where the $1/\sqrt{1-v^{2}}\equiv\gamma$ is identified as the Lorentz factor
from special relativity. Eq.(6.26) holds regardless of the dimension of the
spacetime. Inserting Eq.(6.26) into Eq.(6.25) yields

$\frac{dp_{2}}{dt}\,=\,-\frac{\pi\,T^{2}}{2}\,\sqrt{g_{YM}^{2}N_{c}}\,\frac{p_{2}(t)}{m}\,,$
(6.27)

which is a linearly separable differential equation and easily integrable to
find $p_{2}(t)$. Specifically,

$p_{2}(t)\,=\,p_{2}(0)\,e^{-\,\frac{t}{t_{0}}}\,\,,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{where}\,\,\,\,\,\,\,\,\,t_{0}\,=\,\frac{2}{\pi\,\sqrt{g_{YM}^{2}N_{c}}}\,\frac{m}{T^{2}}\,.$
(6.28)

The expression for the drag force experienced by a probe quark in a thermal
plasma found in Eq.(6.27) agrees with the central result of Gubser [22], and
Herzog et al. [35]. Agreement with the later rests on the identification
$\xi^{\prime}=x_{\text{Herzog}}^{\prime}/l^{2}$ and
$g=-g_{\text{Herzog}}/l^{4}$.

Recall the Langevin Model discussed in section (3) describes a non-
relativistic particle of mass $m$, undergoing Brownian motion in one spatial
dimension. The non-retarded Langevin equation Eq.(3.1) is applicable if – as
in the case presented in this section – the Brownian particle is taken to have
infinite mass with respect to the constituent fluid particles. Since the
infinitely massive probe quark travels at a constant velocity $v$ under the
influence of the external force $K(t)$, Newton’s First Law of Motion ensures
that the random force $F(t)$ present in the system must be of equal and
opposite magnitude ($K(t)=-F(t)$). The Langevin equation is adjusted
accordingly and becomes $\dot{p}(t)=-\gamma_{0}\,p(t)$, where $\gamma_{0}$ is
the friction coefficient. It is therefore apparent, from Eq.(6.27), that in
the non-relativistic limit $v\ll 1$ the friction coefficient in
AdS5-Schwarzschild is given by

$\gamma_{0}^{\text{AdS${}_{5}$}}\,=\,\frac{\pi\,T^{2}}{2m}\,\sqrt{g_{YM}^{2}N_{c}}\,.$
(6.29)

Using the Einstein-Sutherland relation Eq.(3.18) an expression for the
diffusion coefficient can be obtained. Specifically,

$D_{\text{HQ}}^{\text{AdS${}_{5}$}}\,=\,\frac{2}{\pi\,T}\,\frac{1}{\sqrt{g_{YM}^{2}N_{c}}}\,=\,\frac{2\,\beta}{\pi\sqrt{\lambda}}\,,$
(6.30)

where the second equality follows from the ‘t Hooft coupling
$\lambda\,\equiv\,g_{\text{YM}}^{2}\,N_{c}$, and the definition of Hawking
temperature Eq.(4.36).

####  6.2 Generalising to AdSd-Schwarzschild

The result for the drag force experienced by a probe quark in a thermal plasma
found in Eq.(6.27) can be generalised by repeating the calculation presented
in subsection (6.1) in AdSd-Schwarzschild. The AdS-Schwarzschild metric in $d$
dimensions is given by Eq.(4.34), where the blackening factor of the
Schwarzschild black hole situated at the horizon, $h(r;d)$, is defined in
Eq.(4.35). Explicitly, the target spacetime metric $G_{\mu\nu}$ is given by
Eq.(C.2), where $\mu,\nu$ index over the spacetime directions
$\left(t,\,r,\,2,\,...,\,d-1\right)$. As in subsection (6.1), the
reparameterization offered by the gauge choice separates the temporal and
radial coordinates such that the worldsheet parameter space coordinates become
$\tau=t$ and $\sigma=r$, and the mapping functions between the worldsheet and
the target spacetime are specified by $X^{\mu}(t,r)$.

The Hawking temperature of the black-brane in AdS-Schwarzschild scales with
the number of dimensions of the spacetime and – in AdSd-Schwarzschild – is
given by Eq.(4.36). Unchanged as the calculation is generalised to AdSd, the
dynamics of the string are still described by the Nambu-Goto action
(Eq.(4.54)); and the induced metric $g_{ab}$, its inverse $g^{ab}$, and its
determinant $g:=\text{det}(g_{ab})$ are still given by Eqs.(C.4, C.5, C.6)
respectively. Because of this, Eqs.(6.2)-(6.10) all remain true in AdSd-
Schwarzschild.

Consider movement of the test string in AdSd-Schwarzschild, again with a
constant velocity $v$ in one of the transverse directions, the $X^{2}$
direction. Following the same reasoning as for the AdS5 case, Eq.(6.10) can be
solved to yield an expression for $\pi_{\xi}$. Since the blackening factor is
now given by Eq.(4.35), $\pi_{\xi}$ will depend on the number of dimensions of
the spacetime. Rearranging the first equality in Eq.(6.11) leaves

$\pi_{\xi}^{2}\,=\,\frac{r^{4}}{l^{4}}\,v^{2}\,.$ (6.31)

While, for AdSd-Schwarzschild, the second equality in Eq.(6.11) is rearranged
to

$r\,=\,\frac{r_{H}}{(1\,-\,v^{2})^{1/(d-1)}}\,,$ (6.32)

where $h=h(r;d)$ (Eq.(4.35)) is used. Inputting Eq.(6.32) into Eq.(6.31)
results in an appropriate expression for $\pi_{\xi}$,

$\pi_{\xi}\,=\,\pm\,\frac{v}{(1\,-\,v^{2})^{2/(d-1)}}\,\frac{r_{H}^{2}}{l^{2}}\,.$
(6.33)

Plugging this generalised relation for $\pi_{\xi}$ into Eq.(6.10) guarantees
that the square root in Eq.(6.10) is always positive, and $\xi(r)$ is
everywhere real. The $(\pm)$ in Eq.(6.33) determines if the quark is moving in
the $(+)$ or $(-)$ $X^{2}$ direction. Again consider the case where
$\pi_{\xi}>0$. In AdSd-Schwarzschild, Eq.(6.10) becomes121121121Mathematica is
used to help simplify the algebra (see Mathematica Notebook [d]:
DragForce.nb).

$\xi^{\prime}(r)\,=\,\pm\,\frac{l^{2}\,r_{H}^{3}\,r^{d-4}\,v\,\left(1-v^{2}\right)^{-\frac{2}{d-1}}\,\sqrt{\frac{r^{4}\,\left(r_{H}\,\left(v^{2}-1\right)\,r^{d}\,+\,r\,r_{H}^{d}\right)}{-r_{H}\,r^{d+4}\,+\,r_{H}^{5}v^{2}\,r^{d}\,\left(1-v^{2}\right)^{-\frac{4}{d-1}}+r^{5}r_{H}^{d}}}}{r_{H}\,r^{d}\,-\,r\,r_{H}^{d}}\,,$
(6.34)

which reduces down to the familiar Eq.(6.15) for $d=5$. The function
$\xi^{\prime}(r)$ can take on two values: (i) for
$(\xi^{\prime}(r)>0,\,\xi(r)<0)$ the string trails behind its boundary
endpoint which is moving with a velocity $v$ in the $(+)$ $X^{2}$ direction,
and (ii) for $(\xi^{\prime}(r)<0,\,\xi(r)>0)$ the string trails in front of
its boundary endpoint which is moving with a velocity $v$ in the $(+)$ $X^{2}$
direction. Using the expression for $\xi^{\prime}(r)$ in Eq.(6.34) and the
definition for $h(r;d)$ in Eq.(4.35), the Lagrangian density (Eq.(6.6))
becomes

$\mathcal{L}\,:=\,-\sqrt{-g}\,=\,-\,\sqrt{\frac{r^{4}\left(r_{H}\left(v^{2}-1\right)r^{d}\,+\,r\,r_{H}^{d}\right)}{-r_{H}\,r^{d+4}\,+\,r_{H}^{5}\,v^{2}\,r^{d}\,\left(1-v^{2}\right)^{-\frac{4}{d-1}}\,+\,r^{5}\,r_{H}^{d}}}\,.$
(6.35)

To proceed with calculating the drag force $dp_{2}/dt$ note that the canonical
momentum densities $\Pi^{a}_{\mu}(t,\sigma)$ (Eq.(4.7)), the string equations
of motion (Eq.(4.20)), and the definition of the conserved charges associated
with the momentum densities Eq.(6.19) all remain the same in AdSd-
Schwarzschild. As in the $d=5$ case, the string moves only in the $X^{2}$
direction by construction, and its transverse momentum densities are only non-
zero in this direction. The $X^{2}$ component of the spacetime momentum
flowing over some general time interval $\mathcal{I}$ of length $\Delta t$, is
still given by Eq.(6.20) in AdSd-Schwarzschild. For the physical solution
where the string trails behind its boundary endpoint
$(\xi^{\prime}(r)>0,\,\xi(r)<0)$, $dp_{2}/dt=\Pi^{r}_{2}$. According to the
definition of the momentum densities Eq.(4.7),

$\displaystyle\begin{aligned}
\frac{dp_{2}}{dt}\,:=\,\Pi^{r}_{2}\,&=\,-\frac{1}{2\pi\alpha^{\prime}}\,\sqrt{-g}\,g^{rb}\,G_{2\,\nu}\,{\partial_{b}}X^{\nu}\\\\[5.69046pt]
&=\,\frac{1}{2\pi\alpha^{\prime}}\left(\sqrt{\frac{r^{4}\left(r_{H}\left(v^{2}-1\right)r^{d}\,+\,r\,r_{H}^{d}\right)}{-r_{H}\,r^{d+4}\,+\,r_{H}^{5}\,v^{2}\,r^{d}\,\left(1-v^{2}\right)^{-\frac{4}{d-1}}\,+\,r^{5}\,r_{H}^{d}}}\right)^{-\frac{1}{2}}G_{2\,2}\,G_{tt}\,\partial_{r}\xi\\\\[5.69046pt]
&=\,-\frac{1}{2\,\pi\,\alpha^{\prime}}\,\frac{r_{H}^{2}}{l^{2}}\,\frac{v}{\left(1-v^{2}\right)^{\frac{2}{d-1}}}\,,\end{aligned}$
(6.36)

where the constraint equation Eq.(4.12) is used in the first line; while in
the second line, the summation over repeated indices is expanded and Eq.(6.35)
is used. Following a similar calculation to Eq.(6.22), recognize that
$G_{2\,\nu}=G_{2\,2}$ is the only non-zero possibility, Eq.(C.5) can be used
to find the explicit entries of the inverse induced metric $g^{ab}$ and the
ansatz Eq.(6.5) can be used to simplify the derivative terms. The final line
follows from inputting the target spacetime metric entries (Eq.(C.2)) and the
defining relation for $\xi^{\prime}(r)$, Eq.(6.34) (where
$\xi^{\prime}(r)>0$).

The drag force given in Eq.(6.36) reduces down to Eq.(6.22) when $d=5$ – a
necessary consistency check. Using the definition of Hawking temperature
(Eq.(4.36)) to insert a unit factor $\left(T^{2}\times 1/T^{2}\right)$, the
drag force can be rewritten

$\frac{dp_{2}}{dt}\,=\,-\,\frac{8\,\pi\,l^{2}\,T^{2}}{(d-1)^{2}\,\alpha^{\prime}}\,\frac{v}{\left(1-v^{2}\right)^{\frac{2}{d-1}}}\,,$
(6.37)

where the momentum $p_{2}(t)$ is given by Eq.(6.26), which holds regardless of
the dimension of the spacetime. From Eq.(6.37), in the non-relativistic limit
$v\ll 1$, the friction coefficient in AdSd-Schwarzschild is given by

$\gamma_{0}^{\text{AdS${}_{d}$}}\,=\,\frac{8\,\pi\,l^{2}\,T^{2}}{(d-1)^{2}\,\alpha^{\prime}\,m}\,.$
(6.38)

Using the Einstein-Sutherland relation Eq.(3.18) an expression for the
diffusion coefficient in general $d$ dimensions can be obtained. Specifically,

$D_{\text{HQ}}^{\text{AdS${}_{d}$}}(d)\,=\,\frac{(d-1)^{2}\,\alpha^{\prime}}{8\,\pi\,l^{2}\,T}\,=\,\frac{(d-1)^{2}\,\beta}{8\pi\sqrt{\lambda}}\,,$
(6.39)

where the second equality follows from the fundamental string length scale
Eq.(4.65), and the definition of Hawking temperature Eq.(4.36). For $d=3$,
complete agreement is found between this equation and the diffusion
coefficient found in the case of a finite mass heavy quark undergoing Brownian
motion in the thermal plasma (subsection (4.4), Eq.(4.123)) – a comforting
consistency check for this work122122122Eq.(6.39) also agrees with the result
given in Eq.(3.10) of [42]; the results from subsection (3.3) in [35]; the
results from section (3) in [22]; and the result – for $d=5$ – from section
(V) in [38]. The rigorous reader might also be interested in [36], a topical
study which computed an ultra-relativistic quark’s transverse momentum
diffusion..

The heavy quark diffusion coefficient in general $d$ dimensions (Eq.(6.39))
can be compared to the light quark diffusion coefficient in general $d$
dimensions (Eq.(5.127)), to find

$D_{\text{LQ}}^{\text{AdS${}_{d}$}}(d)\,=\,\frac{1}{2}\,D_{\text{HQ}}^{\text{AdS${}_{d}$}}(d)\,,$
(6.40)

which agrees with Eq.(5.62) for $d=3$. The factor of $1/2$ in Eq.(6.40) may
arise through differences in partitioning the worldsheet for the heavy and
light quark test strings. Moerman et al. postulated in [52] that by
introducing a factor $a$ which determines the fraction of the local speed of
light the boundary endpoint of the test string falls at, a general expression
can be found for the mean-squared displacement of said endpoint (and therefore
the diffusion coefficient) which interpolates between the heavy and light
quark results123123123The heavy and light quark set-ups to study Brownian
motion are considered in sections (4) and (5), respectively.. Explicitly, the
string falling endpoint’s mean-squared transverse displacement is given by

$s^{2}_{\text{small}}(t;a;d)\,=\,\frac{1}{\sqrt{\lambda}}\left(\frac{(d-1)\beta}{4\pi}\right)^{2}\ln\left(\frac{2a\beta^{3}}{\pi^{3}(a^{2}-1)^{2}t^{3}}\sinh^{2}\left(\frac{\pi(a+1)t}{\beta}\right)\sinh^{2}\left(\frac{\pi(a-1)t}{\beta}\right)\operatorname{csch}\left(\frac{2\pi
at}{\beta}\right)\right)\,,$ (6.41)

which, at late times, becomes124124124See Mathematica notebook [b]
(BrownianMotion.nb) for a derivation of Eqs.(6.41)-(6.43) – access in appendix
(G).

$s^{2}_{\text{small}}(t;a;d)\,\stackrel{{\scriptstyle(\beta\ll
t)}}{{\longrightarrow}}\,\,\frac{(d-1)^{2}\beta
t}{4\pi\sqrt{\lambda}}\left[1-\frac{a}{2}\right]\,+\,\frac{(d-1)^{2}\beta^{2}}{16\pi^{2}\sqrt{\lambda}}\begin{Bmatrix}4\ln\left(\frac{\beta}{2\pi
t}\right)\,,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{if}\,\,\,\
a=0\\\
\ln\left(\frac{a\beta^{3}}{4\pi^{3}(a^{2}-1)^{2}t^{3}}\right)\,,\,\,\,\,\,\,\,\,\,\text{if}\,\,\,\
0<a<1\\\ \ln\left(\frac{\beta}{4\pi
t}\right)\,,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{if}\,\,\,\
a=1\end{Bmatrix}\,+\,\mathcal{O}\left(\frac{\beta}{t}\right)^{0}\,.$ (6.42)

The diffusion coefficient for a quark in a ($d-1$)-dimensional thermal plasma
can be extracted from Eq.(6.42). Specifically,

$D(a;d)\,=\,\left[1-\frac{a}{2}\,\right]\,\frac{(d-1)^{2}\,\beta}{8\pi\sqrt{\lambda}}\,.$
(6.43)

When $a=0$, Eq.(6.43) reduces to the heavy quark diffusion coefficient,
Eq.(6.39); and when $a=1$, Eq.(6.43) becomes the light quark diffusion
coefficient, Eq.(5.127). Hence Eq.(6.41) provides a natural interpolation
between a test string set-up where the boundary endpoint is held stationary,
and a test string set-up where the boundary endpoint is allowed to fall at the
local speed of light. However, values of $a\notin\\{0,1\\}$ correspond to a
test string whose boundary endpoint is falling at velocity smaller than the
local speed of light. An external force opposing the motion of the falling
string must be present in order for this to be the case125125125The solution
where $a\notin\\{0,1\\}$ is not a physically obtainable string solution in the
current set-up since the Virasoro constraints (Eq.(4.19)) are violated and,
consequently, the late time next-to-leading order behaviour in Eq.(6.42) does
not smoothly interpolate between the light and heavy quark results. This can
be rectified by adding in an external field which will alter the energy-
momentum tensor (Eq.(4.8)) and, by construction, cause $T_{ab}$ to vanish
(which ensures the Virasoro constraints are satisfied).. This might be
realised by the introduction of a flavour D7-brane with a world-volume
electric field on it at the boundary126126126This corresponds to an additional
force acting on the external Brownian particle in the boundary theory whose
motion can now be characterised by the generalised Langevin model (described
in subsection (3.2)).127127127For the case of the heavy quark, adding forced
motion has been studied by de Boer et al. [42], subsection (3.2).. Considering
an external force acting to retard the motion of the falling string’s endpoint
is a natural and interesting extension of this work – a possible starting
point for future research.

###  7 Conclusions and Future Outlook

In this article we present a pedagogic account of applications of the AdS/CFT
correspondence for understanding the dynamical behaviour of probe heavy and
light quarks immersed in a thermal plasma, such as the quark-gluon plasma
created in heavy-ion experiments. As a central result, we give the first self-
consistent derivation of the light quark diffusion coefficient in an arbitrary
number of dimensions, correcting previous errors and giving some of the vaguer
statements found in the literature a rigorous foundation.

To summarise, the gauge/string duality is briefly introduced in section (2).
Particular attention is paid to the throat construction of anti-de Sitter
spacetime as a limit of D3-brane geometry, and the justification of the
AdS/CFT conjecture. Section (3) explored the basic theory of Brownian motion
and Langevin dynamics in the boundary theory. Specifically, particles
undergoing Brownian motion in the absence of an external force can be
described by the non-retarded Langevin equation which is parametrized by two
constants: (i) the friction coefficient $\gamma_{0}$, and (ii) the magnitude
of the random force $\kappa_{0}$. The second fluctuation-dissipation theorem
relates these two constants. It is possible to calculate the mean-squared
displacement (Eq.(3.19)), from which the time dependence of $s^{2}(t)$ is
apparent. At early times, the Brownian particle’s behaviour is proportional to
time and the motion is expected to be ballistic $s(t)\sim t$; while, at late
times, the Brownian particle motion is diffusive, $s(t)\sim\sqrt{t}$. The
cross-over time sets the scale for early and late time behaviour and is given
by $t_{\text{relax}}$, which represents the time it takes for a Brownian
particle which had some initial velocity at $t=0$ to thermalize in the medium.
The section concludes by describing the generalised Langevin model which
adapts the friction term to depend on the past trajectory of the Brownian
particle and takes an additional external force acting on the system into
account. In section (6) it was seen that the generalised Langevin model can be
used in a number of applications: to model heavy quark trailing strings or
light quark strings whose endpoint falls with a velocity that is less than the
local speed of light.

Sections (4), (5) and (6) turn towards the bulk theory. In the bulk theory,
the dual description of these probe quarks are realised as test strings in an
asymptotically anti-de Sitter-Schwarzschild background. Calculations are
computed in AdS-Schwarzschild, and then related to quantities in the boundary
theory using the AdS/CFT dictionary. In section (4) an on-mass-shell external
heavy quark is modelled as a fundamental open string of length $\ell_{0}$
attached at the boundary of anti-de Sitter spacetime and hanging towards the
stretched horizon. At the semiclassical level, the Hawking radiation due to
the Schwarzschild black hole environment excites the modes on the string –
resulting in the string’s boundary endpoint enduring irregular motion. This
motion can be related to the Brownian motion of the external heavy quark in
the boundary gauge theory. The main results of this section include the
derivation of the leading order, static string solution in AdS3-Schwarzschild
(Eq.(4.51)); the transverse equations of motion found by expanding the Nambu-
Goto action up to quadratic order and varying this action with respect to the
transverse string worldsheet coordinates $X^{I}$ (Eq.(4.67)); and from there
the mean-squared displacement of the test string’s boundary endpoint,
$s^{2}(t)$, in AdS3-Schwarzschild (Eq.(4.110)). The cross-over time from early
to late time dynamics is found to be independent of the initial length of the
string $\ell_{0}$ and solely dependant on the Hawking temperature $\beta=1/T$.
As expected, in the early time limit ($t\ll\beta$) the motion is ballistic
(Eq.(4.119)); while in the late time limit ($t\gg\beta$) the motion is
diffusive, and the diffusion coefficient can be extracted from
$s^{2}(t)\rvert_{t\gg\beta}$ (Eq.(4.123)). We further showed that the
diffusion coefficient computed in this way and the friction coefficient
computed in a completely independent manner are exactly related by the
fluctuation dissipation theorem as required for thermalizing motion.

An off-mass-shell external light quark is modelled in section (5) as an open
string, initially stretched between the AdS boundary and just above the
horizon, whose AdS boundary endpoint is released to fall at the local speed of
light. The set-up is termed the Limp Noodle configuration [52]. This initial
set-up is held for an asymptotically long amount of time, allowing for a full
population of thermal modes on the initially stretched string. Using the Bars
et al. method [107, 108], the worldsheet of the Limp Noodle is partitioned
into two regions and the leading order string solution for the string once the
endpoint near the boundary is released is found in AdS3-Schwarzschild
(Eq.(5.40)). Again, the transverse fluctuations on the string excited by the
Schwarzschild black hole are considered. From the equations of motion of these
fluctuations, the string falling endpoint’s mean-squared transverse
displacement $s^{2}(t)$ is calculated (Eq.(5.47)). The limiting cases of
$s^{2}(t)$ are then examined. For the small virtuality case (small string
lengths $\ell_{0}$ compared to the radial position of the black-brane horizon
$r_{H}$), $s_{\text{small}}^{2}(t)$ is analytically evaluated (Eq.(5.58)) and
the early and late time behaviour found to be ballistic and diffusive
respectively. For the arbitrary virtuality case, $s^{2}(t)$ can be
analytically found in the early time limit (Eq.(5.70)); while the behaviour of
arbitrary virtuality quarks at asymptotically late times (i.e. in the near-
horizon region) is found to be encoded in the small virtuality case, i.e.
$s^{2}(t)\rvert_{t\gg\beta}=s_{\text{small}}^{2}(t)$ (Eq.(5.76)). Since
$s_{\text{small}}^{2}(t;d)$ can be solved in any $d\geq 3$ dimensions, this
universality of late times motivates the generalisation of the mean-squared
displacement $s^{2}(t)\rvert_{t\gg\beta}$ to AdSd-Schwarzschild. In terms of
original advancement, subsection (5.3) presents a very important part of this
article: correcting Moerman et al. [52] by presenting the proper method in
which to generalise to $s^{2}(t;d)$. Finally, the diffusion coefficient in
AdSd-Schwarzschild is extricated from $s^{2}(t;d)\rvert_{t\gg\beta}$ and given
in Eq.(5.127) by

$D_{\text{LQ}}^{\text{AdS$d$}}(d)\,=\,\frac{(d-1)^{2}\beta}{16\pi\sqrt{\lambda}}\,,$

which is – to the authors’ knowledge – the first complete, consistent
derivation of the light quark diffusion coefficient in $d$ dimensions.

In section (6), a trailing string in AdS5-Schwarzschild which models an
infinitely massive probe quark moving with a constant velocity $v$ in a
$\mathcal{N}=4$ SYM thermal plasma is considered. By making use of the late-
time behaviour ansatz (Eq.(6.5)), the drag force on the test string is
calculated in the bulk and rewritten – via the AdS/CFT correspondence – in
terms of relevant quantities in the gauge theory (Eq.(6.25)). Because the
external quark is moving with a constant velocity in the boundary theory, it
can be modelled by the Langevin equation adapted such that the external force
$K(t)$ acting on the system is of equal and opposite magnitude to the random
force ($K(t)=−F(t)$). Hence the friction coefficient $\gamma_{0}$ can be read
off from Eq.(6.27), and related – using the Einstein-Sutherland relation – to
the diffusion coefficient in $d=5$ dimensions. In the second subsection, the
drag force calculation is generalised to AdSd-Schwarzschild and the diffusion
coefficient is found to be given in Eq.(6.39) by

$D_{\text{HQ}}^{\text{AdS${}_{d}$}}(d)\,=\,\frac{(d-1)^{2}\,\beta}{8\pi\sqrt{\lambda}}\,,$

which agrees, for $d=3$, with the diffusion coefficient found by studying the
transverse fluctuations on the heavy quark’s test string in section (4)
(Eq.(4.123)).

The heavy and light quark’s diffusion constants in general $d$ dimensions are
related by a factor of a $1/2$ (Eq.(6.40)). This disparity may arise through
the differences in partitioning the worldsheet for the heavy and light quark
test strings. As was briefly discussed towards the end of section (6), in
order to understand the physical origin of the $1/2$ factor an interpolation
between $D_{\text{LQ}}^{\text{AdS$d$}}(d)$ and
$D_{\text{HQ}}^{\text{AdS${}_{d}$}}(d)$ can be sought. This leads to the
realisation that considering test strings with boundary endpoints falling
slower than the local speed of light will only yield physically valid
solutions if an external force is introduced. Considering an external force
acting to retard the motion of the falling string’s endpoint is a natural and
interesting extension of this work, and a possible starting point for future
research. The publications [42, 110, 111, 112] might provide a good
commencement for this exercise.

The main contribution of this article to the field of AdS/CFT calculations is
presenting a definitive, consolidated theoretical derivation of the light and
heavy quark diffusion constants in general $d$ dimensions. The work acts as a
springboard from which students might pursue further research avenues. Some of
these that the authors have considered – besides the addition of an external
electric field – are (i) repeating these calculations using a numerical
framework in order to confirm the analytic results; (ii) examining the
fluctuation-dissipation theorem in the bulk; (iii) considering next-to-leading
order transverse fluctuations on the leading order solution (this would surely
prove difficult as the expanded, near-horizon metric in AdSd-Schwarzschild
would no longer be conformal in ($t,r_{*}$) coordinates); and (iv) considering
different test string configurations to illuminate other aspects of the
thermal plasma. With regards to the latter, one such example is studying the
quantum fluctuations in the non-transverse directions of a trailing string
where the boundary endpoint is allowed to fall. Another excellent example
might be the consideration of a light quark string setup that more closely
models the phenomenologically relevant situation in which a light quark–anti-
quark pair is created highly off-shell and without a pre-population of thermal
modes along the string. Determining the fluctuations on top of the leading
order classical motion of this string as it evolves in time would require
knowledge of the thermal spectrum of metric fluctuations due to the presence
of the black brane.

### Acknowledgements

The authors thank Steven Gubser for the idea which inspired the Limp Noodle
construction. The news of his untimely passing is tragic; he left an indelible
impact on the community. May he rest in peace.

Funding information A.K.M. appreciates the support of the Mandela Rhodes
Foundation, and W.A.H. gratefully acknowledges support from the South African
National Research Foundation and the South Africa-CERN Collaboration.

### Appendix

###  Appendix A Polyakov String Equations of Motion

Working in the static gauge, the string equations of motion are derived in
this appendix by calculating the functional derivative of the Polyakov Action
with respect to the string worldsheet coordinates and setting this variation
to zero. The calculation of the string equations of motion presented here
follows the layout of appendix (A) in Moerman et al.’s exposition on light
quark Brownian motion [52].

Using the definition Eq.(4.7), the Polyakov Action Eq.(4.3) is rewritten in
terms of the canonical momentum densities

$S_{P}\,=\,\frac{1}{2}\,\int_{\mathcal{M}}\,d^{2}\sigma\,\Pi_{\mu}^{a}(t,\sigma)\,\partial_{a}\,X^{\mu}(t,\sigma)\,.$
(A.1)

Determining the functional derivative of $S_{P}$ with respect to $X^{\mu}$ and
setting this variation to vanish, yields

$\displaystyle\begin{aligned}
0\,=\,\delta_{X}S_{P}\,&=\,\frac{1}{2}\,\int_{\mathcal{M}}\,d^{2}\sigma\,\Big{[}\big{(}\delta\Pi_{\mu}^{a}\big{)}\,\partial_{a}X^{\mu}\,+\,\Pi_{\mu}^{a}\,\partial_{a}\big{(}\delta
X^{\mu}\big{)}\Big{]}\\\\[5.69046pt]
&=\,\frac{1}{2}\,\int_{\mathcal{M}}\,d^{2}\sigma\,\Big{[}\big{(}\delta\Pi_{\mu}^{a}\big{)}\,\partial_{a}X^{\mu}\,+\,\partial_{a}\big{(}\Pi_{\mu}^{a}\,\delta
X^{\mu}\big{)}\,-\,\delta
X^{\mu}\,\partial_{a}\big{(}\Pi_{\mu}^{a}\big{)}\Big{]}\,,\end{aligned}$ (A.2)

where the product rule is used in the second line.

The functional derivative of the momentum densities with respect to $X^{\mu}$
gives two terms

$\delta_{X}\Pi_{\mu}^{a}\,=\,-\frac{1}{2\pi\alpha^{\prime}}\,\sqrt{-\gamma}\,\gamma^{ab}\Big{[}G_{\mu\nu}\,{\partial_{b}}\big{(}\delta
X^{\nu}\big{)}\,+\,\partial_{\rho}\,G_{\mu\nu}\,\delta
X^{\rho}{\partial_{b}}X^{\nu}\,\Big{]}\,,$ (A.3)

since, in a curved background, the worldsheet coordinates $X^{\mu}$ as well as
the spacetime metric $G_{\mu\nu}$ (which also depends on $X^{\mu}$) are
varied. Hence the first term in Eq.(A.2),
$\big{(}\delta\,\Pi_{\mu}^{a}\big{)}\,\partial_{a}X^{\mu}$, becomes

$\displaystyle\begin{aligned}
\big{(}\delta_{X}\Pi_{\mu}^{a}\big{)}\,\partial_{a}\,X^{\mu}\,&=\,\frac{-1}{2\pi\alpha^{\prime}}\,\sqrt{-\gamma}\,\gamma^{ab}\,G_{\mu\nu}\,{\partial_{b}}\big{(}\delta
X^{\nu}\big{)}\,\partial_{a}\,X^{\mu}\,+\,\partial_{\rho}\,G_{\mu\nu}\,\delta
X^{\rho}\,\Big{(}\frac{-1}{2\pi\alpha^{\prime}}\,\sqrt{-\gamma}\,\gamma^{ab}\,{\partial_{b}}X^{\nu}\,\Big{)}\,\partial_{a}\,X^{\mu}\\\\[5.69046pt]
&=\,\frac{-1}{2\pi\alpha^{\prime}}\,\sqrt{-\gamma}\,\gamma^{ab}\,G_{\mu\nu}\,{\partial_{b}}X^{\nu}\,\partial_{a}\big{(}\delta
X^{\mu}\big{)}\,+\,\partial_{\rho}\,G_{\mu\nu}\,\delta
X^{\rho}\,\Big{(}\frac{-1}{2\pi\alpha^{\prime}}\,\sqrt{-\gamma}\,\gamma^{ab}\,G_{\gamma\nu}\,{\partial_{b}}X^{\nu}\,\Big{)}\,G^{\gamma\nu}\,\partial_{a}\,X^{\mu}\\\\[5.69046pt]
&=\,\Pi_{\mu}^{a}\,\partial_{a}\big{(}\delta
X^{\mu}\big{)}\,+\,\partial_{\rho}\,G_{\mu\nu}\,\delta
X^{\rho}\,\Pi_{\gamma}^{a}\,G^{\gamma\nu}\,\partial_{a}\,X^{\mu}\\\\[5.69046pt]
&=\,\partial_{a}\big{(}\Pi_{\mu}^{a}\,\delta
X^{\mu}\big{)}\,-\,\delta\,X^{\mu}\,\partial_{a}\big{(}\Pi_{\mu}^{a}\big{)}\,+\,2\,\delta
X^{\rho}\,\Big{(}\frac{1}{2}\,G^{\gamma\nu}\,\partial_{\rho}\,G_{\mu\nu}\Big{)}\,\Pi_{\gamma}^{a}\,\partial_{a}\,X^{\mu}\\\\[5.69046pt]
&=\,\partial_{a}\big{(}\Pi_{\mu}^{a}\,\delta
X^{\mu}\big{)}\,-\,\delta\,X^{\mu}\,\partial_{a}\big{(}\Pi_{\mu}^{a}\big{)}\,+\,2\,\delta
X^{\rho}\,\Big{(}\frac{1}{2}\,G^{\gamma\nu}\big{(}\partial_{\rho}\,G_{\mu\nu}+\partial_{\mu}\,G_{\rho\nu}-\partial_{\nu}\,G_{\rho\mu}\big{)}\Big{)}\,\Pi_{\gamma}^{a}\,\partial_{a}\,X^{\mu}\,,\end{aligned}$
(A.4)

where the definition of the momentum densities Eq.(4.7) is used in the third
line, and the product rule is used in the fourth line. In the final line two
extra terms are added. This can be done since

$\Pi_{\gamma}^{a}\,\partial_{a}\,X^{\mu}\,\big{(}\partial_{\mu}\,G_{\rho\nu}-\partial_{\nu}\,G_{\rho\mu}\big{)}\,=-\frac{1}{2\pi\alpha^{\prime}}\,\sqrt{-\eta}\,G_{\mu\nu}\,\Big{[}\eta^{ab}\,\partial_{b}\,X^{\nu}\,\partial_{a}\,X^{\mu}\big{(}\partial_{\mu}\,G_{\rho\nu}-\partial_{\nu}\,G_{\rho\mu}\big{)}\,\Big{]}\,=\,0\,,$
(A.5)

where the first equality holds in the conformal gauge
($\gamma^{ab}=\eta^{ab}$) – which, as discussed in subsection (4.1), is chosen
to eliminate the Weyl invariance in the Polyakov action. To prove Eq.(A.5),
notice that the Minkowski metric can be expanded

$\eta^{ab}\,\partial_{a}\,X^{\mu}\,\partial_{b}\,X^{\nu}\big{(}\partial_{\mu}\,G_{\rho\nu}-\partial_{\nu}\,G_{\rho\mu}\big{)}\,=\,\partial_{\tau}\,X^{\mu}\,\partial_{\tau}\,X^{\nu}\big{(}\partial_{\mu}\,G_{\rho\nu}-\partial_{\nu}\,G_{\rho\mu}\big{)}\,+\,\partial_{\sigma}\,X^{\mu}\,\partial_{\sigma}\,X^{\nu}\big{(}\partial_{\mu}\,G_{\rho\nu}-\partial_{\nu}\,G_{\rho\mu}\big{)}$

and that for each set of reflective cases ($\mu=1,\,\nu=2$ and
$\mu=2,\,\nu=1$), the terms generated by the first case will cancel those
generated by the second. For the identical case (which doesn’t have a
reflective partner case), i.e. $\mu=\nu=1$ or $\mu=\nu=2$, the bracket
$\big{(}\partial_{\mu}\,G_{\rho\nu}-\partial_{\nu}\,G_{\rho\mu}\big{)}$
vanishes.

The Christoffel symbols are now able to be defined

$\Gamma^{\alpha}_{\mu\nu}\,:=\,\frac{1}{2}\,G^{\alpha\gamma}\big{(}\partial_{\mu}\,G_{\nu\gamma}+\partial_{\nu}\,G_{\mu\gamma}-\partial_{\gamma}\,G_{\mu\nu}\big{)}\,.$
(A.6)

These symbols are a set of numbers which characterise a metric connection. A
metric connection defines precisely how distances are measured on a surface,
by invoking the notion of consistently transporting data in a parallel manner
along a family of curves [96].

With this definition, Eq.(A.4) becomes

$\big{(}\delta_{X}\Pi_{\mu}^{a}\big{)}\,\partial_{a}\,X^{\mu}\,=\,\partial_{a}\big{(}\Pi_{\mu}^{a}\,\delta
X^{\mu}\big{)}\,-\,\delta
X^{\mu}\,\partial_{a}\big{(}\Pi_{\mu}^{a}\big{)}\,+\,2\,\delta
X^{\rho}\,\Gamma^{\gamma}_{\rho\mu}\,\Pi_{\gamma}^{a}\,\partial_{a}X^{\mu}\,.$
(A.7)

Finally, inputting Eq.(A.7) into Eq.(A.2) returns

$\displaystyle\begin{aligned}
0\,=\,\delta_{X}S_{P}\,&=\,\frac{1}{2}\,\int_{\mathcal{M}}\,d^{2}\sigma\,\Big{[}-2\,\delta
X^{\mu}\,\partial_{a}\big{(}\Pi_{\mu}^{a}\big{)}\,+2\,\delta
X^{\rho}\,\Gamma^{\gamma}_{\rho\mu}\,\Pi_{\gamma}^{a}\,\partial_{a}\,X^{\mu}\,+\,2\,\partial_{a}\big{(}\Pi_{\mu}^{a}\,\delta
X^{\mu}\big{)}\Big{]}\\\\[5.69046pt]
&=-\int_{\mathcal{M}}\,d^{2}\sigma\,\delta
X^{\mu}\,\Big{[}\partial_{a}\,\Pi_{\mu}^{a}\,-\,\Gamma^{\gamma}_{\mu\nu}\,\Pi_{\gamma}^{a}\,\partial_{a}X^{\nu}\Big{]}\,+\,\int_{\mathcal{M}}\,d^{2}\sigma\,\partial_{a}\big{(}\Pi_{\mu}^{a}\,\delta
X^{\mu}\big{)}\\\\[5.69046pt] &=-\int_{\mathcal{M}}\,d^{2}\sigma\,\delta
X^{\mu}\,\Big{[}\partial_{a}\,\Pi_{\mu}^{a}\,-\,\Gamma^{\gamma}_{\mu\nu}\,\Pi_{\gamma}^{a}\,\partial_{a}X^{\nu}\Big{]}\,+\,\int_{\partial\mathcal{M}}\,d\sigma^{b}\,\epsilon_{ba}\,\big{(}\Pi_{\mu}^{a}\,\delta
X^{\mu}\big{)}\\\\[5.69046pt] &=-\int_{\mathcal{M}}\,d^{2}\sigma\,\delta
X^{\mu}\,\Big{[}\partial_{a}\,\Pi_{\mu}^{a}\,-\,\Gamma^{\gamma}_{\mu\nu}\,\Pi_{\gamma}^{a}\,\partial_{a}X^{\nu}\Big{]}\,+\,\int_{0}^{\tau_{f}}\,d\tau\,\Big{[}\delta
X^{\mu}\,\Pi_{\mu}^{a}\big{\rvert}^{\sigma=\sigma_{f}}_{\sigma=0}\Big{]}\,,\end{aligned}$
(A.8)

where Stokes’ Theorem in d-dimensions [96] is used in the third line. In the
final line the coordinates on the worldsheet parameter space are chosen to be
$\sigma^{a}=(t,\sigma)^{a}$ where
$\sigma^{a}\in\mathcal{M}=[0,t_{f}]\times[0,\sigma_{f}]$, and one of the
integrals of the last term is simplified by using $\delta
X^{\mu}\rvert_{t\in\\{0,t_{f}\\}}=0$.

Choosing the boundary conditions

$\delta
X^{\mu}\,\Pi_{\mu}^{a}\big{\rvert}^{\sigma=\sigma_{f}}_{\sigma=0}\,=\,0\,,$
(A.9)

the last term in Eq.(A.8) disappears. Hence, the string equations of motion
are given by

$0\,=\,\partial_{a}\,\Pi_{\mu}^{a}\,-\,\Gamma^{\alpha}_{\mu\nu}\,\partial_{a}\,X^{\nu}\,\Pi^{a}_{\alpha}\,=:\,\nabla_{a}\,\Pi^{a}_{\mu}\,,$
(A.10)

which is the string analogue of the geodesic equation for a point particle
[96].

###  Appendix B The Virasoro Constraints and String Equations of Motion in
Isothermal Coordinates

Since the metric of any $(1+1)$-dimensional subspace can be transformed into a
conformally flat metric [107, 108] a new, isothermal coordinate system is
introduced. These isothermal coordinates
$y^{\mu^{\prime}}(\sigma^{+},\sigma^{-})$ are given by Eq.(5.2), while the
corresponding string mapping functions
$Y^{\mu^{\prime}}(\sigma^{+},\sigma^{-})$ are defined in Eq.(5.8). Notice that
light-cone coordinates have been chosen for the parameter space. In this
appendix, the Virasoro constraint equations and the string equations of
motion, Eqs.(4.19, 4.20), are rewritten in terms of
$Y^{\mu^{\prime}}(\sigma^{+},\sigma^{-})$.

In the conformal gauge, the Virasoro constraint equations (Eqs.(4.19)) become

$\displaystyle\begin{aligned}
0\,&=\,G\,\eta_{\mu\nu}\,\partial_{\pm}X^{\mu}\,\partial_{\pm}X^{\nu}\\\\[5.69046pt]
&=\,-G\,\partial_{\pm}X^{0}\,\partial_{\pm}X^{0}\,+\,G\,\partial_{\pm}X^{1}\,\partial_{\pm}X^{1}\\\\[5.69046pt]
&=\,-\frac{G}{2}\,\left[\partial_{\pm}\left(Y^{0^{\prime}}+Y^{1^{\prime}}\right)\,\partial_{\pm}\left(Y^{0^{\prime}}+Y^{1^{\prime}}\right)\right]\,+\,\frac{G}{2}\,\left[\partial_{\pm}\left(Y^{0^{\prime}}-Y^{1^{\prime}}\right)\,\partial_{\pm}\left(Y^{0^{\prime}}-Y^{1^{\prime}}\right)\right]\\\\[5.69046pt]
&=\,-2\,G\partial_{\pm}Y^{0^{\prime}}\,\partial_{\pm}Y^{1^{\prime}}\\\\[5.69046pt]
&=\,\partial_{\pm}Y^{0}\,\partial_{\pm}Y^{1}\,,\end{aligned}$ (B.1)

where $\mu,\nu\in\\{0,1\\}$ are spacetime coordinates,
$\partial_{\pm}:=\frac{\partial}{\partial\sigma^{\pm}}$, and the definition of
the new mapping functions Eq.(5.8) is used in the third line. In the final
line, recognise that $G$ is defined as a non-zero scalar function and that the
indices have been renamed ($\mu^{\prime}=\mu$). This is precisely the Virasoro
constraint equations given in subsection (5.1.1), Eq.(5.13).

The string equations of motion, Eq.(4.20), become

$\partial_{\pm}\,\Pi_{\mu}^{\pm}\,-\,\Gamma^{\alpha}_{\mu\nu}\,\partial_{\pm}\,X^{\nu}\,\Pi^{\pm}_{\alpha}\,=\,0\,,$
(B.2)

in the light-cone coordinate frame. This breaks into two equations of motion,
since $\mu=(0,1)$. For $\mu=0$, the string equations of motion are

$\big{(}\partial_{+}\Pi_{0}^{+}\,-\,\Gamma^{\alpha}_{0\nu}\,\partial_{+}X^{\nu}\,\Pi^{+}_{\alpha}\big{)}\,+\,\big{(}\partial_{-}\Pi_{0}^{-}\,-\,\Gamma^{\alpha}_{0\nu}\,\partial_{-}X^{\nu}\,\Pi^{-}_{\alpha}\big{)}\,=\,0\,.$
(B.3)

The first bracket in Eq.(B.3) can be simplified

$\displaystyle\begin{aligned}
&\partial_{+}\Pi_{0}^{+}\,-\,\Gamma^{\alpha}_{0\nu}\,\partial_{+}X^{\nu}\,\Pi^{+}_{\alpha}\\\\[5.69046pt]
&=\,\partial_{+}\left(-\frac{1}{2\pi\alpha^{\prime}}\,G\,{\partial_{-}}X^{0}\right)\,-\,\Gamma^{\alpha}_{0\nu}\,\partial_{+}X^{\nu}\left(\frac{1}{2\pi\alpha^{\prime}}\,G\,\eta_{\alpha\alpha}\,{\partial_{-}}X^{\alpha}\right)\\\\[5.69046pt]
&=\,\frac{1}{2\pi\alpha^{\prime}}\left[-\partial_{+}\left(G\,{\partial_{-}}X^{0}\right)\,+\,G\eta_{\alpha\alpha}\left(-\Gamma^{\alpha}_{00}\,\partial_{+}X^{0}\,{\partial_{-}}X^{\alpha}\,-\,\Gamma^{\alpha}_{01}\,\partial_{+}X^{1}\,{\partial_{-}}X^{\alpha}\right)\right]\\\\[5.69046pt]
&=\,\frac{1}{2\pi\alpha^{\prime}}\left[-\partial_{+}\left(G\,{\partial_{-}}X^{0}\right)\,+G\left(\Gamma^{0}_{00}\,\partial_{+}X^{0}\,{\partial_{-}}X^{0}-\Gamma^{1}_{00}\,\partial_{+}X^{0}\,{\partial_{-}}X^{1}+\Gamma^{0}_{01}\,\partial_{+}X^{1}\,{\partial_{-}}X^{0}-\Gamma^{1}_{01}\,\partial_{+}X^{1}\,{\partial_{-}}X^{1}\right)\right]\,,\end{aligned}$
(B.4)

where, in the first line, the canonical momentum densities Eq.(5.9) are used
and the spacetime metric simplifies due to the conformal gauge choice (i.e.
$G_{\mu\nu}(x)=G\,\eta_{\mu\nu}$, $G^{\mu\nu}(x)=1/G\,\eta^{\mu\nu}$).
Further, the Christoffel symbols, defined in Eq.(4.21), can be calculated

$\displaystyle\begin{aligned}
&\Gamma^{0}_{00}\,=\,\frac{1}{2\,G}\,\eta^{0\gamma}\left(\eta_{0\gamma}\,\partial_{0}G\,+\,\eta_{0\gamma}\,\partial_{0}G\,-\,\eta_{00}\,\partial_{\gamma}G\right)\,=\,\frac{1}{2\,G}\,\eta^{00}\,\left(-\,\partial_{0}G\,-\,\partial_{0}G\,+\,\partial_{0}G\right)\,=\,\frac{1}{2\,G}\,\left(\partial_{0}G\right)\\\\[5.69046pt]
&\Gamma^{1}_{00}\,=\,\frac{1}{2\,G}\,\eta^{1\gamma}\left(\eta_{0\gamma}\,\partial_{0}G\,+\,\eta_{0\gamma}\,\partial_{0}G\,-\,\eta_{00}\,\partial_{\gamma}G\right)\,=\,\frac{1}{2\,G}\,\eta^{11}\,\left(\partial_{1}G\right)\,=\,\frac{1}{2\,G}\,\left(\partial_{1}G\right)\\\\[5.69046pt]
&\Gamma^{0}_{01}\,=\,\frac{1}{2\,G}\,\eta^{0\gamma}\left(\eta_{1\gamma}\,\partial_{0}G\,+\,\eta_{0\gamma}\,\partial_{1}G\,-\,\eta_{01}\,\partial_{\gamma}G\right)\,=\,\frac{1}{2\,G}\,\eta^{00}\,\left(\partial_{1}G\right)\,=\,\frac{1}{2\,G}\,\left(\partial_{1}G\right)\\\\[5.69046pt]
&\Gamma^{1}_{01}\,=\,\frac{1}{2\,G}\,\eta^{1\gamma}\left(\eta_{1\gamma}\,\partial_{0}G\,+\,\eta_{0\gamma}\,\partial_{1}G\,-\,\eta_{01}\,\partial_{\gamma}G\right)\,=\,\frac{1}{2\,G}\,\eta^{11}\,\left(\partial_{0}G\right)\,=\,\frac{1}{2\,G}\,\left(\partial_{0}G\right)\,.\end{aligned}$
(B.5)

Hence Eq.(B.4) becomes

$\,\frac{1}{2\pi\alpha^{\prime}}\left[-\partial_{+}\left(G\,{\partial_{-}}X^{0}\right)+\frac{1}{2}\left(\partial_{0}G\,\partial_{+}X^{0}\,{\partial_{-}}X^{0}-\partial_{1}G\,\partial_{+}X^{0}\,{\partial_{-}}X^{1}+\partial_{1}G\,\partial_{+}X^{1}\,{\partial_{-}}X^{0}-\partial_{0}G\,\partial_{+}X^{1}\,{\partial_{-}}X^{1}\right)\right]\,,$
(B.6)

where $\partial_{\mu}:=\frac{\partial}{\partial X^{\mu}}$ for $\mu=(0,1)$. The
second bracket in Eq.(B.3) can be similarly simplified

$\displaystyle\begin{aligned}
&\partial_{-}\Pi_{0}^{-}\,-\,\Gamma^{\alpha}_{0\nu}\,\partial_{-}X^{\nu}\,\Pi^{-}_{\alpha}\\\\[5.69046pt]
&=\partial_{-}\left(-\frac{1}{2\pi\alpha^{\prime}}\,G{\partial_{+}}X^{0}\right)\,-\,\Gamma^{\alpha}_{0\nu}\,\partial_{-}X^{\nu}\left(\frac{1}{2\pi\alpha^{\prime}}\,G\eta_{\alpha\alpha}\,{\partial_{+}}X^{\alpha}\right)\\\\[5.69046pt]
&=\frac{1}{2\pi\alpha^{\prime}}\left[-\partial_{-}\left(G{\partial_{+}}X^{0}\right)+G\eta_{\alpha\alpha}\left(-\Gamma^{\alpha}_{00}\,\partial_{-}X^{0}\,{\partial_{+}}X^{\alpha}\,-\,\Gamma^{\alpha}_{01}\,\partial_{-}X^{1}\,{\partial_{+}}X^{\alpha}\right)\right]\\\\[5.69046pt]
&=\frac{1}{2\pi\alpha^{\prime}}\left[-\partial_{-}\left(G{\partial_{+}}X^{0}\right)+G\left(\Gamma^{0}_{00}\,\partial_{-}X^{0}\,{\partial_{+}}X^{0}-\Gamma^{1}_{00}\,\partial_{-}X^{0}\,{\partial_{+}}X^{1}+\Gamma^{0}_{01}\,\partial_{-}X^{1}\,{\partial_{+}}X^{0}-\Gamma^{1}_{01}\,\partial_{-}X^{1}\,{\partial_{+}}X^{1}\right)\right]\\\\[5.69046pt]
&=\frac{1}{2\pi\alpha^{\prime}}\left[-\partial_{-}\left(G{\partial_{+}}X^{0}\right)+\frac{1}{2}\left(\partial_{0}G\,\partial_{-}X^{0}\,{\partial_{+}}X^{0}-\partial_{1}G\,\partial_{-}X^{0}\,{\partial_{+}}X^{1}+\partial_{1}G\,\partial_{-}X^{1}\,{\partial_{+}}X^{0}-\partial_{0}G\,\partial_{-}X^{1}\,{\partial_{+}}X^{1}\right)\right],\end{aligned}$
(B.7)

where the calculated Christoffel symbols (Eq.(B.5)) are used in the final
line. Using the simplifications for the first and second brackets (Eqs.(B.6,
B.7)), the first string equation of motion Eq.(B.3) becomes

$-\partial_{+}\left(G\,{\partial_{-}}X^{0}\right)\,-\,\partial_{-}\left(G\,{\partial_{+}}X^{0}\right)\,+\,\partial_{0}G\,\partial_{+}X^{0}\,{\partial_{-}}X^{0}\,-\,\partial_{0}G\,\partial_{+}X^{1}\,{\partial_{-}}X^{1}\,=\,0\,.$
(B.8)

Following a similar calculation as before, the second string equation of
motion (where $\mu\,=\,1$) is

$\big{(}\partial_{+}\Pi_{1}^{+}\,-\,\Gamma^{\alpha}_{1\nu}\,\partial_{+}X^{\nu}\,\Pi^{+}_{\alpha}\big{)}\,+\,\big{(}\partial_{-}\Pi_{1}^{-}\,-\,\Gamma^{\alpha}_{1\nu}\,\partial_{-}X^{\nu}\,\Pi^{-}_{\alpha}\big{)}\,=\,0\,.$
(B.9)

Again, the first bracket in Eq.(B.9) can be simplified

$\displaystyle\begin{aligned}
&\partial_{+}\Pi_{1}^{+}\,-\,\Gamma^{\alpha}_{1\nu}\,\partial_{+}X^{\nu}\,\Pi^{+}_{\alpha}\\\\[5.69046pt]
&=\,\partial_{+}\left(\frac{1}{2\pi\alpha^{\prime}}\,G\,{\partial_{-}}X^{1}\right)\,-\,\Gamma^{\alpha}_{1\nu}\,\partial_{+}X^{\nu}\left(\frac{1}{2\pi\alpha^{\prime}}\,G\,\eta_{\alpha\alpha}\,{\partial_{-}}X^{\alpha}\right)\\\\[5.69046pt]
&=\,\frac{1}{2\pi\alpha^{\prime}}\left[\partial_{+}\left(G\,{\partial_{-}}X^{1}\right)\,+\,G\eta_{\alpha\alpha}\left(-\Gamma^{\alpha}_{10}\,\partial_{+}X^{0}\,{\partial_{-}}X^{\alpha}\,-\,\Gamma^{\alpha}_{11}\,\partial_{+}X^{1}\,{\partial_{-}}X^{\alpha}\right)\right]\\\\[5.69046pt]
&=\,\frac{1}{2\pi\alpha^{\prime}}\left[\partial_{+}\left(G\,{\partial_{-}}X^{1}\right)\,+\,G\left(\Gamma^{0}_{10}\,\partial_{+}X^{0}\,{\partial_{-}}X^{0}-\Gamma^{1}_{10}\,\partial_{+}X^{0}\,{\partial_{-}}X^{1}+\Gamma^{0}_{11}\,\partial_{+}X^{1}\,{\partial_{-}}X^{0}-\Gamma^{1}_{11}\,\partial_{+}X^{1}\,{\partial_{-}}X^{1}\right)\right]\,,\end{aligned}$
(B.10)

where, in the first line, the canonical momentum densities Eq.(5.9) are used
and the spacetime metric simplifies due to the conformal gauge choice. The
Christoffel symbols now need to be calculated. However first note that the
Christoffel symbols are symmetric on their lower indexes [96], i.e.
$\Gamma^{\gamma}_{\alpha\beta}\,=\,\Gamma^{\gamma}_{\beta\alpha}$. Therefore
$\Gamma^{0}_{10}\,=\,\Gamma^{0}_{01}$ and
$\Gamma^{1}_{10}\,=\,\Gamma^{1}_{01}$ (the latter of which have both been
calculated in Eq.(B.5)). The remaining two Christoffel symbols can be
calculated

$\displaystyle\begin{aligned}
&\Gamma^{0}_{11}\,=\,\frac{1}{2\,G}\,\eta^{0\gamma}\left(\eta_{1\gamma}\,\partial_{1}G\,+\,\eta_{1\gamma}\,\partial_{1}G\,-\,\eta_{11}\,\partial_{\gamma}G\right)\,=\,\frac{1}{2\,G}\,\eta^{00}\,\left(-\,\partial_{0}G\right)\,=\,\frac{1}{2\,G}\,\left(\partial_{0}G\right)\\\\[5.69046pt]
&\Gamma^{1}_{11}\,=\,\frac{1}{2\,G}\,\eta^{1\gamma}\left(\eta_{1\gamma}\,\partial_{1}G\,+\,\eta_{1\gamma}\,\partial_{1}G\,-\,\eta_{11}\,\partial_{\gamma}G\right)\,=\,\frac{1}{2\,G}\,\eta^{11}\,\left(\partial_{1}G\,+\,\partial_{1}G\,-\,\partial_{1}G\right)\,=\,\frac{1}{2\,G}\,\left(\partial_{1}G\right)\,.\end{aligned}$
(B.11)

Hence Eq.(B.10) becomes

$\,\frac{1}{2\pi\alpha^{\prime}}\left[\partial_{+}\left(G\,{\partial_{-}}X^{1}\right)\,+\,\frac{1}{2}\left(\partial_{1}G\,\partial_{+}X^{0}\,{\partial_{-}}X^{0}-\partial_{0}G\,\partial_{+}X^{0}\,{\partial_{-}}X^{1}+\partial_{0}G\,\partial_{+}X^{1}\,{\partial_{-}}X^{0}-\partial_{1}G\,\partial_{+}X^{1}\,{\partial_{-}}X^{1}\right)\right]\,.$
(B.12)

The second bracket in Eq.(B.9) can be similarly simplified

$\displaystyle\begin{aligned}
&\partial_{-}\Pi_{1}^{-}\,-\,\Gamma^{\alpha}_{1\nu}\,\partial_{-}X^{\nu}\,\Pi^{-}_{\alpha}\\\\[5.69046pt]
&=\,\partial_{-}\left(\frac{1}{2\pi\alpha^{\prime}}\,G\,{\partial_{+}}X^{1}\right)\,-\,\Gamma^{\alpha}_{1\nu}\,\partial_{-}X^{\nu}\left(\frac{1}{2\pi\alpha^{\prime}}\,G\,\eta_{\alpha\alpha}\,{\partial_{+}}X^{\alpha}\right)\\\\[5.69046pt]
&=\,\frac{1}{2\pi\alpha^{\prime}}\left[\partial_{-}\left(G\,{\partial_{+}}X^{1}\right)\,+\,G\eta_{\alpha\alpha}\left(-\Gamma^{\alpha}_{10}\,\partial_{-}X^{0}\,{\partial_{+}}X^{\alpha}\,-\,\Gamma^{\alpha}_{11}\,\partial_{-}X^{1}\,{\partial_{+}}X^{\alpha}\right)\right]\\\\[5.69046pt]
&=\,\frac{1}{2\pi\alpha^{\prime}}\left[\partial_{-}\left(G\,{\partial_{+}}X^{1}\right)\,+\,G\left(\Gamma^{0}_{10}\,\partial_{-}X^{0}\,{\partial_{+}}X^{0}-\Gamma^{1}_{10}\,\partial_{-}X^{0}\,{\partial_{+}}X^{1}+\Gamma^{0}_{11}\,\partial_{-}X^{1}\,{\partial_{+}}X^{0}-\Gamma^{1}_{11}\,\partial_{-}X^{1}\,{\partial_{+}}X^{1}\right)\right]\\\\[5.69046pt]
&=\,\frac{1}{2\pi\alpha^{\prime}}\left[\partial_{-}\left(G\,{\partial_{+}}X^{1}\right)+\frac{1}{2}\left(\partial_{1}G\,\partial_{-}X^{0}\,{\partial_{+}}X^{0}-\partial_{0}G\,\partial_{-}X^{0}\,{\partial_{+}}X^{1}\,+\,\partial_{0}G\,\partial_{-}X^{1}\,{\partial_{+}}X^{0}-\partial_{1}G\,\partial_{-}X^{1}\,{\partial_{+}}X^{1}\right)\right],\end{aligned}$
(B.13)

where the calculated Christoffel symbols (Eqs.(B.5, B.11)) are used in the
final line. Using the simplifications for the first and second brackets
(Eqs.(B.12, B.13)), the second string equation of motion Eq.(B.9) becomes

$\partial_{+}\left(G\,{\partial_{-}}X^{1}\right)\,+\,\partial_{-}\left(G\,{\partial_{+}}X^{1}\right)\,+\,\partial_{1}G\,\partial_{+}X^{0}\,{\partial_{-}}X^{0}\,-\,\partial_{1}G\,\partial_{+}X^{1}\,{\partial_{-}}X^{1}\,=\,0\,.$
(B.14)

Now, the two string equations of motion (Eqs.(B.8, B.14)) can be rewritten in
terms of the new string embedding functions
$Y^{\mu^{\prime}}(\sigma^{+},\sigma^{-})$. Combining Eq.(B.8) and Eq.(B.14)
yields

$-\partial_{+}\left(G\,{\partial_{-}}\left(X^{0}-X^{1}\right)\right)\,-\,\partial_{-}\left(G\,{\partial_{+}}\left(X^{0}-X^{1}\right)\right)\,+\,\left(\partial_{0}G+\partial_{1}G\right)\left[\partial_{+}X^{0}\,{\partial_{-}}X^{0}\,-\,\partial_{+}X^{1}\,{\partial_{-}}X^{1}\right]\,=\,0\,,$
(B.15)

where the derivatives are defined as $\partial_{\mu}:=\frac{\partial}{\partial
X^{\mu}}$ for $\mu=(0,1)$. By the chain rule

$\partial_{0}G\,=\,\left(\partial_{0^{\prime}}G+\partial_{1^{\prime}}G\right)\,\,\,\,\text{and}\,\,\,\,\partial_{1}G\,=\,\left(\partial_{0^{\prime}}G-\partial_{1^{\prime}}G\right)\,,$
(B.16)

where $\partial_{\mu^{\prime}}:=\frac{\partial}{\partial Y^{\mu^{\prime}}}$
for $\mu^{\prime}=\left(0^{\prime},1^{\prime}\right)$. Using Eq.(B.16) and the
definition of $Y^{\mu^{\prime}}(\sigma^{+},\sigma^{-})$ (Eq.(5.8)), the
equation of motion Eq.(B.15) becomes

$\displaystyle\begin{aligned}
0&=-\sqrt{2}\left[\partial_{+}\left(G\,{\partial_{-}}Y^{1^{\prime}}\right)+\partial_{-}\left(G\,{\partial_{+}}Y^{1^{\prime}})\right)\right]\,+\,\sqrt{2}\,\partial_{0^{\prime}}G\bigg{[}\,\frac{1}{2}\partial_{+}\left(Y^{0^{\prime}}+Y^{1^{\prime}}\right){\partial_{-}}\left(Y^{0^{\prime}}+Y^{1^{\prime}}\right)\\\\[5.69046pt]
&\,\,\,\,\,\,\,-\frac{1}{2}\partial_{+}\left(Y^{0^{\prime}}-Y^{1^{\prime}}\right){\partial_{-}}\left(Y^{0^{\prime}}-Y^{1^{\prime}}\right)\bigg{]}\\\\[5.69046pt]
&=\,\partial_{+}\left(G\,{\partial_{-}}Y^{1^{\prime}}\right)+\partial_{-}\left(G\,{\partial_{+}}Y^{1^{\prime}}\right)\,+\,\frac{1}{2}\,\partial_{0^{\prime}}G\bigg{[}\,\partial_{+}Y^{0^{\prime}}\partial_{-}Y^{0^{\prime}}+\partial_{+}Y^{0^{\prime}}\partial_{-}Y^{1^{\prime}}+\partial_{+}Y^{1^{\prime}}\partial_{-}Y^{0^{\prime}}+\partial_{+}Y^{1^{\prime}}\partial_{-}Y^{1^{\prime}}\\\\[5.69046pt]
&\,\,\,\,\,\,\,-\partial_{+}Y^{0^{\prime}}\partial_{-}Y^{0^{\prime}}+\partial_{+}Y^{0^{\prime}}\partial_{-}Y^{1^{\prime}}+\partial_{+}Y^{1^{\prime}}\partial_{-}Y^{0^{\prime}}-\partial_{+}Y^{1^{\prime}}\partial_{-}Y^{1^{\prime}}\bigg{]}\\\\[5.69046pt]
&=\,\partial_{+}\left(G\partial_{-}Y^{1}\right)\,+\,\partial_{-}\left(G\partial_{+}Y^{1}\right)\,-\,\left(\partial_{0}G\right)\left[\left(\partial_{+}Y^{0}\right)\left(\partial_{-}Y^{1}\right)\,+\,\left(\partial_{+}Y^{1}\right)\left(\partial_{-}Y^{0}\right)\right]\,,\end{aligned}$
(B.17)

where the indices have been renamed ($\mu^{\prime}=\mu$) in the last line.
This is precisely one of the string equations of motion given in subsection
(5.1.1), Eq.(5.13). In order to find the other string equation of motion,
subtract Eq.(B.8) from Eq.(B.14) to yield

$\displaystyle\begin{aligned}
0&=-\partial_{+}\left(G\,{\partial_{-}}\left(X^{0}+X^{1}\right)\right)\,-\,\partial_{-}\left(G\,{\partial_{+}}\left(X^{0}+X^{1}\right)\right)\,+\,\left(\partial_{0}G-\partial_{1}G\right)\left[\partial_{+}X^{0}\,{\partial_{-}}X^{0}\,-\,\partial_{+}X^{1}\,{\partial_{-}}X^{1}\right]\\\\[5.69046pt]
&-\sqrt{2}\left[\partial_{+}\left(G\,{\partial_{-}}Y^{0^{\prime}}\right)+\partial_{-}\left(G\,{\partial_{+}}Y^{0^{\prime}})\right)\right]\,+\,\sqrt{2}\,\partial_{1^{\prime}}G\bigg{[}\,\frac{1}{2}\partial_{+}\left(Y^{0^{\prime}}+Y^{1^{\prime}}\right){\partial_{-}}\left(Y^{0^{\prime}}+Y^{1^{\prime}}\right)\\\\[5.69046pt]
&\,\,\,\,\,\,\,-\frac{1}{2}\partial_{+}\left(Y^{0^{\prime}}-Y^{1^{\prime}}\right){\partial_{-}}\left(Y^{0^{\prime}}-Y^{1^{\prime}}\right)\bigg{]}\\\\[5.69046pt]
&=\,\partial_{+}\left(G\partial_{-}Y^{0}\right)\,+\,\partial_{-}\left(G\partial_{+}Y^{0}\right)\,-\,\left(\partial_{1}G\right)\left[\left(\partial_{+}Y^{0}\right)\left(\partial_{-}Y^{1}\right)\,+\,\left(\partial_{+}Y^{1}\right)\left(\partial_{-}Y^{0}\right)\right]\,.\end{aligned}$
(B.18)

The Virasoro constraints Eq.(B.1) and the string equations of motion
Eqs.(B.17, B.18) agree with the equations given by Bars et al. in [107, 108]
where $u=Y^{0}$ and $v=Y^{1}$. Further, using the product and chain rules,
Eq.(B.17) becomes

$\displaystyle\begin{aligned}
0&=\left(\partial_{+}G\right)\left(\partial_{-}Y^{1}\right)\,+\,G\partial_{+}\left(\partial_{-}Y^{1}\right)\,+\,\left(\partial_{-}G\right)\left(\partial_{+}Y^{1}\right)\,+\,G\partial_{-}\left(\partial_{+}Y^{1}\right)\\\\[5.69046pt]
&\,\,\,\,\,\,\,-\,\left(\partial_{0}G\right)\left[\left(\partial_{+}Y^{0}\right)\left(\partial_{-}Y^{1}\right)\,+\,\left(\partial_{+}Y^{1}\right)\left(\partial_{-}Y^{0}\right)\right]\\\\[5.69046pt]
&=\left[\left(\partial_{0}G\right)\left(\partial_{+}Y^{0}\right)+\left(\partial_{1}G\right)\left(\partial_{+}Y^{1}\right)\right]\left(\partial_{-}Y^{1}\right)\,+\,\left[\left(\partial_{0}G\right)\left(\partial_{-}Y^{0}\right)+\left(\partial_{1}G\right)\left(\partial_{-}Y^{1}\right)\right]\left(\partial_{+}Y^{1}\right)\,+\,2\,G\partial_{+}\left(\partial_{-}Y^{1}\right)\\\\[5.69046pt]
&\,\,\,\,\,\,\,-\,\left(\partial_{0}G\right)\left[\left(\partial_{+}Y^{0}\right)\left(\partial_{-}Y^{1}\right)\,+\,\left(\partial_{+}Y^{1}\right)\left(\partial_{-}Y^{0}\right)\right]\\\\[5.69046pt]
&=\,G\partial_{+}\partial_{-}Y^{1}\,+\,\left(\partial_{1}G\right)\left(\partial_{+}Y^{1}\right)\left(\partial_{-}Y^{1}\right)\,,\end{aligned}$
(B.19)

and Eq.(B.18) becomes

$\displaystyle\begin{aligned}
0&=\left(\partial_{+}G\right)\left(\partial_{-}Y^{0}\right)\,+\,G\partial_{+}\left(\partial_{-}Y^{0}\right)\,+\,\left(\partial_{-}G\right)\left(\partial_{+}Y^{0}\right)\,+\,G\partial_{-}\left(\partial_{+}Y^{0}\right)\\\\[5.69046pt]
&\,\,\,\,\,\,\,-\,\left(\partial_{1}G\right)\left[\left(\partial_{+}Y^{0}\right)\left(\partial_{-}Y^{1}\right)\,+\,\left(\partial_{+}Y^{1}\right)\left(\partial_{-}Y^{0}\right)\right]\\\\[5.69046pt]
&=\left[\left(\partial_{0}G\right)\left(\partial_{+}Y^{0}\right)+\left(\partial_{1}G\right)\left(\partial_{+}Y^{1}\right)\right]\left(\partial_{-}Y^{0}\right)\,+\,\left[\left(\partial_{0}G\right)\left(\partial_{-}Y^{0}\right)+\left(\partial_{1}G\right)\left(\partial_{-}Y^{1}\right)\right]\left(\partial_{+}Y^{0}\right)\,+\,2\,G\partial_{+}\left(\partial_{-}Y^{0}\right)\\\\[5.69046pt]
&\,\,\,\,\,\,\,-\,\left(\partial_{1}G\right)\left[\left(\partial_{+}Y^{0}\right)\left(\partial_{-}Y^{1}\right)\,+\,\left(\partial_{+}Y^{1}\right)\left(\partial_{-}Y^{0}\right)\right]\\\\[5.69046pt]
&=\,G\partial_{+}\partial_{-}Y^{0}\,+\,\left(\partial_{0}G\right)\left(\partial_{+}Y^{0}\right)\left(\partial_{-}Y^{0}\right)\,,\end{aligned}$
(B.20)

which agrees exactly with the string equations of motion given by Moerman et
al. in [52].

###  Appendix C Energy of a Test String in an AdS-Schwarzschild Background

At $t=0$ the test string set-up for the heavy quark agrees with the test
string set-up for the light quark, since the string is initially static in
both cases – i.e. $X^{I}=0$ (where $I\in(2,\,3,\,...\,,\,d-1)$) at time $t=0$,
with embedding functions
$X^{\mu}(t,\sigma)\,=\,\left(t,r(t,\sigma),0\right)^{\mu}$. The on-mass-shell
heavy quark’s mass or the off-mass-shell light quark’s mass128128128Particles
are on the mass shell, or simply on-mass-shell, if their behaviour satisfies
Einstein’s energy and momentum relation $E^{2}=(pc)^{2}+(2mc)^{2}$. Particles
whose behaviour violates this relation are known as off-mass-shell. In the
case of the light quark it starts at $t=0$ as an off-mass-shell particle
(corresponding to the initial static string), radiates energy as it travels
through the thermal medium (string contracts as the boundary endpoint falls at
the local speed of light), and finally stops radiating as it becomes an on-
mass-shell particle. This appendix is focused on calculating the mass (or
virtuality) of the light quark as an initially off-mass-shell particle; and
the mass of an on-mass-shell heavy quark. can be calculated by finding the
total energy of the relevant test string in AdSd-Schwarzschild. In order to
calculate the string’s total energy, the configuration of the string is chosen
to be stretched between the radial positions $r=0$ and $r=\ell_{0}$. This
choice will simplify the calculation.

The AdSd-Schwarzschild metric in $d$ dimensions is given by

$\displaystyle\begin{aligned}
ds_{d}^{2}&:=G_{\mu\nu}\,dx^{\mu}\,dx^{\nu}\\\\[7.11317pt]
&=G_{tt}\,dt^{2}\,+\,G_{rr}\,dr^{2}\,+\,G_{II}\,d\vec{X}_{I}^{2}\\\\[7.11317pt]
&=\frac{r^{2}}{l^{2}}\,\Big{(}-h(r;d)\,dt^{2}\,+\,d\vec{X}_{I}^{2}\,\Big{)}\,+\,\frac{l^{2}}{r^{2}}\frac{dr^{2}}{h(r;d)}\,,\end{aligned}$
(C.1)

where $t\in[0,\infty)$ is the temporal coordinate, $r\in[0,\infty)$ is the
radial coordinate, and the transverse spatial directions are denoted by
$\vec{X}_{I}=\left(X^{2},X^{3},....,X^{(d-1)}\right)\in\mathbb{R}^{d-2}$.
Further, $l\in\mathbb{R}^{+}$ is the curvature radius of AdSd and $S^{d}$; and
the blackening factor of the Schwarzschild black hole situated at the horizon,
$h(r;d)$, is given by Eq.(4.35). Explicitly, the target spacetime metric
$G_{\mu\nu}$ is given by

$G_{\mu\nu}\,=\,\begin{bmatrix}\Big{(}-\frac{r^{2}}{l^{2}}\,h(r;d)\Big{)}&0&0&0&0&0\\\
0&\Big{(}\frac{l^{2}}{r^{2}}\,\frac{1}{h(r;d)}\Big{)}&0&0&0&0\\\
0&0&\Big{(}\frac{r^{2}}{l^{2}}\Big{)}&0&0&0\\\
0&0&0&\Big{(}\frac{r^{2}}{l^{2}}\Big{)}&\ddots&0\\\ 0&0&0&\ddots&\ddots&0\\\
0&0&0&0&0&\Big{(}\frac{r^{2}}{l^{2}}\Big{)}\end{bmatrix}\,,$ (C.2)

where $\mu,\nu$ index over $\left(t,\,r,\,2,\,...\,,\,d-1\right)$. The induced
worldsheet metric $g_{ab}$, defined in Eq.(4.4), takes the explicit form

$g_{ab}=\begin{bmatrix}g_{tt}&g_{t\sigma}\\\ g_{\sigma
t}&g_{\sigma\sigma}\end{bmatrix}\,,$ (C.3)

since $(t,\sigma)$ are the two parameter space coordinates in the static gauge
($\tau=t$). In order to determine the explicit entries of the induced metric
$g_{ab}$, the definition Eq.(4.4) is used to calculate each component.
Specifically,

$g_{ab}=\begin{bmatrix}G_{tt}\,+\,G_{II}\,\dot{X}_{I}^{2}&G_{II}\,\dot{X}_{I}\,X_{I}^{\prime}\\\
G_{II}\,X_{I}^{\prime}\,\dot{X}_{I}&r^{\prime
2}\,G_{rr}\,+\,G_{II}\,X_{I}^{\prime\,2}\end{bmatrix}\,,$ (C.4)

where $\dot{X}\equiv\partial_{t}X$ and $X^{\prime}\equiv\partial_{\sigma}X$;
$I$ indexes over the transverse directions
$X_{I}=\left(X^{2},X^{3},....,X^{(d-1)}\right)$; and the fact that
$G_{\mu\nu}$ (Eq.(C.2)) is a diagonal matrix is used.

For later use, the inverse metric $g^{ab}$ (where
$g^{ab}\,\equiv\,(g_{ab})^{-1}$) is also written out explicitly

$g^{ab}=\,\frac{1}{\text{det}(g_{ab})}\begin{bmatrix}r^{\prime
2}\,G_{rr}\,+\,G_{II}\,X_{I}^{\prime\,2}&-G_{II}\,\dot{X}_{I}\,X_{I}^{\prime}\\\
-G_{II}\,X_{I}^{\prime}\,\dot{X}_{I}&G_{tt}\,+\,G_{II}\,\dot{X}_{I}^{2}\end{bmatrix}\,.$
(C.5)

The determinant of the induced metric is

$\displaystyle\begin{aligned}
g\,:=\,\text{det}\,(g_{ab})\,&=\,\big{(}r^{\prime
2}\,G_{rr}\,+\,G_{II}\,X_{I}^{\prime\,2}\big{)}\,\big{(}G_{tt}\,+\,G_{II}\,\dot{X}_{I}^{\,2}\big{)}\,-\,\big{(}G^{2}_{II}\,\dot{X}_{I}^{\,2}\,X_{I}^{\prime\,2}\big{)}\\\\[5.69046pt]
&=\,r^{\prime 2}\,G_{rr}\,G_{tt}\,+\,r^{\prime
2}\,G_{rr}\,G_{II}\,\dot{X}_{I}^{\,2}\,+\,G_{tt}\,G_{II}\,X_{I}^{\prime\,2}\,+\,G_{II}^{\,2}\,X_{I}^{\prime\,2}\,\dot{X}_{I}^{\,2}\,-\,G_{II}^{\,2}\,\dot{X}_{I}^{\,2}\,X_{I}^{\prime\,2}\\\\[5.69046pt]
&=\,G_{rr}\,G_{tt}\,\Big{(}r^{\prime 2}\,+\,\frac{r^{\prime
2}}{G_{tt}}\,G_{II}\,\dot{X}_{I}^{\,2}\,+\,\frac{1}{G_{rr}}\,G_{II}\,X_{I}^{\prime\,2}\Big{)}\,.\end{aligned}$
(C.6)

Now that the ground work has been laid, the total energy of the static string
can be calculated

$\displaystyle\begin{aligned}
\text{E}\,&=\,\int_{0}^{\ell_{0}}\,d\sigma\,\Pi^{\tau}_{\,t}\\\\[5.69046pt]
&=\,-\frac{1}{2\pi\alpha^{\prime}}\,\int_{0}^{\ell_{0}}\,d\sigma\,\sqrt{-\gamma}\,\gamma^{\tau
b}\,G_{t\nu}\,\partial_{b}\,X^{\nu}\\\\[5.69046pt]
&=\,-\frac{1}{2\pi\alpha^{\prime}}\,\int_{0}^{\ell_{0}}\,d\sigma\,\sqrt{-g}\,g^{tb}\,G_{tt}\,\partial_{b}\,X^{t}\\\\[5.69046pt]
&=\,-\frac{1}{2\pi\alpha^{\prime}}\,\int_{0}^{\ell_{0}}\,d\sigma\,\sqrt{-g}\,\big{(}g^{tt}\,\partial_{t}\,X^{t}\,+\,g^{t\sigma}\,\partial_{\sigma}\,X^{t}\big{)}\,\bigg{(}-\frac{r^{2}}{l^{2}}\,h(r;d)\bigg{)}\\\\[5.69046pt]
&=\,\frac{1}{2\pi\alpha^{\prime}}\,\int_{0}^{\ell_{0}}\,d\sigma\,\sqrt{-g}\,g^{tt}\,\bigg{(}\frac{r^{2}}{l^{2}}\,h(r;d)\bigg{)}\\\\[5.69046pt]
&=\,\frac{1}{2\pi\alpha^{\prime}}\,\int_{0}^{\ell_{0}}\,d\sigma\,\sqrt{-g}\,\big{(}r^{\prime
2}\,G_{rr}\,+\,G_{II}\,X_{I}^{\prime\,2}\big{)}\,\bigg{(}\frac{r^{2}}{l^{2}}\,h(r;d)\bigg{)}\\\\[5.69046pt]
&=\,\frac{1}{2\pi\alpha^{\prime}}\,\int_{0}^{\ell_{0}}\,d\sigma\,\sqrt{-g}\,r^{\prime
2}\,\bigg{(}\frac{l^{2}}{r^{2}}\,\frac{1}{h(r;d)}\bigg{)}\,\bigg{(}\frac{r^{2}}{l^{2}}\,h(r;d)\bigg{)}\,,\end{aligned}$
(C.7)

where the constraint equation Eq.(4.12) and the static gauge choice ($\tau=t$)
are used in the third line, and Eq.(C.2) is used in the fourth line. The fifth
line follows from remembering the string is static ($X_{I}=0$), and Eq.(C.5)
is used in the sixth line.

Using Eq.(C.6), the total energy of the static string (Eq.(C.7)) becomes

$\displaystyle\begin{aligned}
\text{E}\,&=\,\frac{1}{2\pi\alpha^{\prime}}\,\int_{0}^{\ell_{0}}\,d\sigma\,r^{\prime
2}\,\sqrt{-G_{rr}\,G_{tt}\,\Big{(}r^{\prime 2}\,+\,\frac{r^{\prime
2}}{G_{tt}}\,G_{II}\,\dot{X}_{I}^{\,2}\,+\,\frac{1}{G_{rr}}\,G_{II}\,X_{I}^{\prime\,2}\Big{)}}\\\\[5.69046pt]
&=\,\frac{1}{2\pi\alpha^{\prime}}\,\int_{0}^{\ell_{0}}\,d\sigma\,r^{\prime
2}\,\sqrt{-r^{\prime 2}\,G_{rr}\,G_{tt}}\\\\[5.69046pt]
&=\,\frac{1}{2\pi\alpha^{\prime}}\,\int_{0}^{\ell_{0}}\,d\sigma\,\partial_{\sigma}^{3}r\,,\end{aligned}$
(C.8)

where Eq.(C.6) is used in the first line, the second line follows again from
$X_{I}=0$, and Eq.(C.2) is used the final line.

From Eq.(C.8) it is apparent that the energy of the string is contingent on
how the radial coordinate $r$ depends on the worldsheet parameter space
coordinate $\sigma$. In the static gauge (which is employed in sections (4)
and (5)) the energy of the test string is easily found in AdS3-Schwarzschild
by using the identification Eq.(4.53) and solving the integral in Eq.(C.8).
Since the tortoise coordinate in anti-de Sitter-Schwarzschild spacetime can
only be inverted for $d=3$, an equivalent identification to Eq.(4.53) in AdSd-
Schwarzschild can not be found. However, a different gauge choice can be made.
Taking, for example, the gauge choice129129129This gauge choice is also made
in [22, 35]. in section (6) where $\tau=t$ and $\sigma=r$ are the worldsheet
coordinates, Eq.(C.8) becomes

$\text{E}\,=\,\frac{1}{2\pi\alpha^{\prime}}\,\big{[}\ell_{0}\,-\,0\big{]}\,,$
(C.9)

in AdSd-Schwarzschild. Therefore

$\text{E}^{2}\,\equiv\,m_{q}^{2}\,=\,\frac{\ell_{0}^{2}}{4\pi^{2}\alpha^{\prime
2}}\,,$ (C.10)

where the first equivalence follows from the AdS/CFT correspondence: the
energy of a test string in the bulk theory is equivalent to the mass of the
given probe quark in the boundary theory. Notice that Eq.(C.10) agrees with
Eq.(4.32), which is a reflection of the fact that the spacetime in the bulk
theory is curved precisely along the additional radial direction $r$, and not
along the temporal or transverse directions. Since the boundary theory does
not include this radial direction, from the perspective of the probe quark its
mass will be the same whether the string is in Minkowski or AdS-Schwarzschild
spacetime130130130In an AdS/CFT context, this implies that the energy of the
string is equivalent in Minkowski or AdS-Schwarzschild spacetime.. Considering
that

$\sqrt{\alpha^{\prime}}\,\equiv\,\frac{l}{\lambda^{1/4}}\,,$ (C.11)

where $l\in\mathbb{R}^{+}$ is the radius of curvature of AdSd and
$\lambda=g_{YM}^{2}\,N_{c}$ is the ’t Hooft coupling (see table (1), page 1);
the quark’s mass131131131Remembering this is either the initial, off-mass-
shell mass for the light quark or the on-mass-shell mass for the heavy quark.
becomes

$m_{q}^{2}\,=\,\frac{\lambda\,\ell_{0}^{2}}{4\,\pi^{2}\,l^{4}}\,.$ (C.12)

Alternatively, Eq.(C.12) can be rewritten as

$m_{q}\,=\,\frac{\sqrt{\lambda}\,\ell_{0}}{\beta\,r_{H}}\,=\,\frac{\sqrt{\lambda}}{\beta}\,\frac{(r_{s}+\ell_{0}-r_{s})}{r_{H}}\,=\,\frac{\sqrt{\lambda}}{\beta}\,\left(\tilde{r}_{0}-\frac{r_{s}}{r_{H}}\right)\,\approx\,\frac{\sqrt{\lambda}}{\beta}\,\tilde{r}_{0}\,,$
(C.13)

where the first equality follows from using the definition of the Hawking
temperature Eq.(4.36); the second equality from Eq.(4.89); and the
approximation from the definition of the stretched horizon
$r_{s}=(1+\epsilon)r_{H}$ (where $0<\epsilon\ll 1$). Using the definition of
the Hawking temperature yields $m_{q}=\sqrt{\lambda}\,\tilde{r}_{0}\,T$.
Eq.(C.13) might be a convenient form of the quark mass $m_{q}$ in some cases.

###  Appendix D Nambu-Goto String Equations of Motion for Transverse
Fluctuations

The calculation of the string equations of motion for the transverse
fluctuations presented in this appendix, follows the layout of appendix (B) in
Moerman et al.’s exposition on light quark Brownian motion [52]. Working in
the static gauge, the transverse string equations of motion are derived by
varying the effective action for the transverse fluctuations with respect to
the transverse string worldsheet coordinates, and setting this functional
variation to zero.

Using the definition Eq.(4.7), the effective transverse action Eq.(4.62) can
be written in terms of the transverse canonical momentum densities

$S_{NG}^{(2)}\,=\,\frac{1}{2}\,\int_{\mathcal{M}}\,d^{2}\sigma\,\Pi_{\,I}^{a}(t,\sigma)\,\partial_{a}X^{I}(t,\sigma)\,.$
(D.1)

Determining the functional derivative of $S_{NG}^{(2)}$ with respect to
$X^{I}$ and setting this variation to vanish, results in

$\displaystyle\begin{aligned}
0\,=\,\delta_{X}S_{NG}^{(2)}\,&=\,\frac{1}{2}\,\int_{\mathcal{M}}\,d^{2}\sigma\,\Big{[}\big{(}\delta\Pi_{\,I}^{a}\big{)}\,\partial_{a}\,X^{I}\,+\,\Pi_{\,I}^{a}\,\partial_{a}\big{(}\delta
X^{I}\big{)}\Big{]}\\\\[5.69046pt]
&=\,\frac{1}{2}\,\int_{\mathcal{M}}\,d^{2}\sigma\,\Big{[}\big{(}\delta\Pi_{\,I}^{a}\big{)}\,\partial_{a}\,X^{I}\,+\,\partial_{a}\big{(}\Pi_{\,I}^{a}\,\delta
X^{I}\big{)}\,-\,\partial_{a}\big{(}\Pi_{\,I}^{a}\big{)}\,\delta
X^{I}\Big{]}\,,\end{aligned}$ (D.2)

where the product rule is used in the second line.

The functional derivative of the momentum densities with respect to $X^{I}$ is
given by

$\delta_{X}\Pi_{\,I}^{a}\,=\,-\frac{1}{2\pi\alpha^{\prime}}\,\big{(}\sqrt{-g}\,g^{ab}\,G_{IJ}\big{)}\big{\rvert}_{X^{\mu}_{0}}\,\,{\partial_{b}}\big{(}\delta
X^{J}\big{)}\,,$ (D.3)

where the spacetime metric $G_{\mu\nu}$ does not need to be varied since it is
independent of the transverse directions $X^{I}$ (Eq.(C.2)). Hence the first
term in Eq.(D.2), $\big{(}\delta\Pi_{\,I}^{a}\big{)}\partial_{a}X^{I}$,
becomes

$\displaystyle\begin{aligned}
\big{(}\delta_{X}\Pi_{\,I}^{a}\big{)}\,\partial_{a}\,X^{I}\,&=\left(-\frac{1}{2\pi\alpha^{\prime}}\,\big{(}\sqrt{-g}\,g^{ab}\,G_{IJ}\big{)}\big{\rvert}_{X^{\mu}_{0}}\,{\partial_{b}}\big{(}\delta
X^{J}\big{)}\right)\partial_{a}X^{\mu}\\\\[5.69046pt]
&=\left(-\frac{1}{2\pi\alpha^{\prime}}\,\big{(}\sqrt{-g}\,g^{ab}\,G_{IJ}\big{)}\big{\rvert}_{X^{\mu}_{0}}\,\partial_{a}X^{\mu}\right){\partial_{b}}\big{(}\delta
X^{J}\big{)}\\\\[5.69046pt] &=\,\Pi_{\,J}^{b}\,{\partial_{b}}\big{(}\delta
X^{J}\big{)}\\\\[5.69046pt] &=\,\partial_{b}\big{(}\Pi_{\,J}^{b}\delta
X^{J}\big{)}-\big{(}\partial_{b}\Pi_{\,J}^{b}\big{)}\delta
X^{J}\,,\end{aligned}$ (D.4)

where the definition of the momentum densities Eq.(4.66) are used in the third
line, and the product rule is used in the final line. Inputting Eq.(D.4) into
Eq.(D.2), yields

$\displaystyle\begin{aligned}
0\,=\,\delta_{X}S_{NG}^{(2)}\,&=\,\frac{1}{2}\,\int_{\mathcal{M}}\,d^{2}\sigma\,\Big{[}\partial_{b}\big{(}\Pi_{\,J}^{b}\,\delta
X^{J}\big{)}\,-\,\big{(}\partial_{b}\,\Pi_{\,J}^{b}\big{)}\,\delta
X^{J}\,+\,\partial_{a}\big{(}\Pi_{\,I}^{a}\,\delta
X^{I}\big{)}\,-\,\partial_{a}\big{(}\Pi_{\,I}^{a}\big{)}\,\delta
X^{I}\Big{]}\\\\[5.69046pt] &=\,-\int_{\mathcal{M}}\,d^{2}\sigma\,\delta
X^{I}\big{(}\partial_{a}\Pi_{\,I}^{a}\big{)}\,+\,\int_{\mathcal{M}}\,d^{2}\sigma\,\partial_{a}\big{(}\Pi_{\,I}^{a}\,\delta
X^{I}\big{)}\\\\[5.69046pt] &=\,-\int_{\mathcal{M}}\,d^{2}\sigma\,\delta
X^{I}\big{(}\partial_{a}\Pi_{\,I}^{a}\big{)}\,+\,\int_{\partial\mathcal{M}}\,d\sigma^{b}\,\epsilon_{ba}\,\big{(}\Pi_{\,I}^{a}\,\delta
X^{I}\big{)}\\\\[5.69046pt] &=\,-\int_{\mathcal{M}}\,d^{2}\sigma\,\delta
X^{I}\big{(}\partial_{a}\Pi_{\,I}^{a}\big{)}\,+\,\,\int_{0}^{\tau_{f}}\,d\tau\,\Big{[}\Pi_{\,I}^{a}\,\delta
X^{I}\big{\rvert}^{\sigma=\sigma_{f}}_{\sigma=0}\Big{]}\,,\end{aligned}$ (D.5)

where, in the second line, the indices have been renamed and the terms
grouped; and Stokes’s Theorem in d-dimensions [96] has been used in the third
line. In the final line the coordinates on the worldsheet parameter space are
chosen to be $\sigma^{a}=(t,\sigma)^{a}$ where
$\sigma^{a}\in\mathcal{M}=[0,t_{f}]\times[0,\sigma_{f}]$, and one of the
integrals of the last term is simplified by using $\delta
X^{I}\rvert_{t\in\\{0,t_{f}\\}}=0$.

Choosing the boundary conditions

$\Pi_{\,I}^{a}\,\delta
X^{I}\,\big{\rvert}^{\sigma=\sigma_{f}}_{\sigma=0}\,=\,0\,,$ (D.6)

the last term in Eq.(D.5) disappears. Hence, the string equations of motion
for the transverse fluctuations are given by

$0\,=\,\partial_{a}\,\Pi_{\,I}^{a}\,=\,\nabla_{a}\,\Pi^{a}_{\,I}\,,$ (D.7)

which agrees with Eq.(4.67). The second equality follows because
$\nabla_{a}\Pi^{a}_{\,\mu}:=\partial_{a}\Pi^{a}_{\,\mu}-\Gamma^{\alpha}_{\mu\nu}\,\partial_{a}X^{\nu}\Pi^{a}_{\,\mu}$
in curved spacetime, but the second term vanishes since the Christoffel
symbols are identically zero ($\Gamma^{\alpha}_{IJ}=0$) due to symmetry
between each of the transverse $X^{I}$ directions.

###  Appendix E Canonical Commutation Relations and Normalised Basis

This appendix aims to fix the normalization constant $A_{\omega}$, thereby
completely determining the general solution for the transverse equations of
motion (Eq.(4.92)) discussed in subsection (4.3.2). In order to determine
$A_{\omega}$ the theory is quantized: the scalar field $X(t,\sigma)$ and its
canonically conjugate momentum $P^{t}(t,\sigma)$ are promoted to operators
(Eqs.(4.93, 4.94) respectively), and suitable commutation relations are
imposed

$\displaystyle\begin{aligned}
&\left[\hat{X}(t,\sigma),\,n_{t}\,\hat{P}^{t}(t,\sigma^{\prime})\right]_{\Sigma}\,=\,i\,\delta(\sigma,\sigma^{\prime})\,=\,i\frac{\delta(\sigma-\sigma^{\prime})}{\sqrt{\tilde{g}}\rvert_{\Sigma}}\,,\\\\[5.69046pt]
&\left[\hat{X}(t,\sigma),\hat{X}(t,\sigma^{\prime})\right]_{\Sigma}\,=\,\left[n_{t}\,\hat{P}^{t}(t,\sigma),\,n_{t}\,\hat{P}^{t}(t,\sigma^{\prime})\right]_{\Sigma}\,=\,0\,,\end{aligned}$
(E.1)

where $\Sigma$ is a Cauchy hypersurface in the $x^{\mu}=(t,r)^{\mu}$ part of
spacetime that is chosen to be a constant time surface132132132Giving initial
conditions on this hypersurface determines the future (and past) evolution
uniquely., $\tilde{g}$ is the induced metric on $\Sigma$, and $n_{\mu}$ is the
future pointing normal to $\Sigma$ (where $n_{\mu}=\delta_{\mu
t}/\sqrt{-\tilde{g}_{tt}}$). The commutation relations described in Eq.(E.1)
are known as equal time commutation relations – they encode the expectation
that simultaneous measurements at different points on the string do not
interfere with each other [92].

Additionally, canonical creation and annihilation commutation relations on the
Fourier coefficient operators ($\hat{a}_{\omega},\hat{a}^{\dagger}_{\omega}$)
are enforced:

$\displaystyle\begin{aligned}
&\Big{[}\hat{a}_{\omega},\,\hat{a}_{\omega^{\prime}}^{\dagger}\Big{]}_{\Sigma}\,=\,2\pi\delta(\omega-\omega^{\prime})\,,\\\\[5.69046pt]
&\Big{[}\hat{a}_{\omega},\,\hat{a}_{\omega^{\prime}}\Big{]}_{\Sigma}\,=\,\Big{[}\hat{a}_{\omega}^{\dagger},\,\hat{a}_{\omega^{\prime}}^{\dagger}\Big{]}_{\Sigma}\,=\,0\,.\end{aligned}$
(E.2)

Consistency between the commutation relations Eq.(E.1) and Eq.(E.2) is
required133133133In order to ensure consistent quantization of the theory.. To
this end, the commutator bracket
$\left[\hat{X}(t,\sigma),\,n_{t}\,\hat{P}^{t}(t,\sigma^{\prime})\right]_{\Sigma}$
is calculated from the definitions Eqs.(4.93, 4.94), and the result is equated
to the relevant right hand side of Eq.(E.1). Specifically,

$\displaystyle\begin{aligned}
\left[\hat{X}(t,\sigma),\,n_{t}\hat{P}^{t}(t,\sigma^{\prime})\right]_{\Sigma}&=\hat{X}(t,\sigma)\rvert_{\Sigma}\,\,n_{t}\hat{P}^{t}(t,\sigma^{\prime})\rvert_{\Sigma}\,-\,n_{t}\hat{P}^{t}(t,\sigma^{\prime})\rvert_{\Sigma}\,\,\hat{X}(t,\sigma)\rvert_{\Sigma}\,,\\\\[5.69046pt]
&=-\frac{i}{2\pi\alpha^{\prime}}\,\frac{r^{2}}{l^{2}}\,\frac{\delta_{tt}}{\sqrt{\tilde{g}}\rvert_{\Sigma}}\int_{0}^{\infty}\frac{d\omega\,d\omega^{\prime}}{(2\pi)^{2}}\,\omega\,A_{\omega}\,A_{\omega^{\prime}}\,\,\,\,\times\,\\\\[5.69046pt]
&\,\,\,\,\,\,\,\,\Big{(}\big{[}f_{\omega}(\sigma)e^{-i\omega
t}\hat{a}_{\omega}+f^{\,*}_{\omega}(\sigma)e^{i\omega
t}\hat{a}^{\,\dagger}_{\omega}\big{]}\big{[}f_{\omega}(\sigma^{\prime})e^{-i{\omega^{\prime}}t}\hat{a}_{\omega^{\prime}}-f^{\,*}_{\omega}(\sigma^{\prime})e^{i{\omega^{\prime}}t}\hat{a}^{\,\dagger}_{\omega^{\prime}}\big{]}\\\\[5.69046pt]
&\,\,\,\,\,\,\,\,\,\,\,\,\,-\big{[}f_{\omega}(\sigma^{\prime})e^{-i{\omega^{\prime}}t}\hat{a}_{\omega^{\prime}}-f^{\,*}_{\omega}(\sigma^{\prime})e^{i{\omega^{\prime}}t}\hat{a}^{\,\dagger}_{\omega^{\prime}}\big{]}\big{[}f_{\omega}(\sigma)e^{-i\omega
t}\hat{a}_{\omega}+f^{\,*}_{\omega}(\sigma)e^{i\omega
t}\hat{a}^{\,\dagger}_{\omega}\big{]}\Big{)}\\\\[5.69046pt]
&=-\frac{i}{2\pi\alpha^{\prime}}\,\frac{r^{2}}{l^{2}}\,\frac{1}{\sqrt{\tilde{g}}\rvert_{\Sigma}}\int_{0}^{\infty}\frac{d\omega\,d\omega^{\prime}}{2\pi}\,\omega\,A_{\omega}\,A_{\omega^{\prime}}\,\,\,\,\times\,\\\\[5.69046pt]
&\,\,\,\,\,\,\,\,\Big{(}-f_{\omega}(\sigma)f^{\,*}_{\omega}(\sigma^{\prime})\,e^{i(\omega^{\prime}-\omega)t}\,\delta(\omega-\omega^{\prime})-f_{\omega}(\sigma^{\prime})f^{\,*}_{\omega}(\sigma)\,e^{i(\omega-\omega^{\prime})t}\,\delta(\omega^{\prime}-\omega)\Big{)}\,,\end{aligned}$
(E.3)

where the third line follows because the cross terms
$(\hat{a}_{\omega}\hat{a}_{\omega^{\prime}}-\hat{a}_{\omega^{\prime}}\hat{a}_{\omega})$
and
$(\hat{a}_{\omega}^{\dagger}\hat{a}_{\omega^{\prime}}^{\dagger}-\hat{a}_{\omega^{\prime}}^{\dagger}\hat{a}_{\omega}^{\dagger})$
vanish (using the second commutation relation in Eq.(E.2)). The first
commutation relation in Eq.(E.2) is also used in simplification. Collecting
terms, and converting to $(t,r_{*})$ coordinates using Eq.(4.53), yields

$\displaystyle\begin{aligned}
\therefore\left[\hat{X}(t,\sigma),\,n_{t}\hat{P}^{t}(t,\sigma^{\prime})\right]_{\Sigma}&=-\frac{i}{2\pi\alpha^{\prime}}\frac{r^{2}}{l^{2}}\frac{1}{\sqrt{\tilde{g}}\rvert_{\Sigma}}\int_{0}^{\infty}\frac{d\omega}{2\pi}\omega
A_{\omega}^{2}\left(-f_{\omega}(\sigma)f^{\,*}_{\omega}(\sigma^{\prime})-f_{\omega}(\sigma^{\prime})f^{\,*}_{\omega}(\sigma)\right)\\\\[5.69046pt]
&=\frac{i}{2\pi\alpha^{\prime}}\frac{r_{H}^{2}}{l^{2}}\coth^{2}\left(\frac{r_{H}(r_{s*}+\sigma^{\prime})}{l^{2}}\right)\frac{1}{\sqrt{\tilde{g}}\rvert_{\Sigma}}\int_{0}^{\infty}\frac{d\omega}{2\pi}\omega\,A_{\omega}^{2}\left(f_{\omega}(\sigma)f^{\,*}_{\omega}(\sigma^{\prime})+f_{\omega}(\sigma^{\prime})f^{\,*}_{\omega}(\sigma)\right)\,.\end{aligned}$
(E.4)

Inputting Eqs.(4.84, 4.86, 4.91), the commutator becomes134134134Mathematica
is used to be spared from the tedious multiplication (see Mathematica Notebook
[b]: BrownianMotion.nb).

$\left[\hat{X}(t,\sigma),\,n_{t}\hat{P}^{t}(t,\sigma^{\prime})\right]_{\Sigma}=\frac{1}{\sqrt{\tilde{g}}\rvert_{\Sigma}}\frac{i}{\pi\alpha^{\prime}}\frac{r_{H}^{2}}{l^{2}}\int_{0}^{\infty}\frac{d\omega}{2\pi}\omega
A_{\omega}^{2}\Big{(}e^{-i(\sigma+\sigma^{\prime})\omega}+e^{i(\sigma-\sigma^{\prime})\omega}+e^{i(\sigma+\sigma^{\prime})\omega}+e^{i(\sigma^{\prime}-\sigma)\omega}\Big{)}\,,$
(E.5)

where the near-horizon limit ($r\rightarrow
r_{H}\,\equiv\,r_{s*}\rightarrow-\infty\,\equiv\,\tilde{r}_{0}\rightarrow 1$)
has been taken after multiplication135135135If the commutation relations hold
in the near-horizon limit (i.e. for a specific value of $(t,\sigma)$) then
they hold $\forall\,(t,\sigma)$. In [42] de Boer et al. determine the
normalization constant $A$ by demanding the normalization of the modes through
the Klein-Gordon inner product. The near-horizon limit is taken during this
calculation. In a later work [46], Appendix A, a similar collaboration Atmaja
et al. argue that the inner product contribution from the near-horizon region
can be thought of as the overall inner product. There is a contribution to the
inner product from regions away from the horizon; however, the near-horizon
region is semi-infinite in the tortoise coordinate $r_{*}$ ($r\rightarrow
r_{H}\,\equiv\,r_{s*}\rightarrow-\infty$); and as such the normalization can
be completely fixed by the near-horizon regime. Therefore, the near-horizon
limit can be taken here without concern.. Note that in the near-horizon limit,

$\coth^{2}\left(\frac{r_{H}\,(r_{s*}+\sigma)}{l^{2}}\right)\,\rightarrow\,1,\,\,\,\,\,\,\text{as}\,\,\,\,\,\,r_{s*}\,\rightarrow\,-\infty\,.$
(E.6)

The commutator Eq.(E.5) can be further simplified

$\displaystyle\begin{aligned}
&\left[\hat{X}(t,\sigma),\,n_{t}\hat{P}^{t}(t,\sigma^{\prime})\right]_{\Sigma}\\\\[5.69046pt]
&=\frac{1}{\sqrt{\tilde{g}}\rvert_{\Sigma}}\frac{i}{\pi\alpha^{\prime}}\frac{r_{H}^{2}}{l^{2}}\left(\int_{0}^{\infty}\frac{d\omega}{2\pi}\omega
A_{\omega}^{2}\left[e^{-i(\sigma+\sigma^{\prime})\omega}+e^{-i(\sigma-\sigma^{\prime})\omega}\right]+\int_{0}^{\infty}\frac{d\omega}{2\pi}\omega
A_{\omega}^{2}\left[e^{i(\sigma+\sigma^{\prime})\omega}+e^{i(\sigma-\sigma^{\prime})\omega}\right]\right)\\\\[5.69046pt]
&=\frac{1}{\sqrt{\tilde{g}}\rvert_{\Sigma}}\frac{i}{\pi\alpha^{\prime}}\frac{r_{H}^{2}}{l^{2}}\left(-\int_{\infty}^{0}\frac{d\omega}{2\pi}\omega
A_{\omega}^{2}\left[e^{-i(\sigma+\sigma^{\prime})\omega}+e^{-i(\sigma-\sigma^{\prime})\omega}\right]+\int_{0}^{\infty}\frac{d\omega}{2\pi}\omega
A_{\omega}^{2}\left[e^{i(\sigma+\sigma^{\prime})\omega}+e^{i(\sigma-\sigma^{\prime})\omega}\right]\right)\\\\[5.69046pt]
&=\frac{1}{\sqrt{\tilde{g}}\rvert_{\Sigma}}\frac{i}{\pi\alpha^{\prime}}\frac{r_{H}^{2}}{l^{2}}\left(-\int_{-\infty}^{0}\frac{(-d\omega)}{2\pi}(-\omega)(-A_{\omega}^{2})\left[e^{i(\sigma+\sigma^{\prime})\omega}+e^{i(\sigma-\sigma^{\prime})\omega}\right]+\int_{0}^{\infty}\frac{d\omega}{2\pi}\omega
A_{\omega}^{2}\left[e^{i(\sigma+\sigma^{\prime})\omega}+e^{i(\sigma-\sigma^{\prime})\omega}\right]\right)\\\\[5.69046pt]
&=\frac{1}{\sqrt{\tilde{g}}\rvert_{\Sigma}}\frac{i}{\pi\alpha^{\prime}}\frac{r_{H}^{2}}{l^{2}}\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}\omega
A_{\omega}^{2}\left[e^{i(\sigma+\sigma^{\prime})\omega}+e^{i(\sigma-\sigma^{\prime})\omega}\right]\,.\end{aligned}$
(E.7)

In order for Eq.(E.7) to be consistent with the definition of the commutation
relation Eq.(E.1), the normalization constant needs to be defined as

$A_{\omega}\,:=\,\frac{l}{r_{H}}\,\sqrt{\frac{\pi\alpha^{\prime}}{\omega}}\,=\,\frac{\beta}{2\,\sqrt{\pi\omega}\,\lambda^{1/4}}\,,$
(E.8)

where the second equality follows from using the definition of the AdS radius
of curvature $l$ (Eq.(4.65)), and the relation $l=\beta r_{H}^{2}/2\pi$ from
Eq.(4.36).

To prove that the commutation relations Eq.(E.1) now hold, input the newly
defined $A_{\omega}$ into Eq.(E.7). Explicitly,

$\displaystyle\begin{aligned}
\left[\hat{X}(t,\sigma),\,n_{t}\,\hat{P}^{t}(t,\sigma^{\prime})\right]_{\Sigma}\,&=\frac{1}{\sqrt{\tilde{g}}\rvert_{\Sigma}}\,\frac{i}{\pi\alpha^{\prime}}\,\frac{r_{H}^{2}}{l^{2}}\,\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}\omega\bigg{(}\frac{l^{2}}{r_{H}^{2}}\,\frac{\pi\alpha^{\prime}}{\omega}\bigg{)}\left[e^{i(\sigma+\sigma^{\prime})\omega}\,+\,e^{i(\sigma-\sigma^{\prime})\omega}\right]\,\\\\[5.69046pt]
&=\frac{i}{\sqrt{\tilde{g}}\rvert_{\Sigma}}\,\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}\,\omega\,\,\left[e^{i(\sigma+\sigma^{\prime})\omega}\,+\,e^{i(\sigma-\sigma^{\prime})\omega}\right]\\\\[5.69046pt]
&=\frac{i}{\sqrt{\tilde{g}}\rvert_{\Sigma}}\,\left[\delta(\sigma+\sigma^{\prime})+\delta(\sigma-\sigma^{\prime})\right]\\\\[5.69046pt]
&=\frac{i}{\sqrt{\tilde{g}}\rvert_{\Sigma}}\,\delta(\sigma-\sigma^{\prime})\,,\end{aligned}$
(E.9)

where the third line follows from the Dirac delta function property
$\delta(y-x)=\frac{1}{2\pi}\,\int_{-\infty}^{\infty}dk\,e^{-ik(y-x)}$; and, in
the final line, the first Dirac delta function has disappeared since
$\sigma,\,\sigma^{\prime}\in[0,\sigma_{f}]$. Hence, the scalar field and
conjugate momentum commutation relations (Eq.(E.1)) and the Fourier
coefficient creation and annihilation commutation relations (Eq.(E.7)) are
consistent; proving that the normalization constant $A_{\omega}$ is correctly
given by Eq.(E.8).

###  Appendix F Leading Order Contributions of the Near-Horizon Tortoise
Coordinate

The tortoise coordinate $r_{*}$ is defined in Eq.(4.39). In the near-horizon
limit ($r=(1+\tilde{\epsilon})r_{H}$) the tortoise coordinate can be expanded
and truncated around $\tilde{\epsilon}=0$, in order to yield the near-horizon
tortoise coordinate, $\tilde{\epsilon}_{*}$. This coordinate is given – for
general $d$ dimensions – by

$\displaystyle\begin{aligned}
\tilde{\epsilon}_{*}&:=\frac{l^{2}}{r_{H}^{2}(d-1)}\ln(\tilde{\epsilon})+\frac{(d-4)l^{2}}{2r_{H}^{2}(d-1)}\tilde{\epsilon}+\frac{\left(d^{2}-14d+36\right)l^{2}}{24r_{H}^{2}(d-1)}\tilde{\epsilon}^{2}-\frac{\left(5d^{2}-46d+96\right)l^{2}}{72r_{H}^{2}(d-1)}\tilde{\epsilon}^{3}+\mathcal{O}\left(\tilde{\epsilon}\right)^{4}\\\\[5.69046pt]
&\,=a_{0}\ln(\tilde{\epsilon})+a_{1}\tilde{\epsilon}+a_{2}\tilde{\epsilon}^{2}+a_{3}\tilde{\epsilon}^{3}+\mathcal{O}\left(\tilde{\epsilon}\right)^{4}\,,\end{aligned}$
(F.1)

where $r_{*}=r_{H}\tilde{\epsilon}_{*}$, and $a_{0}$, $a_{1}$, $a_{2}$,
$a_{3}$ are constants. To leading order, the near-horizon AdSd-Schwarzschild
metric is given by

$ds_{d}^{2}=\frac{r_{H}^{2}}{l^{2}}\,d\vec{X}_{I}^{2}\,-\,\frac{(d-1)r_{H}^{2}\tilde{\epsilon}}{l^{2}}\,dt^{2}\,+\,\frac{l^{2}}{(d-1)\tilde{\epsilon}}\,d\tilde{\epsilon}^{2}\,,$
(F.2)

where all terms are $\mathcal{O}(\tilde{\epsilon})$. To be consistent, at
leading order $\tilde{\epsilon}_{*}$ must be considered to
$\mathcal{O}(\tilde{\epsilon})$ – hence, it has the form
$\tilde{\epsilon}_{*}=a_{0}\ln(\tilde{\epsilon})+a_{1}\tilde{\epsilon}+\mathcal{O}\left(\tilde{\epsilon}\right)^{2}$.

The near-horizon tortoise coordinate is used to convert the near-horizon
metric into $(t,\tilde{\epsilon}_{*})$ coordinates, thereby arriving at a
conformally flat description of near-horizon AdSd-Schwarzschild, from which
the leading order string solution and the transverse equations of motion can
be found. Considering Eq.(5.109) it is clear that the metric conversion factor
between ($t,\tilde{\epsilon}$) and ($t,\tilde{\epsilon}_{*}$) coordinates is

$\displaystyle\begin{aligned}
e^{\frac{r_{H}^{2}(d-1)\tilde{\epsilon}_{*}}{l^{2}}}&=e^{\frac{r_{H}^{2}(d-1)}{l^{2}}\left(a_{0}\ln(\tilde{\epsilon})+a_{1}\tilde{\epsilon}+\mathcal{O}\left(\tilde{\epsilon}\right)^{2}\right)}\\\\[5.69046pt]
&=e^{c\,a_{0}\ln(\tilde{\epsilon})}\,e^{c\,a_{1}\tilde{\epsilon}}\,e^{\mathcal{O}\left(\tilde{\epsilon}\right)^{2}}\\\\[5.69046pt]
&=\tilde{\epsilon}^{c\,a_{0}}\left(1+c\,a_{1}\tilde{\epsilon}+\mathcal{O}\left(\tilde{\epsilon}\right)^{2}\right)\left(1+\mathcal{O}\left(\tilde{\epsilon}\right)^{2}\right)\\\\[5.69046pt]
&=\tilde{\epsilon}\left(1+\frac{(d-4)}{2}\tilde{\epsilon}+\mathcal{O}\left(\tilde{\epsilon}\right)^{2}\right)\left(1+\mathcal{O}\left(\tilde{\epsilon}\right)^{2}\right)\\\\[5.69046pt]
&=\tilde{\epsilon}\,+\,\mathcal{O}\left(\tilde{\epsilon}\right)^{2}\,,\end{aligned}$
(F.3)

where the relevant leading order form of $\tilde{\epsilon}_{*}$ is used in the
first equality, the constant $c=r_{H}^{2}(d-1)/l^{2}$ is defined in the second
line, and in the third line the terms $e^{c\,a_{1}\tilde{\epsilon}}$ and
$e^{\mathcal{O}\left(\tilde{\epsilon}\right)^{2}}$ are series expanded around
$\tilde{\epsilon}=0$. While inputting the values for $c$, $a_{0}$, and $a_{1}$
(from Eq.(F.1)) into the fourth line, notice that $c\,a_{0}=1$ by
construction. All terms of $\mathcal{O}\left(\tilde{\epsilon}\right)^{2}$ are
disregarded, in order to be consistent with the leading order near-horizon
AdSd-Schwarzschild metric. Hence, from Eq.(F.3), the inverse near-horizon
tortoise coordinate is given by

$\tilde{\epsilon}:=e^{\frac{r_{H}^{2}(d-1)\tilde{\epsilon}_{*}}{l^{2}}}\,,$
(F.4)

at leading order; while the near-horizon tortoise coordinate is given by

$\tilde{\epsilon}_{*}:=\frac{l^{2}}{r_{H}^{2}(d-1)}\ln(\tilde{\epsilon})\,,$
(F.5)

at leading order. Claims that Eq.(F.5) is the leading order contribution to
the near-horizon tortoise coordinate have been made in [52]. This is, however,
the first time to the authors’ knowledge a proof to this effect (even a
heuristic one) has been provided.

###  Appendix G Accessing the Mathematica Code

The Mathematica code used to generate all the analytic results and plots in
this article is provided as supplementary material.

The code is organised into four comprehensively annotated Mathematica
notebooks containing: (a) the mapping of the heavy and light quark test string
solutions between the worldsheet parameter space and target spacetime
(MappingWorldSheetToTarget.nb); (b) calculations pertaining to the analysis of
heavy and light quark test strings undergoing Brownian motion
(BrownianMotion.nb); (c) the expansion of the AdSd-Schwarzschild metric in the
near-horizon region for the light quark’s test string configuration
(NearHorizonAdSd.nb); and (d) drag force calculations in AdS/CFT
(DragForce.nb).

For readers please always check for the latest versions of the notebooks in
the GitHub repository:

https://github.com/AlexesMes/brownian-motion-of-quarks.

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