4pt 4pt

Asymptotically flat $(1,m,n)$ superstrata:
a farewell to AdS

Willem Govaerts, Robert Walker

Instituut voor Theoretische Fysica, KU Leuven,

Celestijnenlaan 200D, B-3001 Leuven, Belgium

<EMAIL_ADDRESS>, robert.walker @ kuleuven.be

Abstract

Superstrata microstate geometries furnish some of the most successful
laboratories, to date, for probing black hole microstructure in a geometric
setting. This paper extends the $(1,m,n)$ family of superstrata, to allow for
flat R1,5 asymptotics. Previous constructions utilized the decoupling regime,
where the geometry is asymptotically AdS${}_{3}\times$S3. Brief comments are
made on the additional complexity introduced by the flat space coupling, how
this obscures holomorphicity and breaks the consistent truncation to which the
decoupled solutions belong. Holomorphicity and consistent truncation were key
simplifications assisting previous studies of the $(1,m,n)$ superstrata,
undertaken in the decoupling regime. Further, these results open a window for
future projects to determine how previous analysis on the decoupled geometries
extend or are modified once flat space asymptotics are imposed. Since our
universe is almost flat on cosmological scales, this represents progress
towards more phenomenologically relevant microstate geometries. This work can
be considered a continuation of that in [1], where some single mode
superstrata were also coupled to flat space.

###### Contents

  1. 1 Introduction
  2. 2 Six-dimensional $\mathcal{N}=1$ supergravity with two tensor multiplets
    1. 2.1 The theory
    2. 2.2 Useful parametrization of bosonic field content
    3. 2.3 BPS equations
    4. 2.4 Gauge freedom
  3. 3 The full $(1,m,n)$ superstrata solutions
    1. 3.1 “Basic data”
    2. 3.2 The full solutions
    3. 3.3 Regularity considerations
    4. 3.4 Conserved charges
  4. 4 The decoupling limit: asymptotically AdS solutions
    1. 4.1 Holomorphic form
    2. 4.2 Consistent truncation form
  5. 5 The geometry
    1. 5.1 Asymptotically flat region
    2. 5.2 Supertube cap region
    3. 5.3 Long BTZ throat region
  6. 6 Discussion and conclusions
    1. 6.1 Holomorphic considerations
    2. 6.2 Consistent truncation considerations
    3. 6.3 UV/IR mixing considerations
  7. A Constructing the solutions
    1. A.1 First layer ansatz
    2. A.2 Second BPS layer source decomposition
    3. A.3 $S_{1,2}^{(o,+)}$ solutions
    4. A.4 $S_{1,2}^{(s,+)}$ solutions
    5. A.5 $S_{1,2}^{(-)}$ solutions
    6. A.6 Aggregating solution

##  1 Introduction

In the search for a satisfactory description of black hole microstates,
microstate geometries furnish some of the most readily accessible and
instructive laboratories. These geometries provide families of smooth
supergravity solutions, that can arbitrarily closely approximate the geometry
of a black hole outside the horizon, before capping off smoothly at the would
be horizon scale in a multitude of ways. Historically, the D1/D5/P system of
IIB supergravity has been particularly fruitful for constructing new
microstate geometries [2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 11, 12, 13, 14], see
[15, 16, 17] for convenient reviews. Additionally, working in the decoupling
regime, where the geometry is asymptotically AdS, allows insights from field
theory to be leveraged, via application of the AdS/CFT correspondence,
simplifying many analysis.

Although the decoupling limit is extremely useful for building and probing
microstate geometries, it is at odds with the asymptotic geometry of black
holes in our universe. Experimentally, it is well established that on
cosmological scales our universe is very close to being flat [18, 19]. Thus,
it is important to establish:

What features of microstate geometries, in the decoupling limit, are altered
or retained when more realistic asymptotics are imposed?

This paper addresses this question, by explicitly coupling a previously known
(in the decoupling limit) family of microstate geometries to flat space. Then
analyzing the resulting novel features, that may be conceptualised as
“corrections” to the decoupled solutions. It should be noted that although
this represents progress toward more realistic asymptotics, the superstrata
considered here are six-dimensional. Thus, our results are not directly
phenomenologically applicable, but may provide indicative qualitative results
of relevance to phenomenology.

The exemplar for the exploitation of AdS/CFT, for the construction of
microstate geometries, are the superstrata solutions [6, 20, 7, 1, 21, 22, 23,
24, 11, 25, 26, 27, 13]. These solutions are dual to specific states in the
D1/D5 CFT, labelled by three integer mode numbers $(k,m,n)$, coming in two
types111There are now more recent variants, that were not fully considered in
the context of this paper, such as the vector-superstrata of [14].: the
original, [6, 7, 1], and the supercharged, [22]. Further, these states can be
superposed, so that a generic superstratum solution is specified by the choice
of two functions of three variables, accounting for superposition of all
possible modes of each type. The appropriate superpositions are constrained by
demanding regularity of the final geometry, in a procedure which has come to
be known as coiffuring. Further, the superstrata have been exploited in many
precision holographic calculations [28, 29, 30, 31, 32, 33, 34, 35, 36, 37,
38, 39, 40], as well as purely within supergravity, to shed light on potential
resolutions to the information paradox [41, 42].

Unfortunately, due to technical limitations, the full analytic expression for
a generic superstratum solution of two functions of three variables, is not
known. Instead, subfamilies are known and are indexed by the $(k,m,n)$ modes
being considered in the superposition, see [11] for examples. An instructive
family is the $(1,m,n)$ solutions, constructed in [25]. This family rose to
prominence due to the separability of the massless wave equations of its
$(1,0,n)$ and $(1,1,n)$ subfamilies [43, 44], which allowed many explicit
calculations to be performed. This separability was ultimately traced to the
remarkable fact that the $(1,m,n)$ family belongs to a consistent truncation
on a deformed three-sphere [45], and the deformations of the three sphere
preserve some “nice isometries”.

A road map for coupling superstrata to flat space was developed in [1], then
applied to some single mode original superstrata. In this paper we apply this
procedure to find the full $(1,m,n)$ asymptotically flat multimode
superstrata. The resulting geometries are vastly more involved, can no longer
be written cleanly in the holomorphic variables identified in [11] and fall
outside the consistent truncation developed in [25]. Thus, many of “nice
features” that aided in the construction and probing of the decoupled
$(1,m,n)$ superstrata, disappear once they are coupled back to flat space.
This raises the question:

To what extent are the currently accepted results or intuition, in the
microstate geometry program, artefacts of working in the decoupling limit?

Beyond identifying the aforementioned disappearance of nice features and
identifying the potential for analysing UV/IR mixing within the microstate
geometry program, a more complete analysis of this question is left to a
future study. It is entirely possible that most of the intuition developed in
the decoupling limit carries over, just the technical details become an order
of magnitude more complicated.

The remainder of this paper is organized as follows:

  * •

Section 2 provides an introduction to the six-dimensional supergravity, in
which the superstrata are most easily studied. Further, the foundational semi-
linear form of the BPS equations, which aid solution construction, are
presented.

  * •

Section 3 presents the full $(1,m,n)$ solutions and demonstrates their
regularity at key locations. There are sufficient parameters in this solution
to reproduce the decoupled family as well as the asymptotically flat family.
In addition, the technical details involved in this construction are collected
in Appendix A.

  * •

Section 4 presents the decoupled $(1,m,n)$ solutions, that are asymptotically
AdS${}_{3}\times$S3. Introducing the details of both the holomorphic variables
and consistent truncation.

  * •

Section 5 illustrates the geometry of the asymptotically flat $(1,m,n)$
geometries in three illuminating limits: the flat asymptotic region, the
smooth cap at the supertube locus and the long BTZ like throat regime.

  * •

Finally, in section 6 we discuss the results of this paper and make some
concluding remarks on future directions.

##  2 Six-dimensional $\mathcal{N}=1$ supergravity with two tensor multiplets

This section presents the six-dimensional supergravity theory to which the
superstrata belong. Further, the BPS equations are given in their remarkable
semi-linear form. It is this semi-linear form that is at the heart of the
solution generating procedures, which allows for such involved microstate
geometries to be constructed analytically.

###  2.1 The theory

The superstrata are most simply constructed and studied in six-dimensional,
$\mathcal{N}=(1,0)$ supergravity, coupled to two tensor multiplets, identified
in [6, 46, 25]. The bosonic field content of this theory can be organized
into: the metric $g_{\mu\nu}$, two scalars $(\varphi,X)$, and three three-
forms which we label by $G^{I}$ with $I\in\left\\{1,2,4\right\\}$. It is
convenient to introduce the $SO(1,2)$ metric, $\eta^{IJ}$, with non-zero
components:

$\displaystyle\eta^{12}=\eta^{21}=1\qquad\text{and}\qquad\eta^{44}=-2\,,$
(2.1)

which is used to raise and lower $I,J,\cdots$ indices.

The equations of motion for the three forms are given by the twisted self
duality relations: where

$\displaystyle\tensor{M}{{}^{I}_{J}}=\frac{1}{2}e^{\sqrt{2}\varphi}\begin{pmatrix}X^{2}&8&-2\sqrt{2}X\\\
\frac{e^{-2\sqrt{2}\varphi}}{8}\left(2+e^{\sqrt{2}\varphi}X^{2}\right)^{2}&X^{2}&-\frac{X}{2\sqrt{2}}\left(2e^{-\sqrt{2}\varphi}+X^{2}\right)\\\
\frac{X}{\sqrt{2}}\left(2e^{-\sqrt{2}\varphi}+X^{2}\right)&4\sqrt{2}X&-2e^{-\sqrt{2}\varphi}-2X^{2}\end{pmatrix}\,,$
(2.2)

together with the Bianchi identities

$\displaystyle dG^{I}=0\,.$ (2.3)

The remaining equations of motion can be obtained by minimizing the action
constructed from the Lagrangian

$\displaystyle\mathcal{L}=\left[R-\frac{1}{2}g^{\mu\nu}\left(\partial_{\mu}\varphi\partial_{\nu}\varphi+e^{\sqrt{2}\varphi}\partial_{\mu}X\partial_{\nu}X\right)-\frac{1}{6}M_{IJ}\tensor{G}{{}^{I}_{\mu\nu\rho}}G^{J\mu\nu\rho}\right]*1\,.$
(2.4)

###  2.2 Useful parametrization of bosonic field content

It turns out that the BPS equations, consistent with the supersymmetries of
the D1/D5/P system, take a semi-linear form when written in terms of a very
specific parametrization of the bosonic fields introduced in the previous
subsection, [47]. One introduces sets of: functions $(Z_{I},\mathcal{F})$, one
forms $(\beta,\omega)$ and two forms $\Theta^{I}$. Upon the specification of a
hyper-Kähler base $ds_{4}^{2}(\mathcal{B})$, the BPS equations reduce to
solving for this BPS data: $(\beta,Z_{I},\Theta^{I},\mathcal{F},\omega)$.

The most general six-dimensional metric, consistent with the (super)symmetries
of the D1/D5/P system, was considered in [48]:

$\displaystyle ds_{6}^{2}$
$\displaystyle=-\frac{2}{\sqrt{\mathcal{P}}}\,(dv+\beta)\big{(}du+\omega+\tfrac{1}{2}\,\mathcal{F}\,(dv+\beta)\big{)}+\sqrt{\mathcal{P}}\,ds_{4}^{2}(\mathcal{B})\,.$
(2.5)

In this expression, $(u,v)$ are light cone coordinates related to the
$y$-circle222This is the common compact direction that the D1 and D5 branes
wrap. (of circumference $2\pi R_{y}$) by:

$\displaystyle u=\frac{1}{\sqrt{2}}(t-y)\qquad\text{and}\qquad
v=\frac{1}{\sqrt{2}}(t+y)\,.$ (2.6)

The function $\mathcal{P}$ is given in terms if the $Z_{I}$ via:

$\displaystyle\mathcal{P}=\frac{1}{2}\eta^{IJ}Z_{I}Z_{J}\,.$ (2.7)

While, $(\beta,\omega)$ are one forms on ${\cal B}$, and all the BPS data may
have functional dependence on every coordinate except $u$.

The three forms $G^{I}$, can be expanded in terms of a set of “electric”
functions $Z^{I}$ and “magnetic” two forms $\Theta^{I}$, as:

$\displaystyle
G^{I}=d\left[-\frac{1}{2}\frac{\eta^{IJ}Z_{J}}{\mathcal{P}}(du+\omega)\wedge(dv+\beta)\right]+\frac{1}{2}\eta^{IJ}*_{4}\mathcal{D}Z_{J}+\frac{1}{2}(dv+\beta)\wedge\Theta^{I}\,.$
(2.8)

Note that we use $(d,*)$ to refer to the exterior derivative and Hodge star
with respect to the full six-dimensional geometry, (2.5), while $*_{4}$ (and
in the following $d_{4}$) are with respect to $ds_{4}^{2}(\mathcal{B})$.

The remaining bosonic fields $(\varphi,X)$ can also be expressed in terms of
this BPS data, see [49, 46] for further details.

###  2.3 BPS equations

The BPS-equations can be arranged and solved in the semi-linear, layered form:

  * •

Fix a hyper-Kähler base $ds_{4}^{2}(\mathcal{B})$, and choose a $\beta$
satisfying:

$\displaystyle d_{4}\beta=*_{4}d_{4}\beta\,.$ (2.9)

  * •

Find $(Z_{I},\Theta^{I})$ that solve the first layer:

$\displaystyle*_{4}\mathcal{D}\dot{Z}_{I}=-\eta_{IJ}\mathcal{D}\Theta^{J}\,,\qquad\mathcal{D}*_{4}\mathcal{D}Z_{I}=\eta_{IJ}\Theta^{J}\wedge
d\beta\,,\qquad\Theta^{I}=*_{4}\Theta^{I}\,.$ (2.10)

  * •

Find $(\mathcal{F},\omega)$ that solve the second layer:

$\displaystyle(1+*_{4})\mathcal{D}\omega+\mathcal{F}\,d_{4}\beta$
$\displaystyle=Z_{I}\Theta^{I}\,,$ (2.11)
$\displaystyle*_{4}\mathcal{D}*_{4}\left(\dot{\omega}-\frac{1}{2}\mathcal{D}\mathcal{F}\right)$
$\displaystyle=-\frac{1}{4}\eta_{IJ}\left[4\ddot{Z}^{I}Z^{J}+2\dot{Z}^{I}\dot{Z}^{J}-*_{4}\left(\Theta^{I}\wedge\Theta^{J}\right)\right]\,.$
(2.12)

This system is semi-linear, in the sense that, once the base geometry is
fixed, if one finds a set of basic data, $(z_{k,m,n},\vartheta_{k,m,n})$,
satisfying:

$*_{4}{\cal D}\,\dot{{z}}_{k,m,n}=-{\cal D}\,{\vartheta}_{k,m,n},\qquad{\cal
D}*_{4}{\cal D}\,{z}_{k,m,n}={\vartheta}_{k,m,n}\wedge
d\beta,\qquad{\vartheta}_{k,m,n}=*_{4}{\vartheta}_{k,m,n}\leavevmode\nobreak\
,$ (2.13)

then their superposition:

$\displaystyle
Z_{I}=\sum_{k,m,n}b_{k,m,n}z_{k,m,n}\qquad\text{and}\qquad\Theta_{I}=\sum_{k,m,n}b_{k,m,n}\vartheta_{k,m,n}\leavevmode\nobreak\
,$ (2.14)

with constants, $b_{k,m,n}$, will also solve the first layer (2.10). Once the
first layer solution is fixed, the second layer differential equations,
(2.11)-(2.12), are also linear in $(\mathcal{F},\omega)$. The simplification
afforded by organizing the BPS equations in this manner is significant. A
priori, the non-linear interactions between the base, $\mathcal{B}$, and the
data $(\beta,Z_{I},\Theta^{I},\mathcal{F},\omega)$, is extremely difficult to
handle when attempting to find solutions.

###  2.4 Gauge freedom

There is an important gauge transformation of the BPS equations, given by the
residual ability to redefine the $u$ coordinate:

$u\to u+U(x^{i},v)\qquad\Leftrightarrow\qquad\omega\to\omega-
d_{4}U+\dot{U}\,\beta\,,\quad\mathcal{F}\to\mathcal{F}-2\dot{U}\,,$ (2.15)

where the $x^{i}$ are coordinates on $\mathcal{B}$. It is straightforward to
check this transformation leaves the six-dimensional metric (2.5) invariant,
while solutions of the BPS equations, (2.9)-(2.12), are mapped again to
solutions under (2.15). In order to illuminate certain aspects of microstate
geometries, such as their asymptotics, or aid in solution generation, one may
need to identify a convenient gauge.

##  3 The full $(1,m,n)$ superstrata solutions

This section presents the full $(1,m,n)$ superstrata solutions, representing
the main technical result of this paper. These solutions possess enough
tunable parameters to produce both the decoupled asymptotically
AdS${}_{3}\times$S3 family, as well as the asymptotically, R1,5, flat family.
The technical details of how these solutions were constructed, by weaving
together partial results in the literature and novel solutions of the BPS
equations, can be found in Appendix A.

###  3.1 “Basic data”

The superstrata, in six-dimensions, utilize the simplest hyper-Kähler base,
namely flat space. However, it is convenient to adapt the coordinates to the
ring of the underlying supertube333The supertube solution is given by setting
all $b_{k,m,n}=0$ in the following presentation. The superstrata were
originally conjectured to exist in [50], via so called “supertube
transitions,” on top of this basic supertube solution. These transitions
amounts to allowing density variations of the underlying branes and momentum
in the D1/D5 frame.. This is done by introducing spherical bipolar
coordinates, $(r,\theta,\varphi_{1},\varphi_{2})$, in terms of which444Note
that these spherical bipolar coordinates are related to a standard Cartesian
coordinate system $(y^{1},y^{2},y^{3},y^{4})$, for R4, via:
$y^{1}+iy^{2}=\sqrt{a^{2}+r^{2}}\sin\theta\,e^{i\varphi_{1}}\qquad\text{and}\qquad
y^{3}+iy^{4}=r\cos\theta\,e^{i\varphi_{2}}\,.$ Which makes it clear that the
supertube locus is a ring of radius $a$ in the $y^{1}-y^{2}$ plane, while the
center of the geometry is at $(r=0,\theta=0)$. :

$\displaystyle ds_{4}^{2}(\mathcal{B})$
$\displaystyle=\frac{\Sigma}{a^{2}+r^{2}}dr^{2}+\Sigma\,d\theta^{2}+(a^{2}+r^{2})\sin^{2}\theta\,d\varphi_{1}^{2}+r^{2}\cos^{2}\theta\,d\varphi_{2}^{2}\,,$
(3.1)

where:

$\displaystyle\Sigma$ $\displaystyle\equiv r^{2}+a^{2}\cos^{2}\theta\,,$ (3.2)

and $a$ is a positive real number.

The one form $\beta$, is then taken to be the potential for a self-dual
magnetic field on $\mathcal{B}$, with source along the supertube ring
($r=0,\theta=\pi/2$):

$\displaystyle\beta=\frac{R_{y}a^{2}}{\sqrt{2}\Sigma}(\sin^{2}\theta\,d\varphi_{1}-\cos^{2}\theta\,d\varphi_{2})\,.$
(3.3)

The original single-mode superstrata were first constructed in [6, 7, 1], and
utilize the basic data:

$\displaystyle z^{(o)}_{k,m,n}$ $\displaystyle\equiv
R_{y}\frac{\Delta_{k,m,n}}{\Sigma}\cos v_{k,m,n}\leavevmode\nobreak\ ,$ (3.4)
$\displaystyle\vartheta^{(o)}_{k,m,n}$
$\displaystyle\equiv-\sqrt{2}\Delta_{k,m,n}\left[\left((m+n)r\sin\theta+n\left(\frac{m}{k}-1\right)\frac{\Sigma}{r\sin\theta}\right)\Omega^{(1)}\sin
v_{k,m,n}\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad+\left.\left(m\left(\frac{n}{k}+1\right)\Omega^{(2)}+\left(\frac{m}{k}-1\right)n\Omega^{(3)}\right)\cos
v_{k,m,n}\right]\,,$ (3.5)

to solve the first BPS layer. In these expressions
$(\Omega^{(1)},\Omega^{(2)},\Omega^{(3)})$ are a basis of self-dual two-forms
on $\mathcal{B}$, given by

$\displaystyle\Omega^{(1)}$ $\displaystyle\leavevmode\nobreak\
\equiv\leavevmode\nobreak\ \frac{1}{(a^{2}+r^{2})\cos\theta}\,dr\wedge
d\theta+\frac{r\sin\theta}{\Sigma}\,d\varphi_{1}\wedge d\varphi_{2}\,,$ (3.6)
$\displaystyle\Omega^{(2)}$ $\displaystyle\leavevmode\nobreak\
\equiv\leavevmode\nobreak\ \frac{r}{a^{2}+r^{2}}\,dr\wedge
d\varphi_{2}+\tan\theta\,d\theta\wedge d\varphi_{1}\,,$ (3.7)
$\displaystyle\Omega^{(3)}$ $\displaystyle\leavevmode\nobreak\
\equiv\leavevmode\nobreak\ \frac{1}{r}\,dr\wedge
d\varphi_{1}-\cot\theta\,d\theta\wedge d\varphi_{2}\,,$ (3.8)

while:

$\displaystyle\Delta_{k,m,n}$
$\displaystyle\equiv\left(\frac{a}{\sqrt{r^{2}+a^{2}}}\right)^{k}\left(\frac{r}{\sqrt{r^{2}+a^{2}}}\right)^{n}\cos^{m}\theta\sin^{k-m}\theta\,,$
(3.9) $\displaystyle v_{k,m,n}$
$\displaystyle\equiv\frac{\sqrt{2}}{R_{y}}(m+n)v+(k-m)\varphi_{1}-m\varphi_{2}\,.$
(3.10)

One must also impose constraints on the $(k,m,n)$ integers:

$\displaystyle k,n\in\mathbb{Z}^{+}\qquad\text{and}\qquad 0\leq m\leq k\,,$
(3.11)

which can be understood as required to avoid divergences coming from (3.9).

The single-mode supercharged superstrata were initially constructed in [22],
and solve the first BPS layer with the basic data:

$\displaystyle z^{(s)}_{k,m,n}$ $\displaystyle\equiv 0\leavevmode\nobreak\ ,$
(3.12) $\displaystyle\vartheta^{(s)}_{k,m,n}$
$\displaystyle\equiv\sqrt{2}\Delta_{k,m,n}\left[\frac{\Sigma}{r\sin\theta}\Omega^{(1)}\sin
v_{k,m,n}+\left(\Omega^{(2)}+\Omega^{(3)}\right)\cos v_{k,m,n}\right]\,.$
(3.13)

for the integers $(k,m,n)$ with:

$\displaystyle 1\leq m\leq k-1\qquad\text{and}\qquad 1\leq n\,.$ (3.14)

###  3.2 The full solutions

Appendix A gives the technical details for construction of the full $(1,m,n)$
solutions, we now summarize the total solution as discussed in Section A.6,
writing the result as concisely as possible. To do so, it is necessary to
introduce two constants $(c,C_{1})$ and convenient to introduce the decoupling
parameter:

$\epsilon_{i,j}\equiv\frac{C_{1}a^{2}}{2Q_{5}}\left(m_{i}+m_{j}+n_{i}+n_{j}\right)\,.$
(3.15)

The constant $c$ parametrizes a homogeneous solution to the second layer of
the BPS equations. This constant will be fixed in (5.4), as is necessary to
achieve both the asymptotically AdS and flat geometries. The decoupled
solution will correspond to setting the constant $C_{1}=0$, while flat
asymptotics be given by setting $C_{1}=1$. Thus, in the decoupling limit, the
decoupling parameter vanishes.

The solution to the first BPS layer is then given by555The limits of all sums
have been left implicit, explicitly they are restricted by
$m,m_{1},m_{1}\in\\{0,1\\}$ and $n,n_{1},n_{2}\in\\{1,2,3,\cdots\\}$.:

$\displaystyle Z_{1}$
$\displaystyle=C_{1}+\frac{Q_{1}}{\Sigma}+\frac{R_{y}}{2Q_{5}}\sum_{m_{1},m_{2}}\sum_{n_{1},n_{2}}\left(\frac{b_{1,m_{1},n_{1}}b_{1,m_{2},n_{2}}}{1+\epsilon_{1,2}}\right)z^{(o)}_{2,m_{1}+m_{2},n_{1}+n_{2}}\,,$
(3.16) $\displaystyle Z_{2}$ $\displaystyle=C_{1}+\frac{Q_{5}}{\Sigma}\,,$
(3.17) $\displaystyle Z_{4}$
$\displaystyle=\sum_{m,n}b_{1,m,n}z^{(o)}_{1,m,n}\,,$ (3.18)

together with666Keep in mind that
$\vartheta^{(s)}_{2,m_{1}+m_{2},n_{1}+n_{2}}$ is only well defined when
$m_{1}+m_{2}=1$, the sum in (3.20) is correct if one declares that
$\vartheta^{(s)}_{2,0,n_{1}+n_{2}}=\vartheta^{(s)}_{2,2,n_{1}+n_{2}}=0$.:

$\displaystyle\Theta^{1}$ $\displaystyle=0\,,$ (3.19)
$\displaystyle\Theta^{2}$
$\displaystyle=\frac{R_{y}}{2Q_{5}}\sum_{m_{1},m_{2}}\sum_{n_{1},n_{2}}\left(\frac{b_{1,m_{1},n_{1}}b_{1,m_{2},n_{2}}}{1+\epsilon_{1,2}}\right)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\left(\vartheta^{(o)}_{2,m_{1}+m_{2},n_{1}+n_{2}}-\frac{1}{2}(m_{2}-m_{1})\left(n_{2}-n_{1}\right)\vartheta^{(s)}_{2,m_{1}+m_{2},n_{1}+n_{2}}\right)\,,$
(3.20) $\displaystyle\Theta^{4}$
$\displaystyle=-2\sum_{m,n}b_{1,m,n}\vartheta^{(o)}_{1,m,n}\,.$ (3.21)

The second layer solution is given by:

$\mathcal{F}=\frac{1}{a^{2}}\left(-c^{2}+\sum_{m}\sum_{n_{1},n_{2}}b_{1,m,n_{1}}b_{1,m,n_{2}}\Delta_{0,0,2m+n_{1}+n_{2}}\cos
v_{0,0,n_{2}-n_{1}}\right)\,,$ (3.22)

and

$\displaystyle\omega$
$\displaystyle=\omega_{0}+\frac{R_{y}\sin^{2}\theta}{\sqrt{2}\Sigma}c^{2}\,d\varphi_{1}$
(3.23)
$\displaystyle\qquad+\frac{R_{y}}{2\sqrt{2}a^{2}}\sum_{m_{1},m_{2}}\sum_{n_{1},n_{2}}\Delta_{2,m_{1}+m_{2},n_{1}+n_{2}}b_{1,m_{1},n_{1}}b_{1,m_{2},n_{2}}\left(\omega^{m_{1}-m_{2}}_{n_{2}-n_{1}}-\left(\frac{\epsilon_{1,2}}{1+\epsilon_{1,2}}\right)\omega^{m_{1}+m_{2}}_{n_{1}+n_{2}}\right)\,.$

Where the solution for $\omega$ has been broken up into a “subtraction mode”
part:

$\displaystyle\omega^{m_{1}-m_{2}}_{n_{2}-n_{1}}=$
$\displaystyle\frac{2(m_{1}-m_{2})}{\sin 2\theta}d\theta\sin
v_{0,m_{1}-m_{2},n_{1}-n_{2}}$ (3.24)
$\displaystyle-\frac{1}{\Sigma}\left((2-m_{1}-m_{2})(a^{2}+r^{2})d\varphi_{1}-(m_{1}+m_{2})r^{2}d\varphi_{2}\right)\cos
v_{0,m_{1}-m_{2},n_{1}-n_{2}}\,,$

and an “addition mode” part:

$\displaystyle\omega^{m_{1}+m_{2}}_{n_{1}+n_{2}}=\omega^{(1)}+\frac{n_{1}(m_{1}-1)+n_{2}(m_{2}-1)}{(n_{1}+n_{2}+m_{1}+m_{2})}\omega^{(2)},$
(3.25)

with

$\displaystyle\omega^{(1)}$
$\displaystyle=\left(\frac{r}{a^{2}+r^{2}}dr-\tan\theta d\theta\right)\sin
v_{2,m_{1}+m_{2},n_{1}+n_{2}}+\frac{a^{2}\sin^{2}\theta\,d\varphi_{1}-r^{2}\,d\varphi_{2}}{\Sigma}\cos
v_{2,m_{1}+m_{2},n_{1}+n_{2}}\,,$ (3.26) $\displaystyle\omega^{(2)}$
$\displaystyle=-\left(\frac{a^{2}}{r(a^{2}+r^{2})}dr+\frac{2d\theta}{\sin
2\theta}\right)\sin
v_{2,m_{1}+m_{2},n_{1}+n_{2}}-(d\varphi_{1}+d\varphi_{2})\cos
v_{2,m_{1}+m_{2},n_{1}+n_{2}}\,.$ (3.27)

###  3.3 Regularity considerations

The geometry of superstrata are a priori singular at the supertube locus, as
$r\to 0$ and $\theta\to\pi/2$. This is simply illustrated by considering the
$d\varphi_{1}^{2}$ component of the six dimensional metric (2.5), setting:

$r=a\epsilon\,,\qquad\theta=\frac{\pi}{2}-\epsilon\,,$ (3.28)

and expanding about $\epsilon=0$. The result is a term of the form:

$\displaystyle\frac{1}{4\sqrt{Q_{1}Q_{5}}\epsilon^{2}}\left(2Q_{1}Q_{5}-(2a^{2}+c^{2})R_{y}^{2}\right)+\mathcal{O}(\epsilon^{0})\,,$
(3.29)

Thus, regularity at the supertube locus requires the fixing of a regularity
condition:

$Q_{1}Q_{5}=\frac{R_{y}^{2}}{2}(2a^{2}+c^{2})\,.$ (3.30)

Further, the “coiffuring constraints” have already been built into the first
layer BPS data (3.16)-(3.21). As presented in Appendix A, the most general
first layer superposition would introduce the constants
$(d_{k,m,n},f_{k,m,n})$ and utilize the more general:

$\displaystyle Z_{1}$
$\displaystyle=C_{1}+\frac{Q_{1}}{\Sigma}+\frac{R_{y}}{2Q_{5}}\sum_{m_{1},m_{2}}\sum_{n_{1},n_{2}}d_{2,m_{1}+m_{2},n_{1}+n_{2}}z^{(o)}_{2,m_{1}+m_{2},n_{1}+n_{2}}\,,$
(3.31) $\displaystyle\Theta^{2}$
$\displaystyle=\frac{R_{y}}{2Q_{5}}\sum_{m_{1},m_{2}}\sum_{n_{1},n_{2}}\left(d_{2,m_{1}+m_{2},n_{1}+n_{2}}\vartheta^{(o)}_{2,m_{1}+m_{2},n_{1}+n_{2}}+f_{2,m_{1}+m_{2},n_{1}+n_{2}}\vartheta^{(s)}_{2,m_{1}+m_{2},n_{1}+n_{2}}\right)\,.$
(3.32)

However, doing so would introduce singularities along the $\theta=\pi/2$
locus777For instance, one can check this is true for the $d\varphi_{1}$
component of $\omega^{(o,+)}_{m_{1}+m_{2}=0}$ presented in Appendix A.3..
Thus, the coiffuring constraints:

$\displaystyle d_{2,m_{1}+m_{2},n_{1}+n_{2}}$
$\displaystyle=\frac{b_{1,m_{1},n_{1}}b_{1,m_{2},n_{2}}}{1+\epsilon_{1,2}}\,,$
(3.33) $\displaystyle f_{2,m_{1}+m_{2},n_{1}+n_{2}}$
$\displaystyle=-\frac{1}{2}(m_{2}-m_{1})(n_{2}-n_{1})\left(\frac{b_{1,m_{1},n_{1}}b_{1,m_{2},n_{2}}}{1+\epsilon_{1,2}}\right)\,,$
(3.34)

have been implemented.

It was discussed in [11, 25], how the decoupled solution is almost certainly
regular, once the regularity and coiffuring conditions are imposed. Given the
coupling to flat space provides large corrections to the decoupled solution
only in the asymptotic region, where the solution approaches flat space, it
follows that the flat solution is also almost certainly fully regular as well.

###  3.4 Conserved charges

As will be seen in Section 5.1, flat asymptotics are imposed by setting:

$C_{1}=1\qquad\text{and}\qquad c^{2}=\sum_{m,n}b_{1,m,n}^{2}\,.$ (3.35)

Upon implementing these constraints, the conserved charges can be read off
from the asymptotic behavior of the various BPS data, as outlined in Section
5.5 of [1]. The following results for the charges agree exactly with those
calculated using the asymptotically AdS solutions in Appendix D.5 of [25].

The D1 and D5 brane charges, $Q_{1}$, $Q_{5}$, are read off from the $1/r^{2}$
terms in the expansions:

$Z_{1,2}\approx 1+\frac{Q_{1,5}}{r^{2}}\,.$ (3.36)

From (3.16)-(3.17), one sees that the D1, D5 charges are given by $Q_{1}$ and
$Q_{5}$ respectively.

The momentum charge, $Q_{p}$, can be read off from the expansion:

$\mathcal{F}\approx-2Q_{p}/r^{2}+\textit{oscillatory terms}\,.$ (3.37)

From (3.22) one sees that:

$\displaystyle Q_{p}=\frac{1}{2}\sum_{m,n}(m+n)(b_{1,m,n})^{2}\,.$ (3.38)

The independent angular momenta, $J_{L}$, $J_{R}$, can be read off from the
$d\varphi_{1}+d\varphi_{2}$ component of $\beta+\omega$ as:

$\displaystyle\beta_{\varphi_{1}}+\beta_{\varphi_{2}}+\omega_{\varphi_{1}}+\omega_{\varphi_{2}}\approx\frac{\sqrt{2}}{r^{2}}\left(J_{R}-J_{L}\cos
2\theta\right)+\textit{oscillatory terms}\,.$ (3.39)

From (3.3) and (3.23), one then reads off that:

$J_{L}=\frac{R_{y}}{2}a^{2}\qquad\text{and}\qquad
J_{R}=\frac{R_{y}}{2}\left[a^{2}+\sum_{n}(b_{1,1,n})^{2}\right]\,.$ (3.40)

##  4 The decoupling limit: asymptotically AdS solutions

The $(1,m,n)$ solutions in the decoupling regime were first constructed in
[25], heavily relying on the results of [11]. They possesses two key features,
which have facilitated many calculations by supplying organizational
frameworks do deal with the complexity. Namely: holomorphicity and consistent
truncation. These features are introduced in the geometric context888For gauge
field considerations please refer back to [25]., in this section.

The decoupling limit is given simply by setting:

$C_{1}\to 0\,,$ (4.1)

in the full solution presented in the previous section. Equivalently, since
$C_{1}$ only appears packaged in the decoupling parameter of (3.15):

$\epsilon_{i,j}=\frac{C_{1}a^{2}}{2Q_{5}}\left(m_{i}+m_{j}+n_{i}+n_{j}\right)\,,$
(4.2)

or in $Z_{1,2}$, up to oscillating pieces as:

$\displaystyle Z_{1,2}\sim C_{1}+\frac{Q_{1,5}}{\Sigma}\,,$ (4.3)

one could scale:

$a^{2}\ll Q_{5}\qquad\text{and}\qquad C_{1}\ll\frac{Q_{1,5}}{a^{2}+r^{2}}\,.$
(4.4)

These limits are natural since:

  * •

$Q_{1,5}$ being large is required for the supergravity limit to be valid.

  * •

Small $a^{2}$ implies a long BTZ like throat, which an be inferred from the
form of the geometry presented in Section 5.3.

  * •

Considering the regime where: $r^{2}\ll Q_{1,5}/C_{1}$, ensures one is
considering the region well away from where flatness is asserted.

###  4.1 Holomorphic form

For superstrata in the decoupling limit, it was discovered in [11], that the
solutions can be simply expressed in terms of the complex coordinates:

$\chi\equiv\frac{a}{\sqrt{a^{2}+r^{2}}}\sin\theta\,e^{i\varphi_{1}}\,,\qquad\mu\equiv\cot\theta\,e^{i\left(\frac{\sqrt{2}}{R_{y}}v-\varphi_{1}-\varphi_{2}\right)}\,,\qquad\xi\equiv\frac{r}{\sqrt{a^{2}+r^{2}}}e^{i\frac{\sqrt{2}}{R_{y}}v}\,,$
(4.5)

and their complex conjugates.

These complex variables naturally appeared once the coiffuring was completed.
For instance, upon defining the holomorphic function:

$F(\chi,\mu,\xi)\equiv\sum_{k,m,n}b_{k,m,n}\chi^{k}\mu^{m}\xi^{n}\,,$ (4.6)

in the decoupling limit:

$\displaystyle Z_{1}$
$\displaystyle=\frac{Q_{1}}{\Sigma}+\frac{R_{y}^{2}}{4Q_{5}\Sigma}(F^{2}+\bar{F}^{2})\,,$
(4.7) $\displaystyle Z_{4}$
$\displaystyle=\frac{R_{y}}{2\Sigma}(F+\bar{F})\,.$ (4.8)

The other BPS data and the BPS equations themselves, can similarly be written
in these complex coordinates, see [11, 25] for details. The utility of this
perspective was exploited in generating the solutions presented in [11], by
simplifying many calculations. Utilizing the complex coordinates in this way,
may be termed the holomorphic formalism.

The BPS data appearing in the six-dimensional metric, in the decoupling limit,
can be expanded in a holomorhphic form for the $(1,m,n)$ solutions. Define:

$\displaystyle F_{0}(\xi)\equiv\sum_{n}b_{1,0,n}\xi^{n}\qquad\text{and}\qquad
F_{1}(\xi)\equiv\sum_{n}b_{1,1,n}\xi^{n}\,,$ (4.9)

then one can expand:

$\displaystyle\mathcal{F}_{AdS}$
$\displaystyle=\mathcal{F}^{(p)}+c^{2}\mathcal{F}^{(c)}\,,$ (4.10)
$\displaystyle\omega_{AdS}$ $\displaystyle=\frac{4}{\sin
2\theta}\omega_{\mu}^{(p)}\,d\theta+2\left(\omega_{\chi}^{(0)}+\omega_{\chi}^{(p)}+c^{2}\omega_{\chi}^{(p)}\right)\,d\varphi_{1}+2\left(\omega_{\delta}^{(0)}+\omega_{\delta}^{(p)}\right)d\varphi_{2}\,,$
(4.11)

where the “round supertube” part is given by:

$\omega_{\chi}^{(0)}=\frac{\omega_{\delta}^{(0)}}{\left|\mu\right|^{2}}=\frac{R_{y}\left|\chi\right|^{2}}{2\sqrt{2}(1-\left|\chi\right|^{2})}\,,$
(4.12)

the “homogeneous” part by:

$\mathcal{F}^{(c)}=-\frac{1}{a^{2}}\qquad\text{and}\qquad\omega_{\chi}^{(c)}=\frac{R_{y}\left|\chi\right|^{2}}{2\sqrt{2}a^{2}(1-\left|\chi\right|^{2})}\,,$
(4.13)

and the “particular” part by:

$\displaystyle\begin{split}\mathcal{F}^{(p)}&=\frac{1}{a^{2}}\left(\left|F_{0}\right|^{2}+\left|\xi\right|^{2}\left|F_{1}\right|^{2}\right)\,,\\\
\omega_{\mu}^{(p)}&=-\frac{iR_{y}\left|\chi\right|^{2}}{4\sqrt{2}a^{2}}(\mu\bar{F}_{0}F_{1}+\bar{\mu}F_{0}\bar{F}_{1})\,,\end{split}\qquad\begin{split}\qquad\omega_{\chi}^{(p)}&=-\frac{R_{y}}{4\sqrt{2}a^{2}(1-\left|\chi\right|^{2})}(\bar{\chi}\bar{F}_{0}F+\chi
F_{0}\bar{F})\,,\\\
\omega_{\delta}^{(p)}&=\frac{R_{y}\left|\xi\right|^{2}}{4\sqrt{2}a^{2}(1-\left|\chi\right|^{2})}(\chi\mu
F_{1}\bar{F}+\bar{\chi}\bar{\mu}\bar{F}_{1}F)\,.\end{split}$ (4.14)

###  4.2 Consistent truncation form

Due to the consistent truncation identified in [25], the decoupled geometry
can be cast in the form:

$\displaystyle ds_{6}^{2}=\left(\frac{\Delta}{\det
m_{AB}}\right)^{1/2}ds_{3}^{2}+\frac{1}{g_{0}^{2}}\left(\frac{\Delta}{\det
m_{AB}}m_{AB}\right)^{-1/2}\mathcal{D}\mu^{A}\mathcal{D}\mu^{B}\,,$ (4.15)

where the scale of the three sphere is determined by:

$g_{0}=(Q_{1}Q_{5})^{-1/4}\,.$ (4.16)

In (4.15) we have introduced the coordinates, $\mu^{A}$, giving an embedding
of the three sphere into R4:

$\displaystyle\mu^{1}=\sin\theta\sin\varphi_{1}\,,\qquad\mu^{2}=\sin\theta\cos\varphi_{1}\,,\qquad\mu^{3}=\cos\theta\sin\varphi_{2}\,,\qquad\mu^{4}=\cos\theta\cos\varphi_{2}\,,$
(4.17)

the covariant derivative reads,

$\displaystyle\mathcal{D}\mu^{A}=d\mu^{A}-2g_{0}\tilde{A}^{AB}\mu_{B}\,,$
(4.18)

and

$\displaystyle\Delta=m_{AB}\mu^{A}\mu^{B}\,.$ (4.19)

To summarize the truncation data, it is convenient to introduce the auxiliary
real functions:

$\chi_{A}=-\frac{aR_{y}g_{0}^{2}}{\sqrt{2(a^{2}+r^{2})}}\left(iF_{0},\,F_{0},\,-ie^{-\frac{\sqrt{2}}{R_{y}}v}F_{1},\,e^{\frac{\sqrt{2}}{R_{y}}v}F_{1}\right)+c.c.\,.$
(4.20)

One then has the 10 scalars of the symmetric:

$\displaystyle
m_{AB}=\mathds{1}-\frac{1}{4}\begin{pmatrix}\chi_{1}^{2}+\chi_{2}^{2}&0&\chi_{1}\chi_{3}-\chi_{2}\chi_{4}&\chi_{2}\chi_{3}+\chi_{1}\chi_{4}\\\
0&\chi_{1}^{2}+\chi_{2}^{2}&\chi_{2}\chi_{3}+\chi_{1}\chi_{4}&-\chi_{1}\chi_{3}+\chi_{2}\chi_{4}\\\
\chi_{1}\chi_{3}-\chi_{2}\chi_{4}&\chi_{2}\chi_{3}+\chi_{1}\chi_{4}&\chi_{3}^{2}+\chi_{4}^{2}&0\\\
\chi_{2}\chi_{3}+\chi_{1}\chi_{4}&-\chi_{1}\chi_{3}+\chi_{2}\chi_{4}&0&\chi_{3}^{2}+\chi_{4}^{2}\end{pmatrix}\,,$
(4.21)

the three dimensional external metric:

$\displaystyle
ds_{3}^{2}=\frac{R_{y}^{2}g_{0}^{2}}{2}\left[\Omega^{2}ds_{2}^{2}-a^{2}g_{0}^{4}\left(du+dv+\frac{\sqrt{2}}{a^{2}R_{y}g_{0}^{4}}\mathscr{A}\right)^{2}\right]\,,$
(4.22)

where

$\displaystyle
ds_{2}^{2}=\frac{\left|d\xi\right|^{2}}{\left(1-\left|\xi\right|^{2}\right)^{2}}\,,\qquad\Omega^{2}=\frac{1}{2R_{y}^{2}g_{0}^{4}}\left(4-\chi_{A}\chi_{A}\right)\,,\qquad\mathscr{A}=\frac{i}{2}\left(\frac{\xi
d\bar{\xi}-\bar{\xi}d\xi}{1-\left|\xi\right|^{2}}\right)\,.$ (4.23)

While the gauge fields can be expanded as:

$\displaystyle\tilde{A}^{AB}=\frac{1}{\sqrt{2}a^{2}R_{y}g_{0}}\left(A_{1}\eta_{1}^{AB}+A_{2}\eta_{2}^{AB}+A_{3}\eta_{3}^{AB}+\bar{A}_{3}\bar{\eta}_{3}^{AB}\right)\,.$
(4.24)

The terms appearing in the expansion of the gauge fields, (4.24), read:

$\begin{split}\begin{aligned}
A_{1}&=\frac{1}{4}\left(\chi_{1}\chi_{3}-\chi_{2}\chi_{4}\right)\mathbf{d}\,,\\\
A_{2}&=\frac{1}{4}\left(\chi_{1}\chi_{4}+\chi_{2}\chi_{3}\right)\mathbf{d}\,,\end{aligned}\end{split}\qquad\begin{split}\begin{aligned}
A_{3}&=\frac{a^{2}}{2}\,dv-\frac{1}{8}(\chi_{1}^{2}+\chi_{2}^{2}-\chi_{3}^{2}-\chi_{4}^{2})\mathbf{d}\,,\\\
\bar{A}_{3}&=-\left(\frac{a^{2}+2r^{2}}{2}\right)dv+\left(1+\frac{1}{8}\chi_{A}\chi_{A}\right)\mathbf{d}\,,\end{aligned}\end{split}$
(4.25)

where

$\mathbf{d}=\frac{1}{\Omega^{2}}\left[a^{4}(du+dv)+\frac{2r^{2}}{R_{y}^{2}g_{0}^{4}}dv\right]\,,$
(4.26)

and the anti-symmetric $4\times 4$ ‘t Hooft matrices have been introduced:

$\displaystyle\eta_{1}^{AB}$ $\displaystyle=\begin{pmatrix}0&\sigma_{x}\\\
-\sigma_{x}&0\end{pmatrix}\,,\qquad\eta_{2}^{AB}=\begin{pmatrix}0&-\sigma_{z}\\\
\sigma_{z}&0\end{pmatrix}\,,\qquad\eta_{3}^{AB}=\begin{pmatrix}i\sigma_{y}&0\\\
0&i\sigma_{y}\end{pmatrix}\,,$ (4.27) $\displaystyle\bar{\eta}_{1}^{AB}$
$\displaystyle=\begin{pmatrix}0&-i\sigma_{y}\\\
i\sigma_{y}&0\end{pmatrix}\,,\qquad\bar{\eta}_{2}^{AB}=\begin{pmatrix}0&-\mathbb{1}\\\
\mathbb{1}&0\end{pmatrix}\,,\qquad\bar{\eta}_{3}^{AB}=\begin{pmatrix}i\sigma_{y}&0\\\
0&-i\sigma_{y}\end{pmatrix}\,.$ (4.28)

##  5 The geometry

This section elucidates the geometry of the $(1,m,n)$ asymptotically flat
superstrata, by expanding it in three limits:

  * •

The asymptotic region, $r\to\infty$, where the geometry becomes flat.

  * •

The supertube cap region, $r\to 0$ and $\theta=\pi/2$, where the geometry caps
off smoothly, just above where the horizon of the $a\to 0$ black hole would
sit.

  * •

The long BTZ throat region, where the geometry approximates a BTZ black hole
fibered with a deformed three sphere.

###  5.1 Asymptotically flat region

To reach the asymptotically flat region, first set $C_{1}=1$ in the solution
presented in Section 3.2, then expand the metric (2.5), as $r\to\infty$. To
leading order this gives:

$\displaystyle\lim_{r\to\infty}ds_{6}^{2}$
$\displaystyle=\left(\frac{c^{2}-\left|F_{0}^{(\infty)}\right|^{2}-\left|F_{1}^{(\infty)}\right|^{2}}{a^{2}}\right)dv^{2}-2\,du\,dv+dr^{2}+r^{2}d\Omega_{3}^{2}\,.$
(5.1)

Where $d\Omega_{3}^{2}$ is the canonical metric on S3:

$d\Omega_{3}^{2}=d\theta^{2}+\sin^{2}\theta\,d\varphi_{1}^{2}+\cos^{2}\theta\,d\varphi_{2}^{2}\,\,,$
(5.2)

and utilizing (4.9):

$\displaystyle
F_{0,1}^{(\infty)}(v)\equiv\lim_{r\to\infty}F_{0,1}(v,r)=\lim_{\left|\xi\right|\to
1}F_{0,1}(\xi)\,.$ (5.3)

The metric (5.1) is almost flat space, the discrepancy being due to the
$dv^{2}$ term. However, this can be rectified by setting999We now see why it
was prudent to add the homogeneous part to the solution in Section 3.2. It was
required to allow for the correct asymptotics. :

$c^{2}=\sum_{m,n}b_{1,m,n}^{2}\,,$ (5.4)

in order to remove the non-oscillatory RMS part of the $dv^{2}$ term. The
oscillatory part can then be removed by a gauge transformation of the form
(2.15), with101010It is common in the superstrata literature to use a dot to
denote a derivative with respect to $v$, a convention we utilize here.:

$\displaystyle\dot{U}=\frac{1}{2a^{2}}\left(c^{2}-\left|F_{0}^{(\infty)}\right|^{2}-\left|F_{1}^{(\infty)}\right|^{2}\right)\,.$
(5.5)

Then, utilizing the $(t,y)$ coordinates implicitly defined by (2.6), the
asymptotic metric takes the canonically flat form:

$\lim_{r\to\infty}ds_{6}^{2}=-dt^{2}+dy^{2}+dr^{2}+r^{2}d\Omega_{3}^{2}\,.$
(5.6)

###  5.2 Supertube cap region

To illuminate the smooth capping off of the solution at the supertube locus,
we introduce the dimensionless radial variable:

$\rho=r/a\,,$ (5.7)

set $\theta=\pi/2$ and expand about $\rho=0$. We present this expansion with
the sphere contributions factored out as111111We will not dwell on the form of
the fields $\tensor{\mathcal{A}}{{}^{i}_{j}}$.:

$\displaystyle
ds^{2}_{6}=ds_{3}^{2}+\frac{1}{2}\sum_{x^{i}\in\\{\theta,\varphi_{1},\varphi_{2}\\}}\sum_{x^{j}\in\\{t,y,\rho\\}}\left(dx^{i}+\tensor{\mathcal{A}}{{}^{i}_{j}}dx^{j}\right)^{2}\,.$
(5.8)

The result is:

$\displaystyle ds_{3}^{2}$
$\displaystyle=\sqrt{Q_{1}Q_{5}}\left[\frac{d\rho^{2}}{1-\rho^{2}}-\frac{a^{4}R_{y}^{2}}{Q_{1}^{2}Q_{5}^{2}}(1+\rho^{2})dt^{2}+\frac{\rho^{2}}{R_{y}^{2}}\left(dy+\frac{c^{2}R_{y}^{2}}{2Q_{1}Q_{5}}dt\right)^{2}\right]$
(5.9)
$\displaystyle\qquad+\frac{a^{6}R_{y}^{2}C_{1}}{(Q_{1}Q_{5})^{3/2}}\left(\frac{Q_{1}+Q_{5}}{Q_{1}Q_{5}+a^{2}C_{1}(Q_{1}+Q_{5})}\right)dt^{2}$
$\displaystyle\qquad\qquad-\frac{g_{0}^{2}}{2}\left(\left|F_{0}\right|^{2}+\left|F_{1}\right|^{2}\right)\left[\rho^{2}(dt+dy)^{2}+\frac{R_{y}^{2}d\rho^{2}}{(1+\rho^{2})^{2}}\right]+\mathcal{O}(\rho^{2})\,.$

Where all the terms in the third line are $\mathcal{O}(\rho^{2})$, the
corrections that have not been written are quadratic in the $b_{k,m,n}$ and
vanish when $C_{1}\to 0$. The key takeaway from (5.9) is that the flat
corrections, as captured by the second line, only alters the scaling of $t$,
whilst leaving the radius of the $y$-circle unchanged as $\rho\to 0$. Thus the
smooth capping off, in an AdS3 like manner at the supertube locus, given by
the first line in (5.9), is unaltered by the flat space coupling.

A priori, the flat space coupling may also destroy regularity within the
supertube ring, for $\rho=0$ and $\theta<\pi/2$. Since the superstrata
corrections in line three of (5.9) are subleading as $\rho\to 0$, the
regularity analysis inside the ring is identical to that of the round
supertube coupled to flat space. A demonstration of the regularity of the
round supertube can be found in [51, 52]. It is interesting to note that, in
the decoupled solution the circle that caps off is universal to the ring and
its interior, whilst for the asymptotically flat solution it is a non-trivial
combinations of the periodic directions.

###  5.3 Long BTZ throat region

To demonstrate the region in which the geometry approximates a long throated
BTZ black hole, fibred with a three sphere, we first introduce the
dimensionless coordinates:

$(\tilde{t},\tilde{y},\tilde{r})=(Q_{1}Q_{5})^{-1/4}(t,y,r)=g_{0}(t,y,r)\,.$
(5.10)

Then we expand the metric into the form:

$\displaystyle ds_{6}^{2}$ $\displaystyle=\left(\frac{\Delta}{\det
m_{AB}}\right)^{1/2}ds_{3}^{2}+\frac{1}{g_{0}^{2}}\left(\frac{\Delta}{\det
m_{AB}}\right)^{-1/2}\mathcal{D}\mu^{A}m_{AB}\mathcal{D}\mu^{B}$ (5.11)
$\displaystyle\qquad\qquad\qquad\qquad+ds^{2}_{3,C_{1}}+\frac{1}{2}\sum_{x^{i}\in\\{\theta,\varphi_{1},\varphi_{2}\\}}\sum_{x^{j}\in\\{\tilde{t},\tilde{y},\tilde{r}\\}}\left(dx^{i}+\tensor{\mathcal{A}}{{}^{i}_{j}}dx^{j}\right)^{2}\,.$

The first line of (5.11) corresponds to the decoupling limit contributions,
given by setting121212I.e. the asymptotically AdS decoupled solution,
presented in the form of the consistent truncation, as given in Section 4.2.
$C_{1}=0$. While, the second line of (5.11) gives the “flat corrections,”
involving terms that vanish when $C_{1}\to 0$, that are not captured by the
consistent truncation ansatz. We will neglect the terms, mixing the
$(d\tilde{t},d\tilde{y},d\tilde{r})$ and
$(d\theta,d\varphi_{1},d\varphi_{2})$, though one can check that these terms
are perfectly well behaved and don’t materially alter the following story. To
present the results cleanly, it will be useful to define:

$\displaystyle\tilde{F}^{m}_{n_{1},n_{2}}$
$\displaystyle\equiv\frac{1}{2}\sum_{n_{1},n_{2}}b_{1,m,n_{1}}b_{1,m,n_{2}}(n_{1}+n_{2})e^{\frac{i\sqrt{2}}{R_{y}}(n_{2}-n_{1})v}\,.$
(5.12)

Working to first order in $a$, we find that the warp factors expand as:

$\displaystyle\left(\frac{\Delta}{\det m_{AB}}\right)^{\pm 1/2}$
$\displaystyle=1+\mathcal{O}(a^{2})\,,$ (5.13)

the truncated metric expands as:

$\displaystyle ds_{3}^{2}$
$\displaystyle=\sqrt{Q_{1}Q_{5}}\left(\frac{d\tilde{r}^{2}}{\tilde{r}^{2}}-\tilde{r}^{2}\,d\tilde{t}^{2}+\tilde{r}^{2}\,d\tilde{y}^{2}\right)$
$\displaystyle\qquad+\sqrt{Q_{1}Q_{5}}\left\\{\frac{\tilde{F}^{0}_{n_{1}+n_{2}}+\tilde{F}^{1}_{n_{1}+n_{2}}}{\sqrt{2}R_{y}c}-\frac{\sqrt{Q_{1}Q_{5}}}{4R_{y}^{2}c^{4}}\left(\left|F_{0}^{(\infty)}\right|^{2}+\left|F_{1}^{(\infty)}\right|^{2}-c^{2}\right)^{2}\right\\}\left(d\tilde{t}+d\tilde{y}\right)^{2}+\mathcal{O}(a^{2})\,,$
(5.14)

while the flat corrections expand as:

$ds^{2}_{3,C_{1}}=P_{1}(\tilde{r})\left\\{X(v,\tilde{r},\theta,\varphi_{1},\varphi_{2})\left[(d\tilde{t}+d\tilde{y})^{2}-2\tilde{r}^{2}(d\tilde{t}+d\tilde{y})d\tilde{y}\right]+\left(\frac{1}{1-g_{0}^{2}P_{1}(\tilde{r})}\right)\frac{d\tilde{r}^{2}}{\tilde{r}^{2}}\right\\}+\mathcal{O}(a^{2})\,,$
(5.15)

where

$\displaystyle
P_{1}(\tilde{r})=\frac{1}{g_{0}^{2}}\left(1-\frac{1}{\sqrt{(1+C_{1}g_{0}^{2}Q_{1}\tilde{r}^{2})(1+C_{1}g_{0}^{2}Q_{5}\tilde{r}^{2})}}\right)\,,$
(5.16)

$X(v,\tilde{r},\theta,\varphi_{1},\varphi_{2})$ is of
$\mathcal{O}(b_{k,m,n}^{2})$ and appears to have no convenient closed form or
expansion.

To give a sense of the complexity of
$X(v,\tilde{r},\theta,\varphi_{1},\varphi_{2})$, we include its explicit form,
including only modes with $n\in\\{1,2,3\\}$, whence it can be expanded as:

$\displaystyle
X(v,\tilde{r},\theta,\varphi_{1},\varphi_{2})=\tilde{r}^{2}-\frac{1}{2}g_{0}^{2}X_{1}+\frac{R_{y}^{2}g_{0}^{6}}{4}\left(P_{1}g_{0}^{2}-1\right)\left(X_{2}^{2}+X_{3}^{2}+X_{4}^{2}\right)\,,$
(5.17)

where:

$\displaystyle X_{1}$
$\displaystyle=b_{1,0,1}^{2}+2b_{1,0,2}^{2}+3b_{1,0,3}^{2}+2b_{1,1,1}^{2}+3b_{1,1,2}^{3}+4b_{1,1,3}^{3}$
(5.18)
$\displaystyle\qquad+\left[b_{1,0,2}(3b_{1,0,1}+5b_{1,0,3})+b_{1,1,2}(5b_{1,1,1}+7b_{1,1,3})\right]\cos
v_{0,0,1}$
$\displaystyle\qquad\qquad+2\left(2b_{1,0,1}b_{1,0,3}+3b_{1,1,1}b_{1,1,3}\right)\cos
v_{0,0,2}\,,$

and

$\displaystyle X_{2}$
$\displaystyle=\left(b_{1,0,1}b_{1,1,1}+b_{1,0,2}b_{1,1,2}+b_{1,0,3}b_{1,1,3}\right)\cos
v_{0,1,0}+\left(b_{1,0,1}b_{1,1,2}+b_{1,0,2}b_{1,1,3}\right)\cos v_{0,1,1}$
(5.19) $\displaystyle\qquad+b_{1,0,1}b_{1,1,3}\cos
v_{0,1,2}+\left(b_{1,0,2}b_{1,1,1}+b_{1,0,3}b_{1,1,2}\right)\cos
v_{0,1,-1}+b_{1,0,3}b_{1,1,1}\cos v_{0,1,-2}\,,$ $\displaystyle X_{3}$
$\displaystyle=b_{1,1,1}^{2}+b_{1,1,2}^{2}+b_{1,1,3}^{2}-\left(b_{1,0,2}b_{1,0,1}+b_{1,0,2}b_{1,0,3}-b_{1,1,1}b_{1,1,2}-b_{1,1,2}b_{1,1,3}\right)\cos
v_{0,0,1}$
$\displaystyle\qquad+\left(b_{1,1,1}b_{1,1,3}-b_{1,0,1}b_{1,0,3}\right)\cos
v_{0,0,2}\,,$ $\displaystyle X_{4}$
$\displaystyle=\left(b_{1,0,1}b_{1,1,1}+b_{1,0,2}b_{1,1,2}+b_{1,0,3}b_{1,1,3}\right)\sin
v_{0,1,0}+\left(b_{1,0,1}b_{1,1,2}+b_{1,0,2}b_{1,1,3}\right)\sin v_{0,1,1}$
$\displaystyle\qquad+b_{1,0,1}b_{1,1,3}\sin
v_{0,1,2}+\left(b_{1,0,2}b_{1,1,1}+b_{1,0,3}b_{1,1,2}\right)\sin
v_{0,1,-1}+b_{1,0,3}b_{1,1,1}\sin v_{0,1,-2}\,.$

The standard BTZ metric in our coordinates, for the patch outside the horizon,
takes the form:

$ds_{BTZ}^{2}=\sqrt{Q_{1}Q_{5}}\left(\frac{d\tilde{r}^{2}}{\tilde{r}^{2}}-\tilde{r}^{2}\,d\tilde{t}^{2}+\tilde{r}^{2}\,d\tilde{y}^{2}\right)+\alpha(d\tilde{y}+d\tilde{t})^{2}\,,$
(5.20)

where $\alpha$ is a constant upon which the charges depend. Thus, one sees
that external part of the six-dimensional metric:

$ds_{3,\text{external}}^{2}=\left(\frac{\Delta}{\det
m_{AB}}\right)^{1/2}ds_{3}^{2}+ds^{2}_{3,C_{1}}\,,$ (5.21)

is the BTZ metric with two types of corrections:

  * •

$\mathcal{O}(b_{k,m,n}^{2})$ corrections that are oscillatory in $v$.

  * •

Corrections coming from the flat space coupling, that are suppressed by a
factor of $P_{1}$, which ensures these terms vanish in the decoupling limit.

##  6 Discussion and conclusions

Having constructed the full $(1,m,n)$ superstrata in Section 3, considered the
decoupled AdS limit in Section 4 and illustrated various limits of the
asymptotically flat geometries in Section 5, it is now worth revisiting the
two main research questions of this paper:

  * •

What features of microstate geometries, in the decoupling limit, are altered
or retained when more realistic asymptotics are imposed?

  * •

To what extent are the currently accepted results or intuition, in the
microstate geometry program, artefacts of working in the decoupling limit?

To first order, an answer comes from comparing $(\mathcal{F},\omega)$, key
data in the metric expansion (2.5), between the full solution (3.22)-(3.23)
and the decoupled solution, given by setting $\epsilon_{1,2}=0$. One sees
immediately that the “addition modes” in $\omega$, make the full solution’s
geometry vastly more involved than the decoupled solution’s geometry,
especially when many more modes are considered in the superposition.

To second, the explicit implications this additional geometric structure
should be explored. Some straightforward implications demonstrated in this
paper include:

  * •

The full solution is no longer simply written in terms of the holomorphic
formalism.

  * •

The full solution does not belong to the consistent truncation to which the
decoupled solution does.

  * •

Changing the asymptotics has effects on the geometry, even deep within the
long throat.

Beyond these straightforward observations, it is now possible to revisit many
of the calculations previously performed in the decoupling limit, to determine
the corrections due to the flat space coupling. We leave such intriguing
analysis to future projects.

###  6.1 Holomorphic considerations

The holomorphic formalism traces its origins to the coiffuring relations,
required to ensure regularity of the decoupled solution. However, coupling the
solutions to flat space alters the coiffuring relations, see (3.33)-(3.34). So
it is not obvious that the holomorphic formalism survives the transition. The
simplest way to probe this is by considering (3.16):

$Z_{1}=C_{1}+\frac{Q_{1}}{\Sigma}+\frac{R_{y}}{2Q_{5}}\sum_{m_{1},m_{2}}\sum_{n_{1},n_{2}}\left(\frac{b_{1,m_{1},n_{1}}b_{1,m_{2},n_{2}}}{1+\epsilon_{1,2}}\right)z^{(o)}_{2,m_{1}+m_{2},n_{1}+n_{2}}\,,$
(6.1)

where (3.15):

$\epsilon_{i,j}=\frac{C_{1}a^{2}}{2Q_{5}}\left(m_{i}+m_{j}+n_{i}+n_{j}\right)\,.$
(6.2)

When trying to reformulate this in the complex variables, one notes that:

$z^{(o)}_{2,m,n}=\chi^{2}\mu^{m}\xi^{n}+\bar{\chi}^{2}\bar{\mu}^{m}\bar{\xi}^{n}\,,$
(6.3)

but then the $\epsilon_{1,2}$ in (6.1) becomes very cumbersome. This can not
be generated cleanly from some holomorphic functions.

On the other hand, in the decoupling limit, $Z_{1}$ does has the very simple
holomorphic form (4.7):

$Z_{1}=\frac{Q_{1}}{\Sigma}+\frac{R_{y}^{2}}{4Q_{5}\Sigma}(F^{2}+\bar{F}^{2})\,,$
(6.4)

with (4.6):

$F(\chi,\mu,\xi)\equiv\sum_{k=1}^{1}\sum_{m=0}^{1}\sum_{n}b_{k,m,n}\chi^{k}\mu^{m}\xi^{n}\,.$
(6.5)

Thus, it appears that the utility of the holomorphic formalism identified in
[11], was a consequence of working in the decoupling regime. Rather than
reflecting an underlying structure, such as supersymmetry.

###  6.2 Consistent truncation considerations

From the expansion given of the asymptotically flat $(1,m,n)$ solution in the
long BTZ throat region of Section 5.3, it is clear that the full solution can
not belong to a consistent truncation on a three sphere. To see this, consider
the external three dimensional part of the metric (5.21):

$\displaystyle ds_{3,\text{external}}^{2}\equiv\left(\frac{\Delta}{\det
m_{AB}}\right)^{1/2}ds_{3}^{2}+ds^{2}_{3,C_{1}}\,.$ (6.6)

Considering the form of $ds_{3,C_{1}}^{2}$ in (5.15), it is clear that the
dependence on the spherical coordinates, $(\theta,\varphi_{1},\varphi_{2})$,
can not be factored out into a simple “warp factor.” Thus the dynamics of the
sphere can not couple in the requisite way, to the external part of the
geometry, for a consistent truncation to be valid.

Although this removes one of the most powerful supergravity tools for
simplifying analysis, it also opens a window into uncovering new physics. For
instance, it shows that the consistent truncation is missing some supergravity
degrees of freedom, necessary to couple to flat space. Can these degrees of
freedom be identified in some convenient manner? It is conceivable that the
full $(1,m,n)$ solution presented herein could be used to answer such a
question.

###  6.3 UV/IR mixing considerations

An intriguing feature of quantum gravity is the possibility for subtle UV/IR
mixing effects. Wherein the short distance, UV physics, may qualitatively
alter the long distance, IR physics. A result that runs counter to the
effectiveness of re-normalization in quantum field theories, where the UV
degrees of freedom can be discarded.

In the context of microstate geometries, the UV region corresponds to the
asymptotic region, while the IR region corresponds to the region deep down the
long throat, including the cap. With this in mind, consider the conceptual
processes of producing asymptotically flat superstrata:

  1. 1.

Change UV structure, from AdS to flat asymptotics.

  2. 2.

Regularity of superstrata requires altered coiffuring relations.

  3. 3.

Altered coiffuring leads to new structure deep down the long throat and at the
cap.

Thus, the additional structure, relative to the decoupled solutions, presented
in Sections 5.2 and 5.3. Might be suitably interpreted as artifacts of UV/IR
mixing in the microstate geometry program.

## Acknowledgments

We are grateful to Daniel Mayerson and Pierre Heidmann for insightful
discussions on this project. Further, the work of RAW was supported in part by
the Research Foundation - Flanders (FWO).

##  Appendix A Constructing the solutions

This appendix gives details on how the full $(1,m,n)$ asymptotically flat
solutions, of Section 3.2, were generated. Schematically it involves:

  * •

Developing an ansatz for the first BPS layer data, with enough modes to allow
for regularity of the final solution.

  * •

Decomposing the second layer sources into “basic” parts.

  * •

Identifying solutions to the basic second layer sources that already appear in
the literature.

  * •

Solving the second layer for the basic sources that don’t already appear in
the literature.

  * •

Utilizing the semi-linear property of the BPS equations to superpose the basic
solutions.

  * •

Tuning the constants in the first layer ansatz such that the full solution is
regular, a process known as coiffuring.

In practice, for some basic sources, the last three points can only be
completed simultaneously. This is in contrast to the solutions purely in the
decoupled limit, where each step can be completed independently, explicitly.

###  A.1 First layer ansatz

When searching for new multi-mode superstrata, one begins by specifying the
modes in $Z_{4}$, that one would like to consider. In this paper we consider
all possible modes of the form $(1,m,n)$. Since $0\leq m\leq k=1$ for the
original superstrata modes and there are no supercharged modes of this form,
this leaves:

$Z_{4}=\sum_{m=0}^{1}\sum_{n=1}^{\infty}b_{1,m,n}z^{(o)}_{1,m,n}\,.$ (A.1)

In this expansion, the constants $b_{1,m,n}$ have been introduced, these will
be moduli of the final solution. In the remainder of this appendix. the limits
of sums over $m$ and $n$ indices will be left implicit.

After specifying $Z_{4}$, an ansatz for $Z_{2}$ is then postulated that
involves all the “quadratic” modes of those appearing in $Z_{1}$, with
arbitrary coefficients. These quadratic modes will be vital in the coiffuring
process required for regularity, allowing partial cancellation of quadratic
terms coming from $Z_{4}Z_{4}$ and $Z_{4}\Theta^{4}$ like terms in the second
BPS layer sources. Meanwhile, the form of $Z_{2}$ is unchanged for all
superstrata solutions, thus we postulate a full ansatz for the $Z_{I}$ of:

$\displaystyle Z_{1}$
$\displaystyle=C_{1}+\frac{Q_{1}}{\Sigma}+\frac{R_{y}}{2Q_{5}}\sum_{m_{1},m_{2}}\sum_{n_{1},n_{2}}d_{2,m_{1}+m_{2},n_{1}+n_{2}}z^{(o)}_{2,m_{1}+m_{2},n_{1}+n_{2}}\,,$
(A.2) $\displaystyle Z_{2}$ $\displaystyle=C_{1}+\frac{Q_{5}}{\Sigma}\,,$
$\displaystyle Z_{4}$ $\displaystyle=\sum_{m,n}b_{1,m,n}z^{(o)}_{1,m,n}\,.$

The corresponding ansatz for the $\Theta^{I}$ follows directly from the basic
superstrata solutions (3.4)-(3.5) and (3.12)-(3.13), so are given
by131313Given the mode restriction in (3.14), in these sums one must declare
that $\vartheta^{(s)}_{2,0,n_{1}+n_{2}}=\vartheta^{(s)}_{2,2,n_{1}+n_{2}}=0$.
:

$\displaystyle\Theta^{1}$ $\displaystyle=0\,,$ (A.3) $\displaystyle\Theta^{2}$
$\displaystyle=\frac{R_{y}}{2Q_{5}}\sum_{m_{1},m_{2}}\sum_{n_{1},n_{2}}\left(d_{2,m_{1}+m_{2},n_{1}+n_{2}}\vartheta^{(o)}_{2,m_{1}+m_{2},n_{1}+n_{2}}+f_{2,m_{1}+m_{2},n_{1}+n_{2}}\vartheta^{(s)}_{2,m_{1}+m_{2},n_{1}+n_{2}}\right)\,,$
$\displaystyle\Theta^{4}$
$\displaystyle=-2\sum_{m,n}b_{1,m,n}\vartheta^{(o)}_{1,m,n}\,.$

The task now is to solve the second BPS layer, given this first layer ansatz,
while restricting the coefficients $d_{2,m_{1}+m_{2},n_{1}+n_{2}}$ and
$f_{2,m_{1}+m_{2},n_{1}+n_{2}}$ to ensure regularity. This restriction of
coefficients is termed “coiffuring.”

###  A.2 Second BPS layer source decomposition

Given the first layer ansatz (A.2)-(A.3), the sources can be written in terms
of a double sum as:

$\displaystyle\mathcal{S}_{1,2}$
$\displaystyle=\sum_{m_{1},m_{2}}\sum_{n_{1},n_{2}}\left(\mathcal{S}_{1,2}^{(-)}+\mathcal{S}_{1,2}^{(o,+)}+\mathcal{S}_{1,2}^{(s,+)}\right)\,.$
(A.4)

Where the basic sources: $\mathcal{S}_{1,2}^{(-)}$,
$\mathcal{S}_{1,2}^{(o,+)}$ and $\mathcal{S}_{1,2}^{(s,+)}$, have been
introduced. These basic sources depend implicitly on the mode numbers
$(m_{1},m_{2},n_{1},n_{2})$.

The basic sources correspond to; the the “addition mode” terms coming from the
original superstrata contributions in the ansatz:

$\displaystyle\mathcal{S}_{1}^{(o,+)}$
$\displaystyle=\frac{R_{y}}{2}\left[\left(\frac{C_{1}}{Q_{5}}+\frac{1}{\Sigma}\right)d_{2,m_{1}+m_{2},n_{1}+n_{2}}-\frac{1}{\Sigma}b_{1,m_{1},n_{1}}b_{1,m_{2},n_{2}}\right]\vartheta^{(o)}_{2,m_{1}+m_{2},n_{1}+n_{2}}\,,$
$\displaystyle\mathcal{S}_{2}^{(o,+)}$
$\displaystyle=-\frac{1}{R_{y}}(m_{1}+m_{2}+n_{1}+n_{2})^{2}\left[\left(\frac{C_{1}}{Q_{5}}+\frac{1}{\Sigma}\right)d_{2,m_{1}+m_{2},n_{1}+n_{2}}-\frac{1}{\Sigma}b_{1,m_{1},n_{1}}b_{1,m_{2},n_{2}}\right]z^{(o)}_{2,m_{1}+m_{2},n_{1}+n_{2}}\,,$

the “addition mode” terms coming from the supercharged contributions in the
ansatz:

$\displaystyle\mathcal{S}_{1}^{(s,+)}$
$\displaystyle=\frac{R_{y}}{2}\left[\left(\frac{C_{1}}{Q_{5}}+\frac{1}{\Sigma}\right)f_{2,m_{1}+m_{2},n_{1}+n_{2}}+\frac{1}{2\Sigma}(n_{2}-n_{1})(m_{2}-m_{1})b_{1,m_{1},n_{1}}b_{1,m_{2},n_{2}}\right]\vartheta^{(s)}_{2,m_{1}+m_{2},n_{1}+n_{2}}\,,$
$\displaystyle\mathcal{S}_{2}^{(s,+)}$ $\displaystyle=0\,,$

and the “subtraction mode” terms:

$\displaystyle\mathcal{S}_{1}^{(-)}$
$\displaystyle=\frac{R}{\sqrt{2}\Sigma}\Delta_{2,m_{1}+m_{2},n_{1}+n_{2}}b_{1,m_{1},n_{1}}b_{1,m_{2},n_{2}}$
$\displaystyle\qquad\left\\{\left[(m_{1}-m_{2}+n_{1}-n_{2})r\sin\theta\vphantom{\frac{R}{2}}+((m_{1}-1)n_{1}-(m_{2}-1)n_{2})\frac{\Sigma}{r\sin\theta}\right]\Omega^{(1)}\sin
v_{0,m_{1}-m_{2},n_{1}-n_{2}}\right.$
$\displaystyle\qquad\qquad\left.+\left[(m_{1}(1+n_{1})+m_{2}(1+n_{2}))\Omega^{(2)}+((m_{1}-1)n_{1}+(m_{2}-1)n_{2})\Omega^{(3)}\right]\cos
v_{0,m_{1}-m_{2},n_{1}-n_{2}}\right\\}\,,$
$\displaystyle\mathcal{S}_{2}^{(-)}$
$\displaystyle=\frac{1}{R_{y}}\Delta_{2,2m_{1},2n_{1}}b_{1,m_{1},n_{1}}b_{1,m_{2},n_{2}}$
$\displaystyle\qquad\left\\{\frac{1}{\Sigma}(m_{2}-m_{1}+n_{2}-n_{1})^{2}+2\left[\frac{(1-m_{1})(1-m_{2})n_{1}n_{2}}{r^{2}\sin^{2}\theta}+\frac{m_{1}m_{2}(1+n_{1})(1+n_{2})}{(a^{2}+r^{2})\cos^{2}\theta}\right]\right\\}z^{(o)}_{0,m_{2}-m_{1},n_{2}-n_{1}}\,.$

###  A.3 $S_{1,2}^{(o,+)}$ solutions

When solving the second layer with the “addition mode” original basic source:

$\displaystyle\mathcal{S}_{1}^{(o,+)}$
$\displaystyle=\frac{R_{y}}{2}\left[\left(\frac{C_{1}}{Q_{5}}+\frac{1}{\Sigma}\right)d_{2,m_{1}+m_{2},n_{1}+n_{2}}-\frac{1}{\Sigma}b_{1,m_{1},n_{1}}b_{1,m_{2},n_{2}}\right]\vartheta^{(o)}_{2,m_{1}+m_{2},n_{1}+n_{2}}\,,$
$\displaystyle\mathcal{S}_{2}^{(o,+)}$
$\displaystyle=-\frac{1}{R_{y}}(m_{1}+m_{2}+n_{1}+n_{2})^{2}\left[\left(\frac{C_{1}}{Q_{5}}+\frac{1}{\Sigma}\right)d_{2,m_{1}+m_{2},n_{1}+n_{2}}-\frac{1}{\Sigma}b_{1,m_{1},n_{1}}b_{1,m_{2},n_{2}}\right]z^{(o)}_{2,m_{1}+m_{2},n_{1}+n_{2}}\,,$

it is convenient to work with the instances $m_{1}+m_{2}=0,1,2$ separately. In
each instance the solution can be adapted from those presented in [1].

When $m_{1}+m_{2}=0$, the solution is given by:

$\displaystyle\mathcal{F}_{m_{1}+m_{1}=0}^{(o,+)}$ $\displaystyle=0\,,$
$\displaystyle\mathcal{\omega}_{m_{1}+m_{1}=0}^{(o,+)}$
$\displaystyle=-\frac{C_{1}R_{y}}{4\sqrt{2}Q_{5}}(n_{1}+n_{2})d_{2,0,n_{1}+n_{2}}\Delta_{2,0,n_{1}+n_{2}}$
$\displaystyle\qquad\qquad\qquad\left[\sin
v_{2,0,n_{1}+n_{2}}\left(\frac{dr}{r}-\tan\theta\,d\theta\right)+\cos
v_{2,0,n_{1}+n_{2}}\,d\varphi_{2}\right]$
$\displaystyle\qquad+\frac{R_{y}}{2\sqrt{2}}\left(d_{2,0,n_{1}+n_{2}}-b_{1,0,n_{1}}b_{1,0,n_{2}}\right)\Delta_{2,0,n_{1}+n_{2}}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\left[\frac{2}{a^{2}\sin
2\theta}\sin v_{2,0,n_{1}+n_{2}}d\theta+\frac{1}{\Sigma}\cos
v_{2,0,n_{1}+n_{2}}\left(d\varphi_{1}+\frac{r^{2}}{a^{2}}(d\varphi_{1}-d\varphi_{2})\right)\right]\,.$

When $m_{1}+m_{2}=1$, the solution is known when the coiffuring condition:

$d_{2,m_{1}+m_{2},n_{1}+n_{2}}=\frac{b_{1,m_{1},n_{1}}b_{1,m_{2},n_{2}}}{1+\epsilon_{1,2}}\,,$
(A.5)

is imposed. With this coiffuring constraint:

$\displaystyle\mathcal{F}^{(o,+)}_{m_{1}+m_{2}=1}$ $\displaystyle=0\,,$
$\displaystyle\mathcal{\omega}^{(o,+)}_{m_{1}+m_{2}=1}$
$\displaystyle=-\frac{C_{1}R_{y}}{8\sqrt{2}Q_{5}}\Delta_{2,1,n_{1}+n_{2}}\left(\frac{b_{1,m_{1},n_{1}}b_{1,m_{2},n_{2}}}{1+\epsilon_{1,2}}\right)$
$\displaystyle\qquad\left\\{\sin
v_{2,1,n_{1}+n_{2}}\left(\frac{a^{2}(n_{1}+n_{2})+2(1+n_{1}+n_{2})r^{2}}{a^{2}+r^{2}}\frac{dr}{r}+2((n_{1}+n_{2})\cos
2\theta-2\sin^{2}\theta)\frac{d\theta}{\sin 2\theta}\right)\right.$
$\displaystyle\qquad\qquad\left.+\frac{1}{\Sigma}\cos
v_{2,1,n_{1}+n_{2}}\left((a^{2}(2+n_{1}+n_{2})\sin^{2}\theta+(n_{1}+n_{2})(a^{2}+r^{2}))d\varphi_{1}\right.\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\left.\left.+(a^{2}(n_{1}+n_{2})\cos^{2}\theta-(2+n_{1}+n_{2})r^{2})d\varphi_{2}\right)\right\\}\,.$

Finally, when $m_{1}+m_{2}=2$, the solution is given by:

$\displaystyle\mathcal{F}^{(o,+)}_{m_{1}+m_{2}=2}$ $\displaystyle=0\,,$
$\displaystyle\mathcal{\omega}^{(o,+)}_{m_{1}+m_{2}=2}$
$\displaystyle=-\frac{C_{1}R_{y}}{4\sqrt{2}Q_{5}}(2+n_{1}+n_{2})d_{2,2,n_{1}+n_{2}}\Delta_{2,0,2+n_{1}+n_{2}}\cot^{2}\theta$
$\displaystyle\qquad\qquad\qquad\left[\sin
v_{2,2,n_{1}+n_{2}}\left(\frac{dr}{r}+\left(1+\frac{a^{2}}{r^{2}}\right)\cot\theta\,d\theta\right)-\left(1+\frac{a^{2}}{r^{2}}\right)\cos
v_{2,2,n_{1}+n_{2}}\,d\varphi_{1}\right]$
$\displaystyle\qquad-\frac{R_{y}}{2\sqrt{2}}(d_{2,2,n_{1}+n_{2}}-b_{1,1,n_{1}}b_{1,1,n_{2}})\Delta_{2,2,n_{1}+n_{2}}$
$\displaystyle\qquad\qquad\qquad\qquad\left[\frac{2}{a^{2}\sin 2\theta}\sin
v_{2,2,n_{1}+n_{2}}\,d\theta-\frac{1}{\Sigma}\cos
v_{2,2,n_{1}+n_{2}}\left(\left(1+\frac{r^{2}}{a^{2}}\right)d\varphi_{1}-\frac{r^{2}}{a^{2}}d\varphi_{2}\right)\right]\,.$

###  A.4 $S_{1,2}^{(s,+)}$ solutions

When solving the second layer with the “addition mode” supercharged basic
source:

$\displaystyle\mathcal{S}_{1}^{(s,+)}$
$\displaystyle=\frac{R_{y}}{2}\left[\left(\frac{C_{1}}{Q_{5}}+\frac{1}{\Sigma}\right)f_{2,m_{1}+m_{2},n_{1}+n_{2}}+\frac{1}{2\Sigma}(n_{2}-n_{1})(m_{2}-m_{1})b_{1,m_{1},n_{1}}b_{1,m_{2},n_{2}}\right]\vartheta^{(s)}_{2,m_{1}+m_{2},n_{1}+n_{2}}\,,$
$\displaystyle\mathcal{S}_{2}^{(s,+)}$ $\displaystyle=0\,,$

one need only consider the case when $m_{1}+m_{2}=1$, due to (3.14).

The solution to this source can not be found in the literature, but instead
has to be solved for directly. Implementing a solution method directly
paralleling that of [1], one can find the solution:

$\displaystyle\mathcal{F}^{(s,+)}_{m_{1}+m_{2}=1}$ $\displaystyle=0\,,$
$\displaystyle\mathcal{\omega}^{(s,+)}_{m_{1}+m_{2}=1}$
$\displaystyle=-\frac{C_{1}R_{y}}{4\sqrt{2}Q_{5}}f_{2,m_{1}+m_{2},n_{1}+n_{2}}\Delta_{2,m_{1}+m_{2},n_{1}+n_{2}}$
$\displaystyle\qquad\left[\sin
v_{2,m_{1}+m_{2},n_{1}+n_{2}}\left(\frac{a^{2}}{a^{2}+r^{2}}\frac{dr}{r}+\frac{2}{\sin
2\theta}d\theta\right)+\cos
v_{2,m_{1}+m_{2},n_{1}+n_{2}}(d\varphi_{1}+d\varphi_{2})\right]\,,$

provided the imposition of the coiffuring constraint:

$f_{2,m_{1}+m_{2},n_{1}+n_{2}}=-\frac{1}{2}(m_{2}-m_{1})(n_{2}-n_{1})\left(\frac{b_{1,m_{1},n_{1}}b_{1,m_{2},n_{2}}}{1+\epsilon_{1,2}}\right)\,.$
(A.6)

###  A.5 $S_{1,2}^{(-)}$ solutions

Consider solving the second layer with the “subtraction mode” basic source:

$\displaystyle\mathcal{S}_{1}^{(-)}$
$\displaystyle=\frac{R}{\sqrt{2}\Sigma}\Delta_{2,m_{1}+m_{2},n_{1}+n_{2}}b_{1,m_{1},n_{1}}b_{1,m_{2},n_{2}}$
$\displaystyle\qquad\left\\{\left[(m_{1}-m_{2}+n_{1}-n_{2})r\sin\theta\vphantom{\frac{R}{2}}+((m_{1}-1)n_{1}-(m_{2}-1)n_{2})\frac{\Sigma}{r\sin\theta}\right]\Omega^{(1)}\sin
v_{0,m_{1}-m_{2},n_{1}-n_{2}}\right.$
$\displaystyle\qquad\qquad\left.+\left[(m_{1}(1+n_{1})+m_{2}(1+n_{2}))\Omega^{(2)}+((m_{1}-1)n_{1}+(m_{2}-1)n_{2})\Omega^{(3)}\right]\cos
v_{0,m_{1}-m_{2},n_{1}-n_{2}}\right\\}\,,$
$\displaystyle\mathcal{S}_{2}^{(-)}$
$\displaystyle=\frac{1}{R_{y}}\Delta_{2,2m_{1},2n_{1}}b_{1,m_{1},n_{1}}b_{1,m_{2},n_{2}}$
$\displaystyle\qquad\left\\{\frac{1}{\Sigma}(m_{2}-m_{1}+n_{2}-n_{1})^{2}+2\left[\frac{(1-m_{1})(1-m_{2})n_{1}n_{2}}{r^{2}\sin^{2}\theta}+\frac{m_{1}m_{2}(1+n_{1})(1+n_{2})}{(a^{2}+r^{2})\cos^{2}\theta}\right]\right\\}z^{(o)}_{0,m_{2}-m_{1},n_{2}-n_{1}}\,.$

Since $C_{1}$ does not appear in this source, the solution can be obtained
directly from the asymptotically AdS $(1,m,n)$ solution constructed in [25]:

$\displaystyle\mathcal{F}^{(-)}$
$\displaystyle=\frac{1}{a^{2}}b_{1,m_{1},n_{1}}b_{1,m_{2},n_{2}}\left(\delta_{m_{1}+m_{2},0}\Delta_{0,0,n_{1}+n_{2}}+\delta_{m_{1}+m_{2},2}\delta_{0,0,2+n_{1}+n_{2}}\right)\cos
v_{0,m_{2}-m_{1},n_{2}-n_{1}}\,,$ $\displaystyle\mathcal{\omega}^{(-)}$
$\displaystyle=\frac{R_{y}}{\sqrt{2}a^{2}}b_{1,m_{1},n_{1}}b_{1,m_{2},n_{2}}\Delta_{2,m_{1}+m_{2},n_{1}+n_{2}}\left[(m_{2}-m_{1})\sin
v_{0,m_{2}-m_{1},n_{2}-n_{1}}\frac{d\theta}{\sin 2\theta}\right.$
$\displaystyle\qquad\left.-\frac{1}{2\Sigma}\left((a^{2}+r^{2})(2\delta_{m_{1}+m_{2},0}+\delta_{m_{1}+m_{2},1})d\varphi_{1}-(\delta_{m_{1}+m_{2},1}+2\delta_{m_{1}+m_{2},2})r^{2}d\varphi_{2}\right)\cos
v_{0,m_{2}-m_{1},n_{2}-n_{1}}\right]\,.$

###  A.6 Aggregating solution

Given the form of the second layer source decomposition (A.4):

$\displaystyle\mathcal{S}_{1,2}$
$\displaystyle=\sum_{m_{1},m_{2}}\sum_{n_{1},n_{2}}\left(\mathcal{S}_{1,2}^{(-)}+\mathcal{S}_{1,2}^{(o,+)}+\mathcal{S}_{1,2}^{(s,+)}\right)\,.$
(A.7)

Utilizing the linearity of the second BPS layer, the results of the preceding
subsections can be summed, to give the full second layer solution. This
results in a solution for $\mathcal{F}$ of the form:

$\displaystyle\mathcal{F}$
$\displaystyle=\sum_{m_{1},m_{2}}\sum_{n_{1},n_{2}}\mathcal{F}^{(-)}\,,$ (A.8)

and a solution for $\omega$ of the form:

$\displaystyle\omega$
$\displaystyle=\sum_{m_{1},m_{2}}\sum_{n_{1},n_{2}}\left(\omega^{(-)}+\omega^{(o,+)}+\omega^{(s,+)}\right)\,.$
(A.9)

Where $(\mathcal{F}^{(-)},\omega^{(-)},\omega^{(o,+)},\omega^{(s,+)})$, for
the various $(m_{1},m_{2})$ allowed in the $(1,m,n)$ superposition, are
presented in Sections A.3 through A.5.

Further, in constructing these solutions, the coiffuring constraints (A.5) and
(A.6):

$\displaystyle d_{2,m_{1}+m_{2},n_{1}+n_{2}}$
$\displaystyle=\frac{b_{1,m_{1},n_{1}}b_{1,m_{2},n_{2}}}{1+\epsilon_{1,2}}\,,$
(A.10) $\displaystyle f_{2,m_{1}+m_{2},n_{1}+n_{2}}$
$\displaystyle=-\frac{1}{2}(m_{2}-m_{1})(n_{2}-n_{1})\left(\frac{b_{1,m_{1},n_{1}}b_{1,m_{2},n_{2}}}{1+\epsilon_{1,2}}\right)\,,$
(A.11)

must be implemented. The ultimate validity of these coiffuring relations comes
from showing that the final geometry is regular, which is demonstrated/argued
in the body of this paper.

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