$\textstyle\partial_{t}^{2}\mathbf{D}+\nabla\times\frac{1}{\mu}\nabla\times\frac{1}{\epsilon}\mathbf{D}=F,\qquad\mathbf{D}(0,\cdot)=\mathbf{D}_{0},\
\
\partial_{t}\mathbf{D}(0,\cdot)=\mathbf{D}_{1},\quad\nabla\cdot\mathbf{D}_{0}=\nabla\cdot\mathbf{D}_{1}=\nabla\cdot
F=0.$ (6.9)

Recall that the solution to the above problem is given by

$\mathbf{D}(t)=\cos(t\sqrt{H})\mathbf{D}_{0}+\sin(t\sqrt{H})H^{-1/2}\mathbf{D}_{1}+H^{-1/2}\int_{0}^{t}\sin((t-s)\sqrt{H})F(s)ds.$
(6.10)

We denote by $\mathbf{D}^{h}$ the solution to the problem with $F=0$ and by
$\mathbf{D}^{i}$ that one with $\mathbf{D}_{0}=\mathbf{D}_{1}=0$.

For $F=0$, the conditional Strichartz estimate in Proposition 6.2 and the
smoothing estimates in Propositions 5.5 and 5.6 easily yield the Strichartz
inequality for $\mathbf{D}^{h}$. The usual $TT^{*}$ argument then allows us to
bound $\partial_{t}\mathbf{D}^{i}$ and $\sqrt{H}\mathbf{D}^{i}$ in
$L^{p}\dot{H}^{-2/p}_{q}$. To replace here $\sqrt{H}$ by $\nabla$, one would
need a variant of Lemma 5.3 in $\dot{H}^{-2/p}_{q}$ which should require a
substantial effort. We by-pass this difficulty by means of a modified $TT^{*}$
argument that also uses ideas from Proposition 5.5. This is possible since we
only have to control an error term in $L^{2}L^{2}_{-1/2-}$ arising from
Proposition 6.2. Here it turns out to be enough to estimate $\nabla H^{-1/2}$
in $L^{2}_{-1/2-}$ just using Lemma 5.3.

######  Theorem 6.3.

Under the assumptions of Proposition 5.6, Then the solution $\mathbf{D}(t,x)$
to problem (6.9) satisfies for any wave admissible $(p,q)$ and $(r,s)$ the
estimate

$\||D|^{-\frac{2}{p}}D_{t,x}\mathbf{D}\|_{L^{p}L^{q}}\lesssim\|\nabla\mathbf{D}_{0}\|_{L^{2}}+\|\mathbf{D}_{1}\|_{L^{2}}+\||D|^{\frac{2}{r}}F\|_{L^{r^{\prime}}L^{s^{\prime}}}.$
(6.11)

###### Proof.

As in the proof of Proposition 5.4 we can recast (6.9) in the form (6.8).
Since the conditions in Proposition 6.2 are satisfied, estimate (6.2) yields

$\||D|^{-\frac{2}{p}}D_{t,x}\mathbf{D}\|_{L^{p}L^{q}}\lesssim\|\nabla\mathbf{D}_{0}\|_{L^{2}}+\|\mathbf{D}_{1}\|_{L^{2}}+\||D|^{\frac{2}{r}}F\|_{L^{r^{\prime}}L^{s^{\prime}}}+\|D_{t,x}\mathbf{D}\|_{L^{2}L^{2}(|x|\leq
R_{0}+1)}.$

for some fixed radius $R_{0}\geq 1$. The last term is bounded by a constant
times

$\|D_{t,x}\mathbf{D}^{h}\|_{L^{2}L^{2}_{-1/2-}}+\|D_{t,x}\mathbf{D}^{i}\|_{L^{2}L^{2}_{-1/2-}}.$

In view of (6.10), Propositions 5.5 and 5.6 and (5.13) yield

$\|D_{t,x}\mathbf{D}^{h}\|_{L^{2}L^{2}_{-1/2-}}\lesssim\|\nabla\mathbf{D}_{0}\|_{L^{2}}+\|\mathbf{D}_{1}\|_{L^{2}}.$

We thus have shown (6.11) for $F=0$.

Consider now the case $F\neq 0$. To complete the proof it is sufficient to
prove the estimate

$\left\|D_{t,x}\int_{0}^{t}H^{-\frac{1}{2}}e^{i(t-s)\sqrt{H}}F(s)ds\right\|_{Z}\leq
C\|F\|_{E^{*}}$

where $F$ is divergence free and we set $E=L^{r}\dot{H}^{-2/r}_{s}$ and
$Z=L^{2}L^{2}_{-1/2-}.$ First, we notice that by the Christ–Kiselev Lemma this
estimate follows from the analogous unretarded one (since $r>2$)

$\left\|D_{t,x}\int H^{-\frac{1}{2}}e^{i(t-s)\sqrt{H}}F(s)ds\right\|_{Z}\leq
C\|F\|_{E^{*}}.$

Next, we split $D_{t,x}H^{-\frac{1}{2}}\int
e^{i(t-s)\sqrt{H}}F(s)ds=D_{t,x}e^{it\sqrt{H}}H^{-1/2}\int
e^{-is\sqrt{H}}F(s)ds$ and we recall from Propositions 5.5 and 5.6 the
inequalities

$\|e^{it\sqrt{H}}f\|_{Z}\lesssim\|f\|_{L^{2}},\qquad\|\nabla
e^{it\sqrt{H}}f\|_{Z}\lesssim\|\nabla f\|_{L^{2}}.$

Combined with (5.13), these estimates yield

$\displaystyle\left\|D_{t,x}\int
H^{-\frac{1}{2}}e^{i(t-s)\sqrt{H}}F(s)ds\right\|_{Z}$
$\displaystyle\lesssim\left\|\int
e^{-is\sqrt{H}}F(s)ds\right\|_{L^{2}}+\left\|\nabla H^{-\frac{1}{2}}\int
e^{-is\sqrt{H}}F(s)ds\right\|_{L^{2}}$
$\displaystyle\lesssim\left\|\int\cos(s\sqrt{H})F(s)ds\right\|_{L^{2}}+\left\|\int\sin(s\sqrt{H})F(s)ds\right\|_{L^{2}}.$
(6.12)

In the first part of the proof we have seen

$\|D_{t,x}H^{-\frac{1}{2}}\sin(t\sqrt{H})f\|_{E}\lesssim\|f\|_{L^{2}},\qquad\|D_{t,x}\cos(t\sqrt{H})f\|_{E}\lesssim\|\nabla
f\|_{L^{2}}.$

Considering only $D_{t}$, the first inequality implies

$\|\cos(t\sqrt{H})f\|_{E}\lesssim\|f\|_{L^{2}},$

while the second one gives

$\|\sin(t\sqrt{H})f\|_{E}=\|D_{t}\cos(t\sqrt{H})H^{-\frac{1}{2}}f\|_{E}\lesssim\|\nabla
H^{-\frac{1}{2}}f\|_{L^{2}}\lesssim\|f\|_{L^{2}}$

using again (5.13). Applying the dual estimates we see that both terms in
(6.12) can be estimated by $\|F\|_{E^{*}}$, and this concludes the proof.

∎

The above theorem now easily implies our main result.

###### Proof of Theorem 1.1.

In view of (1.6), the main results for $\mathbf{D}$ is an immediate
consequence of the above theorem. The magnetic field $\mathbf{B}$ solves (1.7)
which is of the same form as equation (1.6) for $\mathbf{D}$ except that
$\epsilon$ and $\mu$ are interchanged. So Theorem 6.3 is true with
$\mathbf{B}$ instead of $\mathbf{D}$ if we replace the condition on second
derivatives in (2) of Proposition 5.6 by

$|D^{2}\epsilon|\lesssim\langle
x\rangle^{-2-\delta},\qquad|D^{2}\mu|\lesssim\langle
x\rangle^{-\frac{5}{2}-\delta}.$

Theorem 1.1 for $\mathbf{B}$ again follows easily taking into account (1.7)
and Lemma 6.1.

By Lemma 6.1 we can replace $\mathbf{D}$ by $\mathbf{E}$ in Theorem 1.1, with
the divergence conditions
$\nabla\cdot(\epsilon\mathbf{E}_{0})=\nabla\cdot(\epsilon\mathbf{E}_{1})=0$.
One can pass from $\mathbf{B}$ to $\mathbf{H}$ in same way as from
$\mathbf{D}$ to $\mathbf{E}$. ∎

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Janus and RG interfaces in three-dimensional gauged supergravity II: General
$\alpha$

Michael Gutperle and Charlie Hultgreen-Mena

Mani L. Bhaumik Institute for Theoretical Physics
Department of Physics and Astronomy
University of California, Los Angeles, CA 90095, USA

###### Abstract

In this paper, we continue the study of Janus and RG-flow interfaces in three
dimensional supergravity continuing the work presented in [1]. We consider
$\mathcal{N}=8$ gauged supergravity theories which have a $\mathcal{N}=(4,4)$
$AdS_{3}$ vacuum with $D^{1}(2,1;\alpha)\times D^{1}(2,1;\alpha)$ symmetry for
general $\alpha$. We derive the BPS flow equations and find numerical
solutions. Some holographic quantities such as the entanglement entropy are
calculated.

##  1 Introduction

Janus solutions provide a holographic description of interface conformal field
theories. Generally, the solutions are constructed by considering an
$\text{AdS}_{d}$ slicing of a higher dimensional space where the other fields
depend non-trivially on the slicing coordinate(s). For example the original
Janus solution [2], deforms the $\text{AdS}_{5}\times S^{5}$ vacuum of type
IIB and is given by an $\text{AdS}_{4}$ slicing where the dilation depends
non-trivially on a single slicing coordinate and approaches two different
values on the two boundary components. The solution is dual to an interface of
$N=4$ super Yang-Mills theory where the coupling $g_{\rm YM}$ jumps across a
co-dimension one interface [3]. More general Janus solutions preserving
supersymmetry were constructed as $\text{AdS}_{4}\times S^{2}\times S^{2}$
space warped over a Riemann surface [4]. These solutions are dual to
supersymmetric interface theories in $N=4$ SYM [5, 6, 7]. For other Janus
solutions in ten and eleven dimensions, see e.g. [8, 9, 10, 11]. In general,
constructing such solutions is quite difficult due to the fact that the
supersymmetry variations, as well the equations of motion, depend on more than
one warping coordinate and the resulting equations are nonlinear partial
differential equations. A useful approach is to construct Janus solutions in
lower dimensional gauged supergravities (see for example [12, 13, 14, 15, 16,
17, 18, 19, 20, 21, 22]). Such solutions are often easier to obtain, can be
uplifted to ten or eleven dimensions or can be used to explore qualitative
features of Janus solution in a bottom-up approach.

In lower dimensional gauged supergravities it is often the case that in
addition to a maximally supersymmetric AdS vacuum there are extrema with a
reduced amount of supersymmetry. One of the aims of the present paper is to
construct holographic Janus solutions which correspond to RG interfaces [23],
between different AdS vacua111See [24, 25, 26] for other examples of
holographic RG-flow interfaces.. This paper is a continuation of the work
presented [1], which considered three-dimensional $\mathcal{N}=8$ gauged
supergravity with $n=4$ vector multiplets, first discussed in [27]. This
theory has an $\text{AdS}_{3}$ vacuum with maximal $\mathcal{N}=(4,4)$
supersymmetry as well as two families of $\text{AdS}_{3}$ vacua with
$\mathcal{N}=(1,1)$ supersymmetry [28]. The gauged supergravity has a
parameter $\alpha$ on which the embedding tensor for the gauged supergravity
depends. For this theory the dual superconformal algebra of the
$\mathcal{N}=(4,4)$ vacuum is given by the “large” superconformal algebra
$D^{1}(2,1;\alpha)\times D^{1}(2,1;\alpha)$, and the three-dimensional
supergravity is believed to be a truncation of M-theory on
$\text{AdS}_{3}\times S^{3}\times S^{3}\times S^{1}$ [29, 30, 31, 32]. In the
previous paper we considered the special case of $\alpha=1$ for which the
explicit expressions become simpler. Here we will analyze the case for general
$\alpha$, using both analytical and numerical methods.

The structure of this paper is as follows: In section 2 we review the three
dimensional gauged supergravity with $n=4$ vector multiplets used in the
paper. We consider three truncations where the gauge fields as well as some
scalars can consistently be set to zero and fix the $N=(1,1)$ vacua for
general $\alpha$. In section 3 we derive the BPS flow equations for an
$AdS_{2}$ sliced Janus ansatz, this generalizes and streamlines the discussion
of [1]. In section 4 we derive the flow equations and integrate them
numerically for the three truncations. For the second and third truncations
where $N=(1,1)$ AdS vacua exists we present examples of RG-flow interfaces. In
section 5 we use the solutions to calculate some holographic observables. In
particular we determine the masses of the fluctuating scalars around the
$N=(1,1)$ vacua, the mass squared is positive and quite large, which means
that the scalar fluctuations around fixed point are repulsive in the UV. This
implies that the initial conditions have to be fine tuned in order to reach
the fixed point. We discuss our results and possible directions for future
research in section 6.

##  2 Three-dimensional $\mathcal{N}=8$ gauged supergravity

In this section, we review the $\mathcal{N}=8$ gauged supergravity first
constructed in [27] mainly following the conventions of [1]. The bosonic field
content consists of a graviton $g_{\mu\nu}$, Chern-Simons gauge fields
$B^{\mathcal{M}}_{\mu}$, and scalars fields living in a
$G/H=\operatorname{SO}(8,n)/\operatorname{SO}(8)\times\operatorname{SO}(n)$
coset, which has $8n$ degrees of freedom before gauging. The scalar fields are
parametrized by a $G$-valued matrix $L(x)$ in the vector representation, which
transforms under $H$ and the gauge group $G_{0}\subseteq G$ by right and left
multiplication of group elements respectively.

$\displaystyle L(x)\to g_{0}(x)L(x)h^{-1}(x)$ (2.1)

for $g_{0}\in G_{0}$ and $h\in H$. The Lagrangian is invariant under such
transformations. In this paper we use the following index conventions:

  * •

$I,J,\dotsc=1,2,\dotsc,8$ for $\operatorname{SO}(8)$.

  * •

$r,s,\dotsc=9,10,\dotsc,n+8$ for $\operatorname{SO}(n)$.

  * •

$\bar{I},\bar{J},\dotsc=1,2,\dotsc,n+8$ for $\operatorname{SO}(8,n)$.

  * •

$\mathcal{M},\mathcal{N},\dotsc$ for generators of $\operatorname{SO}(8,n)$.

Let the generators of $G$ be
$\\{t^{\mathcal{M}}\\}=\\{t^{\bar{I}\bar{J}}\\}=\\{X^{IJ},X^{rs},Y^{Ir}\\}$,
where $Y^{Ir}$ are the noncompact generators. Explicitly, the generators of
the vector representation are given by

$\displaystyle\tensor{(t^{\bar{I}\bar{J}})}{{}^{\bar{K}}_{\bar{L}}}=\eta^{\bar{I}\bar{K}}\delta^{\bar{J}}_{\bar{L}}-\eta^{\bar{J}\bar{K}}\delta^{\bar{I}}_{\bar{L}}$
(2.2)

where $\eta^{\bar{I}\bar{J}}=\operatorname{diag}(++++++++-\cdots)$ is an
$\operatorname{SO}(8,n)$-invariant tensor. These generators satisfy the
typical $\operatorname{SO}(8,n)$ commutation relations,

$\displaystyle[t^{\bar{I}\bar{J}},t^{\bar{K}\bar{L}}]=2\quantity(\eta^{\bar{I}[\bar{K}}t^{\bar{L}]\bar{J}}-\eta^{\bar{J}[\bar{K}}t^{\bar{L}]\bar{I}})$
(2.3)

The gauging of the supergravity is characterized by an embedding tensor
$\Theta_{\mathcal{M}\mathcal{N}}$ (which has to satisfy various identities
[33] in order to define a consistent theory) that determines which isometries
are gauged, the coupling to the Chern-Simons fields, and additional terms in
the supersymmetry transformations and action depending on the gauge coupling
$g$. We will look at the particular case in [28] where $n\geq 4$ and the
gauged subgroup is the $G_{0}=\operatorname{SO}(4)\times\operatorname{SO}(4)$
subset of the $\operatorname{SO}(8)\subset\operatorname{SO}(8,n)$. The
embedding tensor has the non vanishing entries,222We use the conventions
$\varepsilon_{1234}=\varepsilon_{5678}=1$.

$\displaystyle\Theta_{\bar{I}\bar{J},\bar{K}\bar{L}}=\begin{cases}\alpha\varepsilon_{\bar{I}\bar{J}\bar{K}\bar{L}}&\text{if
}\bar{I},\bar{J},\bar{K},\bar{L}\in\\{1,2,3,4\\}\\\
\varepsilon_{\bar{I}\bar{J}\bar{K}\bar{L}}&\text{if
}\bar{I},\bar{J},\bar{K},\bar{L}\in\\{5,6,7,8\\}\\\
0&\text{otherwise}\end{cases}$ (2.4)

Note that the gauging depends on a real parameter $\alpha$. As discussed in
[28], the maximally supersymmetric $\text{AdS}_{3}$ vacuum has an isometry
group,

$\displaystyle D^{1}(2,1;\alpha)\times D^{1}(2,1;\alpha)$ (2.5)

which corresponds to the family of “large” superconformal algebras of the dual
SCFT. In this paper we generalize the analysis of [1] where the case
$\alpha=1$ was considered to the case of general $\alpha$. Note that in the
spacial case $\alpha=1$ the super algebra becomes more familiar
$D^{1}(2,1;1)=\operatorname{OSp}(4|2)$.

From the embedding tensor, the $G_{0}$-covariant currents can be obtained,

$\displaystyle
L^{-1}(\partial_{\mu}+g\Theta_{\mathcal{M}\mathcal{N}}B_{\mu}^{\mathcal{M}}t^{\mathcal{N}})L=\frac{1}{2}\mathcal{Q}^{IJ}_{\mu}X^{IJ}+\frac{1}{2}\mathcal{Q}^{rs}_{\mu}X^{rs}+\mathcal{P}^{Ir}_{\mu}Y^{Ir}$
(2.6)

It is convenient to define the
$\tensor{\mathcal{V}}{{}^{\mathcal{M}}_{\mathcal{A}}}$ tensors,

$\displaystyle
L^{-1}t^{\mathcal{M}}L=\tensor{\mathcal{V}}{{}^{\mathcal{M}}_{\mathcal{A}}}t^{\mathcal{A}}=\frac{1}{2}\tensor{\mathcal{V}}{{}^{\mathcal{M}}_{IJ}}X^{IJ}+\frac{1}{2}\tensor{\mathcal{V}}{{}^{\mathcal{M}}_{rs}}X^{rs}+\tensor{\mathcal{V}}{{}^{\mathcal{M}}_{Ir}}Y^{Ir}$
(2.7)

and the $T$-tensor,

$\displaystyle
T_{\mathcal{A}|\mathcal{B}}=\Theta_{\mathcal{M}\mathcal{N}}\tensor{\mathcal{V}}{{}^{\mathcal{M}}_{\mathcal{A}}}\tensor{\mathcal{V}}{{}^{\mathcal{N}}_{\mathcal{B}}}$
(2.8)

The $T$-tensor is used to construct the tensors $A_{1,2,3}$ which will appear
in the scalar potential and the supersymmetry transformations,

$\displaystyle A_{1}^{AB}$
$\displaystyle=-\frac{1}{48}\Gamma^{IJKL}_{AB}T_{IJ|KL}$ $\displaystyle
A_{2}^{A\dot{A}r}$
$\displaystyle=-\frac{1}{12}\Gamma^{IJK}_{A\dot{A}}T_{IJ|Kr}$ $\displaystyle
A_{3}^{\dot{A}r\dot{B}s}$
$\displaystyle=\frac{1}{48}\delta^{rs}\Gamma^{IJKL}_{\dot{A}\dot{B}}T_{IJ|KL}+\frac{1}{2}\Gamma^{IJ}_{\dot{A}\dot{B}}T_{IJ|rs}$
(2.9)

where $A,B$ and $\dot{A},\dot{B}$ are $\operatorname{SO}(8)$-spinor indices.
Our conventions for the $\operatorname{SO}(8)$ Gamma matrices are presented in
appendix A.

Here we choose the spacetime signature $\eta^{ab}=\operatorname{diag}(+--)$ as
mostly negative. The bosonic Lagrangian and scalar potential are given by

$\displaystyle e^{-1}\mathcal{L}_{\rm bos}$
$\displaystyle=-\frac{1}{4}R+\frac{1}{4}\mathcal{P}_{\mu}^{Ir}\mathcal{P}^{Ir\,\mu}+V-\frac{1}{4}e^{-1}\varepsilon^{\mu\nu\rho}g\Theta_{\mathcal{M}\mathcal{N}}B_{\mu}^{\mathcal{M}}\quantity(\partial_{\nu}B_{\rho}^{\mathcal{N}}+\frac{1}{3}g\Theta_{\mathcal{K}\mathcal{L}}\tensor{f}{{}^{\mathcal{N}\mathcal{K}}_{\mathcal{P}}}B_{\nu}^{\mathcal{L}}B_{\rho}^{\mathcal{P}})$
$\displaystyle V$
$\displaystyle=\frac{1}{4}g^{2}\quantity(A^{AB}_{1}A^{AB}_{1}-\frac{1}{2}A^{A\dot{A}r}_{2}A^{A\dot{A}r}_{2})$
(2.10)

The supersymmetry variations take the following form

$\displaystyle\delta\chi^{\dot{A}r}$
$\displaystyle=\frac{1}{2}i\Gamma^{I}_{A\dot{A}}\gamma^{\mu}\varepsilon^{A}\mathcal{P}^{Ir}_{\mu}+gA^{A\dot{A}r}_{2}\varepsilon^{A}$
$\displaystyle\delta\psi^{A}_{\mu}$
$\displaystyle=\quantity(\partial_{\mu}\varepsilon^{A}+\frac{1}{4}\omega_{\mu}^{ab}\gamma_{ab}\varepsilon^{A}+\frac{1}{4}\mathcal{Q}^{IJ}_{\mu}\Gamma^{IJ}_{AB}\varepsilon^{B})+igA^{AB}_{1}\gamma_{\mu}\varepsilon^{B}$
(2.11)

The Einstein equations of motion are

$\displaystyle
R_{\mu\nu}-\mathcal{P}^{Ir}_{\mu}\mathcal{P}^{Ir}_{\nu}-4Vg_{\mu\nu}=0$ (2.12)

and the gauge field equations of motion are

$\displaystyle
e\mathcal{P}^{Ir\,\lambda}\Theta_{\mathcal{Q}\mathcal{M}}\tensor{\mathcal{V}}{{}^{\mathcal{M}}_{Ir}}=\varepsilon^{\lambda\mu\nu}\quantity(\Theta_{\mathcal{Q}\mathcal{M}}\partial_{\mu}B^{\mathcal{M}}_{\nu}+\frac{1}{6}gB^{\mathcal{M}}_{\mu}B^{\mathcal{K}}_{\nu}\quantity(\Theta_{\mathcal{M}\mathcal{N}}\Theta_{\mathcal{K}\mathcal{L}}\tensor{f}{{}^{\mathcal{N}\mathcal{L}}_{\mathcal{Q}}}+2\Theta_{\mathcal{M}\mathcal{N}}\tensor{f}{{}^{\mathcal{L}\mathcal{N}}_{\mathcal{K}}}\Theta_{\mathcal{L}\mathcal{Q}}))$
(2.13)

###  2.1 The $n=4$ case

The smallest number of matter multiplets where multiple supersymmetric vacua
exist is $n=4$. The symmetries of the theory are a local
$G_{0}=\operatorname{SO}(4)\times\operatorname{SO}(4)$ and a global
$\operatorname{SO}(n)$ with $n=4$. Consequently, the scalar potential only
depends on $8\cdot 4-3\cdot 6=14$ fields out of the original $32$. Moreover, a
further consistent truncation outlined in [28] is performed where the coset
representative depends only on eight of the fourteen scalars.

(2.14)

We will not display the general form of the tensors $A_{1}$ and $A_{2}$
defined in (2) here. The scalar potential has terms up to order $\alpha^{2}$.

$\displaystyle g^{-2}V$ $\displaystyle={1\over 2}+{1\over
4}\sum_{i}x_{i}^{2}-{1\over 4}\sum_{i<j<k}x_{i}^{2}x_{j}^{2}x_{k}^{2}-{1\over
2}\prod_{i}x_{i}^{2}+\alpha\left(-\prod_{i}x_{i}y_{i}+\prod_{i}\sqrt{1+x_{i}^{2}+y_{i}^{2}}\right)$
$\displaystyle+\alpha^{2}\left({1\over 2}+{1\over 4}\sum_{i}y_{i}^{2}-{1\over
4}\sum_{i<j<k}y_{i}^{2}y_{j}^{2}y_{k}^{2}-{1\over 2}\prod_{i}y_{i}^{2}\right)$
(2.15)

where all indices run form 1 to 4 unless otherwise indicated and we used the
following defined scalar fields

$\displaystyle x_{i}$ $\displaystyle=\cos p_{i}\sinh q_{i}~{},\qquad
y_{i}=\sin p_{i}\sinh q_{i}$ (2.16)

The $\mathcal{Q}_{\mu}$ and $\mathcal{P}_{\mu}$ currents do not depend on
$\alpha$, excluding the
$g\Theta_{\mathcal{M}\mathcal{N}}B^{\mathcal{M}}_{\mu}\tensor{\mathcal{V}}{{}^{\mathcal{N}}_{\mathcal{A}}}$
term, they are given by

$\displaystyle\mathcal{Q}^{IJ}_{\mu}$
$\displaystyle=\quantity(\smallmatrixquantity{0&0&0&0&\cosh
q_{1}\partial_{\mu}p_{1}&0&0&0\\\ 0&0&0&0&0&\cosh
q_{2}\partial_{\mu}p_{2}&0&0\\\ 0&0&0&0&0&0&\cosh
q_{3}\partial_{\mu}p_{3}&0\\\ 0&0&0&0&0&0&0&\cosh q_{4}\partial_{\mu}p_{4}\\\
-\cosh q_{1}\partial_{\mu}p_{1}&0&0&0&0&0&0&0\\\ 0&-\cosh
q_{2}\partial_{\mu}p_{2}&0&0&0&0&0&0\\\ 0&0&-\cosh
q_{3}\partial_{\mu}p_{3}&0&0&0&0&0\\\ 0&0&0&-\cosh
q_{4}\partial_{\mu}p_{4}&0&0&0&0})_{IJ}$ $\displaystyle\mathcal{Q}^{rs}_{\mu}$
$\displaystyle=0$ $\displaystyle\mathcal{P}^{Ir}_{\mu}$
$\displaystyle=\quantity(\smallmatrixquantity{\sinh
q_{1}\partial_{\mu}p_{1}&0&0&0\\\ 0&\sinh q_{2}\partial_{\mu}p_{2}&0&0\\\
0&0&\sinh q_{3}\partial_{\mu}p_{3}&0\\\ 0&0&0&\sinh
q_{4}\partial_{\mu}p_{4}\\\ \partial_{\mu}q_{1}&0&0&0\\\
0&\partial_{\mu}q_{2}&0&0\\\ 0&0&\partial_{\mu}q_{3}&0\\\
0&0&0&\partial_{\mu}q_{4}})_{Ir}$ (2.17)

Using these matrices, we can check that the combination
$\mathcal{P}^{Ir}_{\mu}\tensor{\mathcal{V}}{{}^{JK}_{Ir}}$ vanishes whenever
the indices $J,K\in\\{1,2,3,4\\}$ or $J,K\in\\{5,6,7,8\\}$. This implies that
there is no source for $B^{\mathcal{M}}_{\mu}$ in the gauge field equation of
motion (2.13), so it is consistent to set $B^{\mathcal{M}}_{\mu}=0$. We will
make this choice from now on.

The kinetic term for the scalars in the action (2) can be expressed in terms
of the $x_{i}$ and $y_{i}$ using the relations (2.16) and take the form

$\displaystyle\frac{1}{4}\mathcal{P}_{\mu}^{Ir}\mathcal{P}^{Ir\,\mu}=-{1\over
4}\sum_{i=1}^{4}{1\over
1+x_{i}^{2}+y_{i}^{2}}\Big{(}(1+y_{i}^{2})(\partial_{\mu}\partial^{\mu}x_{i}-2x_{i}y_{i}\partial_{\mu}x_{i}\partial^{\mu}y_{i}^{\prime}+(1+x_{i}^{2})\partial
y_{i}\partial^{\mu}y_{i}\Big{)}$ (2.18)

This expression will be needed for determining masses of the fluctuations of
the scalar fields around the supersymmetric vacua.

###  2.2 Truncations and supersymmetric $\text{AdS}_{3}$ vacua

In order to make our analysis more tractable, we make further truncations to
reduce the number of independent scalar fields. Below we consider three
consistent truncations, which together explore the $\text{AdS}_{3}$ vacua with
$\mathcal{N}=(4,4)$ and $\mathcal{N}=(1,1)$ supersymmetry. All of the results
are generalizations of the $\alpha=1$ case discussed in [1].

####  2.2.1 Truncation 1

The first truncation is given by denoting $q_{1}=q$, $p_{1}=p$ and setting all
remaining $q_{i}=p_{i}=0$ for $i=2,3,4$. The scalar potential is

$\displaystyle V=\frac{g^{2}}{4}\left(2(1+\alpha^{2})+4\alpha\cosh
q+(\cos^{2}p+\alpha^{2}\sin^{2}p)\sinh^{2}q\right)$ (2.19)

The $\mathcal{N}=(4,4)$ vacuum is given by setting $q=0$ and the vacuum
potential is $V_{0}={1\over 2}g^{2}(1+\alpha)^{2}$. In the $x,y$ coordinates
the $\mathcal{N}=(4,4)$ vacuum is given by $x_{i}=y_{i}=0$. This is the only
supersymmetric vacuum for this truncation. We note that for the choice
$\alpha=1$ the potential is independent of the scalar field $p$. We note we
will chose $g=1/(1+\alpha)$ in order to set the potential at the $N=(4,4)$
vacuum to be $V_{0}={1\over 2}$, which corresponds to a unit radius $AdS_{3}$.

####  2.2.2 Truncation 2

The second truncation is given by setting all the $q$ and $p$ equal
respectively, i.e. $q_{i}=q$, $p_{i}=p$ for $i=1,2,3,4$. The scalar potential
becomes

$\displaystyle V$ $\displaystyle={g^{2}\over
2}\Big{\\{}(1-\cos^{2}p\sinh^{2}q)(1+\cos^{2}p\sinh^{2}q)^{3}+\alpha^{2}(1-\sin^{2}p\sinh^{2}q)(1+\sin^{2}p\sinh^{2}q)^{3}$
$\displaystyle\quad+\alpha(2+4\sinh^{2}q+2\sinh^{4}q-2\sin^{4}p\cos^{4}p\sinh^{8}q)\Big{\\}}$
(2.20)

or in terms of the $x,y$ fields, the potential will take the following form

$\displaystyle V$ $\displaystyle={g^{2}\over
2}\Big{\\{}(x^{2}-1)(x^{2}+1)^{3}+\alpha^{2}(y^{2}-1)(y^{2}+1)^{3}+2\alpha\big{(}2x^{2}(1+y^{2})+(1+y^{2})^{2}-x^{4}(y^{4}-1)\big{)}\Big{\\}}$
(2.21)

As before the $\mathcal{N}=(4,4)$ vacuum is given by $q=0$ or $x=y=0$. There
are $\mathcal{N}=(1,1)$ vacua which are located at

$\displaystyle x$ $\displaystyle=\pm{1\over
6\sqrt{3}\alpha}\left(-12\alpha^{2}-{2^{2\over 3}Y^{1\over
3}(Y+2\alpha^{2}(-3+8\alpha))\over(3+2\alpha)}-{2^{1\over 3}y^{2\over
3}(Y+2\alpha(-18+\alpha(-15+8\alpha)))\over(3+2\alpha)^{2}}\right)^{1\over 2}$
$\displaystyle y$ $\displaystyle=\pm{1\over 3}\Big{(}-1+{2^{2\over 3}Y^{1\over
3}\over\alpha}+{4\;2^{1\over 3}(3+2\alpha)\over Y^{1\over 3}}\Big{)}^{1\over
2}$ (2.22)

where $Y$ is given by

$\displaystyle Y=3i\alpha^{3\over
2}\sqrt{96+3\alpha(61+32\alpha)}-\alpha^{2}(9+16\alpha)$ (2.23)

The central charge of the dual CFT is related to the AdS radius and the value
of the potential $V_{0}$ at its minimum

$\displaystyle c$ $\displaystyle={R_{AdS}\over
4G_{N}}={1\over\sqrt{2V_{0}}}{1\over 4G_{N}}$ (2.24)

Choosing $g=1/(1+\alpha)$ sets the AdS radius of the $N=(4,4)$ vacuum to one
and the ratio of the central charge of the $N=(4,4)$ to the $N=(1,1)$ vacuum
as a function of $\alpha$ becomes

$\displaystyle{c_{N=(1,1)}\over
c_{N=(4,4)}}={1\over\sqrt{2V_{0}^{N=(1,1)}(\alpha)}}$ (2.25)

The expressions derived (2.2.2) are not very illuminating and we present a
plot of the ratio of the central charges for the two vacua in the figure 1. It
is interesting to note that the ratio of central charges is minimized for the
special value $\alpha=1$.

Figure 1: Ration of central charges for the $N=(1,1)$ and $N=(4,4)$ vacua.

####  2.2.3 Truncation 3

The third truncation is given by setting the first three $q$ and $p$ equal,
i.e. $q_{i}=q$, $p_{i}=p$ for $i=1,2,3$, and setting the remaining
$q_{4}=p_{4}=0$. The scalar potential is

$\displaystyle V$ $\displaystyle={g^{2}\over
4}\Big{(}2+3\cos^{2}p\sinh^{2}q-\cos^{6}p\sinh^{6}q+4\alpha\cosh^{3}q+\alpha^{2}(2+3\sin^{2}p\sinh^{2}q-\sin^{6}p\sinh^{6}q)\Big{)}$
(2.26)

or in the $x,y$ variables

$\displaystyle V$ $\displaystyle={g^{2}\over
4}\Big{(}(2+3x^{2}-x^{6})+4\alpha(1+x^{2}+y^{2})^{3\over
2}+\alpha^{2}(2+3y^{2}-y^{6})\Big{)}$ (2.27)

The $\mathcal{N}=(4,4)$ vacuum is given by $q=0$ or $x=y=0$ as before, and
$\mathcal{N}=(1,1)$ vacua can be determined by finding the extrema for the
potential (2.27) away from the origin.

$\displaystyle y$ $\displaystyle=\pm\sqrt{{1\over
2}}\left(1+\varepsilon(\alpha)\sqrt{1-{4\over X^{1\over 3}}+{2X^{1\over
3}\over 3\alpha^{2}}}+\sqrt{2+{4\over X^{1\over 3}}-{2X^{1\over 3}\over
3\alpha^{2}}+{2(\alpha^{2}-4)\over\alpha^{2}\sqrt{1-{4\over X^{1\over
3}}+{2X^{1\over 3}\over 3\alpha^{2}}}}}\right)^{1\over 2}$ $\displaystyle x$
$\displaystyle=\pm\left({\alpha^{2}\over
4}(y^{4}-1)^{4}-1-y^{2}\right)^{1\over 2}$ (2.28)

where we used the abriviation

$\displaystyle
X=3\alpha^{2}\big{(}9-9\alpha^{2}+\sqrt{81-138\alpha^{2}+81\alpha^{4}}\big{)}$
(2.29)

The $\varepsilon(\alpha)$ is a sign which selects a branch of the solutions
which gives real $x,y$ depending on $\alpha$ and we have
$\varepsilon(\alpha)=+1$ for $\alpha<2$ and $\varepsilon(\alpha)=-1$ for
$\alpha>2$. We can plot the ratio of the central charges which is given by
(2.25), determined from the potential (2.27).

Figure 2: Ration of central charges for the $N=(1,1)$ and $N=(4,4)$ vacua.

We note that the qualitative behavior of the ratio for truncation 2 and 3 is
very similar, in particular the central charge for the $N=(1,1)$ vacuum is
minimized at $\alpha=1$.

##  3 Janus flow equations

In this section we will derive the BPS flow equations, expanding on the
construction in our previous paper [1]. The Janus ansatz for the bosonic
fields is give by

$\displaystyle\differential{s^{2}}$
$\displaystyle=e^{2B(u)}\quantity(\frac{\differential{t^{2}}-\differential{z^{2}}}{z^{2}})-\differential{u}^{2}~{},\qquad
q_{i}=q_{i}(u)~{},\qquad p_{i}=p_{i}(u)$ (3.1)

The Chern-Simons gauge fields is set to zero $B^{\mathcal{M}}_{\mu}=0$. We
will check that the source term on the right hand side of the gauge field
equation of motion (2.13) is zero for the solutions considered in this paper.

The gravitino supersymmetry variation $\delta\psi^{A}_{\mu}=0$ is

$\displaystyle 0$
$\displaystyle=\partial_{t}\varepsilon+\frac{1}{2z}\gamma_{0}\quantity(\gamma_{1}-B^{\prime}e^{B}\gamma_{2}+2ige^{B}A_{1})\varepsilon$
$\displaystyle 0$
$\displaystyle=\partial_{z}\varepsilon+\frac{1}{2z}\gamma_{1}\quantity(-B^{\prime}e^{B}\gamma_{2}+2ige^{B}A_{1})\varepsilon$
$\displaystyle 0$
$\displaystyle=\partial_{u}\varepsilon+\frac{1}{4}\mathcal{Q}_{u}^{IJ}\Gamma^{IJ}\varepsilon+ig\gamma_{2}A_{1}\varepsilon$
(3.2)

where we have suppressed the $\operatorname{SO}(8)$-spinor indices of
$\varepsilon^{A}$ and $A_{1}^{AB}$. The spin-${1\over 2}$ variation
$\smash{\delta\chi^{\dot{A}r}=0}$ is

$\displaystyle\quantity(-\frac{i}{2}\Gamma^{I}\mathcal{P}_{u}^{Ir}\gamma_{2}+gA_{2}^{r})_{A\dot{A}}\varepsilon^{A}=0~{},\quad\quad
r=9,10,\dotsc,8+n$ (3.3)

The matrix $A_{1}$ defined in (2) has eigenvectors

$\displaystyle A_{1}^{AB}n_{\pm}^{(i)B}=\pm w_{i}n_{\pm}^{(i)A}~{},\quad\quad
i=1,2,3,4$ (3.4)

For a supersymmetric $AdS_{3}$ vacuum the eigenvalue $w_{i}$ is related to the
value of the potential evaluated at the vacuum via

$\displaystyle w^{2}_{\rm vac}={V_{\rm vac}\over 2g^{2}}$ (3.5)

and the associated eigenvectors $n_{\pm}^{(i)}$ determine the supersymmetries
of the vacuum. For the $\mathcal{N}=(4,4)$ vacuum the $w_{i},i=1,\cdots 4$ all
satisfy (3.5) and hence the vacuum preserves eight supersymmetries. For the
$\mathcal{N}=(1,1)$ vacuum only one of the four $n_{\pm}^{(i)}$ and $w_{i}$
satisfies (3.5). In the following we drop the index (i) to denote the
supersymmetric eigenvalue $w$ and the eigenvector $n_{\pm}^{A}$.

The general ansatz for unbroken supersymmetry $\varepsilon^{A}$ for the Janus
solution is given by

$\displaystyle\varepsilon^{A}=\big{(}f_{+}n_{+}+f_{-}n_{-}^{A}\big{)}\zeta_{+}+\big{(}g_{+}n_{+}^{A}+g_{-}n_{-}^{A}\big{)}\zeta_{-}$
(3.6)

where $\zeta_{\pm}$ are Killing spinors for a unit radius $AdS_{2}$

$\displaystyle D_{\mu}\zeta_{\eta}=i{\eta\over
2}\gamma_{\mu}\zeta_{\eta}~{},\quad\quad\mu=t,z,\quad\eta=\pm 1$ (3.7)

###  3.1 Gravitino variation

The $t,z$ components of the gravitino variation can be expressed as follows by
using the properties of the $\text{AdS}_{2}$ Killing spinors,

$\displaystyle 0$
$\displaystyle=i\quantity{\big{(}f_{+}n_{+}^{A}+f_{-}n_{-}^{A}\big{)}\zeta_{+}-\big{(}g_{+}n_{+}^{A}+g_{-}n_{-}^{A}\big{)}\zeta_{-}}$
(3.8)
$\displaystyle\quad+iB^{\prime}e^{B}i\gamma_{2}\Big{\\{}\big{(}f_{+}n_{+}^{A}+f_{-}n_{-}^{A}\big{)}\zeta_{+}+\big{(}g_{+}n_{+}^{A}+g_{-}n_{-}^{A}\big{)}\zeta_{-}\Big{\\}}$
$\displaystyle\quad+2igwe^{B}\Big{\\{}\big{(}f_{+}n_{+}^{A}-f_{-}n_{-}^{A}\big{)}\zeta_{+}+\big{(}g_{+}n_{+}^{A}-g_{-}n_{-}^{A}\big{)}\zeta_{-}\Big{\\}}$
(3.9)

Using $i\gamma_{2}\zeta_{\eta}=\zeta_{-\eta}$ and the linear independence of
the $n_{\pm}$ and $\zeta_{\pm}$, one obtains a set of equations,

$\displaystyle f_{+}+B^{\prime}e^{B}g_{+}+2gwe^{B}f_{+}$ $\displaystyle=0$
$\displaystyle-g_{+}+B^{\prime}e^{B}f_{+}+2gwe^{B}g_{+}$ $\displaystyle=0$
$\displaystyle f_{-}+B^{\prime}e^{B}g_{-}-2gwe^{B}f_{-}$ $\displaystyle=0$
$\displaystyle-g_{-}+B^{\prime}e^{B}f_{-}-2gwe^{B}g_{-}$ $\displaystyle=0$
(3.10)

It is convenient to define the following expressions

$\displaystyle\gamma(u)=\sqrt{1-{e^{-2B}\over
4g^{2}w^{2}}}~{},\quad\quad\sqrt{1-\gamma^{2}(u)}={e^{-B}\over 2gw}$ (3.11)

The equations (3.1) can then be solved by

$\displaystyle f_{+}={\sqrt{1-\gamma^{2}}-1\over\gamma}g_{+}~{},\qquad
f_{-}={\sqrt{1-\gamma^{2}}+1\over\gamma}g_{-}$ (3.12)

if the integrability condition

$\displaystyle B^{\prime}$ $\displaystyle=\pm\sqrt{4g^{2}w^{2}-e^{-2B}}$
$\displaystyle=\pm 2gw\gamma$ (3.13)

is satisfied. This equations provides us with a differential equation for the
metric factor $B$.

###  3.2 Spin ${1\over 2}$ variation

The spin-${1\over 2}$ variation (3.3) takes the following form of a projector

$\displaystyle\Big{(}M^{AB}i\gamma_{2}+\delta^{AB}\Big{)}\varepsilon^{B}=0$
(3.14)

where

$\displaystyle
M^{(r)}_{AB}=-\frac{1}{2g}\Big{(}\Gamma^{I}\mathcal{P}_{u}^{Ir}(A_{2}^{r})^{-1}\Big{)}^{T}_{AB}$
(3.15)

Note that there is a projector for each $r$, which all have to be satisfied
and the resulting flow equations are mutually consistent for a supersymmetric
Janus solution to exist. This analysis will be performed for the particular
truncations presented in section 2.2.

Inserting $\varepsilon^{A}$ is given by (3.6) and into the spin ${1\over 2}$
projector gives

$\displaystyle 0$
$\displaystyle=\big{(}f_{+}n_{+}^{A}+f_{-}n_{-}^{A}\big{)}\zeta_{+}+\big{(}g_{+}n_{+}^{A}+g_{-}n_{-}^{A}\big{)}\zeta_{-}$
$\displaystyle\quad+M^{AB}i\gamma_{2}\Big{\\{}\big{(}f_{+}n_{+}^{B}+f_{-}n_{-}^{B}\big{)}\zeta_{+}+\big{(}g_{+}n_{+}^{B}+g_{-}n_{-}^{B}\big{)}\zeta_{-}\Big{\\}}$
(3.16)

We have dropped the index $r$ for notational convenience. Using the fact that
the two dimensional Killing spinors are orthogonal we can project (3.2) onto
the $n_{\pm}$ and $\zeta_{\pm}$ components. This produces four equations

$\displaystyle f_{+}n_{+}^{2}+M_{++}g_{+}+M_{+-}g_{-}$ $\displaystyle=0$
$\displaystyle g_{+}n_{+}^{2}+M_{++}f_{+}+M_{+-}f_{-}$ $\displaystyle=0$
$\displaystyle f_{-}n_{-}^{2}+M_{+-}g_{+}+M_{--}g_{-}$ $\displaystyle=0$
$\displaystyle g_{+}n_{-}^{2}+M_{+-}f_{+}+M_{--}f_{-}$ $\displaystyle=0$
(3.17)

where we denoted $n_{\pm}^{2}=n_{\pm}^{A}n_{\pm}^{A}$ and we define

$\displaystyle M_{++}=n_{+}^{A}M^{AB}n_{+}^{B}~{},\qquad
M_{--}=n_{-}^{A}M^{AB}n_{-}^{B}~{},\qquad
M_{+-}=M_{-+}=n_{+}^{A}M^{AB}n_{-}^{B}$ (3.18)

If there is more than one $n_{\pm}$ (as in truncation 1) one has to choose
linear combinations for which $M_{\pm\pm},M_{\pm\mp}$ take the same form for
all $n^{(i)}_{\pm}$, which is a consistency condition. Using (3.12) it can be
shown that equations (3.2) can only be satisfied if we have

$\displaystyle M_{++}=\gamma n_{+}^{2}~{},\qquad M_{--}=-\gamma
n_{-}^{2}~{},\qquad
M_{+-}=M_{-+}=\sqrt{1-\gamma^{2}}\sqrt{n_{+}^{2}n_{-}^{2}}$ (3.19)

In all cases we consider, the $M_{--}$ equation is automatically satisfied if
the $M_{++}$ is satisfied. Hence (3.19) provides two independent equations. It
follows from (3.15) that these equations are linear in the first derivatives
of the scalar fields and provide the BPS flow equations for the scalars. The
complete set of flow equations is given by these equations and the flow
equation for the metric factor (3.1), coming from the gravitino variation.

##  4 Janus and RG-flow solutions

In this section we obtain the flow equations and solve them numerically for
the three truncations considered in this paper. Since the first truncation
does not have $N=(1,1)$ vacua the BPS flows will correspond to Janus solutions
interpolating between $N=(4,4)$ vacua. For the two other truncations we find
Janus as well as RG-flow interface solutions.

###  4.1 Truncation 1

The matrix $A_{1}$ for this truncation is given by

$\displaystyle A_{1}=\left(\begin{array}[]{cccccccc}0&0&0&a&0&0&b&0\\\
0&0&-a&0&0&0&0&b\\\ 0&-a&0&0&-b&0&0&0\\\ a&0&0&0&0&-b&0&0\\\
0&0&-b&0&0&0&0&-a\\\ 0&0&0&-b&0&0&a&0\\\ b&0&0&0&0&a&0&0\\\
0&b&0&0&-a&0&0&0\\\ \end{array}\right)$ (4.9)

where

$\displaystyle a={1\over 2}\cos p(\alpha+\cosh q),\quad b={1\over 2}\sin
p(1+\alpha\cosh q)$ (4.10)

The eigenvalue of $A_{1}$ are $\pm w_{0}$ which is given by

$\displaystyle w_{0}=\sqrt{a^{2}+b^{2}}={1\over 2}\sqrt{\cos^{2}p(\alpha+\cosh
q)^{2}+\sin^{2}p(1+\alpha\cosh q)^{2}}$ (4.11)

The eigenvectors are given by

$\displaystyle n^{(1)}_{\pm}=\left(\begin{array}[]{c}0\\\ {a\pm w_{0}\over
b}\\\ {-a\mp w_{0}\over b}\\\ 0\\\ 1\\\ 0\\\ 0\\\ 1\end{array}\right),\quad
n^{(2)}_{\pm}=\left(\begin{array}[]{c}0\\\ {-a\pm w_{0}\over b}\\\ {-a\pm
w_{0}\over b}\\\ 0\\\ -1\\\ 0\\\ 0\\\ 1\end{array}\right),\quad
n^{(3)}_{\pm}=\left(\begin{array}[]{c}{-a\pm w_{0}\over b}\\\ 0\\\ 0\\\ {a\mp
w_{0}\over b}\\\ 0\\\ 1\\\ 1\\\
0\end{array}\right),\quad,n^{(4)}_{\pm}=\left(\begin{array}[]{c}{a\pm
w_{0}\over b}\\\ 0\\\ 0\\\ {a\pm w_{0}\over b}\\\ 0\\\ -1\\\ 1\\\
0\end{array}\right),$ (4.44)

The matrix $M^{AB}$ defined in (3.15) takes the following form for the
truncation 1

$\displaystyle M$
$\displaystyle=\left(\begin{array}[]{cccccccc}0&0&0&m_{1}&0&0&-m_{2}&0\\\
0&0&-m_{1}&0&0&0&0&-m_{2}\\\ 0&-m_{1}&0&0&m_{2}&0&0&0\\\
m_{1}&0&0&0&0&0&m_{2}&0\\\ 0&0&m_{2}&0&0&0&0&-m_{1}\\\
0&0&0&m_{2}&0&0&m_{1}&0\\\ -m_{2}&0&0&0&0&m_{1}&0&0\\\
0&-m_{2}&0&0&-m_{1}&0&0&0\\\ \end{array}\right)$ (4.53)

with

$\displaystyle m_{1}$ $\displaystyle={\alpha\sin p\;p^{\prime}+\cos p\csch
q\;q^{\prime}\over g(\cos^{2}p+\alpha^{2}\sin^{2}p)},\quad\quad m_{2}={\cos
p\;p^{\prime}-\alpha\sin p\csch q\;q^{\prime}\over
g(\cos^{2}p+\alpha^{2}\sin^{2}p)},$ (4.54)

(a)

(b)

(c) (d) Figure 3: (a)-(c) plots of $p,q,B$ respectively, (d) parametric plot
of the Janus flow in the $x,y$ variables. The initial conditions are
$q(0)=1.0$ and $p(0)=1.5$ and $\alpha=2.3$.

Using the definitions (4.10) and (4.11) the flow equations (3.19) for the
scalars $p,q$ and the metric function $B$ (3.1) can be written relatively
compactly

$\displaystyle p^{\prime}$ $\displaystyle={g\over
w_{0}}\Big{(}\alpha(a\gamma-b\sqrt{1-\gamma^{2}})\sin
p-(b\gamma+a\sqrt{1-\gamma^{2}})\cos p\Big{)}$ $\displaystyle q^{\prime}$
$\displaystyle={g\sinh q\over
w_{0}}\Big{(}\alpha(b\gamma+a\sqrt{1-\gamma^{2}})\sin
p+(a\gamma-b\sqrt{1-\gamma^{2}})\cos p\Big{)}$ $\displaystyle B^{\prime}$
$\displaystyle=\pm\sqrt{4g^{2}w_{0}^{2}-e^{-2B}}$ (4.55)

This system of ordinary differential equations can only be integrated
numerically. We will choose the coordinate $u$ such that the turning point of
the metric function where $B^{\prime}(u)=0$ is located at $u=0$. We then use
the BPS equations (4.1) to determine $p^{\prime}(0)$, $q^{\prime}(0)$ and
$B(0)$ for a given $q(0)$ and $p(0)$. We then integrate the equations of
motion following from the variation of the Lagrangian (2). This means that all
our solutions depend on two initial conditions $q(0)$ and $p(0)$. We have
given an illustrative example of the flows we can obtain in figure 4.1.

###  4.2 Truncation 2

The matrix $A_{1}$ for this truncation is given by

$\displaystyle A_{1}=\left(\begin{array}[]{cccccccc}0&0&0&{a+c\over
4}&0&0&0&0\\\ 0&-{c\over 2}&{-a+c\over 4}&0&b&0&0&b\\\ 0&{-a+c\over
4}&-{c\over 2}&0&-b&0&0&-b\\\ {a+c\over 4}&0&0&0&0&0&0&0\\\ 0&b&-b&0&{c\over
2}&0&0&{-a+c\over 4}\\\ 0&0&0&0&0&0&{a+c\over 4}&0\\\ 0&0&0&0&0&{a+c\over
4}&0&0\\\ 0&b&-b&0&{-a+c\over 4}&0&0&{c\over 2}\\\ \end{array}\right)$ (4.64)

where

$\displaystyle a$
$\displaystyle=2\cos^{4}p(\alpha\cosh^{4}q)+2\sin^{4}p(1+\alpha\cosh^{4}q)$
$\displaystyle b$ $\displaystyle=\sin p\cos p\cosh
q\big{(}\cos^{2}p(\alpha+\cosh^{2}q)-\sin^{2}p(1+\alpha\cosh^{2}q)\big{)}$
$\displaystyle c$ $\displaystyle=(1+\alpha)\cosh^{2}q\sin^{2}2p$ (4.65)

The eigenvectors $n^{(1)}_{\pm}$ of $A_{1}$ with eigenvalues $\pm w_{0}$
corresponding to the unbroken $N=(1,1)$ supersymmetries are given by

$\displaystyle n^{(1)}_{\pm}=\left(\begin{array}[]{c}0\\\ {a-3c\pm 4w_{0}\over
8b}\\\ -{a-3c\pm 4w_{0}\over 8b}\\\ 0\\\ 1\\\ 0\\\ 0\\\
1\end{array}\right),\quad w_{0}={1\over 4}\sqrt{64b^{2}+(3c-a)^{2}}$ (4.74)

We have checked that the extremum (2.2.2) does satisfy the supersymmetry
condition (3.5) for the $w_{0}$ defined above and hence corresponds to an AdS
vacuum with $N=(1,1)$ supersymmetry. The rest of the eigenvectors of $A_{1}$
do not have eigenvalues which satisfy the supersymmetry condition (3.5) for
the $N=(1,1)$ vacuum. We chose the initial conditions the same way as in
section 4.1.

(a)

(b)

(c) (d) Figure 4: Truncation 2: (a)-(c) plots of $p,q,B$ respectively, (d)
parametric plot of the Janus flow in the $x,y$ variables, the $N=(4,4)$ vacuum
is at the origin and the dots denote the locations of the $N=(1,1)$ vacua.
Blue: Janus between $N=(4,4)$ vacua, red: Janus between $N=(1,1)$ vacua,
green: RG-Janus between $N=(4,4)$ and $N=(1,1)$. We have set $\alpha=1.2$ for
these examples.

In figure 4.2 we display examples of solutions to the flow equations
representing Janus flows between $N=(4,4)$ vacua, $N=(1,1)$ vacua and RG-flow
Janus solutions between $N=(4,4)$ and $N=(1,1)$ vacua. We note that the flows
involving the $N=(1,1)$ vacua are a new feature of the truncation. As
discussed in section 5.1 the $N=(1,1)$ is a repulsive fixed point of the flow
and to obtain the numerical solutions one has to fine tune the initial
conditions at the turning point to approach the $N=(1,1)$ vacuum. This implies
that choosing an initial $p(0)$, the initial $q(0)$ for which an RG-flow
solution exists is fixed (if such a numerical solution exists). A third kind
of low solution corresponds to a Janus solution interpolating between
$N=(1,1)$ vacua, since both vacua are repulsive such solutions only exist for
a discrete set of initial conditions.

###  4.3 Truncation 3

The matrix $A_{1}$ for this truncation is given by

$\displaystyle A_{1}=\left(\begin{array}[]{cccccccc}0&0&0&{b+c\over
2}&0&{a+d\over 2}&0&0\\\ 0&-{c}&{-b+c\over 2}&0&{-a+d\over 2}&0&0&d\\\
0&{-b+c\over 4}&-{c}&0&-d&0&0&{a-d\over 2}\\\ {b+c\over
2}&0&0&0&0&0&{-a-d\over 2}&0\\\ 0&{-a+d\over 2}&-d&0&{c}&0&0&{-b+c\over 2}\\\
{a+d\over 2}&0&0&0&0&0&{b+c\over 2}&0\\\ 0&0&0&{-a-d\over 2}&0&{b+c\over
2}&0&0\\\ 0&d&{a-d\over 2}&0&{-b+c\over 2}&0&0&{c}\\\ \end{array}\right)$
(4.83)

where

$\displaystyle a$ $\displaystyle=\sin^{3}p(1+\alpha\cosh^{3}q)$ $\displaystyle
b$ $\displaystyle=\cos^{3}p(\alpha+\cosh^{3}q)$ $\displaystyle c$
$\displaystyle=\sin^{2}p\cos p\cosh q(1+\alpha\cosh q)$ $\displaystyle d$
$\displaystyle=\cos^{2}p\sin p\cosh q(\alpha+\cosh q)$ (4.84)

(a)

(b)

(c) (d) Figure 5: Truncation 3: (a)-(c) plots of $p,q,B$ respectively, (d)
parametric plot of the Janus flow in the $x,y$ variables, the $N=(4,4)$ vacuum
is at the origin and the dots denote the locations of the $N=(1,1)$ vacua.
Blue: Janus between $N=(4,4)$ vacua, red: Janus between $N=(1,1)$ vacua,
green: RG-Janus between $N=(4,4)$ and $N=(1,1)$. We have set $\alpha=1.2$ for
these examples.

The eigenvectors $n^{(1)}_{\pm}$ of $A_{1}$ with eigenvalues $\pm w_{0}$
corresponding to the unbroken $N=(1,1)$ supersymmetries are given by

$\displaystyle n^{(1)}_{\pm}=\left(\begin{array}[]{c}0\\\ -{b-3c\pm
2w_{0}\over a-3d}\\\ {b-3c\pm 2w_{0}\over a-3d}\\\ 0\\\ 1\\\ 0\\\ 0\\\
1\end{array}\right),\quad w_{0}={1\over 2}\sqrt{(a-3d)^{2}+(b-3c)^{2}}$ (4.93)

The rest of the eigenvectors of $A_{1}$ do not have eigenvalues which satisfy
the supersymmetry condition (3.5) for the $N=(1,1)$ vacuum. Note that all of
them reduce to the ones of truncation 1 for the $N=(4,4)$ vacuum.

In figure 4.3 we display a sample of solutions to the flow equations
representing Janus flows between $N=(4,4)$ vacua, $N=(1,1)$ vacua and RG-flow
Janus solutions between an $N=(4,4)$ and $N=(1,1)$ vacuum. We note that the
solutions behave qualitatively similar to the ones displayed for truncation 2.

##  5 Holographic calculations

In this section we will perform some holographic calculations for the
solutions obtained in the section 4. In particular we will calculate the
masses for the fluctuations of the scalar fields around the $N=(4,4)$ and
$N=(1,1)$ vacua. This will allow us to identify the dimensions of the dual
operators which are turned on in the flows. One of the results is that for
truncation 2 and 3 the mass squared of the fluctuations are positive,
corresponding to operators with scaling dimensions $\Delta>2$. Since the
behavior near the AdS vacuum is given by

$\displaystyle\lim_{\varepsilon\to
0}\phi\sim\bar{\phi}+c_{1}\varepsilon^{\Delta}+c_{2}\varepsilon^{2-\Delta}+\cdots$
(5.1)

where $\varepsilon\to 0$ corresponds to approaching the AdS boundary, the
initial conditions have to be fine tuned in order to make the repulsive term
$c_{2}\varepsilon^{2-\Delta}$ very small. In addition we consider the
entanglement entropy of a symmetric region around the defect [34, 35, 37, 38,
36] and give a prescription to obtain the defect entropy (or g-factor) [39].

###  5.1 Operator spectrum

The $N=(4,4)$ vacuum has $q_{i}=0,i=1,2,3,4$. Since the kinetic terms for
$p_{i}$ are vanishing the $x_{i},y_{i}$ defined in (2.16) are better suited to
analyze the fluctuations. Expanding around the $x_{i}=y_{i}=0$ vacuum one
finds for the quadratic term of the fluctuations.

$\displaystyle{1\over e}{\cal L_{(2)}}={1\over
4}\sum_{i}\Big{(}\partial_{\mu}\delta x_{i}\partial^{\mu}\delta
x_{i}+\partial_{\mu}\delta y_{i}\partial^{\mu}\delta
y_{i}\Big{)}+{g_{c}^{2}\over 4}\sum_{i}\Big{(}(1+2\alpha)\delta
x_{i}^{2}+\alpha(\alpha+2)\delta y_{i}^{2}\Big{)}$ (5.2)

From which we can read off the masses of the scalar fluctuations. The masses
the determine the conformal scaling dimensions

$\displaystyle\Delta_{\pm}=1\pm\sqrt{1+m^{2}R^{2}}$ (5.3)

where $R$ is the AdS radius of the vacuum. Setting $g_{c}=1/(1+\alpha)$ to
obtain a unit radius $AdS_{3}$ for the $N=(4,4)$ vacuum and the standard
AdS/CFT relation the conformal dimensions of the dual operators are displayed
in table 1.

$N=(4,4)$ | $m^{2}$ | $\Delta_{+}$ | $\Delta_{-}$
---|---|---|---
$\delta x_{i}$ | $-{1+2\alpha\over(1+\alpha)^{2}}$ | ${1+2\alpha\over 1+\alpha}$ | ${1\over 1+\alpha}$
$\delta y_{i}$ | $-{\alpha(2+\alpha)\over(1+\alpha)^{2}}$ | ${2+\alpha\over 1+\alpha}$ | ${\alpha\over 1+\alpha}$
Table 1: Mass and conformal dimensions of scalar fluctuations for the
$N=(4,4)$ vacuum

Note that $\Delta_{+}$ gives the scaling dimension of the dual operator in the
standard quantization which takes values between $1<\Delta_{+}<2$ for
$\alpha>0$, whereas $\Delta_{-}$ corresponds to the alternative quantization
and $0<\Delta_{-}<1$ for $\alpha>0$. Supersymmetric flows are related to the
standard quantization which we will adapt in the following [40]. We note that
the $N=(4,4)$ vacuum is attractive since both $x$ and $y$ are dual to
operators with $\Delta<2$ and the initial conditions do not have to be fine
tuned for (5.1) to approach the vacuum value.

For truncations 2 and 3 we can determine the scaling dimensions of the
operators at the $N=(1,1)$ vacuum by expanding the scalar action around the
vacuum to second order and diagonalizing the resulting scalar Lagrangian. The
resulting expressions are quite unwieldy and we present the plots of the
scaling dimensions of the two modes as a function of $\alpha$ in figure 6. We
note that the scaling dimensions are larger than 2 and hence the $N=(1,1)$
corresponds to a repulsive fixed point and the initial conditions have to be
fine-tuned.

(a)

(b)

Figure 6: (a) Conformal dimension of operator dual to scalar fluctuations
around the $N=(1,1)$ vacuum for truncation 2, (b) same for truncation 3.

###  5.2 Holographic entanglement entropy

The Ruy-Takayanagi prescription [41] relates holographic entanglement entropy
to the area of a minimal surface in the bulk which when approaching the $AdS$
boundary ends at the border of the entangling surface. For a three dimensional
static bulk spacetime this corresponds to a geodesic in the bulk which
terminates at the ends of the entangling interval on the boundary. For the
$AdS_{2}$ sliced metric (3.1) and an entangling surface which is symmetric
about the defect and of length $2L$, such a geodesic is simply parameterized
by $u$ and constant $z=L$. The entanglement entropy is then given by

$\displaystyle S_{EE}(L)={1\over
4G_{N}}\int_{u_{-\infty}}^{u_{+\infty}}du={1\over
4G_{N}}\big{(}u_{-\infty}-u_{+\infty}\big{)}$ (5.7)

Where $u_{\pm\infty}$ will be related to an UV Fefferman-Graham cutoff in the
following, we will generalize the derivation of [34, 35] to the case of an RG-
flow interface where the AdS radius and hence the central charge take
different values on both sides of the interface. The asymptotic behavior of
the metric is determined by the metric function $B(u)$ as $u\to\pm\infty$

$\displaystyle\lim_{u\to\pm\infty}B(u)=\pm{u\over
R_{\pm}}+\ln\lambda_{\pm}-\ln 2+o({1\over u})$ (5.8)

In the two asymptotic regions we can define a Fefferman-Graham coordinate
system by defining a new coordinates $\hat{u}_{\pm}$

$\displaystyle u\to\pm\infty:\quad u$ $\displaystyle=R_{\pm}\hat{u}_{\pm}\mp
R_{\pm}\ln\lambda_{\pm}+o({1\over u})$ (5.9)

and then the coordinates $\zeta_{\pm},\eta$

$\displaystyle u\to+\infty:$ $\displaystyle\quad\quad
e^{-2\hat{u}_{+}}={1\over
4}{\zeta_{+}^{2}\over\eta^{2}}+o(\zeta_{+}^{4}),\quad\quad
z=\eta\Big{(}1+{1\over 2}{\zeta_{+}^{2}\over\eta^{2}}\Big{)}+o(\zeta_{+}^{4})$
$\displaystyle u\to-\infty:$
$\displaystyle\quad\quad\;e^{2\hat{u}_{-}}={1\over
4}{\zeta_{-}^{2}\over\eta^{2}}+o(\zeta_{-}^{4}),\quad\quad
z=-\eta\Big{(}1+{1\over
2}{\zeta_{-}^{2}\over\eta^{2}}\Big{)}+o(\zeta_{-}^{4})$

This expansion is valid for $\eta>>\zeta_{\pm}$, i.e. if we consider an
entanglement interval which is far away from the interface. In this limit the
metric becomes

$\displaystyle\zeta_{\pm}\to 0$ $\displaystyle\quad\quad
ds^{2}=R_{\pm}^{2}\Big{(}{-d\zeta_{\pm}^{2}-d\eta^{2}+dt^{2}\over\zeta_{\pm}^{2}}\Big{)}+o(1)$
(5.11)

It follows that $R_{\pm}$ defined in (5.8), corresponds to the asymptotic AdS
radius and the left and right side of the interface respectively and a
Fefferman-Graham cutoff is given by setting $\zeta_{\pm}=\varepsilon$. For the
entanglement region located at $z=L$ in follows from (5.11) that the FG cutoff
is related to the $u_{\pm}$ cutoff as follows

$\displaystyle u_{\pm}=\mp R_{\pm\infty}\ln\left({1\over 2}{\varepsilon\over
L}\right)\mp R_{\pm}\ln\lambda_{\pm}$ (5.12)

Plugging this into (5.7) gives the entanglement entropy

$\displaystyle S_{EE}$ $\displaystyle={1\over
4G_{N}}\Big{(}(R_{+}+R_{-})\ln{L\over
2\varepsilon}-R_{+}\ln\lambda_{+}-R_{-}\ln\lambda_{-}\Big{)}$
$\displaystyle={c_{+}+c_{-}\over 6}\ln{L\over 2\varepsilon}-{c_{+}\over
6}\ln\lambda_{+}-{c_{-}\over 6}\ln\lambda_{-}+o(\varepsilon)$ (5.13)

the constant term gives the boundary entropy

$\displaystyle g=-{c_{+}\over 6}\ln\lambda_{+}-{c_{-}\over 6}\ln\lambda_{-}$
(5.14)

Where $c_{\pm}$ is the central charge for the two CFTs on either side of the
RG interface. The g-factor is given by the second and third term in (5.2). For
a Janus interface we have $c_{+}=c_{-}=c$, whereas the central charges differ
on both sides of the interface for a RG-flow interface. It is straightforward
to determine the $R_{\pm}$ and $\ln\lambda_{\pm}$ by numerically fitting the
metric functions (see plot (c) in figures 4.2 and 4.3) to determine the slope
and the intercept (5.8) in the limit of large $|u|$. We will give an example
of numerical results by presenting the g-factors as for the RG-interface
between the $N=(4,4)$ vacuum and a $N=(1,1)$ vacuum in truncation 2. As
discussed in section 4.2 there exists a unique RG-flow interface for a choice
on initial condition $p_{0}$. In figure 7 we present the g-factor as a
function of the initial condition for a particular value of $\alpha=1.4$.

(a)

(b)

Figure 7: (a) Plot of boundary entropy for RG-flow interface for truncation 2,
as a function of initial condition $p_{0}$ at the turning point for
$\alpha=1.4$. (b) Illustration of RG-flows for some initial values of $p_{0}$.

##  6 Discussion

In this paper we found holographic interface solution in three dimensional
gauged supergravity theories. An important feature of these theories is that
they have AdS vacua which preserve $N=(1,1)$ supersymmetry in addition to the
$N=(4,4)$ AdS vacuum. This feature allows us to find solutions which
correspond to interfaces between two $N=(4,4)$ vacua on both side, $N=(1,1)$
on both sides, as well as RG-flow interfaces which have a $N=(4,4)$ on one
side and $N=(1,1)$ vacuum on the other. We derived BPS flow equations which
are three first order nonlinear differential equations for the two scalars
$p,q$ which are non zero in the truncations as well as the warp factor $B$ of
the $AdS_{2}$ slicing. By using the freedom to shift the warping coordinate
$u$ by a constant we can choose the initial conditions for the flow as the
value of $p$ and $q$ at the turning point of the warp factor, where
$B^{\prime}=0$. In fact we use the BPS equations to determine the initial
conditions for the second order equation motion following from the variation
of the action. The numerical accuracy of the solution is tested by checking
the BPS equations away from the point where the initial conditions were fixed.

The $N=(1,1)$ extrema are repulsive fixed points of the flow and hence the
initial condition have to be fine tuned using a shooting method. This is
possible by fixing one scalar initial condition and varying the other in order
to come closer and closer to the $N=(1,1)$ vacuum in the flow. Our results
indicate that the qualitative behavior of the solutions for general $\alpha$
is quite similar to the behavior of the $\alpha=1$ solutions obtained in [1].
In addition we have considered entanglement entropy for the Janus and RG-flow
solutions. Since for the RG-flow solutions the central charges and hence AdS
radii are different on both sides of the interface and one has to carefully
consider the UV cut-off. It is possible to determine the g-function or
interface entropy from the numerical solution by a linear fit of the warp
factor $B$.

We have considered truncations of the scalars to two nonzero scalars $q$ and
$p$ (or $x$ and $y$), it would be interesting to generalize this since it
would then be possible to consider more complicated flows between different
$N=(1,1)$ vacua. It would also be interesting to investigate the solutions we
have found can be lifted and have a representation in $\text{AdS}_{3}\times
S^{3}\times S^{3}\times S^{1}$ holography. It would also be interesting to see
whether the prescription for the interface entropy can be applied to other
examples of RG-flow interfaces. We leave these interesting questions for
future work.

## Acknowledgements

The work of M. G. was supported, in part, by the National Science Foundation
under grant PHY-19-14412. The authors are grateful to Kevin Chen for initial
collaboration. The authors are grateful to the Mani L. Bhaumik Institute for
Theoretical Physics for support.

##  Appendix A $\operatorname{SO}(8)$ Gamma matrices

The formulation of the $N=8$ three dimensional gauged supergravity utilizes
with $8\times 8$ Gamma matrices $\Gamma^{I}_{A\dot{A}}$ and their transposes
$\Gamma^{I}_{\dot{A}A}$, they satisfy the Clifford algebra,

$\displaystyle\Gamma^{I}_{A\dot{A}}\Gamma^{J}_{\dot{A}B}+\Gamma^{J}_{A\dot{A}}\Gamma^{I}_{\dot{A}B}=2\delta^{IJ}\delta_{AB}$
(A.1)

Explicitly, we use the basis as given in Green-Schwarz-Witten [42],

$\displaystyle\Gamma^{8}_{A\dot{A}}$ $\displaystyle=1\otimes 1\otimes 1~{},$
$\displaystyle\Gamma^{1}_{A\dot{A}}$ $\displaystyle=i\sigma_{2}\otimes
i\sigma_{2}\otimes i\sigma_{2}$ $\displaystyle\Gamma^{2}_{A\dot{A}}$
$\displaystyle=1\otimes\sigma_{1}\otimes i\sigma_{2}~{},$
$\displaystyle\Gamma^{3}_{A\dot{A}}$ $\displaystyle=1\otimes\sigma_{3}\otimes
i\sigma_{2}$ $\displaystyle\Gamma^{4}_{A\dot{A}}$
$\displaystyle=\sigma_{1}\otimes i\sigma_{2}\otimes 1~{},$
$\displaystyle\Gamma^{5}_{A\dot{A}}$ $\displaystyle=\sigma_{3}\otimes
i\sigma_{2}\otimes 1$ $\displaystyle\Gamma^{6}_{A\dot{A}}$
$\displaystyle=i\sigma_{2}\otimes 1\otimes\sigma_{1}~{},$
$\displaystyle\Gamma^{7}_{A\dot{A}}$ $\displaystyle=i\sigma_{2}\otimes
1\otimes\sigma_{3}$ (A.2)

The matrices $\Gamma^{IJ}_{AB}$, $\Gamma^{IJ}_{\dot{A}\dot{B}}$ and similar
are defined as antisymmetrized products of $\Gamma$s with the appropriate
indices contracted. For example,

$\displaystyle\Gamma^{IJ}_{AB}\equiv\frac{1}{2}(\Gamma^{I}_{A\dot{A}}\Gamma^{J}_{\dot{A}B}-\Gamma^{J}_{A\dot{A}}\Gamma^{I}_{\dot{A}B})$
(A.3)

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