Elastic Lattices Inspired by Ulam-Warburton Cellular Automaton

Hasan B. Al Ba’ba’a

|  Department of Mechanical Engineering, Union College, Schenectady, NY 12308,
USA
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|<EMAIL_ADDRESS>

Periodic lattices have been predominantly explored for decades, owing to their
peculiar vibrational behavior. On the other hand, certain types of aperiodic
lattices have enabled new phenomena that may not be otherwise attainable in
periodic ones. In this paper, a new class of aperiodic lattices inspired by
cellular automaton is introduced. Cellular automaton was originally developed
as a machine replication algorithm and it has been intensively explored in
computer science. These algorithms yield structures that are not necessarily
periodic, yet follow well-defined rules that lead to interesting aperiodic
patterns. The concept is utilized here to build elastic lattices following
such rules, and Ulam-Warburton Cellular Automaton (UWCA) is demonstrated as an
example. Resulted UWCA lattices are shown to exhibit unique dynamical
properties, including symmetric eigenfrequency spectra, repeated natural
frequencies of large multiplicity, and the emergence of strongly localized
corner modes. It is envisioned that computer-algorithm-inspired lattices may
unlock new wave phenomena that could outperform existing lattice designs.

Keywords

Cellular automaton, Ulam-Warburton, natural frequencies, mode multiplicity,
corner modes, fractal structures.

Engineered structural materials, often referred to as Phononic Materials
(PMs), have revolutionized the design of vibratory structures. The word
“phononic” has become customarily associated with periodicity in structural
design, i.e., structures built from a self-repeating block called unit cells
[1]. In contrast to naturally occurring materials and composites, PMs exhibit
an exceptional ability to enable a plethora of intriguing and unconventional
phenomena in both acoustic and elastic media [1, 2]. Among the prime examples
are frequency bandgaps [3, 4, 5, 6] (i.e., frequency ranges of blocked wave
propagation), single or double negative materials [7, 8], and high refractive
index [9, 10]. As such, several potential applications of such PM systems have
recently emerged. Examples include, but are not limited to, noise control and
low-transmission of sound [11, 12, 13], flow stabilization [14, 15], acoustic
logic gates [16], energy harvesting [17], cloaking [18, 19], topologically
protected wave guiding [20, 21, 22], non-reciprocal wave transmission [23, 24,
25], among others [26, 27].

In elastic systems, a particularly interesting class of PMs is lattices, which
are defined as structures built from simple interconnected elastic elements
like rod, beams, and plates. Being relatively lightweight and exhibiting
unique mechanical properties, lattices have enabled a variety of phenomena in
wave propagation domain, such as wave directionality [28], energy harvesting
[29], cloaking [30], and zero-energy modes [31, 32]. In the lattice design
front, optimization techniques have been applied, for instance, to allow for
better bandgap engineering [33, 34] and topological protection [35]. More
related to this study is fractal-inspired designs of lattices’ unit cell. As a
case in point, lattices with Koch fractal design have demonestrated some level
of control over bandgaps by using different number of iterations of the
fractal, such that the frequency at which the first bandgap opens decreases as
the fractal iteration number increases [36]. Another study has shown direct
influence of fractal design on altering the elastic constant tensor, which in
turn leads to different elastic response and generated bandgaps, where the
latter is greatly affected by the fractal-cut symmetry of the unit cell [37].
Fibonacci-based structures have also been studied, yielding interesting
patterns in their eigenfrequencies/bandgaps, which have been correlated with
the Fibonacci sequence [38, 39]. Finally, bioinspired designs (comprising
nacre, spider-webs, chohlea, moth-wings, and fractal structural shapes) have
recently garnered research interest, and numerous studies in this domain have
been encapsulated in a review by Dal Poggetto [40].

Beyond advancements in structural design, the embodiment of smart materials in
lattices has proven useful in enabling exceptional features, including (i)
self-healing to improve their life span [41], (ii) magnetoactive capabilities
to control lattice spacing via remotely applied magnatic fields [42, 43],
(iii) remarkable stretchablility by means of elastomers [44, 45], (iv) thermal
tunability of bandgaps by leveraging temperature dependence of elastic moduli
[46], and (v) concurrent vibration suppression and energy harvesting via
piezoelectric ceramics [47].

Figure 1: Elastic square lattice inspired by Ulam-Warbutron Cellular Automaton
(UWCA), showing the first seven generations. The initial (zeroth) generation
starts as a two-dimensional infinite monatomic lattice with pinned (or “OFF”)
masses $m$, interconnected via springs of stiffness $k$. The first generation
is enabled by turning the central mass “ON”, which is interpreted here as
unpinning the central mass so it is free to move. Subsequent generations are
determined by unpinning the masses that share a spring connection with exactly
one unpinned (or an “ON”) mass, as dictated by the UWCA algorithm. Close-ups
of the first, fourth, and seventh generations are provided for reference.

While periodic lattices have been a focal point of research for a few decades,
the notion of aperiodicity and exploring new ways for manipulating and
controlling waves is becoming rather appealing. As a case in point, quasi-
periodicity has been shown to enable topologically protected edge states with
various edge modes’ localization [48, 49]. Additionally, the concept of
topological pumping, where a two-dimensional structure is adiabatically
modulated to gradually pump a signal from one corner to an opposite one, has
been experimentally and theoretically achieved [50, 51]. Quasi-periodicity has
also enabled the concept of near zero modes called floppy modes, which are a
class of topological modes in Maxwell lattices [52]. Analogously, the concept
of rainbow trapping, achieved by an array of resonators with a change in the
system parameters that are not necessarily periodic (e.g., Ref. [53]), has
been used for vibration isolation [54], and energy harvesting [55]. Owing to
the promise of lattices that are unique in design and not necessarily
periodic, and the need for novel approaches to control elastic/acoustic waves,
I take inspiration from Cellular Automaton (CA) to introduce a new class of
lattice designs. By definition, CA is a discrete computation model that has
been used in computer science for many applications, thanks to its inventor
von Neumann who developed it in the 1950s [56]. Since its inception, CA has
been adopted by many fields of study. For instance, a biological object can be
considered a quantifiable entity by numbers, and these numbers can relay how
the different objects can self-replicate and influence each other. Having CA
as a universal computational methodology for machine self-replication [57,
58], such algorithms are utilized to computationally model and understand such
biological entities and living organisms, such as protein, RNA, and DNA [59].
In physics, CA has been influential in studying many systems, including (i)
the behavior of sand piles subject to a blow of wind [60], (ii) computational
development of traffic flow problems [61], and (iii) statistical mechanics
[62]. More pertinent studies to this work are (i) the use of hybrid CA, a
computational method originally developed to understand the structural bones
adaptation [63], to optimally design piezoelectric patches anchored to
vibrational skin for maximizing its power output [64], and (ii) the
characterization of fracture behavior in elastic bodies under dynamical
loading via continuous-discontinuous CA [65].

For the sake of illustration, I demonstrate here a lattice design based on
Ulam-Warburton cellular automaton (UWCA), named after the contributions of
Ulam and Warburton [66], not to mention its popularity in mathematical circles
[67, 68, 66]. The UWCA algorithm can be explained via an infinite sheet of
packed square cells. These square cells can be in a state of “ON” or “OFF”
(may also be interpreted as “1” or “0” or “dead” or “alive” cells). Initially,
a pristine set of squares is set at an “OFF” state, and one single square is
turned “ON” in its center. The rule for UWCA mandates that the neighboring
“OFF” squares turn “ON” only if they have a single edge shared with an “ON”
square. If an “OFF” square cell shares two or more edges with “ON” cells, that
square does not get activated (i.e., turned “ON”). Every stage the pattern
grows is called a generation [67] (shall be assigned the variable $n_{g}$
henceforth), and researchers have studied these patterns to predict the growth
after a known number of generations [68]. An additional rule for the
considered Ulam-Warburton CA is that if an “OFF” cell becomes “ON” at any
given generation, it stays “ON” throughout future generations. Note that UWCA
rules can be applied to other regular shapes, like a hexagon or triangle [67].

To demonstrate UWCA in elastic lattices, an elastic square monatomic lattice,
constituting identical masses $m$ that are interconnected via identical
springs $k$ and exhibit out-of-plane motion, is considered [1]. Implementing
the rules of UWCA requires defining “OFF” and “ON” states in the context of a
vibratory square lattice. To this end, the “OFF” state is interpreted as a
“pinned” mass, i.e., a mass that is constrained from motion, and vice versa
for the “ON” state. Following UWCA rules, a monatomic lattice with all masses
at the “OFF” state (i.e., pinned) is the starting point, and the mass at the
lattice’s center is then turned “ON” (i.e., unpinned). Moving forward with an
increasing number of UWCA generations, interesting patterns grow, having more
degrees of freedom after each generation due to the increasing number of
unpinned masses. Schematics of the first seven generations of the UWCA lattice
are depicted in Figure 1.

Figure 2: (a) The natural frequency spectra of elastic lattices built
following the first eleven generations of UWCA. A schematic of the free (or
“ON”) masses for each generation is depicted for reference. (b) The eigenvalue
spectra of the UWCA lattice for an increasing number of generations, showing
symmetry about $\lambda=4$. Among those eigenvalues, there are corner modes
(marked in orange for reference), which occur at odd numbers of generations,
starting at seven and increasing by increments of four. (c) The degrees of
freedom $n$ versus the number of generation $n_{g}$ is shown, demonstrating a
considerable increase in the degrees of freedom $n$ as UWCA pattern grows with
higher generations $n_{g}$.

To understand the UWCA lattice dynamics, the natural modes are computed at
different numbers of generations. To do so, I start by stating the equations
of motion for the UWCA lattice. After assembling the stiffness and mass
matrices at a given generation, and denoting $n$ as the lattice’s degrees of
freedom, the equations of motion can be written in the matrix form as:

$\mathbf{M}\ddot{\mathbf{u}}+\mathbf{K}{\mathbf{u}}=\mathbf{f}$ (1)

where $\mathbf{M}$ and $\mathbf{K}$ are the mass and stiffness matrices,
respectively, while

$\mathbf{u}=\begin{Bmatrix}u_{1}&u_{2}&\dots&u_{n}\end{Bmatrix}^{\text{T}}$
(2a)
$\mathbf{f}=\begin{Bmatrix}f_{1}&f_{2}&\dots&f_{n}\end{Bmatrix}^{\text{T}}$
(2b)

are the displacement and the forcing vectors, respectively. The degrees of
freedom are numbered starting from left to right in the bottom row of the
lattice and the process is repeated row by row until its top row. Given the
discrete nature of the lattices, and uniform values of masses, the mass matrix
is a constant diagonal matrix $\mathbf{M}=m\mathbf{I}$, where $\mathbf{I}$ is
an identity matrix of size $n\times n$. The stiffness matrix $\mathbf{K}$ is
assembled based on the connectivity of the masses, which is different for
various generations of UWCA (Examples of the stiffness matrix are provided in
the supplementary materials). A common characteristic of $\mathbf{K}$ in all
generations of UWCA lattice is the constant diagonal elements of $4k$, which
will facilitate the discussion on its eigenvalues shortly. Introducing
$\mathbf{M}^{-1}\mathbf{K}=\omega_{0}^{2}\mathbf{D}$, where
$\omega_{0}=\sqrt{k/m}$, and applying the harmonic solution in the absence of
forces, Equation (1) reduces to $\mathbf{D}\mathbf{u}=\lambda\mathbf{u}$,
i.e., an eigenvalue problem with eigenvalues
$\lambda=\omega^{2}/\omega_{0}^{2}$ and $\omega$ being the frequency.

Solving the eigenvalue problem using MATLAB, the eigenvalues (and thus the
natural frequencies) are calculated for different generations $n_{g}$, as
shown in Figure 2(a) for the first eleven generations. As seen in the figure,
the results are depicted as a three-dimensional plot, where the axes denote
the natural frequency (normalized in units of $\omega_{0}$), the number of
degrees of freedom ($n$) in a logarithmic scale of base 10, and, the number of
generation ($n_{g}$), respectively. It is observed that the natural
frequencies are bounded between $\omega/\omega_{0}\approx 1.2$ and
$\omega/\omega_{0}\approx 2.6$, owing to the discrete nature of the lattice.
While an upper cutoff frequency is a known characteristic of discrete
lattices, the lower cutoff frequency in the UWCA lattice appears to be a
byproduct of pinning, thus resembling a lattice with an elastic foundation
[69, 22, 70]. Peculiarly, a signature of UWCA lattice dynamics is the repeated
natural frequencies (the flat points in the figure), emerging at various
frequencies (particularly at $\omega/\omega_{0}=2$), and their multiplicity
depends on the overall shape of the structure at a given generation.

If the eigenvalues $\lambda$ are plotted instead of the natural frequencies
$\omega$, interesting patterns are noted, as in Figure 2(b). It is noticed
that all eigenvalues are distributed symmetrically about $\lambda=4$. This
becomes rather clear when the eigenvalues are shifted down by 4, given that
$\mathbf{D}$ has constant diagonal elements of the same magnitude, stemming
from the constant diagonal elements of $\mathbf{K}$. This process results in a
spectrum where every eigenvalue $\lambda$ has a negative counterpart
$-\lambda$, and being symmetric about zero instead of 4. Such symmetry in
eigenvalues is akin to particle-hole symmetry in condensed matter systems
[71]. Finally, the degrees of freedom $n$ versus the number of generation
$n_{g}$ is depicted in Figure 2(c), showing a substantial increase in the
degrees of freedom $n$ as UWCA pattern grows with higher generations $n_{g}$.

Interestingly, some of the repeated natural frequencies observed in Figures
2(a,b) correspond to strong localization at the lattice’s corners, known as
corner modes, which have witnessed a spurt in research interest recently [72,
73, 74, 75, 76]. These strongly-localized corner modes in UWCA lattices emerge
at $\omega/\omega_{0}=\sqrt{3}$ and $\omega/\omega_{0}=\sqrt{5}$ (or
$\lambda=3$ and $\lambda=5$) and only materialize when the generation number
is according to this formula:

$n_{g}=4r-1$ (3)

where $r$ is an integer equal or larger than 2. As such, the number of
generations $n_{g}$ having these special corner modes must be odd numbers that
start with seven at $r=2$ and increase by four hereafter. At such generations,
the UWCA lattice has corners with a perpendicular-sign ($\perp$) shape, such
that six masses stem equally in three directions. Each physical corner of the
described shape corresponds to two repeated natural modes per frequency (i.e.,
two modes for each $\omega/\omega_{0}=\sqrt{3}$ and
$\omega/\omega_{0}~{}=~{}\sqrt{5}$).

Figure 3: The normalized displacement response of UWCA lattice at the first
eight generations with corners of $\perp$ shape. Corner modes are excited at a
frequency near $\omega=\sqrt{5}\omega_{0}$ (or $\lambda=5$) by applying a
force at the junction connecting the $\perp$ shaped corner with the bulk of
the lattice, which are colored as red for ease of reference. The bar chart in
the middle of the figure shows the number of corners of $\perp$ shape versus
the integer $r$. The number of corners differs with various values of $r$ (and
generation number $n_{g}=4r-1$), but they reset to 4 whenever $r=2^{z}$, where
$z$ is an integer equal or larger than 1. At such values, the number of
corners reaches a local maximum.

In Figure 3, the response of UWCA lattice near $\omega/\omega_{0}=\sqrt{5}$
(to avoid numerical singularity at $\sqrt{5}$) are shown for eight successive
generations having corner modes ($r=2$ through $r=9$), where the displacement
is normalized and its magnitude is reflected by the color legend. To excite
the corner modes, forces are applied at the conjunction of the $\perp$ shapes
with the lattice’s bulk, and are marked as red circles for ease of reference.
As can be seen in the figure, the bulk is nearly motionless, while the corner
masses respond to the excitations near $\omega/\omega_{0}=\sqrt{5}$.
Regardless of the value of integer $r$, the corner masses oscillate out of
phase, as illustrated in the close-up of one of the corners in $r=8$ case. If
the excitation is near $\omega/\omega_{0}=\sqrt{3}$ instead, i.e., the other
natural frequency that exhibits corner modes, then the corner masses shall
move in phase.

It is crucial to shed light on the number of corners of $\perp$ shape with
different values of $r$. In the middle of Figure 3, the number of corners for
a swept range of $r$ is depicted, for which the following observations are
drawn:

  1. 1.

The number of corners does not necessarily increase with increasing $r$.

  2. 2.

Whenever $r=2^{z}+1$, corresponding to $n_{g}=2^{z+2}$, with $z$ being an
integer greater than 1, the number of corners resets to four. Observe, for
instance, $r=5$ and $r=9$ cases (corresponding to $z=1$ and $z=2$,
respectively), which are the successive integers of $r=2^{2}=4$ and
$r=2^{3}=8$, respectively.

  3. 3.

Following the previous point, the number of corners locally maximizes at
$n_{g}=2^{z+2}-1$ right before it abruptly drops to 4 at the subsequent
generation, i.e., $n_{g}=2^{z+2}$.

  4. 4.

It is worth mentioning that the local maximum number of corners seems to grow
with increasing $z$ (and, thus, increasing $r$) and each new maximum is triple
the previous one as deduced from the sequence 4, 12, 36, 108, and 324 in the
figure (corresponding to $r=2^{1},2^{2},2^{3},2^{4},\text{and},2^{5}$,
respectively).

In summary, this paper introduced a new class of aperiodic lattices inspired
by Ulam-Warburton Cellular Automaton (UWCA). The algorithm of UWCA starts with
infinite packed squares that are initially “OFF”, and the squares are then
turned “ON” following specific rules. Such an algorithm results in aperiodic
structures of an increasing number of “ON” squares that grow larger with
higher numbers of generations. Identical procedure is applied here to an
infinite elastic monatomic lattice by interpreting the “OFF” (“ON”) state as
pinned (unpinned) mass. Lattices built through UWCA algorithm are demonstrated
to exhibit unique dynamical properties (e.g., corner modes), as revealed by
their eigenfrequency spectra.

Future directions of this research can include (1) exploring various types of
CA algorithms for unique lattice designs and characteristics, (2) the study of
damping characteristics to observe whether these lattice structures give rise
to amplified damping (e.g., metadamping [77, 78, 79, 80]), (3) investigating
the effective elastic and inertial properties of these novel lattices,
especially that lattices are known to be unique in that regard [81], (4)
incorporating materials with tunable mechanical properties in their design
(e.g., magnetoactive materials [42]), and (5) extending the same concept to
acoustic lattices or LC-circuit networks. Work is in progress to explore
similar classes of structures, called toothpick sequences [68], and determine
their unique vibrational traits.

## References

## References

  * [1] Mahmoud I Hussein, Michael J Leamy and Massimo Ruzzene “Dynamics of phononic materials and structures: Historical origins, recent progress, and future outlook” In _Applied Mechanics Reviews_ 66.4 American Society of Mechanical Engineers Digital Collection, 2014
  * [2] Yan-Feng Wang et al. “Tunable and active phononic crystals and metamaterials” In _Applied Mechanics Reviews_ 72.4 American Society of Mechanical Engineers, 2020, pp. 040801
  * [3] Z. Liu et al. “Locally resonant sonic materials” In _Science_ 289.5485 American Association for the Advancement of Science, 2000, pp. 1734–1736
  * [4] H Al Ba’ba’a, M Nouh and T Singh “Formation of local resonance band gaps in finite acoustic metamaterials: A closed-form transfer function model” In _Journal of Sound and Vibration_ 410 Elsevier, 2017, pp. 429–446
  * [5] Mahmoud I Hussein, Gregory M Hulbert and Richard A Scott “Dispersive elastodynamics of 1D banded materials and structures: analysis” In _Journal of sound and vibration_ 289.4-5 Elsevier, 2006, pp. 779–806
  * [6] Mahmoud I Hussein, Gregory M Hulbert and Richard A Scott “Dispersive elastodynamics of 1D banded materials and structures: design” In _Journal of Sound and Vibration_ 307.3-5 Elsevier, 2007, pp. 865–893
  * [7] Victor Manuel García-Chocano et al. “Quasi-two-dimensional acoustic metamaterial with negative bulk modulus” In _Physical Review B_ 85.18 APS, 2012, pp. 184102
  * [8] Rogelio Gracia-Salgado, Daniel Torrent and José Sánchez-Dehesa “Double-negative acoustic metamaterials based on quasi-two-dimensional fluid-like shells” In _New Journal of Physics_ 14.10 IOP Publishing, 2012, pp. 103052
  * [9] Zixian Liang and Jensen Li “Extreme acoustic metamaterial by coiling up space” In _Physical review letters_ 108.11 APS, 2012, pp. 114301
  * [10] Y Cheng et al. “Ultra-sparse metasurface for high reflection of low-frequency sound based on artificial Mie resonances” In _Nature materials_ 14.10 Nature Publishing Group, 2015, pp. 1013
  * [11] Z Yang et al. “Acoustic metamaterial panels for sound attenuation in the 50–1000 Hz regime” In _Applied Physics Letters_ 96.4 AIP, 2010, pp. 041906
  * [12] Ni Sui et al. “A lightweight yet sound-proof honeycomb acoustic metamaterial” In _Applied Physics Letters_ 106.17 AIP Publishing, 2015, pp. 171905
  * [13] Yi Liao et al. “Research on Low-Frequency Noise Control of Automobiles Based on Acoustic Metamaterial” In _Materials_ 15.9 MDPI, 2022, pp. 3261
  * [14] M.. Hussein, S. Biringen, O.. Bilal and A. Kucala “Flow stabilization by subsurface phonons” In _Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences_ 471.2177 The Royal Society, 2015, pp. 20140928 DOI: 10.1098/rspa.2014.0928
  * [15] Armin Kianfar and Mahmoud I Hussein “Phononic-subsurface flow stabilization by subwavelength locally resonant metamaterials” In _New Journal of Physics_ 25.5 IOP Publishing, 2023, pp. 053021
  * [16] Jian-Ping Xia et al. “Programmable coding acoustic topological insulator” In _Advanced Materials_ 30.46 Wiley Online Library, 2018, pp. 1805002
  * [17] Guobiao Hu et al. “Acoustic-elastic metamaterials and phononic crystals for energy harvesting: A review” In _Smart Materials and Structures_ IOP Publishing, 2021
  * [18] Lucian Zigoneanu, Bogdan-Ioan Popa and Steven A Cummer “Three-dimensional broadband omnidirectional acoustic ground cloak” In _Nature materials_ 13.4 Nature Publishing Group, 2014, pp. 352
  * [19] Hussein Nassar, YY Chen and GL Huang “Isotropic polar solids for conformal transformation elasticity and cloaking” In _Journal of the Mechanics and Physics of Solids_ 129 Elsevier, 2019, pp. 229–243
  * [20] Cheng He et al. “Acoustic topological insulator and robust one-way sound transport” In _Nature Physics_ 12.12 Nature Publishing Group, 2016, pp. 1124
  * [21] Raj Kumar Pal and Massimo Ruzzene “Edge waves in plates with resonators: An elastic analogue of the quantum valley Hall effect” In _New Journal of Physics_ 19.2 IOP Publishing, 2017, pp. 25001 DOI: 10.1088/1367-2630/aa56a2
  * [22] H Al Ba’ba’a, X Zhu and Q Wang “Enabling novel dispersion and topological characteristics in mechanical lattices via stable negative inertial coupling” In _Proceedings of the Royal Society A_ 477.2252 The Royal Society Publishing, 2021, pp. 20200820
  * [23] Romain Fleury et al. “Sound isolation and giant linear nonreciprocity in a compact acoustic circulator” In _Science_ 343.6170 American Association for the Advancement of Science, 2014, pp. 516–519
  * [24] M.. Attarzadeh and M. Nouh “Non-reciprocal elastic wave propagation in 2D phononic membranes with spatiotemporally varying material properties” In _Journal of Sound and Vibration_ 422 Elsevier, 2018, pp. 264–277 DOI: 10.1016/j.jsv.2018.02.028
  * [25] MA Attarzadeh, J Callanan and M Nouh “Experimental observation of nonreciprocal waves in a resonant metamaterial beam” In _Physical Review Applied_ 13.2 APS, 2020, pp. 021001
  * [26] Steven A Cummer, Johan Christensen and Andrea Alù “Controlling sound with acoustic metamaterials” In _Nature Reviews Materials_ 1.3 Nature Publishing Group, 2016, pp. 16001
  * [27] Shuang Chen et al. “A review of tunable acoustic metamaterials” In _Applied Sciences_ 8.9 Multidisciplinary Digital Publishing Institute, 2018, pp. 1480
  * [28] A Srikantha Phani, J Woodhouse and NA Fleck “Wave propagation in two-dimensional periodic lattices” In _The Journal of the Acoustical Society of America_ 119.4 AIP Publishing, 2006, pp. 1995–2005
  * [29] Stefano Gonella, Albert C. To and Wing Kam Liu “Interplay between phononic bandgaps and piezoelectric microstructures for energy harvesting” In _Journal of the Mechanics and Physics of Solids_ 57.3, 2009, pp. 621–633 DOI: 10.1016/j.jmps.2008.11.002
  * [30] Hussein Nassar, YY Chen and GL Huang “Polar metamaterials: a new outlook on resonance for cloaking applications” In _Physical review letters_ 124.8 APS, 2020, pp. 084301
  * [31] TC Lubensky et al. “Phonons and elasticity in critically coordinated lattices” In _Reports on Progress in Physics_ 78.7 IOP Publishing, 2015, pp. 073901
  * [32] Rongyu Xia et al. “Microtwist homogenization of three-dimensional Pyrochlore lattices on zero modes and mechanical polarization” In _Journal of the Mechanics and Physics of Solids_ 155 Elsevier, 2021, pp. 104564
  * [33] Ole Sigmund and Jakob Søndergaard Jensen “Systematic design of phononic band–gap materials and structures by topology optimization” In _Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences_ 361.1806 The Royal Society, 2003, pp. 1001–1019
  * [34] Osama R. Bilal and Mahmoud I. Hussein “Ultrawide phononic band gap for combined in-plane and out-of-plane waves” In _Physical Review E - Statistical, Nonlinear, and Soft Matter Physics_ 84.6 APS, 2011, pp. 65701 DOI: 10.1103/PhysRevE.84.065701
  * [35] Zongliang Du, Hui Chen and Guoliang Huang “Optimal quantum valley Hall insulators by rationally engineering Berry curvature and band structure” In _Journal of the Mechanics and Physics of Solids_ 135 Elsevier, 2020, pp. 103784
  * [36] Pengcheng Zhao, Kai Zhang and Zichen Deng “Elastic wave propagation in lattice metamaterials with Koch fractal” In _Acta Mechanica Solida Sinica_ 33 Springer, 2020, pp. 600–611
  * [37] Valentin Kunin et al. “Static and dynamic elastic properties of fractal-cut materials” In _Extreme Mechanics Letters_ 6 Elsevier, 2016, pp. 103–114
  * [38] Zhilin Hou, Fugen Wu and Youyan Liu “Acoustic wave propagating in one-dimensional Fibonacci binary composite systems” In _Physica B: Condensed Matter_ 344.1-4 Elsevier, 2004, pp. 391–397
  * [39] Andrea Bacigalupo, Maria Laura De Bellis and Marcello Vasta “Design of tunable hierarchical waveguides based on Fibonacci-like microstructure” In _International Journal of Mechanical Sciences_ 224 Elsevier, 2022, pp. 107280
  * [40] Vinícius F Dal Poggetto “Bioinspired acoustic metamaterials: From natural designs to optimized structures” In _Frontiers in Materials_ 10 Frontiers Media SA, 2023, pp. 1176457
  * [41] Kunhao Yu et al. “Additive manufacturing of self-healing elastomers” In _NPG Asia Materials_ 11.1 Nature Publishing Group, 2019, pp. 1–11
  * [42] Kunhao Yu, Nicholas X Fang, Guoliang Huang and Qiming Wang “Magnetoactive acoustic metamaterials” In _Advanced Materials_ 30.21 Wiley Online Library, 2018, pp. 1706348
  * [43] Kyung Hoon Lee et al. “Magnetoactive acoustic topological transistors” In _Advanced Science_ 9.18 Wiley Online Library, 2022, pp. 2201204
  * [44] Yanhui Jiang and Qiming Wang “Highly-stretchable 3D-architected mechanical metamaterials” In _Scientific reports_ 6.1 Nature Publishing Group, 2016, pp. 1–11
  * [45] Yanchu Zhang et al. “Mechanics of stretchy elastomer lattices” In _Journal of the Mechanics and Physics of Solids_ 159 Elsevier, 2022, pp. 104782
  * [46] Chaitanya Nimmagadda and Kathryn H Matlack “Thermally tunable band gaps in architected metamaterial structures” In _Journal of Sound and Vibration_ 439 Elsevier, 2019, pp. 29–42
  * [47] Ying Li et al. “Design of mechanical metamaterials for simultaneous vibration isolation and energy harvesting” In _Applied Physics Letters_ 111.25 AIP Publishing, 2017
  * [48] Matheus IN Rosa, Yuning Guo and Massimo Ruzzene “Exploring topology of 1D quasiperiodic metastructures through modulated LEGO resonators” In _Applied Physics Letters_ 118.13 AIP Publishing LLC, 2021, pp. 131901
  * [49] Mohit Gupta and Massimo Ruzzene “Dynamics of quasiperiodic beams” In _Crystals_ 10.12 MDPI, 2020, pp. 1144
  * [50] Emanuele Riva, Matheus IN Rosa and Massimo Ruzzene “Edge states and topological pumping in stiffness-modulated elastic plates” In _Physical Review B_ 101.9 APS, 2020, pp. 094307
  * [51] Matheus IN Rosa, Raj Kumar Pal, José RF Arruda and Massimo Ruzzene “Edge states and topological pumping in spatially modulated elastic lattices” In _Physical review letters_ 123.3 APS, 2019, pp. 034301
  * [52] Di Zhou, Leyou Zhang and Xiaoming Mao In _Physical Review X_ 9.2 APS, 2019, pp. 021054
  * [53] Jacopo M De Ponti et al. “Selective mode conversion and rainbow trapping via graded elastic waveguides” In _Physical Review Applied_ 16.3 APS, 2021, pp. 034028
  * [54] Jun Zhang et al. “Rainbow zigzag metamaterial beams as broadband vibration isolators for beam-like structures” In _Journal of Sound and Vibration_ 530 Elsevier, 2022, pp. 116945
  * [55] Biao Wang, Yang Huang, Weijian Zhou and Zhengbao Yang “Metamaterial beam for flexural wave resonance rainbow trapping and piezoelectric energy harvesting” In _Journal of Applied Physics_ 129.6 AIP Publishing LLC, 2021, pp. 064505
  * [56] Xuewei Li, Jinpei Wu and Xueyan Li “Theory of practical cellular automaton” Springer, 2018
  * [57] Palash Sarkar “A brief history of cellular automata” In _Acm computing surveys (csur)_ 32.1 ACM New York, NY, USA, 2000, pp. 80–107
  * [58] Jarkko Kari “Theory of cellular automata: A survey” In _Theoretical computer science_ 334.1-3 Elsevier, 2005, pp. 3–33
  * [59] Parimal Pal Chaudhuri, Soumyabrata Ghosh, Adip Dutta and Somshubhro Pal Choudhury “A New Kind of Computational Biology: Cellular Automata Based Models for Genomics and Proteomics” Springer, 2018
  * [60] Gianpiero Cattaneo, Marziale Comito and Daniela Bianucci “Sand piles: from physics to cellular automata models” In _Theoretical Computer Science_ 436 Elsevier, 2012, pp. 35–53
  * [61] Sven Maerivoet and Bart De Moor “Cellular automata models of road traffic” In _Physics reports_ 419.1 Elsevier, 2005, pp. 1–64
  * [62] Stephen Wolfram “Statistical mechanics of cellular automata” In _Reviews of modern physics_ 55.3 APS, 1983, pp. 601
  * [63] Andrés TOVAR et al. “Topology optimization using a hybrid cellular automaton method with local control rules” In _Journal of mechanical design_ 128.6 American Society of Mechanical Engineers, 2006, pp. 1205–1216
  * [64] Soobum Lee and Andrés Tovar “Topology optimization of piezoelectric energy harvesting skin using hybrid cellular automata” In _Journal of Mechanical Design_ 135.3 American Society of Mechanical Engineers, 2013, pp. 031001
  * [65] Fei Yan, Peng-Zhi Pan, Xia-Ting Feng and Shao-Jun Li “The continuous-discontinuous cellular automaton method for elastodynamic crack problems” In _Engineering Fracture Mechanics_ 204 Elsevier, 2018, pp. 482–496
  * [66] Mike Warburton “Ulam-Warburton Automaton-Counting Cells with Quadratics” In _arXiv preprint arXiv:1901.10565_ , 2019
  * [67] Stanislaw Ulam “On some mathematical problems connected with patterns of growth of figures” In _Proceedings of symposia in applied mathematics_ 14.14, 1962, pp. 215–224 American Mathematical Society Providence, RI, USA
  * [68] David Applegate, Omar E Pol and Neil JA Sloane “The toothpick sequence and other sequences from cellular automata” In _arXiv preprint arXiv:1004.3036_ , 2010
  * [69] H. Al Ba’ba’a, M. Nouh and T. Singh “Pole distribution in finite phononic crystals: Understanding Bragg-effects through closed-form system dynamics” In _The Journal of the Acoustical Society of America_ , 2017 DOI: 10.1121/1.5001513
  * [70] S Nandi, T Singh and M Nouh “Uncertainty quantification of tunable elastic metamaterials using polynomial chaos” In _Journal of Applied Physics_ 127.1 AIP Publishing, 2020
  * [71] Emil Prodan et al. “Dynamical Majorana edge modes in a broad class of topological mechanical systems” In _Nature Communications_ 8, 2017 DOI: 10.1038/ncomms14587
  * [72] Marc Serra-Garcia et al. “Observation of a phononic quadrupole topological insulator” In _Nature_ 555.7696 Nature Publishing Group UK London, 2018, pp. 342–345
  * [73] Haoran Xue et al. “Acoustic higher-order topological insulator on a kagome lattice” In _Nature materials_ 18.2 Nature Publishing Group UK London, 2019, pp. 108–112
  * [74] Haiyan Fan et al. “Elastic Higher-Order Topological Insulator with Topologically Protected Corner States” In _Physical Review Letters_ 122.20, 2019, pp. 204301 DOI: 10.1103/PhysRevLett.122.204301
  * [75] Hrishikesh Danawe, Heqiu Li, Hasan Al Ba’ba’a and Serife Tol “Existence of corner modes in elastic twisted kagome lattices” In _Physical Review B_ 104.24 APS, 2021, pp. L241107
  * [76] Hasan B Al Ba’ba’a “Zero-frequency corner modes in mechanical graphene” In _Wave Motion_ 127 Elsevier, 2024, pp. 103291
  * [77] Mahmoud I Hussein and Michael J Frazier “Metadamping: An emergent phenomenon in dissipative metamaterials” In _Journal of sound and vibration_ 332.20 Elsevier, 2013, pp. 4767–4774
  * [78] D. DePauw, H. Al Ba’ba’a and M. Nouh “Metadamping and energy dissipation enhancement via hybrid phononic resonators” In _Extreme Mechanics Letters_ 18 Elsevier, 2018, pp. 36–44 DOI: 10.1016/j.eml.2017.11.002
  * [79] Clémence L. Bacquet et al. “Metadamping: Dissipation Emergence in Elastic Metamaterials” In _Advances in Applied Mechanics_ 51 Elsevier, 2018, pp. 115–164 DOI: 10.1016/bs.aams.2018.09.001
  * [80] Hasan Al Ba’ba’a, Zhenkun Lin and Serife Tol “Metadamping enhancement and tunability via scissor-like electromechanical metamaterials” In _Journal of Applied Physics_ 130.18 AIP Publishing LLC, 2021, pp. 184901
  * [81] Stefano Gonella “Symmetry of the phononic landscape of twisted kagome lattices across the duality boundary” In _Physical Review B_ 102.14 APS, 2020, pp. 140301


$\displaystyle\big{[}\bar{J}^{+},\bar{J}^{-}\big{]}=2\bar{J}^{3}\,,\qquad\big{[}\bar{J}^{3},\bar{J}^{+}\big{]}=\bar{J}^{+}\,,\qquad\big{[}\bar{J}^{3},\bar{J}^{-}\big{]}=-\bar{J}^{-}\,.$

The left quadratic Casimir operator is
$J^{2}=\frac{1}{2}\big{(}J^{+}J^{-}+J^{-}J^{+}\big{)}+\big{(}J^{3}\big{)}^{2}$,
and likewise for $\bar{J}^{2}$. A state in a representation with principal
quantum number $j_{\text{su}}$ has $J^{2}$ eigenvalue
$j_{\text{su}}(j_{\text{su}}+1)$.

Degree $k$ scalar harmonics live in the
$(j_{\text{su}},\bar{j}_{\text{su}})=(k/2,k/2)$ representation of
$SU(2)_{L}\times SU(2)_{R}$. We denote these by $Y_{k}^{m,\bar{m}}$, and
$(m,\bar{m})$ are the spin charges under $(J^{3},\bar{J}^{3})$. They satisfy
the following Laplace equation:

$\Box_{S_{3}}Y^{m_{1},\bar{m}_{2}}_{k}=-k(k+2)Y^{m_{1},\bar{m}_{2}}_{k}\,.$
(A.4)

Denoting the volume of S3 by $\Omega_{3}=2\pi^{2}$, we use normalized
spherical harmonics

$\begin{split}\int
Y_{k_{1}}^{*m_{1},\bar{m}_{1}}Y_{k_{2}}^{m_{2},\bar{m}_{2}}\;=\;\Omega_{3}\,\delta_{k_{1},k_{2}}\delta^{m_{1},m_{1}}\delta^{\bar{m}_{1},\bar{m}_{2}}\,.\end{split}$
(A.5)

One can generate the degree $k$ scalar spherical harmonic wavefunctions acting
with the lowering operators in (A.2) on the highest-weight wavefunctions,
which are

$\begin{split}Y_{k}^{\pm\frac{k}{2},\pm\frac{k}{2}}=\sqrt{k+1}\sin^{k}\theta
e^{\pm ik\phi}\,.\end{split}$ (A.6)

The degree $k=1,2$ normalized scalar spherical harmonics are given by:

$\begin{split}Y_{1}^{+\frac{1}{2},+\frac{1}{2}}=\sqrt{2}\sin\theta\,e^{i\phi}\quad&,\quad
Y_{1}^{+\frac{1}{2},-\frac{1}{2}}=\sqrt{2}\cos\theta\,e^{i\psi}\,,\\\
Y_{1}^{-\frac{1}{2},+\frac{1}{2}}=-\sqrt{2}\cos\theta\,e^{-i\psi}\quad&,\quad
Y_{1}^{-\frac{1}{2},-\frac{1}{2}}=\sqrt{2}\sin\theta\,e^{-i\phi}\,;\end{split}$
(A.7)

$\begin{split}Y_{2}^{+1,+1}&=\sqrt{3}\sin^{2}\theta\,e^{2i\phi}\quad,\quad
Y_{2}^{+1,0}=\sqrt{6}\sin\theta\cos\theta\,e^{i(\phi+\psi)}\quad,\quad
Y_{2}^{+1,-1}=\sqrt{3}\cos^{2}\theta e^{2i\psi}\,,\\\
Y_{2}^{0,+1}&=-\sqrt{6}\sin\theta\cos\theta\,e^{i(\phi-\psi)}\quad,\quad
Y_{2}^{0,0}=-\sqrt{3}\cos 2\theta\,\quad,\quad
Y_{2}^{0,-1}=\sqrt{6}\sin\theta\cos\theta\,e^{-i(\phi-\psi)}\\\
Y_{2}^{-1,+1}&=\sqrt{3}\cos^{2}\theta\,e^{-2i\psi}\quad,\quad
Y_{2}^{-1,0}=-\sqrt{6}\sin\theta\cos\theta\,e^{-i(\phi+\psi)}\quad,\quad
Y_{2}^{-1,-1}=\sqrt{3}\sin^{2}\theta\,e^{-2i\phi}\,.\end{split}$ (A.8)

We also define the following triple overlap,

$\int
Y^{M_{k},\bar{M}_{k}}_{k}Y^{m_{1},\bar{m}_{1}}_{k_{1}}\Big{(}Y^{m_{2},\bar{m}_{2}}_{k_{2}}\Big{)}^{*}=\Omega_{3}\,\,a^{M_{k},\bar{M}_{k}}_{(m_{1},\bar{m}_{1})(m_{2},\bar{m}_{2})}\,.$
(A.9)

####  A.1.2 Vector harmonics

Degree $k$ left vector harmonics live in the
$(j_{\text{su}},\bar{j}_{\text{su}})=(\frac{k+1}{2},\frac{k-1}{2})$
representation of $SU(2)_{L}\times SU(2)_{R}$. We denote these by
$Y^{m,\bar{m}}_{\mathrm{\scriptscriptstyle{L}},k}$ where
$\mathrm{\scriptscriptstyle{L}}$ stands for left; we shall suppress the label
$\mathrm{\scriptscriptstyle{L}}$ when we write explicit $S^{3}$ vector indices
(which will be denoted by $a,b$). Similarly, degree $k$ right vector harmonics
have $(j_{\text{su}},\bar{j}_{\text{su}})=(\frac{k-1}{2},\frac{k+1}{2})$ and
we denote them by $Y^{m,\bar{m}}_{\mathrm{\scriptscriptstyle{R}},k}$. We shall
use $Y^{m,\bar{m}}_{v,k}$ to denote a vector harmonic which can be either
right or left. Vector harmonics satisfy

$\nabla^{2}_{S^{3}}Y^{m,\bar{m}}_{v,k}=-(k^{2}+2k-1)Y^{m,\bar{m}}_{v,k}\,,\qquad
D^{a}(Y^{m,\bar{m}}_{v,k})_{a}=0\,,$ (A.10)

where $\nabla^{2}_{S^{3}}=g^{ab}D_{a}D_{b}$ and where $D_{a}$ is the covariant
derivative with Levi-Civita connection.

One could generate the $(1,0)$ and $(0,1)$ vector harmonics by dualizing
(A.2), i.e. through $V_{a}=g_{ab}V^{b}$. We choose a different normalization
for these harmonics by imposing

$\begin{split}\int(Y_{1}^{\hat{a}A})_{a}^{*}(Y_{1}^{\hat{b}B})^{a}\;=\;\Omega_{3}\,\delta^{\hat{a},\hat{b}}\delta^{A,B}\,,\end{split}$
(A.11)

where $\hat{a},\hat{b}=\pm,0$ denote the range of $m$ and $A,B=\pm$ denotes
left/right vector harmonics. With this choice, the degree 1 vector spherical
harmonics expressed as one-forms are

$\begin{split}Y_{1}^{++}&=\frac{1}{\sqrt{2}}e^{i(\phi+\psi)}\,\left[-i\,d\theta+\sin\theta\cos\theta\,d(\phi-\psi)\right]\,,\\\
Y_{1}^{-+}&=\frac{1}{\sqrt{2}}e^{-i(\phi+\psi)}\,\left[i\,d\theta+\sin\theta\cos\theta\,d(\phi-\psi)\right]\,,\\\
Y_{1}^{0+}&=-\cos^{2}\theta\,d\psi-\sin^{2}\theta\,d\phi\,,\\\
Y_{1}^{+-}&=\frac{1}{\sqrt{2}}e^{i(\phi-\psi)}\left[i\,d\theta-\sin\theta\cos\theta\,d(\phi+\psi)\right]\,,\\\
Y_{1}^{--}&=-\frac{1}{\sqrt{2}}e^{-i(\phi-\psi)}\left[i\,d\theta+\sin\theta\cos\theta\,d(\phi+\psi)\right]\,,\\\
Y_{1}^{0-}&=\cos^{2}\theta\,d\psi-\sin^{2}\theta\,d\phi\,.\\\ \end{split}$
(A.12)

One can generate a degree $k$ vector harmonic using the $SU(2)$ tensor product
decomposition

$(\frac{k}{2},\frac{k}{2})\otimes(1,0)=(\frac{k}{2}+1,\frac{k}{2})\oplus(\frac{k}{2},\frac{k}{2})\oplus(\frac{k}{2}-1,\frac{k}{2})\,.$
(A.13)

The highest weight state of $(\frac{k}{2}+1,\frac{k}{2})$ is obtained by
simply multiplying the highest weight states of $(\frac{k}{2},\frac{k}{2})$
and that of $(1,0)$ (the Clebsch-Gordan coefficient in this case is always
one). One can then generate all the descendants by acting with the lowering
operator $J^{-}$. In order to satisfy the generalized normalization condition

$\begin{split}\int(Y^{m_{1},\bar{m}_{1}}_{v,k_{1}})_{a}^{*}(Y^{m_{2},\bar{m}_{2}}_{v,k_{2}})^{a}\;=\;\Omega_{3}\,\delta^{k_{1},k_{2}}\delta^{m_{1},m_{2}}\,\delta^{\bar{m}_{1},\bar{m}_{2}}\,,\end{split}$
(A.14)

one can use the $SU(2)$ algebra to write

$Y^{\frac{k+1}{2}-m,\frac{k-1}{2}-\bar{m}}_{\mathrm{\scriptscriptstyle{L}},k}=\sqrt{\frac{(k+1-m)!}{(k+1)!m!}}\sqrt{\frac{(k-1-\bar{m})!}{(k-1)!\bar{m}!}}\Big{[}(J^{-})^{m}(\bar{J}^{-})^{\bar{m}},Y^{\frac{k+1}{2},\frac{k-1}{2}}_{\mathrm{\scriptscriptstyle{L}},k}\Big{]}\,.$
(A.15)

All the previous discussion proceeds analogously for right vector harmonics.
We define the following triple integral:

$\begin{split}\int
Y_{k}^{(m_{k},\bar{m_{k}})}\big{(}Y_{1}^{a-}\big{)}_{\mu}\big{(}Y_{1}^{b+}\big{)}^{\mu}\;=\;\Omega_{3}\>\\!f^{(k)}_{(m_{k},\bar{m_{k}})ab}\,.\end{split}$
(A.16)

The explicit value of the components of $f^{(k)}_{(m_{k},\bar{m_{k}})ab}$,
defined in (A.16), that have been used in this paper are

$\begin{split}f^{(2)}_{(0,0)00}\,=\,\frac{1}{\sqrt{3}}\,,\hskip
20.00003ptf^{(2)}_{(1,1)--}\,=\,\frac{1}{\sqrt{3}}\,,\hskip
20.00003ptf^{(2)}_{(\pm 1,\pm 1)00}\,=\,0\,.\end{split}$ (A.17)

One also has
$\,\epsilon_{abc}D^{b}(Y^{m,\bar{m}}_{\mathrm{\scriptscriptstyle{L}},k})^{c}=(k-1)(Y^{m,\bar{m}}_{\mathrm{\scriptscriptstyle{L}},k})_{a}\,$
and
$\,\epsilon_{abc}D^{b}(Y^{m,\bar{m}}_{\mathrm{\scriptscriptstyle{R}},k})^{c}=-(k-1)(Y^{m,\bar{m}}_{\mathrm{\scriptscriptstyle{R}},k})_{a}\,$.

####  A.1.3 Useful definitions

Due to the non-linearity of the gauge-invariant combinations at higher order,
one must project products of harmonics into harmonics of higher order. The
following definitions are useful:

  1. (I)

$(Y^{I})^{\mu}Y^{J}=\frac{E^{IJK}}{\Lambda_{K}}D^{\mu}Y^{K}+f^{IJK}(Y^{K})^{\mu}$

  2. (II)

$\epsilon_{\mu\nu\rho}D^{\sigma}Y^{I}D_{\sigma}D_{\rho}Y^{J}=c_{IJK}^{s}\epsilon_{\mu\nu\rho}D^{\rho}Y^{K}+c_{IJK}^{v}D_{[\nu}Y^{K}_{\mu]}$

  3. (III)

$Y^{\sigma,I}\epsilon_{\mu\nu\rho}D_{\sigma}D^{\rho}Y^{J}=g_{IJK}^{s}\epsilon_{\mu\nu\rho}D^{\rho}Y^{K}+g_{IJK}^{v}D_{[\nu}Y^{K}_{\mu]}$

  4. (IV)

$2D_{[\mu}D^{\rho}Y^{I}\epsilon_{\nu]\rho\sigma}D^{\sigma}Y^{J}=n_{IJK}^{s}\epsilon_{\mu\nu\rho}D^{\rho}Y^{K}+n_{IJK}^{v}D_{[\nu}Y^{K}_{\mu]}$

  5. (V)

$2D_{[\mu}(Y^{I})^{\rho}\epsilon_{\nu]\rho\sigma}D^{\sigma}Y^{J}=p_{IJK}^{s}\epsilon_{\mu\nu\rho}D^{\rho}Y^{K}+p_{IJK}^{v}D_{[\nu}Y^{K}_{\mu]}$

where $I,J,K$ are the multi-indices defined below Eq. (3.11), and where on the
right-hand side the index $K$ is summed over.

###  A.2 Harmonics on AdS3

####  A.2.1 Scalar harmonics

The spherical harmonics are a representation of the AdS3 isometry group
$SO(2,2)\simeq SL(2,\mathbb{R})\times SL(2,\mathbb{R})$. We use coordinates
where the line element reads:

$ds^{2}_{AdS_{3}}=-(\tilde{r}^{2}+1)d\tilde{t}^{2}+\frac{d\tilde{r}^{2}}{\tilde{r}^{2}+1}+\tilde{r}^{2}d\tilde{y}^{2}\,.$
(A.18)

In our conventions $\epsilon_{\tilde{t}\tilde{r}\tilde{y}}$=1. The generators
of the isometry group are given by:

$\displaystyle L_{\pm 1}$ $\displaystyle=ie^{\pm
i(\tilde{t}+\tilde{y})}\Big{(}{}-\frac{1}{2}\frac{\tilde{r}}{\sqrt{\tilde{r}^{2}+1}}\partial_{\tilde{t}}-\frac{1}{2}\frac{\sqrt{\tilde{r}^{2}+1}}{r}\partial_{\tilde{y}}\pm\frac{i}{2}\sqrt{r^{2}+1}\,\partial_{\tilde{r}}\Big{)}\,,\qquad
L_{0}=\frac{i}{2}\big{(}\partial_{\tilde{t}}+\partial_{\tilde{y}}\big{)}\,,$
(A.19) $\displaystyle\bar{L}_{\pm 1}$ $\displaystyle=ie^{\pm
i(\tilde{t}-\tilde{y})}\Big{(}-\frac{1}{2}\frac{\tilde{r}}{\sqrt{\tilde{r}^{2}+1}}\partial_{\tilde{t}}+\frac{1}{2}\frac{\sqrt{\tilde{r}^{2}+1}}{\tilde{r}}\partial_{\tilde{y}}\pm\frac{i}{2}\sqrt{\tilde{r}^{2}+1}\,\partial_{\tilde{r}}\Big{)}\,,\qquad\bar{L}_{0}=\frac{i}{2}(\partial_{\tilde{t}}-\partial_{\tilde{y}})\,$

and they respect the algebra

$\big{[}L_{0},L_{\pm}\big{]}=\mp L_{\pm}\hskip
20.00003pt\big{[}L_{1},L_{-1}\big{]}=2L_{0}\hskip
20.00003pt\big{[}\bar{L}_{0},\bar{L}_{\pm}\big{]}=\mp\bar{L}_{\pm}\hskip
20.00003pt\big{[}\bar{L}_{1},\bar{L}_{-1}\big{]}=2\bar{L}_{0}\,.$ (A.20)

The quadratic Casimirs are
$L^{2}=\frac{1}{2}\big{(}L_{1}L_{-1}+L_{-1}L_{1}\big{)}-\big{(}L_{0}\big{)}^{2}$
and the corresponding antiholomorphic operator $\bar{L}^{2}$. For a state of
$L^{2}$ quantum number $j_{\text{sl}}$, we have $L^{2}\lvert
j_{\text{sl}}\rangle=-j_{\text{sl}}(j_{\text{sl}}-1)\lvert
j_{\text{sl}}\rangle.$

Scalar harmonics have $j_{\text{sl}}=\bar{j}_{\text{sl}}$. We introduce for
convenience $l=2j_{\text{sl}}$. We introduce scalar harmonics $B_{l}^{(\pm)}$,
where the superscript $\pm$ denotes the positive and negative frequency modes.
$B^{(+)00}_{l}$ is the lowest-weight state in the discrete series
representation $D^{+}$, and $B^{(-)00}_{l}$ is the highest-weight state in the
discrete series representation $D^{-}$. For ease of language we refer to these
both as highest-weight states. These harmonics solve the following Laplace
equation:

$\Box_{AdS_{3}}B_{l}^{(\pm)}=l(l-2)B_{l}^{(\pm)}$ (A.21)

We then have

$B^{(\pm)}_{l}=\frac{e^{\mp il\tilde{t}}}{\sqrt{\tilde{r}^{2}+1}^{l}}\,,\qquad
L_{0}B^{(\pm)}_{l}=\bar{L}_{0}B^{(\pm)}_{l}=\pm\frac{l}{2}B^{(\pm)}_{l}\,.$
(A.22)

The fact that these are highest-weight states can be seen as follows:

$\big{[}L_{1},B^{(+)}_{l}\big{]}=\big{[}\bar{L}_{1},B^{(+)}_{l}\big{]}=0\qquad\big{[}L_{-1},B^{(-)}_{l}\big{]}=\big{[}\bar{L}_{-1},B^{(-)}_{l}\big{]}=0\,.$
(A.23)

####  A.2.2 Vector Harmonics

We now introduce vector harmonics on AdS3. Those that live in the
representation labeled by $L^{2}$ quantum numbers
$(j_{\text{sl}},\bar{j}_{\text{sl}})=(\frac{l-2}{2},\frac{l}{2})$ will be
denoted by $B_{\mathrm{\scriptscriptstyle{R}},l}^{(\pm)}$, where
$\mathrm{\scriptscriptstyle{R}}$ stands for right. Analogously, there is also
the left representation with quantum numbers
$(j_{\text{sl}},\bar{j}_{\text{sl}})=(\frac{l}{2},\frac{l-2}{2})$, and we will
denote it by $B_{\mathrm{\scriptscriptstyle{L}},l}^{(\pm)}$. The vector
harmonics satisfy the following Laplace equation

$(d\delta+\delta d)B_{v,l}^{(\pm)}=(l-2)^{2}B_{v,l}^{(\pm)}\,.$ (A.24)

Here the subscript $v$ stands for vector, meaning that the formula applies to
both left and right vector harmonics. The $l=0$ vector harmonics are the
Killing one forms, which can be obtained can be obtained dualizing Eq. (A.19),

$\displaystyle L_{0}$
$\displaystyle=-\frac{i}{2}\big{(}(\tilde{r}^{2}+1)d\tilde{t}-\tilde{r}^{2}d\tilde{y}\big{)}\,,$
(A.25) $\displaystyle L_{\pm}$ $\displaystyle=e^{\pm
i(\tilde{t}+\tilde{y})}\frac{\tilde{r}}{2\sqrt{\tilde{r}^{2}+1}}\big{(}\mp\frac{d\tilde{r}}{\tilde{r}}+i(\tilde{r}^{2}+1)(d\tilde{t}-d\tilde{y})\big{)}\,,$
$\displaystyle\bar{L}_{0}$
$\displaystyle=-\frac{i}{2}\big{(}(\tilde{r}^{2}+1)d\tilde{t}+\tilde{r}^{2}d\tilde{y}\big{)}\,,$
$\displaystyle\bar{L}_{\pm}$ $\displaystyle=e^{\pm
i(\tilde{t}-\tilde{y})}\frac{\tilde{r}}{2\sqrt{\tilde{r}^{2}+1}}\big{(}\mp\frac{d\tilde{r}}{\tilde{r}}+i(\tilde{r}^{2}+1)(d\tilde{t}+d\tilde{y})\big{)}\,.$

One can generate degree $l$ vector harmonics, as in the $S^{3}$ case, by
multiplying scalar harmonics with the one-forms in Eq. (A.25) and exploiting
the $SL(2,\mathbb{R})$ tensor product decomposition. For concreteness, let us
consider right vector harmonics, which live in the representation in the first
term of the following direct sum:

$\displaystyle\Big{(}\frac{l}{2},\frac{l}{2}\Big{)}\otimes\big{(}-1,0\big{)}=\Big{(}\frac{l-2}{2},\frac{l}{2}\Big{)}\oplus\Big{(}\frac{l}{2},\frac{l}{2}\Big{)}\oplus\Big{(}\frac{l+2}{2},\frac{l}{2}\Big{)}\,.$
(A.26)

Let us focus on the $B^{(+)}$ modes. It is important to note that the scalar
harmonic $B^{(+)}_{l}$ is a lowest weight state; in order to obtain the lowest
weight of the vectorial representation we need to take the product with
$L_{1}$, and one has:

$\displaystyle\begin{aligned}
&B^{(+)}_{\mathrm{\scriptscriptstyle{R}},l}=B^{(+)}_{l}\otimes
L_{1}\,,\qquad\big{[}L_{0},B^{(+)}_{\mathrm{\scriptscriptstyle{R}},l}\big{]}=\frac{l-2}{2}B^{(+)}_{\mathrm{\scriptscriptstyle{R}},l}\,,\\\
&\big{[}\bar{L}_{0},B^{(+)}_{\mathrm{\scriptscriptstyle{R}},l}\big{]}=\frac{l}{2}B^{(+)}_{\mathrm{\scriptscriptstyle{R}},l}\,,\qquad\big{[}L_{1},B^{(+)}_{\mathrm{\scriptscriptstyle{R}},l}\big{]}=\big{[}\bar{L}_{1},B^{(+)}_{\mathrm{\scriptscriptstyle{R}},l}\big{]}=0\,.\end{aligned}$
(A.27)

Analogous relations hold for the left vector harmonics. Vector harmonics on
AdS3 are also eigenstates of the following operator, where $\star$ is the
Hodge star on global AdS3, (A.18):

$\star
dB^{(\pm)}_{\mathrm{\scriptscriptstyle{R}},l}=(l-2)B^{(\pm)}_{\mathrm{\scriptscriptstyle{R}},l}\,,\qquad\star
dB^{(\pm)}_{\mathrm{\scriptscriptstyle{L}},l}=-(l-2)B^{(\pm)}_{\mathrm{\scriptscriptstyle{L}},l}\,.$
(A.28)

##  Appendix B Extremal 3-point functions

In this Appendix we will derive the scalar chiral primary operators of
dimension two in the single-particle basis. The vanishing of extremal three-
point functions built with the operator $\tilde{O}_{2}$ in (4.16) was
discussed in Section (4.1) of [31]; here we shall focus on the operators
$\tilde{\Omega}$ and $\tilde{\Sigma}_{3}$ defined in Eq. (4.17). These
operators involve a mixing between the single-trace operators $\Sigma_{3}$,
$\Omega$ and the double-trace operators $(\Sigma_{2}\cdot\Sigma_{2})$,
$(J\cdot\bar{J})$ and $(O\cdot O)$. Let us first discuss the mixing matrix
between the single-traces [28]. The single-particle operators take the form

$\tilde{\Omega}\;\sim\;a_{1}\frac{\Sigma_{3}}{N^{\frac{3}{2}}}+a_{2}\frac{\Omega}{N^{\frac{1}{2}}}\,,\qquad\tilde{\Sigma}_{3}\;\sim\;b_{1}\frac{\Sigma_{3}}{N^{\frac{3}{2}}}+b_{2}\frac{\Omega}{N^{\frac{1}{2}}}\,,$
(B.1)

where $\sim$ means that we are retaining only the single-trace contributions,
and where we have included factors of $N$ such that each term contributes at
order $N^{0}$ to the norm of the respective single-particle operator at large
$N$. As we shall discuss below, at large $N$, the multi-trace contribution to
the norm of the single-particle operator is subleading. Thus, the
orthonormality conditions

$\langle\tilde{\Sigma}_{3}^{++}\tilde{\Sigma}_{3}^{--}\rangle\;=\;\langle\tilde{\Omega}^{++}\tilde{\Omega}^{--}\rangle\;=\;1\,,\qquad\langle\tilde{\Sigma}_{3}^{++}\tilde{\Omega}^{--}\rangle\;=\;0\,,$
(B.2)

give three constraints on the four coefficients $a_{1}$, $a_{2}$, $b_{1}$ and
$b_{2}$. In order to completely fix the mixing matrix in Eq. (B.1) we need a
fourth constraint, which was derived in [28] from comparison with non-extremal
supergravity correlators. The result is

$\tilde{\Omega}\;\sim\;\frac{\sqrt{3}}{2}\left(\frac{\Sigma_{3}}{N^{\frac{3}{2}}}+\frac{\Omega}{N^{\frac{1}{2}}}\right)\,,\qquad\tilde{\Sigma}_{3}\;\sim\;\frac{3}{2}\left(\frac{\Sigma_{3}}{N^{\frac{3}{2}}}-\frac{\Omega}{3N^{\frac{1}{2}}}\right)\,.$
(B.3)

We now turn to the mixing of multi-trace operators in the single-particle
basis. As we shall see, having fixed the single-trace mixing in (B.3), the
multi-trace admixtures can then be completely fixed by CFT computations. By
contrast, if we had not fixed the single-trace mixing in (B.3) and performed
the following steps with the general admixture (B.1), we would not have enough
constraints to determine the mixing.

Let us consider the most general linear combination allowed by the quantum
numbers that can give rise to the single-particle operators:

$\alpha\frac{\Sigma_{3}}{N^{3/2}}+\beta\frac{\Omega}{N^{1/2}}+\gamma(\Sigma_{2}\cdot\Sigma_{2})+\delta(J\cdot\bar{J})+\epsilon(O\cdot
O)^{++}\,.$ (B.4)

The two sets of values for the coefficients $\alpha$ and $\beta$ are given by
Eq. (B.3). We now fix the two sets of coefficients $\gamma$, $\delta$ and
$\epsilon$ by imposing that all extremal three-point functions containing the
operator (B.4) and operators of lower dimension vanish. Equivalently, this
imposes that the operator (B.4) is orthogonal to all multi-trace operators, as
discussed around Eq. (4.3).

There are three non-trivial such correlators, corresponding respectively to
the last three terms in (B.4). We work at leading order in large $N$. For ease
of notation we will suppress the standard space-time dependence in the
following correlators.

  * •

The first constraint follows from imposing that the three-point function
containing the operator (B.4) and two $O^{--}$ operators vanishes. Since
$O^{--}$ belongs to the untwisted sector of the theory, the contributions
coming from the twist operators in (B.4) are trivially zero; moreover the
contribution of the multi-trace $(J\bar{J})$ is subleading at large $N$. We
obtain

$\langle\Big{(}\beta\frac{\Omega^{++}}{N^{1/2}}+\epsilon(O\cdot
O)^{++}\Big{)}O^{--}O^{--}\rangle\;=\;\beta N^{1/2}+2\epsilon N\;=\;0\,,$
(B.5)

where we have used the definitions of $O$, $\Omega$ and $(O\cdot O)$ in Eqs.
(2.8), (2.9) and (2.12).

  * •

Let us now consider the current insertions: again, since they carry no twist,
the twist operators in Eq. (B.4) do not contribute to the correlator; the
contribution of the double-trace operator $(O\cdot O)^{++}$ is subleading at
large $N$. The computation leads to

$\langle\Big{(}\beta\frac{\Omega^{++}}{N^{1/2}}+\delta(J\cdot\bar{J})^{++}\Big{)}J^{-}\bar{J}^{-}\rangle\;=\;\beta
N^{1/2}+\delta N\;=\;0\,,$ (B.6)

where we have used the definitions of $J$, $\bar{J}$, $\Omega$ and
$(J\cdot\bar{J})$ in Eqs. (2.6), (2.9) and (2.12).

  * •

The final constraint comes from inserting two twist operators
$\Sigma_{2}^{--}$. There are three leading-order contributions to the extremal
three-point function, from the operators $\Sigma_{3}^{++}$,
$(\Sigma_{2}\cdot\Sigma_{2})^{++}$ and $\Omega^{++}$ in (B.4). First, the
contraction with $\Sigma_{3}^{++}$ gives at large $N$

$\Big{\langle}\sum_{r<s<t}\big{(}\sigma^{++}_{(rst)}+\sigma^{++}_{(rts)}\big{)}\sum_{a<b}\sigma_{(ab)}^{--}\sum_{c<d}\sigma_{(cd)}^{--}\,\Big{\rangle}\;=\;\frac{3N^{3/2}}{4}\,.$
(B.7)

The combinatorics works as follows. The choice of the copies $r,s,t$ gives
$N\choose 3$ and, for each choice, there are two inequivalent cycles. Thus the
twist-three operator can glue three strands out of $N$ in $2{N\choose 3}$
ways. For concreteness let us suppose that the orientation of the 3 copies
$r,s,t$ is chosen to be $(123)$. Then there remains the freedom to take the
first $\sigma_{2}$ to be the 2-cycle $(12)$, $(23)$ or $(13)$ (this fixes the
second $\sigma_{2}$ to be $(23)$, $(13)$ and $(12)$ respectively): this gives
an additional factor of three. Eq. (B.7) follows from combining this
combinatorial factor with the building block

$\big{\langle}\sigma^{++}_{(123)}\sigma_{(12)}^{--}\sigma_{(23)}^{--}\big{\rangle}\;=\;\frac{3}{4}\,,$
(B.8)

derived in [71, Eq. (5.25)].

Second, the contribution from the double-trace
$(\Sigma_{2}\cdot\Sigma_{2})^{++}$ in (B.4) is

$\Big{\langle}\frac{2}{N^{2}}\sum_{(r<s)\neq(p<q)}\sigma_{(rs)}^{++}\sigma_{(pq)}^{++}\sum_{a<b}\sigma_{(ab)}^{--}\sum_{c<d}\sigma_{(cd)}^{--}\Big{\rangle}\;=\;N^{2}\,.$
(B.9)

The computation goes as follows. The choice of the copies $r,s$ and $p,q$
gives the combinatorial factors $N(N-1)/2$ and $(N-2)(N-3)/2$ respectively;
moreover, one can perform two inequivalent Wick contractions, which give an
extra factor of two. Taking into account the normalization of the multi-trace
operator, one obtains Eq. (B.9).

The third contribution comes from the operator $\Omega^{++}$ in (B.4). One
could evaluate this amplitude either by using the techniques developed in [78,
71], or by exploiting Ward identities that relate different $n$-point
functions in the same R-symmetry multiplet together with some results obtained
in [31]; we shall use the latter method.

We first use the permutation property of the correlator (see e.g. [91, Eq.
(2.2.48)]) to reorder the operators as

$\displaystyle\langle\Omega^{++}\,\Sigma_{2}^{--}\,\Sigma_{2}^{--}\rangle$
$\displaystyle=\langle\Sigma_{2}^{--}\,\Omega^{++}\,\Sigma_{2}^{--}\rangle\,.$
(B.10)

We next write the operator $\Sigma_{2}^{--}$ as
$[J^{-}_{0}\bar{J}^{-}_{0},\Sigma_{2}^{++}]$ and move the current modes onto
the other operators, to obtain

$\displaystyle\langle\Sigma_{2}^{--}\,\Omega^{++}\,\Sigma_{2}^{--}\rangle$
$\displaystyle=\big{\langle}\Sigma_{2}^{++}\,\big{(}2\Omega^{00}\big{)}\,\Sigma_{2}^{--}\big{\rangle}=\frac{N^{2}}{4}\,.$
(B.11)

The last equality is obtained by spectral flowing the correlator to the RR
sector. On doing so we notice that $\Omega$ carries no twist so that the twist
operators need to act on the same copies, which introduces a combinatorial
factor of ${N\choose 2}$. We then use [31, Eq. (5.40)].

Combining Eqs. (B.5)–(B.7), (B.9), and (B.11), we find that the required
vanishing of the three-point functions considered imposes the three
constraints

$\gamma\,=\,-\frac{3\alpha}{4N^{1/2}}-\frac{\beta}{4N^{1/2}}\,,\qquad\delta\,=\,-\frac{\beta}{N^{1/2}}\,,\qquad\epsilon\,=\,-\frac{\beta}{2N^{1/2}}\,.$
(B.12)

Combining these constraints with the two sets of values of $\alpha$ and
$\beta$ in Eq. (B.3), we obtain

$\displaystyle\tilde{\Sigma}_{3}^{++}$
$\displaystyle\,=\,\frac{3}{2}\left[\left(\frac{\Sigma_{3}^{++}}{N^{\frac{3}{2}}}-\frac{\Omega^{++}}{3N^{\frac{1}{2}}}\right)+\frac{1}{N^{\frac{1}{2}}}\left(-\frac{2}{3}(\Sigma_{2}\cdot\Sigma_{2})^{++}+\frac{1}{6}(O\cdot
O)^{++}+\frac{1}{3}(J\cdot\bar{J})^{++}\right)\right]\,,$ (B.13)
$\displaystyle\tilde{\Omega}^{++}$
$\displaystyle\,=\,\frac{\sqrt{3}}{2}\left[\left(\frac{\Sigma_{3}^{++}}{N^{\frac{3}{2}}}+\frac{\Omega^{++}}{N^{\frac{1}{2}}}\right)+\frac{1}{N^{\frac{1}{2}}}\left(-(\Sigma_{2}\cdot\Sigma_{2})^{++}-\frac{1}{2}(O\cdot
O)^{++}-(J\cdot\bar{J})^{++}\right)\right],$

as written in the main text in Eq. (4.17). Recall that the factors of $N$
multiplying the single-trace operators are such that these terms all
contribute at leading order (order $N^{0}$) to the norm of the operators. By
contrast, since the double-trace operators are already unit normalized, the
admixture factors of $1/\sqrt{N}$ imply for instance that the double-trace
contribution to the norm of the single-particle operators is subleading at
large $N$.

##  Appendix C Type IIB supergravity ansatz and BPS equations

In this appendix we record for completeness the general solution to Type IIB
supergravity compactified on T4 that is 1/8-BPS, carries D1-D5-P charges, and
is invariant on T4 [67, (E.7)]:

$\displaystyle ds^{2}_{10}$ $\displaystyle\leavevmode\nobreak\
=\leavevmode\nobreak\
\sqrt{\alpha}\,ds^{2}_{6}+\sqrt{\frac{Z_{1}}{Z_{2}}}\,d\hat{s}^{2}_{4}\,,$
$\displaystyle ds^{2}_{6}$ $\displaystyle\leavevmode\nobreak\
=\leavevmode\nobreak\ -\frac{2}{\sqrt{{\cal
P}}}\,(dv+\bm{\beta})\,\Big{[}du+\omega+\frac{\mathcal{F}}{2}(dv+\bm{\beta})\Big{]}+\sqrt{{\cal
P}}\,ds^{2}_{4}\,,$ $\displaystyle e^{2\Phi}$
$\displaystyle\leavevmode\nobreak\ =\leavevmode\nobreak\
\frac{Z_{1}^{2}}{{\cal P}}\,,$ $\displaystyle B$
$\displaystyle\leavevmode\nobreak\ =\leavevmode\nobreak\ -\frac{Z_{4}}{{\cal
P}}\,(du+\omega)\wedge(dv+\bm{\beta})+a_{4}\wedge(dv+\bm{\beta})+\gamma_{4}\,,$
$\displaystyle C_{0}$ $\displaystyle\leavevmode\nobreak\ =\leavevmode\nobreak\
\frac{Z_{4}}{Z_{1}}\,,$ (C.1) $\displaystyle C_{2}$
$\displaystyle\leavevmode\nobreak\ =\leavevmode\nobreak\ -\frac{Z_{2}}{{\cal
P}}\,(du+\omega)\wedge(dv+\bm{\beta})+a^{1}\wedge(dv+\bm{\beta})+\gamma_{2}\,,$
$\displaystyle C_{4}$ $\displaystyle\leavevmode\nobreak\ =\leavevmode\nobreak\
\frac{Z_{4}}{Z_{2}}\,\widehat{\mathrm{vol}}_{4}-\frac{Z_{4}}{{\cal
P}}\,\gamma_{2}\wedge(du+\omega)\wedge(dv+\bm{\beta})+x_{3}\wedge(dv+\bm{\beta})\,,$
$\displaystyle C_{6}$ $\displaystyle\leavevmode\nobreak\ =\leavevmode\nobreak\
\widehat{\mathrm{vol}}_{4}\wedge\left[-\frac{Z_{1}}{{\cal
P}}\,(du+\omega)\wedge(dv+\bm{\beta})+a^{2}\wedge(dv+\bm{\beta})+\gamma_{1}\right],$

where

$\alpha\;=\;\frac{Z_{1}Z_{2}}{Z_{1}Z_{2}-Z_{4}^{2}}\leavevmode\nobreak\
,\qquad\leavevmode\nobreak\ \leavevmode\nobreak\ {\cal
P}\;=\;Z_{1}Z_{2}-Z_{4}^{2}\,.$ (C.2)

Here $d\hat{s}^{2}_{4}$ denotes the flat metric on $T^{4}$, and
$\widehat{\mathrm{vol}}_{4}$ denotes the corresponding volume form. This
ansatz contains all fields that are known to arise from world-sheet
calculations of the backreaction of D1-D5-P bound states invariant on ${\cal
M}$ [92, 93].

The BPS equations for this ansatz have the following structure. The base
metric, $ds^{2}_{4}$, and the one-form $\bm{\beta}$ satisfy non-linear
equations known as the “zeroth layer”. Having solved these initial equations,
the remaining BPS equations are organized into two further layers of linear
equations [67, 94].

We denote the exterior differential on the spatial base ${\cal B}$ by
$\tilde{d}$, and following [95] we introduce

$\mathcal{D}\;\equiv\;\tilde{d}-\bm{\beta}\wedge\frac{\partial}{\partial
v}\,.$ (C.3)

In the current work we consider only solutions in which the four-dimensional
base space is flat $\mathbb{R}^{4}$, and in which $\bm{\beta}$ is independent
of $v$. The BPS equation for $\bm{\beta}$ is then

$d\bm{\beta}\;=\;*_{4}d\bm{\beta}\,,$ (C.4)

where $*_{4}$ stands for the flat $\mathbb{R}^{4}$ Hodge dual.

To write the remaining layers of BPS equations in a covariant form, we make
the rescaling $(Z_{4},a_{4},\gamma_{4})\to(Z_{4},a_{4},\gamma_{4})/\sqrt{2}$
for the remainder of this appendix (and only here). We introduce the $SO(1,2)$
Minkowski metric $\eta_{ab}$ ($a=1,2,4$) in the null form

$\eta_{12}\leavevmode\nobreak\ =\leavevmode\nobreak\
\eta_{21}\leavevmode\nobreak\ =\leavevmode\nobreak\ 1\,,\qquad\eta_{44}=-1\,.$
(C.5)

This metric is used to raise and lower $a,b$ indices. We introduce the two-
forms $\Theta^{1}$, $\Theta^{2}$, $\Theta^{4}$ as follows:121212The relation
to the notation in [21] is that
$\Theta^{1}_{\mathrm{here}}=\Theta_{1}^{\mathrm{there}}$,
$\Theta^{2}_{\mathrm{here}}=\Theta_{2}^{\mathrm{there}}$,
$(1/\sqrt{2})\Theta_{4}^{\mathrm{here}}=\Theta_{4}^{\mathrm{there}}$.

$\Theta^{b}\leavevmode\nobreak\ \equiv\leavevmode\nobreak\
\mathcal{D}a^{b}+\eta^{bc}\>\\!\dot{\gamma}_{c}\;.$ (C.6)

We then have

${\cal P}\leavevmode\nobreak\ \equiv\leavevmode\nobreak\
{\textstyle\frac{1}{2}}\eta^{ab}Z_{a}Z_{b}\leavevmode\nobreak\
=\leavevmode\nobreak\ Z_{1}Z_{2}-{\textstyle\frac{1}{2}}Z_{4}^{2}\,.$ (C.7)

The “first layer” of the BPS equations is then

$\displaystyle*_{4}D\dot{Z}_{a}\leavevmode\nobreak\ =\leavevmode\nobreak\ $
$\displaystyle\eta_{ab}D\Theta^{b}\,,\qquad D*_{4}DZ_{a}\leavevmode\nobreak\
=\leavevmode\nobreak\ -\eta_{ab}\Theta^{b}\\!\wedge
d\bm{\beta}\,,\qquad\Theta^{a}\leavevmode\nobreak\ =\leavevmode\nobreak\
*_{4}\Theta^{a}\,,$ (C.8)

while the “second layer” is given by

$\displaystyle
D\omega+*_{4}D\omega+\mathcal{F}\,d\bm{\beta}\leavevmode\nobreak\
=\leavevmode\nobreak$ $\displaystyle Z_{a}\Theta^{a}\,,$ (C.9)
$\displaystyle*_{4}D*_{4}\\!\Bigl{(}\dot{\omega}-{\textstyle\frac{1}{2}}\,D\mathcal{F}\Bigr{)}\leavevmode\nobreak\
=\leavevmode\nobreak$ $\displaystyle\ddot{\cal
P}-{\textstyle\frac{1}{2}}\eta^{ab}\dot{Z}_{a}\dot{Z}_{b}-{\textstyle\frac{1}{4}}\eta_{ab}*_{4}\\!\Theta^{a}\wedge\Theta^{b}\,.$

##  Appendix D Gauge-fixed holographic dictionary

In Section 4.5 we discussed the holographic dictionary that relates the
expectation value of single-particle operators at dimension one and scalar
single-particle operators at dimension two in term of gauge-invariant
combination of KK fields. This formulation has the advantage of being general
and manifestly gauge invariant: it can be used to probe any microstate
solution in any coordinate system. From a practical point of view, however, it
is useful to formulate the dictionary in a preferred gauge, as this makes it
more manageable to use. In this appendix we study the dictionary in such a
gauge.

In this appendix we consider the general class of solutions for which the
four-dimensional base metric $ds_{4}^{2}$ in (3.2), (C) is flat
$\mathbb{R}^{4}$. This class includes all solutions discussed in this paper.
For solutions in which the base metric is not flat, one should instead use the
gauge-invariant formulation of the dictionary in Eq. (4.24).

We choose coordinates such that the flat $\mathbb{R}^{4}$ base metric takes
the form:

$ds^{2}_{4}=dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta d\phi^{2}+\cos^{2}\theta
d\psi^{2})\,.$ (D.1)

In general, the functions $Z_{1},Z_{2}$ and $Z_{4}$ obey a Poisson-type
equation, where the $\Theta_{i}$ play the role of sources; see Eq. (C.8). For
$1/4$-BPS solutions, one has $\Theta_{1}=\Theta_{2}=\Theta_{4}=0$, and so the
$Z_{a}$ are harmonic. For all the ($1/8$-BPS) superstrata we deal with in the
present work, the first term in the large $r$ expansion of the source
$\Theta_{a}\wedge d\beta$ is of order $1/r^{k}$ where $k\geq 7$.

For convenience we perform the transformation in Eq. (3.10) and reabsorb the
overall factor $\sqrt{Q_{1}Q_{5}}$ in the metric. We thus define the following
expansion131313Note that this equation is more general than [31, Eq. (3.6)],
which applies only to solutions in which the $Z_{a}$ are harmonic. [27, 6]:

$\displaystyle\tilde{Z}_{1}$
$\displaystyle=\frac{Q_{1}}{\tilde{r}^{2}}\left(1+\sum_{k=1}^{2}\sum_{m_{k},\bar{m}_{k}=-k/2}^{k/2}f^{\,(m_{k},\bar{m}_{k})}_{1k}\,\frac{Y_{k}^{m_{k},\bar{m}_{k}}}{\tilde{r}^{k}}+O(\tilde{r}^{-3})\right)\,,$
(D.2) $\displaystyle\tilde{Z}_{2}$
$\displaystyle=\frac{Q_{5}}{\tilde{r}^{2}}\left(1+\sum_{k=1}^{2}\sum_{m_{k},\bar{m}_{k}=-k/2}^{k/2}f^{k\,(m_{k},\bar{m}_{k})}_{5k}\,\frac{Y_{k}^{m_{k},\bar{m}_{k}}}{\tilde{r}^{k}}+O(\tilde{r}^{-3})\right)\,,$
$\displaystyle\tilde{Z}_{4}$
$\displaystyle=\frac{\sqrt{Q_{1}Q_{5}}}{\tilde{r}^{2}}\left(\sum_{k=1}^{2}\sum_{m_{k},\bar{m}_{k}=-k/2}^{k/2}\mathcal{A}_{k}^{(m_{k},\bar{m}_{k})}\,\frac{Y_{k}^{m_{k},\bar{m}_{k}}}{\tilde{r}^{k}}+O(\tilde{r}^{-3})\right)\,,$
$\displaystyle\tilde{A}$
$\displaystyle=\frac{1}{\tilde{r}^{2}}\,\sum_{a=1}^{3}(a_{a+}Y_{1}^{a+}+a_{a-}Y_{1}^{a-})+O(\tilde{r}^{-3})\,,\qquad\mathcal{F}=-\frac{2Q_{p}}{a_{0}\tilde{r}^{2}}+O(r^{-3})\,,\\!\\!\\!\\!\\!\\!\\!\\!$

where we have denoted with $Y_{k}^{m_{k},\bar{m}_{k}}$ the scalar harmonics on
$S^{3}$ of degree $k$ and with $Y_{1}^{a\pm}$ the vector harmonics of degree 1
(see Appendix A), and where $a_{0}$ was defined below (3.10). We are still
left with the freedom of choosing the origin of the coordinate system on the
flat base. We fix this redundancy by requiring

$f_{11}^{(m_{1},\bar{m}_{1})}+f^{(m_{1},\bar{m}_{1})}_{51}=0\,,$ (D.3)

which corresponds to placing the origin at the center of mass of the D1-D5
system. The choices in Eqs. (D.2) and (D.3) imply that, at order $k=1$, all
fields in (3.12) that can be set to zero by a gauge transformation vanish.
With these choices, the expansion of the bulk quantities in the dictionary
(4.24) in terms of (D.2) yields the gauge-fixed single-particle fields in
supergravity:

$\displaystyle\Big{[}s^{(6)(\alpha,\dot{\alpha})}_{k=1}\Big{]}$
$\displaystyle=-2\sqrt{2}f_{51}^{(\alpha,\dot{\alpha})}\,,\qquad\Big{[}s^{(6)(a,\dot{a})}_{k=2}\Big{]}=\sqrt{\frac{3}{2}}(f_{12}^{(a,\dot{a})}-f_{52}^{(a,\dot{a})})\,,$
(D.4) $\displaystyle\Big{[}s^{(7)(\alpha,\dot{\alpha})}_{k=1}\Big{]}$
$\displaystyle=2\sqrt{2}\mathcal{A}_{1}^{(\alpha,\dot{\alpha})}\,,\qquad\leavevmode\nobreak\
\;\Big{[}s^{(7)(a,\dot{a})}_{k=2}\Big{]}=\sqrt{6}(\mathcal{A}_{2}^{(a,\dot{a})})\,,$
$\displaystyle\Big{[}\tilde{\sigma}^{(a,\dot{a})}_{k=2}\Big{]}$
$\displaystyle=-\frac{1}{\sqrt{2}}\big{(}f_{12}^{(a,\dot{a})}+f_{52}^{(a,\dot{a})}\big{)}+2\sqrt{2}\big{(}f_{51}^{(\alpha,\dot{\alpha})}f_{51}^{(\beta,\dot{\beta})}+\mathcal{A}_{1}^{(\alpha,\dot{\alpha})}\mathcal{A}_{1}^{(\beta,\dot{\beta})}\big{)}a^{(a,\dot{a})}_{(\alpha,\dot{\alpha})(\beta,\dot{\beta})}$
$\displaystyle\,\quad+4\sqrt{2}a^{c+}a^{d-}f^{1}_{(a,\dot{a})cd}\,,$
$\displaystyle\Big{[}A^{a(\pm)}_{k=1}\Big{]}$ $\displaystyle=-2a^{-a,\pm}\,,$

where $a^{(a,\dot{a})}_{(\alpha,\dot{\alpha})(\beta,\dot{\beta})}$ and
$f^{1}_{(a,\dot{a})cd}$ are the triple overlap coefficients defined in (A.9)
and (A.16).

Let us compare the dictionary for operators of dimension two given in Eq.
(4.24) with the one given in [31]. First, we note that the dictionary for the
operator $\tilde{O}_{2}$ given in [31, Eq.s (5.35)–(5.36)] is already
formulated in the single-particle basis.

Next, we take [31, Eqs. (5.33)–(5.34)] and rotate them to express them in
terms of the geometric quantities $g_{a,\dot{a}}$, $\tilde{g}_{a,\dot{a}}$. We
note that the operator dual to $g_{a,\dot{a}}$ coincides with
$\tilde{\Sigma}_{3}$, thus in this rotated basis the holographic dictionary
for $g_{a,\dot{a}}$ is given in terms of a single-particle operator.

By contrast, the operator dual to $\tilde{g}_{a,\dot{a}}$ is not yet
$\tilde{\Omega}$. The dictionary for $\tilde{g}_{a,\dot{a}}$ at this point
reads:

$\displaystyle\tilde{g}_{a,\dot{a}}$
$\displaystyle=\sqrt{2}\Big{(}-\big{(}f_{12}^{(a,\dot{a})}+f_{52}^{(a,\dot{a})}\big{)}+8a^{c+}a^{d-}f^{1}_{(a,\dot{a})cd}\Big{)}$
(D.5)
$\displaystyle=-(-1)^{a+\dot{a}}\sqrt{6}\bigg{[}\frac{1}{N^{3/2}}\langle\Sigma_{3}^{a\dot{a}}\rangle+\frac{1}{N^{3/2}}\Big{(}\langle\Omega^{a\dot{a}}\rangle-\frac{1}{3}\langle(\Sigma_{2}\cdot\Sigma_{2})^{a\dot{a}}\rangle-\langle(J\cdot\bar{J})^{a\dot{a}}\rangle+\frac{1}{6}\langle(O\cdot
O)^{a\dot{a}}\rangle\Big{)}\bigg{]}.$

We now observe that the right-hand side of the first line of (D.5) differs
from the second-last line of Eq. (D.4) through the terms that involve $f_{51}$
and ${\cal A}_{1}$. These are gauge-fixed versions of gauge-invariant
quantities. The holographic dictionary for $f_{51}$ and ${\cal A}_{1}$ is
obtained by combining Eqs. (D.4) and (4.24). We can thus improve on the
dictionary (D.5) by adding the terms in $f_{51}$ and ${\cal A}_{1}$ to both
sides of the equation. Doing so results precisely in the dictionary given in
Eq. (D.4). This demonstrates the consistency of the results of [31] and the
present work, and is a non-trivial check of the independent methods used in
the two works.

## References

  * [1] O. Lunin and S. D. Mathur, “AdS/CFT duality and the black hole information paradox,” Nucl. Phys. B623 (2002) 342–394, arXiv:hep-th/0109154.
  * [2] O. Lunin, J. M. Maldacena, and L. Maoz, “Gravity solutions for the D1-D5 system with angular momentum,” arXiv:hep-th/0212210.
  * [3] O. Lunin, “Adding momentum to D1-D5 system,” JHEP 04 (2004) 054, arXiv:hep-th/0404006.
  * [4] S. Giusto, S. D. Mathur, and A. Saxena, “Dual geometries for a set of 3-charge microstates,” Nucl. Phys. B701 (2004) 357–379, arXiv:hep-th/0405017.
  * [5] S. Giusto, S. D. Mathur, and A. Saxena, “3-charge geometries and their CFT duals,” Nucl. Phys. B710 (2005) 425–463, arXiv:hep-th/0406103.
  * [6] I. Kanitscheider, K. Skenderis, and M. Taylor, “Fuzzballs with internal excitations,” JHEP 06 (2007) 056, arXiv:0704.0690 [hep-th].
  * [7] S. D. Mathur and D. Turton, “Microstates at the boundary of AdS,” JHEP 05 (2012) 014, arXiv:1112.6413 [hep-th].
  * [8] O. Lunin, S. D. Mathur, and D. Turton, “Adding momentum to supersymmetric geometries,” Nucl.Phys. B868 (2013) 383–415, arXiv:1208.1770 [hep-th].
  * [9] S. Giusto, O. Lunin, S. D. Mathur, and D. Turton, “D1-D5-P microstates at the cap,” JHEP 1302 (2013) 050, arXiv:1211.0306 [hep-th].
  * [10] A. Strominger and C. Vafa, “Microscopic Origin of the Bekenstein-Hawking Entropy,” Phys. Lett. B379 (1996) 99–104, arXiv:hep-th/9601029.
  * [11] J. Breckenridge, R. C. Myers, A. Peet, and C. Vafa, “D-branes and spinning black holes,” Phys.Lett. B391 (1997) 93–98, arXiv:hep-th/9602065 [hep-th].
  * [12] E. J. Martinec and S. Massai, “String Theory of Supertubes,” JHEP 07 (2018) 163, arXiv:1705.10844 [hep-th].
  * [13] E. J. Martinec, S. Massai, and D. Turton, “String dynamics in NS5-F1-P geometries,” JHEP 09 (2018) 031, arXiv:1803.08505 [hep-th].
  * [14] E. J. Martinec, S. Massai, and D. Turton, “Little Strings, Long Strings, and Fuzzballs,” JHEP 11 (2019) 019, arXiv:1906.11473 [hep-th].
  * [15] E. J. Martinec, S. Massai, and D. Turton, “Stringy Structure at the BPS Bound,” JHEP 12 (2020) 135, arXiv:2005.12344 [hep-th].
  * [16] D. Bufalini, S. Iguri, N. Kovensky, and D. Turton, “Black hole microstates from the worldsheet,” arXiv:2105.02255 [hep-th].
  * [17] I. Bena, S. Giusto, R. Russo, M. Shigemori, and N. P. Warner, “Habemus Superstratum! A constructive proof of the existence of superstrata,” JHEP 05 (2015) 110, arXiv:1503.01463 [hep-th].
  * [18] I. Bena, E. Martinec, D. Turton, and N. P. Warner, “Momentum Fractionation on Superstrata,” JHEP 05 (2016) 064, arXiv:1601.05805 [hep-th].
  * [19] I. Bena, S. Giusto, E. J. Martinec, R. Russo, M. Shigemori, D. Turton, and N. P. Warner, “Smooth horizonless geometries deep inside the black-hole regime,” Phys. Rev. Lett. 117 no. 20, (2016) 201601, arXiv:1607.03908 [hep-th].
  * [20] I. Bena, E. Martinec, D. Turton, and N. P. Warner, “M-theory Superstrata and the MSW String,” JHEP 06 (2017) 137, arXiv:1703.10171 [hep-th].
  * [21] I. Bena, S. Giusto, E. J. Martinec, R. Russo, M. Shigemori, D. Turton, and N. P. Warner, “Asymptotically-flat supergravity solutions deep inside the black-hole regime,” JHEP 02 (2018) 014, arXiv:1711.10474 [hep-th].
  * [22] I. Bena, D. Turton, R. Walker, and N. P. Warner, “Integrability and Black-Hole Microstate Geometries,” JHEP 11 (2017) 021, arXiv:1709.01107 [hep-th].
  * [23] S. D. Mathur and D. Turton, “Oscillating supertubes and neutral rotating black hole microstates,” JHEP 1404 (2014) 072, arXiv:1310.1354 [hep-th].
  * [24] I. Bena, S. F. Ross, and N. P. Warner, “On the Oscillation of Species,” JHEP 1409 (2014) 113, arXiv:1312.3635 [hep-th].
  * [25] I. Bena, S. F. Ross, and N. P. Warner, “Coiffured Black Rings,” Class.Quant.Grav. 31 (2014) 165015, arXiv:1405.5217 [hep-th].
  * [26] M. Shigemori, “Superstrata,” Gen. Rel. Grav. 52 no. 5, (2020) 51, arXiv:2002.01592 [hep-th].
  * [27] I. Kanitscheider, K. Skenderis, and M. Taylor, “Holographic anatomy of fuzzballs,” JHEP 04 (2007) 023, arXiv:hep-th/0611171.
  * [28] M. Taylor, “Matching of correlators in AdS(3) / CFT(2),” JHEP 06 (2008) 010, arXiv:0709.1838 [hep-th].
  * [29] S. Giusto, E. Moscato, and R. Russo, “AdS3 holography for 1/4 and 1/8 BPS geometries,” JHEP 11 (2015) 004, arXiv:1507.00945 [hep-th].
  * [30] J. Garcia i Tormo and M. Taylor, “One point functions for black hole microstates,” Gen. Rel. Grav. 51 no. 7, (2019) 89, arXiv:1904.10200 [hep-th].
  * [31] S. Giusto, S. Rawash, and D. Turton, “AdS3 holography at dimension two,” JHEP 07 (2019) 171, arXiv:1904.12880 [hep-th].
  * [32] N. Ceplak, R. Russo, and M. Shigemori, “Supercharging Superstrata,” JHEP 03 (2019) 095, arXiv:1812.08761 [hep-th].
  * [33] P. Heidmann and N. P. Warner, “Superstratum Symbiosis,” JHEP 09 (2019) 059, arXiv:1903.07631 [hep-th].
  * [34] P. Heidmann, D. R. Mayerson, R. Walker, and N. P. Warner, “Holomorphic Waves of Black Hole Microstructure,” JHEP 02 (2020) 192, arXiv:1910.10714 [hep-th].
  * [35] D. R. Mayerson, R. A. Walker, and N. P. Warner, “Microstate Geometries from Gauged Supergravity in Three Dimensions,” arXiv:2004.13031 [hep-th].
  * [36] I. R. Klebanov and E. Witten, “AdS / CFT correspondence and symmetry breaking,” Nucl. Phys. B 556 (1999) 89–114, arXiv:hep-th/9905104.
  * [37] K. Skenderis and M. Taylor, “Kaluza-Klein holography,” JHEP 05 (2006) 057, arXiv:hep-th/0603016 [hep-th].
  * [38] S. D. Mathur, “The fuzzball proposal for black holes: An elementary review,” Fortsch. Phys. 53 (2005) 793–827, arXiv:hep-th/0502050.
  * [39] K. Skenderis and M. Taylor, “The fuzzball proposal for black holes,” Phys. Rept. 467 (2008) 117–171, arXiv:0804.0552 [hep-th].
  * [40] S. D. Mathur, “Black Holes and Beyond,” Annals Phys. 327 (2012) 2760–2793, arXiv:1205.0776 [hep-th].
  * [41] I. Bena and N. P. Warner, “Resolving the Structure of Black Holes: Philosophizing with a Hammer,” arXiv:1311.4538 [hep-th].
  * [42] O. Aharony, S. S. Gubser, J. M. Maldacena, H. Ooguri, and Y. Oz, “Large N field theories, string theory and gravity,” Phys. Rept. 323 (2000) 183–386, arXiv:hep-th/9905111.
  * [43] J. R. David, G. Mandal, and S. R. Wadia, “Microscopic formulation of black holes in string theory,” Phys. Rept. 369 (2002) 549–686, arXiv:hep-th/0203048.
  * [44] G. Arutyunov and S. Frolov, “Some cubic couplings in type IIB supergravity on AdS(5) x S**5 and three point functions in SYM(4) at large N,” Phys. Rev. D 61 (2000) 064009, arXiv:hep-th/9907085.
  * [45] E. D’Hoker, D. Z. Freedman, S. D. Mathur, A. Matusis, and L. Rastelli, “Extremal correlators in the AdS / CFT correspondence,” arXiv:hep-th/9908160 [hep-th].
  * [46] G. Arutyunov and S. Frolov, “On the correspondence between gravity fields and CFT operators,” JHEP 04 (2000) 017, arXiv:hep-th/0003038 [hep-th].
  * [47] E. D’Hoker, J. Erdmenger, D. Z. Freedman, and M. Perez-Victoria, “Near extremal correlators and vanishing supergravity couplings in AdS / CFT,” Nucl. Phys. B 589 (2000) 3–37, arXiv:hep-th/0003218.
  * [48] 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.
  * [49] L. I. Uruchurtu, “Next-next-to-extremal Four Point Functions of N=4 1/2 BPS Operators in the AdS/CFT Correspondence,” JHEP 08 (2011) 133, arXiv:1106.0630 [hep-th].
  * [50] L. Rastelli and X. Zhou, “How to Succeed at Holographic Correlators Without Really Trying,” JHEP 04 (2018) 014, arXiv:1710.05923 [hep-th].
  * [51] F. Aprile, J. Drummond, P. Heslop, and H. Paul, “Double-trace spectrum of $N=4$ supersymmetric Yang-Mills theory at strong coupling,” Phys. Rev. D 98 no. 12, (2018) 126008, arXiv:1802.06889 [hep-th].
  * [52] F. Aprile, J. Drummond, P. Heslop, H. Paul, F. Sanfilippo, M. Santagata, and A. Stewart, “Single Particle Operators and their Correlators in Free $\mathcal{N}=4$ SYM,” arXiv:2007.09395 [hep-th].
  * [53] J. M. Maldacena, “The large N limit of superconformal field theories and supergravity,” Adv. Theor. Math. Phys. 2 (1998) 231–252, arXiv:hep-th/9711200.
  * [54] C. Vafa, “Gas of D-branes and Hagedorn density of BPS states,” Nucl. Phys. B463 (1996) 415–419, arXiv:hep-th/9511088 [hep-th].
  * [55] S. Deger, A. Kaya, E. Sezgin, and P. Sundell, “Spectrum of D = 6, N = 4b supergravity on AdS(3) x S(3),” Nucl. Phys. B536 (1998) 110–140, arXiv:hep-th/9804166 [hep-th].
  * [56] F. Larsen, “The Perturbation spectrum of black holes in N=8 supergravity,” Nucl. Phys. B536 (1998) 258–278, arXiv:hep-th/9805208 [hep-th].
  * [57] J. de Boer, “Six-dimensional supergravity on S**3 x AdS(3) and 2-D conformal field theory,” Nucl. Phys. B548 (1999) 139–166, arXiv:hep-th/9806104 [hep-th].
  * [58] F. Larsen and E. J. Martinec, “U(1) charges and moduli in the D1-D5 system,” JHEP 06 (1999) 019, arXiv:hep-th/9905064.
  * [59] L. Eberhardt, M. R. Gaberdiel, and R. Gopakumar, “Deriving the AdS3/CFT2 correspondence,” JHEP 02 (2020) 136, arXiv:1911.00378 [hep-th].
  * [60] M. R. Gaberdiel, R. Gopakumar, B. Knighton, and P. Maity, “From Symmetric Product CFTs to ${\rm AdS}_{3}$,” arXiv:2011.10038 [hep-th].
  * [61] M. Shigemori, “Counting superstrata,” Journal of High Energy Physics 2019 no. 10, (Oct, 2019) . http://dx.doi.org/10.1007/JHEP10(2019)017.
  * [62] S. G. Avery, “Using the D1D5 CFT to Understand Black Holes,” arXiv:1012.0072 [hep-th].
  * [63] M. Baggio, J. de Boer, and K. Papadodimas, “A non-renormalization theorem for chiral primary 3-point functions,” JHEP 07 (2012) 137, arXiv:1203.1036 [hep-th].
  * [64] B. Chakrabarty, D. Turton, and A. Virmani, “Holographic description of non-supersymmetric orbifolded D1-D5-P solutions,” JHEP 11 (2015) 063, arXiv:1508.01231 [hep-th].
  * [65] M. Shigemori, “Interpolating between multi-center microstate geometries,” arXiv:2105.11639 [hep-th].
  * [66] L. Romans, “Selfduality for Interacting Fields: Covariant Field Equations for Six-dimensional Chiral Supergravities,” Nucl. Phys. B 276 (1986) 71.
  * [67] S. Giusto, L. Martucci, M. Petrini, and R. Russo, “6D microstate geometries from 10D structures,” Nucl.Phys. B876 (2013) 509–555, arXiv:1306.1745 [hep-th].
  * [68] V. Balasubramanian, J. de Boer, E. Keski-Vakkuri, and S. F. Ross, “Supersymmetric conical defects: Towards a string theoretic description of black hole formation,” Phys. Rev. D64 (2001) 064011, arXiv:hep-th/0011217.
  * [69] J. M. Maldacena and L. Maoz, “De-singularization by rotation,” JHEP 12 (2002) 055, arXiv:hep-th/0012025.
  * [70] A. Jevicki, M. Mihailescu, and S. Ramgoolam, “Gravity from CFT on S**N(X): Symmetries and interactions,” Nucl.Phys. B577 (2000) 47–72, arXiv:hep-th/9907144 [hep-th].
  * [71] O. Lunin and S. D. Mathur, “Three-point functions for M(N)/S(N) orbifolds with N = 4 supersymmetry,” Commun. Math. Phys. 227 (2002) 385–419, arXiv:hep-th/0103169.
  * [72] M. Mihailescu, “Correlation functions for chiral primaries in d = 6 supergravity on ads3 × s3,” Journal of High Energy Physics 2000 no. 02, (Jan, 2000) 007–007. http://dx.doi.org/10.1088/1126-6708/2000/02/007.
  * [73] G. Arutyunov, A. Pankiewicz, and S. Theisen, “Cubic couplings in D = 6 N=4b supergravity on AdS(3) x S**3,” Phys. Rev. D 63 (2001) 044024, arXiv:hep-th/0007061.
  * [74] P. Yang, Y. Jiang, S. Komatsu, and J.-B. Wu, “D-branes and Orbit Average,” arXiv:2103.16580 [hep-th].
  * [75] P. McFadden and K. Skenderis, “Holographic non-gaussianity,” Journal of Cosmology and Astroparticle Physics 2011 no. 05, (May, 2011) 013–013. http://dx.doi.org/10.1088/1475-7516/2011/05/013.
  * [76] M. Bruni, S. Matarrese, S. Mollerach, and S. Sonego, “Perturbations of spacetime: gauge transformations and gauge invariance at second order and beyond,” Classical and Quantum Gravity 14 no. 9, (Sep, 1997) 2585–2606. http://dx.doi.org/10.1088/0264-9381/14/9/014.
  * [77] J. Hansen and P. Kraus, “Generating charge from diffeomorphisms,” JHEP 0612 (2006) 009, arXiv:hep-th/0606230 [hep-th].
  * [78] O. Lunin and S. D. Mathur, “Correlation functions for M(N)/S(N) orbifolds,” Commun. Math. Phys. 219 (2001) 399–442, arXiv:hep-th/0006196.
  * [79] A. Galliani, S. Giusto, E. Moscato, and R. Russo, “Correlators at large c without information loss,” Journal of High Energy Physics 2016 no. 9, (Sep, 2016) . http://dx.doi.org/10.1007/JHEP09(2016)065.
  * [80] A. Galliani, S. Giusto, and R. Russo, “Holographic 4-point correlators with heavy states,” JHEP 10 (2017) 040, arXiv:1705.09250 [hep-th].
  * [81] A. Bombini, A. Galliani, S. Giusto, E. Moscato, and R. Russo, “Unitary 4-point correlators from classical geometries,” Eur. Phys. J. C78 no. 1, (2018) 8, arXiv:1710.06820 [hep-th].
  * [82] A. Bombini and A. Galliani, “AdS3 four-point functions from $\frac{1}{8}$-BPS states,” arXiv:1904.02656 [hep-th].
  * [83] S. Giusto, R. Russo, and C. Wen, “Holographic correlators in AdS3,” JHEP 03 (2019) 096, arXiv:1812.06479 [hep-th].
  * [84] L. Rastelli, K. Roumpedakis, and X. Zhou, “Ads3× s3 tree-level correlators: hidden six-dimensional conformal symmetry,” Journal of High Energy Physics 2019 no. 10, (Oct, 2019) . http://dx.doi.org/10.1007/JHEP10(2019)140.
  * [85] S. Giusto, R. Russo, A. Tyukov, and C. Wen, “Holographic correlators in AdS3 without Witten diagrams,” JHEP 09 (2019) 030, arXiv:1905.12314 [hep-th].
  * [86] S. Giusto, R. Russo, A. Tyukov, and C. Wen, “The CFT6 origin of all tree-level 4-point correlators in AdS${}_{3}\times S^{3}$,” Eur. Phys. J. C 80 no. 8, (2020) 736, arXiv:2005.08560 [hep-th].
  * [87] F. Aprile and M. Santagata, “Two-particle spectrum of tensor multiplets coupled to $AdS_{3}\times S^{3}$ gravity,” arXiv:2104.00036 [hep-th].
  * [88] S. Giusto, M. R. R. Hughes, and R. Russo, “The Regge limit of AdS3 holographic correlators,” JHEP 11 (2020) 018, arXiv:2007.12118 [hep-th].
  * [89] N. Ceplak and M. R. R. Hughes, “The Regge limit of AdS3 holographic correlators with heavy states: towards the black hole regime,” arXiv:2102.09549 [hep-th].
  * [90] N. Ceplak, S. Giusto, M. R. R. Hughes, and R. Russo, “Holographic correlators with multi-particle states,” arXiv:2105.04670 [hep-th].
  * [91] S. Ribault, “Conformal field theory on the plane,” arXiv:1406.4290 [hep-th].
  * [92] W. Black, R. Russo, and D. Turton, “The Supergravity fields for a D-brane with a travelling wave from string amplitudes,” Phys.Lett. B694 (2010) 246–251, arXiv:1007.2856 [hep-th].
  * [93] S. Giusto, R. Russo, and D. Turton, “New D1-D5-P geometries from string amplitudes,” JHEP 11 (2011) 062, arXiv:1108.6331 [hep-th].
  * [94] I. Bena, S. Giusto, M. Shigemori, and N. P. Warner, “Supersymmetric Solutions in Six Dimensions: A Linear Structure,” JHEP 1203 (2012) 084, arXiv:1110.2781 [hep-th].
  * [95] J. B. Gutowski, D. Martelli, and H. S. Reall, “All supersymmetric solutions of minimal supergravity in six dimensions,” Class. Quant. Grav. 20 (2003) 5049–5078, arXiv:hep-th/0306235.