#### Supersymmetric Janus solutions of dyonic ISO(7)-gauged \( \mathcal{N} \) = 8 supergravity

HJE
ISO(7)-gauged = 8 supergravity
Minwoo Suh 0 1 2
0 supergravity. We mostly
1 Daegu 41566 , Korea
2 Department of Physics, Kyungpook National University
We study supersymmetric Janus solutions of dyonic ISO(7)-gauged N = 8 nd Janus solutions owing to 3d N = 8 SYM phase which is the worldvolume theory on D2-branes and non-conformal. There are also solutions owing from the critical points which are dual to 3d SCFTs from deformations of the D2-brane theory.
AdS-CFT Correspondence; Supergravity Models
1 Introduction
2 Dyonic ISO(7)-gauged N = 8 supergravity
3 The SU(3)-invariant Janus solutions
4 The G2-invariant Janus solutions
5 Conclusions
A The equations of motion
B The SO(4)-invariant truncation
well studied in N = 4 super Yang-Mills theory. The Janus eld theories from N
were constructed in type IIB supergravity in [5{8] or in gauged N = 8 supergravity in ve
dimensions in [9] and uplifted to type IIB supergravity in [10].
Although Janus eld theories from ABJM theory are not known, the Janus solutions
are proposed in eleven-dimensional supergravity [11{14]. The ABJM Janus solutions are
also studied in gauged N
= 8 supergravity in four dimensions, and some of them are
uplifted to eleven-dimensions [15, 16]. There are more examples of Janus solutions in
four-dimensional gauged supergravity [17{19]. Lately, Janus solution was studied in F (4)
gauged supergravity in six dimensions, and was proposed to be dual to codimension one
defect in 5d superconformal eld theories [20].
In this paper, we study Janus solutions dual to 3d SCFTs from deformations of
D2brane theory via dyonic ISO(7)-gauged N = 8 supergravity. The well-known examples of
the AdS/CFT correspondence on D3-, M2- and M5-branes involve anti-de Sitter spacetime
as a near horizon geometry of the corresponding branes [1]. There are corresponding large
N conformal eld theories dual to the AdS geometries, and they are N = 4 super
YangMills theory, ABJM theory and 6d (
2,0
) theory, respectively. When it comes to D2-branes,
the near horizon geometry of a stack of D2-branes is not an AdS spacetime, and its dual
gauge theory is 3d N = 8 supersymmetric Yang-Mills theory which is non-conformal.
However, by adding a Chern-Simons term, 3d N = 8 SYM
ows to a conformal xed
{ 1 {
point with N = 2, U(
1
)R SU(3) symmetry [21, 22]. Otherwise, by adding a mass term
to one of the chiral scalars, it ows to another conformal xed point with N = 3, SO(4)
symmetry [23]. Recently, gravity duals of these 3d SCFTs from D2-branes were discovered
in dyonic ISO(7)-gauged N = 8 supergravity [24].
Dyonic ISO(7)-gauged N = 8 supergravity is a consistent truncation of massive type
IIA supergravity on six-sphere [25{27]. The Romans mass corresponds to the magnetic
gauge coupling of dyonic ISO(7)-gauged N = 8 supergravity. The scalar potential of the
theory has four known superymmetric critical points, and they are N = 2 SU(3) U(
1
),
N = 1 SU(3), N = 1 G2, and N = 3 SO(4) critical points. The N = 2 SU(3) U(
1
) and
N = 3 SO(4) critical points are dual to the 3d SCFTs from the deformations of D2-brane
theory discussed in the previous paragraph. The xed point AdS solutions were uplifted to
massive type IIA supergravity [26], and holographic RG
ows between critical points were
studied [28]. The gravitational free energies [25, 29, 30] and spin-2 spectrum [31, 32] were
calculated and matched with the eld theory results.
In the usual supergravity theories, most of the RG
ows and Janus solutions are
attracted to critical points which are dual to conformal eld theories e.g. N
= 4 super
Yang-Mills theory, ABJM theory, and 6d (
2,0
) theory. On the other hand, in dyonic
ISO(7)gauged N = 8 supergravity, most of the RG
ows and Janus solutions are attracted to a
non-conformal phase which is dual to 3d N = 8 SYM on the worldvolume of D2-branes.
Therefore, as we will see, usual Janus solutions in dyonic ISO(7)-gauged N = 8 supergravity
do not exhibit AdS-behavior. On the other hand, if we ne-tune initial values of the scalar
elds, we obtain Janus solutions staying at a critical point for a while and then moving
away to the non-conformal 3d SYM phase.
In section 2, we review the N = 1, Z2 SO(3)-invariant truncation of dyonic
ISO(7)gauged N
= 8 supergravity. In section 3, we study Janus solutions in the N
= 2,
SU(3) U(
1
)-invariant truncation. In section 4, we study Janus solutions in the N = 1,
G2-invariant truncation. We conclude in section 5. In appendix A we present the
equations of motion from the truncations we consider. In appendix B, we present the N = 1,
SO(4)-invariant truncation.
2
Dyonic ISO(7)-gauged N
= 8 supergravity
We begin by considering the N = 1, Z2 SO(3)-invariant truncation of dyonic
ISO(7)gauged N = 8 supergravity. This truncation was studied in detail in appendix A of [25],
and we review it in this section. There are three complex scalar elds, 1
, 2 and
terms of canonical N = 1 formulation, the scalar action is given by
S =
1
1)2
1 + 2
( 2
2)2
2 + 6
3)2
P :
(2.1)
K =
3 log
i( 1
1)
log
i( 2
2)
3 log
i( 3
3) ;
(2.2)
{ 2 {
N = 3
N = 2
N = 1
N = 1
SU(3) U(
1
)
SO(4)
G2
SU(3)
31=2
24=3
31=2
2
51=231=2
27=3
31=251=2
22
1
21=3
0
1
27=3
31=2
22
Then, we de ne the complex superpotential,1
and the real superpotential,
The scalar potential is obtained from
W = p2eK=2V ;
W 2 = jWj2 :
P = 2
"
4
3
1)
2)
The scalar potential has all four known supersymmetric critical points of dyonic
ISO(7)gauged N = 8 supergravity [25], and they are listed in table 1. The ratio of electric and
magnetic gauge couplings, g and m, respectively, is denoted by c = m=g.
The N = 2 SU(3)-, N = 1 SO(4)-, and N = 1 G2-invariant truncations are obtained
as sub-truncations by identifying the complex scalar elds as, [25],
SU(3) truncation:
SO(4) truncation:
G2 truncation:
1 = t ;
1 =
1 =
3 = t ;
2 =
3 = t ;
2 =
2 = u ;
3 = u ;
where we introduce a parametrization in real scalar elds,
t =
+ ie ' ;
u =
+ ie
:
1There is an additional factor 2 in the complex superpotential de ned in (2.3) of [28] compared to (3.27)
is at t; u ! +i 1, which corresponds to ;
= 0 and ';
!
When obtaining a scalar potential of the sub-truncations, one has to rst perform the
di erentiations in (2.7), and then identify the scalar elds by (2.8). Otherwise, one has to
employ the scalar potential formula from the canonical N = 1 formulation, [25],
h
P = eK
K p q (D p V)(D p V)
i
3VV ;
where the Kahler covariant derivative is
(2.10)
(2.11)
t0 =
u0 =
i
i
;
{ 4 {
3
The SU(3)-invariant Janus solutions
The scalar action for the SU(3)-invariant truncation, [25, 28], is obtained by (2.8),
S =
1
The Kahler potential is and the holomorphic superpotential is The scalar potential is obtained from
K =
3 log
i(t
t)
4 log [ i(u
u)] ;
V = 2g t2 + 6tu2 + 2m :
P = 2
"
4
3
(t
The scalar potential has N = 2 SU(3) U(
1
)-, N = 1 SU(3)-, and N = 1 G2-invariant
critical points, and more non-supersymmetric critical points.
For Janus solutions, we consider the AdS3-sliced domain wall for background,
ds2 = e2A(r)d2AdS3 + dr2 :
Then, we solve the supersymmetry variations of fermionic elds on the curved background,
as it was done on the Lorentzian AdS3-domain walls in [15] or on the Euclidean S3-domain
walls in [33]. The supersymmetry equations for the complex scalar elds,
(3.1)
(3.2)
(3.3)
(3.4)
(3.5)
(3.6)
A0 =
r
W 2
e 2A
l
2
;
are obtained where l is the radius of AdS and
=
1. The constant,
=
1, is not related
to the signs in (3.7). Reversing the sign of
merely generates a solution re ected in the
r-coordinate. In terms of the real scalar elds, the supersymmetry equations are
(3.7)
(3.8)
'0 +
0 +
Now we numerically obtain supersymmetric Janus solutions of the SU(3)-invariant
truncation.2 We closely follow the method employed in [15]. We set l = 1, g = 1, m = 1,
and
= +1 for all numerical solutions in this paper. As the supersymmetry equation
for the warp factor, (3.7), involves a square root, it involves a branch cut. In order to
obtain smooth Janus solutions, we should choose the upper sign for r > 0 and lower sign
for r < 0, [9, 10, 15, 20]. In order to avoid dealing with branch cuts, instead of the
rstorder supersymmetry equations, we solve the second-order equations of motion which are
presented in appendix A. To numerically solve the second-order equations of motion, we
should specify the initial conditions for the scalar elds, ', , , , the warp factor, A, and
their derivatives. We impose a smoothness condition at the interface,
Because of the smoothness condition, if we x initial conditions for the scalar elds, f'(0),
(0), (0), (0)g, initial condition for the warp factor, A(0) is automatically determined
through the supersymmetry equation for A(r) in (3.7). Similarly, f'0(0), 0(0), 0(0), 0(0)g
are also determined through the supersymmetry equations for the scalar
elds in (3.8).
Finally, we have only four free parameters, f'(0), (0), (0), (0)g, and when they are
xed, the rest of the parameters are xed with the smoothness condition, (3.9), and the
supersymmetry equations. When we obtain numerical solutions of the equations of motion,
we numerically check whether they satisfy the supersymmetry equations.
2In SO(8)-gauged N = 8 supergravity, the SU(3) U(
1
) U(
1
) sector was considered to study Janus
solutions in [15]. In dyonic ISO(7)-gauged N = 8 supergravity, one can turn o
the scalar
eld, , and
obtain the SU(3)
U(
1
)-invariant truncation [24, 25]. However, it is not consistent to turn o the scalar eld,
, and there is no SU(3) U(
1
) U(
1
)-invariant truncation. One can see this by examining the equation of
motion for . Moreover, it is group theoretically impossible as ISO(7) does not have an SU(3) U(
1
) U(
1
)
subgroup.
A0(0) = 0:
(3.9)
{ 5 {
and N = 1, SU(3) critical points are denoted by yellow and green dots, respectively. In the left
plot, the 3d N = 8 SYM phase is located at
plot of the superpotential. Following [34], in order to have both N = 2, SU(3) U(
1
) and
N = 1, SU(3) critical points depicted on contour plots, we replace some scalar elds by
or
= 2 + ( 1
= 2 + ( 1
'2
2) '2
1
, 2 are the values of the scalar elds at the N = 2, SU(3) critical point, and
'1,
1
, 1 at the N = 1, G2 critical point. In
gure 1, the contour plot with (3.10) is
presented on the left and (3.11) on the right. Regular, singular and critical-point Janus
solutions are depicted by blue, green and red lines, respectively.
First class of solutions are regular Janus solutions. Typical initial conditions give
solutions attracted to 3d N = 8 SYM phase which is non-conformal.3 Therefore, warp
factors of the solutions do not show AdS-behavior of constant slope. To the best of our
knowledge, they are the rst examples of non-conf ormal Janus solutions. We plot a typical
regular Janus solution by blue lines in gures 1 and 2.
The more interesting class of solutions arises as we approach the critical point. As we
have four scalar elds, we x initial conditions of three scalar elds, (0) =
3This is the main di erence between the solutions here and the known Janus solutions of four- and
vedimensional gauged supergravity where the solutions are mostly attracted to the conf ormal points which
are dual to 3d ABJM [15] and 4d N = 4 SYM [9, 10], respectively. They show constant slope behavior of
AdS solutions all along the ow.
{ 6 {
-40
-40
40 r
40
r
-40
-20
-40
-20
φ
-1
-2
-3
-4
f'(0) = ' + 2:9
, by their critical point values, and we have only one initial condition to x, '(0).
When '(0) < 'cr, we obtain the rst class of solutions, i.e. regular Janus solutions. We
found the value of 'cr to be
' + 1:6
When we ne-tune '(0) = 'cr, numerical solution starts approaching the critical point,
stays at the critical point for a while, and then move away to the 3d SYM phase. While
the solution is staying at the critical point, the warp factor exhibits the AdS-behavior with
a constant slope. The warp factor is given by
A(r)
W r;
(3.12)
(3.13)
where the slope, W , is the value of the superpotential at the critical point. We plot
a typical critical-point solution by red lines in
gures 1 and 2. On the contour plot in
gure 1, this solution takes the steepest descent from the SU(3) U(
1
) critical point to the
SYM phase.4
Regularity of the solution owing from the SU(3) U(
1
) critical point can be seen from
two aspects. First, as one can see from
gure 2 where the solution is plotted in red, the
numerical solution is well-de ned though all range of r. The singular solution we are about
to present is not well-de ned at some
nite r and diverges. Second, on the contour plot
of the superpotential in
1 [28].5 Any other direction at in nity corresponds to a singularity.
gure 1, the point dual to 3d N = 8 SYM phase is located at
The solution ows to the location of 3d N = 8 SYM phase, which is regular.
4The three classes of solutions we found are very much analogous to the solutions found in SO(8)-gauged
N = 8 supergravity in
gures 7 and 8 of [15]. Unlike our case, in SO(8)-gauged N = 8 supergravity, the
critical points exist in pairs: they are Z2-symmetric to each other.
5In the parametrization of [28], the 3d N = 8 SYM phase corresponds to z ! 1, 12 ! 1.
{ 7 {
denoted by green lines in
gures 1 and 2. Singular solutions diverge at
nite r. On the
contour plot in gure 1, they ow to in nity which is not the 3d SYM phase.
We have also obtained analogous solutions around the N = 1, SU(3) critical point.
However, as they are similar to the solutions we obtained, we do not present them in
the paper.
The analysis of the G2-invariant Janus solutions parallels the one in the previous section,
and we will be brief on details.
The scalar action for the G2-invariant truncation, [25], is obtained by (2.8),
Now we obtain numerical solutions in the same way we obtained the SU(3)-invariant
solutions. As we will see in detail, there are three classes of solutions: regular Janus
{ 8 {
S =
1
The Kahler potential is and the holomorphic superpotential is The scalar potential is obtained by
K =
7 log
i(t
t) ;
V = 14gt3 + 2m :
P = 2
"
4
7
(t
3W 2 :
The scalar potential has N = 1, G2-invariant critical point, and more non-supersymmetric
critical points. The 3d N = 8 SYM phase is at
The supersymmetry equations for the complex scalar eld,
and for the warp factor,
are obtained where
=
1. In terms of the real scalar elds, the supersymmetry equations
are
t0 =
i
e A
l
A0
K
A0 =
r
W 2
e 2A
l
2
;
0 +
4
7
l
k
k
= 0 :
(4.1)
(4.2)
(4.3)
(4.4)
(4.5)
(4.6)
(4.7)
is denoted by a yellow dot. The 3d N = 8 SYM phase is located at
g (Red), and f'(0) = ' + 2:0
solutions, singular solutions, and as a special case of regular Janus solutions, there are
Janus solutions owing from a critical point. The solutions are depicted on contour plot
of the superpotential in gure 3.
First class of solutions are regular Janus solutions. Typical initial conditions give
solutions attracted to 3d N = 8 SYM phase which is non-conformal. We plot a typical
solution by blue lines in gures 3 and 4.
The more interesting class of solutions arises as we approach the critical point. We x
(0) =
, the value at the critical point. When '(0) < 'cr, we obtain the rst class of
solutions, i.e. regular Janus solutions. We found the value of 'cr to be
' + 1:4
stays at the critical point for a while, and then move away to the 3d SYM phase. While
the solution is staying at the critical point, the warp factor exhibits the AdS-behavior with
a constant slope. The warp factor is given by
where the slope, W , is the value of the superpotential at the critical point. We plot a typical
critical-point solution by red lines in
gures 3 and 4. On the contour plot in
gure 3, this
solution takes the steepest descent from the G2 critical point to the SYM phase.
Lastly, when '(0) > 'cr, we start getting singular solutions. A singular solution is
denoted by green lines in gures 3 and 4.
HJEP04(218)9
A(r)
W r;
(4.9)
5
Conclusions
In this paper, we numerically obtained supersymmetric Janus solutions in dyonic
ISO(7)gauged N = 8 supergravity. Unlike the Janus solutions known in four-and ve-dimensional
gauged supergravity, our Janus solutions are mostly attracted to the non-conformal 3d
N = 8 SYM phase, which is the worldvolume theory on D2-branes. We also discovered
a number of solutions which stay at a critical point for a while and then move to the
SYM phase.
In dyonic ISO(7)-gauged N = 8 supergravity, there is also the N = 3, SO(4)
critical point. Interestringly, this critical point is non-supersymmetric in the SO(4)-invariant
truncation we consider. It is because the SO(4)-invariant gravitino becomes massive at
the critical point and breaks supersymmetry. If we consider the full N = 8 supergravity,
there are three gravitinos outside the SO(4)-invariant sector, and hence the critical point
is N = 3 supersymmetric, [25, 28]. Therefore, this critical point is not captured by the
superpotential or the supersymmetry equations of our truncations. For this reason, we
could not study Janus solutions of this critical point. It will be interesting to study Janus
solutions of this critical point by employing a larger truncation preserving supersymmetry
of this critical point. We present N = 1, SO(4)-invariant truncation in appendix B for
completeness.
Acknowledgments
We are grateful to Hyojoong Kim and Krzysztof Pilch for helpful discussions and
communications. We would like to thank Adolfo Guarino for explaining his works, and
Hyojoong Kim and Nakwoo Kim for collaboration on related subjects. We also would like
to thank Adolfo Guarino and Krzysztof Pilch for comments on the preprint. This work
was supported by the National Research Foundation of Korea under the grant
NRF2017R1D1A1B03034576.
The equations of motion
We present the equations of motion in this appendix. For the SU(3)-invariant truncation,
the equations of motion are given by
2A00 + 3A0A0 + 2
3A0A0 + 2
1
3
l
l
e 2A =
e 2A =
2
2
)
3
2
2
)
The SO(4)-invariant truncation
The scalar action for the SO(4)-invariant truncation, [25, 28], is obtained by (2.8),
S =
1
and the holomorphic superpotential is
The scalar potential is obtained from6
K =
6 log
i(t
t)
log [ i(u
u)] ;
V = 2g 4t2 + 3t2u + 2m :
P = 2
"
2
3
(t
4(u
The scalar potential has N = 1 G2- and N = 3 SO(4)-invariant critical points, and more
non-supersymmetric critical points.
6We have corrected some misprints in appendix C.1 of [28].
3
2
7
2
1
3
l
l
e 2A =
e 2A =
For the G2-invariant truncation, the equations of motion are given by
7
2
2
)
2A00 + 3A0A0 + 2
3A0A0 + 2
2
7 '0'0 +
This article is distributed under the terms of the Creative Commons
Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].
interface conformal eld theory, Phys. Rev. D 71 (2005) 066003 [hep-th/0407073] [INSPIRE].
HJEP04(218)9
[3] E. D'Hoker, J. Estes and M. Gutperle, Interface Yang-Mills, supersymmetry and Janus,
Nucl. Phys. B 753 (2006) 16 [hep-th/0603013] [INSPIRE].
[4] D. Gaiotto and E. Witten, Janus Con gurations, Chern-Simons Couplings, And The
theta-Angle in N = 4 Super Yang-Mills Theory, JHEP 06 (2010) 097 [arXiv:0804.2907]
[5] D. Bak, M. Gutperle and S. Hirano, A Dilatonic deformation of AdS5 and its eld theory
dual, JHEP 05 (2003) 072 [hep-th/0304129] [INSPIRE].
[6] E. D'Hoker, J. Estes and M. Gutperle, Ten-dimensional supersymmetric Janus solutions,
Nucl. Phys. B 757 (2006) 79 [hep-th/0603012] [INSPIRE].
[7] E. D'Hoker, J. Estes and M. Gutperle, Exact half-BPS Type IIB interface solutions. I. Local
solution and supersymmetric Janus, JHEP 06 (2007) 021 [arXiv:0705.0022] [INSPIRE].
[8] E. D'Hoker, J. Estes and M. Gutperle, Exact half-BPS Type IIB interface solutions. II. Flux
solutions and multi-Janus, JHEP 06 (2007) 022 [arXiv:0705.0024] [INSPIRE].
[9] A. Clark and A. Karch, Super Janus, JHEP 10 (2005) 094 [hep-th/0506265] [INSPIRE].
[10] M. Suh, Supersymmetric Janus solutions in ve and ten dimensions, JHEP 09 (2011) 064
[arXiv:1107.2796] [INSPIRE].
[11] E. D'Hoker, J. Estes, M. Gutperle and D. Krym, Exact Half-BPS Flux Solutions in
M-theory. I: Local Solutions, JHEP 08 (2008) 028 [arXiv:0806.0605] [INSPIRE].
[12] E. D'Hoker, J. Estes, M. Gutperle and D. Krym, Exact Half-BPS Flux Solutions in M-theory
II: Global solutions asymptotic to AdS7
S4, JHEP 12 (2008) 044 [arXiv:0810.4647]
[13] E. D'Hoker, J. Estes, M. Gutperle and D. Krym, Janus solutions in M-theory, JHEP 06
(2009) 018 [arXiv:0904.3313] [INSPIRE].
[14] E. D'Hoker, J. Estes, M. Gutperle and D. Krym, Exact Half-BPS Flux Solutions in M-theory
III: Existence and rigidity of global solutions asymptotic to AdS4
S7, JHEP 09 (2009) 067
[arXiv:0906.0596] [INSPIRE].
JHEP 06 (2014) 058 [arXiv:1311.4883] [INSPIRE].
[15] N. Bobev, K. Pilch and N.P. Warner, Supersymmetric Janus Solutions in Four Dimensions,
[16] K. Pilch, A. Tyukov and N.P. Warner, N = 2 Supersymmetric Janus Solutions and Flows:
From Gauged Supergravity to M-theory, JHEP 05 (2016) 005 [arXiv:1510.08090] [INSPIRE].
[17] P. Karndumri, Supersymmetric Janus solutions in four-dimensional N = 3 gauged
supergravity, Phys. Rev. D 93 (2016) 125012 [arXiv:1604.06007] [INSPIRE].
JHEP 12 (2017) 018 [arXiv:1709.09204] [INSPIRE].
[hep-th/0411077] [INSPIRE].
(2007) 056 [arXiv:0704.3740] [INSPIRE].
HJEP04(218)9
(2015) 020 [arXiv:1509.02526] [INSPIRE].
071 [arXiv:1509.07117] [INSPIRE].
(2016) 168 [arXiv:1605.09254] [INSPIRE].
a-maximization, JHEP 01 (2016) 048 [arXiv:1507.05817] [INSPIRE].
[1] J.M. Maldacena , The large N limit of superconformal eld theories and supergravity , Int. J.
[2] A.B. Clark , D.Z. Freedman , A. Karch and M. Schnabl , Dual of the Janus solution: An [18] P. Karndumri , Supersymmetric deformations of 3D SCFTs from tri-sasakian truncation , Eur. Phys. J. C 77 ( 2017 ) 130 [arXiv: 1610 .07983] [INSPIRE].
[19] P. Karndumri and K. Upathambhakul , Supersymmetric RG ows and Janus from type-II orbifold compacti cation , Eur. Phys. J. C 77 ( 2017 ) 455 [arXiv: 1704 .00538] [INSPIRE].
[20] M. Gutperle , J. Kaidi and H. Raj , Janus solutions in six-dimensional gauged supergravity , [21] J.H. Schwarz , Superconformal Chern-Simons theories , JHEP 11 ( 2004 ) 078 [22] D. Gaiotto and X. Yin , Notes on superconformal Chern-Simons-Matter theories , JHEP 08 [23] S. Minwalla , P. Narayan , T. Sharma , V. Umesh and X. Yin , Supersymmetric States in Large N Chern-Simons-Matter Theories , JHEP 02 ( 2012 ) 022 [arXiv: 1104 .0680] [INSPIRE].
[24] A. Guarino , D.L. Ja eris and O. Varela , String Theory Origin of Dyonic N = 8 Supergravity and Its Chern-Simons Duals , Phys. Rev. Lett . 115 ( 2015 ) 091601 [arXiv: 1504 .08009] [25] A. Guarino and O. Varela , Dyonic ISO(7) supergravity and the duality hierarchy , JHEP 02 [26] A. Guarino and O. Varela , Consistent N = 8 truncation of massive IIA on S6 , JHEP 12 [27] O. Varela , AdS4 solutions of massive IIA from dyonic ISO(7) supergravity , JHEP 03 ( 2016 ) [28] A. Guarino , J. Tarrio and O. Varela , Romans-mass-driven ows on the D2-brane , JHEP 08 [29] M. Fluder and J. Sparks , D2-brane Chern-Simons theories : F-maximization = [31] Y. Pang and J. Rong , Evidence for the Holographic dual of N = 3 Solution in Massive Type IIA , Phys. Rev. D 93 ( 2016 ) 065038 [arXiv: 1511 .08223] [INSPIRE].
[32] Y. Pang , J. Rong and O. Varela , Spectrum universality properties of holographic