Line and surface defects in 5D $N=2$ SCFT from matter-coupled $F(4)$ gauged
supergravity

Parinya Karndumri

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

E-mail<EMAIL_ADDRESS>
###### Abstract

We study supersymmetric solutions of matter-coupled $F(4)$ gauged supergravity
in the forms of $AdS_{2}\times S^{3}$\- and $AdS_{3}\times S^{2}$-sliced
domain walls with a non-vanishing two-form field from the supergravity
multiplet. These two types of solutions holographically describe conformal
line and surface defects within five-dimensional $N=2$ SCFTs, respectively. In
the case of $SO(3)\times SO(3)$ gauge group obtained by coupling $F(4)$
supergravity to three vector multiplets, we find charged domain wall solutions
describing holographic RG flows between two supersymmetric $AdS_{6}$ vacua
with a running two-form field supporting the line and surface defects. For a
particular case of one vector multiplet with $SO(3)\times U(1)$ gauge group,
we find solutions describing RG flows from an $AdS_{6}$ vacuum to physically
acceptable singular geometries in the presence of defects. This class of
solutions can be uplifted to type IIB theory using a consistent truncation
obtained from $SO(5,5)$ exceptional field theory with half-maximal structures.

##  1 Introduction

One of many interesting results that can be holographically studied via the
AdS/CFT correspondence [1, 2, 3] is the study of conformal defects within
higher-dimensional superconformal field theories (SCFTs), see for example [4,
5, 6, 7, 8, 9, 10, 11, 12, 13, 14]. In particular, for SCFTs in five and six
dimensions for which there is no Lagrangian description available, various
insights into strongly-coupled dynamics have come mainly from the holographic
study of the proposed gravity duals. Up to now, most of the interesting
results have arisen fruitfully from studying solutions of lower-dimensional
gauged supergravity. A number of complications present in the higher-
dimensional computations become more traceable in the framework of gauged
supergravity. Furthermore, a number of holographic computations are also
simpler in gauged supergravity setting. In some cases, the solutions can be
uplifted to ten or eleven dimensions via consistent truncations generating new
holographic solutions in the context of string/M-theory. In other cases for
which embedding in string/M-theory is presently not known, various types of
holographic solutions within gauged supergravity are still worth studying.
In this paper, we are interested in conformal line and surface defects within
five-dimensional $N=2$ SCFTs in the framework of matter-coupled $F(4)$ gauged
supergravity [15, 16]. These defects preserve one- and two-dimensional
conformal symmetry subgroup of the $SO(2,5)$ conformal symmetry in five
dimensions. The corresponding supergravity solutions can be constructed by
taking the metric ansatz in the form of curved domain walls with
$AdS_{2}\times S^{3}$ and $AdS_{3}\times S^{2}$ slicing, respectively. These
types of solutions have been considered in the context of pure $F(4)$ gauged
supergravity, constructed in [17], in [18] and [19], see also a more general
solution in [20]. In this case, the solutions can be uplifted to massive type
IIA theory using the consistent truncation constructed in [21], for other
holographic studies of conformal defects in five-dimensional SCFTs see [22,
23, 24].
In this work, we generalize these two classes of solutions to the matter-
coupled $F(4)$ gauged supergravity. We will consider two gauged supergravities
obtained from coupling pure or minimal $F(4)$ gauged supergravity to three and
one vector multiplets. The former leads to $SO(4)\sim SO(3)_{R}\times SO(3)$
gauge group admitting two supersymmetric $AdS_{6}$ vacua with $SO(3)_{R}\times
SO(3)$ and $SO(3)_{\textrm{diag}}$ symmetries [25], see also [26, 27, 28, 29]
for other hologrpahic solutions obtained from this gauged supergravity.
$SO(3)_{R}$ denotes the R-symmetry while the second $SO(3)$ factor arises from
the symmetry of the three vector multiplets. On the other hand, the latter
gauged supergravity leads to $SO(3)_{R}\times U(1)$ gauge group. The $U(1)$
factor is just the abelian gauge symmetry of the single vector multiplet under
which all the other fields are uncharged. A number of supersymmetric Janus
solutions and $AdS_{6}$ black holes from this gauged supergravity has been
studied more recently in [30, 31, 32]. For similar holographic solutions
describing conformal surface defects in seven-dimensional gauged
supergravities, see [33, 34, 35, 36].
In the case of $SO(3)_{R}\times SO(3)$ $F(4)$ gauged supergravity, we consider
line and surface defect solutions in the presence of two five-dimensional
conformal fixed points dual to the two supersymmetric $AdS_{6}$ vacua. This
type of solutions has not previously appeared since the pure $F(4)$ gauged
supergravity admits only one supersymmetric $AdS_{6}$ vacuum. The solutions
interpolate between the two aforementioned $AdS_{6}$ vacua with a running two-
form field, generalizing the holographic RG flow studied in [25] by including
the two-form field. For $SO(3)_{R}\times U(1)$ gauged supergravity, it has
been shown in [37] that this gauged supergravity can be obtained from a
consistent truncation of type IIB theory on a product of a two-sphere and a
Riemann surface $(S^{2}\times\Sigma)$. In this case, the solutions are similar
to those studied in [18, 19] and interpolate between an $AdS_{6}$ vacuum and
singular geometries due to the non-existence of the second supersymmetric
$AdS_{6}$ critical point of the scalar potential. We will also uplift the
resulting solutions to type IIB theory using the truncation formulae given in
[37].
The paper is organized as follows. We give a very brief review of matter-
coupled $F(4)$ gauged supergravity as constructed in [15, 16] in section 2. A
number of new solutions describing line and surface defects in the forms of
$AdS_{2}\times S^{3}$\- and $AdS_{3}\times S^{2}$-sliced domain walls are
given respectively in sections 3 and 4. We also give uplifted solutions to
type IIB theory for the case of one vector multiplet and $SO(3)\times U(1)$
gauge group. We give some conclusions and comments in section 5. In the
appendices, we give some details on the derivation of the BPS equations and
collect relevant formulae in the consistent truncation of type IIB theory on
$S^{2}\times\Sigma$.

##  2 Matter coupled $N=(1,1)$ gauged supergravity in six dimensions

We first review the structure of matter-coupled $F(4)$ gauged supergravity in
six dimensions coupled to $n$ vector multiplets. The supergravity and $n$
vector multiplets have the component fields

$\left(e^{\hat{{\mu}}}_{\mu},\psi^{A}_{\mu},A^{\alpha}_{\mu},B_{\mu\nu},\chi^{A},\sigma\right)\qquad\textrm{and}\qquad(A_{\mu},\lambda_{A},\phi^{\alpha})^{I}\,.$
(1)

We use the metric signature $(-+++++)$ with space-time and tangent space
(flat) indices denoted respectively by $\mu,\nu=0,\ldots,5$ and
$\hat{\mu},\hat{\nu}=0,\ldots,5$. The bosonic fields are the graviton
$e^{\hat{\mu}}_{\mu}$, a two-form field $B_{\mu\nu}$, $4+n$ vector fields
$A^{\Lambda}=(A^{\alpha}_{\mu},A^{I})$, with $\alpha=0,1,2,3$ and
$I=1,2,\ldots,n$, together with the dilaton $\sigma$ and $4n$ scalars
$\phi^{\alpha I}$. The fermionic fields are given by two gravitini
$\psi^{A}_{\mu}$, two spin-$\frac{1}{2}$ fields $\chi^{A}$ and $2n$ gaugini
$\lambda^{I}_{A}$ with indices $A,B,\ldots=1,2$ denoting the fundamental
representation of $SU(2)_{R}\sim USp(2)_{R}\sim SO(3)_{R}$ R-symmetry. We will
also split $\alpha,\beta$ indices as $\alpha=(0,r)$ with $r,s,\ldots=1,2,3$
being $SU(2)_{R}$ adjoint indices.
The dilaton and $4n$ scalars from the vector multiplets are described by
$\mathbb{R}^{+}\times SO(4,n)/SO(4)\times SO(n)$ coset with $\mathbb{R}^{+}$
corresponding to the dilaton. The second factor can be parametrized by a coset
representative ${L^{\Lambda}}_{\underline{\Sigma}}$ transforming under the
global $SO(4,n)$ and local $SO(4)\times SO(n)$ symmetries by left and right
multiplications respectively with indices
$\Lambda,\underline{\Sigma}=0,\ldots,n+3$. We can also split the local index
as $\underline{\Sigma}=(\alpha,I)=(0,r,I)$ leading to various components of
the coset representative of the form

${L^{\Lambda}}_{\underline{\Sigma}}=({L^{\Lambda}}_{\alpha},{L^{\Lambda}}_{I}).$
(2)

The inverse of ${L^{\Lambda}}_{\underline{\Sigma}}$ will be denoted by
${(L^{-1})^{\underline{\Lambda}}}_{\Sigma}=({(L^{-1})^{\alpha}}_{\Sigma},{(L^{-1})^{I}}_{\Sigma})$.
$SO(4,n)$ indices are raised and lowered by the invariant tensor

$\eta_{\Lambda\Sigma}=\eta^{\Lambda\Sigma}=(\delta_{\alpha\beta},-\delta_{IJ}).$
(3)

The bosonic Lagrangian of the matter-coupled gauged supergravity can be
written as

$\displaystyle e^{-1}\mathcal{L}$ $\displaystyle=$
$\displaystyle\frac{1}{4}R-e\partial_{\mu}\sigma\partial^{\mu}\sigma-\frac{1}{4}P^{I\alpha}_{\mu}P^{\mu}_{I\alpha}-\frac{1}{8}e^{-2\sigma}\mathcal{N}_{\Lambda\Sigma}\widehat{F}^{\Lambda}_{\mu\nu}\widehat{F}^{\Sigma\mu\nu}-\frac{3}{64}e^{4\sigma}H_{\mu\nu\rho}H^{\mu\nu\rho}$
$\displaystyle-V-\frac{1}{64}e^{-1}\epsilon^{\mu\nu\rho\sigma\lambda\tau}B_{\mu\nu}\left(\eta_{\Lambda\Sigma}\widehat{F}^{\Lambda}_{\rho\sigma}\widehat{F}^{\Sigma}_{\lambda\tau}+mB_{\rho\sigma}\widehat{F}^{\Lambda}_{\lambda\tau}\delta^{\Lambda
0}+\frac{1}{3}m^{2}B_{\rho\sigma}B_{\lambda\tau}\right)$

with $e=\sqrt{-g}$. The corresponding field strength tensors are defined by

$\displaystyle\widehat{F}^{\Lambda}=F^{\Lambda}-m\delta^{\Lambda 0}B,\qquad
F^{\Lambda}=dA^{\Lambda}+\frac{1}{2}{f^{\Lambda}}_{\Sigma\Gamma}A^{\Sigma}\wedge
A^{\Gamma},\qquad H=dB\,.$ (5)

${f^{\Lambda}}_{\Sigma\Gamma}$ are structure constants of the gauge group. We
also note the convention on differential forms used in [15, 16]

$F^{\Lambda}=F^{\Lambda}_{\mu\nu}dx^{\mu}\wedge
dx^{\nu}\qquad\textrm{and}\qquad H=H_{\mu\nu\rho}dx^{\mu}\wedge dx^{\nu}\wedge
dx^{\rho}$ (6)

which are different from the usual convention.
The vielbein on $SO(4,n)/SO(4)\times SO(n)$ denoted by
$P^{I\alpha}_{\mu}=P^{I\alpha}_{x}\partial_{\mu}\phi^{x}$, $x=1,\ldots,4n$,
can be obtained from the left-invariant 1-form

${\Omega^{\underline{\Lambda}}}_{\underline{\Sigma}}={(L^{-1})^{\underline{\Lambda}}}_{\Pi}\nabla{L^{\Pi}}_{\underline{\Sigma}}\qquad\textrm{with}\qquad\nabla{L^{\Lambda}}_{\underline{\Sigma}}={dL^{\Lambda}}_{\underline{\Sigma}}-f^{\phantom{\Gamma}\Lambda}_{\Gamma\phantom{\Lambda}\Pi}A^{\Gamma}{L^{\Pi}}_{\underline{\Sigma}}\,.$
(7)

as

${P^{I}}_{\alpha}=({P^{I}}_{0},{P^{I}}_{r})=(\Omega^{I}_{\phantom{a}0},\Omega^{I}_{\phantom{a}r}).$
(8)

The remaining components of
${\Omega^{\underline{\Lambda}}}_{\underline{\Sigma}}$ are identified as the
$SO(4)\times SO(n)$ composite connections
$(\Omega^{rs},\Omega^{r0},\Omega^{IJ})$. The symmetric scalar matrix
$\mathcal{N}_{\Lambda\Sigma}$ appearing in the vector kinetic term is defined
by

$\displaystyle\mathcal{N}_{\Lambda\Sigma}=L_{\Lambda\alpha}{(L^{-1})^{\alpha}}_{\Sigma}-L_{\Lambda
I}{(L^{-1})^{I}}_{\Sigma}=(\eta LL^{T}\eta)_{\Lambda\Sigma}\,.$ (9)

The scalar potential is given by

$\displaystyle V$ $\displaystyle=$
$\displaystyle-e^{2\sigma}\left[\frac{1}{36}A^{2}+\frac{1}{4}B^{i}B_{i}+\frac{1}{4}\left(C^{I}_{\phantom{s}t}C_{It}+4D^{I}_{\phantom{s}t}D_{It}\right)\right]+m^{2}e^{-6\sigma}\mathcal{N}_{00}$
(10) $\displaystyle-
me^{-2\sigma}\left[\frac{2}{3}AL_{00}-2B^{i}L_{0i}\right].$

We are only interested in compact gauge goups of the form $SU(2)_{R}\times
G_{c}$ with $G_{c}\subset SO(n)$ and
${f^{\Lambda}}_{\Sigma\Gamma}=(g_{1}\epsilon_{rst},g_{2}C_{IJK})$. $C_{IJK}$
are structure constants of $G_{c}$. Various components of fermion-shift
matrices are defined by

$\displaystyle A$ $\displaystyle=$ $\displaystyle\epsilon^{rst}K_{rst},\qquad
B^{i}=\epsilon^{ijk}K_{jk0},$ (11) $\displaystyle C^{\phantom{ts}t}_{I}$
$\displaystyle=$ $\displaystyle\epsilon^{trs}K_{rIs},\qquad D_{It}=K_{0It}$
(12)

in which

$\displaystyle K_{rst}$ $\displaystyle=$ $\displaystyle
g_{1}\epsilon_{lmn}L^{l}_{\phantom{r}r}(L^{-1})_{s}^{\phantom{s}m}L_{\phantom{s}t}^{n}+g_{2}C_{IJK}L^{I}_{\phantom{r}r}(L^{-1})_{s}^{\phantom{s}J}L_{\phantom{s}t}^{K},$
$\displaystyle K_{rs0}$ $\displaystyle=$ $\displaystyle
g_{1}\epsilon_{lmn}L^{l}_{\phantom{r}r}(L^{-1})_{s}^{\phantom{s}m}L_{\phantom{s}0}^{n}+g_{2}C_{IJK}L^{I}_{\phantom{r}r}(L^{-1})_{s}^{\phantom{s}J}L_{\phantom{s}0}^{K},$
$\displaystyle K_{rIt}$ $\displaystyle=$ $\displaystyle
g_{1}\epsilon_{lmn}L^{l}_{\phantom{r}r}(L^{-1})_{I}^{\phantom{s}m}L_{\phantom{s}t}^{n}+g_{2}C_{IJK}L^{I}_{\phantom{r}r}(L^{-1})_{I}^{\phantom{s}J}L_{\phantom{s}t}^{K},$
$\displaystyle K_{0It}$ $\displaystyle=$ $\displaystyle
g_{1}\epsilon_{lmn}L^{l}_{\phantom{r}0}(L^{-1})_{I}^{\phantom{s}m}L_{\phantom{s}t}^{n}+g_{2}C_{IJK}L^{I}_{\phantom{r}0}(L^{-1})_{I}^{\phantom{s}J}L_{\phantom{s}t}^{K}\,.$
(13)

Supersymmetry transformations of the fermionic fields are given by

$\displaystyle\delta\psi_{\mu A}$ $\displaystyle=$ $\displaystyle
D_{\mu}\epsilon_{A}-\frac{1}{24}\left(Ae^{\sigma}+6me^{-3\sigma}(L^{-1})_{00}\right)\epsilon_{AB}\gamma_{\mu}\epsilon^{B}$
(14)
$\displaystyle-\frac{1}{8}\left(B_{t}e^{\sigma}-2me^{-3\sigma}(L^{-1})_{t0}\right)\gamma^{7}\sigma^{t}_{AB}\gamma_{\mu}\epsilon^{B}$
$\displaystyle+\frac{i}{16}e^{-\sigma}\left[\epsilon_{AB}(L^{-1})_{0\Lambda}\gamma_{7}+\sigma^{r}_{AB}(L^{-1})_{r\Lambda}\right]F^{\Lambda}_{\nu\lambda}(\gamma_{\mu}^{\phantom{s}\nu\lambda}-6\delta^{\nu}_{\mu}\gamma^{\lambda})\epsilon^{B}$
$\displaystyle+\frac{i}{32}e^{2\sigma}H_{\nu\lambda\rho}\gamma_{7}({\gamma_{\mu}}^{\nu\lambda\rho}-3\delta_{\mu}^{\nu}\gamma^{\lambda\rho})\epsilon_{A},$
$\displaystyle\delta\chi_{A}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\gamma^{\mu}\partial_{\mu}\sigma\epsilon_{AB}\epsilon^{B}+\frac{1}{24}\left[Ae^{\sigma}-18me^{-3\sigma}(L^{-1})_{00}\right]\epsilon_{AB}\epsilon^{B}$
(15)
$\displaystyle-\frac{1}{8}\left[B_{t}e^{\sigma}+6me^{-3\sigma}(L^{-1})_{t0}\right]\gamma^{7}\sigma^{t}_{AB}\epsilon^{B}$
$\displaystyle-\frac{i}{16}e^{-\sigma}\left[\sigma^{r}_{AB}(L^{-1})_{r\Lambda}-\epsilon_{AB}(L^{-1})_{0\Lambda}\gamma_{7}\right]F^{\Lambda}_{\mu\nu}\gamma^{\mu\nu}\epsilon^{B}$
$\displaystyle-\frac{i}{32}e^{2\sigma}H_{\nu\lambda\rho}\gamma_{7}\gamma^{\nu\lambda\rho}\epsilon_{A},$
$\displaystyle\delta\lambda^{I}_{A}$ $\displaystyle=$ $\displaystyle
P^{I}_{ri}\gamma^{\mu}\partial_{\mu}\phi^{i}\sigma^{r}_{\phantom{s}AB}\epsilon^{B}+P^{I}_{0i}\gamma^{7}\gamma^{\mu}\partial_{\mu}\phi^{i}\epsilon_{AB}\epsilon^{B}-\left(2i\gamma^{7}D^{I}_{\phantom{s}t}+C^{I}_{\phantom{s}t}\right)e^{\sigma}\sigma^{t}_{AB}\epsilon^{B}$
(16)
$\displaystyle+2me^{-3\sigma}(L^{-1})^{I}_{\phantom{ss}0}\gamma^{7}\epsilon_{AB}\epsilon^{B}-\frac{i}{2}e^{-\sigma}(L^{-1})^{I}_{\phantom{s}\Lambda}F^{\Lambda}_{\mu\nu}\gamma^{\mu\nu}\epsilon_{A}$

with ${\sigma^{rA}}_{B}$ being Pauli matrices. $SU(2)_{R}$ fundamental indices
$A,B$ can be raised and lowered by $\epsilon^{AB}$ and $\epsilon_{AB}$ with
the convention $T^{A}=\epsilon^{AB}T_{B}$ and $T_{A}=T^{B}\epsilon_{BA}$.
Finally, the covariant derivative of $\epsilon_{A}$ is given by

$D_{\mu}\epsilon_{A}=\partial_{\mu}\epsilon_{A}+\frac{1}{4}\omega_{\mu}^{ab}\gamma_{ab}\epsilon_{A}+\frac{i}{2}\sigma^{r}_{AB}\left[\frac{1}{2}\epsilon^{rst}\Omega_{\mu
st}-i\gamma_{7}\Omega_{\mu r0}\right]\epsilon^{B}\,.$ (17)

##  3 Line defects from matter-coupled $F(4)$ gauged supergravtiy

In this section, we consider solutions of the matter-coupled $F(4)$ gauged
supergravity which are holographically dual to conformal line defects within
$N=2$ SCFTs in five dimensions. The metric ansatz can be written as

$ds^{2}=e^{2f(r)}ds^{2}_{AdS_{2}}+dr^{2}+e^{2h(r)}ds^{2}_{S^{3}}\,.$ (18)

This metric takes the form of a curved $AdS_{2}\times S^{3}$-sliced domain
wall supported by a non-vanishing two-form field. We will also split the six-
dimensional coordinates as $x^{\mu}=(x^{\alpha},r,\theta^{i})$ for
$\alpha=0,1$ and $i=1,2,3$. Following [18], we will take the ansatz for the
two-form field to be

$B=b(r)\textrm{vol}_{AdS_{2}}\,.$ (19)

We will not consider solutions with non-vanishing vector fields in this paper.
As pointed out in [19], in this case, the action of supersymmetry
transformations on $SU(2)_{R}$ indices $A,B$ is trivial, so we can combine the
two chiralities of the Symplectic-Majorana-Weyl spinors into a Dirac spinor. A
suitable ansatz for the Killing spinors takes the form

$\epsilon=\epsilon^{+}+\epsilon^{-}$ (20)

with

$\displaystyle\epsilon^{+}=iY(r)\left[\cos\theta(r)\chi^{+}\otimes\epsilon_{0}+\sin\theta(r)\chi^{+}\otimes\sigma_{3}\epsilon_{0}\right]\otimes\eta,$
$\displaystyle\epsilon^{-}=Y(r)\left[\sin\theta(r)\chi^{-}\otimes\epsilon_{0}-\cos\theta(r)\chi^{-}\otimes\sigma_{3}\epsilon_{0}\right]\otimes\eta\,.$
(21)

$\epsilon_{0}$ is a two-component constant spinor while $\eta$ and
$\chi^{\pm}$ are respectively Killing spinors on $AdS_{2}$ and $S^{3}$
satisfying

$\hat{\nabla}_{\alpha}\chi^{\pm}=\pm\frac{i}{2L}\rho_{\alpha}\chi^{\mp}\qquad\textrm{and}\qquad\widetilde{\nabla}_{i}\eta=\frac{i}{2R}\tilde{\gamma}_{i}\eta\,.$
(22)

$L$ and $R$ denote the radii of $AdS_{2}$ and $S^{3}$, respectively.
Due to some differences in the convention for $F(4)$ gauged supergravity of
[17] used in [18] and that of [15, 16] used here, we choose a slightly
different choice of gamma matrices

$\displaystyle\gamma^{\alpha}=\rho^{\alpha}\otimes\sigma_{2}\mathbb{I}_{2},\qquad\gamma^{r}=\rho_{*}\otimes\sigma_{2}\otimes\mathbb{I}_{2},$
$\displaystyle\gamma^{i}=-\mathbb{I}_{2}\otimes\sigma_{1}\otimes\tilde{\gamma}^{i},\qquad\gamma_{7}=-i\mathbb{I}_{2}\otimes\sigma_{3}\otimes\mathbb{I}_{2}$
(23)

which are related to those used in [18] by the relation
$\gamma^{\mu}=-i\Gamma_{*}\Gamma^{\mu}$ with $\Gamma_{*}=i\gamma_{7}$. In the
above equations, the chirality matrix on $AdS_{2}$ is defined by
$\rho_{*}=\rho^{0}\rho^{1}$ with $\rho^{2}_{*}=\mathbb{I}_{2}$.
We will consider this type of solutions in matter-coupled $F(4)$ gauged
supergravity for two different gauge groups. The first gauge group is given by
$SO(3)_{R}\times SO(3)$ with the gauge structure constants

${f^{\Lambda}}_{\Pi\Sigma}=(g_{1}\epsilon_{rst},g_{2}\epsilon_{IJK})$ (24)

with $g_{1}$ and $g_{2}$ being the corresponding gauge coupling constants for
the two $SO(3)$ factors. This gauge group can be embedded in the case of
$n\geq 3$ vector multiplets. For simplicity, we will consider only the
simplest case of $n=3$ vector multiplets. The resulting gauged supergravity
admits two supersymmetric $AdS_{6}$ vacua with $SO(3)_{R}\times SO(3)$ and
$SO(3)_{\textrm{diag}}$ symmetries. Both of these vacua are maximally
supersymmetric preserving the full sixteen supercharges.
The other gauge group under consideration here is $SO(3)_{R}\times U(1)\sim
SU(2)_{R}\times U(1)$ obtained by coupling pure $F(4)$ gauged supergravity of
[17] to one vector multiplet. The corresponding gauge structure constants are
given by

${f^{\Lambda}}_{\Pi\Sigma}=g_{1}\epsilon_{rst}\,.$ (25)

It should be noted that the $U(1)$ factor is the usual abelian gauge symmetry
of the additional vector field from the single vector multiplet as in the
ungauged supergravity. Accordingly, all the other fields are uncharged under
this $U(1)$ subgroup. It has been shown in [37] that this matter-coupled
$F(4)$ gauged supergravity can be embedded in type IIB theory via a consistent
truncation obtained from $SO(5,5)$ exceptional field theory with half-maximal
structures.

###  3.1 Line defects with $SO(3)_{\textrm{diag}}$ symmetry

We begin with the case of $F(4)$ gauged supergravity coupled to three vector
multiplets with gauge group $SO(3)_{R}\times SO(3)$. We are interested in
solutions with $SO(3)_{\textrm{diag}}$ symmetry. There is one singlet scalar
from $SO(4,3)/SO(4)\times SO(3)$ coset [25]. With the non-compact generators
of $SO(4,3)$ written as

$\displaystyle Y_{\alpha I}=e^{\alpha,I+3}+e^{I+3,\alpha}$ (26)

with

$(e^{\Lambda\Sigma})_{\Gamma\Pi}=\delta^{\Lambda}_{\Gamma}\delta^{\Sigma}_{\Pi},\qquad\Lambda,\Sigma,\Gamma,\Pi=0,\ldots,6,$
(27)

the $SO(3)_{\textrm{diag}}$ invariant coset representative is given by

$L=e^{\phi(Y_{11}+Y_{22}+Y_{33})}\,.$ (28)

As identified in [25], there are two supersymmetric $AdS_{6}$ vacua with
$SO(3)_{R}\times SO(3)$ and $SO(3)_{\textrm{diag}}$ symmetry in this sector.
Since we are interested in solutions interpolating between these asymptotic
$AdS_{6}$ geometries, it is useful to repeat these two vacua here

$\displaystyle\textrm{I}:$
$\displaystyle\phi=0,\qquad\sigma=\frac{1}{4}\ln\left[\frac{3m}{g_{1}}\right],$
(29) $\displaystyle
V_{0}=-20m^{2}\left(\frac{g_{1}}{3m}\right)^{\frac{3}{2}},\qquad\ell_{\textrm{I}}=\frac{1}{2m}\left(\frac{3m}{g_{1}}\right)^{\frac{3}{4}},$
$\displaystyle\textrm{II}:$
$\displaystyle\phi=\frac{1}{2}\ln\left[\frac{g_{1}+g_{2}}{g_{2}-g_{1}}\right],\qquad\sigma=\frac{1}{4}\ln\left[\frac{3m\sqrt{g_{2}^{2}-g_{1}^{2}}}{g_{1}g_{2}}\right],$
(30) $\displaystyle
V_{0}=-20m^{2}\left[\frac{g_{1}g_{2}}{3m\sqrt{g_{2}^{2}-g_{1}^{2}}}\right]^{\frac{3}{2}},\qquad\ell_{\textrm{II}}=\frac{1}{2m}\left(\frac{3m\sqrt{g_{2}^{2}-g_{1}^{2}}}{g_{1}g_{2}}\right)^{\frac{3}{4}}\,.$

$V_{0}$ and $\ell$ are the cosmological constant and the $AdS_{6}$ radius with
$\ell^{2}=-\frac{5}{V_{0}}$. We can also choose $g_{1}=3m$ to set $\sigma=0$
at the $SO(3)_{R}\times SO(3)$ critical point I.
We are now in a position to analyze relevant BPS equations. By imposing the
projection condition on the Killing spinor of the form

$\sigma_{2}\epsilon_{0}=\epsilon_{0},$ (31)

we find the following BPS equations

$\displaystyle f^{\prime}$ $\displaystyle=$
$\displaystyle\frac{1}{4L}\left[e^{-f}\tan 2\theta+Le^{-3\sigma}\sec
2\theta\left[m-3m\cos 4\theta\right.\right.$ (32)
$\displaystyle\qquad\left.\left.-2e^{4\sigma}(g_{1}\cosh^{3}\phi-
g_{2}\sinh^{3}\phi)\right]\right],$ $\displaystyle h^{\prime}$
$\displaystyle=$ $\displaystyle\frac{1}{4L}\left[e^{-f}\tan
2\theta+Le^{-3\sigma}\sec 2\theta\left[m\cos 4\theta-3m\right.\right.$ (33)
$\displaystyle\qquad\left.\left.-2e^{4\sigma}(g_{1}\cosh^{3}\phi-
g_{2}\sinh^{3}\phi)\right]\right],$ $\displaystyle b^{\prime}$
$\displaystyle=$ $\displaystyle-\frac{2}{L}e^{f-5\sigma}\sec
2\theta\left[e^{3\sigma}+2e^{f}L\sin 2\theta\left[2m\right.\right.$ (34)
$\displaystyle\qquad\left.\left.+e^{4\sigma}(g_{2}\sinh^{3}\phi-
g_{1}\cosh^{3}\phi)\right]\right],\qquad$ $\displaystyle\sigma^{\prime}$
$\displaystyle=$ $\displaystyle-\frac{1}{4L}\sec 2\theta\left[e^{-f}\sin
2\theta+mLe^{-3\sigma}(5+\cos 4\theta)\right.$ (35)
$\displaystyle\qquad\left.-2Le^{\sigma}(g_{1}\cosh^{3}\phi-
g_{2}\sinh^{3}\phi)\right],$ $\displaystyle Y^{\prime}$ $\displaystyle=$
$\displaystyle-\frac{Y}{8L}\sec 2\theta\left[e^{-f}\sin
2\theta+mLe^{-3\sigma}(1-3\cos 4\theta)\right.$ (36)
$\displaystyle\qquad\left.-2Le^{\sigma}(g_{1}\cosh^{3}\phi-
g_{2}\sinh^{3}\phi)\right],$ $\displaystyle\theta^{\prime}$ $\displaystyle=$
$\displaystyle me^{-3\sigma}\sin 2\theta,$ (37) $\displaystyle 0$
$\displaystyle=$ $\displaystyle\left[\phi^{\prime}-e^{\sigma}\sinh
2\phi(g_{1}\cosh\phi-g_{2}\sinh\phi)\right]\cos\theta,$ (38) $\displaystyle 0$
$\displaystyle=$ $\displaystyle\left[\phi^{\prime}+e^{\sigma}\sinh
2\phi(g_{1}\cosh\phi-g_{2}\sinh\phi)\right]\sin\theta$ (39)

together with two algebraic constraints

$\displaystyle b$ $\displaystyle=$
$\displaystyle\frac{e^{f+\sigma}}{Lm}-2e^{2f-2\sigma}\sin 2\theta,$ (40)
$\displaystyle R$ $\displaystyle=$
$\displaystyle\frac{1}{2}e^{h}\left[Le^{-f}\sec 2\theta-2e^{\sigma}\tan
2\theta(g_{1}\cosh^{3}\phi-g_{2}\sinh^{3}\phi)\right].$ (41)

The detail on the derivation of these equations is given in appendix A. From
equations (38) and (39), we immediately see that we need to set $\sin\theta=0$
or $\cos\theta=0$ in order to have a non-trivial scalar profile from the
vector multiplets. The two choices are equivalent and related by interchanging
$r$ with $-r$. For definiteness, we will set $\cos\theta=0$ which leads to
$f=h$. More precisely, either $\sin\theta=0$ or $\cos\theta=0$ give
$h^{\prime}=f^{\prime}$ or $f=h+c$ for a constant $c$. However, we are
interested in solutions with an asymptotically locally $AdS_{6}$ geometry with
$f=h\sim\frac{r}{\ell}$ for $AdS_{6}$ radius $\ell$. We can then set $c=0$
and, with all these results, obtain a simpler set of BPS equations

$\displaystyle f^{\prime}$ $\displaystyle=$
$\displaystyle\frac{1}{2}e^{-3\sigma}\left[m+e^{4\sigma}(g-1\cosh^{3}\phi-
g_{2}\sinh^{3}\phi)\right],$ (42) $\displaystyle\sigma^{\prime}$
$\displaystyle=$
$\displaystyle\frac{3}{2}e^{-3\sigma}m+\frac{1}{2}e^{\sigma}\left(g_{1}\cosh^{3}\phi-
g_{2}\sinh^{3}\phi\right),$ (43) $\displaystyle\phi^{\prime}$ $\displaystyle=$
$\displaystyle-e^{\sigma}\sin 2\phi(g_{1}\cosh\phi-g_{2}\sinh\phi),$ (44)
$\displaystyle b^{\prime}$ $\displaystyle=$
$\displaystyle\frac{2e^{f-2\sigma}}{L},\qquad
Y^{\prime}=\frac{1}{2}Yf^{\prime}\,.$ (45)

The two algebraic constraints also simplify considerably

$\displaystyle b=\frac{e^{f+\sigma}}{Lm}\qquad\textrm{and}\qquad L=2R\,.$ (46)

It can be straightforwardly verified that the BPS equations are compatible
with these two algebraic constraints and satisfy the second-order field
equations.
We note that the two-form field and the warp factor $f$ do not appear in other
equations, so we can firstly solve for $\sigma$ and $\phi$ and eventually find
$b$ and $f$. Solving for $\sigma$ in terms of $\phi$ as in [25], we find

$\sigma=\frac{1}{4}\ln\left[\frac{\sigma_{0}+3m\coth\phi+3m\tanh\phi}{g_{1}\textrm{csch}\phi-
g_{2}\textrm{sech}\phi}\right].$ (47)

As $\phi\rightarrow 0$, we see that
$\sigma\sim\frac{1}{4}\ln\left[\frac{3m}{g_{1}}\right]$ which is the value of
the dilaton at $AdS_{6}$ critical point I. We will determine the integration
constant $\sigma_{0}$ by requiring that the solution interpolate between
$AdS_{6}$ critical points I and II. This is achieved by setting

$\sigma_{0}=-\frac{3m(g_{1}^{2}+g_{2}^{2})}{g_{1}g_{2}}$ (48)

which gives

$\sigma=\frac{1}{4}\ln\left[\frac{3m\cosh\phi}{g_{1}}-\frac{3m\sinh\phi}{g_{2}}\right].$
(49)

Similarly, we find

$\displaystyle f$ $\displaystyle=$
$\displaystyle\frac{1}{4}\ln(g_{1}\cosh\phi-
g_{2}\sinh\phi)-\frac{1}{3}\ln\sinh 2\phi$ (50)
$\displaystyle+\frac{1}{12}\left[6m\cosh
2\phi-\frac{3m(g_{1}^{2}+g_{2}^{2})\sinh 2\phi}{g_{1}g_{2}}\right],$
$\displaystyle b$ $\displaystyle=$
$\displaystyle\frac{3^{\frac{1}{3}}}{2^{\frac{1}{4}}L}\left[\frac{g_{1}^{2}+g_{2}^{2}-2g_{1}g_{2}\coth\phi-
g_{1}g_{2}\tanh\phi}{g_{1}g_{2}m^{2}}\right]^{\frac{1}{3}},$ (51)
$\displaystyle Y$ $\displaystyle=$ $\displaystyle e^{\frac{f}{2}}\,.$ (52)

Finally, we can use these results to find solution for $\phi$. By changing to
a new radial coordinate $\tilde{r}$ given by
$\frac{d\tilde{r}}{dr}=e^{\sigma}$, we can solve equation (44) to obtain

$\displaystyle 2g_{1}g_{2}(\tilde{r}-\tilde{r}_{0})$ $\displaystyle=$
$\displaystyle
g_{2}\ln\cosh\frac{\phi}{2}-g_{2}\ln\sinh\frac{\phi}{2}-2g_{1}\tan^{-1}\tanh\frac{\phi}{2}$
(53)
$\displaystyle-2\sqrt{g_{2}^{2}-g_{1}^{2}}\tanh^{-1}\left[\frac{g_{2}g_{1}\tanh\frac{\phi}{2}}{\sqrt{g_{2}^{2}-g_{1}^{2}}}\right].$

As $\phi\rightarrow 0$, for $g_{1}=3m$, we find $f\sim 2mr$ with
$\tilde{r}\sim r$ and

$\displaystyle\phi\sim e^{-6mr},\qquad\sigma\sim e^{-6mr},\qquad b\sim
e^{2mr}$ (54)

or in terms of the coordinate $z=e^{-\frac{r}{\ell_{\textrm{I}}}}$

$\displaystyle\phi\sim\sigma\sim z^{3}\qquad\textrm{and}\qquad b\sim
z^{-1}\,.$ (55)

Similarly, as
$\phi\rightarrow\frac{1}{2}\ln\left[\frac{g_{2}+g_{1}}{g_{2}-g_{1}}\right]$,
we find

$\displaystyle\sigma\sim z^{3},\qquad\phi\sim z^{-3},\qquad b\sim z^{-1}$ (56)

with $z=e^{-\frac{r}{\ell_{\textrm{II}}}}$.
By linearizing the two-form field equation as in [20], we find that $b$ is
dual to an operator of dimension $\Delta_{b}=4$ at both asymptotic $AdS_{6}$
geometries. Accordingly, the defect solution involves turning on a source of a
dimension-$4$ operator dual to the two-form field. The solution is then a
charged domain wall describing a holographic RG flow between two conformal
fixed points of five-dimensional $N=2$ SCFT in the presence of a conformal
line defect. It could be interesting to identify the field theory dual of this
solution.

###  3.2 Line defects with $SO(2)$ symmetry

We now look at another holographic solution describing line defects with
$SO(2)$ symmetry. In this case, we consider $F(4)$ gauged supergravity coupled
to one vector multiplet with $SO(3)\sim SO(3)_{R}$ gauge group. The procedure
is the same as in the previous case, so we will mainly give the results with
less detail. By considering $SO(2)_{R}\subset SO(3)_{R}$ invariant scalars
from $SO(4,1)/SO(4)$ coset described by the coset representative

$L=e^{\phi_{0}Y_{01}+\phi_{2}Y_{31}}$ (57)

and using the gauge structure constants given in (25), we find the scalar
potential

$V=-e^{-6\sigma}\left[g_{1}^{2}e^{8\sigma}+4g_{1}me^{4\sigma}\cosh\phi_{2}-m^{2}\right]$
(58)

which admits only one supersymmetric $AdS_{6}$ vacuum at

$\phi_{2}=0,\qquad\sigma=\frac{1}{4}\ln\left[\frac{3m}{g_{1}}\right],\qquad\ell^{2}=\frac{3\sqrt{3}}{4g_{1}\sqrt{g_{1}m}}\,.$
(59)

As in the previous case, consistency of the BPS equations requires
$\phi_{0}=0$. Therefore, we have set $\phi_{0}=0$ in the above equations for
simplicity.
With the same ansatze for the remaining fields as in the previous case, we
obtain the relevant BPS equations given in appendix A. In particular, we
require either $\sin\theta=0$ or $\cos\theta=0$ for non-trivial scalars from
the vector multiplet similar to the solutions considered in the previous
analysis. With $\cos\theta=0$, we find the following BPS equations

$\displaystyle f^{\prime}$ $\displaystyle=$ $\displaystyle
h^{\prime}=\frac{1}{2}e^{-3\sigma}(m+e^{4\sigma}g_{1}\cosh\phi_{2}),$ (60)
$\displaystyle\sigma$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(3me^{-3\sigma}-g_{1}e^{\sigma}\cosh\phi_{2}\right),\qquad
Y^{\prime}=\frac{1}{2}Yf^{\prime},$ (61) $\displaystyle b^{\prime}$
$\displaystyle=$
$\displaystyle\frac{2e^{f-2\sigma}}{L},\qquad\phi^{\prime}_{2}=-2g_{1}e^{\sigma}\sinh\phi_{2}$
(62)

with the solution given by

$\displaystyle
f=h=-\frac{1}{3}\ln\sinh\phi_{2}+\frac{1}{12}\ln(\sigma_{0}\sinh\phi_{2}+3m\cosh\phi_{2}),$
(63)
$\displaystyle\sigma=\frac{1}{4}\ln\left[\frac{(\sigma_{0}+3m\coth\phi_{2})\sinh\phi_{2}}{g_{1}}\right],$
(64) $\displaystyle
b=\frac{(\sigma_{0}+3m\coth\phi_{2})^{\frac{1}{3}}}{mLg_{1}^{\frac{1}{4}}},\qquad
Y=e^{\frac{f}{2}},$ (65) $\displaystyle
2g_{1}(\tilde{r}-\tilde{r}_{0})=\ln\cosh\frac{\phi_{2}}{2}-\ln\sinh\frac{\phi_{2}}{2}$
(66)

for a new radial coordinate $\tilde{r}$ defined by
$\frac{d\tilde{r}}{dr}=e^{\sigma}$.
In this case, there is only one asymptotically locally supersymmetric
$AdS_{6}$ vacuum at $\phi_{2}=\sigma=0$ for $g_{1}=3m$. It can also be seen
that the above solution gives $\sigma\sim 0$ as $\phi_{2}\sim 0$ for any
values of the integration constant $\sigma_{0}$. The asymptotic form of the
solution is given by

$\displaystyle\sigma\sim\phi_{2}\sim e^{-6mr}\sim z^{3},\qquad f\sim-\ln
z,\qquad b\sim z^{-1}$ (67)

for $z=e^{-2mr}=e^{-\frac{r}{\ell}}$. However, unlike the previous solution,
there is a singularity at a finite value of the radial coordinate $\tilde{r}$
at which the scalars diverge. There are a number of possibilities depending on
the values of $\sigma_{0}$. We first consider when $\phi_{2}\rightarrow\infty$
for which the $\phi_{2}$ solution becomes

$\phi_{2}\sim-\ln[g_{1}(\tilde{r}-\tilde{r}_{0})].$ (68)

For $\sigma_{0}=-3m$, we find that the solution near the singularity takes the
form of

$\displaystyle\sigma\sim-\frac{1}{4}\phi_{2}\sim\frac{1}{4}\ln[g_{1}(\tilde{r}-\tilde{r}_{0})],\qquad
f\sim\frac{5}{12}\ln[g_{1}(\tilde{r}-\tilde{r}_{0})],$ $\displaystyle
b\sim[g_{1}(\tilde{r}-\tilde{r}_{0})]^{\frac{2}{3}}\rightarrow 0\,.$ (69)

For $\sigma_{0}\neq-3m$, we find

$\displaystyle\sigma\sim-\frac{1}{4}\ln[g_{1}(\tilde{r}-\tilde{r}_{0})],\qquad
f\sim\frac{1}{4}\ln[g_{1}(\tilde{r}-\tilde{r}_{0})],$ $\displaystyle
b\sim(\sigma_{0}+3m)^{\frac{1}{3}}\sim\textrm{constant}\,.$ (70)

With the scalar potential given in (58), both of these behaviors lead to
$V\rightarrow-\infty$. Accordingly, these singularities are physically
acceptable by the criterion given in [38].
Other singular behaviors of the solution are given by
$\phi_{2}\rightarrow-\infty$ with

$\phi_{2}\sim\ln[g_{1}(\tilde{r}-\tilde{r}_{0})].$ (71)

In this case, we find, for $\sigma_{0}=3m$,

$\sigma\sim\frac{1}{4}\ln[g_{1}(\tilde{r}-\tilde{r}_{0})],\qquad
f\sim\frac{5}{12}\ln[g_{1}(\tilde{r}-\tilde{r}_{0})],\qquad b\rightarrow 0$
(72)

while for $\sigma_{0}\neq 3m$, the solution becomes

$\displaystyle\sigma\sim-\frac{1}{4}\ln[g_{1}(\tilde{r}-\tilde{r}_{0})],\qquad
f\sim\frac{1}{4}\ln[g_{1}(\tilde{r}-\tilde{r}_{0})],$ $\displaystyle
b\sim(3m-\sigma_{0})^{\frac{1}{3}}\sim\textrm{constant}\,.$ (73)

These are also acceptable since $V\rightarrow-\infty$ near the singularities.
By the classification of supersymmetric $AdS_{6}$ vacua arising from
consistent truncations of string/M-theory given in [37], the matter-coupled
$F(4)$ gauged supergravity coupled to one vector multiplet under consideration
here can be embedded in type IIB theory via a consistent truncation on
$S^{2}\times\Sigma$ with $\Sigma$ being a Riemann surface. We collect relevant
formulae for uplifting the six-dimensional solution given above to ten
dimensions in appendix B. The resulting ten-dimensional fields are given by

$\displaystyle d\hat{s}^{2}$ $\displaystyle=$
$\displaystyle\frac{8c_{6}(3\rho)^{\frac{5}{4}}|dk|^{\frac{3}{2}}}{g_{1}^{2}\bar{\Delta}^{\frac{3}{4}}}\left[\frac{g_{1}^{2}}{6\rho|dk|^{2}}\left[e^{2f}(ds^{2}_{AdS_{2}}+ds^{2}_{S^{3}})+e^{-2\sigma}d\tilde{r}^{2}\right]\right.$
(74)
$\displaystyle+e^{2\sigma}\left(d\tilde{\theta}^{2}+\sin^{2}\tilde{\theta}d\tilde{\phi}^{2}+w^{2}-\frac{1}{\rho^{2}}p_{a}p_{b}n^{a}n^{b}\right)+\frac{\widetilde{\Delta}e^{-2\sigma}}{\rho^{2}|dk|^{2}}dk^{a}dp_{a}$
$\displaystyle\left.+\frac{3}{\rho}e^{-2\sigma}n_{a}\cosh\phi_{2}\cos\tilde{\theta}dp^{a}\right],$
$\displaystyle H^{ab}$ $\displaystyle=$
$\displaystyle\frac{|dk|p^{a}p^{b}e^{4\sigma}}{\sqrt{3\rho\bar{\Delta}}}+\sqrt{\frac{3\rho}{\bar{\Delta}}}\left(\sinh\phi_{2}\partial_{c}k^{a}\partial^{c}p^{b}+\cosh\phi_{2}\cos\tilde{\theta}\,n^{ac}\partial_{c}p^{b}\right),$
(75) $\displaystyle C^{a}_{(2)}$ $\displaystyle=$
$\displaystyle-\frac{24c_{6}}{g_{1}^{2}}\left[(k^{a}+L^{a})\textrm{vol}_{\tilde{S}^{2}}+\frac{|dk|^{2}}{\bar{\Delta}}\cosh\phi_{2}\sin^{2}\tilde{\theta}\sigma^{a}\wedge
d\tilde{\phi}\right]$ (76)
$\displaystyle-4c_{6}b(\tilde{r})p^{a}\textrm{vol}_{AdS_{2}},$ $\displaystyle
F_{(5)}$ $\displaystyle=$ $\displaystyle F_{(2,3)}+F_{(3,2)}+*_{10}F_{(2,3)}$
(77)

with the $2\times 2$ matrix $H^{ab}$ describing the ten-dimensional dilaton
$\Phi$ and axion $C_{0}$ as shown in (166). Various form fields appearing in
the self-dual $5$-form field strength are defined by

$\displaystyle F_{(3,2)}$ $\displaystyle=$
$\displaystyle-\frac{288c_{6}^{2}|dk|}{g_{1}^{2}}\left\\{\frac{\rho|dk|}{\bar{\Delta}}b(\tilde{r})\wedge\left[(\sinh^{2}\phi_{2}-\cosh^{2}\phi_{2}\cos^{2}\tilde{\theta})\textrm{vol}_{\tilde{S}^{2}}\right.\right.$
$\displaystyle\left.\phantom{\frac{|d|}{\bar{\Delta}}}\left.-\cosh\phi_{2}\sin^{2}\tilde{\theta}w\wedge
d\tilde{\phi}\right]+b^{\prime}(\tilde{r})e^{f-4\sigma}\textrm{vol}_{S^{3}}\wedge\textrm{vol}_{\Sigma}\right\\},$
$\displaystyle F_{(2,3)}$ $\displaystyle=$
$\displaystyle\frac{96c_{6}^{2}|dk|^{2}}{g_{1}^{2}\bar{\Delta}}e^{4\sigma}\textrm{vol}_{AdS_{2}}\wedge\left[\rho^{a}p_{a}\wedge\textrm{vol}_{\tilde{S}^{2}}\right.$
(78)
$\displaystyle\left.-p_{a}p_{b}n^{ac}\partial_{c}k^{b}\cosh\phi_{2}\sin^{2}\tilde{\theta}d\tilde{\theta}\wedge\textrm{vol}_{\Sigma}\right]$

with $F_{(3,2)}=*_{10}F_{(3,2)}$. Various quantities appearing in these
equations are defined in appendix B. For clarity, we have explicitly shown the
argument of the function $b(\tilde{r})$ in the six-dimensional two-form field.
From the ten-dimensional metric $d\hat{s}^{2}$, we find that

$\hat{g}_{00}=-\frac{4c_{6}(3\rho\bar{\Delta})^{\frac{1}{4}}e^{2f}}{|dk|^{\frac{1}{2}}}\,.$
(79)

For $\phi_{2}\rightarrow\infty$, we find

$\displaystyle\hat{g}_{00}\sim e^{-\frac{1}{3}\phi_{2}}\rightarrow
0\qquad\textrm{and}\qquad\hat{g}_{00}\sim\textrm{constant}$ (80)

for $\sigma_{0}=-3m$ and $\sigma_{0}\neq-3m$, respectively. Similarly, as
$\phi_{2}\rightarrow-\infty$, the six-dimensional solution gives

$\displaystyle\hat{g}_{00}\sim e^{\frac{1}{3}\phi_{2}}\rightarrow
0\qquad\textrm{and}\qquad\hat{g}_{00}\sim\textrm{constant}$ (81)

for $\sigma_{0}=3m$ and $\sigma_{0}\neq 3m$, respectively. These results imply
that all the singular behaviors in the six-dimensional solution are physically
acceptable in type IIB theory by the criterion of [39]. Consequently, the
solution should holographically describe a conformal line defect within five-
dimensional $N=2$ SCFT dual to the $AdS_{6}$ vacuum (59).

##  4 Surface defects from matter-coupled $F(4)$ gauged supergravtiy

We now move to another type of solutions with the metric ansatz given by

$ds^{2}=e^{2f(r)}dx^{2}_{AdS_{3}}+dr^{2}+e^{2h(r)}ds^{2}_{S^{2}}\,.$ (82)

This takes a similar form to the case of line defects, but the slicing is now
given by $AdS_{3}\times S^{2}$. The solutions then preserve two-dimensional
conformal symmetry leading to a holographic description of conformal surface
defects within $N=2$ SCFTs in five dimensions.
The six-dimensional coordinates in this case are split as
$x^{\mu}=(x^{\alpha},r,\theta^{i})$ with $\alpha=0,1,2$ and $i=1,2$. To
support the curvature of the slicing, we again turn on the two-form field of
the form

$B=b(r)\textrm{vol}_{S^{2}}\,.$ (83)

The Killing spinors are taken to be

$\epsilon=Y\eta\otimes\left[\cos\theta\chi^{+}\otimes\epsilon_{0}+i\sin\theta\gamma_{*}\chi^{+}\otimes\sigma_{3}\epsilon_{0}-i\cos\theta\gamma_{*}\chi^{-}\otimes\sigma_{3}\epsilon_{0}+\sin\theta\chi^{-}\epsilon_{0}\right]$
(84)

in which the Killing spinors $\eta$ and $\chi^{\pm}$ on $AdS_{3}$ and $S^{2}$
respectively satisfy

$\hat{\nabla}_{\alpha}=\frac{1}{2L}\rho_{\alpha}\eta\qquad\textrm{and}\qquad\tilde{\nabla}_{i}\chi^{\pm}=\frac{1}{2R}\gamma_{*}\tilde{\gamma}_{i}\chi^{\mp}\,.$
(85)

$Y$ and $\theta$ are functions of $r$ as in the previous section. In this
case, the explicit representation of gamma matrices is chosen to be

$\displaystyle\gamma^{\alpha}=\rho^{\alpha}\otimes\mathbb{I}_{2}\otimes\sigma,\qquad\gamma^{r}=-(\mathbb{I}_{2}\otimes\gamma_{*}\otimes\sigma_{1}),$
$\displaystyle\gamma^{i}=-(\mathbb{I}_{2}\otimes\tilde{\gamma}^{i}\otimes\sigma_{1}),\qquad\gamma_{7}=i\mathbb{I}_{2}\otimes\mathbb{I}_{2}\otimes\sigma_{3}$
(86)

with $\gamma_{*}=i\tilde{\gamma}^{1}\tilde{\gamma}^{2}$ and
$\gamma_{*}^{2}=\mathbb{I}_{2}$.
We now impose the following projection condition

$(\gamma_{*}\otimes\sigma_{1})(\chi^{\pm}\otimes\epsilon_{0})=\pm\chi^{\pm}\otimes\epsilon_{0}$
(87)

which is equivalent to

$\chi^{\pm}\otimes\sigma_{2}\epsilon_{0}=\mp
i\gamma_{*}\chi^{\pm}\otimes\sigma_{3}\epsilon_{0}$ (88)

and find the BPS equations as in the case of line defect solutions. The detail
on this is given in appendix A. The relevant analysis also closely follows
that performed in the previous section, so we will only give the main results
with less detail to avoid repetition. In particualr, in all the following
solutions, the gaugino variations restrict the phase of the Killing spinors to
satisfy the conditions $\sin\theta=0$ or $\cos\theta=0$ as in the previous
section.

###  4.1 Surface defects with $SO(3)_{\textrm{diag}}$ symmetry

We begin with the solution interpolating between two supersymmetric $AdS_{6}$
vacua with an $SO(3)_{\textrm{diag}}$ singlet scalar given in (28). We will
set $\sin\theta=0$ which again gives $h(r)=f(r)$ and the following BPS
equations

$\displaystyle f^{\prime}$ $\displaystyle=$
$\displaystyle\frac{1}{2}e^{-3\sigma}\left[m+e^{4\sigma}(g_{1}\cosh^{3}\phi-
g_{2}\sinh^{3}\phi)\right],$ (89) $\displaystyle\sigma^{\prime}$
$\displaystyle=$
$\displaystyle\frac{1}{2}e^{-3\sigma}\left[3m-e^{4\sigma}(g_{1}\cosh^{3}\phi-
g_{2}\sinh^{3}\phi)\right],$ (90) $\displaystyle b^{\prime}$ $\displaystyle=$
$\displaystyle-\frac{4e^{f-2\sigma}}{L},\qquad
Y^{\prime}=\frac{1}{2}Yf^{\prime},$ (91) $\displaystyle\phi^{\prime}$
$\displaystyle=$ $\displaystyle-e^{\sigma}\sinh 2\phi(g_{1}\cosh\phi-
g_{2}\sinh\phi)$ (92)

together with two algebraic constraints

$b=-\frac{2e^{f+\sigma}}{Lm}\qquad\textrm{and}\qquad R=2L\,.$ (93)

As in the previous section, we have verified that the resulting BPS equations
are compatible with these constraints and the corresponding field equations
for all the fields. By the same procedure as in the previous section, we find
the solution

$\displaystyle f$ $\displaystyle=$
$\displaystyle\frac{1}{4}\ln(g_{1}\cosh\phi-
g_{2}\sinh\phi)-\frac{1}{3}\ln\sinh 2\phi$
$\displaystyle+\frac{1}{12}\left[6m\cosh
2\phi-\frac{3m(g_{1}^{2}+g_{2}^{2})\sinh 2\phi}{g_{1}g_{2}}\right],$
$\displaystyle\sigma$ $\displaystyle=$
$\displaystyle\frac{1}{4}\ln\left[\frac{3m\cosh\phi}{g_{1}}-\frac{3m\sinh\phi}{g_{2}}\right],$
(94) $\displaystyle b$ $\displaystyle=$
$\displaystyle-\frac{2^{\frac{3}{4}}3^{\frac{1}{3}}}{Lm^{\frac{2}{3}}}\left[\frac{g_{1}^{2}+g_{2}^{2}-2g_{1}g_{2}\coth\phi-
g_{1}g_{2}\tanh\phi}{g_{1}g_{2}m^{2}}\right]^{\frac{1}{3}},$ (95)
$\displaystyle Y$ $\displaystyle=$ $\displaystyle e^{\frac{f}{2}},$
$\displaystyle 2g_{1}g_{2}(\tilde{r}-\tilde{r}_{0})$ $\displaystyle=$
$\displaystyle
g_{2}\ln\cosh\frac{\phi}{2}-g_{2}\ln\sinh\frac{\phi}{2}-2g_{1}\tan^{-1}\tanh\frac{\phi}{2}$
(96)
$\displaystyle-2\sqrt{g_{2}^{2}-g_{1}^{2}}\tanh^{-1}\left[\frac{g_{2}g_{1}\tanh\frac{\phi}{2}}{\sqrt{g_{2}^{2}-g_{1}^{2}}}\right]$

after choosing an appropriate integration constant for $\sigma$. This solution
is the same up to a numerical factor in $b$ as in the case of line defects.
All the asymptotic behaviors are essentially the same, so we will not repeat
these again. In particular, we just note that there is a source of a
dimension-$4$ operator dual to the two form field turned on along the flow.

###  4.2 Surface defects with $SO(2)$ symmetry

Similarly, we can look for solutions holographically dual to surface defects
with $SO(2)_{R}\subset SO(3)_{R}$ symmetry for $F(4)$ gauged supergravity
coupled to one vector multiplet with $SO(3)_{R}\times U(1)$ gauge group.
Repeating the same analysis leads to the BPS equations

$\displaystyle f^{\prime}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[e^{-3\sigma}m+g_{1}e^{\sigma}\cosh\phi_{2}\right],$
(97) $\displaystyle\sigma^{\prime}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[3me^{-3\sigma}-g_{1}e^{\sigma}\cosh\phi_{2}\right],$
(98) $\displaystyle b^{\prime}$ $\displaystyle=$
$\displaystyle-\frac{4e^{f-2\sigma}}{L},\qquad
Y^{\prime}=\frac{1}{2}Yf^{\prime},$ (99) $\displaystyle\phi_{2}^{\prime}$
$\displaystyle=$ $\displaystyle-2g_{1}e^{\sigma}\sinh\phi_{2}$ (100)

and

$b=-\frac{2e^{f+\sigma}}{Lm}\qquad\textrm{and}\qquad L=2R\,.$ (101)

As in the previous solution, the solution is the same as that of the line
defects up to a numerical factor in the two-form field. All the possible
singularities are also the same as well as the uplifted solutions in type IIB
theory with the six-dimensional metric replaced by (82) and
$(\textrm{vol}_{AdS_{2}},\textrm{vol}_{S^{3}})$ by
$(\textrm{vol}_{AdS_{3}},\textrm{vol}_{S^{2}})$. We will not repeat all these
results again.

##  5 Conclusions

We have studied solutions of matter-coupled $F(4)$ gauged supergravity in the
form of $AdS_{2}\times S^{3}$\- and $AdS_{3}\times S^{2}$-sliced domain walls.
Unlike the solutions considered in [18] and [19], the presence of scalar
fields from the vector multiplets imposes a condition that constrains the warp
factors of $AdS_{2}$ ($AdS_{3}$) and $S^{3}$ ($S^{2}$) to be equal and fixes
the phase function of the Killing spinors to be constant. Therefore, the
resulting solutions take the form of “charged” domain walls as pointed out in
[18] and [19]. This is very similar to the solution describing surface defects
in $N=(1,0)$ SCFT in six dimensions found in matter-coupled seven-dimensional
$N=2$ gauged supergravity studied in [34]. This could arise from the fact that
in both cases, the scalars from vector multiplets do not couple to the two-
form and three-form fields from the supergravity multiplet. It would be
interesting to find the field theory description of this result.
We have considered two $F(4)$ gauged supergravities coupled to three and one
vector multiplets with $SO(3)_{R}\times SO(3)$ and $SO(3)_{R}\times U(1)$
gauge groups. The resulting solutions should holographically describe
conformal line and surface defects within $N=2$ SCFTs in five dimensions with
a source of a dimension-$4$ operator dual to the two-form field turned on
along the flows between two conformal fixed points and flows from a conformal
fixed point to non-conformal phases. In the case of $SO(3)_{R}\times U(1)$
gauge group, the higher-dimensional origin as a consistent truncation of type
IIB theory on $S^{2}\times\Sigma$ is known. We have also given the
corresponding ten-dimensional solutions by uplifting the six-dimensional
solutions of $F(4)$ gauged supergravity.
It would be interesting to identify line and surface defects, in terms of
position dependent deformations, dual to the solutions found here in the
framework of five-dimensional SCFTs. It is also interesting to use holographic
renormalization to compute the expectation values of the defect operators and
boundary stress tensor as done in [20] for the case of line defect solution
found in [18] and match with the field theory results. Finally, finding type
IIB solutions describing possible brane configurations, of which the uplifted
solutions given here describe low-energy regimes as in the case of pure $F(4)$
gauged supergravity studied in [18] and [19], is also worth considering. We
note that although the six-dimensional solutions for both line and surface
defects are the same, the uplifted solutions could correspond to different
brane configurations in type IIB theory. In particular, the uplifted solution
for line defects would involve F1-D1-branes, due to the presence of two-form
fluxes along $AdS_{2}$, and D3-branes corresponding to a non-trivial four-form
field on top of the 5-brane web dual to the SCFT in five dimensions [40]. On
the other hand, we expect the brane configuration corresponding to the surface
defects to involve only D3-branes without F1-D1-branes since the electric part
of the two-form field does not appear in the uplifted ten-dimensional
solution.
Acknowledgement
This work is funded by National Research Council of Thailand (NRCT) and
Chulalongkorn University under grant N42A650263. The author would like to
thank Emanuel Malek and Christoph F. Uhlemann for helpful correspondences.

##  Appendix A Derivation of the BPS equations

In this appendix, we give more detail on the derivation of the relevant BPS
equations for obtaining solutions of line and surface defects. The $SO(2)_{R}$
sector in the case of one vector multiplet with $SU(2)_{R}\times U(1)$ gauge
group is effectively the same as $SO(2)_{R}\times SO(2)$ invariant sector of
$F(4)$ gauged supergravity coupled to three vector multiplets. Furthermore,
both the $SO(3)_{\textrm{diag}}$ and $SO(2)_{R}\times SO(2)$ sectors in the
latter can be obtained from the $SO(2)_{\textrm{diag}}\subset SO(2)_{R}\times
SO(2)$ invariant sector. We will accordingly give the BPS equations in this
$SO(2)_{\textrm{diag}}$ sector and subsequently choose suitable subtruncations
of interest in each case.
As in [28] and [29], the coset representative in the $SO(2)_{\textrm{diag}}$
sector is given by

$L=e^{\phi_{0}Y_{03}}e^{\phi_{1}(Y_{11}+Y_{22})}e^{\phi_{2}Y_{33}}e^{\phi_{3}(Y_{12}-Y_{21})}\,.$
(102)

We also need to set $\phi_{3}=0$ in order to consistently truncate out all the
gauge fields as pointed out in [29]. To obtain solutions for
$SO(3)_{\textrm{diag}}$ singlet scalars, we simply set $\phi_{0}=0$ and
$\phi_{2}=\phi_{1}=\phi$ while the solutions with $SO(2)_{R}$ symmetry in the
case of one vector multiplet and $SU(2)_{R}\times U(1)$ gauge group are
recovered by setting $\phi_{1}=0$.

###  A.1 Line defects

From the metric (18), we will choose the following choice of vielbein

$e^{\hat{\alpha}}=e^{f}\hat{e}^{\hat{\alpha}},\qquad e^{\hat{r}}=dr,\qquad
e^{\hat{i}}=e^{h}\tilde{e}^{\hat{i}}$ (103)

with $\hat{e}^{\hat{\alpha}}$ and $\tilde{e}^{\hat{i}}$ being the vielbeins on
$AdS_{2}$ and $S^{3}$, respectively. We then find non-vanishing components of
the spin connection

$\displaystyle{\omega^{\hat{\alpha}}}_{\hat{\beta}}={\hat{\omega}^{\hat{\alpha}}}_{\phantom{\hat{\alpha}}\hat{\beta}},\qquad{\omega^{\hat{\alpha}}}_{\hat{r}}=f^{\prime}e^{\hat{\alpha}},\qquad{\omega^{\hat{i}}}_{\hat{j}}={\tilde{\omega}^{\hat{i}}}_{\phantom{\hat{i}}\hat{j}},\qquad{\omega^{\hat{i}}}_{\hat{r}}=h^{\prime}e^{\hat{i}}$
(104)

with ${\hat{\omega}^{\hat{\alpha}}}_{\phantom{\hat{\alpha}}\hat{\beta}}$ and
${\tilde{\omega}^{\hat{i}}}_{\phantom{\hat{i}}\hat{j}}$ being the spin
connections on $AdS_{2}$ and $S^{3}$, respectively.
For convenience, we recall that the ansatz for the Killing spinors is given by

$\epsilon=Y\left[i\cos\theta\chi^{+}\otimes\epsilon_{0}+i\sin\theta\chi^{+}\otimes\sigma_{3}\epsilon_{0}+\sin\theta\chi^{-}\otimes\epsilon_{0}-\cos\theta\chi^{-}\otimes\sigma_{3}\epsilon\right]\otimes\eta$
(105)

subject to the projection condition

$\sigma_{2}\epsilon_{0}=\epsilon_{0}\,.$ (106)

We also note that this projector implies

$\sigma_{1}\epsilon_{0}=i\sigma_{3}\epsilon_{0}$ (107)

which is a useful relation in the following computation.
To find the BPS equations, we use the ansatze for all the bosonic fields given
in section 3 together with the above projectors in the supersymmetry
transformations of fermions and separately solve the resulting conditions for
the four components $\chi^{\pm}\otimes\epsilon_{0}\otimes\eta$ and
$\chi^{\pm}\otimes\sigma_{3}\epsilon_{0}\otimes\eta$. We first consider
$\delta\chi^{A}=0$ equations. The $\chi^{+}\otimes\epsilon_{0}\otimes\eta$ and
$\chi^{-}\otimes\sigma_{3}\epsilon_{0}\otimes\eta$ components give the same
condition of the form

$\left[\frac{1}{2}\sigma^{\prime}+(N_{0}-iN_{3})\right]\cos\theta+\left[\frac{1}{8}me^{-\sigma-2f}{(L^{-1})^{0}}_{0}b-\frac{1}{16}e^{2\sigma-2f}b^{\prime}\right]\sin\theta=0$
(108)

while for $\chi^{+}\otimes\sigma_{3}\epsilon_{0}\otimes\eta$ and
$\chi^{-}\otimes\epsilon_{0}\otimes\eta$ components, we find

$\left[-\frac{1}{2}\sigma^{\prime}+(N_{0}-iN_{3})\right]\sin\theta+\left[\frac{1}{8}me^{-\sigma-2f}{(L^{-1})^{0}}_{0}b+\frac{1}{16}e^{2\sigma-2f}b^{\prime}\right]\cos\theta=0\,.$
(109)

In these equations, we have defined the following quantities

$\displaystyle N_{0}$ $\displaystyle=$
$\displaystyle\frac{1}{8}e^{-3\sigma}\cosh\phi_{0}\left[6m+g_{2}e^{4\sigma}\sinh\phi_{2}(\cosh
2\phi_{1}-1)\right]$
$\displaystyle-\frac{1}{8}g_{1}e^{\sigma}\cosh\phi_{2}(\cosh 2\phi_{1}+1),$
$\displaystyle N_{3}$ $\displaystyle=$
$\displaystyle\frac{1}{8}e^{-3\sigma}\sinh\phi_{0}\left[e^{4\sigma}g_{2}(\cosh
2\phi_{1}-1)-6m\sinh\phi_{2}\right]$ (110)

and ${(L^{-1})^{0}}_{0}=\cosh\phi_{0}$. From these equations, we immediately
see that consistency requires $N_{3}=0$ which in turns leads to $\phi_{0}=0$.
Accordingly, we will set $\phi_{0}=0$ and ${(L^{-1})^{0}}_{0}=1$ from now on.
We then move to the gravitino variations $\delta\psi^{A}_{r}$ which give

$\left[Y^{\prime}-S_{0}Y\right]\cos\theta+\left[Y\theta^{\prime}+\frac{1}{8}me^{-\sigma-2f}bY-\frac{1}{16}e^{2\sigma-2f}b^{\prime}Y\right]\sin\theta=0$
(111)

for $\chi^{+}\otimes\epsilon_{0}\otimes\eta$ and
$\chi^{-}\otimes\sigma_{3}\epsilon_{0}\otimes\eta$ components and

$\left[Y^{\prime}+S_{0}Y\right]\sin\theta+\left[Y\theta^{\prime}+\frac{1}{8}me^{-\sigma-2f}bY+\frac{1}{16}e^{2\sigma-2f}b^{\prime}Y\right]\cos\theta=0$
(112)

for $\chi^{+}\otimes\sigma_{3}\epsilon_{0}\otimes\eta$ and
$\chi^{-}\otimes\epsilon_{0}\otimes\eta$ components. In these equations,
$S_{0}$ is given by

$\displaystyle S_{0}$ $\displaystyle=$
$\displaystyle-\frac{1}{8}g_{1}e^{\sigma}\cosh\phi_{2}(\cosh 2\phi_{1}+1)$
(113)
$\displaystyle-\frac{1}{8}e^{-3\sigma}\left[2m-g_{2}e^{4\sigma}\sinh\phi_{2}(\cosh
2\phi_{1}-1)\right].$

Taking into account the Killing spinor equations on $AdS_{2}$ satisfying (22),
we find the following BPS equations from $\delta\psi^{A}_{\alpha}$

$\displaystyle\left[\frac{1}{2}f^{\prime}-S_{0}\right]\cos\theta-\left[\frac{1}{2L}e^{-f}-\frac{3}{8}e^{-\sigma-2f}b-\frac{1}{16}e^{2\sigma-2f}b^{\prime}\right]\sin\theta=0,$
(114)
$\displaystyle\left[\frac{1}{2}f^{\prime}+S_{0}\right]\sin\theta+\left[\frac{1}{2L}e^{-f}-\frac{3}{8}e^{-\sigma-2f}b+\frac{1}{16}e^{2\sigma-2f}b^{\prime}\right]\cos\theta=0,$
(115)

for
$(\chi^{+}\otimes\epsilon_{0}\otimes\eta,\chi^{-}\otimes\sigma_{3}\epsilon_{0}\otimes\eta)$
and
$(\chi^{+}\otimes\sigma_{3}\epsilon_{0}\otimes\eta,\chi^{-}\otimes\epsilon_{0}\otimes\eta)$
components, respectively
Similarly, $\delta\psi^{A}_{i}$ conditions give

$\displaystyle\left[\frac{1}{2}h^{\prime}+S_{0}\right]\sin\theta-\left[\frac{1}{2R}e^{-h}-\frac{1}{8}e^{-\sigma-2f}b+\frac{1}{16}e^{2\sigma-2f}b^{\prime}\right]\cos\theta=0,$
(116)
$\displaystyle\left[\frac{1}{2}h^{\prime}-S_{0}\right]\cos\theta-\left[\frac{1}{2R}e^{-h}+\frac{1}{8}e^{-\sigma-2f}b+\frac{1}{16}e^{2\sigma-2f}b^{\prime}\right]\sin\theta=0,$
(117)

Finally, we consider the gaugino variations $\delta\lambda^{AI}$. From
$\delta\lambda^{1A}$ and $\delta\lambda^{2A}$, we obtain the followng
conditions

$\displaystyle(\phi^{\prime}_{1}+M_{1})\cos\theta=0\qquad\textrm{and}\qquad(\phi^{\prime}_{1}-M_{1})\sin\theta=0$
(118)

while $\delta\lambda^{3A}$ gives

$\displaystyle(\phi^{\prime}_{2}+M_{3})\cos\theta=0\qquad\textrm{and}\qquad(\phi^{\prime}_{2}-M_{3})\sin\theta=0$
(119)

with

$\displaystyle M_{1}=-e^{\sigma}\sinh
2\phi_{1}(g_{1}\cosh\phi_{2}-g_{2}\sinh\phi_{2}),$ $\displaystyle
M_{3}=g_{2}e^{\sigma}\cosh\phi_{2}(\cosh
2\phi_{1}-1)-g_{1}e^{\sigma}\sinh\phi_{2}(\cosh 2\phi_{1}+1).$ (120)

We can now solve equations (108) to (117) and find the following BPS equations

$\displaystyle f^{\prime}$ $\displaystyle=$
$\displaystyle\frac{1}{4L}\left[e^{-f}\tan
2\theta+L(3N_{0}+5S_{0}-3(N_{0}-S_{0})\cos 4\theta)\sec 2\theta\right],$ (121)
$\displaystyle h^{\prime}$ $\displaystyle=$ $\displaystyle\frac{1}{2L}\sec
2\theta\left[e^{-f}\cos\theta\sin\theta+4S_{0}L+(S_{0}-N_{0})L\sin^{2}2\theta\right],$
(122) $\displaystyle b^{\prime}$ $\displaystyle=$
$\displaystyle-\frac{2}{L}e^{f-2\sigma}\sec
2\theta\left[1+2e^{f}L(S_{0}+3N_{0})\sin 2\theta\right],$ (123)
$\displaystyle\sigma^{\prime}$ $\displaystyle=$ $\displaystyle\frac{1}{2L}\sec
2\theta\left[L(N_{0}-S_{0})\sin^{2}2\theta-4N_{0}L-e^{-f}\cos\theta\sin\theta\right],$
(124) $\displaystyle Y^{\prime}$ $\displaystyle=$
$\displaystyle\frac{Y}{8L}e^{-f}\sec 2\theta\left[Le^{f}(5S_{0}+3N_{0})+\sin
2\theta-3Le^{f}(N_{0}-S_{0})\cos 4\theta\right],\quad$ (125)
$\displaystyle\theta^{\prime}$ $\displaystyle=$
$\displaystyle(N_{0}-S_{0})\sin 2\theta$ (126)

with two algebraic constraints

$\displaystyle b$ $\displaystyle=$
$\displaystyle\frac{e^{f+\sigma}}{Lm}\left[1+2e^{f}L(S_{0}-N_{0})\sin
2\theta\right],$ (127) $\displaystyle R$ $\displaystyle=$
$\displaystyle\frac{1}{2}e^{h}\left[Le^{-f}\sec 2\theta+2(3S_{0}+N_{0})\tan
2\theta\right].$ (128)

These constraints arise from the fact that there are more equations than
unknown functions. In particular, equations (108) to (117) lead to three
different expressions for $b^{\prime}$. Consequently, consistency among these
leads to two algebraic constraints. We also note that these equations reduce
to those considered in [18] for $\phi_{1}=\phi_{2}=0$ up to some redefinition
of variables.

###  A.2 Surface defects

In the case of surface defects with an $AdS_{3}\times S^{2}$-sliced metric
given by (82), we use the following vielbein

$e^{\hat{\alpha}}=e^{f}\hat{e}^{\hat{\alpha}},\qquad e^{\hat{r}}=dr,\qquad
e^{\hat{i}}=e^{h}\tilde{e}^{\hat{i}}$ (129)

and the spin connection

$\displaystyle{\omega^{\hat{\alpha}}}_{\hat{\beta}}={\hat{\omega}^{\hat{\alpha}}}_{\phantom{\hat{\alpha}}\hat{\beta}},\qquad{\omega^{\hat{\alpha}}}_{\hat{r}}=f^{\prime}e^{\hat{\alpha}},\qquad{\omega^{\hat{i}}}_{\hat{j}}={\tilde{\omega}^{\hat{i}}}_{\phantom{\hat{i}}\hat{j}},\qquad{\omega^{\hat{i}}}_{\hat{r}}=h^{\prime}e^{\hat{i}}$
(130)

In this case, $(\hat{e}^{\hat{\alpha}},\tilde{e}^{\hat{i}})$ and
$({\hat{\omega}^{\hat{\alpha}}}_{\phantom{\hat{\alpha}}\hat{\beta}},{\tilde{\omega}^{\hat{i}}}_{\phantom{\hat{i}}\hat{j}})$
are respectively the corresponding vielbeins and spin connections on $AdS_{3}$
and $S^{2}$ with $\alpha=0,1,2$ and $i=1,2$.
The Killing spinors are of the form

$\epsilon=Y\eta\otimes\left[\cos\theta\chi^{+}\otimes\epsilon_{0}+i\sin\theta\gamma_{*}\chi^{+}\otimes\sigma_{3}\epsilon_{0}-i\cos\theta\gamma_{*}\chi^{-}\otimes\sigma_{3}\epsilon_{0}+\sin\theta\chi^{-}\otimes\epsilon_{0}\right]$
(131)

with $\epsilon_{0}$ being a two-component constant spinor. $\eta$ and
$\chi^{\pm}$ are respectively Killing spinors on $AdS_{3}$ and $S^{2}$
satisfying the conditions given in (85). By imposing the projection condition

$(\gamma_{*}\otimes\sigma_{1})(\chi^{\pm}\otimes\epsilon_{0})=\pm\chi^{\pm}\otimes\epsilon_{0}$
(132)

which also implies

$\chi^{\pm}\otimes\sigma_{2}\epsilon_{0}=\mp
i\gamma_{*}\chi^{\pm}\otimes\sigma_{3}\epsilon_{0},$ (133)

we can determine all the BPS equations by the same procedure as in the case of
line defects shown previously. In this case, we separately solve the resulting
conditions for the four components $\eta\otimes\chi^{\pm}\otimes\epsilon_{0}$
and $\eta\otimes\gamma_{*}\chi^{\pm}\otimes\sigma_{3}\epsilon_{0}$. As in the
previous case, it turns out that consistency requires $\phi_{0}=0$.
The conditions obtained from $\delta\chi^{A}$ and $\delta\psi^{A}_{\mu}$ are
given by

$\displaystyle\left[\frac{1}{2}\sigma^{\prime}-N_{0}\right]\cos\theta+\left[\frac{1}{8}me^{-\sigma-2h}+\frac{1}{16}e^{2\sigma-2h}b^{\prime}\right]\sin\theta=0,$
(134)
$\displaystyle\left[\frac{1}{2}\sigma^{\prime}+N_{0}\right]\sin\theta+\left[\frac{1}{8}me^{-\sigma-2h}b-\frac{1}{16}e^{2\sigma-2h}b^{\prime}\right]\cos\theta=0,$
(135)
$\displaystyle\left[Y^{\prime}+S_{0}Y\right]\cos\theta-\left[Y\theta^{\prime}+\frac{1}{8}me^{-\sigma-2h}bY+\frac{1}{16}e^{2\sigma-2h}Yb^{\prime}\right]\sin\theta=0,$
(136)
$\displaystyle\left[Y^{\prime}-S_{0}Y\right]\sin\theta+\left[Y\theta^{\prime}-\frac{1}{8}me^{-\sigma-2h}bY+\frac{1}{16}e^{2\sigma-2h}Yb^{\prime}\right]\cos\theta=0,$
(137)
$\displaystyle\left[\frac{1}{2}f^{\prime}+S_{0}\right]\cos\theta+\left[\frac{1}{2L}e^{-f}-\frac{1}{8}me^{-\sigma-2h}b+\frac{1}{16}e^{2\sigma-2h}b^{\prime}\right]\sin\theta=0,$
(138)
$\displaystyle\left[\frac{1}{2}f^{\prime}-S_{0}\right]\sin\theta-\left[\frac{1}{2L}e^{-f}+\frac{1}{8}me^{-\sigma-2h}b+\frac{1}{16}e^{2\sigma-2h}b^{\prime}\right]\cos\theta=0,$
(139)
$\displaystyle\left[\frac{1}{2}h^{\prime}+S_{0}\right]\cos\theta+\left[\frac{1}{2R}e^{-h}+\frac{3}{8}me^{-\sigma-2h}b-\frac{1}{16}e^{2\sigma-2h}b^{\prime}\right]\sin\theta=0,$
(140)
$\displaystyle\left[\frac{1}{2}h^{\prime}-S_{0}\right]\sin\theta+\left[\frac{1}{2R}e^{-h}+\frac{3}{8}me^{-\sigma-2h}b+\frac{1}{16}e^{2\sigma-2h}b^{\prime}\right]\cos\theta=0\,.$
(141)

$N_{0}$ and $S_{0}$ are the same as in the previous case and given in (110)
and (113). These equations can be solved and lead to the following BPS
equations

$\displaystyle f^{\prime}$ $\displaystyle=$
$\displaystyle-\frac{1}{2L}\left[e^{-f}\tan 2\theta+4LS_{0}\cos
2\theta+2L(N_{0}+S_{0})\sin 2\theta\tan 2\theta\right],$ (142) $\displaystyle
h^{\prime}$ $\displaystyle=$ $\displaystyle-\frac{1}{2L}e^{-f}\sec
2\theta\left[\sin 2\theta+e^{f}L(N_{0}-S_{0})\cos
4\theta-e^{f}L(N_{0}-5S_{0})\right],\quad$ (143) $\displaystyle b^{\prime}$
$\displaystyle=$
$\displaystyle-\frac{1}{L}e^{-f+2h-2\sigma}\left[4-8Le^{f}(N_{0}-S_{0})\sin
2\theta\right],$ (144) $\displaystyle\sigma^{\prime}$ $\displaystyle=$
$\displaystyle\frac{1}{2L}\left[\left(e^{-f}+2L(N_{0}+S_{0})\sin
2\theta\right)\tan 2\theta+4LN_{0}\cos 2\theta\right],$ (145) $\displaystyle
Y^{\prime}$ $\displaystyle=$ $\displaystyle-\frac{1}{4L}\left[e^{-f}Y\tan
2\theta\left(1+2Le^{f}(N_{0}+S_{0})\sin 2\theta\right)+4LS_{0}Y\cos
2\theta\right],$ (146) $\displaystyle\theta^{\prime}$ $\displaystyle=$
$\displaystyle(S_{0}-N_{0})\sin 2\theta$ (147)

together with two algebraic constraints

$\displaystyle b$ $\displaystyle=$
$\displaystyle-\frac{2}{Lm}e^{\sigma-f+2h}\sec
2\theta\left[1+2e^{f}L(N_{0}+S_{0})\sin 2\theta\right],$ (148) $\displaystyle
R$ $\displaystyle=$ $\displaystyle\frac{e^{f-2h}L\cos
2\theta}{2+2Le^{f}(N_{0}+3S_{0})\sin 2\theta}\,.$ (149)

As in the previous case, for $\phi_{1}=\phi_{2}=0$, we recover the BPS
equations studied in [19] for pure $F(4)$ gauged supergravity.
Finally, the variations $\delta\lambda^{AI}$ give

$\displaystyle(\phi^{\prime}_{1}-M_{1})\cos\theta=0,\qquad(\phi^{\prime}_{1}+M_{1})\sin\theta=0,$
(150)
$\displaystyle(\phi^{\prime}_{2}+M_{3})\cos\theta=0,\qquad(\phi^{\prime}_{2}-M_{3})\sin\theta=0$
(151)

with $M_{1}$ and $M_{3}$ given in (120).

##  Appendix B Consistent truncation of type IIB theory on $S^{2}\times\Sigma$

In this appendix, we collect relevant formulae for uplifting solutions of six-
dimensional $F(4)$ gauged supergravity coupled to one vector multiplet to type
IIB theory. In particular, we mainly consider $SO(2)_{R}$ singlet sector of
the $F(4)$ gauged supergravity with two scalars denoted by $\phi_{0}$ and
$\phi_{2}$ and the two-form field. Although the solutions found in this paper
only involve non-vanishing $\phi_{2}$, we give all the formulae of the
truncation for both $\phi_{0}$ and $\phi_{2}$ non-vanishing. The result can be
useful in other applications such as uplifting the Janus solution found in
[30] by setting the two-form field to zero.
Consistent truncations of type IIB theory on a product of a two-sphere and a
Riemann surface $S^{2}\times\Sigma$ are classified by two holomorphic
functions $f^{a}$, $a=1,2$, written as

$f^{a}=-p^{a}+ik^{a}\,.$ (152)

These holomorphic functions must satisfy the following consistency conditions

$i\partial f^{a}\bar{\partial}\bar{f}_{a}\geq
0\qquad\textrm{and}\qquad\rho\geq 0$ (153)

with $\rho$ defined via

$d\rho=-p_{a}dk^{a}\,.$ (154)

Indices $a,b$ can be raised and lowered by the convention
$p_{a}=p^{b}\epsilon_{ba}$ and $k^{a}=\epsilon^{ab}k_{b}$. To avoid confusion
with the $S^{2}$ inside the six-dimensional metric of the solutions for
surface defects, we will denote the two-sphere in $S^{2}\times\Sigma$ by
$\tilde{S}^{2}$. Coordinates on $\Sigma$ and $\tilde{S}^{2}$ are denoted
respectively by $x^{a}=(x^{1},x^{2})$ and $y^{i}$, $i=1,2,3$, satisfying
$y^{i}y^{j}\delta_{ij}=1$. The latter can be chosen to be

$y^{1}=\sin\tilde{\theta}\cos\tilde{\varphi},\qquad
y^{2}=\sin\tilde{\theta}\sin\tilde{\varphi},\qquad
y^{3}=\cos\tilde{\theta}\,.$ (155)

The complex coordinate $z$ on $\Sigma$ is defined by $z=x^{1}+ix^{2}$.
The four scalars from $SO(4,1)/SO(4)$ can be parametrized by $m_{\Lambda}$,
$\Lambda=0,1,2,3,4$, satisfying

$m_{\Lambda}\eta^{\Lambda\Sigma}m_{\Sigma}=-1\qquad\textrm{and}\qquad
m^{\Lambda}m^{\Sigma}=\delta^{\alpha\beta}{L^{\Lambda}}_{\alpha}{L^{\Sigma}}_{\beta}-\eta^{\Lambda\Sigma}$
(156)

with ${L^{\Lambda}}_{\underline{\Lambda}}$ being the coset representative used
in the main text. We have renamed various indices used in [37] in order to
match with the $F(4)$ gauged supergravity in [15] and [16]. In particular, it
is useful to note the relation

$X=e^{\sigma},\qquad B_{(2)}=-B,\qquad
R=\frac{\sqrt{2}}{m}=\frac{3\sqrt{2}}{g_{1}}$ (157)

with $R$ being the $\tilde{S}^{2}$ radius. Recall that $SO(2)_{R}$ invariant
coset representative is given by

$L=e^{\phi_{0}Y_{01}+\phi_{2}Y_{04}},$ (158)

we find

$m_{\Lambda}=(\sinh\phi_{0},0,0,\cosh\phi_{0}\cosh\phi_{2},\cosh\phi_{0}\sinh\phi_{2}).$
(159)

With all these, we can write down the ten-dimensional metric and other fields
in type IIB theory as follows

$\displaystyle d\hat{s}^{2}$ $\displaystyle=$
$\displaystyle\frac{8}{g_{1}}c_{6}(3\rho)^{\frac{5}{4}}\left(\frac{|dk|^{2}}{\bar{\Delta}}\right)^{\frac{3}{4}}\left[\frac{g_{1}^{2}\bar{\Delta}}{6\rho|dk|^{2}}ds^{2}+\frac{\widetilde{\Delta}e^{-2\sigma}}{\rho^{2}|dk|^{2}}dk^{a}dp_{a}\right.$
(160)
$\displaystyle+e^{2\sigma}\left(d\tilde{\theta}^{2}+\sin^{2}\tilde{\theta}d\tilde{\varphi}^{2}+w^{2}-\frac{1}{\rho^{2}}p_{a}p_{b}n^{a}n^{b}\right)$
$\displaystyle\left.-\frac{3}{\rho}e^{-2\sigma}n_{a}(\sinh\phi_{0}dk^{a}-\cosh\phi_{0}\cosh\phi_{2}\cos\tilde{\theta}dp^{a})\right],$
$\displaystyle H^{ab}$ $\displaystyle=$
$\displaystyle\frac{e^{4\sigma}p^{a}p^{b}|dk|}{\sqrt{3\rho\bar{\Delta}}}+\sqrt{\frac{3\rho}{\bar{\Delta}}}\left[\cosh\phi_{0}\sinh\phi_{2}\partial_{c}k^{a}\partial^{c}p^{b}\right.$
(161)
$\displaystyle\left.+n^{ac}(\cosh\phi_{0}\cosh\phi_{2}\cos\tilde{\theta}\partial_{c}p^{b}-\sinh\phi_{0}\partial_{c}k^{b})\right],$
$\displaystyle C^{a}_{(2)}$ $\displaystyle=$
$\displaystyle-\frac{24c_{6}}{g_{1}^{2}}\textrm{vol}_{\tilde{S}^{2}}(k^{a}+L^{a})-\frac{24c_{6}}{g_{1}^{2}}\frac{|dk|^{2}}{\bar{\Delta}}\cosh\phi_{0}\cosh\phi_{2}\sin^{2}\tilde{\theta}\sigma^{a}\wedge
d\tilde{\varphi}-4c_{6}p^{a}B,$ $\displaystyle F_{(5)}$ $\displaystyle=$
$\displaystyle F_{(2,3)}+F_{(3,2)}+F_{(4,1)}$ (163)

with

$\displaystyle F_{(2,3)}$ $\displaystyle=$
$\displaystyle\frac{288c_{6}^{2}}{g_{1}^{3}}|dk|\left[\frac{1}{3}\frac{g_{1}\rho|dk|}{\bar{\Delta}}e^{4\sigma}p_{a}B\wedge\rho^{a}\wedge\textrm{vol}_{\tilde{S}^{2}}-\frac{|dk|}{\bar{\Delta}}\cosh\phi_{0}\cosh\phi_{2}\sin^{2}\tilde{\theta}\times\right.$
$\displaystyle\left.\left(2g_{1}\rho|dk|\sinh\phi_{0}+\frac{1}{3}g_{1}e^{4\sigma}p_{a}p_{b}n^{ac}\partial_{c}k^{b}\right)B\wedge\textrm{vol}_{\Sigma}\wedge
d\tilde{\varphi}\right],$ $\displaystyle F_{(3,2)}$ $\displaystyle=$
$\displaystyle\frac{288c_{6}^{2}}{g_{1}^{2}}\frac{\rho^{2}|dk|^{2}}{\bar{\Delta}}dB\wedge\left[\cosh\phi_{0}\cosh\phi_{2}\sin^{2}\tilde{\theta}w\wedge
d\tilde{\varphi}\right.$
$\displaystyle\left.-(\cosh^{2}\phi_{0}\sinh^{2}\phi_{2}-\sinh^{2}\phi_{0}-\cosh^{2}\phi_{0}\cosh^{2}\phi_{2}\cos^{2}\tilde{\theta})\textrm{vol}_{\tilde{S}^{2}}\right]$
$\displaystyle-\frac{288c_{6}^{2}|dk|}{g_{1}^{2}}e^{-4\sigma}*_{6}dB\wedge\textrm{vol}_{\Sigma}$
$\displaystyle=$ $\displaystyle*_{10}F_{(3,2)},$ $\displaystyle F_{(4,1)}$
$\displaystyle=$ $\displaystyle*_{10}F_{(2,3)}\,.$ (164)

Various quantities appearing in the above equations are defined by

$\displaystyle\bar{\Delta}$ $\displaystyle=$ $\displaystyle
e^{4\sigma}|dk|p_{a}p_{b}\left[\cosh\phi_{0}\sinh\phi_{2}\partial_{c}k^{a}\partial^{c}p^{b}\right.$
$\displaystyle\left.+n^{ac}(\cosh\phi_{0}\cosh\phi_{2}\cos\tilde{\theta}\partial_{c}p^{b}-\sinh\phi_{0}\partial_{c}k^{b})\right]$
$\displaystyle+3\rho|dk|^{2}(\cosh^{2}\phi_{0}\sinh^{2}\phi_{2}-\sinh^{2}\phi_{0}-\cosh^{2}\phi_{0}\cosh^{2}\phi_{2}\cos^{2}\tilde{\theta}),$
$\displaystyle\widetilde{\Delta}$ $\displaystyle=$ $\displaystyle
3\rho\cosh\phi_{0}\sinh\phi_{2}|dk|^{2}+e^{4\sigma}|dk|p_{a}p_{b}\partial_{c}k^{a}\partial^{c}p^{b},$
$\displaystyle w$ $\displaystyle=$
$\displaystyle\frac{1}{3\rho^{2}}p_{a}\sigma^{a}-\cosh\phi_{0}\cosh\phi_{2}\sin\tilde{\theta}d\tilde{\theta},$
$\displaystyle\sigma^{a}$ $\displaystyle=$ $\displaystyle
3\rho(\cosh\phi_{0}\sinh\phi_{2}n^{a}-\sinh\phi_{0}dp^{a}-\cosh\phi_{0}\cosh\phi_{2}\cos\tilde{\theta}dk^{a})$
$\displaystyle-e^{4\sigma}p^{a}p_{b}*_{2}n^{b},$ $\displaystyle|dk|$
$\displaystyle=$
$\displaystyle\frac{1}{2}\partial_{a}k_{b}\partial^{a}k^{b},\qquad
n^{a}=\frac{1}{2}\mathfrak{g}\partial
f^{a}d\bar{z}+\frac{1}{2}\bar{\mathfrak{g}}\bar{\partial}\bar{f}^{a}dz,$
$\displaystyle L^{a}$ $\displaystyle=$
$\displaystyle\frac{e^{4\sigma}\rho|dk|p_{b}}{\bar{\Delta}}\left[\cosh\phi_{0}\sinh\phi_{2}\partial_{c}k^{b}\partial^{c}p^{a}\right.$
$\displaystyle\left.+n^{bc}(\cosh\phi_{0}\cosh\phi_{2}\cos\tilde{\theta}\partial_{c}k^{a}-\sinh\phi_{0}\partial_{c}k^{a})\right],$
$\displaystyle\rho^{a}$ $\displaystyle=$
$\displaystyle\cosh\phi_{0}\sinh\phi_{2}dk^{a}-\cosh\phi_{0}\cosh\phi_{2}\cos\tilde{\theta}n^{a}-\sinh\phi_{0}*_{2}n^{a}$
(165)

with $*_{2}$ being the Hodge duality on the flat $\Sigma$. $c_{6}$ is a
constant, and $\mathfrak{g}$ is a $U(1)$ phase.
The ten-dimensional dilaton $\Phi$ and axion $C_{0}$ are encoded in the matrix
$H_{ab}$ as

$H_{ab}=\frac{1}{\textrm{Im}\,\tau}\left(\begin{array}[]{cc}|\tau|^{2}&\textrm{Re}\,\tau\\\
\textrm{Re}\,\tau&1\end{array}\right)$ (166)

with $\tau=e^{\Phi}+iC_{0}$.

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