# On a field tensor for gravity and electromagnetism

M<EMAIL_ADDRESS>
Faculty of Technology, Natural Sciences and Maritime Sciences
University of South-Eastern Norway
Porsgrunn, Norway

###### Abstract

We show that a three rank Lanczos type tensor field is an appropriate choice
to describe relativistic electromagnetic and gravitational effects. More
precisely, we identify the irreducible field-decompositions of this tensor as
gravitaional and electromagnetic fields. A set of divergence equations are
proposed as field equations for the unified field.

##  1 Introduction

In the early to mid 1900 a number of articles were published on the
unification of electromagnetism and gravitation. This program of unification
has been put under the umbrella term Unified Field Theories (UFTs) — see [4]
for a comprehensive review. But due to the remarkable achievment of Quantum
Field theory in unifying the nuclear and electromagnetic forces, the UFT
program has been replaced by the pursuit of a theory of Quantum Gravity. Since
spinors are needed in the description of fermions [13], it is essential for a
unified field theory to admit spinor structure in order to be a viable theory
for the description of e.g. electrons. Geroch has shown in [2] that it is a
necessary and sufficient condition for a non-compact space time to admit a
spinor structure if it carries a global field of orthonormal tetrads. The
frame formalism also reflect the role of observers in physics, and is thus a
natural formalism both in classical relativity and quantum field theory [12].
Furthermore, due to the nonlinearity of the Einstein equations, a metric
distributional solution describing a point particle is not possible in general
relativity [3]. We refer to [10] for a review of the use of distributions in
general relativity. On the other hand, the Maxwell equations do admit a
solution representing a charged point particle. In the present work we explore
the possibility of a theory which both admits a spinor structure — by
employing a global tetrad field — and whose field equations are linear with
respect to the sources and field tensor, in striking similarity with the
Maxwell equations. We remark that we do not make use of the spinor structure
in the present article. A proper investigation of the spinorial equations and
detailed analysis of the spinor fields will be published elsewhere.

##  2 Geometric considerations

Let $(\mathcal{M},{\bm{g}})$ denote a spacetime represented by a 4-dimensional
manifold, $\mathcal{M}$, with a Lorentzian metric ${\bm{g}}$. The motion of
particles of some matter filling spacetime give rise to a natural splitting by
constructing frames comoving with the flow lines of the particles. This has
the further advantage that it does not require a foliation of $\mathcal{M}$.
We shall denote the tangent vector to the flow lines as ${\bm{u}}$ satisfying

${\bm{g}}({\bm{u}},{\bm{u}})=-1.$

At each point $p\in\mathcal{M}$ the frame field $\\{{\bm{e}}_{a}\\}$ is such
that

${\bm{g}}({\bm{e}}_{a},{\bm{e}}_{b})=\eta_{ab},$

where $\eta_{ab}$ are the frame components of the Minkowski metric. The frames
$\\{{\bm{e}}_{a}\\}$ give rise to a co-frame, $\\{\mathbf{\omega}^{a}\\}$
satisfying

$\langle{{\bm{e}}_{a},\mathbf{{\bm{\omega}}}^{b}\rangle}=\delta_{a}{}^{b}.$

In the following all indices will be given in terms of the frame and co-frame
unless otherwise stated. The metric tensor give rise to a natural connection
$\mathbf{\nabla}$ such that $\mathbf{\nabla}{\bm{g}}=0$, which is the metric
compatibility condition. In terms of the frames, this condition takes the form

$\Gamma_{a}{}^{b}{}_{c}\eta_{bd}+\Gamma_{a}{}^{b}{}_{d}\eta_{bc}=0,$ (1)

where the frame connection coefficients are defined by the directional
derivative along the direction of the frame indices

$\nabla_{a}{\bm{e}}_{b}=\Gamma_{a}{}^{c}{}_{b}{\bm{e}}_{c},\qquad\nabla_{a}=\langle{{\bm{e}}_{a},\mathbf{\nabla}\rangle}.$

Thus, for a two rank tensor $\bm{\Omega}$ we have that the frame components of
its derivative is given by,

$\nabla_{a}\Omega_{bc}=e_{c}[\Omega_{bc}]-\Gamma_{a}{}^{d}{b}\Omega_{dc}-\Gamma_{a}{}^{d}{c}\Omega_{bd}$

. Furthermore, if the connection $\mathbf{\nabla}$ is torsion-free, we have
that

$\Sigma_{a}{}^{c}{}_{b}=0,$ (2)

where the frame components of the torsion tensor are defined by

$\Sigma_{a}{}^{c}{}_{b}{\bm{e}}_{c}=\left[{\bm{e}}_{a},{\bm{e}}_{b}\right]+\left(\Gamma_{a}{}^{c}{}_{b}-\Gamma_{b}{}^{c}{}_{a}\right){\bm{e}}_{c}.$

The commutation of the connection may be expressed in terms of the Riemann
curvature tensor and the torsion tensor

$\displaystyle\nabla_{[a}\nabla_{b]}v^{c}=R^{c}{}_{dab}v^{d}+\Sigma_{a}{}^{d}{}_{b}\nabla_{d}v^{c},$
$\displaystyle\nabla_{[a}\nabla_{b]}w_{c}=-R^{d}{}_{cab}w_{d}+\Sigma_{a}{}^{d}{}_{b}\nabla_{d}w_{c}.$

The frame components of the Riemann curvature tensor is given by

$R^{c}{}_{dab}=\partial_{a}\Gamma_{b}{}^{c}{}_{d}-\partial_{b}\Gamma_{a}{}^{c}{}_{d}+\Gamma_{f}{}^{c}{}_{d}(\Gamma_{b}{}^{f}{}_{a}-\Gamma_{a}{}^{f}{}_{b})+\Gamma_{b}{}^{f}{}_{d}\Gamma_{a}{}^{c}{}_{f}-\Gamma_{a}{}^{f}{}_{d}\Gamma_{b}{}^{c}{}_{f}-\Sigma_{a}{}^{f}{}_{b}\Gamma_{f}{}^{c}{}_{d}$
(3)

—see [11] for details. The Riemann tensor has all the usual symmetries, and it
satisfies the Bianchi identity for a general connection

$\displaystyle
R^{d}{}_{[cab]}+\nabla_{[a}\Sigma_{b}{}^{d}{}_{c]}+\Sigma_{[a}{}^{e}{}_{b}\Sigma_{c]}{}^{d}{}_{e}=0,$
(4)
$\displaystyle\nabla_{[a}R^{d}{}_{|e|bc]}+\Sigma_{[a}{}^{f}{}_{b}R^{d}{}_{|e|c]f}=0.$
(5)

Furthermore, we recall that the Riemann tensor admits the _irreducible
decomposition_

$\displaystyle
R^{c}{}_{dab}=C^{c}{}_{dab}+2(\delta^{c}{}_{[a}L_{b]d}-\eta_{d[a}L_{b]}{}^{c}),$
(6)

with $C^{c}{}_{dab}$ the components of the _Weyl tensor_ and

$S_{ab}\equiv R_{ab}-\frac{1}{6}R\eta_{ab}$ (7)

denotes the components of the _Schouten tensor_. The connection
$\mathbf{\nabla}$ is called the Levi-Civita connection of ${\bm{g}}$ if it
satisfies (1) and (2). In what follows we will assume the connection to be
Levi-Civita.

### A projection formalism

At each point in the spacetime manifold $\mathcal{M}$ the flow lines give rise
to a tangent space which can be split into parts in the direction of
${\bm{u}}$ and those orthogonal. This means that without implying a foliation,
we may decompose every tensor defined at each point $p\in\mathcal{M}$ into its
orthogonal and timelike part. This may be done by contracting with
$\mathbf{u}$ and the projector defined as

$h_{a}{}^{b}\equiv\eta_{a}{}^{b}+u_{a}u^{b},\qquad{\bm{u}}=u^{a}\mathbf{e}_{a}.$

Thus, a tensor $T_{ab}$ may be split into its time-like, mixed and space-like
parts given, respectively, by

$T_{00}=u^{a}u^{b}T_{ab},\qquad T^{\prime}_{0c}=u^{a}h^{b}{}_{c}T_{ab},\qquad
T^{\prime}_{cd}=h^{a}{}_{c}h^{b}{}_{d}T_{ab},$

where ′ denotes that the free indices left are spatial —e.g.
$T^{\prime}_{a0}u^{a}=0$. Decomposing $\mathbf{\nabla u}$ we obtain

$\nabla_{a}u^{b}=\chi_{a}{}^{b}-u_{a}a^{b},$ (8)

where $\chi_{a}{}^{b}$ and $a^{b}$ are the components of the Weingarten tensor
and 4-acceleration, respectively, defined by

$\chi_{a}{}^{b}\equiv h_{a}{}^{c}\nabla_{c}u^{b},\qquad a^{b}\equiv
u^{c}\nabla_{c}u^{b}.$

We split $\chi_{ab}$ into its symmetric, tracefree part and antisymmetric part
— i.e we have,

$\chi_{(ab)}-\frac{1}{3}h_{ab}\chi\equiv\sigma_{ab},\qquad\chi_{[ab]}\equiv\omega_{{\bm{a}}{\bm{b}}}.$

In the literature (e.g. see [12] p.217) $\chi$, $\sigma_{ab}$ and
$\omega_{ab}$ is called, respectively, the expansion, shear and the twist of
the congruence with four velocity ${\bm{u}}$. The decomposition (8) now takes
the form,

$\nabla_{a}u^{b}=\sigma_{a}{}^{b}+\frac{1}{3}h_{a}{}^{b}\chi+\omega_{a}{}^{b}-u_{a}a^{b}.$
(9)

The decomposition of the four volume is

$\epsilon_{abcd}=-2\left(u_{[a}\epsilon_{b]cd}-\epsilon_{ab[c}u_{d]}\right),\qquad\epsilon_{bcd}=\epsilon_{abcd}u^{a}.$

Given a tensor $T_{abc}$ which is antisymmetric in its two last indices, we
may construct the electric and magnetic parts with respect to $\mathbf{u}$. In
frame indices this is, respectively, defined by

$E_{cd}\equiv T_{abe}h_{c}{}^{a}h_{d}{}^{b}u^{e},\qquad B_{cd}\equiv
T^{\ast}{}_{abe}h_{c}{}^{a}h_{d}{}^{b}u^{e},$

where the Hodge dual operator, denoted by ∗, is defined by

$T^{\ast}{}_{abe}\equiv-\frac{1}{2}\epsilon^{mn}{}_{be}T_{amn},$

and has the property that

$T^{\ast\ast}{}_{abc}=-T_{abc}.$

Depending on the symmetries and rank of the tensor, the above definition for
electric and magnetic decomposition may vary slightly. Central for our
discussion is that $E_{ab}$ and $B_{ab}$ are spatial and symmetric.

##  3 The field tensor

We consider the rank three tensor ${\bm{Z}}$ (hereafter called the Z-tensor)
with the following symmetries,

$Z_{[abc]}=0,\qquad Z_{abc}=Z_{a[bc]}.$

It can be readily shown that the first symmetry property implies that

$Z_{cab}=2Z_{[ba]c}.$ (10)

The Hodge dual of the Z-tensor ${\bm{Z}}^{\ast}$ is defined in the customary
way by,

$Z^{\ast}{}_{abc}\equiv-\frac{1}{2}\epsilon_{bc}{}^{de}Z_{ade}.$

The frame fields ${\bm{e}}_{a}$ provide a natural 1+3 decomposition of
${\bm{Z}}$ and ${\bm{Z}}^{\ast}$ into parts in the direction of and orthogonal
to the flow ${\bm{u}}$. This is obtained by using the projector ${\bm{h}}$ as
described in Section 2.The decomposition read,

$\displaystyle
Z_{abc}=-2\eta_{a[b}P_{c]}+\epsilon_{bc}{}^{d}\Phi_{ad}+2u_{[b}\Psi_{c]a}-\epsilon^{d}{}_{bc}u_{a}Q_{d}+2\epsilon^{d}{}_{a[c}u_{b]}Q_{d},$
(11a) $\displaystyle
Z^{\ast}{}_{amn}=\epsilon_{mnb}u_{a}P^{b}-2\epsilon_{ab[m}u_{n]}P^{b}+2\Phi_{a[m}u_{n]}+\epsilon_{mnb}\Psi_{a}{}^{b}+2\eta_{a[n}Q_{m]},$
(11b)

where we have defined,

$\Psi_{ab}\equiv Z_{(a^{\prime}b^{\prime})0},\qquad\Phi_{ab}\equiv
Z^{\ast}_{(a^{\prime}b^{\prime})0},\qquad P_{a}\equiv Z_{a00},\qquad
Q_{a}\equiv Z^{\ast}_{a00}.$

The tensors $\Psi_{ab}$ and $\Phi_{ab}$ are by definition symmetric tensors
defined on the orthogonal space of ${\bm{u}}$ —i.e. one has that

$\Psi_{ac}u^{a}=0,\qquad\Phi_{ac}u^{a}=0.$

Furthermore, since $\epsilon_{abc}$, $\Psi_{ab}$ and $\Phi_{ab}$ are spatial
fields, it is readily shown that

$P_{0}=Q_{0}=0.$

The traces of the Z-tensor and its dual are,

$\displaystyle Z^{a}{}_{ba}=3P_{b}+\Psi u_{b},\qquad Z^{a}{}_{b}{}^{b}=0,$
(12) $\displaystyle Z^{\ast}{}^{a}{}_{ba}=3Q_{b}-\Phi u_{b},\qquad
Z^{\ast}{}^{a}{}_{b}{}^{b}=0,$ (13)

where,

$\Psi\equiv\Psi^{a}{}_{a},\qquad\Phi\equiv\Phi^{a}{}_{a}.$

The first trace in (12) implies that

$Z^{a}{}_{0a}=-\Psi,$ (14)

and the first trace in (13) together with the first symmetry property implies
that

$Z^{\ast}{}^{a}{}_{0a}=\Phi=0.$ (15)

######  Lemma 1.

Let ${\bm{Z}}$ be a tensor of rank 3 with antisymmetry about two neighbouring
indices. Then ${\bm{Z}}$ has the symmetry property $Z_{[abc]}=0$ and the dual
field $Z^{\ast}_{(a^{\prime}b^{\prime})0}$ has vanishing trace.

We make the further assumption that $\Psi=0$ — i.e we have that

$Z^{a}{}_{0a}=Z^{\ast}{}^{a}{}_{0a}=0.$

######  Remark 1.

The assumption that $\Psi=0$ is motivated by the fact that we want to relate
the fields $\bm{\Psi}$ and $\bm{\Phi}$ to the electric and magnetic part of
the Weyl tensor. Observe that our assumption is a weaker constraint than the
Lanczos algebraic gauge — e.g see [9], [5],

$Z^{a}{}_{ba}=0.$

In fact, the Lanczos gauge violate our assumption that the fields ${\bm{P}}$
and $\bm{\Psi}$ represents pure electric and gravitational fields,
respectively, and can thus not be related in such a way as this gauge implies
— see equation (12).

######  Remark 2.

Observe that the absence of electric and magnetic fields is a necessary
condition for the Z-tensor to be a a Cotton tensor.

##  4 Finding the field equations for the Z-tensor

In the theory we propose, both gravity and electromagnetism is represented in
terms of a field on space time. The geometry of $\mathcal{M}$ will be given by
the frame components, rather than the metric, and the connection coefficients
as outlined in the introduction. Equations for the frame and the connection is
given by the choice of propagation — e.g. Fermi propagation — and the
definition of the Riemann and the torsion tensor. For more details on the
geometric equations, the reader is referred to [7], [1] and [8]. In what
follows we shall focus the discussion on the fields presented in the previous
section — i.e. $\bm{\Psi}$, $\bm{\Phi}$, ${\bm{P}}$ and ${\bm{Q}}$. These will
be taken as the fundamental fields, from which we may construct the unified
field tensor ${\bm{Z}}$. We thus seek a set of equations for ${\bm{Z}}$ which
will reduce to the relativistic Maxwell equations in the limit of no
gravitational field, and the Bianchi equations in the limit of no
electromagnetic fields. We begin with the Maxwell equations.

We observe that due to the symmetry of ${\bm{Z}}$ and ${\bm{Z}}^{\ast}$, it is
natural to define the two rank anti-symmetric tensors ${\bm{F}}$ and
${\bm{F}}^{\ast}$ as follows,

$F_{ab}\equiv u^{a}Z_{abc},\qquad
F^{\ast}{}_{ab}\equiv\dfrac{1}{2}\epsilon_{ab}{}^{mn}F_{mn}=u^{a}Z^{\ast}{}_{abc}.$

Using the decomposition of the Z-tensor, it is readily shown that

$F_{ab}=u_{b}P_{a}-u_{a}P_{b}+\epsilon_{abc}Q^{c},$

which is the right form of the Faraday tensor with $P_{a}$ and $Q_{a}$ as the
electric and magnetic fields respectively. The Maxwell equations are then
given by

$\displaystyle\nabla^{b}F_{ab}$ $\displaystyle=j_{c}$ (16a)
$\displaystyle\nabla^{b}F^{\ast}{}_{ab}$ $\displaystyle=0,$ (16b)

which may be formulated as evolution and constraint equations for the electric
and magnetic fields — i.e.

$\displaystyle u^{a}h_{mb}\nabla_{a}E^{b}-\epsilon_{mab}\nabla^{b}B^{a}$
$\displaystyle=-a^{a}\epsilon_{mab}B^{b}+J^{a}h_{ma}+E^{a}\chi_{am}-E_{m}\chi^{a}{}_{a}.$
(17a) $\displaystyle\nabla_{a}E^{a}$
$\displaystyle=a^{a}E_{a}+u^{a}J_{a}-\epsilon_{abc}B^{a},$ (17b)
$\displaystyle u^{b}h^{d}{}_{a}\nabla_{b}B^{a}$
$\displaystyle=a^{b}E^{a}\epsilon^{d}{}_{ba}+B^{b}\chi_{b}{}^{d}-B^{d}\chi^{b}{}_{b}-\epsilon^{d}{}_{ba}\nabla^{a}E^{b}\chi^{bc},$
(17c) $\displaystyle\nabla_{b}B^{b}$
$\displaystyle=a^{b}B_{b}+E^{b}\epsilon_{bac}\chi^{ac}.$ (17d)

We now turn to consider equations for the gravitational field. It is customary
to here study solutions to the Einstein field equations — i.e

$R_{ab}-\dfrac{1}{2}Rg_{ab}=\tau_{ab}.$ (18)

But as we are seeking a theory where the geometry is given by the frame
components and the gravitational field is represented by the irreducible
components of the Weyl tensor, we will use the Bianchi identity (5) as field
equations. In this formalism the Einstein equations takes on the form of
constraint equations — see equation (20). Thus, the unknowns for the
gravitational field will be the electric $E_{ab}$ and magnetic $B_{ab}$ part
of the Weyl tensor — i.e we consider the equations

$\displaystyle
u^{a}h_{m}{}^{c}h_{n}{}^{d}\nabla_{a}E_{cd}+\epsilon_{mdc}h_{n}{}^{a}\nabla^{d}B_{a}{}^{c}$
$\displaystyle=-2a^{a}B_{(m}{}^{c}\epsilon_{n)ac}-2E_{mn}\chi^{a}{}_{a}-E_{ac}h_{mn}\chi^{ac}$
$\displaystyle+2E_{na}\chi^{a}{}_{m}+E_{ma}\chi_{n}{}^{a}-\tfrac{1}{2}u^{a}h_{m}{}^{c}h_{n}{}^{d}\nabla_{a}S_{cd}$
$\displaystyle+\tfrac{1}{2}u^{a}h_{m}{}^{c}h_{n}{}^{d}\nabla_{d}S_{ac}$ (19a)
$\displaystyle\nabla_{a}E_{d}{}^{a}$
$\displaystyle=a^{a}E_{da}+E_{ac}u_{d}\chi^{ac}-\epsilon_{dcf}B_{a}{}^{f}\chi^{ac}-\epsilon_{acf}B_{d}{}^{f}$
$\displaystyle\chi^{ac}-\tfrac{1}{2}u^{a}u^{c}\nabla_{c}S_{da}+\tfrac{1}{2}u^{a}u^{c}\nabla_{d}S_{ac},$
(19b) $\displaystyle
u^{a}h_{l}{}^{c}h_{n}{}^{d}\nabla_{a}B_{cd}-\epsilon_{dc(n}h_{l)}{}^{a}\nabla^{d}E_{a}{}^{c}$
$\displaystyle=2a^{a}E_{(n}{}^{c}\epsilon_{l)ac}-2B_{ln}\chi^{a}{}_{a}-B_{ac}h_{ln}\chi^{ac}$
$\displaystyle+2\chi^{a}{}_{(l}B_{n)a}+B_{a(n}\chi_{l)}{}^{a}+\tfrac{1}{2}\epsilon_{cd(n}h_{l)}{}^{a}\nabla^{d}S_{a}{}^{c},$
(19c) $\displaystyle h_{n}{}^{a}\nabla_{c}B_{a}{}^{c}$
$\displaystyle=a^{a}B_{na}-E_{c}{}^{d}\epsilon_{nad}\chi^{ac}+2E_{a}{}^{d}\epsilon_{ncd}\chi^{ac}$
$\displaystyle+\tfrac{1}{2}\epsilon_{ncd}u^{a}\nabla^{d}S_{a}{}^{c}$ (19d)

where $S_{ab}$ is the Schouten tensor and defined in the customary way — see
equation (7). If the Einstein equations are assumed, then the Schouten tensor
is related to the Energy-momentum tensor $\tau_{ab}$ according to

$S_{ab}=\tau_{ab}-\tfrac{1}{3}\tau^{c}{}_{c}\ g_{ab}.$ (20)

Thus, a solution $(E_{ab},B_{ab})$ of the evolution equations (19a) and (19c),
satisfying the constraint equaitons (19b) and (19d), together with equation
(20) is equivalent to a metric solution of the Einstein equations (18) for a
given energy momentum tensor $\tau_{ab}$ — again the reader is referred to [6]
for more details.

Observe that $Z_{abc}$ contains all the fields necessary for a description of
both gravity and electromagnetism. That is, the spatial fields
($P_{a},Q_{a},\Psi_{ab},\Phi_{ab}$) has the correct rank, trace and symmetry
to represent ($E_{a},B_{a},E_{ab},B_{ab}$), respectively. The strategy to find
the correct field equations for the unified field tensor $Z_{abc}$ is to
compare the proposed equations so that they reduce to the Maxwell equations
and Bianchi equations in the case of no gravity and electromagnetism,
respectively. That is, we must construct the equations such that $\bm{\Psi}$
and $\bm{\Phi}$ will be a solution of equations (19a) - (19d) when
$P_{a}=Q_{a}=0$. Similarly, $P_{a},Q_{a}$ is requiered to be a solution of
equations (17a) - (17d) in the limit of $\Psi_{ab}=\Phi_{ab}=0$.

Due to the form of the decomposition of ${\bm{Z}}$, we propose field equations
on the form,

$\displaystyle\nabla^{b}Z_{abc}$ $\displaystyle=T_{ac},$ (21a)
$\displaystyle\nabla^{b}Z^{\ast}{}_{abc}$ $\displaystyle=A_{ac}.$ (21b)

Note that as a consequence of the antisymmetry in the Z-tensor and the
symmetry of the Ricci tensor, it follows that

$\nabla^{c}T_{ac}=\nabla^{c}A_{ac}=0.$ (22)

######  Remark 3.

For generality we shall not impose symmetry about the indices $\\{a,c\\}$, so
as to make $\bm{T}$ and $\bm{A}$ symmetric tensors. But strictly speaking such
an assumption should be made in order to study the equations on the form that
most resembles the Bianchi equations and the relativistic Maxwell equations.
Furthermore, this will make the tensors $\bm{A}$ and $\bm{T}$ divergence free.

In what follows, we will show that there exists tensors $A_{ab}$ and $T_{ab}$
such that the proposed field equations encompass the relativistic Maxwell
equations as well as the Bianchi equations. Recall that any tensor ${\bm{T}}$
may be decomposed in parts orthogonal and parallel to the four velocity
${\bm{u}}$ according to,

$T_{ab}=T_{a^{\prime}b^{\prime}}+T_{a^{\prime}0}u_{b}+T_{0b^{\prime}}u_{a}+T_{00}u_{a}u_{b}.$

We consider first the spatial components of the field equations — i.e

$\displaystyle h_{m}{}^{a}h_{n}{}^{c}\nabla^{b}Z_{abc}$
$\displaystyle=T_{m^{\prime}n^{\prime}}$ (23a) $\displaystyle
h_{m}{}^{a}h_{n}{}^{c}\nabla^{b}Z{}^{\ast}_{abc}$
$\displaystyle=A_{m^{\prime}n^{\prime}}.$ (23b)

Using the decomposition of $Z_{abc}$ and $Z^{\ast}{}_{abc}$ (23a) and (23b)
are equivalent to,

$\displaystyle
u^{a}h_{m}{}^{b}h_{n}{}^{c}\nabla_{a}\Psi_{bc}+\epsilon_{mbc}h_{n}{}^{a}\nabla^{c}\Phi_{a}{}^{b}$
$\displaystyle=-a_{n}P_{m}+a^{a}\epsilon_{nab}\Phi_{m}{}^{b}+a^{a}\epsilon_{mab}\Phi_{n}{}^{b}-2\Psi_{mn}\chi^{a}{}_{a}$
(24a) $\displaystyle-
h_{mn}\Psi_{ab}\chi^{ab}+2\Psi_{na}\chi^{a}{}_{m}+\epsilon_{mna}Q^{a}\chi^{b}{}_{b}-\epsilon_{nab}Q^{a}\chi^{b}{}_{m}$
$\displaystyle+\epsilon_{mab}Q^{a}\chi^{b}{}_{n}+\Psi_{ma}\chi_{n}{}^{a}-h_{m}{}^{b}h_{n}{}^{c}P^{a}\nabla_{a}h_{bc}-h_{mn}\nabla_{a}P^{a}$
$\displaystyle+\epsilon_{mnb}u^{a}\nabla_{a}Q^{b}-\tfrac{1}{2}u^{a}h_{m}{}^{b}h_{n}{}^{c}\nabla_{a}S_{bc}+h_{n}{}^{a}P_{m}\nabla_{b}h_{a}{}^{b}$
$\displaystyle+h_{ma}h_{nb}\nabla^{b}P^{a}+\tfrac{1}{2}u^{a}h_{m}{}^{b}h_{n}{}^{c}\nabla_{c}S_{ab},$
$\displaystyle
u^{a}h_{m}{}^{b}h_{n}{}^{c}\nabla_{a}\Phi_{bc}-\epsilon_{mbc}h_{n}{}^{a}\nabla^{c}\Psi_{a}{}^{b}$
$\displaystyle=-2a^{a}\epsilon_{mab}\Psi_{n}{}^{b}+a_{n}Q_{m}-2\Phi_{mn}\chi^{a}{}_{a}-h_{mn}\Phi_{ab}\chi^{ab}$
(24b)
$\displaystyle+2\Phi_{ma}\chi^{a}{}_{n}+\epsilon_{mna}P^{a}\chi^{b}{}_{b}-\epsilon_{nab}P^{a}\chi^{b}{}_{m}+\epsilon_{mab}P^{a}\chi^{b}{}_{n}$
$\displaystyle+\Phi_{na}\chi_{m}{}^{a}+h_{m}{}^{b}h_{n}{}^{c}Q^{a}\nabla_{a}h_{bc}+\epsilon_{mnb}u^{a}\nabla_{a}P^{b}+\epsilon_{mnb}\nabla_{a}\Psi^{ab}$
$\displaystyle+h_{mn}\nabla_{a}Q^{a}-h_{n}{}^{a}Q_{m}\nabla_{b}h_{a}{}^{b}-h_{ma}h_{nb}\nabla^{b}Q^{a}-\tfrac{1}{2}\epsilon_{mbc}h_{n}{}^{a}\nabla^{c}S_{a}{}^{b}$

where we have defined,

$\displaystyle h_{mc}h_{na}T^{ac}$ $\displaystyle\equiv
a^{a}\epsilon_{nac}\Phi_{m}{}^{c}-\Psi_{mn}\chi^{a}{}_{a}-h_{mn}\Psi_{ac}\chi^{ac}+\Psi_{na}\chi^{a}{}_{m}$
(25a)
$\displaystyle+\Psi_{ma}\chi_{n}{}^{a}-\tfrac{1}{2}u^{a}h_{m}{}^{c}h_{n}{}^{d}\nabla_{a}S_{cd}+\tfrac{1}{2}u^{a}h_{m}{}^{c}h_{n}{}^{d}\nabla_{d}S_{ac}$
$\displaystyle A^{ac}h_{mc}h_{na}$ $\displaystyle\equiv
a^{a}\epsilon_{mac}\Psi_{n}{}^{c}+\Phi_{mn}\chi^{a}{}_{a}+h_{mn}\Phi_{ac}\chi^{ac}+\Phi_{na}\chi^{a}{}_{m}-2\Phi_{ma}\chi^{a}{}_{n}$
(25b)
$\displaystyle-\Phi_{na}\chi_{m}{}^{a}-\epsilon_{mnc}\nabla_{a}\Psi^{ac}+\tfrac{1}{2}\epsilon_{mcd}h_{n}{}^{a}\nabla^{d}S_{a}{}^{c}$

Thus the spatial components of ${\bm{T}}$ and ${\bm{A}}$ are determined by the
assumption that in the absence of electromagnetic fields, equations (23a) and
(23b) reduce to equations (19a) and (19d), respectively, under the
identifications $\Psi_{ab}=E_{ab}$ and $\Phi_{ab}=-B_{ab}$. Next we consider
mixed components. $T_{a^{\prime}0}$ and $A_{a^{\prime}0}$ are obtained by
comparing with the Bianchi constraint equations. We consider the equations

$\displaystyle h^{a}{}_{d}u^{c}\nabla^{c}Z_{abc}$
$\displaystyle=h^{a}{}_{d}u^{c}T_{ac},$ (26a) $\displaystyle
h^{a}{}_{d}u^{c}\nabla^{b}Z{}^{\ast}_{abc}$
$\displaystyle=h^{a}{}_{d}u^{c}A_{ac}.$ (26b)

Again, using the decomposition of the Z tensor and its dual, (26a) and (26b)
are equivalent to

$\displaystyle h_{n}{}^{a}\nabla_{b}\Psi_{a}{}^{b}$
$\displaystyle=a^{a}\Psi_{na}+\epsilon_{nbc}\Phi_{a}{}^{c}\chi^{ab}+\epsilon_{abc}\Phi_{n}{}^{c}\chi^{ab}-P^{a}\chi_{na}$
(27a)
$\displaystyle-\tfrac{1}{2}u^{a}u^{b}h_{n}{}^{c}\nabla_{b}S_{ac}+\epsilon_{nab}\nabla^{b}Q^{a}+\tfrac{1}{2}u^{a}u^{b}h_{n}{}^{c}\nabla_{c}S_{ab},$
$\displaystyle h_{n}{}^{a}\nabla_{b}\Phi_{a}{}^{b}$
$\displaystyle=a^{a}\Phi_{na}-2\epsilon_{nbc}\Psi_{a}{}^{c}\chi^{ab}+\epsilon_{nac}\Psi_{b}{}^{c}\chi^{ab}+Q^{a}\chi_{na}$
(27b)
$\displaystyle+\epsilon_{nab}\nabla^{b}P^{a}-\tfrac{1}{2}\epsilon_{nbc}u^{a}\nabla^{c}S_{a}{}^{b},$
(27c)

where we have defined

$\displaystyle h^{b}{}_{d}u^{a}T_{ba}$
$\displaystyle\equiv\epsilon_{dcf}\Phi_{a}{}^{f}\chi^{ac}-\tfrac{1}{2}u^{a}u^{c}\nabla_{c}S_{da}+\tfrac{1}{2}u^{a}u^{c}\nabla_{d}S_{ac},$
(28a) $\displaystyle h^{b}{}_{d}u_{a}A_{n}{}^{a}$ $\displaystyle\equiv
2\epsilon_{ncd}\Psi_{a}{}^{d}\chi^{ac}-\epsilon_{nad}\Psi_{c}{}^{d}\chi^{ac}-\epsilon_{acd}\Psi_{n}{}^{d}\chi^{ac}$
(28b) $\displaystyle+\tfrac{1}{2}\epsilon_{ncd}u^{a}\nabla^{d}S_{a}{}^{c}.$

The other mixed components $T_{0b^{\prime}}$ and $A_{0b^{\prime}}$ are
determined by comparing with the relativistic Maxwell equations in the limit
of no gravitational fields:

$\displaystyle h^{c}{}_{d}u^{a}\nabla^{b}Z_{abc}$
$\displaystyle=h^{c}{}_{d}u^{a}T_{ac},$ (29a) $\displaystyle
h^{c}{}_{d}u^{a}\nabla^{b}Z{}^{\ast}_{abc}$
$\displaystyle=h^{c}{}_{d}u^{a}A_{ac}.$ (29b)

The decomposed equations are given by

$\displaystyle u^{a}h_{mb}\nabla_{a}P^{b}-\epsilon_{mab}\nabla^{b}Q^{a}$
$\displaystyle=J^{a}h_{ma}-a^{a}\Psi_{ma}-a^{a}\epsilon_{mab}Q^{b}+P^{a}\chi_{am}$
(30a) $\displaystyle-
P_{m}\chi^{a}{}_{a}+\epsilon_{mac}\Phi_{b}{}^{c}\chi^{ab},$ $\displaystyle
u^{a}h_{mb}\nabla_{a}Q^{b}+\epsilon_{mab}\nabla^{b}P^{a}$
$\displaystyle=a^{a}\epsilon_{mab}P^{b}+a^{a}\Phi_{ma}+Q^{a}\chi_{am}-Q_{m}\chi^{a}{}_{a}$
$\displaystyle+\epsilon_{mac}\Psi_{b}{}^{c}\chi^{ab},$ (30b)

where,

$\displaystyle u^{a}h_{mb}T_{a}{}^{b}$
$\displaystyle=-J^{a}h_{ma}+a^{a}\epsilon_{mab}Q^{b}-P^{a}\chi_{am}+P_{m}\chi^{a}{}_{a},$
(31a) $\displaystyle A^{ba}u_{b}h^{d}{}_{a}$
$\displaystyle=-a^{b}\epsilon^{d}{}_{ba}P^{a}-Q^{b}\
\chi_{b}{}^{d}+Q^{d}\chi^{b}{}_{b}.$ (31b)

Finally, we find $T_{00}$ and $A_{00}$ by using the electromagnetic divergence
equations. Thus, we consider the equations,

$\displaystyle u^{c}u^{a}\nabla^{b}Z_{abc}$ $\displaystyle=u^{c}u^{a}T_{ac},$
(32a) $\displaystyle u^{c}u^{a}\nabla^{b}Z{}^{\ast}_{abc}$
$\displaystyle=u^{c}u^{a}A_{ac}.$ (32b)

Again, by the decomposition of the Z-tensor and its dual, these are equivalent
to the divergence equations

$\displaystyle\nabla_{a}P^{a}$
$\displaystyle=u^{a}J_{a}+a^{a}P_{a}-\Psi_{ab}\chi^{ab}-\epsilon_{abc}Q^{a}\chi^{bc},$
(33a) $\displaystyle\nabla_{a}Q^{a}$
$\displaystyle=a^{a}Q_{a}+\Phi_{ab}\chi^{ab}+\epsilon_{abc}P^{a}\chi^{bc}.$
(33b)

As before, we have in the above equations defined,

$\displaystyle u^{a}u^{b}T_{ab}$
$\displaystyle=-u^{a}J_{a}+\epsilon_{abc}Q^{a}\chi^{bc}$ (34a) $\displaystyle
A^{ba}u_{a}u_{b}$ $\displaystyle=-\epsilon_{bac}P^{b}\chi^{ac}.$ (34b)

We have thereby shown that if $\bm{A}$ and ${\bm{T}}$ are given by,

$\displaystyle T_{ab}$
$\displaystyle=-u_{a}u_{b}u^{m}J_{m}-u_{a}J^{m}h_{bm}+a^{m}\epsilon_{amn}\Phi_{b}{}^{n}+a^{m}\epsilon_{bmn}u_{a}Q^{n}$
$\displaystyle\quad+\Psi_{bm}\chi_{a}{}^{m}-u_{a}P^{m}\chi_{mb}+\Psi_{am}\chi^{m}{}_{b}+u_{a}P_{b}\chi^{m}{}_{m}-\Psi_{ab}\chi^{m}{}_{m}$
$\displaystyle\quad+\epsilon_{anc}u_{b}\Phi_{m}{}^{c}\chi^{mn}-h_{ab}\Psi_{mn}\chi^{mn}+\epsilon_{mnc}u_{a}u_{b}Q^{m}\chi^{nc}+\tfrac{1}{2}u_{b}u^{m}u^{n}h_{a}{}^{c}\nabla_{c}S_{mn}$
$\displaystyle\quad-\tfrac{1}{2}u^{m}h_{a}{}^{n}h_{b}{}^{c}\nabla_{m}S_{nc}-\tfrac{1}{2}u_{b}u^{m}u^{n}h_{a}{}^{c}\nabla_{n}S_{mc}+\tfrac{1}{2}u^{m}h_{a}{}^{n}h_{b}{}^{c}\nabla_{n}S_{mc},$
(35a) $\displaystyle A_{ab}$
$\displaystyle=-a^{m}\epsilon_{bmn}u_{a}P^{n}+a^{m}\epsilon_{bmn}\Psi_{a}{}^{n}-\Phi_{am}\chi_{b}{}^{m}-u_{a}Q^{m}\chi_{mb}$
$\displaystyle\quad-2\Phi_{bm}\chi^{m}{}_{a}+\Phi_{am}\chi^{m}{}_{b}+\Phi_{ab}\chi^{m}{}_{m}+u_{a}Q_{b}\chi^{m}{}_{m}$
$\displaystyle\quad+h_{ab}\Phi_{mn}\chi^{mn}-\epsilon_{mnc}u_{b}\Psi_{a}{}^{c}\chi^{mn}+2\epsilon_{anc}u_{b}\Psi_{m}{}^{c}\chi^{mn}-\epsilon_{amc}u_{b}\Psi_{n}{}^{c}\chi^{mn}$
$\displaystyle\quad-\epsilon_{mnc}u_{a}u_{b}P^{m}\chi^{nc}+\tfrac{1}{2}\epsilon_{anc}u_{b}u^{m}\nabla^{c}S_{m}{}^{n}+\tfrac{1}{2}\epsilon_{bnc}h_{a}{}^{m}\nabla^{c}S_{m}{}^{n}+\epsilon_{abn}\nabla_{m}\Psi^{mn},$
(35b)

then there exists a solution of the field equations (21a) and (21b), which are
also solutions to the Bianchi equations and the relativistic Maxwell equations
under appropriate limits. Then, they will also be a solution of the Einstein
equations if the constraint equation (20) is imposed. Observe that the
divergence of $\bm{\Psi}$ in equation (35b) will vanish if symmetry of
${\bm{A}}$ and ${\bm{T}}$ is assumed. Since the tensors ${\bm{T}}$ and
${\bm{A}}$ act as sources for the field tensor, it is worth mentioning that it
is perturbations of the four velocity and the Schouten tensor which is
responsible for a non-vanishing source. That is, a solution
$(\bm{e}_{a},\bm{\Gamma})$ to the geometric equations, determines a solution
to the field equations (21a) and (21b). Wee see in this formalism that the
Einstein equation is only a particular solution for a specific choice of
geometry — i.e. the Ricci tensor and scalar takes a specific form according to
the matter distribution. In the theory proposed here, the perturbations of the
Schouten tensor and frame components creates a matter distribution in space
time which in turn produces gravitational and electromagnetic fields.

##  5 Discussion

It has been shown that it is possible to interpret $\bm{\Psi}$, $\bm{\Phi}$,
${\bm{P}}$ and ${\bm{Q}}$ as the gravitational and electromagnetic fields,
respectively. Although there remains work to be done on the interpretations of
these equations as well as the relation to the Einstein-Maxwell equations, we
have shown that the tensor ${\bm{Z}}$ can be considered a viable candidate for
a unified field theory where the tensors $\bm{T}$ and $\bm{A}$ are the sources
— see equations (21a) and (21b) — and the field equations are first order
divergence equations, in striking similarity to the Maxwell equations. Due to
the existence of a global tetrad field it is natural to consider the spinorial
formulation of the equations. This would be of interest for a possible quantum
description as well as a more lucid interpretation of the equations. Another
interesting further study would be the existence of solutions representing a
charged point particle. The similarity of the equations with the Maxwell
equations may suggest that such a solution exists and makes sense. But observe
that although the field equations resembles the form of the Maxwell equations,
there are derivatives in the sources which may create complications.

## References

  * [1] H. Friedrich.  Evolution equations for gravitating ideal fluid bodies in general relativity.  Physical Review D, 57(4):2317–2322, Feb. 1998.
  * [2] R. Geroch.  Spinor Structure of Space‐Times in General Relativity. I.  Journal of Mathematical Physics, 9(11):1739–1744, 10 2003.
  * [3] R. Geroch and J. Traschen.  Strings and other distributional sources in general relativity.  Phys. Rev. D, 36:1017–1031, Aug 1987.
  * [4] H. Goenner.  On the history of unified field theories.  Living Reviews in Relativity, 7, 2004.
  * [5] C. Lanczos.  Lagrangian multiplier and riemannian spaces.  Rev. Mod. Phys., 21:497–502, Jul 1949.
  * [6] B. D. Normann and I. Brevik.  General Bulk-Viscous Solutions and Estimates of Bulk Viscosity in the Cosmic Fluid.  arXiv e-prints, page arXiv:1601.04519, Jan. 2016.
  * [7] M. Normann and J. Valiente Kroon.  Evolution equations for a wide range of Einstein-matter systems.  arXiv e-prints, page arXiv:2005.14678, May 2020.
  * [8] D. Pugliese and J. Valiente-Kroon.  On the evolution equations for ideal magnetohydrodynamics in curved spacetime.  General Relativity and Gravitation, 44, 2012.
  * [9] M. D. Roberts.  The physical interpretation of the lanczos tensor.  Il Nuovo Cimento B Series 11, 110(10):1165–1176, Oct 1995.
  * [10] R. Steinbauer and J. A. Vickers.  The use of generalized functions and distributions in general relativity.  Classical and Quantum Gravity, 23(10):R91, apr 2006.
  * [11] J. A. Valiente Kroon.  Conformal Methods in General Relativity.  Cambridge University Press, 2016.
  * [12] R. M. Wald.  General Relativity.  Chicago Univ. Pr., Chicago, USA, 1984.
  * [13] R. S. Ward and R. O. Wells, Jr.  Linear field theories, page 241–262.  Cambridge Monographs on Mathematical Physics. Cambridge University Press, 1990.


Janus solutions in three-dimensional ${\cal N}=8$ gauged supergravity

Kevin Chen and Michael Gutperle

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

###### Abstract

Janus solutions are constructed in $d=3$, ${\cal N}=8$ gauged supergravity. We
find explicit half-BPS solutions where two scalars in the
$\operatorname{SO}(8,1)/\operatorname{SO}(8)$ coset have a nontrivial profile.
These solutions correspond on the CFT side to an interface with a position-
dependent expectation value for a relevant operator and a source which jumps
across the interface for a marginal operator.

##  1 Introduction

Janus configurations are solutions of supergravity theories which are dual to
interface CFTs. The original solution [1] was obtained by considering a
deformation of $\operatorname{AdS}_{5}\times S^{5}$ in type IIB supergravity
where the dilaton has a nontrivial profile with respect to the slicing
coordinate of an $\operatorname{AdS}_{4}$ slicing of $\operatorname{AdS}_{5}$.
Subsequently, many more Janus solutions have been found in many different
settings. One may distinguish two kinds of solutions: First, there are top-
down constructions of Janus solutions in ten-dimensional type IIB or eleven-
dimensional M-theory which preserve half of the supersymmetry. Such solutions
are generically constructed by considering a warped product of
$\operatorname{AdS}$ and sphere factors over a two-dimensional Riemann surface
with boundary (see e.g. [2, 3, 4, 5]). Second, there are solutions of gauged
supergravities in lower dimensions with various amounts of broken and unbroken
supersymmetries (see e.g. [6, 7, 8, 9, 10, 11, 12, 13, 14]). Solutions of the
second kind are useful since holographic calculations of quantities such as
the entanglement entropy, sources and expectation values of operators, and
correlation functions in the Janus background are easier to perform in the
lower-dimensional supergravity. In many cases, such solutions can be
constructed as consistent truncations, which can be lifted to solutions of
ten- or eleven-dimensional supergravity.

In the present paper, we consider a particular example of the second approach.
We construct Janus solutions in three-dimensional $\mathcal{N}=8$ gauged
supergravity. Such theories are naturally related to
$\operatorname{AdS}_{3}\times S^{3}\times M_{4}$ compactifications of type
IIB, where $M_{4}$ is either $T_{4}$ or $K3$. We consider one of the simplest
nontrivial settings where we find solutions which preserve eight of the
sixteen supersymmetries of the $\operatorname{AdS}_{3}$ vacuum, where only two
scalars in the coset have a nontrivial profile. One interesting feature of
these solutions is that one scalar is dual to a marginal operator with
dimension $\Delta=2$ where the source terms have different values on the two
sides of the interface. This behavior is the main feature of the original
Janus solution [1, 15]. On the other hand, the second scalar is dual to a
relevant operator with dimension $\Delta=1$ with a vanishing source term and a
position-dependent expectation value. This behavior is a feature of the Janus
solution in M-theory [5].

The structure of the paper is as follows: in section 2 we review
$\mathcal{N}=8$ gauged supergravity in three dimensions, and in section 3 we
construct the half-BPS Janus solutions and investigate some of their
properties using the AdS/CFT dictionary, including the calculation of the
holographic entanglement entropy. We discuss some generalizations and
directions for future research in section 4. Some technical details are
relegated to appendix A.

##  2 $d=3$, $\mathcal{N}=8$ gauged supergravity

In the following, we will use the notation and conventions of [16]. The scalar
fields of $d=3$, $\mathcal{N}=8$ gauged supergravity are parameterized by a
$G/H=\operatorname{SO}(8,n)/\quantity\big(\operatorname{SO}(8)\times\operatorname{SO}(n))$
coset, which has $8n$ independent scalar degrees of freedom. This theory can
be obtained by a truncation of six-dimensional $\mathcal{N}=(2,0)$
supergravity on $\operatorname{AdS}_{3}\times S^{3}$ coupled to $n_{T}\geq 1$
tensor multiplets, where $n_{T}=n-3$. The cases $n_{T}=5$ and $21$ correspond
to compactifications of ten-dimensional type IIB on $T^{3}$ and $K3$,
respectively. See [17] for a discussion of consistent truncations of six-
dimensional $\mathcal{N}=(1,1)$ and $\mathcal{N}=(2,0)$ using exceptional
field theory.

For future reference, 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 non-compact generators. Explicitly, the generators of
the vector representation are given by

$\displaystyle\tensor{(t^{\bar{I}\bar{J}})}{{}^{\bar{K}}_{\bar{L}}}=\eta^{\bar{I}\bar{K}}\delta^{\bar{J}}_{\bar{L}}-\eta^{\bar{J}\bar{K}}\delta^{\bar{I}}_{\bar{L}}$
(2.1)

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

$\displaystyle[t^{\bar{I}\bar{J}},t^{\bar{K}\bar{L}}]=2\quantity(\eta^{\bar{I}[\bar{K}}t^{\bar{L}]\bar{J}}-\eta^{\bar{J}[\bar{K}}t^{\bar{L}]\bar{I}})$
(2.2)

The scalars fields can be 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

$\displaystyle L(x)\longrightarrow g_{0}(x)L(x)h^{-1}(x)$ (2.3)

for $g_{0}\in G_{0}$ and $h\in H$. The Lagrangian is invariant under such
transformations. We can pick a
$\operatorname{SO}(8)\times\operatorname{SO}(n)$ gauge to put the coset
representative into symmetric gauge,

$\displaystyle L=\exp(\phi_{Ir}Y^{Ir})$ (2.4)

for scalar fields $\phi_{Ir}$. The
$\tensor{\mathcal{V}}{{}^{\mathcal{M}}_{\mathcal{A}}}$ tensors are defined by

$\displaystyle
L^{-1}t^{\mathcal{M}}L=\tensor{\mathcal{V}}{{}^{\mathcal{M}}_{\mathcal{A}}}t^{\mathcal{A}}=\frac{1}{2}\tensor{\mathcal{V}}{{}^{\mathcal{M}}_{IJ}}X^{IJ}+\frac{1}{2}\tensor{\mathcal{V}}{{}^{\mathcal{M}}_{rs}}X^{rs}+\tensor{\mathcal{V}}{{}^{\mathcal{M}}_{Ir}}Y^{Ir}$
(2.5)

The gauging of the supergravity is accomplished by introducing Chern-Simons
gauge fields $B^{\mathcal{M}}_{\mu}$ and choosing an embedding tensor
$\Theta_{\mathcal{M}\mathcal{N}}$ (which has to satisfy various identities
[18]) 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 couplings. In the following, we will make one of
the simplest choices and gauge a $G_{0}=\operatorname{SO}(4)$ subset of
$\operatorname{SO}(8)$. Explicitly, we further divide the $I,J$ indices into

  * –

$i,j,\dotsc=1,2,3,4$ for $G_{0}=\operatorname{SO}(4)$.

  * –

$\bar{\imath},\bar{\jmath},\dotsc=5,6,7,8$ for the remaining ungauged
$\operatorname{SO}(4)\subset\operatorname{SO}(8)$.

The embedding tensor we will employ in the following has the non-zero entries

$\displaystyle\Theta_{IJ,KL}=\varepsilon_{ijk\ell}$ (2.6)

As this is totally antisymmetric, the trace is $\theta=0$. As discussed in
[16], this choice of embedding tensor produces a supersymmetric
$\operatorname{AdS}_{3}$ ground state with

$\displaystyle\operatorname{SU}(2|1,1)_{L}\times\operatorname{SU}(2|1,1)_{R}$
(2.7)

super-algebra of isometries. From the embedding tensor, the $G_{0}$-covariant
currents can be obtained,

$\displaystyle
L^{-1}(\partial_{\mu}+g\Theta_{\mathcal{M}\mathcal{N}}B_{\mu}^{\mathcal{M}}t^{\mathcal{N}})L=\frac{1}{2}\mathcal{Q}^{IJ}_{\mu}X^{IJ}+\frac{1}{2}\mathcal{Q}^{rs}_{\mu}X^{rs}+\mathcal{P}^{Ir}_{\mu}Y^{Ir}$
(2.8)

It is convenient to define the $T$-tensor,

$\displaystyle
T_{\mathcal{A}|\mathcal{B}}=\Theta_{\mathcal{M}\mathcal{N}}\tensor{\mathcal{V}}{{}^{\mathcal{M}}_{\mathcal{A}}}\tensor{\mathcal{V}}{{}^{\mathcal{N}}_{\mathcal{B}}}$
(2.9)

as well as the tensors $A_{1,2,3}$ which will appear in the potential and the
supersymmetry transformations.

$\displaystyle A_{1}^{AB}$
$\displaystyle=-\frac{1}{48}\Gamma^{IJKL}_{AB}T_{IJ|KL}$ $\displaystyle
A_{2}^{A\dot{A}r}$
$\displaystyle=-\frac{1}{12}\Gamma^{IJK}_{A\dot{A}}T_{IJ|Kr}$ $\displaystyle
A_{3}^{\dot{A}r\dot{B}s}$
$\displaystyle=\frac{1}{48}\delta^{rs}\Gamma^{IJKL}_{\dot{A}\dot{B}}T_{IJ|KL}+\frac{1}{2}\Gamma^{IJ}_{\dot{A}\dot{B}}T_{IJ|rs}$
(2.10)

$A,B$ and $\dot{A},\dot{B}$ are $\operatorname{SO}(8)$-spinor indices and our
conventions for the $\operatorname{SO}(8)$ Gamma matrices are presented in
appendix A.1.

We take the spacetime signature $\eta^{ab}=\operatorname{diag}(+--)$ to be
mostly negative. The bosonic Lagrangian is

$\displaystyle e^{-1}\mathcal{L}$
$\displaystyle=-\frac{1}{4}R+\frac{1}{4}\mathcal{P}_{\mu}^{Ir}\mathcal{P}^{\mu\,Ir}+W-\frac{1}{4}e^{-1}\varepsilon^{\mu\nu\rho}g\Theta_{\mathcal{M}\mathcal{N}}B_{\mu}^{\mathcal{M}}\quantity(\partial_{\nu}B_{\rho}^{\mathcal{N}}+\frac{1}{3}g\Theta_{\mathcal{K}\mathcal{L}}\tensor{f}{{}^{\mathcal{N}\mathcal{K}}_{\mathcal{P}}}B_{\nu}^{\mathcal{L}}B_{\rho}^{\mathcal{P}})$
$\displaystyle W$
$\displaystyle=\frac{1}{4}g^{2}\quantity(A^{AB}_{1}A^{AB}_{1}-\frac{1}{2}A^{A\dot{A}r}_{2}A^{A\dot{A}r}_{2})$
(2.11)

The SUSY transformations are

$\displaystyle\delta\chi^{\dot{A}r}$
$\displaystyle=\frac{1}{2}i\Gamma^{I}_{A\dot{A}}\gamma^{\mu}\varepsilon^{A}\mathcal{P}^{Ir}_{\mu}+gA^{A\dot{A}r}_{2}\varepsilon^{A}$
$\displaystyle\delta\psi^{A}_{\mu}$
$\displaystyle=\quantity(\partial_{\mu}\varepsilon^{A}+\frac{1}{4}\omega_{\mu}^{ab}\gamma_{ab}\varepsilon^{A}+\frac{1}{4}\mathcal{Q}^{IJ}_{\mu}\Gamma^{IJ}_{AB}\varepsilon^{B})+igA^{AB}_{1}\gamma_{\mu}\varepsilon^{B}$
(2.12)

###  2.1 The $n=1$ case

In this section we will consider the $n=1$ theory, i.e. the scalar fields lie
in a $\operatorname{SO}(8,1)/\operatorname{SO}(8)$ coset. The reason for this
is that the resulting expressions for the supersymmetry transformations and
BPS conditions are compact and everything can be worked out in detail.
Furthermore, we believe that this case illustrates the important features of
more general solutions.

As the index $r=9$ takes only one value in this case, the scalar fields in the
coset representative (2.4) are denoted by $\phi_{i}\equiv\phi_{i9}$ for
$i=1,2,\dotsc,8$. We define the following quantities for notational
convenience,

$\displaystyle\Phi^{2}$
$\displaystyle\equiv\phi_{I}\phi_{I}=\phi_{1}^{2}+\phi_{2}^{2}+\phi_{3}^{2}+\phi_{4}^{2}+\phi_{5}^{2}+\phi_{6}^{2}+\phi_{7}^{2}+\phi_{8}^{2}$
$\displaystyle\phi^{2}$
$\displaystyle\equiv\phi_{i}\phi_{i}=\phi_{1}^{2}+\phi_{2}^{2}+\phi_{3}^{2}+\phi_{4}^{2}$
$\displaystyle\bar{\phi}^{2}$
$\displaystyle\equiv\phi_{\bar{\imath}}\phi_{\bar{\imath}}=\phi_{5}^{2}+\phi_{6}^{2}+\phi_{7}^{2}+\phi_{8}^{2}$
(2.13)

The components of the $\tensor{\mathcal{V}}{{}^{\mathcal{M}}_{\mathcal{A}}}$
tensor are, with no summation over repeated indices and $I,J,K,L$ being unique
indices,

$\displaystyle\tensor{\mathcal{V}}{{}^{IJ}_{IJ}}$
$\displaystyle=1+(\phi_{I}^{2}+\phi_{J}^{2})\frac{\cosh\Phi-1}{\Phi^{2}}$
$\displaystyle\tensor{\mathcal{V}}{{}^{IJ}_{IK}}$
$\displaystyle=\phi_{J}\phi_{K}\frac{\cosh\Phi-1}{\Phi^{2}}$
$\displaystyle\tensor{\mathcal{V}}{{}^{IJ}_{KL}}$ $\displaystyle=0$
$\displaystyle\tensor{\mathcal{V}}{{}^{I9}_{I9}}$
$\displaystyle=\cosh\Phi-\phi_{I}^{2}\frac{\cosh\Phi-1}{\Phi^{2}}$
$\displaystyle\tensor{\mathcal{V}}{{}^{I9}_{J9}}$
$\displaystyle=-\phi_{I}\phi_{J}\frac{\cosh\Phi-1}{\Phi^{2}}$
$\displaystyle\tensor{\mathcal{V}}{{}^{IJ}_{I9}}$
$\displaystyle=\tensor{\mathcal{V}}{{}^{I9}_{IJ}}=\phi_{J}\frac{\sinh\Phi}{\Phi}$
$\displaystyle\tensor{\mathcal{V}}{{}^{IJ}_{K9}}$
$\displaystyle=\tensor{\mathcal{V}}{{}^{K9}_{IJ}}=0$ (2.14)

The $u$-components of the $\mathcal{Q}^{IJ}_{\mu}$ and $\mathcal{P}^{I}_{\mu}$
tensors are

$\displaystyle\mathcal{Q}_{u}^{IJ}$
$\displaystyle=(\phi_{I}^{\prime}\phi_{J}-\phi_{I}\phi_{J}^{\prime})\frac{\cosh\Phi-1}{\Phi^{2}}+g\Theta_{\mathcal{M}\mathcal{N}}B^{\mathcal{M}}_{u}\mathcal{V}^{\mathcal{N}}_{IJ}$
$\displaystyle\mathcal{P}_{u}^{I}$
$\displaystyle=\phi_{I}^{\prime}\frac{\sinh\Phi}{\Phi}-\phi_{I}\Phi^{\prime}\frac{\sinh\Phi-\Phi}{\Phi^{2}}+g\Theta_{\mathcal{M}\mathcal{N}}B^{\mathcal{M}}_{u}\mathcal{V}^{\mathcal{N}}_{I9}$
(2.15)

where the prime ${}^{\prime}\equiv\partialderivative*{u}$ denotes the
derivative with respect to $u$. The terms involving the gauge field have
different forms depending on whether $I,J$ are in $i$ or $\bar{\imath}$.

$\displaystyle\Theta_{\mathcal{M}\mathcal{N}}B^{\mathcal{M}}_{u}\mathcal{V}^{\mathcal{N}}_{ij}$
$\displaystyle=\varepsilon_{ijk\ell}\quantity[\frac{1}{2}B^{k\ell}_{u}\quantity(1+(\phi_{i}^{2}+\phi_{j}^{2})\frac{\cosh\Phi-1}{\Phi^{2}})+\quantity(\phi_{i}B^{ik}_{u}\phi_{\ell}+\phi_{j}B^{jk}_{u}\phi_{\ell})\frac{\cosh\Phi-1}{\Phi^{2}}]$
$\displaystyle\Theta_{\mathcal{M}\mathcal{N}}B^{\mathcal{M}}_{u}\mathcal{V}^{\mathcal{N}}_{i\bar{\imath}}$
$\displaystyle=\frac{1}{2}\varepsilon_{ijk\ell}\phi_{\bar{\imath}}\phi_{j}B^{k\ell}_{u}\frac{\cosh\Phi-1}{\Phi^{2}}$
$\displaystyle\Theta_{\mathcal{M}\mathcal{N}}B^{\mathcal{M}}_{u}\mathcal{V}^{\mathcal{N}}_{\bar{\imath}\bar{\jmath}}$
$\displaystyle=0$
$\displaystyle\Theta_{\mathcal{M}\mathcal{N}}B^{\mathcal{M}}_{u}\mathcal{V}^{\mathcal{N}}_{i9}$
$\displaystyle=\frac{1}{2}\varepsilon_{ijk\ell}\phi_{j}B^{k\ell}_{u}\frac{\sinh\Phi}{\Phi}$
$\displaystyle\Theta_{\mathcal{M}\mathcal{N}}B^{\mathcal{M}}_{u}\mathcal{V}^{\mathcal{N}}_{\bar{\imath}9}$
$\displaystyle=0$ (2.16)

The $T$-tensor has non-zero components

$\displaystyle T_{ij|k\ell}$
$\displaystyle=\varepsilon_{ijk\ell}\quantity(\phi^{2}\frac{\cosh\Phi-1}{\Phi^{2}}+1)$
$\displaystyle T_{ij|k\bar{\imath}}$
$\displaystyle=\varepsilon_{ijk\ell}\phi_{\ell}\phi_{\bar{\imath}}\frac{\cosh\Phi-1}{\Phi^{2}}$
$\displaystyle T_{ij|k9}$
$\displaystyle=\varepsilon_{ijk\ell}\phi_{\ell}\frac{\sinh\Phi}{\Phi}$ (2.17)

Taking $\varepsilon_{1234}=1$, we can use the $T$-tensor to compute

$\displaystyle A_{1}^{AB}$
$\displaystyle=-\frac{1}{2}\Gamma^{1234}_{AC}\Bigg{[}\quantity(\phi^{2}\frac{\cosh\Phi-1}{\Phi^{2}}+1)\delta_{CB}+(\Gamma^{i}_{C\dot{A}}\phi_{i})(\Gamma^{\bar{\imath}}_{\dot{A}B}\phi_{\bar{\imath}})\frac{\cosh\Phi-1}{\Phi^{2}}\Bigg{]}$
$\displaystyle A_{2}^{A\dot{A}}$
$\displaystyle=-\frac{1}{2}\Gamma^{1234}_{AB}(\Gamma^{i}_{B\dot{A}}\phi_{i})\frac{\sinh\Phi}{\Phi}$
$\displaystyle A_{3}^{\dot{A}\dot{B}}$
$\displaystyle=-A_{1}^{AB}\delta_{A\dot{A}}\delta_{B\dot{B}}$ (2.18)

Note that $A_{1}^{AB}=A_{1}^{BA}$ and

$\displaystyle A_{1}^{AC}A_{1}^{BC}$
$\displaystyle=\frac{1}{4}\delta_{AB}\quantity(\frac{\phi^{2}\sinh^{2}\Phi}{\Phi^{2}}+1)$
$\displaystyle A_{2}^{A\dot{A}}A_{2}^{B\dot{A}}$
$\displaystyle=\frac{1}{4}\delta_{AB}\frac{\phi^{2}\sinh^{2}\Phi}{\Phi^{2}}$
(2.19)

so the scalar potential (2.11) becomes

$\displaystyle
W=\frac{g^{2}}{4}\quantity(\frac{\phi^{2}\sinh^{2}\Phi}{\Phi^{2}}+2)$ (2.20)

##  3 Half-BPS Janus solutions

In this section, we construct Janus solutions which preserve eight of the
sixteen supersymmetries of the $\operatorname{AdS}_{3}$ vacuum. Our strategy
is to use an $\operatorname{AdS}_{2}$ slicing of $\operatorname{AdS}_{3}$ and
make the scalar fields as well as the metric functions only dependent on the
slicing coordinate. One complication is given by the presence of the gauge
fields; due to the Chern-Simons action, the only consistent Janus solution
will have vanishing field strength. We show that the gauge fields can be
consistently set to zero for our solutions.

###  3.1 Janus ansatz

We take the Janus ansatz for the metric, scalar fields and Chern-Simons gauge
fields,

$\displaystyle\differential{s^{2}}$
$\displaystyle=e^{2B(u)}\quantity(\frac{\differential{t^{2}}-\differential{z^{2}}}{z^{2}})-e^{2D(u)}\differential{u}^{2}$
$\displaystyle\phi_{I}$ $\displaystyle=\phi_{I}(u)$ $\displaystyle
B^{\mathcal{M}}$ $\displaystyle=B^{\mathcal{M}}(u)\differential{u}$ (3.1)

The $\operatorname{AdS}_{3}$ vacuum solution given by $\phi_{I}\equiv 0$ and
$e^{B}=e^{D}=L\sec u$ has a curvature radius related to the coupling constant
by $L^{-1}=g$. The spin connection 1-forms are

$\displaystyle\omega^{01}$ $\displaystyle=\frac{\differential{t}}{z}$
$\displaystyle\omega^{02}$
$\displaystyle=-\frac{B^{\prime}e^{B-D}}{z}\differential{t}$
$\displaystyle\omega^{12}$
$\displaystyle=-\frac{B^{\prime}e^{B-D}}{z}\differential{z}$ (3.2)

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

$\displaystyle 0$
$\displaystyle=\partial_{t}\varepsilon+\frac{1}{2z}\gamma_{0}\quantity(\gamma_{1}-B^{\prime}e^{B-D}\gamma_{2}+2ige^{B}A_{1})\varepsilon$
$\displaystyle 0$
$\displaystyle=\partial_{z}\varepsilon+\frac{1}{2z}\gamma_{1}\quantity(-B^{\prime}e^{B-D}\gamma_{2}+2ige^{B}A_{1})\varepsilon$
$\displaystyle 0$
$\displaystyle=\partial_{u}\varepsilon+\frac{1}{4}\mathcal{Q}_{u}^{IJ}\Gamma^{IJ}\varepsilon+ige^{D}\gamma_{2}A_{1}\varepsilon$
(3.3)

where we have suppressed the $\operatorname{SO}(8)$-spinor indices. As shown
in appendix A.2, the integrability conditions are

$\displaystyle 0$
$\displaystyle=\quantity(1-(2ge^{B}A_{1})^{2}+(B^{\prime}e^{B-D})^{2})\varepsilon$
$\displaystyle 0$
$\displaystyle=2ige^{B}\quantity(A_{1}^{\prime}-\frac{1}{4}[A_{1},\mathcal{Q}_{u}^{IJ}\Gamma^{IJ}])\varepsilon+\quantity(-\derivative{u}(B^{\prime}e^{B-D})+(2ge^{B}A_{1})^{2}e^{D-B})\gamma_{2}\varepsilon$
(3.4)

The first integrability condition gives a first-order equation which must be
true for all $\varepsilon$, using the replacement for $A_{1}^{2}$ in (2.19),

$\displaystyle
0=1-g^{2}e^{2B}\quantity(\frac{\phi^{2}\sinh^{2}\Phi}{\Phi^{2}}+1)+(B^{\prime}e^{B-D})^{2}$
(3.5)

The derivative of this simplifies the second integrability condition to

$\displaystyle
0=\quantity(A_{1}^{\prime}-\frac{1}{4}[A_{1},\mathcal{Q}_{u}^{IJ}\Gamma^{IJ}])\varepsilon+\frac{ige^{D}}{4B^{\prime}}\derivative{u}\quantity(\frac{\phi^{2}\sinh^{2}\Phi}{\Phi^{2}})\gamma_{2}\varepsilon$
(3.6)

The BPS equation $\smash{\delta\chi^{\dot{A}}=0}$ is

$\displaystyle\quantity(-\frac{i}{2}e^{-D}\Gamma^{I}\mathcal{P}_{u}^{I}\gamma_{2}+gA_{2})_{A\dot{A}}\varepsilon^{A}=0$
(3.7)

When $gA_{2}^{2}\neq 0$, this equation can be rearranged into the form of a
projector

$\displaystyle 0$
$\displaystyle=\quantity(iM_{AB}\gamma_{2}+\delta_{AB})\varepsilon^{A}$ (3.8)

where $M_{AB}$ is given by

$\displaystyle M_{AB}$
$\displaystyle=\frac{e^{-D}}{g}\frac{\Phi}{\phi^{2}\sinh\Phi}(\Gamma^{I}_{A\dot{A}}\mathcal{P}_{u}^{I})(\Gamma^{i}_{\dot{A}C}\phi_{i})\Gamma^{1234}_{CB}$
(3.9)

For consistency of the projector, we must have

$\displaystyle M_{AB}M_{BC}=\delta_{AC}$ (3.10)

As $M^{2}=1$, every generalized eigenvector of rank $\geq 2$ is automatically
an eigenvector, so $M$ is diagonalizable and has eight eigenvectors with
eigenvalues $\pm 1$. $M$ is traceless as it is a sum of products of 2 or 4
Gamma matrices, so it has an equal number of $+1$ and $-1$ eigenvectors. The
operator $iM_{AB}\gamma_{2}$ in the projector (3.8) squares to one and is
traceless, and projects onto an eight-dimensional space of unbroken
supersymmetry generators. If this is the only projection imposed on the
solution, it will be half-BPS and hence preserve eight of the sixteen
supersymmetries of the vacuum.

The condition $M^{2}=1$ gives an equation first-order in derivatives of
scalars.

$\displaystyle
M^{2}=\quantity(\frac{e^{-D}\Phi}{g\phi^{2}\sinh\Phi})^{2}\Big{(}$
$\displaystyle\phi^{2}(-\mathcal{P}_{u}^{i}\mathcal{P}_{u}^{i}+\mathcal{P}_{u}^{\bar{\imath}}\mathcal{P}_{u}^{\bar{\imath}})-2\phi^{2}(\Gamma^{\bar{\imath}}\mathcal{P}_{u}^{\bar{\imath}})(\Gamma^{i}\mathcal{P}_{u}^{i})$
$\displaystyle+2(\mathcal{P}_{u}^{j}\phi_{j})(\Gamma^{\bar{\imath}}\mathcal{P}_{u}^{\bar{\imath}}+\Gamma^{i}\mathcal{P}_{u}^{i})(\Gamma^{k}\phi_{k})\big{)}$
(3.11)

For this to be proportional to the identity, we need all
$\Gamma^{\bar{\imath}}\Gamma^{i}$ and $\Gamma^{i}\Gamma^{j}$ terms to vanish.
Vanishing of the latter requires us to impose the condition

$\displaystyle\mathcal{P}^{i}_{u}\phi_{j}=\mathcal{P}^{j}_{u}\phi_{i}$ (3.12)

As the ratio $\mathcal{P}_{u}^{i}/\phi_{i}$ is the same for all $i$, this
implies

$\displaystyle\sum_{i}\mathcal{P}_{u}^{i}\phi_{i}=\sum_{i}\frac{\mathcal{P}_{u}^{i}}{\phi_{i}}\phi_{i}^{2}=\frac{\mathcal{P}_{u}^{1}}{\phi_{1}}\phi^{2}\qquad\implies\qquad-\phi^{2}\mathcal{P}_{u}^{i}+\phi_{i}\sum_{j}\mathcal{P}_{u}^{j}\phi_{j}=0$
(3.13)

This means that imposing Eq. (3.12) also ensures that the
$\Gamma^{\bar{\imath}}\Gamma^{i}$ terms vanish. Note that

$\displaystyle\sum_{i}\mathcal{P}_{u}^{i}\mathcal{P}_{u}^{i}=\sum_{i}\frac{\mathcal{P}_{u}^{i}}{\phi_{i}}\frac{\mathcal{P}_{u}^{i}}{\phi_{i}}\phi_{i}^{2}=\quantity(\frac{\mathcal{P}_{u}^{1}}{\phi_{1}})^{2}\phi^{2}$
(3.14)

so the $M^{2}=1$ condition becomes

$\displaystyle
M^{2}=\quantity(\frac{e^{-D}\Phi}{g\phi^{2}\sinh\Phi})^{2}\phi^{2}(\mathcal{P}_{u}^{i}\mathcal{P}_{u}^{i}+\mathcal{P}_{u}^{\bar{\imath}}\mathcal{P}_{u}^{\bar{\imath}})=1$
(3.15)

We now give the argument why the Chern-Simons gauge fields can be set to zero.
Since we demand that the $B^{\mathcal{M}}_{\mu}$ only has a component along
the $u$ direction and only depends on $u$, the field strength vanishes,
consistent with the equation of motion coming from the variation of the Chern-
Simons term in the action (2.11) with respect to the gauge field. However,
there is another term which contains the gauge field, namely the kinetic term
of the scalars via (2.15). For the gauge field to be consistently set to zero,
we have to impose

$\displaystyle\left.{\delta\mathcal{L}\over\delta
B^{k\ell}_{u}}\right|_{B^{\mathcal{M}}_{u}=0}=0$ (3.16)

For the Janus ansatz, we find

$\displaystyle\left.{\delta\mathcal{L}\over\delta
B^{k\ell}_{u}}\right|_{B^{\mathcal{M}}_{u}=0}=eg\varepsilon_{ijk\ell}\mathcal{P}^{i\,u}\phi_{j}{\sinh\Phi\over\Phi}$
(3.17)

which indeed vanishes due to Eq. (3.12) imposed by the half-BPS condition.

For a half-BPS solution, the second integrability condition (3.6) should be
identical to the projector (3.8). Indeed, we have the simplification

$\displaystyle A_{1}^{\prime}-\frac{1}{4}$
$\displaystyle[A_{1},\mathcal{Q}_{u}^{IJ}\Gamma^{IJ}]=-\frac{1}{2}\frac{\phi^{2}\sinh^{2}\Phi}{\Phi^{2}}M^{\top}$
(3.18)

so the Gamma matrix structures of the two equations match. Equating the
remaining scalar magnitude gives us an equation for the metric factor $e^{B}$,

$\displaystyle-B^{\prime}=\derivative{u}\ln\frac{\phi\sinh\Phi}{\Phi}$ (3.19)

We can now solve for the metric. Let us define

$\displaystyle\alpha(u)\equiv\frac{\phi\sinh\Phi}{\Phi}$ (3.20)

and set the integration constant for $B$ to be

$\displaystyle e^{B}=\frac{|C|}{g\alpha}$ (3.21)

Plugging this into the first integrability condition (3.5) and picking the
gauge $e^{-D}\equiv g$, we have a first-order equation for $\alpha$,

$\displaystyle 0=\alpha^{2}-C^{2}(\alpha^{2}+1-\alpha^{\prime 2}/\alpha^{2})$
(3.22)

The solution depends on the value of $C\in[0,1]$ and up to translations in $u$
is

$\displaystyle\alpha$ $\displaystyle=e^{\pm u}$ $\displaystyle\text{if }C=1$
$\displaystyle\alpha$ $\displaystyle=\frac{|C|}{\sqrt{1-C^{2}}}\sech u$
$\displaystyle\text{if }0\leq C<1$ (3.23)

We will take the case $0\leq C<1$. This implies that the metric is

$\displaystyle\differential{s^{2}}=g^{-2}\quantity[(1-C^{2})\cosh^{2}u\quantity(\frac{\differential{t^{2}}-\differential{z^{2}}}{z^{2}})-\differential{u^{2}}]$
(3.24)

The choice $C=0$ corresponds to the $\operatorname{AdS}_{3}$ vacuum.

###  3.2 $\phi_{4},\phi_{5}$ truncation

We have yet to fully solve the half-BPS conditions (3.12) and (3.15). For
simplicity, let us consider the case where only $\phi_{4},\phi_{5}$ are non-
zero and the other scalars are identically zero, which trivially satisfies Eq.
(3.12). It turns out that the important features of the Janus solution are
captured by this truncation.

We introduce the following abbreviations

$\displaystyle\Phi^{2}$ $\displaystyle=\phi_{4}^{2}+\phi_{5}^{2}$
$\displaystyle\phi$ $\displaystyle=|\phi_{4}|$ $\displaystyle\bar{\phi}$
$\displaystyle=|\phi_{5}|$ (3.25)

Let us define

$\displaystyle\beta(u)$ $\displaystyle\equiv\frac{\phi_{5}\sinh\Phi}{\Phi}$
(3.26)

so that

$\displaystyle\alpha^{2}+\beta^{2}$ $\displaystyle=\sinh^{2}\Phi$
$\displaystyle\mathcal{P}^{4}_{u}$
$\displaystyle=\alpha^{\prime}+\alpha\Phi^{\prime}\frac{1-\cosh\Phi}{\sinh\Phi}$
$\displaystyle\mathcal{P}^{5}_{u}$
$\displaystyle=\beta^{\prime}+\beta\Phi^{\prime}\frac{1-\cosh\Phi}{\sinh\Phi}$
(3.27)

Plugging these into Eq. (3.15) simplifies to

$\displaystyle\alpha^{\prime 2}+\beta^{\prime
2}-\frac{(\alpha^{\prime}\alpha+\beta^{\prime}\beta)^{2}}{1+\alpha^{2}+\beta^{2}}$
$\displaystyle=\alpha^{2}$ (3.28)

This can be rearranged into a first-order equation in
$f\equiv\beta/\sqrt{1+\alpha^{2}}$,

$\displaystyle f^{\prime}=\frac{\alpha^{2}/C}{1+\alpha^{2}}\sqrt{1+f^{2}}$
(3.29)

where a sign ambiguity from taking a square-root has been absorbed into $C$,
which is now extended to $C\in(-1,1)$. Using the explicit solution (3.23) for
$\alpha$, by noting that

$\displaystyle\derivative{u}\tanh^{-1}(C\tanh
u)=\frac{C\sech^{2}u}{1-C^{2}\tanh^{2}u}=\frac{\alpha^{2}/C}{1+\alpha^{2}}$
(3.30)

the general solution is

$\displaystyle f(u)$ $\displaystyle=\frac{\sinh p+C\cosh p\tanh
u}{\sqrt{1-C^{2}\tanh^{2}u}}$ $\displaystyle\beta(u)$
$\displaystyle=\frac{1}{\sqrt{1-C^{2}}}(\sinh p+C\cosh p\tanh u)$ (3.31)

for some constant $p\in\mathbb{R}$. For later convenience, we also redefine
$C=\tanh q$ for $q\in\mathbb{R}$.

In summary, we have solved for the scalars $\phi_{4},\phi_{5}$ implicitly
through the functions $\alpha,\beta$,

$\displaystyle\frac{|\phi_{4}|\sinh\Phi}{\Phi}$ $\displaystyle=|\sinh q|\sech
u$ $\displaystyle\frac{\phi_{5}\sinh\Phi}{\Phi}$ $\displaystyle=\sinh p\cosh
q+\cosh p\sinh q\tanh u$ (3.32)

for real constants $p,q$. Note that the reflection $\phi_{4}\to-\phi_{4}$ also
gives a valid solution. We have explicitly checked that the Einstein equation
and scalar equations of motion are satisfied.

The $\phi_{4}$ scalar goes to zero at $u=\pm\infty$ as it is a massive scalar
degree of freedom, and has a sech-like profile near the defect. The $\phi_{5}$
scalar interpolates between two boundary values at $u=\pm\infty$, and has a
tanh-like profile. The constant $p$ is related to the boundary values of the
$\phi_{5}$ scalar, as we can note that

$\displaystyle\phi_{5}(\pm\infty)=p\pm q$ (3.33)

The constant $q$ is then related to the jump value of the $\phi_{5}$ scalar.
The defect location $u=0$ can also be freely translated to any point along the
axis. Below is a plot of the solution for the choice $(p,q)=(0,1)$.

Figure 1: Plot of $\phi_{4}$ and $\phi_{5}$ for $(p,q)=(0,1)$

###  3.3 Holography

In our AdS-sliced coordinates, the boundary is given by the two
$\operatorname{AdS}_{2}$ components at $u=\pm\infty$, which are joined
together at the $z=0$ interface. Using $C=\tanh q$, the metric (3.24) becomes

$\displaystyle\differential{s^{2}}=g^{-2}\quantity[\sech^{2}q\cosh^{2}u\quantity(\frac{\differential{t^{2}}-\differential{z^{2}}}{z^{2}})-\differential{u^{2}}]$
(3.34)

Note that this is not $\operatorname{AdS}_{3}$ unless $q=0$, which corresponds
to the vacuum solution with all scalars vanishing. The spacetime is, however,
asymptotically $\operatorname{AdS}_{3}$. In the limit of $u\to\pm\infty$, the
$\sech^{2}q$ can be eliminated from the leading $e^{\pm 2u}$ term in the
metric (3.34) by a coordinate shift. We will present the asymptotic mapping to
a Fefferman-Graham (FG) coordinate system below. In the following, we will set
the $\operatorname{AdS}$ length scale to unity for notational simplicity, i.e.
$g\equiv 1$.

According to the AdS/CFT correspondence, the mass $m^{2}$ of a supergravity
scalar field in $d=3$ is related to the scaling dimension $\Delta$ of the dual
CFT operator by

$\displaystyle m^{2}=\Delta(\Delta-2)$ (3.35)

This relation comes from the linearized equations of motion for the scalar
field near the asymptotic $\operatorname{AdS}_{3}$ boundary. Expanding the
supergravity action (2.11) to quadratic order around the
$\operatorname{AdS}_{3}$ vacuum shows that the $\phi_{4}$ field has mass
$m^{2}=-1$, so the dual operator is relevant with $\Delta=1$ and saturates the
Breitenlohner-Freedman (BF) bound [19]. Note that we choose the standard
quantization [20], which is the correct one for a supersymmetric solution. The
$\phi_{5}$ field is massless, so the dual CFT operator is marginal with
scaling dimension $\Delta=2$.

In FG coordinates,111The $\operatorname{AdS}_{3}$ metric in Poincaré
coordinates is
$\differential{s^{2}}=\frac{-\differential{\rho^{2}}+\differential{t^{2}}-\differential{x^{2}}}{\rho^{2}}$
and is related to the AdS-sliced metric by the coordinate change
$\displaystyle z$ $\displaystyle=\sqrt{x^{2}+\rho^{2}}$ $\displaystyle\sinh u$
$\displaystyle=x/\rho$ the general expansion for a scalar field near the
asymptotic $\operatorname{AdS}_{3}$ boundary at $\rho=0$ is

$\displaystyle\phi_{\Delta=1}$
$\displaystyle\sim\psi_{0}\,\rho\ln\rho+\phi_{0}\,\rho+\cdots$
$\displaystyle\phi_{\Delta\neq 1}$
$\displaystyle\sim\tilde{\phi}_{0}\,\rho^{2-\Delta}+\tilde{\phi}_{2}\,\rho^{\Delta}+\cdots$
(3.36)

Since $\phi_{\Delta=1}$ saturates the BF bound, holographic renormalization
and the holographic dictionary are subtle due to the presence of the logarithm
[21]. As we show below for the solution (3.32), there is no logarithmic term
present and $\phi_{0}$ can be identified with the expectation value of the
dual operator [21, 22]. For the massless field $\phi_{\Delta=2}$, we can
identify $\tilde{\phi}_{0}$ with the source and $\tilde{\phi}_{2}$ with the
expectation value of the dual operator.

It is difficult to find a global map which puts the metric (3.34) in FG form.
Here, we limit our discussion to the coordinate region away from the defect,
where we take $u\to\pm\infty$ and keep $z$ finite [23, 24]. This limit probes
the region away from the interface on the boundary. The coordinate change
suitable for the $u\to\infty$ limit can be expressed as a power series,

$\displaystyle z$ $\displaystyle=x+\frac{\rho^{2}}{2x}+\mathcal{O}(\rho^{4})$
$\displaystyle e^{u}$ $\displaystyle=\cosh
q\quantity(\frac{2x}{\rho}+\frac{\rho}{2x}+\mathcal{O}(\rho^{3}))$ (3.37)

The metric becomes

$\displaystyle\differential{s^{2}}$
$\displaystyle=\frac{1}{\rho^{2}}\quantity[-\differential{\rho}^{2}+\quantity(1-\frac{\rho^{2}\tanh^{2}q}{2x^{2}})(\differential{t}^{2}-\differential{x^{2}})+\mathcal{O}(\rho^{3})]$
(3.38)

In the $u\to-\infty$ limit, the asymptotic form of the metric is the same and
the coordinate change is (3.37) with the replacements $e^{u}\to e^{-u}$ and
$x\to-x$.

Using this coordinate change, the expansions of the scalar fields near the
boundary is

$\displaystyle|\phi_{4}|$ $\displaystyle=|\tanh
q|\frac{p+\tilde{q}}{\sinh(p+\tilde{q})}\cdot\frac{\rho}{|x|}+\mathcal{O}(\rho^{3})$
$\displaystyle\phi_{5}$
$\displaystyle=(p+\tilde{q})-\frac{1}{2\sinh(p+\tilde{q})}\quantity(\frac{p+\tilde{q}}{\sinh(p+\tilde{q})}\tanh^{2}q+\frac{\sinh
p\tanh{\tilde{q}}}{\cosh q})\cdot\frac{\rho^{2}}{x^{2}}+\mathcal{O}(\rho^{4})$
(3.39)

where $\tilde{q}\equiv qx/|x|$ (see appendix A.3 for details). The defect is
located on the boundary at $x=0$. We can see that the relevant operator
corresponding to $\phi_{4}$ has no term proportional to $\rho\ln\rho$ in the
expansion. This implies that the source is zero and the dual operator has a
position-dependent expectation value. The marginal operator corresponding to
$\phi_{5}$ has a source term which takes different values on the two sides of
the defect, corresponding to a Janus interface where the modulus associated
with the marginal operator jumps across the interface.

Another quantity which can be calculated holographically is the entanglement
entropy for an interval $A$ using the Ryu-Takanayagi prescription [25],

$\displaystyle S_{\rm EE}={{\rm Length}(\Gamma_{A})\over 4G_{N}^{(3)}}$ (3.40)

where $\Gamma_{A}$ is the minimal curve in the bulk which ends on $\partial
A$.

There are two qualitatively different choices for location of the interval in
an interface CFT, as shown in figure 2. First, the interval can be chosen
symmetrically around the defect [26, 27]. The minimal surface for such a
symmetric interval is particularly simple in the $\operatorname{AdS}$-sliced
coordinates (3.34), and is given by $z=z_{0}$ and $u\in(-\infty,\infty)$. The
regularized length is given by

$\displaystyle{\rm
Length}(\Gamma_{A})=\int\differential{u}=u_{\infty}-u_{-\infty}$ (3.41)

We can use (3.37) to relate the FG cutoff $\rho=\varepsilon$, which furnishes
the UV cutoff on the CFT side, to the cutoff $u_{\pm\infty}$ in the
$\operatorname{AdS}$-sliced metric,

$\displaystyle
u_{\pm\infty}=\pm\quantity\big(-\log\varepsilon+\log(2z_{0})+\log(\cosh q))$
(3.42)

Putting this together and using the expression for the central charge in terms
of $G_{N}^{(3)}$ gives

$\displaystyle S_{\rm EE}={c\over 3}\log{2z_{0}\over\varepsilon}+{c\over
3}\log(\cosh q)$ (3.43)

Figure 2: (a) The entagling surface $A$ is symmetric around the interface
${\cal I}$, (b) The entagleing surface $A$ is ends at the interface ${\cal I}$

Note that the first logarithmically divergent term is the standard expression
for the entanglement entropy for a CFT without an interface present [28],
since $2z_{0}$ is the length of the interval. The constant term is universal
in the presence of an interface and can be interpreted as the defect entropy
(sometimes called g-factor [29]) associated with the interface.

Second, we can consider an interval which lies on one side of the interface
and borders the interface [30, 31]. As shown in [32], the entangling surface
is located at $u=0$ and the entanglement entropy for an interval of length $l$
bordering the interface is given by

$\displaystyle S^{\prime}_{\rm EE}={c\over 6}\sech{q}\log{l\over\varepsilon}$
(3.44)

###  3.4 All scalars

For completeness, we also present the general solution with all $\phi_{I}$
scalars turned on. Let us define

$\displaystyle\alpha_{i}(u)$
$\displaystyle\equiv\frac{\phi_{i}\sinh\Phi}{\Phi}$ $\displaystyle i$
$\displaystyle=1,2,3,4$ $\displaystyle\beta_{\bar{\imath}}(u)$
$\displaystyle\equiv\frac{\phi_{\bar{\imath}}\sinh\Phi}{\Phi}$
$\displaystyle\bar{\imath}$ $\displaystyle=5,6,7,8$ (3.45)

As a consequence of Eq. (3.12), the ratio $\phi_{i}^{\prime}/\phi_{i}$ is the
same for all $i$ so all the $\phi_{i}$ scalars are proportional to each other.
In other words, we have $\alpha_{i}=n_{i}\alpha$ for constants $n_{i}$
satisfying $n_{i}n_{i}=1$, where $\alpha$ is given in Eq. (3.23). Then Eq.
(3.15) becomes

$\displaystyle\alpha^{\prime
2}+\beta_{\bar{\imath}}^{\prime}\beta_{\bar{\imath}}^{\prime}-\frac{(\alpha^{\prime}\alpha+\beta_{\bar{\imath}}^{\prime}\beta_{\bar{\imath}})^{2}}{1+\alpha^{2}+\beta_{\bar{\imath}}\beta_{\bar{\imath}}}=\alpha^{2}$
(3.46)

We can note that there exists a family of solutions where all
$\beta_{\bar{\imath}}$ functions satisfy

$\displaystyle\beta_{\bar{\imath}}=n_{\bar{\imath}}\beta$ (3.47)

for some function $\beta$ and constants $n_{\bar{\imath}}$ satisfying
$n_{\bar{\imath}}n_{\bar{\imath}}=1$. When this is the case, Eq. (3.46) then
further simplifies to

$\displaystyle\alpha^{\prime 2}+\beta^{\prime
2}-\frac{(\alpha^{\prime}\alpha+\beta^{\prime}\beta)^{2}}{1+\alpha^{2}+\beta^{2}}=\alpha^{2}$
(3.48)

which has already been solved in the previous section. We can prove that these
are the only solutions to Eq. (3.46) which satisfy the equations of motion.
The scalar dependence of the Lagrangian is

$\displaystyle e^{-1}\mathcal{L}$
$\displaystyle\supset-\frac{g^{2}}{4}\mathcal{P}^{I}_{u}\mathcal{P}^{I}_{u}+W$
$\displaystyle=-\frac{g^{2}}{4}\quantity(\alpha^{\prime
2}+\beta_{\bar{\imath}}^{\prime}\beta_{\bar{\imath}}^{\prime}-\frac{(\alpha^{\prime}\alpha+\beta_{\bar{\imath}}^{\prime}\beta_{\bar{\imath}})^{2}}{1+\alpha^{2}+\beta_{\bar{\imath}}\beta_{\bar{\imath}}}-(\alpha^{2}+2))$
(3.49) If we write the $\beta_{\bar{\imath}}$ in spherical coordinates, where
we call the radius $\beta$, this becomes
$\displaystyle=-\frac{g^{2}}{4}\quantity(\alpha^{\prime 2}+\beta^{\prime
2}+\beta^{2}K^{2}-\frac{(\alpha^{\prime}\alpha+\beta^{\prime}\beta)^{2}}{1+\alpha^{2}+\beta^{2}}-(\alpha^{2}+2))$
(3.50)

where $K^{2}$ is the kinetic energy of the angular coordinates.222Explicitly,
let $K^{2}=\theta^{\prime 2}+\sin^{2}\theta\,\phi^{\prime
2}+\sin^{2}\theta\sin^{2}\phi\,\psi^{\prime 2}$. We can treat $\alpha,\beta$,
and the three angles as the coordinates of this Lagrangian. The equation of
motion from varying the Lagrangian with respect to $\alpha$ will only involve
$\alpha$ and $\beta$ and their derivatives. Plugging-in (3.23) for $\alpha$,
satisfying this equation of motion fixes the form of $\beta$ to be what was
found previously in Eq. (3.31). This means that Eq. (3.46) simplifies to
$\beta^{2}K^{2}=0$ and the three angles must be constant.

Therefore, the general solution is

$\displaystyle\frac{\phi\sinh\Phi}{\Phi}$ $\displaystyle=|\sinh q|\sech u$
$\displaystyle\beta$ $\displaystyle=\sinh p\cosh q+\cosh p\sinh q\tanh u$
$\displaystyle\phi_{i}$ $\displaystyle=n_{i}\phi\qquad,\quad n_{i}n_{i}=1$
$\displaystyle\frac{\phi_{\bar{\imath}}\sinh\Phi}{\Phi}$
$\displaystyle=n_{\bar{\imath}}\beta\qquad,\quad
n_{\bar{\imath}}n_{\bar{\imath}}=1$ (3.51)

##  4 Discussion

In this paper, we have presented Janus solutions for $d=3$, ${\cal N}=8$
gauged supergravity. We constructed the simplest solutions with the smallest
number of scalars, namely the $\operatorname{SO}(n,1)/\operatorname{SO}(8)$
coset. The solutions we found have only two scalars displaying a nontrivial
profile. One scalar is dual to a marginal operator $O_{2}$ with scaling
dimension $\Delta=2$ and the other scalar is dual to a relevant operator
$O_{1}$ with scaling dimension $\Delta=1$. We used the holographic
correspondence to find the dual CFT interpretation of these solutions. It is
given by a superconformal interface, with a constant source of the operator
$O_{2}$ which jumps across the interface. For the operator $O_{1}$, the source
vanishes but there is an expectation value which depends on the distance from
the interface. It would be interesting to study whether half-BPS Janus
interfaces which display these characteristics can be constructed in the two-
dimensional $\mathcal{N}=(4,4)$ SCFTs.

We considered solutions for the $\operatorname{SO}(n,1)/\operatorname{SO}(8)$
coset, but these solutions can be trivially embedded into the
$\operatorname{SO}(8,n)/\quantity\big(\operatorname{SO}(8)\times\operatorname{SO}(n))$
cosets with $n>1$. Constructing solutions with more scalars with nontrivial
profiles is in principle possible, but the explicit expressions for the
quantities involved in the BPS equations are becoming very complicated. We
also believe that the $n=1$ case already illustrates the important features of
the more general $n>1$ cosets. Another possible generalization is given by
considering more general gaugings. One important example is given by replacing
the embedding tensor (2.6) with

$\displaystyle\Theta_{IJ,KL}=\varepsilon_{ijk\ell}+\alpha\varepsilon_{\bar{\imath}\bar{\jmath}\bar{k}\bar{\ell}}$
(4.1)

This is a deformation produces an $\operatorname{AdS}_{3}$ vacuum which is
dual to a SCFT with a large $D^{1}(2,1;\alpha)\times D^{1}(2,1;\alpha)$
superconformal algebra. As discussed in [16], this gauging is believed to be a
truncation type II supergravity compactified on $\operatorname{AdS}_{3}\times
S^{3}\times S^{3}\times S^{1}$ [33, 34]. It should be straightforward to adapt
the methods for finding solutions developed in the present paper to this case.

We calculated the holographic defect entropy for our solution. It would be
interesting to investigate whether this quantity can be related to the Calabi
diastasis function following [35, 36]. For this identification to work we
would have to consider the case $n=2$ for which the scalar coset is a Kähler
manifold.

We leave these interesting questions for future work.

## Acknowledgements

We would like to thank Matteo Vicino for collaboration at the initial stages
of this work and Per Kraus for useful conversations. The work of M. G. was
supported, in part, by the National Science Foundation under grant
PHY-19-14412. K. C. and M. G. are grateful to the Mani L. Bhaumik Institute
for Theoretical Physics for support.

##  Appendix A Technical details

In this appendix, we present various technical details which are used in the
main part of the paper.

###  A.1 $\operatorname{SO}(8)$ Gamma matrices

We are working with $8\times 8$ Gamma matrices $\Gamma^{I}_{A\dot{A}}$ and
their transposes $\Gamma^{I}_{\dot{A}A}$, which 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 in [37],

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

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

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

###  A.2 Integrability conditions

For BPS equations of the form

$\displaystyle\partial_{t}\varepsilon$
$\displaystyle=-\frac{1}{2z}\gamma_{0}\quantity\big(\gamma_{1}+f(u)+g(u)\gamma_{2})\varepsilon$
$\displaystyle\partial_{z}\varepsilon$
$\displaystyle=-\frac{1}{2z}\gamma_{1}\quantity\big(f(u)+g(u)\gamma_{2})\varepsilon$
$\displaystyle\partial_{u}\varepsilon$
$\displaystyle=\quantity\big(F(u)+G(u)\gamma_{2})\varepsilon$

where $f,g,F,G$ are matrices acting on $\varepsilon$ that commute with
$\gamma_{a}$, the integrability conditions are

$\displaystyle t,z:\qquad 0$
$\displaystyle=(1+f^{2}+g^{2})\varepsilon+[f,g]\gamma_{2}\varepsilon$
$\displaystyle t,u:\qquad 0$
$\displaystyle=(f^{\prime}+[f,F]-\\{g,G\\})\varepsilon+(g^{\prime}+[g,F]+\\{f,G\\})\gamma_{2}\varepsilon$
$\displaystyle z,u:\qquad\phantom{0}$ $\displaystyle\qquad\text{same as for
}t,u$

###  A.3 Scalar asymptotics

The asymptotic expansions of the $\phi_{4}$ and $\phi_{5}$ scalar fields, as
given in (3.32), in the limits $u\to\pm\infty$ are

$\displaystyle|\phi_{4}|=$ $\displaystyle\;2|\sinh q|\frac{p\pm q}{\sinh(p\pm
q)}e^{\mp u}$ $\displaystyle-\frac{2|\sinh q|}{\sinh^{2}(p\pm
q)}\quantity(\frac{p\pm q}{\sinh(p\pm q)}(\sinh^{2}p+\sinh^{2}q)\pm 2\sinh
p\sinh q)e^{\mp 3u}+\mathcal{O}(e^{\mp 5u})$ $\displaystyle\phi_{5}=$
$\displaystyle\;(p\pm q)-\frac{2}{\sinh(p\pm q)}\quantity(\frac{p\pm
q}{\sinh(p\pm q)}\sinh^{2}q\pm\sinh p\sinh q)e^{\mp 2u}+\mathcal{O}(e^{\mp
4u})$ (A.4)

## References

  * [1] D. Bak, M. Gutperle and S. Hirano, _A Dilatonic deformation of AdS(5) and its field theory dual_ , _JHEP_ 05 (2003) 072, [hep-th/0304129].
  * [2] E. D’Hoker, J. Estes and M. Gutperle, _Exact half-BPS Type IIB interface solutions. I. Local solution and supersymmetric Janus_ , _JHEP_ 06 (2007) 021, [0705.0022].
  * [3] E. D’Hoker, J. Estes and M. Gutperle, _Interface Yang-Mills, supersymmetry, and Janus_ , _Nucl. Phys. B_ 753 (2006) 16–41, [hep-th/0603013].
  * [4] E. D’Hoker, J. Estes, M. Gutperle and D. Krym, _Exact Half-BPS Flux Solutions in M-theory. I: Local Solutions_ , _JHEP_ 08 (2008) 028, [0806.0605].
  * [5] E. D’Hoker, J. Estes, M. Gutperle and D. Krym, _Janus solutions in M-theory_ , _JHEP_ 06 (2009) 018, [0904.3313].
  * [6] A. Clark and A. Karch, _Super Janus_ , _JHEP_ 10 (2005) 094, [hep-th/0506265].
  * [7] N. Bobev, K. Pilch and N. P. Warner, _Supersymmetric Janus Solutions in Four Dimensions_ , _JHEP_ 06 (2014) 058, [1311.4883].
  * [8] M. Suh, _Supersymmetric Janus solutions in five and ten dimensions_ , _JHEP_ 09 (2011) 064, [1107.2796].
  * [9] M. Chiodaroli, E. D’Hoker, Y. Guo and M. Gutperle, _Exact half-BPS string-junction solutions in six-dimensional supergravity_ , _JHEP_ 12 (2011) 086, [1107.1722].
  * [10] M. Gutperle, J. Kaidi and H. Raj, _Janus solutions in six-dimensional gauged supergravity_ , _JHEP_ 12 (2017) 018, [1709.09204].
  * [11] K. Pilch, A. Tyukov and N. P. Warner, _$\mathcal{N}=2$ Supersymmetric Janus Solutions and Flows: From Gauged Supergravity to M Theory_, _JHEP_ 05 (2016) 005, [1510.08090].
  * [12] P. Karndumri, _Supersymmetric Janus solutions in four-dimensional N=3 gauged supergravity_ , _Phys. Rev. D_ 93 (2016) 125012, [1604.06007].
  * [13] N. Bobev, F. F. Gautason, K. Pilch, M. Suh and J. Van Muiden, _Janus and J-fold Solutions from Sasaki-Einstein Manifolds_ , _Phys. Rev. D_ 100 (2019) 081901, [1907.11132].
  * [14] N. Bobev, F. F. Gautason, K. Pilch, M. Suh and J. van Muiden, _Holographic Interfaces in N=4 SYM: Janus and J-folds_ , 2003.09154.
  * [15] A. Clark, D. Freedman, A. Karch and M. Schnabl, _Dual of the Janus solution: An interface conformal field theory_ , _Phys. Rev. D_ 71 (2005) 066003, [hep-th/0407073].
  * [16] H. Nicolai and H. Samtleben, _N=8 matter coupled AdS(3) supergravities_ , _Phys. Lett. B_ 514 (2001) 165–172, [hep-th/0106153].
  * [17] H. Samtleben and O. Sarioglu, _Consistent $S^{3}$ reductions of six-dimensional supergravity_, _Phys. Rev. D_ 100 (2019) 086002, [1907.08413].
  * [18] B. de Wit, I. Herger and H. Samtleben, _Gauged locally supersymmetric D = 3 nonlinear sigma models_ , _Nucl. Phys. B_ 671 (2003) 175–216, [hep-th/0307006].
  * [19] P. Breitenlohner and D. Z. Freedman, _Positive Energy in anti-De Sitter Backgrounds and Gauged Extended Supergravity_ , _Phys. Lett. B_ 115 (1982) 197–201.
  * [20] I. R. Klebanov and E. Witten, _AdS / CFT correspondence and symmetry breaking_ , _Nucl. Phys. B_ 556 (1999) 89–114, [hep-th/9905104].
  * [21] E. Witten, _Multitrace operators, boundary conditions, and AdS / CFT correspondence_ , hep-th/0112258.
  * [22] D. Marolf and S. F. Ross, _Boundary Conditions and New Dualities: Vector Fields in AdS/CFT_ , _JHEP_ 11 (2006) 085, [hep-th/0606113].
  * [23] I. Papadimitriou and K. Skenderis, _Correlation functions in holographic RG flows_ , _JHEP_ 10 (2004) 075, [hep-th/0407071].
  * [24] K. Jensen and A. O’Bannon, _Holography, Entanglement Entropy, and Conformal Field Theories with Boundaries or Defects_ , _Phys. Rev. D_ 88 (2013) 106006, [1309.4523].
  * [25] S. Ryu and T. Takayanagi, _Holographic derivation of entanglement entropy from AdS/CFT_ , _Phys. Rev. Lett._ 96 (2006) 181602, [hep-th/0603001].
  * [26] T. Azeyanagi, A. Karch, T. Takayanagi and E. G. Thompson, _Holographic calculation of boundary entropy_ , _JHEP_ 03 (2008) 054, [0712.1850].
  * [27] M. Chiodaroli, M. Gutperle and L.-Y. Hung, _Boundary entropy of supersymmetric Janus solutions_ , _JHEP_ 09 (2010) 082, [1005.4433].
  * [28] P. Calabrese and J. L. Cardy, _Entanglement entropy and quantum field theory_ , _J. Stat. Mech._ 0406 (2004) P06002, [hep-th/0405152].
  * [29] I. Affleck and A. W. Ludwig, _Universal noninteger ’ground state degeneracy’ in critical quantum systems_ , _Phys. Rev. Lett._ 67 (1991) 161–164.
  * [30] K. Sakai and Y. Satoh, _Entanglement through conformal interfaces_ , _JHEP_ 12 (2008) 001, [0809.4548].
  * [31] E. M. Brehm and I. Brunner, _Entanglement entropy through conformal interfaces in the 2D Ising model_ , _JHEP_ 09 (2015) 080, [1505.02647].
  * [32] M. Gutperle and J. D. Miller, _Entanglement entropy at holographic interfaces_ , _Phys. Rev. D_ 93 (2016) 026006, [1511.08955].
  * [33] J. de Boer, A. Pasquinucci and K. Skenderis, _AdS / CFT dualities involving large 2-D N=4 superconformal symmetry_ , _Adv. Theor. Math. Phys._ 3 (1999) 577–614, [hep-th/9904073].
  * [34] S. Gukov, E. Martinec, G. W. Moore and A. Strominger, _The Search for a holographic dual to AdS(3) x S**3 x S**3 x S**1_ , _Adv. Theor. Math. Phys._ 9 (2005) 435–525, [hep-th/0403090].
  * [35] C. P. Bachas, I. Brunner, M. R. Douglas and L. Rastelli, _Calabi’s diastasis as interface entropy_ , _P_ hys. Rev. D 90 (2014) no.4, 045004[1311.2202].
  * [36] E. D’Hoker and M. Gutperle, _Holographic entropy and Calabi‘s diastasis_ , _J_ HEP 10 (2014), 093, [1406.5124]
  * [37] M. B. Green, J. Schwarz and E. Witten, _SUPERSTRING THEORY. VOL. 1: INTRODUCTION_.  Cambridge Monographs on Mathematical Physics. 7, 1988.