New supersymmetric $AdS_{6}$ black holes from matter-coupled $F(4)$ gauged
supergravity

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 $AdS_{6}$ black holes from matter-coupled $N=(1,1)$
gauged supergravity coupled to three vector multiplets and $SO(3)\times SO(3)$
gauge group. This gauged supergravity admits two supersymmetric $AdS_{6}$
vacua preserving all supersymmetries with $SO(3)\times SO(3)$ and
$SO(3)_{\textrm{diag}}$ symmetries. By considering a truncation to
$SO(2)_{\textrm{diag}}\subset SO(3)_{\textrm{diag}}$ invariant sector, we find
a number of new supersymmetric $AdS_{2}\times\mathcal{M}_{4}$ solutions by
performing a topological twist along $\mathcal{M}_{4}$. For $\mathcal{M}_{4}$
being a product of two Riemann surfaces $\Sigma\times\widetilde{\Sigma}$ and a
Kahler four-cycle, the twist is implemented by $SO(2)_{\textrm{diag}}$ gauge
field while, for $\mathcal{M}_{4}$ given by a Clayley four-cycle, the twist is
performed by turning on $SO(3)_{\textrm{diag}}$ gauge fields. We give
numerical black hole solutions interpolating between
$AdS_{2}\times\mathcal{M}_{4}$ near horizon geometries and asymptotically
locally $AdS_{6}$ vacua. Among these solutions, there are solutions
interpolating between both of the supersymmetric $AdS_{6}$ vacua and near
horizon geometries. The solutions can also be interpreted as holographic RG
flows from five-dimensional SCFTs to superconformal quantum mechanics.

##  1 Introduction

The study of supersymmetric asymptotically AdS black holes in various
dimensions has attracted much attention since the understanding of the
microscopic origin of the Bekenstein-Hawking entropy has been acquired by the
computation of topologically twisted indices in the dual field theories [1, 2,
3], see also [4]-[16] for results on different types of black holes in various
dimensions. The procedure has also been generalized to black strings in five
dimensions in [17]. In general, the black hole solutions interpolate between
an asymptotically $AdS_{d+1}$ space and a near horizon geometry
$AdS_{2}\times\mathcal{M}_{d-1}$ with $\mathcal{M}_{d-1}$ being the event
horizon of the black holes. According to the AdS/CFT correspondence [18],
these solutions also describe holographic RG flows across dimensions from
$d$-dimensional SCFTs dual to $AdS_{d+1}$ in the UV to superconformal quantum
mechanics dual to the $AdS_{2}$ space in the IR via twisted compactifications
on $\mathcal{M}_{d-1}$.
In six dimensions, only the half-maximal non-chiral $N=(1,1)$ or $F(4)$ gauged
supergravity constructed in [19], see also [20, 21] for the matter-coupled
$F(4)$ gauged supergravity, admits supersymmetric $AdS_{6}$ vacua, see [22]
for general conditions on the existence of supersymmetric $AdS_{6}$ vacua. The
corresponding $AdS_{6}$ black hole solutions with the near horizon geometry of
the form $AdS_{2}\times\mathcal{M}_{4}$ from pure $F(4)$ gauged supergravity
has been considered in [23] and more recently in [24] and [25]. More general
solutions in the matter-coupled $F(4)$ gauged supergravity have appeared later
in [26] and [27]. In [26], the $F(4)$ gauged supergravity is coupled to one
vector multiplet leading to $SO(3)\times U(1)$ gauge group with all the fields
being uncharged under the $U(1)$ factor. A more general $F(4)$ gauged
supergravity coupled to three vector multiplets has been considered in [27].
The resulting gauged supergravity has $SO(3)\times SO(3)$ gauge group.
However, the black hole solutions with $AdS_{2}\times H^{2}\times H^{2}$ and
$AdS_{2}\times\mathcal{M}_{4}$ for $\mathcal{M}_{4}$ being a Kahler four-cycle
given in [27] have been found by considering a truncation to $SO(2)\times
SO(2)$ invariant sector. Within this sector, the effect of the second $SO(3)$
factor in the gauge group is invisible. In particular, the gauge coupling
constant for this factor does not appear at all in the BPS equations.
In this work, we look for more interesting supersymmetric $AdS_{6}$ black hole
solutions within this matter-coupled $F(4)$ gauged supergravity with three
vector multiplets and $SO(3)\times SO(3)$ gauge group. This gauged
supergravity has been first studied in [28], and further in [29, 30], in which
two $N=(1,1)$ supersymmetric $AdS_{6}$ vacua with $SO(3)\times SO(3)$ and
$SO(3)_{\textrm{diag}}$ symmetries have been found. By further reduce the
residual symmetry to $SO(2)_{\textrm{diag}}$, we will look for black holes
with hear horizon geometries of the form $AdS_{2}\times\mathcal{M}_{4}$ for
$\mathcal{M}_{4}$ being a product of two Riemann surfaces or a Kahler four-
cycle. We will see that, for negatively curved $\mathcal{M}_{4}$ spaces, there
exist a number of supersymmetric black hole solutions in asymptotically
$AdS_{6}$ spaces described by the two aforementioned $AdS_{6}$ vacua. In
addition, we will also consider the black hole solutions with
$\mathcal{M}_{4}$ given by a Cayley four-cycle by performing a topological
twist using $SO(3)_{\textrm{diag}}$ gauge fields. Although this has already
been studied in [27], we indeed find new supersymmetric $AdS_{6}$ black hole
solutions interpolating between the near horizon geometry and the two
supersymmetric $AdS_{6}$ vacua.
The paper is organized as follows. We give a brief review of the matter-
coupled $F(4)$ gauged supergravity in section 2. We then consider the $F(4)$
gauged supergravity coupled to three vector multiplets with $SO(3)\times
SO(3)$ gauge group and review two known supersymmetric $AdS_{6}$ vacua. By
truncating to $SO(2)_{\textrm{diag}}$ invariant sector, we study
$AdS_{2}\times\Sigma\times\widetilde{\Sigma}$ and find a new class of
$AdS_{2}\times H^{2}\times H^{2}$ solutions in section 3. A number of
numerical black hole solutions are also given. We subsequently extend the
anlysis to the case of $AdS_{2}\times\mathcal{M}_{4}$ solutions with
$\mathcal{M}_{4}$ given by a Kahler four-cycle in section 4. We also consider
the case of $\mathcal{M}_{4}$ being a Cayley four-cycle by performing a
topological twist using $SO(3)_{\textrm{diag}}$ gauge fields. In this case, we
find a new $AdS_{6}$ black hole solution in addition to the solution found
previously in [27]. We finally give some conclusions and comments in section
5. In the appendix, all the bosonic field equations of the matter-coupled
$F(4)$ gauged supergravity are recorded.

##  2 Matter coupled $N=(1,1)$ gauged supergravity in six dimensions

In this section, we review the structure of matter-coupled $F(4)$ gauged
supergravity in six dimensions obtained by gauging the half-maximal $N=(1,1)$
supergravity coupled to vector multiplets. We mostly follow the conventions of
the original construction in [20, 21] but with the metric signature
$(-+++++)$. We firstly review the general matter-coupled $F(4)$ gauged
supergravity coupled to an arbitrary number $n$ of vector multiplets and
finally consider the case of $n=3$ and $SO(3)\times SO(3)$ gauge group.

###  2.1 General structure of matter-coupled $F(4)$ gauged supergravity

The $N=(1,1)$ supergravity multiplet in six dimensions consists of the
following component fields

$\left(e^{a}_{\mu},\psi^{A}_{\mu},A^{\alpha}_{\mu},B_{\mu\nu},\chi^{A},\sigma\right).$

The bosonic fields are the graviton $e^{a}_{\mu}$, a two-form field
$B_{\mu\nu}$, four vector fields $A^{\alpha}_{\mu},\,\alpha=0,1,2,3$, and the
dilaton $\sigma$. Space-time and tangent space or flat indices are denoted
respectively by $\mu,\nu=0,\ldots,5$ and $a,b=0,\ldots,5$. The fermionic
fields are given by two gravitini $\psi^{A}_{\mu}$ and two spin-$\frac{1}{2}$
fields $\chi^{A}$ with indices $A,B,\ldots=1,2$ denoting the fundamental
representation of $SU(2)_{R}\sim USp(2)_{R}\sim SO(3)_{R}$ R-symmetry.
Following [20] and [21], we also introduce the $SU(2)_{R}$ adjoint indices
$r,s,\ldots=1,2,3$ according to the split of indices $\alpha=(0,r)$.
The vector multiplets are described by the field content

$(A_{\mu},\lambda_{A},\phi^{\alpha})^{I},\qquad I=1,2,\ldots,n$

consisting of $n$ vectors $A^{I}_{\mu}$, $2n$ gaugini $\lambda^{I}_{A}$, and
$4n$ scalars $\phi^{\alpha I}$ parametrizing $SO(4,n)/SO(4)\times SO(n)$ coset
manifold. All spinor fields $\chi^{A}$, $\psi^{A}_{\mu}$ and $\lambda_{A}$ as
well as the supersymmetry parameter $\epsilon^{A}$ are eight-component pseudo-
Majorana spinors. In addition, the $4+n$ vector fields from both the gravity
and vector multiplets will be collectively denoted by
$A^{\Lambda}=(A^{\alpha},A^{I})$.
Including the dilaton, there are $4n+1$ scalar fields described by
$\mathbb{R}^{+}\times SO(4,n)/SO(4)\times SO(n)$ coset with $\mathbb{R}^{+}$
corresponding to the dilaton. The $4n$ vector multiplet scalars can be
parametrized by a coset representative ${L^{\Lambda}}_{\underline{\Sigma}}$
transforming under the global $SO(4,n)$ and local $SO(4)\times SO(n)$
symmetries by left and right multiplications respectively with indices
$\Lambda,\underline{\Sigma}=0,\ldots,n+3$. We can also split the index
$\underline{\Sigma}$ transforming under the local $SO(4)\times SO(n)$ as
$\underline{\Sigma}=(\alpha,I)=(0,r,I)$. Accordingly, the coset representative
can be written as

${L^{\Lambda}}_{\underline{\Sigma}}=({L^{\Lambda}}_{\alpha},{L^{\Lambda}}_{I}).$
(1)

The inverse of ${L^{\Lambda}}_{\underline{\Sigma}}$ will be denoted by
${(L^{-1})^{\underline{\Lambda}}}_{\Sigma}=({(L^{-1})^{\alpha}}_{\Sigma},{(L^{-1})^{I}}_{\Sigma})$.
$SO(4,n)$ indices will be raised and lowered by the invariant tensor

$\eta_{\Lambda\Sigma}=\eta^{\Lambda\Sigma}=(\delta_{\alpha\beta},-\delta_{IJ}).$
(2)

We are interested in gauging a compact subgroup of $SO(4,n)$ of the form
$G=SO(3)\times G_{c}$ in which the $SO(3)$ factor is identified with the
R-symmetry $SO(3)_{R}\sim SU(2)_{R}$. This $SO(3)$ is gauged by three vector
fields $A^{r}$ within the gravity multiplet. $G_{c}$ is a compact subgroup of
$SO(n)$ gauged by the vector fields in vector multiplets with
$\textrm{dim}\,(G_{c})\leq n$. The structure constants of the gauge algebra
${f^{\Lambda}}_{\Sigma\Gamma}$ appearing in the Lie algebra of the gauge
generators $T_{\Lambda}$ as

$\left[T_{\Lambda},T_{\Sigma}\right]={f^{\Gamma}}_{\Lambda\Sigma}T_{\Gamma}\qquad\textrm{with}\qquad
f_{[\Lambda\Sigma\Gamma]}=0$ (3)

can be chosen to be $(\epsilon_{rst},C_{IJK})$ with $C_{IJK}$ being the
structure constants of $G_{c}$.
In addition to the gauging of $G\subset SO(4,n)$, there is also a massive
deformation of the two-form field needed for the existence of $AdS_{6}$ vacua.
With both of these deformations taken into account, the Lagrangian for
$N=(1,1)$ gauged supergravity can be written as

$\displaystyle e^{-1}\mathcal{L}$ $\displaystyle=$
$\displaystyle\frac{1}{4}R-e\partial_{\mu}\sigma\partial^{\mu}\sigma-\frac{1}{4}P^{I\alpha}_{\mu}P^{\mu}_{I\alpha}-\frac{1}{8}e^{-2\sigma}\mathcal{N}_{\Lambda\Sigma}\widehat{F}^{\Lambda}_{\mu\nu}\widehat{F}^{\Sigma\mu\nu}-\frac{3}{64}e^{4\sigma}H_{\mu\nu\rho}H^{\mu\nu\rho}$
$\displaystyle-V-\frac{1}{64}e^{-1}\epsilon^{\mu\nu\rho\sigma\lambda\tau}B_{\mu\nu}\left(\eta_{\Lambda\Sigma}\widehat{F}^{\Lambda}_{\rho\sigma}\widehat{F}^{\Sigma}_{\lambda\tau}+mB_{\rho\sigma}\widehat{F}^{\Lambda}_{\lambda\tau}\delta^{\Lambda
0}+\frac{1}{3}m^{2}B_{\rho\sigma}B_{\lambda\tau}\right)$

with $e=\sqrt{-g}$. Various field strength tensors are defined by

$\displaystyle\widehat{F}^{\Lambda}=F^{\Lambda}-m\delta^{\Lambda 0}B,\qquad
F^{\Lambda}=dA^{\Lambda}+\frac{1}{2}{f^{\Lambda}}_{\Sigma\Gamma}A^{\Sigma}\wedge
A^{\Gamma},\qquad H=dB\,.$ (5)

It is also useful to note the convention on components of form fields used in
[20, 21]

$F^{\Lambda}=F^{\Lambda}_{\mu\nu}dx^{\mu}\wedge
dx^{\nu}\qquad\textrm{and}\qquad H=H_{\mu\nu\rho}dx^{\mu}\wedge dx^{\nu}\wedge
dx^{\rho}\,.$ (6)

In particular, these lead to for example
$F^{\Lambda}_{\mu\nu}=\frac{1}{2}\left(\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}+\frac{1}{2}{f^{\Lambda}}_{\Sigma\Gamma}A^{\Sigma}_{\mu}A^{\Gamma}_{\nu}\right)$.
The scalar kinetic term is written in terms of the vielbein on
$SO(4,n)/SO(4)\times SO(n)$ denoted by
$P^{I\alpha}_{\mu}=P^{I\alpha}_{x}\partial_{\mu}\phi^{x}$, $x=1,\ldots,4n$.
This vielbein together with the $SO(4)\times SO(n)$ composite connections
$(\Omega^{rs},\Omega^{r0},\Omega^{IJ})$ are encoded in the left-invariant
1-form

${\Omega^{\underline{\Lambda}}}_{\underline{\Sigma}}={(L^{-1})^{\underline{\Lambda}}}_{\Pi}\nabla{L^{\Pi}}_{\underline{\Sigma}}\qquad\textrm{with}\qquad\nabla{L^{\Lambda}}_{\underline{\Sigma}}={dL^{\Lambda}}_{\underline{\Sigma}}-f^{\phantom{\Gamma}\Lambda}_{\Gamma\phantom{\Lambda}\Pi}A^{\Gamma}{L^{\Pi}}_{\underline{\Sigma}}\,.$
(7)

This leads to the vielbein with the following identification

$P^{I}_{\phantom{s}\alpha}=(P^{I}_{\phantom{a}0},P^{I}_{\phantom{a}r})=(\Omega^{I}_{\phantom{a}0},\Omega^{I}_{\phantom{a}r}).$
(8)

The symmetric scalar matrix $\mathcal{N}_{\Lambda\Sigma}$ appearing in the
kinetic term of the vector fields is defined by

$\displaystyle\mathcal{N}_{\Lambda\Sigma}=L_{\Lambda\alpha}{(L^{-1})^{\alpha}}_{\Sigma}-L_{\Lambda
I}{(L^{-1})^{I}}_{\Sigma}=(\eta LL^{T}\eta)_{\Lambda\Sigma}\,.$ (9)

As usual, gaugings lead to the scalar potential and modified supersymmetry
transformations of fermions by a number of fermion-shift matrices. The
explicit form of the scalar potential reads

$\displaystyle V$ $\displaystyle=$
$\displaystyle-e^{2\sigma}\left[\frac{1}{36}A^{2}+\frac{1}{4}B^{i}B_{i}+\frac{1}{4}\left(C^{I}_{\phantom{s}t}C_{It}+4D^{I}_{\phantom{s}t}D_{It}\right)\right]+m^{2}e^{-6\sigma}\mathcal{N}_{00}$
(10) $\displaystyle-me^{-2\sigma}\left[\frac{2}{3}AL_{00}-2B^{i}L_{0i}\right]$

with $\mathcal{N}_{00}$ being the $00$ component of
$\mathcal{N}_{\Lambda\Sigma}$. Supersymmetry transformation rules for all the
fermionic fields are given by

$\displaystyle\delta\psi_{\mu A}$ $\displaystyle=$ $\displaystyle
D_{\mu}\epsilon_{A}-\frac{1}{24}\left(Ae^{\sigma}+6me^{-3\sigma}(L^{-1})_{00}\right)\epsilon_{AB}\gamma_{\mu}\epsilon^{B}$
(11)
$\displaystyle-\frac{1}{8}\left(B_{t}e^{\sigma}-2me^{-3\sigma}(L^{-1})_{t0}\right)\gamma^{7}\sigma^{t}_{AB}\gamma_{\mu}\epsilon^{B}$
$\displaystyle+\frac{i}{16}e^{-\sigma}\left[\epsilon_{AB}(L^{-1})_{0\Lambda}\gamma_{7}+\sigma^{r}_{AB}(L^{-1})_{r\Lambda}\right]F^{\Lambda}_{\nu\lambda}(\gamma_{\mu}^{\phantom{s}\nu\lambda}-6\delta^{\nu}_{\mu}\gamma^{\lambda})\epsilon^{B}$
$\displaystyle+\frac{i}{32}e^{2\sigma}H_{\nu\lambda\rho}\gamma_{7}({\gamma_{\mu}}^{\nu\lambda\rho}-3\delta_{\mu}^{\nu}\gamma^{\lambda\rho})\epsilon_{A},$
$\displaystyle\delta\chi_{A}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\gamma^{\mu}\partial_{\mu}\sigma\epsilon_{AB}\epsilon^{B}+\frac{1}{24}\left[Ae^{\sigma}-18me^{-3\sigma}(L^{-1})_{00}\right]\epsilon_{AB}\epsilon^{B}$
(12)
$\displaystyle-\frac{1}{8}\left[B_{t}e^{\sigma}+6me^{-3\sigma}(L^{-1})_{t0}\right]\gamma^{7}\sigma^{t}_{AB}\epsilon^{B}$
$\displaystyle-\frac{i}{16}e^{-\sigma}\left[\sigma^{r}_{AB}(L^{-1})_{r\Lambda}-\epsilon_{AB}(L^{-1})_{0\Lambda}\gamma_{7}\right]F^{\Lambda}_{\mu\nu}\gamma^{\mu\nu}\epsilon^{B}$
$\displaystyle-\frac{i}{32}e^{2\sigma}H_{\nu\lambda\rho}\gamma_{7}\gamma^{\nu\lambda\rho}\epsilon_{A},$
$\displaystyle\delta\lambda^{I}_{A}$ $\displaystyle=$ $\displaystyle
P^{I}_{ri}\gamma^{\mu}\partial_{\mu}\phi^{i}\sigma^{r}_{\phantom{s}AB}\epsilon^{B}+P^{I}_{0i}\gamma^{7}\gamma^{\mu}\partial_{\mu}\phi^{i}\epsilon_{AB}\epsilon^{B}-\left(2i\gamma^{7}D^{I}_{\phantom{s}t}+C^{I}_{\phantom{s}t}\right)e^{\sigma}\sigma^{t}_{AB}\epsilon^{B}$
(13)
$\displaystyle+2me^{-3\sigma}(L^{-1})^{I}_{\phantom{ss}0}\gamma^{7}\epsilon_{AB}\epsilon^{B}-\frac{i}{2}e^{-\sigma}(L^{-1})^{I}_{\phantom{s}\Lambda}F^{\Lambda}_{\mu\nu}\gamma^{\mu\nu}\epsilon_{A}$

where $\sigma^{tC}_{\phantom{sd}B}$ are usual Pauli matrices, and
$\epsilon_{AB}=-\epsilon_{BA}$. $A,B$ indices can be raised and lowered by
$\epsilon^{AB}$ and $\epsilon_{AB}$ with the convention
$T^{A}=\epsilon^{AB}T_{B}$ and $T_{A}=T^{B}\epsilon_{BA}$. The covariant
derivative of $\epsilon_{A}$ is given by

$D_{\mu}\epsilon_{A}=\partial_{\mu}\epsilon_{A}+\frac{1}{4}\omega_{\mu}^{ab}\gamma_{ab}\epsilon_{A}+\frac{i}{2}\sigma^{r}_{AB}\left[\frac{1}{2}\epsilon^{rst}\Omega_{\mu
st}-i\gamma_{7}\Omega_{\mu r0}\right]\epsilon^{B}\,.$ (14)

Various components of the fermion-shift matrices are defined as follows

$\displaystyle A$ $\displaystyle=$ $\displaystyle\epsilon^{rst}K_{rst},\qquad
B^{i}=\epsilon^{ijk}K_{jk0},$ (15) $\displaystyle C^{\phantom{ts}t}_{I}$
$\displaystyle=$ $\displaystyle\epsilon^{trs}K_{rIs},\qquad D_{It}=K_{0It}$
(16)

where

$\displaystyle K_{rst}$ $\displaystyle=$ $\displaystyle
g_{1}\epsilon_{lmn}L^{l}_{\phantom{r}r}(L^{-1})_{s}^{\phantom{s}m}L_{\phantom{s}t}^{n}+g_{2}C_{IJK}L^{I}_{\phantom{r}r}(L^{-1})_{s}^{\phantom{s}J}L_{\phantom{s}t}^{K},$
$\displaystyle K_{rs0}$ $\displaystyle=$ $\displaystyle
g_{1}\epsilon_{lmn}L^{l}_{\phantom{r}r}(L^{-1})_{s}^{\phantom{s}m}L_{\phantom{s}0}^{n}+g_{2}C_{IJK}L^{I}_{\phantom{r}r}(L^{-1})_{s}^{\phantom{s}J}L_{\phantom{s}0}^{K},$
$\displaystyle K_{rIt}$ $\displaystyle=$ $\displaystyle
g_{1}\epsilon_{lmn}L^{l}_{\phantom{r}r}(L^{-1})_{I}^{\phantom{s}m}L_{\phantom{s}t}^{n}+g_{2}C_{IJK}L^{I}_{\phantom{r}r}(L^{-1})_{I}^{\phantom{s}J}L_{\phantom{s}t}^{K},$
$\displaystyle K_{0It}$ $\displaystyle=$ $\displaystyle
g_{1}\epsilon_{lmn}L^{l}_{\phantom{r}0}(L^{-1})_{I}^{\phantom{s}m}L_{\phantom{s}t}^{n}+g_{2}C_{IJK}L^{I}_{\phantom{r}0}(L^{-1})_{I}^{\phantom{s}J}L_{\phantom{s}t}^{K}\,.$
(17)

Finally, we note the convention on space-time gamma matrices $\gamma^{a}$
which is slightly different from those of [20, 21]. $\gamma^{a}$ satisfy the
Clifford algebra

$\\{\gamma^{a},\gamma^{b}\\}=2\eta^{ab},\qquad\eta^{ab}=\textrm{diag}(-1,1,1,1,1,1),$
(18)

and the chirality matrix is defined by
$\gamma_{7}=i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}\gamma^{4}\gamma^{5}$
with $\gamma_{7}^{2}=-\mathbf{1}$.

###  2.2 Matter-coupled $F(4)$ gauged supergravity with $SO(3)\times SO(3)$
gauge group and supersymmetric $AdS_{6}$ vacua

We now consider a specific case of $n=3$ vector multiplets and $SO(3)\times
SO(3)\sim SU(2)\times SU(2)$ gauge group. As previously mentioned, the first
$SO(3)$ is the R-symmetry gauged by $A^{r}_{\mu}$, and the second one gauged
by $A^{I}_{\mu}$, $I=1,2,3$, from the three vector multiplets. With
$C_{IJK}=\epsilon_{IJK}$, the structure constants of the full gauge group are
then given by

${f^{\Lambda}}_{\Pi\Sigma}=(\epsilon_{rst},\epsilon_{IJK}).$ (19)

The explicit parametrization of the scalar fields in $SO(4,3)/SO(4)\times
SO(3)$ coset can be obtained as in [28] by introducing the $7\times 7$
matrices

$(e^{\Lambda\Sigma})_{\Gamma\Pi}=\delta^{\Lambda}_{\Gamma}\delta^{\Sigma}_{\Pi},\qquad\Lambda,\Sigma,\Gamma,\Pi=0,\ldots,6\,.$
(20)

The compact $SO(4)\times SO(3)$ generators are given by

$\displaystyle SO(4):\qquad J^{\alpha\beta}$ $\displaystyle=$ $\displaystyle
e^{\beta,\alpha}-e^{\alpha,\beta},\qquad\alpha,\beta=0,1,2,3,$ $\displaystyle
SO(3):\qquad\tilde{J}^{IJ}$ $\displaystyle=$ $\displaystyle
e^{J+3,I+3}-e^{I+3,J+3},\qquad I,J=1,2,3$ (21)

while non-compact generators can be identified as

$\displaystyle Y_{\alpha I}=e^{\alpha,I+3}+e^{I+3,\alpha}\,.$ (22)

The structure constants given above imply that the $SO(3)\times SO(3)$ gauge
generators are given respectively by $J^{rs}$ and $\tilde{J}^{IJ}$.
This gauged supergravity has been originally studied in [28], and two
supersymmetric $N=(1,1)$ $AdS_{6}$ vacua with $SO(3)\times SO(3)$ and
$SO(3)_{\textrm{diag}}$ symmetries have been identified. For convenience, we
will also give these two vacua here. With the $SO(3)_{\textrm{diag}}$
generated by $J^{rs}+\tilde{J}^{rs}$, the only one singlet scalar from
$SO(4,3)/SO(4)\times SO(3)$ coset corresponds to the non-compact generator
$Y_{11}+Y_{22}+Y_{33}$. The coset representative can be written as

$L=e^{\phi(Y_{11}+Y_{22}+Y_{33})}\,.$ (23)

The resulting scalar potential reads

$\displaystyle V$ $\displaystyle=$
$\displaystyle\frac{1}{16}e^{2\sigma}\left[(g_{1}^{2}+g_{2}^{2})[\cosh
6\phi-9\cosh 2\phi]+8(g_{2}^{2}-g_{1}^{2})+8g_{1}g_{2}\sinh^{3}2\phi\right]$
(24) $\displaystyle+e^{-6\sigma}m^{2}-4e^{-2\sigma}m(g_{1}\cosh^{3}\phi-
g_{2}\sinh^{3}\phi)$

which admits two supersymmetric $AdS_{6}$ critical points given by

$\phi=0,\qquad\sigma=\frac{1}{4}\ln\left[\frac{3m}{g_{1}}\right],\qquad
V_{0}=-20m^{2}\left(\frac{g_{1}}{3m}\right)^{\frac{3}{2}}$ (25)

and

$\displaystyle\phi$ $\displaystyle=$
$\displaystyle\frac{1}{2}\ln\left[\frac{g_{1}+g_{2}}{g_{2}-g_{1}}\right],\qquad\sigma=\frac{1}{4}\ln\left[\frac{3m\sqrt{g_{2}^{2}-g_{1}^{2}}}{g_{1}g_{2}}\right],$
$\displaystyle V_{0}$ $\displaystyle=$
$\displaystyle-20m^{2}\left[\frac{g_{1}g_{2}}{3m\sqrt{g_{2}^{2}-g_{1}^{2}}}\right]^{\frac{3}{2}}\,.$
(26)

The first critical point is $SO(4)\sim SO(3)\times SO(3)$ invariant while the
second one preserves only $SO(3)_{\textrm{diag}}\subset SO(3)\times SO(3)$.
For later convenience, we will refer to these vacua as $AdS_{6}$ critical
point $i$ and $ii$, respectively. We can also set $g_{1}=3m$ to have vanishing
dilaton at critical point $i$.

##  3 Supersymmetric $AdS_{6}$ black holes with
$\Sigma\times\widetilde{\Sigma}$ horizons

We first consider black hole solutions with the near horizon geometry of the
form $AdS_{2}\times\Sigma\times\widetilde{\Sigma}$ for $\Sigma$ and
$\widetilde{\Sigma}$ being Riemann surfaces. The metric ansatz takes the form
of

$ds^{2}=-e^{2f(r)}dt^{2}+dr^{2}+e^{2h(r)}(d\theta^{2}+F_{\kappa}(\theta)^{2}d\phi^{2})+e^{2\tilde{h}(r)}(d\tilde{\theta}^{2}+\tilde{F}_{\tilde{\kappa}}(\tilde{\theta})^{2}d\tilde{\phi}^{2})$
(27)

with the function $F_{\kappa}(\theta)$ given by

$F_{\kappa}(\theta)=\begin{cases}\sin\theta,&\kappa=1\phantom{-}\quad\textrm{for}\quad\Sigma^{2}=S^{2}\\\
\theta,&\kappa=0\phantom{-}\quad\textrm{for}\quad\Sigma^{2}=T^{2}\\\
\sinh\theta,&\kappa=-1\quad\textrm{for}\quad\Sigma^{2}=H^{2}\end{cases}$ (28)

and similarly for $\tilde{F}_{\tilde{\kappa}}(\tilde{\theta})$.
With the following choice of vielbein

$\displaystyle e^{\hat{t}}=e^{f}dt,\qquad e^{\hat{r}}=dr,\qquad
e^{\hat{\theta}}=e^{h}d\theta,$ $\displaystyle
e^{\hat{\phi}}=e^{h}F_{\kappa}(\theta)d\phi,\qquad
e^{\hat{\tilde{\theta}}}=e^{\tilde{h}}d\tilde{\theta},\qquad
e^{\hat{\tilde{\phi}}}=e^{\tilde{h}}\tilde{F}_{\tilde{\kappa}}(\tilde{\theta})d\tilde{\phi},$
(29)

non-vanishing components of the spin connection for the above metric are given
explicitly by

$\displaystyle{\omega^{\hat{t}}}_{\hat{r}}=f^{\prime}e^{\hat{t}},\qquad{\omega^{\hat{\theta}}}_{\hat{r}}=h^{\prime}e^{\hat{\theta}},\qquad{\omega^{\hat{\phi}}}_{\hat{r}}=h^{\prime}e^{\hat{\phi}},\qquad{\omega^{\hat{\tilde{\theta}}}}_{\hat{r}}=\tilde{h}^{\prime}e^{\hat{\tilde{\theta}}}$
$\displaystyle{\omega^{\hat{\tilde{\phi}}}}_{\hat{r}}=\tilde{h}^{\prime}e^{\hat{\tilde{\phi}}},\qquad{\omega^{\hat{\phi}}}_{\hat{\theta}}=\frac{F^{\prime}_{\kappa}(\theta)}{F_{\kappa}(\theta)}e^{-h}e^{\hat{\phi}},\qquad{\omega^{\hat{\tilde{\phi}}}}_{\hat{\tilde{\theta}}}=\frac{\tilde{F}^{\prime}_{\tilde{\kappa}}(\tilde{\theta})}{\tilde{F}_{\tilde{\kappa}}(\tilde{\theta})}e^{-\tilde{h}}e^{\hat{\tilde{\phi}}}$
(30)

with hatted indices being flat indices. Throughout the paper, we will use ′ to
denote $r$-derivatives except for
$F^{\prime}_{\kappa}(\theta)=\frac{dF_{\kappa}(\theta)}{d\theta}$ and
$\tilde{F}^{\prime}_{\tilde{\kappa}}(\tilde{\theta})=\frac{d\tilde{F}_{\tilde{\kappa}}(\tilde{\theta})}{d\tilde{\theta}}$.
We also note some useful relations

$F^{\prime\prime}_{\kappa}(\theta)=-\kappa
F_{\kappa}(\theta)\qquad\textrm{and}\qquad
1-{F^{\prime}_{\kappa}(\theta)}^{2}=\kappa F_{\kappa}(\theta)^{2}$ (31)

which also hold for $\tilde{F}_{\tilde{\kappa}}(\tilde{\theta})$.
In this section, we will consider a truncation to $SO(2)_{\textrm{diag}}$
invariant sector. There are four singlet scalars which, for
$SO(2)_{\textrm{diag}}$ generated by $J^{12}+\tilde{J}^{12}$, can be described
by the coset representative

$L=e^{\phi_{0}Y_{03}}e^{\phi_{1}(Y_{11}+Y_{22})}e^{\phi_{2}Y_{33}}e^{\phi_{3}(Y_{12}-Y_{21})}\,.$
(32)

The resulting scalar potential is rather complicated and will not be needed in
the following analysis. Therefore, we refrain from giving it here.
To preserve some supersymmetry, we perform a topological twist by turning on
the following gauge fields

$A^{3}=aF^{\prime}_{\kappa}(\theta)d\phi+\tilde{a}\tilde{F}^{\prime}_{\tilde{\kappa}}(\tilde{\theta})d\tilde{\phi}\qquad\textrm{and}\qquad
A^{6}=bF^{\prime}_{\kappa}(\theta)d\phi+\tilde{b}\tilde{F}^{\prime}_{\tilde{\kappa}}(\tilde{\theta})d\tilde{\phi}$
(33)

with the condition $g_{1}A^{3}=g_{2}A^{6}$ or equivalently $g_{1}a=g_{2}b$ and
$g_{1}\tilde{a}=g_{2}\tilde{b}$ implementing the $SO(2)_{\textrm{diag}}$
subgroup. We also note the corresponding field strength tensors

$\displaystyle F^{3}=-\kappa aF_{\kappa}(\theta)d\theta\wedge
d\phi-\tilde{\kappa}\tilde{a}\tilde{F}_{\tilde{\kappa}}(\tilde{\theta})d\tilde{\theta}\wedge
d\tilde{\phi},$ $\displaystyle F^{6}=-\kappa bF_{\kappa}(\theta)d\theta\wedge
d\phi-\tilde{\kappa}\tilde{b}\tilde{F}_{\tilde{\kappa}}(\tilde{\theta})d\tilde{\theta}\wedge
d\tilde{\phi}\,.$ (34)

The composite connection can be straightforwardly computed to be

$\Omega_{\mu r0}=0\qquad\textrm{and}\qquad\Omega_{\mu
st}=g_{1}A^{3}_{\mu}(\delta_{s2}\delta_{t1}-\delta_{s1}\delta_{t2})$ (35)

which lead to

$Q_{\mu AB}=\frac{i}{2}\sigma_{rAB}\left[\frac{1}{2}\epsilon^{rst}\Omega_{\mu
st}-i\gamma_{7}\Omega_{\mu
r0}\right]=-\frac{i}{2}g_{1}A^{3}_{\mu}\sigma^{3}_{AB}\,.$ (36)

We also note that only the parts involving gauge fields in the composite
connection have been given above. There are additional contributions along the
$r$-direction due to non-vanishing scalars $\phi_{0}$ and $\phi_{3}$. However,
these do not affect the topological twist, so we have omitted them.
We now consider relevant parts of the supersymmetry transformation
$\delta\psi_{\hat{\phi}A}$ and $\delta\psi_{\hat{\tilde{\phi}}A}$. For
$\delta\psi_{\hat{\phi}A}$, we find

$0=\frac{1}{2}\frac{F^{\prime}_{\kappa}(\theta)}{F_{\kappa}(\theta)}e^{-h}\gamma_{\hat{\phi}\hat{\theta}}\epsilon_{A}-\frac{i}{2}g_{1}a\frac{F^{\prime}_{\kappa}(\theta)}{F_{\kappa}(\theta)}e^{-h}\sigma^{3}_{AB}\epsilon^{B}+\ldots$
(37)

with $\ldots$ refers to other terms independent of $\theta$ and $\phi$. By
imposing a projector and a twist condition

$\gamma_{\hat{\theta}\hat{\phi}}\epsilon_{A}=\mp
i\sigma^{3}_{AB}\epsilon^{B}\qquad\textrm{and}\qquad g_{1}a=\pm 1$ (38)

we can cancel the internal spin connection along $\Sigma$. As a result,
$\delta\psi_{\hat{\theta}A}$ and $\delta\psi_{\hat{\phi}A}$ conditions reduce
to the same BPS equation for the warp factor $h(r)$ as expected. A similar
analysis for $\delta\psi_{\hat{\tilde{\phi}}A}$ gives

$\gamma_{\hat{\tilde{\theta}}\hat{\tilde{\phi}}}\epsilon_{A}=\mp
i\sigma^{3}_{AB}\epsilon^{B}\qquad\textrm{and}\qquad g_{1}\tilde{a}=\pm 1\,.$
(39)

We also note that using the definition of
$\gamma_{7}=i\gamma^{\hat{t}}\gamma^{\hat{r}}\gamma^{\hat{\theta}}\gamma^{\hat{\phi}}\gamma^{\hat{\tilde{\theta}}}\gamma^{\hat{\tilde{\phi}}}$,
the two projectors imply

$\gamma^{\hat{t}\hat{r}}\epsilon_{A}=i\gamma_{7}\epsilon_{A}\,.$ (40)

For the gauge fields given in (33), we cannot consistently set the two-form
field to zero. Using the field equation (111) given in the appendix, we find
that $H_{\mu\nu\rho}=0$ and only $B_{\hat{t}\hat{r}}$ component is non-
vanishing and given by

$m^{2}e^{-2\sigma}\mathcal{N}_{00}B^{\hat{t}\hat{r}}=-\frac{1}{16}\epsilon^{\hat{t}\hat{r}\hat{\rho}\hat{\sigma}\hat{\lambda}\hat{\tau}}\eta_{\Lambda\Sigma}F^{\Lambda}_{\hat{\rho}\hat{\sigma}}F^{\Sigma}_{\hat{\lambda}\hat{\tau}}\,.$
(41)

With
$\epsilon^{\hat{t}\hat{r}\hat{\theta}\hat{\phi}\hat{\tilde{\theta}}\hat{\tilde{\phi}}}=1$,
we find

$B_{\hat{t}\hat{r}}=\frac{1}{8}\frac{\kappa\tilde{\kappa}(a\tilde{a}-b\tilde{b})}{m^{2}\mathcal{N}_{00}}e^{2\sigma-2h-2\tilde{h}}\,.$
(42)

To obtain the BPS equations, we also need to impose a projector involving
$\gamma^{\hat{r}}$

$\gamma^{\hat{r}}\epsilon_{A}=-\epsilon_{A}\,.$ (43)

The sign choice is chosen such that an $AdS_{6}$ vacuum appears at
$r\rightarrow\infty$ in the solutions. It turns out that consistency of the
BPS equations requires $\phi_{0}=0$ as in the solutions studied in [26] and
[27]. With all these, we find the following BPS equations

$\displaystyle\phi^{\prime}_{1}$ $\displaystyle=$
$\displaystyle-e^{\sigma}\textrm{sech}2\phi_{3}\sinh
2\phi_{1}(g_{1}\cosh\phi_{2}-g_{2}\sinh\phi_{2}),$ (44)
$\displaystyle\phi^{\prime}_{2}$ $\displaystyle=$
$\displaystyle-e^{\sigma}\left[g_{2}\cosh\phi_{2}+g_{1}\sinh\phi_{2}+\cosh
2\phi_{1}\cosh 2\phi_{3}(g_{1}\sinh\phi_{2}-g_{2}\cosh\phi_{2})\right]$
$\displaystyle+\frac{1}{2}e^{-\sigma-2h-2\tilde{h}}\left[\kappa
e^{2\tilde{h}}(b\cosh\phi_{2}-a\sinh\phi_{2})-\tilde{\kappa}e^{2h}(\tilde{b}\cosh\phi_{2}-\tilde{a}\sinh\phi_{2})\right],$
$\displaystyle\phi^{\prime}_{3}$ $\displaystyle=$
$\displaystyle-e^{\sigma}\cosh 2\phi_{1}\sinh
2\phi_{3}(g_{1}\cosh\phi_{2}-g_{2}\sinh\phi_{2}),$ (45)
$\displaystyle\sigma^{\prime}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}e^{\sigma}\left[g_{1}\cosh\phi_{2}+g_{2}\sinh\phi_{2}+\cosh
2\phi_{1}\cosh 2\phi_{3}(g_{1}\cosh\phi_{2}-g_{2}\sinh\phi_{2})\right]$ (46)
$\displaystyle+\frac{1}{8}e^{-\sigma-2h-2\tilde{h}}\left[\kappa
e^{2\tilde{h}}(a\cosh\phi_{2}-b\sinh\phi_{2})+\tilde{\kappa}e^{2h}(\tilde{a}\cosh\phi_{2}-\tilde{b}\sinh\phi_{2})\right]$
$\displaystyle+\frac{3}{2}me^{-3\sigma}-\frac{1}{32m}e^{\sigma-2h-2\tilde{h}}\kappa\tilde{\kappa}(b\tilde{b}-a\tilde{a})$
$\displaystyle h^{\prime}$ $\displaystyle=$
$\displaystyle\frac{1}{4}e^{\sigma}\left[g_{1}\cosh\phi_{2}+g_{2}\sinh\phi_{2}+\cosh
2\phi_{1}\cosh 2\phi_{3}(g_{1}\cosh\phi_{2}-g_{2}\sinh\phi_{2})\right]$ (47)
$\displaystyle+\frac{1}{8}e^{-\sigma-2h-2\tilde{h}}\left[3\kappa
e^{2\tilde{h}}(a\cosh\phi_{2}-b\sinh\phi_{2})-\tilde{\kappa}e^{2h}(\tilde{a}\cosh\phi_{2}-\tilde{b}\sinh\phi_{2})\right]$
$\displaystyle+\frac{1}{2}me^{-3\sigma}+\frac{1}{32m}e^{\sigma-2h-2\tilde{h}}\kappa\tilde{\kappa}(b\tilde{b}-a\tilde{a})$
$\displaystyle\tilde{h}^{\prime}$ $\displaystyle=$
$\displaystyle\frac{1}{4}e^{\sigma}\left[g_{1}\cosh\phi_{2}+g_{2}\sinh\phi_{2}+\cosh
2\phi_{1}\cosh 2\phi_{3}(g_{1}\cosh\phi_{2}-g_{2}\sinh\phi_{2})\right]$ (48)
$\displaystyle+\frac{1}{8}e^{-\sigma-2h-2\tilde{h}}\left[3\tilde{\kappa}e^{2h}(\tilde{a}\cosh\phi_{2}-\tilde{b}\sinh\phi_{2})-\kappa
e^{2\tilde{h}}(a\cosh\phi_{2}-b\sinh\phi_{2})\right]$
$\displaystyle+\frac{1}{2}me^{-3\sigma}+\frac{1}{32m}e^{\sigma-2h-2\tilde{h}}\kappa\tilde{\kappa}(b\tilde{b}-a\tilde{a})$
$\displaystyle f^{\prime}$ $\displaystyle=$
$\displaystyle\frac{1}{4}e^{\sigma}\left[g_{1}\cosh\phi_{2}+g_{2}\sinh\phi_{2}+\cosh
2\phi_{1}\cosh 2\phi_{3}(g_{1}\cosh\phi_{2}-g_{2}\sinh\phi_{2})\right]$ (49)
$\displaystyle-\frac{1}{8}e^{-\sigma-2h-2\tilde{h}}\left[\kappa
e^{2\tilde{h}}(a\cosh\phi_{2}-b\sinh\phi_{2})+\tilde{\kappa}e^{2h}(\tilde{a}\cosh\phi_{2}-\tilde{b}\sinh\phi_{2})\right]$
$\displaystyle+\frac{1}{2}me^{-3\sigma}-\frac{3}{32m}e^{\sigma-2h-2\tilde{h}}\kappa\tilde{\kappa}(b\tilde{b}-a\tilde{a})\,.$

In deriving these equations, we have chosen the upper sign choice for the
conditions given in (38) for definiteness. It can also be verified that these
equations are compatible with the second-order field equations. For large $r$
with $f\sim h\sim\tilde{h}\sim r$, these equations admit $AdS_{6}$ vacua given
in (25) and (26) as asymptotic solutions.
We note that for $\phi_{1}=\phi_{3}=0$ and without the $SO(2)_{\textrm{diag}}$
condition among the gauge fields $A^{3}$ and $A^{6}$, we recover the BPS
equations given in [26] and [27] for $SO(2)\times SO(2)$ symmetric black holes
up to some notational and conventional differences. However, in the present
case, the magnetic charges $b$ and $\tilde{b}$ are not independent but related
to $a$ and $\tilde{a}$ via the relations $g_{2}b=g_{1}a$ and
$g_{2}\tilde{b}=g_{1}\tilde{a}$. The solutions to these equations preserve
$\frac{1}{8}$ supersymmetry or $2$ supercharges due to three independent
projectors involving $\gamma_{\hat{\theta}\hat{\phi}}$,
$\gamma_{\hat{\tilde{\theta}}\hat{\tilde{\phi}}}$, and $\gamma^{\hat{r}}$.
At the horizon given by an $AdS_{2}\times\Sigma\times\widetilde{\Sigma}$
critical point, we have the conditions

$\sigma^{\prime}=h^{\prime}=\tilde{h}^{\prime}=\phi_{i}^{\prime}=0,\qquad
i=1,2,3,\qquad\textrm{and}\qquad f^{\prime}=\frac{1}{\ell}$ (50)

with an $AdS_{2}$ radius $\ell$. The constant scalars imply that the
$\gamma^{\hat{r}}$ projector is not needed in the BPS equations, and the
$AdS_{2}\times\Sigma\times\widetilde{\Sigma}$ solutions are $\frac{1}{4}$-BPS
preserving four supercharges.

###  3.1 $AdS_{2}\times\Sigma\times\widetilde{\Sigma}$ vacua

We now look for possible $AdS_{2}\times\Sigma\times\widetilde{\Sigma}$ vacua
from the above BPS equations. The first solution is given by

  * •

$AdS_{2}$ critical point I:

$\displaystyle\phi_{1}=\phi_{3}=0,$ $\displaystyle
h=\frac{1}{2}\ln\left[-\frac{e^{2\sigma}\kappa(a\cosh\phi_{2}-b\sinh\phi_{2})}{4m}\right],$
$\displaystyle\tilde{h}=\frac{1}{2}\ln\left[-\frac{e^{2\sigma}\tilde{\kappa}(\tilde{a}\cosh\phi_{2}-\tilde{b}\sinh\phi_{2})}{4m}\right],$
$\displaystyle\sigma=\frac{1}{4}\ln\left[\frac{m[3(b\tilde{b}-a\tilde{a})-(a\tilde{a}+b\tilde{b})\cosh
2\phi_{2}+(a\tilde{b}+b\tilde{a})\sinh
2\phi_{2}]}{2g_{1}\cosh^{3}\phi_{2}(a-b\tanh\phi_{2})(\tilde{b}\tanh\phi_{2}-\tilde{a})}\right],$
$\displaystyle\phi_{2}=\frac{1}{2}\ln\left[\frac{2\Phi^{\frac{2}{3}}+(3g_{1}+g_{2})\Phi^{\frac{1}{3}}+2g_{1}(3g_{1}+2g_{2})}{(g_{2}-g_{1})\Phi^{\frac{1}{3}}}\right],$
$\displaystyle\frac{1}{\ell}=\left[\frac{8mg_{1}^{3}\cosh\phi_{2}[3(b\tilde{b}-a\tilde{a})-(a\tilde{a}+b\tilde{b})\cosh
2\phi_{2}+(a\tilde{b}+b\tilde{a})\sinh
2\phi_{2}]}{(a-b\tanh\phi_{2})(\tilde{b}\tanh\phi_{2}-\tilde{a})}\right]^{\frac{1}{4}}$
(51)

with

$\Phi=5g_{1}^{3}+5g_{1}^{2}g_{2}+g_{1}g_{2}^{2}+g_{1}(g_{1}+g_{2})\sqrt{g_{2}^{2}-2g_{1}^{2}}\,.$
(52)

To simplify some expressions, we have used the relations $g_{1}a=g_{2}b$ and
$g_{1}\tilde{a}=g_{2}\tilde{b}$ which lead to a less symmetric appearance of
$(a,\tilde{a})$ and $(b,\tilde{b})$. A straightforward analysis shows that
only $\kappa=\tilde{\kappa}=-1$ leads to valid
$AdS_{2}\times\Sigma\times\widetilde{\Sigma}$ solutions.
There is also another $AdS_{2}\times\Sigma\times\widetilde{\Sigma}$ solution
of the form

  * •

$AdS_{2}$ critical point II:

$\displaystyle\phi_{2}=\frac{1}{2}\ln\left[\frac{g_{2}+g_{1}}{g_{2}-g_{1}}\right],$
$\displaystyle
h=\frac{1}{2}\ln\left[-\frac{e^{2\sigma}\kappa(ag_{2}-g_{1}b)}{4m\sqrt{g^{2}_{2}-g_{1}^{2}}}\right],\quad\tilde{h}=\frac{1}{2}\ln\left[-\frac{e^{2\sigma}\tilde{\kappa}(\tilde{a}g_{2}-\tilde{b}g_{1})}{4m\sqrt{g^{2}_{2}-g_{1}^{2}}}\right],$
$\displaystyle\sigma=\frac{1}{4}\ln\left[\frac{m[a\tilde{a}(2g_{2}^{2}-g_{1}^{2})+b\tilde{b}(g_{1}^{2}-g_{2}^{2})-g_{1}g_{2}(a\tilde{b}+\tilde{a}b)]\sqrt{g_{2}^{2}-g_{1}^{2}}}{g_{1}g_{2}(ag_{2}-bg_{1})(\tilde{a}g_{2}-\tilde{b}g_{1})}\right],$
$\displaystyle\phi_{1}=\frac{1}{2}\ln\left[\frac{2(g_{1}^{2}+g_{2}^{2})+\sqrt{2}\sqrt{g_{1}^{4}+6g_{1}^{2}g_{2}^{2}+g_{2}^{4}-(g_{2}^{2}-g_{1}^{2})\cosh
4\phi_{3}}}{2(g_{2}^{2}-g_{1}^{2})\cosh 2\phi_{3}}\right],$
$\displaystyle\frac{1}{\ell}=\left[\frac{16mg_{1}^{3}g_{2}^{3}[a\tilde{a}(2g_{2}^{2}-g_{1}^{2})+b\tilde{b}(g_{2}^{2}-2g_{1}^{2})-g_{1}g_{2}(a\tilde{b}+\tilde{a}b)]}{(g_{2}^{2}-g_{1}^{2})^{\frac{3}{2}}(ag_{2}-bg_{1})(\tilde{a}g_{2}-\tilde{b}g_{1})}\right]^{\frac{1}{4}}\,\,\,$
(53)

with $\phi_{3}$ being a constant. We also note that apart from $\phi_{1}$, all
other functions including the $AdS_{2}$ radius do not depend on the value of
$\phi_{3}$ at the critical point. For a particular value of $\phi_{3}=0$, we
find

$\phi_{1}=\frac{1}{2}\ln\left[\frac{g_{2}+g_{1}}{g_{2}-g_{1}}\right].$ (54)

As in the case of critical point I, it turns out that only
$\kappa=\tilde{\kappa}=-1$ leads to viable solutions. Therefore, there are
only black hole solutions with $AdS_{2}\times H^{2}\times H^{2}$ near horizon
geometries.

###  3.2 Numerical black hole solutions

The BPS equations are too complicated to be solved analytically, so we will
numerically find the black hole solutions interpolating between asymptotically
locally $AdS_{6}$ vacua and near horizon $AdS_{2}\times H^{2}\times H^{2}$
geometries. According to the AdS/CFT correspondence, the solutions also
describe RG flows across dimensions from five-dimensional SCFTs in the UV to
superconformal quantum mechanics in the IR.
We begin with solutions flowing to $AdS_{2}$ critical point I. We will choose
the values of various parameters as follows

$g_{1}=3m\qquad\textrm{and}\qquad m=\frac{1}{2}\,.$ (55)

With these values, the supersymmetric $AdS_{6}$ critical point with
$SO(3)\times SO(3)$ symmetry given in (25) has unit radius. We will also set
$\phi_{1}=\phi_{3}=0$ along the entire solutions. Examples of solutions with
different values of $g_{2}$ are shown in figure 1. We can also compute the
corresponding black hole entropy by using the relation

$S_{\textrm{BH}}=\frac{1}{4G_{\textrm{N}}}e^{2h+2\tilde{h}}\textrm{vol}(\Sigma)\textrm{vol}(\widetilde{\Sigma})$
(56)

with the volume of a genus-$\mathfrak{g}$ Riemann surface given by

$\textrm{vol}(\Sigma_{\mathfrak{g}})=2\pi\eta_{\mathfrak{g}},\qquad\eta_{\mathfrak{g}}=\begin{cases}2|\mathfrak{g}-1|,&\quad\textrm{for}\quad\mathfrak{g}\neq
1\\\ 1,&\quad\textrm{for}\quad\mathfrak{g}=1\end{cases}\,.$ (57)

For the present case of black hole with $H^{2}\times H^{2}$ horizon, we find

$S_{\textrm{BH}}=\frac{\pi^{2}\eta_{\mathfrak{g}}\eta_{\tilde{\mathfrak{g}}}\left[(a\tilde{a}+b\tilde{b})\cosh\phi_{2}+(a\tilde{a}-2b\tilde{b})\textrm{sech}\phi_{2}-(\tilde{a}b+a\tilde{b})\sinh\phi_{2}\right]}{16g_{1}mG_{\textrm{N}}}$
(58)

with the value of $\phi_{2}$ at the horizon given in (51).

(a) Solutions for $\sigma(r)$

(b) Solutions for $\phi_{2}(r)$

(c) Solutions for $h(r)$

(d) Solutions for $\tilde{h}(r)$

(e) Solutions for $f^{\prime}(r)$

Figure 1: Supersymmetric $AdS_{6}$ black holes interpolating between $AdS_{6}$
vacuum $i$ and the near horizon geometry $AdS_{2}\times H^{2}\times H^{2}$
(critical point I) for $g_{2}=4$ (blue), $g_{2}=6$ (red), $g_{2}=8$ (green).

We now move to black hole solutions with the near horizon geometry given by
$AdS_{2}\times H^{2}\times H^{2}$ critical point II. By setting $\phi_{3}=0$
and $g_{2}=4$ with all other parameters take the same values as in the
previous case, we give examples of black hole solutions in figure 2. In this
case, there is a solution that flows directly from $AdS_{6}$ critical point
$i$ to $AdS_{2}\times H^{2}\times H^{2}$ vacuum II (red curve) as well as a
solution interpolating between $AdS_{6}$ critical point $ii$ and
$AdS_{2}\times H^{2}\times H^{2}$ vacuum II (green curve). There are also
solutions interpolating between $AdS_{6}$ critical point $i$ and
$AdS_{2}\times H^{2}\times H^{2}$ vacuum II that flow very close to $AdS_{6}$
vacuum $ii$ (blue and purple curves). By fine tuning the boundary conditions,
we can find a solution interpolating between $AdS_{6}$ critical points $i$ and
$ii$ and $AdS_{2}\times H^{2}\times H^{2}$ vacuum II (cyan curve). In the
solutions for $f^{\prime}(r)$, we have also included the values of
$f^{\prime}(r)$ at various critical points (dashed lines), related to
$AdS_{6}$ and $AdS_{2}$ radii, to clearly illustrate the interpolations among
different vacua. The corresponding black hole entropy is given by

$S_{\textrm{BH}}=\frac{\pi^{2}\eta_{\mathfrak{g}}\eta_{\tilde{\mathfrak{g}}}\left[a\tilde{a}(2g_{2}^{2}-g_{1}^{2})+b\tilde{b}(2g_{1}^{2}-g_{2}^{2})-(a\tilde{b}+b\tilde{a})g_{1}g_{2}\right]}{16g_{1}g_{2}m\sqrt{g_{2}^{2}-g_{1}^{2}}G_{\textrm{N}}}\,.$
(59)

(a) Solutions for $\phi_{1}(r)$

(b) Solutions for $\phi_{2}(r)$

(c) Solutions for $\sigma(r)$

(d) Solutions for $h(r)$

(e) Solutions for $\tilde{h}(r)$

(f) Solutions for $f^{\prime}(r)$

Figure 2: Supersymmetric $AdS_{6}$ black holes interpolating between $AdS_{6}$
vacua $i$ and $ii$ and the near horizon geometry $AdS_{2}\times H^{2}\times
H^{2}$ (critical point II) for $g_{2}$=4.

###  3.3 Solutions with $SO(2)_{R}$ twist

Another possibility of performing the topological twist is to turn on only
$SO(2)_{R}\subset SO(3)_{R}$ gauge field. In this case, there are six singlet
scalars parametrized by the coset representative

$L=e^{\varphi_{1}Y_{01}}e^{\varphi_{2}Y_{02}}e^{\varphi_{3}Y_{03}}e^{\phi_{1}Y_{31}}e^{\phi_{2}Y_{32}}e^{\phi_{3}Y_{33}}\,.$
(60)

Similar to the case of $SO(2)_{\textrm{diag}}$ twist, consistency of the BPS
equations requires $\varphi_{1}=\varphi_{2}=\varphi_{3}=0$. The analysis of
the BPS equations shows that $AdS_{2}$ critical points exist only for
$\phi_{1}=\phi_{2}=\phi_{3}=0$. This gives the same critical point studied in
[24] and [27]. Accordingly, we will not give further detail on this result
here to avoid repetition.

##  4 Supersymmetric $AdS_{6}$ black holes with $\mathcal{M}_{4}$ horizons

In this section, we consider $AdS_{6}$ black holes with a near horizon
geometry of the form $AdS_{2}\times\mathcal{M}_{4}$ for $\mathcal{M}_{4}$
being an Einstein four-manifold. We will consider two types of
$\mathcal{M}_{4}$ namely a Kahler four-cycle and a Cayley four-cycle.

###  4.1 Black holes with Kahler four-cycle horizon

We first consider the case of $\mathcal{M}_{4}$ being a Kahler four-cycle with
a $U(2)\sim U(1)\times SU(2)$ holonomy. We will perform the twist along the
$U(1)$ part by turning on the $SO(2)_{\textrm{diag}}$ gauge field as in the
previous section. The $SO(4,3)/SO(4)\times SO(3)$ coset representative is
still given by (32). We will choose the metric ansatz to be

$ds^{2}=-e^{2f(r)}dt^{2}+dr^{2}+e^{2h(r)}ds^{2}_{\mathcal{M}_{4}}$ (61)

with the metric on $\mathcal{M}_{4}$ given by

$ds^{2}_{\mathcal{M}_{4}}=\frac{1}{f_{\kappa}^{2}(\rho)}\left[d\rho^{2}+\rho^{2}f_{\kappa}(\rho)(\tau_{1}^{2}+\tau^{2}_{2})+\rho^{2}\tau_{3}^{2}\right]$
(62)

for $f_{\kappa}(\rho)=1+\kappa\rho^{2}$. $\tau_{i}$, $i=1,2,3$, are $SU(2)$
left-invariant one-forms with the normalization

$d\tau_{i}=\epsilon_{ijk}\tau_{j}\wedge\tau_{k}\,.$ (63)

With an obvious choice of vielbein

$\displaystyle e^{\hat{t}}=e^{f}dt,\qquad e^{\hat{r}}=dr,\qquad
e^{\hat{1}}=\frac{e^{h}\rho}{\sqrt{f_{\kappa}(\rho)}}\tau_{1},$ $\displaystyle
e^{\hat{2}}=\frac{e^{h}\rho}{\sqrt{f_{\kappa}(\rho)}}\tau_{2},\qquad
e^{\hat{3}}=\frac{e^{h}\rho}{f_{\kappa}(\rho)}\tau_{3},\qquad
e^{\hat{4}}=\frac{e^{h}}{f_{\kappa}(\rho)}d\rho,$ (64)

we can determine non-vanishing components of the spin connection

$\displaystyle{\omega^{\hat{t}}}_{\hat{r}}=f^{\prime}e^{\hat{t}},\qquad{\omega^{\hat{\alpha}}}_{\hat{r}}=h^{\prime}e^{\hat{\alpha}},\quad\hat{\alpha}=1,2,3,4,$
$\displaystyle{\omega^{\hat{1}}}_{\hat{4}}={\omega^{\hat{2}}}_{\hat{3}}=\frac{e^{-h}}{\rho}e^{\hat{1}},\qquad{\omega^{\hat{2}}}_{\hat{4}}={\omega^{\hat{3}}}_{\hat{1}}=\frac{e^{-h}}{\rho}e^{\hat{2}},$
$\displaystyle{\omega^{\hat{1}}}_{\hat{2}}=\frac{e^{-h}}{\rho}(2\kappa\rho^{2}+1)e^{\hat{3}},\qquad{\omega^{\hat{4}}}_{\hat{3}}=\frac{e^{-h}}{\rho}(\kappa\rho^{2}-1)e^{\hat{3}}\,.$
(65)

To implement the topological twist along $\mathcal{M}_{4}$, we turn on the
following gauge fields

$A^{3}=3a\kappa\rho e^{-h}e^{\hat{3}}\qquad\textrm{and}\qquad
A^{6}=3b\kappa\rho e^{-h}e^{\hat{3}}$ (66)

with $g_{2}A^{6}=g_{1}A^{3}$ as in the previous section.
We first consider the $\delta\psi_{\hat{3}A}$ condition

$0=\frac{1}{2}\frac{e^{-h}}{\rho}(2\kappa\rho^{2}+1)\gamma_{\hat{1}\hat{2}}\epsilon_{A}+\frac{1}{2}\frac{e^{-h}}{\rho}(\kappa\rho^{2}-1)\gamma_{\hat{4}\hat{3}}\epsilon_{A}-\frac{3i}{2}g_{1}a\kappa\rho
e^{-h}\sigma^{3}_{AB}\epsilon^{B}+\ldots$ (67)

in which we have used the composite connection given in (36). We now perform
the twist by imposing the projectors

$\gamma_{\hat{1}\hat{2}}\epsilon_{A}=-\gamma_{\hat{3}\hat{4}}\epsilon_{A}=i\sigma^{3}_{AB}\epsilon^{B}$
(68)

along with the twist condition

$g_{1}a=1\,.$ (69)

On the other hand, the conditions $\delta\psi_{\hat{1}A}$ and
$\delta\psi_{\hat{2}A}$ give

$0=\frac{1}{2}\frac{e^{-h}}{\rho}(\gamma_{\hat{1}\hat{4}}+\gamma_{\hat{2}\hat{3}})\epsilon_{A}+\ldots\qquad\textrm{and}\qquad
0=\frac{1}{2}\frac{e^{-h}}{\rho}(\gamma_{\hat{2}\hat{4}}+\gamma_{\hat{3}\hat{1}})\epsilon_{A}+\ldots\,.$
(70)

By the projectors (68), these terms vanish identically, and all the conditions
from $\delta\psi_{\hat{\alpha}A}$ with $\hat{\alpha}=1,2,3,4$ lead to the same
BPS equation for the warp factor $h(r)$.
With the gauge field strength tensors

$F^{3}=6\kappa ae^{-2h}(e^{\hat{1}}\wedge e^{\hat{2}}-e^{\hat{3}}\wedge
e^{\hat{4}})\qquad\textrm{and}\qquad F^{6}=6\kappa be^{-2h}(e^{\hat{1}}\wedge
e^{\hat{2}}-e^{\hat{3}}\wedge e^{\hat{4}}),$ (71)

we find the non-vanishing component of the two-form field

$B_{\hat{t}\hat{r}}=\frac{9}{2}\frac{\kappa^{2}e^{2\sigma-4h}}{m^{2}\mathcal{N}_{00}}(b^{2}-a^{2}).$
(72)

We also note that the projectors in (68) imply

$\gamma^{\hat{t}\hat{r}}\epsilon_{A}=-i\gamma_{7}\epsilon_{A}\,.$ (73)

Together with the $\gamma^{\hat{r}}$ projector (43), we find the following BPS
equations

$\displaystyle\phi_{1}^{\prime}$ $\displaystyle=$
$\displaystyle-e^{\sigma}\textrm{sech}2\phi_{3}\sinh
2\phi_{1}(g_{1}\cosh\phi_{2}-g_{2}\sinh\phi_{2}),$ (74)
$\displaystyle\phi_{2}^{\prime}$ $\displaystyle=$
$\displaystyle-e^{\sigma}\left[g_{1}\sinh\phi_{2}+g_{2}\cosh\phi_{2}+\cosh
2\phi_{1}\cosh 2\phi_{3}(g_{1}\sinh\phi_{2}-g_{2}\cosh\phi_{2})\right]$ (75)
$\displaystyle+6\kappa e^{-\sigma-2h}(b\cosh\phi_{2}-a\sinh\phi_{2}),$
$\displaystyle\phi_{3}^{\prime}$ $\displaystyle=$
$\displaystyle-e^{\sigma}\cosh 2\phi_{1}\sinh
2\phi_{3}(g_{1}\cosh\phi_{2}-g_{2}\sinh\phi_{2}),$ (76)
$\displaystyle\sigma^{\prime}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}e^{\sigma}\left[g_{1}\cosh\phi_{2}+g_{2}\sinh\phi_{2}+\cosh
2\phi_{1}\cosh 2\phi_{3}(g_{1}\cosh\phi_{2}-g_{2}\sinh\phi_{2})\right]$ (77)
$\displaystyle+\frac{3}{2}me^{-3\sigma}+\frac{9}{2}\kappa
e^{-\sigma-2h}(a\cosh\phi_{2}-b\sinh\phi_{2})+\frac{9\kappa^{2}}{8m}e^{\sigma-4h}(b^{2}-a^{2}),$
$\displaystyle h^{\prime}$ $\displaystyle=$
$\displaystyle\frac{1}{4}e^{\sigma}\left[g_{1}\cosh\phi_{2}+g_{2}\sinh\phi_{2}+\cosh
2\phi_{1}\cosh 2\phi_{3}(g_{1}\cosh\phi_{2}-g_{2}\sinh\phi_{2})\right]$ (78)
$\displaystyle+\frac{1}{2}me^{-3\sigma}+\frac{9}{2}\kappa
e^{-\sigma-2h}(a\cosh\phi_{2}-b\sinh\phi_{2})+\frac{9\kappa^{2}}{8m}e^{\sigma-4h}(a^{2}-b^{2}),$
$\displaystyle f^{\prime}$ $\displaystyle=$
$\displaystyle\frac{1}{4}e^{\sigma}\left[g_{1}\cosh\phi_{2}+g_{2}\sinh\phi_{2}+\cosh
2\phi_{1}\cosh 2\phi_{3}(g_{1}\cosh\phi_{2}-g_{2}\sinh\phi_{2})\right]$ (79)
$\displaystyle+\frac{1}{2}me^{-3\sigma}-\frac{9}{2}\kappa
e^{-\sigma-2h}(a\cosh\phi_{2}-b\sinh\phi_{2})-\frac{27\kappa^{2}}{8m}e^{\sigma-4h}(a^{2}-b^{2}).$

From these equations, we find two $AdS_{2}\times\mathcal{M}_{4}$ critical
points:

  * •

$AdS_{2}$ critical point III:

$\displaystyle\phi_{1}$ $\displaystyle=$ $\displaystyle\phi_{3}=0,$
$\displaystyle\phi_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\ln\left[\frac{290g_{1}^{2}-164g_{1}g_{2}-4g_{2}^{2}+(g_{2}-29g_{1})\Phi^{\frac{1}{3}}+2\Phi^{\frac{2}{3}}}{9(g_{2}-g_{1})\Phi^{\frac{1}{3}}}\right],$
$\displaystyle h$ $\displaystyle=$
$\displaystyle\frac{1}{2}\ln\left[\frac{3\kappa
e^{-2\sigma}(b\coth\phi_{2}-a)}{g_{1}}\right],$ $\displaystyle\sigma$
$\displaystyle=$
$\displaystyle\frac{1}{4}\ln\left[\frac{2m(b\coth\phi_{2}-a)}{3g_{1}(b\sinh\phi_{2}-a\cosh\phi_{2})}\right],$
$\displaystyle\frac{1}{\ell}$ $\displaystyle=$
$\displaystyle\frac{1}{2}me^{-3\sigma}+2g_{1}e^{\sigma}\cosh\phi_{2}$ (80)
$\displaystyle+\frac{3(b^{2}-a^{2})g_{1}^{2}e^{4\sigma}+12bmg_{1}\textrm{csch}\phi_{2}(a-b\coth\phi_{2})}{8me^{-\sigma}(a-b\coth\phi_{2})^{2}}$

with

$\displaystyle\Phi$ $\displaystyle=$ $\displaystyle
9\sqrt{3}(g_{1}+g_{2})\sqrt{8g_{2}^{4}+767g_{1}^{2}g_{2}^{2}-1000g_{1}^{4}}$
(81)
$\displaystyle+44g_{2}^{3}+33g_{1}g_{2}^{2}+1689g_{1}^{2}g_{2}-1675g_{1}^{3}\,.$

  * •

$AdS_{2}$ critical point IV:

$\displaystyle\phi_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\ln\left[\frac{g_{2}+g_{1}}{g_{2}-g_{1}}\right],\qquad\kappa=-1,$
$\displaystyle h$ $\displaystyle=$
$\displaystyle\frac{1}{4}\ln\left[\frac{9}{4}\frac{a^{2}(g_{1}^{2}+8g_{2}^{2})+b^{2}(8g_{1}^{2}+g_{2}^{2})-18abg_{1}g_{2}}{mg_{1}g_{2}\sqrt{g_{2}^{2}-g_{1}^{2}}}\right],$
$\displaystyle\sigma$ $\displaystyle=$
$\displaystyle\frac{1}{4}\ln\left[\frac{m\sqrt{g_{2}^{2}-g_{1}^{2}}[a^{2}(g_{1}^{2}+8g_{2}^{2})+b^{2}(8g_{1}^{2}+g_{2}^{2})-18abg_{1}g_{2}]}{9g_{1}g_{2}(bg_{1}-ag_{2})^{2}}\right],$
$\displaystyle\phi_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\ln\left[\frac{2(g_{1}^{2}+g_{2}^{2})+\sqrt{2}\sqrt{g_{1}^{4}+6g_{1}^{2}g_{2}^{2}+g_{2}^{4}+(g_{2}^{2}-g_{1}^{2})^{2}\cosh
4\phi_{3}}}{2(g_{2}^{2}-g_{1}^{2})\cosh 2\phi_{3}}\right],$
$\displaystyle\frac{1}{\ell}$ $\displaystyle=$
$\displaystyle\frac{2^{\frac{7}{4}}g_{1}^{\frac{3}{4}}g_{2}^{\frac{3}{4}}m^{\frac{1}{4}}}{\sqrt{3}(g_{2}^{2}-g_{1}^{2})^{\frac{3}{8}}}$
(82)

with $\phi_{3}$ being a constant. As in the previous section, for
$\phi_{3}=0$, we find

$\phi_{1}=\frac{1}{2}\ln\left[\frac{g_{2}+g_{1}}{g_{2}-g_{1}}\right].$ (83)

We note again that only $\phi_{1}$ depends on the value of $\phi_{3}$.

In both of these critical points, $AdS_{2}\times\mathcal{M}_{4}$ solutions
exist only for $\kappa=-1$ leading to black holes with $\mathcal{M}_{4}^{-}$
horizons. We now give numerical black hole solutions interpolating between
these geometries and supersymmetric $AdS_{6}$ vacua. With
$\phi_{1}=\phi_{3}=0$, $m=\frac{1}{2}$, and $\kappa=-1$, we find examples of
solutions interpolating between $AdS_{6}$ vacua $i$ and
$AdS_{2}\times\mathcal{M}^{-}_{4}$ critical point III with different values of
$g_{2}$ as shown in figure 3. Similar to the black hole solutions with
$H^{2}\times H^{2}$ horizons given in the previous section, we can compute the
black hole entropy as

$\displaystyle S_{\textrm{BH}}$ $\displaystyle=$
$\displaystyle\frac{1}{4G_{\textrm{N}}}e^{4h}\textrm{vol}(\mathcal{M}_{4}),$
(84) $\displaystyle=$
$\displaystyle\frac{27\sinh\phi_{2}(a-b\coth\phi_{2})(a\coth\phi_{2}-b)\textrm{vol}(\mathcal{M}_{4})}{8g_{1}mG_{\textrm{N}}}$

with the value of $\phi_{2}$ at the $AdS_{2}\times\mathcal{M}^{-}_{4}$ horizon
given in (80).

(a) Solutions for $\sigma(r)$

(b) Solutions for $\phi_{2}(r)$

(c) Solutions for $h(r)$

(d) Solutions for $f^{\prime}(r)$

Figure 3: Supersymmetric $AdS_{6}$ black holes interpolating between $AdS_{6}$
vacuum $i$ and the near horizon geometry $AdS_{2}\times\mathcal{M}_{4}^{-}$
(critical point III) for $g_{2}=2$ (red), $g_{2}=4$ (green), $g_{2}=6$ (blue).

Examples of black hole solutions with the near horizon geometry given by
critical point IV are shown in figure 4 for $\phi_{3}=0$ and $g_{2}=4$. As in
the previous section, there exist solutions interpolating between the two
supersymmetric $AdS_{6}$ vacua (critical point $i$ and $ii$) and
$AdS_{2}\times\mathcal{M}^{-}_{4}$ geometry IV. In this case, the entropy of
the black hole is given by

$S_{\textrm{BH}}=\frac{9\left[a^{2}(g_{1}^{2}+g_{2}^{2})+b^{2}(8g_{1}^{2}+g_{2}^{2})-18abg_{1}g_{2}\right]\textrm{vol}(\mathcal{M}_{4})}{16g_{1}g_{2}m\sqrt{g_{2}^{2}-g_{1}^{2}}G_{\textrm{N}}}\,.$
(85)

(a) Solutions for $\phi_{1}(r)$

(b) Solutions for $\phi_{2}(r)$

(c) Solutions for $\sigma(r)$

(d) Solutions for $h(r)$

(e) Solutions for $f^{\prime}(r)$

Figure 4: Supersymmetric $AdS_{6}$ black holes interpolating between $AdS_{6}$
vacua $i$ and $ii$ and the near horizon geometry
$AdS_{2}\times\mathcal{M}^{-}_{4}$ (critical point IV) for $g_{2}=4$.

###  4.2 Black holes with Cayley four-cycle horizon

In this section, we consider black hole solutions with the horizon
$\mathcal{M}_{4}$ given by a Cayley four-cycle with the metric ansatz

$ds^{2}=-e^{2f(r)}dt^{2}+dr^{2}+e^{2h(r)}ds^{2}_{\mathcal{M}_{4}}$ (86)

and

$ds^{2}_{\mathcal{M}_{4}}=d\rho^{2}+F_{\kappa}(\rho)^{2}(\tau_{1}^{2}+\tau_{2}^{2}+\tau_{3}^{2})$
(87)

in which $\tau_{i}$ are $SU(2)$ left-invariant one-forms as in the case of
Kahler four-cycle and $F_{\kappa}(\rho)$ defined as in (28).
In this section, we will perform a twist by turning on $SO(3)_{\textrm{diag}}$
gauge fields and identify this $SO(3)_{\textrm{diag}}$ with the self-dual part
$SU(2)_{+}$ of the $SO(4)\sim SU(2)_{+}\times SU(2)_{-}$ isometry of the four-
cycle. We note that this has already been considered in [27] in which an
$AdS_{2}\times\mathcal{M}_{4}$ solution has been found. However, we do find a
new $AdS_{2}\times\mathcal{M}_{4}$ critical point from the resulting BPS
equations. Therefore, we will repeat this analysis here with more detail since
this might be useful for further study.
With the vielbein chosen as

$e^{\hat{t}}=e^{f}dt,\qquad e^{\hat{r}}=dr,\qquad
e^{\hat{i}}=e^{h}F_{\kappa}(\rho)\tau_{i},\qquad e^{\hat{4}}=e^{h}d\rho,$ (88)

we find non-vanishing components of the spin connection as follows

$\displaystyle{\omega^{\hat{t}}}_{\hat{r}}=f^{\prime}e^{\hat{r}},\qquad{\omega^{\hat{i}}}_{\hat{r}}=h^{\prime}e^{\hat{4}},\qquad{\omega^{\hat{i}}}_{\hat{r}}=h^{\prime}e^{\hat{i}},\qquad\hat{i}=1,2,3,$
$\displaystyle{\omega^{\hat{i}}}_{\hat{4}}=e^{-h}\frac{F^{\prime}_{\kappa}(\rho)}{F_{\kappa}(\rho)}e^{\hat{i}},\qquad{\omega^{\hat{i}}}_{\hat{j}}=\frac{e^{-h}}{F_{\kappa}(\rho)}\epsilon_{\hat{i}\hat{j}\hat{k}}e^{\hat{k}}\,.$
(89)

Using the coset representative for the $SO(3)_{\textrm{diag}}$ singlet scalar
given in (23), we find the composite connection

$Q_{AB}=-\frac{i}{2}g_{1}A^{r}\sigma^{r}_{AB}$ (90)

in which we have used the relation $g_{2}A^{I}=g_{1}\delta^{I}_{r}A^{r}$ for
$r,I=1,2,3$. In order to cancel the internal spin connection on
$\mathcal{M}_{4}$, we choose the gauge fields to be

$A^{r}=a(F^{\prime}_{\kappa}(\rho)+1)\delta^{r}_{i}\tau_{i}\qquad\textrm{and}\qquad
A^{I}=b(F^{\prime}_{\kappa}(\rho)+1)\delta^{I}_{i}\tau_{i}$ (91)

with $g_{2}b=g_{1}a$. We also note that it is possible to begin with three
different magnetic charges $a_{r}$, but the twist condition will subsequently
require all the charges to be equal, see also [27].
We now consider the supersymmetry conditions $\delta\psi_{\hat{i}A}$ which
give

$0=\frac{1}{2}\frac{e^{-h}}{F_{\kappa}(\rho)}\left(F^{\prime}_{\kappa}(\rho)\gamma_{\hat{i}\hat{4}}\epsilon_{A}+\frac{1}{2}\epsilon_{\hat{i}\hat{j}\hat{k}}\gamma_{\hat{j}\hat{k}}\epsilon_{A}\right)-\frac{i}{2}g_{1}A^{r}_{\hat{i}}\sigma^{r}_{AB}\epsilon^{B}+\ldots\,.$
(92)

We then impose the following projectors

$\gamma_{\hat{i}\hat{4}}\epsilon_{A}=\frac{1}{2}\epsilon_{\hat{i}\hat{j}\hat{k}}\gamma_{\hat{j}\hat{k}}\epsilon_{A}=-i\delta^{r}_{i}\sigma^{r}_{AB}\epsilon^{B}$
(93)

and the twist condition

$g_{1}a=-1\,.$ (94)

With all these, $\delta\psi_{\hat{i}A}$ conditions reduce to the same BPS
equation for $h(r)$ obtained from $\delta\psi_{\hat{4}A}$. It should be noted
that there are only three independent projectors leading to $\frac{1}{8}$-BPS
$AdS_{2}\times\mathcal{M}_{4}$ solutions preserving two supercharges. The full
black hole solutions with running scalars will further break supersymmetry to
$\frac{1}{16}$-BPS due to the additional $\gamma^{\hat{r}}$ projector (43).
The gauge field strength tensors in this case are given by

$F^{r}=\kappa a\delta^{r}_{i}e^{-2h}\left(e^{\hat{i}}\wedge
e^{\hat{4}}+\frac{1}{2}\epsilon_{\hat{i}\hat{j}\hat{k}}e^{\hat{j}}\wedge
e^{\hat{k}}\right)$ (95)

together with $F^{I}=\frac{g_{1}}{g_{2}}\delta^{I}_{r}F^{r}$. The two-form
field takes the form

$B_{\hat{t}\hat{r}}=\frac{3}{8}\frac{\kappa^{2}e^{2\sigma-4h}}{m^{2}\mathcal{N}_{00}}(a^{2}-b^{2}).$
(96)

With all these and a useful (not independent) relation

$\gamma^{\hat{t}\hat{r}}\epsilon_{A}=i\gamma_{7}\epsilon_{A},$ (97)

we obtain the BPS equations

$\displaystyle\phi^{\prime}$ $\displaystyle=$ $\displaystyle-e^{\sigma}\sinh
2\phi(g_{1}\cosh\phi-g_{2}\sinh\phi)+\kappa
e^{-\sigma-2h}(a\sinh\phi-b\cosh\phi),$ (98) $\displaystyle\sigma^{\prime}$
$\displaystyle=$
$\displaystyle\frac{3}{2}me^{-3\sigma}-\frac{1}{2}e^{\sigma}(g_{1}\cosh^{3}\phi-
g_{2}\sinh^{3}\phi)+\frac{3}{4}\kappa e^{-\sigma-2h}(b\sinh\phi-a\cosh\phi)$
(99) $\displaystyle+\frac{3}{32m}\kappa^{2}e^{\sigma-4h}(a^{2}-b^{2}),$
$\displaystyle h^{\prime}$ $\displaystyle=$
$\displaystyle\frac{1}{2}e^{\sigma}(g_{1}\cosh^{3}\phi-
g_{2}\sinh^{3}\phi)+\frac{1}{2}me^{-3\sigma}$ (100)
$\displaystyle+\frac{3}{4}\kappa
e^{-\sigma-2h}(b\sinh\phi-a\cosh\phi)+\frac{3}{32m}\kappa^{2}e^{\sigma-4h}(b^{2}-a^{2}),$
$\displaystyle f^{\prime}$ $\displaystyle=$
$\displaystyle\frac{1}{2}e^{\sigma}(g_{1}\cosh^{3}\phi-
g_{2}\sinh^{3}\phi)+\frac{1}{2}me^{-3\sigma}$ (101)
$\displaystyle-\frac{3}{4}\kappa
e^{-\sigma-2h}(b\sinh\phi-a\cosh\phi)-\frac{9}{32m}\kappa^{2}e^{\sigma-4h}(b^{2}-a^{2}).$

There are two $AdS_{2}\times\mathcal{M}_{4}$ critical points to these
equations. The first one is given by

$\displaystyle h$ $\displaystyle=$
$\displaystyle\frac{1}{2}\ln\left[\frac{3\kappa
e^{2\sigma}(a\cosh\phi-b\sinh\phi)}{4m}\right],$ $\displaystyle\sigma$
$\displaystyle=$
$\displaystyle\frac{1}{4}\ln\left[\frac{4m(b\cosh\phi-a\sinh\phi)}{\sinh
2\phi(b\sinh\phi-a\cosh\phi)(g_{1}\cosh\phi-g_{2}\sinh\phi)}\right],$
$\displaystyle\phi$ $\displaystyle=$
$\displaystyle\frac{1}{2}\ln\left[\frac{2\Phi^{\frac{2}{3}}+(13g_{2}-7g_{1})\Phi^{\frac{1}{3}}+2(7g_{1}^{2}-50g_{1}g_{2}+52g_{2}^{2})}{3(g_{1}-g_{2})\Phi^{\frac{1}{3}}}\right],$
$\displaystyle\frac{1}{\ell}$ $\displaystyle=$
$\displaystyle\frac{3}{2}me^{-3\sigma}+\frac{1}{2}e^{\sigma}(g_{1}\cosh^{3}\phi-
g_{2}\sinh^{3}\phi)+\frac{m(a^{2}-b^{2})e^{-3\sigma}}{2(a\cosh\phi-b\sinh\phi)^{2}}\quad$
(102)

with

$\Phi=3\sqrt{3}(g_{2}-g_{1})\sqrt{131g_{1}^{2}g_{2}^{2}-192g_{2}^{4}-2g_{1}^{4}}-17g_{1}^{3}+213g_{2}^{2}g_{2}-537g_{1}g_{2}^{2}+368g_{2}^{3}\,.$
(103)

The solution only exists for $\kappa=-1$. This is the
$AdS_{2}\times\mathcal{M}^{-}_{4}$ solution found in [27] in which the
numerical black hole solution has also been given. Therefore, we will not
discuss this solution any further but only give, for completeness, the
corresponding black hole entropy

$S_{\textrm{BH}}=\frac{9e^{4\sigma}(a\cosh\phi-b\sinh\phi)^{2}\textrm{vol}(\mathcal{M}_{4}^{-})}{64m^{2}G_{\textrm{N}}}$
(104)

with $\phi$ given in (102).
There is another $AdS_{2}\times\mathcal{M}_{4}$ solution that has not been
given in [27]. This critical point takes the form

  * •

$AdS_{2}$ critical point V:

$\displaystyle\phi$ $\displaystyle=$
$\displaystyle\frac{1}{2}\ln\left[\frac{g_{2}+g_{1}}{g_{2}-g_{1}}\right],\quad\sigma=\frac{1}{4}\ln\left[\frac{4m\sqrt{g_{2}^{2}-g_{1}^{2}}}{3g_{1}g_{2}}\right],$
$\displaystyle h$ $\displaystyle=$
$\displaystyle\frac{1}{2}\ln\left[\frac{\sqrt{3}\kappa
mg_{1}a(g_{2}^{2}-g_{1}^{2})^{\frac{3}{4}}}{2(mg_{1}g_{2})^{\frac{3}{2}}}\right],\qquad\frac{1}{\ell}=\frac{2\sqrt{2}m}{3^{\frac{1}{4}}(g_{2}^{2}-g_{1}^{2})^{\frac{3}{8}}}\left(\frac{g_{1}g_{2}}{m}\right)^{\frac{3}{4}}\,.\qquad$
(105)

As in all the other cases, the solution only exists for $\kappa=-1$. Examples
of numerical solutions for $m=\frac{1}{2}$ and $g_{2}=4$ are given in figure
5. The entropy of the black hole is given by

$\displaystyle
S_{\textrm{BH}}=\frac{3a^{2}(g_{2}^{2}-g_{1}^{2})^{\frac{3}{2}}\textrm{vol}(\mathcal{M}_{4})}{16mg_{1}g_{2}^{3}G_{\textrm{N}}}\,.$
(106)

(a) Solutions for $\sigma(r)$

(b) Solutions for $\phi(r)$

(c) Solutions for $h(r)$

(d) Solutions for $f^{\prime}(r)$

Figure 5: Supersymmetric $AdS_{6}$ black holes interpolating between $AdS_{6}$
vacua $i$ and $ii$ and the near horizon geometry
$AdS_{2}\times\mathcal{M}_{4}^{-}$ (critical point V) for $m=\frac{1}{2}$ and
$g_{2}=4$.

We end this section by pointing out that we have also considered topological
twists by $SO(2)_{R}$ and $SO(3)_{R}$ in the cases of $\mathcal{M}_{4}$ being
Kahler four-cycle and Cayley four-cycle, respectively. It turns out that there
are no new $AdS_{2}\times\mathcal{M}_{4}$ solutions apart from those given in
[27] with all scalars in the vector multiplets vanishing.

##  5 Conclusions

We have constructed new supersymmetric $AdS_{6}$ black hole solutions from
$F(4)$ gauged supergravity in six dimensions coupled to three vector
multiplets with $SO(3)\times SO(3)$ gauge group. By considering a truncation
to $SO(2)_{\textrm{diag}}$ singlet sector and performing a topological twist
by the $SO(2)_{\textrm{diag}}$ gauge field, we find a number of black hole
solutions with $AdS_{2}\times H^{2}\times H^{2}$ and
$AdS_{2}\times\mathcal{M}^{-}_{4}$ near horizon geometries with
$\mathcal{M}_{4}^{-}$ being a negatively curved Kahler four-cycle. On the
other hand, by performing a twist by $SO(3)_{\textrm{diag}}$ gauge fields, we
have found a new black hole solution with $AdS_{2}\times\mathcal{M}_{4}^{-}$
near horizon geometry for $\mathcal{M}^{-}_{4}$ being a negatively curved
Cayley four-cycle.
We have also given examples of numerical black hole solutions interpolating
between these near horizon geometries and the two supersymmetric $AdS_{6}$
vacua. Unlike the previously found solutions in [26] and [27], some of the
solutions given in this paper interpolate between the near horizon geometries
and both of the supersymmetric $AdS_{6}$ vacua. According to the AdS/CFT
correspondence, these solutions should describe RG flows across dimensions to
superconformal quantum mechanics arising from twisted compactifications on
$\mathcal{M}_{4}$ of five-dimensional SCFTs dual to the $AdS_{6}$ vacua. We
hope the new black hole solutions given here would be useful in the study of
attractor mechanism in six dimensions and provide a holographic description of
twisted compactifications of the dual five-dimensional SCFTs on four-
manifolds.
It would be interesting to compute the black hole entropy for the solutions
given here from the topologically twisted indices of the dual SCFTs in five
dimensions using the results of [31, 32] possibly with some extension and
modification. It is of particular interest to find possible embedding of the
solutions found here and in [26, 27] in ten or eleven dimensions. Along this
direction, the results of [33] and [34] might be useful. Finding $AdS_{6}$
black hole solutions with the horizons being four-dimensional orbifolds as in
the recent results [35, 36, 37, 38, 39] is also worth considering. We leave
these issues for future works.
Acknowledgement
This work is funded by National Research Council of Thailand (NRCT) and
Chulalongkorn University under grant N42A650263. The author would also like to
thank C. A. for mental supports during hard times in his life.

##  Appendix A Bosonic field equations of matter-coupled $F(4)$ gauged
supergravity

In this appendix, we collect all the bosonic field equations obtained from the
Lagrangian given in (LABEL:Lar). These are given by

$\displaystyle
D_{\mu}D^{\mu}\sigma-\frac{1}{4}e^{-2\sigma}\mathcal{N}_{\Lambda\Sigma}\widehat{F}_{\mu\nu}^{\Lambda}\widehat{F}^{\Sigma\mu\nu}+\frac{3}{16}e^{4\sigma}H_{\mu\nu\rho}H^{\mu\nu\rho}\qquad\qquad$
$\displaystyle-6me^{-6\sigma}\mathcal{N}_{00}+4me^{-2\sigma}\left(\frac{1}{3}AL_{00}-L_{0i}B^{i}\right)\qquad$
$\displaystyle-e^{2\sigma}\left[\frac{1}{18}A^{2}+\frac{1}{2}B^{i}B_{i}+\frac{1}{2}(C_{It}C^{It}+4D_{It}D^{It})\right]$
$\displaystyle=$ $\displaystyle 0,$ (107)
$\displaystyle\frac{1}{2}D_{\mu}P^{\mu
I0}-e^{-2\sigma}L_{0\Lambda}L_{I\Sigma}\widehat{F}^{\Lambda}_{\mu\nu}\widehat{F}^{\Sigma\mu\nu}-2me^{-2\sigma}L_{0r}C_{Ir}\qquad$
$\displaystyle+e^{2\sigma}\left(B_{i}C_{iI}+K_{tIJ}D^{Jt}\right)$
$\displaystyle=$ $\displaystyle 0,$ (108)
$\displaystyle\frac{1}{2}D_{\mu}P^{\mu
Ir}+2me^{-2\sigma}\left(L_{00}C_{Ir}-2\epsilon_{rst}L_{0s}D_{It}\right)\qquad\qquad\qquad$
$\displaystyle-e^{-2\sigma}L_{r\Lambda}L_{I\Sigma}\widehat{F}^{\Lambda}_{\mu\nu}\widehat{F}^{\Sigma\mu\nu}+e^{2\sigma}\left[\frac{1}{3}AC_{Ir}+\epsilon_{rst}(B^{s}D_{It}+C_{Js}K_{tIJ})\right]$
$\displaystyle=$ $\displaystyle 0,$ (109) $\displaystyle
D_{\nu}\left(e^{-2\sigma}\mathcal{N}_{\Lambda\Sigma}\widehat{F}^{\Sigma\nu\mu}\right)+2P^{\mu}_{I\alpha}{(L^{-1})^{I}}_{\Sigma}{{f_{\Lambda}}^{\Sigma}}_{\Pi}{L^{\Pi}}_{\alpha}\qquad\qquad$
$\displaystyle-\frac{1}{8}\epsilon^{\mu\rho\sigma\nu\lambda\tau}H_{\rho\sigma\nu}\left(mB_{\lambda\tau}\delta^{\Lambda
0}+\eta_{\Lambda\Sigma}F^{\Sigma}_{\lambda\tau}\right)$ $\displaystyle=$
$\displaystyle 0,$ (110)
$\displaystyle\frac{3}{8}D_{\rho}\left(e^{4\sigma}H^{\mu\nu\rho}\right)-m^{2}e^{-2\sigma}\mathcal{N}_{00}B^{\mu\nu}+me^{-2\sigma}\mathcal{N}_{0\Sigma}F^{\Sigma\mu\nu}\qquad\qquad$
$\displaystyle-\frac{1}{16}\epsilon^{\mu\nu\rho\sigma\lambda\tau}\left(\eta_{\Lambda\Sigma}F^{\Lambda}_{\rho\sigma}F^{\Sigma}_{\lambda\tau}-2mF^{0}_{\lambda\tau}B_{\rho\sigma}+m^{2}B_{\rho\sigma}B_{\lambda\tau}\right)$
$\displaystyle=$ $\displaystyle 0,$ (111) $\displaystyle
R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R-e^{-2\sigma}\mathcal{N}_{\Lambda\Sigma}\widehat{F}^{\Lambda}_{\mu\rho}\widehat{F}^{\Sigma\phantom{\nu}\rho}_{\phantom{\Sigma}\nu}-4\partial_{\mu}\sigma\partial_{\nu}\sigma\qquad\qquad\qquad\qquad$
$\displaystyle-P^{I\alpha}_{\mu}P_{\nu
I\alpha}-\frac{9}{16}e^{4\sigma}H_{\mu\rho\sigma}{H_{\nu}}^{\rho\sigma}+g_{\mu\nu}\left[\frac{1}{4}e^{-2\sigma}\mathcal{N}_{\Lambda\Sigma}\widehat{F}^{\Lambda}_{\rho\sigma}\widehat{F}^{\Sigma\rho\sigma}\right.\qquad$
$\displaystyle\left.+\frac{3}{32}e^{4\sigma}H_{\rho\sigma\lambda}H^{\rho\sigma\lambda}+2\partial_{\rho}\sigma\partial^{\rho}\sigma+\frac{1}{2}P^{I\alpha}_{\rho}P^{\rho}_{I\alpha}-2V\right]$
$\displaystyle=$ $\displaystyle 0\qquad$ (112)

with

$K_{rIJ}=g_{1}\epsilon_{lmn}{L^{l}}_{r}{(L^{-1})_{I}}^{m}{L^{n}}_{J}+g_{2}C_{LMK}{L^{L}}_{r}{(L^{-1})_{I}}^{M}{L^{K}}_{J}\,.$
(113)

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