#### Supersymmetric RG flows and Janus from type II orbifold compactification

Eur. Phys. J. C
Supersymmetric RG flows and Janus from type II orbifold compactification
Parinya Karndumri 0
Khem Upathambhakul 0
0 String Theory and Supergravity Group, Department of Physics, Faculty of Science, Chulalongkorn University , 254 Phayathai Road, Pathumwan, Bangkok 10330 , Thailand
We study holographic RG flow solutions within four-dimensional N = 4 gauged supergravity obtained from type IIA and IIB string theories compactified on T 6/Z2 × Z2 orbifold with gauge, geometric and non-geometric fluxes. In type IIB non-geometric compactifications, the resulting gauged supergravity has I S O(3) × I S O(3) gauge group and admits an N = 4 AdS4 vacuum dual to an N = 4 superconformal field theory (SCFT) in three dimensions. We study various supersymmetric RG flows from this N = 4 SCFT to N = 4 and N = 1 non-conformal field theories in the IR. The flows preserving N = 4 supersymmetry are driven by relevant operators of dimensions = 1, 2 or alternatively by one of these relevant operators, dual to the dilaton, and irrelevant operators of dimensions = 4 while the N = 1 flows in addition involve marginal deformations. Most of the flows can be obtained analytically. We also give examples of supersymmetric Janus solutions preserving N = 4 and N = 1 supersymmetries. These solutions should describe two-dimensional conformal defects within the dual N = 4 SCFT. Geometric compactifications of type IIA theory give rise to N = 4 gauged supergravity with I S O(3) U (1)6 gauge group. In this case, the resulting gauged supergravity admits an N = 1 AdS4 vacuum. We also numerically study possible N = 1 RG flows to non-conformal field theories in this case.
1 Introduction
Along the line of research within the context of the AdS/CFT
correspondence, the study of holographic RG flows is of
particular interest since the original proposal in [1]. There
is much work exploring this type of holographic solutions
in various space-time dimensions with different numbers of
supersymmetry. In this paper, we will particularly consider
holographic RG flows within three-dimensional
superconformal field theories (SCFTs) using gauged supergravity in four
dimensions. This might give some insight to the dynamics
of strongly coupled SCFTs in three dimensions and related
brane configurations in string/M-theory.
Most of the previously studied holographic RG flows have
been found within the maximal N = 8 gauged supergravities
[2–9]. Many of these solutions describe various deformations
of the N = 8 SCFTs arising from M2-brane world-volume
proposed in [10,11]. Similar study in the case of lower
number of supersymmetry is, however, less known. For example,
a number of RG flow solutions have appeared only recently
in N = 3 and N = 4 gauged supergravities [12–14]. In
this work, we will give more solutions of this type from the
half-maximal N = 4 gauged supergravity.
N = 4 supergravity allows for coupling to an
arbitrary number of vector multiplets. With n vector
multiplets, there are 6n + 2 scalars, 2 from gravity and 6n
from vector multiplets, parametrized by S L(2, R)/S O(
2
) ×
S O(6, n)/S O(
6
) × S O(n) coset. The N = 4 gauged
supergravity has been constructed for a long time in [15,16].
Gaugings constructed in [15] are called electric gaugings since
only electric n + 6 vector fields appearing in the ungauged
Lagrangian gauge a subgroup of S O(6, n). These vector
fields transform as a fundamental representation of S O(6, n).
The scalar potential of the resulting gauged supergravity
constructed in this way does not possess any AdS4 critical points
[17,18]. This is not the case for the construction in [16] in
which non-trivial S L(2, R) phases have been included.
The most general gauging in which both electric vector
fields and their magnetic dual can participate has been
constructed in [19] using the embedding tensor formalism. A
general gauge group is a subgroup of the full duality group
S L(2, R) × S O(6, n) with the vector fields and their
magnetic dual transforming as a doublet of S L(2, R). In this
work, we will consider N = 4 gauged supergravity obtained
from compactifications of type II string theories with various
fluxes given in [20]; for other work along this line, see for
example [21–23].
In [20], the scalar potential arising from flux
compactifications of type IIA and IIB theories on T 6/Z2 × Z2 within
a truncation to S O(
3
) singlet scalars has been considered,
and some AdS4 critical points together with their properties
have been given. For type IIB non-geometric
compactification, the vacuum structure is very rich even with only a few
number of fluxes turned on. Among these vacua, there exists
an N = 4 AdS4 critical point dual to an N = 4 SCFT with
S O(
3
) × S O(
3
) global symmetry. In addition, the full
classification of vacua from type IIA geometric compactification
has also been given. In this case, there exist a number of
stable non-supersymmetric AdS4 critical points as well as an
N = 1 AdS4 vacuum; see [24] for an N = 1 supersymmetric
AdS4 vacuum in massive type IIA theory.
We are particularly interested in N = 4 and N = 1 AdS4
critical points from these two compactifications. They
correspond to N = 4 and N = 1 SCFTs in three dimensions
with global symmetries S O(
3
) × S O(
3
) and S O(
3
),
respectively. We will look for possible supersymmetric
deformations within these two SCFTs in the form of RG flows to
non-conformal phases preserving some amount of
supersymmetry. These deformations are described by supersymmetric
domain walls in the N = 4 gauged supergravity. In the case
of N = 1 SCFT arising from massive type IIA theory,
nonsupersymmetric RG flows to conformal fixed points in the
IR have been recently found in [25].
For type IIB compactification, we will also consider
supersymmetric Janus solutions describing (1 + 1)-dimensional
conformal interfaces in the N = 4 SCFT. This type of
solutions breaks conformal symmetry in three dimensions
but preserves a smaller conformal symmetry on the
lowerdimensional interface. Similar to the RG flow solutions, there
are only a few examples of these solutions within the context
of four-dimensional gauged supergravities [14,26–28], see
also [29–31] for examples of higher-dimensional solutions.
They also play an important role in the holographic study
of interface and boundary CFTs [32,33]. We will give more
examples of these solutions in N = 4 gauged supergravity
obtained from non-geometric flux compactification.
The paper is organized as follows. In Sect. 2, we review
relevant formulas and introduce some notations for N = 4
gauged supergravity in the embedding tensor formalism. In
Sects. 3 and 4, we give a detailed analysis of supersymmetric
RG flow and Janus solutions obtained from non-geometric
type IIB compactification. Similar study of RG flows from
geometric type IIA compactification will be given in Sect.
5. We finally give some conclusions and comments on the
results in Sect. 6. We have also included an Appendix
containing more details on the conventions and the explicit form
of complicated equations.
2 N = 4 gauged supergravity coupled to six vector
multiplets
We first review relevant information and necessary formulas
of four-dimensional N = 4 gauged supergravity which is
the framework we use to find supersymmetric solutions. We
mainly follow the most general gauging of N = 4
supergravity in the embedding tensor formalism given in [19] in which
more details on the construction can be found. N = 4
supersymmetry allows for coupling the supergravity multiplet to
an arbitrary number of vector multiplets. We will begin with
a general formulation of N = 4 gauged supergravity with n
vector multiplets and later specify to the case of six vector
multiplets.
In half-maximal N = 4 supergravity, the supergravity
μ
multiplet consists of the graviton eμˆ , four gravitini ψμi, six
vectors Aμm , four spin- 21 fields χ i and one complex scalar τ
consisting of the dilaton φ and the axion χ . The complex
scalar can be parametrized by S L(2, R)/S O(
2
) coset. The
supergravity multiplet can couple to an arbitrary number n
of vector multiplets, and each vector multiplet contains a
vector field Aμ, four gaugini λi and six scalars φm . Similar
to the dilaton and the axion in the gravity multiplet, the 6n
scalar fields in these vector multiplets can be parametrized
by S O(6, n)/S O(
6
) × S O(n) coset.
We will use the following convention on various indices
appearing throughout the paper. Space-time and tangent
space indices are denoted, respectively, by μ, ν, . . . =
0, 1, 2, 3 and μˆ , νˆ , . . . = 0, 1, 2, 3. The S O(
6
) ∼ SU (
4
)
Rsymmetry indices will be described by m, n = 1, . . . , 6 for
the S O(
6
) vector representation and i, j = 1, 2, 3, 4 for the
S O(
6
) spinor or SU (
4
) fundamental representation. The n
vector multiplets will be labeled by indices a, b = 1, . . . , n.
All fields in the vector multiplets then carry an additional
index in the form of ( Aaμ, λia , φma ). Fermionic fields and
the supersymmetry parameters transform in the fundamental
representation of SU (
4
)R ∼ S O(
6
)R R-symmetry and are
subject to the chirality projections
γ5ψμi = ψμi, γ5χ i = −χ i , γ5λi = λi .
(
1
)
On the other hand, for the fields transforming in the
antifundamental representation of SU (
4
)R , we have
γ5ψμi = −ψμi , γ5χi = χi , γ5λi = −λi .
(
2
)
Gaugings of the matter-coupled N = 4 supergravity
can be efficiently described by using the embedding
tensor. This tensor encodes all the information as regards the
(
7
)
(
8
)
(
9
)
(
10
)
(
11
)
(
12
)
(
13
)
will set all vector fields and ξαM to zero from now on. In
particular, this simplifies the full quadratic constraint to
R
fα R[M N fβ P Q] = 0,
αβ fαM N R fβ P Q
R
= 0.
For electric gaugings, these relations reduce to the usual
Jacobi identity for f M N P = f+M N P as shown in [15,16].
The scalar coset manifold S L(2, R)/S O(
2
) × S O(6, n)/
S O(
6
) × S O(n) can be described by the coset
representative (Vα, VM A) with A = (m, a). The first factor can be
parametrized by
1
Vα = √Imτ
or equivalently by a symmetric 2 × 2 matrix
1
Mαβ = Re(VαVβ∗) = Imτ
|τ |2 Reτ
Reτ 1
.
Note also that Im(VαVβ∗) = αβ . The complex scalar τ can
also be written in terms of the dilaton φ and the axion χ as
τ = χ + i eφ .
For the S O(6, n)/S O(
6
) × S O(n) factor, we introduce
another coset representative VM A transforming by left and
right multiplications under S O(6, n) and S O(
6
) × S O(n),
respectively. From the splitting of the index A = (m, a), we
can write the coset representative as VM A m , VM a ).
= (AVsMatisfies the
Being an element of S O(6, n), the matrix VM
relation
m
ηM N = −VM VN
m
a a
+ VM VN .
As in the S L(2, R)/S O(
2
) factor, we can parametrize the
S O(6, n)/S O(
6
) × S O(n) coset in terms of a symmetric
matrix
m
MM N = VM VN
m
a a
+ VM VN .
We are now in a position to give the bosonic Lagrangian
with the vector fields and auxiliary two-form fields vanishing
1 1 1
e−1L = 2 R + 16 ∂μ MM N ∂μ M M N − 4(Imτ )2 ∂μτ ∂μτ ∗ −V
embedding of any gauge group G0 in the global or
duality symmetry group G = S L(2, R) × S O(6, n) in a G
covariant way. According to the analysis in [19], a
general gauging can be described by two components of the
embedding tensor ξ αM and fαM N P with α = (+, −) and
M, N = (m, a) = 1, . . . , n + 6 denoting fundamental
representations of S L(2, R) and S O(6, n), respectively. The
electric vector fields A+M = ( Aμm , Aaμ), appearing in the
ungauged Lagrangian, and their magnetic dual A−M form
a doublet under S L(2, R) denoted by AαM . A particular
electric-magnetic frame in which the S O(
2
)×S O(6, n)
symmetry, with S O(
2
) ⊂ S L(2, R), is manifest in the action can
always be chosen. In this frame, A+M and A−M have charges
+1 and −1 under this S O(
2
).
In general, a subgroup of both S L(2, R) and S O(6, n) can
be gauged, and the magnetic vector fields can also participate
in the gauging. Furthermore, it has been shown in [17], see
also [18], that purely electric gaugings do not admit AdS4
vacua unless an S L(2, R) phase is included [16]. The latter
is, however, incorporated in the magnetic component f−M N P
[19]. Accordingly, we will consider only gaugings involving
both electric and magnetic vector fields in order to obtain
AdS4 vacua. We will see that gauged supergravities obtained
from type II compactifications are precisely of this form.
The gauge covariant derivative can be written as
Dμ = ∇μ − g AαμM
αM N P tN P + g AμM(α β)γ ξγ M tαβ
where ∇μ is the usual space-time covariant derivative
including the spin connection. tM N and tαβ are S O(6, n) and
S L(2, R) generators which can be chosen as
(tM N )P Q = 2δ[QM ηN ]P , (tαβ )γ δ = 2δ(δα β)γ
with αβ = − βα and +− = 1. ηM N = diag(−1, −1, −1,
−1, −1, −1, 1, . . . , 1) is the S O(6, n) invariant tensor, and
g is the gauge coupling constant that can be absorbed in the
embedding tensor .
The embedding tensor component αM N P can be written
in terms of ξ αM and fαM N P components as
αM N P = fαM N P − ξα[N ηP]M .
To define a consistent gauging, the embedding tensor has to
satisfy a quadratic constraint. This ensures that the gauge
generators
XαM =
αM N P t N P − ξ Mβ tαβ
form a closed algebra.
In this work, we will consider solutions with only the
metric and scalars non-vanishing. In addition, we will consider
gaugings with only fαM N P non-vanishing. Therefore, we
(
3
)
(
4
)
(
5
)
(
6
)
where e is the vielbein determinant. The scalar potential can
be written in terms of scalar coset representatives and the
embedding tensor as
g2
V = 16
fαM N P fβ Q RS M αβ
where M M N is the inverse of MM N , and M M N P Q RS is
defined by
MM N P Q RS = mnpqrs VM m VN nVP pVQ q VR r VS
s
(
15
)
with indices raised by ηM N .
Before giving an explicit parametrization of the scalar
coset, we give fermionic supersymmetry transformations of
N = 4 gauged supergravity which play an important role in
subsequent analyses. These are given by
δψμi = 2Dμ i
2
− 3 g Ai1j γμ j ,
4
δχ i = i αβ Vα DμVβ γ μ i − 3 i g Ai2j j ,
δλia = 2i Va M DμVM i j γ μ j + 2i g A2a ji j .
The fermion shift matrices, appearing in fermionic mass-like
terms in the gauged Lagrangian, are defined by
Ai1j =
Ai2j =
j
A2ai =
αβ (Vα)∗Vkl M VN ik
αβ VαVkl M VN ik VP jl fβ M N P ,
αβ VαV Ma V Nik VP jk fβ M N P
VP jl fβ M N P ,
where VM i j is defined in terms of the ’t Hooft symbols Gimj
and VM m as
VM
i j
1
= 2 VM
m Gimj
and similarly for its inverse
M 1
V i j = − 2 V Mm (Gimj )∗.
V = − 31 Ai1j A1i j + 91 Ai2j A2i j + 21 A2ai j A2ai j .
Gimj convert an index m in vector representation of S O(
6
) to
an anti-symmetric pair of indices [i j ] in the SU (
4
)
fundamental representation. They satisfy the relations
1
Gmi j = −(Gimj )∗ = − 2 i jkl Gkml .
The explicit form of these matrices can be found in the
appendix.
We finally note the expression for the scalar potential
written in terms of A1 and A2 tensors as
It follows that unbroken supersymmetry corresponds to an
eigenvalue of Ai j , α, satisfying V0 = − 32 where V0 is the
α
1
value of the scalar potential at the vacuum, the cosmological
(
14
) constant.
We now consider the case of n = 6 vector multiplets.
Possible gauge groups are accordingly subgroups of S O(
6, 6
)
for ξαM = 0. Following [20], we restrict ourselves to
solutions preserving at least S O(3) subgroup of the full
gauge group. The residual S O(
3
) symmetry is embedded
in S O(
6, 6
) as a diagonal subgroup of S O(3) × S O(
3
) ×
S O(
3
) × S O(
3
) with the four factors of S O(
3
) being
subgroups of S O(
6
)× S O(
6
) ⊂ S O(
6, 6
). The 36 scalars within
S O(
6, 6
)/S O(
6
) × S O(
6
) coset transform as (
6, 6
) under
S O(6) × S O(
6
) compact subgroup. The above embedding
of S O(
3
) × S O(
3
) in S O(
6
) is given by
6 → (
3, 1
) + (
1, 3
).
This implies that the 36 scalars transform as
(
6, 6
) → 4 × (1 + 3 + 5)
(24)
(
25
)
(
26
)
(
27
)
(
28
)
under the unbroken S O(
3
) ∼ [S O(
3
) × S O(
3
) × S O(
3
) ×
S O(
3
)]diag. We see that there are four S O(
3
) singlets. We
will denote these scalars by (ϕ1, ϕ2, χ1, χ2) as in [20]. In
addition, we will also use the explicit parametrization given
in [20]. This gives the coset representative
VM
A
=
eT 0
BeT e−1
⊗ I3
where the two 2 × 2 matrices e and B are defined by
e = e 21 (ϕ1+ϕ2)
1 χ2
0 e−ϕ2
.
Explicitly, the S O(
6, 6
)/S O(6)×S O(
6
) coset representative
consisting of all S O(
3
) singlet scalars is given by
e 21 (ϕ1+ϕ2)
−e 21 (ϕ1+ϕ2)χ1
⎛ 0 0 0 ⎞
= ⎜⎜⎜ ee21 (21ϕ(1ϕ+1 +ϕ2ϕ)2χ) χ1χ22 ee21 (21ϕ(1ϕ−1 −ϕ2ϕ)2χ) 1 e− 21 (ϕ01+ϕ2) −e 21 (ϕ20−ϕ1)χ2 ⎟⎟⎟
⎝ 0 0 e 21 (ϕ2−ϕ1) ⎠
It should also be noted that there are two scalars which are
singlet under S O(
3
) × S O(
3
) ⊂ [S O(
3
) × S O(
3
)]diag ×
[S O(
3
) × S O(
3
)]diag as can be seen by taking the tensor
product of the representation 6 in (
24
) giving rise to two
singlets (
1, 1
) of S O(3)×S O(
3
). These two singlets correspond
to ϕ1 and ϕ2.
(
16
)
(
17
)
(
18
)
(
19
)
(
20
)
(
21
)
(
22
)
(
23
)
VM
A
⊗I3.
Similarly, the S L(2, R)/S O(
2
) coset representative will
be parametrized by
1
R = √2
.
(
34
)
Vα = eϕg/2
χg − i e−ϕg
1
.
where we have defined the scalar metric Ki j , which will play
a role in writing the BPS equations.
The four S O(
3
) singlet scalars in S O(
6, 6
)/S O(6) ×
S O(
6
) correspond to non-compact generators of S O(
2, 2
) ⊂
S O(
6, 6
) that commute with the S O(
3
) symmetry. It is
convenient to split indices M = ( A I ) for A = 1, 2, 3, 4 and
I = 1, 2, 3. This implies that the S O(
6, 6
) fundamental
representation decomposes as (
4, 3
) under S O(
2, 2
) × S O(3).
In terms of ( A I ) indices, the embedding tensor can be written
as
fαM N P = fα AI B J C K =
α ABC I J K
with α ABC =
straints read
α(ABC). In particular, the quadratic
conαβ
α AB C
β DEC = 0,
(α A[B
C
β)D]EC = 0.
The S O(
6, 6
) fundamental indices M, N can also be
decomposed into (m, m¯ ), m, m¯ = 1, 2, . . . , 6. In
connection with the internal manifold T 6/Z2 × Z2, the index m is
used to label the T 6 coordinates and split into (a, i ) such that
a = 1, 3, 5 and i = 2, 4, 6. Similar decomposition is also in
use for m¯ = (a¯ , i¯). All together, indices A, B can be written
as A = (
1, 2, 3, 4
) = (a, i, a¯ , i¯). Indices I, J = 1, 2, 3 label
the three T 2’s inside T 6 ∼ T 2 × T 2 × T 2.
The S O(
6, 6
) invariant tensor ηM N and its inverse are
chosen to be
ηM N = ηM N =
.
This leads to some extra projections on the negative and
positive eigenvalues of ηM N . For example, in order to compute
MM N P Q RS in the scalar potential defined by equation (
15
),
we need to project the second index of VM A by using the
projection matrix
(
31
)
(
32
)
(
33
)
(
29
)
Finally, we will also set the gauge coupling g = 21 as in [20].
3 RG flows from type IIB non-geometric compactification
We begin with a non-geometric compactification of type IIB
theory on T 6/Z2 ×Z2. This involves the fluxes of NS and RR
three-form fields (H3, F3) and non-geometric ( P, Q) fluxes.
This compactification admits a locally geometric description
although it is non-geometric in nature.
From the result of [20], the effective N = 4 gauged
supergravity theory is not unique. In this paper, we will only
consider the gauged supergravity admitting the maximally
supersymmetric N = 4 AdS4 vacuum. In this case, all the gauge
and non-geometric fluxes lead to the following components
of the embedding tensor:
f−i¯ j¯k¯ =
f−i¯ j¯k =
−444 = −λ, f+a¯b¯c¯ =
−244 = −λ, f+ab¯c¯ =
+333 = λ,
+133 = λ
(
35
)
for a constant λ. The first and second lines correspond
to (H3, F3) and ( P, Q) fluxes, respectively. As shown in
[20], the gauge group arising from this embedding tensor is
I S O(
3
) × I S O(
3
) ∼ [S O(
3
) T 3] × [S O(
3
) T 3]. This
gauge group is embedded in S O(
6, 6
) via the S O(
3, 3
) ×
S O(
3, 3
) subgroup.
Using this embedding tensor and the explicit form of the
scalar coset representative given in the previous section, we
find the scalar potential
1 eϕ1−3ϕ2−ϕg λ2[e2ϕ1 − 3e2ϕ2 + 6eϕ1+2ϕ2+ϕg
V = 32
− 18e3ϕ2+ϕg − 3e4ϕ2+2ϕg
− 2e2ϕ1+3ϕ2+ϕg (1 + 3χ12) + 3e2(ϕ1+ϕ2)(χ1 − χ2)2
This potential admits a trivial critical point at which all scalars
vanish. The cosmological constant is given by
3 2
V0 = − 8 λ .
At this critical point, we find the scalar masses as in Table 1.
In the table, we also give the corresponding dimensions of the
√
dual operators. The AdS4 radius is given by L = 2 λ 2 . Note
that we have used different convention for scalar masses from
that used in [20]. The masses given in Table 1 are obtained
by multiplying the masses given in [20] by 3.
This AdS4 critical point preserves N = 4
supersymmetry as can be checked from the Ai1j tensor. It should also
be emphasized here that this critical point has S O(
3
) ×
S O(
3
) symmetry which is the maximal compact subgroup
of I S O(
3
) × I S O(
3
) gauge group.
To set up the BPS equations for finding supersymmetric
RG flow solutions, we first give the metric ansatz
ds2 = e2A(r)dx12,2 + dr 2
where dx12,2 is the flat Minkowski metric in three dimensions.
We will use the Majorana representation for gamma
matrices with all γ μ real and γ5 purely imaginary. This choice
implies that i is a complex conjugate of i . All scalar fields
will be functions of only the radial coordinate r . To solve the
BPS conditions coming from setting δχ i = 0 and δλia = 0,
we need the following projection:
γrˆ i = ei i .
A = ±W, ei
W
= ± W
From the δψμi = 0 conditions for μ = 0, 1, 2, we find
where W = |W|, and denotes the r -derivative. These
equations are obtained by solving real and imaginary parts of
δψμi = 0 separately; see for more details [12,27]. The
superpotential W is defined by
where α is the eigenvalue of Ai1j corresponding to the
unbroken supersymmetry. We will choose a definite sign for the A
equation and ei such that the N = 4 critical point identified
with the N = 4 SCFT in the UV corresponds to r → ∞.
For all scalars non-vanishing, the N = 4 supersymmetry
is broken to N = 1 corresponding to the Killing spinor 1.
The superpotential for this unbroken N = 1 supersymmetry
is given by
(42)
(43)
(44)
(45)
and, as usual, the BPS equations from δχ i = 0 and δλia = 0
can be written as
φi = 2K i j ∂∂φWj .
K i j is the inverse of the scalar kinetic metric defined in (
30
).
The explicit form of these equations is rather complicated
and will not be given here. However, they can be found in
the appendix.
It is also straightforward to check that these BPS equations
solve the second order field equations. Furthermore, there
exist a number of interesting subtruncations keeping some
subsets of these S O(
3
) singlets. We will firstly discuss these
truncations and consider the full S O(
3
) singlet sector at the
end of this section.
3.1 RG flows with N = 4 supersymmetry
We begin with RG flow solutions preserving N = 4
supersymmetry to N = 4 non-conformal field theories in the IR.
The analysis of BPS conditions δψμi = 0, δχ i = 0 and
δλia = 0 shows that there are two possibilities in order to
preserve N = 4 supersymmetry. The first one is to truncate
out ϕ1,2 and χ1,2. The second possibility is to keep only the
three dilatons ϕg and ϕ1,2 by setting χg = χ1,2 = 0.
3.1.1 N = 4 RG flows by relevant deformations
From Table 1, we see that (ϕg, χg) correspond to relevant
deformations by operators of dimensions 1 or 2. The BPS
equations admit a consistent truncation to these two scalars.
By setting ϕ1,2 = χ1,2 = 0, we obtain a set of simple BPS
equations
ϕg
λe− 2 (e2ϕg + e2ϕg χg2 − 1)
ϕg = − 2√2
(1 + eϕg )2 + e2ϕg χg2
ϕg
λe− 2
√2
χg = −
A =
ϕg
λe− 2
4√2
χg
(1 + eϕg )2 + e2ϕg χg2
(1 + eϕg )2 + e2ϕg χg2.
,
,
Since (ϕg, χg) are scalars in S L(2, R)/S O(
2
), they are
S O(
6, 6
) singlets and hence S O(3) × S O(
3
) invariant. All
solutions to these equations then preserve the full S O(
3
) ×
S O(
3
) symmetry. Moreover, equations δλia = 0 are
identically satisfied, and it can be checked that N = 4
supersymmetry is unbroken since equations δχ i = 0 and δψμi = 0
hold for all i satisfying the γr projector (39). We should
ˆ
clarify here the convention on the number of
supersymmetry. In four dimensions, the γrˆ projector reduce the number of
supercharges from 16 to 8. The latter corresponds to N = 4
supersymmetry in three dimensions. On the other hand, the
AdS4 vacuum preserves all 16 supercharges corresponding
to N = 4 superconformal symmetry in three dimensions
containing 8 + 8 = 16 supercharges.
We begin with an even simpler solution with χg = 0
which, from the above equations, is clearly a consistent
truncation. In this case, we end up with the BPS equations
λ ϕg
ϕg = − √ e− 2 (eϕg − 1),
2 2
λ ϕg
√ e− 2 (1 + eϕg ).
A = 4 2
The solution to these equations is easily found to be
ϕg = ln[e 2r√λ2 +C − 1] − ln[e 2r√λ2 +C + 1],
with C being an integration constant. The additive integration
constant for A has been neglected since it can be absorbed
by scaling dx12,2 coordinates. In addition, the constant C can
also be removed by shifting the r coordinate.
At large r , we find, as expected for dual operators of
dimensions = 1, 2,
The solution is singular as r → − 2√λ2C since ϕg → −∞.
Near this singularity, we find
ϕg ∼ A ∼ ln r +
2√2C
λ
.
(46)
(47)
(48)
(49)
(50)
(51)
(52)
(53)
(54)
(55)
(56)
(57)
(58)
(59)
The scalar potential is bounded above with V → −∞.
Therefore, the singularity of this solution is physical by the
criterion given in [34]. The solution then describes an RG
flow from the dual N = 4 SCFT to an N = 4 non-conformal
field theory with unbroken S O(
3
) × S O(
3
) symmetry. The
metric in the IR is given by
ds2 = (λr + 2√2C )2dx12,2 + dr 2
where we have absorbed some constants to d x12,2 coordinates.
We then consider possible flows solution with χg = 0. By
introducing a new variable ρ via
dρ
dr =
χg
1 − C χg +
1 − χg(2C + χg)
we find the following solution to Eqs. (46)–(48):
1
ϕg = − 2 ln[1 − 2C χg − χg2],
A = − ln χg + 41 ln[1 − 2C χg − χg2]
1 − 2C χg − χg2 ,
1
+ 2 ln 1 − C χg +
3
ρλ[1 − χg(2C + χg)] 4
1
= 4(
2
) 4 (C + χg −
1 + C 2 +
×
×2 F1
1 + C 2)
√1 + C 2(C + χg) 43
1 + C 2
1 3 5 χg +
, , ,
4 4 4
√1 + C 2 − C
2√1 + C 2
where 2 F1 is the hypergeometric function.
The solution interpolates between the N = 4 AdS4
vacuum as r → ∞ and a singular geometry at a finite value of r .
There are two possibilities for the IR singularities. The first
one is given by
χg ∼ χ0, ϕg ∼ −2 ln
A ∼ √
χ0
1 + χ0
ln[√2r λ(1 + χ0 ) − 4χ0C ]
2
,
where χ0 is a constant. In this case, we have ϕg → ∞ and
χg → constant as √2λr (1 + χ02) → 4χ0C . It should be
noted here that the constant C in these equations is not the
same as in the full solution given in (57)–(59).
Another possibility is given by
A ∼ ln(√2λr + 4C ).
C
˜
ϕg ∼ 2 ln(√2λr + 4C ), χg ∼ 4C + √2λr
,
In this case, as √2r λ → −4C , we have ϕg → −∞ and
χg → ±∞ depending on the sign of the constant C˜ . Both of
these singularities lead to V → −∞, so they are physical.
3.1.2 N = 4 RG flows by relevant and irrelevant
deformations
We now consider RG flows with N = 4 supersymmetry
with χg = χ1,2 = 0. Recall that ϕ1 and ϕ2 are S O(
3
) ×
S O(
3
) singlets, we still have solutions with S O(
3
) × S O(
3
)
unbroken along the flows. It should also be noted that the
truncation χ1,2 = 0 is consistent only for χg = 0. This
implies that N = 4 supersymmetry does not allow turning
on the operators dual to χg and ϕ1,2 simultaneously. It would
be interesting to see the implication of this in the dual N = 4
SCFT.
In this case, the BPS equations reduce to
ϕ˜1 = ϕ1 − ϕ2, ϕ˜2 = ϕ1 + ϕ2
×(3eϕ2 − eϕ1 − 3e2ϕ2+ϕg + eϕ1+3ϕ2+ϕg ),
λ
√ (eϕ1 − eϕ2 − e2ϕ2+ϕg + eϕ1+3ϕ2+ϕg ),
ϕ1 = 4 2
λ
√ (eϕ2 − eϕ1 − e2ϕ2+ϕg + eϕ1+3ϕ2+ϕg ),
ϕ2 = 4 2
λ
A = − √ e 21 (ϕ1−3ϕ2−ϕg)
8 2
×eϕ1 − 3eϕ2 − 3e2ϕ2+ϕg + eϕ1+3ϕ2+ϕg ).
These equations can be analytically solved by introducing
new variables
in terms of which the BPS equations become
λ
√ e 21 (ϕ˜1+ϕg)(eϕ˜1 − 1),
ϕ˜1 = 2 2
λ
√ e 21 (ϕ˜2−ϕg)(eϕ˜2 − 1),
ϕ˜2 = 2 2
λ ϕg
ϕg = 4√2 e− 2 (3e ϕ˜22 − e 23 ϕ˜2 − 3e 21 ϕ˜1+ϕg + e 23 ϕ˜1+ϕg ),
Near the AdS4 critical point, we have ϕ˜1 ∼ ϕ˜2 ∼ 0, which
requires that C1 = 0. This choice leads to ϕ˜2 = ϕ˜1, which
implies ϕ2 = 0 and ϕg = 0. We see that the flow does not
involve ϕg and is driven purely by an irrelevant operator of
dimension 4 dual to ϕ1. In this case, the N = 4 SCFT dual
to the AdS4 vacuum is expected to appear in the IR. Note
also that equation (69) is consistent for ϕg = 0 if and only if
ϕ˜2 = ϕ˜1 as being the case here.
Finally, we can solve Eq. (67) for ϕ˜1(r )
2λ√r2 = 2e− ϕ˜21 + ln(1 − e− ϕ˜21 ) − ln(1 + e− ϕ˜21 ) + C. (77)
By combining all of these equations, we find that
d A 1 dϕg 3 − eϕ˜1
dϕ˜1 − 2 dϕ˜1 = 2(eϕ˜1 − 1)
d A
1 dϕg 3 − eϕ˜2
dϕ˜2 + 2 dϕ˜2 = 2(eϕ˜2 − 1)
,
which can be solved by the following solution:
3
ϕg = 2 (ϕ˜1 − ϕ˜2) − ln(1 − eϕ˜1 ) + ln(1 − eϕ˜2 ),
A =
ϕg 3
2 − 2 ϕ˜1 + ln(1 − eϕ˜1 ).
In this solution, we have fixed the integration constant for ϕg
to zero since at the AdS4 critical point ϕg = ϕ˜1 = ϕ˜2 = 0.
The integration constant for A is irrelevant.
Combining Eqs. (67) and (68), we obtain after substituting
for ϕg
(60)
(61)
(62)
(63)
(64)
(65)
(66)
(67)
(68)
(69)
(70)
(71)
(72)
(73)
(74)
(75)
(76)
The solution is singular as r → 2√λ2C . Near this singularity,
the solution becomes
2
ϕ˜1 ∼ ϕ˜2 ∼ − 3 ln
3 λr
2 C − 2√2 ,
1 1
A ∼ − 2 ϕ˜1 ∼ 3 ln
3 λr
2 C − 2√2 .
This singularity leads to V → ∞, so the solution is
unphysical.
We end the discussion of this truncation by giving some
comments on possible subtruncations. From Eqs. (67) and
(68), we easily see that setting ϕ˜1 = 0 or ϕ˜2 = 0 is a
consistent truncation. This is equivalent to setting ϕ2 = ±ϕ1. In
this case, the solution is found to be
e−ϕ1 − C1e3ϕ1
ϕg = ± ln
,
4λ√ρ2 = e−ϕ1 + 21 ln(1 − e−ϕ1 ) − 21 ln(1 + e−ϕ1 ) + C
where the new radial coordinate ρ is defined by dρ =
ϕg
e− 2 dr .
We see that in this case ϕg is non-trivial along the flow. In
order to make the solution approach the AdS4 critical point
with ϕg ∼ ϕ1 ∼ 0, we need to choose C1 = 1. This gives
4√2C with ϕ1 → ∞. In
3λ
ϕg = ± ln cosh ϕ1.
The solution is singular for ρ →
this limit, we find
3λρ
ϕg ∼ ±ϕ1 ϕ1 ∼ − ln C − 4√2 ,
1 3λρ
A ∼ 2 ln C − 4√2 .
Both of these singularities lead to V → ∞, so they are also
unphysical.
3.2 RG flows with N = 1 supersymmetry
We now consider a class of RG flow solutions preserving
N = 1 supersymmetry and breaking the S O(
3
) × S O(
3
)
to its diagonal subgroup. This is achieved by turning on the
marginal deformations corresponding to χ1 and χ2 to the
solutions. As in the N = 4 case, there is a consistent
subtruncation to only irrelevant and marginal deformations with
only ϕ1 and χ1 non-vanishing. We will consider this case
first and then look for the most general solutions with all six
S O(
3
) singlet scalars non-vanishing. It should be noted that
the truncation with only ϕ2 and χ2 non-vanishing is not
consistent. This is also an interesting feature to look for in the
dual field theory.
(78)
3.2.1 N = 1 RG flows by marginal and irrelevant
deformations
By setting ϕg = χg = ϕ2 = χ2 = 0 in the BPS equations,
we obtain
,
,
We are not able to analytically solve these equations in full
generality, so we will look for numerical solutions in this
case.
Note that further truncation to only ϕ1 gives rise to the
following BPS equations:
λ λ
√ e ϕ21 (eϕ1 − 1) and A = − √ e ϕ21 (eϕ1 − 3),
ϕ1 = 2 2 4 2
(82)
(83)
(84)
(85)
(87)
(79)
(80)
(81)
where χ0 is a constant. This singularity leads to V → ∞,
implying that it is unphysical. We have in addition checked
this by a numerical analysis which consistently shows a
diverging scalar potential near the singularity.
with the solution
3
A = − 2 ϕ1 + ln(1 − eϕ1 ),
λr
2√2 = 2e− ϕ21 + ln 1 − e− ϕ21
This is nothing but the solution of the previous section for
ϕ˜2 = ϕ˜1. Therefore, we will not further discuss this solution.
For non-vanishing χ1, we need to find the solutions
numerically. An example of these solutions is given in Fig. 1.
The asymptotic behavior of this solution can be
determined from the BPS equations at large ϕ1 as follows:
2
χ1 ∼ χ0, ϕ1 ∼ − 3 ln r λ 2 + 18χ02 − 4C1 ,
1
A ∼ 3 ln r λ 2 + 18χ02 − 4C1
− ln 1 + e− ϕ21 . (86)
After considering various consistent truncations, we end this
section by considering N = 1 RG flow solutions with all
six S O(
3
) singlet scalars turned on. The resulting RG flows
will be driven by all types of possible deformations namely
marginal, irrelevant and relevant. In this case, we need to
use a numerical analysis due to the complexity of the full
set of BPS equations given in the Appendix. Similar to the
analysis of [9], there could be many possible IR singularities
due to the competition between various deformations both
by operators and vacuum expectation values (vev) present
in the UV N = 4 SCFT. Some examples of these solutions
are given in Fig. 2. In the figure, we have given solutions for
three different values of the flux parameter λ for comparison.
From Fig. 2, we see a singularity in the IR end of the
flow while near r → ∞ the flow approaches the UV N = 4
AdS4. The numerical analysis shows that the singularity is
of a bad type according to the criterion of [34] since it leads
to V → ∞.
4 Supersymmetric Janus solutions
In this section, we look at another type of solutions with an
AdS3-sliced domain wall ansatz, obtained by replacing the
flat metric dx12,2 by an AdS3 metric of radius ,
ds2 = e2A(r) e 2ξ dx12,1 + dξ 2
+ dr 2.
(88)
The solution, called Janus solution, describes a
conformal interface of co-dimension one within the SCFT dual
to the AdS4 critical point. This solution breaks the
threedimensional conformal symmetry S O(
2, 3
) to that on the
(1 + 1)-dimensional interface S O(
2, 2
).
In this case, the resulting BPS equations will get modified
compared to the RG flow case. First of all, the analysis of
δψμi = 0 equations requires an additional γξˆ projection
5 r
(89)
(90)
(91)
1
2
3
4
while the γrˆ projector in δχ i = 0 and δλia = 0 equations is
still given by Eq. (39) but with the phase ei modified to
γξˆ i = i κei i
ei
W
= A + iκ e−A
.
A 2
+ 12 e−2A = W 2 .
Furthermore, the integrability conditions for δψi
0ˆ,1ˆ = 0
equations lead to
As expected, these two equations reduce to A = ±W and
ei = WA = ± WW in the limit → ∞.
The constant κ, with κ2 = 1, imposes the chirality
condition on the Killing spinors corresponding to the unbroken
supersymmetry on the (1 + 1)-dimensional interface. The
detailed analysis of these equations can be found for example
in [27]. Unlike the RG flow case, the Killing spinors depend
on both r and ξ coordinates; see for more details [26].
We have seen that the analysis of RG flow solutions with
all six S O(
3
) singlet scalars turned on involves a very
complicated set of BPS equations. Since the BPS equations for
supersymmetric Janus solutions are usually more
complicated than those of the RG flows, we will not perform the
full analysis with all S O(
3
) singlet scalars but truncate the
BPS equations to two consistent truncations, with (ϕg, χg)
and (ϕ1, χ1) non-vanishing. As in other cases studied in
[14,26,27], truncations to only dilatons or scalars without
the axions or pseudoscalars are not consistent with the Janus
BPS equations, or equivalently Janus solutions require
nontrivial pseudoscalars.
4.1 N = 4 Janus solution
We first consider the Janus solution with only the dilaton and
axion in the gravity multiplet non-vanishing. In this case,
the BPS conditions δλia = 0 are automatically satisfied by
(a) Solution for
setting ϕ1,2 = χ1,2 = 0. By using the phase (90) in δχ i = 0
equations and separating real and imaginary parts, we obtain
the following BPS equations:
A ∂ W
ϕg = −4 W ∂ ϕg
− 4κ e−ϕg e−A ∂ W
W ∂ χg
=
−2 A (e2ϕg − 1 + 2χg2e2ϕg ) − 4κ eϕg−Aχg
[(1 + eϕg )2 + χg2e2ϕg ]
,
(92)
=
χg = −4 WA e−2ϕg ∂∂χWg
+ 4κ e−ϕg e−A ∂ W
W ∂ ϕg
2κ e−A−ϕg (e2ϕg − 1 + χ 2e2ϕg ) − 4 χg A
g
[(1 + eϕg )2 + χg2e2ϕg ]
,
− 3λ22 e−ϕg [(1 + eϕg )2 + χg2e2ϕg ]
(93)
(94)
where we have also included the gravitini equations from
(91). In terms of the superpotential
W = 4√λ 2 e− ϕ2g
(1 + eϕg )2 + χ 2e2ϕg ,
g
(95)
these equations take a similar form as in the other
fourdimensional Janus solutions in [14, 26, 27]. These equations
solve all the BPS conditions for any i , i = 1, 2, 3, 4.
Therefore, any solutions to these equations will preserve N = 4
supersymmetry. We solve these equations numerically with
an example of the solutions given in Fig. 3.
From Fig. 3, we see that the solution interpolates between
N = 4 AdS4 vacua at both r → ±∞. This solution is
then interpreted as a (1 + 1)-dimensional conformal interface
within the N = 4 SCFT. The interface preserves N = (
4, 0
)
supersymmetry on the interface due to the sign choice κ = 1,
g r
0.012
0.010
0.008
0.006
0.004
0.002
0.002
20
10
10
and S O(
3
)× S O(
3
) symmetry remains unbroken throughout
the solution.
4.2 N = 1 Janus solution
The truncation keeping only ϕ1 and χ1 is still consistent in
the case of Janus BPS equations. In contrast to the previous
truncation, any solutions to these equations will break N = 4
supersymmetry to N = 1 and preserve only S O(
3
) diagonal
subgroup of the full S O(
3
) × S O(
3
) symmetry of the N = 4
AdS4 vacuum.
The real superpotential for this truncation is given by
λ ϕ1
W = 4 2
√ e 2
in terms of which the BPS equations can be written as
4 A ∂ W 4 κe−ϕ1 e−A ∂ W
ϕ1 = − 3 W ∂ϕ1 − 3 W ∂χ1
2 A (4e2ϕ1 − 3 − 9χ12e2ϕ1 − e2ϕ1 ) − 12κeϕ1−Aχ1
[(eϕ1 − 3)2 + 9χ12e2ϕ1 ]
χ1 = − 43 WA e−2ϕ1 ∂∂χW1 + 43 κe−ϕ1 e−WA ∂∂ϕW1
2κe−A−ϕ1 (3 − 4eϕ1 + e2ϕ1 + 9χ12e2ϕ1 ) − 12 χ1 A
[(eϕ1 − 3)2 + 9χ12e2ϕ1 ]
Unlike the previous case, an intensive numerical search has
not found any solutions interpolating between AdS4 vacua
in the limits r → ±∞. All of the solutions found here are
singular Janus in the sense that they connect singular domain
walls at two finite values of the radial coordinate. We give an
example of these solutions in Fig. 4.
This solution could be interpreted as a conformal
interface between two N = 1 non-conformal phases of the dual
N = 4 SCFT. However, the singularities are of the bad type.
An uplift to type IIB theory would be needed in order to
decide whether the solution is physically acceptable in the
ten-dimensional context.
5 RG flows from type IIA geometric compactification
We now carry out a similar analysis for a geometric
compactification of type IIA theory. The procedure is essentially the
same, so we will omit unnecessary details. In this case, the
compactification only involves gauge (H3, F0, F2, F4, F6)
and geometric (ω) fluxes. However, the fluxes are more
complicated and lead to many components of the embedding
tensor compared to the type IIB case
√6
Hi jk ∼ f−a¯b¯c¯ = −333 = 3 λ,
In the above equations, we have also given the form field
corresponding to each flux component.
The resulting gauged N = 4 supergravity has a
nonsemisimple group I S O(
3
) U (
1
)6 and admits the minimal
N = 1 AdS4 vacuum at which the gauge group is broken
down to S O(
3
) compact subgroup. The corresponding
superpotential for the unbroken N = 1 supersymmetry is given
by
1 r
2.0
The scalar potential can be written in terms of W = |W| as
1 K i j ∂ W ∂ W 3
V = − 2 ∂φi ∂φ j − 4
W 2.
Its explicit form is given in the appendix.
When all scalars vanish, there is an N = 1 AdS4 vacuum
with the cosmological constant
V0 = −λ2.
The six scalars have squared masses as follows:
m2 L2 : 0, −2, 4 ± √
1
6, (47 ±
3
√159).
All of these values are in agreement with [20] after changing
to our convention including a factor of 3.
As in the type IIB case, the BPS equations obtained from
supersymmetry variations can be written as
A = W, ϕi = K i j ∂∂φWj .
However, the resulting equations are much more complicated
than those from type IIB compactification. We will then not
give them in this paper. Furthermore, we have not found any
consistent subtruncation within this set of equations. In the
following, we will only give examples of holographic RG
flows from the N = 1 SCFT dual to the above AdS4 critical
point to non-conformal N = 1 field theories in the IR. These
A r
1.1
numerical solutions are shown in Fig. 5 with three different
values of the flux parameter λ as in the IIB case.
As in the IIB case, we have numerically analyzed the scalar
potential near the singularity and found that it leads to V →
∞, which implies the singularity is unphysical.
6 Conclusions and discussions
We have found many supersymmetric RG flows and
examples of Janus solutions in N = 4 gauged supergravities
obtained from flux compactifications of type II string
theories. These solutions describe supersymmetric deformations
and conformal interfaces within the dual N = 4 and N = 1
SCFTs in three dimensions. Many of the flow solutions have
been obtained analytically which should be useful for further
investigation.
In type IIB non-geometric compactification, the gauged
supergravity has I S O(
3
) × I S O(
3
) gauge group and admits
an N = 4 AdS4 vacuum dual to an N = 4 SCFT with global
symmetry S O(
3
) × S O(
3
). We have found two classes of
supersymmetric RG flows. The first one preserves N = 4
supersymmetry, and the global S O(
3
) × S O(
3
) symmetry
is unbroken. This type of solutions can be obtained by
turning on only the dilaton and axion in the gravity multiplet
dual to relevant operators of dimensions = 1, 2. In this
case, the flows are accordingly driven by relevant operators.
Another possibility for preserving N = 4 supersymmetry
is to truncate out all axions or pseudoscalars. The resulting
RG flows are driven by relevant and irrelevant operators of
dimensions = 1, 2 and = 4, respectively. When the
axions in the vector multplets, corresponding to marginal
deformations, are turned on, the flows break N = 4
supersymmetry to N = 1 and break S O(
3
) × S O(
3
) symmetry to
their S O(
3
) diagonal subgroup. We have given numerically
the flows driven by marginal and irrelevant operators and the
most general deformations in the presence of all types of
deformations, relevant, marginal and irrelevant. It has been
pointed out in [20] that the vacuum structure of type IIB
comg r
0.002
0.001
0.001
0.002
0.003
0.004
0.005
pactification is very rich. The solutions found in this paper
show that the number of supersymmetric deformations of
these vacua is also enormous.
Within this type IIB compactification, we have also given
Janus solutions preserving N = 4 and N = 1
supersymmetry. These correspond to (1 + 1)-dimensional conformal
interfaces preserving S O(
3
) × S O(
3
) and S O(
3
) symmetry.
For the N = 4 solution, we have given a numerical solution
interpolating between AdS4 vacua on the two sides of the
interface, called regular Janus. This solution gives a
holographic dual of a conformal interface in N = 4 SCFT. For
the N = 1 case, we have not found this type of solutions
but the singular Janus, interpolating between N = 1
nonconformal phases of the dual N = 4 SCFT. The situation is
very similar to the N = 1 Janus solutions studied in [14].
It would be interesting to have a definite conclusion about
the existence of regular Janus solutions in these two N = 4
gauged supergravities.
In this non-geometric compactification, it is useful to give
some comments about the holographic interpretation of the
results. Due to its non-geometric nature, the stringy origin of
the N = 4 gauged supergravity is presently not well
understood. This makes the meaning of the resulting solutions in
terms of RG flows in the dual SCFT unclear. However,
working in four-dimensional gauged supergravity has an obvious
advantage in the sense that the whole formulation of N = 4
gauged supergravity is virtually unchanged for all gaugings
from both geometric and non-geometric compactifications.
Therefore, the approach used here can be carried out for all
other gaugings regardless of their higher-dimensional
origins. On the other hand, the full interpretation of the results
in higher-dimensional contexts calls for further study.
Hopefully, the results presented here could be useful along this
line of investigations.
We have also carried out the same analysis in a
geometric compactification of type IIA theory resulting in N = 4
gauged supergravity with I S O(
3
) U (
1
)6. The gauged
supergravity admits an N = 1 AdS4 vacuum dual to an N =
1 SCFT in three dimensions. Due to the lack of further
consistent subtruncation, we have numerically found examples
of holographic RG flows to N = 1 non-conformal field
theories. Similar to the solutions in the type IIB case, the flows
are driven by relevant, marginal and irrelevant operators in a
more complicated manner. It should be pointed out that the
massless scalars dual to marginal deformations considered in
this paper are not the Goldstone bosons corresponding to the
symmetry breaking I S O(
3
) × I S O(
3
) → S O(
3
) × S O(
3
)
and I S O(
3
) U (
1
)6 → S O(
3
). The Goldstone bosons
transform non-trivially under the residual symmetry groups
S O(
3
) × S O(
3
) and S O(
3
) while the massless scalars
considered in the solutions are singlets. Therefore, they are truly
marginal deformations in the N = 1 and N = 4 SCFTs. Note
also that, in the type IIB case, these marginal deformations
break N = 4 supersymmetry in consistent with the fact that
all maximally supersymmetric AdS4 vacua of N = 4 gauged
supergravity have no moduli preserving N = 4
supersymmetry [35].
There are many possibilities for further investigations.
First of all, it would be interesting to identify the N = 1
and N = 4 SCFTs dual to the N = 1 and N = 4 AdS4
vacua. This should allow us to identify the dual operators
driving the RG flows obtained holographically in this paper.
It could be interesting to look for more general Janus
solutions in type IIB compactification with more scalars turned
on and also look for similar solutions in type IIA
compactification. Another direction would be to uplift the solutions
found here to ten dimensions. This could be used to identify
the g00 component of the ten-dimensional metric and checked
whether the unphysical singularities by the criterion of [34]
are physical by the criterion of [36]. Finally, it would be of
particular interest to further explore type IIB
compactification with more general fluxes than those considered in [20].
This could enlarge the solution space of both AdS4 vacua and
their deformations including possible flow solutions between
two AdS4 vacua. We leave these issues for future work.
Acknowledgements This work is supported by The Thailand Research
Fund (TRF) under Grant RSA5980037.
Open Access This article is distributed under the terms of the Creative
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ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
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Appendix A: Useful formulas
In this appendix, we collect all of the conventions about ’t
Hooft symbols, the scalar potential coming from type IIA
geometric compactification and complicated BPS equations
arising from type IIB non-geometric compactification with
all S O(
3
) singlet scalars non-vanishing.
A.1: ’t Hooft symbols
To convert an S O(
6
) vector index m to a pair of
antisymmetric SU (
4
) indices [i j ], we use the following ’t Hooft
symbols:
Gi j
+ e2ϕgχg2 + e2ϕ2χ22(−1 + e2ϕgχg2))
− 6e2(ϕ1+ϕ2)χ1(−e2ϕg(−1 + e2ϕ2)χg
− 2e2(ϕ2+ϕg)χ2χg + e2ϕgχg2 + e2ϕ2χ22(1 + e2ϕgχg2))
− e4ϕ2+2ϕgχ24χg + e4ϕ2χ25(1 + e2ϕgχg2)
+ 2e2ϕ2χ23(1 + eϕ2+ϕg
+ e2ϕgχg2) + χ2(1 + 2eϕ2+ϕg − 2e3ϕ2+ϕg
+ e2ϕ1+ϕ2 )χ22χg + e2ϕ1+4ϕ2 χ25(1 + e2ϕg χ 2)
g
A.3: Scalar potential from type IIA compactification
The scalar potential obtained from a geometric
compactification of type IIA theory is given by
1
V = 192
eϕ1−3ϕ2−ϕg λ2[20e2ϕ1+4ϕ2
+ 5e2(ϕ1+ϕ2+ϕg)(1 + 2√
15χ2) − 10χg )
√
15χ2 − 4
15χg ) + 5[4
15
√
+ 15χ24 − 8χ2χg + 4χg + χ22(22 − 4
2
15χ2 + 10
√
15χ25
15χg − 4χ23(3
√
√
+ 8χ2(3χ1 + χg )(18 + 3
√
+ 4
− 5χ24(−21 + 12
√
15χ1
√
15χg ) + 4[9χ1( 15 + χ1)
√
15χ1 +
+ 6χ1χg + χg2] + χ22[−9 − 40 15χg
√ 2
+ 60[9χ12 − 2χ1( 15 − 3χg ) + χg ]]]].
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