AdS3 vacua realising $\mathfrak{osp}(n|2)$ superconformal symmetry

Niall T<EMAIL_ADDRESS>Anayeli
<EMAIL_ADDRESS>

$a$: Department of Physics, University of Oviedo,
Avda. Federico Garcia Lorca s/n, 33007 Oviedo
and
Instituto Universitario de Ciencias y Tecnologías Espaciales de Asturias
(ICTEA),
Calle de la Independencia 13, 33004 Oviedo, Spain

$b$: Dipartimento di Fisica, Università di Milano–Bicocca,

Piazza della Scienza 3, I-20126 Milano, Italy

and

INFN, sezione di Milano–Bicocca

Abstract

We consider ${\cal N}=(n,0)$ supersymmetric AdS3 vacua of type II supergravity
realising the superconformal algebra $\mathfrak{osp}(n|2)$ for $n>4$. For the
cases $n=6$ and $n=5$, one can realise these algebras on backgrounds that
decompose as foliations of AdS${}_{3}\times\mathbb{CP}^{3}$ ( squashed
$\mathbb{CP}^{3}$ for $n=5$) over an interval. We classify such solutions with
bi-spinor techniques and find the local form of each of them: They only exist
in (massive) IIA and are defined locally in terms of an order 3 polynomial $h$
similar to the AdS7 vacua of (massive) IIA. Many distinct local solutions
exist for different tunings of $h$ that give rise to bounded (or semi
infinite) intervals bounded by physical behaviour. We show that it is possible
to glue these local solutions together by placing D8 branes in the interior of
the interval without breaking supersymmetry, which expands the possibilities
for global solutions immensely. We illustrate this point with some simple
examples. Finally we also show that AdS3 vacua for $n=7,8$ only exist in
$d=11$ supergravity and are all locally AdS${}_{4}\times$S7.

###### Contents

  1. 1 Introduction and summary
  2. 2 SO(2,2)$\times$SO(5) invariant type II supergravity on $\text{AdS}_{3}\times\widehat{\mathbb{CP}}\\!\\!~{}^{3}$
  3. 3 Necessary and sufficient conditions for realising supersymmetry
  4. 4 Classification of $\mathfrak{osp}(n|2)$ AdS3 vacua on $\widehat{\mathbb{CP}}\\!\\!~{}^{3}$ for $n=5,6$
  5. 5 Local analysis of $\mathfrak{osp}(n|2)$ vacua for $n>4$
    1. 5.1  Analysis of $\mathfrak{osp}(6|2)$ vacua
      1. 5.1.1 Local solutions and regularity
    2. 5.2  Analysis of $\mathfrak{osp}(5|2)$ vacua
      1. 5.2.1 Local solutions and regularity
  6. 6 Global solutions with interior D8 branes
    1. 6.1 Some simple examples with internal D8 branes
  7. A Derivation of spinors on $\widehat{\mathbb{CP}}\\!\\!~{}^{3}$
    1. A.1 Killing spinors and vectors on S7= SP(2)/SP(1)
    2. A.2 Reduction to $\mathbb{CP}^{3}$
  8. B The SO(3)${}_{L}\times$SO(3)D invariant ${\cal N}=5$ bi-linears
  9. C Ruling out $\mathfrak{osp}(7|2)$ AdS3 vacua

##  1 Introduction and summary

Warped AdS3 solutions of supergravity in 10 and 11 dimensions, “AdS3 string
vacua”, play an important role in string theory in a wide variety of contexts.
AdS3 appears in the near horizon limit of black-strings solution, so the
embedding of such solutions into higher dimensions enables one to employ
string theory to count the micro states making up the Bekenstein–Hawking
entropy a la Strominger–Vafa [1]. Through the AdS-CFT correspondence they are
dual to the strong coupling limit of CFTs in 2 dimensions. This avatar of the
correspondence promises to be the most fruitful as more powerful techniques
are available to probe CFT2s and there is better understanding of how to
quantise strings on AdS3 than in higher dimensional cases. AdS3 vacua also
commonly appear in duals to compactifications of CFT4 on Riemann surfaces [2,
3, 4, 5, 6, 7, 8, 9], a topic of rekindled interest in recent years with
improved understanding of compactifications on surfaces of non-constant
curvature such as spindles. Some other venues in which AdS3 vacua have played
a prominent role are geometric duals to c-extremisation [10, 11] and dual
descriptions of surface defects in higher dimensional CFTs [12, 13, 14, 15,
16].

Given the above listed wealth of applications, a broad effort towards
classifying supersymmetric AdS3 vacua is clearly well motivated, but at this
time many gaps remain. Generically such AdSd+1 vacua can support the same
superconformal algebras as CFTs in $d$ dimensions. The possible $d=2$
superconformal algebras are far more numerous than their higher dimensional
counterparts, which partially accounts for these gaps. For comparison $d>2$
the possible (simple) superconformal algebras typically333$d=5$ is an
exception with only one possibility, $\mathfrak{f}(4)$ come in series
depending on a parameter which varies as the number of super charges increase;
for instance in $d=3$ one has $\mathfrak{osp}(n|4)$ for CFTs preserving ${\cal
N}=(n,0)$ supersymmetry, where $n=1,...,8$. CFTs in $d=2$ buck this trend,
being consistent with several such series as well as isolated examples such as
$\mathfrak{f}(4)$ and $\mathfrak{g}(3)$ \- see [17] for a classification of
these algebras and [18] for those that can be supported by string vacua. The
focus of this work will be AdS3 vacua supporting the algebra
$\mathfrak{osp}(n|2)$ (the $d=2$ analogue of the $d=3$ algebra).

The ${\cal N}=(n,0)$ superconformal algebra $\mathfrak{osp}(n|2)$ for
arbitrary $n$ was first derived, independently, in [19] and [20] – they are
characterised by an $\mathfrak{so}(n)$ R-symmetry with supercurrents
transforming in the fundamental representation and central charge

$c=\frac{k}{2}\frac{n^{2}+6k-10}{k+n-3}.$ (1.1)

A free field relation was presented in [21] (see also [22]) in terms of a free
scalar, $n$ real fermions, and an SO($n$) current algebra of level $k-1$.
There are in fact many examples of AdS3 vacua realising $\mathfrak{osp}(n|2)$
for $n=1,2$ as these are the unique ways to realise $(1,0)$ and $(2,0)$
superconformal symmetries – see for instance respectively[23, 24, 25, 26, 27,
28, 29, 30, 31, 32] and [33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 11, 43] .
Similarly $n=3$ is unique for ${\cal N}=(3,0)$, examples are more sparse [44,
45, 46], but this is likely not a reflection of their actual rarity. The case
of $n=4$ is in fact a degenerate case of the large ${\cal N}=(4,0)$
superconformal algebra $\mathfrak{d}(2,1,\alpha)$, where the continuous
parameter is tuned to $\alpha=1$ – examples of vacua allowing such a tuning
include [47, 48, 49, 50], there is also a Janus solution preserving
$\mathfrak{osp}(4|2)\oplus\mathfrak{osp}(n|2)$ specifically in [13]. The case
of $n=8$ was addressed in [51] where it was shown that the only solution is
the embedding of AdS3 into AdS${}_{4}\times$S7. The status of AdS3 vacua
realising $\mathfrak{osp}(n|2)$ for $n=5,6,7$ has been up to this time unknown
– a main aim of this work is to fill in this gap, we will now explain the
broad strokes of how we approach this problem.

For the case of $\mathfrak{osp}(7|2)$, with a little group theory [51], it is
not hard to establish that the required $\mathfrak{so}(7)$ R-symmetry can only
be realised geometrically on the co-set SO(7)/G2. The metric on this space is
simply the round 7-sphere, which possesses an SO(8) isometry, but the co-set
also comes equipped with a weak G2 structure with associated 3 and 4-forms
that are invariant under SO(7), but charged under SO(8)/SO(7). If these forms
appear in the fluxes of a solution then only SO(7) is preserved. Our results
prove that all such solutions are locally AdS${}_{4}\times\text{S}^{7}$.

To realise the requisite $\mathfrak{so}(6)$ R-symmetry of
$\mathfrak{osp}(6|2)$ one might naively consider including a 5-sphere in
solutions, however this supports Killing spinors in the 4 of SU(4), which is
not the representation associated to the desired algebra. A space supporting
both the correct isometry and spinors transforming in its fundamental
representation is of course round $\mathbb{CP}^{3}$, as famously exemplified
by the AdS${}_{4}\times\mathbb{CP}^{3}$ vacua of type IIA supergravity dual to
${\cal N}=6$ Chern-Simons matter theory [53]. This is the smallest space with
the desired features, and given that a string vacua has to live in d=10 or 11,
it does not take long to realise that the only additional option is to fiber a
U(1) over $\mathbb{CP}^{3}$. Such solutions were ruled out in type II
supergravity at the level of the equations of motion in [51] and those that
exist in M-theory can always be reduced to IIA444Either the spinors are not
charged under the additional U(1), or some algebra other than
$\mathfrak{osp}(6|2)$ is being realised. As such we seek to classify solutions
of type II supergravity that are foliations of
AdS${}_{3}\times\mathbb{CP}^{3}$ over an interval, we leave the status of d=11
vacua containing similar foliations over Riemann surfaces to be resolved
elsewhere.

For the algebra $\mathfrak{osp}(5|2)$ the 4-sphere is as much of a non-starter
to realise the $\mathfrak{so}(5)$ R-symmetry as the 5-sphere was previously.
One way to realise this algebra is to start with an existing
$\mathfrak{osp}(6|2)$ solution, then orbifold $\mathbb{CP}^{3}$ by one of the
discrete groups $\hat{D}_{k}$ (the binary dihedral group) or $\mathbb{Z}_{2}$
, as discussed in the context of AdS${}_{4}\times\mathbb{CP}^{3}$ in
[57]555see section 3 therein. This however only breaks supersymmetry globally,
locally such solutions still preserve $\mathfrak{osp}(6|2)$. One way, perhaps
the only way666One can realise an $\mathfrak{so}(5)$ R-symmetry on a squashing
of $\text{S}^{3}\rightarrow\text{S}^{7}\rightarrow\text{S}^{4}$, but AdS3
vacua containing this factor only exists in $d=11$ and, when they support
$\mathfrak{osp}(5|2)$, they can always be reduced to IIA within the 7-sphere
resulting in squashed $\mathbb{CP}^{3}$ ($\widehat{\mathbb{CP}}\\!\\!~{}^{3}$)
and preserving ${\cal N}=(5,0)$. to break to $\mathfrak{osp}(5|2)$ locally is
to proceed as follows: If one expresses $\mathbb{CP}^{3}$ as a fibration of S2
over S4 and then pinches the fiber, one breaks the SO(6) isometry down to
SO(5) locally. The 6 of SO(6) branches as $\textbf{5}\oplus\textbf{1}$ under
its SO(5) subgroup thereby furnishing us with both the representation and
R-symmetry that $\mathfrak{osp}(5|2)$ demands. We shall thus also classify
${\cal N}=(5,0)$ AdS3 vacua of type II supergravity on squashed
$\mathbb{CP}^{3}$ by generalising the previous ansatz to included additional
warp factors and SO(5) invariant terms in the flux. We shall in fact classify
both these vacua and those supporting $\mathfrak{osp}(6|2)$, or orbifolds
there of, in tandem as the latter are special cases of the former.

We find two classes of solutions preserving respectively $\mathfrak{osp}(5|2)$
(locally) and $\mathfrak{osp}(6|2)$ superconformal algebras. We also find for
each case that it is possible to construct solutions with bounded internal
spaces, which should provide good dual descriptions of CFTs through the
AdS/CFT correspondence. The existence of backgrounds manifestly realising
exactly the superconformal algebra $\mathfrak{osp}(5|2)$ is interesting in the
light of [52], which claims that all CFTs supporting such global
superconformal algebras experience an enhancement to $\mathfrak{osp}(6|2)$.
Our results cast some doubt on the veracity of the claims of [52], at least
naively. It would be interesting to explore what leads to this apparent
contradiction and whether this can be resolved, that however lies outside the
scope of this work.

The layout of this paper is as follows:

In section 2 we consider AdS3 vacua of type II supergravity that preserve an
SO(5) isometry in terms of squashed $\mathbb{CP}^{3}$, without making
reference to supersymmetry. On symmetry grounds alone we are able to give the
local form that the NS and RR fluxes must take in regular regions of their
internal space, which we found useful when deriving the results in the
subsequent sections.

In section 3 we explain our method for solving the supersymmetry constraints.
We reduce the problem to solving for a single ${\cal N}=1$ sub-sector of the
full $(5,0)$ as the remaining 4 ${\cal N}=1$ sub-sectors are shown to be
implied by this and the action of $\mathfrak{osp}(5|2)$ which the spinors
transform in. This enables us to employ an existing minimally supersymmetric
AdS3 bi-spinor classification [27, 28, 31] to the case at hand.

In section 4 we classify ${\cal N}=(5,0)$ vacua of type II supergravities
realising the algebra $\mathfrak{osp}(5|2)$ in terms of a foliation of AdS3
solutions of type II supergravity that are foliations of
AdS${}_{3}\times\widehat{\mathbb{CP}}\\!\\!~{}^{3}$ over an interval, we are
actually able to find the local form of all of them. They only exist in type
IIA, generically have all possible fluxes turned on and are governed by two
ODEs. The first of these takes the form $h^{\prime\prime\prime}=-2\pi F_{0}$,
where $F_{0}$ is the Romans mass, making $h$ locally an order 3 linear
polynomial highly reminiscent of the AdS7 vacua of [54, 55]. The second ODE
defines a linear function $u$ which essentially controls the squashing of
$\mathbb{CP}^{3}$ and hence the breaking of $\mathfrak{osp}(6|2)$ to
$\mathfrak{osp}(5|2)$ . For generic values of $u$ one has ${\cal N}=(5,0)$
supersymmetry, but if one fixes $u=$ constant this is enhanced to ${\cal
N}=(6,0)$

In section 5 we perform a regularity analysis of the local vacua establishing
exactly what boundary behaviour are possible for the interval. We focus on
${\cal N}=(6,0)$ in section 5.1 where we find that fixing $F_{0}=0$ always
gives rise to AdS${}_{4}\times\mathbb{CP}^{3}$ locally, while for $F_{0}\neq
0$ it is possible to bound the interval at one end with several physical
singularities but that the other end is always at infinite proper distance, at
least when $F_{0}$ is fixed globally. We study the ${\cal N}=(5,0)$ case in
section 5.2 were conversely we find no AdS4 limit and that a globally constant
$F_{0}$ is no barrier to constructing bounded solutions. Many more physical
boundary behaviours are possible in this case.

Up to this point in the paper we have assumed $F_{0}$ is constant, globally it
need only be so piece-wise which allows for D8 branes along the interior of
the interval - we explore this possibility in section 6. We establish under
what conditions such interior D8s are supersymmetric and explain how they can
be used to construct broad classes of globally bonded solutions. We illustrate
the point with some explicit examples. All of this points the way to broad
classes of duals interesting superconformal quiver we shall report on in [63].

The work is supplemented by several appendices. In appendix A we provide
technical details of the construction of spinors on the internal space
transforming in the fundamental representation of SO(5) and SO(6). In appendix
B we present details of the $d=6$ bi-linears that feature during computations
in section 4. Finally in appendix C we additionally show that all
$\mathfrak{osp}(7|2)$ preserving AdS3 vacua experience a local enhancement to
AdS${}_{4}\times$S7 – SO(7) preserving orbifolds of this are a possibility,
but such constructions are AdS4 rather than AdS3 vacua.

##  2 SO(2,2)$\times$SO(5) invariant type II supergravity on
$\text{AdS}_{3}\times\widehat{\mathbb{CP}}\\!\\!~{}^{3}$

In this section we consider the most possible vacua of type II supergravity
that preserve the full SO(2,2)$\times$SO(5) isometries of a warped product
containing $\text{AdS}_{3}$ and a squashed $\mathbb{CP}^{3}$
($\widehat{\mathbb{CP}}\\!\\!~{}^{3}$). Specifically we construct the full set
of SO(5) invariant forms and use them to find the general form of NS and RR
fluxes that are consistent with their source free (magnetic) Bianchi
identities. Let us stress that this section makes no use of supersymmetry only
symmetry, it is none the less useful when we choose to impose the former in
the following sections.

In general AdS3 solutions of type II supergravity admit a decomposition in the
following form

$ds^{2}=e^{2A}ds^{2}(\text{AdS}_{3})+ds^{2}(\text{M}_{7}),~{}~{}~{}H=e^{3A}h_{0}\text{vol}(\text{AdS}_{3})+H_{3},~{}~{}~{}F=f_{\pm}+e^{3A}\text{vol}(\text{AdS}_{3})\star_{7}\lambda(f_{\pm}),$
(2.1)

where $(e^{2A},f,H_{3})$ and the dilaton $\Phi$ have support on M7 so as to
preserve the SO(2,2) symmetry of AdS3. The $d=10$ NS and RR fluxes are $H$ and
$F$ respectively, the latter expressed as a polyform of even/odd degree in
IIA/IIB. The function $\lambda$ acts on a p-form as
$\lambda(X_{p})=(-1)^{[\frac{p}{2}]}X_{p}$ \- this ensures the self duality
constraint $F=\star_{10}\lambda(F)$.

We are interested in solutions where M7 preserve an additional SO(5) isometry
that can be identified with the R-symmetry of the ${\cal N}=(5,0)$
superconformal algebra $\mathfrak{osp}(5|2)$. The 4-sphere comes to mind as
the obvious space realising an SO(5) isometry, however this supports Killing
spinors in the 4 of SP(2), where as we require spinor in the 5 of SO(5) - so
we will need to be more inventive.

The coset space $\mathbb{CP}^{3}$ is a 6 dimensional compact manifold that can
be generated by dimensionally reducing S7 on its Hopf fiber - it appears most
famously in the ${\cal N}=6$ AdS${}_{4}\times\mathbb{CP}^{3}$ solution dual to
Chern-Simons matter theory. The 7-sphere supports spinors transforming in the
8 of SO(8) and the reduction to $\mathbb{CP}^{3}$ preserves the portion of
these preserving the 6 of SO(6). Advantageously $\mathbb{CP}^{3}$ has a
parametrisation as an S2 fibration over S4 that allows a squashing breaking
SO(6)$\to$SO(5) by pinching the fiber - we will refer to this space as
$\widehat{\mathbb{CP}}\\!\\!~{}^{3}$. As the 6 branches as
$\textbf{1}\oplus\textbf{5}$ under SO(5) $\subset$ SO(6) clearly
$\widehat{\mathbb{CP}}\\!\\!~{}^{3}$ supports both the isometry group and
spinors we seek. Embedding this SO(5) invariant space into M7 leads to a
metric ansatz of the form

$\displaystyle ds^{2}(\text{M}_{7})$
$\displaystyle=e^{2k}dr^{2}+ds^{2}(\widehat{\mathbb{CP}}\\!\\!~{}^{3})$ (2.2)
$\displaystyle ds^{2}(\widehat{\mathbb{CP}}\\!\\!~{}^{3})$
$\displaystyle=\frac{1}{4}\bigg{[}e^{2C}\left(d\alpha^{2}+\frac{1}{4}\sin^{2}\alpha(L_{i})^{2}\right)+e^{2D}(Dy_{i})^{2}\bigg{]},~{}~{}~{}~{}~{}Dy_{i}=dy_{i}+\cos^{2}\left(\frac{\alpha}{2}\right)\epsilon_{ijk}y_{j}L_{k},$

where $y_{i}$ are embedding coordinates on the unit radius 2-sphere, $L_{i}$
are a set of SU(2) left invariant forms and
$(e^{2A},e^{2C},e^{2D},e^{2k},\Phi)$ have support on $r$ only.

To write an ansatz for the fluxes on this space we need to construct the SO(5)
invariant forms on $\widehat{\mathbb{CP}}\\!\\!~{}^{3}$. As explained in
appendix A, the S4 base of this fiber bundle contains an
SO(4)$=$SO(3)${}_{L}\times$SO(3)R isometry in the 3-sphere spanned by $L_{i}$.
In the full space SO(3)R is lifted to the diagonal SO(3) formed of SO(3)R and
the SO(3) of the 2-sphere. As such the invariants of SO(5) can be expanded in
a basis of the SO(3)${}_{L}\times$SO(3)D invariants on the S${}^{2}\times$S3
fibration (see for instance [56]), namely

$\displaystyle\omega_{1}$
$\displaystyle=\frac{1}{2}L_{i}y_{i},~{}~{}~{}~{}\omega_{2}^{1}=\frac{1}{2}\epsilon_{ijk}y_{i}Dy_{j}\wedge
Dy_{k},~{}~{}~{}~{}\omega_{2}^{2}=\frac{1}{2}L_{i}\wedge Dy_{i},$
$\displaystyle\omega_{2}^{3}$
$\displaystyle=\frac{1}{2}\epsilon_{ijk}y_{i}L_{j}\wedge
Dy_{k},~{}~{}~{}~{}~{}\omega_{2}^{4}=\frac{1}{8}\epsilon_{ijk}y_{i}L_{j}\wedge
L_{k},$ (2.3)

and wedge products there off, leaving only the $\alpha$ dependence of the
SO(5) invariants to fix via consistency with the remaining
SO(5)/(SO(3)${}_{L}\times$SO(3)D) subgroup.

First off when $e^{2C}=e^{2D}=1$ we regain unit radius round
$\mathbb{CP}^{3}$, which is a Kähler Einstein manifold with an SO(6) invariant
Kähler form $J_{2}$, so we have the following SO(6) invariants on
$\mathbb{CP}^{3}$

$\mathbb{CP}^{3}:~{}e^{2C}=e^{2D}=1~{}~{}~{}\Rightarrow~{}~{}~{}\text{SO(6)
invariants}:~{}~{}J_{2},~{}~{}~{}J_{2}\wedge J_{2},~{}~{}~{}J_{2}\wedge
J_{2}\wedge J_{2}=6\text{vol}(\mathbb{CP}^{3}).$ (2.4)

where specifically

$J_{2}=\frac{1}{4}\bigg{(}\sin\alpha
d\alpha\wedge\omega_{1}-\sin^{2}\alpha\omega_{2}^{4}-\omega_{2}^{1}\bigg{)}.$
(2.5)

It is not hard to show that the remaining SO(5) invariants, which are not
invariant under the full SO(6) of $\mathbb{CP}^{3}$, may be expressed in terms
of the SU(3)-structure spanned by

$\tilde{J}_{2}=\frac{1}{4}\bigg{(}\sin\alpha
d\alpha\wedge\omega_{1}-\sin^{2}\alpha\omega_{2}^{4}+\omega_{2}^{1}\bigg{)},~{}~{}~{}~{}\Omega_{3}=-\frac{1}{8}\sin\alpha\bigg{(}\sin\alpha\omega_{1}+id\alpha\bigg{)}\wedge\bigg{(}\omega^{3}_{2}+i\omega_{2}^{2}\bigg{)},$
(2.6)

These invariant forms obey the following identities

$\displaystyle
J_{2}\wedge\Omega_{3}=\tilde{J}_{2}\wedge\Omega_{3}=0,~{}~{}~{}~{}J_{2}\wedge
J_{2}\wedge
J_{2}=\tilde{J}_{2}\wedge\tilde{J}_{2}\wedge\tilde{J}_{2}=\frac{3i}{4}\Omega_{3}\wedge\overline{\Omega}_{3},~{}~{}~{}~{}J_{2}\wedge
J_{2}+\tilde{J}_{2}\wedge\tilde{J}_{2}=2\tilde{J}_{2}\wedge J_{2},$
$\displaystyle
dJ_{2}=0,~{}~{}~{}~{}d\tilde{J}_{2}=4\text{Re}\Omega_{3},~{}~{}~{}~{}d\text{Im}\Omega_{3}=6\tilde{J}_{2}\wedge
J_{2}-2J_{2}\wedge J_{2},$ (2.7)

and as such, they form a closed set under the exterior product and derivative.
This is all that is needed to construct the fluxes.

The general form of an SO(5) invariant $H_{3}$ obeying $dH_{3}=0$ is given by

$H_{3}=dB_{2},~{}~{}~{}~{}B_{2}=b(r)J_{2}+\tilde{b}(r)\tilde{J}_{2},$ (2.8)

The general SO(5) invariant $f_{\pm}$ obeying $df_{\pm}=H_{3}\wedge f_{\pm}$
can be expressed as

$\displaystyle f_{+}$ $\displaystyle=\bigg{[}F_{0}+c_{1}J_{2}+c_{2}J_{2}\wedge
J_{2}+c_{3}\frac{1}{3!}J_{2}\wedge J_{2}\wedge
J_{2}+d\bigg{(}p(r)\text{Im}\Omega_{3}+q(r)\text{Re}\Omega_{3}\bigg{)}\bigg{]}\wedge
e^{B_{2}},$ $\displaystyle f_{-}$
$\displaystyle=d\bigg{[}a_{1}(r)+a_{2}(r)J_{2}+a_{3}(r)\frac{1}{2}J_{2}\wedge
J_{2}+a_{4}(r)J_{2}\wedge\tilde{J}_{2}+a_{5}(r)\frac{1}{3!}J_{2}\wedge
J_{2}\wedge J_{2}\bigg{]}\wedge e^{B_{2}},$ (2.9)

giving us an SO(5) invariant ansatz for the flux in IIA/IIB which is valid
away from the loci of localised sources777These need to be generalised in
scenarios which allow for sources smeared over all their co-dimensions. In IIA
this depends locally on 4 constants $(F_{0},c_{1},c_{2},c_{3})$ and 4
functions of $r$ $(b,\tilde{b},p,q)$ \- there is an enhancement to SO(6) when
$\tilde{b}=p=q=0$. If we also consider
$d(e^{3A}\star_{7}f_{\pm})=e^{3A}H_{3}\wedge\star_{7}f_{\pm}$ we find we must
in general fix $q=0$. In IIB this depends on 7 functions of $r$, with an
enhancement to SO(6) when $\tilde{b}=a_{4}=0$.

##  3 Necessary and sufficient conditions for realising supersymmetry

In this section we present the method by which we shall impose supersymmetry
on SO(5) invariant ansatz of the previous section.

Geometric conditions for ${\cal N}=(1,0)$ AdS3 solutions with purely magnetic
NS flux (ie $h_{0}=0$) were derived first in massive IIA in [27], then
generalised to IIB in [28] with the assumption that $h_{0}=0$, this assumption
was then relaxed in [31] whose conventions we shall follow. These conditions
are defined in terms of two non vanishing Majorana spinors
$(\hat{\chi}_{1},\hat{\chi}_{2})$ on the internal M7 which without loss of
generality obey

$|\hat{\chi}_{1}|^{2}+|\hat{\chi}_{2}|^{2}=2e^{A},~{}~{}~{}~{}|\hat{\chi}_{1}|^{2}-|\hat{\chi}_{2}|^{2}=ce^{-A},$
(3.1)

for $c$ an arbitrary constant. One can solve these constraints in general in
terms of two unit norm spinors $(\chi_{1},\chi_{2})$ and a point dependent
angle $\theta$ as

$\hat{\chi}_{1}=e^{\frac{A}{2}}\sqrt{1-\sin\theta}\chi_{1},~{}~{}~{}~{}\hat{\chi}_{2}=e^{\frac{A}{2}}\sqrt{1+\sin\theta}\chi_{2},~{}~{}~{}c=-2e^{2A}\sin\theta.$
(3.2)

Plugging this into the necessary and sufficient conditions for supersymmetry
in [31] (see Appendix B therein), we find they become888These do not represent
a set of necessary and sufficient conditions when $\cos\theta=0$. However as
this limit turns off one of $(\hat{\chi}_{1},~{}\hat{\chi}_{2})$ the NS 3-form
is the only flux that can be non trivial. The common NS sector of type II
supergravity is S-dual to classes of IIB solution with the RR 3-form the only
non trivial flux which are contained in the conditions we quote.

$\displaystyle
e^{3A}h_{0}=2me^{2A}\sin\theta,~{}~{}~{}~{}d(e^{2A}\sin\theta)=0,$ (3.3a)
$\displaystyle
d_{H_{3}}(e^{2A-\Phi}\cos\theta\Psi_{\mp})=\pm\frac{1}{8}e^{2A}\sin\theta
f_{\pm},$ (3.3b) $\displaystyle d_{H_{3}}(e^{3A-\Phi}\cos\theta\Psi_{\pm})\mp
2me^{2A-\Phi}\cos\theta\Psi_{\mp}=\frac{e^{3A}}{8}\star_{7}\lambda(f_{\pm}),$
(3.3c) $\displaystyle
e^{A}(\Psi_{\mp},~{}f_{\pm})_{7}=\mp\frac{m}{2}e^{-\Phi}\cos\theta\text{vol}(\text{M}_{7}),$
(3.3d)

where $(\Psi_{\mp},~{}f_{\pm})_{7}$ is the 7-form part of
$\Psi_{\mp}\wedge\lambda(f_{\pm})$ and the real even/odd bi-linears
$\Psi_{\pm}$ are defined via

$\chi_{1}\otimes\chi_{2}^{{\dagger}}=\frac{1}{8}\sum_{n=0}^{7}\frac{1}{n!}\chi_{2}^{{\dagger}}\gamma_{a_{n}...a_{1}}\chi_{1}e^{a_{1}...a_{n}}=\Psi_{+}+i\Psi_{-}$
(3.4)

for $e^{a}$ a vielbein on M7. In the above $m$ is the inverse AdS3 radius, in
particular when $m=0$ we have Mink3 while when $m\neq 0$ its precise value is
immaterial as it can be absorbed into the AdS3 warp factor, thus going forward
we fix

$m=1$ (3.5)

without loss of generality.

In this work we will construct explicit solutions preserving $(5,0)$ and
$(6,0)$ supersymmetries and for the cases of extended supersymmetry
(3.3a)-(3.3d) is not on its own sufficient. If one has ${\cal N}=(n,0)$
supersymmetry one has $n$ independent ${\cal N}=(1,0)$ sub-sectors that
necessarily come with their corresponding $n$ independent bi-linears
$\Psi^{(n)}_{\pm}$. These must all solve (3.3a)-(3.3d) for the same bosonic
fields of supergravity. However the AdS3 vacua we are interested in realise
the superconformal algebra $\mathfrak{osp}(n|2)$ which means the internal
spinors which define these bi-linears transform in the n of $\mathfrak{so}(n)$
while the bosonic fields are $\mathfrak{so}(n)$ singlets. Thus the bi-linears
decompose into parts transforming in irreducible representations of the tensor
product $\textbf{n}\otimes\textbf{n}$. Specifically this contains a singlet
part that is common to all $\Psi^{(n)}_{\pm}$ and a charged part in the
symmetric representation999Of course $\textbf{n}\otimes\textbf{n}$ decomposes
into singlet, symmetric traceless and anti-symmetric representations, however
to see the anti-symmetric representation one would need to construct bi-
linears that mix the internal spinors $\chi_{1}$ and $\chi_{2}$ that belong to
different $(1,0)$ sub-sectors - this is not needed for our purposes.. The
charged parts of $\Psi^{(n)}_{\pm}$ are mapped into each other by taking the
Lie derivative with respect to the SO(n) Killing vectors, and in particular
the bi-linears of a single $(1,0)$ sub-sector + the action of SO(n) is enough
to generate the whole set. Then, since the Lie and exterior derivatives
commute, it follows that if a single pair of bi-linears, $\Psi^{1}_{\pm}$ say,
solve (3.3a)-(3.3d) then they all do.

In summary to know that extended supersymmetry holds on (2.2) it is sufficient
to construct an n-tuplet of spinors that transform in the n of
$\mathfrak{so}(n)$, and then solve (3.3a)-(3.3d) for the $\Psi_{\pm}$
following from any ${\cal N}=(1,0)$ sub-sector whilst imposing that the
bosonic fields are all $\mathfrak{so}(n)$ singlets. In particular this means
that we must solve (3.3a)-(3.3d) under the assumption that the warp factors
and dilaton only depend on $r$ and the fluxes only depend on $r$ and SO(n)
invariant forms. We deal with the bulk of the construction of these SO(n)
spinors in appendix A where we construct spinors in relevant representations
on $\mathbb{CP}^{3}$. Below we present the embedding of these spinors into
(2.2).

${\cal N}=6$ spinors in $d=7$ can be expressed in terms of 4 real functions of
$r$ $(f_{1},f_{2},g_{1},g_{2})$ and the spinors in the 6 of SO(6) on
$\mathbb{CP}^{3}$ in (A.31)

$\displaystyle\chi^{{\cal I}}_{1}$
$\displaystyle=\cos\left(\frac{\beta_{1}+\beta_{2}}{2}\right)\xi^{{\cal
I}}_{6}+i\sin\left(\frac{\beta_{1}+\beta_{2}}{2}\right)\gamma_{7}\xi^{{\cal
I}}_{6},$ $\displaystyle\chi^{{\cal I}}_{2}$
$\displaystyle=\cos\left(\frac{\beta_{1}-\beta_{2}}{2}\right)\xi^{{\cal
I}}_{6}+i\sin\left(\frac{\beta_{1}-\beta_{2}}{2}\right)\gamma_{7}\xi^{{\cal
I}}_{6},$ (3.6)

where ${\cal I}=1,...,6$ and $\beta_{1,2}=\beta_{1,2}(r)$. These are only
valid on round $\mathbb{CP}^{3}$, ie when $e^{2B}=e^{2C}$ and the fluxes
depend on $\mathbb{CP}^{3}$ through the SO(6) invariant 2-form $J_{2}$. We
will not actually make explicit use of these spinors as it turns out that
general class of ${\cal N}=(6,0)$ is actually simply one of 2 branching
classes of solution following from the ${\cal N}=(5,0)$ spinors below.

${\cal N}=5$ spinors in $d=7$ can be decomposed in terms the spinors in the 5
of SO(5) on $\mathbb{CP}^{3}$ in (A.29) and 4 constraints as

$\displaystyle\chi^{\alpha}_{1}$
$\displaystyle=a_{11}(\xi_{5}^{\alpha}+Y_{\alpha}i\gamma_{7}\xi_{0})+b_{11}(i\gamma_{7}\xi_{5}^{\alpha}-Y_{\alpha}\xi_{0})+a_{12}Y_{\alpha}\xi_{0}+ib_{12}Y_{\alpha}\gamma_{7}\xi_{0},$
$\displaystyle\chi^{\alpha}_{2}$
$\displaystyle=a_{21}(\xi_{5}^{\alpha}+Y_{\alpha}i\gamma_{7}\xi_{0})+b_{21}(i\gamma_{7}\xi_{5}^{\alpha}-Y_{\alpha}\xi_{0})+a_{22}Y_{\alpha}\xi_{0}+ib_{22}Y_{\alpha}\gamma_{7}\xi_{0},$
$\displaystyle
a_{11}^{2}+b_{11}^{2}=a_{12}^{2}+b_{12}^{2}=a_{21}^{2}+b_{21}^{2}=a_{22}^{2}+b_{22}^{2}=1.$
(3.7)

where $\alpha=1,...,5$, the 8 parameters $a_{11},b_{11},...$ are all real and
have support on $r$ alone, we have parameterised things in this fashion to
make the unit norm constraints simple. These spinors are valid for squashed
$\mathbb{CP}^{3}$.

Finally a set of ${\cal N}=1$ spinors can also be defined in $d=7$, they are
given by

$\displaystyle\chi^{(0)}_{1}$
$\displaystyle=\cos\left(\frac{\beta_{1}+\beta_{2}}{2}\right)\xi_{0}+i\sin\left(\frac{\beta_{1}+\beta_{2}}{2}\right)\gamma_{7}\xi_{0},$
$\displaystyle\chi^{(0)}_{2}$
$\displaystyle=\cos\left(\frac{\beta_{1}-\beta_{2}}{2}\right)\xi_{0}+i\sin\left(\frac{\beta_{1}-\beta_{2}}{2}\right)\gamma_{7}\xi_{0},$
(3.8)

where again $\beta_{1,2}=\beta_{1,2}(r)$ and these spinors are valid on
squashed $\mathbb{CP}^{3}$. The $0$ superscript refers to the the fact that
these are SO(5) $\subset$ SO(6) singlets. These are in fact nothing more than
the 6th component of (3.6), however unlike the ${\cal N}=(6,0)$ case
$e^{2C}\neq e^{2B}$ and the flux can depend on more than merely $r$ and
$J_{2}$. These spinors can be used to construct ${\cal N}=(1,0)$ AdS3
solutions with SO(5) flavour symmetry, something we will report on elsewhere
[64].

##  4 Classification of $\mathfrak{osp}(n|2)$ AdS3 vacua on
$\widehat{\mathbb{CP}}\\!\\!~{}^{3}$ for $n=5,6$

In this section we classify AdS3 solutions preserving ${\cal N}=(5,0)$
supersymmetry on squashed $\mathbb{CP}^{3}$. Such solutions only exist in type
IIA supergravity and experience an enhancement to ${\cal N}=(6,0)$ when a
function is appropriately fixed. We summarise our results between (4.55) and
(4.58).

We take our representative ${\cal N}=1$ sub-sector to be

$\chi_{1}=\chi^{5}_{1},~{}~{}~{}~{}\chi_{2}=\chi^{5}_{2},$ (4.1)

which has the advantage that the bi-linears decompose in terms of the
SO(3)${}_{L}\times$SO(3)D invariant forms on the S${}^{2}\times$S3 fibration.
We find the $d=7$ bi-linears are given by

$\Psi_{+}=(\underline{{\cal
S}}^{1})_{+}.(\underline{\phi})_{+}+e^{k}(\underline{{\cal
S}}^{2})_{-}.(\underline{\phi})_{-}\wedge
dr,~{}~{}~{}~{}\Psi_{-}=(\underline{{\cal
S}}^{1})_{-}.(\underline{\phi})_{-}+e^{k}(\underline{{\cal
S}}^{2})_{+}.(\underline{\phi})_{+}\wedge dr$ (4.2)

where we define

$(\underline{\phi})_{\pm}=(\phi^{1}_{\pm},~{}\phi^{2}_{\pm},~{}Y_{5}\phi^{3}_{\pm},~{}Y_{5}\phi^{4}_{\pm},~{}Y_{5}\phi^{5}_{\pm},~{}Y_{5}\phi^{6}_{\pm},~{}Y_{5}^{2}\phi^{7}_{\pm},~{}Y_{5}^{2}\phi^{8}_{\pm}),~{}~{}~{}~{}~{}Y_{5}=\cos\alpha,$
(4.3)

for $\phi^{1}_{\pm}$ real even/odd bi-linears on
$\widehat{\mathbb{CP}}\\!\\!~{}^{3}$ decomposing in a basis of (2.3) - their
explicit form is given in (B.3). We also define

$\displaystyle(\underline{{\cal S}}^{1})_{+}$
$\displaystyle=\left(\begin{array}[]{c}a_{11}a_{21}+b_{11}b_{21}\\\
a_{11}b_{21}-a_{21}b_{11}\\\ a_{11}a_{22}+b_{11}b_{22}\\\
a_{11}b_{22}-a_{22}b_{11}\\\ a_{12}a_{21}+b_{12}b_{21}\\\
a_{21}b_{12}-a_{12}b_{21}\\\ a_{12}a_{22}+b_{12}b_{22}\\\
a_{22}b_{12}-a_{12}b_{22}\end{array}\right),~{}~{}~{}~{}~{}(\underline{{\cal
S}}^{2})_{-}=\left(\begin{array}[]{c}a_{21}b_{11}+a_{11}b_{21}\\\
b_{11}b_{22}-a_{11}a_{21}\\\ a_{11}b_{22}+a_{22}b_{11}\\\
b_{11}b_{22}-a_{11}a_{22}\\\ a_{21}b_{12}+a_{12}b_{21}\\\
b_{12}b_{21}-a_{12}a_{21}\\\ a_{22}b_{12}+a_{12}b_{22}\\\
b_{12}b_{22}-a_{12}a_{22}\end{array}\right),$ (4.20)
$\displaystyle(\underline{{\cal S}}^{1})_{-}$
$\displaystyle=\left(\begin{array}[]{c}a_{21}b_{11}-a_{11}b_{21}\\\
a_{11}a_{21}+b_{11}b_{21}\\\ a_{22}b_{11}-a_{11}b_{22}\\\
a_{11}a_{22}+b_{11}b_{22}\\\ a_{21}b_{12}-a_{12}b_{21}\\\
-a_{12}a_{21}-b_{12}b_{21}\\\ a_{22}b_{12}-a_{12}b_{22}\\\
-a_{12}a_{22}-b_{12}b_{22}\end{array}\right),~{}~{}~{}~{}(\underline{{\cal
S}}^{2})_{+}=\left(\begin{array}[]{c}a_{11}a_{21}-b_{11}b_{21}\\\
a_{21}b_{11}+a_{11}b_{21}\\\ a_{11}a_{22}-b_{11}b_{22}\\\
a_{22}b_{11}+a_{11}b_{22}\\\ a_{12}a_{21}-b_{12}b_{21}\\\
a_{21}b_{12}+a_{12}b_{21}\\\ a_{12}a_{22}-b_{12}b_{22}\\\
a_{22}b_{12}+a_{12}b_{22}\end{array}\right).$ (4.37)

We begin by solving the constraints in (3.7) by parametrising the functions of
the spinor ansatz as

$\displaystyle a_{11}+ib_{11}$
$\displaystyle=e^{\frac{i}{2}(X_{1}+X_{3})},~{}~{}~{}~{}a_{12}+ib_{12}=e^{\frac{i}{2}(X_{2}+X_{4})},$
$\displaystyle a_{21}+ib_{21}$
$\displaystyle=e^{\frac{i}{2}(X_{1}-X_{3})},~{}~{}~{}~{}a_{22}+ib_{22}=e^{\frac{i}{2}(X_{2}-X_{4})},$
(4.38)

for $X_{1}...X_{4}$ functions of $r$ only. We shall take the magnetic
component of the NS 3-form as in (2.8) and allow the RR fluxes to depend on
$r$ and all the SO(5) invariant forms, ie

$dr,~{}~{}~{}J_{2},~{}~{}~{}\tilde{J}_{2},~{}~{}~{}\text{Re}\Omega_{3},~{}~{}~{}\text{Im}\Omega_{3},$
(4.39)

and the wedge products one can form out of these. One then proceed to
substitute (4.2) into the necessary conditions for supersymmetry (3.3a)-(3.3d)
to fix the $r$ dependence of the ansatz.

In IIB supergravity there are no solutions: One arrives at a set of algebraic
constraints by solving for the parts of (3.3b)-(3.3c) orthogonal to $dr$ which
without loss of generality fix the phases as

$X_{1}=-\frac{\pi}{2}+2\beta(r),~{}~{}~{}~{}X_{2}=\frac{\pi}{2},~{}~{}~{}~{}\theta=X_{3}=X_{4}=0.$
(4.40)

and several parts of the metric and NS 2-form as

$e^{C}=5e^{A}\sin\beta,~{}~{}~{}~{}e^{D}=5e^{A}\sin\beta\cos\beta,~{}~{}~{}~{}\tilde{b}=0.$
(4.41)

Unfortunately if one then tries to solve the $dr$ dependent terms in (3.3b)
one finds the constraint

$\cos\beta=0,$ (4.42)

which cannot be solved without setting $e^{D}=0$, so no ${\cal N}=(5,0)$ or
$(6,0)$ solutions exist on this space in type IIB.

Moving onto type IIA supergravity: Some conditions one may extract from
(3.3b), which simplify matters considerably going forward, are the following

$\sin\theta=0,~{}~{}~{}~{}\sin X_{1}=-\sin X_{2}=1,$ (4.43)

which we can solve without loss of generality as

$\theta=0~{}~{}~{}~{}\Rightarrow~{}~{}~{}~{}h_{0}=0,~{}~{}~{}~{}X_{1}=-X_{2}=\frac{\pi}{2}.$
(4.44)

We then choose to further refine the phases as

$X_{3}=\beta_{1}+\beta_{2},~{}~{}~{}~{}X_{4}=-\beta_{1}+\beta_{2}.$ (4.45)

Plugging these into (3.3b)-(3.3d) we find the following simple definitions of
various functions in the ansatz

$\displaystyle e^{C}$
$\displaystyle=2e^{A}\sin\beta_{2},~{}~{}~{}~{}e^{D}=2e^{A}\sin(\beta_{1}+\beta_{2}),$
$\displaystyle b^{\prime}$
$\displaystyle=4e^{A+k}+2\partial_{r}(e^{2A}\cos\beta_{1}\sin(\beta_{1}+2\beta_{2})),~{}~{}~{}~{}\tilde{b}=-2e^{2A}\cos(\beta_{1}+2\beta_{2})\sin\beta_{1},$
(4.46)

and the following ODEs that need to be actively solved

$\displaystyle\partial_{r}(e^{3A-\Phi}\sin\beta_{1}\sin\beta_{2})+2me^{2A+k-\Phi}\sin\beta_{1}\cos\beta_{2}=0,$
$\displaystyle\partial_{r}(e^{5A-\Phi}\sin^{2}\beta_{2}\sin(\beta_{1}+\beta_{2}))+me^{4A+k-\Phi}\sin\beta_{2}\sin(\beta_{1}+2\beta_{2})=0,$
$\displaystyle\partial_{r}(e^{2A}\tan\beta_{2})+me^{A+k}(2\tan\beta_{2}\cot(\beta_{1}+\beta_{2})-(\cos\beta_{2})^{-2})=0.$
(4.47)

We also extract expressions for the RR fluxes, though we delay presenting them
explicitly until we have simplified the above. To make progress we find it
useful to use diffeomorphism invariance in $r$ to fix101010The reason for the
factors of $\pi$, taken here and elsewhere without loss of generality, is that
they make the Page charges of the RR fluxes simple.

$e^{A+k}=-\pi,$ (4.48)

and introduce local functions of $r$ $(h,g,u)$ such that

$e^{5A-\Phi}\sin^{2}\beta_{2}\sin(\beta_{1}+\beta_{2})=\pi^{2}h,~{}~{}~{}~{}e^{2A}\tan\beta_{2}=g,~{}~{}~{}~{}\frac{\tan\beta_{2}}{\tan(\beta_{1}+\beta_{2})}=\frac{h^{\prime}+h\frac{u^{\prime}}{u}}{h^{\prime}-h\frac{u^{\prime}}{u}}.$
(4.49)

This simplifies the system of ODEs in (4.47) to

$\displaystyle u^{\prime\prime}$
$\displaystyle=0,~{}~{}~{}~{}g=2\pi\frac{hu}{uh^{\prime}-hu^{\prime}},~{}~{}~{}~{}\tan\beta_{1}=\frac{\text{sign}(uh)u^{\prime}\sqrt{\Delta_{1}}}{u^{\prime}(uh^{\prime}-hu^{\prime})+u^{2}h^{\prime\prime}},~{}~{}~{}~{}\tan\beta_{2}=\frac{\text{sign}(uh)\sqrt{\Delta_{1}}}{uh^{\prime}-hu^{\prime}}$
$\displaystyle\Delta_{1}$
$\displaystyle=2hh^{\prime\prime}u^{2}-(uh^{\prime}-hu^{\prime})^{2}.$ (4.50)

which imply supersymmetry and require $\Delta_{1}>0$. What remains is the
explicit form of the magnetic components of the RR fluxes. These can be
expressed most succinctly in terms of their Page flux avatars, ie
$\hat{f}_{+}=f_{+}\wedge e^{-B_{2}}$, however to compute these we must first
integrate $b^{\prime}$. Combining (4.49), (4.50) and (4.46) we find

$b=-\tilde{b}+4\pi\left(-(r-k)+\frac{uh^{\prime}-hu^{\prime}}{uh^{\prime\prime}}\right),~{}~{}~{}~{}~{}\tilde{b}=-2\pi\frac{u^{\prime}}{h^{\prime\prime}}\left(\frac{h}{u}+\frac{hh^{\prime\prime}-2(h^{\prime})^{2}}{2h^{\prime}u^{\prime}+uh^{\prime\prime}}\right),$
(4.51)

where $k$ is an integration constant. We then find for the magnetic Page
fluxes

$\displaystyle\hat{f}_{0}$
$\displaystyle=F_{0}=-\frac{1}{2\pi}h^{\prime\prime\prime},$
$\displaystyle\hat{f}_{2}$
$\displaystyle=2(h^{\prime\prime}-(r-k)h^{\prime\prime\prime})J,$
$\displaystyle\hat{f}_{4}$
$\displaystyle=-4\pi\bigg{[}(2h^{\prime}+(r-k)(-2h^{\prime\prime}+(r-k)h^{\prime\prime\prime}))J_{2}\wedge
J_{2}+d\left(\frac{hu^{\prime}}{u}\text{Im}\Omega_{3}\right)\bigg{]},$
$\displaystyle\hat{f}_{6}$
$\displaystyle=\frac{16\pi^{2}}{3}(6h-(r-k)(6h^{\prime}+(r-k)(-3h^{\prime\prime}+(r-k)h^{\prime\prime\prime})))J_{2}\wedge
J_{2}\wedge J_{2},$ (4.52)

where we have made extensive use of the conditions derived earlier to simply
these expressions. In order to have a solution we must impose that Bianchi
identities of the RR flux hold (that of the NS 3-form is implied), away from
sources this is equivalent to imposing that $d\hat{f}_{2n}=0$ for $n=0,1,2,3$,
we find

$d\hat{f}_{2n}=-\frac{1}{2\pi}(4\pi)^{n}\frac{1}{n!}(r-k)^{n}h^{\prime\prime\prime\prime}dr\wedge
J_{2}^{n},$ (4.53)

which tells us that the Bianchi identities in regular parts of the internal
space demand

$h^{\prime\prime\prime\prime}=0,$ (4.54)

or in other words that $h$ is an order 3 polynomial (at least locally). This
completes our local derivation of the class of solutions.

In summary the local form of solutions in this class take the following form:
NS sector

$\displaystyle\frac{ds^{2}}{2\pi}$
$\displaystyle=\frac{|hu|}{\sqrt{\Delta_{1}}}ds^{2}(\text{AdS}_{3})+\frac{\sqrt{\Delta_{1}}}{4|u|}\bigg{[}\frac{2}{|h^{\prime\prime}|}\bigg{(}ds^{2}(\text{S}^{4})+\frac{1}{\Delta_{2}}(Dy_{i})^{2}\bigg{)}+\frac{1}{|h|}dr^{2}\bigg{]},$
$\displaystyle e^{-\Phi}$
$\displaystyle=\frac{\sqrt{|u|}|h^{\prime\prime}|^{\frac{3}{2}}\sqrt{\Delta_{2}}}{2\sqrt{\pi}\Delta_{1}^{\frac{1}{4}}},~{}~{}~{}~{}\Delta_{1}=2hh^{\prime\prime}u^{2}-(uh^{\prime}-hu^{\prime})^{2},~{}~{}~{}~{}\Delta_{2}=1+\frac{2h^{\prime}u^{\prime}}{uh^{\prime\prime}},~{}~{}~{}~{}H=dB_{2},$
$\displaystyle B_{2}$
$\displaystyle=4\pi\bigg{[}\left(-(r-k)+\frac{uh^{\prime}-hu^{\prime}}{uh^{\prime\prime}}\right)J_{2}+\frac{u^{\prime}}{2h^{\prime\prime}}\left(\frac{h}{u}+\frac{hh^{\prime\prime}-2(h^{\prime})^{2}}{2h^{\prime}u^{\prime}+uh^{\prime\prime}}\right)\left(J_{2}-\tilde{J}_{2}\right)\bigg{]},$
(4.55)

where $(u,h)$ are functions of $r$ and $k$ is a constant. Note that positivity
of the metric and dilaton holds whenever $\Delta_{1}\geq 0$111111Specifically
reality of the metric demands $\Delta_{1}\geq 0$ and that $(h,u)$ are real.
This in turn implies $hh^{\prime\prime}\geq 0$ and it then follows that
$\Delta_{2}\geq 0$. Note that one needs to use that $hh^{\prime\prime}\geq 0$
to bring the metric to this form.. The $d=10$ RR fluxes are given by

$\displaystyle F_{0}$
$\displaystyle=-\frac{1}{2\pi}h^{\prime\prime\prime},~{}~{}~{}~{}~{}F_{2}=B_{2}F_{0}+2(h^{\prime\prime}-(r-k)h^{\prime\prime\prime})J_{2},$
$\displaystyle F_{4}$ $\displaystyle=\pi
d\left(h^{\prime}+\frac{hh^{\prime\prime}u(uh^{\prime}+hu^{\prime})}{\Delta_{1}}\right)\wedge\text{vol}(\text{AdS}_{3})+B_{2}\wedge
F_{2}-\frac{1}{2}B_{2}\wedge B_{2}F_{0}$
$\displaystyle-4\pi\bigg{[}(2h^{\prime}+(r-k)(-2h^{\prime\prime}+(r-k)h^{\prime\prime\prime}))J_{2}\wedge
J_{2}+d\left(\frac{hu^{\prime}}{u}\text{Im}\Omega_{3}\right)\bigg{]}.$ (4.56)

Solutions within this class are defined locally by 2 ODEs: First supersymmetry
demands

$u^{\prime\prime}=0,$ (4.57)

which must hold globally. Second the Bianchi identities of the fluxes demand
that in regular regions of the internal space

$h^{\prime\prime\prime\prime}=0,$ (4.58)

which one can integrate as

$h=c_{0}+c_{1}r+\frac{1}{2}c_{2}r^{2}+\frac{1}{3!}c_{3}r^{3},$ (4.59)

where $c_{i}$ are integration constants here and elsewhere. However the RHS of
(4.58) can contain $\delta$-function sources globally, as we shall explore in
section 6. Before moving on to analyse solutions within this class it is
important to stress a few things. First off that (4.57) must hold globally,
and given how $u,u^{\prime}$ appear in the class means that really $u$ is
parametrising two branching possibilities - either $u^{\prime}=0$ or
$u^{\prime}\neq 0$.

For the first case notice that if $u^{\prime}=0$ then $u$ actually completely
drops out of the bosonic fields, so its precise value doesn’t matter. Further
the warping of the 4-sphere and fibered 2-sphere becomes equal, making the
metric on $\mathbb{CP}^{3}$ the round one, and only $J_{2}$ now appears in the
fluxes. There is thus an enhancement of the global symmetry of the internal
space to SO(6) - indeed supersymmetry is likewise enhanced to ${\cal
N}=(6,0)$. We shall study this limit in section 5.1.

When $u^{\prime}\neq 0$ then $u$ is an order 1 polynomial, however the class
is invariant under $(r\to r+l,~{}k\to k+l)$ which one can use to set the
constant term in $u$ to zero without loss of generality, the specific value of
the constant $u^{\prime}$ then also drops out of the bosonic fields. Thus for
the second class, preserving only ${\cal N}=(5,0)$ supersymmetry, one can fix
$u=r$ without loss of generality. We shall study this limit in section 5.2.

Having a classes of solutions defined in terms of the ODE
$h^{\prime\prime\prime}=-2\pi F_{0}$ is very reminiscent of AdS7 vacua in
massive IIA, which obey essentially the same constraint [55]. For the $(6,0)$
case the formal similarities become more striking as both this and the AdS7
vacua are of the form AdS${}_{2p+1}\times\mathbb{CP}^{4-p}$ foliated over an
interval in terms of an order 3 polynomial and it’s derivatives - however we
should stress these functions do not appear in the same way in each case. None
the less this apparent series of local solutions does beg the question, what
about AdS${}_{5}\times\mathbb{CP}^{2}$?. Establishing whether this also
exists, and how much if any supersymmetry it may preserve, is beyond the scope
of this work but would be interesting to pursue.

##  5 Local analysis of $\mathfrak{osp}(n|2)$ vacua for $n>4$

In this section we perform a local analysis of the $\mathfrak{osp}(6|2)$ and
$\mathfrak{osp}(5|2)$ AdS3 vacua derived in the previous sections in 5.1 and
5.2. We begin with some comments about the non existence of
$\mathfrak{osp}(n|2)$ AdS3 for $n>6$ and on the generality of classes we do
find.

In the previous section we derived classes of solutions in IIA supergravity
that realise the super conformal algebras $\mathfrak{osp}(n|2)$ for $n=5,6$ on
a warped product space consisting of a foliation of
AdS${}_{3}\times\widehat{\mathbb{CP}}\\!\\!~{}^{3}$ foliated over an interval
such that either SO(5) or SO(6) is preserved. In appendix C we prove that the
case of $n=7$ is locally AdS${}_{4}\times$S7, and the same is proved for the
case of $n=8$ in [51]. Thus one only has true AdS3 solutions for $n=5,6$ (or
lower) and we have found them only in type IIA.

The class of $\mathfrak{osp}(6|2)$ AdS3 vacua we find are exhaustive for type
II supergravities: Spinors transforming in the 6 of $\mathfrak{so}(6)$
necessitate either a round $\mathbb{CP}^{3}$ factor in the metric or
$\mathbb{CP}^{3}$ with a U(1) fibered over it. This latter possibility can
easily be excluded at the level of the equations of motion [51]. For the case
of $\mathfrak{osp}(5|2)$ AdS3 vacua we suspect the same is true, but are not
completely certain of that.

###  5.1  Analysis of $\mathfrak{osp}(6|2)$ vacua

The $\mathfrak{osp}(6|2)$ AdS3 solutions are given by the class of the
previous section specialised to the case $u=1$. We find for the NS sector

$\displaystyle\frac{ds^{2}}{2\pi}$
$\displaystyle=\frac{|h|}{\sqrt{2hh^{\prime\prime}-(h^{\prime})^{2}}}ds^{2}(\text{AdS}_{3})+\sqrt{2hh^{\prime\prime}-(h^{\prime})^{2}}\bigg{[}\frac{1}{4|h|}dr^{2}+\frac{2}{|h^{\prime\prime}|}ds^{2}(\mathbb{CP}^{3})\bigg{]},$
$\displaystyle e^{-\Phi}$
$\displaystyle=\frac{(|h^{\prime\prime}|)^{\frac{3}{2}}}{2\sqrt{\pi}(2hh^{\prime\prime}-(h^{\prime})^{2})^{\frac{1}{4}}},~{}~{}~{}~{}H=dB_{2},~{}~{}~{}~{}B_{2}=4\pi\left(-(r-k)+\frac{h^{\prime}}{h^{\prime\prime}}\right)J_{2},$
(5.1)

and the RR sector

$\displaystyle F_{0}$
$\displaystyle=-\frac{1}{2\pi}h^{\prime\prime\prime},~{}~{}~{}~{}~{}F_{2}=B_{2}F_{0}+2(h^{\prime\prime}-(r-k)h^{\prime\prime\prime})J_{2},$
$\displaystyle F_{4}$ $\displaystyle=\pi
d\left(h^{\prime}+\frac{hh^{\prime}h^{\prime\prime}}{2hh^{\prime\prime}-(h^{\prime})^{2}}\right)\wedge\text{vol}(\text{AdS}_{3})+B_{2}\wedge
F_{2}-\frac{1}{2}B_{2}\wedge B_{2}F_{0}$
$\displaystyle-4\pi(2h^{\prime}+(r-k)(-2h^{\prime\prime}+(r-k)h^{\prime\prime\prime}))J_{2}\wedge
J_{2},$ (5.2)

where $k$ is a constant and $h$ is a function of $r$ obeying
$h^{\prime\prime\prime\prime}=0$ in regular regions of a solution.

####  5.1.1 Local solutions and regularity

There are several distinct physical behaviours one can realise locally by
solving $h^{\prime\prime\prime}=-2\pi F_{0}$ (for $F_{0}=$ constant) in
different ways, in this section we shall explore them.

The distinct local solutions the class contains can be characterised as
follows. First off the domain of $r$ should be ascertained, in principle it
can be one of the following: Periodic, bounded (from above and below), semi
infinite or unbounded. For a well defined AdS3 vacua dual to a $d=2$ CFT $r$
must be one of the first 2. In the case at hand the $r$ dependence of $h$ does
not allow for periodic $r$ so we seek bounded solutions. In general a solution
can be bounded by either a regular zero or a physical singularity.

At a regular zero we must have that $e^{-\Phi}$ and the AdS warp factor
becomes constant. The internal space should then decompose as a direct product
of two sub manifolds with the first tending to the behaviour of a Ricci flat
cone of radius $r$ and the second $r$ independent.

There are many ways to realise physical singularities that bound the space at
some loci. The most simple is with D branes and O-planes: For a generic
solution these objects are characterised by a metric and dilaton which
decompose as

$ds^{2}=\frac{1}{\sqrt{h_{p}}}ds^{2}_{\|}+\sqrt{h_{p}}ds^{2}_{\perp},~{}~{}~{}~{}e^{-\Phi}\propto
h_{p}^{\frac{p-3}{4}}$ (5.3)

where $ds^{2}_{\|}$ is the $p+1$ dimensional metric on the world volume of
this object and $ds^{2}_{\perp}$ is the $9-p$ dimensional metric on its co-
dimensions. We will consider only solutions whose metric is a foliation over
an interval $r$. A Dp brane singularity (for $p<7$) is then signaled by the
leading order behaviour

$ds^{2}_{\perp}\propto
dr^{2}+r^{2}ds^{2}(\text{B}^{8-p}),~{}~{}~{}h_{p}\propto\frac{1}{r^{7-p}}$
(5.4)

for $\text{B}^{8-p}$ the base of a Ricci flat cone (for instance
$\text{B}^{8-p}=\text{S}^{8-p}$). The case of $p=8$ is different as while the
solution is singular at such a loci, the metric neither blows up nor tends to
zero so a D8 brane does not bound the solution ($p=7$ will not be relevant to
our analysis). The Op plane singularity (for $p\neq 7$) on the other hand
yields

$ds^{2}_{\perp}\propto
dr^{2}+\alpha_{0}^{2}ds^{2}(\text{B}^{8-p}),~{}~{}~{}h_{p}\propto r,$ (5.5)

for $\alpha_{0}$ some constant. Our task then is to first establish which of
these behaviours can be realised by the class of solutions in this section,
then whether any of these behaviours can coexist in the same local solution.
Let us reiterate: D-brane and O-plane singularities do not exhaust the
possible physical singularities, indeed we will find a more complicated object
later in this section which we will describe when it becomes relevant.

The most obvious thing one can try is to fix $F_{0}=h^{\prime\prime\prime}=0$,
it is not hard to see that one then has $H_{3}=0$ and $F_{4}$ becomes purely
electric. One can integrate $h$ as

$h=c_{1}+rc_{2}+\frac{1}{2}\tilde{k}r^{2}$ (5.6)

and then upon making the redefinitions

$c_{1}=\frac{\tilde{k}L^{4}+c_{2}^{2}\pi^{2}}{2\tilde{k}\pi^{2}},~{}~{}~{}~{}~{}-r=\frac{c_{2}}{\tilde{k}^{2}}+\frac{L^{2}}{\pi}\sinh
x$ (5.7)

one finds the solution is mapped to

$\displaystyle\frac{ds^{2}}{L^{2}}$
$\displaystyle=\bigg{(}\cosh^{2}xds^{2}(\text{AdS}_{3})+dx^{2}\bigg{)}+4ds^{2}(\mathbb{CP}^{3}),~{}~{}~{}~{}e^{-\Phi}=\frac{\tilde{k}}{2L},$
$\displaystyle F_{2}$
$\displaystyle=2\tilde{k}J_{2},~{}~{}~{}~{}~{}F_{4}=\frac{3}{2}\tilde{k}L^{2}\cosh^{3}x\text{vol}(\text{AdS}_{3})\wedge
dx.$ (5.8)

This is of course AdS${}_{4}\times\mathbb{CP}^{3}$, dual to ${\cal N}=6$,
U(N)${}_{\tilde{k}}\times$U(N)${}_{-\tilde{k}}$ Chern-Simons matter theory
(where $N=2\tilde{k}L^{4}$) [53]. Thus there is only one local solution when
$F_{0}=0$ and it is an AdS4 vacua preserving twice the supersymmetries of
generic solutions within this class. This is the only regular solution
preserving (6,0) supersymmetry.

Next we consider the sort of physical singularities the metric and dilaton in
(5.1) can support for $F_{0}\neq 0$. At the loci of such singularities the
space terminates so the interval spanned by $r$ becomes bounded at one end. We
shall use diffeomorphism invariance to assume this bound is at $r=0$.

First off $\Delta_{1}=2hh^{\prime\prime}-(h^{\prime})^{2}$ appears in the
metric and dilaton where one would expect the warp factor of a co-dimension 7
source to appear. Thus if $\Delta_{1}$ has an order 1 zero at a loci where
$h,h^{\prime\prime}$ have no zero one has the behaviour of O2 planes extended
in AdS3 at the tip of a G2 cone over $\mathbb{CP}^{3}$. We now choose the loci
of this O2 plane to be $r=0$, meaning that the constant part of $\Delta_{1}$
has to vanish which forces

$\text{O2
at}~{}r=0:~{}h=c_{1}+c_{2}r+\frac{c_{2}^{2}}{4c_{1}}r^{2}+\frac{1}{3!}c_{3}r^{3},~{}~{}~{}c_{1},c_{2},c_{3}\neq
0,$ (5.9)

where $r\in\mathbb{R}^{\pm}$ when $\text{sign}(c_{1}c_{4})=\pm 1$.

Another type of singularity this solution is consistent with is a D8/O8 system
of world volume AdS${}_{3}\times\mathbb{CP}^{3}$. Such a singularity is
characterised by O8 brane like behaviour in the metric and dilaton. We realise
this behaviour by choosing $h$ such that $(h^{\prime\prime},\Delta_{1})$ both
have an order 1 zero at a loci where $h$ has no zero. After using
diffeomorphism invariance to place the D8/O8 at $r=0$ this is equivalent to
taking $h$ as

$\text{D8/O8 at}~{}r=0:~{}h=c_{1}+\frac{1}{3!}c_{2}r^{3},~{}~{}~{}c_{1,2}\neq
0.,$ (5.10)

where $r\in\mathbb{R}^{\pm}$ for $\text{sign}(c_{1}c_{2})=\pm 1$.

We are yet to find a D brane configuration, given that we have a
$\mathbb{CP}^{3}$ factor the obvious thing to naively aim for is D2 branes at
the tip of a G2 cone similar to the O2 plane realised above. However the warp
factor of a D2 brane blows up like $\lim_{r\to 0}r^{-5}$ its loci, and this is
not possible to achieve for (5.1) such that $h$ is an order 3 polynomial. It
is however possible to realise a more exotic object: It is well known that if
one takes $d=11$ supergravity on the orbifold
$\mathbb{R}^{1,6}\times\mathbb{C}^{2}/\mathbb{Z}_{\tilde{k}}$ then reduces on
the Hopf fibre of the Lens space (equivalently squashed 3-sphere) inside
$\mathbb{C}^{2}/\mathbb{Z}_{\tilde{k}}$ one generates a D6 brane singularity
in type IIA. One can generate the entire flat space D6 brane geometry by
replacing $\mathbb{C}^{2}/\mathbb{Z}_{\tilde{k}}$ in the above with a Taub-Nut
space and likewise reducing to IIA. One can perform an analogous procedure for
$\mathbb{R}^{1,2}\times\mathbb{C}^{4}/\mathbb{Z}_{\tilde{k}}$, reducing this
time on the Hopf fibration (over $\mathbb{CP}^{3}$) of a squashed 7-sphere.
The resulting solution in IIA takes the from

$ds^{2}=\frac{\sqrt{r}}{{\tilde{k}}}ds^{2}(\text{Mink}_{3})+\frac{1}{4{\tilde{k}}\sqrt{r}}\bigg{(}dr^{2}+4r^{2}ds^{2}(\mathbb{CP}^{3})\bigg{)},~{}~{}~{}e^{-\Phi}=\frac{{\tilde{k}}^{\frac{3}{2}}}{r^{\frac{3}{4}}},~{}~{}~{}F_{2}=2{\tilde{k}}J_{2},~{}~{}~{}r\geq
0,$ (5.11)

and is singular at $r=0$. Notice that the $r$ dependence of the dilaton and
metric is the same as one gets at the loci of flat space D6 branes, but the
co-dimensions no longer span a regular cone as they do in that case, indeed
$dr^{2}+cr^{2}ds^{2}(\mathbb{CP}^{3})$ is only Ricci flat for unit radius
round $\mathbb{CP}^{3}$ when $c=\frac{2}{5}$. It is argued in [53] that the
singularity in (5.11) corresponds to a coincident combination of a KK monopole
and ${\tilde{k}}$ D6 branes (T-dual of an $(1,{\tilde{k}})-5$ brane) that
partially intersect another KK monopole. For simplicity we shall refer to this
rather complicated composite object as a $\widetilde{\text{D6}}$ brane. We can
find the behaviour of this object, now extended in AdS3, within this class –
assuming it is located at $r=0$ one need only tune

$\widetilde{\text{D6}}~{}\text{at}~{}r=0:~{}h=r^{2}(c_{1}+c_{2}r),~{}~{}~{}~{}c_{1,2}\neq
0,$ (5.12)

with the caveat that as this only exists when $F_{0}\neq 0$, we can no longer
lift to $d=11$. Again $r\in\mathbb{R}^{\pm}$ for $\text{sign}(c_{1}c_{2})=\pm
1$.

The above exhausts the physical singularities we are able to identify, we do
however find one final further singularity. By tuning $h=cr^{3}$ for $r,c>0$,
the metric and dilaton then become

$ds^{2}=\frac{\pi}{2\sqrt{3}r}\bigg{(}4r^{2}\left(ds^{2}(\text{AdS}_{3})+ds^{2}(\mathbb{CP}^{3})\right)+3dr^{2}\bigg{)},~{}~{}~{}e^{-\Phi}=\frac{3\sqrt{2}3^{\frac{1}{4}}c}{\sqrt{\pi}}\sqrt{r}$
(5.13)

which is singular about $r=0$ in a way we do not recognise.

All the previously discussed physical singularities bound the interval spanned
by $r$ at one end. In order to have a true AdS3 vacuum we need to bound a
solution between 2 of them separated by a finite proper distance. However
assuming that the space starts at $r=0$ with any of an O2, D8/O8 or
$\widetilde{\text{D6}}$ singularity, we find that that none of the warp
factors appearing in metric or dilaton (5.1), (ie
$(h,\Delta_{1},h^{\prime\prime})$) either blow up or vanish until
$r\to\infty$. For each case (assuming $r\geq 0$ for simplicity) the metric and
dilaton as $r\to\infty$ tend to (5.13) . By computing the curvature invariants
it is possible to show that the metric is actually flat at this loci, hence it
tends to $\text{Mink}_{10}$, however as $e^{\Phi}\to\infty$ the solution is
still singular. Worse still the singularity as $r\to\infty$ is at infinity
proper distance from $r=0$, so in all cases the internal space is semi
infinite.

Naively one might now conclude that there are no $\mathfrak{osp}(6|2)$ AdS3
vacua with bounded internal space (true vacua), however as we will show
explicitly in section 6, this is not the case. The missing ingredient is the
inclusion of D8 branes on the interior of $r$ which allow one to glue local
solutions of $h^{\prime\prime\prime\prime}=0$, depending on different
integration constants, together.

###  5.2  Analysis of $\mathfrak{osp}(5|2)$ vacua

The $\mathfrak{osp}(5|2)$ AdS3 solutions are given by the class of section 4
for $u^{\prime}\neq 0$, one can use diffemorphism invariance to fix $u=r$ for
such solutions without loss of generality. The resulting NS sector takes the
form

$\displaystyle\frac{ds^{2}}{2\pi}$
$\displaystyle=\frac{|hr|}{\sqrt{\Delta_{1}}}ds^{2}(\text{AdS}_{3})+\frac{\sqrt{\Delta_{1}}}{4|r|}\bigg{[}\frac{2}{|h^{\prime\prime}|}\bigg{(}ds^{2}(\text{S}^{4})+\frac{1}{\Delta_{2}}(Dy_{i})^{2}\bigg{)}+\frac{1}{|h|}dr^{2}\bigg{]},$
$\displaystyle e^{-\Phi}$
$\displaystyle=\frac{|h^{\prime\prime}|^{\frac{3}{2}}\sqrt{|r|}\sqrt{\Delta_{2}}}{2\sqrt{\pi}\Delta_{1}^{\frac{1}{4}}},~{}~{}~{}~{}\Delta_{1}=2hh^{\prime\prime}r^{2}-(rh^{\prime}-h)^{2},~{}~{}~{}~{}\Delta_{2}=1+\frac{2h^{\prime}}{rh^{\prime\prime}},~{}~{}~{}~{}H=dB_{2},$
$\displaystyle B_{2}$
$\displaystyle=4\pi\bigg{[}\left(-(r-k)+\frac{rh^{\prime}-h}{rh^{\prime\prime}}\right)J_{2}+\frac{1}{2h^{\prime\prime}}\left(\frac{h}{r}+\frac{hh^{\prime\prime}-2(h^{\prime})^{2}}{2h^{\prime}+rh^{\prime\prime}}\right)\left(J_{2}-\tilde{J}_{2}\right)\bigg{]},$
(5.14)

while the $d=10$ RR fluxes are then given by

$\displaystyle F_{0}$
$\displaystyle=-\frac{1}{2\pi}h^{\prime\prime\prime},~{}~{}~{}~{}~{}F_{2}=B_{2}F_{0}+2(h^{\prime\prime}-(r-k)h^{\prime\prime\prime})J_{2},$
$\displaystyle F_{4}$ $\displaystyle=\pi
d\left(h^{\prime}+\frac{hh^{\prime\prime}r(rh^{\prime}+h)}{\Delta_{1}}\right)\wedge\text{vol}(\text{AdS}_{3})+B_{2}F_{2}-\frac{1}{2}B_{2}\wedge
B_{2}F_{0}$
$\displaystyle-4\pi\bigg{[}(2h^{\prime}+(r-k)(-2h^{\prime\prime}+(r-k)h^{\prime\prime\prime}))J_{2}\wedge
J_{2}+d\left(\frac{h}{r}\text{Im}\Omega_{3}\right)\bigg{]},$ (5.15)

where $h$ is defined as before and $k$ is a constant.

####  5.2.1 Local solutions and regularity

In this section we will explore the physically distinct local
$\mathfrak{osp}(5|2)$ solution that follow from solving
$h^{\prime\prime\prime}=-2\pi F_{0}$ in various ways.

Let us begin by commenting on the massless limit
$F_{0}=h^{\prime\prime\prime}=0$: Unlike the class of $\mathfrak{osp}(6|2)$
solutions the result is no longer locally AdS${}_{4}\times\mathbb{CP}^{3}$ \-
it is instructive to lift the class to $d=11$. We find the metric of the
solution can be written as121212Note that to get to this form one must rescale
the canonical 11’th direction, ie if this is $z$ and $L^{i}_{1,2}$ are defined
as in (A.2) then $\phi_{2}=\frac{2}{h^{\prime\prime}}z$

$\displaystyle\frac{ds_{11}^{2}}{2^{\frac{1}{3}}\pi^{\frac{2}{3}}|h^{\prime\prime}|}$
$\displaystyle=\Delta_{2}^{\frac{1}{3}}\bigg{[}\frac{|h||r|^{\frac{4}{3}}}{\Delta_{1}^{\frac{2}{3}}}ds^{2}(\text{AdS}_{3})+\frac{\Delta_{1}^{\frac{1}{3}}}{|r|^{\frac{2}{3}}}\bigg{(}\frac{1}{4|h|}dr^{2}+\frac{2}{|h^{\prime\prime}|}ds^{2}(\widehat{\text{S}}^{7})\bigg{)}\bigg{]},$
$\displaystyle ds^{2}(\widehat{\text{S}}^{7})$
$\displaystyle=\frac{1}{4}\bigg{[}ds^{2}(\text{S}^{4})+\frac{1}{\Delta_{2}}\left(L_{2}^{i}-\cos^{2}\left(\frac{\alpha}{2}\right)L_{1}^{i}\right)^{2}\bigg{]},$
(5.16)

where $L_{1,2}^{i}$ are two sets of SU(2) left invariant forms defined as in
appendix (A.2), so that the internal space is a foliation of an
SP(2)$\times$SP(1) preserving squashed 7-sphere over an interval. This
enhancement of symmetry is also respected by the $d=11$ flux, to see this we
need to define the SP(2)$\times$SP(1) invariant form on this squashed
7-sphere, fortuitously these were already computed in [51], they are

$\Lambda_{3}^{0}=\frac{1}{8}(L_{2}^{1}+\mathcal{A}^{1})\wedge(L_{2}^{2}+\mathcal{A}^{2})\wedge(L_{2}^{3}+\mathcal{A}^{3}),~{}~{}\tilde{\Lambda}_{3}^{0}=\frac{1}{8}(L_{2}^{i}+\mathcal{A}^{i})\wedge(d\mathcal{A}^{i}+\frac{1}{2}\epsilon^{i}_{jk}\mathcal{A}^{j}\wedge\mathcal{A}^{k}),$
(5.17)

where $\mathcal{A}^{i}=-\cos^{2}\left(\frac{\alpha}{2}\right)L_{1}^{i}$, and
their exterior derivatives. One can then show that the $d=11$ flux decompose
as

$\frac{G_{4}}{\pi}=d\left(h^{\prime}+\frac{rhh^{\prime\prime}(rh^{\prime}+h)}{\Delta_{1}}\right)\wedge\text{vol}(\text{AdS}_{3})+4d\bigg{(}\frac{2(r(h^{\prime})^{2}-h(h^{\prime}+rh^{\prime\prime}))}{r(2h^{\prime}+rh^{\prime\prime})}\Lambda_{3}^{0}+\frac{h}{r}(\Lambda_{3}^{0}-\tilde{\Lambda}_{3}^{0})\bigg{)}.$
(5.18)

For such solutions we can in general integrate $h^{\prime\prime\prime}=0$ in
terms of an order 2 polynomial. As we shall see shortly it is possible to
bound $r$ at one end of the space in several physical ways, but when $F_{0}=0$
it always remains semi-infinite. Given that the massless limit of ${\cal
N}=(6,0)$ is always locally AdS${}_{4}\times\mathbb{CP}^{3}$ in IIA, it is
reasonable to ask whether the massless solutions here approach this
asymptotically. Such a solution preserving ${\cal N}=(8,0)$ was found on this
type of squashing of the 7-sphere in [51] and can be interpreted as a
holographic dual to a surface defect. In this case, as $r\to\infty$ the the
curvature invariants (5.2.1) all vanish, ( for instance $R\sim
r^{-\frac{2}{3}}$). This makes the behaviour at infinite $r$ that of Mink11,
so such an interpretation is not possible here.

Let us now move back to IIA and focus on more generic solutions: By studying
the zeros of $(r,~{}h,~{}h^{\prime\prime},~{}\Delta_{1},~{}\Delta_{2})$ we are
able to identify a plethora of boundary behaviours for $\mathfrak{osp}(5|2)$
solutions. The vast majority we are able to identify as physical and most
exist for arbitrary values of $F_{0}$. We already used up translational
invariance of this class to align $u=r$ so we can non longer assume that $r=0$
is a boundary of solutions in this class, rather we must consider possible
boundaries at $r=0$ and $r=r_{0}$ for $r_{0}\neq 0$ separately.

We have two physical boundary behaviours that only exist for $F_{0}\neq 0$:
The first of these is a regular zero for which the warp factors of AdS3 and S4
become constant while the $(r,\text{S}^{2})$ directions approach the origin of
$\mathbb{R}^{3}$ in polar coordinates. This is given by tuning

$\text{Regular zero
at}~{}r=0:~{}~{}~{}~{}h=c_{1}r+\frac{1}{3!}c_{2}r^{3},~{}~{}~{}c_{1,2}\neq
0,~{}~{}~{}\text{sign}(c_{1}c_{2})=1$ (5.19)

where one can take either of $r\in\mathbb{R}^{\pm}$ . This is the only regular
boundary behaviour that is possible.

Next it is possible to realise a fully localised O6 plane of world volume
$(\text{AdS}_{3},~{}\text{S}^{4})$ at $r=r_{0}$ by tuning

$\text{O6 plane
at}~{}r=r_{0}:~{}~{}~{}~{}h=c_{1}r_{0}+c_{1}(r-r_{0})+\frac{1}{3!}c_{2}(r-r_{0})^{3},~{}~{}~{}c_{1},c_{2},r_{0}\neq
0.$ (5.20)

The domain of $r$ in this case depends more intimately on the tuning of
$c_{1},c_{2},r_{0}$ than we have thus far seen: When $r_{0}<0$ one has
$r\in(-\infty,~{}r_{0}]$ for $\text{sign}(c_{1}c_{2})=1$ while for
$\text{sign}(c_{1}c_{2})=-1$ one finds that $r_{0}\leq r\leq r_{1}<0$ for
$r_{1}=r_{1}(c_{1},c_{2})$. Conversely for $r_{0}>0$,
$\text{sign}(c_{1}c_{2})=1$ implies $r\in[r_{0},~{}\infty)$ while
$\text{sign}(c_{1}c_{2})=-1$ implies $0<r_{1}\leq r\leq r_{0}$.

The remaining boundary behaviour exist whether $F_{0}$ is non trivial or not:
We find the behaviour of D6 branes extended in
$(\text{AdS}_{3},~{}\text{S}^{4})$ by tuning

$\text{D6 brane
at}~{}r=0:~{}~{}~{}~{}h=c_{1}r+\frac{1}{2}c_{2}r^{2}+\frac{1}{3!}c_{3}r^{3},~{}~{}~{}~{}c_{1,2}\neq
0.$ (5.21)

When $\text{sign}(c_{1}c_{2})=\pm 1$ $r=0$ is a lower/upper bound. Given this,
$r$ is also bounded from above/below when $\text{sign}(c_{1}c_{3})=-1$ and is
semi-infinite for $\text{sign}(c_{1}c_{3})=1$ and $c_{3}=0$.

As with the $\mathfrak{osp}(6|2)$ class it is possible to realise a
$\widetilde{\text{D}6}$ singularity (see the discussion below (5.10)), this
time at $r=r_{0}$ by tuning

$\widetilde{\text{D}6}~{}\text{brane
at}~{}r=r_{0}:~{}~{}~{}~{}h=\frac{1}{2}c_{1}(r-r_{0})^{2}+\frac{1}{3!}c_{2}(r-r_{0})^{3},~{}~{}~{}~{}r_{0},c_{2}\neq
0,~{}~{}~{}~{}c_{2}\neq-3\frac{c_{1}}{r_{0}}.$ (5.22)

For $\text{sign}(r_{0}c_{1}c_{2})=1$ the domain of $r$ is semi infinite
bounded from above/below when $\text{sign}(r_{0})=\mp 1$. When
$\text{sign}(r_{0}c_{1}c_{2})=-1$ we find that $r$ is bounded between $r_{0}$
and some constant $r_{1}=r_{1}(r_{0},c_{1},c_{2})$. Given the later behaviour,
when $\text{sign}(r_{0})=\pm 1$ one finds that $r$ is strictly
positive/negative with $r_{0}$ the upper/lower of the 2 bounds when
$|c_{2}r_{0}|>3|c_{1}|$ and the lower/upper when $|c_{2}r_{0}|<3|c_{1}|$.

Next we find the behaviour of an O4 plane extended in AdS${}_{3}\times$S2 by
tuning

$\text{O4}~{}\text{plane
at}~{}r=r_{0}:~{}~{}~{}~{}h=\frac{1}{2}c_{1}\big{(}r_{0}^{2}-r_{0}(r-r_{0})+(r-r_{0})^{2}\big{)}+\frac{1}{3!}c_{2}(r-r_{0})^{3},~{}~{}~{}c_{1},r_{0}\neq
0,~{}~{}~{}c_{2}\neq-\frac{3c_{1}}{r_{0}}.$ (5.23)

In this solution the domain of $r$ has the same qualitative dependence on the
signs of $c_{1},c_{2},r_{0}$ and whether $|c_{2}r_{0}|>3|c_{1}|$ or
$|c_{2}r_{0}|<3|c_{1}|$ as the previous example, though the precise value of
$r_{1}(c_{1},c_{2},r_{0})$ is different.

Likewise we find the behaviour of an O2 plane extended in AdS3 and back-
reacted on a G2 cone whose base is round $\mathbb{CP}^{3}$, this is achieved
by tuning

$\text{O2}~{}\text{plane
at}~{}r=r_{0}:~{}~{}~{}~{}~{}h=2r_{0}c_{1}+\frac{1}{2}c_{1}(r-r_{0})^{2}+\frac{1}{3!}c_{2}(r-r_{0})^{3},~{}~{}~{}~{}r_{0},c_{1}\neq
0,~{}~{}~{}c_{2}\neq-\frac{3c_{1}}{r_{0}},$ (5.24)

where the domain of $r$ is qualitatively related to the parameters as it was
for the $\widetilde{\text{D}6}$.

Finally we find the behaviour of an O2’ plane extended in AdS3 and back-
reacted on a G2 cone whose base is a squashed $\mathbb{CP}^{3}$ (ie
$4ds^{2}(\text{B}^{6})=2ds^{2}(\text{S}^{4})+ds^{2}(\text{S}^{2})$) at
$r=r_{0}$ by tuning

$\text{O2'}~{}\text{plane
at}~{}r=r_{0}:~{}~{}~{}~{}~{}h=2c_{1}r_{0}^{2}+2c_{1}b(r-r_{0})+\frac{1}{2}c_{1}(b-1)^{2}(r-r_{0})^{2}+\frac{1}{3!}c_{2}(r-r_{0})^{3},$
(5.25)

where we must additionally impose $r_{0},c_{1}\neq 0$. This gives behaviour
similar to the O2 plane, ie $e^{2A}\sim(r-r_{0})^{-\frac{1}{2}}$ with the rest
of the warp factors scaling as the reciprocal of this. However in general
$e^{2(C-D)}=\left(\frac{b+1}{b-1}\right)^{2}$ at leading order about $r=r_{0}$
and the internal space only spans a Ricci flat cone for $e^{2(C-D)}=1,2$, with
the former yielding (5.24). As such for the O2’ plane we must additionally
tune

$\left(\frac{b+1}{b-1}\right)^{2}=2,$ (5.26)

which has two solutions $b_{\pm}=3\pm 2\sqrt{2}$ and we must have
$c_{2}r_{0}\neq-12b_{\pm}$. Again the domain of $r$ has the same qualitative
dependence as the $\widetilde{\text{D}}6$, though this time the relevant
equalities that determine whether $r_{0}$ is an upper or lower bound are
$|c_{2}r_{0}|>12b_{\pm}|c_{1}|$ or $|c_{2}r_{0}|<12b_{\pm}|c_{1}|$. This
exhausts the the physical singularities we have been able to identify.

As with the case of $\mathfrak{osp}(6|2)$ vacua we have been able to identify
several local solution for which the domain of $r$ is semi infinite. For these
the metric as $r\to\pm\infty$ is once again flat, but at infinite distance and
with a non constant dilaton. For the $\mathfrak{osp}(5|2)$ solutions however
it is possible to bound the majority of the solutions for suitable tunings of
the parameters on which they depend - this necessitates $F_{0}\neq 0$. A
reasonable question to ask then is which physical singularities can reside in
the same local solution? There are actually 7 distinct local solutions bounded
between two physical singularities, we provide details of these in table 1.
Note that the solution with regular zero is unbounded while the D6 solution
can only be bounded by a singularity of the type given in (5.25), but without
(5.26) being satisfied so is thus non-physical.

In this section, and the preceding one we have analysed the possible local
solutions preserving ${\cal N}=(5,0)$ and $(6,0)$ supersymmetry that follow
from various tunings of the (local) order 3 polynomial $h$. We found many
different possibilities, many of which can give rise to a bounded interval in
the $(5,0)$ case, but non of which do in the (6,0) case. This is not the end
of the story, in this section we have assumed that $F_{0}$ is a constant which
excludes the presence of D8 branes along the interior of the interval. In the
next section we shall relax this assumption allowing for much wider classes of
global solution, and in particular $(6,0)$ solutions with bounded internal
space.

Bound at $r=r_{0}$ | Bound at $r=\tilde{r}_{0}$ | Additional tuning and comments
---|---|---
O6 (5.20) | O4 $\lvert$ O2 $\lvert$ O2’ |  $\begin{array}[]{c}c_{1}=\frac{(\tilde{r}_{0}-r_{0})^{3}(r_{0}^{2}+4r_{0}\tilde{r}_{0}-8\tilde{r}_{0}^{2})}{72\tilde{r}_{0}^{3}}c_{2},\\\ r_{0}=\frac{(\sqrt{3}\sqrt{11+8\sqrt{b_{0}}}-1)\tilde{r}_{0}}{2}\end{array}$ $b_{0}=0~{}\lvert~{}1~{}\lvert~{}2$
D6 (5.21) | Non-physical |  $\begin{array}[]{c}\tilde{r}_{0}:~{}\text{Middle of 3 real roots of }\\\ 8c_{3}^{2}\tilde{r}_{0}^{3}+36c_{2}c_{2}\tilde{r}_{0}^{2}+(27c_{2}^{2}+72c_{1}c_{3})\tilde{r}_{0}+72c_{1}c_{2}=0\end{array}$, (5.25) like but for $e^{2C-2D}>2$
$\widetilde{\text{D}6}$ (5.22) | O2’ | $\begin{array}[]{c}c_{1}=\frac{5-4\sqrt{2}-\sqrt{120+86\sqrt{2}}}{21}c_{2}r_{0}\\\ \tilde{r}_{0}=\frac{\sqrt{4+3\sqrt{2}}r_{0}}{2}\end{array}$
O4 (5.23) | O2 |  $\begin{array}[]{c}c_{1}=c_{2}b_{-}r_{0},\\\ \tilde{r}_{0}=b_{+}r_{0}\end{array}$, $\begin{array}[]{c}b_{\pm}:~{}\pm\text{tive real roots of}\\\ 13b_{+}^{3}-29b_{+}^{2}+10b_{+}=6\\\ 1287b_{-}^{3}+610b_{-}^{2}-1287b_{-}+144=0\end{array}$
O4 (5.23) | O2’ |  $\begin{array}[]{c}c_{1}=c_{2}b_{-}r_{0},\\\ \tilde{r}_{0}=b_{+}r_{0}\end{array}$, $\begin{array}[]{c}b_{\pm}:~{}\pm\text{tive real roots of}\\\ 2b_{+}^{6}-8b_{+}^{5}+40b_{+}^{3}-49b_{+}^{2}+6b_{+}=9\\\ 81b_{-}^{6}-18b_{-}^{5}+37b_{-}^{4}+12b_{-}^{3}-93b_{-}^{2}+54b_{-}=9\end{array}$
O2 (5.24) | O2’ |  $\begin{array}[]{c}c_{1}=c_{2}b_{-}r_{0},\\\ \tilde{r}_{0}=b_{+}r_{0}\end{array}$, $\begin{array}[]{c}b_{\pm}:~{}\pm\text{tive real roots of}\\\ 8b_{+}^{6}-16b_{+}^{5}-8b_{+}^{4}+224b_{+}^{3}-209b_{+}^{2}+26b_{+}=169\\\ 711b_{-}^{6}-2388b_{-}^{5}+2858b_{-}^{4}-972b_{-}^{3}-705b_{-}^{2}+648b_{-}=144\end{array}$
Table 1: A list of distinct local $\mathfrak{osp}(5|2)$ solutions with bounded
internal space. Note that we do not include solutions with the singularities
at $r=r_{0}$ and $r=\tilde{r}_{0}$ inverted, as they are physically
equivalent.

##  6 Global solutions with interior D8 branes

In this section we show that it is possible to glue the local solutions of
section 5 together with D8 branes placed along the interior of the interval
spanned by $r$. This opens the way for constructing many more global solutions
with bounded internal spaces.

In the previous section we studied what types of local ${\cal N}=(5,0)$ and
$(6,0)$ it is possible to realise with various tunings of the order 3
polynomial $h$. The assumption we made before was that $F_{0}=$ constant was
fixed globally, but this is not necessary, all that is actually required is
that $F_{0}$ is piecewise constant. A change in $F_{0}$ gives rise to a delta
function source as

$dF_{0}=\Delta F_{0}\delta(r-r_{0})dr,$ (6.1)

where $\Delta F_{0}$ is the difference between the values $F_{0}$ for
$r>r_{0}$ and $r<r_{0}$. D8 branes give rise to a comparatively mild
singularity for which the Bosonic fields neither blow up nor tend to zero so
do not represent a boundary of a solution, indeed the solution continues
passed them unless they appear coincident to an O8 plane. As one crosses a D8
brane the metric and dilaton and NS 3-form are continuous, but the RR sector
can experience a shift. To accommodate such an object within the class of
solutions of section 4 one needs to do so in terms of $h$, so it can lie along
$r$. If we wish to place a D8 at $r=r_{0}$ one should have

$h^{\prime\prime\prime\prime}=-\Delta
N_{8}\delta(r-r_{0}),~{}~{}~{}~{}~{}\Delta N_{8}=2\pi\Delta F_{0},$ (6.2)

where $\Delta N_{8}$ is the difference in D8 brane charge between $r>r_{0}$
and $r<r_{0}$ and is the charge of the D8 brane source at $r=r_{0}$. While the
conditions that the NS sector should be continuous amounts to demanding the
continuity of

$\displaystyle{\cal N}=(6,0)$
$\displaystyle:~{}~{}~{}~{}(h,(h^{\prime})^{2},h^{\prime\prime}),$ (6.3)
$\displaystyle{\cal N}=(5,0)$
$\displaystyle:~{}~{}~{}~{}(h,h^{\prime},h^{\prime\prime}),$

recall that $u^{\prime\prime}=0$ is a requirement for supersymmetry so $u$
cannot change as one crosses the D8. The source corrected Bianchi identities
in general take the form

$(d-H\wedge)f_{+}=\frac{1}{2\pi}\Delta N_{8}\delta(r-r_{0})dr\wedge e^{{\cal
F}}~{}~{}~{}~{}\Rightarrow~{}~{}~{}~{}~{}d\hat{f}_{+}=\frac{1}{2\pi}\Delta
N_{8}\delta(r-r_{0})dr\wedge e^{2\pi\tilde{f}}$ (6.4)

where ${\cal F}=B_{2}+2\pi\tilde{f}$ for $\tilde{f}$ a world volume gauge
field on the D8 brane and where $\hat{f}_{+}$ are the magnetic Page fluxes,
(4.52) for the solution at hand. If $\tilde{f}$ is non zero then the D8 is
actually part of a bound state involving every brane whose flux receives a
source correction in $d\hat{f}_{+}$. Recalling the Bianchi identities (4.53),
we have for the specific case at hand that

$d\hat{f}_{n}=\frac{1}{2\pi}(4\pi)^{n}\frac{1}{n!}(r-k)^{n}\Delta
N_{8}\delta(r-r_{0})dr\wedge J_{2}^{n}$ (6.5)

We thus see that it is consistent to set the world volume flux on the D8 brane
to zero by tuning $r_{0}$, ie

$\tilde{f}=0~{}~{}~{}\Rightarrow~{}~{}~{}r_{0}=k.$ (6.6)

This means only $F_{0}$ receives a delta function source, the rest vanishing
as $(r-r_{0})^{n}\delta(r-r_{0})\to 0$ for $n>0$. So we have shown that it is
possible to place D8 branes, that do not come as a bound state, at $r=k$ and
solve the Bianchi identities - but how should one interpret this? The origin
of $k$ in our classification was as an integration constant, but one can view
it as the result of performing a large gauge transformation, ie shifting the
NS 2-form by a form that is closed but not exact as $B_{2}\to B_{2}+\Delta
B_{2}$ such that $b_{0}=\frac{1}{(2\pi)^{2}}\int_{\Sigma_{2}}B_{2}$ is
quantised over some 2-cycle $\Sigma_{2}$. Clearly squashed $\mathbb{CP}^{3}$
contains an S2 which $B_{2}$ has support on, that of the fiber, one finds

$\frac{1}{(2\pi)^{2}}\int_{\text{S}^{2}}B_{2}-\frac{1}{(2\pi)^{2}}\int_{\text{S}^{2}}B_{2}\bigg{\lvert}_{k=0}=\frac{k}{4\pi}\int_{\text{S}^{2}}\text{vol}(\text{S}^{2})=k,$
(6.7)

so provided $k$ is quantised131313Note: usually one means integer by
quantised, by in the presence of fractional branes, such as in ABJ [57] which
shares a $\mathbb{CP}^{3}$ in its internal space, it is possible for
parameters such as $k$ to merely be rational. its addition to the minimal
potential giving rise to the NS 3-form is indeed the action of a large gauge
transformation. The key point is that because $k$ follows from a large gauge
transformation, it does not need to be fix globally, indeed in many situations
where $B_{2}$ depends on a function of the internal space it is necessary to
perform such large gauge transformations as one move through the internal
space to bound $b_{0}$ to with some quantised range. We conclude that the
Bianchi identities are consistent with placing D8 branes at quantised $r=k$
loci provided that they are accompanied by the appropriate number of large
gauge transformations of the NS 2-form.

Of course to be able claim a supersymmetric vacua the sources themselves need
to have a supersymmetric embedding, further if this is the case the
integrability arguments of [58, 59] imply that the remaining type II equations
of motion of the Bosonic supergravity are implied by the Bianchi identities we
have already established hold. The existence of a supersymmetric brane
embedding can be phrased in the language of (generalised) calibrations [60]: A
source extended in AdS3 and wrapping some n-cycle $\Sigma$ is supersymmetric
if it obeys the following condition

$\Psi^{\text{cal}}_{n}=8\Psi_{+}\wedge e^{-{\cal
F}}\bigg{\lvert}_{\Sigma}=\sqrt{\det(g+{\cal
F})}d\Sigma\bigg{\lvert}_{\Sigma}$ (6.8)

where $\Psi_{+}$ is the bi-linear appearing in (4.2), $g$ is the metric on the
internal space and where the pull back onto ${\Sigma}$ is understood. For a D8
brane placed along $r$ we take $\tilde{f}=0$, $d\Sigma=\sin^{3}\alpha
d\alpha\wedge\text{vol}(\text{S}^{3})\wedge\text{vol}(\text{S}^{2})$ and
$B_{2}$ defined as in (4.55) – we find

$\displaystyle\Psi^{\text{cal}}_{6}=\frac{\pi^{3}}{2\sqrt{2}u^{3}h^{\prime\prime}\sqrt{hh^{\prime\prime}}\sqrt{\Delta_{2}}}\bigg{[}$
$\displaystyle
u\left(h(2u-r_{k}u^{\prime})-r_{k}uh^{\prime}\right)\left(2h\left(u+r_{k}u^{\prime}\right)-u\left(2h^{\prime}-h^{\prime\prime}r_{k}\right)r_{k}\right)$
$\displaystyle+2\cos^{2}\alpha u^{\prime}\Delta_{1}r_{k}^{3}\bigg{]}d\Sigma,$
$\displaystyle\sqrt{\det(g+{\cal
F})}d\Sigma\bigg{\lvert}_{\Sigma}=\frac{\pi^{3}}{8(uh^{\prime\prime})^{\frac{3}{2}}\sqrt{\Delta_{2}}}$
$\displaystyle\sqrt{2(h-h^{\prime}r_{k})(u-u^{\prime}r_{k})+uh^{\prime\prime}r_{k}^{2}}$
$\displaystyle\times\left(2h(u+u^{\prime}r_{k})-ur_{k}(2h^{\prime}-h^{\prime\prime}r_{k})\right)d\Sigma$
(6.9)

where we use the shorthand $r_{k}=r-k$. It is simple to then show that (6.8)
is indeed satisfied for a D8 brane at $r=k$, and so supersymmetry is
preserved.

In summary we have shown that D8 branes can be placed along the interior of
$r$ at the loci $r=k$, for quantised $k$, without breaking supersymmetry
provided an appropriate large gauge transformation of the NS 2-form is
performed. This allows one to place a potentially arbitrary number of D8
branes along $r$ and use them to glue the various local solutions of section 5
together provided that the continuity of (6.3) holds across each D8. In the
next section we will explicitly show this in action with 2 examples.

###  6.1 Some simple examples with internal D8 branes

In this section we will construct two global solutions with interior D8 branes
and bounded internal spaces, one preserving each of ${\cal N}=(6,0)$ and
$(5,0)$ supersymmetry. Let us stress that this only scratches the surface of
what is possible, we save a more thorough investigation for forthcoming work
[63].

We shall first construct a solution preserving ${\cal N}=(6,0)$, meaning that
we need to impose the continuity of $(h,(h^{\prime})^{2},h^{\prime\prime})$ as
we cross a D8. Probably the simplest thing one can do is to place a stack of
D8 branes at the origin $r=0$ and bound $r$ between D8/O8 brane singularities
which are symmetric about this point. As such we can take $h$ to be globally
defined as

$h=\left\\{\begin{array}[]{l}-c_{1}-\frac{c_{2}}{3!}(r+r_{0})^{3}~{}~{}~{}~{}r<0\\\\[5.69054pt]
-c_{1}-\frac{c_{2}}{3!}(r_{0}-r)^{3}~{}~{}~{}~{}r>0\end{array}\right.$ (6.10)

This bounds the interval to $-r_{0}<r<r_{0}$ between D8/O8 singularities at
$r=\pm r_{0}$ and gives rise to a source for the $F_{0}$ of charge $2c_{2}$,
ie

$h^{\prime\prime\prime\prime}=-2c_{2}\delta(r)~{}~{}~{}~{}\Rightarrow~{}~{}~{}~{}dF_{0}=2c_{2}\frac{1}{2\pi}\delta(r)dr$
(6.11)

The form that the warp factors and metric take for this solution is depicted
in figure 1.

Figure 1: Plot of the warp factors in the metric and dilaton for an ${\cal
N}=(6,0)$ solution with D8 at $r=0$ bounded between D8/O8 singularities at
$r=\pm 3$ with the remaining constants in $h$ tuned as $c_{1}=2$, $c_{2}=5$

Given the Page fluxes in (4.52) (with $u=1$), and that we simply have round
$\mathbb{CP}^{3}$ for this solution for which we can take
$\int_{\mathbb{CP}^{n}}J_{2}^{n}=\pi^{n}$, it is a simple matter to compute
the Page charges of the fluxes over the $\mathbb{CP}^{n}$ sub-manifolds of
$\mathbb{CP}^{3}$. By tuning

$c_{1}=N_{2}-\frac{N_{5}^{3}N_{8}}{6},~{}~{}~{}~{}c_{2}=N_{8},~{}~{}~{}~{}r_{0}=N_{5},$
(6.12)

we find that these are given globally by

$\displaystyle 2\pi F_{0}$ $\displaystyle=N_{8}^{\mp}=\pm
N_{8},~{}~{}~{}~{}-\frac{1}{2\pi}\int_{\mathbb{CP}^{1}}\hat{f}_{2}=N_{5}N_{8},~{}~{}~{}~{}\frac{1}{(2\pi)^{3}}\int_{\mathbb{CP}^{2}}\hat{f}_{4}=\frac{N_{5}^{2}N^{\mp}_{8}}{2},$
$\displaystyle-\frac{1}{(2\pi)^{5}}\int_{\mathbb{CP}^{3}}\hat{f}_{6}$
$\displaystyle=N_{2},~{}~{}~{}~{}~{}-\frac{1}{(2\pi)^{2}}\int_{(r,\mathbb{CP}^{1})}H=N_{5}$
(6.13)

where the $\mp$ superscript indicates that we are on the side of the interior
D8 with $r\in\mathbb{R}^{\mp}$ and we have assumed for simplicity that $k=0$
globally in the NS 2-form.

With the expressions for the brane charges we can compute the holographic
charge via the string frame analogue of the formula presented in [61], namely

$c_{hol}=\frac{3}{2^{4}\pi^{6}}\int_{M_{7}}e^{A-2\Phi}\text{vol}(\text{M}_{7}),$
(6.14)

which gives the leading order contribution to the central charge of the
putative dual CFT. Given the class of solution is section 4 we find this
expression reduces to

$c_{hol}=\frac{1}{2}\int\frac{\Delta_{1}}{u^{2}}dr.$ (6.15)

For the case at hand one then finds that

$c_{hol}=N_{2}N_{5}^{2}N_{8}-\frac{3N_{5}^{5}N_{8}^{2}}{20}.$ (6.16)

The central charge of CFTs with $\mathfrak{osp}(n|2)$ superconformal symmetry
takes the form of (1.1), which in the limit of large level $k$ becomes $c=3k$.
The holographic central charge is not obviously of this form, however that
doesn’t mean it is necessarily not the leading contribution to something that
is141414Such scenarios are actually quite common, see for instance [62].. We
leave recovering this result from a CFT computation for future work.

We will now construct a globally bounded solution with interior D8 branes that
preserves $\mathcal{N}=(5,0)$ \- this time we will be more brief. There are
many options for gluing local solutions together for this less supersymmetric
case. We will choose to place a D8 brane in one of the bounded behaviour we
already found in section 5.2 in the absence of interior D8 branes (see table
1), namely will will insert a D8 in the solution bounded between O6 and O4
places. We remind the reader that we get local solutions containing these
singularities by tuning $h$ as

$\displaystyle h_{\text{O4}}$
$\displaystyle=\frac{1}{2}c_{1}\big{(}r_{0}^{2}-r_{0}(r-r_{0})+(r-r_{0})^{2}\big{)}+\frac{1}{3!}c_{2}(r-r_{0})^{3},$
$\displaystyle h_{\text{O6}}$
$\displaystyle=b_{1}\tilde{r}_{0}+b_{1}(r-\tilde{r}_{0})+\frac{1}{3!}b_{2}(r-\tilde{r}_{0})^{3}.$
(6.17)

where the singularity are located at $(r_{0},\tilde{r}_{0})$ respectively. We
will assume $r_{0},\tilde{r}_{0}>0$ and place a stack of D8s at a point
$r=r_{s}$ between the two O plane loci. The condition that the NS sector
should be continuous in this case amounts to imposing that

$(h_{\text{O4}},h^{\prime}_{\text{O4}},h^{\prime\prime}_{\text{O4}})\bigg{\lvert}_{r=r_{s}}=(h_{\text{O6}},h^{\prime}_{\text{O6}},h^{\prime\prime}_{\text{O6}})\bigg{\lvert}_{r=r_{s}},$
(6.18)

of course we also need the value of $F_{0}$ to change as we cross the D8. It
is indeed possible to solve the continuity condition in this case, which fixes
3 parameters, $(c_{1},c_{2},b_{1})$ say, leaving $(r_{0},\tilde{r}_{0},b_{2})$
as free parameters. A plot of this solution for a choice of
$(r_{0},\tilde{r}_{0},b_{2})$ is given in figure 2.

Figure 2: Plot of the warp factors in the metric and dilaton for an ${\cal
N}=(5,0)$ solution bounded between an O4 plane at $r=2$ and an O6 plane at
$r=8$ with a stack of D8 branes at $r=4$. The remaining parameter is tuned as
$b_{2}=-6$

## Acknowledgements

We thank Yolanda Lozano, Noppadol Mekareeya and Alessandro Tomasiello for
useful discussions. The work of NM is supported by the Ramón y Cajal
fellowship RYC2021-033794-I, and by grants from the Spanish government
MCIU-22-PID2021-123021NB-I00 and principality of Asturias SV-
PA-21-AYUD/2021/52177. AR is partially supported by the INFN grant “Gauge
Theories, Strings and Supergravity” (GSS).

##  Appendix A Derivation of spinors on $\widehat{\mathbb{CP}}\\!\\!~{}^{3}$

In this appendix we derive all spinors transforming in the 5 and 1 of
$\mathfrak{so}(5)$ on squashed $\mathbb{CP}^{3}$. We achieve this by starting
with known spinors in the 3 of $\mathfrak{so}(3)$ and 5 of $\mathfrak{so}(5)$
on the 7-sphere, and then reducing them to $\mathbb{CP}^{3}$.

###  A.1 Killing spinors and vectors on S7= SP(2)/SP(1)

The 7-sphere admits a parametrisation as an SP(2) bundle over SP(1), ie the
SP(2)/SP(1) co-set. For a unit radius 7-sphere this has the metric

$ds^{2}(\text{S}^{7})=\frac{1}{4}\bigg{[}d\alpha^{2}+\frac{1}{4}\sin^{2}\alpha(L_{1}^{i})^{2}+\big{(}L_{2}^{i}-\cos^{2}\left(\frac{\alpha}{2}\right)L_{1}^{i}\big{)}^{2}\bigg{]}$
(A.1)

where we take the following basis of SU(2) Left invariant 1-forms

$L^{1}_{1,2}+iL^{2}_{1,2}=e^{i\psi_{1,2}}(id\theta_{1,2}+\sin\theta_{1,2}d\phi_{1,2}),~{}~{}~{}~{}L^{3}_{1,2}=d\psi_{1,2}+\cos\theta_{1,2}d\phi_{1,2}.$
(A.2)

The 7-sphere admits two sets of Killing spinors obeying the relations

$\nabla_{a}\xi_{\pm}=\pm\frac{i}{2}\gamma_{a}\xi_{\pm}.$ (A.3)

With respect to the vielbein and flat space gamma matrices

$\displaystyle e^{1}$
$\displaystyle=\frac{1}{2}d\alpha,~{}e^{2,3,4}=\frac{1}{4}\sin\alpha
L^{1,2,3}_{1},~{}e^{5,6,7}=L_{2}^{1,2,3}-\cos^{2}\left(\frac{\alpha}{2}\right)L_{1}^{1,2,3},$
$\displaystyle\gamma_{1}$
$\displaystyle=\sigma_{1}\otimes\mathbb{I}_{2}\otimes\mathbb{I}_{2},~{}~{}~{}\gamma_{2,3,4}=\sigma_{2}\otimes\sigma_{1,2,3}\otimes\mathbb{I}_{2}~{}~{}~{}\gamma_{5,6,7}=\sigma_{3}\otimes\mathbb{I}_{2}\otimes\sigma_{1,2,3},$
(A.4)

where $\sigma_{1,2,3}$ are the Pauli-matrices, the Killing spinor equation
(A.3) is solved by

$\displaystyle\xi_{\pm}$ $\displaystyle={\cal M}_{\pm}\xi^{0}_{\pm},$ (A.5)
$\displaystyle{\cal M}_{\pm}$ $\displaystyle=e^{\frac{\alpha}{4}(\pm
i\gamma_{1}+Y)}e^{\mp i\frac{\psi_{1}}{2}\gamma_{7}P_{\mp}}e^{\mp
i\frac{\theta_{1}}{2}\gamma_{6}P_{\mp}}e^{\mp
i\frac{\phi_{1}}{2}\gamma_{7}P_{\mp}}e^{\frac{\psi_{2}}{4}(\pm
i\gamma_{7}+X)}e^{\frac{\theta_{2}}{2}(\gamma_{13}P_{+}\pm\gamma_{6}P_{\pm})}e^{\frac{\phi_{2}}{2}(\gamma_{14}P_{+}\pm
i\gamma_{7}P_{\pm})},$

where $\xi^{0}_{\pm}$ are unconstrained constant spinors and

$P_{\pm}=\frac{1}{2}(\mathbb{I}_{4}\pm\gamma_{1234}),~{}~{}~{}X=\gamma_{14}-\gamma_{23}-\gamma_{56},~{}~{}~{}~{}Y=-\gamma_{25}-\gamma_{36}-\gamma_{47}.$
(A.6)

It was shown in [51] that $\xi_{-}$ transform in the $(\textbf{2},\textbf{4})$
of $\mathfrak{sp}(1)\oplus\mathfrak{sp}(2)$ and $\xi_{+}$ in the
$\textbf{3}\oplus\textbf{5}$ of $\mathfrak{so}(3)\oplus\mathfrak{so}(5)$ \- it
is the latter that will be relevant to us here. Denoting the 3 and 5 as
$\xi^{i}_{3}$ for $i=1,...,3$ and $\xi^{\alpha}_{5}$ for $\alpha=1,...,5$ and
defining the 8 independent supercharges contained in $\xi_{+}$ as

$\xi^{I}_{+}={\cal M_{+}}\hat{\eta}^{I},~{}~{}I=1,...,8,$ (A.7)

where the $I^{th}$ entry of $\hat{\eta}^{I}$ is 1 and the rest zero, these are
given specifically by

$\displaystyle\hat{\xi}^{i}_{3}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}i(-\xi^{5}_{+}+\xi^{8}_{+})\\\
\xi^{5}_{+}+\xi^{8}_{+}\\\ i(\xi^{6}_{+}+\xi^{7}_{+})\\\
\end{array}\right)^{i},~{}~{}~{}\hat{\xi}^{\alpha}_{5}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}-i(\xi^{1}_{+}+\xi^{4}_{+})\\\
\xi^{1}_{+}-\xi^{4}_{+}\\\ -i(\xi^{2}_{+}-\xi^{3}_{+})\\\
\xi^{2}_{+}+\xi^{3}_{+}\\\
\xi^{6}_{+}-\xi^{7}_{+}\end{array}\right)^{\alpha},$ (A.16)

which obey

$\xi^{i{\dagger}}_{3}\xi_{3}^{j}=\delta^{ij},~{}~{}~{}~{}\xi^{\alpha{\dagger}}_{5}\xi_{5}^{\beta}=\delta^{\alpha\beta},~{}~{}~{}~{}\xi^{i{\dagger}}_{3}\xi_{5}^{\beta}=0,$
(A.17)

and are Majorana with respect to the intertwiner
$B=\sigma_{3}\otimes\sigma_{2}\otimes\sigma_{2}$. The specific Killing vectors
that make up the relevant SO(3) and SO(5) in the full space are made up of the
following isometries of the base and fibre metrics

$\displaystyle K^{1L}_{1,2}+iK^{2L}_{1,2}$
$\displaystyle=e^{i\phi_{1,2}}\left(i\partial_{\theta_{1,2}}+\frac{1}{\sin\theta_{1,2}}\partial_{\psi_{1,2}}-\frac{\cos\theta_{1,2}}{\sin\theta_{1,2}}\partial_{\phi_{1,2}}\right),~{}~{}~{}K^{3L}_{1,2}=-\partial_{\phi_{1,2}},$
(A.18a) $\displaystyle K^{1R}_{1,2}+iK^{2R}_{1,2}$
$\displaystyle=e^{i\psi_{1,2}}\left(i\partial_{\theta_{1,2}}+\frac{1}{\sin\theta_{1,2}}\partial_{\phi_{1,2}}-\frac{\cos\theta_{1,2}}{\sin\theta_{1,2}}\partial_{\psi_{1,2}}\right),~{}~{}~{}K^{3R}_{1,2}=\partial_{\psi_{1,2}},$
(A.18b) $\displaystyle\hat{K}^{A}_{\text{SO(5)/SO(4)}}$
$\displaystyle=-(\mu_{A}\partial_{\alpha}+\cot\alpha\partial_{x_{i}}\mu_{A}g_{3}^{ij}\partial_{x_{j}}),~{}~{}~{}A=1,...,4$
(A.18c)

where $\mu_{A}$ are embedding coordinates for the S${}^{3}\subset$ S4,
$g_{3}^{ij}$ is the inverse metric of this 3-sphere and
$x_{i}=(\theta_{1},\phi_{1},\psi_{1})_{i}$, we have specifically

$\mu_{A}=\bigg{(}\sin\left(\frac{\theta_{1}}{2}\right)\cos\left(\frac{\phi_{-}}{2}\right),~{}\sin\left(\frac{\theta_{1}}{2}\right)\sin\left(\frac{\phi_{-}}{2}\right),~{}\cos\left(\frac{\theta_{1}}{2}\right)\cos\left(\frac{\phi_{+}}{2}\right),~{}-\cos\left(\frac{\theta_{1}}{2}\right)\sin\left(\frac{\phi_{+}}{2}\right)\bigg{)}_{A},$

for $\phi_{\pm}=\phi_{1}\pm\psi_{1}$. In terms of the isometries on the base
and fibre we define the following Killing vectors on the 7-sphere

$K^{iD}=K^{iR}_{1}+K^{iR}_{2},~{}~{}~{}~{}K^{A}_{\text{SO(5)/SO(4)}}=\hat{K}^{A}_{\text{SO(5)/SO(4)}}+\cot\left(\frac{\alpha}{2}\right)\mu_{B}(\kappa_{A})^{B}_{i}K^{iR}_{2}.$
(A.19)

where

$\kappa_{1}=\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&-1\\\ 0&-1&0\\\
-1&0&0\end{array}\right),~{}~{}~{}\kappa_{2}=\left(\begin{array}[]{ccc}0&0&1\\\
0&0&0\\\ -1&0&0\\\
0&1&0\end{array}\right),~{}~{}~{}\kappa_{3}=\left(\begin{array}[]{ccc}0&1&0\\\
1&0&0\\\ 0&0&0\\\
0&0&-1\end{array}\right),~{}~{}~{}\kappa_{4}=\left(\begin{array}[]{ccc}1&0&0\\\
0&-1&0\\\ 0&0&1\\\ 0&0&0\end{array}\right).$

The isometry groups in the full space are spanned by

$\displaystyle\text{SO}(3)$ $\displaystyle:~{}K^{iL}_{2}$
$\displaystyle\text{SO}(5)$
$\displaystyle:~{}(K^{iL}_{1},~{}K^{iD},~{}K^{A}_{\text{SO(5)/SO(4)}})$ (A.20)

Another Killing vector on S7 that will be relevant is

$\tilde{K}=y_{i}K^{iR}_{1},~{}~{}~{}~{}~{}y_{i}=(\cos\psi_{2}\sin\theta_{2},~{}\sin\psi_{2}\sin\theta_{2},~{}\cos\theta_{2}).$
(A.21)

In terms of this one can define Killing vectors that together with the SO(5)
Killing vectors span SO(6), namely

$\text{SO(6)/SO(5)}:~{}([K^{A}_{\text{SO(5)/SO(4)}},~{}\tilde{K}],~{}\tilde{K})$
(A.22)

###  A.2 Reduction to $\mathbb{CP}^{3}$

It is possible to rewrite (A.1) as fibration of $\partial_{\phi_{2}}$ over
$\mathbb{CP}^{3}$ as

$\displaystyle ds^{2}(\text{S}^{7})$
$\displaystyle=ds^{2}(\mathbb{CP}^{3})+\frac{1}{4}\left(d\phi_{2}+\cos\theta_{2}d\psi_{2}-\cos^{2}\left(\frac{\alpha}{2}\right)y_{i}L_{1}^{i}\right)^{2},$
$\displaystyle ds^{2}(\mathbb{CP}^{3})$
$\displaystyle=\frac{1}{4}\bigg{[}d\alpha^{2}+\frac{1}{4}\sin^{2}\alpha(L_{1}^{i})^{2}+Dy_{i}^{2}\bigg{]},~{}~{}~{}~{}Dy_{i}=dy_{i}+\cos^{2}\left(\frac{\alpha}{2}\right)\epsilon_{ijk}y_{j}L^{k}_{1}.$
(A.23)

This can be achieved by rotating the 5,6,7 components of the vielbein in (A.1)
by

$e^{i}\to\Lambda^{i}_{~{}j}e^{j},~{}~{}~{}~{}\Lambda=\left(\begin{array}[]{ccc}-\sin\psi_{2}&\cos\psi_{2}&0\\\
-\cos\theta_{2}\cos\psi_{2}&-\cos\theta_{2}\sin\psi_{2}&\sin\theta_{2}\\\
\sin\theta_{2}\cos\psi_{2}&\sin\psi_{2}\sin\theta_{2}&\cos\theta_{2}\end{array}\right).$
(A.24)

The corresponding action on the spinors is defined through the matrix

$\Omega=e^{\frac{\theta_{2}}{2}\gamma_{67}}e^{(\frac{\psi_{2}}{2}+\frac{\pi}{4})\gamma_{56}}.$
(A.25)

The 3 and 5 in the rotated frame then take the form

$\xi_{3}^{i}=\Omega\hat{\xi}^{i}_{3},~{}~{}~{}~{}\xi_{5}^{\alpha}=\Omega\hat{\xi}_{5}^{\alpha}.$
(A.26)

Any component of these spinor multiplets that is un-charged under
$\partial_{\phi_{2}}$ is spinor on $\mathbb{CP}^{3}$, as we have rotated to a
frame where translational invariance in $\phi_{2}$ is manifest, this is
equivalent to choosing the parts of $(\xi_{3}^{i},\xi^{\alpha}_{5})$ that are
independent of $\phi_{2}$. It is not hard to establish that this is all of
$\xi^{\alpha}_{5}$ and $\xi_{3}^{3}$, which being a singlet under SO(5) we now
label

$\xi_{0}=\xi_{3}^{3}.$ (A.27)

The chirality matrix on $\mathbb{CP}^{3}$ is identified as
$\hat{\gamma}=\gamma_{7}$, which is clearly an SO(5) singlet, so we can
construct an additional SO(5) quintuplet and singlet by acting with this.
Additionally we can define a set of embedding coordinates on S4 via

$i\xi_{0}^{{\dagger}}\gamma_{7}\xi^{\alpha}_{5}=Y_{\alpha},~{}~{}~{}~{}Y_{\alpha}=(\sin\alpha\mu_{A},~{}\cos\alpha),$
(A.28)

where $\mu_{A}$ are embedding coordinates on S3 defined in (A.1) - as the left
hand side of this expression is a quintuplet so too are these embedding
coordinates. In summary we have the following Majorana spinors on
$\mathbb{CP}^{3}$ respecting the $\textbf{5}\oplus\textbf{1}$ branching of
SO(6) under its SO(5) subgroup

$\textbf{5}:~{}(\xi^{\alpha}_{5},~{}i\gamma_{7}\xi^{\alpha}_{5},~{}Y_{\alpha}\xi_{0},~{}iY_{\alpha}\gamma_{7}\xi_{0}),~{}~{}~{}~{}~{}\textbf{1}:~{}(\xi_{0},~{}i\gamma_{7}\xi_{0}),$
(A.29)

which can be used to construct ${\cal N}=(5,0)$ and ${\cal N}=(1,0)$ AdS3
solutions respectively. One might wonder if one can generate additional spinor
in the 1 or 5 by acting with the SO(5) invariant forms one can define on
$\mathbb{CP}^{3}$. These are quoted in the main text in (2.5) and (2.6), one
can show that on round unit radius $\mathbb{CP}^{3}$

$\displaystyle\nu^{1}_{2}\xi_{5}^{\alpha}$
$\displaystyle=-2Y_{\alpha}\xi_{0}+i\gamma_{7}\xi_{5}^{\alpha},~{}~{}\nu^{2}_{2}\xi_{5}^{\alpha}=-2Y_{\alpha}\xi_{0},~{}~{}\text{Re}\Omega_{3}\xi_{5}^{\alpha}=-4iY_{\alpha}\xi_{0},~{}~{}\text{Im}\Omega_{3}\xi_{5}^{\alpha}=-4Y_{\alpha}\gamma_{7}\xi_{0},$
$\displaystyle\nu^{1}_{2}\xi_{0}$
$\displaystyle=-i\gamma_{7}\xi_{0},~{}~{}\nu^{2}_{2}\xi_{0}=-2i\xi_{0},~{}~{}\text{Re}\Omega_{3}\xi_{0}=-4\gamma_{7}\xi_{0},~{}~{}\text{Im}\Omega_{3}\xi_{0}=4i\gamma_{7}\xi_{0},$
(A.30)

where $2\nu_{2}^{1}=\tilde{J}_{2}-J_{2}$, $2\nu_{2}^{2}=\tilde{J}_{2}+J_{2}$
and the forms should be understood as acting on the spinors through the
Clifford map
$X_{n}\to\frac{1}{n!}(X_{n})_{a_{1}...a_{n}}\gamma^{a_{1}...a_{n}}$, so (A.29)
are in fact exhaustive. The SO(6) Killing vectors on $\mathbb{CP}^{3}$ are
given by (A.20) and (A.22) but with the $\partial_{\phi_{2}}$ dependence
omitted, the SO(5) vectors are still Killing when one allows the base S4 to
have a different radi to the S2 in the $\mathbb{CP}^{3}$ metric (ie for
squashed $\mathbb{CP}^{3}$) this however breaks the SO(6)/SO(5) isometry.
Finally note that $\xi_{0}$ are actually charged under SO(6)/SO(5), and more
specifically when these isometries are not broken (ie $\mathbb{CP}^{3}$ is not
squashed) then we have the following independent SO(6) sextuplets

$\textbf{6}:~{}\xi^{\cal I}_{6}=\left(\begin{array}[]{c}\xi_{5}^{\alpha}\\\
\xi_{0}\end{array}\right)^{\cal I},~{}~{}~{}~{}\hat{\xi}^{\cal
I}_{6}=\left(\begin{array}[]{c}i\gamma_{7}\xi_{5}^{\alpha}\\\
i\gamma_{7}\xi_{0}\end{array}\right)^{\cal I},$ (A.31)

which can be used to construct ${\cal N}=(6,0)$ AdS3 solutions.

##  Appendix B The SO(3)${}_{L}\times$SO(3)D invariant ${\cal N}=5$ bi-linears

In the main text we will need to construct ${\cal N}=5$ bi-linears on the
space (2.2), the non trivial part of this computation comes from the bi-
linears on squashed $\mathbb{CP}^{3}$ \- in this appendix we shall compute
them.

As explained in the main text it is sufficient to solve the supersymmetry
constraints for an ${\cal N}=1$ sub-sector of the quintuplet of SO(5) spinors
defined on the internal space. A convenient component to work with is the 5th
as this is a singlet under an SO(4) subgroup of SO(5). Specifically with
respect to (2.2) and the discussion below it, $\chi^{5}_{1,2}$ are singlets
with respect to SO(4)=SO(3)${}_{L}\otimes$SO(3)D. As such the bi-linears that
follow from $\chi^{5}_{1,2}$ must decompose in a basis of the
SO(3)${}_{L}\otimes$SO(3)D invariant forms on the S${}^{2}\times$S3 fibration
(2.3) and what one can form from these through taking wedge products. The
$d=7$ spinors $\chi^{5}_{1,2}$ depend on $\hat{\mathbb{CP}}^{3}$ through

$\eta_{\pm}=\xi^{5}_{5}\pm
iY_{5}\hat{\gamma}\xi_{0},~{}~{}~{}~{}Y_{5}\xi_{0},~{}~{}~{}~{}Y_{5}i\hat{\gamma}\xi_{0}$
(B.1)

where these are all defined in the previous appendix - it is the bi-linears we
can construct out of these that will be relevant to us. One can show that

$\displaystyle\eta_{\pm}\otimes\eta_{\pm}^{{\dagger}}$
$\displaystyle=\phi^{1}_{+}\pm
i\phi^{1}_{-},~{}~{}~{}~{}~{}\eta_{\pm}\otimes\eta_{\mp}^{{\dagger}}=\pm\phi^{2}_{+}+i\phi^{2}_{-},$
$\displaystyle\eta_{+}\otimes\xi_{0}^{{\dagger}}$
$\displaystyle=\phi^{3}_{+}+i\phi^{3}_{-},~{}~{}~{}~{}~{}\eta_{-}\otimes(i\hat{\gamma}\xi_{0})^{{\dagger}}=\phi^{3}_{+}-i\phi^{3}_{-},$
$\displaystyle\eta_{+}\otimes(i\hat{\gamma}\xi_{0})^{{\dagger}}$
$\displaystyle=\phi^{4}_{+}+i\phi^{4}_{-},~{}~{}~{}~{}\eta_{-}\otimes\xi_{0}^{{\dagger}}=-\phi^{4}_{+}+i\phi^{4}_{-},$
$\displaystyle\xi_{0}\otimes\eta_{+}^{{\dagger}}$
$\displaystyle=\phi^{5}_{+}+i\phi^{5}_{-},~{}~{}~{}~{}~{}~{}i\hat{\gamma}\xi_{0}\otimes\eta_{-}^{{\dagger}}=\phi^{5}_{+}-i\phi^{5}_{-},$
$\displaystyle i\hat{\gamma}\xi_{0}\otimes\eta_{+}^{{\dagger}}$
$\displaystyle=\phi^{6}_{+}+i\phi^{6}_{-},~{}~{}~{}~{}~{}~{}\xi_{0}\otimes\eta_{-}^{{\dagger}}=-\phi^{6}_{+}+i\phi^{6}_{-},$
$\displaystyle\xi_{0}\otimes\xi_{0}^{{\dagger}}$
$\displaystyle=\phi^{7}_{+}+i\phi_{-}^{7},~{}~{}~{}~{}~{}i\hat{\gamma}\xi_{0}\otimes(i\hat{\gamma}\xi_{0})^{{\dagger}}=\phi^{7}_{+}-i\phi_{-}^{7},$
$\displaystyle i\hat{\gamma}\xi_{0}\otimes\xi_{0}^{{\dagger}}$
$\displaystyle=\phi^{8}_{+}+i\phi^{8}_{-},~{}~{}~{}~{}~{}\xi_{0}\otimes(i\hat{\gamma}\xi_{0})^{{\dagger}}=-\phi^{8}_{+}+i\phi^{8}_{-},$
(B.2)

where $\phi^{1...8}_{\pm}$ are real bi-linears of even/odd form degree, they
take the form

$\displaystyle\phi^{1}_{+}$
$\displaystyle=\frac{1}{8}\sin^{2}\alpha\bigg{(}1-\frac{1}{32}e^{2(C+D)}\sin^{2}\alpha\omega_{2}^{2}\wedge\omega_{2}^{2}+\frac{1}{16}e^{2C}\sin\alpha
d\alpha\wedge\omega_{1}\wedge\big{(}e^{2D}\omega_{2}^{1}-e^{2C}\sin^{2}\alpha\omega^{4}_{2}\big{)}\bigg{)},$
$\displaystyle\phi^{1}_{-}$
$\displaystyle=\frac{1}{64}e^{2C+D}\sin^{4}\alpha\omega_{1}\wedge\omega^{2}_{2},~{}~{}~{}~{}\phi^{2}_{-}=-\frac{1}{64}e^{2C+D}\sin^{3}\alpha\bigg{(}\sin\alpha\omega_{1}\wedge\omega^{3}_{2}+d\alpha\wedge\omega^{2}_{2}\bigg{)}$
$\displaystyle\phi^{2}_{+}$
$\displaystyle=\frac{1}{32}\sin^{2}\alpha\bigg{(}e^{2C}\sin^{2}\alpha\omega^{4}_{2}-e^{2D}\omega^{1}_{2}+e^{2C}\sin\alpha
d\alpha\wedge\omega_{1}\wedge\big{(}1-\frac{1}{32}e^{2(C+D)}\sin^{2}\alpha\omega_{2}^{2}\wedge\omega_{2}^{2}\big{)}\bigg{)},$
$\displaystyle\phi^{3}_{+}$
$\displaystyle=\frac{1}{32}\sin^{2}\alpha\bigg{(}-e^{C+D}\omega^{2}_{2}+\frac{1}{4}e^{3C+D}\sin\alpha
d\alpha\wedge\omega_{1}\wedge\omega_{2}^{3}\bigg{)},$
$\displaystyle\phi^{3}_{-}$
$\displaystyle=\frac{1}{16}\sin\alpha\bigg{(}e^{C}\sin\alpha\omega_{1}\wedge\big{(}1-\frac{1}{32}e^{2(C+D)}\sin^{2}\alpha\omega_{2}^{2}\wedge\omega_{2}^{2}\big{)}+\frac{1}{4}d\alpha\wedge\big{(}e^{C+2D}\omega^{1}_{2}-e^{3C}\sin^{2}\alpha\omega_{2}^{4}\big{)}\bigg{)},$
$\displaystyle\phi^{4}_{+}$
$\displaystyle=-\frac{1}{32}e^{C+D}\sin^{2}\alpha\bigg{(}\omega^{3}_{2}+\frac{1}{4}e^{2C}\sin\alpha
d\alpha\wedge\omega_{1}\wedge\omega^{2}_{2}\bigg{)},~{}~{}~{}~{}\phi^{5}_{+}=\frac{1}{32}e^{C+D}\sin^{2}\alpha\bigg{(}\omega^{2}_{2}+\frac{1}{4}e^{2C}\sin\alpha
d\alpha\wedge\omega_{1}\wedge\omega^{3}_{2}\bigg{)},$
$\displaystyle\phi^{4}_{-}$
$\displaystyle=\frac{1}{16}e^{C}\sin\alpha\bigg{(}\frac{1}{4}\sin\alpha\omega_{1}\wedge\big{(}e^{2D}\omega_{2}^{1}-e^{2C}\sin^{2}\alpha\omega_{2}^{4}\big{)}-d\alpha\wedge\big{(}1-\frac{1}{32}e^{2(C+D)}\sin^{2}\alpha\wedge\omega^{2}_{2}\wedge\omega^{2}_{2}\big{)}\bigg{)},$
$\displaystyle\phi^{5}_{-}$
$\displaystyle=\frac{1}{16}e^{C}\sin\alpha\bigg{(}\frac{1}{4}d\alpha\wedge\big{(}e^{2D}\omega^{1}_{2}-e^{2C}\sin^{2}\alpha\omega_{2}^{4}\big{)}-\sin\alpha\omega_{1}\wedge\big{(}1-\frac{1}{32}e^{2(C+D)}\sin^{2}\alpha\omega_{2}^{2}\wedge\omega_{2}^{2}\big{)}\bigg{)},$
$\displaystyle\phi^{6}_{+}$
$\displaystyle=\frac{1}{32}e^{C+D}\sin^{2}\alpha\bigg{(}\omega^{3}_{2}-\frac{1}{4}e^{2C}\sin\alpha
d\alpha\wedge\omega_{1}\wedge\omega_{2}^{2}\bigg{)},$
$\displaystyle\phi^{6}_{-}$
$\displaystyle=\frac{1}{16}e^{C}\sin\alpha\bigg{(}d\alpha\wedge\big{(}1-\frac{1}{32}e^{2(C+D)}\sin^{2}\alpha\omega^{2}_{2}\wedge\omega^{2}_{2}\big{)}+\frac{1}{4}\sin\alpha\big{(}e^{2D}\omega^{1}_{2}-e^{2C}\sin^{2}\alpha\omega^{4}_{2}\big{)}\bigg{)},$
$\displaystyle\phi^{7}_{+}$
$\displaystyle=\frac{1}{8}\bigg{(}1+\frac{1}{16}e^{2C}\sin\alpha
d\alpha\wedge\omega_{1}\wedge\big{(}-e^{2D}\omega^{1}_{2}+e^{2C}\sin^{2}\alpha\omega^{4}_{2}\big{)}-\frac{1}{32}e^{2(C+D)}\sin^{2}\alpha\omega^{2}_{2}\wedge\omega^{2}_{2}\bigg{)},$
$\displaystyle\phi^{7}_{-}$
$\displaystyle=\frac{1}{64}e^{2C+D}\sin\alpha\bigg{(}d\alpha\wedge\omega^{3}_{2}+\sin\alpha\omega^{1}\wedge\omega^{2}_{2}\bigg{)},~{}~{}~{}~{}\phi^{8}_{-}=\frac{1}{64}e^{2C+D}\sin\alpha\bigg{(}-d\alpha\wedge\omega^{2}_{2}+\sin\alpha\omega_{1}\wedge\omega^{3}_{2}\bigg{)}$
$\displaystyle\phi^{8}_{+}$
$\displaystyle=-\frac{1}{32}\bigg{(}e^{2C}\sin\alpha
d\alpha\wedge\omega_{1}\wedge\big{(}1-\frac{1}{32}e^{2(C+D)}\sin^{2}\alpha\omega_{2}^{2}\wedge\omega_{2}^{2}\big{)}+e^{2D}\omega^{1}_{2}-e^{2C}\sin^{2}\alpha\omega^{4}_{2}\bigg{)}$
(B.3)

##  Appendix C Ruling out $\mathfrak{osp}(7|2)$ AdS3 vacua

In this appendix we shall prove that all ${\cal N}=7$ AdS3 solutions
preserving the algebra $\mathfrak{osp}(7|2)$ are locally AdS${}_{4}\times$S7.

$\mathfrak{osp}(7|2)$ necessitates an SO(7) R-symmetry with spinor
transforming in the 7, there is only one way to achieve this. On needs a round
7-sphere in the metric with fluxes that break its SO(8) isometry to SO(7) in
terms of the weak G2 structure 3-forms one can define. Such an ansatz in type
II can be ruled out a the level of the equations of motion [51], our focus
here then will be on $d=11$ supergravity.

All AdS3 solutions of 11 dimensions supergravity admit a decomposition of
their bosonic fields as

$ds^{2}=e^{2A}ds^{2}(\text{AdS}_{3})+ds^{2}(\text{M}_{8}),~{}~{}~{}~{}G=e^{3A}\text{vol}(\text{AdS}_{3})\wedge
F_{1}+F_{4}$ (C.1)

where $F_{1},F_{4},A$ have support on $\text{M}_{8}$ only. We take AdS3 to
have inverse radius $m$. When a solutions is supersymmetric M8 supports (at
least one) Majorana spinor $\chi$ that one can use to define the following bi-
linears

$\displaystyle 2e^{A}$
$\displaystyle=|\chi|^{2},~{}~{}~{}2e^{A}f=\chi^{{\dagger}}\hat{\gamma}^{(8)}\chi,~{}~{}~{}~{}2e^{A}K=\chi^{{\dagger}}_{+}\gamma^{(8)}_{a}\chi_{-}e^{a},$
$\displaystyle 2e^{A}\Psi_{3}$
$\displaystyle=\frac{1}{3!}\chi^{\dagger}\gamma^{(8)}_{abc}\hat{\gamma}^{(8)}\chi
e^{abc},~{}~{}~{}2e^{A}\Psi_{4}=\frac{1}{4!}\chi^{\dagger}\gamma^{(8)}_{abcd}\chi
e^{abcd}$ (C.2)

where $\gamma^{(8)}_{a}$ are eight-dimensional flat space gamma matrices,
$\hat{\gamma}^{(8)}=\gamma^{(8)}_{12345678}$ is the chirality matrix and
$e^{a}$ is a vielbein on M8. Sufficient conditions for ${\cal N}=1$
supersymmetry to hold can be caste as the following differential conditions
the bi-linears should obey [51]

$\displaystyle d(e^{2A}K)=0,$ (C.3a) $\displaystyle
d(e^{3A}f)-e^{3A}F_{1}-2me^{2A}K=0,$ (C.3b) $\displaystyle
d(e^{3A}\Psi_{3})-e^{3A}(-\star_{8}F_{4}+fF_{4})+2me^{2A}\Psi_{4}=0,$ (C.3c)
$\displaystyle d(e^{2A}\Psi_{4})-e^{2A}K\wedge F_{4}=0,$ (C.3d) $\displaystyle
6\star_{8}dA-2f\star_{8}F_{1}+\Psi_{3}\wedge F_{4}=0,$ (C.3e) $\displaystyle
6e^{-A}m\star_{8}K-6f\star_{8}dA+2\star_{8}F_{1}+\Psi_{3}\wedge\star_{8}F_{4}=0$
(C.3f)

where $\star_{8}$ is the hodge dual on the M8. These conditions do not imply
all of the equations of motion of 11 dimensional supergravity however. For
that to follow one must additionally solve the Bianchi identity and equation
of motion of the 4-form flux $G$. Away from the loci of sources, this amounts
to imposing that

$d(F_{4})=0,~{}~{}~{}~{}d(\star_{8}F_{1})-\frac{1}{2}F_{4}\wedge F_{4}=0.$
(C.4)

The only way to realise the SO(7) R-symmetry that $\mathfrak{osp}(7|2)$
necessitates on a 8d space is to take it to be a foliation of the SO(7)/G2 co-
set over an interval. As explained at greater length in section 6.2 of [51],
the metric on this co-set is the round one, but the flux can depend also
depend on a SO(7) invariant 3-form $\phi^{0}_{3}$ such that (C.1) should be
refined as

$\displaystyle ds^{2}(\text{M}_{8})$
$\displaystyle=e^{2B}ds^{2}(\text{S}^{7})+e^{2k}dr^{2},~{}~{}~{}~{}e^{3A}F_{1}=f_{1}dr,~{}~{}~{}F_{4}=4f_{2}\star_{7}\phi^{0}_{3}+f_{3}dr\wedge\phi^{0}_{3}.$
(C.5)

where $(e^{2A},e^{2B},e^{2k},f_{i})$ are functions of the interval only. The
SO(7) invariants obey the following relations

$d\phi^{0}_{3}=4\star_{7}\phi^{0}_{3},~{}~{}~{}\phi^{0}_{3}\wedge\star_{7}\phi^{0}_{3}=7\text{vol}(\text{S}^{7}),$
(C.6)

ie they define the structure of a manifold of weak G2 holonomy. More
specifically, decomposing

$ds^{2}(\text{S}^{7})=d\alpha^{2}+\sin^{2}\alpha ds^{2}(\text{S}^{6})$ (C.7)

One has

$\displaystyle\phi^{0}_{3}$ $\displaystyle=\sin^{2}\alpha d\alpha\wedge
J_{\text{G}_{2}}+\sin^{3}\\!\alpha\,\text{Re}(e^{-i\alpha}\Omega_{\text{G}_{2}}),~{}~{}~{}\star_{7}\phi^{0}_{3}=-\frac{1}{2}\sin^{4}\\!\alpha\,J_{\text{G}_{2}}\wedge
J_{\text{G}_{2}}-\sin^{3}\\!\alpha\,d\alpha\wedge\text{Im}(e^{-i\alpha}\Omega_{\text{G}_{2}}),$
$\displaystyle J_{\text{G}_{2}}$ $\displaystyle=\frac{1}{2}{\cal
C}_{ijk}Y_{\text{S}^{6}}^{i}dY_{\text{S}^{6}}^{j}\wedge
dY_{\text{S}^{6}}^{k},~{}~{}~{}\Omega_{\text{G}_{2}}=\frac{1}{3!}(1-i\iota_{d\alpha}\star_{6}){\cal
C}_{ijk}dY_{\text{S}^{6}}^{i}\wedge dY_{\text{S}^{6}}^{j}\wedge
dY_{\text{S}^{6}}^{k},$ (C.8)

where $Y_{\text{S}^{6}}^{i}$ are unit norm embedding coordinates for
$\text{S}^{6}$ and ${\cal C}_{ijk}$ are the structure constants defining the
product between the octonions, ie
$o^{i}o^{j}=-\delta^{ij}+\mathcal{C}^{ijk}o_{k}$. The Killing spinors on unit
radius S7 obeying the equation

$\nabla^{(7)}_{a}\xi=\frac{i}{2}\gamma^{(7)}_{a}\xi,$ (C.9)

branch as $\textbf{1}+\textbf{7}$ under the SO(7) subgroup of SO(8), we denote
the portions of $\xi$ that transform in these reps as respectively $\xi^{0}$
and $\xi^{I}_{\textbf{7}}$, they can be extracted from the relations

$\textbf{1}:~{}(\phi^{0}_{3}+\frac{i}{7})\xi=0,~{}~{}~{}\textbf{7}:~{}(\phi^{0}_{3}-i)\xi=0,$
(C.10)

where both the 1 and 7 are Majorana. Acting with the SO(7) invariants on
$\xi^{I}_{\textbf{7}}$ does not generate any additional spinors in the 7, and
we can without loss of generality take

$|\xi^{0}|^{2}=|\xi_{\textbf{7}}^{I}|^{2}=1,~{}~{}~{}\xi^{0{\dagger}}\xi_{\textbf{7}}^{I}=0.$
(C.11)

Thus we only have 1 spinor in the 7 and the most general Majorana spinors we
can write on M8 are $\chi=\sqrt{2}e^{\frac{A}{2}}(\chi_{+}+\chi_{-})$
where151515We define the 8d gamma matrices as
$\gamma^{(8)}_{a}=\sigma_{1}\otimes\gamma^{(7)}_{a}$ for $a=1,..,7$ and
$\gamma^{(8)}_{8}=\sigma_{2}\otimes\mathbb{I}$ where the intertwiner defining
Majorana conjugation is $B^{(8)}=\sigma_{3}\otimes B^{(7)}$. This is the
reason for the form that the interval components of the spinors take.

$\chi_{+}=\left(\begin{matrix}a_{+}\\\
0\end{matrix}\right)\otimes\xi_{\textbf{7}}^{I},~{}~{}~{}\chi_{+}=\left(\begin{matrix}0\\\
ia_{-}\end{matrix}\right)\otimes\xi_{\textbf{7}}^{I}$ (C.12)

where $a_{\pm}$ are real functions subject to $|a_{+}|^{2}+|a_{-}|^{2}=1$ \-
which are clearly rather constrained. The bi-linears of each component of
$\xi_{\textbf{7}}^{I}$ give rise to another 7 weak G2 holonomy 3-forms as

$\xi^{(I)}_{\textbf{7}}\otimes\xi^{(I){\dagger}}_{\textbf{7}}=\frac{1}{8}\big{(}1+i\phi_{3}^{(I)}+\star_{7}\phi_{3}^{(I)}+\text{vol}(\text{S}^{7})\big{)}.$
(C.13)

As $\phi_{3}^{(I)}$ are charged under SO(7) they are clearly all independent
of $\phi^{0}_{3}$, so there is no way to generate the invariant forms in the
flux in (C.5) from (C.3c)-(C.3f), thus we must have

$f_{2}=f_{3}=0,~{}~{}~{}~{}\Rightarrow~{}~{}~{}F_{4}=0.$ (C.14)

This makes the flux purely electric and it is proved in [23], that for all
such solutions AdS3 experiences an enhancement to AdS4. As there is no longer
anything breaking the isometries of the 7-sphere locally, clearly then this
ansatz just leads to local AdS${}_{4}\times$S7 . The only global possibility
beyond the standard ${\cal N}=8$ M2 brane near horizon is an orbifolding of
the 7-sphere that breaks supersymmetry to ${\cal N}=7$ \- in any case this is
certainly in no way an AdS3 vacuum.

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