frequencies. However, by studying the full time dependence within holographic
setup of [229], we overcome the usual restrictions of Floquet theory. Within
the holographic mode, we can study non-equilibrium steady states in
periodically driven anomalous QFTs, specially their stability in dependence on
the applied electric fields. It is tempting to investigate, whether we
observer chaotic behavior in the regime of large amplitudes.

#### Quantum gravity and matter in de-Sitter and the $T\bar{T}$ deformation

In chapter 4, we discussed a framework to study quantum gravity in dS – the
DS/dS correspondence. Although the conjecture has been put on more solid
footing over the last years, there are plenty of open questions. For example,
it would be interesting to understand the exact physical interpretation of the
one-parameter family of entangling surfaces in the dual CFTs. We have some
intuition about the special cases, where the entangling surface corresponds to
the cosmological horizon or is located on the UV brane [276]. However, the
precise interpretation of the one-parameter family of entangling surfaces
which interpolates between those two concepts remains an open question. With
the notion of EE in DS/dS, we set out to derive consistency conditions on the
matter content in dSd+1 but surprisingly any matter obeying the null energy
condition violates this bound! There are several ways out of this dilemma. For
one, the framework may not be well defined. Either it is not possible to
define a theory of quantum gravity with matter on dS or DS/dS is not the
correct framework to model it. On a more confident note, we could conclude
that the bound was simply too strong.

The EE in the nominator, $S_{d+1,\text{global}}$, accounts for the EE on the
entire spatial volume. This EE is twice as large as the EE in the dS static
patch. For a single observer living on the UV brane of dSd the maximum entropy
available corresponds to $S_{\text{dS}}$ and this observer thus does not have
sufficient information to fully reconstruct the higher dimensional dSd+1
geometry. To do so, we require some information from beyond the horizon. This
is a surprising fact but must not necessarily be an inconsistency but may
rather be another fascinating property of the DS/dS correspondence.

For future investigations, we may postulate a less stringent bound

$\frac{S_{d+1,\text{static}}}{S_{\text{dS}}}\leq
1\quad\Leftrightarrow\quad\frac{S_{d+1,\text{global}}}{S_{\text{dS}}}\leq 2,$

which is, however, the minimum consistency requirement we have to demand under
any circumstances. Within this bound, we are requiring the EE between the left
and right CFT within the static patch to be less than the dSd entropy. It
would be interesting, to quantify the impact of this consistency requirement
in the context of constraints on the potential of matter in dS. The one-
parameter family of entangling surfaces which all yield the same EE is
intimately connected to the symmetries of dS and thus not present if we deform
the geometry away from dS. For a Dirichlet wall in dS corresponding to a hard
radial cutoff, the single member of the one-parameter family yield different
EE depending on their turning point in the bulk. This was in part topic of the
EE in the context of $T\bar{T}$ deformed QFTs.

We derived the holographic EE for general subintervals for $T\bar{T}$ deformed
QFTs. It would be very interesting, to also extend the field theory
calculation to general geometries. To realize this, we have to evaluate the
associated partition functions on manifolds with conical defects. Furthermore,
we alluded to the construction of [277], relating the QFTs in terms of the
DS/dS correspondence to the AdS/CFT correspondence by $T\bar{T}$ deformations.
Working out this construction explicitly is an important future direction for
understanding quantum gravity in dS within the framework of holography.

##  Chapter 7 Numerical methods

In this chapter, we review the numerical methods we applied throughout this
thesis in order to solve the various differential equations and eigenvalue
problems efficiently and accurately following [470] closely. For problems in
numerical holography, spectral methods have the big advantage that they solve
the equations of motions globally, i.e. we can simultaneously demand boundary
conditions on both ends of the domain. This is a big advantage compared to
Finite-difference or shooting methods where we have to vary the initial
conditions on one side in order to find the appropriate function values on the
other side of the domain. Another point in favor of applying a spectral method
is their fast convergence rate; we have a very robust and accurate numerical
technique at hand which is not necessarily a lot more complicated than using a
shooting method. Lastly, using spectral methods provides us with very good
control of about the quality and accuracy of the numerical solution which is
essential for solving differential equations numerically. To quote Boyd
himself: “One must watch the convergence of a numerical code as carefully as a
father watching his four year old play near a busy road.”

##  Appendix 7.A (Pseudo)-spectral methods

In order to get an intuition about how spectral methods work, we consider the
differential equation

where $\mathcal{L}$ is a differential operator and $u(x)$ the solution
satisfying $\mathcal{L}\,u(x)-f(x)=0$. We can expand the solution in terms of
a basis $\\{\phi_{n}(x)\\}$ by formally writing
$u(x)=\sum_{n=0}^{\infty}\,c_{n}\phi_{n}(x)$. The basic idea of spectral
methods is to approximate the exact solution $u(x)$ by a finite number $N$ of
basis polynomials $\phi_{n}(x)$

$u(x)\approx u_{N}(x)=\sum\limits_{n=0}^{N}\,c_{n}\,\phi_{n}(x)$ (1)

and find the coefficients $\\{c_{n}\\}_{n=0,\dots,N}$ which minimize the
residuum $R(x;c_{n})$

$R(x;c_{0},c_{1},\dots,c_{N})=\mathcal{L}\,u_{N}(x)-f(x)$ (2)

The first important step is to choose the appropriate set of basis function
and to discretize the coordinates $x_{i}$ in order to make it tractable for
computer algorithms.

Since we know the expressions of the basis functions analytically, we can re-
write their first and second derivative by introducing derivative matrices
$\hat{D}$. In the given basis, the derivative of the basis functions is given
as a linear combination of the basis functions i.e.
$\phi^{\prime}_{m}(x)=\sum_{n=0}^{N}\hat{D}_{mn}\phi_{n}(x),\
\phi^{\prime\prime}_{m}(x)=\sum_{n,l=0}^{N}\hat{D}_{mn}\hat{D}_{nl}\,\phi_{l}(x)$.
With the differentiation matrices, we can rewrite derivatives so that they act
on the coefficients

$\displaystyle
u^{\prime}(x)\,\approx\sum\limits_{n=0}^{N}c_{n}\,\phi_{j}^{\prime}(x)\,=\sum\limits_{n,m=0}^{N}c_{n}\,\hat{D}_{nm}\phi_{m}(x)=\sum\limits_{n=0}^{N}c_{n}^{\prime}\,\phi_{n}(x),$
(3) $\displaystyle
u^{\prime\prime}(x)\approx\sum\limits_{n=0}^{N}c_{n}\,\phi_{j}^{\prime\prime}(x)=\sum\limits_{n,m,l=0}^{N}c_{n}\,\hat{D}_{nm}\hat{D}_{ml}\,\phi_{l}(x)=\sum\limits_{n=0}^{N}c_{n}^{\prime\prime}\,\phi_{n}(x).$
(4)

The difference between spectral- and pseudo-spectral methods is that the
former uses the coefficients in order to minimize the residuum (2) while the
latter uses the function values $u_{i}$ evaluated on the gridpoints $x_{i}$.
In contrast to finite differences, the gridpoints are not necessarily
equidistant but given in terms of the zeros and extrema of the basis functions
$\phi_{n}$. The relation of the function values $u_{i}$ and the coefficients
$c_{i}$ is $u_{i}=\sum_{j=0}^{N}c_{j}\,\phi_{j}(x_{i})$.

The best choice for the basis functions in the homogeneous case are the so-
called Chebychev polynomials given by

$T_{k}(x)=\cos(k\,\arccos(x)).$ (5)

In the Chebychev basis, we can choose between different grids in order to
discretize the coordinate. There are the Chebychev-Radau grids which do not
include either the left or the right boundary of the domain, the Chebychev-
Gaußgrid where both boundaries of the domain are excluded and the Chebychev-
Lobatto which includes boundaries of the domain. In AdS/CFT it is very
convenient when we can impose boundary conditions at conformal boundary and
know the numerical solution there. Note that the boundary and the horizon are
regular singular points to the equations of motion in AdS/CFT and the
differential equation is degenerate there. This means that we technically do
not have to impose boundary conditions there and we can use the Chebychev-
Gaußand Chebychev-Radau grids. Even though these grids are interesting, we
restrict our discussion to the Chebychev-Lobatto grid, in particular since we
want the conformal boundary to be part of the integration domain. The
Chebychev-Lobatto gridpoints are given by

$x_{i}=\cos\varphi_{i}=\cos\frac{\pi\,i}{N}$ (6)

with the differentiation matrix given by

$\hat{D}_{mj}\begin{cases}-\frac{2N^{2}+1}{6},&m=j=N,\\\
\frac{2N^{2}+1}{6},&m=j=0,\\\ -\frac{x_{j}}{2\,(1-x_{j}^{2})},&0<m=j<N,\\\
\frac{\kappa_{m}}{\kappa_{j}}\frac{(-1)^{m-j}}{x_{m}-x_{j}},&m\not=j,\end{cases}$
(7)

where we defined

$\kappa_{j}=\begin{cases}2,&\text{for }j=0,j=N\\\ 1,&\text{for }j=1,\ldots
N-1.\end{cases}$ (8)

The differentiation matrix for the second derivative is simply given by
applying the matrix for the first derivative (7) twice. Note that by choosing
non-equidistant gridpoints, we avoid the Runge-phenomenon which appears in
interpolations on equidistant grids. We demand that the residuum vanishes
exactly on the collocation points $x_{i}$ and the numerical solution is thus
exact on the gridpoints (note that the conformal boundary is one of the
gridpoints). For periodic problems it is best to use a Fourier grid. For
spherical symmetrical problems, we recommend using spherical harmonics as
basis functions for latitude and longitude. In these cases, we refer the
interested reader to [470, 471, 472, 473]. The attentive reader might have
noticed that the Chebychev-Lobatto gridpoints are defined on $x\in[-1,1]$
while the radial coordinate of AdS is usually in the interval $u\in[0,1]$
where the conformal boundary is located at $z=0$. In practice, it is thus
convenient to map the gridpoints and differentiation matrices according to

$z_{i}=\frac{x_{i}+1}{2}\in[0,1],\quad\tilde{D}=\left(\partial_{x}\left(\frac{x+1}{2}\right)\right)^{-1}\hat{D}=2\,\hat{D},\quad\tilde{D}^{2}=\tilde{D}\,\tilde{D}=4\,\hat{D}\,\hat{D}.$
(9)

The corresponding Chebychev polynomials are given by
$T_{k}(z)=\cos(k\,\arccos(2z-1))$.

##  Appendix 7.B Pseudospectral solutions to Boundary Value Problems

After discussing the discretization of differential equations in terms of
spectral methods, we now work out how to solve linear boundary value problems
in practice. The step to solving nonlinear boundary values is formally not
more complicated since the nonlinear are solved by iteratively solving the
linear problem. We discuss them in section 7.D.

#### Boundary conditions

Before we impose boundary conditions on the differential equation, we have to
understand the difference between behavioral and numerical boundary
conditions. Periodicity, for example, is a behavioral boundary condition. We
require that the solution satisfies $u(x)=u(x+2\pi)$ but that does not impose
any specific value on the function $u(x)$ or its derivative. For periodic
functions, we choose sines and cosines as basis functions which obviously
satisfy the periodicity condition. By expanding the functions in terms of this
basis, our numerical solution is automatically periodic and we do not have to
impose any explicit boundary conditions since the periodicity requirement is
inherently satisfied by the basis functions. Even more interestingly, the
differential equation

$z\,(1-z)\,\partial^{2}_{z}u(z)-(z+1)\,\partial_{z}u+5\,u(z)=3,\ z\in[0,1],$
(10)

is singular on both endpoints. However, the Chebychev polynomials are
individually analytic at $z=\\{0,1\\}\ (x=\pm 1)$ and so are linear
combinations of them. This leads to the important insight that we can solve
singular differential equations with an exponential rate of convergence (we
get to the meaning of that in the next section) without imposing any
additional constraints! This is extremely convenient in the context of
AdS/CFT. As we learned, the equations of motions we come across in the context
of the AdS/CFT correspondence have regular singular points on both ends of the
intervals. By solving these equations in terms of a Chebychev basis, we cure
those singularities in a very elegant way and are able to construct numerical
solutions converging exponentially despite the singular endpoints. In
contrast, numerical boundary conditions are of the form
$u(1)=\left.\text{d}u/\text{d}x\right|_{x=0}=5$ and have to be imposed
explicitly.

To impose boundary conditions in terms of spectral methods, we usually follow
the following strategies:

  1. (i)

boundary bordering In the $N\times N$ pseudo-spectral matrix, we replace the
lines corresponding to the two endpoints by the explicit boundary condition.
The remaining $(N-2)\times N$ block corresponds to the collocation conditions
in the inner of the interval.

  2. (ii)

basis recombination We re-formulate the boundary value problem so that the new
boundary conditions in the modified boundary value boundary problem are
homogeneous. In this strategy, we modify the set of basis functions so that
they satisfy the constraints individually.

In practice, we usually have to use (or have the luxury of using) both
strategies. To see how this works in context of the AdS/CFT correspondence, we
can consider a very simple setup: the famous holographic superconductor in a
planar AdS4 Schwarzschild background [68]. The resulting equations of motion
are given by

$\displaystyle
z^{2}f(z)\Psi^{\prime\prime}(z)+z\left(zf^{\prime}(z)-2f(z)\right)\Psi^{\prime}(z)+\Psi(z)\left(\frac{z^{2}\Phi(z)^{2}}{f(z)}+2\right)=0$
(11) $\displaystyle
z^{4}\,\Phi^{\prime\prime}(z)-\frac{2\,z^{2}\,\Phi(z)\Psi(z)^{2}}{f(z)}=0,$
(12)

where $f(z)=1-z^{3}$. Furthermore, the function $\Phi(z)$ behaves as
$\Phi(z)\sim(1-z)\,\Phi_{h}+\dots$ at the horizon. We see that both of the
coupled (nonlinear) differential equations are singular at the endpoints $z=0$
and $z=1$ (since $f(1)=0$). Additionally, we know the solution at the boundary
in terms of a power series

$\displaystyle\Psi(z)=\Psi_{(s)}\,z+\Psi_{(v)}\,z^{2}+\dots,$ (13)
$\displaystyle\Phi(z)=\mu+\rho\,z+\dots,$ (14)

where the leading and subleading mode are identified with the field theory
quantities according to the AdS/CFT dictionary. The precise physical
interpretation is not important for now. Let us formulate the following
boundary value boundary problem: We want to find the solutions of
$\Psi(z),\Phi(z)$ to the differential equations (11) and (12) subject to the
boundary conditions:
$\Psi(0)=0,\,\Psi^{\prime}(0)=0,\,\Phi(0)=\mu,\,\Phi(1)=0$ (and also where
$\Psi^{\prime\prime}\not=c$)111The ambiguity of multiple possible solution is
due the nonlinearity of system of equations.

Naïvely, we could go forward and simply implement the boundary conditions and
try to solve the system. However, there is a more elegant way. Since we know
the solution in terms of a power series at $z=0$ (13), we notice that the re-
defined function $\Psi(z)=z^{2}\,\tilde{\Psi}(z)$ automatically satisfies the
two conditions $\Psi(0)=0,\,\Psi^{\prime}(0)=0$ and we do not have to impose
them explicitly anymore. Similarly, the new function
$\Phi(z)=(1-z)\,\tilde{\Phi}(z)$ automatically implements $\Phi(1)=0$. In
terms of the new functions, the system reads

$\displaystyle\tilde{\Psi}\left(2\,z\,f^{\prime}+\frac{(z-1)^{2}\,z^{2}\,\tilde{\Phi}^{2}}{f}-2f+2\right)+z\left(\left(z\,f^{\prime}+2\,f\right)\tilde{\Psi}+z\,f\,\tilde{\Psi}^{\prime\prime}\right)=0$
(15)
$\displaystyle-(z-1)\,\tilde{\Phi}^{\prime\prime}-2\tilde{\Phi}^{\prime}+\frac{2\,(z-1)\,z^{2}\,\tilde{\Phi}\,\tilde{\Psi}^{2}}{f}=0,$
(16)

where the boundary condition $\tilde{\Phi}(0)=\mu$ is left (and we want to
pick the solutions with $\tilde{\Psi}(0)\not=0$). The new boundary value
problem is still a system of second order differential equations but instead
of four boundary conditions, we have only one boundary condition left! So
where do we get the missing boundary conditions from? The answer is simple and
we already stated it in this chapter: in case the equation of motion
degenerates (in terms of singular points) we do not have to impose any
boundary condition since the choice of Chebychev polynomials which are regular
at the singular points we already imposed the missing boundary conditions in
terms of behavioral boundary conditions. Hence it is sufficient to require
that the residuum vanishes at the respective points (which is a fancy way of
saying, ”Do not do anything” and just impose the equation of motion at the
endpoint).

Note that it also possible to introduce the auxiliary function
$\tilde{\Phi}(z)$ as $\Phi(z)=(1-z)\,(\mu+z\,\tilde{\Phi}(z))$. In this way,
$\mu$ will inevitably appear in the equations of motion. This is no problem as
long as we have a determining equation for $\mu$ (for example, a constraint
equation). In this case, we can add the $\mu$ to the unknown parameters and
solve for the corresponding $(N+1)\times(N+1)$ dimensional matrix (in case of
one unknown function).

All in all, we can write this in a compact way (for a (non)linear problem
depending on one coordinate). Let $n$ be the number of unknown functions with
$I\in\\{1,\dots,n\\}$ and $\xi$ denote additional $m$ additional parameters
with $J\in\\{1,\dots,m\\}$. Then we find the following set of equations (see
eq. (2) for the definition of the residuum) for a given resolution $N$

$\displaystyle R_{0}^{I}(u^{\prime I},u^{I};\,\xi^{J})=0$
$\displaystyle\quad\text{for }z=0,$ (17) $\displaystyle R^{I}(u^{\prime\prime
I},u^{\prime I},u^{I},z;\,\xi^{J})=0$ $\displaystyle\quad\text{for }0<z<1,$
(18) $\displaystyle R^{I}_{1}(u^{\prime\prime I},u^{\prime
I},u^{I},z;\,\xi^{J})=0$ $\displaystyle\quad\text{for }z=1,$ (19)
$\displaystyle\xi^{J}(u^{\prime\prime I},u^{\prime I},u^{I};\,\xi^{J})=0$
$\displaystyle\quad\text{for }0<z<1.$ (20)

The coordinate is discretized according to (9) and all derivatives are
replaced by the differentiation matrices (9). The resulting spectral matrix is
a $(n\cdot N+m)\times(n\cdot N+m)$ matrix. For a pseudo-spectral methods, we
can then determine the numerical values of the solution by solving
$(R,\xi)(u)=0$ for the components $u$. The spectral coefficients may be
determined by inverting (1).

##  Appendix 7.C Convergence, Accuracy & All that

The approximation by $N$ basis polynomials in (1) is only a good approximation
if the coefficients beyond the truncation do not contribute significantly.
Furthermore, we can only make statements about higher coefficients if the
coefficients of our solution consistently fall off for larger indices. There
are various rates of convergence. Instead of a rigorous abstract mathematical
treatment, we will keep the discussion about convergence on a graphical level.
This is more than enough for using spectral methods on a day-to-day basis. The
different rates of convergence are depicted in graphic 1.

Figure 1: Different rates of convergence.

Spectral convergence corresponds to exponential decay of the coefficients. In
a log-linear plot where we plot the absolute value of the coefficients against
the number of basis polynomials a purely exponential decay corresponds to a
straight line with negative slope (geometric, black line). The coefficients
decay according $|a_{n}|\sim\mathcal{O}(c_{1}\,\text{exp}(-\,c_{2}\,N))$. If
the coefficients decay even faster, we speak of supergeometric convergence
($|a_{n}|\sim\mathcal{O}(c_{1}\,\text{exp}(-(N/c_{2})\,\log(N)))$ depicted as
green line in figure 1. In the log-linear plot, the slope of supergeometric
convergence rate gets more and more negative instead of converging to a
straight line. On the contrary, in the case of
$|a_{n}|\sim\mathcal{O}(c_{1}\,\text{exp}(-\,c_{2}\,N^{r}))$ with $r<1$, the
coefficients are decaying slower than the straight line in a log-linear plot
and we refer to the decay rate as subgeometric (depicted in red in figure 1).
Lastly, if the coefficients decay as a polynomial in the number of gridpoints,
the convergence is algebraic. In this case, the curve is bend upwards in a
log-linear plot with slope tending to zero. There is one caveat; the notions
of convergence we just discussed are asymptotic quantities and in practise the
effective rate of convergence might differ from the asymptotic ones.

#### Coordinate mappings

As explained in [470], logarithmic singularities render the asymptotic form of
the spectral coefficients algebraic $(\sim N^{-1}\,p^{N})$. The spectral
solutions, we can construct in presence of those singularities have very poor
convergence. In AdS/CFT, it is very common that logarithms appear in the
asymptotic expansions of the functions at the conformal boundary. One example
is the trace anomaly of the energy momentum tensor in $4+1$ dimensions due to
the presence of a magnetic field. As so often when dealing with problems in
spectral methods, there is a solution by Boyd for it [474]. For functions with
endpoint singularities (remember that $x\in[-1,1]$ while $z\in[0,1]$), where
the expansion reads

$f(x)=c_{1}\,(1-x)^{k}\,\log(1-x)+c_{2}\,(1+x)^{k}\,\log(1+x),$ (21)

the algebraic coordinate mapping

$x=\sin\left(\frac{\pi}{2}\,y\right)$ (22)

leads to a much higher resolution near the singularity an may restore the
degraded rate of convergence. In our case, the functions have endpoint
singularities only at the conformal boundary for example

$f(z)=1+z^{4}\,f_{4}+c_{1}\,\log(z)\,z^{4},$ (23)

we propose the Boyd inspired mapping

$z\mapsto z^{2}$ (24)

which was put forward in my master thesis [24, 23]. This mapping is obtained
by transforming eq. (22) to $z\in[0,1]$ and expand the expression around
$z=0$. The mapping moves the logarithmic divergences to higher order according
to

$z^{n}\,\log(z)\mapsto 2\,z^{2n}\,\log(z)$ (25)

and we find a much improved rate of convergence, as shown in figure 3.

##  Appendix 7.D Nonlinear Ordinary differential equations

Applying (pseudo)-spectral methods to nonlinear equations instead of linear
ones is conceptually not significantly more difficult. In terms of Newton’s
method, the equations we have to solve numerically are still linear. Due to
the ambiguity of the solutions to nonlinear equations, however, we have to
start with a good initial guess of the solution so that the Newton method
converges to the solution we are interested in.

##### Newton-Raphson method

The Newton method is an iterative method in order to find the roots of a
nonlinear equations. It works as follows; for an initial guess close to the
true solution, the Newton method is based on using the derivative to do a step
in direction of the true solution. Let $x_{i}$ be the current approximation of
the solution (for the first step it is the initial guess). Then we may find
derive an expression for the next best solution by using the Taylor expansion

$y=f^{\prime}(x_{i})(x-x_{i})+f(x_{i})$ (26)

The tangent intersects the $x$-axis at $x_{i+1}$ (where $y=0$)

$x_{i+1}=x_{i}-\frac{f(x_{i})}{f^{\prime}(x_{i})}.$ (27)

We generalize this to a nonlinear system of algebraic equations $F(\bm{X})=0$.
$\bm{X}$ is the vector of all fields evaluated on the collocation points
$\bm{X}^{T}=(X_{1}^{1},\ldots,X_{N}^{1},\ldots,X_{N}^{n},\xi^{1},\ldots,\xi^{m})$,
where $N$ is the number of collocation points, $n$ the number of fields and
$m$ the number of additional parameters. Since $\bm{X}$ is a vector the
generalization of the derivative $f^{\prime}(x_{i})$ in eq. (27) will be
matrix valued and we introduce the Jacobian $\hat{J}(\bm{X})=\partial
F(\bm{X})\partial\bm{X}$. With this, the Newton step reads

$\bm{X}_{i+1}=\bm{X}_{i}-[\hat{J}(\bm{X}_{i})]^{-1}\,F(\bm{X}_{i}),$ (28)

where we assume that the Jacobian is invertible. For more complicated
problems, it can be very time consuming to evaluate (and store) the Jacobian.
In theses cases, we recommend introducing auxiliary variables for the
derivatives (instead of evaluating them on the gridpoints with the
differentiation matrices) and compute the Jacobian according to the chain
rule. In this way, we separate zeroth, first and second derivatives acting on
the fields. For a discussion see the appendices in [475, 196].

##  Appendix 7.E Eigenvalue problems and Quasi-Normal modes

Computing the Quasi-Normal modes within the AdS/CFT correspondence is a very
important task since it gives us insight into the linear response regime in
strongly coupled field theories. In general, we have to find solutions to the
linearized equations of motion, subject to infalling conditions at the horizon
and Dirichlet conditions at the conformal boundary. One elegant approach to
impose the infalling condition at the horizon is by means of the coordinate
choice in the gravity theory. If we use infalling Eddington-Finkelstein
coordinates instead of Poincaé coordinates, all fields automatically satisfy
infalling conditions at the horizon. We may collect the system of equations of
motion in terms of powers in $\omega$ (where $\omega$ the QNM frequency)

$\bm{\alpha}[\bm{X}]+\omega\,\bm{\beta}[\bm{X}]+\omega^{2}\,\bm{\gamma}[\bm{X}]=0.$
(29)

The vector $\bm{X}$ includes all fields and $\bm{\alpha},\,\bm{\beta},$ and
${\gamma}$ are differential operators. The eq. (29) is a quadratic eigenvalue
problem with the eigenvalues given by the QNM frequencies. In most cases we
are lucky and $\bm{\gamma}[\bm{X}]$ is zero. Nevertheless, it is always
possible to reduce the quadratic eigenvalue problem (29) to a linear one by
introducing auxiliary fields. For example,
$\bm{\alpha}f+\omega\,\bm{\beta}f+\omega^{2}\,\bm{\gamma}f=0$ may be reduced
to the system

$\bm{\alpha}f+\omega\,(\bm{\beta}f+\bm{\gamma}\tilde{f})=0,\ \
\tilde{f}=\omega f$ (30)

This obviously increases the spectral matrix by $N$ for each auxiliary field
since we have to add the new fields to $\bm{X}$. The resulting linear
eigenvalue problem reads

$\bm{\alpha}[\bm{X}]+\omega\,\bm{\beta}[\bm{X}]=0.$ (31)

In order to solve this problem numerically, we discretize the (radial)
coordinate with eq. (9) and replace all derivatives by the differentiation
matrices (9). The Dirichlet boundary conditions may be implemented by using
the methods explained in 7.B. We redefine the functions by factorizing out an
appropriate power $k$ in z, $f=z^{k}\,\tilde{f}$. In practical applications,
we have constraint equations in addition to the equations of motion222Note
that constraint equations are not present if one works in so-called gauge
invariant variables. Implementing these constraint equations is the biggest
difficulty in obtaining the QNMs (or solving the (non)linear problems
described in the previous section). Usually, it is enough to demand constraint
equations at one of the endpoints and we can replace one of the boundary
condition on the fields with the constraint equation. There is no general way
to determine which boundary conditions may be replaced by the constraint
equations. One big advantage of this approach is that we are computing not
only one but up to $n\cdot N+m$ QNMs at once! ($n$ is the number of fields,
$m$ the number of extra conditions and $N$ the number of gridpoints)

##  Appendix 7.F Time-dependent problems

The pioneering work of studying time-dependent problems in the AdS/CFT
correspondence was done by Chesler and Yaffe [417, 418, 419] (see [420] for a
detailed review). The authors developed the standard approach which is the so-
called characteristic formulation by Bondi and Sachs. The characteristic
formulation has the advantage that the coupled set of partial differential
equations decouples and exhibits a nested structure. In the nested structure,
we can solve the equations successively which simplifies the hard task of
solving coupled partial differential equations – a numerically very costly
task – and hence allows for fast and efficient codes.

##### The numerical routine for nonlinear oscillatory shear tests in AdS

In this section, we display the numerical routine to solve the time-dependent
partial differential equations in chapter 2. This subsection is taken from the
appendix of my publication [3]. We make the following re-definitions of the
metric functions

$\displaystyle A=\frac{1+\tilde{A}\ u}{u^{2}},\quad S=\frac{s_{0}+\tilde{S}\
u}{u},$ $\displaystyle H=h_{0}+\tilde{H}\ u,\quad
d_{+}S=\frac{\widetilde{d_{+}S}}{u^{2}},$ (32) $\displaystyle
d_{+}H=\frac{\dot{h_{0}}}{2}+\widetilde{d_{+}H}\ u.$

In our numerical calculation, we use the Chebyshev discretization with 50-100
grid points for the integration along the radial coordinate.

Following the characteristic feature of the bulk equations of motion in EF
coordinates [420], the numerical recipe to solve the equations of motion is:
(24)-(28):

  1. 1.

We start with a static black hole solution with
$\tilde{H}=0,\tilde{S}=s_{1}(t_{0})$ as an initial configuration and choose
the strain function $\gamma(t)$.

  2. 2.

We check the accuracy of the numerical calculation, by plugging
$\tilde{H},\tilde{S}$ in the constraint equation (24).

  3. 3.

We use the definition of apparent horizon, $d_{+}S(u=1)=0$, as a boundary
condition to calculate $\widetilde{d_{+}S}$ by solving eq. (25).

  4. 4.

Then we solve eq. (26) with one boundary condition for $\widetilde{d_{+}H}$ at
asymptotic region,
$\widetilde{d_{+}H}(u=0)=\tfrac{\dot{h_{0}}\dot{s_{0}}}{s_{0}}+\ddot{h_{0}}$.

  5. 5.

Now we can solve eq. (27) to find $\tilde{A}$ with two boundary conditions:
the first is $\tilde{A}(u=0)=2(s_{1}-\dot{s_{0}})/s_{0}$ and the second we can
find by expanding eq. (28) near the horizon which leads to

$A=-\frac{1}{3}\left({d_{+}H}\right)^{2}\bigg{|}_{u=1}.$ (33)

  6. 6.

By using the definition of operator $d_{+}$, we find
$\dot{\tilde{S}},\dot{\tilde{H}}$. Then we integrate in time using a fourth-
order Runge-Kutta method for the first three time steps and then the fourth
order Adams-Bashforth method, to compute $\tilde{H}(u,t_{0}+\delta t)$ and
$\tilde{S}(t_{0}+\delta t)$ and repeat the same routine from step 2.

To impose the sinusoidal strain, we turn on the amplitude smoothly (in the
spirit of [23, 24]) as

$\gamma(t)=\frac{\gamma_{0}}{2}\left(1+\tanh\left(\frac{t-t_{c}}{w_{c}}\right)\right)\sin\left(2\pi\omega
t\right),$ (34)

where parameters $t_{c},w_{c}$ control how accurate the initial configuration
satisfies the constraint equation (24) and how fast the maximum strain is
reached respectively. In figure (2), we show a concrete example of the
constraint equation (24).

Figure 2: Constraint equation (24) for
$s_{1}|_{t=0}=0,\gamma_{0}=0.1,w_{c}=2,t_{c}=25$ for  $\omega/m=0.32$.

##  Appendix 7.G Convergence plots

#### Chiral transport in strong magnetic fields

In this subsection, we exemplarily present a convergence plot for chapter 3.
In the helicity-1 sector, it is convenient to decouple the fluctuation
equations further by introducing $h_{t\pm}=h_{tx}\pm ih_{ty},$
$h_{z\pm}=h_{xz}\pm ih_{yz},$ and $a_{\pm}=a_{x}\pm ia_{y}.$ In figure 3, we
depict the convergence for $h_{t+},\,h_{z+},\,a_{+}$ without coordinate
mapping (left) and with coordinate mapping (right). We note that with
coordinate mapping, the coefficients fall off geometrically to machine
precision before they reach a plateau.

Figure 3: Convergence in the helicity-1 sector without (left) and with (right)
the coordinate mapping (24). The parameters are
$\gamma=2/\sqrt{3},\,\tilde{B}=109.658,\,\tilde{\mu}=7.5$ and the fluctuations
are $h_{t+},\,h_{z+},\,a_{+}$ (red, green, black).

#### Holographic massive gravity

We checked that our solutions satisfy the equations of motions and all
constraint equations. To check the convergence of the numerical solution, we
monitor the change of the solution for finer discretizations. As depicted in
the l.h.s. of figure 4, the change of the quasi-normal mode frequency decays
exponentially with a growing number of grid points; the same is valid for the
corresponding eigenfunctions. Another check for the numerical method are the
Chebychev-coefficients of the solution, displayed in the r.h.s. of figure 4;
the coefficients decay exponentially, indicating exponential accuracy of our
numerical method.

Figure 4: Left: Moving of the lowest three QNMs with increasing grid size.
Right: Chebychev-coefficients of the eigenfunctions corresponding to the
second lowest QNM.

##  Chapter 8 Holographic massive gravity

## Equations of motions

In this appendix, we provide the equations of motions for the fluctuations
(taken from my publications [2, 3]). We consider the momentum aligned in the
$y$ direction, $\vec{k}=(0,0,k)$. Throughout this appendix, derivatives of the
potential always denote derivatives with respect to the argument, i.e.
$V^{\prime}(Z)\equiv dV(Z)/dZ,\,V^{\prime\prime}(Z)\equiv d^{2}V(Z)/dZ^{2}$
and $V^{\prime}(X)\equiv dV(X)/dX,\,V^{\prime\prime}(X)\equiv
d^{2}V(X)/dX^{2}$. To simplify the notion, we suppress all arguments of the
functions.

#### Transverse sector

In the transverse sector we consider the following set of metric and scalar
fluctuations $\\{\delta\phi_{x},\,h_{tx},\,h_{xy}\\}$, where we assumed radial
gauge $h_{xu}=0$ [2].
V(Z) case

$\displaystyle
V^{\prime}\,\left(\delta\phi_{x}^{\prime}\,\left(f^{\prime}+2\,i\,\omega\right)+f\delta\phi_{x}^{\prime\prime}+h_{tx}^{\prime}\right)+4u^{3}\,V^{\prime\prime}\left(f\,\delta\phi_{x}^{\prime}+h_{tx}+i\,\omega\,\delta\phi_{x}\right)=0$
(1) $\displaystyle
h_{tx}\left(2\,u\,f^{\prime}-6f+k^{2}u^{2}+4m^{2}u^{4}\,V^{\prime}-2m^{2}\,V+6\right)$
$\displaystyle+u\left(-u\,f\,h_{tx}^{\prime\prime}+2f\,h_{tx}^{\prime}+k\,u\,\omega\,h_{xy}+4\,i\,m^{2}\,u^{3}\,\omega\,\delta\phi_{x}^{\prime}\,V^{\prime}-i\,u\,\omega\,h_{tx}^{\prime}\right)=0$
(2) $\displaystyle
u\left(h_{xy}^{\prime}\left(-u\,f^{\prime}+2\,f-2\,i\,u\,\omega\right)-ufh_{xy}^{\prime\prime}-i\,k\,u\,h_{tx}^{\prime}\right)$
$\displaystyle+2h_{xy}\,\left(u\,f^{\prime}-3f-m^{2}\,V+i\,u\,\omega+3\right)+2\,i\,k\,u\,h_{tx}=0$
(3) $\displaystyle
i\,k\,u\,h_{xy}^{\prime}-u\left(4\,m^{2}\,u^{2}\,\delta\phi_{x}^{\prime}\,V^{\prime}+h_{tx}^{\prime\prime}\right)+2h_{tx}^{\prime}=0.$
(4)

V(X) case

$\displaystyle-2(1-u^{2}\,V^{\prime\prime}/V^{\prime})h_{tx}+u\,h_{tx}^{\prime}-i\,k\,u\,h_{xy}-\left(k^{2}\,u+2\,i\,\omega(1-u^{2}\,V^{\prime\prime}/V^{\prime})\right)\,\delta\phi_{x}+u\,f\,\delta\phi_{x}^{\prime\prime}$
$\displaystyle+\left(-2(1-u^{2}\,V^{\prime\prime}/V^{\prime})\,f+u\,(2i\omega+f^{\prime})\right)\delta\phi_{x}^{\prime}\,=0\,;$
(5) $\displaystyle 2\,i\,m^{2}\,u^{2}\omega
V^{\prime}\,\delta\phi_{x}+u^{2}\,k\,\omega\,h_{xy}+(6+k^{2}\,u^{2}-2\,m^{2}(V-u^{2}\,V^{\prime})\,$
$\displaystyle-6f+2uf^{\prime})\,h_{tx}+\left(2\,u\,f-i\,u^{2}\omega\right)h_{tx}^{\prime}-u^{2}\,f\,h_{tx}^{\prime\prime}\,=\,0\,\,;$
(6) $\displaystyle
2i\,k\,u\,h_{tx}-iku^{2}h_{tx}^{\prime}-2\,i\,k\,m^{2}\,u^{2}V^{\prime}\delta\phi_{x}+2h_{xy}\left(3+i\,u\,\omega-3f+uf^{\prime}-\,m^{2}(V-u^{2}V^{\prime})\right)$
$\displaystyle-\left(2i\,u^{2}\,\omega-2uf+u^{2}\,f^{\prime}\right)h_{xy}^{\prime}-u^{2}\,f\,h_{xy}^{\prime\prime}\,=\,0\,;$
(7) $\displaystyle
2\,h_{tx}^{\prime}-u\,h_{tx}^{\prime\prime}-2m^{2}\,u\,V^{\prime}\,\delta\phi_{x}^{\prime}+ik\,u\,h_{xy}^{\prime}\,=\,0.$
(8)

#### Longitudinal sector

In the longitudinal sector we consider the following set of fluctuations in
radial gauge
$\\{h_{x,\bm{s}}=1/2\,(h_{xx}+h_{yy}),\,h_{x,\bm{a}}=1/2\,(h_{xx}-h_{yy}),\,\delta\phi_{y},\,h_{tt},\,h_{ty}\\}$.
V(Z) case

$\displaystyle
f^{\prime}\,\delta\phi_{y}^{\prime}\,V^{\prime}+f\delta\phi_{y}^{\prime\prime}\,V^{\prime}+4u^{3}\,f\delta\phi_{y}^{\prime}\,V^{\prime\prime}-\delta\phi_{y}\left(k^{2}\,V^{\prime}+2u^{3}\,\left(k^{2}\,u-2\,i\,\omega\right)V^{\prime\prime}\right)$
$\displaystyle-i\,k\,h_{xx}\,\left(2u^{4}\,V^{\prime\prime}+V^{\prime}\right)+h_{ty}^{\prime}\,V^{\prime}+2\,i\,\omega\,\delta\phi_{y}^{\prime}\,V^{\prime}+4\,u^{3}\,h_{ty}\,V^{\prime\prime}=0$
(9) $\displaystyle
2h_{ty}\left(-uf^{\prime}+3f-2m^{2}u^{4}V^{\prime}+m^{2}V-3\right)+u(ufh_{ty}^{\prime\prime}+(-2f+i\,u\,\omega)\,h_{ty}^{\prime}+i\,k\,u\,h_{tt}^{\prime}$
$\displaystyle-4\,i\,m^{2}\,u^{3}\omega\delta\phi_{y}\,V^{\prime})+k\,u^{2}\,\omega\,(h_{x,\bm{s}}+h_{x,\bm{a}})-2\,i\,k\,u\,h_{tt}=0$
(10) $\displaystyle
6\,h_{tt}+u\,\left(-u\,f^{\prime}\,h_{x,\bm{s}}^{\prime}+2\,f\,h_{x,\bm{s}}^{\prime}+4\,i\,k\,m^{2}\,u^{3}\,\delta\phi_{y}\left(2\,u^{4}\,V^{\prime\prime}+V^{\prime}\right)-i\,k\,u\,h_{ty}^{\prime}+2\,i\,k\,h_{ty}\right.$
$\displaystyle\left.-8\,m^{2}\,u^{7}\,h_{x,\bm{s}}\,V^{\prime\prime}+2\,i\,h_{x,\bm{s}}\left(\omega+2\,i\,m^{2}\,u^{3}\,V^{\prime}\right)+u\,h_{tt}^{\prime\prime}-4\,h_{tt}^{\prime}-2\,i\,u\,\omega
h_{x,\bm{s}}^{\prime}\right)=0$ (11) $\displaystyle
h_{x,\bm{s}}\,\left(2\,u\,f^{\prime}-6\,f+k^{2}\,u^{2}+4\,m^{2}\,u^{4}\,V^{\prime}-2\,m^{2}\,V+4\,i\,u\,\omega+6\right)-u^{2}\,f^{\prime}\,h_{x,\bm{s}}^{\prime}$
$\displaystyle-u^{2}\,f^{\prime}\,h_{x,\bm{a}}^{\prime}+2\,u\,h_{x,\bm{a}}\,f^{\prime}-u^{2}\,f\,h_{x,\bm{s}}^{\prime\prime}-u^{2}\,f\,h_{x,\bm{a}}^{\prime\prime}+4\,u\,f\,h_{x,\bm{s}}^{\prime}+2\,u\,f\,h_{x,\bm{a}}^{\prime}-6\
f\,h_{x,\bm{a}}$
$\displaystyle+k^{2}\,u^{2}\,h_{x,\bm{a}}-4\,i\,k\,m^{2}\,u^{4}\,\delta\phi_{y}\,V^{\prime}+2\,i\,k\,u\,h_{ty}-2\,m^{2}\,h_{x,\bm{a}}\,V-2\,i\,u^{2}\,\omega\,h_{x,\bm{s}}^{\prime}$
$\displaystyle-2\,i\,u^{2}\,\omega\,h_{x,\bm{a}}^{\prime}-2\,u\,h_{tt}^{\prime}+6\,h_{tt}+2\,i\,u\,\omega
h_{x,\bm{a}}+6\,h_{x,\bm{a}}=0$ (12) $\displaystyle
h_{x,\bm{s}}\left(2uf^{\prime}-6\,f+k^{2}\,u^{2}+4\,m^{2}\,u^{4}\,V^{\prime}-2\,m^{2}\,V+4\,i\,u\,\omega+6\right)-u^{2}f^{\prime}\,h_{x,\bm{s}}^{\prime}+u^{2}\,f^{\prime}\,h_{x,\bm{a}}^{\prime}$
$\displaystyle-2\,u\,h_{x,\bm{a}}\,f^{\prime}-u^{2}\,f\,h_{x,\bm{s}}^{\prime\prime}+u^{2}\,fh_{x,\bm{a}}^{\prime\prime}+4\,u\,f\,h_{x,\bm{s}}^{\prime}-2\,u\,f\,h_{x,\bm{a}}^{\prime}+6\,f\,h_{x,\bm{a}}+k^{2}\,u^{2}\,h_{x,\bm{a}}$
$\displaystyle-4\,i\,k\,m^{2}\,u^{4}\,\delta\phi_{y}\,V^{\prime}-2\,i\,k\,u^{2}\,h_{ty}^{\prime}+6\,i\,k\,u\,h_{ty}+2\,m^{2}\,h_{x,\bm{a}}\,V-2\,i\,u^{2}\,\omega\,h_{x,\bm{s}}^{\prime}$
$\displaystyle+2\,i\,u^{2}\,\omega\,h_{x,\bm{a}}^{\prime}-2\,u\,h_{tt}^{\prime}+6\,h_{tt}-2\,i\,u\,\omega\,h_{x,\bm{a}}-6\,h_{x,\bm{a}}=0$
(13) $\displaystyle
2\,h_{ty}^{\prime}-u\,\left(i\,k\,\left(h_{x,\bm{s}}^{\prime}+h_{x,\bm{a}}^{\prime}\right)+4\,m^{2}\,u^{2}\,\delta\phi_{y}^{\prime}\,V^{\prime}+h_{ty}^{\prime\prime}\right)=0$
(14) $\displaystyle h_{x,\bm{s}}^{\prime\prime}=0.$ (15)

V(X) case

$\displaystyle
uf^{\prime}\,\delta\phi_{y}^{\prime}\,V^{\prime}\,+2\,u^{2}\,f\,\delta\phi_{y}^{\prime}\,V^{\prime\prime}+u\,f\,\delta\phi_{y}^{\prime\prime}\,V^{\prime}-2\,f\,\delta\phi_{y}^{\prime}\,V^{\prime}-k^{2}\,u\,\delta\phi_{y}\,V^{\prime}-k^{2}\,u^{3}\,\delta\phi_{y}\,V^{\prime\prime}$
$\displaystyle+i\,k\,u\,h_{x,\bm{a}}\,V^{\prime}-i\,k\,u^{3}\,h_{x,\bm{s}}\,V^{\prime\prime}+2\,i\,u^{2}\,\omega\,\delta\phi_{y}\,V^{\prime\prime}+u\,h_{ty}^{\prime}\,V^{\prime}+2\,i\,u\,\omega\,\delta\phi_{y}^{\prime}\,V^{\prime}$
$\displaystyle-2\,i\,\omega\,\delta\phi_{y}V^{\prime}-2\,h_{ty}\left(V^{\prime}-u^{2}\,V^{\prime\prime}\right)=0$
(16) $\displaystyle
u(f\left(uf^{\prime}\,h_{x,\bm{s}}^{\prime}-2\,f\,h_{x,\bm{s}}^{\prime}-2\,i\,k\,m^{2}\,u^{3}\,\delta\phi_{y}\,V^{\prime\prime}-u\,h_{tt}^{\prime\prime}+4\,h_{tt}^{\prime}\right)$
$\displaystyle+k\,h_{ty}\left(i\,u\,f^{\prime}-2\,i\,f+2\,u\,\omega\right)+h_{x,\bm{s}}\left(2\,m^{2}\,u^{3}\,f\,V^{\prime\prime}+\omega\,\left(i\,u\,f^{\prime}-2\,i\,f+2\,u\,\omega\right)\right))$
$\displaystyle+h_{tt}\left(u\left(-uf^{\prime\prime}+4\,f^{\prime}+2\,m^{2}\,u\,V^{\prime}\right)-12\,f+k^{2}\,u^{2}-2\,m^{2}\,V-2\,i\,u\,\omega+6\right)=0$
(17) $\displaystyle
2h_{ty}\left(u\left(f^{\prime}+m^{2}\,u\,V^{\prime}\right)-3\,f-m^{2}\,V+3\right)$
$\displaystyle-u\left(u\,f\,h_{ty}^{\prime\prime}-2\,f\,h_{ty}^{\prime}+i\,k\,u\,h_{tt}^{\prime}+k\,u\,\omega\,h_{x,\bm{s}}+k\,u\,\omega\,h_{x,\bm{a}}-2\,i\,m^{2}\,u\,\omega\,\delta\phi_{y}\,V^{\prime}+i\,u\,\omega\,h_{ty}^{\prime}\right)$
$\displaystyle+2\,i\,k\,u\,h_{tt}=0$ (18) $\displaystyle
h_{x,\bm{s}}\left(2\,u\left(f^{\prime}+m^{2}\,u\,V^{\prime}\right)-6\,f+k^{2}\,u^{2}-2\,m^{2}\,V+4\,i\,u\,\omega+6\right)-u^{2}\,f^{\prime}\,h_{x,\bm{s}}^{\prime}$
$\displaystyle-u^{2}\,f^{\prime}\,h_{x,\bm{a}}^{\prime}+2\,u\,h_{x,\bm{a}}\,f^{\prime}-u^{2}\,f\,h_{x,\bm{s}}^{\prime\prime}-u^{2}\,f\,h_{x,\bm{a}}^{\prime\prime}+4\,u\,f\,h_{x,\bm{s}}^{\prime}+2\,u\,f\,h_{x,\bm{a}}^{\prime}-6\,f\,h_{x,\bm{a}}$
$\displaystyle+k^{2}\,u^{2}\,h_{x,\bm{a}}+2\,i\,k\,u\,h_{ty}+2\,m^{2}\,u^{2}\,h_{x,\bm{a}}\,V^{\prime}-2\,m^{2}\,h_{x,\bm{a}}\,V-2\,i\,u^{2}\,\omega\,h_{x,\bm{s}}^{\prime}$
$\displaystyle-2\,i\,u^{2}\,\omega\,h_{x,\bm{a}}^{\prime}-2\,u\,h_{tt}^{\prime}+6h_{tt}(u)+2\,i\,u\,\omega\,h_{x,\bm{a}}+6h_{x,\bm{a}}=0$
(19) $\displaystyle
h_{x,\bm{s}}\left(2\,u\left(f^{\prime}+m^{2}\,u\,V^{\prime}\right)-6\,f+k^{2}\,u^{2}-2\,m^{2}\,V+4\,i\,u\,\omega+6\right)-u^{2}\,f^{\prime}\,h_{x,\bm{s}}^{\prime}+u^{2}\,f^{\prime}\,h_{x,\bm{a}}^{\prime}$
$\displaystyle-2\,u\,h_{x,\bm{a}}\,f^{\prime}-u^{2}\,f\,h_{x,\bm{s}}^{\prime\prime}+u^{2}\,f\,h_{x,\bm{a}}^{\prime\prime}+4\,u\,f\,h_{x,\bm{s}}^{\prime}-2\,u\,fh_{x,\bm{a}}^{\prime}+6\,f\,h_{x,\bm{a}}+k^{2}\,u^{2}\,h_{x,\bm{a}}$
$\displaystyle-4\,i\,k\,m^{2}\,u^{2}\,\delta\phi_{y}\,V^{\prime}-2\,i\,k\,u^{2}\,h_{ty}^{\prime}+6\,i\,k\,u\,h_{ty}-2\,m^{2}\,u^{2}\,h_{x,\bm{a}}\,V^{\prime}+2\,m^{2}\,h_{x,\bm{a}}\,V$
$\displaystyle-2\,i\,u^{2}\,\omega\,h_{x,\bm{s}}^{\prime}+2\,i\,u^{2}\,\omega\,h_{x,\bm{a}}^{\prime}-2\,u\,h_{tt}^{\prime}+6\,h_{tt}-2\,i\,u\,\omega\,h_{x,\bm{a}}-6\,h_{x,\bm{a}}=0$
(20)
$\displaystyle-6\,h_{tt}+u\,(u\,f^{\prime}\,h_{x,\bm{s}}^{\prime}-2\,f\,h_{x,\bm{s}}^{\prime}-2\,i\,k\,m^{2}\,u^{3}\,\delta\phi_{y}\,V^{\prime\prime}+i\,k\,u\,h_{ty}^{\prime}-2\,i\,k\,h_{ty}$
$\displaystyle+2\,m^{2}\,u^{3}\,h_{x,\bm{s}}\,V^{\prime\prime}-u\,h_{tt}^{\prime\prime}+4\,h_{tt}^{\prime}+2\,i\,u\,\omega\,h_{x,\bm{s}}^{\prime}-2\,i\,\omega\,h_{x,\bm{s}})=0$
(21) $\displaystyle
k\,u\left(h_{x,\bm{s}}^{\prime}+h_{x,\bm{a}}^{\prime}\right)-i\,u\,\left(2\,m^{2}\,\delta\phi_{y}^{\prime}\,V^{\prime}+h_{ty}^{\prime\prime}\right)+2\,i\,h_{ty}^{\prime}=0$
(22) $\displaystyle h_{x,\bm{s}}^{\prime\prime}=0.$ (23)

##### Time dependent bulk equations

Using the characteristic formulation the equations of motion are,

$\displaystyle
S{}^{{}^{\prime\prime}}+\frac{2}{z}S{}^{{}^{\prime}}+\frac{{H{}^{{}^{\prime}}}^{2}}{4}S=0,$
(24)
$\displaystyle{d_{+}S}{}^{{}^{\prime}}+\frac{S{}^{{}^{\prime}}}{S}{d_{+}S}=\frac{S}{2z^{2}}\left(-3+m^{2}2^{N}\left(\frac{\cosh(H)}{S^{2}}\right)^{N}\right),$
(25)
$\displaystyle{d_{+}H}{}^{{}^{\prime}}+\frac{S^{{}^{\prime}}}{S}d_{+}H=-\frac{H{}^{{}^{\prime}}{d_{+}S}}{S}-m^{2}\frac{2^{N}N\tanh(H){\cosh^{N}(H)}}{z^{2}{S^{2N}}},$
(26) $\displaystyle
A{}^{{}^{\prime\prime}}+\frac{2}{z}A{}^{{}^{\prime}}=\frac{{d_{+}H}H{}^{{}^{\prime}}}{z^{2}}-\frac{4S{}^{{}^{\prime}}{d_{+}S}}{z^{2}S^{2}}+m^{2}\frac{2^{N+1}N\cosh^{N}(H)}{z^{4}S^{2N}},$
(27) $\displaystyle
4{d_{+}^{2}S}+{d_{+}H}^{2}S+2z^{2}A{}^{{}^{\prime}}d_{+}S=0,$ (28)

where
$d_{+}\mathcal{A}:=\dot{\mathcal{A}}-\frac{A}{z^{2}}\mathcal{A}^{{}^{\prime}}$,
and prime stands for derivative with respect to radial coordinate $z$ and use
dot for derivative with respect to time $t$. Note that
$X={2\cosh(H)}/{S^{2}}$.

We make the following re-definitions of the metric functions

$\displaystyle A=\frac{1+\tilde{A}\ u}{u^{2}},\quad S=\frac{s_{0}+\tilde{S}\
u}{u},\quad H=h_{0}+\tilde{H}\ u,\quad
d_{+}S=\frac{\widetilde{d_{+}S}}{u^{2}},\quad
d_{+}H=\frac{\dot{h_{0}}}{2}+\widetilde{d_{+}H}\ u.$

We find the following near boundary expansion

$\displaystyle A=\frac{1}{u^{2}}+\frac{2(s_{1}-\dot{s_{0}})}{s_{0}\
u}+\left(\frac{s_{1}^{2}}{s_{0}^{2}}-\frac{2\dot{s_{1}}}{s_{0}}-\frac{3\dot{h_{0}}^{2}}{4}\right)+a_{3}u+\mathcal{O}(u^{2}),$
$\displaystyle
S=\frac{s_{0}}{u}+s_{1}-\frac{s_{0}\dot{h_{0}}^{2}}{8}u+\frac{s_{1}\dot{h_{0}}^{2}}{8}u^{2}+\mathcal{O}(u^{3}),\quad
H=h_{0}+\dot{h_{0}}u-\frac{s_{1}\dot{h_{0}}}{s_{0}}u^{2}+h_{3}u^{3}+\mathcal{O}(z^{4}).$

accompanied by the Ward identity,

$\displaystyle\dot{a_{3}}+\frac{3a_{3}\dot{s_{0}}}{s_{0}}-\dot{h_{0}}{}^{2}\left(\frac{\dot{s_{0}}{}^{2}-3s_{1}{}^{2}}{2s_{0}{}^{2}}+\frac{\ddot{s_{0}}}{2s_{0}}\right)-\frac{3\dot{h_{0}}\ddot{h_{0}}\dot{s_{0}}}{2s_{0}}+\frac{3}{8}\dot{h_{0}}{}^{4}-\frac{3}{2}h_{3}\dot{h_{0}}-\frac{1}{2}\dddot{h_{0}}\dot{h_{0}}=0.$

Using standard holographic renormalization, the boundary stress tensor reads

$\displaystyle T_{tt}=-a_{3},$ (30) $\displaystyle
T_{xx}=-\frac{{a_{3}}}{2}-\frac{3\gamma{s_{1}}^{2}\dot{\gamma}}{2\left(1-\gamma^{2}\right)^{3/2}}+\frac{3h_{3}}{2}-\frac{\gamma\left(4\left(\gamma^{2}-1\right)^{2}\dddot{\gamma}+3\left(4\gamma^{2}+1\right)\dot{\gamma}^{3}-16\gamma\left(\gamma^{2}-1\right)\dot{\gamma}\ddot{\gamma}\right)}{8\left(\gamma^{2}-1\right)^{3}}\gamma,$
$\displaystyle
T_{xy}=-\frac{a_{3}}{2}\gamma\\!-\\!\frac{3{s_{1}}^{2}\dot{\gamma}}{2\left(1-\gamma^{2}\right)^{3/2}}\\!+\\!\frac{3{h_{3}}}{2}\\!-\\!\frac{4\left(\gamma^{2}-1\right)^{2}\dddot{\gamma}+3\left(4\gamma^{2}+1\right)\dot{\gamma}^{3}-16\gamma\left(\gamma^{2}-1\right)\dot{\gamma}\ddot{\gamma}}{8\left(\gamma^{2}-1\right)^{3}}.$

The shear stress to the strain $h_{xy}(t)=\gamma(t)$ is given by the
expectation value of $T_{xy}(t)$.

##  Chapter 9 Entanglement entropy and $T\bar{T}$

In this section, we show the entanglement entropy calculation and the
comparison to holography for general dimensions. This appendix is taken from
my publication [7].

#### $d=2$

We have

$\omega_{2}=\frac{1+\bm{s}\sqrt{\eta+\frac{c\,\lambda_{2}}{3\pi\,R^{2}}}}{4\,\lambda_{2}}$
(1)

where we denote the sign of the square root of the $T\bar{T}$ deformation by
$\bm{s}$. With $\omega_{2}$ at hand, we may compute the partition function of
the deformed CFT using eq. (55). As argued in the previous section, we choose
the integration constant so that $\log Z_{S^{2}}(R=0)=0$. This yields

$\log
Z_{S^{2}}=-\frac{1}{3\lambda_{2}}\left(c\bm{s}\,\operatorname{arcsin(h)}\\!\left(\frac{\sqrt{3\,\pi}\,R}{\sqrt{c\,\lambda_{2}}}\right)\lambda_{2}+\eta
R\left(3R\pi+\bm{s}\,\sqrt{3\pi}\sqrt{\eta\,3R^{2}\,\pi+c\,\lambda_{2}}\right)\right).$
(2)

The entanglement entropy for two antipodal points on the sphere follows from
the partition function via eq. (56) (for the negative sign of the square root)

$S_{EE}=\frac{c}{3}\,\operatorname{arcsin(h)}\left(\frac{\sqrt{3\,\pi}\,R}{\sqrt{c\,\lambda_{2}}}\right).$
(3)

In two dimensions, we may calculate the cutoff independent renormalized
entanglement entropy immediately from the knowledge of the derivative of the
partition function. Plugging (1) into eq. (55) and combining eq. (56) and
(65), we find the renormalized entanglement entropy which plays the role of
the running $\mathcal{C}$-function in RG flow as

$S_{\text{R,EE}}=c\left(9\,\eta+\frac{3\,c\,\lambda_{2}}{R^{2}\,\pi}\right)^{-1/2}.$
(4)

##### Comparison to the result from holography

The entanglement entropy from holography is given by eq. (35). In order to
compare to the field theory results, we use the dictionary relating the
holography parameters with the field theory ones. This is done by $4\pi
l/\ell_{p}=c/3$ and $c\,\lambda_{2}=3\,\pi\,L^{2}$

$S_{\text{EE}}=\frac{c}{3}\,\operatorname{arcsin(h)}\left(\frac{\sqrt{3\,\pi}\,R}{\sqrt{c\,\lambda_{2}}}\right),$
(5)

with the corresponding renormalized entanglement given by the $R$-derivative
$\mathcal{C}=dS/dR$

$S_{\text{R,EE}}=c\left(9\,\eta+\frac{3\,c\,\lambda_{2}}{R^{2}\,\pi}\right)^{-1/2}.$
(6)

In $d=2$ dimensions, we find the EE from field theory matches the entanglement
entropy from holography exactly for $\eta=1,\bm{s}=-1$ (AdS) and
$\eta=-1,\bm{s}=-1$ (dS).

#### $d=3$

In higher dimensions, the computation is very similar to $d=2$ and we only
display the relevant steps; we may read off $\omega_{3}$ from eq. (59)

$\omega_{3}=\frac{R^{2}+3\,t_{3}+R\,\bm{s}\,\sqrt{\eta\,R^{2}+6\,t_{3}}}{3\,R^{2}\lambda_{3}}.$
(7)

It is straightforward to determine the corresponding partition function, given
by

$\log
Z_{S^{3}}=-\frac{2\pi^{2}\,(R^{3}+9\,R\,t_{3}+\eta\bm{s}\,(\eta\,R^{2}+6\,t_{3})^{3/2})}{3\,\lambda_{3}}+\eta\,\bm{s}\frac{4\sqrt{6}\,\pi^{2}\,t^{3/2}}{3\,\lambda_{3}}.$
(8)

The second term is chosen to ensure $\log Z(R=0)=0$. Finally, we find with
$\eta^{2}=1$ and eq. (56) and the negative sign of the square root

$S_{EE}=\frac{4\,\pi^{2}\,t_{3}}{\lambda_{3}}\left(-R-\eta\,\sqrt{6t_{3}}+\eta\,\sqrt{\eta\,R^{2}+6\,t_{3}}\right).$
(9)

The scheme independent renormalized entanglement entropy is obtained from the
entanglement entropy by using (65) and reads in $d=3$ dimensions

$S_{\text{R,EE}}=\frac{4\,\pi^{2}\,\eta\,t_{3}^{3/2}}{\lambda_{3}}\left(\sqrt{6}-\frac{6\sqrt{t_{3}}}{\sqrt{\eta\,R^{2}+6\,t_{3}}}\right).$
(10)

##### Comparison to the result from holography

The entanglement entropy from holography is given by eq. (36) and may be
expressed in terms of field theory quantities using eq. (54) (with
$6\,\lambda_{3}=\ell_{p}^{2}\,\sqrt{6\,t_{3}},\,L=\sqrt{6\,t_{3}}$)

$S_{EE}=\frac{4\,\pi^{2}\,t_{3}}{\lambda_{3}}\,\eta\,\left(-\sqrt{6\,t_{3}}+\sqrt{6\,t_{3}+\eta\,R^{2}}\right).$
(11)

We see that the field theory calculation and the results from holography match
up to a scheme dependent area term $\sim-4\,t_{3}\pi^{2}\,R/\lambda_{3}$. We
obtain the exact same contribution from the Wald entropy associated with the
counterterms given in eq. (49) which yields (in field theory variables)
exactly $\sim-4\,t_{3}\pi^{2}\,R/\lambda_{3}$. The entanglement entropies on
both sides match, if the contributions of the counterterms – which have been
added to the field theory side – are also taken into account in the
gravitational theory. Similar to the literature, we may compare scheme
independent quantities aka the renormalized entanglement entropy. From the
entanglement entropy, we immediately obtain the renormalized entanglement
entropy by using eq. (65)

$S_{\text{R,EE}}=\frac{4\,\pi^{2}\,\eta\,t_{3}^{3/2}}{\lambda_{3}}\left(\sqrt{6}-\frac{6\,\sqrt{t_{3}}}{\sqrt{\eta
R^{2}+6\,t_{3}}}\right).$ (12)

We see that the results from holography and field theory perfectly match one
another for the negative sign of the square root $\eta=1,\bm{s}=-1$ (AdS) and
$\eta=-1,\bm{s}=-1$ (dS).

#### $d=4$

In $d=4$ we have using eq. (59)

$\omega_{4}=\frac{3\,\left(R^{2}+8\,t_{6}+R\,\bm{s}\sqrt{\eta\,R^{2}+16\,t_{4}}\right)}{8\,R^{2}\,\lambda_{4}}.$
(13)

We can compute the sphere partition function by integrating with respect to
$R$, where we fix the integration constant by demanding that $\log
Z_{S^{d}}(R=0)=0$

$\displaystyle\log Z_{S^{4}}=$
$\displaystyle-\frac{\pi^{2}}{\lambda_{4}}\left(R\left(R^{3}+16\,R\,t_{4}+R^{2}\,\bm{s}\,\sqrt{\eta\,R^{2}+16\,t_{4}}+\eta\,8\,\bm{s}\,t_{4}\sqrt{\eta\,R^{2}+16\,t_{4}}\right)\right.$
$\displaystyle\left.-128\,\eta\,\bm{s}\,t_{4}^{2}\,\operatorname{arcsin(h)}\left(\frac{R}{4\,\sqrt{t_{4}}}\right)\right).$
(14)

We obtain the entanglement entropy by using the replica trick (56). This gives
us

$S_{4,\text{EE}}=\frac{8\pi^{2}\,t_{4}}{\lambda_{4}}\left(R\left(-R+\eta\,\sqrt{\eta\,R^{2}+16\,t_{4}}\right)-16\,\eta\,t_{4}\,\arcsin(h)\left(\frac{R}{4\,\sqrt{t_{4}}}\right)\right).$
(15)

In $d=4$ dimensions, the renormalized entanglement entropy follows from eq.
(15) with eq. (65)

$S_{\text{R,EE}}=\frac{128\,\pi^{2}\,R^{3}\,t_{4}^{2}}{\lambda_{4}\left(\eta\,R^{2}+16t_{4}\right)^{3/2}}.$
(16)

##### Comparison to the result from holography

In holography, the entanglement entropy in $d=4$ is given by eq. (37) which
reads in field theory quantities by relating
$8\,\lambda_{4}=\ell_{p}^{3}\,\sqrt{16\,t_{4}},\,L=\sqrt{16\,t_{4}}$

$S_{\text{EE}}=\frac{8\,\pi^{2}\,t_{4}}{\lambda_{4}}\,\eta\,\left(R\sqrt{\eta\,R^{2}+16\,t_{4}}-16\,t_{4}\,\operatorname{arcsin(h)}\frac{R}{4\,\sqrt{t_{4}}}\right).$
(17)

Again, this matches exactly our field theory computation up to a scheme
dependent area term $-8\pi^{2}R^{2}t_{4}/\lambda_{4}$ for the negative sign of
the the square root. The area term with the negative sign comes from adding
counterterms to our action. If we also consider the contributions of the
counterterms in the gravitational theory eq. (49), we see that we observe the
exact same term there and thus the results of both sides match. From the
holographic entanglement entropy, we may derive the scheme independent
entanglement entropy using eq. (65)

$S_{\text{R,EE}}=\frac{128\pi^{2}R^{3}t_{4}^{2}}{\lambda_{4}\left(\eta\,R^{2}+16t_{4}\right){}^{3/2}}$
(18)

We see that the renormalized entanglement entropies from field theory and
holography in $d=4$ match perfectly for $\eta=1,\bm{s}=-1$ (AdS) and
$\eta=-1,\bm{s}=-1$ (dS).

#### $d=5$

In $d=5$, the counterterm proportional to $c_{d}^{(3)}$ contributes for the
first time. We find $\omega_{5}$ from eq. (59)

$\omega_{5}=\frac{30\,R^{2}\,t_{5}-225\,t_{5}^{2}\,R^{3}\left(R+\bm{s}\sqrt{\eta
R^{2}+30\,t_{5}}\right)}{5\,R^{4}\,\lambda_{5}}.$ (19)

With $\omega_{5}$, we may compute the partition function by integrating eq.
(55) with respect to $R$ which results in

$\displaystyle\log Z_{S^{5}}=$
$\displaystyle-\frac{\pi^{3}}{5\,\lambda}\,\left(20\,\eta\,R^{2}\,\bm{s}\,t_{5}\sqrt{\eta\,R^{2}+30\,t_{5}}+1200\,\bm{s}\,t_{5}^{2}\left(\sqrt{30\,t_{5}}-\sqrt{\eta\,R^{2}+30\,t_{5}}\right)\right.$
$\displaystyle\left.+\left(2\,R^{5}+50\,R^{3}\,t_{5}-1125\,R\,t_{5}^{2}+2\,R^{4}\,\bm{s}\,\sqrt{\eta\,R^{2}+30\,t_{5}}\right)\right),$
(20)

where we fixed the integration constant so that $\log Z_{S^{5}}(R=0)=0$. The
entanglement entropy follows from the partition function using eq. (56)

$\displaystyle
S_{5,\text{EE}}=\frac{4\pi^{3}\,t_{5}}{\lambda_{5}}\left(-R^{3}\,+45\,R\,t_{5}+\eta\,R^{2}\,\sqrt{\eta\,R^{2}+30\,t_{5}}+60\,t_{5}\left(\sqrt{30\,t_{5}}-\sqrt{\eta\,R^{2}+30\,t_{5}}\right)\right).$
(21)

In $d=5$ dimensions, we may compute the renormalized entanglement entropy
using (65)

$S_{\text{R,EE}}=-\frac{240\,\pi^{3}\,t_{5}^{5/2}\left(900\,t_{5}^{3/2}\\!\\!-30t_{5}\,\sqrt{900\,t_{5}+30\,\eta\,R^{2}}+\eta\,R^{2}\\!\left(45\,\sqrt{t_{5}}-\sqrt{900t_{5}+\eta\,R^{2}}\right)\right)}{\lambda_{5}\left(\eta\,R^{2}+30t_{5}\right)^{3/2}}.$
(22)

##### Comparison to the result from holography

To compare with the field theory result, we rewrite the result from holography
(38) with the dictionary
$10\lambda_{d}=\ell_{p}^{4}\,\sqrt{30\,t_{5}},\,L=\sqrt{30\,t_{5}}$ in field
theory quantities

$S_{\text{EE}}=\frac{4\,\pi^{3}\,t_{5}}{\lambda_{5}}\,\left(2\,(30\,t_{5})^{3/2}+(\eta\,R^{2}-60\,t_{5})\,\sqrt{30\,t_{5}+\eta\,R^{2}}\right).$
(23)

We see that the result from holography matches the calculation from field
theory up to the scheme dependent terms $\sim
4\pi^{3}\,t_{5}/\lambda_{5}\,(-R^{3}\,+45\,R\,t_{5})$. However, taking the
contributions of the counterterms in the gravitational theory into account, we
see find the exact same contribution to the entanglement entropy as observed
in eq. (50). The results of both sides hence match. For the sake of
completeness, we calculate the renormalized entanglement entropy by using eq.
(65)

$S_{\text{R,EE}}=\frac{240\pi^{3}t_{5}^{2}}{\lambda_{5}\left(\eta\,R^{2}+30\,t_{5}\right){}^{3/2}}\left(\sqrt{30\,t_{5}}\,\sqrt{\eta\,R^{2}+30\,t_{5}}\left(30\,t_{5}+\eta\,R^{2}\right)-45\eta
R^{2}t_{5}-900t_{5}^{2}\right),$ (24)

we see that the scheme dependent terms vanish and the results from field
theory and holography agree perfectly for $\eta=1,\bm{s}=-1$ (AdS) and
$\eta=-1,\bm{s}=-1$ (dS).

#### $d=6$

In $d=6$, $\omega_{6}$ is given by eq. (59)

$\omega_{6}=\frac{5\left(R^{4}+24\,R^{2}\,t_{6}-288\,t_{6}^{2}+R^{3}\,\bm{s}\,\sqrt{\eta\,R^{2}+48\,t_{6}}\right)}{12\,R^{4}\,\lambda_{6}}.$
(25)

The partition function follows by inserting eq. (25) into eq. (55) and
integrating with respect to $R$

$\displaystyle\log Z_{S^{6}}=$
$\displaystyle-\frac{4\pi^{3}}{9\lambda_{6}}\left(R\left(R^{5}+36\,R^{3}\,t_{6}+R^{4}\,\bm{s}\sqrt{\eta\,R^{2}+48\,t_{6}}-864\,\bm{s}\,t_{6}^{2}\,\sqrt{-\eta\,R^{2}+48\,t_{6}}\right)\right.$
$\displaystyle\left.-864\,R\,t_{6}^{2}+\eta\,12\,R^{2}\,\bm{s}\,t_{6}\,\sqrt{\eta\,R^{2}+48\,t_{6}}+41472\,\bm{s}\,t_{6}^{3}\,\operatorname{arcsin(h)}\left(\frac{R}{4\sqrt{3\,t_{6}}}\right)\right),$

where we chose the integration constant so that $\log Z_{S^{6}}(R=0)=0$. The
entanglement follows from the partition function by eq. (56)

$\displaystyle S_{EE}=$
$\displaystyle-\frac{16\,\pi^{3}\,t_{6}}{3\lambda_{6}}\left(R\left(R^{3}-48\,R\,t_{6}-\eta\,R^{2}\sqrt{\eta\,R^{2}+48\,t_{6}}+72\,t_{6}\,\sqrt{\eta\,R^{2}+48\,t_{6}}\right)\right.$
$\displaystyle\left.-3456\,t_{6}^{2}\,\operatorname{arcsin(h)}\left(\frac{R}{4\,\sqrt{3\,t_{6}}}\right)\right).$
(26)

In $d=6$ dimensions, the renormalized entanglement entropy reads (using eq.
(65))

$S_{\text{R,EE}}=\frac{18432\,\pi^{3}\,R^{5}\,t_{6}^{3}}{\lambda\left(\eta\,R^{2}+48t_{6}\right)^{5/2}}.$
(27)

##### Comparison to the result from holography

The entanglement entropy from holography (39) reads in $d=6$ in field theory
quantities $12\,\lambda_{6}=\ell_{p}^{5}\,\sqrt{48\,t_{6}}$ and
$L=\sqrt{48\,t_{6}}$

$S_{\text{EE}}=\frac{8\,\pi^{3}\,t_{6}}{3\,\lambda_{6}}\left(R\,\sqrt{48\,t_{6}+\eta\,R^{2}}\,(2\,\eta\,R^{2}-144\,t_{6})+6912t_{6}^{2}\,\operatorname{arcsin(h)}\left(\frac{R}{4\,\sqrt{3\,t_{6}}}\right)\right)\\!.$
(28)

The entanglement entropy from field theory matches the result from holography
up to the usual area term $\sim 16\,\pi^{3}\,t_{6}\,R^{4}/(3\lambda_{6})$ and
a scheme dependent term $\sim 256\,\pi^{3}\,R^{2}t_{6}^{2}/\lambda_{6}$. The
exact same terms arise in the gravitational theory too if we also take the
counterterms into account there. The contributions are calculated in eq. (50)
and are an exact match to the missing terms. We thus conclude that the
entanglement entropies of both sides match. For comparison with similar
results in the literature, we are looking at the renormalized entanglement
entropy in $d=6$. We find a perfect match between field theory and the result
from holography given by

$S_{\text{R,EE}}=\frac{18432\,\pi^{3}\,R^{5}\,t_{6}^{3}}{\lambda_{6}\,\left(\eta\,R^{2}+48\,t_{6}\right){}^{5/2}},$
(29)

for $\eta=1,\bm{s}=-1$ (AdS) and $\eta=-1,\bm{s}=-1$ (dS).

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## Acknowledgments

In the first place, I would like to express my gratitude to my advisor Martin
Ammon for his continuous support and over many years (starting with my
Bachelor thesis). I am deeply grateful for the countless hours of (patient)