# Non-Relativistic Supergeometry in the Moore -Read Fractional Quantum Hall
State

Patricio Salgado-Rebolledo  Department of Theoretical Physics, Wrocław
University of Science and Technology, 50-370 Wrocław, Poland  Université Libre
de Bruxelles and International Solvay Institutes, ULB-Campus Plaine CP231,
B-1050 Brussels, Belgium  Giandomenico Palumbo  School of Theoretical Physics,
Dublin Institute for Advanced Studies, 10 Burlington Road, Dublin 4, Ireland

###### Abstract

The Moore-Read state is one of the most well known non-Abelian fractional
quantum Hall states. It supports non-Abelian Ising anyons in the bulk and a
chiral bosonic and chiral Majorana modes on the boundary. It has been recently
conjectured that these modes are superpartners of each other and described by
a supersymmetric conformal field theory Ma  _et al._ (2021). We propose a non-
relativistic supergeometric theory that is compatible with this picture and
gives rise to an effective spin-3/2 field in the bulk. After breaking
supersymmetry through a Goldstino, the spin-3/2 field becomes massive and can
be seen as the neutral collective mode that characterizes the Moore-Read
state. By integrating out this fermion field, we obtain a purely bosonic
topological action that properly encodes the Hall conductivity, Hall viscosity
and gravitational anomaly. Our work paves the way to the exploration of the
fractional quantum Hall effect through non-relativistic supergeometry.

Introduction:  The Fractional Quantum Hall Effect (FQHE) is a milestone in the
understanding of topological phases of matter Wen (1992). Non-Abelian FQH
states have been investigated since the seminal work by Moore and Read Moore
and Read (1991). The Moore-Read (MR) state is a candidate to describe the
filling fraction $5/2$ and represents the prototypical example of FQH state
that supports non-Abelian anyons in the bulk with a single chiral boson and
chiral Majorana mode propagating on the boundary. In the low-energy regime,
these edge modes are described by a $SU(2)_{2}$ Wess-Zumino-Witten (WZW) model
with chiral central charge $c=1+1/2$, which is associated to a $SU(2)_{2}$
Chern-Simons (CS) theory in the bulk Fradkin  _et al._ (1998); Palumbo and
Pachos (2014); Marzuoli and Palumbo (2012). However, recently it has been
proposed an alternative picture of the boundary theory of the MR state, in
which the fermionic and bosonic modes can be seen as superpartners such that
the effective theory is given by a chiral supersymmetric conformal field
theory (SCFT) Ma  _et al._ (2021); Bae and Lee (2021); Sagi and Santos (2017).
The emergence of supersymmetry in this framework is partially justified by the
possible existence of a propagating massive spin-3/2 mode in the bulk
Bonderson  _et al._ (2011); Möller  _et al._ (2011); Yang  _et al._ (2012).
This collective mode in the MR state has been argued to indeed be the
superpartner Yang  _et al._ (2012); Gromov  _et al._ (2020) of the spin-2 GMP
(Girvin-MacDonald-Plazman) mode Girvin  _et al._ (1986); Cappelli  _et al._
(1993); Iso  _et al._ (1992). The latter can be seen a non-relativistic
massive graviton Golkar  _et al._ (2016); Liu  _et al._ (2018); Liou  _et al._
(2019); Gromov and Son (2017); Nguyen  _et al._ (2018) while the former would
be a massive gravitino (gravitinos carry spin-3/2 in supergravity). We remind
here that without being related to the MR state, supersymmetry has been
previously discussed in a number of works concerning the FQHE Tong and Turner
(2015); Vafa (2016); Ino (2001); Hasebe (2005); Hikida  _et al._ (2009);
Fujita  _et al._ (2009).

In this framework, it is natural to expect that the topological sector of the
supersymmetric model describing the MR state is given by a (2+1)-dimensional
CS theory invariant under some non-relativistic superalgebra similar to three-
dimensional non-relativistic supersymmetry or supergravity Horvathy (1993);
Duval and Horvathy (1994); Andringa  _et al._ (2013); Bergshoeff and Rosseel
(2016); Ozdemir  _et al._ (2019); Concha  _et al._ (2020). If this
supersymmetric picture of the MR state is correct, then one should also be
able to take into account further geometric effects such as the Hall viscosity
Avron  _et al._ (1995); Read (2009); Read and Rezayi (2011); Bradlyn  _et al._
(2012); Hoyos and Son (2012), which is an universal feature of the FQHE and
emerges once the FQH fluid is coupled to a curved background. Several works
have discussed the geometric effective non-relativistic field theory of the
Abelian FQH states Gromov and Son (2017); Bradlyn and Read (2015); Gromov  _et
al._ (2015); Cho  _et al._ (2014); Cappelli and Randellini (2016); Gromov  _et
al._ (2017, 2016); Moroz  _et al._ (2015); Klevtsov and Wiegmann (2015); Golan
_et al._ (2019); Gromov and Abanov (2014); Abanov and Gromov (2014); Geracie
_et al._ (2015); Can  _et al._ (2014); Ferrari and Klevtsov (2014); Dwivedi
and Klevtsov (2019), where the effective action is purely bosonic as well as
its corresponding edge states.

The main goal of our work is to fill the gap between supersymmetry and the
geometric theory for the MR state, by introducing and studying a novel non-
relativistic supergeometric theory that is able to describe both the chiral
boundary states and the topological and geometric bulk features, such as Hall
viscosity and the presence of massive spin-3/2 mode. In order to combine
supersymmetry with a geometric theory compatible with the symmetries of the MR
state, we start by defining a novel supersymmetric extension of the extended
Nappi-Witten algebra, which has been introduced in an our previous work that
deals with the geometry of the Abelian Laughlin state Salgado-Rebolledo and
Palumbo (2021). Through this new algebra, we will define the corresponding
gauge connections that contains both the external and hydrodynamical fields.
The universal topological action is given by the CS term associated to this
connection. However, in order to obtain a massive spin-3/2 mode we will add an
extra gauge-invariant term in the action Palumbo (2020) that introduces a
Goldstino field and breaks supersymmetry, giving rise to the mass term of the
spin-3/2 field. This will allow us to integrate our the fermionic field in the
bulk and obtain a full bosonic geometric theory that encodes the Hall
viscosity response of the system. On the boundary, the effect of adding this
term is that the chiral boson and chiral Majorana modes acquire different
velocities, breaking supersymmetry at the level of the edge theory as well.

Our work offers a non-relativistic supergeometric framework to describe the
universal (topological) features of the MR state in the low-energy regime. It
represents a novel bridge between two apparently uncorrelated research fields,
i.e. non-relativistic supersymmetry and FQHE. Although our supersymmetric
theory is formally different with respect to other non-relativistic
supergravity models presented previously in the high-energy-physics context,
it is compatible with the underlying symmetries of the FQHE and paves the way
for the construction of a more general theory describing dynamical (non-
universal) features of massive gravitons and gravitinos.

Nappi-Witten algebra and unbroken supersymmetry:  In order to build the proper
supergeometric theory for the Moore-Read state, we need first to specify the
underlying algebra compatible with the symmetries of the FQHE. In fact, it is
well known that general relativity, in the first-order formalism, can be seen
an effective gauge theory associated to the Lorentz group. In this formalism
the metric tensor is replaced by the spin connection and the vielbein. In the
(2+1)-dimensional case, both exotic (see, Ref.Palumbo and Pachos (2016) for an
application of exotic gravity in topological phases) and Einstein-Hilbert
gravity, with and without cosmological constant, can be rewritten as a CS
theory where the corresponding gauge connection is a linear combination of the
spin connection and the dreibein (i.e. the equivalent of the vielbein in 2+1
dimensions) Witten (1988a); Achucarro and Townsend (1986). Differently from
these relativistic theories, an effective geometric action for FQHE should
take into account the magnetic translations together with the Galilei
invariance. The latter, although not crucial in the FQHE, has been shown to be
very useful in the construction of non-relativistic effective theories in
several FQH states. In our previous work, we have shown that both magnetic
translations and space rotations are naturally encoded in an extended version
of the Nappi-Witten algebra Nappi and Witten (1993); Figueroa-O’Farrill and
Stanciu (1994), which allowed us to build a non-relativistic geometric theory
for the Laughlin state Salgado-Rebolledo and Palumbo (2021). By following
Ref.Ma  _et al._ (2021), we consider now, besides the previously mentioned
symmetries, also supersymmetry as a symmetry for the Moore-Read state (in the
unbroken supersymmetric phase). In this way, by identifying the correct
supersymmetric generalization of the extended Nappi-Witten algebra we will be
able to construct a corresponding non-relativistic supergeometric theory by
gauging the global symmetries. This non-relativistic supersymmetric algebra
exists and is given by

$\displaystyle\left[\mathcal{J},\mathcal{P}_{a}\right]=\epsilon_{a}^{\;\;b}\mathcal{P}_{b},$
$\displaystyle\left[\mathcal{P}_{a},\mathcal{P}_{b}\right]=-\epsilon_{ab}\mathcal{T},$
(1)
$\displaystyle\left[\mathcal{J},\mathcal{Q}_{\alpha}\right]=\frac{1}{2}\left(\gamma_{0}\right)_{\;\;\alpha}^{\beta}\mathcal{Q}_{\beta},$
$\displaystyle\left\\{\mathcal{Q}_{\alpha},\mathcal{Q}_{\beta}\right\\}=\frac{1}{2}\left(C\gamma_{0}\right)_{\alpha\beta}\mathcal{T},$
$\displaystyle\left[\mathcal{Z},\mathcal{P}_{a}\right]=\beta\epsilon_{a}^{\;\;b}\mathcal{P}_{b},$
$\displaystyle\left[\mathcal{Z},\mathcal{Q}_{\alpha}\right]=\frac{\beta}{2}\left(\gamma_{0}\right)_{\;\;\alpha}^{\beta}\mathcal{Q}_{\beta}.$

where $\beta$ is an arbitrary constant,
$C_{\alpha\beta}=\epsilon_{\alpha\beta}$ is the charge conjugation matrix and
$\gamma_{0}=i\sigma_{2}$. In the bosonic sector $\mathcal{P}_{a}$ ($a=1,2$),
stands for the generator of translations in the two-dimensional plane,
$\mathcal{J}$ is the generator of the spatial rotations, $\mathcal{T}$ is the
Lie algebra generator associated to the external electromagnetic field and
$\mathcal{Z}$ is a bosonic generator associated to the emergent U(1) gauge
field. The fermionic generators $Q_{\alpha}$ ($\alpha=1,2$) are Majorana
supercharges. The first line in (1) represents the Euclidean version of the
original algebra by Nappi and Witten Nappi and Witten (1993), which is
ubiquitous in low-dimensional physics Alvarez  _et al._ (2008); Afshar  _et
al._ (2020); Duval and Horvathy (2000); Horvathy  _et al._ (2005); Palumbo
(2018) and is also related to the Maxwell algebra Schrader (1972); Gomis and
Kleinschmidt (2017) in the sense that it includes magnetic translations. The
case $\beta=0$ yields a supersymmetric extension of the Nappi-Witten symmetry.
When the supercharges $Q_{\alpha}$ vanish, we recover the extended Nappi-
Witten algebra employed for the Laughlin state in Salgado-Rebolledo and
Palumbo (2021). From now on, we will refer to the symmetry (1) as
$\mathfrak{snw}$ algebra.

From this novel supersymmetric algebra, by following the gauge principle, we
can naturally build a corresponding one-form gauge connection

$\mathbb{A}=\omega\mathcal{J}+a\mathcal{Z}+e^{a}\mathcal{P}_{a}+A\mathcal{T}+\mathcal{Q}_{\alpha}\Psi^{\alpha},$
(2)

where $\omega$ is the spin connection, $a$ is an emergent gauge field, $e^{a}$
is the spatial dreibein, $A$ is the external electromagnetic field, and
$\Psi^{\alpha}$ is a fermionic spin-3/2 field. They are all the physical
fields in the effective geometric theory that we will construct in the next
section. Note that our superalgebra includes the central generator
$\mathcal{T}$ in the anti-commutator of the supercharges, in analogy with the
super-Bargmann algebra Puzalowski (1978); de Azcarraga and Ginestar (1991);
Bergman and Thorn (1995). Since the Hamiltonian generator does not appear in
the $\\{Q,Q\\}$ bracket, this algebra does not lead to supersymmetry in the
standard sense. In order to include the Hamiltonian generator, it is necessary
to embed the algebra (1) in a lager non-relativistic superalgebra, as done in
Salgado-Rebolledo and Palumbo (2021) in the purely bosonic case. In the same
way, the supersymmetric generators cannot be understood as the square root of
the spatial translations $\mathcal{P}_{a}$, as it happens in non-relativistic
supergravity Andringa  _et al._ (2013); Bergshoeff and Rosseel (2016); Ozdemir
_et al._ (2019); Concha  _et al._ (2020) or superparticle models Lukierski
_et al._ (2006); Bergshoeff  _et al._ (2014). Thus, the theory that will
follow from gauging $\mathfrak{snw}$ will not define a supergravity theory.

Supergeometric Chern-Simons theory and boundary chiral modes:  In previous
section, we have introduced a novel algebra $\mathfrak{snw}$, defined by the
commutation relations (1), that encodes two-dimensional magnetic translations,
spatial rotations and supersymmetry. Another fundamental feature in the FQHE
is the absence of the time-reversal symmetry, which is typically taken into
account by considering a CS theory, which is a topological field theory that
supports a CFT on the boundary of the system. Because we are mainly interested
in the topological (universal) features of the MR state in the low-energy
regime, a natural choice for the construction of an effective field theory in
terms of the gauge connection (2) is to consider a CS action of the form

$S_{\rm
CS}=-\frac{k}{4\pi}\int_{M}\left\langle\mathbb{A}d\mathbb{A}+\dfrac{2}{3}\mathbb{A}\wedge\mathbb{A}\wedge\mathbb{A}\right\rangle.$
(3)

where $M$ is a three-dimensional spacetime manifold and $k$ is the
corresponding quantized level, which we fix $k=1$ for simplicity. Here,
$\left\langle\cdot,\cdot\right\rangle$ denotes some non-degenerate invariant
bilinear form on the gauge algebra (1), i.e. denoting the generators of (1) by
$\mathcal{G}_{A}=\\{\mathcal{J},\mathcal{Z},\mathcal{P}_{a},\mathcal{T},\mathcal{Q}_{\alpha}\\}$,
with $A$ a collective index, the map
$\left\langle\cdot,\cdot\right\rangle:\mathfrak{snw}\times\mathfrak{snw}\rightarrow\mathbb{R}$
satisfies

$\displaystyle
i)\;\left\langle\mathcal{G}_{A},\mathcal{G}_{B}\right\rangle=\left\langle\mathcal{G}_{A}+[\mathcal{G}_{A},\mathcal{G}_{C}],\mathcal{G}_{B}+[\mathcal{G}_{B},\mathcal{G}_{C}]\right\rangle,$
(4) $\displaystyle ii)\;{\rm
det}\left\langle\mathcal{G}_{A},\mathcal{G}_{B}\right\rangle\neq 0.$

Given an arbitrary gauge symmetry defined by some Lie algebra, the existence
of such invariant form, and thus of a well-defined CS action, is not
guaranteed. For example, the non-degenerate invariant bilinear form on the
Poincaré algebra in three spacetime dimensions Witten (1988b) is a peculiarity
that does not occur in higher dimensions and allowed to define a gauge theory
for three-dimensional Einstein gravity. Similarly, in the non-relativistic
case, the Galilean symmetry does not admit a well-defined invariant form
unless its double central extension is considered, which exists only in three
space-time dimensions and allows to define a Bargmann-invariant non-
relativistic CS gravity theory Papageorgiou and Schroers (2009). Along the
same lines, a remarkable property of the extended supersymmetric Nappi-Witten
algebra $\mathfrak{snw}$ is that it admits a non-degenerate invariant bilinear
form, which reads

$\displaystyle\langle\mathcal{J},\mathcal{J}\rangle=\mu_{0}\,,\qquad\langle\mathcal{P}_{a},\mathcal{P}_{b}\rangle=\mu_{1}\,\delta_{ab}\,,$
(5)
$\displaystyle\langle\mathcal{J},\mathcal{Z}\rangle=\mu_{2},\qquad\langle\mathcal{Q}_{\alpha},\mathcal{Q}_{\beta}\rangle=\mu_{1}\,C_{\alpha\beta},$
$\displaystyle\langle\mathcal{Z},\mathcal{Z}\rangle=\mu_{3},\qquad\langle\mathcal{J},\mathcal{T}\rangle=\beta^{-1}\langle\mathcal{Z},\mathcal{T}\rangle=-\mu_{1},$

where $\mu_{i}$ ($i=0,1,2,3$) are all arbitrary real parameters. Therefore, it
is possible to define a well-defined CS action (3) invariant under gauge
transformations $\delta_{\epsilon}A=dA+[A,\epsilon]$ with $\epsilon$ a local
parameter taking values in $\mathfrak{snw}$. The construction of the CS action
invariant under the symmetry (1) is the first result of our paper. As we will
see in the next sections, such action can be used as an effective action for
the Moore-Read fractional quantum Hall states, provided we associate the
constants $\mu_{i}$ to physical quantities, such as Hall conductivity, Hall
viscosity and chiral central charge.

Without matter currents, the corresponding equations of motions are given by
$\mathbb{F}=0$, where $\mathbb{F}$ is curvature tensor of $\mathbb{A}$, namely

$\mathbb{F}=d\mathbb{A}+\frac{1}{2}[\mathbb{A},\mathbb{A}]=d\omega\mathcal{J}+da\mathcal{Z}+R^{a}\mathcal{P}_{a}+F\mathcal{T}+\mathcal{Q}_{\alpha}D\Psi^{\alpha},$
(6)

where

$\displaystyle R^{a}=de^{a}+\epsilon^{a}_{\;\;b}e^{b}\wedge\left(\omega+\beta
a\right)\,,$ (7) $\displaystyle F=dA-\frac{1}{2}\epsilon_{ab}e^{a}\wedge
e^{b}-\frac{1}{4}\bar{\Psi}_{\alpha}\wedge\left(\gamma_{0}\right)^{\alpha}_{\;\;\beta}\Psi^{\beta},$

and we have defined the covariant derivative of $\Psi^{\alpha}$ as

$D\Psi^{\alpha}=d\Psi^{\alpha}+\frac{1}{2}\left(\omega+\beta
a\right)\wedge\left(\gamma_{0}\right)_{\;\;\beta}^{\alpha}\Psi^{\beta}.$ (8)

By employing the above expressions and defining the conjugate spinor
$\bar{\Psi}_{\alpha}=\Psi^{\beta}C_{\beta\alpha}$, we can rewrite the CS
action in Eq.(3) in terms of the physical fields

$\displaystyle S_{\rm CS}=-\frac{1}{4\pi}\int_{M}\bigg{[}\mu_{0}\,\omega\wedge
d\omega+2\mu_{2}\,a\wedge d\omega+\mu_{3}\,a\wedge da$ (9)
$\displaystyle+\mu_{1}\,e^{a}\wedge R_{a}-2\mu_{1}\,A\wedge
d\left(\omega+\beta a\right)-\mu_{1}\,\bar{\Psi}_{\alpha}\wedge
D\Psi^{\alpha}\bigg{]}\,.$

This a gauge invariant topological action for a manifold without boundary and
will give rise to the topological response of the fractional Hall fluid in
presence of the external fields as we will discuss in full detail in the next
section. We now move our attention to the case where $M$ has a boundary and
consider the dynamical quantum modes induced by this effective topological
action on the spatial boundary of the system. In fact, on a manifold with
boundary a CS action is not gauge invariant anymore and dynamical gapless
degrees of freedom need to appear on the boundary in order to compensate the
gauge anomaly of the bulk Dijkgraaf and Witten (1990). This is the well known
CS/CFT correspondence that holds also in our case as we now show. We start
considering for simplicity a manifold $M=D_{2}\times R$, where $R$ is
associated to the time-like coordinate $t$, while $D_{2}$ is a two-dimensional
disk on which we introduce polar coordinates $(r,\phi)$, such that its
boundary is given by one-dimensional circle $S^{1}$. Thus, the boundary theory
is given by the following chiral super-WZW model Coussaert  _et al._ (1995)
for an element $g$ of the Lie group associated to the $\mathfrak{snw}$
algebra,

$\displaystyle S_{\rm WZW}$ $\displaystyle=\frac{1}{4\pi}\int
dtd\phi\,\left\langle g^{-1}\partial_{+}gg^{-1}\partial_{\phi}g\right\rangle$
(10)
$\displaystyle+\frac{1}{12\pi}\int_{M}\left\langle\left(\tilde{g}^{-1}d\tilde{g}\right)^{3}\right\rangle,$

which is obtained after replacing the local solution of the CS constraint
$\mathbb{F}_{ij}=0$ ($i=\\{r,\phi\\}$), given by
$\mathbb{A}_{i}=g^{-1}\partial_{i}g$ with the boundary condition
$\left(\mathbb{A}_{t}+v\mathbb{A}_{\phi}\right)|_{\partial M}=0$ back in the
action (3). Here, we consider a boundary group element that depends only on
the boundary coordinates, i.e. $g=g(t,\phi)$. The group element $\tilde{g}$ in
(10) is the extension of $g$ to the three-dimensional bulk $M$. In the gauge
$\partial_{r}\mathbb{A}_{\phi}=0$, one finds

$\tilde{g}(t,r,\phi)=g(t,\phi)h(t,r),$ (11)

for some $h(t,r)$. The result (10) then follows from introducing the right
boundary term in the CS action such that $\delta S|_{\rm on-shell}=0$ and
considering $\partial_{t}h|_{\partial M}=0$. Notice that $v$ will represent
the velocity of the chiral modes, and we have defined
$x^{\pm}=(1/2)\left(t\pm(1/v)\phi\right)$ and $\partial_{\pm}=\partial_{t}\pm
v\partial_{\phi}$. In order to find the explicit form of $S_{\rm WZW}$ we look
at the left-invariant Maurer-Cartan form on $\mathfrak{snw}$,
$\Omega=g^{-1}dg$, which satisfies the Maurer-Cartan equation
$d\Omega+\Omega\wedge\Omega=0$. From (11), one can see that the bulk and
boundary left-invariant Maurer Cartan forms are related by
$\tilde{\Omega}=h^{-1}\Omega h+h^{-1}dh$. By employing the $\mathfrak{snw}$
commutation relations (1), the Maurer-Cartan equation can be solved and leads
to

$\Omega=d\theta\mathcal{J}+d\varphi\mathcal{Z}+\Omega_{\mathcal{P}}^{a}\mathcal{P}_{a}+\Omega_{\mathcal{T}}\mathcal{T}+\mathcal{Q}_{\alpha}\Omega_{\mathcal{Q}}^{\alpha},$
(12)

where we have defined

$\displaystyle\Omega_{\mathcal{P}}^{a}$
$\displaystyle=d\sigma^{a}-\epsilon^{a}_{\;\;b}\sigma^{b}(d\theta+\beta
d\varphi),$ (13) $\displaystyle\Omega_{\mathcal{T}}$
$\displaystyle=d\vartheta+\frac{1}{2}\epsilon_{ab}\sigma^{a}\Omega_{\mathcal{P}}^{b}-\frac{1}{4}\bar{\chi}_{\alpha}\left(\gamma_{0}\right)^{\alpha}_{\;\;\beta}d\chi^{\beta},$
$\displaystyle\Omega_{\mathcal{Q}}^{\alpha}$
$\displaystyle=\exp[-(1/2)\left(\theta+\beta\varphi\right)\left(\gamma_{0}\right)^{\alpha}_{\;\;\beta}]d\chi^{\beta},$

where $\chi^{\alpha}$ is a Grassmann-valued spinor. The one-form
$\tilde{\Omega}$ has the same functional form as $\Omega$ in terms of fields
$\tilde{\theta}$, $\tilde{\varphi}$, $\tilde{\sigma}^{a}$ and $\tilde{\chi}$
that include $r$-dependent functions which properly decouple when expressing
the Wess-Zumino term in (10) as a boundary integral.

By putting now all these solutions back into $S_{\rm WZW}$, one finds that
$\vartheta$ is the Lagrange multiplier that enforces the constraint
$\partial_{+}\theta^{\prime}+\beta\partial_{+}\varphi^{\prime}=0$, which
implies $\theta+\beta\varphi=\rho(t)+\lambda(x^{-})$, with $\rho(t)$ and
$\lambda(x^{-})$ arbitrary functions of their arguments. Replacing this
expression in the action and integrating out the field $\sigma^{a}$, we find

$\displaystyle S_{\rm WZW}$ $\displaystyle=\frac{1}{4\pi}\int
dtd\phi\Big{(}\tilde{\mu}\,\partial_{\phi}\varphi\partial_{+}\varphi+\mu_{1}\psi\partial_{+}\psi\Big{)},$
(14)

where we have imposed the extra fermionic boundary condition
$\chi^{1}|_{\partial M}={\rm constant}$ (inspired in the three-dimensional
supergravity analysis Henneaux  _et al._ (2000); Barnich  _et al._ (2017)),
introduced the new field $\psi=(\sqrt{v\partial_{-}\lambda}/2)\chi^{2}$, and
defined $\tilde{\mu}=\mu_{3}-2\beta\mu_{2}+\beta^{2}\mu_{0}$. This chiral CFT
describes the chiral boson and chiral Majorana modes that propagate on the
boundary of the system in agreement with the boundary states of the MR state.
This represents one of the main results of our work. Notice that in the
unbroken supersymmetric limit, both modes have the same velocity Ma  _et al._
(2021). We will indeed analyze the broken supersymmetric phase of our theory
in the next section, where we will introduce a Goldstino field that induces a
mass term to the spin-3/2 fermion.

Broken supersymmetry and topological response:  Supersymmetry breaking can be
obtained by introducing a fermionic Goldstone field, known as Goldstino Volkov
and Akulov (1973). Importantly, this new field induces a mass term for the
spin-3/2 field Deser and Zumino (1977); Dereli and Deser (1978); Deser (1984),
which is expected to be a massive collective mode of the bulk. Here we adopt
the approach proposed in Ref.Palumbo (2020) by one of the authors, where it
has been shown that there exists a non-propagating fermionic field in 2+1
dimensions that is able to induce a spin-3/2 mass term. We interpret this
fermion as a Goldstino within our current framework, described by the action

$S_{\rm\eta}=\frac{\mu_{1}}{4\pi}\int_{M}d\bar{\eta}_{\alpha}\wedge\hat{\gamma}^{\alpha}_{\;\;\beta}\wedge\,d\eta^{\alpha},$
(15)

where $\eta^{\alpha}$ is the Goldstino spinor field describing a neutral
fermion, and $\bar{\eta}_{\alpha}=\eta^{\beta}C_{\beta\alpha}$ is its
conjugate. We have also introduced the matrix
$\hat{\gamma}^{\alpha}_{\;\;\beta}=\tau\left(\gamma_{0}\right)^{\alpha}_{\;\;\beta}$,
where $\tau$ is some fixed non-dynamical clock form defining a Newton-Cartan
structure. The term (15) is therefore a boundary term and does not contribute
to the bulk dynamics of the system. This action is metric independent and so
far does not introduce further propagating degrees of freedom in our theory.
Importantly, it partially resembles the higher-derivative term that appears in
the Volkov-Akulov action for the relativistic Goldstino in 2+1 dimensions
Bansal and Sorokin (2018). We can naturally couple the Goldstino to the
spin-3/2 field as follows

$d\eta^{\alpha}\rightarrow d\eta^{\alpha}-\sqrt{m}\Psi^{\alpha},\hskip
11.38092ptd\bar{\eta}_{\alpha}\rightarrow
d\bar{\eta}_{\alpha}-\sqrt{m}\bar{\Psi}_{\alpha},$ (16)

with $m$ a dimensionful parameter. This real coefficient is indeed the mass of
the spin-3/2 field. In fact, by replacing the above expressions in $S_{\eta}$
we obtain the following term

$S_{m}=\frac{\mu_{1}\,m}{4\pi}\int_{M}\bar{\Psi}_{\alpha}\wedge\hat{\gamma}^{\alpha}_{\;\;\beta}\wedge\Psi^{\beta},$
(17)

which the simplest mass term that can be built for the spin-3/2 field and it
is also parity-odd similarly to the Dirac mass in the spin-1/2 theory Dereli
and Deser (1978); Deser (1984). This term breaks local supersymmetry and is
compatible with the possible existence of a massive spin-3/2 collective mode.
We can now integrate out $\Psi^{\alpha}$ in the bulk action $S_{\rm CS}+S_{m}$
to obtain a purely bosonic effective action that will allow us to derive the
topological response of the MR state. Before integrating out $\Psi^{\alpha}$,
we rescale it $\Psi^{\alpha}\rightarrow(\sqrt{8\pi})\Psi^{\alpha}$ and by
assuming that $m$ is large and positive and neglecting the terms that couple
$\eta^{\alpha}$ and $\Psi^{\alpha}$ (they are proportional to $\sqrt{m}$,
which we consider small compared to $m$), we finally obtain a CS term
$(\mu_{1}\varepsilon/4\pi)(\omega+\beta a)\wedge d(\omega+\beta a)$ (with
$\varepsilon={\rm sign}(m)$) that adds up to the other CS terms in (9)
Alvarez-Gaume  _et al._ (1985); Grimm  _et al._ (2019). Moreover, in order to
have a unique geometric response from the background geometry, we vary the
action with respect to the spatial dreibein $e^{a}$ to obtain the field
equation $R_{a}=0$, which in turn yields the following equation for the
torsion

$T^{a}\equiv de^{a}+\epsilon^{a}_{\;\;b}\wedge
e^{b}\omega=-\beta\epsilon^{a}_{\;\;b}e^{b}\wedge a.$ (18)

This equation allows us to formally express the dreibein in terms of the spin
connection and the field $a$. Thus, the effective action that depends only on
the external fields is obtained by integrating out also the spin-1
hydrodynamic field $a$ such that

$\displaystyle
S[A,\omega]=\displaystyle\frac{1}{4\pi}\int_{M}\Bigg{[}\hat{c}\,\omega\wedge
d\omega+\nu A\wedge d\left(A+2\bar{s}\,\omega\right)\Bigg{]}.$ (19)

Here, for the MR state, $\hat{c}=\nu\bar{s}^{2}-c/12$, where $\nu=1/2$,
$\bar{s}=c=3/2$ are the filling factor, average orbital spin and chiral
central charge, respectively Gromov  _et al._ (2015); Gromov and Son (2017).
The first and third terms in the above action are known as gravitational
Chern-Simons Gromov  _et al._ (2015) and Wen-Zee term Wen and Zee (1992),
respectively. The former is associated to the gravitational anomaly, while the
latter is related to the Hall viscosity $\eta_{H}$ Avron  _et al._ (1995);
Read (2009); Read and Rezayi (2011); Hoyos and Son (2012); Bradlyn  _et al._
(2012), which represents the response of the Hall fluid to shear or strain
Geracie  _et al._ (2017); Bradlyn and Read (2015).

The physical coefficients in the action are obtained by fixing the arbitrary
constants $\mu_{i}$ in the invariant bilinear form (5) as
$\mu_{0}=2\varepsilon\nu\bar{s}+c/12$, $\mu_{1}=2\nu\bar{s}$ and
$\mu_{2}=2\beta\nu\bar{s}(\varepsilon+\bar{s})$ and
$\mu_{3}=2\beta^{2}\nu\bar{s}(\varepsilon+2\bar{s})$ in our theory with
$\varepsilon=1$. Note that the results obtained in Salgado-Rebolledo and
Palumbo (2021) for Laughlin states can be recovered by setting $\varepsilon=0$
(which is compatible with removing the fermion field $\Psi^{\alpha}\rightarrow
0$) together with $\nu=1/k$, $\bar{s}=k/2$, $c=1$ and a particular choice of
$\beta$.

Introducing the Goldstino in the system does not only break supersymmetry in
the bulk, but also at the boundary. Indeed, considering $\tau=dt$ and the
boundary value of spin-3/2 field given by the Maurer-Cartan form
$\Omega_{\mathcal{Q}}^{\alpha}$, one finds that the contribution of the term
(17) to the boundary action reads

$\frac{\mu_{1}mv\partial_{-}\lambda}{16\pi}\int
dtd\phi\;\psi\partial_{\phi}\psi.$ (20)

where, without loss of generality we have assumed $\partial_{-}\lambda$ to be
an arbitrary constant. Thus, the effect of the Goldstino at the boundary is a
shift in the velocity of the chiral Majorana fermion $v\rightarrow
v^{\prime}=v+mv\partial_{-}\lambda/4$, breaking the supersymmetry of (14).
Note that $\beta$ is still arbitrary and, by choosing
$\beta=\sqrt{\mu_{1}/\mu_{0}}$, the boundary theory obtained by adding (20) to
$S_{\rm WZW}$ takes the form

$\displaystyle S_{\rm bdy}$ $\displaystyle=\frac{\nu\bar{s}}{2\pi}\int
dtd\phi\Big{(}\partial_{\phi}\varphi(\partial_{t}+v\partial_{\phi})\varphi+\psi(\partial_{t}+v^{\prime}\partial_{\phi})\psi\Big{)},$
(21)

compatible with the broken supersymmetric phase of the edge theory of the MR
state Ma  _et al._ (2021).

Conclusions and outlook:  Summarizing, in this paper we have proposed a novel
geometric model for the Moore-Read FQH state based on non-relativistic
supergeometry by introducing the novel gauge algebra $\mathfrak{snw}$ and
considering the corresponding CS action. Differently from canonical
supergravity, our theory is not relativistic and naturally generalizes the
non-supergeometric actions introduced in literature for the Abelian FQH
states. On one hand, our topological action naturally encodes a spin-3/2
fermion in the bulk, which is expected to emerge as a collective mode in the
MR state. On the other hand, our theory gives rise to a chiral supersymmetric
CFT that decomposes into a chiral boson and a chiral Majorana fermion.
Finally, we have shown that when supersymmetry is broken due to an effective
Goldstino, the spin-3/2 mode acquires a mass and can be integrated out,
providing the correct topological bosonic action that takes into account both
the Hall conductivity and Hall viscosity. At the same time, supersymmetry is
broken at the boundary, where the effect of adding the Goldstino is to modify
velocity of the chiral Majorana field, recovering the result of Ma  _et al._
(2021). It would be very interesting to embed our supergeometric theory into a
generalized super-bimetric model by following Ref.Gromov and Son (2017) to
include the massive spin-2 GMP mode as the superpartner of the spin-3/2
neutral fermion. Moreover, our supergeometric approach could be important for
the geometrical characterization of hierarchies of FQH states in the second
Landau level build from the MR state Bonderson and Slingerland (2008). As it
happens in the bosonic case Bergshoeff  _et al._ (2018), one could expect that
the right massive wave equation for the spin-3/2 field in the FQHE can be
obtained from a special non-relativistic limit of a spin-3/2 generalization of
the Fierz-Pauli equation in 2+1 dimensions. It would be as well interesting to
embedded our model into a full supersymmetric Newton-Cartan geometry, as done
in Salgado-Rebolledo and Palumbo (2021) in the bosonic case by means of Lie
algebra expansions Hatsuda and Sakaguchi (2003); de Azcarraga  _et al._
(2003); Izaurieta  _et al._ (2006) following the method introduced in Peñafiel
and Salgado-Rebolledo (2019). We leave all these important open points to
future work.

Acknowledgments:  The authors are pleased to acknowledge discussions with Eric
Bergshoeff, Andrea Cappelli, Carlos Hoyos. We thank Dung Nguyen for carefully
reading the manuscript and for pointing out important details that led us to
improve the article. P.S-R. has received funding from the Norwegian Financial
Mechanism 2014-2021 via the National Science Centre (NCN) POLS grant
2020/37/K/ST3/03390. This work was partially supported by FNRS-Belgium
(conventions FRFC PDRT.1025.14 and IISN 4.4503.15), as well as by funds from
the Solvay Family.

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