Non-Relativistic M2-Branes and the

AdS/CFT Correspondence

Neil Lambert111E-mail address<EMAIL_ADDRESS>and Joseph
Smith222E-mail address<EMAIL_ADDRESS>
Department of Mathematics

King’s College London

The Strand

WC2R 2LS, UK

Abstract

A non-relativistic limit of the AdS/CFT correspondence is studied in the
context of M2-branes. On the field theory side this corresponds to a near-BPS
limit of ABJM that localises onto solutions of Hitchin’s equations. It is
shown that the symmetries of the theory include an infinite-dimensional
enhancement of the spatial symmetry algebra corresponding to time-dependent
holomorphic transformations. Taking the limit of the gravitational dual splits
the geometry into three ’large’ directions and eight ’small’ directions, which
for the near-horizon limit of the M2-brane metric has the effect of reducing
the $AdS_{4}$ factor to an $AdS_{2}$ factor. Evidence is presented that the
duality is maintained after the limit.

###### Contents

  1. 1 Introduction
  2. 2 Non-Relativistic Limits of Three-Dimensional Chern-Simons-Matter Theories
  3. 3 Near-BPS Limit of ABJM
    1. 3.1 Scaling Limit
    2. 3.2 Field Theory Symmetries
    3. 3.3 Conserved Currents
  4. 4 Eleven-Dimensional Membrane Newton-Cartan Gravity and the M2-Brane
    1. 4.1 A Brief Review of Non-Relativistic Eleven-Dimensional Supergravity
    2. 4.2 Non-Relativistic Limit of the M2-brane
    3. 4.3 Gravitational Symmetries
      1. 4.3.1 Isometries of the Near-Horizon Geometry
      2. 4.3.2 Form-Field Symmetries
      3. 4.3.3 Boundary Structure and Asymptotic Symmetries
  5. 5 Conclusion
  6. A Fermions
    1. A.1 Fermion Action and Symmetries
    2. A.2 Fermionic Contributions to Conserved Currents
  7. B Equations of Motion
  8. C Non-Relativistic Orbifold Geometry

##  1 Introduction

There has been a recent Renaissance in the study of non-Lorentzian physics and
its applications to relativistic theories. A topic that has garnered
considerable attention is non-relativistic limits of string theory. While this
area has been studied for a long time [1, 2], its development has recently
been spurred by a deeper understanding of the non-relativistic worldsheet
theory and the spacetime geometry to which it couples [3, 4, 5]. The key point
(first put forward in [6]) is that strings naturally couple to non-Lorentzian
manifolds with two distinguished directions, known as String Newton-Cartan
(SNC) geometries, in contrast to the single direction in a standard Newton-
Cartan geometry. In practice this means that the relativistic metric is split
into two pieces, $\tau_{\mu\nu}$ and $h^{\mu\nu}$, where $\tau_{\mu\nu}$ has
two non-zero eigenvalues (one positive and one negative) and $h^{\mu\nu}$ has
eight positive eigenvalues333Note that we are assuming we work in the critical
dimension $D=10$.. As the low energy dynamics of a relativistic string theory
are governed by a supergravity theory a corresponding non-relativistic limit
of this can be taken after decomposing the metric into the corresponding SNC
structures [7, 8]. As in the relativistic case, we can view non-relativistic
string theory as the UV completion of the corresponding non-relativistic
supergravity theory. Since string theory contains more extended objects than
just the fundamental string, it is natural to extend this idea to more general
$p$-brane geometries [9]444Note that limits of $p$-brane geometries can also
be studied in the framework of SNC geometries [10].. In [11] this was applied
to the case of M2-branes in eleven-dimensional supergravity, where the non-
relativistic limit was taken and found to give a gravitational theory for
Membrane Newton-Cartan (MNC) geometries with three distinguished directions.
One may hope that, as for non-relativistic limits of the ten-dimensional
supergravity theories, there is a well-defined non-relativistic M-theory that
serves as this theory’s UV completion.

Another active area of research is the process of obtaining non-Lorentzian
quantum field theories from their Lorentzian counterparts. This includes
directly taking limits of our coordinates and fields, as well as more exotic
methods such as null reductions and variations thereof (see [12] for a recent
review). Our method of interest will be taking a Galilean $c\to\infty$ limit.
Typically the field theories studied in this way are massive, and there is a
well-defined way of finding a non-relativistic limit. In contrast, much less
emphasis has been placed on finding limits of conformal field theories (CFTs);
indeed, a naive non-relativistic limit of a massless field gives a theory with
trivial dynamics. The question of finding an interesting limit for the ABJM
theory was tackled in [13], where a scaling limit was found that gives a non-
relativistic theory with the same amount of supersymmetry as the parent
theory. The dynamics of the theory are non-trivial and, in the simplest case,
can be identified with motion on the moduli space of Hitchin’s equations [14].
While an interesting field theory, there are still open questions as to its
structure. For example, the symmetries of the theory after the limit are
unknown. Also, preservation of supersymmetry requires an ad-hoc field
redefinition to be performed before the limit is taken; it would be beneficial
to have an argument as to what this shift signifies in order to understand the
limit’s physical interpretation.

Given that non-relativistic limits can be taken of both field theories and
gravity, it is interesting to ask whether these ideas can be applied to the
AdS/CFT correspondence: in other words, can non-relativistic limits be taken
on both sides such that the duality is maintained? This has previously been
studied in the context of the duality between four-dimensional $\mathcal{N}=4$
super Yang-Mills and type IIB string theory on $AdS_{5}\times S^{5}$ by taking
a decoupling limit of the field theory that isolates operators near a BPS
bound. These are known as Spin Matrix Theory limits [15, 16, 17]; since the
quantum numbers of the surviving operators satisfy certain relations the
holographic dictionary can be used to translate these into a corresponding
non-relativistic limit of the gravitational background. Our aim is to approach
this question from a different perspective for the duality between the ABJM
theory and M-theory on $AdS_{4}\times S^{7}/\mathbb{Z}_{k}$ [18]. We have
already discussed the non-trivial scaling limit of ABJM found in [13]. As ABJM
describes the low-energy dynamics of a stack of M2-branes on a
$\mathbb{C}^{4}/\mathbb{Z}_{k}$ background, we can reinterpret the scaling as
a non-relativistic limit of the M2-brane spacetime. This turns out to be
exactly of the form required by the MNC limit of eleven-dimensional
supergravity in [11]. As the limit leads to well-defined theories on both
sides there is the potential that the duality is retained. We claim that this
is indeed the case, and our goal in this work is to put forward evidence in
favour of this conclusion.

This paper is organised as follows. In section 2 we take a naive non-
relativistic limit of a Chern-Simons matter theory, which leads to a theory in
which all symmetries can be given arbitrary time-dependence and we have no
dynamics. In section 3 we apply a similar procedure to the ABJM theory, where
we retain dynamics by ensuring that the equations defining half-BPS solutions
are unchanged by the limit. We analyse the symmetries of the resulting theory
and find that the spatial symmetries are enhanced to the two-dimensional
Euclidean conformal algebra with arbitrary time-dependence. These should be
interpreted as redundancies, as evidenced by the corresponding conserved
charge reducing to a boundary term. The analogous limit of the gravitational
dual is studied in section 4, where the appropriate non-relativistic theory is
the membrane Newton-Cartan limit of eleven-dimensional supergravity found in
[11]. We show that the physical symmetries of the field theory are realised in
the gravitational solution, leading us to propose that the duality between the
two is maintained after taking the non-relativistic limit on both sides. Our
findings are summarised in section 5 and avenues for further work are
discussed. We also include supplementary material as appendices. The analysis
of the fermionic terms in the field is performed in appendix A, and the
equations of motion for the field theory are collected in appendix B. In
appendix C we discuss the non-relativistic limit of orbifold geometries.

##  2 Non-Relativistic Limits of Three-Dimensional Chern-Simons-Matter
Theories

Let us consider for illustration a Bosonic Chern-Simons matter theory with
action of the form

$\displaystyle S=-\frac{1}{c}\tr\int
d^{3}x\sqrt{-g}\left(g^{\mu\nu}D_{\mu}\hat{\mathcal{Z}}^{M}D_{\nu}\hat{\bar{\mathcal{Z}}}_{M}+\hat{V}(\hat{\mathcal{Z}}^{M},\hat{\bar{\mathcal{Z}}}_{M})\right)+S_{CS}\
,$ (1)

where

$\displaystyle S_{CS}=\frac{k}{4\pi}\tr\int
d^{3}x\,\varepsilon^{\mu\nu\lambda}\left(A_{\mu}\partial_{\nu}A_{\lambda}-\frac{2i}{3}A_{\mu}A_{\nu}A_{\lambda}\right)\
,$ (2)

is a Chern-Simons term and $\hat{V}$ a potential. Here $c$ is identified with
the speed of light and we work with the Minkowski metric $g_{\mu\nu}={\rm
diag}(-c^{2},1,1)$. For simplicity we assume the scalars take values in
${\mathbb{C}}^{N}$ and the gauge fields are in the adjoint of $U(N)$ so
$D_{\mu}\hat{\mathcal{Z}}^{M}=\partial_{\mu}\hat{\mathcal{Z}}^{M}-i[A_{\mu},\hat{\mathcal{Z}}^{M}]$.

As written this is a relativistic field theory. However if the potential
contains an explicit mass term

$\displaystyle\hat{V}(\hat{\mathcal{Z}}^{M},\hat{\bar{\mathcal{Z}}}_{M})=m^{2}c^{2}\hat{\mathcal{Z}}^{M}\hat{\bar{\mathcal{Z}}}_{M}+V(\hat{\mathcal{Z}}^{M},\hat{\bar{\mathcal{Z}}}_{M})\
,$ (3)

then we can consider a non-relativistic limit by writing

$\displaystyle\hat{\mathcal{Z}}^{M}=e^{-imc^{2}t}\mathcal{Z}^{M}\ .$ (4)

In the $c\to\infty$ limit the action becomes

$\displaystyle S=\tr\int
dtd^{2}x\left(im\left(\mathcal{Z}^{M}D_{t}\bar{\mathcal{Z}}_{M}-D_{t}\mathcal{Z}^{M}\bar{\mathcal{Z}}_{M}\right)-D_{i}\mathcal{Z}^{M}D_{i}\bar{\mathcal{Z}}_{M}-V(\mathcal{Z}^{M},\bar{\mathcal{Z}}^{M})\right)+S_{CS}\
.$ (5)

The resulting equations of motion become that of a Schrödinger-type theory
coupled to a non-Abelian gauge field.

On the other hand if the action is classically invariant under the scale
transformations

$\displaystyle x^{\mu}$ $\displaystyle\to\lambda x^{\mu}$ $\displaystyle
A_{\mu}$ $\displaystyle\to\lambda^{-1}A_{\mu}$
$\displaystyle\hat{\mathcal{Z}}^{M}$
$\displaystyle\to\lambda^{-\tfrac{1}{2}}\hat{\mathcal{Z}}^{M}\ ,$ (6)

then there is no mass term and we can’t take such a non-relativistic limit.
Instead we would like to consider the following rescaling of the spacetime
metric (in complex coordinates $z=x^{1}+ix^{2}$ and setting $c=1$):

$\displaystyle g_{\mu\nu}=\left(\begin{matrix}-1&0&0\\\
0&0&\frac{\omega^{2}}{2}\\\ 0&\frac{\omega^{2}}{2}&0\end{matrix}\right)\ ,$
(7)

which can also be viewed as a rescaling of the spatial coordinates. Since we
are in a conformal field theory this is the same as rescaling time
$x^{0}\to\omega^{-1}x^{0}$ and leaving space unchanged (as well as a suitable
action on the fields). We are interested in the limit $\omega\to 0$ which, in
terms of $x^{0}=ct$, is conformally equivalent to a non-relativistic limit
$c\to\infty$ (but would also include a rescaling of the scalars fields).

We note that

$\displaystyle\sqrt{-g}g^{\mu\nu}={\omega^{2}}\left(\begin{matrix}-1&0&0\\\
0&0&\frac{2}{\omega^{2}}\\\
0&\frac{2}{\omega^{2}}&0\end{matrix}\right)=\left(\begin{matrix}-{\omega^{2}}&0&0\\\
0&0&2\\\ 0&2&0\end{matrix}\right)\ .$ (8)

Thus the limit $\omega\to 0$ is smooth and the action reduces to

$\displaystyle S=-2\tr\int
d^{3}x\left(D\mathcal{Z}^{M}\bar{D}\bar{\mathcal{Z}}_{M}+D\bar{\mathcal{Z}}^{M}\bar{D}\mathcal{Z}_{M}\right)+S_{CS}\
,$ (9)

where $D=D_{z}$, $\bar{D}=D_{\bar{z}}$ and we trivially identify
${\mathcal{Z}}^{M}=\hat{\mathcal{Z}}^{M}$. Note that the Chern-Simons term is
unaffected by this deformation of the metric but the potential term vanishes
as $\sqrt{-g}\to 0$.

The equations of motion are now

$\displaystyle F_{z\bar{z}}$ $\displaystyle=0$ $\displaystyle F_{0z}$
$\displaystyle=\frac{2\pi}{k}\left(Z^{M}D\bar{Z}^{M}-DZ^{M}\bar{Z}^{M}\right)$
$\displaystyle(D\bar{D}+\bar{D}D)\mathcal{Z}^{M}$ $\displaystyle=0\ .$ (10)

The first equation tells us that the spatial gauge field is pure gauge. For
simplicity let us assume there are no topological subtleties and we can simply
take $A_{z}=A_{\bar{z}}=0$. The equations are now just

$\displaystyle-\partial A_{0}$
$\displaystyle=\frac{2\pi}{k}\left(\mathcal{Z}^{M}\partial\bar{\mathcal{Z}}^{M}-\partial\mathcal{Z}^{M}\bar{\mathcal{Z}}^{M}\right)$
$\displaystyle\partial\bar{\partial}\mathcal{Z}^{M}$ $\displaystyle=0$ (11)

A natural class of solutions consists of setting
$\bar{\partial}\mathcal{Z}^{M}=0$ and hence

$\displaystyle A_{0}=\frac{2\pi}{k}\mathcal{Z}^{M}\bar{\mathcal{Z}}^{M}\ .$
(12)

(There is a similar class where $\partial Z^{M}=0$.) Note that the time
dependence has played no role.

More generally this action is invariant under an infinite dimensional group of
diffeomorphisms and Weyl transformations. However in general these are not
symmetries but redundancies in the description. Rather, for a given metric
$g_{\mu\nu}$, the only transformations that lead to symmetries555We will
ignore global symmetries for the time being. are those for which

$\displaystyle\frac{\partial x^{\lambda}}{\partial
x^{\prime\mu}}\frac{\partial x^{\rho}}{\partial
x^{\prime\nu}}g_{\lambda\rho}(x)=\Omega^{2}g_{\mu\nu}(x^{\prime})\ ,$ (13)

for some $\Omega$. For the standard choice of $g=\eta$ this leads to the
conformal group $SO(2,3)$; this remains true for the rescaled metric (7) when
$\omega$ is non-zero, as a field redefinition will set $\omega=1$. However,
for $\omega=0$ the metric is degenerate and we must be more careful. From the
action we see the quantity of interest is $\sqrt{-g}g^{\mu\nu}$, which must
have the transformation

$\displaystyle\sqrt{-g}g^{\mu\nu}{\det\left(\frac{\partial x}{\partial
x^{\prime}}\right)}\frac{\partial x^{\prime\lambda}}{\partial
x^{\mu}}\frac{\partial x^{\prime\rho}}{\partial x^{\nu}}=\sqrt{-g}\Omega
g^{\lambda\rho}\ ,$ (14)

with

$\displaystyle\sqrt{-g}g^{\mu\nu}=\left(\begin{matrix}0&0&0\\\ 0&0&1\\\
0&1&0\end{matrix}\right)\ .$ (15)

Looking at the timelike components we see that $x^{\prime 0}$ can be any
function of $x^{0}$ but cannot depend on $z$ or $\bar{z}$. The other
components restrict $z^{\prime}$ to holomorphic functions of $z$ but arbitrary
functions of $x^{0}$ with

$\displaystyle\Omega={\det\left(\frac{\partial x}{\partial
x^{\prime}}\right)}\frac{\partial z^{\prime}}{\partial
z}\frac{\partial\bar{z}^{\prime}}{\partial\bar{z}}=\frac{\partial
x^{0}}{\partial x^{\prime 0}}\ .$ (16)

Thus, in contrast to the case with an invertible metric, the symmetry group is
infinite dimensional. Indeed it contains the infinite dimensional two-
dimensional conformal group consisting of holomorphic transformations (with
time-dependent transformations) along with one-dimensional diffeomorphisms of
time. This is consistent with the solutions to the equations of motion that we
found above.

We have seen that the dynamics of the theory are totally unconstrained.
Indeed, there is no notion of time. Thus we have arrived at something we can
think of as a non-relativistic topological gauge theory. Although curious,
without any dynamics this is of limited physical interest. We can perform a
similar scaling but also rescale one of the scalar fields, say
$\hat{\mathcal{Z}}^{1}=\omega^{-1}\mathcal{Z}^{1}$. This has the effect that
the $\mathcal{Z}^{1}$ kinetic term remains non-zero and hence we retain some
dynamics. This limit is more complicated than the one we have just discussed
as it introduces divergent terms but these can be eliminated. We turn our
attention to this construction in the next section for the specific case of
M2-branes. As we shall soon see we still find an infinite number of
symmetries.

##  3 Near-BPS Limit of ABJM

###  3.1 Scaling Limit

Let us review the field theory obtained in [13] using the approach of [19].
Our goal is to find a scaling limit of the ABJM theory that ’zooms in’ on a
class of $\frac{1}{2}$-BPS solutions such that the dynamics of the theory
reduces to the Manton approximation of geodesic motion on the moduli space of
solutions [20]. This is achieved by finding a rescaling of both fields and
coordinates under which the BPS equations are invariant and kinetic terms are
suppressed relative to gradient terms. We will only focus on the Bosonic
fields in the main body of the text, with the Fermions discussed in appendix
A.

The ABJM theory [18] is a 3d $\mathcal{N}=6$ superconformal $U(N)\times U(N)$
Chern-Simons matter theory with two $U(N)$ gauge fields $A^{L/R}$ and four
complex scalar fields $\hat{\mathcal{Z}}^{M}$, where $M\in\\{1,2,3,4\\}$ is
the $SU(4)$ R-symmetry index, in the bifundamental of $U(N)\times U(N)$ as its
Bosonic field content. As in section 2 we will work with Hermitian gauge
fields throughout, so the covariant derivative of the scalar fields is given
by

$\hat{D}_{\mu}\hat{\mathcal{Z}}^{M}=\hat{\partial}_{\mu}\hat{\mathcal{Z}}^{M}-i\hat{A}_{\mu}^{L}\hat{\mathcal{Z}}^{M}+i\hat{\mathcal{Z}}^{M}\hat{A}_{\mu}^{R}\
.$ (17)

The action for these fields is

$\displaystyle\hat{S}_{B}=\tr\int d^{3}\hat{x}\Bigg{[}$
$\displaystyle-\hat{D}_{\mu}\hat{\bar{\mathcal{Z}}}_{M}\hat{D}^{\mu}\hat{\mathcal{Z}}^{M}+\frac{k\varepsilon^{\mu\nu\rho}}{4\pi}\bigg{(}\hat{A}_{\mu}^{L}\hat{\partial}_{\nu}\hat{A}_{\rho}^{L}-\frac{2i}{3}\hat{A}_{\mu}^{L}\hat{A}_{\nu}^{L}\hat{A}_{\rho}^{L}-\hat{A}_{\mu}^{R}\hat{\partial}_{\nu}\hat{A}_{\rho}^{R}$
$\displaystyle+\frac{2i}{3}\hat{A}_{\mu}^{R}\hat{A}_{\nu}^{R}\hat{A}_{\rho}^{R}\bigg{)}-\frac{8\pi^{2}}{3k^{2}}\hat{\bar{\Upsilon}}^{K}_{IJ}\hat{\Upsilon}^{IJ}_{K}\Bigg{]}\
,$ (18)

where we have defined

$\hat{\Upsilon}^{MN}_{P}=[\hat{\mathcal{Z}}^{M},\hat{\mathcal{Z}}^{N};\hat{\bar{\mathcal{Z}}}_{P}]-\frac{1}{2}\delta^{M}_{P}[\hat{\mathcal{Z}}^{Q},\hat{\mathcal{Z}}^{N};\hat{\bar{\mathcal{Z}}}_{Q}]+\frac{1}{2}\delta^{J}_{P}[\hat{\mathcal{Z}}^{Q},\hat{\mathcal{Z}}^{M};\hat{\bar{\mathcal{Z}}}_{Q}]\
,$ (19)

using the notation

$[\hat{\mathcal{Z}}^{M},\hat{\mathcal{Z}}^{N};\hat{\bar{\mathcal{Z}}}_{P}]=\hat{\mathcal{Z}}^{M}\hat{\bar{\mathcal{Z}}}_{P}\hat{\mathcal{Z}}^{N}-\hat{\mathcal{Z}}^{N}\hat{\bar{\mathcal{Z}}}_{P}\hat{\mathcal{Z}}^{M}\
.$ (20)

Field configurations preserve a supercharge if the supersymmetry
transformation of the Fermions vanish for the spinor parameter associated with
that supercharge. In the ABJM theory this corresponds to solving the equations

$0=-\gamma^{\mu}\hat{D}_{\mu}\hat{\mathcal{Z}}^{N}\xi_{MN}+\frac{2\pi}{k}[\hat{\mathcal{Z}}^{P},\hat{\mathcal{Z}}^{Q};\hat{\bar{\mathcal{Z}}}_{M}]\xi_{PQ}+\frac{2\pi}{k}[\hat{\mathcal{Z}}^{N},\hat{\mathcal{Z}}^{P};\hat{\bar{\mathcal{Z}}}_{Q}]\xi_{MN}\
,$ (21)

where $\xi_{MN}$ are a set of spinor parameters that are antisymmetric in the
R-symmetry indices and satisfy the reality condition

$\bar{\xi}^{MN}\equiv(\xi_{MN})^{*}=\frac{1}{2}\epsilon^{MNPQ}\xi_{PQ}$ (22)

in a specific basis of the gamma matrices discussed further in appendix A. As
we are interested in $\frac{1}{2}$-BPS solutions666i.e. solutions for which
half the supercharges are preserved. we impose the conditions

$\displaystyle i\gamma^{0}\xi_{1A}$ $\displaystyle=\xi_{1A}\ ,$ (23a)
$\displaystyle i\gamma^{0}\xi_{AB}$ $\displaystyle=-\xi_{AB}\ ,$ (23b)

where the index is $A\in\\{2,3,4\\}$, on the spinor parameters to reduce the
degrees of freedom by half. With this choice the equations (21) are then [21]

$\displaystyle\hat{D}_{i}\hat{\mathcal{Z}}^{A}$ $\displaystyle=0\ ,$ (24a)
$\displaystyle\hat{\bar{D}}\hat{\mathcal{Z}}^{1}$ $\displaystyle=0\ ,$ (24b)
$\displaystyle[\hat{\mathcal{Z}}^{1},\hat{\mathcal{Z}}^{2};\hat{\bar{\mathcal{Z}}}_{2}]$
$\displaystyle=[\hat{\mathcal{Z}}^{1},\hat{\mathcal{Z}}^{3};\hat{\bar{\mathcal{Z}}}_{3}]=[\hat{\mathcal{Z}}^{1},\hat{\mathcal{Z}}^{4};\hat{\bar{\mathcal{Z}}}_{4}]\
,$ (24c) $\displaystyle\hat{D}_{0}\hat{\mathcal{Z}}^{1}$
$\displaystyle=\frac{2\pi
i}{3k}[\hat{\mathcal{Z}}^{1},\hat{\mathcal{Z}}^{A};\hat{\bar{\mathcal{Z}}}_{A}]\
,$ (24d) $\displaystyle\hat{D}_{0}\hat{\mathcal{Z}}^{A}$
$\displaystyle=\frac{2\pi
i}{k}[\hat{\mathcal{Z}}^{1},\hat{\mathcal{Z}}^{A};\hat{\bar{\mathcal{Z}}}_{1}]\
,$ (24e)
$\displaystyle[\hat{\mathcal{Z}}^{1},\hat{\mathcal{Z}}^{A};\hat{\bar{\mathcal{Z}}}_{B}]$
$\displaystyle=0\;(A\neq B)\ ,$ (24f)
$\displaystyle[\hat{\mathcal{Z}}^{A},\hat{\mathcal{Z}}^{B};\hat{\bar{\mathcal{Z}}}_{C}]$
$\displaystyle=0\ ,$ (24g)

which we must supplement with the Gauss’s law constraints

$\displaystyle\frac{ik}{2\pi}\hat{F}^{L}_{12}$
$\displaystyle=\hat{\mathcal{Z}}^{M}\hat{D}_{0}\hat{\bar{\mathcal{Z}}}_{M}-\hat{D}_{0}\hat{\mathcal{Z}}^{M}\hat{\bar{\mathcal{Z}}}_{M}$
$\displaystyle=\frac{4\pi
i}{3k}\left(2\hat{\mathcal{Z}}^{A}[\hat{\bar{\mathcal{Z}}}_{1},\hat{\bar{\mathcal{Z}}}_{A};\hat{\mathcal{Z}}^{1}]-[\hat{\mathcal{Z}}^{1},\hat{\mathcal{Z}}^{A};\hat{\bar{\mathcal{Z}}}_{1}]\hat{\bar{\mathcal{Z}}}_{A}-[\hat{\mathcal{Z}}^{1},\hat{\mathcal{Z}}^{A};\hat{\bar{\mathcal{Z}}}_{A}]\hat{\bar{\mathcal{Z}}}_{1}\right)\
,$ (25a) $\displaystyle\frac{ik}{2\pi}\hat{F}_{12}^{R}$
$\displaystyle=\hat{D}_{0}\hat{\bar{\mathcal{Z}}}_{M}\hat{\mathcal{Z}}^{M}-\hat{\bar{\mathcal{Z}}}_{M}\hat{D}_{0}\hat{\mathcal{Z}}^{M}$
$\displaystyle=\frac{4\pi
i}{3k}\left(2[\hat{\bar{\mathcal{Z}}}_{1},\hat{\bar{\mathcal{Z}}}_{A};\hat{\mathcal{Z}}^{1}]\hat{\mathcal{Z}}^{A}-\hat{\bar{\mathcal{Z}}}_{A}[\hat{\mathcal{Z}}^{1},\hat{\mathcal{Z}}^{A};\hat{\bar{\mathcal{Z}}}_{1}]-\hat{\bar{\mathcal{Z}}}_{1}[\hat{\mathcal{Z}}^{1},\hat{\mathcal{Z}}^{A};\hat{\bar{\mathcal{Z}}}_{A}]\right)\
.$ (25b)

We see that singling out the index $M=1$ in the spinor condition has singled
out the field $\hat{\mathcal{Z}}^{1}$ in the BPS equations.

With the BPS equations in hand, we can now discuss coordinate scalings that
satisfy the criteria discussed above. As ABJM has a scaling symmetry under
which all spacetime coordinates transform homogeneously, a scaling for which
both spatial coordinates scale in the same way can be put into the form

$(\hat{t},\hat{x}^{i})=(t,\omega x^{i})\ ,$ (26)

for a specific choice of $\omega$. For our purposes the scaling limit will
take $\omega\to 0$. If the BPS equations are to be invariant under the
scaling, then it appears that we must take the fields to have the scaling

$\displaystyle\hat{\mathcal{Z}}^{1}(\hat{t},\hat{x})$
$\displaystyle=\mathcal{Z}^{1}(t,x)\ ,$ (27a)
$\displaystyle\hat{\mathcal{Z}}^{A}(\hat{t},\hat{x})$
$\displaystyle=\mathcal{Z}^{A}(t,x)\ ,$ (27b)
$\displaystyle\hat{A}_{0}(\hat{t},\hat{x})$ $\displaystyle=A_{0}(t,x)\ ,$
(27c) $\displaystyle\hat{A}_{i}(\hat{t},\hat{x})$
$\displaystyle=\frac{1}{\omega}A_{i}(t,x)\ ,$ (27d)

which coincides with the scaling limit discussed in section 2. However, this
is inconsistent with invariance of the constraints (25) and, as seen
previously, results in a theory with trivial dynamics. The resolution of this
is to shift the timelike components of the gauge field to

$\displaystyle A^{L}_{0}$
$\displaystyle=\mathcal{A}^{L}_{0}-\frac{2\pi}{k}\mathcal{Z}^{1}\bar{\mathcal{Z}}_{1}\
,$ (28a) $\displaystyle A^{R}_{0}$
$\displaystyle=\mathcal{A}^{R}_{0}-\frac{2\pi}{k}\bar{\mathcal{Z}}_{1}\mathcal{Z}^{1}\
,$ (28b)

as this shifts (24e) to

$0=D_{0}\mathcal{Z}^{A}-\frac{2\pi
i}{k}[\mathcal{Z}^{1},\mathcal{Z}^{A};\bar{\mathcal{Z}}_{1}]=\mathcal{D}_{0}\mathcal{Z}^{A}\
,$ (29)

while leaving $D_{0}\mathcal{Z}^{1}=\mathcal{D}_{0}\mathcal{Z}^{1}$. Using
this, we find that the scaling

$\displaystyle\hat{\mathcal{Z}}^{1}(\hat{t},\hat{x})$
$\displaystyle=\frac{1}{\omega}\mathcal{Z}^{1}(t,x)\ ,$ (30a)
$\displaystyle\hat{\mathcal{Z}}^{A}(\hat{t},\hat{x})$
$\displaystyle=\mathcal{Z}^{A}(t,x)\ ,$ (30b)
$\displaystyle\hat{\mathcal{A}}_{0}(\hat{t},\hat{x})$
$\displaystyle=\mathcal{A}_{0}(t,x)\ ,$ (30c)
$\displaystyle\hat{A}_{i}(\hat{t},\hat{x})$
$\displaystyle=\frac{1}{\omega}A_{i}(t,x)\ ,$ (30d)

leaves both the BPS equations and Gauss’s law invariant.

The application of the scaling limit to the ABJM action was performed in
[13]777The scaling there differs by an overall conformal transformation from
that discussed here, but as this is a symmetry of ABJM the outcome is the
same., and we shall briefly review it for the Bosonic sector of the theory
here. Note that from here onwards we will use $A_{0}$ instead of
$\mathcal{A}_{0}$ for the shifted gauge field to simplify our notation. At
finite $\omega$ the action takes the form

$\hat{S}_{B}=\frac{1}{\omega^{2}}S_{B,-}+S_{B,0}+O(\omega^{2})\ .$ (31a) The
finite term is $\displaystyle S_{B,0}=\tr\int d^{3}x\Bigg{[}$ $\displaystyle
D_{0}\mathcal{Z}^{1}D_{0}\bar{\mathcal{Z}}_{1}-2D\mathcal{Z}^{A}\bar{D}\bar{\mathcal{Z}}_{A}-2\bar{D}\mathcal{Z}^{A}D\bar{\mathcal{Z}}_{A}+\frac{2\pi
i}{k}D_{0}\mathcal{Z}^{A}[\bar{\mathcal{Z}}_{1},\bar{\mathcal{Z}}_{A};\mathcal{Z}^{1}]$
$\displaystyle+\frac{2\pi
i}{k}[\mathcal{Z}^{1},\mathcal{Z}^{A};\bar{\mathcal{Z}}_{1}]D_{0}\bar{\mathcal{Z}}_{A}-\frac{4\pi^{2}}{3k^{2}}[\mathcal{Z}^{A},\mathcal{Z}^{1};\bar{\mathcal{Z}}_{A}][\bar{\mathcal{Z}}_{B},\bar{\mathcal{Z}}_{1};\mathcal{Z}^{B}]$
$\displaystyle+\frac{16\pi^{2}}{3k^{2}}[\mathcal{Z}^{A},\mathcal{Z}^{1};\bar{\mathcal{Z}}_{B}][\bar{\mathcal{Z}}_{A},\bar{\mathcal{Z}}_{1},\mathcal{Z}^{B}]+\frac{8\pi^{2}}{3k^{2}}[\mathcal{Z}^{A},\mathcal{Z}^{B};\bar{\mathcal{Z}}_{1}][\bar{\mathcal{Z}}_{A},\bar{\mathcal{Z}}_{B};\mathcal{Z}^{1}]$
$\displaystyle-\frac{4\pi^{2}}{3k^{2}}[\mathcal{Z}^{1},\mathcal{Z}^{A};\bar{\mathcal{Z}}_{1}][\bar{\mathcal{Z}}_{B},\bar{\mathcal{Z}}_{A};\mathcal{Z}^{B}]-\frac{4\pi^{2}}{3k^{2}}[\mathcal{Z}^{B},\mathcal{Z}^{A};\bar{\mathcal{Z}}_{B}][\bar{\mathcal{Z}}_{1},\bar{\mathcal{Z}}_{A};\mathcal{Z}^{1}]$
$\displaystyle+\frac{ik}{2\pi}\bigg{(}A_{0}^{L}F_{z\bar{z}}^{L}+A_{z}^{L}F_{\bar{z}0}^{L}+A_{\bar{z}}^{L}F_{0z}^{L}+iA_{0}^{L}[A_{z}^{L},A_{\bar{z}}^{L}]-L\to
R\bigg{)}\Bigg{]}\ .$ (31b)

However the divergent term $S_{B,-}$ needs to be managed. Following the shift
in $A_{0}$ one finds $S_{B,-}$ can be written as

$S_{B,-}=-\tr\int
d^{3}x\left(4\bar{D}\mathcal{Z}^{1}D\bar{\mathcal{Z}}_{1}+2\partial(\mathcal{Z}^{1}\bar{D}\bar{\mathcal{Z}}_{1})-2\bar{\partial}(\mathcal{Z}^{1}D\bar{\mathcal{Z}}_{1})\right)\
.$ (32)

The last two terms are a total derivative and can be discarded. From a more
physical perspective they are cancelled by coupling the M2-branes to a
constant background $C_{3}$-field:

$\displaystyle S_{WZ}=-\frac{i}{\omega^{2}}\tr\int dx^{0}\wedge
d\mathcal{Z}^{1}\wedge d\bar{\mathcal{Z}}_{1}\ .$ (33)

This leaves us with simply

$S_{B,-}=-4\tr\int d^{3}x\,\bar{D}\mathcal{Z}^{1}D\bar{\mathcal{Z}}_{1}\ .$
(34)

As the integrand of $S_{B,-}$ is now a squared quantity, we can introduce a
complex auxilliary field $H$ in the bifundamental of $U(N)\times U(N)$ and
perform a Hubbard-Stratonovich transformation to rewrite the divergent piece
as

$\frac{1}{\omega^{2}}S_{B,-}=\tr\int
d^{3}x\left(\bar{D}\mathcal{Z}^{1}\bar{H}+HD\bar{\mathcal{Z}}_{1}+\frac{\omega^{2}}{4}H\bar{H}\right)\
,$ (35)

where the equality holds after imposing the equation of motion of the
auxilliary field. After taking the scaling limit $\omega\to 0$ we reach the
fixed-point action

$\displaystyle S_{B}=\tr\int d^{3}x\Bigg{[}$ $\displaystyle
D_{0}\mathcal{Z}^{1}D_{0}\bar{\mathcal{Z}}_{1}+HD\bar{\mathcal{Z}}_{1}+\bar{D}\mathcal{Z}^{1}\bar{H}-2D\mathcal{Z}^{A}\bar{D}\bar{\mathcal{Z}}_{A}-2\bar{D}\mathcal{Z}^{A}D\bar{\mathcal{Z}}_{A}$
$\displaystyle+\frac{2\pi
i}{k}D_{0}\mathcal{Z}^{A}[\bar{\mathcal{Z}}_{1},\bar{\mathcal{Z}}_{A};\mathcal{Z}^{1}]+\frac{2\pi
i}{k}[\mathcal{Z}^{1},\mathcal{Z}^{A};\bar{\mathcal{Z}}_{1}]D_{0}\bar{\mathcal{Z}}_{A}$
$\displaystyle-\frac{4\pi^{2}}{3k^{2}}[\mathcal{Z}^{A},\mathcal{Z}^{1};\bar{\mathcal{Z}}_{A}][\bar{\mathcal{Z}}_{B},\bar{\mathcal{Z}}_{1};\mathcal{Z}^{B}]+\frac{16\pi^{2}}{3k^{2}}[\mathcal{Z}^{A},\mathcal{Z}^{1};\bar{\mathcal{Z}}_{B}][\bar{\mathcal{Z}}_{A},\bar{\mathcal{Z}}_{1},\mathcal{Z}^{B}]$
$\displaystyle+\frac{8\pi^{2}}{3k^{2}}[\mathcal{Z}^{A},\mathcal{Z}^{B};\bar{\mathcal{Z}}_{1}][\bar{\mathcal{Z}}_{A},\bar{\mathcal{Z}}_{B};\mathcal{Z}^{1}]-\frac{4\pi^{2}}{3k^{2}}[\mathcal{Z}^{1},\mathcal{Z}^{A};\bar{\mathcal{Z}}_{1}][\bar{\mathcal{Z}}_{B},\bar{\mathcal{Z}}_{A};\mathcal{Z}^{B}]$
$\displaystyle-\frac{4\pi^{2}}{3k^{2}}[\mathcal{Z}^{B},\mathcal{Z}^{A};\bar{\mathcal{Z}}_{B}][\bar{\mathcal{Z}}_{1},\bar{\mathcal{Z}}_{A};\mathcal{Z}^{1}]+\frac{ik}{2\pi}\bigg{(}A_{0}^{L}F_{z\bar{z}}^{L}+A_{z}^{L}F_{\bar{z}0}^{L}+A_{\bar{z}}^{L}F_{0z}^{L}$
$\displaystyle+iA_{0}^{L}[A_{z}^{L},A_{\bar{z}}^{L}]-L\to R\bigg{)}\Bigg{]}\
.$ (36)

The term quadratic in $H$ has vanished; it becomes a Lagrange multiplier field
whose effect is to implement the constraint

$\bar{D}\mathcal{Z}^{1}=0\ ,$ (37)

which we recognise as the BPS equation (24b). Note that for brevity we will
write the scalar action as

$\displaystyle S_{\mathcal{Z}}=\tr\int d^{3}x\Bigg{[}$ $\displaystyle
D_{0}\mathcal{Z}^{1}D_{0}\bar{\mathcal{Z}}_{1}+HD\bar{\mathcal{Z}}_{1}+\bar{D}\mathcal{Z}^{1}\bar{H}-2D\mathcal{Z}^{A}\bar{D}\bar{\mathcal{Z}}_{A}-2\bar{D}\mathcal{Z}^{A}D\bar{\mathcal{Z}}_{A}$
$\displaystyle+\frac{2\pi
i}{k}D_{0}\mathcal{Z}^{A}[\bar{\mathcal{Z}}_{1},\bar{\mathcal{Z}}_{A};\mathcal{Z}^{1}]+\frac{2\pi
i}{k}[\mathcal{Z}^{1},\mathcal{Z}^{A};\bar{\mathcal{Z}}_{1}]D_{0}\bar{\mathcal{Z}}_{A}-V\Bigg{]}\
.$ (38)

###  3.2 Field Theory Symmetries

As we have emphasised above it is important and interesting to understand the
symmetries of the action (3.1). As the theory at finite $\omega$ is just a
rewriting of ABJM, it is guaranteed that any ABJM symmetry without terms that
diverge as $\omega\to 0$ will either be a symmetry of the fixed-point action
or act trivially on it. However, when $\omega=0$ the symmetry structure is
much richer.

The spacetime symmetries of the theory can be split into two families. First,
motivated by the form of transformations in the original theory, we consider
infinitesimal transformations of the form

$\displaystyle\hat{t}$ $\displaystyle=t+F(t)\ ,$ (39a) $\displaystyle\hat{z}$
$\displaystyle=z\left(1+F^{\prime}\right)\ ,$ (39b)

in terms of an infinitesimal function $F$ that we leave arbitrary for now.
Taking our fields to have the transformations

$\displaystyle\hat{\mathcal{Z}}^{1}(\hat{t},\hat{z},\hat{\bar{z}})$
$\displaystyle=\left(1-\frac{1}{2}F^{\prime}\right)\mathcal{Z}^{1}(t,z,\bar{z})\
,$ (40a) $\displaystyle\hat{\mathcal{Z}}^{A}(\hat{t},\hat{z},\hat{\bar{z}})$
$\displaystyle=\left(1-\frac{1}{2}F^{\prime}\right)\mathcal{Z}^{A}(t,z,\bar{z})\
,$ (40b) $\displaystyle\hat{H}(\hat{t},\hat{z},\hat{\bar{z}})$
$\displaystyle=\left(\left(1-\frac{3}{2}F^{\prime}\right)H+2zF^{\prime\prime}D_{0}\mathcal{Z}^{1}+zF^{\prime\prime\prime}\mathcal{Z}^{1}\right)(t,z,\bar{z})\
,$ (40c) $\displaystyle\hat{A}_{0}^{L/R}(\hat{t},\hat{z},\hat{\bar{z}})$
$\displaystyle=\left((1-F^{\prime})A_{0}^{L/R}-zF^{\prime\prime}A_{z}^{L/R}-\bar{z}F^{\prime\prime}A_{\bar{z}}^{L/R}\right)(t,z,\bar{z})\
,$ (40d) $\displaystyle\hat{A}_{z}^{L}(\hat{t},\hat{z},\hat{\bar{z}})$
$\displaystyle=\left((1-F^{\prime})A_{z}^{L}+\frac{\pi\bar{z}F^{\prime\prime}}{k}\mathcal{Z}^{1}\bar{\mathcal{Z}}_{1}\right)(t,z,\bar{z})\
,$ (40e) $\displaystyle\hat{A}_{z}^{R}(\hat{t},\hat{z},\hat{\bar{z}})$
$\displaystyle=\left((1-F^{\prime})A_{z}^{R}+\frac{\pi\bar{z}F^{\prime\prime}}{k}\bar{\mathcal{Z}}_{1}\mathcal{Z}^{1}\right)(t,z,\bar{z})\
,$ (40f) $\displaystyle\hat{A}_{\bar{z}}^{L}(\hat{t},\hat{z},\hat{\bar{z}})$
$\displaystyle=\left((1-F^{\prime})A_{\bar{z}}^{L}+\frac{\pi
zF^{\prime\prime}}{k}\mathcal{Z}^{1}\bar{\mathcal{Z}}_{1}\right)(t,z,\bar{z})\
,$ (40g) $\displaystyle\hat{A}_{\bar{z}}^{R}(\hat{t},\hat{z},\hat{\bar{z}})$
$\displaystyle=\left((1-F^{\prime})A_{\bar{z}}^{R}+\frac{\pi
zF^{\prime\prime}}{k}\bar{\mathcal{Z}}_{1}\mathcal{Z}^{1}\right)(t,z,\bar{z})\
,$ (40h)

we find that the change in the action at leading order is

$\delta S=-\frac{1}{2}\tr\int
d^{3}x\,F^{\prime\prime\prime}\mathcal{Z}^{1}\bar{\mathcal{Z}}_{1}\ .$ (41)

This means that the symmetries of this form are described by the function

$F(t)=a+bt+ct^{2}\ .$ (42)

We see that this is an $SO(2,1)$ subgroup of the original spacetime symmetry
group.

The other family of spacetime transformations that we will work with are the
transformations

$\displaystyle\hat{t}$ $\displaystyle=t\ ,$ (43a) $\displaystyle\hat{z}$
$\displaystyle=z+f(z,t)\ ,$ (43b)

for some infinitesimal function $f$. Note that the only requirement we’ve
asked of $f$ is that $\bar{\partial}f=0$. If we take

$\displaystyle\hat{\mathcal{Z}}^{1}(\hat{t},\hat{z},\hat{\bar{z}})$
$\displaystyle=\left(1-\partial f\right)\mathcal{Z}^{1}(t,z,\bar{z})\ ,$ (44a)
$\displaystyle\hat{\mathcal{Z}}^{A}(\hat{t},\hat{z},\hat{\bar{z}})$
$\displaystyle=\mathcal{Z}^{A}(t,z,\bar{z})\ ,$ (44b)
$\displaystyle\hat{H}(\hat{t},\hat{z},\hat{\bar{z}})$
$\displaystyle=\left(H+2f^{\prime}D_{0}\mathcal{Z}^{1}+f^{\prime\prime}\mathcal{Z}^{1}\right)(t,z,\bar{z})\
,$ (44c) $\displaystyle\hat{A}_{0}^{L/R}(\hat{t},\hat{z},\hat{\bar{z}})$
$\displaystyle=\left(A_{0}^{L/R}-f^{\prime}A_{z}^{L/R}-\bar{f}^{\prime}A_{\bar{z}}^{L/R}\right)(t,z,\bar{z})\
,$ (44d) $\displaystyle\hat{A}_{z}^{L}(\hat{t},\hat{z},\hat{\bar{z}})$
$\displaystyle=\left((1-\partial
f)A_{z}^{L}+\frac{\pi}{k}\bar{f}^{\prime}\mathcal{Z}^{1}\bar{\mathcal{Z}}_{1}\right)(t,z,\bar{z})\
,$ (44e) $\displaystyle\hat{A}_{\bar{z}}^{L}(\hat{t},\hat{z},\hat{\bar{z}})$
$\displaystyle=\left((1-\bar{\partial}\bar{f})A_{\bar{z}}^{L}+\frac{\pi}{k}f^{\prime}\mathcal{Z}^{1}\bar{\mathcal{Z}}_{1}\right)(t,z,\bar{z})\
,$ (44f) $\displaystyle\hat{A}_{z}^{R}(\hat{t},\hat{z},\hat{\bar{z}})$
$\displaystyle=\left((1-\partial
f)A_{z}^{R}+\frac{\pi}{k}\bar{f}^{\prime}\bar{\mathcal{Z}}_{1}\mathcal{Z}^{1}\right)(t,z,\bar{z})\
,$ (44g) $\displaystyle\hat{A}_{\bar{z}}^{R}(\hat{t},\hat{z},\hat{\bar{z}})$
$\displaystyle=\left((1-\bar{\partial}\bar{f})A_{\bar{z}}^{R}+\frac{\pi}{k}f^{\prime}\bar{\mathcal{Z}}_{1}\mathcal{Z}^{1}\right)(t,z,\bar{z})\
,$ (44h)

we find that the action is invariant.

There are two interesting symmetry enhancements at play here; the standard
spatial symmetry one expects after taking the non-relativistic limit888That
is, spatial translations, Galilean boosts, and rotations. is enhanced to the
two-dimensional Euclidean conformal algebra, and the spatial transformations
can be made time-dependent. Spatial transformations with arbitrary time-
dependence have been previously studied in massive non-relativistic field
theories coupled to background gauge fields [22, 23]. However, upon adding a
Chern-Simons term for the gauge field these symmetries are lost unless the
massless limit of matter in the theory is taken; this reproduces the non-
dynamical theory discussed in section 2. The novelty here is that introduction
of a matter field with two temporal derivative kinetic term allows us to
retain this structure in a dynamical field theory. It is also this derivative
structure that allows for the infinite-dimensional extension of the spatial
symmetry.

It will be convenient to disentangle the spatial and temporal symmetries. If
we combine the two types of transformations and take

$f=-zF^{\prime}$ (45)

then we find a purely temporal symmetry

$\displaystyle\hat{t}$ $\displaystyle=t+F(t)\ ,$ (46a) $\displaystyle\hat{z}$
$\displaystyle=z\ ,$ (46b)

(with $F$ as in (42)) under which the fields have the transformations

$\displaystyle\hat{\mathcal{Z}}^{1}(\hat{t},\hat{z},\hat{\bar{z}})$
$\displaystyle=\left(1+\frac{1}{2}F^{\prime}\right)\mathcal{Z}^{1}(t,z,\bar{z})\
,$ (47a) $\displaystyle\hat{\mathcal{Z}}^{A}(\hat{t},\hat{z},\hat{\bar{z}})$
$\displaystyle=\left(1-\frac{1}{2}F^{\prime}\right)\mathcal{Z}^{A}(t,z,\bar{z})\
,$ (47b) $\displaystyle\hat{H}(\hat{t},\hat{z},\hat{\bar{z}})$
$\displaystyle=\left(1-\frac{3}{2}F^{\prime}\right)H(t,z,\bar{z})\ ,$ (47c)
$\displaystyle\hat{A}^{L/R}_{0}(\hat{t},\hat{z},\hat{\bar{z}})$
$\displaystyle=(1-F^{\prime})A_{0}^{L/R}(t,z,\bar{z})\ ,$ (47d)
$\displaystyle\hat{A}_{z}^{L/R}(\hat{t},\hat{z},\hat{\bar{z}})$
$\displaystyle=A_{z}^{L/R}(t,z,\bar{z})\ ,$ (47e)
$\displaystyle\hat{A}_{\bar{z}}^{L/R}(\hat{t},\hat{z},\hat{\bar{z}})$
$\displaystyle=A_{\bar{z}}^{L/R}(t,z,\bar{z})\ .$ (47f)

In addition to the spacetime symmetries, we have the R-symmetries

$\displaystyle\hat{\mathcal{Z}}^{1}$
$\displaystyle=e^{i\alpha}\mathcal{Z}^{1}\ ,$ (48a) $\displaystyle\hat{H}$
$\displaystyle=e^{i\alpha}H\ ,$ (48b) $\displaystyle\hat{\mathcal{Z}}^{A}$
$\displaystyle=\mathcal{Z}^{A}\ ,$ (48c)

and

$\displaystyle\hat{\mathcal{Z}}^{1}$ $\displaystyle=\mathcal{Z}^{1}\ ,$ (49a)
$\displaystyle\hat{H}$ $\displaystyle=H\ ,$ (49b)
$\displaystyle\hat{\mathcal{Z}}^{A}$
$\displaystyle=\tensor{\mathcal{R}}{{}^{A}_{B}}\mathcal{Z}^{B}\ $

for $\mathcal{R}\in SU(3)$. We also retain the global $U(1)_{b}\subset
U(N)\times U(N)$ baryon number symmetry

$\displaystyle\hat{\mathcal{Z}}^{1}$ $\displaystyle=e^{i\beta}\mathcal{Z}^{1}\
,$ (49axa) $\displaystyle\hat{H}$ $\displaystyle=e^{i\beta}H\ ,$ (49axb)
$\displaystyle\hat{\mathcal{Z}}^{A}$ $\displaystyle=e^{i\beta}\mathcal{Z}^{A}\
.$ (49axc)

The R-symmetries form a $U(1)_{R}\times SU(3)$ subgroup of the original
$SU(4)$ R-symmetry of ABJM999Note that a $U(1)_{b}$ transformation needs to be
simultaneously performed with the transformation (48) for the latter to be
contained within the original R-symmetry group.. We note that all the
symmetries found in this section can be extended to the full supersymmetric
action, with the Fermionic transformations detailed in appendix A.1.

###  3.3 Conserved Currents

We can find the conserved currents associated with the transformations using
the standard Noether procedure of promoting the transformation parameters to
arbitrary functions. Let us first do this for the temporal symmetry (46). It
will be convenient to work with the modified transformations

$\displaystyle\hat{A}_{z}^{L/R}(\hat{t},\hat{z},\hat{\bar{z}})$
$\displaystyle=\left(A^{L/R}_{z}-\partial FA^{L/R}_{0}\right)(t,z,\bar{z})\ ,$
(49aya) $\displaystyle\hat{A}_{\bar{z}}^{L/R}(\hat{t},\hat{z},\hat{\bar{z}})$
$\displaystyle=\left(A^{L/R}_{\bar{z}}-\bar{\partial}FA^{L/R}_{0}\right)(t,z,\bar{z})\
,$ (49ayb)

after introducing spatial dependence into $F$ in order to maintain gauge-
invariance throughout the calculation. The conserved currents are then

$\displaystyle j^{0}_{(a)}$
$\displaystyle=\tr\left(D_{0}\mathcal{Z}^{1}D_{0}\bar{\mathcal{Z}}_{1}+2D\mathcal{Z}^{A}\bar{D}\bar{\mathcal{Z}}_{A}+2\bar{D}\mathcal{Z}^{A}D\bar{\mathcal{Z}}_{A}+V\right)\
,$ (49aza) $\displaystyle j_{(a)}$
$\displaystyle=\tr\left(HD_{0}\bar{\mathcal{Z}}_{1}-2D_{0}\mathcal{Z}^{A}\bar{D}\bar{\mathcal{Z}}_{A}-2\bar{D}\mathcal{Z}^{A}D_{0}\bar{\mathcal{Z}}_{A}\right)\
,$ (49azb) $\displaystyle\bar{j}_{(a)}$
$\displaystyle=\tr\left(D_{0}\mathcal{Z}^{1}\bar{H}-2D\mathcal{Z}^{A}D_{0}\bar{\mathcal{Z}}_{A}-2D_{0}\mathcal{Z}^{A}D\bar{\mathcal{Z}}_{A}\right)\
,$ (49azc)

for the transformation $F=a$,

$\displaystyle j^{0}_{(b)}$
$\displaystyle=\tr\left(t\bigg{(}D_{0}\mathcal{Z}^{1}D_{0}\bar{\mathcal{Z}}_{1}+2\left(D\mathcal{Z}^{A}\bar{D}\bar{\mathcal{Z}}_{A}+\bar{D}\mathcal{Z}^{A}D\bar{\mathcal{Z}}_{A}\right)+V\bigg{)}-\frac{1}{2}\left(\mathcal{Z}^{1}D_{0}\bar{\mathcal{Z}}_{1}+D_{0}\mathcal{Z}^{1}\bar{\mathcal{Z}}_{1}\right)\right)\
,$ (49baa) $\displaystyle j_{(b)}$
$\displaystyle=\tr\left(t\bigg{(}HD_{0}\bar{\mathcal{Z}}_{1}-2D_{0}\mathcal{Z}^{A}\bar{D}\bar{\mathcal{Z}}_{A}-2\bar{D}\mathcal{Z}^{A}D_{0}\bar{\mathcal{Z}}_{A}\bigg{)}-\mathcal{Z}^{A}\bar{D}\bar{\mathcal{Z}}_{A}-\bar{D}\mathcal{Z}^{A}\bar{\mathcal{Z}}_{A}-\frac{1}{2}H\bar{\mathcal{Z}}_{1}\right)\
,$ (49bab) $\displaystyle\bar{j}_{(b)}$
$\displaystyle=\tr\left(t\bigg{(}D_{0}\mathcal{Z}^{1}\bar{H}-2D_{0}\mathcal{Z}^{A}D\bar{\mathcal{Z}}_{A}-2D\mathcal{Z}^{A}D_{0}\bar{\mathcal{Z}}_{A}\bigg{)}-\mathcal{Z}^{A}D\bar{\mathcal{Z}}_{A}-D\mathcal{Z}^{A}\bar{\mathcal{Z}}_{A}-\frac{1}{2}\mathcal{Z}^{1}\bar{H}\right)\
,$ (49bac)

for the transformation $F=bt$, and

$\displaystyle j^{0}_{(c)}$
$\displaystyle=\tr\left(t^{2}\bigg{(}D_{0}\mathcal{Z}^{1}D_{0}\bar{\mathcal{Z}}_{1}+2\left(D\mathcal{Z}^{A}\bar{D}\bar{\mathcal{Z}}_{A}+\bar{D}\mathcal{Z}^{A}D\bar{\mathcal{Z}}_{A}\right)+V\bigg{)}-tD_{0}\left(\mathcal{Z}^{1}\bar{\mathcal{Z}}_{1}\right)+\mathcal{Z}^{1}\bar{\mathcal{Z}}_{1}\right)\
,$ (49bba) $\displaystyle j_{(c)}$
$\displaystyle=\tr\left(t^{2}\bigg{(}HD_{0}\bar{\mathcal{Z}}_{1}-2D_{0}\mathcal{Z}^{A}\bar{D}\bar{\mathcal{Z}}_{A}-2\bar{D}\mathcal{Z}^{A}D_{0}\bar{\mathcal{Z}}_{A}\bigg{)}-t\left(2\mathcal{Z}^{A}\bar{D}\bar{\mathcal{Z}}_{A}+2\bar{D}\mathcal{Z}^{A}\bar{\mathcal{Z}}_{A}+H\bar{\mathcal{Z}}_{1}\right)\right)\
,$ (49bbb) $\displaystyle\bar{j}_{(c)}$
$\displaystyle=\tr\left(t^{2}\bigg{(}D_{0}\mathcal{Z}^{1}\bar{H}-2D_{0}\mathcal{Z}^{A}D\bar{\mathcal{Z}}_{A}-2D\mathcal{Z}^{A}D_{0}\bar{\mathcal{Z}}_{A}\bigg{)}-t\left(2\mathcal{Z}^{A}D\bar{\mathcal{Z}}_{A}+2D\mathcal{Z}^{A}\bar{\mathcal{Z}}_{A}+\mathcal{Z}^{1}\bar{H}\right)\right)\
,$ (49bbc)

for the transformation $F=ct^{2}$. Note that we have used the equations of
motion to write the currents as above.

We can perform the same calculation for the spatial symmetry (43), where we
similarly modify the gauge field transformations to

$\displaystyle\hat{A}_{z}^{L}(\hat{t},\hat{z},\hat{\bar{z}})$
$\displaystyle=\left((1-\partial
f)A_{z}^{L}-\partial\bar{f}A_{\bar{z}}^{L}+\frac{\pi\bar{f}^{\prime}}{k}\mathcal{Z}^{1}\bar{\mathcal{Z}}_{1}\right)(t,z,\bar{z})\
,$ (49bca) $\displaystyle\hat{A}_{\bar{z}}^{L}(\hat{t},\hat{z},\hat{\bar{z}})$
$\displaystyle=\left((1-\bar{\partial}\bar{f})A_{\bar{z}}^{L}-\bar{\partial}fA_{z}^{L}+\frac{\pi
f^{\prime}}{k}\mathcal{Z}^{1}\bar{\mathcal{Z}}_{1}\right)(t,z,\bar{z})\ ,$
(49bcb)

and analogously for $(A_{z}^{R},A_{\bar{z}}^{R})$. As the action is invariant
for any time-dependent holomorphic function $f(z,t)$, the only non-vanishing
terms after taking $f$ to have arbitrary coordinate dependence take the form

$\delta S=-\int
d^{3}x\left(\bar{\partial}f\,T+\partial\bar{f}\,\bar{T}\right)\ ,$ (49bd)

so the associated conservation laws are

$0=\bar{\partial}T$ (49be)

and its complex conjugate. A brief computation gives

$T=\tr\left(\mathcal{Z}^{1}D\bar{H}+4D\mathcal{Z}^{A}D\bar{\mathcal{Z}}_{A}\right)\
.$ (49bf)

Since the symmetry can be taken to have arbitrary time dependence the current
has no temporal component, so there is no codimension-1 conserved charge
associated with the symmetry. We should interpret the conservation law (49be)
as a constraint we must impose on the physical states of the theory. At first
glance, it appears somewhat strange to have gone from a theory possessing
codimension-1 charges associated with spatial translations and rotations to
one where these transformations are still symmetries of the theory but the
analogous charges vanish. To understand this better it is instructive to
directly perform the limit for the spatial momentum in ABJM,

$\hat{P}_{i}=\tr\int_{\hat{\Sigma}}d^{2}\hat{x}\left(D_{0}\hat{\mathcal{Z}}^{M}D_{i}\hat{\bar{\mathcal{Z}}}_{M}+D_{i}\hat{\mathcal{Z}}^{M}D_{0}\hat{\bar{\mathcal{Z}}}_{M}\right)\
.$ (49bg)

Shifting the temporal component of the gauge field as in (28) and performing
the scaling (30) gives

$\displaystyle\hat{P}_{i}$ $\displaystyle=\frac{1}{\omega}P_{i}\ ,$ (49bha)
$\displaystyle P_{i}$
$\displaystyle=\tr\int_{\Sigma}d^{2}x\bigg{[}D_{0}\mathcal{Z}^{1}D_{i}\bar{\mathcal{Z}}_{1}+D_{i}\mathcal{Z}^{1}D_{0}\bar{\mathcal{Z}}_{1}+\frac{2\pi
i}{k}\bigg{(}[\mathcal{Z}^{1},\mathcal{Z}^{A};\bar{\mathcal{Z}}_{1}]D_{i}\bar{\mathcal{Z}}_{A}$
$\displaystyle\hskip
28.45274pt+D_{i}\mathcal{Z}^{A}[\bar{\mathcal{Z}}_{1},\bar{\mathcal{Z}}_{A};\mathcal{Z}^{1}]\bigg{)}\bigg{]}+O(\omega^{2})\
.$ (49bhb)

Note that since we have scaled the spatial coordinates as $\hat{x}^{i}=\omega
x^{i}$ it is natural for the momentum to have the overall scaling above.
Taking the limit $\omega\to 0$ in $P_{i}$, we can use the constraint (37) to
write the $z$-component of the momentum as

$P_{z}=\tr\int_{\Sigma}d^{2}x\bigg{[}D\mathcal{Z}^{1}D_{0}\bar{\mathcal{Z}}_{1}+\frac{2\pi
i}{k}\bigg{(}[\mathcal{Z}^{1},\mathcal{Z}^{A};\bar{\mathcal{Z}}_{1}]D\bar{\mathcal{Z}}_{A}+D\mathcal{Z}^{A}[\bar{\mathcal{Z}}_{1},\bar{\mathcal{Z}}_{A};\mathcal{Z}^{1}]\bigg{)}\bigg{]}\
,$ (49bi)

which after integrating by parts and again using the constraint is

$\displaystyle P_{z}=\tr\int_{\Sigma}d^{2}x\bigg{[}$
$\displaystyle\partial\left(\mathcal{Z}^{1}D_{0}\bar{\mathcal{Z}}_{1}\right)+i\mathcal{Z}^{1}\bar{\mathcal{Z}}_{1}F^{L}_{0z}-i\bar{\mathcal{Z}}_{1}\mathcal{Z}^{1}F^{R}_{0z}$
$\displaystyle+\frac{2\pi
i}{k}\bigg{(}[\mathcal{Z}^{1},\mathcal{Z}^{A};\bar{\mathcal{Z}}_{1}]D\bar{\mathcal{Z}}_{A}+D\mathcal{Z}^{A}[\bar{\mathcal{Z}}_{1},\bar{\mathcal{Z}}_{A};\mathcal{Z}^{1}]\bigg{)}\bigg{]}\
.$ (49bj)

However, when the equations of motion (49et) are satisfied101010We set the
Fermionic terms to zero here. the final four terms cancel, leaving us with

$P_{z}=i\tr\int_{\partial\Sigma}d\bar{z}\,\mathcal{Z}^{1}D_{0}\bar{\mathcal{Z}}_{1}\
.$ (49bk)

The limit has taken the codimension-1 charge to a codimension-2 charge
evaluated at spatial infinity. We could do the same for the spatial rotation

$\hat{Q}_{ij}=\tr\int_{\hat{\Sigma}}d^{2}\hat{x}\bigg{[}\hat{x}^{i}\left(D_{0}\hat{\mathcal{Z}}^{M}D_{j}\hat{\bar{\mathcal{Z}}}_{M}+D_{j}\hat{\mathcal{Z}}^{M}D_{0}\hat{\bar{\mathcal{Z}}}_{M}\right)-\hat{x}^{j}\left(D_{0}\hat{\mathcal{Z}}^{M}D_{i}\hat{\bar{\mathcal{Z}}}_{M}+D_{i}\hat{\mathcal{Z}}^{M}D_{0}\hat{\bar{\mathcal{Z}}}_{M}\right)\bigg{]}\
.$ (49bl)

Following the same steps as above, we find the conserved charge

$\displaystyle
Q_{z\bar{z}}=-\tr\int_{\partial\Sigma}\left(d\bar{z}\,z\mathcal{Z}^{1}D_{0}\bar{\mathcal{Z}}_{1}+dz\,\bar{z}D_{0}\mathcal{Z}^{1}\bar{\mathcal{Z}}_{1}\right)-i\tr\int_{\Sigma}d^{2}x\left(\mathcal{Z}^{1}D_{0}\bar{\mathcal{Z}}_{1}-D_{0}\mathcal{Z}^{1}\bar{\mathcal{Z}}_{1}\right)\
.$ (49bm)

It appears that in this case we’ve found a non-trivial codimension-1 conserved
charge. However, as we’ll see momentarily this is just the conserved charge
associated with the R-symmetry transformation (48), reflecting the fact that
the transformations of $\mathcal{Z}^{1}$ under the ABJM spatial rotation after
the $\omega\to 0$ scaling and under the rotation in the family of
transformations (44) differ by the action of this global symmetry. More
generally, applying Noether’s theorem for a time-independent transformation of
the form (43) gives the charge

$Q[f]=i\tr\int_{\partial\Sigma}\left(d\bar{z}\,f\mathcal{Z}^{1}D_{0}\bar{\mathcal{Z}}_{1}-dz\,\bar{f}D_{0}\mathcal{Z}^{1}\bar{\mathcal{Z}}_{1}\right)\
.$ (49bn)

Since these take the form of boundary terms they cannot generate a symmetry
transformation on the bulk phase space of the theory. It therefore seems
natural to interpret these transformations as gauge redundancies.

Finally, we give the R-symmetry and baryon number currents. The $U(1)_{R}$
R-symmetry transformation (48) has the current

$\displaystyle j^{0}$
$\displaystyle=i\tr\left(\mathcal{Z}^{1}D_{0}\bar{\mathcal{Z}}_{1}-D_{0}\mathcal{Z}^{1}\bar{\mathcal{Z}}_{1}\right)\
,$ (49boa) $\displaystyle j$
$\displaystyle=-i\tr\left(H\bar{\mathcal{Z}}_{1}\right)\ ,$ (49bob)
$\displaystyle\bar{j}$ $\displaystyle=i\tr\left(\mathcal{Z}^{1}\bar{H}\right)\
,$ (49boc)

and the $SU(3)$ symmetry (49) has

$\displaystyle\tensor{\left(J^{0}\right)}{{}^{A}_{B}}$
$\displaystyle=\frac{2\pi}{k}\tr\left([\mathcal{Z}^{1},\mathcal{Z}^{A};\bar{\mathcal{Z}}_{1}]\bar{\mathcal{Z}}_{B}-\mathcal{Z}^{A}[\bar{\mathcal{Z}}_{1},\bar{\mathcal{Z}}_{B};\mathcal{Z}^{1}]-\frac{2}{3}\delta^{A}_{B}[\mathcal{Z}^{1},\mathcal{Z}^{C};\bar{\mathcal{Z}}_{1}]\bar{\mathcal{Z}}_{C}\right)\
,$ (49bpa) $\displaystyle\tensor{J}{{}^{A}_{B}}$
$\displaystyle=2i\tr\left(\bar{D}\mathcal{Z}^{A}\bar{\mathcal{Z}}_{B}-\mathcal{Z}^{A}\bar{D}\bar{\mathcal{Z}}_{B}-\frac{1}{3}\delta^{A}_{B}\left(\bar{D}\mathcal{Z}^{C}\bar{\mathcal{Z}}_{C}-\mathcal{Z}^{C}\bar{D}\bar{\mathcal{Z}}_{C}\right)\right)\
,$ (49bpb) $\displaystyle\tensor{\bar{J}}{{}^{A}_{B}}$
$\displaystyle=2i\tr\left(D\mathcal{Z}^{A}\bar{\mathcal{Z}}_{B}-\mathcal{Z}^{A}D\bar{\mathcal{Z}}_{B}-\frac{1}{3}\delta^{A}_{B}\left(D\mathcal{Z}^{C}\bar{\mathcal{Z}}_{C}-\mathcal{Z}^{C}D\bar{\mathcal{Z}}_{C}\right)\right)\
.$ (49bpc)

The $U(1)_{b}$ baryon number current is

$\displaystyle j_{b}^{0}$
$\displaystyle=i\tr\left(\mathcal{Z}^{1}D_{0}\bar{\mathcal{Z}}_{1}-D_{0}\mathcal{Z}^{1}\bar{\mathcal{Z}}_{1}-\frac{2\pi
i}{k}\left([\mathcal{Z}^{1},\mathcal{Z}^{A};\bar{\mathcal{Z}}_{1}]\bar{\mathcal{Z}}_{B}-\mathcal{Z}^{A}[\bar{\mathcal{Z}}_{1},\bar{\mathcal{Z}}_{B};\mathcal{Z}^{1}]\right)\right)\
,$ (49bqa) $\displaystyle j_{b}$
$\displaystyle=i\tr\left(2\bar{D}\mathcal{Z}^{A}\bar{\mathcal{Z}}_{B}-2\mathcal{Z}^{A}\bar{D}\bar{\mathcal{Z}}_{B}-iH\bar{\mathcal{Z}}_{1}\right)\
,$ (49bqb) $\displaystyle\bar{j}_{b}$
$\displaystyle=i\tr\left(2D\mathcal{Z}^{A}\bar{\mathcal{Z}}_{B}-2\mathcal{Z}^{A}D\bar{\mathcal{Z}}_{B}+\mathcal{Z}^{1}\bar{H}\right)\
.$ (49bqc)

##  4 Eleven-Dimensional Membrane Newton-Cartan Gravity and the M2-Brane

###  4.1 A Brief Review of Non-Relativistic Eleven-Dimensional Supergravity

We have constructed our field theory by taking a non-relativistic limit of
ABJM. Since ABJM is dual to M-theory on an asymptotically $AdS_{4}\times
S^{7}/\mathbb{Z}_{k}$ background, we would like to try and understand whether
a similar duality holds between the non-relativistic field theory and a non-
relativistic limit of eleven-dimensional supergravity. A limit of the Bosonic
sector of eleven-dimensional supergravity that naturally couples to membranes,
known as eleven-dimensional membrane-Newton-Cartan geometry, was found in
[11]; we will review the key features of their limit before applying it to the
M2-brane metric in the next section. For a general overview of recent progress
in understanding non-relativistic gravity see [24, 25].

The Bosonic field content of eleven-dimensional supergravity is the metric
$\hat{g}_{\mu\nu}$ and a 3-form field $\hat{C}_{3}$. In order to define a non-
relativistic limit of eleven-dimensional supergravity, we choose a vielbein
for our spacetime which we partition into two sets
$\\{\hat{E}^{a},\hat{E}^{I}\\}$, with $a\in\\{0,1,2\\}$ and
$I\in\\{3,4,...,10\\}$. We can then introduce a dimensionless parameter $c$
and redefine our veilbein as

$\displaystyle\hat{E}^{a}$ $\displaystyle=c\hat{\tau}^{a}\ ,$ (49bra)
$\displaystyle\hat{E}^{I}$ $\displaystyle=\frac{1}{\sqrt{c}}\hat{e}^{I}\ .$
(49brb)

When working with a specific metric we will implement such a transformation by
choosing a set of coordinates and introducing the parameter $c$ using a
combination of coordinate and parameter scalings. This means that the
components of our vielbein will generically have $c$-dependence; we’ll assume
that we’ve scaled quantities such that $\hat{\tau}^{a}$ and $\hat{e}^{I}$
admit a well-defined Taylor series in $\frac{1}{c^{3}}$. As will be seen
later, this is the case for the non-relativistic limit of the M2-brane metric.
The corresponding orthonormal frame $\\{E_{a},E_{I}\\}$ must scale as

$\displaystyle E_{a}$ $\displaystyle=\frac{1}{c}\hat{\tau}_{a}\ ,$ (49bsa)
$\displaystyle E_{I}$ $\displaystyle=\sqrt{c}\hat{e}_{I}\ ,$ (49bsb)

so that bases remain dual to each other. In terms of the metric, this means
that we have

$\displaystyle\hat{g}_{\mu\nu}$
$\displaystyle=c^{2}\hat{\tau}_{\mu\nu}+\frac{1}{c}\hat{H}_{\mu\nu}\ ,$
(49bta) $\displaystyle\hat{g}^{\mu\nu}$
$\displaystyle=c\hat{H}^{\mu\nu}+\frac{1}{c^{2}}\hat{\tau}^{\mu\nu}\ ,$
(49btb)

where we’ve defined

$\displaystyle\hat{\tau}_{\mu\nu}$
$\displaystyle=\eta_{ab}\hat{\tau}^{a}_{\mu}\hat{\tau}^{b}_{\nu}\ ,$ (49bua)
$\displaystyle\hat{\tau}^{\mu\nu}$
$\displaystyle=\eta^{ab}\hat{\tau}_{a}^{\mu}\hat{\tau}_{b}^{\nu}\ ,$ (49bub)
$\displaystyle\hat{H}_{\mu\nu}$
$\displaystyle=\delta_{IJ}\hat{e}^{I}_{\mu}\hat{e}^{J}_{\nu}\ ,$ (49buc)
$\displaystyle\hat{H}^{\mu\nu}$
$\displaystyle=\delta^{IJ}\hat{e}_{I}^{\mu}\hat{e}_{J}^{\nu}\ .$ (49bud)

We can now expand all quantities in $\frac{1}{c^{3}}$ using the notation

$\displaystyle\hat{\tau}^{a}$
$\displaystyle=\tau^{a}+\frac{1}{c^{3}}m^{a}+O\left(\frac{1}{c^{6}}\right)\ ,$
(49bva) $\displaystyle\hat{\tau}_{a}$
$\displaystyle=\tau_{a}+O\left(\frac{1}{c^{3}}\right)\ ,$ (49bvb)
$\displaystyle\hat{e}^{I}$ $\displaystyle=e^{I}+O\left(\frac{1}{c^{3}}\right)\
,$ (49bvc) $\displaystyle\hat{e}_{I}$
$\displaystyle=e_{I}+\frac{1}{c^{3}}\Phi_{I}+O\left({\frac{1}{c^{6}}}\right)\
,$ (49bvd)

so the metric and its inverse take the form

$\displaystyle g_{\mu\nu}$
$\displaystyle=c^{2}\tau_{\mu\nu}+\frac{1}{c}\left(H_{\mu\nu}+2\eta_{ab}\tau^{a}_{(\mu}m_{\nu)}^{b}\right)+O\left(\frac{1}{c^{4}}\right)\
,$ (49bwa) $\displaystyle g^{\mu\nu}$
$\displaystyle=cH^{\mu\nu}+\frac{1}{c^{2}}\left(\tau^{\mu\nu}+2\delta^{IJ}e_{I}^{(\mu}\Phi^{\nu)}_{J}\right)+O\left(\frac{1}{c^{5}}\right)\
.$ (49bwb)

At finite $c$ the local symmetry algebra of the vielbein is
$\mathfrak{so}(1,10)$, with factors of $c$ inserted to match the expansion in
the metric. However, as we will ultimately be interested in taking the
$c\to\infty$ limit we will only be interested transformations of our variables
that are independent of $c$. While it’s clear that the
$\mathfrak{so}(1,2)\oplus\mathfrak{so}(8)$ subalgebra consisting of Lorentz
transformations of $\tau^{a}$ and rotations of $e^{I}$ satisfies this
condition, it is less obvious which of the transformations that mix the two
are retained in the limit. Expanding the invertibility condition
$\delta^{\mu}_{\nu}=g^{\mu\rho}g_{\rho\nu}$ in powers of $c$ gives the
relations

$\displaystyle\tau^{a}(e_{I})$ $\displaystyle=0\ ,$ (49bxa) $\displaystyle
e^{I}(\tau_{a})$ $\displaystyle=0\ ,$ (49bxb)
$\displaystyle\tau^{a}(\tau_{b})$ $\displaystyle=\delta^{a}_{b}\ ,$ (49bxc)
$\displaystyle e^{I}(e_{J})$ $\displaystyle=\delta^{I}_{J}\ ,$ (49bxd)
$\displaystyle\delta$ $\displaystyle=\tau_{a}\otimes\tau^{a}+e_{I}\otimes
e^{I}\ ,$ (49bxe)

at the first three orders. It’s easy to see that these are invariant under the
reparameterisations

$\displaystyle e^{\prime I}$ $\displaystyle=e^{I}+\lambda^{I}_{a}\tau^{a}\ ,$
(49bya) $\displaystyle\tau_{a}^{\prime}$
$\displaystyle=\tau_{a}-\lambda^{I}_{a}e_{I}\ ,$ (49byb)

which we recognise as the leading-order term in the non-relativistic expansion
of a local Lorentz boost. These act non-trivially on $H_{\mu\nu}$ and
$\tau^{\mu\nu}$, changing them to

$\displaystyle H^{\prime}_{\mu\nu}$
$\displaystyle=H_{\mu\nu}+2\delta_{IJ}\lambda_{a}^{I}e^{J}_{(\mu}\tau^{a}_{\nu)}+\delta_{IJ}\lambda_{a}^{I}\lambda_{b}^{J}\tau_{\mu}^{a}\tau_{\nu}^{b}\
,$ (49bza) $\displaystyle\tau^{\prime\mu\nu}$
$\displaystyle=\tau^{\mu\nu}-2\eta^{ab}\lambda^{I}_{a}\tau_{b}^{(\mu}e_{I}^{\nu)}+\eta^{ab}\lambda^{I}_{a}\lambda^{J}_{b}e^{\mu}_{I}e^{\nu}_{J}\
.$ (49bzb)

However, also taking the subleading terms to have the transformations

$\displaystyle m^{\prime a}$
$\displaystyle=m^{a}-\lambda^{a}_{I}e^{I}-\frac{1}{2}\lambda^{a}_{I}\lambda^{I}_{b}\tau^{b}\
,$ (49caa) $\displaystyle\Phi^{\prime}_{I}$
$\displaystyle=\Phi_{I}+\lambda^{a}_{I}v_{a}-\frac{1}{2}\lambda^{a}_{I}\lambda^{J}_{a}e_{J}\
,$ (49cab)

we see that both $g_{\mu\nu}$ and $g^{\mu\nu}$ are invariant at subleading
order in $c$. The transformations (49by) and (49ca) form a local invariance of
the system known as a local Galilean boost. The boosts have the unusual
property that they alter quantities with all local indices contracted. This is
a manifestation of the fact that both $H_{\mu\nu}$ and $\tau^{\mu\nu}$ are
sections of quotients of tensor bundles [9] and are therefore only well-
defined up to an equivalence class. The variables $\tau_{\mu\nu}$ and
$H^{\mu\nu}$ are, however, sections of genuine tensor bundles and therefore
don’t transform under the local boosts.

To find the dynamics of the theory in the non-relativistic limit we must
specify a prescription for the expansion of the 3-form field $\hat{C}_{3}$
after the introduction of $c$. This turns out to be highly constrained by the
requirement that divergent terms in the action should cancel on physical
solutions, and forces us to take decompose $\hat{C}_{3}$ as

$\hat{C}_{3}=-\frac{c^{3}}{6}\epsilon_{abc}\tau^{a}\wedge\tau^{b}\wedge\tau^{c}+C_{3}+\frac{1}{c^{3}}\tilde{C}_{3}+O\left(\frac{1}{c^{6}}\right)\
.$ (49cb)

With this, the action takes the schematic form

$\hat{S}_{11d}=c^{3}S_{3}+S_{0}+O\left(\frac{1}{c^{3}}\right)\ ;$ (49cc)

requiring that the divergent piece cancels gives us the constraint

$\displaystyle\Omega
H^{\mu_{1}\nu_{1}}...H^{\mu_{4}\nu_{4}}F_{\nu_{1}...\nu_{4}}$
$\displaystyle=-\frac{1}{144}\epsilon^{\mu_{1}...\mu_{11}}\epsilon_{abc}F_{\mu_{5}...\mu_{9}}\tau_{\mu_{9}}^{a}\tau_{\mu_{10}}^{b}\tau_{\mu_{11}}^{c}\
,$ (49cda) where $F_{4}$ is the field strength of $C_{3}$ and we define
$\Omega$ by $\displaystyle\Omega^{2}$
$\displaystyle=-\frac{1}{3!8!}\epsilon^{\mu_{1}...\mu_{11}}\epsilon^{\nu_{1}...\nu_{11}}\tau_{\mu_{1}\nu_{1}}...\tau_{\mu_{3}\nu_{3}}H_{\mu_{4}\nu_{4}}...H_{\mu_{11}\nu_{11}}\
.$ (49cdb)

When this constraint is imposed the action is finite and the $c\to\infty$
limit can be taken. In this limit the subleading fields $m^{a}$ and $\Phi_{I}$
drop out of the dynamics. However, it turns out that the subleading 3-form
field $\tilde{C}_{3}$ will enter into the action and must be retained. This
means that the transformations (49by), and hence also the corresponding metric
transformations, become local invariances of the theory, as can be verified
from the action upon also taking the 3-forms to have the infinitesimal
transformations

$\displaystyle\delta C_{\mu\nu\rho}$
$\displaystyle=-3\epsilon_{abc}\lambda_{I}^{a}e^{I}_{[\mu}\tau^{b}_{\nu}\tau^{c}_{\rho]}\
,$ (49cea) $\displaystyle\delta\tilde{F}_{\mu\nu\rho\sigma}$
$\displaystyle=-4\lambda^{a}_{I}e^{I}_{[\mu}F_{\nu\rho\sigma]\lambda}\tau_{a}^{\lambda}\
,$ (49ceb)

where we’ve defined $\tilde{F}_{4}=d\tilde{C}_{3}$.

There is some freedom in how $c$ is defined since it will not be present in
the theory after the limit is taken. For instance, redefining $c$ by an
overall scale

$c=\lambda c_{\lambda}$ (49cf)

and taking $c_{\lambda}\to\infty$ will yield the same limit as taking
$c\to\infty$ for any $\lambda>0$, since the equations of motion of the system
arise from the $c$-independent piece of the action. This corresponds to an
emergent dilatation symmetry that takes

$\displaystyle\tau_{\mu\nu}$ $\displaystyle\to\lambda^{2}\tau_{\mu\nu}\ ,$
(49cga) $\displaystyle H^{\mu\nu}$ $\displaystyle\to\lambda H^{\mu\nu}\ .$
(49cgb)

Since $C_{3}$ appears with no overall factor of $c$ it should not transform
under the dilatation, while we must take the subleading field to have the
transformation

$\tilde{F}_{4}\to\lambda^{-3}\tilde{F}_{4}\ .$ (49ch)

In fact, we can go further than this; the action is invariant under the
infinitesimal transformations

$\displaystyle\delta\tau_{\mu\nu}$ $\displaystyle=2\Lambda(x)\tau_{\mu\nu}\ ,$
(49cia) $\displaystyle\delta H^{\mu\nu}$ $\displaystyle=\Lambda(x)H^{\mu\nu}\
,$ (49cib) $\displaystyle\delta\tilde{F}_{4}$
$\displaystyle=-3\Lambda(x)\tilde{F}_{4}\ ,$ (49cic)

for any function $\Lambda(x)$, so the obvious dilatations are enhanced to a
local symmetry of the non-Lorentzian theory.

###  4.2 Non-Relativistic Limit of the M2-brane

We can use the limit discussed in the previous section to find a consistent
non-relativistic M2-brane metric. Let us split our spacetime coordinates into
$(t,z,\bar{z},u,\bar{u},\vec{v})$, where the transverse coordinate $u$ loosely
corresponds to $\mathcal{Z}^{1}$ and $\vec{v}$ to the real and imaginary parts
of $\mathcal{Z}^{A}$ in the field theory. The metric for a stack of M2-branes
extended along $(t,z,\bar{z})$ at $u=\vec{v}=0$ is

$ds^{2}=\hat{\mathcal{H}}^{-\frac{2}{3}}\left(-dt^{2}+dzd\bar{z}\right)+\hat{\mathcal{H}}^{\frac{1}{3}}\left(dud\bar{u}+d\vec{v}\cdot
d\vec{v}\right)\ ,$ (49cj)

where the function $\hat{\mathcal{H}}$ is

$\hat{\mathcal{H}}=1+\frac{\hat{R}^{6}}{\left(u\bar{u}+\vec{v}\cdot\vec{v}\right)^{3}}\
.$ (49ck)

Recall that we took the scalings (26) and (30) for the coordinates and fields
in the field theory. We can use a homogeneous conformal
transformation111111This is a symmetry of ABJM and will therefore give an
equivalent scaling limit. to put this into the form

$\displaystyle\left(t,z,\mathcal{Z}^{1},\mathcal{Z}^{A}\right)\to\left(ct,c^{-\frac{1}{2}}z,c\mathcal{Z}^{1},c^{-\frac{1}{2}}\mathcal{Z}^{A}\right)\
,$ (49cl)

where the scaling limit takes $c\to\infty$. As we are considering a limit of
supergravity in which the Planck length is unchanged, it is natural to use the
field theory scaling to postulate the coordinate transformation

$\left(t,z,u,\vec{v}\right)\to\left(ct,c^{-\frac{1}{2}}z,cu,c^{-\frac{1}{2}}\vec{v}\right)$
(49cm)

for the metric (49cj). If we require that $\hat{\mathcal{H}}$ is non-trivial
in the limit $c\to\infty$ then we must also scale the parameter $\hat{R}$ to

$\hat{R}=cR\ ,$ (49cn)

where $R$ is taken to be finite and independent of $c$. Note that in contrast
to the issues discussed in [10] this limit is physical, corresponding to an
infinite number of M2-branes. After implementing the scaling we find that the
metric becomes

$ds^{2}\to
c^{2}\bigg{[}-\mathcal{H}^{-\frac{2}{3}}dt^{2}+\mathcal{H}^{\frac{1}{3}}dud\bar{u}\bigg{]}+\frac{1}{c}\bigg{[}\mathcal{H}^{-\frac{2}{3}}dzd\bar{z}+\mathcal{H}^{\frac{1}{3}}dv^{s}dv^{s}\bigg{]}\
,$ (49co)

where

$\mathcal{H}=1+\frac{R^{6}}{\left(u\bar{u}+c^{-3}\vec{v}\cdot\vec{v}\right)^{3}}\
.$ (49cp)

We see that as $c\to\infty$ the metric becomes a membrane Newton-Cartan
structure with

$\displaystyle\tau_{\mu\nu}dx^{\mu}\otimes dx^{\nu}$
$\displaystyle=-\left(1+\frac{R^{6}}{\left(u\bar{u}\right)^{3}}\right)^{-\frac{2}{3}}dt\otimes
dt+\frac{1}{2}\left(1+\frac{R^{6}}{\left(u\bar{u}\right)^{3}}\right)^{\frac{1}{3}}\left(du\otimes
d\bar{u}+d\bar{u}\otimes du\right)\ ,$ (49cqa) $\displaystyle
H^{\mu\nu}\frac{\partial}{\partial x^{\mu}}\otimes\frac{\partial}{\partial
x^{\nu}}$
$\displaystyle=2\left(1+\frac{R^{6}}{\left(u\bar{u}\right)^{3}}\right)^{\frac{2}{3}}\left(\partial\otimes\bar{\partial}+\bar{\partial}\otimes\partial\right)+\left(1+\frac{R^{6}}{\left(u\bar{u}\right)^{3}}\right)^{-\frac{1}{3}}\left(\frac{\partial}{\partial\vec{v}}\otimes\frac{\partial}{\partial\vec{v}}\right)\
.$ (49cqb)

From here onwards we will refer to $(t,u,\bar{u})$ as $\tau$ coordinates and
the rest as $H$ coordinates.

In the near-horizon limit, where we take $R$ to be much larger than any other
scale at which we probe the geometry, the spatial and temporal metrics
simplify to

$\displaystyle\tau_{\mu\nu}dx^{\mu}\otimes dx^{\nu}$
$\displaystyle=-\frac{\left(u\bar{u}\right)^{2}dt\otimes
dt}{R^{4}}+\frac{R^{2}}{2u\bar{u}}\left(du\otimes d\bar{u}+d\bar{u}\otimes
du\right)\ ,$ (49cra) $\displaystyle H^{\mu\nu}\frac{\partial}{\partial
x^{\mu}}\otimes\frac{\partial}{\partial x^{\nu}}$
$\displaystyle=\frac{2R^{4}}{\left(u\bar{u}\right)^{2}}\left(\partial\otimes\bar{\partial}+\bar{\partial}\otimes\partial\right)+\frac{u\bar{u}}{R^{2}}\left(\frac{\partial}{\partial\vec{v}}\otimes\frac{\partial}{\partial\vec{v}}\right)\
.$ (49crb)

The geometry defined by the $\tau$ directions is $AdS_{2}\times S^{1}$,
whereas the $H$ geometry is somewhat more exotic and consists of two planes
that grow and shrink as $u$ varies. As discussed in the previous section, we
can introduce the projective inverses

$\displaystyle\tau^{\mu\nu}\frac{\partial}{\partial
x^{\mu}}\otimes\frac{\partial}{\partial x^{\nu}}$
$\displaystyle=-\frac{R^{4}}{\left(u\bar{u}\right)^{2}}\frac{\partial}{\partial
t}\otimes\frac{\partial}{\partial
t}+\frac{2u\bar{u}}{R^{2}}\left(\frac{\partial}{\partial
u}\otimes\frac{\partial}{\partial\bar{u}}+\frac{\partial}{\partial\bar{u}}\otimes\frac{\partial}{\partial
u}\right)\ ,$ (49csa) $\displaystyle H_{\mu\nu}dx^{\mu}\otimes dx^{\nu}$
$\displaystyle=\frac{\left(u\bar{u}\right)^{2}}{2R^{4}}\left(dz\otimes
d\bar{z}+d\bar{z}\otimes dz\right)+\frac{R^{2}d\vec{v}\otimes
d\vec{v}}{u\bar{u}}\ ,$ (49csb)

though we again note that these are only defined up to the local Galilean
boosts (49bz). When we require the use of vielbeins we will take

$\displaystyle\tau^{t}$ $\displaystyle=\frac{u\bar{u}}{R^{2}}dt\ ,$ (49cta)
$\displaystyle\tau^{u}$ $\displaystyle=\frac{Rdu}{u}\ ,$ (49ctb)

and

$\displaystyle e^{z}$ $\displaystyle=\frac{u^{2}}{R^{2}}dz\ ,$ (49cua)
$\displaystyle e^{\vec{v}}$ $\displaystyle=\frac{R}{\sqrt{u\bar{u}}}d\vec{v}\
.$ (49cub)

In order for the limiting metric to be a solution of the eleven-dimensional
non-relativistic theory we must check that its $C$-field admits a
decomposition of the form (49cb) and that the constraint (49cda) is satisfied.
The $C$-field for the relativistic M2 solution is

$\hat{C}_{3}=\hat{\mathcal{H}}^{-1}dt\wedge dx^{1}\wedge dx^{2}+k$ (49cv)

for some constant 3-form $k$ that will drop out of all physical quantities.
Upon taking the scaling (49cm) and the near-horizon limit this becomes

$\hat{C}_{3}=\frac{i}{2}dt\wedge dz\wedge
d\bar{z}\left(\frac{\left(u\bar{u}\right)^{3}}{R^{6}}+\frac{3\left(u\bar{u}\right)^{2}\vec{v}\cdot\vec{v}}{c^{3}R^{6}}+O\left(\frac{1}{c^{6}}\right)\right)+k\
,$ (49cw)

which appears to not match (49cb) due to the absence of a $c^{3}$ term.
However, the volume form in the $\tau$ directions is

$\frac{1}{6}\epsilon_{abc}\tau^{a}\wedge\tau^{b}\wedge\tau^{c}=\frac{i}{2}dt\wedge
du\wedge d\bar{u}\ ,$ (49cx)

so taking the constant 3-form to be

$k=-\frac{i}{2}c^{3}dt\wedge du\wedge d\bar{u}$ (49cy)

we see that the expansion of $\hat{C}_{3}$ takes the desired form. It should
be noted that, up to an overall constant, $k$ is identical to the background
3-form field (33) used to cancel the divergent boundary terms in the field
theory.

From this we find that $F_{4}$ is

$F_{4}=\frac{3i\left(u\bar{u}\right)^{2}}{2R^{6}}\left(ud\bar{u}+\bar{u}du\right)\wedge
dt\wedge dz\wedge d\bar{z}\ ,$ (49cz)

so both sides of (49cda) vanish identically. We therefore expect that the
scaled solution solves the equations of motion of the non-Lorentzian theory.
We can also read off the subleading field strength

$\tilde{F}_{4}=\frac{3i\left(u\bar{u}\right)^{2}}{R^{6}}\left(\vec{v}\cdot\vec{v}\left(\frac{du}{u}+\frac{d\bar{u}}{\bar{u}}\right)+\vec{v}\cdot
d\vec{v}\right)\wedge dt\wedge dz\wedge d\bar{z}\ .$ (49da)

Unfortunately, the supersymmetric extension of the Bosonic theory presented in
[11] is not known, so we can’t discuss whether our solution preserves any
supercharges from the non-Lorentzian perspective. In a way, the conjecture
that there is still a holographic duality between the near-BPS limit field
theory discussed in section 3 and a prospective eleven-dimensional membrane
Newton-Cartan supergravity theory in asymptotically AdS spacetimes (i.e. the
asymptotic behaviour of the metric resembles (49cr)) is a prediction that the
supersymmetric completion of the gravity theory exists, and that the metric
structures (49cr) and 4-form field strengths (49cz) and (49da) form a
maximally supersymmetric solution of the theory.

###  4.3 Gravitational Symmetries

####  4.3.1 Isometries of the Near-Horizon Geometry

An infinitesimal coordinate transformation $\delta x^{\mu}=\xi^{\mu}$ is an
isometry of the Newton-Cartan structure if both $\tau$ and $H$ are unchanged
up to local invariances. Recalling that these are the local Galilean boosts
and dilatations, the transformations need to obey the equations

$\displaystyle 0$
$\displaystyle=\mathcal{L}_{\xi}\tau_{\mu\nu}+2\Lambda\tau_{\mu\nu}\ ,$
(49dba) $\displaystyle 0$
$\displaystyle=\mathcal{L}_{\xi}\tau^{\mu\nu}-2\tau_{a}^{(\mu}H^{\nu)\rho}\lambda_{I}^{a}e^{I}_{\rho}-2\Lambda\tau^{\mu\nu}\
,$ (49dbb) involving $\tau$, and the equations $\displaystyle 0$
$\displaystyle=\mathcal{L}_{\xi}H^{\mu\nu}+\Lambda H^{\mu\nu}\ ,$ (49dbc)
$\displaystyle 0$
$\displaystyle=\mathcal{L}_{\xi}H_{\mu\nu}+2\eta_{ab}\lambda^{a}_{I}e^{I}_{(\mu}\tau^{b}_{\nu)}-\Lambda
H_{\mu\nu}\ ,$ (49dbd)

involving $H$.

Let’s focus for now on (49dba). The mixed $\tau$ and $H$ coordinate components
just impose that $\xi^{t}$ and $\xi^{u}$ are independent of the $H$
coordinates, and the $uu/\bar{u}\bar{u}$ components impose that $\xi^{u}$
cannot depend on $\bar{u}$. The system of equations that we have to solve is
then

$\displaystyle\frac{1}{u}\xi^{u}+\frac{1}{\bar{u}}\xi^{\bar{u}}$
$\displaystyle=\frac{1}{3}\left(\partial_{u}\xi^{u}+\partial_{\bar{u}}\xi^{\bar{u}}-2\partial_{t}\xi^{t}\right)\
,$ (49dca) $\displaystyle\partial_{t}\xi^{u}$
$\displaystyle=\frac{2\left(u\bar{u}\right)^{3}}{R^{6}}\partial_{\bar{u}}\xi^{t}\
,$ (49dcb) $\displaystyle\Lambda$
$\displaystyle=-\frac{1}{3}\left(\partial_{t}\xi^{t}+\partial_{u}\xi^{u}+\partial_{\bar{u}}\xi^{\bar{u}}\right)\
.$ (49dcc)

A bit of work shows that the solution to this is

$\displaystyle\xi^{t}$
$\displaystyle=a+bt+ct^{2}+\frac{cR^{6}}{4\left(u\bar{u}\right)^{2}}-\frac{R^{6}}{4\bar{u}^{2}}\left(\beta+2\gamma
t\right)-\frac{R^{6}}{4u^{2}}\left(\bar{\beta}+2\bar{\gamma}t\right)\ ,$
(49dda) $\displaystyle\xi^{u}$
$\displaystyle=-\frac{b}{2}u-cut+u^{3}\left(\alpha+\beta t+\gamma
t^{2}\right)+\frac{R^{6}\bar{\gamma}}{4u}+i\theta u\ ,$ (49ddb)
$\displaystyle\Lambda$ $\displaystyle=-\left(\alpha+\beta t+\gamma
t^{2}\right)u^{2}-\left(\bar{\alpha}+\bar{\beta}t+\bar{\gamma}t^{2}\right)\bar{u}^{2}+\frac{R^{6}\bar{\gamma}}{4u^{2}}+\frac{R^{6}\gamma}{4\bar{u}^{2}}\
,$ (49ddc)

in terms of real infinitesimal parameters $\\{a,b,c,\theta\\}$ and complex
infinitesimal parameters $\\{\alpha,\beta,\gamma\\}$. The algebra the
transformations form is isomorphic to $\mathfrak{so}(2,3)$, which can be seen
by noting that $\tau$ is conformally equivalent to the flat metric on
$\mathbb{R}^{1,2}$ and the solutions to (49dba) are the conformal Killing
vectors of $\tau$.

We can work through (49dbc) or (49dbd) in a similar way. The only difference
between the two arises in the purely $\tau$ and mixed component equations. In
(49dbc), the purely $\tau$ components vanish and the mixed components impose
the same constraint found in (49dba) that
$\\{\xi^{t},\xi^{u},\xi^{\bar{u}}\\}$ must be independent of the $H$
coordinates. In (49dbd), the purely $\tau$ components of the equation impose
that the $\tau$ components of the 1-forms $\lambda^{a}$ vanish. The mixed
components fix the form of the Galilean boosts required for the
transformations found from (49dba) and allow for any new constants of
integration introduced in $\\{\xi^{z},\vec{\xi}\\}$ to be given arbitrary
dependence on $(t,u,\bar{u})$ through an appropriate choice of the
aforementioned 1-forms. This is consistent with (49dbc) as in that formulation
any constants introduced in the solution of these equations alone are never
subject to derivatives along the $\tau$ directions, so their dependence on
these is arbitrary.

The remaining equations to be solved are

$\displaystyle\partial_{z}\xi^{z}+\partial_{\bar{z}}\xi^{\bar{z}}$
$\displaystyle=-2\left(\frac{1}{u}\xi^{u}+\frac{1}{\bar{u}}\xi^{\bar{u}}\right)+\Lambda\
,$ (49dea) $\displaystyle\partial_{\bar{z}}\xi^{z}$ $\displaystyle=0\ ,$
(49deb) $\displaystyle\partial_{r}\xi^{s}+\partial_{s}\xi^{r}$
$\displaystyle=\delta_{rs}\left(\frac{1}{u}\xi^{u}+\frac{1}{\bar{u}}\xi^{\bar{u}}+\Lambda\right)\
,$ (49dec) $\displaystyle
2\partial_{\bar{z}}\xi^{r}+\left(\frac{u\bar{u}}{R^{2}}\right)^{3}\partial_{r}\xi^{z}$
$\displaystyle=0\ .$ (49ded)

where we have introduced indices $r,s=1,2,3,...,6$ for vectors in
${\mathbb{R}}^{6}$; $\vec{x}\to\xi^{r}$, $\vec{\xi}\to\xi^{r}$. The solutions
to these can be taken to be121212There is some ambiguity in the split of some
terms between $\xi^{z}$ and $\xi^{\bar{z}}$, but this just corresponds to a
particular parameterisation of the function $r$.

$\displaystyle\xi^{z}$ $\displaystyle=\left(b+2ct-3u^{2}\left(\alpha+\beta
t+\gamma
t^{2}\right)-\frac{R^{6}\bar{\gamma}}{4u^{2}}\right)z+\chi+2v^{a}\rho^{a}+irz\
,$ (49dfa) $\displaystyle\xi^{r}$
$\displaystyle=\left(\frac{R^{6}\bar{\gamma}}{4u^{2}}+\frac{R^{6}\gamma}{4\bar{u}^{2}}-\frac{b}{2}-ct\right)v^{r}+\tensor{R}{{}^{r}_{s}}v^{s}+k^{r}+\left(\frac{u\bar{u}}{R^{2}}\right)^{3}\left(\bar{z}\rho^{r}+z\bar{\rho}^{r}\right)\
,$ (49dfb)

where $\\{\chi,\vec{\rho}\\}$ are complex, $\\{r,R,k\\}$ are real, $R$ is
antisymmetric, and all are arbitrary functions of $(t,u,\bar{u})$. However, to
make contact with the field theory we must implement a $\mathbb{Z}_{k}$
orbifold of the $\mathbb{R}^{6}$ parameterised by $\\{\vec{v}\\}$ as discussed
in appendix C. Requiring that the Killing vectors are globally defined on the
orbifold forces us to take

$\vec{k}=\vec{\rho}=0\ ,$ (49dg)

which we shall assume from here onwards, and restricts the allowed rotation
matrices $R$ to an $\mathfrak{su}(3)\oplus\mathfrak{u}(1)$ subalgebra. We can
then read off the required local Galilean boosts from (49dbd), where we find

$\displaystyle\lambda^{t}_{z}$
$\displaystyle=\frac{\bar{u}}{2u}\Big{[}\bar{z}\left(2c-3\bar{u}^{2}\left(\bar{\beta}+2\bar{\gamma}t\right)-i\partial_{t}r\right)+\partial_{t}\bar{\chi}\Big{]}\
,$ (49dha) $\displaystyle\lambda^{t}_{r}$
$\displaystyle=\frac{R^{3}}{\left(u\bar{u}\right)^{\frac{3}{2}}}\Big{[}\partial_{t}\tensor{R}{{}^{r}_{s}}v^{s}-cv^{r}\Big{]}\
,$ (49dhb) $\displaystyle\lambda^{u}_{z}$
$\displaystyle=\frac{\bar{u}^{3}}{R^{3}}\bigg{[}\bar{z}\left(6\bar{u}\left(\bar{\alpha}+\bar{\beta}t+\bar{\gamma}t^{2}\right)+i\partial_{\bar{u}}r-\frac{R^{6}\gamma}{2\bar{u}^{3}}\right)-\partial_{\bar{u}}\bar{\chi}\bigg{]}\
,$ (49dhc) $\displaystyle\lambda^{\bar{u}}_{z}$
$\displaystyle=\frac{u\bar{u}^{2}}{R^{3}}\Big{[}i\bar{z}\partial_{u}r-\partial_{u}\bar{\chi}\Big{]}\
,$ (49dhd) $\displaystyle\lambda^{u}_{r}$
$\displaystyle=\sqrt{\frac{\bar{u}}{u}}\bigg{[}\frac{R^{6}\gamma}{\bar{u}^{3}}v^{r}-2\partial_{\bar{u}}\tensor{R}{{}^{r}_{s}}v^{s}\bigg{]}\
.$ (49dhe)

Finally, we must check that everything computed above is consistent with
(49dbb). The components of the equation vanish when both $\mu$ and $\nu$ are
spatial indices, and when both are temporal the equation is automatically
satisfied as $\tau^{\mu\nu}$ is the inverse of $\tau_{\mu\nu}$ when projected
onto the temporal coordinates. The only non-trivial check that must be
performed is therefore for the mixed components, for which the equation is

$\partial_{\nu}\xi^{r}\tau^{\mu\nu}+\tau_{a}{}^{\mu}H^{rs}e_{s}^{I}\lambda^{a}_{I}=0\
.$ (49di)

It is straightforward to check that every component of this is satisfied for
the spatial Killing vector components (49df) and local boosts (49dh).

####  4.3.2 Form-Field Symmetries

So far we have only determined which coordinate transformations are isometries
of the metric structures. However, this is not enough to determine whether we
have a symmetry of the solution; the 3-form field $C_{3}$ and 4-form field
$\tilde{F}_{4}$ must also be invariant. The transformation of the fields under
the isometries given by the vector field $\xi^{\mu}$ and local boosts
$\lambda^{a}_{I}$ are given by

$\displaystyle\delta C_{3}$
$\displaystyle=\mathcal{L}_{\xi}C_{3}-\frac{1}{2}\epsilon_{abc}\lambda^{a}_{I}e^{I}\wedge\tau^{b}\wedge\tau^{c}+d\sigma_{2}\
,$ (49dja) $\displaystyle\delta\tilde{F}_{4}$
$\displaystyle=\mathcal{L}_{\xi}\tilde{F}_{4}+\lambda_{I}^{a}e^{I}\wedge\iota_{\tau_{a}}F_{4}\
.$ (49djb)

Note that by including $\sigma_{2}\in\Omega^{2}(\mathcal{M})$ we allow for
arbitrary gauge transformations, which are of course a local symmetry of the
theory. The symmetries of the solution are then determined by the conditions
$\delta C_{3}=\delta\tilde{F}_{4}=0$.

We will deal with $C_{3}$ first. The transformations parameterised by
$\\{a,b,\theta\\}$ all leave $C_{3}$ invariant and aren’t associated with any
local boosts, so these isometries are full symmetries of our solution. The
transformation parameterised by $c$ is non-trivial, but the new terms in
$\delta C_{3}$ are exact (and can therefore be absorbed by a particular choice
of $\sigma_{2}$) so this is also a symmetry. Things get more interesting when
we consider the transformation parameterised by $\alpha$. The Lie derivative
and local transformations are given by

$\displaystyle\mathcal{L}_{\xi}C_{3}$
$\displaystyle=-\frac{3i\left(u\bar{u}\right)^{3}}{R^{6}}\left(\alpha
uzdt\wedge du\wedge d\bar{z}+\bar{\alpha}\bar{u}\bar{z}dt\wedge dz\wedge
d\bar{u}\right)\ ,$ (49dka)
$\displaystyle\epsilon_{abc}\lambda^{a}_{I}e^{I}\wedge\tau^{b}\wedge\tau^{c}$
$\displaystyle=\frac{6i(u\bar{u})^{3}}{R^{6}}\left(\alpha uzdt\wedge du\wedge
d\bar{z}+\bar{\alpha}\bar{u}\bar{z}dt\wedge dz\wedge d\bar{u}\right)\ ;$
(49dkb)

as these do not cancel and are not exact we see that $\delta C_{3}\neq 0$, so
$\alpha$ is not a symmetry of $C_{3}$. Similarly, the $\beta$ transformation
induces the terms

$\displaystyle\mathcal{L}_{\xi}C_{3}=\,$
$\displaystyle-\frac{3it\left(u\bar{u}\right)^{3}}{R^{6}}\left(\beta
zudt\wedge du\wedge d\bar{z}+\bar{\beta}\bar{z}\bar{u}dt\wedge dz\wedge
d\bar{u}\right)$
$\displaystyle+\frac{R^{6}}{2}\left(\frac{\bar{\beta}du}{u^{2}}+\frac{\beta
d\bar{u}}{\bar{u}^{3}}\right)\wedge dz\wedge d\bar{z}\ ,$ (49dla)
$\displaystyle\epsilon_{abc}\lambda^{a}_{I}e^{I}\wedge\tau^{b}\wedge\tau^{c}=\,$
$\displaystyle\frac{6it\left(u\bar{u}\right)^{3}}{R^{6}}\left(\beta zudt\wedge
du\wedge d\bar{z}+\bar{\beta}\bar{z}\bar{u}dt\wedge dz\wedge d\bar{u}\right)$
$\displaystyle-\frac{3i}{2}\left(\beta
zu^{2}d\bar{z}+\bar{\beta}\bar{z}\bar{u}^{2}dz\right)\wedge du\wedge d\bar{u}\
,$ (49dlb)

while $\gamma$ induces

$\displaystyle\mathcal{L}_{\xi}C_{3}=\,$
$\displaystyle\frac{it}{2}\left(\gamma
u^{3}d\bar{u}+\bar{\gamma}\bar{u}^{3}du\right)\wedge dz\wedge d\bar{z}$
$\displaystyle+\frac{idt}{4}\wedge\left(\gamma u^{3}\bar{z}dz\wedge
d\bar{u}+\bar{\gamma}\bar{u}^{3}zdu\wedge d\bar{z}\right)\ ,$ (49dma)
$\displaystyle\epsilon_{abc}\lambda^{a}_{I}e^{I}\wedge\tau^{b}\wedge\tau^{c}=\,$
$\displaystyle-3it\left(\gamma
uzd\bar{z}+\bar{\gamma}\bar{u}\bar{z}dz\right)\wedge du\wedge d\bar{u}$
$\displaystyle+\frac{i\left(u\bar{u}\right)^{3}\bar{z}}{R^{6}}\left(6t\bar{u}\bar{\gamma}-\frac{R^{6}\gamma}{2\bar{u}^{3}}\right)dt\wedge
dz\wedge d\bar{u}$
$\displaystyle+\frac{i\left(u\bar{u}\right)^{3}z}{R^{6}}\left(6tu\gamma-\frac{R^{6}\bar{\gamma}}{2u^{3}}\right)dt\wedge
du\wedge d\bar{z}\ ,$ (49dmb)

so neither of these are symmetries of $C_{3}$.

What about the transformations acting solely on the $H$ coordinates? Recalling
that the metric transformations (49db) allowed
$\\{\chi,r,\tensor{R}{{}^{r}_{s}}\\}$ to have arbitrary dependence on the
$\tau$ coordinates, the transformation of $C_{3}$ is

$\displaystyle\mathcal{L}_{\xi}C_{3}=\,$
$\displaystyle\frac{i\left(u\bar{u}\right)^{3}}{2R^{6}}\bigg{(}\partial_{u}\chi
dt\wedge du\wedge d\bar{z}+\partial_{\bar{u}}\chi dt\wedge d\bar{u}\wedge
d\bar{z}$ $\displaystyle+\partial_{u}\bar{\chi}dt\wedge dz\wedge
du+\partial_{\bar{u}}\bar{\chi}dt\wedge dz\wedge d\bar{u}\bigg{)}\ ,$ (49dna)
$\displaystyle\epsilon_{abc}\lambda^{a}_{I}e^{I}\wedge\tau^{b}\wedge\tau^{c}=\,$
$\displaystyle\frac{i}{2}\partial_{t}\left(\bar{\chi}dz+\chi
d\bar{z}\right)\wedge du\wedge d\bar{u}$
$\displaystyle-\frac{i\left(u\bar{u}\right)^{3}}{R^{6}}\bigg{(}\partial_{\bar{u}}\left(\bar{\chi}dz+\chi
d\bar{z}\right)\wedge d\bar{u}\wedge dt$ (49dnb)
$\displaystyle+\partial_{u}\left(\bar{\chi}dz+\chi d\bar{z}\right)\wedge
dt\wedge du\bigg{)}\ ,$

for $\chi$,

$\displaystyle\mathcal{L}_{\xi}C_{3}=\,$
$\displaystyle-\frac{\left(u\bar{u}\right)^{3}}{2R^{6}}\bigg{(}\partial_{u}rdt\wedge
du\wedge d\left(z\bar{z}\right)-\partial_{\bar{u}}rdt\wedge
d\left(z\bar{z}\right)\wedge d\bar{u}\bigg{)}\ ,$
$\displaystyle\epsilon_{abc}\lambda^{a}_{I}e^{I}\wedge\tau^{b}\wedge\tau^{c}=\,$
$\displaystyle\frac{1}{2}\partial_{t}r\left(\bar{z}dz-zd\bar{z}\right)\wedge
du\wedge d\bar{u}$
$\displaystyle-\frac{\left(u\bar{u}\right)^{3}}{R^{6}}\bigg{(}\partial_{\bar{u}}r\left(\bar{z}dz-
zd\bar{z}\right)\wedge d\bar{u}\wedge dt$ (49doa)
$\displaystyle+\partial_{u}r\left(\bar{z}dz-zd\bar{z}\right)\wedge dt\wedge
du\bigg{)}\ ,$

for $r$, and

$\displaystyle\mathcal{L}_{\xi}C_{3}=\,$ $\displaystyle 0\ ,$ (49dpa)
$\displaystyle\epsilon_{abc}\lambda_{I}^{a}e^{I}\wedge\tau^{b}\wedge\tau^{c}=\,$
$\displaystyle\frac{iR^{6}}{\left(u\bar{u}\right)^{3}}\partial_{t}\tensor{R}{{}^{r}_{s}}v^{s}dv^{r}\wedge
du\wedge d\bar{u}$ $\displaystyle-2iv^{s}dv^{r}\wedge
dt\wedge\left(\partial_{u}\tensor{R}{{}^{r}_{s}}du-\partial_{\bar{u}}\tensor{R}{{}^{r}_{s}}d\bar{u}\right)\
,$ (49dpb)

for $\tensor{R}{{}^{r}_{s}}$. We see that these are only symmetries of $C_{3}$
if the functions are taken to be constants. The symmetries of $C_{3}$ are then
the transformations with constant parameters
$\\{a,b,c,\theta,\chi,r,\tensor{R}{{}^{r}_{s}}\\}$. It is straightforward to
check that (49djb) vanishes for each of these, so the symmetry algebra of the
gravitational solution is

$\mathfrak{g}=\mathfrak{so}(1,2)\oplus\mathfrak{u}(1)_{\theta}\oplus\mathfrak{iso}(2)\oplus\mathfrak{su}(3)\oplus\mathfrak{u}(1)_{R}\
.$ (49dq)

####  4.3.3 Boundary Structure and Asymptotic Symmetries

In the previous section we found the exact symmetries of the non-relativistic
eleven-dimensional background. However, in the holographic context we expect
to match global symmetries of our field theory with the asymptotic symmetries
of the gravity dual, that is diffeomorphisms which only need to be symmetries
as we approach the boundary. While this necessarily contains the symmetries
previously discussed, there is the potential for an extended symmetry group to
emerge once these are taken into account. These can either take the form of
new transformations that leave the Newton-Cartan metric structures and 3-form
field invariant at leading order near the boundary, or exact isometries of the
metric structure that are only symmetries of the 3-form field asymptotically.

To discuss the boundary in the (relativistic) $AdS_{4}$ solution we can go to
the Poincaré patch with radial coordinate $\hat{\rho}$ defined by the
coordinate transformation

$\hat{\rho}=\frac{\hat{R}^{3}}{2\hat{r}^{2}}\ ,$ (49dr)

so that the boundary lies at $\hat{\rho}=0$. In our coordinates we have

$\hat{\rho}=\frac{cR^{3}}{2\left(u^{2}+\frac{\vec{v}\cdot\vec{v}}{c^{3}}\right)}\
.$ (49ds)

We can define analogous coordinates for the $AdS_{2}$ in $\tau$ and radial
coordinate in $H$ by

$\displaystyle\rho$ $\displaystyle=\frac{R^{3}}{2u\bar{u}}\ ,$ (49dta)
$\displaystyle\sigma$ $\displaystyle=\frac{R^{3}}{2\vec{v}\cdot\vec{v}}\ ,$
(49dtb)

so

$\hat{\rho}=\frac{c\rho\sigma}{\sigma+\frac{1}{c^{3}}\rho}\ .$ (49du)

At finite $c$ the $AdS_{4}$ boundary $\hat{\rho}=0$ corresponds to $\rho=0$ or
$\sigma=0$. However in the non-relativistic limit $c\to\infty$ we have, for
any non-zero $\sigma$, $\hat{\rho}=c\rho$. Thus to match the isometries found
in the previous section onto the field theory transformations we take the
boundary to be $\rho=0$, which is a subset of the original boundary. The
metric structures and 3-form field in this coordinate system are

$\displaystyle\tau_{\mu\nu}dx^{\mu}\otimes dx^{\nu}$
$\displaystyle=\frac{R^{2}\left(-dt\otimes dt+d\rho\otimes
d\rho\right)}{4\rho^{2}}+R^{2}d\theta\otimes d\theta\ ,$ (49dva)
$\displaystyle H^{\mu\nu}\frac{\partial}{\partial
x^{\mu}}\otimes\frac{\partial}{\partial x^{\nu}}$
$\displaystyle=\frac{8\rho^{2}}{R^{2}}\left(\partial\otimes\bar{\partial}+\bar{\partial}\otimes\partial\right)+\frac{4\sigma^{3}}{\rho
R^{2}}\frac{\partial}{\partial\sigma}\otimes\frac{\partial}{\partial\sigma}+\frac{\sigma}{\rho
R^{2}}g^{-1}_{S^{5}/\mathbb{Z}_{k}}\ ,$ (49dvb) $\displaystyle C_{3}$
$\displaystyle=\frac{iR^{3}}{16\rho^{3}}dt\wedge dz\wedge d\bar{z}\ ,$ (49dvc)

so if we require that the radius of the $S^{5}/\mathbb{Z}_{k}$ factor remains
finite at the boundary we should also take $\sigma$ to zero with the ratio of
$\rho$ and $\sigma$ held fixed. However, these considerations won’t be
relevant to our discussion. We note in passing that as we approach the
boundary the co-metric in the $z\bar{z}$ plane vanishes, which could be a bulk
indication of the ’gauging’ of spatial transformations in the field theory; we
will not pursue this point any further in this work.

We do not possess a complete understanding of the appropriate boundary
conditions (i.e. the fall-off required in the subleading terms) for a solution
of the theory in [11] to be asymptotically of the form we’ve presented, so we
cannot say with certainty when a transformation is an asymptotic symmetry of
our solution. We will therefore be somewhat heuristic and demand that each
component of the change in our fields is subleading in $\rho$ with respect to
the components (49dv) in the boundary coordinate system, since this seems
likely to be a necessary condition for the system to asymptotically approach
the required form.

As mentioned previously, there are two classes of potential asymptotic
symmetries we could consider. The first are transformations that are exact
symmetries of the metric structures but not of the 3-form field but which may
enhance to symmetries at the boundary. However, upon examining the
transformations of $C_{3}$ for each isometry that don’t form symmetries we see
that every transformation contains terms that aren’t subleading with respect
to the original components as we approach the boundary. We can therefore rule
out this class of asymptotic symmetries.

The second are new transformations that asymptotically preserve the metric,
which we may hope also are symmetries of $C_{3}$ at the boundary. While we
won’t be exhaustive in our discussion of these, there is a particularly
important case we must address. In [26] it was proposed that the duals of
theories invariant under an infinite-dimensional extension of the Galilean
conformal algebra should be Newton-Cartan geometries with an $AdS_{2}$ factor
in $\tau$131313The authors of that work were interested solely in non-
relativistic limits of string theory, so $\tau$ was taken to only have two
non-zero eigenvalues. However, since our $\tau$ geometry factorises into
$AdS_{2}\times S^{1}$ the discussion will be similar., with the infinite-
dimensional extension arising from the asymptotic symmetries of the $AdS_{2}$
metric [27, 28]. Such infinite dimensional symmetries are not present in our
field theory, we would like to see what goes wrong with this argument in our
system. Let us examine an adapted version of their argument for these
symmetries for our geometry. We observe that the infinitesimal transformation

$\displaystyle t^{\prime}$
$\displaystyle=t+f(t)+\frac{R^{6}f^{\prime\prime}}{8\left(u\bar{u}\right)^{2}}\
,$ (49dwa) $\displaystyle u^{\prime}$
$\displaystyle=u\left(1-\frac{f^{\prime}}{2}\right)\ ,$ (49dwb)

changes $\tau$ to

$\tau^{\prime}=-\frac{\left(u\bar{u}\right)^{2}}{R^{4}}\left(1+\frac{R^{6}f^{\prime\prime\prime}}{4\left(u\bar{u}\right)^{2}}\right)dt\otimes
dt+\frac{R^{2}}{2u\bar{u}}\left(du\otimes d\bar{u}+d\bar{u}\otimes du\right)\
,$ (49dx)

leaving the geometry invariant up to terms that are subleading as we approach
the boundary. Combining the transformation of $u$ with

$\displaystyle z^{\prime}$ $\displaystyle=z\left(1+f^{\prime}\right)\ ,$
(49dya) $\displaystyle\vec{v}^{\prime}$
$\displaystyle=\vec{v}\left(1-\frac{f^{\prime}}{2}\right)\ ,$ (49dyb)

leaves $H$ invariant, provided we take the local Galilean boost to be

$\displaystyle\lambda^{t}_{z}$
$\displaystyle=\frac{\bar{u}\bar{z}f^{\prime\prime}}{2u}\ ,$ (49dza)
$\displaystyle\lambda^{t}_{r}$
$\displaystyle=-\frac{R^{3}f^{\prime\prime}v^{r}}{2\left(u\bar{u}\right)^{\frac{3}{2}}}\
.$ (49dzb)

With these, a short computation gives the contributions

$\displaystyle\mathcal{L}_{\xi}C_{3}$
$\displaystyle=\frac{i\left(u\bar{u}\right)f^{\prime\prime\prime}}{2}dt\wedge
dz\wedge d\bar{z}-\frac{if^{\prime\prime}}{8}d\left(u\bar{u}\right)\wedge
dz\wedge d\bar{z}\ ,$ (49eaa)
$\displaystyle\epsilon_{abc}\lambda^{a}_{I}e^{I}\wedge\tau^{b}\wedge\tau^{c}$
$\displaystyle=\frac{if^{\prime\prime}}{2}\left(dz+d\bar{z}\right)\wedge
du\wedge
d\bar{u}-\frac{iR^{6}f^{\prime\prime}}{2\left(u\bar{u}\right)^{3}}\vec{v}\cdot
d\vec{v}\wedge du\wedge d\bar{u}\ ,$ (49eab)

to the transformation of $C_{3}$. After rewriting this in terms of $\rho$ we
see the transformation induces new terms in $C_{3}$ that diverge faster than
the background solution as we take $\rho\to 0$. Thus it is reasonable to
assert that this is not an asymptotic symmetry of the theory.

As we find no additional symmetries, we propose that the asymptotic symmetry
algebra is (49dq). Aside from the $\mathfrak{iso}(2)$ factor, these coincide
with the rigid symmetries of the field theory. At first glance the
$\mathfrak{iso}(2)$ factor appears not to match with the enhanced spatial
conformal symmetry we saw there. However, since we interpret the field theory
symmetries as redundancies this is perfectly natural- we’re only interested in
matching the symmetries associated with charges that act non-trivially on the
phase space across both sides of the duality. From this perspective, the
oddity is the appearance of the factor of $\mathfrak{iso}(2)$. For our
conjectured duality to hold, the charges generated by these Killing vectors
should vanish. Showing this goes beyond the scope of this paper, and is a
topic for future work.

##  5 Conclusion

In this paper we have discussed a peculiar non-relativistic limit of the
Chern-Simons-matter theory associated to M2-branes. The resulting theory has
been constructed and analysed before in [29, 13, 30], where it was shown to
maintain the same number of supersymmetries as the parent theory and its
dynamics reduces to motion on a Hitchin moduli space. Here we saw that the
spatial symmetry algebra of the theory is an infinite-dimensional enhancement
of the algebra one would naively expect to obtain from a non-relativistic
limit of ABJM. However, as the associated charge reduces to a boundary term we
interpret these transformations as redundancies of the theory. We also
considered the corresponding limit taken in the dual eleven-dimensional
supergravity theory. This turned out to be a special case of the general
membrane-Newton-Cartan limit given in [11], giving us a solution in which the
$AdS_{4}$ factor in the relativistic theory is reduced to an $AdS_{2}$. The
symmetries of the solution were calculated and found to contain the physical
field theory symmetries, leading us to propose that the duality between the
two theories remains after taking the limit on both sides.

There are some outstanding questions that require further study. In particular
it is important to better understand the boundary in the dual AdS geometry. We
argued above that the field theory resides at the $\rho=0$ boundary of
$AdS_{2}$. While this is a subset of the original $AdS_{4}$ boundary, we have
effectively ignored the section of the boundary at $\sigma=0$. As this lies in
$H$ and not $\tau$ the interpretation of this is not clear. On a related note,
unlike in well-understood AdS duals the non-field theoretic part of the $H$
geometry is non-compact and singular as we approach the $\rho\to 0$ boundary.
Since we have discussed a limit of a well-defined AdS/CFT pair we expect that
the pathologies associated to this can be alleviated. For instance, a
prescription where we consider the boundary $\rho\to 0$ and $\sigma\to 0$ with
their ratio fixed would answer both this question and the one previously
discussed, though it is not obvious that this is the correct thing to
consider. It is desirable to have a clearer understanding of these matters.
Another issue is the physical role of the $\mathfrak{iso}(2)$ algebra symmetry
of the gravitational solution. As discussed in section 4.3.3, for the proposed
duality to hold these transformations should not lead to an action on the
phase space of the theory; a calculation determining whether this is true, as
well a broader discussion of charges in Newton-Cartan-type gravity theories,
is therefore of considerable interest.

We close by discussing some future directions it may be fruitful to pursue. In
the regime $k^{5}\gg N$ M-theory on $AdS_{4}\times S^{7}/{\mathbb{Z}}_{k}$
backgrounds reduces to type IIA string theory on
$AdS_{4}\times{\mathbb{C}P}^{3}$ backgrounds, for which there is a known exact
worldsheet CFT [31, 32]. It is therefore reasonable to ask if there is a non-
relativistic limit of the string $\sigma$-model that is related to the large
$k$ limit of the MNC solution. Though we have only discussed a single AdS/CFT
pairing in this work, we also expect there to be similar non-relativistic
limits for other dualities. The obvious example to discuss would be a limit of
$\mathcal{N}=4$ super Yang-Mills that isolates BPS field configurations, along
with a corresponding limit of type IIB supergravity on $AdS_{5}\times S^{5}$
backgrounds. As this is closely related to the Spin Matrix Theory limits of
$\mathcal{N}=4$ [15] it would be illuminating to see if there is a link
between the two approaches.

## Acknowledgements

We would like to thank Dionysios Anninos and Chris Blair for interesting
discussions. N.L. is supported in part by the STFC consolidated grant
ST/X000753/1. J.S. is supported by the STFC studentship ST/W507556/1

##  Appendix A Fermions

###  A.1 Fermion Action and Symmetries

The Bosonic action (3.1) admits a supersymmetric completion [13] that arises
by taking the non-relativistic limit of the Fermion terms in the ABJM action:
let us review how this works. To find the correct limit we must split the
spinor fields into chiral components with respect to the matrix

$\Gamma=i\gamma^{0}\ ,$ (49eb)

where $\\{\gamma^{\mu}\\}$ are chosen to be real 2x2 matrices satisfying
$\gamma^{0}\gamma^{1}\gamma^{2}=\mathbbm{1}$. In other words, using the
projection operator $P_{\pm}=\frac{1}{2}\left(\mathbbm{1}\pm\Gamma\right)$ we
define

$\hat{\psi}_{M}^{\pm}=P_{\pm}\hat{\psi}^{M}\ .$ (49ec)

As our spinors initially had 2 components, the chiral fields are (single-
component) complex Grassmann-valued fields. It will be convenient to work with
the complex-coordinate gamma matrices

$\gamma_{z}=\frac{1}{2}\left(\gamma_{1}-i\gamma_{2}\right)\ ,$ (49ed)

and its conjugate. We see that these satisfy

$\displaystyle\gamma_{z}\Gamma$ $\displaystyle=\gamma_{z}\ ,$ (49eea)
$\displaystyle\gamma_{\bar{z}}\Gamma$ $\displaystyle=-\gamma_{\bar{z}}\ ,$
(49eeb)

so $\gamma_{z}$ annihilates spinors of negative chirality (and similarly for
$\gamma_{\bar{z}}$). Putting this all together, we can rewrite the ABJM
Fermion terms as

$\displaystyle\hat{S}_{F}=\tr\int d^{3}\hat{x}\bigg{(}$ $\displaystyle
i\hat{\bar{\psi}}^{M,+}\hat{D}_{0}\hat{\psi}_{M}^{+}+i\hat{\bar{\psi}}^{M,-}\hat{D}_{0}\hat{\psi}^{-}_{M}-2i\hat{\bar{\psi}}^{M,+}\hat{D}\hat{\psi}_{M}^{-}-2i\hat{\bar{\psi}}^{M,-}\hat{\bar{D}}\hat{\psi}_{M}^{+}$
$\displaystyle-\frac{2\pi}{k}\hat{\bar{\psi}}^{M,+}[\hat{\psi}_{M}^{+},\hat{\mathcal{Z}}^{N};\hat{\bar{\mathcal{Z}}}_{N}]+\frac{2\pi}{k}\hat{\bar{\psi}}^{M,-}[\hat{\psi}_{M}^{-},\hat{\mathcal{Z}}^{N};\hat{\bar{\mathcal{Z}}}_{N}]+\frac{4\pi}{k}\hat{\bar{\psi}}^{M,+}[\hat{\psi}_{N}^{+},\hat{\mathcal{Z}}^{N};\hat{\bar{\mathcal{Z}}}_{M}]$
$\displaystyle-\frac{4\pi}{k}\hat{\bar{\psi}}^{M,-}[\hat{\psi}_{N}^{-},\hat{\mathcal{Z}}^{N};\hat{\bar{\mathcal{Z}}}_{M}]+\frac{\pi}{k}\epsilon_{MNPQ}\hat{\bar{\psi}}^{M,+}[\hat{\mathcal{Z}}^{P},\hat{\mathcal{Z}}^{Q};\hat{\bar{\psi}}^{N,-}]$
$\displaystyle-\frac{\pi}{k}\epsilon_{MNPQ}\hat{\bar{\psi}}^{M,-}[\hat{\mathcal{Z}}^{P},\hat{\mathcal{Z}}^{Q};\hat{\bar{\psi}}^{N,+}]-\frac{\pi}{k}\epsilon^{MNPQ}\hat{\psi}^{+}_{M}[\hat{\bar{\mathcal{Z}}}_{P},\hat{\bar{\mathcal{Z}}}_{Q};\hat{\psi}^{-}_{N}]$
$\displaystyle+\frac{\pi}{k}\epsilon^{MNPQ}\hat{\psi}^{-}_{M}[\hat{\bar{\mathcal{Z}}}_{P},\hat{\bar{\mathcal{Z}}}_{Q};\hat{\psi}_{N}^{+}]\bigg{)}\
,$ (49ef)

where $\hat{\bar{\psi}}^{M,\pm}$ denotes the Hermitian conjugate of
$\hat{\psi}_{M}^{\pm}$. Under the scaling (26), we take our Fermions to
transform as

$\displaystyle\hat{\psi}^{1,-}(\hat{t},\hat{z},\hat{\bar{z}})$
$\displaystyle=\frac{1}{\omega}\psi^{1,-}(t,z,\bar{z})\ ,$ (49ega)
$\displaystyle\hat{\psi}^{1,+}(\hat{t},\hat{z},\hat{\bar{z}})$
$\displaystyle=\psi^{1,+}(t,z,\bar{z})\ ,$ (49egb)
$\displaystyle\hat{\psi}^{A,-}(\hat{t},\hat{z},\hat{\bar{z}})$
$\displaystyle=\psi^{A,-}(t,z,\bar{z})\ ,$ (49egc)
$\displaystyle\hat{\psi}^{A,+}(\hat{t},\hat{z},\hat{\bar{z}})$
$\displaystyle=\frac{1}{\omega}\psi^{A,+}(t,z,\bar{z})\ .$ (49egd)

Combining this with the transformation (30) of the Bosonic fields leads to a
finite action, so we can take the limit $\omega\to 0$ and find the Fermion
action

$\displaystyle S_{F}=\tr\int d^{3}x\bigg{[}$ $\displaystyle
i\bar{\psi}^{1,-}D_{0}\psi_{1}^{-}-2i\bar{\psi}^{1,+}D\psi_{1}^{-}-2i\bar{\psi}^{1,-}\bar{D}\psi_{1}^{+}+i\bar{\psi}^{A,+}D_{0}\psi^{+}_{A}$
$\displaystyle-2i\bar{\psi}^{A,+}D\psi_{A}^{-}-2i\bar{\psi}^{A,-}\bar{D}\psi_{A}^{+}+\frac{2\pi}{k}\Big{(}2\bar{\psi}^{1,+}[\psi_{1}^{+},\mathcal{Z}^{1};\bar{\mathcal{Z}}_{1}]$
$\displaystyle+\bar{\psi}^{1,-}[\psi_{1}^{-},\mathcal{Z}^{A};\bar{\mathcal{Z}}_{A}]+2\bar{\psi}^{A,-}[\psi_{A}^{-},\mathcal{Z}^{1};\bar{\mathcal{Z}}_{1}]-\bar{\psi}^{A,+}[\psi_{A}^{+},\mathcal{Z}^{B};\bar{\mathcal{Z}}_{B}]$
$\displaystyle+2\bar{\psi}^{1,+}[\psi_{A}^{+},\mathcal{Z}^{A};\bar{\mathcal{Z}}_{1}]-2\bar{\psi}^{1,-}[\psi_{A}^{-},\mathcal{Z}^{A};\bar{\mathcal{Z}}_{1}]+2\bar{\psi}^{A,+}[\psi_{1}^{+},\mathcal{Z}^{1};\bar{\mathcal{Z}}_{A}]$
$\displaystyle-2\bar{\psi}^{A,-}[\psi_{1}^{-},\mathcal{Z}^{1};\bar{\mathcal{Z}}_{A}]+2\bar{\psi}^{A,+}[\psi_{B}^{+},\mathcal{Z}^{B};\bar{\mathcal{Z}}_{A}]\Big{)}$
$\displaystyle+\frac{4\pi}{k}\Big{(}\epsilon_{ABC}\bar{\psi}^{A,+}[\mathcal{Z}^{B},\mathcal{Z}^{1};\bar{\psi}^{C,-}]-\epsilon^{ABC}\psi^{+}_{A}[\bar{\mathcal{Z}}_{B},\bar{\mathcal{Z}}_{1};\psi_{C}^{-}]\Big{)}\bigg{]}\
.$ (49eh)

It is now simple to check that the symmetries of the Bosonic extension extend
to symmetries of $S_{F}$. We see that it is invariant under the spatial
transformations (43) provided we take

$\displaystyle\hat{\psi}_{1}^{-}(\hat{t},\hat{z},\hat{\bar{z}})$
$\displaystyle=\left(1-\bar{\partial}\bar{f}\right)\psi_{1}^{-}(t,z,\bar{z})\
,$ (49eia) $\displaystyle\hat{\psi}_{1}^{+}(\hat{t},\hat{z},\hat{\bar{z}})$
$\displaystyle=\left(\psi_{1}^{+}-\frac{1}{2}\bar{f}^{\prime}\psi_{1}^{-}\right)(t,z,\bar{z})\
,$ (49eib) $\displaystyle\hat{\psi}_{A}^{-}(\hat{t},\hat{z},\hat{\bar{z}})$
$\displaystyle=\left(\psi_{A}^{-}-\frac{1}{2}f^{\prime}\psi_{A}^{+}\right)(t,z,\bar{z})\
,$ (49eic) $\displaystyle\hat{\psi}_{A}^{+}(\hat{t},\hat{z},\hat{\bar{z}})$
$\displaystyle=\left(1-\partial f\right)\psi_{A}^{+}(t,z,\bar{z})\ ,$ (49eid)

and the temporal transformations (46) if

$\displaystyle\hat{\psi}_{1}^{-}(\hat{t},\hat{z},\hat{\bar{z}})$
$\displaystyle=\psi_{1}^{-}(t,z,\bar{z})\ ,$ (49eja)
$\displaystyle\hat{\psi}_{1}^{+}(\hat{t},\hat{z},\hat{\bar{z}})$
$\displaystyle=\left(1-F^{\prime}\right)\psi_{1}^{+}(t,z,\bar{z})\ ,$ (49ejb)
$\displaystyle\hat{\psi}_{A}^{-}(\hat{t},\hat{z},\hat{\bar{z}})$
$\displaystyle=\left(1-F^{\prime}\right)\psi_{A}^{-}(t,z,\bar{z})\ ,$ (49ejc)
$\displaystyle\hat{\psi}_{A}^{+}(\hat{t},\hat{z},\hat{\bar{z}})$
$\displaystyle=\psi_{A}^{+}(t,z,\bar{z})\ .$ (49ejd)

The transformations of the Fermions under the global symmetries are

$\hat{\bar{\psi}}^{A,\pm}=\tensor{\mathcal{R}}{{}^{A}_{B}}\bar{\psi}^{B,\pm}$
(49ek)

for the $SU(3)$ R-symmetry,

$\displaystyle\hat{\psi}_{1}^{\pm}$
$\displaystyle=e^{-\frac{i\alpha}{2}}\psi_{1}^{\pm}\ ,$ (49ela)
$\displaystyle\hat{\psi}_{A}^{\pm}$
$\displaystyle=e^{\frac{i\alpha}{2}}\psi_{A}^{\pm}\ ,$ (49elb)

for the $U(1)_{R}$ R-symmetry, and

$\displaystyle\hat{\psi}_{1}^{\pm}$ $\displaystyle=e^{i\beta}\psi_{1}^{\pm}\
,$ (49ema) $\displaystyle\hat{\psi}_{A}^{\pm}$
$\displaystyle=e^{i\beta}\psi_{A}^{\pm}\ ,$ (49emb)

for the $U(1)_{b}$ baryon number symmetry.

###  A.2 Fermionic Contributions to Conserved Currents

In section 3.3 we determined the contributions to the conserved currents in
the theory solely coming from Bosonic fields- in this appendix we extend this
to include the Fermionic terms.

The temporal transformations (49ej) give the current

$\displaystyle j^{0}_{(a)}$
$\displaystyle=\tr\left(2i\left(\bar{\psi}^{1,+}D\psi_{1}^{-}+\bar{\psi}^{1,-}\bar{D}\psi_{1}^{+}+\bar{\psi}^{A,+}D\psi_{A}^{-}+\bar{\psi}^{A,-}\bar{D}\psi_{A}^{+}\right)+V_{f}\right)\
,$ (49ena) $\displaystyle j_{(a)}$
$\displaystyle=-2i\tr\left(\bar{\psi}^{1,+}D_{0}\psi_{1}^{-}+\bar{\psi}^{A,+}D_{0}\psi_{A}^{-}\right)\
,$ (49enb) $\displaystyle\bar{j}_{(a)}$
$\displaystyle=-2i\tr\left(\bar{\psi}^{1,-}D_{0}\psi_{1}^{+}+\bar{\psi}^{A,-}D_{0}\psi_{A}^{+}\right)\
,$ (49enc)

for $F=a$; it is then easy to see that other currents are just this multiplied
by powers of $t$, i.e.

$\displaystyle j^{\mu}_{(b)}$ $\displaystyle=tj^{\mu}_{(a)}\ ,$ (49eoa)
$\displaystyle j^{\mu}_{(c)}$ $\displaystyle=t^{2}j^{\mu}_{(a)}\ .$ (49eob)

The spatial transformations (49ei) give the holomorphic current

$\displaystyle
T=2i\tr\left(D\bar{\psi}^{A,-}\psi_{A}^{+}-\bar{\psi}^{1,-}D\psi_{1}^{+}\right)\
.$ (49epa)

Finally, the R-symmetry transformations give the currents

$\displaystyle\tensor{\left(J^{0}\right)}{{}^{A}_{B}}$
$\displaystyle=\tr\left(\bar{\psi}^{A,+}\psi_{B}^{+}\right)\ ,$ (49eqa)
$\displaystyle\tensor{\left(J\right)}{{}^{A}_{B}}$
$\displaystyle=\tr\left(\bar{\psi}^{A,+}\psi_{B}^{-}\right)\ ,$ (49eqb)
$\displaystyle\tensor{\left(\bar{J}\right)}{{}^{A}_{B}}$
$\displaystyle=\tr\left(\bar{\psi}^{A,-}\psi_{B}^{+}\right)\ ,$ (49eqc)

for the $SU(3)$ symmetry (49ek),

$\displaystyle j^{0}$
$\displaystyle=\frac{1}{2}\tr\left(\bar{\psi}^{1,-}\psi_{1}^{-}-\bar{\psi}^{A,+}\psi_{A}^{+}\right)\
,$ (49era) $\displaystyle j$
$\displaystyle=\tr\left(\bar{\psi}^{A,+}\psi_{A}^{-}-\bar{\psi}^{1,+}\psi_{1}^{-}\right)\
,$ (49erb) $\displaystyle\bar{j}$
$\displaystyle=\tr\left(\bar{\psi}^{A,-}\psi_{A}^{+}-\bar{\psi}^{1,-}\psi_{1}^{+}\right)\
,$ (49erc)

for the $U(1)_{R}$ symmetry (49el), and the baryon number $U(1)_{b}$ symmetry
(49em) gives

$\displaystyle j_{b}^{0}$
$\displaystyle=-\tr\left(\bar{\psi}^{1,-}\psi_{1}^{-}+\bar{\psi}^{A,+}\psi_{A}^{+}\right)\
,$ (49esa) $\displaystyle j_{b}$
$\displaystyle=2\tr\left(\bar{\psi}^{A,+}\psi_{A}^{-}+\bar{\psi}^{1,+}\psi_{1}^{-}\right)\
,$ (49esb) $\displaystyle\bar{j_{b}}$
$\displaystyle=2\tr\left(\bar{\psi}^{A,-}\psi_{A}^{+}+\bar{\psi}^{1,-}\psi_{1}^{+}\right)\
.$ (49esc)

##  Appendix B Equations of Motion

The equations of motion that come from the actions (3.1) and (A.1) are

$\displaystyle-D_{0}^{2}\mathcal{Z}^{1}+\frac{2\pi
i}{k}\left[\mathcal{Z}^{1},D_{0}\mathcal{Z}^{A};\bar{\mathcal{Z}}_{A}\right]+\frac{2\pi
i}{k}\left[\mathcal{Z}^{A},\mathcal{Z}^{1};D_{0}\bar{\mathcal{Z}}_{A}\right]+\frac{4\pi^{2}}{3k^{2}}\left[\mathcal{Z}^{B},\left[\mathcal{Z}^{A},\mathcal{Z}^{1};\bar{\mathcal{Z}}_{A}\right];\bar{\mathcal{Z}}_{B}\right]$
$\displaystyle-\frac{16\pi^{2}}{3k^{2}}\left[\mathcal{Z}^{B},\left[\mathcal{Z}^{A},\mathcal{Z}^{1};\bar{\mathcal{Z}}_{B}\right];\bar{\mathcal{Z}}_{A}\right]-\frac{8\pi^{2}}{3k^{2}}\left[\mathcal{Z}^{A},\mathcal{Z}^{B};\left[\bar{\mathcal{Z}}_{A},\bar{\mathcal{Z}}_{B};\mathcal{Z}^{1}\right]\right]$
$\displaystyle-\frac{4\pi^{2}}{3k^{2}}\left[\mathcal{Z}^{A},\mathcal{Z}^{1};\left[\bar{\mathcal{Z}}_{B},\bar{\mathcal{Z}}_{A};\mathcal{Z}^{B}\right]\right]-\frac{4\pi^{2}}{3k^{2}}\left[\mathcal{Z}^{1},\left[\mathcal{Z}^{B},\mathcal{Z}^{A};\bar{\mathcal{Z}}_{B}\right];\bar{\mathcal{Z}}_{A}\right]-DH$
$\displaystyle+\frac{4\pi}{k}\\{\mathcal{Z}^{1},\psi_{1}^{+};\bar{\psi}^{1,+}\\}+\frac{4\pi}{k}\\{\mathcal{Z}^{1},\psi_{A}^{-};\bar{\psi}^{A,-}\\}+\frac{4\pi}{k}\\{\mathcal{Z}^{A},\psi^{+}_{A};\bar{\psi}^{1,+}\\}$
$\displaystyle-\frac{4\pi}{k}\\{\mathcal{Z}^{A},\psi_{A}^{-};\bar{\psi}^{1,-}\\}-\frac{4\pi}{k}\epsilon^{ABC}\\{\psi_{A}^{+},\psi_{C}^{-};\bar{\mathcal{Z}}_{B}\\}=0\
,$ (49eta) $\displaystyle 2(D\bar{D}+\bar{D}D)\mathcal{Z}^{A}-\frac{4\pi
i}{k}\left[\mathcal{Z}^{1},D_{0}\mathcal{Z}^{A};\bar{\mathcal{Z}}_{1}\right]-\frac{2\pi
i}{k}\left[D_{0}\mathcal{Z}^{1},\mathcal{Z}^{A};\bar{\mathcal{Z}}_{1}\right]$
$\displaystyle-\frac{2\pi
i}{k}\left[\mathcal{Z}^{1},\mathcal{Z}^{A};D_{0}\bar{\mathcal{Z}}_{1}\right]+\frac{16\pi^{2}}{3k^{2}}\left[\mathcal{Z}^{1},\left[\mathcal{Z}^{A},\mathcal{Z}^{B};\bar{\mathcal{Z}}_{1}\right];\bar{\mathcal{Z}}_{B}\right]$
$\displaystyle-\frac{4\pi^{2}}{3k^{2}}\left[\mathcal{Z}^{1},\mathcal{Z}^{A};\left[\bar{\mathcal{Z}}_{B},\bar{\mathcal{Z}}_{1};\mathcal{Z}^{B}\right]\right]-\frac{4\pi^{2}}{3k^{2}}\left[\mathcal{Z}^{A},\left[\mathcal{Z}^{B},\mathcal{Z}^{1};\bar{\mathcal{Z}}_{B}\right];\bar{\mathcal{Z}}_{1}\right]$
$\displaystyle+\frac{16\pi^{2}}{3k^{2}}\left[\mathcal{Z}^{B},\left[\mathcal{Z}^{A},\mathcal{Z}^{1};\bar{\mathcal{Z}}_{B}\right];\bar{\mathcal{Z}}_{1}\right]+\frac{16\pi^{2}}{3k^{2}}\left[\mathcal{Z}^{1},\mathcal{Z}^{B};\left[\bar{\mathcal{Z}}_{B},\bar{\mathcal{Z}}_{1};\mathcal{Z}^{A}\right]\right]$
$\displaystyle+\frac{4\pi^{2}}{3k^{2}}\left[\mathcal{Z}^{B},\left[\mathcal{Z}^{1},\mathcal{Z}^{A};\bar{\mathcal{Z}}_{1}\right];\bar{\mathcal{Z}}_{B}\right]-\frac{4\pi^{2}}{3k^{2}}\left[\mathcal{Z}^{A},\left[\mathcal{Z}^{1},\mathcal{Z}^{B};\bar{\mathcal{Z}}_{1}\right];\bar{\mathcal{Z}}_{B}\right]$
$\displaystyle-\frac{4\pi^{2}}{3k^{2}}\left[\mathcal{Z}^{1},\left[\mathcal{Z}^{A},\mathcal{Z}^{B};\bar{\mathcal{Z}}_{B}\right];\bar{\mathcal{Z}}_{1}\right]+\frac{4\pi^{2}}{3k^{2}}\left[\mathcal{Z}^{A},\mathcal{Z}^{B};\left[\bar{\mathcal{Z}}_{1},\bar{\mathcal{Z}}_{B};\mathcal{Z}^{1}\right]\right]$
$\displaystyle+\frac{2\pi}{k}\\{\mathcal{Z}^{A},\psi_{1}^{-};\bar{\psi}^{1,-}\\}-\frac{2\pi}{k}\\{\mathcal{Z}^{A},\psi_{B}^{+};\bar{\psi}^{B,+}\\}+\frac{4\pi}{k}\\{\mathcal{Z}^{1},\psi_{1}^{+};\bar{\psi}^{A,+}\\}$
$\displaystyle-\frac{4\pi}{k}\\{\mathcal{Z}^{1},\psi_{1}^{-};\bar{\psi}^{A,-}\\}+\frac{4\pi}{k}\\{\mathcal{Z}^{B},\psi_{B}^{+};\bar{\psi}^{A,+}\\}+\frac{4\pi}{k}\epsilon^{ABC}\\{\psi_{A}^{+},\psi_{C}^{-};\bar{\mathcal{Z}}_{1}\\}=0\
,$ (49etb) $\bar{D}\mathcal{Z}^{1}=0\ ,$ (49etc) $\displaystyle
F_{0z}^{L}-\frac{\pi}{k}\mathcal{Z}^{1}\bar{H}+\frac{2\pi}{k}\bigg{(}\mathcal{Z}^{A}D\bar{\mathcal{Z}}_{A}-D\mathcal{Z}^{A}\bar{\mathcal{Z}}_{A}\bigg{)}-\frac{2\pi
i}{k}\bigg{(}\psi_{1}^{+}\bar{\psi}^{1,-}+\psi_{A}^{+}\bar{\psi}^{A,-}\bigg{)}$
$\displaystyle=0\ ,$ (49etd) $\displaystyle
F_{0z}^{R}-\frac{\pi}{k}\bar{H}\mathcal{Z}^{1}-\frac{2\pi}{k}\bigg{(}\bar{\mathcal{Z}}_{A}D\mathcal{Z}^{A}-D\bar{\mathcal{Z}}_{A}\mathcal{Z}^{A}\bigg{)}+\frac{2\pi
i}{k}\bigg{(}\bar{\psi}^{1,-}\psi_{1}^{+}+\bar{\psi}^{A,-}\psi_{A}^{+}\bigg{)}$
$\displaystyle=0\ ,$ (49ete) $\displaystyle
F_{z\bar{z}}^{L}-\frac{\pi}{k}\bigg{(}\mathcal{Z}^{1}D_{0}\bar{\mathcal{Z}}_{1}-D_{0}\mathcal{Z}^{1}\bar{\mathcal{Z}}_{1}\bigg{)}$
$\displaystyle-\frac{2\pi^{2}i}{k^{2}}\bigg{(}\mathcal{Z}^{A}\left[\bar{\mathcal{Z}}_{1},\bar{\mathcal{Z}}_{A};\mathcal{Z}^{1}\right]-\left[\mathcal{Z}^{1},\mathcal{Z}^{A};\bar{\mathcal{Z}}_{1}\right]\bar{\mathcal{Z}}_{A}\bigg{)}$
$\displaystyle+\frac{i\pi}{k}\bigg{(}\psi_{1}^{-}\bar{\psi}^{1,-}+\psi_{A}^{+}\bar{\psi}^{A,+}\bigg{)}=0\
,$ (49etf) $\displaystyle
F_{z\bar{z}}^{R}+\frac{\pi}{k}\bigg{(}\bar{\mathcal{Z}}_{1}D_{0}\mathcal{Z}^{1}-D_{0}\bar{\mathcal{Z}}_{1}\mathcal{Z}^{1}\bigg{)}$
$\displaystyle+\frac{2\pi^{2}i}{k^{2}}\bigg{(}\bar{\mathcal{Z}}_{A}\left[\mathcal{Z}^{1},\mathcal{Z}^{A};\bar{\mathcal{Z}}_{1}\right]-\left[\bar{\mathcal{Z}}_{1},\bar{\mathcal{Z}}_{A};\mathcal{Z}^{1}\right]\mathcal{Z}^{A}\bigg{)}$
$\displaystyle-\frac{i\pi}{k}\bigg{(}\bar{\psi}^{1,-}\psi_{1}^{-}+\bar{\psi}^{A,+}\psi_{A}^{+}\bigg{)}=0\
,$ (49etg) $\displaystyle
iD_{0}\psi_{1}^{-}-2i\bar{D}\psi_{1}^{+}+\frac{2\pi}{k}[\psi_{1}^{-},\mathcal{Z}^{A};\bar{\mathcal{Z}}_{A}]-\frac{4\pi}{k}[\psi_{A}^{-},\mathcal{Z}^{A};\bar{\mathcal{Z}}_{1}]$
$\displaystyle=0\ ,$ (49eth) $\displaystyle D\psi_{1}^{-}+\frac{2\pi
i}{k}[\psi_{1}^{+},\mathcal{Z}^{1};\bar{\mathcal{Z}}_{1}]+\frac{2\pi
i}{k}[\psi^{+}_{A},\mathcal{Z}^{A};\bar{\mathcal{Z}}_{1}]$ $\displaystyle=0\
,$ (49eti) $\displaystyle
iD_{0}\psi_{A}^{+}-2iD\psi_{A}^{-}-\frac{2\pi}{k}[\psi_{A}^{+},\mathcal{Z}^{B};\bar{\mathcal{Z}}_{B}]+\frac{4\pi}{k}[\psi_{1}^{+},\mathcal{Z}^{1};\bar{\mathcal{Z}}_{A}]$
$\displaystyle+\frac{4\pi}{k}[\psi_{B}^{+},\mathcal{Z}^{B};\bar{\mathcal{Z}}_{A}]+\frac{4\pi}{k}\epsilon_{ABC}[\mathcal{Z}^{B},\mathcal{Z}^{1};\bar{\psi}^{C,-}]$
$\displaystyle=0\ ,$ (49etj) $\displaystyle\bar{D}\psi_{A}^{+}+\frac{2\pi
i}{k}[\psi_{A}^{-},\mathcal{Z}^{1};\bar{\mathcal{Z}}_{1}]-\frac{2\pi
i}{k}[\psi_{1}^{-},\mathcal{Z}^{1};\bar{\mathcal{Z}}_{A}]-\frac{2\pi
i}{k}\epsilon_{ABC}[\mathcal{Z}^{B},\mathcal{Z}^{1};\bar{\psi}^{C,+}]$
$\displaystyle=0\ ,$ (49etk)

where we’ve defined

$\\{A,B;\bar{C}\\}=A\bar{C}B+B\bar{C}A\ .$ (49eu)

To find classical solutions we set all Fermions to zero. The obvious set of
solutions to these are those with

$\displaystyle A_{0}^{L}=A_{0}^{R}$ $\displaystyle=0\ ,$ (49eva)
$\displaystyle A_{z}^{L}=A_{z}^{R}$ $\displaystyle\equiv A_{z}\ ,$ (49evb)
$\displaystyle\partial_{0}A_{z}$ $\displaystyle=0\ ,$ (49evc)
$\displaystyle\partial_{0}\mathcal{Z}^{1}$ $\displaystyle=0\ ,$ (49evd)
$\displaystyle\mathcal{Z}^{A}$ $\displaystyle=w^{A}\mathbbm{1}_{N}\ ,$ (49eve)
$\displaystyle H$ $\displaystyle=0\ ,$ (49evf)

with $v^{A}$ constant, so that the equations reduce to

$\displaystyle\bar{D}\mathcal{Z}^{1}\equiv\bar{\partial}\mathcal{Z}^{1}-i[A_{\bar{z}},\mathcal{Z}^{1}]$
$\displaystyle=0\ ,$ (49ewa) $\displaystyle
F_{z\bar{z}}+\frac{4\pi^{2}i}{k^{2}}w^{A}\bar{w}_{A}[\mathcal{Z}^{1},\bar{\mathcal{Z}}_{1}]$
$\displaystyle=0\ .$ (49ewb)

We recognise these as the Hitchin equations [14]. A more detailed analysis of
the solutions, including time dependence was given in [30] for the gauge group
$SU(2)\times SU(2)$.

##  Appendix C Non-Relativistic Orbifold Geometry

Recall that the dual of ABJM is the near-horizon limit of M2-branes on the
background with transverse space $\mathbb{C}^{4}/\mathbb{Z}_{k}$. To begin we
consider $\mathbb{C}^{4}$ with coordinates $z^{M}$:

$\displaystyle
ds^{2}_{\mathbb{C}^{4}}=dz^{1}d\bar{z}_{1}+dz^{2}d\bar{z}_{2}+dz^{3}d\bar{z}_{3}+dz^{4}d\bar{z}_{4}\
.$ (49ex)

In order to implement the orbifold we note that $S^{2n+1}$ can be realised as
a circle fibration over $\mathbb{CP}^{2n}$. If we introduce the
parameterisation

$z^{M}=r_{M}e^{i\theta_{M}}\ ,$ (49ey)

then the metric takes the form

$ds^{2}_{\mathbb{C}^{4}}=d\tilde{r}^{2}+\tilde{r}^{2}\left(ds^{2}_{\mathbb{CP}^{3}}+\left(d\tilde{\phi}+\tilde{\omega}\right)^{2}\right)\
,$ (49ez)

where $\tilde{r}=\sqrt{r_{1}^{2}+...+r_{4}^{2}}$,
$\tilde{\phi}=\theta_{1}+...+\theta_{4}$ and $d{\tilde{\omega}}$ is the Kähler
form on $\mathbb{CP}^{3}$. The orbifold then takes

$\tilde{\phi}\to\frac{\tilde{\phi}}{k}\ ,$ (49fa)

where we have retained the periodicity of $\tilde{\phi}\sim\tilde{\phi}+2\pi$.

In order to take the non-relativistic limit we must make a distinction between
$z^{1}$ and the other coordinates. It is therefore convenient to write the
metric on $\mathbb{C}^{4}$ as

$ds_{\mathbb{C}^{4}}^{2}=dr_{1}^{2}+r_{1}^{2}d\theta^{2}_{1}+dr^{2}+r^{2}\left(ds_{\mathbb{CP}^{2}}^{2}+\left(d\phi+\omega\right)^{2}\right)\
,$ (49fb)

where $r=\sqrt{r_{2}^{2}+r_{3}^{2}+r_{4}^{2}}$,
$\phi=\theta_{2}+\theta_{3}+\theta_{4}$, and $d\omega$ is the Kähler form on
$\mathbb{CP}^{2}$. We note that this means that

$\tilde{\phi}=\phi+\theta_{1}\ .$ (49fc)

We can then implement the non-relativistic limit of the metric, obtaining the
(partial) Newton-Cartan structures

$\displaystyle\hat{P}_{\mu\nu}dx^{\mu}\otimes dx^{\nu}=$
$\displaystyle\,dr_{1}\otimes dr_{1}+r_{1}^{2}d\theta_{1}\otimes d\theta_{1}\
,$ (49fda) $\displaystyle\hat{Q}^{\mu\nu}\frac{\partial}{\partial
x^{\mu}}\otimes\frac{\partial}{\partial x^{\nu}}=$
$\displaystyle\,\frac{\partial}{\partial r}\otimes\frac{\partial}{\partial
r}+\frac{1}{r^{2}}\Bigg{(}g^{mn}_{(\mathbb{CP}^{2})}\frac{\partial}{\partial
z^{m}}\otimes\frac{\partial}{\partial
z^{n}}-\omega^{i}\left(\frac{\partial}{\partial
z^{m}}\otimes\frac{\partial}{\partial\phi}+\frac{\partial}{\partial\phi}\otimes\frac{\partial}{\partial
z^{m}}\right)$
$\displaystyle+\left(1+\omega^{i}\omega_{i}\right)\frac{\partial}{\partial\phi}\otimes\frac{\partial}{\partial\phi}\Bigg{)}\
,$ (49fdb)

where $\\{z^{m}\\}$ are some local coordinates on $\mathbb{CP}^{2}$ and we’ve
defined $\omega^{m}=g^{mn}_{(\mathbb{CP}^{2})}\omega_{n}$. We see that if we
make the angular coordinate transformation
$(\theta_{1},\phi)\to(\theta_{1},\tilde{\phi})$ the form of both structures is
unchanged, i.e. we just replace $\phi$ with $\tilde{\phi}$ in the expression
for $\hat{Q}$. We can then implement the orbifold, leaving us with

$\displaystyle P_{\mu\nu}dx^{\mu}\otimes dx^{\nu}=$
$\displaystyle\,dr_{1}\otimes dr_{1}+r_{1}^{2}d\theta_{1}\otimes d\theta_{1}\
,$ (49fea) $\displaystyle Q^{\mu\nu}\frac{\partial}{\partial
x^{\mu}}\otimes\frac{\partial}{\partial x^{\nu}}=$
$\displaystyle\,\frac{\partial}{\partial r}\otimes\frac{\partial}{\partial
r}+\frac{1}{r^{2}}\Bigg{(}g^{mn}_{(\mathbb{CP}^{2})}\frac{\partial}{\partial
z^{m}}\otimes\frac{\partial}{\partial
z^{n}}-k\omega^{m}\left(\frac{\partial}{\partial
z^{m}}\otimes\frac{\partial}{\partial\tilde{\phi}}+\frac{\partial}{\partial\tilde{\phi}}\otimes\frac{\partial}{\partial
z^{m}}\right)$
$\displaystyle+k^{2}\left(1+\omega^{m}\omega_{m}\right)\frac{\partial}{\partial\tilde{\phi}}\otimes\frac{\partial}{\partial\tilde{\phi}}\Bigg{)}\
.$ (49feb)

This defines a non-relativistic
$\mathbb{C}\times\mathbb{C}^{3}/\mathbb{Z}_{k}$ geometry, with the orbifold
only acting on the co-metric.

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