# BPS defects in AdS3 supergravity

Gaston Giribet1, , Olivera Mišković2, , Nahuel Yazbek3, , Jorge Zanelli4,5

1 Department of Physics, New York University,
726 Broadway, New York City, 10003, USA.
2 Instituto de Física, Pontificia Universidad Católica de Valparaíso,
Avda. Universidad 330, Curauma, Valparaíso, Chile.
3 Instituto de Matemática Luis A. Santaló, CONICET-UBA,
Ciudad Universitaria, Pabellón I, CABA, C1428ZAA, Argentina.
4 Universidad San Sebastián, General Lagos 1163, Valdivia, Chile.
5 Centro de Estudios Científicos (CECs), Arturo Prat 514, Valdivia, Chile.
gaston.giribet@nyu.eduolivera.miskovic@pucv.clnyazbek<EMAIL_ADDRESS>
###### Abstract

AdS supergravity admits supersymmetric solutions that describe BPS defects.
Here, we investigate such solutions in AdS3 supergravity, which is formulated
as a Chern-Simons theory on $\mathrm{OSp}(2|1)\,\times\,\mathrm{OSp}(2|1)$. We
compute the Killing spinor equation on the BTZ geometry in different ways,
looking for BPS solutions on the entire space of parameters. We focus our
attention on defects that represent geometries with integer angular excesses;
these correspond to specific negative values of the BTZ mass. We compare our
solutions with other results in the literature, finding exact agreement. We
argue that, in the semiclassical limit, the BPS defects can be associated to
degenerate representations of the Virasoro symmetry at the boundary. The case
of non-diagonal representations, describing stationary, non-static defects, is
also discussed.

##  1 Introduction

The BTZ geometry [1] is a solution to Einstein equations with the negative
cosmological constant in $2+1$ dimension which, for a certain range of its two
parameters, describes an asymptotically anti-de Sitter (AdS) [2] black hole.
Being a three-dimensional Einstein space, the BTZ solution is locally
equivalent to AdS3 spacetime itself, which means that the former can be
constructed by identifications from the latter [3]. In particular, this
implies that, locally, the solution is of constant negative curvature, albeit
with a curvature $\delta$-like singularity at center [4, 5].

Similarly to the four-dimensional Kerr solution, the BTZ metric has two
integration constants, $M$ and $J$, which are the Noether charges associated
to the two Killing vectors that generate the $\mathbb{R}\times SO(2)$ isometry
and, consequently, they are interpreted as the mass and the angular momentum,
respectively. For the solution to represent a black hole, the constraints
$M\geq|J|/\ell\geq 0$ must hold, with $\ell$ being the curvature radius of
AdS3 space; extremal black holes correspond to $M=\pm J/\ell>0$. Stationary
BTZ black holes share many geometrical properties with their higher-
dimensional analogs: they exhibit an event horizon and an inner Killing
horizon, along with an ergosphere and a shielded singularity. It also shares
with higher-dimensional black holes their main thermodynamics properties, such
as finite Hawking temperature and a Bekenstein-Hawking entropy that obeys the
area law.

As noted in [6], the existence of these black holes with non-trivial
thermodynamic properties makes three-dimensional Einstein gravity a much more
exciting system, especially when studied in connection to AdS/CFT
correspondence [7]. Besides, the BTZ black hole appears in many other
scenarios: being locally equivalent to AdS3, it is also a solution to
supergravity [8], conformal gravity [9], other Chern-Simons actions [10],
string theory [11], topologically massive gravity [12], higher-curvature
gravity [12, 13], bi-gravity theory [14], and higher-spin theories [15]; see
also [16, 17, 18, 19, 20, 21, 22, 23, 24, 25].

The BTZ geometry describes a stationary black hole only for $M\geq|J|/\ell\geq
0$, but it also presents interesting features in other ranges of parameters.
For example, in the case $M+1/(8G)=J/\ell=0$, with $G$ the Newton constant,
BTZ reduces to global AdS3 spacetime. In the segment $0>M>-1/(8G)$ (and
$M<-|J|/\ell$), the solution can be regarded as point-like massive particles,
i.e., a naked conical singularity similar to the point-like particles of
three-dimensional gravity in flat space [26]. Less studied cases correspond to
the range $M\ell\leq-|J|$, $(M\ell)^{2}-J^{2}>1/(8G)^{2}$, where the solution
describes geometries with angular excesses around the origin. Here, we will be
concerned with the latter; we will consider solutions for the specific
negative values of $M+J/\ell$, such that they exhibit integer angular excesses
and special supersymmetric properties. From the AdS/CFT point of view, in the
semiclassical limit such geometries are associated to special non-normalizable
states in the dual CFT2. Static supersymmetric configurations with negative
$M$ correspond to degenerate representations in the dual CFT2, while
stationary non-static configurations with $M\pm J/\ell<0$ can be identified
with non-diagonal (spinfull) non-integrable representations of the type
studied by Migliaccio and Ribault in [27].

Solutions that represent naked singularities of integer angular excesses have
been recently considered in the literature, especially in connection to
higher-spin theories. In the higher-spin context they were introduced in [25],
where the way to characterize such defects as states with trivial Chern-Simons
holonomy was explained in detail. A clear discussion on the holographic
interpretation of those states appeared in [28], and the identification
between conical solutions and primaries in the $W_{s}$ minimal models that
appear in spin-$s$ theories was revisited in [29]. The role of integer angular
excess solutions for the $s=2$ case was studied in [30] and more recently in
[31]; see also references thereof. Other interesting papers where the defects
and degenerate representations are discussed in the context of three-
dimensional gravity are [32, 33]. There are also interesting works in two
dimensions that, to some extent, are related to this, e.g. [34, 35, 36];
however, the relation to the uplift to dimension three is not totally obvious
to us. Here, we will be concerned with naked singularities of arbitrary values
of the parameters in three-dimensional supergravity. We will show that those
geometries which exhibit integer angular excesses are the only BPS states
appearing in the negative mass sector of the BTZ geometry [4]. It is well-
known that the positive mass sector admits supersymmetric solutions: the
massless BTZ solution $M=J=0$ has two exact supersymmetries, while the
extremal solutions with $M=\pm J/\ell\neq 0$ have only one [8]. These,
together with the AdS3 vacuum, are the only solutions of positive mass with
supersymmetry.

Here we want to investigate the BPS solutions in the negative mass sector: we
will solve the spinor Killing equation in the stationary geometry and look for
globally well-defined solutions that preserve at least one supersymmetry. The
latter would represent BPS solutions with naked singularities. BPS solutions
with naked singularities in AdS supergravity are known to exist in other
models, e.g., the Romans solution of $\mathcal{N}=2$ supergravity in AdS4 is
$\frac{1}{2}$ BPS [37]. Here, we will identify similar solutions in AdS3. We
will study the case of Chern-Simons theory for the
$\mathrm{OSp}(2|1)\times\mathrm{OSp}(2|1)$ supergroup, which realizes a
supersymmetric extension of AdS3 algebra with $\mathcal{N}=2$ supercharges
[43].111As well-known, three-dimensional Einstein gravity in AdS space is
equivalent, at the level of actions, to Chern-Simons gravity for
AdS${}_{3}\simeq\mathrm{SO(2,2)}$. This equivalence includes boundary terms,
where Chern-Simons action provides correct surface terms which make AdS
gravity finite [44]. Negative mass BPS states were also studied in [38] in the
context of (2,0) AdS3 supergravity, where the results in the full agreement
were found. Such BPS defects, on the other hand, can be seen to persist in the
flat space limit [39]. As we will argue, in the AdS3 case these defects are
associated to degenerate representations in the dual CFT2, which in the
semiclassical limit of three-dimensional gravity can be associated to an
effective Liouville field theory [40]; see also [41, 42].

The paper is organized as follows: In section 2, we review the Chern-Simons
theory that describes AdS3 supergravity, and we introduce our notation and
conventions. In section 3, we solve the Killing spinor equation in the
negative mass sector of the BTZ geometry, both for the static case and the
stationary case. We show that BPS solutions describing defects with integer
angular excesses do exist and are distributed all over the $(M,J)$ plane of
the BTZ spectrum. In section 4, we provide an alternative derivation of the
Killing spinor equation and compare our solutions with other results in the
literature. As a final discussion, in section 5 we make some comments about
the holographic description of the BPS defects, whose distribution in the
$(M,J)$ plane can be associated with the semiclassical limit of degenerate
representations of the effective Liouville description at the boundary. We
will also discuss the case of non-diagonal representations, describing
stationary BPS solutions in the bulk.

##  2 Chern-Simons supergravity

The action of three-dimensional supergravity in (A)dS can be written as a
Chern-Simons (CS) theory. The level is $k=\ell/(4G)$, with $\ell$ being the
curvature radius of the space and $G$ is the Newton constant (hereafter we
will take $G=1/8$ unless otherwise stated).

Consider a supersymmetric extension of AdS3 algebra with $\mathcal{N}=2$
supercharges, given by $\mathrm{OSp}(2|1)\times\mathrm{OSp}(2|1)$
superalgebra222The notation used in this text is consistent with [8] and [45].
[43]:

$[J_{a}^{\pm},J_{b}^{\pm}]={{\varepsilon_{ab}}^{c}\,J^{\pm}_{c}}\,,\qquad[J_{a}^{\pm},Q_{\alpha}^{\pm}]=-\frac{1}{2}\,(\Gamma_{a})_{\alpha}^{\
\beta}\,Q_{\beta}^{\pm}\,,\qquad\\{Q_{\alpha}^{\pm},Q_{\beta}^{\pm}\\}=(C\Gamma^{a})_{\alpha\beta}\,J_{a}^{\pm}\,.$
(2.1)

Here, $a,b,c=0,1,2$ are Lorentz indices; $\alpha,\beta=1,2$; and $\pm$ refer
to two commuting copies of the superalgebra. The corresponding gauge field
($A=A_{\mu}dx^{\mu}$) is expressed in terms of the dreibein
($e^{a}=e^{a}_{\mu}dx^{\mu}$), the spin connection
($\omega^{ab}=\omega^{ab}_{\ \mu}dx^{\mu}$) and the algebra generators as
follows

$A=\left(\omega^{a}+\frac{1}{\ell}\,e^{a}\right)J_{a}^{+}+\left(\omega^{a}-\frac{1}{\ell}\,e^{a}\right)J_{a}^{-}+\frac{1}{\sqrt{\ell}}\,\left(\xi_{+}^{\alpha}Q_{\alpha}^{+}+\xi_{-}^{\alpha}Q_{\alpha}^{-}\right)\,,$
(2.2)

where $\omega^{ab}=-\varepsilon^{abc}\,\omega_{c}$,
$\omega_{a}=\frac{1}{2}\,\varepsilon_{abc}\omega^{bc}$ and
$J^{ab}=\varepsilon^{abc}\,J_{c}$,
$J_{a}=-\frac{1}{2}\,\varepsilon_{abc}\,J^{bc}$. The torsion and curvature
2-forms are

$T^{a}=De^{a}\,,\qquad
R_{a}=\frac{1}{2}\,\varepsilon_{abc}R^{bc}=d\omega^{a}+\frac{1}{2}\,\varepsilon_{abc}\omega^{b}\omega^{c}\,,$
(2.3)

respectively. Covariant derivatives act on vectors and spinors as

$De^{a}=de^{a}+\varepsilon^{abc}\omega_{b}e_{c}\,,\qquad
D\psi=d\psi-\frac{1}{2}\,\theta\omega^{a}\Gamma_{a}\psi\,.$ (2.4)

We use the representation of Gamma matrices

$\Gamma_{0}=\mathrm{i}\theta\sigma_{1}\,,\qquad\Gamma_{1}=\theta\sigma_{2}\,,\qquad\Gamma_{2}=\theta\sigma_{3}\,,$
(2.5)

where $\theta=\pm 1$ corresponds to the two inequivalent representations of
the three-dimensional Clifford algebra
$\\{\Gamma_{a},\Gamma_{b}\\}=2\eta_{ab}$. The signature of Minkowski metric is
$\eta_{ab}=\mathrm{diag}(-,+,+)$, and the convention for the Levi-Civita
symbol is

$\varepsilon_{012}=-\varepsilon^{012}=1\,.$ (2.6)

With this notation, the spinorial representation of AdS3 generators becomes

$P_{a}=\frac{1}{2}\,\Gamma_{a}\,,\qquad
J_{a}=\frac{1}{2}\,\theta\,\Gamma_{a}\,,$ (2.7)

because

$J_{ab}=\frac{1}{4}\,[\Gamma_{a},\Gamma_{b}]=-\frac{1}{2}\,\theta\varepsilon_{abc}\,\Gamma^{c}\,.$
(2.8)

It also implies that a covariant derivative acts on a spinor as

$\displaystyle\nabla\psi=\left(d+\frac{1}{2}\,\omega^{ab}J_{ab}+\frac{1}{\ell}\,e^{a}P_{a}\right)\psi=D\psi+\frac{1}{2\ell}\,e^{a}\Gamma_{a}\psi\,,$
(2.9)

where the Lorentz-covariant derivative is given in (2.4). The extra $\theta$
does not affect the vectorial representation of the generators, i.e., the
definition of $De^{a}$.

As regards the spacetime, the BTZ geometry is described by the metric

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-f^{2}dt^{2}+\frac{dr^{2}}{f^{2}}+r^{2}\left(Ndt+d\varphi\right)^{2},$
$\displaystyle f^{2}$ $\displaystyle=$
$\displaystyle-M+\frac{r^{2}}{\ell^{2}}+\frac{J^{2}}{4r^{2}}\,,\qquad
N=-\frac{J}{2r^{2}}\,,$ (2.10)

with $t\in\mathbb{R}$, $r\in\mathbb{R}_{\geq 0}$ and $\varphi\in[0,2\pi]$. As
it is well-known, depending on the parameters $(M,J)$ this geometry
corresponds to:

  * •

$M\geq|J|/\ell\geq 0$ : black hole with the mass $M\geq 0$ and the angular
momentum $J$;

  * •

$M=-1/(8G)$, $J=0$ : globally AdS3 space;

  * •

$M\ell\leq-|J|$, $(M\ell)^{2}-J^{2}<(8G)^{-2}$ : naked singularity with
angular deficit (point-like particle);

  * •

$M\ell\leq-|J|$, $(M\ell)^{2}-J^{2}>(8G)^{-2}$ : naked singularity with
angular excess;

  * •

$|M|\ell<|J|$: over-spinning geometries.

Geometries with zero or negative mass parameter $M\neq-1$ describe topological
defects whose total angle in the plane is $2\pi\left(1-\alpha\right)$. Thus,
the parameter $\alpha$ measures a difference with respect to the space without
a defect, $\alpha=0$ corresponding to the AdS3 space. Furthermore, since the
angular deficit is introduced in a plane by Killing vector identifications,
for it to be a true manifold, a successive repetition of identifications has
always to finish at the same point. A consequence is that the deficit angle,
$\alpha$, has to be a fractional factor of $2\pi$, i.e.
$\alpha/(2\pi)\in\mathbb{Q}$. In the stationary case, two angular deficits
turn out to be associated to two rational numbers.

The coordinate frame where the static defect becomes explicit is

$ds^{2}=-\left(\frac{\rho^{2}}{\ell^{2}}+1\right)\,d\tau^{2}+\frac{d\rho^{2}}{\frac{\rho^{2}}{\ell^{2}}+1}+(1-\alpha)^{2}\rho^{2}d\varphi^{2}\,,$

where $0\leq\varphi\leq 2\pi$ is periodic.

Introducing the coordinates
$(t,r)=\left(\frac{\tau}{1-\alpha},(1-\alpha)\rho\right)$, the metric acquires
the form (2.10),

$ds^{2}=-\left(\frac{r^{2}}{\ell^{2}}+(1-\alpha)^{2}\right)dt^{2}+\frac{dr^{2}}{\frac{r^{2}}{\ell^{2}}+(1-\alpha)^{2}}+r^{2}d\varphi^{2}\,,$
(2.11)

such that we recognize the mass parameter

$M=-(1-\alpha)^{2}\,,\qquad J=0\,.$ (2.12)

Note that the conical defects have a conical singularity at $r=0$ in the
$(x^{1},x^{2})=(r\cos\varphi_{12}\,,r\sin\varphi_{12})$ plane $\Sigma_{12}$,
where $\varphi_{12}=2\pi\left(1-\alpha\right)\varphi$, such that [4]

$R^{ab}+\frac{1}{\ell^{2}}\,e^{a}e^{b}=2\pi\alpha\,\delta(\Sigma_{12})d\Omega_{12}\,J_{12}\,\eta^{[12][ab]}\,,\qquad
T^{a}=0\,,$ (2.13)

where $\eta^{[ab][cd]}=\eta^{ac}\eta^{bd}-\eta^{ad}\eta^{bc}$ is the invariant
tensor.

##  3 Killing spinors

###  3.1 The static case

It is well-known that BTZ black hole exhibits supersymmetry for specific
values of the parameters. More precisely,

  * -

the zero mass black hole ($M=J=0$) has two exact supersymmetries;

  * -

the vacuum of the Neveu-Schwarz sector is given by global AdS3 ($M=-1$,
$J=0$);

  * -

the extremal black holes ($0<M=|J|/\ell$) have only one supersymmetry;

  * -

the generic black holes ($0<M\neq|J|/\ell$) do not have any supersymmetry.

The solution with $M=J=0$ is the ground state of (1,1)-supergravity in AdS3
with Ramond boundary conditions on the spinor fields. Here, we will explore
the extension of this result to the range $M<0$ of the BTZ geometry, cf. [38].

Consider a static naked singularity with a conical excess or defect. Then, the
vielbein and a torsionless spin connection, with the notation for the Levi-
Civita symbol (2.6), can be chosen as follows,

$\begin{array}[]{llllll}e^{0}&=f\,dt\,,&e^{1}&=\dfrac{dr}{f}\,,&e^{2}&=r\,d\varphi\,,\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\omega^{0}&=f\,d\varphi\,,&\omega^{1}&=0\,,&\omega^{2}&=\dfrac{r}{\ell^{2}}\,dt\,,\end{array}$
(3.1)

where we used $f^{\prime}=\frac{r}{f\ell^{2}}$. Then, $\psi$ is a Killing
spinor if it is a globally well-defined solution of the Killing spinor
equation obtained from (2.4); namely, a solution where all fermions vanish,
$\xi=0$, has to be preserved under supersymmetry transformations,

$\delta_{\psi}\xi=0\quad\Rightarrow\quad
D\psi+\frac{1}{2\ell}\,\Gamma_{a}\,e^{a}\psi\equiv
d\psi-\frac{1}{2}\,\Gamma_{a}\left(\theta\omega^{a}-\frac{1}{\ell}\,e^{a}\right)\psi=0\,.$
(3.2)

It corresponds to a set of linear differential equations for two components of
the spinor $\psi$. These equations read

$0=d\psi+\frac{1}{2\ell}\left(dt-\theta\ell
d\varphi\right)\left(f\,\Gamma_{0}-\theta\,\frac{r}{\ell}\,\Gamma_{2}\right)\psi+\frac{1}{2\ell
f}\,\Gamma_{1}\psi\,dr\,,$ (3.3)

or, expressed in components,

$\displaystyle dt$ $\displaystyle:$ $\displaystyle\quad
0=\partial_{t}\psi+\frac{1}{2\ell}\left(f\,\Gamma_{0}-\theta\,\frac{r}{\ell}\,\Gamma_{2}\right)\psi\,,$
(3.4) $\displaystyle dr$ $\displaystyle:$ $\displaystyle\quad
0=\partial_{r}\psi+\frac{1}{2\ell f}\,\Gamma_{1}\psi\,,$ (3.5) $\displaystyle
d\varphi$ $\displaystyle:$ $\displaystyle\quad
0=\partial_{\varphi}\psi-\frac{\theta}{2}\,\left(f\,\Gamma_{0}-\theta\,\frac{r}{\ell}\,\Gamma_{2}\right)\psi\,.$
(3.6)

The equations along $dt$ and $d\varphi$ can be rewritten in terms of the
coordinates

$\displaystyle x^{\pm}$ $\displaystyle=$ $\displaystyle
t\pm\theta\ell\varphi\quad\Leftrightarrow\quad
t=\frac{x_{+}+x_{-}}{2}\,,\quad\varphi=\frac{x_{+}-x_{-}}{2\ell\theta}\,,$
$\displaystyle\partial_{\pm}$ $\displaystyle=$
$\displaystyle\frac{1}{2\ell}\,\left(\ell\partial_{t}\pm\theta\partial_{\varphi}\right)\,,$
(3.7)

where $x^{\pm}$ has dimensions of length in the natural units, such that

$\displaystyle dt+\theta\ell d\varphi$ $\displaystyle:$ $\displaystyle\quad
0=\partial_{+}\psi\quad\Rightarrow\quad\psi=\psi\left(x^{-},r\right)\,,$
$\displaystyle dt-\theta\ell d\varphi$ $\displaystyle:$ $\displaystyle\quad
0=\partial_{-}\psi+\frac{1}{2\ell}\left(f\,\Gamma_{0}-\theta\,\frac{r}{\ell}\,\Gamma_{2}\right)\psi\,.$
(3.8)

Taking second derivative of the last equation and using the identity
$\left(f\,\Gamma_{0}-\theta\,\frac{r}{\ell}\,\Gamma_{2}\right)^{2}=-f^{2}+\frac{r^{2}}{\ell^{2}}=M$,
we arrive at

$\partial_{-}^{2}\psi-\frac{M}{4\ell^{2}}\,\psi=0\,.$ (3.9)

The general solution reads

$\psi=\mathrm{e}^{-\frac{\mathrm{i}}{2\ell}\sqrt{-M}\,x^{-}}\chi_{1}(r)+\mathrm{e}^{\frac{\mathrm{i}}{2\ell}\sqrt{-M}\,x^{-}}\chi_{2}(r)\,.$
(3.10)

We have to plug (3.10) back into the second equation (3.8). Because of the
independence of $\mathrm{e}^{\frac{\mathrm{i}}{2\ell}\sqrt{-M}\,x^{-}}$ and
$\mathrm{e}^{-\frac{\mathrm{i}}{2\ell}\sqrt{-M}\,x^{-}}$, we obtain two
linearly independent algebraic equations for $\chi_{1}$ and $\chi_{2}$, where
the only difference is the sign of $\sqrt{-M}$, so it is enough to solve only
one of them,

$\left(\mathrm{i}\sqrt{-M}+\theta\,\frac{r}{\ell}\,\Gamma_{2}-f\Gamma_{0}\right)\chi_{1}=0\,.$
(3.11)

Using the representation of Gamma matrices (2.5), we solve
$\chi_{1}=\left(\begin{array}[]{c}\lambda_{1}\\\ \rho_{1}\end{array}\right)$
from

$\left(\begin{array}[]{cc}\mathrm{i}\sqrt{-M}+\frac{r}{\ell}&-\mathrm{i}\theta
f\\\ -\mathrm{i}\theta
f&\mathrm{i}\sqrt{-M}-\frac{r}{\ell}\end{array}\right)\left(\begin{array}[]{c}\lambda_{1}\\\
\rho_{1}\end{array}\right)=0\,.$ (3.12)

Note that the determinant of the above matrix vanishes, so there is only one
independent equation; namely

$\chi_{1}=\left(\begin{array}[]{c}\lambda_{1}\\\
-\frac{\mathrm{i\theta}}{f}\,\left(\mathrm{i}\sqrt{-M}+\frac{r}{\ell}\right)\lambda_{1}\end{array}\right).$
(3.13)

Finally, the radial equation (3.5) becomes

$0=\left(\begin{array}[]{cc}\partial_{r}&-\frac{\mathrm{i}\theta}{2\ell f}\\\
\frac{\mathrm{i}\theta}{2\ell
f}&\partial_{r}\end{array}\right)\left(\begin{array}[]{c}\lambda_{1}\\\
-\frac{\mathrm{i}\theta}{f}\,\left(\mathrm{i}\sqrt{-M}+\frac{r}{\ell}\right)\lambda_{1}\end{array}\right)\,.$
(3.14)

The first component yields

$\lambda_{1}(r)=\mathrm{e}^{h(r)}\eta_{1}\,,\qquad\eta_{1}=const\,,$ (3.15)

where $\eta_{1}$ is a constant Grassmann number, and

$\displaystyle h(r)$ $\displaystyle=$
$\displaystyle\frac{1}{2\ell}\int\frac{dr}{f^{2}}\,\left(\mathrm{i}\sqrt{-M}+\frac{r}{\ell}\right)=\frac{1}{2}\,\log\left(-\mathrm{i}\sqrt{-M}+\frac{r}{\ell}\right)\,.$
(3.16)

Thus, the radial part of the spinor becomes

$\mathrm{e}^{h(r)}=\left(-\mathrm{i}\sqrt{-M}+\frac{r}{\ell}\right)^{\frac{1}{2}}.$
(3.17)

Therefore, the spinor components are

$\displaystyle\lambda_{1}(r)$ $\displaystyle=$
$\displaystyle\left(-\mathrm{i}\sqrt{-M}+\frac{r}{\ell}\right)^{\frac{1}{2}}\,\eta_{1}\,,$
$\displaystyle\rho_{1}(r)$ $\displaystyle=$
$\displaystyle-\mathrm{i}\theta\left(\mathrm{i}\sqrt{-M}+\frac{r}{\ell}\right)^{\frac{1}{2}}\,\eta_{1}\,.$
(3.18)

The second component of the matrix equation (3.14) is identically satisfied
and the final solution reads

$\psi=\mathrm{e}^{-\frac{\mathrm{i}}{2\ell}\sqrt{-M}\,x^{-}}\left(\begin{array}[]{c}\left(-\mathrm{i}\sqrt{-M}+\frac{r}{\ell}\right)^{\frac{1}{2}}\\\
-\mathrm{i}\theta\left(\mathrm{i}\sqrt{-M}+\frac{r}{\ell}\right)^{\frac{1}{2}}\end{array}\right)\eta_{1}+\mathrm{e}^{\frac{\mathrm{i}}{2\ell}\sqrt{-M}\,x^{-}}\left(\begin{array}[]{c}\left(\mathrm{i}\sqrt{-M}+\frac{r}{\ell}\right)^{\frac{1}{2}}\\\
-\mathrm{i}\theta\left(-\mathrm{i}\sqrt{-M}+\frac{r}{\ell}\right)^{\frac{1}{2}}\end{array}\right)\eta_{2}\,.$
(3.19)

For large $r$, the spinor behaves as $\sqrt{\frac{r}{\ell}}$ for any $\theta$.
$M=0$ is special in the sense that there are no two independent spinors, as

$\psi=\sqrt{\frac{r}{\ell}}\left(\begin{array}[]{c}1\\\
-\mathrm{i}\theta\end{array}\right)\left(\eta_{1}+\eta_{2}\right).$ (3.20)

Solution (3.19) is globally well-defined when the angular part
$\mathrm{e}^{\pm\frac{\mathrm{i}}{2}\,\sqrt{-M}\varphi}$ has a period $\pi$
(antiperiodic spinor) or $2\pi$ (periodic spinor). Since the period of the
angular coordinate is $2\pi$, it means that $\frac{1}{2}\,\sqrt{-M}$ has to be
an integer or half-integer for a global well-definiteness. In sum,

$\sqrt{-M}=n\in\mathbb{Z}_{\geq 0}\quad\Rightarrow\quad M=-n^{2}\,,$ (3.21)

when the spinor becomes periodic (even $n$) or antiperiodic (odd $n$) in the
angle $\varphi$. Obtained BPS geometries are summarized as follows:

  * •

$n=0$, $M=0$ : massless black hole;

  * •

$n=1$, $M=-1$ : global AdS3 space;

  * •

$n\in\mathbb{Z}_{>1}$, $M=-n^{2}$ : naked singularity with integer angular
excess.

Recall that the massless black hole, $M=0$, is the Ramond vacuum, i.e., with
periodic boundary conditions; the solution $M=-1$ is the Neveu-Schwarz vacuum,
i.e., with anti-periodic boundary conditions, as the space is simply
connected. Remarkably, we found that we also have supersymmetric solutions
with negative values of the mass, $M=-n^{2}$, with $n\in\mathbb{Z}_{>1}$. The
latter are solutions with integer angular excesses around the origin. In
addition to these configurations, we will also find that there exist
stationary, non-static solutions that are $\frac{1}{2}$ BPS. Namely, in the
same way we have the well-known extremal black hole solutions with $M=\pm
J/\ell>0$, there are also supersymmetric solutions with negative values of
$M+J/\ell$. We will study the latter configurations in the following
subsection.

###  3.2 The stationary case

In the stationary case, the vielbein and a torsionless spin connection can be
chosen as

$\begin{array}[]{llllll}e^{0}&=f\,dt\,,&e^{1}&=\dfrac{dr}{f}\,,&e^{2}&=r(d\varphi+N\,dt\,),\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\omega^{0}&=fd\varphi\,,&\omega^{1}&=\dfrac{J}{2fr^{2}}\,dr\,,&\omega^{2}&=\dfrac{r}{\ell^{2}}\,dt-\dfrac{J}{2r}\,d\varphi\,,\end{array}$
(3.22)

where $f(r)$ and $N(r)$ are given in Eq. (2.10), and $(M,J)$ are the mass and
angular momentum of the naked singularity. To compute the spin connection
components, we have used
$f^{\prime}=\frac{r}{f\ell^{2}}-\frac{J^{2}}{4fr^{3}}$.

With the Gamma matrices representation (2.5), the Killing spinor equation
(3.2) now reads

$d\psi+\left[\frac{1}{2\ell}\left(\rule{0.0pt}{12.0pt}f\Gamma_{0}-\left(\frac{J}{2r}+\theta\,\frac{r}{\ell}\right)\Gamma_{2}\right)(dt-\theta\ell
d\varphi)-\frac{1}{2rf}\,\Gamma_{1}\left(\frac{\theta
J}{2r}-\frac{r}{\ell}\right)dr\right]\psi=0\,,$ (3.23)

or, expressed in components,

$\displaystyle dt$ $\displaystyle:$ $\displaystyle\quad
0=\partial_{t}\psi+\frac{1}{2\ell}\,\left[f\,\Gamma_{0}-\Gamma_{2}\left(\frac{J}{2r}+\theta\,\frac{r}{\ell}\right)\right]\psi\,,$
$\displaystyle dr$ $\displaystyle:$ $\displaystyle\quad
0=\partial_{r}\psi-\frac{1}{2rf}\,\Gamma_{1}\left(\dfrac{\theta
J}{2r}-\frac{r}{\ell}\right)\psi\,,$ $\displaystyle d\varphi$ $\displaystyle:$
$\displaystyle\quad
0=\partial_{\varphi}\psi-\frac{\theta}{2}\,\left[f\Gamma_{0}-\Gamma_{2}\left(\frac{J}{2r}+\theta\,\frac{r}{\ell}\right)\right]\psi.$
(3.24)

In the coordinates (3.7), these equations take the form

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\partial_{+}\psi\quad\Rightarrow\quad\psi=\psi\left(x^{-},r\right)\,,$
$\displaystyle 0$ $\displaystyle=$
$\displaystyle\partial_{-}\psi+\frac{1}{2\ell}\,\left[f\Gamma_{0}-\Gamma_{2}\left(\frac{J}{2r}+\theta\,\dfrac{r}{\ell}\right)\right]\psi\,,$
(3.25) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\partial_{r}\psi-\frac{1}{2rf}\,\Gamma_{1}\left(\dfrac{\theta
J}{2r}-\frac{r}{\ell}\right)\psi\,.$

Taking the second derivative in $\partial_{-}$ of the second equation leads to

$\displaystyle\partial_{-}^{2}\psi=\frac{1}{4\ell^{2}}\,\left[f\Gamma_{0}-\Gamma_{2}\left(\theta\,\dfrac{r}{\ell}+\frac{J}{2r}\right)\right]^{2}\psi=\frac{1}{4\ell^{2}}\,\left(M+\frac{\theta
J}{\ell}\right)\psi\,.$ (3.26)

For convenience, let us denote

$\omega=\sqrt{-M-\frac{\theta J}{\ell}}\,,\qquad\omega^{2}\geq
0\,\quad\mbox{or}\quad-M\ell\geq\theta J\,,$ (3.27)

where the positivity of $\omega^{2}$ is because we are interested in naked
singularities; black hole solutions were discussed in [8]. Therefore, we get

$\partial_{-}^{2}\psi+\frac{\omega^{2}}{4}\,\psi=0\,,$ (3.28)

leading to the general solution

$\psi\left(x^{-},r\right)=\mathrm{e}^{-\frac{\mathrm{i}}{2\ell}\,\omega\,x^{-}}\chi_{1}(r)+\mathrm{e}^{\frac{\mathrm{i}}{2\ell}\,\omega\,x^{-}}\chi_{2}(r)\,.$
(3.29)

We still have to impose that the second equation in (3.25) is satisfied.
Focusing on the positive-frequency component (the other one is obtained by
$\omega\rightarrow-\omega$), we write it as

$\left[\mathrm{i}\omega-f\Gamma_{0}+\Gamma_{2}\left(\theta\,\dfrac{r}{\ell}+\frac{J}{2r}\right)\right]\chi_{1}=0\,,$
(3.30)

or, equivalently, as

$\left(\begin{array}[]{cc}\mathrm{i}\omega+\frac{r}{\ell}+\frac{\theta
J}{2r}&-\mathrm{i}\theta f\\\ -\mathrm{i}\theta
f&\mathrm{i}\omega-\frac{r}{\ell}-\frac{\theta
J}{2r}\end{array}\right)\left(\begin{array}[]{c}\lambda_{1}\\\
\rho_{1}\end{array}\right)=0\,.$ (3.31)

The determinant of the system is zero, so there is a non trivial solution;
namely

$\chi_{1}=\left(\begin{array}[]{c}1\\\
-\frac{\mathrm{i}}{f}\left(\mathrm{i}\theta\omega+\theta\,\frac{r}{\ell}+\frac{J}{2r}\right)\end{array}\right)\lambda_{1}\,.$
(3.32)

It only remains to solve the radial equation in (3.25). We write it as

$0=\left(\begin{array}[]{cc}\partial_{r}&\frac{\mathrm{i}}{2rf}\left(\frac{J}{2r}-\theta\,\frac{r}{\ell}\right)\\\
-\frac{\mathrm{i}}{2rf}\left(\frac{J}{2r}-\theta\,\frac{r}{\ell}\right)&\partial_{r}\end{array}\right)\left(\begin{array}[]{c}\lambda_{1}\\\
-\frac{\mathrm{i}}{f}\left(\mathrm{i}\theta\omega+\theta\,\frac{r}{\ell}+\frac{J}{2r}\right)\lambda_{1}\end{array}\right),$
(3.33)

or, equivalently, as

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\partial_{r}\lambda_{1}+\frac{1}{2rf^{2}}\left(\frac{J}{2r}-\theta\,\frac{r}{\ell}\right)\left(\mathrm{i}\theta\omega+\theta\,\frac{r}{\ell}+\frac{J}{2r}\right)\lambda_{1}\,,$
$\displaystyle 0$ $\displaystyle=$
$\displaystyle\frac{1}{2rf}\left(\frac{J}{2r}-\theta\,\frac{r}{\ell}\right)\lambda_{1}+\partial_{r}\left[\frac{1}{f}\left(\mathrm{i}\theta\omega+\theta\,\frac{r}{\ell}+\frac{J}{2r}\right)\lambda_{1}\right].$
(3.34)

The first line can be solved exactly as

$\lambda_{1}(r)=\mathrm{e}^{h(r)}\eta_{1}\,,\qquad\eta_{1}=const\,,$ (3.35)

with $\eta_{1}$ being a constant Grassmann number. The radial part is given by
the function

$\displaystyle
h(r)=\frac{1}{2}\int\frac{dr}{rf^{2}}\left(\frac{r}{\ell}-\frac{\theta
J}{2r}\right)\left(\frac{r}{\ell}+\frac{\theta
J}{2r}+\mathrm{i}\omega\right)=\frac{1}{2}\,\log\left(\frac{r}{\ell}+\frac{\theta
J}{2r}+\mathrm{i}\omega\right)\,,$ (3.36)

where we applied the identity $f^{2}=\left(\frac{r}{\ell}+\frac{\theta
J}{2r}+\mathrm{i}\omega\right)\left(\frac{r}{\ell}+\frac{\theta
J}{2r}-\mathrm{i}\omega\right)$ in order to solve the integral. Therefore, the
radial part has the form

$\mathrm{e}^{h(r)}=\left(\frac{r}{\ell}+\frac{\theta
J}{2r}-\mathrm{i}\omega\right)^{\frac{1}{2}}\,.$ (3.37)

Again, it results proportional to $\sqrt{\frac{r}{\ell}}$ for large $r$.

The last equation to be checked is the second line of (3.34), which is
algebraic, and can be factorized as follows

$0=-\frac{1}{2rf^{3}}\left(\frac{J}{2r}-\frac{r}{\ell}\right)\left[f^{2}-2\left(\frac{r}{\ell}+\frac{J}{2r}\right)\left(\mathrm{i}\omega+\frac{r}{\ell}+\frac{J}{2r}\right)+\left(\mathrm{i}\omega+\frac{r}{\ell}+\frac{J}{2r}\right)^{2}\right].$
(3.38)

Replacing
$f^{2}=\omega^{2}+\frac{J}{\ell}+\frac{r^{2}}{\ell^{2}}+\frac{J^{2}}{4r^{2}}$
in the second factor, we easily observe that it vanishes identically.

The final solution is a $J\neq 0$ generalization of the static one (3.19), and
it reads

$\psi=\mathrm{e}^{-\frac{\mathrm{i}}{2\ell}\omega\,x^{-}}\left(\begin{array}[]{c}\left(\frac{r}{\ell}+\frac{\theta
J}{2r}-\mathrm{i}\omega\right)^{1/2}\\\
-\mathrm{i}\theta\left(\frac{r}{\ell}+\frac{\theta
J}{2r}+\mathrm{i}\omega\right)^{1/2}\end{array}\right)\eta_{1}+\mathrm{e}^{\frac{\mathrm{i}}{2\ell}\omega\,x^{-}}\left(\begin{array}[]{c}\left(\frac{r}{\ell}+\frac{\theta
J}{2r}+\mathrm{i}\omega\right)^{1/2}\\\
-\mathrm{i}\theta\left(\frac{r}{\ell}+\frac{\theta
J}{2r}-\mathrm{i}\omega\right)^{1/2}\end{array}\right)\eta_{2}\,.$ (3.39)

Similarly to the static case (3.21), this solution is globally well-defined
when the spinor is periodic or antiperiodic in the angle $x^{-}$, leading to
the condition

$\omega=n\in\mathbb{Z}_{\geq 0}\quad\Rightarrow\quad M+\frac{\theta
J}{\ell}=-n^{2}\,.$ (3.40)

##  4 Alternative derivations

###  4.1 Computing Killing spinors and separability

Now, let us solve the Killing spinor equation (3.25) in an alternative way
that, in particular, will allow us to make contact with the results of Ref.
[4]; see appendix therein. First, we solve the radial components in (3.25),
considering

$\psi=U(r)\,\Phi(x^{+},x^{-})\,,$ (4.1)

where $\Phi(x^{+},x^{-})$ is an arbitrary spinor and $U(r)$ is the following
invertible matrix,

$U(r)=\mathrm{e}^{-\beta(r)\Gamma_{1}}=\cosh\beta-\Gamma_{1}\sinh\beta\,,$
(4.2)

with the radial part given by333Introducing the variable
$z=\frac{r}{\ell}+\frac{\theta J}{2r}+\mathrm{i}\omega$ in
$f^{2}=z^{2}+\omega^{2}$, the radial integral can be solved as
$\beta=\frac{1}{2}\int\frac{dz}{\sqrt{z^{2}+\omega^{2}}}=\frac{1}{2}\ln\left(z+f\right)$.

$\beta(r)=\int\frac{dr}{2rf}\,\left(\frac{r}{\ell}-\dfrac{\theta
J}{2r}\right)=\frac{1}{2}\ln\left(\frac{r}{\ell}+\dfrac{\theta
J}{2r}+f\right)\,.$ (4.3)

It means that the radial matrix acquires the form

$U(r)=\frac{1+\Gamma_{1}}{2}\left(\frac{r}{\ell}+\dfrac{\theta
J}{2r}+f\right)^{-\frac{1}{2}}+\frac{1-\Gamma_{1}}{2}\left(\frac{r}{\ell}+\dfrac{\theta
J}{2r}+f\right)^{\frac{1}{2}}\,.$ (4.4)

With this result at hand, we can write the other two Killing spinor equation
components in (3.25) as

$\displaystyle 0$ $\displaystyle=$ $\displaystyle
U\,\partial_{+}\Phi(x^{+},x^{-})\quad\Rightarrow\quad\Phi=\Phi\left(x^{-}\right)\,,$
$\displaystyle 0$ $\displaystyle=$
$\displaystyle\partial_{-}\Phi\left(x^{-}\right)+\frac{1}{2\ell}\,U^{-1}\Omega
U\,\Phi\left(x^{-}\right)\,,$ (4.5)

where we defined the radial matrix

$\Omega(r)=f\Gamma_{0}-\Gamma_{2}\left(\frac{J}{2r}+\theta\,\dfrac{r}{\ell}\right)\,,$
(4.6)

with the property

$\det\Omega=-M-\dfrac{\theta J}{\ell}=\omega^{2}\,.$ (4.7)

Using the identity

$\Gamma_{a}\Gamma_{b}=-\theta\varepsilon_{abc}\,\Gamma^{c}\,,$ (4.8)

it is a straightforward computation to show that the matrix $U^{-1}\Omega U$
is $r$-independent; namely

$U^{-1}\Omega
U=\frac{1+\omega^{2}}{2}\,\Gamma_{0}-\frac{1-\omega^{2}}{2}\,\theta\Gamma_{2}\,.$
(4.9)

Moreover, it satisfies

$\left(U^{-1}\Omega U\right)^{2}=-\omega^{2}\,,$ (4.10)

implying that it has eigenvalues $\pm\mathrm{i}\omega$. Its representation is

$U^{-1}\Omega
U=\left(\begin{array}[]{cc}-\frac{1-\omega^{2}}{2}&\mathrm{i}\theta\,\frac{1+\omega^{2}}{2}\\\
\mathrm{i}\theta\,\frac{1+\omega^{2}}{2}&\frac{1-\omega^{2}}{2}\end{array}\right).$
(4.11)

We could solve the second Eq. (4.5) in a diagonal basis and, then, we would
have two solutions with coefficients
$\mathrm{e}^{\pm\frac{\mathrm{i}}{2\ell}\,\omega x^{-}}$, such that their
single-valuedness would again impose the condition (3.40) on the frequency.
However, instead of proceeding in that way, for a simpler comparison with the
previous section, let us solve the second Eq. (4.5) in the present basis.
Because of $U^{-1}\Omega U$ being a constant matrix, we can take its second
derivative in $\partial_{-}$, finding

$0=\partial_{-}^{2}\Phi+\frac{\omega^{2}}{4\ell^{2}}\,\Phi\quad\Rightarrow\quad\Phi\left(x^{-}\right)=\mathrm{e}^{-\frac{\mathrm{i}}{2\ell}\,\omega
x^{-}}\chi_{1}+\mathrm{e}^{\frac{\mathrm{i}}{2\ell}\,\omega x^{-}}\chi_{2}\,,$
(4.12)

where $\chi_{i}$ are constant spinors ($i=1,2$). Denoting
$\chi_{i}=\binom{\eta_{i}}{\rho_{i}}$, Eq. (4.5) becomes

$0=\left(\begin{array}[]{cc}-\mathrm{i}\omega-\frac{1-\omega^{2}}{2}&\mathrm{i}\theta\,\frac{1+\omega^{2}}{2}\\\
\mathrm{i}\theta\,\frac{1+\omega^{2}}{2}&-\mathrm{i}\omega+\frac{1-\omega^{2}}{2}\end{array}\right)\left(\begin{array}[]{c}\eta_{1}\\\
\rho_{1}\end{array}\right)\,,$ (4.13)

and the negative-frequency solution is obtained by the replacement
$\omega\rightarrow-\omega$. The two components are not independent and it is
enough to solve only the first one. We find

$\rho_{1}={-}\mathrm{i}\theta\,\frac{1-\omega^{2}+2\mathrm{i}\omega}{1+\omega^{2}}\,\eta_{1}\,={-}\mathrm{i}\theta\frac{1+\mathrm{i\omega}}{1-\mathrm{i\omega}}\eta_{1}\,.$
(4.14)

The final solution is $\psi=U\Phi$, with

$\Phi=\mathrm{e}^{-\frac{\mathrm{i}}{2\ell}\,\omega
x^{-}}\left(\begin{array}[]{c}1\\\
-\mathrm{i}\theta\,\frac{1+\mathrm{i\omega}}{1-\mathrm{i\omega}}\end{array}\right)\eta_{1}+\mathrm{e}^{\frac{\mathrm{i}}{2\ell}\,\omega
x^{-}}\left(\begin{array}[]{c}1\\\
-\mathrm{i}\theta\,\frac{1-\mathrm{i\omega}}{1+\mathrm{i\omega}}\end{array}\right)\eta_{2}\,,$
(4.15)

and the radial part (4.4) given by

${U(r)=\left(\begin{array}[]{cc}\frac{1}{2}\,\mu^{\frac{1}{2}}+\frac{1}{2}\,\mu^{-\frac{1}{2}}&\frac{\mathrm{i}\theta}{2}\,\mu^{\frac{1}{2}}-\frac{\mathrm{i}\theta}{2}\,\mu^{-\frac{1}{2}}\\\
-\frac{\mathrm{i}\theta}{2}\,\mu^{\frac{1}{2}}+\frac{\mathrm{i}\theta}{2}\,\mu^{-\frac{1}{2}}&\frac{1}{2}\,\mu^{\frac{1}{2}}+\frac{1}{2}\,\mu^{-\frac{1}{2}}\end{array}\right)\,,}$
(4.16)

where $\mu=\frac{r}{\ell}+\frac{\theta J}{2r}+f$. For large $r$, the solution
$\psi$ behaves as $\sqrt{\frac{r}{\ell}}$ and, similarly as before, it has a
common vector $\binom{1}{\mathrm{i}\theta}$ with a $\omega$-dependent factor
multiplying each $\eta_{i}$. This solution is totally equivalent to the
solution found previously: see the following two subsections for more details.

###  4.2 The Coussaert-Hennaux solution

Now, let us compare the solution (4.15) with the one found by Coussaert and
Henneaux (CH) in Ref. [8]. While, in [8], the authors were mainly interested
in the black hole solutions, the derivation above is general enough, so that
it has to match the CH solution after restricting in the correct range of the
space of parameters. Let us plug in the expressions for $U(r)$ and
$\Phi(x^{-})$ in the (positive frequency part of the) Killing spinor using the
notation $A=B^{-1}=\mu^{\frac{1}{2}}$. We get

$\psi_{+}=U(r)\Phi_{+}(x^{-})=\mathrm{e}^{-\frac{\mathrm{i}}{2\ell}\,\omega
x^{-}}\frac{1}{2}\,\left(\begin{array}[]{c}A+B+\left(A-B\right)\,\frac{1+\mathrm{i}\omega}{1-\mathrm{i}\omega}\\\
-\mathrm{i}\theta\left[\left(A-B\right)+\left(A+B\right)\,\frac{1+\mathrm{i}\omega}{1-\mathrm{i}\omega}\right]\end{array}\right)\eta_{1}\,,$
(4.17)

or simply

$\psi_{+}=\mathrm{e}^{-\frac{\mathrm{i}}{2\ell}\,\omega
x^{-}}\left(\begin{array}[]{c}A-\mathrm{i}\omega B\\\
-\mathrm{i}\theta\left(A+\mathrm{i}\omega
B\right)\end{array}\right)\tilde{\eta}_{1}\,,$ (4.18)

and for the negative frequency part

$\psi_{-}=\mathrm{e}^{\frac{\mathrm{i}}{2\ell}\,\omega
x^{-}}\begin{pmatrix}A+\mathrm{i}\omega B\\\
-\mathrm{i}\theta\left(A-\mathrm{i}\omega
B\right)\end{pmatrix}\tilde{\eta}_{2}\,,$ (4.19)

with redefinitions $\tilde{\eta}_{1}=\frac{\eta_{1}}{1-\mathrm{i\omega}}$ and
$\tilde{\eta}_{2}=\frac{\eta_{2}}{1+\mathrm{i\omega}}$.

The CH solution corresponds to the extremal black hole, i.e.,

$M=\frac{\theta J}{\ell}\geq 0\,,$ (4.20)

where $\theta=\pm 1$ is chosen so that the positivity of $M$ is satisfied. In
that case, $f=\frac{r}{\ell}-\frac{\theta J}{2r}$ and
$A=B^{-1}=\sqrt{2}\left(\frac{r}{\ell}\right)^{\frac{1}{2}}$.

We can use this result to make the Killing spinor looks like CH solution.
Indeed, in

$\psi=\psi_{+}+\psi_{-}=\mathrm{e}^{-\frac{\mathrm{i}}{2\ell}\,\omega
x^{-}}\left(\begin{array}[]{c}A-\mathrm{i}\omega B\\\
-\mathrm{i}\theta\left(A+\mathrm{i}\omega
B\right)\end{array}\right)\tilde{\eta}_{1}+\mathrm{e}^{\frac{\mathrm{i}}{2\ell}\,\omega
x^{-}}\begin{pmatrix}A+\mathrm{i}\omega B\\\
-\mathrm{i}\theta\left(A-\mathrm{i}\omega
B\right)\end{pmatrix}\tilde{\eta}_{2}\,,$ (4.21)

we first separate the terms growing and decreasing with $r$ (by $A$ and $B$,
respectively),

$\displaystyle\psi=A\left(\mathrm{e}^{-\frac{\mathrm{i}}{2\ell}\,\omega
x^{-}}\tilde{\eta}_{1}+\mathrm{e}^{\frac{\mathrm{i}}{2\ell}\,\omega
x^{-}}\tilde{\eta}_{2}\right)\begin{pmatrix}1\\\
-\mathrm{i}\theta\end{pmatrix}-\mathrm{i}\omega
B\left(\mathrm{e}^{-\frac{\mathrm{i}}{2\ell}\,\omega
x^{-}}\tilde{\eta}_{1}-\mathrm{e}^{\frac{\mathrm{i}}{2\ell}\,\omega
x^{-}}\tilde{\eta}_{2}\right)\left(\begin{array}[]{c}1\\\
\mathrm{i}\theta\end{array}\right)\,.$ (4.24)

Plugging in $A=B^{-1}=\sqrt{2}\left(\frac{r}{\ell}\right)^{\frac{1}{2}}$ and
introducing the trigonometric functions along
$\Lambda_{1}=\tilde{\eta}_{1}+\tilde{\eta}_{2}$ and
$\Lambda_{2}=\tilde{\eta}_{1}-\tilde{\eta}_{2}$, we get

$\displaystyle\psi$ $\displaystyle=$
$\displaystyle\sqrt{2}\left(\frac{r}{\ell}\right)^{\frac{1}{2}}\left[\Lambda_{1}\cos\left(\frac{\omega
x^{-}}{2\ell}\right)-\mathrm{i}\Lambda_{2}\sin\left(\frac{\omega
x^{-}}{2\ell}\right)\right]\begin{pmatrix}1\\\ -\mathrm{i}\theta\end{pmatrix}$
(4.27)
$\displaystyle+\frac{\mathrm{i}\omega}{\sqrt{2}}\left(\frac{r}{\ell}\right)^{-\frac{1}{2}}\left[-\Lambda_{2}\cos\left(\frac{\omega
x^{-}}{2\ell}\right)+\mathrm{i}\Lambda_{1}\sin\left(\frac{\omega
x^{-}}{2\ell}\right)\right]\left(\begin{array}[]{c}1\\\
\mathrm{i}\theta\end{array}\right)\,.$

Furthermore, the extremal black hole have complex
$\omega=\mathrm{i}\sqrt{2M}$, such that we arrive at

$\displaystyle\psi$ $\displaystyle=$
$\displaystyle\sqrt{2}\left(\frac{r}{\ell}\right)^{\frac{1}{2}}\left[\Lambda_{1}\cosh\left(\sqrt{\frac{M}{2}}\frac{x^{-}}{\ell}\right)+\Lambda_{2}\sinh\left(\sqrt{\frac{M}{2}}\frac{x^{-}}{\ell}\right)\right]\begin{pmatrix}1\\\
-\mathrm{i}\theta\end{pmatrix}$ (4.30)
$\displaystyle+\sqrt{M}\left(\frac{r}{\ell}\right)^{-\frac{1}{2}}\left[\Lambda_{2}\cosh\left(\sqrt{\frac{M}{2}}\frac{x^{-}}{\ell}\right)+\Lambda_{1}\sinh\left(\sqrt{\frac{M}{2}}\frac{x^{-}}{\ell}\right)\right]\left(\begin{array}[]{c}1\\\
\mathrm{i}\theta\end{array}\right)\,.$

Now, we are in conditions to compare this solution we obtained with the CH
Killing spinor found in [8] for the extremal black hole. The latter is

$\displaystyle\psi_{\mathrm{CH}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\frac{r}{\ell}\right)^{\frac{1}{2}}\left[a_{1}\cosh\left(\sqrt{\frac{M}{2}}\frac{x^{-}}{\ell}\right)+a_{2}\sinh\left(\sqrt{\frac{M}{2}}\frac{x^{-}}{\ell}\right)\right]\begin{pmatrix}1\\\
-1\end{pmatrix}$ (4.33)
$\displaystyle+\frac{1}{2}\left(\frac{r}{\ell}\right)^{-\frac{1}{2}}\sqrt{\frac{M}{2}}\left[a_{2}\cosh\left(\sqrt{\frac{M}{2}}\frac{x^{-}}{\ell}\right)+a_{1}\sinh\left(\sqrt{\frac{M}{2}}\frac{x^{-}}{\ell}\right)\right]\left(\begin{array}[]{c}1\\\
1\end{array}\right)$

And we, actually, verify that they completely match, provided one chooses the
normalization as $\Lambda_{i}=\frac{a_{i}}{2\sqrt{2}}$. A difference in the
columns is related to the representation choice: in [8] it is used
$\\{\Gamma_{0}=\mathrm{i}\sigma^{2},\;\Gamma_{1}=\sigma^{1},\;\Gamma_{2}=\sigma^{3}\\}$.

As a consistency check, let us notice that the Killing spinor (3.39), obtained
in a different way, is also equivalent to the CH solution in the extremal
black hole case. Indeed, when $\omega=\mathrm{i}\sqrt{2M}$, it becomes

$\psi=\mathrm{e}^{\sqrt{\frac{M}{2}}\,\frac{x^{-}}{\ell}}\left(\begin{array}[]{c}\left(\frac{r}{\ell}+\frac{\theta
J}{2r}+\sqrt{2M}\right)^{\frac{1}{2}}\\\
-\mathrm{i}\theta\left(\frac{r}{\ell}+\frac{\theta
J}{2r}-\sqrt{2M}\right)^{\frac{1}{2}}\end{array}\right)\eta_{1}+\mathrm{e}^{-\sqrt{\frac{M}{2}}\,\frac{x^{-}}{\ell}}\left(\begin{array}[]{c}\left(\frac{r}{\ell}+\frac{\theta
J}{2r}-\sqrt{2M}\right)^{\frac{1}{2}}\\\
-\mathrm{i}\theta\left(\frac{r}{\ell}+\frac{\theta
J}{2r}+\sqrt{2M}\right)^{\frac{1}{2}}\end{array}\right)\eta_{2}\,.$ (4.34)

Then, noting that $\frac{r}{\ell}+\frac{\theta
J}{2r}\pm\sqrt{2M}=\left[\left(\frac{r}{\ell}\right)^{\frac{1}{2}}\pm\left(\frac{M\ell}{2r}\right)^{\frac{1}{2}}\right]^{2}$
when $M\ell=\theta J$, it can be rewritten as

$\psi=\mathrm{e}^{\sqrt{\frac{M}{2}}\,\frac{x^{-}}{\ell}}\left(\begin{array}[]{c}\left(\frac{r}{\ell}\right)^{\frac{1}{2}}+\left(\frac{M\ell}{2r}\right)^{\frac{1}{2}}\\\
-\mathrm{i}\theta\left[\left(\frac{r}{\ell}\right)^{\frac{1}{2}}-\left(\frac{M\ell}{2r}\right)^{\frac{1}{2}}\right]\end{array}\right)\eta_{1}+\mathrm{e}^{-\sqrt{\frac{M}{2}}\,\frac{x^{-}}{\ell}}\left(\begin{array}[]{c}\left(\frac{r}{\ell}\right)^{\frac{1}{2}}-\left(\frac{M\ell}{2r}\right)^{\frac{1}{2}}\\\
-\mathrm{i}\theta\left[\left(\frac{r}{\ell}\right)^{\frac{1}{2}}+\left(\frac{M\ell}{2r}\right)^{\frac{1}{2}}\right]\end{array}\right)\eta_{2}\,,$
(4.35)

or separating the $\left(\frac{r}{\ell}\right)^{\frac{1}{2}}$ and
$\left(\frac{r}{\ell}\right)^{-\frac{1}{2}}$ contributions,

$\displaystyle\psi$ $\displaystyle=$
$\displaystyle\left(\frac{r}{\ell}\right)^{\frac{1}{2}}\left(\mathrm{e}^{\sqrt{\frac{M}{2}}\,\frac{x^{-}}{\ell}}\eta_{1}+\mathrm{e}^{-\sqrt{\frac{M}{2}}\,\frac{x^{-}}{\ell}}\eta_{2}\right)\left(\begin{array}[]{c}1\\\
-\mathrm{i}\theta\end{array}\right)$ (4.41)
$\displaystyle+\left(\frac{r}{\ell}\right)^{-\frac{1}{2}}\sqrt{\frac{M}{2}}\left(\mathrm{e}^{\sqrt{\frac{M}{2}}\,\frac{x^{-}}{\ell}}\eta_{1}-\mathrm{e}^{-\sqrt{\frac{M}{2}}\,\frac{x^{-}}{\ell}}\eta_{2}\right)\left(\begin{array}[]{c}1\\\
\mathrm{i}\theta\end{array}\right)\,,$

or equivalently, in terms of hyperbolic functions,

$\displaystyle\psi$ $\displaystyle=$
$\displaystyle\left(\frac{r}{\ell}\right)^{\frac{1}{2}}\left[\cosh\left(\sqrt{\frac{M}{2}}\frac{x^{-}}{\ell}\right)\left(\eta_{1}+\eta_{2}\right)+\mathrm{\sinh}\left(\sqrt{\frac{M}{2}}\frac{x^{-}}{\ell}\right)\left(\eta_{1}-\eta_{2}\right)\right]\left(\begin{array}[]{c}1\\\
-\mathrm{i}\theta\end{array}\right)$ (4.47)
$\displaystyle+\left(\frac{r}{\ell}\right)^{-\frac{1}{2}}\sqrt{\frac{M}{2}}\left[\cosh\left(\sqrt{\frac{M}{2}}\frac{x^{-}}{\ell}\right)\left(\eta_{1}-\eta_{2}\right)+\mathrm{\sinh}\left(\sqrt{\frac{M}{2}}\frac{x^{-}}{\ell}\right)\left(\eta_{1}+\eta_{2}\right)\right]\left(\begin{array}[]{c}1\\\
\mathrm{i}\theta\end{array}\right)\,.$

Redefining the constant spinors as $\eta_{1}+\eta_{2}=\frac{a_{1}}{2}$ and
$\eta_{1}-\eta_{2}=\frac{a_{2}}{2}$, the result exactly matches Eq. (4.33).

###  4.3 Equivalence of solutions

To complete the discussion and as a further consistency check of the
calculation, let us show now that the Killing spinors (3.39) and (4.21),
obtained by different procedures, are equivalent for any $M$. To make it
simpler, we first note that both solutions have the form

$\psi=\mathrm{e}^{-\frac{\mathrm{i}}{2\ell}\omega\,x^{-}}\begin{pmatrix}z\\\
-\mathrm{i}\theta\,z^{\ast}\end{pmatrix}\eta_{1}+\mathrm{e}^{\frac{\mathrm{i}}{2\ell}\omega\,x^{-}}\begin{pmatrix}z^{\ast}\\\
-\mathrm{i}\theta\,z\end{pmatrix}\eta_{2}\,,$ (4.48)

where $z$ is a complex number and $z^{\ast}$ is its conjugate. Specifically,
$z_{1}$ in Eq. (3.39) and $z_{2}$ in Eq. (4.21) are

$\displaystyle z_{1}$ $\displaystyle=$
$\displaystyle\left(\frac{r}{\ell}+\frac{\theta
J}{2r}-\mathrm{i}\omega\right)^{\frac{1}{2}}\,,$ $\displaystyle z_{2}$
$\displaystyle=$ $\displaystyle\left(\frac{r}{\ell}+\frac{\theta
J}{2r}+f\right)^{\frac{1}{2}}-\mathrm{i}\omega\left(\frac{r}{\ell}+\frac{\theta
J}{2r}+f\right)^{-\frac{1}{2}}=A-\mathrm{i}\omega B\,.$ (4.49)

Thus, it is enough to prove that $z_{1}^{2}$ and $z_{2}^{2}$ are proportional
to each other. Then, we compute

$\displaystyle z_{1}^{2}$ $\displaystyle=$
$\displaystyle\frac{r}{\ell}+\frac{\theta J}{2r}-\mathrm{i}\omega\,,$
$\displaystyle z_{2}^{2}$ $\displaystyle=$ $\displaystyle
A^{2}-\omega^{2}B^{2}-2\mathrm{i}\omega\,,$ (4.50)

where we used $AB=1$. Using also
$f^{2}=\omega^{2}+\left(\frac{r}{\ell}+\frac{\theta J}{2r}\right)^{2}$, we can
manipulate the real part of $z_{2}^{2}$ as

$\displaystyle A^{2}-\omega^{2}B^{2}=2\left(\frac{r}{\ell}+\frac{\theta
J}{2r}\right)\,,$ (4.51)

such that the complex numbers are manifestly proportional, namely

$z_{2}^{2}=2\left(\frac{r}{\ell}+\frac{\theta
J}{2r}\right)-2\mathrm{i}\omega=2z_{1}^{2}\,.$ (4.52)

Then, we conclude that the Killing spinors (3.39) and (4.21) are equal,
provided the relation between the constant spinors is
$\eta_{i}=\sqrt{2}\tilde{\eta}_{i}$.

##  5 A CFT2 boundary perspective

Before concluding, let us make some comments about how to look at these
special states from the holographic point of view; that is to say, from the
dual CFT2 perspective. We are interested in identifying which representations
of the Virasoro symmetry are those associated to the negative mass BPS
configurations discussed above. As anticipated, in the semiclassical limit,
the static BPS states can be associated to the so-called degenerate
representation of Liouville field theory, which are non-normalizable states of
the CFT2 that contain null descendants in the Verma modulo. But, first, let us
review where the Liouville CFT2 description comes from. Since here we are
dealing with AdS3 supergravity, the right theory to look at would be super-
Liouville; nevertheless, it will be sufficient for us to focus on the bosonic
theory first; we will see below how the super-Liouville leads to the same
results.

In the renowned paper [2], Brown and Henneaux found that the asymptotic
isometries in asymptotically AdS3 spacetimes is generated by two commuting
copies of the Witt algebra, with the associated Noether charges satisfying two
copies of Virasoro algebra with the central charge444In this subsection, we
restore the dependence of the Newton constant $G$.

$c=\frac{3\ell}{2G}\,.$ (5.1)

The extension of the study of the asymptotic dynamics in AdS3 supergravity was
done in [46] and yields an equivalent result. The observation in [2] is often
considered a precursor of AdS/CFT [47], the reason being that, from a modern
perspective, the central charge (5.1) is understood as that of the dual CFT2.
Also, one can identify the conformal dimension of the CFT2 states as coming
from the $L_{0}$ and $\bar{L}_{0}$ generators of the isometry algebra, which
correspond to the spin and the energy of the configuration; namely

$h-\frac{c}{24}=\frac{1}{2}(\ell M+J)\,,\ \ \ \ \
\bar{h}-\frac{c}{24}=\frac{1}{2}(\ell M-J),$ (5.2)

which is equivalent to

$\frac{h+\bar{h}}{\ell}=M+\frac{1}{8G}\,,\ \ \ \ \ h-\bar{h}=J.$ (5.3)

This result enables us to identify the gap in the spectrum of the BTZ black
hole with respect to the AdS3 vacuum $h=\bar{h}=0$, which corresponds to

$M_{0}=-\frac{1}{8G}\,,\ \ \ \ \ J_{0}=0.$ (5.4)

Reciprocally, the configuration $M=J=0$ corresponds to

$h_{0}=\bar{h}_{0}=\frac{c}{24}.$ (5.5)

In [40], van Driel, Coussaert and Henneaux went further in the CFT2
description of the AdS3 asymptotic dynamics and, following a sequence of steps
that includes a gauge fixing and a prescription of boundary conditions that
implement a Hamiltonian reduction [42], they found that the asymptotic
dynamics of Einstein (super-)gravity in AdS3 is governed by a
(super-)Liouville field theory action. Because this equivalence is valid only
at the level of the actions, one should not understand the relation between
three-dimensional gravity and Liouville as holding beyond the semiclassical
limit. In fact, there are many reasons why Liouville field theory should not
be regarded as dual to a sensible quantum gravity theory: the continuous
spectrum and the absence of an $SL(2,\mathbb{C})$-invariant vacuum are
probably the most salient reasons. Nevertheless, nothing prevents us from
investigating to what extent this relation between AdS3 gravity and Liouville
CFT2 can be taken for valid.

Therefore, let us be reminded of some basic aspects of Liouville field theory.
The theory has a central charge given by

$c=1+6Q^{2}\,,\ \ \text{with}\ \ Q=q+q^{-1},$ (5.6)

where $q\in\mathbb{R}$ for $c\geq 25$, and $Q$ is the background charge. The
semiclassical (large $c$) limit of the theory corresponds to $q\to 0$.

Liouville field theory has a continuous spectrum, with normalizable states
having conformal dimension

$h=\bar{h}=\frac{c-1}{24}+\lambda^{2}\,,\ \ \text{with}\ \
\lambda^{2}\in\mathbb{R}_{\geq 0}.$ (5.7)

We notice from (5.7) that the spectrum has a gap, which in the semiclassical
limit reads

$\text{min}(h)=\text{min}(\bar{h})=\frac{c-1}{24}\simeq\frac{1}{4q^{2}};$
(5.8)

cf. (5.5). From (5.4) and (5.8) we read a relation between the gravity
parameter $\ell/G$ and the Liouville variable $q$; namely

$q^{2}=\frac{4G}{\ell}\,.$ (5.9)

In addition to the normalizable states (5.7), the theory has other interesting
lower weight representations. These are the degenerate representations;
namely, non-normalizable, spinless states that contain null descendants in the
Verma modulo and are useful to carry on the bootstrap method in the theory,
for example, to solve correlation functions. The states of the degenerate
representations are of the form

$h_{m,n}=\bar{h}_{m,n}=\frac{c-1}{24}-\frac{1}{4}(mq+nq^{-1})^{2}\,,\ \
\text{with}\ \ m,n\in\mathbb{Z}_{\geq 0}.$ (5.10)

If we again consider the semiclassical limit, we can write the following state
identification

$h_{m,n}=\bar{h}_{m,n}\simeq\frac{c}{24}(1-n^{2})\simeq\frac{\ell}{16G}(1-n^{2})\,,$
(5.11)

which yields the mass and angular momentum of the BPS states,

$M\simeq-\frac{n^{2}}{8G}\,,\qquad J=0\,.$ (5.12)

This means that we can identify the BPS static configurations of negative mass
with the semiclassical limit of Liouville degenerate representations.

Now, let us consider the super-Liouville theory: Virasoro central charge in
the superconformal algebra of super-Liouville theory is

$\hat{c}=1+2Q^{2}\,.$ (5.13)

On the other hand, the central charge $\hat{c}$ is also related to the Chern-
Simons level $k$ as $\hat{c}=4k$. It is useful to compare the notation in [48]
with that in [46]. In order to do so, we can define

$\hat{c}\,=\,\frac{2}{3}\,{c},$ (5.14)

with $c=6k=\frac{3\ell}{2G}$ being the Brown-Henneaux central charge.

Studying the degenerate representations in super-Liouville, we also find

$h_{m,n}+\bar{h}_{m,n}-\frac{c}{12}=\ell M=-\frac{n^{2}\ell}{8G}\,.$ (5.15)

So far, we have discussed spin-zero representations, which correspond to
static geometries. For those states we have identified the BPS configurations
of negative mass with the semiclassical limit of spinless Liouville degenerate
representations. The question arises as to whether such an identification is
also possible for the spinning ($J\neq 0$) configurations. Answering this
question leads us to investigate the non-diagonal representations of CFT2,
which were recently studied in [27]. These representations take the form

$h_{m,n}=\frac{c-1}{24}-\frac{1}{4}(mq+nq^{-1})^{2}\,,\ \ \
\bar{h}_{m,n}=\frac{c-1}{24}-\frac{1}{4}(mq-nq^{-1})^{2},$ (5.16)

with $m,n$ taking (semi-)integer values, cf. [49]. Figure 1 depicts the non-
diagonal representations of [27] for some (finite) values of the central
charge $c$. This yields the spin $h_{m,n}-\bar{h}_{m,n}=mn$ and, in the
semiclassical limit, the condition

$h_{m,n}-\frac{c}{24}=\frac{1}{2}(\ell M+J)\simeq-\frac{n^{2}\ell}{16G},$
(5.17)

which is exactly the BPS condition (3.40) after restoring the factor of $G$.
However, the last approximation in (5.17) requires $h_{m,n}-\bar{h}_{m,n}$ to
be parametrically small relative to $\frac{n^{2}}{q^{2}}$ in the $q\to 0$
limit, and so this fails to represent states with non-zero spin in the
semiclassical limit.

Figure 1: Non-diagonal representations for some specific, finite values of $c$
as a function of $M$ and $J$. Sparse spectrum is observed for large $c$.

It would be interesting to explore the CFT2 realization of spinning BPS states
discussed here in the semiclassical limit from a boundary perspective. This
would require to take in (5.16) $m\sim\hat{m}q^{-2}$, which yields $\ell
M+J\simeq-\frac{(\hat{m}+n)^{2}\ell}{8G}$. This result can be thought of as a
motivation to further study non-diagonal representations of supersymmetric
non-rational CFT2 and generalize the results of [27, 49]. It would also be
interesting to explore the relation between the solutions studied here and
some of the states discussed in Ref. [50]. Understanding the connection with
theories in two dimensions would also be important to seek. We leave these
problems for the future.

### Acknowledgments

The authors thank Glenn Barnich, Laura Donnay, Andrea Campoleoni, Marc
Henneaux, Julio Oliva, Joris Raeymaekers, Sylvain Ribault, and Gustavo Turiaci
for discussions, correspondence and references. This work has been funded in
part by Anillo Grant ANID/ACT210100 and FONDECYT Grants 1231779, 1230492,
1220862 and 1201208.

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