# A spin model for intrinsic antiferromagnetic skyrmions on a triangular
lattice

Amal Aldarawsheh1,2∗ Moritz Sallermann1,3,4 Muayad Abusaa5 and Samir
Lounis1,2∗

Peter Grünberg Institute and Institute for Advanced Simulation,
Forschungszentrum Jülich and JARA, D-52425 Jülich, Germany

Faculty of Physics, University of Duisburg-Essen and CENIDE, 47053 Duisburg,
Germany

RWTH Aachen University, 52056 Aachen, Germany

Science Institute and Faculty of Physical Sciences, University of Iceland, VR-
III, 107 Reykjavík, Iceland

Department of Physics, Arab American University, Jenin, Palestine

*<EMAIL_ADDRESS><EMAIL_ADDRESS>
## Abstract

Skyrmions are prospected as the potential future of data storage due to their
topologically protected spin structures. However, traditional ferromagnetic
(FM) skyrmions experience deflection when driven with an electric current,
hindering their usage in spintronics. Antiferromagnetic (AFM) skyrmions,
consisting of two FM solitons coupled antiferromagnetically, are predicted to
have a zero Magnus force, making them promising candidates for spintronic
racetrack memories. Currently, they have been stabilized in synthetic AFM
structures, i.e. multilayers hosting FM skyrmions, which couple
antiferromagnetically through a non-magnetic spacer, while recent first-
principles simulations predict their emergence in an intrinsic form, within an
row-wise AFM single monolayer of Cr deposited on PdFe bilayer grown on Ir(111)
surfaces. The latter material forms a triangular lattice, where single and
interlinked AFM skyrmions can be stabilized. Here, we explore the minimal
Heisenberg model enabling the occurrence of such AFM solitons and the
underlying phase diagrams by accounting for the interplay between the
Dzyaloshinskii-Moriya and Heisenberg exchange interactions, as well as the
magnetic anisotropy and impact of magnetic field. By providing the fundamental
basis to identify and understand the behavior of intrinsic AFM skyrmions, we
anticipate our model to become a powerful tool for exploring and designing new
topological magnetic materials to conceptualize devices for AFM spintronics.

Keywords: Intrinsic antiferromagnetic skyrmions, spin model, single and
interchained AFM skyrmions, triangular lattice, thermal stability, phase
diagram, antiferromagnetism, topology.

## Introduction

Since their early observation [1, 2, 3, 4], skyrmions, which are magnetic
textures with unique properties, have garnered the attention of the condensed
matter community. They are seen as potential bit representatives for future
spintronic devices due to their nontrivial topological twists and exotic
properties [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21].
Skyrmion-based racetrack memory devices are expected to remarkably reduce the
power consumption in data flow compared to domain walls [22, 23]. However,
ferromagnetic (FM) skyrmions are afflicted with various drawbacks that limit
their optimal utilization such as: their sensitivity to stray fields and
suffering from dipolar interactions [24, 25], in addition to their complex
response to applied currents leading to unwanted deflections [8, 26], which
can become even more elaborated under the presence of defects [27, 26, 24, 28,
29]. In contrast, antiferromagnetic (AFM) skyrmions have several advantages
over their FM counterparts since the stray field cancels out [30, 31]
augmented with an immunity to the Magnus force [32, 33, 34, 35, 36, 37] with
their potential for ultrafast dynamics [38], and ability to overcome defects
[39, 40].

Several recent theoretical studies are inspecting the realization of
individual AFM skyrmions or their periodic arrangement assuming a squared [32,
33, 41, 35, 37], a triangular [42, 43] or a honeycomb lattice [36]. On the
experimental side, synthetic AFM skyrmions were unveiled experimentally in
multilayers, where FM films host regular FM skyrmions with an interfilm
coupling of AFM nature through various spacers [44, 45, 46, 47, 48], while
fractional antiferromagnetic skyrmion lattice was stabilized in MnSc2S4 [49,
50] , and complex topological AFM objects were found in a bulk phase [51]. In
Ref. [52], we predicted the emergence of intrinsic single and interchained AFM
skyrmions on a triangular lattice of row-wise AFM (RW-AFM) Cr layer deposited
on PdFe/Ir(111). The latter substrate became over the last decade a perfect
bed system for a plethora of phenomena pertaining to FM skyrmions [4, 53, 54,
11, 55, 56, 14, 15, 57, 58, 20].

The goal of the current work is to introduce a Heisenberg model that
incorporates the essential magnetic interactions required to produce AFM
skyrmions on a triangular lattice. We perform atomistic spin simulations on
the basis of the Landau-Lifschitz-Gilbert (LLG) equations as implemented in
the Spirit code [59]. We consider the interplay between the exchange
interactions, Dzyaloshinskii-Moriya interactions (DMI), the magnetic
anisotropy and the impact of an external magnetic field to establish the phase
diagrams of the intrinsic AFM skyrmions while inspecting their stability via
simulations based on the geodesic nudged elastic band method (GNEB) [60, 61,
59]. Our model offers a robust approach to comprehend the behavior of AFM
skyrmions in a triangular lattice with the aim of understanding the required
ingredients for their stabilization and to create novel materials and devices
for AFM spintronics.
Methods
In our study, we consider a two dimensional Heisenberg model on a triangular
lattice, equipped with Heisenberg exchange coupling, DMI, the magnetic
anisotropy energy (MAE), and Zeeman term. The energy functional reads as
follows:

$H=-\sum\limits_{<i,j>}J_{ij}\;\textbf{S}_{i}\cdot\textbf{S}_{j}-\sum\limits_{<i,j>}\textbf{D}_{ij}\cdot(\textbf{S}_{i}\times\textbf{S}_{j})-K\sum\limits_{i}(S_{i}^{z})^{2}-\sum\limits_{i}h_{i}S_{i}^{z},$
(1)

where $i$ and $j$ are site indices, each carrying a magnetic moment. S is the
unit vector of the magnetic moment. $J$ is the Heisenberg exchange coupling
strength, being negative for an AFM interaction while D is the DMI vector, and
$K$ the magnetic anisotropy energy per atom favoring an out-of-plane
orientation if positive. $h_{i}=\mu_{i}B$ describes the Zeeman coupling to the
atomic spin moment $\mu$ at site $i$, assuming $\mu=1~{}\mu_{B}$ and an out-
of-plane field. To explore the magnetic properties and emerging complex states
we utilize the Landau-Lifshitz-equation (LLG) as implemented in the Spirit
code [59]. The simulations were carried out with 1002, 2002 and 3002 sites
assuming periodic boundary conditions to model the extended two dimensional
Heisenberg Hamiltonian at zero Kelvin.

## Results

###  0.1 Phase diagrams

We start our investigations by determining the conditions to form a RW-AFM
spin state, which was observed experimentally in Mn/Re(0001) [62, 63]. As
established in Refs. [64, 65] the minimum set of Heisenberg exchange
interactions involves the interactions with first ($J_{1}$) and second
($J_{2}$) nearest neighboring interactions as shown in the upper inset of Fig.
1 a. The formation of the FM skyrmions building up our AFM skyrmions requires,
as demonstrated in Ref. [52], a third nearest neighboring interaction $J_{3}$,
which should mediate a ferromagnetic coupling. Fig. 1 a illustrates the
underlying phase diagram, where we expect four regions that can host either a
Néel, FM, spin spiraling and RW-AFM spin states. The dark blue color indicates
the region of interest, where the magnetic moments are distributed into four
sublattices L1, L2, L3 and L4 as shown in the lower inset of Fig. 1 a. $J_{3}$
mediates the magnetic interaction within each sublattice and must be positive,
i.e. favoring a FM alignment, to enable the stabilization of the RW-AFM state.
If too weak with respect to $J_{1}$ or if it is of antiferromagnetic nature,
either spin spirals or a Néel state are favored depending on the strength of
$J_{2}$. We observe that the RW-AFM configuration occupies a larger phase area
when $J_{2}$ is of AFM nature. In the RW-AFM state, $J_{3}$ is thus positive,
which together with the DMI vector $\textbf{D}_{3}$ that is connecting the
third n.n. similarly to $J_{3}$, enable the formation of sublattice FM
skyrmions. $\textbf{D}_{3}$ lies in-plane and is perpendicular to the bond
connecting neighboring atoms as shown in Supplementary Figure S1. The AFM
interaction among the FM skyrmions is mediated by $J_{1}$ such that the
presence of $J_{2}$ is not requested. As predicted in Ref. [52], the single
AFM skyrmion consists of FM skyrmions present in two sublattices (L1 and L2)
with the other two sublattices remaining collinear, while for the double AFM
skyrmions, the building blocks FM skyrmions reside in each of the four
sublattices (L1, L2, L3 and L4) as illustrated in Figs. 1b-c.

After setting the base for the magnetic interactions needed to realize our AFM
solitons, we inspect the range of parameters ($J_{2}$, $J_{3}$, $D_{3}$ and
$K$) normalised to the absolute value of $J_{1}$, within which the single and
double interchained AFM skyrmions can be stabilized (Figs. 1d-g). The building
blocks of the AFM solitons are FM skyrmions. The region hosting the skyrmions,
color coded in terms of their radius, is sandwiched between the RW-AFM and
stripe domains phases. Thus the impact of the underlying interactions is
similar to what is expected from the FM topological objects. For instance
increasing $J_{3}$ (Figs. 1d-e, with $K/|J_{1}|$ and $D_{3}/|J_{1}|$ equal to
0.024 and 0.03 respectively), which defines the FM interaction among the spins
of the FM skyrmions, or the magnetic anisotropy energy $K$ (Figs. 1f-g, with
$J_{2}/|J_{1}|$ and $J_{3}/|J_{1}|$ equal to -0.2 and 0.2 respectively)
shrinks the size of the spin-texture by ultimately leading to its
annihilation, while the DM interaction $D_{3}$ induces the opposite behavior
(Figs. 1f-g). Interestingly, $J_{2}$ counteracts $J_{3}$ by amplifying the
skyrmion size, which at some point can be deformed into stripe domains. For
completeness, snapshots of skyrmions, labelled from A to L in Fig. 1, are
presented in Supplementary Figure S2.

It is significant to point out that the shape of the AFM skyrmions is
determined by the specific interaction parameters involved. While the
interactions between spins in one sublattice Li, characterized by $J_{3}$ and
$D_{3}$, lead to the formation of a FM skyrmion in that sublattice, the
interactions with spins in the other FM skyrmion hosting sublattice, governed
by $J_{1}$ and $J_{2}$, have a substantial impact on shaping the resulting AFM
skyrmion, as illustrated in Supplementary Figure S3 and discussed in detail in
Supplementary Note 1. Without $J_{2}$, there is an anisotropic cancellation of
contributing magnetic interactions induced by the AFM alignment of the spins
around the skyrmion, which triggers a shape elongation. A finite $J_{2}$ helps
establishing a balance in the magnetic interaction, which reduces the
aforementioned skyrmion asymmetry. However, interlinking of the AFM skyrmions
results in a slight deformation in the overlap region, as the spins in the
overlapping area interact with the noncollinear spins of the second AFM
skyrmion, which is not the case for the other spins on the free side of the
AFM skyrmion.

It is worth mentioning that the size of the single AFM skyrmion is smaller
than those participating in the formation of the interchained magnetic
textures (see for example the radius given in Figs. 1b-c), which impacts on
the details of the phase diagrams. On the one hand, the window in which the
double AFM skyrmions are stabilised while varying $J_{2}$ and $J_{3}$ is
larger than that of the single magnetic objects (Figs.1d-e). On the other
hand, the single skyrmion phase seems wider and shifted to the upper region of
the diagram while tuning $D_{3}$ and $K$.

###  0.2 Response to external magnetic fields

The stability of skyrmions when exposed to an external magnetic field is an
essential aspect for their utilization in future spintronics. Here, we
investigate the response of the single and double AFM skyrmions to a magnetic
field perpendicular to the lattice. Within our model, as theoretically
expected [42, 66, 67, 52], and in contrast to their FM counterparts, the size
of the AFM skyrmions increases with the external magnetic field, until its
magnitude approaches a critical value ($B_{c}$), after which, the skyrmion
deforms into the stripe domain phase. It has been shown that the single and
interlinked AFM skyrmions formed with the realistic interactions among Cr
atoms bear high magnetic fields [52]. At the model level, the critical value
of the normalised magnetic field ($\mu B_{c}/|J1|$) can be enhanced by
increasing the anisotropy magnitude, as depicted in Fig. 2a, for both single
and double AFM skyrmions. In contrast, the DMI lessens the highest magnetic
field survived by the AFM solitons, as shown in Fig. 2c. Various formulas have
been proposed to describe the impact of DMI and anisotropy magnitude on the
radius of the FM skyrmions [68, 69, 12, 70]. Inspired by Ref. [70], and
utilizing the fact that $|J_{1}|>>D_{3},K$, our results on the dependence of
the AFM skyrmion radius $R$ on the anisotropy (Fig. 2b) and DMI (Fig. 2d) when
the external field is switched-off can be fitted with
$R_{0}=a+b\frac{D_{3}}{K}\left(1+c\frac{D_{3}^{2}}{|J_{1}|K}\right)$, where a,
b and c are fitting parameters.

Upon application of the magnetic field, we found that the form proposed in
Ref. [66] has to be amended with a linear field-dependent term. After a Taylor
expansion in the regime where the field is smaller than the rest of the
magnetic interactions, we find
$R=a+b\frac{D_{3}}{K}\left(1+c\frac{D_{3}^{2}}{|J_{1}|K}\right)\left(1+\alpha\frac{B}{|J_{1}|}+\beta\frac{B^{2}}{|J_{1}|^{2}}+\gamma\frac{B^{3}}{|J_{1}|^{3}}\right)$,
where $\alpha$, $\beta$ and $\gamma$ are additional fitting parameters, grasps
reasonably the dependencies reported in Figs. 2e-f (with $D_{3}/|J_{1}|=0.03$
and $K/|J_{1}|=0.023$).

Overall, the magnetic interactions reducing (increasing) the size of the
skyrmions, as the magnetic anisotropy (DMI) does, enable an enhanced (reduced)
stability with respect to an external magnetic field.

###  0.3 AFM skyrmions thermal stability

Now we turn to the stability of the AFM skyrmions against thermal
fluctuations, by calculating the energy barrier which is needed for the
collapse of the single and double interchained AFM skyrmions into the RW-AFM
ground state utilizing the GNEB method [60, 61, 59]. To inspect their
stability, we calculate the energy barrier for both single and double
interlinked AFM skyrmions assuming $J_{2}/|J_{1}|$ = -0.2, $J_{3}/|J_{1}|$ =
0.2, $D_{3}/|J_{1}|$ = 0.03, and $K/|J_{1}|$ = 0.024. The barrier is
determined by the energy difference between the local minimum magnetic state
hosting the AFM skyrmion and its relevant saddle point, which lies on the path
of minimum energy connecting the skyrmion configuration to the RW-AFM ground
state. In the absence of external magnetic field, the double AFM skyrmions
with radius of 1.95 nm, has an energy barrier of 0.67 meV, which translates to
$\approx$ 7.8 K, while for the single AFM skyrmion with radius of 1.6 nm, the
energy barrier is 0.055 meV ($\approx$ 0.64 K). For both cases, the major key
for the stability of the AFM skyrmions is the DMI which contributes with
$\Delta E_{\mathrm{DMI}}$ = 15.66 meV to the energy barrier of the double AFM
skyrmion and 4.33 meV for the single case, while the anisotropy and exchange
interactions prefer the collapse of the AFM solitons by contributing with
$\Delta E_{K}$ = -9.21 meV (-2.53 meV), and $\Delta E_{J}$ = -5.79 meV (-1.71
meV) for double (single) AFM skyrmions. Moreover, we addressed another
important aspect, the impact of the magnetic field, by carrying out a
systematic study with results illustrated in Fig. 3. The thermal stability is
obviously enhanced with the magnetic field, which impacts more efficiently the
double than the single AFM skyrmion (Fig. 3a). For $\mu B/|J_{1}|$ = 1, the
energy barrier of the double (single) AFM skyrmions increased to 0.81 meV
(0.12 meV) $\approx$ 9.4 K (1.3 K). By increasing the magnetic field, the
skyrmions expand (Fig. 3f), which in contrast to the DMI and Zeeman
contributions (Figs. 3c-d) is disfavored by those of the exchange and
anisotropy (Figs. 3b, e). Snapshots of the various states prospected in
defining the energy barriers are presented in Supplementary Figure S4.

Figure 1: Phase diagrams for AFM skyrmions. a The phase diagram for the
Heisenberg model of a triangular lattice without spin-orbit induced
interactions. The impact of magnetic interactions up to the third nearest
neighbors is considered assuming an antiferromagnetic coupling $J_{1}$ among
first nearest neighbors. The upper inset shows the triangular lattice
indicating the considered magnetic exchange interactions while the lower inset
illustrates the four sublattices building up the RW-AFM ground state. b The
single AFM skyrmion is made of two FM skyrmions, each hosted by sublattices L1
and L2, while L3 and L4 remain collinear, whereas the latter can host a second
AFM skyrmion in case of the overlapped skyrmion scenario c. Note that the
separation of the four sublattices in b-c is done for illustration purpose
only since all skyrmions reside in the same layer. Phase diagram showing the
range of interactions $J_{2}/|J_{1}|$ and $J_{3}/|J_{1}|$ at which the single
d and double e AFM skyrmions can be stabilised with $D_{3}/|J_{1}|$ =0.03, and
$K/|J_{1}|$=0.024. The color code indicates the radius of the stabilized AFM
skyrmion. f and g Phase diagrams obtained by changing the $D_{3}/|J_{1}|$
magnitude along with that of $K/|J_{1}|$ while fixing $J_{2}/|J_{1}|$ at -0.2,
and $J_{3}/|J_{1}|$ at 0.2, for single and double AFM skyrmions, respectively.
The letters A-L shown in the diagrams indicate skyrmions which are plotted in
Supplementary Figure S2. Figure 2: AFM skyrmions response to an external
magnetic field. The critical magnetic field $\mu B_{c}/|J_{1}|$ tolerated by
the single (green dots) and double (golden dots) AFM skyrmions as a function
of A the normalized values of the anisotropy $K/|J_{1}|$, and C the normalized
values of DMI $D_{3}/|J_{1}|$. B Increasing $K/|J_{1}|$ shrinks the radii of
both the double and single AFM skyrmions, while D increasing $D_{3}/|J_{1}|$
expands them. E,F Impact of the external magnetic field on the radii of both
types of AFM skyrmions. Figure 3: Thermal stability of the single and double
AFM skyrmions in the presence of an external magnetic field. a The total
energy barrier for double (golden) and single (green) AFM skyrmions as a
function of the normalised value of the external magnetic field. The different
contribution to the energy barrier as shown in b exchange, c DMI, d Zeeman and
e magnetic anisotropy. f The Radii of the single and double AFM skyrmions are
plotted as function of the magnetic field.

## Discussion

Inspired by our recent findings on the emergence of single and interchained
AFM skyrmions on a triangular lattice, we propose here a spin model with the
minimum set of magnetic interactions needed to realize such intriguing
solitons. They form in a RW-AFM state, which can be decomposed into four
sublattices. The exchange interaction within each sublattice, mediating the
coupling between the third n.n., is of FM nature along with the associated DMI
and out-of-plane anisotropy permit the formation of FM skyrmions within the
sublattices. The first n.n. have to be AFM to impose the emergence of AFM
skyrmions. We identified the phase diagrams of the latter entities as well as
their dependencies on the magnitude of the various magnetic interactions and
sensitivity to an external magnetic field.

The identification of AFM spin-textures requires specific experimental
techniques. We expect the recently proposed all-electrical detection based on
the tunneling spin-mixing magnetoresistance (TXMR)[11, 71], with its different
possible modes [20] that can be enhanced by the proper implantation of atomic
defects [15] to be useful for the exploration of AFM states. In this context,
the predicted non-collinear Hall effect[72], the topological spin Hall effect
for antiferromagnets [73] as well as the spin-resolved inelastic electron
scattering approaches could be valuable[74, 75]. Obviously spin-polarized
scanning tunneling microscopy is capable of resolving antiferromagnetic state
via atomic resolution[76, 77] while enormous progress has been made with X-ray
magnetic microscopy [47] and all-optical relaxometry with a scanning quantum
sensor based on a single nitrogen-vacancy (NV) defect in diamond, which were
applied for various synthetic AFM textures, among which skyrmions [46].

We expect our work to facilitate the search and the identification of single
or overlapping AFM skyrmions while contributing to the detailed understanding
of their various properties, which is a corner stone in the field of
topological antiferromagnetism and its potential use in devices for
information technology.

##  1  ACKNOWLEDGEMENTS

This work was supported by the Federal Ministry of Education and Research of
Germany in the framework of the Palestinian-German Science Bridge (BMBF grant
number 01DH16027) and the Deutsche Forschungsgemeinschaft (DFG) through SPP
2137 “Skyrmionics” (Project LO 1659/8-1). The authors gratefully acknowledge
the computing time granted through JARA on the supercomputer JURECA at
Forschungszentrum Jülich.

##  2  AUTHOR CONTRIBUTIONS

S.L. initiated, designed and supervised the project. A.A. performed the
simulations with support and supervision from M.S. A.A., M.S., M.A., and S.L.
discussed the results. A.A. and S.L. wrote the manuscript to which all co-
authors contributed.

##  3  COMPETING INTERESTS.

The authors declare no competing interests.

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11institutetext: Instituto de Ciencias Exactas y Naturales (ICEN), Universidad
Arturo Prat, Avenida Arturo Prat Chacón 2120, 1110939, Iquique,
Chile22institutetext: Departament de Física Quàntica i Astrofísica and
Institut de Ciències del Cosmos (ICCUB), Universitat de Barcelona, Martí i
Franquès 1, E-08028 Barcelona, Spain33institutetext: Science Institute,
University of Iceland, Dunhaga 3, 107 Reykjavík, Iceland44institutetext:
Centro de Estudios Científicos (CECs), Av. Arturo Prat 514, Valdivia,
Chile55institutetext: Universidad San Sebastián, Av. General Lagos 1163,
Valdivia, Chile

# Electric/Magnetic Newton-Hooke and Carroll Jackiw-Teitelboim Gravity

Luis Avilés 2 Joaquim Gomis 3 Diego Hidalgo 4,5 Jorge Zanelli
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We construct the electric and magnetic Newton-Hooke and Carroll Jackiw-
Teitelboim gravity theories using the isomorphism of Newton-Hooke± and (A-)dS
Carroll algebras in $(1+1)$-spacetime dimensions. The starting point is the
non-relativistic and Carroll version of Jackiw-Teitelboim gravity without
restrictions on the geometry studied in Gomis:2020wxp .

††preprint:

##  1 Introduction

There has recently been a surge in interest in non-Lorentzian theories and
their associated geometries. The reasons are several: their applications in
condensed matter physics SachdevBook ; ZaanenBook , non-relativistic
holographic theories Barnich:2009se ; Bagchi:2012cy ; Bagchi:2019xfx ;
Ciambelli:2020eba ; Campoleoni:2022wmf , Einstein gravity and black holes
Blau:2015nee ; Penna:2018gfx ; Donnay:2019jiz ; deBoer:2021jej ; Perez:2021abf
; Perez:2022jpr ; Fuentealba:2022gdx , and also in hydrodynamics
Ciambelli:2018xat ; Ciambelli:2018wre ; Freidel:2022bai .

Non-Lorentzian theories can arise from the non-relativistic (or Galiliean,
i.e. $c\rightarrow\infty$) and Carroll ($c\rightarrow 0$) limits of Lorentzian
theories. In particular, for electromagnetism, Le Bellac and Lévy-Leblond
showed that there exist two different Galilean limits: the so-called
“electric” and “magnetic” limits. Later, the Carroll analogue was derived in
Duval:2014uoa . Also, it has been recently shown that the Hamiltonian
formulation of general relativity admits at least two Carroll limits
Henneaux:2021yzg . The so-called magnetic case is equivalent to the Carroll
theory of gravity, defined through a gauging of the Carroll algebra
Campoleoni:2022ebj .
Alternatively, non-Lorentzian theories can also be built from non-relativistic
and Carroll groups using the tools of gauge theory without performing a limit
process Bergshoeff:2017btm . So far, it is not clear how to identify the
“electric” and “magnetic” analogue theories in this context. However, based on
the results obtained from the Lagrangian formulation of non-Lorentzian field
theories Bergshoeff:2022eog , it is possible to notice that an “electric” non-
relativistic field theory implies the inclusion of extra fields while the
“magnetic” one does not. The opposite is true for the Carroll case; that is,
we have a “magnetic” Carroll field theory when extra fields are included in
the theory and an “electric” Carroll theory when they are not. Here, the
terminology magnetic and electric is based on this fact.
The relationship among the non-relativistic and Carroll theories
Barducci:2018wuj ; Gomis:2022spp  in a new way has also been studied axelericq
. The main idea is that given a non-relativistic or Carroll theory, we can
construct two more models with additional fields restricting the initial
dynamics. This fact will play a very prominent role in this work.
In this note, we further analyze the relation between electric and magnetic
gravities in $(1+1)$-dimensions. In other words, we construct the
corresponding Jackiw-Teitelboim (JT) gravity theories with extra fields
playing the role of Lagrange multipliers. In this case, the Newton-Hooke
algebras and the corresponding Carroll ones are isomorphic. We start by
considering the non-relativistic and Carroll BF gravities introduced in
Gomis:2020wxp  and following the procedure of axelericq  we construct new non-
relativistic and Carroll gravity theories with extra fields that further
restrict the dynamics. We also build new gravity theories without introducing
an $U(1)$ gauge field.

##  2 Basics on non-relativistic JT gravity in the first order formalism

A first-order formulation of JT gravity can be defined by gauging the (A-)dS2
symmetry and constructing a BF theory Fukuyama:1985gg ; Isler:1989hq ;
Chamseddine:1989yz . The gauge algebra is given by $\mathfrak{so}(1,2)$ in the
A-dS case ($\tilde{\Lambda}<0$) and $\mathfrak{so}(2,1)$ for dS
($\tilde{\Lambda}>0$). The generators are based on the Lie algebra111We use
the conventions $\epsilon_{01}=-\epsilon_{10}=1$, and $\eta_{ab}={\rm
diag}(-,+)$ is the two-dimensional Minkowski metric.

$\left[\tilde{J},\tilde{P}_{a}\right]={\epsilon_{a}}^{b}\tilde{P}_{b}\,,\hskip
25.6073pt\left[\tilde{P}_{a},\tilde{P}_{b}\right]=-\tilde{\Lambda}\,\epsilon_{ab}\,\tilde{J}\,,$
(1)

where $\tilde{P}_{a}$ stand for translations, $\tilde{J}$ is the boost
generator, $\tilde{\Lambda}$ is the cosmological constant, and $a=0,1$. This
algebra admits the following most general non-degenerate invariant bilinear
form

$\left\langle\tilde{J},\tilde{J}\right\rangle=\tilde{\gamma}_{0}\,,\hskip
22.76228pt\left\langle\tilde{P}_{a},\tilde{P}_{b}\right\rangle=-\tilde{\gamma}_{0}\,\tilde{\Lambda}\,\eta_{ab}\,,$
(2)

where $\tilde{\gamma}_{0}\neq 0$ is an arbitrary constant. In order to define
a BF theory we consider a scalar field $B$, valued on the (A-)dS2 algebra

$B=\Phi^{a}\,\tilde{P}_{a}+\Phi\,\tilde{J}\,,$ (3)

and a set of one-form gauge fields, $E^{a}=E^{a}_{\mu}\,dx^{\mu}$ and
$\Omega=\Omega_{\mu}\,dx^{\mu}$, corresponding to the zweibein form and the
dual spin connection $\Omega\equiv-\frac{1}{2}\epsilon_{ab}\,\Omega^{ab}$,
respectively. Here the space-time indexes run as $\mu=0,1$. The gauge fields
define the (A-)dS2 connection one form

$A=E^{a}\,\tilde{P}_{a}+\Omega\,\tilde{J}\,,$ (4)

together with the corresponding curvature two-form $F=dA+\left[A,A\right]$,
given by222Wedge product among differential forms is assumed, for instance
$E^{a}E^{b}=E^{a}\wedge E^{b}=-E^{b}E^{a}$.

$\displaystyle F$ $\displaystyle=$ $\displaystyle
R^{a}(\tilde{P})\tilde{P}_{a}+R(\tilde{J})\tilde{J}$ (5) $\displaystyle
R^{a}(\tilde{P})$ $\displaystyle=$ $\displaystyle
dE^{a}-\epsilon^{a}_{\;\;b}\,\Omega E^{b}$ (6) $\displaystyle R(\tilde{J})$
$\displaystyle=$ $\displaystyle
d\Omega-\frac{\tilde{\Lambda}}{2}\epsilon_{ab}E^{a}E^{b}\,.$ (7)

The (A-)dS BF action then reads

$S[B,A]=\int\left\langle B,F\right\rangle=\tilde{\gamma}_{0}\int\left(\Phi
R(\tilde{J})-\tilde{\Lambda}\Phi_{a}R^{a}(\tilde{P})\right)\,.$ (8)

The action (8) is invariant under infinitesimal gauge transformations

$\delta
A=d\tilde{\chi}+[A,\tilde{\chi}]\,,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\delta
B=[B,\tilde{\chi}]\,,$ (9)

where $\tilde{\chi}=\tilde{\lambda}^{a}\tilde{P}_{a}+\tilde{\lambda}\tilde{J}$
is a gauge parameter valued in the (A-)dS2 algebra.

The theory (8) admits a second-order formulation. The proof works as follows:
varying with respect to $\Phi^{a}$ yields $R^{a}(\tilde{P})=0$, which allows
solving for the spin connection in terms of the zweibein and its derivatives.
On the other hand, varying with respect to $\Omega$ gives the equation
$d\Phi-\tilde{\Lambda}\epsilon_{ab}E^{a}\Phi^{b}=0$, which allows expressing
$\Phi^{a}$ in terms of $\Phi$. Plugging back $\Omega$ and and $\Phi^{a}$ into
(8), the BF action takes the form of the JT gravity action. Since the fields
$\\{\Omega,\Phi^{a}\\}$ were algebraically obtained from their own field
equations, there is a dynamical equivalence between the first and the well-
known second-order formulation of JT gravity, where $\Phi$ is the dilaton
field (see, for instance, Pons:2009ch ).

A finite non-relativistic (NR) limit of JT gravity in the first-order
formulation can be defined starting from a BF theory with gauge algebra
(A-)dS${}_{2}\times\mathbb{R}$ Gomis:2020wxp . Thus we extend the (A-)dS2
symmetry (1) by including an Abelian generator $\tilde{Y}$, and its gauge
field $X$. To carry out the contractions, we express the relativistic algebra
generators with a linear combination of new generators that involves a
dimensionless parameter $\varepsilon$. The NR contraction follows from
defining NR one-form gauge fields $\tau$, $e$, $\omega$ and $m$, as

$\displaystyle\tau$ $\displaystyle=$
$\displaystyle\varepsilon\,\left(E^{0}+X\right)\,,~{}~{}~{}~{}~{}~{}~{}e=E^{1}\,,$
(10a) $\displaystyle m$ $\displaystyle=$
$\displaystyle\frac{1}{\varepsilon}\left(E^{0}-X\right)\,,~{}~{}~{}~{}~{}~{}~{}\omega=\frac{1}{\varepsilon}\Omega\,,$
(10b)

and also the definition of the NR scalar fields $\\{\eta,\rho,\phi,\zeta\\}$

$\displaystyle\eta$ $\displaystyle=$
$\displaystyle\frac{\varepsilon}{2}\,\left(\Phi^{0}+\Psi\right)\,,~{}~{}~{}~{}~{}~{}\zeta=\frac{1}{\varepsilon}\,\left(\Phi^{0}-\Psi\right)\,,$
(11a) $\displaystyle\phi$ $\displaystyle=$
$\displaystyle\frac{\Phi}{\varepsilon}\,,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\rho=\Phi^{1}\,.$
(11b)

Replacing Eqs. (10) and Eqs. (11) into the Eq. (8), and defining the NR
cosmological constant $\Lambda_{\scriptscriptstyle NR}$ and the constant
$\gamma_{0}$ as

$\Lambda_{\scriptscriptstyle
NR}=\frac{1}{\varepsilon^{2}}\tilde{\Lambda}\,,\hskip
42.67912pt\gamma_{0}=\varepsilon^{2}\,\tilde{\gamma}_{0}\,,$ (12)

we find, in the the limit $\varepsilon\rightarrow 0$, the NR two-dimensional
gravity theory

$S_{\scriptscriptstyle NR}=\gamma_{0}\int\Bigg{(}\phi R_{\scriptscriptstyle
NR}(G)+\,\Lambda_{\scriptscriptstyle NR}\bigg{(}\eta R_{\scriptscriptstyle
NR}(M)+\,\zeta R_{\scriptscriptstyle NR}(H)-\rho R_{\scriptscriptstyle
NR}(P)\bigg{)}\Bigg{)}\,,$ (13)

where

$\displaystyle R_{\scriptscriptstyle
NR}(H)=d\tau\,,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}R_{\scriptscriptstyle
NR}(P)=de+\omega\tau\,,$ (14a) $\displaystyle R_{\scriptscriptstyle
NR}(G)=d\omega-\Lambda_{\scriptscriptstyle NR}\tau
e\,,~{}~{}~{}~{}~{}~{}~{}~{}~{}R_{\scriptscriptstyle NR}(M)=dm+\omega e\,.$
(14b)

The action (13) is invariant under the extended Newton-Hooke±333 The plus sign
corresponds to the contraction of dS algebra while the minus sign corresponds
to the contraction of AdS algebra. Note that, the space-time symmetries of the
harmonic oscillator form the NH${}_{2}^{+}$ algebra, whereas the inverted
harmonic oscillator has a NH${}_{2}^{-}$ symmetry algebra. algebra, given by

$[G,H]=P\,,~{}~{}~{}~{}~{}~{}~{}[G,P]=M\,,~{}~{}~{}~{}~{}~{}~{}[H,P]=-\Lambda_{\scriptscriptstyle
NR}G\,,$ (15)

where $H,P$ are generators of time and spatial translations, $G$ the Galilean
boost generator, and $M$ is an additional abelian generator. It is worth
mentioning that this action can be alternatively obtained as a BF theory based
on the extended Newton-Hooke (NH) algebra (15), where the one-form gauge
connection $A=A_{\mu}dx^{\mu}$ and the $B$ scalar field are given by

$\displaystyle A$ $\displaystyle=\tau H+eP+\omega G+mM\,,$ (16) $\displaystyle
B$ $\displaystyle=\eta H+\rho P+\phi G+\zeta M\,.$

By considering the NR gauge symmetry parameter $\chi$ valued on the extended
NH algebra as $\chi=\lambda H+\xi P+\Theta G+\Gamma M$, the symmetry
transformations of the gauge fields are given by

$\displaystyle\delta_{\scriptscriptstyle NH}\tau_{\mu}$ $\displaystyle=$
$\displaystyle\partial_{\mu}\lambda\,,$ (17a)
$\displaystyle\delta_{\scriptscriptstyle NH}e_{\mu}$ $\displaystyle=$
$\displaystyle\partial_{\mu}\xi-\tau_{\mu}\Theta+\omega_{\mu}\lambda\,,$ (17b)
$\displaystyle\delta_{\scriptscriptstyle NH}\omega_{\mu}$ $\displaystyle=$
$\displaystyle\partial_{\mu}\Theta+\Lambda_{\scriptscriptstyle
NR}(e_{\mu}\lambda-\tau_{\mu}\xi)\,,$ (17c)
$\displaystyle\delta_{\scriptscriptstyle NH}m_{\mu}$ $\displaystyle=$
$\displaystyle\partial_{\mu}\Gamma+\omega_{\mu}\xi-e_{\mu}\Theta\,,$ (17d)
$\displaystyle\delta_{\scriptscriptstyle NH}\eta$ $\displaystyle=$
$\displaystyle 0\,,$ (17e) $\displaystyle\delta_{\scriptscriptstyle NH}\rho$
$\displaystyle=$ $\displaystyle\lambda\phi-\Theta\eta\,,$ (17f)
$\displaystyle\delta_{\scriptscriptstyle NH}\zeta$ $\displaystyle=$
$\displaystyle\xi\phi-\Theta\rho\,,$ (17g)
$\displaystyle\delta_{\scriptscriptstyle NH}\phi$ $\displaystyle=$
$\displaystyle\Lambda_{\scriptscriptstyle NR}(\lambda\rho-\xi\eta)\,.$ (17h)

Like the relativistic case, we can move on to the second-order formulation of
(13). Indeed, the field equations follow upon varying with $\rho$ and $\eta$,
$R_{\scriptscriptstyle NR}(P)=0$ and $R_{\scriptscriptstyle NR}(M)=0$, allows
to solve the NR spin-connection, we get

$\omega_{\mu}=2\,\tau^{[\alpha}e^{\beta]}\,\left(e_{\mu}\partial_{\alpha}e_{\beta}-\tau_{\mu}\partial_{\alpha}m_{\beta}\right)\,.$
(18)

Then, plugging (18) into (13), we get the magnetic extended NH gravity theory

$\displaystyle S_{\scriptscriptstyle Mag.Ext.NH}=\kappa\int d^{2}x\det{(\tau
e)}\,\phi\,\Bigg{(}\mathcal{R}_{\scriptscriptstyle
NR}-2\Lambda_{\scriptscriptstyle NR}\Bigg{)}\,,$ (19)

where $\kappa=-\gamma_{0}/2,\,\det{(\tau
e)}\equiv-\epsilon^{\mu\nu}\tau_{\mu}e_{\nu}$, and

$\mathcal{R}_{\scriptscriptstyle
NR}=8\tau^{[\mu}e^{\nu]}\partial_{\mu}\left(\tau^{[\alpha}e^{\beta]}\left(e_{\nu}\partial_{\alpha}e_{\beta}-\tau_{\nu}\partial_{\alpha}m_{\beta}\right)\right)\,.$
(20)

Note that theories described by (13) and (19) are not necessarily dynamically
equivalent because the equations solved by (18) are not those obtained by
varying with respect to $\omega_{\mu}$. At any rate, in the sector $d\tau=0$,
the two systems have equivalent field equations and the invariance of (19)
under Newton-Hooke boost symmetry can be confirmed in the sector
$d\tau=0$.444This is similar to the case of first-order gravity in four
dimensions $S=\kappa\int E\wedge E\wedge R(\Omega)$ which is invariant under
local translations only when the torsion vanishes. The action (19) corresponds
to the NR second-order formulation of JT gravity Gomis:2020wxp .

Another singular limit that can be carried out on the relativistic theory (13)
is the Carroll limit. This contraction is obtained from the observation that
the Carroll (A-)dS2 algebra admits a central extension in the commutator of
the Galilean boost and momentum generator,555The existence of this extension
is a unique feature of the two-dimensional case since, unlike the Galilean
case, the Carroll algebra does not admit a non-trivial central extension in
four dimensions. we name this algebra extended Carroll (A-)dS2 algebra. One
can see that this symmetry follows from the extended NH${}_{2}^{\pm}$ algebra
by interchanging the generators $H$ and $P$ and changing the sign of the
cosmological constant $\Lambda_{\scriptscriptstyle NR}$. This fact allows one
to pass from Newton-Hooke to Carroll symmetries. The isomorphism reads as

$\displaystyle{\rm Extended\;\,NH}_{2}^{+}\leftrightarrow{\rm
Extended\;\,Carroll\;AdS}_{2}$ (21) $\displaystyle{\rm
Extended\;\,NH}_{2}^{-}\leftrightarrow{\rm Extended\;\,Carroll\;dS}_{2}\,.$

Then, the extended (A-)dS2 Carroll algebra is defined as

$[G,P]=H\,,~{}~{}~{}~{}~{}~{}~{}[G,H]=M\,,~{}~{}~{}~{}~{}~{}~{}[H,P]=-\Lambda_{\scriptscriptstyle
C}G\,,$ (22)

where $\Lambda_{\scriptscriptstyle C}=-\Lambda_{\scriptscriptstyle NR}$. The
gravity theories invariant under these two algebras are classically related
through the interchange of its fields: $\tau\leftrightarrow e$ and
$\eta\leftrightarrow\rho$. Hence, the Carroll action can be obtained from the
NR one (13), yielding

$S_{\scriptscriptstyle C}=\gamma_{0}\,\int\Bigg{(}\phi R_{\scriptscriptstyle
C}(G)+\Lambda_{\scriptscriptstyle C}\bigg{(}\rho R_{\scriptscriptstyle
C}(M)+\,\zeta R_{\scriptscriptstyle C}(P)-\eta R_{\scriptscriptstyle
C}(H)\bigg{)}\Bigg{)}\,,$ (23)

with the Carroll curvature two-forms given by

$\displaystyle R_{\scriptscriptstyle C}(H)$ $\displaystyle=$ $\displaystyle
d\tau+\omega e\,,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}R_{\scriptscriptstyle
C}(P)=de\,,$ (24) $\displaystyle R_{\scriptscriptstyle C}(G)$ $\displaystyle=$
$\displaystyle d\omega-\Lambda_{\scriptscriptstyle C}\tau
e\,,~{}~{}~{}~{}~{}~{}R_{\scriptscriptstyle C}(M)=dm+\omega\tau\,.$ (25)

The action (23) is invariant under infinitesimal Carroll gauge transformations

$\displaystyle\delta_{\scriptscriptstyle C}\tau_{\mu}$ $\displaystyle=$
$\displaystyle\partial_{\mu}\lambda-\Theta e_{\mu}+\omega_{\mu}\xi,$ (26a)
$\displaystyle\delta_{\scriptscriptstyle C}e_{\mu}$ $\displaystyle=$
$\displaystyle\partial_{\mu}\xi\,,$ (26b)
$\displaystyle\delta_{\scriptscriptstyle C}\omega_{\mu}$ $\displaystyle=$
$\displaystyle\partial_{\mu}\Theta+\Lambda_{\scriptscriptstyle
C}(e_{\mu}\lambda-\tau_{\mu}\xi)\,,$ (26c)
$\displaystyle\delta_{\scriptscriptstyle C}m_{\mu}$ $\displaystyle=$
$\displaystyle\partial_{\mu}\Gamma+\omega_{\mu}\lambda-\tau_{\mu}\Theta\,,$
(26d) $\displaystyle\delta_{\scriptscriptstyle C}\eta$ $\displaystyle=$
$\displaystyle\xi\phi-\Theta\rho\,,$ (26e)
$\displaystyle\delta_{\scriptscriptstyle C}\rho$ $\displaystyle=$
$\displaystyle 0\,,$ (26f) $\displaystyle\delta_{\scriptscriptstyle C}\zeta$
$\displaystyle=$ $\displaystyle\lambda\phi-\Theta\eta\,,$ (26g)
$\displaystyle\delta_{\scriptscriptstyle C}\phi$ $\displaystyle=$
$\displaystyle\Lambda_{\scriptscriptstyle C}(\lambda\rho-\xi\eta)\,.$ (26h)

The action (23) admits a second-order formulation in the sector
$R_{\scriptscriptstyle C}(H)=de=0$. We refer to it as the electric extended
Carroll gravity theory

$S_{\scriptscriptstyle Elec.Ext.C}=\kappa\int d^{2}x\det{(\tau
e)}\,\phi\,\Bigg{(}\mathcal{R}_{\scriptscriptstyle
C}+2\Lambda_{\scriptscriptstyle C}\Bigg{)}\,,$ (27)

where we have defined

$\mathcal{R}_{\scriptscriptstyle
C}=8e^{[\mu}\,\tau^{\nu]}\,\partial_{\mu}\left(e^{[\alpha}\tau^{\beta]}(\tau_{\nu}\partial_{\alpha}\tau_{\beta}-e_{\nu}\partial_{\alpha}m_{\beta})\right)\,.$
(28)

Unlike the relativistic case, we emphasize that the NR second-order actions
(19) and (27) are not dynamically equivalent to their first-order versions.
This is because the equivalence holds only in the $d\tau=0$ and $de=0$
sectors.

##  3 Electric and magnetic (extended) versions of non-relativistic and
Carroll JT gravity

Recent results in non-Lorentzian field theories reveal that one can derive two
different NR/Carroll field theories invariant under the same symmetry group
Henneaux:2021yzg ; Bergshoeff:2022eog . Following the ideas for the flat field
theories of axelericq , we show a way to obtain the so-called
electric/magnetic gravity theories for NR and Carroll JT gravity. By
considering the second-order formalism, the derivation of these gravity
theories suggests the inclusion of Lagrange multipliers to constrain the
geometry and matter.

###  3.1 Electric extended Newton-Hooke gravity theory

Let us start with the extended Carroll version of second-order JT gravity
(27). We will now show how this Carroll gravity theory, which is invariant in
the torsionless sector, can be modified to contain NH symmetry. The
corresponding is given by

$S_{\scriptscriptstyle Elec.Ext.NH}=\kappa\,\int\,d^{2}x\Bigg{(}\det(\tau
e)\,\phi\,\left(\mathcal{R}_{\scriptscriptstyle
C}+2\Lambda_{\scriptscriptstyle
C}\right)+~{}\partial_{\mu}\pi^{[\mu\nu]}a_{\nu}\Bigg{)}\,,$ (29)

where $\pi^{[\mu\nu]}\equiv 4\,\det(\tau e)\,\phi\,e^{[\mu}\tau^{\nu]}$. We
call (29) the electric extended Newton-Hooke gravity theory. The variation of
the action (27) under NH boosts yields a term proportional to an NH invariant
piece. Due to this, one can also apply the procedure proposed in axelericq .
Indeed, by using the NH gauge transformations for the vector fields

$\displaystyle\delta_{\scriptscriptstyle NH}\tau^{\mu}$ $\displaystyle=$
$\displaystyle
e^{\mu}\Theta-\left(e^{\mu}\partial_{\nu}\xi+e^{\mu}\omega_{\nu}\lambda+\tau^{\mu}\partial_{\nu}\lambda\right)\tau^{\nu}$
(30a) $\displaystyle\delta_{\scriptscriptstyle NH}e^{\mu}$ $\displaystyle=$
$\displaystyle-\left(e^{\mu}\partial_{\nu}\xi+e^{\mu}\omega_{\nu}\lambda+\tau^{\mu}\partial_{\nu}\lambda\right)e^{\nu}\,,$
(30b)

one can easily show that the term $\pi^{[\mu\nu]}$ is an extended NH
invariant. It is manifest that the field $a_{\mu}$ acts as a Lagrange
multiplier enforcing the following conditions for the zweibein $e$, the clock
one-form $\tau$ and the scalar field $\phi$

$\partial_{\mu}\pi^{[\mu\nu]}=0\,,$ (31)

which are obtained varying (29) with respect to $a_{\nu}$. One can verify that
the last NR gravity theory (29) is invariant under boosts of the extended NH
algebra if the auxiliary field $a_{\mu}$ transforms as

$\delta_{\scriptscriptstyle
NH}a_{\mu}=2e^{[\alpha}\tau^{\beta]}\left(\partial_{\alpha}m_{\beta}\tau_{\mu}+\partial_{\alpha}e_{\beta}e_{\mu}\right)\Theta-\tau^{\alpha}\partial_{\alpha}\Theta
e_{\mu}\,.$ (32)

It is also possible to modify (19) to be Carroll invariant by including
Lagrange multipliers. In this way, one arrives at

$S_{\scriptscriptstyle Mag.Ext.C}=\kappa\,\int\,d^{2}x\Bigg{(}\det(\tau
e)\,\phi\,\left(\mathcal{R}_{\scriptscriptstyle
NR}-2\Lambda_{\scriptscriptstyle
NR}\right)-\,\partial_{\mu}\pi^{[\mu\nu]}\,b_{\nu}\Bigg{)}\,,$ (33)

where we have introduced the Lagrange multiplier $b_{\mu}$ transforming under
boosts of the extended Carroll algebra as

$\delta_{\scriptscriptstyle
C}b_{\mu}=2\tau^{[\alpha}e^{\beta]}\left(\partial_{\alpha}m_{\beta}e_{\mu}+\partial_{\alpha}\tau_{\beta}\tau_{\mu}\right)\Theta-e^{\alpha}\partial_{\alpha}\Theta\tau_{\mu}\,.$
(34)

We call (33) the magnetic extended Carroll gravity theory. Note that using the
isomorphism between extended NH and (A-)dS Carroll algebras, one can easily
obtain (33) from (29) by interchanging $\tau\leftrightarrow e$. In the same
way, the gauge transformations of the Lagrange multipliers $a_{\mu}$ and
$b_{\mu}$ are related through the same mapping. This new action is invariant
under the extended Carroll algebra.

##  4 Electric and magnetic non-relativistic and Carroll JT gravity

###  4.1 NH Electric version

In $(1+1)$-spacetime dimensions, the Carroll algebra without a central
extension admits a degenerate invariant bilinear form $\langle
P,P\rangle=\gamma_{0}\Lambda_{\scriptscriptstyle C}$. The BF theory that one
can built from the two-dimensional Carroll algebra is

$S_{\scriptscriptstyle C}[\rho,e]=\int\langle
B,F\rangle=\gamma_{0}\Lambda_{\scriptscriptstyle
C}\int\rho\,R_{\scriptscriptstyle C}(P)\,,$ (35)

with $B=\eta H+\rho P+\phi G,F=dA+A^{2},A=\tau H+eP+\omega G$, with $H,P,G$
satisfying the commutation relations (22). As in the previous section, we can
build an action invariant under the complete NH algebra from (35). Demanding
this condition, it is necessary to introduce specific auxiliary fields
ensuring that the last action satisfies this requirement. This action takes
the form

$S_{\scriptscriptstyle NH}=S_{\scriptscriptstyle
C}\,+\,\gamma_{0}\int\Bigg{(}\phi R_{\scriptscriptstyle
NR}(G)+\,\Lambda_{\scriptscriptstyle NR}\Bigg{(}\alpha\eta+\beta
R_{\scriptscriptstyle NR}(H)-\rho\omega\tau\Bigg{)}\Bigg{)}\,,$ (36)

where the NR curvatures $R_{\scriptscriptstyle NR}(H),R_{\scriptscriptstyle
NR}(P)$, and $R_{\scriptscriptstyle NR}(G)$, are defined in Eq. (14). The two-
form $\alpha$ and the scalar $\beta$ are new auxiliary fields. The invariance
of this new action under the unextended NH algebra requires choosing
$\Lambda_{\scriptscriptstyle C}=-\Lambda_{\scriptscriptstyle NR}$, and the
following symmetry transformations for the auxiliary fields

$\displaystyle\delta_{\scriptscriptstyle NH}\alpha$ $\displaystyle=$
$\displaystyle\xi R_{\scriptscriptstyle NR}(G)-\Theta R_{\scriptscriptstyle
NR}(P)$ (37) $\displaystyle\delta_{\scriptscriptstyle NH}\beta$
$\displaystyle=$ $\displaystyle\xi\phi-\Theta\rho\,.$ (38)

We stress that (37) and (38) are not the transformation laws of a BF gauge
theory. They imply that $\alpha$ and $\beta$ are not part of a gauge
connection that allows the construction of a gauge theory.
Unlike in the previous section, for Carroll gravities without the field $m$,
we cannot solve algebraically for $\omega$ in terms of the remaining fields
and their derivatives since the field equations of (36) do not fix a specific
component of $\omega$. Then, we can solve partially and analytically the spin
connection $\omega$ from the field equation $R_{\scriptscriptstyle NR}(P)=0$
by projecting along the $\tau$ direction; we get

$\omega_{\mu}=\omega_{\perp\mu}+\omega_{\parallel\mu}\,,$ (39)

where

$\omega_{\perp\mu}=\left(2\tau^{[\alpha}e^{\beta]}\partial_{\alpha}e_{\beta}\right)e_{\mu}\,,$
(40)

which is clearly orthogonal to $\tau^{\mu}$ since $e_{\mu}\tau^{\mu}=0$. In
the second order formulation, we will interpret the indeterminate part
$\omega_{\parallel\mu}$ as a Lagrange multiplier Bergshoeff:2017btm . In the
torsionless sector $d\tau=0$ with $\eta=0$, and integrating by parts, we find
the electric NH gravity theory

$S_{\scriptscriptstyle{\it Elec.NH}}=\mathring{S}_{\scriptscriptstyle
C}\,+\,\kappa\int
d^{2}x\,\partial_{\mu}\pi^{[\mu\nu]}\,\omega_{\scriptscriptstyle\parallel\nu}\,,$
(41)

with

$\mathring{S}_{\scriptscriptstyle C}=\kappa\int d^{2}x\,\det(\tau
e)\,\phi\left(\mathring{\mathcal{R}}_{\scriptscriptstyle
C}~{}+2\Lambda_{\scriptscriptstyle C}\right)\,,$ (42)

where we introduced the Carroll-invariant Ricci scalar
$\mathring{\mathcal{R}}_{\scriptscriptstyle C}$, written in terms of
$\omega_{\perp\mu}$

$\mathring{\mathcal{R}}_{\scriptscriptstyle
C}:=4\tau^{[\mu}e^{\nu]}\partial_{\mu}\omega_{\perp\nu}=8\tau^{[\mu}e^{\nu]}\partial_{\mu}\left(\tau^{[\alpha}e^{\beta]}\partial_{\alpha}e_{\beta}e_{\nu}\right)\,.$
(43)

The action (41) written in this form allows for efficiently checking the
invariance of each term. In fact, the first term
$\mathring{S}_{\scriptscriptstyle C}$ is Carroll-boost invariant. We refer to
(42) as the electric Carroll gravity action. This theory coincides with (19)
for $m_{\mu}=0$ and identifying $\Lambda_{\scriptscriptstyle
C}=-\Lambda_{\scriptscriptstyle NR}$. Therefore, there exists an equivalence
between magnetic extended NH theory and electric Carroll theory when the extra
$U(1)$ field vanishes.
On the other hand, as in the previous section, the term
$\pi^{[\mu\nu]}=4\,\det(\tau e)\,\phi\,e^{[\mu}\tau^{\nu]}$ is Carroll-boost
invariant. The invariance of $S_{\scriptscriptstyle{\it Elec.NH}}$ under the
NH symmetries implies to that the indeterminate part
$\omega_{\scriptscriptstyle\parallel}$ of the spin connection transforms as

$\delta_{\scriptscriptstyle
NH}\omega_{\parallel\mu}=e^{\alpha}e_{\mu}\partial_{\alpha}\Theta-2\tau^{[\alpha}e^{\beta]}\left(e_{\mu}\partial_{\alpha}\tau_{\beta}+\tau_{\mu}\partial_{\alpha}e_{\beta}\right)\Theta\,.$
(44)

We can interpret $\omega_{\scriptscriptstyle\parallel}$ as a Lagrange
multiplier constraining the geometry and matter of the theory. The variation
of the action $S_{\scriptscriptstyle{\it Elec.NH}}$ with respect to
$\omega_{\scriptscriptstyle\parallel}$ implies the constraint (31) previously
obtained.

###  4.2 Carroll magnetic version

An analogue procedure can be followed to obtain the magnetic version of
Carroll gravity theory via the mapping between NH and Carroll algebras.
Starting with the BF theory based on the degenerate bilinear form $\langle
H,H\rangle=-\gamma_{0}\Lambda_{\scriptscriptstyle NR}$ of the NH algebra, a
new Carroll-invariant theory can be constructed.
The magnetic Carroll gravity theory, obtained by interchanging $\tau$ and $e$
in (41) is given by

$S_{\scriptscriptstyle Mag.C}=\mathring{S}_{\scriptscriptstyle NH}-\kappa\int
d^{2}x\,\partial_{\scriptscriptstyle\mu}\pi^{[\mu\nu]}\,\omega_{\scriptscriptstyle\parallel\nu}\,,$
(45)

where the magnetic NH gravity theory is defined as

$\mathring{S}_{\scriptscriptstyle NH}=\kappa\int d^{2}x\,\det(\tau
e)\,\phi\left(\mathring{\mathcal{R}}_{\scriptscriptstyle
NR}~{}-2\Lambda_{\scriptscriptstyle NR}\right)\,,$ (46)

and the NH invariant Ricci scalar is

$\mathring{\mathcal{R}}_{\scriptscriptstyle
NR}:=8e^{[\mu}\tau^{\nu]}\partial_{\mu}\left(e^{[\alpha}\tau^{\beta]}\partial_{\alpha}\tau_{\beta}\tau_{\nu}\right)\,.$
(47)

The magnetic terminology for $\mathring{S}_{\scriptscriptstyle NH}$ is
justified because it does not present extra fields. Again, we find a
coincidence between the electric extended Carroll theory and magnetic NH
theory for $m_{\mu}=0$ and $\Lambda_{\scriptscriptstyle
NR}=-\Lambda_{\scriptscriptstyle C}$. For this case, the invariance of (45)
under Carroll symmetry implies the following transformation for the
indeterminate part of the spin connection (orthogonal to $\tau^{\mu}$)

$\delta_{\scriptscriptstyle
C}\omega_{\parallel\mu}=\tau^{\alpha}\tau_{\mu}\partial_{\alpha}\Theta-2e^{[\alpha}\tau^{\beta]}\left(\tau_{\mu}\partial_{\alpha}e_{\beta}+e_{\mu}\partial_{\alpha}\tau_{\beta}\right)\Theta\,.$
(48)

Let us mention that the symmetry transformations of the gravity theories found
here do not correspond to the ones found in the previous section imposing
$m_{\mu}=0$.

##  5 Conclusions and Outlook

The mapping between non-relativistic and Carroll field theories axelericq  has
been extended to the case of JT gravity. Here, we have shown that the second
order non-relativistic and Carroll JT gravity (depending on the vielbein
variables $\tau,e,$ and the $U(1)$ vector field $m$ introduced in
Gomis:2020wxp ) can be deformed with two more theories introducing Lagrange
multipliers that restrict the dynamics of the previous theories. However, they
have the same symmetry algebras; the only difference with the gravity theories
introduced in Gomis:2020wxp , is the transformation law for the Lagrange
multipliers. The basis of this construction is due to the isomorphism between
NH and the Carroll algebras.

It will be interesting to know the number of physical degrees of freedom of
the eight theories we have constructed.

Our results suggest a deep connection between the physics of non-relativistic
and Carroll gravities at least in $(1+1)$-dimensions. The extension of our
results to the gravity theories in more dimensions is an interesting path to
follow. The applications to other physical theories would also be useful.

### Acknowledgments

We acknowledge interesting discussions about the relation between Galilean and
Carroll theories with Roberto Casalbuoni and Jose Figueroa-O’Farrill. We also
thank Eric Bergshoeff and Axel Kleinschmidt for discussions in a related
project. The work of J.Z and L.A was partially supported by Fondecyt grants
122086 and 3220805, respectively. The work of JG has been supported in part by
MINECO FPA2016-76005-C2-1-P and PID2019-105614GB-C21 and from the State Agency
for Research of the Spanish Ministry of Science and Innovation through the
Unit of Excellence Maria de Maeztu 2020-203 award to the Institute of Cosmos
Sciences (CEX2019-000918-M). DH was supported by the Icelandic Research Fund
grant 228952-051.

##  Appendix A Symmetry of electric Newton-Hooke JT gravity

In this appendix, we explicitly check the invariance of the action (41) under
the extended Newton-Hooke boosts. To do this, we should note that
$\mathring{S}_{\scriptscriptstyle C}$ can be rewritten, up to boundary terms,
as

$\displaystyle\mathring{S}_{\scriptscriptstyle C}$ $\displaystyle=$
$\displaystyle\kappa\int d^{2}x\,\det(\tau
e)\,\phi\left(\mathring{\mathcal{R}}_{\scriptscriptstyle
C}~{}+2\Lambda_{\scriptscriptstyle C}\right)$ (49) $\displaystyle=$
$\displaystyle\kappa\int d^{2}x\,\det(\tau
e)\,\phi\left(4\tau^{[\mu}e^{\nu]}\partial_{\mu}\omega_{\perp\nu}~{}+2\Lambda_{\scriptscriptstyle
C}\right)$ $\displaystyle=$ $\displaystyle\kappa\int
d^{2}x\partial_{\mu}\pi^{[\mu\nu]}\omega_{\perp\nu}~{}\,+\,2\Lambda_{\scriptscriptstyle
C}\kappa\int d^{2}x\det(\tau e)\phi\,.$

Then, the invariance of $\mathring{S}_{\scriptscriptstyle C}$ is seen as
follows: the second term in the last equation is invariant under NH boost
symmetry ($\delta_{\scriptscriptstyle NH}$) since $\delta_{\scriptscriptstyle
NH}\phi=0$ and $\delta_{\scriptscriptstyle NH}\det(\tau e)=0$. Also, from the
transformation (30) and $\delta_{\scriptscriptstyle NH}\phi=0$ one can show

$\delta_{\scriptscriptstyle NH}\left(\partial_{\mu}\pi^{[\mu\nu]}\right)=0\,.$
(50)

So, under NH boosts of $\mathring{S}_{\scriptscriptstyle C}$ transforms as

$\delta_{\scriptscriptstyle NH}\mathring{S}_{\scriptscriptstyle C}=\kappa\int
d^{2}x\,\partial_{\mu}\pi^{[\mu\nu]}\delta_{\scriptscriptstyle
NH}\omega_{\perp\nu}\,.$ (51)

Therefore, the invariance of the action (41) yields

$\delta_{\scriptscriptstyle NH}S_{\scriptscriptstyle{\it Elec.NH}}=\kappa\int
d^{2}x\,\partial_{\mu}\pi^{[\mu\nu]}\delta_{\scriptscriptstyle
NH}\omega_{\perp\nu}\,+\,\kappa\int
d^{2}x\,\partial_{\mu}\pi^{[\mu\nu]}\,\delta_{\scriptscriptstyle
NH}\omega_{\parallel\nu}\,,$

which is satisfied if

$\delta_{\scriptscriptstyle NH}\omega_{\perp}=-\delta_{\scriptscriptstyle
NH}\omega_{\scriptscriptstyle\parallel}\,.$ (52)

The same procedure can be applied to the other gravity theories.

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