# Supersymmetric solutions from $N=6$ gauged supergravity

Parinya Karndumri  Jakkapat Seeyangnok<EMAIL_ADDRESS>and
<EMAIL_ADDRESS>String Theory and Supergravity Group, Department of
Physics, Faculty of Science, Chulalongkorn University, 254 Phayathai Road,
Pathumwan, Bangkok 10330, Thailand

###### Abstract

We study supersymmetric solutions in four-dimensional $N=6$ gauged
supergravity with $SO(6)$ gauge group. There is a unique $N=6$ supersymmetric
$AdS_{4}$ vacuum with $SO(6)$ symmetry dual to an $N=6$ SCFT in three
dimensions. We find a number of domain walls interpolating between this
$AdS_{4}$ vacuum and singular geometries in the IR with $SO(2)\times SO(4)$,
$U(3)$, $SO(3)$ and $SO(2)\times SO(2)\times SO(2)$ symmetries. The $SO(3)$
case admits $N=6$ or $N=2$ solutions depending on whether the pseudoscalars
are present or not. On the other hand, all the remaining solutions preserve
$N=6$ supersymmetry. These solutions describe RG flows from the $N=6$ SCFT to
non-conformal field theories driven by mass deformations. In particular, the
$SO(2)\times SO(4)$ solution is in agreement with the previously known mass
deformations of the dual $N=6$ SCFT. We also give a supersymmetric Janus
solution with $SO(2)\times SO(4)$ symmetry, describing two-dimensional
conformal defects in the $N=6$ SCFT with unbroken $N=(4,2)$ supersymmetry.
Finally, we find an $N=2$ supersymmetric $AdS_{2}\times H^{2}$ solution with
$SO(2)\times SO(4)$ symmetry and the corresponding domain wall interpolating
between this fixed point and the $AdS_{4}$ vacuum. The solution describes an
$AdS_{4}$ black hole with a magnetic charge and is dual to a twisted
compactification of the $N=6$ SCFT on a hyperbolic space $H^{2}$. We also give
a domain wall interpolating between a locally supersymmetric $AdS_{4}$ and a
curved domain wall with $SO(2)\times SO(2)\times SO(2)$ symmetry dual to an RG
flow across dimensions from the $N=6$ SCFT to a supersymmetric quantum
mechanics.

##  I Introduction

Supersymmetric solutions of gauged supergravities in various space-time
dimensions play an important role in string/M-theory. In the AdS/CFT
correspondence maldacena ; Gubser_AdS_CFT ; Witten_AdS_CFT , these solutions
provide holographic descriptions of strongly coupled systems such as (non)
conformal field theories, conformal defects, $AdS$-black holes and condensed
matter physics. In many cases, solutions of lower-dimensional gauged
supergravities can be uplifted to ten or eleven dimensions via consistent
truncations resulting in complete AdS/CFT dualities in the context of
string/M-theory.
In this paper, we are interested in supersymmetric solutions of four-
dimensional $N=6$ gauged supergravity with $SO(6)$ gauge group, see
Ortin_SUSY_solution  for all time-like supersymmetric solutions in ungauged
$N=6$ supergravity. This has been constructed in Mario_N6  in the embedding
tensor formalism obtained from truncating the maximal $N=8$ gauged
supergravity N8_4D_gauged_embedded , see also de_Wit_Nicolai ;
Gauged_maximal_Henning ; magnetic_charge_gauged . The $N=6$ gauged
supergravity has been shown to admit a unique $N=6$ supersymmetric $AdS_{4}$
vacuum, with the full $SO(6)$ symmetry unbroken, dual to an $N=6$
superconformal field theory (SCFT) in three dimensions. The uniqueness of the
$N=6$ $AdS_{4}$ vacua has also been shown in a recent result on supersymmetric
$AdS$ vacua Severin_Maximal_AdS . It has been pointed out in Mario_N6  that
this $AdS_{4}$ fixed point describes a truncation of type IIA theory on
$CP^{3}$, so the $AdS_{4}$ vacuum can be uplifted to $AdS_{4}\times CP^{3}$
geometry in type IIA theory. This truncation has been studied long ago in
Pope_IIA_CP3  in which the full mass spectrum has also been given, for more
recent studies see for example AdS4_CP3_1 ; AdS4_CP3_2 ; AdS4_CP3_3 ;
AdS4_CP3_4 .
The very first example of the dual $N=6$ SCFT from type IIA theory has been
given in ABJM . In general, SCFTs in three dimensions take the form of Chern-
Simons-Matter (CSM) theories since the usual gauge theories with Yang-Mills
gauge kinetic terms are not conformal. A number of these SCFTs with different
numbers of supersymmetries have already been constructed, see Bena ; BL1 ; BL2
; BL3 ; Gustavsson ; Basu_Harvey ; Schwarz_3D_SCFT ; ABJ ; N5_6_3D_SCFT ; ST ;
BL4 ; LP ; Origin_3_algebra ; Eric_3D_supercon ; Chen ; Eric_M2_brane ;
Eric_Mbrane_SUGRA ; MFM ; BB ; Honda  for an incomplete list. These SCFTs
arise as world-volume theories of M2-branes on various transverse spaces and
play an important role in understanding the dynamics of M2-branes.
Supersymmetric solutions of four-dimensional gauged supergravities are
expected to be very useful for their holographic descriptions at least in the
large-$N$ limit.
Various types of supersymmetric solutions from gauged supergravities have been
considered and given interpretations in terms of the corresponding dual field
theories. We will study these solutions in $N=6$ gauged supergravity beginning
with supersymmetric domain walls interpolating between the $N=6$
supersymmetric $AdS_{4}$ vacuum and singular geometries. The solutions
describe holographic RG flows from the dual $N=6$ SCFT in the UV to non-
conformal phases in the IR arising from mass deformations of the UV $N=6$
SCFT. Similar solutions have been extensively studied in $N=8$ and $N=2$
gauged supergravities, see for example Warner_membrane_flow ;
Warner_M_F_theory_flow ; Warner_higher_Dflow ; Flow_in_N8_4D ; 4D_G2_flow ;
Warner_M2_flow ; Guarino_BPS_DW ; Elec_mag_flows ; Yi_4D_flow . Solutions in
gauged supergravities with $N=3,4,5$ supersymmetries have also been considered
recently in N3_SU2_SU3 ; N3_4D_gauging ; tri-sasakian-flow ; orbifold_flow ;
4D_N4_flows ; N5_flow . This work could hopefully fill the existing gap by
providing a number of supersymmetric solutions in $N=6$ gauged supergravity.
We will also find Janus solutions in the form of $AdS_{3}$-sliced domain walls
interpolating between asymptotically $AdS_{4}$ spaces. These are
holographically dual to two-dimensional conformal defects within the $N=6$
SCFT and break the superconformal symmetry in the three-dimensional bulk to a
smaller one on the two-dimensional surfaces. Solutions of this type in other
four-dimensional gauged supergravities have previously been studied in tri-
sasakian-flow ; orbifold_flow ; N5_flow ; warner_Janus ; N3_Janus ;
Minwoo_4DN8_Janus ; Kim_Janus . Finally, we will look for
$AdS_{2}\times\Sigma^{2}$ geometries with $\Sigma^{2}$ being a Riemann surface
together with solutions interpolating between these backgrounds and the
supersymmetric $AdS_{4}$ vacuum. The solutions describe supersymmetric black
holes in asymptotically $AdS_{4}$ space, and a number of these solutions have
already been studied in other gauged supergravities in N5_flow ; BH_M_theory1
; BH_M_theory2 ; Kim_AdS2 ; AdS4_BH1 ; AdS4_BH2 ; AdS4_BH3 ; AdS4_BH4 ;
AdS4_BH5 ; Guarino_AdS2_1 ; Guarino_AdS2_2 ; Trisasakian_AdS2 . In the dual
field theory, the solutions describe RG flows from the $N=6$ SCFT to
superconformal quantum mechanics in the IR which play a prominent role in
microscopic computation of black hole entropy in asymptotically $AdS_{4}$
spaces, see for example Zaffaroni_BH_entropy ; BH_entropy_benini ;
BH_entropy_Passias . In this context, the superconformal quantum mechanics, or
one-dimensional SCFT, is obtained from the $N=6$ SCFT via twisted
compactifications on $\Sigma^{2}$.
Four-dimensional $N=6$ gauged supergravity has $SO^{*}(12)$ global symmetry
with the maximal compact subgroup $U(6)\sim SU(6)\times U(1)$. There are
thirty scalars encoded in $SO^{*}(12)/U(6)$ coset manifold. The $SO(6)$
gauging of this supergravity can be obtained from a consistent truncation of
the maximal $N=8$ gauged supergravity with $SO(8)$ gauge group. The latter is
in turn a consistent truncation of eleven-dimensional supergravity on $S^{7}$
N8_4D_S7_1 ; N8_4D_S7_2 ; N8_4D_S7_3 ; N8_4D_S7_4 ; N8_4D_S7_5 ; N8_4D_S7_6 .
The $N=6$ gauged supergravity with $SO(6)$ gauge group can accordingly be
uplifted to eleven dimensions via a series of consistent truncations. On the
other hand, the $SO(6)$, $N=6$ gauged supergravity is a consistent truncation
of type IIA theory on $CP^{3}$. Therefore, all the solutions given here have
known higher-dimensional origins and can be embedded in ten- or eleven-
dimensional supergravities. The scalar potential of the $N=6$ gauged
supergravity has been analyzed for a long time in Warner_Potential . More
recently, this gauged supergravity has been rewritten in a more general
setting of the embedding tensor formalism in Mario_N6  in which the fermion-
shift matrices and the scalar potential have been given by truncating the
$N=8$ theory. In this paper, we first complete the task by extending the
truncation to all terms in the bosonic Lagrangian and fermionic supersymmetry
transformations. Both of these are of course a relevant part in the present
analysis.
The paper is organized as follows. In section II, we review four-dimensional
$N=6$ gauged supergravity with $SO(6)$ gauge group in the embedding tensor
formalism. In section III, we study supersymmetric domain wall solutions
describing RG flows in the dual $N=6$ SCFT to non-conformal phases in the IR.
We then turn to supersymmetric Janus solutions in section IV and finally look
for possible supersymmetric $AdS_{2}\times\Sigma^{2}$ solutions together with
flow solutions interpolating between the $AdS_{4}$ vacuum and these geometries
in section V. Conclusions and comments are given in section VI.

##  II $N=6$ gauged supergravity with $SO(6)$ gauge group

We first give a review of $N=6$ gauged supergravity in the embedding tensor
formalism as described in Mario_N6 . We will follow most of the convention in
Mario_N6  but with a mostly plus signature for the space-time metric. The only
supermultiplet in $N=6$ supersymmetry is the gravity multiplet with the field
content

$(e^{\hat{\mu}}_{\mu},\psi_{\mu
A},A^{AB}_{\mu},A^{0}_{\mu},\chi_{ABC},\chi_{A},\phi_{AB}).$ (1)

The component fields correspond to the graviton $e^{\hat{\mu}}_{\mu}$, six
gravitini $\psi_{\mu A}$, sixteen vectors $A^{AB}_{\mu}=-A^{BA}_{\mu}$ and
$A^{0}_{\mu}$, twenty-six spin-$\frac{1}{2}$ fields $\chi_{ABC}=\chi_{[ABC]}$
and $\chi_{A}$ together with fifteen complex scalars $\phi_{AB}=-\phi_{BA}$.
Real and imaginary parts of $\phi_{AB}$ are usually called scalars and
pseudoscalars, respectively.
In this work, space-time and tangent space indices are denoted by
$\mu,\nu,\ldots=0,1,2,3$ and $\hat{\mu},\hat{\nu},\ldots=0,1,2,3$,
respectively. Indices $A,B,\ldots=1,2,\ldots,6$ correspond to the fundamental
representation of $SU(6)$ which is in turn a subgroup of the R-symmetry
$U(6)\sim SU(6)\times U(1)$. The $30$ real scalars within $\phi_{AB}$ are
coordinates of the scalar manifold $SO^{*}(12)/U(6)$ and can be described by
the coset representative in representation $\mathbf{32}$ of $SO^{*}(12)$ of
the form

${\mathcal{V}_{M}}^{\underline{M}}=\mathcal{A}^{\dagger}e^{Y}$ (2)

with the Cayley matrix

$\mathcal{A}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{cc}\mathbb{I}_{16}&i\mathbb{I}_{16}\\\
\mathbb{I}_{16}&-i\mathbb{I}_{16}\\\ \end{array}\right)$ (3)

and

$Y=\left(\begin{array}[]{cccc}0&0_{1\times 15}&0&\phi_{CD}\\\ 0_{15\times
1}&0_{15\times
15}&\phi_{AB}&\frac{1}{2}\epsilon_{ABCDEF}\bar{\phi}^{EF}\phi_{CD}\\\
0&\bar{\phi}^{CD}&0&0_{1\times 15}\\\
\bar{\phi}^{AB}&\frac{1}{2}\epsilon^{ABCDEF}\phi_{EF}&0_{15\times
1}&0_{15\times 15}\\\ \end{array}\right).$ (4)

We also note that $\bar{\phi}^{AB}=(\phi_{AB})^{*}$.
In subsequent analysis, it is useful to define $16\times 16$ submatrices of
${\mathcal{V}_{M}}^{\underline{M}}$ by the following identification

${\mathcal{V}_{M}}^{\underline{M}}=\left(\begin{array}[]{cc}{\bar{h}_{\Lambda}}^{\phantom{\Lambda}\underline{\Lambda}}&h_{\Lambda\underline{\Lambda}}\\\
\bar{f}^{\Lambda\underline{\Lambda}}&{f^{\Lambda}}_{\underline{\Lambda}}\\\
\end{array}\right)$ (5)

in which $\mathbf{f}$, $\mathbf{h}$, $\bar{\mathbf{f}}$ and $\bar{\mathbf{h}}$
satisfy the relations

$\displaystyle(\mathbf{f}\mathbf{f}^{\dagger})^{T}=\mathbf{f}\mathbf{f}^{\dagger},\qquad(\mathbf{h}\mathbf{h}^{\dagger})^{T}=\mathbf{h}\mathbf{h}^{\dagger},\qquad\mathbf{f}\mathbf{h}^{\dagger}-\bar{\mathbf{f}}\mathbf{h}^{T}=i\mathbb{I}_{16},$
$\displaystyle\mathbf{f}^{\dagger}\mathbf{h}-\mathbf{h}^{\dagger}\mathbf{f}=-i\mathbb{I}_{16},\qquad\mathbf{f}^{T}\mathbf{h}-\mathbf{h}^{T}\mathbf{f}=0\,.$
(6)

The inverse of ${\mathcal{V}_{M}}^{\underline{M}}$ can accordingly be written
in terms of $\mathbf{f}$ and $\mathbf{h}$ as

${\mathcal{V}_{\underline{M}}}^{M}=\left(\begin{array}[]{cc}-i{f^{\Lambda}}_{\underline{\Lambda}}&ih_{\Lambda\underline{\Lambda}}\\\
i\bar{f}^{\Lambda\underline{\Lambda}}&-i{\bar{h}_{\Lambda}}^{\phantom{\Lambda}\underline{\Lambda}}\\\
\end{array}\right).$ (7)

The sixteen electric gauge fields $A^{AB}$ and $A^{0}$ combine into
$A^{\Lambda}=(A^{0},A^{AB})$. Together with the magnetic dual $A_{\Lambda}$,
the gauge fields transform as $\mathbf{32}$ representation of $SO^{*}(12)$

$A^{M}=(A^{\Lambda},A_{\Lambda}).$ (8)

Gaugings are efficiently described by the embedding tensor formalism in which
the corresponding gauge generators are defined as

$X_{M}={\theta_{M}}^{n}t_{n}$ (9)

with $t^{n}$ being the $SO^{*}(12)$ generators. ${\theta_{M}}^{n}$ is called
the embedding tensor in term of which the covariant derivative implementing
the minimal coupling of various fields can be written as

$D_{\mu}=\nabla_{\mu}-gA^{M}_{\mu}X_{M}\,.$ (10)

$\nabla_{\mu}$ is the usual space-time covariant derivative including
(possibly) the local $U(6)$ composite connection. The parameter $g$ is the
gauge coupling constant which can be absorbed in the definition of
${\theta_{M}}^{m}$.
In $\mathbf{32}$ representation, with $SO^{*}(12)$ generators
${(t_{n})_{M}}^{N}$, the embedding tensor can be described by the generalized
structure constants

${X_{MN}}^{P}={\theta_{M}}^{n}{(t_{n})_{N}}^{P}\,.$ (11)

To define a proper gauging and preserve the full supersymmetry of the ungauged
theory, the embedding tensor needs to satisfy the so-called linear and
quadratic constraints given respectively by

${X_{(MN}}^{L}\Omega_{P)L}=0\qquad\textrm{and}\qquad{\theta_{M}}^{m}{\theta_{N}}^{n}{f_{mn}}^{p}+{X_{MN}}^{P}{\theta_{P}}^{p}=0$
(12)

with ${f_{mn}}^{p}$ being the $SO^{*}(12)$ structure constants. The former
implies that the embedding tensor ${\theta_{M}}^{m}$ is in the representation
$\mathbf{351}$ of $SO^{*}(12)$ while the latter gives rise to

$[X_{M},X_{N}]=-{X_{MN}}^{P}X_{P}\,.$ (13)

The gauge generators then form a closed subalgebra for which ${X_{MN}}^{P}$
act as the corresponding structure constants.
As usual in gauging a supergravity theory, supersymmetry requires some
modifications to the ungauged Lagrangian and supersymmetry transformations.
These modifications are of first and second order in the gauge coupling
constant and can be written in term of the so-called T-tensor

${T_{\underline{M}\underline{N}}}^{\underline{P}}={\mathcal{V}_{\underline{M}}}^{M}{\mathcal{V}_{\underline{N}}}^{N}{\mathcal{V}_{P}}^{\underline{P}}{X_{MN}}^{P}\,.$
(14)

The bosonic Lagrangian of the $N=6$ gauged supergravity can be written as

$\displaystyle e^{-1}\mathcal{L}$ $\displaystyle=$
$\displaystyle\frac{1}{2}R-\frac{1}{24}P_{\mu ABCD}P^{\mu
ABCD}-\frac{i}{4}\left(\mathcal{N}_{\Lambda\Sigma}F^{+\Lambda}_{\mu\nu}F^{+\Sigma\mu\nu}-\overline{\mathcal{N}}_{\Lambda\Sigma}F^{-\Lambda}_{\mu\nu}F^{-\Sigma\mu\nu}\right)-V\,.$

The scalar kinetic term is given in term of the vielbein
$P_{\mu}^{ABCD}=(P_{\mu ABCD})^{*}$ on the $SO^{*}(12)/U(6)$ coset which is
defined by

$\displaystyle P_{\mu}^{ABCD}$ $\displaystyle=$
$\displaystyle\mathcal{V}^{ABM}D_{\mu}{\mathcal{V}_{M}}^{CD}$ (16)
$\displaystyle=$ $\displaystyle i\left(\bar{f}^{\Lambda
AB}D_{\mu}{\bar{h}_{\Lambda}}^{\phantom{\Lambda}CD}-{\bar{h}_{\Lambda}}^{\phantom{\Lambda}AB}D_{\mu}\bar{f}^{\Lambda
CD}\right).$

The scalar matrix appearing in the gauge kinetic terms is given by

$\mathcal{N}_{\Lambda\Sigma}=-{\bar{h}_{\Lambda}}^{\phantom{\Lambda}\underline{\Lambda}}(f^{-1})_{\underline{\Lambda}\Sigma}$
(17)

with $\overline{\mathcal{N}}_{\Lambda\Sigma}$ being its complex conjugate. The
complex self-dual and anti-self-dual gauge field strengths are defined by

$F^{\pm\Lambda}_{\mu\nu}=\frac{1}{2}\left(F^{\Lambda}_{\mu\nu}\pm\frac{i}{2}\epsilon_{\mu\nu\rho\sigma}F^{\Lambda\rho\sigma}\right)$
(18)

with $F^{\Lambda}_{\mu\nu}$ given by

$F^{\Lambda}_{\mu\nu}=\partial_{\mu}A^{\Lambda}_{\nu}-\partial_{\nu}A_{\mu}^{\Lambda}+{X_{\Gamma\Sigma}}^{\Lambda}A^{\Gamma}_{\mu}A^{\Sigma}_{\nu}\,.$
(19)

The scalar potential is obtained from the fermion-shift matrices as follows

$V=-2S_{AB}S^{AB}+\frac{1}{36}{N_{A}}^{BCD}{N^{A}}_{BCD}+\frac{1}{6}{N_{A}}^{B}{N^{A}}_{B}\,.$
(20)

We note that upper and lower $SU(6)$ indices are related by complex
conjugation. In terms of the various components of the T-tensor with the
spliting of indices $\underline{\Lambda},\underline{\Sigma},\ldots$ as
$(0,[AB])$, we have

$\displaystyle S_{AB}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{2}}{5}{T_{C(A,B)E}}^{CE},\qquad
N_{AB}=-\frac{8\sqrt{2}}{3}{T_{C[A,B]E}}^{CE},$ $\displaystyle{N^{A}}_{B}$
$\displaystyle=$
$\displaystyle-2\sqrt{2}{T_{0,BC}}^{AC},\qquad{N^{A}}_{BCD}=-2\sqrt{2}{T_{[CD,B]E}}^{AE}-\frac{1}{4}\delta^{A}_{[B}N_{CD]}\,.$
(21)

The fermionic supersymmetry transformations, with all fermionic fields
vanishing, are given by

$\displaystyle\delta\psi_{\mu A}$ $\displaystyle=$ $\displaystyle
D_{\mu}\epsilon_{A}-S_{AB}\gamma_{\mu}\epsilon^{B}-\frac{1}{4\sqrt{2}}\hat{F}^{+}_{\rho\sigma
AB}\gamma^{\rho\sigma}\gamma_{\mu}\epsilon^{B},$ (22)
$\displaystyle\delta\chi_{A}$ $\displaystyle=$
$\displaystyle-\frac{1}{4!}\epsilon_{ABCDEF}P^{BCDE}_{\mu}\gamma^{\mu}\epsilon^{F}+{N^{B}}_{A}\epsilon_{B}-\frac{1}{2\sqrt{2}}\hat{F}^{+}_{\mu\nu}\gamma^{\mu\nu}\epsilon_{A},$
(23) $\displaystyle\delta\chi_{ABC}$ $\displaystyle=$ $\displaystyle-P_{\mu
ABCD}\gamma^{\mu}\epsilon^{D}+{N^{D}}_{ABC}\epsilon_{D}-\frac{3}{2\sqrt{2}}\hat{F}^{+}_{\mu\nu[AB}\epsilon_{C]}\,.$
(24)

We note here the chiralities of the fermionic fields

$\gamma_{5}\psi_{\mu A}=-\psi_{\mu
A},\qquad\gamma_{5}\chi_{ABC}=-\chi_{ABC},\qquad\gamma_{5}\chi_{A}=-\chi_{A}$
(25)

with $\psi^{A}_{\mu}$, $\chi^{ABC}$ and $\chi^{A}$ having opposite
chiralities. The tensors $\hat{F}^{+}_{\mu\nu
AB}=(\hat{F}_{\mu\nu}^{-AB})^{*}$ can be obtained from

$\hat{F}^{-AB}_{\mu\nu}={\mathcal{V}_{M}}^{AB}G^{-M}_{\mu\nu}$ (26)

with

$G^{M}_{\mu\nu}=\left(\begin{array}[]{c}F^{\Lambda}_{\mu\nu}\\\
G_{\Lambda\mu\nu}\\\ \end{array}\right)$ (27)

and
$G_{\Lambda\mu\nu}=i\epsilon_{\mu\nu\rho\sigma}\frac{\partial\mathcal{L}}{\partial
F^{\Lambda}_{\rho\sigma}}$. Similarly, we have
$\hat{F}^{+}_{\mu\nu}=({\mathcal{V}_{M}}^{0}G^{-M}_{\mu\nu})^{*}$.
The covariant derivative of $\epsilon_{A}$ is defined by

$D_{\mu}\epsilon_{A}=\partial_{\mu}\epsilon_{A}+\frac{1}{4}{\omega_{\mu}}^{ab}\gamma_{ab}\epsilon_{A}+\frac{1}{2}{Q_{\mu
A}}^{B}\epsilon_{B}\,.$ (28)

The connection ${Q_{\mu A}}^{B}$ is given by

${Q_{\mu A}}^{B}=\frac{2i}{3}\left(h_{\Lambda
AC}\partial_{\mu}\bar{f}^{\Lambda
AB}-{f^{\Lambda}}_{AC}\partial_{\mu}{\bar{h}_{\Lambda}}^{\phantom{\Lambda}BC}\right)-gA^{M}_{\mu}{Q_{MA}}^{B}$
(29)

with ${Q_{MA}}^{B}$ obtained from

${Q_{MAB}}^{CD}={\mathcal{V}_{AB}}^{P}{X_{MP}}^{N}{\mathcal{V}_{N}}^{CD}$ (30)

by the relation ${Q_{MAB}}^{CD}=4\delta^{[C}_{[A}{Q_{MB]}}^{D]}$.
In general, both electric and magnetic gauge fields can participate in the
gaugings leading to many possibilities of viable gauge groups. However, in
this work, we will only consider $SO(6)$ gauge group embedded electrically in
$U(6)\subset SO^{*}(12)$. This gauging only involves electric gauge fields
$A^{AB}$. In this case, we have

${X_{I_{1}J_{1},I_{2}J_{2}}}^{I_{3}J_{3}}=4g\delta^{[I_{3}}_{[I_{1}}\delta_{I_{2}][J_{2}}\delta^{J_{3}]}_{J_{2}]}\qquad\textrm{and}\qquad{{X_{I_{1}J_{1}}}^{I_{3}J_{3}}}_{I_{2}J_{2}}=-{X_{I_{1}J_{1},I_{2}J_{2}}}^{I_{3}J_{3}}$
(31)

with all remaining components vanishing. In particular, there are no
${{X^{\Lambda}}_{M}}^{N}$ components which couple to magnetic gauge fields.
With the splitting of indices $\Lambda,\Sigma,\ldots=(0,[IJ])$, we find from
the definition (14) that

$\displaystyle{T_{AB}}^{CD}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}{f^{I_{1}J}}_{0}({f^{JJ_{1}}}_{AB}{\bar{h}_{I_{1}J_{1}}}^{\phantom{I_{1}J_{1}}CD}+h_{I_{1}J_{1},AB}\bar{f}^{JJ_{1},CD}),$
(32) $\displaystyle{T_{EF,AB}}^{CD}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}{f^{I_{1}J}}_{EF}({f^{JJ_{1}}}_{AB}{\bar{h}_{I_{1}J_{1}}}^{\phantom{I_{1}J_{1}}CD}+h_{I_{1}J_{1},AB}\bar{f}^{JJ_{1},CD}).$
(33)

From these, it is straightforward to obtain all the fermion-shift matrices and
the scalar potential.
In subsequent sections, we will look for various types of supersymmetric
solutions to this $N=6$ gauged supergravity with $SO(6)$ gauge group. It has
been shown in Mario_N6  that this gauged supergravity admits a supersymmetric
$N=6$ $AdS_{4}$ vacuum with the cosmological constant $V_{0}=-48g^{2}$ when
all scalars vanish. According to the AdS/CFT correspondence, this is dual to
an $N=6$ SCFT in three dimensions. We will find solutions that are asymptotic
to this $AdS_{4}$ geometry and can be interpreted as different types of
deformations of the dual $N=6$ SCFT.

##  III Holographic RG flows

We first consider holographic RG flow solutions in the form of domain walls
interpolating between the supersymmetric $AdS_{4}$ vacuum and another
$AdS_{4}$ vacuum (if exists) or a singular geometry. These solutions
correspond respectively to RG flows of the dual UV $N=6$ SCFT to another
conformal fixed point or to a non-conformal phase in the IR.
The metric ansatz is taken to be

$ds^{2}=e^{2A(r)}dx^{2}_{1,2}+dr^{2}$ (34)

with $dx^{2}_{1,2}$ being the flat metric on three-dimensional Minkowski
space. Scalar fields are allowed to depend only on the radial coordinate $r$
with all the other fields set to zero.

###  III.1 Solutions with $SO(2)\times SO(4)$ symmetry

We first consider a simple case of solutions with $SO(2)\times SO(4)$
symmetry. The embedding of $SO(6)$ implies that the scalars $\phi_{AB}$
transform as an adjoint representation of $SO(6)$. There is one singlet scalar
under $SO(2)\times SO(4)\subset SO(6)$ given explicitly by

$\phi_{AB}=\phi(\delta_{A}^{1}\delta^{2}_{B}-\delta^{1}_{B}\delta^{2}_{A}).$
(35)

We will also write

$\phi=\varphi e^{i\zeta}$ (36)

for real scalars $\varphi$ and $\zeta$ depending only on $r$.
By a straightforward computation, we find the tensor $S_{AB}$ of the form

$S_{AB}=2g\cosh\varphi\delta_{AB}=\frac{1}{2}\mathcal{W}\delta_{AB}\,.$ (37)

We have introduced the “superpotential” $\mathcal{W}$ for convenience. In
general, the function $\mathcal{W}$ is related to the eigenvalue of $S_{AB}$
corresponding to the unbroken supersymmetry. In the present case, $S_{AB}$ is
proportional to the identity matrix indicating that the solutions will
preserve either $N=6$ supersymmetry with all $\epsilon_{A}$ non-vanishing or
no supersymmetry at all. Note also that $\mathcal{W}$ has a critical point at
$\varphi=0$ which is the supersymmetric $N=6$ $AdS_{4}$ vacuum mentioned
above.
To solve all the BPS conditions, we will, as in other previous works, impose
the following projector

$\gamma_{\hat{r}}\epsilon_{A}=e^{i\Lambda}\epsilon^{A}$ (38)

for a real function $\Lambda$. Throughout the paper, we will use Majorana
representation for gamma matrices in which all $\gamma^{\mu}$ are real, but
$\gamma_{5}$ is purely imaginary. This implies that $\epsilon_{A}$ and
$\epsilon^{A}$ are related by complex conjugation. Note also that the
projector (38) relates the two chiralities of $\epsilon^{A}$, so the full flow
solutions will preserve only half of the original supersymmetry or twelve
supercharges.
Considering the conditions $\delta\psi_{\mu A}=0$ for $\mu=0,1,2$, we find

$e^{i\Lambda}A^{\prime}-\mathcal{W}=0$ (39)

with ′ denoting $r$-derivatives. This equation gives

$A^{\prime}=\pm|\mathcal{W}|\qquad\textrm{and}\qquad
e^{i\Lambda}=\pm\frac{\mathcal{W}}{|\mathcal{W}|}\,.$ (40)

In what follow, we will write $W=|\mathcal{W}|$ for convenience. We will also
choose the upper signs in order to make the supersymmetric $AdS_{4}$ critical
point correspond to $r\rightarrow\infty$. Since, in this case, the
superpotential is real, we simply have

$A^{\prime}=4g\cosh\varphi\qquad\textrm{and}\qquad e^{i\Lambda}=1\,.$ (41)

The condition $\delta\psi_{rA}=0$ gives the standard Killing spinors of the
domain walls

$\epsilon_{A}=e^{\frac{A}{2}}\epsilon_{A(0)}$ (42)

for spinors $\epsilon_{A(0)}$ satisfying (38).
Using the projection (38) in the variations $\delta\chi_{ABC}$ and
$\delta\chi_{A}$ gives the following BPS equations

$\varphi^{\prime}=-4g\sinh\varphi\qquad\textrm{and}\qquad\zeta^{\prime}=0\,.$
(43)

We have now obtained the BPS equations that solve all the supersymmetry
conditions. It can also be readily verified that these equations imply the
second-order field equations.
We can analytically solve the above BPS equations with the following solution

$\displaystyle 4gr$ $\displaystyle=$
$\displaystyle\ln(1+e^{\varphi})-\ln(1-e^{\varphi}),$ (44) $\displaystyle A$
$\displaystyle=$ $\displaystyle\varphi-\ln(1-e^{2\varphi}).$ (45)

We have neglected the integration constants in these equations since they can
be removed by shifting the radial coordinate and scaling the $x^{0,1,2}$
coordinates, respectively. As $r\rightarrow\infty$, we find that

$\varphi\sim e^{-4gr}\sim e^{-\frac{r}{L}}\qquad\textrm{and}\qquad A\sim
4gr\sim\frac{r}{L}$ (46)

with $L$ being the $AdS_{4}$ radius related to the cosmological constant by

$L=\sqrt{-\frac{3}{V_{0}}}=\frac{1}{4g}\,.$ (47)

We have also taken $g>0$ for convenience.
The behavior of $\varphi$ implies that $\varphi$ is dual to a relevant
operator of dimensions $\Delta=1,2$ in the dual SCFT. In addition, the
solution is singular at $r\rightarrow 0$ with

$\varphi\sim\pm\ln(4gr)\qquad\textrm{and}\qquad A\sim\ln(4gr)\,.$ (48)

We then find that $\varphi\rightarrow\pm\infty$ near the singularity. From the
explicit form of the scalar potential, we have

$V\sim-8g^{2}e^{\pm 2\varphi}\rightarrow-\infty\,.$ (49)

By the criterion given in Gubser_singularity , we conclude that the
singularity is physical. Therefore, the solution decribes an RG flow from the
UV $N=6$ SCFT to a non-conformal phase in the IR. The flow is driven by an
operator of dimensions $\Delta=1,2$ corresponding to scalar or fermion mass
terms in three dimensions. The flow breaks superconformal symmetry but
preserves the full $N=6$ Poincare supersymmetry. Moreover, the R-symmetry
$SO(6)$ is broken to $SO(2)\times SO(4)$ subgroup. This is precisely in
agreement with the field theory result given in N5_6_3D_SCFT . We then expect
the solution to describe mass deformations of the three-dimensional $N=6$
SCFT.

###  III.2 Solutions with $U(3)$ symmetry

We now consider another residual symmetry namely $U(3)\sim SU(3)\times
U(1)\subset SO(6)$. The $U(3)$ generators in the fundamental representation of
$SO(6)$ can be written as

$X=\left(\begin{array}[]{cc}A_{3\times 3}&S_{3\times 3}\\\ -S_{3\times
3}&A_{3\times 3}\\\ \end{array}\right)$ (50)

in which $A_{3\times 3}$ and $S_{3\times 3}$ are anti-symmetric and symmetric
$3\times 3$ matrices, respectively. The matrices $A_{3\times 3}$ generate an
$SO(3)\subset SU(3)$ which is a diagonal subgroup of $SO(3)\times SO(3)\subset
SO(6)$. The $U(1)$ factor corresponds to $S_{3\times 3}=\mathbb{I}_{3}$. There
is only one $U(3)$ singlet scalar given by

$\phi_{AB}=\left(\begin{array}[]{cc}0_{3\times 3}&\phi\mathbb{I}_{3}\\\
-\phi\mathbb{I}_{3}&0_{3\times 3}\\\ \end{array}\right)=\phi J_{AB}\,.$ (51)

The matrix $J_{AB}$ is identified with the Kahler form of $CP^{3}$ on which
the ten-dimensional type IIA theory compactifies Mario_N6 .
By writting $\phi=\varphi e^{i\zeta}$ and repeating the same analysis as in
the previous case, we find the scalar potential

$V=-24g^{2}e^{-2\varphi}(1+e^{4\varphi})$ (52)

which is exactly the same as that given in Mario_N6 . As in the $SO(2)\times
SO(4)$ case, this potential admits an $N=6$ $AdS_{4}$ critical point at
$\varphi=0$ dual to an $N=6$ SCFT in three dimensions.
The matrix $S_{AB}$ is proportional to the identity

$S_{AB}=\frac{1}{2}\mathcal{W}\delta_{AB}$ (53)

with a complex superpotential

$\mathcal{W}=\frac{1}{2}e^{-3\varphi-i\zeta}\left[(e^{6\varphi}+3e^{2\varphi})(1+e^{i\zeta})+(1+e^{4\varphi})(e^{i\zeta}-1)\right].$
(54)

The variations $\delta\chi_{A}$ and $\delta\chi_{ABC}$ lead to

$e^{-i\Lambda}(2\varphi^{\prime}\pm
i\sinh(2\varphi)\zeta^{\prime})=-ge^{-3\varphi}(e^{4\varphi}-1)[1-e^{i\zeta}+e^{2\varphi}(1+e^{i\zeta})]$
(55)

which implies $\zeta^{\prime}=0$. It turns out that $\zeta^{\prime}=0$ is also
required by the field equations. For constant $\zeta=\zeta_{0}$, we have
verified that all the resulting BPS equations are compactible with the field
equations. In the following analysis, we will set $\zeta_{0}=0$ and end up
with the BPS equations

$\varphi^{\prime}=-e^{-\varphi}(e^{4\varphi}-1)\qquad\textrm{and}\qquad
A^{\prime}=ge^{-\varphi}(3+e^{4\varphi}).$ (56)

The solution can be readily obtained

$\displaystyle A$ $\displaystyle=$ $\displaystyle
3\varphi-\ln(1-e^{4\varphi}),$ (57) $\displaystyle 4gr$ $\displaystyle=$
$\displaystyle 2\tan^{-1}e^{\varphi}-\ln(1-e^{\varphi})+\ln(1+e^{\varphi}).$
(58)

As in the previous case, the solution is asymptotic to the supersymmetric
$AdS_{4}$ with $\varphi$ dual to an operator of dimensions $\Delta=1,2$ while
at $r=0$, the solution is singular with

$\varphi\sim\ln(gr)\qquad\textrm{and}\qquad A\sim 3\varphi\sim 3\ln(gr)$ (59)

and

$\varphi\sim-\ln(gr)\qquad\textrm{and}\qquad A\sim-\varphi\sim\ln(gr).$ (60)

Both of these give

$V\sim-24g^{2}e^{\pm 2\varphi}\rightarrow-\infty,$ (61)

so the two singularities are physical. We can accordingly interpret the
solution as a holographic dual of RG flows from the $N=6$ SCFT to non-
conformal phases in the IR. The flow preserves $N=6$ Poincare supersymmetry in
three-dimensions as in the $SO(2)\times SO(4)$ case but breaks the $SO(6)$
R-symmetry to $U(3)$. It would be interesting to identify the corresponding
mass deformations in the dual $N=6$ SCFT similar to the $SO(2)\times SO(4)$
case.

###  III.3 Solutions with $SO(2)\times SO(2)\times SO(2)$ symmetry

To obtain more interesting and more complicated solutions, we consider
solutions with a smaller symmetry namely $SO(2)\times SO(2)\times SO(2)\subset
SO(6)$ symmetry. There are three complex scalars which are singlets under this
$SO(2)\times SO(2)\times SO(2)$. The explicit parametrization of these
singlets can be written as

$\phi_{AB}=\left(\begin{array}[]{ccc}\phi_{1}i\sigma_{2}&0_{2\times
2}&0_{2\times 2}\\\ 0_{2\times 2}&\phi_{2}i\sigma_{2}&0_{2\times 2}\\\
0_{2\times 2}&0_{2\times 2}&\phi_{3}i\sigma_{2}\\\ \end{array}\right)\,.$ (62)

By setting

$\phi_{\alpha}=\varphi_{\alpha}e^{i\zeta_{\alpha}},\qquad\alpha=1,2,3,$ (63)

we find the scalar potential

$V=-16g^{2}[\cosh(2\varphi_{1})+\cosh(2\varphi_{2})+\cosh(2\varphi_{3})].$
(64)

It is clearly seen that this potential admits only one critical point at
$\varphi_{1}=\varphi_{2}=\varphi_{3}=0$ which is the aforementioned $N=6$
$AdS_{4}$ vacuum.
The matrix $S_{AB}$ is given by

$S_{AB}=\frac{1}{2}\left(\begin{array}[]{ccc}\mathcal{W}_{1}\mathbb{I}_{2}&0_{2\times
2}&0_{2\times 2}\\\ 0_{2\times 2}&\mathcal{W}_{2}\mathbb{I}_{2}&0_{2\times
2}\\\ 0_{2\times 2}&0_{2\times 2}&\mathcal{W}_{3}\mathbb{I}_{2}\\\
\end{array}\right)$ (65)

with

$\displaystyle\mathcal{W}_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{2}ge^{-\varphi_{1}-\varphi_{2}-\varphi_{3}}\left[e^{i(\zeta_{1}-\zeta_{2}-\zeta_{3})}(e^{2\varphi_{1}}-1)(e^{2\varphi_{2}}-1)(e^{2\varphi_{3}}-1)\right.$
(66)
$\displaystyle\left.-(1+e^{2\varphi_{1}})(1+e^{2\varphi_{2}})(1+e^{2\varphi_{3}})\right].$

$\mathcal{W}_{2}$ and $\mathcal{W}_{3}$ take a similar form with the phase
$e^{i(\zeta_{1}-\zeta_{2}-\zeta_{3})}$ replaced by
$e^{i(\zeta_{2}-\zeta_{1}-\zeta_{3})}$ and
$e^{i(\zeta_{3}-\zeta_{1}-\zeta_{2})}$, respectively.
It turns out that none of these $\mathcal{W}_{\alpha}$ gives rise to the
superpotential in term of which the scalar potential (64) can be written
unless $\zeta_{1}=\zeta_{2}=\zeta_{3}=0$. This is also implied by the
consistency between the resulting BPS equations and the field equations. We
now set $\zeta_{1}=\zeta_{2}=\zeta_{3}=0$, resulting in $e^{i\Lambda}=\pm 1$,
and obtain the following BPS equations

$\displaystyle\varphi_{1}^{\prime}$ $\displaystyle=$ $\displaystyle-
ge^{-\varphi_{1}-\varphi_{2}-\varphi_{3}}[e^{2(\varphi_{1}+\varphi_{2})}+e^{2(\varphi_{1}+\varphi_{3})}-e^{2(\varphi_{2}+\varphi_{3})}-1],$
(67) $\displaystyle\varphi_{2}^{\prime}$ $\displaystyle=$ $\displaystyle-
ge^{-\varphi_{1}-\varphi_{2}-\varphi_{3}}[e^{2(\varphi_{1}+\varphi_{2})}+e^{2(\varphi_{2}+\varphi_{3})}-e^{2(\varphi_{1}+\varphi_{3})}-1],$
(68) $\displaystyle\varphi_{3}^{\prime}$ $\displaystyle=$ $\displaystyle-
ge^{-\varphi_{1}-\varphi_{2}-\varphi_{3}}[e^{2(\varphi_{1}+\varphi_{3})}+e^{2(\varphi_{2}+\varphi_{3})}-e^{2(\varphi_{1}+\varphi_{2})}-1],$
(69) $\displaystyle A^{\prime}$ $\displaystyle=$ $\displaystyle
ge^{-\varphi_{1}-\varphi_{2}-\varphi_{3}}[e^{2(\varphi_{1}+\varphi_{2})}+e^{2(\varphi_{1}+\varphi_{3})}+e^{2(\varphi_{3}+\varphi_{3})}+1].$
(70)

To find the solution to these equations, we first take a linear combination

$\varphi_{1}^{\prime}+\varphi_{2}^{\prime}=-2ge^{-\varphi_{1}-\varphi_{2}-\varphi_{3}}(e^{2(\varphi_{1}+\varphi_{2})}-1).$
(71)

After changing to a new radial coordinate $\rho$ defined by

$\frac{d\rho}{dr}=e^{-\varphi_{1}-\varphi_{2}-\varphi_{3}},$ (72)

we find

$\varphi_{2}=2g\rho-\varphi_{1}-\frac{1}{2}\ln(e^{4g\rho}+C_{2})$ (73)

for a constant $C_{2}$.
Similarly, taking a linear combination
$\varphi_{1}^{\prime}+\varphi_{3}^{\prime}$ gives rise to

$\varphi_{3}=2g\rho-\varphi_{1}-\frac{1}{2}\ln(e^{4g\rho}+C_{3}).$ (74)

Using these results in equation (67), we find

$\varphi_{1}=\frac{1}{4}\ln\left[\frac{e^{4g\rho}(e^{4g\rho}+C_{1})}{(e^{4g\rho}+C_{2})(e^{4g\rho}+C_{3})}\right].$
(75)

Finally, with all these results, the solution for $A$ is given by

$A=g\rho+\frac{1}{4}\ln(e^{4g\rho}+C_{1})+\frac{1}{4}\ln(e^{4g\rho}+C_{2})+\frac{1}{4}\ln(e^{4g\rho}+C_{3}).$
(76)

We now look at the behavior of the solution as $\varphi_{\alpha}\sim 0$ which
gives $\rho\sim r$ and

$\varphi_{1}\sim\frac{1}{4}(C_{1}-C_{2}-C_{3})e^{-4g\rho},\qquad\varphi_{2,3}\sim-\frac{1}{4}(C_{1}-C_{3,2})e^{-4g\rho},\qquad
A\sim 4g\rho\,.$ (77)

This is the expected behavior of the solution asymptotic to the supersymmetric
$AdS_{4}$ vacuum. As in the previous cases, the solution is singular as
$4g\rho\rightarrow\ln(-C_{\alpha})$. For $C_{1}\neq C_{2}\neq C_{3}$, there
are three possibilities:

  * •

$C_{1}<C_{2,3}$: In this case, the solution is singular when
$4g\rho\rightarrow\ln(-C_{1})$ with

$\displaystyle\varphi_{1}\sim\frac{1}{4}\ln(4g\rho-\tilde{C}_{1}),\qquad\tilde{C}_{1}=\ln(-C_{1}),$
$\displaystyle\varphi_{2,3}\sim-\varphi_{1},\qquad A\sim\varphi_{1}\,.$ (78)

  * •

$C_{2}<C_{1,3}$ or $C_{3}<C_{1,2}$: In this case, we find

$\displaystyle\varphi_{1}\sim-\frac{1}{4}\ln(4g\rho-\tilde{C}_{2,3}),\qquad\tilde{C}_{2,3}=\ln(-C_{2,3}),$
$\displaystyle\varphi_{2,3}\sim\varphi_{1},\qquad A\sim-\varphi_{1}\,.$ (79)

In the first case, we have $\varphi_{1}\rightarrow-\infty$ and
$\varphi_{2,3}\rightarrow\infty$ while in the second case, the solution gives
$\varphi_{1,2,3}\rightarrow\infty$. It can be easily verified that all of
these behaviors lead to $V\rightarrow-\infty$, so the singularities are
physically acceptable. The solution then describes different types of mass
deformations within the dual $N=6$ SCFT to non-conformal phases with
$SO(2)\times SO(2)\times SO(2)$ symmetry. The solution also preserves $N=6$
Poincare supersymmetry as in the previous cases.

###  III.4 Solutions with $SO(3)$ symmetry

We further reduce the residual symmetry to $SO(3)\subset SO(3)\times
SO(3)\subset SO(6)$ generated by the antisymmetric matrices $A_{3\times 3}$ in
the upper-left block of (50). There are three singlet scalars parametrized by

$\phi_{AB}=\left(\begin{array}[]{cc}0_{3\times 3}&0_{3\times 3}\\\ 0_{3\times
3}&\hat{A}_{3\times 3}\\\ \end{array}\right)$ (80)

with

$\hat{A}=\left(\begin{array}[]{ccc}0&\tilde{\phi}_{1}&\tilde{\phi}_{2}\\\
-\tilde{\phi}_{1}&0&\tilde{\phi}_{3}\\\
-\tilde{\phi}_{2}&-\tilde{\phi}_{3}&0\\\ \end{array}\right).$ (81)

By writting $\tilde{\phi}_{\alpha}=\varphi_{\alpha}e^{i\zeta_{\alpha}}$ with

$\varphi_{1}=\Phi\cos\theta,\qquad\varphi_{2}=\Phi\sin\theta\cos\vartheta,\qquad\varphi_{3}=\Phi\sin\theta\sin\vartheta$
(82)

and

$\zeta_{1}=\zeta,\qquad\zeta_{2}=\zeta+\eta,\qquad\zeta_{3}=\zeta+\xi,$ (83)

we find the scalar potential

$\displaystyle V$ $\displaystyle=$
$\displaystyle-g^{2}\left[16\cos^{4}\theta(2+\cosh
2\Phi)+16\cosh^{4}\theta\sin^{4}\theta(2+\cosh 2\Phi)\right.$ (84)
$\displaystyle+16\sin^{4}\theta\sin^{4}\vartheta(2+\cosh
2\Phi)-\cos^{2}\theta\sin^{2}\theta\cos^{2}\vartheta\times$
$\displaystyle(\cosh 4\Phi-8\cos 2\eta\sinh^{4}\Phi-36\cosh
2\Phi-61)+\sin^{2}\theta\sin^{2}\vartheta\times$
$\displaystyle\left[8\sinh^{4}\Phi(\cos^{2}\theta\cos
2\xi+\cos^{2}\vartheta\sin^{2}\theta\cos[2(\eta-\xi)])\right.$
$\displaystyle\left.\left.+(61+36\cosh 2\Phi-\cosh
4\Phi)(\cos^{2}\theta+\cos^{2}\vartheta\sin^{2}\theta)\right]\right].$

In this case, the scalar potential depends on phases of the complex scalars,
and the analysis is more complicated. To make the analysis more traceable, we
will further truncate to two scalars by setting $\vartheta=0$ and
$\xi=-\zeta$. This is equivalent to setting $\tilde{\phi}_{3}=0$. We now begin
with the eigenvalues of the matrix $S_{AB}$. These are of the form, after
diagonalization,

$S^{\textrm{diag}}_{AB}=\textrm{diag}(-2g\cosh\Phi_{\times
4},\frac{1}{2}\mathcal{W}_{+},\frac{1}{2}\mathcal{W}_{-})$ (85)

with $\mathcal{W}_{\pm}$ given by

$\displaystyle\mathcal{W}_{\pm}$ $\displaystyle=$ $\displaystyle 2g(\cos
2\eta+2\sin\eta)\sinh^{4}\frac{\Phi}{2}(\cos 4\theta\sin\eta\pm i\sin
2\theta)$ (86) $\displaystyle-\frac{1}{4}g(3+12\cosh\Phi+\cosh 2\Phi).$

The corresponding eigenvectors are

$\hat{\epsilon}_{\pm}=-\frac{1}{2}\sec 2\theta\left(2\cos\eta\sin
2\theta\mp\sqrt{3+\cos 2\eta+2\cos
4\theta\sin^{2}\eta}\right)\epsilon_{5}+\epsilon_{6}.$ (87)

The scalar kinetic term is given by

$\displaystyle\mathcal{L}_{\textrm{kin}}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}G_{\alpha\beta}{\phi^{\alpha}}^{\prime}{\phi^{\beta}}^{\prime}$
(88) $\displaystyle=$ $\displaystyle-\Phi^{\prime
2}-\sinh^{2}\Phi\theta^{\prime 2}-\frac{1}{4}\sinh^{2}2\Phi\zeta^{\prime
2}-\frac{1}{2}\sin^{2}\theta\sinh^{2}2\Phi\zeta^{\prime}\eta^{\prime}$
$\displaystyle-\frac{1}{4}\sin^{2}\theta\sinh^{2}\Phi(3+\cosh 2\Phi-2\cos
2\theta\sinh^{2}\Phi)\eta^{\prime 2}$

with $\phi^{\alpha}=(\Phi,\theta,\zeta,\eta)$. It is useful to give an
explicit form of the inverse of $G_{\alpha\beta}$ here

$G^{\alpha\beta}=-\frac{1}{2}\left(\begin{array}[]{cccc}1&0&0&0\\\
0&\textrm{csch}^{2}\Phi&0&0\\\
0&0&-\textrm{sech}^{2}\Phi+\textrm{csch}^{2}\Phi\sec^{2}\theta&-\textrm{csch}^{2}\Phi\sec^{2}\theta\\\
0&0&-\textrm{csch}^{2}\Phi\sec^{2}\theta&4\textrm{csc}^{2}2\theta\textrm{csch}\Phi\end{array}\right).$
(89)

The scalar potential can be written in term of the real superpotential
$W=|\mathcal{W}_{+}|=|\mathcal{W}_{-}|$ as

$\displaystyle V$ $\displaystyle=$
$\displaystyle-2G^{\alpha\beta}\frac{\partial
W}{\partial\phi^{\alpha}}\frac{\partial W}{\partial\phi^{\beta}}-3W^{2}$ (90)
$\displaystyle=$ $\displaystyle
g^{2}\left[\cos^{2}\theta\sin^{2}\theta\left(\cosh 4\Phi-8\cos
2\eta\sinh^{4}\Phi-36\cosh 2\Phi-61\right)\right.$
$\displaystyle\left.-4(3+\cos 4\theta)(2+\cosh 2\Phi)\right].$

After setting $\epsilon^{1,2,3,4}=0$ and imposing the projection conditions

$\gamma_{\hat{r}}\epsilon_{\pm}=e^{\pm
i\Lambda}\epsilon^{\pm}\qquad\textrm{with}\qquad e^{\pm
i\Lambda}=\frac{\mathcal{W}_{\pm}}{W},$ (91)

we find the following BPS equations

$\displaystyle\Phi^{\prime}$ $\displaystyle=$
$\displaystyle\frac{1}{16W}g^{2}\left[8\sinh^{3}\Phi\cosh\Phi(\cos 2\eta+2\cos
4\theta\sin^{3}\eta)-30\sin 2\Phi-\sinh 4\Phi\right],$
$\displaystyle\theta^{\prime}$ $\displaystyle=$
$\displaystyle-\frac{1}{W}g^{2}\sin^{2}\eta\sin 4\theta\sinh^{2}\Phi,$ (93)
$\displaystyle\zeta^{\prime}$ $\displaystyle=$
$\displaystyle\frac{2}{W}g^{2}\sin 2\eta\sin^{2}\theta\sinh^{2}\Phi,$ (94)
$\displaystyle\eta^{\prime}$ $\displaystyle=$
$\displaystyle-\frac{2}{W}g^{2}\sin 2\eta\sinh^{2}\Phi,$ (95) $\displaystyle
A^{\prime}$ $\displaystyle=$ $\displaystyle W\,.$ (96)

The flow equations for the scalars can be written in a compact form as

${\phi^{\alpha}}^{\prime}=2G^{\alpha\beta}\frac{\partial
W}{\partial\phi^{\beta}}\,.$ (97)

It can straightforwardly be verified that all these equations satisfy the
second-order field equations. We also see that from these equations, there is
only one supersymmetric critical point, with
$\Phi^{\prime}=\theta^{\prime}=\zeta^{\prime}=\eta^{\prime}=0$, at
$\phi^{\alpha}=0$.
We note that although the superpotential and the scalar potential do not
depend on $\zeta$, $\zeta^{\prime}$ is still non-vanishing due to the mixed
terms between $\zeta$ and $\eta$ in the matrix $G_{\alpha\beta}$. Moreover,
further truncations such as $\eta=0$ or $\theta=0$ will lead to the BPS
equations in the case of $N=6$ supersymmetry with all the six eigenvalues of
$S_{AB}$ leading to

$\mathcal{W}=4g\cosh\Phi\,.$ (98)

This is very similar to the $N=5$ gauged supergravity studied in N5_flow  in
which the differences in the phases of the scalars are crucial for breaking
the original supersymmetry to lower amount.
We now look at the solution to the above equations. Combining $\eta^{\prime}$
and $\theta^{\prime}$ equations gives

$\frac{d\theta}{d\eta}=\frac{1}{4}\sin 4\theta\tan\eta$ (99)

with the solution given by

$\cot 2\theta=C_{1}\cos\eta\,.$ (100)

Similarly, taking the combination between $\zeta^{\prime}$ and $\eta^{\prime}$
equations leads to

$\frac{d\zeta}{d\eta}=-\sin^{2}\theta\,.$ (101)

After using the above solution for $\theta$, we find the solution

$\zeta=\zeta_{0}-\frac{\eta}{2}+\frac{\sqrt{C_{1}^{2}+\sec^{2}\eta}\cos\eta\tan^{1}\frac{\sqrt{2}C_{1}\sin\eta}{\sqrt{2+C_{1}^{2}(1+\cos
2\eta)}}}{\sqrt{4+2C_{1}^{2}(1+\cos 2\eta)}}$ (102)

for constant $\zeta_{0}$.
Combining $\Phi^{\prime}$ and $\eta^{\prime}$ equations by taking into account
all the previous results together with a redefinition

$\tilde{\Phi}=\sinh\Phi,$ (103)

we find

$\frac{d\tilde{\Phi}}{d\eta}=\textrm{csc}2\eta(1+\tilde{\Phi}^{2})\left(\frac{\tilde{\Phi}\tan^{2}\eta}{C_{1}^{2}+\sec^{2}\eta}+\frac{2}{\tilde{\Phi}}\right)$
(104)

whose solution is given by

$\displaystyle\frac{\tilde{\Phi}^{2}}{4}=-\frac{1+C_{1}^{2}\cos^{2}\eta-
C_{2}\sqrt{(1+\cos 2\eta)(2+C_{1}^{2}(1+\cos
2\eta))}}{3+4C_{1}^{2}\cos^{2}\eta+\cos 2\eta-4C_{2}\sqrt{(1+\cos
2\eta)(2+C_{1}^{2}(1+\cos 2\eta))}}\,.$ (105)

Using all these results in $\eta^{\prime}$ equation, we find the solution for
$\eta(r)$ implicitly from

$\displaystyle 8gr$ $\displaystyle=$
$\displaystyle\sinh^{-1}\left[2C_{2}\sqrt{\frac{\Xi-1}{(1+C_{1}^{2})[2+(C_{1}^{2}-4C_{2}^{2})(1+\Xi)]}}\right]$
(106)
$\displaystyle-\tanh^{-1}\sqrt{\frac{(1+C_{1}^{2}-4C_{2}^{2})(1+\Xi)}{\Xi-1}}$

in which we have defined

$\Xi=\cos 2\eta\,.$ (107)

Finally, we can find the solution for $A(\Xi)$ as

$\displaystyle A$ $\displaystyle=$
$\displaystyle\frac{1}{4}(\tanh^{-1}\alpha_{+}-\tanh^{-1}\alpha_{-})-\frac{1}{2}\tanh^{-1}\left[2C_{2}\sqrt{\frac{\Xi+1}{2+C_{1}^{2}(1+\Xi)}}\right]$
(108)
$\displaystyle-\frac{1}{8}\ln\left[4C_{1}^{4}(1+\Xi)^{2}+(3+\Xi)^{2}-4(1+\Xi)[8C_{2}^{2}+C_{1}^{2}(4C_{2}^{2}(1+\Xi)-3-\Xi)]\right]$
$\displaystyle+\frac{1}{4}\ln[2+(1+\Xi)(C_{1}^{2}-4C_{2}^{2})]$

with $\alpha_{\pm}$ defined by

$\alpha_{\pm}=\sqrt{-\frac{2+C_{1}^{2}(1+\Xi)}{(1+\Xi)\left[1+C_{1}^{2}-4C_{2}(2C_{2}+\pm\sqrt{4C_{2}^{2}-C_{1}^{2}-1})\right]}}\,.$
(109)

The solution preserves $N=2$ supersymmetry and breaks the $SO(6)$ R-symmetry
to $SO(3)$. The solution is singular when

$\cos^{2}\eta=-\frac{1}{1+2C_{1}^{2}-8C_{2}^{2}\pm
4C_{2}\sqrt{4C_{2}^{2}-C_{1}^{2}-1}}$ (110)

This gives $\tilde{\Phi}\rightarrow\pm\infty$ or $\Phi\rightarrow\pm\infty$
which in turn leads to

$V\rightarrow g^{2}e^{4|\Phi|}\cos^{2}\theta\sin^{2}\eta\sin^{2}\theta\,.$
(111)

We can see that the scalar potential is unbounded from above,
$V\rightarrow+\infty$, unless $\theta=0$ or $\eta=0$ both of which give the
$N=6$ solution as previously mentioned. Therefore, the IR singularities of the
$N=2$ solutions are unphysical by the criterion of Gubser_singularity .

##  IV Supersymmetric Janus solutions

In this section, we consider supersymmetric Janus solutions in the form of
curved domain walls. The solutions can be obtained from an $AdS_{3}$-sliced
domain wall ansatz

$ds^{2}=e^{2A}(e^{\frac{2\xi}{\ell}}dx^{2}_{1,1}+d\xi^{2})+dr^{2}\,.$ (112)

Since the analysis closely follows that given in warner_Janus , see also
N3_Janus , we will not repeat all the detail here but mainly review relevant
results for deriving the corresponding BPS equations.
Compared to the RG flow case, the BPS equations will be modified by the
curvature of the three-dimensional slices. In addition, as pointed out in
warner_Janus , the existence of Janus solutions requires non-vanishing
pseudoscalars resulting in more complicated set of BPS equations in constrast
to the simple flat domain wall or RG flow case. We begin with the conditions
$\delta\psi^{i}_{\hat{\mu}}=0$ for $\hat{\mu}=0,1$ which give

$A^{\prime}\gamma_{\hat{r}}\epsilon_{A}+\frac{1}{\ell}e^{-A}\gamma_{\hat{\xi}}\epsilon_{A}-\mathcal{W}\epsilon^{A}=0\,.$
(113)

This leads to the following BPS equation

$A^{\prime 2}=W^{2}-\frac{1}{\ell^{2}}e^{-2A}$ (114)

for $W=|\mathcal{W}|$. We still use the $\gamma_{\hat{r}}$ projection given in
(38). Imposing the $\gamma_{\hat{\xi}}$ projection of the form

$\gamma_{\hat{\xi}}\epsilon_{A}=i\kappa e^{i\Lambda}\epsilon^{A}$ (115)

with $\kappa^{2}=1$, we can solve the condition (113) for the
$\gamma_{\hat{r}}$ projector leading to the phase factor

$e^{i\Lambda}=\frac{A^{\prime}}{W}+\frac{i\kappa}{\ell}\frac{e^{-A}}{W}$ (116)

for real $\mathcal{W}$ and

$e^{i\Lambda}=\frac{\mathcal{W}}{A^{\prime}+\frac{i\kappa}{\ell}e^{-A}}$ (117)

for complex $\mathcal{W}$. We also note that the constant $\kappa=\pm 1$
corresponds to the chiralities of the Killing spinors on the two-dimensional
conformal defects described by the $AdS_{3}$-slices. Finally, the conditions
$\delta\psi^{A}_{\hat{\xi}}=0$ and $\delta\psi^{A}_{\hat{r}}=0$ can be solved
to obtain the explicit form of the Killing spinors, see details in
warner_Janus ,

$\epsilon_{A}=e^{\frac{A}{2}+\frac{\xi}{2\ell}+i\frac{\Lambda}{2}}\varepsilon^{(0)}_{A}$
(118)

in which the constant spinors $\varepsilon^{(0)}_{A}$ could possibly have an
$r$-dependent phase and satisfy

$\gamma_{\hat{r}}\varepsilon^{(0)}_{A}=\varepsilon^{(0)A}\qquad\textrm{and}\qquad\gamma_{\hat{\xi}}\varepsilon^{(0)}_{A}=i\kappa\varepsilon^{(0)A}\,.$
(119)

It turns out that among the previously considered cases only $SO(2)\times
SO(4)$ and $SO(3)$ symmetric scalars can possibly possess supersymmetric Janus
solutions. This is mainly a consequence of the consistency in turning on non-
vanishing pseudoscalars. For the $SO(3)$ case, the analysis is highly
complicated as already seen in the case of RG flows considered in the previous
section. Therefore, we will only give the Janus solution with $SO(2)\times
SO(4)$ symmetry. This case is more traceable, and it turns out that the
solution can be analytically obtained.

###  IV.1 Janus solutions with $SO(2)\times SO(4)$ symmetry

In this case, the superpotential is real, so we will use the phase
$e^{i\Lambda}$ from (116). In general, since $\epsilon^{1,2}$ and
$\epsilon^{A}$ with $A=3,4,5,6$ transform differently under $SO(2)\times
SO(4)$ namely as $(\mathbf{2},\mathbf{1})+(\mathbf{1},\mathbf{4})$, the two
sets of Killing spinors can satisfy different projectors. We find that in
order to obtain a consistent set of BPS equations we need to choose opposite
signs of $\kappa$ for $\epsilon^{1,2}$ and $\epsilon^{3,\ldots,6}$. Therefore,
the surface defect will preserve $N=(2,4)$ or $N=(4,2)$ superconformal
symmetry.
With the superpotential

$\mathcal{W}=4g\cosh\varphi,$ (120)

we find the following BPS equations

$\displaystyle\varphi^{\prime}$ $\displaystyle=$
$\displaystyle-\frac{8g^{2}\ell^{2}A^{\prime}e^{2A}\sinh(2\varphi)}{1+\ell^{2}A^{\prime
2}e^{2A}},$ (121) $\displaystyle\zeta^{\prime}$ $\displaystyle=$
$\displaystyle-\frac{16g^{2}\kappa\ell e^{A}}{1+\ell^{2}A^{\prime 2}e^{2A}},$
(122) $\displaystyle A^{\prime 2}+\frac{e^{-2A}}{\ell^{2}}$ $\displaystyle=$
$\displaystyle 16g^{2}\ell^{2}\cosh^{2}\varphi\,.$ (123)

It should be noted that, for $\ell\rightarrow\infty$, these equations reduce
to those of the RG flow studied in the previous section. Furthermore, these
equations take a very similar form to the $SO(4)$ symmetric Janus solution in
$N=5$ gauged supergravity studied in N5_flow .
By taking $\varphi$ as an independent variable, we can solve for $A(\varphi)$
and $\zeta(\varphi)$. The complete solution is given by

$\displaystyle A$ $\displaystyle=$ $\displaystyle C-\ln\sinh\varphi,$ (124)
$\displaystyle\cosh(2\varphi)$ $\displaystyle=$
$\displaystyle\frac{32g^{2}\ell^{2}\tanh^{2}[4g(r-r_{0})]}{16g^{2}\ell^{2}-1},$
(125) $\displaystyle\kappa\tan\zeta$ $\displaystyle=$
$\displaystyle-\sqrt{1-16g^{2}\ell^{2}}\sinh[4g(r-r_{0})]$ (126)

for constants $C$ and $r_{0}$. This solution takes the same form as the
solution given in warner_Janus , N5_flow  and N3_Janus  in $N=8,5,3$ gauged
supergravities. We also note the unbroken supersymmetries on the conformal
defects in these cases as follows: $N=(4,4)$, $N=(4,1)$ and $N=(2,1)$. All of
these solutions should be related by truncations of $N=8$ gauged supergravity
to $N=3$ and $N=5,6$ theories. This indicates that the $N=(4,4)$ Janus
solution of warner_Janus  survives in the truncation to $N=3,5,6$ gauged
supergravities. According to the AdS/CFT correspondence, we then expect the
dual $N=3,5,6$ SCFTs to possess the same two-dimensional conformal defect as
in the $N=8$ SCFT.
We end this section by giving a brief comment on the possible $SO(3)$
symmetric Janus solution. A partial analysis shows that there appears to be no
obstruction in obtaining the BPS equations for Janus solutions in this case as
in a similar analysis of $N=5$ theory in N5_flow . Therefore, we expect a
supersymmetric $N=2$ Janus solution with $SO(3)$ symmetry to exist in $N=6$
gauged supergravity as well. Since the analysis is far more complicated than
the $SO(2)\times SO(4)$ case, we refrain from giving any definite result here.

##  V Supersymmetric $AdS_{4}$ black holes

In this section, we look for supersymmetric $AdS_{4}$ black holes in the form
of curved-domain wall solutions interpolating between a locally asymptotically
$AdS_{4}$ and $AdS_{2}\times\Sigma^{2}$ geometries. The
$AdS_{2}\times\Sigma^{2}$ space describes the near horizon geometry of the
black holes. In the following analysis, we will consider only the cases of
$\Sigma^{2}$ being a two-sphere $(S^{2})$ and a hyperbolic space $(H^{2})$.
The metric ansatz is taken to be

$ds^{2}=-e^{2f(r)}dt^{2}+dr^{2}+e^{2h(r)}(d\theta^{2}+F^{2}(\theta)d\phi^{2})$
(127)

with the function $F(\theta)$ defined by

$F(\theta)=\begin{cases}\sin\theta,&\,\Sigma_{2}=S^{2}\\\
\sinh\theta,&\,\Sigma_{2}=H^{2}\end{cases}\,.$ (128)

It is straightforward to derive non-vanishing components of the spin
connection

$\displaystyle\omega^{\hat{t}\hat{r}}$ $\displaystyle=$ $\displaystyle
f^{\prime}e^{\hat{t}},\qquad\omega^{\hat{\theta}\hat{r}}=h^{\prime}e^{\hat{\theta}},$
$\displaystyle\omega^{\hat{\phi}\hat{r}}$ $\displaystyle=$ $\displaystyle
h^{\prime}e^{\hat{\phi}},\qquad\omega^{\hat{\theta}\hat{\phi}}=\frac{F^{\prime}}{F}e^{-h}e^{\hat{\phi}}$
(129)

with $F^{\prime}(\theta)=\frac{dF}{d\theta}$.
In general, the curvature of the $\Sigma^{2}$ part on the world-volume of the
domain wall will completely break supersymmetry. However, it is well-known
that some amount of supersymmetry can be preserved by performing a topological
twist. This can be achieved by turning on some gauge fields along $\Sigma^{2}$
in such a way that the corresponding spin connection,
$\omega^{\hat{\theta}\hat{\phi}}$ in the above metric, is cancelled. This
turns the covariant derivative of $\epsilon_{A}$ along $\Sigma^{2}$ into a
partial one. The resulting Killing spinors are accordingly given by spinors
that are independent of the $\Sigma^{2}$ coordinates.

###  V.1 $SO(2)\times SO(2)\times SO(2)$ symmetric solutions

We first consider the $SO(2)\times SO(2)\times SO(2)$ twist with the following
ansatz for the gauge fields

$\displaystyle A^{12}=A_{1}(r)dt-p_{1}F^{\prime}(\theta)d\phi,$ $\displaystyle
A^{34}=A_{2}(r)dt-p_{2}F^{\prime}(\theta)d\phi,$ $\displaystyle
A^{56}=A_{3}(r)dt-p_{3}F^{\prime}(\theta)d\phi\,.$ (130)

The constants $p_{i}$, $i=1,2,3$, are identified with magnetic charges. The
corresponding field strengths are given by

$\displaystyle F^{12}=A_{1}^{\prime}dr\wedge dt+\kappa
p_{1}F(\theta)d\theta\wedge d\phi,$ $\displaystyle
F^{34}=A_{2}^{\prime}dr\wedge dt+\kappa p_{2}F(\theta)d\theta\wedge d\phi,$
$\displaystyle F^{56}=A_{3}^{\prime}dr\wedge dt+\kappa
p_{3}F(\theta)d\theta\wedge d\phi\,.$ (131)

We have introduced a parameter $\kappa$ with $\kappa=1,-1$ for $\Sigma^{2}$
being $S^{2}$ or $H^{2}$, respectively. We also note that
$F^{\prime\prime}(\theta)=-\kappa F(\theta)$.
With the $SO(2)\times SO(2)\times SO(2)$ singlet scalars given by (62), we
find non-vanishing components of the composite connection

${Q_{A}}^{B}=2gi\sigma_{2}\otimes\left(\begin{array}[]{ccc}A^{12}&&\\\
&A^{34}&\\\ &&A^{56}\\\ \end{array}\right).$ (132)

With the component ${Q_{\hat{\phi}A}}^{B}$, the spin connection
$\omega^{\hat{\theta}\hat{\phi}}$ can be cancelled by imposing the following
projector

$\gamma_{\hat{\theta}\hat{\phi}}\epsilon_{A}={(i\sigma_{2}\otimes\mathbb{I}_{3})_{A}}^{B}\epsilon_{B}$
(133)

and the twist conditions

$2gp_{1}=2gp_{2}=2gp_{3}=1\,.$ (134)

These conditions imply that $p_{1}=p_{2}=p_{3}$. The twist is then obtained
from the diagonal subgroup $SO(2)_{\textrm{diag}}\subset SO(2)\times
SO(2)\times SO(2)$ as in the pure $N=4$ and $N=5$ gauged supergravities
studied in flow_acrossD_bobev  and N5_flow , respectively. For consistency, we
will also set $A_{3}=A_{2}=A_{1}$.
For all $p_{i}$ non-vanishing, the twists allow all the supersymmetries
corresponding to $\epsilon^{A}$, $A=1,2,\ldots,6$ to be unbroken subject to
the projector (133). We also note some useful relations for deriving the full
set of BPS equations. Using
$\epsilon_{\hat{0}\hat{r}\hat{\theta}\hat{\phi}}=1$ and
$\gamma_{5}\epsilon_{A}=-\epsilon_{A}$, we find

$\gamma^{\hat{0}\hat{r}}\epsilon_{A}=-i\gamma^{\hat{\theta}\hat{\phi}}\epsilon_{A}={(\sigma_{2}\otimes\mathbb{I}_{3})_{A}}^{B}\epsilon_{B}\,.$
(135)

It turns out that, we need to turn on the $SO(2)\sim U(1)$ gauge field of
$U(6)\sim SU(6)\times U(1)$ in order to find a consistent set of BPS
equations. We similarly take the ansatz for this $U(1)$ gauge field to be

$A^{0}=A_{0}(r)dt-p_{0}F^{\prime}(\theta)d\phi\qquad\textrm{and}\qquad
F^{0}=A_{0}^{\prime}(r)dr\wedge dt+\kappa p_{0}d\theta\wedge d\phi\,.$ (136)

We also note that both $A^{0}$ and $A^{IJ}$ can appear in the BPS equations
due to the off-diagonal element of the scalar coset representative. In
particular, we have the relations

$\displaystyle\hat{F}^{+}_{AB}=h_{\Lambda
AB}F^{+\Lambda}=h_{0AB}F^{+0}+\frac{1}{2}h_{IJ,AB}F^{+IJ},$ (137)
$\displaystyle\hat{F}^{+}=h_{\Lambda
0}F^{+\Lambda}=h_{00}F^{+0}+\frac{1}{2}h_{IJ0}F^{+IJ}\,.$ (138)

However, $A^{0}$ does not participate in the twist since $\epsilon_{A}$ are
not charged under the $U(1)$ factor outside $SU(6)$.
It is also useful to define the “central charge” matrix

$\mathcal{Z}_{AB}=-\frac{1}{\sqrt{2}}(\hat{F}^{+}_{\hat{\theta}\hat{\phi}AC}-i\hat{F}^{+}_{\hat{0}\hat{r}AC}){(i\sigma_{2}\otimes\mathbb{I}_{3})^{C}}_{B}\,.$
(139)

In the present case, it turns out that $\mathcal{Z}_{AB}$ is proportional to
the identity matrix, $\mathcal{Z}_{AB}=\mathcal{Z}\delta_{AB}$.
With all these and the projector (38), we find the following BPS equation,
from $\delta\psi_{\hat{\theta}A}$ and $\delta\psi_{\hat{\phi}A}$,

$h^{\prime}e^{i\Lambda}-\mathcal{W}-\mathcal{Z}=0$ (140)

which leads to

$h^{\prime}=\pm|\mathcal{W}+\mathcal{Z}|\qquad\textrm{and}\qquad
e^{i\Lambda}=\pm\frac{\mathcal{W}+\mathcal{Z}}{|\mathcal{W}+\mathcal{Z}|}\,.$
(141)

With the projectors (38) and (135), the condition $\delta\psi_{\hat{0}A}=0$
gives

$(f^{\prime}+2igA_{1})e^{i\Lambda}-\mathcal{W}+\mathcal{Z}=0$ (142)

which implies

$f^{\prime}=\textrm{Re}[e^{-i\Lambda}(\mathcal{W}-\mathcal{Z})]\qquad\textrm{and}\qquad
2gA_{1}=\textrm{Im}[e^{-i\Lambda}(\mathcal{W}-\mathcal{Z})]\,.$ (143)

The latter fixes the time component of the gauge fields. Finally, as in the
case of domain walls and Janus solutions, the condition
$\delta\psi_{\hat{r}A}=0$ determines the $r$-dependence of the Killing spinors
giving rise to $\epsilon_{A}=e^{\frac{f}{2}}\epsilon_{A(0)}$.
Similar to the RG flow case, it turns out that we need to set
$\zeta_{1}=\zeta_{2}=\zeta_{3}=0$ for consistency. This gives real
$\mathcal{W}$ and $\mathcal{Z}$ resulting in $e^{i\Lambda}=\pm 1$ and
$A_{1}=0$. We will also set $A_{0}(r)=0$ for simplicity although it is not
constrained by the previously obtained conditions. In addition, the
compatibility between the BPS equations coming from $\delta\chi_{A}$ and
$\delta\chi_{ABC}$ requires

$p_{0}=\kappa p_{1}\,.$ (144)

We also note that, in this case, the choice $p_{0}=0$ breaks all
supersymmetry. This implies that the $SO(2)\times SO(2)\times SO(2)$ twist
needs to be accompanied by the $U(1)$ gauge field $A^{0}_{\mu}$.
With all these, we find a consistent set of BPS equations given by

$\displaystyle\varphi_{1}^{\prime}$ $\displaystyle=$
$\displaystyle-\frac{\partial|\mathcal{W}+\mathcal{Z}|}{\partial\varphi_{1}}$
(145) $\displaystyle=$
$\displaystyle\frac{1}{2}e^{-\varphi_{1}-\varphi_{2}-\varphi_{3}}\left[2g(1+e^{2(\varphi_{2}+\varphi_{3})}-e^{2(\varphi_{1}+\varphi_{2})}-e^{2(\varphi_{1}+\varphi_{3})})\right.$
$\displaystyle\left.-p_{1}\kappa
e^{-2h+2\varphi_{1}+2\varphi_{2}+\varphi_{3}}\right],$
$\displaystyle\varphi_{2}^{\prime}$ $\displaystyle=$
$\displaystyle-\frac{\partial|\mathcal{W}+\mathcal{Z}|}{\partial\varphi_{2}}$
(146) $\displaystyle=$
$\displaystyle\frac{1}{2}e^{-\varphi_{1}-\varphi_{2}-\varphi_{3}}\left[2g(1-e^{2(\varphi_{2}+\varphi_{3})}-e^{2(\varphi_{1}+\varphi_{2})}+e^{2(\varphi_{1}+\varphi_{3})})\right.$
$\displaystyle\left.-p_{1}\kappa
e^{-2h+2\varphi_{1}+2\varphi_{2}+\varphi_{3}}\right],$
$\displaystyle\varphi_{3}^{\prime}$ $\displaystyle=$
$\displaystyle-\frac{\partial|\mathcal{W}+\mathcal{Z}|}{\partial\varphi_{3}}$
(147) $\displaystyle=$
$\displaystyle\frac{1}{2}e^{-\varphi_{1}-\varphi_{2}-\varphi_{3}}\left[2g(1-e^{2(\varphi_{2}+\varphi_{3})}+e^{2(\varphi_{1}+\varphi_{2})}-e^{2(\varphi_{1}+\varphi_{3})})\right.$
$\displaystyle\left.-p_{1}\kappa
e^{-2h+2\varphi_{1}+2\varphi_{2}+\varphi_{3}}\right],$ $\displaystyle
h^{\prime}$ $\displaystyle=$ $\displaystyle|\mathcal{W}+\mathcal{Z}|$ (148)
$\displaystyle=$
$\displaystyle\frac{1}{2}e^{-\varphi_{1}-\varphi_{2}-\varphi_{3}}\left[2g(1+e^{2(\varphi_{2}+\varphi_{3})}+e^{2(\varphi_{1}+\varphi_{2})}+e^{2(\varphi_{1}+\varphi_{3})})\right.$
$\displaystyle\left.+p_{1}\kappa
e^{-2h+2\varphi_{1}+2\varphi_{2}+\varphi_{3}}\right],$ $\displaystyle
f^{\prime}$ $\displaystyle=$ $\displaystyle|\mathcal{W}-\mathcal{Z}|$ (149)
$\displaystyle=$
$\displaystyle\frac{1}{2}e^{-\varphi_{1}-\varphi_{2}-\varphi_{3}}\left[2g(1+e^{2(\varphi_{2}+\varphi_{3})}+e^{2(\varphi_{1}+\varphi_{2})}+e^{2(\varphi_{1}+\varphi_{3})})\right.$
$\displaystyle\left.-p_{1}\kappa
e^{-2h+2\varphi_{1}+2\varphi_{2}+\varphi_{3}}\right].$

For an $AdS_{2}\times\Sigma^{2}$ fixed point to exist, we require that
$\varphi_{1}^{\prime}=\varphi_{2}^{\prime}=\varphi_{3}^{\prime}=h^{\prime}=0$
and $f^{\prime}\sim\frac{1}{L_{AdS_{2}}}$. It can be easily verified that the
above equations do not admit any $AdS_{2}\times\Sigma^{2}$ fixed points.
Although there is no supersymmetric $AdS_{2}\times\Sigma^{2}$ fixed point, we
are able to analytically obtain the complete solution to these BPS equations.
Since it might be useful for some holographic studies, we will present the
solution here. By changing to a new radial coordinate $\rho$ using the
relation $\frac{d\rho}{dr}=e^{\varphi_{3}}$, we can form the following linear
combinations

$\displaystyle\frac{d}{d\rho}(\varphi_{1}-\varphi_{2})$ $\displaystyle=$
$\displaystyle 2g(e^{\varphi_{2}-\varphi_{1}}-e^{\varphi_{1}-\varphi_{2}})$
(150)
$\displaystyle\textrm{and}\qquad\frac{d}{d\rho}(\varphi_{2}-\varphi_{3})$
$\displaystyle=$ $\displaystyle
2g(e^{\varphi_{1}-\varphi_{2}}-e^{\varphi_{1}+\varphi_{2}-2\varphi_{3}}).$
(151)

The first equation can be solved by

$\varphi_{1}=\ln\left[\frac{e^{\varphi_{2}}(e^{4g\rho}+e^{4g\rho_{0}})}{e^{4g\rho}-e^{4g\rho_{0}}}\right]$
(152)

with an integraton constant $\rho_{0}$. Using this result in the second
equation, we find the solution

$\varphi_{2}=\ln\left[\frac{e^{\varphi_{3}-2g\rho}(e^{4g\rho}-e^{4g\rho_{0}})}{\sqrt{e^{4g\rho}+e^{8g\rho_{0}}+8gC}}\right]$
(153)

with another integration constant $C$.
By treating $f$ and $h$ as functions of $\varphi_{3}$, we find the following
solutions for $f$ and $h$

$\displaystyle f$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\ln\left[e^{4g\rho_{0}}\left(256g^{3}C\tilde{C}-16g\tilde{C}e^{8g\rho_{0}}+\kappa
p_{1}\ln\left[\frac{1+e^{8g(\rho-\rho_{0})}+8gCe^{4g\rho-8g\rho_{0}}}{e^{8g(\rho-\rho_{0})}-1}\right]\right)\right.$
(154) $\displaystyle\left.-8gC\kappa
p_{1}\tanh^{-1}e^{4g(\rho-\rho_{0})}\right]+h,$ $\displaystyle h$
$\displaystyle=$
$\displaystyle\frac{1}{2}\ln\left[e^{4g\rho_{0}}\left(16g\tilde{C}(e^{8g\rho_{0}}-16g^{2}C^{2})-\kappa
p_{1}\ln\left[\frac{1+e^{8g(\rho-\rho_{0})}+8gCe^{4g\rho-8g\rho_{0}}}{e^{8g(\rho-\rho_{0})}-1}\right]\right)\right.$
(155) $\displaystyle\left.\phantom{\frac{1}{e^{8}}}+8gC\kappa
p_{1}\tanh^{-1}e^{4g(\rho-\rho_{0})}\right]+\frac{1}{2}\ln\left[\frac{e^{12g\rho_{0}}(1-e^{8g(\rho-\rho_{0})})^{2}}{8g(e^{8g\rho_{0}}-16g^{2}C^{2})}\right]+\varphi_{3}-4g\rho\,.\qquad\quad$

Finally, the solution $\varphi_{3}(\rho)$ can be given implicitly in the
following equation

$\displaystyle 4C_{0}e^{4g\rho}(e^{8g\rho_{0}}+e^{8g\rho}+8gCe^{4g\rho})$
$\displaystyle=$
$\displaystyle\beta_{0}+\beta_{1}\ln\left[\frac{e^{4g(\rho_{0}-\rho)}+1}{e^{4g(\rho_{0}-\rho)}-1}\right]$
(156)
$\displaystyle+\beta_{2}\ln\left[\frac{e^{8g(\rho-\rho_{0})}-1}{1+e^{8g(\rho-\rho_{0})}+8gCe^{4g\rho}}\right]$

in which $C_{0}$ is a constant, and the coefficients $\beta_{0}$, $\beta_{1}$
and $\beta_{2}$ are defined by

$\displaystyle\beta_{0}$ $\displaystyle=$
$\displaystyle-16g\tilde{C}e^{4\rho_{0}}(16Cg^{2}-e^{8g\rho_{0}})\left[2e^{4\varphi_{3}+8g(\rho+\rho_{0})}+8gCe^{4g\rho}(e^{8g\rho}+e^{8g\rho_{0}})\right.$
(157)
$\displaystyle\left.+(e^{16g\rho_{0}}+e^{16g\rho})(1-e^{4\varphi_{3}})+2e^{8g(\rho+\rho_{0})}\right],$
$\displaystyle\beta_{1}$ $\displaystyle=$ $\displaystyle\left[\frac{\kappa
p_{1}}{2(e^{4g\rho}+e^{4g(3\rho-2\rho_{0})}+8Cge^{8g(\rho-\rho_{0})})}\right]\left[e^{12g\rho}+e^{4g(\rho+2\rho_{0})}+4Cge^{8g\rho}(3+e^{4\varphi_{3}})\right.$
(158)
$\displaystyle\left.+16g^{2}C^{2}e^{4g\rho}(1+e^{8g(\rho-\rho_{0})})-2Cg(e^{4\varphi_{3}}-1)(e^{8g\rho_{0}}+e^{8g(2\rho-\rho_{0})})\right],$
$\displaystyle\beta_{2}$ $\displaystyle=$ $\displaystyle\frac{\kappa
p_{1}[(e^{8g\rho_{0}}+e^{8g\rho}+8Cge^{4g\rho})^{2}-e^{4\varphi_{3}}(e^{8g\rho_{0}}-e^{8g\rho})^{2}]}{4e^{4g(\rho-\rho_{0})}(e^{8g\rho_{0}}+e^{8g\rho}+8Cge^{4g\rho})}\,.$
(159)

Since there is no $AdS_{2}\times\Sigma^{2}$ fixed point in the IR, the
solution describes a flow from the locally supersymmetric $AdS_{4}$ vacuum to
a curved domain wall with world-volume $\mathbb{R}\times\Sigma^{2}$. According
to the AdS/CFT correspondence, the solution is expected to describe an RG flow
from the $N=6$ SCFT in three dimensions to a supersymmetric quantum mechanics
in the IR. The latter arises from the former by a twisted compactification on
$\Sigma^{2}$.

###  V.2 $SO(2)\times SO(4)$ symmetric solutions

We now look at a truncation of the previous result by setting $p_{2}=p_{3}=0$
and $\varphi_{2}=\varphi_{3}=0$. The resulting solutions will preserve
$SO(2)\times SO(4)$ symmetry with the twist performed along the $SO(2)$
factor. In this case, the supersymmetry corresponding to $\epsilon^{3,4,5,6}$
will be broken since we cannot perform a twist along these directions. With
$\epsilon^{3,4,5,6}=0$, we find the BPS equations

$\displaystyle\varphi^{\prime}$ $\displaystyle=$
$\displaystyle\frac{1}{4}e^{-2h-\varphi}[8ge^{2h}-p_{0}+\kappa
p_{1}-e^{2\varphi}(8ge^{2h}+p_{0}+\kappa p_{1})],$ (160) $\displaystyle
h^{\prime}$ $\displaystyle=$
$\displaystyle\frac{1}{4}e^{-2h-\varphi}[8ge^{2h}-p_{0}+\kappa
p_{1}+e^{2\varphi}(8ge^{2h}+p_{0}+\kappa p_{1})],$ (161) $\displaystyle
f^{\prime}$ $\displaystyle=$
$\displaystyle\frac{1}{4}e^{-2h-\varphi}[8ge^{2h}+p_{0}-\kappa
p_{1}+e^{2\varphi}(8ge^{2h}-p_{0}-\kappa p_{1})]$ (162)

in which we have set $\varphi_{1}=\varphi$. We also note that with only the
$SO(2)$ twist, it is not necessary to set $p_{0}=\kappa p_{1}$. However, the
existence of an $AdS_{2}\times\Sigma^{2}$ fixed point requires vanishing
$p_{0}$. For $p_{0}=0$, we find a fixed point

$\varphi=\varphi_{0},\qquad h=\frac{1}{2}\ln\left[-\frac{\kappa
p_{1}}{8g}\right],\qquad L_{AdS_{2}}=\frac{1}{8g\cosh 2\varphi_{0}}\,.$ (163)

for constant $\varphi_{0}$. This is an $AdS_{2}\times H^{2}$ fixed point since
the reality of $h$ implies $\kappa=-1$.
The complete flow solution can be obtained by using the same procedure as in
the previous sections. The resulting solution is given by

$\displaystyle h$ $\displaystyle=$
$\displaystyle\varphi-\ln(1-e^{2\varphi})+C,$ (164) $\displaystyle f$
$\displaystyle=$ $\displaystyle h-2\varphi+\ln[\kappa
p_{1}(1+e^{4\varphi})+2e^{2\varphi}(4g-\kappa p_{1})],$ (165) $\displaystyle
8g(\rho-\rho_{0})$ $\displaystyle=$ $\displaystyle 2\sqrt{\frac{2g}{\kappa
p_{1}-2g}}\tan^{-1}\left[\frac{4g+\kappa
p_{1}(e^{2\varphi}-1)}{2\sqrt{2g(\kappa p_{1}-2g)}}\right]$ (166)
$\displaystyle+\ln\left[\frac{\kappa
p_{1}(1+e^{4\varphi})+2e^{2\varphi}(4g-\kappa
p_{1})}{(1-e^{2\varphi})^{2}}\right]$

in which we have defined the new radial coordinate $\rho$ by
$\frac{d\rho}{dr}=e^{\varphi}$. We have neglected the integration constant of
$f$ by absorbing it in the rescaling of the time coordinate $t$.
Near $r\sim\rho\rightarrow\infty$, we find

$\varphi\sim e^{-4gr},\qquad h\sim f\sim 4gr$ (167)

which gives an asymptotically locally $AdS_{4}$ critical point. On the other
hand, by choosing $\varphi_{0}=\frac{1}{2}\ln\left(1-2\sqrt{-\frac{2g}{\kappa
p_{1}}}\right)$ and $C=-\varphi_{0}$, we find that as
$\varphi\rightarrow\varphi_{0}$

$h\sim\frac{1}{2}\ln\left[-\frac{\kappa
p_{1}}{8g}\right]\qquad\textrm{and}\qquad f\sim
8gr\frac{1-\sqrt{-\frac{2g}{\kappa p_{1}}}}{\sqrt{1-2\sqrt{-\frac{2g}{\kappa
p_{1}}}}}$ (168)

which is the $AdS_{2}\times H^{2}$ fixed point identified above.
It should be noted that, in this case, the solution can be regarded as a
solution of a truncated $N=2$ gauged supergravity. In particular, the solution
with vanishing scalar corresponds to a universal RG flow across dimension of
which the uplifts to M-theory and massive type IIA theory have been
extensively studied in Zaffaroni_BH_entropy .

###  V.3 $U(3)$ symmetric solutions

As a final case, we consider $U(3)$ symmetric solutions with a twist performed
along the $SO(2)\sim U(1)$ factor. The corresponding gauge generator of this
$U(1)$ factor is given by $X_{14}+X_{25}+X_{36}$. We then turn on the
following gauge fields

$\mathcal{A}=A^{14}=A^{25}=A^{36}=A(r)dt-\kappa pF^{\prime}(\theta)d\phi\,.$
(169)

With the $U(3)$ singlet scalar given in (51), we find the composite connection

${Q_{A}}^{B}=2gi\mathcal{A}{(\mathbb{I}_{3}\otimes\sigma_{2})_{A}}^{B}\,.$
(170)

The twist is implemented by imposing

$\gamma_{\hat{\theta}\hat{\phi}}\epsilon_{A}={(\mathbb{I}_{3}\otimes
i\sigma_{2})_{A}}^{B}\epsilon_{B}\qquad\textrm{and}\qquad 2gp=1\,.$ (171)

We also note that, similar to the $SO(2)\times SO(2)\times SO(2)$ twist, all
$\epsilon_{A}$ can be non-vanishing. In addition, we also need non-vanishing
$A^{0}$ which we will again use the ansatz (136). In this case, consistency
requires $p_{0}=-\kappa p$.
As in the RG flow case, we need to set $\zeta=0$ for consistency between the
BPS equations and the field equations. This again results in $A(r)=0$.
Repeating the same analysis as in the previous cases, we find the following
BPS equations

$\displaystyle\varphi^{\prime}$ $\displaystyle=$ $\displaystyle
g(e^{-\varphi}-e^{3\varphi})+\frac{1}{2}\kappa pe^{-2h-3\varphi},$ (172)
$\displaystyle h^{\prime}$ $\displaystyle=$ $\displaystyle
g(3e^{-\varphi}+e^{3\varphi})+\frac{1}{2}\kappa pe^{-2h-3\varphi},$ (173)
$\displaystyle\varphi^{\prime}$ $\displaystyle=$ $\displaystyle
g(3e^{-\varphi}+e^{3\varphi})-\frac{1}{2}\kappa pe^{-2h-3\varphi}\,.$ (174)

It is easily verified that there is no $AdS_{2}\times\Sigma^{2}$ fixed point
in these equations. In this case, we are not able to obtain the analytic flow
solution.

##  VI Conclusions and discussions

In this paper, we have studied $N=6$ gauged supergravity in four dimensions
with $SO(6)$ gauge group. The gauged supergravity can be obtained from a
truncation of the maximal $N=8$ theory with $SO(8)$ gauge group. There is a
unique $N=6$ supersymmetric $AdS_{4}$ vacuum preserving the full $SO(6)$ gauge
symmetry. This can be identified with $AdS_{4}\times CP^{3}$ geometry in type
IIA theory dual to an $N=6$ SCFT in three dimensions. We have found a number
of RG flow solutions with various symmetries from this $N=6$ SCFT to possible
non-conformal phases in the IR. In particular, there is one solution, breaking
the $SO(6)$ R-symmetry to $SO(2)\times SO(4)$, with unbroken $N=6$ Poincare
supersymmetry. This is precisely in agreement with the field theory result on
mass deformations of $N=6$ SCFTs given in N5_6_3D_SCFT . Other solutions
preserve $U(3)$, $SO(3)$ and $SO(2)\times SO(2)\times SO(2)$ symmetries. While
most of the solutions preserve $N=6$ supersymmetry, in the case of $SO(3)$
symmetry, it is possible to find $N=2$ supersymmetric solutions. We have
analytically given all of these solutions and also checked that, except for
the $N=2$ solution, the resulting IR singularities are physical by the
criterion given in Gubser_singularity .
We have also considered more complicated solutions by generalizing the flat
domain walls to the curved ones. In the case of $AdS_{3}$-sliced domain walls,
we have found a supersymmetric Janus solution, describing a two-dimensional
conformal defect within the $N=6$ SCFT, with $SO(2)\times SO(4)$ symmetry and
$N=(2,4)$ supersymmetry on the defect. The resulting solution takes the same
form as those given in $N=8$, $N=5$ and $N=3$ gauged supergravities studied in
warner_Janus , N5_flow  and N3_Janus . We therefore argue that these solutions
are related to the $N=8$ solution by truncations. In order for Janus solutions
to exist, it is necessary that pseudoscalars are non-vanishing as pointed out
in warner_Janus . It turns out that, among the remaining cases studied in this
work, only the $SO(3)$ invariant sector could possibly admit supersymmetric
Janus solutions.
Futhermore, we have studied supersymmetric solutions of the form
$AdS_{2}\times\Sigma^{2}$ and the interpolating solutions between these
geometries and the $N=6$ $AdS_{4}$ vacuum. We have found one $AdS_{2}\times
H^{2}$ fixed point with $SO(2)\times SO(4)$ symmetry from $SO(2)$ twist. The
solution interpolating between this fixed point and the $AdS_{4}$ vacuum
preserves two supercharges while the IR fixed point $AdS_{2}\times H^{2}$ has
four supercharges. Holographically, this solution corresponds to an RG flow
from the $N=6$ SCFT to superconformal quantum mechanics which is useful in
computing black hole entropy along the ling of Zaffaroni_BH_entropy ;
BH_entropy_benini ; BH_entropy_Passias .
For $SO(2)\times SO(2)\times SO(2)$ twist, the BPS equations are more
complicated but admit no $AdS_{2}\times\Sigma^{2}$ fixed point. However, in
this case, we are able to obtain a complete flow solution between the
$AdS_{4}$ critical point to a curved domain wall with world-volume
$\mathbb{R}\times\Sigma^{2}$ in the IR. The solution preserves $N=6$
supersymmetry in three dimensions, or twelve supercharges, and $SO(2)\times
SO(2)\times SO(2)$ symmetry. This should be dual to a twisted compactification
on $\Sigma^{2}$ of the UV $N=6$ SCFT to a supersymmetric quantum mechanics in
the IR. We have also looked for $AdS_{2}\times\Sigma^{2}$ geometries from an
$SO(2)\sim U(1)$ twist in the case of $U(3)$ symmetric solutions, but there do
not exist any $AdS_{2}\times\Sigma^{2}$ fixed points.
Since all the solutions presented here are fully analytic, we hope they could
be useful in the study of gauge/gravity holography and other related aspects.
We also note that most of the structures of the solutions are very similar to
those of the $N=5$ gauged supergravity studied in N5_flow . In particular, the
$N=6$ Poincare supersymmetry in three dimensions is unbroken on the domain
wall solutions if there are no non-vanishing pseudoscalars. Unlike in the
$N=5$ theory, we are not able to find a definite conclusion on whether this is
true in general due to a more complicated scalar coset manifold. However, many
similarities in the structures of various types of supersymmetric solutions
suggest that this should be the case.
There are a number of directions to extend the present work which is clearly
only the first-step in classifying supersymmetric solutions of $N=6$ gauged
supergravity. First of all, it would be interesting to uplift the RG flow
solutions to M-theory via the embedding in $N=8$ gauged supergravity which in
turn can be obtained from a truncation of M-theory on $S^{7}$. The time
component $g_{00}$ of the eleven-dimensional metric can be used to determine
whether the four-dimensional singularities, in particular the $N=2$ case, are
physically acceptable in M-theory using the criterion given in
Maldacena_Nunez_nogo . This would lead to a complete holograhic description of
mass deformations of $N=6$ CSM theory and possible related M-brane
configurations.
We have only considered gauged supergravity with $SO(6)$ gauge group
electrically embedded in the global $SO^{*}(12)$ symmetry. It would be
interesting to study magnetic and dyonic gaugings involving also magnetic
gauge fields. In particular, performing a similar study in the case of $N=6$
gauged supergravity with the electric-magnetic phase $\omega$, see omega_N6
and omega_N8 ; omega_N8_2 , could be of particular interest since in the omega
deformed $N=8$ theory, the structure of vacua and domain walls are much richer
than the electric counterpart, see Guarino_BPS_DW ; omega_vacua_1 ;
omega_vacua_2 ; omega_DW_2  for more detail. In addition, the study of genuine
$N=6$ gaugings which cannot be embedded in the $N=8$ theory is worth
considering. In this case, the gaugings do not satisfy extra quadratic
constraints coming from the truncation of the $N=8$ theory, see a discussion
in twin_SUGRA , so the corresponding solutions cannot be embedded in the
maximal theory.

###### Acknowledgements.

We would like to give a special thank to M. Trigiante for a valuable
correspondence. We also thank C. Panwisawas for his help in finding some
literature. This work is supported by The Thailand Research Fund (TRF) under
grant RSA6280022.

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