_{ri}^{\rm b}\partial_{v}B_{ai}^{\rm
b}\right)+\tilde{\chi}_{1,4}B_{av}^{\rm a}B_{av}^{\rm a}B_{ai}^{\rm
b}\partial_{v}B_{ai}^{\rm b},$ $\displaystyle\tilde{\mathscr{L}}_{2}=$
$\displaystyle\tilde{\chi}_{2,1}\left(B_{rv}^{\rm a}B_{rv}^{\rm b}B_{ai}^{\rm
a}\partial_{v}B_{ri}^{\rm b}-B_{rv}^{\rm a}B_{rv}^{\rm b}B_{ri}^{\rm
a}\partial_{v}B_{ai}^{\rm b}+B_{rv}^{\rm a}B_{av}^{\rm
b}\partial_{v}B_{ri}^{\rm a}B_{ri}^{\rm b}+B_{rv}^{\rm a}B_{av}^{\rm
b}B_{ri}^{\rm a}\partial_{v}B_{ri}^{\rm b}\right)$
$\displaystyle+\tilde{\chi}_{2,2}\left(-B_{rv}^{\rm a}B_{rv}^{\rm
b}\partial_{v}B_{ai}^{\rm a}B_{ai}^{\rm b}+B_{rv}^{\rm a}B_{av}^{\rm
b}\partial_{v}B_{ri}^{\rm a}B_{ai}^{\rm b}+B_{rv}^{\rm a}B_{av}^{\rm
b}B_{ai}^{\rm a}\partial_{v}B_{ri}^{\rm b}\right)$
$\displaystyle+\tilde{\chi}_{2,3}\left(-\frac{1}{4}B_{rv}^{\rm a}B_{av}^{\rm
b}\partial_{v}B_{ai}^{\rm a}B_{ai}^{\rm b}-\frac{1}{4}B_{rv}^{\rm
a}B_{av}^{\rm b}B_{ai}^{\rm a}\partial_{v}B_{ai}^{\rm
b}+\frac{1}{4}B_{av}^{\rm a}B_{av}^{\rm b}B_{ai}^{\rm
a}\partial_{v}B_{ri}^{\rm b}\right.$
$\displaystyle\left.-\frac{1}{4}B_{av}^{\rm a}B_{av}^{\rm b}B_{ri}^{\rm
a}\partial_{v}B_{ai}^{\rm b}\right)+\tilde{\chi}_{2,4}B_{av}^{\rm
a}B_{av}^{\rm b}\partial_{v}B_{ai}^{\rm a}B_{ai}^{\rm b},$
$\displaystyle\tilde{\mathscr{L}}_{3}=$
$\displaystyle\tilde{\chi}_{3,1}B_{ri}^{\rm a}B_{ri}^{\rm a}B_{rj}^{\rm
b}\partial_{v}B_{aj}^{\rm b}+\tilde{\chi}_{3,2}\left(\partial_{v}B_{ri}^{\rm
a}B_{ri}^{\rm a}B_{aj}^{\rm b}B_{aj}^{\rm b}+\partial_{v}B_{ri}^{\rm
a}B_{ai}^{\rm a}B_{rj}^{\rm b}B_{aj}^{\rm b}+\partial_{v}B_{ri}^{\rm
a}B_{ai}^{\rm a}B_{aj}^{\rm b}B_{rj}^{\rm b}\right),$
$\displaystyle\tilde{\mathscr{L}}_{4}=$
$\displaystyle\tilde{\chi}_{4,1}B_{ri}^{\rm a}B_{ri}^{\rm b}B_{rj}^{\rm
a}\partial_{v}B_{aj}^{\rm b}+\tilde{\chi}_{4,2}\left(\partial_{v}B_{ri}^{\rm
a}B_{ri}^{\rm b}B_{aj}^{\rm a}B_{aj}^{\rm b}+\partial_{v}B_{ri}^{\rm
a}B_{ai}^{\rm b}B_{aj}^{\rm a}B_{rj}^{\rm b}+B_{ai}^{\rm
a}\partial_{v}B_{ri}^{\rm b}B_{aj}^{\rm a}B_{rj}^{\rm b}\right),$ (3.57)

where various coefficients are

$\displaystyle\tilde{\chi}_{1,1}=-\tilde{\chi}_{2,1}=-\frac{\pi^{2}\beta}{48},\qquad\tilde{\chi}_{1,2}=-\tilde{\chi}_{2,2}=-\frac{i\zeta(3)\beta}{16\pi},\qquad\tilde{\chi}_{1,3}=-\tilde{\chi}_{2,3}=-\frac{\pi^{2}\beta}{48},$
$\displaystyle\tilde{\chi}_{1,4}=-\tilde{\chi}_{2,4}=\frac{i\zeta(3)\beta}{32\pi},\qquad\,\tilde{\chi}_{3,1}=-\tilde{\chi}_{4,1}=-\frac{\pi^{2}\beta}{8},\qquad\quad\tilde{\chi}_{3,2}=-\tilde{\chi}_{4,2}=\frac{i21\zeta(3)\beta}{16\pi},$
(3.58)

where $\beta$ is the inverse temperature.

We compare our results with recent works on hydrodynamic EFT for charge
diffusion, see e.g. [2, 54, 55, 14, 17, 53]. The main novelty of present work
could be identified as the following two aspects:

First, our effective action is more complete in systematically capturing
nonlinear interactions among noise variable $\varphi_{a}^{\rm a}$, as well as
nonlinear interactions between noise variable $\varphi_{a}^{\rm a}$ and
dynamical variable $\varphi_{r}^{\rm a}$. In our effective action $S_{eff}$,
these are represented by terms cubic and/or quartic in $B_{a\mu}^{\rm a}$,
which are not considered in [53] (see also [54, 55, 14] for U(1) diffusion).
As pointed out in [2], nonlinear interactions of this type cannot be covered
in stochastic formulation of hydrodynamics. Therefore, it will be interesting
to explore non-negligible signatures, generated by these nonlinear
interactions, in hydrodynamic limit of correlators, following the example of
U(1) diffusion [17].

Second, derived within a holographic model, our effective action $S_{eff}$
automatically accounts for both thermal fluctuation and quantum fluctuation,
in contrast with [2, 14, 17, 53] which focused on thermal noise. A quantum
hydrodynamic EFT for U(1) diffusion was considered in [54, 55], which
implemented the dynamical KMS symmetry at quantum level. It would be
interesting to clarify consequences of quantum fluctuations based on quantum
hydrodynamic EFT as constructed here by us and in [54, 55].

Finally, our results perfectly pass through various consistency checks:

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

Basically, this requires the effective action $S_{eff}$ to satisfy

$\displaystyle S_{eff}^{*}[B_{1\mu}^{\rm a}(x),B_{2\mu}^{\rm
a}(x)]=-S_{eff}[B_{2\mu}^{\rm a}(x),B_{1\mu}^{\rm a}(x)]$
$\displaystyle\Leftrightarrow$ $\displaystyle S_{eff}^{*}[B_{r\mu}^{\rm
a}(x),B_{a\mu}^{\rm a}(x)]=-S_{eff}[B_{r\mu}^{\rm a}(x),-B_{a\mu}^{\rm
a}(x)],$ (3.59)

which implies that the effective Lagrangian must have some complex
coefficients. When field contents are real variables (as in our case), the
$Z_{2}$ reflection symmetry (3.59) requires that coefficient of a term
containing even number of $a$-type variables must be purely imaginary, while
coefficient of a term containing odd number of $a$-type variables must be
purely real. This is perfectly satisfied by our results.

$\bullet$ Imaginary part of $S_{eff}$ is non-negative.

This requirement is to ensure that path integral based on effective action
$S_{eff}$ is well-defined. For quadratic Lagrangian $\mathcal{L}_{eff}^{(2)}$,
this is perfectly satisfied: leading order results of $w_{1},w_{2}$ are non-
negative, see (3.40). For cubic Lagrangian $\mathcal{L}_{eff}^{(3)}$, this
requirement does not give any constraint. At quartic order, we collect
positive-definite structures

$\displaystyle\mathcal{L}_{\rm pd}^{(4)}$ $\displaystyle=\chi_{1,2}B_{rv}^{\rm
a}B_{rv}^{\rm a}B_{ai}^{\rm b}B_{ai}^{\rm b}+\chi_{2,2}B_{rv}^{\rm
a}B_{rv}^{\rm b}B_{ai}^{\rm a}B_{ai}^{\rm b}+\chi_{1,5}B_{av}^{\rm
a}B_{av}^{\rm a}B_{ai}^{\rm b}B_{ai}^{\rm b}$
$\displaystyle+\chi_{2,5}B_{av}^{\rm a}B_{av}^{\rm b}B_{ai}^{\rm a}B_{ai}^{\rm
b}+\chi_{3,2}B_{ri}^{\rm a}B_{ri}^{\rm a}B_{aj}^{\rm b}B_{aj}^{\rm
b}+\chi_{4,2}B_{ri}^{\rm a}B_{ri}^{\rm b}B_{aj}^{\rm a}B_{aj}^{\rm b}$
$\displaystyle+\chi_{3,4}B_{ai}^{\rm a}B_{ai}^{\rm a}B_{ai}^{\rm b}B_{ai}^{\rm
b}+\chi_{4,4}B_{ai}^{\rm a}B_{ai}^{\rm b}B_{aj}^{\rm a}B_{aj}^{\rm b},$ (3.60)

where each term (apart of the coefficient) is non-negative for any
configuration of $B_{r\mu}^{\rm a},B_{a\mu}^{\rm a}$. However, our results for
the $\chi$’s in (3.60) are not all positive-definite. It is important to
stress that this does not necessarily mean our results are pathological.
Recall that our procedure of constructing $S_{eff}$ is based on perturbative
expansion around origin of field configuration. Thus, our results are
meaningful when $B_{r\mu}^{\rm a},B_{a\mu}^{\rm a}$ are tiny, in which region
we always have

$\displaystyle\frac{1}{2}w_{1}^{0}B_{av}^{\rm a}B_{av}^{\rm
a}+\frac{1}{2}w_{2}^{0}B_{ai}^{\rm a}B_{ai}^{\rm a}\gg\big{|}\mathcal{L}_{\rm
pd}^{(4)}\big{|},$ (3.61)

which guarantees the path integral to be well-defined. Here,
$w_{1}^{0},w_{2}^{0}$ are leading order results for $w_{1},w_{2}$, see (3.40).
Not surprisingly, the fact that $\mathcal{L}_{\rm pd}^{(4)}$ is not positive-
definite for “big” configuration of $B_{r\mu}^{\rm a},B_{a\mu}^{\rm a}$
implies instabilities of the system, which we believe are related to those
revealed in [68, 69, 70]. It is of great interest to explore the exact
relationship, and particularly construct an EFT for the order parameter
associated with phase transitions investigated in [68, 69, 70], along the line
of [44]. To this end, dynamical variable in the EFT will be identified with
“normalizable modes” of bulk field $C_{\mu}^{\rm a}$, rather than non-
normalizable modes employed here. We leave further investigation on this
subject for future work.

$\bullet$ Time-independent diagonal gauge transformation of $B_{s\mu}$:

$\displaystyle
B_{1v}\to\mathcal{U}(\vec{x})B_{1v}\mathcal{U}^{\dagger}(\vec{x}),\qquad\qquad\qquad\qquad\quad
B_{2v}\to\mathcal{U}(\vec{x})B_{2v}\mathcal{U}^{\dagger}(\vec{x}),$
$\displaystyle
B_{1i}\to\mathcal{U}(\vec{x})B_{1i}\mathcal{U}^{\dagger}(\vec{x})+i\,\mathcal{U}(\vec{x})\partial_{i}\mathcal{U}^{\dagger}(\vec{x}),\qquad
B_{2i}\to\mathcal{U}(\vec{x})B_{2i}\mathcal{U}^{\dagger}(\vec{x})+i\,\mathcal{U}(\vec{x})\partial_{i}\mathcal{U}^{\dagger}(\vec{x}),$
(3.62)

where $\mathcal{U}(\vec{x})=e^{i\alpha^{\rm a}(\vec{x})t^{\rm a}}$ is an
element of the SU(2) group for the boundary theory. Equivalently, the above
symmetry requirement is

$\displaystyle
B_{rv}\to\mathcal{U}(\vec{x})B_{rv}\mathcal{U}^{\dagger}(\vec{x}),\qquad
B_{a\mu}\to\mathcal{U}(\vec{x})B_{a\mu}\mathcal{U}^{\dagger}(\vec{x}),$
$\displaystyle
B_{ri}\to\mathcal{U}(\vec{x})B_{ri}\mathcal{U}^{\dagger}(\vec{x})+i\mathcal{U}(\vec{x})\partial_{i}\mathcal{U}^{\dagger}(\vec{x}),$
(3.63)

which relates certain coefficients in $\mathcal{L}_{eff}^{(3)}$ and
$\mathcal{L}_{eff}^{(4),\,\rm LO}$:

$\displaystyle\lambda_{1}=-i\chi_{1,1}=i\chi_{2,1},\qquad\lambda_{2}=-i\chi_{1,2}=i\chi_{2,2},\qquad\quad\lambda_{3}=-i\chi_{1,3}=i\chi_{2,3}$
$\displaystyle\lambda_{7}=i\chi_{3,1}=-i\chi_{4,1},\qquad\lambda_{8}=2i\chi_{3,2}=-2i\chi_{4,2},\qquad\lambda_{9}=i\chi_{3,3}=-i\chi_{4,3},$
(3.64)

which are perfectly obeyed by holographic results (3.51) and (3.55).

Indeed, with the help of covariant derivative operator
$\mathcal{D}_{i}\equiv\partial_{i}-i[B_{ri},~{}~{}]$ as invented in [53], our
results could be reorganized into a more compact form888We have not considered
putting $\mathcal{L}_{eff}^{(4),\,\rm NLO}$ into a form bearing symmetry
(3.63) transparently, which would require some second and third order
derivative terms that are beyond scope of present work.:

$\displaystyle\mathcal{L}_{eff}^{(2)}+\mathcal{L}_{eff}^{(3)}+\mathcal{L}_{eff}^{(4),\,\rm
LO}=\tilde{\mathcal{L}}_{eff}^{(2)}+\tilde{\mathcal{L}}_{eff}^{(3)}+\tilde{\mathcal{L}}_{eff}^{(4),\,\rm
LO},$ (3.65)

where

$\displaystyle\tilde{\mathcal{L}}_{eff}^{(2)}=$
$\displaystyle\frac{i}{2}B_{av}^{\rm a}w_{1}B_{av}^{\rm
a}+\frac{i}{2}B_{ai}^{\rm a}w_{2}B_{ai}^{\rm
a}+\frac{i}{2}\mathcal{D}_{i}B_{ai}^{\rm a}w_{3}\mathcal{D}_{j}B_{aj}^{\rm
a}+iB_{av}^{\rm a}w_{4}\mathcal{D}_{k}B_{ai}^{\rm a}+B_{av}^{\rm
a}w_{5}B_{rv}^{\rm a}$ $\displaystyle+B_{av}^{\rm
a}w_{6}\mathcal{D}_{i}\partial_{v}B_{ri}^{\rm a}+\mathcal{D}_{i}B_{ai}^{\rm
a}w_{7}B_{rv}^{\rm a}+B_{ai}^{\rm a}w_{8}\partial_{v}B_{ri}^{\rm
a}+\mathcal{D}_{i}B_{aj}^{\rm a}w_{9}\mathcal{F}_{rij}^{\rm a},$ (3.66)
$\displaystyle\tilde{\mathcal{L}}_{eff}^{(3)}=$
$\displaystyle-i\lambda_{1}\epsilon^{\rm abc}B_{rv}^{\rm a}B_{ai}^{\rm
b}\mathcal{D}_{i}B_{rv}^{\rm c}-i\lambda_{2}\epsilon^{\rm abc}(B_{rv}^{\rm
a}B_{ai}^{\rm b}\mathcal{D}_{i}B_{av}^{\rm c}+B_{av}^{\rm a}B_{ai}^{\rm
b}\mathcal{D}_{i}B_{rv}^{\rm c})$ $\displaystyle-i\lambda_{3}\epsilon^{\rm
abc}B_{av}^{\rm a}B_{ai}^{\rm b}\mathcal{D}_{i}B_{av}^{\rm
c}-2i\lambda_{4}\epsilon^{\rm abc}B_{rv}^{\rm a}B_{ai}^{\rm
b}\partial_{v}B_{ri}^{\rm c}-i\lambda_{5}\epsilon^{\rm abc}(B_{rv}^{\rm
a}B_{ai}^{\rm b}\partial_{v}B_{ai}^{\rm c}$ $\displaystyle+B_{av}^{\rm
a}B_{ai}^{\rm b}\partial_{v}B_{ri}^{\rm c})-i\lambda_{6}\epsilon^{\rm
abc}B_{av}^{\rm a}B_{ai}^{\rm b}\partial_{v}B_{ai}^{\rm
c}-i\lambda_{8}\epsilon^{\rm abc}B_{ai}^{\rm a}B_{aj}^{\rm
b}\mathcal{F}_{rij}^{\rm c}$ $\displaystyle-i\lambda_{9}\epsilon^{\rm
abc}B_{ai}^{\rm a}B_{aj}^{\rm b}\mathcal{D}_{i}B_{aj}^{\rm c},$ (3.67)
$\displaystyle\tilde{\mathcal{L}}_{eff}^{(4),\,\rm LO}=$
$\displaystyle\chi_{1,2}B_{rv}^{\rm a}B_{rv}^{\rm a}B_{ai}^{\rm b}B_{ai}^{\rm
b}+(\chi_{1,3}+\chi_{1,4})B_{rv}^{\rm a}B_{av}^{\rm a}B_{ai}^{\rm
b}B_{ai}^{\rm b}+\chi_{1,5}B_{av}^{\rm a}B_{av}^{\rm a}B_{ai}^{\rm
b}B_{ai}^{\rm b}$ $\displaystyle+\chi_{2,2}B_{rv}^{\rm a}B_{rv}^{\rm
b}B_{ai}^{\rm a}B_{ai}^{\rm b}+(\chi_{2,3}+\chi_{2,4})B_{rv}^{\rm
a}B_{av}^{\rm b}B_{ai}^{\rm a}B_{ai}^{\rm b}+\chi_{2,5}B_{av}^{\rm
a}B_{av}^{\rm b}B_{ai}^{\rm a}B_{ai}^{\rm b}$
$\displaystyle+\chi_{3,4}B_{ai}^{\rm a}B_{ai}^{\rm a}B_{aj}^{\rm b}B_{aj}^{\rm
b}+\chi_{4,4}B_{ai}^{\rm a}B_{ai}^{\rm b}B_{aj}^{\rm a}B_{aj}^{\rm b},$ (3.68)

where all the coefficients are constants except for the following ones in
$\tilde{\mathcal{L}}_{eff}^{(2)}$

$\displaystyle
w_{m}=w_{m}^{0}+w_{m}^{1}\partial_{v}+w_{m}^{2,0}\partial_{v}^{2}+w_{m}^{0,2}\mathcal{D}^{2},\qquad
m=1,2,5$ $\displaystyle w_{n}=w_{n}^{0}+w_{n}^{1}\partial_{v},\qquad n=4,7,8.$
(3.69)

In order to achieve (3.66), we have utilized those results for second order
derivative terms for quadratic action of [43], namely, the values for
$w_{m}^{2,0},w_{m}^{0,2}$ ($m=1,2,5$), $w_{n}^{1}$ ($n=4,7,8$), and $w_{9}$.
As already noticed in [53], the “building blocks”, say $B_{a\mu}$,
$\mathcal{D}_{i}B_{a\mu}$, $B_{rv}$, $\mathcal{D}_{i}B_{rv}$,
$\partial_{v}B_{ri}$,
$\mathcal{F}_{rij}\equiv\partial_{i}B_{rj}-\partial_{j}B_{ri}-i[B_{ri},B_{rj}]$,
transform in a simple way under (3.63), schematically
$(\cdots)\to\mathcal{U}(\vec{x})(\cdots)\mathcal{U}^{\dagger}(\vec{x})$. Thus,
the symmetry (3.63) becomes more manifest for (3.65).

$\bullet$ Dynamical KMS symmetry at quantum level.

The proposal of dynamical KMS symmetry [4] is crucial in formulating non-
equilibrium EFT for dissipative fluids. While it is usually implemented in the
classical statistical limit (i.e., capturing only thermal fluctuations), we
find that our effective action $S_{eff}$ derived from holographic model
satisfies dynamical KMS symmetry at quantum level, which implies the boundary
system, although in the large $N_{c}$ limit, is actually a quantum system as
expected. We defer more details to subsection 3.5.

$\bullet$ Current constitutive relation in the mean-field limit.

With the effective action $S_{eff}$, we compute the hydrodynamic constitutive
relation by taking mean-field limit [2, 53]:

$\displaystyle J^{{\rm a}\,\mu}_{\rm mf}\equiv\frac{\delta S_{eff}}{\delta
A_{\mu}^{\rm a}}\bigg{|}_{B_{a\mu}=0}.$ (3.70)

When the SU(2) background electromagnetic field is turned off, the spatial
component of the hydrodynamic current is

$\displaystyle J^{{\rm a}\,i}_{\rm mf}=w_{8}^{0}\partial_{i}\mu^{\rm
a}+i(2\lambda_{4}-\lambda_{1})\epsilon^{\rm abc}\mu^{\rm
b}\partial_{i}\mu^{\rm c}+\tilde{\chi}_{2,1}\left(\mu^{a}\mu^{\rm
b}\partial_{i}\mu^{\rm b}-\mu^{\rm b}\mu^{\rm b}\partial_{i}\mu^{\rm
a}\right),$ (3.71)

where the SU(2) chemical potential is defined as [53]

$\displaystyle\mu\equiv\mathcal{U}^{-1}(\varphi)B_{rv}\mathcal{U}(\varphi)=A_{v}-i\mathcal{U}^{-1}(\varphi)\partial_{v}\mathcal{U}(\varphi)$
(3.72)

Before comparing (3.71) with relevant results of [47, 53], we clarify
similarities/differences between the models studied in [47, 53] and the one
explored here. Apart of backreaction effect, our holographic model is the same
as that of [47]. While non-Abelian SU(2) symmetry is essential in both [47,
53] and present work, the conserved non-Abelian charges are identified as spin
densities in [53], which is different from the flavor charges in [47] and this
work. Now, we turn to compare our result with [47, 53]. Our result (3.71)
slightly differs from that of [53], particularly on the structure of the last
term. However, (3.71) is fully consistent with [47], which derived the current
constitutive relation by fluid-gravity correspondence.

###  3.5 Dynamical KMS symmetry: from boundary to bulk

With the effective action $S_{eff}[B_{r},B_{a}]$, the partition function of
dual boundary theory is represented as a path integral over gapless modes (for
notational simplification we omitted SU(2) flavor indices):

$\displaystyle Z_{\rm
CFT}=\int[D\varphi_{r}][D\varphi_{a}]e^{iS_{eff}[B_{r\mu},B_{a\mu}]}.$ (3.73)

The doubling of degrees of freedom, due to usage of SK formalism, guarantees
systematic inclusion of both fluctuations and dissipations in the boundary
theory. The information of state is reflected in coefficients of effective
action $S_{eff}$. It turns out that a path integral like (3.73) actually
corresponds to quantum field theory of a statistical system, in which both
statistical fluctuations and quantum fluctuations are consistently covered.

When the boundary system is in a thermal state, the KMS condition sets
important constraint on the generating functional $W=-i\log Z_{\rm CFT}$. The
KMS condition can be expressed in terms of $n$-point correlation functions
(i.e., functional derivatives of $W$ with respect to external sources
$A_{s\mu}$), generalizing familiar FDT to nonlinear case [71, 72] (see also
[2]). Obviously, the KMS condition and the generalized nonlinear FDT are valid
at the full quantum level. Within non-equilibrium EFT framework, KMS condition
is guaranteed by the proposal that non-equilibrium effective action $S_{eff}$
shall satisfy dynamical KMS symmetry [3, 4]:

$\displaystyle
S_{eff}[B_{1\mu},B_{2\mu}]=S_{eff}[\widehat{B}_{1\mu},\widehat{B}_{2\mu}],$
(3.74)

where

$\displaystyle\widehat{B}_{1\mu}(-v,-\vec{x})=(-1)^{\eta_{\mu}}B_{1\mu}(v,\vec{x}),\qquad\qquad\widehat{B}_{2\mu}(-v,-\vec{x})=(-1)^{\eta_{\mu}}B_{2\mu}(v-i\beta,\vec{x}),$
(3.75)

where $\beta$ is the inverse temperature, and $(-1)^{\eta_{\mu}}$ is the
eigenvalue of discrete symmetry transformation $\Theta$ (containing time-
reversal $\mathcal{T}$) acting on $B_{\mu}$. Physically, the dynamical KMS
symmetry (3.75) plays the role of imposing microscopic time-reversibility and
local equilibrium [3]. Interestingly, it was discovered that by taking
classical statistical limit [3, 4], the theory (3.73) can be consistently
truncated into a classical statistical theory, in which only statistical
thermal fluctuations survive. To this end, one restores the Planck constant
$\hbar$ by substituting $\beta\to\hbar\beta$ in (3.75) and writes

$\displaystyle B_{1\mu}=B_{r\mu}+\frac{\hbar}{2}B_{a\mu},\qquad
B_{2\mu}=B_{r\mu}-\frac{\hbar}{2}B_{a\mu}.$ (3.76)

Then, taking the limit $\hbar\to 0$, one obtains classical statistical limit
of the dynamical KMS symmetry (3.75)

$\displaystyle\widehat{B}_{r\mu}(-v,-\vec{x})=(-1)^{\eta_{\mu}}B_{r\mu}(v,\vec{x}),\quad\widehat{B}_{a\mu}(-v,-\vec{x})=(-1)^{\eta_{\mu}}\left[B_{a\mu}(v,\vec{x})+i\beta\partial_{0}B_{r\mu}(v,\vec{x})\right].$
(3.77)

In the comprehensive studies of [2, 3, 4], it is indeed the classical
statistical limit (3.77) that was implemented for the construction of
hydrodynamic EFT for dissipative charged fluids. Thus, the effective theory
constructed in [2, 3, 4] covers statistical thermal fluctuations but ignores
quantum ones. Later on, this was refined in [56] which proposed a quantum
hydrodynamic theory (valid at finite $\hbar$ and to all orders in derivatives)
for maximally chaotic systems (see also [54, 55] for the problem of U(1)
diffusion).

Now we turn to the effective action $S_{eff}$ derived within a specific
holographic model. While our derivation is carried out in the large $N_{c}$
limit such that the dual gravity becomes classical, this does not necessarily
mean the dual boundary theory will only capture thermal fluctuations. Thus, we
do not expect our effective action $S_{eff}$ to obey the classical statistical
limit (3.77). Moreover, the derivative expansion adopted in the holographic
derivation corresponds to $\beta$-expansion on the boundary theory. Thus,
instead of the $\hbar$-expansion, it is reasonable to consider
$\beta$-expansion of (3.75)

$\displaystyle\widehat{B}_{r\mu}(-v,-\vec{x})=(-1)^{\eta_{\mu}}\left[B_{r\mu}(v,\vec{x})-\frac{i}{2}\hbar\beta\partial_{0}B_{r\mu}(v,\vec{x})+\frac{i}{4}\hbar^{2}\beta\partial_{0}B_{a\mu}(v,\vec{x})\right],$
$\displaystyle\widehat{B}_{a\mu}(-v,-\vec{x})=(-1)^{\eta_{\mu}}\left[B_{a\mu}(v,\vec{x})+i\beta\partial_{0}B_{r\mu}(v,\vec{x})-\frac{i}{2}\hbar\beta\partial_{0}B_{a\mu}(v,\vec{x})\right],$
(3.78)

which reduces into the classical statistical limit (3.77) once $\hbar\to 0$ is
taken. Given that our $S_{eff}$ is valid to first order in boundary
derivative, in (3.78) we ignored higher powers in $\beta$. The dynamical KMS
symmetry (3.78) puts constraints999For quadratic Lagrangian
$\mathcal{L}_{eff}^{(2)}$, dynamical KMS symmetry has been carefully examined
in [43] beyond hydrodynamic limit. among some coefficients in
$\mathcal{L}_{eff}^{(4),\,\rm LO}$ and $\mathcal{L}_{eff}^{(4),\,\rm NLO}$:

$\displaystyle\tilde{\chi}_{1,1}=-\tilde{\chi}_{2,1}=\frac{i\beta}{2}\chi_{1,2},\qquad\tilde{\chi}_{1,2}=-\tilde{\chi}_{2,2}=\frac{i\beta}{2}\left(\chi_{1,3}+\chi_{1,4}-\frac{1}{4}\chi_{1,1}\right),$
$\displaystyle\tilde{\chi}_{1,3}=-\tilde{\chi}_{2,3}=\frac{i\beta}{2}\chi_{1,2},\qquad\tilde{\chi}_{1,4}=-\tilde{\chi}_{2,4}=-\frac{i\beta}{4}\chi_{1,4},\qquad\tilde{\chi}_{3,1}=-\tilde{\chi}_{4,1}=-i\beta\chi_{3,2},$
$\displaystyle\tilde{\chi}_{3,2}=-\tilde{\chi}_{4,2}=\frac{i\beta}{2}\left(\chi_{3,3}-\frac{1}{4}\chi_{3,1}\right).$
(3.79)

It is then straightforward to check that our holographic results (3.55) and
(3.58) perfectly satisfy (3.79)101010 While KMS-invariance of $S_{eff}^{(2)}$
and $S_{eff}^{(4)}$ is insensitive to the eigenvalue $(-1)^{\eta_{\mu}}$, for
the cubic part $S_{eff}^{(3)}$ to be KMS-invariant, we shall think of the
eigenvalue $(-1)^{\eta_{\mu}}$ to be associated with $\Theta=\mathcal{PT}$,
say $(-1)^{\eta_{\mu}}=+1$ for all components, which shall be compensated by
transpose of matrix-valued generators $t^{\rm a}$.. This implies that
effective theory derived from holographic method does capture both quantum and
thermal fluctuations, as speculated in [29]. Moreover, in contrast with the
hydrodynamic EFT framework, it is impossible to split quantum and thermal
fluctuations for a holographic theory, as seen from (3.78). More precisely, it
is natural to think of $\hbar$-expansion and derivative expansion (i.e.,
$\beta$-expansion) as independent from hydrodynamic EFT perspective, the
derivative expansion is controlled by the combination $\hbar\beta$ for a
holographic theory.

Finally, we would like to understand implication of dynamical KMS symmetry
(3.75) on the bulk dynamics, and hopefully give a holographic interpretation
of (3.75). Recall that we work in the saddle-point approximation for the bulk
theory. Thus, we are motivated to examine properties of partially on-shell
bulk solution under the KMS transformation (3.75). Via time-translational
operator, we may rewrite the KMS transformation (3.75) as [56]

$\displaystyle\widehat{B}_{1\mu}(-v,-\vec{x})=(-1)^{\eta_{\mu}}B_{1\mu}(v,\vec{x}),\qquad\qquad\widehat{B}_{2\mu}(-v,-\vec{x})=(-1)^{\eta_{\mu}}e^{-i\beta\partial_{v}}B_{2\mu}(v,\vec{x}).$
(3.80)

Then, it is direct to check that when boundary theory undergoes KMS
transformation (3.80), the leading order bulk solutions (3.16) and (3.21)
transform analogously

$\displaystyle\widehat{C}_{M}^{{\rm a}(1)~{}\rm
up}(r,-x^{\mu})=(-1)^{\eta_{M}}e^{-i\beta\partial_{v}}C_{M}^{{\rm a}(1)~{}\rm
up}(r,x^{\mu}),$ $\displaystyle\widehat{C}_{M}^{{\rm a}(1)~{}\rm
dw}(r,-x^{\mu})=(-1)^{\eta_{M}}C_{M}^{{\rm a}(1)~{}\rm dw}(r,x^{\mu}).$ (3.81)

Here, the $\mathcal{PT}$-symmetries of the leading order solutions (3.16) and
(3.21), extensively explored in [43], are useful in demonstrating the
transformation property (3.81). It is important to stress that (3.81) is valid
to all orders in boundary derivatives. Indeed, (3.81) amounts to saying that
leading order EOMs (3.2) are invariant under the bulk KMS transformation
(3.81), which is more transparent as viewed in Schwarzschild coordinate
system.

We turn to higher order solution $C_{M}^{{\rm a}(n\geq 2)}$, whose dynamical
EOMs differ from those of $C_{M}^{{\rm a}(1)}$ by the source terms,
$\mathbb{S}_{\mu}^{{\rm a}(1)}=0$ versus $\mathbb{S}_{\mu}^{{\rm a}(n\geq
2)}\neq 0$. Iteratively, one can show that under KMS transformation (3.80),
the source term $\mathbb{S}_{\mu}^{{\rm a}(n\geq 2)}$ changes analogously as
(3.81)

$\displaystyle\widehat{\mathbb{S}}_{\mu}^{{\rm a}(n)\;\rm
up}(r,-x^{\alpha})=(-1)^{\eta_{\mu}}e^{-i\beta\partial_{v}}\mathbb{S}_{\mu}^{{\rm
a}(n)\;\rm up}(r,x^{\alpha}),$ $\displaystyle\widehat{\mathbb{S}}_{\mu}^{{\rm
a}(n)\;\rm dw}(r,-x^{\alpha})=(-1)^{\eta_{\mu}}\mathbb{S}_{\mu}^{{\rm
a}(n)\;\rm dw}(r,x^{\alpha}),\qquad\qquad n=2,3,\cdots,$ (3.82)

which, combined with vanishing AdS boundary conditions for $C_{\mu}^{{\rm
a}(n\geq 2)}$, helps to conclude that under the transformation (3.80), the
higher order bulk solution $C_{M}^{{\rm a}(n\geq 2)}$ changes as

$\displaystyle\widehat{C}_{M}^{{\rm a}(n)\;\rm
up}(r,-x^{\mu})=(-1)^{\eta_{M}}e^{-i\beta\partial_{v}}C_{M}^{{\rm a}(n)\;\rm
up}(r,x^{\mu}),$ $\displaystyle\widehat{C}_{M}^{{\rm a}(n)\;\rm
dw}(r,-x^{\mu})=(-1)^{\eta_{M}}C_{M}^{{\rm a}(n)\;\rm dw}(r,x^{\mu}).$ (3.83)

Eventually, we observe a holographic analogue of the dynamical KMS symmetry
(3.75) satisfied by the boundary effective theory (3.74): the bulk action
$S_{\rm bulk}=S_{0}+S_{\rm ct}$ shall satisfy the following genuine $Z_{2}$
symmetry:

$\displaystyle S_{\rm bulk}[C_{M}^{\rm up},C_{M}^{\rm dw}]=S_{\rm
bulk}[\widehat{C}_{M}^{\rm up},\widehat{C}_{M}^{\rm dw}],$ (3.84)

where

$\displaystyle\widehat{C}_{M}^{\rm
up}(r,-x^{\mu})=(-1)^{\eta_{M}}e^{-i\beta\partial_{v}}C_{M}^{\rm
up}(r,x^{\mu}),\qquad\widehat{C}_{M}^{\rm
dw}(r,-x^{\mu})=(-1)^{\eta_{M}}C_{M}^{\rm dw}(r,x^{\mu}),$ (3.85)

Thus, (3.84) and (3.85) are taken as holographic interpretation of dynamical
KMS symmetry proposal (3.74) and (3.75). The invariance of bulk action under
bulk KMS transformation becomes more transparent if one turns to Fourier
space, so that the exponential factor becomes unity due to delta-function of
four-momenta.

##  4 Summary and discussion

In this work we present a holographic derivation of hydrodynamic EFT for SU(2)
diffusion. To first order in derivative expansion, the effective action is
analytically computed to quartic order in the gauge-invariant objects
$B_{r\mu}^{\rm a}$ and $B_{a\mu}^{\rm a}$. As a non-Gaussian effective theory,
such an EFT contains a complete list of cubic and quartic interactions.
Particularly, the generalized nonlinear FDT is guaranteed via dynamical KMS
symmetry at quantum level. The dynamical KMS symmetry is found to have a bulk
analogue.

The effective action $S_{eff}$ provides a framework to systematically explore
phenomenological consequences of nonlinear interactions as emphasized in
subsection 3.4. One approach towards this end would be to compute loop
diagrams based on the effective action, as in [14, 17, 18, 19, 20]. An
alternative way would be to cast the hydrodynamic effective action into a
stochastic differential equation satisfied by dynamical variable
$\varphi_{r}$. However, as demonstrated in [2], the latter approach works
perfectly only when terms beyond quadratic order in $\varphi_{a}$ are ignored
in the effective action. Physically, this truncation amounts to turning off
non-Gaussian noise, which is usually unavoidable and might be of importance in
realistic systems [2, 17].

It would be interesting to derive Fokker-Planck (FP) type equation for
distribution of the dynamical variable $\varphi_{r}$, which is more suitable
for numerical investigation. In contrast to stochastic Langevin-type equation,
the FP equation describes the probability of finding the system in a certain
configuration and is thus fully deterministic. When first order derivatives in
$S_{eff}^{(3)}$ and $S_{eff}^{(4)}$ are turned off, this derivation can
closely follow the textbook [60] by making an analog of the hydrodynamic EFT
with Hamiltonian formulation of quantum mechanics. By this analog, the task
boils down to looking for “Hamiltonian” $H$ of the hydrodynamic EFT. A more
general method would be to discretize the time and space, and consider the
“restricted” partition function [60], which is identified as the probability
of finding the system in a certain configuration. We leave such a derivation
for a future project.

Finally, based on present work it is possible to go beyond classical treatment
for the bulk theory, and consider loop corrections to tree-level Witten
diagrams of Figure 2. This corresponds to including finite $N_{c}$ correction
in the boundary EFT [73, 74]. Technically, this task will boil down to
assembling bulk-to-boundary propagator (to be read off from leading order
solution $C_{\mu}^{{\rm a}(1)}$) and bulk-to-bulk propagator (encoded in bulk
Green’s functions $G_{\perp,\parallel}$) using proper vertices. We leave this
study as a forthcoming project.

##  Appendix A Yang-Mills equation in different coordinate systems

In this appendix, we collect explicit forms of bulk Yang-Mills equation in two
coordinate systems: the Schwarzschild versus ingoing EF.

From Schwarzschild coordinate system to ingoing EF one, bulk Yang-Mills field
changes as

$\displaystyle\tilde{C}_{t}(r,t,\vec{x})=C_{v}(r,v,\vec{x}),\qquad\tilde{C}_{i}(r,t,\vec{x})=C_{i}(r,v,\vec{x}),$
$\displaystyle\tilde{C}_{r}(r,t,\vec{x})=C_{r}(r,v,\vec{x})+\frac{C_{v}(r,v,\vec{x})}{r^{2}f(r)},$
(A.1)

where tilded quantities correspond to those in Schwarzschild coordinate
system. The gauge choice (2.16) taken throughout this work amounts to imposing
$\tilde{C}_{r}=0$ in Schwarzschild coordinate system. Then, constraint
component of bulk Yang-Mills equation is

$\displaystyle\tilde{\nabla}_{M}\tilde{F}^{{\rm a}Mr}+\epsilon^{\rm
abc}\tilde{C}_{M}^{\rm b}\tilde{F}^{{\rm
c}Mr}=0\Leftrightarrow\nabla_{M}F^{{\rm a}Mr}+\epsilon^{\rm abc}C_{M}^{\rm
b}F^{{\rm c}Mr}=0$ (A.2)

while dynamical EOMs are

$\displaystyle\tilde{\nabla}_{M}\tilde{F}^{{\rm a}Mt}+\epsilon^{\rm
abc}\tilde{C}_{M}^{\rm b}\tilde{F}^{{\rm
c}Mt}=0\Leftrightarrow\nabla_{M}F^{{\rm a}Mv}+\epsilon^{\rm abc}C_{M}^{\rm
b}F^{{\rm c}Mv}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad-\frac{1}{r^{2}f(r)}\left[\nabla_{M}F^{{\rm
a}Mr}+\epsilon^{\rm abc}C_{M}^{\rm b}F^{{\rm c}Mr}\right]=0,$
$\displaystyle\tilde{\nabla}_{M}\tilde{F}^{{\rm a}Mi}+\epsilon^{\rm
abc}\tilde{C}_{M}^{\rm b}\tilde{F}^{{\rm
c}Mi}=0\Leftrightarrow\nabla_{M}F^{{\rm a}Mi}+\epsilon^{\rm abc}C_{M}^{\rm
b}F^{{\rm c}Mi}=0.$ (A.3)

Explicitly, the dynamical EOMs (A.3) in Schwarzschild coordinate system are

$\displaystyle 0=$
$\displaystyle\partial_{r}\left(r^{3}\partial_{r}\tilde{C}_{t}^{\rm
a}\right)+\frac{1}{rf(r)}\partial_{k}\left(\partial_{k}\tilde{C}_{t}^{\rm
a}-\partial_{t}\tilde{C}_{k}^{\rm a}+\epsilon^{\rm abc}\tilde{C}_{k}^{\rm
b}\tilde{C}_{t}^{\rm c}\right)+\frac{1}{rf(r)}\epsilon^{\rm
abc}\tilde{C}_{k}^{\rm b}$
$\displaystyle\times\left(\partial_{k}\tilde{C}_{t}^{\rm
c}-\partial_{t}\tilde{C}_{k}^{\rm c}+\epsilon^{\rm cde}\tilde{C}_{k}^{\rm
d}\tilde{C}_{t}^{\rm e}\right),$ $\displaystyle 0=$
$\displaystyle\partial_{r}\left[r^{3}f(r)\partial_{r}\tilde{C}_{i}^{\rm
a}\right]-\frac{1}{rf(r)}\partial_{t}\left(\partial_{t}\tilde{C}_{i}^{\rm
a}-\partial_{i}\tilde{C}_{t}^{\rm a}+\epsilon^{\rm abc}\tilde{C}_{t}^{\rm
b}\tilde{C}_{i}^{\rm c}\right)-\frac{1}{rf(r)}\epsilon^{\rm
abc}\tilde{C}_{t}^{\rm b}$
$\displaystyle\times\left(\partial_{t}\tilde{C}_{i}^{\rm
c}-\partial_{i}\tilde{C}_{t}^{\rm c}+\epsilon^{\rm cde}\tilde{C}_{t}^{\rm
d}\tilde{C}_{i}^{\rm
e}\right)+\frac{1}{r}\partial_{k}\left(\partial_{k}\tilde{C}_{i}^{\rm
a}-\partial_{i}\tilde{C}_{k}^{\rm a}+\epsilon^{\rm abc}\tilde{C}_{k}^{\rm
b}\tilde{C}_{i}^{\rm c}\right)$ $\displaystyle+\frac{1}{r}\epsilon^{\rm
abc}\tilde{C}_{k}^{\rm b}\left(\partial_{k}\tilde{C}_{i}^{\rm
c}-\partial_{i}\tilde{C}_{k}^{\rm c}+\epsilon^{\rm cde}\tilde{C}_{k}^{\rm
d}\tilde{C}_{i}^{\rm e}\right).$ (A.4)

In ingoing EF coordinate system, (A.3) take the following form

$\displaystyle 0=$
$\displaystyle\partial_{r}\left[r^{3}\left(\partial_{v}C_{r}^{\rm
a}-\partial_{r}C_{v}^{\rm
a}\right)\right]-\frac{r}{f(r)}\partial_{v}\left(\partial_{r}C_{v}^{\rm
a}-\partial_{v}C_{r}^{\rm a}\right)-\frac{1}{rf(r)}$
$\displaystyle\times\partial_{k}\left(\partial_{k}C_{v}^{\rm
a}-\partial_{v}C_{k}^{\rm a}+\epsilon^{\rm abc}C_{k}^{\rm b}C_{v}^{\rm
c}\right)-\frac{\epsilon^{\rm abc}}{rf(r)}C_{k}^{\rm
b}\left(\partial_{k}C_{v}^{\rm c}-\partial_{v}C_{k}^{\rm c}+\epsilon^{\rm
cde}C_{k}^{\rm d}C_{v}^{\rm e}\right),$ $\displaystyle 0=$
$\displaystyle\partial_{r}\left[r^{3}f(r)\left(\partial_{r}C_{i}^{\rm
a}-\partial_{i}C_{r}^{\rm
a}\right)\right]+\partial_{r}\left[r\left(\partial_{v}C_{i}^{\rm
a}-\partial_{i}C_{v}^{\rm a}\right)\right]$
$\displaystyle+r\partial_{v}\left(\partial_{r}C_{i}^{\rm
a}-\partial_{i}C_{r}^{\rm a}+\epsilon^{\rm abc}C_{r}^{\rm b}C_{i}^{\rm
c}\right)+\frac{1}{r}\partial_{j}\left(\partial_{j}C_{i}^{\rm
a}-\partial_{i}C_{j}^{\rm a}+\epsilon^{\rm abc}C_{j}^{\rm b}C_{i}^{\rm
c}\right)$ $\displaystyle+r\epsilon^{\rm abc}C_{r}^{\rm
b}\left(\partial_{v}C_{i}^{\rm c}-\partial_{i}C_{v}^{\rm c}+\epsilon^{\rm
cde}C_{v}^{\rm d}C_{i}^{\rm e}\right)+\frac{1}{r}\epsilon^{\rm abc}C_{j}^{\rm
b}\left(\partial_{j}C_{i}^{\rm c}-\partial_{i}C_{j}^{\rm c}+\epsilon^{\rm
cde}C_{j}^{\rm d}C_{i}^{\rm e}\right).$ (A.5)

When $C_{M}^{\rm a}$ is linearized as in subsection 3.1, the dynamical EOMs
(A.5) turn into a system of linear PDEs:

$\displaystyle\partial_{r}\left(r^{3}\partial_{r}C_{v}^{{\rm
a}(n)}\right)-\partial_{r}\left(r^{3}\partial_{v}C_{r}^{{\rm
a}(n)}\right)+\frac{r}{f(r)}\partial_{r}\partial_{v}C_{v}^{{\rm
a}(n)}-\frac{r}{f(r)}\partial_{v}^{2}C_{r}^{{\rm a}(n)}$
$\displaystyle+\frac{1}{rf(r)}\left({\vec{\partial}}^{\,2}C_{v}^{{\rm
a}(n)}-\partial_{v}\partial_{k}C_{k}^{{\rm a}(n)}\right)=\mathbb{S}_{v}^{{\rm
a}(n)},$ $\displaystyle\partial_{r}\left[r^{3}f(r)\partial_{r}C_{i}^{{\rm
a}(n)}\right]-\partial_{r}\left[r^{3}f(r)\partial_{i}C_{r}^{{\rm
a}(n)}+r\left(\partial_{v}C_{i}^{{\rm a}(n)}-\partial_{i}C_{v}^{{\rm
a}(n)}\right)\right]+r\partial_{r}\partial_{v}C_{i}^{{\rm a}(n)}$
$\displaystyle-r\partial_{v}\partial_{i}C_{r}^{{\rm
a}(n)}+\frac{1}{r}\partial_{k}\left(\partial_{k}C_{i}^{{\rm
a}(n)}-\partial_{i}C_{k}^{{\rm a}(n)}\right)=\mathbb{S}_{i}^{{\rm a}(n)},$
(A.6)

where the source terms $\mathbb{S}_{\mu}^{{\rm a}(n)}$ are products of lower
order solutions (as well as their derivatives). For the first two orders, the
source terms are

$\displaystyle\mathbb{S}_{v}^{{\rm a}(1)}=$ $\displaystyle\mathbb{S}_{i}^{{\rm
a}(1)}=0,$ $\displaystyle\mathbb{S}_{v}^{{\rm a}(2)}=$
$\displaystyle-\frac{1}{rf(r)}\partial_{k}\left(\epsilon^{\rm abc}C_{k}^{\rm
b(1)}C_{v}^{\rm c(1)}\right)-\frac{\epsilon^{\rm abc}}{rf(r)}C_{k}^{\rm
b(1)}\left(\partial_{k}C_{v}^{\rm c(1)}-\partial_{v}C_{k}^{\rm c(1)}\right)$
$\displaystyle\mathbb{S}_{i}^{{\rm a}(2)}=$
$\displaystyle-r\partial_{v}\left(\epsilon^{\rm abc}C_{r}^{\rm b(1)}C_{i}^{\rm
c(1)}\right)-\frac{1}{r}\partial_{k}\left(\epsilon^{\rm abc}C_{k}^{\rm
b(1)}C_{i}^{\rm c(1)}\right)$ $\displaystyle-r\epsilon^{\rm abc}C_{r}^{\rm
b(1)}\left(\partial_{v}C_{i}^{\rm c(1)}-\partial_{i}C_{v}^{\rm
c(1)}\right)-\frac{1}{r}\epsilon^{\rm abc}C_{k}^{\rm
b(1)}\left(\partial_{k}C_{i}^{\rm c(1)}-\partial_{i}C_{k}^{\rm c(1)}\right).$
(A.7)

In parallel, (A.4) are linearized as

$\displaystyle\partial_{r}\left[r^{3}\partial_{r}\tilde{C}_{t}^{{\rm
a}(n)}\right]+\frac{1}{rf(r)}\partial_{k}\left(\partial_{k}\tilde{C}_{t}^{{\rm
a}(n)}-\partial_{t}\tilde{C}_{k}^{{\rm
a}(n)}\right)=\tilde{\mathbb{S}}_{t}^{{\rm a}(n)},$
$\displaystyle\partial_{r}\left[r^{3}f(r)\partial_{r}\tilde{C}_{i}^{{\rm
a}(n)}\right]-\frac{1}{rf(r)}\partial_{t}\left(\partial_{t}\tilde{C}_{i}^{{\rm
a}(n)}-\partial_{i}\tilde{C}_{t}^{{\rm
a}(n)}\right)+\frac{1}{r}\partial_{k}\left(\partial_{k}\tilde{C}_{i}^{{\rm
a}(n)}-\partial_{i}\tilde{C}_{k}^{{\rm
a}(n)}\right)=\tilde{\mathbb{S}}_{i}^{{\rm a}(n)},$ (A.8)

where for the first two orders the source terms are

$\displaystyle\tilde{\mathbb{S}}_{t}^{{\rm a}(1)}=$
$\displaystyle\tilde{\mathbb{S}}_{i}^{{\rm a}(1)}=0,$
$\displaystyle\tilde{\mathbb{S}}_{t}^{{\rm a}(2)}=$
$\displaystyle-\frac{1}{rf(r)}\partial_{k}\left(\epsilon^{\rm
abc}\tilde{C}_{k}^{\rm b(1)}\tilde{C}_{t}^{\rm
c(1)}\right)-\frac{1}{rf(r)}\epsilon^{\rm abc}\tilde{C}_{k}^{\rm
b(1)}\left(\partial_{k}\tilde{C}_{t}^{\rm c(1)}-\partial_{t}\tilde{C}_{k}^{\rm
c(1)}\right),$ $\displaystyle\tilde{\mathbb{S}}_{i}^{{\rm a}(2)}=$
$\displaystyle\frac{1}{rf(r)}\partial_{t}\left(\epsilon^{\rm
abc}\tilde{C}_{t}^{\rm b(1)}\tilde{C}_{i}^{\rm
c(1)}\right)+\frac{1}{rf(r)}\epsilon^{\rm abc}\tilde{C}_{t}^{\rm
b(1)}\left(\partial_{t}\tilde{C}_{i}^{\rm c(1)}-\partial_{i}\tilde{C}_{t}^{\rm
c(1)}\right)$ $\displaystyle-\frac{1}{r}\partial_{k}\left(\epsilon^{\rm
abc}\tilde{C}_{k}^{\rm b(1)}\tilde{C}_{i}^{\rm
c(1)}\right)-\frac{1}{r}\epsilon^{\rm abc}\tilde{C}_{k}^{\rm
b(1)}\left(\partial_{k}\tilde{C}_{i}^{\rm c(1)}-\partial_{i}\tilde{C}_{k}^{\rm
c(1)}\right).$ (A.9)

##  Appendix B More on partially on-shell solution: beyond leading order

In this appendix we elaborate on generic structure of partially on-shell
solution beyond leading order. Recall that the dynamical EOMs at each order in
perturbative expansion differ by source terms. Therefore, we will take the
next-to-leading order correction $C_{\mu}^{{\rm a}(2)}$ as the example, for
which the source terms do not vanish.

We start with the transverse mode $C_{\perp}^{{\rm a}(2)}$, which satisfies a
closed ODE

$\displaystyle\partial_{r}\left[r^{3}f(r)\partial_{r}C_{\perp}^{{\rm
a}(2)}\right]+\Box_{\perp}(\partial_{r};\omega,q^{2})C_{\perp}^{{\rm
a}(2)}=\mathbb{S}_{\perp}^{(2)}(r,k^{\mu}),$ (B.1)

where the operator $\Box_{\perp}(\partial_{r};\omega,q)$ and the source term
can be read off from (A.6) and (A.7) in appendix A. Via Green’s function
method, (B.1) is solved by

$\displaystyle C_{\perp}^{{\rm
a}(2)}(r,k^{\mu})=\int_{\infty_{2}}^{\infty_{1}}G_{\perp}(r,r^{\prime};k^{\mu})\mathbb{S}_{\perp}^{(2)}(r^{\prime},k^{\mu})dr^{\prime},\qquad
r\in(\infty_{2},\infty_{1}),$ (B.2)

where $G_{\perp}(r,r^{\prime};k^{\mu})$ is the Green’s function satisfying

$\displaystyle\partial_{r}\left[r^{3}f(r)\partial_{r}G_{\perp}(r,r^{\prime};k^{\mu})\right]+\Box_{\perp}(\partial_{r};\omega,q^{2})G_{\perp}(r,r^{\prime};k^{\mu})=\delta(r-r^{\prime}).$
(B.3)

Thus, $G_{\perp}$ is indeed the bulk-to-bulk propagator. In order to uniquely
fix $G_{\perp}(r,r^{\prime};k^{\mu})$, we impose two boundary conditions

$\displaystyle G_{\perp}(r=\infty_{1},r^{\prime};k^{\mu})=0,\qquad\qquad
G_{\perp}(r=\infty_{2},r^{\prime};k^{\mu})=0,$ (B.4)

which is convenient since $C_{\perp}^{{\rm a}(2)}$ also vanishes at both AdS
boundaries.

The Green’s function $G_{\perp}(r,r^{\prime};k^{\mu})$ could be constructed
from linearly independent solutions in (3.16). For convenience, we make linear
combination over the two linearly independent solutions presented in (3.16)
and generate two new basis solutions

$\displaystyle Y_{1}(r,k^{\mu})=aC_{\perp}^{\rm ig}(r,k^{\mu})+C_{\perp}^{\rm
ig}(r,\bar{k}^{\mu})e^{i\omega\zeta(r)},$ $\displaystyle
Y_{2}(r,k^{\mu})=C_{\perp}^{\rm ig}(r,k^{\mu})+bC_{\perp}^{\rm
ig}(r,\bar{k}^{\mu})e^{i\omega\zeta(r)},$ (B.5)

where we have analytically continued the linearly independent solutions of
(3.16) so that they are valid over the entire contour. Accordingly, the
function $\zeta(r)$ is

$\displaystyle\zeta(r)=\int_{\infty_{2}}^{r}\frac{dy}{y^{2}f(y)},\qquad
r\in(\infty_{2},\infty_{1}),$ (B.6)

which is multi-valued. The coefficients $a,b$ in (B.5) are fixed by imposing

$\displaystyle Y_{1}(r=\infty_{1})=0,\qquad\qquad Y_{2}(r=\infty_{2})=0,$
(B.7)

which becomes possible since $\zeta(r=\infty_{1})\neq\zeta(r=\infty_{2})$.
Eventually, the Green’s function $G_{\perp}$ is

$\displaystyle G_{\perp}(r,r^{\prime};k^{\mu})=\frac{1}{r^{\prime
3}f(r^{\prime})W(r^{\prime})}\left[Y_{2}(r)Y_{1}(r^{\prime})\theta(r^{\prime}-r)+Y_{1}(r)Y_{2}(r^{\prime})\theta(r^{\prime}-r)\right],$
(B.8)

where $W(r)$ is the Wronskian determinant of $Y_{1}(r),Y_{2}(r)$

$\displaystyle W(r)\equiv
Y_{2}(r)\partial_{r}Y_{1}(r)-Y_{1}(r)\partial_{r}Y_{2}(r)=\frac{2i\pi}{r^{3}f(r)}$
(B.9)

The step function $\theta(r-r^{\prime})$ is defined on the radial contour of
Figure 1, with $r>r^{\prime}$ ($r<r^{\prime}$) understood as counter clockwise
path-ordered relations.

We turn to the longitudinal sector, which involves a system of two modes
$C_{v}^{{\rm a}(2)}$ and $C_{x}^{{\rm a}(2)}$. Thus, the above method based on
Green’s function should be extended appropriately to that based on Green’s
matrix. We advance by rewriting second order ODEs (3.15) for $C_{v}^{{\rm
a}(2)}$ and $C_{x}^{{\rm a}(2)}$ into a system of first order ODEs:

$\displaystyle\partial_{r}X(r,k^{\mu})=M(r,k^{\mu})X(r,k^{\mu})+g(r,k^{\mu})$
(B.10)

where

$\displaystyle X=\left(C_{v}^{{\rm a}(2)},~{}\partial_{r}C_{v}^{{\rm
a}(2)},~{}C_{x}^{{\rm a}(2)},~{}\partial_{r}C_{x}^{{\rm a}(2)}\right)^{\rm T}$
(B.11)

The $4\times 4$ matrix $M$ and column vector $g$ can be directly read off from
second order ODEs (3.15). For simplicity, we will not report their expressions
here. First, we consider the homogeneous part of (B.10)

$\displaystyle\partial_{r}X(r,k^{\mu})=M(r,k^{\mu})X(r,k^{\mu})$ (B.12)

The four linearly independent solutions for (B.12) are presented in (3.21),
which we rewrite here

$\displaystyle X_{1}=\left(C_{v}^{\rm ig}(r,k^{\mu}),~{}\partial_{r}C_{v}^{\rm
ig}(r,k^{\mu}),~{}C_{x}^{\rm ig}(r,k^{\mu}),~{}\partial_{r}C_{x}^{\rm
ig}(r,k^{\mu})\right)^{\rm T},$ $\displaystyle X_{2}=\left(C_{v}^{\rm
ig}(r,\bar{k}^{\mu})e^{i\omega\zeta(r)},~{}\partial_{r}\left(C_{v}^{\rm
ig}(r,\bar{k}^{\mu})e^{i\omega\zeta(r)}\right),~{}C_{x}^{\rm
ig}(r,\bar{k}^{\mu})e^{i\omega\zeta(r)},~{}\partial_{r}\left(C_{x}^{\rm
ig}(r,\bar{k}^{\mu})e^{i\omega\zeta(r)}\right)\right)^{\rm T},$ $\displaystyle
X_{3}=\left(C_{v}^{\rm pg}(r,k^{\mu}),~{}\partial_{r}C_{v}^{\rm
pg}(r,k^{\mu}),~{}C_{x}^{\rm pg}(r,k^{\mu}),~{}\partial_{r}C_{x}^{\rm
pg}(r,k^{\mu})\right)^{\rm T},$ $\displaystyle X_{4}=\left(C_{v}^{\rm
pn}(r,k^{\mu}),~{}\partial_{r}C_{v}^{\rm pn}(r,k^{\mu}),~{}C_{x}^{\rm
pn}(r,k^{\mu}),~{}\partial_{r}C_{x}^{\rm pn}(r,k^{\mu})\right)^{\rm T},$
(B.13)

where as for the transverse sector we have analytically continued all the
linearly independent solutions to the entire radial contour. The linearly
independent solutions (B.13) for homogeneous system (B.12) help to build a
fundamental matrix

$\displaystyle M_{0}(r,k^{\mu})=\left(X_{1},~{}X_{2},~{}X_{3},~{}X_{4}\right)$
(B.14)

which satisfies

$\displaystyle\partial_{r}M_{0}(r,k^{\mu})=M(r,k^{\mu})M_{0}(r,k^{\mu}).$
(B.15)

Then, the general solution for the inhomogeneous system (B.10) is

$\displaystyle
X(r,k^{\mu})=M_{0}(r,k^{m}u)c+M_{0}(r,k^{\mu})\int_{\infty_{2}}^{r}M_{0}^{-1}(r^{\prime},k^{\mu})g(r^{\prime},k^{\mu})dr^{\prime},$
(B.16)

where the column vector $c$ represents the four integration constants, to be
determined by boundary conditions. Here, a subtlety arises from boundary
conditions for $C_{v}^{{\rm a}(2)}$: besides vanishing conditions at the AdS
boundaries, it is also imposed to be zero at the horizon. Thus, the final
solution for longitudinal sector will be piecewise as for the leading order
case.

## Acknowledgements

We would like to thank Gao-Liang Zhou and Tianchun Zhou for helpful
discussions.

## References

  * [1] P. Glorioso, M. Crossley, and H. Liu, “A prescription for holographic Schwinger-Keldysh contour in non-equilibrium systems,” arXiv:1812.08785 [hep-th].
  * [2] M. Crossley, P. Glorioso, and H. Liu, “Effective field theory of dissipative fluids,” JHEP 09 (2017) 095, arXiv:1511.03646 [hep-th].
  * [3] P. Glorioso and H. Liu, “The second law of thermodynamics from symmetry and unitarity,” arXiv:1612.07705 [hep-th].
  * [4] P. Glorioso, M. Crossley, and H. Liu, “Effective field theory of dissipative fluids (II): classical limit, dynamical KMS symmetry and entropy current,” JHEP 09 (2017) 096, arXiv:1701.07817 [hep-th].
  * [5] F. M. Haehl, R. Loganayagam, and M. Rangamani, “The Fluid Manifesto: Emergent symmetries, hydrodynamics, and black holes,” JHEP 01 (2016) 184, arXiv:1510.02494 [hep-th].
  * [6] F. M. Haehl, R. Loganayagam, and M. Rangamani, “Topological sigma models & dissipative hydrodynamics,” JHEP 04 (2016) 039, arXiv:1511.07809 [hep-th].
  * [7] F. M. Haehl, R. Loganayagam, and M. Rangamani, “Effective Action for Relativistic Hydrodynamics: Fluctuations, Dissipation, and Entropy Inflow,” JHEP 10 (2018) 194, arXiv:1803.11155 [hep-th].
  * [8] S. Endlich, A. Nicolis, R. A. Porto, and J. Wang, “Dissipation in the effective field theory for hydrodynamics: First order effects,” Phys. Rev. D 88 (2013) 105001, arXiv:1211.6461 [hep-th].
  * [9] P. Kovtun, G. D. Moore, and P. Romatschke, “Towards an effective action for relativistic dissipative hydrodynamics,” JHEP 07 (2014) 123, arXiv:1405.3967 [hep-ph].
  * [10] A. Nicolis, R. Penco, and R. A. Rosen, “Relativistic Fluids, Superfluids, Solids and Supersolids from a Coset Construction,” Phys. Rev. D 89 no. 4, (2014) 045002, arXiv:1307.0517 [hep-th].
  * [11] M. Harder, P. Kovtun, and A. Ritz, “On thermal fluctuations and the generating functional in relativistic hydrodynamics,” JHEP 07 (2015) 025, arXiv:1502.03076 [hep-th].
  * [12] S. Grozdanov and J. Polonyi, “Viscosity and dissipative hydrodynamics from effective field theory,” Phys. Rev. D 91 no. 10, (2015) 105031, arXiv:1305.3670 [hep-th].
  * [13] H. Liu and P. Glorioso, “Lectures on non-equilibrium effective field theories and fluctuating hydrodynamics,” PoS TASI2017 (2018) 008, arXiv:1805.09331 [hep-th].
  * [14] X. Chen-Lin, L. V. Delacrétaz, and S. A. Hartnoll, “Theory of diffusive fluctuations,” Phys. Rev. Lett. 122 no. 9, (2019) 091602, arXiv:1811.12540 [hep-th].
  * [15] J. Chao and T. Schaefer, “Multiplicative noise and the diffusion of conserved densities,” JHEP 01 (2021) 071, arXiv:2008.01269 [hep-th].
  * [16] M. J. Landry, “Non-equilibrium effective field theory and second sound,” JHEP 04 (2021) 213, arXiv:2008.11725 [hep-th].
  * [17] A. Jain and P. Kovtun, “Late Time Correlations in Hydrodynamics: Beyond Constitutive Relations,” Phys. Rev. Lett. 128 no. 7, (2022) 071601, arXiv:2009.01356 [hep-th].
  * [18] A. Jain, P. Kovtun, A. Ritz, and A. Shukla, “Hydrodynamic effective field theory and the analyticity of hydrostatic correlators,” JHEP 02 (2021) 200, arXiv:2011.03691 [hep-th].
  * [19] N. Sogabe, N. Yamamoto, and Y. Yin, “Positive magnetoresistance induced by hydrodynamic fluctuations in chiral media,” JHEP 09 (2021) 131, arXiv:2105.10271 [hep-th].
  * [20] N. Sogabe and Y. Yin, “Off-equilibrium non-Gaussian fluctuations near the QCD critical point: an effective field theory perspective,” JHEP 03 (2022) 124, arXiv:2111.14667 [nucl-th].
  * [21] J. M. Maldacena, “The Large N limit of superconformal field theories and supergravity,” Adv. Theor. Math. Phys. 2 (1998) 231–252, arXiv:hep-th/9711200.
  * [22] S. S. Gubser, I. R. Klebanov, and A. M. Polyakov, “Gauge theory correlators from noncritical string theory,” Phys. Lett. B 428 (1998) 105–114, arXiv:hep-th/9802109.
  * [23] E. Witten, “Anti-de Sitter space and holography,” Adv. Theor. Math. Phys. 2 (1998) 253–291, arXiv:hep-th/9802150.
  * [24] M. Crossley, P. Glorioso, H. Liu, and Y. Wang, “Off-shell hydrodynamics from holography,” JHEP 02 (2016) 124, arXiv:1504.07611 [hep-th].
  * [25] J. de Boer, M. P. Heller, and N. Pinzani-Fokeeva, “Effective actions for relativistic fluids from holography,” JHEP 08 (2015) 086, arXiv:1504.07616 [hep-th].
  * [26] I. Heemskerk and J. Polchinski, “Holographic and Wilsonian Renormalization Groups,” JHEP 06 (2011) 031, arXiv:1010.1264 [hep-th].
  * [27] T. Faulkner, H. Liu, and M. Rangamani, “Integrating out geometry: Holographic Wilsonian RG and the membrane paradigm,” JHEP 08 (2011) 051, arXiv:1010.4036 [hep-th].
  * [28] S.-J. Sin and Y. Zhou, “Holographic Wilsonian RG Flow and Sliding Membrane Paradigm,” JHEP 05 (2011) 030, arXiv:1102.4477 [hep-th].
  * [29] J. de Boer, M. P. Heller, and N. Pinzani-Fokeeva, “Holographic Schwinger-Keldysh effective field theories,” JHEP 05 (2019) 188, arXiv:1812.06093 [hep-th].
  * [30] C. P. Herzog and D. T. Son, “Schwinger-Keldysh propagators from AdS/CFT correspondence,” JHEP 03 (2003) 046, arXiv:hep-th/0212072.
  * [31] K. Skenderis and B. C. van Rees, “Real-time gauge/gravity duality,” Phys. Rev. Lett. 101 (2008) 081601, arXiv:0805.0150 [hep-th].
  * [32] K. Skenderis and B. C. van Rees, “Real-time gauge/gravity duality: Prescription, Renormalization and Examples,” JHEP 05 (2009) 085, arXiv:0812.2909 [hep-th].
  * [33] E. Barnes, D. Vaman, C. Wu, and P. Arnold, “Real-time finite-temperature correlators from AdS/CFT,” Phys. Rev. D 82 (2010) 025019, arXiv:1004.1179 [hep-th].
  * [34] P. Arnold and D. Vaman, “Jet quenching in hot strongly coupled gauge theories revisited: 3-point correlators with gauge-gravity duality,” JHEP 10 (2010) 099, arXiv:1008.4023 [hep-th].
  * [35] P. Arnold, D. Vaman, C. Wu, and W. Xiao, “Second order hydrodynamic coefficients from 3-point stress tensor correlators via AdS/CFT,” JHEP 10 (2011) 033, arXiv:1105.4645 [hep-th].
  * [36] P. Arnold and D. Vaman, “4-point correlators in finite-temperature AdS/CFT: Jet quenching correlations,” JHEP 11 (2011) 033, arXiv:1109.0040 [hep-th].
  * [37] B. Chakrabarty, J. Chakravarty, S. Chaudhuri, C. Jana, R. Loganayagam, and A. Sivakumar, “Nonlinear Langevin dynamics via holography,” JHEP 01 (2020) 165, arXiv:1906.07762 [hep-th].
  * [38] C. Jana, R. Loganayagam, and M. Rangamani, “Open quantum systems and Schwinger-Keldysh holograms,” JHEP 07 (2020) 242, arXiv:2004.02888 [hep-th].
  * [39] B. Chakrabarty and P. M. Aswin, “Open effective theory of scalar field in rotating plasma,” JHEP 21 (2020) 169, arXiv:2011.13223 [hep-th].
  * [40] R. Loganayagam, K. Ray, and A. Sivakumar, “Fermionic Open EFT from Holography,” arXiv:2011.07039 [hep-th].
  * [41] R. Loganayagam, K. Ray, S. K. Sharma, and A. Sivakumar, “Holographic KMS relations at finite density,” JHEP 03 (2021) 233, arXiv:2011.08173 [hep-th].
  * [42] J. K. Ghosh, R. Loganayagam, S. G. Prabhu, M. Rangamani, A. Sivakumar, and V. Vishal, “Effective field theory of stochastic diffusion from gravity,” JHEP 05 (2021) 130, arXiv:2012.03999 [hep-th].
  * [43] Y. Bu, T. Demircik, and M. Lublinsky, “All order effective action for charge diffusion from Schwinger-Keldysh holography,” JHEP 05 (2021) 187, arXiv:2012.08362 [hep-th].
  * [44] Y. Bu, M. Fujita, and S. Lin, “Ginzburg-Landau effective action for a fluctuating holographic superconductor,” arXiv:2106.00556 [hep-th].
  * [45] T. He, R. Loganayagam, M. Rangamani, and J. Virrueta, “An effective description of momentum diffusion in a charged plasma from holography,” arXiv:2108.03244 [hep-th].
  * [46] Y. Bu and B. Zhang, “Schwinger-Keldysh effective action for a relativistic Brownian particle in the AdS/CFT correspondence,” Phys. Rev. D 104 no. 8, (2021) 086002, arXiv:2108.10060 [hep-th].
  * [47] M. Torabian and H.-U. Yee, “Holographic nonlinear hydrodynamics from AdS/CFT with multiple/non-Abelian symmetries,” JHEP 08 (2009) 020, arXiv:0903.4894 [hep-th].
  * [48] C. Eling, Y. Neiman, and Y. Oz, “Holographic Non-Abelian Charged Hydrodynamics from the Dynamics of Null Horizons,” JHEP 12 (2010) 086, arXiv:1010.1290 [hep-th].
  * [49] Y. Neiman and Y. Oz, “Relativistic Hydrodynamics with General Anomalous Charges,” JHEP 03 (2011) 023, arXiv:1011.5107 [hep-th].
  * [50] C. Hoyos, B. S. Kim, and Y. Oz, “Odd Parity Transport In Non-Abelian Superfluids From Symmetry Locking,” JHEP 10 (2014) 127, arXiv:1404.7507 [hep-th].
  * [51] J. J. Fernandez-Melgarejo, S.-J. Rey, and P. Surówka, “A New Approach to Non-Abelian Hydrodynamics,” JHEP 02 (2017) 122, arXiv:1605.06080 [hep-th].
  * [52] R. M. A. Dantas, F. Peña Benitez, B. Roy, and P. Surówka, “Non-Abelian anomalies in multi-Weyl semimetals,” Phys. Rev. Res. 2 no. 1, (2020) 013007, arXiv:1905.02189 [cond-mat.mes-hall].
  * [53] P. Glorioso, L. V. Delacrétaz, X. Chen, R. M. Nandkishore, and A. Lucas, “Hydrodynamics in lattice models with continuous non-Abelian symmetries,” SciPost Phys. 10 no. 1, (2021) 015, arXiv:2007.13753 [cond-mat.stat-mech].
  * [54] K. Jensen, N. Pinzani-Fokeeva, and A. Yarom, “Dissipative hydrodynamics in superspace,” JHEP 09 (2018) 127, arXiv:1701.07436 [hep-th].
  * [55] K. Jensen, R. Marjieh, N. Pinzani-Fokeeva, and A. Yarom, “An entropy current in superspace,” JHEP 01 (2019) 061, arXiv:1803.07070 [hep-th].
  * [56] M. Blake, H. Lee, and H. Liu, “A quantum hydrodynamical description for scrambling and many-body chaos,” JHEP 10 (2018) 127, arXiv:1801.00010 [hep-th].
  * [57] M. Blake and H. Liu, “On systems of maximal quantum chaos,” JHEP 05 (2021) 229, arXiv:2102.11294 [hep-th].
  * [58] N. Abbasi, “Long-time tails in the SYK chain from the effective field theory with a large number of derivatives,” arXiv:2112.12751 [hep-th].
  * [59] M. Round, “Holographic Renormalisation and the Electroweak Precision Parameters,” Phys. Rev. D 82 (2010) 053002, arXiv:1003.2933 [hep-ph].
  * [60] A. Kamenev, Field Theory of Non-Equilibrium Systems.  Cambridge University Press, 2011.
  * [61] D. Nickel and D. T. Son, “Deconstructing holographic liquids,” New J. Phys. 13 (2011) 075010, arXiv:1009.3094 [hep-th].
  * [62] Y. Bu and M. Lublinsky, “All order linearized hydrodynamics from fluid-gravity correspondence,” Phys. Rev. D 90 no. 8, (2014) 086003, arXiv:1406.7222 [hep-th].
  * [63] Y. Bu and M. Lublinsky, “Linearized fluid/gravity correspondence: from shear viscosity to all order hydrodynamics,” JHEP 11 (2014) 064, arXiv:1409.3095 [hep-th].
  * [64] Y. Bu and M. Lublinsky, “Linearly resummed hydrodynamics in a weakly curved spacetime,” JHEP 04 (2015) 136, arXiv:1502.08044 [hep-th].
  * [65] Y. Bu, M. Lublinsky, and A. Sharon, “Hydrodynamics dual to Einstein-Gauss-Bonnet gravity: all-order gradient resummation,” JHEP 06 (2015) 162, arXiv:1504.01370 [hep-th].
  * [66] Y. Bu, M. Lublinsky, and A. Sharon, “$U(1)$ current from the AdS/CFT: diffusion, conductivity and causality,” JHEP 04 (2016) 136, arXiv:1511.08789 [hep-th].
  * [67] D. T. Son and D. Teaney, “Thermal Noise and Stochastic Strings in AdS/CFT,” JHEP 07 (2009) 021, arXiv:0901.2338 [hep-th].
  * [68] S. S. Gubser and S. S. Pufu, “The Gravity dual of a p-wave superconductor,” JHEP 11 (2008) 033, arXiv:0805.2960 [hep-th].
  * [69] M. Ammon, J. Erdmenger, P. Kerner, and M. Strydom, “Black Hole Instability Induced by a Magnetic Field,” Phys. Lett. B 706 (2011) 94–99, arXiv:1106.4551 [hep-th].
  * [70] Y.-Y. Bu, J. Erdmenger, J. P. Shock, and M. Strydom, “Magnetic field induced lattice ground states from holography,” JHEP 03 (2013) 165, arXiv:1210.6669 [hep-th].
  * [71] E. Wang and U. W. Heinz, “A Generalized fluctuation dissipation theorem for nonlinear response functions,” Phys. Rev. D 66 (2002) 025008, arXiv:hep-th/9809016.
  * [72] D.-f. Hou, E. Wang, and U. W. Heinz, “n point functions at finite temperature,” J. Phys. G 24 (1998) 1861–1868, arXiv:hep-th/9807118.
  * [73] P. Kovtun and L. G. Yaffe, “Hydrodynamic fluctuations, long time tails, and supersymmetry,” Phys. Rev. D 68 (2003) 025007, arXiv:hep-th/0303010.
  * [74] S. Caron-Huot and O. Saremi, “Hydrodynamic Long-Time tails From Anti de Sitter Space,” JHEP 11 (2010) 013, arXiv:0909.4525 [hep-th].