WZ1 | $\frac{1}{2}(i\not{D}^{(1)_{+}}_{1/2})$ |  $0^{-}(Q^{(3)_{+}})$ |  | $0_{\scriptscriptstyle(+)}^{+}(\Delta_{0}^{(1)})$ | $\bf{1}$ | $\frac{3}{2}+\frac{5}{2}+\frac{1}{6}\sqrt{\vphantom{C_{G}}\smash[b]{20C_{g}+81}}$
WZ2 | $\frac{1}{2}(i\not{D}^{(1)_{-}}_{1/2})$ |  $0^{+}_{\scriptscriptstyle(-)}(\Delta_{0}^{(1)})$ |  $\leftrightarrow$ | $0^{-}(Q^{(3)_{-}})$ | $\bf{1}$ | $\frac{3}{2}\pm\bigl{(}-\frac{5}{2}+\frac{1}{6}\sqrt{\vphantom{C_{G}}\smash[b]{20C_{g}+81}}\bigr{)}$
WZ3 | $\frac{1}{2}(i\not{D}_{3/2}^{(1)_{+}})$ |  $0^{+}(\Delta_{L}^{(2)_{+}})$ |  | $0^{-}(Q^{(1)_{+}})$ | $\bf{1}$ | $\frac{3}{2}+\frac{5}{6}+\frac{1}{6}\sqrt{\vphantom{C_{G}}\smash[b]{20C_{g}+1}}$
WZ4 | $\frac{1}{2}(i\not{D}_{3/2}^{(1)^{\prime}_{+}})$ |  $0^{+}(\Delta_{L}^{(2)^{\prime}_{+}})$ |  | $0^{-}(Q^{(1)^{\prime}_{+}})$ | $\bf{5}$ | $\frac{3}{2}+\frac{5}{6}+\frac{1}{6}\sqrt{\vphantom{C_{G}}\smash[b]{20C_{g}+1}}$
WZ5 | $\frac{1}{2}(i\not{D}_{3/2}^{(1)_{-}})$ |  $0^{-}(Q^{(1)_{-}})$ |  | $0^{+}(\Delta_{L}^{(2)_{-}})$ | $\bf{1}$ | $\frac{3}{2}-\frac{5}{6}+\frac{1}{6}\sqrt{\vphantom{C_{G}}\smash[b]{20C_{g}+1}}$
WZ6 | $\frac{1}{2}(i\not{D}_{3/2}^{(1)^{\prime}_{-}})$ |  $0^{-}(Q^{(1)^{\prime}_{-}})$ |  | $0^{+}(\Delta_{L}^{(2)^{\prime}_{-}})$ | $\bf{5}$ | $\frac{3}{2}-\frac{5}{6}+\frac{1}{6}\sqrt{\vphantom{C_{G}}\smash[b]{20C_{g}+1}}$
Table 5: Wess–Zumino supermultiplets for $C_{g}>0$. See Table 4 for
definitions. The upper sign in the WZ2 $E_{0}$ value corresponds to $(D,D,D)$
boundary conditions for $p+q\geq 3$, $(D,N,D)$ for $p+q=2$ and $(N,N,D)$ for
$p+q=1$. The scalar $0^{+}$ and pseudo-scalar $0^{-}$ should change places in
the WZ2 supermultiplet if the lower sign in the $E_{0}$ formula is used, as
indicated by the arrow. The lower sign is only valid for $p+q=2$ and $p+q=1$
and then corresponds to $(N,N,D)$ and $(D,N,D)$ boundary conditions,
respectively. The subscript on $0^{+}_{\scriptscriptstyle(\pm)}$ indicates
which branch of $M^{2}(\Delta_{0})$ is used.

We now turn to the six infinite sequences of Wess–Zumino multiplets with
$C_{g}>0$ in Table 5. The cross diagrams and operator eigenvalues for the
first two Wess–Zumino multiplets in this list are specified by the spin-1/2
entries, see Figure 3 and 181–182. Note that the isometry singlet mode of
$\Delta_{0}$ should be removed from $0^{+}_{\scriptscriptstyle(-)}$, but not
from $0^{+}_{\scriptscriptstyle(+)}$, as explained in Duff:1986hr . Looking at
WZ1–2, this is consistent with the fact that the $(0,0;0)$ eigenmode of
$i\not{D}_{1/2}$ has an eigenvalue given by $i\not{D}_{1/2}^{(1)_{+}}$ and
there is no $(0,0;0)$ eigenmode of $i\not{D}_{1/2}$ with eigenvalue
$i\not{D}_{1/2}^{(1)_{-}}$, see Figure 3.

For the Wess–Zumino multiplets numbered 3 to 6, on the other hand, it is not
possible, with the information obtained in this paper, to group the remaining
twelve cross diagrams, for each of the operators $i\not{D}_{3/2}$,
$\Delta_{L}$ and $Q$, in the appendix of Nilsson:2018lof  into two separate
sets of ${\bm{1}}+{\bm{5}}$. However, by invoking also results from
Duboeuf:2022mam  this can be done.202020Note that the $\Delta$ of
Duboeuf:2022mam  is the lowest conformal dimension of a given supermultiplet
and, hence, shifted compared to the $E_{0}$-value we use here,
$\Delta=E_{0}-1/2$, see Footnote 3. The result is presented in Figure 5. If
one constructs the corresponding eigenmodes, we expect that the crosses below
the boxed crosses cannot uniquely be assigned to the $\bm{1}$ or the $\bm{5}$,
similar to what we found in Section 4.2.

|  | WZ3–4 |  |
---|---|---|---|---
$1$$2$$3$$4$$1$$2$$3$$4$$0$$p$$0$$q$$r=p-4$$\times$$\times$$\times$$\times$$\times$ |  $1$$2$$3$$4$$1$$2$$3$$4$$0$$p$$0$$q$$r=p-2$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$ |  $1$$2$$3$$4$$1$$2$$3$$4$$0$$p$$0$$q$$r=p$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\boxtimes$$\boxtimes$$\boxtimes$$\boxtimes$$\boxtimes$$\boxtimes$$\boxtimes$$\boxtimes$$\boxtimes$$\boxtimes$$\boxtimes$$\boxtimes$ |  $1$$2$$3$$4$$1$$2$$3$$4$$0$$p$$0$$q$$r=p+2$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$ |  $1$$2$$3$$4$$1$$2$$3$$4$$0$$p$$0$$q$$r=p+4$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$
|  | WZ5–6 |  |
$1$$2$$3$$4$$1$$2$$3$$4$$0$$p$$0$$q$$r=p-4$$\times$$\times$$\times$$\times$$\times$ |  $1$$2$$3$$4$$1$$2$$3$$4$$0$$p$$0$$q$$r=p-2$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$ |  $1$$2$$3$$4$$1$$2$$3$$4$$0$$p$$0$$q$$r=p$$\otimes$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\boxtimes$$\boxtimes$$\boxtimes$$\boxtimes$$\boxtimes$$\boxtimes$$\boxtimes$$\boxtimes$$\boxtimes$ |  $1$$2$$3$$4$$1$$2$$3$$4$$0$$p$$0$$q$$r=p+2$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$ |  $1$$2$$3$$4$$1$$2$$3$$4$$0$$p$$0$$q$$r=p+4$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$
Figure 5: The cross diagrams in the appendix of Nilsson:2018lof  that remain
when all other supermultiplets with their respective cross diagrams have been
removed. These twelve diagrams correspond to two sets of $\bm{1}\oplus\bm{5}$.
The displayed division into two different sets, see Table 5, relies on the
results of Duboeuf:2022mam  for small $p$ or $q$. The multiplicities of the
isometry irreps are one for crosses and two for boxes with crosses. The single
encircled cross, in the irrep $(0,0;0)$, is the Page supermultiplet (see
below) containing the mode “coming from nowhere” in the analysis of
Nilsson:2018lof ; Nilsson:2023ctq .

From the general structure of a Wess–Zumino multiplet
$D(\frac{1}{2},E_{0})\oplus D(0^{\pm},E_{0}-\frac{1}{2})\oplus
D(0^{\mp},E_{0}+\frac{1}{2})$, we see that there can be no Neumann boundary
conditions if $E_{0}\geq 2$. For $\frac{3}{2}<E_{0}<2$, the boundary
conditions are $(D,N,D)$ and if all boundary conditions are flipped the fields
fit into a supermultiplet $(\frac{1}{2},0^{\mp},0^{\pm})$ where the scalar and
pseudo-scalar have switched places and the boundary conditions are
$(N,N,D)$.212121The case $E_{0}=3/2$ is identical to the $3/2<E_{0}<2$ one
except that the boundary conditions for the spinor now stay the same if the
boundary conditions for the scalar and pseudo-scalar are flipped.

That only irreps with $C_{g}\geq 4$ occur in Figure 5, except for the isometry
singlet with $C_{g}=0$, implies that Neumann boundary conditions are never
possible in WZ3–6, only in WZ2. This happens for the five irreps with $p+q\leq
2$ and $C_{g}$ values given in Table 6.

Specifically, the $(1/2,0^{+},0^{-})$ WZ2 supermultiplet as given in Table 5
have $(D,D,D)$ boundary conditions for $p+q\geq 3$, $(D,N,D)$ for $p+q=2$ and
$(N,N,D)$ for $p+q=1$. For $p+q=2$ (last three entries in Table 6) and $p+q=1$
(first two entries in Table 6) the boundary conditions of all three fields can
be flipped and they then fit in a $(1/2,0^{-},0^{+})$ supermultiplet with
boundary conditions $(N,N,D)$ and $(D,N,D)$, respectively. Note that the
$E_{0}$ value of the supermultiplet given in the table should be replaced by
$3-E_{0}$ in these cases.

Also for the $C_{g}$ values in the range $12<C_{g}<25$ ($p+q=3$) both
Dirichlet and Neumann boundary conditions are possible in WZ2 but now only for
the scalars $0^{+}$. A complete list of fields with $E_{0}$ in this window is
provided in Table 9 below. After skew-whiffing there seems to be no reason for
choosing one or the other of the boundary conditions for these $0^{+}$ fields.
The supermultiplets in the left-squashed case containing these scalar fields
can, however, only be of the kind $(D,D,D)$.

$(p,q;r)$ | $(0,1;0)$ | $(1,0;1)$ | $(1,1;1)$ | $(0,2;0)$ | $(2,0;2)$
---|---|---|---|---|---
$(d_{2},d_{1})$ | $({\bf 5},{\bf 1})$ | $({\bf 4},{\bf 2})$ | $({\bf 8},{\bf 2})$ | $({\bf 14},{\bf 1})$ | $({\bf 10},{\bf 3})$
$C_{g}$ | 4 | 19/4 | 39/4 | 10 | 12
Table 6: Irreps with $r=p$ and $C_{g}>0$ for which Wess–Zumino multiplets in
WZ2 admit both Dirichlet and Neumann boundary conditions. The implications of
this are explained in the text. Here, $(d_{2},d_{1})$ denotes the dimensions
of the $sp_{2}$\- and $sp_{1}^{C}$-irreps.

It would be interesting to find a way to discriminate between the different
boundary conditions respecting supersymmetry. This could potentially be
achieved by relating the squashed vacuum to the round one with
$\mathcal{N}=8$. In this context, we note that if one squashes the seven-
sphere continuously and follows the spinors in the isometry irrep $(0,0;2)$ of
$G=Sp_{2}\times Sp_{1}^{C}$, which comes from the $Spin(8)$-irrep $(0,0,1,0)$
on the round sphere (see Table 17), the relevant eigenvalue is Nilsson:1983ru

$i\not{D}_{1/2}(0,0;2)=-\frac{\sqrt{7}}{10}\frac{10\lambda^{2}+25}{\sqrt{1+8\lambda^{2}-2\lambda^{4}}}m(\lambda),$
(226)

where

$m^{2}(\lambda)=\frac{1+8\lambda^{2}-2\lambda^{4}}{28\lambda^{2}}$ (227)

comes from demanding that $R=42m^{2}$ for all values of the squashing
parameter $\lambda$. From this, we see that the associated $AdS_{4}$ field
starts out as a fermionic singleton, with
$i\not{D}_{1/2}(0,0;2)=\frac{-7m}{2}$ and $E_{0}=1$, in the round sphere
vacuum ($\lambda^{2}=1$) and ends up as the massless spin-1/2 companion of the
gauge field in the short massless supermultiplet related to the $Sp_{1}^{C}$
isometry after squashing ($\lambda^{2}=1/5$), with
$i\not{D}_{1/2}^{(2)_{-}}(0,0;2)=\frac{-9m}{2}$ and $E_{0}=\frac{3}{2}$.
Hence, this is one example of Higgsing involving a singleton and, thus, also a
change of boundary conditions triggered by squashing.222222The language used
here describes what happens when going from round to squashed, for consistency
with Nilsson:2023ctq ; Nilsson:2018lof , even though the round vacuum has to
be in the IR of the squashed one for any potential RG-flow connecting the two,
see Section 6.

Another relevant point is that there can be exactly marginal singlet operators
that can be used to deform the dual CFT, as discussed in, e.g., Witten:2001ua
. We will return to the issue of marginal operators in Section 5.2 where we
give the full content of marginal single- and multi-trace operators for the
theory discussed in this paper. A perhaps relevant fact here is that the
right-squashed spectrum does not allow for any single- or double-trace
marginal operators. If these considerations are applied to the left-squashed
case, such deformations will break supersymmetry.

As an example where different boundary conditions are possible, we consider
the irrep $(1,0;1)$ (with dimension $(\bm{4},\bm{2})$) in the Einstein-
squashed spectrum. The scalar $0^{+}$ in this multiplet has
$M^{2}(0^{+}_{\scriptscriptstyle(-)})=95m^{2}/9$, from which we see that

$0^{+}_{\scriptscriptstyle(-)}(\Delta_{0}^{(1)})_{(1,0;1)}\colon\quad
E_{0}=\frac{3}{2}\pm\frac{1}{6}\bigl{(}18-4\sqrt{11}{\mkern
2.0mu}\bigr{)}>\frac{1}{2},$ (228)

which is compatible with both Dirichlet ($E_{0}>3/2$) and Neumann
($1/2<E_{0}<3/2$) boundary conditions. For the other two fields in this
Wess–Zumino multiplet we have

$\displaystyle\frac{1}{2}(i\not{D}_{1/2}^{(1)_{-}})_{(1,0;1)}\colon\quad$
$\displaystyle E_{0}=\frac{3}{2}\pm\frac{1}{6}\bigl{(}15-4\sqrt{11}{\mkern
2.0mu}\bigr{)}>1,$ (229) $\displaystyle
0^{-}(Q^{(3)_{-}})_{(1,0;1)}\colon\quad$ $\displaystyle
E_{0}=\frac{3}{2}\pm\frac{1}{6}\bigl{(}4\sqrt{11}-12\bigr{)}>\frac{1}{2},$
(230)

which also allow for both types of boundary conditions. One should note here
that the three fields in a supermultiplet must have the same sign of the
square root term in $E_{0}$, due to supersymmetry, which implies that the
supermultiplets will involve fields with different boundary conditions as
described above.

Remarkably, it is not consistent to assume that all fields will retain their
boundary conditions when squashing the seven-sphere from the round $S^{7}$ to
the Einstein-squashed one, even though the spectrum can be understood in this
way through a Higgs/de-Higgs mechanism Nilsson:2018lof ; Nilsson:2023ctq .
This fact is clear from looking at the irrep $(0,2;0)$ for which all three
fields come from fields with Dirichlet boundary conditions in the round sphere
vacuum which leads to an impossible set of boundary conditions (see Table 6)
in the left-squashed case if supersymmetry is kept intact. In fact, also for
the irrep $(1,0;1)$ it is clear that one cannot keep the boundary conditions
when squashing since the spin-1/2 field comes from the massless field
$(1,0,1,0)$ with $E_{0}=3/2$ in the round spectrum. This field has neither
Dirichlet nor Neumann boundary conditions, as discussed in, e.g.,
Witten:2001ua .

The last supermultiplets to account for are the Wess–Zumino ones with
$C_{g}=0$. There are only two such multiplets, see Table 8, which is clear
from the cross diagrams in Nilsson:2018lof . We have listed the two
Wess–Zumino multiplets that exist for $C_{g}=0$ separately since all the
relevant mode functions can be constructed explicitly as done recently in
Nilsson:2023ctq  and reproduced in Section 4.3 of this paper. The results
obtained there are summarised in Table 7.

Spin | Eigenvalue | Mass | $E_{0}$
---|---|---|---
$\frac{1}{2}$ | $i\not{D}_{1/2}^{(1)_{+}}=\frac{7m}{2}$ | $M=-8m$ | $\frac{11}{2}$
$\frac{1}{2}$ | $i\not{D}_{3/2}^{(1)_{-}}=-\frac{m}{6}$ | $M=\frac{4m}{3}$ | $\frac{13}{6}$
$0^{+}$ | $\Delta_{0}^{(1)}=0$ | $M^{2}_{\scriptscriptstyle(+)}=80m^{2}$ | $6$
$0^{+}$ | $\Delta_{L}^{(2)_{-}}=\frac{28m^{2}}{9}$ | $M^{2}=-\frac{8m^{2}}{9}$ | $\frac{3}{2}\pm\frac{1}{6}$
$0^{-}$ | $Q^{(3)_{+}}=4m$ | $M^{2}=48m^{2}$ | $5$
$0^{-}$ | $Q^{(1)_{-}}=-\frac{2m}{3}$ | $M^{2}=\frac{40m^{2}}{9}$ | $\frac{8}{3}$
Table 7: $AdS_{4}$ spin, operator eigenvalues, masses and $E_{0}$ values of
isometry singlet ($C_{g}=0$) modes and the associated $AdS_{4}$ fields. The
plus subscript on $M^{2}$ indicates the branch of the mass formula in Table 3.
The $\pm$ in the $E_{0}$ column corresponds to different boundary conditions.
Below, we will see that supersymmetry fixes the plus sign.

The fields corresponding to isometry singlet modes (see Table 7) fit into two
Wess–Zumino supermultiplets uniquely, given in Table 8. All three fields in
both these supermultiplets have Dirichlet boundary conditions, which in the
case of the Page multiplet is a consequence of supersymmetry. This is most
easily seen by noting that the spin-1/2 fields have $E_{0}>2$.

After skew-whiffing the situation is a bit more complicated since there is no
supersymmetry in the right-squashed vacuum and several choices of boundary
conditions become possible Nilsson:2023ctq . We shall return to this issue
below.

Mult. | $E_{0}$ | $E_{0}-\frac{1}{2}$ | $E_{0}+\frac{1}{2}$ | $\mathclap{Sp^{C}_{1}}$ |  $E_{0}$ values
---|---|---|---|---|---
$(\text{WZ1})_{0}$ | $\frac{1}{2}({i\not{D}}_{1/2}^{(1)_{+}})$ | $0^{-}(Q^{(3)_{+}})$ | $0_{\scriptscriptstyle(+)}^{+}(\Delta_{0}^{(1)})$ | $\bf{1}$ | $\frac{3}{2}+4=\frac{11}{2}$
Page | $\frac{1}{2}(i\not{D}_{3/2}^{(1)_{-}})$ | $0^{+}(\Delta_{L}^{(2)_{-}})$ | $0^{-}(Q^{(1)_{-}})$ | $\bf{1}$ | $\frac{3}{2}+\frac{2}{3}=\frac{13}{6}$
Table 8: Wess–Zumino supermultiplets with $C_{g}=0$.

While the $(\text{WZ1})_{0}$ value $E_{0}=\frac{3}{2}+4$ fits into the general
$E_{0}$ formula for WZ1 this is not the case for the Page multiplet which does
not fit into any of the $E_{0}$ formulas in Table 5. However, it does
nevertheless belong to WZ5–6 as we will now explain. The reason for the odd
behaviour of the Page multiplet is that it utilises a special property that
arises for $C_{g}=0$ in the $E_{0}$ formulas for all three fields in the
multiplet, similar to the five cases in Table 6. These formulas are of the
form $\frac{3}{2}\pm|\sqrt{\vphantom{C_{G}}\smash[b]{20C_{g}+1}}-X|$ where the
argument of the absolute value is negative only for the singlet isometry
irrep. This leads to the scalar and pseudo-scalar of the Page multiplet
changing places relative to WZ5–6 and the $E_{0}$ value being reflected in
$3/2$ relative to the formula in Table 5.

Note that if the $E_{0}$ value of the spinor in the Page multiplet would have
been less than $2$, the boundary conditions of the fields could have been
flipped and the multiplet would have fitted exactly into WZ5–6 as given in the
table. Attempting this with the actual values of $E_{0}$ leads to a violation
of the unitarity bounds.

The Page WZ-multiplet contains the scalar that was discussed by Page in
Page:1983mke . This scalar is the squashing mode of the $S^{7}$
compactification and was shown by Page to have $M^{2}=-\frac{8}{9}m^{2}$ (just
above the BF bound) in the Einstein-squashed case.

Clearly there are no singletons in the spectrum of the left-squashed seven-
sphere Nilsson:2018lof . The singleton that exists after skew-whiffing to the
right-squashed vacuum Nilsson:2018lof  will be discussed below.

###  5.2 Boundary conditions and marginal operators in the
$\mathcal{N}\\!=\\!1$ and $\mathcal{N}\\!=\\!0$ vacua

subsubsection subsubsection

One reason for constructing the entire operator spectrum of the left-squashed
seven-sphere is that one can use supersymmetry as a consistency check of the
results and then flip its orientation, i.e., perform a so called skew-whiffing
Duff:1986hr , to produce the right-squashed compactification with no
supersymmetry. This solution is known to be BF-stable Duff:1984sv  despite
having no supersymmetry and there should therefore exist, in view of the
swampland $AdS$ swampland conjecture first discussed in Ooguri:2016pdq , some
other decay mode with respect to which the right-squashed vacuum is unstable.

One possible such decay mode is related to special composite operators in the
boundary theory of the $AdS_{4}$ gravity theory and condensation of scalar
composites in the bulk. As discussed in the Introduction (and in the
references cited there), if marginal operators exist ($\Delta=3$), either
single, double- or multi-trace, that may signal an instability of the
$AdS_{4}$ theory. For the gravity theory in the bulk, this corresponds to
looking for composite scalar gauge singlet states with total $E_{0}=3$.

Such trace operators can be of only a few kinds, namely, (schematically)
$\lambda^{2}\phi^{n}$ for $n=0,1,2$ and $\phi^{n}$ for $n=1,\ldots,6$, with
the spinor $\lambda$ and scalar $\phi$ representing both the bulk fields and
their dual single-trace operators (with $\Delta=E_{0}$). Note that in the last
cases in these lists the fields have to be singletons and scalar singletons
are known to exist only in the round sphere case Nilsson:2018lof . There is
also a spinorial singleton with $E_{0}=1$ in the right-squashed spectrum
Nilsson:2018lof  which could play a role here. In fact, in the recent analysis
of the singlet sector of the right-squashed spectrum in Nilsson:2023ctq  the
fields in this sector were found to either give rise to two kinds of marginal
triple-trace operators, of which one involved the fermionic singleton
mentioned above, or to give rise to no marginal operators depending on which
of two possible choices of boundary conditions one imposes on the fields.

Here, we will generalise this analysis to all fields by including also the
ones coming from modes with $C_{g}>0$, i.e., non-singlet modes. These fields
have non-trivial properties even in the left-squashed supersymmetric case
which were analysed in Section 5.1. In Section 5.2, we will therefore perform
a complete analysis of all fields which can have Neumann boundary conditions
in both the left- and right-squashed vacuum, after discussing skew-whiffing
and the right-squashed spectrum in Section 5.2. Marginal operators will be
discussed in detail in Section 5.2. There we will find that, since there exist
several fields in the squashed vacua with $E_{0}\leq\frac{3}{2}$, bound states
containing different fields may give rise to large numbers of marginal
operators depending on the boundary conditions. In fact, one such mixed
example was discussed recently in Nilsson:2023ctq .

#### Skew-whiffing and the spectrum of the right-squashed vacuum

To see the effects on the spectrum of an orientation flip, we should recall
how it is implemented, see for instance Duff:1986hr . The definition of the
linear operators $Q$, $i\not{D}_{1/2}$ and $i\not{D}_{3/2}$ are sensitive to
the orientation and, as explained in Duff:1986hr , the sign of their spectra
are flipped when the orientation is changed.232323This way to implement skew-
whiffing is designed to keep the $AdS_{4}$ the same when flipping, including
the flux determined by $m$. One way to do this is to leave the vielbein, the
epsilon-tensor and the gamma-matrices intact on $S^{7}$ and produce the sign
flip in the volume form by letting the coordinates $y^{i}$ go to $-y^{i}$. The
same effect can be obtained by changing $m\rightarrow-m$ and keeping the
$S^{7}$ intact. Thus this has implications only for the $AdS_{4}$ fields with
spin 3/2, 1/2 and the pseudo-scalars $0^{-}$. In practice, we can implement
this conveniently by only referring to the operators, eigenvalues and cross
diagrams of the left-squashed seven-sphere and flip the signs of the linear
operators in the mass formulas as follows (cf. Table 3):

$\displaystyle\tfrac{3}{2}_{\scriptscriptstyle(\pm)}(i\not{D}_{1/2})\colon\qquad$
$\displaystyle M_{R}=i\not{D}_{1/2}+\frac{7m}{2},$ (231)
$\displaystyle\tfrac{1}{2}_{\scriptscriptstyle(\pm)}(i\not{D}_{1/2})\colon\qquad$
$\displaystyle M_{R}=i\not{D}_{1/2}-\frac{9m}{2},$ (232)
$\displaystyle\tfrac{1}{2}_{\scriptscriptstyle(\pm)}(i\not{D}_{1/2})\colon\qquad$
$\displaystyle M_{R}=-i\not{D}_{3/2}+\frac{3m}{2},$ (233) $\displaystyle
0^{-}_{\scriptscriptstyle(\pm)}(Q)\colon\qquad$ $\displaystyle
M_{R}^{2}=(Q-3m)^{2}-m^{2},$ (234)

where we have introduced the following notation for masses of scalar and
spin-1/2 fields in the left- and right-squashed vacua: We keep the standard
notation $M^{2}$ and $M$ in the cases where the orientation does not matter,
while, when it does matter, we from now on use the notation $M^{2}_{L}$ or
$M_{L}$ and $M^{2}_{R}$ or $M_{R}$ for the left- and right-squashed cases,
respectively. The right-squashed masses have several interesting consequences.
The first one is that the spinor $\eta$, satisfying
$i\not{D}_{1/2}\eta=\frac{7m}{2}\eta$, is no longer a Killing spinor and hence
does not lead to a massless spin-3/2 field in $AdS_{4}$ gravity theory and
supersymmetry is therefore lost. Furthermore, when using this spinor for the
spin-1/2 fields one finds a field with mass $M=-m$ which, if given Neumann
boundary conditions, will have $E_{0}=1$. This spinorial singleton is known
from Nilsson:2018lof  to exist in the right-squashed case (see also
Nilsson:2023ctq ).

#### Possible boundary conditions for the fields in both vacua

We will here tabulate the scalar fields in $AdS_{4}$ in the mass range
$-m^{2}\leq M^{2}\leq 3m^{2}$ and the fermions having $|M|\leq m$. These are
the only masses for which fields with both Dirichlet and Neumann boundary
conditions are possible. Since the boundary conditions affect the conformal
dimension of the dual operator, this is crucial for the analysis of marginal
multi-trace operators in Section 5.2.

To list these mass values we need the $C_{g}$ Casimir

$C_{g}(p,q;r)=\frac{1}{2}(p^{2}+2q^{2}+4p+6q+2pq)+\frac{3}{4}r(r+2).$ (235)

We will specify the irreps using the Dynkin labels $(p,q;r)$ as above. The
dimensions of the $sp_{2}$\- and $sp_{1}$-irreps can be obtained from these as

$\displaystyle d_{2}(p,q)=\frac{1}{6}(p+2q+3)(p+q+2)(p+1)(q+1),$ (236)
$\displaystyle d_{1}(r)=r+1,$ (237)

respectively.

The issue of stability, and as we saw above its connection to the existence of
fields with $E_{0}\leq\frac{3}{2}$, is thus related to when Neumann boundary
conditions are possible on $AdS_{4}$ for scalar and Dirac fields. This is
clear from their energy formulas

$E_{0}(s=0^{\pm})=\frac{3}{2}\pm\frac{1}{2}\sqrt{\frac{M^{2}}{m^{2}}+1}\geq\frac{1}{2},\qquad
E_{0}\biggl{(}s=\frac{1}{2}\biggr{)}=\frac{3}{2}\pm\frac{1}{2}\biggl{|}\frac{M}{m}\biggr{|}\geq
1,$ (238)

where the bounds are due to unitarity. Note also the immediate implication
that for scalars $M^{2}\geq-m^{2}$ which is the BF bound. This implies that
Neumann boundary conditions are possible, implementing also unitarity,
provided that

$\text{Scalars: }-m^{2}<M^{2}\leq 3m^{2},\qquad\text{Spinors: }-m\leq M\leq
m.$ (239)

Inserting the various operator spectra into the formulas for the mass matrices
we can check if there are modes that satisfy these bounds. We might suspect
from experience with the round $S^{7}$ spectrum that such states are quite
rare. In fact, for the round $S^{7}$ the only such fields are the $(2,0,0,0)$
(i.e., ${\bf 35}_{v}$ of $so(8)$) scalars in the $Spin(2,3)$-irrep
$D(1,0^{+})$ and the supersingleton242424Singletons belong to the lower
(minus) branch of the $E_{0}$ formulas for scalars and spin-1/2 fermions but
have a more intricate behaviour near the $AdS_{4}$ boundary than ordinary
fields with Neumann boundary conditions. Often, singletons are considered as
fields confined to the boundary. in $D(\frac{1}{2},0)\oplus D(1,\frac{1}{2})$
in the $so(8)$-irreps $(1,0,0,0)$ and $(0,0,1,0)$ (i.e., ${\bf 8}_{v}$ and
${\bf 8}_{c}$) as shown in Nilsson:2018lof . There are also some fields in the
round spectrum with $E_{0}=\frac{3}{2}$ that hence have neither Dirichlet nor
Neumann boundary conditions. These are scalars from the first massive level in
the irrep $(3,0,0,0)$ (or ${\bf 112}_{v}$) and spin-1/2 fields in the irrep
$(1,0,1,0)$ (or ${\bf 56}_{s}$). There are no pseudo-scalars of any similar
kind. However, we find here (see below) that in the $AdS_{4}$ spectra coming
from the squashed sphere, fields of these kinds are quite abundant. The
exception here is the singletons which exist only in the form of a singlet in
the right-squashed vacuum Nilsson:2018lof ; Nilsson:2023ctq .

We list, for the squashed sphere vacua, the scalar, pseudo-scalar and spinor
fields in the mass ranges specified above, that is, the isometry irreps for
which both Dirichlet and Neumann boundary conditions are possible, in Tables
9, 10 and 11, respectively. The analysis in the left-squashed case will show
that whenever there are two branches (either in the operator eigenvalues or in
$M^{2}$) only the minus branch will have any irreps in these ranges.
Similarly, only the plus branches of eigenvalues appear in the right-squashed
part of the pseudo-scalar and spinor tables. The scalar fields closest to the
BF bound are the ones in the $(p,q;r)=(2,0;2)$ isometry irrep of
$0^{+}_{\scriptscriptstyle(-)}(\Delta_{0}^{(1)})$, both in the left- and
right-squashed case. These have $M^{2}=-(12\sqrt{321}-212)m^{2}/3$, which is
less than one part per mille above the BF bound $-m^{2}$.

Here we note that the singlet spin-3/2 mode, which belong to the spectrum of
$i\not{D}_{3/2}^{(1)_{-}}$, is not present in neither the left- nor right-
squashed part of Table 11 due to it having $E_{0}=13/6>2$ and $E_{0}=7/3>2$,
respectively. Even though the $E_{0}$ formula corresponding to this eigenvalue
in the left-squashed case is
$\frac{3}{2}\pm\frac{1}{6}|-5+\sqrt{\vphantom{C_{G}}\smash[b]{20C_{g}+1}}|$,
where the square root could a-priori make Neumann boundary conditions
possible, Neumann boundary conditions are in fact never allowed by unitarity
due to which values of $C_{g}$ occur in the cross diagram associated to this
eigenvalue.

In the right-squashed vacuum we find the members of the $(\text{WZ1})_{0}$
multiplet that were discussed above and in Nilsson:2023ctq , namely the
singlet ones in $1/2(i\not{D}_{1/2}^{(1)_{+}})$ and $0^{-}(Q^{(3)_{+}})$. The
issue of instability due to possible marginal operators constructed from these
two singlet modes was discussed in Nilsson:2023ctq  with the conclusion that
marginal operators can either appear or not depending on the boundary
conditions of the corresponding fields. We will discuss this further in
Section 5.2.

We might add here that in general it is not possible to demand that a given
eigenmode of an operator should generate fields in the left- and right-
squashed vacua with the same boundary conditions. In the left-squashed vacuum,
we know from Section 5.1 that either the spinors associated with
$i\not{D}_{1/2}^{(1)_{-}}$ or the pseudo-scalars associated with $Q^{(3)_{-}}$
need to have Neumann boundary conditions for the isometry irreps $(0,1;0)$ and
$(1,0;1)$ to respect supersymmetry. However, we see from Tables 10 and 11 that
the fields associated with these modes in the right-squashed vacuum can only
have Dirichlet boundary conditions. Some fields must be allowed to change
boundary conditions also when squashing, as was mentioned already in Section
5.1 where the example of the WZ2 supermultiplet in the irrep $(0,2;0)$ was
given.

The spin-3/2 fields are also affected by skew-whiffing but, in this case, only
what happens to the Killing spinor mode leads to anything interesting and this
was already discussed above.

The scalar fields with $\frac{5}{2}<E_{0}\leq 3$ and the spinor fields with
$2<E_{0}\leq 3$ might also be of interest since they too are dual to relevant
single-trace operators. These have masses satisfying $3m^{2}<M^{2}\leq 8m^{2}$
and $m<|M|\leq 3m$, respectively, and can thus only have Dirichlet boundary
conditions. We list these in Appendix D.

The implications for the issue of possible instabilities due to marginal
operator appearing in the right-squashed vacuum are discussed in Section 5.2.
Having tabulated all fields with small masses we note that some of them have
indeed negative values which is in accord with the assertion of
Andriot:2022brg .

Field | $(p,q;r)$ | $C_{g}$ | $M^{2}/m^{2}$ | $E_{0}$
---|---|---|---|---
$0^{+}_{\scriptscriptstyle(-)}(\Delta_{0}^{(1)})$ | $(0,1;0)$ | $4$ | $\frac{1}{9}\bigl{(}476-36\sqrt{161}{\mkern 2.0mu}\bigr{)}$ | $\frac{3}{2}\pm\frac{1}{6}\bigl{(}18-\sqrt{161}{\mkern 2.0mu}\bigr{)}$
| $(1,0;1)$ | $19/4$ | $\frac{1}{9}\bigl{(}491-144\sqrt{11}{\mkern 2.0mu}\bigr{)}$ | $\frac{3}{2}\pm\frac{1}{3}\bigl{(}9-2\sqrt{11}{\mkern 2.0mu}\bigr{)}$
| $(1,1;1)$ | $39/4$ | $-\frac{1}{3}\bigl{(}24\sqrt{69}-197\bigr{)}$ | $\frac{3}{2}\pm\frac{1}{3}\bigl{(}9-\sqrt{69}{\mkern 2.0mu}\bigr{)}$
| $(0,2;0)$ | $10$ | $-\frac{1}{9}\bigl{(}36\sqrt{281}-596\bigr{)}$ | $\frac{3}{2}\pm\frac{1}{6}\bigl{(}18-\sqrt{281}{\mkern 2.0mu}\bigr{)}$
| $(2,0;2)$ | $12$ | $-\frac{1}{3}\bigl{(}12\sqrt{321}-212\bigr{)}$ | $\frac{3}{2}\pm\frac{1}{6}\bigl{(}18-\sqrt{321}{\mkern 2.0mu}\bigr{)}$
| $(1,2;1)$ | $67/4$ | $-\frac{1}{9}\bigl{(}144\sqrt{26}-731\bigr{)}$ | $\frac{3}{2}\pm\frac{1}{3}\bigl{(}2\sqrt{26}-9\bigr{)}$
| $(0,3;0)$ | $18$ | $0$ | $\frac{3}{2}\pm\frac{1}{2}$
| $(2,1;2)$ | $18$ | $0$ | $\frac{3}{2}\pm\frac{1}{2}$
| $(3,0;3)$ | $87/4$ | $\frac{1}{3}\bigl{(}277-24\sqrt{129}{\mkern 2.0mu}\bigr{)}$ | $\frac{3}{2}\pm\frac{1}{3}\bigl{(}\sqrt{129}-9\bigr{)}$
$0^{+}(\Delta_{L}^{(2)_{-}})$ | $(0,0;0)$ | $0$ | $-\frac{8}{9}$ | $\frac{3}{2}\pm\frac{1}{6}$
Table 9: Scalars that can have both Dirichlet and Neumann boundary conditions. The listed fields and their properties are the same in the left- and the right-squashed vacuum. Note that only the five first rows can respect supersymmetry with Neumann boundary conditions in the left-squashed vacuum, see Section 5.1. All brackets are positive. Field ($LS^{7}$) | $(p,q;r)$ | $C_{g}$ | $M^{2}_{L}/m^{2}$ | $E_{0}$
---|---|---|---|---
$0^{-}(Q^{(2)_{-}})$ | $(1,1;1)$ | $39/4$ | $\frac{5}{9}\bigl{(}67-8\sqrt{61}{\mkern 2.0mu}\bigr{)}$ | $\frac{3}{2}\pm\frac{1}{3}\bigl{(}\sqrt{61}-5\bigr{)}$
| $(0,1;2)$ | $10$ | $\frac{20}{9}\bigl{(}17-\sqrt{249}{\mkern 2.0mu}\bigr{)}$ | $\frac{3}{2}\pm\frac{1}{6}\bigl{(}\sqrt{249}-10\bigr{)}$
$0^{-}(Q^{(3)_{-}})$ | $(0,1;0)$ | $4$ | $-\frac{8}{9}\bigl{(}3\sqrt{161}-37\bigr{)}$ | $\frac{3}{2}\pm\frac{1}{6}\bigl{(}\sqrt{161}-12\bigr{)}$
| $(1,0;1)$ | $19/4$ | $-\frac{1}{9}\bigl{(}96\sqrt{11}-311\bigr{)}$ | $\frac{3}{2}\pm\frac{2}{3}\bigl{(}\sqrt{11}-3\bigr{)}$
| $(1,1;1)$ | $39/4$ | $\frac{1}{3}\bigl{(}137-16\sqrt{69}{\mkern 2.0mu}\bigr{)}$ | $\frac{3}{2}\pm\frac{1}{3}\bigl{(}\sqrt{69}-6\bigr{)}$
| $(0,2;0)$ | $10$ | $\frac{8}{9}\bigl{(}52-3\sqrt{281}{\mkern 2.0mu}\bigr{)}$ | $\frac{3}{2}\pm\frac{1}{6}\bigl{(}\sqrt{281}-12\bigr{)}$
| $(2,0;2)$ | $12$ | $\frac{8}{3}\bigl{(}19-\sqrt{321}{\mkern 2.0mu}\bigr{)}$ | $\frac{3}{2}\pm\frac{1}{6}\bigl{(}\sqrt{321}-12\bigr{)}$
Field ($RS^{7}$) | $(p,q;r)$ | $C_{g}$ | $M^{2}_{R}/m^{2}$ | $E_{0}$
$0^{-}(Q^{(1)_{+}})$ | $(0,1;0)$ | $4$ | $-\frac{8}{9}$ | $\frac{3}{2}\pm\frac{1}{6}$
| $(1,1;1)$ | $39/4$ | $\frac{7}{9}$ | $\frac{3}{2}\pm\frac{2}{3}$
| $(0,2;0)$ | $10$ | $\frac{4}{9}\bigl{(}73-5\sqrt{201}{\mkern 2.0mu}\bigr{)}$ | $\frac{3}{2}\pm\frac{1}{6}\bigl{(}\sqrt{201}-10\bigr{)}$
| $(2,0;2)$ | $12$ | $\frac{4}{9}\bigl{(}83-5\sqrt{241}{\mkern 2.0mu}\bigr{)}$ | $\frac{3}{2}\pm\frac{1}{6}\bigl{(}\sqrt{241}-10\bigr{)}$
| $(3,0;1)$ | $51/4$ | $3$ | $\frac{3}{2}+1$
$0^{-}(Q^{(2)_{+}})$ | $(1,0;1)$ | $19/4$ | $\frac{7}{9}$ | $\frac{3}{2}\pm\frac{2}{3}$
| $(2,0;0)$ | $6$ | $\frac{16}{9}$ | $\frac{3}{2}\pm\frac{5}{6}$
$0^{-}(Q^{(3)_{+}})$ | $(0,0;0)$ | $0$ | $0$ | $\frac{3}{2}\pm\frac{1}{2}$
Table 10: Pseudo-scalars that can have both Dirichlet and Neumann boundary conditions. The list is divided into the ones occurring in the left-squashed vacuum ($LS^{7}$) and those belonging to the right-squashed vacuum ($RS^{7}$). Some details about which $(p,q;r)$ belong to which branch of $0^{-}(Q^{(1)_{\pm}})$ have been collected from Duboeuf:2022mam  and used here. Note that the last entry for $0^{-}(Q^{(1)_{+}})$ is special since the option with a minus sign in $E_{0}$, corresponding to a singleton, is not possible in the $RS^{7}$ vacuum. This follows from the results in Nilsson:2018lof . Note also that Neumann boundary conditions for $0^{-}(Q^{(2)_{-}})$ do not respect supersymmetry in the left-squashed vacuum, see Section 5.1. All brackets are positive. Field ($LS^{7}$) | $(p,q;r)$ | $C_{g}$ | $M_{L}/m$ | $E_{0}$
---|---|---|---|---
$\frac{1}{2}(i\not{D}_{1/2}^{(1)_{-}})$ | $(0,1;0)$ | $4$ | $-\frac{1}{3}\bigl{(}15-\sqrt{161}{\mkern 2.0mu}\bigr{)}$ | $\frac{3}{2}\pm\frac{1}{6}\bigl{(}15-\sqrt{161}{\mkern 2.0mu}\bigr{)}$
| $(1,0;1)$ | $19/4$ | $-\frac{1}{3}\bigl{(}15-4\sqrt{11}{\mkern 2.0mu}\bigr{)}$ | $\frac{3}{2}\pm\frac{1}{6}\bigl{(}15-4\sqrt{11}{\mkern 2.0mu}\bigr{)}$
| $(1,1;1)$ | $39/4$ | $\frac{1}{3}\bigl{(}2\sqrt{69}-15\bigr{)}$ | $\frac{3}{2}\pm\frac{1}{6}\bigl{(}2\sqrt{69}-15\bigr{)}$
| $(0,2;0)$ | $10$ | $\frac{1}{3}\bigl{(}\sqrt{281}-15\bigr{)}$ | $\frac{3}{2}\pm\frac{1}{6}\bigl{(}\sqrt{281}-15\bigr{)}$
| $(2,0;2)$ | $12$ | $\frac{1}{3}\bigl{(}\sqrt{321}-15\bigr{)}$ | $\frac{3}{2}\pm\frac{1}{6}\bigl{(}\sqrt{321}-15\bigr{)}$
$\frac{1}{2}(i\not{D}_{1/2}^{(2)_{-}})$ | $(2,0;0)$ | $6$ | $0$ | $\frac{3}{2}$
| $(0,0;2)$ | $6$ | $0$ | $\frac{3}{2}$
| $(1,1;1)$ | $39/4$ | $\frac{1}{3}\bigl{(}2\sqrt{61}-13\bigr{)}$ | $\frac{3}{2}\pm\frac{1}{6}\bigl{(}2\sqrt{61}-13\bigr{)}$
| $(0,1;2)$ | $10$ | $\frac{1}{3}\bigl{(}\sqrt{249}-13\bigr{)}$ | $\frac{3}{2}\pm\frac{1}{6}\bigl{(}\sqrt{249}-13\bigr{)}$
Field ($RS^{7}$) | $(p,q;r)$ | $C_{g}$ | $M_{R}/m$ | $E_{0}$
$\frac{1}{2}(i\not{D}_{1/2}^{(1)_{+}})$ | $(0,0;0)$ | $0$ | $-1$ | $\frac{3}{2}-\frac{1}{2}$
| $(0,1;0)$ | $4$ | $\frac{1}{3}\bigl{(}\sqrt{161}-12\bigr{)}$ | $\frac{3}{2}\pm\frac{1}{6}\bigl{(}\sqrt{161}-12\bigr{)}$
| $(1,0;1)$ | $19/4$ | $\frac{4}{3}\bigl{(}\sqrt{11}-3\bigr{)}$ | $\frac{3}{2}\pm\frac{2}{3}\bigl{(}\sqrt{11}-3\bigr{)}$
$\frac{1}{2}(i\not{D}_{1/2}^{(2)_{+}})$ | $(1,0;1)$ | $19/4$ | $-\frac{2}{3}$ | $\frac{3}{2}\pm\frac{1}{3}$
| $(2,0;0)$ | $6$ | $-\frac{1}{3}$ | $\frac{3}{2}\pm\frac{1}{6}$
| $(1,1;1)$ | $39/4$ | $\frac{2}{3}\bigl{(}\sqrt{61}-7\bigr{)}$ | $\frac{3}{2}\pm\frac{1}{3}\bigl{(}\sqrt{61}-7\bigr{)}$
| $(0,1;2)$ | $10$ | $\frac{1}{3}\bigl{(}\sqrt{249}-14\bigr{)}$ | $\frac{3}{2}\pm\frac{1}{6}\bigl{(}\sqrt{249}-14\bigr{)}$
| $(2,0;2)$ | $12$ | $1$ | $\frac{3}{2}+\frac{1}{2}$
| $(2,1;0)$ | $12$ | $1$ | $\frac{3}{2}+\frac{1}{2}$
Table 11: Spinors that can have both Dirichlet and Neumann boundary
conditions. The special entries are of two kinds: 1) The one in the $LS^{7}$
part of the table with $E_{0}=\frac{3}{2}$, which have neither Dirichlet nor
Neumann boundary conditions (see Breitenlohner:1982jf ). 2) The first and last
entries in the $RS^{7}$ part of the table where only one sign in $E_{0}$ is
given as dictated by the fact that the only singleton that exists belongs to
the $RS^{7}$ vacuum and is a singlet Nilsson:2018lof . Note that Neumann
boundary conditions for $\frac{1}{2}(i\not{D}_{1/2}^{(2)_{-}})$ do not respect
supersymmetry in the left-squashed vacuum, see Section 5.1. All brackets are
positive.

#### Marginal operators in the dual CFTs

Having obtained a detailed understanding of the spectrum of the non-
supersymmetric right-squashed vacuum in the previous subsection, including the
ambiguities due to the many possible choices of boundary conditions, we can
now analyse the question of marginal operators.

As a preamble we consider the left-squashed spectrum of fields and list the
possible marginal operators. It should be noted that these operators cannot,
of course, lead to instabilities since stability is in this case guaranteed by
supersymmetry. However, despite being part of supermultiplets, the boundary
conditions of the fields involved, and hence their $E_{0}$ values, are not
uniquely determined as we saw in the previous subsections.

The scalar $sp_{2}\times sp_{1}^{C}$-singlet marginal operators, depending on
the boundary conditions and even for boundary conditions that do not respect
supersymmetry, in the left-squashed case are given in Table 12. With boundary
conditions preserving supersymmetry, the only scalar singlet marginal
operators are those guaranteed by supersymmetry, formed from the spinors with
$E_{0}=3/2$ from the massless vector supermultiplets.

#$\bm{1}_{g}^{p}$ | Field 1: $E_{0}$ | … |
---|---|---|---
$3^{+}$ | $0^{+}_{\scriptscriptstyle(-)}(\Delta_{0}^{(1)})_{(2,1;2)}^{-}\colon\;1$ | $0^{+}_{\scriptscriptstyle(-)}(\Delta_{0}^{(1)})_{(2,1;2)}^{-}\colon\;1$ | $0^{+}_{\scriptscriptstyle(-)}(\Delta_{0}^{(1)})_{(2,1;2)}^{-}\colon\;1$
$1^{+}$ | $0^{+}_{\scriptscriptstyle(-)}(\Delta_{0}^{(1)})_{(0,3;0)}^{-}\colon\;1$ | $0^{+}_{\scriptscriptstyle(-)}(\Delta_{0}^{(1)})_{(2,1;2)}^{-}\colon\;1$ | $0^{+}_{\scriptscriptstyle(-)}(\Delta_{0}^{(1)})_{(2,1;2)}^{-}\colon\;1$
$2^{-}$ | $0^{+}_{\scriptscriptstyle(-)}(\Delta_{0}^{(1)})_{(2,1;2)}^{-}\colon\;1$ | $0^{-}(Q^{(3)_{-}})_{(1,1;1)}^{-}\colon\;\frac{7}{2}-\sqrt{\frac{23}{3}}$ | $0^{+}_{\scriptscriptstyle(-)}(\Delta_{0}^{(1)})_{(1,1;1)}^{-}\colon\;\sqrt{\frac{23}{3}}-\frac{3}{2}$
$1^{-}$ | $0^{+}_{\scriptscriptstyle(-)}(\Delta_{0}^{(1)})_{(0,3;0)}^{-}\colon\;1$ | $0^{-}(Q^{(3)_{-}})_{(1,1;1)}^{-}\colon\;\frac{7}{2}-\sqrt{\frac{23}{3}}$ | $0^{+}_{\scriptscriptstyle(-)}(\Delta_{0}^{(1)})_{(1,1;1)}^{-}\colon\;\sqrt{\frac{23}{3}}-\frac{3}{2}$
$1^{-}$ | $0^{+}_{\scriptscriptstyle(-)}(\Delta_{0}^{(1)})_{(2,1;2)}^{-}\colon\;1$ | $0^{-}(Q^{(3)_{-}})_{(2,0;2)}^{-}\colon\;\frac{7}{2}-\frac{1}{6}\sqrt{321}$ | $0^{+}_{\scriptscriptstyle(-)}(\Delta_{0}^{(1)})_{(2,0;2)}^{-}\colon\;\frac{1}{6}\sqrt{321}-\frac{3}{2}$
$1^{+}$ | $\frac{1}{2}(i\not{D}_{1/2}^{(2)_{-}})_{(2,0;0)}\colon\;\frac{3}{2}$ | $\frac{1}{2}(i\not{D}_{1/2}^{(2)_{-}})_{(2,0;0)}\colon\;\frac{3}{2}$ |
$1^{+}$ | $\frac{1}{2}(i\not{D}_{1/2}^{(2)_{-}})_{(0,0;2)}\colon\;\frac{3}{2}$ | $\frac{1}{2}(i\not{D}_{1/2}^{(2)_{-}})_{(0,0;2)}\colon\;\frac{3}{2}$ |
Table 12: Possible scalar $g=sp_{2}\times sp_{1}^{C}$ singlet marginal
operators with parity $p$ in $LS^{7}$ depending on the boundary conditions.
The subscripts after the eigenvalues indicate the isometry irreps $(p,q;r)$
while the superscript indicates the boundary conditions when applicable ($+$
for Dirichlet, $-$ for Neumann). The leftmost column gives the number of
$g$-singlets in the composite. Note that only the ones containing two spinors
are compatible with the boundary conditions that preserve $\mathcal{N}=1$
supersymmetry.

Turning to the more interesting case of the right-squashed non-supersymmetric
vacuum, there is a lot more freedom in choosing boundary conditions leading to
a much longer list of marginal operators. We present these in Table 13. There
are several features of this table that deserve a comment. We use here the
language of the boundary $CFT_{3}$ theory although the notation is from the
bulk theory.

First, we note that in the right-squashed vacuum there are no marginal single-
or double-trace operators,252525This fact might be of some importance, see for
instance Giombi:2017mxl  for the role of such operators. while both marginal
triple- and quadruple-trace operators occur. Secondly, as observed in
Nilsson:2023ctq , in the singlet sector the two possible boundary conditions
for the pseudo-scalar generate either two marginal operators or none (see
Table 13). This phenomenon generalises to the entire spectrum. In other words,
instead of choosing the Neumann boundary conditions on which almost all of
Table 13 is based, we can use Dirichlet boundary conditions when possible and
thereby eliminate all of the marginal operators. With such boundary
conditions, the theory cannot be afflicted by any instability issues related
to scalar singlet marginal operators.

Although they seem possible, it remains to be seen if such choices of boundary
conditions can be made freely or if they are constrained in some way. In any
case, although this may seem like a step towards proving stability it is just
a small one since there may exist a number of other decay modes we are still
ignorant about.

#$\bm{1}_{g}^{p}$ | Field 1: $E_{0}$ | … |
---|---|---|---
$1^{+}$ | $0^{-}(Q^{(2)_{+}})_{(2,0;0)}^{-}\colon\;\frac{2}{3}$ | $0^{-}(Q^{(2)_{+}})_{(2,0;0)}^{-}\colon\;\frac{2}{3}$ | $0^{+}(\Delta_{L}^{(2)_{-}})_{(0,0;0)}^{+}\colon\;\frac{5}{3}$
$1^{-}$ | $0^{-}(Q^{(2)_{+}})_{(2,0;0)}^{-}\colon\;\frac{2}{3}$ | $0^{-}(Q^{(2)_{+}})_{(2,0;0)}^{-}\colon\;\frac{2}{3}$ | $0^{-}(Q^{(1)_{+}})_{(0,1;0)}^{+}\colon\;\frac{5}{3}$
$1^{+}$ | $0^{-}(Q^{(1)_{+}})_{(1,1;1)}^{-}\colon\;\frac{5}{6}$ | $0^{-}(Q^{(1)_{+}})_{(1,1;1)}^{-}\colon\;\frac{5}{6}$ | $0^{+}(\Delta_{L}^{(2)_{-}})_{(0,0;0)}^{-}\colon\;\frac{4}{3}$
$1^{-}$ | $0^{-}(Q^{(1)_{+}})_{(1,1;1)}^{-}\colon\;\frac{5}{6}$ | $0^{-}(Q^{(1)_{+}})_{(1,1;1)}^{-}\colon\;\frac{5}{6}$ | $0^{-}(Q^{(1)_{+}})_{(0,1;0)}^{-}\colon\;\frac{4}{3}$
$1^{-}$ | $0^{-}(Q^{(1)_{+}})_{(1,0;1)}^{-}\colon\;\frac{5}{6}$ | $0^{-}(Q^{(1)_{+}})_{(1,1;1)}^{-}\colon\;\frac{5}{6}$ | $0^{-}(Q^{(1)_{+}})_{(0,1;0)}^{-}\colon\;\frac{4}{3}$
$1^{+}$ | $0^{-}(Q^{(1)_{+}})_{(1,0;1)}^{-}\colon\;\frac{5}{6}$ | $0^{-}(Q^{(1)_{+}})_{(1,0;1)}^{-}\colon\;\frac{5}{6}$ | $0^{+}(\Delta_{L}^{(2)_{-}})_{(0,0;0)}^{-}\colon\;\frac{4}{3}$
$1^{-}$ | $0^{-}(Q^{(1)_{+}})_{(1,0;1)}^{-}\colon\;\frac{5}{6}$ | $0^{-}(Q^{(1)_{+}})_{(1,0;1)}^{-}\colon\;\frac{5}{6}$ | $0^{-}(Q^{(1)_{+}})_{(0,1;0)}^{-}\colon\;\frac{4}{3}$
$1^{-}$ | $0^{-}(Q^{(3)_{+}})_{(0,0;0)}^{-}\colon\;1$ | $0^{-}(Q^{(3)_{+}})_{(0,0;0)}^{-}\colon\;1$ | $0^{-}(Q^{(3)_{+}})_{(0,0;0)}^{-}\colon\;1$
$1^{-}$ | $0^{-}(Q^{(3)_{+}})_{(0,0;0)}^{-}\colon\;1$ | $0^{+}_{\scriptscriptstyle(-)}(\Delta_{0}^{(1)})_{(0,3;0)}^{-}\colon\;1$ | $0^{+}_{\scriptscriptstyle(-)}(\Delta_{0}^{(1)})_{(0,3;0)}^{-}\colon\;1$
$1^{-}$ | $0^{-}(Q^{(3)_{+}})_{(0,0;0)}^{-}\colon\;1$ | $0^{+}_{\scriptscriptstyle(-)}(\Delta_{0}^{(1)})_{(2,1;2)}^{-}\colon\;1$ | $0^{+}_{\scriptscriptstyle(-)}(\Delta_{0}^{(1)})_{(2,1;2)}^{-}\colon\;1$
$1^{+}$ | $0^{+}_{\scriptscriptstyle(-)}(\Delta_{0}^{(1)})_{(0,3;0)}^{-}\colon\;1$ | $0^{+}_{\scriptscriptstyle(-)}(\Delta_{0}^{(1)})_{(2,1;2)}^{-}\colon\;1$ | $0^{+}_{\scriptscriptstyle(-)}(\Delta_{0}^{(1)})_{(2,1;2)}^{-}\colon\;1$
$3^{+}$ | $0^{+}_{\scriptscriptstyle(-)}(\Delta_{0}^{(1)})_{(2,1;2)}^{-}\colon\;1$ | $0^{+}_{\scriptscriptstyle(-)}(\Delta_{0}^{(1)})_{(2,1;2)}^{-}\colon\;1$ | $0^{+}_{\scriptscriptstyle(-)}(\Delta_{0}^{(1)})_{(2,1;2)}^{-}\colon\;1$
$1^{+}$ | $0^{-}(Q^{(2)_{+}})_{(2,0;0)}^{-}\colon\;\frac{2}{3}$ | $0^{-}(Q^{(2)_{+}})_{(2,0;0)}^{-}\colon\;\frac{2}{3}$ |
| $0^{-}(Q^{(2)_{+}})_{(2,0;0)}^{-}\colon\;\frac{2}{3}$ | $0^{-}(Q^{(3)_{+}})_{(0,0;0)}^{-}\colon\;1$ |
$1^{-}$ | $0^{-}(Q^{(2)_{+}})_{(2,0;0)}^{-}\colon\;\frac{2}{3}$ | $0^{-}(Q^{(2)_{+}})_{(2,0;0)}^{-}\colon\;\frac{2}{3}$ |
| $0^{-}(Q^{(2)_{+}})_{(2,0;0)}^{-}\colon\;\frac{2}{3}$ | $0^{+}_{\scriptscriptstyle(-)}(\Delta_{0}^{(1)})_{(0,3;0)}^{-}\colon\;1$ |
$8^{+}$ | $0^{-}(Q^{(2)_{+}})_{(2,0;0)}^{-}\colon\;\frac{2}{3}$ | $0^{-}(Q^{(2)_{+}})_{(2,0;0)}^{-}\colon\;\frac{2}{3}$ |
| $0^{-}(Q^{(1)_{+}})_{(1,1;1)}^{-}\colon\;\frac{5}{6}$ | $0^{-}(Q^{(1)_{+}})_{(1,1;1)}^{-}\colon\;\frac{5}{6}$ |
$4^{+}$ | $0^{-}(Q^{(2)_{+}})_{(2,0;0)}^{-}\colon\;\frac{2}{3}$ | $0^{-}(Q^{(2)_{+}})_{(2,0;0)}^{-}\colon\;\frac{2}{3}$ |
| $0^{-}(Q^{(1)_{+}})_{(1,1;1)}^{-}\colon\;\frac{5}{6}$ | $0^{-}(Q^{(2)_{+}})_{(1,0;1)}^{-}\colon\;\frac{5}{6}$ |
$3^{+}$ | $0^{-}(Q^{(2)_{+}})_{(2,0;0)}^{-}\colon\;\frac{2}{3}$ | $0^{-}(Q^{(2)_{+}})_{(2,0;0)}^{-}\colon\;\frac{2}{3}$ |
| $0^{-}(Q^{(2)_{+}})_{(1,0;1)}^{-}\colon\;\frac{5}{6}$ | $0^{-}(Q^{(2)_{+}})_{(1,0;1)}^{-}\colon\;\frac{5}{6}$ |
$1^{-}$ | $0^{-}(Q^{(3)_{+}})_{(0,0;0)}^{-}\colon\;1$ | $\frac{1}{2}(i\not{D}_{1/2}^{(1)_{+}})_{(0,0;0)}^{-}\colon\;1$ | $\frac{1}{2}(i\not{D}_{1/2}^{(1)_{+}})_{(0,0;0)}^{-}\colon\;1$
Table 13: Possible scalar $g=sp_{2}\times sp_{1}^{C}$ singlet marginal
operators with parity $p$ in $RS^{7}$ depending on the boundary conditions,
with notation as in Table 12. Note that the lines without a number in the
leftmost column are continuation lines.

##  6 Conclusions and comments

In this paper, we have reported on technical progress on the computation of
the eigenvalue spectra of operators on the squashed seven-sphere relevant in
the compactification of eleven-dimensional supergravity. The full isometry
irrep spectrum was derived in Nilsson:2018lof , including the structure of the
$AdS_{4}$ supermultiplet spectrum for the left-squashed vacuum. With the
results of Karlsson:2021oxd  and this paper on the solution of the spin-3/2
equation, these supermultiplets can be associated with values of $E_{0}$,
pinpointing the $Spin(2,3)$-irrep $D(E_{0},s)$, thus providing a full
understanding of the left-squashed spectrum.262626Note that, as explained in
Section 5.1, we use results of Duboeuf:2022mam  to resolve some ambiguities
that would otherwise appear for the supermultiplets denoted WZ3–6 for small
$p$ or $q$. This is the case also after skew-whiffing to the right-squashed
non-supersymmetric vacuum, whose spectrum becomes fully determined up to a
number of ambiguities connected to the possibility to choose either Dirichlet
or Neumann boundary conditions for a number of fields. Some ambiguities of
this kind exist also in the supersymmetric left-squashed case. These results
are summarised in Tables 4–13 of Section 5.

This progress was made possible by deriving more powerful versions of some key
formulas used in solving operator equations on coset manifolds, see
Karlsson:2021oxd  and Section 2, here the squashed $S^{7}$ represented by the
coset $Sp_{2}\times Sp_{1}^{C}/Sp_{1}\times Sp_{1}$. Using these refined
formulas, in particular 38 and 40, we finally managed to solve also the
spin-3/2 equation. These results are summarised in Section 3.3 but they were
reported on already in Karlsson:2021oxd  where some additional details can be
found compared to the presentation in this paper.

The structure of the results is to a large extent as expected apart from some
surprising features: We found degeneracies in the spectrum of two-forms, in
the sense that different mode functions have identical operator eigenvalues.
We demonstrated this explicitly by constructing all eigenmodes of the operator
$\Delta_{2}$, providing a direct means of calculating the eigenvalues. These
novel results are presented in Section 4. This kind of degeneracy appears in
(almost) all supermultiplets that involve fields associated to the eigenmodes
of the Lichnerowicz operator on the squashed seven-sphere, as was shown in
Section 5.

We also found that supersymmetry alone is not enough to determine the boundary
conditions for a number of supermultiplets. This issue was discussed in
Section 5 where the supermultiplet structure was presented in full detail.

The implications for the skew-whiffed non-supersymmetric right-squashed seven-
sphere compactification are discussed at the end of the previous section.
Here, the boundary condition ambiguities appear for many fields which implies
that one can find a large number of scalar singlet marginal operators in the
boundary $CFT_{3}$. This is of interest since the right-squashed case is BF-
stable and has, in fact, not yet been shown to be unstable, although this is
predicted by the swampland $AdS$ swampland conjecture Ooguri:2016pdq ;
Freivogel:2016qwc . However, in order to determine the fate of the right-
squashed vacuum one must analyse the corresponding $\beta$ functions to
determine whether a marginal operator leads to an instability or not. We leave
this for a future study. Interestingly enough, there are multiple sets of
boundary conditions for which no scalar singlet marginal operators exists.
This phenomenon was recently observed in Nilsson:2023ctq  to be a property of
the singlet sector of the spectrum.272727Generally, we believe that Dirichlet
boundary conditions can always be chosen in non-supersymmetric vacua such that
the only possible scalar singlet marginal operators are single-trace operators
or multi-trace operators dual to composites containing a singleton or fields
with $E_{0}=3/2$. Another property of the space of singlet scalar marginal
operators found in Section 5.2 is that there are none that are either single-
or double-trace. The implications of this fact might be worth investigating.

The squashed spectra clearly also imply that the boundary theories contain a
large number of relevant operators which can generate RG-flows to other CFTs.
For instance, a double-trace operator constructed as the square of a single-
trace operator dual to a scalar field with Neumann boundary conditions
generates a flow that leads, in the infrared, to the dual scalar field having
Dirichlet boundary conditions Witten:2001ua . It would be interesting to
investigate whether there are RG-flows connecting the boundary conditions
preserving supersymmetry in the left-squashed vacuum. Another interesting
potential RG-flow is one that corresponds to a domain-wall interpolating
between either of the squashed vacua and the round $S^{7}$ vacuum in the bulk,
as discussed, but not found explicitly, in Ahn:1999dq ; Campos:2000yu  (see
also, e.g., Witten:2001ua ; Papadimitriou:2007sj ; Giombi:2017mxl ) or one
that connects the left- and right-squashed vacua.

The operator corresponding to the field associated with the squashing mode
(the $\Delta_{L}$-row in Table 7) is relevant in the squashed vacuum and
irrelevant in the round vacuum Nilsson:2023ctq . This indicates that an RG-
flow connecting the two would have to start from the squashed vacuum in the UV
and flow to the round one in the IR. Another argument for this is provided by
the F-theorem Jafferis:2010un ; Jafferis:2011zi ; Closset:2012vg , which
states that the $S^{3}$ free energy decreases along RG-trajectories.
Holographically, the free energy is given by282828It should be understood that
this, and what follows, is only valid to leading order in $N$ or
$L^{2}/G_{N}$.

$\mathcal{F}_{\\!S^{3}}=I_{EAdS}=\frac{\pi L^{2}}{2G_{N}},$ (240)

where $I_{EAdS}$ is the regularised on-shell action of Euclidean $AdS$,
$L^{2}$ the $AdS$ curvature radius and $G_{N}$ the four-dimensional Newton’s
constant. By dimensional reduction, we have
$G_{N}\propto\ell_{\mathrm{Pl}}{}^{9}/\operatorname{Vol}(M_{7})$ where
$\ell_{\mathrm{Pl}}$ is the eleven-dimensional Planck length. The flux
quantisation condition gives the number of M2-branes as
$N\propto\operatorname{Vol}(M_{7})/(\ell_{\mathrm{Pl}}{}^{6}L)$. Putting this
together to eliminate $\ell_{\mathrm{Pl}}$, we get
$\mathcal{F}_{\\!S^{3}}\propto
N^{3/2}L^{7/2}/\sqrt{\operatorname{Vol}(M_{7})}$. Lastly, computing the volume
of the squashed $S^{7}$ and fixing the proportionality constant through the
well-known round $S^{7}$ answer or by properly keeping track of all numerical
factors in the quantisation condition and the dimensional reduction (see,
e.g., Jafferis:2011zi ), we arrive at (c.f. Ahn:1999dq ; Page:1983mke ;
Campos:2000yu )

$\mathcal{F}_{\\!S^{3}}^{{\mkern
2.0mu}(\mathrm{round})}=\frac{\pi\sqrt{2}}{3}N^{3/2},\qquad\quad\mathcal{F}_{\\!S^{3}}^{{\mkern
2.0mu}(\mathrm{squashed})}=\frac{\pi\sqrt{2}}{3}\sqrt{\frac{5^{5}}{3^{7}}}\,N^{3/2}.$
(241)

Note that the latter is about $20{\mkern 2.0mu}\%$ larger than the former,
consistent with the above assertion that the RG-flow, if it exists, goes from
squashed in the UV to round in the IR.

It would also be interesting to investigate if the marginal operators found in
this paper can be used to argue holographically for a manifold of non-
supersymmetric dual CFTs, in line with Witten:2001ua , which requires
computing the $\beta$ functions of the marginal operators. However, no solid
conclusions can be drawn without incorporating $1/N$ or finite $N$ effects,
which currently seems out of reach.

The methods and results of this paper can also be used to analyse
compactifications on $S^{7}/Z_{k}$. Some $AdS/CFT$ aspects in this context are
discussed in, e.g., Ooguri:2008dk ; Murugan:2016aty .

From the results of this paper it is clear that all values of $E_{0}$ in the
squashed vacua are of the form

$E_{0}=\biggl{(}\frac{3}{2}\pm
A\biggr{)}+\frac{1}{6}\sqrt{\vphantom{C_{G}}\smash[b]{20C_{g}+X}},\qquad
X=1,\,9,\,49,\,\text{or}\,81,$ (242)

with a constant $A$ that is also restricted to a small number of special
rational values. The simplicity of this result is striking, especially in view
of how the various values of $E_{0}$ arose, potentially containing three
layers of nested square roots, one each coming from the operator eigenvalues,
the mass formulas and finally from the formulas for $E_{0}(s)$ as functions of
$M$ or $M^{2}$. Even more intriguing is the fact that also the operator
eigenvalue spectra are given by similar formulas, either with or without a
square root term. From Section 3 we see that the general forms of the
eigenvalues of operators quadratic in derivatives are

$\kappa^{2}=\frac{m^{2}}{9}(20C_{g}+B),\qquad\kappa^{2}=\frac{m^{2}}{9}\Bigl{(}20C_{g}+\tilde{A}\pm\tilde{B}\sqrt{\vphantom{C_{G}}\smash[b]{20C_{g}+X}}\Bigr{)},$
(243)

while for linear operators, the eigenvalues are all of the form

$\kappa=\frac{m}{3}\Bigl{(}\pm\hat{A}\pm\sqrt{\vphantom{C_{G}}\smash[b]{20C_{g}+X}}\Bigr{)}.$
(244)

It is interesting to note, however, that all eigenvalues of the form
$\kappa^{2}=\frac{m^{2}}{9}(20C_{g}+\tilde{A}\pm\tilde{B}\sqrt{\vphantom{C_{G}}\smash[b]{20C_{g}+X}})$
can be expressed as a square minus a constant term proportional to $m^{2}$
which immediately eliminates the square root in the corresponding formula for
$M^{2}$, as was noticed already in Ekhammar:2021gsg . In fact, looking at the
table of mass matrices in Section 2, we realise that the same phenomenon will
appear when using the mass formulas in the expressions for $E_{0}$. The reason
behind the universal simple results for $E_{0}$ is then clear. This might
nevertheless point towards the existence of a more direct derivation of
$E_{0}$.

In this context we should mention that the expressions for $E_{0}$ for the
various supermultiplets was recently derived by entirely different methods in
Duboeuf:2022mam , giving in addition a refined version of the results in terms
of a universal formula for $E_{0}$. Interestingly enough, these methods do not
involve solving the operator equations on the squashed seven-sphere but
instead rely on ExFT methods. The results of this paper partially overlap with
those of Duboeuf:2022mam , in particular, the $E_{0}$ results that were
derived explicitly for finitely many levels in Duboeuf:2022mam . Note also
that the many ambiguities in the choice of Dirichlet or Neumann boundary
conditions we have found are not discussed in Duboeuf:2022mam .

We end with an observation concerning the degeneracies mentioned above. There
are three pairs of identical supermultiplets, all containing a scalar sector
governed by $\Delta_{L}$, one pair of $1^{+}$ and two pairs of Wess–Zumino
multiplets. We know from explicit calculations of the spectrum of
$\Delta_{2}$-eigenmodes in Section 4, which gives also $E_{0}$, that an
infinite set of pairs of different eigenmodes (with $r=p$ or $r=p\pm 2$) with
identical isometry irreps appear with the same eigenvalues and, thus, also the
same $E_{0}$ for the corresponding fields. We do not have an explanation for
this feature but hope to come back to it in a future publication.

Instead, we point out some facts related to the operator $\Delta_{L}$ which is
involved in all these cases. It is known from the work of House and Micu
House:2004pm  that $\Delta_{L}$ is related to a first order differential
operator, here denoted as $Q_{L}$, by

$\Delta_{L}-3m^{2}=(Q_{L}+m)^{2},$ (245)

which implies

$M^{2}(0^{+(2)})=\Delta_{L}-4m^{4}=(Q_{L}+m)^{2}-m^{2}\ \implies\
E_{0}(0^{+(2)},Q_{L})=\frac{3}{2}\pm\frac{1}{2}\biggl{|}\frac{Q_{L}}{m}+1\biggr{|}.$
(246)

Note that this seems to imply that $\Delta_{L}\geq 3m^{2}$ which, however,
must be supplemented by a precise statement about which modes of the two
operators the House–Micu equation is valid for. This is a rather non-trivial
issue as we will indicate below.

One might interpret this to mean that the three known eigenvalues of
$\Delta_{L}$ could be used twice, that is, that we should be able to use the
positive and negative parts of the spectrum of $Q_{L}$ when taking the square
root of the House–Micu formula. Thus we consider

$\displaystyle\Delta_{L}^{(1)}=\frac{m^{2}}{9}(20C_{g}+36)$
$\displaystyle\implies
Q_{L}=-m\pm\frac{m}{3}\sqrt{\vphantom{C_{G}}\smash[b]{20C_{g}+9}},$ (247)
$\displaystyle\Delta_{L}^{(2)_{+}}=\frac{m^{2}}{9}\Bigl{(}20C_{g}+32+4\sqrt{\vphantom{C_{G}}\smash[b]{20C_{g}+1}}\Bigr{)}$
$\displaystyle\implies
Q_{L}=-m\pm\biggl{(}\frac{2}{3}m+\frac{m}{3}\sqrt{\vphantom{C_{G}}\smash[b]{20C_{g}+1}}\biggr{)},$
(248)
$\displaystyle\Delta_{L}^{(2)_{-}}=\frac{m^{2}}{9}\Bigl{(}20C_{g}+32-4\sqrt{\vphantom{C_{G}}\smash[b]{20C_{g}+1}}\Bigr{)}$
$\displaystyle\implies
Q_{L}=-m\pm\biggl{|}-\frac{2}{3}m+\frac{m}{3}\sqrt{\vphantom{C_{G}}\smash[b]{20C_{g}+1}}\biggr{|}.$
(249)

However, this interpretation has a number of problems. First,
$Q_{L}=-m\pm\frac{m}{3}\sqrt{\vphantom{C_{G}}\smash[b]{20C_{g}+9}}$ is not a
possible set of eigenvalues for an operator like $Q$ acting on transverse
three-forms as we saw in Section 3. One issue here is that although the irrep
${\bm{2}7}$ describes both the symmetric second rank tensors and part of the
three-forms, the transversality conditions act differently on these tensors
making the House–Micu formula difficult to apply. For instance, at least one
of the sets of suggested $Q_{L}$-eigenvalues in 247 could instead be relevant
for non-transverse three-form modes. Furthermore, only one of the two signs in
248 and 249 give existing $Q$-eigenvalues and these are precisely the
eigenvalues of $Q$ that correspond to eigenmodes with only a $\bm{27}$-part
and that are used by the pseudo-scalars in the supermultiplets using
$\Delta_{L}^{(2)_{\pm}}$, i.e., WZ3–6. So, $Q$ does not resolve the analogous
degeneracies of $\Delta_{3}$ and since these $Q$-eigenvalues, which should
also be $Q_{L}$-eigenvalues, have cross diagrams that already cover all
isometry irreps of these supermultiplets, it seems like $Q_{L}$ cannot resolve
the degeneracies either.

## Acknowledgements

J.K. thanks Junho Hong, Emanuel Malek, Henning Samtleben, Thomas Van Riet and,
in particular, Nikolay Bobev for useful comments and discussions. The second
author, B.E.W.N., is grateful to Mike Duff and Chris Pope for many discussions
and collaboration at an early stage of this work. We are also grateful to Igor
Klebanov and the referee for discussions and comments on parts of the
manuscript. J.K. is supported by the Research Foundation - Flanders (FWO)
doctoral fellowship 1171823N and, in part, by the FWO research grants G092617N
and G0H9318N. B.E.W.N. is partly supported by the Wilhelm and Martina Lundgren
Foundation.

##  Appendix A Conventions on octonions, gamma-matrices and $Sp_{1}^{C}$
Killing vectors

The octonions are elements of a non-associative algebra spanned by a real unit
$1$ and octonionic imaginary units $o_{a}$, where $a=1,\ldots,7$, with
multiplication

$o_{a}o_{b}=-\delta_{ab}+a_{ab}{}^{c}o_{c},$ (250)

where $a_{abd}$ (index lowered by $\delta_{cd}$) are the completely
antisymmetric octonionic structure constants with independent components

$a_{abc}=1\quad\text{for }abc=123,257,365,437,761,642,415,$ (251)

in a particular basis. By splitting the index as $a=(\hat{i},0,i)$, where
$\hat{i}=(\hat{1},\hat{2},\hat{3})=(1,2,3)$ and $i=(5,6,7)$, the non-zero
components of $a_{abc}$ can conveniently be summarised as follows292929The
indices $i,j,k$ should be shifted to $(1,2,3)$ to make sense of 252.

$a_{\hat{i}\hat{j}\hat{k}}=\epsilon_{ijk},\qquad
a_{ij\hat{k}}=-\epsilon_{ijk},\qquad a_{0i\hat{j}}=-\delta_{ij},$ (252)

in terms of the standard totally antisymmetric $\epsilon_{ijk}$ and the
Kronecker delta $\delta_{ij}$. The Hodge dual of $a_{abc}$, denoted
$c_{abcd}$, is

$c_{abcd}=\frac{1}{6}\epsilon_{abcdefg}{\mkern 2.0mu}a^{efg},$ (253)

where the seven-dimensional epsilon tensor is totally antisymmetric with
$\epsilon_{1234567}=1$. This implies that $c_{abcd}$ has non-zero components

$c_{0ijk}=\epsilon_{ijk},\qquad c_{0\hat{i}\hat{j}k}=-\epsilon_{ijk},\qquad
c_{\hat{i}\hat{j}}{}^{kl}=-2\delta_{ij}^{kl}.$ (254)

It is sometimes convenient to express the seven-dimensional gamma-matrices
$\Gamma_{a}$ in terms of $a_{abc}$, $c_{abcd}$ and the Killing spinor of the
squashed seven-sphere, $\eta$, satisfying303030The octonions can be identified
with $Spin(7)$ spinors and the gamma-matrices $\Gamma_{a}$ realised through
multiplication by $o_{a}$. $G_{2}$ is the automorphism group of the octonions
or, equivalently, the subgroup of $Spin(7)$ leaving the unit octonion $1$,
identified with the spinor $\eta$, invariant. See, e.g., Karlsson:2021oxd  for
more details on this perspective and how to derive some of the following
relations using it.

$D_{a}\eta=-\frac{i}{2}m{\mkern 2.0mu}\Gamma_{a}\eta,\qquad\bar{\eta}\eta=1.$
(255)

The octonionic conventions above lead then to the relations

$a_{abc}=i\bar{\eta}\Gamma_{abc}\eta,\qquad
c_{abcd}=-\bar{\eta}\Gamma_{abcd}\eta,$ (256)

which are consistent with the gamma-matrices satisfying their defining
relations

$\\{\Gamma_{a},\Gamma_{b}\\}=2\delta_{ab},\qquad\Gamma_{abcdefg}=i\epsilon_{abcdefg}\bf{1}.$
(257)

The fact that the squashed seven-sphere is a _weak_ $G_{2}$ manifold is then
encoded in the equation

$D_{a}a_{bcd}=m{\mkern 2.0mu}c_{abcd},$ (258)

which is a direct consequence of the above relations with the octonionic
structure constants and the Killing spinor equation.

The consistency of these conventions can be verified by explicitly writing out
the seven $8\times 8$ gamma-matrices, with spinor indices $A,B,C,\ldots$
taking the eight values $A=(\hat{0},a)$, as follows:313131Note that vector
$a,b,c,...$ and spinor $A,B,C,...$ indices are raised and lowered by
$\delta_{ab}$ and $\delta_{AB}$ (and their inverses), respectively.

$(\Gamma_{a})_{B}{}^{C}\colon\qquad(\Gamma_{a})_{b}{}^{c}=ia_{ab}{}^{c},\quad(\Gamma_{a})_{b}{}^{\hat{0}}=-i\delta_{ab}.$
(259)

Note that all seven $\Gamma_{a}$ are purely imaginary and antisymmetric and
that they satisfy 257.

To derive and simplify the components of $\Gamma_{ab}$ and $\Gamma_{abc}$ we
need some structure constant identities. We list these and, for the
convenience of the reader, some other structure constant identities that are
used heavily throughout the paper. We have

$\displaystyle a_{abc}{\mkern 2.0mu}a^{cde}=2\delta_{ab}^{de}+c_{ab}{}^{de},$
(260) $\displaystyle a_{abc}{\mkern 2.0mu}a^{bcd}=6\delta_{a}^{d},$ (261)
$\displaystyle a_{abc}{\mkern 2.0mu}a^{abc}=42.$ (262)

The analogous identities involving $c_{abcd}$ are

$\displaystyle c_{abcd}{\mkern 2.0mu}c^{defg}=a_{abc}{\mkern
2.0mu}a^{efg}-9c_{[ab}{}^{[ef}\delta_{c]}^{g]}-6\delta_{abc}^{efg},$ (263)
$\displaystyle c_{abcd}{\mkern
2.0mu}c^{cdef}=8\delta_{ab}^{ef}+2c_{ab}{}^{ef},$ (264) $\displaystyle
c_{abcd}{\mkern 2.0mu}c^{bcde}=-24\delta_{a}^{e},$ (265) $\displaystyle
c_{abcd}{\mkern 2.0mu}c^{abcd}=168.$ (266)

Some identities involving contractions between the $a_{abc}$ and $c_{abcd}$
structure constants are:

$\displaystyle c_{abcd}{\mkern 2.0mu}a^{def}=6a_{[ab}{}^{[e}\delta_{c]}^{f]},$
(267) $\displaystyle c_{abcd}{\mkern 2.0mu}a^{cde}=4a_{ab}{}^{e},$ (268)
$\displaystyle c_{abcd}{\mkern 2.0mu}a^{bcd}=0.$ (269)

Two other identities that have been used in this work are

$\displaystyle\delta_{ab}$
$\displaystyle=\frac{1}{4!}a_{a}{}^{cd}a_{b}{}^{ef}c_{cdef},$ (270)
$\displaystyle a_{[ab}{}^{[d}a_{c]}{}^{ef]}$
$\displaystyle=\frac{1}{3}a_{abc}a^{def}-2c_{[ab}{}^{[de}\delta_{c]}^{f]}.$
(271)

These and the following identities may be verified directly by using the
components in the basis given above. By using the Fierz identity of the
$G_{2}$-invariant $\eta$,

$\Gamma_{a}\eta\bar{\eta}\Gamma^{a}=1-\eta\bar{\eta},$ (272)

one can also prove 260–270 more elegantly.

Using the octonionic definition of the gamma-matrices $(\Gamma_{a})_{B}{}^{C}$
above we can express also the antisymmetric matrices $(\Gamma_{ab})_{CD}$ this
way. We find

$(\Gamma_{ab})_{C}{}^{D}\colon\qquad(\Gamma_{ab})^{cd}=-c_{ab}{}^{cd}+2\delta_{ab}^{cd},\quad(\Gamma_{ab})_{c}{}^{\hat{0}}=-a_{abc}.$
(273)

Finally, for the symmetric matrices $(\Gamma_{abc})_{DE}$ we have

$\displaystyle(\Gamma_{abc})_{D}{}^{E}\colon\qquad$
$\displaystyle(\Gamma_{abc})^{de}=-ic_{[ab}{}^{ef}a_{c]f}{}^{d}-ia_{[ab}{}^{e}\delta_{c]}^{d}-2ia_{[ab}{}^{d}\delta_{c]}^{e},$
$\displaystyle(\Gamma_{abc})_{d}{}^{\hat{0}}=-ic_{abcd},$
$\displaystyle(\Gamma_{abc})_{\hat{0}}{}^{\hat{0}}=-ia_{abc}.$ (274)

The first one of these expressions can be simplified further using the
identity (note the order of the indices)

$c_{[ab|d|}{}^{f}a_{c]f}{}^{e}=6\bigl{(}\delta_{[a}^{[c^{\prime}}a_{bd]}{}^{e]}\delta_{cc^{\prime}}\bigr{)}\big{|}_{[abc]}.$
(275)

Then, the required symmetry property in $de$ becomes clear:

$(\Gamma_{abc})^{de}=i\bigl{(}a_{abc}\delta^{de}-6\delta^{(d}_{[a}a_{bc]}{}^{e)}\bigr{)}.$
(276)

Finally, we see that $\operatorname{Tr}(\Gamma^{abc})=0$ as it must.

The relations above show that this representation of the gamma-matrices is
consistent with the definition of the structure constants $a_{abc}$ in terms
of the Killing spinor $\eta$. To clarify this further we can solve the Killing
spinor equation and get an explicit form of this spinor. We know from the
left-squashed $S^{7}$ compactification that the holonomy is $G_{2}$ and hence
that the Killing spinor equation has one solution. Normalising it to
$\bar{\eta}\eta=1$ we for now assume that, in the above basis, $\eta$ has only
one non-zero real constant component. Then

$\displaystyle
D_{a}\eta_{B}=-\frac{i}{2}m(\Gamma_{a})_{B}{}^{C}\eta_{C}\colon$
$\displaystyle B=b\colon\quad$
$\displaystyle\partial_{a}\eta_{b}+\frac{1}{4}\omega_{ade}\bigl{(}(\Gamma^{de})_{b}{}^{c}\eta_{c}+(\Gamma^{de})_{b}{}^{\hat{0}}\eta_{\hat{0}}\bigr{)}=-\frac{i}{2}m\bigl{(}(\Gamma_{a})_{b}{}^{c}\eta_{c}+(\Gamma_{a})_{b}{}^{{\hat{0}}}\eta_{\hat{0}}\bigr{)}$
$\displaystyle B=\hat{0}\colon\quad$
$\displaystyle\partial_{a}\eta_{\hat{0}}+\frac{1}{4}\omega_{ade}(\Gamma^{de})_{\hat{0}}{}^{c}\eta_{c}=-\frac{i}{2}m(\Gamma_{a})_{\hat{0}}{}^{c}\eta_{c}$
(277)

It is clear from these two equations that a spinor $\eta$ with only a constant
non-zero component $\eta_{\hat{0}}$ has a chance to be a solution. Assuming
this to be the case, the second equation is solved and the first becomes the
single condition on the spin connection

$\frac{1}{4}\omega_{a}{}^{de}a_{bde}=\frac{m}{2}\delta_{ab}\quad\implies\quad\hat{\omega}_{abc}=\frac{m}{3}a_{abc},$
(278)

where the hat indicates that there might be (and are in case of the squashed
seven-sphere) other terms that vanish when contracted with $a_{abc}$ this way.
This condition is in fact satisfied but the easiest way to verify the assumed
form of the Killing spinor is instead to consider the integrability condition

$W_{ab}{}^{cd}\Gamma_{cd}\eta=0,$ (279)

where $W_{ab}{}^{cd}$ is the Weyl tensor and the only linearly independent
solution is

$\eta_{A}=(1,0,....,0).$ (280)

Note that this explicit form of $\eta$ and the gamma-matrices is consistent
with $\bar{\eta}\Gamma_{abc}\eta=-ia_{abc}$. One can check that this $\eta$
solves not only the integrability condition but also the Killing spinor
equation on the left-squashed seven-sphere but not on the right-squashed one
Awada:1982pk ; Duff:1983ajq . That the integrability condition 279 has exactly
one independent solution is equivalent to the fact that the holonomy of the
Killing spinor equation is $G_{2}$, i.e.,
$\frac{1}{4}W_{ab}{}^{cd}\Gamma_{cd}$ generates the Lie algreba $g_{2}$.

There are also identities where structure constants are contracted with the
$Sp_{1}^{C}$ Killing vector components

$s_{a}{}^{j}=(s_{\hat{i}}{}^{j},\;s_{0}{}^{j},\;s_{i}{}^{j})=\biggl{(}\frac{1}{\sqrt{5}}\delta^{j}_{i},\;0,\;0\biggr{)}.$
(281)

The $Sp_{1}^{C}$ Killing vectors are used heavily in the mode calculations in
Section 4. From the explicit realisation of $a_{abc}$ and $c_{abcd}$ given
above one finds the following identities:

$\displaystyle
a_{a}{}^{bc}s_{b}{}^{i}s_{c}{}^{j}=\frac{1}{\sqrt{5}}\epsilon^{ij}{}_{k}s_{a}{}^{k},$
(282) $\displaystyle
c_{ab}{}^{cd}s_{c}{}^{i}s_{d}{}^{j}=\frac{1}{\sqrt{5}}\epsilon^{ij}{}_{k}a_{ab}{}^{c}s_{c}{}^{k}-2s_{[a}{}^{i}s_{b]}{}^{j},$
(283) $\displaystyle
s_{[a|}{}^{i}s^{d}{}_{i}a_{d|bc]}=\frac{1}{15}a_{abc}+\frac{2\sqrt{5}}{3}\epsilon_{ijk}s_{a}{}^{i}s_{b}{}^{j}s_{c}{}^{k},$
(284)

In addition, there is an important derivative relation given in Nilsson:1983ru
which in our notation reads

$D_{a}s_{b}{}^{i}=\frac{1}{2\sqrt{5}}a_{ab}{}^{c}s_{c}{}^{i}-3\epsilon^{i}{}_{jk}s_{a}{}^{j}s_{b}{}^{k}.$
(285)

##  Appendix B $G_{2}$ and $H$ projection operators and Casimirs

Projection operators picking out the various $G_{2}$-irreps occurring in the
decompositions of two-forms, three-forms, spinors and vector-spinors on
$S^{7}$ under $Spin(7)\rightarrow G_{2}$ are, for the convenience of the
reader, given below (the other $so(7)$-irreps of interest are also
$g_{2}$-irreps and so do not need to be decomposed). For spinors, one- and
two-forms, we also give the projection operators onto the $H$-irreps occurring
when decomposing $Spin(7)\rightarrow G_{2}\rightarrow H$. This is relevant for
Section 4, in particular Section 4.2 since $C_{h}$ is not proportional to the
identity on $\bm{14}_{g_{2}}\subset\bm{21}_{so(7)}$. In Section B.2, we give
our conventions for the Casimirs of the various groups.

###  B.1 Projection operators

#### One-forms

The one-form irrep $\bm{7}$ of $so(7)$ does not decompose when breaking
$so(7)$ to $g_{2}$. However, when breaking to $h=sp_{1}^{A}\oplus
sp_{1}^{B+C}\subset g_{2}$, it decomposes as
$\bm{7}\rightarrow(0,2)\oplus(1,1)$. Since $s_{a}{}^{i}s^{b}{}_{i}$ is a
$H$-singlet and has rank three when seen as an operator acting on one-forms,
we have

$(P_{(0,2)})_{a}{}^{b}=5s_{a}{}^{i}s^{b}{}_{i},$ (286)

where we have fixed the coefficient by demanding $P_{(0,2)}{}^{2}=P_{(0,2)}$.
It follows that the projector onto $(1,1)$ is

$(P_{(1,1)})_{a}{}^{b}=\delta_{a}^{b}-5s_{a}{}^{i}s^{b}{}_{i}.$ (287)

#### Two-forms

The two-form irrep $\bm{21}$ decomposes into $\bm{7}\oplus\bm{14}$ under
$g_{2}$. Since $a_{a}{}^{bc}$ can be viewed as an intertwiner
$\bm{21}\to\bm{7}$, we have

$\displaystyle(P_{7})_{ab}{}^{cd}=\frac{1}{6}a_{ab}{}^{e}a^{cde}=\frac{1}{6}(2\delta_{ab}^{cd}+c_{ab}{}^{cd}),$
(288)
$\displaystyle(P_{14})_{ab}{}^{cd}=(1-P_{7})_{ab}{}^{cd}=\frac{1}{6}(4\delta_{ab}^{cd}-c_{ab}{}^{cd}).$
(289)

From this, we see that

$c_{ab}{}^{cd}=(4P_{7}-2P_{14})_{ab}{}^{cd},$ (290)

which is often useful in computations.

When restricting to $H$, the $\bm{7}$ decomposes as described in the one-form
case above and the $\bm{14}$ decomposes into $(0,2)\oplus(2,0)\oplus(1,3)$.
The projection operators projecting onto the $H$-irreps in $\bm{7}$ can be
constructed using $a_{abc}$ and the projectors from the one-form case. For
instance, we see that (c.f. 286)

$(P_{(0,2)\subset\bm{7}})_{ab}{}^{cd}=\frac{5}{6}a_{ab}{}^{e}s_{e}{}^{i}s^{f}{}_{\\!i}{\mkern
2.0mu}a_{f}{}^{cd}=75(P_{7}\varsigma P_{7})_{ab}{}^{cd},$ (291)

where

$\varsigma_{ab}{}^{cd}=s_{[a|}{}^{i}s^{[c|}{}_{i}{\mkern
2.0mu}s_{|b]}{}^{j}s^{|d]}_{j}$ (292)

is a useful building block for constructing the projectors. One can verify
that all four $g_{2}$-blocks of $\varsigma_{ab}{}^{cd}$, viewed as a linear
operator acting on two-forms, that appear when decomposing
$\bm{21}\to\bm{7}\oplus\bm{14}$ have rank three. Since the $7\times 7$ block
is proportional to the projector $(P_{(0,2)\subset\bm{7}})_{ab}{}^{cd}$, the
off-diagonal blocks intertwine $(0,2)\subset\bm{7}$ and $(0,2)\subset\bm{14}$
and the $14\times 14$ block is proportional to
$(P_{(0,2)\subset\bm{14}})_{ab}{}^{cd}$. Working out the normalisation, we
find

$(P_{(0,2)\subset\bm{14}})_{ab}{}^{cd}=\frac{75}{2}(P_{14}\varsigma
P_{14})_{ab}{}^{cd}.$ (293)

Another useful building block is (c.f. 287)

$(\sigma-5\varsigma)_{ab}{}^{cd}=(\delta_{[a|}^{[c|}-5s_{[a|}{}^{i}s^{[c|}{}_{i})s_{|b]}{}^{j}s^{|d]}_{j},\quad\text{where}\quad\sigma_{ab}{}^{cd}=\delta_{[a|}^{[c|}s_{|b]}{}^{i}s^{|d]}_{i}.$
(294)

The off-diagonal $g_{2}$-blocks of $(\sigma-5\varsigma)_{ab}{}^{cd}$ vanish
while the $7\times 7$ and $14\times 14$ blocks have ranks four and eight which
are the dimensions of $(1,1)\subset\bm{7}$ and $(1,3)\subset\bm{14}$,
respectively. Hence, by demanding that the projectors square to themselves to
fix the normalisation, we find

$(P_{(1,1)\subset\bm{7}})_{ab}{}^{cd}=10P_{7}(\sigma-5\varsigma)_{ab}{}^{cd}P_{7},\quad(P_{(1,3)\subset\bm{14}})_{ab}{}^{cd}=10P_{14}(\sigma-5\varsigma)_{ab}{}^{cd}P_{14}.$
(295)

The last projector is simply

$(P_{(2,0)\subset\bm{14}})_{ab}{}^{cd}=(P_{14}-P_{(0,2)\subset\bm{14}}-P_{(1,3)\subset\bm{14}})_{ab}{}^{cd}.$
(296)

These projection operators can be used to get an expression for the Weyl
tensor. Specifically, by using 38 on two-forms we find that

$W_{ab}{}^{cd}=\frac{1}{5}(4P_{14}+8P_{(0,2)\subset\bm{14}}-10P_{(1,3)\subset\bm{14}})_{ab}{}^{cd}\\\
=\frac{1}{5}P_{14}(4-100\sigma+800\varsigma)_{ab}{}^{cd}P_{14}.$ (297)

In computations, we sometimes find it useful to rewrite this as

$W_{ab}{}^{cd}=\frac{4}{5}\delta_{ab}^{cd}-20\sigma_{ab}{}^{cd}+160\varsigma_{ab}{}^{cd}+12(P_{7}\sigma)_{ab}{}^{cd}-120(P_{7}\varsigma)_{ab}{}^{cd}-60(\varsigma
P_{7})_{ab}{}^{cd}.$ (298)

#### Three-forms

The three-form irrep $\bm{35}$ of $so(7)$ decomposes into
$\bm{1}\oplus\bm{7}\oplus\bm{27}$ when restricted to $g_{2}$. Clearly,
$a_{abc}$ and $c_{a}{}^{bcd}$ can be viewed as intertwiners from three-forms
to scalars and one-forms, respectively, which allows us to easily construct
the projection operators onto $\bm{1}$ and $\bm{7}$. The projector onto the
$\bm{27}$ is then obtained by subtracting the previous two from the identity.
Fixing the normalisations through $P^{2}=P$, this gives

$\displaystyle(P_{1})_{abc}{}^{def}=\frac{1}{42}a_{abc}a^{def},$ (299)
$\displaystyle(P_{7})_{abc}{}^{def}=-\frac{1}{24}c_{abc}{}^{g}c_{g}{}^{def}=\frac{1}{24}(6\delta_{abc}^{def}+9\delta_{[a}^{[d}c_{bc]}{}^{ef]}-a_{abc}a^{def}),$
(300)
$\displaystyle(P_{27})_{abc}{}^{def}=(1-P_{1}-P_{7})_{abc}{}^{def}=\frac{1}{56}(42\delta_{abc}^{def}-21\delta_{[a}^{[d}c_{bc]}{}^{ef]}+a_{abc}a^{def}).$
(301)

#### Spin 1/2

The spinor $\bm{8}$ of $so(7)$ splits into a singlet $\bm{1}$, the Killing
spinor, and a $\bm{7}$ when restricted to $g_{2}$. The projection operators
are

$P_{1}=\eta\bar{\eta},\qquad P_{7}=\Gamma_{a}\eta\bar{\eta}\Gamma^{a},$ (302)

where $\eta$ is the normalised Killing spinor ($\bar{\eta}\eta=1$) discussed
in detail in Appendix A.

The projectors can be written in terms of gamma-matrices as

$P_{1}=\frac{1}{8}+\frac{i}{48}a_{abc}\Gamma^{abc},\qquad
P_{7}=\frac{7}{8}-\frac{i}{48}a_{abc}\Gamma^{abc},$ (303)

where we have used that $a_{abc}=i\bar{\eta}\Gamma_{abc}\eta$. This is seen by
computing the traces $\operatorname{Tr}(P\Gamma^{(n)})$ where $n=0,1,2,3$
indicates the number of indices on $\Gamma$. The above proves the Fierz
identity $\Gamma_{a}\eta\bar{\eta}\Gamma^{a}=1-\eta\bar{\eta}$ which is
nothing but $P_{1}+P_{7}=1$.

Lastly, we note that the projectors onto the $H$-irreps appearing in the
decomposition of the spinor under $so(7)\rightarrow h$ are immediately
obtained from the above as

$P_{(0,2)}=\Gamma^{a}\eta(5s_{a}{}^{i}s^{b}{}_{i})\bar{\eta}\Gamma_{b},\qquad
P_{(1,1)}=\Gamma^{a}\eta(\delta_{a}^{b}-5s_{a}{}^{i}s^{b}{}_{i})\bar{\eta}\Gamma_{b}.$
(304)

#### Spin 3/2

Gamma-traceless vector-spinors, or spin-3/2 fields, transform in the $\bm{48}$
of $so(7)$, which decomposes into $\bm{7}\oplus\bm{14}\oplus\bm{27}$ under
$so(7)\rightarrow g_{2}$. From 107–110 one can read off the projection
operators (with supressed spinor indices)

$\displaystyle(P_{7})_{a}{}^{b}=\frac{1}{6}(6\delta_{a}^{b}\eta+ia_{a}{}^{bc}\Gamma_{c}\eta)\bar{\eta},$
(305)
$\displaystyle(P_{14})_{a}{}^{b}=-(P_{14})_{ac}{}^{db}\Gamma^{c}\eta\bar{\eta}\Gamma_{d},$
(306)
$\displaystyle(P_{27})_{a}{}^{b}=\delta_{(a}^{b}\Gamma^{c}\eta\bar{\eta}\Gamma_{c)},$
(307)

where $(P_{14})_{ac}{}^{db}$ on the second line is the two-form projection
operator.

###  B.2 Casimir conventions

In this subsection, we give our conventions for the Casimirs of the groups
appearing in the analysis. First, the isometry group ($G=Sp_{2}\times
Sp_{1}^{C}$) Casimir in the master formula for $\Delta$, 40, is

$C_{g}(p,q;r)=-T_{A}T^{A}=2C_{sp_{2}}(p,q)+3C_{\smash[t]{sp_{1}^{C}}}(r),$
(308)

where $(p,q)$ and $(r)$ are the Dynkin labels of $sp_{2}$ and $sp_{1}^{C}$,
respectively, $C_{sp_{2}}$ and $C_{sp_{1}}$ are given in Table 14 and, as in
Section 2, we raise and lower $G$-indices using $-\kappa_{AB}/6$ and its
inverse, where $\kappa_{AB}$ is the negative definite Cartan–Killing form. The
Casimir of the isotropy group, $H=Sp_{1}^{A}\times Sp_{1}^{B+C}$, is

$C_{h}=-T_{i}T^{i}=2C_{sp_{1}^{A}}+\frac{6}{5}C_{\smash[t]{sp_{1}^{B+C}}},$
(309)

where $H$-indices are raised and lowered using the restriction of
$-\kappa_{AB}/6$ to $h$ and its inverse.

The Casimirs of $Spin(7)$ and $G_{2}$ also appear in 40. These are defined by

$C_{so(7)}=-\Sigma_{ab}\Sigma^{ab},\qquad
C_{g_{2}}=-(P_{14})_{abcd}\Sigma^{ab}\Sigma^{cd}.$ (310)

Formulas for their eigenvalues on irreps in terms of Dynkin labels are
provided in Table 14.

Casimir | Adjoint
---|---
$C_{sp_{1}}(r)=\frac{1}{4}{\mkern 2.0mu}r(r+2)$ | $C_{sp_{1}}(2)=2$
$C_{sp_{2}}(p,q)=\frac{1}{4}{\mkern 2.0mu}p(p+2q+4)+\frac{1}{2}{\mkern 2.0mu}q(q+3)$ | $C_{sp_{2}}(2,0)=3$
$C_{g_{2}}(p,q)=\frac{1}{3}{\mkern 2.0mu}p(p+3q+5)+q(q+3)$ | $C_{g_{2}}(0,1)=4$
$C_{so(7)}(p,q,r)=\frac{1}{2}{\mkern 2.0mu}p(p+2q+r+5)+q(q+r+4)+\frac{3}{8}{\mkern 2.0mu}r(r+6)$ | $C_{so(7)}(0,1,0)=5$
Table 14: Casimir eigenvalues on irreps specified by Dynkin labels with the
adjoint representation given explicitly.

##  Appendix C Non-transverse modes and two-form mode scalar products

In this appendix we give the non-transverse eigenmodes of $\Delta_{1}$ and
$\Delta_{2}$ that, just like the transverse ones, are obtained by
diagonalising 156 and 197. We also provide the two-form scalar products that
are needed to derive the cross diagrams in Figure 4 and below. In
Nilsson:2018lof , the isometry irrep spectra are first derived for the
transverse and non-transverse modes combined by decomposing the
$Spin(7)$-irreps into $H$-irreps and, then, the non-transverse pieces are
subtracted to arrive at the transverse irrep spectra. Deriving the cross
diagrams for the non-transverse modes and combining these with the transverse
ones of Section 4 thus provides a consistency check of this last step in
Nilsson:2018lof  upon comparison.

From 156, we immediately see that the non-transverse one-form eigenmode is
generated by

$\tilde{\mathcal{Y}}^{(2)}_{a}=\mathcal{Y}^{(1)}_{a}=\check{D}_{a}$ (311)

with eigenvalue

$\Delta_{1}^{(2)}=C_{g}$ (312)

and norm

$\|\tilde{\mathcal{Y}}^{(2)}\phi\|^{2}=C_{g}^{\phi}\|\phi\|^{2}.$ (313)

Hence, this non-transverse eigenmode exists for $(p,q;r)=(p,q;p)$ with
$(p,q)\neq(0,0)$ corresponding to the cross diagram in Figure 6. Note that the
cross-diagrams in Figures 2 and 6 combined exactly match the isometry irrep
spectrum of (transverse and non-transverse) one-froms obtained from the
$(0,2)$ and $(1,1)$ $H$-diagrams in Nilsson:2018lof .323232Recall that
$\bm{7}\to(0,2)\oplus(1,1)$ when $Spin(7)\to H$. This provides the first
consistency check of the type described above.

$\Delta_{1}^{(2)}$
---
$1$$2$$3$$4$$1$$2$$3$$4$$0$$p$$0$$q$$r=p$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$
Figure 6: Non-transverse one-form cross diagrams. Each cross corresponds to an
isometry irrep $(p,q;r)$ of non-transverse eigenmodes of $\Delta_{1}$
generated by the differential operator in 311 and with the eigenvalue in 312.

Turning to two-forms, the non-transverse $\Delta_{2}$-eigenmodes obtained by
diagonalising 197 are generated by the following differential operators with
associated eigenvalues:

$\displaystyle\begin{aligned}
\tilde{\mathcal{Y}}^{(4)i}_{ab}&=\frac{1}{20}\Bigl{(}15\Delta
C-13\pm\sqrt{\vphantom{C_{G}}\smash[b]{20C_{g}+49}}\Bigr{)}\mathcal{Y}^{(2)i}_{ab}+\frac{3}{2\sqrt{5}}\Bigl{(}13\mp\sqrt{\vphantom{C_{G}}\smash[b]{20C_{g}+49}}\Bigr{)}\mathcal{Y}^{(3)i}_{ab}\\\
&\phantom{{}={}}+\frac{1}{2\sqrt{5}}\Bigl{(}12\mp\sqrt{\vphantom{C_{G}}\smash[b]{20C_{g}+49}}\Bigr{)}\mathcal{Y}^{(4)i}_{ab}-\frac{\sqrt{5}}{2}\mathcal{Y}^{(5)i}_{ab}+\mathcal{Y}^{(6)i}_{ab},\end{aligned}$
(314)
$\displaystyle\Delta_{2}^{(4)_{\pm}}=\frac{m^{2}}{9}\Bigl{(}20C_{g}+14\pm
2\sqrt{\vphantom{C_{G}}\smash[b]{20C_{g}+49}}\Bigr{)}.$ (315)

To compute the norms of these non-transverse eigenmodes, and also for the
norms of the transverse eigenmodes in 207–210 leading to Figure 4, we need the
scalar products of the basic building blocks $\mathcal{Y}^{(\text{1--
7})}_{ab}$ from 190–192. Hence, we compute

$\displaystyle\langle\mathcal{Y}^{(1)}\phi,\mathcal{Y}^{(1)}\phi\rangle=6C_{g}^{\phi}\|\phi\|^{2},$
(316)
$\displaystyle\langle\mathcal{Y}^{(1)}\phi,\mathcal{Y}^{(2)i}s_{i}\phi\rangle=6C_{\smash[t]{sp_{1}^{C}}}^{\phi}\|\phi\|^{2},$
(317)
$\displaystyle\langle\mathcal{Y}^{(1)}\phi,\mathcal{Y}^{(3)i}s_{i}\phi\rangle=\frac{2}{\sqrt{5}}C_{\smash[t]{sp_{1}^{C}}}^{\phi}\|\phi\|^{2},$
(318)
$\displaystyle\langle\mathcal{Y}^{(1)}\phi,\mathcal{Y}^{(4)i}s_{i}\phi\rangle=-\frac{3}{\sqrt{5}}C_{\smash[t]{sp_{1}^{C}}}^{\phi}\|\phi\|^{2},$
(319)
$\displaystyle\langle\mathcal{Y}^{(1)}\phi,\mathcal{Y}^{(5)i}s_{i}\phi\rangle=4\langle\mathcal{Y}^{(1)}\phi,\mathcal{Y}^{(4)i}s_{i}\phi\rangle,$
(320)
$\displaystyle\langle\mathcal{Y}^{(1)}\phi,\mathcal{Y}^{(6)i}s_{i}\phi\rangle=\frac{27}{10}C_{\smash[t]{sp_{1}^{C}}}^{\phi}\|\phi\|^{2},$
(321)
$\displaystyle\langle\mathcal{Y}^{(1)}\phi,\mathcal{Y}^{(7)ij}s_{ij}\phi\rangle=-\frac{1}{6}C_{\smash[t]{sp_{1}^{C}}}^{\phi}\bigl{(}4C_{\smash[t]{sp_{1}^{C}}}^{\phi}-3\bigr{)}\|\phi\|^{2},$
(322)
$\displaystyle\langle\mathcal{Y}^{(2)i}\phi,\mathcal{Y}^{(2)j}\varphi\rangle=\frac{6}{5}\langle\phi,\delta^{ij}\varphi\rangle,$
(323)
$\displaystyle\langle\mathcal{Y}^{(2)i}\phi,\mathcal{Y}^{(3)j}\varphi\rangle=\frac{2}{5\sqrt{5}}\langle\phi,\delta^{ij}\varphi\rangle,$
(324)
$\displaystyle\langle\mathcal{Y}^{(2)i}\phi,\mathcal{Y}^{(4)j}\varphi\rangle=\frac{1}{\sqrt{5}}\langle\phi,\epsilon^{ijk}s_{k}\varphi\rangle,$
(325)
$\displaystyle\langle\mathcal{Y}^{(2)i}\phi,\mathcal{Y}^{(5)j}\varphi\rangle=4\langle\mathcal{Y}^{(2)i}\phi,\mathcal{Y}^{(4)j}\varphi\rangle,$
(326)
$\displaystyle\langle\mathcal{Y}^{(2)i}\phi,\mathcal{Y}^{(6)j}\varphi\rangle=\frac{1}{5}C_{g}^{\phi}\langle\phi,\delta^{ij}\varphi\rangle+\langle\phi,s^{ij}\varphi\rangle,$
(327)
$\displaystyle\langle\mathcal{Y}^{(2)i}\phi,\mathcal{Y}^{(7)jk}s_{k}\varphi\rangle=-\frac{1}{10}C_{\smash[t]{sp_{1}^{C}}}^{\phi}\langle\phi,\delta^{ij}\varphi\rangle-\frac{1}{12}\langle\phi,\epsilon^{ijk}s_{k}\varphi\rangle+\frac{1}{30}\langle\phi,s^{ij}\varphi\rangle,$
(328)
$\displaystyle\langle\mathcal{Y}^{(3)i}\phi,\mathcal{Y}^{(3)j}\varphi\rangle=\frac{2}{25}\langle\phi,\delta^{ij}\varphi\rangle,$
(329)
$\displaystyle\langle\mathcal{Y}^{(3)i}\phi,\mathcal{Y}^{(4)j}\varphi\rangle=\frac{1}{5}\langle\phi,\epsilon^{ijk}s_{k}\varphi\rangle,$
(330)
$\displaystyle\langle\mathcal{Y}^{(3)i}\phi,\mathcal{Y}^{(5)j}\varphi\rangle=\frac{2}{\sqrt{5}}\langle\mathcal{Y}^{(2)i}\phi,\mathcal{Y}^{(4)j}\varphi\rangle-2\langle\mathcal{Y}^{(3)i}\phi,\mathcal{Y}^{(4)j}\varphi\rangle,$
(331)
$\displaystyle\langle\mathcal{Y}^{(3)i}\phi,\mathcal{Y}^{(6)j}\varphi\rangle=\frac{1}{\sqrt{5}}C_{\smash[t]{sp_{1}^{C}}}^{\phi}\langle\phi,\delta^{ij}\varphi\rangle+\frac{1}{\sqrt{5}}\langle\phi,s^{ij}\varphi\rangle,$
(332)
$\displaystyle\langle\mathcal{Y}^{(3)i}\phi,\mathcal{Y}^{(7)jk}s_{k}\varphi\rangle=-\frac{1}{10\sqrt{5}}C_{\smash[t]{sp_{1}^{C}}}^{\phi}\langle\phi,\delta^{ij}\varphi\rangle-\frac{1}{12\sqrt{5}}\langle\phi,\epsilon^{ijk}s_{k}\varphi\rangle+\frac{1}{30\sqrt{5}}\langle\phi,s^{ij}\varphi\rangle,$
(333)
$\displaystyle\langle\mathcal{Y}^{(4)i}\phi,\mathcal{Y}^{(4)j}\varphi\rangle=\frac{1}{10}C_{g}^{\phi}\langle\phi,\delta^{ij}\varphi\rangle-\frac{1}{4}\langle\phi,\epsilon^{ijk}s_{k}\varphi\rangle+\frac{1}{2}\langle\phi,s^{ij}\varphi\rangle,$
(334)
$\displaystyle\langle\mathcal{Y}^{(4)i}\phi,\mathcal{Y}^{(5)j}\varphi\rangle=-\frac{2}{5}\langle\phi,\epsilon^{ijk}s_{k}\varphi\rangle,$
(335) $\displaystyle\begin{aligned}
\langle\mathcal{Y}^{(4)i}\phi,\mathcal{Y}^{(6)j}\varphi\rangle&=\frac{1}{20\sqrt{5}}\bigl{(}C_{g}^{\phi}-30C_{\smash[t]{sp_{1}^{C}}}^{\phi}\bigr{)}\langle\phi,\delta^{ij}\varphi\rangle+\frac{1}{8\sqrt{5}}\bigl{(}4C_{g}^{\phi}-9\bigr{)}\langle\phi,\epsilon^{ijk}s_{k}\varphi\rangle\\\
&\phantom{{}={}}-\frac{11}{4\sqrt{5}}\langle\phi,s^{ij}\varphi\rangle,\end{aligned}$
(336) $\displaystyle\begin{aligned}
\langle\mathcal{Y}^{(4)i}\phi,\mathcal{Y}^{(7)jk}s_{k}\varphi\rangle&=-\frac{3}{20\sqrt{5}}C_{\smash[t]{sp_{1}^{C}}}^{\phi}\langle\phi,\delta^{ij}\varphi\rangle-\frac{1}{12\sqrt{5}}\bigl{(}3C_{\smash[t]{sp_{1}^{C}}}^{\phi}-2\bigr{)}\langle\phi,\epsilon^{ijk}s_{k}\varphi\rangle\\\
&\phantom{{}={}}-\frac{11}{30\sqrt{5}}\langle\phi,s^{ij}\varphi\rangle,\end{aligned}$
(337)
$\displaystyle\langle\mathcal{Y}^{(5)i}\phi,\mathcal{Y}^{(5)j}\varphi\rangle=8\langle\mathcal{Y}^{(4)i}\phi,\mathcal{Y}^{(4)j}\varphi\rangle+2\langle\mathcal{Y}^{(4)i}\phi,\mathcal{Y}^{(5)j}\varphi\rangle,$
(338)
$\displaystyle\langle\mathcal{Y}^{(5)i}\phi,\mathcal{Y}^{(6)j}\varphi\rangle=2\langle\mathcal{Y}^{(4)i}\phi,(3P_{7}-1)\mathcal{Y}^{(6)j}\varphi\rangle,$
(339)
$\displaystyle\langle\mathcal{Y}^{(5)i}\phi,\mathcal{Y}^{(7)jk}s_{k}\varphi\rangle=2\langle\mathcal{Y}^{(4)i}\phi,(3P_{7}-1)\mathcal{Y}^{(7)jk}s_{k}\varphi\rangle,$
(340) $\displaystyle\begin{aligned}
\langle\mathcal{Y}^{(6)i}\phi,\mathcal{Y}^{(6)j}\varphi\rangle&=\frac{1}{200}C_{g}^{\phi}\bigl{(}20C_{g}^{\phi}-39\bigr{)}\langle\phi,\delta^{ij}\varphi\rangle-\frac{1}{80}\bigl{(}36C_{g}^{\phi}-171\bigr{)}\langle\phi,\epsilon^{ijk}s_{k}\varphi\rangle\\\
&\phantom{{}={}}+\frac{1}{40}\bigl{(}20C_{g}^{\phi}-3\bigr{)}\langle\phi,s^{ij}\varphi\rangle,\end{aligned}$
(341) $\displaystyle\begin{aligned}
\langle\mathcal{Y}^{(6)i}\phi,\mathcal{Y}^{(7)jk}s_{k}\varphi\rangle&=-\frac{1}{200}C_{\smash[t]{sp_{1}^{C}}}^{\phi}\bigl{(}10C_{g}^{\phi}-47\bigr{)}\langle\phi,\delta^{ij}\varphi\rangle\\\
&\phantom{{}={}}-\frac{1}{120}\bigl{(}5C_{g}^{\phi}-19C_{\smash[t]{sp_{1}^{C}}}^{\phi}+7\bigr{)}\langle\phi,\epsilon^{ijk}s_{k}\varphi\rangle\\\
&\phantom{{}={}}+\frac{1}{300}\bigl{(}5C_{g}^{\phi}-100C_{\smash[t]{sp_{1}^{C}}}^{\phi}+214\bigr{)}\langle\phi,s^{ij}\varphi\rangle,\end{aligned}$
(342) $\displaystyle\begin{aligned}
\langle\mathcal{Y}^{(7)ij}\phi,\mathcal{Y}^{(7)}_{kl}\varphi\rangle&=\frac{1}{50}\bigl{(}C_{g}^{\phi}-5C_{\smash[t]{sp_{1}^{C}}}^{\phi}\bigr{)}\langle\phi,\delta^{\\{i}_{\\{k}\delta^{j\\}}_{l\\}}\varphi\rangle+\frac{9}{100}\langle\phi,\delta^{\\{i}_{\\{k}\epsilon_{l\\}}{}^{j\\}m}s_{m}\varphi\rangle\\\
&\phantom{{}={}}-\frac{1}{10}\langle\phi,\delta^{\\{i}_{\\{k}s_{l\\}}{}^{j\\}}\varphi\rangle,\end{aligned}$
(343)

where

$\displaystyle
P_{7}\mathcal{Y}^{(6)i}_{ab}=\frac{1}{6}\mathcal{Y}^{(1)}_{ab}s^{i}-\frac{1}{6}\mathcal{Y}^{(2)i}_{ab}\check{\Box}+\frac{1}{2}\epsilon^{i}{}_{jk}\mathcal{Y}^{(2)j}_{ab}s^{k}+\frac{1}{6\sqrt{5}}\mathcal{Y}^{(4)i}_{ab}+\frac{1}{12\sqrt{5}}\mathcal{Y}^{(5)i}_{ab},$
(344) $\displaystyle
P_{7}\mathcal{Y}^{(7)ij}_{ab}=\frac{1}{6}\mathcal{Y}^{(2)\\{i}_{ab}s^{j\\}}.$
(345)

To compute scalar products and norms like $\langle
P_{r}\mathcal{Y}^{m}\phi,P_{r}\mathcal{Y}^{(n)}\phi\rangle$, we need to use
151 in cases where the differential operators are not $sp_{1}^{C}$-singlets.
For the $\bm{5}_{sp_{1}^{C}}$ case, we need some $sp_{1}^{C}$ identities, such
as

$\displaystyle\epsilon^{(i}{}_{kl}\delta^{j)}_{(m}\delta_{n)}^{k}\epsilon^{(m|p}{}_{r}\mathcal{Y}_{pq}s^{|n)qr}s^{l}$
$\displaystyle=\frac{1}{3}Y^{ij}s^{2}+\frac{1}{4}\delta^{ij}\mathcal{Y}^{kl}s_{kl}-\frac{1}{4}\epsilon^{(i}{}_{kl}\mathcal{Y}^{j)k}s^{l}$
$\displaystyle\phantom{{}={}}-\frac{1}{6}\mathcal{Y}^{k(i}s_{k}{}^{j)}(7+3s^{2})+\frac{3}{2}\epsilon^{(i|k}{}_{m}\mathcal{Y}_{kl}s^{|j)lm}+\frac{1}{2}\mathcal{Y}_{kl}s^{ijkl},$
(346)

since 151 requires four successive applications of the Casimir

$\displaystyle C_{\smash[t]{sp_{1}^{C}}}\mathcal{Y}^{ij}\phi$
$\displaystyle=-(T_{k}T^{k}\mathcal{Y}^{ij})\phi-\mathcal{Y}^{ij}(T_{k}T^{k}\phi)-2(T_{k}\mathcal{Y}^{ij})(T^{k}\phi)$
$\displaystyle=\bigl{(}C_{\smash[t]{sp_{1}^{C}}}^{\phi}+6\bigr{)}\mathcal{Y}^{ij}\phi-4\epsilon^{(i}{}_{kl}\mathcal{Y}^{j)k}s^{l},$
(347)

where the $6$ comes from $C_{\smash[t]{sp_{1}^{C}}}(\bm{5})=6$. This reduces
the computation of $\langle
P_{r}\mathcal{Y}^{m}\phi,P_{r}\mathcal{Y}^{(n)}\phi\rangle$ to 316–343 with
$\varphi$ substituted by one of

$\delta_{ij}\phi,\qquad\epsilon_{ijk}s^{k}\phi,\qquad s_{ij},$ (348)

in the $\bm{3}_{sp_{1}^{C}}$ cases (of the form
$\langle\mathcal{Y}^{i}\phi,\mathcal{Y}^{\prime j}\varphi\rangle$) and one of

$\delta_{(i}^{(k}\delta_{j)}^{l)}\phi,\quad\epsilon_{(i}{}^{(k|m|}\delta_{j)}^{l)}s_{m}\phi,\quad\delta_{(i}^{(k}s_{j)}{}^{l)}\phi,\quad\epsilon_{(i}{}^{(k|m|}s_{j)m}{}^{l)}\phi,\quad
s_{ij}{}^{kl}\phi,$ (349)

in the $\bm{5}_{sp_{1}^{C}}$ cases (of the form
$\langle\mathcal{Y}^{ij}\phi,\mathcal{Y}^{\prime}_{kl}\varphi\rangle$).
Lastly, to put the above together, we need identities like 171 in the
$\bm{3}_{sp_{1}^{C}}$ case and, for instance,

$\langle\phi,s_{l}{}^{j}s_{ij}{}^{il}\phi\rangle=-C_{\smash[t]{sp_{1}^{C}}}^{\phi}\biggl{(}\frac{5}{36}-\frac{23}{54}C_{\smash[t]{sp_{1}^{C}}}^{\phi}+\frac{4}{9}\bigl{(}C_{\smash[t]{sp_{1}^{C}}}^{\phi}\bigr{)}^{2}\biggr{)}\|\phi\|^{2},$
(350)

in the $\bm{5}_{sp_{1}^{C}}$ case.

Calculating the norms of the non-transverse $\Delta_{2}$-eigenmodes using the
above scalar products, we arrive at the cross-diagrams in Figure 7. Note that
the cross diagrams in Figures 7 and 4 combined exactly match the complete two-
form isometry irrep spectrum obtained by combining the relevant $H$-diagrams
in Nilsson:2018lof , providing a non-trivial consistency check.333333Recall
that $\bm{21}\to(1,1)\oplus 2\times(0,2)\oplus(2,0)\oplus(1,3)$ when
$Spin(7)\to H$.

$\Delta_{2}^{(4)_{-}}$ | $\Delta_{2}^{(4)_{+}}$
---|---
$1$$2$$3$$4$$1$$2$$3$$4$$0$$p$$0$$q$$r=p+2$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$ |  $1$$2$$3$$4$$1$$2$$3$$4$$0$$p$$0$$q$$r=p+2$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$
$1$$2$$3$$4$$1$$2$$3$$4$$0$$p$$0$$q$$r=p$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$ |  $1$$2$$3$$4$$1$$2$$3$$4$$0$$p$$0$$q$$r=p$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$
$1$$2$$3$$4$$1$$2$$3$$4$$0$$p$$0$$q$$r=p-2$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$ |  $1$$2$$3$$4$$1$$2$$3$$4$$0$$p$$0$$q$$r=p-2$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$
Figure 7: Non-transverse two-form cross diagrams. Each cross corresponds to an
isometry irrep $(p,q;r)$ of non-transverse eigenmodes of $\Delta_{2}$, listed
in 314, with associated eigenvalues given in 315.

##  Appendix D Spin-0 fields with $\frac{5}{2}<E_{0}\leq 3$ and spin-1/2
fields with $2<E_{0}\leq 3$

We start by listing the bosonic and fermionic fields in the $AdS_{4}$ spectrum
of the round $S^{7}$ with $E_{0}\leq 3$ Duff:1986hr . These are, apart from
the massless graviton (with $E_{0}=3$), spin-1 gauge fields (with $E_{0}=2$)
and massive $1^{-}$ vector fields from the levels $n=1$ and $n=2$ (with
$E_{0}=\frac{5}{2}$ and $E_{0}=3$), the scalar, pseudo-scalar and spin-1/2
fermion fields listed in Tables 15 and 16.

| $0^{+}_{\scriptscriptstyle(-)}(\Delta_{0})$ | $0^{-}_{\scriptscriptstyle(-)}(Q)$
---|---|---
$n$ |  $Spin(8)$ irrep | $M^{2}$ | $E_{0}$ | B.c. |  $Spin(8)$ irrep | $M^{2}$ | $E_{0}$ | B.c.
$-1$ | $(1,0,0,0)=\bm{8}_{v}$ | $3m^{2}$ | $\frac{1}{2}$ | sing. |  |  |  |
$0$ | $(2,0,0,0)=\bm{35}_{v}$ | $0$ | $1$ | Neu. | $(0,0,2,0)=\bm{35}_{c}$ | $0$ | $2$ | Dir.
$1$ | $(3,0,0,0)=\bm{112}_{v}$ | $-m^{2}$ | $\frac{3}{2}$ | deg. | $(1,0,2,0)=\bm{112}_{cv}$ | $3m^{2}$ | $\frac{5}{2}$ | Dir
$2$ | $(4,0,0,0)$ | $0$ | $2$ | Dir. | $(2,0,2,0)$ | $8m^{2}$ | $3$ | Dir.
$3$ | $(5,0,0,0)$ | $3m^{2}$ | $\frac{5}{2}$ | Dir. |  |  |  |
$4$ | $(6,0,0,0)$ | $8m^{2}$ | $3$ | Dir. |  |  |  |
Table 15: Scalar and pseudo-scalar fields with $E_{0}\leq 3$ in the $AdS_{4}$ spectrum based on the round $S^{7}$ where $n$ is the level number. All massless (pseudo-)scalar fields in the spectrum are present in this list. These fields are dual to relevant operators in the boundary $CFT_{3}$. We also specify whether the fields have singleton ($E_{0}=\frac{1}{2}$), Neumann ($\frac{1}{2}<E_{0}<\frac{3}{2}$), degenerate ($E_{0}=\frac{3}{2}$) or Dirichlet ($E_{0}>\frac{3}{2}$) boundary conditions. | $\frac{1}{2}_{\scriptscriptstyle(-)}(i\not{D}_{1/2})$ | $\frac{1}{2}_{\scriptscriptstyle(-)}(i\not{D}_{3/2})$
---|---|---
$n$ |  $Spin(8)$ irrep | $M$ | $E_{0}$ | B.c. |  $Spin(8)$ irrep | $M$ | $E_{0}$ | B.c.
$-1$ | $(0,0,1,0)=\bm{8}_{c}$ | $-m$ | $1$ | sing. |  |  |  |
$0$ | $(1,0,1,0)=\bm{56}_{s}$ | $0$ | $\frac{3}{2}$ | deg. |  |  |  |
$1$ | $(2,0,1,0)$ | $m$ | $2$ | Dir. | $(0,1,1,0)$ | $-3m$ | $3$ | Dir.
$2$ | $(3,0,1,0)$ | $2m$ | $\frac{5}{2}$ | Dir. |  |  |  |
$3$ | $(4,0,1,0)$ | $3m$ | $3$ | Dir. |  |  |  |
Table 16: Spin-1/2 fermion fields with $E_{0}\leq 3$ in the $AdS_{4}$ spectrum
based on the round $S^{7}$ where $n$ is the level number. All massless
spin-1/2 fields in the spectrum are present in this list. These fields are
dual to relevant operators in the boundary $CFT_{3}$. We also specify whether
the fields have singleton ($E_{0}=1$), Neumann ($1<E_{0}<\frac{3}{2}$),
degenerate ($E_{0}=\frac{3}{2}$) or Dirichlet ($E_{0}>\frac{3}{2}$) boundary
conditions.

To facilitate the comparison between the squashed and the round cases, we
present the decomposition of some of the $Spin(8)$ irreps under
$Spin(8)\rightarrow Sp_{2}\times Sp_{1}^{C}$, for both left- and right-
squashing when this is relevant (indicated by $L$ and $R$ on the arrow), in
Table 17. Recall from, e.g., Duff:1986hr  that the left- and right-squashings
can be defined by how the three eight-dimensional irreps of $Spin(8)$ behave.
While ${\bf 8}_{v}=(1,0,0,0)\rightarrow(1,0;1)=({\bf 4},{\bf 2})$ is true in
both cases the irreps ${\bf 8}_{c}=(0,0,1,0)$ and ${\bf 8}_{s}=(0,0,0,1)$
interchange their behaviour when skew-whiffed, see Table 17.

$Spin(8)$ | $\longrightarrow$ | $Sp_{2}\times Sp_{1}^{C}$
---|---|---
${\bf 8}_{v}=(1,0,0,0)$ | $\overset{\phantom{L}}{\longrightarrow}$ | $(1,0;1)=({\bf 4},{\bf 2})$
${\bf 35}_{v}=(2,0,0,0)$ | $\overset{\phantom{L}}{\longrightarrow}$ | $(2,0;2)\oplus(0,1;0)=({\bf 10},{\bf 3})\oplus({\bf 5},{\bf 1})$
${\bf 112}_{v}=(3,0,0,0)$ | $\overset{\phantom{L}}{\longrightarrow}$ | $(3,0;3)\oplus(1,1;1)=({\bf 20},{\bf 4})\oplus({\bf 16},{\bf 2})$
${\bf 294}_{v}=(4,0,0,0)$ | $\overset{\phantom{L}}{\longrightarrow}$ | $(4,0;4)\oplus(2,1;2)\oplus(0,2;0)=({\bf 35}^{\prime},{\bf 5})\oplus({\bf 35},{\bf 3})\oplus({\bf 14},{\bf 1})$
${\bf 8}_{c}=(0,0,1,0)$ | $\overset{L}{\longrightarrow}$ | $(0,1;0)\oplus(0,0;2)=({\bf 5},{\bf 1})\oplus({\bf 1},{\bf 3})$
${\bf 8}_{s}=(0,0,0,1)$ | $\overset{L}{\longrightarrow}$ | $(1,0;1)=({\bf 4},{\bf 2})$
${\bf 56}_{s}=(1,0,1,0)$ | $\overset{L}{\longrightarrow}$ | $(1,1;1)\oplus(1,0;3)\oplus(1,0;1)=({\bf 16},{\bf 2})\oplus({\bf 4},{\bf 4})\oplus({\bf 4},{\bf 2})$
${\bf 8}_{c}=(0,0,1,0)$ | $\overset{R}{\longrightarrow}$ | $(1,0;1)=({\bf 4},{\bf 2})$
${\bf 8}_{s}=(0,0,0,1)$ | $\overset{R}{\longrightarrow}$ | $(0,1;0)\oplus(0,0;2)=({\bf 5},{\bf 1})\oplus({\bf 1},{\bf 3})$
${\bf 56}_{s}=(1,0,1,0)$ | $\overset{R}{\longrightarrow}$ | $(2,0;2)\oplus(2,0;0)\oplus(0,1;2)\oplus(0,0;0)=$
|  | $({\bf 10},{\bf 3})\oplus({\bf 10},{\bf 1})\oplus({\bf 5},{\bf 3})\oplus({\bf 1},{\bf 1})$
Table 17: Decompositions of some $Spin(8)$-irreps into $Sp_{2}\times
Sp_{1}^{C}$-irreps. The behaviour of the irreps ${\bf 8}_{c}=(0,0,1,0)$ and
${\bf 8}_{s}=(0,0,0,1)$ define how the decompositions work in the left- and
right-squashed seven-sphere vacua.

In the case of the $AdS_{4}$ spectra related to the squashed sphere, we here
list the scalar and pseudo-scalar fields in the range $\frac{5}{2}<E_{0}\leq
3$, see Table 18, which complements Tables 9 and 10 to provide a full account
of all such fields with $E_{0}\leq 3$. These fields have masses between
$M^{2}=3m^{2}$ and $M^{2}=8m^{2}$ and can thus only have Dirichlet boundary
conditions. The analogous fermions have $2<E_{0}\leq 3$ and masses in the
range $m<|M|\leq 3m$. These fields can be found in Tables 19 and 20,
complementing LABEL:{tab:D_or_N:spinors}.

Field | $(p,q;r)$ | $C_{g}$ | $M^{2}/m^{2}$ | $E_{0}$
---|---|---|---|---
$0^{+}_{\scriptscriptstyle(-)}(\Delta_{0}^{(1)})$ | $(1,3;1)$ | $103/4$ | $\frac{911}{9}-8\sqrt{149}$ | $\frac{3}{2}+\frac{1}{3}\bigl{(}\sqrt{149}-9\bigr{)}$
| $(2,2;2)$ | $26$ | $\frac{916}{9}-4\sqrt{601}$ | $\frac{3}{2}+\frac{1}{6}\bigl{(}\sqrt{601}-18\bigr{)}$
| $(0,4;0)$ | $28$ | $\frac{956}{9}-4\sqrt{641}$ | $\frac{3}{2}+\frac{1}{6}\bigl{(}\sqrt{641}-18\bigr{)}$
| $(3,1;3)$ | $115/4$ | $\frac{971}{9}-16\sqrt{41}$ | $\frac{3}{2}+\frac{1}{3}\bigl{(}2\sqrt{41}-9\bigr{)}$
Field ($LS^{7}$) | $(p,q;r)$ | $C_{g}$ | $M_{L}^{2}/m^{2}$ | $E_{0}$
$0^{-}(Q^{(1)_{-}})$ | $(0,0;0)$ | $0$ | $\frac{40}{9}$ | $\frac{3}{2}+\frac{7}{6}$
| $(0,2;0)$ | $10$ | $\frac{16}{9}\bigl{(}16-\sqrt{201}{\mkern 2.0mu}\bigr{)}$ | $\frac{3}{2}+\frac{1}{6}\bigl{(}\sqrt{201}-8\bigr{)}$
| $(2,0;2)$ | $12$ | $\frac{8}{9}\bigl{(}37-2\sqrt{241}{\mkern 2.0mu}\bigr{)}$ | $\frac{3}{2}+\frac{1}{6}\bigl{(}\sqrt{241}-8\bigr{)}$
$0^{-}(Q^{(2)_{-}})$ | $(2,1;0)$ | $12$ | $\frac{40}{9}$ | $\frac{3}{2}+\frac{7}{6}$
| $(3,0;1)$ | $51/4$ | $\frac{5}{9}\bigl{(}79-16\sqrt{19}{\mkern 2.0mu}\bigr{)}$ | $\frac{3}{2}+\frac{1}{3}\bigl{(}2\sqrt{19}-5\bigr{)}$
| $(1,0;3)$ | $55/4$ | $\frac{55}{9}$ | $\frac{3}{2}+\frac{4}{3}$
$0^{-}(Q^{(3)_{-}})$ | $(1,2;1)$ | $67/4$ | $\frac{1}{9}\bigl{(}551-96\sqrt{26}{\mkern 2.0mu}\bigr{)}$ | $\frac{3}{2}+\frac{2}{3}\bigl{(}\sqrt{26}-3\bigr{)}$
| $(0,3;0)$ | $18$ | $8$ | $\frac{3}{2}+\frac{3}{2}$
| $(2,1;2)$ | $18$ | $8$ | $\frac{3}{2}+\frac{3}{2}$
Field ($RS^{7}$) | $(p,q;r)$ | $C_{g}$ | $M^{2}_{R}/m^{2}$ | $E_{0}$
$0^{-}(Q^{(1)_{+}})$ | $(4,0;0)$ | $16$ | $\frac{4}{9}\bigl{(}103-5\sqrt{321}{\mkern 2.0mu}\bigr{)}$ | $\frac{3}{2}+\frac{1}{6}\bigl{(}\sqrt{321}-10\bigr{)}$
| $(1,2;1)$ | $67/4$ | $\frac{1}{9}\bigl{(}427-80\sqrt{21}{\mkern 2.0mu}\bigr{)}$ | $\frac{3}{2}+\frac{1}{3}\bigl{(}2\sqrt{21}-5\bigr{)}$
| $(0,3;0)$ | $18$ | $8$ | $\frac{3}{2}+\frac{3}{2}$
| $(2,1;2)$ | $18$ | $8$ | $\frac{3}{2}+\frac{3}{2}$
$0^{-}(Q^{(2)_{+}})$ | $(1,1;1)$ | $39/4$ | $\frac{1}{9}\bigl{(}299-32\sqrt{61}{\mkern 2.0mu}\bigr{)}$ | $\frac{3}{2}+\frac{1}{3}\bigl{(}\sqrt{61}-4\bigr{)}$
| $(0,1;2)$ | $10$ | $\frac{16}{9}\bigl{(}19-\sqrt{249}{\mkern 2.0mu}\bigr{)}$ | $\frac{3}{2}+\frac{1}{6}\bigl{(}\sqrt{249}-8\bigr{)}$
| $(2,0;2)$ | $12$ | $8$ | $\frac{3}{2}+\frac{3}{2}$
| $(2,1;0)$ | $12$ | $8$ | $\frac{3}{2}+\frac{3}{2}$
$0^{-}(Q^{(3)_{+}})$ | $(0,1;0)$ | $4$ | $\frac{4}{9}\bigl{(}47-3\sqrt{161}{\mkern 2.0mu}\bigr{)}$ | $\frac{3}{2}+\frac{1}{6}\bigl{(}\sqrt{161}-6\bigr{)}$
| $(1,0;1)$ | $19/4$ | $\frac{1}{9}\bigl{(}203-48\sqrt{11}{\mkern 2.0mu}\bigr{)}$ | $\frac{3}{2}+\frac{1}{3}\bigl{(}2\sqrt{11}-3\bigr{)}$
Table 18: Scalars and pseudo-scalars with $\frac{5}{2}<E_{0}\leq 3$ in both the left- and right-squashed vacuum. Note that the properties of the scalar fields are the same in the two vacua. Field ($LS^{7}$) | $(p,q;r)$ | $C_{g}$ | $M_{L}/m$ | $E_{0}$
---|---|---|---|---
$\frac{1}{2}(i\not{D}_{1/2}^{(1)_{-}})$ | $(1,2;1)$ | $67/4$ | $\frac{1}{3}\bigl{(}4\sqrt{26}-15\bigr{)}$ | $\frac{3}{2}+\frac{1}{6}\bigl{(}4\sqrt{26}-15\bigr{)}$
| $(0,3;0)$ | $18$ | $2$ | $\frac{3}{2}+1$
| $(2,1;2)$ | $18$ | $2$ | $\frac{3}{2}+1$
| $(3,0;3)$ | $87/4$ | $\frac{1}{3}\bigl{(}2\sqrt{129}-15\bigr{)}$ | $\frac{3}{2}+\frac{1}{6}\bigl{(}2\sqrt{129}-15\bigr{)}$
$\frac{1}{2}(i\not{D}_{1/2}^{(2)_{-}})$ | $(2,1;0)$ | $12$ | $\frac{4}{3}$ | $\frac{3}{2}+\frac{2}{3}$
| $(3,0;1)$ | $51/4$ | $\frac{1}{3}\bigl{(}4\sqrt{19}-13\bigr{)}$ | $\frac{3}{2}+\frac{1}{6}\bigl{(}4\sqrt{19}-13\bigr{)}$
| $(1,0;3)$ | $55/4$ | $\frac{5}{3}$ | $\frac{3}{2}+\frac{5}{6}$
| $(0,2;2)$ | $16$ | $\frac{1}{3}\bigl{(}3\sqrt{41}-13\bigr{)}$ | $\frac{3}{2}+\frac{1}{6}\bigl{(}3\sqrt{41}-13\bigr{)}$
| $(1,2;1)$ | $67/4$ | $\frac{1}{3}\bigl{(}8\sqrt{6}-13\bigr{)}$ | $\frac{3}{2}+\frac{1}{6}\bigl{(}8\sqrt{6}-13\bigr{)}$
| $(2,1;2)$ | $18$ | $\frac{1}{3}\bigl{(}\sqrt{409}-13\bigr{)}$ | $\frac{3}{2}+\frac{1}{6}\bigl{(}\sqrt{409}-13\bigr{)}$
| $(1,1;3)$ | $75/4$ | $\frac{1}{3}\bigl{(}2\sqrt{106}-13\bigr{)}$ | $\frac{3}{2}+\frac{1}{6}\bigl{(}2\sqrt{106}-13\bigr{)}$
| $(3,1;1)$ | $79/4$ | $\frac{1}{3}\bigl{(}2\sqrt{111}-13\bigr{)}$ | $\frac{3}{2}+\frac{1}{6}\bigl{(}2\sqrt{111}-13\bigr{)}$
| $(2,2;0)$ | $20$ | $\frac{1}{3}\bigl{(}\sqrt{449}-13\bigr{)}$ | $\frac{3}{2}+\frac{1}{6}\bigl{(}\sqrt{449}-13\bigr{)}$
$\frac{1}{2}(i\not{D}_{3/2}^{(1)_{-}})$ | $(0,0;0)$ | $0$ | $\frac{4}{3}$ | $\frac{3}{2}+\frac{2}{3}$
$\frac{1}{2}(i\not{D}_{3/2}^{(3)_{-}})$ | $(1,0;1)$ | $19/4$ | $-\frac{7}{3}$ | $\frac{3}{2}+\frac{7}{6}$
| $(2,0;0)$ | $6$ | $-\frac{8}{3}$ | $\frac{3}{2}+\frac{4}{3}$
$\frac{1}{2}(i\not{D}_{3/2}^{(4)_{-}})$ | $(1,1;1)$ | $39/4$ | $-\frac{1}{3}\bigl{(}2\sqrt{61}-7\bigr{)}$ | $\frac{3}{2}+\frac{1}{6}\bigl{(}2\sqrt{61}-7\bigr{)}$
| $(0,1;2)$ | $10$ | $-\frac{1}{3}\bigl{(}\sqrt{249}-7\bigr{)}$ | $\frac{3}{2}+\frac{1}{6}\bigl{(}\sqrt{249}-7\bigr{)}$
Table 19: Spinor fields with $2<E_{0}\leq 3$ in the left-squashed vacuum. Field ($RS^{7}$) | $(p,q;r)$ | $C_{g}$ | $M_{R}/m$ | $E_{0}$
---|---|---|---|---
$\frac{1}{2}(i\not{D}_{1/2}^{(1)_{+}})$ | $(1,1;1)$ | $39/4$ | $\frac{2}{3}\bigl{(}\sqrt{69}-6\bigr{)}$ | $\frac{3}{2}+\frac{1}{3}\bigl{(}\sqrt{69}-6\bigr{)}$
| $(0,2;0)$ | $10$ | $\frac{1}{3}\bigl{(}\sqrt{281}-12\bigr{)}$ | $\frac{3}{2}+\frac{1}{6}\bigl{(}\sqrt{281}-12\bigr{)}$
| $(2,0;2)$ | $12$ | $\frac{1}{3}\bigl{(}\sqrt{321}-12\bigr{)}$ | $\frac{3}{2}+\frac{1}{6}\bigl{(}\sqrt{321}-12\bigr{)}$
| $(1,2;1)$ | $67/4$ | $\frac{4}{3}\bigl{(}\sqrt{26}-3\bigr{)}$ | $\frac{3}{2}+\frac{2}{3}\bigl{(}\sqrt{26}-3\bigr{)}$
| $(0,3;0)$ | $18$ | $3$ | $\frac{3}{2}+\frac{3}{2}$
| $(2,1;2)$ | $18$ | $3$ | $\frac{3}{2}+\frac{3}{2}$
$\frac{1}{2}(i\not{D}_{1/2}^{(2)_{+}})$ | $(0,2;2)$ | $16$ | $\frac{1}{3}\bigl{(}3\sqrt{41}-14\bigr{)}$ | $\frac{3}{2}+\frac{1}{6}\bigl{(}3\sqrt{41}-14\bigr{)}$
| $(3,0;1)$ | $51/4$ | $\frac{2}{3}\bigl{(}2\sqrt{19}-7\bigr{)}$ | $\frac{3}{2}+\frac{1}{3}\bigl{(}2\sqrt{19}-7\bigr{)}$
| $(1,2;1)$ | $67/4$ | $\frac{2}{3}\bigl{(}4\sqrt{6}-7\bigr{)}$ | $\frac{3}{2}+\frac{1}{3}\bigl{(}4\sqrt{6}-7\bigr{)}$
| $(2,1;2)$ | $18$ | $\frac{1}{3}\bigl{(}\sqrt{409}-14\bigr{)}$ | $\frac{3}{2}+\frac{1}{6}\bigl{(}\sqrt{409}-14\bigr{)}$
| $(1,1;3)$ | $75/4$ | $\frac{2}{3}\bigl{(}\sqrt{106}-7\bigr{)}$ | $\frac{3}{2}+\frac{1}{3}\bigl{(}\sqrt{106}-7\bigr{)}$
| $(3,1;1)$ | $79/4$ | $\frac{2}{3}\bigl{(}\sqrt{111}-7\bigr{)}$ | $\frac{3}{2}+\frac{1}{3}\bigl{(}\sqrt{111}-7\bigr{)}$
| $(2,2;0)$ | $20$ | $\frac{1}{3}\bigl{(}\sqrt{449}-14\bigr{)}$ | $\frac{3}{2}+\frac{1}{6}\bigl{(}\sqrt{449}-14\bigr{)}$
| $(3,0;3)$ | $87/4$ | $\frac{8}{3}$ | $\frac{3}{2}+\frac{4}{3}$
| $(4,0;2)$ | $22$ | $\frac{1}{3}\bigl{(}\sqrt{489}-14\bigr{)}$ | $\frac{3}{2}+\frac{1}{6}\bigl{(}\sqrt{489}-14\bigr{)}$
| $(0,3;2)$ | $24$ | $3$ | $\frac{3}{2}+\frac{3}{2}$
$\frac{1}{2}(i\not{D}_{3/2}^{(1)_{-}})$ | $(0,0;0)$ | $0$ | $\frac{5}{3}$ | $\frac{3}{2}+\frac{5}{6}$
$\frac{1}{2}(i\not{D}_{3/2}^{(1)_{+}})$ | $(0,1;0)$ | $4$ | $-\frac{5}{3}$ | $\frac{3}{2}+\frac{5}{6}$
$\frac{1}{2}(i\not{D}_{3/2}^{(2)_{+}})$ | $(1,1;1)$ | $39/4$ | $-\frac{2}{3}\bigl{(}\sqrt{51}-3\bigr{)}$ | $\frac{3}{2}+\frac{1}{3}\bigl{(}\sqrt{51}-3\bigr{)}$
$\frac{1}{2}(i\not{D}_{3/2}^{(3)_{+}})$ | $(0,0;2)$ | $6$ | $-3$ | $\frac{3}{2}+\frac{3}{2}$
| $(2,0;0)$ | $6$ | $-3$ | $\frac{3}{2}+\frac{3}{2}$
Table 20: Spinor fields with $2<E_{0}\leq 3$ in the right-squashed vacuum.

##  Appendix E Derivation of the supermultiplets

section section

In this appendix, we explain, in detail, how one can associate $E_{0}$ values
and cross diagrams to each of the supermultiplets as summarized in Tables 4, 5
and 8.

In the process of explaining this we also obtain information of how to assign
unique operator eigenvalues to the various cross diagrams, or single crosses
in some special cases. Based on the results of Nilsson:2018lof ;
Karlsson:2021oxd  we here use supersymmetry to carry this analysis as far as
possible but a second goal is to correlate these results with the ones
obtained in Section 4 where we provide a direct connection between cross
diagrams and operator eigenvalues. This is particularly important for the
novel results concerning $\Delta_{2}$ which will give new important insights
into the structure of supermultiplets. Thus, we will use the notation for
cross diagrams and their eigenvalues introduced in Section 4 and in some cases
compare them to the cross diagrams listed in the appendix of Nilsson:2018lof .
Alternatively, one can start from the results of Sections 3 and 4 and arrive
at the same results without using Nilsson:2018lof  except for the cross
diagrams of the supermultiplets only associated to mode types that were not
analysed in Section 4, specifically the results in Figure 8 below.

In what follows, we do not look for numerical coincidences that could give
rise to exceptional supermultiplets for small $p$ or $q$. Although we do not
present the details, we have verified that no such exceptions are possible
when accounting for the full spectrum of supermultiplets.

###  E.1 Spin-2 supermultiplets

We start from the top of the list of supermultiplets, 221–225 above, and
follow it downwards. The first one is thus the spin $2^{+}$ supermultiplets

$D(E_{0},2^{+}:\Delta_{0})\oplus
D(E_{0}-\tfrac{1}{2},\tfrac{3}{2}:i\not{D}_{1/2})\oplus
D(E_{0}+\tfrac{1}{2},\tfrac{3}{2}:i\not{D}_{1/2})\oplus
D(E_{0},1^{+}:\Delta_{2}),$ (351)

where we have included the operators responsible for the different spectra.
The values of $E_{0}$ are computed from the operator eigenvalues as follows:
Using Table 3 and 131 we get $M^{2}(2^{+})=\Delta_{0}$ with eigenvalues
$\Delta_{0}^{(1)}=\frac{20m^{2}}{9}C_{g}$ which gives (see Table 2)

$E_{0}(2^{+})=\frac{3}{2}+\frac{1}{2}\sqrt{\left(\frac{M}{m}\right)^{2}+9}=\frac{3}{2}+\frac{1}{2}\sqrt{\frac{20}{9}C_{g}+9}=\frac{3}{2}+\frac{1}{6}\sqrt{\vphantom{C_{G}}\smash[b]{20C_{g}+81}},$
(352)

applicable to all isometry irreps in the cross diagram of $\Delta_{0}$, that
is, all irreps $(p,q;r)$ with $p=r\geq 0$, $q\geq 0$ (see Nilsson:2018lof  or
Figure 1). There is, however, a special case namely the short massless
supergravity multiplet in the irrep $(0,0;0)$. Looking more carefully at the
cross diagrams for the different fields in the spin $2^{+}$ supermultiplets
one finds that the cross diagrams of Nilsson:2018lof  are related as
follows343434The signs appearing as indices on the 3/2 components refer to the
sign in their energies $E_{0}\pm\frac{1}{2}$.

$\displaystyle 2^{+}\colon[\Delta_{0}](1),\
(3/2)_{\scriptscriptstyle(-)}\colon[i\not{D}_{1/2}](1)=[\Delta_{0}](1),\
(3/2)_{\scriptscriptstyle(+)}\colon[i\not{D}_{1/2}](2)=[\Delta_{0}](1)-(0,0;0),$
$\displaystyle 1^{+}\colon[\Delta_{2}](2+3-1)=[\Delta_{0}](1)-(0,0;0),$ (353)

where the bracket $[...]$ enclosing the operator indicates that it refers to a
cross diagram and not an eigenvalue. Here $[i\not{D}_{1/2}](1)$ refers to the
first cross diagram in the list of $i\not{D}_{1/2}$ in the appendix of
Nilsson:2018lof  while $[\Delta_{2}](2+3-1)$ means that one should add the
$\Delta_{2}$ cross diagrams number 2 and 3 and subtract diagram number 1. Thus
we see that the last two fields in the supermultiplets have cross diagrams
that do not contain the irrep $(0,0;0)$ which therefore gives rise to a short
supermultiplet, the supergravity multiplet. In particular this means that the
Dirac eigenvalues are associated with cross diagrams as follows

$\displaystyle
i\not{D}_{1/2}(1)=\frac{m}{2}+\frac{m}{3}\sqrt{\vphantom{C_{G}}\smash[b]{20C_{g}+81}}\quad\implies\quad
E_{0}=\frac{3}{2}-\frac{1}{2}+\frac{1}{6}\sqrt{\vphantom{C_{G}}\smash[b]{20C_{g}+81}},$
(354) $\displaystyle
i\not{D}_{1/2}(2)=\frac{m}{2}-\frac{m}{3}\sqrt{\vphantom{C_{G}}\smash[b]{20C_{g}+81}}\quad\implies\quad
E_{0}=\frac{3}{2}+\frac{1}{2}+\frac{1}{6}\sqrt{\vphantom{C_{G}}\smash[b]{20C_{g}+81}},$
(355)

which follows from the old results of Nilsson:1983ru  or the results in this
paper.353535Note that the sign in front of the first term cannot be deduced
from the eigenvalue calculations in Section 3 of this paper, which are based
on computing $(i{\not{D}}_{1/2})^{2}$, but follows here instead from
implementing supersymmetry. The mode calculation in Section 4, on the other
hand, does provide the sign.

Similarly, we conclude that, in analogy with
$\Delta_{0}(1)=\frac{m^{2}}{9}\,20C_{g}$, we have

$\Delta_{2}(2+3-1)=\frac{m^{2}}{9}(20C_{g}+72)\quad\implies\quad
E_{0}(1^{+})=\frac{3}{2}+\frac{1}{6}\sqrt{\vphantom{C_{G}}\smash[b]{20C_{g}+81}},$
(356)

with the same $E_{0}$ as for the spin $2^{+}$ component, as required by
supersymmetry.

Using the notation of Section 4 we can summarize the result as follows: The
four fields in the $2^{+}$ supersmultiplet make use of the following cross
diagrams and eigenvalues

$2^{+}(\Delta_{0}^{(1)}),\quad\frac{3}{2}(i\not{D}_{1/2}^{(1)_{+}}),\quad\frac{3}{2}(i\not{D}_{1/2}^{(1)_{-}}),\quad
1^{+}(\Delta_{2}^{(1)}),$ (357)

given in Figures 1, 3 and 4. It is interesting to note that the combination of
diagrams from the appendix of Nilsson:2018lof  appearing in
$[\Delta_{2}](2+3-1)=[\Delta_{0}](1)-(0,0;0)$ comes out automatically in the
mode calculation presented in Section 4 as $\Delta_{2}^{(1)}$. This will
happen many times in the following when comparing these two approaches but
from now on we do not give any details on this issue. We should mention that
apart from combining entire cross diagrams it is also necessary in some cases
to move columns or rows, or just single crosses, between diagrams in
Nilsson:2018lof  in order to generate the ones that come out of the mode
function analysis in Section 4. This is to be expected since the analysis of
Nilsson:2018lof  only provides the mutliplicity of the isometry irreps for the
various operators; the split into separate cross diagrams in Nilsson:2018lof
is to some extent arbitrary.

###  E.2 Spin-3/2 supermultiplets

Turning to the next set of supermultiplets, the six spin-3/2 ones, we have

$D(E_{0},\tfrac{3}{2}:i\not{D}_{1/2})\oplus
D(E_{0}-\tfrac{1}{2},1^{\pm}:\Delta_{2\,\text{or}\,1})\oplus
D(E_{0}+\tfrac{1}{2},1^{\mp}:\Delta_{1\,\text{or}\,2})\oplus
D(E_{0},\tfrac{1}{2}:i\not{D}_{3/2}).$ (358)

Here we must relate each of the two spin 1 parity assignments to three cross
diagrams, a fact that follows by checking how supersymmetry works starting
from the six cross diagrams for the spin-3/2 component. We conclude
immediately that these are determined by the six Dirac cross diagrams not
already used above, all associated to the eigenvalues with a
$\sqrt{\vphantom{C_{G}}\smash[b]{20C_{g}+49}}$. The reason for this is that we
must use up all the Dirac modes associated to spin-3/2 fields, which is the
first component of the spin-3/2 supermultiplet.

All the forty $i\not{D}_{3/2}$ cross diagrams in the appendix of
Nilsson:2018lof  must appear a single time each when looking at the entire
spectrum. The fact that the spin-1/2 fields in the spin-3/2 supermultiplets
are related to the $i\not{D}_{3/2}$ and not $i\not{D}_{1/2}$ is a consequence
of supersymmetry which uniquely gives this answer.

From the results derived in Nilsson:1983ru  (or in Section 4 of this paper) we
have

$i\not{D}_{1/2}(3,5,7)=-\frac{m}{6}-\frac{m}{3}\sqrt{\vphantom{C_{G}}\smash[b]{20C_{g}+49}}\
\implies\
E_{0}(3/2)=\frac{3}{2}+\frac{5}{6}+\frac{1}{6}\sqrt{\vphantom{C_{G}}\smash[b]{20C_{g}+49}},$
(359)

where the first spin-1 field in the supermultiplet is $1^{-}$ (see below), and

$i\not{D}_{1/2}(4,6,8)=-\frac{m}{6}+\frac{m}{3}\sqrt{\vphantom{C_{G}}\smash[b]{20C_{g}+49}}\
\implies\
E_{0}(3/2)=\frac{3}{2}-\frac{5}{6}+\frac{1}{6}\sqrt{\vphantom{C_{G}}\smash[b]{20C_{g}+49}}$
(360)

where the first spin-1 field is $1^{+}$ (see below). In the notation of
Section 4 these are, respectively, denoted $i\not{D}_{1/2}^{(2)_{-}}$ and
$i\not{D}_{1/2}^{(2)_{+}}$, see Figure 3.

To establish the results for the spin-1 components is here a bit more
intricate than for the $2^{+}$ multiplets above. The spin 1 content of these
six branches (cross diagrams) is three each of $1^{+}$ and $1^{-}$. However,
the six $\Delta_{1}$ cross diagrams are used twice since the mass square
operator has two square root branches, see Table 3. The second occurrence of
these six sets of cross diagrams will appear in the six sets of $1^{-}$
supermultiplets discussed below. The two sets of $\pm$ signs gives four
different sets of $E_{0}$ values and all four are needed for everything to
work out. To see this we recall the derivation of the $E_{0}(1^{-})$ values
here.

To obtain the $E_{0}(1^{-})$ values we start from Table 2,

$E_{0}(1^{-})=\frac{3}{2}+\frac{1}{2}\sqrt{M^{2}/m^{2}+1},$ (361)

where $M^{2}$ has two branches(see Table 3)363636Recalling from Duff:1986hr
that $\Delta_{1}\geq 7m^{2}$ we see that $\sqrt{\Delta_{1}+4m^{2}}\geq 3m$.

$M^{2}(1^{-}_{\scriptscriptstyle(\pm)})=\Delta_{1}+12m^{2}\pm
6m\sqrt{\Delta_{1}+4m^{2}}=(\sqrt{\Delta_{1}+4m^{2}}\pm 3m)^{2}-m^{2},$ (362)

where also $\Delta_{1}$ has two branches, denoted $\Delta_{1}^{(1)_{\pm}}$ in
132,

$\Delta_{1}=\frac{m^{2}}{9}\bigl{(}20C_{g}+14\pm
2\sqrt{\vphantom{C_{G}}\smash[b]{20C_{g}+49}}\bigr{)}=\frac{m^{2}}{9}\bigl{(}\sqrt{\vphantom{C_{G}}\smash[b]{20C_{g}+49}}\pm
1\bigr{)}^{2}-4m^{2}.$ (363)

Inserting the last expression into the one immediately above it gives

$M^{2}(1^{-})=\frac{m^{2}}{9}\bigl{(}\sqrt{\vphantom{C_{G}}\smash[b]{20C_{g}+49}}\pm
1\pm 9\bigr{)}^{2}-m^{2}.$ (364)

The sign choice associated with $\Delta_{1}$ is the one that must be
correlated to the different cross diagrams as will be done shortly.

The four choices of signs in 364 give $-10,-8,8,10$ to be added to the square
root in $M^{2}$. Using $10$ and $-10$ we get373737Note that
$\sqrt{\vphantom{C_{G}}\smash[b]{20C_{g}+49}}\geq 10$ for all irreps in the
relevant Dirac cross diagrams.

$E_{0}(1^{-})=\frac{3}{2}\pm\frac{5}{3}+\frac{1}{6}\sqrt{\vphantom{C_{G}}\smash[b]{20C_{g}+49}},$
(365)

while $8$ and $-8$ give

$E_{0}(1^{-})=\frac{3}{2}\pm\frac{4}{3}+\frac{1}{6}\sqrt{\vphantom{C_{G}}\smash[b]{20C_{g}+49}}.$
(366)

The first set of $E_{0}(1^{-})$ values will be needed in the spin $1^{-}$
supermultiplets as demonstrated below while the second set fit into the spin
$3/2$ ones analysed here. This last statement follows since the values to be
added to $\frac{3}{2}$ are $\pm\frac{5}{6}$ which arise as
$\frac{4}{3}-\frac{1}{2}=\frac{5}{6}$ and
$-\frac{4}{3}+\frac{1}{2}=-\frac{5}{6}$ when relating the different values of
$E_{0}$ in the multiplet. We therefore conclude that the content of the six
3/2 supermultiplets is as given above in terms of the numbered Dirac cross
diagrams.

Note that the six cross diagrams of $\Delta_{1}$ are identical to the
$\sqrt{\vphantom{C_{G}}\smash[b]{20C_{g}+49}}$ Dirac ones. This is clear from
Figures 2 and 3 which also correlates the signs in their operator
eigenvalues.383838The same result is obtained by supersymmetry alone, namely
that the $\Delta_{1}$ cross diagrams numbered 1, 3 and 5 in Nilsson:2018lof
have eigenvalues from the minus branch and diagrams 2, 4 and 6 from the plus
branch. This assignment of signs must be a consistent choice also in the
$1^{-}$ supermultiplets which it is as we will see below.

The $1^{+}$ field in this supermultiplet has to come from modes with
eigenvalues given by $\Delta_{2}^{(2)_{\pm}}$ due to the
$\sqrt{\vphantom{C_{G}}\smash[b]{20C_{g}+49}}$. For these, we have

$\Delta_{2}^{(2)_{\pm}}=\frac{m^{2}}{9}\Bigl{(}20C_{g}+44\pm
4\sqrt{\vphantom{C_{G}}\smash[b]{20C_{g}+49}}\Bigr{)}\ \implies\
E_{0}(1^{+})=\frac{3}{2}\pm\frac{2}{6}+\frac{1}{6}\sqrt{\vphantom{C_{G}}\smash[b]{20C_{g}+49}}.$
(367)

Hence, the fields associated to $\Delta_{2}^{(2)_{\pm}}$ fit into the
supermultiplet containing the spin-3/2 field associated to
$i\not{D}_{1/2}^{(2)_{\mp}}$. Note that the corresponding cross diagrams in
Figures 3 and 4 agree.

The $i\not{D}_{3/2}$-eigenvalues that should be used for the spin $1/2$
components to fit into these six spin $3/2$ supermultiplets are (see 145)

$i\not{D}_{3/2}^{(3)\pm}=\frac{m}{3}\biggl{(}\frac{1}{2}\pm\sqrt{\vphantom{C_{G}}\smash[b]{20C_{g}+49}}\biggr{)}\
\implies\
E_{0}(1/2)=\frac{3}{2}\pm\frac{5}{6}+\frac{1}{6}\sqrt{\vphantom{C_{G}}\smash[b]{20C_{g}+49}},$
(368)

where the eigenvalues with a plus and a minus are associated with,
respectively, the same cross diagrams as those with eigenvalues
$i\not{D}_{1/2}^{(2)-}$ and $i\not{D}_{1/2}^{(2)+}$ (see 142 and Figure 3).

We can summarise the situation for the six spin $3/2$ supermultiplets, using
the notation of Section 4, as follows (ordered as
$E_{0},E_{0}-\frac{1}{2},E_{0}+\frac{1}{2},E_{0}$)

$\displaystyle\frac{3}{2}_{1}\colon\quad\frac{3}{2}(i\not{D}_{1/2}^{(2)_{-}}),\
1^{+}(\Delta_{2}^{(2)_{+}}),\
1^{-}_{\scriptscriptstyle(+)}(\Delta_{1}^{(1)_{-}}),\
\frac{1}{2}(i\not{D}_{3/2}^{(3)_{+}}),$ (369)
$\displaystyle\frac{3}{2}_{2}\colon\quad\frac{3}{2}(i\not{D}_{1/2}^{(2)_{+}}),\
1^{-}_{\scriptscriptstyle(-)}(\Delta_{1}^{(1)_{+}}),\
1^{+}(\Delta_{2}^{(2)_{-}}),\ \frac{1}{2}(i\not{D}_{3/2}^{(3)_{-}}).$ (370)

We emphasise that since for the spin $3/2$ operator $i\not{D}_{3/2}$ we have
no independent way to relate eigenvalues to cross diagrams (as was done in
Section 4 for $i\not{D}_{1/2}$, $\Delta_{1}$ and $\Delta_{2}$) we have here
relied entirely on supersymmetry and used the cross diagrams of
Nilsson:2018lof .

###  E.3 The $1^{-}$ supermultiplets

The six sets of $1^{-}$ supermultiplets are of the form

$D(E_{0},1^{-}:\Delta_{1})\oplus
D(E_{0}-\tfrac{1}{2},\tfrac{1}{2}:i\not{D}_{1/2\,\text{or}\,3/2})\oplus
D(E_{0}+\tfrac{1}{2},\tfrac{1}{2}:i\not{D}_{3/2\,\text{or}\,1/2})\oplus
D(E_{0},0^{-}:Q),$ (371)

where $i\not{D}_{1/2}$ and $i\not{D}_{3/2}$ occur in a similar way to how in
the previous case there were three supermultiplets with spin-1 fields coming
from $\Delta_{1}$ and $\Delta_{2}$. That this happens here is a direct
consequence of supersymmetry and is easily established using the known
eigenvalues of the various operators.

The eigenvalues of the $3+3$ cross diagrams of $\Delta_{1}$ were in this case
identified above as the ones coming from 10 and -10 in the expression for
$E_{0}(1^{-})$ in 365. Thus

$\displaystyle\Delta_{1}^{(1)_{+}}\colon\
E_{0}(1^{-}_{\scriptscriptstyle(+)})=\frac{3}{2}+\frac{5}{3}+\frac{1}{6}\sqrt{\vphantom{C_{G}}\smash[b]{20C_{g}+49}},$
(372) $\displaystyle\Delta_{1}^{(1)_{-}}\colon\
E_{0}(1^{-}_{\scriptscriptstyle(-)})=\frac{3}{2}-\frac{5}{3}+\frac{1}{6}\sqrt{\vphantom{C_{G}}\smash[b]{20C_{g}+49}},$
(373)

where the cross diagrams associated to the two sets of
$\Delta_{1}^{(1)\pm}$-eigenvalues are specified in Figure 2.

The last component of these supermultiplets is the pseudo-scalar $0^{-}$.
Using the $\sqrt{\vphantom{C_{G}}\smash[b]{20C_{g}+49}}$ eigenvalues of $Q$ in
139

$Q^{(2)\pm}=\frac{m}{3}\bigl{(}1\pm\sqrt{\vphantom{C_{G}}\smash[b]{20C_{g}+49}}\bigr{)},$
(374)

and the $0^{-}$ mass operator from Table 3

$M^{2}(0^{-})=(Q+3m)^{2}-m^{2},$ (375)

we find as expected that all six cross diagrams satisfy

$E_{0}(0^{-})=E_{0}(1^{-}).$ (376)

Finally, we must check that also the spin-1/2 fields fit into these $1^{-}$
supermultiplets. It is interesting to note that we need to use both
$i\not{D}_{1/2}$ and $i\not{D}_{3/2}$ eigenvalues for this to work. Let us
start with the spin-1/2 energies $E_{0}=\frac{3}{2}\pm\frac{1}{2}|M/m|$ where
the minus sign cannot occur in these supermultiplets due to the unitarity
bound $E_{0}(1)\geq 2$. The mass related to the $i\not{D}_{1/2}$ and
$i\not{D}_{3/2}$ operators are (see Table 3, 142 and 146)

$\displaystyle
M(1/2,i\not{D}_{1/2}^{(2)_{\pm}})=-i\not{D}_{1/2}-\frac{9m}{2}=-\frac{13m}{3}\mp\frac{m}{3}\sqrt{\vphantom{C_{G}}\smash[b]{20C_{g}+49}},$
(377) $\displaystyle
M(1/2,i\not{D}_{3/2}^{(4)_{\pm}})=i\not{D}_{3/2}+\frac{3m}{2}=\frac{7m}{3}\pm\frac{m}{3}\sqrt{\vphantom{C_{G}}\smash[b]{20C_{g}+49}},$
(378)

where we should note that it is the second set of
$\sqrt{\vphantom{C_{G}}\smash[b]{20C_{g}+49}}$ eigenvalues of $i\not{D}_{3/2}$
that must be used here. This produces the following two sets of energy values
(for Dirichlet boundary conditions)

$\displaystyle
E_{0}(1/2,i\not{D}_{1/2}^{(2)_{\pm}})=\frac{3}{2}\mp\frac{13}{6}+\frac{1}{6}\sqrt{\vphantom{C_{G}}\smash[b]{20C_{g}+49}},$
(379) $\displaystyle
E_{0}(1/2,i\not{D}_{3/2}^{(4)_{\pm}})=\frac{3}{2}\pm\frac{7}{6}+\frac{1}{6}\sqrt{\vphantom{C_{G}}\smash[b]{20C_{g}+49}}.$
(380)

Hence the supermultiplets are (ordered as
$(E_{0},E_{0}-1/2,E_{0}+1/2,E_{0})$), in terms of the notation of Section 4,

$\displaystyle 1^{-}_{1}\colon\quad
1_{\scriptscriptstyle(+)}^{-}(\Delta_{1}^{(1)_{+}}),\
\frac{1}{2}(i\not{D}_{3/2}^{(4)_{+}}),\
\frac{1}{2}(i\not{D}_{1/2}^{(2)_{+}}),\ 0^{-}(Q^{(2)_{+}}),$ (381)
$\displaystyle 1^{-}_{2}\colon\quad
1_{\scriptscriptstyle(-)}^{-}(\Delta_{1}^{(1)_{-}}),\
\frac{1}{2}(i\not{D}_{1/2}^{(2)_{-}}),\
\frac{1}{2}(i\not{D}_{3/2}^{(4)_{-}}),\ 0^{-}(Q^{(2)_{-}}),$ (382)

where as before the lower sign in the bracket
$1^{-}_{\scriptscriptstyle(\pm)}$ indicates which branch of $M^{2}$ is used.
These results for the cross diagrams imply that there are two short gauge
multiplets when $C_{g}=6$, and hence $E_{0}=2$, connected to the two adjoint
irreps $(2,0;0)$ and $(0,0;2)$ of the isometry group $Sp_{2}\times
Sp^{C}_{1}$.

###  E.4 The $1^{+}$ supermultiplets

So far, $\mathcal{N}=1$ supersymmetry, together with the operator eigenvalues
derived in previous works and the result for the spin-3/2 operator obtained in
Karlsson:2021oxd , have been sufficient to uniquely fix all relations between
operator eigenvalues and isometry irreps (that is, cross diagrams). This nice
state of affairs will no longer persist when we now turn to the eight spin
$1^{+}$ supermultiplets, each with the content

$D(E_{0},1^{+}:\Delta_{2})\oplus
D(E_{0}-\tfrac{1}{2},\tfrac{1}{2}:i\not{D}_{3/2})\oplus
D(E_{0}+\tfrac{1}{2},\tfrac{1}{2}:i\not{D}_{3/2})\oplus
D(E_{0},0^{+}:\Delta_{L}),$ (383)

where the potential ambiguity for the last entry, the $0^{+}$ scalars, which
could be given by either $\Delta_{0}$ or $\Delta_{L}$ is settled by
supersymmetry.

It is clear that the remaining eight cross diagrams for $\Delta_{2}$ in Figure
4 account for the ${\bf 3}$ and ${\bf 5}$ spin $1^{+}$ components of these
supermultiplets. These must therefore all be associated with the eigenvalues
135 and energies

$\Delta_{2}=\frac{m^{2}}{9}\,20C_{g}\quad\implies\quad
E_{0}(1^{+},\Delta_{2})=\frac{3}{2}+\frac{1}{6}\sqrt{\vphantom{C_{G}}\smash[b]{20C_{g}+9}}.$
(384)

Now we understand that the spin-1/2 fields must be associated with
$i\not{D}_{3/2}$ since the eigenvalues of $i\not{D}_{1/2}$ do not contain
$\sqrt{\vphantom{C_{G}}\smash[b]{20C_{g}+9}}$.

Similarly, the scalar fields $0^{+}$ are related to $\Delta_{L}$-eigenvalues
136 and the energies

$\Delta_{L}=\frac{m^{2}}{9}\,(20C_{g}+36)\quad\implies\quad
E_{0}(0^{+},\Delta_{L})=\frac{3}{2}+\frac{1}{6}\sqrt{\vphantom{C_{G}}\smash[b]{20C_{g}+9}}.$
(385)

The unexpected feature that appears here is that neither of the two sets of
eigenvalues, related to the ${\bf 3}$ and ${\bf 5}$ cross diagrams, have a
negative square root branch (cf. $\Delta_{2}^{(2)_{\pm}}$). This means that
there is a degeneracy in the spectrum since different modes in the same
isometry irrep must pairwise be associated with the same operator eigenvalue.
This must happen for all the fields in these $1^{+}$ supermultiplets whose
irreps are of the kind $(p,q;r)$ with either $r=p$ or $r=p\pm 2$. As we will
discover below when discussing the Wess–Zumino supermultiplets, the ones
containing scalar fields related to $\Delta_{L}$, and only these ones, will
display the same phenomenon.

This being concluded here, we should now recall the results of Section 4 on
the $\Delta_{2}$ eigenmodes and their eigenvalues which support this
conclusion. In fact, in Section 4 we identified explicitly the different mode
functions that have the same $\Delta_{2}$-eigenvalues,
$\Delta_{2}=\frac{m^{2}}{9}\,20C_{g}$, and the modes are precisely the ones in
the ${\bm{3}}$ and ${\bm{5}}$ cross diagrams given above. The notation used in
Section 4 to make this clear was to denote eigenvalues associated with the
mode functions (and thus the cross diagrams) related to ${\bm{3}}$ by