Holographic RG flows and symplectic deformations of $N=4$ gauged supergravity

Parinya Karndumri

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

E-mail<EMAIL_ADDRESS>

###### Abstract

We study four-dimensional $N=4$ gauged supergravity with $SO(4)\times
SO(4)\sim SO(3)\times SO(3)\times SO(3)\times SO(3)$ gauge group in the
presence of symplectic deformations. There are in general four electric-
magnetic phases corresponding to each $SO(3)$ factor, but two phases of the
$SO(3)$ factors embedded in the $SO(6)$ R-symmetry are fixed. One phase can be
set to zero by $SL(2,\mathbb{R})$ transformations. The second one gives
equivalent theories for any non-vanishing values and can be set to
$\frac{\pi}{2}$ resulting in gauged supergravities that admit $N=4$
supersymmetric $AdS_{4}$ vacua. The remaining two phases are truely
deformation parameters leading to different $SO(4)\times SO(4)$ gauged
supergravities. As in the $\omega$-deformed $SO(8)$ maximal gauged
supergravity, the cosmological constant and scalar masses of the $AdS_{4}$
vacuum at the origin of the scalar manifold with $SO(4)\times SO(4)$ symmetry
do not depend on the electric-magnetic phases. We find $N=1$ holographic RG
flow solutions between $N=4$ critical points with $SO(4)\times SO(4)$ and
$SO(3)_{\textrm{diag}}\times SO(3)\times SO(3)$ or $SO(3)\times
SO(3)_{\textrm{diag}}\times SO(3)$ symmetries. We also give $N=2$ and $N=1$ RG
flows from these critical points to various non-conformal phases. However,
contrary to the $\omega$-deformed $SO(8)$ gauged supergravity, there exist
non-trivial supersymmetric $AdS_{4}$ critical points only for particular
values of the deformation parameters within the scalar sectors under
consideration.

##  1 Introduction

The discovery of a new family of maximal gauged supergravities in [1], see
also [2, 3, 4], called $\omega$-deformed gauged supergravity has led to
various interesting consequences. The new $\omega$-deformed $SO(8)$ gauged
supergravity is obtained from a symplectic deformation of the original $SO(8)$
gauged supergravity constructed in [5]. In the context of the AdS/CFT
correspondence [6, 7, 8], the $\omega$-deformed $SO(8)$ gauged supergravity
admits a much richer structure of supersymmetric $AdS_{4}$ vacua and other
holographic solutions such as holographic RG flows and Janus solutions [1, 9,
10, 11, 12, 13, 14, 15].
An extension of symplectic deformations to $N\geq 2$ gauged supergravities has
been considered in [16] in which some examples of symplectically deformed
$N=2$ and $N=4$ gauged supergravities have been given. In the present paper,
we are interested in symplectic deformations of $N=4$ gauged supergravity with
$SO(4)\times SO(4)\sim SO(3)\times SO(3)\times SO(3)\times SO(3)$ gauge group.
Generally, there can be four deformation parameters or electric-magnetic
phases for the four $SO(3)$ factors, see also [17, 18] for an eariler
construction of $N=4$ gauged supergravity with these phases. In the notation
of [16], these phases are denoted by $\alpha_{0}$, $\alpha$, $\beta_{1}$ and
$\beta_{2}$. $\alpha_{0}$ can be set to zero by $SL(2,\mathbb{R})$
transformations. In addition, all values of $\alpha>0$ lead to equivalent
gauged supergravities and can be set to $\frac{\pi}{2}$. The remaining phases
$\beta_{1}$ and $\beta_{2}$ constitute free deformation parameters of the
$SO(4)\times SO(4)$ gauged supergravity. With these two phases, we expect to
find a rich structure of vacua and other interesting holographic solutions as
in the $\omega$-deformed $SO(8)$ gauged supergravity.
The $SO(4)\times SO(4)$ gauged supergravity with particular values of
$\alpha_{0}=\beta_{1}=0$ and $\alpha=\beta_{2}=\frac{\pi}{2}$ has been
considered previously in [19, 20, 21, 22]. In particular, a number of
supersymmetric $AdS_{4}$ vacua, holographic RG flows, Janus solutions and
$AdS_{4}$ black holes have been found in [20, 21, 22]. In the present paper,
we will consider $SO(4)\times SO(4)$ gauge group with arbitrary values of
$\beta_{1}$ and $\beta_{2}$. We will mainly look for supersymmetric $AdS_{4}$
vacua and holographic RG flows interpolating between these vacua or from
$AdS_{4}$ critical points to singular geometries in the IR. The former
describe RG flows between conformal fixed points in the dual $N=4$ Chern-
Simons-matter (CSM) theories in three dimensions, see for example [23, 24, 25,
26, 27, 28, 29, 30, 31], while the latter correspond to RG flows from a
conformal fixed point to non-conformal phases. We will consider
$SO(3)_{\textrm{diag}}\times SO(3)_{\textrm{diag}}$, $SO(2)\times SO(2)\times
SO(2)\times SO(2)$ and $SO(3)_{\textrm{diag}}\times SO(3)$ scalar sectors.
It turns out that, unlike the $\omega$-deformed $SO(8)$ gauged supergravity,
there do not exist any supersymmetric $AdS_{4}$ critical points apart from
those identified previously in [20] or critical points related to these at
least within the aforementioned scalar sectors. However, we do find new
classes of holographic RG flows with $N=1$ and $N=2$ supersymmetries. In
particular, some of the $N=1$ solutions describe RG flows between $N=4$
critical points with $SO(4)\times SO(4)$ and $SO(3)_{\textrm{diag}}\times
SO(3)\times SO(3)$ or $SO(3)\times SO(3)_{\textrm{diag}}\times SO(3)$
symmetries. To the best of the author’s knowledge, these are the first
examples of holographic RG flows between conformal fixed points preserving
$N=1$ supersymmetry in the framework of $N=4$ gauged supergravity, see [32]
for examples of $N=1$ RG flows to non-conformal phases, and should further
extend the list of known $N=4$ and $N=2$ solutions given in [20] and [21], see
[12, 13, 14, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45] for an
incomplete list of similar solutions in other four-dimensional gauged
supergravities.
It should be pointed out that the $N=4$ gauged supergravity under
consideration here has currently no known higher dimensional origins as in the
case of $\omega$-deformed $SO(8)$ gauged supergravity. Accordingly, the
complete holographic interpretation in string/M-theory framework is
unavailable. However, it is still useful to have holographic solutions in
lower dimensional gauged supergravities, and with recent developments in
double field theory formalism, particularly the result of [46], the embedding
of $SO(4)\times SO(4)$ gauged supergravity in higher dimensions could be
achieved.
The paper is organized as follows. In section 2, we review the general
structure of $N=4$ gauged supergravity in the embedding tensor formalism
together with symplectic deformations of $SO(4)\times SO(4)$ gauge group. The
truncations to $SO(3)_{\textrm{diag}}\times SO(3)_{\textrm{diag}}$,
$SO(2)\times SO(2)\times SO(2)\times SO(2)$ and $SO(3)_{\textrm{diag}}\times
SO(3)$ singlet scalars are considered in sections 3, 4 and 5, respectively. In
these scalar sectors, we will focus on $AdS_{4}$ vacua and possible RG flow
solutions between these vacua and RG flows to singular geometries. We end the
paper by giving some conclusions and comments on the results in section 6.
Useful formulae and details on relevant BPS equations can be found in the
appendix.

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

We first review the general structure of $N=4$ gauged supergravity coupled to
vector multiplets in the embedding tensor formalism [47], see also [48] for an
eariler construction. In four dimensions, there are two types of $N=4$
supermultiplets, the gravity and vector multiplets, with the following field
content

$(e^{\hat{\mu}}_{\mu},\psi^{i}_{\mu},A_{\mu}^{m},\chi^{i},\tau)$ (1)

and

$(A^{a}_{\mu},\lambda^{ia},\phi^{ma}).$ (2)

The component fields in the gravity multiplet are given by the graviton
$e^{\hat{\mu}}_{\mu}$, four gravitini $\psi^{i}_{\mu}$, six vectors
$A_{\mu}^{m}$, four spin-$\frac{1}{2}$ fields $\chi^{i}$ and one complex
scalar $\tau$ while those in a vector multiplet are given by a vector field
$A_{\mu}$, four gaugini $\lambda^{i}$ and six scalars $\phi^{m}$.

It is useful to note the convention for various types of indices here. Indices
$\mu,\nu,\ldots=0,1,2,3$ and $\hat{\mu},\hat{\nu},\ldots=0,1,2,3$ are
respectively space-time and tangent space (flat) indices while
$m,n=1,\ldots,6$ and $i,j=1,2,3,4$ indices describe fundamental
representations of $SO(6)_{R}$ and $SU(4)_{R}$ R-symmetry. The vector
multiplets are labeled by indices $a,b=1,\ldots,n$. From both the gravity and
vector multiplets, there are $6+n$ vector fields
$A^{+M}=(A^{m}_{\mu},A^{a}_{\mu})$. These are called electric gauge fields and
appear in the ungauged Lagrangian with the usual Yang-Mills kinetic term.
Indices $M,N,\ldots=1,2,\ldots,6+n$ denote fundamental representation of
$SO(6,n)$. Together with the magnetic dual $A^{-M}$, the resulting $2(6+n)$
vector fields form a doublet under $SL(2,\mathbb{R})$ and will be denoted by
$A^{\alpha M}$ with $\alpha=(+,-)$ being an index of $SL(2,\mathbb{R})$
fundamental representation.
All fermionic fields and supersymmetry parameters transform in fundamental
representation of $SU(4)_{R}\sim SO(6)_{R}$ and are subject to the chirality
projections

$\gamma_{5}\psi^{i}_{\mu}=\psi^{i}_{\mu},\qquad\gamma_{5}\chi^{i}=-\chi^{i},\qquad\gamma_{5}\lambda^{i}=\lambda^{i}$
(3)

while those transforming in anti-fundamental representation of $SU(4)_{R}$
satisfy

$\gamma_{5}\psi_{\mu i}=-\psi_{\mu
i},\qquad\gamma_{5}\chi_{i}=\chi_{i},\qquad\gamma_{5}\lambda_{i}=-\lambda_{i}\,.$
(4)

The complex scalar $\tau$ consists of the dilaton $\phi$ and the axion $\chi$
which parametrize $SL(2,\mathbb{R})/SO(2)$ coset manifold. This
$SL(2,\mathbb{R})/SO(2)$ can be described by the coset representative
$\mathcal{V}_{\alpha}$ of the form

$\mathcal{V}_{\alpha}=e^{\frac{\phi}{2}}\left(\begin{array}[]{c}\chi+ie^{\phi}\\\
1\\\ \end{array}\right).$ (5)

with

$\tau=\chi+ie^{\phi}\,.$ (6)

Similarly, the $6n$ scalars $\phi^{ma}$ parametrize $SO(6,n)/SO(6)\times
SO(n)$ coset manifold with the coset representative denoted by
$\mathcal{V}_{M}^{\phantom{M}A}$. Under the global $SO(6,n)$ and local
$SO(6)\times SO(n)$ symmetries, $\mathcal{V}_{M}^{\phantom{M}A}$ transforms by
left and right multiplications, respectively. Accordingly, the $SO(6)\times
SO(n)$ index $A$ can be split as $A=(m,a)$ resulting in the following
components of the coset representative

$\mathcal{V}_{M}^{\phantom{M}A}=(\mathcal{V}_{M}^{\phantom{M}m},\mathcal{V}_{M}^{\phantom{M}a}).$
(7)

The matrix $\mathcal{V}_{M}^{\phantom{M}A}$ satisfies the relation

$\eta_{MN}=-\mathcal{V}_{M}^{\phantom{M}m}\mathcal{V}_{N}^{\phantom{M}m}+\mathcal{V}_{M}^{\phantom{M}a}\mathcal{V}_{N}^{\phantom{M}a}$
(8)

with $\eta_{MN}=\textrm{diag}(-1,-1,-1,-1,-1,-1,1,\ldots,1)$ being the
$SO(6,n)$ invariant tensor. The inverse of $\mathcal{V}_{M}^{\phantom{M}A}$
will be denoted by
${\mathcal{V}_{A}}^{M}=({\mathcal{V}_{m}}^{M},{\mathcal{V}_{a}}^{M})$.
All possible gaugings of the aforementioned matter-coupled $N=4$ supergravity
are encoded in the embedding tensor [47]. $N=4$ supersymmetry allows only two
non-vanishing components of the embedding tensor denoted by $\xi^{\alpha M}$
and $f_{\alpha MNP}$. A given gauge group $G_{0}\subset SL(2,\mathbb{R})\times
SO(6,n)$ can be embedded in both $SL(2,\mathbb{R})$ and $SO(6,n)$ and can be
gauged by either electric or magnetic vector fields or combinations thereof.
We also note that each magnetic vector field must be accompanied by an
auxiliary two-form field in order to remove the extra degrees of freedom. The
embedding tensor also needs to satisfy the quadratic constraint in order for
the resulting gauge generators to form a closed subalgebra of
$SL(2,\mathbb{R})\times SO(6,n)$.
In this paper, we are mainly interested in gauge groups admitting
supersymmetric $AdS_{4}$ vacua. As shown in [49], see also [17, 18] for an
earlier result, this requires the gauge groups to be embedded solely in
$SO(6,n)$ and gauged by both electric and magnetic vector fields. This implies
that both electric and magnetic components of $f_{\alpha MNP}$ must be non-
vanishing and $\xi^{\alpha M}=0$. Accordingly, we will set $\xi^{\alpha M}$ to
zero from now on. Furthermore, since we will study supersymmetric $AdS_{4}$
vacua and domain wall solutions that involve only the metric and scalar
fields, we will also set all vector and fermionic fields to zero.
With all these, the bosonic Lagrangian can be written as

$e^{-1}\mathcal{L}=\frac{1}{2}R+\frac{1}{16}\partial_{\mu}M_{MN}\partial^{\mu}M^{MN}-\frac{1}{4(\textrm{Im}\tau)^{2}}\partial_{\mu}\tau\partial^{\mu}\tau^{*}-V$
(9)

where $e=\sqrt{-g}$ is the vielbein determinant. The scalar potential is given
in terms of the scalar coset representative and the embedding tensor by

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

We also note that $f_{\alpha MNP}$ include the gauge coupling constants.
The symmetric matrix $M_{MN}$ is defined by

$M_{MN}=\mathcal{V}_{M}^{\phantom{M}m}\mathcal{V}_{N}^{\phantom{M}m}+\mathcal{V}_{M}^{\phantom{M}a}\mathcal{V}_{N}^{\phantom{M}a}$
(11)

with $M^{MN}$ denoting its inverse. The tensor $M^{MNPQRS}$ is obtained from

$M_{MNPQRS}=\epsilon_{mnpqrs}\mathcal{V}_{M}^{\phantom{M}m}\mathcal{V}_{N}^{\phantom{M}n}\mathcal{V}_{P}^{\phantom{M}p}\mathcal{V}_{Q}^{\phantom{M}q}\mathcal{V}_{R}^{\phantom{M}r}\mathcal{V}_{S}^{\phantom{M}s}$
(12)

by raising indices with $\eta^{MN}$. Finally, $M^{\alpha\beta}$ is the inverse
of the symmetric $2\times 2$ matrix $M_{\alpha\beta}$ defined by

$M_{\alpha\beta}=\textrm{Re}(\mathcal{V}_{\alpha}\mathcal{V}^{*}_{\beta}).$
(13)

Fermionic supersymmetry transformations are given by

$\displaystyle\delta\psi^{i}_{\mu}$ $\displaystyle=$ $\displaystyle
2D_{\mu}\epsilon^{i}-\frac{2}{3}A^{ij}_{1}\gamma_{\mu}\epsilon_{j},$ (14)
$\displaystyle\delta\chi^{i}$ $\displaystyle=$
$\displaystyle-\epsilon^{\alpha\beta}\mathcal{V}_{\alpha}D_{\mu}\mathcal{V}_{\beta}\gamma^{\mu}\epsilon^{i}-\frac{4}{3}iA_{2}^{ij}\epsilon_{j},$
(15) $\displaystyle\delta\lambda^{i}_{a}$ $\displaystyle=$ $\displaystyle
2i\mathcal{V}_{a}^{\phantom{a}M}D_{\mu}\mathcal{V}_{M}^{\phantom{M}ij}\gamma^{\mu}\epsilon_{j}-2iA_{2aj}^{\phantom{2aj}i}\epsilon^{j}$
(16)

with the fermion shift matrices defined by

$\displaystyle A_{1}^{ij}$ $\displaystyle=$
$\displaystyle\epsilon^{\alpha\beta}(\mathcal{V}_{\alpha})^{*}\mathcal{V}_{kl}^{\phantom{kl}M}\mathcal{V}_{N}^{\phantom{N}ik}\mathcal{V}_{P}^{\phantom{P}jl}f_{\beta
M}^{\phantom{\beta M}NP},$ $\displaystyle A_{2}^{ij}$ $\displaystyle=$
$\displaystyle\epsilon^{\alpha\beta}\mathcal{V}_{\alpha}\mathcal{V}_{kl}^{\phantom{kl}M}\mathcal{V}_{N}^{\phantom{N}ik}\mathcal{V}_{P}^{\phantom{P}jl}f_{\beta
M}^{\phantom{\beta M}NP},$ $\displaystyle A_{2ai}^{\phantom{2ai}j}$
$\displaystyle=$
$\displaystyle\epsilon^{\alpha\beta}\mathcal{V}_{\alpha}{\mathcal{V}_{a}}^{M}{\mathcal{V}_{ik}}^{N}\mathcal{V}_{P}^{\phantom{P}jk}f_{\beta
MN}^{\phantom{\beta MN}P}\,.$ (17)

$\mathcal{V}_{M}^{\phantom{M}ij}$ and ${\mathcal{V}_{ij}}^{M}$ are defined in
terms of the ’t Hooft symbols $G^{ij}_{m}$ as

$\mathcal{V}_{M}^{\phantom{M}ij}=\frac{1}{2}\mathcal{V}_{M}^{\phantom{M}m}G^{ij}_{m}$
(18)

and

${\mathcal{V}_{ij}}^{M}=-\frac{1}{2}{\mathcal{V}_{m}}^{M}(G^{ij}_{m})^{*}\,.$
(19)

The explicit representation of $G^{ij}_{m}$ used in this paper is given in the
appendix. It is also useful to note that upper and lower $i,j,\ldots$ indices
are related by complex conjugation.

###  2.1 $SO(4)\times SO(4)$ gauge group and symplectic deformations

In this work, we only consider $SO(4)\times SO(4)\sim SO(3)\times SO(3)\times
SO(3)\times SO(3)$ gauge group. The embedding of this gauge group into the
global symmetry $SO(6,n)$ requires at least $n=6$ vector multiplets. We will
only consider the minimal case of $n=6$. All four factors of $SO(3)$’s are
embedded in $SO(6,6)$ via the maximal compact subgroup $SO(6)_{R}\times SO(6)$
with $SO(6)_{R}\sim SU(4)_{R}$ being the R-symmetry. Components of the
embedding tensor describing the full gauge group are given by the $SO(3)$
structure constant for each $SO(3)$ factor. The existence of supersymmetric
$AdS_{4}$ vacua requires the first two $SO(3)$ factors embedded in $SO(6)_{R}$
to be gauged differently, one factor electrically gauged and the other
magnetically gauged.
In general, each $SO(3)$ factor can acquire a non-trivial $SL(2,\mathbb{R})$
phase resulting in symplectic deformations of a particular $SO(4)\times SO(4)$
gauging such as purely electric gauged $SO(4)\times SO(4)$ gauge group [16],
see also [17, 18] for an earlier consideration of this deformation. To give an
explicit form of the embedding tensor, it is convenient to split the $SO(6,6)$
fundamental index as $M=(\hat{m},\tilde{m},\hat{a},\tilde{a})$ for
$\hat{m},\tilde{m},\hat{a},\tilde{a}=1,2,3$. The embedding tensor for
symplectically deformed $SO(4)\times SO(4)$ gauging as given in [46] can be
written, in the notation of [16], as

$\displaystyle
f_{+\hat{m}\hat{n}\hat{p}}=-g_{0}\cos\alpha_{0}\epsilon_{\hat{m}\hat{n}\hat{p}},\qquad
f_{-\hat{m}\hat{n}\hat{p}}=g_{0}\sin\alpha_{0}\epsilon_{\hat{m}\hat{n}\hat{p}},$
$\displaystyle
f_{+\tilde{m}\tilde{n}\tilde{p}}=-g\cos\alpha\epsilon_{\tilde{m}\tilde{n}\tilde{p}},\qquad
f_{-\tilde{m}\tilde{n}\tilde{p}}=g\sin\alpha\epsilon_{\tilde{m}\tilde{n}\tilde{p}},$
$\displaystyle
f_{+\hat{a}\hat{b}\hat{c}}=h_{1}\cos\beta_{1}\epsilon_{\hat{a}\hat{b}\hat{c}},\qquad
f_{-\hat{a}\hat{b}\hat{c}}=h_{1}\sin\beta_{1}\epsilon_{\hat{a}\hat{b}\hat{c}},$
$\displaystyle
f_{+\tilde{a}\tilde{b}\tilde{c}}=h_{2}\cos\beta_{2}\epsilon_{\tilde{a}\tilde{b}\tilde{c}},\qquad
f_{-\tilde{a}\tilde{b}\tilde{c}}=h_{2}\sin\beta_{2}\epsilon_{\tilde{a}\tilde{b}\tilde{c}}\,.$
(20)

The constants $\alpha_{0}$, $\alpha$, $\beta_{1}$ and $\beta_{2}$ are the
electric-magnetic phases while $g_{0}$, $g$, $h_{1}$ and $h_{2}$ are the
corresponding gauge coupling constants for each $SO(3)$ factor.

A particular case of $\alpha_{0}=\beta_{1}=0$ and
$\alpha=\beta_{2}=\frac{\pi}{2}$, after a redefinition of gauge coupling
constants, has been considered in [19, 20, 21]. For later convenience, we will
call the $SO(4)\times SO(4)$ gauge group with this particular choice of phases
“undeformed” $SO(4)\times SO(4)$ gauge group. It has been pointed out in [16]
that by gauge fixing the $SL(2,\mathbb{R})$ symmetry, we can set
$\alpha_{0}=0$. In addition, all the gaugings with $\alpha>0$ are equivalent
to the gauge group with $\alpha=\frac{\pi}{2}$ up to a shift of gauge
invariant theta terms and a redefinition of the axion. We will set
$\alpha_{0}=0$ but keep $\alpha$ generic to keep track of the effects of
symplectic deformations. We also note that if all the phases are not $0$ or
$\frac{\pi}{2}$, see [16] for possible ranges of these phases, all four
$SO(3)$ factors are dyonically gauged by both electric and magnetic vector
fields since all $f_{\pm MNP}$ are non-vanishing. For convenience, we will
introduce the notation $SO(3)_{0}$, $SO(3)_{\alpha}$, $SO(3)_{1}$ and
$SO(3)_{2}$ for these four $SO(3)$ factors. $SO(3)_{0}\times SO(3)_{\alpha}$
and $SO(3)_{1}\times SO(3)_{2}$ are embedded in $SO(6)_{R}$ and $SO(6)$,
respectively.

We also note that for particular values of the electric-magnetic phases

$\alpha_{0}=0,\qquad\alpha=\frac{\pi}{2},\qquad\beta=\frac{\pi}{2}-2\omega,\qquad\beta_{2}=-2\omega$
(21)

with $\omega\in[0,\frac{\pi}{8}]$ and $g=-g_{0}=h_{1}=h_{2}$, the resulting
$N=4$ gauged supergravity is a truncation of the $\omega$-deformed $SO(8)$
maximal gauged supergravity constructed in [1].

###  2.2 Parametrization of scalar manifold and BPS equations

Since we are mainly interested in holographic RG flow solutions in the form of
supersymmetric domain walls, an explicit parametrization of the scalar
manifold $SO(6,6)/SO(6)\times SO(6)$ is crucial. To give the
$SO(6,6)/SO(6)\times SO(6)$ coset representative, we first define $SO(6,6)$
generators in the fundamental representation by

$(t_{MN})_{P}^{\phantom{P}Q}=2\delta^{Q}_{[M}\eta_{N]P}\,.$ (22)

The $SO(6,6)$ non-compact generators are then given by

$Y_{ma}=t_{m,a+6}\,.$ (23)

To make things more manageable, we will only consider particular truncations
of the full $36$-dimensional coset to submanifolds with a few scalars non-
vanishing. The truncations we will consider contain singlet scalars under
$SO(4)_{\textrm{diag}}\sim SO(3)_{\textrm{diag}}\times SO(3)_{\textrm{diag}}$,
$SO(2)\times SO(2)\times SO(2)\times SO(2)$ and $SO(3)_{\textrm{diag}}\times
SO(3)$.

To find supersymmetric domain wall solutions, we use the standard metric
ansatz

$ds^{2}=e^{2A(r)}dx^{2}_{1,2}+dr^{2}$ (24)

with $dx^{2}_{1,2}$ being the metric on three-dimensional Minkowski space. The
only remaining non-vanishing fields are given by scalars. To preserve the
isometry of $dx_{1,2}^{2}$, scalar fields can depend only on the radial
coordinate $r$.
Supersymmetric solutions can be found by considering solutions to the BPS
equations obtained by setting fermionic supersymmetry transformations to zero.
With the metric ansatz (24), the variations of gravitini along $\mu=0,1,2$
directions give

$A^{\prime}\gamma_{\hat{r}}\epsilon^{i}-\frac{2}{3}A^{ij}_{1}\epsilon_{j}=0\,.$
(25)

To proceed, we will use Majorana representation for space-time gamma matrices
with all $\gamma^{\hat{\mu}}$ real and $\gamma_{5}$ purely imaginary. Left and
right chiralities of fermions are then related to each other by complex
conjugation. The symmetric matrix $A_{1}^{ij}$ can be diagonalized with
eiganvalues denoted by $\mathcal{A}_{i}$. In general, in the presence of
unbroken supersymmetry, some or all of these eigenvalues will give rise to the
superpotential in terms of which the scalar potential can be written. Let
$\hat{\alpha}$ be the eigenvalue of the Killing spinors $\epsilon^{\hat{i}}$
corresponding to the unbroken supersymmetry. We can rewrite the above equation
as

$A^{\prime}\gamma^{\hat{r}}\epsilon^{\hat{i}}-\frac{2}{3}\hat{\alpha}\epsilon_{\hat{i}}=0\,.$
(26)

To proceed further, we impose the following projector

$\gamma^{\hat{r}}\epsilon^{\hat{i}}=e^{i\Lambda}\epsilon_{\hat{i}}$ (27)

with an $r$-dependent phase $\Lambda$. This projector relates the two
chiralities of $\epsilon^{\hat{i}}$ breaking half of the supersymmetry. A
domain wall solution is then half-supersymmetric. By defining the
superpotential

$\mathcal{W}=\frac{2}{3}\hat{\alpha},$ (28)

we obtain the BPS condition

$A^{\prime}e^{i\Lambda}-\mathcal{W}=0$ (29)

which implies

$A^{\prime}=\pm W$ (30)

and

$e^{i\Lambda}=\pm\frac{\mathcal{W}}{W}$ (31)

for $W=|\mathcal{W}|$.
Repeating the same procedure for $\delta\psi^{i}_{\hat{r}}$, we find a
differential condition on the Killing spinors

$2\partial_{r}\epsilon^{\hat{i}}-\mathcal{W}\gamma_{\hat{r}}\epsilon_{\hat{i}}=0\,.$
(32)

With the condition (26), we find

$\epsilon^{\hat{i}}=e^{\frac{A}{2}}\epsilon_{0}^{\hat{i}}$ (33)

for constant spinors $\epsilon_{0}^{\hat{i}}$. Finally, using the
$\gamma_{\hat{r}}$ projector (27) in the variations $\delta\chi^{i}$ and
$\delta\lambda^{i}_{a}$, we can determine all the BPS equations for scalars.
In subsequent sections, we will find explicit solutions for various residual
symmetries and different numbers of unbroken supersymmetries.

##  3 $SO(3)_{\textrm{diag}}\times SO(3)_{\textrm{diag}}$ sector

We begin with a simple case of $SO(3)_{\textrm{diag}}\times
SO(3)_{\textrm{diag}}$ singlet scalars. This sector has been considered in the
undeformed $SO(4)\times SO(4)$ gauge group in [20]. In this work, we will
consider effects of arbitrary electric-magnetic phases. There are two singlets
from $SO(6,6)/SO(6)\times SO(6)$ coset corresponding to the non-compact
generators

$\hat{Y}_{1}=Y_{11}+Y_{22}+Y_{33}\qquad\textrm{and}\qquad\hat{Y}_{2}=Y_{44}+Y_{55}+Y_{66}\,.$
(34)

Accordingly, the coset representative can be written as

$\mathcal{V}=e^{\phi_{1}\hat{Y}_{1}}e^{\phi_{2}\hat{Y}_{2}}\,.$ (35)

The dilaton and axion are also $SO(3)_{\textrm{diag}}\times
SO(3)_{\textrm{diag}}$ singlets since these scalars are singlets under the
full $SO(6,6)$ global symmetry. Therefore, the $SO(3)_{\textrm{diag}}\times
SO(3)_{\textrm{diag}}$ sector consists of $4$ scalars.

###  3.1 Supersymmetric $AdS_{4}$ vacua

We first look at possible supersymmetric $AdS_{4}$ vacua within the
$SO(3)_{\textrm{diag}}\times SO(3)_{\textrm{diag}}$ sector. The scalar
potential is given by

$\displaystyle V$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left[2g_{0}^{2}e^{-\phi}\cosh^{2}\phi_{1}(\cosh
2\phi_{1}-2)+8gg_{0}\cosh^{3}\phi_{1}\cosh^{3}\phi_{2}\sin\alpha\right.$ (36)
$\displaystyle+8gh_{1}\cosh^{3}\phi_{2}\sin(\alpha+\beta_{1})\sinh^{3}\phi_{1}-8g_{0}h_{2}\sin\beta_{2}\cosh^{3}\phi_{1}\sinh^{3}\phi_{2}$
$\displaystyle+8h_{1}h_{2}\sin(\beta_{1}-\beta_{2})\sinh^{3}\phi_{1}\sinh^{3}\phi_{2}+e^{-\phi}g_{0}h_{1}\sinh^{3}2\phi_{1}(\cos\beta_{1}-\sin\beta_{1}\chi)$
$\displaystyle+2e^{-\phi}g^{2}\cosh^{4}\phi_{2}(\cosh
2\phi_{2}-2)(\cos^{2}\alpha+\sin^{2}\alpha e^{2\phi}+\chi\sin
2\alpha+\sin^{2}\alpha\chi^{2})$
$\displaystyle+2e^{-\phi}h_{1}^{2}\sinh^{4}\phi_{1}(\cos^{2}\beta_{1}+e^{2\phi}\sin^{2}\beta_{1}-\chi\sin
2\beta_{1}+\sin^{2}\beta_{1}\chi^{2})\times$ $\displaystyle(\cosh
2\phi_{1}+2)+e^{-\phi}gh_{2}\sinh^{3}2\phi_{2}\left[\cos\alpha\cos\beta_{2}-e^{2\phi}\sin\alpha\sin\beta_{2}\right.$
$\displaystyle\left.+\sin(\alpha-\beta_{2})\chi-\sin\alpha\sin\beta_{2}\chi^{2}\right]+2e^{-\phi}h_{2}^{2}\sinh^{4}\phi_{2}(\cosh
2\phi_{2}+2)$
$\displaystyle\left.(\cos^{2}\beta_{2}+e^{2\phi}\sin^{2}\beta_{2}-\sin
2\beta_{2}+\sin^{2}\beta_{2}\chi^{2})\right].$

As in the undeformed $SO(4)\times SO(4)$ gauge group, it turns out that
$A_{1}^{ij}$ tensor is proportional to the identity matrix with the four equal
eigenvalues given by

$\displaystyle\mathcal{A}$ $\displaystyle=$
$\displaystyle-\frac{3}{4}e^{-\frac{\phi}{2}}\left[h_{1}\sinh^{3}\phi_{1}(\cos\beta_{1}+ie^{\phi}\sin\beta_{1})-ih_{2}\sinh^{3}\phi_{2}(\cos\beta_{2}+ie^{\phi}\sin\beta_{2})\right.$
(37)
$\displaystyle-(h_{1}\sin\beta_{1}\sinh^{3}\phi_{1}+ig\sin\alpha\cosh^{3}\phi_{2}-ih_{2}\sin\beta_{2}\sinh^{3}\phi_{2})\chi$
$\displaystyle\left.+g_{0}\cosh^{3}\phi_{1}-g\cosh^{3}\phi_{2}(i\cos\alpha+e^{\phi}\sin\alpha)\right].$

We now look for possible supersymmetric $AdS_{4}$ vacua. We begin with a
simple case of $\phi_{1}=\phi_{2}=0$. It can be straightforwardly verified
that this choice satisfies all the BPS conditions provided that

$\phi=\ln\left[-\frac{g_{0}}{g\sin\alpha}\right]\qquad\textrm{and}\qquad\chi=-\frac{\cos\alpha}{\sin\alpha}\,.$
(38)

This leads to a supersymmetric $AdS_{4}$ vacuum preserving $N=4$ supersymmetry
and the full $SO(4)\times SO(4)$ gauge symmetry with the corresponding
cosmological constant given by

$V_{0}=3gg_{0}\sin\alpha\,.$ (39)

The $AdS_{4}$ radius is given by

$L=\sqrt{-\frac{3}{V_{0}}}=\sqrt{-\frac{1}{gg_{0}\sin\alpha}}\,.$ (40)

It should be noted that $V_{0}<0$ since the reality condition on $\phi$
implies $gg_{0}\sin\alpha<0$. We also note that for $\alpha=0$, the $AdS_{4}$
vacuum does not exist. This is in agreement with the fact that $\alpha=0$
together with the previous choice of $\alpha_{0}=0$ imply that the $SO(3)_{0}$
and $SO(3)_{\alpha}$ are both electrically gauged leading to no supersymmetric
$AdS_{4}$ vacua [49].
We can bring this $SO(4)\times SO(4)$ vacuum to the origin of the scalar
manifold by shifting the dilaton and axion or equivalently choosing

$\alpha=\frac{\pi}{2}\qquad\textrm{and}\qquad g_{0}=-g\,.$ (41)

This simply realizes the general result of [16] that any values of $\alpha>0$
lead to physically equivalent theories up to a redefinition of the axion.
Therefore, this $N=4$ $AdS_{4}$ vacuum is the same as that of the undeformed
$SO(4)\times SO(4)$ gauged supergravity considered in [20]. Moreover, the
scalar masses turn out to be independent of all electric-magnetic phases with
all scalar masses equal $m^{2}L^{2}=-2$. This result is similar to the
maximally supersymmetric $AdS_{4}$ vacuum at the origin of the scalar manifold
in $\omega$-deformed $SO(8)$ $N=8$ gauged supergravity.
To look for other supersymmetric vacua, it is more convenient to first analyze
the resulting BPS conditions. The conditions arising from
$\delta\lambda^{i}_{a}$ reduce to the following two equations

$\displaystyle
e^{-i\Lambda}\phi_{1}^{\prime}+\frac{1}{2}e^{-\frac{\phi}{2}}\sinh
2\phi_{1}\left[h_{1}\sinh\phi_{1}(ie^{\phi}\sin\beta_{1}-\cos\beta_{1})\right.$
$\displaystyle\left.+h_{1}\sin\beta_{1}\sinh\phi_{1}\chi-
g_{0}\cosh\phi_{1}\right]$ $\displaystyle=$ $\displaystyle 0,\qquad$ (42)
$\displaystyle
e^{-i\Lambda}\phi_{2}^{\prime}+\frac{i}{2}e^{-\frac{\phi}{2}}\sinh
2\phi_{2}\left[h_{2}\sin\beta_{2}\sinh\phi_{2}(\chi+ie^{\phi})-h_{2}\cos\beta_{2}\sinh\phi_{2}\right.$
$\displaystyle\left.+g\cosh\phi_{2}[\cos\alpha+\sin\alpha(\chi+ie^{\phi})]\right]$
$\displaystyle=$ $\displaystyle 0\,.\qquad$ (43)

In these equations and in the following analysis, we choose an upper sign
choice in (30) and (31) for definiteness. This also identifies the trivial
$SO(4)\times SO(4)$ critical point in the limit $r\rightarrow\infty$ in the RG
flow solutions.
At the vacua with constant $\phi_{1}$ and $\phi_{2}$, we have
$\phi_{1}^{\prime}=\phi_{2}^{\prime}=0$, and consistency of the above two
equations imposes the following conditions

$\displaystyle h_{1}e^{\phi}\sin\beta_{1}\sinh\phi_{1}\sinh 2\phi_{1}$
$\displaystyle=$ $\displaystyle 0,$ $\displaystyle e^{-\frac{\phi}{2}}\sinh
2\phi_{1}\left[h_{1}\sin\beta_{1}\sinh\phi_{1}\chi-
h_{1}\sinh\phi_{1}\cos\beta_{1}-g_{0}\cosh\phi_{1}\right]$ $\displaystyle=$
$\displaystyle 0,$ $\displaystyle e^{\frac{\phi}{2}}\sinh
2\phi_{2}(h_{2}\sin\beta_{2}\sinh\phi_{2}-g\cosh\phi_{2})$ $\displaystyle=$
$\displaystyle 0,$ $\displaystyle e^{\frac{\phi}{2}}\sinh
2\phi_{2}\left[(h_{2}\sin\beta_{2}\sinh\phi_{2}-g\cosh\phi_{2})\chi-
h_{2}\cos\beta_{2}\sinh\phi_{2}\right]$ $\displaystyle=$ $\displaystyle 0$
(44)

after setting $\alpha=\frac{\pi}{2}$. All these conditions imply that for
$\phi_{1}\neq 0$ or $\phi_{2}\neq 0$, $AdS_{4}$ vacua are possible only for

$\beta_{1}=0\qquad\textrm{and}\qquad\beta_{2}=\frac{\pi}{2}\,.$ (45)

Accordingly, non-trivial $N=4$ supersymmetric $AdS_{4}$ vacua with at least
$SO(3)_{\textrm{diag}}\times SO(3)_{\textrm{diag}}$ symmetry only exist in the
undeformed $SO(4)\times SO(4)$ gauge group considered in [20]. We also note
that for both $\phi_{1}$ and $\phi_{2}$ non-vanishing, the residual symmetry
is give by $SO(3)\times SO(3)$ while for one of them vanishing, the vacua
preserve larger $SO(4)\times SO(3)$ or $SO(3)\times SO(4)$ symmetries.

Further analysis also shows that consistency of the full BPS equations from
$\delta\lambda^{a}_{i}=0$ conditions with $r$-dependent scalars also requires
(45) for any values of $\alpha=0,\frac{\pi}{2}$. This also implies that apart
from the solutions found in [20] no supersymmetric domain walls or RG flows
with at least $SO(3)_{\textrm{diag}}\times SO(3)_{\textrm{diag}}$ symmetry
exist in the symplectically deformed $SO(4)\times SO(4)$ gauge group.
Therefore, in $SO(3)_{\textrm{diag}}\times SO(3)_{\textrm{diag}}$ sector, no
new $AdS_{4}$ vacua and holographic RG flows interpolating between them exist
apart from those already given in [20].

###  3.2 Holographic RG flows and supersymmetric domain walls

We end this section by giving supersymmetric domain wall solutions with
$\phi_{1}=\phi_{2}=0$. The solutions can be considered to be solutions of pure
$N=4$ gauged supergravity with $SO(3)_{0}\times SO(3)_{\alpha}$ gauge group.
Although all values of $\alpha>0$ give physically equivalent gauged
supergravities, we keep $\alpha$ to be arbitrary here for generality of the
expressions. With $\phi_{1}=\phi_{2}=0$, the scalar potential reads

$\displaystyle V$ $\displaystyle=$ $\displaystyle 4\left(\frac{\partial
W}{\partial\phi}\right)^{2}+4e^{2\phi}\left(\frac{\partial
W}{\partial\chi}\right)^{2}-3W^{2}$ (46) $\displaystyle=$ $\displaystyle
2gg_{0}\sin\alpha-\frac{1}{2}g^{2}\sin^{2}\alpha
e^{\phi}-\frac{1}{2}e^{-\phi}(g^{2}\cos^{2}\alpha+g_{0}^{2})$
$\displaystyle-\frac{1}{2}e^{-\phi}\chi(g^{2}\sin
2\alpha+g^{2}\sin^{2}\alpha\chi)$

with the superpotential given by

$\mathcal{W}=\frac{1}{2}e^{-\frac{\phi}{2}}(ig\cos\alpha-
g_{0}+e^{\phi}g\sin\alpha+ig\sin\alpha\chi).$ (47)

With this superpotential, we find the following BPS equations

$\displaystyle A^{\prime}$ $\displaystyle=$ $\displaystyle W$ (48)
$\displaystyle=$
$\displaystyle\frac{1}{2}e^{-\frac{\phi}{2}}\sqrt{g^{2}(\cos\alpha+\sin\alpha\chi)^{2}+(g_{0}-ge^{\phi}\sin\alpha)^{2}},$
$\displaystyle\phi^{\prime}$ $\displaystyle=$ $\displaystyle-4\frac{\partial
W}{\partial\phi}$ (49) $\displaystyle=$
$\displaystyle\frac{e^{-\frac{\phi}{2}}\left[g_{0}^{2}+(\cos^{2}\alpha-e^{2\phi\sin^{2}\alpha})g^{2}g^{2}\sin\alpha\chi(2\cos\alpha+\sin\alpha\chi)\right]}{\sqrt{(g_{0}-e^{\phi}g\sin\alpha)^{2}+(g\cos\alpha+g\sin\alpha\chi)^{2}}},$
$\displaystyle\chi^{\prime}$ $\displaystyle=$
$\displaystyle-4e^{2\phi}\frac{\partial W}{\partial\chi}$ (50)
$\displaystyle=$
$\displaystyle-\frac{2e^{\frac{3\phi}{2}}g^{2}\sin\alpha(\cos\alpha+\sin\alpha\chi)}{\sqrt{(g_{0}-e^{\phi}g\sin\alpha)^{2}+(g\cos\alpha+g\sin\alpha\chi)^{2}}}\,.$

Combining $A^{\prime}$ and $\phi^{\prime}$ equations with $\chi^{\prime}$
equation, we obtain the solutions for $A$ and $\phi$ as functions of $\chi$

$\displaystyle A$ $\displaystyle=$
$\displaystyle\frac{1}{4}\ln\left[4g_{0}^{2}-C_{0}\chi-2g^{2}(1+\chi^{2})+[C_{0}\chi+2g^{2}(\chi^{2}-1)]\cos
2\alpha\right.$ (51) $\displaystyle\left.-(C_{0}+4g^{2}\chi)\sin
2\alpha\right]$
$\displaystyle-\frac{1}{2}\ln\left[4\sqrt{2}g_{0}\sqrt{4g_{0}^{2}-C_{0}\chi-2g^{2}(1+\chi^{2})+[C_{0}\chi+2g^{2}(\chi^{2}-1)]\cos
2\alpha}\right.$ $\displaystyle\left.\overline{-(C_{0}+4g^{2}\chi)\sin
2\alpha}+\sqrt{2}\left\\{8g_{0}^{2}+C_{0}\chi(\cos 2\alpha-1)-C_{0}\sin
2\alpha\right\\}\right],$ $\displaystyle\phi$ $\displaystyle=$
$\displaystyle\frac{1}{2}\ln\left[\frac{1}{2g^{2}}\textrm{csc}^{2}\alpha\left\\{2g_{0}^{2}-g^{2}(1+\cos
2\alpha)-C_{0}\sin\alpha\cos\alpha\right.\right.$ (52)
$\displaystyle\left.\phantom{\frac{1}{2}}\left.-\sin\alpha\chi(4g^{2}\cos\alpha+C_{0}\sin\alpha)\right\\}-\chi^{2}\right]$

with an integration constant $C_{0}$. We have neglected an additive
integration constant for $A$ which can be removed by rescaling coordinates of
$dx^{2}_{1,2}$. Finally, by changing to a new radial coordinate $\rho$ defined
by

$\frac{dr}{d\rho}=e^{-\frac{\phi}{2}}\sqrt{(g_{0}-g\sin\alpha
e^{\phi})^{2}+(g\cos\alpha+g\sin\alpha\chi)^{2}},$ (53)

we obtain the solution for $\chi$ of the form

$\displaystyle 2gg_{0}\sin\alpha(\rho-\rho_{0})$ $\displaystyle=$
$\displaystyle\ln\left[8g_{0}^{2}+C_{0}\chi(\cos 2\alpha-1)-C_{0}\sin
2\alpha\phantom{\sqrt{g_{0}^{2}}}\right.$ (54)
$\displaystyle\left.+4g_{0}\sqrt{4g_{0}^{2}-2(\chi+\cot\alpha)\sin^{2}\alpha[C_{0}+2g^{2}(\chi+\cot\alpha)]}\right]$
$\displaystyle+\ln\left[\frac{\sin\alpha}{2\sqrt{2}gg_{0}(\chi+\cot\alpha)}\right]$

with another integration constant $\rho_{0}$ which can also be removed by
shifting the coordinate $\rho$. For $\alpha=\frac{\pi}{2}$, this is the
holographic RG flow from a three-dimensional $N=4$ SCFT to a non-conformal
field theory in the IR given in [20]. However, the $\chi(\rho)$ solution has
not been given. Accordingly, the present result should fill this gap.
For $\alpha=0$, the BPS equations simplify considerably to

$A^{\prime}=\frac{1}{2}e^{-\frac{\phi}{2}}\sqrt{g^{2}+g_{0}^{2}}\qquad\textrm{and}\qquad\phi^{\prime}=e^{-\frac{\phi}{2}}\sqrt{g^{2}+g_{0}^{2}}$
(55)

together with $\chi^{\prime}=0$. The solution takes a simple form

$\phi=2\ln\left[\frac{1}{2}\sqrt{g^{2}+g_{0}^{2}}(r-r_{0})\right]\qquad\textrm{and}\qquad
A=\frac{1}{2}\phi$ (56)

with constant $\chi=\chi_{0}$. This gives a half-supersymmetric domain wall
vacuum of $SO(3)_{0}\times SO(3)_{\alpha=0}$ gauged supergravity.

##  4 $SO(2)\times SO(2)\times SO(2)\times SO(2)$ sector

The $SO(2)\times SO(2)\times SO(2)\times SO(2)$ sector of the undeformed
$SO(4)\times SO(4)$ gauged supergravity has been considered recently in [21]
in which a number of holographic RG flows and Janus solutions have been found.
In the present paper, we will consider the same sector with electric-magnetic
phases.

As shown in [21], there are four $SO(2)\times SO(2)\times SO(2)\times SO(2)$
singlet scalars from $SO(6,6)/SO(6)\times SO(6)$ corresponding to non-compact
generators $Y_{33}$, $Y_{36}$, $Y_{63}$ and $Y_{66}$ in terms of which the
coset representative can be written as

$\mathcal{V}=e^{\phi_{1}Y_{33}}e^{\phi_{2}Y_{36}}e^{\phi_{3}Y_{63}}e^{\phi_{4}Y_{66}}\,.$
(57)

Together with the dilaton and axion from the gravity multiplet, there are six
scalars in the $SO(2)\times SO(2)\times SO(2)\times SO(2)$ sector. The kinetic
term for these six scalars takes the form

$\displaystyle\mathcal{L}_{\textrm{kin}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}G_{rs}{\Phi^{r}}^{\prime}{\Phi^{s}}^{\prime}$ (58)
$\displaystyle=$ $\displaystyle-\frac{1}{4}(\phi^{\prime
2}+e^{-2\phi}\chi^{\prime 2})-\frac{1}{16}\left[6+\cosh
2(\phi_{2}-\phi_{3})\right.$ $\displaystyle\left.+\cosh
2(\phi_{2}+\phi_{3})+2\cosh 2\phi_{4}(\cosh 2\phi_{2}\cosh
2\phi_{3}-1)\right]\phi^{\prime 2}_{1}$
$\displaystyle-\cosh\phi_{2}\cosh\phi_{4}\sinh\phi_{3}\sinh\phi_{4}\phi^{\prime}_{1}\phi^{\prime}_{2}-\cosh\phi_{3}\cosh\phi_{4}\sinh\phi_{2}\sinh\phi_{4}\phi^{\prime}_{1}\phi^{\prime}_{3}$
$\displaystyle+\sinh\phi_{2}\sinh\phi_{3}\phi^{\prime}_{1}\phi^{\prime}_{4}-\frac{1}{2}\cosh^{2}\phi_{4}\phi^{\prime
2}_{2}-\frac{1}{2}\cosh^{2}\phi_{4}\phi^{\prime 2}_{3}-\frac{1}{2}\phi^{\prime
2}_{4}$

in which we have introduced a symmetric matrix $G_{rs}$ and a notation
$\Phi^{r}=(\phi,\chi,\phi_{1},\phi_{2},\phi_{3},\phi_{4})$, with
$r,s=1,2,\ldots,6$ for later convenience.

The resulting scalar potential is given by

$\displaystyle V$ $\displaystyle=$
$\displaystyle-\frac{1}{4}e^{-\phi}[g^{2}(1+\cos
2\alpha)+2g_{0}^{2}+2g^{2}\sin\alpha\chi(2\cos\alpha+\sin\alpha\chi)]-\frac{1}{2}e^{\phi}g^{2}\sin^{2}\alpha$
(59)
$\displaystyle+2gg_{0}\sin\alpha\cosh\phi_{1}\cosh\phi_{2}\cosh\phi_{3}\cosh\phi_{4}\,.$

In this case, the phases $\beta_{1}$ and $\beta_{2}$ do not appear in the
scalar potential. In addition, as in the undeformed $SO(4)\times SO(4)$ gauge
group, the potential admits only a trivial $AdS_{4}$ critical point at
$\phi=\chi=\phi_{1}=\phi_{2}=\phi_{3}=\phi_{4}=0$ for $\alpha=\frac{\pi}{2}$
and $g_{0}=-g$.

Since all values of $\alpha>0$ are equivalent to $\alpha=\frac{\pi}{2}$, we
also see that in this sector, no new results arise from the symplectic
deformation. However, the RG flow solutions considered in [21] are obtained
only in a subtrucation with $SO(2)\times SO(2)\times SO(2)\times SO(3)$ or
$SO(2)\times SO(2)\times SO(3)\times SO(2)$ symmetries corresponding to
setting $\phi_{2}=\phi_{4}=0$ or $\phi_{1}=\phi_{3}=0$, respectively. In the
present paper, we will consider the most general solutions in the full
$SO(2)\times SO(2)\times SO(2)\times SO(2)$ sector with all scalars non-
vanishing.

In this case, the $A_{1}^{ij}$ tensor is diagonal and takes the form of

$A_{1}^{ij}=\textrm{diag}(\mathcal{A}_{-},\mathcal{A}_{+},\mathcal{A}_{+},\mathcal{A}_{-})$
(60)

with the two eigenvalues given by

$\displaystyle\mathcal{A}_{\pm}$ $\displaystyle=$
$\displaystyle\frac{1}{4}e^{-\frac{\phi}{2}}\left[3\cosh\phi_{4}[g\cosh\phi_{3}(e^{\phi}\sin\alpha+i\cos\alpha)\pm
ig_{0}\sinh\phi_{1}\sinh\phi_{3}]\right.$
$\displaystyle\left.-3g_{0}\cosh\phi_{1}(\cosh\phi_{2}\mp
i\sinh\phi_{2}\sinh\phi_{4})+3ig\sin\alpha\cosh\phi_{3}\cosh\phi_{4}\chi\right].$

$\mathcal{A}_{-}$ and $\mathcal{A}_{+}$ eigenvalues correspond to unbroken
$N=2$ supersymmetry with the Killing spinors given by $\epsilon^{1,4}$ and
$\epsilon^{2,3}$, respectively. The two choices are equivalent, and we will
choose $\epsilon^{1,4}$ as Killing spinors for definiteness.

With $\epsilon^{2,3}=0$ and the superpotential of the form

$\displaystyle\mathcal{W}$ $\displaystyle=$
$\displaystyle\frac{2}{3}\mathcal{A}_{-}$ (62) $\displaystyle=$
$\displaystyle\frac{1}{2}e^{-\frac{\phi}{2}}\left[\cosh\phi_{4}[g\cosh\phi_{3}(e^{\phi}\sin\alpha+i\cos\alpha)-g_{0}\sinh\phi_{1}\sinh\phi_{3}]\right.$
$\displaystyle\left.-g_{0}\cosh\phi_{1}(\cosh\phi_{2}+i\sinh\phi_{2}\sinh\phi_{4})+ig\sin\alpha\cosh\phi_{3}\cosh\phi_{4}\chi\right],\quad$

we find that all the BPS equations can be written collectively as

${\Phi^{r}}^{\prime}=2G^{rs}\frac{\partial W}{\partial\Phi^{s}}\,.$ (63)

$G^{rs}$ is the inverse of the scalar matrix $G_{rs}$ which is in turn given
by

$G_{rs}=\begin{pmatrix}-\frac{1}{2}&0&0&0&0&0\\\
0&-\frac{1}{2}e^{-2\phi}&0&0&0&0\\\
0&0&\square&\Delta_{1}&\Delta_{2}&\Delta_{3}\\\
0&0&\Delta_{1}&-\cosh^{2}\phi_{4}&0&0\\\
0&0&\Delta_{2}&0&-\cosh^{2}\phi_{4}&0\\\ 0&0&\Delta_{3}&0&0&-1\end{pmatrix}$
(64)

with

$\displaystyle\square$ $\displaystyle=$ $\displaystyle\frac{1}{8}\left[2\cosh
2\phi_{4}(1-\cosh 2\phi_{2}\cosh 2\phi_{3})-\cosh 2(\phi_{2}-\phi_{3})\right.$
$\displaystyle\left.-\cosh 2(\phi_{2}+\phi_{3})-6\right],$
$\displaystyle\Delta_{1}$ $\displaystyle=$
$\displaystyle-\cosh\phi_{2}\cosh\phi_{4}\sinh\phi_{3}\sinh\phi_{4},$
$\displaystyle\Delta_{2}$ $\displaystyle=$
$\displaystyle-\cosh\phi_{3}\cosh\phi_{4}\sinh\phi_{2}\sinh\phi_{4},$
$\displaystyle\Delta_{3}$ $\displaystyle=$
$\displaystyle\sinh\phi_{2}\sinh\phi_{3}\,.$ (65)

We also note that the scalar potential can be written as

$V=-2G^{rs}\frac{\partial W}{\partial\Phi^{r}}\frac{\partial
W}{\partial\Phi^{s}}-3W^{2}\,.$ (66)

For $\alpha=\frac{\pi}{2}$ and $g_{0}=-g$, the explicit form of the BPS
equations reads

$\displaystyle W\phi^{\prime}$ $\displaystyle=$
$\displaystyle-g^{2}\cosh\phi_{3}\cosh\phi_{4}(\cosh\phi_{1}\cosh\phi_{2}+e^{\phi}\cosh\phi_{3}\cosh\phi_{4})$
(67)
$\displaystyle+\frac{1}{2}e^{-\phi}g^{2}\left[\cosh\phi_{4}(\sinh\phi_{1}\sinh\phi_{3}+\cosh\phi_{3}\chi)+\cosh\phi_{1}\sinh\phi_{2}\sinh\phi_{4}\right]^{2}$
$\displaystyle+\frac{1}{2}e^{-\phi}g^{2}(\cosh\phi_{1}\cosh\phi_{2}+e^{\phi}\cosh\phi_{3}\cosh\phi_{4})^{2},$
$\displaystyle W\chi^{\prime}$ $\displaystyle=$
$\displaystyle-e^{\phi}g^{2}\cosh\phi_{3}\cosh\phi_{4}\times$ (68)
$\displaystyle(\cosh\phi_{4}\sinh\phi_{1}\sinh\phi_{3}+\cosh\phi_{1}\sinh\phi_{2}\sinh\phi_{4}+\cosh\phi_{3}\cosh\phi_{4}\chi),$
$\displaystyle W\phi_{1}^{\prime}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}e^{-\phi}g^{2}\left[e^{\phi}\cosh\phi_{4}\textrm{sech}\phi_{2}\textrm{sech}\phi_{3}\sinh\phi_{1}+\cosh\phi_{1}\tanh\phi_{3}\chi+\cosh\phi_{1}\times\right.$
(69)
$\displaystyle\left.(\textrm{sech}^{2}\phi_{3}\sinh\phi_{1}-e^{\phi}\sinh\phi_{4}\tanh\phi_{2}\tanh\phi_{3}+\sinh\phi_{1}\tanh^{2}\phi_{3})\right],$
$\displaystyle W\phi_{2}^{\prime}$ $\displaystyle=$
$\displaystyle-\frac{1}{16}e^{-\phi}g^{2}\left[e^{\phi}\cosh\phi_{1}[6+2\cosh
2\phi_{3}+\cosh 2(\phi_{3}-\phi_{4})-2\cosh 2\phi_{4}\right.$ (70)
$\displaystyle+\cosh
2(\phi_{3}+\phi_{4})]\textrm{sech}\phi_{3}\textrm{sech}\phi_{4}\sinh\phi_{2}-8e^{\phi}\sinh\phi_{1}\sinh\phi_{4}\tanh\phi_{3}$
$\displaystyle\left.+4\cosh^{2}\phi_{1}\sinh
2\phi_{2}+8\cosh\phi_{1}\cosh\phi_{2}\textrm{sech}\phi_{3}\tanh\phi_{4}\chi\right],$
$\displaystyle W\phi_{3}^{\prime}$ $\displaystyle=$
$\displaystyle\frac{1}{16}e^{-\phi}g^{2}\left[\sinh
2(\phi_{1}-\phi_{3})+(2-4e^{2\phi})\sinh 2\phi_{3}-\sinh
2(\phi_{1}+\phi_{3})\right.$ (71)
$\displaystyle-8e^{\phi}\cosh\phi_{1}\textrm{sech}\phi_{4}\sinh\phi_{3}(\textrm{sech}\phi_{2}+\cosh^{2}\phi_{4}\sinh\phi_{2}\tanh\phi_{2})$
$\displaystyle\left.+8e^{\phi}\sinh\phi_{1}\sinh\phi_{4}\tanh\phi_{2}-4\chi(2\cosh
2\phi_{3}\sinh\phi_{1}+\sinh 2\phi_{3}\chi)\right],$ $\displaystyle
W\phi_{4}^{\prime}$ $\displaystyle=$
$\displaystyle\frac{1}{8}e^{-\phi}g^{2}\left[\sin 2\phi_{4}(\cosh
2\phi_{1}-2e^{2\phi}\cosh^{2}\phi_{3}-\cosh
2\phi_{3}\sinh^{2}\phi_{1})\right.$ (72)
$\displaystyle-2e^{\phi}\cosh\phi_{1}\textrm{sech}\phi_{2}\textrm{sech}\phi_{3}\sinh\phi_{4}(\cosh
2\phi_{2}+\cosh 2\phi_{3})$
$\displaystyle-4\cosh\phi_{4}\sinh\phi_{1}(2\cosh\phi_{1}\cosh\phi_{4}\sinh\phi_{2}\sinh\phi_{3}+e^{\phi}\tanh\phi_{2}\tanh\phi_{3})$
$\displaystyle-2\chi(\sinh\phi_{1}\sinh 2\phi_{3}\sinh
2\phi_{4}+\cosh^{2}\phi_{3}\sinh 2\phi_{4}\chi)-\cosh^{2}\phi_{1}\cosh
2\phi_{2}\times$ $\displaystyle\left.\sinh
2\phi_{4}-4\chi\cosh\phi_{1}\sinh\phi_{2}(\cosh\phi_{3}\cosh
2\phi_{4}+\sinh\phi_{3}\tanh\phi_{3})\right].$

We numerically solve these equations with some examples of possible solutions
given in figure 1.

(a) $\phi(r)$ solution

(b) $\chi(r)$ solution

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

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

(e) $\phi_{3}(r)$ solution

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

(g) $A(r)$ solution

(h) Scalar potential

Figure 1: Examples of $N=2$ RG flows from the $N=4$ SCFT with $SO(4)\times
SO(4)$ symmetry in the UV to non-conformal phases in the IR with $g=1$ and
$\alpha=\frac{\pi}{2}$.

The solutions interpolate between the supersymmetric $AdS_{4}$ vacuum with
$SO(4)\times SO(4)$ symmetry and singular geometries with diverging scalars.
In all solutions, we see that $V\rightarrow-\infty$ implying that all
singularities are physical by the criterion given in [50]. Indeed, we find the
scalar potential, for $\alpha=\frac{\pi}{2}$ and $g_{0}=-g$,

$V=-\frac{1}{2}e^{-\phi}g^{2}(1+e^{2\phi}+\chi^{2}+4e^{\phi}\cosh\phi_{1}\cosh\phi_{2}\cosh\phi_{3}\cosh\phi_{4})$
(73)

which is always bounded from above. Therefore, for any diverging behaviors of
scalar fields, all possible solutions in $SO(2)\times SO(2)\times SO(2)\times
SO(2)$ truncation are physically acceptable and describe RG flows from the
dual $N=4$ SCFT to various non-conformal phases in the IR. Using the scalar
masses $m^{2}L^{2}=-2$ and asymptotic behaviors near the $AdS_{4}$ critical
point

$\phi\sim\chi\sim\phi_{i}\sim e^{-gr}\sim e^{-\frac{r}{L}},\qquad i=1,2,3,4,$
(74)

we can determine that the flows are driven by relevant operators of dimensions
$\Delta=1,2$ and preserve $N=2$ supersymmetry and $SO(2)\times SO(2)\times
SO(2)\times SO(2)$ symmetry along the flows. This extends the results of [21],
in which only particular truncations to $SO(2)\times SO(2)\times SO(3)\times
SO(2)$ or $SO(2)\times SO(2)\times SO(2)\times SO(3)$ singlet scalars have
been considered, to the full $SO(2)\times SO(2)\times SO(2)\times SO(2)$
sector.

##  5 $SO(3)_{\textrm{diag}}\times SO(3)$ sector

In this section, we look at another scalar sector with residual symmetry
$SO(3)_{\textrm{diag}}\times SO(3)$. Since this sector has not previously been
studied, we will give the construction in more detail than the previous two
cases. The $36$ scalars in $SO(6,6)/SO(6)\times SO(6)$ coset transform under
the $SO(6)\times SO(6)$ compact subgroup as $(\mathbf{6},\mathbf{6})$. We
further decompose each $SO(6)$ factor into the $SO(3)\times SO(3)$ subgroup
under which the fundamental representation $\mathbf{6}$ decomposes as
$(\mathbf{3},\mathbf{1})+(\mathbf{1},\mathbf{3})$. The $36$ scalar fields then
transform under $SO(3)\times SO(3)\times SO(3)\times SO(3)$ as

$(\mathbf{3},\mathbf{3},\mathbf{1},\mathbf{1})+(\mathbf{3},\mathbf{1},\mathbf{1},\mathbf{3})+(\mathbf{1},\mathbf{3},\mathbf{3},\mathbf{1})+(\mathbf{1},\mathbf{3},\mathbf{1},\mathbf{3}).$
(75)

By taking a diagonal subgroup of the first three $SO(3)$ factors, we find

$\displaystyle(\mathbf{3}\times\mathbf{3},\mathbf{1})+(\mathbf{3},\mathbf{3})+(\mathbf{3}\times\mathbf{3},\mathbf{1})+(\mathbf{3},\mathbf{3})$
$\displaystyle\rightarrow$ $\displaystyle
2\times[(\mathbf{1},\mathbf{1})+(\mathbf{3},\mathbf{1})+(\mathbf{5},\mathbf{1})+(\mathbf{3},\mathbf{3})].$
(76)

Accordingly, there are two $SO(3)_{\textrm{diag}}\times SO(3)$ singlets
corresponding to the two $(\mathbf{1},\mathbf{1})$ representations. These two
singlets correspond to the following $SO(6,6)$ non-compact generators

$\hat{Y}_{1}=Y_{11}+Y_{22}+Y_{33}+Y_{44}\qquad\textrm{and}\qquad\hat{Y}_{3}=Y_{51}+Y_{62}+Y_{73}+Y_{84}\,.$
(77)

If we consider an even smaller $SO(3)_{\textrm{diag}}$ residual symmetry,
there are two additional singlets obtained from the last representation
$(\mathbf{3},\mathbf{3})$ in (76). These singlets correspond to the non-
compact generators

$\hat{Y}_{2}=Y_{15}+Y_{26}+Y_{37}+Y_{48}\qquad\textrm{and}\qquad\hat{Y}_{4}=Y_{55}+Y_{66}+Y_{77}+Y_{88}\,.$
(78)

We also note that $\hat{Y}_{1}$ and $\hat{Y}_{4}$ are
$SO(3)_{\textrm{diag}}\times SO(3)_{\textrm{diag}}$ singlets considered in
section 3. The coset representative for $SO(3)_{\textrm{diag}}$ sector can
then be written as

$\mathcal{V}=e^{\phi_{1}\hat{Y}_{1}}e^{\phi_{2}\hat{Y}_{2}}e^{\phi_{3}\hat{Y}_{3}}e^{\phi_{4}\hat{Y}_{4}}\,.$
(79)

However, it turns out that the resulting scalar potential and BPS equations
are highly complicated. Therefore, we refrain from giving the complete
analysis of this sector but simply note that the $A^{ij}_{1}$ tensor takes the
form

$A^{ij}_{1}=\textrm{diag}(\mathcal{A},\mathcal{B},\mathcal{B},\mathcal{B}).$
(80)

To make the analysis more traceable, we will perform further truncation to
$SO(3)_{\textrm{diag}}\times SO(3)$ singlet scalars by setting
$\phi_{2}=\phi_{4}=0$. Although this subtruncation leads to simpler
expressions for the results, there are still some new interesting features.
For simplicity of the results, in this section, we will set
$\alpha=\frac{\pi}{2}$ and $g_{0}=-g$. The $A^{ij}_{1}$ tensor for the
subtruncation still takes the form (80) with the eigenvalues given by

$\displaystyle\mathcal{A}$ $\displaystyle=$
$\displaystyle\frac{3}{4}e^{-\frac{\phi}{2}}\left[g(\cosh\phi_{1}+i\sinh\phi_{1}\sinh\phi_{3})^{3}-h_{1}\cos\beta_{1}(\sinh\phi_{1}+i\cosh\phi_{1}\sinh\phi_{3})^{3}\right.$
(81)
$\displaystyle+e^{\phi}[g\cosh^{3}\phi_{3}+h_{1}\sin\beta_{1}(i\sinh\phi_{1}-\cosh\phi_{1}\sinh\phi_{3})^{3}]$
$\displaystyle\left.+[h_{1}\sin\beta_{1}(\sinh\phi_{1}+i\cosh\phi_{1}\sinh\phi_{3})^{3}+ig\cosh^{3}\phi_{3}]\chi\right],$
$\displaystyle\mathcal{B}$ $\displaystyle=$
$\displaystyle\frac{3}{4}e^{-\frac{\phi}{2}}\left[g(\cosh\phi_{1}\sinh^{2}\phi_{1}\sinh^{2}\phi_{3}-i\cosh^{2}\phi_{1}\sinh\phi_{1}\sinh\phi_{3}\right.$
(82)
$\displaystyle+\cosh^{3}\phi_{1}+e^{\phi}\cosh^{3}\phi_{3}-i\sinh^{3}\phi_{1}\sinh^{3}\phi_{3}+i\cosh^{3}\phi_{3}\chi)$
$\displaystyle-
h_{1}(\sinh\phi_{1}-i\cosh\phi_{1}\sinh\phi_{3})^{2}(\sinh\phi_{1}+i\cosh\phi_{1}\sinh\phi_{3})\times$
$\displaystyle\left.(\cos\beta_{1}+ie^{\phi}\sin\beta_{1}-\sin\beta_{1}\chi)\right].$

We find that the first eigenvalue $\mathcal{A}$ gives rise to the
superpotential of the form

$\displaystyle\mathcal{W}$ $\displaystyle=$
$\displaystyle\frac{1}{2}e^{\frac{\phi}{2}}\left[g\cosh^{3}\phi_{3}+h_{1}\sin\beta_{1}(i\sinh\phi_{1}-\cosh\phi_{1}\sinh\phi_{3})^{3}\right]$
$\displaystyle+\frac{1}{2}e^{-\frac{\phi}{2}}\left[g(\cosh\phi_{1}+i\sinh\phi_{1}\sinh\phi_{3})^{3}-(\sinh\phi_{1}+i\cosh\phi_{1}\sinh\phi_{3})^{3}\times\right.$
$\displaystyle\left.h_{1}\cos\beta_{1}\right]+\frac{1}{2}e^{-\frac{\phi}{2}}\left[ig\cosh^{3}\phi_{3}+h_{1}\sin\beta_{1}(\sinh\phi_{1}+i\cosh\phi_{1}\sinh\phi_{3})^{3}\right]\chi\,.$

Accordingly, RG flow solutions will preserve only $N=1$ supersymmetry. It
should be noted that for $\phi_{3}=0$ or $\phi_{1}=0$, the two eigenvalues
$\mathcal{A}$ and $\mathcal{B}$ are equal leading to an enhanced $N=4$
supersymmetry.

The scalar potential can be written in terms of the superpotential as follows

$V=4\left(\frac{\partial
W}{\partial\phi}\right)^{2}+4e^{2\phi}\left(\frac{\partial
W}{\partial\chi}\right)^{2}+\frac{2}{3}\textrm{sech}^{2}\phi_{3}\left(\frac{\partial
W}{\partial\phi_{1}}\right)^{2}+\frac{2}{3}\left(\frac{\partial
W}{\partial\phi_{3}}\right)^{2}-3W^{2}\,.$ (84)

The explicit form of $V$ is given in the appendix. We note that only
$\beta_{1}$ appears in the results. We can also make another subtruncation by
setting $\phi_{1}=\phi_{3}=0$ in which only $\beta_{2}$ appears. This gives
similar results with $(\phi_{1},\phi_{3})$ and $(\phi_{2},\phi_{4})$ together
with $\beta_{1}$ and $\beta_{2}$ interchanged. On the other hand, if we
consider the full $SO(3)_{\textrm{diag}}$ sector, both $\beta_{1}$ and
$\beta_{2}$ appear in the scalar potential and the superpotential.

From the superpotential given above, we have not found any non-trivial
supersymmetric $AdS_{4}$ critical points for arbitrary values of $\beta_{1}$.
However, there are two $AdS_{4}$ vacua for particular values of $\beta_{1}=0$
and $\beta_{1}=\frac{\pi}{2}$. These are given by

$\displaystyle i:\,\,\,\beta_{1}=0;\qquad$
$\displaystyle\phi_{3}=\chi=0,\qquad\phi_{1}=\frac{1}{2}\ln\left[\frac{h_{1}+g}{h_{1}-g}\right],$
(85)
$\displaystyle\phi=-\frac{1}{2}\ln\left[1-\frac{g^{2}}{h_{1}^{2}}\right],\qquad
V_{0}=-\frac{3g^{2}h_{1}}{\sqrt{h_{1}^{2}-g^{2}}}$

and

$\displaystyle ii:\,\,\,\beta_{1}=\frac{\pi}{2};\qquad$
$\displaystyle\phi_{1}=\chi=0,\qquad\phi_{3}=\frac{1}{2}\ln\left[\frac{h_{1}+g}{h_{1}-g}\right],$
(86)
$\displaystyle\phi=\frac{1}{2}\ln\left[1-\frac{g^{2}}{h_{1}^{2}}\right],\qquad
V_{0}=-\frac{3g^{2}h_{1}}{\sqrt{h_{1}^{2}-g^{2}}}\,.$

Both of these critical points preserve $N=4$ supersymmetry since the two
eigenvalues of $A^{ij}_{1}$ are degenerate for $\phi_{1}=0$ or $\phi_{3}=0$ as
previously mentioned. Critical point $i$ preserves
$SO(3)_{\textrm{diag}}\times SO(3)_{\alpha}\times SO(3)_{2}$ with the
$SO(3)_{\textrm{diag}}$ being a diagonal subgroup of $SO(3)_{0}\times
SO(3)_{1}$. This is not a new $AdS_{4}$ critical point but one of the $N=4$
critical points identified in [20]. On the other hand, critical point $ii$
preserves $SO(3)_{0}\times SO(3)_{\textrm{diag}}\times SO(3)_{2}$ with the
$SO(3)_{\textrm{diag}}$ being a diagonal subgroup of $SO(3)_{\alpha}\times
SO(3)_{1}$. These two critical points appear to be related to each other by
interchanging scalars and $SO(3)$ factors within the unbroken symmetry as well
as a sign flip in the dilaton. However, the two vacua correspond to different
values of electric-magnetic phase $\beta_{1}$.

By the same procedure as in the previous sections, we find the BPS equations
of the form

$\displaystyle\phi^{\prime}$ $\displaystyle=$ $\displaystyle-4\frac{\partial
W}{\partial\phi},\qquad\chi^{\prime}=-4e^{2\phi}\frac{\partial
W}{\partial\chi},$ $\displaystyle\phi_{1}^{\prime}$ $\displaystyle=$
$\displaystyle-\frac{2}{3}\textrm{sech}^{2}\phi_{3}\frac{\partial
W}{\partial\phi_{1}},\qquad\phi_{3}^{\prime}=-\frac{2}{3}\frac{\partial
W}{\partial\phi_{3}},\qquad A^{\prime}=W\,.$ (87)

The explicit form of these equations is rather long and given in the appendix.

###  5.1 $N=4$ holographic RG flows

We begin with holographic RG flow solutions preserving $N=4$ supersymmetry
obtained by truncating out the axion $\chi$ together with one of the two
scalars $\phi_{1}$ and $\phi_{3}$. Although the solution interpolating between
the trivial critical point and critical point $i$ has already been given in
[20], it is useful to repeat it here in the present convention for the sake of
comparison with the $N=1$ solutions given later on.

With $\phi_{3}=\chi=0$, the BPS equations reduce to

$\displaystyle\phi^{\prime}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}e^{-\frac{\phi}{2}}\left[4ge^{\phi}+h_{1}\sinh
3\phi_{1}-3h_{1}\sinh\phi_{1}-g(3\cosh\phi_{1}+\cosh 3\phi_{1})\right],\qquad$
(88) $\displaystyle\phi_{1}^{\prime}$ $\displaystyle=$
$\displaystyle-e^{-\frac{\phi}{2}}\cosh\phi_{1}\sinh\phi_{1}(g\cosh\phi_{1}-h_{1}\sinh\phi_{1}),$
(89) $\displaystyle A^{\prime}$ $\displaystyle=$
$\displaystyle\frac{1}{2}e^{-\frac{\phi}{2}}(ge^{\phi}+g\cosh^{3}\phi_{1}-h_{1}\sinh^{3}\phi_{1}).$
(90)

The solution can be found by first combining $\phi^{\prime}$ and
$\phi^{\prime}_{1}$ equation and finding the solution for $\phi$ as a function
of $\phi_{1}$. The result is given by

$\phi=\ln\left[\frac{g\cosh\phi_{1}-h_{1}\sinh\phi_{1}}{g\cosh
2\phi_{1}+C_{1}\sinh 2\phi_{1}}\right].$ (91)

We readily see that $\phi\rightarrow 0$ as $\phi_{1}\rightarrow 0$. To make
the solution end at the IR fixed point given by critical point $i$, we choose
the integration constant to be

$C_{1}=-\frac{g^{2}+h_{1}^{2}}{2}$ (92)

resulting in

$\phi=-\ln\left[\cosh\phi_{1}-\frac{g}{h_{1}}\sinh\phi_{1}\right].$ (93)

Similarly, we find the solution for $A$ as follows

$A=\frac{1}{2}\ln(g\sinh\phi_{1}-h_{1}\cosh\phi_{1})+\ln(g\cosh\phi_{1}-h_{1}\sinh\phi_{1})-\ln\sinh
2\phi_{1}\,.$ (94)

Finally, using the previous results and changing to a new radial coordinate
$\rho$ given by $\frac{d\rho}{dr}=e^{-\frac{\phi}{2}}$, we find

$gh_{1}(\rho-\rho_{0})=h_{1}\ln\coth\frac{\phi_{1}}{2}+2\sqrt{h_{1}^{2}-g^{2}}\tanh^{-1}\left[\frac{g\tanh\frac{\phi_{1}}{2}-h_{1}}{\sqrt{h_{1}^{2}-g^{2}}}\right]-2\tan^{-1}\tanh\frac{\phi_{1}}{2}$
(95)

with $\rho_{0}$ being an integration constant. Near the UV and IR fixed
points, the asymptotic behaviors of the scalar fields are given respectively
by

$\phi\sim\phi_{1}\sim e^{-gr}\sim e^{-\frac{r}{L}},\qquad L=\frac{1}{g}$ (96)

and

$\phi\sim e^{-\frac{r}{L_{i}}},\qquad\phi_{1}\sim e^{\frac{r}{L_{i}}},\qquad
L_{i}=\frac{1}{g}\left(1-\frac{g^{2}}{h_{1}^{2}}\right)^{\frac{1}{4}}\,.$ (97)

Accordingly, the flow is driven by relevant operators of dimensions
$\Delta=1,2$. In the IR, the operator dual to $\phi_{1}$ becomes irrelevant
with dimension $\Delta=4$ while that dual to $\phi$ is still relevant.

Similarly, by the same procedure, we can find a flow solution interpolating
between the trivial $AdS_{4}$ critical point and ciritcal point $ii$. In this
case, the BPS equations read

$\displaystyle\phi_{3}^{\prime}$ $\displaystyle=$
$\displaystyle-e^{\frac{\phi}{2}}\cosh\phi_{3}\sinh\phi_{3}(g\cosh\phi_{3}-h_{1}\sinh\phi_{3}),\quad$
(98) $\displaystyle\phi^{\prime}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}e^{\frac{\phi}{2}}[g(2\cosh\phi_{3}+\cosh
3\phi_{3})+h_{1}(3\sinh\phi_{3}-\sinh 3\phi_{3})-4ge^{-\phi}],$ (99)
$\displaystyle A^{\prime}$ $\displaystyle=$
$\displaystyle\frac{1}{2}e^{-\frac{\phi}{2}}[g+e^{\phi}(g\cosh^{3}\phi_{3}-h_{1}\sinh^{3}\phi_{3})]$
(100)

with the solution given by

$\displaystyle\phi$ $\displaystyle=$
$\displaystyle\ln\left[\cosh\phi_{3}-\frac{g}{h_{1}}\sinh\phi_{3}\right],$
(101) $\displaystyle A$ $\displaystyle=$
$\displaystyle\frac{1}{2}\ln(g\sinh\phi_{3}-h_{1}\cosh\phi_{3})+\ln(g\cosh\phi_{3}-h_{1}\sinh\phi_{3})$
(102) $\displaystyle-\ln\sinh 2\phi_{3},$ $\displaystyle g(\rho-\rho_{0})$
$\displaystyle=$ $\displaystyle\ln(g^{2}+h_{1}^{2}-2gh_{1}\coth 2\phi_{3})$
(103)

with $\rho$ defined as in the previous case. The asymptotic behaviors and
holographic interpretations are also similar. Furthermore, the solution for
$\phi_{3}$ can be rewritten in a similar form as (95) by changing to another
radial coordinate $\eta$ given by $\frac{d\eta}{d\rho}=e^{\phi}$ resulting in

$\displaystyle gh_{1}(\eta-\eta_{0})$ $\displaystyle=$ $\displaystyle
h_{1}\ln\coth\frac{\phi_{3}}{2}+2\sqrt{h_{1}^{2}-g^{2}}\tanh^{-1}\left[\frac{g\tanh\frac{\phi_{3}}{2}-h_{1}}{\sqrt{h_{1}^{2}-g^{2}}}\right]$
(104) $\displaystyle-2\tan^{-1}\tanh\frac{\phi_{3}}{2}\,.$

In these two solutions, we see that the operators dual to $\phi$ and
$\phi_{1}$ break conformal symmetry but preserve $N=4$ Poincare supersymmetry
in three dimensions.

###  5.2 $N=1$ holographic RG flows

We now consider holographic RG flows in the full $SO(3)_{\textrm{diag}}\times
SO(3)$ sector. We first point out that setting $\chi=0$ still gives
$\mathcal{A}\neq\mathcal{B}$ resulting in $N=1$ supersymmetry. However,
truncating out only $\chi$ is not consistent with the BPS equations given in
(87) unless $\phi_{1}=0$ or $\phi_{3}=0$. Accordingly, $N=1$ RG flow solutions
to critical points $i$ or $ii$ involve all scalars in the
$SO(3)_{\textrm{diag}}\times SO(3)$ sector. This makes finding the solutions
more difficult, so we will numerically give some examples of possible
solutions.

Using the BPS equations given in the appendix, we find an RG flow solution
interpolating between the trivial $N=4$ fixed point with $SO(4)\times SO(4)$
symmetry and critcal point $i$ as shown in figure 2. Since all scalars have
the same mass $m^{2}L^{2}=-2$ at the $SO(4)\times SO(4)$ critical point, the
flow is again driven by relevant operators of dimensions $\Delta=1,2$. In the
IR, using the scalar masses given in [20], we find that $\phi_{1}$ is dual to
an irrelevant operator of dimension $\Delta=4$, but $\phi$, $\chi$ and
$\phi_{3}$ are dual to relevant operators of dimensions $\Delta=1,2$. Unlike
the $N=4$ solutions given above, in addition to breaking conformal symmetry,
turning on the operators dual to $\chi$ and $\phi_{3}$ along the flow further
breaks the $N=4$ Poincare supersymmetry to $N=1$. However, at the IR fixed
point, the conformal symmetry is restored, and the supersymmetry is enhanced
to $N=4$ due to the vanishing of $\chi$ and $\phi_{3}$.

(a) $\phi(r)$ solution

(b) $\chi(r)$ solution

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

(d) $\phi_{3}(r)$ solution

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

Figure 2: An $N=1$ RG flow from the $N=4$ SCFT with $SO(4)\times SO(4)$
symmetry to an $N=4$ conformal fixed point in the IR with
$SO(3)_{\textrm{diag}}\times SO(3)\times SO(3)$ symmetry for $\beta_{1}=0$,
$g=1$ and $h_{1}=2$.

A similar $N=1$ flow solution from the $SO(4)\times SO(4)$ fixed point to
critical point $ii$ can also be found. This is shown in figure 3. For other
values of the phase $\beta_{1}$, we have not found any non-trivial $AdS_{4}$
critical points. Examples of RG flows from the $SO(4)\times SO(4)$ fixed point
to non-conformal phases are given in figure 4. There are also RG flows from
$AdS_{4}$ critical points $i$ and $ii$ to non-conformal phases. Examples of
these solutions are given in figures 5 and 6. Unlike the $N=2$ RG flows given
in the previous section, these $N=1$ RG flows turn out to be unphysical
according to the criterion of [50] due to $V\rightarrow\infty$ as seen from
the figure. It could be interesting to see whether these singularities are
physical in the (if any) uplifted solutions to ten or eleven dimensions.

(a) $\phi(r)$ solution

(b) $\chi(r)$ solution

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

(d) $\phi_{3}(r)$ solution

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

Figure 3: An $N=1$ RG flow from the $N=4$ SCFT with $SO(4)\times SO(4)$
symmetry to an $N=4$ conformal fixed point in the IR with $SO(3)\times
SO(3)_{\textrm{diag}}\times SO(3)$ symmetry for $\beta_{1}=\frac{\pi}{2}$,
$g=1$ and $h_{1}=2$.

(a) $\phi(r)$ solution

(b) $\chi(r)$ solution

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

(d) $\phi_{3}(r)$ solution

(e) $A(r)$ solution

(f) Scalar potential

Figure 4: Examples of $N=1$ RG flows from the $N=4$ SCFT with $SO(4)\times
SO(4)$ symmetry in the UV to non-conformal phases in the IR for different
values of the electric-magnetic phase $\beta_{1}=0$ (red), $\frac{\pi}{6}$
(green), $\frac{\pi}{4}$ (blue), $\frac{\pi}{3}$ (purple),$\frac{\pi}{2}$
(pink) with $g=1$ and $h_{1}=2$.

(a) $\phi(r)$ solution

(b) $\chi(r)$ solution

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

(d) $\phi_{3}(r)$ solution

(e) $A(r)$ solution

(f) Scalar potential

Figure 5: Examples of $N=1$ RG flows from the $N=4$ SCFT with
$SO(3)_{\textrm{diag}}\times SO(3)\times SO(3)$ symmetry (critical point $i$)
to non-conformal phases in the IR with $\beta_{1}=0$, $g=1$ and $h_{1}=2$.

(a) $\phi(r)$ solution

(b) $\chi(r)$ solution

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

(d) $\phi_{3}(r)$ solution

(e) $A(r)$ solution

(f) Scalar potential

Figure 6: Examples of $N=1$ RG flows from the $N=4$ SCFT with $SO(3)\times
SO(3)_{\textrm{diag}}\times SO(3)$ symmetry (critical point $ii$) to non-
conformal phases in the IR with $\beta_{1}=\frac{\pi}{2}$, $g=1$ and
$h_{1}=2$.

##  6 Conclusions and discussions

In this paper, we have studied symplectically deformed $N=4$ gauged
supergravity with $SO(4)\times SO(4)\sim SO(3)\times SO(3)\times SO(3)\times
SO(3)$ gauge group with two independent electric-magnetic phases. We have
considered three scalar sectors invariant under $SO(3)_{\textrm{diag}}\times
SO(3)_{\textrm{diag}}$, $SO(2)\times SO(2)\times SO(2)\times SO(2)$ and
$SO(3)_{\textrm{diag}}\times SO(3)$ subgroups of $SO(4)\times SO(4)$. Similar
to the $\omega$-deformed $SO(8)$ maximal gauged supergravity, for the trivial
supersymmetric $AdS_{4}$ vacuum at the origin of the scalar manifold, the
cosmological constant and scalar masses are independent from the electric-
magnetic phases. However, unlike the $\omega$-deformed $SO(8)$ gauged
supergravity, it turns out that other $AdS_{4}$ critical points are the same
or related to those identified previously in [20] for the “undeformed”
$SO(4)\times SO(4)$ gauge group. Although we have not found any genuinely new
supersymmetric $AdS_{4}$ vacua, we have given a large number of new
holographic RG flows preserving $N=2$ and $N=1$ supersymmetries.

The $N=2$ solutions decribe holographic RG flows from the dual $N=4$ CSM
theory to non-conformal phases in the IR driven by relevant operators of
dimensions $\Delta=1,2$. The $SO(4)\times SO(4)$ symmetry is broken down to
$SO(2)\times SO(2)\times SO(2)\times SO(2)$ along the flows through the IR
phases. We have found that all these singular solutions describe physical RG
flows in the dual field theories since the singularities are physically
acceptable by the criterion of [50]. Moreover, we have also shown that all
non-conformal flows within this sector are physical in the sense that all
types of singularities lead to the scalar potential that is bounded from
above. These solutions also generalize those given recently in [21] within an
$SO(2)\times SO(2)\times SO(3)\times SO(2)$ subtruncation. However, in this
$SO(2)\times SO(2)\times SO(2)\times SO(2)$ sector, no nontrivial $AdS_{4}$
critical points appear, so there are no RG flows between confornal fixed
points. In addition, no non-trivial electric-magnetic phases appear in the
analysis. Given that all values of the phase $\alpha>0$ give rise to
equivalent gauged supergravities, this sector is essentially the same as the
undeformed $SO(4)\times SO(4)$ gauged supergravity considered in [20] and
[21].

For $N=1$ solutions within $SO(3)_{\textrm{diag}}\times SO(3)$ sector, the
phase $\beta_{1}$ for an $SO(3)$ factor in the vector multiplets appears. We
have found three $N=4$ supersymmetric $AdS_{4}$ critical points with one of
them being the trivial $AdS_{4}$ critical point. The remaining two non-trivial
critical points exist for particular values of $\beta_{1}=0$ and
$\beta_{1}=\frac{\pi}{2}$. The first one preserves
$SO(3)_{\textrm{diag}}\times SO(3)\times SO(3)$ symmetry identified in [20].
The second one with $SO(3)\times SO(3)_{\textrm{diag}}\times SO(3)$ symmetry
is very similar to the first critical point and should be related by electric-
magnetic duality. We have studied $N=1$ supersymmetric RG flows between the
trivial $AdS_{4}$ vacuum to these two non-trivial critical points similar to
the $N=4$ RG flows given previously in [20]. These flows are driven by
relevant operators of dimensions $\Delta=1,2$ and preserve $N=1$ supersymmetry
along the flows. At both the UV and IR fixed points, the supersymmetry
enhances to $N=4$. For other values of the phase $\beta_{1}$, we have not
found any non-trivial $AdS_{4}$ critical point. An intensed numerical search
suggests that there are no other supersymmetric $AdS_{4}$ vacua in this
sector. We have also given a number of holographic RG flows from $AdS_{4}$
critical points to various types of non-conformal phases. Unlike the $N=2$
solutions, it turns out that all these flows are unphysical.
Similar to the $\omega$-deformed $SO(8)$ gauged supergravity, the $N=4$ gauged
supergravity considered here currently has no higher dimensional origin. It
would be interesting to find the embedding of this gauged supergravity in ten
or eleven dimensions. The relevant consistent truncation ansatze could be
obtained by using double field theory at $SL(2)$ angles developed in [46]
similar to the embedding of half-maximal gauged supergravities in higher
dimensions studied in [51, 52, 53, 54, 55, 56]. These could be used to uplift
the solutions given here to ten/eleven dimensions resulting in a complete
AdS4/CFT3 holography in the framework of string/M-theory. In particular, the
unphysical singularities of $N=1$ non-conformal RG flows might be resolved in
ten/eleven dimensions and give rise to genuine gravity duals of three-
dimensional field theories. It would also be interesting to identify the dual
$N=4$ SCFTs and relevant deformations dual to the solutions given in this
paper. In addition, the $SO(3)_{\textrm{diag}}$ sector in which both the
phases $\beta_{1}$ and $\beta_{2}$ appear deserves further study and might
lead to new $AdS_{4}$ vacua. Finally, other types of solutions such as Janus
solutions and $AdS_{4}$ black holes are also worth considering.
Acknowledgement
This work is funded by National Research Council of Thailand (NRCT) and
Chulalongkorn University under grant N42A650263. The author would like to
thank G. Inverso for a useful correspondence and T. Assawasowan for
collaboration in a related project.

##  Appendix A Useful formulae

In this appendix, we give some formulae used in the main text in particular
the convention on ’t Hooft matrices and explicit forms of the scalar potential
and BPS equations in $SO(3)_{\textrm{diag}}\times SO(3)$ sector.

###  A.1 ’t Hooft matrices

We use the following representation of the ’t Hooft matrices

$\displaystyle G_{1}^{ij}$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{cccc}0&1&0&0\\\ -1&0&0&0\\\ 0&0&0&1\\\
0&0&-1&0\\\ \end{array}\right],\qquad
G_{2}^{ij}=\left[\begin{array}[]{cccc}0&0&1&0\\\ 0&0&0&-1\\\ -1&0&0&0\\\
0&1&0&0\\\ \end{array}\right],$ (113) $\displaystyle G_{3}^{ij}$
$\displaystyle=$ $\displaystyle\left[\begin{array}[]{cccc}0&0&0&1\\\
0&0&1&0\\\ 0&-1&0&0\\\ -1&0&0&0\\\ \end{array}\right],\qquad
G_{4}^{ij}=\left[\begin{array}[]{cccc}0&i&0&0\\\ -i&0&0&0\\\ 0&0&0&-i\\\
0&0&i&0\\\ \end{array}\right],$ (122) $\displaystyle G_{5}^{ij}$
$\displaystyle=$ $\displaystyle\left[\begin{array}[]{cccc}0&0&i&0\\\
0&0&0&i\\\ -i&0&0&0\\\ 0&-i&0&0\\\ \end{array}\right],\qquad
G_{6}^{ij}=\left[\begin{array}[]{cccc}0&0&0&i\\\ 0&0&-i&0\\\ 0&i&0&0\\\
-i&0&0&0\\\ \end{array}\right].$ (131)

These matrices satisfy the relations

$G_{mij}=(G^{ij}_{m})^{*}=\frac{1}{2}\epsilon_{ijkl}G^{kl}_{m}\,.$ (132)

###  A.2 Scalar potential in $SO(3)_{\textrm{diag}}\times SO(3)$ sector

The scalar potential in $SO(3)_{\textrm{diag}}\times SO(3)$ sector is
explicitly given by

$\displaystyle V$ $\displaystyle=$
$\displaystyle\frac{1}{64}g^{2}e^{-\phi}\left[(3+\cosh 2\phi_{1}+2\cosh
2\phi_{3}\sinh^{2}\phi_{1})^{2}(2\cosh 2\phi_{3}\sinh^{2}\phi_{1}\right.$
(133) $\displaystyle\left.+\cosh
2\phi_{2}-3)-16\sinh^{3}\phi_{1}\sinh^{3}2\phi_{3}\chi+32\cosh^{4}\phi_{3}\chi^{2}(\cosh
2\phi_{3}-2)\right]$
$\displaystyle+\frac{1}{64}e^{-\phi}h_{1}^{2}(2\cosh^{2}\phi_{1}\cosh
2\phi_{3}+\cosh 2\phi_{1}-3)^{2}(3+2\cosh^{2}\phi_{1}\cosh 2\phi_{3}$
$\displaystyle+\cosh
2\phi_{1})(\cos^{2}\beta_{1}+\sin^{2}\beta_{1}(e^{2\phi}+\chi^{2})-\sin
2\beta_{1}\chi)-2g^{2}\cosh^{3}\phi_{1}\cosh^{3}\phi_{3}$
$\displaystyle-\frac{1}{4}e^{-\phi}gh_{1}\left[8\cos\beta_{1}\cosh^{3}\phi_{1}\cosh^{6}\phi_{3}\sinh^{3}\phi_{1}-8e^{\phi}\sin\beta_{1}\sinh^{3}\phi_{3}\right.$
$\displaystyle+8e^{\phi}\cosh^{3}\phi_{3}(e^{\phi}\sin\beta_{1}\cosh^{3}\phi_{1}\sinh^{3}\phi_{3}-\cos\beta_{1}\sinh^{3}\phi_{1})$
$\displaystyle-\chi(\sin\beta_{1}\cosh^{6}\phi_{3}\sinh^{3}2\phi_{1}+\cos\beta_{1}\cosh^{3}\phi_{1}\sinh^{3}2\phi_{3})$
$\displaystyle\left.+\sin\beta_{1}\cosh^{3}\phi_{1}\sinh^{3}2\phi_{3}\chi^{2}\right]+\frac{1}{2}g^{2}e^{\phi}\cosh^{4}\phi_{3}(\cosh
2\phi_{3}-2).$

###  A.3 BPS equations in $SO(3)_{\textrm{diag}}\times SO(3)$ sector

In this section, we collect all the BPS equations in
$SO(3)_{\textrm{diag}}\times SO(3)$ sector. These are given by

$\displaystyle W\phi^{\prime}$ $\displaystyle=$
$\displaystyle\frac{1}{2}g^{2}e^{-\phi}\left[(\cosh^{2}\phi_{1}+\sinh^{2}\phi_{1}\sinh^{2}\phi_{3})^{3}+\cosh^{6}\chi^{2}-e^{2\phi}\cosh^{6}\phi_{3}\phantom{\frac{1}{4}}\right.$
(134)
$\displaystyle\left.+\left(6\cosh^{2}\phi_{1}\cosh^{3}\phi_{3}\sinh\phi_{1}\sinh\phi_{3}-\frac{1}{4}\sinh^{3}\phi_{1}\sinh^{3}2\phi_{3}\right)\chi\right]$
$\displaystyle+\frac{1}{256}h_{1}^{2}e^{-\phi}\left[1+\cos
2\beta_{1}-2e^{2\phi}\sin^{2}\beta_{1}+2\sin\beta_{1}\chi(\sin\beta_{1}\chi-2\cos\beta_{1})\right]\times$
$\displaystyle(2\cosh^{2}\phi_{1}\cosh 2\phi_{3}+\cosh
2\phi_{1}-3)^{3}+\frac{1}{64}gh_{1}e^{-\phi}\left[-2\cos\beta_{1}\cosh^{6}\phi_{3}\times\right.$
$\displaystyle\sinh
6\phi_{1}+8(\cosh^{2}\phi_{1}\sinh^{3}2\phi_{3}-24\cosh^{3}\phi_{3}\sinh^{2}\phi_{1}\sinh\phi_{3})\times$
$\displaystyle
e^{2\phi}\sin\beta_{1}\cosh\phi_{1}+\chi\left[-6\sin\beta_{1}\sinh
2\phi_{1}\sinh^{2}2\phi_{3}(3+\cosh 4\phi_{1})\right.$
$\displaystyle+8\cos\beta_{1}\cosh^{3}\phi_{1}\sinh^{3}2\phi_{3}+\cosh^{2}\phi_{3}\left\\{\sin\beta_{1}\sinh^{3}2\phi_{1}(\cosh
4\phi_{3}\right.$ $\displaystyle\left.+28\cosh
2\phi_{3}-21)-96\cos\beta_{1}\cosh\phi_{1}\sinh^{2}\phi_{1}\sinh
2\phi_{3}\right\\}$
$\displaystyle\left.-64\sin\beta_{1}\cosh\phi_{1}\cosh^{3}\phi_{3}\sinh\phi_{3}\chi(\cosh^{2}\phi_{1}\sinh^{2}\phi_{3}-3\sinh^{2}\phi_{1})\right]$
$\displaystyle\left.+\frac{3}{4}\cos\beta_{1}\cosh^{2}\phi_{3}\sinh
2\phi_{1}(\cosh 4\phi_{3}+68\cosh 2\phi_{3}-61)\right],$ $\displaystyle
W\chi^{\prime}$ $\displaystyle=$ $\displaystyle
h_{1}^{2}e^{\phi}\sin\beta_{1}(\cos\beta_{1}-\sin\beta_{1}\chi)\left(\sinh^{2}\phi_{1}+\cosh^{2}\phi_{1}\sinh^{2}\phi_{3}\right)^{3}$
$\displaystyle-g^{2}e^{\phi}\cosh^{3}\phi_{3}\left(3\cosh^{2}\phi_{1}\sinh\phi_{1}\sinh\phi_{3}-\sinh^{3}\phi_{1}\sinh^{3}\phi_{3}+\cosh^{3}\chi\right)$
$\displaystyle+\frac{1}{8}gh_{1}e^{\phi}\cosh\phi_{1}\left[16\sin\beta_{1}\sinh\phi_{3}\cosh^{3}\phi_{3}\chi(\cosh^{2}\phi_{1}\sinh^{2}\phi_{3}-3\sinh^{2}\phi_{1})\right.$
$\displaystyle-\cosh^{2}\phi_{1}\cosh^{2}\phi_{3}\left\\{\sin\beta_{1}\sinh^{3}\phi_{1}(\cosh
4\phi_{3}+28\cosh 2\phi_{3}-21)\right.$
$\displaystyle\left.+8\cos\beta_{1}\cosh\phi_{3}\sinh^{3}\phi_{3}\right\\}+\frac{3}{2}\sinh\phi_{1}\sinh
2\phi_{3}\times$
$\displaystyle\left.\\{8\cos\beta_{1}\cosh^{2}\phi_{3}\sinh\phi_{1}+(3+\cosh
4\phi_{1})\sin\beta_{1}\sinh 2\phi_{3}\\}\right],$ (136) $\displaystyle
W\phi_{1}^{\prime}$ $\displaystyle=$
$\displaystyle\frac{1}{32}g^{2}e^{-\phi}\left[\left\\{\sinh\phi_{1}\sinh
2\phi_{1}\sinh 4\phi_{3}-(\cosh\phi_{1}+7\cosh 3\phi_{1})\sinh
2\phi_{3}\right\\}\chi\right.$ (137) $\displaystyle+e^{\phi}\left\\{(3\cosh
3\phi_{3}-7\cosh\phi_{3})\sinh
3\phi_{1}-4\cosh^{3}\phi_{3}\sinh\phi_{1}\right\\}$
$\displaystyle\left.-16\cosh^{5}\phi_{1}\sinh\phi_{1}-4\sinh^{3}2\phi_{1}\sinh^{2}\phi_{3}-16\cosh\phi_{1}\sinh^{5}\phi_{1}\sinh^{4}\phi_{3}\right]$
$\displaystyle-\frac{1}{128}h_{1}^{2}e^{-\phi}\left\\{1+\cos
2\beta_{1}+2e^{2\phi}\sin^{2}\beta_{1}+2\sin\beta_{1}\chi(\sin\beta_{1}\chi-2\cos\beta_{1})\right\\}\times$
$\displaystyle\sinh 2\phi_{1}(2\cosh^{2}\phi_{1}\cosh 2\phi_{3}+\cosh
2\phi_{1}-3)^{2}$
$\displaystyle+\frac{1}{16}gh_{1}e^{-\phi}\left[\cos\beta_{1}\left\\{\cosh^{2}\phi_{1}\cosh
2\phi_{1}\sinh^{2}\phi_{1}(\cosh 4\phi_{3}+8\cosh 2\phi_{3}-1)\right.\right.$
$\displaystyle\left.-8\sinh^{2}\phi_{3}(\cosh^{6}\phi_{1}+\sinh^{6}\phi_{1})\sinh^{2}\phi_{3}\right\\}-8e^{\phi}\cosh\phi_{1}\cosh\phi_{3}\times$
$\displaystyle\left\\{\cos\beta_{1}\sinh^{2}\phi_{1}(\cosh
2\phi_{3}-2)+\cos\beta_{1}\cosh^{2}\phi_{1}\sinh^{2}\phi_{3}\right.$
$\displaystyle\left.+\sin\beta_{1}\cosh 2\phi_{1}\sinh\phi_{1}\sinh
2\phi_{3}\right\\}-e^{2\phi}\sin\beta_{1}\sinh\phi_{1}\times$
$\displaystyle\\{3+7\cosh 2\phi_{1}\sinh 2\phi_{3}-\cosh^{2}\phi_{1}\sinh
4\phi_{3}\\}-\frac{1}{16}\sin\beta_{1}\chi\cosh 2\phi_{1}\times$
$\displaystyle\left\\{58+6\cosh 4\phi_{1}+8\cosh 2\phi_{3}(\cosh
4\phi_{1}-9)+4\cosh 4\phi_{3}\sinh^{2}2\phi_{1}\right\\}$
$\displaystyle+\sin\beta_{1}\sinh\phi_{1}\chi^{2}\left\\{\cosh^{2}\phi_{1}\sinh
4\phi_{4}-\sinh 2\phi_{3}(3+7\cosh 2\phi_{1})\right\\}$
$\displaystyle\left.+\cos\beta_{1}\sinh\phi_{1}\sinh 2\phi_{3}\chi(3+7\cosh
2\phi_{1}-2\cosh^{2}\phi_{1}\cosh 2\phi_{3})\right],$ $\displaystyle
W\phi_{3}^{\prime}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}e^{-\phi}\cosh\phi_{3}\left[\sinh\phi_{3}\left\\{2e^{2\phi}\cosh^{4}\phi_{3}+e^{\phi}\cosh\phi_{1}\cosh\phi_{3}[2\cosh^{2}\phi_{1}\right.\right.$
$\displaystyle\left.+\sinh^{2}\phi_{1}(1-5\cosh
2\phi_{3})]+2\sinh^{2}\phi_{1}(\cosh^{2}\phi_{1}+\sinh^{2}\phi_{1}\sinh^{2}\phi_{3})^{2}\right\\}$
$\displaystyle+2\cosh\phi_{3}\chi\left\\{\cosh^{3}\phi_{3}\sinh\phi_{3}\chi-\cosh
2\phi_{3}\sinh^{3}\phi_{1}\sinh^{2}\phi_{3}\right.$
$\displaystyle\left.\left.-\cosh^{2}\phi_{1}\sinh\phi_{1}(1-2\cosh
2\phi_{3})\right\\}\right]-\frac{1}{2}h_{1}^{2}e^{-\phi}\cosh^{2}\phi_{1}\cosh\phi_{3}\sinh\phi_{3}\times$
$\displaystyle(\sinh^{2}\phi_{1}+\cosh^{2}\phi_{1}\sinh^{2}\phi_{3})^{2}\left[e^{2\phi}\sin^{2}\beta_{1}+(\cos\beta_{1}-\sin\beta_{1}\chi)^{2}\right]$
$\displaystyle+\frac{1}{256}gh_{1}e^{-\phi}\cosh\phi_{3}\left[32e^{2\phi}\sin\beta_{1}\cosh\phi_{1}\left\\{4\cosh\phi_{3}\sinh^{2}\phi_{1}\times\right.\right.$
$\displaystyle\left.(1-2\cosh 2\phi_{3})+\cosh^{2}\phi_{1}\sinh\phi_{3}\sinh
4\phi_{3}\right\\}-2e^{\phi}\left\\{33\sin\beta_{1}\right.$
$\displaystyle\left.+\sin\beta_{1}[40\cosh^{2}\phi_{1}\cosh
4\phi_{3}\sinh^{2}\phi_{1}-\cosh 4\phi_{1}+4\cosh 2\phi_{3}(\cosh
4\phi_{1}-9)]\right.$
$\displaystyle\left.+8\cos\beta_{1}\sinh\phi_{1}[(1-3\cosh 2\phi_{1})\sinh
2\phi_{3}+5\cosh^{2}\phi_{1}\sinh 4\phi_{3}]\right\\}$
$\displaystyle+\cos\beta_{1}\sinh 2\phi_{1}\left\\{2\sinh\phi_{3}(31+8\cosh
4\phi_{1}\cosh^{4}\phi_{3})-67\sinh 3\phi_{3}\right.$
$\displaystyle\left.-\sinh
5\phi_{3}\right\\}+\chi\left\\{\sin\beta_{1}\sinh\phi_{3}[-8\cosh^{4}\phi_{3}\sinh
6\phi_{1}\right.$ $\displaystyle+(9+140\cosh 2\phi_{3}+3\cosh 4\phi_{3})\sinh
2\phi_{1}]+64\cos\beta_{1}\cosh\phi_{1}\times$ $\displaystyle\left\\{2\cosh
3\phi_{3}\sinh^{2}\phi_{1}-\cosh^{2}\phi_{1}(\cosh\phi_{3}+\cosh
3\phi_{3})\sinh^{2}\phi_{3}\right\\}$
$\displaystyle\left.\left.+4\sin\beta_{1}\chi\left\\{4\cosh^{3}\phi_{1}\cosh
5\phi_{3}+(5\cosh\phi_{1}-9\cosh 3\phi_{1})\cosh
3\phi_{3}\right\\}\right\\}\right]$

together with an equation for the metric function $A^{\prime}=W$.

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