geometry. This background has trivial vielbeins and only the longitudinal
component of the three-form is non-zero. Let the 11-dimensional coordinates be
$x^{\mu}=(x^{A},x^{a})$ with $A=0,1,2$ and $a=3,\dots,10$. Then we specify the
geometry by:

$\tau^{A}{}=\mathrm{d}x^{A}\,,\quad e_{a}=\partial_{a}\,,\quad
c_{012}=\Phi(x)\,,\quad\lambda_{abcd}=0\,.$ (121)

The solution coming from U-duality will have $\Phi(x)=\Phi(x^{a})$ depending
on transverse directions only, however for the moment we take (121) as an
ansatz inspired by duality or uplift, and remain agnostic about whether we can
also have longitudinal coordinate dependence. The expectation is that
$\Phi(x)$ will serve as a Newtonian potential, as we previously discussed and
following expectations from [19].

All the intrinsic torsion constraints vanish, and the only non-trival
components of a spin connection are

$\omega_{\mu}{}^{Aa}=\tfrac{1}{3}\delta^{A}{}_{\mu}\partial^{a}\Phi\,,$ (122)

so that the only non-trivial curvature is:

$R_{\mu\nu}{}^{Aa}=\tfrac{2}{3}\delta^{A}_{[\nu}\partial_{\mu]}\partial^{a}\Phi\Rightarrow\breve{R}_{Ab}{}^{Aa}=-\partial_{b}\partial^{a}\Phi\,.$
(123)

Taking the trace of $\breve{R}_{Ab}{}^{Aa}$ gives the Poisson equation. This
is the only non-trivial equation of motion and dictates that the function
$\Phi$ should be harmonic in the transverse space:

$\partial^{a}\partial_{a}\Phi=0\,.$ (124)

This is the expected behaviour for a Newtonian gravitational potential, and is
consistent with the ten-dimensional uplift as well as the U-duality to known
relativistic solutions. Note that the equations of motion do not constrain the
longitudinal dependence of $\Phi$, so in principle we may allow $\Phi$ to have
arbitrary dependence on the coordinates $x^{A}$.

In the theory with half-maximal supersymmetry, we can immediately solve the
Killing spinor equations (111). They reduce for the background (121) to the
Killing spinor equations for flat spacetime considered in the previous
subsection, with the imposition of $\epsilon_{-}=0$. Accordingly, there is
again an infinite family of solutions

$\epsilon_{+}=\theta_{+}(x^{A})\,,\qquad\eta_{-}=0\,,\qquad\rho_{+A}=-\partial_{A}\epsilon_{+}\,,$
(125)

where $\gamma^{A}\partial_{A}\theta_{+}(x^{B})=0$.

Now let us consider the theory with maximal supersymmetry. We now have to take
into account the curvature constraints (93b), in particular that
$D_{a}\breve{R}_{A(b}{}^{A}{}_{c)}=0$, which requires the stronger condition

$\partial_{a}\partial_{b}\partial_{c}\Phi=0\,,$ (126)

on the function $\Phi$. We write the solution in terms of arbitrary functions
of the longitudinal coordinates as

$\Phi=M_{ab}(x^{A})x^{a}x^{b}+N_{a}(x^{A})x^{a}+C(x^{A})\,,$ (127)

where $M_{ab}$ is symmetric and also, in order to satisfy the Poisson
equation, traceless i.e. $\delta^{ab}M_{ab}=0$, The Killing spinor equations
(110) are:

$\partial_{a}\epsilon_{-}=0\,,\quad\partial_{A}\epsilon_{-}+\gamma_{A}\eta_{-}=0\,,\quad\partial_{a}\epsilon_{+}-\tfrac{1}{2}\gamma_{a}\eta_{-}=0\,,\quad\partial_{A}\epsilon_{+}+\rho_{+A}+\tfrac{1}{6}\partial_{a}\Phi\gamma_{A}\gamma^{a}\epsilon_{-}=0\,,$
(128)

the first three of which appeared in the case of the flat background.
Accordingly we can write

$\epsilon_{+}=\gamma_{a}x^{a}\Theta_{+}(x^{A})+\theta_{+}(x^{A})\,,$ (129)

where now from the gamma trace of the final equation we obtain

$\gamma^{A}\partial_{A}(\gamma_{a}x^{a}\Theta_{+}+\theta_{+})+M_{ab}x^{a}\gamma^{b}\epsilon_{-}+\tfrac{1}{2}N_{a}\gamma^{a}\epsilon_{-}=0\,.$
(130)

Taking $\partial_{a}$ of this equation we get
$\gamma^{A}\gamma_{a}\partial_{A}\Theta_{+}+M_{ab}\gamma^{b}\epsilon_{-}=0$,
and contracting with $\gamma^{a}$ we find
$\gamma^{A}\partial_{A}\Theta_{+}=0$, using that $M_{ab}$ is symmetric and
traceless. This in turn implies

$M_{ab}\gamma^{b}\epsilon_{-}=0\,.$ (131)

Given that $M_{ab}$ is symmetric, we can diagonalise it using an orthogonal
matrix, which will not affect the Clifford algebra obeyed by the $\gamma^{a}$.
Then as long as $M_{ab}$ is not identically zero we find
$\gamma^{a}\epsilon_{-}=0$ for some $a$, which means that $\epsilon_{-}=0$.
The latter implies $\eta_{-}=0$, hence $\partial_{a}\epsilon_{+}=0$, and so
$\Theta_{+}=0$. In this case we find again the flat background solutions

$M_{ab}\neq
0:\quad\epsilon_{-}=0\,,\quad\epsilon_{+}=\theta_{+}(x^{A})\,,\quad\eta_{-}=0\,,\quad\rho_{+A}=-\partial_{A}\theta_{+}\,,\quad\gamma^{A}\partial_{A}\theta_{+}=0\,.$
(132)

This solution has $16$ independent Killing spinors.

If on the other hand, $\Phi$ is only linear in the transverse coordinates, so
$M_{ab}=0$ identically, then we still need to solve

$\gamma^{A}\partial_{A}\theta_{+}+\tfrac{1}{2}N_{a}(x^{A})\gamma^{a}\epsilon_{-}=0\,.$
(133)

If $N_{a}$ is in fact constant and not an arbitrary set of functions of the
longitudinal coordinates, one solution is

$M_{ab}=0:\quad\epsilon_{-}=\text{constant}\,\quad\epsilon_{+}=-\tfrac{1}{6}\gamma_{A}x^{A}N_{a}\gamma^{a}\epsilon_{-}+\epsilon_{+}^{\prime}\,,\quad\eta_{-}=0\,,\quad\rho_{+A}=0\,,$
(134)

with $\epsilon_{+}^{\prime}$ constant. This solution has 32 independent
Killing spinors.

In the maximally supersymmetric non-relativistic supergravity, the solution
for the Newton potential $\Phi$ satisfies stronger constraints than just the
Poisson equation and consequently does not constitute a supersymmetrisation of
standard Newton-Cartan gravity. Surprisingly, it thus seems that there is no
maximally supersymmetric version of 11-dimensional Newton-Cartan gravity. Note
that examples of supersymmetrisations of Newton-Cartan gravity in lower
dimensions and with less supersymmetry have been found in 3 and 10 dimensions
(see [3] for a recent overview).

###  6.4 Lambert-Smith holographic M2 background

An interesting solution to the bosonic part of the non-relativistic
supergravity theory was obtained recently in [10]. In this paper, the authors
take an MNC limit of an M2 background solution, where the longitudinal
directions of the limit only overlap with the time direction of the
longitudinal directions of the solution. This orientation can be illustrated
as follows:

| $0$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | ${10}$
---|---|---|---|---|---|---|---|---|---|---|---
MNC limit | $\times$ | $\times$ | $\times$ | – | – | – | – | – | – | – | –
M2 solution | $\times$ | – | – | $\times$ | $\times$ | – | – | – | – | – | –

This limit involved rescaling the coordinates $(x^{0},x^{1},x^{2})$
longitudinally (i.e. by a factor of $c$) and the coordinates
$x^{3},\dots,x^{10}$ transversally (by a factor of $c^{-1/2}$). The harmonic
function of the M2 brane solution was $\mathcal{H}=1+R^{6}/r^{6}$, with
$r^{2}=(x^{1})^{2}+(x^{2})^{2}+(x^{5})^{2}+\dots+(x^{10})^{2}$ and in order to
obtain a non-trivial solution the constant $R$ was rescaled as $R\rightarrow
cR$.

The latter rescaling is not part of the definition of the MNC limit that we
have used to obtain the limit of the theory itself. With just the rescaling of
the coordinates, the limit of the M2 solution gives a trivial background. The
additional rescaling of $R$ (which could potentially be viewed as sending the
number of M2 branes sourcing the geometry to infinity) ensures that the limit
produces a background geometry with a non-trivial profile. This additional
rescaling can be viewed as a manipulation required to engineer the
relativistic M2 solution into a form from which the MNC limit can be taken. It
is not however guaranteed a priori that this manipulation guarantees that the
limit of the relativistic solution is a solution of the limit of the
relativistic theory, which must be checked independently.

It was argued in [10] that the resulting MNC geometry in its near horizon
limit is holographically dual to a non-relativistic limit of the theory on
multiple membranes. As such it should be viewed as analogous to (and indeed a
limit of) the maximally supersymmetric $\text{AdS}_{4}\times S^{7}$ solution
of the relativistic 11-dimensional supergravity. Similar rescalings were used
for other brane solutions in other non-relativistic limits in [11, 12], again
with holographic interpretations: it would be interesting to further explore
the physical meaning of this mechanism (and compare to the discussion in [26,
27]).

Here we simply accept the solution of [10] as a geometry of potential interest
for the 11-dimensional non-relativistic supergravity. In fact, we will
generalise the background of [10] and obtain an infinite family of
supersymmetric backgrounds. Let us introduce some (flat) indices adapted to
the M2/MNC limit configuration:

$A=(0,\alpha)\,,\quad\alpha=1,2\,,\quad a=(m,I)\,,\quad m=3,4\,,\quad
I=5,\dots,10\,.$ (135)

Below we will denote the corresponding curved indices with a hat. (Coordinates
$x^{\mu}$ and derivatives $\partial_{\mu}$ will be written ‘naturally’ with
unhatted indices; however, this does _not_ denote contraction with a
vielbein.) We take the bosonic fields to be:

$\begin{split}\tau^{0}&=\mathcal{H}^{-1/3}\mathrm{d}x^{0}\,,\quad\tau^{\alpha}=\mathcal{H}^{1/6}\mathrm{d}x^{\alpha}\,,\quad
e^{m}=\mathcal{H}^{-1/3}\mathrm{d}x^{m}\,,\quad
e^{I}=\mathcal{H}^{1/6}\mathrm{d}x^{I}\,,\\\
c_{(3)}&=\mathcal{H}^{-1}\mathrm{d}x^{0}\wedge\mathrm{d}x^{3}\wedge\mathrm{d}x^{4}\,,\quad\lambda_{abcd}=0\,.\end{split}$
(136)

The powers of the function $\mathcal{H}=\mathcal{H}(x^{1},x^{2})$ are
determined from the relativistic M2 solution. We make the obvious choice for
the inverse vielbein pair. To compare with the results of [10], we note that
the coordinates used in [10] are related to the ones chosen here via
$(u,\bar{u})\leftrightarrow(x^{1},x^{2})$,
$(z,\bar{z})\leftrightarrow(x^{3},x^{4})$ and
$v\leftrightarrow(x^{5},\dots,x^{10})$. Moreover, the solution of [10] is
given by (136) with $\mathcal{H}$ given by
$\mathcal{H}=1+R^{6}/((x^{1})^{2}+(x^{2})^{2})^{3}$. The near horizon limit of
this solution is then obtained simply by dropping the constant term $1$ in
$\mathcal{H}$.

The field strength is

$f_{(4)}=-\partial_{\alpha}\mathcal{H}^{-1}\mathrm{d}x^{0}\wedge\mathrm{d}x^{\alpha}\wedge\mathrm{d}x^{3}\wedge\mathrm{d}x^{4}\,.$
(137)

It follows that the only non-zero field strength components are those of
$f_{ABab}$, and the only non-zero torsion components are those of
$T_{AB}{}^{C}$, with:

$f_{0\alpha
mn}=-\mathcal{H}^{5/6}\partial_{\alpha}\mathcal{H}^{-1}\epsilon_{mn}\,,\quad
T_{0\alpha}{}^{0}=\tfrac{1}{3}\mathcal{H}^{-7/6}\partial_{\alpha}\mathcal{H}\,,\quad
T_{\beta\gamma}{}^{\alpha}=\tfrac{1}{6}2\mathcal{H}^{-7/6}\delta^{\gamma}_{[\alpha}\partial_{\beta]}\mathcal{H}\,.$
(138)

Thus this background satisfies all intrinsic torsion constraints (37) imposed
in the theory with maximal supersymmetry. The non-zero components of the spin
connections are:

$\displaystyle\omega_{\mu}{}^{AB}:\quad$
$\displaystyle\omega_{\hat{0}}{}^{0\alpha}=-\tfrac{1}{3}\mathcal{H}^{-3/2}\partial^{\alpha}\mathcal{H}\,,\quad\omega_{\hat{\alpha}}{}^{\beta\gamma}=\tfrac{1}{6}\mathcal{H}^{-1}2\delta^{[\beta}_{\alpha}\partial^{\gamma]}\mathcal{H}\,,$
(139a) $\displaystyle\omega_{\mu}{}^{ab}:\quad$
$\displaystyle\omega_{\hat{\alpha}}{}^{34}=\tfrac{1}{2}\mathcal{H}^{-1}\epsilon_{\alpha\beta}\partial^{\beta}\mathcal{H}\,,$
(139b) $\displaystyle\omega_{\mu}{}^{Aa}:\quad$
$\displaystyle\omega_{\hat{m}}{}^{\alpha
n}=\mathcal{H}^{-3/2}(\tfrac{1}{3}\delta^{n}_{m}\partial^{\alpha}\mathcal{H}+\tfrac{1}{2}\epsilon^{\alpha\beta}\epsilon_{m}{}^{n}\partial_{\beta}\mathcal{H})\,,\quad\omega_{\hat{I}}{}^{\alpha
J}=-\tfrac{1}{6}\mathcal{H}^{-1}\delta_{I}^{J}\partial^{\alpha}\mathcal{H}\,,$
(139c)

with $b_{\mu}$ vanishing. Using these expressions, we compute the curvatures
appearing in the constraints of the non-relativistic supergravity with maximal
supersymmetry. The only non-zero well-defined curvatures are
$\breve{R}_{AB}{}^{ab}$ and $\breve{R}_{Aa}{}^{Ab}$, with:

$\breve{R}_{\alpha\beta}{}^{mn}=-\tfrac{1}{2}\epsilon_{\alpha\beta}\epsilon^{mn}\mathcal{H}^{-1/3}\partial_{\gamma}\partial^{\gamma}\log\mathcal{H}\,,\quad\breve{R}_{A(a}{}^{A}{}_{b)}=\tfrac{3}{8}\mathcal{H}^{-1/3}\partial_{\alpha}\partial^{\alpha}\log\mathcal{H}\begin{pmatrix}1_{2}&0\\\
0&-\tfrac{1}{3}1_{6}\end{pmatrix}\,.$ (140)

Note that $\breve{R}_{Aa}{}^{Aa}=0$, which means the Poisson equation is
trivially satisfied. The vanishing of the intrinsic torsions as well as all
other curvature conditions means that all the equations of motion are
satisfied _with no restrictions on the function $\mathcal{H}(x^{1},x^{2})$_.
This means that the background (136) is a solution of the bosonic theory for
$\mathcal{H}$ an arbitrary function of the longitudinal spatial coordinates.
It is then also automatically a solution of the theory with half-maximal
supersymmetry.

In order for it to be a solution of the theory with maximal supersymmetry,
from the constraints (93b), we need to impose that
$D_{c}\breve{R}_{A(a}{}^{A}{}_{b)}=0$. It follows from the form of the
$\mathrm{SO}(8)$ spin connection (139b) and the curvature in (140) that this
is the case.

Let us make a remark about the possibility of imposing stronger conditions on
this geometry. At the end of appendix C, we mention that instead of imposing
the constraints that the transverse covariant derivatives of $\lambda_{abcd}$,
$\breve{r}_{+a}$ and $\breve{R}_{A(b}{}^{A}{}_{c)}$ must vanish, we could
actually require that these tensors vanish themselves, which from the
variation of $\breve{r}_{+a}$ would also require $\breve{R}_{AB}{}^{ab}=0$
(when covariant derivatives are taken one obtains a possible constraint
$D_{c}\breve{R}_{AB}{}^{ab}=0$ which is equivalent to a Bianchi identity by
other constraints, and satisfied by the curvature in (140)). Requiring both
curvatures in (140) to vanish identically amounts to requiring that
$\log\mathcal{H}$ must be harmonic in two dimensions, i.e.
$\log\mathcal{H}\propto\log((x^{1})^{2}+(x^{2})^{2})$, or

$\mathcal{H}(x^{1},x^{2})=A((x^{1})^{2}+(x^{2})^{2})^{\beta}$ (141)

for arbitrary power $\beta$ and some (dimensionful) constant $A$. The solution
of [10] in the near horizon limit has $\beta=-3$ and $A=R^{6}$.

We now seek to solve the Killing spinor equations of the maximal
supersymmetric theory for this background. We restrict to ‘near horizon’
solutions with $\mathcal{H}$ given by (141), and set $A=1$ for simplicity. The
Killing spinor equations are given in equation (110). The Killing spinor
equations (111) of the half-maximally supersymmetric theory are for this
background simply the $\epsilon_{-}=0$ truncation. For the ansatz we have
made, we find that (110) in components leads to:

$\displaystyle 0$
$\displaystyle=\partial_{0}\epsilon_{+}-\tfrac{1}{6}\mathcal{H}^{-1/2}\partial_{\alpha}\log\mathcal{H}\gamma_{0}\gamma^{\alpha}\epsilon_{+}+\mathcal{H}^{-1/3}\rho_{+0}\,,$
(142a) $\displaystyle 0$
$\displaystyle=\partial_{\alpha}\epsilon_{+}+\epsilon_{\alpha\beta}\partial^{\beta}\log\mathcal{H}(\tfrac{1}{12}\gamma_{12}+\tfrac{1}{4}\gamma_{34})\epsilon_{+}+\mathcal{H}^{1/6}\rho_{+\alpha}\,,$
(142b) $\displaystyle 0$
$\displaystyle=\partial_{m}\epsilon_{+}+\mathcal{H}^{-1/2}\tfrac{1}{2}(\tfrac{1}{3}\partial^{\alpha}\log\mathcal{H}\gamma_{\alpha}\gamma_{m}+\tfrac{1}{2}\epsilon^{\alpha\beta}\epsilon_{mn}\partial_{\beta}\log\mathcal{H}\gamma_{\alpha}\gamma^{n})\epsilon_{-}-\tfrac{1}{2}\mathcal{H}^{-1/3}\gamma_{m}\eta_{-}\,,$
(142c) $\displaystyle 0$
$\displaystyle=\partial_{I}\epsilon_{+}-\tfrac{1}{12}\partial^{\alpha}\log\mathcal{H}\gamma_{\alpha}\gamma_{I}\epsilon_{-}-\tfrac{1}{2}\mathcal{H}^{1/6}\gamma_{I}\eta_{-}\,,$
(142d) $\displaystyle 0$
$\displaystyle=\partial_{0}\epsilon_{-}-\tfrac{1}{6}\mathcal{H}^{-1/2}\partial_{\alpha}\log\mathcal{H}\gamma_{0}\gamma^{\alpha}\epsilon_{-}+\mathcal{H}^{-1/3}\gamma_{0}\eta_{-}\,,$
(142e) $\displaystyle 0$
$\displaystyle=\partial_{\alpha}\epsilon_{-}+\epsilon_{\alpha\beta}\partial^{\beta}\log\mathcal{H}(\tfrac{1}{12}\gamma_{12}+\tfrac{1}{4}\gamma_{34})\epsilon_{-}+\mathcal{H}^{1/6}\gamma_{\alpha}\eta_{-}\,,$
(142f) $\displaystyle 0$ $\displaystyle=\partial_{m}\epsilon_{-}\,,\quad
0=\partial_{I}\epsilon_{-}\,.$ (142g)

It’s convenient to define polar coordinates $x^{1}=r\cos\theta$,
$x^{2}=r\sin\theta$ and take $\mathcal{H}=r^{2\beta}$. Then the Killing spinor
equations become

$\displaystyle 0$
$\displaystyle=\partial_{0}\epsilon_{+}-\tfrac{\beta}{3}r^{-\beta-1}\gamma_{0}\gamma_{1}\exp(-\gamma_{0}\theta)\epsilon_{+}+r^{-2\beta/3}\rho_{+0}\,,$
(143a) $\displaystyle 0$
$\displaystyle=\partial_{r}\epsilon_{+}+r^{\beta/3}(\cos\theta\rho_{+1}+\sin\theta\rho_{+2})\,,$
(143b) $\displaystyle 0$
$\displaystyle=\partial_{\theta}\epsilon_{+}-2\beta(\tfrac{1}{12}\gamma_{12}+\tfrac{1}{4}\gamma_{34})\epsilon_{+}+r^{\beta/3+1}(\cos\theta\rho_{+2}-\sin\theta\rho_{+1})\,,$
(143c) $\displaystyle 0$
$\displaystyle=\partial_{m}\epsilon_{+}-r^{-\beta-1}\beta\left(\tfrac{1}{3}\gamma_{m}\gamma_{1}-\tfrac{1}{2}\epsilon_{mn}\gamma^{n}\gamma_{2}\right)\exp(\gamma_{0}\theta)\epsilon_{-}-\tfrac{1}{2}r^{-2\beta/3}\gamma_{m}\eta_{-}\,,$
(143d) $\displaystyle 0$
$\displaystyle=\partial_{I}\epsilon_{+}+\tfrac{1}{2}r^{\beta/3}\gamma_{I}\left(\tfrac{\beta}{3}r^{-1-\beta/3}\gamma_{1}\exp(\gamma_{0}\theta)\epsilon_{-}-\eta_{-}\right)\,,$
(143e) $\displaystyle 0$
$\displaystyle=\partial_{0}\epsilon_{-}-r^{-2\beta/3}\gamma_{0}\left(\tfrac{\beta}{3}r^{-1-\beta/3}\gamma_{1}\exp(\gamma_{0}\theta)\epsilon_{-}-\eta_{-}\right)\,,$
(143f) $\displaystyle 0$
$\displaystyle=\partial_{r}\epsilon_{-}+r^{\beta/3}\gamma_{1}\exp(\gamma_{0}\theta)\eta_{-}\,,$
(143g) $\displaystyle 0$
$\displaystyle=\partial_{\theta}\epsilon_{-}-2\beta(\tfrac{1}{12}\gamma_{12}+\tfrac{1}{4}\gamma_{34})\epsilon_{-}+r^{\beta/3+1}\gamma_{2}\exp(\gamma_{0}\theta)\eta_{-}\,,$
(143h) $\displaystyle 0$ $\displaystyle=\partial_{m}\epsilon_{-}\,,\quad
0=\partial_{I}\epsilon_{-}\,.$ (143i)

Here we used extensively the fact that the spinors have definite chirality
under the longitudinal projectors $\Pi_{\pm}=\tfrac{1}{2}(1\pm\gamma_{012})$,
so in particular
$\gamma_{1}\gamma_{2}\epsilon_{\pm}=\mp\gamma_{0}\epsilon_{\pm}$.

Now, suppose we attempted to find a solution of these equations with the
Stuckelberg transformation parameters set to zero, $\rho_{+A}=0=\eta_{-}$.
Then we would have $\partial_{r}\epsilon_{+}=0$, $\partial_{r}\epsilon_{-}=0$.
Differentiating the $\mu=0$ equations, (143a) and (143f) would then imply that
$\epsilon_{\pm}=0$ if $\beta\neq-1$. If $\beta=-1$ the solutions of the
$\mu=\theta$ equations, (143c) and (143d) would give
$\epsilon_{\pm}=\exp(\pm\tfrac{1}{6}\gamma_{0}\theta)\exp(-\tfrac{1}{2}\gamma_{34}\theta)\tilde{\epsilon}(x^{0},x^{m},x^{I})$.
However from the $\mu=0$ equations we would find
$\partial_{0}\tilde{\epsilon}_{\pm}=-\tfrac{1}{3}\gamma_{0}\gamma_{1}\exp(\mp\tfrac{2}{3}\gamma_{0}\theta)\tilde{\epsilon}_{\pm}$,
where the left-hand side is independent of $\theta$ but the right-hand side is
not. So we cannot find a solution in this case.

We therefore have to allow for non-vanishing Stuckelberg parameters.
Inspecting the above equations, a simplifying possibility is to assume
$\partial_{0}\epsilon_{-}=0$ or equivalently $\partial_{I}\epsilon_{+}=0$,
which determines $\eta_{-}$ from (143e) or (143f). Substituting this value of
$\eta_{-}$ into the other equations, it is straightforward to obtain the
following solution:

$\displaystyle\epsilon_{+}$
$\displaystyle=\epsilon_{+}^{\prime}+\epsilon_{+}^{\prime\prime}\,,$ (144a)
$\displaystyle\epsilon_{+}^{\prime}$
$\displaystyle=r^{\beta/3}\exp(-\tfrac{2\beta}{3}\gamma_{0}\theta)\tilde{\epsilon}_{++}+r^{\beta/3}\exp(\tfrac{\beta}{3}\gamma_{0}\theta)\tilde{\epsilon}_{+-}\,,$
(144b) $\displaystyle\epsilon_{+}^{\prime\prime}$ $\displaystyle=\beta
r^{-1-4\beta/3}\exp([1+\beta]\gamma_{0}\theta)x^{m}\gamma_{m}\gamma_{1}\tilde{\epsilon}_{--}\,,$
(144c) $\displaystyle\epsilon_{-}$
$\displaystyle=r^{-\beta/3}\tilde{\epsilon}_{-+}+r^{-\beta/3}\exp(\beta\gamma_{0}\theta)\tilde{\epsilon}_{--}\,,$
(144d) $\displaystyle\rho_{+0}$
$\displaystyle=\gamma_{0}\gamma_{1}\tfrac{\beta}{3}r^{-1-\beta/3}\exp(-\gamma_{0}\theta)(\epsilon_{+}^{\prime}+\epsilon_{+}^{\prime\prime})\,,$
(144e) $\displaystyle\rho_{+1}$
$\displaystyle=r^{-1-\beta/3}\left(-\tfrac{\beta}{3}\cos\theta\epsilon_{+}^{\prime}+\tfrac{\beta}{3}\sin\theta\gamma_{0}\epsilon_{+}^{\prime\prime}+(1+\tfrac{4\beta}{3})\exp(\gamma_{0}\theta)\epsilon_{+}^{\prime\prime}\right)\,,$
(144f) $\displaystyle\rho_{+2}$
$\displaystyle=r^{-1-\beta/3}\left(-\tfrac{\beta}{3}\sin\theta\epsilon^{\prime}_{+}-\tfrac{\beta}{3}\cos\theta\gamma_{0}\epsilon_{+}^{\prime\prime}-(1+\tfrac{4\beta}{3})\gamma_{0}\exp(\gamma_{0}\theta)\epsilon_{+}^{\prime\prime}\right)\,,$
(144g) $\displaystyle\eta_{-}$
$\displaystyle=\tfrac{\beta}{3}r^{-1-\beta/3}\gamma_{1}\exp(\theta\gamma_{0})\epsilon_{-}\,,$
(144h)

where $\tilde{\epsilon}_{+\pm}$ and $\tilde{\epsilon}_{-\pm}$ are constant and
obey $\gamma_{034}\tilde{\epsilon}_{+\pm}=\pm\tilde{\epsilon}_{+\pm}$,
$\gamma_{034}\tilde{\epsilon}_{-\pm}=\pm\tilde{\epsilon}_{-\pm}$. Therefore
this is a solution involving 32 independent Killing spinors. We have not
demonstrated that this is the unique solution to the Killing spinor equations.
Nonetheless, its existence demonstrates that we can view the family of
solutions given by (136) and (141) as supersymmetric solutions to the non-
relativistic eleven-dimensional supergravity, when we impose the constraints
compatible with either maximal or half-maximal supersymmetry (in the latter
case, the Killing spinor solution is simply the truncation
$\tilde{\epsilon}_{-\pm}=0$).

It would be interesting to further explore the geometry of this family of
solutions. In particular, requiring the above Killing spinor solution to be
globally defined may restrict the allowed values of $\beta$. It would also be
of interest to see if the solutions other than the case of [10] have an origin
as a limit of a relativistic background, with or without a holographic
interpretation. We leave this as interesting future work.

##  7 Discussion

In this paper, we have explored the membrane Newton-Cartan non-relativistic
limit of 11-dimensional supergravity. This limit has thrown up various
surprises.

We have found that we have to impose constraints, compatible either with
maximal or half-maximal supersymmetry. In the maximal case, this leads to a
surprisingly complicated series of constraints – which remarkably do not
trivialise the theory, but admit an infinite family of supersymmetric
solutions including the putative holographic M2 near horizon non-relativistic
geometry of [10].

In the case with half-maximal supersymmetry, we found an unusual realisation
of supersymmetry. The attractive feature of this scenario is that it imposes
fewer constraints on the theory – in particular no constraints on bosonic
curvatures are needed. This avoids the surprising tension between the maximal
supersymmetry constraints and the possibility of having a Newton potential of
standard form in the solution of section 6.3. However, the supersymmetry
algebra (107) with parameters (108) no longer closes into diffeomorphisms.
Therefore it is not clear to what extent this should be viewed as supergravity
in the standard sense.

In the case with maximal supersymmetry, our results show at the linearised
level that there is a consistent multiplet of constraints defining a
consistent supermultiplet realizing a representation of an underlying non-
relativistic superalgebra. In order to complete this multiplet at the non-
linear level it would be beneficial to reformulate the linearised results of
this paper in terms of constrained superfields on a suitable non-relativistic
superspace. For earlier discussions of such a non-relativistic superspace and
superfields in lower dimensions, see e.g. [28, 29, 30, 31, 32].

It is interesting to note that the underlying geometry of the theories
considered in this paper, constitutes a generalization of a Newton-Cartan
geometry with three new features. Firstly, it includes a three-form (instead
of a one-form) and a tangent space splitting in 3 longitudinal and 8
transversal directions (instead of a time and spatial directions). Secondly,
it has local conformal symmetry built in and finally, it also incorporates
supersymmetry. This geometry exhibits several features that are already
present in $2$-brane Galilean geometry (that does not include a three-form,
nor a type of superconformal symmetry), such as the presence of intrinsic
torsion and undetermined connection components [33]. In the 2-brane Galilean
case, these can both be studied and classified via Spencer cohomology. It
would be interesting to investigate whether these techniques can be extended
to the present situation.

We have not shown that the non-relativistic limit that we defined in this
paper is the unique membrane non-relativistic limit. For instance, the ansatz
for the gravitino expansion (9) is not the most general one that is covariant
with respect to SO(1,2) and SO(8) transformations. Modifying the ansatz would
alter the structure of the divergences that appear in the supersymmetry
transformations, and hence potentially lead to limits with different (ideally
simpler) sets of constraints. We explored some possible expansions along these
lines by decomposing the gravitino further (e.g. introducing different
$c$–rescalings of the gamma trace and gamma traceless parts of the
longitudinal and transverse projections) but came to the preliminary
conclusion that this did not lead to significant simplifications in the
structure of the theory. We nevertheless cannot exclude that there exist other
field redefinitions that lead to different consistent limits as well.

The standard dimensional reduction procedure should lead to a non-relativistic
supergravity with $\mathcal{N}=2$ supersymmetry. We can dimensionally reduce
either along a direction longitudinal to the membrane Newton-Cartan limit or
transverse to the limit. This should lead to two distinct non-relativistic
supersymmetric IIA supergravity theories, one with an underlying string
Newton-Cartan geometry and the other with a D2 brane Newton-Cartan geometry,
extending the bosonic sectors obtained by these reductions in [20]. The latter
has recently appeared in [6] and been termed M$2$T geometry there. T-duality
of the SNC type IIA would then connect to the IIB case, whose bosonic sector
was obtained in [34].

In addition, the naive expectation would be that the longitudinal reduction to
type IIA SNC supergravity could be truncated to $\mathcal{N}=1$ in order to
match the $\mathcal{N}=1$ SNC supergravity of [18]. However, this is not the
case: reducing and truncating, we find a different $\mathcal{N}=1$ expansion
and set of constraints. This can be demonstrated through a representation
theory argument, which traces the rearrangement and truncation of the
$\mathbf{32}\otimes\mathbf{11}$ fermionic gravitino components in both cases.
The truncation from eleven-dimensional supergravity to minimal supergravity in
ten dimensions follows a specific process: first, reduce to type IIA in ten
dimensions, then truncate the right-handed gravitino ($\mathbf{16}^{v}_{R}$)
and the left-handed dilatino ($\mathbf{16}_{L}^{s}$). We apply this same basic
procedure in the non-relativistic setting, based on the observation that both
should result from a limit. However, we will demonstrate that these procedures
do not commute. When first truncating to $\mathcal{N}=1$ and taking the limit
as described in [18], we obtain:

$\displaystyle\mathbf{32}^{v}\quad\to\quad\mathbf{16}_{L}^{v}\oplus\mathbf{16}_{R}^{s}\quad\to\quad\big{(}\mathbf{8}_{L+}^{v}\oplus\mathbf{8}_{R+}^{s}\big{)}\oleft\big{(}\mathbf{8}_{L-}^{v}\oplus\mathbf{8}_{R-}^{s}\big{)}\,.$
(145)

Here, $v/s$ denote gravitino/dilatino, $L/R$ indicate eigenvalues under
$\gamma_{2}$, and $+/-$ represent eigenvalues under $\gamma_{01}$.
Importantly, $+/-$ also signify behavior under boosts. We use the notation
$X_{+}\oleft X_{-}$ to indicate that under boosts, $X_{+}\to X_{-}\to 0$.
Conversely, if we begin with the decomposition introduced in this paper, we
arrive at:

$\displaystyle\mathbf{32}^{v}\quad\to\quad\mathbf{16}_{+}^{v}\oleft\mathbf{16}_{-}^{v}\quad\to\quad\big{(}\mathbf{8}_{L+}^{v}\oplus\mathbf{8}_{R-}^{s}\big{)}\oleft\big{(}\mathbf{8}_{L-}^{v}\oplus\mathbf{8}_{R+}^{s}\big{)}\,.$
(146)

Note that the sublabels on $\mathbf{16}_{\pm}^{v}$ indicate eigenvalues of
$\gamma_{012}$, whereas the $+/-$ on
$\mathbf{8}_{L\pm}^{v}/\mathbf{8}_{R\mp}^{s}$ indicates eigenvalues of
$\gamma_{01}$. This result does not align with the representation structure of
the $\mathcal{N}=1$ limit, as the $\gamma_{01}$-chiralities of the dilatini
$\mathbf{8}^{s}_{R\mp}$ do not align with the boost structure as they did
above. This discrepancy can be traced back to the observation that decomposing
the spinors into $\gamma_{012}=\gamma_{01}\gamma_{2}$ chiralities effectively
couples the $\gamma_{2}$ and $\gamma_{01}$ chiralities in ten dimensions. We
have attempted to find an alternative, twisted truncation procedure but have
been unsuccessful thus far. The intriguing question of how to relate maximal
and half-maximal multiplets remains open for future investigation. Potentially
the known ten-dimensional $\mathcal{N}=1$ theory simply does not embed
directly into type IIA but should be viewed as a truncation of heterotic
supergravity (a bosonic non-relativistic limit of which has been studied in
[35]), which may require some Hořava-Witten dimensional reduction on an
interval to obtain from the 11-dimensional theory.

In addition to the membrane Newton-Cartan limit considered here, the eleven-
dimensional supergravity should also admit a non-relativistic limit associated
with the M5 brane, with six longitudinal and five transverse directions. This
would lead to a further non-uniqueness of non-relativistic eleven-dimensional
supergravity.

All these possible non-relativistic limits, and their ten-dimensional
reductions, should be related by dualities, e.g. the membrane and five-brane
decoupling limits are U-dual to each other. The membrane Newton-Cartan limit
considered in this paper is U-dual to the null reduction or DLCQ of M-theory,
which leads to the Matrix theory description. See [6] for a modern overview of
the non-relativistic duality web. It would be interesting to explore further
the implications of these observations for our understanding of M-theory and
its decoupling limits in curved supergravity backgrounds.

Finally, apart from constructing non-relativistic supergravity theories we
also considered solutions. In particular, we found that the recent solution
found by Lambert and Smith in [10] solves the eleven-dimensional non-
relativistic supergravity theory constructed in this paper. This solution, in
its near horizon limit, is a particular non-relativistic limit of
AdS${}_{4}\times S^{7}$ that constitutes a holographic dual to a decoupling
limit of ABJM theory which describes dynamics on the Hitchin moduli space. As
such it was conjectured that this solution would be maximally supersymmetric
which indeed we confirmed. Surprisingly, we found that the Lambert-Smith
solution is a particular member of a larger class of solutions. It would be
interesting to explore the consequences of this larger set of solutions. It
was further suggested in [12] that some of the supersymmetries of the
supergravity background dual to the non-relativistic limit of ABJM may be non-
physical. Our Killing spinor solution indicates that this background does
admit 32 supersymmetries, at least locally. It would be interesting to further
study this and other brane solutions of the non-relativistic 11-dimensional
supergravity, in search of other surprises.

## Acknowledgements

The work of CB is supported through the grants CEX2020-001007-S and
PID2021-123017NB-I00, funded by MCIN/AEI/10.13039/501100011033 and by ERDF A
way of making Europe. The work of JR is supported by the Croatian Science
Foundation project IP-2022-10-5980 “Non-relativistic supergravity and
applications”. Nordita is supported in part by NordForsk. We would like to
thank Martin Cederwall, Neil Lambert, Rishi Mouland, Niels Obers, Luca Romano,
Ingmar Saberi, Joseph Smith, and Ziqi Yan for useful discussions. Without the
M.O.N.S.T.E.R., this paper would not have been finished. The authors have
incurred debts of hospitality to numerous institutes where parts of this work
have been done: this includes KU Leuven (EB, CB, JL, JR, not all at the same
time); the Erwin Schrödinger Institute in Vienna during the program “ESI
workshop on Carrollian physics and holography” (EB, JL, JR); the
Rijksuniversiteit Groningen (JL, JR); the Rudjer Bošković Institute, Zagreb
during the Mini-symposium on “Physics and Geometry” 2022 (EB) and 2024 (EB,
CB); Nordita (EB, CB, JR); the Instituto de Física Teórica UAM/CSIC (EB); the
Mainz Institute for Theoretical Physics during the MITP scientific program
“Higher structures, gravity, and fields” (EB); and the Vrije Universiteit
Brussel (CB).

##  Appendix A Technical details

###  A.1 Gamma matrix identities

#### Useful identities

The product of all gamma matrices (with upper indices) is $\gamma^{0123\dots
10}=+1$. Hence

$\gamma^{\hat{a}_{1}\dots\hat{a}_{p}}=\tfrac{(-1)^{p(p-1)/2}}{(11-p)!}\epsilon^{\hat{a}_{1}\dots\hat{a}_{p}\hat{b}_{1}\dots\hat{b}_{11-p}}\gamma_{\hat{b}_{1}\dots\hat{b}_{11-p}}\,.$
(147)

We split $\hat{a}=(A,a)$. The epsilon tensors with flat indices are
$\epsilon_{ABC}$ with $\epsilon_{012}=+1$ and $\epsilon^{ABC}$ with
$\epsilon^{012}=-1$, as well as $\epsilon^{a_{1}\dots a_{8}}$, and
$\epsilon_{a_{1}\dots a_{8}}$ which are numerically identical. The following
relationships are then useful:

$\gamma_{ABC}=\epsilon_{ABC}\gamma_{012}\,,\quad\gamma_{AB}=\epsilon_{ABC}\gamma^{C}\gamma_{012}\,,$
(148) $\gamma^{a_{1}\dots
a_{p}}=\tfrac{1}{8-p!}(-1)(-1)^{\tfrac{1}{2}p(p-1)}\epsilon^{a_{1}\dots
a_{p}b_{1}\dots b_{8-p}}\gamma_{b_{1}\dots b_{8-p}}\gamma_{012}\,.$ (149)

The longitudinal projectors $\Pi_{\pm}=\tfrac{1}{2}(1\pm\gamma_{012})$ are
such that

$\gamma_{A}\Pi_{\pm}=\Pi_{\pm}\gamma_{A}\,,\quad\gamma_{a}\Pi_{\pm}=\Pi_{\mp}\gamma_{a}\,,$
(150)

If $\chi_{\pm}\equiv\Pi_{\pm}\chi$, then
$\bar{\chi}_{\pm}=\bar{\chi}\Pi_{\pm}$ so conjugates are projected in the same
way. Accordingly one has

$\bar{\chi}\gamma_{A_{1}\dots A_{q}a_{1}\dots
a_{p}}\psi=\begin{cases}\bar{\chi}_{+}\gamma_{A_{1}\dots A_{q}a_{1}\dots
a_{p}}\psi_{+}+\bar{\chi}_{-}\gamma_{A_{1}\dots A_{q}a_{1}\dots
a_{p}}\psi_{-}\,,&p\,\text{even}\,,\\\ \bar{\chi}_{+}\gamma_{A_{1}\dots
A_{q}a_{1}\dots a_{p}}\psi_{-}+\bar{\chi}_{-}\gamma_{A_{1}\dots
A_{q}a_{1}\dots a_{p}}\psi_{+}\,,&p\,\text{odd}\,.\end{cases}$ (151)

A useful identity relating the spinorial projectors and the transverse tangent
space self-duality projectors is:

$\gamma^{abcd}\Pi_{\pm}=\tfrac{1}{4!}P^{(\mp)}{}^{abcd}{}_{efgh}\gamma^{efgh}\,.$
(152)

#### Fierz identities

From the relativistic 11-dimensional Fierz identities [21] we can derive the
following Fierz identities for spinors in the non-relativistic theory,
according to their chiralities under the projectors $\Pi_{\pm}$. For
$\lambda_{1}$ and $\lambda_{3}$ with the same chirality the Fierz identity is:

$\begin{split}(\bar{\lambda}_{1}\lambda_{2})\lambda_{3}&=-\tfrac{1}{32}\Big{[}2(\bar{\lambda}_{1}\lambda_{3})\lambda_{2}+2(\bar{\lambda}_{1}\gamma^{A}\lambda_{3})\gamma_{A}\lambda_{2}\\\
&\qquad\qquad+2\tfrac{1}{2}(\bar{\lambda}_{1}\gamma^{ab}\lambda_{3})\gamma_{ba}\lambda_{2}+2\tfrac{1}{2}(\bar{\lambda}_{1}\gamma^{A}\gamma^{ab}\lambda_{3})\gamma_{A}\gamma_{ba}\lambda_{2}\\\
&\qquad\qquad+\tfrac{1}{4!}(\bar{\lambda}_{1}\gamma^{abcd}\lambda_{3})\gamma_{dcba}\lambda_{2}+\tfrac{1}{4!}(\bar{\lambda}_{1}\gamma^{Aabcd}\lambda_{3})\gamma_{dcbaA}\lambda_{2}\Big{]}\,.\end{split}$
(153)

For $\lambda_{1}$ and $\lambda_{3}$ with the opposite chirality the Fierz
identity is:

$\begin{split}(\bar{\lambda}_{1}\lambda_{2})\lambda_{3}&=-\tfrac{1}{16}\Big{[}(\bar{\lambda}_{1}\gamma^{a}\lambda_{3})\gamma_{a}\lambda_{2}+(\bar{\lambda}_{1}\gamma^{Aa}\lambda_{3})\gamma_{aA}\lambda_{2}\\\
&\qquad\qquad+\tfrac{1}{3!}(\bar{\lambda}_{1}\gamma^{abc}\lambda_{3})\gamma_{cba}\lambda_{2}+\tfrac{1}{3!}(\bar{\lambda}_{1}\gamma^{Aabc}\lambda_{3})\gamma_{cbaA}\lambda_{2}\Big{]}\,.\end{split}$
(154)

###  A.2 Expansion of supersymmetry transformations of the fermions

Starting from the relativistic supersymmetry transformations (2), making the
expansions (4) and (9) of the bosonic and fermionic fields, one finds that the
supersymmetry transformations can be expanded as
$\delta\Psi_{\mu}=\sum_{n=-3}^{2}c^{3n/2}(X_{3n/2})_{\mu}\epsilon$ in terms of
particular operators $(X_{m})_{\mu}$. Those appearing with integer powers
$m=(3,0,-3)$ of $c$ map between spinors of the same chirality, while those
with half-integer powers $m=(3/2,-3/2,-9/2)$ change chirality. After using the
equation of motion symmetry to remove divergent terms from the variation of
$\psi_{+\mu}$ acccording to (32) and hence (34), a straightforward but tedious
calculation shows that the finite part of the supersymmetry variations of the
fermions take the form:

$\begin{split}\delta^{\prime}_{0}\psi_{+\mu}&=(X_{0})_{\mu}\epsilon_{+}+(X_{-3/2}){}_{\mu}\epsilon_{-}-\tfrac{1}{12}\tau^{A}{}_{\mu}\gamma_{A}{\boldsymbol{\ell}}\epsilon_{+}-\tfrac{1}{8}e^{a}{}_{\mu}{\boldsymbol{\ell}}\gamma_{a}\epsilon_{-}\,,\\\
\delta^{\prime}_{0}\psi_{-\mu}&=(X_{0})_{\mu}\epsilon_{-}+(X_{3/2})_{\mu}\epsilon_{+}\,,\end{split}$
(155)

where the explicit forms of the operators that appear are as follows:

$\displaystyle(X_{0})_{\mu}$
$\displaystyle=\mathbb{I}\,\partial_{\mu}+\tfrac{1}{4}\hat{\omega}(\tau)_{\mu}{}^{AB}\gamma_{AB}$
$\displaystyle\qquad+\tfrac{1}{4}(\hat{\omega}(e)_{\mu}{}_{ab}-\tfrac{1}{12}\tau_{A\mu}\epsilon^{ABC}\hat{f}^{\prime}_{BCab}\gamma_{012}+\tfrac{1}{3}\tau^{A}{}_{\mu}\hat{f}^{\prime}_{ABab}\gamma^{B}-\tfrac{1}{3}e_{a}{}_{\mu}\hat{T}_{bA}{}^{A}\gamma_{012})\gamma^{ab}$
$\displaystyle\qquad+\tfrac{1}{6}e^{a}{}_{\mu}\hat{T}_{aA}{}^{A}\gamma_{012}-\tfrac{1}{12}\tau^{A}{}_{\mu}\gamma_{A}\boldsymbol{Z}+\tfrac{1}{12}\tfrac{1}{3!}e^{a}{}_{\mu}\hat{f}_{Abcd}(\gamma_{a}{}^{bcd}-6\delta^{b}_{a}\gamma^{cd})\gamma^{A}\,,$
(156a) $\displaystyle(X_{3/2})_{\mu}\epsilon_{+}$
$\displaystyle=\tau_{A\mu}\Big{(}\frac{1}{2}\gamma_{B}({\hat{T}}_{a}{}^{(AB)}+\tfrac{1}{3}\eta^{AB}{\hat{T}}{}_{aC}{}^{C})\gamma^{a}+\tfrac{1}{4}(\eta^{AB}-\tfrac{1}{3}\gamma^{A}\gamma^{B})\boldsymbol{\hat{f}}_{B}\Big{)}\epsilon_{+}$
$\displaystyle\qquad+e^{a}{}_{\mu}\Big{(}-\tfrac{1}{2}\gamma_{a}[-\tfrac{1}{12}\boldsymbol{\hat{f}}^{(-)}+\tfrac{1}{6}\gamma_{a}{\boldsymbol{\hat{T}}^{A}}\gamma_{A}]+\tfrac{1}{2}\hat{T}_{ab}{}^{A}\gamma^{b}\gamma_{A}-\tfrac{1}{8}\boldsymbol{\hat{f}}^{(+)}\gamma_{a}\Big{)}\epsilon_{+}\,,$
(156b) $\displaystyle(X_{-3/2})_{\mu}\epsilon_{-}$
$\displaystyle=\Big{(}\tfrac{1}{2}\hat{R}_{\mu}{}^{Aa}\gamma_{Aa}+e^{a}{}_{\mu}\tfrac{1}{4}\hat{f}^{\prime}_{bcAB}(\tfrac{1}{24}\gamma_{a}\gamma^{bc}-\tfrac{1}{8}\gamma^{bc}\gamma_{a})\gamma^{AB}+e^{a}{}_{\mu}\tfrac{1}{24}(\gamma_{a}\boldsymbol{Z}-3\boldsymbol{Z}\gamma_{a})$
$\displaystyle\qquad+\tau^{A}{}_{\mu}(\tfrac{1}{6}\hat{f}^{\prime}_{012a}\gamma^{a}\gamma_{A}+\tfrac{1}{6}\boldsymbol{Z}_{A}-\tfrac{1}{12}\boldsymbol{Z}_{B}\gamma_{A}{}^{B})\Big{)}\epsilon_{-}\,.$
(156c)

Here boldface symbols denote contraction with transverse gamma matrices, and

$\begin{split}\hat{\omega}(\tau)_{\mu}{}^{AB}&=\hat{T}_{\mu}{}^{[AB]}-\tfrac{1}{2}\tau^{C}{}_{\mu}\hat{T}^{AB}{}_{C}\,,\,\\\
\hat{\omega}(e)_{\mu}{}^{ab}&=\hat{\Omega}_{\mu}{}^{[ab]}-\tfrac{1}{2}e_{c\mu}\hat{\Omega}^{abc}+\tfrac{1}{2}\tau^{A}{}_{\mu}\bar{\psi}_{+}{}^{a}\gamma_{A}\psi_{+}{}^{b}\,,\\\
\hat{R}_{\mu}{}^{Aa}&=\tfrac{1}{2}\hat{\Omega}_{\mu}{}^{Aa}-\tfrac{1}{2}e_{\mu
b}\hat{\Omega}^{Aab}+\tfrac{1}{2}\tau^{B}{}_{\mu}(-\bar{\psi}_{+}{}^{a}\gamma^{A}\psi_{+B}+\bar{\psi}_{+}{}^{A}\gamma_{B}\psi_{+}{}^{a})-\tfrac{1}{2}\bar{\psi}_{+}{}^{a}\gamma^{A}\psi_{+b}e^{b}{}_{\mu}\end{split}$
(157a) while the field strength combinations are:
$\begin{split}\hat{f}_{abcd}&\equiv
f_{abcd}-6\bar{\psi}_{-[a}\gamma_{bc}\psi_{-d]}\,,\\\ \hat{f}_{Aabc}&\equiv
f_{Aabc}-3\bar{\psi}_{-A}\gamma_{[ab}\psi_{-c]}+6\bar{\psi}_{+[a}\gamma_{|A|b}\psi_{-c]}\,,\\\
\hat{f}^{\prime}_{ABab}&\equiv
f_{ABab}-4(\bar{\psi}_{+[A}\gamma_{B][a}\psi_{-b]}+\bar{\psi}_{-[A}\gamma_{B][a}\psi_{+b]})-\bar{\psi}_{+a}\gamma_{AB}\psi_{+b}-\bar{\psi}_{-A}\gamma_{ab}\psi_{-B}\,,\\\
\hat{f}^{\prime}_{ABCa}&\equiv
f_{ABCa}+6\bar{\psi}_{+[A}\gamma_{B|a|}\psi_{-C]}-3\bar{\psi}_{+[A}\gamma_{BC]}\psi_{+a}\,,\\\
\end{split}$ (157b) and also some purely fermionic objects appear:
$Z_{abcd}\equiv-6\bar{\psi}_{+[a}\gamma_{bc}\psi_{+d]}\,,\quad
Z_{Aabc}\equiv-3\bar{\psi}_{+A}\gamma_{[ab}\psi_{+c]}\,,\quad
Z_{ABab}\equiv-\bar{\psi}_{+A}\gamma_{ab}\psi_{+B}\,.$ (157c)

To simplify the above expressions, we make field-dependent Stuckelberg
transformations with the following parameters:

$\begin{split}\rho_{A+}&=\tfrac{1}{12}(\epsilon_{ABC}\hat{f}^{\prime
BC}{}_{ab}-\tfrac{1}{3}\hat{f}^{\prime}_{ABab}\gamma^{B})\gamma^{ab}\epsilon_{+}-\tfrac{1}{6}(\boldsymbol{Z}_{A}-\tfrac{1}{2}\boldsymbol{Z}_{B}\gamma_{A}{}^{B})\epsilon_{-}\\\
&\qquad-\tfrac{1}{2}\left(\bar{\psi}_{+}{}_{(A}\gamma_{B)}\psi_{+}^{a}-\tfrac{1}{3}\bar{\psi}_{+C}\gamma^{C}\psi_{+}^{a}\eta_{AB}\right)\gamma^{B}\gamma_{a}\epsilon_{-}\,,\\\
\eta_{-}&=-\tfrac{1}{3}\hat{T}_{aA}{}^{A}\gamma^{a}\epsilon_{+}+\tfrac{1}{12}\boldsymbol{Z}\epsilon_{-}-\tfrac{1}{24}\gamma^{BC}\hat{f}^{\prime}_{BCab}\gamma^{ab}\epsilon_{-}\,.\end{split}$
(158)

This then leads to the supersymmetry transformations in the form (39) stated
and used in the main text. Note that the purely fermionic second line of the
parameter $\rho_{A+}$ in (158) ensures that it is indeed the supercovariant
combination $\hat{\omega}_{\mu}{}^{Aa}$ that appear in the final expressions
(39), while $\hat{\omega}_{\mu}{}^{AB}$ and $\hat{\omega}_{\mu}{}^{ab}$ come
out automatically. In particular, this means that the following correct
supercovariant field strengths appear (in place of the quantities
$\hat{f}^{\prime}$ which come out originally from the expansion):

$\displaystyle\hat{f}_{ABab}$ $\displaystyle\equiv
f_{ABab}-4(\bar{\psi}_{+[A}\gamma_{B][a}\psi_{-b]}+\bar{\psi}_{-[A}\gamma_{B][a}\psi_{+b]})-\bar{\psi}_{-A}\gamma_{ab}\psi_{-B}-2\bar{\psi}_{+a}\gamma_{AB}\psi_{+b}\,,$
(159a) $\displaystyle\hat{f}_{ABCa}$ $\displaystyle\equiv
f_{ABCa}+6\bar{\psi}_{+[A}\gamma_{B|a|}\psi_{-C]}-6\bar{\psi}_{+[A}\gamma_{BC]}\psi_{+a}\,.$
(159b)

###  A.3 Gravitino kinetic terms at subleading order and super-Poisson
equations

In this appendix, we show how to extract useful information from the limit of
the gravitino kinetic term, treating the background non-relativistic geometry
(described by $\tau^{A}{}_{\mu}$ and $e^{a}{}_{\mu}$) as being fixed. With
this assumption, we consider the kinetic term:

$\begin{split}\mathcal{L}_{\text{RS}}&=-2\sqrt{|g|}\bar{\Psi}_{\mu}\gamma^{\mu\nu\rho}\partial_{\nu}\Psi_{\rho}\\\
&=-2\Omega\bar{\Psi}_{\mu}\big{(}c^{1/2}e^{\mu}{}_{a}e^{\nu}{}_{b}e^{\rho}{}_{c}\gamma^{abc}+3c^{-1}e^{[\mu}{}_{a}e^{\nu}{}_{b}\tau^{\rho]}{}_{A}\gamma^{abA}\\\
&\qquad\qquad+3c^{-5/2}\tau^{[\mu}{}_{A}\tau^{\nu}{}_{B}e^{\rho]}{}_{a}\gamma^{ABa}+c^{-4}\tau^{\mu}{}_{A}\tau^{\nu}{}_{B}\tau^{\rho}{}_{C}\gamma^{ABC}\big{)}\partial_{\nu}\Psi_{\rho}\,.\end{split}$
(160)

Using the gravitino expansion (9) and integrating by parts

$\begin{split}\Omega^{-1}\mathcal{L}_{\text{RS}}&=-4\bar{\psi}_{+a}\gamma^{abc}\partial_{b}\psi_{-a}-4\bar{\psi}_{-A}\gamma^{abA}\partial_{a}\psi_{-b}+2\bar{\psi}_{-a}\gamma^{abA}(\partial_{A}\psi_{-b}-\partial_{b}\psi_{-A})\\\
&\quad+c^{-3}\Big{[}-4\bar{\psi}_{+A}\gamma^{abA}\partial_{a}\psi_{+b}+2\bar{\psi}_{+a}\gamma^{abA}\partial_{A}\psi_{+b}\\\
&\qquad\qquad-4\bar{\psi}_{+A}\gamma^{ABa}(\partial_{B}\psi_{-a}-\partial_{a}\psi_{-B})-4\bar{\psi}_{+a}\gamma^{ABa}\partial_{A}\psi_{-B}\\\
&\qquad\qquad-2\bar{\psi}_{-A}\gamma^{ABC}\partial_{B}\psi_{-C}\Big{]}+\mathcal{O}(c^{-6})\,.\end{split}$
(161)

Varying according to (53), we find the following expansion for the equations
of motion:

$\begin{split}\mathcal{E}(\psi_{+})^{a}&=-4\gamma^{abc}\partial_{b}\psi_{-c}-4c^{-3}\left(\gamma^{ABa}\partial_{A}\psi_{-B}+\gamma^{abA}(\partial_{b}\psi_{+A}-\partial_{A}\psi_{+b})\right)\,,\\\
\mathcal{E}(\psi_{+})^{A}&=-4c^{-3}\left(\gamma^{abA}\partial_{a}\psi_{+b}+\gamma^{ABa}(\partial_{B}\psi_{-a}-\partial_{a}\psi_{-B})\right)\,,\\\
\mathcal{E}(\psi_{-})^{a}&=-4\left(\gamma^{abc}\partial_{b}\psi_{+c}+\gamma^{abA}(\partial_{b}\psi_{-A}-\partial_{A}\psi_{-b})\right)+\mathcal{O}(c^{-3})\,,\\\
\mathcal{E}(\psi_{-})^{A}&=-4\gamma^{abA}\partial_{a}\psi_{-b}+c^{-3}\left(4\gamma^{ABa}(\partial_{a}\psi_{+B}-\partial_{B}\psi_{+a})-4\gamma^{ABC}\partial_{B}\psi_{-C}\right)\,.\end{split}$
(162)

The subleading parts cannot in general be expressed in terms of the covariant
determined gravitino curvatures (83). However, for the dynamics we are only
interested in extracting the equations of motion conjugate to the Stuckelberg
variation, which gives equations that start at order $c^{-3}$. Firstly, using
$\delta\psi_{-A}=\gamma_{A}\eta_{-}$,
$\delta\psi_{+a}=-\tfrac{1}{2}\gamma_{a}\eta_{-}$ we find the combination:

$(-\gamma_{A}\mathcal{E}(\psi_{-})^{A}-+\tfrac{1}{2}\gamma_{a}\mathcal{E}(\psi_{-})^{a})\big{|}_{c^{-3}}=6\left(\gamma^{a}\gamma^{A}(\partial_{A}\psi_{+a}-\partial_{+a}\psi_{+A})-2\gamma^{AB}\partial_{B}\psi_{-C}\right)\,,$
(163)

from which we infer, referring to the definitions (83), that we are obtaining
here the linearisation of the covariant super-Poisson equation (58a). For the
other Stuckelberg symmetry, we have $\delta\psi_{+A}=\rho_{+A}$, and we have

$(\mathcal{E}(\psi_{+})^{A}-\tfrac{1}{3}\gamma^{A}\gamma_{B}\mathcal{E}(\psi_{+})^{B})\big{|}_{c^{-3}}=4\gamma^{a}(\partial^{A}\psi_{-a}-\partial_{a}\psi_{-}^{A}-\tfrac{1}{3}\gamma^{A}\gamma^{B}(\partial_{B}\psi_{-a}-\partial_{a}\psi_{-B})$
(164)

from which we infer that we are obtaining here the linearisation of the
covariant super-Poisson equation (58b).

###  A.4 Supersymmetry transformation of Lagrange multiplier

In this appendix, we discuss the supersymmetry transformation of
$\lambda_{abcd}$, which was determined in (43) to have the form:

$\begin{split}\delta_{Q}\lambda_{abcd}&=c^{3}\left(\delta^{\prime}_{\epsilon}\hat{f}^{(-)}_{abcd}-\tfrac{1}{24}\bar{\epsilon}_{+}\gamma_{abcd}\gamma_{A}\,{\mathcal{E}}(\psi_{+})^{A}-\tfrac{1}{16}\bar{\epsilon}_{-}\gamma_{e}\gamma_{abcd}\,{\mathcal{E}}(\psi_{+})^{e}\right)\\\
&\quad-\tfrac{1}{2}P^{-}{}_{abcd}{}^{efgh}\bar{\psi}_{+e}\gamma_{fg}(\delta^{\prime}_{\epsilon}\psi_{+h})\,,\end{split}$
(165)

where $\mathcal{E}(\psi_{+})^{\mu}$ denotes the equation of motion of
$\psi_{+\mu}$. The supersymmetry transformation
$\delta^{\prime}_{\epsilon}\psi_{+\mu}$ is defined in (33) and (34). The
supersymmetry transformation $\delta^{\prime}_{\epsilon}\psi_{-\mu}$ is
defined in (33) at order $c^{3}$ and at order $c^{0}$ is given in (155).
Acting on the bosons, $\delta^{\prime}_{\epsilon}$ coincides at order $c^{0}$
with the transformations $\delta_{Q}$ given in (24); below we will also use
the subleading transformation acting on $c_{\mu\nu\rho}$, given by

$\delta^{\prime}_{-3}c_{\mu\nu\rho}=3\bar{\epsilon}_{+}\gamma_{ab}\psi_{+[\mu}e^{a}{}_{\nu}e^{b}{}_{\rho]}\,.$
(166)

Let’s first consider the terms at order $c^{3}$ in (165). This starts with the
finite variation of $\hat{f}^{(-)}_{abcd}$, which can be obtained by
projecting

$\delta^{\prime}_{\epsilon}\hat{f}_{abcd}\big{|}_{c^{0}}=6\bar{\epsilon}_{-}\gamma_{[ab}\breve{r}_{-cd]}+(\text{constraints})\,,$
(167)

as follows from a short calculation, where we omit writing terms which vanish
either when imposing the initial geometric constraints needed for maximal or
half-maximal supersymmetry. The gravitino curvature $\breve{r}_{-ab}$ is
defined in (83) and coincides with that $\mathring{r}_{-ab}$ obtained in the
equations of motion derived from the limit of the action. We similarly know
that the variation $\delta^{\prime}_{\epsilon}\psi_{+h}$ at order $c^{3}$ has
the form (33) and so also vanishes when we impose these constraints. We then
need to consider the terms involving the equations of motion
$\mathcal{E}(\psi_{+})^{\mu}$ of $\psi_{+\mu}$ at order $c^{0}$, which were
obtained directly by varying the non-relativistic action and are recorded in
(54). We easily see that all terms contributing at order $c^{3}$ from these
equations of motion and involving the bosonic intrinsic torsions vanish when
the constraints are applied, such that

$\left(-\tfrac{1}{24}\bar{\epsilon}_{+}\gamma_{abcd}\gamma_{A}\,{\mathcal{E}}(\psi_{+})^{A}-\tfrac{1}{16}\bar{\epsilon}_{-}\gamma_{e}\gamma_{abcd}\,{\mathcal{E}}(\psi_{+})^{e}\right)\big{|}_{c^{0}}=\tfrac{1}{8}\bar{\epsilon}_{-}\gamma_{e}\gamma_{abcd}\gamma^{efg}\breve{r}_{-fg}+(\text{constraints})\,.$
(168)

The sum of the gravitino curvature terms in (167) and (168) then turns out to
vanish after some gamma matrix algebra. We conclude that the supersymmetry
transformation of $\lambda_{abcd}$ is guaranteed to be finite simply by the
initial constraints (37) or (38) (with the further implied constraint
$f_{abcd}=0$) imposed for maximal or half-maximal supersymmetry.

Now we have to work out the the finite part of the transformation. To simplify
matters, we again assume that the vielbein are fixed in order that we can
focus solely on terms involving derivatives of $\psi_{\pm\mu}$. We then invoke
covariance to upgrade these to covariant expressions. The cost of doing so is
that we may miss contributions involving the intrinsic torsions (65). These
are all set to zero in the maximal supersymmetry case, so this procedure
should be at least unambiguous there.

Proceeding in this manner, using (166) and the subleading equations of motion
(162), we find

$\displaystyle\delta^{\prime}_{\epsilon}\hat{f}_{abcd}\big{|}_{c^{-3}}$
$\displaystyle\sim
6\bar{\epsilon}_{+}\gamma_{[ab}r_{+cd]}+12\partial_{[c}\bar{\epsilon}_{+}\gamma_{ab}\psi_{+d]}\,,$
(169)
$\displaystyle-\tfrac{1}{24}\bar{\epsilon}_{+}\gamma_{abcd}\gamma_{A}\,{\mathcal{E}}(\psi_{+})^{A}\big{|}_{c^{-3}}$
$\displaystyle\sim\tfrac{1}{6}\bar{\epsilon}_{+}\gamma_{abcd}\left(\tfrac{3}{2}\gamma^{ef}r_{+ef}+2\gamma^{A}\gamma^{e}r_{-Ae}\right)\,,$
(170)
$\displaystyle-\tfrac{1}{16}\bar{\epsilon}_{-}\gamma_{e}\gamma_{abcd}\,{\mathcal{E}}(\psi_{+})^{e}\big{|}_{c^{-3}}$
$\displaystyle\sim\tfrac{1}{4}\bar{\epsilon}_{-}\gamma^{e}\gamma_{abcd}\gamma^{A}r_{+Ae}\,,$
(171)

where at the order we are working at, $r_{\pm\mu\nu}\sim
2\partial_{[\mu}\psi_{\pm\nu]}$. The term involving a derivative of
$\epsilon_{+}$ cancels against the finite variation of
$\delta^{\prime}_{\epsilon}\psi_{+h}$ coming from the final term of (165):
this is the explicit realisation of the statement made in the main text that
it is $\lambda_{abcd}$ defined via (42) by adding this term which is
supercovariant. Inserting the above expressions into (165) and using a slew of
gamma matrix identities we ultimately find a transformation consistent with a
covariant transformation of the form:

$\delta_{Q}\lambda_{abcd}=\tfrac{1}{4}\bar{\epsilon}_{-}\gamma^{e}\gamma_{abcd}\breve{r}_{+e}+2\bar{\epsilon}_{+}\gamma_{[abc}{}^{e}\breve{r}_{+d]e}+\dots\,,$
(172)

where the ellipsis denotes the possibility that there are modifications to
this expression missed by the above (essentially linear) analysis, involving
bosonic intrinsic torsion tensors.

###  A.5 The dependent fermionic connections

In this appendix, we explain how to solve for the fermionic gauge fields
$\varphi_{-\mu}$ and $\varphi_{+A\mu}$ introduced in (81). To do so, let’s
rewrite the gravitino curvatures (80) as

$\begin{split}r_{-\mu\nu}=2\,D_{[\mu}\psi_{-\nu]}+2\,\gamma_{A}\tau^{A}{}_{[\mu}\varphi_{-\nu]}\,,\quad
r_{+\mu\nu}=2\,\tilde{D}_{[\mu}\psi_{+\nu]}-\gamma_{a}e^{a}{}_{[\mu}\varphi_{-\nu]}+2\,\tau^{A}{}_{[\mu}\varphi_{+A\nu]}\,,\end{split}$
(173)

where we define the following bosonic covariant derivative

$\tilde{D}_{[\mu}\psi_{+\nu]}=2\,D_{[\mu}\psi_{+\nu]}-\tfrac{1}{6}\boldsymbol{\lambda}\tau^{A}{}_{[\mu}\gamma_{A}\psi_{+\nu]}-\tfrac{1}{4}\boldsymbol{\lambda}e^{e}{}_{[\mu}\gamma_{e}\psi_{-\nu]}\,,$
(174)

Below we use the shorthand notation $(D\psi)_{-\mu\nu}\equiv
2D_{[\mu}\psi_{-\nu]}$ and $(\tilde{D}\psi)_{+\mu\nu}\equiv
2\tilde{D}_{[\mu}\psi_{+\nu]}$. Before we start, note that since by definition
$\gamma^{A}\varphi_{+A\mu}=0$ we have

$\gamma^{A}\varphi_{+[AB]}+\gamma^{A}\varphi_{+(AB)}=0\,,\quad\gamma^{AB}\varphi_{+AB}=\eta^{AB}\varphi_{+AB}\,.$
(175)

To solve for $\varphi_{-\mu}$ we use the following conventional constraints:

$r_{-AB}=0\,,\quad\gamma^{A}r_{-Aa}=0\,.$ (176)

Here we have $3+8=11$ constraints and $11$ unknowns (not counting spinor
degrees of freedom). There is a unique result

$\varphi_{-\mu}=\tau^{A}{}_{\mu}\,\big{(}\gamma^{B}(D\psi)_{-AB}-\tfrac{1}{4}\,\gamma_{A}\gamma^{BC}(D\psi)_{-BC}\big{)}-\frac{1}{3}\,e^{a}{}_{\mu}\gamma^{A}(D\psi)_{-Aa}\,.$
(177)

To solve for $\varphi_{+A\mu}$ we use the following conventional constraints:

$r_{+Aa}-\tfrac{1}{3}\gamma_{A}\gamma^{B}r_{+Ba}=0\,,\quad r_{+AB}=0\,.$ (178)

Counting $\gamma^{A}\varphi_{+A\mu}=0$ as a constraint, we have $3\times
11=33$ unknowns and $(3-1)\times 8+3+11=30$ constraints. Hence we obtain a
non-unique answer:

$\begin{split}\varphi_{+A\mu}&=-\tfrac{1}{2}\,\tau^{B}{}_{\mu}(\tilde{D}\psi)_{+AB}+\tau^{B}{}_{\mu}\varphi_{+(AB)}\\\
&\quad-e^{a}{}_{\mu}\big{(}(\tilde{D}\psi)_{+Aa}+\tfrac{1}{2}\,\gamma_{a}\varphi_{-A}-\tfrac{1}{3}\gamma_{A}\gamma^{B}((\tilde{D}\psi)_{+Ba}+\tfrac{1}{2}\,\gamma_{a}\varphi_{-B})\big{)}\,.\end{split}$
(179)

The symmetric part $\varphi_{+(AB)}$ naively gives 6 undetermined components,
but the gamma trace condition amounts to three conditions on this quantity:

$\gamma^{A}\varphi_{+(AB)}=\tfrac{1}{2}\gamma^{A}(\tilde{D}\psi)_{+AB}\,.$
(180)

So in the end there are three unknowns left over, as expected. Following from
(180), we also have

$\eta^{AB}\varphi_{+(AB)}=-\tfrac{1}{2}\gamma^{AB}(\tilde{D}\psi)_{+AB}\,,\quad\gamma^{B}\varphi_{+AB}=\gamma^{B}(\tilde{D}\psi)_{+BA}\,.$
(181)

Note that the three curvature components not used in the above are
$\gamma^{a}r_{+Aa}$ which can be evaluated as

$\gamma^{a}r_{+Aa}=\gamma_{A}\gamma^{B}(\tfrac{1}{3}\gamma^{a}(\tilde{D}\psi)_{+Ba}-\tfrac{1}{2}\varphi_{-B})\,,$
(182)

and so do not help to solve for the undetermined components of
$\varphi_{+A\mu}$.

##  Appendix B Bianchi identities

###  B.1 Bosonic Bianchi identities

In this appendix, we work out the bosonic Bianchi identities for an MNC
geometry obeying the following conventional constraints:

$\displaystyle\mathcal{T}^{A}$
$\displaystyle=\mathrm{d}\tau^{A}+\omega^{A}{}_{B}\tau^{B}-b\tau^{A}=\frac{1}{2}\,e^{a}e^{b}\,T_{ab}{}^{A}+e^{a}\tau_{B}\,T_{a}{}^{\\{AB\\}}\,,$
(183) $\displaystyle\mathcal{T}^{a}$
$\displaystyle=\mathrm{d}e^{a}+\omega^{a}{}_{b}e^{b}+\frac{1}{2}\,be^{a}-\omega_{A}{}^{a}\tau^{A}=0\,,$
(184) $\displaystyle F_{4}$
$\displaystyle=\mathrm{d}c_{3}+\tfrac{1}{2}\,\epsilon_{ABC}\omega^{A}{}_{a}e^{a}\tau^{B}\tau^{C}=\frac{1}{4!}\,e^{a}e^{b}e^{c}e^{d}\,f_{abcd}+\frac{1}{3!}\,\tau^{A}e^{a}e^{b}e^{c}\,f_{Aabc}\,,$
(185)

where in particular we do not assume that the right-hand sides vanish. For
convenience we have adopted differential form notation, with wedge products
implicit.

#### Algebraic BIs

These constraints can be easily seen to imply the following algebraic Bianchi
identities:

$\displaystyle R^{A}{}_{B}\tau^{B}-R\,\tau^{A}$
$\displaystyle=\frac{1}{2}\,e^{a}e^{b}\,DT_{ab}{}^{A}+e^{a}\tau_{B}\,DT_{a}{}^{\\{AB\\}}-e^{a}\mathcal{T}_{B}T_{a}{}^{\\{AB\\}}\,,$
(186) $\displaystyle
R^{a}{}_{b}e^{b}+\frac{1}{2}\,R\,e^{a}-R_{A}{}^{a}\tau^{A}$
$\displaystyle=0\,,$ (187)
$\displaystyle\frac{1}{2}\,\epsilon_{ABC}R^{A}{}_{a}e^{a}\tau^{B}\tau^{C}$
$\displaystyle=\frac{1}{4!}\,e^{a}e^{b}e^{c}e^{d}\,Df_{abcd}$
$\displaystyle\quad+\frac{1}{3!}\,\tau^{A}e^{a}e^{b}e^{c}\,Df_{Aabc}+\frac{1}{3!}\,\mathcal{T}^{A}e^{a}e^{b}e^{c}\,f_{Aabc}\,.$
(188)

Note that the curvatures here contain explicit torsion contributions that are
in principle not straightforward to figure out; see (72) for an example where
we worked out the explicit contributions. Since the starting point was
covariant it is guaranteed that everything re-collects into covariant
expressions appropriately. Expanding the implicit differential forms in terms
of $\tau^{A}$ and $e^{a}$, we read component forms of the Bianchi identities
as follows.

Completely transversal:

$\displaystyle 0$
$\displaystyle=3\,D_{[a}T_{bc]}{}^{A}-3\,T_{[abB}T_{c]}{}^{\\{AB\\}}\,,$ (189)
$\displaystyle 3\,R_{[ab}{}^{d}{}_{c]}+\frac{3}{2}\,R_{[ab}\delta^{d}{}_{c]}$
$\displaystyle=0\,,$ (190) $\displaystyle 0$
$\displaystyle=5\,D_{[a}f_{bcde]}+10\,T_{[ab}{}^{A}f_{Acde]}\,.$ (191)

One longitudinal:

$\displaystyle R_{ab}{}^{A}{}_{B}-R_{ab}\delta^{A}{}_{B}$
$\displaystyle=D_{B}T_{ab}{}^{A}+2\,D_{[a}T_{b]}{}^{\\{AB\\}}+2\,T_{[a\\{BC\\}}T_{b]}{}^{\\{AC\\}}\,,$
(192) $\displaystyle
2\,R_{A[a}{}^{b}{}_{c]}+R_{A[a}\delta^{b}{}_{c]}-R_{acA}{}^{b}$
$\displaystyle=0\,,$ (193) $\displaystyle 0$
$\displaystyle=D_{A}f_{abcd}+4\,D_{[a}f_{Abcd]}-4\,T_{[a\\{AB\\}}f^{B}{}_{bcd]}\,.$
(194)

Two longitudinal:

$\displaystyle 2\,R_{a[A}{}^{B}{}_{C]}-2\,R_{a[A}\delta^{B}{}_{C]}$
$\displaystyle=2\,\eta_{D[A}D_{C]}T_{a}{}^{\\{BD\\}}\,,$ (195) $\displaystyle
R_{AB}{}^{a}{}_{b}+\frac{1}{2}\,R_{AB}\delta^{a}{}_{b}-2\,R_{b[AB]}{}^{a}$
$\displaystyle=0\,,$ (196) $\displaystyle 3\,R_{[ab}{}^{A}{}_{c]}$
$\displaystyle=-\epsilon^{ABC}D_{B}f_{Cabc}\,.$ (197)

Three longitudinal:

$\displaystyle 3\,R_{[AB}{}^{C}{}_{D]}-3\,R_{[AB}\delta^{C}{}_{D]}$
$\displaystyle=0\,,$ (198) $\displaystyle 3\,R_{[ABC]}{}^{a}$
$\displaystyle=0\,,$ (199) $\displaystyle R_{A[a}{}^{A}{}_{b]}$
$\displaystyle=0\,.$ (200)

These can be further simplified into irreps by contracting indices and
combining equations appropriately. This gives rise to a number of useful
identities

$\displaystyle R_{ab}$ $\displaystyle=-\frac{1}{3}\,D_{A}T_{ab}{}^{A}\,,$
(201) $\displaystyle\epsilon^{A}{}_{BC}R_{ab}{}^{BC}$
$\displaystyle=-\epsilon^{AB}{}_{C}\big{(}D_{B}T_{ab}{}^{C}+2\,T_{[a\\{BD\\}}T_{b]}{}^{\\{CD\\}}\big{)}\,,$
(202) $\displaystyle R_{aABC}+2\,\eta_{A[B}R_{aC]}$
$\displaystyle=2\,\delta_{[B}{}^{D}D_{C]}T_{a\\{AD\\}}\,.$ (203)

#### Differential BIs

Given the definitions of the curvature two-forms we can also derive associated
differential Bianchi identities. These are given by:

$\displaystyle D(T)R=0\,,$ $\displaystyle D(T)R^{AB}=0\,,$ $\displaystyle
D(T)R^{ab}=0\,,$ $\displaystyle D(T)R^{Aa}=0\,.$ (204)

The covariant derivative used here contains connections according to the index
structure. The expression of the boost connection is more complicated
$D(T)R^{Aa}=\mathrm{d}R^{Aa}+\omega^{A}{}_{B}R^{Bb}+\tfrac{3}{2}\,bR^{Aa}+\omega^{a}{}_{b}R^{Ab}-\omega^{Ba}R^{A}{}_{B}-\omega^{Ab}R^{a}{}_{b}-\tfrac{3}{2}\,\omega^{Aa}R+\Delta(T)$.
We have also absorbed potential torsion terms $\Delta(T)$ in the definition of
$D(T)$ without being explicit, since we effectively only need the case where
torsion is set to zero. As above, we can project the Bianchi identities along
directions longitudinal and transverse to the foliation. In practice, we only
use

$\displaystyle
D_{a}\breve{R}_{AB}{}^{bc}+2\,D_{[A}\breve{R}_{B]a}{}^{bc}=0\,.$ (205)

###  B.2 Fermionic Bianchi identities

We summarize the conventional constraints for the superconformal gauge fields,
in form notation, as:

$\displaystyle r_{-}$
$\displaystyle=\tfrac{1}{2}\,e^{a}e^{b}\,\breve{r}_{-ab}+\tau^{A}e^{a}\,\breve{r}_{-Aa}\,,$
(206) $\displaystyle r_{+}$
$\displaystyle=\tfrac{1}{2}\,e^{a}e^{b}\,\breve{r}_{+ab}+\tfrac{1}{3}\,\tau^{A}e^{a}\,\gamma_{A}\breve{r}_{+a}\,,$
(207)

where $r_{\pm}$ are defined in (80). These equations are invariant under the
following shift of the superconformal gauge field:

$\displaystyle\varphi_{+A\mu}\to\varphi_{+A\mu}+\tau^{B}{}_{\mu}\tilde{f}_{+(AB)}\,,$
(208)

where $\tilde{f}_{+(AB)}$ is defined to manifestly satisfy
$\gamma^{B}\tilde{f}_{+(AB)}=0$ and thus also $\eta^{AB}\tilde{f}_{+(AB)}=0$.
This shift ambiguity captures the fact that we are three conventional
constraints short of the number of connection components. This is in complete
analogy to the bosonic shift ambiguity (3.1). The superconformal curvature
two-forms $S_{-\mu\nu}$ and $S_{+A\mu\nu}$, defined in (86), naturally
transform under this fermionic shift as $S_{+A}\to
S_{+A}-\tau^{B}\,D\tilde{f}_{+(AB)}$. This means that not all components of
the superconformal curvature components $S_{+A\mu\nu}$ are unambiguous under
the fermionic shift. In order to further analyse these, we decompose the 48
components $S_{+ABa}$, each carrying sixteen fermionic components themselves,
into the irreducible representations $S_{+[AB]a}$, $S_{+\\{AB\\}a}$ and
$\eta^{AB}S_{+ABa}$. The gamma-tracelessness condition $\gamma^{A}S_{+ABa}=0$
turns into the statement
$\gamma^{B}S_{+[AB]a}=\gamma^{B}S_{+\\{AB\\}a}+\tfrac{1}{3}\gamma_{A}\eta^{BC}S_{+BCa}$
which effectively subtracts $\mathbf{24}$ components. Note, however, that as a
consequence we also have $\gamma^{AB}S_{+ABa}=\eta^{AB}S_{+ABa}$—meaning we
should read the gamma traceless condition as $\mathbf{16}\oplus\mathbf{8}$
conditions. We subtract the $\mathbf{16}$ from the $S_{+\\{AB\\}a}$ meaning we
count it effectively as $\mathbf{24}$ components and use the $\mathbf{8}$ to
read the trace as a dependent expression. This is useful because the remaining
components $S_{+\\{AB\\}a}$ are then the ones that are ambiguously defined
with respect to the shift $\tilde{f}_{+(AB)}$. A similar analysis can be done
for the completely longitudinal components $S_{+ABC}$. In summary, we can
separate the $\mathbf{110}$ components into

$\displaystyle\mathrm{well-defined}~{}(\mathbf{80}):$ $\displaystyle
S_{+Aab}~{}(\mathbf{56})\,,$ $\displaystyle S_{+[AB]a}~{}(\mathbf{24})\,,$
$\displaystyle\epsilon^{ABC}S_{+ABC}~{}(\mathbf{1})\,,$
$\displaystyle\mathrm{ill-defined}~{}(\mathbf{29}):$ $\displaystyle
S_{+(AB)C}~{}(\mathbf{5})\,,$ $\displaystyle
S_{+\\{AB\\}a}~{}(\mathbf{24})\,.$ (209)

#### Algebraic BIs

The algebraic Bianchi identities are:

$\displaystyle-\tau^{A}\gamma_{A}S_{-}$
$\displaystyle=\tfrac{1}{2}\,e^{a}e^{b}\,\widehat{D}\breve{r}_{-ab}+\tau^{A}e^{a}\,\widehat{D}\breve{r}_{-Aa}+\mathcal{T}^{A}e^{a}\breve{r}_{-Aa}\,,$
(210a) $\displaystyle\tfrac{1}{2}\,e^{a}\gamma_{a}S_{-}-\tau^{A}S_{+A}$
$\displaystyle=\tfrac{1}{2}\,e^{a}e^{b}\,\widehat{D}\breve{r}_{+ab}+\tfrac{1}{3}\,\tau^{A}e^{a}\,\gamma_{A}\widehat{D}\breve{r}_{+a}+\tfrac{1}{3}\,\mathcal{T}^{A}e^{a}\,\gamma_{A}\breve{r}_{+a}$
$\displaystyle\quad{-\tfrac{1}{12}\,\tau^{A}\boldsymbol{\lambda}\gamma_{A}r_{+}-\tfrac{1}{8}\,e^{a}\boldsymbol{\lambda}\gamma_{a}r_{-}}\,,$
(210b)

where we’ve omitted $\mathcal{O}(\lambda^{2})$ terms for simplicity. The
covariant derivatives are defined as
$\widehat{D}r_{-}=Dr_{-}-\mathcal{T}^{A}\gamma_{A}\varphi_{-}$ and
$\widehat{D}r_{+}=Dr_{+}-\mathcal{T}^{A}\varphi_{+A}$. Projecting this gives
rise to the following constraints in terms of components.

Completely transversal:

$\displaystyle 0$
$\displaystyle=3\,\widehat{D}_{[a}\breve{r}_{-bc]}+3\,\breve{r}_{-A[a}T_{bc]}{}^{A}\,,$
(211a) $\displaystyle 3\,\gamma_{[a}S_{-bc]}$
$\displaystyle=6\,\widehat{D}_{[a}\breve{r}_{+bc]}+2\,T_{[ab}{}^{A}\gamma_{A}\breve{r}_{+c]}{-\tfrac{3}{8}\boldsymbol{\lambda}\gamma_{[a}\breve{r}_{-bc]}}\,.$
(211b)

One longitudinal:

$\displaystyle S_{-ab}$
$\displaystyle=-\tfrac{1}{3}\,\gamma_{A}\big{(}\widehat{D}^{A}\breve{r}_{-ab}-2\,T_{[a}{}^{\\{AB\\}}\breve{r}_{-Bb]}\big{)}\,,$
(212a) $\displaystyle S_{+Aab}+\gamma_{[a}S_{-Ab]}$
$\displaystyle=-\widehat{D}_{A}\breve{r}_{+ab}+\tfrac{2}{3}\,\gamma_{A}\widehat{D}_{[a}\breve{r}_{+b]}$
$\displaystyle\quad+\tfrac{2}{3}\,T_{[a\\{AB\\}}\gamma^{B}\breve{r}_{+b]}{-\tfrac{1}{24}\,\gamma_{A}\boldsymbol{\lambda}\breve{r}_{+ab}}\,.$
(212b)

Two longitudinal:

$\displaystyle 2\,\gamma_{[A}S_{-B]b}$
$\displaystyle=-2\,\widehat{D}_{[A}\breve{r}_{-B]b}\,,$ (213a) $\displaystyle
2\,S_{+[AB]a}-\tfrac{1}{2}\,\gamma_{a}S_{-AB}$
$\displaystyle=\tfrac{2}{3}\,\gamma_{[A}\widehat{D}_{B]}\breve{r}_{+a}{-\tfrac{1}{18}\,\gamma_{AB}\boldsymbol{\lambda}\breve{r}_{+a}}\,.$
(213b)

Three longitudinal:

$\displaystyle 3\,\gamma_{[A}S_{-BC]}$ $\displaystyle=0\,,$ (214a)
$\displaystyle S_{+[ABC]}$ $\displaystyle=0\,.$ (214b)

All the components appearing in these expressions are unambiguously defined
with respect to both the bosonic and the fermionic shift ambiguities. Upon
imposing level two constraints
$\breve{r}_{-ab}=0=\breve{r}_{+ab}=\breve{r}_{-Aa}$ we can further simplify
these expressions to find

$\displaystyle S_{-ab}=0\,,$ $\displaystyle S_{-Aa}=0\,,$
$\displaystyle\gamma^{AB}S_{-AB}=0\,,$ $\displaystyle S_{+Aab}=0\,,$
$\displaystyle S_{+[ABC]}=0\,,$ $\displaystyle 2\,D_{[a}\breve{r}_{+b]}=0\,,$
(215)

which further implies

$\displaystyle\gamma^{AB}S_{+ABa}+\tfrac{2}{3}\,\gamma^{A}D_{A}\breve{r}_{+a}-{\tfrac{1}{6}\,\boldsymbol{\lambda}\breve{r}_{+a}=0}\,.$
(216)

#### Differential BIs

The superconformal curvatures (86) also satisfy differential Bianchi
identities. These follow simply from the definition and can be written as
three-form constraints

$\displaystyle DS_{-}=0\,,$ $\displaystyle DS_{+A}=0\,,$ (217)

where the definition of the derivatives includes second order bosonic
curvatures. These can be projected to give rise to the following constraints
on the components of the superconformal curvatures that are not set to zero by
the algebraic Bianchi identities (B.2):

$\displaystyle D_{a}S_{-AB}=0\,,$ $\displaystyle D_{[a}S_{+[AB]b]}=0\,.$ (218)

##  Appendix C The tower of constraints for maximal supersymmetry

In this appendix, we discuss the derivation of the tower of constraints
present when we impose the bosonic constraints (37) compatible with maximal
supersymmetry. The full set of constraints must be invariant under all
symmetries of the theory. This means in particular that we have to vary (37)
under supersymmetry (and superconformal transformations) and require that the
variation itself vanishes, using both (37) and new fermionic constraints. Then
we have to repeat this process with the new constraints that appear, until we
have determined all possible constraints.

We start with the supersymmetry transformations of the constraints (37). Let
us refer to these as the Level 1 constraints. We find:

$\displaystyle\delta_{Q}\hat{T}_{ab}{}^{A}$
$\displaystyle\approx\bar{\epsilon}_{-}\gamma^{A}\breve{r}_{-ab}\,,$ (219a)
$\displaystyle\delta_{Q}\hat{T}_{a}{}^{\\{AB\\}}$
$\displaystyle\approx-\bar{\epsilon}_{-}\gamma^{\\{A}\breve{r}_{-}{}^{B\\}}{}_{a}\,,$
(219b) $\displaystyle\delta_{Q}\hat{f}_{abcd}$ $\displaystyle\approx
6\,\bar{\epsilon}_{-}\gamma_{[ab}\breve{r}_{-cd]}\,,$ (219c)
$\displaystyle\delta_{Q}\hat{f}_{Aabc}$
$\displaystyle\approx+3\,\bar{\epsilon}_{+}\gamma_{A[a}\breve{r}_{-bc]}-3\,\bar{\epsilon}_{-}\gamma_{A[a}\breve{r}_{+bc]}-3\,\bar{\epsilon}_{-}\gamma_{[ab}\breve{r}_{-c]A}\,.$
(219d)

Here we have written $\approx$ to mean that these equations are true up to the
imposition of the constraints (37). For instance, what we explicitly find is
that:

$\begin{split}\delta_{Q}\hat{T}_{ab}{}^{A}&=\bar{\epsilon}_{-}\gamma^{A}\breve{r}_{-ab}+2\,\hat{T}_{c[a}{}^{A}(\bar{\epsilon}_{+}\gamma^{c}\psi_{-b]}+\bar{\epsilon}_{-}\gamma^{c}\psi_{+b]})+2\bar{\epsilon}_{-}\gamma_{B}\psi_{-[a}\hat{T}_{b]}{}^{\\{AB\\}}\\\
&\qquad-\tfrac{1}{12}(\bar{\epsilon}_{-}\gamma^{A}\gamma_{[a}\boldsymbol{\hat{f}}^{(-)}\psi_{+b]}+\bar{\psi}_{-[a}\gamma^{A}\gamma_{b]}\boldsymbol{\hat{f}}^{(-)}\epsilon_{+})\approx\bar{\epsilon}_{-}\gamma^{A}\breve{r}_{-ab}\,.\end{split}$
(220)

We remark that this shows again that $\hat{f}^{(-)}{}_{abcd}$, which did not
appear in the original divergent terms in the supersymmetry transformation,
should also be included as a constraint in (37).

We thus find from the variations (219) that we should impose the following
Level 2 constraints as superpartners of the bosonic constraints (37):

$\breve{r}_{-ab}=0\,,\quad\breve{r}_{-Aa}=0\,,\quad\breve{r}_{+ab}=0\,.$ (221)

Note that not all curvatures appearing in (83) are set to zero. Next, we must
vary (221). On general grounds we expect to find the well-defined bosonic
curvatures $\breve{R}$ appearing as new constraints.

A tempting shortcut at this point is to resort to a linearised calculation.
This allows the curvature combinations that must appear in the variation to be
easily identified from their leading terms, using covariance to conclude that
the full variation must follow.

This way one gets: $\displaystyle\delta_{Q}\breve{r}_{-ab}$
$\displaystyle\approx\tfrac{1}{4}\breve{R}_{ab}{}^{cd}\gamma_{cd}\epsilon_{-}\,,\quad\delta_{Q}\breve{r}_{-Aa}\approx\frac{1}{4}\breve{R}_{Ba}{}^{bc}(\delta_{A}^{B}-\tfrac{1}{3}\gamma_{A}\gamma^{B})\gamma_{bc}\epsilon_{-}\,,$
(222a) $\displaystyle\delta_{Q}\breve{r}_{+ab}$
$\displaystyle\approx\tfrac{1}{4}\breve{R}_{ab}{}^{cd}\gamma_{cd}\epsilon_{+}+\left(-\breve{R}_{A[ab]c}\gamma^{c}-\tfrac{1}{12}\breve{R}_{A[a}{}^{cd}\gamma_{b]}\gamma_{cd}\right)\gamma^{A}\epsilon_{-}-\tfrac{1}{4}D_{[a}{\boldsymbol{\lambda}}\gamma_{b]}\epsilon_{-}\,,$
(222b)

after using the Bianchi identities of appendix B.1.

The issue with the linearised calculation is that it may miss terms. For
instance, suppose we started only with the knowledge that $\hat{T}_{ab}{}^{A}$
was a constraint. Then a linearised calculation would show the result (219)
from which we would conclude that $\hat{r}_{-ab}$ appears as a new constraint.
However, the explicit variation (220) also contains, at the non-linear level,
other bosonic constraints, namely $\hat{T}_{a}{}^{\\{AB\\}}$ and
$\hat{f}^{(-)}_{abcd}$. This means that in principle we could find
$\breve{r}_{+a}$ appearing in the variation of the other gravitino curvatures,
which would imply that we have to set this curvature to zero too and add it to
the level 2 fermionic constraints (221).

An involved calculation of the non-linear variation101010Albeit omitting
higher order terms involving $\lambda_{abcd}$. of $\breve{r}_{-ab}$,
$\breve{r}_{-Aa}$ and $\breve{r}_{+ab}$, taking into account the supersymmetry
transformation of the bosonic connections and the higher order supercovariant
completions of the bosonic curvatures, shows that in fact their variation
takes exactly the form (222), modulo imposing the level 1 and 2 constraints
(37) and (221) – up to terms involving the gamma trace
$\gamma^{a}\breve{r}_{+a}$, which appears in term bilinear in fermions. We
therefore also require

$\gamma^{a}\breve{r}_{+a}\approx 0\,,$ (223)

which is in fact one of the super-Poisson equations. Let us discuss the
variation of this equation. A linearised calculation gives

$\displaystyle\delta_{Q}\breve{r}_{+a}\approx\tfrac{1}{4}\,D_{a}\boldsymbol{\lambda}\epsilon_{+}+\tfrac{1}{4}\,\gamma^{AB}\breve{R}_{AB}{}^{bc}\big{(}\delta_{ab}\gamma_{c}-\tfrac{1}{8}\gamma_{a}\gamma_{bc}\big{)}\epsilon_{-}+\tfrac{3}{2}\,\breve{R}_{A(a}{}^{A}{}_{b)}\gamma^{b}\epsilon_{-}\,,$
(224)

from which we infer

$\delta_{Q}(\gamma^{a}\breve{r}_{+a})\approx\tfrac{1}{4}\,\gamma^{a}D_{a}\boldsymbol{\lambda}\epsilon_{+}+\tfrac{3}{2}\breve{R}_{Aa}{}^{Aa}\epsilon_{-}\,.$
(225)

We note in passing that the bosonic Poisson equation is
$\breve{R}_{Aa}{}^{Aa}-\tfrac{1}{4!}\lambda_{abcd}\lambda^{abcd}=0$, and so
this must be what appears in the $\epsilon_{-}$ term in the non-linear
completion of (225). The remaining term in this variation implies that
$D_{[a}\lambda_{bcde]}=0$ (or equivalently using anti-self-duality
$D_{a}\lambda^{abcd}=0$). This corresponds, modulo the level 1 constraints, to
the equation of motion (51) of the three-form. We also note that under
Stuckelberg transformations,
$\delta_{S}\breve{r}_{+a}=-\tfrac{1}{4}\,\boldsymbol{\lambda}\gamma_{a}\eta_{-}$,
so that $\delta_{S}\gamma^{a}\breve{r}_{+a}=0$.

Returning to (222), we then pick out the following Level 3 constraints:

$\breve{R}_{ab}{}^{cd}=0\,,\quad\breve{R}_{Aa}{}^{bc}=0\,,\quad
D_{a}\lambda_{bcde}=0\,,$ (226)

the last of which follows from combining $D_{[a}\lambda_{bcde]}=0$ with
$D_{[a}\lambda_{b]cde}=0$, coming from equation (222). Next, a linearised
variation of the level 3 constraints then produces:

$\displaystyle\delta_{Q}\breve{R}_{ab}{}^{cd}\approx
0\,,\quad\delta_{Q}\breve{R}_{Aa}{}^{bc}\approx-\bar{\epsilon}_{-}\gamma_{A}\gamma^{[b}D^{c]}\breve{r}_{+a}\,,\quad\delta_{Q}D_{a}\lambda_{b_{1}\cdots
b_{4}}\approx\frac{1}{4}\,\bar{\epsilon}_{-}\gamma^{c}\gamma_{b_{1}\cdots
b_{4}}D_{a}\breve{r}_{+c}\,,$ (227a)

on using the Bianchi identities $S_{-ab}\approx 0\approx S_{-Aa}\approx
S_{+Aab}\approx D_{[a}\breve{r}_{+b]}$. This leads us to the solitary new
Level 4 constraint:

$D_{a}\breve{r}_{+b}\approx 0\,.$ (228)

The linearised variation of this, using (224), leads to the following new
Level 5 constraint:

$D_{a}\breve{R}_{A(b}{}^{A}{}_{c)}=0\,.$ (229)

(We also appear to find $D_{a}\breve{R}_{AB}{}^{bc}=0$, however, this is zero
identically due to the differential Bianchi identity (205).) Varying (229) at
the linearised level, we find

$\displaystyle\delta_{Q}\breve{R}_{A(a}{}^{A}{}_{b)}$
$\displaystyle\approx\bar{\epsilon}_{-}\gamma^{AB}\gamma_{(a}S_{+ABb)}+\frac{1}{3}\,\bar{\epsilon}_{-}\gamma^{A}\gamma_{(a}D_{A}\breve{r}_{+b)}\,.$
(230)

after using Bianchi identities. We can then infer that

$\delta_{Q}D_{a}\breve{R}_{A(b}{}^{A}{}_{c)}\approx\tfrac{3}{2}\bar{\epsilon}_{-}D_{c}\gamma_{(a}\gamma^{AB}S_{+ABb)}\,,$
(231)

using the Bianchi identity (216). This suggests a possible further constraint:

$D_{a}\gamma^{AB}S_{+ABb}=0\,.$ (232)

However, this is identically satisfied at the linearised level by using the
derivative of the same Bianchi identity (216) together with the level 3 and 4
constraints. We also have that at the non-linearised level that if we hit
(216) with $D_{a}$, write $D_{a}D_{A}=[D_{a},D_{A}]+D_{A}D_{a}$, rewrite the
commutator in terms of bosonic curvatures and apply constraints, we find:

$D_{a}\gamma^{AB}S_{+ABb}+\tfrac{2}{3}R_{aA}\gamma^{A}\breve{r}_{+b}\approx
0\,.$ (233)

Here the curvature components, $R_{Aa}$ can be seen to act as a gauge field
for the special conformal transformations acting as shifts of $b_{A}$, as
noted in footnote 8, such that the whole expression (233) is independent of
shifts of the $b_{A}$ and thus a well defined quantity. We expect that working
at non-linear order the ‘full’ definition of the covariant derivative $D_{a}$
appearing in (232) absorbs these $R_{Aa}$ terms: in that case (232) is
automatically satisfied as a consequence of the Bianchi identity (216) and
previous constraints. At the linearised level, where we can simply ignore the
$R_{aA}$ term in (233), we can conclude however that the tower of constraints
terminates at level 5.

Let us summarise our findings. The variation of the constraints (37) required
to have a maximally supersymmetric non-relativistic 11-dimensional
supergravity leads to a complicated tower of constraints, given in (221),
(226), (228) and (229). The variation of (229) leads to a possible constraint
(232), however at least at linear level this appears to be zero by Bianchi. We
are fairly confident that this signals that we have a finite set of
constraints.

The presence of a constraint involving a covariant derivative of a curvature
(therefore a third derivative bosonic equation) may seem unusual. Instead of
imposing $D_{e}\lambda_{abcd}$ at level 3, we could directly require the
stronger constraint $\lambda_{abcd}$. This then directly leads to the
following set of constraints, which amount to lifting the covariant
derivatives at level 4 and 5:

$\lambda_{abcd}=0=\breve{r}_{+a}=\breve{R}_{A(b}{}^{A}{}_{c)}=\breve{R}_{AB}{}^{bc}\,.$
(234)

Using the Bianchi identity (216) we infer that $\gamma^{AB}S_{+ABa}\approx 0$
directly and hence the variation of $\breve{R}_{A(a}{}^{A}{}_{b)}$ vanishes.
However, we also have to compute the variation of $\breve{R}_{AB}{}^{bc}$.
This gives:

$\displaystyle\delta\breve{R}_{AB}{}^{bc}$
$\displaystyle\approx+\tfrac{1}{2}\,\bar{\epsilon}_{-}\gamma^{bc}S_{-AB}\approx\tfrac{1}{4}\bar{\epsilon}_{-}\gamma^{bc}\gamma^{a}S_{+[AB]a}\,,$
(235)

using a Bianchi identity (213b) at the second step. Therefore we need to add
$S_{-AB}=0$ (or equivalently $\gamma^{a}S_{+[AB]a}=0$) to the set (234). We
would then need to vary this additional fermionic constraint. On general
grounds it should lead to a derivative $D\breve{R}$ of a bosonic curvature.
However, all such curvatures have already been set to zero, so any such
constraint will automatically be satisifed. The possibility (234) is therefore
a valid set of sufficient constraints.

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