\section{Introduction}
\label{sec:intro}

The aim of this paper is to analyze the properties of 2-point correlation functions
of finite-temperature strongly coupled (deconfined) gauge field theories defined on spacetimes with general spatial topology,
and admit a dual black hole description via the AdS/CFT correspondence.
Of specific interest is the dependence of these
properties on the topology of the space on which the gauge theory is defined, and its temperature.
We will see that the topology is indeed encoded
in the properties of the correlation function and its poles, corresponding to quasinormal modes in the
dual bulk spacetime: the sound of topology. In
particular, the asymptotic ``slope'' of the poles encodes information about the spatial
topology and the temperature, but is ``universal'' otherwise, i.e. it doesn't depend on the type of operators
considered.

One motivation for this work is the fact that important aspects of the strong coupling dynamics of
gauge field theories can depend crucially on the topology of space-time.
For instance, the confinement-deconfinement phase transition is a property of strongly coupled thermal conformal gauge theories defined on a space-time with
spherical spatial topology. The theory is always in the deconfined phase in the planar and hyperbolic cases. 
Thus, one may hope that insights 
to the  confinement-deconfinement phase transition mechanism can be gained by analyzing
the strong coupling dynamics on different  spatial topologies. This can be done 
by using the gravitational dual description.
A second motivation is the recently studied correspondence between gravitational dynamics and strongly coupled condensed matter systems. One can naturally envision in and out of equilibrium condensed matter systems realized on different spatial
topologies, where our study can be of much relevance.
A third motivation, perhaps a more remote one, is the relevance of the interplay between topology and field theory dynamics to
astrophysical/cosmological setups.

We distinguish three classes of spatial topology \footnote{In the spherical case, a Hawking-Page phase transition
exists, which is interpreted in terms of the AdS/CFT correspondence as a transition between a higher
temperature deconfined phase and a lower temperature confined phase (see~\cite{Witten:1998qj},
~\cite{Witten:1998zw}). Phase transitions might exist also for the flat and hyperbolic cases, e.g. the
transition between the black hole solution at a higher temperature and the Horowitz-Myers ``AdS
soliton'' at a lower temperature (see \cite{Surya:2001vj}). }: The spherical,
the flat and
the hyperbolic (The relevant thermodynamic quantities and their dependence on the topology are given in
Appendix~\ref{app:thermodynamicquant}).
Consider a gauge theory
defined on a $(d-1)$-dimensional spacetime with the topology of
$ \mathbb{R}\times\Omega_{d-2}^{FT} $,
where $ \Omega_{d-2}^{FT} $  is a constant curvature $ (d-2) $-dimensional Riemannian manifold of
arbitrary topology\footnote{We assume that $ d\ge 4 $.}.
The two basic scales that characterize the theory are the temperature $ T $ and the scalar curvature
of the $ \Omega_{d-2}^{FT} $ manifold $ R_\Omega^{FT} $.
In the analysis we will consider two different regimes:
\begin{enumerate}
\item The regime of $ \frac{\sqrt{|R_\Omega^{FT}|}}{T} \to 0 $: In this case the curvature of the
$ \Omega_{d-2}^{FT} $ manifold may essentially be neglected, so we expect all physical
quantities to behave as they do in the case of a flat ($ R_\Omega^{FT} = 0 $) space. Therefore,
in this limit one does not distinguish between different topologies (aside from the known changes
of the Laplace operator spectra between different topologies). This limit includes
the hydrodynamic limit\footnote{The hydrodynamic condition $ \frac{L_s^{FT}}{T} \to 0$
requires that $ \frac{\sqrt{|R_\Omega^{FT}|}}{T} \to 0 $ since the minimal eigenvalue
$ L_{s,min}^{FT} $ is of the same order as $ \sqrt{|R_\Omega^{FT}|} $.}:
$ \frac{\omega}{T},\frac{L_s^{FT}}{T} \to 0 $, where $ \omega $ denotes the frequency and  $ L_s^{FT} $ is the spatial Laplace
operator eigenvalue.
\item The regime of $ \frac{\sqrt{|R_\Omega^{FT}|}}{T} = \text{fixed} $: In this case it will be
possible to distinguish between the different topologies. In particular we will study the
limit of $ \frac{L_s^{FT}}{T} = \text{fixed} $ and $ \frac{\omega}{T} \gg 1 $, where analytical
asymptotic expressions will be derived for the quasinormal modes and the 2-point correlation functions.

\end{enumerate}
Note that the $ \frac{\sqrt{|R_\Omega^{FT}|}}{T} \gg 1 $ regime does not exist in the black hole 
description in the case of spherical topology (since the temperature has a finite minimal value) - it
corresponds to a different (confining) phase on the field theory side. Nor does it exist in the case of 
flat topology (for which $ R_\Omega^{FT} = 0 $).
While this regime does exist for the hyperbolic case, there is nothing separating it qualitatively 
from the $ \frac{\sqrt{|R_\Omega^{FT}|}}{T} = \text{fixed} $ regime in the context of this paper.
For these reasons, we won't consider this regime separately here.

The main new results of this work are (see Appendix~\ref{app:notations} for a list of notations
and definitions):
\begin{itemize}
\item In the asymptotic limit, where $ \frac{L_s^{FT}}{T} \sim \text{fixed} $ and
$ \frac{\omega}{T} \gg 1 $, the following holds:
\begin{itemize}
\item For all types of topologies and all the temperatures , the quasinormal
frequencies (and therefore the poles of the field theory correlators) are of the form
\begin{equation}
\lambda_n = \lambda_0 + n\Delta\lambda \ ,
\end{equation}
where $ \Delta\lambda $ depends on the topology and temperature.
\item In this limit we derive asymptotic expressions for the field theory
2-point retarded correlation functions of the operators dual to the different perturbation
types. These are given
in Equations~\ref{eq:asymptcftcorrelatorscalarevend} and~\ref{eq:asymptcftcorrelatorscalaroddd} for the
scalar perturbation modes,
in Equations~\ref{eq:asymptcftcorrelatorlongvectorevend} and~\ref{eq:asymptcftcorrelatorlongvectoroddd}
for the longitudinal vector perturbation modes,
and in Equations~\ref{eq:asymptcftcorrelatortransvectorevend}
and~\ref{eq:asymptcftcorrelatortransvectoroddd} for the transverse vector perturbation modes.

\end{itemize}

\item In the hyperbolic case there is
a specific temperature $ T_c $, where the bulk ``black hole'' solution has no singularity, and is isometric to
AdS. In this special case the following holds:
\begin{itemize}
\item At $ T=T_c $ we derive exact expressions for the field theory 2-point retarded correlation
functions of the operators dual to the different perturbation types. These are given
in Equations~\ref{eq:scalarexactcorrelatorintceven} and~\ref{eq:scalarexactcorrelatorintcodd}
for the scalar perturbation modes,
in Equations~\ref{eq:longvectorexactcorrelatorintceven} and~\ref{eq:longvectorexactcorrelatorintcodd}
for the longitudinal vector perturbation modes
and in Equations~\ref{eq:transvectorexactcorrelatorintceven}
and~\ref{eq:transvectorexactcorrelatorintcodd}
for the transverse vector perturbation modes.

\item For $ T>T_c $, the asymptotic $ \Delta\lambda $ has a real component $ \re(\Delta\lambda)>0 $.
As $ T\to T_c $, $ \Delta\lambda $ becomes imaginary according to
Equation~\ref{eq:asymptparamsaroundtcde4b} for $ d=4 $ (as previously calculated in \cite{Koutsoumbas:2006xj}),
and according to Equation~\ref{eq:asymptparamsaroundtcdg4b} for $ d>4 $.
\end{itemize}

\end{itemize}

The paper is organized as follows.
In Section~\ref{sec:quasinormalmodes}, we will review some known results
regarding quasinormal modes (formulated in our notation).
We will calculate the QNM equations for massless scalar and
vector perturbations, the exact solutions for the ``special'' case mentioned above (the hyperbolic case
with the specific temperature $ T_c $), the approximate quasinormal frequencies in the asymptotic limit
($ \frac{L_s^{FT}}{T} = \text{fixed} $ and $ \frac{\omega}{T} \gg 1 $) and the diffusion mode in the
hydrodynamic limit. We will discuss the dependence of the asymptotic quasinormal modes on the topology
and the temperature, in particular in the hyperbolic case around the temperature $ T_c $, and demonstrate
it using some numerical results.
In Section~\ref{sec:cftcorrelators}, we will use AdS/CFT dictionary to calculate the 2-point correlation
functions of the dual field theory. We will obtain exact expressions for the hyperbolic case with
$ T=T_c $ and approximate expressions in the asymptotic limit for general topology and temperature.
We will compare the asymptotic expressions with exact numerical solutions.
Section~\ref{sec:discussion} is devoted to a discussion. Details of calculations are outlined in the appendices.

\section{Quasi-Normal Modes}
\label{sec:quasinormalmodes}

\subsection{QNM Definition}
\label{subsec:qnmdefinition}

Quasinormal modes of black holes or black branes are defined as the late-time oscillation modes of the
black hole metric and the fields coupled to it, satisfying certain boundary conditions. Put
differently, these are the eigenmodes of the linearized equations of motion over the black hole
background. The boundary conditions are specified at the black hole horizon and at spatial infinity. The boundary condition
at the horizon is chosen so that the solution corresponds to  a wave ingoing into the horizon, while
the boundary condition choice at spatial infinity depends on the asymptotic nature of the spacetime.

Choosing the incoming wave solution gives the boundary condition:
\begin{equation}
\left.\psi\right|_{z=0} \sim z^{-\frac{i\omega}{C}} \ ,
\end{equation}
where $ z=0 $ corresponds to the BH horizon and $ z=1 $ corresponds to spatial infinity.
As for the boundary condition at spatial infinity, different boundary conditions have been
investigated for different asymptotic spacetime geometries. In the context of AdS/CFT, the relevant
boundary condition is the one that corresponds to the poles of the retarded correlator of the
CFT operators dual to the investigated field. For the fields discussed here, this condition
amounts to the Dirichlet boundary condition (see~\cite{Son:2002sd},~\cite{Berti:2009kk}
and~\cite{Nunez:2003eq}):
\begin{equation}
\left.\psi\right|_{z=1}=0 \ .
\end{equation}

\subsection{QNM Equations}
\label{subsec:qnmequations}

We consider the d-dimensional black hole metric given by a metric of the form:
\begin{equation}
\label{eq:rmetric}
\,\mathrm{d} s_{bulk}^2 = -f(r)\,\mathrm{d} t^2 + \frac{1}{f(r)} \,\mathrm{d} r^2 + r^2 \,\mathrm{d}\Omega_{d-2}^2 \ ,
\end{equation}
where
$
\,\mathrm{d}\Omega_{d-2}^2 = \sum_{i,j} (g_\Omega)_{ij} \,\mathrm{d} x^i \,\mathrm{d} x^j
$
is the inner metric of a constant curvature, (d-2)-dimensions Riemannian manifold with of same
topology as the $ \Omega_{d-2}^{FT} $ manifold.

From the vacuum Einstein equation with a cosmological constant
one can deduce the form of $ f(r) $ to be
\begin{equation}
f(r) = k + \frac{r^2}{R^2} - \frac{r_0^{d-3}}{r^{d-3}} \ ,
\end{equation}
where $ R $ is related to the cosmological constant by
$
R^2 = -\frac{(d-2)(d-1)}{2\Lambda}
$
and $ k $ is related to the scalar curvature $ R_\Omega $ of the manifold $ \Omega_{d-2} $
by
(see, for example, \cite{Birmingham:1998nr})
$
k = \frac{R_\Omega}{(d-2)(d-3)}
$.
In terms of these coordinates, the horizon of the black hole is located at
$ r=r_+ $ where $ f(r_+)=0 $, while the boundary is located at $ r\to\infty $.

After making the transformation to the coordinate $ z = 1-\frac{r_+}{r} $,
the metric takes the form:
\begin{equation}
\label{eq:zmetric}
\,\mathrm{d} s^2 = -\frac{\rho^2 \tilde{g}(z)}{(1-z)^2} \,\mathrm{d} t^2 + \frac{r_+^2}{\rho^2\tilde{g}(z)(1-z)^2}\,\mathrm{d} z^2 + \frac{r_+^2}{(1-z)^2}\,\mathrm{d} \Omega_{d-2}^2 \ ,
\end{equation}
where $ \rho \equiv \frac{r_+}{R} $, $ K \equiv \frac{k}{\rho^2} $ and
\begin{equation}
\label{eq:gzexp}
\tilde{g}(z) \equiv 1 + K(1-z)^2 - (1+K)(1-z)^{d-1} \ .
\end{equation}
In terms of the new coordinate $ z $, the horizon is located at $ z=0 $ while the boundary is located at $ z=1 $.

Let us assume that the spectrum of the scalar and vector Laplace operator $ \Delta_\Omega $\footnote{In
the context of this work we shall define the Laplace operator as: $ \Delta \equiv \,\mathrm{\delta}\,\mathrm{d} $ where
$ \,\mathrm{d} $ and $ \,\mathrm{\delta} $ are the exterior derivative and codifferential operators respectively.}
defined on the manifold $ \Omega_{d-2} $ is given by
\begin{align}
\Delta_\Omega H_{L^2}(x) & = L_s^2 H_{L^2}(x) \\
\Delta_\Omega \boldsymbol{A}_{L^2}(x) & = L_v^2 \boldsymbol{A}_{L^2}(x)
\end{align}
respectively.
In order for a QNM  corresponding to the frequency $ \omega $ and the above Laplace operator eigenvalue to exist, the corresponding boundary conditions problem should have a non-trivial solution. Introducing the dimensionless parameters
$
\lambda \equiv \frac{\omega r_+}{\rho^2}
$, $
q_s \equiv \frac{L_s}{\rho}
$, $
q_v \equiv \frac{L_v}{\rho}
$ and $
C \equiv \tilde{g}'(0)
$,
we find the following equations (see Appendix~\ref{app:qnmequationsderivation} for a detailed
derivation):
\begin{itemize}
\item For a (massless, minimally coupled) scalar QNM, the equation is
\begin{equation}
\label{eq:scalareq}
(1-z)^{d-2}\partial_z\left[ \frac{\tilde{g}(z)}{(1-z)^{d-2}} \partial_z \psi \right] +
\left[ \frac{\lambda^2}{\tilde{g}(z)} - q_s^2 \right] \psi = 0 \ ,
\end{equation}
along with the boundary conditions:
\begin{equation}
\label{eq:scalarbc}
\left.\psi\right|_{z=0} \sim z^{-\frac{i\lambda}{C}}
\qquad
\left.\psi\right|_{z=1} = 0 \ .
\end{equation}
\item For a ``longitudinal'' vector QNM, the equation is
\begin{equation}
\label{eq:longvectoreq}
\partial_z \left[ \tilde{g}(z)(1-z)^{d-4} \partial_z \left( \frac{1}{(1-z)^{d-4}} \psi \right) \right] + \left[ \frac{\lambda^2}{\tilde{g}(z)} - q_s^2 \right] \psi = 0 \ ,
\end{equation}
along with the boundary conditions:
\begin{multline}
\label{eq:longvectorbc}
\left.\psi\right|_{z=0} \sim z^{-\frac{i\lambda}{C}}\\
\left. (1-z)^{d-4}\partial_z\left[ \frac{1}{(1-z)^{d-4}} \psi \right] \right|_{z=1} = \left.\left( \partial_z \psi + \frac{d-4}{1-z}\psi \right)\right|_{z=1} = 0 \ ,
\end{multline}
or \emph{equivalently} the equation
\begin{multline}
\label{eq:longvectoreq2}
(1-z)^{d-4}\partial_z\left[ \frac{\tilde{g}(z)}{(1-z)^{d-4}} \partial_z \psi \right]\\
+ \frac{q_s^2}{\lambda^2-q_s^2 \tilde{g}(z)} \tilde{g}(z)\partial_z\tilde{g}(z) \partial_z\psi + \left[ \frac{\lambda^2}{\tilde{g}(z)} - q_s^2 \right] \psi =0 \ ,
\end{multline}
along with the boundary conditions:
\begin{equation}
\label{eq:longvectorbc2}
\left.\psi\right|_{z=0} \sim z^{-\frac{i\lambda}{C}}
\qquad
\left.\psi\right|_{z=1} = 0 \ .
\end{equation}
\item For a ``transverse'' vector QNM, the equation is
\begin{equation}
\label{eq:transvectoreq}
(1-z)^{d-4}\partial_z\left[ \frac{\tilde{g}(z)}{(1-z)^{d-4}} \partial_z \psi \right] +
\left[ \frac{\lambda^2}{\tilde{g}(z)} - q_v^2 \right] \psi = 0 \ ,
\end{equation}
along with the boundary conditions:
\begin{equation}
\label{eq:transvectorbc}
\left.\psi\right|_{z=0} \sim z^{-\frac{i\lambda}{C}}
\qquad
\left.\psi\right|_{z=1} = 0 \ .
\end{equation}
\end{itemize}

It can be easily seen from the above equations that if $ \psi(z) $ is as a solution for
any of the equations (satisfying the corresponding boundary conditions) with frequency
$ \lambda $, then $ \psi^*(z) $  is a solution for the same equation (with the same
boundary conditions) with frequency $ -\lambda^* $. This means that the QNM frequencies
are always symmetric with respect to the imaginary axis.

A detailed derivation of the QNM Equations can be found in Appendix~\ref{app:qnmequationsderivation}.

\subsection{Exact solutions for the \texorpdfstring{$K=-1$}{K=-1} case}
\label{subsec:qnmexactsolutionsfortc}

In the case where the $ \Omega_{d-2} $ manifold is hyperbolic, we consider the special state
corresponding to the temperature:
\begin{equation}
T_C \equiv \frac{\sqrt{|k|}}{2\pi R} = \frac{1}{2\pi}\sqrt{\frac{|R_\Omega^{FT}|}{(d-2)(d-3)}} \ ,
\end{equation}
for which $ K=-1 $, and the horizon radius is given by $ \rho_C = \sqrt{|k|} $.
In this case, the QNM equations
(Equations~\ref{eq:scalareq}, \ref{eq:longvectoreq} and~\ref{eq:transvectoreq}) can be solved
analytically. This has been done in~\cite{Birmingham:2006zx},~\cite{Aros:2002te}
and~\cite{LopezOrtega:2007vu}. Here we summarize these solutions in our notation.

Making the transformation:
\begin{align}
w &\equiv\tilde{g}(z) = 1-(1-z)^2 \\
\psi &\equiv w^\gamma (1-w)^\delta \phi
\end{align}
(where $ \gamma $ and $ \delta $ are defined in Appendix~\ref{app:exactsolutionsfortcderivation}
for each perturbation type), one obtains the hypergeometric equation. The solution to the
equation satisfying the incoming-wave boundary condition at the horizon is
$\sb{2}F_1\left(a,b;c;w\right)$, where $ a $, $ b $ and $ c $ are given in
Appendix~\ref{app:exactsolutionsfortcderivation} for each case.

Applying the QNM boundary condition at the AdS boundary we get for the scalar and transverse vector
cases:
\begin{equation}
\sb{2}F_1\left(a,b;c;1\right)=\frac{\Gamma(c)\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)}=0 \ ,
\end{equation}
and for the longitudinal vector case (and using known formula fot the hypergeometric
function, taking into account $ c-a-b = \frac{d-5}{2} $) we get:
\begin{multline}
\left.(1-w)^{\frac{d-3}{2}}\partial_w\left[\frac{1}{(1-w)^\frac{d-5}
{2}}\sb{2}F_1(a,b;c;w)\right]\right|_{w=1} \\
= \frac{(c-a)(c-b)}{c}\sb{2}F_1(a,b;c+1;1) \\
= \frac{(c-a)(c-b)}{c} \frac{\Gamma(c+1)\Gamma(c-a-b+1)}{\Gamma(c-a+1)\Gamma(c-b+1)}
= 0  \ .
\end{multline}
The solution for which is given by $c-a=-n$ or $c-b=-n$ where $n\in\mathbb{Z}$ and $n\geq0$,
so that the quasi-normal frequencies are:
\begin{itemize}
\item For the massless scalar case:
\begin{equation}
\lambda_n = -\frac{i}{2}(d+1) \pm \sqrt{q^2-\frac{1}{4}(d-3)^2} -2ni \ .
\end{equation}
\item For the longitudinal vector case:
\begin{equation}
\lambda_n=-\frac{i}{2}(d-3)\pm\sqrt{q^2-\frac{1}{4}(d-3)^2}-2ni \ .
\end{equation}
\item For the transverse vector case:
\begin{equation}
\lambda_n = -\frac{i}{2}(d-1) \pm \sqrt{q^2-\frac{1}{4}(d-5)^2} -2ni \ .
\end{equation}
\end{itemize}

\subsection{QNM Asymptotics (for \texorpdfstring{$ K>-1 $}{K>-1})}
\label{subsec:qnmasymoptotics}

For the case $ K>-1 $, one may find an approximate analytical expression for the $n$-th QNM frequency
for $ n\to\infty $ - that is, in the asymptotic limit of $ \frac{L_s^{FT}}{T} = \text{fixed} $ and
$ \frac{\omega}{T} \gg 1 $. This has been done in~\cite{Natario:2004jd} using the ``monodromy'' method.
Here we summarize this calculation using our notations.
The general stages of the calculation are as follows:
\begin{enumerate}

\item We bring each of the QNM equations to a Schr\"odinger form, with some effective potentials,
using appropriate coordinate and function transformations.
\item We find the poles of these effective potentials on the complex plane, and write approximate forms
for the equations in the vicinity of the poles.
\item We solve the approximate equations around each pole using Bessel functions.
\item Using the asymptotic limit of $ |\lambda| \gg 1 $, we replace the Bessel functions with their
asymptotic forms.
\item We match the solutions on each region of the complex plane, and apply the appropriate boundary
conditions, to get equations for values of $ \lambda $ that allow for non-trivial solutions.
\item We solve the equations analytically and obtain an asymptotic expression of the form:
\begin{equation}
\lambda_n = \lambda_0 + n\Delta\lambda \ .
\end{equation}

\end{enumerate}

\subsubsection{Calculation of Asymptotic QNMs}

Starting from the QNM Equations, we perform the following transformation:
\begin{equation}
\psi(z) = (1-z)^{\frac{\alpha}{2}} \phi(z)
\end{equation}
\begin{equation}
\tilde{z} \equiv \int \frac{1}{\tilde{g}(z)} \,\mathrm{d} z = \sum_{k=1}^{d-1} \gamma_k \ln(z-z_k) + \tilde{z}_0 \ ,
\end{equation}
where:
\begin{equation}
\alpha=
\begin{cases}
d-2 & \text{for scalar},\\
d-4 & \text{for vector}
\end{cases} \ ,
\end{equation}
$ z_k $ ($k=1,\ldots, d-1 $) are the zeros of the polynomial $ \tilde{g}(z) $ and\footnote{The branch cuts are chosen here so that $ 0\leq\arg(z)<2\pi $.}
\begin{equation}
\gamma_k = \lim_{z\to z_k} \frac{z-z_k}{\tilde{g}(z)} = \frac{1}{\tilde{g}'(z_k)}
\end{equation}
\begin{equation}
\tilde{z}_0 = - \sum_{k=1}^{d-1} \gamma_k \ln(1-z_k)
\end{equation}
(so that $ \tilde{z}(z=1) = 0 $).
After this transformation the equations take the Schr\"odinger form:
\begin{equation}
\partial_{\tilde{z}}^2\phi + \left[ \lambda^2-V(\tilde{z}) \right]\phi = 0 \ ,
\end{equation}
where the effective potential $ V(z) $ is given for each case in Appendix~\ref{app:effectivepotentials}.

Next we turn to the calculation of the asymptotic QNMs using the monodromy method. We first note that
the asymptotic QNM frequencies satisfy (see~\cite{Cardoso:2004up} and~\cite{Natario:2004jd})
\begin{equation}
\im(\lambda\tilde{z}_0)=0 \ .
\end{equation}

Looking at the anti-Stokes lines, defined by
\begin{equation}
\im(\lambda\tilde{z}) = 0 \ ,
\end{equation}
we choose two of them to be the contour for the calculation\footnote{The choice of branch cuts so
that $ 0\leq\arg(z)<2\pi $ makes sure that the contours don't pass through a branch cut. Choosing
$ 0<\arg(z)\leq 2\pi $ instead would give the complex conjugates of these contours.}.
We then proceed by developing the equation, and finding its solutions, around points along the
chosen contour (the boundary and $ z\to\infty $ ) and ``matching'' the solutions. The chosen
contour is shown in Figure~\ref{fig:contour}\footnote{The assumed shape of the anti-Stokes lines
is only true in the case of $ K>-1 $.}.

\begin{figure}[htbp]
\begin{center}
\subfloat[$z$ plane contour]{%
\label{fig:zcontour}\includegraphics[width=8cm]{zcontour.pdf}}
\hspace{1cm}
\subfloat[$\tilde{z}$ plane contour]{%
\label{fig:ztcontour}\includegraphics[width=8cm]{ztcontour.pdf}}
\end{center}
\caption{Chosen contour for the monodromy calculation: \protect\subref{fig:zcontour} In the $z$ plane
and \protect\subref{fig:ztcontour} In the $ \tilde{z} $ plane.}
\label{fig:contour}
\end{figure}

We first look at the equation at $ z\to\infty $ or $ \tilde{z}\to\tilde{z}_0 $. As shown in
Appendix~\ref{app:effectivepotentials}, the equation at $ \tilde{z}\to\tilde{z}_0 $ is approximately
\begin{equation}
\partial_{\tilde{z}}^2\phi + \left[\lambda^2-\frac{j_\infty^2-1}{4(\tilde{z}-\tilde{z}_0)^2}\right] = 0 \ .
\end{equation}
The solution to this equation can be written as:
\begin{equation}
\phi \approx A_+ P_+(\tilde{z}-\tilde{z}_0) + A_- P_-(\tilde{z}-\tilde{z}_0) \ .
\end{equation}
In the non-scalar case where $ j_\infty \neq 0 $ (and more generally $ j_\infty $ isn't an even integer),
\begin{equation}
P_\pm(\tilde{z}-\tilde{z}_0) \equiv \sqrt{2\pi\lambda(\tilde{z}-\tilde{z}_0)}J_{\pm\frac{j_\infty}{2}}\left(\lambda(\tilde{z}-\tilde{z}_0)\right) \ ,
\end{equation}
where $ J_{\pm\frac{j_\infty}{2}} $ are the Bessel functions.
The fact that asymptotically\footnote{Another assumption is that
$ |\lambda| \gg |q|$, so that the effect of the effective potential can be neglected in the regions
between the the ``special'' points ($ z\rightarrow0 $, $ z\rightarrow1 $, $ z\rightarrow\infty $)}
$ |\lambda| \gg 1 $ allows us to replace the Bessel functions with their corresponding asymptotic
forms.
For direction (1) (as appears in Figure~\ref{fig:contour}), we have $ \lambda(\tilde{z}-\tilde{z}_0) > 0 $, so
that
\begin{multline}
\label{eq:monoasympinftyplus}
\phi \approx
\left[A_+ \mathrm{e}^{i(-\lambda\tilde{z}_0-\alpha_+)} + A_- \mathrm{e}^{i(-\lambda\tilde{z}_0-\alpha_-)}\right] \mathrm{e}^{i\lambda\tilde{z}}\\
+ \left[A_+ \mathrm{e}^{i(\lambda\tilde{z}_0+\alpha_+)} + A_- \mathrm{e}^{i(\lambda\tilde{z}_0+\alpha_-)} \right] \mathrm{e}^{-i\lambda\tilde{z}}
\ ,
\end{multline}
where
\begin{equation}
\alpha_\pm = \frac{\pi}{4}(1 \pm j_\infty) \ .
\end{equation}
For direction (2) we have $ \lambda(\tilde{z}-\tilde{z}_0) < 0 $, so that
\begin{multline}
\label{eq:monoasympinftymin}
\phi \approx
\left[A_+\mathrm{e}^{i(-\lambda\tilde{z}_0+3\alpha_+)} + A_-\mathrm{e}^{i(-\lambda\tilde{z}_0+3\alpha_-)}\right]\mathrm{e}^{i\lambda\tilde{z}}\\
+\left[A_+\mathrm{e}^{i(\lambda\tilde{z}_0+\alpha_+)} + A_-\mathrm{e}^{i(\lambda\tilde{z}_0+\alpha_-)}\right]\mathrm{e}^{-i\lambda\tilde{z}} \ .
\end{multline}

Next we look at the equation at the boundary. As shown in Appendix~\ref{app:effectivepotentials}, the
equation at $ \tilde{z}\to 0 $ is approximately
\begin{equation}
\partial_{\tilde{z}}^2\phi + \left[\lambda^2-\frac{j_1^2-1}{4(\tilde{z}-\tilde{z}_0)^2}\right] = 0 \ .
\end{equation}
The solution is therefore:
\begin{equation}
\phi \approx B_+ P_+(\tilde{z}) + B_- P_-(\tilde{z})\\
= B_+\sqrt{2\pi\lambda\tilde{z}} J_{\frac{j_1}{2}}(\lambda\tilde{z})
+ B_-\sqrt{2\pi\lambda\tilde{z}} J_{-\frac{j_1}{2}}(\lambda\tilde{z}) \ .
\end{equation}
We now apply the boundary conditions at the boundary ($ z=1 $, $ \tilde{z}=0 $) to this solution. In all cases (the scalar, transverse vector and longitudinal vector) we end up with the condition
\begin{equation}
B_- = 0 \ .
\end{equation}
Next we again assume $ |\lambda| \gg 1 $ so that the Bessel function can be replaced with its asymptotic form. For direction (2), we have $ \lambda\tilde{z} > 0 $, so that
\begin{equation}
\label{eq:monoasympboundary}
\phi \approx 2B_+\cos(\lambda\tilde{z}-\beta_+)
= \left[B_+ \mathrm{e}^{-i\beta_+}\right]\mathrm{e}^{i\lambda\tilde{z}} + \left[B_+ \mathrm{e}^{i\beta_+}\right]\mathrm{e}^{-i\lambda\tilde{z}} \ ,
\end{equation}
where
\begin{equation}
\beta_+ = \frac{\pi}{4}(1+j_1) \ .
\end{equation}

Finally from the boundary conditions at the horizon ($ z=0 $, $ \tilde{z}\to -\infty $) we have (for line (1) as it appears in Figure~\ref{fig:contour})
\begin{equation}
\label{eq:monoasymphorizon}
\phi \sim \mathrm{e}^{-i\lambda\tilde{z}} \ .
\end{equation}

Now we ``match'' the solutions on lines (1) and (2).
In the non-scalar case, in which $ j_\infty \ne 0 $ so that $ \alpha_+ \ne \alpha_- $.
In this case we have for line (1) (from Equations~\ref{eq:monoasympinftyplus} and~\ref{eq:monoasymphorizon})
\begin{equation}
\label{eq:monoeq1}
A_+\mathrm{e}^{i(-\lambda\tilde{z}_0-\alpha_+)} + A_-\mathrm{e}^{i(-\lambda\tilde{z}_0-\alpha_-)} = 0 \ .
\end{equation}
For line (2) we have (from Equations~\ref{eq:monoasympinftymin} and~\ref{eq:monoasympboundary})
\begin{align}
A_+\mathrm{e}^{i(-\lambda\tilde{z}_0+3\alpha_+)} + A_-\mathrm{e}^{i(-\lambda\tilde{z}_0+3\alpha_-)} &= B_+\mathrm{e}^{-i\beta_+}\\
A_+\mathrm{e}^{i(\lambda\tilde{z}_0+\alpha_+)} + A_-\mathrm{e}^{i(\lambda\tilde{z}_0+\alpha_-)} &= B_+\mathrm{e}^{i\beta_+} \ .
\end{align}

The condition for having a non-trivial solution for this system of equations is:
\begin{equation}
\tan(\lambda\tilde{z}_0-\beta_+)
= \frac{\mathrm{e}^{i3\alpha_+}\sin(\alpha_+) - \mathrm{e}^{i3\alpha_-}\sin(\alpha_-)}{\mathrm{e}^{i3\alpha_+}\cos(\alpha_+) - \mathrm{e}^{i3\alpha_-}\cos(\alpha_-)} \ .
\end{equation}
Finally, we obtain the result\footnote{Obviously, $ -\lambda_n^* $ would also be a solution.
Choosing the branch cuts so that $ 0<\arg(z)\leq 2\pi $ would give this solution instead.}:
\begin{equation}
\label{eq:asymptoticlambda}
\lambda_n = \lambda_0 + n\Delta\lambda \ ,
\end{equation}
where
\begin{equation}
\lambda_0 = \frac{\beta_+ + \arctan\left[\frac{\mathrm{e}^{i3\alpha_+}\sin(\alpha_+) - \mathrm{e}^{i3\alpha_-}\sin(\alpha_-)}{\mathrm{e}^{i3\alpha_+}\cos(\alpha_+) - \mathrm{e}^{i3\alpha_-}\cos(\alpha_-)}\right]}{\tilde{z}_0}
\qquad
\Delta\lambda = \frac{\pi}{\tilde{z}_0} \ .
\end{equation}

In the scalar case, in which $ j_\infty = 0 $, we have:
\begin{align}
P_+(\tilde{z}-\tilde{z}_0) &\equiv \sqrt{2\pi\lambda(\tilde{z}-\tilde{z}_0)}J_0\left(\lambda(\tilde{z}-\tilde{z}_0)\right)\\
P_-(\tilde{z}-\tilde{z}_0) &\equiv \sqrt{2\pi\lambda(\tilde{z}-\tilde{z}_0)}Y_0\left(\lambda(\tilde{z}-\tilde{z}_0)\right) \ .
\end{align}
In a calculation similar to the non-scalar case, we again obtain the asymptotic form
in Equation~\ref{eq:asymptoticlambda},
where
\begin{equation}
\lambda_0 = \frac{\beta_+ + \frac{\pi}{4}-\frac{i}{2}\ln(2)}{\tilde{z}_0}
\qquad
\Delta\lambda = \frac{\pi}{\tilde{z}_0} \ .
\end{equation}

\subsubsection{Asymptotic Dependence on Topology and Temperature}

From the results of the calculation in this section we see that the asymptotic parameters of the
QNM frequencies (namely $ \lambda_0 $ and $ \Delta\lambda $) of each perturbation type depend on
the $ \Omega_{d-2} $ manifold topology and the temperature through the parameter $ \tilde{z}_0 $,
which depends only on $ d $ and $ K $ (where $ K $ contains the dependency on both the topology and
the temperature). In fact, the asymptotic slope of the frequencies ($ \Delta\lambda $) depends only
on the spatial topology and the temperature - it is the same for all types of perturbations
(even for types that aren't discussed here, such as tensorial perturbations):
\begin{equation}
\Delta\lambda = \frac{\pi}{\tilde{z}_0} \ .
\end{equation}
The dependence of $ \tilde{z}_0 $ on $ K $ is plotted in Figure~\ref{fig:zt0valuesd4} for $ d=4 $ and in
Figure~\ref{fig:zt0valuesd5} for $ d=5 $.
In the case of spherical ($ K>0 $) and flat ($ K=0 $) topologies, the parameter $ \tilde{z}_0 $, and therefore
the slope $ \Delta\lambda $, always has an imaginary part and a real part (for all temperature values).
However in the hyperbolic case, at the temperature $ T_c $ (as defined in
Section~\ref{sec:intro}) where $ K=-1 $, the slope becomes completely
imaginary\footnote{In dimensions $ d=4 $ and $ d=5 $, $ \re\tilde{z}_0 \ge 0 $ below the temperature $ T_c $,
so that the asymptotic slope of the QNM frequencies remains imaginary below this temperature as well.}
(as seen in the exact results given in Subsection~\ref{subsec:qnmexactsolutionsfortc} for this case).
Moreover, it can be clearly seen in the Figures~\ref{fig:zt0valuesd4} and~\ref{fig:zt0valuesd5} that
$ \tilde{z}_0 (K) $ (and therefore $ \Delta\lambda(T) $) is not smooth at this temperature.

\begin{figure}[htbp]
\centering
\includegraphics[width=10cm]{Zt0Valuesd4.pdf}
\caption{The dependence of the parameter $ \tilde{z}_0 $ on the parameter $ K $ on the $ \tilde{z}_0 $ complex
plane, in $ d=4 $ dimensions.}
\label{fig:zt0valuesd4}
\end{figure}

\begin{figure}[htbp]
\centering
\includegraphics[width=10cm]{Zt0Valuesd5.pdf}
\caption{The dependence of the parameter $ \tilde{z}_0 $ on the parameter $ K $ on the $ \tilde{z}_0 $ complex
plane, in $ d=4 $ dimensions.}
\label{fig:zt0valuesd5}
\end{figure}

In the following we study two specific cases: the case of $ K=0 $, and the case of $ K\to -1$
($ k<0 $,$ T\to T_c $).

\paragraph{The case of $ K=0 $}

Let us first review the case of $ K=0 $ (where the $ \Omega_{d-2} $ manifold is ``flat''),
described for example in \cite{Natario:2004jd} as the limiting case of large spherical black holes.
In this case, the roots of $ \tilde{g}(z) $ are analytically known and we can get an accurate expression for the
asymptotic QNM frequencies (and most importantly the gap $ \Delta\lambda $).

Assuming that $ K=0 $, we get for $ \tilde{g}(z) $:
\begin{equation}
\tilde{g}(z) = 1-(1-z)^{d-1} \ .
\end{equation}
Its roots are given by:
\begin{equation}
z_k = 1 - \mathrm{e}^{i2\pi\frac{k-1}{d-1}} \qquad k=1,\ldots,(d-1) \ .
\end{equation}

After calculating the residues and using the definition of $ \tilde{z}_0 $, we obtain the
results:
\begin{equation}
\tilde{z}_0 = -\frac{\pi}{d-1}\mathrm{e}^{-i\frac{\pi}{d-1}}\frac{1}{\sin\left(\frac{\pi}{d-1}\right)}
\end{equation}
\begin{equation}
\Delta\lambda = \frac{\pi}{\tilde{z}_0} = -(d-1)\mathrm{e}^{i\frac{\pi}{d-1}}\sin\left(\frac{\pi}{d-1}\right) \ .
\end{equation}
For example, for $ d=4 $ we get $ \Delta\lambda = -\frac{3\sqrt{3}}{4}-i\frac{9}{4} $,
and for $ d=5 $ we get $ \Delta\lambda = -2(1+i) $.

\paragraph{The case of $ K\to -1 $}

Here we consider the case of $ K\to -1 $ ($ T\to T_c $) from above (meaning $ K>-1 $). Define:
\begin{equation}
K\equiv -1+\epsilon \qquad \epsilon>0 \ ,
\end{equation}
so that
\begin{equation}
\tilde{g}(z) = 1-(1-\epsilon)(1-z)^2-\epsilon(1-z)^{d-1} \ .
\end{equation}
Notice also that from the discussion in Section~\ref{sec:intro} and Appendix~\ref{app:thermodynamicquant}
we can relate $ \epsilon $ to the temperature difference by:
\begin{equation}
\epsilon = \frac{2}{d-2} \frac{\Delta T}{T_c} \ .
\end{equation}

We first find an approximation for the roots of $ \tilde{g}(z) $. For $ K=-1 $ ($ \epsilon=0 $) we have:
\begin{equation}
\tilde{g}(z) = 1-(1-z)^2 = 0
\qquad\Rightarrow\qquad
z_1=0, z_2=2 \ .
\end{equation}
In this case there are only \emph{two} roots. When $ K>-1 $, there are $ d-1 $ roots. Two of them will
be close to the two roots of the $ K=-1 $ case, while the others go to $ \infty $ as $ K\to -1 $.

Developing the roots and residues around $ z=0 $ and $ z=2 $ to first order in $ \epsilon $,
we get:
\begin{align}
z_1 & =0 \\
\gamma_1 &\approx \frac{1}{2+(d-3)\epsilon} \approx \frac{1}{2}-\frac{d-3}{4}\epsilon \\
z_2 &\approx 2+\frac{1+(-1)^d}{2}\epsilon \\
\gamma_2 &\approx -\frac{1}{2}-\frac{(-1)^d(d-2)+1}{4}\epsilon \ .
\end{align}

Approximating the other roots we get:
\begin{align}
1-z_k &\approx \epsilon^{-\frac{1}{d-3}} \mathrm{e}^{i\left(\frac{\pi}{d-3}+\frac{2\pi}{d-3}j\right)} \\
\gamma_k &\approx -\frac{1}{d-3}\epsilon^{\frac{1}{d-3}}\mathrm{e}^{-i\left(\frac{\pi}{d-3}+\frac{2\pi}{d-3}j\right)} \ ,
\end{align}
where $ j \equiv d-k-1 = 0,\ldots,d-4 $.

We put these values into the definition for $ \tilde{z}_0 $, and after some calculations we obtain the following results:\\
For the case $ d=4 $:
\begin{empheq}[box=\fbox]{align}
\label{eq:asymptparamsaroundtcde4a}
\tilde{z}_0 &\approx \frac{i\pi}{2} + \epsilon\ln\epsilon\\
\label{eq:asymptparamsaroundtcde4b}
\Delta\lambda &= \frac{\pi}{\tilde{z}_0} \approx -2i + \frac{4}{\pi}\epsilon\ln\epsilon \ ,
\end{empheq}
And for the case $ d>4 $:
\begin{empheq}[box=\fbox]{align}
\label{eq:asymptparamsaroundtcdg4a}
\tilde{z}_0 &\approx \frac{i\pi}{2} - \frac{\pi}{d-3}\frac{1}{\sin\left(\frac{\pi}{d-3}\right)}\epsilon^{\frac{1}{d-3}}\\
\label{eq:asymptparamsaroundtcdg4b}
\Delta\lambda &= \frac{\pi}{\tilde{z}_0} \approx -2i - \frac{4}{d-3}\frac{1}{\sin\left(\frac{\pi}{d-3}\right)}\epsilon^{\frac{1}{d-3}} \ .
\end{empheq}
For example, for $ d=5 $ we get
\begin{equation}
\Delta\lambda \approx -2i-2\sqrt{\epsilon} \ .
\end{equation}

\subsection{Hydrodynamic Approximation}

Taking the hydrodynamic limit of $ \lambda,q \to 0 $ ($ \frac{\omega}{T},\frac{L_s^{FT}}{T} \to 0 $,
$ \frac{\sqrt{|R_\Omega^{FT}|}}{T} \to 0 $ is required) in the appropriate QNM equations, one may
calculate the hydrodynamic constants of the dual gauge theory (see
~\cite{Son:2007vk},~\cite{Kovtun:2005ev}).
Since the longitudinal vector mode is coupled to charge density fluctuations in the gauge theory, its
QNM spectrum in the hydrodynamic limit has to contain a mode corresponding to the charge diffusion mode
in the gauge theory. Here we review the derivation of the diffusion mode from the QNM Equation
in our notation.

Starting from Equation~\ref{eq:longvectoreq} for a longitudinal vector QNM, let us
define a new function $ \psi' = z^{-\frac{i\lambda}{C}}\psi $, and get a new ``shifted''
equation:
\begin{multline}
\label{eq:longvectoreqshifted}
\left( \partial_z-\frac{i\lambda}{Cz} \right)
\left[ \tilde{g}(z) (1-z)^{d-4} \left(\partial_z-\frac{i\lambda}{Cz}\right) \left( \frac{1}{(1-z)^{d-4}}\psi \right) \right]\\
+ \left[ \frac{\lambda^2}{\tilde{g}(z)}-q^2 \right] \psi = 0 \ ,
\end{multline}
with boundary conditions
\begin{equation}
\left. \psi \right|_{z=0} \sim \text{const.}
\qquad
\left. (1-z)^{d-4} \left(\partial_z-\frac{i\lambda}{Cz}\right) \left(\frac{1}{(1-z)^{d-4}}\psi\right) \right|_{z=1} = 0 \ .
\end{equation}

For the hydrodynamic approximation we shall assume $ \lambda \sim q^2 \ll 1 $, and develop $ \psi $ in
orders of $ \lambda $ :
\begin{equation}
\psi = \psi^{(0)} + \psi^{(1)} + \psi^{(2)} + \ldots \ .
\end{equation}
We put this into Equation~\ref{eq:longvectoreqshifted}, equate each order to 0 and solve the equation
with the corresponding boundary conditions in terms of the higher order solutions.

For the 0-order we get the general solution:
\begin{equation}
\psi^{(0)}
\sim C_0\left[ \frac{1}{C}\ln z + \text{const.} \right] + D_0 (1-z)^{d-4} \ ,
\end{equation}
since $ \left. \tilde{g}(z) \right|_{z=0} \sim Cz $.
From the boundary condition at $ z=0 $ we get
\begin{equation}
C_0 = 0 \ ,
\end{equation}
so that
\begin{equation}
\psi^{(0)} = D_0 (1-z)^{d-4} \ ,
\end{equation}
while the boundary conditions at $ z=1 $ are automatically fulfilled.

Next we turn to the $ \lambda^1 $ order, for which we get the solution:
\begin{multline}
\psi^{(1)} = \frac{i\lambda D_0}{C}(1-z)^{d-4} \ln z
-(1-z)^{d-4} \frac{q^2 D_0}{d-3} \int \frac{1-z}{\tilde{g}(z)} \,\mathrm{d} z\\
+C_1 (1-z)^{d-4} \int \frac{1}{\tilde{g}(z) (1-z)^{d-4}} \,\mathrm{d} z
+D_1 (1-z)^{d-4} \ .
\end{multline}
Applying the boundary condition at $ z=0 $ we get
\begin{equation}
\left.\psi^{(1)}\right|_{z=0} \sim \frac{i\lambda D_0}{C}\ln z
-\frac{q^2 D_0}{(d-3)C}\ln z
+\frac{C_1}{C}\ln z + D_1 \sim \text{const.} \ ,
\end{equation}
from which we get the condition
\begin{equation}
\label{eq:hydrocond1}
\left( i\lambda - \frac{q^2}{d-3} \right)D_0 + C_1 = 0 \ .
\end{equation}
From the boundary condition at $ z=1 $ we have
\begin{multline}
\left. (1-z)^{d-4}\partial_z\left( \frac{1}{(1-z)^{d-4}}\psi^{(1)} \right) \right|_{z=1}
-\frac{i\lambda}{C}\left. \psi^{(0)} \right|_{z=1} \\
= \frac{i\lambda D_0}{C}\delta_{d,4} + C_1 - \frac{i\lambda D_0}{C}\delta_{d,4}
= C_1 = 0 \ .
\end{multline}
Putting this into Equation~\ref{eq:hydrocond1} we get to the conclusion
\begin{equation}
\lambda = -\frac{i}{d-3} q^2 \ ,
\end{equation}
which is a (normalized) diffusion relation, with the diffusion constant equal to $ \frac{1}{d-3} $,
or, in terms of $ \omega $ and $ L_\mathbf{s}^{FT} $:
\begin{equation}
\omega \approx -iD \left(L_\mathbf{s}^{FT}\right)^2
\qquad
D = \frac{d-1}{d-3} \frac{1}{4\pi T} \ .
\end{equation}
This is true for any dimension $ d \ge 4 $ and any $ \Omega_{d-2} $ topology and temperature for whom
the hydrodynamic condition can be fulfilled. As expected, the leading term in this approximation
doesn't depend on the topology of $ \Omega_{d-2} $.

\subsection{Numerical Calculation of QNMs}

In the following we present some results of exact numerical calculations of the QNM spectra in several
cases in order to illustrate the dependence on topology and temperature. The spectra have been
calculated using the method outlined in Appendix~\ref{app:numericalmethods}.

Figures~\ref{fig:qnmresultsforseveralkscalard4q0K10m05}
and~\ref{fig:qnmresultsforseveralklongvectord5q141K10m05} contain the exact QNM spectra for the cases
of spherical ($ K=1 $), flat ($ K=0 $) and hyperbolic ($ K=-0.5 $) topologies, for the scalar and
longitudinal vector cases. The dependence of the asymptotic slope of the spectrum on the parameter $ K $
can be easily seen in these figures (as well as its independence of other parameters such as the spatial
mode or perturbation type). The (pure imaginary) hydrodynamic mode can also be seen in the spectra of
the longitudinal vector perturbations.

\begin{figure}[htbp]
\begin{minipage}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{QNMResultsForSeveralKScalard4q0K10m05.pdf}
\caption{Numerically calculated QNM frequencies for the case of scalar perturbations with
$ d=4 $,$ K = 1 , 0 , -0.5 $ and $ q_\mathbf{s} = 0 $.}
\label{fig:qnmresultsforseveralkscalard4q0K10m05}
\end{minipage}
\hfill
\begin{minipage}{0.49\textwidth}
\includegraphics[width=\textwidth]{QNMResultsForSeveralKLongVectord5q141K10m05.pdf}
\caption{Numerically calculated QNM frequencies for the case of longitudinal vector perturbations with
$ d=5 $,$ K = 1 , 0 , -0.5 $ and $ q_\mathbf{s} = \sqrt{2} $.}
\label{fig:qnmresultsforseveralklongvectord5q141K10m05}
\end{minipage}
\end{figure}


The numerical results for $ -1.1 < K < -0.9 $, as illustrated in
Figures~\ref{fig:qnmresultsforseveralkscalard4q1} and~\ref{fig:qnmresultsforseveralklongvectord5q1p5},
demonstrate the phenomenon discussed in Sections~\ref{subsec:qnmexactsolutionsfortc}
and~\ref{subsec:qnmasymoptotics} for an hyperbolic $ \Omega_{d-2} $ manifold: At $ T=T_c $ ($ K=-1 $) the
asymptotic QNM frequency gap ($ \Delta\lambda $) becomes imaginary, and for $ d=4,5 $ it remains
imaginary for $ T<T_c $ ($ K<-1 $).

\begin{figure}[htbp]
\begin{minipage}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{QNMResultsForSeveralKScalard4q1.pdf}
\caption{Numerically calculated QNM frequencies for the case of scalar perturbations with
$ d=4 $,$ -1.05 < K < -0.9 $ and $ q_\mathbf{s} = 1 $.}
\label{fig:qnmresultsforseveralkscalard4q1}
\end{minipage}
\hfill
\begin{minipage}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{QNMResultsForSeveralKLongVectord5q1p5.pdf}
\caption{Numerically calculated QNM frequencies for the case of longitudinal vector perturbations with
$ d=5 $,$ -1.1 < K < -0.9 $ and $ q_\mathbf{s} = 1.5 $.}
\label{fig:qnmresultsforseveralklongvectord5q1p5}
\end{minipage}
\end{figure}

\clearpage

\section{CFT Correlators}
\label{sec:cftcorrelators}

\subsection{CFT Correlators From Holographic Principle}
\label{subsec:cftcorrelatorsfromhol}

Via the AdS/CFT corresondence, a black hole (or black brane) background in
the bulk spacetime, such as the ones discussed here, corresponds to a dual CFT at finite temperature in the deconfined phase.
The correspondence  allows one to calculate the 2-point correlation function of CFT operators
by calculating the on-shell action of the bulk field dual to that operator.

The general prescription for calculating the dual CFT correlators from the AdS/CFT correspondence
is discussed
in~\cite{Maldacena:1997re},~\cite{Witten:1998qj},~\cite{Gubser:1998bc}, and~\cite{Aharony:1999ti}.
In the case of Minkowski spacetime, the calculation of the Minkowski correlators from AdS/CFT
requires some additional subtleties, as explained in~\cite{Son:2002sd}.
In particular, the retarded green function defined as:
\begin{equation}
G^R\left(t,x;t',x'\right) = -i\theta(t-t')\left\langle\left[O(t,x),O(t',x')\right]\right\rangle
\end{equation}
can be calculated by choosing the incoming-wave boundary condition at the black hole horizon for
the solution of the classical EOM in the bulk spacetime (see
Subsection~\ref{subsec:qnmdefinition}).

One consequence of the above prescription is the fact that the QNM frequency spectrum of the
bulk field (as defined in Subsection~\ref{subsec:qnmdefinition}) comprises the poles of the retarded
correlator of the dual CFT operator. Therefore all of the results of
Section~\ref{sec:quasinormalmodes} serve to teach about the poles of the retarded correlators of
the theories dual to the discussed black holes. The purpose of this section is the expansion of
those results from knowledge of the poles to expressions for the correlators themselves.


\subsection{General Formulae For Correlators}
\label{subsec:generalformulaeforcorrelators}

The background spacetime metric on which the field theory dual to the bulk black hole
is defined will be given by:
\begin{equation}
\label{eq:cftmetricnoR}
\,\mathrm{d} s_{FT}^2 = -\,\mathrm{d} t^2 + \,\mathrm{d}\Omega_{FT,d-2}^{2} \ ,
\end{equation}
where
$
\,\mathrm{d}\Omega_{FT,d-2}^2 = \sum_{i,j}(g_\Omega^{FT})_{ij}\,\mathrm{d} x_{FT}^i \,\mathrm{d} x_{FT}^j
$
is the inner metric of the $ \Omega_{d-2}^{FT} $ manifold.
This metric is conformally related
to the bulk background metric at its boundary, with the conformal factor chosen to cancel the
divergence in the bulk metric. The relation between the metrics is then given by
(see~\cite{Emparan:1999gf})
\begin{equation}
g_{\mu\nu}^{FT} = \lim_{r\to\infty}\left(\frac{R^2}{r^2}g_{\mu\nu}^{bulk}\right) \ ,
\end{equation}
or:
\begin{equation}
\label{eq:cftmetric}
\,\mathrm{d} s_{FT}^2 = -\,\mathrm{d} t^2 + R^2 \,\mathrm{d}\Omega_{d-2}^2 \ .
\end{equation}
Comparing this with Equation~(\ref{eq:cftmetricnoR}) we get the relation between the field theory
manifold and the bulk manifold $ \,\mathrm{d}\Omega_{FT,d-2}^2 = R^2\,\mathrm{d}\Omega_{d-2}^2 $, and therefore
the relation between the corresponding curvatures -
$
R_\Omega^{FT} = \frac{R_\Omega}{R^2}
$
and
$
k_{FT} \equiv \frac{R_\Omega^{FT}}{(d-2)(d-3)} = \frac{k}{R^2}
$,
and the relation between the corresponding Laplace operator eigenvalues
$
L_s^{FT} = \frac{L_s}{R}
$.

Using the AdS/CFT prescription for a massless scalar bulk field and a gauge vector bulk field, we
may find expressions for the retarded correlation functions of the dual field theory operators.
This is done by integrating by parts the expressions for the action of these fields,
using the EOMs they satisfy to get an expression that depends only on the boundary values of these
fields and taking the derivative of the expression with respect to the the boundary values.
We find the following expressions:
\begin{itemize}

\item For a massless scalar field, the correlation function is given by:
\begin{equation}
\label{eq:scalarcftcorrelator}
G^R(\omega,\mathbf{s})
= \left.-2C_s\frac{r_+^{d-1}}{R^d}\frac{1}{(1-z)^{d-2}}
\partial_z\widehat{\psi}_{\omega,\mathbf{s}}(z)\right|_{z\to 1} \ ,
\end{equation}
where $ \widehat{\psi}_{\omega,\mathbf{s}} $ satisfies the scalar QNM equation
(Equation~\ref{eq:scalareq}) along with the boundary conditions:
\begin{equation}
\left.\widehat{\psi}_{\omega,\mathbf{s}}\right|_{z=0} \sim z^{-\frac{i\lambda}{C}}
\qquad
\left.\widehat{\psi}_{\omega,\mathbf{s}}\right|_{z\to 1} = 1 \ .
\end{equation}
$ C_s $ is the appropriate normalization constant for the bulk scalar field (see Appendix C).
\item For the longitudinal component of a vector field, the correlation function is given
by:
\begin{equation}
\label{eq:longvectorcftcorrelator}
G_{tt}^R(\omega,\mathbf{s})
= \left.4C_v\frac{r_+^{d-3}}{R^{d-2}}\frac{1}{(1-z)^{d-4}}
\widehat{\psi}_{\omega,\mathbf{s}}\right|_{z\to 1}
\end{equation}
\begin{align}
\label{eq:longvectorcftcorrelator2}
G_{t\|}^R(\omega,\mathbf{s}) &= i\frac{\lambda}{q_\mathbf{s}}G_{tt}^R(\omega,\mathbf{s}) \\
\label{eq:longvectorcftcorrelator3}
G_{\|\|}^R(\omega,\mathbf{s}) &= \frac{\lambda^2}{q_\mathbf{s}^2}G_{tt}^R(\omega,\mathbf{s}) \ ,
\end{align}
where $ \widehat{\psi}_{\omega,\mathbf{s}} $ satisfies the longitudinal vector QNM equation
(Equation~\ref{eq:longvectoreq}) along with the boundary conditions:
\begin{equation}
\left.\widehat{\psi}_{\omega,\mathbf{s}}\right|_{z=0} \sim z^{-\frac{i\lambda}{C}}
\quad
\left.\widehat{\chi}_{\omega,\mathbf{s}}\right|_{z\to 1}
= \left.\frac{1}{q_\mathbf{s}^2}\tilde{g}(z)(1-z)^{d-4}\partial_z
\left[\frac{1}{(1-z)^{d-4}}\widehat{\psi}_{\omega,\mathbf{s}}\right]\right|_{z\to 1} = 1 \ .
\end{equation}
$ C_v $ is the appropriate normalization constant for the bulk vector field (see Appendix C).
\item For the transverse component of a vector field, the correlation function is given by:
\begin{equation}
\label{eq:transvectorcftcorrelator}
G_{\bot\bot}^R(\omega,\mathbf{v})\\
= \left.-4C_v\frac{r_+^{d-3}}{R^{d-2}}\frac{1}{(1-z)^{d-4}}
\partial_z\widehat{\psi}_{\omega,\mathbf{v}}(z)\right|_{z\to 1} \ ,
\end{equation}
where $ \widehat{\psi}_{\omega,\mathbf{v}} $ satisfies the transverse vector QNM equation
(Equation~\ref{eq:transvectoreq}) along with the boundary conditions:
\begin{equation}
\left.\widehat{\psi}_{\omega,\mathbf{v}}\right|_{z=0} \sim z^{-\frac{i\lambda}{C}}
\qquad
\left.\widehat{\psi}_{\omega,\mathbf{v}}\right|_{z\to 1} = 1 \ .
\end{equation}

\end{itemize}

Several notes about taking the limit $ z\to 1 $:
\begin{enumerate}
\item The expression $ \partial_z\widehat{\psi}_{\omega,\mathbf{s}}(z) $ in the limit $ z\to 1 $ is
calculated by normalizing $ \psi $ at $ z=1-\epsilon $, differentiating it and then taking
$ \epsilon\to 1 $, so that:
\begin{equation}
\left.\partial_z\widehat{\psi}_{\omega,\mathbf{s}}(z)\right|_{z\to 1}
= \left.\frac{\partial_z\psi_{\omega,\mathbf{s}}(z)}{\psi_{\omega,\mathbf{s}}(z)}\right|_{z\to 1} \ .
\end{equation}
\item When taking the limit $ z\to 1 $ , one must drop the contact terms - the terms that are polynomials in $ \omega $ and $ L_\mathbf{s} $ and diverge as $ z\to 1 $ (These terms are
removed by process of renormalization). This can be done by developing
$ \partial_z\widehat{\psi}_{\omega,\mathbf{s}}(z) $ in orders of $ 1-z $, and then dropping all terms
up to the lowest non-contact-term order.
\end{enumerate}

Note also that the spacetime coordinates correlators for the vector perturbations are related to the
above longitudinal and transverse components by the following relations:
\begin{eqnarray}
G_{tt}^R(t,x;t',x') &=& \int\frac{\,\mathrm{d}\omega}{2\pi}\sum_\mathbf{s}
G_{tt}^R(\omega,\mathbf{s})H_\mathbf{s}^{FT*}(x)H_\mathbf{s}^{FT}(x')\mathrm{e}^{-i\omega(t'-t)}\\
G_{ti}^R(t,x;t',x') &=&  \int\frac{\,\mathrm{d}\omega}{2\pi}\sum_\mathbf{s}
G_{t\|}^R(\omega,\mathbf{s})H_\mathbf{s}^{FT*}(x)\frac{\partial_i H_\mathbf{s}^{FT}(x')}
{L_\mathbf{s}^{FT}}\mathrm{e}^{-i\omega(t'-t)} \\
G_{ij}^R(t,x;t',x') &=&  \int\frac{\,\mathrm{d}\omega}{2\pi}\sum_\mathbf{s}
G_{\|\|}^R(\omega,\mathbf{s})\frac{\partial_i H_\mathbf{s}^{FT*}(x)}
{L_\mathbf{s}^{FT}}\frac{\partial_j H_\mathbf{s}^{FT}(x')}
{L_\mathbf{s}^{FT}}\mathrm{e}^{-i\omega(t'-t)} \nonumber\\
&& + \int\frac{\,\mathrm{d}\omega}{2\pi}\sum_\mathbf{v}
G_{\bot\bot}^R(\omega,\mathbf{v})\tilde{A}_{\mathbf{v},i}^{FT*}(x)
\tilde{A}_{\mathbf{v},j}^{FT}(x')\mathrm{e}^{-i\omega(t'-t)} \ .
\end{eqnarray}

A detailed derivation of these expressions can be found in Appendix~\ref{app:corrformulaederivation}.

\subsection{Exact Correlators for the \texorpdfstring{$K=-1$}{K=-1} Case}
\label{subsec:cftexactcorrelatorsfortc}

As in Subsection~\ref{subsec:qnmexactsolutionsfortc}, in the $K=-1$ case (where the $\Omega_{d-2}$
manifold is hyperbolic and $T=T_c$), analytical expressions for the CFT correlators may be found.
This section is dedicated to the calculation of the correlators in this case.

\subsubsection{Scalar Correlator}

Continuing from the transformation defined in
Appendix~\ref{app:exactsolutionsfortcderivation}, the expression in
Equation~\ref{eq:scalarcftcorrelator} can be written in terms of $ \phi $ and $ w $:
\begin{equation}
\label{eq:scalarcftcorrelatorinw}
G^R(\omega,\mathbf{s})
= \left.-4C_s\frac{r_+^{d-1}}{R^d}\frac{1}{(1-w)^\frac{d-3}{2}}
\frac{\partial_w\phi_{\omega,\mathbf{s}}}{\phi_{\omega,\mathbf{s}}}\right|_{w\to 1}
\end{equation}
(where a contact term has been dropped).

As explained in Appendix~\ref{app:exactsolutionsfortcderivation}, the solution to
the EOM with an incoming-wave boundary condition at the horizon is
$ \phi_{\omega,\mathbf{s}} = {}_2F_1\left(a,b;c;w\right)  $,
where $ a,b $ and $ c $ are given by Equations~\ref{eq:scalartransexactsolutionfortcparams1}
and~\ref{eq:scalartransexactsolutionfortcparams2}. Define:
\begin{equation}
\Delta \equiv c-a-b = \frac{d-1}{2} \ .
\end{equation}

There are now two possible cases:
\begin{enumerate}
\item $ d $ is even, so that $ \Delta $ is non-integer. In this case, the hypergeometric
function satisfies the following connection formula:
\begin{multline}
\label{eq:hypergeometricevenconnectionformula}
\phi_{\omega,\mathbf{s}}(w) = {}_2F_1\left(a,b;c;w\right) \\
= \frac{\Gamma(c)\Gamma(\Delta)}{\Gamma(c-a)\Gamma(c-b)}\,{}_2F_1\left(a,b;1-\Delta;1-w\right)\\
+ \frac{\Gamma(c)\Gamma(-\Delta)}{\Gamma(a)\Gamma(b)}(1-w)^\Delta
\,{}_2F_1\left(c-a,c-b;\Delta+1;1-w\right) \ .
\end{multline}
Defining the coefficients:
\begin{align}
A(\omega,\mathbf{s}) &=  \frac{\Gamma(c)\Gamma(\Delta)}{\Gamma(c-a)\Gamma(c-b)} \\
B(\omega,\mathbf{s}) &=  \frac{\Gamma(c)\Gamma(-\Delta)}{\Gamma(a)\Gamma(b)} \ ,
\end{align}
we get:
\begin{equation}
\left.\frac{\partial_w\phi_{\omega,\mathbf{s}}}{\phi_{\omega,\mathbf{s}}}\right|_{w=1-\epsilon}
\approx -\Delta\frac{B(\omega,\mathbf{s})}{A(\omega,
\mathbf{s})}\epsilon^{\Delta-1}\\
= -\Delta \frac{\Gamma(-\Delta)}{\Gamma(\Delta)} \frac{\Gamma(c-a)\Gamma(c-b))}
{\Gamma(a)\Gamma(b)} \epsilon^{\Delta-1} \ .
\end{equation}
Putting this into Equation~\ref{eq:scalarcftcorrelatorinw} we get:
\begin{equation}
\label{eq:scalarexactcorrelatorintceven}
\boxed{
G^R(\omega,\mathbf{s})
= 2(d-1)C_s \frac{r_+^{d-1}}{R^d}\frac{\Gamma(-\Delta)}{\Gamma(\Delta)}
\frac{\Gamma(a+\Delta)}{\Gamma(a)}\frac{\Gamma(b+\Delta)}{\Gamma(b)} \ .
}
\end{equation}

\item $ d $ is odd, so that $ \Delta $ is an integer. In this case, the hypergeometric
function satisfies\footnote{Here and in related formulae, $ \psi(z) $ represents the Digamma function,
defined by : $ \psi(z) \equiv \frac{\Gamma'(z)}{\Gamma(z)}$.}:
\begin{multline}
\label{eq:hypergeometricoddconnectionformula}
\phi_{\omega,\mathbf{s}}(w) = {}_2F_1\left(a,b;c;w\right) \\
= \frac{\Gamma(c)\Gamma(\Delta)}{\Gamma(a+\Delta)\Gamma(b+\Delta)}
\sum_{n=0}^{\Delta-1}\frac{(a)_n (b)_n}{n!(1-\Delta)_n}(1-w)^n \\
- \frac{\Gamma(c)}{\Gamma(a)\Gamma(b)}(w-1)^\Delta\sum_{n=0}^{\infty}
\frac{(a+\Delta)_n(b+\Delta)_n}{n!(n+\Delta)!}(1-w)^n [\ln(1-w)\\
+\psi(a+\Delta+n)+\psi(b+\Delta+n)-\psi(n+1)-\psi(n+\Delta+1)] \ ,
\end{multline}
so that\footnote{The terms $ \psi(1) $ and $ \psi(\Delta+1) $ have been dropped from $ B $ because
after dividing by A they amount to a contact term.}:
\begin{align}
A(\omega,\mathbf{s}) &=  \frac{\Gamma(c)\Gamma(\Delta)}{\Gamma(a+\Delta)\Gamma(b+\Delta)} \\
B(\omega,\mathbf{s}) &=  \frac{\Gamma(c)}{\Gamma(a)\Gamma(b)}\frac{(-1)^{\Delta+1}}{(\Delta)!}
\left[\psi(a+\Delta)+\psi(b+\Delta)\right]
\end{align}

\begin{equation}
\left.\frac{\partial_w\phi_{\omega,\mathbf{s}}}{\phi_{\omega,\mathbf{s}}}\right|_{w=1-\epsilon}
\approx -\frac{(-1)^{\Delta+1}}{\Gamma^2(\Delta)}(a)_\Delta(b)_\Delta
\left[\psi(a+\Delta)+\psi(b+\Delta)\right]\epsilon^{\Delta-1} \ .
\end{equation}

Putting this into Equation~\ref{eq:scalarcftcorrelatorinw} we get:
\begin{equation}
\label{eq:scalarexactcorrelatorintcodd}
\boxed{
G^R(\omega,\mathbf{s})
= 4C_s\frac{r_+^{d-1}}{R^d}\frac{(-1)^{\Delta+1}}{\Gamma^2(\Delta)}(a)_\Delta(b)_\Delta
\left[\psi(a+\Delta)+\psi(b+\Delta)\right] \ .
}
\end{equation}

\end{enumerate}

\subsubsection{Vector Correlators}

The calculation of the exact vector correlators in the $ K=-1 $ case is similar to the scalar case:
We write the expressions for the correlators in Equations~\ref{eq:longvectorcftcorrelator}
and~\ref{eq:transvectorcftcorrelator} in terms of the variables defined in
Appendix~\ref{app:exactsolutionsfortcderivation} and calculate the limits using the exact solutions.
A detailed derivation is given in Appendix~\ref{app:exactvectorcorrelatorsderivationfortc}, and here
we quote the final results.

For the longitudinal vector mode, we define:
\begin{equation}
\Delta \equiv \frac{d-5}{2} \ .
\end{equation}
For the case of even $ d $ ($ \Delta $ is non-integer), we obtain the expression:
\begin{equation}
\label{eq:longvectorexactcorrelatorintceven}
\boxed{
G_{tt}^R(\omega,\mathbf{s})
= \frac{4C_v}{d-5}\frac{r_+^{d-3}}{R^{d-2}}q_\mathbf{s}^2
\frac{\Gamma(-\Delta)}{\Gamma(\Delta)}\frac{\Gamma(a+\Delta)}{\Gamma(a)}
\frac{\Gamma(b+\Delta)}{\Gamma(b)} \ .
}
\end{equation}
For the case of odd $ d $ ($ \Delta $ is an integer), we obtain the expression:
\begin{equation}
\label{eq:longvectorexactcorrelatorintcodd}
\boxed{
G_{tt}^R(\omega,\mathbf{s})
= 2C_v\frac{r_+^{d-3}}{R^{d-2}}q_\mathbf{s}^2\frac{(-1)^{\Delta+1}}{\Gamma^2(\Delta+1)}
(a)_\Delta(b)_\Delta\left[\psi(a+\Delta)+\psi(b+\Delta)\right] \ .
}
\end{equation}
The other components of the longitudinal mode correlator can be calculated from
Equations~\ref{eq:longvectorcftcorrelator2} and~\ref{eq:longvectorcftcorrelator3}.

For the transverse vector mode, we define:
\begin{equation}
\Delta \equiv \frac{d-3}{2} \ .
\end{equation}
The caluclation gives for even $ d $:
\begin{equation}
\label{eq:transvectorexactcorrelatorintceven}
\boxed{
G_{\bot\bot}^R(\omega,\mathbf{v})
= 4(d-3)C_v \frac{r_+^{d-3}}{R^{d-2}}\frac{\Gamma(-\Delta)}{\Gamma(\Delta)}
\frac{\Gamma(a+\Delta)}{\Gamma(a)}\frac{\Gamma(b+\Delta)}{\Gamma(b)} \ ,
}
\end{equation}
and for odd $ d $:
\begin{equation}
\label{eq:transvectorexactcorrelatorintcodd}
\boxed{
G_{\bot\bot}^R(\omega,\mathbf{v})
= 8C_v\frac{r_+^{d-3}}{R^{d-2}}\frac{(-1)^{\Delta+1}}{\Gamma^2(\Delta)}(a)_\Delta(b)_\Delta
\left[\psi(a+\Delta)+\psi(b+\Delta)\right] \ .
}
\end{equation}

\subsection{Asymptotic Correlator Expressions (for \texorpdfstring{$ K>-1 $}{K>-1} )}
\label{subsec:cftasymptoticcorrelatorexpressions}

Continuing Subsection~\ref{subsec:qnmasymoptotics}, the goal of this subsection is to find an approximate
expression for the correlators of the CFT operators coupled to the scalar and vector fields in the
bulk spacetime for large values of the frequency, where $ |\lambda| \gg q $ (corresponding to
$ \frac{L_s^{FT}}{T} = \text{fixed} $ and $ \frac{\omega}{T} \gg 1 $)
and $ \im\lambda < 0 $. The general stages of the calculation are as follows:
\begin{enumerate}

\item We start from the effective potentials and their poles calculated in
Subsection~\ref{subsec:qnmasymoptotics}.
\item We solve the approximate QNM equations around each pole using Bessel functions.
\item Using the asymptotic limit of $ |\lambda| \gg 1 $, we replace the Bessel functions with their
asymptotic forms.
\item We match the solutions on each region of the complex plane, and apply the boundary condition at
the horizon, to get an expression for the solution near the boundary up to a multiplicative constant.
\item We use the approximate solution near the boundary and the expressions for the retarded correlators
from Subsection~\ref{subsec:generalformulaeforcorrelators} to obtain an asymptotic expression for the
correlators.

\end{enumerate}

\subsubsection{Calculation of Asymptotic Correlators for the Scalar Case}
\label{subsubsec:cftasymptoticcorrelatorexpressionsscalar}

Proceeding from the definitions in Subsection~\ref{subsec:qnmasymoptotics}, the expression in
Equation~\ref{eq:scalarcftcorrelator} can be written in terms of $ \phi $ and $ \tilde{z} $
(after dropping the contact term and using the fact that $ \tilde{z}\approx z-1 $ for
$ z\to 1 $):
\begin{equation}
\label{eq:scalarcftcorrelatorinzt}
G^R(\omega,\mathbf{s}) =  \left.-2C_s\frac{r_+^{d-1}}{R^d}\frac{(-1)^d}{\tilde{z}^{d-2}}
\frac{\partial_{\tilde{z}}\phi_{\omega,\mathbf{s}}(\tilde{z})}{\phi_{\omega,\mathbf{s}}(\tilde{z})}\right|_{\tilde{z}\to 0} \ .
\end{equation}

We continue in a method similar to the one applied in Subsection~\ref{subsec:qnmasymoptotics}:
find asymptotic solutions around the poles of the effective potential and ``match'' the
solutions\footnote{
While in the general case, where $ \im(\lambda\tilde{z}_0)\neq 0 $, the anti-Stokes lines
emanating from $ \tilde{z}=0 $ and $ \tilde{z}_0 $ don't align, the ``matching'' of the solutions in
the region (2) is still possible since there are no Stokes lines between these two
anti-Stokes lines in this region, so that the asymptotic solution remains
the same between them.
}. As before, we assume that $ |\lambda|\gg 1 $ and $ |\lambda|\gg q $.
We also assume that $ \lambda $ is in the 3rd quadrant of the complex plane
($ \re\lambda<0 $, $ \im\lambda<0 $).
The solutions around $ \tilde{z}\to\tilde{z}_0 $ ($ z\to\infty $) are given by:
\begin{align}
P_+(\tilde{z}-\tilde{z}_0) &\equiv \sqrt{2\pi\lambda(\tilde{z}-\tilde{z}_0)}J_0\left(\lambda(\tilde{z}-\tilde{z}_0)\right)\\
P_-(\tilde{z}-\tilde{z}_0) &\equiv \sqrt{2\pi\lambda(\tilde{z}-\tilde{z}_0)}Y_0\left(\lambda(\tilde{z}-\tilde{z}_0)\right) \ .
\end{align}
Replacing the Bessel functions with their asymptotic forms gives for direction (1):
\begin{equation}
\label{eq:monoasympinftyplus2}
\phi \approx
\left[A_+\mathrm{e}^{i(-\lambda\tilde{z}_0-\frac{\pi}{4})}+A_-\mathrm{e}^{i(-\lambda\tilde{z}_0-\frac{3\pi}{4})}\right]\mathrm{e}^{i\lambda\tilde{z}}\\
+\left[A_+\mathrm{e}^{i(\lambda\tilde{z}_0+\frac{\pi}{4})}-A_-\mathrm{e}^{i(\lambda\tilde{z}_0-\frac{\pi}{4})}\right]\mathrm{e}^{-i\lambda\tilde{z}} \ ,
\end{equation}
and for direction (2):
\begin{equation}
\label{eq:monoasympinftymin2}
\phi \approx
\left[A_+\mathrm{e}^{i(-\lambda\tilde{z}_0+\frac{3\pi}{4})}-3A_-\mathrm{e}^{i(-\lambda\tilde{z}_0+\frac{\pi}{4})}\right]\mathrm{e}^{i\lambda\tilde{z}}\\
+\left[A_+\mathrm{e}^{i(\lambda\tilde{z}_0+\frac{\pi}{4})}-A_-\mathrm{e}^{i(\lambda\tilde{z}_0-\frac{\pi}{4})}\right]\mathrm{e}^{-i\lambda\tilde{z}} \ .
\end{equation}
From the boundary condition at the horizon ($ z=0 $, $ \tilde{z}\to -\infty $) we have:
\begin{equation}
\phi \approx \mathrm{e}^{-i\lambda\tilde{z}} \ .
\end{equation}

There are now two possible cases:
\begin{enumerate}
\item $ d $ is even. In this case the solution around $ \tilde{z}\to 0 $ is approximately:
\begin{equation}
\phi \approx B_+ P_+(\tilde{z}) + B_- P_-(\tilde{z})\\
= B_+\sqrt{2\pi\lambda\tilde{z}} J_{\frac{j_1}{2}}(\lambda\tilde{z})
+ B_-\sqrt{2\pi\lambda\tilde{z}} J_{-\frac{j_1}{2}}(\lambda\tilde{z}) \ ,
\end{equation}
where $ j_1 = d-1 $ (so that $ \frac{j_1}{2} $ is non-integer). Replacing the Bessel functions
with their asymptotic form, we have:
\begin{equation}
\phi
\approx \left[B_+\mathrm{e}^{-i\beta_+} + B_-\mathrm{e}^{-i\beta_-}\right]\mathrm{e}^{i\lambda\tilde{z}} +
 \left[B_+\mathrm{e}^{i\beta_+} + B_-\mathrm{e}^{i\beta_-}\right]\mathrm{e}^{-i\lambda\tilde{z}} \ ,
\end{equation}
where $ \beta_\pm \equiv \frac{\pi}{4}(1 \pm j_1)$.
We now match the solutions on lines (1) and (2). We have for line (1):
\begin{equation}
\label{eq:cftmonoeq3}
A_+\mathrm{e}^{i(-\lambda\tilde{z}_0-\frac{\pi}{4})}+A_-\mathrm{e}^{i(-\lambda\tilde{z}_0-\frac{3\pi}{4})} = 0
\qquad\Rightarrow\qquad
A_- = -iA_+ \ .
\end{equation}
For the section (2) we have:
\begin{align}
\label{eq:cftmonoeq4}
A_+\mathrm{e}^{i(-\lambda\tilde{z}_0+\frac{3\pi}{4})}-3A_-\mathrm{e}^{i(-\lambda\tilde{z}_0+\frac{\pi}{4})}
&= B_+\mathrm{e}^{-i\beta_+} + B_-\mathrm{e}^{-i\beta_-}\\
\label{eq:cftmonoeq5}
A_+\mathrm{e}^{i(\lambda\tilde{z}_0+\frac{\pi}{4})}-A_-\mathrm{e}^{i(\lambda\tilde{z}_0-\frac{\pi}{4})}
&= B_+\mathrm{e}^{i\beta_+} + B_-\mathrm{e}^{i\beta_-} \ .
\end{align}
Putting Equation~\ref{eq:cftmonoeq3} into Equations~\ref{eq:cftmonoeq4} and~\ref{eq:cftmonoeq5}
we get:
\begin{align}
B_+\mathrm{e}^{-i\theta_+} + B_-\mathrm{e}^{-i\theta_-} &= 2iA_+ \\
B_+\mathrm{e}^{i\theta_+} + B_-\mathrm{e}^{i\theta_-} &= 4iA_+ \ ,
\end{align}
where $ \theta_\pm \equiv \lambda\tilde{z}_0 - \frac{\pi}{4} - \beta_\pm $.
Solving for $ B_\pm $ we can get an expression for $ \frac{B_+}{B_-} $:
\begin{equation}
\frac{B_+}{B_-}
= \frac{\left|\begin{array}{cc}
2 & \mathrm{e}^{-i\theta_-} \\
4 & \mathrm{e}^{i\theta_-}
\end{array}\right|}
{\left|\begin{array}{cc}
\mathrm{e}^{-i\theta_+} & 2 \\
\mathrm{e}^{i\theta_+} & 4
\end{array}\right|}
= -i^{d-1}\frac{\mathrm{e}^{2i\theta_-}-2}{\mathrm{e}^{2i\theta_-}+2}
\end{equation}
(here we used the fact that $ d $ is even).

Next we evaluate the correlator. Developing $ \phi $ around $ \tilde{z}=0 $ we have:
\begin{multline}
\label{eq:asymptoticcorrelatorsphiaround0even}
\phi
\approx B_+\sqrt{2\pi\lambda\tilde{z}} J_{\frac{j_1}{2}}(\lambda\tilde{z})
+ B_-\sqrt{2\pi\lambda\tilde{z}} J_{-\frac{j_1}{2}}(\lambda\tilde{z}) \\
\approx B_-\sqrt{2\pi\lambda\tilde{z}}(\frac{1}{2}\lambda\tilde{z})^{-\frac{j_1}{2}}
\sum_{k=0}^\infty \frac{(-\frac{1}{4}\lambda^2\tilde{z}^2)^k}{k!\Gamma(-\frac{j_1}{2}+k+1)} \\
+ B_+\sqrt{2\pi\lambda\tilde{z}}(\frac{1}{2}\lambda\tilde{z})^{\frac{j_1}{2}}
\sum_{k=0}^\infty \frac{(-\frac{1}{4}\lambda^2\tilde{z}^2)^k}{k!\Gamma(\frac{j_1}{2}+k+1)} \ .
\end{multline}
Defining the coefficients:
\begin{align}
A(\omega,\mathbf{s}) &= \frac{\sqrt{2\pi}\lambda^{-\Delta+\frac{1}{2}}}
{2^{-\Delta}\Gamma(-\Delta+1)} B_- \\
B(\omega,\mathbf{s}) &= \frac{\sqrt{2\pi}\lambda^{\Delta+\frac{1}{2}}}
{2^{\Delta}\Gamma(\Delta+1)} B_+
\end{align}
(where $ \Delta\equiv \frac{j_1}{2} = \frac{d-1}{2} $),
we get:
\begin{equation}
\left.\frac{\partial_{\tilde{z}}\phi_{\omega,\mathbf{s}}}{\phi_{\omega,\mathbf{s}}}\right|_{\tilde{z}=\epsilon}
= (2\Delta)\frac{B(\omega,\mathbf{s})}{A(\omega,\mathbf{s})}\epsilon^{2\Delta-1}
\approx (d-1)\left(\frac{i\lambda}{2}\right)^{d-1}\frac{\Gamma(-\Delta)}{\Gamma(\Delta)}
\frac{\mathrm{e}^{2i\theta_-}-2}{\mathrm{e}^{2i\theta_-}+2}\epsilon^{d-2} \ .
\end{equation}
And finally putting this into Equation~\ref{eq:scalarcftcorrelatorinzt}:
\begin{equation}
G^R(\omega,\mathbf{s}) \approx  -2(d-1)C_s\frac{r_+^{d-1}}{R^d}
\frac{\Gamma(-\Delta)}{\Gamma(\Delta)}\left(\frac{i\lambda}{2}\right)^{d-1}
\frac{\mathrm{e}^{2i\theta_-}-2}{\mathrm{e}^{2i\theta_-}+2}
\end{equation}
(Note that this expression is true up to the addition of contact terms).
This asymptotic formula is only true for $ \re\lambda<0 $. In order to get an expression for
$ \re\lambda>0 $, we can make use of the symmetry properties of the correlator\footnote{
This property can also be deduced from the QNM equations themselves.}:
\begin{equation}
G^R(\omega,\mathbf{s}) = G^{R*}(-\omega^*,\mathbf{s}) \ .
\end{equation}
We get for $ \re\lambda>0 $:
\begin{equation}
G^R(\omega,\mathbf{s}) \\
= -2(d-1)C_s\frac{r_+^{d-1}}{R^d}
\frac{\Gamma(-\Delta)}{\Gamma(\Delta)}\left(\frac{i\lambda}{2}\right)^{d-1}
\frac{\mathrm{e}^{2i\overline{\theta_-}}-2}{\mathrm{e}^{2i\overline{\theta_-}}+2} \ ,
\end{equation}
where we define:
\begin{equation}
\overline{\theta_-}(\lambda) \equiv -\theta_-^*(-\lambda*)
= \lambda\tilde{z}_0^*-\frac{\pi}{4}(d-3) \ .
\end{equation}
Looking at the asymptotic behaviour of the expressions for $ \re\lambda<0 $ and $ \re\lambda>0 $,
we may unify them into one asymptotic expression:
\begin{equation}
G^R(\omega,\mathbf{s}) \approx  2(d-1)C_s\frac{r_+^{d-1}}{R^d}
\frac{\Gamma(-\Delta)}{\Gamma(\Delta)}\left(\frac{i\lambda}{2}\right)^{d-1}
\frac{\mathrm{e}^{2i\theta_-}-2}{\mathrm{e}^{2i\theta_-}+2}\,
\frac{\mathrm{e}^{2i\overline{\theta_-}}-2}{\mathrm{e}^{2i\overline{\theta_-}}+2} \ .
\end{equation}
Finally, using $ \lambda=\frac{\omega R^2}{r_+} $, the asymptotic expression may be written:
\begin{equation}
\label{eq:asymptcftcorrelatorscalarevend}
\boxed{
G^R(\omega,\mathbf{s}) \approx  2(d-1)C_s R^{d-2}
\frac{\Gamma(-\Delta)}{\Gamma(\Delta)}\left(\frac{i\omega}{2}\right)^{d-1}
\frac{\mathrm{e}^{2i\theta_-}-2}{\mathrm{e}^{2i\theta_-}+2}\,
\frac{\mathrm{e}^{2i\overline{\theta_-}}-2}{\mathrm{e}^{2i\overline{\theta_-}}+2} \ .
}
\end{equation}

\item $ d $ is odd. In this case the solution around $ \tilde{z}\to 0 $ is approximately:
\begin{equation}
\phi \approx B_+ P_+(\tilde{z}) + B_- P_-(\tilde{z})\\
= B_+\sqrt{2\pi\lambda\tilde{z}} J_{\frac{j_1}{2}}(\lambda\tilde{z})
+ B_-\sqrt{2\pi\lambda\tilde{z}} Y_{\frac{j_1}{2}}(\lambda\tilde{z}) \ ,
\end{equation}
where $ j_1 = d-1 $ (so that $ \frac{j_1}{2} $ is an integer). Replacing the Bessel functions
with their asymptotic form, we have:
\begin{equation}
\phi
\approx \left[(B_+-iB_-)\mathrm{e}^{-i\beta_+}\right]\mathrm{e}^{i\lambda\tilde{z}} +
\left[(B_++iB_-)\mathrm{e}^{i\beta_+}\right]\mathrm{e}^{-i\lambda\tilde{z}} \ ,
\end{equation}
where $ \beta_+ = \frac{\pi}{4}(1+j_1) = \frac{\pi}{4}d $.
We again match the solutions on lines (1) and (2).
For (1) we again have:
\begin{equation}
\label{eq:cftmonoeq6}
A_+\mathrm{e}^{i(-\lambda\tilde{z}_0-\frac{\pi}{4})}+A_-\mathrm{e}^{i(-\lambda\tilde{z}_0-\frac{3\pi}{4})} = 0 \ .
\end{equation}
For (2):
\begin{align}
\label{eq:cftmonoeq7}
A_+\mathrm{e}^{i(-\lambda\tilde{z}_0+\frac{3\pi}{4})}-3A_-\mathrm{e}^{i(-\lambda\tilde{z}_0+\frac{\pi}{4})}
&= (B_+-iB_-)\mathrm{e}^{-i\beta_+}\\
\label{eq:cftmonoeq8}
A_+\mathrm{e}^{i(\lambda\tilde{z}_0+\frac{\pi}{4})}-A_-\mathrm{e}^{i(\lambda\tilde{z}_0-\frac{\pi}{4})}
&= (B_++iB_-)\mathrm{e}^{i\beta_+} \ .
\end{align}
Putting Equation~\ref{eq:cftmonoeq6} into Equations~\ref{eq:cftmonoeq7} and~\ref{eq:cftmonoeq8}
we get (defining $ \theta_+ \equiv \lambda\tilde{z}_0 - \frac{\pi}{4} - \beta_+ $):
\begin{align}
(B_+-iB_-)\mathrm{e}^{i\theta_+} &= 4iA_+ \\
(B_++iB_-)\mathrm{e}^{-i\theta_+} &= 2iA_+ \ .
\end{align}
Solving for $ B_\pm $ we can get an expression for $ \frac{B_+}{B_-} $:
\begin{equation}
\frac{B_+}{B_-}
= \frac{\left|\begin{array}{cc}
4 & -i\mathrm{e}^{i\theta_+} \\
2 & i\mathrm{e}^{-i\theta_+}
\end{array}\right|}
{\left|\begin{array}{cc}
\mathrm{e}^{i\theta_+} & 4 \\
\mathrm{e}^{-i\theta_+} & 2
\end{array}\right|}
= 2i\frac{\mathrm{e}^{2i\theta_+}}{\mathrm{e}^{2i\theta_+}-2}-i \ .
\end{equation}

Next we evaluate the correlator. Developing $ \phi $ around $ \tilde{z}=0 $ we have (we again
define $ \Delta\equiv \frac{j_1}{2} = \frac{d-1}{2} $):
\begin{equation}
\label{eq:asymptoticcorrelatorsphiaround0odd}
\phi
\approx B_+\sqrt{2\pi\lambda\tilde{z}} J_{\Delta}(\lambda\tilde{z})
+ B_-\sqrt{2\pi\lambda\tilde{z}} Y_{\Delta}(\lambda\tilde{z}) \\
\end{equation}
Defining the coefficients:
\begin{equation}
A(\omega,\mathbf{s}) = -\frac{\sqrt{2\pi}\lambda^{-\Delta+\frac{1}{2}}}{\pi 2^{-\Delta}}
(\Delta-1)!\,B_-
\end{equation}
\begin{multline}
B(\omega,\mathbf{s}) = \frac{2\sqrt{2\pi}\lambda^{\Delta+\frac{1}{2}}}
{\pi 2^\Delta\Gamma(\Delta+1)}\ln\left(-\frac{\lambda}{2}\right)\,B_- \\
-\frac{\sqrt{2\pi}\lambda^{\Delta+\frac{1}{2}}}{\pi 2^\Delta(\Delta)!}
\left[\psi(1)+\psi(\Delta+1)\right]\,B_-
+\frac{\sqrt{2\pi}\lambda^{\Delta+\frac{1}{2}}}{2^\Delta \Gamma(\Delta+1)}\,B_+ \ ,
\end{multline}
we again have:
\begin{equation}
\left.\frac{\partial_{\tilde{z}}\phi_{\omega,\mathbf{s}}}{\phi_{\omega,\mathbf{s}}}\right|_{\tilde{z}=\epsilon}
\approx (2\Delta)\frac{B(\omega,\mathbf{s})}{A(\omega,\mathbf{s})}\epsilon^{2\Delta-1} \ .
\end{equation}
And finally after putting this into Equation~\ref{eq:scalarcftcorrelatorinzt}, and again making
use of the symmetry properties of the correlator we obtain:
\begin{empheq}[box=\fbox]{multline}
\label{eq:asymptcftcorrelatorscalaroddd}
G^R(\omega,\mathbf{s}) \approx
4C_s R^{d-2}\frac{(-1)^{\Delta+1}}{\Gamma^2(\Delta)}
\left(\frac{i\omega}{2}\right)^{d-1} \\
\left[2\pi i\frac{\mathrm{e}^{2i\theta_+}}{\mathrm{e}^{2i\theta_+}-2}
- 2\pi i\frac{\mathrm{e}^{2i\overline{\theta_+}}}{\mathrm{e}^{2i\overline{\theta_+}}-2}
+ 2\ln\left(\frac{i\lambda}{2}\right)\right] \ ,
\end{empheq}
where:
\begin{equation}
\overline{\theta_+}(\lambda) \equiv -\theta_+^*(-\lambda^*) = \lambda\tilde{z}_0^*
+\frac{\pi}{4}(d+1) \ .
\end{equation}

\end{enumerate}

\subsubsection{Calculation of Asymptotic Correlators for the Vector Case}
\label{subsubsec:cftasymptoticcorrelatorexpressionsvector}

The calculation of asymptotic expressions for the vector correlators is similar to the scalar case.
A detailed derivation is given in Appendix~\ref{app:cftasymptoticcorrelatorexpressionsvector}, and
here we quote the final results.

For the longitudinal vector mode, we define:
\begin{equation}
\Delta \equiv \frac{d-5}{2} \ .
\end{equation}
We distinguish between two possible cases\footnote{Note that for the case of vector gauge field
perturbations with $ d=4 $, the effective potential for both modes is $ V(z) = q^2 \tilde{g}(z) $, and has a pole
of degree $ -\frac{3}{2} $ at $ \tilde{z}\to\tilde{z}_0 $. In this case the method applied here can't be used since
the equation is not of the Bessel type around the pole $ \tilde{z}_0 $, and since $ q $ obviously can't be
neglected (neglecting it results in an an expression with no poles). This case therefore requires
a separate and more complete treatment, which shall not be investigated here. This section will
therefore focus on the case of $ d\geq 5 $.}:
\begin{enumerate}

\item $ d $ is even. In this case we obtain the asymptotic expression:
\begin{empheq}[box=\fbox]{multline}
\label{eq:asymptcftcorrelatorlongvectorevend}
G_{tt}^R(\omega,\mathbf{s})
\approx \frac{4C_v}{d-5}R^{d-4}
(L_\mathbf{s}^{FT})^2\frac{\Gamma(-\Delta)}{\Gamma(\Delta)}\left(\frac{i\omega}{2}\right)^{d-5}\\
\frac{\mathrm{e}^{2i\theta_-}+2\cos\left(\frac{\pi}{d-2}\right)}
{\mathrm{e}^{2i\theta_-}-2\cos\left(\frac{\pi}{d-2}\right)}
\frac{\mathrm{e}^{2i\overline{\theta_-}}+2\cos\left(\frac{\pi}{d-2}\right)}
{\mathrm{e}^{2i\overline{\theta_-}}-2\cos\left(\frac{\pi}{d-2}\right)} \ ,
\end{empheq}
where:
\begin{align}
\theta_- &\equiv \lambda\tilde{z}_0 + \frac{\pi}{4}(d-7) \\
\overline{\theta_-}(\lambda) &\equiv -\theta_-^*(-\lambda^*)
= \lambda\tilde{z}_0^*-\frac{\pi}{4}(d-7) \ .
\end{align}

\item $ d $ is odd. In this case we obtain the asymptotic expression:
\begin{empheq}[box=\fbox]{multline}
\label{eq:asymptcftcorrelatorlongvectoroddd}
G_{tt}^R(\omega,\mathbf{s})
\approx 2C_v R^{d-4}(L_\mathbf{s}^{FT})^2\frac{(-1)^{\Delta+1}}{\Gamma^2(\Delta+1)}
\left(\frac{i\omega}{2}\right)^{d-5} \\
\left[2\pi i\frac{\mathrm{e}^{2i\theta_+}}{\mathrm{e}^{2i\theta_+} + 2\cos\left(\frac{\pi}{d-2}\right)}
- 2\pi i\frac{\mathrm{e}^{2i\overline{\theta_+}}}{\mathrm{e}^{2i\overline{\theta_+}} + 2\cos\left(\frac{\pi}{d-2}\right)}
+ 2\ln\left(\frac{i\lambda}{2}\right)\right] \ ,
\end{empheq}
where:
\begin{align}
\theta_+ &\equiv \lambda\tilde{z}_0 - \frac{\pi}{4}(d-3) \\
\overline{\theta_+}(\lambda) &\equiv -\theta_+^*(-\lambda*)
= \lambda\tilde{z}_0^*+\frac{\pi}{4}(d-3) \ .
\end{align}

\end{enumerate}
The other components of the longitudinal mode correlator can be calculated from
Equations~\ref{eq:longvectorcftcorrelator2} and~\ref{eq:longvectorcftcorrelator3}.

For the transverse vector mode, we define:
\begin{equation}
\Delta \equiv \frac{d-3}{2} \ .
\end{equation}
We again distinguish between two possible cases:
\begin{enumerate}

\item $ d $ is even. In this case we obtain the asymptotic expression:
\begin{empheq}[box=\fbox]{multline}
\label{eq:asymptcftcorrelatortransvectorevend}
G_{\bot\bot}^R(\omega,\mathbf{v})
\approx 4(d-3)C_v R^{d-4}
\frac{\Gamma(-\Delta)}{\Gamma(\Delta)}\left(\frac{i\omega}{2}\right)^{d-3}\\
\frac{\mathrm{e}^{2i\theta_-}-2\cos\left(\frac{\pi}{d-2}\right)}
{\mathrm{e}^{2i\theta_-}+2\cos\left(\frac{\pi}{d-2}\right)}
\frac{\mathrm{e}^{2i\overline{\theta_-}}-2\cos\left(\frac{\pi}{d-2}\right)}
{\mathrm{e}^{2i\overline{\theta_-}}+2\cos\left(\frac{\pi}{d-2}\right)} \ ,
\end{empheq}
where:
\begin{align}
\theta_- &\equiv \lambda\tilde{z}_0 + \frac{\pi}{4}(d-5) \\
\overline{\theta_-}(\lambda) &\equiv -\theta_-^*(-\lambda*)
= \lambda\tilde{z}_0^*-\frac{\pi}{4}(d-5) \ .
\end{align}

\item $ d $ is odd. In this case we obtain the asymptotic expression:
\begin{empheq}[box=\fbox]{multline}
\label{eq:asymptcftcorrelatortransvectoroddd}
G_{\bot\bot}^R(\omega,\mathbf{v})
\approx 8C_v R^{d-4}
\frac{(-1)^{\Delta+1}}{\Gamma^2(\Delta)}
\left(\frac{i\omega}{2}\right)^{d-3}\\
\left[2\pi i\frac{\mathrm{e}^{2i\theta_+}}{\mathrm{e}^{2i\theta_+} - 2\cos\left(\frac{\pi}{d-2}\right)}
- 2\pi i\frac{\mathrm{e}^{2i\overline{\theta_+}}}{\mathrm{e}^{2i\overline{\theta_+}} - 2\cos\left(\frac{\pi}{d-2}\right)}
+ 2\ln\left(\frac{i\lambda}{2}\right)\right] \ ,
\end{empheq}
where:
\begin{align}
\theta_+ &\equiv \lambda\tilde{z}_0 - \frac{\pi}{4}(d-1) \\
\overline{\theta_+}(\lambda) &\equiv -\theta_+^*(-\lambda*)
= \lambda\tilde{z}_0^*+\frac{\pi}{4}(d-1) \ .
\end{align}

\end{enumerate}

Notice that, taking into account the differences in the definition of $ \Delta $ and $ \theta_\pm $
between the longitudinal and the transverse perturbation modes:
\begin{align}
\Delta^{\text{long}} &= \Delta^{\text{trans}} - 1 \\
\theta_\pm^{\text{long}} &= \theta_\pm^{\text{trans}} \pm \frac{\pi}{2} \ ,
\end{align}
the asymptotic expressions for $ G_{\bot\bot}^R(\omega,\mathbf{v}) $ and
$ G_{\|\|}^R(\omega,\mathbf{s}) $ are identical. This is expected and serves as a check for our
calculations, since in this limit we have neglected the spatial mode $ L_{\mathbf{s}/\mathbf{v}}^{FT} $,
so that the differences between the longitudinal and the transverse modes are negligible.

\subsection{Numerical Calculation of Correlators}

In the following we present some results of exact numerical calculations of the gauge theory
retarded correlation functions for several cases, and compare them to the asymptotic analytical
expressions (as discussed in Subsection~\ref{subsec:cftasymptoticcorrelatorexpressions}).
The correlation functions have been calculated using the method outlined in Appendix~\ref{app:numericalmethods}.

The following calculations were made for a degree of $ N=200 $ and $ z_0 = 0.6 $ (see
Appendix~\ref{app:numericalmethods} for details):

\begin{itemize}

\item The case of scalar perturbation in $ d=4 $ bulk dimensions, with $ K=1 $ (spherical boundary
topology) and $ q_\mathbf{s} = 0 $ (s-wave). The results are given in
Figures~\ref{fig:correlationfunctionresultspolesscalard4K1q0},
~\ref{fig:correlationfunctionresultscirclescalard4K1q0R20},
~\ref{fig:correlationfunctionresultscontourscalard4K1q0}
and~\ref{fig:correlationfunctionresultsspectralscalard4K1q0}.

\begin{figure}[htbp]
\centering
\subfloat[
]{
\label{fig:correlationfunctionresultspolesscalard4K1q0}
\includegraphics[width=0.4\textwidth]{CorrelationFunctionResultsPolesScalard4K1q0.pdf}
}
\subfloat[
]{
\label{fig:correlationfunctionresultscirclescalard4K1q0R20}
\includegraphics[width=0.56\textwidth]{CorrelationFunctionResultsCircleScalard4K1q0R20.pdf}
}\\
\subfloat[
]{
\label{fig:correlationfunctionresultscontourscalard4K1q0}
\includegraphics[width=0.56\textwidth]{CorrelationFunctionResultsContourScalard4K1q0.pdf}
}
\subfloat[
]{
\label{fig:correlationfunctionresultsspectralscalard4K1q0}
\includegraphics[width=0.4\textwidth]{CorrelationFunctionResultsSpectralScalard4K1q0.pdf}
}
\caption{Numerically calculated ``normalized'' retarded correlation function for the case of scalar
perturbations with $ d=4 $,$ K=1 $ and $ q_\mathbf{s} = 0 $. The plots show:
\protect\subref{fig:correlationfunctionresultspolesscalard4K1q0}
The absolute value of the
``normalized'' correlation function along the analytically calculated asymptotic line of QNM frequencies,
given by the parameters: $ \lambda_0 = -2.7206 - 2.7205 i \quad \Delta\lambda = -1.9691 - 2.3502 i $,
\protect\subref{fig:correlationfunctionresultscirclescalard4K1q0R20}
the real parts of both the
numerically calculated and the analytical asymptotic ``normalized'' correlation functions on a circle in
the complex plane of radius $ |\lambda| = 20 $,
\protect\subref{fig:correlationfunctionresultscontourscalard4K1q0}
a contour plot of the real part
of the numerically calculated ``normalized'' correlation function in the 4-th quadrant of the complex plane,
\protect\subref{fig:correlationfunctionresultsspectralscalard4K1q0}~the spectral function associated with $ G^R $ along the real frequency axis, in units of 
$ -C_s\frac{r_+^{d-1}}{R^d} $.
}
\end{figure}



\item The case of longitudinal vector perturbations in $ d=5 $ bulk dimensions, with $ K=0 $
(flat boundary topology) and $ q_\mathbf{s} = 0.6 $. The results are given in
Figures~\ref{fig:correlationfunctionresultspoleslongvectord5K0q06},
~\ref{fig:correlationfunctionresultscirclelongvectord5K0q06R20},
~\ref{fig:correlationfunctionresultscontourlongvectord5K0q06}
and~\ref{fig:correlationfunctionresultsspectrallongvectord5K0q06}.

\begin{figure}[htbp]
\centering
\subfloat[]{
\label{fig:correlationfunctionresultspoleslongvectord5K0q06}
\includegraphics[width=0.4\textwidth]{CorrelationFunctionResultsPolesLongVectord5K0q06.pdf}
}
\subfloat[]{
\label{fig:correlationfunctionresultscirclelongvectord5K0q06R20}
\includegraphics[width=0.56\textwidth]{CorrelationFunctionResultsCircleLongVectord5K0q06R20.pdf}
}\\
\subfloat[]{
\label{fig:correlationfunctionresultscontourlongvectord5K0q06}
\includegraphics[width=0.56\textwidth]{CorrelationFunctionResultsContourLongVectord5K0q06.pdf}
}
\subfloat[]{
\label{fig:correlationfunctionresultsspectrallongvectord5K0q06}
\includegraphics[width=0.4\textwidth]{CorrelationFunctionResultsSpectralLongVectord5K0q06.pdf}
}
\caption{Numerically calculated ``normalized'' retarded correlation function for the case of longitudinal
vector perturbations with $ d=5 $,$ K=0 $ and $ q_\mathbf{s} = 0.6 $. The plots show:
\protect\subref{fig:correlationfunctionresultspoleslongvectord5K0q06}
The absolute value of
the ``normalized'' correlation function along the analytically calculated asymptotic line of QNM
frequencies, given by the parameters: $ \lambda_0 = 0 \quad \Delta\lambda = -2 - 2 i $,
\protect\subref{fig:correlationfunctionresultscirclelongvectord5K0q06R20}
the real parts of
both the numerically calculated and the analytical asymptotic ``normalized'' correlation functions on a
circle in the complex plane of radius $ |\lambda| = 20 $,
\protect\subref{fig:correlationfunctionresultscontourlongvectord5K0q06}
a contour plot of
the real part of the numerically calculated ``normalized'' correlation function in the 4-th quadrant of
the complex plane,
\protect\subref{fig:correlationfunctionresultsspectrallongvectord5K0q06}~the spectral function associated with $ G_{tt}^R $ along the real frequency axis, in units of 
$ -C_v\frac{r_+^{d-3}}{R^{d-2}} $.}
\end{figure}


\item The case of transverse vector perturbations in $ d=5 $ bulk dimensions, with $ K=-0.5 $
(hyperbolic boundary topology) and $ q_\mathbf{v} = 1 $. The results are given in
Figures~\ref{fig:correlationfunctionresultspolestransvectord5Km05q1},
~\ref{fig:correlationfunctionresultscircletransvectord5Km05q1R20},
~\ref{fig:correlationfunctionresultscontourtransvectord5Km05q1}
and~\ref{fig:correlationfunctionresultsspectraltransvectord5Km05q1}.

\begin{figure}[htbp]
\centering
\subfloat[]{
\label{fig:correlationfunctionresultspolestransvectord5Km05q1}
\includegraphics[width=0.4\textwidth]{CorrelationFunctionResultsPolesTransVectord5Km05q1.pdf}
}
\subfloat[]{
\label{fig:correlationfunctionresultscircletransvectord5Km05q1R20}
\includegraphics[width=0.56\textwidth]{CorrelationFunctionResultsCircleTransVectord5Km05q1R20.pdf}
}\\
\subfloat[]{
\label{fig:correlationfunctionresultscontourtransvectord5Km05q1}
\includegraphics[width=0.56\textwidth]{CorrelationFunctionResultsContourTransVectord5Km05q1.pdf}
}
\subfloat[]{
\label{fig:correlationfunctionresultsspectraltransvectord5Km05q1}
\includegraphics[width=0.4\textwidth]{CorrelationFunctionResultsSpectralTransVectord5Km05q1.pdf}
}
\caption{Numerically calculated ``normalized'' retarded correlation function for the case of transverse
vector perturbations with $ d=5 $,$ K=-0.5 $ and $ q_\mathbf{v} = 1 $. The plots show:
\protect\subref{fig:correlationfunctionresultspolestransvectord5Km05q1}
The absolute value
of the ``normalized'' correlation function along the analytically calculated asymptotic line of QNM
frequencies, given by the parameters:
$ \lambda_0 = -1.4142 - 2.0000 i \quad \Delta\lambda = -1.4142 - 2.0000 i $,
\protect\subref{fig:correlationfunctionresultscircletransvectord5Km05q1R20}
the real parts of
both the numerically calculated and the analytical asymptotic ``normalized'' correlation functions on a
circle in the complex plane of radius $ |\lambda| = 20 $,
\protect\subref{fig:correlationfunctionresultscontourtransvectord5Km05q1}
a contour plot of
the real part of the numerically calculated ``normalized'' correlation function in the 4-th quadrant of
the complex plane,
\protect\subref{fig:correlationfunctionresultsspectraltransvectord5Km05q1}~the spectral function associated with $ G_{\bot\bot}^R $ along the real frequency axis, in units of 
$ -C_v\frac{r_+^{d-3}}{R^{d-2}} $.}
\end{figure}



\end{itemize}

For the sake of better visual representation of the results, we define the
``normalized'' retarded correlators $ I(\lambda) $ by ``normalizing out'' the power and logarithmic
behaviour of the correlators, keeping only the part of the
correlators that behaves as $ O(1) $ for $ |\lambda| \to \infty $\footnote{See 
Appendix~\ref{app:numericalmethods} for a more precise definition.}. These ``normalized'' correlators
have poles at the QNM frequencies, and approach a constant on each side of the QNM frequencies line for
frequencies far enough away from that line.

Figures~\ref{fig:correlationfunctionresultspolesscalard4K1q0},
~\ref{fig:correlationfunctionresultspoleslongvectord5K0q06}
and~\ref{fig:correlationfunctionresultspolestransvectord5Km05q1} show the value of
$ |I(\lambda_0 + s\Delta\lambda)| $ as a function of $ s $, where
$ \lambda_0 $ and $ \Delta\lambda $ are the analytical asymptotic parameters of the QNM spectrum
as calculated in Subsection~\ref{subsec:qnmasymoptotics} and $ s $ is a positive, real parameter. As
expected, we find the poles of the correlation function approximately at integer values of $ s $.

Figures~\ref{fig:correlationfunctionresultscirclescalard4K1q0R20},
~\ref{fig:correlationfunctionresultscirclelongvectord5K0q06R20}
and~\ref{fig:correlationfunctionresultscircletransvectord5Km05q1R20} show the value of
$ \re I\left(\lambda_R\mathrm{e}^{-i\theta}\right) $ as a function of $ \theta $, where $ \lambda_R $
is some constant radius in the frequency complex plane (in this case $ \lambda_R = 20 $). As expected,
the numerically calculated values are close to the ones of the analytical asymptotic expressions
calculated in Subsection~\ref{subsec:cftasymptoticcorrelatorexpressions}.

Figures~\ref{fig:correlationfunctionresultscontourscalard4K1q0},
~\ref{fig:correlationfunctionresultscontourlongvectord5K0q06}
and~\ref{fig:correlationfunctionresultscontourtransvectord5Km05q1} show a contour plot of
$ \re I(\lambda) $ in the 4-th quadrant of the complex frequency plane. Both the poles at the QNM
frequencies and the step-function-like behaviour of the ``normalized'' correlation function can be
clearly seen in the plots.

Figures~\ref{fig:correlationfunctionresultsspectralscalard4K1q0},
~\ref{fig:correlationfunctionresultsspectrallongvectord5K0q06}
and~\ref{fig:correlationfunctionresultsspectraltransvectord5Km05q1} 
show the values of the appropriate spectral functions on the real frequency axis
($ \rho(\lambda) $ in the case of scalar perturbations, $ \rho_{tt}(\lambda) $ in the case of
longitudinal vector perturbations and $ \rho_{\bot\bot}(\lambda) $ in the case of transverse vector
perturbations). At large frequency values, these functions approach the analytical asymptotic expressions
and are proportional to $ \lambda^s $ (where $ s $ is the appropriate exponent for the perturbation
type), as expected from conformal symmetry considerations. In the longitudinal vector case 
(Figure~\ref{fig:correlationfunctionresultsspectrallongvectord5K0q06}),
the spectral function oscillates at lower frequencies, due to the existence of the hydrodynamic 
diffusion pole.

\clearpage

\section{Discussion}
\label{sec:discussion}

In this work we studied some of the properties of the retarded correlation function of composite
operators of finite temperature gauge field theories with dual gravity descriptions in the large
coupling limit. Of specific interest was the dependence of these
properties on the topology of the space on which the gauge theory is defined, and its temperature.
This goal was achieved by performing exact analytical calculations for specific cases where it was
possible (the hyperbolic case with $ T=T_c $), by performing numerical calculations and by obtaining
approximate expressions for the large frequency limit ($ \frac{L_s^{FT}}{T} = \text{fixed} $ and
$ \frac{\omega}{T} \gg 1 $).

In the large frequency limit we saw that in all cases (all topologies, temperatures and perturbation
types) the correlation functions exhibit the same kind of asymptotic pole structure, with a constant
``gap'' between two consecutive poles. Moreover, we saw that asymptotically these poles are produced by
Bose-Einstein-like factors in the correlation functions, with some complex parameter multiplying the
frequency.

We have also seen that the spatial topology on which the gauge theory is defined is indeed noticeable
in the properties of the correlation function and its poles (corresponding to quasinormal modes in the
dual bulk spacetime), in the case where $ \frac{\sqrt{|R_\Omega^{FT}|}}{T} = \text{fixed} $. In
particular, the asymptotic ``slope'' of the correlator poles encodes information about the spatial
topology and the temperature, but is ``universal'' otherwise - it doesn't depend on the type of operators
considered.
In the spherical (positive curvature) and flat (zero curvature) cases, this slope always has a real part
and an imaginary part. In the spherical case, it depends on the relation between the spatial curvature
and the temperature, while in the flat case it's constant (as expected, since in this case the
temperature is the only scale in the theory, and the correlator poles are necessarily proportional
to it).
In the hyperbolic (negative curvature) case, the slope has a real part for temperatures above $ T_c $,
which goes to $ 0 $ in a non-smooth way for $ T\to T_c $ (In $ d = 4,5 $ dimensions the real part is
also $ 0 $ for temperatures below $ T_c $).
At $ T=T_c $ the slope is completely imaginary, and exact analytical expressions can be found for the
correlation functions and their poles. Interestingly, the poles ``gap'' in this case, and indeed the
general form of the correlation functions, is exactly what one would expect to find for the retarded
correlation function of composite operators in a free thermal field theory (the poles ``gap'' in this
case corresponds to the Matsubara frequency ``gap'' - see~\cite{Hartnoll:2005ju}).

Several open issues are left for further investigation:
\begin{itemize}

\item This work has dealt with uncharged bulk black hole solutions, but we can also expand the scope
and discuss charged black holes, that correspond to gauge theories with non-zero chemical potentials
(see \cite{Son:2006em}).
The background of the perturbations in such solutions contains both a black
hole metric and a non-zero gauge field. We can then ask - how does the charge change the asymptotic
properties of the gauge theory correlation functions and their dependence on the spatial topology?

\item As noted in Subsubsection~\ref{subsubsec:cftasymptoticcorrelatorexpressionsvector}, the method we have
employed for calculating the asymptotic expressions for the correlation functions doesn't work as is
for the vector perturbation modes in $ d=4 $ bulk dimensions. This case requires a different treatment
using complex WKB analysis, although numerical calculations show that our conclusions regarding the
asymptotic ``slope'' of the correlator poles remain true in this case as well.

\item While the asymptotic structure of the correlators (and their poles) has a geometrical meaning
in the gravity (bulk) side, it is unclear if it has an interpretation in the strongly coupled dual
field theory. Weak coupling calculations don't seem to exhibit such a structure (see
~\cite{Hartnoll:2005ju}). The asymptotic poles ``gap'' in the large coupling limit depends only on the
spatial topology and the temperature, suggesting it may have an interpretation involving them, but it
can't be trivially deduced from the symmetries of the theory alone.

\item Considering the properties of the correlation functions in the hyperbolic case for $ T\to T_c $,
does this temperature have any importance in the dual gauge theory? In~\cite{Shen:2007xk}, it is
suggested that in $ d=4 $, the behaviour of the QNM frequencies slope near this temperature is related
to the known phase transition between the topological (hyperbolic) black hole and the MTZ solution
(a hyperbolic black hole with scalar hair - see~\cite{Martinez:2004nb},~\cite{Myung:2008ze}).
It is not clear, however, what is the interpretation of
such a phase transition in the gauge theory side, and whether the argument extends to higher dimensions.

\end{itemize}


\section*{Acknowledgements}
The work is supported in part by the Israeli Science Foundation center
of excellence.

