Interior analysis, stretched technique and bubbling geometries

Qiuye Jia1, Hai Lin2,3

_Mathematical Sciences Institute, Australian National University, Canberra ACT
2601, Australia_

_Shing-Tung Yau Center and School of Mathematics, Southeast University,
Nanjing 210096, China _

_Yau Mathematical Sciences Center, Tsinghua University, Beijing 100084, China
_

We perform a detailed analysis of quarter BPS bubbling geometries with AdS
asymptotics and their corresponding duality relations with their dual states
in the quantum field theory side, among other aspects. We derive generalized
Laplace-type equations with sources, obtained from linearized Monge-Ampere
equations, and used for asymptotically AdS geometry. This enables us to obtain
solutions specific to the asymptotically AdS context. We conduct a thorough
analysis of boundary conditions and explore the stretched technique where
boundary conditions are imposed on a stretched surface. These boundary
conditions include grey droplets. This stretched technique is naturally used
for the superstar, where we place grey droplet boundary conditions on the
stretched surface. We also perform a coarse-graining of configurations and
analyze the symplectic forms on the configuration space and their coarse-
graining.

##  1 Introduction

The gauge-gravity correspondence has revealed remarkable relations and
equivalences between ordinary quantum systems on one side and quantum gravity
on the other side, e.g. [1, 2, 3] and other related aspects. In the
correspondence, the spacetime geometries are emergent from the dynamics of
ordinary quantum systems on the dual side.

On the gravity side, there are giant graviton branes [4]-[10]. The
corresponding states in the Hilbert space of the quantum field theory are
explicitly mapped to the states in the gravity side. The dual large operators
as well as their operator bases have been illuminated [7]-[10]. These are
brane-like operators, which are holographically dual to branes on the gravity
side. Moreover, these branes are associated to bubbling geometries, in the
bulk side. Both giant graviton branes and bubbling geometries have string
excitations. Analyses in the field theory side show that these different
states live in the same Hilbert space. See also related interesting
discussions, e.g. [11]-[18].

These smooth bubbling geometries correspond to excited states in the Hilbert
space of the dual quantum field theory. The states in the Hilbert space of the
quantum field theory are explicitly mapped to the states in the gravity side.
The Hilbert spaces of states in both sides are isomorphic to each other. The
configuration spaces of the droplet configurations have been analyzed and
illuminated in, e.g. [19, 20, 21, 22]. Various types of degrees of freedom on
the gravity side or on the string theory side, can be described by
corresponding operators on the quantum field theory side. Moreover, different
operator bases can be transformed into each other by a change of basis.

There are various operator bases for large operators, corresponding to heavy
states in the gravity. The large operators includes those describe giant
gravitons and emergent geometries, among other things. The large operators
also describe further excitations on these heavy excited states. The
correlation functions between light operators and large operators can also be
computed. The large operators can be expanded systematically in terms of
representation bases. For more information along these ideas and aspects, see
e.g. [23]-[35]. Various bases can be transformed into each other. These heavy
operators correspond to heavy excited states in the Hilbert space of the
quantum gravity side.

One can also compute the interactions of these objects in the gravity side, by
computing correlation functions of their dual operators. For example, the
correlation functions of giant gravitons and light fields have been computed
from gravity and from gauge theory sides, and they match with each, e.g.
[36]-[43]. The operators used in the computations are Schur operators and
their superpositions. It has been very useful computing correlation functions
involving large operators, e.g. [44, 45, 43], for strong gravitation in AdS,
among other things. Transformations between bases are useful for computations,
including those aspects related to amplitudes of the gravity side. These heavy
operators involve emergent geometries in the dual quantum gravity system.
These excited states of geometries live in the same quantum Hilbert space. One
may further deform the Hamiltonians while keeping structures in the Hilbert
space. The set-up on the field theory side is very calculable, and this
enables us to go into further and more sophisticated regimes of the duality.

Other examples or classes of bubbling geometries with quarter BPS and eighth
BPS cases have been analyzed in e.g. [46]-[55], e.g. in the $N$=4 SYM example,
among other things. There are other less-supersymmetric deformations, e.g.
[56] and related discussions. One can obtain the symplectic form for the
configuration of the quarter BPS geometries, by using the same procedure as in
[21, 22, 57]. Moreover, by discrete identification of base space, one can
obtain orientifold cases, e.g. [58]-[63] and their related aspects. Various
other aspects have also been discussed and addressed, e.g. [64]-[73]. There
are many other related analysis and aspects. See also discussions on related
giant gravitons, e.g. [74, 75, 76] and their closely related aspects.

The correlators with heavy operators have been considered from various
perspectives. One can also compute HHLL correlators, where heavy H operators
are dual to background bubbling geometry and light L operators are dual to
light fields in AdS. Heavy means relatively heavier than the lighter one.
Fluctuations of light fields on these backgrounds can also be captured by
those types of very interesting correlators. Very interesting and similar
types of correlators involving heavy and light operators have been computed
recently in e.g. [77, 78, 79]. HHLL correlators can also capture the string
degrees of freedom in bubbling geometries, e.g. [80, 81, 82, 83] and related
references.

Heavy geometries are also dual to Wilson operators and surface operators.
There are also bubbling geometries for Wilson operators, e.g. [84, 94] and for
surface operators, e.g. [95, 99], as examples, and various their related
references. There are many interesting and closely related phenomena, among
these cases. In the context of Wilson operators, the involved path integral
can be computed by powerful localization formalism. The localization
techniques are also very useful for aspects of local operators, e.g. [27, 100,
101] and related references. It would be nice to have an unified understanding
of various interconnected aspects.

Here, we analyze in more details the quarter BPS bubbling geometries and their
properties and duality relations with their dual states on the quantum field
theory side, among many other aspects. In Sec. 2, we obtain generalized
Laplace-type equations around reference geometries. In Sec. 3, we analyze
solutions with AdS asymptotics. In Sec. 4, we analyze in details solutions to
the general equations with boundary conditions. In Sec. 5, we analyze the
stretched technique where we place boundary conditions on the stretched
surface. In Sec. 5.2 and 5.3, we make coarse graining of configurations, with
grey droplet boundary conditions, and analyze coarse-graining for superstar
geometries, among other things. Finally in Sec. 6, we conclude and discuss
closely related aspects. More details of computations are included in
Appendices A to F.

##  2 Linearized equation with sources

###  2.1 Linearized equation on reference geometries

We consider smooth quarter BPS bubbling geometries as described in the
Introduction section. We derive a linearized equation in 5d (parametrized by
$y,z_{i},z_{\bar{j}},i,j=1,2$). Here, in the same way, there are multiple
droplets in quarter BPS bubbling geometries. The 10D metric is

$\displaystyle
ds_{10}^{2}=-h^{-2}(dt+\omega)^{2}+h^{2}((Z+\frac{1}{2})^{-1}2h_{i\bar{j}}dz^{i}d\bar{z}^{\bar{j}}+dy^{2})+y(e^{G}d\Omega_{3}^{2}+e^{-G}(d\psi+A)^{2}),$
$\displaystyle h^{-2}=2y\cosh G,\qquad
h_{i\bar{j}}=\partial_{i}\partial_{\bar{j}}K.$ (2.1)

We have
$Z\equiv\frac{1}{2}\mathrm{\tanh}G=-\frac{1}{2}y\partial_{y}\frac{1}{y}\partial_{y}K$.
The four-dimensional base metric $h_{i\bar{j}}$ is Kahler, and satisfies a
Monge-Ampere type equation, and an auxiliary condition, see also [46, 47],

$\displaystyle\log\det h_{i\bar{j}}=\log(Z+\frac{1}{2})+n\eta\log
y+\frac{1}{y}(2-n\eta)\partial_{y}K+D(z_{i},\bar{z}_{\bar{j}}),$
$\displaystyle(1+\ast_{4})\partial\bar{\partial}D=\frac{4}{y^{2}}(1-n\eta)\partial\bar{\partial}K.$
(2.2)

The range of $y$ is $0\leq y<\infty.$

Consider the case $n\eta=1$, $D=\mathrm{const}$. In this case $A$ may be set
to zero. The solutions for $AdS_{5}\times S^{5}$ are in e.g. (6.19) of [46].
Other smooth solutions are characterized by smooth droplet boundary
conditions. The equation, for $n\eta=1$, $D=\mathrm{const.}$, is

$\displaystyle\det\partial_{i}\partial_{\bar{j}}K$ $\displaystyle=$
$\displaystyle(Z+\frac{1}{2})e^{\frac{1}{y}\partial_{y}K+\frac{1}{2}\log
y^{2}+D},$ $\displaystyle Z$ $\displaystyle\equiv$
$\displaystyle-\frac{1}{2}y\partial_{y}\frac{1}{y}\partial_{y}K.$ (2.3)

Here $D$ is a constant and can be related to an irrelevant integration
constant. $\frac{1}{2}\log y^{2}+D$ has no $z_{i},z_{\bar{j}}$ dependence and
can be associated to an integration constant for $K$. The RHS only involves
derivative with respect to $y^{2}$, and the LHS only involves derivative with
respect to $z_{i},z_{\bar{j}}$. There also exist scaling transformations and
shift transformations for the Kahler potential [49, 46].

At $y=0$, we have $Z-\frac{1}{2}=-1$ for black droplets where $S^{3}$ shrinks
to zero smoothly. The $S^{3}$ combines with a radial coordinate to form
locally flat space when shrinking smoothly. At $y=0$, we have
$Z-\frac{1}{2}=0$ for white droplets where $S^{1}$ shrinks to zero smoothly.
The $S^{1}$ combines with a radial coordinate to form locally flat space when
shrinking smoothly. White droplets are the complementary regions of the black
droplets. The ten dimensional geometry is smooth [46]. There are quantized
flux quanta through various interior cycles in the smooth geometry, and the
geometries have complicated topologies and are everywhere smooth [46, 49, 47].

There is certain complementarity between white droplets and black droplets in
the interior region. When the black droplets become smaller, the white
droplets in between them may become bigger and vice versa, since in the
interior droplet space, the white droplets are the complementary regions of
the black droplets. When you define one region, the other is its complement,
and vice versa. In fact, the white region can be defined as the complement of
the black region. We may redefine a function
$u(z)_{y=0}=\frac{1}{2}-Z(z,y)_{y=0}$, and then $u(z)$ is 1 on black regions
and 0 on white regions. These smooth geometries correspond to pure states of
the theory.

Consider the solution for $AdS_{5}\times S^{5}$ as $K_{(0)}$. We want to get a
special family of other solutions by adding sources $\delta_{i}\ $on the $y=0$
space (parametrized by $z_{i},z_{\bar{j}},i,j=1,2$), on top of the
$AdS_{5}\times S^{5}$ background. This idea has not been proposed before. We
superimpose these sources for the function $Z$.

The solution for $AdS_{5}\times S^{5}$ is as follows. Here
$r^{2}=|z_{1}|^{2}+|z_{2}|^{2}$.
$K_{(0)}^{\prime}=\partial_{r^{2}}K_{(0)}={B/}2{r^{2}}$, which can be
integrated to give the Kahler potential

$K_{(0)}(r^{2},y)=\frac{1}{2}\int^{r^{2}}\frac{{B(r^{2},y)}}{{r^{2}}}d(r^{2}).$
(2.4)
$B=\frac{1}{2}(r^{2}-y^{2}-1)+\sqrt{\frac{1}{4}(r^{2}+y^{2}-1)^{2}+y^{2}}.$
(2.5)

The Kahler potential is

$\displaystyle K_{(0)}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\frac{1}{2}(r^{2}+y^{2}+1)+\sqrt{\frac{1}{4}(r^{2}+y^{2}-1)^{2}+y^{2}}\right)$
$\displaystyle-\frac{1}{2}\log\left(\frac{1}{2}(r^{2}+y^{2}+1)+\sqrt{\frac{1}{4}(r^{2}+y^{2}-1)^{2}+y^{2}}\right)$
$\displaystyle-\frac{1}{2}y^{2}\log\left(\frac{1}{2}(-r^{2}+y^{2}+1)+\sqrt{\frac{1}{4}(r^{2}+y^{2}-1)^{2}+y^{2}}\right)+\frac{1}{2}y^{2}\log(y).$

For more details, see Appendix A.

Now consider a family of new solutions

$K=K_{(0)}+sK_{(1)},$ (2.7)

where $s$ is a small parameter. The $y=0$ is a four-dimensional droplet space.
We cover the source $\delta_{i}$ by a four-dimensional ball-neighborhood
$B_{i}$. We use an enclosure surface which is along the $y$ direction and ends
on this four-dimensional ball-neighborhood. The above solution is good for the
region outside these enclosure surfaces; these regions are where the change of
the metric due to added sources are small; The norm of
$s\partial_{i}\partial_{\bar{j}}K_{(1)}$ is much smaller than the norm of
$\partial_{i}\partial_{\bar{j}}K_{(0)}$ in these regions. The white region for
$Z=\frac{1}{2},y=0$ on the droplet space is now
$R^{4}\backslash\cup_{i}B_{i}$. The region $B_{i}$ are sources with
$Z=-\frac{1}{2},y=0$. In other words, $B_{i}$ are ‘$Z=-\frac{1}{2}$’ sources
of black regions. We make droplet configurations as the collections of these
‘$Z=-\frac{1}{2}$’ sources. The $B_{i}$ are domains on the $y=0$ space.

Now we derive 5d linearized equation for $K_{(1)}$, from (2.3), for the region
outside the enclosure surfaces. We keep terms up to $o(s)$ terms. The order
$o(s^{0})$ terms cancel precisely, due to the reference geometry. We have:

$\displaystyle\det\partial_{i}\partial_{\bar{j}}K$ $\displaystyle=$
$\displaystyle\partial_{1}\partial_{\bar{1}}K\partial_{2}\partial_{\bar{2}}K-\partial_{1}\partial_{\bar{2}}K\partial_{2}\partial_{\bar{1}}K$
(2.8) $\displaystyle=$
$\displaystyle\det\partial_{i}\partial_{\bar{j}}K_{(0)}$
$\displaystyle+s\left(\partial_{1}\partial_{\bar{1}}K_{(0)}\right)\partial_{2}\partial_{\bar{2}}K_{(1)}+s\left(\partial_{2}\partial_{\bar{2}}K_{(0)}\right)\partial_{1}\partial_{\bar{1}}K_{(1)}$
$\displaystyle-s\left(\partial_{1}\partial_{\bar{2}}K_{(0)}\right)\partial_{2}\partial_{\bar{1}}K_{(1)}-s\left(\partial_{2}\partial_{\bar{1}}K_{(0)}\right)\partial_{1}\partial_{\bar{2}}K_{(1)}$

and

$\displaystyle\frac{1}{2}(1-4y^{2}\partial_{y^{2}}\partial_{y^{2}}K)e^{2\partial_{y^{2}}K+\frac{1}{2}\log
y^{2}+D}$ (2.9) $\displaystyle=$
$\displaystyle\frac{1}{2}(1-4y^{2}\partial_{y^{2}}\partial_{y^{2}}K_{(0)})e^{\frac{1}{2}\log
y^{2}+D}e^{2\partial_{y^{2}}K_{(0)}}$
$\displaystyle+s[(1-4y^{2}\partial_{y^{2}}\partial_{y^{2}}K_{(0)})\partial_{y^{2}}K_{(1)}-2y^{2}\partial_{y^{2}}\partial_{y^{2}}K_{(1)}]e^{2\partial_{y^{2}}K_{(0)}+\frac{1}{2}\log
y^{2}+D}.$

Hence (denoting $V=K_{(1)}$),

$\displaystyle\left(\partial_{2}\partial_{\bar{2}}K_{(0)}\right)\partial_{1}\partial_{\bar{1}}V+\left(\partial_{1}\partial_{\bar{1}}K_{(0)}\right)\partial_{2}\partial_{\bar{2}}V$
(2.10)
$\displaystyle-\left(\partial_{1}\partial_{\bar{2}}K_{(0)}\right)\partial_{2}\partial_{\bar{1}}V-\left(\partial_{2}\partial_{\bar{1}}K_{(0)}\right)\partial_{1}\partial_{\bar{2}}V$
$\displaystyle=$ $\displaystyle e^{2\partial_{y^{2}}K_{(0)}+\frac{1}{2}\log
y^{2}+D}[(1-4y^{2}\partial_{y^{2}}\partial_{y^{2}}K_{(0)})\partial_{y^{2}}V-2y^{2}\partial_{y^{2}}\partial_{y^{2}}V].$

This form of linearized equation can also be written as

$\displaystyle\left(\partial_{2}\partial_{\bar{2}}K_{(0)}\right)\partial_{1}\partial_{\bar{1}}V+\left(\partial_{1}\partial_{\bar{1}}K_{(0)}\right)\partial_{2}\partial_{\bar{2}}V-\left(\partial_{1}\partial_{\bar{2}}K_{(0)}\right)\partial_{2}\partial_{\bar{1}}V-\left(\partial_{2}\partial_{\bar{1}}K_{(0)}\right)\partial_{1}\partial_{\bar{2}}V$
$\displaystyle+2\left(\det\partial_{i}\partial_{\bar{j}}K_{(0)}\right)\left(\frac{y^{2}}{(Z+\frac{1}{2})_{(0)}}\partial_{y^{2}}\partial_{y^{2}}V-\partial_{y^{2}}V\right)=0.$
(2.11)

This is a generalized Laplace-like equation. In the above, we have
$V=K_{(1)},K=K_{(0)}+sK_{(1)}.$

This equation (2.11) holds, starting from any smooth quarter BPS geometry with
its corresponding Kahler potential denoted $K_{(0)}$, as a reference geometry,
then the linearized equation on the reference geometry in the regions outside
the enclosure surfaces is (2.11). This is the linearized technique.

###  2.2 Adding sources and superimposing sources for the function $Z$

For the linearized equation, $Z-\frac{1}{2}$ has a superimposition structure,
whose boundary value at $y=0$, is $-1$ on the black droplets and $0$ on the
white droplets.

Because $Z\equiv-2y^{2}\partial_{y^{2}}\partial_{y^{2}}K$, the new solution
$K\ $also induces new solution for $Z$, as

$Z=Z_{(0)}+V_{(1)}.$ (2.12)

In other words,

$Z-\frac{1}{2}=Z_{(0)}-\frac{1}{2}+V_{(1)}.$ (2.13)

$V_{(1)}:=Z_{(1)}=-2sy^{2}\partial_{y^{2}}\partial_{y^{2}}K_{(1)}$. Hence for
the solution of the linearized equation, the boundary value of $V_{(1)}~{}$at
$y=0$, is $-1$ on the black droplets and $0$ on the white droplets.

We denote $V=V_{(1)}:=Z_{(1)}$. The $Z_{(1)}$ satisfies the same form of
Laplace-like equation:

$\displaystyle\left(\partial_{2}\partial_{\bar{2}}K_{(0)}\right)\partial_{1}\partial_{\bar{1}}V+\left(\partial_{1}\partial_{\bar{1}}K_{(0)}\right)\partial_{2}\partial_{\bar{2}}V-\left(\partial_{1}\partial_{\bar{2}}K_{(0)}\right)\partial_{2}\partial_{\bar{1}}V-\left(\partial_{2}\partial_{\bar{1}}K_{(0)}\right)\partial_{1}\partial_{\bar{2}}V$
$\displaystyle+2\left(\det\partial_{i}\partial_{\bar{j}}K_{(0)}\right)\left(\frac{y^{2}}{Z_{(0)}+\frac{1}{2}}\partial_{y^{2}}\partial_{y^{2}}V-\partial_{y^{2}}V\right)=0.$
(2.14)

Then we place boundary condition at $y=0$, by adding above ‘$Z=-\frac{1}{2}$’
sources as discussed in Sec. 2.1, and solve $V=Z_{(1)}\ $using the kernel of
this equation. The sources are approximated as delta function sources for
‘$Z=-\frac{1}{2}$’ sources. In other words, the RHS is 0 for $y>0$, and has
‘$Z=-\frac{1}{2}$’ sources at $y=0$. We have the identity:

$\int_{B_{i}}(-1)d^{4}z_{i}^{\prime}=(-1)v_{s,i}=-\int_{B_{i}}v_{s,i}\delta^{(4)}(z-z_{i}^{\prime})d^{4}z_{i}^{\prime}\
.$ (2.15)

Here $z$ denotes $(z_{1},z_{\bar{1}},z_{2},z_{\bar{2}})$. Here $i$ denotes the
$i$-th sources $\delta_{i}$. Hence

$Z_{(1)}(z,0)|_{B_{i}}=-\int_{B_{i}}\delta^{(4)}(z-z_{i}^{\prime})d^{4}z_{i}^{\prime}=-1.$
(2.16)

$\delta_{i}$ is at the center of $B_{i}$.

In short, we solve the equation for the kernel $f(z,y;z^{\prime})$:

$\displaystyle\left(\partial_{2}\partial_{\bar{2}}K_{(0)}\right)\partial_{1}\partial_{\bar{1}}f+\left(\partial_{1}\partial_{\bar{1}}K_{(0)}\right)\partial_{2}\partial_{\bar{2}}f-\left(\partial_{1}\partial_{\bar{2}}K_{(0)}\right)\partial_{2}\partial_{\bar{1}}f-\left(\partial_{2}\partial_{\bar{1}}K_{(0)}\right)\partial_{1}\partial_{\bar{2}}f$
$\displaystyle+2\left(\det\partial_{i}\partial_{\bar{j}}K_{(0)}\right)\left(\frac{y^{2}}{Z_{(0)}+\frac{1}{2}}\partial_{y^{2}}\partial_{y^{2}}f-\partial_{y^{2}}f\right)=0\
;$ $\displaystyle f(z,0;z_{i}^{\prime})=\delta^{(4)}(z-z_{i}^{\prime})$ (2.17)

in which the second equation is the boundary condition at the source locations
$z=z_{i}^{\prime}$. Because in linearized equation, we can put
‘$Z=-\frac{1}{2}$’ sources in rather general locations, we can superimpose the
sources at different generic separate locations labelled by $i$. In other
words, $V_{(1)}$ has a superimposition structure for superimposing the
‘$Z=-\frac{1}{2}$’ loci.

One can obtain the kernel of this equation, which we denote as

$f(z_{i},z_{\bar{j}},y;z_{i}^{\prime},z_{\bar{j}}^{\prime}):=f(z,y;z^{\prime}),$
(2.18)

which is a solution of (2.17). Here $z$ denotes
$(z_{1},z_{\bar{1}},z_{2},z_{\bar{2}})$. Hence

$Z(z,y)=Z_{(0)}(z,y)-\sum_{i=1}^{n}\int
f(z,y;z_{i}^{\prime})d^{4}z_{i}^{\prime}.$ (2.19)

Here $i$ denotes the $i$-th sources. The $f(z,y;z^{\prime})\ $denotes the
kernel.

We see that the above scheme is perfectly nice, since we can make both $s$ and
the domain size $v_{s,i}$ of $B_{i}$ small, with a scaling relation between
them. Hence $s$ is related to the size of the enclosure surface of each small
droplet.

Denote the quanta of each small droplet as $q_{i}$. The total number of flux
quanta of small droplets are $q_{t}=\sum\limits_{i}q_{i}$. Denoting
$q=\mathrm{max}\\{q_{i}\\}$, the parameter is approximately of order $s\sim
o(ql^{2}N^{-1})$. In order to linearize on the reference geometry e.g.
$AdS_{5}\times S^{5}$, we also have $q_{i}N^{-1}\ll 1$, and $q_{i}$ can still
be large although it is much smaller than $N$. Hence $qN^{-1}\ll 1$. The
dilute distribution limit in Appendix B means $q_{i}\ll N$. This is in the
same regime as the linearized technique.

We can also consider the cases of grey droplets. Grey droplets mean that the
boundary value of $\frac{1}{2}-Z$ is not 1 but smaller than 1. These are
discussed in Sec. 5.3.

###  2.3 Relation between the dilute distribution regime and linearized
equation with sources

We can also consider dilute distributions of small droplets. The idea of
dilute distribution and extreme-ratio limit, and near droplet geometry has
been discussed in [46]. We do not repeat the description here, and we give the
analysis in Appendix B. The quanta for each small droplet $q_{i}$ is much
smaller than the total quanta $N$, i.e. $q_{i}\ll N$; and this is referred to
as the extreme-ratio limit. Dilute means that the droplets are isolated from
each other.

Starting from the extreme-ratio limit, and with one central droplet much
larger than other remote small droplets, we enclose the small droplets. The
region outside the enclosures are described by the linearized equation, which
are linearized on the reference geometry. The reference geometry for the
linearization, in the simplest case, is the central big droplet. We also
linearize on other smooth droplet background, generalized from the central
droplet background. The source domain in the linearized technique, is resolved
by small smooth droplets, as in e.g. B.1 of Appendix B. Near each small black
droplet, that is near $y=0$,$~{}Z=-\frac{1}{2}$, they satisfy the smooth
droplet boundary condition.

##  3 Simplification of the equations

###  3.1 Derivation of the simplified equations

For the asymptotic AdS space at the asymptotia, we have

$Z_{(0)}=\frac{r^{2}+y^{2}-l^{2}}{2\sqrt{(r^{2}+y^{2}-l^{2})^{2}+4y^{2}l^{2}}}.$
(3.1)

Here $r^{2}=|z_{1}|^{2}+|z_{2}|^{2}$, and $l$ is a parameter (and sometimes
for simplicity are set to $l=1$ in appropriate unit). One have that
$Z_{(0)}\equiv-2y^{2}\partial_{y^{2}}\partial_{y^{2}}K_{(0)}$. This is a
single central droplet.

Let’s further make simplification of equation (2.17). Now we consider
$(r^{2}+y^{2}-l^{2})^{2}\gg y^{2}l^{2}$ approximation, and

$\displaystyle Z_{(0)}-\frac{1}{2}$ $\displaystyle=$
$\displaystyle-\frac{y^{2}l^{2}}{(r^{2}+y^{2}-l^{2})^{2}}+o(l^{4})$ (3.2)
$\displaystyle=$
$\displaystyle-\frac{y^{2}l^{2}}{(r^{2}+y^{2})^{2}}+o(l^{4}),$
$\displaystyle\frac{1}{Z_{(0)}+\frac{1}{2}}$ $\displaystyle=$ $\displaystyle
1+\frac{y^{2}l^{2}}{(r^{2}+y^{2}-l^{2})^{2}}+o(l^{4})$ (3.3) $\displaystyle=$
$\displaystyle 1+\frac{y^{2}l^{2}}{\left(r^{2}+y^{2}\right)^{2}}+o(l^{4}).$

Hence

$\displaystyle\left(\frac{y^{2}}{Z_{(0)}+\frac{1}{2}}\partial_{y^{2}}\partial_{y^{2}}V-\partial_{y^{2}}V\right)$
(3.4) $\displaystyle=$
$\displaystyle\frac{1}{4}y^{3}\partial_{y}\frac{1}{y^{3}}\partial_{y}V+\frac{l^{2}y^{2}}{(r^{2}+y^{2})^{2}}y^{2}\partial_{y^{2}}\partial_{y^{2}}V+O(l^{4}).$

The equation (2.17) is thus further simplified:

$\displaystyle\left(\partial_{2}\partial_{\bar{2}}K_{(0)}\right)\partial_{1}\partial_{\bar{1}}f+\left(\partial_{1}\partial_{\bar{1}}K_{(0)}\right)\partial_{2}\partial_{\bar{2}}f-\left(\partial_{1}\partial_{\bar{2}}K_{(0)}\right)\partial_{2}\partial_{\bar{1}}f-\left(\partial_{2}\partial_{\bar{1}}K_{(0)}\right)\partial_{1}\partial_{\bar{2}}f$
$\displaystyle+\frac{1}{2}\left(\det\partial_{i}\partial_{\bar{j}}K_{(0)}\right)\left(y^{3}\partial_{y}\frac{1}{y^{3}}\partial_{y}f+\frac{4l^{2}y^{2}}{(r^{2}+y^{2})^{2}}y^{2}\partial_{y^{2}}\partial_{y^{2}}f\right)=0\
;$ $\displaystyle
f(z,y;z_{i}^{\prime})|_{z=z_{i}^{\prime},y=\epsilon}=\delta^{(4)}(z-z_{i}^{\prime})$
(3.5)

where

$K_{(0)}=\frac{1}{2}\left(r^{2}+y^{2}\right)-\frac{1}{2}l^{2}\log\left(r^{2}+y^{2}\right)-\frac{1}{4}y^{2}\log(y^{2}).$
(3.6)

$Z_{(1)}(z,y)$ is denoted $V(z,y)$ in the above.

One may use the method of kernel which may be denoted as

$f(z,y;z^{\prime}).$ (3.7)

Then we make detailed analysis of simplified equations. Denote
$z_{1}=x_{1}+ix_{2},z_{2}=x_{3}+ix_{4}$. We have

$\partial_{1}\partial_{\bar{1}}=\frac{1}{4}(\partial_{x_{1}}^{2}+\partial_{x_{2}}^{2}),~{}~{}~{}\partial_{\bar{1}}=\frac{1}{2}(\partial_{x_{1}}+i\partial_{x_{2}}),~{}\partial_{1}=\frac{1}{2}(\partial_{x_{1}}-i\partial_{x_{2}}).$

$l^{2}$ is a parameter in this class of solutions. Here, we solve the
linearized equation up to order $O(l^{2})$. Denoting
$T=|z_{1}|^{2}+|z_{2}|^{2}+y^{2}$,

$\displaystyle\partial_{2}\partial_{\bar{2}}K_{(0)}$ $\displaystyle=$
$\displaystyle
1/2-\frac{l^{2}}{2}\frac{1}{T}+\frac{l^{2}}{2}\frac{|z_{2}|^{2}}{T^{2}},~{}~{}\partial_{1}\partial_{\bar{1}}K_{(0)}=1/2-\frac{l^{2}}{2}\frac{1}{T}+\frac{l^{2}}{2}\frac{|z_{2}|^{2}}{T^{2}},$
$\displaystyle\partial_{1}\partial_{\bar{2}}K_{(0)}$ $\displaystyle=$
$\displaystyle\frac{l^{2}}{2}\frac{\bar{z}_{1}z_{2}}{T^{2}},~{}~{}~{}~{}~{}~{}~{}~{}\partial_{2}\partial_{\bar{1}}K_{(0)}=\frac{l^{2}}{2}\frac{z_{1}\bar{z}_{2}}{T^{2}},$
$\displaystyle\det\partial_{i}\partial_{\bar{j}}K_{(0)}$ $\displaystyle=$
$\displaystyle
1/4-\frac{l^{2}}{2T}+\frac{l^{2}(|z_{1}|^{2}+|z_{2}|^{2})}{4T^{2}}.$ (3.8)

Hence,

$\displaystyle\left(\partial_{2}\partial_{\bar{2}}K_{(0)}\right)\partial_{1}\partial_{\bar{1}}V+\left(\partial_{1}\partial_{\bar{1}}K_{(0)}\right)\partial_{2}\partial_{\bar{2}}V-\left(\partial_{1}\partial_{\bar{2}}K_{(0)}\right)\partial_{2}\partial_{\bar{1}}V-\left(\partial_{2}\partial_{\bar{1}}K_{(0)}\right)\partial_{1}\partial_{\bar{2}}V$
(3.9)
$\displaystyle+\frac{1}{2}\left(\det\partial_{i}\partial_{\bar{j}}K_{(0)}\right)\left(y^{3}\partial_{y}\frac{1}{y^{3}}\partial_{y}V+\frac{4l^{2}y^{2}}{(r^{2}+y^{2})^{2}}y^{2}\partial_{y^{2}}\partial_{y^{2}}V\right)$
$\displaystyle=$ $\displaystyle(D_{0}+D_{2})V,$

where

$\displaystyle D_{0}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\partial_{1}\partial_{\bar{1}}+\frac{1}{2}\partial_{2}\partial_{\bar{2}}+\frac{1}{8}\left(y^{3}\partial_{y}\frac{1}{y^{3}}\partial_{y}\right)$
(3.10) $\displaystyle=$
$\displaystyle\frac{1}{8}(\sum_{i=1}^{4}\partial_{x_{i}}^{2}+y^{3}\partial_{y}\frac{1}{y^{3}}\partial_{y}),$

and

$\displaystyle D_{2}$
$\displaystyle=-\frac{l^{2}}{2}\frac{\bar{z}_{1}z_{2}}{T^{2}}\partial_{\bar{z}_{1}}\partial_{z_{2}}-\frac{l^{2}}{2}\frac{z_{1}\bar{z}_{2}}{T^{2}}\partial_{z_{1}}\partial_{\bar{z}_{2}}+(-\frac{l^{2}}{2}\frac{1}{T}+\frac{l^{2}}{2}\frac{|z_{2}|^{2}}{T^{2}})\partial_{z_{1}}\partial_{\bar{z}_{1}}+(-\frac{l^{2}}{2}\frac{1}{T}+\frac{l^{2}}{2}\frac{|z_{1}|^{2}}{T^{2}})\partial_{z_{2}}\partial_{\bar{z}_{2}}$
$\displaystyle+(-\frac{l^{2}}{2T}+\frac{l^{2}(|z_{1}|^{2}+|z_{2}|^{2})}{4T^{2}})\frac{1}{2}y^{3}\partial_{y}\frac{1}{y^{3}}\partial_{y}+\frac{l^{2}y^{4}}{2T^{2}}\partial_{y^{2}}\partial_{y^{2}},$
(3.11)

and $T=|z_{1}|^{2}+|z_{2}|^{2}+y^{2}.$

We write $f=f_{0}+f_{2}$, and the solution is of the form

$f(z,y;z^{\prime})=\frac{cy^{4}}{\left(|z_{1}-z_{1}^{\prime}|^{2}+|z_{2}-z_{2}^{\prime}|^{2}+y^{2}\right)^{4}}+f_{2}.$
(3.12)

Near $y=0$, $f$ approaches 4d delta function, of 4d coordinates
$(z-z^{\prime})$, with some coefficient. Finite small $y^{2}$ plays the role
of smoothing of the delta function.

In the below section, we can also perform a regularization of $f_{0}$ to place
the delta function at $y=\epsilon$.

We have

$Z\left(z,y\right)-\frac{1}{2}=Z_{(0)}\left(z,y\right)-\frac{1}{2}-\int_{\cup_{i}B_{i}}f(z,y;z^{\prime})d^{4}z^{\prime}~{}.$
(3.13)

In other words, the solution $Z\left(z,y\right)$ to the linearized equation is
in terms of the integral of the kernel $f(z,y;z^{\prime})$. In order to get
the solution for $Z\left(z,y\right)$, we need to solve $f(z,y;z^{\prime})$,
which is a function of $l^{2}$. We integrate over domain denoted $B_{i}$, to
smooth-out and straighten the distribution labeled by $i$. The
$f(z,y;z^{\prime})$ is a nontrivial function of $l^{2}$, since the Laplace-
type equation contains $l^{2}$ as a parameter.

###  3.2 Summary of equations and boundary conditions

The differential operator $D$ is

$D=D_{0}+D_{2},$ (3.14)

and $D=D(l^{2})$ is a function of $l^{2}.$

$D_{0}=\frac{1}{8}(\sum_{i=1}^{4}\partial_{x_{i}}^{2}+y^{3}\partial_{y}\frac{1}{y^{3}}\partial_{y}),$
(3.15) $\displaystyle D_{2}$
$\displaystyle=-\frac{l^{2}}{2}\frac{\bar{z}_{1}z_{2}}{T^{2}}\partial_{\bar{z}_{1}}\partial_{z_{2}}-\frac{l^{2}}{2}\frac{z_{1}\bar{z}_{2}}{T^{2}}\partial_{z_{1}}\partial_{\bar{z}_{2}}+(-\frac{l^{2}}{2}\frac{1}{T}+\frac{l^{2}}{2}\frac{|z_{2}|^{2}}{T^{2}})\partial_{z_{1}}\partial_{\bar{z}_{1}}+(-\frac{l^{2}}{2}\frac{1}{T}+\frac{l^{2}}{2}\frac{|z_{1}|^{2}}{T^{2}})\partial_{z_{2}}\partial_{\bar{z}_{2}}$
$\displaystyle+(-\frac{l^{2}}{2T}+\frac{l^{2}(|z_{1}|^{2}+|z_{2}|^{2})}{4T^{2}})\frac{1}{2}y^{3}\partial_{y}\frac{1}{y^{3}}\partial_{y}+\frac{l^{2}y^{4}}{2T^{2}}\partial_{y^{2}}\partial_{y^{2}}.$
(3.16)

The kernel $f=f(z,y;z^{\prime})$ is

$f(z,y;z^{\prime})=f_{0}(z,y;z^{\prime})+f_{2}(z,y;z^{\prime}),$ (3.17)

where $f_{0}$ is of the form in (3.12). Here, $y=0$ is the boundary and $y>0$
is the bulk.

We can perform a regularization, by considering $y=\epsilon$ instead of $y=0$,
where $\epsilon$ is a small regularization parameter.

These are the equation and boundary behavior near $y=\epsilon$. Since
$D=D(l^{2})$ is a function of $l^{2}$, $f$ is also a function of $l^{2}$. The
equation can be written as

$\displaystyle(D_{0}+D_{2})(f_{0}+f_{2})$ $\displaystyle=$ $\displaystyle
0,~{}~{}~{}~{}y>\epsilon,$ (3.18)
$\displaystyle\left(f_{0}+f_{2}\right)(z,y;z^{\prime})|_{y=\epsilon}$
$\displaystyle=$ $\displaystyle\delta^{(4)}(z-z^{\prime}).$ (3.19)

We will solve $f=f_{0}+f_{2}$. The equation is simplified to

$\displaystyle D_{0}f_{0}=0,~{}D_{0}f_{2}+D_{2}f_{0}=0,~{}~{}~{}y>\epsilon,$
(3.20)
$\displaystyle\left(f_{0}+f_{2}\right)(z,y;z^{\prime})|_{y=\epsilon}=\delta^{(4)}(z-z^{\prime}).$
(3.21)

We keep up to $o(l^{2})$ orders.

Hence the equation for $f_{2}$ is

$\frac{1}{8}(\sum_{i=1}^{4}\partial_{x_{i}}^{2}+y^{3}\partial_{y}\frac{1}{y^{3}}\partial_{y})f_{2}=-D_{2}\left(\frac{cy^{4}}{\left(|z_{1}-z_{1}^{\prime}|^{2}+|z_{2}-z_{2}^{\prime}|^{2}+y^{2}\right)^{4}}\right).$
(3.22)

The behavior of $f=f_{0}+f_{2}$ near $y=\epsilon$ is, in general,

$\lim_{y\rightarrow\epsilon}f(z,y;z^{\prime})=\delta^{(4)}(z-z^{\prime}).$
(3.23)

In the later sections, we will discuss both the cases when the delta function
behavior is placed at $y=0$ and at $y=\epsilon$ respectively.

##  4 Equations in Fourier space and derivation of global solutions

In this section, we analyze the general equation for the function $f$, and
give solutions for asymptotic AdS backgrounds. The solutions come with
parameters which are related to boundary conditions and asymptotia.

Denote the momentum variable dual to $x_{i}$ by $k_{i}$, then
$\sum_{i=1}^{4}\partial_{x_{i}}^{2}$ is turned into multiplication by
$-\sum_{i=1}^{4}k_{i}^{2}$, and we denote it as $-k^{2}$. We denote
$s_{i}=x_{i}-x_{i}^{\prime}$, and the norm of $s_{i}$ is denoted $s$. The norm
of $k_{i}$ is denoted $k$.

We denote $\mu=y^{2}\ $and we have

$D_{2}f_{0}=D_{2}(\frac{cy^{4}}{(s^{2}+y^{2})^{4}})=-cl^{2}\mu^{2}(\mu+s^{2})^{-6}.$
(4.1)

We have

$\frac{1}{8}(\sum_{i=1}^{4}\partial_{x_{i}}^{2}+4\mu\partial_{\mu}^{2}-4\partial_{\mu})f_{2}=-D_{2}(\frac{cy^{4}}{(|z_{1}-z_{1}^{\prime}|^{2}+|z_{2}-z_{2}^{\prime}|^{2}+y^{2})^{4}}).$
(4.2)

We can perform radial Fourier transforms in 4d. The $k$ is the conjugate
variable to $s$. Denote the radial Fourier transform of $f(s,\mu)$ as

$F[f](k,\mu)=F(k,\mu).$ (4.3)

Then

$f=\int\frac{d^{4}k}{(2\pi)^{4}}F(k,\mu)e^{ik\cdot x},$ (4.4) $F(k,\mu)=\int
d^{4}xf(s,\mu)e^{-ik\cdot x}.$ (4.5)

We denote $F_{0}(k,\mu):=F[f_{0}](k,\mu)$ and $F_{2}(k,\mu):=F[f_{2}](k,\mu)$.
The inverse Fourier transform is denoted $F^{-1}[\cdot].$

According to Hankel transform and its inverse Hankel transform,

$kF(k,\mu)=(2\pi)^{2}\int_{0}^{\infty}J_{1}(ks)sf(s,\mu)sds;$ (4.6)
$sf(s,\mu)=(2\pi)^{2}\int_{0}^{\infty}J_{1}(ks)kF(k,\mu)kdk.$ (4.7)

From above, we have

$(\partial_{s}^{2}+3s^{-1}\partial_{s}+4\mu\partial_{\mu}^{2}-4\partial_{\mu})f_{2}=8cl^{2}\mu^{2}(\mu+s^{2})^{-6}.$
(4.8)

The $f_{0}$ satisfies the equation

$(\partial_{s}^{2}+3s^{-1}\partial_{s}+4\mu\partial_{\mu}^{2}-4\partial_{\mu})f_{0}=0,~{}~{}~{}~{}y>0.$
(4.9)

Hence we have,

$(\partial_{s}^{2}+3s^{-1}\partial_{s}+4\mu\partial_{\mu}^{2}-4\partial_{\mu})f=8cl^{2}\mu^{2}(\mu+s^{2})^{-6},~{}\
$ (4.10)

where $f=f_{0}+f_{2}$. The $f_{0}$ is the piece annihilated by the
differential operator on the left-hand side; while $f_{2}$ when acted by the
differential operator, has a source term on the right-hand side.

We consider both the cases, when the boundary condition is at $y=0$; and when
the boundary condition is at $y=\epsilon$. The latter can be considered as a
regularization of the former.

We can obtain the Fourier transform for the equation for $f(s,y)$. Multiply
both sides of (4.10) by $\mu$:

$\mu(\partial_{s}^{2}+3s^{-1}\partial_{s}+4\mu\partial_{\mu}^{2}-4\partial_{\mu})f=8cl^{2}\mu^{3}(\mu+s^{2})^{-6}.$
(4.11)

We have the Fourier transform
$F[\frac{y^{6}}{\left(y^{2}+s^{2}\right)^{6}}]=\xi y^{2}k^{4}K_{4}(ky)$, where
$\xi=\frac{4\pi^{2}}{3840}$. We also have

$F\left[\mu(\partial_{s}^{2}+3s^{-1}\partial_{s}+4\mu\partial_{\mu}^{2}-4\partial_{\mu})\right]=-\mu
k^{2}+4\mu^{2}\partial_{\mu}^{2}-4\mu\partial_{\mu}\ ,$ (4.12)
$F[f(s,\mu)]=F[f](k,\mu)=F(k,\mu)\ .$ (4.13)

So we have the momentum space version of the equation (4.10):

$\left(-y^{2}k^{2}+y^{2}\partial_{y}^{2}-3y\partial_{y}\right)F(k,y)=8\xi
cy^{2}l^{2}k^{4}K_{4}(ky)$ (4.14)

since

$4\mu\partial_{\mu}^{2}-4\partial_{\mu}=\partial_{y}^{2}-3y^{-1}\partial_{y}\
.$ (4.15)

This equation does not assume using the regularization at $y=\epsilon$. For
simplicity we may also denote $\epsilon$ as $y_{c}$ in this section. The
equation itself does not know about the limiting parameter $\epsilon$.

We can solve the solution by expansion using modified Bessel functions
$K_{n}(ky)$. Note

$\left(-\mu
k^{2}+4\mu^{2}\partial_{\mu}^{2}-4\mu\partial_{\mu}\right)\frac{y^{2}K_{2}(ky)}{K_{2}(ky_{c})}=0.$
(4.16)

We get

$F(k,y)=F_{0}(k,y)+F_{2}(k,y)=c_{1}\frac{y^{2}K_{2}(ky)}{y_{c}^{2}K_{2}(ky_{c})}+c_{2}\frac{y^{2}l^{2}K_{4}(ky)}{y_{c}^{4}K_{4}(ky_{c})}.$
(4.17)

Then we analyze more details of the coefficients $c_{1},c_{2}$.

Note that the modified Bessel functions $K_{n}(ky)$ themselves have recurrence
relations

$K_{n+1}(ky)-K_{n-1}(ky)=\frac{2n}{ky}K_{n}(ky).$ (4.18)

Note we do not consider $I_{n}(ky)$ since for large $y$, $I_{n}(ky)\sim
e^{ky}/\sqrt{ky}$, while $K_{n}(ky)\sim e^{-ky}/\sqrt{ky}$. They have the
properties:

$\mathrm{when}\ y_{c}\rightarrow 0,\ \
K_{n}(ky_{c})\rightarrow\frac{\Gamma(n)}{2}(\frac{2}{ky_{c}})^{n},$ (4.19)

e.g.

$\mathrm{when}~{}y_{c}\rightarrow
0,~{}\frac{1}{y_{c}^{4}K_{4}(ky_{c})}\rightarrow\frac{k^{4}}{48},$ (4.20)
$\frac{1}{y_{c}^{2}K_{2}(ky_{c})}\rightarrow\frac{k^{2}}{2}.$ (4.21)

The above (4.14) implies

$F_{2}(k,y)=\frac{2}{3}\xi cl^{2}y^{2}k^{4}K_{4}(ky)\simeq 32\xi
c\frac{y^{2}l^{2}K_{4}(ky)}{y_{c}^{4}K_{4}(ky_{c})}.$ (4.22)

Here the $\simeq$ sign means “equal when $y_{c}\rightarrow 0$”. The (4.16)
implies

$F_{0}(k,y)=\frac{c_{1}}{2}y^{2}k^{2}K_{2}(ky)\simeq
c_{1}\frac{y^{2}K_{2}(ky)}{y_{c}^{2}K_{2}(ky_{c})}.$ (4.23)

In the regularization case, we have families of solutions with parameters
$y_{c}>0$, and since $y_{c}$ is a cut-off, it is hence tunable. We can also
take$~{}y_{c}\rightarrow 0^{+}$ limit.

The combination $F_{0}(k,y)+F_{2}(k,y)\ $approaches 1, when $y$ approaches
$y_{c}$. This means that $c_{1}+\frac{c_{2}l^{2}}{y_{c}^{2}}=1$. The condition
corresponds to the boundary value of $\frac{1}{2}-Z(z,y)$ to be 1, at the
black regions in the interior end, which originates from the smooth droplet
boundary conditions.

We can equivalently write:

$F(k,y)\simeq
c_{1}\frac{y^{2}K_{2}(ky)}{y_{c}^{2}K_{2}(ky_{c})}+c_{2}\frac{y^{2}l^{2}K_{4}(ky)}{y_{c}^{4}K_{4}(ky_{c})},\
\ \ $ (4.24) $F(k,y)\simeq
c_{1}\frac{1}{2}k^{2}y^{2}K_{2}(ky)+c_{2}\frac{1}{48}k^{4}l^{2}y^{2}K_{4}(ky),\
\ $ (4.25)

where $c_{1}=1-c_{2}\frac{l^{2}}{y_{c}^{2}}$.

The function $f$ may be written as

$\displaystyle f$ $\displaystyle=$ $\displaystyle f_{0}+f_{2}=F^{-1}[F[f]]$
(4.26) $\displaystyle\simeq$
$\displaystyle\frac{y^{2}}{y_{c}^{2}}\int\frac{d^{4}k}{(2\pi)^{4}}\left(\frac{c_{1}K_{2}(ky)}{K_{2}(ky_{c})}+\frac{c_{2}l^{2}K_{4}(ky)}{y_{c}^{2}K_{4}(ky_{c})}\right)e^{ik\cdot(z-z^{\prime})}.$

##  5 Stretched limit and its corresponding solutions in Fourier space

###  5.1 General methods of stretched limit

Here we consider another solution technique, the stretched surface limit. This
is similar to the idea in [57]. The stretched technique can be naturally
combined with the technique in Sec. 2 and Sec. 3. The linearized technique put
small enclosure surfaces near each small droplets. The stretched surface
limit, put a single surface at $y=y_{c}$. The small enclosure surfaces are all
inside the stretched surface. The stretched surface can hence be viewed as a
big and universal enclosure surface.

In the stretched surface limit, the central droplet is a special case, and one
can push-forward the central droplet distribution onto the stretched surface
at $y=y_{c}$. The change with respective to the $y_{c}$ variable is analogous
to renormalization group transformation. We transport the data from a smaller
$y$ surface to a bigger $y$ surface, and this procedure is a coarse-graining
over the high-momentum modes on the stretched surface. All the droplets at
$y=0$ are enclosed by $y=y_{c}$ stretched surface, as a big enclosure surface.

When we combine both techniques, the solutions are asymptotically AdS space,
whose leading terms are (3.6), hence a coarse-graining description of them
also describes superstar like geometries and their deformations. The boundary
condition for superstars is a coarse-grained density and hence a grey droplet
density, see Sec. 5.3.1.

For asymptotic AdS background, we have an interior cut-off $y_{c}$, and the
equation is

$\left(-y^{2}k^{2}+y^{2}\partial_{y}^{2}-3y\partial_{y}\right)F(k,y)=\frac{c_{2}}{4}l^{2}y^{2}k^{4}K_{4}(ky).$
(5.1)

For asymptotic AdS,

$F(k,y)\simeq
c_{1}\frac{y^{2}K_{2}(ky)}{y_{c}^{2}K_{2}(ky_{c})}+c_{2}\frac{y^{2}l^{2}K_{4}(ky)}{y_{c}^{4}K_{4}(ky_{c})},\
\ \ \ y\geq y_{c}$ (5.2)

and $c_{1}+c_{2}\frac{l^{2}}{y_{c}^{2}}=\frac{1}{1+\nu}$. Hence,

$\displaystyle c_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{1+\nu}-c_{2}\frac{l^{2}}{y_{c}^{2}},$ (5.3)
$\displaystyle c_{2}/c_{1}$ $\displaystyle=$
$\displaystyle\gamma\quad=\quad\frac{\frac{1}{1+\nu}-c_{1}}{c_{1}}\frac{y_{c}^{2}}{l^{2}},$
(5.4)

where $0\leq\frac{1}{1+\nu}\leq 1$. The solutions have parameters $y_{c}$,
$l$, $c_{1}$ and $\nu$. When $y_{c}^{2}$ is increased to be larger, $\gamma$
tends to be larger. For coarse-grained grey droplet, this means that the
boundary value is not 1, but $\frac{1}{1+\nu}$. The condition corresponds to
the boundary value of $\frac{1}{2}-Z(z,y)$ to be $\frac{1}{1+\nu}$, at the
black regions in the interior end. This describes situations of superstar
geometries, see Sec. 5.3. Note $\nu=0$ is a special case, where the charge of
superstar goes to zero. The function contains parameter $y_{c}$, hence we also
denote it as $F(k,y)_{y_{c}}$.

Now the data on $y=y_{c}$ corresponds to a grey distribution, and they
correspond to coarse-grained microstates of two-charge superstar. This idea is
similar to Sec. 4 and 5 of [57]. Hence the boundary data is $\frac{1}{1+\nu}$
instead of 1, and see the analysis in Sec. 5.3. The $y=y_{c}$ can be viewed as
a stretched surface, where the droplet distribution data are pushed-forward or
mapped from the interior to this stretched surface, and hence the droplet data
is coarse-grained on the stretched surface. The $l$ is a parameter for the
asymptotic AdS radius, and $\nu$ is a parameter related to the total charge of
superstar.

The smooth geometries correspond to smooth droplet boundary conditions in the
interior, i.e. $y=0$. We can push-forward or map the droplet density data to a
stretched surface at $y=y_{c}$. Now we consider a distribution of droplets,
resulting in a coarse distribution in the $y=y_{c}$ stretched droplet space.
The $z$ are four-dimensional coordinates. We define
$u(z,y)=\frac{1}{2}-Z(z,y)$. We place a coarse distribution at $y=y_{c}$
stretched droplet space, and $u(z,y)|_{y=y_{c}}=u_{0}(z,y_{c})$ is the coarse
distribution function at $y=y_{c}$ stretched droplet space. We use the
convolution with the kernel to push-forward this distribution to the bulk. We
call this stretched droplet space limit, as “stretched limit”. Hence

$u(z,y)=\int_{\cup_{i}C_{i}}f(z,z^{\prime},y)_{y_{c}}u_{0}(z^{\prime},y_{c})d^{4}z^{\prime},$
(5.5)

where $f(z,z^{\prime},y)_{y_{c}}=F^{-1}[F(k,y)_{y_{c}}]$. Here
$u_{0}(z^{\prime},y_{c})$ includes the distribution of all the droplets. The
Fourier transform, as well as Mellin transform, has nice convolution
properties,

$F[u(z,y)]=F(k,y)F[u_{0}(z,y_{c})].$ (5.6)

The metric function by the stretched droplet space can be written as, in the
stretched limit

$Z\left(z,y\right)-\frac{1}{2}=-\int_{\cup_{i}C_{i}}f(z,z^{\prime},y)_{y_{c}}u_{0}(z^{\prime},y_{c})d^{4}z^{\prime}~{}.$
(5.7)

The $C_{i}$ are domains on $y=y_{c}$ stretched surface. Here
$u_{0}(z^{\prime},y_{c})$ has included the distribution (denoted $C_{0}$ in
$\cup_{i}C_{i}$) of the central droplet pushed-forward to the $y=y_{c}$
stretched surface. The stretched surface include all the droplets inside. The
Fourier transform of the metric function can be written as in the stretched
limit

$F\left[\frac{1}{2}-Z\left(z,y\right)\right](k,y)=F(k,y)F[u_{0}(z,y_{c})].$
(5.8)

Note $y_{c}$ is a parameter. Changing the $y_{c}$ infinitesimally is changing
the scale of coarse-graining and is analogous to renormalization group
transformation.

Hence the Fourier transform of the metric function can be written as, in the
stretched limit

$F\left[\frac{1}{2}-Z\left(z,y\right)\right](k,y)=\frac{y^{2}}{y_{c}^{2}}\left(\frac{c_{1}K_{2}(ky)}{K_{2}(ky_{c})}+\frac{c_{2}l^{2}K_{4}(ky)}{y_{c}^{2}K_{4}(ky_{c})}\right)F[u_{0}(z,y_{c})].$
(5.9)

The metric function by the stretched droplet space can be written as, in the
stretched limit

$\displaystyle Z\left(z,y\right)-\frac{1}{2}$ $\displaystyle=$
$\displaystyle-\int_{\cup_{i}C_{i}}f(z,z^{\prime},y)_{y_{c}}u_{0}(z^{\prime},y_{c})d^{4}z^{\prime}~{}$
$\displaystyle\simeq$
$\displaystyle-\int_{\cup_{i}C_{i}}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{y^{2}}{y_{c}^{2}}\left(\frac{c_{1}K_{2}(ky)}{K_{2}(ky_{c})}+\frac{c_{2}l^{2}K_{4}(ky)}{y_{c}^{2}K_{4}(ky_{c})}\right)e^{ik\cdot(z-z^{\prime})}u_{0}(z^{\prime},y_{c})d^{4}z^{\prime}.$

These (LABEL:eqn_66) are in the stretched limit, where the boundary condition
is placed at the stretched surface with a density $u_{0}(z^{\prime},y_{c})$.
The droplets on the stretched surface are grey droplets, whose distribution
are described by $u_{0}(z^{\prime},y_{c}).$

The above stretched limit (LABEL:eqn_66), is naturally used for the superstar,
which has a natural cut-off [57]. One can get coarse graining of distributions
for the two-charge superstar. We place the above stretched surface at
$y=y_{0}$ for superstar, and this is similar to [57]. The coarse graining of
these smooth geometries leads to e.g. the superstar geometry, which is an
effective geometry. Low energy observers see an averaged metric, which is an
effective geometry.

We can also perform coarse-graining at a coarse-graining scale
$k_{0}\sim\frac{1}{y_{0}}$, i.e. at $y=y_{0}$. For coarse-graining and
renormalization group flow purpose, we use a coarse-graining function
$P_{IR}(k)$,

$P_{IR}(k)\simeq F(k,y_{0})\simeq
c_{1}\frac{1}{2}k^{2}y_{0}^{2}K_{2}(ky_{0})+c_{2}\frac{1}{48}k^{4}l^{2}y_{0}^{2}K_{4}(ky_{0})$
(5.11)

which is a smooth filter reducing the UV modes higher than scale $y_{0}^{-1}$.
This is a momentum filter function, as similar to [57]. We can transport, in
other words, push-forward the data from $y=0$ outwards to $y=y_{0}$ surface.
See also Appendix E and (E.13), for the use of $P_{IR}(k)$ in the coarse-
graining.

###  5.2 Relation between linearized technique, regularization technique, and
stretched limit

The linearized technique is, a priori, independent of regularization
technique, and stretched limit. It can be naturally combined with
regularization technique and with stretched technique.

The regularization technique is encountered when we place boundary conditions
at $y=\epsilon$. It is a regularization for the case $\epsilon$ goes to zero.
The boundary conditions are those origante from smooth droplet boundary
conditions.

The stretched technique, is different from the regularization technique. The
two can be naturally combined. The stretched technique is placing $y=y_{c}$
stretched surface. The boundary conditions are more general than the
regularization technique. We can place grey droplet boundary conditions. It
can include two-charge superstar spacetimes and other spacetimes including
spacetime foams, e.g. related to [102, 103, 104, 20]. Changing $y_{c}$ is
analogous to renormalization group transformation; and we can also see this
from the combination of $y_{c}k$, e.g. scaling up $y_{c}$ is related to
scaling smaller $k$. The bulk geometry, automatically and efficiently,
geometrized this renormalization group process.

###  5.3 Relation to superstar

####  5.3.1 Relation to superstar and general aspects

Here we analyze the coarse density for superstar for the boundary condition.
The asymptotics of superstar are analyzed in Appendix C, where we have shown
that the dilute distribution of small droplets, in the asymptotic region
matches with the asymptotic expansion of the superstar. Hence we consider a
coarse-grained description.

Consider a stretched surface or stretched horizon for superstar, at $y=y_{0}$.
The density $u_{0}(z,y_{0})$ defines a coarse distribution, and this defines
also a coarse-grained distribution of microstates. It has a distribution on
stretched surface. The distribution is related to Young tableau profiles.

The two-charge superstars have also been analyzed in [46], and the $Z$
function for superstar is

$\displaystyle\frac{1}{2}-Z$ $\displaystyle=$
$\displaystyle\frac{1}{1+Q_{2}+Q_{3}+{\tilde{r}}^{2}+{Q_{2}\,Q_{3}}/{\tilde{r}^{2}}+\cot^{2}\theta[({\tilde{r}}^{2}+Q_{3})\sin^{2}\alpha+({\tilde{r}}^{2}+Q_{2})\cos^{2}\alpha]}\
.$

We work in unit $l=1$. Note the “$1$” on the right hand side correspond to
$l^{2}$, when restored. Consider the identification [46] that
$y={\tilde{r}}\sin\theta$,$~{}r={\tilde{r}}\cos\theta$, and
$r^{2}={\tilde{r}}^{2}-y^{2}$, and if $y$ is small, then the density is
concentrated in the small $r$ region, due to the $\cot^{2}\theta$ term. When
close to small ${\tilde{r}}^{2}$ region, the superstar geometry is replaced by
smooth microstate geometries; there we use a small $y^{2}$ expansion to
extract the droplet distributions, e.g. (C.11).

Hence we assume the density is within $|z_{1}|^{2}\leqslant r_{1,0}^{2}$,
$|z_{2}|^{2}\leqslant r_{2,0}^{2}$. The coarse density corresponds to the
coarse-grained density of droplets for superstars. The fine density is the
distribution of smooth droplet configurations of the smooth microstate
geometries. Here, for superstar we consider the coarse density. This is
similar to limit shape of Young tableaux (YT). For the moment, we first assume
small charge regimes $Q_{i}\ll l^{2}$.

Note that in the half BPS case, if one redefines the function
$-z(x,y)\rightarrow z(x,y)$ there [10], the system is the same. Here we have
chosen a convention to use, and here we use $Z(z,y)$ instead of using
$-Z(z,y)$. With this convention, $0\leq(\frac{1}{2}-Z)_{y_{0}}(z)\leq 1$,

$u(z)_{y_{0}}=(\frac{1}{2}-Z)_{y_{0}}(z).$ (5.13)

Then we simplify

$\displaystyle u(z)_{y_{0}}$ $\displaystyle=$
$\displaystyle(\frac{1}{2}-Z)|_{y=y_{0}}(z)\approx\frac{1}{1+Q_{2}+Q_{3}}$
(5.14) $\displaystyle\simeq$ $\displaystyle\frac{1}{1+\nu},$ (5.15)

where $\nu$ is related to the total charge of superstar. We can have a small
${Q_{3}}$ limit. The small ${Q_{3}}$ limit of (5.14) goes over to the results
similar to [20] and their related dicussions. Among other things, mass, radius
and charge relations can be produced by entropic maximization related methods
[20]. See also discussions on related aspects, e.g. [108, 109]. Grey droplets
refer to that the boundary value of $\frac{1}{2}-Z$ is not 1 but smaller than
1. The coarse-grained distributions are grey droplets. Here, $\nu$ is related
to the slopes of YTs.

One can describe the subsystem here by two chemical potentials and an
effective canonical temperature, in canonical ensembles and grand canonical
ensembles, depending on changes of variables and Legendre transformations. The
two chemical potentials conjugate to the row lengths of two YTs
$r_{1},r_{2}~{}$for the $Y$ and $X$ fields. We also refer to these two Young
tableaux as YT1 and YT2. The YTs of this kind have been constructed in e.g.
[30] and their related references [49]. These YTs can also be seen from the
charge relation. One of the simplest cases are operators of the form in e.g.
[49]. See also related construction [35, 23]111The shapes of Young tableaux
have been considered in e.g. [20, 57, 105, 106, 107] and their related
references.. These constructions are related by a change of basis. In the next
subsection 5.3.2, we also consider a similar coherent state version of the YT
states.

Negative ${Q_{3}}$ or ${Q_{2}}$, can make $(\frac{1}{2}-Z)_{y_{0}}(z)>1$,
where there are closed time-like curves. The case of negative charges has been
discussed in e.g. [110, 30, 103, 49] and their related references and aspects.

We hence have a 4d density $u_{0}(z)$ on the stretched surface $y=y_{0}$, and
one transports $u_{0}(z,y_{0})$ to $u(z,y)$ at a larger radial location $y$.
This idea has been considered in [57]; and the set-up here is a
generalization. We have

$u\left(z,y\right)=\int_{C}f(z,z^{\prime},y)_{y_{0}}u_{0}(z^{\prime},y_{0})d^{4}z^{\prime}\
,$ (5.16)

where $f(z,z^{\prime},y)_{y_{0}}=F^{-1}[F(k,y)_{y_{0}}]$ and $C$ is the domain
where $u_{0}\simeq(1+\nu)^{-1}$. For simplicity, consider equal charge case,
$C$ is $r^{2}\leq r_{0}^{2}$, where $r_{0}^{2}\simeq l^{2}(1+\nu)$. One can
generate deformations around superstar by deforming the density from
$u_{0}(z,y_{0})$ to

$u_{0}(z,y_{0})+\delta u_{0}(z,y_{0}),$ (5.17)

which is a general distribution with general variations. This deformed density
is transported to a larger radial location $y$. See also appendix E and
(E.13), for the use of coarse-graining procedure.

####  5.3.2 Relation to the symplectic form

In the quantum field theory side, there are many different but equivalent
bases for the quarter and eighth BPS operators. These bases can be transformed
to each. A way to describe excitations is to use their collective variables,
and this includes coherent states of collective excitations. One of them is by
coherent state basis, written in terms of displacer

$\mathrm{exp}(\sum\limits_{n,m}N_{n,m}^{-1}\Lambda_{n,m}a_{n,m}^{\dagger})\left|\Omega\right\rangle.$
(5.18)

Here $\left|\Omega\right\rangle$ is a reference state, which can also be a
vacuum state e.g. $\left|0\right\rangle\ $as a special case. The
$\Lambda_{n,m}$ is coherent state amplitude and eigenvalue of $a_{n,m}$ on the
coherent states. $a_{n,m}^{\dagger}$ adds a column of length $n$ to YT1 and a
column of length $m$ to YT2. The $a_{n,m}^{\dagger}$ is analogous to the
product of two determinant operators or subdeterminant operators with lengths
$n$ and $m$. These can be viewed as the generalization of the coherent states
in [57]. By the Berry curvature formalism, the symplectic form is

$\Omega=\sum\limits_{n,m}\delta\Lambda_{n,m}\wedge\delta\bar{\Lambda}_{n,m}.$
(5.19)

One can perform canonical transformations by changes of variables, to the
symplectic forms, and other symplectic forms using equivalent operator-bases
are equivalent to this symplectic form, by canonical transformations. The
symplectic forms constructed by different bases, including operator bases and
geometric bases [21, 22, 57], can be transformed to each other by changes of
variables and canonical transformations. These are equivalent to the
symplectic form derived from gravity side, by a canonical transformation.

Further, we consider deformations around this distribution, which are $\delta
u_{0}(z)$, and we analyzed them in 5.3.1. The deformed density
$u_{0}(z)+\delta u_{0}(z)$ is a general distribution with general variations.

The dilation of white regions decreases the density for the black regions, and
squeezing of white regions increases the density for the black regions, as
they are complementary regions as explained in Sec. 2. Now, dilation of white
regions is related to adding YT columns.

Denote $c_{1,i},c_{2,i}\ $as the number of columns with length $i$, for the 2
YTs respectively. $a_{n,m}^{\dagger}$ adds a column of length $n$ to YT1 and a
column of length $m$ to YT2. These operations make shape deformations of the
two YTs. Let $x$ label rows, $y_{1},y_{2}$ label columns, [20, 57]. The
typical tableaux are described by the shape functions $y_{1}(x),y_{2}(x)$, and
the slopes are the derivatives,

$u_{0}\simeq\frac{1}{1+y_{1}^{\prime}+y_{2}^{\prime}}.$ (5.20)

This is associated to (5.15). Consider deformation from $u_{0}\ $to
$u_{0}+\delta u(r^{2})$. The change of the shape functions are $\delta
y_{1}(x),\delta y_{2}(x)$ and

$y_{1}(x_{1})+x_{1}+y_{2}(x_{2})+x_{2}=\frac{r^{2}}{2}.$ (5.21)

By variation principle we have

$\delta
u(r^{2})\simeq-\frac{1}{(1+y_{1}^{\prime}+y_{2}^{\prime})^{2}}\left(\delta
y_{1}^{\prime}(x_{0,1})+\delta y_{2}^{\prime}(x_{0,2})\right).$ (5.22)

Now we make an approximation that the YTs are approximately triangular, so
that $y_{1}(x),y_{2}(x)$ are approximately linear in $x$, and hence we can
neglect the$\ y_{1}^{\prime\prime}(x),y_{2}^{\prime\prime}(x)$ terms which are
approximately zero.

Observe that $y_{1}(x)\simeq\omega_{1}x$ where
$\omega_{1}\simeq\frac{N_{c,1}}{N}$, and $y_{2}(x)\simeq\omega_{2}x$ where
$\omega_{2}\simeq\frac{N_{c,2}}{N}$. Since $\delta
y^{\prime}\left(x_{0}\right)$ means changing the number of columns at $x_{0}$,
and since columns are viewed as holes, reducing the holes increases particle
density, and vice versa. Hence $\delta y_{1}^{\prime}(x_{0,1})+\delta
y_{2}^{\prime}(x_{0,2})=\delta\langle c_{1,N-x_{0,1}}\rangle+\delta\langle
c_{2,N-x_{0,2}}\rangle$. Denote $\omega=\omega_{1}+\omega_{2}$,

$\delta u(r^{2})\simeq-\frac{1}{(1+\omega)^{2}}\left(\delta\langle
c_{1,N-x_{0,1}}\rangle+\delta\langle c_{2,N-x_{0,2}}\rangle\right).$ (5.23)

Hence$\ \delta u_{n,m}\sim\delta\langle c_{1,N-x_{0,1}}\rangle+\delta\langle
c_{2,N-x_{0,2}}\rangle$. For the operators with appropriate normalization of
two-point functions, we have that $\alpha_{n,m}$ is eigenvalue of$\ a_{n,m}.$
We have that$\ c_{1,N-x_{0}}+c_{2,N-x_{0}}\sim\langle
a_{n,m}^{\dagger}a_{n,m}\rangle$ and hence $u_{n,m}\sim|\alpha_{n,m}|^{2}$.
Writing by using variable $\alpha_{n,m}=|\alpha_{n,m}|e^{i\pi_{n,m}}$, we have
that the canonical conjugate variable to $u_{n,m}$ is the canonical
$\pi_{n,m}$. Hence by canonical transformations, we have that

$\Omega\sim\int\delta u(r)\wedge\delta\pi(r)\sim\sum_{n,m}\delta
u_{n,m}\wedge\delta\pi_{n,m}\sim\sum_{n,m}\delta\Lambda_{n,m}\wedge\delta\bar{\Lambda}_{n,m}.$
(5.24)

These different writings of symplectic forms are related by canonical
transformations, and changes of variables, including Fourier transforms. These
modes can also be captured by computing correlators. In Appendix E, we also
discuss the symplectic forms in special cases. It would be good to understand
better the canonical transformations between symplectic forms obtained by
operator bases and geometric bases.

##  6 Discussion

We obtained generalized Laplace-type equations with sources, from linearized
Monge-Ampere equations, on a general reference geometry and for the
asymptotically AdS geometry. This led us to obtain asymptotically AdS
solutions. One may have further simplifications for above cases. They can be
used in combination with a stretched limit, where boundary conditions can be
placed at the stretched surface. These boundary conditions include grey
droplets.

The above stretched limit, e.g. (LABEL:eqn_66), is naturally used for the
superstar, which has a natural cut-off [57]. We place grey droplet boundary
conditions on the stretched surface. The two-charge superstar is the extremal
case of the near-extremal black hole. The superstar corresponds to grey
droplet boundary conditions on the stretched surface. On the gravity side,
geometric quantization and coarse-graining can be naturally combined, and this
has also been performed in [57]. The coarse gaining of the density function,
leads to the coarse-graining of symplectic forms.

For coarse-graining purpose, we primarily use the momentum space version and
the associated Fourier transforms. One may also use Mellin transforms to
analyze the solutions. We can alternatively solve the global solution by
Mellin transforms. We can also perform the analysis using the methods of [66,
67]. Among many other aspects, the solution techniques and their various
relations have been summarized in Sec. 2.3, Sec. 5.2, and Appendix F.

The IR part of these geometries [10], with a cutoff away from the asymptotic
boundary, are also dual to the IR of the irrelevant deformation of SCFTs. The
IR region of the geometry, in particular, is for the IR region of the geometry
dual to irrelevant deformation preserving quarter BPS supersymmetry, and half
BPS supersymmetry, respectively [111, 112, 113]. See also related aspects
[114]. These deformations are deformations by BPS irrelevant operators. It
would be good to understand these aspects in more details.

There are many hierarchies of scales in smooth bubbling geometries, partly due
to large ratios of different sizes of different coexisting droplets or
bubbles, and partly due to that the spectra of the QFT side have rich
phenomena as well as complexity; and there can be many intermediate scales in
between the fundamental string scale and AdS scale. The coarse-graining scale
is a scale that separates the high energy with the low energy. The black hole
microstructure can appear and be evident at scales that are much larger than
the planck scale, e.g. [57, 115].

The systems can be put inside top-down string theory and are UV complete. The
string theory describes hard modes very sophisticatedly, because of its UV
completeness. On the other hand, there are no UV or IR divergences in full
string theory, e.g. [116]. Soft modes in the bulk are included in the
description in the dual field theory side, e.g. [57, 115]. See also related
discussion on soft modes related to various stretched horizons [115]. Our work
has also implications for soft modes on the stretched horizons, as discussed
in e.g. Appendix E.

Infalling massless particles can be trapped into the interior regions in
droplet space without escaping [117], which is similar to entering into a
maze. The behavior is similar to behavior in fuzzballs. These scenarios as
well as our related discussions, are closely related to aspects of microstate
geometries and microstructure of fuzzballs and blackholes, e.g. [118, 124].
Various insights and effects of microstate geometries have been unraveled. See
also related discussions [125, 126] on chaotic behaviors.

The superstar [110] is the extremal case of the near-extremal black hole.
There is a near-extremality parameter which is related to the near-BPS energy.
These are related to near-BPS states, e.g. [108, 109]. One can obtain non-BPS
excitations as further excitations on top of BPS excitations. Near BPS states
can also describe near BPS black holes [130, 129, 108, 109], and are related
to various giant configurations. The string configurations [127, 128] on
giants are relevant. One of the very useful methods for near-BPS states, is
the spin-matrix theory, see [132, 134] and related references. See also
related aspects and methods [131]. Another one of very useful methods for
near-BPS states, are restricted Schurs of giant graviton branes, e.g. [135,
136] and related references. They also include BPS states in the moduli space.

String probes in these background can be conveniently computed as they are
globally smooth. Excitations, for example string excitations as well as light
field excitations on the bubbling geometries, have been analyzed in e.g. [88,
82, 137], and their related references.

## Acknowledgments

We would like to thank V. Balasubramanian, A. Belin, I. Bena, D. Berenstein,
J. Caetano, P. Caputa, Y.J. He, Z. He, J. Hou, L.Y. Hung, A. Jevicki, R. de
Mello Koch, M.K. Kim, S. Komatsu, Y. Jiang, Y.Z. Li, J. Liu, H. Lu, O. Lunin,
J. Maldacena, J. Nian, C. Nunez, T. Okazaki, T. Okuda, K. Papadodimas, M.
Paulos, S. Ramgoolam, L. Rastelli, V.S. Rychkov, K. Skenderis, J. Simon, M.
Taylor, N. Warner, for related discussions in the past, and R. Suzuki for
crucial comments and correspondences. The work was supported in part by
National Key R&D Program of China grant 2020YFA0713000, by Grant No.
TH-533310008, by Overseas high-level talents program, by Fundamental Research
Funds for the Central Universities of China, by Grant No. 4007012320 and by
Grant No. 3207012204.

##  Appendix A AdS background solution and expansion

Here we explain how to use the $AdS_{5}\times S^{5}$ as a reference geometry,
for the purpose of performing linearized technique, among other things.

The potential with the AdS radius $l$ restored is ($l$ was sometimes set as 1
in [46] for simplicity and it can be restored by dimensional analysis or
dimensional counting)

$\displaystyle K_{(0)}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\frac{1}{2}(r^{2}+y^{2}+l^{2})+\sqrt{\frac{1}{4}(r^{2}+y^{2}-l^{2})^{2}+y^{2}l^{2}}\right)$
$\displaystyle-\frac{1}{2}l^{2}\log\left(\frac{1}{2}(r^{2}+y^{2}+l^{2})+\sqrt{\frac{1}{4}(r^{2}+y^{2}-l^{2})^{2}+y^{2}l^{2}}\right)$
$\displaystyle-\frac{1}{2}y^{2}\log\left(\frac{1}{2}(-r^{2}+y^{2}+l^{2})+\sqrt{\frac{1}{4}(r^{2}+y^{2}-l^{2})^{2}+y^{2}l^{2}}\right)+\frac{1}{2}y^{2}\log(y).$

Now we consider $r^{2}+y^{2}>l^{2}$ (this is outside and away from the central
droplet) and expand the potential in powers of $l^{2}$. We have

$\sqrt{\frac{1}{4}(r^{2}+y^{2}-l^{2})^{2}+y^{2}l^{2}}=\frac{1}{2}(r^{2}+y^{2}-l^{2})+\frac{y^{2}}{(r^{2}+y^{2})}l^{2}+O(l^{4}).$
(A.2)

Hence

$\displaystyle K_{(0)}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(r^{2}+y^{2}+\frac{y^{2}}{(r^{2}+y^{2})}l^{2}\right)$
(A.3)
$\displaystyle-\frac{1}{2}l^{2}\log\left(r^{2}+y^{2}+\frac{y^{2}}{(r^{2}+y^{2})}l^{2}\right)$
$\displaystyle-\frac{1}{2}y^{2}\log\left(y^{2}+\frac{y^{2}}{(r^{2}+y^{2})}l^{2}\right)+\frac{1}{2}y^{2}\log(y)+O(l^{4}).$

The $o(l^{2})$ term $\frac{y^{2}}{2(r^{2}+y^{2})}l^{2}$ from the first term
and the $o(l^{2})$ term $-\frac{y^{2}}{2(r^{2}+y^{2})}l^{2}$ from the second
log term precisely cancel, and we thus have

$\displaystyle K_{(0)}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(r^{2}+y^{2}\right)-\frac{1}{2}l^{2}\log\left(r^{2}+y^{2}\right)-\frac{1}{2}y^{2}\log\left(y^{2}\right)+\frac{1}{2}y^{2}\log(y)+O(l^{4})$
(A.4) $\displaystyle=$
$\displaystyle\frac{1}{2}\left(r^{2}+y^{2}\right)-\frac{1}{2}l^{2}\log\left(r^{2}+y^{2}\right)-\frac{1}{4}y^{2}\log(y^{2})+O(l^{4}).$

The first three terms are most important terms, and remaining higher order
terms are of order $O(l^{4})$.

##  Appendix B Multi droplets, dilute droplets, and the dilute distribution
regime

We consider isolated small smooth droplets, which are very far from each
other. Near the isolated small round droplet, the geometry locally approaches
that of smaller AdS space, with $l^{2}$ parameter effectively rescaled to
$h_{i}^{2}$. The small droplet is dual to giants, near each small isolated
droplet.

We have

$\displaystyle Z$ $\displaystyle=$
$\displaystyle\frac{|z_{1}|^{2}+|z_{2}|^{2}+y^{2}-L_{0}^{2}}{2\sqrt{(|z_{1}|^{2}+|z_{2}|^{2}+y^{2}-L_{0}^{2})^{2}+4y^{2}L_{0}^{2}}}$
(B.1)
$\displaystyle+\sum\limits_{i=2}^{n}\left(\frac{|z_{1}-s_{1i}|^{2}+|z_{2}-s_{2i}|^{2}+y^{2}-h_{i}^{2}}{2\sqrt{(|z_{1}-s_{1i}|^{2}+|z_{2}-s_{2i}|^{2}+y^{2}-h_{i}^{2})^{2}+4y^{2}h_{i}^{2}}}-\frac{1}{2}\right),$

in the dilute distribution regime. The extreme-ratio regime is $h_{i}^{2}\ll
L^{2}$, for $i=2,...,n$. And $h_{1}^{2}=L_{0}^{2}.$ The droplets are well-
separated. We also include the central droplet, whose center is denoted as
$s_{11}=s_{21}=0$. We can also write this expression for $Z-\frac{1}{2}$ which
has a superimposition structure. The $V_{(1)}=Z_{(1)}=Z-Z_{0}$ also has a
superimposition structure. Locally near the small droplet, the local behavior
is similar to the local behavior of the central droplet. Locally, the small
droplet is dual to$~{}n_{i}=h_{i}^{2}N/L^{2}$ three-brane giants. The quanta
for each small droplet $q_{i}$ is much smaller than the total quanta $N$, i.e.
$q_{i}\ll N$; and this is referred to as the extreme-ratio limit. Dilute means
that the droplets are isolated from each other.

Now we expand in the regime $r^{2}+y^{2}\gg h_{i}^{2}$. The Kahler potential
is

$\displaystyle K$ $\displaystyle\approx$
$\displaystyle\sum\limits_{i=1}^{n}\frac{1}{2}\left(|z_{1}-s_{1i}|^{2}+|z_{2}-s_{2i}|^{2}+y^{2}\right)$
(B.2)
$\displaystyle-\sum\limits_{i=1}^{n}\frac{1}{2}h_{i}^{2}\log\left(|z_{1}-s_{1i}|^{2}+|z_{2}-s_{2i}|^{2}+y^{2}\right)-\frac{1}{4}y^{2}\log(y^{2})+c.$

where $s_{11}=s_{21}=0$. Hence we have

$\displaystyle K$ $\displaystyle\approx$
$\displaystyle\sum\limits_{i=1}^{n}\frac{1}{2}\left(|z_{1}|^{2}+|z_{2}|^{2}+y^{2}\right)$
(B.3)
$\displaystyle-\sum\limits_{i=1}^{n}\frac{1}{2}h_{i}^{2}\log\left(|z_{1}-s_{1i}|^{2}+|z_{2}-s_{2i}|^{2}+y^{2}\right)$
$\displaystyle-\frac{1}{4}y^{2}\log(y^{2})+c.$

$h_{i}^{2}$ is proportional to the total number of eigenvalues $n_{i}$ in the
droplet $i$. The droplet is an eigenvalue cluster.

With appropriate rescaling,

$\displaystyle
K|_{y=0}\sim\frac{1}{2}\left(|z_{1}|^{2}+|z_{2}|^{2}\right)-\sum\limits_{i=1}^{n}\frac{1}{2}n_{i}\log\left(|z_{1}-s_{1i}|^{2}+|z_{2}-s_{2i}|^{2}\right)+c.$
(B.4)

This behaves as a potential that a probe eigenvalue labeled by $z_{i}$, feels
in the presence of other eigenvalues, in which there is a logarithmic
potential. We have $\sum\limits_{i=1}^{n}n_{i}=N$. The relation between
$n_{i},h_{i}$ is $n_{i}\sim h_{i}^{2}$. There is also a confining potential
$|z_{i}|^{2}$, and the confining potential in the dual field theory side has
been originally derived from dual field theory dynamics [28], see also e.g.
[147]. The droplet configuration has multiple equivalent interpretations from
the dual field theory side. See also [147]. The droplet configurations in the
gravity side correspond to eigenvalue clusters in the field theory side.222See
also various related discussions e.g. [138]-[150] and their related aspects in
diverse or different contexts. There are rich dynamics, phases, and strong-
coupling phenomena that are inter-connected.

This is in the dilute distribution limit, and the extra isolated eigenvalues,
away from the central droplet, are well-separated. There can be more general
regimes than the regime we consider here. The potential on the droplet space,
e.g. (B.4) is related to the logarithm of the wavefunction inner-product of
appropriate matrix wavefunction basis of the field theory side. On the other
hand, matrix wavefunction bases may not be unique, and there may also be other
bases by changes of variables.

We further confirm the observations in [28, 104] and their related aspects.
Our case here is a special case of [28, 104].

##  Appendix C Superstar and asymptotic expansion

The two-charge superstar have also been analyzed in [46]. The symbols
$r,{\tilde{r}}$ in Sec. 6.2.1 of [46] are denoted as ${\tilde{r},}r$ here, due
to that we use symbol $r$ for the droplet space. The $Z$ function for
superstar is

$\displaystyle Z$ $\displaystyle=$
$\displaystyle\frac{1}{2}-\frac{1}{1+Q_{2}+Q_{3}+{\tilde{r}}^{2}+{Q_{2}\,Q_{3}}/{\tilde{r}^{2}}+\cot^{2}\theta[({\tilde{r}}^{2}+Q_{3})\sin^{2}\alpha+({\tilde{r}}^{2}+Q_{2})\cos^{2}\alpha]}\
.$

We work in unit $l=1$. Note the “$1$” on the right hand side correspond to
$l^{2}$, when restored.

The analysis is that we expand from the point of view of asymptotic region.
Consider the identification for large ${\tilde{r}}$ and

$y={\tilde{r}}\sin\theta,\ r={\tilde{r}}\cos\theta.$ (C.2)

We have $r^{2}={\tilde{r}}^{2}-y^{2}$ and$\
\sin^{2}\theta=\frac{y^{2}}{{\tilde{r}}^{2}}$. Consider equal charge case for
simplicity, $Q_{2}=Q_{3}=Q$. We have

$\displaystyle Z$ $\displaystyle=$
$\displaystyle\frac{1}{2}-\frac{\sin^{2}\theta}{\sin^{2}\theta\left(1+2Q+{\tilde{r}}^{2}+{Q}^{{2}}/{\tilde{r}}^{2}\right)+\cos^{2}\theta({\tilde{r}}^{2}+Q)}$
(C.3) $\displaystyle=$
$\displaystyle\frac{1}{2}-\frac{\sin^{2}\theta}{{\tilde{r}}^{2}+Q+\sin^{2}\theta(1+Q)+\sin^{2}\theta{Q}^{{2}}/{\tilde{r}}^{2}}.$

We use $1/{\tilde{r}}$ expansion. The large ${\tilde{r}}$ expansion is

$Z\simeq\frac{1}{2}-\frac{\sin^{2}\theta}{{\tilde{r}}^{2}}+\frac{\sin^{2}\theta(Q+\sin^{2}\theta(1+Q))}{{\tilde{r}}^{4}}+...$
(C.4)

and

$\frac{Q+\sin^{2}\theta(1+Q)}{{\tilde{r}}^{2}}=\frac{Q}{{\tilde{r}}^{2}}+(1+Q)\frac{y^{2}}{{\tilde{r}}^{4}}.$
(C.5)

To extract the distribution, we have then,

$Z-\frac{1}{2}\simeq-\frac{y^{2}l^{2}}{(r^{2}+y^{2})^{2}}\left(1-\frac{Q}{r^{2}+y^{2}}\right)+o(\frac{1}{(r^{2}+y^{2})^{4}}),$
(C.6)

with asymptotic AdS radius $l$ restored. The superstar metric and smooth
geometries differ slightly in the interior region approaching the stretched
horizon. We expand in the exterior region.

We use a technique [49] that we expand using $1/(r^{2}+y^{2})$. We can expand
using $r^{2}+y^{2},$ by the dilute distribution configuration, e.g. (B.1),

$\displaystyle\frac{r^{2}+y^{2}-h_{i}^{2}}{2\sqrt{(r^{2}+y^{2}-h_{i}^{2})^{2}+4y^{2}h_{i}^{2}}}-\frac{1}{2}$
(C.7) $\displaystyle=$
$\displaystyle-\frac{y^{2}h_{i}^{2}}{(r^{2}+y^{2})^{2}}+o(h_{i}^{4}).$

It has both $r^{2}+y^{2}$ dependence and angular $y^{2}/r^{2}$ dependence. We
have

$\displaystyle-\frac{y^{2}h_{i}^{2}}{(|z_{1}-s_{1i}|^{2}+|z_{2}-s_{2i}|^{2}+y^{2})^{2}}$
(C.8) $\displaystyle\simeq$
$\displaystyle-\frac{y^{2}h_{i}^{2}}{(r^{2}+y^{2})^{2}}\left(1-\frac{2(|s_{1i}|^{2}+|s_{2i}|^{2})}{r^{2}+y^{2}}\right)+o(\frac{1}{(r^{2}+y^{2})^{4}})$
$\displaystyle\simeq$
$\displaystyle-\frac{y^{2}h_{i}^{2}}{{\tilde{r}}^{4}}\left(1-\frac{2(|s_{1i}|^{2}+|s_{2i}|^{2})}{{\tilde{r}}^{2}}\right)+o(\frac{1}{{\tilde{r}}^{8}}).$

Hence by (C.6) and (C.8),

$\displaystyle-\frac{y^{2}l^{2}}{{\tilde{r}}^{4}}\left(1-\frac{Q}{{\tilde{r}}^{2}}\right)$
(C.9) $\displaystyle\simeq$
$\displaystyle\sum\limits_{i}-\frac{y^{2}h_{i}^{2}}{{\tilde{r}}^{4}}\left(1-\frac{2(|s_{1i}|^{2}+|s_{2i}|^{2})}{{\tilde{r}}^{2}}\right).$

We include central droplet $i=0$, and $h_{0}^{2}=l_{0}^{2}$. Hence by (C.9),
$\sum\limits_{i}h_{i}^{2}=l^{2}$. Hence

$l^{2}Q\simeq\sum\limits_{i}2h_{i}^{2}(|s_{1i}|^{2}+|s_{2i}|^{2}).$ (C.10)

Note this is in the dilute distribution configuration limit. The small
droplets are far away from the central droplet, $|s_{1i}|^{2}+|s_{2i}|^{2}\gg
1.$

We assumed equal charge, and when we restore $Q_{2}$ and $Q_{3}$, we find,

$\frac{1}{2}l^{2}Q_{2}\simeq\sum\limits_{i}2h_{i}^{2}|s_{1i}|^{2},\ \ \ \
\frac{1}{2}l^{2}Q_{3}\simeq\sum\limits_{i}2h_{i}^{2}|s_{2i}|^{2}.$ (C.11)

Note the convention of labels $2,3$ in “$Q_{2,3}$” in [46] are labels $1,2$
here in this paper. One can also consider equal size cases where
$h_{i}^{2}=h^{2}$ are the same for each droplets, and then take a continuous
distribution limit, under which the droplet density becomes grey
distributions.

The geometries are asymptotic to $AdS_{5}\times S^{5}$. We set the $AdS$
radius parameter to be $l$ for this $AdS_{5}\times S^{5}$ geometry at the
asymptotia.

##  Appendix D Properties of $F_{+}(k,y)$ and a new approximate solution

Denote the solution for “$y=0$ boundary” case as $F_{+}(k,y)$. We can have
approximate solution as follows:

$F_{+}(k,y)\simeq
c_{1}\frac{1}{2}k^{2}y^{2}K_{2}(ky)+\frac{c_{2}l^{2}}{y_{c}^{2}+y^{2}}\frac{1}{48}k^{4}y^{4}K_{4}(ky),\
\ \ y\geq 0.$ (D.1)

Note the above “$y=y_{c}$ boundary” solution (4.25) is:

$F(k,y)\simeq
c_{1}\frac{1}{2}k^{2}y^{2}K_{2}(ky)+c_{2}\frac{1}{48}k^{4}l^{2}y^{2}K_{4}(ky).$
(D.2)

When $y\rightarrow 0$:

$F_{+}(k,y)\simeq
c_{1}\frac{1}{2}k^{2}y^{2}K_{2}(ky)+\frac{c_{2}l^{2}}{y_{c}^{2}}\frac{1}{48}k^{4}y^{4}K_{4}(ky),\
\ \ 10^{-1}y_{c}>y\geq 0.$ (D.3)

$F_{+}(k,y)|_{y=0}=c_{1}+\frac{c_{2}l^{2}}{y_{c}^{2}}=1$. This means it
satisfies the boundary condition placed at $y=0$, approaching delta function.

When $y>10y_{c}$:

$F_{+}(k,y)\simeq
c_{1}\frac{1}{2}k^{2}y^{2}K_{2}(ky)+c_{2}\frac{1}{48}k^{4}l^{2}y^{2}K_{4}(ky),\
\ \ y>10y_{c}.$ (D.4)

This becomes (D.2). Hence (D.1) is an approximate solution for “$y=0$
boundary” case, satisfying the boundary condition placed at $y=0$.

##  Appendix E Relation to the symplectic form and discussion

We have observed similar phenomena encountered in [57] in the context of
deformation modes in smooth bubbling geometries and their behaviors under
coarse-graining. We first work with smooth droplet configurations, and later
perform coarse-graining.

The coordinate $z_{1}=q_{1}+ip_{1},z_{2}=q_{2}+ip_{2}$, can be identified with
the holomorphic coordinate associated to the degree of freedom for a probe
eigenvalue in the field theory side. For related discussions and equivalent
interpretations, see e.g. [27]. This can be identified as one particle phase
space. Hence one can identify the 4d droplet space coordinate as one particle
phase space. We can also write
$z_{1}=r_{1}e^{i\phi},z_{2}=r_{2}e^{i\varphi},r^{2}=|z_{1}|^{2}+|z_{2}|^{2}$.
We can denote $|z_{1}|=r\cos\chi$, $|z_{2}|=r\sin\chi$, and
$0\leq\chi\leq\frac{\pi}{2}$, $\sin\chi\geq 0,\cos\chi\geq 0$.

Here we mainly use coherent states and Berry curvature formalism. One can also
use coadjoint orbit formalism, coherent state formalism, and other formalisms
[151]. It may also be interesting to use collective field theory. The
calculations can be simplified and unified by the Berry curvature formalism.

As discussed in e.g. [57] and Sec. 5.3.2 of this paper, $u(z)$ is density, and
we define $\pi(z)$ as the canonical conjugate momentum to the density $u(z)$.
The symplectic form is

$\boldsymbol{\Omega}=\int_{r<R}d^{4}z\,\boldsymbol{\delta}\pi(z)\boldsymbol{\delta}u(z).$
(E.1)

We use canonical quantization. There is a natural Poisson structure induced by
the two-dimensional quantum harmonic oscillator [151].

Consider the action of coherent state operators on excitations of a reference
state denoted by $|\psi_{0}\rangle$. The resulting state can be expressed as

$|\psi\rangle=e^{iT_{\pi}}|\psi_{0}\rangle,\;\;\;T_{\pi}=\int\prod_{i=1,2}\frac{dp_{i}^{\prime}dq_{i}^{\prime}}{2\pi\hbar}\pi(q_{i}^{\prime},p_{i}^{\prime})u(q_{i}^{\prime},p_{i}^{\prime}).$
(E.2)

The change of the state is given by

$\delta|\psi\rangle=i\int\prod_{i=1,2}\frac{dp_{i}^{\prime}dq_{i}^{\prime}}{2\pi\hbar}\delta\pi(q_{i}^{\prime},p_{i}^{\prime})u(q_{i}^{\prime},p_{i}^{\prime})|\psi_{0}\rangle.$
(E.3)

This characterizes the coadjoint orbit of $\psi_{0}$, which gives the orbit of
the coadjoint action. The symplectic form above can also be expressed as
$\hbar$ times the Berry curvature. This has also been discussed in the context
of gauge/gravity correspondence by [152] in general. We have, see also [152]
and [57], hence that

$\Omega=\hbar
F_{Berry}=i\hbar\left(\langle\delta_{1}\psi|\delta_{2}\psi\rangle-\langle\delta_{2}\psi|\delta_{1}\psi\rangle\right).$
(E.4)

Hence,

$\Omega=\int\prod_{i=1,2}\frac{dp_{i}^{\prime}dq_{i}^{\prime}}{2\pi\hbar}\delta\pi(q_{i}^{\prime},p_{i}^{\prime})\delta
u(q_{i}^{\prime},p_{i}^{\prime}).$ (E.5)

We work in the natural units. Our methods here are similar to those of [57].
We consider concentric shell model, e.g. [49]; we consider the black regions
are along the radial directions in concentric shell configurations. The shells
are labeled by $i$. The $u$ depend on the droplet pattern. We have

$u(r)=\sum_{i}(-1)^{i}\Theta(r_{i}-r),$ (E.6)

where $\Theta(x)$ denotes step function. The $i$ is $i$-th droplet shell. For
$i$ even, the black droplet region is between $r_{i}$ and $r_{i-1}$. The
droplet spacing along $r$ is denoted as $\epsilon.\ $The change of density
localize on droplet edge $i.$ This is a choice of basis for deformation modes.
Consider the change of the density from $u$ to $u+\delta u$, and

$\delta u(r,\phi,\varphi)=\sum_{i}(-1)^{i}\delta
r_{i}(\phi,\varphi)\delta(r_{i}-r).$ (E.7)

We have $u_{i}(\phi,\varphi)$ as

$\delta u(r,\phi,\varphi)=\epsilon\sum_{i}r_{i}\delta
u_{i}(\phi,\varphi)\frac{1}{r_{i}}\delta(r-r_{i}).$ (E.8)

The change of density $\boldsymbol{\delta}u(r,\phi,\varphi)$ is decomposed
into $\delta u_{i}(\phi,\varphi)$. We than have

$\delta r_{i}(r,\phi,\varphi)=(-1)^{i}\epsilon\delta u_{i}(r,\phi,\varphi).$
(E.9)

We then get $\partial_{\phi}\delta\pi(r_{i},\phi,\varphi)=\cos^{2}\chi
r_{i}\delta r_{i}$,
$\partial_{\varphi}\delta\pi(r_{i},\phi,\varphi)=\sin^{2}\chi r_{i}\delta
r_{i}$, and hence

$\delta\pi(r,\phi,\varphi)\sim\frac{1}{4\pi}\int
d\phi^{\prime}d\varphi^{\prime}(\mathrm{Sign}(\phi-\phi^{\prime})\cos^{2}\chi+\mathrm{Sign}(\varphi-\varphi^{\prime})\sin^{2}\chi)\sum_{i}\epsilon
r_{i}(-1)^{i}\delta
u(r_{i},\phi^{\prime},\varphi^{\prime})H_{\epsilon}(r-r_{i}).$ (E.10)

Here $H_{\epsilon}(r-r_{i})$ is a smooth function covering the $i$-th shell;
it’s centered around $r_{i}$ with width $\epsilon$, and its value is 1 in
between $r_{i}-\frac{1}{2}\epsilon$ and $r_{i}+\frac{1}{2}\epsilon$, and drops
to zero outside this range. We have $r_{i}=i\epsilon$, and hence $e^{i\pi
r_{i}/\epsilon}H_{\epsilon}(r-r_{i})=(-1)^{i}H_{\epsilon}(r-r_{i})$. Hence,

$\displaystyle\Omega$ $\displaystyle\sim$
$\displaystyle-\frac{1}{16\pi^{3}}\int r^{2}dr\int\sin\chi\cos\chi d\chi d\phi
d\varphi
d\phi^{\prime}d\varphi^{\prime}\boldsymbol{\delta}u(r,\phi,\varphi)(\mathrm{Sign}(\phi-\phi^{\prime})\cos^{2}\chi+\mathrm{Sign}(\varphi-\varphi^{\prime})\sin^{2}\chi)$
(E.11) $\displaystyle\sum_{i}\epsilon r_{i}e^{i\pi
r/\epsilon}\boldsymbol{\delta}u(r_{i},\phi^{\prime},\varphi^{\prime})H_{\epsilon}(r-r_{i}).$

Here we observed a similar phenomena. Decompose the density variation by

$\boldsymbol{\delta}u(r,\phi,\varphi)=\boldsymbol{\delta}A(r,\phi,\varphi)+\frac{e^{-i\pi
r/\epsilon}}{\epsilon}\boldsymbol{\delta}B(r,\phi,\varphi),$ (E.12)

and one get the highly-oscillating terms
$\boldsymbol{\delta}A\boldsymbol{\delta}A$ and
$\boldsymbol{\delta}B\boldsymbol{\delta}B$, and the remaining non-oscillating
term $\boldsymbol{\delta}A\boldsymbol{\delta}B$.

The symplectic form is integrated and one can perform a coarse-graining. By
using the coarse-graining function (5.11), we obtain, in the same way,

$\boldsymbol{\delta}u_{IR}(x)=\int
d^{4}yP_{IR}(x,y)\boldsymbol{\delta}u(y)\sim\boldsymbol{\delta}A(x).$ (E.13)

This is the emergent new soft mode on the stretched horizon. We have $k\leq
y_{0}^{-1}<k_{\epsilon}=\frac{1}{\epsilon}$. This is because $P_{IR}(k)$ is
very small for $k>y_{0}^{-1}$, and $k_{\epsilon}=\frac{1}{\epsilon}$ is in the
range of being removed, hence the highly oscillatory part is removed, for the
coarse-grained data. For observer who has coarse-grained data on the scale
much larger than $\epsilon$, he or she sees the effect of nonoscillating
$\boldsymbol{\delta}A\boldsymbol{\delta}B$. The resulting phenomena and
related conclusions are similar to [57], hence we refer the readers to [57]
for detailed expositions.

##  Appendix F Overview and summary of solution techniques

The solutions come with different techniques. There is an overall solution
space of the smooth geometries. Different solution techniques describe various
regimes of the solution space.

One technique is the solutions of linearized Monge-Ampere equations with
sources. The linearized equations are Laplace-type equations with sources. The
core or source of the linearized solutions can be viewed as the limit of
collapsing of isolated small droplets. For linearized solutions with sources,
the solution space, is approximately isomorphic to $q_{t}$ points on
$C^{2}\backslash\\{0\\}$.

Another technique is dilute distribution and extreme-ratio solutions, where
there are multiple small droplets and can include the central droplet. The
blowing-down of these solutions along the cycles of smooth small droplets is
related to the solutions of linearized equations with sources.

Another technique is the wavy deformation of droplets [21, 22], which include
general cases of wavy deformation of a general set of droplets. These
solutions are also related to volume preserving deformations. These solutions
are used to obtain the symplectic forms of the configuration space of general
solutions, including general droplets.

Another technique is the stretched droplet space solutions, where arbitrary
coarse distribution are placed at a stretched surface $y=y_{c}$ away from the
$y=0$ surface. The linearized technique can be combined with this technique,
because we can place this stretched surface outside all the enclosure surfaces
of each droplets.

## References

  * [1] N. Beisert, C. Ahn, L. F. Alday, Z. Bajnok, J. M. Drummond, L. Freyhult, N. Gromov, R. A. Janik, V. Kazakov and T. Klose, et al. “Review of AdS/CFT Integrability: An Overview,” Lett. Math. Phys. 99 (2012), 3 [arXiv:1012.3982 [hep-th]].
  * [2] M. Rangamani and T. Takayanagi, “Holographic Entanglement Entropy,” Lect. Notes Phys. 931 (2017), 1, 2017, [arXiv:1609.01287 [hep-th]].
  * [3] G. T. Horowitz and J. Polchinski, “Gauge/gravity duality,” In: D. Oriti, Approaches to quantum gravity, Cambridge University Press [gr-qc/0602037].
  * [4] J. McGreevy, L. Susskind and N. Toumbas, “Invasion of the giant gravitons from Anti-de Sitter space,” JHEP 06 (2000), 008 [arXiv:hep-th/0003075 [hep-th]].
  * [5] M. T. Grisaru, R. C. Myers and O. Tafjord, “SUSY and goliath,” JHEP 08 (2000), 040 [arXiv:hep-th/0008015 [hep-th]].
  * [6] A. Hashimoto, S. Hirano and N. Itzhaki, “Large branes in AdS and their field theory dual,” JHEP 08 (2000), 051 [arXiv:hep-th/0008016 [hep-th]].
  * [7] S. Corley, A. Jevicki and S. Ramgoolam, “Exact correlators of giant gravitons from dual N=4 SYM theory,” Adv. Theor. Math. Phys. 5 (2002), 809-839 [arXiv:hep-th/0111222 [hep-th]].
  * [8] D. Berenstein, “A Toy model for the AdS / CFT correspondence,” JHEP 07 (2004), 018 [arXiv:hep-th/0403110 [hep-th]].
  * [9] V. Balasubramanian, M. Berkooz, A. Naqvi and M. J. Strassler, “Giant gravitons in conformal field theory,” JHEP 04 (2002), 034 [arXiv:hep-th/0107119 [hep-th]].
  * [10] H. Lin, O. Lunin and J. M. Maldacena, “Bubbling AdS space and 1/2 BPS geometries,” JHEP 10 (2004), 025 [arXiv:hep-th/0409174 [hep-th]].
  * [11] D. Gaiotto and J. H. Lee, “The Giant Graviton Expansion,” [arXiv:2109.02545 [hep-th]].
  * [12] Y. Imamura, “Analytic continuation for giant gravitons,” PTEP 2022 (2022) no.10, 103B02 [arXiv:2205.14615 [hep-th]].
  * [13] K. Budzik and D. Gaiotto, “Giant gravitons in twisted holography,” [arXiv:2106.14859 [hep-th]].
  * [14] S. Murthy, “Unitary matrix models, free fermions, and the giant graviton expansion,” [arXiv:2202.06897 [hep-th]].
  * [15] D. S. Eniceicu, “Comments on the Giant-Graviton Expansion of the Superconformal Index,” [arXiv:2302.04887 [hep-th]].
  * [16] J. T. Liu and N. J. Rajappa, “Finite N indices and the giant graviton expansion,” [arXiv:2212.05408 [hep-th]].
  * [17] S. Choi, S. Kim, E. Lee and J. Lee, “From giant gravitons to black holes,” [arXiv:2207.05172 [hep-th]].
  * [18] M. Beccaria and A. Cabo-Bizet, “Large black hole entropy from the giant brane expansion,” [arXiv:2308.05191 [hep-th]].
  * [19] D. Berenstein and A. Miller, “Superposition induced topology changes in quantum gravity,” JHEP 11 (2017), 121 [arXiv:1702.03011 [hep-th]].
  * [20] V. Balasubramanian, J. de Boer, V. Jejjala and J. Simon, “The Library of Babel: On the origin of gravitational thermodynamics,” JHEP 12 (2005), 006 [arXiv:hep-th/0508023 [hep-th]].
  * [21] L. Grant, L. Maoz, J. Marsano, K. Papadodimas and V. S. Rychkov, “Minisuperspace quantization of ’Bubbling AdS’ and free fermion droplets,” JHEP 08 (2005), 025 [arXiv:hep-th/0505079 [hep-th]].
  * [22] L. Maoz and V. S. Rychkov, “Geometry quantization from supergravity: The Case of Bubbling AdS,” JHEP 08 (2005), 096 [arXiv:hep-th/0508059 [hep-th]].
  * [23] R. Bhattacharyya, S. Collins and R. de Mello Koch, “Exact Multi-Matrix Correlators,” JHEP 03 (2008), 044 [arXiv:0801.2061 [hep-th]].
  * [24] R. Bhattacharyya, R. de Mello Koch and M. Stephanou, “Exact Multi-Restricted Schur Polynomial Correlators,” JHEP 06 (2008), 101 [arXiv:0805.3025 [hep-th]].
  * [25] R. de Mello Koch, P. Diaz and N. Nokwara, “Restricted Schur Polynomials for Fermions and integrability in the su(2$|$3) sector,” JHEP 03 (2013), 173 [arXiv:1212.5935 [hep-th]].
  * [26] T. W. Brown, P. J. Heslop and S. Ramgoolam, “Diagonal multi-matrix correlators and BPS operators in N=4 SYM,” JHEP 02 (2008), 030 [arXiv:0711.0176 [hep-th]].
  * [27] D. Berenstein and S. Wang, “BPS coherent states and localization,” JHEP 08 (2022), 164 [arXiv:2203.15820 [hep-th]].
  * [28] D. Berenstein, “Large N BPS states and emergent quantum gravity,” JHEP 01 (2006), 125 [arXiv:hep-th/0507203 [hep-th]].
  * [29] A. Dhar, G. Mandal and M. Smedback, “From gravitons to giants,” JHEP 03 (2006), 031 [arXiv:hep-th/0512312 [hep-th]].
  * [30] Y. Kimura and S. Ramgoolam, “Branes, anti-branes and brauer algebras in gauge-gravity duality,” JHEP 11 (2007), 078 [arXiv:0709.2158 [hep-th]].
  * [31] R. de Mello Koch, J. Smolic and M. Smolic, “Giant Gravitons - with Strings Attached (II),” JHEP 09 (2007), 049 [arXiv:hep-th/0701067 [hep-th]].
  * [32] P. Caputa, R. de Mello Koch and P. Diaz, “A basis for large operators in N=4 SYM with orthogonal gauge group,” JHEP 03 (2013), 041 [arXiv:1301.1560 [hep-th]].
  * [33] R. Suzuki, “Three-point functions in $\mathcal{N}$ = 4 SYM at finite Nc and background independence,” JHEP 05 (2020), 118 [arXiv:2002.07216 [hep-th]].
  * [34] J. Pasukonis and S. Ramgoolam, “Quivers as Calculators: Counting, Correlators and Riemann Surfaces,” JHEP 04 (2013), 094 [arXiv:1301.1980 [hep-th]].
  * [35] C. Lewis-Brown and S. Ramgoolam, “Quarter-BPS states, multi-symmetric functions and set partitions,” JHEP 03 (2021), 153 [arXiv:2007.01734 [hep-th]].
  * [36] A. Bissi, C. Kristjansen, D. Young and K. Zoubos, “Holographic three-point functions of giant gravitons,” JHEP 06 (2011), 085 [arXiv:1103.4079 [hep-th]].
  * [37] D. Bak, B. Chen and J. B. Wu, “Holographic Correlation Functions for Open Strings and Branes,” JHEP 06 (2011), 014 [arXiv:1103.2024 [hep-th]].
  * [38] P. Caputa, R. de Mello Koch and K. Zoubos, “Extremal versus Non-Extremal Correlators with Giant Gravitons,” JHEP 08 (2012), 143 [arXiv:1204.4172 [hep-th]].
  * [39] H. Lin, “Giant gravitons and correlators,” JHEP 12 (2012), 011 [arXiv:1209.6624 [hep-th]].
  * [40] Y. Jiang, S. Komatsu and E. Vescovi, “Structure constants in $\mathcal{N}$ = 4 SYM at finite coupling as worldsheet g-function,” JHEP 07 (2020), 037 [arXiv:1906.07733 [hep-th]].
  * [41] G. Chen, R. de Mello Koch, M. Kim and H. J. R. Van Zyl, “Absorption of closed strings by giant gravitons,” JHEP 10 (2019), 133 [arXiv:1908.03553 [hep-th]].
  * [42] P. Yang, Y. Jiang, S. Komatsu and J. B. Wu, “D-branes and orbit average,” SciPost Phys. 12 (2022) no.2, 055 [arXiv:2103.16580 [hep-th]].
  * [43] A. Holguin and W. W. Weng, “Orbit averaging coherent states: holographic three-point functions of AdS giant gravitons,” JHEP 05 (2023), 167 [arXiv:2211.03805 [hep-th]].
  * [44] J. Abajian, F. Aprile, R. C. Myers and P. Vieira, “Holography and correlation functions of huge operators: spacetime bananas,” [arXiv:2306.15105 [hep-th]].
  * [45] A. Holguin, “1/2 BPS structure constants and random matrices,” JHEP 12 (2023), 046 [arXiv:2305.06390 [hep-th]].
  * [46] B. Chen, S. Cremonini, A. Donos, F. L. Lin, H. Lin, J. T. Liu, D. Vaman and W. Y. Wen, “Bubbling AdS and droplet descriptions of BPS geometries in IIB supergravity,” JHEP 10 (2007), 003 [arXiv:0704.2233 [hep-th]].
  * [47] O. Lunin, “Brane webs and 1/4-BPS geometries,” JHEP 09 (2008), 028 [arXiv:0802.0735 [hep-th]].
  * [48] H. Lin, “Studies on 1/4 BPS and 1/8 BPS geometries,” [arXiv:1008.5307 [hep-th]].
  * [49] Y. Kimura and H. Lin, “Young diagrams, Brauer algebras, and bubbling geometries,” JHEP 01 (2012), 121 [arXiv:1109.2585 [hep-th]].
  * [50] J. P. Gauntlett, N. Kim and D. Waldram, “Supersymmetric AdS(3), AdS(2) and Bubble Solutions,” JHEP 04 (2007), 005 [arXiv:hep-th/0612253 [hep-th]].
  * [51] N. Kim, “AdS(3) solutions of IIB supergravity from D3-branes,” JHEP 01 (2006), 094 [arXiv:hep-th/0511029 [hep-th]].
  * [52] E. Gava, G. Milanesi, K. S. Narain and M. O’Loughlin, “1/8 BPS states in AdS/CFT,” JHEP 05 (2007), 030 [arXiv:hep-th/0611065 [hep-th]].
  * [53] J. T. Liu, H. Lu, C. N. Pope and J. F. Vazquez-Poritz, “Bubbling AdS black holes,” JHEP 10 (2007), 030 [arXiv:hep-th/0703184 [hep-th]].
  * [54] Z. W. Chong, H. Lu and C. N. Pope, “BPS geometries and AdS bubbles,” Phys. Lett. B 614 (2005), 96-103 [arXiv:hep-th/0412221 [hep-th]].
  * [55] O. Lunin, “1/2-BPS states in M theory and defects in the dual CFTs,” JHEP 10 (2007), 014 [arXiv:0704.3442 [hep-th]].
  * [56] J. T. Liu, D. Vaman and W. Y. Wen, “Bubbling 1/4 BPS solutions in type IIB and supergravity reductions on S*n x S*n,” Nucl. Phys. B 739 (2006), 285-310 [arXiv:hep-th/0412043 [hep-th]].
  * [57] V. Balasubramanian, D. Berenstein, A. Lewkowycz, A. Miller, O. Parrikar and C. Rabideau, “Emergent classical spacetime from microstates of an incipient black hole,” JHEP 01 (2019), 197 [arXiv:1810.13440 [hep-th]].
  * [58] S. Mukhi and M. Smedback, “Bubbling orientifolds,” JHEP 08 (2005), 005 [arXiv:hep-th/0506059 [hep-th]].
  * [59] B. Fiol, B. Garolera and G. Torrents, “Exact probes of orientifolds,” JHEP 09 (2014), 169 [arXiv:1406.5129 [hep-th]].
  * [60] C. Lewis-Brown and S. Ramgoolam, “BPS operators in $\mathcal{N}=4$ $SO(N)$ super Yang-Mills theory: plethysms, dominoes and words,” JHEP 11 (2018), 035 [arXiv:1804.11090 [hep-th]].
  * [61] A. Holguin and S. Wang, “Giant gravitons, Harish-Chandra integrals, and BPS states in symplectic and orthogonal $\mathcal{N}$ = 4 SYM,” JHEP 10 (2022), 078 [arXiv:2206.00020 [hep-th]].
  * [62] S. Giombi and B. Offertaler, “Wilson loops in $\mathcal{N}$ = 4 SO(N) SYM and D-branes in AdS5 × $\mathbb{R}$$\mathbb{P}$5,” JHEP 10 (2021), 016 [arXiv:2006.10852 [hep-th]].
  * [63] P. Caputa, R. de Mello Koch and P. Diaz, “Operators, Correlators and Free Fermions for SO(N) and Sp(N),” JHEP 06 (2013), 018 [arXiv:1303.7252 [hep-th]].
  * [64] S. Bhattacharyya, S. Minwalla and K. Papadodimas, “Small Hairy Black Holes in $AdS_{5}$x$S^{5}$,” JHEP 11 (2011), 035 [arXiv:1005.1287 [hep-th]].
  * [65] H. K. Kunduri and J. Lucietti, “Near-horizon geometries of supersymmetric AdS(5) black holes,” JHEP 12 (2007), 015 [arXiv:0708.3695 [hep-th]].
  * [66] K. Skenderis and M. Taylor, “Anatomy of bubbling solutions,” JHEP 09 (2007), 019 [arXiv:0706.0216 [hep-th]].
  * [67] A. Christodoulou and K. Skenderis, “Holographic Construction of Excited CFT States,” JHEP 04 (2016), 096 [arXiv:1602.02039 [hep-th]].
  * [68] J. Armas, N. A. Obers and A. V. Pedersen, “Null-Wave Giant Gravitons from Thermal Spinning Brane Probes,” JHEP 10 (2013), 109 [arXiv:1306.2633 [hep-th]].
  * [69] G. Kemp and S. Ramgoolam, “BPS states, conserved charges and centres of symmetric group algebras,” JHEP 01 (2020), [arXiv:1911.11649 [hep-th]].
  * [70] J. B. Geloun and S. Ramgoolam, “The quantum detection of projectors in finite-dimensional algebras and holography,” JHEP 05 (2023), 191 [arXiv:2303.12154 [quant-ph]].
  * [71] V. Balasubramanian, B. Czech, K. Larjo and J. Simon, “Integrability versus information loss: A Simple example,” JHEP 11 (2006), 001 [arXiv:hep-th/0602263 [hep-th]].
  * [72] V. Balasubramanian, B. Czech, K. Larjo, D. Marolf and J. Simon, “Quantum geometry and gravitational entropy,” JHEP 12 (2007), 067 [arXiv:0705.4431 [hep-th]].
  * [73] V. Balasubramanian, B. Czech, V. E. Hubeny, K. Larjo, M. Rangamani and J. Simon, “Typicality versus thermality: An Analytic distinction,” Gen. Rel. Grav. 40 (2008), 1863-1890 [arXiv:hep-th/0701122 [hep-th]].
  * [74] I. Biswas, D. Gaiotto, S. Lahiri and S. Minwalla, “Supersymmetric states of N=4 Yang-Mills from giant gravitons,” JHEP 12 (2007), 006 [arXiv:hep-th/0606087 [hep-th]].
  * [75] G. Mandal and N. V. Suryanarayana, “Counting 1/8-BPS dual-giants,” JHEP 03 (2007), 031 [arXiv:hep-th/0606088 [hep-th]].
  * [76] S. Kim and K. M. Lee, “1/16-BPS Black Holes and Giant Gravitons in the AdS(5) x S*5 Space,” JHEP 12 (2006), 077 [arXiv:hep-th/0607085 [hep-th]].
  * [77] H. Paul, E. Perlmutter and H. Raj, “Exact large charge in $\mathcal{N}$ = 4 SYM and semiclassical string theory,” JHEP 08 (2023), 078 [arXiv:2303.13207 [hep-th]].
  * [78] J. Caetano, S. Komatsu and Y. Wang, “Large Charge ’t Hooft Limit of $\mathcal{N}=4$ Super-Yang-Mills,” [arXiv:2306.00929 [hep-th]].
  * [79] Y. Z. Li, Z. F. Mai, H. Lu, “Holographic OPE Coefficients from AdS Black Holes with Matters,” JHEP 09 (2019), 001 [arXiv:1905.09302 [hep-th]].
  * [80] R. de Mello Koch, T. K. Dey, N. Ives and M. Stephanou, “Correlators Of Operators with a Large R-charge,” JHEP 08 (2009), 083 [arXiv:0905.2273 [hep-th]].
  * [81] D. Berenstein and A. Holguin, “Open giant magnons on LLM geometries,” JHEP 01 (2021), 080 [arXiv:2010.02236 [hep-th]].
  * [82] H. Y. Chen, D. H. Correa and G. A. Silva, “Geometry and topology of bubble solutions from gauge theory,” Phys. Rev. D 76 (2007), 026003 [arXiv:hep-th/0703068 [hep-th]].
  * [83] R. de Mello Koch, J. H. Huang and L. Tribelhorn, “Exciting LLM Geometries,” JHEP 07 (2018), 146 [arXiv:1806.06586 [hep-th]].
  * [84] S. Yamaguchi, “Bubbling geometries for half BPS Wilson lines,” Int. J. Mod. Phys. A 22 (2007), 1353-1374 [arXiv:hep-th/0601089 [hep-th]].
  * [85] O. Lunin, “On gravitational description of Wilson lines,” JHEP 0606 (2006) 026 [hep-th/0604133].
  * [86] J. Gomis and F. Passerini, “Holographic Wilson Loops,” JHEP 08 (2006), 074 [arXiv:hep-th/0604007 [hep-th]].
  * [87] E. D’Hoker, J. Estes and M. Gutperle, “Gravity duals of half-BPS Wilson loops,” JHEP 0706 (2007) 063 [arXiv:0705.1004 [hep-th]].
  * [88] J. Aguilera-Damia, D. H. Correa, F. Fucito, V. I. Giraldo-Rivera, J. F. Morales and L. A. Pando Zayas, “Strings in Bubbling Geometries and Dual Wilson Loop Correlators,” JHEP 12 (2017), 109 [arXiv:1709.03569 [hep-th]].
  * [89] S. Yamaguchi, “Wilson loops of anti-symmetric representation and D5-branes,” JHEP 05 (2006), 037 [arXiv:hep-th/0603208 [hep-th]].
  * [90] S. A. Gentle, M. Gutperle and C. Marasinou, “Entanglement entropy of Wilson surfaces from bubbling geometries in M-theory,” JHEP 08 (2015), 019 [arXiv:1506.00052 [hep-th]].
  * [91] T. Okuda, “A Prediction for bubbling geometries,” JHEP 01 (2008), 003 [arXiv:0708.3393 [hep-th]].
  * [92] S. Giombi and V. Pestun, “Correlators of local operators and 1/8 BPS Wilson loops on S*2 from 2d YM and matrix models,” JHEP 10 (2010), 033 [arXiv:0906.1572 [hep-th]].
  * [93] S. A. Hartnoll and S. P. Kumar, “Higher rank Wilson loops from a matrix model,” JHEP 08 (2006), 026 [arXiv:hep-th/0605027 [hep-th]].
  * [94] J. Gomis, S. Matsuura, T. Okuda and D. Trancanelli, “Wilson loop correlators at strong coupling: From matrices to bubbling geometries,” JHEP 08 (2008), 068 [arXiv:0807.3330 [hep-th]].
  * [95] S. Gukov and E. Witten, “Rigid Surface Operators,” Adv. Theor. Math. Phys. 14 (2010) no.1, 87-178 [arXiv:0804.1561 [hep-th]].
  * [96] J. Gomis and S. Matsuura, “Bubbling surface operators and S-duality,” JHEP 06 (2007), 025 [arXiv:0704.1657 [hep-th]].
  * [97] S. A. Gentle, M. Gutperle and C. Marasinou, “Holographic entanglement entropy of surface defects,” JHEP 04 (2016), 067 [arXiv:1512.04953 [hep-th]].
  * [98] H. Lin and J. M. Maldacena, “Fivebranes from gauge theory,” Phys. Rev. D 74 (2006), 084014 [arXiv:hep-th/0509235 [hep-th]].
  * [99] N. Drukker, J. Gomis and S. Matsuura, “Probing N=4 SYM With Surface Operators,” JHEP 10 (2008), 048 [arXiv:0805.4199 [hep-th]].
  * [100] Y. Asano, G. Ishiki and S. Shimasaki, “Emergent bubbling geometries in gauge theories with SU(2$|$4) symmetry,” JHEP 09 (2014), 137 [arXiv:1406.1337 [hep-th]].
  * [101] Y. Asano, G. Ishiki, T. Okada and S. Shimasaki, “Exact results for perturbative partition functions of theories with SU(2$|$4) symmetry,” JHEP 02 (2013), 148 [arXiv:1211.0364 [hep-th]].
  * [102] V. Jejjala, D. Minic, Y. J. Ng and C. H. Tze, “Turbulence and Holography,” Class. Quant. Grav. 25 (2008), 225012 [arXiv:0806.0030 [hep-th]].
  * [103] M. M. Caldarelli, D. Klemm and P. J. Silva, “Chronology protection in anti-de Sitter,” Class. Quant. Grav. 22 (2005), 3461-3466 [arXiv:hep-th/0411203 [hep-th]].
  * [104] D. Berenstein and R. Cotta, “A Monte-Carlo study of the AdS/CFT correspondence: An Exploration of quantum gravity effects,” JHEP 04 (2007), 071 [arXiv:hep-th/0702090 [hep-th]].
  * [105] J. Simon, “Correlations vs connectivity in R-charge,” JHEP 10 (2018), 048 [arXiv:1805.11279 [hep-th]].
  * [106] D. Berenstein and K. Yan, “The endpoint of partial deconfinement,” [arXiv:2307.06122 [hep-th]].
  * [107] A. Ahmadain, A. Frenkel, K. Ray and R. M. Soni, “Boundary Description of Microstates of the Two-Dimensional Black Hole,” [arXiv:2210.11493 [hep-th]].
  * [108] V. Balasubramanian, J. de Boer, V. Jejjala and J. Simon, “Entropy of near-extremal black holes in AdS(5),” JHEP 05 (2008), 067 [arXiv:0707.3601 [hep-th]].
  * [109] R. Fareghbal, C. N. Gowdigere, A. E. Mosaffa and M. M. Sheikh-Jabbari, “Nearing Extremal Intersecting Giants and New Decoupled Sectors in N = 4 SYM,” JHEP 08 (2008), 070 [arXiv:0801.4457 [hep-th]].
  * [110] R. C. Myers and O. Tafjord, “Superstars and giant gravitons,” JHEP 11 (2001), 009 [arXiv:hep-th/0109127 [hep-th]].
  * [111] C. Cordova, T. T. Dumitrescu and K. Intriligator, “Multiplets of Superconformal Symmetry in Diverse Dimensions,” JHEP 03 (2019), 163 [arXiv:1612.00809 [hep-th]].
  * [112] J. Caetano, W. Peelaers and L. Rastelli, “Maximally supersymmetric RG flows in 4D and integrability,” JHEP 12 (2021), 119 [arXiv:2006.04792 [hep-th]].
  * [113] C. Cordova, T. T. Dumitrescu and K. Intriligator, “Deformations of Superconformal Theories,” JHEP 11 (2016), 135 [arXiv:1602.01217 [hep-th]].
  * [114] D. Roychowdhury, “Matrix model correlators from non-Abelian T-dual of $AdS_{5}\times S^{5}$,” [arXiv:2310.10210 [hep-th]].
  * [115] Y. Nomura, “Spacetime and Universal Soft Modes — Black Holes and Beyond,” Phys. Rev. D 101 (2020), 066024 [arXiv:1908.05728 [hep-th]].
  * [116] E. Witten, “Superstring Perturbation Theory Revisited,” [arXiv:1209.5461 [hep-th]].
  * [117] D. Berenstein, E. Maderazo, R. Mancilla and A. Ramirez, “Chaotic LLM billiards,” [arXiv:2305.19321 [hep-th]].
  * [118] D. R. Mayerson, “Fuzzballs and Observations,” Gen. Rel. Grav. 52 (2020) no.12, 115 [arXiv:2010.09736 [hep-th]].
  * [119] S. D. Mathur, Class. Quant. Grav. 26 (2009), 224001 [arXiv:0909.1038 [hep-th]].
  * [120] M. Bianchi, D. Consoli and J. F. Morales, “Probing Fuzzballs with Particles, Waves and Strings,” JHEP 06 (2018), 157 [arXiv:1711.10287 [hep-th]].
  * [121] B. Chakrabarty, S. Rawash and D. Turton, “Shockwaves in black hole microstate geometries,” JHEP 02 (2022), 202 [arXiv:2112.08378 [hep-th]].
  * [122] P. Heidmann, N. Speeney, E. Berti and I. Bah, “Cavity effect in the quasinormal mode spectrum of topological stars,” Phys. Rev. D 108 (2023) 024021 [arXiv:2305.14412 [gr-qc]].
  * [123] F. Bacchini, D. R. Mayerson, B. Ripperda, J. Davelaar, H. Olivares, T. Hertog and B. Vercnocke, “Fuzzball Shadows: Emergent Horizons from Microstructure,” Phys. Rev. Lett. 127 (2021), 171601 [arXiv:2103.12075 [hep-th]].
  * [124] I. Bena, E. J. Martinec, S. D. Mathur and N. P. Warner, “Snowmass White Paper: Micro- and Macro-Structure of Black Holes,” [arXiv:2203.04981 [hep-th]].
  * [125] R. de Mello Koch, E. Gandote and A. L. Mahu, “Scrambling in Yang-Mills,” JHEP 01 (2021), 058 [arXiv:2008.12409 [hep-th]].
  * [126] T. McLoughlin, R. Pereira and A. Spiering, “Quantum chaos in perturbative super-Yang-Mills Theory,” SciPost Phys. 14 (2023) no.3, 049 [arXiv:2011.04633 [hep-th]].
  * [127] D. Berenstein and A. Holguin, “String junctions suspended between giants,” JHEP 11 (2022), 085 [arXiv:2202.11729 [hep-th]].
  * [128] A. Holguin, “Giant Gravitons Intersecting at Angles from Integrable Spin Chains,” [arXiv:2111.05981 [hep-th]].
  * [129] K. Behrndt, M. Cvetic and W. A. Sabra, “Nonextreme black holes of five-dimensional N=2 AdS supergravity,” Nucl. Phys. B 553 (1999), 317-332 [arXiv:hep-th/9810227 [hep-th]].
  * [130] K. Behrndt, A. H. Chamseddine and W. A. Sabra, “BPS black holes in N=2 five-dimensional AdS supergravity,” Phys. Lett. B 442 (1998), 97-101 [arXiv:hep-th/9807187 [hep-th]].
  * [131] H. Lin, “Coherent state excitations and string-added coherent states in gauge-gravity correspondence,” Nucl. Phys. B 986 (2023), 116066 [arXiv:2206.06524 [hep-th]].
  * [132] S. Baiguera, T. Harmark and Y. Lei, “Spin Matrix Theory in near $\frac{1}{8}$-BPS corners of $\mathcal{N}$ = 4 super-Yang-Mills,” JHEP 02 (2022), 191 [arXiv:2111.10149 [hep-th]].
  * [133] S. Baiguera, T. Harmark and Y. Lei, “The Panorama of Spin Matrix theory,” JHEP 04 (2023), 075 [arXiv:2211.16519 [hep-th]].
  * [134] S. Baiguera, T. Harmark, Y. Lei and N. Wintergerst, “Symmetry structure of the interactions in near-BPS corners of $\mathcal{N}=4$ super-Yang-Mills,” JHEP 04 (2021), 029 [arXiv:2012.08532 [hep-th]].
  * [135] R. de Mello Koch and S. Ramgoolam, “A double coset ansatz for integrability in AdS/CFT,” JHEP 06 (2012), 083 [arXiv:1204.2153 [hep-th]]
  * [136] W. Carlson, R. de Mello Koch and H. Lin, “Nonplanar Integrability,” JHEP 03 (2011), 105 [arXiv:1101.5404 [hep-th]].
  * [137] R. de Mello Koch, M. Kim and H. J. R. Van Zyl, “Integrable Subsectors from Holography,” JHEP 05 (2018), 198 [arXiv:1802.01367 [hep-th]].
  * [138] D. E. Berenstein, M. Hanada and S. A. Hartnoll, “Multi-matrix models and emergent geometry,” JHEP 02 (2009), 010 [arXiv:0805.4658 [hep-th]].
  * [139] E. Kiritsis, “Dissecting the string theory dual of QCD,” Fortsch. Phys. 57 (2009), 396-417 [arXiv:0901.1772 [hep-th]].
  * [140] A. Donos, A. Jevicki and J. P. Rodrigues, “Matrix model maps in AdS/CFT,” Phys. Rev. D 72 (2005), 125009 [arXiv:hep-th/0507124 [hep-th]].
  * [141] D. H. Correa and M. Wolf, “Shaping up BPS States with Matrix Model Saddle Points,” J. Phys. A 43 (2010), 465402 [arXiv:1007.5284 [hep-th]].
  * [142] U. Gursoy, S. A. Hartnoll, T. J. Hollowood and S. P. Kumar, “Topology change and new phases in thermal N=4 SYM theory,” JHEP 11 (2007), 020 [arXiv:hep-th/0703100 [hep-th]].
  * [143] S. Dutta and R. Gopakumar, “Free fermions and thermal AdS/CFT,” JHEP 03 (2008), 011 [arXiv:0711.0133 [hep-th]].
  * [144] D. Berenstein and R. Cotta, “Aspects of emergent geometry in the AdS/CFT context,” Phys. Rev. D 74 (2006), 026006 [arXiv:hep-th/0605220 [hep-th]].
  * [145] T. Ishii, G. Ishiki, S. Shimasaki and A. Tsuchiya, “Fiber Bundles and Matrix Models,” Phys. Rev. D 77 (2008), 126015 [arXiv:0802.2782 [hep-th]].
  * [146] G. Mandal and T. Morita, “Quantum quench in matrix models: Dynamical phase transitions, Selective equilibration and the Generalized Gibbs Ensemble,” JHEP 10 (2013), 197 [arXiv:1302.0859 [hep-th]].
  * [147] R. de Mello Koch, D. Gossman, L. Nkumane and L. Tribelhorn, “Eigenvalue Dynamics for Multimatrix Models,” Phys. Rev. D 96 (2017) no.2, 026011 [arXiv:1608.00399 [hep-th]].
  * [148] A. Holguin and S. Wang, “Multi-matrix correlators and localization,” [arXiv:2307.03235 [hep-th]].
  * [149] H. Lin, “Coherent state operators, giant gravitons, and gauge-gravity correspondence,” Annals Phys. 451 (2023), 169248 [arXiv:2212.14002 [hep-th]].
  * [150] D. Berenstein, R. Cotta and R. Leonardi, “Numerical tests of AdS/CFT at strong coupling,” Phys. Rev. D 78 (2008), 025008 [arXiv:0801.2739 [hep-th]].
  * [151] M. Puta, “Hamiltonian Mechanical Systems and Geometric Quantization,” Mathematics and its applications, v. 260, 1994.
  * [152] A. Belin, A. Lewkowycz and G. Sarosi, “The boundary dual of the bulk symplectic form,” Phys. Lett. B 789 (2019), 71-75 [arXiv:1806.10144 [hep-th]].