Holographic $\frac{1}{2}$-BPS surface defects in ABJM

Yolanda<EMAIL_ADDRESS>Niall T.
<EMAIL_ADDRESS>Nicolò
<EMAIL_ADDRESS>Anayeli<EMAIL_ADDRESS>

a: Department of Physics, University of Oviedo, Avda. Federico Garcia Lorca
s/n, 33007 Oviedo

and

b: Instituto Universitario de Ciencias y Tecnologías Espaciales de Asturias
(ICTEA),
Calle de la Independencia 13, 33004 Oviedo, Spain

c: Department of Physics, Ben-Gurion University of the Negev, Be’er-Sheva
84105, Israel

d: Dipartimento di Fisica, Università degli studi di Milano–Bicocca,
and INFN, Sezione di Milano–Bicocca,
Piazza della Scienza 3, 20126 Milano, Italy

Abstract

We study the class of $\text{AdS}_{3}\times\mathbb{CP}^{3}$ solutions to
massive Type IIA supergravity with $\mathfrak{osp}(6|2)$ superconformal
algebra recently constructed in [1]. These solutions are foliations over an
interval preserving $\mathcal{N}=(0,6)$ supersymmetry in two dimensions, that
in the massless limit can be mapped to the
$\text{AdS}_{4}\times\mathbb{CP}^{3}$ solution of ABJM/ABJ. We show that in
the massive case extra NS5-D8 branes, that we interpret as $\frac{1}{2}$-BPS
surface defect branes within the ABJ theory, backreact in the geometry and
turn one of the 3d field theory directions onto an energy scale, generating a
flow towards a 2d CFT. We construct explicit quiver field theories that we
propose flow in the IR to the $(0,6)$ SCFTs dual to the solutions. Finally, we
show that the $\text{AdS}_{3}$ solutions realise geometrically, in terms of
large gauge transformations, an extension to the massive case of Seiberg
duality in ABJ theories proposed in the literature.

###### Contents

  1. 1 Introduction
  2. 2 The class of ${\cal N}=(0,6)$ AdS${}_{3}\times\mathbb{CP}^{3}$ solutions to massive IIA
    1. 2.1 The massless case: The ABJM/ABJ solution
    2. 2.2 Analysis of global solutions with D8 branes
  3. 3 A new class of $\mathcal{N}=(0,4)$ $\text{AdS}_{3}$ solutions to Type IIB
  4. 4 The dual field theory
    1. 4.1 Hanany-Witten brane set-up
    2. 4.2 Building blocks
    3. 4.3 2d quivers
      1. 4.3.1 Completion of the quivers
    4. 4.4 Connection with Seiberg duality
    5. 4.5 A comment on the central charge
      1. 4.5.1 The central charge as a product of electric and magnetic charges
  5. 5 Conclusions

##  1 Introduction

The study of the AdS/CFT correspondence in low dimensions has been the subject
of intense research in the last years, particularly regarding the
classification of new solutions. This study is prompted by the key role played
by $\text{AdS}_{3}$ and $\text{AdS}_{2}$ spaces as near horizon geometries of
extremal black strings/holes, and the unique possibility the AdS/CFT
correspondence offers for their microscopical description. More recently, low
dimensional holography has also found fruitful applications in the description
of conformal defects in higher dimensional CFTs. These defects are typically
realised as operator insertions that reduce the conformality and the number of
supersymmetries of the higher dimensional theory where they are embedded. When
the number of defects is large they backreact in the geometry and if some
conformality is still preserved give rise to lower dimensional AdS geometries.
The AdS/CFT correspondence formulated in these lower dimensional spaces offers
a useful framework to study these defects, especially when they are strongly
interacting.

The variety of low dimensional AdS backgrounds is especially rich, given the
many possibilities for the geometry of their internal spaces and the large
variety of superconformal algebras that exist in low dimensions. Many new low
dimensional AdS solutions have been reported in the last years whose
construction has profited from the development of powerful techniques
involving bispinors and G-structures. In this paper we will concentrate on
$\text{AdS}_{3}/\text{CFT}_{2}$ holography. In these dimensions new results
have been reported in [2]-[36], plus the already mentioned [1], that will be
the focus of this paper.

Remarkably, some of these classification efforts have been accompanied by
interesting investigations of the dual CFTs. Notable examples where this has
been possible are the $\text{AdS}_{3}/\text{CFT}_{2}$ pairs constructed in [2,
5, 10, 16, 17, 18, 19, 23, 25, 26, 29, 32, 36], which provide explicit
settings for the microscopical description of black strings. This has been
addressed in [37, 38, 32]. Notably, most known pairs rely on
$\mathcal{N}=(n,n)$ or $(0,n)$ with $n\leq 4$ supersymmetry.

In this paper we will investigate $\text{AdS}_{3}/\text{CFT}_{2}$ holography
with $(0,6)$ supersymmetry, taking as our starting point the new class of
solutions with $\mathcal{N}=(0,6)$ supersymmetries constructed recently in
[1]. Therefore this work opens up the interesting new possibility of
investigating the $\text{AdS}_{3}/\text{CFT}_{2}$ correspondence in less
standard supersymmetric settings111See also [83] which points the way towards
similar constructions in 1 dimension less. A second class of solutions with
$\mathcal{N}=(0,5)$ supersymmetries was also constructed in [1], and it would
be interesting to further extend the analysis in this paper to that class. The
solutions with $\mathcal{N}=(0,6)$ supersymmetries involve a three-dimensional
complex projective space in their internal geometry, so they share many
properties with the ABJM/ABJ geometry [39, 40]. The connection with the
ABJM/ABJ theories will indeed play a key role in the identification of their
dual CFTs. As we will see, this class of solutions can be regarded as an
extension of the ABJM/ABJ solution of massless Type IIA supergravity to the
massive case. In this extension one of the external directions becomes an
energy scale, generating a flow towards an $\text{AdS}_{3}$ space. Remarkably,
the Seiberg duality proposed in [41], in a particular massive generalisation
of the ABJM/ABJ theory, is explicitly realised in these solutions as a large
gauge transformation, in the spirit of [42, 43].

The paper is organised as follows. In section 2 we summarise the main
properties of the solutions with $\mathcal{N}=(0,6)$ supersymmetry reported in
[1], and focus on the construction of local solutions that can be glued
together with D8-branes. In this section we elaborate on the massless limit,
briefly addressed in [1]. In this limit a change of variables allows to relate
the direction interpreted as an energy scale to the radius of
$\text{AdS}_{4}$, and the solutions reduce to the
$\text{AdS}_{4}\times\mathbb{CP}^{3}$ solution of Type IIA supergravity, dual
to the ABJM/ABJ theory. We study in detail how the free energy of the ABJM/ABJ
theory arises and review some aspects of the theory, such as the realisation
of Seiberg duality [40, 43], that will be useful later on in our study of the
massive case.

In section 3 we construct a new class of solutions to Type IIB supergravity by
acting with Abelian T-duality along one of the directions of the
$\mathbb{CP}^{3}$, preserving $\mathcal{N}=(0,4)$ supersymmetry. We show that
they reduce in the massless limit to the $\text{AdS}_{4}$ solution to Type IIB
supergravity with $\mathcal{N}=4$ supersymmetry T-dual to
$\text{AdS}_{4}\times\mathbb{CP}^{3}$, that constitute the basis for the brane
construction of the ABJM/ABJ theory.

In section 4, in analogy with the $\text{AdS}_{4}$ case, we use the Type IIB
solutions to propose the brane set-up where the 2d dual CFTs should be
realised. We argue that this brane intersection preserves $\mathcal{N}=(0,3)$
supersymmetry in 2d, that, we propose, should be enhanced to
$\mathcal{N}=(0,6)$ in the IR, in analogy with what happens in the ABJM/ABJ
theory, albeit with half the number of supersymmetries. We show that extra
NS5-D7 branes render the brane intersection of the ABJM/ABJ theory two
dimensional and half-supersymmetric. These branes find in this way an
interpretation as $\frac{1}{2}$-BPS surface defect branes within the ABJM/ABJ
theory. This connects our findings with those in [44], where D4-D8 (D5-D7 in
Type IIB) defect branes were introduced as probe branes in the ABJM theory, in
order to realise edge states in the Fractional Quantum Hall Effect. Our study
shows that similar defects can be introduced in a way that preserves half of
the supersymmetries and a $\mathfrak{osp}(6|2)$ superconformal subalgebra.

Specifically, the brane intersection that we propose consists on a brane box
model in which D3-branes are bounded between two types of NS5-branes and bound
states of NS5 and D5 branes, rotated with respect to each other, displayed
along two field theory directions. This brane box model can be viewed as a
$\mathcal{N}=(0,4)$ brane box model of the type constructed in [45], in which
some of the NS5 and D5 branes have been rotated with respect to each other. We
argue that the rotation reduces the SO(4)R R-symmetry group of the brane box
construction in [45] to SO(3)R, and thereby the supersymmetry from
$\mathcal{N}=(0,4)$ to $\mathcal{N}=(0,3)$, with the extra SO$(3)$ remaining
as a global symmetry. We summarise the field content of the
$\mathcal{N}=(0,4)$ brane box models studied in [45], to which we add the
effect of the rotation. This renders some of the scalars living in the
hypermultiplets in the Hanany-Okazaki brane boxes massive. As a result of our
study we are able to associate a 2d quiver field theory to our brane
intersection. Interestingly, we show that these quiver field theories realise
a “massive” extension of Seiberg duality in ABJ proposed in [41]. This is
realised in terms of a brane creation effect when NS5-branes are crossed along
the direction that became an energy scale in the 2d theory. This links Seiberg
duality to large gauge transformations, albeit in one dimension less, in the
spirit of [42, 43]. We end the section with the computation of the holographic
central charge, which, as we will see, includes higher derivative corrections.
This represents a fairly non-trivial prediction for the central charge of the
dual $(0,6)$ SCFTs. Finally, we point out an interesting relation between the
holographic central charge and a product of electric an magnetic charges
computed from the solutions, in the spirit of [46, 47, 48]. These references
generalised the proposal in [49], showing that the central charge in the
algebra of symmetry generators of $\text{AdS}_{2}$ with an electric field is
proportional to the square of the electric field, to fully-fledged
$\text{AdS}_{2}$ string theory set-ups. Our result shows that this proposal
extends to AdS backgrounds with other dimensionalities and supersymmetries,
and therefore suggests a more general interpretation of the central charge in
terms of products of electric and magnetic charges. We conclude in section 5
with a summary of our results and a discussion of the main open problems that
will be interesting to further investigate.

##  2 The class of ${\cal N}=(0,6)$ AdS${}_{3}\times\mathbb{CP}^{3}$ solutions
to massive IIA

In this section we explore the new class of solutions originally found in [1].
Our main aim is to understand what possible global solutions can be
constructed by gluing local solutions together with D8 branes. We begin by
reviewing some results of [1] that we will make use of later in this section.

In [1] the local form of AdS3 solutions in massive IIA preserving the
superconformal algebra $\mathfrak{osp}(6|2)$ were derived. Their NS sector can
be expressed as

$\displaystyle\frac{ds^{2}}{2\pi}$
$\displaystyle=\frac{|h|}{\sqrt{2hh^{\prime\prime}-(h^{\prime})^{2}}}ds^{2}(\text{AdS}_{3})+\sqrt{2hh^{\prime\prime}-(h^{\prime})^{2}}\bigg{[}\frac{1}{4|h|}dr^{2}+\frac{2}{|h^{\prime\prime}|}ds^{2}(\mathbb{CP}^{3})\bigg{]},$
$\displaystyle e^{-\Phi}$
$\displaystyle=\frac{(|h^{\prime\prime}|)^{\frac{3}{2}}}{2\sqrt{\pi}(2hh^{\prime\prime}-(h^{\prime})^{2})^{\frac{1}{4}}},~{}~{}~{}~{}H_{3}=dB_{2},~{}~{}~{}~{}B_{2}=4\pi\left(-(r-l)+\frac{h^{\prime}}{h^{\prime\prime}}\right)J.$
(2.1)

Where $J$ is the Kahler form on $\mathbb{CP}^{3}$ which has unit radius, $l$
is an integer (whose role is clarified below) and $h$ is a function of $r$. A
convenient parametrisation of $\mathbb{CP}^{3}$ is as a foliation of $T^{1,1}$
over an interval, with respect to this we have

$\displaystyle ds^{2}(\mathbb{CP}^{3})$
$\displaystyle=d\xi^{2}+\frac{1}{4}\cos^{2}\xi
ds^{2}(\text{S}^{2}_{1})+\frac{1}{4}\sin^{2}\xi
ds^{2}(\text{S}^{2}_{2})+\frac{1}{4}\sin^{2}\xi\cos^{2}\xi(d\psi+\eta_{1}+\eta_{2})^{2},~{}~{}~{}d\eta_{i}=-\text{vol}(\text{S}^{2}_{i}),$
$\displaystyle J$
$\displaystyle=\frac{1}{4}\sin^{2}\xi\text{vol}(\text{S}^{2}_{2})-\frac{1}{4}\cos^{2}\xi\text{vol}(\text{S}^{2}_{1})-\frac{1}{2}\sin\xi\cos\xi
d\xi\wedge(d\psi+\eta_{1}+\eta_{2}),$ (2.2)

where $(\xi,\psi)$ have periods $(\frac{1}{2}\pi,4\pi)$ and the 2-spheres have
unit radius.

The fluxes making up the RR sector can be compactly expressed in terms of the
NS 2-form potential as

$\displaystyle F_{0}$
$\displaystyle=-\frac{1}{2\pi}h^{\prime\prime\prime},~{}~{}~{}~{}F_{2}=B_{2}F_{0}+2(h^{\prime\prime}-(r-l)h^{\prime\prime\prime})J,$
$\displaystyle F_{4}$ $\displaystyle=\pi
d\left(h^{\prime}+\frac{hh^{\prime}h^{\prime\prime}}{2hh^{\prime\prime}-(h^{\prime})^{2}}\right)\wedge\text{vol}(\text{AdS}_{3})+B_{2}\wedge
F_{2}-\frac{1}{2}B_{2}\wedge B_{2}F_{0}$
$\displaystyle-4\pi(2h^{\prime}+(r-l)(-2h^{\prime\prime}+(r-l)h^{\prime\prime\prime}))J\wedge
J.$ (2.3)

In regular regions of a solution the Bianchi identities imply that $F_{0}$ is
constant, we thus have the ODE

$h^{\prime\prime\prime}=-2\pi F_{0}.$ (2.4)

Formally this is very similar to the class of AdS7 solutions found in [50], as
presented in [51]. Of the remaining Bianchi identities, clearly that of
$H_{3}$ is implied while those of the RR fields are implied when (2.4) holds.
To this it is useful to introduce the following magnetic Page fluxes

$\hat{f}=e^{-B_{2}}\wedge f$ (2.5)

for $f=f_{0}+f_{2}+f_{4}+f_{6}$ where $F_{6}=-\star F_{4}$ and $f$ means the
magnetic part of $F=\sum_{n=0}^{5}F_{2n}$. We find these take the form

$\displaystyle\hat{f}_{2}$
$\displaystyle=2\left(h^{\prime\prime}-(r-l)h^{\prime\prime\prime}\right)J,$
$\displaystyle\hat{f}_{4}$
$\displaystyle=-4\pi\left(2h^{\prime}+(r-l)((r-l)h^{\prime\prime\prime}-2h^{\prime\prime})\right)J\wedge
J,$ $\displaystyle\hat{f}_{6}$
$\displaystyle=\frac{16\pi^{2}}{3}\left(6h-(r-l)(6h^{\prime}+(r-l)((r-l)h^{\prime\prime\prime}-3h^{\prime\prime}))\right)J\wedge
J\wedge J.$ (2.6)

In regular regions of a solution these must be closed for the Bianchi
identities of (2.3) to hold and vice-versa. It is a simple matter to show that

$d\hat{f}_{2n}=-\frac{1}{2\pi}(4\pi)^{n}(r-l)^{n}h^{\prime\prime\prime\prime}\frac{1}{n!}dr\wedge
J^{n},$ (2.7)

so the magnetic Page fluxes are indeed closed in regular parts of a solution
by (2.4).

When $F_{0}$ is constant (2.4) implies $h$ is an order 3 polynomial that
depends on 4 parameters $c_{1,2,3,4}$. Depending on how these are tuned the
domain of the interval $r$ can differ and distinct physical behaviours can be
realised. Assuming $F_{0}\neq 0$ and that $r$ is bounded at one end, which one
can assume to be at $r=0$ without loss of generality, in [1] 3 singular yet
physical boundary behaviours were identified: namely that associated to a
D8/O8 system of world volume AdS${}_{3}\times\mathbb{CP}^{3}$, an O2 plane
extended in AdS3 and backtracked on a G2 cone of base $\mathbb{CP}^{3}$ and
the singularity associated to the $d=3$ KK monopole one gets when reducing the
embedding of the hyper-Kahler 8-manifolds constructed in [52] into $d=11$
supergravity to Type IIA, albeit extended in AdS3 rather than
$\text{Mink}_{3}$. These objects are supersymmetric wherever one places them
in the space. To realise them at $r=0$ one must tune

$\displaystyle\text{D8/O8}:~{}~{}~{}~{}h=c_{1}+c_{2}r^{3},~{}~{}~{}~{}c_{1,2}\neq
0$ (2.8a)
$\displaystyle\text{O2}:~{}~{}~{}~{}h=c_{1}+c_{2}r+\frac{c_{2}^{2}}{4c_{1}}r^{2}+c_{3}r^{3},~{}~{}~{}~{}c_{1,2,3}\neq
0$ (2.8b)
$\displaystyle\text{Monopole}:~{}~{}~{}~{}~{}h=(c_{1}+c_{2}r)r^{2},~{}~{}~{}c_{1,2}\neq
0.$ (2.8c)

However the domain of $r$ for all of these local solutions is semi-infinite,
with $r=0$ an infinite proper distance from $r=\infty$. When $F_{0}=0$ one can
show that one always locally recovers AdS${}_{4}\times\mathbb{CP}^{3}$. We
explore this massless case in some detail in section 2.1.

As stressed in [1] globally $F_{0}$ needs only be piece-wise constant. If one
defines $r=r_{0}$ to be a locus where $F_{0}$ is discontinuous, then in the
neighbourhood of $r_{0}$ one has

$dF_{0}=\Delta F_{0}\delta(r-r_{0}),$ (2.9)

indicating the presence of a stack of D8-branes of charge $2\pi\Delta F_{0}$,
where $\Delta F_{0}$ is the difference between the constant values $F_{0}$
takes for $r>r_{0}$ and $r<r_{0}$. Of course for this really to be true the NS
sector should be consistent with this behaviour. A D8 brane gives rise to a
comparatively mild singularity for which all components of the metric and the
dilaton neither blow nor tend to zero. They should however both be continuous
as one crosses a D8 brane, which one can show amounts to imposing the
continuity of

$(h,(h^{\prime})^{2},h^{\prime\prime})$ (2.10)

as one crosses its loci. $h^{\prime\prime\prime}$ should of course be
discontinuous or there is no D8. The NS 2-form should also be continuous
modulo large gauge transformations, which is something else one needs to
impose.

Another point is that one cannot expect to arbitrarily place branes in some
supersymmetric solution without breaking supersymmetry. However such a D8
brane, with no world-volume flux turned on, has been shown to preserve the
full supersymmetry of this background when placed at

$r_{0}=l.$ (2.11)

Like wise the source corrected Bianchi identities of such an object are
implied by (2.7) - only $dF_{0}$ experiences such a correction, the remaining
$d\hat{f}_{2n}$ terms vanish as $(r-l)^{n}\delta(r-l)\to 0$ for $n=1,2,3$. So
one is free to make $F_{0}$ discontinuous at $r=l$ and the result is a
supersymmetric stack of D8 branes. Naively this sounds like one can only place
a single D8 brane stack on the interior of $r$, but one must appreciate that
$l$ does not need to be fixed globally. Indeed one can view the difference
between $B_{2}$ with $l=0$ and the generic $B_{2}$ as a large gauge
transformation, provided $l$ is an integer. Thus one can place arbitrary
numbers of D8 branes along the interior of $r$, provided they are placed at
integer points along $r$ and accompanied by the appropriate number of large
gauge transformations of $B_{2}$.

In [1] the above facts where used to construct a simple globally bounded
solution, by gluing 2 copies of (2.8a) together with a single D8 brane stack.
In section 2.2 we shall perform a more general analysis of possible global
solutions, however before doing so it will be instructive to first consider
the massless case in the next section.

Central charge:

A closed expression for the calculation of the holographic central charge
applicable to the solutions was given in [53, 54], extending the computation
in [55] to more general backgrounds. In this reference the holographic central
charge is computed for a metric and dilaton

$ds^{2}_{10}=a(\zeta,\vec{\theta})(dx_{1,d}^{2}+b(\zeta)d\zeta^{2})+g_{ij}(\zeta,\vec{\theta})d\theta^{i}d\theta^{j},\qquad\Phi=\Phi(\zeta,\vec{\theta}),$
(2.12)

from

$c_{hol}=\frac{3d^{d}}{G_{N}}\frac{b(\zeta)^{d/2}(\hat{H})^{\frac{2d+1}{2}}}{(\hat{H}^{\prime})^{d}},$
(2.13)

where

$\hat{H}=\Bigl{(}\int
d\vec{\theta}e^{-2\Phi}\sqrt{\text{det}[g_{ij}]a(\zeta,\vec{\theta})^{d}}\Bigr{)}^{2}.$
(2.14)

This gives the simple expression

$c_{hol}=\frac{1}{2}\int dr\;(2hh^{\prime\prime}-(h^{\prime})^{2})$ (2.15)

for the solutions in (2).

###  2.1 The massless case: The ABJM/ABJ solution

As shown in [1] in the massless limit the ABJM/ABJ solution [39, 40] is
recovered. To see this we can parametrise

$h(r)=Q_{2}-Q_{4}r+\frac{1}{2}Q_{6}r^{2},$ (2.16)

where $Q_{2,4,6}$ are constants whose significance will become clear shortly.
Then one can easily check that the change of variables

$\sinh{\mu}=\frac{Q_{6}\,r-Q_{4}}{\sqrt{2Q_{2}Q_{6}-Q_{4}^{2}}}$ (2.17)

gives rise to the ABJM/ABJ metric and dilaton

$\displaystyle
ds^{2}=\frac{4\pi\sqrt{2Q_{2}Q_{6}-Q_{4}^{2}}}{Q_{6}}\,\Bigl{(}\frac{1}{4}ds^{2}(\text{AdS}_{4})+ds^{2}(\mathbb{CP}^{3})\Bigr{)},$
(2.18) $\displaystyle
ds^{2}(\text{AdS}_{4})=d\mu^{2}+\cosh^{2}{\mu}\,ds^{2}(\text{AdS}_{3}),\quad
e^{-\Phi}=\frac{Q_{6}^{\frac{3}{2}}}{2\sqrt{\pi}(2Q_{2}Q_{6}-Q_{4}^{2})^{\frac{1}{4}}}.$
(2.19)

The radius of $\mathbb{CP}^{3}$ is thus given by

$L=\Bigl{(}\frac{32\pi^{2}}{Q_{6}^{2}}(Q_{2}Q_{6}-\frac{1}{2}Q_{4}^{2})\Bigr{)}^{1/4}.$
(2.20)

Therefore there is just one local solution when $F_{0}=0$ and it is an AdS4
vacuum preserving twice the supersymmetries of generic solutions within this
class. This is actually the only regular solution also.

One can now use the Page fluxes in (2.6) to compute the various brane charges,
leading to

$\displaystyle\frac{1}{2\pi}\int_{\mathbb{CP}^{1}}\hat{f}_{2}=Q_{6},~{}~{}~{}~{}\frac{1}{(2\pi)^{3}}\int_{\mathbb{CP}^{2}}\hat{f}_{4}$
$\displaystyle=Q_{4},~{}~{}~{}~{}\frac{1}{(2\pi)^{5}}\int_{\mathbb{CP}^{3}}\hat{f}_{6}=Q_{2}.$
(2.21)

Thus the constants in (2.16), $Q_{p}$, are identified with the Page charges of
$D_{p}$ branes, for $p=2,4,6$. In turn, in this case one finds $H_{3}=0$ and
$\hat{f}_{4}=-f_{2}\wedge B_{2}$. The NS5 and the D4 branes are thus not
physical, since they annihilate to nothing and only their fluxes remain [40].
$B_{2}$ is given by

$B_{2}=-4\pi\frac{Q_{4}}{Q_{6}}J,\qquad\text{such that}\qquad
b=-\frac{Q_{4}}{Q_{6}},$ (2.22)

and the discrete holonomy in the construction of [40] arises. As discussed
there this fractional charge is associated to the fact that in the presence of
$Q_{6}$ units of $f_{2}$ flux a NS5-brane wrapped on the $\mathbb{CP}^{2}$
must have $Q_{4}$ D4-branes wrapped on the $\mathbb{CP}^{1}$ ending on it.

Table 1 summarises the brane set-up associated to the ABJM/ABJ solution in its
Type IIB realisation [39]222After a T-duality along the Hopf fibre of the S3
included in the $T^{1,1}$ in the $\mathbb{CP}^{3}$ has been performed (see the
next section).. The D3-branes stretch on the $\psi$-circle, intersecting one
NS5’-brane and one $(1,k)$ 5’-brane333As we recall below, the numbers of
branes in the field theory do not actually coincide with the Page charges
computed from the solution. This explains our different notation for the
charges in this discussion., extended along $(x^{0},x^{1},r)$ and the
$[3,7]_{\theta}$, $[4,8]_{\theta}$, $[5,9]_{\theta}$ directions, with
$\tan{\theta}=k$. On top of this there are $M$ fractional branes stretched
just along one segment of the circle, which as mentioned above are not actual
branes since they are unstable. The brane system preserves $\mathcal{N}=3$
supersymmetry in 3d, and this is enhanced to $\mathcal{N}=6$ in the IR [39].

branes | $x^{0}$ | $x^{1}$ | $r$ | $x^{3}$ | $x^{4}$ | $x^{5}$ | $\psi$ | $x^{7}$ | $x^{8}$ | $x^{9}$
---|---|---|---|---|---|---|---|---|---|---
$N\,\mbox{$\mathrm{D}$}3$ | $\times$ | $\times$ | $\times$ | $-$ | $-$ | $-$ | $\times$ | $-$ | $-$ | $-$
$\mbox{$\mathrm{NS}$}5^{\prime}$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $-$ | $-$ | $-$ | $-$
$(1,k)5^{\prime}$ | $\times$ | $\times$ | $\times$ | $\cos{\theta}$ | $\cos{\theta}$ | $\cos{\theta}$ | $-$ | $\sin{\theta}$ | $\sin{\theta}$ | $\sin{\theta}$

Table 1: Brane intersection describing the ABJM brane system.
$(x^{0},x^{1},r)$ span the 3d field theory. The NS5′ brane and D5′ brane at
$\psi=\pi$ are rotated the same angle on the $[3,7]$, $[4,8]$ and $[5,9]$
directions. The brane intersection preserves 6 Poincaré supersymmetries
($\mathcal{N}=3$ in 3d).

In the massless limit a large gauge transformation $b\rightarrow b+1$ maps the
theory onto itself, and therefore the associated change in the quantised
charges, given by equations (2.37) for $Q_{8}=0$,

$\displaystyle Q_{2}\rightarrow Q_{2}-Q_{4}+\frac{1}{2}Q_{6},$ $\displaystyle
Q_{4}\rightarrow Q_{4}-Q_{6},$ (2.23)

must be a symmetry of the theory. Indeed, as shown in [43], these
transformations generate Seiberg dualities relating the IR behaviour of
different 3d $\mathcal{N}=3$ Chern-Simons matter theories. In order to see
this it is necessary to recall that due to the non-trivial topology of the
$\mathbb{CP}^{3}$ the quantised charges emerging from the solutions do not
coincide with the numbers of branes in the field theory. This happens because
the fractional worldvolume field strength that is needed to cancel the Freed-
Witten anomaly [56] associated to the $\mathbb{CP}^{2}$, being a non-spin
manifold, induces charge on the branes wrapped on the $\mathbb{CP}^{2}$. On
top of this there is a non-vanishing contribution of higher curvature terms
for these branes [57]. Adding these contributions the following linear
relations between the quantised charges and the numbers of D2, D4 and D6
branes, that we have denoted by $N,M$ and $k$, respectively, are found [43]

$\displaystyle Q_{2}=N+\frac{k}{12},$ $\displaystyle Q_{4}=M-\frac{k}{2},$
$\displaystyle Q_{6}=k.$ (2.24)

Substituting these relations in (2.22) the shift of $B_{2}$ by $2\pi J$
required to cancel the Freed-Witten anomaly naturally arises. Moreover, one
can see that the transformations (2.23) translate into

$N\rightarrow N+k-M,\qquad M\rightarrow M-k,$ (2.25)

for the field theory parameters. These transformations map the 3d Chern-Simons
matter theory with gauge group U$(N+M)_{k}\times$U$(N)_{-k}$ to the one with
gauge group U$(N)_{k}\times$U$(N-M+k)_{-k}$, and successively. These are the
Seiberg duality transformations that relate the IR behaviour of the different
Chern-Simons matter theories found in [40, 43]. One can check that the central
charge computed using the expression (2.15), that resorts in the massless case
to

$c_{hol}=\frac{1}{2}(2Q_{2}Q_{6}-Q_{4}^{2})=Nk-\frac{1}{2}M(M-k)-\frac{1}{24}k^{2},$
(2.26)

per unit $r$-interval, where we have used the relations (2.24) to write it in
terms of the field theory charges, is indeed invariant under the
transformations (2.23) and (2.25). Note that in order to recover the right
scaling of the central charge of the 3d theory we have to multiply the
previous expression by the radius squared of $\text{AdS}_{4}$444Divided by
$2\pi$, in order to match the different conventions used for the 2d and 3d
metrics.. This yields

$c_{hol}^{(3d)}=\frac{(2Q_{2}Q_{6}-Q_{4}^{2})^{3/2}}{Q_{6}}=\frac{L^{6}Q_{6}^{2}}{64\pi^{3}}=\frac{1}{k}\Bigl{(}2Nk-M(M-k)-\frac{1}{12}k^{2}\Bigr{)}^{3/2},$
(2.27)

using again (2.24). This matches the result obtained for the ABJM solution
using equations (2.12)-(2.14) for $d=2$ and the metric and dilaton given by
(2.18),(2.19). This expression is interpreted as the free energy of the 3d CFT
in the supergravity approximation. However, given that we have used the
shifted charges given by (2.24) we obtain the planar free energy (at large ’t
Hooft parameter) with higher derivative sub-leading corrections [57, 58]. As
in these references we can use the Maxwell charge for the D2-branes, given by,

$Q_{2}^{M}=Q_{2}+b\,Q_{4}+\frac{1}{2}b^{2}\,Q_{6}=N-\frac{1}{24}k-\frac{1}{2}\frac{M^{2}}{k}+\frac{1}{2}M,$
(2.28)

to write

$c_{hol}^{(3d)}=2\sqrt{2k}(Q_{2}^{M})^{3/2}=2^{3/2}\,k^{2}\,{\hat{\lambda}}^{3/2},$
(2.29)

with $\hat{\lambda}=\frac{Q_{2}^{M}}{k}$ the shifted ’t Hooft coupling
parameter found in [59]. This expression was shown to agree with the strong
coupling expansion of the free energy computed from the matrix model
calculation of the path integral on S3 [59, 60, 61, 62]. Note that in three
dimensions the useful role played by conformal anomaly coefficients in even
dimensional CFTs in measuring the number of degrees of freedom is now played
by matrix integral calculations of the path integral, using supersymmetric
localization methods [63].

###  2.2 Analysis of global solutions with D8 branes

In this section we perform an analysis of the global solutions it is possible
to construct by placing D8-branes along the interior of $r$. Our aim is not to
be completely exhaustive but rather to shed light on the wide variety of
possibilities, and assemble the necessary building blocks for their
construction.

In order to construct broad classes of solutions we shall segment the interval
spanned by $r$ into intervals of unit length with $h$ taking the form

$h_{l}(r)=Q_{2}^{l}-Q_{4}^{l}(r-l)+\frac{1}{2}Q_{6}^{l}(r-l)^{2}-\frac{1}{6}Q_{8}^{l}(r-l)^{3},$
(2.30)

in the interval $r\in[l,l+1]$ for $Q_{2},...Q_{8}$ constants. We shall assume
that in the $l$’th interval

$B_{2}=4\pi\left(-(r-l)+\frac{h_{l}^{\prime}}{h_{l}^{\prime\prime}}\right)J,$
(2.31)

so that when $Q_{8}^{l}\neq Q_{8}^{l-1}$, the resulting stack of D8 branes at
$r=l$ is supersymmetric. Consistency requires that as we cross between the 2
adjacent intervals at $r=l$ we have that

$b=\frac{1}{4\pi^{2}}\int_{\mathbb{CP}^{1}}B_{2},$ (2.32)

is such that $b\rightarrow b+z$ for $z\in\mathbb{Z}$, so that a large gauge
transformation is performed creating $z$ units of NS5 brane charge. Again the
constants appearing in $h_{l}$, $Q_{2}^{l}$, $Q_{4}^{l}$, $Q_{6}^{l}$ and
$Q_{8}^{l}$ are the Page charges associated to D2, D4, D6 and D8 branes in
each interval, ie using (2.3) one finds

$\displaystyle 2\pi F_{0}$
$\displaystyle=Q_{8}^{l},~{}~{}~{}~{}\frac{1}{2\pi}\int_{\mathbb{CP}^{1}}\hat{f}_{2}=Q_{6}^{l},$
$\displaystyle\frac{1}{(2\pi)^{3}}\int_{\mathbb{CP}^{2}}\hat{f}_{4}$
$\displaystyle=Q_{4}^{l},~{}~{}~{}~{}\frac{1}{(2\pi)^{5}}\int_{\mathbb{CP}^{3}}\hat{f}_{6}=Q_{2}^{l},$
(2.33)

in the $l$’th cell.

Metric reality:
The first thing we need to ensure is that the warp factors of the metric are
real in each cell. The metric is real and positive when
$-(h^{\prime})^{2}+2h^{\prime\prime}h\geq 0$, with equality only possible at
the upper or lower bound of $r$ in a global solution. In a generic unit cell
one should have $-(h^{\prime})^{2}+2h^{\prime\prime}h>0$ which implies
$hh^{\prime\prime}>0$, in what follows we shall assume this is solved as

$h>0,~{}~{}~{}~{}h^{\prime\prime}>0,$ (2.34)

without loss of generality. It is possible to show that
$-(h^{\prime})^{2}+2h^{\prime\prime}h>0$ across the $l$’th cell if the
inequality holds at $r=l$ and $r=l+1$. This amounts to imposing one of two
sets of constraints on the charges

$\displaystyle Q^{l}_{2}\geq 0,~{}~{}~{}~{}Q^{l}_{6}\geq
0,~{}~{}~{}~{}Q^{l}_{8}\leq
0,~{}~{}~{}~{}2Q^{l}_{2}Q^{l}_{6}\geq(Q^{l}_{4})^{2},$ (2.35a) $\displaystyle
Q^{l}_{2}\geq 0,~{}~{}~{}~{}Q^{l}_{6}>Q^{l}_{8}\geq
0,~{}~{}~{}~{}2(Q^{l}_{6}-Q^{l}_{8})(Q^{l}_{2}+\frac{1}{2}Q^{l}_{6}-Q^{l}_{4}-\frac{1}{6}Q^{l}_{8})\geq(Q^{l}_{4}-Q^{l}_{6}+\frac{Q^{l}_{8}}{2})^{2},$
(2.35b)

in generic cells, which both leave the sign of $Q_{4}$ arbitrary.

Metric continuity:
Next we need to impose the continuity conditions (2.10) as we cross each D8
brane at the intersections of the cells. To impose the continuity constraints
we must distinguish between two cases

$h^{\prime}_{l-1}(l)=\pm h^{\prime}_{l}(l).$ (2.36)

Let us first assume $h^{\prime}_{l-1}(l)=h^{\prime}_{l}(l)$; the continuity
conditions for $h_{l}$, $h^{\prime}_{l}$, $h^{\prime\prime}_{l}$ across
intervals force the conditions

$\displaystyle
Q_{2}^{l}=Q_{2}^{l-1}-Q_{4}^{l-1}+\frac{1}{2}Q_{6}^{l-1}-\frac{1}{6}Q_{8}^{l-1},$
$\displaystyle Q_{4}^{l}=Q_{4}^{l-1}-Q_{6}^{l-1}+\frac{1}{2}Q_{8}^{l-1},$
$\displaystyle Q_{6}^{l}=Q_{6}^{l-1}-Q_{8}^{l-1},$ (2.37)

between the $(l-1)^{th}$ and $l^{th}$ cells. We note that if (2.35a) holds in
the $(l-1)^{th}$ cell the continuity conditions impose that it also holds in
the $l^{th}$ cell provided that $Q^{l}_{8}<0$. Conversely if (2.35b) holds in
the $(l+1)^{th}$ cell it is guaranteed to also hold in the $l^{th}$ if
$Q^{l}_{8}>0$. We thus see if one imposes $\eqref{eq:simplesol}$ in a single
cell it is guaranteed to also hold in each subsequent cell to the right of it
satisfying $Q^{l}_{8}\leq 0$. An analogous statement also holds for cells
satisfying $Q^{l}_{8}\geq 0$ to the left of a cell satisfying (2.35b).

Using that

$B_{2}^{l}=4\pi\frac{-Q_{4}^{l}+\frac{1}{2}Q_{8}^{l}(r-l)^{2}}{Q_{6}^{l}-Q_{8}^{l}(r-l)}\,J,$
(2.38)

in the $[l,l+1]$ interval, one can check that $b^{l}(l)=b^{l-1}(l)+1$, meaning
that the shift in $B_{2}$ is indeed a large gauge transformation as we cross
the D8 at $r=l$, and so 1 unit of NS5-brane charge is created as we do so.
Note that the number of NS5-branes at $r=l$ is still fixed by the numbers of
D4 and D6 branes in such interval. Thus, the D4-branes keep being fractional
branes as in the massless case.

We can use the relation (2.37) to express the charges in the $l$’th cell in
terms of those in some other $\tilde{l}$’th cell, with $\tilde{l}<l$; assuming
$h^{\prime}_{l-1}(l)=h^{\prime}_{l}(l)$ is satisfied across all D8 branes, we
find

$\displaystyle Q_{2}^{l}=Q_{2}^{\tilde{l}}-\Delta
lQ_{4}^{\tilde{l}}+\frac{1}{2}(\Delta
l)^{2}Q_{6}^{\tilde{l}}-\frac{1}{6}\sum_{i=1}^{\Delta
l}(1-3i+3i^{2})Q_{8}^{l-i},$ $\displaystyle Q_{4}^{l}=Q_{4}^{\tilde{l}}-\Delta
lQ_{6}^{\tilde{l}}+\frac{1}{2}\sum_{i=1}^{\Delta l}(2i-1)Q_{8}^{l-i},$
$\displaystyle Q_{6}^{l}=Q_{6}^{\tilde{l}}-\sum_{i=1}^{\Delta l}Q_{8}^{l-i},$
(2.39)

where $\Delta l=l-\tilde{l}$.

If we instead assume $h^{\prime}_{l-1}(l)=-h^{\prime}_{l}(l)$ things are a bit
more complicated. The continuity conditions impose the same constraints on
$(Q_{2}^{l},Q_{6}^{l})$ as in (2.37) but that of $Q_{4}^{l}$ is modified as

$Q_{4}^{l}=-(Q_{4}^{l-1}-Q_{6}^{l-1}+\frac{1}{2}Q_{8}^{l-1}),$ (2.40)

which now implies that

$b^{l}(l)-b^{l-1}(l)=\frac{2Q_{4}^{l-1}-Q_{6}^{l-1}}{Q_{6}^{l-1}-Q_{8}^{l-1}}.$
(2.41)

This is by no means automatically a large gauge transformation, for that we
must impose that the LHS of the above is an integer, which has consequences
for the charges in both the $l$’th and $(l-1)$’th cells, namely

$\displaystyle 2Q_{4}^{l}$ $\displaystyle=-(z-1)Q_{6}^{l},$ $\displaystyle
2Q_{4}^{l-1}$ $\displaystyle=(z+1)Q_{6}^{l-1}-zQ_{8}^{l-1},$ (2.42)

for $z\in\mathbb{Z}$ \- note that these are not independent, they imply each
other given (2.40). We again find that if (2.35a) holds in the $(l-1)$’th cell
it also holds in the $l$’th cell when $Q_{8}^{l}\leq 0$ and that if (2.35b)
holds in the $l$’th then it also holds in the $(l-1)$’th cell when
$Q_{8}^{l-1}\geq 0$.

We can again sum the recursion relations, this time assuming that
$h^{\prime}_{l-1}(l)=h^{\prime}_{l}(l)$ is satisfied across all D8s except the
one at $r=l$. We find

$\displaystyle Q_{2}^{l}=Q_{2}^{\tilde{l}}-\Delta
lQ_{4}^{\tilde{l}}+\frac{1}{2}(\Delta
l)^{2}Q_{6}^{\tilde{l}}-\frac{1}{6}\sum_{i=1}^{\Delta
l}(1-3i+3i^{2})Q_{8}^{l-i},$ $\displaystyle
Q_{4}^{l}=\frac{1}{2}(1-z)(Q_{6}^{\tilde{l}}-\sum_{i=1}^{\Delta
l}Q^{l-i}_{8}),$ $\displaystyle Q_{6}^{l}=Q_{6}^{\tilde{l}}-\sum_{i=1}^{\Delta
l}Q_{8}^{l-i},$ (2.43)

which is subject to the additional constraint

$2Q_{4}^{\tilde{l}}-(2\Delta l+z-1)Q_{6}^{\tilde{l}}+\sum_{i=2}^{\Delta
l}(2(i-1)+z)Q_{8}^{l-i}+zQ^{l-1}_{8}=0,$ (2.44)

which one can take to define, for instance, $Q^{l-1}_{8}$. The most general
configuration one can construct will have
$h^{\prime}_{l-1}(l)=-h^{\prime}_{l}(l)$ at an arbitrary number of cell
intersections, with each such intersections being separated by an arbitrary
number of cells whose intersections obey
$h^{\prime}_{l-1}(l)=+h^{\prime}_{l}(l)$ \- clearly this leads to infinite
possibilities. One can always compute the charges in a given cell of an
arbitrary solution by combining the results of (2.39) and (2.43).

Simple global solutions:
To construct global solutions we need to decide how the space is going to
start and end: If we choose to bound the space from below we can do so without
loss of generality in the $0$’th cell at $r=0$, we must choose one of the
profiles in (2.8a)-(2.8c) so one achieves this with a physical singularity.
Each of these choices amounts to tuning the charges in the $0$’th cell as

$\displaystyle\text{D8/O8 at
}r=0:~{}~{}~{}~{}~{}~{}~{}~{}Q_{4}^{0}=Q_{6}^{0}=0,$ (2.45)
$\displaystyle\text{O2 at
}r=0:~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}2Q_{2}^{0}Q_{6}^{0}=(Q_{4}^{0})^{2},$
$\displaystyle\text{Monopole at }r=0:~{}~{}~{}~{}Q_{2}^{0}=Q^{0}_{4}=0,$

where in each case $Q_{8}^{0}<0$ and $Q_{6}^{0},Q_{2}^{0}\geq 0$ ensures a
well defined metric throughout the first cell, and so also in every subsequent
cell for which $Q^{l}_{8}<0$. As fixing $F_{0}=0$ in our class makes it
locally AdS${}_{4}\times\mathbb{CP}^{3}$, as explored earlier, clearly we can
also choose to begin the space with an AdS4 boundary at $r=-\infty$ by simply
fixing $Q^{l}_{8}=0$ in every cell below some lower bound $l=l_{0}$ were we
begin placing D8 brane sources. This requires

$\text{AdS}_{4}~{}\text{boundary at:
}r=-\infty:~{}~{}~{}~{}~{}~{}~{}Q^{l}_{8}=0,~{}~{}~{}\text{for}~{}~{}~{}l<l_{0},$
(2.46)

one also needs to arrange for $2Q_{2}^{l}Q_{6}^{l}>(Q_{4}^{l})^{2}$ below
$r=l_{0}$, but then provided that we keep $Q^{l}_{8}\leq 0$ for $l\geq l_{0}$
we are then guaranteed to have a well defined metric.

Each of the previous options allows us to continue the space arbitrarily
towards $r=+\infty$, placing D8 branes at arbitrary integer values of $r$ as
we go as long as Q${}^{l}_{8}\leq 0$ in each cell. At some point though we
need to arrange for the space to terminate physically, one option is to keep
$Q^{l}_{8}<0$ for $l=0,...,P-1$ and place a final D8 brane at $r=P$. In this
scenario by far the easiest physical option to end the space is to have an
AdS4 boundary at $r=+\infty$, as all this requires one to do is fix

$\text{AdS}_{4}~{}\text{boundary at:
}r=+\infty,~{}~{}~{}~{}~{}~{}~{}Q^{l}_{8}=0,~{}~{}~{}l\geq P,$ (2.47)

but this is not the only option at our disposal. We can instead choose to
terminate the space in the $P$’th cell at $r=P+1$, for which we must again
select a profile as in (2.8a)-(2.8c), but such that the singularity lies in
the appropriate place, which amounts to taking $h$ in the final cell to be of
the form

$h=Q^{P+1}_{2}-Q^{P+1}_{4}(r-P-1)+\frac{1}{2}Q^{P+1}_{6}(r-P-1)^{2}-\frac{1}{3!}Q^{P}_{8}(r-P-1)^{3},$
(2.48)

where $Q^{P+1}_{2,4,6}$ are the Page charges in the would be $(P+1)$’th cell,
related to those in $P$’th cell by (2.37). The distinct boundary behaviours
are given by tuning

$\displaystyle\text{D8/O8 at
}r=P+1:~{}~{}~{}~{}~{}~{}~{}~{}Q^{P+1}_{4}=Q^{P+1}_{6}=0,$ (2.49)
$\displaystyle\text{O2 at
}r=P+1:~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}2Q^{P+1}_{2}Q^{P+1}_{6}=(Q^{P+1}_{4})^{2},$
$\displaystyle\text{Monopole at }r=P+1:~{}~{}~{}~{}Q^{P+1}_{2}=Q^{P+1}_{4}=0,$

where we still require that $Q^{P+1}_{2},Q^{P+1}_{6}>0$ as before, but now
$Q_{8}^{P}>0$ in each case. Notice that these imply the conditions (2.35b), so
one can equally well work backwards from $r=P+1$, gluing on cells with
$Q_{8}^{l}>0$ and have guaranteed metric positivity when the continuity
constraints hold - we shall return to this point shortly. Under the assumption
that $h^{\prime}$ does not change sign as we cross the D8 at $r=P$, it does
not take long to establish that terminating at a D8/O8 is impossible, ie one
finds

$2Q^{P+1}_{2}+\frac{1}{3}(1+3P+6P^{2})Q_{6}^{0}=\sum_{i=1}^{P}(2P-i)iQ_{8}^{P-i},$
(2.50)

which is impossible to solve as the LHS is positive definite while the RHS is
negative definite. Ending with a monopole under these assumptions is possible,
but only when $Q^{0}_{4}>0$, we find we must tune the charges as

$\displaystyle Q_{4}^{0}$
$\displaystyle=(P+1)Q_{6}^{0}-\frac{1}{2}Q_{8}^{P}-\frac{1}{2}\sum_{i=1}^{P}(2i+1)Q_{8}^{P-i},$
$\displaystyle 6Q_{2}^{0}$
$\displaystyle=3(P+1)^{2}Q_{6}^{0}-(2+3P)\sum_{i=0}^{P}Q^{i}_{8}+\sum_{i=1}^{P}i(i-(2P+1))Q_{8}^{P-i},$
$\displaystyle Q_{6}^{0}$
$\displaystyle>\sum_{i=0}^{P}Q^{i}_{8}=Q^{P}_{8}+\sum_{i=1}^{P}Q^{i-1}_{8},$
(2.51)

where $Q_{4}^{0}>0$ follows from these and the fact that we must have
$\tilde{Q}^{P}_{6}>0$. It is also possible to end the space with a O2 plane in
this fashion, but solving
$2\tilde{Q}_{2}^{P}\tilde{Q}_{6}^{P}=(\tilde{Q}_{4}^{P})^{2}$ given (2.39)
leads to rather complicated expressions we will omit. We conclude that if
$Q^{l}_{8}\leq 0$ for $l=0,...,P-1$ and $Q^{P}_{8}\neq 0$ then when
$h^{\prime}$ does not change sign at $r=P$ the possible global solutions are

$(\text{AdS}_{4},~{}\text{O2})\to(\text{AdS}_{4},~{}\text{O2},~{}\text{Monopole}),$
(2.52)

with an arbitrary number of D8 branes on the interior of the $r$ interval.

If $h^{\prime}$ does change sign at $r=P$ it becomes possible to terminate the
space with a D8/O8 singularity, this imposes

$\displaystyle Q^{0}_{6}$ $\displaystyle=\sum_{i=0}^{P}Q_{8}^{i}\geq
0,~{}~{}~{}Q_{2}^{0}\geq
PQ^{0}_{4}+\frac{1-3P^{2}}{6}\sum_{i=1}^{P}Q_{8}^{i}+3\sum_{i=1}^{P}i(i-1)Q_{8}^{i-1},$
$\displaystyle Q^{0}_{4}$
$\displaystyle=-Q_{8}^{P}+\frac{1}{2}(1+2P)\sum_{i=1}^{P}Q^{i}_{8}-\sum_{i=1}^{P}iQ^{P-1}_{8},~{}~{}~{}~{}z=0$
(2.53)

which is compatible with all physical options for the lower bound of the space
except the monopole. It is also impossible to end the space with a monopole
when $h^{\prime}$ changes sign at $r=P$ as the necessary positivity
constraints on the charges cannot be satisfied. Finally, although we will
again not give the details due to their complexity, it is possible to end the
space with an O2 in this fashion. We thus find that when $Q^{l}_{8}\leq 0$ for
$l=0,...,P-1$ and $Q^{P}_{8}\neq 0$, then if $h^{\prime}$ does change sign at
$r=P$

$(\text{D8/O8},~{}\text{AdS}_{4},~{}\text{O2})\to(\text{D8/O8},~{}\text{AdS}_{4},~{}\text{O2})$
(2.54)

are the possible global solutions.

Symmetric profiles:
In the previous examples of global solutions we only allowed for one cell with
$Q^{l}_{8}>0$. To end this section we will consider the more symmetric case of
an even number of positive and negative charges.

One can begin the space at $r=0$ with any of the behaviours in (2.45), or
arrange $Q^{0}_{8}$ to be the first non zero values of $Q^{l}_{8}$ such that
we have an AdS4 boundary at $r=-\infty$. Either way, we continue the space in
a well defined manor up to $r=P$ so long as $Q^{l}_{8}<0$ for $l=0,...,P-1$
and we impose (2.35a) in the $0$’th cell. The idea is to then essentially glue
the profile for $h$ to itself in the $P$’th cell: To this end it is useful to
fix the charges in the $P$’th cell as

$\displaystyle Q^{P}_{2}$
$\displaystyle=Q^{P-1}_{2}-Q^{P-1}_{4}+\frac{1}{2}Q^{P-1}_{6}-\frac{1}{3!}Q^{P-1}_{8},~{}~{}~{}~{}Q^{P}_{8}=-Q^{P-1}_{8}$
$\displaystyle Q^{P}_{4}$
$\displaystyle=-Q^{P-1}_{4}+Q^{P-1}_{6}-\frac{1}{2}Q^{P-1}_{8},~{}~{}~{}~{}Q_{6}^{P}=Q^{P-1}_{6}-Q^{P-1}_{8},$
(2.55)

which satisfies the continuity conditions at $r=P$ with $h^{\prime}$ changing
sign at this point. This tuning has the benefit of automatically solving the
metric reality constraints: As $Q^{P}_{8}\geq 0$ the appropriate constraints
are (2.35b), which under (2.55) become

$\displaystyle Q_{8}^{P-1}\leq 0,~{}~{}~{}~{}Q^{P-1}_{6}\geq
0,~{}~{}~{}Q^{P-1}_{2}\geq
0,~{}~{}~{}~{}2Q_{2}^{P-1}Q_{6}^{P-1}\geq(Q^{P-1}_{4})^{2},$ $\displaystyle
Q_{2}^{P-1}-Q_{4}^{P-1}+\frac{1}{2}Q_{6}^{P-1}-\frac{1}{3!}Q^{P-1}_{8}\geq 0,$
(2.56)

the first line of which is just the (2.35a) version of the metric reality
conditions for the $(P-1)$’th cell, while the second line is implied by these.
If we now consider the metric continuity constraints (2.37) (ie without
$h^{\prime}$ changing sign) at $r=P+1$ we find that under (2.55) they are
mapped to

$Q^{P+1}_{2}=Q^{P-1}_{2},~{}~{}~{}~{}~{}Q^{P+1}_{4}=-Q^{P-1}_{4},~{}~{}~{}~{}~{}Q^{P+1}_{6}=Q^{P-1}_{6}$
(2.57)

If we further identify $Q^{P+1}_{8}=-Q^{P-2}_{8}$ then the metric reality
constraints of the $(P+1)$’th cell are implied by those of the $(P-2)$’th
cell. We can continue in this fashion to construct a well defined global
solution by simply identifying the charges in the $(P+n)$’th cell as

$Q^{P+n}_{2}=Q^{P-n}_{2},~{}~{}~{}~{}~{}Q^{P+n}_{4}=-Q^{P-n}_{4},~{}~{}~{}~{}~{}Q^{P+n}_{6}=Q^{P-n}_{6},~{}~{}~{}~{}~{}Q^{P+n}_{8}=-Q^{P-n-1}_{8}.,$
(2.58)

for $n=1,...,P-1$. In such symmetric profiles $h^{\prime}$ changes sign at the
D8 at $r=P$, and only at this D8. We thus also need to ensure that the
transformation of $B_{2}$ across $r=P$ is a large gauge transformation, which
amounts to imposing

$2Q_{4}^{0}-(2(P-1)+z-1)Q_{6}^{0}+\sum_{i=2}^{P-1}(2(i-1)+z)Q_{8}^{P-i}+zQ^{P-1}_{8}=0,$
(2.59)

which we can take as the definition of $Q_{8}^{P-1}$, which one can always
ensure is negative by suitably choosing the integer $z$. In the $(2P-1)$’th
cell we have $n=P-1$ and by using (2.37), the profile in this cell can then be
written as

$h=Q^{0}_{2}-(-Q^{0}_{4})(r-2P)+\frac{1}{2}Q^{0}_{6}(r-2P)^{2}-\frac{1}{3!}(-Q^{0}_{8})(r-2P)^{3}$
(2.60)

making clear that if we begin the space at $r=0$ with one of the behaviours in
(2.45), then it ends at $r=2P$ in the same fashion - likewise an AdS4 lower
boundary will interpolate to an upper boundary AdS4. It is thus possible to
construct global solutions interpolating between the following boundary
behaviours in this way

$\text{D8/O8}\to\text{D8/O8},~{}~{}~{}~{}\text{Monopole}\to\text{Monopole},~{}~{}~{}~{}\text{O2}\to\text{O2},~{}~{}~{}~{}\text{AdS}_{4}\to\text{AdS}_{4}.$
(2.61)

It is also not hard to further generalise - for instance one can separate the
negative and positive $Q_{8}^{l}$ by a region with $Q^{l}_{8}=0$ of arbitrary
length in a similar fashion, or one could take a portion of a symmetric
profile then glue it to a local AdS4 region. Let us stress that there is much
more that one can do than we have discussed explicitly, however this section
does provide the necessary building blocks to construct all these
possibilities.

Central charge:
Finally, let us comment on the central charge of generic solutions: The
holographic central charge (2.15) clearly depends on how we start and end the
space, but we can make some general comments: The expression (2.15) can be
compartmented into its contribution in each cell, with $h_{l}$ given by
equation (2.30) one finds the contribution between $r=l$ and $r=l+1$ to be

$c^{l}_{hol}=\frac{1}{2}\int_{l}^{l+1}dr\Bigl{(}2h_{l}h_{l}^{\prime\prime}-(h_{l}^{\prime})^{2}\Bigr{)}=2Q_{2}^{l}Q_{6}^{l}-(Q_{4}^{l})^{2}-Q_{8}^{l}\,(Q_{2}^{l}-\frac{1}{3}Q_{4}^{l}+\frac{1}{12}Q_{6}^{l}-\frac{1}{60}Q_{8}^{l}).$
(2.62)

If the space is bounded at finite proper distance then computing the central
charge just amounts to summing the $c^{l}_{hol}$ contributions from each cell
which represents a non-trivial prediction for the central charge of the 2d
CFTs dual to the solutions (2). If one or both of the boundaries are AdS4 then
in principle the same is true, albeit now there are an infinite number of
cells. We note that for the symmetric profiles discussed earlier we have

$c^{P+n}=c^{P-n-1},~{}~{}n=0,...,P-1~{}~{}~{}~{}\Rightarrow~{}~{}~{}~{}\sum_{l=0}^{2P-1}c^{l}=2\sum_{l=0}^{P-1}c^{l}.$
(2.63)

We will come back to the analysis of the central charge for a concrete choice
of solution in section 4.

##  3 A new class of $\mathcal{N}=(0,4)$ $\text{AdS}_{3}$ solutions to Type
IIB

In this section we present a new class of Type IIB solutions consisting on
AdS${}_{3}\times$S${}^{2}\times$S${}^{2}\times$S1 geometries foliated over two
intervals. These solutions preserve the $\mathfrak{osp}(4|2)$ superconformal
algebra and are generated by acting with Abelian T-duality on the class of
AdS${}_{3}\times\mathbb{CP}^{3}$ solutions in (2).

The $\mathbb{CP}^{3}$ space has a natural U(1) isometry on which one can act
by T-duality - in terms of the parametrisation of (2.2) this is
$\partial_{\psi}$. Following the standard Buscher’s procedure leads to a dual
metric of the form

$\begin{split}&\frac{ds^{2}}{2\pi}=\frac{h}{\sqrt{2hh^{\prime\prime}-(h^{\prime})^{2}}}\,ds^{2}(\text{AdS}_{3})+\frac{2h^{\prime\prime}}{\sqrt{2hh^{\prime\prime}-(h^{\prime})^{2}}}\,\left[\left(\frac{h^{\prime}}{h^{\prime\prime}}-r\right)d\xi-\frac{d\psi}{2\pi\sin\xi\cos\xi}\right]^{2}\\\
&\qquad+\frac{{\sqrt{2hh^{\prime\prime}-(h^{\prime})^{2}}}}{4h}\,dr^{2}+\frac{2{\sqrt{2hh^{\prime\prime}-(h^{\prime})^{2}}}}{h^{\prime\prime}}\left(d\xi^{2}+\frac{1}{4}\cos^{2}\xi
ds^{2}(\text{S}_{1}^{2})+\frac{1}{4}\sin^{2}\xi
ds^{2}(\text{S}_{2}^{2})\right)\,.\end{split}$ (3.1)

The dualisation of the rest of the NS-NS sector555For simplicity of notation
we set $l=0$ in (2) and (2.3). leads to the following expressions for the
dilaton, the $B_{2}$ field and its field-strength, $H_{3}=dB_{2}$,

$\begin{split}&e^{\Phi}=\frac{2}{\sin\xi\cos\xi\,h^{\prime\prime}}\,,\\\
&B_{2}=\pi\left(\frac{h^{\prime}}{h^{\prime\prime}}-r\right)\,\bigl{[}-\cos^{2}\xi\,\text{vol}(\text{S}_{1}^{2})+\sin^{2}\xi\,\text{vol}(\text{S}_{2}^{2})\bigr{]}+\left(\eta_{1}+\eta_{2}\right)\wedge
d\psi\,,\\\
&H_{3}=2\pi\sin\xi\cos\xi\left[\left(\frac{h^{\prime}}{h^{\prime\prime}}-r\right)d\xi-\frac{d\psi}{2\pi\sin\xi\cos\xi}\right]\wedge\bigl{[}\text{vol}(\text{S}_{1}^{2})+\text{vol}(\text{S}_{2}^{2})\bigr{]}\\\
&\qquad+\frac{\pi
h^{\prime}h^{\prime\prime\prime}}{(h^{\prime\prime})^{2}}\,dr\wedge\bigl{[}\cos^{2}\xi\,\text{vol}(\text{S}_{1}^{2})-\sin^{2}\xi\,\text{vol}(\text{S}_{2}^{2})\bigr{]}\,.\end{split}$
(3.2)

Finally, the R-R fluxes are given by

$\begin{split}&F_{1}=\sin\xi\cos\xi\,\left(h^{\prime\prime}-rh^{\prime\prime\prime}\right)d\xi-\frac{h^{\prime\prime\prime}}{2\pi}\,d\psi\,,\\\
&F_{3}=\pi\sin\xi\cos\xi\,F_{3,\xi\psi}\wedge\bigl{[}\cos^{2}\xi\,\text{vol}(\text{S}_{1}^{2})-\sin^{2}\xi\,\text{vol}(\text{S}_{2}^{2})\bigr{]}\,,\\\
&F_{5}=2\pi^{2}\sin\xi\cos\xi
F^{e}_{5,\xi\psi}\wedge\text{vol}(\text{AdS}_{3})\wedge
dr+\pi^{2}\sin^{3}\xi\cos^{3}\xi\,F^{m}_{5,\xi\psi}\wedge\text{vol}(\text{S}_{1}^{2})\wedge\text{vol}(\text{S}_{2}^{2})\,,\end{split}\,$
(3.3)

where we introduced the quantities $F_{3,\xi\psi}$, $F^{e}_{5,\xi\psi}$ and
$F^{m}_{5,\xi\psi}$ given by

$\begin{split}&F_{3,\xi\psi}=\frac{h^{\prime}(h^{\prime\prime}+rh^{\prime\prime\prime})-r(h^{\prime\prime})^{2}}{h^{\prime\prime}}\,d\xi-\frac{(h^{\prime\prime})^{2}-h^{\prime}h^{\prime\prime\prime}}{h^{\prime\prime}}\,\frac{d\psi}{2\pi\sin\xi\cos\xi}\,,\\\
&F^{e}_{5,\xi\psi}=\left(rh^{\prime}-2h-\frac{hh^{\prime}(h^{\prime}-rh^{\prime\prime})}{2hh^{\prime\prime}-(h^{\prime})^{2}}\right)^{\prime}d\xi+\frac{1}{2}\,\left(3h^{\prime}+\frac{(h^{\prime})^{3}}{2hh^{\prime\prime}-(h^{\prime})^{2}}\right)^{\prime}\frac{d\psi}{2\pi\sin\xi\cos\xi}\,,\\\
&F^{m}_{5,\xi\psi}=\frac{-6h(h^{\prime\prime})^{2}+3(h^{\prime})^{2}h^{\prime\prime}+r(h^{\prime})^{2}h^{\prime\prime\prime}}{(h^{\prime\prime})^{2}}\,d\xi+\frac{(h^{\prime})^{2}h^{\prime\prime\prime}}{(h^{\prime\prime})^{2}}\,\frac{d\psi}{2\pi\sin\xi\cos\xi}\,.\\\
\end{split}$ (3.4)

The function $h$ satisfies the same master equation as the
AdS${}_{3}\times\mathbb{CP}^{3}$ Type IIA solutions given by (2), namely

$h^{\prime\prime\prime}=-2\pi F_{0}\,,$ (3.5)

where in Type IIB $F_{0}$ is now associated to the charge of D7 branes, with a
discontinuous $F_{0}$ leading to sources smeared along the $\psi$ direction.

As for its dual partner (2), finding a brane intersection reproducing the
background (LABEL:IIBmetric) as a near-horizon geometry is particularly
challenging. To this end one can try, at least, to differentiate the defect
branes, associated to the back-reacted AdS3 geometry, and the “mother branes”,
associated to the ambient theory, including the defect. As we already argued
in the previous section the mother theory is ABJM. On the supergravity side,
this is realised when one fixes $F_{0}=0$ in (2). An analogous AdS4 limit can
also be obtained in Type IIB, which is actually the context in which the ABJM
brane intersection was first discussed: This consists of a stack of D3 branes
stretched between NS5’ and $(1,k)5^{\prime}$ branes, as depicted in Table 1.
These constitute the mother branes. In the next section we will propose a
concrete brane intersection consisting on NS5-D7 defect branes intersecting
the ABJM bound state breaking the isometries of the
AdS${}_{4}\times\mathbb{CP}^{3}$ vacuum. Our concrete brane picture will be
obtained by studying the Page charges.

The main elements that make the construction of the exact brane solution
associated to our backgrounds particularly intricate is that it is not
possible to extract the AdS4 vacuum as a local, asymptotic limit of the
AdS${}_{3}\times\text{I}$ solutions. Rather, as explained in the previous
section, it arises through a global gluing procedure. This is the
manifestation of the fact that the back-reaction of the defect branes
irredeemably breaks the AdS4 vacuum geometry. This implies that we cannot
study the equations of motion of defect and mother branes independently, as it
can be done for other simpler AdS3 solutions. Another difficulty is the
presence of fractional branes. These are produced by dualising $H_{3}$ and the
magnetic 4-flux in (2), which were defined along 2- and 4-cycles within the
$\mathbb{CP}^{3}$ space. From the Type IIA perspective these fluxes were
associated to fractional D4 branes.

In the remainder of this section we identify the sub-class of Type IIB
solutions associated to the ABJM theory. Fixing

$h^{\prime\prime\prime}=0\,~{}~{}~{}~{}\Rightarrow~{}~{}~{}~{}h=c_{2}+c_{4}r+\frac{c_{6}}{2}\,r^{2}$
(3.6)

again implies that the $(\text{AdS}_{3},r)$ directions become AdS4, as
explained in section 2.1. This is equivalent to taking the T-dual of the
AdS${}_{4}\times\mathbb{CP}^{3}$ solution. In this limit supersymmetry is
enhanced and the background (LABEL:IIBmetric) takes the form of an
$\mathcal{N}=4$ solution with topology
$\mbox{$\mathrm{AdS}$}_{4}\times\text{S}^{2}\times\text{S}^{2}\times\Sigma_{2}$,
which were classified in [64]. Specifically, we find the metric

$\begin{split}\frac{ds^{2}}{2\pi}&=\frac{\sqrt{2c_{2}c_{6}-c_{4}^{2}}}{2c_{6}}\,\left(ds^{2}(\text{AdS}_{4})+4d\xi^{2}+\cos^{2}\xi
ds^{2}(\text{S}_{1}^{2})+\sin^{2}\xi ds^{2}(\text{S}_{2}^{2})\right)\\\
&\quad+\frac{2c_{4}^{2}}{c_{6}\sqrt{2c_{2}c_{6}-c_{4}^{2}}}\,\bigl{(}d\xi-\frac{c_{6}}{2\pi
c_{4}\sin\xi\cos\xi}\,d\psi\bigr{)}^{2}.\end{split}$ (3.7)

From this expression we can extract the 2d metric over the Riemann surface
$\Sigma_{2}$, parametrised by the coordinates $(\xi,\psi)$. We observe that
this surface is an annulus.

The rest of the NS-NS sector boils down to

$\begin{split}&e^{\Phi}=\frac{2}{c_{6}\,\sin\xi\cos\xi}\,,\\\ &B_{2}=\frac{\pi
c_{4}}{c_{6}}\,\bigl{[}-\cos^{2}\xi\,\text{vol}(\text{S}_{1}^{2})+\sin^{2}\xi\,\text{vol}(\text{S}_{2}^{2})\bigr{]}+\left(\eta_{1}+\eta_{2}\right)\wedge
d\psi\,,\\\ &H_{3}=\left(\frac{2\pi
c_{4}}{c_{6}}\sin\xi\cos\xi\,d\xi-d\psi\right)\wedge\bigl{(}\text{vol}(\text{S}_{1}^{2})+\text{vol}(\text{S}_{2}^{2})\bigr{)}\,,\end{split}$
(3.8)

while the RR fluxes read

$\begin{split}&F_{1}=c_{6}\,\sin\xi\cos\xi\,d\xi\,,\\\
&F_{3}=\frac{c_{6}}{2}\,\left(\frac{2\pi
c_{4}}{c_{6}}\sin\xi\cos\xi\,d\xi-d\psi\right)\wedge\bigl{[}\cos^{2}\xi\,\text{vol}(\text{S}_{1}^{2})-\sin^{2}\xi\,\text{vol}(\text{S}_{2}^{2})\bigr{]}\,,\\\
&F_{5}=-\frac{3\pi}{2}\sqrt{2c_{2}c_{6}-c_{4}^{2}}\,\text{vol}(\text{AdS}_{4})\wedge\left(\frac{2\pi
c_{4}}{c_{6}}\sin\xi\cos\xi d\xi-d\psi\right)\\\
&\qquad+\frac{3\pi^{2}}{c_{6}}\sqrt{2c_{2}c_{6}-c_{4}^{2}}\sin^{3}\xi\cos^{3}\xi\,d\xi\wedge\text{vol}(\text{S}_{1}^{2})\wedge\text{vol}(\text{S}_{2}^{2}).\end{split}$
(3.9)

We conclude by pointing out that in order to reproduce the ABJM vacuum in the
form discussed in [39], we have to perform the following coordinate
transformation within the Riemann surface,

$\psi\to\psi+\frac{2\pi c_{4}}{c_{6}}\sin\xi\cos\xi\,d\xi\,.$ (3.10)

The effect of this transformation is to turn off the components of $H_{3}$,
$F_{3}$ and $F_{5}$ in (LABEL:NSNSABJMlimit) and (LABEL:RRABJMlimit)
orthogonal to $\psi$. These flux terms are pure gauge and they are typically
produced when one T-dualises a solution with an exact $B_{2}$ field, like our
Type IIA solution (2).

##  4 The dual field theory

In this section we take the first steps towards identifying the field theory
dual to the solutions in the general massive case. As in [39, 40] we propose a
brane set-up in Type IIB string theory. We argue that this brane set-up
preserves $\mathcal{N}=(0,3)$ supersymmetry, that should be enhanced to
$(0,6)$ in the IR. The mechanism at work should be a 2d realisation of the
supersymmetry enhancement that occurs in the ABJM/ABJ theory, albeit with half
the supersymmetries. In this case the enhancement is to manifest $(0,4)$
supersymmetry in Type IIB, consistently with the number of supersymmetries
preserved by the Type IIB $\text{AdS}_{3}$ solutions just discussed. This
becomes $(0,6)$ when the T-duality is undone to recover the Type IIA
description, again in agreement with the number of supersymmetries preserved
by the Type IIA solutions.

###  4.1 Hanany-Witten brane set-up

Let us first propose the brane set-up. On top of the D2, KK and D6 branes (or
D3, NS5′ and D5′-branes in the Type IIB description) of the massless case,
depicted in Table 1, there are now additional D8 and NS5 branes (or D7 and NS5
branes in the Type IIB description), as well as D4 (or D5) fractional branes.
A natural interpretation of the $\text{AdS}_{3}\times\mathbb{CP}^{3}$
solutions is then as the backreacted geometries that arise when a large number
of D8-NS5 defect branes are introduced in the ABJ theory, in a way that
preserves $\mathcal{N}=(0,6)$ supersymmetry, that is, half of the
supersymmetries, and a OSp$(6|2)$ superconformal group. The NS5-branes have
the effect of bounding the D2 and D6 branes along the domain wall direction of
the D8 branes.

In what follows we will consider the brane intersection in Type IIB, after a
T-duality is performed along the Hopf fibre of the S3 contained in the
$\mathbb{CP}^{3}$, in the parametrisation (2.2). As discussed in section 3,
the T-duality breaks the supersymmetry from $\mathcal{N}=(0,6)$ to
$\mathcal{N}=(0,4)$. In the brane intersection the 8 (Poincaré)
supersymmetries preserved by the D3-NS5′-D5′ branes of the ABJM set-up prior
to the rotation of the D5′-branes are broken by a half by the NS5-branes,
giving rise to $(0,4)$ supersymmetry in 2d666This was shown explicitly in
[65].. The D7-branes are added without breaking any further supersymmetries.
As in ABJM, once the rotation of the D5′-branes is taken into account
supersymmetry should be further broken to $(0,3)$, if the branes are rotated
the same angle along the $[3,7]$, $[4,8]$ and $[4,9]$ directions with respect
to the NS5′-branes777Otherwise the supersymmetries would be reduced to
$(0,2)$..

The Type IIB brane configuration is depicted in Table 2.

branes | $x^{0}$ | $x^{1}$ | $r$ | $x^{3}$ | $x^{4}$ | $x^{5}$ | $\psi$ | $x^{7}$ | $x^{8}$ | $x^{9}$
---|---|---|---|---|---|---|---|---|---|---
$\mbox{$\mathrm{D}$}3$ | $\times$ | $\times$ | $\times$ | $-$ | $-$ | $-$ | $\times$ | $-$ | $-$ | $-$
$\mbox{$\mathrm{NS}$}5^{\prime}$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $-$ | $-$ | $-$ | $-$
$(1,k)5^{\prime}$ | $\times$ | $\times$ | $\times$ | $\cos{\theta}$ | $\cos{\theta}$ | $\cos{\theta}$ | $-$ | $\sin{\theta}$ | $\sin{\theta}$ | $\sin{\theta}$
$\mbox{$\mathrm{D}$}7$ | $\times$ | $\times$ | $-$ | $\times$ | $\times$ | $\times$ | $-$ | $\times$ | $\times$ | $\times$
$\mbox{$\mathrm{NS}$}5$ | $\times$ | $\times$ | $-$ | $-$ | $-$ | $-$ | $\times$ | $\times$ | $\times$ | $\times$

Table 2: Brane intersection describing the ABJ brane system in Type IIB with
D7-NS5 “edge” states. The intersection describes a brane box with D3 colour
branes extended along the $r$ and $\psi$ directions. The brane intersection
should preserve $\mathcal{N}=(0,3)$ supersymmetry in 2d.

In this brane set-up the D7-NS5 defect branes create a domain wall in the 3d
theory living in the D3-NS5′-$(1,k)5^{\prime}$ branes. This is similar to the
D4-D8 defect branes introduced in [44] as probe defect branes in ABJM, in
order to realise edge states in the Fractional Quantum Hall Effect. The effect
of these branes in the probe brane approximation was to modify the ranks and
levels of the gauge groups of the Chern-Simons theory living in the D3-branes.
Instead, our study shows that for a large number of NS5-D8 defect branes the
resulting field theory becomes two dimensional, while still preserving one
half of the supersymmetries and a subgroup of the superconformal group.

In more detail, the brane set-up depicted in Table 2 describes a brane box
model in which $N_{l}$ D3-branes are bounded between NS5-branes along the $r$
direction, and between the NS5′ brane located at $\psi=0$ and the $(1,k_{l})$
5′ branes (a bound state of 1 NS5′ and $k_{l}$ D5′ branes) located at
$\psi=\pi$ along the $\psi$-direction. This number becomes $N_{l}+M_{l}$
between the $(1,k_{l})$ 5′-branes and the NS5′-brane at $\psi=2\pi$, due to
extra fractional D5-branes created between these branes. We have depicted the
brane picture in Figure 1.

Figure 1: Brane configuration associated to the
$\text{AdS}_{3}\times\mathbb{CP}^{3}$ solutions. $N_{l}$ D3-branes stretch
between $\psi=0$ and $\pi$ and $N_{l}+M_{l}$ between $\psi=\pi$ and $2\pi$.

In this configuration the number of D5′-branes bounded with the NS5′ brane at
$\psi=\pi$, $k_{l}$, changes as one moves along the $r$-direction and
therefore also the rotation angle $\theta_{l}$ between both types of
branes888Even if we have not indicated this explicitly in the drawing for the
sake of clarity., whose tangent is given by $\tan{\theta_{l}}=\Delta k_{l}$.
This happens due to a brane creation effect when the D7 flavour branes are
crossed. We have depicted this part of the brane set-up in Figure 2. This
subsystem describes a D5′-D7-NS5 brane intersection with five common
worldvolume directions, $k_{l}$ D5′-branes stretched between NS5-branes and
$\Delta q_{l}$ D7 flavour branes located at each $r\in[l,l+1]$ interval. As we
will see in the next subsection the number of D5′-branes created across
intervals is the one that corresponds to balanced 5d quivers, as described in
[66].

Figure 2: Brane creation effect for the D5′-NS5-D7 subsystem of the brane set-
up.

The whole configuration is thus a brane-box model in which D3 colour branes
are stretched between NS5-branes along the $r$-direction and between NS5′ and
$(1,k_{l})$ 5’-branes along the $\psi$-direction, with the number of
D5′-branes changing across intervals due to the presence of the D7-branes. On
top of this, D5 fractional branes are created in the $\psi\in[\pi,2\pi]$
interval when the $k_{l}$ D5′-branes are crossed.

In the next subsection we analyse the field content associated to this
configuration and propose a concrete quiver field theory that describes it.

###  4.2 Building blocks

As in 3d, the $\mathcal{N}=(0,3)$ gauge theory living in the D3-branes is
expected to have the same field content as that of an $\mathcal{N}=(0,4)$
gauge theory, except for the deformations introduced by the rotations of the
5-branes, responsible for the breaking of the supersymmetry to
$\mathcal{N}=(0,3)$. Specifically, the $(0,4)$ theory has SO(4) R-symmetry,
that should reduce to
SU(2)${}_{R}=\text{diag}(\text{SU}(2)\times\text{SU}(2))$ for the $(0,3)$
theory. In 3d the $\mathcal{N}=4$ Coulomb branch combines with the
$\mathcal{N}=4$ Higgs branch such that $\mathcal{N}=3$ supersymmetry remains.
In 2d $(0,4)$ theories there is no Coulomb branch, because the scalars live
now in (twisted) adjoint hypermultiplets. The mixing between the Coulomb and
Higgs branches should then be replaced by a combination between the twisted
and untwisted $(0,4)$ hypermultiplets, resulting in $(0,3)$ supersymmetry.
Unfortunately, since not much is known about 2d $(0,3)$ theories, in this
subsection we will take a “phenomenological” approach, hoping that our
discussion will stimulate more detailed investigations.

We start recalling the field content of the $(0,4)$ brane box models studied
in [45], to which we will add the rotation of the D5′-branes relative to the
NS5′-branes.

The brane box model studied in [45] consists on a D3-NS5-NS5′-D5-D5′ brane
intersection in which the branes are oriented as depicted in Table 3, with the
whole brane system preserving large $\mathcal{N}=(0,4)$ supersymmetry.

branes | $x^{0}$ | $x^{1}$ | $r$ | $x^{3}$ | $x^{4}$ | $x^{5}$ | $\psi$ | $x^{7}$ | $x^{8}$ | $x^{9}$
---|---|---|---|---|---|---|---|---|---|---
$\mbox{$\mathrm{D}$}3$ | $\times$ | $\times$ | $\times$ | $-$ | $-$ | $-$ | $\times$ | $-$ | $-$ | $-$
$\mbox{$\mathrm{NS}$}5^{\prime}$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $-$ | $-$ | $-$ | $-$
$\mbox{$\mathrm{D}$}5^{\prime}$ | $\times$ | $\times$ | $\times$ | $-$ | $-$ | $-$ | $-$ | $\times$ | $\times$ | $\times$
$\mbox{$\mathrm{NS}$}5$ | $\times$ | $\times$ | $-$ | $-$ | $-$ | $-$ | $\times$ | $\times$ | $\times$ | $\times$
$\mbox{$\mathrm{D}$}5$ | $\times$ | $\times$ | $-$ | $\times$ | $\times$ | $\times$ | $\times$ | $-$ | $-$ | $-$

Table 3: Brane intersection describing the D3-brane box model studied in [45],
where we have denoted $r$ and $\psi$ the field theory directions, as in the
brane set-up depicted in Table 2.

In this set-up the SO(3)${}_{345}\times$SO(3)789 rotation group is identified
with the SO(4)R R-symmetry. In our case the NS5′ and the D5′ branes are
located at the same position in $\psi$ and are rotated the same angle on the
$[3,7]$, $[4,8]$ and $[5,9]$ directions. These rotations render some of the
scalars of the brane box massive [67, 68], as we discuss below.

Prior to the rotation of the D5′-branes the quantisation of the open strings
with ends on the D3-branes in the brane box model gives rise to four types of
$\mathcal{N}=(0,4)$ multiplets [45]:

  * •

When the end-points of the string lie on the same stack of D3-branes the
projections induced by both the NS5 and NS5′ branes leave behind a $(0,4)$
vector multiplet.

  * •

When the end-points of the string lie on two different stacks of D3-branes
separated by an NS5-brane the degrees of freedom along the
$(x^{7},x^{8},x^{9})$ directions are fixed, leaving behind the scalars
associated to the $(x^{3},x^{4},x^{5})$ directions, which together with the
$A_{r}$ component of the gauge field give rise to a $(0,4)$ twisted
hypermultiplet in the bifundamental representation. The hypermultiplets are
twisted because the scalars are charged with respect to the SO(3)345 subgroup
of the R-symmetry group.

  * •

When the end-points of the string lie on two different stacks of D3-branes
separated by an NS5′-brane the degrees of freedom along the
$(x^{3},x^{4},x^{5})$ directions are fixed, leaving behind the scalars
associated to the $(x^{7},x^{8},x^{9})$ directions, which together with the
$A_{\psi}$ component of the gauge field give rise to a $(0,4)$ hypermultiplet
in the bifundamental representation. The hypermultiplets are untwisted because
the scalars are charged with respect to the SO(3)789 subgroup of the
R-symmetry group.

  * •

When the end-points of the string lie on two different stacks of D3-branes
separated by both an NS5 and an NS5′ brane all the scalars are fixed, leaving
behind the fermionic mode associated to a bifundamental $(0,2)$ Fermi
multiplet.

We have depicted the building blocks of the $(0,4)$ quivers just described in
Figure 3. In this quiver circles denote $(0,4)$ vector multiplets, black lines
$(0,4)$ twisted hypermultiplets, grey lines $(0,4)$ hypermultiplets and grey
dashed lines $(0,2)$ Fermi multiplets.

Figure 3: General quiver associated to D3-branes stretched between NS5 and
NS5’ branes in two perpendicular directions. Circles denote $(0,4)$ vector
multiplets, black lines $(0,4)$ twisted hypermultiplets, grey lines $(0,4)$
hypermultiplets and grey dashed lines $(0,2)$ Fermi multiplets.

To the previous matter content we have to add the multiplets that come from
the D5′ flavour branes999Since the D5-branes are fractional branes.. These
branes contribute with $(0,4)$ fundamental hypermultiplets to the D3-branes
lying in the same $r$-interval, and with $(0,2)$ fundamental Fermi multiplets
to the D3-branes in adjacent $r$-intervals. On top of this we need to add the
contribution of the D7-branes, which being completely orthogonal to the
D3-branes just contribute with $(0,2)$ fundamental Fermi multiplets to the
D3-branes lying in the same $r$-interval.

###  4.3 2d quivers

Taking into account the field content arising from the different branes in the
brane box configuration just reviewed we can now build up the quiver
associated to our particular brane set-up. We have depicted it in Figure 4.

Figure 4: Quiver diagram associated to our brane set-up. Black dashed lines
denote $(0,4)$ Fermi multiplets.

Since now $\psi$ is a compact direction, there are two $(0,4)$ hypermultiplets
connecting the $N_{l}$ and $N_{l}+M_{l}$ gauge groups at each $r$-interval,
that originate from the open strings that connect the $N_{l}$ D3 and
$N_{l}+M_{l}$ D3 branes across either one of the NS5′ branes. Similarly, there
are two $(0,2)$ Fermi multiplets connecting $N_{l}$ D3-branes with
$N_{l^{\prime}}+M_{l^{\prime}}$ D3 branes in adjacent $r$-intervals, depending
on the NS5′-brane crossed by the open strings. We have denoted these $(0,4)$
Fermi-multiplets with black dashed lines. In each $r$-interval the two $(0,4)$
bifundamental hypermultiplets connecting the gauge nodes with ranks $N_{l}$
and $N_{l}+M_{l}$ combine onto a $(0,6)$ bifundamental hypermultiplet. On top
of this we have the $(0,4)$ fundamental hypermultiplets and $(0,2)$
fundamental Fermi multiplets associated to the open strings that connect the
D3-branes with the D5′-branes and the D7-branes. In all, our quiver field
theory consists on a sequence of U$(N_{l})$ and U$(N_{l}+M_{l})$ gauge groups
with the field content of $(0,4)$ vector multiplets connected to each other by
$(0,6)$ bifundamental hypermultiplets, with extra $k_{l}$ fundamental or
antifundamental $(0,4)$ hypermultiplets101010Taking into account that due to
the different relative positions of the NS5′-branes at $\psi=0,2\pi$ compared
to the $(1,k_{l})$ 5-branes at $\psi=\pi$, the flavour groups contribute with
fundamentals to the $N_{l}$ gauge nodes and with antifundamentals to the
$N_{l}+M_{l}$ gauge nodes.. Note that at this point we have not yet taken into
account the rotation between the D5′ and the NS5′-branes. We will do this
after we discuss gauge anomaly cancellation.

Taking into account the contribution of the different multiplets to the gauge
anomaly, summarised in Table 4,

Multiplet | Contribution
---|---
(0,4) hyper | 1
(0,4) vector | -2N
(0,2) Fermi | $-\frac{1}{2}$
Table 4: Contribution to the gauge anomaly of the different multiplets that
couple in the quiver depicted in Figure 4.

we find that the following condition must be satisfied for each node in the
quiver

$2M_{l}-M_{l-1}-M_{l+1}+k_{l}-\frac{1}{2}k_{l-1}-\frac{1}{2}k_{l+1}-\frac{1}{2}\Delta
q_{l}=0.$ (4.1)

We proceed now with checking whether this is satisfied by the concrete quivers
associated to our solutions.

In doing this we first need to recall that due to the effects of the Freed-
Witten anomaly and the higher curvature terms the numbers of branes in the
brane set-up are not the same as the quantised charges built from the
solutions, as discussed already for the massless case in section 2.1. In the
massive case the introduction of D8(D7)-branes in the different intervals
produces further changes to the ones already summarised in the massless case,
given by (2.24). As shown in [41], the introduction of D8(D7)-branes produces
a jump on the field theory parameters due to the brane creation effect, but in
non-trivial topologies introduces as well extra D4(D5)-brane charge coming
from the half-integer component of the $B_{2}$ flux needed to cancel the
Freed-Witten anomaly and the higher curvature contributions. Through an
analogous calculation to the one that shows that a D6-brane wrapped on
$\mathbb{CP}^{3}$ carries 1/12 units of D2-brane charge, as a result of the
sum of a contribution from a Chern-Simons term and a higher curvature term,
one finds that a D8-brane domain wall carries 1/12 units of D4-brane charge
dissolved in its worldvolume. Putting all this together one finds that the
field theory D2, D4, D6 and D8 brane charges in the different $r$-intervals,
denoted by $(N_{l},M_{l},k_{l},q_{l})$, are related to the
$(Q_{2}^{l},Q_{4}^{l},Q_{6}^{l},Q_{8}^{l})$ quantised charges computed in
equations (2.33) through

$\displaystyle Q_{2}=N+\frac{k}{12},$ $\displaystyle
Q_{4}=M-\frac{k}{2}+\frac{q}{12},$ $\displaystyle Q_{6}=k,$ $\displaystyle
Q_{8}=-q,$ (4.2)

where we have used Type IIA notation to make the comparison with the
corresponding expressions in the massless case easier. These relations were
already found in [41] in a non-supersymmetric massive extension of the ABJ
theory considered therein. Using these relations one can then see that the
transformations (2.37) give rise to the following transformations for the
numbers of D3 (D2), D5 (D4) and D5′ (D6) branes across intervals

$\displaystyle N_{l}=N_{l-1}-M_{l-1}+k_{l-1},$ (4.3) $\displaystyle
M_{l}+\frac{q_{l}}{12}=M_{l-1}+\frac{q_{l-1}}{12}-k_{l-1},$ (4.4)
$\displaystyle k_{l}=k_{l-1}+q_{l-1},$ (4.5)

where we recall that even if in our configuration the D5-branes are fractional
branes, their numbers are still changing across intervals. Consistently with
this, one can check that the condition (4.4) is the same needed to ensure that
an NS5-brane is created across intervals, following the same reasoning around
equation (2.38), now in terms of the numbers of D4(D5), NS5 and D8(D7)-branes.
Note that the change in the D4(D5)-brane charge is due, on the one hand, to
the jump produced by the D8(D7)-brane domain wall, and, on the other, to the
D4(D5)-brane charge dissolved in its worldvolume, due to the topological CS
and higher curvature terms. The change in the D4(D5)-brane charge due to the
brane creation effect is the shift by $-k_{l-1}$.

As we already mentioned, we can understand the change of the D5′-brane charge
across intervals by looking at the 5d field theory living in the D5′-D7-NS5
brane subsystem of the brane set-up. The D7-branes introduced between
consecutive NS5-branes at $r=l,l+1$ create $k_{l}$ D5′-branes stretched
between the NS5-branes, satisfying the condition of balanced 5d quivers [66]

$2k_{l}=k_{l-1}+k_{l+1}+\Delta q_{l},$ (4.6)

with $\Delta q_{l}=q_{l-1}-q_{l}$ the number of D7-branes in the $[l,l+1]$
interval, responsible for the jump in the charge from $q_{l-1}$ to $q_{l}$.
This is precisely as implied by equation (4.5). As we mentioned, the
D5′-branes become however flavour in the complete brane intersection, being
orthogonal to the D3-branes along the $\psi$-direction.

In turn, the D5-brane charge generated by brane creation, that is, excluding
the charge dissolved in the D7-branes, satisfies

$2M_{l}=M_{l-1}+M_{l+1}-\Delta k_{l},$ (4.7)

with $\Delta k_{l}=k_{l-1}-k_{l}$. This condition shows that the number of D5
fractional branes changes across intervals due to the D5′-branes that are
being created at each $r$ interval because of the D7-branes.

At this point we can now proceed with checking equation (4.1). Substituting
(4.6) this condition becomes

$2M_{l}-M_{l-1}+M_{l+1}=0,$ (4.8)

which clearly is only satisfied by our quantised charges when $\Delta
k_{l}=0$, due to equation (4.7). Therefore, as expected, we find that only in
the massless case, when the quivers are truly $(0,4)$ supersymmetric111111They
are actually $\mathcal{N}=3$ in 3d, as discussed in section 2.1., the
condition for gauge anomaly cancellation for $(0,4)$ quivers is satisfied.
Instead, for our solutions the D5′-branes are rotated with respect to the
NS5′-branes, and this renders the scalar fields transverse to the D5′-branes
massive [68, 67]. The mass of these fields is proportional to the tangent of
the rotation angle between the branes, given at each interval by
$\tan{\theta_{l}}=\Delta k_{l}$, which is precisely the extra term in equation
(4.7). This shows that in the $(0,3)$ theory, when the massive scalar fields
are not taken into account in the calculation, gauge anomaly cancellation is
indeed satisfied. This provides a non-trivial consistency check for our
proposed quivers and their preserved supersymmetries.

Having checked the consistency of our proposal we construct in the next
subsection a concrete completion of the quivers depicted in Figure 4.

####  4.3.1 Completion of the quivers

In order for our proposed quivers to describe well-defined 2d field theories
we have to properly account for how they start and terminate along the $r$
non-compact direction. We will do this by embedding them onto the 3d ABJ
theory, that is, by ending the space with two massless regions that are
locally $\text{AdS}_{4}\times\mathbb{CP}^{3}$, as described in subsection 2.1.
As discussed in the previous section, the simplest way to do this
geometrically is to only have non trivial $Q^{8}_{l}<0$ for $1\leq r<P$
assuming continuity conditions of the type (2.39) throughout. The result is
depicted in Figure 5.

Figure 5: Quiver completed by two AdS4 regions for $r<1$ and and $r\geq P$ .

This shows a quiver like the one depicted in Figure 4 terminating with a
massless regions without D8-branes.

In the massless regions at both ends $r$ becomes part of the external space
and a 3d theory is deconstructed, their behaviour is analogous so we will only
describe the first region explicitly: Here $k_{0}$ turns on a Chern-Simons
level for the 3d gauge groups with ranks $N_{0}$ and $N_{0}\pm M_{0}$, with
opposite signs due to the relative positions of the NS5′ at $\psi=0,2\pi$
compared to the $(1,k_{0})$ 5-branes at $\psi=\pi$. Additionally the two
$(0,4)$ Fermi multiplets that were connecting gauge nodes across intervals
combine with the $(0,6)$ hypermultiplet connecting the $N_{0}$ and $N_{0}\pm
M_{0}$ gauge nodes, to form a $(6,6)$ hypermultiplet. In turn, the $(0,4)$
twisted hypermultiplets connecting the $N_{0}$ and $N_{1}$ nodes and the
$N_{0}\pm M_{0}$ and $N_{1}\pm M_{1}$ nodes turn onto adjoint twisted
hypermultiplets for each of the $N_{0}$ and $N_{0}\pm M_{0}$ gauge nodes. They
then combine with the respective $(0,4)$ vector multiplets to form $(4,4)$
vector multiplets. This is completed by the two gauge nodes being connected
wit the $k_{0}$ flavour group with $(4,4)$ fundamental multiplets. Therefore,
we find that in the massless regions supersymmetry is enhanced to 8 Poincaré
supercharges. The rotations between the D5′ and NS5′-branes, not taken into
account in this analysis, finally render the number of supersymmetries equal
to 6 Poincaré supercharges, equivalent to $\mathcal{N}=3$ in 3d, as in the
ABJM/ABJ theory.

We show in the next subsection that the mappings across intervals given by
equations (4.3)-(4.5) find an interesting interpretation in relation with
Seiberg duality in 3d Chern-Simons matter theories.

###  4.4 Connection with Seiberg duality

The transformations of the field theory charges given by (4.3)-(4.5)

$N\rightarrow N-M+k,\qquad M\rightarrow M-k,\qquad k\rightarrow k+q$ (4.9)

due to brane creation across intervals (which in the case of the D4-branes
requires subtracting the charge dissolved in the D8-branes) are precisely the
extension of the Seiberg dualities in ABJM/ABJ discussed in [40, 43] to the
non-supersymmetric U$(N+M)_{k}\times$U$(N)_{-k+q}$ Chern-Simons matter
theories proposed in [41]. The particular set up in [41] is a non-
supersymmetric one obtained by a D8-brane deformation of the ABJM/ABJ theory,
in which the D8-brane is a non-supersymmetric domain wall embedded into
$\text{AdS}_{4}\times\mathbb{CP}^{3}$ spanning $\text{AdS}_{4}$ and a five
dimensional subspace of $\mathbb{CP}^{3}$. Treating the D8-branes as probes it
was shown that the deformation leaves the metric unchanged and interpolates
between the ABJM/ABJ background and the non-supersymmetric
AdS${}_{4}\times\mathbb{CP}^{3}$ background of massive IIA found in [69]. A
generalisation of the brane creation effects discussed in [43] to include the
D8-branes led them to propose precisely the transformations above as the
generalisation of Seiberg duality to the massive case. These transformations
map non-supersymmetric Chern-Simons matter theories with gauge groups
U$(N+M)_{k}\times$U$(N)_{-k+q}$ and U$(N)_{k+q}\times$U$(N-M+k)_{-k}$ onto
each other121212Still, in the absence of supersymmetry it is not possible to
conclude that these theories should be equivalent in the IR..

Our results suggest that when the D8-branes are embedded in the 3d theory in a
supersymmetric way, which implies adding as well NS5 branes and fractional
D4-branes, one of the 3d field theory directions (the $r$ direction in the
parametrisation (2.17)-(2.18)) turns into an energy scale, and generates a
flow towards a 2d CFT. Geometrically the backreaction of the D4-D8-NS5 defect
branes gives rise to a $(0,6)$ supersymmetric
$\text{AdS}_{3}\times\mathbb{CP}^{3}$ geometry where $r$ becomes part of the
internal space. In this geometric setting it is now possible to perform large
gauge transformations along the $r$ direction that create new branes, that
precisely realise the extension of Seiberg duality to the massive case
proposed in [41].

###  4.5 A comment on the central charge

The holographic central charge of the 2d $(0,6)$ SCFTs dual to the solutions
with $h$-functions given by (2.30) can be obtained substituting in the general
expression (2.15). For the completed quivers depicted in Figure 5 we obtain

$c_{hol}=\frac{1}{2}\sum_{l=0}^{P}\Bigl{(}2N_{l}k_{l}-M_{l}^{2}+M_{l}k_{l}-\frac{1}{12}k_{l}^{2}+q_{l}(N_{l}-\frac{1}{2}M_{l}+\frac{5}{12}k_{l}-\frac{13}{720}q_{l})\Bigr{)}.$
(4.10)

This constitutes a very non-trivial prediction on the field theory side. One
can check that in the massless limit (4.10) reduces to equation (2.26), later
“reconverted” onto a 3d expression in (2.27) (and shown to agree with the
field theory results using localization [59, 60, 61, 62]). Note that since we
have used the shifted values of the charges this expression includes higher
derivative corrections. This implies that the predictions on the field theory
side should go beyond the planar approximation.

On the field theory side the central charge can be computed as the central
extension of the $\mathfrak{osp}(6|2)$ superconformal algebra, given by [70]

$c_{R}=\frac{k(3k+13)}{k+3},$ (4.11)

where $k$ is the level of the algebra. For superconformal algebras with
$(0,2)$ supersymmetry, the level of the algebra can be computed from the
U$(1)_{R}$ R-symmetry anomaly, using that (see for instance [2])

$k=\text{Tr}[\gamma_{3}Q_{R}^{2}],$ (4.12)

where $Q_{R}$ is the R-charge under the U$(1)_{R}$ R-symmetry group, and the
trace is over all Weyl fermions in the theory. Given that the R-symmetry
anomaly is a ’t Hooft anomaly, the calculation can be performed in the weakly
coupled UV description of the field theory. We have argued that in our case we
have $(0,3)$ supersymmetry, so this expression, in principle, does not apply.
Still, we can check that it is possible to recover the leading terms in
equation (4.10) when $q=0$, that is, when the field theory is $(0,4)$
supersymmetric, and therefore expression (4.12) can be used. Indeed, recalling
the R-charges of the fermions in the different multiplets, summarised in Table
5,

Multiplet | chirality | R-charge
---|---|---
(0,4) hyper | R.H. | -1
(0,4) twisted hyper | R.H. | 0
(0,4) vector | L.H. | 1
(0,2) Fermi | L.H. | 0
Table 5: R-charges and chiralities of fermions in $(0,4)$ multiplets.

we find

$k=\sum_{l=0}^{P}\Bigl{(}2N_{l}k_{l}+M_{l}k_{l}-M_{l}^{2}\Bigr{)},$ (4.13)

which matches the first three terms in (4.10) upon suitable normalisation. In
turn, the terms in (4.10) proportional to the mass should originate field
theoretically from genuine $(0,3)$ multiplets, whose contribution to the
central charge has not, to our knowledge, been worked out in the literature.
This remains as an interesting open problem that deserves further
investigation. On more general grounds, one could expect that a mixing between
the SU(2)R R-symmetry and the SU(2) global symmetry occurs in the
determination of the infrared R-current, which should then be computed via
c-extremisation [71, 72]. Expression (4.10) constitutes in this sense a very
useful result that can be used as a lead for the field theory investigations.

Finally, in the remainder of this subsection we show that the holographic
central charge can be obtained as a product of electric and magnetic charges
associated to the solutions. This generalises the results in [46, 47, 48] for
$\text{AdS}_{2}$ backgrounds with $\mathcal{N}=4$ supersymmetries to
$\text{AdS}_{3}$, with different number of supersymmetries. This suggests that
the findings in [46, 47, 48] should hold more broadly.

####  4.5.1 The central charge as a product of electric and magnetic charges

Following [46, 47, 48] we define a density of electric and magnetic charges
from the electric and magnetic components of the Page fluxes of a given
background, as

$\displaystyle\mathcal{Q}_{p}^{e}=\frac{\hat{f}^{e}_{p+2}}{(2\pi)^{p}},\qquad\mathcal{Q}_{p}^{m}=\frac{\hat{f}^{m}_{8-p}}{(2\pi)^{7-p}},$
(4.14)

where, as in these references, the electric density will require a
regularisation, as it involves an integration over the $\text{AdS}_{3}$
subspace.

The product of electric and magnetic densities computed as

$\mathcal{Q}=\int\sum_{k=0}^{4}(-1)^{k+1}\mathcal{Q}_{(2k)}^{e}\mathcal{Q}_{(2k)}^{m}=\frac{1}{(2\pi)^{7}}\int\sum_{k=1}^{4}(-1)^{k+1}\hat{f}^{e}_{2k+2}\hat{f}^{m}_{8-2k},$
(4.15)

gives,

$\mathcal{Q}=\int\frac{\text{vol}(\text{AdS}_{3})}{6\pi}\wedge\frac{\text{vol}(\mathbb{CP}^{3})}{\pi^{3}/6}\wedge
dr\left(2hh^{\prime\prime}-(h^{\prime})^{2}+\partial_{r}\left(hh^{\prime}-\frac{h^{\prime\prime\prime}h^{3}}{2hh^{\prime\prime}-(h^{\prime})^{2}}\right)\right),$
(4.16)

upon substitution of the magnetic Page fluxes given by (2.6) and the
corresponding electric components, given by131313For simplicity of notation we
set $r=l$.

$\displaystyle\begin{aligned}
&\hat{f}^{e}_{4}=\frac{-h\pi}{2hh^{\prime\prime}\\!-\\!(h^{\prime})^{2}}\left(\frac{(h^{\prime})^{3}h^{\prime\prime\prime}}{2hh^{\prime\prime}\\!-\\!(h^{\prime})^{2}}\\!-\\!3(h^{\prime\prime})^{2}\right)\text{vol}(\text{AdS}_{3})\\!\wedge\\!dr,\\\\[5.69054pt]
&\hat{f}^{e}_{6}=\frac{4h\pi^{2}}{2hh^{\prime\prime}\\!-\\!(h^{\prime})^{2}}\left(\frac{2h(h^{\prime})^{2}h^{\prime\prime\prime}}{2hh^{\prime\prime}\\!-\\!(h^{\prime})^{2}}\\!-\\!3h^{\prime}h^{\prime\prime}\right)\text{vol}(\text{AdS}_{3})\\!\wedge\\!J\\!\wedge\\!dr,\\\
&\hat{f}^{e}_{8}=\frac{-8h\pi^{3}}{2hh^{\prime\prime}\\!-\\!(h^{\prime})^{2}}\left(\frac{4h^{\prime}h^{\prime\prime\prime}h^{2}}{2hh^{\prime\prime}\\!-\\!(h^{\prime})^{2}}\\!-\\!2(hh^{\prime\prime}\\!+\\!(h^{\prime})^{2})\right)\text{vol}(\text{AdS}_{3})\\!\wedge\\!J\\!\wedge\\!J\\!\wedge\\!dr,\\\
&\hat{f}^{e}_{10}=\frac{32h\pi^{4}}{2hh^{\prime\prime}\\!-\\!(h^{\prime})^{2}}\left(\frac{8h^{3}h^{\prime\prime\prime}}{3(2hh^{\prime\prime}\\!-\\!(h^{\prime})^{2})}\\!-\\!2hh^{\prime}\right)\text{vol}(\text{AdS}_{3})\\!\wedge\\!J\wedge\\!J\wedge\\!J\\!\wedge\\!dr.\end{aligned}$

As one can see, this expression is proportional up to a boundary
term141414Note that using (2.30) and the boundary conditions $h(0)=h(P)=0$ the
boundary term does not contribute to the final result for the holographic
central charge. As already mentioned, in order to find a finite result the
volume of $\text{AdS}_{3}$ has to be conveniently regularised., to our
expression (2.15) for the holographic central charge. In [46, 47, 48] an
analogous result was obtained for various $\text{AdS}_{2}$ backgrounds with
$\mathcal{N}=4$ supersymmetries, and it was interpreted as a generalisation of
the proposal in [49], showing that the central charge in the algebra of
symmetry generators of $\text{AdS}_{2}$ with an electric field is proportional
to the square of the electric field. In these references it was shown that the
proposal in [49] could be extended to fully-fledged $\text{AdS}_{2}$ string
theory set-ups, where it holds for the different branes involved in terms of
their electric and magnetic charges. Our result in this section shows that the
proposal in [49] is not exclusive to $\text{AdS}_{2}$, but holds in more
general settings involving AdS backgrounds with different dimensionalities.
This points at a possible relation to the calculation of the holographic
central charge in terms of the regularised on-shell supergravity action. Work
is in progress to try to shed more light on this issue.

##  5 Conclusions

In this paper we have made progress towards the understanding of
$\text{AdS}_{3}/\text{CFT}_{2}$ holography with $\mathcal{N}=(0,6)$
supersymmetry. Taking as our starting point the recent local solutions
constructed in [1], for which the internal space is a foliation of a
$\mathbb{CP}^{3}$ over an interval, and thus bear a close resemblance with the
ABJM/ABJ solution, we have proposed a brane set-up and, associated to it, a
quiver field theory emerging from the quantisation of the open strings.
According to our proposal this field theory should flow in the IR to the 2d
CFT dual to the $\text{AdS}_{3}\times\mathbb{CP}^{3}$ solutions. We have
argued that, in full similarly with the 3d case, supersymmetry should be
enhanced from $\mathcal{N}=(0,3)$ for the brane intersection to
$\mathcal{N}=(0,6)$ for the SCFT that emerges in the IR. A difference with the
3d case is that both our solutions and associated field theories preserve half
of the supersymmetries of the 3d case. This is consistent with an
interpretation of the solutions as describing $\frac{1}{2}$-BPS backreacted
surface defects within the ABJM/ABJ theory. These defects consist on D8-NS5
branes that reduce the supersymmetries of the ABJ brane set-up by a half and
the superconformal algebra to $\mathfrak{osp}(6|2)$. This could represent a
conformal embedding of the D4-D8 defects in ABJM proposed in [44], in order to
realise edge states in the Fractional Quantum Hall Effect151515Albeit with
fractional D4-branes.. In the absence of D8-branes our proposed brane
configuration can also be seen as the intersection of the ABJM brane set-up,
preserving $\mathcal{N}=3$ in 3d, with the D2-NS5-D4 brane intersection
studied in [67, 68], also $\mathcal{N}=3$ supersymmetric in 3d161616See also
the closely related brane intersections studied in [73, 74], realising Chern-
Simons matter theories of quiver type.. The resulting D2-KK-D6-D4-NS5
intersection would preserve $\mathcal{N}=(0,3)$ supersymmetry in 2d, and these
supersymmetries would be left unbroken when the D8-branes are introduced. An
alternative interpretation of the $\text{AdS}_{3}\times\mathbb{CP}^{3}$
solutions would thus be as holographic duals to the 2d CFTs to which the field
theories living in these brane intersections flow in the IR. Our proposed 2d
quivers manifestly realise the embedding of the defect branes within the
quiver field theory associated to the ABJM/ABJ theory. It would be interesting
to compute observables such as one point correlation functions and the
displacement operator [8], that further confirm our defect interpretation.

Interestingly, we have shown that large gauge transformations in the brane
set-up induce the generalisation to the massive case of Seiberg duality in
ABJM/ABJ theories [40, 43] proposed in [41]. This shows that, as in other
theories [42, 43], Seiberg duality can be understood geometrically in terms of
large gauge transformations. Remarkably, we have obtained the same
transformations as the ones proposed in [41] for non-supersymmetric 3d Chern-
Simons matter theories. This suggests, on the one hand, that the extension of
Seiberg duality in ABJM/ABJ to the massive case proposed in [41] could hold
more generally, and, on the other, that Seiberg duality in the Gaiotto-
Tomasiello theories [69] ($\mathcal{N}=3$ in 3d) could be realised as a large
gauge transformation at the expense of turning one of the field theory
directions onto an energy scale, inducing a flow across dimensions to a 2d CFT
with $\mathcal{N}=(0,6)$ supersymmetry and an
$\text{AdS}_{3}\times\mathbb{CP}^{3}$ dual.

The latter might suggest that $\text{AdS}_{4}$ solutions to massive Type IIA
supergravity with $\mathcal{N}=3$ supersymmetry might not exist beyond the
perturbative small mass limit taken in [75]. We would like to stress that in
our solutions the mass parameter can be of the same order as the rest of
quantised charges, so the solutions are in this sense fully non-perturbative,
and so are their proposed 2d dual CFTs, as opposed to the small deformations
of the ABJM/ABJ theory studied in [69]. Note as well that the 3d CFTs studied
in [69] were constructed under the assumption that the (small) massive
deformations should give rise to different fixed point theories in the same
number of dimensions, namely three. It would be interesting to investigate the
small mass limit of our solutions and see whether they bear any connection
with the perturbative solutions found in [75].

With respect to the central charge we have made a concrete prediction for the
$(0,6)$ SCFTs dual to the $\text{AdS}_{3}\times\mathbb{CP}^{3}$ solutions that
include sub-leading corrections. However, we have not been able to check this
result against a field theory calculation once the rotation of the branes is
taken into account and the supersymmetry is reduced to $(0,3)$. As mentioned
in the main text, we are unaware of a general result in the literature that
relates the level of the superconformal algebra with the R-symmetry anomaly
for $(0,3)$ supersymmetry. We hope that our, in this sense, phenomenological,
results, stimulate further investigations in this direction. It is likely that
the studies in [76, 77] of 2d $(0,1)$ field theories can be successfully used
for this purpose.

More generally, it would be interesting to relate the (holographic) central
charge computed in this paper to the exact results for the free energy of the
small deformations of the ABJM/ABJ theory with $\mathcal{N}=3$ supersymmetry
studied in [78, 79, 80, 81], along the lines of our discussion for the
massless case.

## Acknowledgements

We would like to thank Antonio Amariti and Carlos Nunez for useful
discussions. YL and NM are partially supported by the grant from the Spanish
government MCIU-22-PID2021-123021NB-I00. The work of NM is also supported by
the Ramón y Cajal fellowship RYC2021-033794-I. The work of NP is supported by
the Israel Science Foundation (grant No. 741/20) and by the German Research
Foundation through a German-Israeli Project Cooperation (DIP) grant
“Holography and the Swampland”. The work of AR is partially supported by the
INFN grant Gauge and String Theory (GAST) and by the INFN-UNIMIB contract,
number 125370/2022.

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