# An Effective Theory Of Anomalous Momentum Diffusion From Holography

Jewel K. Ghosh⋆#, M. Arshad Momen⋆#†
⋆ Independent University Bangladesh (IUB), Bashundhara RA, Dhaka 1229,
Bangladesh
# Center for Computational and Data Sciences, Independent University,
Bangladesh, Bashundhara RA, Dhaka 1229, Bangladesh
† University of Dhaka, Dhaka-1000, Bangladesh
Email<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract:

We consider a $U(1)$ Maxwell-Chern-Simons theory in $5$-dimensions, and
analyze the vector perturbations around a classical charged black-brane
background. We solve the equations of motion for these perturbations in a
derivative expansion. By computing the boundary current, we find that time and
spatial derivatives can be interpreted as the induced electric and magnetic
field respectively, and the Chern-Simons term contributes to a nonzero
divergence of the boundary current which indicates a quantum anomaly. Using
holography, we construct a two-derivative effective action for the vector
perturbations. By complexifying the radial coordinate, and using appropriate
transformation, we construct the full solution on the complexified bulk
contour. By computing the on-shell action for the full Schwinger-Keldysh
geometry, we obtain the Keldysh functional. We find that the single boundary
on-shell action mixes parity, whereas the Keldysh functional does not depend
on the Chern-Simons term up to the quadratic orders in derivative expansion.

††dedicated: Dedicated to Mrs. Shitali Ghosh, beloved mother of Jewel K.
Ghosh.

##  1 Introduction

The Anti-de Sitter (AdS)/Conformal Field Theory (CFT) correspondence [1] (for
review see [2, 3]) relates a Quantum Field Theory in $d$-dimensions to a
gravitational theory defined on higher dimensions. In the appropriate limit,
the gravitational theory is approximated by classical General Relativity,
where metric of the latter describes the “bulk”. Different fields in the bulk
theory correspond to different operators of the dual quantum field theory. The
correspondence is further strengthened by the equality of the partition
functions as described by the GKPW rule [4, 5].
After the initial discovery, there have been several extensions in different
directions. Bulks which are not AdS everywhere have been studied. A particular
class of example is asymptotically AdS space-time, as the name suggests the
bulk asymptotes to AdS space-time near the boundary, though it can differ from
AdS in the interior. It can have horizon, consequently, such black hole space-
time carries a non-trivial Hawking temperature. On the dual side, the
asymptotically AdS black hole corresponds to an equilibrium quantum field
theory at a finite temperature.
To study non-equilibrium properties of quantum field theories, a natural tool
is the Schwinger-Keldysh closed time path formalism [6, 7]. Instead of
defining the action on a positive real line for time, in the Schwinger-Keldysh
formalism, the time runs forward and then backward in time forming a closed
path. This is because of the fact that ket and bra in a density matrix evolves
in opposite directions in time. In holography, the early attempts to capture
the Schwinger-Keldysh physics was accomplished in [8], where the authors used
Kruskal diagram to identify the field doubling as required by the Schwinger-
Keldysh formalism. Recently, a prescription [9] has been proposed to capture
the Schwinger-Keldysh physics from the gravity dual. In this proposal the bulk
geometry is complexified encircling the horizon. A cartoon of this geometry is
shown in Fig. 1.

\begin{overpic}[scale={.4}]{Figures/Contour} \put(18.0,20.0){$r_{+}$}
\put(100.0,25.0){$\infty+i\epsilon$} \put(100.0,9.0){$\infty-i\epsilon$}
\end{overpic} Figure 1: Schwinger-Keldysh contour requires complexifying the
bulk geometry.

Two segments of the bulk contour correspond to the two different segments of
the Schwinger-Keldysh closed path, and we specify the sources at the
boundaries corresponding to each segment. Using this prescription, several
studies have been conducted for different kinds of systems [10, 11, 12, 13,
14, 15, 16, 17, 18, 19, 20, 21].
Anomaly is an important concept in quantum field theory which is defined to be
any classical symmetry of the Lagrangian that is broken by quantum mechanical
effects [22, 23]. Although the classical Lagrangian is invariant under a
symmetry, the measure of the path integral may not be invariant of the
symmetry under consideration, and hence the divergencelessness of the
corresponding current does not hold quantum mechanically. Different aspects of
anomaly in holography have been extensively studied in the literature [24, 25,
26, 27, 28].
The purpose of this paper is to understand anomaly in Schwinger-Keldysh
physics. As we have mentioned before, different bulk fields correspond to
different boundary operators. In particular, a bulk gauge field described by a
$U(1)$ Maxwell theory is dual to a boundary current. Different aspects of the
bulk gauge field have been extensively studied in the literature [29, 30, 31],
and provided some early results emerging from holography. To describe anomaly
in holography, a standard approach is to add a Chern-Simons term. In total,
this corresponds to the Maxwell-Chern-Simons theory for a $U(1)$ gauge field.
Holographically, this gauge field is dual to an anomalous current on the
boundary quantum field theory. We study perturbation around the classical
background, and then find an effective action for this perturbation. In the
other way around, we derive a two-derivative action whose dynamics is dual to
an anomalous current on the dual strongly coupled quantum field theory. After
finding the solution of perturbation equation, we use an appropriate
transformation to find the solution for the whole holographic Schwinger-
Keldysh contour. By constructing the on-shell action, we find the Keldysh
functional holographically.
The structure of this paper is as follows. In section 2, we describe the setup
of the system that we study. We also describe some of the important properties
of the background there. In section 3, we analyze the perturbation around the
background fields. The perturbation is classified as vector sector, and we
show that it is the vector perturbation that gives rise to interesting
dynamics and affects the Chern-Simons term in the Lagrangian. We solve the
perturbation equation in a derivative expansion, and the steps are spelled out
in that section. We also construct a $2$-derivative action variation of which
gives rise to the vector perturbation equations of motion. In section 4, we
compute the on-shell action, and renormalize it with appropriate counter-term.
Section 5 is devoted for the Schwinger-Keldysh calculation. First we construct
the full solution on the Schwinger-Keldysh radial contour, and then compute
the Keldysh functional.

##  2 Setup

In this work, we consider the Maxwell-Chern-Simons theory for a $U(1)$ gauge
field in a $5$-dimensional space-time. The action we consider is 111Our
notation is the following: beginning small Latin letters (a,b,c…) will denote
the bulk indices, Greek letters denote the boundary indices, middle small
Latin letters (i,j,m…) will denote the spatial indices. ,222Convention for the
Levi-Civita tensor is
$\epsilon^{abcef}=-\frac{\varepsilon^{abcef}}{\sqrt{-g}}$, where
$\varepsilon^{rtxyz}=1$.

$S=-\int
d^{5}x\sqrt{-g}\left(F_{ab}F^{ab}+\frac{\alpha}{4}\epsilon^{abcde}A_{a}F_{bc}F_{ef}\right).$
(1)

Here

$A=A_{a}dx^{a}$ (2)

is the $U(1)$ gauge field and

$F=dA=\frac{1}{2}F_{ab}dx^{a}\wedge dx^{b}$ (3)

is the corresponding field-strength. The first part of the action (1) is the
usual Maxwell part, whereas the second part is the Chern-Simons term with
$\alpha$ being the corresponding coupling. Note that the dimension of $\alpha$
is $-\frac{3}{2}$, and the theory is perturbatively non-renormalizable.
We are interested in studying an anomalous current in a quantum field theory
at a finite temperature. Holographically, the minimum ingredients we need a
black-brane metric, and a non-trivial gauge field. To accomplish this, we take
the following background [32]

$\displaystyle ds^{2}=-r^{2}f(r)dv^{2}+2dvdr+r^{2}\delta_{ij}dx^{i}dx^{i},$
(4) $\displaystyle A=-\xi(r)dv,$ (5)

where

$\displaystyle
f(r)=1-\left(1+Q^{2}\right)\left(\frac{r_{+}}{r}\right)^{4}+Q^{2}\left(\frac{r_{+}}{r}\right)^{6},$
(6) $\displaystyle\xi(r)=\sqrt{\frac{3}{2}}Q\frac{r_{+}^{3}}{r^{2}}.$ (7)

The metric is written in terms of ingoing Eddington-Finkelstein coordinates.
The transformation that brings from the radial to Eddington-Finkelstein
coordinates is described in Appendix A. These field configurations satisfy the
background equations:

$\displaystyle\nabla_{b}F^{ba}-\frac{3\alpha}{16}\epsilon^{abcde}F_{bc}F_{de}=0.$
(8)

The blackening factor $f(r)$ asymptotes to $1$ when $r\rightarrow\infty$. Near
$r\rightarrow\infty$, the metric (4) becomes:

$ds^{2}\approx-r^{2}dr^{2}+2dvdr+r^{2}\delta_{ij}dx^{i}dx^{j}$ (9)

which is the AdS5 metric written in Eddington-Finkelstein coordinates.
$r=\infty$ is the conformal boundary of the space-time, and near boundary, the
metric asymptotes to the $5$-dimensional Anti-de Sitter space-time.
The metric (6) describes a black-brane which is charged under the gauge field.
By setting $Q=0$, we obtain the uncharged case, which is the pure black-brane
metric. The blackening factor has a zero when $r=r_{+}$ which corresponds to a
horizon. Temperature corresponding to this black-brane is given by

$T=\frac{2-Q^{2}}{2\pi b}$ (10)

where we have defined

$b=\frac{1}{r_{+}}.$ (11)

Setting $Q=\sqrt{2}$, the Hawking temperature vanishes. This emerges from the
fact that, at this particular value of $Q$, there is a double zero of the
blackening factor. As a consequence of the presence of a double zero, there
will be an AdS2 factor in the near horizon geometry. This corresponds to an
extremal case. In general, we will call (4) to be Anti-de Sitter-Reissner-
Nordström (AdS-RN) black-brane metric.
Equations (6)-(7) constitute the background for our system. In this work, we
do not take gravity to be dynamical. That means we consider the metric to be
the background, and any back-reaction on this will be ignored. In other words,
we consider the probe limit, and the probe gauge field is described by the
action (1). Variation of the action gives

$\displaystyle\delta S=$ $\displaystyle 4\int_{M}d^{5}x\sqrt{-g}\ \delta
A_{a}\left(\nabla_{b}F^{ba}-\frac{3\alpha}{16}\epsilon^{abcde}F_{bc}F_{de}\right)$
$\displaystyle-4\int_{\partial M}d^{4}x\sqrt{-\gamma}\ n_{a}\delta
A_{b}\left(F^{ab}+\frac{\alpha}{4}\epsilon^{abcde}A_{c}F_{de}\right)$ (12)

where $M$ is the bulk manifold and $\partial M$ is its boundary with outward
normal vector $n_{a}$. The gauge field satisfies the equation of motion

$\nabla_{b}F^{ba}-\frac{3\alpha}{16}\epsilon^{abcde}F_{bc}F_{de}=0.$ (13)

One can easily check that the background (6)-(7) satisfy Eq. (13). In the
following section we will study the fluctuation of the gauge field.

##  3 Vector perturbation

In this section, we study the perturbation of the gauge field on top of the
background given by Eq. (7). The perturbation can be classified as vector or
scalar depending on their transformation under $SO(3)$ corresponding to the
$3$-dimensional spatial part of the metric (6). Since we are considering a
spin-1 field, the perturbation can be either vector or scalar. We start with
the vector part, and comment on the scalar part in Appendix C.
In the vector sector, we perturb the gauge field333$\int_{\omega,k}$ is the
shorthand notation for $\int\frac{d\omega}{2\pi}\frac{d^{3}k}{(2\pi)^{3}}$

$a_{m}=r\int_{\omega,k}e^{-i\omega
v}\left[\Psi_{E}(r,k,\omega)V_{m}^{E}(\vec{k},\vec{x})+\Psi_{O}(r,k,\omega)V_{m}^{O}(\vec{k},\vec{x})\right].$
(14)

where $V_{m}^{E,O}$ are the vector harmonics satisfying [33, 34]

$\nabla^{2}V_{m}^{E,O}+k^{2}V_{m}^{E,O}=0,\ \ \partial^{m}V_{m}^{E,O}=0.$ (15)

Here $\nabla^{2}=\partial^{m}\partial_{m}$ is the $3$-dimensional Laplacian
corresponding to the spatial part of the metric. The superscript refer to even
(E) and odd (O) based on their behavior under parity transformations
$\vec{x}\rightarrow-\vec{x},\ \vec{k}\rightarrow-\vec{k}$. These vector
harmonics have the following property:

$\displaystyle\partial_{m}V_{n}^{E}-\partial_{n}V_{m}^{E}=-ik\epsilon_{mnp}V_{p}^{O},$
(16)
$\displaystyle\partial_{m}V_{n}^{O}-\partial_{n}V_{m}^{O}=ik\epsilon_{mnp}V_{p}^{E}.$
(17)

These equations are consistent in the sense that by using Eq. (15), one
equation maps the other. Note the sign difference on the right hand side of
Eqs. (16)-(17). This will manifest itself in the Chern-Simons contribution of
the fluctuation equation which we describe below.
We use the following normalization condition

$\displaystyle\int d^{3}\vec{x}\
V^{E}_{m}\left(\vec{k}_{1},\vec{x}\right)V_{E}^{m}\left(\vec{k}_{2},\vec{x}\right)=\left(2\pi\right)^{3}\delta^{(3)}\left(\vec{k}_{1}+\vec{k}_{2}\right),$
(18) $\displaystyle\int d^{3}\vec{x}\
V^{O}_{m}\left(\vec{k}_{1},\vec{x}\right)V_{O}^{m}\left(\vec{k}_{2},\vec{x}\right)=-\left(2\pi\right)^{3}\delta^{(3)}\left(\vec{k}_{1}+\vec{k}_{2}\right),$
(19) $\displaystyle\int d^{3}\vec{x}\
V^{O}_{m}\left(\vec{k}_{1},\vec{x}\right)V_{E}^{m}\left(\vec{k}_{2},\vec{x}\right)=0.$
(20)

This is consistent with Eqs. (16)-(17). Although we did not use any specific
representation of the vector harmonics, a particular representation is given
by [33]

$\displaystyle
V^{E}_{m}\left(\vec{k},\vec{x}\right)=\frac{1}{kk_{\perp}}e^{i\vec{k}\cdot\vec{x}}\left(k_{\perp}^{2}\quad-
k_{x}k_{y}\quad-k_{x}k_{z}\right)$ (21) $\displaystyle
V^{O}_{m}\left(\vec{k},\vec{x}\right)=\frac{1}{k_{\perp}}e^{i\vec{k}\cdot\vec{x}}\left(0\quad-
k_{z}\quad k_{y}\right)$ (22)

where

$\vec{k}=\left(k_{x}\quad k_{y}\quad k_{z}\right),\ \
k_{\perp}=\sqrt{k_{y}^{2}+k_{z}^{2}},\ \ \text{and}\ \
k=\sqrt{k_{x}^{2}+k_{y}^{2}+k_{z}^{2}}.$ (23)

Using this representation one can check all the statements made above.
Including the perturbation, in the vector sector, the gauge field and the
field-strength tensors are:

$\displaystyle A=-\xi
dv+\int_{\omega,k}\left(r\Psi_{E}V_{j}^{E}+r\Psi_{O}V_{j}^{O}\right)e^{-i\omega
v}dx^{j}$ (24) $\displaystyle F=-\xi^{\prime}dr\wedge
dv+\int_{\omega,k}\left[\left\\{\frac{d}{dr}\left(r\Psi_{E}\right)V_{j}^{E}+\frac{d}{dr}\left(r\Psi_{O}\right)V_{j}^{O}\right\\}e^{-i\omega
v}dr\wedge dx^{j}\right.$ $\displaystyle\qquad\qquad\qquad\qquad\qquad-i\omega
r\left(\Psi_{E}V_{j}^{E}+\Psi_{O}V_{j}^{O}\right)e^{-i\omega v}dv\wedge
dx^{j}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\left.-\frac{ikr}{2}\epsilon_{mnp}\left(\Psi_{E}V^{O}_{p}-\Psi_{O}V^{E}_{p}\right)e^{-i\omega
v}dx^{m}\wedge dx^{n}\right]$ (25)

Using these, we can obtain the equation of motion for the vector fluctuation.
They are:

$\displaystyle\frac{d}{dr}\left[r^{3}f\frac{d}{dr}\left(r\Psi_{E}\right)\right]=\left(k^{2}+3i\omega
r\right)\Psi_{E}+2i\omega r^{2}\frac{d\Psi_{E}}{dr}+\frac{3i\alpha
k\xi^{\prime}r}{2}\Psi_{O},$ (26)
$\displaystyle\frac{d}{dr}\left[r^{3}f\frac{d}{dr}\left(r\Psi_{O}\right)\right]=\left(k^{2}+3i\omega
r\right)\Psi_{O}+2i\omega r^{2}\frac{d\Psi_{O}}{dr}-\frac{3i\alpha
k\xi^{\prime}r}{2}\Psi_{E}.$ (27)

A quick look at the Eqs. (26) and (27) shows that a complex combination of the
perturbations $\Psi_{E,O}$ will satisfy decoupled radial equations. These two
equations got coupled because of the presence of the Chern-Simons term. As we
mentioned before, the Chern-Simon term mixes the parity and effectively acts
as a coupling between the even and odd sector. This is reflected in the
equations (26)-(27). Note also the sign difference in front of the Chern-
Simons coupling in Eqs. (26)-(27). This is because of Eqs. (16)-(17), since
they have opposite sign in the right hand side.

###  3.1 A two-derivative action for the vector perturbation

In the previous section we had obtained the equation of motion for the vector
fluctuations, which obeys a second order differential equation. It is
therefore instructive to find a two-derivative action whose variation will
yield the fluctuation equation.
There are several ways to obtain this action. The straightforward procedure is
to start with the action (1) and expand it upto quadratic order in the
fluctuation $\Psi_{E,O}$. The quadratic part is the desired action. Performing
this exercise we find:

$\displaystyle S^{(2)}=-2\int_{\omega,k}\int
dr\left[r^{3}f\left\\{\frac{d}{dr}\left(r\Psi_{E}\right)\frac{d}{dr}\left(r\Psi_{E}^{(-)}\right)-\frac{d}{dr}\left(r\Psi_{O}\right)\frac{d}{dr}\left(r\Psi_{O}^{(-)}\right)\right\\}\right.$
$\displaystyle+i\omega
r^{2}\left\\{\Psi_{E}^{(-)}\frac{d}{dr}\left(r\Psi_{E}\right)-\Psi_{E}\frac{d}{dr}\left(r\Psi_{E}^{{(-)}}\right)-\Psi_{O}^{{(-)}}\frac{d}{dr}\left(r\Psi_{O}\right)+\Psi_{O}\frac{d}{dr}\left(r\Psi_{O}^{(-)}\right)\right\\}$
$\displaystyle\left.+k^{2}r\left(\Psi_{E}\Psi_{E}^{(-)}-\Psi_{O}\Psi_{O}^{(-)}\right)\right]$
$\displaystyle+ik\alpha\int_{\omega,k}\int
dr\left[\frac{d}{dr}\left\\{r^{2}\xi\left(\Psi_{E}\Psi_{O}^{(-)}+\Psi_{E}^{(-)}\Psi_{O}\right)\right\\}-3r^{2}\xi^{\prime}\left(\Psi_{E}\Psi_{O}^{(-)}+\Psi_{E}^{(-)}\Psi_{O}\right)\right]$
(28)

where
$\Psi^{{(-)}}_{E,O}\left(\omega,k\right)=\Psi_{E,O}\left(-\omega,k\right)|_{\vec{k}\rightarrow-\vec{k}}$.
Note that, the $\alpha$-independent term arises from parity preserving Maxwell
part, while the $\alpha$-dependent Chern-Simons term mixes parity. The total
derivative part in the last line of Eq. (28) comes from the Chern-Simons part,
and being a total derivative term, this does not play a role in the equation
of motion. It will, however, contribute to the on-shell action. We will
elaborate more on this in section 4. Starting from the action (28), a
straight-forward exercise reveals that the equation of motion for $\Psi_{E}$
and $\Psi_{O}$ are indeed Eqs. (26)-(27).

###  3.2 Solution of perturbation equations in the hydrodynamic expansion

In section 3, we have found the equations for the vector fluctuation. Since
the equations got coupled, this is unlikely that a closed form exact solution
can be found. Instead, we work in the gradient expansion. Since the time and
$\vec{x}$ dependence is exponential, a gradient expansion is also equivalent
to small $\omega$ and $k$ expansion. This is what we refer to the hydrodynamic
expansion.
In the hydrodynamic expansion, we look for solution in the form

$\displaystyle\Psi_{E}=\Psi_{E}^{(0)}+\left(\omega\Psi_{E}^{(\omega)}+k\Psi_{E}^{(k)}\right)+\left(\omega^{2}\Psi_{E}^{(\omega^{2})}+k^{2}\Psi_{E}^{(k^{2})}+\omega
k\Psi_{E}^{(\omega k)}\right)+\cdots$ (29)
$\displaystyle\Psi_{O}=\Psi_{O}^{(0)}+\left(\omega\Psi_{O}^{(\omega)}+k\Psi_{O}^{(k)}\right)+\left(\omega^{2}\Psi_{O}^{(\omega^{2})}+k^{2}\Psi_{O}^{(k^{2})}+\omega
k\Psi_{O}^{(\omega k)}\right)+\cdots$ (30)

where the superscript denotes the order of the perturbative expansion. For the
pure Maxwell Lagrangian, no odd power of $k$ emerged. But due the presence of
Chern-Simons term, odd power of $k$ can be non-trivial. This is a novel
feature in the Chern-Simons case.
We insert the expansions (29) and (30) into Eqs. (26)-(27). Since the right
hand side of Eqs. (26)-(27) is higher order in derivative than the left hand
side, equating at each order we have equations of the form:

$\displaystyle\frac{d}{dr}\left[r^{3}f\frac{d}{dr}\left(r\Psi_{E}^{\left(\omega^{m}k^{n}\right)}\right)\right]=J_{E}^{\left(\omega^{m}k^{n}\right)}(r),$
(31)
$\displaystyle\frac{d}{dr}\left[r^{3}f\frac{d}{dr}\left(r\Psi_{O}^{\left(\omega^{m}k^{n}\right)}\right)\right]=J_{O}^{\left(\omega^{m}k^{n}\right)}(r)$
(32)

where the right hand side represents the source that is built from the
solutions of previous orders. The only exception is the zeroth order for which
the equation is sourceless, i.e the right hand side is zero.
Let us now take the equations at each order, in which the zeroth order
equation is:

$\frac{d}{dr}\left[r^{3}f\frac{d}{dr}\left(r\Psi_{E,O}^{\left(0\right)}\right)\right]=0.$
(33)

This is a second order differential equation, integration of which contains
two integration constants. As we want to impose regularity at the horizon
$r=r_{+}$, one integration constant has to be set to zero. Here regularity
guarantees the analyticity of the quantity at the horizon. Therefore, the
solution to the zeroth order equation is

$\Psi_{E,O}^{\left(0\right)}=\frac{C_{E,O}}{r}.$ (34)

Following the usual holographic dictionary, $C_{E,O}$ will be identified as
the source on the boundary.
For non-zero order, we can write a general expression for the solution of
equations Eqs. (31)-(32). Again, these are second order differential
equations, which require two boundary conditions. We demand regularity at the
horizon. We also require that $r\Psi_{E,O}$ goes to $C_{E,O}$ at the boundary.
Since we have already demanded that $r\Psi_{E,O}^{(0)}$ goes to $C_{E,O}$ at
the boundary, this implies that $r\Psi_{E,O}^{(\omega^{m}k^{n})}$ where both
$m,n\neq 0$, goes to zero at the boundary. After imposing these two
conditions, some manipulations yield the following general expression for the
solution:

$\displaystyle\Psi_{E,O}^{\left(\omega^{m}k^{n}\right)}$
$\displaystyle=-b^{2}\frac{1}{br}\int_{br}^{\infty}d\eta\
\frac{1}{\eta^{3}f\left(\frac{\eta}{b}\right)}\int_{1}^{\eta}d\kappa\
J_{(E,O)}^{\omega^{m}k^{n}}\left(\frac{\kappa}{b}\right)$ (35)
$\displaystyle=-b^{2}\frac{1}{br}\int_{br}^{\infty}d\eta\
\frac{1}{\eta^{3}f\left(\frac{\eta}{b}\right)}\left\\{F^{\left(\omega^{m}k^{n}\right)}(\eta)-F^{\left(\omega^{m}k^{n}\right)}(1)\right\\}$
(36) $\displaystyle=b^{2}\frac{1}{br}\int_{br}^{\infty}d\eta\
\frac{F^{\left(\omega^{m}k^{n}\right)}(1)}{\eta^{3}f\left(\frac{\eta}{b}\right)}-b^{2}\frac{1}{br}\int_{br}^{\infty}d\eta\
\frac{F^{\left(\omega^{m}k^{n}\right)}(\eta)}{\eta^{3}f\left(\frac{\eta}{b}\right)},$
(37)

where we have defined

$F^{\left(\omega^{m}k^{n}\right)}(\eta)=\int_{1}^{\eta}d\kappa\
J_{(E,O)}^{\omega^{m}k^{n}}\left(\frac{\kappa}{b}\right).$ (38)

The first term in Eq. (37) is universal, and can be integrated to find:

$\displaystyle\int_{br}^{\infty}d\eta\frac{1}{\eta^{3}f\left(\frac{\eta}{b}\right)}=$
$\displaystyle-\frac{i\pi\left(-2Q^{2}+2Z+1\right)}{4\left(Q^{2}-2\right)Z}-\frac{\left(1-2Q^{2}\right)\tanh^{-1}\left(\frac{2b^{2}r^{2}+1}{Z}\right)}{2\left(Q^{2}-2\right)Z}$
$\displaystyle-\frac{\log\left(b^{4}r^{4}+b^{2}r^{2}-Q^{2}\right)-2\log\left(1-b^{2}r^{2}\right)}{4\left(Q^{2}-2\right)}$
(39)

where we have defined $Z=\sqrt{1+4Q^{2}}$. Near $r=\infty$, this has an
asymptotic expansion:

$\int_{br}^{\infty}d\eta\frac{1}{\eta^{3}f\left(\frac{\eta}{b}\right)}\underset{r\rightarrow\infty}{=}\frac{1}{2(br)^{2}}+\frac{1+Q^{2}}{6(br)^{6}}+\mathcal{O}\left(\frac{1}{(br)^{7}}\right).$
(40)

The main difficulty to find an exact solution comes from the second part of
Eq. (37). However, often we need an expansion near the boundary located at
$r=\infty$. We circumvent the difficulty by expanding the integrand of the
second term in Eq. (37) near $r=\infty$, and then perform the integration. One
might worry that the constant term might be different in this way of finding
the near boundary expansion. This is fixed since we set the constant term for
non-zero order to be zero, which is manifest from the upper limit of Eq. (35).
Now we perform the integrations when possible to find the solution, and then
find the near boundary expansion. In the case where the integrations are not
tractable, we perform the strategy outline above to obtain the near boundary
expansion.
The details of the calculations are presented in Appendix B. Here, we record
the main results. For the linear order, we could find the exact solutions.
They are:

$\displaystyle\Psi^{(\omega)}_{E,O}(r)=-ib^{2}\frac{C_{E,O}}{br}F_{1}(br),$
(41)
$\displaystyle\Psi^{(k)}_{E,O}(r)=\pm\left(\frac{3}{2}\right)^{3/2}i\alpha
C_{O,E}Qb^{2}\frac{1}{br}F_{2}(br),$ (42)

where we have defined

$\displaystyle
F_{1}(br)=\int_{br}^{\infty}d\eta\frac{\eta-1}{\eta^{3}f\left(\frac{\eta}{b}\right)},$
(43) $\displaystyle
F_{2}(br)=\int_{br}^{\infty}d\eta\frac{\eta^{2}-1}{\eta^{5}f\left(\frac{\eta}{b}\right)}.$
(44)

Note that Eq. (42) originates from the Chern-Simons terms. In the absence of
Chern-Simons term or for uncharged solution, this term would not be present.
The integrations becomes intractable for quadratic terms. However, by the
strategy outlined above we have obtained the near boundary
$r\rightarrow\infty$ expansion. They are:

$\displaystyle\Psi_{E,O}^{(\omega^{2})}=\frac{b^{3}}{4}C_{E,O}\left\\{\frac{4G_{1}-1}{(br)^{3}}+\frac{2\ln(br)}{(br)^{3}}\right\\}+\frac{b^{3}}{2}C_{E,O}\frac{1}{(br)^{4}}+\mathcal{O}\left(\frac{1}{r^{7}}\right),$
(45) $\displaystyle\Psi_{E,O}^{(k^{2})}=-b^{3}C_{E,O}\frac{27\alpha^{2}\ln
2+16\ln(br)-16G_{2}+8}{32(br)^{3}}+\mathcal{O}\left(\frac{1}{r^{7}}\right)$
(46) $\displaystyle\Psi_{E,O}^{(\omega k)}=\pm
b^{3}\left(\frac{3}{2}\right)^{\frac{1}{2}}\alpha
C_{O,E}\left\\{\frac{f_{3}(Q)}{(br)^{3}}+\frac{3Q}{4(br)^{4}}-\frac{3Q}{8(br)^{6}}\right\\}+\mathcal{O}\left(\frac{1}{r^{7}}\right)$
(47)

where $G_{1}$ and $G_{2}$ are the values of integrals $\int
d\kappa\sqrt{\kappa}\frac{d}{d\kappa}\left\\{\sqrt{\kappa}F_{1}(\kappa)\right\\}$
and $G_{2}$ is the value of $\int
d\kappa\left\\{\frac{1}{\kappa}-\frac{27\alpha^{2}Q^{2}}{4\kappa^{3}}F_{2}(\kappa)\right\\}$
respectively evaluated at $\kappa=1$ which is the horizon. The function
$f_{3}(Q)$ is defined in Eq. (126). This vanishes at $Q=0$.
Now we have all the ingredients to write the near boundary expansion of the
fluctuation. We find

$\displaystyle\frac{\Psi_{E,O}}{b}=$ $\displaystyle
C_{E,O}\frac{1}{br}-ib\omega
C_{E,O}\left\\{\frac{1}{(br)^{2}}-\frac{1}{2(br)^{3}}+\frac{1+Q^{2}}{5(br)^{6}}+\mathcal{O}\left(\frac{1}{(br)^{7}}\right)\right\\}$
$\displaystyle\pm ibk\left(\frac{3}{2}\right)^{3/2}\alpha
C_{O,E}Q\left\\{\frac{1}{(br)^{2}}-\frac{1}{2(br)^{3}}+\frac{1+Q^{2}}{5(br)^{6}}+\mathcal{O}\left(\frac{1}{(br)^{7}}\right)\right\\}$
$\displaystyle+\frac{b^{2}\omega^{2}}{4}C_{E,O}\left\\{\left(\frac{4G_{1}-1}{(br)^{3}}+\frac{2\ln(br)}{(br)^{3}}\right)+\frac{1}{2(br)^{4}}+\mathcal{O}\left(\frac{1}{(br)^{7}}\right)\right\\}$
$\displaystyle-b^{2}k^{2}C_{E,O}\left\\{\frac{27\alpha^{2}\ln
2+16\ln(br)-16G_{2}+8}{32(br)^{3}}+\mathcal{O}\left(\frac{1}{(br)^{7}}\right)\right\\}$
$\displaystyle\pm b^{2}\omega k\left(\frac{3}{2}\right)^{\frac{1}{2}}\alpha
C_{O,E}\left\\{\frac{f_{3}(Q)}{(br)^{3}}+\frac{3Q}{4(br)^{4}}-\frac{3Q}{8(br)^{6}}+\mathcal{O}\left(\frac{1}{(br)^{7}}\right)\right\\}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\mathcal{O}\left(\omega^{3},k^{3},\omega
k^{2},k^{2}\omega\right).$ (48)

We will require this expression for various holographic calculations. In the
next section, we compute the on-shell action, and then renormalize it.

##  4 Quadratic on-shell action

In this section, we will obtain the on-shell action. Following the AdS/CFT
dictionary, this on-shell action corresponds to the generating functional of
the dual quantum field theory. We start with the bare action, and then proceed
to renormalize by using the holographic renormalization.

###  4.1 Bare action

The bare action can be obtained in a couple of ways. One can start from the
first variation of the action recorded in Eq. (12), and impose the on-shell
condition. The latter kills the first line of Eq. (12) and we find:

$S^{(1)}_{\text{on-shell}}=-4\int_{\partial M}d^{4}x\sqrt{-\gamma}\
n_{a}\delta
A_{b}\left(F^{ab}+\frac{\alpha}{4}\epsilon^{abcde}A_{c}F_{de}\right).$ (49)

Here $n_{a}$ is the outward unit normal to the regulated boundary defined via
$r=\Lambda$, where $\Lambda\rightarrow\infty$ for the boundary, and
$\gamma_{\mu\nu}$ is the induced metric on the boundary. A straight-forward
calculation gives

$n_{a}=\left(\frac{1}{r\sqrt{f}},0,\vec{0}\right).$ (50)

Since the variation of the gauge field does not change the shape of the
boundary, the normal vector is also unchanged. A variation of Eq. (49) gives

$S^{(2)}_{\text{on-shell}}=-2\int_{\partial M}d^{4}x\sqrt{-\gamma}n_{a}\delta
A_{b}\left\\{\delta F^{ab}+\frac{\alpha}{4}\epsilon^{abcde}\left(\delta
A_{c}\bar{F}_{de}+\bar{A}_{c}\delta F_{de}\right)\right\\}$ (51)

where a over-bar denotes the background (unperturbed) value.
Alternatively, one can start with the action (28). There will be some total
derivative terms which become the boundary terms. In either way we find a
simple result

$\displaystyle S^{(2)}_{\text{on-shell}}=$
$\displaystyle-2\int_{\omega,k}r^{4}f\left\\{\Psi_{E}\frac{d}{dr}\left(r\Psi_{E}^{(-)}\right)-\Psi_{O}\frac{d}{dr}\left(r\Psi_{O}^{(-)}\right)\right\\}$
$\displaystyle+i\alpha
r^{2}\xi\int_{\omega,k}k\left(\Psi_{E}\Psi_{O}^{(-)}+\Psi_{E}^{(-)}\Psi_{O}\right).$
(52)

The first line comes from the Maxwell part, and the second line is the Chern-
Simons contribution. Once again, the Maxwell part is parity preserving whereas
the Chern-Simons part mixes parity. As this is a quadratic action for the
perturbations, a suitable diagonalied modes involving the mixing of the modes
$\Psi_{E,O}$ will produce decoupled equation as we had commented earlier. This
action is the generating functional of the dual QFT via the AdS/CFT
correspondence.

###  4.2 Holographic renormalization

The on-shell action is divergent. To remove the divergences we need
appropriate counter-term. As described in [35], the required counter-term is:

$S_{\text{ct}}=u\int
d^{4}x\sqrt{-\gamma}\hat{F}_{(0)\mu\nu}\hat{F}^{(0)\mu\nu}$ (53)

where the bulk metric is written in domain wall coordinates as:

$ds^{2}=du^{2}+\gamma_{\mu\nu}dx^{\mu}dx^{\nu}.$ (54)

In Eq. (53), $\hat{F}_{(0)\mu\nu}$ is the leading term in the near boundary
expansion. Now we need a relationship between the coordinates $r$ and $u$ near
the boundary.
Following Appendix A, a coordinate transformation

$du=\frac{dr}{r\sqrt{f(r)}}$ (55)

takes the metric from the Eddington-Finkelstein coordinates to the domain wall
coordinates. Near the boundary $f(r)\rightarrow 1$, therefore the required
coordinate transformation is:

$u=\ln(mr)+\mathcal{O}\left(\frac{1}{r}\right).$ (56)

where $m$ is introduced for dimensional reason. Since the only dimension we
have is the temperature, we set the scheme by choosing $m=b$. Therefore, the
required counter-term is:

$S_{\text{ct}}=\ln(br)\int
d^{4}x\sqrt{-\gamma}\hat{F}_{(0)\mu\nu}\hat{F}^{(0)\mu\nu}.$ (57)

Since this action is quadratic in the field-strength tensor, the
renormalization is required starting from the second order in fluctuation.
We can extract $\hat{F}_{\mu\nu}$ from Eq. (25). Explicitly,

$\displaystyle\hat{F}=-\int_{\omega,k}\left[\frac{}{}i\omega
r\left(\Psi_{E}V_{j}^{E}+\Psi_{O}V_{j}^{O}\right)e^{-i\omega v}dv\wedge
dx^{j}\right.$
$\displaystyle\left.+\frac{ikr}{2}\epsilon_{mnp}\left(\Psi_{E}V^{O}_{p}-\Psi_{O}V^{E}_{p}\right)e^{-i\omega
v}dx^{m}\wedge dx^{n}\right].$ (58)

From here we can extract:

$\displaystyle\hat{F}_{vj}=-\int_{\omega,k}\frac{}{}i\omega
r\left(\Psi_{E}V_{j}^{E}+\Psi_{O}V_{j}^{O}\right)e^{-i\omega v},$ (59)
$\displaystyle\hat{F}_{mn}=-\int_{\omega,k}ikr\epsilon_{mnp}\left(\Psi_{E}V^{O}_{p}-\Psi_{O}V^{E}_{p}\right)e^{-i\omega
v}.$ (60)

Using these, a straightforward calculation yields

$S_{\text{ct}}=-2r^{2}\ln(br)\int_{\omega,k}\left(\frac{\omega^{2}}{\sqrt{f}}-k^{2}\sqrt{f}\right)\left(\Psi_{E}\Psi_{E}^{(-)}-\Psi_{O}\Psi_{O}^{(-)}\right).$
(61)

Note the the counter-term starts from the quadratic order in the hydrodynamic
expansion.

###  4.3 Renormalized action

Now we have all the ingredients to compute the renormalized on-shell action.
The latter is given by:

$S^{\text{ren}}_{\text{on-shell}}=S^{(2)}_{\text{on-shell}}+S_{\text{ct}}.$
(62)

Using Eqs. (52) and (61) we find the following expression for the renormalized
on-shell action

$\displaystyle S^{\text{ren}}_{\text{on-shell}}=$
$\displaystyle\int_{\omega,k}\left[-2r^{4}f\left\\{\Psi_{E}\frac{d}{dr}\left(r\Psi_{E}^{(-)}\right)-\Psi_{O}\frac{d}{dr}\left(r\Psi_{O}^{(-)}\right)\right\\}+i\alpha
kr^{2}\xi\left(\Psi_{E}\Psi_{O}^{(-)}+\Psi_{E}^{(-)}\Psi_{O}\right)\right.$
$\displaystyle\left.-2r^{2}\ln(br)\left(\frac{\omega^{2}}{\sqrt{f}}-k^{2}\sqrt{f}\right)\left(\Psi_{E}\Psi_{E}^{(-)}-\Psi_{O}\Psi_{O}^{(-)}\right)\right].$
(63)

This will be the generating functional on the QFT side via the AdS/CFT
correspondence. We can observe that both the Maxwell and the counter-term
parts of the renormalized on-shell action (63) are parity preserving, but the
Chern-Simons part mixes parity.
We insert the near boundary expansion of $\Psi_{E,O}$ into Eq. (63). Since
$\xi\sim\frac{1}{r^{2}}$ and
$\Psi_{E,O}\sim\left\\{\frac{1}{r}+\mathcal{O}\left(\frac{1}{r^{2}}\right)\right\\}$,
the Chern-Simons part does not contribute on-shell action directly. It does,
however, through the solution. Therefore, all the divergences come from the
Maxwell term, which get canceled by the counter-term. After the cancellation
we get:

$\displaystyle S_{\text{on-
shell}}^{\text{ren}}=\frac{3\sqrt{6}iC_{E}C_{O}Q\alpha}{b}k+\frac{C_{E}^{2}-C_{O}^{2}}{8}\left\\{32G_{1}\omega^{2}+\left(16G_{2}-27\alpha^{2}\ln
2\right)k^{2}\right\\}$
$\displaystyle+\mathcal{O}\left(\omega^{3},\omega^{2}k,\omega
k^{2},k^{3}\right),$ (64)

where $G_{1}$ and $G_{2}$ are defined in Eqs. (115) and (121) respectively. In
Eq. (64) all the terms proportional to $\alpha$ are coming from the Chern-
Simons term. Once again, the parity mixing term begets from the Chern-Simons
term. Consequently, the even-odd correlator is non-trivial in this case.

###  4.4 The boundary current

To get more insight about the derivative expansion, in this section, we will
compute the boundary current. According to the standard AdS/CFT prescription,
the boundary current is given by [36, 37]:

$\langle
J^{\mu}\rangle=\sqrt{-\gamma}n_{r}\left(F^{r\mu}+\frac{\alpha}{4}\varepsilon^{r\mu\nu\rho\sigma}A_{\nu}F_{\rho\sigma}\right).$
(65)

Using the results from previous section, this can be written as

$\langle
J^{\mu}\rangle=r^{3}\left(F^{r\mu}+\frac{\alpha}{4}\varepsilon^{r\mu\nu\rho\sigma}A_{\nu}F_{\rho\sigma}\right).$
(66)

Using different expressions and after some algebra we find:

$\displaystyle\langle
J^{v}\rangle=r^{3}\left[\xi^{\prime}+\frac{i\alpha}{2r}\int_{\omega_{1},\vec{k}_{1}}\int_{\omega_{2},\vec{k}_{2}}k_{2}\left\\{\Psi_{O}^{(1)}\Psi_{E}^{(2)}\vec{V}_{O}^{(1)}\cdot\vec{V}_{O}^{(2)}-\Psi_{E}^{(1)}\Psi_{O}^{(2)}\vec{V}_{E}^{(1)}\cdot\vec{V}_{E}^{(2)}\right.\right.$
$\displaystyle\qquad\qquad\qquad\left.\left.+\Psi_{E}^{(1)}\Psi_{E}^{(2)}\vec{V}_{E}^{(1)}\cdot\vec{V}_{O}^{(2)}-\Psi_{O}^{(1)}\Psi_{O}^{(2)}\vec{V}_{O}^{(1)}\cdot\vec{V}_{E}^{(2)}\right\\}e^{-i(\omega_{1}+\omega_{2})v}\right]$
(67) $\displaystyle\langle
J^{m}\rangle=r^{3}\left[\int_{\omega,k}\left\\{\left(f\frac{d}{dr}\left(r\Psi_{E}\right)-\frac{i\omega}{r}\Psi_{E}\right)V^{m}_{E}+\left(f\frac{d}{dr}\left(r\Psi_{O}\right)-\frac{i\omega}{r}\Psi_{O}\right)V^{m}_{O}\right\\}e^{-i\omega
v}\right.$
$\displaystyle\qquad\qquad-\frac{\alpha}{4r^{3}}\left\\{-2ir\xi\int_{\omega,k}k\left(\Psi_{E}V^{m}_{O}-\Psi_{O}V^{m}_{E}\right)e^{-i\omega
v}\right.$
$\displaystyle\qquad\qquad-2ir^{2}\int_{\omega_{1},k_{1}}\int_{\omega_{2},k_{2}}\epsilon^{mnp}\omega_{1}\left(\Psi_{E}^{(1)}\Psi_{E}^{(2)}V_{n}^{E(1)}V_{p}^{E(2)}+\Psi_{O}^{(1)}\Psi_{O}^{(2)}V_{n}^{O(1)}V_{p}^{O(2)}\right.$
$\displaystyle\qquad\qquad\left.\left.\left.+\Psi_{E}^{(1)}\Psi_{O}^{(2)}V_{n}^{E(1)}V_{p}^{O(2)}+\Psi_{O}^{(1)}\Psi_{E}^{(2)}V_{n}^{O(1)}V_{p}^{E(2)}\right)e^{-i(\omega_{1}+\omega_{2})v}\right\\}\right].$
(68)

To see the implications of the hydrodynamic expansion, we compute the
divergence of the boundary current. After some algebra, we find:

$\displaystyle\langle\partial_{\mu}J^{\mu}\rangle$ $\displaystyle=2\alpha
r^{2}\int_{\omega_{1},k_{1}}\int_{\omega_{2},k_{2}}\omega_{1}k_{2}\left\\{\Psi_{O}^{(1)}\Psi_{E}^{(2)}\vec{V}_{O}^{(1)}\cdot\vec{V}_{O}^{(2)}-\Psi_{E}^{(1)}\Psi_{O}^{(2)}\vec{V}_{E}^{(1)}\cdot\vec{V}_{E}^{(2)}\right.$
$\displaystyle\left.+\Psi_{E}^{(1)}\Psi_{E}^{(2)}\vec{V}_{E}^{(1)}\cdot\vec{V}_{O}^{(2)}-\Psi_{O}^{(1)}\Psi_{O}^{(2)}\vec{V}_{O}^{(1)}\cdot\vec{V}_{E}^{(2)}\right\\}e^{-i(\omega_{1}+\omega_{2})v}.$
(69)

This can be succinctly written

$\displaystyle\langle\partial_{\mu}J^{\mu}\rangle=-\frac{\alpha}{4}\epsilon^{\mu\nu\rho\sigma}\hat{F}_{\mu\nu}\hat{F}_{\rho\sigma}=-\alpha\hat{F}\wedge\hat{F}.$
(70)

That means inserting expression of $\hat{F}$ from Eq. (58) into Eq. (70), we
can retrieve Eq. (69). This is the expression of familiar ABJ anomaly in
Quantum Field Theory [38, 39, 40]. Note that there was no electric field or
magnetic field in the background. It’s the fluctuation that induces an
effective electric $(\sim\omega)$ and magnetic field $(\sim k)$. Therefore, we
can interpret the current $J^{\mu}$ to be anomalous where the symmetry is
broken by the gradient expansion. The Chern-Simons term in the action takes
care of this symmetry breaking.

##  5 Schwinger-Keldysh contour and the influence functional

In the previous section we have used the ingoing Eddington-Finkelstein
coordinates. To construct the full Schwinger-Keldysh solution we should also
patch the the outgoing Hawking solution. The procedure to obtain the outgoing
solution from the ingoing solution is presented below [12].
The metric we started with is:

$ds^{2}=2dvdr-r^{2}f(r)dv^{2}+r^{2}dx_{i}dx^{i}\ .$ (71)

This was written in ingoing Eddington-Finkelstein coordinates. We define a new
coordinate $\chi$ by the following equation:

$dr=\frac{i\beta}{2}r^{2}f(r)d\chi$ (72)

where

$\beta=\frac{2\pi b}{\left[2-Q^{2}\right]}$ (73)

is the inverse Hawking temperature. The metric (71) can be written as:

$ds^{2}=\frac{i\beta}{2}\left(r^{2}f\right)^{2}dv\left(i\beta d\chi-
dv\right)+r^{2}dx_{i}dx^{i}.$ (74)

From here this is manifest that:

$dv\to i\beta d\chi-dv=dv^{\prime},\ \omega\to-\omega\ .$ (75)

preserves the form of the metric. This coordinate transformation will be
referred as the Debye time reversal, and it takes from the ingoing to the
outgoing Eddington-Finkelstein coordinates. Under this time reversal the
vector perturbations do not change since they only concern the spatial
coordinates. Therefore the full solution is given by:

$\displaystyle\Psi_{E}^{SK}=\Psi_{E}^{in}|_{\left\\{C_{E}\rightarrow
C_{\text{in}}^{E},C_{O}\rightarrow
C_{\text{in}}^{O}\right\\}}+e^{-\omega\beta\chi}\Psi_{E}^{out}|_{\left\\{C_{E}\rightarrow
C_{\text{out}}^{E},C_{O}\rightarrow C_{\text{out}}^{O}\right\\}},$ (76)
$\displaystyle\Psi_{O}^{SK}=\Psi_{O}^{in}|_{\left\\{C_{E}\rightarrow
C_{\text{in}}^{E},C_{O}\rightarrow
C_{\text{in}}^{O}\right\\}}+e^{-\omega\beta\chi}\Psi_{O}^{out}|_{\left\\{C_{E}\rightarrow
C_{\text{out}}^{E},C_{O}\rightarrow C_{\text{out}}^{O}\right\\}}.$ (77)

Here $\Psi^{in}_{E,O}$ are the solutions that we have constructed so far in
the ingoing Eddington-Finkelstein coordinates. $\Psi^{out}_{E,O}$ is
constructed from $\Psi^{in}_{E,O}$ by taking $\omega\rightarrow-\omega$. We
can also write the coefficients $C_{in}^{E,O},C_{out}^{E,O}$ in terms of the
Schwinger-Keldysh basis $C_{L}^{E,O},C_{R}^{E,O}$. Holographically this
amounts to complexifying the bulk geometry as Fig. 1 shows. From Eq. (72), by
a straightforward complex integration by using Cauchy integral formula we can
write:

$\chi(\infty+i\epsilon)=0,\quad\chi(\infty-i\epsilon)=1.$ (78)

The loci $\chi=0$ and $\chi=1$ will be denoted as the left and right
boundaries respectively. We can specify the sources at the left and right
boundaries which we denote as the left/right basis. In the left/right basis we
have the following equations:

$\displaystyle C_{in}^{E,O}+C_{out}^{E,O}=C_{L}^{E,O},$ (79) $\displaystyle
C_{in}^{E,O}+C_{out}^{E,O}e^{-\omega\beta}=C_{R}^{E,O}.$ (80)

Solution of these is given by:

$\displaystyle
C_{in}^{E,O}=f_{BE}(\omega)\left(C_{R}^{E,O}e^{\beta\omega}-C_{L}^{E,O}\right),\quad
C_{out}^{E,O}=f_{BE}(\omega)\left(C_{L}^{E,O}-C_{R}^{E,O}\right)e^{\beta\omega},$
(81)

where

$f_{BE}(\omega)=\frac{1}{e^{\beta\omega}-1}$ (82)

is the Bose-Einstein factor.
We will now proceed to calculate the on-shell action (63) for the full
Schwinger-Keldysh solution. For the complexified geometry, there are two
boundaries located at $\chi=0$ and $\chi=1$. To get the full on-shell action
we consider the individual on-shell actions, and subtract. We subtract because
the unit normals are directed opposite at the boundaries. That means, the
Schwinger-Keldysh functional is given by:

$S_{\text{on-shell}}^{\text{SK}}=S_{\text{on-
shell}}^{\text{ren}}|_{\chi=1}-S_{\text{on-shell}}^{\text{ren}}|_{\chi=0}.$
(83)

To better understand, we calculate the on-shell actions at the respective
boundaries. We find:

$\displaystyle S_{\text{on-
shell}}^{\text{ren}}|_{\chi=0}=\int_{\omega,k}\left[\frac{3\sqrt{6}i}{b}\left(C^{E}_{\text{in}}+C^{E}_{\text{out}}\right)\left(C^{O}_{\text{in}}+C^{O}_{\text{out}}\right)\alpha
Qk+\frac{1}{8}\left\\{\left(C^{E}_{\text{in}}+C^{E}_{\text{out}}\right)^{2}\right.\right.$
$\displaystyle\qquad\qquad\qquad\qquad\left.-\left(C^{O}_{\text{in}}+C^{O}_{\text{out}}\right)^{2}\right\\}\left\\{32G_{1}\omega^{2}+\left(16G_{2}-27\alpha^{2}\ln
2\right)k^{2}\right\\}$
$\displaystyle\qquad\qquad\qquad\qquad\left.\frac{}{}+\mathcal{O}\left(\omega^{3},\omega^{2}k,\omega
k^{2},k^{3}\right)\right]$ (84) $\displaystyle S_{\text{on-
shell}}^{\text{ren}}|_{\chi=1}=\int_{\omega,k}\left[\frac{3\sqrt{6}i}{b}\left(C^{E}_{\text{in}}+C^{E}_{\text{out}}\right)\left(C^{O}_{\text{in}}+C^{O}_{\text{out}}\right)\alpha
Qk+\frac{1}{8}\left\\{\left(C^{E}_{\text{in}}+C^{E}_{\text{out}}\right)^{2}\right.\right.$
$\displaystyle\qquad\qquad\qquad\qquad\left.-\left(C^{O}_{\text{in}}+C^{O}_{\text{out}}\right)^{2}\right\\}\left\\{32G_{1}\omega^{2}+\left(16G_{2}-27\alpha^{2}\ln
2\right)k^{2}\right\\}$
$\displaystyle\qquad\qquad\qquad\qquad\left.\frac{}{}+\frac{4i}{b}\left(C^{E}_{\text{in}}C^{E}_{\text{out}}-C^{O}_{\text{in}}C^{O}_{\text{out}}\right)\beta\omega^{2}+\mathcal{O}\left(\omega^{3},\omega^{2}k,\omega
k^{2},k^{3}\right)\right].$ (85)

From here, we can obtain

$S_{\text{on-
shell}}^{\text{SK}}=\int_{\omega,k}\left[\frac{4i}{b}\left(C^{E}_{\text{in}}C^{E}_{\text{out}}-C^{O}_{\text{in}}C^{O}_{\text{out}}\right)\beta\omega^{2}+\mathcal{O}\left(\omega^{3},\omega^{2}k,\omega
k^{2},k^{3}\right)\right].$ (86)

We can express this in terms of left/right basis by using Eq. (80). We find

$\displaystyle S_{\text{on-
shell}}^{\text{SK}}=\int_{\omega,k}\left[-\frac{4ie^{\beta\omega}}{b\left(e^{\beta\omega}-1\right)^{2}}\left\\{C^{E}_{L}\left(C^{E}_{L}-C^{E}_{R}\right)-C^{O}_{L}\left(C^{O}_{L}-C^{O}_{R}\right)+\left[C^{O}_{R}\left(C^{O}_{L}-C^{O}_{R}\right)\right.\right.\right.$
$\displaystyle\qquad\qquad\qquad\left.\left.\left.-C^{E}_{R}\left(C^{E}_{L}-C^{E}_{R}\right)e^{\beta\omega}\right]\beta\omega^{2}\right\\}+\frac{}{}\mathcal{O}\left(\omega^{3},\omega^{2}k,\omega
k^{2},k^{3}\right)\right]$ (87)

We can also express this in terms of Keldysh (average/difference) basis

$C^{E,O}_{A}=\frac{1}{2}\left(C^{E,O}_{L}+C^{E,O}_{R}\right),\ \
C^{E,O}_{D}=\left(C^{E,O}_{L}-C^{E,O}_{R}\right).$ (88)

In terms of this basis we can write Eq. (87) as

$\displaystyle S_{\text{on-
shell}}^{\text{SK}}=\int_{\omega,k}\left[-\frac{2ie^{\beta\omega}}{b\left(e^{\beta\omega}-1\right)^{2}}\left\\{C^{E}_{D}\left(2C^{E}_{A}+C^{E}_{D}\right)-C^{O}_{D}\left(2C^{O}_{A}+C^{O}_{D}\right)+\left[C^{O}_{D}\left(2C^{O}_{A}-C^{O}_{D}\right)\right.\right.\right.$
$\displaystyle\left.\left.\left.-C^{E}_{D}\left(2C^{E}_{A}-C^{E}_{D}\right)\right]e^{\beta\omega}\right\\}\beta\omega^{2}+\frac{}{}\mathcal{O}\left(\omega^{3},\omega^{2}k,\omega
k^{2},k^{3}\right)\right].$ (89)

This vanishes when we set $C^{E}_{D}=C^{O}_{D}=0$ as expected for a Keldysh
functional. The important piece of information we can extract is that
contribution from the Chern-Simons term gets canceled. Hence, the Schwinger-
Keldysh functional is insensitive to the anomaly, at least up to the order
that we considered here. Although we have not computed the higher order terms
in derivative expansion, we believe the cancellation of anomaly will also
persist for the higher order. This statement, however, requires careful
scrutiny checks which we keep as a future work.

##  6 Conclusion

In this work we have studied the Maxwell-Chern-Simons theory in a
$5$-dimensional space-time: a charged black-brane which asymptotes to a AdS5
near $r=\infty$ which is the boundary. The gauge field only has time
component. Via the AdS/CFT correspondence, we are considering a current in the
dual field theory side with a non-trivial chemical potential which is the
difference between the gauge field value evaluated at the boundary and
horizon.
We have considered perturbation of the gauge field. This perturbation can be
classified as vector and scalar modes depending on their transformation
properties with respect to the spatial $\mathbb{R}^{3}$ metric. In Appendix C,
we have shown that the Chern-Simons term does not contribute for scalar
perturbation. Therefore, the non-trivial perturbation comes from the vector
sector only. This is further classified as even or odd depending on their
transformation under parity.
We have found the equations of motion for the vector fluctuations. These
equations are different for the even and odd sector. Unlike the Maxwell
sector, the presence of the Chern-Simons term mixes this even and odd
perturbation. To solve these equations, we have performed a derivative
expansion which we call the hydrodynamic expansion. The details of this
procedure is presented in Appendix B. While in the first order we have found
an exact form of the solution, in the second order the integrations become
intractable. For this case, we have performed an asymptotic expansion of the
integrand, and then we integrate. This procedure enables us to find the
asymptotic expansion of the formal solution.
We have found a quadratic effective action (28) variation of which gives the
vector equations of motion. This come from the Maxwell-Chern-Simons action,
and then expanding it to the quadratic order. While the Maxwell part of the
action is parity preserving, the Chern-Simons part mixes parity. Later, we
show that the expectation value of the divergence of the boundary current is
non-zero, and therefore the current is anomalous. We interpret the time
derivative as the induced “electric field”, and the spatial derivative to be
induced “magnetic field”. Therefore the anomaly is reminiscent of ABJ anomaly
of quantum field theory.
We have computed the on-shell action which via the AdS/CFT correspondence is
the generating functional of the dual current. We have found that parity
preserving part comes from Maxwell term. The Chern-Simons term mixes parity,
and gives rises to the even-odd correlator.
The bulk geometry can be complexified to find the Schwinger-Keldysh
functional. We have used a complex radial contour, with the end point
boundaries corresponding to the left and right sector of the Schwinger-Keldysh
functional. Normal to these boundaries are oppositely oriented, and after the
appropriate subtraction of the on-shell actions, we have obtained the
Schwinger-Keldysh functional. In the Keldysh basis (average/difference basis),
we have found that the Schwinger-Keldysh functional becomes zero when we set
the difference source to be zero. This is an expected feature of Keldysh
functional. Upon the subtraction of the on-shell actions, we found that the
Chern-Simons term gets canceled. This implies that the Schwinger-Keldysh
function is evanescent to the anomalous term up to the order we have
considered, and we expect that this persists for the higher order terms as
well.
There are several interesting arenas to explore as future works. In this work,
we have not considered the backreaction of the gauge field on the metric. The
effects of this gauge field on metric is an interesting problem that we want
to study in near future. Depending on the parity of the gauge field, there
should be two different dispersion relations. It would be intereseting to
study these questions together with the Schwinger-Keldysh analysis. We wish to
report the answers to this question in near future.
Another study that we wish to pursue is how to address the question of vacuum
decay or phase transition in the context of Schwinger-Keldysh physics. Some of
the work have been done, and presented in [6, 7], but the effect of strong
coupling is not properly understood. Here holography comes into the role, and
numerous works have been done under the name Coleman-de Luccia (CdL) tunneling
(see [41] for a recent work). It would be interesting to understand the vacuum
decay via CdL instanton in the Schwinger-Keldysh formalism. We hope to report
this in a future work.

## Acknowledgments

We would like to thank Karl Landsteiner for clarifying issues with the
holographic renormalization to us. We also express our gratitude to R.
Loganayagam for illuminating discussions. We also thank Irfan Sadat Ameen for
checking some of the calculations at initial stages of this work.
JKG dedicates this paper to his beloved mother Mrs. Shitali Ghosh for her
unconditional love, and lifelong support.

##  Appendix A Coordinate Transformations

In this appendix, we will find a coordinate transformation between radial and
Eddington-Finkelstein coordinates. First we write the AdS-black-brane metric
in radial coordinates as follows:

$\displaystyle ds^{2}$
$\displaystyle=\frac{d\rho^{2}}{h(\rho)a^{2}(\rho)}+a^{2}(\rho)\left[-h(\rho)dt^{2}+d\vec{x}^{2}\right]$
$\displaystyle=a^{2}(\rho)h(\rho)\left[-dt^{2}+\frac{d\rho^{2}}{a^{4}(\rho)h^{2}(\rho)}\right]+a^{2}(\rho)d\vec{x}^{2}\
.$ (90)

The tortoise coordinate is defined as

$\frac{d\rho}{a^{2}(\rho)h(\rho)}=dr_{\star}\ .$ (91)

In terms of this coordinate, the above metric can be written as

$ds^{2}=a^{2}(\rho)h(\rho)\left(-dt+dr_{\star}\right)\left(dt+dr_{\star}\right)+a^{2}(\rho)d\vec{x}^{2}\
.$ (92)

Now we define

$dt+dr_{\star}=dv,\quad d\rho=dr\ .$ (93)

In terms of these coordinates, the above metric can be written as

$ds^{2}=2dv\ dr-a^{2}(r)h(r)dv^{2}+a^{2}(r)d\vec{x}^{2}\ .$ (94)

Setting $a(r)=r$ and $h(r)=f(r)$, we can retrieve the metric (4) that we
started with.

##  Appendix B Details of finding the solutions in derivative expansion

In this appendix we will present the details of the calculation to find the
solutions in the derivative expansion. Two equations that we intend to solve
are Eqs. (26) and (27). We solve these in a derivative expansion as a double
series in $\omega$ and $k$ given by Eqs. (29)-(30).
Zeroth order
For the zeroth order, there is no source term present in the right hand side
of Eqs. (31)-(32). We also impose regularity at the horizon. Imposing this, we
find that

$\Psi^{(O)}_{E,O}=\frac{C_{E,O}}{r}$ (95)

where $C_{E}$ and $C_{O}$ are the integration constants.
Order $\omega$
At order $\omega$, we need to solve the following equations:

$\displaystyle\frac{d}{dr}\left[r^{3}f\frac{d}{dr}\left(r\Psi_{E}^{\left(\omega\right)}\right)\right]=2ir^{1/2}\frac{d}{dr}\left[r^{3/2}\Psi_{E}^{(0)}\right],$
(96)
$\displaystyle\frac{d}{dr}\left[r^{3}f\frac{d}{dr}\left(r\Psi_{O}^{\left(\omega\right)}\right)\right]=2ir^{1/2}\frac{d}{dr}\left[r^{3/2}\Psi_{O}^{(0)}\right].$
(97)

Since the zeroth order solution was:

$\Psi_{E,O}^{(0)}(r)=\frac{C_{E,O}}{r},$ (98)

the source is:

$J_{E,O}^{(\omega)}(r)=iC_{E,O}.$ (99)

Therefore the order $\omega$ solution is:

$\Psi_{E,O}^{(\omega)}(r)=-ib^{2}\frac{C_{E,O}}{br}F_{1}(br)$ (100)

where

$F_{1}(br)=\int_{br}^{\infty}d\eta\frac{\eta-1}{\eta^{3}f\left(\frac{\eta}{b}\right)}.$
(101)

This can be integrated to find:

$\displaystyle
F_{1}(br)=\left[\frac{1}{\sqrt{2}\left(Q^{2}-2\right)Z}\left\\{\left(\frac{\left(Z+1\right)Q^{2}-Z+1}{\sqrt{1-Z}}\right)\right.\right.$
$\displaystyle\tan^{-1}\left(\frac{\sqrt{2}\eta}{\sqrt{1-Z}}\right)+\left(\frac{\left(Z-1\right)Q^{2}-Z-1}{\sqrt{Z+1}}\right)+\left.\tan^{-1}\left(\frac{\sqrt{2}\eta}{\sqrt{Z+1}}\right)\right\\}$
$\displaystyle+\frac{1}{4\left(Q^{2}-2\right)Z}\left\\{-Z\log\left[4\left(\eta^{4}+\eta^{2}-Q^{2}\right)\right]\left.+\left(1-2Q^{2}\right)\log\left(\frac{2\eta^{2}-Z+1}{2\eta^{2}+Z+1}\right)\right\\}\right.$
$\displaystyle\left.+\frac{\log(\eta+1)}{Q^{2}-2}\right]_{br}^{\infty}.$ (102)

Although this looks complicated, the near boundary expansion is quite simple!
The near boundary $r\rightarrow\infty$ expansion is:

$\displaystyle\Psi_{E,O}^{(\omega)}(r)=-iC_{E,O}b^{2}\left[\frac{1}{(br)^{2}}-\frac{1}{2(br)^{3}}+\frac{1+Q^{2}}{5(br)^{6}}-\frac{1+Q^{2}}{6(br)^{7}}+\mathcal{O}\left(\frac{1}{(br)^{8}}\right)\right].$
(103)

#### Order $k$

At order $k$, we have these two equations

$\displaystyle\frac{d}{dr}\left[r^{3}f\frac{d}{dr}\left(r\Psi_{E}^{\left(k\right)}\right)\right]=\frac{3i\alpha\xi^{\prime}r}{2}\Psi_{O}^{(0)},$
(104)
$\displaystyle\frac{d}{dr}\left[r^{3}f\frac{d}{dr}\left(r\Psi_{O}^{\left(k\right)}\right)\right]=-\frac{3i\alpha\xi^{\prime}r}{2}\Psi_{E}^{(0)}.$
(105)

By following the same procedure we can find the solution:

$\Psi^{(k)}_{E,O}(r)=\pm\left(\frac{3}{2}\right)^{3/2}i\alpha
C_{O,E}Qb^{2}\frac{1}{br}F_{2}(br)$ (106)

where

$F_{2}(br)=\int_{br}^{\infty}d\eta\frac{\eta^{2}-1}{\eta^{5}f\left(\frac{\eta}{b}\right)}.$
(107)

This can be integrated to find:

$\displaystyle F_{2}(br)$
$\displaystyle=-\frac{1}{Z}\left[\tanh^{-1}\left(\frac{1+2\eta^{2}}{Z}\right)\right]_{br}^{\infty}$
$\displaystyle=\frac{1}{Z}\left\\{\frac{i\pi}{2}+\tanh^{-1}\left(\frac{1+2(br)^{2}}{Z}\right)\right\\}.$
(108)

Therefore

$\Psi^{(k)}_{E,O}(r)=\pm\left(\frac{3}{2}\right)^{3/2}\frac{i\alpha
C_{O,E}Qb^{2}}{Z}\frac{1}{br}\left\\{\frac{i\pi}{2}+\tanh^{-1}\left(\frac{1+2(br)^{2}}{Z}\right)\right\\}.$
(109)

The near boundary $r\rightarrow\infty$ expansion is

$\displaystyle\Psi^{(k)}_{E,O}(r)=\pm\left(\frac{3}{2}\right)^{3/2}i\alpha
C_{O,E}Qb^{2}\left[\frac{1}{2(br)^{3}}-\frac{1}{4(br)^{5}}+\frac{1+Q^{2}}{6(br)^{7}}+\mathcal{O}\left(\frac{1}{(br)^{8}}\right)\right].$
(110)

Quadratic order
Now we will move to find the second order solutions. These will be determined
by solving equations (26)-(27) where the source term is composed from the
first order solutions. For the solutions of second order, we could not find a
closed form expression. However, since we need only the near boundary
expansion, by using the strategy mentioned in the main text, we can obtain the
near boundary expansions. We will elucidate this below.

#### Order $\omega^{2}$

At this order, the source term is

$J_{E,O}^{(\omega^{2})}(r)=i\left(3r\Psi_{E,O}^{(\omega)}+2r^{2}\frac{d\Psi_{E,O}^{(\omega)}}{dr}\right).$
(111)

After some manipulations, this can be written as:

$J_{E,O}^{(\omega^{2})}(r)=2bC_{E,O}\left(br\right)^{1/2}\frac{d}{d(br)}\left\\{\left(br\right)^{1/2}F_{1}(br)\right\\}.$
(112)

Then applying the general formula (35) we can write:

$\Psi_{E,O}^{(\omega^{2})}=-2b^{3}C_{E,O}\frac{1}{(br)}\int_{br}^{\infty}d\eta\frac{1}{\eta^{3}f\left(\eta/b\right)}\int_{1}^{\eta}d\kappa\
\sqrt{\kappa}\frac{d}{d\kappa}\left\\{\sqrt{\kappa}F_{1}(\kappa)\right\\}$
(113)

Near the boundary $r\rightarrow\infty$, the function behaves as:

$\displaystyle\Psi_{E,O}^{(\omega^{2})}=\frac{b^{3}}{4}C_{E,O}\left\\{\frac{4G_{1}-1}{(br)^{3}}+\frac{2\ln(br)}{(br)^{3}}\right\\}+\frac{b^{3}}{2}C_{E,O}\frac{1}{(br)^{4}}$
$\displaystyle+\frac{(1+Q^{2})b^{3}}{360}C_{E,O}\left\\{\frac{120G_{1}-77}{(br)^{7}}+\frac{60\ln(br)}{(br)^{7}}\right\\}+\mathcal{O}\left(\frac{1}{(br)^{8}}\right)$
(114)

where $G_{1}$ is the value of the integral $\int
d\kappa\sqrt{\kappa}\frac{d}{d\kappa}\left\\{\sqrt{\kappa}F_{1}(\kappa)\right\\}$
evaluated at $\kappa=1$. This is given by

$\displaystyle G_{1}$
$\displaystyle=-\frac{1}{4Z}\coth^{-1}\left(\frac{3}{Z}\right)-\frac{1}{16Q\left(Q^{2}-2\right)}\left[\sqrt{2}\pi\left\\{-i\sqrt{\frac{1}{Z}+\frac{1}{Z^{2}}}+\sqrt{\frac{Z-1}{Z^{2}}}\right.\right.$
$\displaystyle+\sqrt{Z-1}+i\sqrt{Z+1}-Q^{2}\left(i\sqrt{\frac{1}{Z}+\frac{1}{Z^{2}}}+\sqrt{Z-1}+i\sqrt{Z+1}\right.$
$\displaystyle\left.\left.-\sqrt{\frac{Z-1}{Z^{2}}}\right)\right\\}+Q\ln
16-\left.\frac{}{}+2Q\left(Q^{2}-2\right)\left(\log\left(3-Z\right)+\log\left(Z+3\right)\right)\right].$
(115)

#### Order $k^{2}$

The source term at this order is given by:

$J_{E,O}^{(k^{2})}=\Psi_{E,O}^{(0)}\pm\frac{3i\alpha\xi^{\prime}r}{2}\Psi_{O,E}^{(k)}.$
(116)

Using different expressions, we can write the this source as:

$J_{E,O}^{(k^{2})}=C_{E,O}b\left\\{\frac{1}{br}-\frac{27\alpha^{2}Q^{2}}{4(br)^{3}}F_{2}(br)\right\\}.$
(117)

Inserting this into the general expression for the solution (35) we can write:

$\Psi_{E,O}^{(k^{2})}=-C_{E,O}b^{3}\frac{1}{br}\int_{br}^{\infty}d\eta\frac{1}{\eta^{3}f\left(\frac{\eta}{b}\right)}\int_{1}^{\eta}d\kappa\left\\{\frac{1}{\kappa}-\frac{27\alpha^{2}Q^{2}}{4\kappa^{3}}F_{2}(\kappa)\right\\}$
(118)

This can be simplified a bit further

$\Psi_{E,O}^{(k^{2})}=-C_{E,O}b^{3}\frac{1}{br}\int_{br}^{\infty}d\eta\frac{\ln\eta}{\eta^{3}f\left(\frac{\eta}{b}\right)}+\frac{27C_{E,O}b^{3}\alpha^{2}Q^{2}}{4br}\int_{br}^{\infty}d\eta\frac{1}{\eta^{3}f\left(\frac{\eta}{b}\right)}\int_{1}^{\eta}d\kappa\frac{F_{2}(\kappa)}{\kappa^{3}}.$
(119)

Using the techniques outlined in the main text we can extract the near
boundary $r\rightarrow\infty$ expansion

$\displaystyle\Psi_{E,O}^{(k^{2})}=$ $\displaystyle
C_{E,O}\frac{-27\alpha^{2}\ln
2-16\ln(br)+16G_{2}-8}{32r^{3}}+\frac{C_{E,O}}{576b^{4}r^{7}}\left\\{96G_{2}\left(Q^{2}+1\right)-16\left(Q^{2}+1\right)\right.$
$\displaystyle\left.-81\alpha^{2}\left[Q^{2}(1+\ln 4)+\ln
4\right]-96\left(Q^{2}+1\right)\ln(br)\right\\}+\mathcal{O}\left(\frac{1}{(br)^{8}}\right)$
(120)

where $G_{2}$ is the value of $\int
d\kappa\left\\{\frac{1}{\kappa}-\frac{27\alpha^{2}Q^{2}}{4\kappa^{3}}F_{2}(\kappa)\right\\}$
evaluated at $\kappa=1$. Explicitly, this is given by

$\displaystyle G_{2}$ $\displaystyle=\frac{27\alpha^{2}}{32Z}\left[2i\pi
Q^{2}+4Q^{2}\tanh^{-1}\left(\frac{3}{Z}\right)-2\coth^{-1}\left(\frac{3}{Z}\right)\right.$
$\displaystyle\left.Z\left\\{\ln\left(3-Z\right)+\ln\left(Z+3\right)\right\\}\right].$
(121)

#### Order $\omega k$

At this order the source term is

$J_{E,O}^{(\omega
k)}=3ir\Psi_{E,O}^{(k)}+2ir^{2}\frac{d\Psi_{E,O}^{(k)}}{dr}\pm\frac{3i\alpha\xi^{\prime}r}{2}\Psi_{O,E}^{(\omega)}.$
(122)

Using different expressions above and after some manipulations we can write:

$J_{E,O}^{(\omega k)}=\mp 2\left(\frac{3}{2}\right)^{3/2}C_{O,E}\alpha
Qb\left[(br)^{\frac{1}{2}}\frac{d}{d(br)}\left\\{(br)^{\frac{1}{2}}F_{2}(br)\right\\}+\frac{F_{1}(br)}{(br)^{3}}\right].$
(123)

Then we can write from the general solution (35)

$\Psi_{E,O}^{(\omega k)}=\pm
2\left(\frac{3}{2}\right)^{\frac{3}{2}}C_{O,E}\alpha
Qb^{3}\frac{1}{br}\int_{br}^{\infty}d\eta\frac{1}{\eta^{3}f\left(\frac{\eta}{b}\right)}\int_{1}^{\eta}d\kappa\left[\sqrt{\kappa}\frac{d}{d\kappa}\left\\{\sqrt{\kappa}F_{2}(\kappa)\right\\}+\frac{F_{1}(\kappa)}{\kappa^{3}}\right].$
(124)

Again, using the method outlined before we can extract the near boundary
asymptotics:

$\displaystyle\Psi_{E,O}^{(\omega k)}=\pm
b^{3}\left(\frac{3}{2}\right)^{\frac{1}{2}}\alpha
C_{O,E}\left\\{\frac{f_{3}(Q)}{(br)^{3}}+\frac{3Q}{4(br)^{4}}-\frac{3Q}{8(br)^{6}}\right\\}+\mathcal{O}\left(\frac{1}{r^{7}}\right)$
(125)

where:

$\displaystyle
f_{3}(Q)=-\frac{3}{32\left(Q^{2}-2\right)Z^{2}}\left[16G_{3}Q\left(Q^{2}-2\right)Z^{2}+\sqrt{2}\pi\left\\{-\sqrt{-1+Z}\right.\right.$
$\displaystyle-\sqrt{Z^{2}\left(-1+Z\right)}+Q\left(i\sqrt{-1+Z}-i\sqrt{Z^{2}(-1+Z)}-\sqrt{1+Z}\right.$
$\displaystyle+\sqrt{Z^{2}(1+Z)}+2Q\left(-2\sqrt{-1+Z}+Q\left(2i\sqrt{-1+Z}+i\sqrt{Z^{2}(-1+Z)}\right.\right.$
$\displaystyle\left.\left.\left.\left.\left.-2\sqrt{1+Z}+\sqrt{Z^{2}(1+Z)}\right)\right)\right)\right\\}-4QZ^{2}\ln
2\right]$ (126)

and $G_{3}$ is the value of the integral $\int
d\kappa\left[3\sqrt{\kappa}\frac{d}{d\kappa}\left\\{\sqrt{\kappa}F_{2}(\kappa)\right\\}+9\frac{F_{1}(\kappa)}{\kappa^{3}}\right]$
evaluated at $\kappa=1$. The explicit expression of $G_{3}$ can be found by
straight-forward calculation. But this is long, and not important for the
result. So we refrain to record this here.

##  Appendix C Scalar sector

In this appendix we will study the scalar sector. First we write the
perturbation as:

$\displaystyle
a_{r}=\int\frac{d\omega}{2\pi}\frac{d^{3}k}{(2\pi)^{3}}\Phi_{r}(r,\omega,k)\
S\left(\vec{k},\vec{x}\right),$ (127) $\displaystyle
a_{v}=\int\frac{d\omega}{2\pi}\frac{d^{3}k}{(2\pi)^{3}}\Phi_{t}(r,\omega,k)\
S\left(\vec{k},\vec{x}\right),$ (128) $\displaystyle
a_{i}=r\int\frac{d\omega}{2\pi}\frac{d^{3}k}{(2\pi)^{3}}\Phi_{S}(r,\omega,k)\
\partial_{i}S\left(\vec{k},\vec{x}\right).$ (129)

Here $S(\vec{k},\vec{x})$ is a scalar harmonic with the following property:

$\left(\nabla^{2}+k^{2}\right)S=0.$ (130)

The field-strength tensor is given by:

$\displaystyle F$
$\displaystyle=2\left\\{-\xi^{\prime}+\left(i\omega\Phi_{r}+\Phi_{t}^{\prime}\right)e^{-i\omega
v}S\right\\}dr\wedge
dv+2\left(-\Phi_{r}+\Phi_{S}+r\Phi_{S}^{\prime}\right)e^{-i\omega
v}\partial_{i}S\ dr\wedge dx^{i}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad-2\left(i\omega
r\Phi_{s}+\Phi_{t}\right)e^{-i\omega v}\partial_{i}S\ dv\wedge dx^{i}.$ (131)

From this expression this is apparent that

$F\wedge F=0,$ (132)

and consequently the Chern-Simons term does not contribute at all in the
scalar sector. The equations for the fluctuation is therefore:

$\partial_{a}\left(\sqrt{-g}F^{ab}\right)=0$ (133)

where we take the linear order part. This has been analyzed in [14].

## References

  * [1] J. M. Maldacena, _The Large N limit of superconformal field theories and supergravity_ , _Adv. Theor. Math. Phys._ 2 (1998) 231 [hep-th/9711200].
  * [2] O. Aharony, S. S. Gubser, J. M. Maldacena, H. Ooguri and Y. Oz, _Large N field theories, string theory and gravity_ , _Phys. Rept._ 323 (2000) 183 [hep-th/9905111].
  * [3] J. Casalderrey-Solana, H. Liu, D. Mateos, K. Rajagopal and U. A. Wiedemann, _Gauge/String Duality, Hot QCD and Heavy Ion Collisions_. Cambridge University Press, 2014, 10.1017/CBO9781139136747, [1101.0618].
  * [4] S. S. Gubser, I. R. Klebanov and A. M. Polyakov, _Gauge theory correlators from noncritical string theory_ , _Phys. Lett. B_ 428 (1998) 105 [hep-th/9802109].
  * [5] E. Witten, _Anti-de Sitter space and holography_ , _Adv. Theor. Math. Phys._ 2 (1998) 253 [hep-th/9802150].
  * [6] E. A. Calzetta and B.-L. B. Hu, _Nonequilibrium Quantum Field Theory_ , Cambridge Monographs on Mathematical Physics. Cambridge University Press, 2008, 10.1017/CBO9780511535123.
  * [7] A. Kamenev, _Field Theory of Non-Equilibrium Systems_. Cambridge University Press, 2011, 10.1017/CBO9781139003667.
  * [8] C. P. Herzog and D. T. Son, _Schwinger-Keldysh propagators from AdS/CFT correspondence_ , _JHEP_ 03 (2003) 046 [hep-th/0212072].
  * [9] P. Glorioso, M. Crossley and H. Liu, _A prescription for holographic Schwinger-Keldysh contour in non-equilibrium systems_ , 1812.08785.
  * [10] B. Chakrabarty, J. Chakravarty, S. Chaudhuri, C. Jana, R. Loganayagam and A. Sivakumar, _Nonlinear Langevin dynamics via holography_ , _JHEP_ 01 (2020) 165 [1906.07762].
  * [11] R. Loganayagam, K. Ray and A. Sivakumar, _Fermionic Open EFT from Holography_ , 2011.07039.
  * [12] C. Jana, R. Loganayagam and M. Rangamani, _Open quantum systems and Schwinger-Keldysh holograms_ , _JHEP_ 07 (2020) 242 [2004.02888].
  * [13] R. Loganayagam, K. Ray, S. K. Sharma and A. Sivakumar, _Holographic KMS relations at finite density_ , _JHEP_ 03 (2021) 233 [2011.08173].
  * [14] 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 [2012.03999].
  * [15] T. He, R. Loganayagam, M. Rangamani and J. Virrueta, _An effective description of momentum diffusion in a charged plasma from holography_ , _JHEP_ 01 (2022) 145 [2108.03244].
  * [16] T. He, R. Loganayagam, M. Rangamani, A. Sivakumar and J. Virrueta, _The timbre of Hawking gravitons: an effective description of energy transport from holography_ , 2202.04079.
  * [17] T. He, R. Loganayagam, M. Rangamani and J. Virrueta, _An effective description of charge diffusion and energy transport in a charged plasma from holography_ , 2205.03415.
  * [18] Y. Bu, T. Demircik and M. Lublinsky, _All order effective action for charge diffusion from Schwinger-Keldysh holography_ , _JHEP_ 05 (2021) 187 [2012.08362].
  * [19] Y. Bu and B. Zhang, _Schwinger-Keldysh effective action for a relativistic Brownian particle in the AdS/CFT correspondence_ , _Phys. Rev. D_ 104 (2021) 086002 [2108.10060].
  * [20] Y. Bu, X. Sun and B. Zhang, _Holographic Schwinger-Keldysh field theory of SU(2) diffusion_ , 2205.00195.
  * [21] A. Banerjee, T. Mitra and A. Mukhopadhyay, _Correlation functions of the Bjorken flow in the holographic Schwinger-Keldysh approach_ , 2207.00013.
  * [22] R. A. Bertlmann, _Anomalies in quantum field theory_. 1996.
  * [23] K. Fujikawa and H. Suzuki, _Path integrals and quantum anomalies_. 2004, 10.1093/acprof:oso/9780198529132.001.0001.
  * [24] K. Fukushima, D. E. Kharzeev and H. J. Warringa, _The Chiral Magnetic Effect_ , _Phys. Rev. D_ 78 (2008) 074033 [0808.3382].
  * [25] D. E. Kharzeev, _The Chiral Magnetic Effect and Anomaly-Induced Transport_ , _Prog. Part. Nucl. Phys._ 75 (2014) 133 [1312.3348].
  * [26] K. Landsteiner, _Notes on Anomaly Induced Transport_ , _Acta Phys. Polon. B_ 47 (2016) 2617 [1610.04413].
  * [27] D. E. Kharzeev and J. Liao, _Chiral magnetic effect reveals the topology of gauge fields in heavy-ion collisions_ , _Nature Rev. Phys._ 3 (2021) 55 [2102.06623].
  * [28] J. K. Ghosh, S. Grieninger, K. Landsteiner and S. Morales-Tejera, _Is the chiral magnetic effect fast enough?_ , _Phys. Rev. D_ 104 (2021) 046009 [2105.05855].
  * [29] G. Policastro, D. T. Son and A. O. Starinets, _The Shear viscosity of strongly coupled N=4 supersymmetric Yang-Mills plasma_ , _Phys. Rev. Lett._ 87 (2001) 081601 [hep-th/0104066].
  * [30] G. Policastro, D. T. Son and A. O. Starinets, _From AdS / CFT correspondence to hydrodynamics_ , _JHEP_ 09 (2002) 043 [hep-th/0205052].
  * [31] P. Kovtun, D. T. Son and A. O. Starinets, _Holography and hydrodynamics: Diffusion on stretched horizons_ , _JHEP_ 10 (2003) 064 [hep-th/0309213].
  * [32] N. Banerjee, J. Bhattacharya, S. Bhattacharyya, S. Dutta, R. Loganayagam and P. Surowka, _Hydrodynamics from charged black branes_ , _JHEP_ 01 (2011) 094 [0809.2596].
  * [33] S. S. Gubser and S. S. Pufu, _Master field treatment of metric perturbations sourced by the trailing string_ , _Nucl. Phys. B_ 790 (2008) 42 [hep-th/0703090].
  * [34] H. Kodama and A. Ishibashi, _Master equations for perturbations of generalized static black holes with charge in higher dimensions_ , _Prog. Theor. Phys._ 111 (2004) 29 [hep-th/0308128].
  * [35] K. Landsteiner, E. Megias, L. Melgar and F. Pena-Benitez, _Holographic Gravitational Anomaly and Chiral Vortical Effect_ , _JHEP_ 09 (2011) 121 [1107.0368].
  * [36] J. Zaanen, Y.-W. Sun, Y. Liu and K. Schalm, _Holographic Duality in Condensed Matter Physics_. Cambridge Univ. Press, 2015.
  * [37] M. Ammon and J. Erdmenger, _Gauge/gravity duality: Foundations and applications_. Cambridge University Press, Cambridge, 4, 2015.
  * [38] S. L. Adler, _Axial vector vertex in spinor electrodynamics_ , _Phys. Rev._ 177 (1969) 2426.
  * [39] J. S. Bell and R. Jackiw, _A PCAC puzzle: $\pi^{0}\to\gamma\gamma$ in the $\sigma$ model_, _Nuovo Cim. A_ 60 (1969) 47.
  * [40] M. E. Peskin and D. V. Schroeder, _An Introduction to quantum field theory_. Addison-Wesley, Reading, USA, 1995.
  * [41] J. K. Ghosh, E. Kiritsis, F. Nitti and L. T. Witkowski, _Revisiting Coleman-de Luccia transitions in the AdS regime using holography_ , _JHEP_ 09 (2021) 065 [2102.11881].