Conformal Carrollian Spin-3 Gravity in 3d

Iva Lovrekovic

Institute for Theoretical Physics, Technische Universität Wien,
Wiedner Hauptstrasse, 8-10,
1040, Wien, Austria

###### Abstract

We consider ultra-relativistic limit of the spin-2 and spin-3 conformal
gravity theories in three dimensions, which leads to conformal Carrollian
spin-2 gravity and its spin-3 analog. We also comment on the generalization of
the result to arbitrary spin. The holographic examples show generalization in
comparison with non-conformal Carrollian gravity.

##  1 Introduction

The motivation for the research of conformal Carrollian Spin-3 Gravity in 3d
comes from several sides. Conformal symmetry has been studied from number of
aspects, as a symmetry that brings extra simplifications, defines gravity
models as a toy models using which one can resolve otherwise unresolvable
issues, or as a symmetry that plays a key role in the Physics at the Planck’s
energy scale [1].

Carroll symmetries have been studied due to the connection with asymptotic
symmetries of flat-space times described by Bondi-Metzner-Sachs (BMS) group
and their relation with near horizon boundary conditions [2, 3]. BMS
symmetries are isomorphic to conformal Carrollian symmetries [4, 5], and there
is also a very interesting picture which relates soft theorems and asymptotic
symmetries in QFT in asymptotically flat space-times with memory effects.
Review of advances in this field can be found in [6]. Carroll algebra can be
obtained as an ultra-relativistic limit of Poincare algebra, and similarly
conformal version follows from conformal algebra [7]. The $c\to 0$ limit is
depicted in a light cone structure which has a light cone collapsed into a
line [8]. The Carrollian space-times are formed in future and past null-
infinity and conformal Carroll algebra can be viewed as conformal extension of
BMS algebra. In the near horizon boundary conditions one obtains BMS algebra
as a composite in the terms of infinite copies of Heisenberg algebra.

Carroll symmetries have recently been studied in relation with higher spin
theories [9] with the aim of filling the gap of the less developed field of
the higher spin extensions of the Poincare Algebra in higher than three
dimensions. In three dimensions the situation is better, where number of
constructions has been considered [10, 11, 12]. At present, there is a handful
of complete higher spin gravity models that have actions and avoid the non-
locality issue: (a) $3d$ massless [10, 13, 14]; (b) $4d$ conformal [15, 16];
(c) $4d$ chiral [17] and (d) $3d$ conformal [18].

Higher spin gravities in three dimensions can be constructed consistently
while containing a finite number of higher spin-fields. In fact, one can
construct infinite number of such theories [19]. In the relativistic theory,
this is allowed by considering the action of Chern-Simons form which has also
been used in construction of the non-conformal Carrollian theories [20]. Since
higher spin theories usually contain infinite number of higher spin fields and
only several consistent theories are known, it is our aim to use one of them
[19] and investigate what can the ultra-relativistic limit teach us.

In the construction of the gauge theories based on the Carrollian algebras,
the ultra-relativistc limit of General relativity in the first order
formulation gives equations of motion which are not sufficient to find all the
components of the connection fields expressed in terms of the other fields.
Those components play a role of Lagrange multipliers for the constraints on
the geometry [21], and carry a physical interpretation in terms of radiative
degrees of freedom for gravity at null-distances [22, 23]. Which is what we
can expect to appear here as well.

The paper is structured as follows: In the first section we review the
construction using which we obtain spin-2 and spin-3 cases of conformal
gravities, we review conformal gravity in three dimensions, i.e. linearization
of the theory and the ultra-relativistic limit. In the third section we
analyze the ultra-relativistic limit of the spin-2 case while in the fourth
section we analyze the spin-3 case and comment generalization to arbitrary
spin. In the fifth section we consider the holography of the theories, and in
the sixth section we conclude.

##  2 Construction

We study Chern-Simons action

$\displaystyle S[\omega]$ $\displaystyle=\int{\mathrm{Tr}\,}\left[\omega\wedge
d\omega+\frac{2}{3}\omega\wedge\omega\wedge\omega\right]\,.$ (2.1)

as a gauge theory for the Carollian conformal algebra. Knowing the higher spin
extension of $so(3,2)$, (i.e. algebras that have $so(3,2)$ as a subalgebra and
also contain other nontrivial representations of $so(3,2)$), we can take the
ultra-relativistic limit that gives conformal Carrollian algebra and consider
their generalization to the higher spin.

To construct finite spectrum of higher spin fields (called Fradkin-Tseytlin
fields [24]) one needs to take a nontrivial finite-dimensional irreducible
representation $V$ of $so(3,2)$, such as an irreducible tensor or a spin-
tensor. Then one needs to evaluate $U(so(d,2))$ in $V$, which means multiply
the generators 111Here, indices $A,B,...=0,...,4$ denote the indices of
$so(3,2)$ while $\eta_{AB}$ denotes invariant metric. $T_{AB}=-T_{BA}$ of
$so(3,2)$ in given representation and determine the generated algebra [19]. In
general, this algebra is denoted with $\mathfrak{hs}(V)$. Here we are going to
consider spin-2 and spin-3 fields. For that case, one needs to take the vector
representation, which is denoted by one-cell Young diagram
\begin{picture}(10.0,10.0){}\end{picture} . Then, the spectrum of the
corresponding algebra,
$\mathfrak{hs}(\parbox{10.0pt}{\begin{picture}(10.0,10.0){}\end{picture}})$ is
the following:

$\displaystyle\mathfrak{hs}(\parbox{10.0pt}{\begin{picture}(10.0,10.0){}\end{picture}})$
$\displaystyle=\parbox{10.0pt}{\begin{picture}(10.0,10.0){}\end{picture}}\otimes\parbox{10.0pt}{\begin{picture}(10.0,10.0){}\end{picture}}=\parbox{10.0pt}{\begin{picture}(10.0,20.0){}\end{picture}}\oplus\parbox{20.0pt}{\begin{picture}(20.0,10.0){}\end{picture}}\oplus\bullet.$
(2.2)

This algebra is a matrix algebra with $(3+2)^{2}$ generators $t{}_{A}{}^{B}\,$
which are decomposed with respect to $so(3,2)$. The $gl_{3+2}$ commutation
relations

$\displaystyle[t{}_{A}{}^{B}\,,t{}_{C}{}^{D}\,]$
$\displaystyle=-\delta{}_{A}{}^{D}\,t{}_{C}{}^{B}\,+\delta{}_{C}{}^{B}\,t{}_{A}{}^{D}\,\,.$
(2.3)

with the irreducible generators $T_{AB}=-T_{BA}$, $S_{AB}=S_{BA}$ and $R$, in
the $so(3,2)$ base

$\displaystyle T_{AB}$ $\displaystyle=t_{A|B}-t_{B|A}\,,$ $\displaystyle
S_{AB}$
$\displaystyle=t_{A|B}+t_{B|A}-\frac{2}{d+2}t{}_{C}{}^{C}\,\eta_{AB}\,,$
$\displaystyle R=t{}_{C}{}^{C}\,\,,$ (2.4)

lead to the commutation relations

$\displaystyle[T_{AB},T_{CD}]$
$\displaystyle=\eta_{BC}T_{AD}-\eta_{AC}T_{BD}-\eta_{BD}T_{AC}+\eta_{AD}T_{BC}\,,$
(2.5a) $\displaystyle[T_{AB},S_{CD}]$
$\displaystyle=\eta_{BC}S_{AD}-\eta_{AC}S_{BD}+\eta_{BD}S_{AC}-\eta_{AD}S_{BC}\,,$
(2.5b) $\displaystyle[S_{AB},S_{CD}]$
$\displaystyle=\eta_{BC}T_{AD}+\eta_{AC}T_{BD}+\eta_{BD}T_{AC}+\eta_{AD}T_{BC}\,.$
(2.5c)

Here $R$ commutes with everything because it is associated with $1$ in
$gl(V)$. So it can be truncated away. One obtains a theory that contains two
fields:

$\displaystyle\omega$ $\displaystyle=\omega^{A,B}T_{AB}+\omega^{AB}S_{AB}$
(2.6)

$T_{AB}$ is the conformal graviton and $S_{AB}$ is a field similar to the
spin-three partially-massless field. From the bilinear form of $gl_{n}$

$\displaystyle{\mathrm{Tr}\,}(t{}_{A}{}^{B}\,,t{}_{C}{}^{D}\,)$
$\displaystyle=\delta{}_{A}{}^{D}\,\delta{}_{C}{}^{B}\,$ (2.7)

one obtains invariant bilinear forms for $so(3,2)$

$\displaystyle{\mathrm{Tr}\,}(T_{AB},T_{CD})$
$\displaystyle=2(\eta_{AD}\eta_{CB}-\eta_{BD}\eta_{CA})$ (2.8)
$\displaystyle{\mathrm{Tr}\,}(S_{AB},S_{CD})$
$\displaystyle=2(\eta_{AD}\eta_{CB}+\eta_{BD}\eta_{CA}-\frac{2}{d+2}\eta_{AB}\eta_{CD})$
(2.9)

Before proceeding to linearizing a theory and taking the ultra-relativistic
limit, let us first remember conformal gravity as a gauge theory in three
dimensions, studied in [25]. Fixing the commutation relations of the $so(3,2)$
conformal algebra (for $a=0,1,2$)

$\displaystyle[D,P^{a}]$ $\displaystyle=-P^{a}\,,$
$\displaystyle[J^{ab},P^{c}]$ $\displaystyle=P^{a}\eta^{bc}-P^{b}\eta^{ac}\,,$
(2.10a) $\displaystyle[D,K^{a}]$ $\displaystyle=K^{a}\,,$
$\displaystyle[J^{ab},K^{c}]$ $\displaystyle=K^{a}\eta^{bc}-K^{b}\eta^{ac}\,,$
(2.10b) $\displaystyle[P^{a},K^{b}]$ $\displaystyle=-J^{ab}+\eta^{ab}D\,,$
$\displaystyle[J^{ab},J^{cd}]$
$\displaystyle=J^{ad}\eta^{bc}-J^{ac}\eta^{bd}-J^{bd}\eta^{ac}+J^{bc}\eta^{ad},,$
(2.10c)

for $P^{a}$ translations, $K^{a}$ special conformal tranformations, $J^{ab}$
Lorentz boosts and $D$ dilatation, we can write the connection for the
$so(3,2)$ as

$\displaystyle\omega$
$\displaystyle=\frac{1}{2}\varpi^{a,b}J_{ab}+e^{a}P_{a}+f^{a}K_{a}+bD.$ (2.11)

The action, is a standard Chern-Simons action

$\displaystyle S[\omega]$ $\displaystyle=\int{\mathrm{Tr}\,}\left[\omega\wedge
d\omega+\frac{2}{3}\omega\wedge\omega\wedge\omega\right]\,$ (2.12)

where the curvature $F=d\omega+\frac{1}{2}[\omega,\omega]$ is set to zero by
the equations. The components of the curvature give

$\displaystyle F[P^{a}]$ $\displaystyle=\nabla e^{a}-b\wedge e^{a},$ (2.13a)
$\displaystyle F[D]$ $\displaystyle=\nabla b+e_{m}\wedge f^{m},$ (2.13b)
$\displaystyle F[J^{a,b}]$ $\displaystyle=R^{a,b}-e^{a}\wedge
f^{b}+e^{b}\wedge f^{a},$ (2.13c) $\displaystyle F[K^{a}]$
$\displaystyle=\nabla f^{a}+b\wedge f^{a},$ (2.13d)

where we denote the Lorentz covariant derivative with $\nabla=d+\varpi$. In
this notation Riemann two-form is given by
$R^{a,b}=d\omega^{a,b}+\omega{}^{a,}{}_{c}\,\wedge\omega^{c,b}$. To show how
one obtains the theory when $\omega$ from the action (2.12) is only a spin
connection and $e_{\mu}{}^{a}$ only dynamical variable, we have to solve the
equations and fix the gauge conditions. Our gauge parameter is a $so(3,2)$
algebra valued 0-form,

$\displaystyle\Xi$
$\displaystyle=\frac{1}{2}\eta^{a,b}L_{ab}+\xi^{a}P_{a}+\zeta^{a}K_{a}+\rho D$
(2.14)

which gives

$\displaystyle\delta e^{a}$ $\displaystyle=\nabla\xi^{a}-\rho\wedge
e^{a}-b\wedge\xi^{a}+\eta^{a,}{}_{b}\wedge e^{a},$ (2.15a) $\displaystyle\
\delta b$ $\displaystyle=\nabla\rho+\xi_{m}\wedge f^{m}+e_{m}\wedge\zeta^{m},$
(2.15b) $\displaystyle\delta\omega^{a,b}$
$\displaystyle=\nabla\eta^{a,b}-\xi^{a}\wedge f^{b}+\xi^{b}\wedge
f^{a}-e^{a}\wedge\zeta^{b}+e^{b}\wedge\zeta^{a},$ (2.15c) $\displaystyle\delta
f^{a}$ $\displaystyle=\nabla\zeta^{a}+b\wedge\zeta^{a}+\rho\wedge
f^{a}+\eta^{a,}{}_{b}\wedge f^{a}\,.$ (2.15d)

If we assume that the dreibein $e^{a}_{{\mu}}$ is invertible, we can fix the
gauge to set $b=0$. Now, one can write
$g_{\mu\nu}=e^{a}_{\mu}e^{b}_{\nu}\eta_{ab}$ for the conformal metric. From
the equation $F[D]=0$ we obtain that $f^{a}_{\mu}e_{a\nu}$ is symmetric. The
equation $F[P^{a}]=0$ we recognize as the standard torsion constraint, which
defines $\varpi^{a,b}$ using $e^{a}$. The following equation $F[J^{a,b}]=0$
defines that $f^{a}_{\mu}e_{a\nu}$ is the Schouten tensor, while the
$F[K^{a}]=0$ sets Cotton tensor $C_{\mu\nu}=0$. That equation is the only
equation which is dynamical. If we plug these solutions for the fields back
into the action (2.12) we obtain the action of the same form, where the only
dynamical variable is $e^{a}$.

Here, we want to determine the fields of this conformal gravity theory for the
conformal graviton and a spin three field in an ultrarelativistic limit, i.e.
when we have conformal Carrollian theory. To find the spectrum of the fields,
one has to linearize a theory over the Minkowski vacuum. We choose it to be
$\omega_{0}=h^{a}P_{a}$ for $h^{a}=h_{\mu}{}^{a}dx^{\mu}$, where the
background metric is $h_{\mu}{}^{a}=\delta_{\mu}{}^{a}$. The equations at the
linear order and the linearized gauge symmetries are

$\displaystyle d\omega+\omega_{0}\wedge\omega+\omega\wedge\omega_{0}=0,$
$\displaystyle\delta\omega=d\xi+[\omega_{0},\xi].$ (2.16)

Here, $\omega_{0}$ is the background field, while $\omega$ field is the Lie
algebra valued one-form. It is valued in one of the algebras of the so(3,2)
irreducible modules obtained from the decomposition of the $\mathfrak{hs}(V)$.
In our case it will be valued in the ultra-relativistic limit of the algebra
we consider: in the ultra-relativistic limit of the conformal algebra and in
the ultra-relativistic limit of the extension of the conformal algebra for
spin-three field described above.

For the generators $t^{\Lambda}$ the equations (2.16), with
$\omega=\omega^{\Lambda}t_{\Lambda}$ and $\xi=\xi^{\Lambda}t_{\Lambda}$, will
give

$\displaystyle
d\omega^{\Lambda}t_{\Lambda}+h^{a}\wedge\omega^{\Lambda}[P_{a},t_{\Lambda}]=0$
$\displaystyle\delta\omega^{\Lambda}t_{\Lambda}=d\xi^{\Lambda}t_{\Lambda}+h^{a}\wedge\xi^{\Lambda}[P_{a},t_{\Lambda}]$
(2.17)

where $t^{\Lambda}$ denotes the generators of the so(3,2) irreducible modules
$T_{AB}$, $S_{AB}$ etc., $\Lambda$ denotes indices in the light cone
coordinates $\Lambda=a,+,-$, and $\eta^{AB}$ is corresponding metric with
$\eta_{+-}=\eta_{-+}=1$.

To take the ultra-relativistic limit we are going to follow the procedure of
taking Inönü-Wigner (IW) contraction [26]. In our terminology we do this as
follows. We denote the Lie algebra from which we start with $\mathfrak{g}$,
while $\mathfrak{h}$ is a subalgebra. The decomposition
$\mathfrak{g}=\mathfrak{h}+\mathfrak{i}$ denotes a direct sum of vector
spaces. The generators of the ideal $\mathfrak{i}$, are rescaled with
contraction parameter $\epsilon$, such that
$\mathfrak{i}\rightarrow\epsilon\mathfrak{i}$. The commutation relations then
take the form

$\displaystyle[\mathfrak{h},\mathfrak{h}]=\mathfrak{h}\,,$
$\displaystyle[\mathfrak{h},\mathfrak{i}]=\frac{1}{\epsilon}\mathfrak{h}+\mathfrak{i}\,,$
$\displaystyle[\mathfrak{i},\mathfrak{i}]=\frac{1}{\epsilon^{2}}\mathfrak{h}+\frac{1}{\epsilon}\mathfrak{i}$
(2.18)

which means that when taking $\epsilon\to\infty$, one obtains well defined
limit. For this to hold, $\mathfrak{h}$ needs to be subalgebra of
$\mathfrak{g}$. If one had
$[\mathfrak{h},\mathfrak{h}]=\mathfrak{h}+\epsilon\mathfrak{i}$ the limit when
$\epsilon\to\infty$ would not be well defined.

##  3 Spin-2

Let us focus on the first case with a Young module
\begin{picture}(10.0,10.0){}\end{picture} for conformal graviton when
$t_{ab}=-t_{ab}$. The dictionary between the generators $T_{AB}$ for the
conformal graviton and generators of translations, special conformal
transformations, dilatation and Lorentz boosts is $P_{a}=T_{a+}$,
$K_{a}=T_{a-}$, $D=-T_{+-}$, $L_{ab}=T_{ab}$ respectively. To obtain the
generators in the ultra-relativistic limit from the generators $t^{\Lambda}$,
we perform an IW contraction [26] of the underlying algebra, in a way that
leads to the conformal Carrollian algebra.

We introduce space-time splitting of indices where $a=\\{0,i;i=1,2\\}$. That
allows us to introduce the notation

$\displaystyle P_{0}\equiv H\,,$ $\displaystyle K_{0}\equiv K_{0}\,,$
$\displaystyle J_{0i}\equiv B_{i}$ (3.1)

When we choose the subalgebra $\mathfrak{h}$ and ideal $\mathfrak{i}$ as

$\displaystyle\mathfrak{h}=\\{P_{i},K_{i},D,J_{ij}\\}\,,$
$\displaystyle\mathfrak{i}=\\{H,K_{0},B_{i}\\}$ (3.2)

starting from the conformal algebra (2.10c), we obtain the algebra

$\displaystyle[D,H]$ $\displaystyle=-H\,,$ $\displaystyle[D,K_{0}]$
$\displaystyle=K_{0}\,,$ (3.3a) $\displaystyle[B^{j},P^{k}]$
$\displaystyle=H\eta^{jk}\,,$ $\displaystyle[B^{j},K^{k}]$
$\displaystyle=K^{0}\eta^{jk}$ (3.3b) $\displaystyle[H,K^{j}]$
$\displaystyle=-B^{j}\,,$ $\displaystyle[P^{i},K^{0}]$
$\displaystyle=B^{i}\,,$ (3.3c) $\displaystyle[J^{ij},B^{l}]$
$\displaystyle=B^{i}\eta^{jl}-B^{j}\eta^{il},$ (3.3d)

i.e., conformal Carrollian algebra [7]. The Lie algebra valued one-form now
becomes

$\displaystyle\omega$ $\displaystyle=e^{i}P_{i}+\tau
H+\tfrac{1}{2}\omega^{i,j}J_{ij}+\beta^{j}B_{j}-bD+f^{i}K_{i}+\kappa K_{0}$
(3.4)

with an analogous decomposition of the gauge parameter

$\displaystyle\xi$
$\displaystyle=e_{i}\xi^{i+}+\tau\xi^{0+}+\tfrac{1}{2}\omega_{i,j}\xi^{i,j}+\beta_{j}\xi^{0j}+b\xi^{+-}+f_{i}\xi^{i-}+\kappa\xi^{0-}.$
(3.5) The linearized equations of the conformal Carrollian gravity and the
gauge symmeteries read $\displaystyle D$ $\displaystyle:$ $\displaystyle db-
h_{m}\wedge f^{m}$ $\displaystyle=0\,,$ $\displaystyle\delta\omega^{+-}$
$\displaystyle=d\xi^{+-}-h_{m}\xi^{m-}\,,$ (3.6a) $\displaystyle P_{i}$
$\displaystyle:$ $\displaystyle de^{i}+h_{m}\wedge\omega^{m,i}-h^{i}\wedge b$
$\displaystyle=0\,,$ $\displaystyle\delta e^{i}$
$\displaystyle=d\xi^{i+}+h_{m}\xi^{m,i}-h^{i}\xi^{+-}\,,$ (3.6b)
$\displaystyle H$ $\displaystyle:$ $\displaystyle d\tau-
h_{i}\wedge\beta^{i}-h^{0}\wedge b$ $\displaystyle=0\,,$
$\displaystyle\delta\tau$
$\displaystyle=d\xi^{0+}+h_{i}\xi^{0,i}-h^{0}\xi^{+-}\,,$ (3.6c)
$\displaystyle t_{ij}$ $\displaystyle:$ $\displaystyle
d\omega^{i,j}-h^{i}\wedge f^{j}+h^{j}\wedge f^{i}$ $\displaystyle=0\,,$
$\displaystyle\delta\omega^{i,j}$
$\displaystyle=d\xi^{i,j}-h^{i}\xi^{j-}+h^{j}\xi^{i-}\,,$ (3.6d)
$\displaystyle B_{j}$ $\displaystyle:$ $\displaystyle
d\beta^{j}+h^{j}\wedge\kappa-h^{0}\wedge f^{j}$ $\displaystyle=0\,,$
$\displaystyle\delta\beta^{j}$
$\displaystyle=d\xi^{0,j}-h^{0}\xi^{j-}+h^{j}\xi^{0-}\,,$ (3.6e)
$\displaystyle K_{i}$ $\displaystyle:$ $\displaystyle df^{i}$
$\displaystyle=0\,,$ $\displaystyle\delta f^{i}$ $\displaystyle=d\xi^{i-}\,.$
(3.6f) $\displaystyle t_{0-}$ $\displaystyle:$ $\displaystyle d\kappa$
$\displaystyle=0\,,$ $\displaystyle\delta\kappa$ $\displaystyle=d\xi^{0-}\,.$
(3.6g)

To find the spectrum of the theory one has to solve this system of equations.
From (3.6a) the gauge invariance allows to fix $b_{m}$ just like in three
dimensions. That defines $\xi^{-}_{m}=\partial_{m}\xi^{+-}$, and makes
$b_{m}=0$. The equation (3.6a) demands that $f_{mn}=f_{nm}$ is symmetric
field. The components $m,0$ of equation (3.6a)
$E[t_{+-}]^{m0}=-h^{ma}f{}^{0}{}_{a}+\partial^{m}b{}^{0}=0$ give

$\displaystyle f^{0}{}^{m}=\partial^{m}b^{0}.$ (3.7)

The gauge transformations (3.6b) allow to fix antisymmetric part of
$e{}_{[m|n]}=0$ and its trace $e{}_{m}{}^{m}=0$ using
$\xi_{m},{}^{n}=-\partial_{m}\xi^{+n}+h_{m}{}^{n}\xi^{+-}$ and
$\xi^{+-}=\frac{1}{2}\partial_{m}\xi^{+m}$ respectively, and the remaining
symmetric part we call $e_{(m,n)}\equiv\phi_{mn}$. The corresponding equation

$\displaystyle
E[t_{i+}]^{mni}=-\omega^{mni}+\omega^{nmi}+\partial^{m}\phi^{ni}-\partial^{n}\phi^{mi}$
(3.8)

can be split into symmetric, antisymmetric and hook part. The only non-trivial
part, hook part, of the equation defines

$\displaystyle\omega^{jm,n}$
$\displaystyle=\partial^{m}\phi^{nj}-\partial^{n}\phi^{mj}\,,$
$\displaystyle\omega^{jm,}{}_{j}=\partial_{j}\phi^{mj}$ (3.9)

while the antisymmetric and symmetric part are exactly equal to zero. The
$m,0$ component of (3.6b) does not allow for any more gauge fixing. The
symmetric part of the equation gives $b^{0}=\frac{1}{2}\partial_{i}e^{0i}$
while the antisymmetric part leads to

$\displaystyle\omega^{0im}=\tfrac{1}{2}(-\partial^{i}e^{0}{}^{m}+\partial^{m}e^{0}{}^{i}).$
(3.10)

From the gauge transformation (3.6c) one can fix $\tau_{m}{}=0$ with
$\xi^{0,}{}_{m}=\partial_{m}\xi^{0+}$. The $m,n$ component of the
corresponding equation $E[t_{0+}]^{mn}=-\beta^{nm}+\beta^{mn}=0$ makes the
field $\beta^{m,n}$ symmetric, while the $m,0$ component defines
$\partial^{m}\tau^{0}{}=\beta^{0}{}^{m}$.

The gauge (3.6d) has all the components of the gauge parameters fixed, which
means the equation is

$\displaystyle
E[t_{ij}]^{mn}=-h^{nj}f{}^{mi}+h^{ni}f{}^{mj}+h^{mj}f{}^{ni}-h^{mi}f{}^{nj}+\partial^{m}\omega^{nij}-\partial^{n}\omega^{mij}=0.$
(3.11)

The symmetric part of the equation exactly vanishes and does not give any
conditions or definitions of the field. The antisymmetric part of the equation
as well vanishes when the definition of $\omega^{ni,j}$ is used. The Riemann
component of the equation can be contracted with the $h_{ij}$ to define

$\displaystyle f^{i}{}_{i}=-\partial_{j}\omega_{i}{}^{ij}.$ (3.12)

The Riemann part itself gives equation constraint on the geometry which we
express in terms of the fields $\beta^{ij},\tau^{0},\phi^{ij}$, in (7.1) the
appendix.

The $m,0$ components of the equation define

$\displaystyle
E[t_{ij}]^{m0}=-\partial^{0}{}\omega^{mij}-f^{0}{}^{j}\eta^{mi}+f^{0}{}^{i}\eta^{mj}+\partial^{m}\omega^{0}{}^{ij}=0$
(3.13)

whose symmetric part is zero, and in the hook part we use $f^{0j}$,
$\omega^{0ij}$ and $\omega^{mij}$ from (3.7),(3.10), (3.9) respectively, and
$\tau_{m}=0$ to obtain second constraint on the geometry

$\displaystyle-\partial_{0}{}\partial^{i}\phi^{jm}+2\partial_{0}{}\partial^{j}\phi^{im}-\partial_{0}{}\partial^{m}\phi^{ij}=0$
(3.14)

Contraction of the constraint with $h_{im}$ gives condition

$\displaystyle\partial_{0}{}\partial_{i}\phi^{ji}=0$ (3.15)

which can be used later if necessary for simplifications. The antisymmetric
part, after insertion of $\omega^{ijl}$, and the value for $\omega^{0}{}^{im}$
turns out to be exact equation.

The equation (3.6e) still has one gauge parameter to fix. One may define
$\beta_{m}{}^{m}=0$ using $\xi^{0-}=-\partial_{m}\xi^{0,m}+h^{0}\xi^{m-}$. The
symmetric and antisymmetric part of (3.6e) will identically vanish, while the
hook part defines

$\displaystyle\kappa^{n}=\partial_{m}\beta^{n}{}^{m}.$ (3.16)

The $m0$ component of the equation

$\displaystyle
E[B^{i}]^{m0}=\partial^{m}\beta^{0}{}^{i}-\partial^{0}\beta^{m}{}^{i}-h^{mi}\kappa^{0}+h_{0}{}^{0}f^{mi}=0$
(3.17)

has antisymmetric part which makes relation
$\partial^{m}\beta_{0}{}^{i}-\partial^{i}\beta_{0}{}^{m}=0$ symmetric, and
symmetric part which allows us to define

$\displaystyle\kappa^{0}{}=-\partial_{k}\partial^{l}\phi_{l}{}^{k}+\partial^{m}\partial_{m}\tau^{0}{}$
(3.18)

the 0-th component of $\kappa_{\mu}$ field in terms of the $\tau^{0}$ and
$\phi^{mn}$. It also allows to find $f^{mn}$

$\displaystyle
f^{mn}=\partial_{0}{}\beta^{mn}+h^{mn}\partial_{k}\partial^{k}\tau^{0}{}-h^{mn}\partial_{l}\partial_{k}\phi^{kl}-\partial^{m}\partial^{n}\tau^{0}{}$
(3.19)

in terms of the fields $\phi^{mn}$, $\tau^{0}$ and $\beta^{mn}$. One can
notice that all the fields can be determined in terms of the components of the
$e_{\mu}{}^{a}$ and $\tau_{\mu}$. Where the remaining component of the
$e_{\mu}{}^{a}$ is symmetric and traceless component while $\tau_{\mu}$ are
the components that belong to H generator, i.e. both originated from
relativistic $P_{a}$ generator of translations. The undetermined field
$\beta^{mn}$ is symmetric and traceless field.

The equations for the $mn$ and $m0$ components of the fields belonging to
special conformal transformations generator $K^{i}$, (3.6f) read

$\displaystyle
2\partial_{0}{}\partial^{m}\beta^{ni}+\eta^{in}(\partial^{m}\partial_{a}\partial^{a}\tau^{0}{}-\partial^{m}\partial_{b}\partial_{a}\phi^{ab})-2\partial_{0}{}\partial^{n}\beta^{mi}-\eta^{im}(\partial^{n}\partial_{a}\partial^{a}\tau^{0}{}+\partial^{n}\partial_{b}\partial_{a}\phi^{ab})$
$\displaystyle=0,$ (3.20a)
$\displaystyle-\partial_{0}\partial_{0}\beta^{mi}+h^{im}\partial_{0}{}(-\partial_{a}\partial^{a}\tau^{0}{}+\partial_{b}\partial_{a}\phi^{ab})+\partial_{0}{}\partial^{i}\partial^{m}\tau^{0}{}+\tfrac{1}{2}\partial^{i}\partial^{m}\partial_{a}e^{0}{}^{a}$
$\displaystyle=0$ (3.20b)

and equations for the $m,n$ and $m,0$ components of the generator for the
special conformal transformations in the 0 direction (3.6g) are

$\displaystyle\partial^{m}\partial_{e}\beta^{ne}-\partial^{n}\partial_{e}\beta^{me}$
$\displaystyle=0$ (3.21)
$\displaystyle-\partial_{0}{}\partial_{e}\beta^{me}+\partial^{m}\partial_{e}\partial^{e}\tau^{0}{}-\partial^{m}\partial_{j}\partial_{e}\phi^{ej}$
$\displaystyle=0$ (3.22)

respectively. The latter can be combined with a (3.20a) into constraint on the
field $\beta^{mn}$,
$2\partial_{0}\partial^{m}\beta^{ni}+\eta^{in}\partial_{0}\partial_{l}\beta^{ml}-2\partial_{0}\partial^{n}\beta^{mi}-\eta^{im}\partial_{0}\partial_{l}\beta^{nl}=0$.
Taking the trace and the constraint $\beta^{m}{}_{m}=0$ one obtains that
$\partial_{0}\partial_{l}\beta^{ml}=0$ vanishes. Inserting that into (3.22)
gives relation between
$\partial^{m}\partial^{l}\partial_{l}\tau^{0}=\partial_{m}\partial_{j}\partial_{l}\phi^{lj}$.
This, combined with $\delta_{im}$ contraction of the equation (7.1) from the
appendix gives dynamical equation for the field $\phi^{mn}$

$\displaystyle\partial_{m}\partial_{i}\partial^{i}\phi^{jn}-\partial^{m}\partial_{i}\partial^{j}\phi^{ni}-\partial^{m}\partial_{i}\partial^{n}\phi^{ji}=0.$
(3.23)

We write the list of gauge fields in the Table (1) below, we also give a list
of gauge transformations and gauge parameters in the appendix in Table (3)

Generator | Gauge field | Gauge parameter
---|---|---
$D$ | $b^{m}=0$ | $\xi^{+-}$
| $b^{0}=\frac{1}{2}\partial_{i}e^{0i}$ |
$P_{i}$ | $e^{m}{}^{i}\equiv\phi^{mi}$ | $\xi^{i+}$
| $e^{0}{}^{i}$ |
$H$ | $\tau^{m}=0$ | $\xi^{0+}$
| $\tau^{0}$ |
$J_{ij}$ | $\omega^{jm,n}=\partial^{m}\phi^{nj}-\partial^{n}\phi^{mj}$ | $\xi^{i,j}$
| $\omega{}^{0mi}-\partial^{0}{}\phi^{mi}-b^{0}{}\eta^{mi}+\partial^{m}e^{0}{}^{i}=0$ |
$B_{i}$ | $\beta^{m}{}^{i}$ | $\xi^{0,i}$
| $\beta^{0}{}^{i}=\partial^{i}\tau^{0}$ |
$K_{i}$ | $f^{mn}=\partial_{0}{}\beta^{mn}+h^{mn}\partial_{k}\partial^{k}\tau^{0}{}-h^{mn}\partial_{l}\partial_{k}\phi^{kl}-\partial^{m}\partial^{n}\tau^{0}{}$ | $\xi^{i-}$
| $f^{0m}=\frac{1}{2}\partial^{m}\partial_{i}e^{0i}$ |
$K_{0}$ | $\kappa^{n}=\partial_{m}\beta^{mn}$ | $\xi^{0-}$
| $\kappa^{0}=-\partial_{k}\partial^{l}\phi_{l}{}^{k}+\partial^{m}\partial_{m}\tau^{0}$ |
Table 1:  List of gauge fields with their corresponding generators and gauge
parameters

We have obtained four remaining fields $\phi^{mn},e^{0i},\beta^{mn},\tau^{0}$,
and two free gauge parameters $\xi^{m+},\xi^{0+}$ in terms of which the others
are expressed. We also got one dynamical equation for the field $\phi^{mn}$,
and constraints on the geometry which did not play a role in determination of
the dynamical equation or expressing the other gauge fields. Earlier analyses
of the Carrollian theories showed that such constraints play a role of
constraints for Lagrange multipliers in the form of the undetermined fields
[20, 21].

##  4 Spin-3

We consider the irreducible Young module of $so(3,2)$
\begin{picture}(20.0,10.0){}\end{picture} , which describes spin-3 field
$t_{ab}=t_{ba}$. As in previous example of conformal graviton, to find the
solution of the linearized equations and fix suitably gauge symmetries, we
have to find the IW contraction analogous to the contraction above. The
commutation relations of the generator of translations and the generators of
spin three algebra are

$\displaystyle[P_{a},t_{c+}]$ $\displaystyle=-\eta_{ac}t_{++}\,,$
$\displaystyle[P_{a},t_{c-}]$
$\displaystyle=t_{ac}+\frac{1}{2}t{}_{m}{}^{m}\,\,,$
$\displaystyle[P_{a},t_{--}]$ $\displaystyle=2t_{a-}\,,$ (4.1)
$\displaystyle[P_{a},t_{++}]$ $\displaystyle=0\,,$
$\displaystyle[P_{a},t_{cd}]$
$\displaystyle=-\eta_{ac}t_{d+}-\eta_{ad}{\boldsymbol{t}}_{c+}\,.$ (4.2)

For the conformal graviton case, it was necessary to choose $H,K_{0}$ and
$B_{i}$ to be elements of $\mathfrak{i}$, so here we proceed in the same way,
choosing the generators $SH,SK_{0},SB_{i}$ to be elements of $\mathfrak{i}$
while the remaining generators build $\mathfrak{h}$

$\displaystyle\mathfrak{h}$
$\displaystyle=\\{SP_{j},SK_{j},t_{++},t_{--},S_{jk},SC\\}$
$\displaystyle\mathfrak{i}=\\{SH,SK_{0},SB_{i}\\}.$ (4.3)

The dictionary we use between the symmetric $t^{\Lambda}$ generators $S^{AB}$
for the spin-3 case, and the generators denoted here is
$t^{j+}=SP^{j},t^{j-}=SK^{j},t^{jk}=SJ^{jk},t^{00}=SC$, also
$t^{0+}=SH,t^{0-}=SK^{0},t^{0i}=SB^{i}$. The letter $S$ in front of what one
can recognize as generators in conformal Carrollian algebra, denote that they
originate from the symmetric generators of $t^{\Lambda}$. The algebra that we
obtain after taking the limit reads

$\displaystyle[P_{i},SP_{j}]$ $\displaystyle=-\eta_{ij}t_{++}\,,$
$\displaystyle[P_{i},SK_{j}]$
$\displaystyle=SJ_{ij}+\tfrac{1}{2}(SJ{}_{k}{}^{k}\,+SC)\eta_{ij}\,,$
$\displaystyle[P_{i},t_{--}]$ $\displaystyle=2SK_{i-}\,,$
$\displaystyle[P_{i},t_{++}]$ $\displaystyle=0\,,$
$\displaystyle[P_{i},SJ_{jk}]$
$\displaystyle=-\eta_{ij}SP_{k}-\eta_{ik}SP_{j}\,,$ $\displaystyle[P_{i},SH]$
$\displaystyle=0\,,$ $\displaystyle[P_{i},SK_{0}]$ $\displaystyle=SB_{i}\,,$
$\displaystyle[P_{i},SB_{k}]$ $\displaystyle=-\eta_{ik}SH\,,$
$\displaystyle[H,t_{++}]$ $\displaystyle=0\,,$ $\displaystyle[H,t_{--}]$
$\displaystyle=2SK_{0}\,,$ $\displaystyle[H,SK_{j}]$ $\displaystyle=B_{j}\,,$
$\displaystyle[H,t_{--}]$ $\displaystyle=2SK_{0}\,,$
$\displaystyle[H,SJ_{jk}]$ $\displaystyle=0\,.$ $\displaystyle[H,SP_{j}]$
$\displaystyle=0\,$

so that one can write the algebra valued one-form $\omega$ and zero-form $\xi$
as

$\displaystyle\omega$ $\displaystyle=se^{i}SP_{i}+s\tau
SH+\tfrac{1}{2}s\omega^{ij}SJ_{ij}+s\beta^{i}SB_{i}+\gamma
SC+\frac{1}{2}\omega^{++}t_{++}+\frac{1}{2}\omega^{--}t_{--}+sf^{i}K_{i}+s\kappa
K_{0}\,,$ (4.4) $\displaystyle\xi$ $\displaystyle=se^{i}\xi_{i+}+s\tau
S\xi^{0+}+\tfrac{1}{2}s\omega_{ij}\xi^{ij}+s\beta_{i}\xi^{0i}+\gamma\xi^{00}+\frac{1}{2}\omega^{++}\xi^{++}+\frac{1}{2}\omega^{--}\xi^{--}+sf_{i}\xi^{i-}+s\kappa\xi^{0-}\,.$
(4.5)

The system of linearized equations and gauge symmetries reads

$\displaystyle t_{++}$ $\displaystyle:$ $\displaystyle
d\omega^{++}-2h_{k}\wedge se^{k}$ $\displaystyle=0\,,$
$\displaystyle\delta\omega^{++}$ $\displaystyle=d\xi^{++}-2h_{k}\xi^{k+}\,,$
(4.6a) $\displaystyle SP_{i}$ $\displaystyle:$ $\displaystyle
dse^{i}-h_{l}\wedge\omega^{li}$ $\displaystyle=0\,,$ $\displaystyle\delta
se^{i}$ $\displaystyle=d\xi^{i+}-h_{l}\xi^{li}\,,$ (4.6b) $\displaystyle SH$
$\displaystyle:$ $\displaystyle ds\tau-h_{i}\wedge
s\beta^{i}-h^{0}\wedge\gamma$ $\displaystyle=0\,,$ $\displaystyle\delta s\tau$
$\displaystyle=d\xi^{0+}-h_{l}\xi^{l0}-h^{0}\xi^{00}\,,$ (4.6c) $\displaystyle
SJ_{ij}$ $\displaystyle:$ $\displaystyle d\omega^{ij}+h^{(i}\wedge
sf^{j)}+h_{l}\wedge sf^{l}\eta^{ij}$ $\displaystyle=0\,,$
$\displaystyle\delta\omega^{ij}$
$\displaystyle=d\xi^{ij}+h^{(i}\xi^{j)-}+h_{l}\xi^{l-}\eta^{ij}\,,$ (4.6d)
$\displaystyle SB_{j}$ $\displaystyle:$ $\displaystyle ds\beta^{j}+h^{0}\wedge
sf^{j}+h^{j}\wedge s\kappa$ $\displaystyle=0\,,$ $\displaystyle\delta
s\beta^{j}$ $\displaystyle=d\xi^{0j}+h^{0}\xi^{j-}+h^{j}\xi^{0-}\,,$ (4.6e)
$\displaystyle SC$ $\displaystyle:$ $\displaystyle d\gamma+h^{i}\wedge sf_{i}$
$\displaystyle=0\,,$ $\displaystyle\delta\gamma$
$\displaystyle=d\xi^{00}+h^{i}\xi_{i}{}^{-}\,,$ (4.6f) $\displaystyle SK_{i}$
$\displaystyle:$ $\displaystyle dsf^{i}+h^{i}\wedge\omega^{--}$
$\displaystyle=0\,,$ $\displaystyle\delta sf^{i}$
$\displaystyle=d\xi^{i-}+h^{i}\xi^{--}\,,$ (4.6g) $\displaystyle SK_{0}$
$\displaystyle:$ $\displaystyle ds\kappa+h^{0}\wedge\omega^{--}$
$\displaystyle=0\,,$ $\displaystyle\delta s\kappa$
$\displaystyle=d\xi^{0-}+h^{0}\xi^{--}\,,$ (4.6h) $\displaystyle st_{--}$
$\displaystyle:$ $\displaystyle d\omega^{--}$ $\displaystyle=0\,,$
$\displaystyle\delta\omega^{--}$ $\displaystyle=d\xi^{--}\,.$ (4.6i)

From the first gauge condition (4.6a) we see that we can fix
$\omega^{m}{}^{++}=0$ to be equal to zero, by setting
$\xi^{m}{}^{+}=\frac{1}{2}\partial^{m}\xi^{++}$. Since the component of the
background field $h_{k0}$ vanishes, the field $\omega^{0}{}^{++}$ remains
unfixed, and its gauge transformation is set to
$\delta\omega^{0}{}^{++}=\partial^{0}\xi^{++}$. The m and n component of the
equation require from the field $se^{mn}=se^{nm}$ to be symmetric, and $m0$
component defines

$\displaystyle se^{0m}=\frac{1}{2}\partial^{m}\omega^{0++}$ (4.7)

From the equation and gauge condition (4.6b) we can determine that the
remaining symmetric part of the field $se^{(mi)}=0$ is zero, similarly as in
non-relativistic case, by fixing
$\xi^{mi}=\partial^{m}\xi^{i+}=\frac{1}{2}\partial^{m}\partial^{i}\xi^{++}$.
The zero-th component gauge condition fixes gauge transformation of $\delta
se^{0i}=\partial^{0}\xi^{i+}=\frac{1}{2}\partial_{0}\partial^{i}\xi^{++}$. The
symmetric and antisymmetric $mn$ component of the equation are zero, and only
the hook component gives the condition $\omega^{mni}=\omega^{nmi}$ which makes
$\omega^{mni}\equiv\phi^{mni}$ totally symmetric. The $m0$ component’s
symmetric and antisymmetric part fix $\omega^{0mi}$ field in terms of the
$\partial^{i}se^{0m}$

$\displaystyle\omega^{0mi}=\frac{1}{2}(\partial^{m}se^{0i}+\partial^{i}se^{0m}),$
$\displaystyle\partial^{m}se^{0i}=\partial^{i}se^{0m}.$ (4.8)

and yield this to be a symmetric tensor.

From the gauge condition (4.6c) we can already see the second gauge parameter
which will not fix any field components $\xi^{0+}$. The remaining two gauge
parameters, $\xi^{0m}=\partial^{m}\xi^{0+}$ and
$\xi^{00}=\partial^{0}\xi^{0+}$ fix $s\tau^{m}=0$ and $s\tau^{0}=0$
respectively. The $mn$ component of the equation define
$s\beta^{mn}=s\beta^{nm}$ to be symmetric field and
$s\beta^{0m}=h^{00}\gamma^{m}$ the zero-th component to be equal to field
$\gamma^{m}$. The gauge condition and the equation from the $SJ_{ij}$
generator (4.6d) determine the gauge parameter
$\xi^{i-}=-\frac{2}{5}\partial_{m}\xi^{im}$ and the trace of the field
$\phi_{m}{}^{im}=0$. One can write the gauge transformation for the two
dimensional field $\phi^{mni}$ as

$\displaystyle\delta\omega^{mij}=\frac{1}{2}\partial^{m}\partial^{j}\partial^{i}\xi^{++}-\frac{1}{10}(\partial^{l}\partial^{j}\partial_{l}\xi^{++}h_{m}{}^{i}+\partial^{l}\partial^{i}\partial_{l}\xi^{++}h_{m}{}^{j})-\frac{1}{5}\partial_{l}\partial^{m}\partial^{l}\xi^{++}\eta^{ij}.$
(4.9)

Due to fixing the trace of $sf_{i}{}^{i}=0$ with
$\xi^{--}=-\frac{1}{2}\partial_{i}\xi^{i-}$, the only non-vanishing component,
hook component of the equation gives $\partial_{m}\omega_{i}{}^{im}=0$, and

$\displaystyle sf^{nm}=-\partial_{i}\phi^{mni}.$ (4.10)

Without taking contractions, equation (4.6d) gives equation for the
$\phi^{mni}$ field

$\displaystyle\tfrac{1}{6}(-\eta^{mn}\partial_{a}\phi^{ija}+\eta^{jn}\partial_{a}\phi^{ima}+\eta^{jm}\partial_{a}\phi^{ina}-\eta^{in}\partial_{a}\phi^{jma}-\eta^{im}\partial_{a}\phi^{jna}$
$\displaystyle+\eta^{ij}\partial_{a}\phi^{mna}+\partial^{i}\phi^{jmn}-3\partial^{j}\phi^{imn}+\partial^{m}\phi^{ijn}+\partial^{n}\phi^{ijm})=0.$
(4.11)

The $m0$ component of the equation have only hook and symmetric part of the
equation. They can be solved to determine

$\displaystyle
sf^{0j}=-\frac{2}{5}\partial_{i}\omega^{0ji}=-\frac{1}{5}\partial_{i}\partial^{i}\partial^{j}\omega^{0++}.$
(4.12)

From (4.6e) gauge condition we fix $s\beta^{i}{}_{i}=0$ with $\xi^{0-}$, we
have thus

$\displaystyle s\kappa^{m}=\partial_{j}s\beta^{mj}.$ (4.13)

The $m0$ component gives $\delta s\beta^{0j}=\partial_{0}\xi^{0j}+\xi^{j-}$
and $s\kappa^{0}=\frac{1}{2}\partial_{j}s\beta^{0j}$. Components $mn$ and $m0$
from (4.6g) give

$\displaystyle\omega^{n--}=-\partial_{i}sf^{ni}=\partial_{i}\partial_{j}\phi^{nij}$
$\displaystyle\omega^{0--}{}=-\frac{1}{2}\partial_{i}sf^{0i}=\frac{1}{10}\partial_{i}\partial^{i}\partial_{j}\partial^{j}\omega^{0++}.$
(4.14)

The $mn$ component of (4.6h) is exactly satisfied, while $m0$ component gives
constraint on the $s\beta^{ij}$ and $\phi^{ijk}$ fields
$-\frac{1}{2}\partial^{m}\partial_{j}s\beta^{0j}+\partial_{0}\partial_{l}s\beta^{ml}-\partial_{i}\partial_{j}\phi^{imj}=0$.
The equation (4.6i) gives condition on the $\phi^{mni}$ field
$\epsilon_{mn}\partial_{i}\partial_{j}\phi^{inj}=0$ and condition on
$\omega_{0}{}^{++}$,
$\frac{1}{10}\partial^{m}\partial_{i}\partial^{i}\partial_{l}\partial^{l}\omega_{0}{}^{++}-\partial_{0}\partial_{i}\partial_{j}\phi^{mij}=0$.
Using the constraint from (4.6i) and from (4.6h) one obtains
$\epsilon_{mn}\partial^{m}\partial_{j}s\beta^{0j}=0$.

To present the results transparently, we write the gauge fields with
corresponding gauge parameters and generators in the Table (2) below, while
the list of linearized gauge transformations and expressions for gauge
parameters, we give in the appendix, in the Table (4).

Generator | Gauge field | Gauge parameter
---|---|---
$St_{++}$ | $\omega_{m}{}^{++}=0$ | $\xi^{++}$
| $\omega_{0}^{++}$ |
$SP_{i}$ | $se_{mn}=0$ | $\xi^{i+}$
| $se^{0m}=\frac{1}{2}\partial_{m}\omega_{0}{}^{++}$ |
$SH$ | $s\tau^{m}=0$ | $\xi^{0+}$
| $s\tau^{0}=0$ |
$SJ_{ij}$ | $\omega^{mni}=\phi^{mni}$ | $\xi^{ij}$
| $\omega{}^{0mi}=\frac{1}{2}\partial^{m}\partial^{i}\omega_{0}{}^{++}$ |
$SB_{i}$ | $s\beta^{m}{}^{i}$ | $\xi^{0i}$
| $s\beta^{0}{}^{i}=\gamma_{m}$ |
$SK_{i}$ | $sf^{mn}=-\partial_{i}\phi^{mni}$ | $\xi^{i-}$
| $sf^{0m}=-\frac{1}{5}\partial_{i}\partial^{i}\partial^{m}\omega_{0}{}^{++}$ |
$SK_{0}$ | $s\kappa^{n}=-\partial_{m}s\beta^{mn}$ | $\xi^{0-}$
| $s\kappa^{0}=-\frac{1}{2}\partial_{j}s\beta^{0j}$ |
$St_{--}$ | $\omega_{n}{}^{--}=\partial_{i}\partial_{j}\phi^{nij}$ | $\xi^{--}$
| $\omega_{0}{}^{--}=\frac{1}{10}\partial_{i}\partial^{i}\partial_{j}\partial^{j}\omega_{0}{}^{++}$ |
$SC$ | $\gamma_{m}$ | $\xi^{00}$
| $\gamma_{0}$ |
Table 2: List of gauge fields with corresponding generators and gauge
parameters for the spin-3 conformal gravity in ultra-relativistic limit

Similarly like in the previous case, we have obtained one dynamical field
$\phi^{mni}$ with corresponding dynamical equation and four additional
remaining fields $\omega^{0++},s\beta^{mi}$, $\gamma_{m}$ and $\gamma_{0}$. We
also obtained constraints on the geometry, which should play similar role as
in the spin-2 case. Analogously one may expect that some of them correspond to
radiative degrees of freedom for gravity at null-infinity.

Comment on generalization to spin-s.

To determine the spectrum of the theory for the arbitrary spin, one first has
to perform IW contraction of the given algebra. For the spin-s, the algebra
prior to contraction is going to be defined by the Young tableaux with the s-1
boxes in the first row and s-t boxes in the second row. The generators that
need to be set in the $\mathfrak{h}$, for analogous consideration as in spin-2
and spin-3 cases here, are those that have 1 and 2 components, and those that
appear in the definition of traces. That way one makes sure that the algebra
$\mathfrak{h}$ is closed. The generators in the 0-th direction have to be part
of $\mathfrak{i}$. After solving the system of equations, one can expect to
obtain the lower dimensional dynamical spin-s field, and minimally two
remaining fields which are undetermined, but are related with constraints on
the geometry, and play a role of radiative degrees of freedom at null-
infinity. One should also expect to obtain two remaining gauge parameters,
that determine linear gauge transformations of the remaining fields.

##  5 Conformal Carroll Spin-2 and Spin-3 Gravity

Conformal Carrol gravity is defined by the Chern-Simons action (2.12) and the
connection (3.4). We want to see what kind of generalizations one obtains from
conformal Carroll algebra in comparison to the Carroll algebra.Carroll gravity
for the flat background has a line element

$\displaystyle ds^{2}=\rho^{2}d\phi^{2}+d\rho^{2}$ (5.1)

defined by $e^{a}e^{b}\delta_{ab}$ where the corresponding zweibeins take the
form $e_{\phi}^{1}=\rho,e_{\rho}^{2}=1$ and $e_{\rho}^{2}=e_{\phi}^{2}=0$.
Here, $\rho$ denotes radial and $\phi$ angular coordinate periodic in $2\pi$,
$\phi\sim\phi+2\pi$. The time component is $\tau=dt$. Following the known
procedure [27, 28, 29] we partially fix the radial gauge and consider the
connection in the form

$\displaystyle\omega=b^{-1}(\rho)(d+\mathcal{\omega}(t,\phi))b(\rho).$ (5.2)

We allow for the connection on the boundary to depend on $t$ and $\phi$
coordinates, but not on $\rho$ coordinate. It is convenient to impose the
following boundary conditions

$\displaystyle\omega_{\phi}^{(s_{2})}$
$\displaystyle=B_{i}+\mathcal{H}(t,\phi)H+\mathcal{P}_{i}(t,\phi)P_{i}+\mathcal{J}_{ij}(t,\phi)J_{ij}+\mathcal{K}_{i}(t,\phi)K_{i}+\mathcal{K}_{0}(t,\phi)K_{0}+\mathcal{D}(t,\phi)D$
(5.3a) $\displaystyle\omega_{t}^{(s_{2})}$ $\displaystyle=\mu(t,\phi)H$ (5.3b)

denoting with $(s_{2})$ that we are considering spin-2 case. We allow for the
functions from $\omega_{\phi}^{(s_{2})}$ to vary at the boundary, while
variation of the chemical potential $\delta\mu(t,\phi)=0$ is fixed.

To find the metric formulation for these boundary conditions we define the
group element which needs to be valued in the algebra, $b(\rho)=e^{\rho
P_{2}}$. When the generator in the group element commutes with generators from
the boundary conditions into generator of translations, one obtains
contribution to zweibein. Using the Baker-Cambell-Hausdorff formula, we can
determine connection (5.2) and read out the components of the zweibein and
$\tau$ to be

$\displaystyle e_{\phi}^{1}$
$\displaystyle=\mathcal{J}_{12}(t,\phi)\rho+\mathcal{P}_{1}(t,\phi)\,,$
$\displaystyle e_{\phi}^{2}=\mathcal{P}_{2}(t,\phi)-\mathcal{D}(t,\phi)\rho$
(5.4a) $\displaystyle e_{\rho}^{1}$ $\displaystyle=0\,,$ $\displaystyle
e_{\rho}^{2}=1$ (5.4b) $\displaystyle\tau_{t}$ $\displaystyle=\mu(t,\phi)\,,$
$\displaystyle\tau_{\phi}=\rho+\mathcal{H}(t,\phi)$ (5.4c)

That leads to the line element

$\displaystyle
ds^{2}=\left((\mathcal{P}_{2}(t,\phi)+\mathcal{D}(t,\phi)\rho)^{2}+(\mathcal{P}_{1}(t,\phi)+\mathcal{J}_{12}(t,\phi)\rho)^{2}\right)d\phi^{2}+2(\mathcal{P}_{2}(t,\phi)+\mathcal{D}(t,\phi)\rho)d\phi
d\rho+d\rho^{2}$ (5.5)

and

$\displaystyle d\tau=(\rho+\mathcal{H}(t,\phi))d\phi+\mu(t,\phi)dt$ (5.6)

which allows for the two additional functions to appear in the two dimensional
line element in conformal Carroll gravity (5.5), compared to similar line
element in Carroll gravity [20]. In general, conformal Carroll gravity is
expected to allow for the additional functions in the line element compared to
Carroll gravity. However, even with these additional functions, one cannot
eliminate the shift in the $\phi$ direction, the same as in Carroll gravity.

We wonder which boundary conditions could be imposed in the spin-3 case such
that we do not have shift in the $\phi$ component in the (5.6). Following the
same procedure as above, we partially fix to radial gauge and write the
connection in the form (5.2) which can depend on $t$ and $\phi$ on the
boundary, but not on coordinate $\rho$. The connection $\omega(t,\phi)$ is now
of the form
$\omega(t,\phi)=\omega(t,\phi)^{(s_{2})}+\omega(t,\phi)^{(s_{3})}$. Where
$(s_{2})$ represents boundary conditions for conformal gravity (spin-2 case),
i.e. equations (5.3a) and (5.3b). The additional minimal boundary condition
that we impose due to spin-3 is

$\displaystyle\omega_{\phi}^{(s_{3})}=\mathcal{S}(t,\phi)t_{++}$ (5.7)

and the group element $b(\rho)$ that we choose is $b(\rho)=e^{\rho
P_{2}+a_{2}\rho SP_{2}+a_{3}B_{1}}$. The components of the metric for such
extension of conformal Carroll gravity will have components

$\displaystyle e_{\phi}{}^{1}$
$\displaystyle=(\mathcal{P}_{1}(t,\phi)+\mathcal{J}_{12}(t,\phi)-a_{3}\mathcal{P}_{1}(t,\phi))$
$\displaystyle e_{\phi}=\mathcal{P}_{2}(t,\phi)+\mathcal{D}(t,\phi)\rho$
(5.8a) $\displaystyle e_{\rho}{}^{1}$ $\displaystyle=0$ $\displaystyle
e_{\rho}^{2}=1$ (5.8b) $\displaystyle\tau_{\phi}$
$\displaystyle=\mathcal{H}(t,\phi)+\rho+2a_{2}\mathcal{S}(t,\phi)\rho+a_{3}\mathcal{P}_{1}(t,\phi)$
$\displaystyle\tau_{t}=\mu(t,\phi)$ (5.8c)

from which we see that suitable condition on $\mathcal{P}_{1}$ and
$\mathcal{S}$ will lead to disappearance of the shift in the $\phi$ direction.
For that we had to not only add (5.7) minimal boundary condition, but also
modify the group element $b(\rho)$.

##  6 Conclusion

We have studied IW contraction of the algebras corresponding to the Young
modules \begin{picture}(10.0,20.0){}\end{picture} and
\begin{picture}(20.0,10.0){}\end{picture} from the $so(3,2)$ decomposition of
$hs(\parbox{10.0pt}{\begin{picture}(10.0,10.0){}\end{picture}})$. First of
which described conformal graviton and the second spin-3 field. The choice of
IW contraction that we took, led in the first case to the conformal Carrollian
algebra, ultra-relativistic limit of the conformal algebra. Based on this
algebra we have constructed an ultra-relativistic gravity theory starting from
the Chern-Simons action. For the spin-3 case, we have conducted IW contraction
along the same lines as for the conformal spin-2 graviton, followed by the
construction of the gravity theory based on Chern-Simons action. It would be
conceivable to think of it as a possible spin-3 generalization of the
conformal Carroll gravity.

We have also studied holography of the conformal Carroll gravity imposing the
suitable set of boundary conditions. In the metric formulation this boundary
conditions led to the generalized line element compared to the one obtained
from Carroll gravity. The line element contained the $d\tau$ part with a shift
in the $\phi$ direction which was not possible to remove within Carroll or
conformal Carroll gravity. Considering the holography of spin-3 theory, we
managed to find suitable projector and boundary conditions which eliminate
this shift.

As a future direction of study it would be interesting to consider the spin-s
generalization of the map between the conformal Carroll algebras and the
higher spin fields in the metric formulation. It would be especially
interesting to see if formulation of three dimensional conformal higher spin
gravity in this limit would give additional information on the restriction of
interacting vertices in higher spin theory.

## Acknowledgments

The author would like to thank Jan Rosseel, Daniel Grumiller, Romain Ruzziconi
and Adrien Fiorucci for the discussions. This work was made possible by the
Hertha Firnberg grant T 1269-N by Austrian Science fund (FWF).

##  7 Appendix

###  7.1 Conformal gravity example

Inserting the $f^{im}$ (3.19) into the first constraint from the equation
(3.11) one obtains

$\displaystyle\tfrac{1}{2}(-2\partial_{0}{}\beta^{jn}\eta^{im}+2\partial_{0}{}\beta^{jm}\eta^{in}+2\partial_{0}{}\beta^{in}\eta^{jm}-2\partial_{0}{}\beta^{im}\eta^{jn}+4\eta^{in}\eta^{jm}\partial_{a}\partial^{a}\tau^{0}{}$
$\displaystyle-4\eta^{im}\eta^{jn}\partial_{a}\partial^{a}\tau^{0}{}-4\eta^{in}\eta^{jm}\partial_{b}\partial_{a}\phi^{ab}+4\eta^{im}\eta^{jn}\partial_{b}\partial_{a}\phi^{ab}+\partial^{i}\partial^{m}\phi^{jn}+2\eta^{jn}\partial^{i}\partial^{m}\tau^{0}{}$
$\displaystyle-\partial^{i}\partial^{n}\phi^{jm}-2\eta^{jm}\partial^{i}\partial^{n}\tau^{0}{}-\partial^{j}\partial^{m}\phi^{in}-2\eta^{in}\partial^{j}\partial^{m}\tau^{0}{}+\partial^{j}\partial^{n}\phi^{im}+2\eta^{im}\partial^{j}\partial^{n}\tau^{0}{}$
$\displaystyle+\partial^{m}\partial^{i}\phi^{jn}-\partial^{m}\partial^{j}\phi^{in}-\partial^{n}\partial^{i}\phi^{jm}+\partial^{n}\partial^{j}\phi^{im})=0$
(7.1)

###  7.2 Gauge parameters and gauge transformations

In this Table 3 we present the gauge parameters and linearized gauge
transformations for the spin-2 case

Generator | Gauge | linearized gauge transformation | Gauge parameter
---|---|---|---
| field |  |
$D$ | $b^{m}$ | $\delta b^{m}=0$ | $\xi^{+-}=\frac{1}{2}\partial_{l}\xi^{l+}$
| $b^{0}$ | $\delta b_{0}=\frac{1}{2}\partial_{0}\partial_{l}\xi^{l+}$ |
$P_{i}$ | $e^{m}{}^{i}$ | $\delta e^{mi}=\frac{1}{2}(\partial^{m}\xi^{i+}+\partial^{i}\xi^{m+})-\frac{1}{2}h^{mi}\partial_{l}\xi^{l+}$ | $\xi^{i+}$
| $e^{0}{}^{i}$ | $\delta e^{0i}=\partial^{0}\xi^{i+}$ |
$H$ | $\tau^{m}$ | $\delta\tau_{m}=0$ | $\xi^{0+}$
| $\tau^{0}$ | $\delta\tau^{0}=\partial^{0}\xi^{0+}-\frac{1}{2}h_{0}{}^{0}\partial_{l}\xi^{l+}$ |
$J_{ij}$ | $\omega^{jm,n}$ | $\delta\omega^{mi,j}=\frac{1}{2}\partial^{m}(\partial^{j}\xi^{i+}-\partial^{i}\xi^{j+})$ | $\xi^{i,j}=\frac{1}{2}(\partial^{j}\xi^{i+}-\partial^{i}\xi^{j+})$
|  |  $-\frac{1}{2}\partial_{l}(h^{mi}\partial^{j}\xi^{l+}-h^{jm}\partial^{i}\xi^{l+})$ |
| $\omega{}^{0mi}$ | $\delta\omega^{0i,j}=\frac{1}{2}\partial^{0}(\partial^{j}\xi^{i+}-\partial^{i}\xi^{j+})$ |
$B_{i}$ | $\beta^{m}{}^{i}$ | $\delta\beta^{mj}=-\partial^{m}\partial^{j}\xi^{0+}+\frac{1}{2}h^{jm}\partial_{l}\partial^{l}\xi^{0+}$ | $\xi^{0,i}=\partial^{i}\xi^{0+}$
| $\beta^{0}{}^{i}$ | $\delta\beta^{0j}=-\partial^{0}\partial^{j}\xi^{0+}+\frac{1}{2}h_{0}{}^{0}\partial^{j}\partial_{l}\xi^{l+}$ |
$K_{i}$ | $f^{mn}$ | $\delta f^{mi}=\frac{1}{2}\partial^{m}\partial^{i}\partial_{l}\xi^{l+}$ | $\xi^{i-}=-\frac{1}{2}\partial^{i}\partial_{l}\xi^{l+}$
| $f^{0m}$ | $\delta f^{0i}=\frac{1}{2}\partial^{0}\partial^{i}\partial_{l}\xi^{l+}$ |
$K_{0}$ | $\kappa^{n}$ | $\delta\kappa_{m}=-\frac{1}{2}\partial_{m}\partial_{l}\partial^{l}\xi^{0+}$ | $\xi^{0-}=-\frac{1}{2}\partial_{m}\partial^{m}\xi^{0+}$
| $\kappa^{0}$ | $\delta\kappa^{0}=-\frac{1}{2}\partial^{0}\partial_{l}\partial^{l}\xi^{0+}$ |
Table 3: List of linearized gauge transformations of the fields and gauge
parameters

and in the Table 4 for the spin-3 case

Generator | Gauge | Linearized gauge transformations | Gauge parameter
---|---|---|---
| field |  |
$St_{++}$ | $\omega_{m}{}^{++}$ | $\delta\omega^{m}{}^{++}=0$ | $\xi^{++}$
| $\omega_{0}^{++}$ | $\delta\omega_{0}^{++}=\partial_{0}\xi^{++}$ |
$SP_{i}$ | $se_{mn}$ | $\delta se_{mn}=0$ | $\xi^{m+}=\frac{1}{2}\partial^{m}\xi^{++}$
| $se^{0m}$ | $\delta se^{0i}=\frac{1}{2}\partial^{0}\partial^{i}\xi^{++}$ |
$SH$ | $s\tau^{m}$ | $\delta s\tau_{m}=0$ | $\xi^{0+}$
| $s\tau^{0}$ | $\delta s\tau_{0}=0$ |
$SJ_{ij}$ | $\omega^{mni}$ | $\delta\omega^{mij}=\frac{1}{2}\partial^{m}\partial^{j}\partial^{i}\xi^{++}+C\partial_{l}\partial^{m}\partial^{l}\xi^{++}\eta^{ij}$ | $\xi^{mi}=\frac{1}{2}\partial^{m}\partial^{i}\xi^{++}$
|  |  $+\frac{C}{2}(\partial^{l}\partial^{j}\partial_{l}\xi^{++}h_{m}{}^{i}+\partial^{l}\partial^{i}\partial_{l}\xi^{++}h_{m}{}^{j})$ |
| $\omega{}^{0mi}$ | $\delta\omega{}^{0mi}=\frac{1}{2}\partial^{0}\partial^{i}\partial^{m}\xi^{++}$ |
$SB_{i}$ | $\beta^{m}{}^{i}$ | $\delta\beta^{m}{}^{i}=\partial^{m}\partial^{i}\xi^{0+}-\frac{1}{2}h^{mi}\partial^{l}\partial_{l}\xi^{++}$ | $\xi^{0m}=\partial^{m}\xi^{0+}$
| $\beta^{0}{}^{i}$ | $\delta\beta^{0}{}^{i}=\partial^{0}\partial^{i}\xi^{0+}+Ch_{0}{}^{0}\partial^{i}\partial^{l}\partial_{l}\xi^{++}$ |
$SK_{i}$ | $sf^{mn}$ | $\delta sf^{mi}=C\partial^{m}\partial_{l}\partial^{l}\partial^{i}\xi^{++}-\frac{C}{2}h^{mi}\partial_{j}\partial_{n}\partial^{n}\partial^{j}\xi^{++}$ | $\xi^{i-}=C\partial_{m}\partial^{m}\partial^{i}\xi^{++}$
| $sf^{0i}$ | $\delta sf^{0i}=C\partial^{0}\partial_{l}\partial^{l}\partial^{i}\xi^{++}$ |
$SK_{0}$ | $s\kappa^{n}$ | $\delta s\kappa^{n}=-\frac{1}{2}\partial^{n}\partial_{l}\partial^{l}\xi^{0+}$ | $\xi^{0-}=-\frac{1}{2}\partial^{m}\partial_{m}\xi^{0+}$
| $s\kappa^{0}$ | $\delta s\kappa^{0}=-\frac{1}{2}\partial^{0}\partial_{l}\partial^{l}\xi^{0+}-\frac{C}{2}h_{0}^{0}\partial_{i}\partial_{m}\partial^{m}\partial^{i}\xi^{++}$ |
$St_{--}$ | $\omega_{n}{}^{--}$ | $\delta\omega^{n--}=-\frac{C}{2}\partial^{n}\partial_{i}\partial_{m}\partial^{m}\partial^{i}\xi^{++}$ | $\xi^{--}=-\frac{C}{2}\partial_{i}\partial_{m}\partial^{m}\partial^{i}\xi^{++}$
| $\omega_{0}{}^{--}$ | $\delta\omega^{0--}=\frac{1}{10}\partial^{0}\partial_{i}\partial_{m}\partial^{m}\partial^{i}\xi^{++}$ |
$St^{00}$ | $\gamma_{m}$ | $\delta\gamma_{m}=\partial_{m}\partial^{0}\xi^{0+}+C\partial_{l}\partial^{l}\partial_{m}\xi^{++}$ | $h_{0}{}^{0}\xi^{00}=\partial_{0}\xi^{0+}$
| $\gamma^{0}$ | $\delta\gamma^{0}=\partial^{0}\partial^{0}\xi^{0+}$ |
Table 4: List of linearized gauge transformations and gauge parameters for the
ultra-relativistic limit of the spin-3 conformal gravity, $C=-\frac{1}{5}$

## References

  * [1] G. ’t Hooft, Local Conformal Symmetry: the Missing Symmetry Component for Space and Time, 1410.6675.
  * [2] H. Afshar, S. Detournay, D. Grumiller, W. Merbis, A. Perez, D. Tempo and R. Troncoso, Soft Heisenberg hair on black holes in three dimensions, Phys. Rev. D 93 (2016), no. 10 101503 [1603.04824].
  * [3] H. Afshar, D. Grumiller, W. Merbis, A. Perez, D. Tempo and R. Troncoso, Soft hairy horizons in three spacetime dimensions, Phys. Rev. D 95 (2017), no. 10 106005 [1611.09783].
  * [4] A. Bagchi, Correspondence between Asymptotically Flat Spacetimes and Nonrelativistic Conformal Field Theories, Phys. Rev. Lett. 105 (2010) 171601 [1006.3354].
  * [5] C. Duval, G. W. Gibbons and P. A. Horvathy, Conformal Carroll groups and BMS symmetry, Class. Quant. Grav. 31 (2014) 092001 [1402.5894].
  * [6] A. Strominger, Lectures on the Infrared Structure of Gravity and Gauge Theory, 1703.05448.
  * [7] A. Bagchi, A. Mehra and P. Nandi, Field Theories with Conformal Carrollian Symmetry, JHEP 05 (2019) 108 [1901.10147].
  * [8] J. Hartong, Gauging the Carroll Algebra and Ultra-Relativistic Gravity, JHEP 08 (2015) 069 [1505.05011].
  * [9] A. Campoleoni and S. Pekar, Carrollian and Galilean conformal higher-spin algebras in any dimensions, 2110.07794.
  * [10] M. Blencowe, A Consistent Interacting Massless Higher Spin Field Theory in $D$ = (2+1), Class.Quant.Grav. 6 (1989) 443.
  * [11] A. Campoleoni, Higher Spins in D = 2+1, 1110.5841.
  * [12] H. Afshar, A. Bagchi, R. Fareghbal, D. Grumiller and J. Rosseel, Spin-3 Gravity in Three-Dimensional Flat Space, Phys. Rev. Lett. 111 (2013), no. 12 121603 [1307.4768].
  * [13] A. Campoleoni, S. Fredenhagen, S. Pfenninger and S. Theisen, Asymptotic symmetries of three-dimensional gravity coupled to higher-spin fields, JHEP 1011 (2010) 007 [1008.4744].
  * [14] M. Henneaux and S.-J. Rey, Nonlinear $W_{infinity}$ as Asymptotic Symmetry of Three-Dimensional Higher Spin Anti-de Sitter Gravity, JHEP 1012 (2010) 007 [1008.4579].
  * [15] A. Y. Segal, Conformal higher spin theory, Nucl. Phys. B664 (2003) 59–130 [hep-th/0207212].
  * [16] A. A. Tseytlin, On limits of superstring in AdS(5) x S**5, Theor. Math. Phys. 133 (2002) 1376–1389 [hep-th/0201112]. [Teor. Mat. Fiz.133,69(2002)].
  * [17] D. Ponomarev and E. D. Skvortsov, Light-Front Higher-Spin Theories in Flat Space, J. Phys. A 50 (2017), no. 9 095401 [1609.04655].
  * [18] C. N. Pope and P. K. Townsend, Conformal Higher Spin in (2+1)-dimensions, Phys. Lett. B225 (1989) 245–250.
  * [19] M. Grigoriev, I. Lovrekovic and E. Skvortsov, New Conformal Higher Spin Gravities in $3d$, JHEP 01 (2020) 059 [1909.13305].
  * [20] E. Bergshoeff, D. Grumiller, S. Prohazka and J. Rosseel, Three-dimensional Spin-3 Theories Based on General Kinematical Algebras, JHEP 01 (2017) 114 [1612.02277].
  * [21] E. Bergshoeff, J. Gomis, B. Rollier, J. Rosseel and T. ter Veldhuis, Carroll versus Galilei Gravity, JHEP 03 (2017) 165 [1701.06156].
  * [22] Y. Herfray, Carrollian manifolds and null infinity: A view from Cartan geometry, 2112.09048.
  * [23] Y. Herfray, Asymptotic shear and the intrinsic conformal geometry of null-infinity, J. Math. Phys. 61 (2020), no. 7 072502 [2001.01281].
  * [24] E. S. Fradkin and A. A. Tseytlin, Conformal Supergravity, Phys. Rept. 119 (1985) 233–362.
  * [25] J. H. Horne and E. Witten, Conformal Gravity in Three-dimensions as a Gauge Theory, Phys. Rev. Lett. 62 (1989) 501–504.
  * [26] E. Inonu and E. P. Wigner, On the Contraction of groups and their represenations, Proc. Nat. Acad. Sci. 39 (1953) 510–524.
  * [27] A. Campoleoni, S. Fredenhagen and S. Pfenninger, Asymptotic W-symmetries in three-dimensional higher-spin gauge theories, JHEP 09 (2011) 113 [1107.0290].
  * [28] M. Banados, T. Brotz and M. E. Ortiz, Boundary dynamics and the statistical mechanics of the (2+1)-dimensional black hole, Nucl. Phys. B 545 (1999) 340–370 [hep-th/9802076].
  * [29] D. Grumiller and M. Riegler, Most general AdS3 boundary conditions, JHEP 10 (2016) 023 [1608.01308].