01-12/2023

Carroll Fermions

Eric A. Bergshoeff${}^{\,a}$, Andrea Campoleoni${}^{\,b}$, Andrea
Fontanella${}^{\,c,d}$,
Lea Mele${}^{\,b}$ and Jan Rosseel${}^{\,e}$

a Van Swinderen Institute, University of Groningen
Nijenborgh 4, 9747 AG Groningen, The Netherlands
e.a.bergshoeff[at]rug.nl

b Service de Physique de l’Univers, Champs et Gravitation
Université de Mons – UMONS, 20 place du Parc, 7000 Mons, Belgium
andrea.campoleoni[at]umons.ac.be
lea.mele[at]umons.ac.be

c Perimeter Institute for Theoretical Physics,
Waterloo, Ontario, N2L 2Y5, Canada
d School of Mathematics $\&$ Hamilton Mathematics Institute,
Trinity College Dublin, Ireland
andrea.fontanella[at]tcd.ie

e Division of Theoretical Physics, Rudjer Bošković Institute,
Bijenička 54, 10000 Zagreb, Croatia
Jan.Rosseel[at]irb.hr

Using carefully chosen projections, we consider different Carroll limits of
relativistic Dirac fermions in any spacetime dimensions. These limits define
Carroll fermions of two types: electric and magnetic. The latter type
transforms as a reducible but indecomposable representation of the Carroll
group. We also build action principles for all Carroll fermions we introduce;
in particular, in even dimensions we provide an action principle for a minimal
magnetic Carroll fermion, having the same number of components as a Dirac
spinor. We then explore the coupling of these fermions to magnetic Carroll
gravity in both its first-order and second-order formulations.

###### Contents

  1. 1 Introduction
  2. 2 Carroll Fermions
    1. 2.1 Carroll transformations of fermion fields
    2. 2.2 Electric and magnetic Carroll fermions from a Lagrangian perspective
    3. 2.3 Carroll limit of Dirac’s action in Hamiltonian form
  3. 3 Coupling to Carroll gravity
    1. 3.1 Carroll geometry
    2. 3.2 Coupling Carroll fermions to gravity in the first-order formulation
    3. 3.3 Carroll limit of fermions coupled to gravity in the second-order formulation
  4. 4 Conclusions
  5. A Conventions

##  1 Introduction

The Carroll group refers to the particular Inönü-Wigner contraction of the
Poincaré group that is physically interpreted as its ultra-relativistic limit,
in which the speed of light $c$ is taken to 0. Originally studied by Lévy-
Leblond [1] and Gupta [2], Carroll symmetries have recently received attention
in the context of flat space holography (see, e.g., the recent reviews [3, 4])
and in investigations of the physics of black hole horizons [5]. This renewed
interest stems from the fact that the conformal extension of the Carroll group
is the so-called BMS group that describes the asymptotic symmetries of flat
spacetime at null infinity [6] and that any null hypersurface is described by
a manifold whose structure group is the Carroll group. Carroll symmetries also
naturally arise in the tensionless limit of string theory; see, e.g., [7, 8]
and references therein.

Motivated by these applications of Carroll symmetry, various authors have
recently considered the construction and study of Carroll invariant field
theories [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24]. Such
field theories are usually obtained by taking a $c\rightarrow 0$ limit of
relativistic ones and this can typically be done in two different ways.
Correspondingly, two different types of Carroll field theories, often dubbed
‘electric’ and ‘magnetic’, are obtained in the limit. To illustrate these
electric and magnetic Carroll limits, let us follow [12, 13] and consider a
relativistic free real scalar $\Phi$ with mass $M$, whose Lagrangian is given
in Hamiltonian form by

$\displaystyle\mathcal{L}=\Pi_{\Phi}\partial_{t}\Phi-\frac{c^{2}}{2}\Pi_{\Phi}^{2}-\frac{1}{2}\partial_{a}\Phi\partial^{a}\Phi\mp\frac{(Mc)^{2}}{2}\Phi^{2}\,,$
(1.1)

where $\partial_{a}$ denotes the derivatives with respect to the spatial
coordinates and $\Pi_{\Phi}$ is the momentum conjugate to $\Phi$. We have
furthermore reinstated the speed of light $c$, by setting the time-like
coordinate equal to $ct$, and left the sign in front of the mass term
arbitrary, so that we allow the scalar to be tachyonic. The Lagrangian of the
real free electric Carroll scalar is then obtained as the $c\rightarrow 0$
limit of (1.1) (with the upper sign for the mass term), after having first
redefined $\Pi_{\Phi}$, $\Phi$ and $M$ as

$\displaystyle\Pi_{\Phi}=\frac{\pi}{c}\,,\qquad\qquad\Phi=c\,\phi\,,\qquad\qquad
M=\frac{m}{c^{2}}\,.$ (1.2)

Explicitly, one finds

$\displaystyle\mathcal{L}_{\text{el.
scalar}}=\pi\partial_{t}\phi-\frac{1}{2}\pi^{2}-\frac{m^{2}}{2}\phi^{2}\,,\quad\text{or
equivalently}\quad\mathcal{L}_{\text{el.
scalar}}=\frac{1}{2}\left(\partial_{t}\phi\right)^{2}-\frac{m^{2}}{2}\phi^{2}\,,$
(1.3)

where the second Lagrangian is obtained from the first one by eliminating the
auxiliary field $\pi$ using its equation of motion. Alternatively, one can
first perform the redefinitions

$\displaystyle\Pi_{\Phi}=\pi\,,\qquad\qquad\Phi=\phi\,,\qquad\qquad
M=\frac{m}{c}\,,$ (1.4)

before taking the $c\rightarrow 0$ limit in (1.1). For the tachyonic scalar
(i.e., with the lower sign in front of the mass term in (1.1)), this gives a
Lagrangian for a magnetic Carroll scalar:

$\displaystyle\mathcal{L}_{\text{magn.
scalar}}=\pi\partial_{t}\phi-\frac{1}{2}\partial_{a}\phi\partial^{a}\phi+\frac{m^{2}}{2}\phi^{2}\,,$
(1.5)

Unlike the electric theory, which can be written in terms of a single scalar
$\phi$, the magnetic theory involves two scalar fields $\pi$, $\phi$. These
transform in a reducible indecomposable representation of the Carroll group,
since under Carroll boosts the scalar $\pi$ transforms to $\phi$, but not vice
versa [25].

The purpose of this paper is to discuss an analogue of the above electric and
magnetic limits for fermion fields. Electric and magnetic Carroll fermion
field theories have been derived using various limit procedures (e.g., by
considering the $c\to 0$ limit of the equations of motion or in a Lagrangian
or Hamiltonian formalism) in the literature before [11, 15, 23].111An
interesting bottom-up construction of Carroll fermion field theories appeared
in [18, 19] (see also [26]). In this approach, one does not start from a $c\to
0$ limit of relativistic fermions, but instead constructs spinor
representations of the homogeneous Carroll group, starting from degenerate
Clifford algebras [27, 28, 29]. Here, we aim to provide a systematic analysis
of the possible limits to obtain recently constructed, as well as novel
Lagrangians for both electric and magnetic Carroll fermions, i.e., Lagrangians
whose field equations can be viewed as ‘square roots’ of the equations of
motion of electric and magnetic Carroll scalars. The systematic nature of our
treatment also clarifies some subtleties that arise when considering Carroll
limits of fermion field theories.

The procedure presented in section 2 of this paper in particular elucidates
how Lagrangians for electric and magnetic Carroll fermions can be obtained
from different limits of a single relativistic starting point, not unlike how
the electric and magnetic scalar Lagrangians (1.3) and (1.5) correspond to
different limits of (1.1). Due to the first-order nature of fermion field
Lagrangians, achieving this is slightly more subtle than in the scalar field
case. Indeed, since the Klein-Gordon Lagrangian is of second order in
derivatives, $\Phi$ and its conjugate momentum $\Pi_{\Phi}$ in (1.1) are two
independent phase-space variables. They can thus naturally be rescaled
independently (as in (1.2) and (1.4)), giving rise to two different Carroll
limits. By contrast, since a relativistic fermion $\Psi$ obeys a first-order
equation of motion, its conjugate momentum $\Pi_{\Psi}$ is no longer an
independent variable, but obeys a primary second-class constraint that says
that $\Pi_{\Psi}$ is proportional to $\Psi^{\dagger}$. Any rescaling of $\Psi$
then fixes a corresponding rescaling of $\Pi_{\Psi}$ and consequently only one
regular222Rescaling $\Pi_{\Psi}$ and $\Psi$ in a way that violates the primary
constraint that relates them can lead to a second Carroll limit. We discuss
such a limit in subsection 2.3, but we consider it as singular, since the
resulting theory contains twice as many fermionic degrees of freedom as the
original relativistic starting point, due to the fact that in this limit
$\Pi_{\Psi}$ and $\Psi$ are treated independently. Carroll limit of the Dirac
action can be defined that turns out to be an electric one.

In order to be able to take both electric and magnetic limits in a regular
manner, we start in subsection 2.2 from an off-diagonal Lagrangian for two
relativistic Dirac spinors. We furthermore decompose these two fermion fields
in two spinors that each constitute a representation of the spatial rotation
subgroup of the Lorentz group. The four spinors thus obtained give enough
freedom to perform the independent rescalings needed to take not only an
electric Carroll limit, but also a magnetic one that features an
indecomposable reducible spinor representation of the Carroll group. While
these limits can be taken in arbitrary dimensions, they do not always lead to
a minimal result, in the sense that sometimes one can perform a further
consistent truncation that still gives a non-trivial theory for an electric or
magnetic Carroll fermion. For instance, in even dimensions, a consistent
truncation on our result for the magnetic Carroll limit leads to a novel
action333While this action has to our knowledge not yet appeared in the
literature, the field equations stemming from it have been considered before
in the massless case in [11]. for a minimal magnetic Carroll fermion, with as
many components as a single Dirac spinor. In addition to describing the two
Carroll limits in a Lagrangian formalism involving two spinors, we will in
subsection 2.3 also outline how the resulting theories can be obtained via a
singular limit of the usual Dirac action in Hamiltonian form, along the lines
already explored in [23, 30].

Once a detailed description of Carroll limits for spinors in flat space is
achieved, the extension to fermions in curved spacetime and coupled to gravity
does not present particular difficulties and we illustrate this in two
particular cases of electric and magnetic limits in section 3. In particular,
we discuss the coupling of the electric and minimal magnetic Carroll fermions
to a first-order formulation of Carroll gravity, called magnetic Carroll
gravity, emphasizing the role of the spin-connection field. We show that, in
passing to a second-order formulation, only magnetic Carroll fermions lead to
quartic interactions for fermions in the action. Taking the limit directly in
a second-order formulation of General Relativity coupled to fermions, leads to
an additional divergent term in the action. Accepting this leading order term
as the basic invariant leads to a second-order formulation of Carroll gravity,
called electric Carroll gravity, that is independent of the spin-connection
and does not contain any surviving fermions. Alternatively, one may control
this divergent term by applying a Hubbard-Stratonovich transformation after
which the sub-leading terms basically lead to the same coupling of magnetic
Carroll gravity to Carroll fermions as found before.

##  2 Carroll Fermions

In this section, we will discuss how electric and magnetic Carroll fermions
arise as limits of relativistic fermions, in which the speed of light $c$ is
sent to zero. We will first consider the $c\rightarrow 0$ limit of the Lorentz
transformation rules of a relativistic Dirac spinor in any spacetime
dimensions. We will show that this limit can be taken in two different ways,
leading to Carroll transformation rules of electric and magnetic Carroll
fermions respectively. Next, in subsection 2.2, we will consider two different
ways of taking the $c\rightarrow 0$ limit on an off-diagonal Lagrangian for
two Dirac spinors and show that, after performing suitable consistent
truncations, this leads to Lagrangians for electric and magnetic Carroll
fermions.444In the electric case, the truncation is allowed in any dimension
$D$, whereas in the magnetic case, the truncation is allowed only in even $D$,
as it requires the use of the $\Gamma_{\star}$ matrix defined in Appendix A.
Finally, in subsection 2.3 we will reconsider the magnetic Carroll fermion
action from a Hamiltonian point of view, as a singular limit of the usual
Dirac action in Hamiltonian form.

###  2.1 Carroll transformations of fermion fields

To obtain the transformation rules of fermionic fields under (homogeneous)
Carroll symmetries, we start from a relativistic Dirac spinor $\Psi$ in
$D$-dimensional Minkowski spacetime with inertial coordinates $X^{A}$,
$A=0,1,...,D-1$. Under infinitesimal Lorentz transformations (with parameters
$\Lambda^{AB}=-\Lambda^{BA}$), $\Psi$ transforms as follows:

$\delta\Psi=\Xi^{A}\frac{\partial\Psi}{\partial
X^{A}}-\frac{1}{4}\Lambda^{AB}\Gamma_{AB}\Psi\,,$ (2.1)

with

$\displaystyle\delta X^{A}\equiv X^{\prime
A}-X^{A}=-\Xi^{A}\,,\qquad\qquad\Xi^{A}=\Lambda^{A}{}_{B}\,X^{B}\,.$ (2.2)

The Carroll limit formally consists of sending the speed of light $c$ to
0.555Strictly speaking, one should first introduce a dimensionless parameter
$\omega$ via the redefinition $c\to\omega c$ and then take the limit
$\omega\to 0$. It is however convenient to first define $\tilde{c}=1/c$, which
has opposite dimension, and instead take $\tilde{c}\to\infty$. The Carroll
limit is then defined by first performing the following redefinition of the
coordinates and parameters:

$\displaystyle X^{0}=\frac{t}{\tilde{c}}\,,\qquad\qquad
X^{a}=x^{a}\,,\qquad\qquad\Lambda^{ab}=\lambda^{ab}\,,\qquad\qquad\Lambda^{0a}=\frac{1}{\tilde{c}}\lambda^{0a}\,,$
(2.3)

and next taking the limit $\tilde{c}\to\infty$. Applying this procedure to
(2.2) leads to the usual transformations of the spacetime coordinates under
spatial rotations and Carroll boosts with parameters $\lambda^{ab}$ and
$\lambda^{0a}$ respectively,

$\displaystyle\delta t\equiv
t^{\prime}-t=-\xi^{0}\,,\qquad\qquad\qquad\qquad\delta x^{a}\equiv x^{\prime
a}-x^{a}=-\xi^{a}\,,$ $\displaystyle\text{with }\ \
\xi^{0}=\lambda^{0}{}_{a}\,x^{a}\ \ \text{ and }\ \
\xi^{a}=\lambda^{a}{}_{b}\,x^{b}\,.$ (2.4)

The $\tilde{c}\rightarrow\infty$ limit, applied to the Lorentz transformation
rule (2.1) after performing the redefinitions (2.3), gives the following
transformation rule of a fermion $\psi\equiv\Psi$:

$\delta\psi=\xi^{0}\frac{\partial\psi}{\partial
t}+\xi^{a}\frac{\partial\psi}{\partial
x^{a}}-\frac{1}{4}\lambda^{ab}\Gamma_{ab}\psi\,.$ (2.5)

We note that only spatial rotations appear in the spin part of the
transformation rule of the above Carroll fermion. As we will see in the next
subsection, this transformation behaviour is characteristic of an electric
Carroll fermion. It turns out that there exists a second way of taking the
limit that leads to a transformation that contains Carroll boosts in its spin
part and that is the relevant one for magnetic Carroll fermions. This limit
uses the following orthogonal projection operators666Note that for even $D$
one can also split a Dirac spinor in another way that preserves spatial
rotations by using the projectors
$\tilde{P}_{\pm}=\frac{1}{2}\big{(}\mathds{1}\pm\Gamma^{0}\Gamma_{\star}\big{)}$,
with $\Gamma_{\star}$ defined in (A.3). However, it turns out that using these
${\tilde{P}}_{\pm}$ projectors leads to similar results that are mapped to the
ones using the $P_{\pm}$ projectors via the gamma matrix isomorphism
$\Gamma^{A}\to\pm i\Gamma^{A}\Gamma_{\star}$.

$\displaystyle
P_{\pm}\equiv\frac{1}{2}\big{(}\mathds{1}\pm\mathrm{i}\Gamma^{0}\big{)}\,,\qquad\qquad\text{obeying
}\ P_{\pm}P_{\pm}=P_{\pm}\,,\ \ P_{\pm}P_{\mp}=0\,,\ \
P_{\pm}^{\dagger}=P_{\pm}\,.$ (2.6)

Using these projectors, one can split a Dirac spinor $\Psi$ into two
independent components $\Psi_{\pm}\equiv P_{\pm}\Psi$, each of which
transforms covariantly with respect to spatial rotations. We then have the
freedom to redefine these two projections differently when defining a Carroll
limit. More precisely, we make the following redefinition:

$\Psi_{\pm}=\tilde{c}^{\pm 1/2+\epsilon}\,P_{\pm}\,\psi_{\pm}\,,$ (2.7)

where $\epsilon$ is a free parameter. Note that this redefinition is
invertible:

$\psi_{\pm}=\tilde{c}^{\mp 1/2-\epsilon}\,P_{\pm}\,\Psi_{\pm}\,.$ (2.8)

Substituting the redefinitions (2.3) and (2.7) into the Lorentz transformation
rule (2.1) we obtain the following transformation rule of a projected Carroll
fermion:

$\displaystyle\delta\psi_{+}$ $\displaystyle=$
$\displaystyle\xi^{0}\frac{\partial\psi_{+}}{\partial
t}+\xi^{a}\frac{\partial\psi_{+}}{\partial
x^{a}}-\frac{1}{4}\lambda^{ab}\Gamma_{ab}\psi_{+}\,,$ (2.9a)
$\displaystyle\delta\psi_{-}$ $\displaystyle=$
$\displaystyle\xi^{0}\frac{\partial\psi_{-}}{\partial
t}+\xi^{a}\frac{\partial\psi_{-}}{\partial
x^{a}}-\frac{1}{4}\lambda^{ab}\Gamma_{ab}\psi_{-}-\frac{1}{2}\lambda^{0a}\Gamma_{0a}\psi_{+}\,.$
(2.9b)

Note that the Carroll boosts now appear non-trivially in the spin part of this
transformation rule. The projected spinors $\psi_{\pm}$ form a reducible but
indecomposable representation of the homogeneous Carroll group (that consists
of spatial rotations and Carroll boosts).

###  2.2 Electric and magnetic Carroll fermions from a Lagrangian perspective

Having defined the two limits leading to a Carroll fermion and a projected
Carroll fermion, we now take analogous limits at the level of the Lagrangian.
It is convenient to use as starting point the following off-diagonal
Lagrangian for two Dirac spinors $\Psi$ and $\mathbf{X}$:

$\mathcal{L}_{\text{off-
diag}}=\bar{\mathbf{X}}\Gamma^{A}\partial_{A}\Psi-\frac{M}{\tilde{c}}\bar{\mathbf{X}}\Psi+\text{h.c.}\,,$
(2.10)

where $M$ is a complex parameter with the dimension of mass and where the
Dirac conjugate is defined by
$\bar{\Psi}\equiv\mathrm{i}\Psi^{\dagger}\Gamma^{0}$.777More details about our
notation and conventions can be found in Appendix A. Both Dirac spinors $\Psi$
and $\mathbf{X}$ transform under Lorentz transformations as in equations (2.1)
and (2.2). As it is, the Lagrangian (2.10) describes both tachyonic and non-
tachyonic modes, due to the fact the parameter $M$ can be chosen to be
complex.888This is similar to the bosonic case, where the starting point of
certain Carroll limits also requires a tachyonic mode. An additional feature
in the fermionic case is that, after diagonalisation, one of the two spinors
has a negative definite kinetic term. While this Lagrangian may seem
unconventional, we note that in the next subsection we will argue that our
results on its Carrollian limits can also be reproduced from a more
conventional starting point, namely using a Dirac-like action in Hamiltonian
form.

It is clear that when we define a limit in which we naïvely scale all
components of the Dirac spinors as $\mathbf{X}\to\tilde{c}^{\alpha}\mathbf{X}$
and $\Psi\to\tilde{c}^{\beta}\Psi$, the terms in the action with a time
derivative will always dominate the terms with a spatial derivative. In
analogy to the bosonic case, we will call such a limit an Electric Carroll
limit. To create more freedom in performing rescalings of the spinors we
introduce the following projected spinors:

$\Psi_{\pm}=P_{\pm}\Psi\,,\qquad\qquad\qquad\mathbf{X}_{\pm}=P_{\pm}\mathbf{X}\,,$
(2.11)

where $P_{\pm}$ are the projectors that we introduced in (2.6). The four
spinors $\Psi_{\pm}$ and $\mathbf{X}_{\pm}$, which we are free to scale
independently, give us sufficient freedom to define a second limit that leads
to an action resembling the magnetic scalar case. We will call this limit a
Magnetic Carroll limit. Below we discuss these two different limits in more
detail.

The Electric Carroll limit

The Electric Carroll limit is defined by the rescaling (2.3) together with the
following rescaling of the four spinors:

$\displaystyle\Psi_{+}$ $\displaystyle=$
$\displaystyle\sqrt{\tilde{c}}\,\tilde{c}^{\epsilon}\psi_{+}\,,\qquad\qquad\Psi_{-}=\frac{1}{\sqrt{\tilde{c}}}\,\tilde{c}^{\epsilon}\psi_{-}\,,$
(2.12a) $\displaystyle\mathbf{X}_{+}$ $\displaystyle=$
$\displaystyle\sqrt{\tilde{c}}\,\tilde{c}^{\epsilon}\chi_{+}\,,\qquad\qquad\mathbf{X}_{-}=\frac{1}{\sqrt{\tilde{c}}}\,\tilde{c}^{\epsilon}\chi_{-}\,.$
(2.12b)

We take $\epsilon=-1$, in order to have no overall rescaling of the
Lagrangian. Furthermore, we rescale the complex parameter $M$ as
follows:999Note that we work in the convention where $M$ has the dimension of
a mass parameter, and therefore different values of $\alpha$ lead to
parameters $m$ with different dimensions. In order to avoid spurious notation,
we will keep calling this parameter $m$.

$\displaystyle M=\tilde{c}^{\alpha}m\,,$ (2.13)

and we take $\alpha=2$. By taking this electric Carroll limit, the spinors
$\chi_{-}$ and $\psi_{-}$ drop out and we obtain the following off-diagonal
Lagrangian in terms of $\chi_{+}$ and $\psi_{+}$:

$\displaystyle\mathcal{L}_{\text{off-
diag}}=\bar{\chi}_{+}\Gamma^{0}\dot{\psi}_{+}-m\bar{\chi}_{+}\psi_{+}+\text{h.c.}\
.$ (2.14)

We use a notation where $\dot{\psi}\equiv\partial\psi/\partial t$. This action
is invariant under the following Carroll transformations:101010Since it
describes a free field theory, we expect that the Lagrangian (2.14) has
additional symmetries, like it happens in the bosonic case for Carrollian
scalars, both in two dimensions [31, 32] and generic $D$ [11, 20, 33, 34].

$\displaystyle\delta\psi_{+}$ $\displaystyle=$
$\displaystyle\xi^{0}\dot{\psi}_{+}+\xi^{a}\partial_{a}\psi_{+}-\frac{1}{4}\lambda_{ab}\Gamma^{ab}\psi_{+}\,,$
(2.15a) $\displaystyle\delta\chi_{+}$ $\displaystyle=$
$\displaystyle\xi^{0}\dot{\chi}_{+}+\xi^{a}\partial_{a}\chi_{+}-\frac{1}{4}\lambda_{ab}\Gamma^{ab}\chi_{+}\,,$
(2.15b)

that correspond to those in (2.5). These transformation rules show that it is
consistent to perform the following truncation

$\displaystyle\chi_{+}=\psi_{+}\,,$ (2.16)

after which we obtain the following electric Carroll Lagrangian in diagonal
form [15, 19]

$\displaystyle\mathcal{L}_{\text{electric\ Carroll,\
1}}=2\bar{\psi}_{+}\Gamma^{0}\dot{\psi}_{+}-2\,\mathfrak{Re}(m)\bar{\psi}_{+}\psi_{+}\
,$ (2.17)

where $\mathfrak{Re}(m)=\frac{1}{2}(m+m^{*})$ is a real mass parameter.

The purpose of having introduced the parent Lagrangian (2.10) is to unify the
electric limit with the magnetic limit discussed below. However, if one wants
to just obtain the electric Carroll Lagrangian (2.17), one can just start from
the Dirac Lagrangian, corresponding to the truncation $\mathbf{X}=\Psi$ in the
parent Lagrangian (2.10). One next redefines $\Psi={\tilde{c}}^{-1/2}\psi$ and
takes the Carroll limit. After taking this limit, one makes a further
truncation by setting $\psi_{-}=0$, thus obtaining the same electric Carroll
action (2.17). In section 3, we will show how a curved space analogue of this
way of taking the electric Carroll limit leads to an action of an electric
Carroll fermion coupled to Carroll gravity.

The Magnetic Carroll limit

The Magnetic Carroll limit is defined by the rescalings (2.3) together with
the following ‘twisted’ rescalings of the four spinors $\Psi_{\pm}$ and
$\mathbf{X}_{\pm}$:

$\displaystyle\Psi_{+}$ $\displaystyle=$
$\displaystyle\sqrt{\tilde{c}}\,\tilde{c}^{\epsilon}\psi_{+}\,,\qquad\qquad\Psi_{-}=\frac{1}{\sqrt{\tilde{c}}}\,\tilde{c}^{\epsilon}\psi_{-}\,,$
(2.18a) $\displaystyle\mathbf{X}_{+}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{\tilde{c}}}\,\tilde{c}^{\epsilon}\chi_{+}\,,\qquad\qquad\mathbf{X}_{-}=\sqrt{\tilde{c}}\,\tilde{c}^{\epsilon}\chi_{-}\,.$
(2.18b)

We now take $\epsilon=-1/2$, in order to have no overall rescaling of the
Lagrangian. Furthermore, we rescale the complex parameter $M$ as in (2.13)
with $\alpha=2$. By taking this magnetic limit, all four spinor projected
components survive, and we obtain the following action:

$\begin{split}\mathcal{L}_{\text{off-
diag}}=\bar{\chi}_{+}\Gamma^{0}\dot{\psi}_{+}+\bar{\chi}_{-}\Gamma^{0}\dot{\psi}_{-}+\bar{\chi}_{-}\Gamma^{a}\partial_{a}\psi_{+}-m(\bar{\chi}_{+}\psi_{+}+\bar{\chi}_{-}\psi_{-})+\text{h.c.}\
.\end{split}$ (2.19)

This action is invariant under the following Carroll transformations:

$\displaystyle\delta\psi_{+}$ $\displaystyle=$
$\displaystyle\xi^{0}\dot{\psi}_{+}+\xi^{a}\partial_{a}\psi_{+}-\frac{1}{4}\lambda_{ab}\Gamma^{ab}\psi_{+}\,,$
(2.20a) $\displaystyle\delta\psi_{-}$ $\displaystyle=$
$\displaystyle\xi^{0}\dot{\psi}_{-}+\xi^{a}\partial_{a}\psi_{-}-\frac{1}{4}\lambda_{ab}\Gamma^{ab}\psi_{-}-\frac{1}{2}\lambda_{0a}\Gamma^{0a}\psi_{+}\,,$
(2.20b) $\displaystyle\delta\chi_{+}$ $\displaystyle=$
$\displaystyle\xi^{0}\dot{\chi}_{+}+\xi^{a}\partial_{a}\chi_{+}-\frac{1}{4}\lambda_{ab}\Gamma^{ab}\chi_{+}-\frac{1}{2}\lambda_{0a}\Gamma^{0a}\chi_{-}\,,$
(2.20c) $\displaystyle\delta\chi_{-}$ $\displaystyle=$
$\displaystyle\xi^{0}\dot{\chi}_{-}+\xi^{a}\partial_{a}\chi_{-}-\frac{1}{4}\lambda_{ab}\Gamma^{ab}\chi_{-}\,.$
(2.20d)

These transformation rules show that in even dimensions we can make the
following truncations:111111Note that we give here two truncations. One could
also impose only one of the two truncations. We will not consider this non-
minimal case further here.

$\displaystyle\chi_{\pm}=\Gamma_{\star}\psi_{\mp}\,,$ (2.21)

where we have made use of the $\Gamma_{\star}$ matrix121212Further properties
of this $\Gamma_{\star}$ matrix can be found in Appendix A.

$\Gamma_{\star}=(-\mathrm{i})^{\frac{D}{2}+1}\Gamma^{0}\Gamma^{1}\cdots\Gamma^{D-1}$
(2.22)

to connect the different projections. Upon making this truncation, we obtain
the following minimal Lagrangian:131313There is also a second way of obtaining
the magnetic Carroll 1 Lagrangian (2.2). This consists in replacing the off-
diagonal mass term in the parent Lagrangian (2.10) with the following diagonal
one  $\mathcal{L}_{\text{mass-
diag}}=-i\frac{M}{2\tilde{c}}\left(\bar{\Psi}\Gamma_{\star}\Psi-\bar{\mathbf{X}}\Gamma_{\star}\mathbf{X}\right)\,,$
where here $M$ is a real parameter. Then, by applying the same magnetic
rescaling defined above, and by rescaling $M=\tilde{c}^{2}\,m$, one gets
(2.2).

$\displaystyle\mathcal{L}_{\text{magnetic\ Carroll,\
1}}=2\bar{\psi}_{-}\Gamma^{0}\Gamma_{\star}\dot{\psi}_{+}+2\bar{\psi}_{+}\Gamma^{0}\Gamma_{\star}\dot{\psi}_{-}+2\bar{\psi}_{+}\Gamma^{a}\Gamma_{\star}\partial_{a}\psi_{+}+$
$\displaystyle\hskip
93.89418pt+\,2i\,\mathfrak{Im}(m)(\bar{\psi}_{+}\Gamma_{\star}\psi_{-}+\bar{\psi}_{-}\Gamma_{\star}\psi_{+})\
,$ (2.23)

where $\mathfrak{Im}(m)=-\frac{i}{2}(m-m^{*})$ is a real mass parameter. In
the massless case, the equations of motion stemming from this Lagrangian
correspond to those obtained in [11] by taking a limit of the Dirac equation
but, to our knowledge, an action principle involving this minimal projected
Carroll fermion has not appeared before.

One can obtain a second magnetic Carroll Lagrangian with a different mass term
by inserting a $\Gamma_{\star}$ in the mass term of the initial Lagrangian
(2.10).141414A similar trick does not work in the electric case since in there
we only have a kinetic term involving two spinors of the same chirality. By
inserting a $\Gamma_{\star}$ in the mass term, one introduces a spinor with
opposite chirality that acts as a Lagrange multiplier which on-shell sets to
zero the action. Also, as explained in footnote 6, inserting a
$\Gamma_{\star}$ into the projection operators leads to Lagrangians that are
equivalent to the Lagrangians we obtained using projection operators without a
$\Gamma_{\star}$. This leads to the following alternative mass term:

$\displaystyle\mathcal{L}_{\text{mass}}=-\frac{M}{\tilde{c}}\bar{\mathbf{X}}\Gamma_{\star}\Psi+\text{h.c.}\,.$
(2.24)

After taking the magnetic limit defined above, and rescaling the mass
parameter $M$ as in (2.13) with $\alpha=1$, this mass term becomes

$\displaystyle\mathcal{L}_{\text{mass}}=-m\,\bar{\chi}_{-}\Gamma_{\star}\psi_{+}+\text{h.c.}\,,$
(2.25)

which after performing the truncation (2.21) reduces to the mass term of a
second magnetic Carroll Lagrangian:151515As we mentioned for the magnetic
Carroll 1 Lagrangian, there is also a second way of obtaining the magnetic
Carroll 2 Lagrangian (2.26). The diagonal mass term to consider in this case
is  $\mathcal{L}_{\text{mass-
diag}}=-\frac{M}{2\tilde{c}}\left(\bar{\Psi}\Psi-\bar{\mathbf{X}}\mathbf{X}\right)\,,$
and the Carroll limit requires to rescale $M=\tilde{c}\,m$.

$\displaystyle\hskip-5.69046pt\mathcal{L}_{\text{ magnetic\ Carroll,\
2}}=2\bar{\psi}_{-}\Gamma^{0}\Gamma_{\star}\dot{\psi}_{+}+2\bar{\psi}_{+}\Gamma^{0}\Gamma_{\star}\dot{\psi}_{-}+2\bar{\psi}_{+}\Gamma^{a}\Gamma_{\star}\partial_{a}\psi_{+}-2\,\mathfrak{Re}(m)\bar{\psi}_{+}\psi_{+}\,.$
(2.26)

It is also possible to obtain the two magnetic Carroll actions (2.2) and
(2.26) by taking the limit directly on a Lagrangian depending only on a single
Dirac spinor $\Psi$. A way to do it, is to either truncate the parent
Lagrangian (2.10) or the same Lagrangian but with modified mass term (2.24) by
imposing $\mathbf{X}_{\pm}=\Gamma_{\star}\Psi_{\mp}$. This leads to the
following ‘tachyonic Dirac’161616That these two Lagrangians describe tachyonic
fermions is seen by multiplying their equations of motion with the operators
$\Gamma^{A}\Gamma_{\star}\partial_{A}-\mathrm{i}(M/\tilde{c})\Gamma_{\star}$
and $\Gamma^{A}\Gamma_{\star}\partial_{A}+(M/\tilde{c})$ respectively. This
leads to the Klein-Gordon equation $(\Box+M^{2}/\tilde{c}^{2})\Psi=0$ for a
tachyon. Lagrangians

$\displaystyle\mathcal{L}=\bar{\Psi}\Gamma^{A}\Gamma_{\star}\partial_{A}\Psi-\mathrm{i}\frac{M}{\tilde{c}}\bar{\Psi}\Gamma_{\star}\Psi\,,\qquad\text{and}\qquad\mathcal{L}=\bar{\Psi}\Gamma^{A}\Gamma_{\star}\partial_{A}\Psi-\frac{M}{\tilde{c}}\bar{\Psi}\Psi\,,$
(2.27)

where the mass parameter $M$ is now assumed to be real. Then one needs to
rescale $\Psi_{\pm}$ as in (2.18a) and take the Carroll limit to recover (2.2)
and (2.26). In section 3, we will obtain the action of a magnetic Carroll
fermion, coupled to Carroll gravity, as a curved space analogue of this way of
taking the magnetic Carroll limit.

This concludes our discussion of Carroll fermions and Carroll limits. We have
summarized some of our findings in Table 1.

Model | $\alpha$ | $\epsilon$ | equation
---|---|---|---
electric Carroll | $2$ | $-1$ | (2.17)
magnetic Carroll 1 | $2$ | $-1/2$ | (2.2)
magnetic Carroll 2 | $1$ | $-1/2$ | (2.26)
Table 1: Summary of the minimal electric and magnetic Carroll Lagrangians. The
number $\alpha$ is defined as in equation (2.13), while the number $\epsilon$
is defined in (2.7). The last column indicates the equation number where the
expressions for the different Lagrangians can be found.

###  2.3 Carroll limit of Dirac’s action in Hamiltonian form

We now show that the previous magnetic Carrollian Lagrangian (2.2) can also be
recovered by considering a limit of the Dirac Lagrangian

$\mathcal{L}=\frac{1}{2}\,\bar{\Psi}\Gamma^{A}\partial_{A}\Psi-\frac{1}{2}\,\partial_{A}\bar{\Psi}\Gamma^{A}\Psi-\frac{\mathfrak{Re}(M)}{\tilde{c}}\bar{\Psi}\Psi$
(2.28)

rewritten in Hamiltonian form, where the notation $\mathfrak{Re}(M)$
anticipates that later we will introduce an imaginary mass parameter. To
proceed, one can introduce the conjugate momenta

$\Pi\equiv\frac{\partial\mathcal{L}}{\partial\dot{\Psi}}=\frac{\tilde{c}}{2}\bar{\Psi}\Gamma^{0},\qquad\bar{\Pi}\equiv\frac{\partial\mathcal{L}}{\partial\dot{\bar{\Psi}}}=-\frac{\tilde{c}}{2}\Gamma^{0}\Psi,$
(2.29)

that satisfy the conjugation relations
$\bar{\Pi}=i\Gamma^{0}\Pi^{\dagger},\Pi=i\bar{\Pi}^{\dagger}\Gamma^{0}$. The
Dirac action can then be rewritten as

$\mathcal{L}[\Psi,\Pi,\lambda]=\Pi\dot{\Psi}+\dot{\bar{\Psi}}\bar{\Pi}-\mathcal{H}+\left(\Pi-\frac{\tilde{c}}{2}\bar{\Psi}\Gamma^{0}\right)\lambda+\bar{\lambda}\left(\bar{\Pi}+\frac{\tilde{c}}{2}\Gamma^{0}\Psi\right),$
(2.30)

where we have introduced Lagrange multipliers $\lambda$ and $\bar{\lambda}$
that are related via Dirac conjugation
($\bar{\lambda}=i\lambda^{\dagger}\Gamma^{0}$), and where the Hamiltonian
density is given by

$\mathcal{H}=-\frac{1}{2}\bar{\Psi}\Gamma^{a}\partial_{a}\Psi+\frac{1}{2}\partial_{a}\bar{\Psi}\Gamma^{a}\Psi+\frac{\mathfrak{Re}(M)}{\tilde{c}}\bar{\Psi}\Psi.$
(2.31)

The equations of motion of the Lagrange multipliers correspond to the
following second class constraints for the canonical variables.

$\zeta_{1}\equiv\Pi-\frac{\tilde{c}}{2}\bar{\Psi}\Gamma^{0}\approx
0,\quad\zeta_{2}\equiv\bar{\Pi}+\frac{\tilde{c}}{2}\Gamma^{0}\Psi\approx 0.$
(2.32)

Equivalently, performing the following redefinition of the Lagrange multiplier
$\lambda$

$\lambda=\lambda^{\prime}+\left(-\frac{1}{\tilde{c}}\Gamma^{0}\Gamma^{a}\partial_{a}+\frac{1}{\tilde{c}^{2}}\mathfrak{Re}(M)\Gamma^{0}\right)\Psi+\frac{i}{\tilde{c}^{2}}\mathfrak{Im}(M)\Gamma^{0}\Psi,$
(2.33)

in the Lagrangian (2.30), one can rewrite the latter as

$\mathcal{L}[\Psi,\Pi,\lambda^{\prime}]=\Pi\dot{\Psi}+\dot{\bar{\Psi}}\bar{\Pi}-\mathcal{H}^{\prime}+\left(\Pi-\frac{\tilde{c}}{2}\bar{\Psi}\Gamma^{0}\right)\lambda^{\prime}+\bar{\lambda}^{\prime}\left(\bar{\Pi}+\frac{\tilde{c}}{2}\Gamma^{0}\Psi\right),$
(2.34)

with Hamiltonian

$\mathcal{H}^{\prime}=\frac{1}{\tilde{c}}\Pi\Gamma^{0}\Gamma^{a}\partial_{a}\Psi+\frac{1}{\tilde{c}}\partial_{a}\bar{\Psi}\Gamma^{a}\Gamma^{0}\bar{\Pi}-\frac{M}{\tilde{c}^{2}}\Pi\Gamma^{0}\Psi+\frac{M^{*}}{\tilde{c}^{2}}\bar{\Psi}\Gamma^{0}\bar{\Pi}.$
(2.35)

Note that the field redefinition (2.33) contains a new real parameter,
$\mathfrak{Im}(M)$, that has been introduced so as to obtain a mass term with
the same structure as that in (2.10), depending on a complex mass parameter
$M$. The Hamiltonian (2.31) and the one-parameter family of Hamiltonians
(2.35), parameterized by $\mathfrak{Im}(M)$, are clearly equivalent for any
finite value of $\tilde{c}$. The parameter $\mathfrak{Im}(M)$ does not play
any role at finite $\tilde{c}$ and it can always be eliminated by undoing the
field redefinition (2.33). On the other hand, different rewritings of the
Hamiltonian will lead to different Carroll actions because we will define the
magnetic limit so that $\lambda$ disappears when sending $\tilde{c}\to\infty$.

Since one is allowed to eliminate $\Pi$ and $\bar{\Pi}$ using the algebraic
equations of motion (2.32), one readily sees that the Lagrangian (2.34) (or
(2.30)) is equivalent to the usual Dirac Lagrangian (2.28). We can then
equally well consider the magnetic Carroll limit of the Lagrangian (2.34) with
$\mathcal{H}^{\prime}$ given by (2.35). As in the previous section, we
introduce the projected spinors

$\Psi_{\pm}=P_{\pm}\Psi,\quad\Pi_{\pm}=\Pi
P_{\pm},\quad\lambda_{\pm}^{\prime}=P_{\pm}\lambda^{\prime},\quad
P_{\pm}=\frac{1}{2}(\mathds{1}\pm i\Gamma^{0})\,,$ (2.36)

and we take the $\tilde{c}\rightarrow\infty$ limit of (2.34) after imposing
the following rescalings of the canonical variables

$\displaystyle\Pi_{+}$ $\displaystyle=\frac{1}{\tilde{c}}\pi_{+},$
$\displaystyle\Psi_{+}$ $\displaystyle=\tilde{c}\,\psi_{+},$ (2.37a)
$\displaystyle\Pi_{-}$ $\displaystyle=\pi_{-},$ $\displaystyle\Psi_{-}$
$\displaystyle=\psi_{-},$ (2.37b)

as well as the rescalings

$\lambda_{+}^{\prime}=\frac{1}{\tilde{c}^{3+\epsilon}}\tilde{\lambda}_{+},\quad\lambda_{-}^{\prime}=\frac{1}{\tilde{c}^{2+\epsilon}}\tilde{\lambda}_{-},\quad\textrm{with}\
\epsilon\geq 0.$ (2.38)

We further rescale the complex mass parameter as $M=\tilde{c}^{2}\,m$. The
rescalings (2.37) have been chosen so as to keep the kinetic term
$\Pi\dot{\Psi}+\dot{\bar{\Psi}}\bar{\Pi}$ untouched in the limit, in agreement
with the strategy adopted in the bosonic case in [12]. Due to the
disappearance of the Lagrange multipliers in the limit, the second class
constraints $\zeta_{1}$ and $\zeta_{2}$ are lost and the fields $\Pi$ and
$\Psi$ become independent. In the Carroll limit, we thus effectively doubled
the number of fields with respect to the original Dirac action and in the
massive case we introduced a new parameter, the imaginary part of $M$.

In the limit $\tilde{c}\rightarrow\infty$, the Lagrangian (2.34) with ${\cal
H}^{\prime}$ given in (2.35) takes the form

$\begin{split}\mathcal{L}&=\pi_{+}\dot{\psi}_{+}+\pi_{-}\dot{\psi}_{-}+\dot{\bar{\psi}}_{+}\bar{\pi}_{+}+\dot{\bar{\psi}}_{-}\bar{\pi}_{-}-\pi_{-}\Gamma^{0}\Gamma^{a}\partial_{a}\psi_{+}-\partial_{a}\bar{\psi}_{+}\Gamma^{a}\Gamma^{0}\bar{\pi}_{-}\\\
&\phantom{=}+m\pi_{+}\Gamma^{0}\psi_{+}+m\pi_{-}\Gamma^{0}\psi_{-}-m^{*}\bar{\psi}_{+}\Gamma^{0}\bar{\pi}_{+}-m^{*}\bar{\psi}_{-}\Gamma^{0}\bar{\pi}_{-},\end{split}$
(2.39)

which corresponds to the Lagrangian (2.19), that we obtained starting from the
two-fermion relativistic Lagrangian (2.10), upon performing the identification

$\pi_{\pm}=\bar{\chi}_{\pm}\Gamma^{0}.$ (2.40)

In even spacetime dimensions, the same truncation as in (2.21) can then be
introduced to describe a minimal magnetic theory. The approaches presented in
this subsection and in the previous one are therefore completely equivalent if
one wishes to produce an action principle for a magnetic Carroll theory.

To conclude, we wish to point out that one can also start from the Lagrangian
(2.30) with the Hamiltonian written in the form (2.31) and rescale the fields
as

$\Pi=\pi,\qquad\Psi=\psi,\qquad\lambda=\frac{1}{\tilde{c}^{2+\epsilon}}\tilde{\lambda},\quad\textrm{with}\
\epsilon\geq 0,$ (2.41)

therefore avoiding the splitting of the fields into irreducible $SO(D-1)$
representations as in (2.37). In the limit $\tilde{c}\rightarrow\infty$, with
the mass rescaled as $M=\tilde{c}m$, one obtains the following Carroll
Lagrangian for the two spinors $\psi$ and $\bar{\pi}$ [23]:

$\mathcal{L}=\pi\dot{\psi}+\dot{\bar{\psi}}\bar{\pi}+\bar{\psi}\Gamma^{a}\partial_{a}\psi-\mathfrak{Re}(m)\bar{\psi}\psi,$
(2.42)

Its equations of motion have a structure similar to that of a magnetic Carroll
scalar:

$\dot{\bar{\pi}}=\Gamma^{a}\partial_{a}\psi-\mathfrak{Re}(m)\psi,\qquad\dot{\psi}=0\,.$
(2.43)

In spite of the homogeneous rescaling of both $SO(D-1)$ irreducible components
of each spinor, the presence of two spinors thus still allows for a magnetic
behaviour. The option to perform a truncation leading to a system with the
same number of components as a single relativistic Dirac fermion is however
less evident in this form. Still, it can be recovered also in this setup by
introducing a suitable splitting of the spinors in $SO(D-1)$ irreducible
components after the limit. The kinetic term of (2.42) corresponds indeed to
that in our non-minimal magnetic Carrollian Lagrangian (2.19) upon performing
the field redefinition

$\psi=\psi_{+}+\chi_{-},\qquad\pi=(\bar{\chi}_{+}+\bar{\psi}_{-})\Gamma^{0},$
(2.44)

which is legitimate since after the limit the spinors $\psi$ and $\pi$ are
independent. On the other hand, we note that the mass term has a different
form with respect to that in (2.19) and, in particular, if one implements the
truncation (2.21) it vanishes.

##  3 Coupling to Carroll gravity

In this section, we will discuss the coupling of Carroll fermions to
Carrollian gravity (see [16, 17] for discussions on the coupling of Carrollian
scalars to arbitrary but curved Carrollian backgrounds and [35] for their
coupling to Carrollian gravity). This requires a Cartan formulation of Carroll
geometry that can be viewed as a special instance of the so-called $p$-brane
non-Lorentzian geometries, studied recently in [36].171717More precisely,
there exists a duality [37, 38] that relates Carroll geometry to $(D-2)$-brane
Galilean geometry that was discussed in detail in [36]. We will review this
description of Carroll geometry in section 3.1. Coupling fermions to General
Relativity can be done in two ways: either in a first-order Palatini
formulation or directly in the second-order formulation. The difference
between these two is that in the Palatini formulation, passing to the second-
order formalism leads to quartic fermion terms, stemming from fermion bilinear
torsion contributions, that are not present when the coupling is performed
directly in the second-order formulation. In this section, we will discuss
Carrollian analogues of these two inequivalent ways of coupling fermions to
gravity. In section 3.2, we will apply a Carroll limit to General Relativity,
coupled to a Dirac and tachyonic Dirac fermion in the Palatini formulation, to
obtain an analogous first-order coupling of an electric and magnetic Carroll
fermion to so-called magnetic Carroll gravity [35, 12, 39, 40]. We will also
comment on the passage to the second-order formulation explored, e.g., in [41,
42, 13, 43, 22]. In section 3.3, we will discuss the Carroll limit of General
Relativity, coupled to a (tachyonic) Dirac spinor, directly in the second-
order formalism.

###  3.1 Carroll geometry

In order to couple Carroll fermions to gravity, a suitable notion of spin-
connection fields for Carroll geometry is needed. A convenient way to
introduce these spin-connections is by considering a Carroll limit of the
Cartan formulation of Lorentzian geometry. Recall that in the latter, the
metric is encoded in the solder form $E_{\mu}{}^{A}$ (whose inverse
$E_{A}{}^{\mu}$ is the Vielbein) that transforms under local Lorentz
transformations with parameters $\Lambda^{AB}=-\Lambda^{BA}$ as

$\displaystyle\delta E_{\mu}{}^{A}$
$\displaystyle=-\Lambda^{A}{}_{B}E_{\mu}{}^{B}\,.$ (3.1)

A metric-compatible connection is described by a spin-connection field
$\Omega_{\mu}{}^{AB}=-\Omega_{\mu}{}^{BA}$, whose Lorentz transformation rule
is given by

$\displaystyle\delta\Omega_{\mu}{}^{AB}=\partial_{\mu}\Lambda^{AB}-2\,\Lambda^{[A}{}_{C}\Omega_{\mu}{}^{|C|B]}\,,$
(3.2)

and that obeys the first Cartan structure equations:

$T_{\mu\nu}{}^{A}=2\,\partial_{[\mu}E_{\nu]}{}^{A}+2\,\Omega_{[\mu}{}^{AB}E_{\nu]B}\qquad\qquad\text{where
}T_{\mu\nu}{}^{A}\text{ is the torsion tensor}\,.$ (3.3)

Note that these constitute as many equations as there are spin-connection
components and that each of these equations contains a spin-connection
component. As a consequence, the first Cartan structure equations (3.3) can be
solved for all components of $\Omega_{\mu}{}^{AB}$ in terms of the (inverse)
Vielbein and the torsion tensor components in a unique fashion:

$\displaystyle\Omega_{\mu}{}^{AB}=E^{[A|\nu|}\left(2\,\partial_{[\mu}E_{\nu]}{}^{B]}-T_{\mu\nu}{}^{B]}\right)-\frac{1}{2}E_{\mu
C}\,E^{A\nu}\,E^{B\rho}\left(2\,\partial_{[\nu}E_{\rho]}{}^{C}-T_{\nu\rho}{}^{C}\right)\,.$
(3.4)

In order to obtain a Cartan formulation of Carroll geometry, we perform the
following rescalings

$\displaystyle E_{\mu}{}^{0}$
$\displaystyle=\frac{1}{\tilde{c}}\tau_{\mu}\,,\qquad$ $\displaystyle
E_{\mu}{}^{a}$ $\displaystyle=e_{\mu}{}^{a}\,,\qquad$ $\displaystyle
E_{0}{}^{\mu}$ $\displaystyle=\tilde{c}\tau^{\mu}\,,\qquad$ $\displaystyle
E_{a}{}^{\mu}$ $\displaystyle=e_{a}{}^{\mu}\,,$
$\displaystyle\Omega_{\mu}{}^{ab}$
$\displaystyle=\omega_{\mu}{}^{ab}\,,\qquad$
$\displaystyle\Omega_{\mu}{}^{a0}$
$\displaystyle=\frac{1}{\tilde{c}}\omega_{\mu}{}^{a0}\,,\qquad$ $\displaystyle
T_{\mu\nu}{}^{0}$ $\displaystyle=\frac{1}{\tilde{c}}t_{\mu\nu}{}^{0}\,,\qquad$
$\displaystyle T_{\mu\nu}{}^{a}$ $\displaystyle=t_{\mu\nu}{}^{a}\,,$
$\displaystyle\Lambda^{ab}$ $\displaystyle=\lambda^{ab}\,,\qquad$
$\displaystyle\Lambda^{0a}$ $\displaystyle=\frac{1}{\tilde{c}}\lambda^{0a}\,,$
(3.5)

in the transformation rules (3.1), (3.2) and first Cartan structure equations
(3.3) and take the $\tilde{c}\rightarrow\infty$ limit. This leads to a
Carrollian solder form $(\tau_{\mu},e_{\mu}{}^{a})$ and Vielbein
$(\tau^{\mu},e_{a}{}^{\mu})$ that transform under local spatial rotations
(with parameters $\lambda^{ab}=-\lambda^{ba}$) and Carroll boosts (with
parameters $\lambda^{0a}$) as

$\displaystyle\delta\tau_{\mu}=-\lambda^{0}{}_{a}\,e_{\mu}{}^{a}\,,\qquad\delta
e_{\mu}{}^{a}=-\lambda^{a}{}_{b}\,e_{\mu}{}^{b}\,,\qquad\delta\tau^{\mu}=0\,,\qquad\delta
e_{a}{}^{\mu}=\lambda^{0}{}_{a}\,\tau^{\mu}-\lambda_{a}{}^{b}\,e_{b}{}^{\mu}\,.$
(3.6)

Here and in the following, we have freely raised and lowered the $a$, $b$
indices with a Euclidean Kronecker-delta metric. We will frequently use the
Carroll Vielbein $(\tau^{\mu},e_{a}{}^{\mu})$ to turn curved $\mu$, $\nu$
indices of form fields into flat $0$, $a$ indices; e.g., for a one-form
$X_{\mu}$ and a two-form $X_{\mu\nu}$ we define:

$\displaystyle X_{0}\equiv\tau^{\mu}X_{\mu}\,,\ \ \ X_{a}\equiv
e_{a}{}^{\mu}X_{\mu}\,,\qquad\qquad
X_{0a}\equiv\tau^{\mu}e_{a}{}^{\nu}X_{\mu\nu}\,,\ \ \ X_{ab}\equiv
e_{a}{}^{\mu}e_{b}{}^{\nu}X_{\mu\nu}\,.$ (3.7)

The rescalings (3.1) and $\tilde{c}\rightarrow\infty$ limit furthermore give
rise to Carrollian spin-connection fields $\omega_{\mu}{}^{ab}$ and
$\omega_{\mu}{}^{0a}$ for spatial rotations and Carroll boosts that transform
as

$\displaystyle\delta\omega_{\mu}{}^{ab}$
$\displaystyle=\partial_{\mu}\lambda^{ab}-2\lambda^{[a}{}_{c}\,\omega_{\mu}{}^{|c|b]}\,,\qquad\qquad\delta\omega_{\mu}{}^{0a}=\partial_{\mu}\lambda^{0a}-\lambda^{a}{}_{b}\,\omega_{\mu}{}^{0b}-\lambda^{0}{}_{b}\,\omega_{\mu}{}^{ba}\,.$
(3.8)

The $\tilde{c}\rightarrow\infty$ limit of the Lorentzian first Cartan
structure equations (3.3) corresponds to the following Carrollian analogue:

$\displaystyle\tau_{\mu\nu}+2\omega_{[\mu}{}^{0a}e_{\nu]a}=t_{\mu\nu}{}^{0}\qquad\qquad$
$\displaystyle\text{with}\ \ \tau_{\mu\nu}\equiv
2\partial_{[\mu}\tau_{\nu]}\,,$ (3.9a) $\displaystyle
e_{\mu\nu}{}^{a}+2\omega_{[\mu}{}^{ab}e_{\nu]b}=t_{\mu\nu}{}^{a}\qquad\qquad$
$\displaystyle\text{with}\ \ e_{\mu\nu}{}^{a}\equiv
2\partial_{[\mu}e_{\nu]}{}^{a}\,,$ (3.9b)

where $t_{\mu\nu}{}^{0}\equiv t_{\mu\nu}{}^{\rho}\tau_{\rho}$ and
$t_{\mu\nu}{}^{a}\equiv t_{\mu\nu}{}^{\rho}e_{\rho}{}^{a}$. Contrary to what
happens in Lorentzian geometry, it is no longer true that each of these first
Cartan structure equations contains a spin-connection component. In
particular, multiplying both sides of (3.9b) with $\tau^{\mu}e_{b}{}^{\nu}$
and symmetrizing in $(ab)$, one finds

$\displaystyle e_{0(a,b)}=t_{0(a,b)}\,,$ (3.10)

where $e_{0b,a}\equiv\tau^{\mu}e_{b}{}^{\nu}e_{\mu\nu a}$ and
$t_{0b,a}\equiv\tau^{\mu}e_{b}{}^{\nu}t_{\mu\nu a}$. One thus sees that
setting components of $t_{0(a,b)}$ equal to zero no longer solely amounts to
specifying a particular connection, but additionally leads to differential
constraints on $e_{\mu}{}^{a}$ and thus constraints on the geometry of the
underlying manifold. Such geometric constraints are called intrinsic torsion
constraints and the components of $t_{0(a,b)}$ are referred to as intrinsic
torsion components. Intrinsic torsion constraints that are covariant with
respect to local spatial rotations and Carroll boosts can be imposed in four
different ways, leading to four distinct types of Carroll geometry, called
Carroll 1 – Carroll 4:

Carroll 1: all intrinsic torsion tensors are non-zero.

Carroll 2: $t_{0a}{}^{a}=0$.

Carroll 3: $t_{0\\{a,b\\}}=0$.

Carroll 4: $t_{0a}{}^{a}=t_{0\\{a,b\\}}=0$.

Here, $t_{0\\{a,b\\}}$ denotes the symmetric traceless part (in $(ab)$) of
$t_{0a,b}$. The different constraints that correspond to Carroll 2, 3 and 4
geometry have a clear geometric interpretation. For instance, the Carroll 2
constraint implies that the $(D-1)$-form $\epsilon_{a_{1}\cdots
a_{D-1}}e_{[\mu_{1}}{}^{a_{1}}\cdots e_{\mu_{D-1}]}{}^{a_{D-1}}$ is closed and
this closure can be physically interpreted as stating that there exists a
notion of absolute spatial volumes. The Carroll 3 constraint can be rephrased
as saying that $\tau^{\mu}$ is a conformal Killing vector with respect to the
spatial metric $h_{\mu\nu}=e_{\mu}{}^{a}e_{\nu a}$. We refer to [36] for
further details.

While the number of Carrollian first Cartan structure equations (3.9) equals
the number of Carrollian spin-connection components, the presence of intrinsic
torsion components implies that not all of them can be used to solve for
components of $\omega_{\mu}{}^{ab}$ and $\omega_{\mu}{}^{0a}$. Not all spin-
connection components can thus be expressed in terms of the Carrollian solder
form, Vielbein and torsion tensors $t_{\mu\nu}{}^{0}$, $t_{\mu\nu}{}^{a}$.
Explicitly, one finds that the components

$\displaystyle\omega^{(a|0|b)}\equiv e^{(a|\mu}\omega_{\mu}{}^{0|b)}$ (3.11)

can not be solved from the first Cartan structure equations (3.9). We will
refer to them as the independent spin-connection components. The remaining
components will be called dependent and by solving them from (3.9) one finds
that they can be expressed in terms of the Carrollian solder form, Vielbein
and torsion tensors as follows181818To prevent confusion, we use a notation in
which we put a comma between the indices to indicate which two indices form
(the projection of) an anti-symmetric pair, e.g., we write
$t_{0}{}^{a,0}=\tau^{\mu}e^{a\nu}t_{\mu\nu}{}^{0}$ to clarify that the first
two indices (0 and $a$) are those of an anti-symmetric pair.

$\displaystyle\omega_{0}{}^{ab}$
$\displaystyle=\omega_{0}{}^{ab}(\tau,e)-t_{0}{}^{[a,b]}\,,\qquad\qquad\quad$
$\displaystyle\omega_{c}{}^{ab}$
$\displaystyle=\omega_{c}{}^{ab}(e)-t_{c}{}^{[a,b]}+\frac{1}{2}t^{ab}{}_{c}\,,$
$\displaystyle\omega_{0}{}^{0a}$
$\displaystyle=\omega_{0}{}^{0a}(\tau,e)+t_{0}{}^{a,0}\,,\qquad\qquad\quad$
$\displaystyle\omega^{[a|0|b]}$
$\displaystyle=\omega^{[a|0|b]}(\tau,e)+\frac{1}{2}t^{ab,0}\,,$ (3.12)

with

$\displaystyle\omega_{0}{}^{ab}(\tau,e)$
$\displaystyle=e_{0}{}^{[a,b]}\,,\qquad\qquad\qquad$
$\displaystyle\omega_{c}{}^{ab}(e)$
$\displaystyle=e_{c}{}^{[a,b]}-\frac{1}{2}e^{ab}{}_{c}\,,$
$\displaystyle\omega_{0}{}^{0a}(\tau,e)$
$\displaystyle=-\tau_{0}{}^{a}\,,\qquad\qquad\qquad$
$\displaystyle\omega^{[a|0|b]}(\tau,e)$
$\displaystyle=-\frac{1}{2}\tau^{ab}\,.$ (3.13)

The appearance of intrinsic torsion and independent spin-connection components
is not restricted to Carroll geometry but is a generic feature of non-
Lorentzian geometry and has been studied in full generality in [36].

Let us end this section by recalling that the Lorentzian first Cartan
structure equations (3.3) for zero torsion ($T_{\mu\nu}{}^{A}=0$) correspond
to the equations of motion of $\Omega_{\mu}{}^{AB}$ of the Einstein-Hilbert
action in the first-order formulation

$\displaystyle S_{\text{EH}}=\frac{1}{16\pi
G_{N}}\int\mathrm{d}^{D}x\,EE_{A}{}^{\mu}E_{B}{}^{\nu}R_{\mu\nu}{}^{AB}\,,$
(3.14) $\displaystyle\text{with}\ \ \
R_{\mu\nu}{}^{AB}=2\,\partial_{[\mu}\Omega_{\nu]}{}^{AB}+2\,\Omega_{[\mu}{}^{A}{}_{C}\Omega_{\nu]}{}^{CB}\,.$

Likewise, their zero torsion ($t_{\mu\nu}{}^{0}=0=t_{\mu\nu}{}^{a}$)
Carrollian analogue (3.9) can be derived as equations of motion for
$\omega_{\mu}{}^{ab}$ and $\omega_{\mu}{}^{0a}$ of the following first-order
‘magnetic Carroll gravity’ action [35, 39, 40]:

$\displaystyle S_{\text{Carr. Grav.}}=\frac{1}{16\pi
G_{C}}\int\mathrm{d}^{D}x\,e\Big{(}e_{a}{}^{\mu}e_{b}{}^{\nu}R_{\mu\nu}(J)^{ab}+2\tau^{\mu}e_{a}{}^{\nu}R_{\mu\nu}(C)^{0a}\Big{)}\,,$
(3.15) $\displaystyle\text{with}\ \ \
R_{\mu\nu}(J)^{ab}=2\,\partial_{[\mu}\omega_{\nu]}{}^{ab}+2\,\omega_{[\mu}{}^{a}{}_{c}\,\omega_{\nu]}{}^{cb}\,,$
$\displaystyle\text{and}\ \ \
\,R_{\mu\nu}(C)^{0a}=2\,\partial_{[\mu}\omega_{\nu]}{}^{0a}+2\,\omega_{[\mu}{}^{ab}\,\omega_{\nu]}{}^{0}{}_{b}\,.$

This action is obtained by using the rescalings (3.1) (along with
$G_{N}=G_{C}/\tilde{c}$) in (3.14) and taking the $\tilde{c}\rightarrow\infty$
limit. It is interesting to note that the spin-connection components
$\omega^{(a|0|b)}$ only appear linearly in it and that they assume the role of
Lagrange multipliers for the Carroll 4 constraints [35, 36, 40]. Variation of
(3.15) with respect to $\omega_{\mu}{}^{ab}$ and $\omega_{\mu}^{0a}$ leads to

$\displaystyle\delta{S}_{\text{Carr. Grav.}}$ $\displaystyle=\frac{1}{8\pi
G_{C}}\int\mathrm{d}^{D}x\,e\Bigg{[}\delta\omega_{\mu}{}^{ab}\left(-e_{a}{}^{\mu}t_{0b}{}^{0}+e_{a}{}^{\mu}t_{bc}{}^{c}+\frac{1}{2}e_{c}{}^{\mu}t_{ab}{}^{c}+\frac{1}{2}\tau^{\mu}t_{ab}{}^{0}\right)$
$\displaystyle\qquad+\delta{\omega}_{\mu}{}^{0a}\left(-e_{a}{}^{\mu}t_{0b}{}^{b}+e_{b}{}^{\mu}t_{0a}{}^{b}+\tau^{\mu}t_{ab}{}^{b}\right)\Bigg{]}\,.$
(3.16)

From this, one sees that the Carrollian first Cartan structure equations (3.9)
for zero torsion are derived from (3.15) as the equations of motion of the
spin-connections.

In the next section, we will discuss the coupling of both electric and
magnetic Carroll fermions to this first-order Carroll gravity action.

###  3.2 Coupling Carroll fermions to gravity in the first-order formulation

In General Relativity, fermionic matter can be coupled to the Einstein-Hilbert
action in the first-order formulation, by adding a Dirac Lagrangian suitably
coupled to the Vielbein and first-order spin-connection $\Omega_{\mu}{}^{AB}$:

$\displaystyle
S=S_{\text{EH}}+\int\mathrm{d}^{D}x\,E\left[-\frac{1}{2}\bar{\Psi}E_{A}{}^{\mu}\Gamma^{A}\big{(}\partial_{\mu}\Psi+\frac{1}{4}\Omega_{\mu}{}^{BC}\Gamma_{BC}\Psi\big{)}+\frac{M}{2\tilde{c}}\bar{\Psi}\Psi+\text{h.c.}\right]\,.$
(3.17)

The first Cartan structure equations that result as equations of motion of
$\Omega_{\mu}{}^{AB}$ then receive torsion contributions in the form of
fermion bilinears. When passing to the second-order formulation, by replacing
the spin-connections $\Omega_{\mu}{}^{AB}$ in (3.17) by their dependent
expressions (3.4), these fermion bilinear torsion components give rise to an
action that includes quartic fermion terms. In this section, we will
investigate the analogous coupling of Carroll fermions to magnetic Carroll
gravity (3.15) in the first-order formulation and we will comment on the
passage to the second-order formulation. We will discuss the cases of electric
and magnetic Carroll fermions in turn.

Let us first focus on the electric case. By applying the rescalings (3.1)
(along with $\Psi=\psi$ and $M=m\tilde{c}^{2}$) to (3.17) and taking the
$\tilde{c}\rightarrow\infty$ limit, we obtain the action of an electric
Carroll fermion, coupled to magnetic Carroll gravity in the first-order
formalism:191919This action was already considered in [35].

$\displaystyle S=S_{\text{Carr.
Grav.}}+\int\mathrm{d}^{D}x\,e\left[-\frac{1}{2}\bar{\psi}\Gamma^{0}\tau^{\mu}D_{\mu}\psi+\frac{m}{2}\bar{\psi}\psi+\text{h.c.}\right]\,,$
(3.18) $\displaystyle\text{with}\ \
D_{\mu}\psi=\partial_{\mu}\psi+\frac{1}{4}\omega_{\mu}{}^{ab}\Gamma_{ab}\psi\,.$

The equations of motion of $\omega_{\mu}{}^{ab}$ and $\omega_{\mu}{}^{0a}$
that follow from (3.18) correspond to the Carrollian first Cartan structure
equations (3.9) with

$\displaystyle t_{0a}{}^{0}=0\,,\quad\quad
t_{ab}{}^{0}=\frac{\kappa_{C}^{2}}{2}\bar{\psi}\Gamma^{0}\Gamma_{ab}\psi\,,\quad\quad
t_{\mu\nu}{}^{a}=0\,,\qquad\text{with }\ \ \kappa_{C}^{2}=8\pi G_{C}\,.$
(3.19)

The first-order coupling of an electric Carroll fermion to General Relativity
thus introduces bilinear fermion torsion.

To pass to the second-order formulation, one replaces the dependent spin-
connection components $\omega_{\mu}{}^{ab}$, $\tau^{\mu}\omega_{\mu}{}^{0a}$
and $e^{[a|\mu}\omega_{\mu}{}^{0|b]}$ in the action (3.18) by their explicit
expressions (3.1) (with the torsion tensor components replaced by (3.19)).
This leads, after partial integration, to the following action

$\displaystyle S$
$\displaystyle=\int\mathrm{d}^{D}x\,e\Big{[}\frac{1}{2\kappa_{C}^{2}}\Big{(}\omega_{b}{}^{ab}(e)\omega_{ca}{}^{c}(e)-\omega_{abc}(e)\omega^{bac}(e)-2\omega_{0}{}^{0a}(\tau,e)\omega_{ba}{}^{b}(e)$
$\displaystyle\qquad\qquad-2\omega_{[a|0|b]}(\tau,e)\omega_{0}{}^{ab}(\tau,e)+2\omega^{(a|0|b)}e_{0(a,b)}-2\omega_{a}{}^{0a}e_{0a}{}^{a}\Big{)}$
$\displaystyle\qquad\qquad+\Big{\\{}-\frac{1}{2}\bar{\psi}\Gamma^{0}\tilde{D}_{0}\psi+\frac{m}{2}\bar{\psi}\psi+\mathrm{h.c.}\Big{\\}}\Big{]}\,,$
(3.20)

with

$\displaystyle\tilde{D}_{0}\psi\equiv\tau^{\mu}\partial_{\mu}\psi+\frac{1}{4}\omega_{0}{}^{ab}(\tau,e)\Gamma_{ab}\psi\,.$
(3.21)

Note that, the bilinear torsion (3.19) does not lead to quartic fermion terms,
when passing to the second-order formulation, in contrast to what happens for
a Dirac fermion coupled to General Relativity. The same observation was
reached in [30] while linking the first-order formulation of Carroll gravity
coupled to Dirac fermions to its Hamiltonian formulation.

To discuss the coupling of magnetic Carroll fermions to Carroll gravity, we
assume that the spacetime dimension $D$ is even (and $>2$) and start from the
second of the two actions (2.27) for a tachyonic Dirac spinor, coupled to the
Einstein-Hilbert action in the first-order formulation:

$\displaystyle
S=S_{\text{EH}}+\int\mathrm{d}^{D}x\,E\left[-\frac{1}{2}\bar{\Psi}E_{A}{}^{\mu}\Gamma^{A}\Gamma_{\star}\big{(}\partial_{\mu}\Psi+\frac{1}{4}\Omega_{\mu}{}^{BC}\Gamma_{BC}\Psi\big{)}+\frac{M}{2\tilde{c}}\bar{\Psi}\Psi+\text{h.c.}\right]\,.$
(3.22)

By splitting $\Psi$ into $\Psi_{\pm}$, using the rescalings (3.1) (along with
$\Psi_{+}=\tilde{c}^{1/2}\psi_{+}$, $\Psi_{-}=\tilde{c}^{-1/2}\psi_{-}$ and
$M=\tilde{c}\,m$) to (3.22) and taking the $\tilde{c}\rightarrow\infty$ limit,
the following action for a minimal magnetic Carroll fermion coupled to
magnetic Carroll gravity in the first-order formalism is obtained:

$\displaystyle S=S_{\text{Carr.
Grav.}}+\int\mathrm{d}^{D}x\,e\,\Big{[}-\frac{1}{2}\bar{\psi}_{+}\Gamma^{0}\Gamma_{\star}\tau^{\mu}D_{\mu}{\psi}_{-}-\frac{1}{2}\bar{\psi}_{-}\Gamma^{0}\Gamma_{\star}\tau^{\mu}D_{\mu}{\psi}_{+}$
$\displaystyle\qquad\qquad-\frac{1}{2}\bar{\psi}_{+}\Gamma^{a}\Gamma_{\star}e_{a}{}^{\mu}D_{\mu}\psi_{+}+\frac{m}{2}\bar{\psi}_{+}\psi_{+}+\mathrm{h.c.}\Big{]}\,,$
(3.23)

where

$\displaystyle
D_{\mu}\psi_{+}\equiv\partial_{\mu}\psi_{+}+\frac{1}{4}\omega_{\mu}{}^{ab}\Gamma_{ab}\psi_{+}\,,\qquad
D_{\mu}\psi_{-}\equiv\partial_{\mu}\psi_{-}+\frac{1}{4}\omega_{\mu}{}^{ab}\Gamma_{ab}\psi_{-}+\frac{1}{2}\omega_{\mu}{}^{0a}\Gamma_{0a}\psi_{+}\,.$
(3.24)

In this case, the torsion tensor components of the Carrollian first Cartan
structure equations (3.9) that arise as equations of motion of
$\omega_{\mu}{}^{ab}$ and $\omega_{\mu}{}^{0a}$ are given by

$\displaystyle t_{0a}{}^{0}$ $\displaystyle=0\,,\qquad\quad$ $\displaystyle
t_{ab}{}^{0}$
$\displaystyle=\frac{\kappa_{C}^{2}}{2}\bar{\psi}_{+}\Gamma^{0}\Gamma_{ab}\Gamma_{\star}\psi_{-}+\frac{\kappa_{C}^{2}}{2}\bar{\psi}_{-}\Gamma^{0}\Gamma_{ab}\Gamma_{\star}\psi_{+}\,,$
$\displaystyle t_{0b}{}^{a}$ $\displaystyle=0\,,\qquad\quad$ $\displaystyle
t_{bc}{}^{a}$
$\displaystyle=\frac{\kappa_{C}^{2}}{2}\bar{\psi}_{+}\Gamma^{a}{}_{bc}\Gamma_{\star}\psi_{+}\,.$
(3.25)

Going to the second-order formulation, by replacing $\omega_{\mu}{}^{ab}$,
$\tau^{\mu}\omega_{\mu}{}^{0a}$ and $e^{[a|\mu}\omega_{\mu}{}^{0|b]}$ in (3.2)
by their expressions (3.1) in terms of the Carroll solder form, Vielbein and
torsion tensor components (3.2) now leads (after partial integration) to the
following action:

$\displaystyle S$
$\displaystyle=\int\mathrm{d}^{4}x\,e\Big{[}\frac{1}{2\kappa_{C}^{2}}\Big{(}\omega_{b}{}^{ab}(e)\omega_{ca}{}^{c}(e)-\omega_{abc}(e)\omega^{bac}(e)-2\omega_{0}{}^{0a}(\tau,e)\omega_{ba}{}^{b}(e)$
$\displaystyle\qquad\qquad-2\omega_{[a|0|b]}(\tau,e)\omega_{0}{}^{ab}(\tau,e)+2\omega^{(a|0|b)}e_{0(a,b)}-2\omega_{a}{}^{0a}e_{0a}{}^{a}\Big{)}$
$\displaystyle\qquad\qquad-\frac{1}{2}\left\\{\bar{\psi}_{+}\Gamma^{0}\Gamma_{\star}\tilde{D}_{0}\psi_{-}+\bar{\psi}_{-}\Gamma^{0}\Gamma_{\star}\tilde{D}_{0}\psi_{+}+\bar{\psi}_{+}\Gamma^{a}\Gamma_{\star}\tilde{D}_{a}\psi_{+}+\mathrm{h.c.}\right\\}$
$\displaystyle\qquad\qquad+m\bar{\psi}_{+}\psi_{+}+\frac{\kappa_{C}^{2}}{32}\left(\bar{\psi}_{+}\Gamma^{abc}\Gamma_{\star}\psi_{+}\right)\left(\bar{\psi}_{+}\Gamma_{abc}\Gamma_{\star}\psi_{+}\right)\Big{]}\,,$
(3.26)

with

$\displaystyle\tilde{D}_{0}\psi_{+}$
$\displaystyle\equiv\tau^{\mu}\partial_{\mu}\psi_{+}+\frac{1}{4}\omega_{0}{}^{ab}(\tau,e)\Gamma_{ab}\psi_{+}\,,\qquad\tilde{D}_{a}\psi_{+}\equiv
e_{a}{}^{\mu}\partial_{\mu}\psi_{+}+\frac{1}{4}\omega_{a}{}^{bc}(e)\Gamma_{bc}\psi_{+}\,,$
$\displaystyle\tilde{D}_{0}\psi_{-}$
$\displaystyle\equiv\tau^{\mu}\partial_{\mu}\psi_{-}+\frac{1}{4}\omega_{0}{}^{ab}(\tau,e)\Gamma_{ab}\psi_{-}+\frac{1}{2}\omega_{0}{}^{0a}(\tau,e)\Gamma_{0a}\psi_{+}\,.$
(3.27)

Unlike what happened for the electric Carroll fermion, the bilinear fermion
torsion (3.2) does lead to quartic fermion terms when one writes the first-
order action (3.2) in second-order form.

For both the electric and magnetic Carroll fermion, the spatial parts
$e_{b}{}^{\mu}\omega_{\mu}{}^{0a}$ of the boost spin-connection is not
involved in the coupling to gravity. From (3.1), one then sees that adding
each of these first-order fermion Lagrangians to (3.15) does not lead to
sources for the $t_{0a}{}^{b}$ components of the torsion. The resulting system
thus still obeys the Carroll 4 geometry constraints, for which all intrinsic
torsion components $t_{0(a,b)}$ are equal to zero. The remark made below
(3.15), namely that the $\omega^{(a|0|b)}$ spin-connection components are
Lagrange multipliers for the Carroll 4 constraints, is thus unaffected by the
coupling to fermions and this is clearly visible in the second order
Lagrangians (3.2) and (3.2).

###  3.3 Carroll limit of fermions coupled to gravity in the second-order
formulation

In the previous section, we obtained first-order actions for an electric and
magnetic Carroll fermion, coupled to Carroll gravity, by taking a Carroll
limit of (3.17) and (3.22). This limit can also be taken directly in the
second-order formulation in which the spin-connection $\Omega_{\mu}{}^{AB}$ is
no longer independent, but instead given by the Levi-Civita spin-connection
$\Omega_{\mu}{}^{AB}(E)$:

$\displaystyle\Omega_{\mu}{}^{AB}(E)=2\,E^{[A|\nu|}\,\partial_{[\mu}E_{\nu]}{}^{B]}-E_{\mu
C}\,E^{A\nu}\,E^{B\rho}\,\partial_{[\nu}E_{\rho]}{}^{C}\,.$ (3.28)

Upon applying the rescalings of $E_{\mu}{}^{A}$/$E_{A}{}^{\mu}$ given in
(3.1), the components of
$\Omega_{C}{}^{AB}(E)=E_{C}{}^{\mu}\Omega_{\mu}{}^{AB}(E)$ can be expanded in
powers of $\tilde{c}$ as follows:

$\displaystyle\Omega_{0}{}^{ab}(E)$
$\displaystyle=\tilde{c}\,\omega_{0}{}^{ab}(\tau,e)+\mathcal{O}\left(\tilde{c}^{-1}\right)\,,\qquad\qquad$
$\displaystyle\Omega_{c}{}^{ab}(E)$ $\displaystyle=\omega_{c}{}^{ab}(e)\,,$
$\displaystyle\Omega_{0}{}^{0a}(E)$
$\displaystyle=\omega_{0}{}^{0a}(\tau,e)\,,\qquad\qquad$
$\displaystyle\Omega^{a,0b}(E)$
$\displaystyle=\tilde{c}\,t^{0(a,b)}+\tilde{c}^{-1}\,\omega^{[a|0|b]}(\tau,e)\,.$
(3.29)

Note that the $\tilde{c}$-power of the leading-order term in the expansions of
$\Omega_{0}{}^{ab}(E)$, $\Omega_{c}{}^{ab}(E)$ and $\Omega_{0}{}^{0a}(E)$
agrees with that of the rescalings (3.1) of the corresponding spin-connection
components $\Omega_{C}{}^{AB}=E_{C}{}^{\mu}\Omega_{\mu}{}^{AB}$ in the first-
order formalism. This is however not true for $\Omega^{a,0b}(E)$, for which
the $\tilde{c}$-power suggested by the rescalings (3.1) only appears at
subleading order. Compared to the rescaling of $\Omega^{a,0b}$ in the first-
order formulation, the $\tilde{c}$-expansion of $\Omega^{a,0b}(E)$ contains an
additional ‘divergent’ $\mathcal{O}(\tilde{c})$-contribution that is given by
the intrinsic torsion components (3.10).

Let us then start from (3.17) and (3.22) in the second-order formulation
(i.e., with $\Omega_{\mu}{}^{AB}$ replaced by $\Omega_{\mu}{}^{AB}(E)$). After
setting $G_{N}=G_{C}/\tilde{c}$, rescaling $E_{\mu}{}^{A}$/$E_{A}{}^{\mu}$ as
in (3.1) and $\Psi$ and $M$ as in the electric, resp. magnetic case of the
previous subsection, the $\tilde{c}\rightarrow\infty$ limit of (3.17), resp.
(3.22) is given by the leading order term in their $\tilde{c}$-expansion.
These leading order terms are of order $\mathcal{O}(\tilde{c}^{2})$, instead
of order $\mathcal{O}(\tilde{c}^{0})$ as was the case for the analogous
expansions in the first-order formulation, due to the divergent
$\mathcal{O}(\tilde{c})$-contribution in the expansion (3.3) of
$\Omega^{a,0b}(E)$. In particular, one gets for both (3.17) and (3.22) the
following expansion:

$\displaystyle
S\propto\tilde{c}^{2}\,\int\mathrm{d}^{D}x\,e\,\left(t_{0}{}^{\\{a,b\\}}t_{0\\{a,b\\}}-\frac{D-2}{D-1}t_{0a}{}^{a}t_{0b}{}^{b}\right)+\mathcal{O}(\tilde{c}^{0})\,.$
(3.30)

One thus sees that no fermion terms survive the $\tilde{c}\rightarrow\infty$
limit, whose result is the so-called ‘electric Carroll gravity’ theory [41]
(see also [42, 12, 43]):

$\displaystyle S_{\text{El. Carr. Grav.}}$ $\displaystyle=\frac{1}{16\pi
G_{C}}\int\mathrm{d}^{D}x\,e\,\left(t_{0}{}^{\\{a,b\\}}t_{0\\{a,b\\}}-\frac{D-2}{D-1}t_{0a}{}^{a}t_{0b}{}^{b}\right)\,.$
(3.31)

It is possible to take a Carroll limit in the second-order formalism that
results in theories that include fermion contributions and that closely
resemble (3.2) and (3.2). To do this, one applies a Hubbard-Stratonovich
transformation to the leading order term of (3.30), i.e., one introduces
auxiliary fields $\chi^{ab}=\chi^{\\{ab\\}}$ and $\chi$ to rewrite this term
in the classically equivalent form

$\displaystyle\int\mathrm{d}^{D}x\,e\,\left(\tilde{c}^{\alpha}\chi^{ab}t_{0\\{a,b\\}}-\tilde{c}^{\beta}\frac{D-2}{D-1}\chi
t_{0a}{}^{a}-\frac{\tilde{c}^{2\alpha-2}}{4}\chi^{\\{ab\\}}\chi_{\\{ab\\}}+\frac{\tilde{c}^{2\beta-2}}{4}\frac{D-2}{D-1}\chi^{2}\right)\,,$
(3.32)

where $\alpha$, $\beta\leq 0$. Applying this transformation with the choice
$\alpha=0=\beta$, one finds that the resulting action has the following term
at leading order:

$\displaystyle S$
$\displaystyle=\int\mathrm{d}^{D}x\,e\Big{[}\frac{1}{2\kappa_{C}^{2}}\Big{(}\omega_{b}{}^{ab}(e)\omega_{ca}{}^{c}(e)-\omega_{abc}(e)\omega^{bac}(e)-2\omega_{0}{}^{0a}(\tau,e)\omega_{ba}{}^{b}(e)$
$\displaystyle\
-2\omega_{[a|0|b]}(\tau,e)\omega_{0}{}^{ab}(\tau,e)+\chi^{ab}e_{0\\{a,b\\}}-\frac{D-2}{D-1}\chi
e_{0a}{}^{a}\Big{)}-\frac{1}{2}\left\\{\bar{\psi}\Gamma^{0}\tilde{D}_{0}\psi+\mathrm{h.c.}\right\\}+m\bar{\psi}\psi\Big{]}\,.$
(3.33)

For the action (3.22), one gets:

$\displaystyle S$
$\displaystyle=\int\mathrm{d}^{D}x\,e\Big{[}\frac{1}{2\kappa_{C}^{2}}\Big{(}\omega_{b}{}^{ab}(e)\omega_{ca}{}^{c}(e)-\omega_{abc}(e)\omega^{bac}(e)-2\omega_{0}{}^{0a}(\tau,e)\omega_{ba}{}^{b}(e)$
$\displaystyle\
-2\omega_{[a|0|b]}(\tau,e)\omega_{0}{}^{ab}(\tau,e)+\chi^{ab}e_{0\\{a,b\\}}-\frac{D-2}{D-1}\chi
e_{0a}{}^{a}\Big{)}-\frac{1}{2}\Big{\\{}\bar{\psi}_{+}\Gamma^{0}\Gamma_{\star}\tilde{D}_{0}\psi_{-}$
$\displaystyle\
+\bar{\psi}_{-}\Gamma^{0}\Gamma_{\star}\tilde{D}_{0}\psi_{+}+\bar{\psi}_{+}\Gamma^{a}\Gamma_{\star}\tilde{D}_{a}\psi_{+}+\mathrm{h.c.}\Big{\\}}+m\bar{\psi}_{+}\psi_{+}\Big{]}\,.$
(3.34)

Upon identifying $\chi^{ab}$ and $\chi$ with $2\omega^{\\{a|0|b\\}}$ and
$2\omega_{a}{}^{0a}$, one sees that this way of taking the limit reproduces
the results (3.2) and (3.2) of the limit of the first-order formulation, apart
from the quartic fermion terms in (3.2).

One can also apply the Hubbard-Stratonovich transformation (3.32) with
different values of $\alpha$ and $\beta$. Taking $\alpha=0$ and e.g.,
$\beta=-2$, leads to the actions (3.3) and (3.3) without the $\chi
e_{0a}{}^{a}$ term. Only the auxiliary field $\chi^{\\{ab\\}}$ is retained in
the limit and it acts as a Lagrange multiplier for the Carroll 3 constraint.
Likewise, taking $\beta=0$ and e.g., $\alpha=-2$ leads to the actions (3.3)
and (3.3) without the $\chi^{ab}e_{0\\{a,b\\}}$ term and the remaining field
$\chi$ plays the role of Lagrange multiplier for the Carroll 2 constraint.
Finally, in the above scenarios, we applied the Hubbard-Stratonovich
transformation to both leading order terms of (3.30). By using it more
selectively, one can obtain two electric Carroll gravity theories, other than
the one of (3.31). In particular, applying it to only the
$t_{0}{}^{\\{a,b\\}}t_{0\\{a,b\\}}$ leading order term of (3.30) leads in the
$\tilde{c}\rightarrow\infty$ limit to an electric Carroll gravity action
proportional to

$\displaystyle\int\mathrm{d}^{D}x\,e\,t_{0a}{}^{a}t_{0b}{}^{b}\,.$ (3.35)

Likewise, an action proportional to

$\displaystyle\int\mathrm{d}^{D}x\,e\,t_{0}{}^{\\{a,b\\}}t_{0\\{a,b\\}}\,,$
(3.36)

is obtained by applying the Hubbard-Stratonovich trick to only the
$t_{0a}{}^{a}t_{0b}{}^{b}$ leading order contribution of (3.30). One can even
apply a further Hubbard-Stratonovich transformation to each of these, to
obtain actions proportional to

$\displaystyle\int\mathrm{d}^{D}x\,e\,\chi
t_{0a}{}^{a}\,,\qquad\int\mathrm{d}^{D}x\,e\,\chi^{\\{ab\\}}t_{0\\{a,b\\}}\,,$
(3.37)

that just consist of Lagrange multiplier terms that impose the Carroll 2 and
Carroll 3 constraints respectively.

##  4 Conclusions

In this work we showed how to obtain electric and magnetic Carroll fermions by
taking Carroll limits of relativistic Dirac fermions. We first studied the
possible transformations of spinor fields under the homogeneous Carroll group
and then we identified dynamical models implementing the various options.
Similarly to the bosonic case, the electric theory can be defined simply by
considering the limit of vanishing speed of light of the Dirac action.
Magnetic limits are instead subtler: in odd spacetime dimensions they require
to double the number of spinors in the relativistic theory. Due to this
subtlety, and in order to allow for easier comparisons with the existing
literature, we derived our magnetic actions in two complementary ways. We
first started from an off-diagonal two-fermion action and then we rederived
the same result starting from a singular limit of the Dirac action rewritten
in Hamiltonian form. In the second approach the doubling of spinorial
variables is induced by the loss of second class constraints in the limit. We
however stressed that in even dimensions a truncation of the magnetic theory
is possible, leading to a magnetic Carroll fermion with the same number of
components as a single Dirac fermion. In this case, one can also avoid the
doubling of fermionic variables to begin with and define the Carroll limit
starting from the tachyonic action (2.27).

Similar limits of General Relativity have been considered before leading to
electric [41] and magnetic [35, 12, 39, 40] Carroll gravity theories. By
combining the two limits we obtained expressions for electric and magnetic
Carroll fermions coupled to magnetic Carroll gravity. It would be interesting
to see whether, using different techniques, electric Carrol gravity can also
be coupled to fermions.

We note that the results on Carroll limits can be generalized to similar
Galilei limits of both scalars [44], fermions [23] and gravity [35]. This
leads to similar notions such as electric and magnetic Galilei fermions and
electric and magnetic Galilei gravity. For instance, the expression for
electric Galilei gravity is given by the expression

$\displaystyle S_{\text{el. Galilei. grav.}}=\frac{1}{16\pi
G_{G}}\int\mathrm{d}^{4}x\,e\,t^{ab}t_{ab}\,,$ (4.1)

where
$t_{ab}=e_{a}{}^{\mu}e_{b}{}^{\nu}(\partial_{\mu}\tau_{\nu}-\partial_{\nu}\tau_{\mu})$
and $\tau_{\mu}$ is the clock function.

An undesirable feature of our limit technique is that it requires the
existence of a Dirac fermion. This excludes for instance applications to
supergravity in ten and eleven spacetime dimensions. This obstacle can be
circumvented by generalizing our particle limit to extended objects where the
$\Gamma_{0}$ occurring in the projection operator is replaced by a gamma
matrix with indices in all directions longitudinal to the extended object
[45]. For instance, in ten and eleven dimensions where the corresponding
supergravity theories do not contain Dirac spinors one could define string and
membrane projection operators as follows:

$P_{\pm}=\frac{1}{2}(1\pm\Gamma_{01})\,:\textrm{strings}\hskip
28.45274pt\textrm{and}\hskip
28.45274ptP_{\pm}=\frac{1}{2}(1\pm\Gamma_{012})\,:\textrm{membranes}$ (4.2)

These projection operators are consistent with Majorana-Weyl spinors in 10D
and Majorana spinors in 11D.

An advantage of our limit is that the rescalings with $\tilde{c}$ guarantee
that our final expressions are invariant under global scalings of the fields.
This implies that the Carroll symmetry is extended to a conformal Carroll
symmetry. This leads to interesting connections with the BMS symmetry that
plays such a prominent role in flat space holography and celestial holography.
In this context, it is of interest to note that the free Carroll fermion
models we constructed can be used to construct infinite dimensional algebras
of the type $w_{1+\infty}$ in the same way as this has been done before in the
relativistic case, see e.g. [46, 47]. It would be interesting to see whether
there exists electric and magnetic versions of these infinite-dimensional
algebras.

Last but not least, given that we have electric (magnetic) Carroll scalars and
fermions, it is natural to consider supersymmetric combinations. We hope to
report on this in a future work [48] (see also [23]).

## Acknowledgments

We thank Nicolas Boulanger, Dario Francia, Laurent Freidel, Marios P.
Petropoulos for discussions. AC is a research associate of the Fonds de la
Recherche Scientifique – FNRS. The work of AC and LM was supported in part by
FNRS under Grants FC.36447 and F.4503.20. This research was supported in part
by Perimeter Institute for Theoretical Physics. Research at Perimeter
Institute is supported in part by the Government of Canada through the
Department of Innovation, Science and Economic Development and by the Province
of Ontario through the Ministry of Colleges and Universities. During the last
part of this work, AF was supported by SFI and the Royal Society under the
grant number RFF$\backslash$EREF$\backslash$210373\. AF thanks Lia for her
permanent support. EB thanks Perimeter Institute for hospitality. AC, EB and
JR thank the organisers of the workshops “Strings, Branes and Gravitational
Waves” (Solvay Institutes, Brussels, 22/8/2023), “Quantum Gravity, Strings and
the Swampland” (Corfu, 12–19/9/2023) and of the “3rd Carroll workshop”
(Aristotle University of Thessaloniki, 2–6/10/2023) for hospitality and for
the opportunity to present a preliminary version of the results of this paper.

##  Appendix A Conventions

We use capital Latin indices $A$, $B$, $\cdots(=0,1,\cdots,D-1)$ to denote
inertial Minkowski coordinates. These indices will often be split in a time-
like one, denoted by 0, and spatial ones, denoted by lowercase Latin indices
$a$, $b$, $\cdots(=1,\cdots,D-1$). The Minkowski metric $\eta_{AB}$ used in
this paper has mostly plus signature:
$\eta_{AB}=\mathrm{diag}(-1,+1,+1,\cdots,+1)$.

We take the $\Gamma$-matrices to obey the following Clifford algebra relation:

$\displaystyle\\{\Gamma_{A},\Gamma_{B}\\}=2\eta_{AB}\mathds{1}\,.$ (A.1)

They satisfy the following Hermiticity properties:

$\displaystyle\left(\Gamma^{0}\right)^{\dagger}=-\Gamma^{0}\,,\qquad\left(\Gamma^{a}\right)^{\dagger}=\Gamma^{a}\,,\qquad\qquad(a=1,2,\cdots,D-1)\,.$
(A.2)

We define the matrix $\Gamma_{\star}$ in terms of the product of all
$\Gamma$-matrices as:

$\displaystyle\Gamma_{\star}\equiv(-\mathrm{i})^{\frac{D}{2}+1}\Gamma^{0}\Gamma^{1}\cdots\Gamma^{D-1}\,.$
(A.3)

It obeys the following properties:

$\displaystyle\Gamma_{\star}^{\dagger}=\Gamma_{\star}\,,\qquad\qquad\Gamma_{\star}^{2}=1\,.$
(A.4)

In $D=4$, $\Gamma_{*}=\mathrm{i}\Gamma^{0}\Gamma^{1}\Gamma^{2}\Gamma^{3}$
corresponds to the matrix that is usually called $\Gamma_{5}$.

The Dirac conjugate of a spinor is defined as

$\displaystyle\bar{\Psi}\equiv\mathrm{i}\Psi^{\dagger}\Gamma^{0}\,.$ (A.5)

The complex conjugate of a product of (Grassmann-valued) spinor components
$\varepsilon_{\alpha}$, $\psi_{\beta}$ is defined as the product of their
complex conjugates in reverse order:

$\displaystyle(\varepsilon_{\alpha}\psi_{\beta})^{*}=\psi_{\beta}^{*}\varepsilon_{\alpha}^{*}\,.$
(A.6)

We note the following useful commutation properties of the projectors
$P_{\pm}$ introduced in (2.6)

$\displaystyle P_{\pm}\Gamma^{0}=\Gamma^{0}P_{\pm}\,,\qquad
P_{\pm}\Gamma^{a}=\Gamma^{a}P_{\mp}\,,\qquad
P_{\pm}\Gamma_{\star}=\Gamma_{\star}P_{\mp}\,.\qquad$ (A.7)

Furthermore, projected spinors obey the following properties:

$\displaystyle\bar{\Psi}_{\pm}=\bar{\Psi}_{\pm}P_{\pm}\,,\qquad\qquad\qquad\Gamma^{0}\Psi_{\pm}=\mp\mathrm{i}\Psi_{\pm}\,.$
(A.8)

## References

  * [1] J.-M. Lévy-Leblond, “Une nouvelle limite non-relativiste du groupe de Poincaré”, Annales de l’I.H.P. Phys. Theor. 3, 1 (1965), http://eudml.org/doc/75509.
  * [2] N. D. Sen Gupta, “On an Analogue of the Galileo Group”, Nuovo Cim. 54, 512 (1966).
  * [3] S. Pasterski, M. Pate and A.-M. Raclariu, “Celestial Holography”, arxiv:2111.11392, in: “Snowmass 2021”.
  * [4] L. Donnay, “Celestial holography: An asymptotic symmetry perspective”, arxiv:2310.12922.
  * [5] L. Donnay and C. Marteau, “Carrollian Physics at the Black Hole Horizon”, Class. Quant. Grav. 36, 165002 (2019), arxiv:1903.09654.
  * [6] C. Duval, G. W. Gibbons and P. A. Horvathy, “Conformal Carroll groups and BMS symmetry”, Class. Quant. Grav. 31, 092001 (2014), arxiv:1402.5894.
  * [7] A. Bagchi, S. Chakrabortty and P. Parekh, “Tensionless Strings from Worldsheet Symmetries”, JHEP 1601, 158 (2016), arxiv:1507.04361.
  * [8] E. Casali and P. Tourkine, “On the null origin of the ambitwistor string”, JHEP 1611, 036 (2016), arxiv:1606.05636.
  * [9] C. Duval, G. W. Gibbons, P. A. Horvathy and P. M. Zhang, “Carroll versus Newton and Galilei: two dual non-Einsteinian concepts of time”, Class. Quant. Grav. 31, 085016 (2014), arxiv:1402.0657.
  * [10] E. Bergshoeff, J. Gomis and G. Longhi, “Dynamics of Carroll Particles”, Class. Quant. Grav. 31, 205009 (2014), arxiv:1405.2264.
  * [11] A. Bagchi, A. Mehra and P. Nandi, “Field Theories with Conformal Carrollian Symmetry”, JHEP 1905, 108 (2019), arxiv:1901.10147.
  * [12] M. Henneaux and P. Salgado-Rebolledo, “Carroll contractions of Lorentz-invariant theories”, JHEP 2111, 180 (2021), arxiv:2109.06708.
  * [13] J. de Boer, J. Hartong, N. A. Obers, W. Sybesma and S. Vandoren, “Carroll Symmetry, Dark Energy and Inflation”, Front. in Phys. 10, 810405 (2022), arxiv:2110.02319.
  * [14] B. Chen, R. Liu and Y.-f. Zheng, “On higher-dimensional Carrollian and Galilean conformal field theories”, SciPost Phys. 14, 088 (2023), arxiv:2112.10514.
  * [15] A. Bagchi, D. Grumiller and P. Nandi, “Carrollian superconformal theories and super BMS”, JHEP 2205, 044 (2022), arxiv:2202.01172.
  * [16] D. Rivera-Betancour and M. Vilatte, “Revisiting the Carrollian scalar field”, Phys. Rev. D 106, 085004 (2022), arxiv:2207.01647.
  * [17] S. Baiguera, G. Oling, W. Sybesma and B. T. Søgaard, “Conformal Carroll scalars with boosts”, SciPost Phys. 14, 086 (2023), arxiv:2207.03468.
  * [18] A. Banerjee, S. Dutta and S. Mondal, “Carroll fermions in two dimensions”, Phys. Rev. D 107, 125020 (2023), arxiv:2211.11639.
  * [19] A. Bagchi, A. Banerjee, R. Basu, M. Islam and S. Mondal, “Magic fermions: Carroll and flat bands”, JHEP 2303, 227 (2023), arxiv:2211.11640.
  * [20] X. Bekaert, A. Campoleoni and S. Pekar, “Carrollian conformal scalar as flat-space singleton”, Phys. Lett. B 838, 137734 (2023), arxiv:2211.16498.
  * [21] B. Chen, R. Liu, H. Sun and Y.-f. Zheng, “Constructing Carrollian Field Theories from Null Reduction”, arxiv:2301.06011.
  * [22] J. de Boer, J. Hartong, N. A. Obers, W. Sybesma and S. Vandoren, “Carroll stories”, JHEP 2309, 148 (2023), arxiv:2307.06827.
  * [23] K. Koutrolikos and M. Najafizadeh, “Super Carrollian and Super Galilean Field Theories”, arxiv:2309.16786.
  * [24] L. Ciambelli, “Dynamics of Carrollian Scalar Fields”, arxiv:2311.04113.
  * [25] E. Bergshoeff, J. Figueroa-O’Farrill and J. Gomis, “A non-lorentzian primer”, SciPost Phys. Lect. Notes 69, 1 (2023), arxiv:2206.12177.
  * [26] M. B. Stakenborg, “Carroll limit of the Dirac Lagrangian”, master thesis Utrecht University (2023), https://studenttheses.uu.nl/handle/20.500.12932/44588.
  * [27] H. C. Lee, “On Clifford algebras and their representations”, Annals Math. 49, 760 (1948).
  * [28] J. A. Brooke, “A Galileian formulation of spin. I. Clifford algebras and spin groups”, J. Math. Phys. 19, 952 (1978).
  * [29] J. A. Brooke, “A Galileian formulation of spin. II. Explicit realizations.”, J. Math. Phys. 21, 617 (1980).
  * [30] L. Mele, “Carrollian fermions coupled to gravity”, master thesis Université de Mons (2023), https://orbi.umons.ac.be/handle/20.500.12907/46498.
  * [31] P.-x. Hao, W. Song, X. Xie and Y. Zhong, “BMS-invariant free scalar model”, Phys. Rev. D 105, 125005 (2022), arxiv:2111.04701.
  * [32] A. Bagchi, A. Banerjee, S. Dutta, K. S. Kolekar and P. Sharma, “Carroll covariant scalar fields in two dimensions”, JHEP 2301, 072 (2023), arxiv:2203.13197.
  * [33] L. Bidussi, J. Hartong, E. Have, J. Musaeus and S. Prohazka, “Fractons, dipole symmetries and curved spacetime”, SciPost Phys. 12, 205 (2022), arxiv:2111.03668.
  * [34] O. Kasikci, M. Ozkan and Y. Pang, “Carrollian origin of spacetime subsystem symmetry”, Phys. Rev. D 108, 045020 (2023), arxiv:2304.11331.
  * [35] E. Bergshoeff, J. Gomis, B. Rollier, J. Rosseel and T. ter Veldhuis, “Carroll versus Galilei Gravity”, JHEP 1703, 165 (2017), arxiv:1701.06156.
  * [36] E. Bergshoeff, J. Figueroa-O’Farrill, K. van Helden, J. Rosseel, I. Rotko and T. ter Veldhuis, “$p$-brane Galilean and Carrollian Geometries and Gravities”, arxiv:2308.12852.
  * [37] A. Barducci, R. Casalbuoni and J. Gomis, “Confined dynamical systems with Carroll and Galilei symmetries”, Phys. Rev. D 98, 085018 (2018), arxiv:1804.10495.
  * [38] E. Bergshoeff, J. M. Izquierdo and L. Romano, “Carroll versus Galilei from a Brane Perspective”, JHEP 2010, 066 (2020), arxiv:2003.03062.
  * [39] J. Figueroa-O’Farrill, E. Have, S. Prohazka and J. Salzer, “The gauging procedure and carrollian gravity”, JHEP 2209, 243 (2022), arxiv:2206.14178.
  * [40] A. Campoleoni, M. Henneaux, S. Pekar, A. Pérez and P. Salgado-Rebolledo, “Magnetic Carrollian gravity from the Carroll algebra”, JHEP 2209, 127 (2022), arxiv:2207.14167.
  * [41] M. Henneaux, “Geometry of Zero Signature Space-times”, Bull. Soc. Math. Belg. 31, 47 (1979).
  * [42] J. Hartong, “Gauging the Carroll Algebra and Ultra-Relativistic Gravity”, JHEP 1508, 069 (2015), arxiv:1505.05011.
  * [43] D. Hansen, N. A. Obers, G. Oling and B. T. Søgaard, “Carroll Expansion of General Relativity”, SciPost Phys. 13, 055 (2022), arxiv:2112.12684.
  * [44] C. Batlle, J. Gomis, L. Mezincescu and P. K. Townsend, “Tachyons in the Galilean limit”, JHEP 1704, 120 (2017), arxiv:1702.04792.
  * [45] J. Gomis, K. Kamimura and P. K. Townsend, “Non-relativistic superbranes”, JHEP 0411, 051 (2004), hep-th/0409219.
  * [46] C. N. Pope, “Lectures on W algebras and W gravity”, hep-th/9112076, in: “Summer School in High-energy Physics and Cosmology”, 827–867p.
  * [47] E. Bergshoeff, B. de Wit and M. A. Vasiliev, “The Structure of the superW(infinity) (lambda) algebra”, Nucl. Phys. B 366, 315 (1991).
  * [48] E. Bergshoeff, A. Fontanella and J. Rosseel, “In progress”.