# Modulation Instability and Wavenumber Bandgap Breathers in a Time Layered
Phononic Lattice

Christopher Chong  Department of Mathematics, Bowdoin College, Brunswick,
Maine 04011  Brian Kim  Department of Mechanical and Civil Engineering,
California Institute of Technology, Pasadena, CA 91125, USA.  Evelyn Wallace
Department of Mathematics, Bowdoin College, Brunswick, Maine 04011  Chiara
Daraio  Department of Mechanical and Civil Engineering, California Institute
of Technology, Pasadena, CA 91125, USA.

###### Abstract

We demonstrate the existence of wavenumber bandgap (q-gap) breathers in a
time-periodic phononic lattice. These breathers are localized in time and
periodic in space, and are the counterparts to the classical breathers found
in space-periodic systems. We derive an exact condition for modulation
instability that leads to the opening of wavenumber bandgaps. The q-gap
breathers become more narrow and larger in amplitude as the wavenumber goes
further into the bandgap. In the presence of damping, these structures acquire
a non-zero, oscillating tail. The experiment and model exhibit qualitative
agreement.

##  I Introduction

The classical discrete breather is defined as a spatially localized, time-
periodic solution of a nonlinear lattice differential equation. They are a
fundamental structure found in many platforms, including photonics, phononics,
and in electrical systems, see [1, 2, 3] for comprehensive reviews. One
mechanism through which breathers can manifest is the modulation instability
(MI) of plane waves in spatially periodic lattices [4]. Such breathers have a
frequency that falls into a spectral gap [1].

A natural counterpart to the fundamentally important discrete breather is the
so-called wavenumber bandgap (q-gap) breather. It is localized in time,
periodic in space, and has wavenumber that falls into a q-gap. Q-gap breathers
represent a newer class of solutions and are distinct from q-breathers, which
are localized in wavenumber and periodic in time [5]. Since q-gap breathers
have the roles of time and space switched when compared to classical
breathers, it is natural to consider lattices that are time varying (instead
of space varying). Recent advances in experimental platforms for time varying
systems makes the topic particularly relevant, including studies in photonic
[6, 7, 8, 9], electric [10, 11, 12], and phononic systems [13, 14, 15, 16, 17,
18, 19, 15, 20, 21, 22, 23, 24, 25]. Time localization can also arise via
mechanisms other than a wavenumber bandgap, such as zero-wavenumber gain
modulation instability [26]. The so-called Akhmediev breathers of the
nonlinear Schrödinger equation [27], and its discrete integrable counter-part,
the Ablowtiz-Ladik lattice [28], are also localized in time. They do not have
wavenumber in a q-gap, and hence are distinct from q-gap breathers. Very
recently, it was shown theoretically that time-localized solitons with
wavenumbers in the q-gap exist in a time-periodic photonic crystal [29].
Breathers in time varying phononic systems, however, remain unexplored.

In the present letter, we employ numerical, analytical, and experimental
approaches to explore q-gap breathers and the modulation instability leading
to the opening of wavenumber bandgaps. Q-gap breathers could be exploited for
the creation of phononic frequency combs [30, 31, 32], energy harvesting
applications [33, 34], or acoustic signal processing [35].

##  II Experimental and Model Set-up

The particular phononic lattice under consideration is a system of repelling
magnetic masses with grounding stiffness controlled by electrical coils driven
applied voltage signals that are given by a time-periodic step function. The
experimental setup is adapted form the platform developed in [15, 25]. The
chain is composed of $N-1$ ring magnets (K&J Magnetic, Inc., P/N R848) lined
with sleeve bearings (McMaster-Carr P/N 6377K2) comprising the uniform masses,
arranged with alternating polarity on a smooth rod (McMaster-Carr P/N
8543K28). Electromagnetic coils (APW Company SKU: FC-6489) are fixed
concentrically around the equilibrium positions of each of the innermost eight
masses, such that they may exert a restoring force on each mass proportional
to the current induced by applied voltage step-function (Aglient 33220A, Accel
Instruments TS250-2). The velocity of each mass is measured using laser
Doppler vibrometer (Polytec CLV-2534), repeating experiments to a construct
full velocity field for the lattice. Figure 1 shows a schematic of the
experimental setup.

Figure 1: Schematic of the modulated magnetic lattice.

The system is modeled as Fermi-Pasta-Ulam-Tsingou type lattice [15, 25]

$\displaystyle
M\ddot{u}_{n}+k(t)\,u_{n}+c\dot{u}_{n}=F(u_{n}-u_{n-1})-F(u_{n+1}-u_{n})$ (1)

where $u_{n}$ is the displacement of the $n$th ring magnet from its
equilibrium position, where the equilibrium distance between adjacent magnets
is $d=0.0334$ m. The indices run from $n=1\ldots N-1$ and we consider fixed
boundary conditions $u_{0}(t)=u_{N}(t)=0$. All ring magnets have uniform mass
$M=0.0097$ kg. Dissipative forces are modeled with a phenomenological viscous
damping term $c\frac{d{u}_{n}}{dt}$, where the damping coefficient $c=0.15$
Ns/m is determined empirically by matching the simulated and experimental
spatial decay of the velocity amplitude envelope of waves traveling through
the lattice [25]. The coupling force term is defined using the repulsive
magnetic force between neighboring masses. The experimentally measured force-
distance relation between neighboring masses is fit with a dipole-dipole
approximation, as in [15], which is given by $F(x)=A(d+x)^{-\alpha}$, where
$x$ is the center-to-center distance between masses with $A=9.044\times
10^{-7}\text{Nm${}^{4}$}$ and $\alpha=4$. The current resulting from a
periodic step function voltage applied to the electromagnetic coils induces a
magnetic field that provides a grounding stiffness modulation of the form

$k(t)=\left\\{\begin{array}[]{cc}k_{a}&0\leq t<\tau T\\\ k_{b}&\tau T\leq
t<T\end{array}\right.\,,\quad k(t)=k(t+T)$ (2)

The step values $k_{a},k_{b}$ and duty-cycle $0<\tau<1$ ($[\tau]=\text{s}$)
are parameters. Unless otherwise stated, we use $k_{a}=0$, $k_{b}=150$ N/m and
$\tau=0.5$ s. We will use the modulation frequency $f_{\rm mod}=1/T$ Hz as the
main system parameter to be varied.

##  III Modulation Instability

To find breathers, we first need to determine the wavenumber bandgap. This is
achieved by computing the stability of plane waves (i.e., the modulation
stability) of the linearized model

$M\ddot{u}_{n}=K(u_{n-1}-2u_{n}+u_{n+1})-k(t)u_{n}-c\dot{u}_{n},$ (3)

where $K=\alpha Ad^{\alpha-1}$. For time-independent stiffness $(k(t)=0)$ the
undamped ($c=0$) linear equation has the dispersion relationship $\omega_{\rm
disp}^{2}(q)=4K/M\sin^{2}(q/2)$ such that the linear spectrum extends from
$[0,\sqrt{K/M}]$ and all plane waves are stable.

(a) (b)

Figure 2:  (a) Plot of $f=\omega/(2\pi)$ where $\omega$ is the imaginary part
of the Floquet exponent (blue curve) and $\sigma$, the real part of the
exponent (red curve) for an infinite lattice. The black dots show the
corresponding values for a finite sized lattice with $N=11$. The modulation
frequency is $f_{\rm mod}=1/T=45$ Hz. The shaded gray region indicates the
region of instability (i.e., the wavenumber bandgap). (b) Plot of the function
$G_{D}$ from Eq. (7) in the infinite lattice (solid blue line) and in the
$N=11$ lattice (solid black dots). The plane wave is unstable when
$|G_{D}|>1$, which is highlighted by the gray region. The inset shows the
corresponding Floquet multipliers in the complex plane. There are two
multipliers lying outside the unit circle (also shown) demonstrating the
instability of a general solution.

In the case of time-dependent stiffness $(k(t)\neq 0)$ a gap in the wavenumber
axis $q$ is possible. For general time-periodic stiffness $k(t)$ with period
$T=2\pi/\omega_{\rm mod}$, a wavenumber bandgap will open where the dispersion
curve $\omega_{\rm disp}(q)$ intersects itself when translated by an integer
multiple of half the modulation frequency $\omega_{\rm mod}/2$ [36]. The
advantage of considering $k(t)$ to be a periodic step function is that the
modified dispersion relation can be computed exactly. Making the ansatz
$u_{n}(t)=X_{m}(n)\cdot\Theta_{m}(t)$, one finds upon substitution into Eq.
(3) and enforcing Dirichlet boundary conditions that the eigenfunctions are
$X_{m}(n)=\sin\left(q_{m}n\right)$ where the wavenumber is $q_{m}=m\pi/N$ with
$m=1,\ldots,N-1$. In the infinite lattice, $q\in[0,\pi]$. The associated
eigenvalues are $\lambda_{m}=\sin^{2}\left(q_{m}/2\right)$. The temporal part
$\Theta(t)$ satisfies

$\displaystyle M\ddot{\Theta}_{m}$ $\displaystyle=$
$\displaystyle-(\lambda_{m}K+k(t))\Theta_{m}-c\dot{\Theta}_{m}.$ (4)

We can obtain an exact solution of this equation (and dispersion relation and
stability condition), by adapting a procedure carried out in the context of an
undamped Kronig-Penney photonic lattice [37, 38]. The general solution of Eq.
(4) will be a superposition of functions of the form
$\Theta_{m}(t)=H_{m}(t)e^{\mu_{m}t}$ where $H(t)$ has period $T$ and
$\mu_{m}=\sigma_{m}+i\omega_{m}$ is the Floquet exponent where
$\sigma_{m},\omega_{m}\in\mathbb{R}$. The Floquet multiplier is
$e^{\mu_{m}T}$. The waveform associated to the wavenumber $q_{m}$ will be
stable if $\sigma_{m}\leq 0$, or equivalently, if the Floquet multiplier has
modulus less than unity, $|e^{\mu_{m}T}|\leq 1$. Substitution of
$\Theta_{m}(t)=H_{m}(t)e^{\mu_{m}t}$ into equation (4) and demanding that
$H_{m}(t)$ and $\dot{H}_{m}(t)$ are continuous at $t=0$ and $t=\tau$ leads to
the following equations (detailed in the Supplemental Material)

$\displaystyle G$ $\displaystyle=$
$\displaystyle\cosh\left[\left(\sigma_{m}+\frac{c}{2M}\right)T\right]\cos(\omega_{m}T)$
(5) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\sinh\left[\left(\sigma_{m}+\frac{c}{2M}\right)T\right]\sin(\omega_{m}T)$
(6)

where $G$ depends on the wavenumber and system parameters, but not the Floquet
exponent $\mu_{m}$:

$\displaystyle G\equiv$
$\displaystyle-\frac{s(k_{a})^{2}+s(k_{b})^{2}}{2s(k_{a})s(k_{b})}\sin(s(k_{a})\tau
T)\sin(s(k_{b})(1-\tau)T)$ $\displaystyle+\cos(s(k_{a})\tau
T)\cos(s(k_{b})(1-\tau)T)$

where $s(k)=\sqrt{4M\left(\lambda_{m}\cdot K+k\right)-c^{2}}/\left(2M\right)$.
These equations allow for the exact computation of the Floquet exponents
$\mu_{m}=\sigma_{m}+i\omega_{m}$. An example plot is shown in Fig. 2(a). If
$|G|\leq 1$ then $\sigma_{m}=-c/(2M)$ and $\omega_{m}=\cos^{-1}(G)/T$. In this
case the underlying solution is stable. If $\pm G>1$ then $\omega_{m}=(3\pm
1)\pi/(2T)$, implying that the imaginary part of the Floquet exponent is an
integer multiple of half the modulation frequency. In this case, the real part
of the Floquet exponent is $\sigma_{m}=\pm\cosh^{-1}\left(\mp
G\right)/T-c/(2M)$ which implies the following condition for stability,

$\left|G_{D}\right|\leq 1,\qquad G_{D}\equiv
G\,\mathrm{sech}\left(\frac{cT}{2M}\right).$ (7)

Note that this expression is exact and gives an efficient way to check for
stability via direct substitution of the system parameters into $G_{D}$ and
simply checking the inequality. A plot of $G_{D}$ is shown in Fig. 2(b). The
linear stability predictions agree well with experimental observations
(examples given in the Supplemental Material). The set of wavenumbers where
$|G_{D}|>1$ make up the so-called wavenumber bandgap. The edges can be found
by solving $G_{D}=\pm 1$. See the gray regions of Fig. 2 for example
wavenumber bandgaps.

(a) (b) (c)

Figure 3: Wavenumber bandgap breather for $f_{\rm mod}=45$ Hz. (a) Intensity
plot of velocity after initializing the unstable plane wave with wavenumber
$q_{5}$ with $c=0$. (b) Time series of the velocity of node 6 of panel (a).
The dashed line shows the best-fit envelope (in the least square sense). The
red dots are the solution sampled every $2T$ seconds. The insets show the
spatial Fourier transform before (left) and after (right) the turning point.
(c) Plot of the Poincaré map of the solution shown as red dots in panel (b).
The yellow squares and blue circles correspond to the same simulation with
$c=0.075$ Ns/m and $c=0.15$ Ns/m, respectively.

##  IV Wavenumber Bandgap Breathers

To draw analogy to the classic breathers of space-periodic systems, we first
consider Eq. (1) without damping $(c=0)$. To generate a wavenumber bandgap
breather, Eq. (1) is initialized with an unstable plane wave, e.g.,
$u_{n}(0)=a\sin\left(q_{m}n\right)$, where $q_{m}\in(q_{\ell},q_{r})$ where
$(q_{\ell},q_{r})$ is the wavenumber bandgap, $0<a\ll 1$, and the initial
velocity $v_{n}(0)$ is determined via Eq. (4). An example is shown in Fig. 3
with $f_{\rm mod}=45$ Hz, $a=10^{-4}$ and wavenumber $q_{5}=5\pi/11$, which
falls into the gap $(q_{\ell},q_{r})\approx(0.93,1.52)$. The Floquet exponent
associated to this wavenumber is
$\mu_{5}=\sigma_{5}+i\omega_{5}=4.8107+i\omega_{\rm mod}/2$. The corresponding
Floquet multiplier is shown in the inset of Fig. 2(b) (the smaller of the two
multipliers lying outside the unit circle).

The solution initially grows exponentially with growth rate given by
$\sigma_{5}$, due to the modulation instability, but reaches a turning point
and then decays with the rate $-\sigma_{5}$. This happens uniformly within the
lattice, as shown by the intensity plot in Fig. 3(a). Figure 3(b) shows the
time series of the velocity of the 6th node, i.e., $\dot{u}_{6}(t)=v_{6}(t)$
(solid blue curve). Both panels (a) and (b) demonstrate that the dynamics are
localized in time. Spatial periodicity of the solution is imposed by
construction due to the finite length of the lattice with zero boundary
conditions. The role of space and time have been switched when compared to the
classic breathers of space-periodic systems. Thus, the solution shown in Fig.
3 is the so-called called wavenumber bandgap breather. Motivated by the fact
that the envelope of a breather of a space-periodic FPUT lattice is described
by a soliton of the Nonlinear Schrödinger (NLS) equation (in the limit of the
temporal frequency approaching the band edge from within the spectral gap), we
fit the velocity profile $v_{6}$ with a function of the form
$\beta_{1}\mathrm{sech}\left(\sigma_{5}(t-\beta_{2})\right)$ where $\beta_{j}$
are fitting parameters and $\sigma_{5}$ is the real part of the Floquet
exponent. See the gray dashed line of Fig. 3(b). The good agreement between
the velocity profile and the fit envelope function confirms that the
growth/decay rate is indeed given by the real part of the associated Floquet
exponent, in this case $\sigma_{5}$. To better understand the mechanism behind
the formation of the wavenumber bandgap breather, we construct a Poincaré map
of the dynamics by sampling the solution with the frequency associated with
the breather, namely $\omega_{\rm mod}/2$. This corresponds to twice the
period of the modulation. Thus, the map will be of the form
$\mathbf{F}_{j}(\mathbf{u}^{0})=\mathbf{u}(2Tj)$, where $j$ is an integer, $T$
is the period of $k(t)$, and
$\mathbf{u}=(u_{1},u_{2},\ldots,u_{N-1},v_{1},v_{2},\ldots,v_{N-1})\in\mathbb{R}^{2(N-1)}$
is vector valued solution of Eq. (1) with initial value $\mathbf{u}^{0}$. As
in the simulation shown in Fig. 3(b), the initial value of the map is given by
an unstable plane wave (e.g., with wavenumber $q_{5}$). The red dots of Fig.
3(b) show values of the map $\mathbf{F}$ that correspond to $v_{6}$. The red
dots of Fig. 3(c) show values of $\mathbf{F}$ in the $(u_{6},v_{6})$ phase
plane. The eigenvector corresponding to the unstable (stable) Floquet exponent
$\mu_{5}$ ($-\mu_{5}$) is shown in red (blue). The gray line of Fig. 3(c) is
obtained by repeatedly generating the map $\mathbf{F}$ in the $(u_{6},v_{6})$
plane for various (small) multiples of the initial value $\mathbf{u}^{0}$. The
origin is a saddle type fixed point, and the trajectory forms a near
homoclinic orbit, made possible by the nonlinearity of the system. The orbit
is not exactly homoclinic, since it does not approach the origin via the
stable eigenvector as $t\rightarrow\infty$. Indeed, as can also be inferred
from Fig. 3(b), the solution does not decay to zero, but rather it experiences
small oscillations. This is due to the existence of other modes in the system
(e.g., ones with associated multipliers lying on the unit circle), which are
excited during the dynamic evolution. The insets of Fig. 3(b) show a
normalized spatial Fourier transform of the signal before (the left inset) and
after (the right inset) the maximum velocity is attained. In particular, the
quantity $|\hat{v}|/|\max\hat{v}|$ is shown against the wavenumber, where
$\hat{v}(q)=\frac{2}{N-1}\sum_{n}v_{n}(t)\sin(nq)$ where $t=0.66$s and
$t=1.02$s are the times used to compute the transform before and after the
turning point, respectively. Before the turning point the only prominent
wavenumber is the one associated to the initial value (in this case
$q_{5}=5\pi/11$). After the turning point, there is an additional mode excited
that lies outside the wavenumber bandgap (in this case $q_{3}=3\pi/11$). It is
this mode that is primarily responsible for the non-zero oscillations at the
tail of the breather. If one imposes the additional criterion that a breather
must have tails decaying to zero, then strictly speaking, the structure found
here would be a generalized breather, since the orbit is not exactly
homoclinic. Classic breathers with tails that do not decay to zero are common
in non-integrable systems, and have sometimes been referred to as generalized
breathers [39]. Over longer time windows, the amplitude of the signal can grow
again (leading to a repeated appearance of breathers), but eventually the
structure typically breaks down, leading to chaotic type dynamics for long-
time simulations. Similar observations have been made for k-gap solitons in
photonic systems [29]. The existence of perfectly homoclinic solutions in this
system is an important question, but it lies beyond the scope of the present
article. Inspection of the long time dynamics, as well as breathers for other
parameter values, are provided in the Supplemental Material. We now generate a
family of wavenumber bandgap breathers parameterized by the distance the
underlying wavenumber is from the band edge $q_{r}$. This is a natural
parameter to consider, as the distance to the bandedge determines breather
width and amplitude in space-periodic systems [1]. Keeping all parameters
fixed, but gradually varying the modulation frequency $f_{\rm mod}$ has the
effect of shifting the bandgap in the wavenumber axis. Thus, we fix the
wavenumber ($q_{5}$ in this case), whose distance to the right edge will
increase as the modulation frequency is increased. For the breather shown in
Fig. 3 the distance to the edge is $\Delta q=q_{r}-q_{5}\approx 0.09$. The red
dots of Fig. 4(a) show the amplitude of the breather vs. $\Delta
q=q_{r}(f_{\rm mod})-q_{5}$. The amplitude is computed as the maximum velocity
of the 6th node, i.e., $\max_{t}\|v_{6}(t)\|$. For $\Delta q>0.4$ the
breathers do not form a coherent localization, like in Fig. 3. Similar
simulations with wavenumber near the left edge of the bandgap ($q_{5}\approx
q_{\ell}$) did not lead to the robust formation of breathers. The amplitude
data is fit with a function of the form $\gamma_{1}\Delta q^{\gamma_{2}}$,
with the best fit values being $\gamma_{1}=3.34$ and $\beta_{2}=0.57$ (the
solid line in Fig. 4(a)). This is consistent with the trend found for classic
breathers in space-periodic systems where it is well known that the breather
amplitude grows like $\mathcal{O}\left(\sqrt{\Delta\omega}\right)$, where
$\Delta\omega$ is the difference between the breather frequency and the edge
of the frequency spectrum [1].

(a) (b) (c)

Figure 4: (a) Plot of the breather amplitude for the $c=0$ (red dots) and
$c=0.15$ Ns/m (blue circles) simulations vs. distance of the wavenumber to the
bandedge, $\Delta q$. The lines show the best fit function of the form
$\beta_{1}\Delta q^{\beta_{2}}$. The markers with error bars are the measured
amplitude from the experiment. The solid red squares and open blue squares
show the theoretical HWHM for the $c=0$ and $c=0.15$ Ns/m cases, respectively.
The black triangles are the experimentally measured HWHM. (b) Time series
example for the $c=0.15$ Ns/m simulation with $\Delta q=0.147$. The dashed
line shows the best-fit envelope of the solution up until the maximum. The
insets show the spatial Fourier transform before (left) and after (right) the
turning point. (c) Same as panel (b), but for the experiment.

Now that we have established the existence of wavenumber bandgap breathers in
Eq. (1), we now consider the role of damping, which will bring us closer to
the experimentally relevant situation. Breathers in experiments will always be
a dissipative analog of breathers in lossless models. Thus, we modify our
definition of a breather to account for dissipative effects. To motivate our
modified definition, we repeat the simulations shown in Fig. 3 but with a
nonzero damping parameter. The yellow square (blue circle) markers of Fig.
3(c) show the orbit with a damping parameter of $c=0.075$ ($c=0.15$) Ns/m. The
orbit starts close to being homoclinic, but the dynamics are attracted to a
stable fixed point (i.e., a time-periodic orbit of the original system with
period $2T$). Note that the time-dependent stiffness term in the model (with
the $k(t)$ coefficient) acts as an effective gain term to balance loss. The
orbit in the damped system experiences an initial exponential growth and a
turning point, like the lossless breather, but rather than approaching near
zero amplitude, the dynamics tend to a stable fixed point. Thus, the left tail
of the “damped breather” (in the time domain) is much like a lossless
breather, whose amplitude is slightly lower due to the presence of damping.
The right tail of the “damped breather” approaches a periodic oscillation,
whose amplitude is not necessarily small relative to the amplitude of the
breather. See Fig. 4(b) for an example time series with $c=0.15$ Ns/m. An
alternative classification for the structure found in the damped system would
be a wavenumber bandgap “front”, since the solution is heteroclinic, as it
connects the unstable zero fixed point to a non-zero stable fixed point.
Important for the present study, however, is that the transition between the
zero and nonzero state is approximately described by the lossless breather
(i.e., the near homoclinic orbit). We measure the amplitude of the structure
in the same way we measured the breather amplitude in the lossless system,
i.e., $\max_{t}\|v_{6}(t)\|$. We repeat this for various modulation
frequencies (and hence $\Delta q$ values) for the damping value $c=0.15$ Ns/,
which is shown as the blue circle markers in Fig. 4(a). The qualitative
amplitude trend is similar to the lossless case, but the amplitude is
decreased. The amplitude data in the damped case is also fit with a function
of the form $\gamma_{1}\Delta q^{\gamma_{2}}$, with the best fit values being
$\gamma_{1}=2.62$ and $\beta_{2}=0.59$.

We now turn to the experimental construction of wavenumber bandgap breathers
(using the dissipative definition of a breather defined above). To excite a
plane wave with a particular wavenumber (in this case $q_{5}$) the unmodulated
system is driven with the frequency $\omega_{\rm disp}(q_{5})$. Once the
desired plane wave is excited, the initial driving is turned off and the
modulation is turned on simultaneously. Like in the damped simulation, the
amplitude will initially grow, reach a turning point, decay, but eventually
approach a periodic orbit (or exhibit chaotic behavior). An example
experimental time series is shown in Fig. 4(c). Notice the qualitative
agreement to the theoretical prediction shown in Fig. 4(b). Since the band
edge can only be computed in the infinite lattice, we use the relation $\Delta
q=q_{r}(f_{\rm mod})-q_{5}$, where $q_{r}(f_{\rm mod})$ is found analytically.
In general, there will be a mistuning between the experimental and model
dynamics for a fixed modulation frequency. We account for this mistuning by
measuring the modulation frequency the mode $q_{5}$ becomes unstable in the
model and experiment. The difference between the experimental and theoretical
critical modulation frequencies, $\delta f_{\rm mod}$, is then added to the
experimental modulation frequencies when calculating the distance to the band
edge, namely $\Delta q=q_{r}(f_{\rm mod}-\delta f_{\rm mod})-q_{5}$. Thus,
$\Delta q=0$ corresponds to the frequency at which $q_{5}$ becomes unstable
for both experiment and model. The mode associated to $q_{5}$ is considered
unstable when its corresponding amplitude in Fourier space exceeds a noise
threshold (details given in the Supplemental Material). With these definitions
in place, we now measure the amplitude of the breather as a function of
$\Delta q$, see the markers with error bars in Fig. 4(a). The qualitative
amplitude trend agrees with the theoretical prediction. Although the time
series between the theoretical prediction for $c=0.15$ Ns/m and the experiment
agree qualitatively (compare panels (b) and (c) of Fig. 4) the model
underestimates the amplitude. This can be partially explained by the fact that
other modes (including unstable ones) in the experiment besides $q_{5}$ are
excited. Inspection of the spatial Fourier transform of the experiment shows
there are always traces of additional modes, even before the turning point of
the structure. The insets of Fig. 4(c) show the spatial Fourier transform
before ($t=1.6$ s) and after ($t=2.2$ s) the turning point (more details of
the Fourier analysis of the experiment is given in the Supplemental Material).

Finally, we measure the width of the structures using the half-width at half
maximum (HWHM) metric. The HWHM is given by $t_{\rm max}-t_{\rm half}$, where
$t_{\rm max}$ is the time the maximum is attained and $t_{\rm half}$ is the
time where the trajectory first attains half the maximum value. The
experimentally measured values are shown as the black triangles in Fig. 4(a).
The width of the structure can be predicted theoretically using the real part
of the Floquet exponent. In particular, assuming the envelope of the breather
follows the form $a\mathrm{sech}(\sigma t)$ (which we demonstrated previously
was a reasonable assumption), we can compute the HWHM as
$\mathrm{sech}^{-1}(1/2)/\sigma_{5}$. The prediction in the lossless ($c=0$)
and damped ($c=0.15$ Ns/m) cases are shown as red solid squares and open blue
squares, respectively, in Fig. 4(a). For sufficiently large values of $\Delta
q$, there is good agreement between theory and experiment, and both show the
structure becomes more narrow as the wavenumber goes deeper into the gap. This
is consistent with classic breathers in space-periodic systems.

##  V Conclusion

We explored a new class of solutions, phononic wavenumber bandgap breathers
and their damped analogues, in a time-varying lattice. Several avenues of
inquiry follow naturally from this study, including the possible existence of
genuine q-gap breathers (i.e., with both tails decaying to zero), an
analytical description of the breather profiles via a multiple-scale analysis,
and the exploration of such structures in higher spatial dimensions.

_Acknowledgement_. This material is based upon work supported by the US
National Science Foundation under Grant Nos. EFRI-1741565 (C.D.) and
DMS-2107945 (C.C. and E.W.). The authors are grateful for discussions with
P.G. Kevrekidis.

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$(K^{(11)})^{M}$ defines a Killing vector that can be time-like or null - for
AdS3 it will be null. These should obey

$\displaystyle d\Omega^{(11)}$ $\displaystyle=\iota_{K^{(11)}}G,$ (A.6)
$\displaystyle d\Sigma^{(11)}$
$\displaystyle=\iota_{K^{(11)}}\star_{11}G-\Omega^{(11)}\wedge G,$ (A.7)
$\displaystyle\star_{11}dK^{(11)}$
$\displaystyle=-\frac{2}{3}\Omega^{(11)}\wedge\star_{11}G+\frac{1}{3}\Sigma^{(11)}\wedge
G,$ (A.8)

for ${\mathcal{N}}=1$ to hold. To extract AdS3 conditions from this we first
decompose the 11d gamma matrices as

$\Gamma_{\mu}=e^{A}\gamma^{(3)}_{\mu}\otimes\hat{\gamma},~{}~{}~{}\Gamma_{\mu}=\mathbb{I}\otimes\gamma_{a}$
(A.9)

and take the intertwiner defining Majorana conjugation as
$\epsilon^{c}=B^{11}\epsilon^{*}$ to be $B^{11}=\mathbb{I}\otimes B$ such that
$\gamma_{a}^{*}=B^{-1}\gamma_{a}B$, $BB^{*}=\mathbb{I}$. We parameterise AdS3
as

$ds^{2}(\text{AdS}_{3})=e^{2mr}(-dt^{2}+dx^{2})+dr^{2}$ (A.10)

which means the ( Mink2 respecting part of) the AdS3 spinor is

$\zeta=e^{\frac{m}{2}r}\left(\begin{array}[]{c}1\\\ 0\end{array}\right),$
(A.11)

in the obvious frame, from which one can easily derive bi-linears on AdS3 \-
we skip the details. On M8 we define the bi linears

$2e^{A}f=\chi^{{\dagger}}\hat{\gamma}\chi,~{}~{}~{}2e^{A}K_{a}=\chi^{{\dagger}}\gamma_{a}\chi,~{}~{}~{}2e^{A}(\Psi_{3})_{abc}=\chi^{{\dagger}}\gamma_{abc}\hat{\gamma}\chi,~{}~{}~{}2e^{A}(\Psi_{4})_{abcd}=\chi^{{\dagger}}\gamma_{abcd}\chi.$
(A.12)

Obviously these can be further decomposed in terms of an angle, a G2 structure
3-form and a component of the vielbein on M8 as in [10] \- but this refinement
does not make the conditions easier to work with, so we will not peruse it
here. One can then show that the 11d bi-linears decompose in terms of the 8d
ones and data on AdS3 as

$\displaystyle K^{(11)}$ $\displaystyle=2e^{A}(e^{t}-e^{x}),$ (A.13)
$\displaystyle\Omega^{(11)}$
$\displaystyle=2e^{A}(e^{t}-e^{x})\wedge(K+fe^{r}),$ (A.14)
$\displaystyle\Sigma^{(11)}$
$\displaystyle=2e^{A}(e^{t}-e^{x})\wedge(-e^{r}\wedge\Psi_{3}+\Psi_{4}),$
(A.15)

where $e^{t,x,r}$ are the parts of the 11d vielbein along the AdS3 directions,
not the unwarped AdS3 ones. Plugging these into (A.6) we find $\mathcal{N}=1$
supersymmetry for AdS3 requires

$\displaystyle d(e^{2A}K)=0,$ (A.16) $\displaystyle
d(e^{3A}f)-e^{3A}F_{1}+2me^{2A}K=0,$ (A.17) $\displaystyle
d(e^{3A}\Psi_{3})-e^{3A}(-\star_{8}F_{4}+fF_{4})-2me^{2A}\Psi_{4}=0,$ (A.18)

which are all direct analogues of 3 of the 6 [10] conditions, but we also have

$\displaystyle d(e^{2A}\Psi_{4})-e^{2A}K\wedge F_{4}=0,$ (A.19) $\displaystyle
6\star_{8}dA-2f\star_{8}F_{1}+\Psi_{3}\wedge F_{4}=0,$ (A.20) $\displaystyle
6m\text{Vol}(\text{M}_{8})-e^{A}(2K\wedge\star_{8}F_{1}+\Psi_{4}\wedge
F_{4})=0$ (A.21)

which appear rather different, but should be equivalent to the final 3
conditions in [10] (they are more similar to the AdS2 conditions in [57]
derived in a similar fashion) - these are much easier to work with for our
purposes. Sufficiency of our conditions follows from the sufficiency of (A.6).
One can show that these supersymmetry conditions imply the remaining equations
of motion when the magnetic parts of

$dG=0,~{}~{}~{}~{}d\star_{11}G+\frac{1}{2}G\wedge G=0$ (A.22)

are imposed by hand - the electric components along with the rest of the EOM
being implied by supersymmetry and these.

The map relating our bi linears to [10] is

$\displaystyle e^{2A}$
$\displaystyle=e^{2\Delta},~{}~{}~{}~{}ds^{2}(\text{M}_{8})=e^{2\Delta}ds^{2}(\tilde{\text{M}}_{8}),~{}~{}~{}F_{1}=\tilde{f},~{}~{}~{}F_{4}=e^{3\Delta}F,~{}~{}~{}m=\frac{1}{2}\tilde{m}$
(A.23) $\displaystyle f$
$\displaystyle=\sin\zeta,~{}~{}~{}K=e^{\Delta}\cos\zeta\tilde{K},~{}~{}~{}\Psi_{3}=-e^{3\Delta}\cos\zeta\Phi_{3},~{}~{}~{}\Psi_{4}=e^{4\Delta}Y$
(A.24)

where we add tildes to objects defined in [10] when we use the same symbol in
our classification.

##  Appendix B Bi-spinors on spheres

In this appendix we present bi-spinors on various spheres that appear in the
computations of the main text. In order to do this it is helpful to consider
the even and odd dimensional spheres separately, as the former can actually be
presented in terms of the latter.

###  B.1 Odd Spheres

It is well known that odd dimensional spheres support a Sasaki–Einstein
structure. Specifically on an S2n+1 one can define a 1-form $V_{n}$, 2-form
$J_{n}$ and holomorphic $n$-form $\Omega_{(n)}$ such that

$dV_{n}=2J_{n},~{}~{}~{}~{}d\Omega_{(n)}=i(n+1)V_{n}\wedge\Omega_{(n)},$ (B.1)

where this also holds for S1, albeit with $dV_{0}=J_{0}=0$ which reflects the
fact that there is no Kahler–Einstein base over which to define a cone in this
case. For $n>0$ one can always parametise S2n+1 as a U(1) bundle over
$\mathbb{CP}^{n}$ such that the metric becomes

$ds^{2}(\text{S}^{2n+1})=(d\psi_{n}+\eta_{n})^{2}+ds^{2}(\mathbb{CP}^{n}),~{}~{}~{}d\eta_{n}=2J_{n}$
(B.2)

where $\partial_{\psi_{n}}$ is a U(1) isometry of period $2\pi$, $J_{n}$ is a
Kahler form on $\mathbb{CP}^{n}$, $\eta_{n}$ has support on the base only and
the metric on $\mathbb{CP}^{n}$ can be defined recursively as

$ds^{2}(\mathbb{CP}^{n})=d\theta_{n}^{2}+\sin^{2}\theta_{n}ds^{2}(\mathbb{CP}^{n-1})+\sin^{2}\theta_{n}\cos^{2}\theta_{n}(d\psi_{n-1}+\eta_{n-1})^{2},$
(B.3)

where $\eta_{n}=\sin^{2}\theta_{n}(d\psi_{n-1}+\eta_{n-1})$. For the lowest
order case, $n=1$ we have

$ds^{2}(\mathbb{CP}^{1})=\frac{1}{4}ds^{2}(\text{S}^{2})=\frac{1}{4}(d\theta_{1}^{2}+\sin\theta_{1}^{2}d\phi_{1}^{2}),~{}~{}~{}~{}\eta_{1}=\cos\theta_{1}d\phi_{1}.$
(B.4)

Using these definitions is it simple to establish that
Ricci$(\mathbb{CP}^{n})=2(n+1)g(\mathbb{CP}^{n})$ and so
Ricci$(\text{S}^{2n+1})=2ng(\text{S}^{2n+1})$, as it should for a unit radius
sphere. In general the forms $(V,J,\Omega_{n})$ are in fact not uniquely
defined: First off clearly if $\Omega_{n}$ satisfies (B.1), then it also does
when multiplied by a constant phase. Second, modulo such constant phases there
are $2^{n+1}$ distinct forms $(V,J,\Omega_{n})$ on S2n+1 that solve (B.1). To
see this more clearly consider the case of S3; here the metric can be arranged
in two equivalent Hopf fibrations U(1) $\rightarrow$
S${}^{3}\hookrightarrow~{}\mathbb{CP}^{1}$, namely

$\frac{1}{4}(d\theta_{1}^{2}+\sin^{2}\theta_{1}d\phi_{1}^{2})+(d\psi_{1}+\cos\theta_{1}d\phi_{1})^{2}=\frac{1}{4}d\theta_{1}^{2}+\sin^{2}\theta_{1}d\psi_{1}^{2}+(\frac{1}{2}d\phi_{1}+\cos\theta_{1}d\psi_{1})^{2},$
(B.5)

and as the metric is insensitive to the sign of $(V_{n},J_{n})$, this is
suggestive of 4 distinct Sasaki Einstein structures on the 3-sphere,
specifically

$\begin{array}[]{c|c|c}V_{1}&J_{1}&\Omega_{(1)}\\\
\hline\cr\hline\cr\pm(d\psi_{1}+\frac{1}{2}\cos\theta_{1}d\phi_{1})&\mp\frac{1}{4}\sin\theta_{1}d\theta_{1}\wedge
d\phi_{1}&\frac{1}{2}e^{\pm 2i\psi_{1}}(\pm
id\theta_{1}+\sin\theta_{1}d\phi_{1})\\\
\hline\cr\pm(\frac{1}{2}d\phi_{1}+\cos\theta_{1}d\psi_{1})&\mp\frac{1}{2}\sin\theta_{1}d\theta_{1}\wedge
d\psi_{1}&-e^{\pm i\phi_{1}}(\pm
i\frac{1}{2}d\theta_{1}+\sin\theta_{1}d\psi_{1})\end{array}$ (B.6)

which all solve (B.1) for the same parametrisation of the metric. More
generally the recursion relation of (B.3) essentially ensures that as one
moves from considering $n\to n+1$ the number of metric rearrangements of the
type (B.5) one can perform doubles, giving the $2^{n+1}$ distinct
Sasaki–Einstein structures on S2n+1. We shall not present these explicitly as
for our purposes it is more important that these $2^{n+1}$ structures are in
one to one correspondence with the number of supercharges that the
$(2n+1)$-sphere preserves.

####  B.1.1 Killing spinors on S2n+1

On the unit norm $(2n+1)$-sphere one can define two types of Killing spinor
$\xi_{\pm}$ that solve the Killing spinor equations

$\nabla_{a}\xi_{\pm}=\pm\frac{i}{2}\gamma_{a}\xi_{\pm},$ (B.7)

where $\gamma_{a}$ are gamma matrices on S2n+1 with each of $\xi_{+}$ and
$\xi_{-}$ preserving $2^{n}$ supercharges for $n>0$, or 2 when $n=0$, we will
not consider this case further here as it does not feature in the main text.
For the 3-sphere, when $n=1$, we can set our flat space gamma matrices equal
to the Pauli matrices as $\gamma_{a}=\sigma_{a}$, then in Hopf fibration frame
$e^{a}=(\frac{1}{2}d\theta_{1},\frac{1}{2}\sin\theta_{1}d\phi_{1},d\psi_{1}+\frac{1}{2}\cos\theta_{1}d\phi_{1})^{a}$
one finds that

$\xi^{\text{S}^{3}}_{+}=e^{\frac{i}{2}\theta_{1}\sigma_{1}}e^{\frac{i}{2}\phi_{1}\sigma_{3}}\xi^{0}_{+},~{}~{}~{}~{}\xi^{\text{S}^{3}}_{-}=e^{-\frac{i}{2}\psi_{1}\sigma_{3}}\xi^{0}_{-}$
(B.8)

where $\xi^{0}_{\pm}$ are arbitrary constant $d=2$ spinors that span the $4$
supercharges on the 3-sphere. For higher dimensional spheres one can exploit
the recursion relation for the metric of $\mathbb{CP}^{n}$ (B.3). Decomposing
$a=(\hat{a},\psi_{n-1},\beta_{n},\psi_{n})$, where $\hat{a}$ span
$\mathbb{CP}^{n-1}$ one may define a frame on S2n+1 as

$e^{\hat{a}}=\sin\theta_{n}e^{\hat{a}}_{\mathbb{CP}^{n-1}},~{}~{}~{}e^{\psi_{n-1}}=\sin\theta_{n}\cos\theta_{n}(d\psi_{n-1}+\eta_{n-1}),~{}~{}~{}e^{\theta_{n}}=d\theta_{n},~{}~{}~{}e^{\psi_{n}}=(d\psi_{n}+\eta_{n}),$
(B.9)

and a corresponding basis of gamma matrices

$\displaystyle\gamma_{\hat{a}}$
$\displaystyle=\sin\theta_{n}\gamma^{\mathbb{CP}^{n-1}}_{\hat{a}}\otimes\sigma_{1},~{}~{}~{}\gamma_{\psi_{n-1}}=\sin\theta_{n}\cos\theta_{n}\hat{\gamma}_{\mathbb{CP}^{n-1}}\otimes\sigma_{1},$
$\displaystyle\gamma_{\theta_{n}}$
$\displaystyle=\mathbb{I}_{2(n-1)}\otimes\sigma_{2},~{}~{}~{}\gamma_{\psi_{n}}=\mathbb{I}_{2(n-1)}\otimes\sigma_{3},$
(B.10)

where $\hat{\gamma}_{\mathbb{CP}^{n-1}}=\mathbb{I}_{2(n-2)}\otimes\sigma_{3}$
is the chirality matrix on $\mathbb{CP}^{n-1}$. In terms of this frame the
Killing spinors on S2n+1 are then given by the recursive expression

$\displaystyle\xi^{\text{S}^{2n+1}}_{\pm}$
$\displaystyle=e^{\frac{\psi_{n}}{2}(\not{J}_{n}\pm
i\gamma_{\psi_{n}})}\Big{(}P^{\mathbb{CP}^{n-1}}_{\pm}\otimes e^{\pm
i\beta_{n}\sigma_{2}}+P^{\mathbb{CP}^{n-1}}_{\mp}\otimes\mathbb{I}_{2}\Big{)}Q^{2n+1},$
$\displaystyle Q^{2n+1}$ $\displaystyle=\xi^{\text{S}^{2n-1}}_{+}\otimes
q^{0}_{-}+\xi^{\text{S}^{2n-1}}_{-}\otimes
q^{0}_{+},~{}~{}~{}P^{\mathbb{CP}^{n-1}}_{\pm}=\frac{1}{2}(\mathbb{I}_{2(n-1)}\pm\hat{\gamma}_{\mathbb{CP}^{n-1}}),~{}~{}~{}\sigma_{3}q^{0}_{\pm}=\pm
q^{0}_{\pm},$ (B.11)

where $q^{0}=q^{0}_{+}+q^{0}_{-}$ is an arbitrary $d=2$ constant spinor -
hence each of $\xi^{\text{S}^{2n+1}}_{\pm}$ preserve $2^{n}$ supercharges, for
a total of $2^{n+1}$ on $\text{S}^{2n+1}$.

####  B.1.2 Bi-spinors on S2n+1

We established in the previous section that the Killing spinors on S2n+1
$\xi_{\pm}$ each preserve $2^{n}$ supercharges, as such one can in general
decompose them in an orthonormal basis as

$\xi_{\pm}=\sum_{a=1}^{2^{n}}c_{a,\pm}\xi_{a,\pm},~{}~{}~{}~{}\xi^{{\dagger}}_{a,\pm}\xi_{b,\pm}=\delta_{ab}$
(B.12)

with each of $\xi_{a,\pm}$ preserving a single supercharge and $c_{a,\pm}$
arbitrary constants. Either by solving (B.7) explicitly as in the previous
section, or by making use of Fierz identities, one can then show that each
supercharge defines bi spinors of the form

$\displaystyle\Phi_{a,\pm}=\xi_{a,\pm}\otimes\xi^{{\dagger}}_{a,\pm}$
$\displaystyle=\frac{1}{2^{n}}(1\pm V_{a,\pm})\wedge
e^{-iJ_{a,\pm}},~{}~{}~{}~{}\tilde{\Phi}_{a,\pm}=\xi_{a,\pm}\otimes\xi^{{\dagger}c}_{a,\pm}=\frac{1}{2^{n}}(1\pm
V_{a,\pm})\wedge\Omega_{a,\pm},$ (B.13)
$\displaystyle\xi_{a,\pm}^{c}\otimes\xi^{c{\dagger}}_{a,\pm}$
$\displaystyle=(-1)^{{n\,\text{deg}}}(\Phi_{a,\pm})^{*},\qquad\qquad\qquad\,\,\,\,\xi_{a,\pm}^{c}\otimes\xi^{{\dagger}}_{a,\pm}=(-1)^{n\,{\text{deg}}+\frac{n}{2}(n+1)}(\tilde{\Phi}_{a,\pm})^{*}$

where we have defined the function deg which takes the form degree, assumed
without loss of generality $|\xi_{a,\pm}|=1$ and defined Majorana conjugation
as $\xi^{c}=B_{2n+1}\xi^{*}$ with intertwiner such that
$B_{2n+1}B_{2n+1}^{*}=(-1)^{\frac{n}{2}(n+1)}$,
$B_{2n+1}\gamma_{a}B_{2n+1}^{-1}=(-1)^{n}\gamma^{*}_{a}$. We additionally have
that

$\pm\frac{1}{n!}V_{a,\pm}\wedge(J_{a,\pm})^{n}=\text{vol}(\text{S}^{2n+1}),$
(B.14)

with $\text{vol}(\text{S}^{2n+1})$ the only SO(2n+2) invariant form on S2n+1.
As the notation suggests these bi-spinors give rise to the $2^{n+1}$
independent Sasaki-Einstein structures mentioned earlier - we have dropped the
$n$ subscript on the forms to ease notation. In the main text we are
ultimately interested in solving differential bi spinor relations in 7 or 8
dimensions that will contain $(\Phi_{a,\pm},\tilde{\Phi}_{a,\pm})$ as part of
wedge products with other bi spinors. For a given $n$, all
$(V_{a,\pm},J_{a,\pm},\Omega_{a,\pm})$ obey the same differential conditions
(B.1), but $(\Phi_{a,+},\tilde{\Phi}_{a,+})$ and
$(\Phi_{a,-},\tilde{\Phi}_{a,-})$ do contain some sign differences. So, it
should be clear that if our higher dimensional conditions are solved by
$(\Phi_{a,+},\tilde{\Phi}_{a,+})$, when $a=1$, then they are also solved for
any $a$, but it need not necessarily follow from this that
$(\Phi_{a,-},\tilde{\Phi}_{a,-})$ also solve these higher dimensional
conditions. As such, from here on we will drop the $a$ index in $\xi_{a,\pm}$
but keep the $\pm$, with the understanding that $\xi_{\pm}$ is representative
of any of the $\pm$ supercharges supported by S2n+1, we thus define the bi-
spinors

$\Phi_{\pm}=\frac{1}{2^{n}}\xi_{\pm}\otimes\xi^{{\dagger}}_{\pm}=(1\pm
V_{\pm})\wedge
e^{-iJ_{\pm}},~{}~{}~{}~{}\tilde{\Phi}_{\pm}=\frac{1}{2^{n}}\xi_{\pm}\otimes\xi^{{\dagger}c}_{\pm}=(1\pm
V_{\pm})\wedge\Omega_{\pm,n}$ (B.15)

which should be understood as representing any of the $2^{n}$ supercharges
contained in $\xi_{\pm}$. A word of caution is in order for the case of the
7-sphere where it is possible to take a Majorana basis for the killing spinor.
When one does this, one ends up with $2^{2n}$, G2 structures, but these can
also be derived simply in terms of the above bi-linears.

###  B.2 Even Spheres

Killing spinors on S2(n+1) $(n=0,1...)$ also obey a condition of the form
(B.7), however one can now define a chirality matrix in $2(n+1)$ dimensions
$\hat{\Gamma}$ relating $\xi^{2(n+1)}_{\pm}$ as
$\xi^{2(n+1)}_{-}=\hat{\Gamma}\xi^{2(n+1)}_{+}$, so that $\xi^{2(n+1)}_{\pm}$
are no longer independent. As such we shall take Killing spinors on even
dimensional spheres to obeys

$\nabla_{A}\xi=\frac{i}{2}\Gamma_{A}\xi,$ (B.16)

for $\Gamma_{A}$ a $2(n+1)$ dimensional gamma matrix - which has $2^{(n+1)}$
independent solutions each parameterising a distinct supercharge preserved on
S2(n+1) \- notice this is the same number that S2n+1 preserves, this is no
accident as each supercharge on an even-dimensional sphere can be mapped to a
supercharge on an odd sphere of one less dimension by expressing S2(n+1) as a
foliation of S2n+1 over an interval. Explicitly if one takes the even sphere
metric of unit radius to be

$ds^{2}(\text{S}^{2(n+1)})=d\alpha^{2}+\sin^{2}\alpha
ds^{2}(\text{S}^{2n+1}),$ (B.17)

which suggests splitting the gamma-matrix index as $A=(a,\alpha)$, then one
define a basis for the Clifford algebra in $2(n+1)$ dimensions as

$\displaystyle\Gamma_{1}$
$\displaystyle=\sin\alpha\sigma_{1},~{}~{}~{}~{}\Gamma_{\alpha}=\sigma_{2},~{}~{}~{}~{}\text{for}~{}n=0,$
(B.18) $\displaystyle\Gamma^{2n+1}_{a}$
$\displaystyle=\sin\alpha\gamma_{a}\otimes\sigma_{1},~{}~{}~{}~{}\Gamma_{\alpha}=\mathbb{I}\otimes\sigma_{2},~{}~{}~{}\text{for}~{}n>0$
(B.19)

with $\gamma_{a}$ gamma matrices on the unit $2n+1$-sphere and $\sigma_{i}$
the Pauli matrices, we take the chirality matrix to be
$\hat{\Gamma}=\mathbb{I}\otimes\sigma_{3}$. With respect to this basis it is
simple to show that (B.16) is solved when $\xi$ is any linear combination of

$\varphi_{\pm}=\xi_{\pm}\otimes e^{\frac{i}{2}\alpha\sigma_{2}}\theta_{\mp},$
(B.20)

with $\theta_{\pm}$ constant 2d spinors obeying
$\sigma_{3}\theta_{\pm}=\pm\theta_{\pm}$ whenever $\xi_{\pm}$ solves (B.7) on
S2n+1. Note that (B.20) also holds for $n=0$, i.e. for the 2-sphere - one
merely needs to take the S1 “spinors” to be the scalars
$\xi_{\pm}=e^{\pm\frac{i}{2}\phi}$, where $\phi\sim\phi+2\pi$ spans S1. As
$\xi^{2n+1}_{\pm}$ each contain $2^{n}$ supercharges, we then have a total of
$2^{n+1}$ on S2(n+1). To define the Majorana conjugate on S2(n+1) we must
choose an intertwiner, there are actually two consistent with the intertwiner
on S2n+1 that we parametrise as

$B^{(1)}_{2(n+1)}=B_{2n+1}\otimes\sigma_{1}^{n+1},~{}~{}~{}~{}B^{(2)}_{2(n+1)}=(-1)^{n+1}i\hat{\Gamma}B^{(1)}_{2(n+1)}$
(B.21)

(the $B_{2n+1}$ factor should be dropped on S2) which are unitary matrix such
that

$B^{(k)}_{2(n+1)}B^{(k)*}_{2(n+1)}=(-1)^{\frac{n}{2}(n+1)+(k-1)(n+1)},~{}~{}~{}B^{(k)}_{2(n+1)}\Gamma_{A}(B^{(k)}_{2(n+1)})^{-1}=(-1)^{n+k-1}\Gamma_{A}^{*}$
(B.22)

where the index $k=1,2$. Of course in principle it should make no physical
difference which choice of intertwiner one takes in (B.21), however one choice
may be better suited to embed a particular sphere into a higher dimensional
solution. In terms of these definition we can then compute the bi-spinors on
S2(n+1), fixing

$\theta_{+}=\left(\begin{array}[]{c}i\\\
0\end{array}\right),~{}~{}~{}~{}\theta_{-}=\left(\begin{array}[]{c}0\\\
1\end{array}\right)$ (B.23)

without loss of generality, we find

$\displaystyle\Phi_{\pm}$
$\displaystyle=\varphi_{\pm}\otimes\varphi_{\pm}^{{\dagger}}=\frac{1}{2^{n+1}}\Big{(}1+\sin\alpha_{\pm}V_{\pm}\wedge(\sin\alpha_{\pm}+id\sin\alpha_{\pm})\Big{)}\wedge
e^{-i\sin^{2}\alpha_{\pm}J_{\pm}},$ $\displaystyle\Phi_{\hat{\Gamma}\pm}$
$\displaystyle=\hat{\Gamma}\varphi_{\pm}\otimes\varphi_{\pm}^{{\dagger}}=-\frac{1}{2^{n+1}}\Big{(}\cos\alpha_{\pm}+id\cos\alpha_{\pm}\wedge(1+V_{\pm})\Big{)}\wedge
e^{-i\sin^{2}\alpha_{\pm}J_{\pm}},$ $\displaystyle\tilde{\Phi}^{(1)}_{\pm}$
$\displaystyle=\varphi_{\pm}\otimes\varphi_{\pm}^{c_{1}{\dagger}}=\frac{1}{2^{n+1}}\Big{(}\sin\alpha_{\pm}(1+V_{\pm})-id\sin\alpha_{\pm}\Big{)}\wedge(\sin^{n}\alpha_{\pm}\Omega_{\pm}),$
$\displaystyle\tilde{\Phi}^{(1)}_{\hat{\Gamma}\pm}$
$\displaystyle=\hat{\Gamma}\varphi_{\pm}\otimes\varphi_{\pm}^{c_{1}{\dagger}}=\frac{1}{2^{n+1}}i\Big{(}d\alpha_{\pm}+\sin\alpha_{\pm}V_{\pm}\wedge(i\cos\alpha_{\pm}+d\cos\alpha_{\pm})\Big{)}\wedge(\sin^{n}\alpha_{\pm}\Omega_{\pm}),$
$\displaystyle\tilde{\Phi}^{(2)}_{\pm}$
$\displaystyle=\varphi_{\pm}\otimes\varphi_{\pm}^{c_{2}{\dagger}}=\frac{1}{2^{n+1}}\Big{(}d\alpha_{\pm}+\sin\alpha_{\pm}V_{\pm}\wedge(i\cos\alpha_{\pm}-d\cos\alpha_{\pm}\Big{)}\wedge(\sin^{n}\alpha_{\pm}\Omega_{\pm}),$
$\displaystyle\tilde{\Phi}^{(2)}_{\hat{\Gamma}\pm}$
$\displaystyle=\hat{\Gamma}\varphi_{\pm}\otimes\varphi_{\pm}^{c_{2}{\dagger}}=-\frac{1}{2^{n+1}}i\Big{(}\sin\alpha_{\pm}(-1+V_{\pm})-id\sin\alpha_{\pm}\Big{)}\wedge(\sin^{n}\alpha_{\pm}\Omega_{\pm}),$
(B.24)

where we define $\varphi^{c_{k}}=B_{2(n+1)}^{(k)}\varphi^{*}$ and

$\alpha_{+}=\alpha,~{}~{}~{}~{}\alpha_{-}=\pi-\alpha,$ (B.25)

which has absorbed all the sign changes that appear between the bi linears
following from $\xi_{+}$ and $\xi_{-}$ (B.15). The other possible bispinors
can be computed in terms of these ones using these identities

$\displaystyle\varphi_{\pm}^{c_{k}}\otimes\varphi_{\pm}^{c_{k}{\dagger}}=(-)^{(n+k-1)\text{deg}}(\Phi_{\pm})^{*},\qquad\qquad\qquad\,\,\,\hat{\Gamma}\varphi_{\pm}^{c_{k}}\otimes\varphi_{\pm}^{c_{k}{\dagger}}=(-)^{(n+k-1)\text{deg}}(\Phi_{\hat{\Gamma}\pm})^{*},$
(B.26)
$\displaystyle\varphi_{\pm}^{c_{k}}\otimes\varphi_{\pm}^{{\dagger}}=(-)^{(\frac{n}{2}+k-1)(n+1)+(n+k-1)\text{deg}}(\tilde{\Phi}^{(k)}_{\pm})^{*},\quad\hat{\Gamma}\varphi_{\pm}^{c_{k}}\otimes\varphi_{\pm}^{{\dagger}}=(-)^{(\frac{n}{2}+k-1)(n+1)+(n+k-1)\text{deg}}(\tilde{\Phi}^{(k)}_{\hat{\Gamma}\pm})^{*},$

and also by recalling that, for generic spinors $\varphi_{1},\varphi_{2}$ we
have

$\hat{\Gamma}\varphi_{1}\otimes\varphi_{2}^{{\dagger}}=(-)^{\text{deg}}\varphi_{1}\otimes(\hat{\Gamma}\varphi_{2})^{{\dagger}},\qquad
B^{(k)}_{2(n+1)}\hat{\Gamma}(B^{(k)}_{2(n+1)})^{-1}=(-)^{n+1}\hat{\Gamma}^{*},$
(B.27)

one may infer the remaining bilinears.

Finally we note that the volume form of S2(n+1) is given by

$\frac{1}{n!}d\alpha_{\pm}\wedge(\sin\alpha_{\pm}V_{\pm})\wedge(\sin^{2}\alpha_{\pm}J_{\pm})^{n}=\sin^{2n+1}\alpha\,d\alpha\wedge\text{vol}(\text{S}^{2n+1})=\text{vol}(\text{S}^{2(n+1)}).$
(B.28)

Again we are ultimately interested in solving higher dimensional bi spinor
relations that will contain the S2(n+1) bi-spinors. As for a given $n$,
$(\alpha_{\pm},V_{\pm},J_{\pm},\Omega_{\pm})$ all act in the same way under
exterior differentiation, and this time give rise to
$\text{vol}($S${}^{2(n+1)})$ (which can appear in the fluxes of a supergravity
solution) in the same fashion, it actually makes no difference if we choose
$+$ or $-$ in (B.2) so we drop this index in the main text for even spheres.

##  Appendix C Lifting isometries

In this section we will summarize some results about fiber-bundles as
presented in [58]. We will assume that the fibers of our Riemannian submersion
are totally geodesics, so that using Hermann theorem we get that they are
isometric and the bundle is a $G$-bundle where $G$ is the isometry group of
the fiber. Let’s parameterize such a Lie algebra by the following set of
Killing vectors on the fiber

$K_{F}^{a}=K_{F}^{ai}\partial_{y_{i}},\qquad[K_{F}^{a},K_{F}^{b}]=f^{ab}_{\quad
c}K_{F}^{c}$ (C.1)

where $\\{y_{i}\\}$ are coordinates on the fiber. At the end of this section
we will see some explicit examples.

In order to define a covariant exterior derivative $D$ on the whole fiber-
bundle we have to introduce the connection $A^{i}=A_{a}K_{F}^{ai}$, which is
nothing but a one-form on the base which takes value in the Lie algebra of
$G$. Specifically if $x^{m}$ are the coordinates on the base then
$A_{a}=A_{am}(x)dx^{m}$. $D$ acts on the fiber coordinates as

$Dy^{i}=dy^{i}+A_{a}K_{F}^{ai}\,$ (C.2)

while it acts on the base coordinate as the usual external derivative. Using
these definitions we can write the fiber-bundle metric as following:

$ds^{2}=(g_{B})(x)_{mn}dx^{m}dx^{n}+(g_{F})(y)_{ij}Dy^{i}Dy^{j}\,,$ (C.3)

where $g_{B}$ is the metric on the base while $g_{F}$ the one on the fiber;
recall that $K_{F}^{a}$ are Killing vectors of $(g_{F})_{ij}dy^{i}dy^{j}$, but
not necessarily of $ds^{2}$.

Let us now denote $K_{B}$ to be a Killing vector of the base metric
$(g_{B})_{mn}(x)dx^{m}dx^{n}$. We would like to understand under which
conditions such an isometry can be uplifted to a symmetry of the whole space.
We will be agnostic about the form of the lift, and we will simply assume that
it is $K=K_{B}-\lambda^{i}\partial_{y^{i}}$ where
$\lambda^{i}=\lambda^{i}(x,y)$. What we have to calculate is therefore the
condition under which

$\mathcal{L}_{K}ds^{2}=0.$ (C.4)

Since the base metric is already isometric we have

$\begin{split}\mathcal{L}_{K}ds^{2}&=-\lambda^{k}\partial_{y^{k}}((g_{F})_{ij})Dy^{i}Dy^{j}+2(g_{F})_{ij}Dy^{j}\mathcal{L}_{K}(Dy^{i}).\end{split}$
(C.5)

Let us now expand the final term of this expression; using
$\mathcal{L}_{V}=\iota_{V}d+d\iota_{V}$, we find

$\begin{split}\mathcal{L}_{K}(Dy^{i})=&\mathcal{L}_{K_{B}-\lambda^{k}\partial_{y^{k}}}(dy^{i}+A_{a}K_{F}^{ai})=-d\lambda^{i}+\mathcal{L}_{K_{B}}A_{a}K_{F}^{ai}-A_{a}\lambda^{k}\partial_{y^{k}}K_{F}^{ai}\\\
=&\mathcal{L}_{K_{B}}A_{a}K_{F}^{ai}-\partial_{x^{m}}\lambda^{i}dx^{m}-A_{a}(\lambda^{k}\partial_{y^{k}}K_{F}^{ai}-K_{F}^{ak}\partial_{y^{k}}\lambda^{i})-\partial_{y^{k}}\lambda^{i}Dy^{k}.\end{split}$
(C.6)

Using this result we get

$\begin{split}\mathcal{L}_{K}ds^{2}=&-\mathcal{L}_{\lambda^{k}\partial_{y^{k}}}((g_{F})_{ij})Dy^{i}Dy^{j}\\\
&+2(g_{F})_{ij}Dy^{j}(\mathcal{L}_{K_{B}}A_{a}K_{F}^{ai}-\partial_{x^{m}}\lambda^{i}dx^{m}-A_{a}(\lambda^{k}\partial_{y^{k}}K_{F}^{ai}-K_{F}^{ak}\partial_{y^{k}}\lambda^{i}));\end{split}$
(C.7)

since the first line is independent of the last, in order to make it vanishes
we have to impose that $\lambda^{k}\partial_{y^{k}}$ is a Killing vector for
the fiber metric, which means that $\lambda^{k}=\lambda_{a}(x)K_{F}^{ak}$. The
remaining part of the metric Lie derivative reads

$\begin{split}\mathcal{L}_{K}ds^{2}=&2(g_{F})_{ij}Dy^{j}(\mathcal{L}_{K_{B}}A_{a}K_{F}^{ai}-d(\lambda_{a})K_{F}^{ai}-A_{a}\lambda_{b}(K_{F}^{bk}\partial_{y^{k}}K_{F}^{ai}-K_{F}^{ak}\partial_{y^{k}}K_{F}^{bi}))\\\
=&2(g_{F})_{ij}Dy^{j}(\mathcal{L}_{K_{B}}A_{a}-d\lambda_{a}-f^{bc}{}_{a}\lambda_{b}A_{c})K_{F}^{ai}\,.\end{split}$
(C.8)

So what we get is that $K_{B}$ can be uplifted to an isometry of the fiber-
bundle if and only if $\mathcal{L}_{K_{B}}$ acts on $A_{a}$ as a gauge
transformation:

$\mathcal{L}_{K_{B}}A_{a}=d\lambda_{a}+f^{bc}{}_{a}\lambda_{b}A_{c}\,.$ (C.9)

It is not hard to show that this argument can in-fact be generalised to
metrics of the form

$ds^{2}=e^{2C_{1}(x,y)}(g_{B})(x)_{mn}dx^{m}dx^{n}+e^{2C_{2}(x,y)}(g_{F})(y)_{ij}Dy^{i}Dy^{j},~{}~{}~{}Dy_{i}=dy_{i}+A_{a}K_{F}^{ai}$
(C.10)

where in addition to (C.9), one should also impose $\mathcal{L}_{K}C_{1,2}=0$,
so that these functions are invariants of the lifted isometry.

In the following we will consider two particularly interesting cases.

###  C.1 Abelian case

In this case we have that the condition (C.9) can be rephrase as the statement
that the field $F=dA$ is a singlet under the action of the base isometries,
indeed

$0=d\mathcal{L}_{K_{B}}A=\mathcal{L}_{K_{B}}F.$ (C.11)

###  C.2 Fibration of coset manifolds and Lie groups

Let’s start our discussion by considering a group manifold $G$. On a group
manifold we have that the isometry group $G_{L}\times G_{R}$ is generated by
the Killing vectors $K^{a}_{L,R}$ defined as the dual of the right- and left-
invariant one form232323Notice that in this notation $(K^{a}_{F})_{L}$
generates left translations but it is right-invariant, and vice-versa.:

$\omega^{L}_{a}=-\text{Tr}(t_{a}g^{-1}dg),\qquad\omega^{R}_{a}=-\text{Tr}(t_{a}dgg^{-1}),\qquad
K^{a}_{L,R}\cdot\omega^{R,L}_{b}=\delta^{a}_{b},\qquad\omega^{R}_{a}=D_{a}{}^{b}\omega^{L}_{b}$
(C.12)

where $t_{a}\in\text{Lie}(G)$242424We are using the convention
$\delta_{ab}=-\text{Tr}(t_{a}t_{b})$. and $D_{ab}(g)$ is the adjoint action of
$G$: $D_{ab}(g)=-\text{Tr}(g^{-1}t_{a}gt_{b})$ .

Now let’s consider a coset defined by the right quotient M$=G/H$, again we can
define left-invariant forms $\omega^{L}_{\alpha}$, where $\alpha$ is an index
in the lie algebra of $G/H$, using the restriction of the left action of $G$
to $G/H$. The isometry group now reduces to $G$ and it is generated by the
vectors $K^{a}$

$K^{a}=D^{a\alpha}(\omega^{L})_{\alpha}^{\mu}\partial_{\mu}\,,\qquad
K^{a}\cdot\omega_{b}=\delta^{a}_{b}$ (C.13)

where the forms $\omega_{a}$ dual to the Killing vectors are the Maurer-Cartan
forms for the coset. Notice that even if $K^{a}$ are defined starting from a
the right-invariant Killing vectors inherited from the Lie group, the presence
of the adjoint action makes them respect the same algebra of the left ones,
i.e. with an opposite sign in front of the structure constant.

Let’s consider now two identical coset manifold Mx and My of the same group
manifold $G$, so that they preserve (at least) an isometry group equal to $G$.
Now suppose we want to fiber My over Mx so that part the isometries of the
base are preserved. In order to do so let’s consider, as in (C.2), the
following coordinates on the fiber

$Dy^{i}=dy^{i}+A_{a}K^{ai}_{y}\,,$ (C.14)

and let’s take the connection $A_{a}$ to be proportional to the Maurer-Cartan
forms over $M_{x}$, i.e. $A_{a}=c\omega_{a}$, with $c$ constant252525Really we
mean constant with respect to the fiber bundle coordinates. If the bundle is
embedded into some large space then allowing $c$ to depend on the remaining
coordinates changes nothing of what follows, so that

$\mathcal{L}_{K^{a}_{x}}A_{b}=-f_{b}{}^{ac}A_{c}.$ (C.15)

Now, if we want to uplift $\mathcal{L}_{K^{a}_{x}}$ we have to define Killing
vectors $K^{a}$ such that
$\mathcal{L}_{K^{a}}=\mathcal{L}_{K^{a}_{x}-\lambda_{d}K^{d}_{y}}$ where
$\lambda_{d}$ has to satisfy (C.9). If we assume that $\lambda_{d}$ is
constant then this is equivalent to

$\mathcal{L}_{K^{a}_{x}}A_{b}=f^{dc}{}_{b}\lambda_{d}A_{c}.$ (C.16)

Comparing this last equation with (C.15) we get that we have a solution for
every $K^{a}_{x}$ if $\lambda_{d}=-\delta^{a}_{d}$ provided that the Lie group
$G$ is semisimple (and therefore $f^{ab}{}_{c}$ are completely antisymmetric
in a certain basis). Thus we get that the isometry, spanned by
$K^{a}=K^{a}_{x}+K^{a}_{y}$ is nothing but the diagonal action of $G$ acting
on both Mx and My. Notice that we can perform an almost identical computation
by considering the base manifold as the Lie group $G$ and $A_{a}$ the left- or
right-invariant forms on it.

A particularly interesting case is when M$=$Sk since every sphere can be
obtained as a quotient (and we have also the particular case
S${}^{3}\\!=$SU(2)). In this case it is convenient to express the algebra
index a as a couple of antisimmetrised indices [mn] on the embedding
coordinates of the spheres in $R^{k+1}$. Using this reparametrisation we have

$K^{[mn]i}=g^{ij}x^{[m}\partial_{j}x^{n]},\qquad A_{[mn]}=x_{[m}dx_{n]}.$
(C.17)

In the particular case $k=2$ one usually rewrite this couple of indices back
to a single one using the three-dimensional Levi-Civita symbol.

##  Appendix D On the Sp(2)$\times$Sp(1) preserving squashed S7

In this appendix we discuss the fibration of S3 over S4 that preserves an
Sp(2)$\times$Sp(1) isometry. This manifold can be viewed as a squashed S7, and
was introduced in [49] in the context of AdS4 vacua of M-theory with weak
G2-structure internal space. The metric can be expressed in terms of two sets
of SU(2) left invariant forms $(L^{i}_{1},L^{i}_{2})$ $i=1,2,3$ which obey

$dL^{i}_{1,2}=\frac{1}{2}\epsilon_{ijk}L^{j}_{1,2}\wedge L^{k}_{1,2}$ (D.1)

as

$ds^{2}=\frac{1}{4}\bigg{[}d\alpha^{2}+\frac{1}{4}\sin^{2}\alpha(L_{1}^{i})^{2}+\lambda^{2}\big{(}L_{2}^{i}-\cos^{2}\left(\frac{\alpha}{2}\right)L_{1}^{i}\big{)}^{2}\bigg{]}$
(D.2)

where $\lambda$ is a constant squashing parameter262626When embedded into a
higher dimensional space $\lambda$ can become a function of the condimensions
of the squashed S7 with the unit radius round 7-sphere recovered when
$\lambda=1$ 272727The fibration becomes topologically trivial in this limit,
which is made more obvious if on rearranges (D.2) with $\lambda=1$ as
$\frac{d\alpha^{2}}{4}+\frac{1}{4}\cos^{2}\left(\frac{\alpha}{2}\right)(L_{1}^{i})^{2}+\frac{1}{4}\sin^{2}\left(\frac{\alpha}{2}\right)(L_{2}^{i}-L_{1}^{i})^{2}.$
(D.3) There exists an SU(2) transformation, mapping $L^{i}_{2}\to
L_{2}^{i}-L_{1}^{i}$, so the fibration is clearly trivial.. The Killing
vectors corresponding to the Sp(2)$\times$Sp(1) isometry on the
S${}^{4}\times$S3 fibration can be constructed in the fashion of the previous
section, in terms of the isometries of the base S4 and fiber S3, i.e. Sp(2)
$\equiv$ SO(5) = SO(5)/SO(4)$\times$SO(4) and SO(4)=SU(2)${}_{L}\times$SU(2)R
respectively. Choosing a specific parameteriation for the invariant forms

$L^{1}_{a}+iL^{2}_{a}=e^{i\psi_{a}}(id\theta_{a}+\sin\theta_{a}d\phi_{a}),~{}~{}~{}~{}L^{3}_{a}=d\psi_{a}+\cos\theta_{a}d\phi_{a},~{}~{}~{}~{}a=1,2,$
(D.4)

which is compatible with the following complex emedding coordinates for the
7-sphere

$\displaystyle Z^{1}$
$\displaystyle=e^{\frac{i}{2}(\phi_{2}-\psi_{1})}\sin\left(\frac{\alpha}{2}\right)\sin\left(\frac{\theta_{2}}{2}\right),~{}~{}Z^{2}=ie^{-\frac{i}{2}(\phi_{2}+\psi_{1})}\sin\left(\frac{\alpha}{2}\right)\cos\left(\frac{\theta_{2}}{2}\right),$
$\displaystyle Z^{3}$
$\displaystyle=e^{\frac{i}{2}(\phi_{1}+\phi_{2}-\psi_{1}-\psi_{2})}\cos\left(\frac{\alpha}{2}\right)(e^{i\psi_{1}}\cos\left(\frac{\theta_{1}}{2}\right)\sin\left(\frac{\theta_{2}}{2}\right)-e^{i\psi_{2}}\cos\left(\frac{\theta_{2}}{2}\right)\sin\left(\frac{\theta_{1}}{2}\right)),$
$\displaystyle Z^{4}$
$\displaystyle=ie^{-\frac{i}{2}(\phi_{1}-\phi_{2}+\psi_{1}+\psi_{2})}\cos\left(\frac{\alpha}{2}\right)(e^{i\psi_{1}}\sin\left(\frac{\theta_{1}}{2}\right)\sin\left(\frac{\theta_{2}}{2}\right)+e^{i\psi_{2}}\cos\left(\frac{\theta_{1}}{2}\right)\cos\left(\frac{\theta_{2}}{2}\right))$
(D.5)

the Killing vectors of the SO(4) isometries appearing in the base and fibre
are

$\displaystyle K^{1L}_{a}+iK^{2L}_{a}$
$\displaystyle=e^{i\phi_{a}}\left(i\partial_{\theta_{a}}+\frac{1}{\sin\theta_{a}}\partial_{\psi_{a}}-\frac{\cos\theta_{a}}{\sin\theta_{a}}\partial_{\phi_{a}}\right),~{}~{}~{}K^{3L}_{a}=-\partial_{\phi_{a}},$
(D.6) $\displaystyle K^{1R}_{a}+iK^{2R}_{a}$
$\displaystyle=e^{i\psi_{a}}\left(i\partial_{\theta_{a}}+\frac{1}{\sin\theta_{a}}\partial_{\phi_{a}}-\frac{\cos\theta_{a}}{\sin\theta_{a}}\partial_{\psi_{a}}\right),~{}~{}~{}K^{3R}_{a}=\partial_{\psi_{a}}$
(D.7)

while for the SO(5)/SO(4) Killing vectors on the base we have

$K^{A}_{SO(5)/SO(4)}=-(\mu_{A}\partial_{\alpha}+\cot\alpha\partial_{y_{i}}\mu_{A}g_{3}^{ij}\partial_{y_{j}}),~{}~{}~{}A=1,...,4$
(D.8)

where $\mu_{A}$ are embedding coordinates for the 3-sphere spanned by
$K^{i}_{1}$, $g_{3}^{ij}$ is its metric and
$y_{i}=(\theta_{1},\phi_{1},\psi_{1})_{i}$, specifically we have taken

$\mu_{A}=\bigg{(}\sin\left(\frac{\theta_{1}}{2}\right)\cos\left(\frac{\phi_{-}}{2}\right),~{}\sin\left(\frac{\theta_{1}}{2}\right)\sin\left(\frac{\phi_{-}}{2}\right),~{}\cos\left(\frac{\theta_{1}}{2}\right)\cos\left(\frac{\phi_{+}}{2}\right),~{}-\cos\left(\frac{\theta_{1}}{2}\right)\sin\left(\frac{\phi_{+}}{2}\right)\bigg{)}_{A},$

where $\phi_{\pm}=\phi_{1}\pm\psi_{1}$.

In the full space (D.2) some of the isometries (D.6)-(D.8) are broken, some
are unaffected and some get lifted. Clearly as we fiber in terms of the left
invariant forms, the Killing vectors (D.6) are still preserved by the fiber
bundle. As 6 of the 7 directions of (D.2) form an S${}^{3}\times$S3 fibration
of the type described in appendix C.2, the base SO(3)R is lifted as

$K^{i}_{D}=K^{iR}_{1}+K^{iR}_{2}.$ (D.9)

which is a diagonal, the anti diagonal $K^{iR}_{1}-K^{iR}_{2}$ is broken. The
base SO(5)/SO(4) on the other hand is lifted in a new fashion as

$\hat{K}^{A}_{SO(5)/SO(4)}=K^{A}_{SO(5)/SO(4)}+\cot\left(\frac{\alpha}{2}\right)y_{B}(\kappa_{A})^{B}_{i}K^{iR}_{2}$
(D.10)

where

$\kappa_{1}=\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&-1\\\ 0&-1&0\\\
-1&0&0\end{array}\right),~{}~{}~{}\kappa_{2}=\left(\begin{array}[]{ccc}0&0&1\\\
0&0&0\\\ -1&0&0\\\
0&1&0\end{array}\right),~{}~{}~{}\kappa_{3}=\left(\begin{array}[]{ccc}0&1&0\\\
1&0&0\\\ 0&0&0\\\
0&0&-1\end{array}\right),~{}~{}~{}\kappa_{4}=\left(\begin{array}[]{ccc}1&0&0\\\
0&-1&0\\\ 0&0&1\\\ 0&0&0\end{array}\right).$

It is not hard to confirm that (D.10) does indeed obey (C.9) as required. We
note as a possible point of interest that the matrices $\kappa_{A}$ give rise
to a 3d basis of $\mathfrak{so}$(3) as

$\kappa^{T}_{A}\kappa_{B}=\left(\begin{array}[]{cccc}\mathbb{I}_{3}&\Lambda_{3}&-\Lambda_{2}&\Lambda_{1}\\\
-\Lambda_{3}&\mathbb{I}_{3}&\Lambda_{3}&\Lambda_{2}\\\
\Lambda_{2}&-\Lambda_{1}&\mathbb{I}_{3}&\Lambda_{3}\\\
-\Lambda_{1}&-\Lambda_{2}&-\Lambda_{3}&\mathbb{I}_{3}\end{array}\right)_{AB},~{}~{}~{}[\Lambda_{i},\Lambda_{j}]=-\epsilon_{ijk}\Lambda_{k}.$
(D.11)

The Killing vectors spanning Sp(2)$\times$Sp(1) on the squashed 7-sphere are
then

$\text{Sp(2)}:~{}(K^{iL}_{1},K^{i}_{D},\hat{K}^{A}_{SO(5)/SO(4)}),~{}~{}~{}~{}\text{Sp(1)}:~{}K^{iL}_{2}.$
(D.12)

###  D.1 Killing spinors on the round S7

In the obvious frame that (D.2) suggests (i.e.
$e^{1}=\frac{1}{2}d\alpha,~{}e^{2,3,4}=\frac{1}{4}\sin\alpha L^{1,2,3},...)$,
the corresponding 7-sphere Killing spinors, i.e. (D.2) in the $\lambda=1$
limit, are given by

$\displaystyle\xi_{\pm}$ $\displaystyle={\cal M}_{\pm}\xi^{0}_{\pm},$ (D.13)
$\displaystyle{\cal M}_{\pm}$ $\displaystyle=e^{\frac{\alpha}{4}(\pm
i\gamma_{1}+Y)}e^{\mp i\frac{\psi_{1}}{2}\gamma_{7}P_{\mp}}e^{\mp
i\frac{\theta_{1}}{2}\gamma_{6}P_{\mp}}e^{\mp
i\frac{\phi_{1}}{2}\gamma_{7}P_{\mp}}e^{\frac{\psi_{2}}{4}(\pm
i\gamma_{7}+X)}e^{\frac{\theta_{2}}{2}(\gamma_{13}P_{+}\pm\gamma_{6}P_{\pm})}e^{\frac{\phi_{2}}{2}(\gamma_{14}P_{+}\pm
i\gamma_{7}P_{\pm})}$

where $\gamma_{a}$ are a basis of flat 7d gamma matrices, specifically

$\gamma_{1}=\sigma_{1}\otimes\mathbb{I}_{2}\otimes\mathbb{I}_{2},~{}~{}~{}\gamma_{2,3,4}=\sigma_{2}\otimes\sigma_{1,2,3}\otimes\mathbb{I}_{2}~{}~{}~{}\gamma_{5,6,7}=\sigma_{3}\otimes\mathbb{I}_{2}\otimes\sigma_{1,2,3}\,,$
(D.14)

$\xi^{0}_{\pm}$ are unconstrained constant spinors and

$P_{\pm}=\frac{1}{2}(\mathbb{I}_{4}\pm\gamma_{1234}),~{}~{}~{}X=\gamma_{14}-\gamma_{23}-\gamma_{56},~{}~{}~{}~{}Y=-\gamma_{25}-\gamma_{36}-\gamma_{47}.$
(D.15)

We remind the reader that $\xi_{\pm}$ solve the Killing spinor equations
$\nabla_{a}\xi_{\pm}=\pm\frac{i}{2}\gamma_{a}\xi_{\pm}$. Note that this is not
the spinor preserved on the internal space of the AdS4 squashed 7-sphere
solution in [49], this preserves just $\mathcal{N}=1$ supersymmetry and solves
the equation $\nabla_{a}\xi=s\frac{i}{2}\gamma_{a}\xi$ for $\lambda^{-2}=5$,
$s^{2}\lambda^{-2}=9$ only, it is in fact constant in the above frame.

We want to establish the transformation properties of the Killing spinors of
this parameterisation of the 7-sphere under the Killing vectors which are
preserved by the squashing. It is not hard to establish that for any
parametrisation of the 7-sphere, the 8 independent $\pm$ Killing spinors each
transform in the 8 of SO(8), we thus define

$\xi^{I}_{\pm}={\cal
M_{\pm}}\hat{\eta}^{I},~{}~{}~{}\hat{\eta}^{I}=\sum_{J}\delta^{IJ}$ (D.16)

Interestingly the two octuplets $\xi^{I}_{\pm}$ behave quite differently under
the squashing. For $\xi^{I}_{+}$ we have 8$\to$3$\oplus$5. To see this it
useful to define the following triplet and quintuplet

$\displaystyle\xi^{i}_{3}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}i(-\xi^{5}_{+}+\xi^{8}_{+})\\\
\xi^{5}_{+}+\xi^{8}_{+}\\\ i(\xi^{6}_{+}+\xi^{7}_{+})\\\
\end{array}\right)^{i},~{}~{}~{}\xi^{\alpha}_{5}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}-i(\xi^{1}_{+}+\xi^{4}_{+})\\\
\xi^{1}_{+}-\xi^{4}_{+}\\\ -i(\xi^{2}_{+}-\xi^{3}_{+})\\\
\xi^{2}_{+}+\xi^{3}_{+}\\\
\xi^{6}_{+}-\xi^{7}_{+}\end{array}\right)^{\alpha},$ (D.25)

which one can check are Majorana. One can establish what symmetries these
objects are charged under by computing the spinoral Lie derivative with
respect to the appropriate Killing vectors: For a spinor $\eta$ this is
defined as

${\cal
L}_{K}\eta=K^{a}\nabla_{a}\eta+\frac{1}{4}\nabla_{a}K_{b}\gamma^{ab}\eta,$
(D.26)

where $K$ is a Killing vector. We find the following transformation properties
for the Killing vectors preserved by the squashing

$\displaystyle\mathcal{L}_{K^{iL}_{2}}\xi^{j}_{3}$
$\displaystyle=-\epsilon_{ijk}\xi^{k}_{3},~{}~{}~{}\mathcal{L}_{K^{iL}_{1}}\xi^{j}_{3}=\mathcal{L}_{K^{i}_{D}}\xi^{j}_{3}=\mathcal{L}_{K^{A}_{SO(5)/SO(4)}}\xi^{j}_{3}=0,~{}~{}~{}~{}\mathcal{L}_{K^{iL}_{2}}\xi^{\alpha}_{5}=0$
(D.27) $\displaystyle\mathcal{L}_{K^{iL}_{1}}\xi^{\alpha}_{5}$
$\displaystyle=\left(\begin{array}[]{c|c}T^{iL}_{1}&0_{4}\\\ \hline\cr
0^{T}_{4}&0\end{array}\right)^{\alpha}_{~{}\beta}\xi_{5}^{\beta},~{}~{}\mathcal{L}_{K^{i}_{D}}\xi^{\alpha}_{5}=\left(\begin{array}[]{c|c}T^{i}_{D}&0_{4}\\\
\hline\cr
0^{T}_{4}&0\end{array}\right)^{\alpha}_{~{}\beta}\xi_{5}^{\beta},~{}~{}\mathcal{L}_{K^{A}_{SO(5)/SO(4)}}\xi^{\alpha}_{5}=\left(\begin{array}[]{c|c}0_{4\times
4}&-c_{A}\\\ \hline\cr
c_{A}^{T}&0\end{array}\right)^{\alpha}_{~{}\beta}\xi_{5}^{\beta},$ (D.34)

where the A${}^{\text{th}}$ entry of $c_{A}$ is 1 with the rest zero and where

$T^{iL}_{1}=\frac{i}{2}(\sigma_{1}\otimes\sigma_{2},~{}\sigma_{2}\otimes\mathbb{I}_{2},~{}\sigma_{3}\otimes\sigma_{2})^{i},~{}~{}~{}T^{i}_{D}=-\frac{i}{2}(\sigma_{2}\otimes\sigma_{1},~{}\sigma_{2}\otimes\sigma_{3},~{}\mathbb{I}_{2}\otimes\sigma_{2})^{i}$
(D.35)

such that $(T^{iL}_{1},T^{i}_{D})$ span $\mathfrak{so}$(4) (which
$\xi^{5}_{5}$ is a singlet under) and together with the matrices involving
$c_{A}$ give a fundamental representation of $\mathfrak{so}$(5). This confirms
that $\xi^{j}_{3}$ and $\xi^{\alpha}_{5}$ indeed transform in the fundamentals
of SO(3) and SO(5) respectively, and are singlets under the opposing groups.
We thus conclude that the $\xi^{I}_{+}$ Killing spinors are relevant for
solutions preserving ${\cal N}=(3,0)$ and $(5,0)$, but not $(8,0)$.

The $\xi^{I}_{-}$ spinors behave quite differently, they do not decompose into
separate multiplets instead they transform in the (4,2) of
$\mathfrak{sp}$(2)$\oplus\mathfrak{sp}$(1). This fact is made considerably
more obvious by defining

$\xi^{I}_{(4,2)}=\left(\begin{array}[]{c}\xi^{1}_{-}\\\ \xi^{2}_{-}\\\
\xi^{3}_{-}\\\ \xi^{4}_{-}\\\ -\xi^{5}_{-}\\\ -\xi^{7}_{-}\\\ -\xi^{6}_{-}\\\
-\xi^{8}_{-}\end{array}\right)^{I}$ (D.36)

which transforms under the Killing vectors the squashing preserves as

$\displaystyle\mathcal{L}_{K^{iL}_{2}}\xi^{I}_{(4,2)}$
$\displaystyle=\big{[}\mathbb{I}_{4}\otimes(\frac{i}{2}\sigma_{i})\big{]}^{I}_{~{}J}\xi^{J}_{(4,2)},~{}~{}~{}\mathcal{L}_{K^{iL}_{1}}\xi^{I}_{(4,2)}=\big{[}(0_{2\times
2}\oplus\frac{i}{2}\sigma_{i})\otimes\mathbb{I}_{2}\big{]}^{I}_{~{}J}\xi^{J}_{(4,2)},$
$\displaystyle\mathcal{L}_{K^{i}_{D}}\xi^{I}_{(4,2)}$
$\displaystyle=\big{[}(\frac{i}{2}\sigma_{i}\oplus 0_{2\times
2})^{*}\otimes\mathbb{I}_{2}\big{]}^{I}_{~{}J}\xi^{J}_{(4,2)},~{}~{}~{}\mathcal{L}_{K^{A}_{SO(5)/SO(4)}}\xi^{I}_{(4,2)}=\big{[}\Sigma_{SO(5)/SO(4)}^{A}\otimes\mathbb{I}_{2}\big{]}^{I}_{~{}J}\xi^{J}_{(4,2)}$
(D.37)

for

$\Sigma_{SO(5)/SO(4)}^{A}=\frac{i}{2}(-\sigma_{2}\otimes\sigma_{2},~{}\sigma_{2}\otimes\sigma_{1},~{}-\sigma_{1}\otimes\mathbb{I}_{2},~{}\sigma_{2}\otimes\sigma_{3})^{A}.$
(D.38)

Clearly $\frac{i}{2}\sigma_{i}$ spans $\mathfrak{sp}(1)=\mathfrak{su}(2)$,
while $((\frac{i}{2}\sigma_{i})^{*}\oplus 0_{2\times 2},~{}0_{2\times
2}\oplus\frac{i}{2}\sigma_{i},~{}\Sigma_{SO(5)/SO(4)}^{A})$ give 10
independent anti-Hermitian 4$\times$4 matrices formed of anti-Hermitian
2$\times$2 blocks, so span $\mathfrak{sp}(2)$ \- as every component of
$\xi^{I}_{(4,2)}$ transforms under both these groups we conclude that it does
indeed transform in the (4,2) of $\mathfrak{sp}$(2)$\oplus\mathfrak{sp}$(1).
As such we see that only the $\xi^{I}_{-}$ Killing spinors are suitable for
realising $\mathcal{N}=(8,0)$ on the squashed 7-sphere.

###  D.2 Generalized Killing spinors on the squashed S7

The discussion in the previous section perfectly matches the group theoretical
argument in equation (2.11), which predicted that there is a Killing spinor
multiplet on the S7 that transforms in the (4,2), while the other two
multiplets are given by acting on this one with the squashed S7 invariant
forms $\Lambda^{0}_{4},\tilde{\Lambda}_{0}^{4}$, as already discussed in
section 6.3.

Notice however that such multiplets cannot be made of Killing spinors on the
squashed S7 since such condition would force the manifold to be Einstein,
which is not in general the case. What we have to do is therefore to
generalize the Killing spinor equation in such a way that it preserves the
multiplet, i.e. if $\xi_{-}$ solves the equation also $\mathcal{L}_{K}\xi_{-}$
is a solution, where $K$ is a generic Killing vector of Sp(2)$\times$Sp(1).
Requiring this property forces us to use just invariant forms on the squashed
S7 to deform the equation, which in general reads:

$\begin{split}\nabla_{a}\xi_{-}=&-\frac{i}{2}\Big{[}\gamma_{a}+\left(a_{1}\Lambda^{0}_{4}-\frac{a_{2}}{2}\tilde{\Lambda}^{0}_{4}\right)\gamma_{a}+\gamma_{a}\left(a_{3}\Lambda^{0}_{4}-\frac{a_{4}}{2}\tilde{\Lambda}^{0}_{4}\right)\\\
+&\frac{1}{2}\left(a_{5}\Lambda^{0}_{4}\gamma_{a}\tilde{\Lambda}_{0}^{4}+a_{6}\tilde{\Lambda}^{0}_{4}\gamma_{a}\Lambda^{0}_{4}+a_{7}\tilde{\Lambda}^{0}_{4}\gamma_{a}\tilde{\Lambda}^{0}_{4}+a_{8}\Lambda^{0}_{4}\gamma_{a}\Lambda^{0}_{4}\right)\Big{]}\xi_{-},\end{split}$
(D.39)

since we don’t have to consider higher powers of the invariant forms thanks to
the relations

$(\Lambda^{0}_{4})^{2}=1,\quad\Lambda^{0}_{4}\tilde{\Lambda}^{0}_{4}=\tilde{\Lambda}^{0}_{4}\Lambda^{0}_{4}=\tilde{\Lambda}^{0}_{4},\quad(\tilde{\Lambda}_{0}^{4})^{2}=6+6\Lambda^{0}_{4}-4\tilde{\Lambda}^{0}_{4}.$
(D.40)

Let’s now prove that $(\mathcal{L}_{K}\xi_{-})$ is also a solution if we
restrict ourselves to consider (D.39). Following [58] for the usual Killing
spinor case, we have:

$\begin{split}\nabla_{a}(\mathcal{L}_{K}\xi_{-})=&\mathcal{L}_{K}\nabla_{a}\xi_{-}+\partial_{a}K^{b}\nabla_{b}\xi_{-}\\\
=&\mathcal{L}_{K}\Big{[}-\frac{i}{2}\left(\gamma_{a}+\left(a_{1}\Lambda^{0}_{4}-\frac{a_{2}}{2}\tilde{\Lambda}^{0}_{4}\right)\gamma_{a}+\dots\right)\xi_{-}\Big{]}\\\
+&\partial_{a}K^{b}\Big{[}-\frac{i}{2}\left(\gamma_{b}+\left(a_{1}\Lambda^{0}_{4}-\frac{a_{2}}{2}\tilde{\Lambda}^{0}_{4}\right)\gamma_{b}+\dots\right)\xi_{-}\Big{]}\,.\end{split}$
(D.41)

Now, thanks to the fact that the action of $\mathcal{L}_{K}$ on the invariant
forms is zero, we have that $\mathcal{L}_{K}$ just acts on the free-index
gamma matrix $\gamma_{a}$. The commutator between $\mathcal{L}_{K}$ and
$\gamma_{a}$ can be written has following:

$[\mathcal{L}_{K},\gamma_{a}]=(\nabla_{a}K_{b}-\nabla_{[a}K_{b]}-\partial_{a}K_{b})\gamma^{b}=(\nabla_{(a}K_{b)}-\partial_{a}K_{b})\gamma^{b})=\partial_{a}K^{b}\gamma_{b}$
(D.42)

so that we are left with

$\nabla_{a}(\mathcal{L}_{K}\xi_{-})=-\frac{i}{2}\left(\gamma_{a}+\left(a_{1}\Lambda^{0}_{4}-\frac{a_{2}}{2}\tilde{\Lambda}^{0}_{4}\right)\gamma_{a}+\dots\right)\mathcal{L}_{K}\xi_{-}$
(D.43)

which is what we wanted to prove.

In order to fix the coefficients $a_{i}$ in (D.39) we can use the following
equation:

$\gamma^{b}[\nabla_{a},\nabla_{b}]=\frac{1}{2}\gamma^{b}R_{ab},\qquad
R_{ab}=2(5+2\lambda^{2})g_{ab}-2(-1+5\lambda^{2})g_{ab}|_{\lambda=1}$ (D.44)

where $g_{ab}$ is the metric on the squashed S7 as defined in (D.2) while
$g_{ab}|_{\lambda=1}$ is the metric on the round S7. It is particularly easy
to notice from the explicit expression of the Ricci that the manifold is
Einstein for $\lambda^{2}=1,1/5$. Plugging (D.39) inside (D.44) we find a
consistency condition which is satisfied for

$a_{1}=a_{2}=a_{3}=a_{4}=\frac{a_{7}}{2}=\frac{\lambda-1}{2\lambda},\quad
a_{5}=a_{6}=\lambda+\frac{1}{2\lambda}-\frac{3}{2},\quad a_{8}=0.$ (D.45)

An explicit computation shows that the multiplets generated by
$\\{\xi_{-},\Lambda^{0}_{4}\xi_{-},\tilde{\Lambda}^{0}_{4}\xi_{-}\\}$ all
satisfy (D.39) with the choice of parameters (D.45)282828Notice that the
condition (D.44) does not depend on our multiplet $\xi_{-}$ and indeed we can
find solution to this equation with a different choice of parameters respect
to (D.45). However the (4,2) multiplets will not solve these generalized
Killing spinor equations..

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