Strange higher-spin topological systems in 3D

Nicolas Boulanger, Andrea Campoleoni, Victor Lekeu and Evgeny Skvortsov

Service de Physique de l’Univers, Champs et Gravitation, Université de Mons –
UMONS,
20 place du Parc, 7000 Mons, Belgium

###### Abstract

Motivated by the generation of action principles from off-shell dualisation,
we present a general class of free, topological theories in three dimensional
Minkowski spacetime that exhibit higher-spin gauge invariance. In the spin-two
case, we recover a dual reformulation of the triplet system already known,
while the higher-spin systems that we obtain seem to be new. They are
associated with wild quivers. We study in which situations these exotic (or
strange) higher-spin models can be extended to dS3 and AdS3 backgrounds,
revealing that the flat limit of such models, when they exist, admits a one-
parameter freedom. Interactions are studied in the simplest higher-spin case
featuring spin-2 and spin-3 fields. We then give several higher-spin
generalizations of these strange systems.

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###### Contents

  1. 1 Introduction
  2. 2 Propagating case with a spin two field in 3D
    1. 2.1 From higher dualisation to a family of models
    2. 2.2 Dualisation of the spin-1 field and link with a spin-2 triplet
    3. 2.3 Deformation to (A)dS
  3. 3 Family of spin-2/spin-3 topological systems
    1. 3.1 A family of models
    2. 3.2 Dualisation of the spin-2 field
    3. 3.3 First-order reformulation and non-Abelian deformation
    4. 3.4 Systems in (A)dS backgrounds
  4. 4 Chern-Simons formulation and Generalizations
    1. 4.1 Higher spin Quivers
    2. 4.2 Strange higher spin systems
  5. 5 Conclusions
  6. A Dictionary between spinor and vector notation
  7. B Some definitions about quivers

##  1 Introduction

Recently in [1], two of the authors obtained seemingly new action principles
for totally-symmetric tensors on a three-dimensional Minkowski background that
enjoy gauge invariance of the higher-spin type. The procedure for constructing
such action principles followed from the off-shell higher dualisation scheme
of [2]; see, e.g., [3, 4] for recent applications of this procedure in
different contexts. Due to the importance of higher-spin topological systems
in three dimensions, see e.g. [5, 6, 7] and their subsequent developments, we
believe it is important to clarify the status of the systems found in [1],
which all share the property of describing parity-invariant systems in spite
of an explicit dependence on the Levi-Civita symbol in their actions.

One of the motivations of the present paper is therefore to understand to
which extend these actions are giving new, free topological theories in three
dimensions that exhibit higher-spin gauge invariance. The field content and
gauge transformation laws severely restrict the possible actions, but field
redefinitions and dualisations may change the form of the actions.

Before turning our attention to topological systems, we first consider the
simplest case of higher dualisation studied in [1], where the original action
(hence its dual) describe propagating degrees of freedom in three dimensions.
The starting model is Maxwell theory, and in the dual action $S[h_{ab},A_{a}]$
the fields consist of a vector $A_{a}$ and a traceful rank-two tensor
$h_{ab}\,$. As we show in section 2, the dual action $S[h_{ab},A_{a}]$ of [1]
reproduces the spin-two triplet system already known before [8, 9, 10], after
appropriate local field redefinitions. On the other hand, to the best of our
knowledge, the higher-spin topological systems presented in [1] never appeared
before in the literature.

In the class of 3D topological systems that we consider, in section 3 we first
focus on the simplest higher-spin system of [1], that features symmetric
tensor gauge fields of rank two and three, and show that it actually belongs
to a one-parameter family of action principles, with the additional freedom of
choosing the relative sign between the canonical kinetic terms of the two
fields involved. By an abuse of terminology, we henceforth refer to these
systems as spin-3/spin-2 metric-like systems, because of their containing
rank-two and three symmetric tensor gauge fields $h_{ab}$ and
$\varphi_{abc}\,$.

Then, we reformulate the one-parameter family of spin-3/spin-2 metric-like
systems in first order, frame-like form. For this, following the strategy
first applied in three dimensions in [11], one has to modify the field content
in such a way that both the spin-2 and spin-3 sectors are described off-shell
by a pair of one-forms valued in the spin-1 and spin-2 representations of the
Lorentz group $SO(1,2)$, respectively. This is in accordance with the argument
explained in [12] that any (non)interacting topological system in 3D can be
formulated as a Chern-Simons theory. The resulting actions, however, differ
from the flat-space spin-2/spin-3 Chern-Simons actions of [13, 14], mainly
because in our case the spin-2 connection is not associated with a
$sl(2,\mathbb{R})$ subalgebra of a full gauge algebra. After having performed
the frame-like reformulation of the one-parameter family of spin-3/spin-2
metric-like systems in flat space, we study their possible non-Abelian Chern-
Simons extensions and their deformation to the (anti) de Sitter, (A)dS3
background.

We find that the spin-3/spin-2 models in flat space do not admit any non-
Abelian Chern-Simons deformation. On the other hand, we discover that, in
(A)dS3 background, there are very few spin-3/spin-2 models with the same set
of fields and gauge parameters as in flat space. For the flat limit of the
action in (A)dS3 to exactly reproduce the one-parameter family of actions in
flat space, one has to perform a redefinition of the fields and gauge
parameters before sending the cosmological constant to zero. The operations of
performing field redefinitions and flat limit do not commute. The field
redefinition matrices depend on the parameters $z\,$ and $\gamma$ that label
the family of action principles in flat space and the relative sign of the
canonical kinetic terms, and on the sign of the cosmological constant through
a square root. As a result, depending on the values of the constants $z\,$ and
$\gamma\,$ some flat models admit a deformation to AdS3, while other flat
models admit a deformation to dS3 space only.

Then, we show how to generalise these spin-2/spin-3 models to spins $s>3$ and
higher multiplicities of higher-spin fields in the spectrum, both in flat and
(A)dS3 spaces. These models are related to quivers of the wild type, for which
a full classification is not available in the general case. While in flat
space this prevents us from classifying all inequivalent models, in (A)dS3 the
semi-simple nature of the spacetime isometry algebra allows for a
classification, given a spectrum of fields.

##  2 Propagating case with a spin two field in 3D

Here we discuss the simplest action $S[h_{ab},A_{a}]$ studied in [1], that
results from the higher dualisation of a Maxwell field in three dimensions and
that features the tensor fields $A_{a}$ and $h_{ab}\,$, of rank one and two,
respectively. The tensor $h_{ab}$ is symmetric and traceful. Differently from
all the systems that we will analyse later in this paper, the action
$S[h_{ab},A_{a}]$ is not topological: its equations of motion describe the
propagation of a scalar degree of freedom, as it is the case for a Maxwell
field in 3D. Still, the structure of the action $S[h_{ab},A_{a}]$ and of the
gauge transformations that leave it invariant bear similarities with the
metric-like actions and the gauge transformations of the topological higher-
spin systems that we will analyse in section 3. Moreover, the action resulting
from the higher dualisation of Maxwell’s theory belongs to a one-parameter
family of inequivalent actions, as is the case for its higher-spin
counterpart.

###  2.1 From higher dualisation to a family of models

We first review the higher dualisation of a massless vector field in a
spacetime of arbitrary dimension presented in [15, 1]. The starting point is
Maxwell’s action, up to boundary terms that we neglect:

$\displaystyle S[A_{a}]=-\frac{1}{2}\,\int
d^{n}x\left(\partial_{a}A_{b}\,\partial^{a}A^{b}-\partial_{a}A^{a}\,\partial_{b}A^{b}\right).$
(2.1)

One can then introduce a parent action depending on two fields
$(Y^{ab|c}\,,P_{ab})\,$ that have no symmetries under permutations of their
indices, apart from the antisymmetry $Y^{ab|c}=-Y^{ba|c}\,$:

$\displaystyle S[Y^{ab|c},P_{ab}]=\int
d^{n}x\left(P_{ab}\,\partial_{c}Y^{ca|b}-\tfrac{1}{2}\,P_{ab}\,P^{ab}+\tfrac{1}{2}\,P_{a}{}^{a}\,P_{b}{}^{b}\right).$
(2.2)

Extremising with respect to $Y^{ab|c}$ imposes $P_{ab}=\partial_{a}A_{b}$
that, when substituted inside the parent action, reproduces Maxwell’s action
(2.1). On the other hand, $P_{ab}$ is an auxiliary field. Solving for it
inside the parent action yields the dual action

$\displaystyle S[Y^{ab|c}]=\int
d^{n}x\left(\tfrac{1}{2}\,\partial_{c}Y^{ca|b}\,\partial_{d}Y^{d}{}_{a|b}-\tfrac{1}{2(n-1)}\,\partial_{a}\,Y^{ab|}{}_{b}\,\,\partial_{c}\,Y^{cd|}{}_{d}\right),$
(2.3)

invariant under the gauge transformations

$\displaystyle\delta
Y^{ab|}{}_{c}=\delta^{[a}{}_{c}\,\partial^{b]}\epsilon+\partial_{d}\psi^{abd}{}_{c}\;,\qquad\psi^{abd}{}_{c}=\psi^{[abd]}{}_{c}\;.$
(2.4)

Here and below, indices enclosed by a pair of (square) brackets denote a
(anti)symmetrisation with strength one, where dividing by the number of terms
in the (anti)symmetrisation is understood. The $GL(n,\mathbb{R})$ irreducible
decomposition of $Y^{ab|}{}_{c}$ reads
$Y^{ab|}{}_{c}=X^{ab}{}_{c}+2\,\delta_{c}{}^{[a}Z^{b]}\,$, where
$X^{ab}{}_{a}\equiv 0\,$. In dimension $n=3\,$, the above decomposition
amounts to

$\displaystyle
Y^{ab|}{}_{c}=\varepsilon^{abc}\,h_{cd}+2\,\delta_{c}{}^{[a}Z^{b]}\,,\qquad
h_{ab}=h_{ba}\;.$ (2.5)

Sticking to the dimension $n=3\,$, the dual action in terms of $h_{ab}$ and
$Z_{a}$ reads

$\displaystyle S[h_{ab},Z_{a}]=\int
d^{3}x\left(-\tfrac{1}{2}\,\partial_{a}h_{bc}\,\partial^{a}h^{bc}+\tfrac{1}{2}\,\partial_{a}h_{bc}\,\partial^{b}h^{ac}+\tfrac{1}{2}\,\varepsilon^{bcd}\,F_{cd}\,\partial^{a}h_{ab}+\tfrac{1}{4}\,F^{ab}\,F_{ab}\right),$
(2.6)

where $F_{ab}=2\,\partial_{[a}Z_{b]}\,$. The above action is invariant under
the gauge transformations

$\displaystyle\delta h_{ab}=2\,\partial_{(a}\xi_{b)}\;,\qquad\delta
Z_{a}=\partial_{a}\epsilon+\varepsilon_{abc}\,\partial^{b}\xi^{c}\;.$ (2.7)

It is also possible to dualise the vector field $Z_{a}$ into a scalar field
$\phi\,$, following the standard procedure that we will discuss in section
2.2. After dualisation, the action reads [1]

$\displaystyle S[h_{ab},\varphi]=\int
d^{3}x\left[-\tfrac{1}{2}\,\partial_{a}h_{bc}\,\partial^{a}h^{bc}+\partial_{a}h^{ab}\,\partial^{c}h_{bc}+2\,\partial_{a}\varphi\,(\partial^{a}\varphi+\partial_{b}h^{ab})\right]\;,$
(2.8)

where the field $h_{ab}$ transforms as in (2.7) and the scalar $\varphi$
transforms as $\delta\varphi=-\partial^{a}\xi_{a}\,$. After the field
redefinition $\varphi=\frac{1}{2}\,(\phi+h)$ that combines the trace of the
field $h_{ab}$ with the new scalar field $\phi$, one obtains the equivalent
action

$\displaystyle S[h_{ab},\phi]=\int d^{3}x$
$\displaystyle[-\tfrac{1}{2}\,\partial_{a}h_{bc}\,\partial^{a}h^{bc}+\tfrac{1}{2}\,\partial_{a}h\,\partial^{a}h-\partial_{a}h\,\partial_{b}h^{ab}+\partial_{a}h^{ab}\,\partial^{c}h_{bc}$
$\displaystyle\qquad+\tfrac{1}{2}\,\partial_{a}\phi\,\partial^{a}\phi+\partial_{a}\phi(\partial_{b}h^{ab}-\partial^{a}h)]\;,$
(2.9)

where the scalar field $\phi$ does not transform and the field $h_{ab}$ still
transforms as in (2.7). The terms quadratic in the first derivative of
$h_{ab}$ reproduce the (massless) Fierz-Pauli Lagrangian. Extremising the
action with respect to both fields and combining the field equation for $\phi$
with the trace of the field equation for $h_{ab}$ yields the following set of
equations:

$\displaystyle\Box\phi=0\;,\quad\Box
h-\partial^{a}\partial^{b}h_{ab}=0\;,\quad
R_{ab}:=\partial_{ab}h-2\,\partial_{(a}\partial^{c}h_{b)c}+\Box
h_{ab}=\partial_{a}\partial_{b}\phi\;.$ (2.10)

The first two equations taken alone would lead to a doubling of degrees of
freedom corresponding to two scalars propagating in 3 dimensions. However, the
“wrong” relative sign for the kinetic terms of the two fields inside the
action (2.6) or its dual (2.1) is responsible for the third equation in
(2.10), which identifies the curvatures of the two fields and reduces the
degrees of freedom to a single scalar, in agreement with the starting point
for a Maxwell field in 3D. Indeed, by construction, the higher dualisation
procedure does not change the number of degrees of freedom. See [1] for more
discussions and generalisations to higher dimensions.

The action (2.6) resulting from the higher dualisation of a Maxwell field in
3D is, in fact, a member of the following one-parameter family of actions,

$\displaystyle S_{1}[h_{ab},Z_{a}]=\frac{1}{2}\int d^{3}x\,\big{(}$
$\displaystyle-\partial_{a}h_{bc}\partial^{a}h^{bc}+(\alpha+2)\,\partial\cdot
h_{a}\partial\cdot h^{a}-(\alpha+1)\,\partial^{a}h\left[\,2\,\partial\cdot
h_{a}-\partial_{a}h\,\right]$
$\displaystyle-\tfrac{\alpha}{2}\,F_{ab}F^{ab}-\alpha\,\varepsilon_{abc}\,\partial\cdot
h^{a}F^{bc}\big{)}\,,$ (2.11)

which is invariant under the gauge transformations (2.7). The action (2.6) is
recovered for $\alpha=-1\,$. Notice that, although all these actions exhibit
the antisymmetric Levi-Civita symbol, they are invariant under both parity and
time-reversal transformations, under which the fields transform as
$A_{a}\mapsto A_{a}$ and $h_{ab}\mapsto-h_{ab}\,$. The sign flip in the latter
transformation can be understood recalling that the field $h_{ab}$ first
appeared in (2.5) contracted with an antisymmetric 3D symbol. The same
transformations can then be postulated for all members of the one-parameter
family of action principles.

The family of actions (2.11) provides a simple example of couplings between
free fields in Minkowski space induced by $\varepsilon$-terms, that we further
explore in section 3. In this case, for $\alpha=0\,$ the vector field
disappears from the action (2.11), that reduces to the Fierz-Pauli action in
3D Minkowski space. This leads to a discontinuity in the number of propagating
degrees of freedom, since the higher dualisation procedure preserves the
number of propagating degrees of freedom, while the Fierz-Pauli action does
not propagate any degrees of freedom in three dimensions.

###  2.2 Dualisation of the spin-1 field and link with a spin-2 triplet

As anticipated in the previous section, since the action (2.11) only depends
on $A_{a}$ via its field strength, the vector field can be dualised into a
scalar [1]. To this end, one considers the antisymmetric tensor $F_{ab}$ as an
independent field in place of the curl of $Z_{a}\,$ that appears in
$S_{1}[h_{ab},Z_{a}]\,$ (see Eq. (2.11)), and one constructs the following
action:

$S_{\textrm{parent}}[h_{ab},F_{ab},\varphi]=S_{1}[h_{ab},F_{ab}]+\alpha\int
d^{3}x\,\varepsilon_{abc}\,\varphi\,\partial^{a}F^{bc}\,,$ (2.12)

where we fixed the normalisation so as to simplify some of the ensuing
formulae. Extremising the new action with respect to $\varphi$ one recovers
the Bianchi identity for the field strength as an equation of motion:

$\frac{\delta
S_{\textrm{parent}}}{\delta\varphi}=0\quad\Rightarrow\quad\partial_{[a}F_{bc]}=0\quad\Rightarrow\quad
F_{ab}=\partial_{a}Z_{b}-\partial_{b}Z_{a}\,.$ (2.13)

Extremising the parent action with respect to $F_{ab}$ one obtains instead

$\frac{\delta S_{\textrm{parent}}}{\delta F^{ab}}=0\quad\Rightarrow\quad
F_{ab}=-\,\varepsilon_{abc}\left(2\,\partial^{c}\varphi+\partial\cdot
h^{c}\right),$ (2.14)

and substituting this algebraic relation into the action gives

$\begin{split}S_{0}[h_{ab},\varphi]=\frac{1}{2}\int
d^{3}x\,\big{(}&-\partial_{a}h_{bc}\partial^{a}h^{bc}+2\,\partial\cdot
h_{a}\partial\cdot h^{a}-(\alpha+1)\,\partial^{a}h\left[\,2\,\partial\cdot
h_{a}-\partial_{a}h\,\right]\\\
&-4\alpha\left[\,\partial_{a}\varphi\,\partial^{a}\varphi-\varphi\,\partial\cdot\partial\cdot
h\,\right]\big{)}\,.\end{split}$ (2.15)

This action is invariant under

$\delta
h_{ab}=2\,\partial_{(a}\xi_{b)}\,,\qquad\delta\varphi=-\,\partial\cdot\xi\,.$
(2.16)

In fact, it is easy to check that the above action (2.15) is gauge invariant
in arbitrary dimension. For $\alpha=-1$ one recovers the action (2.28) of [1]
which we reproduced above in (2.8). For the same value of $\alpha$, the action
also corresponds to that of a spin-2 triplet [8, 9, 10] after the elimination
of the field with an algebraic equations of motion. The action of a spin-2
triplet, reviewed, e.g., in [16, 17], indeed reads

$S_{\textrm{triplet}}=\int
d^{n}x\left(-\tfrac{1}{2}\,\partial_{a}h_{bc}\partial^{a}h^{bc}+2\,\partial\cdot
h_{a}\,{\cal C}^{a}+2\,\partial\cdot{\cal C}\,{\cal D}+\partial_{a}{\cal
D}\,\partial^{a}{\cal D}-{\cal C}_{a}{\cal C}^{a}\right)$ (2.17)

and it is invariant under

$\delta h_{ab}=2\,\partial_{(a}\xi_{b)}\,,\qquad\delta{\cal
C}_{\mu}=\Box\xi_{a}\,,\qquad\delta{\cal D}=\partial\cdot\xi\,.$ (2.18)

The equation of motion for ${\cal C}_{a}$ is algebraic:

${\cal C}_{a}=\partial\cdot h_{a}-\partial_{a}{\cal D}\,.$ (2.19)

Substituting it into the action (2.17) one gets

$S_{\textrm{triplet}}=\int
d^{n}x\left(-\tfrac{1}{2}\,\partial_{a}h_{bc}\partial^{a}h^{bc}+\partial\cdot
h_{a}\partial\cdot h^{a}+2\left[\,\partial_{a}{\cal D}\,\partial^{a}{\cal
D}+{\cal D}\,\partial\cdot\partial\cdot h\,\right]\right),$ (2.20)

that is, when $D=3$, the action (2.15) with ${\cal D}=-\varphi$ and
$\alpha=-1$.

Alternatively, as pointed out in [18], the action (2.20) can also be obtained
starting from the Maxwell-like action

$S_{\textrm{M-L}}=\int
d^{n}x\left(-\tfrac{1}{2}\,\partial_{a}h_{bc}\partial^{a}h^{bc}+\partial\cdot
h_{a}\partial\cdot h^{a}\right),$ (2.21)

with a traceful $h_{ab}$.111Considering the same action with a traceless
$h_{ab}$ along the lines of [19] gives instead an action equivalent to the
Fierz-Pauli one. A similar pattern applies to higher-spin fields: Maxwell-like
actions for traceless fields are equivalent to Fronsdal ones [19], while the
same actions for traceful fields are equivalent to higher-spin triplet systems
[18]. Reducible spectra with less propagating fields can also be obtained by
imposing the vanishing of only some traces of the fields [20]. This action is
invariant under

$\delta
h_{ab}=2\,\partial_{(a}\xi_{b)}\quad\textrm{with}\quad\partial\cdot\xi=0\,.$
(2.22)

The differential constraint on the gauge parameter can however be eliminated
via the Stueckelberg shift

$h_{ab}\to h_{ab}-2\,\partial_{(a}\theta_{b)}\,,$ (2.23)

where the new field transforms as $\delta\theta_{a}=\xi_{a}$. The resulting
action can only depend on $\theta_{a}$ via its divergence: introducing the
field ${\cal D}=\partial\cdot\theta$ gives back the action (2.20).

In conclusion, the indecomposable system obtained from the higher dualisation
of the Maxwell action in three dimensions corresponds to the dualisation of
the already known indecomposable system given by the triplet [8, 9, 10], in
its simplified version involving only two fields [16, 17, 18]. The dualisation
substituting the triplet’s scalar with a vector is, obviously, only possible
in three dimensions. On the other hand, the one-parameter family of actions
(2.15) and, consequently, the action (2.20) for $\alpha=-1$, can be formulated
in any space-time dimensions.

###  2.3 Deformation to (A)dS

The triplet system can be deformed to (A)dS [21, 22, 17, 23, 24, 18];
similarly, the action (2.15) involving a scalar besides the rank-two field
admits a deformation to (A)dS for any value of the parameter $\alpha$. The
action

$\begin{split}S_{0}=\frac{1}{2}\int
d^{3}x\,\big{(}&-\nabla_{\\!a}h_{bc}\nabla^{a}h^{bc}+2\,\nabla\cdot
h_{a}\nabla\cdot h^{a}-(\alpha+1)\,\nabla^{a}h\left[\,2\,\nabla\cdot
h_{a}-\nabla_{\\!a}h\,\right]\\\
&-4\alpha\left[\,\nabla_{\\!a}\varphi\,\nabla^{a}\varphi-\varphi\,\nabla\cdot\nabla\cdot
h\,\right]-2\,\sigma\lambda^{2}\left[h_{ab}h^{ab}+\alpha\,h^{2}-4\alpha\,\varphi^{2}\right]\big{)}\,,\end{split}$
(2.24)

where $\nabla$ denotes the (A)dS covariant derivative while we parameterize
the cosmological constant as $\Lambda=-\sigma\lambda^{2}$, is indeed invariant
under

$\delta
h_{ab}=2\,\nabla_{\\!(a}\xi_{b)}\,,\qquad\delta\varphi=-\,\nabla\cdot\xi\,.$
(2.25)

On the other hand, it is not possible to preserve a deformation of the gauge
symmetry (2.7) of the action (2.11) involving a vector besides the rank-two
field. The most general action giving back (2.11) in the flat, $\lambda\to 0$
limit is

$\displaystyle S=\frac{1}{2}\int\\!d^{3}x\,\big{(}$
$\displaystyle-\nabla_{\\!a}h_{bc}\nabla^{a}h^{bc}+(\alpha+2)\,\nabla\cdot
h_{a}\nabla\cdot h^{a}-(\alpha+1)\left[\,2\,\nabla\cdot
h_{a}\nabla^{a}h-\nabla_{\\!a}h\nabla^{a}h\,\right]$
$\displaystyle-\tfrac{\alpha}{2}\,F_{ab}F^{ab}-\alpha\,\varepsilon_{abc}\nabla\cdot
h^{a}F^{bc}+\lambda\,a_{1}\,Z^{a}\nabla\cdot
h_{a}+\lambda\,a_{2}\,h\nabla\cdot Z$ (2.26)
$\displaystyle+\sigma\lambda^{2}\left[\,m_{1}^{2}\,h_{ab}h^{ab}+m_{2}^{2}\,h^{2}+m_{3}^{2}\,Z_{a}Z^{a}\,\right]\big{)}\,,$

and one can consider gauge transformations of the type

$\delta
h_{ab}=2\,\nabla_{\\!(a}\xi_{b)}+\sigma\lambda\,k_{1}\,g_{ab}\epsilon\,,\qquad\delta
A_{a}=\partial_{a}\epsilon+\varepsilon_{abc}\partial^{b}\xi^{c}+\sigma\lambda\,k_{2}\,\xi_{a}\,.$
(2.27)

Still, it is not possible to preserve the gauge symmetries generated by
$\xi_{a}$ and $\epsilon$ for any choice of the coefficients.222Choosing
$a_{1}=a_{2}=0$, $m_{1}^{2}=2(\alpha-1)$, $m_{2}^{2}=-2\alpha$,
$m_{3}^{2}=4\alpha$ together with $k_{1}=k_{2}=0$ allows one to preserve the
gauge symmetry generated by $\xi_{a}$, while the one generated by $\epsilon$
is broken by the non-vanishing mass-like term for the vector. This result
anticipates some subtlelties in the deformation to (A)dS of the action
principles directly given or suggested by the higher dualisation procedure
that we shall encounter in the following sections, although we shall discuss
this issue mainly in the frame-like reformulation of our new models.

##  3 Family of spin-2/spin-3 topological systems

In this section, we analyse in details the simplest exotic model with higher-
spin gauge symmetry, a model that one obtains from the higher dualisation of a
massless spin-2 field in three-dimensional Minkowski spacetime. We first show
that it actually belongs to a one-parameter family of inequivalent exotic
models in flat space, whose spectrum of fields consists of the pair
$(h_{ab},\varphi_{abc})$ of traceful, symmetric tensors. We also deform these
flat-space spin-2/spin-3 models to the (A)dS3 background and show that the
one-parameter freedom disappears. In other words, there exists only a discrete
number of spin-2/spin-3 models in (A)dS${}_{3}\,$. Retrospectively, this means
that there is a one-parameter freedom in taking the flat limit of the models
in (A)dS${}_{3}\,$, at least at the level of the equations of motion.

###  3.1 A family of models

It turns out that the action found in [1] from the dualization of the Fierz-
Pauli action in three dimensions is a member of a the following family of
actions for the traceful, symmetric tensors $h_{ab}$ and
$\varphi_{abc}\,$,333The symbols $h$ and $\varphi_{a}\,$ denote, respectively,
the trace of the tensors $h_{bc}$ and $\varphi_{abc}\,$.

$\displaystyle S[\varphi_{abc},h_{ab}]=\frac{1}{2}\int d^{3}\\!x\,\Big{(}$
$\displaystyle\;a_{0}\,\partial_{a}\varphi_{bcd}\,\partial^{a}\varphi^{bcd}+a_{1}\,\partial^{a}\varphi^{b}\,\partial^{c}\varphi_{abc}+a_{2}\,\partial_{a}\varphi^{abc}\,\partial^{d}\varphi_{bcd}$
$\displaystyle+a_{3}\,\partial_{a}\varphi_{b}\,\partial^{a}\varphi^{b}+a_{4}\,\partial_{a}\varphi^{a}\,\partial^{b}\varphi_{b}$
$\displaystyle+b_{0}\,\partial_{a}h_{bc}\,\partial^{a}h^{bc}+b_{1}\,\partial_{a}h\,\partial^{a}h+b_{2}\,\partial^{a}h_{ab}\,\partial_{c}h^{bc}+b_{3}\,\partial^{a}h\,\partial_{c}h_{a}{}^{c}$
$\displaystyle+c_{1}\,\varepsilon_{pqr}\,\partial^{a}h_{a}{}^{p}\,\partial^{q}\varphi^{r}+c_{2}\,\varepsilon_{apq}\,\partial^{b}h^{ac}\,\partial^{p}\varphi^{q}{}_{bc}\;\Big{)}\,.$
(3.1)

These actions are invariant under gauge transformations of the form

$\displaystyle\delta\varphi_{abc}$
$\displaystyle=3\,\partial_{(a}\xi_{bc)}-3x\,\varepsilon_{(a}{}^{pq}\,\eta_{bc)}\partial_{p}\epsilon_{q}\;,$
(3.2a) $\displaystyle\delta h_{ab}$
$\displaystyle=2\,\partial_{(a}\epsilon_{b)}-2z\,\varepsilon_{pq(a}\partial^{p}\xi^{q}{}_{b)}\;,$
(3.2b)

where the gauge parameter $\xi_{ab}$ is symmetric and traceless and the
parameters $x$ and $z$ are fixed by the requirement of gauge invariance,
leading to an interesting one-parameter family of models. This is done in
several steps:

  * •

First of all, we assume that the spectrum of fields indeed involves a genuine
rank-3 symmetric tensor $\varphi_{abc}\,$, with its traceless part appearing
in the action: this means that the parameters $a_{0}$, $a_{1}$, $a_{2}$ and
$c_{2}$ cannot all vanish. Under this assumption, we find that $b_{0}$ is
never zero: we can therefore choose to fix it to $\pm 1/2$ by rescaling the
$h_{ab}$ field and possibly by flipping the sign of the whole action. We will
write $\gamma$ for this sign choice: thus, $b_{0}=\gamma/2\,$.

  * •

Next, one finds that $x=0$ if and only if $z=0$: in that case, there is no
mixing in the gauge transformations. As a result, one gets $c_{1}=c_{2}=0$,
i.e. no terms in the Lagrangian mixing the two fields. The action then reduces
to the sum (or difference) of the Fronsdal action for spin $3$ and the Fierz-
Pauli action for spin $2$ with some arbitrary relative sign $\gamma\,$. In
what follows, we will therefore assume $x\neq 0$ and $z\neq 0\,$, implying
that at least $c_{1}$ or $c_{2}$ is nonvanishing: there is a genuine mixing in
the action between the two fields.

  * •

We then use the freedom of rescaling the $\varphi_{abc}$ field. The generic
case is $a_{0}\neq 0$: we can then fix $a_{0}=-1$ (possibly by again flipping
the sign of the whole action), and $\gamma$ is the relative sign between the
$\varphi_{abc}$ and $h_{ab}$ kinetic terms. All the other parameters are then
fixed in terms of $\gamma$ and the parameter $z$: from the requirement of
gauge invariance, one finds

$x=-\frac{2\gamma z}{9\left(3\gamma z^{2}-2\right)}\;,$ (3.3)

and

$\displaystyle a_{0}$ $\displaystyle=-1\,,\quad a_{1}=7\gamma z^{2}-6\,,\quad
a_{2}=3-2\gamma z^{2}\,,$ (3.4a) $\displaystyle a_{3}$
$\displaystyle=-\frac{1}{d}\left(49z^{4}-75\gamma z^{2}+27\right)\,,\quad
a_{4}=-\frac{1}{4d}\left(172z^{4}-195\gamma z^{2}+54\right)\,,$ (3.4b)
$\displaystyle b_{0}$ $\displaystyle=\frac{\gamma}{2}\,,\quad
b_{1}=-\frac{\gamma}{2d}\left(8\gamma z^{2}-9\right)\,,\quad
b_{2}=-\frac{3\gamma}{d}\left(4\gamma z^{2}-3\right),$ (3.4c) $\displaystyle
b_{3}$ $\displaystyle=\frac{\gamma}{d}\left(8\gamma z^{2}-9\right)\,,\quad
c_{1}=-\frac{2\gamma z}{d}\left(14\gamma z^{2}-9\right)\,,\quad c_{2}=2\gamma
z\,,$ (3.4d)

where the denominator $d$ appearing in several terms is $d=16\gamma
z^{2}-9\,$. This is a genuine one-parameter family of inequivalent actions, as
all the freedom of field rescalings has been used up. The action of [1]
describing a higher dualisation of the Fierz-Pauli action is recovered for
$\gamma=+1$ and $z=-1\,$ (hence $x=2/9$). As observed in [1], a wrong relative
sign between the kinetic terms is a characteristic of actions obtained by the
higher-dualisation procedure. Note that for $\gamma=+1$, there are values for
the parameter $z$ where (3.3) or (3.4) diverge. The spectrum changes at those
values: the case $z=\pm 3/4$ (where $d=0$) corresponds, after multiplying the
action by a global factor of $d$, to the case $a_{0}=a_{1}=a_{2}=c_{2}=0$
where the traceless part of $\varphi_{abc}$ drops out of the action.
Similarly, the case $z=\pm\sqrt{2/3}$ where (3.3) diverges corresponds to
removing the usual gauge transformation of the spin two field, i.e., the first
term in $\delta h_{ab}\,$. We therefore exclude these cases.

  * •

We finally discuss the remaining exotic case $a_{0}=0$, where the usual
kinetic term $\partial_{a}\varphi_{bcd}\,\partial^{a}\varphi^{bcd}$ for the
spin 3 field is absent. This is an isolated point: we fix the normalisation of
$\varphi_{abc}$ by $a_{2}=-1$, and the solution reads

$\displaystyle a_{0}$ $\displaystyle=0\,,\quad a_{1}=\frac{7}{2}\,,\quad
a_{2}=-1\,,\quad a_{3}=-\frac{49}{32}\,,\quad a_{4}=-\frac{43}{32}\,,$ (3.5a)
$\displaystyle b_{0}$ $\displaystyle=\frac{1}{2}\,,\quad
b_{1}=-\frac{1}{4}\,,\quad b_{2}=-\frac{3}{4}\,,\quad b_{3}=\frac{1}{2}\,,$
(3.5b) $\displaystyle c_{1}$ $\displaystyle=-\frac{7}{4\sqrt{2}}\,,\quad
c_{2}=\sqrt{2}\,,$ (3.5c) $\displaystyle x$
$\displaystyle=-\frac{2\sqrt{2}}{27}\,,\quad z=\frac{1}{\sqrt{2}}\,.$ (3.5d)

###  3.2 Dualisation of the spin-2 field

Inside the above one-parameter action, in the generic case with $a_{0}=-1\,$,
the spin-2 field $h_{ab}$ appears only through its curl

$\displaystyle\omega_{abc}(h):=\partial_{a}h_{bc}-\partial_{b}h_{ac}\;.$ (3.6)

Indeed, one finds that

$\displaystyle S[\varphi_{abc},h_{ab}]=\frac{1}{2}\int d^{3}x\Big{[}$
$\displaystyle-\partial_{a}\varphi_{bcd}\,\partial^{a}\varphi^{bcd}+a_{1}\,\partial^{a}\varphi^{b}\,\partial^{c}\varphi_{abc}+a_{2}\,\partial_{a}\varphi^{abc}\,\partial^{d}\varphi_{bcd}$
$\displaystyle+a_{3}\,\partial_{a}\varphi_{b}\,\partial^{a}\varphi^{b}+a_{4}\,\partial_{a}\varphi^{a}\,\partial_{b}\varphi^{b}+\tfrac{\gamma}{4}\,\omega^{abc}(h)\,\omega_{abc}(h)+\tfrac{\beta}{2d}\,\omega^{ab\,}{}_{b}(h)\,\omega_{ac\,}{}^{c}(h)$
$\displaystyle+\,\varepsilon_{abc}\,\omega^{ab\,}{}_{d}(h)\left(\tfrac{\mu}{d}\,\partial^{d}\varphi^{c}+\nu\,\partial^{e}\varphi_{e}{}^{cd}\right)\Big{]}\;,$
(3.7)

where the constants $a_{1}\,$, $a_{2}\,$, $a_{3}$, and $a_{4}$ take the same
values as in (3.4) while

$\displaystyle\beta=9\gamma-8z^{2}\;,\quad d=16\gamma
z^{2}-9\;,\quad\mu=z(14z^{2}-9\gamma)\;,\quad\nu=-\gamma\,z\;.$ (3.8)

As a result, one can dualise the spin-2 field $h_{ab}$ and trade it for a
vector field $A_{a}$, in analogy with what we did for the spin-1 field in
section 2.2. This is achieved by introducing the parent action in the usual
way, with the coupling
$\varepsilon_{abc}\,\omega^{ab}{}_{d}\,\partial^{c}A^{d}\,$, where now
$\omega_{abc}$ is an independent field satisfying the following algebraic
symmetries:

$\displaystyle\omega_{bac}\equiv-\omega_{abc}\;,\qquad\omega_{[abc]}\equiv
0\;.$ (3.9)

The field $\omega_{abc}$ is auxiliary. One can solve for it in terms of the
fields $\varphi_{abc}$ and $A_{a}$ by using its own field equations. Upon
substituting the corresponding expression of the auxiliary field
$\omega_{abc}$ inside the parent action
$S[\varphi_{abc},\omega_{abc},A_{a}]\,$, one finds the action

$\displaystyle S[\varphi_{abc},A_{a}]=\int d^{3}x\,\Big{[}$
$\displaystyle\,\tfrac{8z^{2}}{9}\,\partial^{a}A^{b}\,\partial_{a}A_{b}+k_{1}\,\partial^{a}A^{b}\,\partial_{b}A_{a}$
$\displaystyle+k_{2}\,\partial_{a}\varphi_{b}\,\partial^{b}A^{a}+k_{3}\,\partial_{a}\varphi_{b}\,\partial^{a}A^{b}-2z\,\partial^{c}\varphi_{abc}\,\partial^{a}A^{b}$
$\displaystyle+k_{4}\,\partial_{a}\varphi^{a}\,\partial_{b}\varphi^{b}+k_{5}\,\partial_{b}\varphi_{c}\,\partial^{b}\varphi^{c}+\tfrac{3}{2}\,\partial_{a}\varphi^{abc}\,\partial^{d}\varphi_{bcd}$
$\displaystyle+k_{6}\,\partial^{c}\varphi^{b}\,\partial^{d}\varphi_{bcd}-\tfrac{1}{2}\,\partial_{d}\varphi_{abc}\,\partial^{d}\varphi^{abc}\Big{]}\;$
(3.10)

for some definite values of the six parameters $\\{k_{i}\\}_{i=1,\ldots,6}\,$
that are functions of the parameters $z$ and $\gamma\,$, and that we will not
need to specify here.

The above action $S[\varphi_{abc},A_{a}]$ is invariant under

$\displaystyle\delta\varphi_{abc}$
$\displaystyle=3\,\partial_{(a}\xi_{bc)}-3\,x\,\varepsilon_{(a}{}^{pq}\,\eta_{bc)}\partial_{p}\epsilon_{q}\;,\quad
x=-\tfrac{2\gamma z}{9\left(3\gamma z^{2}-2\right)}\;,$ (3.11a)
$\displaystyle\delta A_{a}$
$\displaystyle=\alpha\,\partial^{b}\xi_{ab}+\beta\,\epsilon_{abc}\,\partial^{b}\epsilon^{c}\;,\quad\alpha=z\,\tfrac{56\,\gamma\,z^{2}-27}{32\,z^{2}-18\,\gamma}\;,\qquad\beta=\tfrac{280\,\gamma\,z^{4}-423\,z^{2}+162\,\gamma}{864\,z^{4}-1062\,\gamma\,z^{2}+324}\;.$
(3.11b)

We note that the gauge transformation of $A_{a}$ is not proportional to the
gauge transformation of the trace $\varphi_{a}\,$ and that there is no real
value for $z$ such that the parameter $\beta$ would vanish. The vector field
is thus independent of the spin-3 field. One can perform the field
redefinition

$\displaystyle\phi_{abc}$
$\displaystyle:=\varphi_{abc}+\zeta\;\eta_{(ab}A_{c)}\;,\qquad\zeta=-\tfrac{12\left(48\,\gamma\,z^{5}-59\,z^{3}+18\,\gamma\,z\right)}{840\,z^{6}-1829\,\gamma\,z^{4}+1332\,z^{2}-324\,\gamma}$
(3.12)

that leads to a transformation law where the vector gauge parameter
$\epsilon_{a}\,$ drops out:

$\displaystyle\delta\phi_{abc}$
$\displaystyle=3\,\partial_{(a}\xi_{bc)}-\tau\;\eta_{(ab}\partial^{d}\xi_{c)d}\;,$
(3.13)

for some value of the parameter $\tau\,$ we do not need to display here. The
point is that the vector field $A_{a}$ still transforms with the parameter
$\epsilon_{a}\,$, therefore it is not possible to have a set of independent
fields $\\{A_{a},\varphi_{abc}\\}\,$ both of which being inert under the
$\epsilon_{a}$ gauge transformations, showing that the action principle (3.2)
cannot be recast into a spin-$3$ triplet system. Moreover, a spin-$3$ triplet
propagates a scalar degree of freedom even in $3D$, while any member of our
family of actions is topological. We shall make this manifest in section 3.3
by showing that the action (3.7) can be rewritten in Chern-Simons-like form.

Therefore, to the best of our knowledge, with the action (3.7) we have a
genuinely new action principle for a topological system involving a spin-3 and
a spin-2 gauge fields or, equivalently, a spin-3 and a spin-1 fields, if one
chooses to dualise the spin-2 field into a spin-1 field, as we have done in
this section. In the latter case, one can also notice the absence of the Levi-
Civita symbol in the action (3.2). Moreover, the Levi-Civita symbol and the
gauge parameter $\epsilon_{a}$ only enter the gauge transformations (3.11) via
the combination $\epsilon_{abc}\,\partial^{b}\epsilon^{c}$ that can be traded
for a divergenceless vector. In analogy with what we observed for the action
(2.15), this suggests the option to define an action with the same field
content and similar gauge transformations also in a Minkowski background of
arbitrary dimension. Indeed, the action

$\begin{split}S[\varphi_{abc},A_{a}]=\frac{1}{2}\int
d^{n}x\Big{(}&-\partial_{a}\varphi_{bcd}\,\partial^{a}\varphi^{bcd}+\tilde{a}_{1}\,\partial^{a}\varphi^{b}\,\partial^{c}\varphi_{abc}+\tilde{a}_{2}\,\partial_{a}\varphi^{abc}\,\partial^{d}\varphi_{bcd}\\\
&+\tilde{a}_{3}\,\partial_{a}\varphi_{b}\,\partial^{a}\varphi^{b}+\tilde{a}_{4}\,\partial_{a}\varphi^{a}\,\partial_{b}\varphi^{b}\\\
&+n(n+1)\partial_{a}A_{b}\partial^{a}A^{b}+\tilde{b}_{1}\,\partial_{a}A_{b}\partial^{b}A^{a}\\\
&+\tilde{c}_{1}\,\partial^{c}\varphi_{abc}\partial^{a}A^{b}+\tilde{c}_{2}\,\partial_{a}\varphi_{b}\partial^{a}A^{b}+\tilde{c}_{3}\,\partial_{a}\varphi_{b}\partial^{b}A^{a}\Big{)}\end{split}$
(3.14)

is invariant under

$\displaystyle\delta\varphi_{abc}=3\left(\partial_{(a}\widehat{\xi}_{bc)}+\eta_{(ab}\Lambda_{c)}\right),\quad\delta
A_{a}=\sqrt{3}\left(\alpha\,\partial\cdot\widehat{\xi}_{a}+\tfrac{(n+2)\alpha\pm
2}{2}\,\Lambda_{a}\right),\quad\partial\cdot\Lambda=0\;,$ (3.15)

for any value of the space-time dimension $n$ provided that

$\displaystyle\tilde{a}_{1}$ $\displaystyle=-3\left(2\pm
n\,\alpha\right),\quad\tilde{a}_{2}=3\,,\quad\tilde{a}_{3}=\tfrac{3}{4}\left(n\,\alpha\left((n+1)\alpha\pm
4\right)+4\right),$ (3.16a) $\displaystyle\tilde{a}_{4}$
$\displaystyle=\tfrac{3}{8}\left(n\,\alpha\left((n-2)\alpha\pm
4\right)+4\right),\quad\tilde{b}_{1}=\tfrac{1}{2}\,n(n-2)\,,$ (3.16b)
$\displaystyle\tilde{c}_{1}$ $\displaystyle=\pm
2\sqrt{3}\,n\,,\quad\tilde{c}_{2}=-\sqrt{3}\,n\left((n+1)\alpha\pm
2\right),\quad\tilde{c}_{3}=-\tfrac{\sqrt{3}\,n}{2}\left((n-2)\alpha\pm
2\right).$ (3.16c)

We fixed the normalisation of the fields by conventionally fixing the
coefficients in front of the terms
$\partial_{a}\varphi_{bcd}\,\partial^{a}\varphi^{bcd}$ and
$\partial_{a}A_{b}\partial^{a}A^{b}$, taking into account that gauge
invariance requires them to have opposite sign. Notice that we obtained a one-
parameter family of actions and that the field $A_{b}$ cannot be gauged away
because the gauge parameter $\Lambda_{a}$ is divergenceless, while $A_{a}$ is
an arbitrary vector. To the best of our knowledge, the action (3.2), which has
the same field content of a spin-3 triplet but displays a different gauge
symmetry, was never studied before it will be interesting to analyze its
spectrum for $n>3$.

In the following we will reformulate the new action principle (3.7) in an
Abelian Chern-Simons-like form. After we have done it, we will be able to
study its possible non-Abelian deformations and to generalise it to many new
topological systems in both flat and (A)dS3 backgrounds.

###  3.3 First-order reformulation and non-Abelian deformation

We now investigate the first-order formulation of the family of models (3.1),
in terms of one-forms $(e^{a},\omega^{ab})$ for the spin-2 field and
$(E^{ab},\Omega^{ab,c})$ for the spin-3 field, in agreement with the strategy
first developed in [25, 11]. In particular, we will recover the gauge
transformations (3.2) for the fields $h_{ab}$ and $\varphi_{abc}$ after a
Lorentz-like partial gauge fixing, and exhibit an Abelian Chern-Simons-like
action for these models, in accordance with the general discussion in [12]. We
recall that these models are defined around the Minkowski three-dimensional
background. In Section 3.4 we will consider deformations to the (A)dS3
background.

#### Gauge transformations.

A general ansatz for the variations of the one-forms $e^{a}=e_{c}{}^{a}\,{\rm
d}x^{c}$ and $E^{ab}=E_{c}{}^{ab}\,{\rm d}x^{c}$ is

$\displaystyle\delta E_{a,bc}$
$\displaystyle=\partial_{a}\xi_{bc}-\alpha_{bc,a}+x\,(\eta_{bc}\tilde{\Lambda}_{a}-3\eta_{a(b}\tilde{\Lambda}_{c)})\;,$
(3.17a) $\displaystyle\delta e_{a,b}$
$\displaystyle=\partial_{a}\epsilon_{b}-\Lambda_{ab}+2z\,\tilde{\alpha}_{ab}\;,$
(3.17b)

where

$\tilde{\alpha}_{ab}=\tfrac{1}{2}\,\varepsilon_{apq}\,\alpha_{b}{}^{p,q}\;,\qquad\tilde{\Lambda}_{a}=\tfrac{1}{2}\,\varepsilon_{abc}\,\Lambda^{bc}\;.$
(3.18)

Here, $E_{a,bc}$ is symmetric and traceless in $b,c$ with no other symmetry
involving the index $a\,$. The parameter $\alpha_{bc,a}$ is a traceless hook
in the symmetric convention:

$\displaystyle\alpha_{bc,a}$
$\displaystyle=\alpha_{(bc),a}\;,\qquad\alpha_{(bc,a)}=0\;,\qquad\eta^{bc}\alpha_{bc,a}=0=\eta^{ab}\alpha_{bc,a}\;.$
(3.19)

The Lorentz parameter $\Lambda_{ab}$ is antisymmetric. These parameters are
those appearing in the gauge transformations of the connections
$\omega^{bc}=\omega_{a}{}^{bc}\,{\rm d}x^{a}$ and
$\Omega^{bc,d}=\Omega_{a}{}^{bc,d}\,{\rm d}x^{a}$:

$\displaystyle\delta\omega_{a}{}^{bc}=\partial_{a}\Lambda^{bc}\,,\quad\delta\Omega_{a}{}^{bc,d}=\partial_{a}\alpha^{bc,d}\,.$
(3.20)

From the transformations (3.17), it is clear that we can use $\alpha$ and
$\Lambda$ to gauge-fix to zero the corresponding components of the frame-like
fields $e^{a}$ and $E^{ab}\,$, i.e., the traceless hook part of $E_{a,bc}$ and
the antisymmetric part of $e_{a,b}\,$. One calls such a gauge the Lorentz-like
gauge. Residual gauge transformations then have to satisfy $\delta
E_{a,bc}\big{|}_{\text{traceless hook}}=0$ and $\delta e_{[a,b]}=0\,$: this
gives

$\displaystyle\alpha_{bc,a}^{\text{res.}}$
$\displaystyle=\partial_{a}\xi_{bc}\big{|}_{\text{traceless
hook}}=\partial_{a}\xi_{bc}-\partial_{(a}\xi_{bc)}+\frac{1}{3}\left(\eta_{bc}\,\partial\\!\cdot\\!\xi_{a}-\eta_{a(b}\,\partial\\!\cdot\\!\xi_{c)}\right)\,,$
(3.21) $\displaystyle\Lambda_{ab}^{\text{res.}}$
$\displaystyle=\partial_{[a}\epsilon_{b]}\,.$ (3.22)

Notice that there is no entanglement of gauge parameters here: indeed,
$\Lambda$ only appears through pure trace terms in $\delta E_{a,bc}$ so is not
involved in its traceless part. Similarly, because $\alpha$ is traceless, we
have $\tilde{\alpha}_{[ab]}=0$ identically and therefore $\alpha$ does not
appear in $\delta e_{[a,b]}\,$.

After the Lorentz-like gauge-fixing, the Fronsdal and Fierz-Pauli fields

$\varphi_{abc}:=3\,E_{(a,bc)}\,,\qquad h_{ab}:=2\,e_{(a,b)}$ (3.23)

transform as

$\displaystyle\delta\varphi_{abc}=3\,\partial_{(a}\xi_{bc)}-3x\,\varepsilon_{pq(a}\,\eta_{bc)}\partial^{p}\epsilon^{q}\;,\quad\delta
h_{ab}=2\,\partial_{(a}\epsilon_{b)}-2z\,\varepsilon_{pq(a}\partial^{p}\xi^{q}{}_{b)}\;,$
(3.24)

making contact with the original gauge transformations (3.2).

In differential form notation (that we will use from now on), the gauge
transformations (3.17)–(3.20) read444Repeated covariant or contravariant
indices are implicitly symmetrised with strength one.

$\displaystyle\delta E^{aa}$ $\displaystyle={\rm
d}\xi^{aa}-h_{b}\,\alpha^{aa,b}+x\,(\eta^{aa}\,h^{b}\,\tilde{\Lambda}_{b}-3h^{a}\tilde{\Lambda}^{a})$
$\displaystyle={\rm
d}\xi^{aa}+\tfrac{4}{3}\,h_{b}\,\varepsilon^{abc}\,\tilde{\alpha}_{c}{}^{a}-3x\,(h^{a}\tilde{\Lambda}^{a}-\tfrac{1}{3}\,\eta^{aa}\,h^{b}\,\tilde{\Lambda}_{b})\,,$
(3.25a) $\displaystyle\delta\Omega^{aa}$ $\displaystyle={\rm
d}\tilde{\alpha}^{aa}$ (3.25b)

for the spin 3 sector, and

$\displaystyle\delta e^{a}$ $\displaystyle={\rm
d}\xi^{a}+h_{b}\,\Lambda^{ab}+2z\,h_{b}\,\tilde{\alpha}^{ab}$
$\displaystyle={\rm
d}\xi^{a}-\varepsilon^{abc}\,h_{b}\,\tilde{\Lambda}_{c}+2z\,h_{b}\,\tilde{\alpha}^{ab}\;,$
(3.26a) $\displaystyle\delta\omega^{a}$ $\displaystyle={\rm
d}\tilde{\Lambda}^{a}$ (3.26b)

for the spin 2 sector. We have dualised the connection one-forms, similarly to
(3.18): $\omega_{a}=\tfrac{1}{2}\,\varepsilon_{abc}\,\omega^{bc}\;$,
$\Omega_{ab}=\tfrac{1}{2}\,\varepsilon_{apq}\,\Omega_{b}{}^{p,q}\;$. These
objects carry less indices and we will use them exclusively in what follows.
The one-forms $h^{a}$ are the background dreibeins for Minkowski space: e.g.,
in Cartesian coordinates they read $h^{a}=\delta^{a}{}_{b}\,{\rm d}x^{b}$.

#### First-order action for the strange topological system.

The first-order action invariant under the gauge transformations (3.25)–(3.26)
is

$\displaystyle S[e^{a},\omega^{a},E^{aa},\Omega^{aa}]=\int_{M_{3}}$
$\displaystyle\Big{[}\omega_{a}({\rm
d}e^{a}-\frac{1}{2}\,\varepsilon^{apq}\,h_{p}\,\omega_{q})+2z\;\omega_{a}h_{b}\,\Omega^{ab}$
$\displaystyle\quad+\frac{2z}{3x}\;\Omega_{aa}({\rm
d}E^{aa}+\frac{2}{3}\,\varepsilon^{apq}\,h_{p}\,\Omega_{q}{}^{a})\Big{]}\;.$
(3.27)

This action can be rewritten in the form

$S[e^{a},\omega^{a},E^{aa},\Omega^{aa}]=\frac{1}{2}\int_{M_{3}}\Big{[}\omega_{a}R^{a}(e)+e_{a}R^{a}(\omega)+\frac{2z}{3x}\left(\Omega_{ab}R^{ab}(E)+E_{ab}R^{ab}(\Omega)\right)\Big{]}\,,$
(3.28)

where the invariant curvatures read

$\displaystyle R^{aa}(E)$ $\displaystyle={\rm
d}E^{aa}+\tfrac{4}{3}\,h_{p}\,\varepsilon^{pqa}\,\Omega_{q}{}^{a}-3x\,(h^{a}\omega^{a}-\tfrac{1}{3}\,\eta^{aa}\,h^{b}\,\omega_{b})\;,$
$\displaystyle R^{aa}(\Omega)$ $\displaystyle={\rm d}\Omega^{aa}\;,$ (3.29a)
$\displaystyle R^{a}(e)$ $\displaystyle={\rm
d}e^{a}-\varepsilon^{abc}\,h_{b}\,\omega_{c}+2z\,h_{b}\,\Omega^{ab}\;,$
$\displaystyle R^{a}(\omega)$ $\displaystyle={\rm d}\omega^{a}\;.$ (3.29b)

These curvatures satisfy the Bianchi identities

$\displaystyle 0$ $\displaystyle\equiv{\rm
d}R^{aa}(E)+\tfrac{4}{3}\,h_{p}\,\varepsilon^{pqa}\,R_{q}{}^{a}(\Omega)-3x\,(h^{a}R^{a}(\omega)-\tfrac{1}{3}\,\eta^{aa}\,h^{b}\,R_{b}(\omega))\;,$
$\displaystyle 0$ $\displaystyle\equiv{\rm d}R^{aa}(\Omega)\;,$ (3.30a)
$\displaystyle 0$ $\displaystyle\equiv{\rm
d}R^{a}(e)-\;\varepsilon^{abc}\,h_{b}\,R_{c}(\omega)\;+2z\,h_{b}\,R^{ab}(\Omega)\;,$
$\displaystyle 0$ $\displaystyle\equiv{\rm d}R^{a}(\omega)\;.$ (3.30b)

The relative factor $2z/3x$ between the spin two and spin three parts of the
action (3.28) is necessary for gauge invariance. The field equations obtained
from the above action simply read

$\displaystyle R^{a}(e)=0\;,\quad R^{a}(\omega)=0\;,\quad R^{aa}(E)=0\;,\quad
R^{aa}(\Omega)=0\;.$ (3.31)

As it is clear from the form (3.3) of the action, the connections $\omega^{a}$
and $\Omega^{aa}$ are auxiliary fields: they can be expressed in terms of the
frame-like fields $e^{a}$ and $E^{aa}$ by solving their field equations
algebraically. The first-order action principle, upon expressing the auxiliary
fields in terms of $e^{a}$ and $E^{aa}$, then gives a second-order action
principle for the latter fields which is (3.1) in the case $x,z\neq 0$.

In particular, in the special case where $z=-1$ and $\gamma=+1\,$, therefore
$x=2/9\,$, we have shown that the Abelian Chern-Simons-like action (3.3)
reproduces the metric-like action obtained in [1] by performing an off-shell
higher-dualisation of three-dimensional linearised gravity around Minkowski
background, as expected from [12].

#### Non-Abelian deformations.

Now that we have reformulated the spin-3/spin-2 systems in first-order form,
we are ready to study their non-Abelian deformations. It turns out that,
unfortunately, there is none.

We search for a nonlinear extension of (3.25)–(3.26) and (3.3) in the form

$\displaystyle S[e^{a},\omega^{a},E^{aa},\Omega^{aa}]=\int_{M_{3}}{\rm
Tr}(\tfrac{1}{2}\,A{\rm d}A+\tfrac{1}{3}\,A^{3})\;.$ (3.32)

From the structure of the action (3.3), it is clear that the relevant
connection 1-form is

$\displaystyle
A=\omega^{a}\,J_{a}+(h^{a}+e^{a})\,P_{a}+\Omega^{ab}\,J_{ab}+E^{ab}\,P_{ab}\;.$
(3.33)

It is also clear that, up to a normalisation, we have the standard Killing
form [13, 14]:

$\displaystyle{\rm Tr}(J_{a}P_{b})=\eta_{ab}\;,\qquad{\rm
Tr}(J_{ab}P_{cd})=\frac{z}{3x}\,(\eta_{ac}\eta_{bd}+\eta_{ad}\eta_{bc}-\tfrac{2}{3}\,\eta_{ab}\eta_{cd})\;.$
(3.34)

We search for a nonlinear extension of the action (3.3), which entails finding
a non-Abelian algebra for the $8+8$ generators
$\\{J_{a},J_{ab},P_{a},P_{ab}\\}\,$. From the linearised action and the gauge
transformations laws (3.25)–(3.26), we can already read off some of the
commutation relations, those that imply the generators of the background
connection $A_{0}=h^{a}\,P_{a}\,$:

$\displaystyle[P_{a},J_{b}]$
$\displaystyle=-\,\varepsilon_{abc}\,P^{c}-3x\,P_{ab}\;,$
$\displaystyle[P_{a},P_{b}]=0\;,$ (3.35) $\displaystyle[P_{a},J_{bc}]$
$\displaystyle=2z\left(\eta_{a(b}\,P_{c)}-\tfrac{1}{3}\,\eta_{bc}\,P_{a}\right)+\tfrac{4}{3}\,\varepsilon_{a(b}{}^{m}\,P_{c)m}\;,$
$\displaystyle[P_{a},P_{bc}]=0\;.$ (3.36)

This is our initial datum. We must now parametrise all the other commutators
and constrain them via the Jacobi identities. The idea of the proof is to
write down the most general Ansatz for the other commutators and check whether
the Jacobi identities are satisfied. There are a priori twenty Jacobi
identities to be checked. We find that at least one of these identities cannot
be satisfied, for all nonzero values of the parameters $x$ and $z\,$. There is
therefore no non-Abelian deformation of the theory.

Notice that, had we found a Lie algebra, the resulting non-Abelian Chern-
Simons action would have been an exotic higher-spin extension of 3D gravity,
in the sense that the spin-2 sector would not have been a consistent
truncation of the full theory, differently from the higher-spin theories
considered in, e.g., [11, 5, 6, 13, 14]. In other words, the generators
$\\{P_{a},J_{a}\\}$ would not have formed a subalgebra, as is clear from the
commutator $[P_{a},J_{b}]=-\,\varepsilon_{abc}\,P^{c}-3x\,P_{ab}\,$, since the
parameter $x$ is nonzero.

###  3.4 Systems in (A)dS backgrounds

In this section, we first classify the most general gauge-invariant, first-
order field equations for the fields $(e^{a},\omega^{a},E^{aa},\Omega^{aa})$
in (A)dS${}_{3}\,$ that can be cast as zero-curvature conditions. We find
that, in (A)dS${}_{3}\,$, it is always possible to perform field
redefinitions555Such redefinitions are not available in Minkowski space since
the fields have different dimensions; in (A)dS3, a dimensionful parameter (the
cosmological constant) is available to resolve the mismatch. within the spin-2
sector $(e^{a},\omega^{a})$ and the spin-3 sector $(E^{aa},\Omega^{aa})\,$ in
such a way as to produce seven inequivalent models, six of which are defined
in AdS3 and one in dS3. An action admitting these field equations upon
variation can then be constructed. This is in sharp contrast with the
continuous one parameter (and a sign) family of actions (3.28) and equations
(3.31) in flat space.

We then consider the flat limit of the field equations and actions in
(A)dS${}_{3}\,$ and investigate whether one can reproduce the family (3.28)
and (3.31) of models in flat space. For this, one first has to perform field
redefinitions in both spin sectors $(e^{a},\omega^{a})$ and
$(E^{aa},\Omega^{aa})\,$, bringing in the free parameter $z\,$, before sending
the cosmological constant to zero. We find that this is always possible for
the field equations, while the limit of the action can be defined only when
the free parameter $z$ assumes a finite set of numerical values. Equivalently,
it means that the family of flat space actions (3.28) admits a deformation to
(A)dS3 only when the free parameter $z$ assumes some very specific values:
only a finite number of members of the family admit a deformation to
(A)dS${}_{3}\,$.

Since we are dealing with topological systems, the difference between the
Minkowski and (A)dS3 backgrounds reflects the crucial difference in the
representation theory of the corresponding isometry algebras. The finite-
dimensional, non-unitary representations of the Poincaré algebra are much more
numerous than those of the isometry algebras of the (A)dS3 backgrounds [26,
27, 28, 29, 30].

#### Conventions.

Going to (A)dS3 background amounts to considering the Lorentz-covariant
derivative one-form valued operator $\nabla=h^{a}\,\nabla_{a}$ such that

$\displaystyle\nabla^{2}V^{a}=-\sigma\lambda^{2}\;h^{a}\,h_{b}\,V^{b}\;.$
(3.37)

The sign parameter $\sigma$ is such that $\sigma=1$ corresponds to anti-de
Sitter and $\sigma=-1$ to de Sitter, and the $h^{a}$ are now the vielbeins of
(A)dS. In these conventions, the one-forms $h^{a}$ have a dimension of length,
while $\lambda$ has the dimension of mass. The cosmological constant is
$\Lambda=-\sigma\,\lambda^{2}\,$.

#### Gauge transformations and curvatures.

The most general gauge transformations and corresponding gauge-invariant
curvatures for the spectrum of fields considered in the previous section are

$\displaystyle\delta e^{a}$ $\displaystyle=\nabla\xi^{a}+\lambda
x_{1}\,\varepsilon^{abc}\,h_{b}\,\xi_{c}+x_{2}\,\varepsilon^{abc}\,h_{b}\,\tilde{\Lambda}_{c}+\lambda
x_{3}\,h_{b}\xi^{ab}+x_{4}\,h_{b}\,\tilde{\alpha}^{ab}\;,$ (3.38a)
$\displaystyle\delta\omega^{a}$
$\displaystyle=\nabla\tilde{\Lambda}^{a}+\lambda^{2}x_{5}\,\varepsilon^{abc}h_{b}\xi_{c}+\lambda
x_{6}\,\varepsilon^{abc}\,h_{b}\,\tilde{\Lambda}_{c}+\lambda^{2}x_{7}\,h_{b}\,\xi^{ab}+\lambda
x_{8}\,h_{b}\,\tilde{\alpha}^{ab}\;,$ (3.38b) $\displaystyle\delta E^{aa}$
$\displaystyle=\nabla\xi^{aa}+\lambda
x_{9}\,(h^{a}\xi^{a}-\tfrac{1}{3}\,\eta^{aa}\,h^{b}\,\xi_{b})+x_{10}\,(h^{a}\tilde{\Lambda}^{a}-\tfrac{1}{3}\,\eta^{aa}\,h^{b}\,\tilde{\Lambda}_{b})$
$\displaystyle\qquad+\lambda
x_{11}\,h_{b}\,\varepsilon^{abc}\,\xi_{c}{}^{a}+x_{12}\,h_{b}\,\varepsilon^{abc}\,\tilde{\alpha}_{c}{}^{a}\;,$
(3.38c) $\displaystyle\delta\Omega^{aa}$
$\displaystyle=\nabla\tilde{\alpha}^{aa}+\lambda^{2}x_{13}\,(h^{a}\xi^{a}-\tfrac{1}{3}\,\eta^{aa}\,h^{b}\,\xi_{b})+\lambda
x_{14}\,(h^{a}\tilde{\Lambda}^{a}-\tfrac{1}{3}\,\eta^{aa}\,h^{b}\,\tilde{\Lambda}_{b})$
$\displaystyle\qquad+\lambda^{2}x_{15}\,h_{b}\,\varepsilon^{abc}\,\xi_{c}{}^{a}+\lambda
x_{16}\,h_{b}\,\varepsilon^{abc}\,\tilde{\alpha}_{c}{}^{a}\;,$ (3.38d)

and

$\displaystyle R^{a}(e)$ $\displaystyle=\nabla e^{a}+\lambda
x_{1}\,\varepsilon^{abc}\,h_{b}\,e_{c}+x_{2}\varepsilon^{abc}\,h_{b}\,\omega_{c}+\lambda
x_{3}\,h_{b}E^{ab}+x_{4}\,h_{b}\,\Omega^{ab}\;,$ (3.39a) $\displaystyle
R^{a}(\omega)$
$\displaystyle=\nabla\omega^{a}+\lambda^{2}x_{5}\,\varepsilon^{abc}h_{b}e_{c}+\lambda
x_{6}\,\varepsilon^{abc}\,h_{b}\,\omega_{c}+\lambda^{2}x_{7}\,h_{b}\,E^{ab}+\lambda
x_{8}\,h_{b}\,\Omega^{ab}\;,$ (3.39b) $\displaystyle R^{aa}(E)$
$\displaystyle=\nabla E^{aa}+\lambda
x_{9}\,(h^{a}e^{a}-\tfrac{1}{3}\,\eta^{aa}\,h^{b}\,e_{b})+x_{10}\,(h^{a}\omega^{a}-\tfrac{1}{3}\,\eta^{aa}\,h^{b}\,\omega_{b})$
$\displaystyle\qquad+\lambda
x_{11}\,h_{b}\,\varepsilon^{abc}\,E_{c}{}^{a}+x_{12}\,h_{b}\,\varepsilon^{abc}\,\Omega_{c}{}^{a}\;,$
(3.39c) $\displaystyle R^{aa}(\Omega)$
$\displaystyle=\nabla\Omega^{aa}+\lambda^{2}x_{13}\,(h^{a}e^{a}-\tfrac{1}{3}\,\eta^{aa}\,h^{b}\,e_{b})+\lambda
x_{14}\,(h^{a}\omega^{a}-\tfrac{1}{3}\,\eta^{aa}\,h^{b}\,\omega_{b})$
$\displaystyle\qquad+\lambda^{2}x_{15}\,h_{b}\,\varepsilon^{abc}\,E_{c}{}^{a}+\lambda
x_{16}\,h_{b}\,\varepsilon^{abc}\,\Omega_{c}{}^{a}\;.$ (3.39d)

The Bianchi identities take the same form: for example,

$0\equiv\nabla R^{a}(e)+\lambda
x_{1}\,\varepsilon^{abc}\,h_{b}\,R_{c}(e)+x_{2}\,\varepsilon^{abc}\,h_{b}\,R_{a}(\omega)+\lambda
x_{3}\,h_{b}R^{ab}(E)+x_{4}\,h_{b}\,R^{ab}(\Omega)\,.$ (3.40)

It is useful to rewrite the gauge transformations in matrix form

$\displaystyle\delta\begin{pmatrix}\lambda e^{a}\\\ \omega^{a}\end{pmatrix}$
$\displaystyle=\nabla\begin{pmatrix}\lambda\xi^{a}\\\
\tilde{\Lambda}^{a}\end{pmatrix}+\lambda
A\,\varepsilon^{abc}h_{b}\begin{pmatrix}\lambda\xi_{c}\\\
\tilde{\Lambda}_{c}\end{pmatrix}+\lambda
B\,h_{b}\begin{pmatrix}\lambda\xi^{ab}\\\ \tilde{\alpha}^{ab}\end{pmatrix}\;,$
(3.41a) $\displaystyle\delta\begin{pmatrix}\lambda E^{aa}\\\
\Omega^{aa}\end{pmatrix}$
$\displaystyle=\nabla\begin{pmatrix}\lambda\xi^{aa}\\\
\tilde{\alpha}^{aa}\end{pmatrix}+\lambda
C\,\left(h^{a}\delta^{a}_{b}-\tfrac{1}{3}\eta^{aa}h_{b}\right)\begin{pmatrix}\lambda\xi^{b}\\\
\tilde{\Lambda}^{b}\end{pmatrix}+\lambda
D\,\varepsilon^{abc}h_{b}\begin{pmatrix}\lambda\xi_{c}{}^{a}\\\
\tilde{\alpha}_{c}{}^{a}\end{pmatrix}\;,$ (3.41b)

with the matrices, only involving dimensionless coefficients, explicitly given
by

$A=\begin{pmatrix}x_{1}&x_{2}\\\ x_{5}&x_{6}\end{pmatrix}\,,\quad
B=\begin{pmatrix}x_{3}&x_{4}\\\ x_{7}&x_{8}\end{pmatrix}\,,\quad
C=\begin{pmatrix}x_{9}&x_{10}\\\ x_{13}&x_{14}\end{pmatrix}\,,\quad
D=\begin{pmatrix}x_{11}&x_{12}\\\ x_{15}&x_{16}\end{pmatrix}\,.$ (3.42)

The requirement of gauge invariance of the curvatures gives sixteen quadratic
equations on the parameters $x_{i}\,$. In matrix form, they read

$\displaystyle A^{2}-\tfrac{5}{6}\,BC$ $\displaystyle=\sigma I\,,$ (3.43a)
$\displaystyle\tfrac{1}{2}\,D^{2}+CB$ $\displaystyle=2\sigma I\,,$ (3.43b)
$\displaystyle-\tfrac{3}{2}\,DC+CA$ $\displaystyle=0\,,$ (3.43c)
$\displaystyle-\tfrac{3}{2}\,BD+AB$ $\displaystyle=0\,.$ (3.43d)

We look for solutions that mix the spin-2 and spin-3 sectors, implying that
the matrices $B$ and $C$ cannot simultaneously vanish.

We can of course redefine the fields and gauge parameters in each sector,

$\displaystyle\begin{pmatrix}\lambda e^{a}\\\
\omega^{a}\end{pmatrix}=M\begin{pmatrix}\lambda e^{\prime a}\\\ \omega^{\prime
a}\end{pmatrix}\,,\quad\begin{pmatrix}\lambda\xi^{a}\\\
\tilde{\Lambda}^{a}\end{pmatrix}=M\begin{pmatrix}\lambda\xi^{\prime a}\\\
\tilde{\Lambda}^{\prime a}\end{pmatrix}\;,$ (3.44a)
$\displaystyle\begin{pmatrix}\lambda E^{aa}\\\
\Omega^{aa}\end{pmatrix}=N\begin{pmatrix}\lambda E^{\prime aa}\\\
\Omega^{\prime aa}\end{pmatrix}\,,\quad\begin{pmatrix}\lambda\xi^{aa}\\\
\tilde{\alpha}^{aa}\end{pmatrix}=N\begin{pmatrix}\lambda\xi^{\prime aa}\\\
\tilde{\alpha}^{\prime aa}\end{pmatrix}\;,$ (3.44b)

with $M,N$ arbitrary $GL(2,\mathbb{R})$ matrices. Then, for the primed fields
and gauge parameters, the gauge transformations take the same form (3.41),
with primed matrices given by

$A^{\prime}=M^{-1}AM\,,\quad B^{\prime}=M^{-1}BN\,,\quad
C^{\prime}=N^{-1}CM\,,\quad D^{\prime}=N^{-1}DN\,.$ (3.45)

Note that this transformation leaves equations (3.43) invariant, as it should.
Therefore, two solutions of the matrix equations (3.43) that differ by a
transformation of the above form must be regarded as equivalent. Also note
that, if $(A,B,C,D)$ is a solution of the system (3.43), then so is
$(-A,-B,-C,-D)\,$. Notice that the transformation
$(A,B,C,D)\mapsto(A,-B,-C,D)\,$, that is a symmetry of the system (3.43), can
be generated by a $GL(2,\mathbb{R})\times GL(2,\mathbb{R})$ transformation
with $(M,N)=(\mathbb{I},-\mathbb{I})\,$.

The algebraic problem at hand – classifying matrices $A$, $B$, $C$ $D$
satisfying (3.43) up to the equivalences (3.45) – can be viewed as a quiver
representation problem. The quiver in our case is

and contains two vertices that each correspond to a two-dimensional space
$\mathbb{R}^{2}$; the left vertex for the spin-2 sector and the right vertex
for the spin-3 sector. The four edges correspond to the maps $A$, $B$, $C$,
$D$ from a vector space to another. Basic definitions about quivers can be
found in reference [31] and appendix B.

This a wild quiver, since the underlying graph

is neither Dynkin nor Euclidean, i.e. does not correspond to a simply-laced
simple Lie algebra or their affine extensions. Representations of wild quivers
are not classified in general; however, in this particular case a full
classification can be achieved because of the small dimensions and number of
matrices involved. More generally, the problem in (A)dS3 is indeed equivalent
to identifying a finite-dimensional representation of the (anti)-de Sitter
algebra reproducing the set of fields at hand. This is done in what follows.
The flat limit will then be studied, exhibiting a one-parameter freedom.

#### Classification of the solutions.

Using $GL(2,\mathbb{R})\times GL(2,\mathbb{R})$ transformations generated by
the matrices $M$ and $N\,$, the matrices $A$ and $D$ can be put in one of the
following three real Jordan forms:

$\begin{pmatrix}\lambda_{1}&~{}0\\\
0&~{}\lambda_{2}\end{pmatrix}\,,\quad\begin{pmatrix}\lambda&~{}1\\\
0&~{}\lambda\end{pmatrix}\quad\text{or}\quad\begin{pmatrix}\mu&~{}-\nu\\\
\nu&~{}\mu\end{pmatrix}\,,$ (3.46)

where we insist that all the entries are real. The last of those three forms
is similar to a complex diagonal matrix with complex conjugate eigenvalues
$\mu\pm i\nu\,$.

A detailed analysis shows that there are two general classes of solutions.

  1. 1.

The four matrices $A,B,C$ and $D$ are all real diagonal (first form in
(3.46)). This requires $\sigma=1\,$, the AdS3 background.

  2. 2.

In the second case, they are all antisymmetric (third form in (3.46) with
$\mu=0$). This is possible only in the dS3 background, i.e. $\sigma=-1$.

In the flat case, which formally amounts to taking $\sigma=0\,$, it is easy to
show that we exactly recover the results of the section 3.3, formulae (3.25)
and (3.26).

Fully diagonal case: If the four matrices are diagonal, then the fields
$(e^{a},E^{aa})$ and $(\omega^{a},\Omega^{aa})$ form two separate
systems.666Recall that to reach the form (3.46) for the matrices entering the
gauge transformations (3.41) we allowed redefinitions of fields and
parameters, see eq. (3.44). For the spin-2 sector, for instance, the fields in
each separate system can originate from linear combinations of the non-linear
vielbein and spin-connection. Writing any of those generically as
$(f^{a},F^{aa})$, they have gauge transformations of the form

$\displaystyle\delta f^{a}$
$\displaystyle=\nabla\epsilon^{a}+\lambda\,a\,\varepsilon^{abc}\,h_{b}\,\epsilon_{c}+\lambda\,b\,h_{b}\,\epsilon^{ab}\;,$
(3.47a) $\displaystyle\delta F^{aa}$
$\displaystyle=\nabla\epsilon^{aa}+\lambda\,c\,\left(h^{a}\,\delta^{a}_{b}-\tfrac{1}{3}\,\eta^{aa}\,h_{b}\right)\epsilon^{b}+\lambda\,d\,\varepsilon^{abc}\,h_{b}\epsilon_{c}{}^{a}.$
(3.47b)

Here, the parameters $a$, $b$, $c$, $d$ are real numbers (the diagonal
elements of the corresponding matrices) constrained to satisfy

$\displaystyle a^{2}-\tfrac{5}{6}\,bc$ $\displaystyle=\sigma\,,$ (3.48a)
$\displaystyle\tfrac{1}{2}\,d^{2}+cb$ $\displaystyle=2\sigma\,,$ (3.48b)
$\displaystyle-\tfrac{3}{2}\,dc+ca$ $\displaystyle=0\,,$ (3.48c)
$\displaystyle-\tfrac{3}{2}\,bd+ab$ $\displaystyle=0\,.$ (3.48d)

These equations only admit solutions in AdS3, i.e. for $\sigma=+1$. This
conclusion is reached from an analysis of the free equations of motion, but it
anticipates the option to define interacting higher-spin gauge theories in
AdS3 from the sum of two non-Abelian Chern-Simons actions, that stems from the
structure of the isometry algebra of AdS3. The latter is not simple,
$so(2,2)\cong sl(2,\mathbb{R})\oplus sl(2,\mathbb{R})$, and this allows one,
e.g., to rewrite the Einstein Hilbert action as the difference of two
$sl(2,\mathbb{R})$ actions [32, 33]. The two spin-2 fields belonging to the
two separate systems discussed above are thus the analogues of the two
connections entering the two $sl(2,\mathbb{R})$ actions of [32, 33] or the
generalisations thereof studied, e.g., in [34].

There are solutions of (3.48) where the spin $2$ and spin $3$ sectors of the
system do not mix:

$a=1\,,\quad b=0\,,\quad c=0\,,\quad d=\pm 2\,.$ (3.49)

This case corresponds to the free limit of a $sl(3,\mathbb{R})$ action.
Combining the two separate systems as, e.g., in [6] one obtains a model that
can be deformed into an interacting higher-spin theory described by a
$sl(3,\mathbb{R})\oplus sl(3,\mathbb{R})$ Chern-Simons theory.777In this
context the sign freedom in (3.49) has a neat interpretation: one can indeed
introduce the connections $f^{a}_{\pm}=\omega^{a}\pm\lambda e^{a}$ for the
spin-2 sector and then choose to define the corresponding connections for the
spin-3 sector either as $f^{aa}_{\pm}=\omega^{aa}\pm\lambda e^{aa}$ or as
$f^{aa}_{\pm}=\omega^{aa}\mp\lambda e^{aa}$. The latter two options
correspond, respectively, to $d=2$ and $d=-2$ in (3.47). In this case, the
spin-2 sector is associated with a $so(2,2)$ subalgebra of the full gauge
algebra. The spin-2 fields thus precisely correspond to the linearization of
the two $sl(2,\mathbb{R})$ connections of [32, 33, 34]. These models admit a
consistent truncation to Einstein’s gravity, contrary to the possible
interacting theories based on more exotic setups in which different spins
already mix in the free theory.

The more interesting solution from our current perspective has both $b$ and
$c$ nonvanishing and mixes the two sectors (spin-2 and spin-3) of the system
$(f^{a},F^{aa})$. It corresponds to

$a=\frac{3}{2}\,,\quad b=1\,,\quad c=\frac{3}{2}\,,\quad d=1\,.$ (3.50)

When two systems $(f^{a}_{(i)},F^{aa}_{(i)})\,$, $i=1,2\,$, are considered
simultaneously so as to reconstruct vielbeins and spin-connections, we can
combine the solutions (3.49) or (3.50) for each system with a relative sign.
In the ensuing analysis we focus on the cases in which at least one of the two
systems $(f^{a}_{(i)},F^{aa}_{(i)})$ mixes the sectors with different spins.
We thus exclude the well-studied case leading to $sl(3,\mathbb{R})\oplus
sl(3,\mathbb{R})$ interacting theories or its generalisations with non-
vanishing torsions [35] (both corresponding to $B=C=0$ in (3.41), but
involving different relative coefficients between the two sectors). Taking
into account (3.49), (3.50) and the relative sign introduced by the
combination of the two systems, there exist only six inequivalent solutions
for the matrices $A,B,C$ and $D$ in the case where they are all real diagonal
and $B$, $C$ are different from zero. They are explicitly given by

$A_{1}=\tfrac{3}{2}\begin{pmatrix}1&~{}0\;\\\ 0&~{}\eta\;\end{pmatrix}\,,\quad
B_{1}=\begin{pmatrix}1&~{}0\\\ 0&~{}\eta\end{pmatrix}\,,\quad
C_{1}=\tfrac{3}{2}\begin{pmatrix}1&~{}0\\\ 0&~{}\eta\end{pmatrix}\,,\quad
D_{1}=\begin{pmatrix}1&~{}0\;\\\ 0&~{}\eta\;\end{pmatrix}\quad(\eta=\pm
1\,,\;\sigma=+1)$ (3.51)

when the two systems both mix spin $2$ and spin $3$, and

$A_{2}=\begin{pmatrix}\tfrac{3}{2}&~{}0\;\\\
0&~{}\eta_{1}\;\end{pmatrix}\,,\quad B_{2}=\begin{pmatrix}1&~{}0\\\
0&~{}0\end{pmatrix}\,,\quad C_{2}=\begin{pmatrix}\tfrac{3}{2}&~{}0\\\
0&~{}0\end{pmatrix}\,,\quad D_{2}=\begin{pmatrix}1&~{}0\;\\\
0&~{}2\eta_{2}\;\end{pmatrix}\quad(\eta_{i}=\pm 1\,,\;\sigma=+1)$ (3.52)

when only one of them does, say the first one $(e^{a},E^{aa})$. We recall that
these solutions only exist in AdS3 space, $\sigma=1$.

Antisymmetric case: Apart from the fully diagonal, real cases presented above,
the only other real solution of (3.43) with $B$ and $C$ different from zero is

$A_{3}=\tfrac{3}{2}\begin{pmatrix}0&~{}1\;\\\ -1&~{}0\;\end{pmatrix}\,,\quad
B_{3}=\begin{pmatrix}0&~{}1\;\\\ -1&~{}0\end{pmatrix}\,,\quad
C_{3}=\tfrac{3}{2}\begin{pmatrix}0&~{}1\;\\\ -1&~{}0\;\end{pmatrix}\,,\quad
D_{3}=\begin{pmatrix}0&~{}1\;\\\ -1&~{}0\;\end{pmatrix}\quad(\sigma=-1)\,.$
(3.53)

This solution exists only in the dS background, $\sigma=-1$. This completes
the classification of the real solutions with mixing of the system of
equations (3.43).

Note that, using $GL(2,\mathbb{R})\times GL(2,\mathbb{R})$ transformations
generated by the matrices $M$ and $N\,$, the solution (3.51) in the case
$\eta=-1$ and the solution (3.53) can be brought in a unified anti-diagonal
form, which has the advantage of being valid for _both_ signs of the
cosmological constant:

$A_{0}=\begin{pmatrix}0&~{}1\;\\\
\frac{9\sigma}{4}&~{}0\;\end{pmatrix}\,,\quad B_{0}=\begin{pmatrix}0&~{}1\\\
\frac{3\sigma}{2}&~{}0\end{pmatrix}\,,\quad C_{0}=\begin{pmatrix}0&~{}1\\\
\frac{3\sigma}{2}&~{}0\end{pmatrix}\,,\quad D_{0}=\begin{pmatrix}0&~{}1\\\
\sigma&~{}0\end{pmatrix}\,.$ (3.54)

This form does not cover the case $\eta=+1$ of solution (3.51), or solution
(3.52).

#### Gauge-invariant action in (A)dS.

We look for an action in the form

$S[e^{a},\omega^{a},E^{aa},\Omega^{aa}]=\frac{1}{2\lambda}\int_{M_{3}}\left[\begin{pmatrix}\lambda
e_{a}&\;&\omega_{a}\end{pmatrix}G\begin{pmatrix}\lambda R^{a}(e)\\\
R^{a}(\omega)\end{pmatrix}+\begin{pmatrix}\lambda
E_{aa}&\;&\Omega_{aa}\end{pmatrix}H\begin{pmatrix}\lambda R^{aa}(E)\\\
R^{aa}(\Omega)\end{pmatrix}\right]\,.$ (3.55)

The $2\times 2$ matrices $G$ and $H$ should be symmetric and non-degenerate.
The constraints arising from gauge invariance of the action read

$\displaystyle A^{T}G-GA$ $\displaystyle=0$ (3.56a) $\displaystyle GB+C^{T}H$
$\displaystyle=0$ (3.56b) $\displaystyle D^{T}H-HD$ $\displaystyle=0\,,$
(3.56c)

with solutions related by a transformation of the form

$G^{\prime}=M^{T}GM\,,\quad H^{\prime}=N^{T}HN$ (3.57)

being considered equivalent (since they differ by the field redefinition
(3.44)). We now discuss the solutions for $G$ and $H$ corresponding to the
solutions for the matrices $A$, $B$, $C$, $D$ found above.

Fully diagonal case: For the six solutions (3.51) and (3.52) in AdS3, one can
show that, by the action of residual transformations by matrices $M$ and
$N\,$, the matrices $G$ and $H$ can be taken to be diagonal. Therefore, we can
again look at subsystems $(f^{a},F^{aa})$ in isolation.888While this is
certainly true for the free theory, we stress that not all combinations may
allow one to introduce non-linear deformations. For instance, in the case of
AdS3 gravity, one introduces the $sl(2,\mathbb{R})$-valued connections
$f^{a}_{\pm}=\omega^{a}\pm\lambda e^{a}$. The different dependence on the
vielbein has an impact on the signs entering the linearized gauge
transformations that read $\delta
f^{a}_{\pm}=\nabla\lambda_{\pm}\pm\varepsilon^{ab}{}_{c}h_{b}\lambda^{c}_{\pm}$,
thus suggesting the need for a given relative sign to allow for a non-linear
completion. The equations (3.56) then reduce to the single constraint
$gb+ch=0$ for the diagonal coefficients (written here $g$ and $h$
generically).

When the system $(f^{a},F^{aa})$ does not mix the spin $2$ and spin $3$ fields
(solution (3.49)), we get the sum of two decoupled actions. The coefficients
$g$ and $h$ can be rescaled by a field redefinition (and/or an overall factor
in the action), leaving only a relative sign: $g=1$, $h=\pm 1$. The action
explicitly reads

$S[f^{a},F^{aa}]=\frac{1}{2}\int_{M_{3}}\left(f_{a}R^{a}(f)\pm
F_{aa}R^{aa}(F)\right)\,,$ (3.58)

with gauge invariance and curvatures

$\displaystyle\delta f^{a}$
$\displaystyle=\nabla\epsilon^{a}+\lambda\,\varepsilon^{abc}\,h_{b}\,\epsilon_{c}\;,$
$\displaystyle\quad R^{a}(f)$ $\displaystyle=\nabla
f^{a}+\lambda\,\varepsilon^{abc}\,h_{b}\,f_{c}\;,$ (3.59a)
$\displaystyle\delta F^{aa}$ $\displaystyle=\nabla\epsilon^{aa}\pm
2\lambda\,\varepsilon^{abc}\,h_{b}\epsilon_{c}{}^{a}\;,$ $\displaystyle\quad
R^{aa}(F)$ $\displaystyle=\nabla F^{aa}\pm
2\lambda\,\varepsilon^{abc}\,h_{b}F_{c}{}^{a}\;.$ (3.59b)

When the system $(f^{a},F^{aa})$ mixes the spin $2$ and spin $3$ sectors
(solution (3.50)), one finds $g=1$ and $h=-\tfrac{2}{3}\,$. The action is then

$S[f^{a},F^{aa}]=\frac{1}{2}\int_{M_{3}}\left(f_{a}R^{a}(f)-\tfrac{2}{3}F_{aa}R^{aa}(F)\right)\,,$
(3.60)

with gauge invariance and curvatures

$\displaystyle\delta f^{a}$
$\displaystyle=\nabla\epsilon^{a}+\tfrac{3}{2}\lambda\,\varepsilon^{abc}\,h_{b}\,\epsilon_{c}+\lambda\,h_{b}\,\epsilon^{ab}\;,$
(3.61a) $\displaystyle\delta F^{aa}$
$\displaystyle=\nabla\epsilon^{aa}+\tfrac{3}{2}\lambda\,\left(h^{a}\,\delta^{a}_{b}-\tfrac{1}{3}\,\eta^{aa}\,h_{b}\right)\epsilon^{b}+\lambda\,\varepsilon^{abc}\,h_{b}\epsilon_{c}{}^{a}\;,$
(3.61b) $\displaystyle R^{a}(f)$ $\displaystyle=\nabla
f^{a}+\tfrac{3}{2}\lambda\,\varepsilon^{abc}\,h_{b}\,f_{c}+\lambda\,h_{b}\,F^{ab}\;,$
(3.61c) $\displaystyle R^{aa}(F)$ $\displaystyle=\nabla
F^{aa}+\tfrac{3}{2}\lambda\,\left(h^{a}\,\delta^{a}_{b}-\tfrac{1}{3}\,\eta^{aa}\,h_{b}\right)f^{b}+\lambda\,\varepsilon^{abc}\,h_{b}F_{c}{}^{a}\;.$
(3.61d)

We can now put two such systems together, corresponding to $(e^{a},E^{aa})$
and $(\omega^{a},\Omega^{aa})$, possibly with relative signs: the matrices $G$
and $H$ are then

$G_{1}=\begin{pmatrix}1&~{}0\\\ 0&~{}\tau\end{pmatrix}\,,\quad
H_{1}=-\frac{2}{3}\begin{pmatrix}1&~{}0\\\ 0&~{}\tau\end{pmatrix}$ (3.62)

for (3.51) and

$G_{2}=\begin{pmatrix}1&~{}0\\\ 0&~{}\tau_{1}\end{pmatrix}\,,\quad
H_{2}=\begin{pmatrix}-\tfrac{2}{3}&~{}0\\\ 0&~{}\tau_{2}\end{pmatrix}$ (3.63)

for solution (3.52), with independent signs $\tau$, $\tau_{1}$, $\tau_{2}=\pm
1\,$. This whole discussion is valid in AdS3 space only ($\sigma=+1$).

Antisymmetric case: The remaining case is that of solution (3.53), which is
valid in dS3 space only ($\sigma=-1$). As explained above, using a
transformation generated by appropriate matrices $M$ and $N$, this case is
covered by the solution (3.54) for $\sigma=-1$. Therefore, the matrices
$G_{3}$ and $H_{3}$ corresponding to the solution (3.53) are not presented
explicitly since they can be obtained from the matrices $G_{0}$ and $H_{0}$
written in (3.64) below.

Off-diagonal case: We now consider the solution (3.54), which is valid in both
dS3 and AdS3 and covers the case (3.51) with $\eta=-1$ (in AdS3) as well as
the case (3.53) (in dS3). With the matrices $A_{0}$, $B_{0}$, $C_{0}$ and
$D_{0}$ of (3.54) considered above, the solution of equations (3.56) is given
by

$G_{0}=\begin{pmatrix}0&~{}1\\\ 1&~{}0\end{pmatrix}=-H_{0}\,,$ (3.64)

up to the action of matrices $M$ and $N$ that leave (3.54) invariant. These
matrices $G_{0}$ and $H_{0}$ reproduce the standard symplectic structures
$e\,{\rm d}\omega$ and $E\,{\rm d}\Omega\,$, respectively.

To conclude, we have found seven inequivalent systems mixing the spin-2 and
spin-3 gauge transformations in (A)dS3 backgrounds. Of those seven solutions,
two can be unified in a form (3.54) – (3.64) that is valid for both signs of
the cosmological constant. The other five exist in AdS3 space only. A
representation-theoretic argument for the existence of a discrete family of
systems will be explained in section 4.

In what follows we first consider the flat limit of the field equations, and
then, the flat limit of the corresponding actions.

#### Flat limit I: curvatures and gauge transformations.

To recover the flat limit (3.25) – (3.26) of the curvatures and gauge
transformations, some parameters are fixed as functions of $x$ and $\gamma\,$:

$x_{2}=-1\,,\quad x_{4}=2z\,,\quad x_{10}=-3x=\frac{2\gamma z}{3\left(3\gamma
z^{2}-2\right)}\,,\quad x_{12}=\frac{4}{3}\,.$ (3.65)

At first sight, it appears that these values for the above four parameters do
not comply with any of the solutions presented above. However, the values
(3.65) can be reached by acting on the simple solution (3.54) with a
$GL(2,\mathbb{R})\times GL(2,\mathbb{R})$ transformation of the form (3.45)
with $z$-dependent matrices $M$ and $N$ given by

$M=\begin{pmatrix}\frac{3}{\sqrt{2}}\,\Delta&\;&z\vspace{0.1cm}\\\
-\frac{9\sigma}{4}\,z&\;&-\frac{3}{\sqrt{2}}\,\Delta\end{pmatrix}\,,\quad
N=\begin{pmatrix}0&\;&1\\\ \frac{3\sigma}{4}&\;&0\end{pmatrix}\,,$ (3.66)

where $\Delta$ is the square root

$\Delta=\sqrt{\gamma\sigma\left(2\gamma z^{2}-1\right)}\,.$ (3.67)

Note that we should be looking at _real_ solutions for the parameters
$x_{i}\,$. The reality of $\Delta$ then determines whether the field equations
(3.31) around Minkowski space can be extended to dS3 ($\sigma=-1$) and/or AdS3
($\sigma=1$):

  * •

If $\gamma=+1$, we have $\Delta=\sqrt{\sigma z^{2}\left(2z^{2}-1\right)}\,$:
the model can be extended to dS3 when $z^{2}<1/2\,$, to AdS3 for
$z^{2}>1/2\,$, and to both when $z^{2}=1/2\,$. In particular, the original
action of [1] corresponds to $z=-1$ and therefore can only be continued to
AdS${}_{3}\,$, not to dS${}_{3}\,$.

  * •

If $\gamma=-1\,$, we have $\Delta=\sqrt{\sigma z^{2}\left(2z^{2}+1\right)}\,$:
these models can only be deformed to AdS${}_{3}\,$.

Equivalently, this means that the field equations (3.31) obtained from the
one-parameter family of actions (3.28) can be reached from the simple solution
(3.54) in dS3 or AdS3 (depending on the values of $\gamma$ and $z$ discussed
above) by _first_ performing a $GL(2,\mathbb{R})\times GL(2,\mathbb{R})$ field
redefinition of the model in curved space, _and then_ taking the flat limit.

To conclude this discussion, let us also cover the isolated case (3.5) with
$a_{0}=0$, which corresponds to

$x_{2}=-1\,,\quad x_{4}=2z=\sqrt{2}\,,\quad
x_{10}=-3x=\frac{2\sqrt{2}}{9}\,,\quad x_{12}=\frac{4}{3}\,.$ (3.68)

This model can only be continued to AdS ($\sigma=+1$). To reach the values
(3.68) from the simple solution (3.54), the matrices $M$ and $N$ can be taken
as

$M=\begin{pmatrix}0&\;&-\frac{3}{\sqrt{2}}\\\
\frac{27}{4\sqrt{2}}&\;&0\end{pmatrix}\,,\quad N=\begin{pmatrix}-2&\;&1\\\
\frac{3}{4}&\;&-\frac{3}{2}\end{pmatrix}\,.$ (3.69)

#### Flat limit II: action.

The previous discussion only applies to the gauge transformations and
curvatures, i.e. at the level of equations of motion. As we shall see, the
existence of an action is much more constrained.

In the action (3.55), we have adjusted the powers of $\lambda$ such that the
terms appearing in (3.28) come with $\lambda^{0}$. Then, the terms
$e_{a}R^{a}(e)$ and $E_{aa}R^{aa}(E)$ come with $\lambda^{1}$ and vanish as
$\lambda\to 0$, while the terms $\omega_{a}R^{a}(\omega)$ and
$\Omega_{aa}R^{aa}(\Omega)$ come with $\lambda^{-1}$ and are singular in the
flat limit. Therefore, the action (3.55) has a smooth flat limit if the
bottom-right entry of the matrices $G$ and $H$ vanishes: $G_{22}=0=H_{22}\,$.

To recover the action (3.28) in the flat limit, we should therefore impose

$G_{22}=0=H_{22}\,,\quad G_{12}=1=G_{21}\,,\quad
H_{12}=\frac{2z}{3x}=H_{21}\;,$ (3.70)

in addition to the conditions (3.65) (one can also impose the opposite of the
values in (3.70), since the global sign of the action is of no relevance
here). Remarkably, the system then only admits solutions for specific values
of the product $xz\,$:

$xz=-\frac{2}{3}\,,\;-\frac{2}{15}\,,\;\frac{2}{45}\,,\;\frac{2}{9}\,.$ (3.71)

Of those values, only $xz=2/9$ is possible in both dS3 ($\sigma=-1$) and AdS3
($\sigma=1$) spaces. Recalling that $x=-\tfrac{2\gamma z}{9\left(3\gamma
z^{2}-2\right)}\,$, the equality $xz=\frac{2}{9}$ implies that
$\gamma\,z^{2}=\frac{1}{2}\,$, which in turn implies that $\gamma=+1$ and
$z=\pm\frac{1}{\sqrt{2}}\,$. The other values of $xz$ correspond to solutions
in AdS3 space only. The dual system of [1] has $xz=-2/9$, hence cannot be
deformed to (A)dS3.

These solutions can most efficiently be described by exhibiting the matrices
$M$ and $N$ that can be used to reach them from some elementary solution
presented above.

  * •

In both dS3 and AdS3, the solution with $xz=2/9$ can be reached from the
simple solution (3.54)– (3.64) by acting with the matrices

$M=\begin{pmatrix}-1&\;&0\\\ 0&\;&1\end{pmatrix}\,,\quad
N=-z\begin{pmatrix}\frac{3}{2}&\;&0\\\ 0&\;&2\end{pmatrix}\,.$ (3.72)

  * •

The other values are in the orbit of the rather strange solution
(3.52)–(3.63), with signs $\tau_{1}=-1$ and $\tau_{2}=+1$ for the matrices $G$
and $H$. The different choices of signs $\eta_{1}$ and $\eta_{2}$ provide the
different values for the product $xz\,$:

$xz=-\frac{2}{3(3-2\eta_{1})(2\eta_{2}-1)}\in\\{-\frac{2}{3}\,,\;-\frac{2}{15}\,,\;\frac{2}{45}\,,\;\frac{2}{9}\\}\;,$
(3.73)

(in particular, $\eta_{1}=-\eta_{2}=1$ provides another inequivalent solution
with $xz=2/9$). The matrices $M$ and $N$ are

$M=\begin{pmatrix}-\sqrt{\frac{3-2\eta_{1}}{2}}&\;&\sqrt{\frac{2}{3-2\eta_{1}}}\\\
0&\;&-\sqrt{\frac{2}{3-2\eta_{1}}}\end{pmatrix}\,,\quad
N=z\sqrt{3-2\eta_{1}}\begin{pmatrix}\frac{3(2\eta_{2}-1)}{2\sqrt{2}}&\;&-\sqrt{2}\\\
0&\;&-\frac{2}{\sqrt{3}}\end{pmatrix}\,.$ (3.74)

We therefore conclude that there is only a discrete set of values of the free
parameter $z$ such that the action (3.28) in Minkowski spacetime admits a
deformation to (A)dS${}_{3}\,$. Of those values, only $\gamma=+1$ and
$z=\pm\frac{1}{\sqrt{2}}\,$ can be deformed to both dS and AdS; the resulting
actions are in the orbit of the system described by the matrices
(3.54)–(3.64).

##  4 Chern-Simons formulation and Generalizations

In sections 2 and 3 a number of peculiar theories with and without propagating
degrees of freedom was discussed, some of which were given a Chern-Simons-like
formulation in section 3. According to [12] all three-dimensional higher spin
theories without propagating degrees of freedom (called topological here) are
equivalent to Chern-Simons theories. The goal of the present section is to
develop a formalism to construct topological higher spin models in order to
explain the examples of section 3 and to find generalizations thereof.

###  4.1 Higher spin Quivers

We would like to describe the space of all topological higher spin theories in
$3D$.999Some of the results below may apply to non-topological systems, e.g.,
topologically massive gravity (contrary to its name, it is not topological in
the sense of having propagating degrees of freedom) and its various
generalizations and higher spin extensions. See [36] for more detail on these
cases. We will consider only manifestly Lorentz covariant theories and,
therefore, assume that all fields can be decomposed into a number of Lorentz
(spin)-tensors. This means that we will consider fields that carry various
representations of $so(1,2)\sim sl_{2}$, i.e. we will have a set of fields

$\displaystyle\Phi^{\alpha_{1}\cdots\alpha_{N}|{\mathcal{I}}}(x)\equiv\Phi^{\alpha(N)|{\mathcal{I}}}(x)$
(4.1)

that are symmetric spin-tensors with indices $\alpha_{1},\ldots,\alpha_{N}$,
which may carry some additional label ${\mathcal{I}}$ — whose range of values
may depend on $N$ — to be able to distinguish different fields valued in the
same $sl_{2}$-module. The fields can be $p$-forms with $p=0,1,2,3$. The
fermions correspond to odd $N$ and are Grassmann odd, which is irrelevant for
the free equations.

#### Relevant algebras.

Lorentz symmetry, i.e. $sl_{2}(\mathbb{R})$, is always manifest in our
approach. Massless fields fall into representations of a larger algebra: the
Poincaré algebra $iso(2,1)$ in flat space, $sl_{2}(\mathbb{C})\cong so(3,1)$
in de Sitter space and $sl(2,\mathbb{R})\oplus sl(2,\mathbb{R})\cong so(2,2)$
in anti-de Sitter space. Let $(j_{1},j_{2})$ denote the
$(2j_{1}+1)(2j_{2}+1)$-dimensional representation of the (anti)-de Sitter
algebra. (Partially)-Massless fields in (anti)-de Sitter space require
representations that are nontrivially charged under the diagonal Lorentz
algebra $sl(2,\mathbb{R})$ contained in the (anti)-de Sitter algebra [11, 37,
6, 5]. In order to cover partially-massless fields [38, 39, 40, 41, 42] one
needs [12] to include representations of type $(N,M)$, $MN\neq 0$ charged
under both algebras, i.e., to consider fields
$\Phi^{\alpha_{1}...\alpha_{N},{\dot{\alpha}}_{1}...{\dot{\alpha}}_{M}|{\mathcal{I}}}(x)$.
(Un)dotted indices are those of $sl(2,\mathbb{R})\oplus sl(2,\mathbb{R})$ or
of $sl_{2}(\mathbb{C})$. The massless case corresponds to $(M,0)$, $(0,M)$. To
make the link to the metric-like formulation of (partially)-massless fields
one has to consider conjugated pairs $(M,N)\oplus(N,M)$. In what follows, we
only keep the Lorentz symmetry manifest. In particular, this is the only
option for the Poincaré case.

Finite-dimensional representations of all algebras mentioned above save for
$iso(2,1)$ are completely reducible. It is the Poincare case that is tricky
and admits a lot of strange topological higher spin systems. It is worth
adding that a classification of finite-dimensional representations of
$iso(2,1)$ is not available. Some examples and references can be found in
section 3 of [43].

#### General topological systems.

It is useful to pack $\Phi^{\alpha(N)|{\mathcal{I}}}(x)$ into a generating
function by contracting the $\alpha$’s with an auxiliary spinor $y_{\alpha}$:

$\displaystyle\Phi(y|x)$ $\displaystyle=\sum_{N,{\cal
I}}\tfrac{1}{N!}\,y_{\alpha_{1}}\cdots
y_{\alpha_{N}}\,\Phi^{\alpha_{1}\cdots\alpha_{N}|{\cal I}}(x)\,.$ (4.2)

Different representations of $sl_{2}$ belong to the eigenspaces of the Euler
operator $N=y^{\alpha}\frac{\partial}{\partial y^{\alpha}}\,$. To allow for
multiplicity, accounted by the index ${\mathcal{I}}$, we assume that for each
eigenvalue of $N$, the field $\Phi(y)$ takes values in some vector space
$V_{N}$.

Let us assume that $\mathcal{M}_{3}$ is a three-dimensional space equipped
with a dreibein $h^{\alpha\beta}$ and a compatible spin-connection and, hence,
we have a Lorentz covariant derivative $\nabla$. The most general topological
system we can write for $\Phi$s all having the same form degree
is:101010Similar systems of varying degree of generality have already been
considered in the literature [44, 36].

$\displaystyle\nabla\Phi$ $\displaystyle=Q\Phi\,,$ (4.3)

where the most general horizontal differential $Q$ reads

$\displaystyle Q\Phi$
$\displaystyle=[\alpha(N)h^{\alpha\alpha}y_{\alpha}y_{\alpha}+\beta(N)h^{\alpha\alpha}\partial_{\alpha}\partial_{\alpha}+\gamma(N)h^{\alpha\alpha}y_{\alpha}\partial_{\alpha}]\Phi\,,$
(4.4)

where $\partial_{\alpha}\equiv\frac{\partial}{\partial y^{\alpha}}$. Here
$\alpha_{N},\beta_{N},\gamma_{N}$ are linear maps

$\displaystyle\alpha_{N}:\ V_{N-2}\rightarrow V_{N}\,,\qquad\beta_{N}:\
V_{N+2}\rightarrow V_{N}\,,\qquad\gamma_{N}:\ V_{N}\rightarrow V_{N}\,,$ (4.5)

i.e. they are matrices that depend (including the size) on
$N=y^{\alpha}\partial_{\alpha}$. For the system to be topological, the
covariant derivative $D=\nabla-Q$ has to be nilpotent. The same condition
implies it is gauge invariant under $\delta\omega=D\xi$, where $\xi$ are zero-
forms taking values in the same collection of vector spaces $V_{N}$. The
nilpotency condition gives a number of conditions:

$\displaystyle(N-2)\alpha(N)\gamma(N-2)=(N+2)\gamma(N)\alpha(N)\,,$ (4.6a)
$\displaystyle(N-2)\gamma(N-2)\beta(N-2)=(N+2)\beta(N-2)\gamma(N)\,,$ (4.6b)
$\displaystyle-(N-1)\alpha(N)\beta(N-2)+\gamma(N)\gamma(N)+(N+3)\beta(N)\alpha(N+2)=\sigma\lambda^{2}\mathbf{1}\,.$
(4.6c)

We assumed that
$\nabla^{2}=\sigma\,\lambda^{2}H^{\alpha\alpha}y_{\alpha}\partial_{\alpha}\mathbf{1}_{N}$,
where $\lambda^{2}$ is the cosmological constant and $\mathbf{1}_{N}$ is the
identity map on $V_{N}$. The system has natural automorphisms that originate
from linear field redefinitions $\Phi\rightarrow A_{N}\Phi$, where
$A_{N}:V_{N}\rightarrow V_{N}$ is an automorphism of $V_{N}$:

$\displaystyle\alpha_{N}$ $\displaystyle\rightarrow
A^{-1}_{N}\alpha_{N}A_{N-2}\,,$ $\displaystyle\beta_{N}$
$\displaystyle\rightarrow A^{-1}_{N}\beta_{N}A_{N+2}\,,$
$\displaystyle\gamma_{N}$ $\displaystyle\rightarrow
A^{-1}_{N}\gamma_{N}A_{N}\,.$ (4.7)

Therefore, the system corresponds to a quiver with certain additional
restrictions given by (4.6). The quiver is

which should be extended down to the minimal value of $N$ and up to the
maximal, possibly infinite, value of $N$.

In particular, the case studied in the previous section 3 corresponds to two
sectors with spin 2 and 3, respectively, hence with vector spaces $V_{2}$ and
$V_{4}$ of dimension two each, for the two one-form fields in each sector.
Associated with these two vector spaces, we therefore have the matrices
$\alpha_{4}\,$, $\beta_{2}\,$, $\gamma_{2}$ and $\gamma_{4}\,$. These four
$2\times 2$ matrices correspond to the matrices $A\,$, $B\,$, $C\,$ and $D\,$
of section 3. Finally, equations (4.6) correspond to the conditions (3.43)
found in that section, and the matrices $A_{2}$ and $A_{4}$ appearing in (4.7)
correspond to the matrices $M$ and $N$ of (3.45).

It is useful to rescale the maps as

$\displaystyle\alpha_{N}$ $\displaystyle=\frac{\bar{\alpha}_{N}}{N-1}\,,$
$\displaystyle\beta_{N}$ $\displaystyle=\frac{\bar{\beta}_{N}}{N+3}\,,$
$\displaystyle\gamma_{N}$ $\displaystyle=\frac{\bar{\gamma}_{N}}{N(N+2)}\,,$
(4.8)

and define $\bar{\alpha}$, $\bar{\beta}$ and $\bar{\gamma}$ as the maps that
act on the corresponding $V_{N}$. Relations (4.6) can be summarized as

$\displaystyle[\bar{\gamma},\bar{\alpha}]$ $\displaystyle=0\,,$
$\displaystyle[\bar{\gamma},\bar{\beta}]$ $\displaystyle=0\,,$
$\displaystyle-\tfrac{1}{N+1}[\bar{\alpha},\bar{\beta}]+\tfrac{1}{N^{2}(N+2)^{2}}\bar{\gamma}^{2}$
$\displaystyle=\sigma\lambda^{2}\mathbf{1}\,.$ (4.9)

In general the topological system looks intractable — it corresponds to a
quiver of the wild type. In practice this means that we cannot just
diagonalize or Jordanize the matrices with the help of automorphisms as there
are too few of them and the reduced form does not have any reasonable
classification. However, the quiver is supplemented with Eqs. (4.9).
Altogether, they imply that the total space of
$\Phi^{\alpha(N)|{\mathcal{I}}}(x)$ forms a representation of the space-time
symmetry algebra. In the (anti)-de Sitter case all finite-dimensional
representations are completely reducible and, hence, the wildness of the
quiver plays no role. It is the Poincare case that presents a problem. To
illustrate the formalism let us consider some well-known examples.

#### Example: same spin.

For a set of same spin fields we have $\alpha=\beta=0$ and the quiver is

(4.10)

It corresponds to a matrix $\gamma_{N}$ up to conjugation,
$\gamma_{N}\rightarrow A^{-1}_{N}\gamma_{N}A_{N}$ and the classification is
well-known: indecomposable representations are given by Jordan blocks (or real
Jordan blocks in the real case). Jordan cells of size greater than two are not
nilpotent and, hence, do not satisfy (4.6c) for any $\lambda$.

As is well-known [11, 37] and as we already recalled in section 3, a single
massless field in (A)dS3 or flat space can be described (in the sense of being
equivalent to the Fronsdal approach) by two one-forms taking values in some
finite-dimensional irreducible representation of the $sl_{2}$ Lorentz algebra.
Therefore $\dim V_{N}=2$ and $N=2s-2$. The matrix $\gamma_{N}$ can be chosen
as

$\displaystyle\textrm{(A)dS}_{3}$ $\displaystyle:\
\gamma_{N}=\begin{pmatrix}0&1\\\ \sigma\lambda^{2}&0\end{pmatrix},$ Minkowski
$\displaystyle:\ \gamma_{N}=F\equiv\begin{pmatrix}0&1\\\ 0&0\end{pmatrix}.$
(4.11)

Accordingly, a spin-$s$ massless field in flat space is described by

$\displaystyle de^{\alpha(2s-2)}$
$\displaystyle=h{}^{\alpha}{}_{\beta}\,\wedge\omega^{\beta\alpha(2s-3)}\,,$
$\displaystyle d\omega^{\alpha(2s-2)}$ $\displaystyle=0\,,$ (4.12)

where $\gamma_{N}=F$ is manifested by the way the fields mix with each other.
It is invariant under

$\displaystyle\delta e^{\alpha(2s-2)}$
$\displaystyle=d\xi^{\alpha(2s-2)}-h{}^{\alpha}{}_{\beta}\,\wedge\eta^{\beta\alpha(2s-3)}\,,$
$\displaystyle\delta\omega^{\alpha(2s-2)}$
$\displaystyle=d\eta^{\alpha(2s-2)}\,.$ (4.13)

In the (A)dS3 case one gets an additional term in the r.h.s. of the second
equation in (4.12):

$\displaystyle de^{\alpha(2s-2)}$
$\displaystyle=h{}^{\alpha}{}_{\beta}\,\wedge\omega^{\beta\alpha(2s-3)}\,,$
$\displaystyle d\omega^{\alpha(2s-2)}$
$\displaystyle=\sigma\lambda^{2}\,h{}^{\alpha}{}_{\beta}\,\wedge
e^{\beta\alpha(2s-3)}\,.$ (4.14)

As we discussed extensively in section 3, in the anti-de Sitter case the
system can be diagonalized by mapping it to

$\displaystyle dA^{\alpha(2s-2)}$ $\displaystyle=+\lambda
h{}^{\alpha}{}_{\beta}\,\wedge A^{\beta\alpha(2s-3)}\,,$ $\displaystyle
dB^{\alpha(2s-2)}$ $\displaystyle=-\lambda h{}^{\alpha}{}_{\beta}\,\wedge
B^{\beta\alpha(2s-3)}\,.$ (4.15)

#### Example: diagonalizable case.

Let us assume that we managed to diagonalize all $\alpha$, $\beta$, $\gamma$
simultaneously or, at least, various matrix products give the same matrix for
each equation so that we can check the overall coefficients only. Now, the
system reduces to a simple scalar equation. We assume that the module consists
of $sl(2,\mathbb{R})$-tensors with ranks from $n_{1}$ to $n_{2}$ in steps of
two:

$\displaystyle
T_{\alpha(n_{1})}\,,T_{\alpha(n_{1}+2)}\,,\ldots\,,T_{\alpha(n_{2})}\,.$
(4.16)

In this case the general solution reads [44, 36]:

$\displaystyle\sigma(n)$
$\displaystyle=\frac{-\sigma\lambda^{2}\left(n^{2}-n_{1}^{2}\right)\left((n_{2}+2)^{2}-n^{2}\right)}{4n^{2}\left(n^{2}-1\right)}\,,$
(4.17) $\displaystyle\gamma(n)$ $\displaystyle=\frac{\gamma_{0}\lambda
n_{1}(n_{2}+2)}{n(n+2)}\,,$ $\displaystyle\gamma_{0}=\pm 1\,,$ (4.18)

where $\sigma(N)=\alpha(N)\beta(N-2)$, which is the combination invariant
under rescalings of the fields.

#### Example: partially-massless fields.

The simplest example for which the solution above is relevant is the case of
partially-massless fields [41, 42], where we can choose

$\displaystyle\alpha_{n}$ $\displaystyle=\sigma(n)\,\mathrm{Id}_{2}\,,$
$\displaystyle\beta_{n}$ $\displaystyle=\mathrm{Id}_{2}\,,$
$\displaystyle\gamma_{n}$ $\displaystyle=\gamma(n)\begin{pmatrix}0&1\\\
1&0\end{pmatrix}\,.$ (4.19)

The partially-massless system written in the usual basis reads schematically

$\displaystyle\begin{aligned} \nabla\omega^{k}&=\gamma
e^{k}+\alpha\omega^{k-2}+\beta\omega^{k+2}\,,\\\ \nabla
e^{k}&=\gamma\omega^{k}+\alpha e^{k-2}+\beta e^{k+2}\,,\end{aligned}$ (4.20)

where we indicated without the explicit coefficients the contributions from
the $\alpha$, $\beta$, $\gamma$ matrices and the number of
$sl(2,\mathbb{R})$-indices that the fields carry. With the solution (4.17),
the maximal spin is $2s-2=n_{2}$ and the minimal one is $2(s-t)=n_{1}$, where
$t$ is the depth of partially–masslessness. While $\alpha$ and $\beta$ are
already diagonal, one can also diagonalize $\gamma$ to get two decoupled
systems.

In the (A)dS3 case due to the complete reducibility we can map a
partially–massless system that contains spins from $s-t$ to $s$ to two
irreducible connections, $\omega^{\alpha(2s-t-1),{\dot{\alpha}}(t-1)}$ and
$\omega^{\alpha(t-1),{\dot{\alpha}}(2s-t-1)}$ of $sl(2,\mathbb{R})\oplus
sl(2,\mathbb{R})$. Therefore, any topological system in (A)dS3 consists of
(partially)-massless fields and nothing else.

#### Free actions.

Given the equations of motion of a topological system, we can also ask whether
they can be derived from an action principle. Let us define a gauge-invariant
curvature as $R=(\nabla-Q)\omega$ and take

$\displaystyle S$
$\displaystyle=\int\langle\omega^{{\mathcal{I}}}|G_{{\mathcal{I}}{\mathcal{J}}}(N)|R^{{\mathcal{J}}}\rangle\,,$
(4.21)

where the conjugate is defined by $y_{\alpha}\rightarrow\partial_{\alpha}$,
$\partial_{\alpha}\rightarrow-y_{\alpha}$ (it swaps the order of $y$ and
$\partial$) and, possibly, by complex conjugation as well. The gauge
invariance can be checked via

$\displaystyle\delta S$
$\displaystyle=\langle\nabla\xi^{{\mathcal{I}}}-Q{}^{{\mathcal{I}}}{}_{{\mathcal{K}}}\,\xi^{{\mathcal{K}}}|G_{{\mathcal{I}}{\mathcal{J}}}(N)|R^{{\mathcal{J}}}\rangle=$
$\displaystyle=-\langle\xi^{{\mathcal{I}}}|G_{{\mathcal{I}}{\mathcal{J}}}(N)|QR^{{\mathcal{J}}}\rangle-\langle\xi^{{\mathcal{K}}}|Q^{\dagger}{}{}^{{\mathcal{I}}}{}_{{\mathcal{K}}}\,G_{{\mathcal{I}}{\mathcal{J}}}(N)|R^{{\mathcal{J}}}\rangle\,,$

where we integrated by parts and used the Bianchi identities $\nabla R\equiv
QR$. The gauge invariance imposes:

$\displaystyle
G_{{\mathcal{K}}{\mathcal{I}}}(N)\beta{}^{{\mathcal{I}}}{}_{{\mathcal{J}}}\,(N)+G_{{\mathcal{I}}{\mathcal{J}}}(N+2)\alpha{}^{{\mathcal{I}}}{}_{{\mathcal{K}}}\,(N+2)$
$\displaystyle=0\,,$ (4.22a) $\displaystyle
G_{{\mathcal{K}}{\mathcal{I}}}(N)\gamma{}^{{\mathcal{I}}}{}_{{\mathcal{J}}}\,(N)-G_{{\mathcal{I}}{\mathcal{J}}}(N)\gamma{}^{{\mathcal{I}}}{}_{{\mathcal{K}}}\,(N)$
$\displaystyle=0\,.$ (4.22b)

In the special case studied in Section 3, the corresponding equations are
(3.56).

These conditions imply that $Q$ is self-adjoint with respect to the bilinear
product defined by (4.21). We require $G$ be symmetric,
$G_{{\mathcal{I}}{\mathcal{J}}}=G_{{\mathcal{J}}{\mathcal{I}}}$, and non-
degenerate for the variation to reduce to

$\displaystyle\delta S$
$\displaystyle=2\langle\delta\omega^{{\mathcal{I}}}|G_{{\mathcal{I}}{\mathcal{J}}}(N)|R^{{\mathcal{J}}}\rangle\,.$
(4.23)

The equations of motion are equivalent to the desired $R^{{\mathcal{I}}}=0$.
For the diagonalizable case, $\sigma_{N}$ is a scale invariant combination of
$\alpha_{N}$ and $\beta_{N-2}$, but the action principle requirement fixes the
relative normalization for $\alpha$ and $\beta$:

$\displaystyle(\beta_{N})^{2}$
$\displaystyle=-\frac{\sigma_{N+2}G_{N+2}}{G_{N}}\,.$ (4.24)

One can choose $G_{N}=1$, which makes the action the simplest. In particular,
this gives an action for partially-massless fields in $3D$ [42], which is not
a simple adaptation of [41].

#### Example: massless fields.

For a single massless field in Minkowski space we can take

$\displaystyle G$ $\displaystyle=K\,,$ $\displaystyle K=\begin{pmatrix}0&1\\\
1&0\end{pmatrix}\,,$ (4.25)

which leads to

$\displaystyle S$ $\displaystyle=\langle
e|R(\omega)\rangle+\langle\omega|R(e)\rangle\,.$ (4.26)

For a single massless field in (A)dS3 the action has exactly the same form,
but the curvatures have a $\lambda^{2}$-correction. Now one can perform a
linear change of variables and get

$\displaystyle\gamma_{N}$ $\displaystyle=\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}\,,$ (4.27)

which leads to two decoupled actions, i.e., the matrix $K$ becomes numerically
equal to $\gamma_{N}\,$. Once the actions decouple, the relative coefficient
can be made arbitrary, see e.g. [33, 34, 35].

###  4.2 Strange higher spin systems

We are now equipped with all the necessary machinery to generalize the
examples of section 3. First, let us consider the topological system of
coupled spin-two and -three fields in Minkowski space of section 3. Indeed,
from the gauge transformations (3.2) we see that the first derivative of one
field enters the gauge transformations of the other. As done explicitly in
section 3.3 in the frame-like formulation, such terms can be obtained in two
steps: 1) one fixes the local Lorentz symmetry (4.13) with parameter
$\eta^{\alpha(2s-2)}$ and the Lorentz gauge parameter $\eta^{\alpha(2s-2)}$
gets expressed as the first derivative of the Fronsdal parameter
$\xi^{\alpha(2s-2)}$; 2) the Lorentz gauge parameter $\eta^{\alpha(2s-2)}$
enters the transformations of the vielbein of the other field in the system
and the other way around.

In spinor notation111111The dictionary between vector and spinor notation is
given in Appendix A., the gauge transformations of section 3.3 are the form

$\displaystyle\delta e^{\alpha\alpha}$
$\displaystyle=d\eta^{\alpha\alpha}+h{}^{\alpha}{}_{\beta}\,\chi^{\alpha\beta}+h_{\beta\beta}\rho^{\alpha\alpha\beta\beta}\,,$
(4.28a) $\displaystyle\delta\omega^{\alpha\alpha}$
$\displaystyle=d\chi^{\alpha\alpha}\,,$ (4.28b) $\displaystyle\delta
e^{\alpha\alpha\alpha\alpha}$
$\displaystyle=d\xi^{\alpha\alpha\alpha\alpha}+h{}^{\alpha}{}_{\beta}\,\rho^{\alpha\alpha\alpha\beta}+h^{\alpha\alpha}\chi^{\alpha\alpha}\,,$
(4.28c) $\displaystyle\delta\omega^{\alpha\alpha\alpha\alpha}$
$\displaystyle=d\rho^{\alpha\alpha\alpha\alpha}\,.$ (4.28d)

More generally, for any two neighbouring spins the following system is
consistent

(4.29)

In the case of section 3 we need fields with $N=2,4$, but for any $N$ a
representation of the quiver can be chosen to be

$\displaystyle\gamma_{N}$ $\displaystyle=\gamma_{N+2}=\beta_{N}=F\,,$
$\displaystyle\alpha_{N+2}$ $\displaystyle=qF\,,$ $\displaystyle
F=\begin{pmatrix}0&1\\\ 0&0\end{pmatrix}.$ (4.30)

These matrices are spin-independent, which is a particular solution. With the
help of $GL(2)\times GL(2)$ transformations we can reach
$\gamma_{N}=\gamma_{N+2}=\beta_{N}=F$, but $\alpha_{N+2}=qF$, where $q$ is a
genuine parameter of this representation of the Poincaré algebra.
Alternatively, we can choose $\gamma_{N}=\gamma_{N+2}=\alpha_{N+2}=F$ and
$\beta_{N}=qF$. Therefore, we have found a family of finite-dimensional
representations of $iso(2,1)$ that depend on one free parameter.

Upon gauge fixing and going to the metric-like formalism one finds
schematically

$\displaystyle\delta\phi_{s}$
$\displaystyle=\partial\xi_{s-1}+\eta\epsilon\partial\xi_{s-2}\,,$
$\displaystyle\delta\phi_{s-1}$
$\displaystyle=\partial\xi_{s-2}+\epsilon\partial\xi_{s-1}\,.$ (4.31)

The field content, both frame-like and metric-like, matches the one required
to describe a depth-$2$ partially-massless field. Indeed, there is a smooth
deformation to (A)dS3 with

$\displaystyle\left(\begin{array}[]{cc}0&1\\\
\frac{(N+4)^{2}\lambda^{2}}{(N+2)^{2}}&0\\\ \end{array}\right)\,,$
$\displaystyle\left(\begin{array}[]{cc}0&1\\\
\frac{N^{2}\lambda^{2}}{(N+2)^{2}}&0\\\ \end{array}\right)\,,$
$\displaystyle\left(\begin{array}[]{cc}0&1\\\
\frac{N(N+4)\lambda^{2}}{2N^{2}}&0\\\ \end{array}\right)\,,$
$\displaystyle\left(\begin{array}[]{cc}0&-\frac{4}{N(N+4)}\\\
-\frac{4\lambda^{2}}{(N+2)^{2}}&0\\\ \end{array}\right)\,,$ (4.40)

for the same matrices $\gamma_{N}\,,\gamma_{N+2}\,,\beta_{N}\,,\alpha_{N+2}$.
Up to a simple linear $GL(2)\times GL(2)$ transformation of the fields the
system is equivalent to the canonical form of the partially-massless system
(4.19). There is no free parameter, of course. As it was mentioned around
(4.24), the kinetic matrix can be chosen to be $N$-independent. In a different
form the action can be found in [42].

Coming back to the case of $N=2,4$ of section 3, the existence of the various
different systems found in (A)dS3 can be understood from the following
representation theory argument. Recall that $(j_{1},j_{2})$ denotes an
irreducible representation of $so(2,2)$ or $sl(2,\mathbb{C})\cong so(1,3)$ of
dimension $(2j_{1}+1)(2j_{2}+1)$ and $(j)$ denotes the dimension-$(2j+1)$
irreducible representation of the Lorentz subalgebra $sl_{L}(2,\mathbb{R})$.
There are several ways to produce a one-form gauge field valued in a given
Lorentz spin-$(j)$ representation within an (A)dS3 one-form field
$\omega^{\alpha(N),\dot{\alpha}(M)}\,$ transforming in the representation
$(N/2,M/2)$ of the (A)dS3 isometry algebra. Sticking to the case studied in
section 3, there are many solutions since from the $sl_{L}(2,\mathbb{R})$
representations $2\times(1)$ and $2\times(2)$ corresponding to the generators
$(P_{a},J_{a},P_{aa},J_{aa})$, one can think of various possible combinations
among the set $\\{(1,0),(0,1),(2,0),(0,2),(3/2,1/2),(1/2,3/2)\\}$ of (A)dS3
representations. Although the first four representations in this set are
simple in their decomposition with respect to the Lorentz subalgebra, the
latter two branch into the direct sum $(2)\oplus(1)\,$. The usage of the
representations $(1,0)$ and $(2,0)$ (or their conjugates) to represent two
one-form gauge fields means that the corresponding spin-$2$ and spin-$3$
fields do not talk to each other, since they correspond to the two generators
$(J_{a},J_{aa})$ of the same $SL(3,\mathbb{R})$ group factor. As we already
commented below Eq. (3.49), in the free theory the spin-2 and spin-3 sectors
do not mix in this situation. One can also have systems without any
straightforward metric-like interpretation, e.g.
$(1,0)\oplus(2,0)\oplus(3/2,1/2)$, or $2\times(3/2,1/2)$. For example, the
system (3.51) with $\eta=+1$ corresponds to $(3/2,1/2)\oplus(3/2,1/2)\,$. The
partially-massless systems in $(A)dS_{3}$ correspond to
$(3/2,1/2)\oplus(1/2,3/2)$; the two decoupled systems to $(1,0)\oplus(2,0)$
and $(1,0)\oplus(0,2)$; the four $AdS_{3}$-models with one pair of fields
decoupled correspond to $(3/2,1/2)\oplus X$, where $X$ is any combination of
$(1,0)$ or $(0,1)$ with $(2,0)$ or $(0,2)$.

#### Possible interactions.

Since any interacting topological theory has to be of Chern-Simons form [12],
in order to find interactions we have to identify the fields associated with
the generators of some Lie algebra that has a non-degenerate invariant
bilinear form. Let us go back to the simplest case with spin-two and -three
fields. After we moved to (A)dS3 we are looking for an algebra that has two
generators $T^{\alpha(4)}$ and $T^{\alpha(2)}$ (when decomposed into a sum of
Lorentz modules) plus, possibly, additional generators associated to other
fields that might be required to obtain a non-Abelian algebra. The two
generators $T^{\alpha(4)}$ and $T^{\alpha(2)}$ can be understood as coming
from a single partially-massless generator $T^{\alpha(3),{\dot{\alpha}}(1)}$
and its conjugate $T^{\alpha(1),{\dot{\alpha}}(3)}$ – if we do not consider
models where one pair of fields forms $(1,0)\oplus(2,0)$ and can be unified by
one $sl(3,\mathbb{R})$.121212We use here and below the notation
$T^{\alpha(2j_{1}),{\dot{\alpha}}(2j_{2})}$ to simply denote the corresponding
representation of $sl(2,\mathbb{R})\oplus sl(2,\mathbb{R})$.

One way to get a simple finite dimensional higher spin algebra131313By a
higher spin algebra we simply mean a Lie algebra that contains a given space-
time symmetry algebra as a subalgebra and decomposes into irreducible modules
that are larger than the adjoint ones, where the latter are associated with
some higher spin fields.  is to take an irreducible module $V$ of the space-
time symmetry algebra and consider $gl(V)=u(1)\oplus sl(V)$. One can also
apply this construction to a module $V$ that is a direct sum of irreducible
modules. In order to get the required spectrum from a higher spin algebra of
$gl(V)$-type we can take $V=T^{\alpha(2)}\oplus T^{\alpha,{\dot{\alpha}}}$ of
the (anti)-de Sitter algebra, i.e. we cannot consider just a single
irreducible representation of $sl(2,\mathbb{R})$. The full spectrum is then
given by the tensor product $V\otimes V^{*}$ and reads $2\times
T^{\alpha(2),\dot{\alpha}(2)}\oplus T^{\alpha(3),\dot{\alpha}}\oplus
T^{\alpha,\dot{\alpha}(3)}\oplus 2\times T^{\alpha,\dot{\alpha}}\oplus
T^{\alpha(2)}\oplus T^{\dot{\alpha}(2)}\oplus T$. This seems to be the most
minimal extension that admits interactions and contains the spin-two
subsector. It is clear that there is no algebra that contains only the
generators $T^{\alpha(3),{\dot{\alpha}}(1)}\oplus
T^{\alpha(1),{\dot{\alpha}}(3)}$, which a posteriori explains the no-go result
of section 3. Indeed, there is no $V$ such that $V\otimes V^{*}$ gives just
$(3/2,1/2)\oplus(1/2,3/2)\oplus(0,0)$, which can be seen by enumerating a
handful of low-spin representations $V$. We postpone to a future work the
analysis of the model based on these fields in AdS3 and their flat limit.

#### Simple generalization.

The system above has an obvious generalization to a topological system that
covers a range of spins. We can extend the system by duplicating the nodes and
defining $\alpha\sim\beta\sim\gamma\sim F$. It will always be consistent since
$FF=0$. The (first few levels of the) gauge transformations look schematically
as

$\displaystyle\delta\phi_{s}$
$\displaystyle=\partial\xi_{s-1}+\eta\epsilon\partial\xi_{s-2}\,,$ (4.41a)
$\displaystyle\delta\phi_{s-1}$
$\displaystyle=\partial\xi_{s-2}+\epsilon\partial\xi_{s-1}+\eta\epsilon\partial\xi_{s-3}\,,$
(4.41b) $\displaystyle\delta\phi_{s-2}$
$\displaystyle=\partial\xi_{s-3}+\epsilon\partial\xi_{s-2}+\eta\epsilon\partial\xi_{s-4}\,,$
(4.41c)

and can extend down to any spin $s\geq 1$. This system has more free
parameters: there is one parameter per each $\alpha$ (or $\beta$), which gives
other examples of finite-dimensional representations of $iso(2,1)$.

An (A)dS3-deformation of such a system is a partially-massless field that
originates from $T^{\alpha(2s-2-k),{\dot{\alpha}}(k)}$ and
$T^{\alpha(k),{\dot{\alpha}}(2s-2-k)}$. After decomposing with respect to the
diagonal Lorentz algebra, its top spin component is $T^{\alpha(2s-2)}$ and the
lowest one is $T^{\alpha(2s-2-2k)}$. For the same reason as before, this
system does not admit interactions unless we extend it with more fields,
assuming the deformation of interactions has to be smooth in the cosmological
constant.

#### Even stranger systems.

Another interesting example is a system that contains fields of spins $2$,
$2$, $3$, $4$ or, more generally, $s$, $s$, $s+1$, $s+2$. It corresponds to
the following quiver

(4.42)

with a representation given by

$\displaystyle\alpha_{N+2}=\beta_{N}=\gamma_{N+2}=\gamma_{N}=F\,,$ (4.43)
$\displaystyle\gamma_{N-2}=\begin{pmatrix}F&0\\\
0&F\end{pmatrix}\,,\qquad\alpha_{N}=\begin{pmatrix}F&F\end{pmatrix}\,,\qquad\beta_{N-2}=\begin{pmatrix}F\\\
F\end{pmatrix}\,.$ (4.44)

Here-above, we provide an obvious generalization to any spin. The action can
be written with

$\displaystyle G_{N-2}$ $\displaystyle=\begin{pmatrix}K&0\\\
0&K\end{pmatrix}\,,$ $\displaystyle-G_{N}=G_{N+2}=K\,.$ (4.45)

The AdS deformation leads to $T^{\alpha(4),{\dot{\alpha}}(2)}$ and
$T^{\alpha(2)}$, i.e. it is a partially-massless field and a massless one that
are decoupled from each other.

It is easy to generalize this example to more complicated systems. We can
begin with any number $k$ of $V_{N-2i}$, $i=0,...,k$ that are even
dimensional. $\gamma_{N-2i}$ can be block diagonal made of $F$, and $\alpha$,
$\beta$ can mix them. Nilpotency of $F$ ensures that the system is consistent.
Some free parameters can be introduced in the same way as before.

##  5 Conclusions

In this paper, we studied some higher-spin systems in three-dimensional
Minkowski space originally found in [1], following the higher off-shell
dualisation procedure proposed originally in [2]. We found several
generalisations of these systems in flat space. Firstly, we found that these
strange higher-spin actions in flat space admit a one-parameter extension.
Secondly, we found that for some discrete values of the parameter in the
action, these system could be deformed to the (A)dS3 background. Thirdly, we
found various generalisations of these models to larger spectra of fields
having spin even higher than three, both in flat and (A)dS3.

At the free level, an interesting mathematical problem we have encountered is
the classification of finite-dimensional representations of Poincaré algebra,
since each of such representations defines one of our free topological
systems. To the best of our knowledge this problem remains unsolved.

Concerning possible interactions at the action level, once the flat space
system is assumed to have a smooth deformation to (A)dS3 the powerful theorems
on the representation theory of (semi)-simple Lie algebras are at our
disposal. Free topological systems in (A)dS3 are simpler to study as compared
to topological systems in flat space, since we know all the finite-dimensional
representations of the (A)dS3 isometry algebras. As we discuss in section 4.2,
all metric-like topological (A)dS3 systems contain (partially)-massless fields
and nothing else. Then, as far as interactions are concerned, the (A)dS3
background also makes the search for interactions simpler, since we know which
spectrum of fields to introduce in order to have a $gl(V)$ associative matrix
algebra, out of which a Lie algebra is obtained by taking the commutator, the
trace operation being the trace of matrices in $End(V)$. It remains to be seen
if there are genuine interacting topological theories in flat space, i.e.
those that do not admit any deformations to (A)dS3.

Since we found some (A)dS3 models that are not analytical in the cosmological
constant, it is not yet clear whether there could be some non-Abelian theory
in (A)dS3 that would be non-analytical in the cosmological constant, hence
admitting no flat limit. We leave this for future investigations. The simplest
spin-2/spin-3 models (3.54) – (3.64) that we found in (A)dS3 admit a smooth
flat limit and therefore cannot allow for a non-Abelian deformation that would
be analytical in the cosmological constant.

As another possible outlook, it would be interesting to look at the asymptotic
symmetries of the new topological higher-spin systems we found. In AdS such
analysis should fit within the extension of the asymptotics of massless
higher-spin fields [5, 6, 45] modulo possible generalisations of the
“standard” boundary conditions along the lines, e.g., of [46]. On the other
hand, in flat space the variety of inequivalent bulk symplectic structure that
we identified should lead to a rich landscape of higher-spin asymptotic
symmetries beyond those discussed, e.g., in [13, 14, 47, 48] and references
therein.

### Acknowledgements

We are grateful to Antoine Bourget for useful discussions on quiver
representations. We would also like to thank Gaston Giribet, Stam Nicolis and
Massimo Porrati for discussions.

AC and ES are research associates of the Fund for Scientific Research – FNRS,
Belgium. This work was partially supported by the FNRS through the grants No.
FC.36447, No. F.4503.20, and No. T.0022.19. The work of ES was supported by
the European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation programme (grant agreement No 101002551). The work of
VL was funded by the European Union’s Horizon 2020 research and innovation
programme under the Marie Skłodowska-Curie grant agreement No 101034383.

##  Appendix A Dictionary between spinor and vector notation

We introduce a basis of three real, symmetric, $2\times 2$ matrices
$\tau^{a}=(\tau^{a}_{\alpha\beta})=(\tau^{a}_{\beta\alpha})$ and raise (lower)
the indices according to
$q^{\alpha}=\epsilon^{\alpha\beta}\,q_{\beta}=\epsilon^{\alpha\beta}\,q^{\gamma}\epsilon_{\gamma\beta}\,$.
The three $\tau^{a}$ matrices obey the orthogonality and completeness
relations

$\tau^{a}{}_{\alpha\beta}\,\tau^{b}{}^{\alpha\beta}=-2\,\eta^{ab}\,,\quad\tau^{a}{}_{\alpha\beta}\,\tau_{a}{}^{\gamma\delta}=-2\,\delta_{(\alpha}^{\gamma}\delta_{\beta)}^{\delta}\,,$
(A.1)

where we use the mostly plus convention $(\eta_{ab})=\;$diag$(-1,+1,+1)\,$
together with

$\tau^{a}_{\alpha\beta}\,\tau^{b}{}^{\beta\gamma}=-\eta^{ab}\,\delta^{\gamma}_{\alpha}+\epsilon^{abc}\,\tau_{c\,\alpha}{}^{\gamma}\;.$
(A.2)

In particular, one has

$\epsilon^{abc}=-\tfrac{1}{2}\,{\rm Tr}(\tau^{a}\tau^{b}\tau^{c})\;,\qquad{\rm
where}\qquad\epsilon^{012}=1\;.$ (A.3)

The dictionary between vector and spinor notation is

$V^{a}=\tau^{a}_{\alpha\beta}\,V^{\alpha\beta}\qquad\Leftrightarrow\qquad
V^{\alpha\beta}=-\tfrac{1}{2}\,\tau_{a}^{\alpha\beta}\,V^{a}\;.$ (A.4)

Therefore, associated with the transformation laws in vector notation

$\displaystyle\delta e^{a}$ $\displaystyle={\rm
d}\xi^{a}+\alpha\,\epsilon^{abc}\,h_{b}\,\Lambda_{c}+\beta\,h_{b}\,\alpha^{ab}\;,$
(A.5) $\displaystyle\delta e^{aa}$ $\displaystyle={\rm
d}\xi^{aa}+\gamma\,h_{b}\,\epsilon^{abc}\,\alpha_{c}{}^{a}+\sigma\,(h^{a}\Lambda^{a}-\tfrac{1}{3}\,\eta^{aa}\,h_{b}\,\Lambda^{b})\;,$
(A.6)

one has, respectively, the following transformations

$\displaystyle\delta e^{\alpha\alpha}$ $\displaystyle={\rm
d}\xi^{\alpha\alpha}+2\,\alpha\,h^{\beta\alpha}\,\Lambda^{\alpha}{}_{\beta}-2\,\beta\,h_{\beta\beta}\,\alpha^{\alpha\alpha\beta\beta}\;,$
(A.7) $\displaystyle\delta e^{\alpha(4)}$ $\displaystyle={\rm
d}\xi^{\alpha(4)}+2\,\gamma\,h^{\beta\alpha}\,\alpha_{\beta}{}^{\alpha(3)}+\sigma\,h^{\alpha\alpha}\,\Lambda^{\alpha\alpha}\;.$
(A.8)

From the definition of the operator $Q\,$ in (4.4), we have

$\displaystyle\delta e^{\alpha\alpha}$
$\displaystyle=\nabla\xi^{\alpha\alpha}+\beta(2)_{12}\,h_{\beta\beta}\rho^{\alpha\alpha\beta\beta}+2\,\gamma(2)_{12}h_{\beta}{}^{\alpha}\chi^{\alpha\beta}$
(A.9) $\displaystyle\delta\omega^{\alpha\alpha}$
$\displaystyle=\nabla\chi^{\alpha\alpha}+\beta(2)_{21}\,h_{\beta\beta}\xi^{\alpha\alpha\beta\beta}+2\,\gamma(2)_{21}h_{\beta}{}^{\alpha}\xi^{\alpha\beta}$
(A.10) $\displaystyle\delta e^{\alpha(4)}$
$\displaystyle=\nabla\xi^{\alpha(4)}+12\,\alpha(4)_{12}\,h^{\alpha\alpha}\chi^{\alpha\alpha}+4\,\gamma(4)_{12}h_{\beta}{}^{\alpha}\rho^{\alpha(3)\beta}$
(A.11) $\displaystyle\delta\omega^{\alpha(4)}$
$\displaystyle=\nabla\rho^{\alpha(4)}+12\,\alpha(4)_{21}\,h^{\alpha\alpha}\xi^{\alpha\alpha}+4\,\gamma(4)_{21}h_{\beta}{}^{\alpha}\xi^{\alpha(3)\beta}\;$
(A.12)

for some $2\times 2$ matrices $\beta(2)\,$, $\gamma(2)\,$, $\gamma(4)\,$ and
$\alpha(4)\,$.

On the other hand, doing the translation between vector and spinor notation,
we find that, associated with the transformation laws in spinor notation

$\displaystyle\delta e^{\alpha\alpha}$ $\displaystyle={\rm
d}\xi^{\alpha\alpha}+2\,\alpha\,h^{\beta\alpha}\,\Lambda^{\alpha}{}_{\beta}-2\,\beta\,h_{\beta\beta}\,\alpha^{\alpha\alpha\beta\beta}\;,$
(A.13) $\displaystyle\delta e^{\alpha(4)}$ $\displaystyle={\rm
d}\xi^{\alpha(4)}+2\,\gamma\,h^{\beta\alpha}\,\alpha_{\beta}{}^{\alpha(3)}+\sigma\,h^{\alpha\alpha}\,\Lambda^{\alpha\alpha}\;,$
(A.14)

there corresponds the following transformations in the vector notation:

$\displaystyle\delta e^{a}$ $\displaystyle={\rm
d}\xi^{a}+\alpha\,\epsilon^{abc}\,h_{b}\,\Lambda_{c}+\beta\,h_{b}\,\alpha^{ab}\;,$
(A.15) $\displaystyle\delta e^{aa}$ $\displaystyle={\rm
d}\xi^{aa}+\gamma\,h_{b}\,\epsilon^{abc}\,\alpha_{c}{}^{a}+\sigma\,(h^{a}\Lambda^{a}-\tfrac{1}{3}\,\eta^{aa}\,h_{b}\,\Lambda^{b})\;.$
(A.16)

From the above dictionary and (3.41), we find the following identification of
$2\times 2$ matrices:

$\gamma(2)=-A\,,\quad 2\gamma(4)=-D\,,\quad
12\alpha(4)=C\,,\quad\beta(2)=-2B\,.$ (A.17)

Then the equations (3.43) and (4.6) are in perfect agreement. With the further
identification

$G=-2G(2)\;,\quad H=\tfrac{1}{12}G(4)\;,$ (A.18)

equations (3.56) and (4.22) agree as well.

##  Appendix B Some definitions about quivers

We recall verbatim from [31] some definitions and results about quivers and
their representations that we refer to in the main body of the paper:

  * •

A quiver $\vec{Q}$ is a directed graph; formally it can be described by a set
of vertices $I\,$, a set of edges $\Omega\,$, and two maps
$s,t:\Omega\rightarrow I$ which assign to every edge its source and target,
respectively. One can also think of a quiver $\vec{Q}$ as a graph $Q$ along
with an orientation, i.e., choosing for each edge of $Q\,$, which of the two
endpoints is the source and which is the target. It is assumed that the set of
edges and vertices are finite and that $\vec{Q}$ is connected. A
representation of a quiver $\vec{Q}$ is the following collection of data:

    * –

For every vertex $i\in I\,$, vector space $V_{i}$ over the field
$\mathbb{K}\,$;

    * –

For every edge $h\in\Omega\,$, $h:i\rightarrow j\,$, a linear operator
$x_{h}:V_{i}\rightarrow V_{j}\,$ .

For the quivers considered in this paper, the operators $x_{h}$ also have to
satisfy certain quadratic relations such as (3.43) or (4.6). A simple example
of a quiver is given by the Jordan quiver:

$\displaystyle\vec{Q}_{J}=\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
5.5pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{}{{}{{}{{}{{}{{}}{}{{}{{}}}}}}}{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{}{{}{{}}{}{{}}{}{{}}{}{{}}{}{{}{{}}{}{{}{{}{{}{{}}{}{{}}{}{{}}}}}}}{\hbox{\kern
3.2935pt\raise-2.99599pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{}{{}}{{}{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}{}}{}}}}\ignorespaces{}\ignorespaces}}}}\ignorespaces$
(B.3)

and a representation of this quiver is the pair $(V,x)$ where $V$ is a vector
space over the field $\mathbb{K}$ and $x:V\rightarrow V$ is a linear map.
Classifying the representations of $\vec{Q}_{J}$ is equivalent to classifying
matrices up to similarity.

  * •

A quiver $\vec{Q}$ is called _Dynkin_ iif the underlying graph $Q$ is one of
the following graphs:

(B.6) (B.9) (B.12) (B.15) (B.18)

These are the Dynkin diagrams of simply-laced, finite-dimensional, simple Lie
algebras over the complex numbers.

  * •

A quiver $\vec{Q}$ is called _Euclidean_ iif the underlying graph $Q$ is one
of the following graphs:

(B.23) (B.26) (B.29) (B.32) (B.35)

These are the Dynkin diagrams of simply-laced affine Kac-Moody algebras.

The following results are quoted in [31], with references to the proofs given
therein:

  * •

Quivers can be _tame_ or _wild_ ; formal definitions of these concepts can be
found in chapter 7 of [31] and will not be reproced here. However, a simple
characterization holds [31, theorem 7.47]:

Theorem: Let $\vec{Q}$ be a connected quiver.

    1. (1)

If $\vec{Q}$ is Dynkin or Euclidean, then it is tame;

    2. (2)

If $\vec{Q}$ is neither Dynkin nor Euclidean, then it is wild.

Representations of tame quivers have been classified. On the other hand, the
representations of wild quivers are not known in general. The quivers
considered in this paper are wild.

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