Supersymmetric $AdS_{4}$ black holes from matter-coupled $N=3,4$ gauged
supergravities

Parinya Karndumri

String Theory and Supergravity Group, Department of Physics, Faculty of
Science, Chulalongkorn University, 254 Phayathai Road, Pathumwan, Bangkok
10330, Thailand

E-mail<EMAIL_ADDRESS>

###### Abstract

We study supersymmetric $AdS_{4}$ black holes in matter-coupled $N=3$ and
$N=4$ gauged supergravities in four dimensions. In $N=3$ theory, we consider
$N=3$ gauged supergravity coupled to three vector multiplets and $SO(3)\times
SO(3)$ gauge group. The resulting gauged supergravity admits two $N=3$
supersymmetric $AdS_{4}$ vacua with $SO(3)\times SO(3)$ and $SO(3)$
symmetries. We find an $AdS_{2}\times H^{2}$ solution with $SO(2)\times SO(2)$
symmetry and an analytic solution interpolating between this geometry and the
$SO(3)\times SO(3)$ symmetric $AdS_{4}$ vacuum. For $N=4$ gauged supergravity
coupled to six vector multiplets with $SO(4)\times SO(4)$ gauge group, there
exist four supersymmetric $AdS_{4}$ vacua with $SO(4)\times SO(4)$,
$SO(4)\times SO(3)$, $SO(3)\times SO(4)$ and $SO(3)\times SO(3)$ symmetries.
We find a number of $AdS_{2}\times S^{2}$ and $AdS_{2}\times H^{2}$ geometries
together with the solutions interpolating between these geometries and all,
but the $SO(3)\times SO(3)$, $AdS_{4}$ vacua. These solutions provide a new
class of $AdS_{4}$ black holes with spherical and hyperbolic horizons dual to
holographic RG flows across dimensions from $N=3,4$ SCFTs in three dimensions
to superconformal quantum mechanics within the framework of four-dimensional
gauged supergravity.

##  1 Introduction

String/M-theory has provided a number of insights to various aspects of
quantum gravity for many decades. In particular, a resolution for a long-
standing problem of black hole entropy has been proposed in [1]. After this
pioneering work, many other papers followed and clarified the issues of
microscopic entropy of asymptotically flat black holes. For asymptotically
$AdS_{4}$ black holes, a concrete result on the corresponding microscopic
entropy, using AdS/CFT correspondence [2, 3, 4], has appeared recently in [5,
6, 7], see also [8, 9, 10, 11, 12].
On the gravity side, an important ingredient along this line is $AdS_{4}$
black hole solutions interpolating between asymptotic $AdS_{4}$ and
$AdS_{2}\times\Sigma^{2}$ spaces with $\Sigma^{2}$ being a Riemann surface.
The latter describes the geometry of the black hole horizon with the values of
scalars determined by the attractor mechanism. These solutions holographically
describe RG flows across dimensions from three-dimensional SCFTs, dual to the
$AdS_{4}$ vacua, to superconformal quantum mechanics, dual to the $AdS_{2}$
factor of the horizons. The latter is obtained from twisted compactifications
of the former which play an important role in computing Bekenstein-Hawking
entropy of the black holes via twisted indices.
In this paper, we are interested in supersymmetric $AdS_{4}$ black holes with
the horizon geometry $AdS_{2}\times S^{2}$ and $AdS_{2}\times H^{2}$ with
$S^{2}$ and $H^{2}$ being a two-sphere and a two-dimensional hyperbolic space,
respectively. We will work in matter-coupled $N=3$ and $N=4$ gauged
supergravities. This type of solutions has been extensively studied in $N=2$
gauged supergravity for a long time [13, 14, 15, 16, 17, 18, 19], see also
[20] for some results in $N=8$ gauged supergravity. Similar studies in other
gauged supergravities have appeared only recently in [21, 22, 23, 24]. In
particular, a study of $AdS_{2}\times\Sigma^{2}$ solutions in $N=3$ with only
magnetic charges has been initiated in [21]. We will extend this result by
performing a more systematic analysis and including a possible dyonic
generalization. We will consider a particular case of $N=3$ gauged
supergravity coupled to three vector multiplets with a compact $SO(3)\times
SO(3)$ gauge group. We will see that only one magnetic $AdS_{2}\times H^{2}$
solution with $SO(2)\times SO(2)$ symmetry exists. This is very similar to
solutions in $N=5$ and $N=6$ gauged supergravities given in [23] and [24].
For $N=4$ case, we will consider $N=4$ gauged supergravity coupled to six
vector multiplets with $SO(4)\times SO(4)$ gauge group. Unlike the $N=3$
theory with a purely electric gauging, any $N=4$ supergravity that admits
supersymmetric $AdS_{4}$ vacua must be dyonically gauged [25]. In this case,
apart from an $AdS_{2}\times H^{2}$ solution similar to $N=3,5,6$ gauged
supergravities, there exist a number of supersymmetric $AdS_{2}\times S^{2}$
and $AdS_{2}\times H^{2}$ solutions. It should also be pointed out that some
$AdS_{2}\times\Sigma^{2}$ solutions in $N=4$ gauged supergravity obtained from
a truncation of eleven-dimensional supergravity have also been found in [22].
However, in that case, the gauge group is of non-semisimple form, and the
resulting BPS equations are highly complicated. In the present work, we
provide a number of much simpler examples of supersymmetric $AdS_{4}$ black
holes in $N=4$ gauged supergravity. In particular, the two-form fields
required by the consistency of incorporating magnetic gauge fields can be
truncated out in the present case.
The paper is organized as follows. In section 2, we will review the structure
of $N=3$ gauged supergravity after translating the original construction in
group manifold approach to the usual formulae in space-time. This is followed
by a general analysis of relevant BPS equations for finding supersymmetric
$AdS_{4}$ black hole solutions. An $AdS_{2}\times H^{2}$ solution with
$SO(2)\times SO(2)$ symmetry together with the full flow solution
interpolating between this fixed point and the supersymmetric $AdS_{4}$ vacuum
with $SO(3)\times SO(3)$ symmetry are also given. Similar analysis is then
performed in section 3 in which we will find a number of $AdS_{2}\times S^{2}$
and $AdS_{2}\times H^{2}$ fixed points and solutions interpolating between
them and supersymmetric $AdS_{4}$ vacua with various unbroken symmetries in
$N=4$ gauged supergravity. We end the paper by giving conclusions and comments
on the results in section 4.

##  2 $AdS_{4}$ black holes from $N=3$ gauged supergravity

In this section, we consider matter-coupled $N=3$ gauged supergravity and
possible supersymmetric $AdS_{4}$ black holes. We begin with a review of $N=3$
gauged supergravity and the analysis of relevant BPS equations. These are
followed by the explicit solutions at the end of the section.

###  2.1 Matter-coupled $N=3$ gauged supergravity

We now give a description of $N=3$ gauged supergravity coupled to $n$ vector
multiplets. This has been constructed by the geometric group manifold approach
in [26], see also [27, 28]. However, the final form of the space-time
Lagrangian has not been given, and the supersymmetry transformations of
fermions have been given in a rather implicit form. We will first collect all
these ingredients and specify to the case of $n=3$ vector multiplets later on.
The interested reader can find a more detailed construction and some
discussions on the structure of the scalar manifold and electric-magnetic
duality in [26]. We will mostly follow the notations of [26] but in a mostly
plus signature for the space-time metric and a slightly different convention
for the gauge fields.
For $N=3$ supersymmetry in four dimensions, there are two types of
supermultiplets, the gravity and vector multiplets. The former consists of the
following component fields

$(e^{a}_{\mu},\psi_{\mu A},A^{A}_{\mu},\chi).$

$e^{a}_{\mu}$ is the graviton, and $\psi_{\mu A}$ are three gravitini. Space-
time and tangent space indices will be denoted by $\mu,\nu,\ldots$ and
$a,b,\ldots$, respectively. The gravity multiplet also contains three vector
fields $A^{A}_{\mu}$ with indices $A,B,\ldots=1,2,3$ denoting the fundamental
representation of the $SU(3)_{R}$ part of the full $SU(3)_{R}\times U(1)_{R}$
R-symmetry. There is also an $SU(3)_{R}$ singlet spinor field $\chi$.
$N=3$ supersymmetry allows the gravity multiplet to couple to an arbitrary
number of vector multiplets, the only matter fields in this case. The
component fields in a vector multiplet are given by the following field
content

$(A_{\mu},\lambda_{A},\lambda,z_{A})$

consisting of a vector field $A_{\mu}$, four spinor fields $\lambda$ and
$\lambda_{A}$ which are respectively singlet and triplet of $SU(3)_{R}$, and
three complex scalars $z_{A}$ in the fundamental of $SU(3)_{R}$. We will use
indices $i,j,\ldots=1,\ldots,n$ to label each vector multiplet.
The fermionic fields are subject to the chirality projection conditions

$\psi_{\mu A}=\gamma_{5}\psi_{\mu
A},\qquad\chi=\gamma_{5}\chi,\qquad\lambda_{A}=\gamma_{5}\lambda_{A},\qquad\lambda=-\gamma_{5}\lambda\,.$
(1)

These also imply $\psi_{\mu}^{A}=-\gamma_{5}\psi_{\mu}^{A}$ and
$\lambda^{A}=-\gamma_{5}\lambda^{A}$ for the corresponding conjugate spinors.
In the matter-coupled supergravity with $n$ vector multiplets, there are $3n$
complex scalar fields $z_{A}^{\phantom{A}i}$ parametrizing the coset space
$SU(3,n)/SU(3)\times SU(n)\times U(1)$. These scalars are conveniently
described by the coset representative
$L_{\Lambda}^{\phantom{\Lambda}\underline{\Sigma}}$. The coset representative
transforms under the global $G=SU(3,n)$ and the local $H=SU(3)\times
SU(n)\times U(1)$ symmetries by left and right multiplications, respectively.
The $SU(3,n)$ indices $\Lambda,\Sigma,\ldots$ will take values $1,\ldots,n+3$.
On the other hand, it is convenient to split the $SU(3)\times SU(n)\times
U(1)$ indices $\underline{\Lambda},\underline{\Sigma},\ldots$ as $(A,i)$. We
can then write the coset representative as

$L_{\Lambda}^{\phantom{\Lambda}\underline{\Sigma}}=(L_{\Lambda}^{\phantom{\Lambda}A},L_{\Lambda}^{\phantom{\Lambda}i}).$
(2)

The $n+3$ vector fields from both the gravity and vector multiplets are
combined into $A^{\Lambda}_{\mu}=(A^{A}_{\mu},A^{i}_{\mu})$. These are called
electric vector fields that appear in the Lagrangian with the usual Yang-Mills
(YM) kinetic terms. Accompanied by the corresponding magnetic dual
$A_{\Lambda\mu}$, these vector fields transform in the fundamental
representation $\mathbf{n+3}$ of the global symmetry group $SU(3,n)$, also
called the duality group.
For the gaugings of the matter-coupled $N=3$ supergravity, we will follow the
original result of [26] since the complete modern approach using the embedding
tensor has not been worked out so far. For general gaugings obtained from the
embedding tensor formalism, both electric and magnetic gauge fields can
participate in the gaugings. The construction of [26], called electric
gaugings, with only electric vector fields becoming the gauge fields results
in gauge groups that only account for a smaller class of all possible
gaugings. All gauge groups considered in [26] are subgroups of $SO(3,n)$ which
is the electric subgroup of the full global symmetry $SU(3,n)$.
After gauging a particular subgroup $G_{0}$ of $SO(3,n)\subset SU(3,n)$, the
corresponding non-abelian gauge field strengths are given by

$F^{\Lambda}=dA^{\Lambda}+{f_{\Sigma\Gamma}}^{\Lambda}A^{\Sigma}\wedge
A^{\Gamma}$ (3)

where ${f_{\Lambda\Sigma}}^{\Gamma}$ denote the structure constants of the
gauge group. The gauge generators $T_{\Lambda}$ satisfy

$\left[T_{\Lambda},T_{\Sigma}\right]=f_{\Lambda\Sigma}^{\phantom{\Lambda\Sigma}\Gamma}T_{\Gamma}\,.$
(4)

Indices $\Lambda,\Sigma,\ldots$ can be raised and lowered by the $SU(3,n)$
invariant tensor

$J_{\Lambda\Sigma}=J^{\Lambda\Sigma}=(\delta_{AB},-\delta_{ij})$ (5)

which will become the Killing form of the gauge group $G_{0}$. In order for
the gaugings to be consistent with supersymmetry, the structure constants
$f_{\Lambda\Sigma\Gamma}$ need to satisfy the following constraint

$f_{\Lambda\Sigma\Gamma}=f_{\Lambda\Sigma}^{\phantom{\Lambda\Sigma}\Delta}J_{\Delta\Gamma}=f_{[\Lambda\Sigma\Gamma]}$
(6)

which is equivalent to the linear constraint in the embedding tensor
formalism. Some examples of possible gauge groups are $SO(3)\times H_{n}$,
$SO(3,1)\times H_{n-3}$ and $SO(2,2)\times H_{n-3}$ with $H_{n}$ being an
$n$-dimensional compact subgroup of $SO(n)\subset SU(n)$. These gaugings
together with possible supersymmetric $AdS_{4}$ vacua and domain walls have
already been studied in [33].
With the fermion mass terms and the scalar potential included as required by
supersymmetry, the bosonic Lagrangian of the $N=3$ gauged supergravity can be
written as

$\displaystyle e^{-1}\mathcal{L}$ $\displaystyle=$
$\displaystyle\frac{1}{4}R-\frac{1}{4}P_{\mu}^{iA}P^{\mu}_{Ai}-a_{\Lambda\Sigma}F^{+\Lambda}_{\mu\nu}F^{+\Sigma\mu\nu}-\bar{a}_{\Lambda\Sigma}F^{-\Lambda}_{\mu\nu}F^{-\Sigma\mu\nu}-V\,.$
(7)

This Lagrangian is obtained from translating the first-order Lagrangian in the
geometric group manifold approach given in [26] to the usual space-time
Lagrangian. We have also multiplied the whole Lagrangian by a factor of $3$
resulting in a factor of $3$ in the scalar potential given below as compared
to that given in [26].
The self-dual and antiself-dual field strengths are defined by

$F^{\pm\Lambda}_{ab}=\frac{1}{2}\left(F_{ab}^{\Lambda}\mp\frac{i}{2}\epsilon_{abcd}F^{\Lambda
cd}\right)$ (8)

which satisfy the following relations

$\frac{1}{2}\epsilon_{abcd}F^{\pm\Lambda cd}=\pm
iF_{ab}^{\pm\Lambda}\qquad\textrm{and}\qquad
F_{ab}^{\pm\Lambda}=(F_{ab}^{\mp\Lambda})^{*}.$ (9)

To write down the explicit form of the scalar matrix $a_{\Lambda\Sigma}$ in
terms of the coset representative, we first identify various components of the
coset representative as

${L_{\Lambda}}^{\underline{\Sigma}}=\left(\begin{array}[]{cc}{L_{A}}^{B}&{L_{A}}^{i}\\\
{L_{j}}^{B}&{L_{j}}^{i}\\\ \end{array}\right).$ (10)

The symmetric matrix $a_{\Lambda\Sigma}$ can be written as

$a_{\Lambda\Sigma}=({\mathbf{f}^{\dagger}}^{-1}\mathbf{h}^{\dagger})_{\Lambda\Sigma}$
(11)

in which the matrices
${\mathbf{f}_{\Lambda}}^{\underline{\Sigma}}=({L_{\Lambda}}^{A},({L_{\Lambda}}^{i})^{*})$
and
$\mathbf{h}_{\Lambda\underline{\Sigma}}=-(J\mathbf{f}J)_{\Lambda\underline{\Sigma}}$
are given explicitly by

${\mathbf{f}_{\Lambda}}^{\underline{\Sigma}}=\left(\begin{array}[]{cc}{L_{A}}^{B}&({L_{A}}^{i})^{*}\\\
{L_{j}}^{B}&({L_{j}}^{i})^{*}\\\
\end{array}\right)\qquad\textrm{and}\qquad\mathbf{h}_{\Lambda\underline{\Sigma}}=\left(\begin{array}[]{cc}{L_{A}}^{B}&-({L_{A}}^{i})^{*}\\\
-{L_{j}}^{B}&({L_{j}}^{i})^{*}\\\ \end{array}\right).$ (12)

The scalar kinetic terms are written in terms of the vielbein on the
$SU(3,n)/SU(3)\times SU(n)\times U(1)$ obtained from the Maurer-Cartan one-
form

$\Omega_{\underline{\Lambda}}^{\phantom{\Lambda}\underline{\Pi}}=(L^{-1})_{\underline{\Lambda}}^{\phantom{\Lambda}\Sigma}dL_{\Sigma}^{\phantom{\Sigma}\underline{\Pi}}+(L^{-1})_{\underline{\Lambda}}^{\phantom{\Lambda}\Sigma}{f_{\Sigma\Omega}}^{\Gamma}A^{\Omega}L_{\Gamma}^{\phantom{\Gamma}\underline{\Pi}}$
(13)

via the components

$P_{i}^{\phantom{i}A}=\Omega_{i}^{\phantom{i}A}=(\Omega_{A}^{\phantom{A}i})^{*}\,.$
(14)

We also note that the upper and lower indices of $SU(3)$ and $SU(n)$ are
related by complex conjugation. Since ${L_{\Lambda}}^{\underline{\Sigma}}$ is
an element of $SU(3,n)$, the inverse
${(L^{-1})_{\underline{\Lambda}}}^{\Sigma}$ satisfies the following relation

$(L^{-1})_{\underline{\Lambda}}^{\phantom{\Lambda}\Sigma}=J_{\underline{\Lambda}\underline{\Pi}}J^{\Sigma\Delta}(L_{\Delta}^{\phantom{\Delta}\underline{\Pi}})^{*}\,.$
(15)

The composite connections $Q_{A}^{\phantom{A}B}$, $Q_{i}^{\phantom{i}j}$ and
$Q$ for the $SU(3)\times SU(n)\times U(1)$ local symmetry are given by

$\Omega_{A}^{\phantom{A}B}=Q_{A}^{\phantom{A}B}-n\delta^{B}_{A}Q\qquad\textrm{and}\qquad\Omega_{i}^{\phantom{i}j}=Q_{i}^{\phantom{i}j}+3\delta^{j}_{i}Q$
(16)

with $Q_{A}^{\phantom{A}A}=Q_{i}^{\phantom{i}i}=0$.
The scalar potential is given by

$\displaystyle V$ $\displaystyle=$
$\displaystyle-2S_{AB}S^{AB}+\frac{2}{3}\mathcal{U}_{A}\mathcal{U}^{A}+\frac{1}{6}\mathcal{N}_{iA}\mathcal{N}^{iA}+\frac{1}{6}\mathcal{M}^{iB}_{\phantom{iB}A}\mathcal{M}_{iB}^{\phantom{iB}A}$
(17) $\displaystyle=$
$\displaystyle\frac{1}{8}|C_{iA}^{\phantom{iA}B}|^{2}+\frac{1}{8}|C_{i}^{\phantom{A}PQ}|^{2}-\frac{1}{4}\left(|C_{A}^{\phantom{A}PQ}|^{2}-|C_{P}|^{2}\right)$

with $C_{P}=-C_{PM}^{\phantom{PM}M}$. Various components of the fermion-shift
matrices are defined in terms of the “boosted” structure constants

${C^{\underline{\Lambda}}}_{\underline{\Pi}\underline{\Gamma}}={L_{\Lambda}}^{\underline{\Lambda}}{(L^{-1})_{\underline{\Pi}}}^{\Pi}{(L^{-1})_{\underline{\Gamma}}}^{\Gamma}{f_{\Pi\Gamma}}^{\Lambda}\qquad\textrm{and}\qquad{C_{\underline{\Lambda}}}^{\underline{\Pi}\underline{\Gamma}}=J_{\underline{\Lambda}\underline{\Lambda}^{\prime}}J^{\underline{\Pi}\underline{\Pi}^{\prime}}J^{\underline{\Gamma}\underline{\Gamma}^{\prime}}{(C^{\underline{\Lambda}^{\prime}}}_{\underline{\Pi}^{\prime}\underline{\Gamma}^{\prime}})^{*}$
(18)

as

$\displaystyle S_{AB}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left(\epsilon_{BPQ}C_{A}^{\phantom{A}PQ}+\epsilon_{ABC}C_{M}^{\phantom{M}MC}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{8}\left(C_{A}^{\phantom{A}PQ}\epsilon_{BPQ}+C_{B}^{\phantom{A}PQ}\epsilon_{APQ}\right),$
$\displaystyle\mathcal{U}^{A}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}C_{M}^{\phantom{A}MA},\qquad\mathcal{N}_{iA}=-\frac{1}{2}\epsilon_{APQ}C_{i}^{\phantom{A}PQ},$
$\displaystyle\mathcal{M}_{iA}^{\phantom{iA}B}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\delta_{A}^{B}C_{iM}^{\phantom{iM}M}-2C_{iA}^{\phantom{iA}B}).$
(19)

Finally, the fermionic supersymmetry transformations obtained from the
rheonomic parametrization of the fermionic curvatures are given by111We also
note an additional factor of $\frac{1}{2}$ in the gauge field strengths due to
different conventions for differential forms, namely
$F^{\Lambda}_{\textrm{here}}=\frac{1}{2}F^{\Lambda}_{\mu\nu}dx^{\mu}\wedge
dx^{\nu}$ while
$F^{\Lambda}_{\textrm{\cite[cite]{[\@@bibref{}{N3_Ferrara}{}{}]}}}=F^{\Lambda}_{\mu\nu}dx^{\mu}\wedge
dx^{\nu}$.

$\displaystyle\delta\psi_{\mu A}$ $\displaystyle=$ $\displaystyle
D_{\mu}\epsilon_{A}-\epsilon_{ABC}G^{B}_{\mu\nu}\gamma^{\nu}\epsilon^{C}+S_{AB}\gamma_{\mu}\epsilon^{B},$
(20) $\displaystyle\delta\chi$ $\displaystyle=$
$\displaystyle-\frac{1}{4}G^{A}_{\mu\nu}\gamma^{\mu\nu}\epsilon_{A}+\mathcal{U}^{A}\epsilon_{A},$
(21) $\displaystyle\delta\lambda_{i}$ $\displaystyle=$ $\displaystyle-
P_{i\mu}^{\phantom{i}A}\gamma^{\mu}\epsilon_{A}+\mathcal{N}_{iA}\epsilon^{A},$
(22) $\displaystyle\delta\lambda_{iA}$ $\displaystyle=$ $\displaystyle-
P_{i\mu}^{\phantom{i}B}\gamma^{\mu}\epsilon_{ABC}\epsilon^{C}-\frac{1}{2}G_{i\mu\nu}\gamma^{\mu\nu}\epsilon_{A}+\mathcal{M}_{iA}^{\phantom{iA}B}\epsilon_{B}\,.$
(23)

The covariant derivative for $\epsilon_{A}$ is defined by

$D\epsilon_{A}=d\epsilon_{A}+\frac{1}{4}\omega^{ab}\gamma_{ab}\epsilon_{A}+Q_{A}^{\phantom{A}B}\epsilon_{B}+\frac{1}{2}nQ\epsilon_{A}\,.$
(24)

The field strengths appearing in the supersymmetry transformations are given
by

$\displaystyle G^{A}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\textrm{Re}\,a_{\Lambda\Sigma}{L_{\Sigma}}^{A}F^{+\Lambda}_{\mu\nu}=M^{AB}(L^{-1})_{B}^{\phantom{B}\Lambda}F^{+}_{\Lambda\mu\nu},$
(25) $\displaystyle G^{i}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\textrm{Re}\,a_{\Lambda\Sigma}({L_{\Sigma}}^{i})^{*}F^{+\Lambda}_{\mu\nu}=-M^{ij}(L^{-1})_{j}^{\phantom{j}\Lambda}F^{-}_{\Lambda\mu\nu}$
(26)

where $M^{ij}$ and $M^{AB}$ are respectively inverse matrices of

$M_{ij}=(L^{-1})_{i}^{\phantom{j}\Lambda}(L^{-1})_{j}^{\phantom{j}\Pi}J_{\Lambda\Pi}\qquad\textrm{and}\qquad
M_{AB}=(L^{-1})_{A}^{\phantom{A}\Lambda}(L^{-1})_{B}^{\phantom{B}\Pi}J_{\Lambda\Pi}\,.$
(27)

###  2.2 BPS equations for supersymmetric $AdS_{4}$ black holes

We now look at the BPS equations for supersymmetric $AdS_{4}$ black holes with
the near horizon geometry given by $AdS_{2}\times\Sigma^{2}$. The metric
ansatz is taken to be

$ds^{2}=-e^{2f(r)}dt^{2}+dr^{2}+e^{2h(r)}(d\theta^{2}+F(\theta)^{2}d\phi^{2})$
(28)

with $F(\theta)$ defined by

$F(\theta)=\sin\theta\qquad\textrm{and}\qquad F(\theta)=\sinh\theta$ (29)

for $\Sigma^{2}=S^{2}$ and $\Sigma^{2}=H^{2}$, respectively. The functions
$f(r)$ and $h(r)$ together with all other non-vanishing fields only depend on
the radial coordinate $r$. With the following choice of vielbein

$e^{\hat{t}}=e^{f}dt,\qquad e^{\hat{r}}=dr,\qquad
e^{\hat{\theta}}=e^{h}d\theta,\qquad e^{\hat{\phi}}=e^{h}F(\theta)d\phi,$ (30)

it is straightforward to compute non-vanishing components of the spin
connection

$\displaystyle\omega^{\hat{t}\hat{r}}$ $\displaystyle=$ $\displaystyle
f^{\prime}e^{\hat{t}},\qquad\omega^{\hat{\theta}\hat{r}}=h^{\prime}e^{\hat{\theta}},$
$\displaystyle\omega^{\hat{\phi}\hat{r}}$ $\displaystyle=$ $\displaystyle
h^{\prime}e^{\hat{\phi}},\qquad\omega^{\hat{\theta}\hat{\phi}}=\frac{F^{\prime}(\theta)}{F(\theta)}e^{-h}e^{\hat{\phi}}\,.$
(31)

For clarity, we have used the values of flat indices as
$a,b,\ldots,=(\hat{t},\hat{r},\hat{\theta},\hat{\phi})$.
In the present paper, we are interested in a simple $N=3$ gauged supergravity
coupled to $n=3$ vector multiplets with a compact gauge group $SO(3)\times
SO(3)$. The non-vanishing components of $f_{\Lambda\Sigma\Gamma}$ are given by

$f_{\Lambda\Sigma}^{\phantom{\Lambda\Sigma}\Gamma}=(g_{1}\epsilon_{ABC},g_{2}\epsilon_{ijk}).$
(32)

We also recall that the $SO(3)\times SO(3)$ gauge group is electrically gauged
with the corresponding gauge fields being the vector fields appearing in the
ungauged Lagrangian with YM kinetic terms. To avoid confusion, we will call
the first $SO(3)$ factor $SO(3)_{R}$ since this factor is embedded in
$SU(3)_{R}$ R-symmetry.
To preserve some amount of supersymmetry, we implement a topological twist by
turning an $SO(2)\sim U(1)\subset SO(3)_{R}\subset SU(3)_{R}$ gauge field
along $\Sigma^{2}$. In addition, we can also turn on an $SO(2)\subset SO(3)$
gauge field from the second $SO(3)$ factor. We will choose these gauge fields
to be $A^{3}_{\mu}$ and $A^{6}_{\mu}$ with the following ansatz

$\displaystyle
A^{\Lambda}=\tilde{q}^{\Lambda}(r)dt-p^{\Lambda}(r)F^{\prime}(\theta)d\phi,\qquad\Lambda=3,6,$
(33)

for $F^{\prime}(\theta)=\frac{dF(\theta)}{d\theta}$. The corresponding field
strengths are given by

$F^{\Lambda}=dA^{\Lambda}=\tilde{q}^{\Lambda^{\prime}}dr\wedge
dt-p^{\Lambda^{\prime}}F^{\prime}(\theta)dr\wedge d\phi+\kappa
p^{\Lambda}F(\theta)d\theta\wedge d\phi\,.$ (34)

Throughout the paper, we will use ′ to denote a derivative with respect to the
radial coordinate $r$ with an exception for
$F^{\prime}(\theta)=\frac{dF(\theta)}{d\theta}$. In this equation, we have
also introduced a parameter $\kappa$ via the relation
$F^{\prime\prime}(\theta)=-\kappa F(\theta)$ with $\kappa=1$ and $\kappa=-1$
for $\Sigma^{2}=S^{2}$ and $\Sigma^{2}=H^{2}$, respectively. Imposing the
Bianchi’s identity $DF^{\Lambda}=0$ implies $p^{\Lambda^{\prime}}=0$, so
$p^{\Lambda}$ are constant and will be identified with magnetic charges.
It is useful to recall the definition of electric and magnetic charges given
by

$q_{\Lambda}=\frac{1}{4\pi}\int_{\Sigma^{2}}G_{\Lambda}\qquad\textrm{and}\qquad
p^{\Lambda}=\frac{1}{4\pi}\int_{\Sigma^{2}}F^{\Lambda}$ (35)

with $G_{\Lambda}=\frac{\delta S}{\delta F^{\Lambda}}$. To further fix the
ansatz for the gauge fields, we consider the Lagrangian for the gauge fields

$\mathcal{L}_{\textrm{gauge}}=-\frac{1}{2}R_{\Lambda\Sigma}*F^{\Lambda}\wedge
F^{\Sigma}+\frac{1}{2}I_{\Lambda\Sigma}F^{\Lambda}\wedge F^{\Sigma}$ (36)

in which we have rewritten the relevant terms in the Lagrangian (7) in
differential form language. We have also used the following definition

$R_{\Lambda\Sigma}=\textrm{Re}\,a_{\Lambda\Sigma}\qquad\textrm{and}\qquad
I_{\Lambda\Sigma}=\textrm{Im}\,a_{\Lambda\Sigma}\,.$ (37)

From the above Lagrangian, we find

$G_{\Lambda}=-R_{\Lambda\Sigma}*F^{\Sigma}+I_{\Lambda\Sigma}F^{\Sigma}$ (38)

which, together with the above definition of $(q_{\Lambda},p^{\Lambda})$ and
$F^{\Lambda}_{\theta\phi}=\kappa p^{\Lambda}F(\theta)$, leads to

$F^{\Lambda}_{\hat{t}\hat{r}}=-e^{-2h}R^{\Lambda\Sigma}(\kappa
I_{\Sigma\Gamma}p^{\Gamma}+q_{\Sigma}).$ (39)

We have written the inverse of $R_{\Lambda\Sigma}$ as $R^{\Lambda\Sigma}$. For
later convenience, we also note the Maxwell equations obtained from the
Lagrangian (7)

$D_{\nu}(R_{\Lambda\Sigma}F^{\Sigma\mu\nu}+\frac{1}{2}I_{\Lambda\Sigma}e^{-1}\epsilon^{\mu\nu\rho\sigma}F^{\Sigma}_{\rho\sigma})={{P^{\mu}}_{A}}^{i}{(L^{-1})_{i}}^{\Sigma}{f_{\Lambda\Sigma}}^{\Gamma}{L_{\Gamma}}^{A}\,.$
(40)

This can be rewritten in form language as

$*DG_{\Lambda}=P^{Ai}{(L^{-1})_{i}}^{\Sigma}{f_{\Lambda\Sigma}}^{\Gamma}{L_{\Gamma}}^{A}$
(41)

with $P^{Ai}=P^{Ai}_{\mu}dx^{\mu}$. It should also be emphasized that the
left-hand side is related to a radial derivative of electric charges via the
definition in (35). Therefore, in general, electric charges are not conserved
if the YM currents are non-vanishing as also pointed out in [17].

We are now in a position to perform the analysis of BPS equations. The
analysis is closely parallel to that in $N=2$ gauged supergravity given in
[14] and [17]. We will work in Majorana representation with all $\gamma^{a}$
real but
$\gamma_{5}=i\gamma^{\hat{t}}\gamma^{\hat{r}}\gamma^{\hat{\theta}}\gamma^{\hat{\phi}}$
purely imaginary. In this representation, the two chiral components
$\epsilon_{A}$ and $\epsilon^{A}$ of the Killing spinors are related to each
other by complex conjugation. In addition, in all of the solutions considered
in this work, we assume that the Killing spinors depend only on the radial
coordinate $r$. We are only interested in solutions with $SO(2)\times SO(2)$
and $SO(2)_{\textrm{diag}}$ symmetries, but in this section, we will consider
the general structure of the BPS equations.
We begin with the BPS equation from the variation $\delta\psi_{\hat{\phi}A}$
given by

$0=\frac{1}{2}h^{\prime}\gamma_{\hat{\phi}\hat{r}}\epsilon_{A}+\frac{1}{2}e^{-h}\frac{F^{\prime}}{F}\gamma_{\hat{\phi}\hat{\theta}}\epsilon_{A}+{\Omega_{\hat{\phi}A}}^{B}\epsilon_{B}+S_{AB}\epsilon^{B}-\epsilon_{ABC}G^{B}_{\hat{\phi}\hat{\theta}}\gamma^{\hat{\theta}}\epsilon^{C}\,.$
(42)

The matrix $S_{AB}$ is symmetric and can be diagonalized. The corresponding
eigenvalues will lead to the superpotential $\mathcal{W}$ in terms of which
the scalar potential can be written. We then write, without summation on $A$,

$S_{AB}=-\frac{1}{2}\mathcal{W}_{A}\delta_{AB}$ (43)

in which $\mathcal{W}_{A}$ denote eigenvalues of $S_{AB}$. It is also useful
to define the central charge matrix as

$\mathcal{Z}_{AB}=-2\epsilon_{ABC}G^{C}_{\hat{\theta}\hat{\phi}}=-2\epsilon_{ABC}M^{CD}{(L^{-1})_{D}}^{\Lambda}(\kappa
p^{\Lambda}e^{-2h}-i\tilde{q}^{\Lambda^{\prime}}e^{-f}).$ (44)

We now impose the following projector

$\gamma^{\hat{r}}\epsilon^{A}=e^{-i\Lambda}\delta^{AB}\epsilon_{B}\qquad\textrm{or}\qquad\gamma^{\hat{r}}\epsilon_{A}=e^{i\Lambda}\delta_{AB}\epsilon^{B}$
(45)

and rewrite equation (42) as

$0=\gamma_{\hat{\phi}}\left[h^{\prime}e^{i\Lambda}\delta_{AB}-\mathcal{W}_{A}\delta_{AB}-Z_{AB}\gamma_{\hat{\phi}\hat{\theta}}\right]\epsilon^{B}+\gamma_{\hat{\phi}}\left[\frac{F^{\prime}(\theta)}{F(\theta)}e^{-h}\gamma_{\hat{\phi}\hat{\theta}}\epsilon_{A}+2g_{1}{\epsilon_{AC}}^{B}A^{C}_{\hat{\phi}}\epsilon_{B}\right].$
(46)

We have used the explicit form of
${\Omega_{\hat{\phi}_{A}}}^{B}=g_{1}{\epsilon_{ACD}}\delta^{DB}A^{C}_{\hat{\phi}}\epsilon_{B}$
which is valid for both cases we are interested in. We now notice that only
the terms in the second bracket of equation (46) depend on $\theta$.
Therefore, these terms must cancel against each other, and using the gauge
field ansatz (33), we find that

$\gamma_{\hat{\phi}\hat{\theta}}\epsilon_{A}=2g_{1}{\epsilon_{AC}}^{B}p^{C}\epsilon_{B}\,.$
(47)

Since only $p^{3}$ is non-vanishing, we find that the supersymmetry
corresponding to $\epsilon^{3}$ must be broken. Imposing the twist condition

$2g_{1}p^{3}=1$ (48)

and writting $\epsilon_{AB3}=\epsilon_{\hat{A}\hat{B}}$ for
$\hat{A},\hat{B}=1,2$ and $\epsilon_{12}=1$, we obtain the following projector

$\gamma_{\hat{\theta}\hat{\phi}}\epsilon_{\hat{A}}={\epsilon_{\hat{A}}}^{\hat{B}}\epsilon_{\hat{B}}\,.$
(49)

In this analysis, we have written
$\epsilon^{A}=(\epsilon^{\hat{A}},\epsilon^{3})$. We also remark that indices
$\hat{A},\hat{B},\ldots$ of $\epsilon_{\hat{A}\hat{B}}$ and
$\epsilon^{\hat{A}\hat{B}}$ are simply raised and lowered by the Kronecker
delta $\delta_{\hat{A}\hat{B}}$ and $\delta^{\hat{A}\hat{B}}$.
Using the projector (49) in the first bracket of (46) with $\epsilon^{3}=0$,
we find the BPS condition

$(h^{\prime}e^{i\Lambda}-\mathcal{W}_{\hat{A}})\delta_{\hat{A}\hat{B}}-\mathcal{Z}_{\hat{A}\hat{C}}{\epsilon^{\hat{C}}}_{\hat{B}}=0\,.$
(50)

In general, $\mathcal{W}_{\hat{A}}$ for a particular value of $\hat{A}$ gives
the superpotential corresponding to the eigenvalue of $S_{\hat{A}\hat{B}}$
along the directions of the Killing spinors $\epsilon^{\hat{A}}$. We will
simply denote this eigenvalue by $\mathcal{W}$. Moreover, it turns out that in
the cases we will consider, only $G^{3}_{\mu\nu}$ is non-vanishing. We then
find that

$\mathcal{Z}_{\hat{A}\hat{C}}{\epsilon^{\hat{C}}}_{\hat{B}}=\mathcal{Z}\delta_{\hat{A}\hat{B}}$
(51)

in which we have defined a complex number $\mathcal{Z}$ sometimes called the
“central charge” as

$\mathcal{Z}=2M^{3A}{(L^{-1})_{A}}^{\Lambda}(\kappa
p^{\Lambda}e^{-2h}-i\tilde{q}^{\Lambda^{\prime}}e^{-f}).$ (52)

With all these, we finally obtain the BPS equation from
$\delta\psi_{\hat{\phi}A}$

$h^{\prime}e^{i\Lambda}=\mathcal{W}+\mathcal{Z}$ (53)

which implies

$h^{\prime}=\pm|\mathcal{W}+\mathcal{Z}|\qquad\textrm{and}\qquad
e^{i\Lambda}=\pm\frac{\mathcal{W}+\mathcal{Z}}{|\mathcal{W}+\mathcal{Z}|}\,.$
(54)

Using all of the results previously obtained, we can perform a similar
analysis for $\delta\psi_{\hat{\theta}A}$. This results, as expected, in the
same BPS equations given in (54).
We now move to the variation $\delta\psi_{\hat{t}\hat{A}}$ of the form

$\displaystyle
0=\frac{1}{2}f^{\prime}\gamma_{\hat{t}}\gamma_{\hat{r}}\epsilon_{\hat{A}}+{\mathcal{A}_{\hat{t}\hat{A}}}^{\hat{B}}\epsilon_{\hat{B}}+\epsilon_{\hat{A}\hat{C}}G^{3}_{\hat{t}\hat{r}}\gamma^{\hat{r}}\epsilon^{\hat{C}}+S_{\hat{A}\hat{B}}\gamma_{\hat{t}}\epsilon^{\hat{B}}$
(55)

with

${\mathcal{A}_{\hat{t}A}}^{B}=-g_{1}\epsilon_{\hat{A}\hat{B}}A^{3}_{\hat{t}}\,.$
(56)

We then impose another projector

$\gamma^{\hat{t}}\epsilon^{\hat{A}}=ie^{-i\Lambda}\epsilon^{\hat{A}\hat{B}}\epsilon_{\hat{B}}\,.$
(57)

It should be noted that this is not an independent projector since it is
implied by the $\gamma_{\hat{r}}$ and $\gamma_{\hat{\theta}\hat{\phi}}$
projectors given in (45) and (49) by the relation
$\gamma_{5}\epsilon^{\hat{A}}=-\epsilon^{\hat{A}}$.
We note here that the central charge matrix can also be written as

$\mathcal{Z}_{AB}=-2i\epsilon_{ABC}G^{C}_{\hat{t}\hat{r}}=2i\epsilon_{ABC}M^{CD}{(L^{-1})_{D}}^{\Lambda}(\tilde{q}^{\Lambda^{\prime}}e^{-f}+i\kappa
p^{\Lambda}e^{-2h}).$ (58)

With all the previous results, we can write equation (55) as

$\left[f^{\prime}-e^{-i\Lambda}(\mathcal{W}+\mathcal{Z})\right]\epsilon_{\hat{A}\hat{B}}-2ig_{1}\epsilon_{\hat{A}\hat{B}}A^{3}_{\hat{t}}=0$
(59)

which implies

$\displaystyle f^{\prime}$ $\displaystyle=$
$\displaystyle\textrm{Re}\,\left[e^{-i\Lambda}(\mathcal{W}-\mathcal{Z})\right],$
(60) $\displaystyle\textrm{and}\qquad g_{1}A^{3}_{\hat{t}}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\textrm{Im}\,\left[e^{-i\Lambda}(\mathcal{W}-\mathcal{Z})\right].$
(61)

The second equation fixes the form of $A^{3}_{t}$.
Finally, we consider the variation $\delta\psi_{\hat{r}A}$ which gives

$\epsilon^{\prime}_{\hat{A}}-\frac{1}{2}e^{-i\Lambda}(\mathcal{W}-\mathcal{Z})\epsilon_{\hat{A}}+\frac{3}{2}Q_{r}\epsilon_{\hat{A}}+{Q_{r\hat{A}}}^{\hat{B}}\epsilon_{\hat{B}}=0\,.$
(62)

In all the cases we will consider, it turns out that $Q_{r}=0$ and
${Q_{r\hat{A}}}^{\hat{B}}=0$. Using $\delta\psi_{\hat{t}\hat{A}}=0$ equation,
we can rewrite this equation as

$\epsilon^{\prime}_{\hat{A}}=\frac{1}{2}(f^{\prime}-2ig_{1}A^{3}_{\hat{t}})\epsilon_{\hat{A}}$
(63)

which gives

$\epsilon_{\hat{A}}=e^{\frac{f}{2}-i\int
g_{1}A^{3}_{\hat{t}}dr}\epsilon^{(0)}_{\hat{A}}$ (64)

with $\epsilon^{(0)}_{\hat{A}}$ being $r$-independent spinors subject to the
projectors

$\gamma_{\hat{r}}\epsilon^{(0)}_{\hat{A}}=\delta_{\hat{A}\hat{B}}\epsilon^{(0)\hat{B}}\qquad\textrm{and}\qquad\gamma_{\hat{\theta}\hat{\phi}}\epsilon^{(0)}_{\hat{A}}={\epsilon_{\hat{A}}}^{\hat{B}}\epsilon^{(0)}_{\hat{B}}\,.$
(65)

Consistency with the projector (45) leads to a flow equation for the phase
$\Lambda$

$\Lambda^{\prime}+2ig_{1}A^{3}_{\hat{t}}=0\,.$ (66)

Since all scalars depend only on the radial coordinate $r$, the BPS equations
obtained from $\delta\chi$, $\delta\lambda_{i}$ and $\delta\lambda^{A}_{i}$
only involve $\gamma_{\hat{r}}$. By using the projector (45) and phase factor
in (54) in these variations, we eventually obtain flow equations for scalars.
Before giving the solutions, we end this section with the conditions for the
near horizon geometry $AdS_{2}\times\Sigma^{2}$

$f^{\prime}=\frac{1}{L_{AdS_{2}}},\qquad
h^{\prime}=0,\qquad({z_{i}}^{A})^{\prime}=0$ (67)

meaning that the function $h$ and all scalars are constant, and $f$ is linear
in $r$ in this limit. We will also choose an upper sign choice in (54) for
definiteness.

###  2.3 Solutions with $SO(2)\times SO(2)$ symmetry

We now consider supersymmetric solutions to the BPS equations with the general
structure given in the previous section. We begin with explicit
parametrization of the $SU(3,3)/SU(3)\times SU(3)\times U(1)$ coset manifold.
It is convenient to introduce a basis for $GL(6,\mathbb{R})$ matrices

$(e_{\Lambda\Sigma})_{\Pi\Gamma}=\delta_{\Lambda\Pi}\delta_{\Sigma\Gamma}\,.$
(68)

With the structure constants given in (32), the $SO(3)_{R}\times SO(3)$ gauge
generators are given by

$(T^{(1)}_{A})_{\Pi}^{\phantom{\Pi}\Gamma}=f_{A\Pi}^{\phantom{\Lambda\Pi}\Gamma}\qquad\textrm{and}\qquad(T^{(2)}_{i})_{\Pi}^{\phantom{\Pi}\Gamma}=f_{i+3,\Pi}^{\phantom{i+3,\Pi}\Gamma}\,.$
(69)

The residual $SO(2)\times SO(2)$ symmetry is generated by $T^{(1)}_{3}$ and
$T^{(2)}_{3}$. There are two singlet scalars corresponding to the following
$SU(3,3)$ non-compact generators

$\hat{Y}_{1}=e_{36}+e_{63}\qquad\textrm{and}\qquad\hat{Y}_{2}=ie_{63}-ie_{36}\,.$
(70)

The coset representative can be written as

$L=e^{\phi_{1}\hat{Y}_{1}}e^{\phi_{2}\hat{Y}_{2}}\,.$ (71)

In this case, the YM currents vanish, so the electric charges are constant.
The scalar potential is given by

$V=-\frac{1}{2}g_{1}^{2}e^{-2\phi_{1}}\left[e^{2\phi_{1}}+\cosh
2\phi_{2}(1+e^{4\phi_{1}})\right].$ (72)

This potential admits a unique $N=3$ supersymmetric vacuum at
$\phi_{1}=\phi_{2}=0$ with the cosmological constant
$V_{0}=-\frac{3}{2}g_{1}^{2}$. The $AdS_{4}$ radius is given by the relation

$L_{AdS_{4}}=\sqrt{-\frac{3}{2V_{0}}}=\frac{1}{g_{1}}$ (73)

in which we have taken $g_{1}>0$ for convenience. We also note that truncating
all vector multiplets out gives rise to pure $N=3$ gauged supergravity with
$SO(3)_{R}$ gauge group and cosmological constant $-\frac{3}{2}g_{1}^{2}$
constructed in [29] and [30], see also a more recent result [31] in which pure
$N=3$ gauged supergravity is embedded in massive type IIA theory.
The matrix $S_{AB}$ is given by

$S_{AB}=-\frac{1}{2}\textrm{diag}(\mathcal{W}_{1},\mathcal{W}_{1},\mathcal{W}_{2})$
(74)

in which $\mathcal{W}_{1}$ and $\mathcal{W}_{2}$ are given by

$\displaystyle\mathcal{W}_{1}$ $\displaystyle=$ $\displaystyle
g_{1}\cosh\phi_{1}\cosh\phi_{2},$ $\displaystyle\mathcal{W}_{2}$
$\displaystyle=$ $\displaystyle
g_{1}(\cosh\phi_{1}\cosh\phi_{2}+i\sinh\phi_{1}\sinh\phi_{2}).$ (75)

It turns out that only $\mathcal{W}_{2}$ gives the superpotential in terms of
which the scalar potential (72) can be written as, see more detail in [33],

$V=-\frac{1}{2\cosh^{2}2\phi_{2}}\left(\frac{\partial
W_{2}}{\partial\phi_{1}}\right)^{2}-\frac{1}{2}\left(\frac{\partial
W_{2}}{\partial\phi_{2}}\right)^{2}-\frac{3}{2}W^{2}_{2}$ (76)

with $W_{2}=|\mathcal{W}_{2}|$. In this case, the supersymmetry associated
with $\epsilon^{1,2}$, which are relevant to the present work, is broken. For
$\phi_{2}=0$, $\mathcal{W}_{1}$ can give rise to the superpotential leading to
unbroken supersymmetry along $\epsilon^{1,2}$, and in this case,
$\mathcal{W}_{1}$ and $\mathcal{W}_{2}$ are equal. We then set $\phi_{2}=0$ in
the following analysis. We will also write
$\mathcal{W}=\mathcal{W}_{1}=\mathcal{W}_{2}$ and $\phi=\phi_{1}$. In
addition, it is worth noting that setting pseudo-scalars, corresponding to
imaginary parts of the complex scalars ${z_{A}}^{i}$, to zero always gives
$I_{\Lambda\Sigma}=0$. This implies that the components
$F^{\Lambda}_{\hat{t}\hat{r}}$ are given only in terms of electric charges and
vanish for purely magnetic solutions.
With $\epsilon^{3}=0$, we find that $\delta\chi=0$ and $\delta\lambda_{i}=0$
identically. By using the coset representative (71) with $\phi_{2}=0$, we find
a consistent BPS equation for $\phi$ from $\delta\lambda^{A}_{i}$ provided
that one of these two conditions is satified

$q_{3}=q_{6}=0\qquad\textrm{or}\qquad p^{6}=q_{6}=0\,.$ (77)

The first one corresponds to a purely magnetic case while the second one is a
dyonic case with only $q_{3}$ and $p^{3}$ non-vanishing.
Setting $q_{3}=q_{6}=0$ and using the BPS equations given in the previous
section, we find the following set of BPS equations

$\displaystyle f^{\prime}$ $\displaystyle=$
$\displaystyle\frac{1}{2}e^{-\phi-2h}\left[g_{1}e^{2h}-g_{1}e^{2\phi}(p^{3}-p^{6})\kappa+g_{1}e^{2(\phi+h)}-\kappa(p^{3}+p^{6})\right],$
(78) $\displaystyle h^{\prime}$ $\displaystyle=$
$\displaystyle|\mathcal{W}+\mathcal{Z}|$ (79) $\displaystyle=$
$\displaystyle\frac{1}{2}e^{-\phi-2h}\left[g_{1}e^{2h}+g_{1}e^{2\phi}(p^{3}-p^{6})\kappa+g_{1}e^{2(\phi+h)}+\kappa(p^{3}+p^{6})\right],$
$\displaystyle\phi^{\prime}$ $\displaystyle=$
$\displaystyle-\frac{\partial|\mathcal{W}+\mathcal{Z}|}{\partial\phi}$ (80)
$\displaystyle=$
$\displaystyle\frac{1}{2}e^{-\phi-2h}\left[g_{1}e^{2h}-g_{1}e^{2\phi}(p^{3}-p^{6})\kappa-
g_{1}e^{2(\phi+h)}+\kappa(p^{3}+p^{6})\right].$

We note that both $\mathcal{W}$ and $\mathcal{Z}$ are real giving rise to
$e^{i\Lambda}=\pm 1$. The existence of $AdS_{2}\times\Sigma^{2}$ fixed points
requires $p^{6}=0$. In this case, we find the fixed point given by

$\phi=\phi_{0},\qquad h=\frac{1}{2}\ln\left[-\frac{\kappa
p^{3}}{g_{1}}\right],\qquad
f^{\prime}=\frac{1}{L_{AdS_{2}}}=2g_{1}\cosh\phi_{0}$ (81)

for a constant $\phi_{0}$. For real $h$ and $2g_{1}p^{3}=1>0$, we need to take
$\kappa=-1$, so this is an $AdS_{2}\times H^{2}$ fixed point.
For $p^{6}=q_{6}=0$, we find

$\mathcal{W}+\mathcal{Z}=\frac{1}{2}e^{-\phi-2h}(1+e^{2\phi})(e^{2h}g_{1}+\kappa
p^{3}+iq_{3})$ (82)

leading to the BPS equations

$\displaystyle f^{\prime}$ $\displaystyle=$
$\displaystyle\frac{e^{-\phi-2h}(1+e^{2\phi})(e^{4h}g_{1}^{2}-q_{3}^{3}-(p^{3})^{2})}{2\sqrt{(e^{2h}g_{1}+\kappa
p^{3})^{2}+q_{3}^{2}}},$ (83) $\displaystyle h^{\prime}$ $\displaystyle=$
$\displaystyle|\mathcal{W}+\mathcal{Z}|=\frac{1}{2}e^{-\phi-2h}(1+e^{2\phi})\sqrt{(\kappa
p^{3}+g_{1}e^{2h})^{2}+q_{3}^{2}},$ (84) $\displaystyle\phi^{\prime}$
$\displaystyle=$
$\displaystyle-\frac{\partial|\mathcal{W}+\mathcal{Z}|}{\partial\phi}=-\frac{1}{2}e^{-\phi-2h}(e^{2\phi}-1)\sqrt{(\kappa
p^{3}+g_{1}e^{2h})^{2}+q_{3}^{2}}$ (85)

together with

$\tilde{q}^{3}=-\frac{q_{3}e^{-\phi+f}(1+e^{2\phi})}{2\sqrt{(e^{2h}g_{1}+\kappa
p^{3})^{2}+q_{3}^{2}}}$ (86)

which fixes the time component of the gauge field ansatz. We also note that
upon using the BPS equations for $f^{\prime}$, $h^{\prime}$ and
$\phi^{\prime}$, we find

$\tilde{q}^{3^{\prime}}=\frac{1}{2}q_{3}e^{-2\phi+f-2h}(1+e^{4\phi})$ (87)

in agreement with the gauge field ansatz given in (39).
The existence of $AdS_{2}\times\Sigma^{2}$ fixed points requires $q_{3}=0$.
This can be clearly seen from the condition $h^{\prime}=0$. With $q_{3}=0$,
the $AdS_{2}\times\Sigma^{2}$ fixed point is just the $AdS_{2}\times H^{2}$
vacuum given in (81). We then find that all supersymmetric black hole
solutions will be magnetically charged without any dyonic generalization.
We now look for a solution interpolating between the supersymmetric $AdS_{4}$
vacuum and this $AdS_{2}\times H^{2}$ critical point. To find the relevant
solution, we can further set $p^{6}=0$ and $q_{3}=0$ in the two sets of the
BPS equations. In this case, the two sets lead to the same BPS equations which
we repeat here for convenience

$\displaystyle f^{\prime}$ $\displaystyle=$
$\displaystyle\frac{1}{2}e^{-\phi-2h}(1+e^{2\phi})(g_{1}e^{2h}-\kappa p^{3}),$
(88) $\displaystyle h^{\prime}$ $\displaystyle=$
$\displaystyle\frac{1}{2}e^{-\phi-2h}(1+e^{2\phi})(g_{1}e^{2h}+\kappa p^{3}),$
(89) $\displaystyle\phi^{\prime}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}e^{-\phi-2h}(e^{2\phi}-1)(g_{1}e^{2h}+\kappa
p^{3}).$ (90)

These equations are very similar to those given in $N=5$ and $N=6$ gauged
supergravities studied in [23] and [24]. By a similar analysis, we can obtain
an analytic solution

$\displaystyle h$ $\displaystyle=$ $\displaystyle\phi-\ln(1-e^{2\phi}),$ (91)
$\displaystyle f$ $\displaystyle=$ $\displaystyle\ln[\kappa
p^{3}(1+e^{4\phi})+(g_{1}-2\kappa p^{3})e^{2\phi}]-\ln(1-e^{2\phi})-\phi,$
(92) $\displaystyle 2g_{1}\rho$ $\displaystyle=$ $\displaystyle\ln[\kappa
p^{3}(1+e^{4\phi})+(g_{1}-2\kappa p^{3})e^{2\phi}]-2\ln(1-e^{2\phi})$ (93)
$\displaystyle+2\sqrt{\frac{g_{1}}{4\kappa
p^{3}-g_{1}}}\tan^{-1}\left[\frac{g_{1}+2\kappa
p^{3}(e^{2\phi}-1)}{\sqrt{g_{1}(4\kappa p^{3}-g_{1})}}\right]$

with the new radial coordinate $\rho$ defined by $\frac{d\rho}{dr}=e^{-\phi}$.
As $\phi\sim 0$, we find that the solution becomes

$f\sim h\sim g_{1}r\qquad\textrm{and}\qquad\phi\sim e^{-g_{1}\rho}\sim
e^{-g_{1}r}\sim e^{-\frac{r}{L_{AdS_{4}}}}$ (94)

which is an asymptotically locally $AdS_{4}$ preserving the full $N=3$
supersymmetry. On the other hand, for $\phi\sim\phi_{0}$ with

$\phi_{0}=\frac{1}{2}\ln\left[\frac{2\kappa
p^{3}-g_{1}+\sqrt{g_{1}(g_{1}-4\kappa p^{3})}}{2\kappa p^{3}}\right],$ (95)

the solution approaches the $AdS_{2}\times H^{2}$ fixed point with

$h\sim\frac{1}{2}\ln\left[-\frac{\kappa p^{3}}{g_{1}}\right],\qquad\phi\sim
e^{-\frac{r}{L_{AdS_{2}}}},\qquad f\sim\frac{r}{L_{AdS_{2}}}$ (96)

for $L_{AdS_{2}}=\frac{1}{g_{1}}\sqrt{\frac{\kappa p^{3}}{4\kappa
p^{3}-g_{1}}}$.
We end this section by a comment on the solution of pure $N=3$ gauged
supergravity in which we set $\phi=0$ for the entire solution. This simply
gives the following solution

$h=\frac{1}{2}\ln\left[\frac{e^{2g_{1}(r-r_{0})}-\kappa
p^{3}}{g_{1}}\right]\quad\textrm{and}\quad
f=2g_{1}r-\frac{1}{2}\ln(e^{2g_{1}(r-r_{0})}-\kappa p^{3})$ (97)

for a constant $r_{0}$. This solution can be embedded in massive type IIA
theory via $S^{6}$ truncation given in [31]. Alternatively, this solution can
also be embedded in eleven dimensions using a consistent truncation on a
trisasakian manifold given in [32].

###  2.4 Solutions with $SO(2)_{\textrm{diag}}$ symmetry

We now consider solutions with $SO(2)_{\textrm{diag}}$ symmetry generated by
$T^{(1)}_{3}+T^{(2)}_{3}$. There are six singlet scalars corresponding to the
following non-compact generators

$\displaystyle\bar{Y}_{1}$ $\displaystyle=$ $\displaystyle
e_{36}+e_{63},\qquad\bar{Y}_{2}=-ie_{36}+ie_{63},$ $\displaystyle\bar{Y}_{3}$
$\displaystyle=$ $\displaystyle
e_{25}+e_{52}+e_{14}+e_{41},\qquad\bar{Y}_{4}=-ie_{25}+ie_{52}-ie_{14}+ie_{41},$
$\displaystyle\bar{Y}_{5}$ $\displaystyle=$ $\displaystyle
e_{15}+e_{51}-e_{24}-e_{42},\qquad\bar{Y}_{6}=-ie_{15}+ie_{51}+ie_{24}-ie_{42}\,.$
(98)

The coset representative is given by

$L=e^{\phi_{1}\bar{Y}_{1}}e^{\phi_{2}\bar{Y}_{2}}e^{\phi_{3}\bar{Y}_{3}}e^{\phi_{4}\bar{Y}_{4}}e^{\phi_{5}\bar{Y}_{5}}e^{\phi_{6}\bar{Y}_{6}}\,.$
(99)

In this case, the scalar potential turns out to be highly complicated. We
refrain from giving its explicit form here, but it is useful to note that
there are two supersymmetric $AdS_{4}$ vacua, see more detail in [33]. The
first one is the $N=3$ supersymmetric $AdS_{4}$ vacuum with all scalars
vanishing and the full $SO(3)\times SO(3)$ gauge group unbroken. This is the
same as the $AdS_{4}$ critical point mentioned in the previous section. The
second one is another $N=3$ $AdS_{4}$ critical point with
$SO(3)_{\textrm{diag}}\subset SO(3)\times SO(3)$ symmetry given by

$\phi_{1}=\pm\phi_{3}=\frac{1}{2}\ln\left[\frac{g_{2}-g_{1}}{g_{1}+g_{2}}\right],\qquad
V_{0}=-\frac{3}{2}\frac{g_{1}^{2}g_{2}^{2}}{g_{2}^{2}-g_{1}^{2}}$ (100)

with all other scalars vanishing. We now repeat the same analysis as in the
previous case with an additional condition $g_{2}A^{6}=g_{1}A^{3}$
implementing the $SO(2)_{\textrm{diag}}$ subgroup. This condition results in
the same component ${Q_{\hat{\phi}A}}^{B}$ as in the $SO(2)\times SO(2)$ case,
so the twist can be performed by the same procedure. We will not repeat all
the details here to avoid repetition.
As in the previous case, it turns out that all pseudo-scalars must be
truncated out in order to preserve supersymmetry along $\epsilon^{1}$ and
$\epsilon^{2}$. Therefore, we need to set $\phi_{2}=\phi_{4}=\phi_{6}=0$.
Consistency for the scalar equations also requires all electric charges to
vanish resulting in a real phase $e^{i\Lambda}=\pm 1$. We will accordingly set
$q_{\Lambda}=0$ and obtain the following BPS equations

$\displaystyle f^{\prime}$ $\displaystyle=$
$\displaystyle\frac{1}{16g_{2}}e^{-\phi_{1}}\left[g_{2}e^{-2\phi_{3}-2\phi_{5}}(1+e^{4\phi_{3}})(1+e^{4\phi_{5}})[(g_{1}+g_{2})e^{2\phi_{1}}+g_{1}-g_{2}]\right.$
(101) $\displaystyle\left.+4e^{-2h}[(g_{1}-g_{2})(e^{2h}g_{2}+2\kappa
p^{3})e^{2\phi_{1}}+(g_{1}+g_{2})(e^{2h}g_{2}-2\kappa p^{3})]\right],$
$\displaystyle h^{\prime}$ $\displaystyle=$
$\displaystyle\frac{1}{16g_{2}}e^{-\phi_{1}}\left[g_{2}e^{-2\phi_{3}-2\phi_{5}}(1+e^{4\phi_{3}})(1+e^{4\phi_{5}})[(g_{1}+g_{2})e^{2\phi_{1}}+g_{1}-g_{2}]\right.$
(102) $\displaystyle\left.+4e^{-2h}[(g_{1}-g_{2})(e^{2h}g_{2}-2\kappa
p^{3})e^{2\phi_{1}}+(g_{1}+g_{2})(e^{2h}g_{2}+2\kappa p^{3})]\right],$
$\displaystyle\phi_{1}^{\prime}$ $\displaystyle=$
$\displaystyle-\frac{1}{16g_{2}}e^{-\phi_{1}}\left[g_{2}e^{-2\phi_{3}-2\phi_{5}}(1+e^{4\phi_{3}})(1+e^{4\phi_{5}})[(g_{1}+g_{2})e^{2\phi_{1}}+g_{2}-g_{1}]\right.$
(103) $\displaystyle\left.-4e^{-2h}[(g_{1}+g_{2})(e^{2h}g_{2}+2\kappa
p^{3})-(g_{1}-g_{2})(e^{2h}g_{2}-2\kappa p^{3})e^{2\phi_{1}}]\right],$
$\displaystyle\phi_{3}^{\prime}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\textrm{sech}2\phi_{5}\sinh
2\phi_{3}(g_{1}\cosh\phi_{1}+g_{2}\sinh\phi_{1}),$ (104)
$\displaystyle\phi_{5}^{\prime}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\cosh 2\phi_{3}\sinh
2\phi_{5}(g_{1}\cosh\phi_{1}+g_{2}\sinh\phi_{1}).$ (105)

We also note that these equations can be written more compactly as

$\displaystyle f^{\prime}=|\mathcal{W}-\mathcal{Z}|,\qquad
h^{\prime}=|\mathcal{W}+\mathcal{Z}|,\qquad\phi_{1}^{\prime}=-\frac{\partial|\mathcal{W}+\mathcal{Z}|}{\partial\phi_{1}},$
$\displaystyle\phi_{3}^{\prime}=-\frac{1}{2}\textrm{sech}^{2}2\phi_{5}\frac{\partial|\mathcal{W}+\mathcal{Z}|}{\partial\phi_{3}},\qquad\phi_{5}^{\prime}=-\frac{1}{2}\frac{\partial|\mathcal{W}+\mathcal{Z}|}{\partial\phi_{5}}\,.$
(106)

For $AdS_{2}\times\Sigma^{2}$ fixed points to exist, we immediately see from
$\phi_{3}^{\prime}$ and $\phi_{5}^{\prime}$ equations that there are two
possibilities; $\phi_{3}=\phi_{5}=0$ or
$\phi_{1}=\frac{1}{2}\ln\left[\frac{g_{2}-g_{1}}{g_{2}+g_{1}}\right]$.
However, both of these choices do not lead to any $AdS_{2}\times\Sigma^{2}$
fixed point, so there are no supersymmetric $AdS_{4}$ black holes with
$SO(2)_{\textrm{diag}}$ symmetry.
At this point, it should be noted that similar BPS equations have been
considered in [21] with more vector multiplets ($n=8$), and a number of
$AdS_{2}\times\Sigma^{2}$ fixed points have been given. A truncation of that
results to three vector multiplets can be performed resulting in the BPS
equations given above. It is worth pointing out here that there is a sign
error in the BPS equations considered in [21] regarding to the contribution of
the gauge fields to the supersymmetry transformations. The corresponding
equations from the present analysis are correct and compatible with the
second-order field equations. Therefore, the $AdS_{2}\times\Sigma^{2}$ fixed
points with $SO(2)_{\textrm{diag}}\times SO(2)$ symmetry found in [21] do not
exist.

##  3 $AdS_{4}$ black holes from $N=4$ gauged supergravity

In this section, we repeat the same analysis as in the previous section for
matter-coupled $N=4$ gauged supergravity. Unlike the $N=3$ gauged supergravity
considered in the previous section, gaugings of $N=4$ supergravity that can
give rise to supersymmetric $AdS_{4}$ vacua need to be dyonic, involving both
electric and magnetic vector fields. However, there always exists a symplectic
frame in which the resulting gaugings are purely electric. As in the previous
section, we will begin with a review of $N=4$ gauged supergravity coupled to
$n$ vector multiplets.

###  3.1 Matter-coupled $N=4$ gauged supergravity

Unlike the $N=3$ gauged supergravity, $N=4$ gauged supergravity has completely
been constructed in the embedding tensor formalism in [34]. We will mainly
follow the construction and notation used in [34].
Similar to the $N=3$ theory, $N=4$ supersymmetry in four dimensions only
allows for the graviton and vector multiplets. Unlike $N=3$ supersymmetry, the
graviton multiplet in $N=4$ supersymmetry does contain scalars with the full
field content given by

$(e^{\hat{\mu}}_{\mu},\psi^{i}_{\mu},A_{\mu}^{m},\chi^{i},\tau).$ (107)

The component fields are given by the graviton $e^{\hat{\mu}}_{\mu}$, four
gravitini $\psi^{i}_{\mu}$, six vectors $A_{\mu}^{m}$, four spin-$\frac{1}{2}$
fields $\chi^{i}$ and one complex scalar $\tau$ parametrizing the
$SL(2,\mathbb{R})/SO(2)$ coset. In this case, indices $m,n=1,\ldots,6$ and
$i,j=1,2,3,4$ respectively describe the vector and chiral spinor
representations of the $SO(6)_{R}\sim SU(4)_{R}$ R-symmetry. The former is
equivalent to a second-rank anti-symmetric tensor representation of
$SU(4)_{R}$. Furthermore, in this section, we denote flat space-time indices
by $\hat{\mu},\hat{\nu},\ldots$ to avoid confusion with indices labeling the
vector multiplets to be introduced later.
As in the $N=3$ theory, the supergravity multiplet can couple to an arbitrary
number $n$ of vector multiplets. Each vector multiplet will be labeled by
indices $a,b=1,\ldots,n$ and contain the following field content

$(A^{a}_{\mu},\lambda^{ia},\phi^{ma})$ (108)

corresponding to vector fields $A^{a}_{\mu}$, gaugini $\lambda^{ia}$ and
scalars $\phi^{ma}$. The $6n$ scalar fields can be described by
$SO(6,n)/SO(6)\times SO(n)$ coset. We also note the well-known fact that the
field contents of the vector multiplet in $N=3$ and $N=4$ supersymmetries are
the same.
All fermionic fields and supersymmetry parameters that transform in the
fundamental representation of $SU(4)_{R}$ R-symmetry are subject to the
chirality projections

$\gamma_{5}\psi^{i}_{\mu}=\psi^{i}_{\mu},\qquad\gamma_{5}\chi^{i}=-\chi^{i},\qquad\gamma_{5}\lambda^{i}=\lambda^{i}\,.$
(109)

Similarly, the conjugate fields transforming in the anti-fundamental
representation of $SU(4)_{R}$ satisfy

$\gamma_{5}\psi_{\mu i}=-\psi_{\mu
i},\qquad\gamma_{5}\chi_{i}=\chi_{i},\qquad\gamma_{5}\lambda_{i}=-\lambda_{i}\,.$
(110)

The most general gaugings of the matter-coupled $N=4$ supergravity can be
efficiently described by the embedding tensor $\Theta$. There are two
components of the embedding tensor $\xi^{\alpha M}$ and $f_{\alpha MNP}$ with
$\alpha=(+,-)$ and $M,N=(m,a)=1,\ldots,n+6$ denoting respectively fundamental
representations of $SL(2,\mathbb{R})\times SO(6,n)$ global symmetry. The
electric vector fields $A^{M+}=(A^{m}_{\mu},A^{a}_{\mu})$ together with their
magnetic dual $A^{M-}$, collectively denoted by $A^{M\alpha}$, form a doublet
of $SL(2,\mathbb{R})$. The existence of $AdS_{4}$ vacua requires $\xi^{\alpha
M}=0$ [25], so we will consider gaugings with only $f_{\alpha MNP}$ non-
vanishing and set $\xi^{\alpha M}$ to zero from now on.
The embedding tensor implements the minimal coupling to various fields via the
covariant derivative

$D_{\mu}=\nabla_{\mu}-gA_{\mu}^{M\alpha}f_{\alpha M}^{\phantom{\alpha
M}NP}t_{NP}$ (111)

where $\nabla_{\mu}$ is the space-time covariant derivative including
(possibly) the spin connections. $t_{MN}$ denote $SO(6,n)$ generators which
can be chosen as

$(t_{MN})_{P}^{\phantom{P}Q}=2\delta^{Q}_{[M}\eta_{N]P},$ (112)

with $\eta_{MN}=\textrm{diag}(-1,-1,-1,-1,-,1-,1,1,1,\ldots,1)$ being the
$SO(6,n)$ invariant tensor. The gauge coupling constant $g$ can also be
absorbed in the definition of the embedding tensor $f_{\alpha MNP}$.
In addition to $\xi^{\alpha M}=0$, the existence of $AdS_{4}$ vacua requires
the gaugings to be dyonic involving both electric and magnetic vector fields.
In this case, both $A^{M+}$ and $A^{M-}$ enter the Lagrangian, and $f_{\alpha
MNP}$ with $\alpha=\pm$ are non-vanishing. Consistency requires the presence
of two-form fields when magnetic vector fields are included. In the case of
$\xi^{\alpha M}=0$, the two-forms transform as an anti-symmetric tensor under
$SO(6,n)$ and will be denoted by $B^{MN}_{\mu\nu}=B^{[MN]}_{\mu\nu}$. The two-
forms are also needed to define covariant gauge field strengths given by

$\mathcal{H}^{M\pm}=dA^{M\pm}-\frac{1}{2}\eta^{MQ}f_{\alpha
QNP}A^{N\alpha}\wedge A^{P\pm}\pm\frac{1}{2}\eta^{MQ}f_{\mp QNP}B^{NP}\,.$
(113)

In particular, for non-vanishing $f_{-MNP}$ the electric field strengths
$\mathcal{H}^{M+}$ acquire a contribution from the two-form fields.
The scalar coset manifold $SL(2,\mathbb{R})/SO(2)$ in the graviton multiplet
can be described by a coset representative

$\mathcal{V}_{\alpha}=\frac{1}{\sqrt{\textrm{Im}\tau}}\left(\begin{array}[]{c}\tau\\\
1\\\ \end{array}\right)$ (114)

or equivalently by a symmetric matrix

$M_{\alpha\beta}=\textrm{Re}(\mathcal{V}_{\alpha}\mathcal{V}^{*}_{\beta})=\frac{1}{\textrm{Im}\tau}\left(\begin{array}[]{cc}|\tau|^{2}&\textrm{Re}\tau\\\
\textrm{Re}\tau&1\\\ \end{array}\right).$ (115)

We also note the relation
$\textrm{Im}(\mathcal{V}_{\alpha}\mathcal{V}^{*}_{\beta})=\epsilon_{\alpha\beta}$.
The complex scalar $\tau$ can in turn be written in terms of the dilaton
$\phi$ and the axion $\chi$ as

$\tau=\chi+ie^{\phi}\,.$ (116)

For the $SO(6,n)/SO(6)\times SO(n)$ coset from vector multiplets, we introduce
the coset representative $\mathcal{V}_{M}^{\phantom{M}A}$ transforming by left
and right multiplications under $SO(6,n)$ and $SO(6)\times SO(n)$,
respectively. The $SO(6)\times SO(n)$ index will be split as $A=(m,a)$
according to which the coset representative can be written as

$\mathcal{V}_{M}^{\phantom{M}A}=(\mathcal{V}_{M}^{\phantom{M}m},\mathcal{V}_{M}^{\phantom{M}a}).$
(117)

Being an element of $SO(6,n)$, the matrix $\mathcal{V}_{M}^{\phantom{M}A}$
satisfies the relation

$\eta_{MN}=-\mathcal{V}_{M}^{\phantom{M}m}\mathcal{V}_{N}^{\phantom{M}m}+\mathcal{V}_{M}^{\phantom{M}a}\mathcal{V}_{N}^{\phantom{M}a}\,.$
(118)

The $SO(6,n)/SO(6)\times SO(n)$ coset can also be parametrized in terms of a
symmetric matrix defined by

$M_{MN}=\mathcal{V}_{M}^{\phantom{M}m}\mathcal{V}_{N}^{\phantom{M}m}+\mathcal{V}_{M}^{\phantom{M}a}\mathcal{V}_{N}^{\phantom{M}a}$
(119)

with a manifest $SO(6)\times SO(n)$ invariance.
The bosonic Lagrangian of the $N=4$ gauged supergravity for $\xi^{\alpha M}=0$
is given by

$\displaystyle e^{-1}\mathcal{L}$ $\displaystyle=$
$\displaystyle\frac{1}{2}R+\frac{1}{16}\mathcal{D}_{\mu}M_{MN}\mathcal{D}^{\mu}M^{MN}-\frac{1}{4(\textrm{Im}\tau)^{2}}\partial_{\mu}\tau\partial^{\mu}\tau^{*}-V$
(120) $\displaystyle-\frac{1}{4}\textrm{Im}\,\tau
M_{MN}\mathcal{H}^{M+}_{\mu\nu}\mathcal{H}^{N+\mu\nu}-\frac{1}{8}\textrm{Re}\,\tau
e^{-1}\epsilon^{\mu\nu\rho\sigma}\eta_{MN}\mathcal{H}^{M+}_{\mu\nu}\mathcal{H}^{N+}_{\rho\sigma}$
$\displaystyle-\frac{1}{2}\left[f_{-MNP}A^{M-}_{\mu}A^{N+}_{\nu}\partial_{\rho}A^{P-}_{\sigma}+\frac{1}{4}f_{\alpha
MNR}f_{\beta
PQS}\eta^{RS}A^{M\alpha}_{\mu}A^{N+}_{\nu}A^{P\beta}_{\rho}A^{Q-}_{\sigma}\right.$
$\displaystyle-\frac{1}{4}f_{-MNP}B^{NP}_{\mu\nu}\left(2\partial_{\rho}A^{M-}_{\sigma}-\frac{1}{2}\eta^{MS}f_{\alpha
SQR}A^{Q\alpha}_{\rho}A^{R-}_{\sigma}\right)$
$\displaystyle\left.-\frac{1}{16}f_{+MNR}f_{-PQS}\eta^{RS}B^{MN}_{\mu\nu}B^{PQ}_{\rho\sigma}\right]e^{-1}\epsilon^{\mu\nu\rho\sigma}$

where $e$ is the vielbein determinant.
The scalar potential is given by

$\displaystyle V$ $\displaystyle=$
$\displaystyle\frac{g^{2}}{16}\left[f_{\alpha MNP}f_{\beta
QRS}M^{\alpha\beta}\left[\frac{1}{3}M^{MQ}M^{NR}M^{PS}+\left(\frac{2}{3}\eta^{MQ}-M^{MQ}\right)\eta^{NR}\eta^{PS}\right]\right.$
(121) $\displaystyle\left.-\frac{4}{9}f_{\alpha MNP}f_{\beta
QRS}\epsilon^{\alpha\beta}M^{MNPQRS}\right]$

where $M^{MN}$ is the inverse of $M_{MN}$, and $M^{MNPQRS}$ is defined by

$M_{MNPQRS}=\epsilon_{mnpqrs}\mathcal{V}_{M}^{\phantom{M}m}\mathcal{V}_{N}^{\phantom{M}n}\mathcal{V}_{P}^{\phantom{M}p}\mathcal{V}_{Q}^{\phantom{M}q}\mathcal{V}_{R}^{\phantom{M}r}\mathcal{V}_{S}^{\phantom{M}s}$
(122)

with indices raised by $\eta^{MN}$. The covariant derivative of $M_{MN}$ is
defined by

$\mathcal{D}M_{MN}=dM_{MN}+2A^{P\alpha}\eta^{QR}f_{\alpha QP(M}M_{N)R}\,.$
(123)

The magnetic vectors and two-form fields do not have kinetic terms. They are
auxiliary fields and enter the Lagrangian through topological terms. The
corresponding field equations give rise to the duality relation between two-
forms and scalars and the electric-magnetic duality between $A^{M+}$ and
$A^{M-}$, respectively. The field equations resulting from varying the
Lagrangian with respect to $A^{M\pm}_{\mu}$ and $B^{MN}_{\mu\nu}$ are given by

$\displaystyle\eta_{MN}*\mathcal{D}\mathcal{H}^{N-}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}{f_{+MP}}^{N}M_{NQ}\mathcal{D}M^{QP},$ (124)
$\displaystyle\eta_{MN}*\mathcal{D}\mathcal{H}^{N+}$ $\displaystyle=$
$\displaystyle\frac{1}{4}{f_{-MP}}^{N}M_{NQ}\mathcal{D}M^{QP},$ (125)
$\displaystyle\mathcal{H}^{M-}$ $\displaystyle=$
$\displaystyle\textrm{Im}\,\tau
M^{MN}\eta_{NP}*\mathcal{H}^{P+}-\textrm{Re}\,\tau\mathcal{H}^{M+}$ (126)

written in differential form language for computational convenience. By
substituting $\mathcal{H}^{M-}$ from (126) in (124), we obtain the usual Yang-
Mills equations for $\mathcal{H}^{M+}$ while equation (125) simply gives the
relation between the Hodge dual of the three-form field strengths and the
scalars due to the usual Bianchi identity of the gauge field strengths defined
by

$\mathcal{F}^{M\pm}=dA^{M\pm}-\frac{1}{2}\eta^{MQ}f_{\alpha
QNP}A^{N\alpha}\wedge A^{P\pm}\,.$ (127)

The supersymmetry transformations of fermionic fields are given by

$\displaystyle\delta\psi^{i}_{\mu}$ $\displaystyle=$ $\displaystyle
2D_{\mu}\epsilon^{i}-\frac{2}{3}gA^{ij}_{1}\gamma_{\mu}\epsilon_{j}+\frac{i}{4}(\mathcal{V}_{\alpha})^{*}{\mathcal{V}_{M}}^{ij}\mathcal{H}^{M\alpha}_{\nu\rho}\gamma^{\nu\rho}\gamma_{\mu}\epsilon_{j},$
(128) $\displaystyle\delta\chi^{i}$ $\displaystyle=$
$\displaystyle-\epsilon^{\alpha\beta}\mathcal{V}_{\alpha}D_{\mu}\mathcal{V}_{\beta}\gamma^{\mu}\epsilon^{i}-\frac{4}{3}igA_{2}^{ij}\epsilon_{j}+\frac{1}{2}\mathcal{V}_{\alpha}{\mathcal{V}_{M}}^{ij}\mathcal{H}^{M\alpha}_{\mu\nu}\gamma^{\mu\nu}\epsilon_{j},$
(129) $\displaystyle\delta\lambda^{i}_{a}$ $\displaystyle=$ $\displaystyle
2i\mathcal{V}_{a}^{\phantom{a}M}D_{\mu}\mathcal{V}_{M}^{\phantom{M}ij}\gamma^{\mu}\epsilon_{j}-2igA_{2aj}^{\phantom{2aj}i}\epsilon^{j}-\frac{1}{4}\mathcal{V}_{\alpha}\mathcal{V}_{Ma}\mathcal{H}^{M\alpha}_{\mu\nu}\gamma^{\mu\nu}\epsilon^{i}$
(130)

with the fermion shift matrices defined by

$\displaystyle A_{1}^{ij}$ $\displaystyle=$
$\displaystyle\epsilon^{\alpha\beta}(\mathcal{V}_{\alpha})^{*}\mathcal{V}_{kl}^{\phantom{kl}M}\mathcal{V}_{N}^{\phantom{N}ik}\mathcal{V}_{P}^{\phantom{P}jl}f_{\beta
M}^{\phantom{\beta M}NP},$ $\displaystyle A_{2}^{ij}$ $\displaystyle=$
$\displaystyle\epsilon^{\alpha\beta}\mathcal{V}_{\alpha}\mathcal{V}_{kl}^{\phantom{kl}M}\mathcal{V}_{N}^{\phantom{N}ik}\mathcal{V}_{P}^{\phantom{P}jl}f_{\beta
M}^{\phantom{\beta M}NP},$ $\displaystyle A_{2ai}^{\phantom{2ai}j}$
$\displaystyle=$
$\displaystyle\epsilon^{\alpha\beta}\mathcal{V}_{\alpha}{\mathcal{V}_{a}}^{M}{\mathcal{V}_{ik}}^{N}\mathcal{V}_{P}^{\phantom{P}jk}f_{\beta
MN}^{\phantom{\beta MN}P}$ (131)

where $\mathcal{V}_{M}^{\phantom{M}ij}$ is defined in terms of the ‘t Hooft
matrices $G^{ij}_{m}$ and $\mathcal{V}_{M}^{\phantom{M}m}$ as

$\mathcal{V}_{M}^{\phantom{M}ij}=\frac{1}{2}\mathcal{V}_{M}^{\phantom{M}m}G^{ij}_{m}$
(132)

and similarly for its inverse

${\mathcal{V}_{ij}}^{M}=-\frac{1}{2}{\mathcal{V}_{m}}^{M}(G^{ij}_{m})^{*}\,.$
(133)

We note that $G^{ij}_{m}$ satisfy the relations

$G_{mij}=(G^{ij}_{m})^{*}=\frac{1}{2}\epsilon_{ijkl}G^{kl}_{m}\,.$ (134)

We will choose the explicit form of these matrices as follows

$\displaystyle G_{1}^{ij}$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{cccc}0&1&0&0\\\ -1&0&0&0\\\ 0&0&0&1\\\
0&0&-1&0\\\ \end{array}\right],\qquad
G_{2}^{ij}=\left[\begin{array}[]{cccc}0&0&1&0\\\ 0&0&0&-1\\\ -1&0&0&0\\\
0&1&0&0\\\ \end{array}\right],$ (143) $\displaystyle G_{3}^{ij}$
$\displaystyle=$ $\displaystyle\left[\begin{array}[]{cccc}0&0&0&1\\\
0&0&1&0\\\ 0&-1&0&0\\\ -1&0&0&0\\\ \end{array}\right],\qquad
G_{4}^{ij}=\left[\begin{array}[]{cccc}0&i&0&0\\\ -i&0&0&0\\\ 0&0&0&-i\\\
0&0&i&0\\\ \end{array}\right],$ (152) $\displaystyle G_{5}^{ij}$
$\displaystyle=$ $\displaystyle\left[\begin{array}[]{cccc}0&0&i&0\\\
0&0&0&i\\\ -i&0&0&0\\\ 0&-i&0&0\\\ \end{array}\right],\qquad
G_{6}^{ij}=\left[\begin{array}[]{cccc}0&0&0&i\\\ 0&0&-i&0\\\ 0&i&0&0\\\
-i&0&0&0\\\ \end{array}\right].$ (161)

The covariant derivative of $\epsilon^{i}$ is given by

$D_{\mu}\epsilon^{i}=\partial_{\mu}\epsilon^{i}+\frac{1}{4}{\omega_{\mu}}^{\hat{\mu}\hat{\nu}}\gamma_{\hat{\mu}\hat{\nu}}\epsilon^{i}+{Q_{\mu
j}}^{i}\epsilon^{j}\,.$ (162)

Finally, it should be noted that the scalar potential can be written in terms
of $A_{1}$ and $A_{2}$ tensors as

$V=-\frac{1}{3}A^{ij}_{1}A_{1ij}+\frac{1}{9}A^{ij}_{2}A_{2ij}+\frac{1}{2}A_{2ai}^{\phantom{2ai}j}A_{2a\phantom{i}j}^{\phantom{2a}i}$
(163)

which is usually referred to as supersymmetric Ward’s identity. We also recall
that upper and lower $i,j,\ldots$ indices are related by complex conjugation.
We end this section by giving some relations which are very useful in deriving
the BPS equations in subsequent analysis. With the explicit form of
$\mathcal{V}_{\alpha}$ given in (114) and equation (126), it is
straightforward to derive the following identities

$\displaystyle
i\mathcal{V}_{\alpha}{\mathcal{V}_{M}}^{ij}\mathcal{H}^{M\alpha}_{\mu\nu}\gamma^{\mu\nu}$
$\displaystyle=$
$\displaystyle-(\mathcal{V}_{-})^{-1}{\mathcal{V}_{M}}^{ij}\mathcal{H}^{M+}_{\mu\nu}\gamma^{\mu\nu}(1-\gamma_{5}),$
(164) $\displaystyle
i\mathcal{V}_{\alpha}{\mathcal{V}_{M}}^{a}\mathcal{H}^{M\alpha}_{\mu\nu}\gamma^{\mu\nu}$
$\displaystyle=$
$\displaystyle-(\mathcal{V}_{-})^{-1}{\mathcal{V}_{M}}^{a}\mathcal{H}^{M+}_{\mu\nu}\gamma^{\mu\nu}(1+\gamma_{5}),$
(165) $\displaystyle
i(\mathcal{V}_{\alpha})^{*}{\mathcal{V}_{M}}^{ij}\mathcal{H}^{M\alpha}_{\mu\nu}\gamma^{\mu\nu}\gamma_{\rho}$
$\displaystyle=$
$\displaystyle(\mathcal{V}_{-})^{-1}{\mathcal{V}_{M}}^{ij}\mathcal{H}^{M+}_{\mu\nu}\gamma^{\mu\nu}\gamma_{\rho}(1-\gamma_{5})$
(166)

in which we have used the following relations for the $SO(6,n)$ coset
representative [35]

$\displaystyle\eta_{MN}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\epsilon_{ijkl}{\mathcal{V}_{M}}^{ij}{\mathcal{V}_{N}}^{kl}+{\mathcal{V}_{M}}^{a}{\mathcal{V}_{N}}^{a},\qquad{\mathcal{V}_{M}}^{a}{\mathcal{V}_{ij}}^{M}=0,$
$\displaystyle{\mathcal{V}_{M}}^{ij}{\mathcal{V}_{kl}}^{M}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}(\delta^{i}_{k}\delta^{j}_{l}-\delta^{i}_{l}\delta^{j}_{k}),\qquad{\mathcal{V}_{M}}^{a}{\mathcal{V}_{b}}^{M}=\delta^{a}_{b}\,.$
(167)

It should be noted that these relations are slightly different from those
given in [34] due to a different convention on $\mathcal{V}_{\alpha}$ in terms
of the scalar $\tau$ namely $\mathcal{V}_{\alpha}$ used in this paper
satisfies $\mathcal{V}_{+}/\mathcal{V}_{-}=\tau$ while that used in [34] gives
$\mathcal{V}_{+}/\mathcal{V}_{-}=\tau^{*}$.

###  3.2 Solutions with $SO(2)\times SO(2)\times SO(2)\times SO(2)$ symmetry

In this paper, we are interested in $N=4$ gauged supergravity with $n=6$
vector multiplets and $SO(4)\times SO(4)\sim SO(3)\times SO(3)\times
SO(3)\times SO(3)$ gauge group. The corresponding embedding tensor takes the
following form [36]

$\displaystyle f_{+\hat{m}\hat{n}\hat{p}}$ $\displaystyle=$ $\displaystyle
g_{1}\epsilon_{\hat{m}\hat{n}\hat{p}},\qquad
f_{+\hat{a}\hat{b}\hat{c}}=\tilde{g}_{1}\epsilon_{\hat{a}\hat{b}\hat{c}},$
$\displaystyle f_{-\tilde{m}\tilde{n}\tilde{p}}$ $\displaystyle=$
$\displaystyle g_{2}\epsilon_{\tilde{m}\tilde{n}\tilde{p}},\qquad
f_{-\tilde{a}\tilde{b}\tilde{c}}=\tilde{g}_{2}\epsilon_{\tilde{a}\tilde{b}\tilde{c}}\,.$
(168)

We have used the convention on the $SO(6,6)$ index
$M=(m,a)=(\hat{m},\tilde{m},\hat{a},\tilde{a})$ with $\hat{m}=1,2,3$,
$\tilde{m}=4,5,6$, $\hat{a}=7,8,9$ and $\tilde{a}=10,11,12$. The two $SO(4)$
factors are electrically and magnetically embedded in $SO(6,6)$ and will be
denoted by $SO(4)_{+}\times SO(4)_{-}$. In terms of the $SO(3)$ factors
corresponding to the embedding tensor in (168), we will write the gauge group
as $SO(3)_{+}\times SO(3)_{-}\times SO(3)_{+}\times SO(3)_{-}$ with the first
two factors embedded in the $SU(4)_{R}\sim SO(6)_{R}$.
We now consider solutions preserving $SO(2)\times SO(2)\times SO(2)\times
SO(2)$ symmetry. To proceed further, we first give an explicit parametrization
of the $SO(6,6)/SO(6)\times SO(6)$ coset. The scalar sector of $SO(2)\times
SO(2)\times SO(2)\times SO(2)$ singlets have already been studied recently in
[37]. We will mostly take various results from [37] in which more details can
be found. By using $SO(6,6)$ generators in the fundamental representation of
the form given in (112), we can identify the $SO(6,6)$ non-compact generators
as

$Y_{ma}=t_{m,a+6}\,.$ (169)

There are four $SO(2)\times SO(2)\times SO(2)\times SO(2)$ singlet scalars
from the $SO(6,6)/SO(6)\times SO(6)$ coset. With the $SO(2)\times SO(2)\times
SO(2)\times SO(2)$ generators chosen to be $X_{+3}$, $X_{-6}$, $X_{+9}$ and
$X_{-12}$, the non-compact generators corresponding to these singlets are
given by $Y_{33}$, $Y_{36}$, $Y_{63}$ and $Y_{66}$ in terms of which the coset
representative can be written as

$\mathcal{V}=e^{\phi_{1}Y_{33}}e^{\phi_{2}Y_{36}}e^{\phi_{3}Y_{63}}e^{\phi_{4}Y_{66}}\,.$
(170)

Together with the dilaton and axion, there are six scalars in the $SO(2)\times
SO(2)\times SO(2)\times SO(2)$ sector. The scalar potential for these singlet
scalars is given by

$V=-\frac{1}{2}e^{-\phi}(g_{1}^{2}+e^{2\phi}g_{2}^{2}+g_{2}^{2}\chi^{2})-2g_{1}g_{2}\cosh\phi_{1}\cosh\phi_{2}\cosh\phi_{3}\cosh\phi_{4}$
(171)

which admits a unique $AdS_{4}$ critical point at

$\phi=\ln\left[\frac{g_{1}}{g_{2}}\right]\qquad\textrm{and}\qquad\phi_{1}=\phi_{2}=\phi_{3}=\phi_{4}=\chi=0$
(172)

with the cosmological constant and $AdS_{4}$ radius given by

$V_{0}=-3g_{1}g_{2}\qquad\textrm{and}\qquad
L=\sqrt{-\frac{3}{V_{0}}}=\frac{1}{\sqrt{g_{1}g_{2}}}\,.$ (173)

This $AdS_{4}$ vacuum preserves $N=4$ supersymmetry and the full $SO(4)\times
SO(4)$ symmetry. We can also choose $g_{2}=g_{1}=g$, by shifting the dilaton,
to make the dilaton vanish at this critical point. Holographic RG flows and
Janus solutions in this sector have been extensively studied in [37]. In the
present work, we look for supersymmetric $AdS_{4}$ black holes with the
horizons of $AdS_{2}\times\Sigma^{2}$ geometry. The analysis is parallel to
the $N=3$ case considered in the previous section with some modifications to
incorporate the magnetic gauge fields. Similar analyses can be found in [18,
19, 38] and [22] in the contexts of $N=2$ and $N=4$ gauged supergravities,
respectively. We will closely follow the procedure in [22].
We first consider the ansatz for $SO(2)\times SO(2)\times SO(2)\times SO(2)$
gauge fields of the form

$\displaystyle A^{M+}$ $\displaystyle=$ $\displaystyle
A^{M}_{t}dt-p^{M}F^{\prime}(\theta)d\phi,\qquad M=3,6,9,12$ (174)
$\displaystyle A^{M-}$ $\displaystyle=$ $\displaystyle\tilde{A}^{M}_{t}dt-
e_{M}F^{\prime}(\theta)d\phi\,.$ (175)

We also note that the gauge fields participating in the $SO(4)\times SO(4)$
gauging are given by $A^{3+}$, $A^{6-}$, $A^{9+}$ and $A^{12-}$ while the
above ansatz includes all of their electric-magnetic duals. The ansatz for
relevant two-form fields is given by

$\displaystyle B^{12}$ $\displaystyle=$ $\displaystyle
b_{3}(r)F(\theta)d\theta\wedge d\phi,\qquad
B^{45}=b_{6}(r)F(\theta)d\theta\wedge d\phi,$ $\displaystyle B^{78}$
$\displaystyle=$ $\displaystyle b_{9}(r)F(\theta)d\theta\wedge d\phi,\qquad
B^{10,11}=b_{12}(r)F(\theta)d\theta\wedge d\phi\,.$ (176)

The metric ansatz is still given by (28). In addition, to avoid some confusion
and make various expressions less cumbersome, we will denote the magnetic
charges with a subscript, $p^{M}=(p_{3},p_{6},p_{9},p_{12})$.
With the embedding tensor (168), it is straightforward to compute the
covariant gauge field strengths

$\displaystyle\mathcal{H}^{3+}$ $\displaystyle=$ $\displaystyle
A^{3^{\prime}}_{t}dr\wedge dt+\kappa p_{3}F(\theta)d\theta\wedge d\phi,$
$\displaystyle\mathcal{H}^{3-}$ $\displaystyle=$
$\displaystyle\tilde{A}^{3^{\prime}}_{t}dr\wedge dt+(\kappa
e_{3}-g_{1}b_{3})F(\theta)d\theta\wedge d\phi,$
$\displaystyle\mathcal{H}^{6+}$ $\displaystyle=$ $\displaystyle
A^{6^{\prime}}_{t}dr\wedge dt+(\kappa
p_{6}+\tilde{g}_{1}b_{6})F(\theta)d\theta\wedge d\phi,$
$\displaystyle\mathcal{H}^{6-}$ $\displaystyle=$
$\displaystyle\tilde{A}^{6^{\prime}}_{t}dr\wedge dt+\kappa
e_{6}F(\theta)d\theta\wedge d\phi,$ $\displaystyle\mathcal{H}^{9+}$
$\displaystyle=$ $\displaystyle A^{9^{\prime}}_{t}dr\wedge dt+\kappa
p_{9}F(\theta)d\theta\wedge d\phi,$ $\displaystyle\mathcal{H}^{9-}$
$\displaystyle=$ $\displaystyle\tilde{A}^{9^{\prime}}_{t}dr\wedge dt+(\kappa
e_{9}-g_{2}b_{9})F(\theta)d\theta\wedge d\phi,$
$\displaystyle\mathcal{H}^{12+}$ $\displaystyle=$ $\displaystyle
A^{12^{\prime}}_{t}dr\wedge dt+(\kappa
p_{12}+\tilde{g}_{2}b_{12})F(\theta)d\theta\wedge d\phi,$
$\displaystyle\mathcal{H}^{12-}$ $\displaystyle=$
$\displaystyle\tilde{A}^{12^{\prime}}_{t}dr\wedge dt+\kappa
e_{12}F(\theta)d\theta\wedge d\phi\,.$ (177)

In this $SO(2)\times SO(2)\times SO(2)\times SO(2)$ sector, it turns out that
all components of YM current are zero

${f_{\pm MP}}^{N}M_{NQ}DM^{QP}=0\,.$ (178)

Equations (124) and (125) then imply that $D\mathcal{H}^{M\pm}=0$. Therefore,
we find that all the fields $b_{i}(r)$ and electric charges $e_{i}$ are
constant.
As pointed out in [37], supersymmetric solutions with $SO(2)\times SO(2)\times
SO(2)\times SO(2)$ symmetry can arise from two possibilities,
$\chi=\phi_{2}=\phi_{3}=0$ or $\chi=\phi_{1}=\phi_{4}=0$. For definiteness, we
will choose the first possibility. Choosing the second one results in
relabeling the scalars. With $\textrm{Re}\,\tau=\chi=0$, equations (126) gives

$\displaystyle A^{3^{\prime}}_{t}$ $\displaystyle=$ $\displaystyle
e^{f-\phi-2h}\left[(\kappa e_{3}-g_{1}b_{3})\cosh 2\phi_{1}+(\kappa
e_{9}-g_{2}b_{9})\sinh 2\phi_{1}\right],$ $\displaystyle A^{6^{\prime}}_{t}$
$\displaystyle=$ $\displaystyle\kappa e^{f-\phi-2h}(e_{6}\cosh
2\phi_{4}+e_{12}\sinh 2\phi_{4}),$ $\displaystyle A^{9^{\prime}}_{t}$
$\displaystyle=$ $\displaystyle-e^{f-\phi-2h}\left[(\kappa
e_{9}-g_{2}b_{9})\cosh 2\phi_{1}+(\kappa e_{3}-g_{1}b_{3})\sinh
2\phi_{1}\right],$ $\displaystyle A^{12^{\prime}}_{t}$ $\displaystyle=$
$\displaystyle-\kappa e^{f-\phi-2h}(e_{12}\cosh 2\phi_{4}+e_{6}\sinh
2\phi_{4}),$ $\displaystyle\tilde{A}^{3^{\prime}}_{t}$ $\displaystyle=$
$\displaystyle-\kappa e^{f+\phi-2h}(p_{3}\cosh 2\phi_{1}+p_{9}\sinh
2\phi_{1}),$ $\displaystyle\tilde{A}^{6^{\prime}}_{t}$ $\displaystyle=$
$\displaystyle-e^{f+\phi-2h}\left[(\kappa p_{6}+\tilde{g}_{1}b_{6})\cosh
2\phi_{4}+(\kappa p_{12}+\tilde{g}_{2}b_{12})\sinh 2\phi_{4}\right],$
$\displaystyle\tilde{A}^{9^{\prime}}_{t}$ $\displaystyle=$
$\displaystyle\kappa e^{f+\phi-2h}(p_{9}\cosh 2\phi_{1}+p_{3}\sinh
2\phi_{1}),$ $\displaystyle\tilde{A}^{12^{\prime}}_{t}$ $\displaystyle=$
$\displaystyle e^{f+\phi-2h}\left[(\kappa p_{12}+\tilde{g}_{2}b_{12})\cosh
2\phi_{4}+(\kappa p_{6}+\tilde{g}_{1}b_{6})\sinh 2\phi_{4}\right].$ (179)

All these relations fix the ansatz for the $H^{M\alpha}_{0r}$ components of
the field strengths in terms of scalars and various charges.
We now consider topological twists along $\Sigma^{2}$. The scalar coset
representative (170) gives the composite connection of the form

${Q_{\mu
i}}^{j}=\frac{1}{2}g_{1}A^{3+}_{\mu}{(i\sigma_{2}\otimes\sigma_{1})_{i}}^{j}+\frac{1}{2}g_{2}A^{6-}_{\mu}{(\sigma_{1}\otimes
i\sigma_{2})_{i}}^{j}$ (180)

with $\sigma_{a}$, $a=1,2,3$, are usual Pauli matrices. To perform a twist, we
consider relevant terms in the variation $\delta\psi^{i}_{\hat{\phi}}$ of the
form

$\frac{1}{2}e^{-h}\frac{F^{\prime}(\theta)}{F(\theta)}\left[\gamma_{\hat{\phi}\hat{\theta}}\delta_{j}^{i}-g_{1}p_{3}{(i\sigma_{2}\otimes\sigma_{1})_{j}}^{i}-g_{2}e_{6}{(\sigma_{1}\otimes
i\sigma_{2})_{j}}^{i}\right]\epsilon^{j}=0\,.$ (181)

There are a few possibilities to satisfy this condition. These are given by
the following two main categories:

  * •

$N=4$ twists: By setting either $p_{3}=0$ or $e_{6}=0$, all four
$\epsilon^{i}$ can be non-vanishing. These two choices lead to the following
twist conditions and projectors

$\displaystyle e_{6}=0;$ $\displaystyle\qquad
g_{1}p_{3}=1,\qquad\gamma_{\hat{\theta}\hat{\phi}}\epsilon^{i}={(i\sigma_{2}\otimes\sigma_{1})^{i}}_{j}\epsilon^{j},$
(182) $\displaystyle p_{3}=0;$ $\displaystyle\qquad
g_{2}e_{6}=1,\qquad\gamma_{\hat{\theta}\hat{\phi}}\epsilon^{i}={(\sigma_{1}\otimes\sigma_{2})^{i}}_{j}\epsilon^{j}\,.$
(183)

We will refer to these two cases as $N=4$ twists which have a similar
structure to the $N=3$ theory.

  * •

$N=2$ twists: By using the relation

$(\sigma_{3}\otimes\sigma_{3})(\sigma_{1}\otimes
i\sigma_{2})=(\sigma_{1}\otimes
i\sigma_{2})(\sigma_{3}\otimes\sigma_{3})=i\sigma_{2}\otimes\sigma_{1},$ (184)

we can rewrite the condition (181) as

$\gamma_{\hat{\phi\hat{\theta}}}\epsilon^{i}=\left[g_{1}p_{3}{(\sigma_{3}\otimes\sigma_{3})_{j}}^{k}+g_{2}e_{6}\delta_{j}^{k}\right]{(\sigma_{1}\otimes
i\sigma_{2})_{k}}^{i}\epsilon^{j}\,.$ (185)

This can be solved by imposing the following conditions

$g_{1}p_{3}+g_{2}e_{6}=1,\qquad\gamma_{\hat{\theta}\hat{\phi}}\epsilon^{i}={(\sigma_{1}\otimes
i\sigma_{2})^{i}}_{j}\epsilon^{j},\qquad{(\sigma_{3}\otimes\sigma_{3})^{i}}_{j}\epsilon^{j}=\epsilon^{i}\,.$
(186)

The last projector simply sets $\epsilon^{2}=\epsilon^{3}=0$ reducing half of
the original supersymmetry. Accordingly, we will call this case $N=2$ twists.

We also note that the situation is very similar to $AdS_{5}$ black strings in
five-dimensional $N=4$ gauged supergravity considered in [39]. In addition,
the two possibilities of $N=4$ twists correspond to the $H$-twist and
$C$-twist of the dual $N=4$ SCFT in three dimensions considered in [40].
By a similar analysis performed in the $N=3$ theory, we find a general
structure of the BPS equations given by

$\displaystyle h^{\prime}=|\mathcal{W}+\mathcal{Z}|\qquad\textrm{and}\qquad
f^{\prime}=\textrm{Re}\,[e^{-i\Lambda}(\mathcal{W}-\mathcal{Z})]$ (187)

together with an algebraic constraint

$g_{1}A^{3}_{t}+g_{2}A^{6}_{t}=e^{f}\textrm{Im}\,[e^{-i\Lambda}(\mathcal{W}-\mathcal{Z})].$
(188)

In these equations, $\mathcal{W}$ is the superpotential obtained from the
eigenvalue of the $A^{ij}_{1}$ tensor along the Killing spinors, and
$\mathcal{Z}$ is the central charge as in the previous section. We have also
imposed the following projector

$\gamma_{\hat{r}}\epsilon_{i}=e^{i\Lambda}\delta_{ij}\epsilon^{j}\qquad\textrm{with}\qquad
e^{i\Lambda}=\frac{\mathcal{W}+\mathcal{Z}}{|\mathcal{W}+\mathcal{Z}|}\,.$
(189)

Using this projector in the supersymmetry transformations $\delta\chi^{i}$ and
$\delta\lambda^{i}_{a}$ leads to the BPS equations for scalars in the gravity
and vector multiplets, respectively.

####  3.2.1 Solutions with $N=4$ twists

We begin with the case of $N=4$ twist by $A^{3+}$. In addition to setting
$e_{6}=0$, unbroken $N=4$ supersymmetry also requires

$b_{6}=b_{12}=e_{12}=p_{6}=p_{12}=0\,.$ (190)

Moreover, consistency of the scalar equations imposes further conditions of
the form

$e_{3}=e_{9}=b_{3}=b_{9}=0\,.$ (191)

All these lead to the following set of consistent BPS equations

$\displaystyle f^{\prime}$ $\displaystyle=$
$\displaystyle|\mathcal{W}-\mathcal{Z}|$ (192) $\displaystyle=$
$\displaystyle\frac{1}{2}e^{-\frac{\phi}{2}}\left[g_{2}\cosh\phi_{1}+e^{\phi}g_{2}\cosh\phi_{4}-\kappa
e^{\phi-2h}(p_{3}\cosh\phi_{1}+p_{9}\sinh\phi_{1})\right],\qquad$
$\displaystyle h^{\prime}$ $\displaystyle=$
$\displaystyle|\mathcal{W}+\mathcal{Z}|$ (193) $\displaystyle=$
$\displaystyle\frac{1}{2}e^{-\frac{\phi}{2}}\left[g_{2}\cosh\phi_{1}+e^{\phi}g_{2}\cosh\phi_{4}+\kappa
e^{\phi-2h}(p_{3}\cosh\phi_{1}+p_{9}\sinh\phi_{1})\right],\qquad$
$\displaystyle\phi_{1}^{\prime}$ $\displaystyle=$
$\displaystyle-2\frac{\partial|\mathcal{W}+\mathcal{Z}|}{\partial\phi_{1}}$
(194) $\displaystyle=$
$\displaystyle-e^{-2h-\frac{\phi}{2}}\left[e^{\phi}\kappa(p_{3}\sinh\phi_{1}+p_{9}\cosh\phi_{1})+e^{2h}g_{1}\sinh\phi_{1}\right],$
$\displaystyle\phi_{4}^{\prime}$ $\displaystyle=$
$\displaystyle-2\frac{\partial|\mathcal{W}+\mathcal{Z}|}{\partial\phi_{4}}=-g_{2}e^{\frac{\phi}{2}}\sinh\phi_{4},$
(195) $\displaystyle\phi^{\prime}$ $\displaystyle=$
$\displaystyle-4\frac{\partial|\mathcal{W}+\mathcal{Z}|}{\partial\phi}$ (196)
$\displaystyle=$ $\displaystyle
e^{-\frac{\phi}{2}}\left[g_{1}\cosh\phi_{1}-e^{\phi-2h}(g_{2}e^{2h}\cosh\phi_{4}+\kappa
p_{3}\cosh\phi_{1}+\kappa p_{9}\sinh\phi_{1})\right].\qquad$

However, there do not exist any $AdS_{2}\times\Sigma^{2}$ fixed points in
these equations.
We then look at the case of $N=4$ twist by $A^{6-}$ in which consistency
similarly requires the following conditions

$b_{3}=b_{9}=e_{3}=e_{9}=p_{9}=b_{6}=b_{12}=p_{6}=p_{12}=0\,.$ (197)

The BPS equations are given by

$\displaystyle f^{\prime}$ $\displaystyle=$
$\displaystyle\frac{1}{2}e^{-\frac{\phi}{2}}\left[g_{1}\cosh\phi_{1}+e^{\phi}g_{2}\cosh\phi_{4}-\kappa
e^{-2h}(e_{6}\cosh\phi_{4}+e_{12}\sinh\phi_{4})\right],\qquad$ (198)
$\displaystyle h$ $\displaystyle=$
$\displaystyle\frac{1}{2}e^{-\frac{\phi}{2}}\left[g_{1}\cosh\phi_{1}+e^{\phi}g_{2}\cosh\phi_{4}+\kappa
e^{-2h}(e_{6}\cosh\phi_{4}+e_{12}\sinh\phi_{4})\right],\qquad$ (199)
$\displaystyle\phi_{1}^{\prime}$ $\displaystyle=$ $\displaystyle-
g_{1}e^{-\frac{\phi}{2}}\sinh\phi_{1},$ (200) $\displaystyle\phi_{4}^{\prime}$
$\displaystyle=$ $\displaystyle-e^{-2h-\frac{\phi}{2}}[(e^{2h+\phi}+\kappa
e_{6})\sinh\phi_{4}+\kappa e_{12}\cosh\phi_{4}],$ (201)
$\displaystyle\phi^{\prime}$ $\displaystyle=$ $\displaystyle
e^{-2h-\frac{\phi}{2}}\left[e^{2h}g_{1}\cosh\phi_{1}+(\kappa
e_{6}-g_{2}e^{2h+\phi})\cosh\phi_{4}+\kappa e_{12}\sinh\phi_{4}\right]$ (202)

which do not admit any $AdS_{2}\times\Sigma^{2}$ fixed points as in the case
of $A^{3+}$ twist.

####  3.2.2 Solutions with $N=2$ twists

We now move to a more interesting and more complicated case of $N=2$ twists by
both $A^{3+}$ and $A^{6-}$. The resulting BPS conditions are much more
involved than those in the previous case. However, we are able to find a
number of solutions for special values of electric and magnetic charges.

  * •

Solutions from pure $N=4$ gauged supergravity

We will begin with a simple case of pure $N=4$ gauged supergravity with
$\phi_{1}=\phi_{4}=0$ and $A^{9+}=A^{12-}=0$.
In this case, the constraint (188) requires $e_{3}=p_{6}=0$, and we find

$\displaystyle\mathcal{W}$ $\displaystyle=$
$\displaystyle\frac{1}{2}e^{-\frac{\phi}{2}}[g_{1}+g_{2}e^{\phi}+ig_{2}\chi],$
(203) $\displaystyle\mathcal{Z}$ $\displaystyle=$
$\displaystyle\frac{1}{2}e^{-\frac{\phi}{2}-2h}\kappa[e_{6}+p_{3}e^{\phi}+ip_{3}\chi].$
(204)

We then find the following BPS equations

$\displaystyle\chi^{\prime}$ $\displaystyle=$
$\displaystyle-4e^{2\phi}\frac{\partial|\mathcal{W}+\mathcal{Z}|}{\partial\chi}=-\frac{e^{-4h+\phi}(\kappa
g_{2}e^{2h}+p_{3})^{2}\chi}{|\mathcal{W}+\mathcal{Z}|},$ (205)
$\displaystyle\phi^{\prime}$ $\displaystyle=$
$\displaystyle-4\frac{\partial|\mathcal{W}+\mathcal{Z}|}{\partial\phi}$ (206)
$\displaystyle=$
$\displaystyle\frac{e^{-4h-\phi}}{2|\mathcal{W}+\mathcal{Z}|}\left[(e_{6}+\kappa
g_{1}e^{2h})^{2}-(\kappa g_{2}e^{2h}+p_{3})^{2}(e^{2\phi}-\chi^{2})\right],$
$\displaystyle h^{\prime}$ $\displaystyle=$
$\displaystyle|\mathcal{W}+\mathcal{Z}|,$ (207) $\displaystyle f^{\prime}$
$\displaystyle=$
$\displaystyle\frac{e^{-4h-\phi}}{4|\mathcal{W}+\mathcal{Z}|}\left[e^{4h}(g_{1}+g_{2}e^{\phi})^{2}-(e_{6}+p_{3}e^{\phi})^{2}+(e^{4h}g_{2}^{2}-p_{3}^{2})^{2}\chi^{2}\right]$
(208)

with

$|\mathcal{W}+\mathcal{Z}|=\frac{1}{2}e^{-2h-\frac{\phi}{2}}\sqrt{[e^{2h}(g_{1}+e^{\phi}g_{2})+\kappa
e_{6}+\kappa p_{3}e^{\phi}]^{2}+(e^{2h}g_{2}+\kappa p_{3})^{2}\chi^{2}}\,.$
(209)

From these equations, we find an $AdS_{2}\times H^{2}$ fixed point given by

$h=\frac{1}{2}\ln\left[-\frac{\kappa p_{3}}{g_{2}}\right],\qquad
L_{AdS_{2}}=\frac{1}{g_{1}e^{-\frac{\phi_{0}}{2}}+g_{2}e^{\frac{\phi_{0}}{2}}}$
(210)

for constants $\phi=\phi_{0}$ and $\chi=\chi_{0}$ provided that
$g_{2}e_{6}=g_{1}p_{3}$. We note that for $\chi=0$, the above BPS equations
and the $AdS_{2}\times H^{2}$ fixed point are the same as those considered in
[41] with an appropriate change of symplectic frame to purely electric $SO(4)$
gauge group. We have slightly generalized the equations in [41] by including a
non-vanishing axion. We now give the flow solutions interpolating between the
$AdS_{4}$ vacuum and the $AdS_{2}\times H^{2}$ geometry. Before giving
explicit solutions, we first simplify the expressions by setting $g_{2}=g_{1}$
according to which the twist condition gives $p_{3}=e_{6}=\frac{1}{2g_{1}}$.
For $\chi=0$ and $\kappa=-1$, we find a much simpler set of BPS equations

$\displaystyle\phi^{\prime}$ $\displaystyle=$
$\displaystyle-e^{-2h-\frac{\phi}{2}}(e^{\phi}-1)(e^{2h}g_{1}-p_{3}),$ (211)
$\displaystyle h^{\prime}$ $\displaystyle=$
$\displaystyle\frac{1}{2}e^{-2h-\frac{\phi}{2}}(1+e^{\phi})(e^{2h}g_{1}-p_{3}),$
(212) $\displaystyle f^{\prime}$ $\displaystyle=$
$\displaystyle\frac{1}{2}e^{-2h-\frac{\phi}{2}}(1+e^{\phi})(e^{2h}g_{1}+p_{3}).$
(213)

These equations take a very similar form to those of $N=5,6$ gauged
supergravities and $N=3$ gauged supergravity given in the previous section. We
then expect that the resulting solutions are related to each other by
truncations of $N=6$ gauged supergravity to gauged supergravities with lower
amounts of supersymmetry. The solution is given by

$\displaystyle g_{1}(r-r_{0})$ $\displaystyle=$
$\displaystyle\tanh^{-1}\sqrt{\frac{1+\cosh\phi}{2}}$ (214)
$\displaystyle-2\sqrt{\frac{p_{3}}{g_{1}+4p_{3}}}\tanh^{-1}\sqrt{\frac{2p_{3}(1+\cosh\phi)}{g_{1}+4p_{3}}},$
$\displaystyle h$ $\displaystyle=$
$\displaystyle\frac{\phi}{2}-\ln(1-e^{\phi}),$ (215) $\displaystyle f$
$\displaystyle=$
$\displaystyle\ln\left[p_{3}(1+e^{2\phi})-(g_{1}+2p_{3})e^{\phi}\right]-\ln(1-e^{\phi})-\frac{\phi}{2}\,.$
(216)

This solution flows to the $AdS_{2}\times H^{2}$ fixed point (210) for
$\phi_{0}$ given by

$\phi_{0}=\ln\left[\sqrt{\frac{g_{1}(g_{1}+4p_{3})}{4p_{3}^{2}}}+\frac{g_{1}}{2p_{3}}+1\right].$
(217)

For $\chi\neq 0$, we have the BPS equations

$\displaystyle f^{\prime}$ $\displaystyle=$
$\displaystyle\frac{1}{2}e^{-2h-\frac{\phi}{2}}(e^{2h}g_{1}+p_{3})\sqrt{(1+e^{\phi})^{2}+\chi^{2}},$
(218) $\displaystyle h^{\prime}$ $\displaystyle=$
$\displaystyle\frac{1}{2}e^{-2h-\frac{\phi}{2}}(e^{2h}g_{1}-p_{3})\sqrt{(1+e^{\phi})^{2}+\chi^{2}},$
(219) $\displaystyle\phi^{\prime}$ $\displaystyle=$
$\displaystyle\frac{e^{-2h-\frac{\phi}{2}}(e^{2h}g_{1}-p_{3})(1-e^{2\phi}+\chi^{2})}{\sqrt{(1+e^{\phi})^{2}+\chi^{2}}},$
(220) $\displaystyle\chi^{\prime}$ $\displaystyle=$
$\displaystyle-\frac{2e^{-2h+\frac{3\phi}{2}}(e^{2h}g_{1}-p_{3})\chi}{\sqrt{(1+e^{\phi})^{2}+\chi^{2}}}$
(221)

with the solution given by

$\displaystyle\phi$ $\displaystyle=$
$\displaystyle\frac{1}{2}\ln(1-\chi^{2}+C_{0}\chi),$ (222) $\displaystyle f$
$\displaystyle=$ $\displaystyle\ln(e^{2h}g_{1}-p_{3})-h,$ (223) $\displaystyle
h$ $\displaystyle=$
$\displaystyle\frac{1}{8}\ln\left[\frac{1+C_{0}\chi-\chi^{2}}{\chi^{4}}\right]+\frac{1}{4}\ln\left[2+C_{0}\chi+2\sqrt{1+C_{0}\chi-\chi^{2}}\right]$
(224)

for a constant $C_{0}$. However, we are not able to find an analytic solution
for $\chi(r)$. The solution flows to the $AdS_{2}\times H^{2}$ fixed point if

$\chi_{0}=\frac{g_{1}\tilde{C}[\tilde{C}^{2}-2p_{3}^{2}+g_{1}C_{0}\tilde{C}]}{2(g_{1}^{2}\tilde{C}^{2}+p_{3}^{4})}$
(225)

with $\tilde{C}=\sqrt{g_{1}^{2}(4+C_{0}^{2})+4p_{3}^{2}}$.

  * •

Solutions from matter-coupled $N=4$ gauged supergravity

We now consider solutions from matter-coupled $N=4$ gauged supergravity with
$\phi_{1},\phi_{4}\neq 0$. Consistency for setting $\phi_{2}=\phi_{3}=0$ in
$\delta\lambda_{a}^{i}$ conditions also requires setting $A^{9+}=0$. The
residual symmetry of the solutions in this case is then enhanced to
$SO(2)\times SO(2)\times SO(3)\times SO(2)$. With all these, we find two sets
of BPS equations consistent with the constraint (188). These are given by

$\displaystyle i$ $\displaystyle:$ $\displaystyle\qquad\chi=\phi_{2}=0,\qquad
e_{3}=p_{6}=p_{12}=0,$ (226) $\displaystyle ii$ $\displaystyle:$
$\displaystyle\qquad\chi=\phi_{4}=0,\qquad e_{3}=p_{6}=e_{12}=0\,.$ (227)

$\square$ Case $i$:
In this case, we find the following BPS equations

$\displaystyle f^{\prime}$ $\displaystyle=$
$\displaystyle\frac{1}{4}e^{-2h-\frac{\phi}{2}-\phi_{4}}\left[e^{\phi}[e^{2h}g_{2}(1+e^{2\phi_{4}})-2\kappa
p_{3}e^{\phi_{4}}]\right.$ (228)
$\displaystyle\left.+2g_{1}e^{2h+\phi_{4}}+\kappa
e_{12}(1-e^{2\phi_{4}})-\kappa e_{6}(1+e^{2\phi_{4}})\right],$ $\displaystyle
h^{\prime}$ $\displaystyle=$
$\displaystyle\frac{1}{4}e^{-2h-\frac{\phi}{2}-\phi_{4}}\left[e^{\phi}[e^{2h}g_{2}(1+e^{2\phi_{4}})+2\kappa
p_{3}e^{\phi_{4}}]\right.$ (229)
$\displaystyle\left.+2g_{1}e^{2h+\phi_{4}}-\kappa
e_{12}(1-e^{2\phi_{4}})+\kappa e_{6}(1+e^{2\phi_{4}})\right],$
$\displaystyle\phi^{\prime}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}e^{-2h-\frac{\phi}{2}-\phi_{4}}\left[e^{2h+\phi}g_{2}-2g_{1}e^{2h+\phi_{4}}+e^{2h+\phi+2\phi_{4}}g_{2}\right.$
(230)
$\displaystyle\left.+\kappa(e_{12}-e_{6})-\kappa(e_{12}+e_{6})e^{2\phi_{4}}+2\kappa
p_{3}e^{\phi+\phi_{4}}\right],$ $\displaystyle\phi_{4}^{\prime}$
$\displaystyle=$
$\displaystyle-\frac{1}{2}e^{-2h-\frac{\phi}{2}-\phi_{4}}\left[e^{2h+\phi}g_{2}(e^{2\phi_{4}}-1)+\kappa(e_{12}-e_{6})+\kappa(e_{12}+e_{6})e^{2\phi_{4}}\right].\quad$
(231)

There is a family of $AdS_{2}\times\Sigma^{2}$ fixed points given by

$\displaystyle\phi$ $\displaystyle=$
$\displaystyle\ln\left[\frac{(1+e^{2\phi_{4}})[e_{12}(1+e^{2\phi_{4}})+e_{6}(e^{2\phi_{4}}-1)]}{2p_{3}(e^{2\phi_{4}}-1)}\right]-\phi_{4},$
$\displaystyle h$ $\displaystyle=$
$\displaystyle\frac{\phi_{4}}{2}-\frac{1}{2}\ln\left[-\frac{g_{2}(1+e^{2\phi_{4}})}{2\kappa
p_{3}}\right],$ $\displaystyle\phi_{4}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\ln\left[\frac{2g_{1}p_{3}-e_{6}g_{2}+\sqrt{e_{12}^{2}g_{2}^{2}+4g_{1}p_{3}(g_{1}p_{3}-g_{2}e_{6})}}{g_{2}(e_{12}+e_{6})}\right].$
(232)

It can be verified that for appropriate values of the parameters, this
critical point is valid for both $\kappa=1$ and $\kappa=-1$ resulting in a
class of $AdS_{2}\times S^{2}$ and $AdS_{2}\times H^{2}$ geometries. Since
$p_{12}=0$ in this case, the solutions carry only electric charges of
$A^{12-}$.
Examples of solutions interpolating between $AdS_{4}$ and $AdS_{2}\times
H^{2}$ vacua with

$g_{2}=g_{1}=1,\qquad p_{3}=\frac{3}{2},\qquad\kappa=-1$ (233)

and $e_{12}=1,2,3$ are shown in figure 1. We also note that the value of
$e_{6}$ is fixed by the twist condition $g_{1}(p_{3}+e_{6})=1$.
A number of interpolating solutions between $AdS_{4}$ and $AdS_{2}\times
S^{2}$ critical points are shown in figure 2 with the following numerical
values

$g_{2}=g_{1}=1,\qquad p_{3}=-2,\qquad\kappa=1$ (234)

and $e_{12}=4,6,8$.
$\square$ Case $ii$:

(a) $\phi_{4}(r)$ solution

(b) $\phi(r)$ solution

(c) $h(r)$ solution

(d) $f^{\prime}(r)$ solution

Figure 1: Supersymmetric $AdS_{4}$ black holes with $AdS_{2}\times H^{2}$
horizon for $g_{2}=g_{1}=1$, $p_{3}=\frac{3}{2}$, $\kappa=-1$ and
$e_{12}=1(\textrm{red}),2(\textrm{green}),3(\textrm{blue})$.

(a) $\phi_{4}(r)$ solution

(b) $\phi(r)$ solution

(c) $h(r)$ solution

(d) $f^{\prime}(r)$ solution

Figure 2: Supersymmetric $AdS_{4}$ black holes with $AdS_{2}\times S^{2}$
horizon for $g_{2}=g_{1}=1$, $p_{3}=-2$, $\kappa=1$ and
$e_{12}=4(\textrm{red}),6(\textrm{green}),8(\textrm{blue})$.

In this case, the solutions carry magnetic charges of $A^{12-}$, and the
resulting BPS equations are given by

$\displaystyle f^{\prime}$ $\displaystyle=$
$\displaystyle\frac{1}{4}e^{-2h-\frac{\phi}{2}-\phi_{2}}\left[e^{2h}(g_{1}+g_{1}e^{2\phi_{2}}+2g_{2}e^{\phi+\phi_{2}})-2\kappa
e_{6}e^{\phi_{2}}\right.$ (235) $\displaystyle\left.+\kappa
e^{\phi}(p_{12}-p_{3})-\kappa(p_{12}+p_{3})e^{\phi+\phi_{2}}\right],$
$\displaystyle h^{\prime}$ $\displaystyle=$
$\displaystyle\frac{1}{4}e^{-2h-\frac{\phi}{2}-\phi_{2}}\left[e^{2h}(g_{1}+g_{1}e^{2\phi_{2}}+2g_{2}e^{\phi+\phi_{2}})+2\kappa
e_{6}e^{\phi_{2}}\right.$ (236) $\displaystyle\left.-\kappa
e^{\phi}(p_{12}-p_{3})+\kappa(p_{12}+p_{3})e^{\phi+\phi_{2}}\right],$
$\displaystyle\phi^{\prime}$ $\displaystyle=$
$\displaystyle\frac{1}{2}e^{-2h-\frac{\phi}{2}-\phi_{2}}\left[e^{2h}g_{1}(1+e^{2\phi_{2}})-2g_{2}e^{2h+\phi+\phi_{2}}+2\kappa
e_{6}e^{\phi_{2}}\right.$ (237)
$\displaystyle\left.+\kappa(p_{12}-p_{3})e^{\phi}-\kappa(p_{12}+p_{3})e^{2\phi_{4}+\phi}\right],$
$\displaystyle\phi_{2}^{\prime}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}e^{-\frac{\phi}{2}-\phi_{2}}\left[g_{1}(e^{2\phi_{2}}-1)+\kappa
e^{-2h+\phi}[p_{12}-p_{3}+(p_{12}+p_{3})e^{2\phi_{2}}]\right].$ (238)

From these equations, we find a family of $AdS_{2}\times\Sigma^{2}$ fixed
points given by

$\displaystyle\phi$ $\displaystyle=$
$\displaystyle\ln\left[\frac{2e_{6}e^{2\phi_{2}}(e^{2\phi_{2}}-1)}{(1+e^{2\phi_{2}})[p_{12}-p_{3}+(p_{12}+p_{3})e^{2\phi_{2}}]}\right],$
$\displaystyle h$ $\displaystyle=$
$\displaystyle\frac{\phi_{2}}{2}-\frac{1}{2}\ln\left[-\frac{g_{1}(1+e^{2\phi_{2}})}{2\kappa
e_{6}}\right],$ $\displaystyle\phi_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\ln\left[\frac{2e_{6}g_{2}-g_{1}p_{3}+\sqrt{4e_{6}^{2}g_{2}^{2}+g_{1}^{2}p_{12}^{2}-4e_{6}p_{3}g_{1}g_{2}}}{g_{1}(p_{12}+p_{3})}\right].$
(239)

Similar to the previous case, both $AdS_{2}\times S^{2}$ and $AdS_{2}\times
H^{2}$ geometries are possible depending on the values of various parameters.
Examples of flow solutions from the $AdS_{4}$ vacuum to $AdS_{2}\times H^{2}$
fixed points with

$g_{2}=g_{1}=1,\qquad p_{3}=\frac{1}{4},\qquad\kappa=-1$ (240)

and $p_{12}=1,2,3$ are given in figure 3. For flow solutions to $AdS_{2}\times
S^{2}$ fixed points, we give some representative solutions for $p_{12}=3,6,9$
and

$g_{2}=g_{1}=1,\qquad p_{3}=2,\qquad\kappa=1$ (241)

in figure 4.

(a) $\phi_{2}(r)$ solution

(b) $\phi(r)$ solution

(c) $h(r)$ solution

(d) $f^{\prime}(r)$ solution

Figure 3: Supersymmetric $AdS_{4}$ black holes with $AdS_{2}\times H^{2}$
horizon for $g_{2}=g_{1}=1$, $p_{3}=\frac{1}{4}$, $\kappa=-1$ and
$p_{12}=1(\textrm{red}),2(\textrm{green}),3(\textrm{blue})$.

(a) $\phi_{2}(r)$ solution

(b) $\phi(r)$ solution

(c) $h(r)$ solution

(d) $f^{\prime}(r)$ solution

Figure 4: Supersymmetric $AdS_{4}$ black holes with $AdS_{2}\times S^{2}$
horizon for $g_{2}=g_{1}=1$, $p_{3}=2$, $\kappa=1$ and
$p_{12}=3(\textrm{red}),6(\textrm{green}),9(\textrm{blue})$.

###  3.3 Solutions with $SO(2)_{\textrm{diag}}\times SO(2)_{\textrm{diag}}$
symmetry

In this section, we repeat the same analysis for a smaller residual symmetry
$SO(2)_{\textrm{diag}}\times SO(2)_{\textrm{diag}}$. As we will see, a new
feature is the appearance of a number of non-trivial supersymmetric $AdS_{4}$
vacua. All of these vacua are not new but have recently been found in [42] to
which we refer for more details. Since the analysis of
$SO(2)_{\textrm{diag}}\times SO(2)_{\textrm{diag}}$ singlet scalars has not
previously appeared, we will give more detail than the $SO(2)\times
SO(2)\times SO(2)\times SO(2)$ sector considered in the previous section.
We begin with the scalars from $SO(6,6)/SO(6)\times SO(6)$ coset which
contains six singlets corresponding to the following non-compact generators

$\displaystyle Y_{11},\qquad Y_{11}+Y_{22},\qquad Y_{12}-Y_{21},\qquad
Y_{66},\qquad Y_{44}+Y_{55},\qquad Y_{45}-Y_{54}\,.$ (242)

The coset representative can be then written as

$\mathcal{V}=e^{\phi_{1}Y_{11}}e^{\phi_{2}(Y_{11}+Y_{22})}e^{\phi_{3}(Y_{12}-Y_{21})}e^{\phi_{4}Y_{66}}e^{\phi_{5}(Y_{44}+Y_{55})}e^{\phi_{6}(Y_{45}-Y_{54})}\,.$
(243)

With this coset representative, scalar kinetic terms are given by

$\displaystyle e^{-1}\mathcal{L}_{\textrm{kin}}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}({\phi^{\prime}}^{2}-e^{-2\phi}{\chi^{\prime}}^{2})-\frac{1}{2}[{\phi_{1}^{\prime}}^{2}+(1+\cosh
4\phi_{3}){\phi_{2}^{\prime}}^{2}+2{\phi_{3}^{\prime}}^{2}]$ (244)
$\displaystyle-\frac{1}{2}[{\phi_{4}^{\prime}}^{2}+(1+\cosh
4\phi_{6}){\phi_{5}^{\prime}}^{2}+2{\phi_{6}^{\prime}}^{2}].$

The tensor $A_{1}^{ij}$ is proportional to the identity matrix of which the
four-fold degenerate eigenvalue gives the superpotential only for $\chi=0$.
Since the complete expressions are much more complicated and will not play any
important role in subsequent analysis, we will only give the potential and
superpotential for the case of $\chi=0$. These are given respectively by

$\displaystyle V$ $\displaystyle=$
$\displaystyle\frac{1}{8}\left[g_{1}^{2}\cosh^{2}\phi_{2}[\cosh
2(\phi_{1}-\phi_{2})+\cosh 2(\phi_{1}+\phi_{2})-2\cosh
2\phi_{1}-4]\times\right.$ (245)
$\displaystyle\cosh^{2}\phi_{2}(\cosh\phi-\sin\phi)+g_{2}^{2}\cosh^{2}\phi_{5}(\cosh\phi+\sinh\phi)\times$
$\displaystyle[\cosh 2(\phi_{4}-\phi_{5})+\cosh 2(\phi_{4}+\phi_{5})-2\cosh
2\phi_{4}-4]$ $\displaystyle+\tilde{g}_{1}^{2}[\cosh
2(\phi_{1}-\phi_{2})+\cosh 2(\phi_{1}+\phi_{2})+2\cosh 2\phi_{1}-4]\times$
$\displaystyle(\cosh\phi-\sinh\phi)\sinh^{2}\phi_{2}+16g_{2}\tilde{g}_{1}\cosh\phi_{4}\cosh^{2}\phi_{5}\sinh\phi_{1}\times$
$\displaystyle\sinh^{2}\phi_{2}-2g_{1}\tilde{g}_{1}(\cosh\phi-\sinh\phi)\sinh
2\phi_{1}\sinh^{2}\phi_{2}$ $\displaystyle+\tilde{g}_{2}^{2}[\cosh
2(\phi_{4}-\phi_{5})+\cosh 2(\phi_{4}+\phi_{5})+2\cosh
2\phi_{4}-4]\sinh^{2}\phi_{5}\times$
$\displaystyle(\cosh\phi+\sinh\phi)+16g_{1}\tilde{g}_{2}\cosh\phi_{1}\cosh^{2}\phi_{2}\sinh\phi_{4}\sinh^{2}\phi_{5}$
$\displaystyle-16g_{1}g_{2}\cosh\phi_{1}\cosh^{2}\phi_{2}\cosh\phi_{4}\cosh^{2}\phi_{5}$
$\displaystyle+16\tilde{g}_{1}\tilde{g}_{2}\sinh\phi_{1}\sinh^{2}\phi_{2}\sinh\phi_{4}\sinh^{2}\phi_{5}$
$\displaystyle\left.-2g_{2}\tilde{g}_{2}e^{\phi}\sinh
2\phi_{4}\sinh^{2}2\phi_{5}\right]$

and

$\displaystyle W$ $\displaystyle=$
$\displaystyle\frac{1}{4}e^{-\frac{\phi}{2}}\left[g_{1}\cosh\phi_{1}(1+\cosh
2\phi_{2}\cosh 2\phi_{3})+\tilde{g}_{1}\sinh\phi_{1}(1-\cosh 2\phi_{2}\cosh
2\phi_{3})\right.$ $\displaystyle\left.g_{2}e^{\phi}\cosh\phi_{4}(1+\cosh
2\phi_{5}\cosh 2\phi_{6})+\tilde{g}_{2}e^{\phi}\sinh\phi_{4}(1-\cosh
2\phi_{5}\cosh 2\phi_{6})\right].$

It is straightforward to verify that the superpotential admits the following
four supersymmetric $AdS_{4}$ vacua

I $\displaystyle:$
$\displaystyle\qquad\phi_{\alpha}=0,\quad\alpha=1,2,\ldots,6,\quad\phi=\ln\left[\frac{g_{1}}{g_{2}}\right],\quad
V_{0}=-3g_{1}g_{2},\quad$ (247) II $\displaystyle:$
$\displaystyle\qquad\phi_{\alpha}=0,\quad\alpha=1,2,3,6,\qquad\phi_{4}=\pm\phi_{5}=\frac{1}{2}\ln\left[\frac{\tilde{g}_{2}+g_{2}}{\tilde{g}_{2}-g_{2}}\right],$
(248)
$\displaystyle\qquad\phi=\frac{1}{2}\ln\left[\frac{g_{1}^{2}(\tilde{g}_{2}^{2}-g_{2}^{2})}{g_{2}^{2}\tilde{g}_{2}^{2}}\right],\qquad
V_{0}=-\frac{3g_{1}g_{2}\tilde{g}_{2}}{\sqrt{\tilde{g}_{2}^{2}-g_{2}^{2}}},$
III $\displaystyle:$
$\displaystyle\qquad\phi_{\alpha}=0,\quad\alpha=3,4,5,6,\qquad\phi_{1}=\pm\phi_{2}=\frac{1}{2}\ln\left[\frac{\tilde{g}_{1}+g_{1}}{\tilde{g}_{1}-g_{1}}\right],$
(249)
$\displaystyle\qquad\phi=-\frac{1}{2}\ln\left[\frac{g_{2}^{2}(\tilde{g}_{1}^{2}-g_{1}^{2})}{g_{1}^{2}\tilde{g}_{1}^{2}}\right],\qquad
V_{0}=-\frac{3g_{1}g_{2}\tilde{g}_{1}}{\sqrt{\tilde{g}_{1}^{2}-g_{2}^{1}}},$
IV $\displaystyle:$
$\displaystyle\qquad\phi_{3}=\phi_{6}=0,\qquad\phi_{1}=\pm\phi_{2}=\frac{1}{2}\ln\left[\frac{\tilde{g}_{1}+g_{1}}{\tilde{g}_{1}-g_{1}}\right],$
(250)
$\displaystyle\qquad\phi_{4}=\pm\phi_{5}=\frac{1}{2}\ln\left[\frac{\tilde{g}_{2}+g_{2}}{\tilde{g}_{2}-g_{2}}\right],\qquad\phi=\ln\left[\frac{g_{1}\tilde{g}_{1}}{g_{2}\tilde{g}_{2}}\sqrt{\frac{\tilde{g}_{2}^{2}-g_{2}^{2}}{\tilde{g}_{1}^{2}-g_{1}^{2}}}\right],$
$\displaystyle\qquad
V_{0}=-\frac{3g_{1}g_{2}\tilde{g}_{1}\tilde{g}_{2}}{\sqrt{(\tilde{g}_{2}^{2}-g_{2}^{2})(\tilde{g}_{1}^{2}-g_{1}^{2})}}\,.$

All of these vacua have already been found in [42], but we repeat them here
for later convenience. We also note the unbroken gauge symmetries for these
solutions which are given respectively by $SO(4)\times SO(4)$, $SO(4)\times
SO(3)$, $SO(3)\times SO(4)$ and $SO(3)\times SO(3)$.
To find supersymmetric $AdS_{4}$ black hole solutions, we now turn to the
analysis of Yang-Mills equations. To implement the
$SO(2)_{\textrm{diag}}\times SO(2)_{\textrm{diag}}$ symmetry, we impose the
following conditions on the gauge fields

$g_{1}A^{3+}=-\tilde{g}_{1}A^{9+}\qquad\textrm{and}\qquad
g_{2}A^{6-}=-\tilde{g}_{2}A^{12-}$ (251)

which lead to the same composite connection given in (180). Therefore, the
twist conditions and relevant projectors are the same.
Unlike the $SO(2)\times SO(2)\times SO(2)\times SO(2)$ case, the YM currents
are non-vanishing in this case. From equation (124), we find

$\displaystyle D\mathcal{H}^{3-}$ $\displaystyle=$
$\displaystyle\frac{1}{2}g_{1}(\cosh 2\phi_{2}\sinh
4\phi_{3}\phi_{2}^{\prime}-2\sinh 2\phi_{2}\phi_{3}^{\prime})*dr,$ (252)
$\displaystyle D\mathcal{H}^{9-}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\tilde{g}_{1}(\cosh 2\phi_{2}\sinh
4\phi_{3}\phi_{2}^{\prime}-2\sinh 2\phi_{2}\phi_{3}^{\prime})*dr$ (253)

which, from the ansatz of the gauge fields, imply that $b_{3}$ and $b_{9}$ are
constant and

$\phi_{2}=0\qquad\textrm{or}\qquad\phi_{3}=0\,.$ (254)

Similarly, equation (125) gives

$\displaystyle D\mathcal{H}^{6+}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}g_{2}(\cosh 2\phi_{5}\sinh
4\phi_{6}\phi_{5}^{\prime}-2\sinh 2\phi_{5}\phi_{6}^{\prime})*dr,$ (255)
$\displaystyle D\mathcal{H}^{12+}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\tilde{g}_{2}(\cosh 2\phi_{5}\sinh
4\phi_{6}\phi_{5}^{\prime}-2\sinh 2\phi_{5}\phi_{6}^{\prime})*dr$ (256)

which lead to constant $b_{6}$ and $b_{12}$ together with

$\phi_{5}=0\qquad\textrm{or}\qquad\phi_{6}=0\,.$ (257)

We also note that the radial component of the composite connection is given by

${Q_{ri}}^{j}=-\cosh\phi_{3}\sinh\phi_{3}\phi_{2}^{\prime}{(i\sigma_{2}\otimes\sigma_{1})_{i}}^{j}-\cosh\phi_{6}\sinh\phi_{6}\phi_{5}^{\prime}{(\sigma_{1}\otimes
i\sigma_{2})_{i}}^{j}$ (258)

which identically vanishes whenever $\phi_{2}=0$ or $\phi_{3}=0$ and
$\phi_{5}=0$ or $\phi_{6}=0$. In order to find solutions interpolating between
supersymmetric $AdS_{4}$ vacua identified above, we will choose a definite
choice

$\phi_{3}=\phi_{6}=0\,.$ (259)

We then consider equation (126). Equations for $\mathcal{H}^{3-}$ and
$\mathcal{H}^{9-}$ give

$\displaystyle\tilde{A}_{t}^{3^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{\kappa p_{3}}{\tilde{g}_{1}}e^{\phi+f-2h}(g_{1}\sinh
2\phi_{1}-\tilde{g}_{1}\cosh 2\phi_{1}),$ (260)
$\displaystyle\tilde{A}_{t}^{9^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{\kappa
p_{3}}{\tilde{g}_{1}}e^{\phi+f-2h}(\tilde{g}_{1}\sinh 2\phi_{1}-g_{1}\cosh
2\phi_{1})$ (261)

together with

$A_{t}^{3^{\prime}}=-\frac{\tilde{g}_{1}}{g_{1}}A_{t}^{9^{\prime}}=\frac{g_{1}e^{f-\phi-2h}(\kappa
e_{3}-g_{1}b_{3})}{\tilde{g}_{1}\cosh 2\phi_{1}-g_{1}\sinh
2\phi_{1}}=\frac{\tilde{g}_{1}e^{f-\phi-2h}(\kappa
e_{9}-g_{2}b_{9})}{g_{1}\cosh 2\phi_{1}-\tilde{g}_{1}\sinh 2\phi_{1}}\,.$
(262)

For $\tilde{g}_{1}\neq g_{1}$ which is needed for the existence of non-trivial
$AdS_{4}$ vacua, the last equation implies

$e_{3}=e_{9}=b_{3}=b_{9}=0$ (263)

which in turn gives

$A_{t}^{3^{\prime}}=A_{t}^{9^{\prime}}=0\,.$ (264)

Similarly, equations for $\mathcal{H}^{6-}$ and $\mathcal{H}^{12-}$ give

$p_{12}=p_{6}=b_{6}=b_{12}=0\qquad\textrm{and}\qquad\tilde{A}^{6^{\prime}}_{t}=\tilde{A}^{12^{\prime}}_{t}=0$
(265)

together with

$\displaystyle A_{t}^{6^{\prime}}$ $\displaystyle=$ $\displaystyle\frac{\kappa
e_{6}}{\tilde{g}_{2}}e^{f-2h-\phi}(\tilde{g}_{2}\cosh 2\phi_{4}-g_{2}\sinh
2\phi_{4}),$ (266) $\displaystyle A_{t}^{12^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{\kappa e_{6}}{\tilde{g}_{2}}e^{f-2h-\phi}(g_{2}\cosh
2\phi_{4}-\tilde{g}_{2}\sinh 2\phi_{4}).$ (267)

With $\chi=\phi_{3}=\phi_{6}=0$, we find that both $\mathcal{W}$ and
$\mathcal{Z}$ are real and given by

$\displaystyle\mathcal{W}$ $\displaystyle=$
$\displaystyle\frac{1}{2}e^{-\frac{\phi}{2}}(g_{1}\cosh\phi_{1}\cosh^{2}\phi_{2}-\tilde{g}_{1}\sinh\phi_{1}\sinh^{2}\phi_{2})$
(268)
$\displaystyle+\frac{1}{2}e^{\frac{\phi}{2}}(g_{2}\cosh\phi_{4}\cosh^{2}\phi_{5}-\tilde{g}_{2}\sinh\phi_{4}\sinh^{2}\phi_{5}),$
$\displaystyle\mathcal{Z}$ $\displaystyle=$
$\displaystyle-\frac{\kappa}{2\tilde{g}_{1}\tilde{g_{2}}}e^{-2h-\frac{\phi}{2}}\left[e^{\phi}p_{3}\tilde{g}_{2}(\tilde{g}_{1}\cosh\phi_{1}-g_{1}\sinh\phi_{1})\right.$
(269)
$\displaystyle\left.+e_{6}\tilde{g}_{1}(g_{2}\sinh\phi_{4}-\tilde{g}_{2}\cosh\phi_{4})\right].$

It can be readily verified that critical points I, II, III, and IV are
critical points of $\mathcal{W}$ as expected for supersymmetric vacua.
As in the previous case, there are two possible topological twists, $N=4$ and
$N=2$ twists. The $N=4$ twists do not give rise to any
$AdS_{2}\times\Sigma^{2}$ fixed points, so we will only give the results on
$N=2$ twists. Since both $\mathcal{W}$ and $\mathcal{Z}$ are real, we find the
phase $e^{i\Lambda}=\pm 1$, and the BPS equations are given by

$\displaystyle f^{\prime}$ $\displaystyle=$
$\displaystyle|\mathcal{W}-\mathcal{Z}|$ (270) $\displaystyle=$
$\displaystyle\frac{1}{2}e^{-\frac{\phi}{2}}\left[g_{1}\cosh\phi_{1}\cosh^{2}\phi_{2}-\tilde{g}_{1}\sinh\phi_{1}\sinh^{2}\phi_{2}\right.$
$\displaystyle\left.+g_{2}e^{\phi}\cosh\phi_{4}\cosh^{2}\phi_{5}-\tilde{g}_{2}e^{\phi}\sinh\phi_{4}\sinh^{2}\phi_{5}\right]$
$\displaystyle-\kappa\frac{e^{-\frac{\phi}{2}-2h}}{\tilde{g}_{1}\tilde{g}_{2}}\left[e_{6}\tilde{g}_{1}(\tilde{g}_{2}\cosh\phi_{4}-g_{2}\sinh\phi_{4})-\right.$
$\displaystyle\left.e^{\phi}\tilde{g}_{2}p_{3}(\tilde{g}_{1}\cosh\phi_{1}-g_{1}\sinh\phi_{1})\right],$
$\displaystyle h^{\prime}$ $\displaystyle=$
$\displaystyle|\mathcal{W}+\mathcal{Z}|$ (271) $\displaystyle=$
$\displaystyle\frac{1}{2}e^{-\frac{\phi}{2}}\left[g_{1}\cosh\phi_{1}\cosh^{2}\phi_{2}-\tilde{g}_{1}\sinh\phi_{1}\sinh^{2}\phi_{2}\right.$
$\displaystyle\left.+g_{2}e^{\phi}\cosh\phi_{4}\cosh^{2}\phi_{5}-\tilde{g}_{2}e^{\phi}\sinh\phi_{4}\sinh^{2}\phi_{5}\right]$
$\displaystyle+\kappa\frac{e^{-\frac{\phi}{2}-2h}}{\tilde{g}_{1}\tilde{g}_{2}}\left[e_{6}\tilde{g}_{1}(\tilde{g}_{2}\cosh\phi_{4}-g_{2}\sinh\phi_{4})\right.$
$\displaystyle\left.-e^{\phi}\tilde{g}_{2}p_{3}(\tilde{g}_{1}\cosh\phi_{1}-g_{1}\sinh\phi_{1})\right],$
$\displaystyle\phi^{\prime}$ $\displaystyle=$
$\displaystyle-4\frac{\partial|\mathcal{W}+\mathcal{Z}|}{\partial\phi}$ (272)
$\displaystyle=$ $\displaystyle
e^{-\frac{\phi}{2}}\left[g_{1}\cosh\phi_{1}\cosh^{2}\phi_{2}-\tilde{g}_{1}\sinh\phi_{1}\sinh^{2}\phi_{2}\right.$
$\displaystyle\left.+e^{\phi}(\tilde{g}_{2}\sinh\phi_{4}\sinh^{2}\phi_{5}-g_{2}\cosh\phi_{4}\cosh^{2}\phi_{5})\right]$
$\displaystyle+\frac{\kappa}{\tilde{g}_{1}\tilde{g}_{2}}e^{-2h-\frac{\phi}{2}}\left[e^{\phi}\tilde{g}_{2}p_{3}(\tilde{g}_{1}\cosh\phi_{1}-g_{1}\sinh\phi_{1})\right.$
$\displaystyle\left.+e_{6}\tilde{g}_{1}(\tilde{g}_{2}\cosh\phi_{4}-g_{2}\sinh\phi_{4})\right],$
$\displaystyle\phi_{1}^{\prime}$ $\displaystyle=$
$\displaystyle-2\frac{\partial|\mathcal{W}+\mathcal{Z}|}{\partial\phi_{1}}$
(273) $\displaystyle=$ $\displaystyle
e^{-\frac{\phi}{2}}(\tilde{g}_{1}\cosh\phi_{1}\sinh^{2}\phi_{2}-g_{1}\cosh^{2}\phi_{2}\sinh\phi_{1})$
$\displaystyle+\frac{\kappa
p_{3}}{\tilde{g}_{1}}e^{-2h+\frac{\phi}{2}}(\tilde{g}_{1}\sinh\phi_{1}-g_{1}\cosh\phi_{1}),$
$\displaystyle\phi_{2}^{\prime}$ $\displaystyle=$
$\displaystyle-\frac{\partial|\mathcal{W}+\mathcal{Z}|}{\partial\phi_{2}}$
(274) $\displaystyle=$ $\displaystyle
e^{-\frac{\phi}{2}}\cosh\phi_{2}\sinh\phi_{2}(\tilde{g}_{1}\sinh\phi_{1}-g_{1}\cosh\phi_{1}),$
$\displaystyle\phi_{4}^{\prime}$ $\displaystyle=$
$\displaystyle-2\frac{\partial|\mathcal{W}+\mathcal{Z}|}{\partial\phi_{4}}$
(275) $\displaystyle=$ $\displaystyle
e^{\frac{\phi}{2}}(\tilde{g}_{2}\cosh\phi_{4}\sinh^{2}\phi_{5}-g_{2}\cosh^{2}\phi_{5}\sinh\phi_{4})$
$\displaystyle-\frac{\kappa
e_{6}}{\tilde{g}_{2}}e^{-2h-\frac{\phi}{2}}(\tilde{g}_{2}\sinh\phi_{4}-g_{2}\cosh\phi_{4}),$
$\displaystyle\phi_{5}^{\prime}$ $\displaystyle=$
$\displaystyle-\frac{\partial|\mathcal{W}+\mathcal{Z}|}{\partial\phi_{5}}$
(276) $\displaystyle=$ $\displaystyle
e^{\frac{\phi}{2}}\cosh\phi_{5}\sinh\phi_{5}(\tilde{g}_{2}\sinh\phi_{4}-g_{2}\cosh\phi_{4}).$

From $\phi_{2}^{\prime}$ and $\phi_{5}^{\prime}$ equations, we immediately see
that there are four possibilities for $AdS_{2}\times\Sigma^{2}$ fixed points
to exist:

$\displaystyle i$ $\displaystyle:$ $\displaystyle\qquad\phi_{2}=\phi_{5}=0,$
$\displaystyle ii$ $\displaystyle:$
$\displaystyle\qquad\phi_{2}=0\quad\textrm{and}\quad\phi_{4}=\frac{1}{2}\ln\left[\frac{\tilde{g}_{2}+g_{2}}{\tilde{g}_{2}-g_{2}}\right],$
$\displaystyle iii$ $\displaystyle:$
$\displaystyle\qquad\phi_{5}=0\quad\textrm{and}\quad\phi_{1}=\frac{1}{2}\ln\left[\frac{\tilde{g}_{1}+g_{1}}{\tilde{g}_{1}-g_{1}}\right]$
$\displaystyle iv$ $\displaystyle:$
$\displaystyle\qquad\phi_{1}=\frac{1}{2}\ln\left[\frac{\tilde{g}_{1}+g_{1}}{\tilde{g}_{1}-g_{1}}\right]\quad\textrm{and}\quad\phi_{4}=\frac{1}{2}\ln\left[\frac{\tilde{g}_{2}+g_{2}}{\tilde{g}_{2}-g_{2}}\right].$
(277)

These coincide with the values of scalars at supersymmetric $AdS_{4}$ vacua I,
II, III and IV. However, the last possibility does not lead to any
$AdS_{2}\times\Sigma^{2}$ fixed points. We then consider only the remaining
three cases:

  * •

$i$: In this case, we set $\phi_{2}=\phi_{5}=0$ and find an
$AdS_{2}\times\Sigma^{2}$ fixed point given by

$\displaystyle h$ $\displaystyle=$
$\displaystyle\frac{1}{2}\phi+\frac{1}{2}\ln\left[\frac{\kappa
p_{3}(\tilde{g}_{1}-g_{1}\coth\phi_{1})}{g_{1}\tilde{g}_{1}}\right],$
$\displaystyle\phi$ $\displaystyle=$
$\displaystyle\frac{1}{2}\ln\left[\frac{e_{6}g_{1}\tilde{g}_{1}(g_{2}\coth\phi_{4}-\tilde{g}_{2})}{p_{3}g_{2}\tilde{g}_{2}(\tilde{g}_{1}-g_{1}\coth\phi_{1})}\right],$
$\displaystyle\phi_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\ln\left[\frac{g_{1}(g_{2}\cosh
2\phi_{4}-\tilde{g}_{2}\sinh 2\phi_{4})}{g_{2}(\tilde{g}_{1}-g_{1})}\right.$
$\displaystyle\left.+\frac{\sqrt{g_{2}^{2}(\tilde{g}_{1}^{2}-g_{1}^{2})+g_{1}^{2}(g_{2}\cosh
2\phi_{4}-\tilde{g}_{2}\sinh
2\phi_{4})^{2}}}{g_{2}(\tilde{g}_{1}-g_{1})}\right],$ $\displaystyle\phi_{4}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\ln\left[\frac{e_{6}^{2}g_{2}^{4}\tilde{g}_{1}^{2}\tilde{g}_{2}+2e_{6}g_{1}g_{2}^{3}\tilde{g}_{1}^{2}\tilde{g}_{2}p_{3}+g_{1}^{4}\tilde{g}_{2}^{3}p_{3}^{2}+g_{2}\sqrt{X}}{(\tilde{g}_{2}-g_{2})(g_{1}^{4}\tilde{g}_{2}^{2}p_{3}^{2}-e_{6}^{2}g_{2}^{4}\tilde{g}_{1}^{2})}\right]$
(278)

for

$\displaystyle X$ $\displaystyle=$ $\displaystyle
e_{6}^{4}g_{2}^{8}\tilde{g}_{1}^{4}+4e_{6}^{3}g_{1}g_{2}^{5}\tilde{g}_{1}^{4}\tilde{g}_{2}^{2}p_{3}+2e_{6}^{2}g_{1}^{2}g_{2}^{2}\tilde{g}_{1}^{2}\tilde{g}_{2}^{2}[2g_{2}^{2}\tilde{g}_{1}^{2}-g_{1}^{2}(g_{2}^{2}-2\tilde{g}_{2}^{2})]p_{3}^{2}$
(279)
$\displaystyle+4e_{6}g_{1}^{5}g_{2}\tilde{g}_{1}^{2}\tilde{g}_{2}^{4}p_{3}^{3}+g_{1}^{8}\tilde{g}_{2}^{4}p_{3}^{4}\,.$

  * •

$ii$: In this case, we have $\phi_{2}=0$ and

$\displaystyle\phi_{4}$ $\displaystyle=$
$\displaystyle\phi_{5}=\frac{1}{2}\ln\left[\frac{\tilde{g}_{2}+g_{2}}{\tilde{g}_{2}-g_{2}}\right],$
$\displaystyle h$ $\displaystyle=$
$\displaystyle\frac{1}{2}\ln\left[\frac{\kappa
p_{3}e^{\phi}(\tilde{g}_{1}-g_{1}\coth\phi_{1})}{g_{1}\tilde{g}_{1}}\right],$
$\displaystyle\phi$ $\displaystyle=$
$\displaystyle\ln\left[\frac{\sqrt{\tilde{g}_{2}^{2}-g_{2}^{2}}\left[2g_{1}p_{3}(g_{1}\cosh
2\phi_{1}-\tilde{g}_{1}\sinh
2\phi_{1})+\sqrt{2g_{1}p_{3}Y}\right]}{4g_{2}\tilde{g}_{2}p_{3}(g_{1}\cosh\phi_{1}-\tilde{g}_{1}\sinh\phi_{1})}\right],$
$\displaystyle\phi_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\ln\left[\frac{2e_{6}g_{2}\tilde{g}_{1}+g_{1}\tilde{g}_{1}p_{3}+\sqrt{4e_{6}^{2}g_{2}^{2}\tilde{g}_{1}^{2}+4e_{6}g_{1}g_{2}\tilde{g}_{1}^{2}p_{3}+g^{4}_{1}p_{3}^{2}}}{g_{1}p_{3}(g_{1}-\tilde{g}_{1})}\right]$
(280)

with

$\displaystyle Y$ $\displaystyle=$ $\displaystyle
g_{1}p_{3}(\tilde{g}_{1}^{2}+g_{1}^{2})\cosh
4\phi_{1}-4g_{1}\tilde{g}_{1}\sinh 2\phi_{1}(e_{6}g_{2}+g_{1}p_{3}\cosh
2\phi_{1})$ (281)
$\displaystyle+g_{1}^{3}p_{3}-\tilde{g}_{1}^{2}(4e_{6}g_{2}+g_{1}p_{3})+4e_{6}g_{2}\tilde{g}_{1}^{2}\cosh
2\phi_{1}\,.$

  * •

$iii$: For this final possibility, we have $\phi_{5}=0$ and

$\displaystyle\phi_{1}$ $\displaystyle=$
$\displaystyle\phi_{2}=\frac{1}{2}\ln\left[\frac{\tilde{g}_{1}+g_{1}}{\tilde{g}_{1}-g_{1}}\right],$
$\displaystyle h$ $\displaystyle=$
$\displaystyle\frac{1}{2}\ln\left[\frac{\kappa
e_{6}e^{-\phi}[g_{2}(1+e^{2\phi_{4}})+\tilde{g}_{2}(1-e^{2\phi_{4}})]}{g_{2}\tilde{g}_{2}(e^{2\phi_{4}}-1)}\right],$
$\displaystyle\phi$ $\displaystyle=$
$\displaystyle\ln\left[\sqrt{e_{6}g_{2}(e_{6}g_{2}^{3}+2g_{1}\tilde{g}_{2}^{2}p_{3}-2g_{1}\tilde{g}_{2}^{2}p_{3}\cosh
2\phi_{4}+2g_{1}g_{2}\tilde{g}_{2}p_{3}\sinh 2\phi_{4})}\right.$
$\displaystyle\left.\phantom{\sqrt{g_{2}^{3}}}+e_{6}g_{2}^{2}\right]+\ln\left[\frac{\tilde{g}_{1}e^{\phi_{4}}(\coth\phi_{4}-1)}{2g_{2}\tilde{g}_{2}p_{3}\sqrt{\tilde{g}_{1}^{2}-g_{1}^{2}}}\right],$
$\displaystyle\phi_{4}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\ln\left[\frac{\tilde{g}_{2}(e_{6}g_{2}+2g_{1}p_{3})+\sqrt{e_{6}^{2}g_{2}^{4}+4e_{6}g_{1}g_{2}\tilde{g}_{2}^{2}p_{3}+4g_{1}^{2}\tilde{g}_{2}^{2}p_{3}^{2}}}{e_{6}g_{2}(g_{2}-\tilde{g}_{2})}\right].$
(282)

In each case, we have not explicitly given the expressions for $L_{AdS_{2}}$
due to their complexity. These can be obtained from $f^{\prime}$ equation by
using the values of the other fields at the fixed points. We have verified
that all the above three cases indeed lead to valid $AdS_{2}\times\Sigma^{2}$
fixed points in each case. This will also be clearly seen later in numerical
analyses.
For critical point $i$, we obtain only $AdS_{2}\times H^{2}$ solutions with
$\kappa=-1$. Examples of solutions interpolating between the supersymmetric
$AdS_{4}$ critical point I and these $AdS_{2}\times H^{2}$ geometries are
shown in figure 5 for $g_{2}=g_{1}=1$, $\tilde{g}_{1}=2g_{1}$,
$\tilde{g}_{2}=3g_{2}$ and $p_{3}=-3,-3.00000025,-3.005$. The reason for
choosing values of $p_{3}$ very close to each other is for the convenience in
the presentation. The numerical plots for solutions in which the values of
$p_{3}$ are widely separated are very far from each other.
For critical point $ii$, we have found only $AdS_{2}\times H^{2}$ solutions as
in critical point $i$. An example of the solutions interpolating between
supersymmetric $AdS_{4}$ critical points I and II and an $AdS_{2}\times H^{2}$
geometry with $g_{2}=g_{1}=1$, $\tilde{g}_{1}=2g_{1}$, $\tilde{g}_{2}=3g_{2}$
and $p_{3}=-3$ is shown in figure 6. We have set $\phi_{2}=0$ along the entire
solution. We also note that the solution indeed exhibits an intermediate
$AdS_{4}$ critical point II with the value $\phi=-0.05889$ given by the chosen
values of various parameters in this solution.
Unlike the previous two cases, in critical point $iii$, we only find
$AdS_{2}\times S^{2}$ solutions. An example of flow solutions is shown in
figure 7 with $g_{2}=g_{1}=1$, $\tilde{g}_{1}=2g_{1}$, $\tilde{g}_{2}=3g_{2}$
and $p_{3}=3$. Along the entire flow, we have set $\phi_{5}=0$. As in the flow
solution to $AdS_{2}\times H^{2}$ critical point $ii$, the solution exhibits
an intermediate $AdS_{4}$ critical point III with $\phi=0.143841$, so the
solution interpolates between $AdS_{4}$ critical points I and II and
$AdS_{2}\times S^{2}$ geometry in the IR. The solutions in this case and the
flow to critical point $ii$ are similar to solutions describing RG flows
across dimensions in half-maximal gauged supergravities in five, six and seven
dimensions [39, 43, 44, 45]. Moreover, there also exist solutions that flow
directly from $AdS_{4}$ critical point I to these $AdS_{2}\times S^{2}$ and
$AdS_{2}\times H^{2}$ fixed points. We will not give these solutions here
since they are similar to the solutions in $SO(2)\times SO(2)\times
SO(2)\times SO(2)$ case without non-trivial $AdS_{4}$ vacua.
We end this section by noting that there do not exist any
$AdS_{2}\times\Sigma^{2}$ fixed points for case $iv$ discussed above.
Therefore, there are no flow solutions from the supersymmetric $AdS_{4}$
vacuum IV to $AdS_{2}\times\Sigma^{2}$ geometries in the IR. This is in line
with the $N=3$ gauged supergravity studied in the previous section in which no
$AdS_{2}\times\Sigma^{2}$ fixed points exist for RG flows involving the non-
trivial $N=3$ $AdS_{4}$ critical point with $SO(3)$ symmetry. On the other
hand, as we have seen above, $AdS_{2}\times\Sigma^{2}$ critical points $ii$
and $iii$ do exist and are connected to non-trivial $AdS_{4}$ critical points
II and III. However, the latter do not have an analogue in the case of $N=3$
gauged supergravity.

(a) $\phi_{1}(r)$ solution

(b) $\phi_{4}(r)$ solution

(c) $\phi(r)$ solution

(d) h(r) solution

(e) $f^{\prime}(r)$ solution

Figure 5: Supersymmetric $AdS_{4}$ black holes with $AdS_{2}\times H^{2}$
horizon $(i)$ for $g_{2}=g_{1}=1$, $\tilde{g}_{1}=2g_{1}$,
$\tilde{g}_{2}=3g_{2}$ and
$p_{3}=-3(\textrm{red}),-3.00000025(\textrm{blue}),-3.005(\textrm{green})$.

(a) $\phi_{1}(r)$ solution

(b) $\phi_{4}(r)$ solution

(c) $\phi_{5}(r)$ solution

(d) $\phi(r)$ solution

(e) $h(r)$ solution

(f) $f^{\prime}(r)$ solution

Figure 6: A supersymmetric $AdS_{4}$ black hole with $AdS_{2}\times H^{2}$
horizon $(ii)$ for $g_{2}=g_{1}=1$, $\tilde{g}_{1}=2g_{1}$,
$\tilde{g}_{2}=3g_{2}$ and $p_{3}=-3$.

(a) $\phi_{1}(r)$ solution

(b) $\phi_{2}(r)$ solution

(c) $\phi_{4}(r)$ solution

(d) $\phi(r)$ solution

(e) $h(r)$ solution

(f) $f^{\prime}(r)$ solution

Figure 7: A supersymmetric $AdS_{4}$ black hole with $AdS_{2}\times S^{2}$
horizon $(iii)$ for $g_{2}=g_{1}=1$, $\tilde{g}_{1}=2g_{1}$,
$\tilde{g}_{2}=3g_{2}$ and $p_{3}=3$.

##  4 Conclusions and discussions

We have studied a number of supersymmetric black hole solutions in
asymptotically $AdS_{4}$ space from matter-coupled $N=3$ and $N=4$ gauged
supergravities. In $N=3$ theory, we have found an $AdS_{2}\times H^{2}$
solution with $SO(2)\times SO(2)$ symmetry. We have also given a complete
solution interpolating between $SO(3)\times SO(3)$ symmetric $AdS_{4}$ vacuum
and this $AdS_{2}\times H^{2}$ geometry with a non-vanishing scalar. The
resulting solution has a very similar structure to those given in $N=5,6$
gauged supergravities. The solution with vanishing scalars is a solution of
pure $N=3$ gauged supergravity and can be embedded in massive type IIA theory
using the result of [31]. We have also shown that there are no $AdS_{4}$ black
hole solutions with $SO(2)_{\textrm{diag}}$ symmetry. Therefore, in $N=3$
gauged supergravity under consideration here, it is clear that there are no
other solutions.
Although we have considered only a particular case of three vector multiplets,
it has been shown in [46] that the $SO(3)_{R}\subset SU(3)_{R}$ symmetry must
be gauged in order for the gaugings to admit a supersymmetric $AdS_{4}$
vacuum. This is also an essential part in performing topological twists since
the gravitini and Killing spinors are charged exclusively under this symmetry
or a diagonal subgroup with parts of the symmetry of vector multiplets.
Therefore, even with extra vector multiplets and possibly larger gauge groups,
the structure of the topological twists should be the same and eventually
leads to a similar conclusion.
In pure $N=4$ gauged supergravity, we have recovered an $AdS_{2}\times H^{2}$
solution studied in [41]. However, we have included a non-vanishing axion and
given the interpolating solutions between this geometry and the supersymmetric
$AdS_{4}$ vacuum. For matter-coupled $N=4$ gauged supergravity, we have found
a number of $AdS_{2}\times S^{2}$ and $AdS_{2}\times H^{2}$ solutions with
$SO(2)\times SO(2)\times SO(3)\times SO(2)$ symmetry. We have also given
various examples of numerical solutions interpolating between these geometries
and the $AdS_{4}$ vacuum with $SO(4)\times SO(4)$ symmetry. The BPS equations
are very complicated, and we are not able to completely carry out the
analysis. However, we have given a number of possible $AdS_{4}$ black hole
solutions with both spherical and hyperbolic horizons. We note that unlike
$N=5$ and $N=6$ gauged supergravities, there exist matter multiplets in $N=4$
theory, and the two $SO(2)$ factors involving in the twists are not
necessarily equal though related, see the twist condition in (186). This gives
a weaker constraint on the charges and leaves more freedom to find
$AdS_{2}\times\Sigma^{2}$ solutions. This is also supported by the fact that,
when restricted to the case of pure $N=4$ gauged supergravity, the charges of
$A^{3+}$ and $A^{6-}$ must be equal, and only one $AdS_{2}\times H^{2}$
solution which is an analogue of similar solutions in $N=5,6$ theories exists.
We have also found $AdS_{2}\times S^{2}$ and $AdS_{2}\times H^{2}$ solutions
with $SO(2)_{\textrm{diag}}\times SO(2)_{\textrm{diag}}$ symmetry. Similar to
the $N=3$ theory, in this case, we have performed a complete analysis and
classified all possible supersymmetric $AdS_{2}\times\Sigma^{2}$ solutions
with the aforementioned residual symmetry at least for the case of six vector
multiplets. In this case, apart from the trivial $AdS_{4}$ critical point with
the full $SO(4)\times SO(4)$ symmetry, there exist additional three
supersymmetric $AdS_{4}$ vacua with $SO(4)\times SO(3)$, $SO(3)\times SO(4)$
and $SO(3)\times SO(3)$ symmetries. Except for the last critical point, we
have found black hole solutions interpolating between these vacua and
$AdS_{2}\times S^{2}$ and $AdS_{2}\times H^{2}$ geometries. We hope all these
solutions could be useful in black hole physics and holographic studies of
twisted compactifications of $N=3$ and $N=4$ SCFTs in three dimensions on a
Riemann surface.
It is interesting to look for more general solutions in the $SO(2)\times
SO(2)\times SO(2)\times SO(2)$ case in particular solutions carrying both
electric and magnetic charges of the same gauge fields. In this paper, we have
given only some representative examples of the possible solutions which carry
either electric or magnetic charges of a given gauge field. Another direction
is to find an embedding of the solutions given here in string/M-theory.
Solutions in pure $N=3$ and $N=4$ gauged supergravities can be embedded in ten
and eleven dimensions using consistent truncations given respectively in [31,
32] and [47]. It would be useful to find similar embedding for the solutions
in matter-coupled gauged supergravities. It could also be of particular
interest to study the dual three-dimensional $N=3,4$ SCFTs with topological
twists and compute microscopic entropy of the black holes. Finally, it would
be interesting to study similar solutions in other gauged supergravities such
as $\omega$-deformed $N=8$ gauged supergravity and $N=4$ truncation of massive
type IIA on $S^{6}$ given in [48] and [49], respectively.
Acknowledgement
This work is supported by The Thailand Research Fund (TRF) under grant
RSA6280022.

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