# Conductivity and Shear Viscosity of $arcsin$-Yang-Mills AdS Black Brane

Mehdi Sadeghi and S. M. Moosavi Khansari
Department of Physics, School of Sciences,
Ayatollah Boroujerdi University, Boroujerd, Iran Corresponding author:
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this paper, a non-abelian $arcsin$-Yang-Mills AdS black brane solution is
introduced. Then, the color non-abelian direct current (DC) conductivity and
shear viscosity to entropy density ratio of this model is calculated using
fluid-gravity duality. Our results show that the Kovtun, Son and Starinets
(KSS) bound is saturated and is exactly equal to $\frac{1}{4\pi}$ but the
color conductivity bound is violated for this model. Also, our outcomes
recover the Yang-Mills AdS black brane when the coupling of Yang-Mills and
gravity fields approaches zero.

PACS numbers: 11.10.Jj, 11.10.Wx, 11.15.Pg, 11.25.Tq

Keywords: Black brane, AdS/CFT duality, the ratio of Shear viscosity to
entropy density, DC Conductivity

##  1 Introduction

Models of non-linear electrodynamics (NED) have been introduced to address
physical problems in various areas of physics. The initial Big Bang
singularity and early-time inflation in cosmology, the infinite
electromagnetic energy in Maxwell’s electrodynamics, and an effective
description of the properties of certain condensed matter materials and
optical media in solid state are examples of these problems. Some examples of
these non-linear electrodynamics were introduced in [1], [2], [3], [4], [5].
In the early stages of the Universe’s creation, electromagnetic fields and
gravitational fields were very strong, and non-linear effects played a very
important role. Therefore, studying non-linear theories is worthwhile. In this
paper, we consider the $arcsin$-Yang-Mills as a non-linear model of quantum
chromodynamics (QCD) in AdS space-time.
The AdS/CFT duality [6]-[7] states that the weakly coupled gravity theory in
an asymptotically anti-de-Sitter space-time is dual to a strongly coupled
field theory living on the AdS boundary. The physics of gravitational bulk can
be translated into the boundary field theory through the dictionary of this
duality.
The fluid-gravity duality [8, 9, 10, 11] is resulted from the AdS/CFT duality
in the low energy limit. Hydrodynamics is a good description of the theory
only in the hydrodynamic limit or low energy limit. Therefore, the theory on
the boundary in the AdS/CFT duality is effectively described by hydrodynamics
in this limit. The equations of motion up to the first order of derivative
expansion for a relativistic conformal charge fluid are stated by conservation
laws as follows.

$\displaystyle\nabla_{\mu}J^{\mu}=0\,\,\,\,\,,\,\,\nabla_{\mu}T^{\mu\nu}=0,$
(1) $\displaystyle J^{\mu}=n\,u^{\mu}-\sigma
TP^{\mu\nu}\partial_{\nu}(\frac{\mu}{T})$ $\displaystyle
T^{\mu\nu}=(\rho+p)u^{\mu}u^{\nu}+pg^{\mu\nu}-\sigma^{\mu\nu},$
$\displaystyle\sigma^{\mu\nu}={P^{\mu\alpha}P^{\nu\beta}}[\eta(\nabla_{\alpha}u_{\beta}+\nabla_{\beta}u_{\alpha})+(\zeta-\frac{2}{3}\eta)g_{\alpha\beta}\nabla.u]$
$\displaystyle P^{\mu\nu}=g^{\mu\nu}+u^{\mu}u^{\nu},$

where $n$, $\sigma$, $\eta$, $\zeta$, $\sigma^{\mu\nu}$ and $P^{\mu\nu}$ are
charge density, charge conductivity, shear viscosity, bulk viscosity, shear
tensor and projection operator, respectively [8, 9, 10, 11].
Transport coefficients provide us with additional information about the theory
on the boundary. In this paper, we calculate the color non-abelian DC
conductivity and the ratio of shear viscosity to entropy density by Kubo
formula [12]-[13].
There are universal bounds on the shear viscosity to entropy density ratio and
conductivity. KSS bound [14] ,$\frac{\eta}{s}\geq\frac{1}{4\pi}$, is saturated
for Einstein-Hilbert gravity but it is violated for higher derivative gravity
[15],[16, 17] and massive gravity [18],[19],[20], Horndeski theories[21] and
deformed AdS $5$–Schwarzschild black brane [22]. The conductivity bound,
$\sigma\geq 1$, is violated for breaking of translation symmetry model [23], a
specific type of non-linear electrodynamics [24] and non-abelian Einstein-
Born-Infeld AdS black brane[25]. We want to investigate whether these bounds
are preserved in our model or not.
By calculating $\frac{\eta}{s}$ and the color non-abelian direct current
conductivity in this paper, we aim to study the dual field theory of the non-
linear $arcsin$-Yang-Mills AdS black brane.

##  2  Non-linear $arcsin$-Yang-Mills AdS black brane

The action of Einstein non-linear $arcsin$-Yang-Mills theory is given by
($8\pi G=1$)[26],

$\displaystyle S=\int
d^{4}x\sqrt{-g}\bigg{[}R-2\Lambda-\frac{q_{1}}{q_{2}}arcsin\big{(}q_{2}\mathcal{F}\big{)}\bigg{]},$
(2)

where $R$ is the Ricci scalar, $\Lambda=-\frac{3}{L^{2}}$ the cosmological
constant, $L$ the AdS radius,
$\mathcal{F}=F^{(a)}_{\mu\alpha}F^{(a)\mu\alpha}$ is Yang-Mills invariant.
$F^{(a)}_{\mu\nu}$ is the field strength of the Yang-Mills potential
$A^{(a)}_{\nu}$,

$\displaystyle
F^{(a)}_{\mu\nu}=\partial_{\mu}A^{(a)}_{\nu}-\partial_{\nu}A^{(a)}_{\mu}+f^{(a)}_{(b)(c)}[A^{(b)}_{\mu},A^{(c)}_{\nu}],$
(3)

in which the structure constants $f^{(a)}_{(b)(c)}$ are equal to 1,
$A^{(a)}_{\nu}$’s are the Yang-Mills potentials where $q_{1}$ and $q_{2}$ are
generally arbitrary. $q_{1}$ is dimensional parameter with the dimension of
(length)4.
For finding the black brane solution of our model, we consider the following
ansatz:

$ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+\frac{r^{2}}{L^{2}}(dx^{2}+dy^{2}).$
(4)

The ansatz of gauge potential for our model is cosidered as,

${\bf{A}}=\frac{1}{\sqrt{2}}h(r)dt\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}.$
(5)

$\sigma_{3}=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}$ is the gauge group of the
gauge field which is written in terms of the diagonal generator of the
$SL(2,C)$.
Einstein’s equation is obtained by varying the action (2) with respect to the
metric $g_{\mu\nu}$ as,

$R_{\mu\nu}-\tfrac{1}{2}g_{\mu\nu}R+\Lambda g_{\mu\nu}=\kappa
q_{1}T_{\mu\nu},$ (6)

where,

$T_{\mu\nu}=\tfrac{1}{2q_{2}}arcsin\big{(}\frac{q_{2}}{4}\mathcal{F}\big{)}g_{\mu\nu}-\tfrac{F_{\mu}^{(a)\alpha}F_{\nu\alpha}{}^{(a)}}{2\sqrt{1-\frac{1}{16}q_{2}^{2}\mathcal{F}^{2}}}.$
(7)

The non-linear $arcsin$-Yang-Mills’ equation is is obtained by varying the
action (2) with respect to the gauge field $A^{(a)}_{\mu}$,

$\displaystyle\nabla_{\mu}\bigg{(}\frac{q_{1}F^{(a)\mu\nu}}{2\sqrt{1-\frac{1}{4}q_{2}^{2}\mathcal{F}^{2}}}\bigg{)}=0.$
(8)

The $(t,t)$ component of Einstein’s field equations is determined by utilizing
Eq.(6) as follows,

$\displaystyle f+\Lambda
r^{2}+rf^{\prime}+\frac{q_{1}r^{2}}{2q_{2}}arcsin(\frac{1}{2}q_{2}h^{\prime
2})-\frac{q_{1}h^{\prime 2}}{\sqrt{4+q_{2}^{2}h^{\prime 4}}}=0,$ (9)

for $q_{1}=0$ we have,

$f+\Lambda r^{2}+rf^{\prime}=0.$ (10)

The other component of Einstein’s equations is the same as Eq.(9) because
there is an unknown function ,$f(r)$, in the ansatz Eq.(4).
$h(r)$ is found by using Eq.(8) and Eq.(4) as follows,

$\displaystyle h(r)$
$\displaystyle=\frac{\sqrt{2}}{Qq_{2}}\int^{r}du\sqrt{\sqrt{u^{8}+Q^{4}q_{2}^{2}}-u^{4}}$
(11)
$\displaystyle=\frac{\sqrt{2}}{3Qq_{2}}\bigg{[}-2^{\frac{7}{4}}Q\sqrt{q_{2}}\bigg{(}\big{(}r^{8}+Q^{4}q_{2}^{2}\big{)}^{\frac{1}{2}}-r^{4}\bigg{)}^{\frac{1}{4}}$
$\displaystyle
F\bigg{(}\frac{1}{8},\frac{3}{4};\frac{9}{8};\frac{(r^{4}-\sqrt{r^{8}+Q^{4}q_{2}^{2}})^{2}}{Q^{4}q_{2}^{2}}\bigg{)}+r\sqrt{\frac{r^{8}+Q^{4}q_{2}^{2}}{\sqrt{r^{8}+Q^{4}q_{2}^{2}}-r^{4}}}$
$\displaystyle-\frac{r^{5}}{\sqrt{r^{8}+Q^{4}q_{2}^{2}}-r^{4}}\bigg{]}+C,$

and $F(a,b;c;z)$ is the hyper-geometric function.
$f(r)$ is found by substituting Eq.(11) in Eq.(9) and solving Eq.(9) as below,

$f(r)=\frac{2m}{r}+\frac{r^{2}}{L^{2}}+\frac{q_{1}}{2r}\int^{r}u^{2}\bigg{(}\frac{2h^{\prime}(u)^{2}}{\sqrt{4+q_{2}^{2}h^{\prime}(u)^{4}}}-\frac{1}{q_{2}}arcsin\big{(}\frac{1}{2}q_{2}h^{\prime}(u)^{2}\big{)}\bigg{)}du,$
(12)

where,

$h^{\prime}(u)=\frac{\sqrt{2}}{Qq_{2}}\sqrt{\sqrt{u^{8}+Q^{4}q_{2}^{2}}-u^{4}}.$
(13)

##  3 Conductivity and Shear Viscosity

The AdS/CFT duality can be used to describe strongly coupled non-perturbative
quantum systems in terms of weakly coupled gravity quantities. For example, a
field $\phi$ in gravity is dual to an operator $\mathcal{O}_{\phi}$ on the CFT
side. The conductivity is related to the retarded correlation function of
currents at zero momentum through the Green-Kubo formula [12] as follows,

$\sigma^{ij}_{ab}(k_{\mu})=\frac{1}{\omega}\int dt\theta(t)e^{i\omega
t}<[J^{i}_{a}(t,\mathbb{K}),J^{j}_{b}(t,\mathbb{K})]>=-\mathop{\lim}\limits_{\omega\to
0}\frac{1}{\omega}\Im G^{ij}(k_{\mu}).$ (14)

So the retarded two-point current correlator of CFTs at finite temperature is
required for calculating conductivity.
The gauge potential, $A^{(a)}_{\mu}$, on the bulk theory is dual to the
current, $J^{i}_{\mu}$, on the boundary theory. Therefore, the gauge potential
must be perturbed as $\tilde{A}_{x}=\tilde{A}_{x}(r)e^{-i\omega t}$ on bulk of
AdS. Where $a,b$ indices refer to $SL(2,C)$ group symmetry.
We insert the gauge potential perturbation into the action Eq.(2) and expand
the resulting action up to the second order of the perturbed part. We have,

$\displaystyle S^{(2)}=-\int
d^{4}x\frac{4q_{1}}{f\sqrt{1-q_{2}^{2}h^{\prime}(r)^{4}}}\Bigg{[}-f^{2}\left((\partial_{r}\tilde{A}_{x}^{(1)})^{2}+(\partial_{r}\tilde{A}_{x}^{(2)})^{2}+(\partial_{r}\tilde{A}_{x}^{(3)})^{2}\right)$
$\displaystyle+\big{(}\omega^{2}+h^{2}\big{)}\bigg{(}(\tilde{A}_{x}^{(1)})^{2}+(\tilde{A}_{x}^{(2)})^{2}\bigg{)}+\omega^{2}(\tilde{A}_{x}^{(3)})^{2}\Bigg{]}.$
(15)

The equations of motion for perturbed gauge fields are obtained by varying the
action $S^{(2)}$ (3) with respect to $\tilde{A}_{x}^{(1)}$,
$\tilde{A}_{x}^{(2)}$, $\tilde{A}_{x}^{(3)}$, respectively, as the following,

$\displaystyle
f\left(f\tilde{A}_{x}^{(1)^{\prime}}\right)^{\prime}+\tilde{A}_{x}^{(1)}\left(h^{2}+\omega^{2}\right)+\frac{2q_{2}^{2}f(r)^{2}h^{\prime}(r)^{3}h^{\prime\prime}(r)\tilde{A}_{x}^{(1)^{\prime}}}{1-q_{2}^{2}h^{\prime}(r)^{4}}=0.$
(16) $\displaystyle
f\left(f\tilde{A}_{x}^{(2)^{\prime}}\right)^{\prime}+\tilde{A}_{x}^{(2)}\left(h^{2}+\omega^{2}\right)+\frac{2q_{2}^{2}f(r)^{2}h^{\prime}(r)^{3}h^{\prime\prime}(r)\tilde{A}_{x}^{(2)^{\prime}}}{1-q_{2}^{2}h^{\prime}(r)^{4}}=0.$
(17) $\displaystyle
f\left(f\tilde{A}_{x}^{(3)^{\prime}}\right)^{\prime}+\omega^{2}\
\tilde{A}_{x}^{(3)}-\frac{2q_{2}^{2}f(r)^{2}h^{\prime}(r)^{3}h^{\prime\prime}(r)\tilde{A}_{x}^{(3)^{\prime}}}{1-q_{2}^{2}h^{\prime}(r)^{4}}=0.$
(18)

Eq.(16) is the same as Eq.(17).
Now we need to find the solutions of Eq.(16) and Eq.(18). First, we solve them
near the event horizon.
Since the condition $\tilde{A}_{x}^{(a)}(r_{h})=0$ holds, and given the
relation $f\sim f^{\prime}(r_{h})(r-r_{h})$, we consider the following ansatz:

$\displaystyle\tilde{A}_{x}^{(a)}\sim(r-r_{h})^{z_{a}}\,,\qquad a=1,2,3$ (19)

where,

$\displaystyle z_{1}=z_{2}=\pm i\frac{\sqrt{\omega^{2}+h(r_{h})^{2}}}{4\pi
T},$ (20) $\displaystyle z_{3}=\pm i\frac{\omega}{4\pi T}.$ (21)

The Hawking temperature of the black brane $T$ is given as follows,

$T=\frac{1}{2\pi}\Big{[}\frac{1}{\sqrt{g_{rr}}}\frac{d}{dr}\sqrt{-g_{tt}}\Big{]}\Bigg{|}_{r=r_{h}}=\frac{f^{\prime}(r_{h})}{4\pi}.$
(22)

For general solution of $\tilde{A}_{x}^{(a)}$, we consider the following
ansatz,

$\displaystyle\tilde{A}_{x}^{(1)}=\tilde{A}^{(1)}_{\infty}\Big{(}\frac{-3f}{\Lambda
r^{2}}\Big{)}^{z_{1}}\Big{(}1+i\omega b_{1}(r)+\cdots\Big{)},$ (23)
$\displaystyle\tilde{A}_{x}^{(2)}=\tilde{A}^{(2)}_{\infty}\Big{(}\frac{-3f}{\Lambda
r^{2}}\Big{)}^{z_{2}}\Big{(}1+i\omega b_{2}(r)+\cdots\Big{)},$ (24)
$\displaystyle\tilde{A}_{x}^{(3)}=\tilde{A}^{(3)}_{\infty}\Big{(}\frac{-3f}{\Lambda
r^{2}}\Big{)}^{z_{3}}\Big{(}1+i\omega b_{3}(r)+\cdots\Big{)},$ (25)

where $\tilde{A}^{(a)}_{\infty}$ represents the value of fields at the
boundary, $z_{i}$’s correspond to the negative signs in Eq.(20) and Eq.(21)
and $r$ ranges from the event horizon to the boundary.
$b_{3}(r)$ in Eq.(25) is as follows,

$b_{3}(r)=C_{3}+\int^{r}\frac{\left(C_{4}+\int^{u_{2}}Y(u_{1})du_{1}\right)\sqrt{1-q_{2}^{2}h^{\prime}(u_{2})^{4}}}{f(u_{2})}du_{2},$
(26)

where $C_{3}$ and $C_{4}$ are integration constants and $Y(u_{1})$ is as
follows,

$Y(u_{1})=-\frac{u_{1}^{2}f^{\prime\prime}(u_{1})-2u_{1}f^{\prime}(u_{1})+2f(u_{1})}{u_{1}^{2}\left(1-q_{2}^{2}h^{\prime}(u_{1})^{4}\right)^{1/2}}+\frac{2q_{2}^{2}u_{1}\left(u_{1}f^{\prime}(u_{1})-2f(u_{1})\right)h^{\prime}(u_{1})^{3}h^{\prime\prime}(u_{1})}{u_{1}^{2}\left(1-q_{2}^{2}h^{\prime}(u_{1})^{4}\right)^{3/2}}.$
(27)

The solution should be regular on the event horizon. Therefore, $C_{4}$ is
determined by demanding the application of regularity to $h_{3}(r)$ at the
horizon as,

$C_{4}=-\int^{r_{h}}Y(u_{1})du_{1}.$ (28)

Considering the solution of $\tilde{A}_{x}^{(3)}$ in Eq.(3) and the variation
with respect to $\tilde{A}^{(3)}_{\infty}$, Green’s function [12] can be
obtained using Witten’s prescription [27] as follows,

$\displaystyle
G_{xx}^{(33)}(\omega,\vec{0})=-i\omega\frac{rf^{\prime}(r)+f(r)\left(rb_{3}^{\prime}(r)-2\right)}{4\pi
rT}\bigg{|}_{r\to\infty}.$ (29)

The conductivity is as following,

$\displaystyle\sigma_{xx}^{(33)}=\frac{rf^{\prime}(r)+f(r)\left(rb_{3}^{\prime}(r)-2\right)}{4\pi
rT},$ (30)

by substituting $b_{3}(r)$ in above equation and using Eq.(14) we have,

$\displaystyle\sigma_{xx}^{(33)}=\sqrt{1-q_{2}^{2}h^{\prime}(r_{h})^{4}}.$
(31)

It shows that the conductivity bound [28] is violated for the non-linear
$arcsin$-Yang-Mills AdS black brane theory. In the limit of $q_{2}\to 0$, we
have,

$\displaystyle\sigma_{xx}^{(33)}=1.$ (32)

It can be easily shown that $\sigma_{xx}^{(11)}=\sigma_{xx}^{(22)}=0$.
The ratio of shear viscosity to entropy density is another important quantity
in fluid-gravity duality. This ratio is proportional to the inverse squared
coupling of the theory on the boundary. We want to calculate this ratio using
the formula introduced in the [15].

$\frac{\eta}{s}=\frac{1}{4\pi}\phi(r_{h})^{2},$ (33)

where $\phi$ represents the perturbed part of the metric.
Therefore, we perturb the metric as $g_{\mu\nu}\to
g_{\mu\nu}+\frac{r^{2}}{L^{2}}\phi(r)$ 111The components of $t$ and $z$ in
$\phi(t,r,z)$ are disappeared by Fourier transforms. and put it into action,
expanding the action up to the second order of $\phi$. Finally, the equation
of motion of $\phi(r)$ is obtained by varying the resulting action with
respect to $\phi(r)$ as,

$\displaystyle\phi^{\prime}f^{\prime}+f\phi^{\prime\prime}=0.$ (34)

The solution of $\phi(r)$ is as follows,

$\displaystyle\phi(r)=C_{5}+C_{6}\int^{r}\frac{1}{f(u)}du.$ (35)

We consider solution of $\phi(r)$ near the event horizon as the following,

$\displaystyle\phi(r)=C_{5}+\frac{C_{6}}{4\pi T}\log(r-r_{h}).$ (36)

By demanding the regularity of $\phi(r)$ on the event horizon. We have,

$\displaystyle C_{6}=0.$ (37)

By applying normalization on solution Eq.(35) so $C_{5}=1$, we have,

$\displaystyle\phi(r)=1.$ (38)

Finally, the shear viscosity to entropy density ratio is given by,

$\frac{\eta}{s}=\frac{1}{4\pi}\phi(r_{h})^{2}=\frac{1}{4\pi}.$ (39)

The $\frac{\eta}{s}$ is satisfied for dual to strongly correlated systems in
this universal relation as $\frac{\eta}{s}\geq\frac{1}{4\pi}$.
Our results show that the KSS bound is saturated for the non-linear
$arcsin$-Yang-Mills AdS black brane.

##  4 Results and discussion

We introduced a non-abelian $arcsin$-Yang-Mills black brane in AdS spacetime.
Conductivity and $\frac{\eta}{s}$ are two important quantities that are
calculated via AdS/CFT duality. Since the shear viscosity to entropy density
ratio is proportional to the inverse square of the coupling. We conclude that
the coupling of this duality in our model is the same as the field theory dual
of Einstein-Hilbert gravity, and the conductivity bound, $\sigma\geq 1$ 222 We
consider $\frac{1}{e^{2}}=1$, is violated in our model. It indicates the
existence of non-linear effects on this bound. Regarding conductivity, it is
important to emphasize that the probe limit is assumed. Otherwise, in this
model, the DC conductivity would be infinite, and no bounds would be violated
at all. We mention the bounds in real fluids [29] as an application example to
convince ourselves that this is not a holographic game. Also, these quantities
can be confirmed as experimental data for string theory.

##  5 Conclusions

In this paper, the $\frac{\eta}{s}$ and $\sigma$ of the non-linear
$arcsin$-Yang-Mills AdS black brane are calculated using fluid-gravity
duality. We conclude some information about our field theory dual of this
solution. The shear viscosity to entropy density ratio tells us that the
coupling of the field theory side is the same as the dual of Einstien-Hilbert
gravity on AdS spacetime. Also, the conductivity bound is violated for current
model. Our results show that the non-abelian Ohm’s law is diagonal and the
conductivity is scalar from spatial and gauge group indices.In the following,
we mention some future works on this model.
This model can be studied in different theories of gravity such as Horendeski,
massive gravity, scalar-tensor gravity, bigravity and higher derivative
gravities. Hydrodynamic aspects and stability of these solutions can be
discussed.
The solution of this model in higher dimensions is an interesting subject and
the shadow of this solution can be investigated.
Considering the bounds on thermal conductivity [30] and thermal diffusivity
[31],[32] for this model are another interesting subjects, too.

Acknowledgment Authors would like to thank Reza Saadati, Komeil Babaei,
Mojtaba Shahbazi, Farmarz Rahamni and the referee of INJP for useful comments
and suggestions.

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