Holography of New Conformal Higher Spin Gravities in $\boldsymbol{3d}$

Iva Lovrekovica

a Institute for Theoretical Physics, TU Wien,
Wiedner Hauptstr. 8-10, 1040 Vienna, Austria

###### Abstract

We study the holography of the new conformal higher spin theories imposing
general boundary conditions and the near horizon boundary conditions. General
boundary conditions lead to the asymptotic symmetry algebra which is a loop
algebra of the underlying gauge algebra. The near horizon boundary conditions
give u(1) currents at the boundary. For each of the boundary conditions we
consider an example of conformal graviton, and its combination with a vector,
and a spin-3 field, respectively. The near horizon boundary conditions lead to
three dimensional Ba$\tilde{n}$ados-Teitelboim-Zanelli (BTZ)-like black hole
which can have up to four real eigenvalues, Lobachevsky solution, and to
generalisation of that BTZ-like solution. We verify this result independently
obtaining it from the Cotton tensor. We also consider the classification of
the one parameter subgroups of the SO(3,2) and comment on the classification
of the obtained solutions.

###### Contents

  1. 1 Introduction
  2. 2  New Conformal Higher Spin Theories
    1. 2.1 Two Simple Higher Spin Algebras
  3. 3 General boundary conditions
    1. 3.1 Conformal spin-s field
    2. 3.2 Conformal spin-2 and spin-3 field
    3. 3.3 Conformal spin-2 and spin-1 field
  4. 4 Near horizon boundary conditions
    1. 4.1 so(3,2): conformal spin-2 field
    2. 4.2 sl(4): conformal spin-2 and spin-1 field
    3. 4.3 sl(5): conformal spin-2 and spin-3 field
  5. 5 One parameter subgroups of SO(3,2)
    1. 5.1 Special cases from classification
      1. 5.1.1 Case $I_{d}$
      2. 5.1.2 Case $I_{b}$
      3. 5.1.3 Case V
  6. 6 Conclusion
  7. 7 Acknowledgements
  8. 8 Appendix
    1. 8.1 Appendix A: Holography of Chern-Simons action
      1. 8.1.1 Gauge fixing
    2. 8.2 Appendix B
    3. 8.3 Appendix C
    4. 8.4 Appendix D
    5. 8.5 Appendix E
    6. 8.6 Appendix F
    7. 8.7 Type $I_{a}$
    8. 8.8 Type $I_{b}$
    9. 8.9 Type $I_{c}$
    10. 8.10 Type $I_{d}$
    11. 8.11 Type $II_{a}$
    12. 8.12 Type $II_{b}$
    13. 8.13 Type $II_{c}$
    14. 8.14 Type $III_{a}$
    15. 8.15 Type $III_{b}$
    16. 8.16 Type $III_{c}$
    17. 8.17 Type $IV$
    18. 8.18 Type $V$
    19. 8.19 Appendix G

##  1 Introduction

Theory of conformal higher spins (CHS) has been known for long time [1]. These
original CHS are known as symmetric transverse traceless fields with arbitrary
number of derivatives and spins [2, 3]. Their number in these theories is
infinite. The work by Pope and Townsend [2], that analized that infinite tower
of higher spins has predicted there should be a construction with infinite
number of theories and finite number of higher spin fields in each of the
theories. This is new conformal higher spin gravity theory (CHiSGRA) [4].

When we consider a theory in the bulk of the three dimensional manifold, in
the framework of Anti-de Sitter/Conformal Field Theory (AdS/CFT) [5]
correspondence we can obtain the information about the theory on the boundary
of the manifold. The general approach considers ADM (Arnowitt-Deser-Misner)
split into 2+1, where one coordinate is holographic. Quantities with
holographic coordinate taken to its limiting value and evaluated in the bulk
theory, will correspond to the quantities calculated in the theory on the
boundary in two dimensions. The symmetry conserved at the boundary is called
asymptotic symmetry, and the conditions imposed, which conserve that symmetry,
are boundary conditions (bcs). For example one can demand specific value of
the fields at the boundary as a boundary condition, or that the leading order
term of the metric takes AdS form. That makes metric asymptotically AdS.

Three dimensional higher-spin (HS) theories are usually based on the Chern-
Simons action with underlying gauge algebra. Chern-Simons action allows for
incorporating infinite tower of higher spin fields. Consistent truncation of
infinite tower of fields then describes a finite number of interacting
nonpropagating spin-s fields (for s=2,3,…,N) [6]. Different higher spin black
holes require extended asymptotic behaviour that incorporates chemical
potential related to the global charges. This modified set of bcs accommodates
the respective black hole solutions with higher spin fields [7, 8, 9, 10].

The program of the holographic analysis for the three dimensional HS gauge
theories started with [11, 12], i.e. generalization of the Brown and Henneaux
work [13] for the massless higher spins [14, 15]. It showed that two copies of
Virasoro algebra with central extension and a central charge dependent on the
AdS radius, turn into two copies of the $\mathcal{W}$ algebra with the central
charge as for the pure gravity [16]. The large N limit of the $SL(N)\times
SL(N)$ Chern-Simons theories can be considered as one-parameter family of
higher-spin gauge theories.

The further steps of holographyic analysis of Chern-Simons theory with
$sl(2)\oplus sl(2)$ (3d Einstein gravity) gauge group included general bcs and
near horizon bcs. The general bcs will lead to two times more independent
functions than there is generators in the underlying gauge algebra. On the
example of Einstein gravity in three dimensions, that will be twelve
independent functions (six charges and six chemical potentials), where only
charges are allowed to vary. The charges define the asymptotic symmetry
algebra (ASA) which gives two copies of the affine $sl(2)_{k}$ algebra [17].
In conformal gravity in three dimensions with $so(3,2)$ underlying gauge
algebra one has twenty independent functions. If chemical potentials are set
to zero, ten functions remain. The modification of conformal gravity by adding
the vector field leads to five more functions which can vary. The addition of
the spin three field to the conformal gravity results with altogether 24
freely varying functions and independent charges. These bcs for the theories
in new conformal HiSGRA will in asymptotics have loop algebra of the
underlying gauge algebras of each theory. They can be further specified into
obtaining specific solutions.

Near horizon bcs [18] lead to configuration called black flower. Its
asymptotics contains number of affine $u(1)$ charges which commute with
Hamiltonian, and correspond to the soft hair [19]. Higher spin extension of
these bcs considers bosonic higher spin fields coupled to gravity in $AdS_{3}$
[15]. Near horizon holography of CHiSGRA in 3d [4] will analogously give
charges denoted with soft hair. These charges are going to depend on the
choice of the embedding. For the so(3,2) they are going to correspond to
dilatation and Lorentz rotation generators, as they will be represented with
diagonal matrices. For higher spin generalization after mapping of the spin-s
conformal algebra to $sl(N)$ algebra, they are going to correspond to $L_{0}$
and $W_{0}^{s}$ generators.

Here, we consider general bcs and near horizon bcs of CHiSGRA in three
dimensions, and the classification of the solutions from these bcs according
to one parameter subgroups of so(3,2) algebra.

First we review the new CHS theories in the section 2. Experts in higher spin
gravity can skip this section and proceed to section 3. In section 3 we review
holography of CS theory and study the general bcs with the fixed potential.
The structure of the bcs is generalized to an arbitrary spin. We show this for
conformal gravity (CG), CG with a vector field, CG with a spin three field,
and the most general case we can consider. The readers that are familiar with
holography, can proceed immediately to section 4.

In section 4, we consider near horizon bcs for the spin-s of new CHS theories,
and for the specific examples of CG, CG with a vector field, and CG with a
spin three field. For specific cases we construct a map between the generators
in the conformal ”PKLD” basis of gauge algebra and the generators in the ”WL”
fundamental representation. This basis allows for the simpler implementation
of the near horizon bcs. The $u(1)$ currents of the resulting algebra
correspond to the levels determined by the higher spin field added in the
action. When cast in the metric formulation, these bcs lead to the
generalization of well known solutions in three dimensional gravity. For the
rotating black hole in 3d we find generalization for 3d CG, and comment on
higher spin generalizations.

In section 5, we analyze CG in more detail by studying the one parameter
subgroups of the $so(3,2)$ group. The solutions of CG can be classified
according to the discrete group generated by the set of Killing vectors. The
spin connection defined with that Killing vector possesses distinct set of
eigenvalues and Casimir invariants. In comparison to $so(2,2)$ group,
$so(3,2)$ gives two additional Killing vectors that can be used to generate
distinct types of solutions. We also write the examples for types of solutions
in classification and clarify how our earlier holographic solutions obtained
from near horizon bcs fit in this classification. Section 6 summarises and
concludes our results.

##  2  New Conformal Higher Spin Theories

The construction of the new CHiSGRA is based on the Chern-Simons action. In
the standard Chern-Simons action

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

the field $\omega$ is Lie algebra valued one form. When the Lie algebra is
$so(3,2)$, the action is equivalent to CG action for $\omega$ valued in
Lorentz group [20]. The action (2.1) based on $so(3,2)$ group admits higher
spin extension [2] that leads to a theory with infinite tower of higher spin
fields. Based on the (2.1) action and $so(3,2)$ embedding group, (as predicted
in [2]) one can define an infinite number of theories, new CHiSGRA, each with
finite number of higher spin fields [4]. These higher spin fields consist of
Fradkin-Tseytlin fields and conformal partially massless fields. Fradkin-
Tseytlin fields are symmetric traceless tensors $\phi_{a_{1}...a_{s}}$ with
linearized gauge symmetries

$\displaystyle\delta\phi_{a_{1}...a_{s}}=\partial_{a_{1}}\xi_{a_{2}...a_{s}}+\text{permutations}-\text{traces}.$
(2.2)

With conformal partially massless fields we refer to the symmetric traceless
tensors that have higher derivatives in their gauge transformations

$\displaystyle\delta\phi_{a_{1}...a_{s}}=\partial_{a_{1}}...\partial_{a_{t}}\xi_{a_{t+1}...a_{s}}+\text{permutations}-\text{traces}.$
(2.3)

The latter can be also understood as boundary values for the real partially
massless fields in dimension higher AdS space. This kind of fields are going
to appear as a result of gauging the higher spin algebras constructed below.
We are going to need a higher spin extension of $so(3,2)$, i.e. algebras that
have $so(3,2)$ as a subalgebra and also contain various other nontrivial
representations of $so(3,2)$.

The theory has so far been studied from the perspective of non-relativistic
and ultra-relativistic limit in which the authors considered the spectra of
the theory and their comparison with the known Galilean and Carroll theories
in three dimensions [21, 22]. The work on Carroll limit also considers
holographic aspect of the theory and specific metric solutions which indicate
non-vanishing entropy. With this work being developed in parallel the charges
and asymptotic structure for specific similar cases can be read out.

###  2.1 Two Simple Higher Spin Algebras

##### Finite-dimensional algebras.

Algebras with a finite spectrum of Fradkin-Tseytlin fields are constructed by
taking any nontrivial finite-dimensional irreducible representation $V$ of
$so(3,2)$, for example an irreducible tensor or a spin-tensor. After that we
evaluate $U(so(d,2))$ in $V$, i.e. multiply the generators111Indices
$A,B,...=0,...,4$ are the indices of $so(3,2)$ with the invariant metric
denoted $\eta_{AB}$. ${\boldsymbol{T}}_{AB}=-{\boldsymbol{T}}_{BA}$ of
$so(3,2)$ in this representation and identify the algebra that they generate,
which we denote with $\mathfrak{hs}(V)$. That algebra is associative by
construction. Since $V$ is irreducible the algebra is the algebra of all
matrices of size $\dim V$, i.e. $\mathrm{End}(V)$. Therefore
$\mathfrak{hs}(V)=\mathrm{End}(V)$ comes equipped with a specific embedding of
$so(3,2)$ generators via ${\boldsymbol{T}}_{AB}$ by construction. This algebra
can be understood as the quotient $U(so(3,2))/I$ by a two-sided ideal $I$ that
is the annihilator of $V$, i.e. $I$ contains all polynomials in
${\boldsymbol{T}}_{AB}$ that vanish on $V$. The next step is decomposition of
$\mathfrak{hs}(V)$ into irreducible $so(3,2)$-modules — this decomposition, as
is shown below, determines the spectrum of Fradkin-Tseytlin fields. We are
interested in the holography of the action (2.1).

To summarize, there are infinite-many higher spin extensions of $so(3,2)$ that
are labelled by various irreducible $so(3,2)$-modules:222Various peculiarities
of higher derivative theories of type $\square^{k}\Phi(x)=0$ were noticed in
[23]. It would be interesting to see if the finite-dimensional higher spin
algebras can be explained from the CFT point of view.

$\displaystyle\mathfrak{hs}(V)=\mathrm{End}(V)=V\otimes V^{*}\,.$ (2.4)

Spin-tensor representations $V$ are also allowed.333For example, a toy-model
was studied in [2], where $\mathfrak{hs}$ was taken to be $so(4,2)$. This can
be understood as $sl(V)$ for $V$ being the spinor representation of $so(3,2)$.
Further extensions can be obtained by tensoring $\mathfrak{hs}(V)$ with the
matrix algebra $\mathrm{Mat}_{N}$, which turns the otherwise abelian spin-one
field associated with $1$ of $\mathrm{End}(V)$ into the Yang-Mills
field.444One can also project onto the symmetric and anti-symmetric parts of
the tensor product $V\otimes V$ (as a matter of fact $V\sim V^{*}$ for finite-
dimensional representations of $so(3,2)$), the only restriction being for it
to contain the adjoint, i.e. $so(3,2)$ itself. In this way one can define
higher spin analogs of $so(V)$ and $sp(V)$.  Note, that
$\mathfrak{hs}(V)=gl(V)=sl(V)\oplus u(1)$ as a Lie algebra and the spin-one
field always decouples unless the Yang-Mills groups are added. Since the
finite-dimensional higher spin algebras result from $\mathrm{End}(V)$ they are
equipped with a canonical trace operation ${\mathrm{Tr}\,}$, which allows us
to write down the Chern-Simons action.

##### Example 1.

As the simplest example, let us take the vector representation, which is
denoted by one-cell Young diagram \begin{picture}(10.0,10.0){}\end{picture} .
The corresponding algebra,
$\mathfrak{hs}(\parbox{10.0pt}{\begin{picture}(10.0,10.0){}\end{picture}})$,
has the following spectrum:555When treated as a higher spin algebra in
$AdS_{4}$ this spectrum corresponds to a spin-two field, partially-massless
depth-$3$ spin-three field and to a spin-one field.

$\displaystyle\mathfrak{hs}(\parbox{10.0pt}{\begin{picture}(10.0,10.0){}\end{picture}})$
$\displaystyle=\parbox{10.0pt}{\begin{picture}(10.0,10.0){}\end{picture}}\otimes\parbox{10.0pt}{\begin{picture}(10.0,10.0){}\end{picture}}=\parbox{10.0pt}{\begin{picture}(10.0,20.0){}\end{picture}}\oplus\parbox{20.0pt}{\begin{picture}(20.0,10.0){}\end{picture}}\oplus\bullet\,.$
(2.5)

The algebra is just the matrix algebra with $(d+2)^{2}$ generators
${\boldsymbol{t}}{}_{A}{}^{B}\,$ decomposed with respect to $so(d,2)$ (we keep
the discussion $d$-dimensional here, otherwise, $d=3$). The $gl_{d+2}$
commutation relations are

$\displaystyle[{\boldsymbol{t}}{}_{A}{}^{B}\,,{\boldsymbol{t}}{}_{C}{}^{D}\,]$
$\displaystyle=-\delta{}_{A}{}^{D}\,{\boldsymbol{t}}{}_{C}{}^{B}\,+\delta{}_{C}{}^{B}\,{\boldsymbol{t}}{}_{A}{}^{D}\,\,.$
(2.6)

In the $so(d,2)$ base, the irreducible generators are
${\boldsymbol{T}}_{AB}=-{\boldsymbol{T}}_{BA}$,
${\boldsymbol{S}}_{AB}={\boldsymbol{S}}_{BA}$ and ${\boldsymbol{R}}$, i.e.

$\displaystyle{\boldsymbol{T}}_{AB}$
$\displaystyle={\boldsymbol{t}}_{A|B}-{\boldsymbol{t}}_{B|A}\,,$
$\displaystyle{\boldsymbol{S}}_{AB}$
$\displaystyle={\boldsymbol{t}}_{A|B}+{\boldsymbol{t}}_{B|A}-\frac{2}{d+2}{\boldsymbol{t}}{}_{C}{}^{C}\,\eta_{AB}\,,$
$\displaystyle{\boldsymbol{R}}={\boldsymbol{t}}{}_{C}{}^{C}\,\,.$ (2.7)

The commutation relations read

$\displaystyle[{\boldsymbol{T}}_{AB},{\boldsymbol{T}}_{CD}]$
$\displaystyle=\eta_{BC}{\boldsymbol{T}}_{AD}-\eta_{AC}{\boldsymbol{T}}_{BD}-\eta_{BD}{\boldsymbol{T}}_{AC}+\eta_{AD}{\boldsymbol{T}}_{BC}\,,$
(2.8a) $\displaystyle[{\boldsymbol{T}}_{AB},{\boldsymbol{S}}_{CD}]$
$\displaystyle=\eta_{BC}{\boldsymbol{S}}_{AD}-\eta_{AC}{\boldsymbol{S}}_{BD}+\eta_{BD}{\boldsymbol{S}}_{AC}-\eta_{AD}{\boldsymbol{S}}_{BC}\,,$
(2.8b) $\displaystyle[{\boldsymbol{S}}_{AB},{\boldsymbol{S}}_{CD}]$
$\displaystyle=\eta_{BC}{\boldsymbol{T}}_{AD}+\eta_{AC}{\boldsymbol{T}}_{BD}+\eta_{BD}{\boldsymbol{T}}_{AC}+\eta_{AD}{\boldsymbol{T}}_{BC}\,,$
(2.8c)

while ${\boldsymbol{R}}$ commutes with everything since it is associated with
$1$ in $gl(V)$. We can truncate it away! As a result our theory contains two
fields:

$\displaystyle\omega$
$\displaystyle=\omega^{A,B}{\boldsymbol{T}}_{AB}+\omega^{AB}{\boldsymbol{S}}_{AB}$
(2.9)

The first one is the conformal graviton and the second one is the similar to
the partially-massless spin-three case. It is studied in the paper. In flat
space background we find a metric-like field with the following gauge
symmetries

$\displaystyle\delta\phi^{abc}$
$\displaystyle=\partial^{a}\partial^{b}\partial^{c}\xi-\frac{1}{5}(\eta^{ab}\partial^{c}\square\xi+\eta^{ac}\partial^{b}\square\xi+\eta^{ac}\partial^{b}\square\xi)\,.$
(2.10)

The bilinear form

$\displaystyle{\mathrm{Tr}\,}({\boldsymbol{t}}{}_{A}{}^{B}\,,{\boldsymbol{t}}{}_{C}{}^{D}\,)$
$\displaystyle=\delta{}_{A}{}^{D}\,\delta{}_{C}{}^{B}\,$ (2.11)

from which we easily get for $so(d,2)$

$\displaystyle{\mathrm{Tr}\,}({\boldsymbol{T}}_{A,B},{\boldsymbol{T}}_{C,D})$
$\displaystyle=2(\eta_{AD}\eta_{CB}-\eta_{BD}\eta_{CA})$ (2.12)
$\displaystyle{\mathrm{Tr}\,}({\boldsymbol{S}}_{A,B},{\boldsymbol{S}}_{C,D})$
$\displaystyle=2(\eta_{AD}\eta_{CB}+\eta_{BD}\eta_{CA}-\frac{2}{d+2}\eta_{AB}\eta_{CD})$
(2.13)

##### Example 2.

We can take $V$ to be the spinor of $so(3,2)$ then the resulting algebra is
nothing but $so(4,2)$. Let us simply write its commutation relations

$\displaystyle[{\boldsymbol{T}}_{AB},{\boldsymbol{T}}_{CD}]$
$\displaystyle=\eta_{BC}{\boldsymbol{T}}_{AD}-\eta_{AC}{\boldsymbol{T}}_{BD}-\eta_{BD}{\boldsymbol{T}}_{AC}+\eta_{AD}{\boldsymbol{T}}_{BC}\,,$
(2.14a) $\displaystyle[{\boldsymbol{T}}_{AB},{\boldsymbol{U}}_{C}]$
$\displaystyle=\eta_{BC}{\boldsymbol{U}}_{A}-\eta_{AC}{\boldsymbol{U}}_{B}\,,$
(2.14b) $\displaystyle[{\boldsymbol{U}}_{A},{\boldsymbol{U}}_{C}]$
$\displaystyle={\boldsymbol{T}}_{AC}$ (2.14c)

So the gauge field is

$\displaystyle\omega$
$\displaystyle=\omega^{A,B}{\boldsymbol{T}}_{AB}+\omega^{A}{\boldsymbol{U}}_{A}$
(2.15)

and the second field describes a conformal analog of the partially-massless
spin-two field:

$\displaystyle\delta\phi_{ab}=\partial_{a}\partial_{b}\xi-\frac{1}{3}\eta_{ab}\square\xi\,.$
(2.16)

When we represent the generators of this basis, we refer to to corresponding
matrices as written in PKLD basis. The name comes from the common use of
letters
${\boldsymbol{P}}_{a},{\boldsymbol{K}}_{a},{\boldsymbol{L}}_{ab},{\boldsymbol{D}}$,
for
$t_{a+}\equiv{\boldsymbol{P}}_{a},t_{ab}\equiv{\boldsymbol{L}}_{ab},t_{a-}\equiv{\boldsymbol{K}}_{a},t_{+-}\equiv-{\boldsymbol{D}}$
to refer to generators of translations, special conformal transformations,
Lorentz rotations and dilatations respectively, which appear in each of the
examples of the new CHiSGRA theories.

##  3 General boundary conditions

The Chern-Simons action can be put in the canonical form on the manifold with
the topology $\Sigma\times\mathbb{R}$ where $\Sigma$ is a two dimensional
manifold on the boundary and $\mathbb{R}$ denotes one dimensional manifold.
$\Sigma$ is usually considered as a disc with coordinates $\rho$ and $\phi$.
The action can be written in the 2+1 decomposition and calculation of charges
follows standard procedure of holographic analysis for Chern-Simons action,
which is briefly reviewed in the Appendix A. We are going to partially fix to
radial gauge, which means that we can write the field $\omega$ as

$\displaystyle\omega=b(\rho)^{-1}(d+\Omega(t,\phi))b(\rho)$ (3.1)

and group element $b$ depends on the radial coordinate. The particular choice
of bcs is then going to be done by choosing $\Omega(t,\phi)$, and when we want
to translate the solution to metric formulation, it will depend on the precise
form of the group element $b$. Earlier works that focused on the $sl(2)\times
sl(2)$ and generalizations to $sl(N)\times sl(N)$, have analyzed specific bcs,
fixing for example $\omega_{-}=0$ (for $x_{-}=t-\phi$) and choosing radial
gauge, or focusing on bcs that reproduce asymptotically $AdS_{3}$, BTZ black
hole, or massless higher spin generalizations [24, 11, 25, 26]. Here, when we
write the bcs as (3.1) we will focus on the generalization of the latter for
conformal and higher spin conformal cases.

Field dependence. In our picture of Chern-Simons theory, beside gauge
transformations, there is an invariance under diffeomorphisms, and other
symmetries depending on the conformal HiSGRA that we are considering. For
example, for conformal graviton, there would appear an additional invariance
under Lorentz transformations, special conformal transformations and
dilatation. For the conformal graviton and higher spin field, in addition
there would be invariance under higher spin symmetries. This allows that there
is an independent set of conserved charges due to that symmetry. The parametrs
of the gauge transformations responsible for these symmetries are
$\xi^{a}=-\xi^{i}\omega_{i}^{a}$ for $\xi^{i}$ an arbitrary vector. The
corresponding charge $Q$ can be written as

$\displaystyle\delta
Q=\frac{k}{2\pi}\int\xi^{i}\omega^{i}\delta\omega_{j}dx^{j}$ (3.2)

Taking into account gauge conditions one obtains the algebra of global charges

$\displaystyle\\{Q(\xi^{i}),Q(\zeta^{j})\\}^{*}=Q([\xi,\zeta]^{i})+\frac{k\alpha^{2}}{2\pi}\int\xi^{r}\partial_{\phi}\zeta^{r}d\phi$
(3.3)

where $\omega_{r}^{a}\equiv\alpha^{a}$ is a constant element valued in the Lie
algebra, and $\xi^{r},\zeta^{r}$ are $r$ components of the vectors $\xi^{i}$
and $\zeta^{i}$. Using the Fourier modes, one can write a charge as

$\displaystyle Q(\xi)=\sum_{n}T_{n}\xi^{n}$ (3.4)

and the modes with

$\displaystyle\xi^{n}=\frac{1}{4\pi}\int d\phi\xi(\phi)e^{in\phi}$ (3.5)

The procedure that looks for ASA then requires these modes to be inserted into
the (3.3). In the general case for full set of generators, and with no special
conditions, this leads to the loop algebra of the underlying gauge algebra. In
such case, one can by choosing the right group element, obtain any of the
particular metrics that were originally obtained by a stricter set of bcs
[27]. When the bcs are already restricted, the described procedure leads to
some specific algebra of charges. We will see this on the example of near
horizon bcs.

###  3.1 Conformal spin-s field

We are going to follow the procedure in [17], where general $AdS_{3}$ bcs for
general relativity are considered. As we said, considerations of holography in
$AdS_{3}$ usually assume 3D manifold $\mathcal{M}$ topologically as a
cylinder. The radial coordinate is denoted as $\rho\in\mathbb{R}^{+}$, angular
coordinate is $\phi\sim\phi+2\pi$, while the time coordinate along cylinder is
denoted with $t\in\mathbb{R}$. The standard basis of our generators has been
discussed above, and for our new gravity theories they are
${\boldsymbol{t}}_{A(k),B(l)}$. Where ${\boldsymbol{t}}_{A(k),B(l)}$ is
general generator from which we can write the symmetric, antisymmetric and
scale generators. That means that ${\boldsymbol{t}}_{A(k),B(l)}$ is going to
be one of the main ingredients used to define $\Omega(t,\phi)$ in (3.1). In
this case the group element $b\in so(3,2)$ and $\delta b=0$. We also assume
$b$ is $b(\rho)$ as we mentioned above. The ansatz for $\Omega(t,\phi)$, is

$\displaystyle\Omega$
$\displaystyle=\mathcal{\tau}^{A(k),B(l)}(t,\phi){\boldsymbol{t}}_{A(k),B(l)}$
(3.6)

$\Omega\in so(3,2)$ and $\delta\Omega\neq 0$. The independent sets of
functions in (3.6) are denoted by $\mathcal{\tau}^{A(k),B(l)}$, which we call
charges and allow them to vary

$\displaystyle\delta\Omega$
$\displaystyle=\delta\mathcal{\tau}^{A(k),B(l)}(t,\phi){\boldsymbol{t}}_{A(k),B(l)}.$
(3.7)

If we were to fix some of these conditions, equations of motion would impose
additional conditions on charges. We expect that these bcs lead to integrable
canonical boundary charges and well-defined variational principle. Combining
(3.1) and (3.6) we obtain

$\displaystyle\omega=b(\rho)^{-1}(d+\Omega(t,\phi))b(\rho)=b(\rho)^{-1}(d+\mathcal{\tau}^{A(k),B(l)}(t,\phi){\boldsymbol{t}}_{A(k),B(l)})b(\rho).$
(3.8)

Asymptotic symmetry algebra.  To find the ASA, we have to consider
transformations

$\displaystyle\delta_{\xi}\omega=d\xi+[\omega,\xi]=\mathcal{O}(\delta\omega)$
(3.9)

which preserve above bcs. For this we have to split the gauge parameter $\xi$
into $so(3,2)$ components

$\displaystyle\xi=b^{-1}[\xi_{i}^{A(k),B(l)}{\boldsymbol{t}}_{A(k),B(l)}]b.$
(3.10)

Here $i$ denotes the gauge parameter which distinguishes between different
generators $t_{A(k),B(l)}$ to which it is associated. One needs to find the
restriction on the functions $\xi_{i}$ that conserve the transformation (3.9).
Then it is possible to determine the infinitesimal changes of the functions
$\mathcal{{\boldsymbol{t}}}_{A(k),B(l)}$ under transformations that preserve
bcs (3.1), (3.8). The variation of the canonical boundary charges which is
background independent [28, 25, 24, 29] is then equal to

$\displaystyle\delta Q[\xi]=\frac{k}{2\pi}\oint\left<\xi\delta\omega\right>.$
(3.11)

Functional integration of (3.11) gives a result for the canonical boundary
charges. Infinitesimal transformation and canonical boundary charge yield ASA
in canonical realization, using (3.3),(3.4) and (3.5).

Since we had an expansion of gauge field and a gauge parameter in arbitrary
spin-s generator of the given algebra, one can see from the equation (3.11)
that the contribution to the background independent boundary charge is going
to be

$\displaystyle\delta Q$ $\displaystyle=\frac{k}{2\pi}\oint
d\phi\xi_{1}^{A(l),B(k)}\delta\mathcal{\tau}^{C(m),D(n)}\left<{\boldsymbol{t}}_{A(l),B(k)},{\boldsymbol{t}}_{C(m),D(n)}\right>.$
(3.12)

Here $\left<{\boldsymbol{t}}_{A(l),B(k)},{\boldsymbol{t}}_{C(m),D(n)}\right>$
is $\text{Tr}({\boldsymbol{t}}_{A(l),B(k),{\boldsymbol{t}}_{C(m),D(n)}})$, two
specific examples for spin-2 and spin-3 we have seen in (2.12) and (2.13). If
we considered all the theories from spin-2 to arbitrary spin, the charges
would have contribution from generators from antisymmetric spin-2
${\boldsymbol{T}}_{AB}$ up to arbitrary spin ${\boldsymbol{t}}_{A(m),B(n)}$.
The algebra that would give closed form for the
${\boldsymbol{t}}^{m}_{A(l),B(k)}$ and ${\boldsymbol{t}}^{n}_{C(m),D(n)}$ is
not yet known. Once the algebra is known, after integration and the
decomposition of the gauge parameters and the fields in the Fourier
decomposition at the boundary, that charge would lead to the loop algebra of
the underlying gauge algebra.

The practical side of this bcs is that from them, one can expect to obtain
further bcs with more restrictions.

In the following subsections, we consider examples 1 and 2 that include
conformal spin-2 and spin-3 field, and conformal spin-2 and vector field
respectively. Similarly as in the section 2.1.

###  3.2 Conformal spin-2 and spin-3 field

The generators of the conformal spin-2 and spin-3 we denote with
${\boldsymbol{T}}_{AB}$ and ${\boldsymbol{S}}_{AB}$ respectively. As already
indicated in the last paragraph of the previous chapter, the generators are
split as ${\boldsymbol{t}}_{ab}={\boldsymbol{L}}_{ab}$,
${\boldsymbol{t}}_{a+}={\boldsymbol{P}}_{a}$,
${\boldsymbol{t}}_{a-}={\boldsymbol{K}}_{a}$ and
${\boldsymbol{t}}_{+-}={\boldsymbol{D}}$. From the symmetric generators we
denote them as ${\boldsymbol{S}}_{ab}=s_{ab}$, ${\boldsymbol{S}}_{a+}=s_{a+}$,
${\boldsymbol{S}}_{a-}=s_{a-}$, ${\boldsymbol{S}}_{+-}=s_{+-}$ 666In the
literature [21, 22] these generators are also written as
${\boldsymbol{S}}_{ab}=s_{ab}=SL_{ab}$, ${\boldsymbol{S}}_{a+}=s_{a+}=SP_{a}$,
${\boldsymbol{S}}_{a-}=s_{a-}=SK_{a}$, ${\boldsymbol{S}}_{+-}=s_{+-}=SD$ to
denote their origin and refer to them as symmetric version of $so(3,2)$
generators of PKLD basis. Here the letters $\\{A,B,..\\}$ refer to lightcone
coordinates with metric $\eta_{AB}$ on the lightcone, and
$\eta_{+-}=\eta_{-+}=1$. The letters $\\{a,b..\\}$ refer to Minkowski
coordinates. If we write $\Omega(t,\phi)$ in (3.6) as

$\displaystyle\Omega$
$\displaystyle=\mathcal{T}^{AB}(t,\phi){\boldsymbol{T}}_{AB}+\mathcal{S}^{AB}(t,\phi){\boldsymbol{S}}_{AB}$
(3.13)

which is related to example 1 from section 2, using (3.1), by

$\displaystyle\omega=\omega^{A,B}{\boldsymbol{T}}_{AB}+\omega^{AB}{\boldsymbol{S}}_{AB}=b(\rho)^{-1}(d+\mathcal{T}^{AB}(t,\phi){\boldsymbol{T}}_{AB}+\mathcal{S}^{AB}(t,\phi){\boldsymbol{S}}_{AB})b(\rho)$
(3.14)

we will have two independent sets of functions (charges) in the above
conditions, and allow them to vary.

$\displaystyle\delta\Omega$
$\displaystyle=\delta\mathcal{T}^{AB}(t,\phi){\boldsymbol{T}}_{AB}+\delta\mathcal{S}^{AB}(t,\phi){\boldsymbol{S}}^{AB}$
(3.15)

The charges $\mathcal{T}_{AB}$ when split are also referred to as
$\mathcal{T}_{ab}=\omega_{ab}$, $\mathcal{T}_{a+}=e_{a}$,
$\mathcal{T}_{a-}=\sigma_{a}$, $\mathcal{T}_{+-}=b$. Which makes
distinguishable part that comes from three dimensional conformal gravity [20].
Fixing the above conditions (3.13) and (3.15), results in additional
conditions on charges via equations of motion. We need that the transformation
(3.9) preserves bcs. First we expand the gauge parameter $\xi$ into $so(3,2)$
components

$\displaystyle\xi=b^{-1}[\xi_{1}^{A,B}{\boldsymbol{T}}_{AB}+\xi_{2}^{AB}{\boldsymbol{S}}_{AB}]b.$
(3.16)

Here $\xi^{A,B}_{1}$ and $\xi^{AB}_{2}$ are functions that conserve the
transformation (3.9). Evaluating (3.9) gives the infinitesimal changes of the
functions $\mathcal{T}_{AB},\mathcal{S}_{AB},\mathcal{R}$.

Inserting the values for (3.16) and (3.15) into (3.11) functional integration
of (3.11) gives a result for the variation of canonical boundary charges

$\displaystyle\delta Q$ $\displaystyle=\frac{k}{2\pi}\oint
d\phi\left(\xi_{1}^{AB}\delta\mathcal{T}^{CD}\left<T_{AB},{\boldsymbol{T}}_{CD}\right>+\xi_{2}^{AB}\delta\mathcal{S}^{CD}\left<{\boldsymbol{S}}_{AB},S_{CD}\right>\right).$
(3.17)

It is evaluated using (2.12) and (2.13), and after integration it leads to

$\displaystyle Q$ $\displaystyle=\frac{k}{2\pi}\oint
d\phi\left(2\xi_{1}^{AB}\mathcal{T}^{CD}\left(\eta_{AD}\eta_{CB}-\eta_{BD}\eta_{CA}\right)\right.$
(3.18)
$\displaystyle\left.+\xi_{2}^{AB}\mathcal{S}^{CD}\left(\eta_{AD}\eta_{CB}+\eta_{BD}\eta_{CA}-\frac{2}{d+2}\eta_{AB}\eta_{CD}\right)\right).$

Decomposing the gauge parameters and the fields at the boundary in the Fourier
modes gives the ASA

$\displaystyle[{\boldsymbol{T}}_{AB}^{n},{\boldsymbol{T}}_{CD}^{m}]$
$\displaystyle=\eta_{BC}{\boldsymbol{T}}_{AD}^{n+m}-\eta_{AC}{\boldsymbol{T}}_{BD}^{n+m}-\eta_{BD}{\boldsymbol{T}}_{AC}^{n+m}+\eta_{AD}{\boldsymbol{T}}_{BC}^{n+m}-knTr({\boldsymbol{T}}_{AB}{\boldsymbol{T}}_{CD})\delta_{n+m,0}$
(3.19) $\displaystyle[{\boldsymbol{T}}_{AB}^{n},{\boldsymbol{S}}_{CD}^{m}]$
$\displaystyle=\eta_{BC}{\boldsymbol{S}}_{AD}^{n+m}-\eta_{AC}{\boldsymbol{S}}_{BD}^{n+m}-\eta_{BD}{\boldsymbol{S}}_{AC}^{n+m}+\eta_{AD}{\boldsymbol{S}}_{BC}^{n+m}$
(3.20) $\displaystyle[{\boldsymbol{S}}_{AB}^{n},{\boldsymbol{S}}_{CD}^{m}]$
$\displaystyle=\eta_{BC}{\boldsymbol{T}}_{AD}^{n+m}+\eta_{AC}{\boldsymbol{T}}_{BD}^{n+m}+\eta_{BD}{\boldsymbol{T}}_{AC}^{n+m}+\eta_{AD}{\boldsymbol{T}}_{BC}^{n+m}+knTr({\boldsymbol{S}}_{AB}{\boldsymbol{S}}_{CD})\delta_{n+m,0}$
(3.21)

If we had a vector field the calculation and the result would be similar, with
the difference that we would obtain vanishig central charge for the
commutation relations of vector field and fields appearing with multiplication
of symmetric or antisymmetric generators. The practical side of this bcs is
that from them, one can expect to obtain entire set of possible solutions by
imposing further specific restrictions.

###  3.3 Conformal spin-2 and spin-1 field

The PKLD basis of generators for the example 2 in (2.15) will lead to the
ansatz for the function (3.6)
$\Omega(t,\phi)=\Omega_{\text{CG}}(t,\phi)+\Omega_{\text{V}}(t,\phi)$ for

$\displaystyle\Omega_{\text{CG}}$
$\displaystyle=e^{a}(t,\phi){\boldsymbol{P}}_{a}+\omega^{a}_{b}(t,\phi){\boldsymbol{L}}^{b}_{a}+\sigma^{a}(t,\phi){\boldsymbol{K}}_{a}+\phi(t,\phi){\boldsymbol{D}}.$
(3.22) $\displaystyle\Omega_{\text{V}}$
$\displaystyle=\omega^{+}(t,\phi){\boldsymbol{U}}_{+}+\omega^{-}(t,\phi){\boldsymbol{U}}_{-}+\omega^{a}(t,\phi){\boldsymbol{U}}_{a}.$
(3.23)

We denote the independent functions which we allow to vary with
$e_{a},\omega_{ab},\sigma_{a},\phi,\omega^{+},\omega^{-},\omega^{a}$, i.e. the
CG part is denoted as in the previous example, while the new functions belong
to a vector field. The gauge parameter is

$\displaystyle\xi_{\text{CG}}$
$\displaystyle=b^{-1}[\xi_{1}^{a}(t,\phi){\boldsymbol{P}}_{a}+\xi_{2}^{a}{}_{b}(t,\phi){\boldsymbol{L}}^{b}_{a}+\xi_{3}^{a}(t,\phi){\boldsymbol{K}}_{a}+\xi_{4}(t,\phi){\boldsymbol{D}}]b$
(3.24) $\displaystyle\xi_{\text{V}}$
$\displaystyle=b^{-1}[\xi_{5}^{+}(t,\phi){\boldsymbol{U}}_{+}+\xi_{6}{}^{-}(t,\phi){\boldsymbol{U}}_{-}+\xi_{7}^{a}(t,\phi){\boldsymbol{U}}_{a}]b.$
(3.25)

Inserting the gauge parameter and variation of charges in the formula (3.11)
we obtain:

$\displaystyle\delta Q_{\text{CG+V}}$ $\displaystyle=\frac{k}{2\pi}\oint
d\phi\left(\xi_{1}^{a}\delta\sigma^{b}\left<{\boldsymbol{P}}_{a}{\boldsymbol{K}}_{b}\right>+\xi_{3}^{a}\delta
e^{b}\left<{\boldsymbol{K}}_{a}{\boldsymbol{P}}_{b}\right>+\xi_{2}^{ab}\delta\omega^{cd}\left<{\boldsymbol{L}}_{ab}{\boldsymbol{L}}_{cd}\right>+\xi_{4}\delta\phi\left<DD\right>\right.$
$\displaystyle+\left.\xi_{5}^{+}\delta\omega^{-}\left<{\boldsymbol{U}}_{+}{\boldsymbol{U}}_{-}\right>+\xi_{6}^{-}\delta\omega^{+}\left<{\boldsymbol{U}}_{-}{\boldsymbol{U}}_{+}\right>+\xi_{7}^{a}\delta\omega^{b}\left<{\boldsymbol{U}}_{a}{\boldsymbol{U}}_{b}\right>\right)$
(3.26) $\displaystyle=\frac{k}{2\pi}\oint
d\phi\left(-2\xi_{1}^{a}\delta\sigma_{a}-2\xi_{3}^{a}\delta
e_{a}-4\xi_{2}^{ab}\delta\omega_{ab}+2\xi_{4}\delta\phi\right.$
$\displaystyle\left.+2\xi_{5}^{+}\delta\omega^{-}+2\xi_{6}^{-}\delta\omega^{+}+2\xi_{7}^{a}\delta\omega^{b}\right).$
(3.27)

While the functional integration of (3.27), gives canonical boundary charges

$\displaystyle Q_{\text{CG+V}}$ $\displaystyle=\frac{k}{2\pi}\oint
d\phi\left(-2\xi_{1}^{a}\sigma_{a}-2\xi_{3}^{a}e_{a}-4\xi_{2}^{ab}\omega_{ab}+2\xi_{4}\phi+2\xi_{5}^{+}\omega^{-}+2\xi_{6}^{-}\omega^{+}+2\xi_{7}^{a}\omega^{b}\right).$
(3.28)

Using the expression for the charges (3.3) and appropriate Fourier
decomposition $\xi^{a}$ and
$\sigma_{a},e_{a},\omega_{ab},\phi,\omega^{+},\omega^{-},\omega^{a}$ at the
boundary, one can notice that the algebra will be a loop algebra of the
generators that define the action

$\displaystyle[{\boldsymbol{D}}^{n},{\boldsymbol{P}}_{a}^{m}]$
$\displaystyle=-{\boldsymbol{P}}^{n+m}_{a}\,,$
$\displaystyle[{\boldsymbol{L}}_{ab}^{n},{\boldsymbol{P}}_{c}^{m}]$
$\displaystyle={\boldsymbol{P}}_{a}^{n+m}\eta_{bc}-{\boldsymbol{P}}_{b}^{n+m}\eta_{ac}\,,$
(3.29a) $\displaystyle[{\boldsymbol{D}}^{n},{\boldsymbol{K}}^{m}_{a}]$
$\displaystyle={\boldsymbol{K}}^{n+m}_{a}\,,$
$\displaystyle[{\boldsymbol{L}}_{ab}^{n},{\boldsymbol{K}}_{c}^{m}]$
$\displaystyle={\boldsymbol{K}}_{a}^{n+m}\eta_{bc}-{\boldsymbol{K}}_{b}^{n+m}\eta_{ac}\,,$
(3.29b) $\displaystyle[{\boldsymbol{P}}_{a}^{n},{\boldsymbol{K}}_{b}^{m}]$
$\displaystyle=-{\boldsymbol{L}}_{ab}^{n+m}+\eta_{ab}{\boldsymbol{D}}^{n+m}+2kn\eta_{ab}\delta^{n+m,0}\,,$
(3.29c) $\displaystyle[{\boldsymbol{L}}_{ab}^{n},{\boldsymbol{L}}_{cd}^{m}]$
$\displaystyle=\eta_{bc}{\boldsymbol{L}}_{ad}^{n+m}-\eta_{ac}{\boldsymbol{L}}_{bd}^{n+m}-\eta_{bd}{\boldsymbol{L}}_{ac}^{n+m}+\eta_{ad}{\boldsymbol{L}}_{bc}^{n+m}+knTr[{\boldsymbol{L}}_{ab},{\boldsymbol{L}}_{cd}]\delta^{n+m,0}\,,$
(3.29d)

The part of the algebra with the vector filed will be

$\displaystyle[{\boldsymbol{T}}^{n}_{AB},{\boldsymbol{U}}^{m}_{C}]$
$\displaystyle=\eta_{BC}{\boldsymbol{U}}^{n+m}_{A}-\eta_{AC}{\boldsymbol{U}}^{n+m}_{B}\,,$
(3.30a) $\displaystyle[{\boldsymbol{U}}^{n}_{A},{\boldsymbol{U}}^{m}_{C}]$
$\displaystyle={\boldsymbol{T}}^{n+m}_{AC}+2kn\eta_{AC}\delta^{n+m,0}.$
(3.30b)

Here we used the identification from section 2.1. The above consideration
automatically shows the result that one would obtain considering only CG.

In the following section we will consider near horizon bcs. On the example of
conformal gravity, one will be able to see that from the entire set of
generators for the ansatz taken in (3.22) we will restrict ourselves to those
whose matrices are in our basis diagonal. On the other examples we will see
that first we have to find the map from the generators in our basis PKLD to
the generators in WL basis in which the near bcs are easier to impose. The map
will actually select the right coefficients and generators to be kept in (3.6)
777 As a possible continuation of the research one can study classification of
the bcs according to subalgebras of so(3,2) in Patera et al. [Patera:1976my].

##  4 Near horizon boundary conditions

The following set of bcs that we consider are near horizon bcs. Their name is
inspired by a fact that they lead to the black hole solutions when the
underlying gauge group of CS theory is $sl(N,\mathbb{R})\oplus
sl(N,\mathbb{R})$, which is represented in WL representation.

When the underlying gauge algebra of the theory is $sl(n,\mathbb{R})\oplus
sl(n,\mathbb{R})$, the connections $\Omega$ and $\overline{\Omega}$ take value
in the generators $L_{0}$ and $W_{0}^{(s)}$ of Cartan subalgebra of the gauge
group. These generators have diagonal matrices in the fundamental
representation which we call WL representation (according the names of the
generators). $\Omega$ is then defined by

$\displaystyle\Omega(n)=(\mathcal{K}d\phi+\mu
dt)L_{0}+\sum_{s=3}^{n}(\mathcal{K}_{s}d\phi+\mu_{s}dt)W_{0}^{(s)}$ (4.1)

where and $\overline{\Omega}$ is defined analogously. Arbitrary functions of
time and angle in (4.1) define the dynamical fields $\mathcal{K}$ and
$\mathcal{K}_{s}$ while $\mu$ and $\mu_{s}$ are chemical potentials. If we
compare this with definition of $\Omega$ in the previous section, we see that
here we write explicitly the $d\phi$ and $dt$ part. While in the previous
section we set chemical potential to zero. Concretely, in the
$(\ref{genbcs1})$ the generators ${\boldsymbol{t}}_{A,B}$ are selected such
that they form $L_{0}$ and $W_{0}^{s}$ in (4.1).

The new CHS theory, i.e. its corresponding infinite set of theories, we can
write as

$\displaystyle\mathfrak{hs}\Big{|}_{so(3,2)}=\bigoplus_{s=0}^{\infty}\bigoplus_{k=0}^{k=q-1}\parbox{90.0pt}{{\begin{picture}(90.0,20.0)\put(0.0,0.0){\begin{picture}(70.0,10.0)\put(0.0,0.0){\line(1,0){70.0}}\put(0.0,0.0){\line(0,1){10.0}}\put(70.0,10.0){\line(-1,0){70.0}}\put(70.0,10.0){\line(0,-1){10.0}}{\put(35.0,1.0){{$s-2k-1$}}}\end{picture}}\put(0.0,10.0){\begin{picture}(90.0,10.0)\put(0.0,0.0){\line(1,0){90.0}}\put(0.0,0.0){\line(0,1){10.0}}\put(90.0,10.0){\line(-1,0){90.0}}\put(90.0,10.0){\line(0,-1){10.0}}{\put(45.0,1.0){{$s-1$}}}\end{picture}}\end{picture}}}\,$
(4.2)

Here, we denoted that decomposition is done according to $so(3,2)$ algebra.
For each $s$ we obtain an algebra which leads to corresponding theory. Each
such algebra is isomorphic to one copy of a certain $sl(n,\mathbb{R})$
algebra. Knowing this $n$, we can write the bcs as in (4.1), with a difference
that using this scenario we will not have a second copy of $sl(n,\mathbb{R})$
and corresponding $\overline{\Omega}$. In (4.1), the values that $n$ can take
are determined by
$n=\sqrt{\frac{1}{6}\left(2s_{1}+3\right)\left(s_{1}-s_{2}+1\right)\left(s_{1}+s_{2}+2\right)\left(2s_{2}+1\right)+1}$.
Here $s_{1}$ is the number of boxes in the first row of the Young diagram in
(4.2), while $s_{2}$ is the number of boxes in the second row. To use the near
horizon bcs in the WL representation, we need to know how to write WL
generators in terms of PKLD generators. We find an explicit map from the PKLD
basis on the examples of conformal graviton and a vector field, whose algebra
corresponds to $sl(4)$ algebra, and for the graviton and a spin three field.
The algebra of the graviton and a spin three field corresponds to $sl(5)$
algebra. For the conformal graviton itself, whose algebra is $so(3,2)$ there
is no $n$ for which one obtains isomorphism to $sl(n)$ algebra, so we consider
this example by choosing diagonal generators in PKLD basis.

In comparison to most general bcs, fields $\mathcal{K}_{s}$ would correspond
to a certain linear combination of the fields $\tau^{A(k),B(l)}$. The chemical
potentials are assumed not to vary and they are fixed at the boundary. (Notice
that in comparison to the general bcs, we have simply chosen suitable sets of
$\tau^{A(k),B(l)}$ and imposed such restrictions that only terms with $L_{0}$
and $W_{0}^{(s)}$ are kept.) The equations of motion for the fields read

$\displaystyle\partial_{t}\mathcal{K}=\partial_{\phi}\mu,$
$\displaystyle\partial_{t}\mathcal{K}_{3}=\partial_{\phi}\mu_{3},$
$\displaystyle...$
$\displaystyle\partial_{t}\mathcal{K}_{s}=\partial_{\phi}\mu_{s}.$ (4.3)

The form (4.1) is maintained by the asymptotic symmetries defined by
parameters valued in a Lie algebra. The gauge parameters take the form

$\displaystyle\epsilon=\xi L_{0}+\sum_{s=3}^{n}\xi_{s}W_{0}^{(s)}$ (4.4)

for $\partial_{t}\xi=\partial_{t}\xi_{3}=...=\partial_{t}\xi_{s}=0$ if the
fields transform as

$\displaystyle\delta\mathcal{K}=\partial_{\phi}\xi,$
$\displaystyle\delta\mathcal{K}_{3}=\partial_{\phi}\xi_{3},$
$\displaystyle...$
$\displaystyle\delta\mathcal{K}_{s}=\partial_{\phi}\xi_{s}.$ (4.5)

One can write the canonical generators of the asymptotic symmetries as

$\displaystyle\mathcal{Q}[\eta,..,\eta_{(s)},n]=-\frac{k}{4\pi}\int
d\phi\left(\xi(\phi)\mathcal{K}(\phi)+\sum_{s=3}^{n}\alpha_{s}\xi_{s}(\phi)\mathcal{K}_{a}(\phi))\right)$
(4.6)

for

$\displaystyle\alpha^{(n)}_{s}=\frac{48(s-1)!^{4}}{(2s-1)!(2s-2)!}\prod_{i=2}^{s-1}(n^{2}-i^{2}).$
(4.7)

The charges define the algebra

$\displaystyle\\{\mathcal{K}(\phi),\mathcal{K}(\phi^{\prime})\\}$
$\displaystyle=-\frac{4\pi}{k}\delta^{\prime}(\phi-\phi^{\prime})$ (4.8a)
$\displaystyle\\{\mathcal{K}_{s}(\phi),\mathcal{K}_{s}(\phi^{\prime})\\}$
$\displaystyle=-\frac{4\pi}{\alpha_{s}k}\delta^{\prime}(\phi-\phi^{\prime}).$
(4.8b)

If we expand the charges in the Fourier modes using

$\displaystyle\mathcal{K(\phi)}=\frac{2}{k}\sum_{n=-\infty}^{\infty}J_{n}e^{in\phi},$
$\displaystyle\mathcal{K}_{3}(\phi)=\frac{2}{\alpha_{4}k}\sum_{n=-\infty}^{\infty}J^{(3)}_{n}e^{in\phi},$
$\displaystyle...$
$\displaystyle\mathcal{K}_{s}(\phi)=\frac{2}{\alpha_{s}k}\sum_{n=-\infty}^{\infty}J^{(s)}_{n}e^{in\phi},$
(4.9)

the algebra (4.8) gives ASA

$\displaystyle i\\{J_{n},J_{m}\\}$ $\displaystyle=\frac{k}{2}n\delta_{m+n,0}$
$\displaystyle
i\\{J^{(s)}_{n},J^{(s)}_{m}\\}=\frac{\alpha_{s}k}{2}n\delta_{m+n,0}\delta_{s,s}$
(4.10a) $\displaystyle i\\{J^{(3)}_{n},J^{(3)}_{m}\\}$
$\displaystyle=\frac{\alpha_{3}k}{2}n\delta_{m+n,0}$ $\displaystyle
i\\{J^{(s-t)}_{n},J^{(s)}_{m}\\}=0.$ (4.10b) $\displaystyle...$ (4.10c)

For arbitrary $t\neq s$, this algebra is an algebra of $u(1)$ currents of
levels $\frac{1}{2}k$, … $\frac{1}{2}\alpha_{s}k$.

###  4.1 so(3,2): conformal spin-2 field

If we apply the near horizon bcs to the concrete example of conformal
graviton, we have to identify the diagonal generators in PKLD basis. To
express the algebra of the $so(3,2)$

$\displaystyle[{\boldsymbol{D}},{\boldsymbol{P}}^{a}]$
$\displaystyle=-{\boldsymbol{P}}^{a}\,,$
$\displaystyle[{\boldsymbol{L}}^{ab},{\boldsymbol{P}}^{c}]$
$\displaystyle={\boldsymbol{P}}^{a}\eta^{bc}-{\boldsymbol{P}}^{b}\eta^{ac}\,,$
(4.11a) $\displaystyle[{\boldsymbol{D}},{\boldsymbol{K}}^{a}]$
$\displaystyle={\boldsymbol{K}}^{a}\,,$
$\displaystyle[{\boldsymbol{L}}^{ab},{\boldsymbol{K}}^{c}]$
$\displaystyle={\boldsymbol{K}}^{a}\eta^{bc}-{\boldsymbol{K}}^{b}\eta^{ac}\,,$
(4.11b) $\displaystyle[{\boldsymbol{P}}^{a},{\boldsymbol{K}}^{b}]$
$\displaystyle=-{\boldsymbol{L}}^{ab}+\eta^{ab}{\boldsymbol{D}}\,,$
$\displaystyle[{\boldsymbol{L}}^{ab},{\boldsymbol{L}}^{cd}]$
$\displaystyle={\boldsymbol{L}}^{ad}\eta^{bc}+\text{three more}\,,$ (4.11c)

we use standard representation in terms of $\gamma$ matrices. PKLD basis for
$sl(4)$ group is described in Appendix C. There, we represent $sl(4)$ group in
WL and PKLD bases of generators, and find and a map from PKLD to WL basis.
Since $so(3,2)$ algebra is subalgebra of $sl(4)$, one can also read out the
representation for $so(3,2)$. To impose the near horizon bcs we notice that we
can have two diagonal generators. One of them is the ${\boldsymbol{D}}$
generator, while another one is generator of Lorentz rotations, which we call
${\boldsymbol{L}}_{d}$ to point out that it is represented by diagonal matrix.
For the gauge field $\omega=b^{-1}(d+\Omega)b$ we can write the ansatz for the
connection

$\displaystyle\Omega=(\mathcal{K}(\phi,t)d\phi+\mu(\phi,t)dt){\boldsymbol{D}}+(\mathcal{K}_{d}(\phi,t)+\mu_{d}(\phi,t)dt){\boldsymbol{L}}_{d}.$
(4.12)

Since the exact form of the group element $b$ is not relevant for the analysis
in Chern-Simons form when $\delta b=0$ we keep it to be a function of $\rho$,
$b=b(\rho)$. Here we defined the charges as $\mathcal{K}$ and
$\mathcal{K}_{d}$ which are allowed to vary,

$\displaystyle\delta\Omega_{\phi}$
$\displaystyle=\delta\mathcal{K}(\phi,t){\boldsymbol{D}}+\delta\mathcal{K}_{d}(\phi,t){\boldsymbol{L}}_{d}$
(4.13) $\displaystyle\delta\Omega_{t}$ $\displaystyle=0$ (4.14)

while the variation of the chemical potentials $\mu$ and $\mu_{d}$ vanishes.
The equations of motion lead to expression as in (4.3), familiar for the
chosen gauge and bcs

$\displaystyle\partial_{t}\mathcal{K}$ $\displaystyle=\partial_{\phi}\mu,$
$\displaystyle\partial_{t}\mathcal{K}_{d}=\partial_{\phi}\mu_{d}.$ (4.15)

The parameter $\epsilon$ which is valued in the Lie algebra and keeps the
asymptotic symmetries takes the form

$\displaystyle\epsilon=\xi{\boldsymbol{D}}+\xi_{d}{\boldsymbol{L}}_{d}$ (4.16)

where $\partial_{t}\xi=\partial_{t}\xi_{d}=0$. The fields $\mathcal{K}$ and
$\mathcal{K}_{d}$ then transform as

$\displaystyle\delta\mathcal{K}=\partial_{\phi}\xi,$
$\displaystyle\delta\mathcal{K}_{d}=\partial_{\phi}\xi_{d}.$ (4.17)

Using the variation of the canonical boundary charge (3.11) one can obtain

$\displaystyle\delta Q[\epsilon]=\frac{k}{2\pi}\oint
d\phi(\xi\delta\mathcal{K}\langle{\boldsymbol{D}}{\boldsymbol{D}}\rangle+\xi_{d}\delta\mathcal{K}_{d}\langle{\boldsymbol{L}}_{d}{\boldsymbol{L}}_{d}\rangle)$
(4.18)

and the conserved charge that is associated to the asymptotic symmetries

$\displaystyle Q[\xi,\xi_{d}]$ $\displaystyle=\frac{k}{2\pi}\oint
d\phi(\xi(\phi)\mathcal{K}(\phi)\langle{\boldsymbol{D}}{\boldsymbol{D}}\rangle+\xi_{d}(\phi)\mathcal{K}_{d}(\phi)\langle{\boldsymbol{L}}_{d}{\boldsymbol{L}}_{d}\rangle)$
$\displaystyle=\frac{k}{2\pi}\oint
d\phi(2\xi(\phi)\mathcal{K}(\phi)+\frac{3}{2}\xi_{d}(\phi)\mathcal{K}_{d}(\phi)).$
(4.19)

Calculating Poisson brackets of the charges $\delta_{Y}Q[X]=\\{Q[X],Q[Y]\\}$
and the variation of the fields $\mathcal{K}$ and $\mathcal{K}_{d}$ one can
obtain the algebra

$\displaystyle\\{\mathcal{K}(\phi),\mathcal{K}(\phi)\\}$
$\displaystyle=\frac{\pi}{k}\delta(\phi-\phi^{\prime})$ (4.20)
$\displaystyle\\{\mathcal{K}_{d}(\phi),\mathcal{K}_{d}(\phi)\\}$
$\displaystyle=\frac{4\pi}{3k}\delta(\phi-\phi^{\prime}).$ (4.21)

Using the expansion of the charges in the Fourier modes

$\displaystyle\mathcal{K(\phi)}=\frac{1}{k}\sum_{n=-\infty}^{\infty}J_{n}e^{in\phi},$
$\displaystyle\mathcal{K}_{d}(\phi)=\frac{4}{3k}\sum_{n=-\infty}^{\infty}J^{(d)}_{n}e^{in\phi}$
(4.22)

we obtain for the ASA

$\displaystyle i\\{J_{n},J_{m}\\}$ $\displaystyle=kn\delta_{m+n,0}$
$\displaystyle i\\{J^{(d)}_{n},J^{(d)}_{m}\\}=\frac{3k}{4}n\delta_{m+n,0}.$
(4.23)

which as well corresponds to $u(1)$ currents.

For the spherically symmetric solution and certain set of assumptions
satisfied by the near horizon bcs, the entropy [30, 31, 32] is given by

$\displaystyle S=-\frac{k_{N}}{\pi}Im\beta\int d\phi
Tr(\Omega_{\phi}\Omega_{t}).$ (4.24)

To evaluate this, we have to use the definition for $\Omega$ (4.12) the
definition for trace of $\langle{\boldsymbol{D}}{\boldsymbol{D}}\rangle$ and
$\langle{\boldsymbol{L}}_{d}{\boldsymbol{L}}_{d}\rangle$ matrices. The entropy
for this bcs is going to be given by

$\displaystyle S=-2\frac{k_{N}}{\pi}Im\left[\beta\int
d\phi(\mu\mathcal{K}-\mu_{d}\mathcal{K}_{d})\right].$ (4.25)

The chemical potentials we can evaluate as it was done in [27]. Here we define
the contractible circle $\tau=it$ and Hawking temperature $T=1/\beta$. For the
near horizon bcs one can find the chemical potentials from the holonomy around
the contractible circle [10]

$\displaystyle e^{\int\Omega_{t}}=-\mathds{1}$ (4.26)

For the $\Omega_{t}$ from (4.12) one obtains a solution

$\displaystyle\mu=-\frac{2}{\beta}(m+n+1)\pi,$
$\displaystyle\mu_{d}=\frac{2}{\beta}(m-n)\pi$ (4.27)

Which from $S=-k_{N}\beta(\mu\mathcal{K}-\mu_{d}\mathcal{K}_{d})$ leads to

$\displaystyle
S=-2k_{N}\left[-(m+n+1)\pi\mathcal{K}-(m-n)\pi\mathcal{K}_{d}\right]$ (4.28)

We can notice that when the chemical potential for the dilatation is zero, it
means $m=-n-1$, in which case the entropy becomes
$S=-2k_{N}(2n+1)\pi\mathcal{K}_{d}$. Since this corresponds to the generator
that is within Poincaré algebra, for $n=0$, one can think of it as a charge
that corresponds to $L_{0}$ generator of one half of the $sl(2)\oplus sl(2)$
of Einstein gravity in three dimensions. Or, depending on the embedding of the
$sl(2)\oplus sl(2)$, it can be a charge corresponding to linear combination of
$L_{0}^{+}$ and $L_{0}^{-}$ generators (for $x^{+}=t+\phi$ and
$x^{-}=t-\phi$.). The branch with entropy corresponding to the charges of the
generators $L_{0}^{+}$ and $L_{0}^{-}$ in Einstein gravity, is the branch of
higher spin black flowers that is continuously connected to BTZ black hole.

so(3,2)–(simplest) example 1. To calculate the metric formulation we follow
the prescription for the near horizon bcs. The bcs $\omega$ are defined by

$\displaystyle\omega=b^{-1}[b_{1}{\boldsymbol{D}}+b_{2}{\boldsymbol{L}}_{d}]b+b^{-1}\partial_{\rho}bd\rho$
(4.29)

for $b_{1}=\mu dt+\mathcal{K}d\phi$, $b_{2}=\mu_{d}dt+\mathcal{K}_{d}d\phi$.
Here, $\mu,\mu_{d},\mathcal{K},\mathcal{K}_{d}$ are functions of ($t,\phi$).
Calculating $\omega$ yields

$\displaystyle\omega=\Omega+\left[Y,\Omega\right]+\frac{1}{2}\left[Y,\left[Y,\Omega\right]\right]+\frac{1}{6}\left[Y,\left[Y,\left[Y,\Omega\right]\right]\right]+...+b^{-1}\partial_{\rho}bd\rho$
(4.30)

which give terms up to order $\rho^{3}$. If we collect the coefficients of the
generators ${\boldsymbol{P}}_{a}$ we can identify the components of the
dreibein, up to denoted term in the expansion. When we evaluate entire
$\omega$ (which we do here) we can obtain full metric.

Within this method, first we have to choose $b$, which has to belong to group
as described above, as $b=e^{Y}$. We also refer to $b$ as a projector, because
it projects the solution from the first order formulation to metric
formulation. We define
$Y=\rho{\boldsymbol{P}}_{y}+{\boldsymbol{P}}_{x}+\tilde{c}_{1}{\boldsymbol{D}}$.
Note that we use the basis in which the generators are defined using
$\\{x,y,t\\}$ coordinates. If we assume that in transition to a group that
describes the motion on a cylinder we consider only linearly combined
generators, and do not take any contractions of a group, we can use this
notation. We also think of the generators, as linearly combined generators in
WL basis. In Einstein gravity in three dimensions, a gauge theory of
$sl(2)\oplus sl(2)$, the metric is defined as
$g_{\mu\nu}={\mathrm{Tr}\,}\left((A_{\mu}-\overline{A}_{\mu})(A_{\nu}-\overline{A}_{\nu})\right)$.
The definition of the metric in conformal gravity is taken to be
$g_{\mu\nu}=e_{\mu}{}^{a}e_{\nu}{}^{b}\eta_{ab}$ for $e_{\mu}^{a}$ read out
from fields associated with generator of translations. (We use representation
of matrices with definition of $\eta_{ab}=diag(1,1,-1)$, and generators
represented by matrices in Appendix C.) Here, we consider the above defined
metric $g_{\mu\nu}$, which leads to the dreibein

$\displaystyle e^{a}_{\mu}=c_{1}\left(\begin{array}[]{ccc}0&1&0\\\
-\mathcal{K}&-\mathcal{K}\rho&\mathcal{K}_{d}\rho\\\
-\mu&-\mu\rho&\mu_{d}\rho\\\ \end{array}\right)$ (4.34)

which after identification defines a metric. To more easily transform a metric
to a desired form we assume constant fields and chemical potentials. Bringing
the metric to ADM form it leads to BTZ-like metric, with $g_{rt}=g_{r\phi}=0$,
$g_{\phi\phi}=r^{2}$ and

$\displaystyle g_{rr}$
$\displaystyle=\frac{r^{2}}{\left(\mathcal{K}^{2}-r^{2}\right)\left(\mathcal{K}_{d}^{2}-r^{2}\right)}$
(4.35a) $\displaystyle g_{\phi t}$
$\displaystyle=\frac{f_{2}^{\prime}(t)\left(\mathcal{K}\mu\left(r^{2}-\mathcal{K}_{d}^{2}\right)+\mathcal{K}_{d}\mu_{3}\left(\mathcal{K}^{2}-r^{2}\right)\right)}{\mathcal{K}_{d}^{2}-\mathcal{K}^{2}}+r^{2}f_{1}^{\prime}(t)$
(4.35b) $\displaystyle g_{tt}$
$\displaystyle=\frac{2f_{2}^{\prime}(t)\bigl{(}f_{1}^{\prime}(t)\left(\mathcal{K}\mu\left(r^{2}-\mathcal{K}_{d}^{2}\right)+\mathcal{K}_{d}\mu_{3}\left(\mathcal{K}^{2}-r^{2}\right)\right)-f_{2}^{\prime}(t)\left(\mu^{2}(\mathcal{K}_{d}^{2}-r^{2})-\mu_{3}^{2}\left(\mathcal{K}^{2}-r^{2}\right)\right)\bigr{)}}{{\mathcal{K}^{2}-\mathcal{K}_{d}^{2}}}$
$\displaystyle+r^{2}f_{1}^{\prime}(t){}^{2}$ (4.35c)

Here $c_{1}=\frac{-1+e^{\tilde{c}_{1}}}{\tilde{c}_{1}}$ and $f_{1}$ and
$f_{2}$ are functions of $t$ such that $f_{i}(t)=\tilde{f_{i}}((-1+e)t)$ for
$i=1,2$, and we rescaled chemical potentials with $(e-1)$. The curvature
$R=-\frac{6}{c_{1}^{2}}$ for the solution (4.35) points to an anti-de Sitter
solution. Fixing the functions $f_{1}$ to $t$ and $f_{2}$ to $t/(-1+e)$ the
components $g_{\phi t}$ and $g_{tt}$ become

$\displaystyle g_{\phi t}$
$\displaystyle=\frac{\left(\mathcal{K}\mu\left(r^{2}-\mathcal{K}_{d}^{2}\right)+\mathcal{K}_{d}\mu_{3}\left(\mathcal{K}^{2}-r^{2}\right)\right)}{\mathcal{K}_{d}^{2}-\mathcal{K}^{2}}+r^{2}$
(4.36a) $\displaystyle g_{tt}$
$\displaystyle=\frac{2\bigl{(}\left(\mathcal{K}\mu\left(r^{2}-\mathcal{K}_{d}^{2}\right)+\mathcal{K}_{d}\mu_{3}\left(\mathcal{K}^{2}-r^{2}\right)\right)-\left(\mu^{2}(\mathcal{K}_{d}^{2}-r^{2})-\mu_{3}^{2}\left(\mathcal{K}^{2}-r^{2}\right)\right)\bigr{)}}{{\mathcal{K}^{2}-\mathcal{K}_{d}^{2}}}+r^{2}$
(4.36b)

Switching off both chemical potentials would lead to degenerate metric with
zero determinant. From $g_{rr}$ one can see that solution has two real
eigenvalues which would indicate that it belongs to the particular class of
the metric which is $I_{b}$ according to the classification described in the
Section 5.

To compare the solution with the case where entropy is proportional to
$\mathcal{K}_{d}$ when, $\mu=0$, we take $\mu$ to be zero in the solution
above. Fixing the chemical potential $\mu_{d}$ to
$\mu_{d}=\frac{\mathcal{K}^{2}-\mathcal{K}^{2}_{d}}{\mathcal{K}_{d}}$ and
$t\rightarrow\frac{\mathcal{K}_{d}}{\mathcal{K}}t$ we obtain the AdS solution
exactly of the BTZ form. We have one horizon which is $r_{1}=\mathcal{K}_{d}$
while another is $r_{2}=\mathcal{K}$. Setting $r_{+}=r_{1}$ and $r_{-}=r_{2}$
in the standard BTZ metric [33] depicts the solution.

so(3,2): conformal graviton – example 2. In this example we examine the
influence of $\rho$ dependency multiplying each of the generators in the
projector $b$. One of the interesting forms of the metric we can obtain in
this case is when
$Y=a(\rho){\boldsymbol{D}}+b(\rho){\boldsymbol{P}}_{y}+c(\rho){\boldsymbol{P}}_{x}$.
The dreibein becomes then

$\displaystyle
e_{\mu}{}^{a}=\frac{(-1+e^{a(\rho)})}{a(\rho)}\left(\begin{array}[]{ccc}c^{\prime}(\rho)+c(\rho)a^{\prime}(\rho)\left(\frac{1}{e^{a(\rho)}-1}-\frac{1}{a(\rho)}\right)&b^{\prime}(\rho)+b(\rho)a^{\prime}(\rho)\left(\frac{1}{e^{a(\rho)}-1}-\frac{1}{a(\rho)}\right)&0\\\
-\mathcal{K}c(\rho)&-\mathcal{K}b(\rho)&\mathcal{K}_{d}b(\rho)\\\ -\mu
c(\rho)&-\mu b(\rho)&\mu_{d}b(\rho)\\\ \end{array}\right)$ (4.40)

and gives the metric $g_{ij}$ with non-vanishing terms

$\displaystyle g_{rr}=$
$\displaystyle\frac{1}{{a(\rho)^{2}c_{1}(\rho)}}\left[\left(b(\rho)\left(c_{1}(\rho)-1\right)a^{\prime}(\rho)-a(\rho)c_{1}(\rho)b^{\prime}(\rho)\right){}^{2}\right.$
$\displaystyle\left.+\left(c(\rho)\left(c_{1}(\rho)-1\right)a^{\prime}(\rho)-a(\rho)c_{1}(\rho)c^{\prime}(\rho)\right){}^{2}\right]$
(4.41a) $\displaystyle g_{r\phi}=$
$\displaystyle-\frac{c_{1}(\rho)\mathcal{K}}{a(\rho)}\left[b(\rho)\left(a(\rho)c_{1}(\rho)b^{\prime}(\rho)-b(\rho)\left(c_{1}(\rho)-1\right)a^{\prime}(\rho)\right)\right.$
$\displaystyle\left.+c(\rho)\left(a(\rho)c_{1}(\rho)c^{\prime}(\rho)-c(\rho)\left(c_{1}(\rho)-1\right)a^{\prime}(\rho)\right)\right]$
(4.41b) $\displaystyle g_{rt}=$
$\displaystyle-\frac{c_{1}(\rho)\mu}{a(\rho)}\left[b(\rho)\left(a(\rho)c_{1}(\rho)b^{\prime}(\rho)-b(\rho)\left(c_{1}(\rho)-1\right)a^{\prime}(\rho)\right)\right.$
$\displaystyle\left.+c(\rho)\left(a(\rho)c_{1}(\rho)c^{\prime}(\rho)-c(\rho)\left(c_{1}(\rho)-1\right)a^{\prime}(\rho)\right)\right]$
(4.41c) $\displaystyle g_{\phi\phi}=$ $\displaystyle
c_{1}(\rho)^{2}\left(b(\rho)^{2}(\mathcal{K}^{2}-\mathcal{K}_{d}^{2})+\mathcal{K}^{2}c(\rho)^{2}\right)$
(4.41d) $\displaystyle g_{\phi t}=$ $\displaystyle
c_{1}(\rho)^{2}\left(b(\rho)^{2}(\mathcal{K}\mu-\mathcal{K}_{d}\mu_{d})+\mathcal{K}\mu
c(\rho)^{2}\right)$ (4.41e) $\displaystyle g_{tt}=$ $\displaystyle
c_{1}(\rho)^{2}\left(b(\rho)^{2}(\mu^{2}-\mu^{2}_{d})+\mu^{2}c(\rho)^{2}\right)$
(4.41f)

for $c_{1}(\rho)=\frac{e^{a(\rho)}-1}{a(\rho)}$ arbitrary parameter. The above
metric contains the fields and chemical potentials with their initial
dependency on the coordinates. Setting them to constant values lets us find
suitable choices of the $c(\rho)$ and $b(\rho)$ that reduce the metric to one
of the simpler examples. Here, we determine the function $b(\rho)$ by the
requirement to get $\rho^{2}d\phi^{2}$. And transform the coordinates
$\phi\rightarrow\phi+\frac{\mu_{d}\left[2(a(\rho)+\log(a(\rho)))-\log\left(a(\rho)^{2}(\rho^{2}-\mathcal{K}_{d}^{2}c_{1}(\rho)^{2}c(\rho)^{2})\right)\right]}{2\mathcal{K}_{d}\mu-2k\mu_{d}}+c_{2}(T)$
and
$t\rightarrow\frac{-\mathcal{K}_{d}\left[2(a(\rho)+\log(a(\rho)))-\log\left(a(\rho)^{2}(\rho^{2}-\mathcal{K}_{d}^{2}c_{1}(\rho)^{2}c(\rho)^{2})\right)\right]}{2\mathcal{K}_{d}\mu-2k\mu_{d}}+c_{1}(T)$.
We further rescale the metric with numerator in $g_{\rho\rho}/\rho^{2}$ and
one more time fix that $g_{\phi\phi}=\rho^{2}$, this time using the function
$c(\rho)$. We simplify the result by setting $a(\rho)=1$, which leads to
metric

$\displaystyle
ds^{2}=f_{\rho\rho}(\rho)d\rho^{2}+f_{tt}(\rho)dt^{2}-2\rho^{2}n(\rho)dtd\phi+\rho^{2}d\phi^{2}$
(4.42) for $\displaystyle f_{tt}(\rho)$
$\displaystyle=\frac{1}{\mathcal{K}^{2}-\mathcal{K}^{2}_{d}}\left[(\mathcal{K}^{2}-\mathcal{K}_{d}^{2})\left(\mathcal{K}^{2}\mu_{d}^{2}-\mathcal{K}_{d}^{2}\mu^{2}+\rho^{2}(\mu^{2}-\mu_{d}^{2})\right)\right.$
(4.43a)
$\displaystyle\left.-2q_{2}(\mathcal{K}_{d}\mu-\mathcal{K}\mu_{d})\left(-\mathcal{K}^{2}\mathcal{K}_{d}\mu_{d}+\mathcal{K}\mathcal{K}_{d}\rho
q_{1}\left(\rho
q_{1}-2\right)(\mathcal{K}_{d}\mu-\mathcal{K}\mu_{d})+\mathcal{K}\mu(\mathcal{K}_{d}^{2}-\rho^{2})+\mathcal{K}_{d}\rho^{2}\mu_{d}\right)\right.$
$\displaystyle\left.+\rho^{2}q_{2}^{2}(\mathcal{K}_{d}\mu-\mathcal{K}\mu_{d})^{2}+\rho
q_{1}(\mathcal{K}^{2}-\mathcal{K}_{d}^{2})\left(\rho
q_{1}-2\right)(\mathcal{K}^{2}\mu^{2}_{d}-\mathcal{K}^{2}_{d}\mu^{2})\right]$
$\displaystyle f_{\rho\rho}(\rho)$
$\displaystyle=\frac{\rho^{2}}{\left(\mathcal{K}\rho
q_{1}-\mathcal{K}-\rho\right)\left(\mathcal{K}\rho
q_{1}-\mathcal{K}+\rho\right)\left(\mathcal{K}_{d}\rho
q_{1}-\mathcal{K}_{d}-\rho\right)\left(\mathcal{K}_{d}\rho
q_{1}-\mathcal{K}_{d}+\rho\right)}$ (4.43b) $\displaystyle n(\rho)$
$\displaystyle=\frac{\rho\left(\mathcal{K}\mathcal{K}_{d}q_{1}\left(\rho
q_{1}-2\right)-\rho
q_{2}\right)(\mathcal{K}\mu_{d}-\mathcal{K}_{d}\mu)+\rho^{2}(\mathcal{K}\mu-\mathcal{K}_{d}\mu_{d})+\mathcal{K}\mathcal{K}_{d}(\mathcal{K}\mu_{d}-\mathcal{K}_{d}\mu)}{\rho^{2}(\mathcal{K}_{d}\mu-\mathcal{K}\mu_{d})}$
(4.43c)

while the $f_{tt}(\rho)=\rho^{2}n(\rho)^{2}-1/f_{\rho\rho}(\rho)$, and $q_{1}$
and $q_{2}$ are free parameters. The parameter $q_{2}$ we obtained by fixing
the function $c_{2}(T)=q_{2}T$, while we fixed the function $c_{1}$ to
$c_{1}(T)=\frac{\mathcal{K}_{d}^{2}-\mathcal{K}^{2}}{\mathcal{K}\mu_{d}-\mathcal{K}_{d}\mu}T$.
The solution satisfies Cotton tensor, however its Ricci scalar is not
constant,
$R=-6+\frac{2q_{1}}{\rho^{3}}\left[\left(-2\mathcal{K}^{2}\left(\mathcal{K}_{d}^{2}+\rho^{2}\right)+\rho
q_{1}\left(3\rho^{2}\left(\mathcal{K}^{2}+\mathcal{K}_{d}^{2}\right)+\mathcal{K}^{2}\mathcal{K}_{d}^{2}\rho
q_{1}\left(4-3\rho
q_{1}\right)+\mathcal{K}^{2}\mathcal{K}_{d}^{2}\right)-2\mathcal{K}_{d}^{2}\rho^{2}\right)\right]$.
Taking $\rho\to\infty$ the Ricci scalar goes to value determined by the fields
and an arbitrary constant,
$R=-6\left(\mathcal{K}^{2}q_{1}^{2}-1\right)\left(\mathcal{K}_{d}^{2}q_{1}^{2}-1\right)$.
The metric written in this form has two real eigenvalues for general values of
$\mathcal{K}$, $\mathcal{K}_{d}$ and $q_{1}$. For the special case when
$\mathcal{K},\mathcal{K}_{d},q_{1}>0$, $\mathcal{K}q_{1}>1$ and
$\mathcal{K}_{d}q_{1}>1$ we have four positive real radii. These are visible
from (4.43b). If we fix the $b(\rho)$ in (4.41) to get $\rho^{2}$ for the
$g_{\phi\phi}$ term in the metric, transform the coordinates such that
$g_{\rho\phi}$ and $g_{\rho t}$ vanish, we can fix
$c(\rho)=1/(-1+e^{a}(\rho))$ and $a(\rho)=1+\frac{c_{1}}{\rho}$ to get the
similar form of the metric. Suitable choice of parameters in such metric can
also be chosen to give four positive radii. The metric gives Cotton tensor
equal to zero. It would be interesting to further analyze this metric.

If in the above metric, we set $\mathcal{K}=0$, $q_{2}=\frac{\mu_{2}}{\mu}$
and rescale a metric with some constant $a_{1}$, after absorption of $a_{1}$
to $\phi$ and $a_{1}^{2}$ to $t$ coordinates, we recover the three dimensional
analog [34] of the four dimensional MKR (Mannheim-Kazanas-Riegert) solution
[Mannheim:1988dj]. The metric is then
$ds^{2}=dr^{2}/f(r)-f(r)dt^{2}+r^{2}d\phi^{2}$, for
$f(r)=a_{1}^{2}\mathcal{K}_{d}^{2}-2a_{1}^{2}\mathcal{K}_{d}^{2}q_{1}r+a_{1}^{2}(\mathcal{K}_{d}^{2}q_{1}^{2}-1)r^{2}$.
Depending on the choice of $\mathcal{K}_{d}$, $q_{1}$ and $a_{1}$, the
solution can contain general form of eigenvalue with real and imaginary part,
contained in the class $I_{a}$.

Since we are considering conformal graviton example, here we do not have an
option to consider higher spin solution. These metric examples are more
interesting from the perspective of conformal gravity in three dimensions. We
also elaborate on this metric in the Appendix B.

From the general form of the metric we can also obtain de Sitter solution with
one positive horizon, which belongs to the $I_{d}$ class in our
classification. Similar kind of metrics have been analyzed in a different
context in [35]. They considered backreaction of quantum fields as
perturbation in conical $dS_{3}$, which allowed a formulation of a black hole
in the $dS_{3}$ spacetime.

so(3,2): conformal graviton – example 3. In this example we analyze how does
changing the dependency on $\rho$ in front of a generator in the projector $b$
influence the metric. We set $a(\rho)$ and $b(\rho)$ in (4.41) to one, rescale
the metric with $(-1+e)$, and then consider it while keeping only $c(\rho)$.

The Ricci scalar of this metric is $-2$ which shows that the metric can be
brought to more familiar Lobachevsky form. By transformation of the
coordinates, rescaling of the fields one obtains the metric that generalizes
the Lobachevsky form in [36]

$\displaystyle
ds^{2}=\frac{dr^{2}}{r^{2}-Br+A^{2}+C}-dt^{2}+2Adtd\phi+(r^{2}-Br)d\phi^{2}.$
(4.44)

If we use the function $c(\rho)$ so that $c(\rho)=q_{2}+q_{1}\rho$ we can
transform $t$ and $\phi$ to set $g_{\rho\phi}=g_{\rho t}=0$ and so that the
generalized metric takes the form

$\displaystyle
ds^{2}=\frac{\mathcal{K}^{2}q_{1}^{2}}{\mathcal{K}q_{1}\rho\left(2\sqrt{\mathcal{K}_{d}^{2}-\mathcal{K}^{2}}+\mathcal{K}q_{1}\rho\right)+\mathcal{K}_{d}^{2}}d\rho^{2}-dt^{2}+\frac{2\mathcal{K}_{d}}{\mathcal{K}q_{1}}dtd\phi+\left(\frac{2\rho\sqrt{\mathcal{K}_{d}^{2}-k^{2}}}{\mathcal{K}q_{1}}+\rho^{2}\right)d\phi^{2}.$
(4.45a)

In this notation we have
$B=-\frac{2\sqrt{\mathcal{K}_{d}^{2}-\mathcal{K}^{2}}}{\mathcal{K}q_{1}}$ and
$A=\frac{\mathcal{K}_{d}}{\mathcal{K}q_{1}}$, for $C=0$. Comparing the
solution to (4.44), for $q_{1}=1$ and: $\mathcal{K}_{d}=0$ we recover global
Lobachevsky; for $\mathcal{K}_{d}=\mathcal{K}=0$ we obtain Poincaré
Lobachevsky; and for $\mathcal{K}=\mathcal{K}_{d}$ rotating Lobachevsky
metric.

If we choose the $a(\rho)$ so that
$\frac{\left(e^{a(\rho)}-1\right)^{2}}{a(\rho)^{2}}$ in front of genertor of
dilatations, ${\boldsymbol{D}}$ is a constant, the Ricci scalar gets rescaled
by that factor.
Interesting to notice is that starting from
$Y=b(\rho){\boldsymbol{P}}_{y}+{\boldsymbol{D}}+c(\rho){\boldsymbol{P}}_{x}$
and $\mu=0$ the dreibein becomes combination of above (4.45a) example, and the
simplest example 1. One obtains the Ricci scalar as a function of the
$b(\rho)$ and $c(\rho)$. The choice $b(\rho)=1$ and $c(\rho)$ free as in the
previous example, leads to the metric that has Ricci scalar
$-\frac{2}{c_{1}^{2}}$ while the choice $c(\rho)=1$ and $b(\rho)$ free leads
to $R=-\frac{6}{c_{1}^{2}}$ Ricci scalar, as in the example 1.

###  4.2 sl(4): conformal spin-2 and spin-1 field

We continue the analysis on the theory of conformal graviton and a vector
field. The algebra of conformal graviton and a vector field in the PKLD basis
is isomorphic to $sl(4)$ algebra. It consists of the conformal algebra,
algebra of the vector field (2.14b), and their commutation relations (2.14c),
or explicitly (4.11) and

$\displaystyle[{\boldsymbol{P}}_{a},{\boldsymbol{t}}_{+}]$
$\displaystyle=0\,,$
$\displaystyle[{\boldsymbol{P}}_{a},{\boldsymbol{t}}_{-}]$
$\displaystyle={\boldsymbol{t}}_{a}\,,$
$\displaystyle[{\boldsymbol{P}}_{a},{\boldsymbol{t}}_{c}]$
$\displaystyle=-\eta_{ac}{\boldsymbol{t}}_{+}\,,$
$\displaystyle[{\boldsymbol{L}}_{ab},{\boldsymbol{t}}_{+}]$
$\displaystyle=0\,,$
$\displaystyle[{\boldsymbol{L}}_{ab},{\boldsymbol{t}}_{c}]$
$\displaystyle=\eta_{bc}{\boldsymbol{t}}_{a}-\eta_{ac}{\boldsymbol{t}}_{b}\,,$
$\displaystyle[{\boldsymbol{L}}_{ab},{\boldsymbol{t}}_{-}]$
$\displaystyle=0\,,$
$\displaystyle[{\boldsymbol{K}}_{a},{\boldsymbol{t}}_{+}]$
$\displaystyle={\boldsymbol{t}}_{a}\,,$
$\displaystyle[{\boldsymbol{K}}_{a},{\boldsymbol{t}}_{b}]$
$\displaystyle=-\eta_{ab}{\boldsymbol{t}}_{-}\,,$
$\displaystyle[{\boldsymbol{K}}_{a},{\boldsymbol{t}}_{-}]$ $\displaystyle=0$
$\displaystyle[{\boldsymbol{D}},{\boldsymbol{t}}_{+}]$
$\displaystyle=-{\boldsymbol{t}}_{+}\,,$
$\displaystyle[{\boldsymbol{D}},{\boldsymbol{t}}_{a}]$ $\displaystyle=0\,,$
$\displaystyle[{\boldsymbol{D}},{\boldsymbol{t}}_{-}]$
$\displaystyle={\boldsymbol{t}}_{-}$
$\displaystyle[{\boldsymbol{t}}_{+},{\boldsymbol{t}}_{-}]$
$\displaystyle=-{\boldsymbol{D}}\,,$
$\displaystyle[{\boldsymbol{t}}_{a},{\boldsymbol{t}}_{+}]$
$\displaystyle={\boldsymbol{P}}_{a}\,,$
$\displaystyle[{\boldsymbol{t}}_{a},{\boldsymbol{t}}_{-}]$
$\displaystyle={\boldsymbol{K}}_{a},$
$\displaystyle[{\boldsymbol{t}}_{a},{\boldsymbol{t}}_{b}]$
$\displaystyle={\boldsymbol{L}}_{ab}.$ (4.46)

The diagonal matrices in the WL basis are given by linear combination of
matrices in PKLD basis. To study near horizon bcs and be able to compare this
with the earlier works it is more convenient to consider the matrices in WL
basis. In WL basis, $sl(4)$ algebra closes the commutation relations

$\displaystyle[L_{n},L_{m}]=(n-m)L_{n+m}$
$\displaystyle[L_{n},W_{m}^{(s)}]=((s-1)n-m)W_{n+m}^{(s)}$ (4.47)

where the indices in $W^{4}_{m}$ and $W^{3}_{n}$ are $m=0,\pm 1,\pm 2,\pm 3$
and $n=0,\pm 1,\pm 2$ respectively. The generators $L_{n}$ are generators of
$sl_{2}$ algebra. The details about the representation of generators are given
in the Appendix C.

Once we know the map between the generators in PKLD basis and the WL basis
(Appendix C), we can define the near horizon bcs by specifying $n=4$ in the
general case (4.1)

$\displaystyle\Omega(n)=(\mathcal{K}d\phi+\mu
dt)L_{0}+(\mathcal{K}_{3}d\phi+\mu_{3}dt)W_{0}^{(3)}+(\mathcal{K}_{4}d\phi+\mu_{4}dt)W_{0}^{(4)}.$
(4.48)

The $\epsilon$ is defined from (4.4)

$\displaystyle\epsilon=\xi L_{0}+\xi_{3}W_{0}^{(3)}+\xi_{4}W_{0}^{(4)}.$
(4.49)

By defining $n=4$ in (4.6) we can also read out the charge, which is

$\displaystyle Q[\xi,\xi_{3},\xi_{4}]=-\frac{k}{4\pi}\int
d\phi\left(\xi(\phi)\mathcal{K}(\phi)+\alpha_{3}\xi_{3}(\phi)\mathcal{K}_{3}(\phi)+\alpha_{4}\xi_{4}(\phi)\mathcal{K}_{4}(\phi)\right).$
(4.50)

ASA, is algebra (4.10) for charges $\mathcal{K},\mathcal{K}_{3}$ and
$\mathcal{K}_{4}$, and corresponding $u(1)$ currents are of levels
$\frac{1}{2}k$, $\frac{1}{2}\alpha_{3}k$ and $\frac{1}{2}\alpha_{4}k$.

To evaluate the entropy of this solution, we use our definition for $\Omega$
(4.1) as above, map between the coordinates (8.32,8.54,8.85) and (8.149), and
(2.12) and (2.13) or the definition for trace of $\langle L_{0}L_{0}\rangle$
and $\langle W_{0}^{(s)}W_{0}^{(s)}\rangle$ matrices. This leads to

$\displaystyle S$ $\displaystyle=-\frac{k_{N}}{\pi}Im\left[\beta\int
d\phi\left[5\mu\mathcal{K}+16\mu^{(3)}\mathcal{K}_{0}^{(3)}+\frac{36}{5}\mu^{(4)}\mathcal{K}_{0}^{(4)}\right]\right].$
$\displaystyle=-2k_{N}\beta\left[5\mu\mathcal{K}+16\mu^{(3)}\mathcal{K}_{0}^{(3)}+\frac{36}{5}\mu^{(4)}\mathcal{K}_{0}^{(4)}\right].$
(4.51)

In this case we have fields of spin 1 and spin 2. We can evaluate the chemical
potentials in the similar way as for $so(3,2)$

$\displaystyle\mu=\frac{2\pi(l-m-2n-1)}{5\beta}$
$\displaystyle\mu_{3}=\frac{\pi(l+n+1)}{2\beta},$
$\displaystyle\mu_{4}=\frac{\pi(2l+3m+n+3)}{3\beta}.$ (4.52)

Which gives for the entropy

$\displaystyle S=-\pi
k_{N}\frac{4}{5\beta}\left(5\mu\mathcal{K}(l-m-2n-1)+6\mu_{5}\mathcal{K}_{0}^{(4)}(2l+3m+n+3)+20\mu_{3}\mathcal{K}_{0}^{(3)}(l+n+1)\right).$
(4.53)

We know from the map between the PKLD and WM bases that $\mathcal{K}$ is
charge for the generator that is linear combination of dilatations and one of
the vector generators. Since dilatations are not part of Poincaré group we set
$\mu$ to zero. The same is true for the $\mathcal{K}_{0}^{(4)}$ so we set
$\mu_{4}$ to zero too. That leads to relation $m=-1-l$ and $n=l$. In this case
the solution that comes from one of the generators of Poincaré algebra, would
be for the charge $\mathcal{K}_{3}$. The corresponding generator is
proportional to ${\boldsymbol{L}}_{d}$ (from $so(3,2)$ example). For
$\mu=\mu_{4}=0$ the entropy is $S=-16\pi k_{N}(1+2l)\mathcal{K}_{0}^{(3)}$.
This is the entropy that up to a factor of proportionality agrees with
solution in $so(3,2)$ case, when the remaining generator corresponds to
$L_{d}$, (as it should). $\mathcal{K}_{3}$ is associated to a charge from
generator proportional to ${\boldsymbol{L}}_{d}$. The vector charge requires
certain linear combination of the $L_{0}$ and $W_{0}^{(4)}$ generators. To
find particular solutions requires particular embedding of $sl(2)\oplus sl(2)$
in $sl(4)$.

To look at the explicit examples we consider the metric formulation. For that
we specify the group element $b$. Since we have to identify the dreibein
$e_{\mu}^{a}$ from
$\omega_{\mu}=e_{\mu}^{a}{\boldsymbol{P}}_{a}+\tilde{\omega}_{\mu}^{a}{\boldsymbol{L}}_{a}+\sigma_{\mu}^{a}{\boldsymbol{K}}_{a}+\phi_{\mu}{\boldsymbol{D}}+\omega_{\mu}^{+}{\boldsymbol{t}}_{+}+\omega_{\mu}^{-}{\boldsymbol{t}}_{-}+\omega_{\mu}^{a}{\boldsymbol{t}}_{a}$
we are interested in coefficients of ${\boldsymbol{P}}_{a}$ as they define
components of $e_{\mu}^{a}$ matrix. If we denote $b_{1}=\mathcal{K}d\phi+\mu
dt,b_{2}=\mathcal{K}^{(3)}d\phi+\mu^{(3)}dt,b_{3}=\mathcal{K}^{(4)}d\phi+\mu^{(4)}dt$,
in our decomposition we can write

$\displaystyle\omega=b^{-1}[b_{1}L_{0}+b_{2}V_{0}^{(3)}+b_{3}V_{0}^{(4)}]b+b^{-1}\partial_{\rho}bd\rho.$
(4.54)

If we use group element $b$ built from
$Y=\rho{\boldsymbol{P}}_{y}+{\boldsymbol{P}}_{x}+\tilde{c}_{1}{\boldsymbol{D}}$
we obtain the dreibein

$\displaystyle e_{\mu a}=c_{1}\left(\begin{array}[]{ccc}0&1&0\\\
\frac{-2}{5}\left(5\mathcal{K}-3\mathcal{K}_{4}\right)&\frac{-2}{5}\left(5\mathcal{K}-3\mathcal{K}_{4}\right)\rho&-4\mathcal{K}_{3}\rho\\\
\frac{-2}{5}\left(5\mu-3\mu_{4}\right)&\frac{-2}{5}\left(5\mu-3\mu_{4}\right)\rho&-4\mu_{3}\rho\\\
\end{array}\right)$ (4.58)

where $c_{1}\equiv\frac{(-1+e^{\tilde{c}_{1}})}{\tilde{c}_{1}}$. The dreibein
is of the same form as the dreibein in the $so(3,2)$ case, with the
identification
$\mathcal{K}(so(3,2))=-\frac{2}{5}[5\mathcal{K}(sl(4))-3\mathcal{K}_{4}(sl(4))]$,
$\mathcal{K}_{d}(so(3,2))=-4\mathcal{K}_{3}(sl(4))$,
$\mu(so(3,2))=-\frac{2}{5}[5\mu(sl(4))-3\mu(sl(4))]$,
$\mu_{d}(so(3,2))=-4\mu_{3}(sl(4))$. If we compare the procedure to get
dreibein for $so(3,2)$ and $sl(4)$ case we can notice the similarity and the
difference. The difference in the reading out the coefficients appears in the
terms that do not multiply $e_{\mu a}$ in the case of $sl(4)$, more precisely,
the difference appears in the additional terms with ${\boldsymbol{t}}_{+}$ and
${\boldsymbol{t}}_{t}$ of
$\omega_{\mu}=e_{\mu}^{a}{\boldsymbol{P}}_{a}+\tilde{\omega}_{\mu}^{a}{\boldsymbol{L}}_{a}+\sigma_{\mu}^{a}{\boldsymbol{K}}_{a}+\phi_{\mu}{\boldsymbol{D}}+\omega_{\mu}^{+}{\boldsymbol{t}}_{+}+\omega_{\mu}^{-}{\boldsymbol{t}}_{-}+\omega_{\mu}^{a}{\boldsymbol{t}}_{a}$.
We can see from the algebra (4.46) which term we have to include into group
element to get contribution to the metric. These are the generators which
commute into generator of translations. This kind of modification cannot be
obtained in the so(3,2) case. Additional contribution to metric will appear
when group element $b$ for $sl(4)$ is formed using ${\boldsymbol{t}}_{+}$
generator. The bcs contain the generator ${\boldsymbol{t}}_{t}$ which appears
in $L_{0}$ and $V_{0}^{(4)}$, so that group element with
${\boldsymbol{t}}_{+}$ will lead to a term that can be adapted independently
on the translation generators.

Using $g_{\mu\nu}=\eta^{ab}e_{\mu a}e_{\nu b}$ leads to the metric with Ricci
scalar $-\frac{6}{c_{1}^{2}}$. After transformation of the coordinates in the
ADM form and setting $\rho=r$ the metric reads exactly as (4.35) with above
defined changes in fields and chemical potentials. Similarly, one can further
conduct analogous analysis as in the simplest example $so(3,2)$.

When we set the functions $f_{1}(t)\rightarrow t$ and $f_{2}(t)\rightarrow t$
and the chemical potentials $\mu=\mu_{4}=0$ and
$\mu_{3}=\frac{(\mathcal{K}-\mathcal{K}_{4})^{2}-\mathcal{K}_{3}^{2}}{\mathcal{K}_{3}}$
we obtain the metric

$\displaystyle ds^{2}$
$\displaystyle=\frac{c_{1}^{2}r^{2}dr^{2}}{(\mathcal{K}_{3}^{2}-r^{2})\left((\mathcal{K}_{4}-\mathcal{K})^{2}-r^{2}\right)}+dt^{2}\left(\mathcal{K}_{3}^{2}+(\mathcal{K}_{4}-\mathcal{K})^{2}-r^{2}\right)+$
$\displaystyle+2\mathcal{K}_{3}(-\mathcal{K}_{4}+\mathcal{K})dtd\phi+r^{2}d\phi^{2}$
(4.59)

that has the form of the BTZ metric. One can notice that the existence of the
second horizon depends on the behaviour of the $\mathcal{K}_{4}$ and
$\mathcal{K}$. The chemical potential $\mu_{3}$ got absorbed during the
redefinition of the coordinates, and brought to the fact that resulting metric
depends only on the fields $\mathcal{K}$ and $\mathcal{K}_{i}$, $i=3,4$.

###  4.3 sl(5): conformal spin-2 and spin-3 field

Similarly to two previous examples, here as well we first find a map from our
generators in $PKLD$ basis to $WL$ basis. The sl(5) algebra in $WL$ basis is
going to correspond to the algebra of conformal graviton and a spin three
field in our $PKLD$ basis. From the definitions (2.6) and (2.7) we can find
the 5x5 matrices for the conformal graviton and spin three field which can be
mapped to a $WL$ representation convenient for imposing near horizon bcs. The
matrices satisfy the condition $2SD=-(SL_{xx}+SL_{yy}+SL_{tt})$, and the
commutation relations

$\displaystyle[{\boldsymbol{t}}_{a+},s_{b+}]$ $\displaystyle=-\eta_{ab}s_{++}$
$\displaystyle[{\boldsymbol{t}}_{a+},s_{++}]$ $\displaystyle=0$
$\displaystyle[{\boldsymbol{t}}_{a+},s_{cd}]$
$\displaystyle=-\eta_{ac}s_{+d}-\eta_{ad}s_{+c}$ (4.60a)
$\displaystyle[{\boldsymbol{t}}_{a+},s_{--}]$ $\displaystyle=2s_{a-}$
$\displaystyle[{\boldsymbol{t}}_{a+},s_{b-}]$
$\displaystyle=-\eta_{ab}s_{+-}+s_{ab}$
$\displaystyle[{\boldsymbol{t}}_{ab},s_{c+}]$
$\displaystyle=\eta_{ab}s_{a+}-\eta_{ac}s_{b+}$ (4.60b)
$\displaystyle[{\boldsymbol{t}}_{ab},s_{++}]$ $\displaystyle=0$
$\displaystyle[{\boldsymbol{t}}_{ab},s_{--}]$ $\displaystyle=0$
$\displaystyle[{\boldsymbol{t}}_{ab},s_{c-}]$
$\displaystyle=\eta_{bc}s_{a-}-\eta_{ac}s_{b-}$ (4.60c)
$\displaystyle[{\boldsymbol{t}}_{+-},s_{a+}]$ $\displaystyle=s_{+a}$
$\displaystyle[{\boldsymbol{t}}_{+-},s_{ab}]$ $\displaystyle=0$
$\displaystyle[{\boldsymbol{t}}_{+-},s_{++}]$ $\displaystyle=2s_{++}$ (4.60d)
$\displaystyle[{\boldsymbol{t}}_{+-},s_{--}]$ $\displaystyle=-2s_{--}$
$\displaystyle[{\boldsymbol{t}}_{+-},s_{a-}]$ $\displaystyle=-s_{-a}$
$\displaystyle[{\boldsymbol{t}}_{a-},s_{b+}]$
$\displaystyle=-\eta_{ab}s_{-+}+s_{ab}$ (4.60e)
$\displaystyle[{\boldsymbol{t}}_{a-},s_{++}]$ $\displaystyle=2s_{a+}$
$\displaystyle[{\boldsymbol{t}}_{a-},s_{--}]$ $\displaystyle=0$
$\displaystyle[{\boldsymbol{t}}_{a-},s_{cd}]$
$\displaystyle=-\eta_{ac}s_{-d}-\eta_{ad}s_{-c}$ (4.60f)
$\displaystyle[{\boldsymbol{t}}_{a-},s_{b-}]$ $\displaystyle=-\eta_{ab}s_{--}$
(4.60g) $\displaystyle[{\boldsymbol{t}}_{ab},s_{cd}]$
$\displaystyle=\eta_{bc}s_{ad}-\eta_{ac}s_{bd}+\eta_{bd}s_{ac}-\eta_{ad}s_{bc}$
(4.61)

and for $[{\boldsymbol{S}}_{AB},{\boldsymbol{S}}_{AB}]$

$\displaystyle[s_{++},s_{a+}]$ $\displaystyle=0$
$\displaystyle[s_{++},s_{ab}]$ $\displaystyle=0$
$\displaystyle[s_{++},s_{++}]$ $\displaystyle=0$ (4.62a)
$\displaystyle[s_{++},s_{--}]$ $\displaystyle=4{\boldsymbol{t}}_{+-}$
$\displaystyle[s_{++},s_{c-}]$ $\displaystyle=2{\boldsymbol{t}}_{+c}$
$\displaystyle[s_{a-},s_{c+}]$
$\displaystyle=\eta_{ac}{\boldsymbol{t}}_{-+}+{\boldsymbol{t}}_{ac}$ (4.62b)
$\displaystyle[s_{a-},s_{cd}]$
$\displaystyle=\eta_{ac}{\boldsymbol{t}}_{-d}+\eta_{ad}{\boldsymbol{t}}_{-c}$
$\displaystyle[s_{a-},s_{c-}]$ $\displaystyle=0$
$\displaystyle[s_{a-},s_{--}]$ $\displaystyle=0$ (4.62c)
$\displaystyle[s_{ab},s_{cd}]=\eta_{bc}{\boldsymbol{t}}_{ad}+\eta_{ac}{\boldsymbol{t}}_{bd}+\eta_{bd}{\boldsymbol{t}}_{ac}+\eta_{ad}{\boldsymbol{t}}_{bc}.$
(4.63)

The representation of the matrices in (4.60), (4.61),(4.62), (4.63) and (4.11)
in $PKLD$ basis with the $\eta_{ab}=diag(\pm 1,1,1)$ is given in the Appendix
D. We use the $\eta_{ab}=diag(1,1,1)$ in further calculations, and in the
transformation map from PKLD basis to WL basis.

Similarly like for the $sl(4)$ case we can define the near horizon bcs by
specifying $n=5$ in the general case (4.1), and from (4.4), we read out the
gauge parameter $\epsilon$. By defining $n=5$ we can also read out the charge
from (4.6) which is

$\displaystyle Q[\xi,\xi_{3},\xi_{4},\xi_{5}]=-\frac{k}{4\pi}\int
d\phi\left(\xi(\phi)\mathcal{K}(\phi)+\alpha_{3}\xi_{3}(\phi)\mathcal{K}_{3}(\phi)+\alpha_{4}\xi_{4}(\phi)\mathcal{K}_{4}(\phi)+\alpha_{5}\xi_{5}(\phi)\mathcal{K}_{5}(\phi)\right).$
(4.64)

The algebra of charges $J_{n},J_{n}^{(3)},J_{n}^{(4)}$ and $J_{n}^{(5)}$
defines ASA of the form (4.10) which corresponds to currents of the levels
$\frac{1}{2}k$, $\frac{1}{2}\alpha_{3}k$, $\frac{1}{2}\alpha_{4}k$, and
$\frac{1}{2}\alpha_{5}k$.

Calculation of entropy for these bcs is going to use $\Omega$ (4.1) with $n=5$
and the definition for trace of $\langle L_{0}L_{0}\rangle$ and $\langle
W_{0}^{(s)}W_{0}^{(s)}\rangle$ matrices. This leads to

$\displaystyle S=-\frac{k_{N}}{\pi}Im\left[\beta\int
d\phi\left(\mathcal{K}\mu_{0}\langle
L_{0}L_{0}\rangle+\mathcal{K}_{3}\mu_{3}\langle
W_{0}^{(3)}W_{0}^{(3)}\rangle+\mathcal{K}_{4}\mu_{4}\langle
W_{0}^{(4)}W_{0}^{(4)}\rangle+\mathcal{K}_{5}\mu_{5}\langle
W_{0}^{5}W_{0}^{(5)}\rangle\right)\right].$ (4.65)

Using that
$tr(W_{0}^{(s)}W_{0}^{(s)})=\frac{n(n^{2}-1)}{12}\alpha_{s}=n(n^{2}-1)\frac{4(s-1)!^{4}}{(2s-1)!(2s-2)!}\prod_{i=2}^{s-1}(n^{2}-i^{2})$
we obtain

$\displaystyle S=-\frac{k_{N}}{\pi}Im\left[\beta\int
d\phi\left(10\mathcal{K}\mu_{0}+56\mathcal{K}_{3}\mu_{3}+\frac{288}{5}\mathcal{K}_{4}\mu_{4}+\frac{1152}{35}\mathcal{K}_{5}\mu_{5}\right)\right],$
(4.66) $\displaystyle
S=-2k_{N}\beta\left(10\mathcal{K}\mu_{0}+56\mathcal{K}_{3}\mu_{3}+\frac{288}{5}\mathcal{K}_{4}\mu_{4}+\frac{1152}{35}\mathcal{K}_{5}\mu_{5}\right).$
(4.67)

Similarly as in the $sl(4)$ and $so(3,2)$ case we calculate the chemical
potentials

$\displaystyle\mu_{0}=\frac{(p-k-2m-3n)\pi}{5\beta},$
$\displaystyle\mu_{3}=\frac{3(n+p)-k}{14\beta}\pi,$
$\displaystyle\mu_{4}=\frac{(2k+4m+n+3p)\pi}{12\beta},$
$\displaystyle\mu_{5}=\frac{5(2k+n+p)\pi}{24\beta}.$ (4.68)

The entropy becomes

$\displaystyle S$
$\displaystyle=-2k_{N}\pi\bigl{[}2(p-k-2m-3n)\mathcal{K}+4\left(3(n+p)-k\right)\mathcal{K}_{3}+\frac{24}{5}(2k+4m+n+3p)\mathcal{K}_{4}\bigr{.}$
$\displaystyle\bigl{.}+\frac{48}{7}(2k+n+p)\mathcal{K}_{5}\bigr{]}.$ (4.69)

If we look at the map between the $WL$ and $PKLD$ matrices, we are going to
see that each of the diagonal generators in $WL$ basis contains generator $D$
and diagonal generators $SL_{ii}$. Since the generators $SL_{ii}$ do not
belong to Poincaré algebra, and they are in linear combination of each of the
diagonal $WL$ generators, we cannot use this embedding to compare the entropy
with those for $sl(4)$ and $so(3,2)$ case with the only remaining generator
$L_{d}$. We can try to compare with the $sl(4)$ scenario when the remaining
generator is $L_{0}$. For $sl(4)$ in that case the entropy is
$S=-\frac{20}{3}k_{N}\pi(1+2m)\mathcal{K}$, while for the entropy in the
$sl(5)$ case we get $S=-40k_{N}\pi m\mathcal{K}$. The difference is present
due to a different embedding. The generator $L_{0}$ in $sl(5)$ scenario is a
linear combination of PKLD generators, and not proportional only to generator
$D$.

To consider metric formulation we need to determine the group element $b$. For
the consistency with the earlier examples, we write the group element in terms
of the generators ${\boldsymbol{P}}_{i}$ and a diagonal generator in the PKLD
basis. If we consider
$b=\rho{\boldsymbol{P}}_{y}+{\boldsymbol{P}}_{t}+\tilde{c}_{1}{\boldsymbol{D}}$
we obtain degenerate dreibein. Reason is the representation of the dilatation
generator. To obtain the example of the same form as in $sl(4)$ and $so(3,2)$,
we have to take the group element
$b=a_{2}\rho{\boldsymbol{P}}_{x}+a_{1}{\boldsymbol{P}}_{y}+a_{3}\rho{\boldsymbol{P}}_{t}+{\boldsymbol{S}}{\boldsymbol{L}}_{tt}$

The elements of dreibein for that group element are
$e_{\rho}^{\tilde{x}}=a_{2}$,
$e_{\rho}^{\tilde{y}}=0,e_{\rho}^{\tilde{t}}=\frac{1}{2}a_{3}\sinh{2}$ and

$\displaystyle e_{i}^{\tilde{x}}$
$\displaystyle=\frac{1}{70}a_{2}\rho\left(35X^{i}-210X^{i}_{3}+252X^{i}_{4}-120X^{i}_{5}\right)$
(4.70a) $\displaystyle e_{i}^{\tilde{y}}$
$\displaystyle=a_{1}\bigl{(}-\frac{X^{i}}{2}+3X^{i}_{3}+\frac{12X^{i}_{5}}{7}-\frac{18X^{i}_{4}}{5}\bigr{)}$
(4.70b) $\displaystyle e_{i}^{\tilde{t}}$
$\displaystyle=-\frac{\left(e^{2}-1\right)a_{3}\rho\left(35\left(4e^{2}-3\right)X^{i}-210X^{i}_{3}+84\left(1+2e^{2}\right)X^{i}_{4}-120X^{i}_{5}\right)}{140e^{2}}.$
(4.70c)

Here we denote $i=\\{\phi,t\\}$, for $i=\phi$ and $t$ respectively we have,
$X^{\phi}=\mathcal{K},X_{3}^{\phi}=\mathcal{K}_{3},X^{t}=\mu,X_{3}^{t}=\mu_{3}$,
etc.. This dreibein is very similar to (8.294) for the example analyzed in the
Appendix E. The metric that dreibein (4.70) leads to, is conformally flat,
Einstein space, with Ricci scalar $R=-\frac{6}{a_{1}^{2}}$. The metric
contains all the chemical potentials and fields, where fields can be dependent
on $\phi$ coordinate. For simplicity we set all chemical potentials and fields
to zero beside $\mu_{5}$, $\mathcal{K}_{5}$ and $\mathcal{K}$ keeping them
both constant. Such metric can be transformed to a form
$ds^{2}=f_{rr}dr^{2}+f_{tt}dt^{2}+f_{t\phi}dtd\phi+r^{2}\phi^{2}$ for the

$\displaystyle f_{rr}$
$\displaystyle=\frac{r^{2}}{\left(r^{2}-(\mathcal{K}-\mathcal{K}_{5})^{2}\right)\left(\left(\mathcal{K}_{5}-\mathcal{K}q_{1}\right){}^{2}+r^{2}\right)}$
(4.71a) $\displaystyle f_{tt}$
$\displaystyle=\frac{\left(-4\mathcal{K}q_{2}\mathcal{K}_{5}\left(\mathcal{K}^{2}q_{1}-\mathcal{K}q_{3}(2\mathcal{K}_{5}+\mu_{5})+\mathcal{K}_{5}(\mathcal{K}_{5}+\mu_{5})\right)+4\mathcal{K}^{2}q_{2}^{2}r^{2}+r^{2}\left(-\mathcal{K}q_{3}+\mathcal{K}_{5}+\mu_{5}\right){}^{2}\right)}{4\left(4\mathcal{K}^{2}q_{2}^{2}+\left(\mathcal{K}_{5}-\mathcal{K}q_{3}\right){}^{2}\right)}$
(4.71b) $\displaystyle f_{t\phi}$
$\displaystyle=\frac{\mu_{5}\left(2\mathcal{K}q_{3}-2\mathcal{K}_{5}\right)\left((\mathcal{K}-\mathcal{K}_{5})^{2}-r^{2}\right)}{\left(\mathcal{K}_{5}-kq_{1}\right){}^{2}+(\mathcal{K}-\mathcal{K}_{5})^{2}}+\mu_{5}(\mathcal{K}_{5}-\mathcal{K})+r^{2}$
(4.71c)

for $q_{1}=-3+4e^{2},q_{2}=-1+e^{2},q_{3}=-1+2e^{2}$. The parameters from the
group element have been set to $a_{1}=1,a_{2}=\frac{1}{2e^{2}}$ and
$a_{3}=\frac{1}{-1+e^{2}}$. The fields and the chemical potential have been
suitably rescaled. The reason that we get the Euler’s number is that in
choosing the group element $b$ we allowed for the generators multiplied by
constant terms, and not only those multiplied by function of $\rho$. If we
kept function of $\rho$ we would obtain the terms with $e^{\rho}$ in the
metric, which would lead to $\cosh(\rho)$ and $\sinh(\rho)$ functions. And we
would have to consider the perturbative form of the Hausdorff formula. Keeping
the dependency on $\rho$ only on selected places in the group element $b$ and
allowing for the generators multiplied by constants, we obtain the terms
multiplied by $e$ in the metric. We can see from (4.71a) that metric has one
real and one imaginary eigenvalue, it belongs to $I_{d}$ class of one
parameter subgroups of $SO(3,2)$.

If we take into account the chemical potential and fields we kept for the
solution, we obtain for the entropy
$S=-2k_{N}\pi\frac{48}{7}(2k+n+p)\mathcal{K}_{5}=-2k_{N}\pi\frac{48}{3}k\mathcal{K}_{5}$.
Setting the field $\mathcal{K}_{5}$ to zero and rescaling the metric with -1
lands us in the dS space where we have a metric with cosmological horizon.
Choosing the right parameter $b$ for the $sl(5)$ case, can also lead to
Lobachevsky metric, which we describe in the Appendix E.

Here, we can also calculate the spin-3 field for the dreibein (4.70) using
$\phi_{\mu\nu\rho}=e_{\mu}{}^{i}e_{\nu}{}^{j}e_{\rho}{}^{k}{\mathrm{Tr}\,}[T_{i}T_{j}T_{k}]$.
One obtains

$\displaystyle\varphi_{(\rho\phi
t)}=-\frac{3}{70e^{4}}\left[\left(e^{2}-1\right)\rho\left(7\mathcal{K}_{3}\left(5\mu+6\mu_{4}\right)+4\mathcal{K}_{5}\left(5\mu+6\mu_{4}\right)\right.\right.$
$\displaystyle\left.\left.-5\mathcal{K}\left(7\mu_{3}-7\mu_{4}+4\mu_{5}\right)-\mathcal{K}_{4}\left(35\mu+42\mu_{3}+24\mu_{5}\right)\right)\right]$
(4.72)

The brackets denote symmetry under permutation of indices. We use, for
consistency, the Lorentz generators for generators over which a trace is
taken. To be able to consistently treat the conformal scenario with higher
spins and for the spin-2 we have to define the embedding of $sl(2)\oplus
sl(2)$ in $so(3,2)$ so that we can continue with analogous treatment in higher
spin cases, or we need to consider invariant bilinear of the entire $so(3,2)$
to define the metric, and not just Lorentz generators (standardly used for
such definitions in conformal gravity).

##  5 One parameter subgroups of SO(3,2)

Summary of the classification.  To analyze the one parameter subgroups of
G$\equiv$ SO(3,2) we follow closely the analysis of SO(2,2) given in [33]. In
the following table (1) we write the different classes of metrics depending on
the Killing vectors they conserve and 1. and 3. Casimir invariant which define
the class. The Killing vectors in the table (1) are determined from
$\omega^{ab}J_{ab}$, for $\omega^{ab}$ a spin connection. Its derivation in
terms of the eigenvalues is described in Appendix F where one can also find
the procedure to obtain this classification.

Type | Killing vector | 1\. Casimir | 3\. Casimir
---|---|---|---
$I_{a}$ | $b(J_{01}+J_{23})-a(J_{03}+J_{12})$ | $4(b^{2}-a^{2})$ | $4(a^{3}-3ab^{2})$
$I_{b}$ | $\lambda_{2}J_{03}+\lambda_{1}J_{12}$ | $-2(\lambda_{1}^{2}+\lambda_{2}^{2})$ | $2(\lambda_{1}^{3}+\lambda_{2}^{3})$
$I_{c}$ | $b_{1}J_{01}+b_{2}J_{23}$ | $2(b_{1}^{2}+b_{2}^{2})$ | 0
$I_{d}$ | $bJ_{01}+\lambda J_{03}$ | $2(b^{2}-\lambda^{2})$ | $2\lambda(b_{1}^{2}-\lambda^{2})$
$II_{a}$ | $J_{01}-J_{02}-J_{13}+J_{23}-\lambda(J_{03}+J_{12})$ | $-4\lambda^{2}$ | $4\lambda^{2}(3+\lambda)$
$II_{b}$ | $(b-1)J_{01}+(b+1)J_{32}+J_{02}-J_{13}$ | $4b^{2}$ | 0
$III_{a+}$ | $-J_{13}+J_{23}$ | 0 | 0
$III_{a-}$ | $-J_{01}+J_{02}$ | 0 | 0
V | $\frac{1}{4}(-J_{01}+J_{03}-J_{12}-J_{23})+J_{04}+J_{24}$ | 0 | -2
Table 1: One-parameter subgroups of SO(3,2)

$J_{ab}$ are Killing vectors of SO(3,2) defined with

$\displaystyle J_{ab}=x_{b}\frac{\partial}{\partial
x_{a}}-x_{a}\frac{\partial}{\partial x_{b}}$ (5.1)

for $x^{a}=(u,v,x,y,z)$. Definition of the most general Killing vector is
therefore $\frac{1}{2}\omega^{ab}J_{ab}$ while $\omega^{ab}=-\omega^{ba}$.
$a,b,b_{1},b_{2},\lambda_{1}$ and $\lambda_{2}$ are real numbers defined by
eigenvalues of $\omega^{ab}$ according to which classification is made.
$\lambda_{1}$ and $\lambda_{2}$ are real eigenvalues, while $a$, $b$, $b_{1}$,
and $b_{2}$, come from complex eigenvalues. See Appendix F for details.

Requirement for appearance of black hole. Each Killing vector has an one
parameter subgroup of isometries of AdS space. For the Killing vector $\xi$
this is written as

$\displaystyle P\rightarrow e^{t\xi}P$ (5.2)

These mappings for $t=2n\pi$ for n integer, define identification subgroup.

The quotient space that we obtain by identifying the points of a given orbit
that belong to the identification subgroup, inherits a well defined metric
from AdS space. The quotient space remains the solution of the equations that
we are considering.

Due to the identification process, the curves that are on the same orbit and
which join two points from AdS space, are closed in the quotient space. These
closed curves should not be timelike or null, in order for the quotient space
to have valid causal structure. A necessary condition for these curves not to
be closed timelike curves is

$\displaystyle\xi\cdot\xi>0$ (5.3)

In general this condition is not sufficient to guarantee absence of closed
timelike curves (CTC), however in this case it is [33].

In some regions, the Killing vectors that appear in identifications and which
give rise to black hole solutions, are timelike or null. These regions have to
be cut out from the AdS space, for the identifications to be allowed. We call
this new space AdS’. It is geodesically incomplete because there are geodesics
that go beyond $\xi\cdot\xi>0$ to $\xi\cdot\xi<0$. The surface $\xi\cdot\xi=0$
arises as a singularity in the causal structure, as continuation beyond this
surface would result in the CTCs. Therefore, that surface is considered as a
true singularity in the quotient space.

###  5.1 Special cases from classification

####  5.1.1 Case $I_{d}$

Let us consider the Killing vector for the type $I_{d}$ of the classification

$\displaystyle\xi=b_{1}J_{01}+\lambda
J_{03}=b_{1}(v\partial_{u}-u\partial_{v})+\lambda(y\partial_{u}+u\partial_{y})$
(5.4)

The condition $\xi\cdot\xi>0$ then gives for the coordinates $(u,v,x,y,z)$ and
vector $\xi_{a}=(b_{1}v+\lambda y,-b_{1}u,0,\lambda u,0)$ the following

$\displaystyle\xi\cdot\xi=\lambda^{2}u^{2}-b_{1}^{2}y^{2}-(b_{1}v+\lambda
y^{2})$ (5.5)
$\displaystyle\frac{\lambda^{2}}{b_{1}^{2}}u^{2}-y^{2}-(v+\frac{\lambda}{b_{1}}y)^{2}>0$
(5.6)

We can consider what happens in special cases for different values of
parameters $\lambda$ and $b_{1}$.

  1. 1.

$\frac{\lambda}{b_{1}}=1$ in this special case one has
$u^{2}-y^{2}>(v+y)^{2}$. This region reminds of the outer regions of the BTZ.

  2. 2.

For $\lambda>>b_{1}$ one obtains $u^{2}-y^{2}>0$, which would remind of the
intermediate region of BTZ, however in this case there is no $l^{2}$ that
would limit the $u^{2}-y^{2}$ from above.

  3. 3.

For $b_{1}>>1$ and small u, one can notice $y^{2}+v^{2}<0$. This region is
unphysical, which is expected since the imaginary value dominates. The norm of
the Killing vector in this region is smaller than zero.

Transformation for the $I_{d}$ case.
The vector which defines the $I_{d}$ class of solutions is written in the form
$\xi_{d}=b_{1}J_{01}+\lambda J_{03}$. The compactification from the
$-du^{2}-dv^{2}+dx^{2}+dy^{2}+dz^{2}$ to the three dimensions which will leave
the vector $\partial_{\phi}$ conserved leads to three partial differential
equations from transforming $b_{1}J_{01}+\lambda J_{03}$ from $(r,t,\phi)$ to
$(u,v,x,y,z)$ coordinates

$\displaystyle\left(b_{1}(v\partial_{u}-u\partial_{v})+\lambda(y\partial_{u}+u\partial_{y})\right)r(u,v,x,y,z)=0$
(5.7)
$\displaystyle\left(b_{1}(v\partial_{u}-u\partial_{v})+\lambda(y\partial_{u}+u\partial_{y})\right)t(u,v,x,y,z)=0$
(5.8)
$\displaystyle\left(b_{1}(v\partial_{u}-u\partial_{v})+\lambda(y\partial_{u}+u\partial_{y})\right)\phi(u,v,x,y,z)=1.$
(5.9)

The solution of these equations in the most general form allows for the
coordinates $r$ and $t$ to transform into

$\displaystyle r\rightarrow f_{r}\left(x,z,\frac{\lambda
v+b_{1}y}{b_{1}},\frac{(b_{1}^{2}+\lambda^{2})v^{2}+b_{1}^{2}u^{2}+2b_{1}\lambda
vy}{2}\right)$ (5.10a) $\displaystyle t\rightarrow
f_{t}\left(x,z,\frac{\lambda
v+b_{1}y}{b_{1}},\frac{(b_{1}^{2}+\lambda^{2})v^{2}+b_{1}^{2}u^{2}-2b_{1}\lambda
vy}{2}\right)$ (5.10b)

while the third partial differential equation defines that the conserved
Killing vector is proportional to $\partial_{\phi}$ with the transformation

$\displaystyle\phi\rightarrow\frac{i\log(-2b_{1}i\sqrt{b_{1}^{2}-\lambda^{2}}u+2b_{1}(b_{1}v+\lambda
y))}{\sqrt{b_{1}^{2}-\lambda^{2}}}+f_{\phi}\left(x,z,\frac{\lambda
v+b_{1}y}{b_{1}},\frac{(b_{1}^{2}+\lambda^{2})v^{2}+b_{1}^{2}u^{2}-2b_{1}\lambda
vy}{2}\right).$ (5.11)

Since we need the inverse transformation $(u,v,x,y,z)\rightarrow(r,t,\phi)$ to
apply to $du^{2}+dv^{2}-dx^{2}-dy^{2}-dz^{2}$ we need to invert the above
relations. For the BTZ black hole for the outer region, the transformations
analogous to (5.10) and (5.11) can be written as

$\displaystyle r$
$\displaystyle=\frac{\sqrt{l^{2}r_{+}^{2}-(r_{-}^{2}-r_{+}^{2})(u-x)(u+x)}}{l}$
(5.12a) $\displaystyle\phi$
$\displaystyle=\frac{r_{+}\sinh^{-1}\left(\frac{v}{\sqrt{l^{2}+u^{2}-x^{2}}}\right)+r_{-}\cosh^{-1}\left(\frac{u}{\sqrt{u^{2}-x^{2}}}\right)}{r_{-}^{2}-r_{+}^{2}},$
(5.12b) $\displaystyle t$
$\displaystyle=\frac{r_{-}\sinh^{-1}\left(\frac{v}{\sqrt{l^{2}+u^{2}-x^{2}}}\right)+r_{+}\cosh^{-1}\left(\frac{u}{\sqrt{u^{2}-x^{2}}}\right)}{r_{-}^{2}-r_{+}^{2}}$
(5.12c)

for $r_{+}$ and $r_{-}$ outer and inner horizon respectively. Since the
transformation (5.11) is chosen to give $\xi\propto\partial_{\phi}$ its form
is determined. One needs to remember that logarithm can be expressed in terms
of the $arcsinh$ or $arccosh$ in which case we could think of transformation
(5.11) as analogous to transformation (5.12) for $\phi$ (while setting $r_{+}$
or $r_{-}$) to zero. The function under the inverse hyperbolic function would
be of the form determined by (5.11). However one needs to be careful in this
case because the $\phi$ transforms into imaginary part with logarithm of a
complex number.

We consider one robust example in which we choose a simple functions $f_{r}$,
$f_{t}$ and $f_{\phi}$ which we invert and find a coordinates that embedd a
metric. This is described in Appendix G. Taking $z\rightarrow f(r,t,\phi)$,
$\phi\rightarrow i\phi$ and demanding that metric after transformation (8.417)
has Ricci scalar equal to -6, one can find functions that define AdS metric.
As a peculiarity of the metric we consider a case when $z\rightarrow it$, and
$\phi\rightarrow i\phi$. The metric that one obtains is

$\displaystyle ds^{2}$
$\displaystyle=-\frac{dr^{2}}{b_{1}^{4}l^{2}+2b_{1}^{2}r}+\frac{2\sqrt{b_{1}^{2}-\lambda^{2}}drd\phi}{b_{1}^{2}}+d\phi^{2}\left(r\left(\frac{2\lambda^{2}}{b_{1}^{2}}-2\right)-\lambda^{2}t^{2}\right)+\frac{2\lambda^{2}tdtd\phi}{\sqrt{b_{1}^{2}-\lambda^{2}}}+\frac{\lambda^{2}dt^{2}}{\lambda^{2}-b_{1}^{2}}$
(5.13)

We can further transform the metric (5.13) using the transformation
$\phi\rightarrow\frac{\log\left(b_{1}^{2}\lambda^{2}f_{1}(r,T)^{2}+2r(b_{1}^{2}-\lambda^{2})\right)}{2\sqrt{b_{1}^{2}-\lambda^{2}}}$
and $t\rightarrow
f_{1}(r,T)=\sqrt{r}\frac{\sqrt{-b_{1}^{2}t-2b_{1}^{2}+2\lambda^{2}}}{b_{1}\lambda}$
obtaining eventually

$\displaystyle
ds^{2}=\frac{l^{2}dr^{2}}{2b_{1}^{2}l^{2}r+4r^{2}}-\frac{rdt^{2}}{2T\left(b_{1}^{2}(T+2)-2\lambda^{2}\right)}+rTd\phi^{2}.$
(5.14)

The metric (5.14) is a solution of the Einstein gravity with cosmological
constant. It can be transformed using
$r\rightarrow\frac{v^{2}}{f(v,\tilde{t})}$ and $T\rightarrow f(v,\tilde{t})$,
for
$f(v,\tilde{t})=\frac{2v^{2}\left(\lambda^{2}-b_{1}^{2}\right)\sec^{2}\left(\left(\lambda^{2}-b_{1}^{2}\right)(\tilde{t}-c_{2})\right)}{b_{1}^{2}\left(b_{1}^{2}l^{2}+v^{2}\sec^{2}\left(\left(\lambda^{2}-b_{1}^{2}\right)(\tilde{t}-c_{2})\right)-\lambda^{2}l^{2}-v^{2}\right)}$,
and $c_{2}$ integration constant, into

$\displaystyle
ds^{2}=-d\tilde{t}^{2}\left(-b_{1}^{2}l^{2}+\lambda^{2}l^{2}+v^{2}\right)\left(\lambda^{2}-b_{1}^{2}\right)+\frac{l^{2}dv^{2}}{-b_{1}^{2}l^{2}+\lambda^{2}l^{2}+v^{2}}+v^{2}d\phi^{2}.$
(5.15)

We can notice the solution that we get can have positive or negative value for
$|-b_{1}^{2}+\lambda^{2}|$ depending on value of $b_{1}$ and $\lambda$. This
situation captures the metrics of the form (4.71) when
$\mathcal{K}=\mathcal{K}_{5}$ and $\mu_{5}$ is chosen such that
$g_{\tilde{t}\phi}$ vanishes. The Killing vectors are $\partial_{\phi}$ and
$\partial_{\tilde{t}}$.

Thinking from the aspect of the solutions of

####  5.1.2 Case $I_{b}$

To transform from the SO(3,2) embedding space

$\displaystyle ds^{2}=du^{2}+dv^{2}-dx^{2}-dy^{2}-dz^{2}$ (5.16)

with

$\displaystyle u^{2}+v^{2}-x^{2}-y^{2}=l^{2}$ (5.17)

to the three dimensional space with scaling property we use the standard
transformations

$\displaystyle u$ $\displaystyle=l\cosh(\mu)\sin(\lambda),$ $\displaystyle
v=l\cosh(\mu)\cos(\lambda),$ $\displaystyle x=l\sinh(\mu)\cos(\theta)$ (5.18)
$\displaystyle y$ $\displaystyle=l\sinh(\mu)\sin(\theta)$ $\displaystyle z=l$
(5.19)

If we keep $l$ as a free varying component the contribution $dl^{2}$ which
appears from the components $(u,v,x,y)$ cancels with the addition of
$dz^{2}=dl^{2}$ in comparison to the SO(2,2) case. Leading to the metric

$\displaystyle
ds^{2}=l^{2}\left(-\cosh(\mu)^{2}d\lambda^{2}+d\mu^{2}+\sinh(\mu)^{2}d\theta^{2}\right)$
(5.20)

where $l$ is not a constant. The analysis $\xi\cdot\xi$ is similar like in the
SO(2,2) case.

If we choose the coordinates as

$\displaystyle u$ $\displaystyle=l\sqrt{F(r)}\sinh(r_{2}\phi-r_{1}t)$ (5.21)
$\displaystyle v$ $\displaystyle=l\sqrt{G(r)}\cosh(r_{2}t-r_{1}\phi)$ (5.22)
$\displaystyle x$ $\displaystyle=l\sqrt{F(r)}\cosh(r_{2}\phi-r_{1}t)$ (5.23)
$\displaystyle y$ $\displaystyle=l\sqrt{G(r)}\sinh(r_{2}t-r_{1}\phi)$ (5.24)
$\displaystyle z$ $\displaystyle=l\sqrt{G(r)-F(r)}$ (5.25)

for

$\displaystyle F(r)$
$\displaystyle=\frac{r^{2}-r_{1}^{2}}{r_{2}^{2}-r_{1}^{2}}$ (5.26)
$\displaystyle G(r)$
$\displaystyle=\frac{r^{2}-r_{2}^{2}}{r_{2}^{2}-r_{1}^{2}}$ (5.27)

we obtain the metric which is similar to the BTZ black hole in the so(2,2)
case however $l$ is not constant. The $dl^{2}$ term here cancels due to the
term coming from z component. The entire metric is multiplied with $l$

$\displaystyle
ds^{2}=l^{2}\left(\frac{r^{2}dr^{2}}{(r^{2}-r_{1}^{2})(r^{2}-r_{2}^{2}))}+(-r^{2}+r_{1}^{2}+r_{2}^{2})dt^{2}-2r_{1}r_{2}dtd\phi+r^{2}d\phi^{2}\right)$
(5.28)

leading to conformal version of the familiar line element that is used to
define BTZ black hole upon the identification $\phi\rightarrow\phi+2k\pi$
along the $\phi$ direction.

####  5.1.3 Case V

The Killing vector of the type V can be written as

$\displaystyle\xi_{a}=\frac{1}{4}\left(-v+y+4z\right)\partial_{u}+\frac{u-x}{4}\partial_{v}+\frac{u+x}{4}\partial_{y}-\frac{1}{4}(v+y+4x)\partial_{x}+(u-x)\partial_{z}$
(5.29)

and it satisfies the condition $\xi\cdot\xi>0$ for

$\displaystyle\xi_{a}\xi^{a}=4u^{2}-7ux+v(2x+y+2z)+2x(4x+y)-2yz-4z^{2}>0$
(5.30)

Using the analogous steps as for the case $I_{d}$ we find the transformation
to $(r,t,\phi)$ coordinates. We solve the transformation equations not
requiring conservation of particular Killing vector. Instead we show the
conserved Killing vectors on the obtained solution. Transformations

$\displaystyle-4v+z=r,$ $\displaystyle-15u^{2}+(v-4z)^{2}+15x^{2}+15y^{2}=t,$
$\displaystyle 2(4u^{2}-7ux+4x^{2}+y(v-4z)=\phi$ (5.31)

can be solved for $u,v,z,y$. One of the solutions gives
$v\rightarrow-\frac{4r}{15},z\to-\frac{r}{15},u\to\frac{1}{8}\left(7x-\sqrt{8\phi-15x^{2}}\right),\\\
y\to-\frac{1}{4}\sqrt{\frac{16t}{15}-\frac{7}{2}x\sqrt{8\phi-15x^{2}}-\frac{15x^{2}}{2}+2\phi}$.
We set $x=f(t)=t$ which defines a metric. Changing
$\phi\rightarrow\frac{\psi}{8}-\frac{15}{8}\tau^{2}$,
$t\rightarrow\frac{\sqrt{-\tau+\psi}}{\sqrt{30}}$ and
$\psi\rightarrow\zeta+\tau$ we obtain
$ds^{2}=\frac{dr^{2}}{15}+g_{\tau\zeta}d\tau
d\zeta+g_{\tau\tau}d\tau^{2}+g_{\zeta\zeta}d\zeta^{2}$

$\displaystyle g_{\tau\zeta}$
$\displaystyle=\frac{1}{f(\zeta,\tau)}(30\zeta\tau-7\zeta-\sqrt{30}\tau)$
(5.32a) $\displaystyle g_{\tau\tau}$
$\displaystyle=\frac{1}{f(\zeta,\tau)}(\sqrt{30}\zeta-15\zeta^{2})$ (5.32b)
$\displaystyle g_{\zeta\zeta}$
$\displaystyle=\frac{1}{15f(\zeta,\tau)}(15(7-15\tau)\tau-16)$ (5.32c)

for
$f(\zeta,\tau)=-210\sqrt{30}\zeta\tau+\zeta\left(64\sqrt{30}-225\zeta\right)+450\tau^{2}.$
(We considered the branch where $-\tau+\psi>0$.) Such obtained metric solves
the Cotton tensor, and it has constant Ricci scalar. Conserved Killing vectors
are $\partial_{\tau}$ and $\partial_{\zeta}$ for vanishing conformal factor.
The Einstein tensor points to a solution with vanishing pressure, and non-
vanishing energy density. To find the general form of solutions that appear in
Case V, one would have to solve system of equations analogous to those in Case
$I_{d}$ and find the general dependence of $(r,t,\phi)$ on $(u,v,x,y,z)$
coordinates. Inverting those would hint at the embedding that would lead to
more general solution.

##  6 Conclusion

We have studied the holography of the CHiSGRA theories imposing the general
and the near horizon bcs. For the holography of the general bcs we obtained as
an ASA a loop algebra of underlying gauge algebra. Selecting a specific set of
generators among these, we defined a near horizon bcs, which led to ASA that
corresponds to $u(1)$ currents.

Imposing the near horizon bcs, we obtain the solutions that can be placed in
the classes studied under the classification of the one parameter subgroups of
SO(3,2). More precisely, once the near horizon bcs are imposed to the example
with conformal graviton, suitable choice of the group element in the
transformation from the first order to the metric formulation can lead to
different solutions, depending on the choice of the projector $b$. As examples
we obtain solutions with (i) four real eigenvalues, (ii) one real and one
imaginary eigenvalue which belongs to class $I_{d}$, (iii) Lobachevsky
solution, (iv) solution that generalizes BTZ to conformal scenario, and (v)
general solution which depends on three arbitrary functions of $\rho$, and
(vi) simplest example that reduces to BTZ. We verify (iv) by solving the
Cotton tensor equations of motion which we show in Appendix B. Simplified form
of metric (v) reproduces 3 dimensional analog of the MKR solution.

For the near horizon bcs of the $sl(4)$ we show the example that is similar to
solution (vi) of conformal graviton. The modification comes from number of
charges in the bcs. The distinction to $so(3,2)$ case will appear when the
group element $b$ contains the generator which commutes into the generator of
translation mutliplied by an arbitrary function of $\rho$. That will allow for
the appearance of the additional term in the metric.

As an example of the metric formulation for the near horizon bcs of conformal
graviton and a spin three field defined from the sl(5) algebra we obtain the
solution from the $I_{d}$ class. For the same bcs, we find as well a spin-3
field. We also calculate the entropies of these solutions, and look for the
conditions under which they are connected to the known results. The solutions
of the conformal gravity in three dimensions have been significantly less
studied in comparison to those of Einstein gravity so we provide the analysis
of the SO(3,2) one parameter subgroups. The analysis goes along the lines of
[33] where the solutions of the Einstein gravity in three dimensions were
classified in the SO(2,2) one parameter subgroups. The classification of the
SO(3,2) in comparison to the analysis of the SO(2,2) is complemented with the
two additional classes which do not appear in the SO(2,2). For the two new
classes, which were denoted with case $I_{d}$ and case $V$ we consider
specific examples starting from the embedding
$du^{2}+dv^{2}-dx^{2}-dy^{2}-dz^{2}$ of SO(3,2) group. Example from the
$I_{d}$ class led to a solution which can take a real or imaginary eigenvalue.
The solution however, which follows geometric description is still lacking. As
visible from Table 1, the definition of the Killing vector for the $I_{b}$ and
$I_{c}$ classes, contains different components in Killing vectors associated
with eigenvalue. The class $I_{d}$ contains the $0$ component in both Killing
vector which complicates the construction of the geometric solution analogous
to the BTZ construction for the $I_{b}$ class. Provided solution is a specific
example, while the most general representative of the $I_{d}$ class is yet to
be found.

##  7 Acknowledgements

I would like to thank Daniel Grumiller and Evgeny Skvortsov for the
discussions and suggestions in development of this work, and for the comments
on the draft. This work was supported by the Hertha Firnberg grant T 1269-N of
the Austrian Science Fund FWF.

##  8 Appendix

###  8.1 Appendix A: Holography of Chern-Simons action

Let us review the general approach to holography of the Chern-Simons action in
the vielbein formulation.

The action can be written in 2+1 decomposition as

$\displaystyle
I=\frac{k}{4\pi}\int_{\mathbb{R}}dt\int_{\Sigma}d^{2}x\epsilon^{ij}g_{\mu\nu}(\dot{\omega}^{\mu}_{i}\omega_{j}^{\nu}+\omega_{t}^{\mu}F_{ij}^{\nu})+B(\partial\Sigma)$
(8.1)

For $B$ the boundary term that depends on the bcs and needs to be included to
ensure that the action is gauge invariant. The curvature $F_{ij}^{\mu}$ is
$F_{ij}^{\mu}=\partial_{i}\omega_{j}^{\mu}-\partial_{j}\omega_{i}^{\mu}+f^{\mu}_{\nu\rho}\omega^{\nu}\omega^{\rho}$
and the $f^{\mu}_{\nu\rho}$ are the structure constants of the Lie algebra.
The indices $\mu,\nu$ are raised and lowered with the metric $g_{\mu\nu}$
which is invertible. $\omega_{t}^{\mu}$ is treated as a Lagrange mutliplier
and the fields $\omega_{i}^{\mu}$ are dynamical fields which satisfy Poisson
brackets. The variation of the action with respect to the $\omega_{t}^{\mu}$
gives the constraint
$G_{a}\equiv\frac{k}{4\pi}g_{\mu\nu}\epsilon^{ij}F_{ij}^{\nu}$ which satisfies
Poisson bracket algebra

$\displaystyle\\{G_{\mu}(x),G_{\nu}(y)\\}=f^{\rho}_{\mu\nu}G_{\rho}(x)\delta(x,y)$
(8.2)

so $G_{\mu}$ are generators of gauge transformations which act on the phase
space. The smeared generators
$G(\xi^{\mu})=\int_{\Sigma}\xi^{\mu}G_{\mu}+Q(\xi)$ give central extension of
the algebra of the global charges. We assume that the parameter $\xi$ does not
depend on the $\omega_{i}$ in the interior, such that the variation of the
smeared generator leads to

$\displaystyle\delta
G=\frac{k}{2\pi}\int_{\Sigma}\epsilon^{ij}\xi_{\mu}D_{i}\delta\omega_{j}^{\mu}+\delta
Q(\xi)$ (8.3)

for
$D_{i}v^{\mu}\equiv\partial_{i}v^{\mu}+f^{\mu}_{\nu\rho}\omega_{i}^{\nu}v^{\rho}$
covariant derivative. Here, $v^{\mu}$ is an arbitrary vector field.
Integration by parts of the first term leads to the

$\displaystyle\delta
G=-\frac{k}{2\pi}\int_{\Sigma}\epsilon^{ij}D_{i}\xi_{\mu}\delta\omega_{j}^{\nu}+\frac{k}{2\pi}\int_{\partial\Sigma}\xi_{\mu}\delta\omega_{k}^{\mu}dx^{k}+\delta
Q.$ (8.4)

The functional derivative of G then becomes well defined and the surface terms
cancel out if we demand that the variation of the charge is given by

$\displaystyle\delta
Q=-\frac{k}{2\pi}\int_{\partial\Sigma}\xi_{\mu}\delta\omega_{i}^{\mu}dx^{i}.$
(8.5)

The Poisson bracket of the two smeared generators, with the parameters $\xi$
and $\lambda$

$\displaystyle\\{G(\xi),G(\lambda)\\}=-\frac{k}{2\pi}\int_{\Sigma}D\xi_{a}\wedge
D\lambda^{a}$ (8.6)

can be integrated by parts to obtain

$\displaystyle\\{G(\xi),G(\lambda)\\}=\int_{\Sigma}[\xi,\lambda]^{a}G_{a}+\frac{k}{2\pi}\int_{\partial\Sigma}\xi_{a}D\lambda^{a}$
(8.7)

For $[\xi,\lambda]^{a}=f^{a}_{bc}\xi^{b}\lambda^{c}$ commutator of the
algebra.

To obtain the smeared generator one needs to add the surface term to the first
term on the right to make it differentiable. That means the second term on the
right hand side, the surface term, needs be the same as the charge which
regularizes the first term up to a central extension. Therefore, one can write

$\displaystyle\frac{k}{2\pi}\int\xi_{a}D\lambda^{a}=Q(\sigma)+K(\xi,\lambda)$
(8.8)

for the Q defined in (8.5), and $\sigma$ a function of $\lambda$ and $\xi$.
Its precise form will define the algebra of the global charges, which depends
on the bcs. The second term on the right hand side of (8.8) is the central
extension and it does not depend on the variables varied on the boundary.

Combining the (8.8), (8.7) replacing the constraints with their charges and
promoting the Poisson bracket into Dirac bracket one obtains

$\displaystyle\\{Q(\lambda),Q(\xi)\\}^{*}=Q(\sigma)+K(\xi,\lambda).$ (8.9)

We are going to consider the simple bcs for which one can integrate the above
expression (8.5), and parameter $\xi_{\mu}$ depends linearly on the fields

$\displaystyle\xi^{i}=-\xi^{i}_{\mu}\omega^{\mu}.$ (8.10)

####  8.1.1 Gauge fixing

The bcs that we are going to consider here, are going to require choosing the
radial component as

$\displaystyle\omega_{\rho}=b(\rho)^{-1}\partial_{\rho}b(\rho)$ (8.11)

where group element $b$ depends only on radial coordinate. This has been
referred to as partial fixing to radial gauge [27] and it has been standardly
used in the analysis of Chern-Simons action because it simplifies the later
analysis, and it is also a choice that is always possible [11]. In the both
cases of bcs that we are going to impose, we are going to use

$\displaystyle\omega=b(\rho)^{-1}(d+\Omega(t,\phi))b(\rho)$ (8.12)

form of the gauge field.

###  8.2 Appendix B

Here, we compare our solution for $so(3,2)$ conformal graviton in example 3,
to most general solution of Cotton tensor, of the form
$ds^{2}=\frac{dr^{2}}{f(r)}+(r^{2}n(r,t)^{2}-f(r))dt^{2}-2r^{2}n(r,t)dtd\phi+r^{2}d\phi^{2}$.
The solution reads

$\displaystyle f(r)$
$\displaystyle=c_{3}r^{2}+\frac{c_{2}{}^{2}}{4r^{2}}+c_{1}\left(\frac{c_{2}}{r}-c_{4}r\right)+c_{1}{}^{2}-\frac{c_{2}c_{4}}{2}$
(8.13a) $\displaystyle f(t)$ $\displaystyle=r^{2}n(r,t)^{2}-f(r)$ (8.13b)
$\displaystyle=\left(c_{3}{}^{2}+(-1+2c_{5})c_{3}+c_{5}{}^{2}\right)r^{2}+c_{1}(-2c_{3}+c_{4}-2c_{5})r+\frac{1}{2}c_{2}(-2c_{3}+c_{4}-2c_{5})$
(8.13c) $\displaystyle n(r,t)$
$\displaystyle=\frac{1}{2r^{2}}(-2(c_{3}+c_{5})r^{2}+2c_{1}r+c_{2})$ (8.13d)

and it is equivalent to the solution of the example 2 for the

$\displaystyle c_{1}$ $\displaystyle\to-2\mathcal{K}\mathcal{K}_{d}q_{1}$
(8.14a) $\displaystyle c_{2}$ $\displaystyle\to 2\mathcal{K}_{d}\mathcal{K}$
(8.14b) $\displaystyle c_{3}$
$\displaystyle\to\mathcal{K}^{2}\mathcal{K}_{d}^{2}q_{1}^{4}-\mathcal{K}^{2}q_{1}^{2}-\mathcal{K}_{d}^{2}q_{1}^{2}+1$
(8.14c) $\displaystyle c_{4}$
$\displaystyle\to-\frac{\mathcal{K}^{2}+\mathcal{K}_{d}^{2}-2\mathcal{K}^{2}\mathcal{K}_{d}^{2}q_{1}^{2}}{\mathcal{K}\mathcal{K}_{d}}$
(8.14d) $\displaystyle c_{5}$
$\displaystyle\to-\mathcal{K}^{2}\mathcal{K}_{d}^{2}q_{1}^{4}+q_{1}^{2}\left(\mathcal{K}^{2}-\mathcal{K}\mathcal{K}_{d}+\mathcal{K}_{d}^{2}\right)+\frac{(\mathcal{K}-\mathcal{K}_{d})(\mu+\mu_{d})}{\mathcal{K}_{d}\mu-\mathcal{K}\mu_{d}}+q_{2}.$
(8.14e)

Here, the function $f(r)$ agrees with the one from the example 2 for
$\frac{\sqrt{2\mathcal{K}^{2}+\mathcal{K}_{d}^{2}+\mathcal{K}_{d}}}{2\mathcal{K}\mathcal{K}_{d}}$.
The function $f(r)$ has null points for the

$\displaystyle
r_{1\pm}\to\frac{c_{1}\left(c_{4}-\sqrt{c_{4}{}^{2}-4c_{3}}\right)\pm\sqrt{2\left(c_{4}{}^{2}-\sqrt{c_{4}{}^{2}-4c_{3}}c_{4}-2c_{3}\right)c_{1}{}^{2}+4c_{2}c_{3}\left(c_{4}-\sqrt{c_{4}{}^{2}-4c_{3}}\right)}}{4c_{3}}$
(8.15a) $\displaystyle
r_{2\pm}\to\frac{c_{1}\left(c_{4}+\sqrt{c_{4}{}^{2}-4c_{3}}\right)\pm\sqrt{2\left(c_{4}\left(c_{4}+\sqrt{c_{4}{}^{2}-4c_{3}}\right)-2c_{3}\right)c_{1}{}^{2}+4c_{2}c_{3}\left(c_{4}+\sqrt{c_{4}{}^{2}-4c_{3}}\right)}}{4c_{3}}$
(8.15b)

which shows that one obtained the solution which can have complex eigenvalues,
depending on the coefficients $c_{1},c_{2},...,c_{5}$. That makes it belong to
type $I_{a}$ from the classification according to one parameter subgroups of
$so(3,2)$.

###  8.3 Appendix C

The $sl(n)$ generators are in our convention represented by NxN matrices

$\displaystyle(L_{1})_{jk}=-\sqrt{j(N-j)}\delta_{j+1,k}$ (8.16)
$\displaystyle(L_{-1})_{jk}=\sqrt{k(N-k)}\delta_{j,k+1}$ (8.17)
$\displaystyle(L_{0})_{jk}=\frac{1}{2}(N+1-2j)\delta_{j,k}$ (8.18)

where $j,k=1,..,N$ ($j+1,k+1=2,..N+1$) while the remaining $W$ generators are
defined by

$\displaystyle
W_{m}^{(s)}=2(-1)^{s-m-1}\frac{(s+m-1)!}{(2s-2)!}[L_{-1},[L_{-1}...[L_{-1},(L_{1})^{s-1}]..]]$
(8.19)

with $s-m-1$ as a number of commutations. This basis we call WL basis.

In following we define $V_{m}^{4}\equiv W^{4}_{m}$ and $V_{n}^{3}\equiv
W^{3}_{n}$. With the help of basis of generators for the principal embedding
of spin-4 gravity [27], $sl(2)$ generators

$\displaystyle L_{0}$
$\displaystyle=\frac{1}{2}\left(\begin{array}[]{cccc}3&0&0&0\\\ 0&1&0&0\\\
0&0&-1&0\\\ 0&0&0&-3\end{array}\right),$ $\displaystyle
L_{-1}=\left(\begin{array}[]{cccc}0&0&0&0\\\ \sqrt{3}&0&0&0\\\ 0&2&0&0\\\
0&0&\sqrt{3}&0\end{array}\right),$ $\displaystyle
L_{1}=\left(\begin{array}[]{cccc}0&-\sqrt{3}&0&0\\\ 0&0&-2&0\\\
0&0&0&-\sqrt{3}\\\ 0&0&0&0\\\ \end{array}\right),$ (8.32)

quintet

$\displaystyle V^{3}_{0}$ $\displaystyle=\left(\begin{array}[]{cccc}2&0&0&0\\\
0&-2&0&0\\\ 0&0&-2&0\\\ 0&0&0&2\\\ \end{array}\right),$ $\displaystyle
V^{3}_{1}=\left(\begin{array}[]{cccc}0&2\sqrt{3}&0&0\\\ 0&0&0&0\\\
0&0&0&-2\sqrt{3}\\\ 0&0&0&0\\\ \end{array}\right),$ $\displaystyle
V^{3}_{-1}=\left(\begin{array}[]{cccc}0&0&0&0\\\ -2\sqrt{3}&0&0&0\\\
0&0&0&0\\\ 0&0&2\sqrt{3}&0\\\ \end{array}\right),$ (8.45) $\displaystyle
V^{3}_{2}$ $\displaystyle=\left(\begin{array}[]{cccc}0&0&4\sqrt{3}&0\\\
0&0&0&4\sqrt{3}\\\ 0&0&0&0\\\ 0&0&0&0\\\ \end{array}\right),$ $\displaystyle
V^{3}_{-2}=\left(\begin{array}[]{cccc}0&0&0&0\\\ 0&0&0&0\\\ 4\sqrt{3}&0&0&0\\\
0&4\sqrt{3}&0&0\\\ \end{array}\right)$ (8.54)

and septet

$\displaystyle V^{4}_{2}$
$\displaystyle=\left(\begin{array}[]{cccc}0&0&-2\sqrt{3}&0\\\
0&0&0&2\sqrt{3}\\\ 0&0&0&0\\\ 0&0&0&0\\\ \end{array}\right),$ $\displaystyle
V^{4}_{-2}=\left(\begin{array}[]{cccc}0&0&0&0\\\ 0&0&0&0\\\
-2\sqrt{3}&0&0&0\\\ 0&2\sqrt{3}&0&0\\\ \end{array}\right),$ (8.63)
$\displaystyle V^{4}_{0}$
$\displaystyle=\left(\begin{array}[]{cccc}-\frac{3}{5}&0&0&0\\\
0&\frac{9}{5}&0&0\\\ 0&0&-\frac{9}{5}&0\\\ 0&0&0&\frac{3}{5}\\\
\end{array}\right),$ $\displaystyle
V^{4}_{1}=\frac{-4}{5}\left(\begin{array}[]{cccc}0&\sqrt{3}&0&0\\\ 0&0&-3&0\\\
0&0&0&\sqrt{3}\\\ 0&0&0&0\\\ \end{array}\right),$ $\displaystyle
V^{4}_{-1}=\frac{4}{5}\left(\begin{array}[]{cccc}0&0&0&0\\\ \sqrt{3}&0&0&0\\\
0&-3&0&0\\\ 0&0&\sqrt{3}&0\\\ \end{array}\right),$ (8.76) $\displaystyle
V^{4}_{-3}$ $\displaystyle=\left(\begin{array}[]{cccc}0&0&0&0\\\ 0&0&0&0\\\
0&0&0&0\\\ 12&0&0&0\end{array}\right),$ $\displaystyle
V^{4}_{3}=\left(\begin{array}[]{cccc}0&0&0&-12\\\ 0&0&0&0\\\ 0&0&0&0\\\
0&0&0&0\end{array}\right),$ (8.85)

and generators in our PKLD basis

$\displaystyle P_{t}$
$\displaystyle=\left(\begin{array}[]{cccc}0&0&-\frac{1}{2}&0\\\
0&0&0&\frac{1}{2}\\\ 0&0&0&0\\\ 0&0&0&0\\\
\end{array}\right),P_{y}=\left(\begin{array}[]{cccc}0&0&\frac{1}{2}&0\\\
0&0&0&\frac{1}{2}\\\ 0&0&0&0\\\ 0&0&0&0\\\
\end{array}\right),P_{x}=\left(\begin{array}[]{cccc}0&0&0&1\\\
0&0&-\frac{1}{4}&0\\\ 0&0&0&0\\\ 0&0&0&0\\\
\end{array}\right),D=\left(\begin{array}[]{cccc}-\frac{1}{2}&0&0&0\\\
0&-\frac{1}{2}&0&0\\\ 0&0&\frac{1}{2}&0\\\ 0&0&0&\frac{1}{2}\\\
\end{array}\right)$ (8.102) $\displaystyle K_{t}$
$\displaystyle=\left(\begin{array}[]{cccc}0&0&0&0\\\ 0&0&0&0\\\ -1&0&0&0\\\
0&1&0&0\\\ \end{array}\right),K_{y}=\left(\begin{array}[]{cccc}0&0&0&0\\\
0&0&0&0\\\ -1&0&0&0\\\ 0&-1&0&0\\\
\end{array}\right),K_{x}=\left(\begin{array}[]{cccc}0&0&0&0\\\ 0&0&0&0\\\
0&2&0&0\\\ -\frac{1}{2}&0&0&0\\\
\end{array}\right),L_{ty}=\left(\begin{array}[]{cccc}-\frac{1}{2}&0&0&0\\\
0&\frac{1}{2}&0&0\\\ 0&0&\frac{1}{2}&0\\\ 0&0&0&-\frac{1}{2}\\\
\end{array}\right)$ (8.119) $\displaystyle L_{tx}$
$\displaystyle=\left(\begin{array}[]{cccc}0&1&0&0\\\ \frac{1}{4}&0&0&0\\\
0&0&0&1\\\ 0&0&\frac{1}{4}&0\\\
\end{array}\right),L_{yx}=\left(\begin{array}[]{cccc}0&-1&0&0\\\
\frac{1}{4}&0&0&0\\\ 0&0&0&1\\\ 0&0&-\frac{1}{4}&0\\\
\end{array}\right),t_{+}=\left(\begin{array}[]{cccc}0&0&0&-1\\\
0&0&-\frac{1}{4}&0\\\ 0&0&0&0\\\ 0&0&0&0\\\
\end{array}\right),t_{-}=\left(\begin{array}[]{cccc}0&0&0&0\\\ 0&0&0&0\\\
0&-2&0&0\\\ -\frac{1}{2}&0&0&0\\\ \end{array}\right)$ (8.136) $\displaystyle
t_{t}$ $\displaystyle=\left(\begin{array}[]{cccc}0&1&0&0\\\
-\frac{1}{4}&0&0&0\\\ 0&0&0&1\\\ 0&0&-\frac{1}{4}&0\\\
\end{array}\right),t_{y}=\left(\begin{array}[]{cccc}0&-1&0&0\\\
-\frac{1}{4}&0&0&0\\\ 0&0&0&1\\\ 0&0&\frac{1}{4}&0\\\
\end{array}\right),t_{x}=\left(\begin{array}[]{cccc}-\frac{1}{2}&0&0&0\\\
0&\frac{1}{2}&0&0\\\ 0&0&-\frac{1}{2}&0\\\ 0&0&0&\frac{1}{2}\\\
\end{array}\right)$ (8.149)

which closes algebra of the vector field and conformal spin-2, for the
$\eta_{ab}=diag(-1,1,1)$ for (t,y,x), we can determine the map

$\displaystyle{\boldsymbol{L}}_{-1}$ $\displaystyle\to
2\left(\frac{1}{4}{\boldsymbol{K}}_{x}+\sqrt{3}\left({\boldsymbol{L}}_{tx}-{\boldsymbol{t}}_{t}\right)-\frac{1}{4}{\boldsymbol{t}}_{-}\right),$
$\displaystyle{\boldsymbol{L}}_{0}\to\left(2{\boldsymbol{D}}+{\boldsymbol{t}}_{x}\right),$
(8.150a) $\displaystyle{\boldsymbol{L}}_{1}$
$\displaystyle\to\frac{1}{2}\left(-\sqrt{3}{\boldsymbol{L}}_{tx}+8{\boldsymbol{P}}_{x}+8t_{+}-\sqrt{3}{\boldsymbol{t}}_{t}\right),$
$\displaystyle V^{3}{}_{-2}\to-4\sqrt{3}{\boldsymbol{K}}_{y},$ (8.150b)
$\displaystyle V^{3}{}_{0}$ $\displaystyle\to-4{\boldsymbol{L}}_{ty},$
$\displaystyle
V^{3}{}_{1}\to-\sqrt{3}({\boldsymbol{L}}_{yx}+{\boldsymbol{t}}_{y}),$ (8.150c)
$\displaystyle V^{3}{}_{2}$ $\displaystyle\to 8\sqrt{3}{\boldsymbol{P}}_{y},$
$\displaystyle
V^{4}{}_{-3}\to-12\left({\boldsymbol{K}}_{x}+{\boldsymbol{t}}_{-}\right),$
(8.150d) $\displaystyle V^{4}{}_{-2}$ $\displaystyle\to
2\sqrt{3}{\boldsymbol{K}}_{t},$ $\displaystyle
V^{4}{}_{-1}\to\frac{1}{5}\left(-3{\boldsymbol{K}}_{x}+8\sqrt{3}\left({\boldsymbol{L}}_{yx}-{\boldsymbol{t}}_{x}\right)+3{\boldsymbol{t}}_{-}\right),$
(8.150e) $\displaystyle V^{4}{}_{0}$
$\displaystyle\to\frac{6}{5}\left(2{\boldsymbol{t}}_{x}-{\boldsymbol{D}}\right),$
$\displaystyle
V^{4}{}_{1}\to\frac{2}{5}\left(-\sqrt{3}{\boldsymbol{L}}_{tx}-12{\boldsymbol{P}}_{x}-12{\boldsymbol{t}}_{+}-\sqrt{3}{\boldsymbol{t}}_{t}\right),$
(8.150f) $\displaystyle V^{4}{}_{2}$ $\displaystyle\to
4\sqrt{3}{\boldsymbol{P}}_{t},$ $\displaystyle V^{4}{}_{3}\to
6\left({\boldsymbol{t}}_{+}-{\boldsymbol{P}}_{x}\right)$ (8.150g)
$\displaystyle V^{3}{}_{-1}$ $\displaystyle\to
4\sqrt{3}({\boldsymbol{t}}_{y}-{\boldsymbol{L}}_{yx}).$ (8.150h)

The matrices (8.149) are related to standard construction using $\gamma$
matrices $\gamma_{1},\gamma_{2},\gamma_{3},\gamma_{4},\gamma_{5}$ in Majorana
basis via following. We obtain 9 linearly independent matrices for

$\displaystyle v_{1}$ $\displaystyle=[\gamma_{1},\gamma_{4}]/2,$
$\displaystyle v_{2}=[\gamma_{2},\gamma_{3}]/2,$ $\displaystyle
v_{3}=[\gamma_{5},\gamma_{1}]/2,$ $\displaystyle
v_{4}=[\gamma_{1},\gamma_{3}]/2,$ $\displaystyle
v_{5}=[\gamma_{2},\gamma_{4}]/2,$ $\displaystyle v_{6}$
$\displaystyle=[\gamma_{2},\gamma_{5}]/2,$ $\displaystyle
v_{7}=[\gamma_{1},\gamma_{2}]/2,$ $\displaystyle
v_{8}=[\gamma_{3},\gamma_{4}]/2,$ $\displaystyle
v_{9}=[\gamma_{3},\gamma_{5}]/2,$ (8.151)

multiplying which we construct additional 6 ones

$\displaystyle v_{10}$ $\displaystyle=v_{7}.v_{9},$ $\displaystyle
v_{11}=v_{7}.v_{8},$ $\displaystyle v_{12}=v_{5}.v_{6},$ $\displaystyle
v_{13}=v_{2}.v_{12},$ $\displaystyle v_{14}=v_{4}.v_{12},$ $\displaystyle
v_{15}=v_{7}.v_{12}.$ (8.152)

Together they define the (8.149)

$\displaystyle P_{t}$ $\displaystyle=\frac{v_{1}}{4}-\frac{v_{8}}{4},$
$\displaystyle P_{y}=\frac{v_{3}}{4}+\frac{v_{9}}{4}$ $\displaystyle
P_{x}=\frac{1}{16}\left(-3v_{2}-3v_{7}-5\left(v_{13}+v_{15}\right)\right)$
(8.153a) $\displaystyle D$ $\displaystyle=-\frac{v_{4}}{2},$ $\displaystyle
K_{t}=\frac{1}{2}\left(v_{1}+v_{8}\right)$ $\displaystyle
K_{x}=\frac{1}{8}\left(3v_{2}-3v_{7}-5v_{13}+5v_{15}\right)$ (8.153b)
$\displaystyle K_{y}$ $\displaystyle=\frac{1}{2}\left(v_{3}-v_{9}\right)$
$\displaystyle L_{ty}=-\frac{v_{12}}{2}$ $\displaystyle
t_{+}=\frac{1}{16}\left(5v_{2}+5v_{7}+3\left(v_{13}+v_{15}\right)\right)$
(8.153c) $\displaystyle L_{tx}$
$\displaystyle=\frac{1}{8}\left(3v_{5}-5v_{10}\right)$ $\displaystyle
L_{yx}=\frac{1}{8}\left(5v_{11}-3v_{6}\right)$ $\displaystyle
t_{-}=\frac{1}{8}\left(-5v_{2}+5v_{7}+3v_{13}-3v_{15}\right)$ (8.153d)
$\displaystyle t_{t}$ $\displaystyle=\frac{1}{8}\left(5v_{5}-3v_{10}\right)$
$\displaystyle t_{y}=\frac{1}{8}\left(3v_{11}-5v_{6}\right)$ $\displaystyle
t_{x}=-\frac{v_{14}}{2}$ (8.153e)

###  8.4 Appendix D

Using the identification $s_{ab}\equiv SL_{ab},s_{a+}\equiv
SP_{a},s_{a-}\equiv SK_{a},SD=s_{+-}$ the matrices that satisfy the
commutation relations (4.60), (4.61),(4.62), (4.63) and (4.11) for the
$\eta_{ab}=diag(\pm 1,1,1)$ can be written as:

$\displaystyle p_{x}$ $\displaystyle=\left(\begin{array}[]{ccccc}0&0&\mp
1&0&0\\\ 0&0&0&0&0\\\ 0&1&0&0&0\\\ 0&0&0&0&0\\\ 0&0&0&0&0\\\
\end{array}\right),p_{y}=\left(\begin{array}[]{ccccc}0&0&0&-1&0\\\
0&0&0&0&0\\\ 0&0&0&0&0\\\ 0&1&0&0&0\\\ 0&0&0&0&0\\\
\end{array}\right),p_{t}=\left(\begin{array}[]{ccccc}0&0&0&0&-1\\\
0&0&0&0&0\\\ 0&0&0&0&0\\\ 0&0&0&0&0\\\ 0&1&0&0&0\\\ \end{array}\right),$
(8.169) $\displaystyle k_{x}$
$\displaystyle=\left(\begin{array}[]{ccccc}0&0&0&0&0\\\ 0&0&\mp 1&0&0\\\
1&0&0&0&0\\\ 0&0&0&0&0\\\ 0&0&0&0&0\\\
\end{array}\right),k_{y}=\left(\begin{array}[]{ccccc}0&0&0&0&0\\\
0&0&0&-1&0\\\ 0&0&0&0&0\\\ 1&0&0&0&0\\\ 0&0&0&0&0\\\
\end{array}\right),k_{t}=\left(\begin{array}[]{ccccc}0&0&0&0&0\\\
0&0&0&0&-1\\\ 0&0&0&0&0\\\ 0&0&0&0&0\\\ 1&0&0&0&0\\\
\end{array}\right),D=-\left(\begin{array}[]{ccccc}1&0&0&0&0\\\ 0&-1&0&0&0\\\
0&0&0&0&0\\\ 0&0&0&0&0\\\ 0&0&0&0&0\\\ \end{array}\right),$ (8.190)
$\displaystyle L_{xy}$ $\displaystyle=\left(\begin{array}[]{ccccc}0&0&0&0&0\\\
0&0&0&0&0\\\ 0&0&0&1&0\\\ 0&0&\mp 1&0&0\\\ 0&0&0&0&0\\\
\end{array}\right),L_{xt}=\left(\begin{array}[]{ccccc}0&0&0&0&0\\\
0&0&0&0&0\\\ 0&0&0&0&1\\\ 0&0&0&0&0\\\ 0&0&\mp 1&0&0\\\
\end{array}\right),L_{yt}=\left(\begin{array}[]{ccccc}0&0&0&0&0\\\
0&0&0&0&0\\\ 0&0&0&0&0\\\ 0&0&0&0&1\\\ 0&0&0&-1&0\\\ \end{array}\right)$
(8.206)

$\displaystyle SP_{x}=$ $\displaystyle\left(\begin{array}[]{ccccc}0&0&\pm
1&0&0\\\ 0&0&0&0&0\\\ 0&1&0&0&0\\\ 0&0&0&0&0\\\ 0&0&0&0&0\\\
\end{array}\right),SP_{y}=\left(\begin{array}[]{ccccc}0&0&0&1&0\\\
0&0&0&0&0\\\ 0&0&0&0&0\\\ 0&1&0&0&0\\\ 0&0&0&0&0\\\
\end{array}\right),SP_{t}=\left(\begin{array}[]{ccccc}0&0&0&0&1\\\
0&0&0&0&0\\\ 0&0&0&0&0\\\ 0&0&0&0&0\\\ 0&1&0&0&0\\\
\end{array}\right),SK_{x}=\left(\begin{array}[]{ccccc}0&0&0&0&0\\\ 0&0&\pm
1&0&0\\\ 1&0&0&0&0\\\ 0&0&0&0&0\\\ 0&0&0&0&0\\\ \end{array}\right),$ (8.227)
$\displaystyle SK_{y}$ $\displaystyle=\left(\begin{array}[]{ccccc}0&0&0&0&0\\\
0&0&0&1&0\\\ 0&0&0&0&0\\\ 1&0&0&0&0\\\ 0&0&0&0&0\\\
\end{array}\right),SK_{t}=\left(\begin{array}[]{ccccc}0&0&0&0&0\\\
0&0&0&0&1\\\ 0&0&0&0&0\\\ 0&0&0&0&0\\\ 1&0&0&0&0\\\
\end{array}\right),SD=\left(\begin{array}[]{ccccc}\frac{3}{5}&0&0&0&0\\\
0&\frac{3}{5}&0&0&0\\\ 0&0&-\frac{2}{5}&0&0\\\ 0&0&0&-\frac{2}{5}&0\\\
0&0&0&0&-\frac{2}{5}\\\
\end{array}\right),SL_{xy}=\left(\begin{array}[]{ccccc}0&0&0&0&0\\\
0&0&0&0&0\\\ 0&0&0&1&0\\\ 0&0&\pm 1&0&0\\\ 0&0&0&0&0\\\ \end{array}\right),$
(8.248) $\displaystyle SL_{xt}$
$\displaystyle=\left(\begin{array}[]{ccccc}0&0&0&0&0\\\ 0&0&0&0&0\\\
0&0&0&0&1\\\ 0&0&0&0&0\\\ 0&0&\pm 1&0&0\\\
\end{array}\right),SL_{yt}=\left(\begin{array}[]{ccccc}0&0&0&0&0\\\
0&0&0&0&0\\\ 0&0&0&0&0\\\ 0&0&0&0&1\\\ 0&0&0&1&0\\\
\end{array}\right),s_{++}=\left(\begin{array}[]{ccccc}0&2&0&0&0\\\
0&0&0&0&0\\\ 0&0&0&0&0\\\ 0&0&0&0&0\\\ 0&0&0&0&0\\\
\end{array}\right),s_{--}=\left(\begin{array}[]{ccccc}0&0&0&0&0\\\
2&0&0&0&0\\\ 0&0&0&0&0\\\ 0&0&0&0&0\\\ 0&0&0&0&0\\\ \end{array}\right),$
(8.269) $\displaystyle SL_{xx}$
$\displaystyle=\left(\begin{array}[]{ccccc}\mp\frac{2}{5}&0&0&0&0\\\
0&\mp\frac{2}{5}&0&0&0\\\ 0&0&\pm\frac{8}{5}&0&0\\\ 0&0&0&\mp\frac{2}{5}&0\\\
0&0&0&0&\mp\frac{2}{5}\\\
\end{array}\right),SL_{yy}=\left(\begin{array}[]{ccccc}-\frac{2}{5}&0&0&0&0\\\
0&-\frac{2}{5}&0&0&0\\\ 0&0&-\frac{2}{5}&0&0\\\ 0&0&0&\frac{8}{5}&0\\\
0&0&0&0&-\frac{2}{5}\\\
\end{array}\right),SL_{tt}=\left(\begin{array}[]{ccccc}-\frac{2}{5}&0&0&0&0\\\
0&-\frac{2}{5}&0&0&0\\\ 0&0&-\frac{2}{5}&0&0\\\ 0&0&0&-\frac{2}{5}&0\\\
0&0&0&0&\frac{8}{5}\\\ \end{array}\right)$ (8.285)

They also satisfy the condition $2SD=-(SL_{xx}+SL_{yy}+SL_{tt})$. The map to
write the WL basis for sl(5) example in the terms of the PKLD matrices (8.206)
and (8.285) is

$\displaystyle L_{-1}$
$\displaystyle\to\text{SL}_{\text{--}}-\sqrt{\frac{3}{2}}L_{\text{xy}}-L_{\text{yt}}+\sqrt{\frac{3}{2}}P_{x}+\sqrt{\frac{3}{2}}\text{SL}_{\text{xy}}+\text{SL}_{\text{yt}}+\sqrt{\frac{3}{2}}\text{SP}_{x},$
(8.286a) $\displaystyle L_{0}$
$\displaystyle\to\frac{1}{2}\left(-D+\frac{7}{2}\text{SL}_{\text{tt}}+\frac{5}{2}\text{SL}_{\text{yy}}+\frac{3}{2}\text{SL}_{\text{xx}}\right),$
(8.286b) $\displaystyle L_{1}$
$\displaystyle\to-\text{SL}_{\text{++}}+\sqrt{\frac{3}{2}}K_{x}-\sqrt{\frac{3}{2}}L_{\text{xy}}-L_{\text{yt}}-\sqrt{\frac{3}{2}}\text{SK}_{t}-\sqrt{\frac{3}{2}}\text{SL}_{\text{xy}}-\text{SL}_{\text{yt}},$
(8.286c)