New $\text{AdS}_{2}/\text{CFT}_{1}$ pairs from $\text{AdS}_{3}$ and monopole
bubbling

Andrea<EMAIL_ADDRESS>Yolanda<EMAIL_ADDRESS>Niall T<EMAIL_ADDRESS>

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

and

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

Abstract

We present general results on generating $\text{AdS}_{2}$ solutions to Type II
supergravity from $\text{AdS}_{3}$ solutions via U(1) and SL(2) T-dualities.
We focus on a class of Type IIB solutions with small $\mathcal{N}=4$
supersymmetry, that we show can be embedded into a more general class of
solutions obtained by double analytical continuation from $\text{AdS}_{3}$
geometries with small $\mathcal{N}=(0,4)$ supersymmetry constructed in the
literature. We then start the analysis of the superconformal quantum mechanics
dual to the $\mathcal{N}=4$ backgrounds focusing on a subclass of
$\text{AdS}_{2}\times\text{S}^{3}\times\mathbb{T}^{3}$ solutions foliated over
a Riemann surface. We show that the associated supersymmetric quantum
mechanics describes monopole bubbling in 4d $\mathcal{N}=2$ supersymmetric
gauge theories living in D3-D7 branes, as previously discussed in the
literature. Therefore, we propose that our solutions provide a geometrical
description via holography of monopole bubbling in 4d $\mathcal{N}=2$ SCFTs.
We check our proposal with the computation of the central charge.

###### Contents

  1. 1 Introduction
  2. 2 Generating AdS2 solutions from U(1) and SL(2) T-duality on AdS3
  3. 3 New classes of small ${\cal N}=4$ AdS2 solutions in Type IIB
    1. 3.1  A small ${\cal N}=(0,4)$ AdS3 class in IIA and its SL(2) T-duality
    2. 3.2 Two solutions on AdS${}_{2}\times\text{S}^{3}\times\mathbb{T}^{3}\times\Sigma_{2}$
      1. 3.2.1 AdS2 case with $hg\propto 1$
      2. 3.2.2 AdS2 case with $h=$ constant
    3. 3.3 Embedding into a general class of AdS${}_{2}\times\text{S}^{3}\times\mathbb{T}^{3}\times\Sigma_{2}$ solutions
      1. 3.3.1 Recovering the $hg\propto 1$ case
      2. 3.3.2 Recovering the $h=$ constant case
  4. 4 Field theory analysis
    1. 4.1 Hanany-Witten brane set-up and quantised charges
    2. 4.2 Baryon vertex interpretation
    3. 4.3 Quiver quantum mechanics
      1. 4.3.1 Computation of the central charge
    4. 4.4 ’t Hooft defect interpretation
  5. 5 Conclusions
  6. A Conventions
  7. B AdS3, its isometries and Killing spinors
  8. C Deriving the SL(2) T-duality dual fields
  9. D Proving SL(2) T-duality preserves supersymmetry
    1. D.1 Supersymmetry constraints for AdS3 solutions
    2. D.2 U(1) T-duality on the Hopf fiber of AdS3
    3. D.3 SL(2) T-duality on AdS3
  10. E Details of the quiver construction

##  1 Introduction

The $\text{AdS}_{2}/\text{CFT}_{1}$ correspondence plays a key role in the
microscopical description of extremal black holes, $\text{AdS}_{2}$ being part
of the geometry that appears in their near horizon limit in any dimension. It
presents however important conceptual problems that have to do mainly with the
non-connected nature of the $\text{AdS}_{2}$ boundary and the fact that
$\text{AdS}_{2}$ gravity is non-dynamical.

In recent years new examples of $\text{AdS}_{2}$ geometries with different
supersymmetries have been constructed [1, 2, 3, 4, 5, 6, 7, 8] (see also [9,
10]) that show that when extra fluxes are added $\text{AdS}_{2}$ gravity
ceases to be non-dynamical, and allows for a non-vanishing central extension
from which the black hole entropy can be computed. These examples extend the
$\text{AdS}_{2}$ geometries with constant electric fields proposed early on in
[11, 12, 13, 14, 15]. In some of these constructions the dual superconformal
quantum mechanics (SCQM) was also identified, and it was possible to check the
conjecture in [13], namely, that a consistent quantum gravity on
$\text{AdS}_{2}$ should be dual to a chiral half 2d CFT, with explicit
realisations. Indeed, in these constructions the supersymmetric quantum
mechanics from which the SCQMs arise in the IR are compactifications of 2d
(0,4) SCFTs, and have the same superconformal algebras associated to them.
This has been used to compute the central charge of the SCQMs, and perfect
agreement has been found with the corresponding holographic expressions. The
proposed SCQMs thus provide explicit scenarios where the entropy of extremal
black holes can be computed microscopically.

Another useful application of the $\text{AdS}_{2}/\text{CFT}_{1}$
correspondence is to the holographic description of superconformal line
defects in higher dimensional CFTs [16, 17, 18, 19, 6, 7, 8]. Geometrically, a
sign that an $\text{AdS}_{2}$ solution may be describing a superconformal line
defect is that it flows asymptotically locally to a higher dimensional AdS
background, dual far from the defects to the higher dimensional CFT in which
they are embedded. In some of these constructions it has been shown that the
quiver quantum mechanics dual to the $\text{AdS}_{2}$ solutions can be
embedded within the quivers that describe the higher dimensional SCFTs [6, 7].
In some instances the defects are described by quivers living in brane boxes
[7], with the latter finding in this way a powerful holographic realisation.

In this paper we present general results on the construction of
$\text{AdS}_{2}$ solutions to Type II supergravity via U(1) and SL(2)
T-dualities. We then exploit these to construct new classes of small
$\mathcal{N}=4$ solutions in Type II supergravity and study the dual field
theory description of a subclass of these solutions, for which the SCQM arises
once more upon compactification of a 2d (0,4) SCFT. The intricacy of the brane
set-up associated to these solutions reveals an interesting defect
interpretation in terms of baryon vertices and ’t Hooft loops within higher
dimensional CFTs.

The paper is organised as follows. We start in section 2 presenting general
results on the generation of $\text{AdS}_{2}$ solutions in Type II
supergravity from $\text{AdS}_{3}$ solutions via U(1) and SL(2) T-dualities,
paying special attention to the conditions for preservation of supersymmetry.
This section is supplemented by appendices A-D. In section 3 we focus on the
particular classes of solutions to Type IIB supergravity preserving
$\mathcal{N}=4$ supersymmetry obtained from the
$\text{AdS}_{3}\times\text{S}^{3}\times\text{M}_{4}$ solutions to massive IIA
with $\mathcal{N}=(0,4)$ supersymmetry and SU(3) structure constructed in
[20]. We show that the solutions with a 3-torus isometry can be embedded into
a more general class of solutions, that we construct via double analytical
continuation from the $\text{AdS}_{3}$ solutions in Type IIB with
$\mathcal{N}=(0,4)$ supersymmetry constructed in [21]. Then in section 4 we
move to the field theory description of a subclass of the previous solutions,
consisting on $\text{AdS}_{2}\times\text{S}^{3}\times\mathbb{T}^{3}$
foliations over a 2d Riemann surface, parametrised by two non-compact
directions $(\rho,r)$. We show that the associated quantum mechanics lives in
D1-branes stretched between NS5-branes along the $\rho$ direction, with D3, D5
and D7 branes contributing with flavour groups to the resulting (0,4) quantum
mechanics, that, we propose, flows in the IR to the SCQMs dual to the
solutions. Making contact with similar 1d quivers proposed in the literature
in the description of bubbling dyonic monopoles in 4d $\mathcal{N}=2$
supersymmetric field theories living in D3-D7 systems [22, 23, 24, 25], we
identify our quiver constructions as describing the bubbling sector of
vanishing effective magnetic charge in these 4d theories. In turn, the
disposition of the branes along the $r$ direction allows us to interpret the
F1-strings present in the brane set-up in relation to baryon vertices in the
5d theory living in the D5-NS5-D7 subset of branes. We check our holographic
proposal with the computation of the central charge on both sides of the
AdS/CFT correspondence. On the field theory side we compute the central charge
from the R-symmetry anomaly, using the same expression valid in two
dimensions. We find perfect agreement with the holographic result. Appendix E
contains technical details of the construction of the 1d quivers discussed in
section 4.

##  2 Generating AdS2 solutions from U(1) and SL(2) T-duality on AdS3

In this section we present some general results on generating AdS2 solutions
of Type II supergravity from AdS3 solutions via U(1) and SL(2) T-dualities. In
particular we present the general form of such solutions and explain how they
preserve supersymmetry. The content of this section is extensively
supplemented by the technical appendices A-D where the results are derived.

An AdS3 solution of Type II supergravity can in general be decomposed in the
form

$\displaystyle ds^{2}$
$\displaystyle=e^{2A}ds^{2}(\text{AdS}_{3})+ds^{2}(\text{M}_{7}),$
$\displaystyle H^{(10)}$
$\displaystyle=c_{0}\text{vol}(\text{AdS}_{3})+H,~{}~{}~{}~{}F=f_{\pm}+e^{3A}\text{vol}(\text{AdS}_{3})\wedge\star_{7}\lambda
f_{\pm},$ (2.1)

where $(e^{2A},f_{\pm},H)$ and the dilaton $\Phi$ have support on
$\text{M}_{7}$, $c_{0}$ is a constant and the upper/lower signs are taken in
IIA/IIB. We shall be interested in a subset of these for which

$c_{0}=0,~{}~{}~{}~{}H^{(10)}=H,$ (2.2)

ie those with purely magnetic NS 3-form flux, otherwise the potential for
$H^{(10)}$ will not be SL(2) invariant and we will be unable to perform a non-
Abelian T-duality transformation on it444Note that an electric component of
the NS flux does not pose a problem for U(1) T-duality on the Hopf fiber of
AdS3, however the dual solution would not have a round AdS2 factor - this
would instead appear with a U(1) fibred over it in the dual solution.. The
Bianchi identities the fluxes should obey (away form the loci of sources)
reduce to the $d=7$ conditions

$dH=0,~{}~{}~{}(d-H\wedge)f_{\pm}=0,~{}~{}~{}~{}(d-H\wedge)(e^{3A}\star_{7}\lambda(f_{\pm}))=0.$
(2.3)

When an AdS3 solution preserves at least ${\cal N}=1$ supersymmetry it admits
at least a pair of Majorana spinors $(\chi_{1},\chi_{2})$ on M7 such that one
can define a pair of $d=10$ Majorana-Weyl spinors decomping as

$\epsilon_{1}=\zeta\otimes\theta_{+}\otimes\chi_{1},~{}~{}~{}~{}~{}\epsilon_{2}=\zeta\otimes\theta_{\mp}\otimes\chi_{2},$
(2.4)

where $\zeta$ is a Killing spinor on unit radius AdS3 obeying the equation

$\nabla^{\text{AdS}_{3}}_{\mu}\zeta=\frac{s}{2}\gamma_{\mu}\zeta,~{}~{}~{}\mu=0,1,2~{}~{}~{}~{}s^{2}=1.$
(2.5)

For such spinors the necessary conditions for supersymmetry ultimately reduce
to a set of $d=7$ conditions involving only $(\chi_{1},\chi_{2})$ (see
(D.8a)-(D.8f)). The specific sign $s$ takes is related to the type of chiral
algebra one has on the boundary of AdS3, ${\cal N}=(1,0)$ or ${\cal N}=(0,1)$.
Which corresponds to which sign is a matter of convention and the literature
has no set standard. The remaining terms $\theta_{\pm}$ are 2 dimensional
vectors which account for $d=10$ chirality, again the upper/lower signs are
taken in Type IIA/IIB - further details can be found in appendix D.1. Of
course a generic AdS3 solution can preserve ${\cal N}=(p,q)$ supersymmetry for
$p,q$ integers such that $p+q\leq 8$ [26]. In this case one will actually have
$p$ independent versions of (2.4) coupled to Killing spinors on AdS3 that obey
(2.5) for one sign and a further $q$ versions with AdS3 Killing spinors
obeying the opposite sign - each of these comes equipped with an independent
$(\chi_{1},\chi_{2})$ pair.

It is possible to generate an AdS2 solution from an AdS3 background by
performing a U(1) T-duality on the Hopf fiber of AdS3. The result of doing
this to (2) with $c_{0}=0$ is the following

$\displaystyle ds^{2}_{\mathfrak{u}(1)}$
$\displaystyle=\frac{e^{2A}}{4}ds^{2}(\text{AdS}_{2})+e^{-2A}dr^{2}+ds^{2}(\text{M}_{7}),~{}~{}~{}~{}e^{-\Phi_{\mathfrak{u}(1)}}=e^{-\Phi+A},$
$\displaystyle H_{\mathfrak{u}(1)}$
$\displaystyle=-\frac{s}{2}r\text{vol}(\text{AdS}_{2})+H,~{}~{}~{}~{}F_{\mathfrak{u}(1)}=t(f_{\pm}\wedge
dr\mp\frac{1}{4}e^{3A}\text{vol}(\text{AdS}_{2})\wedge\star_{7}\lambda
f_{\pm}),$ (2.6)

where $t^{2}=1$ and the upper/lower signs refer to whether the original
solution was in IIA/IIB, with the dual solution presented above in IIB/IIA.
The vector $\partial_{r}$ is Killing with respect to the entire solution, and
if M7 was bounded the generated AdS2 solution will be likewise bounded if $r$
is periodically identified. It is a simple matter to show that the Bianchi
identities of the the dual flux

$(d-H_{\mathfrak{u}(1)}\wedge)F_{\mathfrak{u}(1)}=0,~{}~{}~{}dH_{\mathfrak{u}(1)}=0,$
(2.7)

are implied by (2.3). Likewise the remaining equations of motion of the
bosonic supergravity fields (A.4) are implied by those following from (2).

If the original solution was supersymmetric the dual solution preserves all of
the chiral supercharges of a single chirality, determined by the choice $s=1$
or $s=-1$ so if the original solution was ${\cal N}=(p,q)$ supersymmetric the
dual can preserve one of ${\cal N}=p$ or ${\cal N}=q$ supersymmetry. The dual
$d=10$ Killing spinors share the same internal spinors as (2.4), they take the
form

$\epsilon^{\mathfrak{u}(1)}_{1}=\zeta_{2}\otimes\theta_{+}\otimes\chi_{1},~{}~{}~{}~{}\epsilon_{2}^{\mathfrak{u}(1)}=-t\gamma_{\underline{r}}\zeta_{2}\otimes\theta_{\pm}\otimes\chi_{2},$
(2.8)

where $\zeta_{2}$ are Killing spinors on unit radius AdS2 obeying

$\nabla^{\text{AdS}_{2}}_{\mu}\zeta_{2}=\frac{s}{2}\gamma_{\mu}\zeta_{2},~{}~{}~{}\mu=0,1~{}~{}~{}~{}s^{2}=1,$
(2.9)

and there exists frames where $\zeta=\zeta_{2}$.

Generating AdS2 solutions from AdS3 with Hopf fiber T-duality is an old idea,
a more novel way to generate AdS2 solutions is to instead utilise SL(2) non-
Abelian T-duality, as recently done in [3, 4]. For completeness we spell out
the process of how this is done in appendix C. The result of applying an SL(2)
T-duality on (2) with $c_{0}=0$ is the following solution in Type IIB/IIA

$\displaystyle ds^{2}_{\mathfrak{sl}(2)}$
$\displaystyle=\frac{e^{2A}}{4\Delta}ds^{2}(\text{AdS}_{2})+e^{-2A}dr^{2}+ds^{2}(\text{M}_{7}),~{}~{}~{}~{}e^{-\Phi_{\mathfrak{sl}(2)}}=e^{-\Phi+A}r\sqrt{\Delta},$
$\displaystyle H_{\mathfrak{sl}(2)}$
$\displaystyle=-\frac{s}{2\Delta}r\text{vol}(\text{AdS}_{2})+H,~{}~{}~{}~{}~{}\Delta=1-\frac{e^{4A}}{4r^{2}}$
$\displaystyle F_{\mathfrak{sl}(2)}$
$\displaystyle=t\bigg{[}f_{\pm}\wedge(r-\frac{se^{4A}}{8\Delta}\text{vol}(\text{AdS}_{2}))\wedge
dr\mp\frac{e^{3A}}{2}(-s+\frac{r}{2\Delta}\text{vol}(\text{AdS}_{2}))\wedge\star_{7}\lambda
f_{\pm}\bigg{]},$ (2.10)

where again $t^{2}=1$. Unlike the case of U(1) T-duality, $\partial_{r}$ is
not an isometry of (2.10) and cannot be periodically identified. In fact as
with SU(2) T-duality the dual $r$ coordinate is now a semi-infinite interval:
As $r\to\infty$ the metric exhibits no regular zeros or singularites so $r$ is
not bounded from above, conversely as $r$ approaches loci for which $\Delta=0$
the behaviour of OF1 planes (which are the S-dual of O1 planes) is recovered
[4]. Again one can show that the Bianchi identities of the fluxes

$(d-H_{\mathfrak{sl}(2)}\wedge)F_{\mathfrak{sl}(2)}=0,~{}~{}~{}dH_{\mathfrak{sl}(2)}=0,$
(2.11)

are implied by (2.3), and that the rest of the Type II equations of motion
(A.4) are implied by the reduced $d=7$ conditions following from (2).

An interesting fact emerges while comparing (2.10) to (2.6), given that
$\Delta\to 1$ as $r\to\infty$. It quickly becomes clear that the NS sector of
the U(1) T-dual is recovered from that of the SL(2) T-dual as

$\lim_{r\to\infty}(ds^{2}_{\mathfrak{sl}(2)},~{}H_{\mathfrak{sl}(2)},~{}re^{\Phi_{\mathfrak{sl}(2)}})=(ds^{2}_{\mathfrak{u}(1)},~{}H_{\mathfrak{u}(1)},~{}e^{\Phi_{\mathfrak{u}(1)}}).$
(2.12)

A similar map can be obtained for the RR sector, when weighted by the dilaton,
ie

$\lim_{r\to\infty}e^{\Phi_{\mathfrak{sl}(2)}}F_{\mathfrak{sl}(2)}=e^{\Phi_{\mathfrak{u}(1)}}F_{\mathfrak{u}(1)},$
(2.13)

holds in general. Similar observations have been made for SU(2) T-duality
[27].

If the original AdS3 solution is supersymmetric, like the U(1) case, the dual
solution can preserve all the spinors corresponding to a given chirality
determined by the sign of $s$. The dual Killing spinors take the form

$\displaystyle\epsilon^{\mathfrak{sl}(2)}_{1}$
$\displaystyle=(N_{+}+N_{-}\gamma_{\underline{r}})\zeta_{2}\otimes\theta_{+}\otimes\chi_{1},~{}~{}~{}~{}\epsilon^{\mathfrak{sl}(2)}_{2}=-t(N_{+}-N_{-}\gamma_{\underline{r}})\gamma_{\underline{r}}\zeta_{2}\otimes\theta_{\pm}\otimes\chi_{2},$
$\displaystyle N_{\pm}$
$\displaystyle=\frac{1}{2\Delta^{\frac{1}{4}}}(\sqrt{\Delta_{+}}\pm\sqrt{\Delta_{-}}),~{}~{}~{}~{}\Delta_{\pm}=1\pm\frac{e^{2A}s}{2r},$
(2.14)

where again $\zeta_{2}$ are Killing spinors on AdS2 obeying (2.9).

In the next section we shall utilise the technology presented in this section
to generate a new class of AdS2 solutions in Type II supergravity preserving
small ${\cal N}=4$ supersymmetry.

##  3 New classes of small ${\cal N}=4$ AdS2 solutions in Type IIB

In section 3.1 we derive a new class of small ${\cal N}=4$ AdS2 solutions of
Type II supergravity via SL(2) T-duality. In section 3.2 we present two
solutions within this class whose geometries are warped products of
$\text{AdS}_{2}\times\text{S}^{3}\times\mathbb{T}^{3}\times\Sigma_{2}$. We
then derive a general class of solutions with this topology in section 3.3
with a view towards “completing” the SL(2) T-duals within global solutions
with bounded internal space in the future. As a first step we confirm that
they do indeed lie within this broader class.

###  3.1  A small ${\cal N}=(0,4)$ AdS3 class in IIA and its SL(2) T-duality

We are interested in studying the SL(2) T-duality of a class of small ${\cal
N}=(0,4)$ AdS3 solutions in massive IIA first derived in [20]. The general
class has a NS sector of the form

$\displaystyle ds^{2}$
$\displaystyle=\frac{q}{\sqrt{h}}\left(ds^{2}(\text{AdS}_{3})+ds^{2}(\text{S}^{3})\right)+g\sqrt{h}dx_{i}^{2}+\frac{gd\rho^{2}}{\sqrt{h}},\qquad
e^{-\Phi}=\frac{h^{\frac{3}{4}}}{\sqrt{g}},$ $\displaystyle H$
$\displaystyle=\partial_{\rho}(gh)dx_{1}\wedge dx_{2}\wedge
dx_{3}-\frac{1}{2}\epsilon_{ijk}\partial_{x_{i}}gdx_{j}\wedge dx_{k}\wedge
d\rho,$ (3.1)

where $i=1,2,3$ with Einstein summation conventions assumed, $g=g(x_{i})$,
$h=h(\rho,x_{i})$, $q$ is a constant and $\epsilon_{ijk}$ is the flat space
Levi-Civita symbol. All possible $d=10$ RR fluxes are non trivial and take the
form

$\displaystyle F_{0}$
$\displaystyle=\frac{\partial_{\rho}h}{g},~{}~{}~{}F_{2}=-\frac{1}{2}\epsilon_{ijk}\partial_{x_{i}}hdx_{j}\wedge
dx_{k},~{}~{}~{}F_{4}=2q\left(\text{vol}(\text{AdS}_{3})+\text{vol}(\text{S}^{3})\right)\wedge
d\rho.$ (3.2)

The Bianchi identities of the NS and RR fluxes, away from the loci of sources
require that we impose that

$d(\frac{\partial_{\rho}h}{g})=0,~{}~{}~{}~{}\partial_{x_{i}}^{2}g+\partial_{\rho}^{2}(gh)=0,~{}~{}~{}~{}\partial_{x_{i}}^{2}h+F_{0}\partial_{\rho}(gh)=0.$
(3.3)

If these requirements are satisfied then (3.1) and (3.2) always yield a
solution of the full Type IIA equations of motion preserving at least ${\cal
N}=(0,4)$, however this is actually enhanced to (4,4) when
$h=\text{constant}$.

With the recipe presented in the previous section it is a simple matter to
compute the SL(2) T-dual of the above class of solutions, we need only put it
in the form of (2) to read off the answer. We have

$\displaystyle e^{A}$
$\displaystyle=\frac{\sqrt{q}}{h^{\frac{1}{4}}},~{}~{}~{}~{}ds^{2}(\text{M}_{7})=\frac{q}{\sqrt{h}}ds^{2}(\text{S}^{3})+g\left(\sqrt{h}dx_{i}^{2}+\frac{d\rho^{2}}{\sqrt{h}}\right),\qquad
e^{-\Phi}=\frac{h^{\frac{3}{4}}}{\sqrt{g}},$ $\displaystyle f_{+}$
$\displaystyle=\frac{\partial_{\rho}h}{g}-\frac{1}{2}\epsilon_{ijk}\partial_{x_{i}}hdx_{j}\wedge
dx_{k}+2q\text{vol}(\text{S}^{3})\wedge d\rho+2qghdx_{1}\wedge dx_{2}\wedge
dx_{3}\wedge\text{vol}(\text{S}^{3}),$ (3.4)

with $H$ simply given as in (3.1). The sign appearing in the AdS3 Killing
spinor equation (2.5) that [20] takes is

$s=1,$ (3.5)

so we should choose this sign for small ${\cal N}=4$ to be preserved in the
dual solution. Just to make a concrete choice we additionally fix $t=-1$.

The SL(2) T-dual solution in Type IIB supergravity then has the following NS
sector

$\displaystyle ds^{2}$
$\displaystyle=\frac{q}{\sqrt{h}}\left(\frac{1}{2\Delta}ds^{2}(\text{AdS}_{2})+ds^{2}(\text{S}^{3})\right)+\frac{\sqrt{h}}{q}dr^{2}+g\sqrt{h}dx_{i}^{2}+\frac{gd\rho^{2}}{\sqrt{h}},\qquad
e^{-\Phi}=r\sqrt{\frac{qh\Delta}{g}},$ $\displaystyle H$
$\displaystyle=-\frac{1}{2\Delta}r\text{vol}(\text{AdS}_{2})+\partial_{\rho}(gh)dx_{1}\wedge
dx_{2}\wedge dx_{3}-\frac{1}{2}\epsilon_{ijk}\partial_{x_{i}}gdx_{j}\wedge
dx_{k}\wedge d\rho,$ (3.6)

where now $\Delta=1-\frac{q^{2}}{4r^{2}h}$. The RR sector on the other hand is
given by

$\displaystyle F_{1}$
$\displaystyle=qd\rho-\frac{\partial_{\rho}h}{g}rdr,~{}~{}~{}F_{5}=qd\left((1-2\Delta)r^{2}\right)\wedge\text{vol}(\text{S}^{3})\wedge
d\rho$
$\displaystyle+\frac{q}{2\Delta}\text{vol}(\text{AdS}_{2})\wedge\bigg{[}rghdx_{1}\wedge
dx_{2}\wedge dx_{3}-\frac{q}{8}\epsilon_{ijk}\partial_{x_{i}}\log hdr\wedge
dx_{j}\wedge dx_{k}\bigg{]},$ (3.7) $\displaystyle F_{3}$
$\displaystyle=-\frac{q}{2\Delta}\text{vol}(\text{AdS}_{2})\wedge\left(rd\rho-\frac{q\partial_{\rho}h}{4gh}dr\right)-qghdx_{1}\wedge
dx_{2}\wedge dx_{3}+\frac{r}{2}\epsilon_{ijk}\partial_{x_{i}}hdr\wedge
dx_{j}\wedge dx_{k}.$

As with the original solution, the Bianchi identities and remaining Type II
equations of motion hold when (3.3) or its source corrected equivalent do.

We will now focus on the SL(2) T-duals of two solutions found and studied in
[20] for which concrete CFT duals have been proposed.

###  3.2 Two solutions on
AdS${}_{2}\times\text{S}^{3}\times\mathbb{T}^{3}\times\Sigma_{2}$

Two interesting classes of AdS2 solutions arise by imposing that $x_{i}$ span
a 3-torus which the warp factors respect the isometries of. Thus we restrict
to $h=h(\rho)$ and $g=g(\rho)$. One then has that the last of (3.3) reduces to
simply

$F_{0}\partial_{\rho}(gh)=0,$ (3.8)

representing a branching of possible solutions. Either $h\propto g^{-1}$ or
$F_{0}=0$ which requires $h=$ constant and that $g$ is locally a linear
function - we present these respective cases in sections 3.2.1 and 3.2.2.

####  3.2.1 AdS2 case with $hg\propto 1$

For the first case we solve the whole of (3.3) as

$\displaystyle h=\sqrt{u},\qquad g=\frac{c}{\sqrt{u}},~{}~{}~{}~{}u=u(\rho),$
(3.9)

where $u$ is a linear function (ie $u^{\prime\prime}=0$ locally) and $c$ is a
constant. As a consequence we have $\partial_{\rho}(gh)=0$ and so the NS
3-form becomes purely electric. Globally in massive IIA this tuning of $(h,g)$
gives rise to AdS${}_{3}\times\text{S}^{3}\times\mathbb{T}^{3}$ foliated over
an interval bounded between D8/O8 singularities and with D8 branes along the
interior.

The SL(2) T-dual of the solution described above takes the following form

$\displaystyle ds^{2}$
$\displaystyle=\frac{q}{u^{\frac{1}{4}}}\bigg{(}\frac{1}{4\Delta_{1}}ds^{2}(\text{AdS}_{2})+ds^{2}(\text{S}^{3})+cds^{2}(\mathbb{T}^{3})\bigg{)}+u^{\frac{1}{4}}\left(\frac{dr^{2}}{q}+\frac{cd\rho^{2}}{u}\right),~{}~{}~{}e^{-\Phi}=r\sqrt{\frac{qu\Delta_{1}}{c}},$
$\displaystyle H$
$\displaystyle=dB,~{}~{}~{}B=-\frac{r}{2\Delta_{1}}\text{vol}(\text{AdS}_{2}),~{}~{}~{}~{}\Delta_{1}=1-\frac{q^{2}}{4r^{2}\sqrt{u}}.$
(3.10)

As there is a well defined NS 2-form for this solution we find it convenient
to express the RR sector in terms of their Page flux avatars defined as

$\hat{F}=e^{-B}\wedge F,$ (3.11)

which are closed if $(d-H\wedge)F=0$. We find the following

$\displaystyle F_{1}$
$\displaystyle=qd\rho-\frac{ru^{\prime}}{2c}dr,~{}~{}~{}\hat{F}_{3}=-\frac{r^{2}u^{\prime}}{4c}\text{vol}(\text{AdS}_{2})\wedge
dx-
qc\text{vol}(\mathbb{T}^{3}),~{}~{}~{}\hat{F}_{5}=2qr\text{vol}(\text{S}^{3})\wedge
dr\wedge d\rho,$ $\displaystyle\hat{F}_{7}$
$\displaystyle=\text{vol}(\text{S}^{3})\wedge\bigg{(}qr^{2}\text{vol}(\text{AdS}_{2})\wedge
dr\wedge
d\rho+cq\text{vol}(\mathbb{T}^{3})\wedge\left(2rdr-\frac{q}{2}d(u^{-\frac{1}{2}})\right)\bigg{)},$
$\displaystyle\hat{F}_{9}$
$\displaystyle=cqr^{2}\text{vol}(\text{AdS}_{2})\wedge\text{vol}(\text{S}^{3})\wedge\text{vol}(\mathbb{T}^{3})\wedge
dr,$ (3.12)

where of course $\hat{F}_{1}=F_{1}$. One has a solution away from the loci of
sources so long as $u^{\prime\prime}=0$, making $u$ linear. Globally however
$u$ needs only be piecewise linear with the discontinuities in $u^{\prime}$
giving rise to delta function sources in $u^{\prime\prime}$, which are D7
branes smeared over $r$. Such objects behave like D8 branes, ie the metric and
dilaton neither go to zero nor blow up at their loci, as such they can be
placed along the interior of the interval spanned by $\rho$.

####  3.2.2 AdS2 case with $h=$ constant

The second way to solve (3.8) is to fix $F_{0}=0$ which, given that the
solution must also respect the isometries of $\mathbb{T}^{3}$ means we shall
fix

$h=h_{0},$ (3.13)

for $h_{0}$ a constant. We then have that (3.3) reduces to

$\displaystyle g^{\prime\prime}=0,$ (3.14)

away from the loci of sources. In massless IIA this tuning of $(h,g)$ gives
rise to another solution with
AdS${}_{3}\times\text{S}^{3}\times\mathbb{T}^{3}$ foliated over an interval.
This time instead of D8 branes there are NS5 branes, that are smeared over all
but one of their co-dimensions, placed along the interior of the interval and
giving rise to delta function source corrections to (3.14).

The SL(2) T-dual solution has a NS sector of the following form

$\displaystyle ds^{2}$
$\displaystyle=\frac{1}{\sqrt{h_{0}}}\left(\frac{q}{4\Delta_{2}}ds^{2}(\text{AdS}_{2})+qds^{2}(\text{S}^{3})\right)+\frac{\sqrt{h_{0}}}{q}dr^{2}+g\left(\sqrt{h_{0}}ds^{2}(\mathbb{T}^{3})+\frac{1}{\sqrt{h_{0}}}d\rho^{2}\right),$
$\displaystyle e^{-\Phi}$
$\displaystyle=\sqrt{\frac{h_{0}kq\Delta_{2}}{g}},~{}~{}~{}H=-\frac{1}{2}d(\frac{r}{\Delta_{2}})\wedge\text{vol}(\text{AdS}_{2})+h_{0}g^{\prime}\text{vol}(\mathbb{T}^{3}),~{}~{}~{}~{}\Delta_{2}=1-\frac{q^{2}}{4h_{0}r^{2}}.$
(3.15)

Note that $H$ no longer has a well defined NS 2-form so using (3.11) to
simplify the expressions of the RR sector is no longer appropriate. The $d=10$
RR fluxes are given by

$\displaystyle F_{1}$
$\displaystyle=qd\rho,~{}~{}~{}F_{3}=-q\left(\frac{r}{2\Delta_{2}}\text{vol}(\text{AdS}_{2})\wedge
d\rho+h_{0}g\text{vol}(\text{T}^{3})\right),$ $\displaystyle F_{5}$
$\displaystyle=qr\left(\frac{h_{0}g}{2\Delta_{2}}\text{vol}(\text{AdS}_{2})\wedge\text{vol}(\mathbb{T}^{3})+2\text{vol}(\text{S}^{3})dr\wedge
d\rho\right).$ (3.16)

In section 4 we study the SCQM dual to this class of solutions, as the
starting point for a more general study of SCQMs dual to $\text{AdS}_{2}$
solutions with $\mathcal{N}=4$ supersymmetries that would include the
solutions with $hg\propto 1$ and the broader class that we construct in the
next subsection. We argue that the solutions in this subsection describe
geometrically the bubbling of monopoles in 4d $\mathcal{N}=2$ D3-D7 theories.

###  3.3 Embedding into a general class of
AdS${}_{2}\times\text{S}^{3}\times\mathbb{T}^{3}\times\Sigma_{2}$ solutions

An ever present issue with applying non-Abelian generalisations of U(1)
T-duality to the AdS/CFT correspondence is that they give rise to solutions
with an unbounded internal space. SL(2) T-duality is no different, the T-dual
coordinate $r$ is only bounded at one end. This is often a sign that the
putative dual CFT/CQM is ill defined, although it may also indicate that the
geometry is dual to a defect in a higher dimensional CFT. Significant progress
on both the geometry and CFT side has been made towards making sense of such
geometries in recent years. The CFT dual of a non-Abelian T-dual geometry can
often be viewed as an infinite linear quiver that one can make finite, by
instead choosing to terminate it at a flavour node [28, 27, 29, 30, 31, 32, 4,
7]. To realise such a method geometrically one needs to embed the non-Abelian
T-dual geometry into a broader class of solutions and essentially glue another
solution onto it at finite $r$, such that the interval spanned by $r$ becomes
bounded and the dual CFT well defined. This has been successfully achieved in
several contexts. With this in mind in this section we are interested in
embedding the solutions of sections 3.2.1 and 3.2.2 into a more general class
of AdS${}_{2}\times\text{S}^{3}\times\mathbb{T}^{3}\times\Sigma_{2}$ solutions
preserving small ${\cal N}=4$ supersymmetry. To our knowledge such a class
does not currently exist, so here we shall derive one by double Wick rotating
(a subclass of) a broad class of solutions on
$\text{AdS}_{3}\times\text{S}^{2}$ found in [21].

The relevant class of AdS${}_{3}\times\text{S}^{2}\times\text{M}_{5}$
solutions is presented in section 3.3 of [21]. The internal manifold M5 is
spanned by coordinates $(y,x,z_{1},z_{2},z_{3})$, with the former 2
coordinates fibered over the latter. Generically the class is quite
complicated but undergoes a pronounced simplification if one assumes that
$\partial_{z_{i}}$ are isometries of the solutions spanning a 3-torus. It is
then possible to perform a coordinate transformation in the $(x,y)$ directions
to diagonalise the metric. Let us describe the process briefly, note that
$(\lambda,g_{0},h_{0},u_{0})$ refer to functions defined in [21] (the latter 2
not containing the subscript zero in that work). After imposing that
$\partial_{z_{i}}$ are isometries one first performs the coordinate
transformation

$\displaystyle x\rightarrow\tilde{x}=x,\qquad y\rightarrow
4\tilde{y}=4\tilde{y}(y,x)$ (3.17)

in such a way that

$\lambda=\frac{\partial\tilde{y}}{\partial x},\qquad
g_{0}=\frac{\partial\tilde{y}}{\partial y},$ (3.18)

which leads to a diagonal metric. One then redefines

$\displaystyle
g_{0}=\frac{4}{K},~{}~{}~{}~{}h_{0}=4S,~{}~{}~{}~{}u_{0}=v,~{}~{}~{}~{}m=1,$
(3.19)

where $K=K(x,y)$, $S=S(x,y)$ and $v=v(x)$. Finally we send
$\tilde{y}\rightarrow y$ to uncluttered notation.

The resulting class of small ${\cal N}=(0,4)$ AdS3 solutions has a NS sector
of the following form

$\displaystyle ds^{2}$
$\displaystyle=\frac{v}{\sqrt{KS}}\bigg{(}ds^{2}(\text{AdS}_{3})+\frac{1}{\Xi}ds^{2}(\text{S}^{2})\bigg{)}+\frac{\sqrt{SK}}{v}dx^{2}+\frac{1}{\sqrt{SK}}dy^{2}+\sqrt{\frac{S}{K}}ds^{2}(\mathbb{T}^{3}),$
$\displaystyle e^{-\Phi}$
$\displaystyle=K\sqrt{\frac{S\Xi}{v}},~{}~{}~{}~{}H=\frac{1}{2}d\left(\frac{vv^{\prime}}{4KS\Xi}-x\right)\wedge\text{vol}(\text{S}^{2})+\frac{\partial_{y}S}{K}\text{vol}(\mathbb{T}^{3}),~{}~{}~{}~{}\Xi=1+\frac{v^{\prime
2}}{4KS}.$ (3.20)

It supports the following non trivial $d=10$ RR fluxes

$\displaystyle F_{1}$
$\displaystyle=\partial_{x}Kdy-\frac{1}{v}\partial_{y}(KS)dx,~{}~{}~{}~{}F_{5}=(1+\star_{10})f_{5}$
$\displaystyle F_{3}$
$\displaystyle=\left(-\frac{v^{\prime}\partial_{y}(KS)}{8KS\Xi}dx+\frac{vv^{\prime}\partial_{x}\log
Kdy}{8S\Xi}-\frac{1}{2}Kdy\right)\wedge\text{vol}(\text{S}^{2})-\partial_{x}S\text{vol}(\mathbb{T}^{3}),$
$\displaystyle f_{5}$
$\displaystyle=\frac{1}{8KS\Xi}\left(4KS^{2}-v^{2}v^{\prime}\partial_{x}\left(\frac{S}{v}\right)\right)\text{vol}(\mathbb{T}^{3})\wedge\text{vol}(\text{AdS}_{2}).$
(3.21)

One has a solution whenever the Bianchi identities of the fluxes hold, in the
absence of sources these reduce to the following PDEs

$\frac{\partial_{y}S}{K}=\text{constant}=b,~{}~{}~{}\partial_{x}^{2}K+\frac{1}{v}\partial_{y}^{2}(SK)=0,~{}~{}~{}\partial_{x}^{2}S+\frac{b}{v}\partial_{y}(SK)=0,~{}~{}~{}v^{\prime\prime}=0.$
(3.22)

Notice that this is a very similar system of PDEs to that of the massive IIA
class in (3.3) - here it is now $b$, which is related to the NS flux through
$\mathbb{T}^{3}$, that plays the role that $F_{0}$ previously did.

We now need to double Wick rotate the above class to get to a class of
solutions on AdS${}_{2}\times\text{S}^{3}\times\mathbb{T}^{3}\times\Sigma_{2}$
\- this is achieved as follows: First we wick rotate the coordinates of AdS2
and S3 so that they become AdS3 and S2 respectively. The rules are

$\displaystyle ds^{2}(\text{AdS}_{3})$ $\displaystyle\rightarrow-
ds^{2}(\text{S}^{3}),\qquad ds^{2}(\text{S}^{2})\rightarrow-
ds^{2}(\text{AdS}_{2}),$ $\displaystyle\text{vol}(\text{AdS}_{3})$
$\displaystyle\rightarrow-\text{vol}(\text{S}^{3}),\qquad\text{vol}(\text{S}^{2})\rightarrow-i\text{vol}(\text{AdS}_{2}).$
(3.23)

In addition to this one needs to analytically continue the various functions
appearing in the class as

$\displaystyle v\rightarrow iv,\qquad K\rightarrow-iK,\qquad S\rightarrow-
iS,\qquad x\rightarrow ix,$ (3.24)

so that the metric has the correct signature and is real. Finally we
supplement the procedure by sending

$\displaystyle e^{-\Phi}$ $\displaystyle\rightarrow-e^{-\Phi},\qquad
F\rightarrow-F,$ (3.25)

which ensures that the dilaton is real. When the dust settles we arrive at a
class of AdS2 solutions with the following NS sector

$\displaystyle ds^{2}$
$\displaystyle=\frac{v}{\sqrt{KS}}\left(\frac{1}{4\tilde{\Xi}}ds^{2}(\text{AdS}_{2})+ds^{2}(\text{S}^{3})\right)+\sqrt{\frac{S}{K}}ds^{2}(\mathbb{T}^{3})+\frac{\sqrt{KS}}{v}dx^{2}+\sqrt{\frac{K}{S}}dy^{2},$
$\displaystyle e^{-\Phi}$
$\displaystyle=\frac{K\sqrt{S}\sqrt{\tilde{\Xi}}}{\sqrt{v}},~{}~{}~{}H=-\frac{1}{2}d\left(\frac{vv^{\prime}}{4KS\tilde{\Xi}}+x\right)\wedge\text{vol}(\text{AdS}_{2})+\frac{1}{K}\partial_{y}S\text{vol}(\mathbb{T}^{3}),~{}~{}~{}\tilde{\Xi}=1-\frac{v^{\prime
2}}{4KS}$ (3.26)

and the following $d=10$ RR fluxes

$\displaystyle F_{1}$
$\displaystyle=\partial_{x}Kdy-\frac{1}{v}\partial_{y}(KS)dx,~{}~{}~{}~{}F_{5}=(1+\star_{10})f_{5}$
$\displaystyle F_{3}$
$\displaystyle=\left(\frac{v^{\prime}\partial_{y}(KS)}{8KS\tilde{\Xi}}dx-\frac{vv^{\prime}\partial_{x}\log
Kdy}{8S\tilde{\Xi}}-\frac{1}{2}Kdy\right)\wedge\text{vol}(\text{AdS}_{2})-\partial_{x}S\text{vol}(\mathbb{T}^{3}),$
$\displaystyle f_{5}$
$\displaystyle=\frac{1}{8KS\tilde{\Xi}}\left(4KS^{2}+v^{2}v^{\prime}\partial_{x}\left(\frac{S}{v}\right)\right)\text{vol}(\text{AdS}_{2})\wedge\text{vol}(\mathbb{T}^{3}).$
(3.27)

One has a solution to the Type IIB equations of motion whenever the Bianchi
identities of the fluxes hold, these still reduce to the PDEs in (3.22) away
from the loci of sources. This class of AdS2 solutions should preserve small
${\cal N}=4$ supersymmetry like the AdS3 class we generated it from.

In the next sections we show how to recover the solutions of section 3.2.1 and
3.2.2 from this more general class of AdS2 solutions. This actually also
serves as evidence to our claim that the general class preserves small ${\cal
N}=4$ supersymmetry, because these two specific solutions certainly do.

####  3.3.1 Recovering the $hg\propto 1$ case

The solution of section 3.2.1 is recovered from (3.26)-(3.27) by fixing

$\displaystyle
S=cqv_{0}(x),~{}~{}~{}~{}K=\frac{q}{c}p(y)v_{0}(x),~{}~{}~{}~{}v=q^{2}v_{0},$
(3.28)

where the functions $v_{0},p$ obey

$\partial_{x}^{2}v_{0}=0,~{}~{}~{}~{}\partial_{y}^{2}p=0,$ (3.29)

which solves all of (3.22) for $b=0$. To precisely reproduce the form of the
solution in 3.2.1 one needs to do an implicit coordinate transformation such
that

$p(y)=\sqrt{u(\rho)}.$ (3.30)

The resulting solution is actually a mild generalisation of the solution in
section 3.2.1 which depends on the linear function $v_{0}$. Specifically if
one fixes $v_{0}=x$ one recovers the solution of section 3.2.1, but if one
fixes $v_{0}=$ constant one instead recovers its U(1) T-dual analogue. Such
hybrid T-dual/non-Abelian T-dual solutions were previously found in [33].

####  3.3.2 Recovering the $h=$ constant case

Likewise the solution of section 3.2.2 is recovered from (3.26)-(3.27) by
fixing

$\displaystyle
S=q\sqrt{Q(y)}v_{0}(x),~{}~{}~{}~{}K=\frac{h_{0}qv_{0}(x)}{\sqrt{Q(y)}},~{}~{}~{}~{}v=q^{2}v_{0}(x),$
(3.31)

where

$\partial_{x}^{2}v_{0}=0,~{}~{}~{}\partial_{y}Q=2bh_{0},$ (3.32)

which solves (3.22). To get to the precise form of the solution in section
3.2.2 one needs to change coordinates from $y$ to $\rho$ in such a way that

$\displaystyle Q(y)=h_{0}^{2}g(\rho)^{2}$ (3.33)

which requires that one identifies

$b=h_{0}\partial_{\rho}g.$ (3.34)

Again the result of doing this is actually a T-dual/non-Abelian T-dual hybrid
of the solution we seek: Fixing $v_{0}=x$ recovers the solution of section
3.2.2 while $v_{0}=$ constant gives rise to its U(1) T-dual analogue.

##  4 Field theory analysis

In this section we study the 1d SCFTs dual to the
$\text{AdS}_{2}\times\text{S}^{3}\times\mathbb{T}^{3}\times\Sigma_{2}$
solutions with $F_{0}=0$ constructed in section 3.2.2, as a first step towards
a more general analysis of the field theories dual to our broader class of
$\mathcal{N}=4$ solutions, in particular the ones constructed in section
3.2.1. We construct explicit quivers that we conjecture flow in the IR to the
SCQMs dual to the solutions, and check our proposal with the computation of
the field theory and holographic central charges. We propose an interpretation
of the solutions as holographic duals of monopole bubbling in 4d
$\mathcal{N}=2$ theories living in D3-D7 branes, and of baryon vertices in 5d
$\mathcal{N}=1$ theories living in D5-NS5-D7 branes. These two interpretations
are based on the disposition of the branes that make the brane set-up
associated to the solutions along the two non-compact directions of the 2d
Riemann surface.

###  4.1 Hanany-Witten brane set-up and quantised charges

Even if we have not constructed the brane solution whose near horizon geometry
is described by the geometry specified by equations (3.2.2) and (3.16)555This
is a difficult task when a non-Abelian T-duality transformation is involved
(see [34, 35])., one can see that the brane intersection shown in Table 1 is
consistent with both the fluxes and the preserved supersymmetries of the
solutions. As it typically happens the $x^{i}$ coordinates will mix upon
taking the near horizon limit to give rise to the radius of AdS, the two non-
compact directions $r$ and $\rho$ and the coordinates along the
$\text{S}^{3}$. In this configuration the D1-branes play the role of colour
branes, and $x^{4}$ of field theory direction, and the F1-strings stretch
along $x^{5}$. As we will show below once the near horizon limit is taken the
D1-branes will effectively stretch along the $\rho$ direction and the
F1-strings along $r$. We will also see that the D1-branes carry as well
electric charge, which allows one to interpret them as baryon vertices for the
D7-branes. The same interpretation is found for the D3-branes with respect to
the D5-branes. We can indeed check in Table 1 that these branes have the right
relative orientations that allow to infer this. Alternatively, as we will
discuss, the brane set-up can be interpreted as describing the super conformal
quantum mechanics associated to the bubbling of dyonic monopoles in the 4d
theory living in the intersection of the D3 and D7 branes, in a brane scenario
in which the D3-D7 branes and the D1-F1 strings are put inside a 5-brane web.

| $t$ | $z_{1}$ | $z_{2}$ | $z_{3}$ | $x^{4}$ | $x^{5}$ | $x^{6}$ | $x^{7}$ | $x^{8}$ | $x^{9}$
---|---|---|---|---|---|---|---|---|---|---
D1 | x |  |  |  | x |  |  |  |  |
D3 | x | x | x | x |  |  |  |  |  |
D5 | x |  |  |  | x |  | x | x | x | x
D7 | x | x | x | x |  |  | x | x | x | x
NS5 | x |  |  |  |  | x | x | x | x | x
F1 | x |  |  |  |  | x |  |  |  |
Table 1: $\frac{1}{8}$-BPS brane intersection underlying the $\mathcal{N}=4$
AdS2 solutions constructed in section 3.2.2. $t$ is the time direction where
the 1d dual CFT lives; ($z_{1}$, $z_{2}$, $z_{3}$) parametrise the
$\mathbb{T}^{3}$; $x^{4}$ is the field theory direction; $x^{5}$ is the
direction where the F1-strings lie, and ($x^{6},x^{7},x^{8},x^{9}$) are the
directions associated to the isometries of the S3.

Let us start our discussion with the computation of the quantised charges. To
this end we find it helpful to decompose the NS 3-form into its electric and
magnetic parts as

$H=H^{e}+H^{m},~{}~{}~{}~{}H^{e}=-\frac{1}{2}d\left(\frac{r}{1-\frac{q^{2}}{4h_{0}^{2}r^{2}}}\right)\wedge\text{vol}(\text{AdS}_{2}),~{}~{}~{}~{}H^{m}=h_{0}g^{\prime}\text{vol}(\mathbb{T}^{3}).$
(4.1)

We start analysing the D7-D5-NS5 brane subsystem. The quantised charge
associated to the D7-branes is obtained integrating the magnetic component of
$F_{1}$, as

$Q_{D7}=\int f_{1}=q\int d\rho.$ (4.2)

To proceed we need to define the domain of integration of the
$\rho$-direction, which is in turn related to the way we define the Page
fluxes in our backgrounds. Indeed, the standard definition of Page fluxes
given by (3.11) cannot be used for our solutions, since there is no globally
defined magnetic $B$-field. Instead, we note that the definition666We use
$f_{p}$ to denote $p$-form RR magnetic fluxes, and an $m$ superscript to
denote magnetic components.

$\hat{f}_{p}=f_{p}+H^{m}\wedge C^{m}_{p-3},$ (4.3)

is equivalent to (3.11) for the magnetic part of the Page fluxes. This
definition of the Page fluxes has the non-trivial implication that instead of
having Dp-branes being created as NS5-branes are crossed, which is the usual
way the Hanany-Witten brane creation effect is understood at the level of the
fluxes, it describes Dp-branes being created as D(p+2)-branes are crossed. In
particular, for the D5-branes we have

$\hat{f}_{3}=f_{3}+H^{m}\,C_{0}.$ (4.4)

From this expression we see that $\hat{f}_{3}$ is sensitive to gauge
transformations of $C_{0}$. In order to carefully account for these we demand
that $C_{0}$ lies in the range $C_{0}\in[0,q]$. In order to accomplish this we
need to take

$C_{0}=q(\rho-k)\qquad\text{for}\qquad\rho\in[k,k+1].$ (4.5)

Then, given that the D7-brane charge is obtained computing

$Q_{D7}^{(k)}=\int_{k}^{k+1}dC_{0},$ (4.6)

this imposes that $q$ D7-branes are created in this interval. This clarifies
the role played by the large gauge transformations of $C_{0}$, as generating a
strong coupling realisation of the Hanany-Witten brane creation effect by
which D7-branes, instead of NS5-branes, are positioned along the field theory
direction. In turn, the large gauge transformation in (4.5) modifies
$\hat{f}_{3}$ as $\hat{f}_{3}\rightarrow\hat{f}_{3}-qkH^{m}$ such that

$\hat{f}_{3}=qh_{0}\Bigl{(}g^{\prime}(\rho-k)-g\Bigr{)}\text{vol}(\mathbb{T}^{3})\qquad\text{in}\qquad\rho\in[k,k+1].$
(4.7)

Recall that away from the loci of sources the Bianchi identities require that
$g^{\prime\prime}=0$, which makes $g$ a linear function. However globally we
need only impose that $g$ is continuous, while $g^{\prime}$ is allowed to have
discontinuities that give rise to delta function sources in
$g^{\prime\prime}$, signaling the presence of smeared NS5-branes at
$\rho_{k}=k$. Therefore, we take $g$ piecewise linear such that

$g_{k}=\frac{1}{2\pi
h_{0}}\Bigl{(}\alpha_{k}+\beta_{k}(\rho-k)\Bigr{)}\qquad\text{for}\qquad\rho\in[k,k+1].$
(4.8)

Here $\alpha_{k}$ and $\beta_{k}$ are integer numbers, since they are related
to the numbers of D5 and NS5-branes in the $k$th interval, according to

$\displaystyle Q_{D5}^{(k)}$ $\displaystyle=$
$\displaystyle-\frac{1}{(2\pi)^{2}}\int_{\mathbb{T}^{3}}\hat{f}_{3}=q\alpha_{k},$
(4.9) $\displaystyle Q_{NS5}^{(k)}$ $\displaystyle=$
$\displaystyle\frac{1}{(2\pi)^{2}}\int_{\mathbb{T}^{3}}H^{m}=\beta_{k}.$
(4.10)

As well defined global solutions should have a continuous metric and dilaton
this amounts to requiring that $\alpha_{k},\beta_{k}$ must satisfy

$\alpha_{k+1}=\alpha_{k}+\beta_{k}=\sum_{j=0}^{k}\beta_{j}.$ (4.11)

The value of $\beta_{k}$ is not fixed however, indeed we allow it to change
between intervals, this means that in the vicinity of $\rho=k$ we have

$g^{\prime\prime}=\frac{1}{2\pi h_{0}}(\beta_{k}-\beta_{k-1})\delta(\rho-k)$
(4.12)

and so we have source NS5 branes with charge $\beta_{k}-\beta_{k-1}$ at
$\rho=k$ that are extended in AdS${}_{3}\times\text{S}^{3}$ and smeared over
$\mathbb{T}^{3}$. Note that this does not induce any source terms in the RR
sector as should be clear from examining (3.16), which is independent of
$g^{\prime}$. Next we need to care about how the space ends on the
$\rho$-direction. We choose to take $\rho\in[0,P+1]$ and impose that at both
ends $g$ vanishes. This implies that in the $[0,1]$ and $[P,P+1]$ intervals we
must have

$g_{0}=\frac{\beta_{0}}{2\pi h_{0}}\rho,\qquad g_{P}=\frac{1}{2\pi
h_{0}}\Bigl{(}\alpha_{P}+\beta_{P}(\rho-P)\Bigr{)},$ (4.13)

with $\beta_{P}=-\alpha_{P}$. The behaviour close to both ends of the space is
then that of an ONS5 orientifold fixed plane (S-dual of an O5) that is smeared
on the $\mathbb{T}^{3}$.

Summarising our results so far, we have seen that there are $q\alpha_{k}$
D5-branes stretched between $q$ D7-branes located at $\rho_{k}=k$,
$\rho_{k+1}=k+1$ with $\beta_{k}$ perpendicular NS5-branes in each
$[\rho_{k},\rho_{k+1}]$ interval.

Let us analyse now the D1-D3-F1 subsector of the brane set-up. As we have
mentioned, the D1 and D3-branes carry electric charges, which allows us to
interpret them as baryon vertices. These charges are obtained integrating the
Page fluxes associated to the electric components of ${F}_{3}$ and ${F}_{5}$
in (3.16), defining these is a little subtle in this case. The electric part
of $H$ in (4.1) allows for a globally defined NS 2-form, given by

$B^{e}=-\frac{1}{2}\frac{r}{1-\frac{q^{2}}{4h_{0}r^{2}}}\,\text{vol}(\text{AdS}_{2}).$
(4.14)

We need to define some combinations of the fluxes whose integrals maybe be
interpreted as charges, as such they should be closed (away from sources). It
is not hard to see that if one decomposes the magnetic fluxes as777We use an
$e$ superscript to denote electric components.

$F^{e}_{p}=dC^{e}_{p-1}-H^{m}\wedge C^{e}_{p-3}+B^{e}\wedge f_{p-2}$ (4.15)

for $C^{e}_{p}$ some electric RR potentials, we get the correct Bianchi
identity for the electric part of the flux, ie $dF^{e}_{p}=H^{m}\wedge
F^{e}_{p-2}+H^{e}\wedge f_{p-2}$. This suggests defining the electric Page
fluxes as

$\hat{F}_{p}^{e}=F_{p}^{e}+H^{m}\wedge C^{e}_{p-3}-B^{e}\wedge f_{p-2},\\\ $
(4.16)

which are closed (away from sources) by construction. In turn, the F1-strings
are electrically charged with respect to $B$, with charge

$Q_{F1}^{e}=\frac{1}{(2\pi)^{2}}\int_{\text{AdS}_{2}}B^{e},$ (4.17)

with $B^{e}$ given by (4.14). One can easily see that regularising the volume
of $\text{AdS}_{2}$ as $\text{Vol}(\text{AdS}_{2})=4\pi$ and dividing the $r$
direction in intervals of length $2\pi$ a F1-string lies at each such
interval. This can be linked to the condition that the integral of $B^{e}$
lies in the fundamental region. In this case a large gauge transformation of
gauge parameter $n$ must be performed for $r\in[2n\pi,2(n+1)\pi]$, such that

$B^{e}=\left(-\frac{1}{2}\frac{r}{1-\frac{q^{2}}{4h_{0}r^{2}}}+n\pi\right)\text{vol}(\text{AdS}_{2}),$
(4.18)

and an F1-string is created. We identify the beginning of the space at
$r_{0}=\frac{q}{2\sqrt{h_{0}}}$ with $2\pi$, consistently with the fact that,
as we showed in section 2, the behaviour of the metric and dilaton at this
point is that associated to an OF1-plane. This fixes
$h_{0}=\frac{q^{2}}{16\pi^{2}}$.

Substituting in (4.16) we then find in the interval $r\in[2n\pi,2(n+1)\pi]$
that

$\displaystyle\hat{F}_{3}^{e}$ $\displaystyle=$ $\displaystyle-n\pi
q\,d\rho\wedge\text{vol}(\text{AdS}_{2}),~{}~{}~{}\Rightarrow~{}~{}~{}C_{2}^{e}=-2\pi
nq(\rho-l)$ $\displaystyle\hat{F}_{5}^{e}$ $\displaystyle=$
$\displaystyle\frac{1}{2}nq(g+(\rho-l)g^{\prime})\text{vol}(\text{AdS}_{2})\wedge\text{vol}(\mathbb{T}^{3})$
(4.19)

where $l$ is an integration constant. Given this we find

$Q_{D1}^{e(n)}=-\frac{1}{(2\pi)^{2}}\int\hat{F}_{3}^{e}=nq=nQ_{D7}^{(k)},$
(4.20)

meaning that there are $nq$ units of electric D1 charge within the cell
$r\in[2n\pi,2(n+1)\pi]$, thus for consistency we should constrain

$0<-C_{2}^{e}<nq~{}~{}~{}~{}\Rightarrow~{}~{}~{}~{}l=k.$ (4.21)

We then have that the electric D3 brane charge is given by

$Q_{D3}^{e(n)}=\frac{1}{(2\pi)^{4}}\int\hat{F}_{5}^{e}=nq\alpha_{k}=nQ_{D5}^{(k)}.$
(4.22)

The brane set-up associated to these quantised charges is depicted in Figure
1.

Figure 1: Brane set-up in the $r$ direction for $\rho$ constant, in units of
$q$. The numbers of D7 and D5 branes at each interval are given by their
respective magnetic charges. Instead, for the numbers of D1 and D3 branes we
give their electric charges (computed in (4.20) and (4.22)) as these are the
ones that play a role in their interpretation as baryon vertices.

Clearly, as $r$ is a non-compact direction we need to care about its global
definition. This will be important later on when we compute the central charge
associated to the solutions. We choose to complete the solutions in this
direction by glueing them to themselves, at a certain point
$r_{P^{\prime}+1}=2(P^{\prime}+1)\pi$. The completed brane set-up is depicted
in Figure 2.

Figure 2: Completed brane set-up in the $r$ direction for $\rho$ constant, in
units of $q$. The branes in blue are used to glue the brane set-up with
itself.

Specifically, the solutions are extended beyond the
$[2\pi,2(P^{\prime}+1)\pi]$ interval by gluing a second solution onto it which
takes the form of (3.2.2)-(3.16), but with $r\rightarrow 4(P^{\prime}+1)\pi-r$
and for which $r\in[2(P^{\prime}+1)\pi,2(2P^{\prime}+1)\pi]$. One can easily
check that the metric and fluxes of this global completion are continuous
across $r=2(P^{\prime}+1)\pi$. The
$r\in[2(P^{\prime}+1)\pi,2(2P^{\prime}+1)\pi]$ part of the interval is
likewise divided in $P^{\prime}$ intervals of length $2\pi$ where F1-strings
are created, associated to the large gauge transformations of the $B^{e}$
field, that we complete as

$B^{e}=\left\\{\begin{array}[]{ccrcl}\Bigl{(}-\frac{1}{2}\frac{r^{3}}{r^{2}-(2\pi)^{2}}+n\pi\Bigr{)}\text{vol}(\text{AdS}_{2})\,,\quad
r\in[2\pi,2(P^{\prime}+1)\pi]\\\\[5.69054pt]
\Bigl{(}-\frac{1}{2}\frac{(4(P^{\prime}+1)\pi-r)^{3}}{(4(P^{\prime}+1)\pi-r)^{2}-(2\pi)^{2}}+(4(P^{\prime}+1)-n)\pi\Bigr{)}\text{vol}(\text{AdS}_{2})\,,\quad
r\in[2(P^{\prime}+1)\pi,2(2P^{\prime}+1)\pi]\end{array}\right.$ (4.23)

Note that with this completion of the solutions a new singularity associated
to an OF1-plane arises at $r=2(2P^{\prime}+1)\pi$, where the space now ends.

###  4.2 Baryon vertex interpretation

Our previous analysis shows that there is one F1-string extended in $r$ for
$r\in[2n\pi,2(n+1)\pi]$ for $n=1,2,\dots 2P^{\prime}$. At each of these
intervals there are $nq$ units of D1-brane electric charge and $nq\alpha_{k}$
units of D3-brane electric charge for $n=1,2,\dots,P^{\prime}+1$, as well as
$(4(P^{\prime}+1)-n)q$ units of D1-brane electric charge and
$(4(P^{\prime}+1)-n)q\alpha_{k}$ units of D3-brane electric charge for
$n=P^{\prime}+1,P^{\prime}+2,\dots,2P^{\prime}+1$. Recalling the existence of
the following couplings in the WZ actions of the D1 and D3 branes,

$S_{D1}=T_{1}\int f_{1}\wedge A_{t},\qquad S_{D3}=T_{3}\int\hat{f}_{3}\wedge
A_{t},$ (4.24)

where $A_{t}$ is the electric component of the Born-Infeld vector field, we
find that a D1-brane extended in $\rho$ between $[\rho_{k},\rho_{k+1}]$ and
located at a fixed position in the $r\in[2n\pi,2(n+1)\pi]$ interval behaves as
a baryon vertex for the D7-branes, since it carries $Q_{D1}^{e}=nq=nQ_{D7}$
units of F1-string charge if $n=1,2,\dots,P^{\prime}+1$, and
$Q_{D1}^{e}=(4(P^{\prime}+1)-n)Q_{D7}$ if
$n=P^{\prime}+1,\dots,2P^{\prime}+1$. Similarly, a D3-brane wrapped on the
$\mathbb{T}^{3}$ and located at a fixed position in the
$r\in[2n\pi,2(n+1)\pi]$ interval carries $Q_{D3}^{e}=nq\alpha_{k}=nQ_{D5}$
units of F1-string charge if $n=1,2,\dots,P^{\prime}+1$ and
$Q_{D3}^{e}=(4(P^{\prime}+1)-n)Q_{D5}$ if
$n=P^{\prime}+1,\dots,2P^{\prime}+1$, and therefore behaves as a baryon vertex
for the D5-branes. Indeed, the relative orientation between the D1 and the D7
branes and between the D3 and the D5 branes in the brane set-up is the one
that allows to create F1-strings stretched between the D1 and the D7 branes
and between the D3 and the D5 branes, as depicted in Figure 3.

Figure 3: Wilson loops in the $Q_{D7}$-th and $Q_{D5}$-th antisymmetric
representations of U($Q_{D1}$) and U($Q_{D3}$), respectively.

As we have mentioned, the location of the branes along the $r$ direction is
the one depicted in Figure 2. As explained in detail in [2] this brane
configuration can be related by a combination of a T-duality, an S-duality,
successive Hanany-Witten moves and a further T-duality to the brane set-up
depicted in Figure 4.

Figure 4: Hanany-Witten brane set-up equivalent to the brane configuration in
Figure 2, in units of $q$.

After these transformations the analogy with the description of half-BPS
Wilson loops in antisymmetric representations labelled by the Young tableau
depicted in Figure 5, proposed in [36, 37], is evident.

Figure 5: Young tableau labelling the irreducible representations of U(N).

In these references if was shown that a Wilson loop in the
$(l_{1},l_{2},\dots,l_{M})$ antisymmetric representation is realised by a
configuration of stacks of branes separated a distance $L$ from the colour
branes with $(l_{1},l_{2},\dots,l_{M})$ F1-strings stretched between the
stacks. This is completely analogous to our configuration in Figure 4. In our
case we encounter the particular situation in which the sum of the F1-strings
stretched between each D1 (D3) and the flavour D7 (D5) branes coincides with
the rank of the gauge group of the D1 (D3) branes, which implies that the
Wilson lines are in the fundamental representation of the gauge groups.
Therefore, the D1-D3 branes behave as baryon vertices for the D7-D5 branes.

Our analysis in this subsection suggests that the AdS2 solutions constructed
in section 3.2.2 could find an interpretation as backreacted D1-D3-F1 baryon
vertices in the 5d $\mathcal{N}=1$ theory living in a D5-D7-NS5 brane
intersection. Indeed, our discussion so far mimics the interpretation of the
AdS2 solutions found in [2, 3, 4, 6, 7], as dual to baryon vertices in 5d
Sp(N), 4d $\mathcal{N}=4$ and 4d $\mathcal{N}=2$ superconformal field
theories. However, a key difference with our construction in this paper is
that the AdS2 solutions discussed in these references asymptote locally to the
$\text{AdS}_{6}$ or $\text{AdS}_{5}$ solutions dual to the higher dimensional
SCFTs where the baryon vertices are embedded, while this is not the case for
our AdS2 solutions. Based on this it is likely that in our construction in
this paper the branes could play a different role, a possibility that we will
further discuss in section 4.4, where we will see that they allow for an
alternative interpretation in terms of dyonic monopoles in the 4d theory
living in D3-D7 branes.

###  4.3 Quiver quantum mechanics

In the previous subsection we computed the electric charges of the D1 and D3
branes, which are the relevant ones for the construction of Wilson lines or
baryon vertices. In this subsection we are interested instead in the
construction of the quiver quantum mechanics dual to our solutions, for which
the relevant charges are the magnetic ones.

Analogously to our discussion for the D7-D5-NS5 brane subsystem, the D1 and
D3-branes are magnetically charged with respect to the Page fluxes defined by

$\hat{f}_{7}=f_{7}+H^{m}\wedge C_{4}^{m},\qquad\hat{f}_{5}=f_{5}+H^{m}\wedge
C_{2}^{m}.$ (4.25)

These expressions show that the numbers of D1 and D3 branes are sensible to
gauge transformations of $C_{4}$ and $C_{2}$, respectively. In particular, we
have for our background

$\hat{f}_{5}=dC_{4}^{m}=2qr\text{vol}(\text{S}^{3})\wedge dr\wedge d\rho$
(4.26)

and, upon integration

$C_{4}^{m}=2qr\rho\text{vol}(\text{S}^{3})\wedge dr.$ (4.27)

Taking $\rho\in[k,(k+1)]$ a large gauge transformation of parameter $k$, such
that

$C_{4}^{m}=2qr(\rho-k)\wedge\text{vol}(\text{S}^{3})\wedge dr,$ (4.28)

creates a number of D3-branes in this interval given by

$Q_{D3}^{(k)}=q\left\\{\begin{array}[]{ccrcl}n+\frac{1}{2},\quad\text{for}\quad
n=1,2,\dots,P^{\prime}\\\ 4(P^{\prime}+1)-n-\frac{1}{2},\quad\text{for}\quad
n=P^{\prime}+1,\dots,2P^{\prime}\,.\end{array}\right.$ (4.29)

Similarly, substituting in $\hat{f}_{7}$ above we find,

$\hat{f}_{7}=2qh_{0}r\Bigl{(}g-g^{\prime}(\rho-k)\Bigr{)}dr\wedge\text{vol}(\mathbb{T}^{3})\wedge\text{vol}(\text{S}^{3}),$
(4.30)

from which we compute

$Q_{D1}^{(k)}=\frac{1}{(2\pi)^{6}}\int\hat{f}_{7}=q\alpha_{k}.\left\\{\begin{array}[]{ccrcl}n+\frac{1}{2},\quad\text{for}\quad
n=1,2,\dots,P^{\prime}\\\ 4(P^{\prime}+1)-n-\frac{1}{2},\quad\text{for}\quad
n=P^{\prime}+1,\dots,2P^{\prime}\,.\end{array}\right.$ (4.31)

Combining with our results for the magnetic charges for the D7 and D5 branes,
reported in subsection 4.1, the picture that arises is a non-perturbative
setting in which $q\alpha_{k}$ D5-branes are stretched between stacks of $q$
D7-branes located at $\rho_{k}=k$ and $\rho_{k+1}=(k+1)$, and
$q(n+\frac{1}{2})\alpha_{k}\equiv qp\alpha_{k}$ D1-branes are stretched
between stacks of $qp$ D3-branes located at the same positions. This is so for
$r\in[2n\pi,2(n+1)\pi]$ intervals with $n=1,2\dots,P^{\prime}$, with the
obvious changes implied by equations (4.29) and (4.31) in the intervals with
$n=P^{\prime}+1,\dots,2P^{\prime}$.

In this non-perturbative brane scenario D1-branes are created when D3-branes
are crossed, while D5-branes are created when D7-branes are crossed. These are
non-perturbative realisations of Hanany-Witten brane set-ups, in which Dp-
branes are stretched between D(p+2)-branes located at fixed positions in the
field theory direction with orthogonal NS5-branes lying between the stacks of
D(p+2)-branes. In our case the D5-branes will finally contribute with flavour
groups, leaving just the D1-branes stretched between D3-branes as the ones
giving rise to colour groups. Note that the configuration consisting on
D1-branes stretched between D3-branes is precisely the Type IIB description of
smooth monopoles in the 4d theory living in D3 or D3-D7 brane systems,
according to the Nahm construction [38]. Our solutions describe naturally this
type of configurations, because of the definition of Page fluxes that needs to
be taken in the absence of a globally well-defined $B$ field. We will come
back to this discussion in subsection 4.4 when we relate our SCQM to the
bubbling of singular loops in the 4d theory living in D3-D7 branes.

From the previous quantised (magnetic) charges we can now proceed with the
construction of the quiver that describes the supersymmetric quantum mechanics
that, we propose, flows in the IR to the SCQM dual to our solutions. In order
to extract it we need to account for the ordering of the NS5-branes along the
$\rho$-direction, together with the net number of D1-branes ending on them and
the number of orthogonal D3-branes lying between NS5-branes. Similarly, in
order to account for the number of D5-branes in each interval we need to
compute the net number of them ending on NS5-branes together with the number
of orthogonal D7-branes between NS5-branes. The massless modes that give rise
to the quiver quantum mechanics arise then from the open strings that connect
the D-branes in the same interval between NS5-branes or adjacent ones.

Given that in our brane set-up, depicted in Figure 6, the D1 and D5 branes
stretch, respectively, between D3 and D7 branes with NS5-branes orthogonal to
them, we need to perform a series of Hanany-Witten moves that create D1 and D5
branes stretching between NS5-branes, with orthogonal D3 and D7 branes lying
between them.

Figure 6: Brane set-up along the $\rho$ direction, for $r$ constant, in units
of $q$.

This is explained in detail in Appendix E, following closely [20], where the
same analysis was carried out for the D2-D4-NS5 brane system underlying the
$\text{AdS}_{3}$ solutions studied therein. We will find the same field
content as in the 2d D2-D4-NS5 brane system discussed in [20] with the
D2-branes replaced by D1 (or wrapped D5) branes and the D4-branes by D3 (or
wrapped D7) branes, with 1d $\mathcal{N}=4$ multiplets described in terms of
2d (0,4) multiplets in the standard way.

The results in Appendix E show that the D5-D7-NS5 and D1-D3-NS5 subsystems of
the brane set-up are described by the (4,4) quivers depicted in Figures 7 and
8.

Figure 7: Quiver associated to the D5-D7-NS5 brane subsystem for $r$ constant.
Circles denote (4,4) vector multiplets and black lines (4,4) bifundamental
hypermultiplets. The gauge groups with ranks $\alpha_{k}$, with
$k=1,\dots,P-1$ to the left of the gauge group with rank $\alpha_{P}$ have
U(1) flavour symmetries. The gauge group with rank $\alpha_{P}$ has U(2)
flavour symmetry. The rest of gauge groups do not have attached any flavours.
We have used units of $q$. Figure 8: Quiver associated to the D1-D3-NS5 brane
subsystem for $r$ constant. Circles denote (4,4) vector multiplets and black
lines (4,4) bifundamental hypermultiplets. The gauge groups with ranks
$p\alpha_{k}$, with $k=1,\dots,P-1$ to the left of the gauge group with rank
$p\alpha_{P}$ have U(1) flavour symmetries. The gauge group with rank
$p\alpha_{P}$ has U(2) flavour symmetry. The rest of gauge groups do not have
attached any flavours. We have used units of $q$.

Our next step is to couple these two quivers to each other. This will reduce
the supersymmetries to (0,4). The new massless modes that arise are the ones
associated to the open strings stretched between the D1 and the D5 and D7
branes, given by:

  * •

D1-D5 strings: Strings with one end on D1-branes and the other end on
orthogonal D5-branes in the same interval between NS5-branes contribute with
fundamental (0,4) twisted hypermultiplets, associated to the motion of the
string along the $(x^{6},x^{7},x^{8},x^{9})$ directions, which are charged
under the R-symmetry. Strings with one end on D1-branes and the other end on
D5-branes in adjacent intervals between NS5-branes contribute with fundamental
(0,2) Fermi multiplets, since all the scalars are fixed.

  * •

D1-D7 strings: Strings with one end on D1-branes and the other end on
D7-branes in the same interval contribute with fundamental (0,2) Fermi
multiplets. This can be seeing more concretely by noting that the set-up is
T-dual to the D0-D8 system.

These new massless modes thus render the quiver (0,4) (or $\mathcal{N}=4$ in
1d) supersymmetric. The resulting quiver is depicted in Figure 9.

Figure 9: Quiver associated to the D1-D3-D5-D7-NS5 brane system for $r$
constant. Circles denote (4,4) vector multiplets, black lines (4,4)
bifundamental hypermultiplets, red lines (0,4) bifundamental twisted
hypermultiplets and dashed lines bifundamental (0,2) Fermi multiplets. The
gauge groups associated to the D5-branes have become flavour groups since
these branes are much heavier than the D1-branes. One can check that all gauge
nodes are balanced (ie $2N_{c}=N_{f}$, with $N_{c}$ the number of colours and
$N_{f}$ the number of flavours) but the ones with ranks $p\alpha_{k}$, with
$k=1,2,\dots,P-1$ to the left of the gauge group with rank $p\alpha_{P}$. We
have used units of $q$.

Our proposal is that this quiver describes a supersymmetric quantum mechanics
that flows in the IR to the SCQM dual to our solutions. Similar quivers to the
ones constructed in this subsection have been proposed in the literature in
order to describe the bubbling sector of singular ’t Hooft monopoles in 4d
$\mathcal{N}=2$ theories living in D3-D7 branes, in particular in the presence
of D5-branes (see [25]). We will come back to this discussion in subsection
4.4. Previous to that we will check our proposal with the computation of the
central charge.

####  4.3.1 Computation of the central charge

As a check of our proposal we proceed now to the computation of the field
theory and holographic central charges. We start with the field theory
calculation.

It is well-known that an $\mathcal{N}=4$ quantum mechanics can be described in
terms of 2d $(0,4)$ multiplets, and that both theories share the same
superconformal algebra. Therefore, we can use that the superconformal algebra
relates the (right-moving) central charge to the R-symmetry anomaly (the level
of the superconformal R-symmetry) to compute the central charge also in one
dimension. The relation is that

$c_{R}=6(n_{hyp}-n_{vec}),$ (4.32)

where $n_{hyp}$ stands for the number of (0,4) untwisted hypermultiplets and
$n_{vec}$ for the number of (0,4) vector multiplets. Indeed, this expression
has been successfully888In the sense that it agrees with the corresponding
holographic results. used in previous computations of the central charge of
$\mathcal{N}=4$ SCQMs (see [1, 2, 3, 4, 6, 7]). For the quiver depicted in
Figure 9 this gives,

$n_{hyp}^{(n)}=q^{2}p^{2}\,\Bigl{(}2\sum_{k=1}^{\alpha_{P}-1}k(k+1)+\sum_{k=1}^{P-1}\alpha_{k}+2\alpha_{P}\Bigr{)}$
(4.33)

and

$n_{vec}^{(n)}=q^{2}p^{2}\,\Bigl{(}2\sum_{k=1}^{\alpha_{P}-1}k^{2}+\alpha_{P}^{2}\Bigr{)},$
(4.34)

with $p=(n+\frac{1}{2})$, and finally

$c_{R}^{(n)}=6q^{2}p^{2}\,\sum_{k=1}^{P}\alpha_{k}.$ (4.35)

This gives the contribution to the central charge of the
$r\in[2n\pi,2(n+1)\pi]$ interval with $n=1,2,\dots,P^{\prime}$. Adding these
contributions plus the ones with
$n=P^{\prime}+1,P^{\prime}+2,\dots,2P^{\prime}$, for which $p\rightarrow
2(P^{\prime}+1)-p$, we find

$c_{R}=q^{2}\Bigl{(}4P^{\prime 3}+12P^{\prime
2}+11P^{\prime}\Bigr{)}\sum_{k=1}^{P}\alpha_{k}\sim 4q^{2}P^{\prime
3}\sum_{k=1}^{P}\alpha_{k}$ (4.36)

to leading order in $P^{\prime}$.

Let us proceed now with the computation of the holographic central charge. We
use that [1]

$c_{hol}=\frac{3}{4\pi G_{N}}\int d^{8}\xi\,e^{-2\Phi}\sqrt{\text{det}g_{8}},$
(4.37)

which gives for our solutions

$c_{hol}=q^{2}\Bigl{(}4P^{\prime 3}+12P^{\prime
2}\Bigr{)}\sum_{k=1}^{P}\alpha_{k}\sim 4q^{2}P^{\prime
3}\sum_{k=1}^{P}\alpha_{k},$ (4.38)

to leading order in $P^{\prime}$. In fact, the quantity that needs to be
compared to the holographic central charge is

$c_{hol}\longleftrightarrow\frac{c_{L}+c_{R}}{2}$ (4.39)

with

$c_{L}=c_{R}+\text{Tr}\gamma^{3},$ (4.40)

where the trace is over the Weyl fermions in the theory and $\gamma^{3}$ is
the chirality matrix in 2d. The (4,4) multiplets in the quiver quantum
mechanics depicted in Figure 9 contain the same number of left-handed and
right-handed multiplets. Therefore, the only ones that contribute to the
previous expression are the ones in (0,4) multiplets, right-handed, and the
ones in (0,2) Fermi multiplets, left-handed. We obtain

$\text{Tr}\gamma^{3}=-2q^{2}\sum_{n=1}^{P^{\prime}}(n+\frac{1}{2})\sum_{k=1}^{P-1}\alpha_{k}=-q^{2}P^{\prime}(P^{\prime}+2)\sum_{k=1}^{P-1}\alpha_{k}.$
(4.41)

Putting this result together with the field theory computation (4.36) we get

$\frac{c_{L}+c_{R}}{2}=q^{2}\Bigl{(}4P^{\prime 3}+\frac{23}{2}P^{\prime
2}+10P^{\prime}\Bigr{)}\sum_{k=1}^{P}\alpha_{k}+\frac{1}{2}q^{2}P^{\prime}(P^{\prime}+2)\alpha_{P}\sim
4q^{2}P^{\prime 3}\sum_{k=1}^{P}\alpha_{k},$ (4.42)

to leading order in $P^{\prime}$. Therefore, we find perfect agreement between
the field theory and holographic results, to leading order in $P^{\prime}$.

###  4.4 ’t Hooft defect interpretation

Quiver quantum mechanics like the ones discussed in the previous subsection
have been used in the literature in localization computations of monopole
bubbling contributions [39] to supersymmetric ’t Hooft loops, in 4d
$\mathcal{N}=2$ supersymmetric gauge theories [22, 23, 24, 25]. In these
calculations the contribution of the monopole bubbling sector is computed as
the index of a supersymmetric quantum mechanics, defined from the string
theory realisation of the ’t Hooft loop bubbling sector.

In 4d D3-D7 systems smooth monopoles are realised by D1-branes stretched
between the D3-branes, according to the Nahm construction [38]. In turn,
singular ’t Hooft monopoles are realised by introducing spatially transverse
NS5-branes [40, 23], which source $\pm\frac{1}{2}$ units of magnetic charge in
the worldvolume of the D3-branes (see Figure 10).

Figure 10: Brane realisation of singular ’t Hooft monopoles in 4d D3-brane
systems.

Since these D1-branes have Dirichlet and Neumann boundary conditions at
opposite ends they do not carry any degrees of freedom, as expected from ’t
Hooft operators. Monopole bubbling occurs when the smooth monopoles are
absorbed by the ’t Hooft defect, decreasing its effective magnetic charge [40,
23]. In the brane setting this occurs when D1-branes stretched between
adjacent D3-branes coincide with NS5-branes [23, 22]. The SQM from which the
bubbling sector is computed arises then as the low energy theory on the
D1-branes of the brane set-up, depicted in Table 2.

| $t$ | $x^{1}$ | $x^{2}$ | $x^{3}$ | $x^{4}$ | $x^{5}$ | $x^{6}$ | $x^{7}$ | $x^{8}$ | $x^{9}$
---|---|---|---|---|---|---|---|---|---|---
D3 | x | x | x | x |  |  |  |  |  |
D7 | x | x | x | x |  |  | x | x | x | x
NS5 | x |  |  |  |  | x | x | x | x | x
D1 | x |  |  |  | x |  |  |  |  |
Table 2: Brane set-up for the bubbling sector of 4d $\mathcal{N}=2$ D3-D7
theories.

This has been supported by localization computations that show that the
monopole bubbling contribution to the partition function of the 4d theory is
obtained as the Witten index of the SQM living in the D1-branes [22, 24, 25].

Interestingly, it was shown in [25] that in order to correctly account for the
bending of the NS5-branes caused by the branch cut of the D7-branes, extra
D5-branes need to be introduced in the previous brane set-up, along with
F1-strings. The D5-branes introduce new fields in the SQM, coming from the
D1-D5 open strings, that also contribute to the Witten index. The introduction
of the D5 and F1 branes renders the whole brane intersection realising the SQM
S-duality symmetric, such that it can describe at the same time the bubbling
of ’t Hooft loops, associated to the NS5-D1 subsector, of Wilson loops, coming
from the D5-F1 branes, or of dyonic loops. It was in fact argued in [25] that
the bubbling sector computed from the SQM is associated to dyonic loops, given
the presence of both D1 and F1-strings in the brane configuration. The brane
set-up that arises when the D5-branes and the F1-strings are added is the one
depicted in Table 3, which is no other than the brane intersection proposed in
Table 1 as underlying our $\text{AdS}_{2}$ solutions.

| $x^{0}$ | $x^{1}$ | $x^{2}$ | $x^{3}$ | $x^{4}$ | $x^{5}$ | $x^{6}$ | $x^{7}$ | $x^{8}$ | $x^{9}$
---|---|---|---|---|---|---|---|---|---|---
D3 | x | x | x | x |  |  |  |  |  |
D7 | x | x | x | x |  |  | x | x | x | x
NS5 | x |  |  |  |  | x | x | x | x | x
D1 | x |  |  |  | x |  |  |  |  |
D5 | x |  |  |  | x |  | x | x | x | x
F1 | x |  |  |  |  | x |  |  |  |
Table 3: Brane set-up for the bubbling sector of 4d $\mathcal{N}=2$ D3-D7
theories including D5 and F1 branes [25].

It is thus natural to expect that our solutions may provide a geometrical
description of the bubbling of dyonic loops in 4d $\mathcal{N}=2$ D3-D7
systems.

Further to this, it was shown in [22] that the bubbling sector of ’t Hooft
loops in the 4d $\mathcal{N}=2$ SU(N) theory living in D3-branes in an Omega
background is described by quiver quantum mechanics consisting on N-1 balanced
quivers999That satisfy that $N_{f}=2N_{c}$, with $N_{f}$ the number of
flavours that couple to the gauge node and $N_{c}$ its rank. of length the
number of NS5-branes in each interval between two D3-branes, connected by N-2
unbalanced gauge nodes. The N D3-branes of the 4d theory contribute to these
quivers with flavour groups, that couple to the unbalanced gauge nodes. The
detailed structure of the unbalanced quivers contains the information about
the particular bubbling sector described by the quiver QM, as carefully
explained in [22]. These constructions were then extended in [24] to include
$N_{f}$ D7-branes in the 4d SU(N) gauge theory. Interestingly, in reference
[25] these quivers were improved to account for the bending of the NS5-branes
caused by the D7-branes. In this completed brane set-up the D3 and D7 branes
are interpreted as sitting inside a 5-brane web. The resulting QM living in
the completed brane set-up is a (0,4) supersymmetric quiver theory built out
of (4,4) vector multiplets coming from each gauge node, (4,4) bifundamentals
coming from D1-D1 branes across NS5-branes, twisted (0,4) bifundamentals
coming from D1-D5 strings ((0,2) bifundamentals if across an NS5-brane) and
(0,2) bifundamental Fermi multiplets coming from D1-D7 strings. This is
exactly the field content of the quiver quantum mechanics depicted in Figure
9, which can thus be interpreted as describing dyonic loops in 4d
$\mathcal{N}=2$ theories living in D3-D7 branes.

Let us analyse in a bit more detail this interpretation. In the bubbling
description the number of NS5-branes between stacks of D3-branes has to be an
even integer number, for them to induce an integer magnetic charge in the
worldvolume of the D3-branes (see for instance [25]). This fixes $\beta_{k}\in
2\mathbb{Z}$, $p\alpha_{k}\in\mathbb{Z}$, and therefore the number of
D1-branes in each interval to an integer number. Second, we can identify the
precise bubbling sector that our quiver quantum mechanics is describing. For
this we can follow the analysis in [22] for the D1-D3-NS5 brane subsystem,
where the information about the bubbling sector is encoded. Comparison with
the quivers proposed in [22] shows that our quiver in Figure 9 would describe
the bubbling sector in which all the magnetic charge of the defect has been
screened101010The sector with ${\bf v}=\text{diag}(0,\dots,0)$ in the notation
of [22]., of the 4d SU($qp(P+1)$) gauge theory living in the D3-branes, with
$N_{f}=q(P+1)$ flavour groups coming from the D7-branes, in a given
$r\in[2n\pi,2(n+1)\pi]$ interval. One can check that our quiver consists
indeed of $P$ balanced sub-quivers of length $\beta_{k}$, with
$k=0,1,\dots,\beta_{P-1}$, separated by $P-1$ unbalanced ones111111Note that
our quivers are scaled by a factor $qp$ compared to the ones in [22]., to
which the flavour groups introduced by the D3-D7 strings couple. Note that in
our quantum mechanics the condition for a gauge node to be balanced is
inherited from the 2d theory, where the cancellation of the gauge anomaly
imposes that the number of (0,2) Fermi multiplets connected to a given gauge
node must be twice the number of (0,4) hypermultiplets that couple to this
node.

A related question to address is whether the 4d theory living in the D3-D7
branes is conformal. Our construction gives
$N_{c}=qp(P+1)=(n+\frac{1}{2})N_{f}$, and therefore differs from the condition
$N_{c}=\frac{1}{2}N_{f}$ required by conformal invariance. However, one can
note that it is possible to cancel the term proportional to $n$ by adding a
worlvolume flux with instanton number $\int_{\mathbb{R}^{4}}F\wedge
F=8\pi^{2}n$, that would render the 4d theory conformally invariant.

The interpretation of the D1-D3-NS5 subsystem of the brane set-up as
associated to ’t Hooft loops becomes clearer when one notices that this part
of the brane set-up is just the S-dual of the F1-D3-D5 subsector used to
describe Wilson loops (or baryon vertices). Indeed, dualising the Born-Infeld
vector field in the worldvolume of the D3-branes to its electric-magnetic dual
$\tilde{A}$, one finds a coupling

$S_{D3}=T_{3}\int B\wedge d{\tilde{A}}=-T_{3}\int H\wedge{\tilde{A}},$ (4.43)

that shows that a D1-string with electric charge $nq$ induces $nq\beta_{k}$
D3-brane electric charge. Adding the contributions of all $\rho$-intervals one
finds a total $Q_{D3}^{e}=nq\alpha_{k}$ electric charge, as found in (4.22).
This is illustrated in Figure 11, which can be obtained from the D1-D3-NS5
sector of the brane set-up depicted in Figure 6 after a sequence of Hanany-
Witten moves, where in this case we use electric instead of magnetic charges
to count the numbers of D3 and D1 branes.

Figure 11: D1-D3-NS5 sector of the brane set-up depicted in Figure 6 after a
sequence of Hanany-Witten moves, where in this case we use electric instead of
magnetic charges for the D3 and D1 branes.

The D3-branes thus find a dual interpretation: as baryon vertices together
with the F1 and the D5 branes, in the $r$-space, and as ’t Hooft lines
together with the D1 and the NS5-branes in the $\rho$-space. The ’t Hooft line
is again in the completely antisymmetric representation of
U($qn\alpha_{1}$)$\times$ U($qn\alpha_{2}$) $\times\dots$ U($qn\alpha_{P})$,
and could thus be interpreted as a sort of magnetic baryon vertex. The reader
can check the similarity between Figure 11 and the D3-F1-NS5 sector of Figure
4, up to the different global completions used for the $\rho$ and $r$
directions.

Our analysis in this subsection suggests an interpretation of the
$\text{AdS}_{2}$ solutions constructed in section 3.2.2 as geometric duals of
SCQMs describing the bubbling sector of vanishing effective magnetic charge in
4d $\mathcal{N}=2$ SCFTs living in D3-D7 systems. It would be interesting to
investigate whether the broader solutions constructed in section 3.3 could
geometrically describe more general bubbling sectors. Note that based on this
interpretation one would expect that the solutions (3.2.2)-(3.16) asymptoted
locally to the $\text{AdS}_{5}$ spacetime dual to the 4d SCFT living in the
D3-D7 branes. This is however not the case for our class of solutions. Indeed,
in the two different interpretations that we have discussed the D3-branes
either play the role of background branes, in their interpretation in this
section, or defect branes, in the baryon vertex interpretation discussed in
section 4.2. Therefore, compared to the simpler configurations studied in [2,
3, 6, 7], in this case there is no clear distinction between defect and
background branes, and one should not expect that a higher dimensional AdS
space, that is typically identified as the near horizon geometry of the
background branes, arises asymptotically locally from the solutions, in
agreement with our findings. It would be interesting to construct the explicit
brane intersection that underlies the solutions to shed further light on their
possible defect interpretation.

Related to our previous observation, it is worth mentioning that the brane
intersection given in Table 1 can more generally be regarded as a coupled
5d-4d-1d system, from which one can compute the bubbling sectors of ’t Hooft
loops in the 4d theory living in the D3-D7 branes but also the instanton
sectors of the 5d theory living in the 5-brane web. The latter approach was
the one taken in [41], where the same brane set-up depicted in Table 1 was
proposed for the study of loop operators in the 5d theory living on D5-NS5
branes. In this setting the D3-branes introduce Wilson lines, together with
the F1-strings, and S-dual Wilson lines, together with the D1-branes, that in
this case connect the D3-branes with the NS5-branes of the 5-brane web. Note
that the condition for the 5d theory to flow to a conformal point in the UV,
given by $2N_{c}\geq N_{f}+c_{cl}$ is satisfied by the D5-D7-NS5 subsector of
our quiver mechanics. Connecting to our discussion in the previous paragraph,
it is worth stressing that in this interpretation the D3-branes would play the
role of defect branes in both the instanton and baryon vertex interpretations,
with the D5-D7-NS5 branes playing the role of background branes also in both
interpretations. Thus, one could expect that the AdS6 geometry arising in the
near horizon of the 5-brane web may emerge asymptotically locally from the
solutions. As we discussed in section 4.2 this is however not the case. As
mentioned above the explicit construction of the brane intersection underlying
the solutions would be very enlightening in clarifying this issue.

##  5 Conclusions

In this paper we have presented general results on the generation of
$\text{AdS}_{2}$ solutions to Type II supergravities from $\text{AdS}_{3}$
backgrounds using U(1) and SL(2) T-dualities, paying special attention to the
conditions for preservation of supersymmetry. We have discussed in detail the
solutions that arise from the $\text{AdS}_{3}\times\text{S}^{3}$ solutions
with $\mathcal{N}=(0,4)$ supersymmetries and SU(3) structure studied in [20],
in particular to the ones constructed via SL(2) T-duality with $h$ constant.
We have shown that these solutions belong to the more general class obtained
from the $\text{AdS}_{3}\times S^{2}$ solutions to Type IIB supergravity
constructed in [21] through a double analytical continuation.

We have initiated the study of the field theory duals to the new
$\text{AdS}_{2}$ solutions focusing on the previously alluded to SL(2) T-dual,
consisting on $\text{AdS}_{2}\times\text{S}^{3}\times\mathbb{T}^{3}$
foliations over a 2d Riemann surface, parametrised by two non-compact
directions $(\rho,r)$. We have proposed that these solutions are dual to SCQMs
arising in the IR from supersymmetric quantum mechanics living in D1-branes
extended in the $\rho$ direction between NS5-branes, with flavour groups
arising from D3, D5 and D7 branes. We have seen that the F1-strings also
present in the brane set-up play the role, together with the D3-branes, of
baryon vertices for the D5-branes, and, together with the D1-branes, of baryon
vertices for the D7-branes. This structure is revealed by looking at the
position of the branes along the $r$ direction. The D3-D5-F1 and D1-D7-F1
brane subsystems are indeed displayed exactly as in the D3-D5-F1 brane
configurations studied in [36, 37], describing Wilson lines in antisymmetric
representations in 4d $\mathcal{N}=4$ SYM. As encountered in previous examples
of $\text{AdS}_{2}$ solutions associated to baryon vertex configurations [2,
3, 4, 6, 7], one can associate the solutions to D5-D7-NS5 brane intersections,
where a 5d $\mathcal{N}=1$ theory lives, in which one dimensional defects are
introduced. In the IR the gauge symmetry on the D5-D7-NS5 system becomes
global, turning them from colour to flavour branes, with the defect branes
becoming the new colour branes of the backreacted geometry. In our
construction in this paper, however, the presence of the second non-compact
direction in the geometry (not present in the examples in the previous
references), playing the role of field theory direction along which the
D1-branes extend, renders the D1-branes the only remaining colour branes of
the configuration.

The intricate brane set-up associated to the solutions unveils, moreover, a
similar structure along the $\rho$-direction, to the one arising along the
$r$-direction, with the D1-branes playing the same role as the F1-strings now
in conjunction with the D3-branes and the NS5-branes, to describe, in this
case, ’t Hooft monopoles in the 4d theory living on D3-D7 branes. Quiver
quantum mechanics like the ones we have explicitly constructed have been
proposed in the literature in the description of monopole bubbling in 4d
$\mathcal{N}=2$ supersymmetric theories living in these 4d intersections [22,
23, 24, 25]. These constructions involve the D1-D3-NS5-D7 subsystem of the
brane set-up associated to the solutions. Interestingly, in order to account
for the bending of the NS5-branes due to the branch cuts introduced by the
D7-branes it was shown in [25] that D5-branes (and F1-strings) need also be
introduced in the description, such that the final brane set-up must be
precisely the one underlying our solutions. Our $\text{AdS}_{2}$ solutions
thus find a natural interpretation as geometrical descriptions of monopole
bubbling in 4d $\mathcal{N}=2$ D3-D7 systems. As discussed in the paper, the
bubbling sector described by our quantum mechanics would be the one of
vanishing effective magnetic charge. It is likely that the more general
solutions presented in section 3 would provide the geometrical description of
more general bubbling sectors. It would be interesting to investigate this
further.

## Acknowledgments

We are supported by grants from the Spanish government
MCIU-22-PID2021-123021NB-I00 and Principality of Asturias SV-
PA-21-AYUD/2021/52177. The work of NM is also supported by the Ramón y Cajal
fellowship RYC2021-033794-I.

##  Appendix A Conventions

We follow democratic like conventions of [42] for Type II.

The bosonic sector of Type II supergravity consists of the NS-NS and RR
sectors. The former containing the metric, dilaton $\Phi$ and NS 3-form $H$.
The later can be expressed in term of a polyform $F$ such that

$F=\left\\{\begin{array}[]{l}F_{0}+F_{2}+F_{4}+F_{6}+F_{8}+F_{10}~{}~{}~{}~{}\text{IIA}\\\\[5.69054pt]
F_{1}+F_{3}+F_{5}+F_{7}+F_{9}~{}~{}~{}~{}\text{IIB}\end{array}\right.$ (A.1)

The polyform is subject to a self duality constraint which halfs its degrees
of freedom, namely

$F=\star\lambda(F),$ (A.2)

where $\lambda(C_{k})=(-1)^{[\frac{k}{2}]}C_{k}$. The Hodge dual is defined in
terms of the $d=10$ vielbein $e^{\underline{M}}$ as

$\star
e^{\underline{M}_{1}...\underline{M}_{k}}=\frac{1}{(d-k)!}\epsilon_{\underline{M}_{k+1}...\underline{M}_{d-k}}^{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\underline{M}_{1}...\underline{M}_{k}}e^{\underline{M}_{k+1}...\underline{M}_{d-k}},$
(A.3)

where the $d=10$ indices $M$ are curved and $\underline{M}$ are flat.

A solution to type II supergravity is one that solves the following equations
of motion

$\displaystyle
d_{H}F=0,~{}~{}~{}~{}dH=0,~{}~{}~{}~{}d(e^{-2\Phi}\star_{10}H)-\frac{1}{2}(F,F)_{8}=0,$
$\displaystyle
2R^{(10)}-H^{2}-8e^{\Phi}(\nabla^{(10)})^{2}e^{-\Phi}=0,~{}~{}~{}R^{(10)}_{AB}+2\nabla^{(10)}_{A}\nabla^{(10)}_{B}\Phi-\frac{1}{2}H^{2}_{AB}-\frac{e^{\Phi}}{4}(F)^{2}_{AB}=0,$
(A.4)

where $(F,F)_{8}$ is the 8-form part of $F\wedge\lambda(F)$ and we refer the
reader to [42] for further details. In the presence of sources these equations
get modified.

A solution of Type II supergravity preserves supersymmetry if the following
spinorial conditions can be solved for non trivial $d=10$ Majorana-Weyl
spinors $\epsilon_{1,2}$

$\displaystyle\left(\nabla^{(10)}_{M}-\frac{1}{4}H_{M}\right)\epsilon_{1}+\frac{e^{\Phi}}{16}F\Gamma_{M}\epsilon^{2}=0,$
(A.5a)
$\displaystyle\left(\nabla^{(10)}_{M}+\frac{1}{4}H_{M}\right)\epsilon_{2}\pm\frac{e^{\Phi}}{16}\lambda(F)\Gamma_{M}\epsilon^{1}=0,$
(A.5b)
$\displaystyle\left(\nabla^{(10)}-\frac{1}{4}H-d\Phi\right)\epsilon_{1}=0,$
(A.5c)
$\displaystyle\left(\nabla^{(10)}+\frac{1}{4}H-d\Phi\right)\epsilon_{2}=0,$
(A.5d)

where the Clifford map is assumed and we define the spin covariant derivative
as

$\nabla_{M}=\partial_{M}+\frac{1}{4}\omega_{M}{}^{\underline{P}\underline{Q}}\Gamma_{\underline{P}\underline{Q}},~{}~{}~{}~{}de^{\underline{M}}+\omega^{\underline{M}}{}_{\underline{N}}\wedge
e^{\underline{N}}=0.$ (A.6)

The upper/lower signs are taken in IIA/IIB and for the chirality matrix
defined as $\hat{\Gamma}=\Gamma^{\underline{0}...\underline{9}}$ we have that

$\hat{\Gamma}\epsilon_{1}=\epsilon_{1},~{}~{}~{}~{}\hat{\Gamma}\epsilon_{2}=\mp\epsilon_{2}.$
(A.7)

We also have the useful identities

$\hat{\Gamma}C_{k}=\star\lambda(C_{k}),~{}~{}~{}~{}~{}\hat{\Gamma}C_{k}=-\lambda(\star
C_{k}).$ (A.8)

Throughout the next sections we shall make use of a 3+7 split of the $d=10$
gamma matrices such that

$\displaystyle\Gamma_{\underline{\mu}}$
$\displaystyle=\gamma_{\underline{\mu}}\otimes\sigma_{3}\otimes\mathbbm{1}_{8},\qquad$
$\displaystyle\mu={0,1,2},$ $\displaystyle\Gamma_{a}$
$\displaystyle=\mathbbm{1}_{2}\otimes\sigma_{1}\otimes\gamma_{a},\qquad$
$\displaystyle a={1,..,7},~{}~{}~{}~{}i\gamma_{1...7}=\mathbbm{1}_{8}$ (A.9)

where
$\gamma_{\underline{\mu}}=(i\sigma_{2},\sigma_{1},\sigma_{3})_{\underline{\mu}}$
and are thus real. The $d=10$ intertwiner defining Majorana conjugation as
$\epsilon^{c}=B^{(10)}\epsilon^{*}$ and chirality matrix are then

$B^{(10)}=\mathbbm{1}_{2}\otimes\sigma_{3}\otimes
B,~{}~{}~{}~{}B^{-1}\gamma_{a}B=-\gamma_{a}^{*},~{}~{}~{}~{}\hat{\Gamma}=-\mathbbm{1}_{2}\otimes\sigma_{2}\otimes\mathbbm{1}_{8}$
(A.10)

##  Appendix B AdS3, its isometries and Killing spinors

In this appendix we explore some details of the isometries of AdS3, the
Killing vectors it supports and what they are charged under. This will be used
in the following appendices.

One can express the Hopf fibration of unit radius AdS3 as

$ds^{2}(\text{AdS}_{3})=\frac{1}{4}ds^{2}(\text{AdS}_{2})+\frac{1}{4}(dr+V)^{2},~{}~{}~{}~{}dV=s\text{vol}(\text{AdS}_{2}),~{}~{}~{}~{}s^{2}=1,$
(B.1)

where $\partial_{r}$ is an isometry. A useful parameterisation for what
follows is to take

$ds^{2}(\text{AdS}_{2})=-\cosh^{2}xdt^{2}+dx^{2},~{}~{}~{}V=-s\sinh xdt.$
(B.2)

AdS3 has a global SO(2,2) symmetry with 2 distinct SL(2) subgroups we shall
label as SL(2)L/R with $L/R$ standing for left and right. We define the
generators of $\mathfrak{sl}(2)$ to be

$\tau_{i}=\frac{1}{2}(i\sigma_{2},\sigma_{1},\sigma_{3})_{i},$ (B.3)

for $\sigma_{1,2,3}$ the Pauli matrices. These obey the Lie bracket relation

$[\tau_{i},\tau_{j}]=f_{ij}^{~{}~{}k}\tau_{k},$ (B.4)

where the structure constants can be easily computed and indices are raised
and lowered with the standard mostly positive Minkowki3 metric
$\eta=\text{Diag}(-1,1,1)$. We define an element of SL(2) to be

$G=e^{-st\tau_{1}}e^{-sx\tau_{2}}e^{-sr\tau_{3}},$ (B.5)

with left/right invariant 1-forms on AdS3 defined in terms of this as

$L^{i}=-2s\text{Tr}\bigg{[}\tau^{i}G^{-1}dG\bigg{]},~{}~{}~{}R^{i}=-2s\text{Tr}\bigg{[}\tau^{i}dGG^{-1}\bigg{]},$
(B.6)

which both span AdS3 as

$ds^{2}(\text{AdS}_{3})=\frac{1}{4}L_{i}L^{i}=\frac{1}{4}R_{i}R^{i},$ (B.7)

and obey the following differential relation

$dL^{i}=\frac{s}{2}f^{i}_{~{}jk}L^{j}\wedge
L^{k},~{}~{}~{}~{}dR^{i}=-\frac{s}{2}f^{i}_{~{}jk}R^{j}\wedge R^{k}.$ (B.8)

The vectors dual to the left/right invariant forms define the SL(2)R/L Killing
vectors as

$K_{R/L}^{i}=\frac{1}{4}(L/R)^{i}_{\mu}(g_{\text{AdS}_{3}})^{\mu\nu}\partial_{\mu}.$
(B.9)

with the factor of $\frac{1}{4}$ taken such that
$(K_{L/R}^{i})^{\mu}((R/L)^{j})_{\mu}=\eta^{ij}$, we also have the relations

${\cal L}_{K^{i}_{L/R}}(L/R)^{j}=0,~{}~{}~{}~{}{\cal
L}_{K^{i}_{L/R}}(R/L)^{j}=(+/-)f^{ij}_{~{}~{}k}(R/L)^{k}.$ (B.10)

With respect to our specific parameterisation the Killing vectors are

$\displaystyle K_{R}^{1}\pm K_{R}^{2}$ $\displaystyle=e^{\pm sr}(s\tanh
x\partial_{r}+\operatorname{sech}x\partial_{t}-\partial_{x}),~{}~{}~{}~{}K_{R}^{3}=\partial_{r},$
$\displaystyle K_{L}^{3}+iK_{L}^{2}$
$\displaystyle=e^{ist}(\operatorname{sech}x\partial_{r}-s\tanh x\partial
t+i\partial_{x}),~{}~{}~{}~{}K_{L}^{1}=-\partial_{t},$ (B.11)

so in particular it SL(2)R that includes the U(1) isometry corresponding to
$\partial_{r}$.

AdS3 supports two physically distinct types of spinors that transform as
doublets with respect to one of SL(2)L/R whilst also being singlets with
respect to SL(2)R/L. To see this let us define the following frame maintaining
the Hopf fibration structure,

$e^{\underline{\mu}}_{3}=\frac{1}{2}(\cosh xdt,dx,(dr+V))^{\underline{\mu}},$
(B.12)

where an underscore indicates that an index is flat. We shall define the
independent spinors on (unit radius) AdS3 to solve the Killing spinor
equations

$\nabla_{\mu}\zeta_{R/L}=(+/-)\frac{s}{2}\gamma_{\mu}\zeta_{R/L}.$ (B.13)

One can show with respect to the above frame that these constraints are solved
by

$\zeta_{R}=e^{\frac{s}{2}x\sigma_{1}}e^{\frac{s}{2}ti\sigma_{2}}\zeta_{R}^{0},~{}~{}~{}~{}\zeta_{L}=e^{-\frac{s}{2}r\sigma_{3}}\zeta_{L}^{0},$
(B.14)

where $\zeta^{0}_{R/L}$ are arbitrary constant spinors, they need only be real
for $\zeta_{R/L}$ to be Majorana in our conventions. In terms of these we can
define our spinor doublets: Defining unit norm constant spinors
$\zeta^{0}_{\pm}$ such that $\sigma_{3}\zeta^{0}_{\pm}=\pm\zeta^{0}_{\pm}$ we
have that

$\zeta^{A}_{R}=\left(\begin{array}[]{c}e^{\frac{s}{2}x\sigma_{1}}e^{\frac{s}{2}ti\sigma_{2}}\zeta^{0}_{+}\\\
e^{\frac{s}{2}x\sigma_{1}}e^{\frac{s}{2}ti\sigma_{2}}\zeta^{0}_{-}\end{array}\right)^{A},~{}~{}~{}~{}~{}\zeta^{A}_{L}=\left(\begin{array}[]{c}e^{-\frac{s}{2}r\sigma_{3}}\zeta^{0}_{+}\\\
e^{-\frac{s}{2}r\sigma_{3}}\zeta^{0}_{-}\end{array}\right)^{A}$ (B.15)

are such that

${\cal L}_{K^{i}_{R/L}}\zeta^{A}_{R/L}=0,~{}~{}~{}~{}~{}{\cal
L}_{K^{i}_{R/L}}\zeta^{A}_{L/R}=(-/+)s(\tau_{i})^{A}_{~{}B}\zeta^{B}_{L/R}.$
(B.16)

We thus find that it is $\zeta_{R}$ that is a singlet with respect to the
isometry $\partial_{r}$, and likewise the rest of SL(2)R, spanned by
$K^{i}_{R}$. An important thing to also appreciate is that the Killing spinor
$\zeta_{R}$ additionally solves the Killing spinor equation on round AdS2,
namely

$\nabla_{\mu_{2}}^{\text{AdS}_{2}}\zeta_{R}=\frac{s}{2}\gamma_{\mu_{2}}\zeta_{R},$
(B.17)

when the frame is $e^{\underline{\mu}_{2}}_{2}=(\cosh
xdt,dx)^{\underline{\mu}_{2}}$ and
$\underline{\mu}=(\underline{\mu}_{2},\underline{r})$.

In the main text we are interested in performing T-duality transformations on
the isometries of AdS3. Previous results, [43, 44] on the preservation of
supersymmetry under U(1) and SU(2) T-dualities suggest that T-dualising on
AdS3 will only lead to a solution that preserves supersymmety when its spinors
are singlets with respect to the dualisation isometry. Thus any AdS3 solution
of Type II supergravity that preserves supersymmetry better be compatible with
the spinors $\zeta_{R}$ for supersymmetry to be preserved after T-dualising on
$\partial_{r}$. For SL(2) T-duality we have two options for dualisation
isometry, we shall choose to dualise on SL(2)R as this contains the U(1) of
the Hopf fiber. Thus far our discussion on spinors has been with respect to
the frame (B.12), note however that this is not the appropriate frame for
SL(2) T-duality, for that the vielbein should also respect an SL(2) isometry.
We shall take

$e^{\underline{\mu}}_{3,\text{SL}(2)}=\frac{1}{2}(R^{1},~{}R^{2},~{}R^{3})^{\underline{\mu}}.$
(B.18)

We could also use $L^{i}$ to span the frame, however we choose the right SL(2)
to relate to the case of U(1) T-duality. The difference between the 2 frames
is simply a Lorentz transformation, specifically

$e^{\underline{\mu}}_{3,\text{SL}(2)}={\cal
R}^{\underline{\mu}}_{~{}\underline{\nu}}e^{\underline{\nu}}_{3},~{}~{}~{}~{}{\cal
R}=\left(\begin{array}[]{ccc}1&0&0\\\ 0&\cos t&s\sin t\\\ 0&-s\sin t&\cos
t\end{array}\right)\left(\begin{array}[]{ccc}\cosh x&0&s\sinh x\\\ 0&1&0\\\
s\sinh x&0&\cosh x\end{array}\right).$ (B.19)

Under such a transformation the spinors also transform as $\zeta\to S\zeta$
where $S$ solves the equation

$S^{-1}\gamma^{\underline{\mu}}S={\cal
R}^{\underline{\mu}}_{~{}\underline{\nu}}\gamma^{\underline{\nu}},$ (B.20)

we find

$S=e^{-\frac{s}{2}ti\sigma_{2}}e^{-\frac{s}{2}x\sigma_{1}}$ (B.21)

and so in the frame (B.18) we have

$\zeta_{R,\text{SL}(2)}=S\zeta_{R}=\zeta_{R}^{0},~{}~{}~{}~{}\zeta_{L,\text{SL}(2)}=S\zeta_{L}=e^{-\frac{s}{2}ti\sigma_{2}}e^{-\frac{s}{2}x\sigma_{1}}e^{-\frac{s}{2}r\sigma_{3}}\zeta_{L}^{0}=G\zeta_{L}^{0}.$
(B.22)

##  Appendix C Deriving the SL(2) T-duality dual fields

In this appendix we shall derive the dual NS and RR fields that one gets after
performing an SL(2) non-Abelian T-duality transformation on an AdS3 solution
of the form (2). The procedure for doing this in the presence of non trivial
RR fields was first worked out in [45] with a more detailed, and easier to
follow, account given by [46]. Both of these works focus on SU(2) T-duality,
but the SL(2) case is rather analogous so we shall be brief here and refer the
reader to [46] for more details. The reader can also check [4] where SL(2)
T-duality was applied to
$\text{AdS}_{3}\times\text{S}^{3}\times\mathbb{T}^{3}$ and a briefer account
on the content of this section, especially on the part regarding the
transformation of the spinors, was also given.

The dual NS sector is computed via a non Abelian Buscher procedure: One first
defines a sigma model with worldvolume coordinates $\sigma_{\pm}$ in terms of
the NS fields with components along the SL(2) isometry directions. For a
solution of the form (2), with trivial electric NS flux we have

$ds^{2}=g_{ij}R^{i}R^{j}+....,~{}~{}~{}~{}B=....,~{}~{}~{}~{}dB=H,~{}~{}~{}~{}g_{ij}=\frac{e^{2A}}{4}\eta_{ij}$
(C.1)

where … have no components along the SL(2) directions and are thus spectators
to the Buscher procedure. The Lagrangian of our sigma model is then simply

${\cal L}_{0}=g_{ij}R^{i}_{+}R^{j}_{-},$ (C.2)

where $R^{i}_{\pm}=-2s\text{Tr}\bigg{[}\tau^{i}\partial_{\pm}GG^{-1}\bigg{]}$
for $G\in$ SL(2) and $\tau_{i}$ the generators of $\mathfrak{sl}(2)$,
consistent with the previous section.

The next step of the Buscher procedure is to rewrite ${\cal L}_{0}$ in a
locally equivalent form. For that one gauges the SL(2) isometry by introducing
SL(2) gauge fields ${\cal A}_{\pm}=({\cal A}_{\pm})^{i}\tau_{i}$ such that

$\partial_{\pm}G\to D_{\pm}G=\partial_{\pm}G-{\cal A}_{\pm}G,$ (C.3)

but one additionally adds a Lagrange multiplier term to the action of the form
$\text{Tr}(v{\cal F}_{\pm})$, where ${\cal F}_{\pm}$ is the non-Abelian field
strength of ${\cal A}_{\pm}$ \- this term setting ${\cal F}_{\pm}=0$ on shell.
A dual sigma model is then generated if one integrates out the gauge fields.
This theory generically depends on the 3 coordinates packaged within $G$ and
the Lagrange multipliers $v_{i}$, and one needs to gauge fix such that only 3
of these remain. The simplest way to do this is to set

$G=\mathbbm{1}.$ (C.4)

The Lagrangian of the resulting dual sigma model is then given by

$\hat{{\cal
L}}=\partial_{-}v_{i}(M^{-1})^{ij}\partial_{-}v_{j},~{}~{}~{}~{}M_{ij}=g_{ij}+sf_{ij}^{~{}~{}k}v_{k},$
(C.5)

where $f_{ij}^{~{}~{}k}$ are the structure constants of SL(2). From this the
dual metric and NS flux can be read off as

$d\hat{s}^{2}=dv_{i}(M^{-1})^{(ij)}dv_{j}+....,~{}~{}~{}~{}~{}\hat{B}=\frac{1}{2}(M^{-1})^{[ij]}dv_{i}\wedge
dv_{j}+...$ (C.6)

where $...$ are the spectator parts of the original metric and NS 2-form. To
compute the dual dilaton one needs to also consider quantum effects. These
lead to

$e^{-\hat{\Phi}}=ce^{-\Phi}\sqrt{\det M},$ (C.7)

where $c$ is an arbitrary constant which can be set to any value by rescaling
$g_{s}$.

The dual RR sector can be computed as in [45]. One first observes that the
dual sigma model defines a pair of canonical frames. In the seed geometry, we
had that the $d=10$ vielbein along the AdS3 could be expressed in an SL(2)
invariant way as

$e^{i}=\frac{e^{A}}{2}R^{i},$ (C.8)

while the Buscher procedure defines 2 frames in the dual solutions

$e^{i}_{+}=\frac{e^{A}}{2}(M^{-T})^{ij}dv_{j},~{}~{}~{}~{}e^{i}_{-}=\frac{e^{A}}{2}(M^{-1})^{ij}dv_{j}.$
(C.9)

These are related by a Lorentz transformation

$e_{+}=\Lambda e_{-},~{}~{}~{}~{}~{}\Lambda=M^{-T}M.$ (C.10)

As usual one can generate an action on spinors from the above
$\epsilon\to\Omega\epsilon$ by solving the equation

$\Omega^{-1}\Gamma^{\underline{M}}\Omega=\Lambda^{\underline{M}}_{~{}\underline{N}}\Gamma^{\underline{N}}$
(C.11)

where $\Gamma^{\underline{M}}$ are a basis of $d=10$ flat index gamma
matrices. The main contribution of [45] was to realise (through analogy with
[43] ) that it is through the action of $\Omega$ that the dual RR sector is
generated. It is possible to show that $\Omega$ is given by

$\Omega=\frac{e^{3A}}{8\sqrt{\det
M}}(s\Gamma^{\underline{0}\underline{1}\underline{2}}+\frac{4}{e^{2A}}v_{i}\Gamma^{\underline{i-1}}),$
(C.12)

where we have fixed an arbitrary sign. The dual RR fluxes can then be
extracted from the following polyform identities

$e^{\hat{\Phi}}\hat{F}=te^{\Phi}F\Omega^{-1},~{}~{}~{}~{}t^{2}=1,$ (C.13)

where $\Omega^{-1}$ acts on F through the Clifford map.

Let us now examine the symmetries of the dual solution. To this end we find it
convenient to fix

$v_{3}=-\frac{1}{2}sr\sinh x,~{}~{}~{}~{}v_{2}=\frac{1}{2}rs\cosh x\sin
t,~{}~{}~{}~{}v_{3}=\frac{1}{2}\cos t\cosh x,~{}~{}~{}~{}c=4.$ (C.14)

The dual NS sector then becomes

$\displaystyle d\hat{s}^{2}$
$\displaystyle=\frac{e^{2A}}{4\Delta}ds^{2}(\text{AdS}_{2})+e^{-2A}dr^{2}+...,~{}~{}~{}~{}\hat{B}=-\frac{sr}{2\Delta}\text{vol}(\text{AdS}_{2})+....,~{}~{}~{}~{}e^{-\hat{\Phi}}=e^{-\Phi+A}r\sqrt{\Delta},$
$\displaystyle ds^{2}(\text{AdS}_{2})$
$\displaystyle=-\cosh^{2}xdt^{2}+dx^{2},~{}~{}~{}~{}\text{vol}(\text{AdS}_{2})=\cosh
xdt\wedge dx.$ (C.15)

So we see that the NS sector respects the SL(2) isometry of AdS2. To compute
the dual RR fluxes we use that

$te^{\Phi-\hat{\Phi}}\Omega^{-1}:~{}~{}1\to
t(r-\frac{se^{4A}}{8\Delta}\text{vol}(\text{AdS}_{2}))\wedge
dr,~{}~{}~{}~{}e^{3A}\text{vol}(\text{AdS}_{3})\to
t\frac{e^{3A}}{2}(-s+\frac{r}{2\Delta}\text{vol}(\text{AdS}_{2})).$ (C.16)

We thus find the dual $d=10$ flux is

$\hat{F}=t\bigg{[}f_{\pm}\wedge(r-\frac{se^{4A}}{8\Delta}\text{vol}(\text{AdS}_{2}))\wedge
dr\mp\frac{e^{3A}}{2}(-s+\frac{r}{2\Delta}\text{vol}(\text{AdS}_{2}))\wedge\star_{7}\lambda
f_{\pm}\bigg{]},$ (C.17)

which also respects the isometries of AdS2.

Finally before moving on to address the issue of supersymmetry let us give a
frame that also respects the isometries of AdS2, as $e^{i}_{\pm}$ do not.
Defining $\hat{e}=\hat{\cal R}e_{+}$ for

$\hat{\cal R}=\left(\begin{array}[]{ccc}\cosh Y&-\sinh Y&0\\\ -\sinh Y&\cosh
Y&0\\\ 0&0&1\end{array}\right){\cal
R}^{-1},~{}~{}~{}~{}Y=\log\left(\frac{\Delta_{-}}{\Delta_{+}}\right),~{}~{}~{}~{}\Delta_{\pm}=1\pm\frac{e^{2A}s}{2r},$
(C.18)

where $\cal R$ is defined in (B.19), we arrive at

$\hat{e}^{i}=(\frac{e^{A}}{2\sqrt{\Delta}}\cosh
xdt,~{}\frac{e^{A}}{2\sqrt{\Delta}}dx,~{}e^{-A}dr)^{i}.$ (C.19)

The corresponding action on $d=10$ spinors (where $S$ is defined in (B.21)) is

$\hat{S}=e^{-\frac{1}{2}Y\Gamma_{\underline{0}\underline{1}}}S^{-1}\otimes\mathbbm{1}_{16},$
(C.20)

which will be useful in the next section.

##  Appendix D Proving SL(2) T-duality preserves supersymmetry

In this appendix we shall prove that SL(2) T-duality preserves supersymmetry.
We begin by deriving the reduced $d=7$ constraints for AdS3 vacua, which will
be necessary for proving that supersymmetry of the dual solutions is implied
by that of the original solutions.

###  D.1 Supersymmetry constraints for AdS3 solutions

In this appendix we derive reduced $d=7$ spinorial conditions that AdS3 vacua
of the form (2) must obey to be supersymmetric. As in the main text we focus
on those with purely magnetic NS 3-form, ie those for which $c_{0}=0$.

Given our choice of gamma matrices in (A.9) the $d=10$ spinors should
decompose as

$\displaystyle\epsilon_{1}$
$\displaystyle=\zeta\otimes\theta_{+}\otimes\chi_{1},\quad\epsilon_{2}=\zeta\otimes\theta_{\mp}\otimes\chi_{2},$
(D.1)

where $\zeta$ are real Killing spinors on AdS3 obeying the relation

$\displaystyle\nabla_{\mu}^{\text{AdS}_{3}}\zeta=\frac{s}{2}\gamma_{\mu}\zeta,~{}~{}~{}~{}s^{2}=1,$
(D.2)

and $\chi_{1,2}$ are independent Majorana spinors on $\text{M}_{7}$. The
objects $\theta_{\pm}$ are auxiliary 2d vectors which are needed to make the
dimensionality of the $d=10$ spinors correct when decomposed in terms of 3 and
7 dimensional ones. They are defined as

$\displaystyle\theta_{+}=\frac{1}{\sqrt{2}}\begin{pmatrix}1\\\
-i\end{pmatrix},\qquad\theta_{-}$
$\displaystyle=\frac{1}{\sqrt{2}}\begin{pmatrix}1\\\ i\end{pmatrix},$ (D.3)

such that $\epsilon_{1,2}$ are indeed Majorana-Weyl. The $\pm$ signs indicate
$d=10$ chirality and so as before the upper/lower signs are taken in Type
IIA/IIB. We also have the following identities which we will make much use of
in the coming computations

$\sigma_{1}\theta_{\pm}=\mp
i\theta_{\mp},~{}~{}~{}~{}\sigma_{3}\theta_{\pm}=\theta_{\mp},~{}~{}~{}~{}\sigma_{1}\sigma_{3}\theta_{\pm}=\pm
i\theta_{\pm}.$ (D.4)

Our task now is to insert our ansatz for the spinor into the necessary
conditions for supersymmetry (A.5a)-(A.5d). To achieve this we must first
decompose the $d=10$ spin covariant derivative in terms of the factors that
appear in the warped product. The curved space analogue of
$\Gamma_{\underline{\mu}}$ in (A.9) is

$\Gamma_{\mu}=e^{A}\gamma_{\mu}\otimes\sigma_{3}\otimes\mathbbm{1}_{8},$ (D.5)

as such we find that

$\displaystyle\nabla_{\mu}^{(10)}$
$\displaystyle=\nabla_{\mu}^{\text{AdS}_{3}}+\frac{1}{2}\Gamma_{\mu}dA,\quad$
$\displaystyle\mu={0,1,2},$ $\displaystyle\nabla_{a}^{(10)}$
$\displaystyle=\nabla_{a},\quad$ $\displaystyle a={3,..,9},$ (D.6)

where $\nabla_{a}$ is the spin covariant derivative on M7. Making use of (D.2)
and (D.4) and the identities

$F=(\mathbb{I}_{32}+\hat{\Gamma})f_{\pm},~{}~{}~{}~{}\lambda(F)=(\mathbb{I}_{32}\mp\hat{\Gamma})\lambda(f_{\pm}),$
(D.7)

which follow from (A.8), we find that (A.5a)-(A.5d) reduce to the following
$d=7$ conditions

$\displaystyle(se^{-A}-idA)\chi_{1}+\frac{1}{4}e^{\Phi}\beta_{\pm}f_{\pm}\chi_{2}=0,$
(D.8a) $\displaystyle(se^{-A}\pm
idA)\chi_{2}+\frac{1}{4}e^{\Phi}\beta_{\pm}^{*}\lambda(f_{\pm})\chi_{1}=0,$
(D.8b)
$\displaystyle(\nabla_{a}-\frac{1}{4}H_{a})\chi_{1}+\frac{1}{8}e^{\Phi}i\beta_{\pm}^{*}f_{\pm}\gamma_{a}\chi_{2}=0,$
(D.8c)
$\displaystyle(\nabla_{a}+\frac{1}{4}H_{a})\chi_{2}-\frac{1}{8}e^{\Phi}i\beta_{\pm}^{*}\lambda(f_{\pm})\gamma_{a}\chi_{1}=0,$
(D.8d)
$\displaystyle\left(\frac{3}{2}se^{-A}-i(\frac{3}{2}dA+\nabla+\frac{i}{4}H-d\Phi)\right)\chi_{1}=0,$
(D.8e) $\displaystyle\left(\frac{3}{2}se^{-A}\pm
i(\frac{3}{2}dA+\nabla+\frac{1}{4}H-d\Phi)\right)\chi_{2}=0.$ (D.8f)

where $\beta_{\pm}$ is a constant, specifically

$\displaystyle\beta_{+}=$ $\displaystyle 1,\quad\beta_{-}=i.$ (D.9)

###  D.2 U(1) T-duality on the Hopf fiber of AdS3

In this appendix we show that U(1) T-duality on the Hopf fiber of AdS3
preserves supersymmetry.

As first established in [43], U(1) T-duality preserves the portion of the
supercharges in the orginal solution that are not charged under the U(1). When
one performs a Hopf fiber T-duality on AdS3 then, we should identify $\zeta$
in (D.1) with the SL(2)R invariant Killing spinor $\zeta_{R}$ in (B.14). Given
an AdS3 solution of the form (2), in the conventions of (B) the result of
T-dualising on the Hopf fiber isometry $\partial_{r}$ is the following

$\displaystyle d\hat{s}^{2}$
$\displaystyle=\frac{e^{2A}}{4}ds^{2}(\text{AdS}_{2})+e^{-2A}dr^{2}+ds^{2}(\text{M}_{7}),~{}~{}~{}~{}e^{-\hat{\Phi}}=e^{-\Phi+A},$
$\displaystyle\hat{H}$
$\displaystyle=-\frac{s}{2}r\text{vol}(\text{AdS}_{2})+H,~{}~{}~{}~{}\hat{F}=t(f_{\pm}\wedge
dr\mp\frac{1}{4}e^{3A}\text{vol}(\text{AdS}_{2})\wedge\star_{7}\lambda
f_{\pm}),$ (D.10)

where again $s^{2}=t^{2}=1$. Here the dual RR flux is generated via
$e^{\hat{\Phi}}\hat{F}=e^{\Phi}F\Gamma_{\underline{r}}$ where we have split
the $\mu$ index as $\mu=(\mu_{2},r)$ where $\mu_{2}=0,1$ and runs over the
AdS2 directions.

A main result of [43] is the derivation of the precise form of the dual
Killing spinors in terms of the original one, when U(1) T-duality is
performed. The map is

$\epsilon_{1}\to\hat{\epsilon}_{1}=\epsilon_{1},~{}~{}~{}~{}\epsilon_{2}\to\hat{\epsilon}_{2}=\tilde{t}\Gamma_{\underline{r}}\epsilon_{2},~{}~{}~{}\tilde{t}^{2}=1,$
(D.11)

where we shall fix $\tilde{t}$ momentarily. For this to hold we must be in a
canonical frame in which the vielbein decomposes as a U(1) fibration over a
9-d base, ie we take the $d=10$ vielbein in the original solution to be

$e^{\underline{\mu}}_{10}=e^{A}e^{\underline{\mu}}_{3},~{}~{}~{}~{}e^{\underline{a}}_{10}=e^{\underline{a}}$
(D.12)

where $e^{\underline{\mu}}_{3}$ is defined in (B.12) and $e^{a}$ is a vielbein
on $\text{M}_{7}$. The dual frame on the other hand is

$\hat{e}^{\underline{\mu}_{2}}_{10}=\frac{e^{A}}{2}e^{\underline{\mu}_{2}}_{2},~{}~{}~{}~{}\hat{e}^{\underline{r}}_{10}=e^{-A}dr,~{}~{}~{}~{}\hat{e}^{\underline{a}}_{10}=e^{\underline{a}}$
(D.13)

where $e^{\underline{\mu}}_{2}$ is a vielbein on unit radius AdS2 (ie one can
take $e^{\underline{\mu}_{2}}_{2}=(\cosh xdt,dx)^{\underline{\mu}_{2}}$).

Our task is once more to derive the reduced $d=7$ conditions that follow from
considering (A.5a)-(A.5d) on the dual solution - note that T-duality maps a
solution in IIA to one in IIB so one needs to be careful with the signs. We
find that the $d=10$ covariant derivative now decomposes as

$\displaystyle\nabla_{\mu_{2}}^{(10)}$
$\displaystyle=\nabla_{\mu_{2}}^{\text{AdS}_{2}}+\frac{1}{2}\Gamma_{\mu_{2}}dA,~{}~{}~{}~{}\nabla_{r}^{(10)}=\partial_{r}-\frac{1}{2}\Gamma_{r}dA,~{}~{}~{}~{}\nabla_{a}^{(10)}=\nabla_{a}.$
(D.14)

The dual $d=10$ spinors decompose as

$\hat{\epsilon}_{1}=\zeta\otimes\theta_{+}\otimes\chi_{1},~{}~{}~{}~{}~{}\hat{\epsilon}_{2}=\Gamma_{\underline{r}}\epsilon_{2}=\gamma_{\underline{r}}\zeta\otimes\theta_{\pm}\otimes\chi_{2},$
(D.15)

if we then consider (A.5c) we find it decomposes as

$2e^{-A}\gamma^{\mu_{2}}\nabla^{\text{AdS}_{2}}_{\mu_{2}}\zeta\otimes\theta_{-}\otimes\chi_{1}-\zeta\otimes\theta_{-}\otimes\bigg{[}\frac{s}{2}e^{-A}+i(\nabla+\frac{3}{2}dA-\frac{1}{4}H-d\Phi)\bigg{]}\chi_{1}=0,$
(D.16)

which reproduces (D.8e) if

$\nabla_{\mu_{2}}^{\text{AdS}_{2}}\zeta=\frac{s}{2}\gamma_{\mu_{2}}\zeta,$
(D.17)

but this is precisely what the SL(2)R singlet spinor on AdS3 does obey. If we
now consider (A.5a) along the internal directions we find it reduces to

$(\nabla_{a}-\frac{1}{4}H_{a})\epsilon_{1}-t\tilde{t}\frac{e^{\Phi}}{16}F\Gamma_{a}\epsilon_{2}=0$
(D.18)

which reproduces (D.8c) if we fix

$t\tilde{t}=-1.$ (D.19)

Given this one finds that inserting the dual solution into the rest of
(A.5a)-(A.5d) reproduces all of (D.8a)-(D.8f). The key point however is that
only one of the 2 types of Killing spinors that AdS3 supports is preserved by
the duality, what we referred to as $\zeta_{R}$ in appendix B \- the other is
always projected out, ie what we earlier called $\zeta_{L}$ does not obey
(D.17).

###  D.3 SL(2) T-duality on AdS3

In this appendix we show that SL(2) T-duality performed on the SL(2)R isometry
of AdS3 preserves supersymmetry.

By analogy with the previous appendix and the results of [44] for SU(2)
T-duality, we make the following ansatz for how SL(2) T-duality acts on the
spinors of an AdS3 solution

$\hat{\epsilon}_{1}=\epsilon_{1},~{}~{}~{}~{}\hat{\epsilon}_{2}=\tilde{t}\Omega\epsilon_{2},$
(D.20)

where $\Omega$ is defined in (C.12). Like with U(1) T-duality, SL(2)R
T-duality has a canonical frame, namely the vielbein of the original solution
should take the form

$e^{\underline{\mu}}_{10}=e^{A}e^{\underline{\mu}}_{3,SL(2)},~{}~{}~{}~{}e^{\underline{a}}_{10}=e^{\underline{a}},$
(D.21)

where $e^{\underline{\mu}}_{3,SL(2)}$ is defined in (B.18). Importantly this
differs from the frame of U(1) T-duality in the previous appendix by a Lorentz
transformation as described in appendix B. This means that if we take our
original Killing spinors to be (D.1) and identify $\zeta=\zeta_{R}$ in the
U(1) T-duality frame, they are now given by

$\epsilon_{1}=S\zeta\otimes\theta_{+}\otimes\chi_{1},~{}~{}~{}~{}\epsilon_{2}=S\zeta\otimes\theta_{\mp}\otimes\chi_{2}$
(D.22)

in the frame (D.21). Likewise, as explained towards the end of appendix C, in
the canonical frame of SL(2) T-duality the isometries of AdS2 are not
manifest, to reach such a frame one needs to do another Lorentz tranformation
that will also act on (D.20). A dual frame that respects the isometries of
AdS2 has the $d=10$ vielbein

$\hat{e}^{\underline{\mu}_{2}}_{10}=\frac{e^{A}}{2\sqrt{\Delta}}e^{\underline{\mu}_{2}}_{2},~{}~{}~{}~{}\hat{e}^{\underline{r}}_{10}=e^{-A}dr,~{}~{}~{}~{}\hat{e}^{\underline{a}}_{10}=e^{\underline{a}},$
(D.23)

where we remind the reader of the following functions

$\Delta=1-\frac{e^{4A}}{4r^{2}},~{}~{}~{}~{}~{}\Delta_{\pm}=1\pm\frac{e^{2A}s}{2r}$
(D.24)

The combined actions of the Lorentz transformations required to reach this
frame mean that our dual spinors now take the form

$\displaystyle\hat{\epsilon}_{1}$
$\displaystyle=(N_{+}+N_{-}\gamma_{\underline{r}})\zeta\otimes\theta_{+}\otimes\chi_{1},~{}~{}~{}~{}\hat{\epsilon}_{2}=\tilde{t}\Gamma_{\underline{r}}(N_{+}-N_{-}\gamma_{\underline{r}})\zeta\otimes\theta_{\mp}\otimes\chi_{2},$
$\displaystyle N_{\pm}$
$\displaystyle=\frac{1}{2\Delta^{\frac{1}{4}}}(\sqrt{\Delta_{+}}\pm\sqrt{\Delta_{-}}),$
(D.25)

where the action of (C.20) is what maps the spinor between the frame of (D.21)
and (D.23), the respective $S$ and $S^{-1}$ terms canceling in the process.
The terms appearing in the $d=10$ supersymmetry conditions can also be written
in terms of $\Delta,\Delta_{\pm}$, after applying the Clifford map these
become

$\displaystyle\hat{H}$
$\displaystyle=H-2se^{-A}\Gamma^{\underline{0}\underline{1}\underline{r}}+\frac{2}{\Delta_{+}-\Delta_{-}}d\log(\Delta)\Gamma^{\underline{0}\underline{1}},$
$\displaystyle d\hat{\Phi}$ $\displaystyle=d\Phi-
dA-\frac{1}{2}d\log(\Delta)-s(\Delta_{+}-\Delta_{-})e^{-A}\Gamma^{\underline{r}},$
$\displaystyle e^{\hat{\Phi}}\hat{F}$
$\displaystyle=t\frac{e^{\Phi}}{\sqrt{\Delta}}(1+\Gamma)\left(f_{\pm}\Gamma^{\underline{r}}-\frac{1}{2}(\Delta_{+}-\Delta_{-})f_{\pm}\Gamma^{\underline{0}\underline{1}\underline{r}}\right).$
$\displaystyle e^{\hat{\Phi}}\lambda(\hat{F})$
$\displaystyle=t\frac{e^{\Phi}}{\sqrt{\Delta}}(1\pm\Gamma)\left(\Gamma^{\underline{r}}+\frac{1}{2}(\Delta_{+}-\Delta_{-})\Gamma^{\underline{0}\underline{1}\underline{r}}\right)\lambda(f_{\pm}).$
(D.26)

In this case the the spin covariant derivative decomposes as

$\nabla_{\mu_{2}}^{(10)}=\nabla_{\mu_{2}}^{\text{AdS}_{2}}+\frac{1}{2}\Gamma_{\mu_{2}}dA-\frac{1}{4}\Gamma_{\mu_{2}}d(\log(\Delta)),~{}~{}~{}~{}\nabla_{r}^{(10)}=\partial_{r}-\frac{1}{2}\Gamma_{r}dA,~{}~{}~{}~{}\nabla_{a}^{(10)}=\nabla_{a}.$
(D.27)

We also have the following useful identities

$\displaystyle d(N_{+}\pm
N_{-}\gamma_{\underline{r}})=\mp\frac{1}{2(\Delta_{+}-\Delta_{-})}d\log(\Delta)\gamma_{\underline{r}}(N_{+}\pm
N_{-}\gamma_{\underline{r}}),$ $\displaystyle
2N_{\pm}^{2}=\frac{1}{\sqrt{\Delta}}\pm 1,\qquad
N_{+}N_{-}=\frac{\Delta_{+}-\Delta_{-}}{4\sqrt{\Delta}}.$ (D.28)

We can now proceed with reducing the $d=10$ Killing spinor equations as
before. We first consider (A.5c) and find that the terms involving
$\log(\Delta)$ and $\partial_{r}$ mutually cancel leaving

$\displaystyle\bigg{(}\Gamma^{\mu_{2}}\nabla_{\mu_{2}}^{\text{AdS}_{2}}+\frac{3}{2}dA+\nabla^{\chi_{1}}-\frac{1}{4}H+\frac{s}{2}e^{-A}\Gamma^{\underline{0}\underline{1}\underline{r}}-d\Phi+s(\Delta_{+}-\Delta_{-})e^{-A}\Gamma^{\underline{r}}\bigg{)}\hat{\epsilon}^{1}=0,$
(D.29)

where $\nabla^{\chi_{1}}$ indicates that $\nabla$ only acts on $\chi_{1}$ in
this expression, not $N_{\pm}$. Through (D.3) it is possible to establish that

$\displaystyle\left((N_{+}-N_{-}\gamma_{\underline{r}})2\sqrt{\Delta}se^{-A}+s(\Delta_{+}-\Delta_{-})\gamma_{\underline{r}}(N_{+}+N_{-}\gamma_{\underline{r}})\right)=2se^{-A}(N_{+}+N_{-}\gamma_{\underline{r}}),$
(D.30)

so that (D.29) decomposes as

$\displaystyle
2\sqrt{\Delta}e^{-A}(N_{+}-N_{-}\gamma_{\underline{r}})(\gamma^{\mu_{2}}\nabla^{\text{AdS}_{2}}_{\mu_{2}}-s)\zeta\otimes\theta_{-}\otimes\chi_{1}$
$\displaystyle+(N_{+}+N_{-}\gamma_{\underline{r}})\zeta\otimes\theta_{-}\otimes\left(\frac{3}{2}se^{-A}-i(\frac{3}{2}dA+\nabla-\frac{1}{4}H-d\Phi)\right)\chi_{1}=0,$
(D.31)

where the first line vanishes if $\zeta=\zeta_{R}$ and the second reproduces
(D.8e). Assuming from now that indeed $\zeta=\zeta_{R}$, so that (D.17) holds,
with an analogous computation we find that (A.5d) reduces to

$\displaystyle\gamma_{\underline{r}}(N_{+}-\gamma_{\underline{r}}N_{-})\zeta\otimes\theta_{\mp}\otimes\left(\frac{3}{2}se^{-A}\pm
i(\frac{3}{2}dA+\nabla+\frac{1}{4}H-d\Phi)\right)\chi_{2}=0,$ (D.32)

which likewise reproduces (D.8f).

Turning our attention now to the part of (A.5a) along the internal directions
we find, after extensive use of the identities in (D.3), that it reduces to

$\displaystyle\left(N_{+}+N_{-}\gamma_{\underline{r}}\right)\zeta\otimes\theta_{+}\left(\left(\nabla_{a}-\frac{1}{4}H_{a}\right)\chi_{1}-t\tilde{t}\frac{e^{\Phi}}{8}i\beta_{\pm}^{*}f_{\pm}\gamma_{a}\chi_{2}\right)=0,$
(D.33)

which like the U(1) case reproduces (D.8c) if we fix

$t\tilde{t}=-1.$ (D.34)

Given this the remaining components of (A.5a), along respectively $r$ and
$\mu_{2}$ can be massaged to

$\displaystyle\Gamma_{\underline{r}}\left(N_{+}+N_{-}\gamma_{\underline{r}}\right)\zeta\otimes\theta_{+}\otimes\left(\left(e^{-A}s-idA\right)\chi_{1}+\frac{e^{\Phi}}{4}\beta_{\pm}f_{\pm}\chi_{2}\right)=0,$
(D.35)
$\displaystyle\Gamma_{\mu_{2}}\left(N_{+}(2-\sqrt{\Delta})-\gamma_{\underline{r}}N_{-}(2+\sqrt{\Delta})\right)\zeta\otimes\theta_{-}\otimes\left(\left(e^{-A}s-idA\right)\chi_{1}+\frac{1}{4}e^{\Phi}\beta_{\pm}f_{\pm}\chi_{2}\right)=0,$

which both reproduce (D.8a). Note that simplifying the second of these is more
challenging and the following identity is useful

$\displaystyle
d\log(\Delta)=\frac{\Delta-1}{\Delta}\left(4dA-2se^{-A}(\Delta_{+}-\Delta_{-})\Gamma^{\underline{r}}\right).$
(D.36)

Finally we consider (A.5b). Through similar means to before it is possible to
reduce this along each of $a,r,\mu_{2}$ to the form

$\displaystyle(N_{+}-\gamma_{\underline{r}}N_{-})\zeta\otimes\theta_{\pm}\otimes\left(\left(\nabla_{a}+\frac{1}{4}H_{a}\right)\chi_{2}-\frac{e^{\Phi}}{8}i\beta_{\pm}^{*}\lambda(f_{\pm})\gamma_{a}\chi_{1}\right)=0,$
$\displaystyle\Gamma_{r}(N_{+}-\gamma_{\underline{r}}N_{-})\zeta\otimes\theta_{\pm}\otimes\left((se^{-A}\pm
idA)\chi^{2}+\frac{1}{4}e^{\Phi}\beta_{\pm}^{*}\lambda(f_{\pm})\chi_{1}\right)=0,$
$\displaystyle\Gamma_{\mu_{2}}(N_{+}-\gamma_{\underline{r}}N_{-})\zeta\otimes\theta_{\pm}\otimes\left((se^{-A}\pm
idA)\chi^{2}+\frac{1}{4}e^{\Phi}\beta_{\pm}^{*}\lambda(f_{\pm})\chi_{1}\right)=0,$
(D.37)

which reproduce (D.8b) and (D.8d).

We have thus established that performing an SL(2)R T-duality on a
supersymmetric AdS3 solution that supports spinors invariant under SL(2)R
preserves supersymmetry. Just as with the U(1) case any spinors charged under
SL(2)R will be projected out.

##  Appendix E Details of the quiver construction

In this appendix we present some details of the construction of the 1d quivers
discussed in section 4.

As discussed in the main text, our Hanany-Witten brane set-up consists on
D-branes stretched between D(p+2)-branes, with orthogonal NS5-branes lying
between them, instead of the more conventional scenario in which the Dp-branes
are stretched between NS5-branes, with orthogonal D(p+2)-branes between them,
which allows to directly read the quiver field theory. In this situation in
order to obtain the quiver field theory we need to perform Hanany-Witten moves
that produce D-branes stretched between NS5-branes with D(p+2)-branes
orthogonal to them. We do this using that the D5-D7-NS5 and D1-D3-NS5 brane
subsystems in our brane set-up are related by T-duality to the D3-D5-NS5 brane
intersection studied in [47], and also to the D2-D4-NS5 two dimensional brane
intersection underlying the
$\text{AdS}_{3}\times\text{S}^{3}\times\mathbb{T}^{3}\times{\cal I}$ solutions
to Type IIA constructed in [20], from which our solutions are derived through
SL(2) T-duality. We will then follow closely [20] where the results in [47]
were used to describe the 2d quiver field theory living in this brane
intersection. In this appendix we basically rewrite the analysis therein
particularised to the D5-D7-NS5 and D1-D3-NS5 subsystems of the brane set-up
depicted in Table 1.

We start discussing the D5-D7-NS5 brane subsystem. Following [20] we use the
following definitions for the linking numbers

$\displaystyle l_{i}=n_{i}+L_{i}^{NS5},\qquad\text{for the D7-branes}$
$\displaystyle\hat{l}_{j}=-\hat{n}_{j}+R_{j}^{D7},\qquad\text{for the
NS5-branes},$

where $n_{i}$ is the number of D5-branes ending on the $i$th D7-brane from the
right minus the number of D5-branes ending on it from the left, $\hat{n}_{j}$
is the same quantity for the $j$th NS5-brane, $L_{i}^{NS5}$ is the number of
NS5-branes lying on the left of the $i$th D7-brane, and $R_{j}^{D7}$ is the
number of D7-branes lying on the right of the $j$th NS5-brane.

We proceed by computing the number of D5-branes that end on the left of a
collection of D7-branes and on the right of a collection of NS5-branes, given
by $N=\sum_{i=1}^{p}l_{i}=\sum_{j=1}^{\hat{p}}\hat{l}_{j}$, with $p$ and
$\hat{p}$ the numbers of D7 and NS5 branes. Here the partition
$N=\sum_{j=1}^{\hat{p}}\hat{l}_{j}$ has to be such that
$\hat{l}_{1}\geq\hat{l}_{2}\geq\dots\hat{l}_{\hat{p}}$. To read the quiver we
consider this partition plus the partition $N=\sum_{s=1}^{r}M_{s}q_{s}$,
constructed from a list of positive integers satisfying $q_{1}\geq
q_{2}\geq\dots q_{r}$, chosen such that the number of terms in the
decomposition that are equal or bigger than a given integer $j$, that we
denote as $m_{j}$, satisfy that

$\sum_{j=1}^{i}m_{j}\geq\sum_{j=1}^{i}\hat{l}_{j},\quad\text{for}\,\,\,i=1,\dots,\hat{p}.$
(E.1)

$M_{s}$ is then the number of times each integer $q_{s}$ appears in the
partition. From these two partitions the ranks of the gauge groups of the
quiver are computed as

$N_{i}=\sum_{j=1}^{i}(m_{j}-\hat{l}_{j}).$ (E.2)

In turn, $M_{s}$ give the ranks of the fundamental matter groups that couple
to each of the gauge groups. A very detailed account of this construction can
be found in [48].

We now apply this procedure to the brane set-up associated to our solutions,
that we terminate with $\beta_{P}$ anti NS5-branes at the end of the space, at
$\rho=P+1$. For the D5-D7-NS5 subsystem the resulting brane set-up is the one
depicted in Figure 12.

Figure 12: Brane set-up for the D5-D7-NS5 branes along the $\rho$ direction,
for $r$ constant, in units of $q$.

From this configuration the linking numbers are read exactly as in [20],
substituting the D2 for D5-branes and the D4 for D7-branes. Doing this, we
find for the D7-branes

$l_{i}=\sum_{r=0}^{i-2}\beta_{r}+2\beta_{i-1},\quad i=1,\dots,P$ (E.3)

and for the NS5-branes

$\displaystyle\hat{l}_{1}=\hat{l}_{2}=\dots=\hat{l}_{\beta_{0}}=P,$
$\displaystyle\hat{l}_{\beta_{0}+1}=\hat{l}_{\beta_{0}+2}=\dots=\hat{l}_{\beta_{0}+\beta_{1}}=P-1,$
$\displaystyle\hskip 113.81102pt\vdots$
$\displaystyle\hat{l}_{\beta_{0}+\beta_{1}+\dots+\beta_{P-3}+1}=\hat{l}_{\beta_{0}+\beta_{1}+\dots+\beta_{P-3}+2}=\dots=\hat{l}_{\beta_{0}+\beta_{1}+\dots+\beta_{P-2}}=2,$
$\displaystyle\hat{l}_{\beta_{0}+\beta_{1}+\dots+\beta_{P-2}+1}=\dots=\hat{l}_{\beta_{0}+\beta_{1}+\dots+\beta_{P-1}}=1,$
$\displaystyle\hat{l}_{\beta_{0}+\beta_{1}+\dots+\beta_{P-1}+1}=\dots=\hat{l}_{\beta_{0}+\beta_{1}+\dots+\beta_{P-1}+\beta_{P}}=1.$
(E.4)

From the linking numbers we construct the total number of D5-branes ending on
D7-branes on the left and NS5-branes on the right. This is given by

$N=\sum_{i=1}^{P}l_{i}=\sum_{j=1}^{\beta_{0}+\dots+\beta_{P}}\hat{l}_{j}=\sum_{k=0}^{P-1}(P-k+1)\beta_{k}.$
(E.5)

Now, from $N$ we define the two partitions that will allow us to read the
quiver CFT. The NS5-branes in our brane set-up are ordered such that
$\hat{l}_{1}\geq\hat{l}_{2}\geq\dots\geq\hat{l}_{\hat{\beta_{0}+\dots+\beta_{P}}}$.
These linking numbers define then one of the two partitions,
$N=\sum_{j=1}^{\beta_{0}+\dots+\beta_{P}}\hat{l}_{j}$. In turn, for the
D7-branes we take

$N=\underbrace{\beta_{0}}+\underbrace{\beta_{0}+\beta_{1}}+\underbrace{\beta_{0}+\beta_{1}+\beta_{2}}+\dots+\underbrace{\beta_{0}+\beta_{1}+\dots+\beta_{P-2}}+2\underbrace{(\beta_{0}+\beta_{1}+\dots+\beta_{P-1})}$
(E.6)

from which we find

$\displaystyle m_{1}=m_{2}=\dots=m_{\beta_{0}}=P+1,$ $\displaystyle
m_{\beta_{0}+1}=\dots=m_{\beta_{0}+\beta_{1}}=P,$ $\displaystyle\hskip
113.81102pt\vdots$ $\displaystyle
m_{\beta_{0}+\beta_{1}+\dots+\beta_{P-3}+1}=\dots=m_{\beta_{0}+\beta_{1}+\dots+\beta_{P-2}}=3,$
$\displaystyle
m_{\beta_{0}+\beta_{1}+\dots+\beta_{P-2}+1}=\dots=m_{\beta_{0}+\beta_{1}+\dots+\beta_{P-1}}=2.$
(E.7)

These numbers satisfy the condition (E.1) $\forall
i=1,\dots,(\beta_{0}+\dots+\beta_{P})$. We then find for the ranks of the
gauge groups

$\displaystyle N_{1}=m_{1}-\hat{l}_{1}=P+1-P=1,\quad
N_{2}=N_{1}+m_{2}-\hat{l}_{2}=2,\quad\dots\quad N_{\beta_{0}}=\beta_{0},$
$\displaystyle N_{\beta_{0}+1}=\beta_{0}+1,\quad\dots\quad
N_{\beta_{0}+\beta_{1}+\dots+\beta_{P-1}}=\beta_{0}+\beta_{1}+\dots+\beta_{P-1},$
(E.8)

to then start decreasing

$N_{\beta_{0}+\beta_{1}+\dots\beta_{P-1}+1}=\beta_{0}+\beta_{1}+\dots+\beta_{P-1}-1,\quad\dots\quad,N_{\beta_{0}+\beta_{1}+\dots\beta_{P-1}+\beta_{P}-1}=1.$
(E.9)

That is, the ranks of the gauge groups increase in units of 1 till the value
$\beta_{0}+\beta_{1}+\dots+\beta_{P-1}$ is reached, to then start decreasing,
again in units of one, till the gauge group of rank 1 is reached,
corresponding to the D5-branes stretched between the last pair of NS5-branes.

Finally, from the partition (E.6) we have that

$M_{\beta_{0}}=M_{\beta_{0}+\beta_{1}}=\dots=M_{\beta_{0}+\beta_{1}+\dots+\beta_{P-2}}=1,\quad
M_{\beta_{0}+\beta_{1}+\dots+\beta_{P-1}}=2.$ (E.10)

This implies that the gauge groups with ranks $\beta_{0}=\alpha_{1}$,
$\beta_{0}+\beta_{1}=\alpha_{2}$, till
$\beta_{0}+\dots+\beta_{P-2}=\alpha_{P-1}$ have U(1) flavour groups, while the
gauge group with rank $\beta_{0}+\beta_{1}\dots+\beta_{P-1}=\alpha_{P}$ has
flavour group U(2). The rest of gauge groups have no flavour groups attached.
The resulting quiver is depicted in Figure 13.

Figure 13: Quiver associated to the D5-D7-NS5 brane subsystem for $r$
constant, written in terms of 2d (4,4) multiplets. Circles denote (4,4) vector
multiplets and black lines (4,4) bifundamental hypermultiplets. The gauge
groups with ranks $\alpha_{k}$, with $k=1,\dots,P-1$ to the left of the gauge
group with rank $\alpha_{P}$ have U(1) flavour symmetries. The gauge group
with rank $\alpha_{P}$ has U(2) flavour symmetry. The rest of gauge groups do
not have attached any flavours. We have taken units of $q$.

In this quiver we have decomposed the 5d $\mathcal{N}=1$ vector multiplets and
hypermultiplets in terms of 1d (4,4) multiplets, which are the appropriate
ones to describe the D1-D3-NS5 subsystem and therefore the final quantum
mechanics that arises in our brane intersection. Indeed, it is clear that the
quiver associated to the D1-D3-NS5 brane subsystem of the brane set-up is
obtained from the one describing the D5-D7-NS5 subsystem scaling it by
$p=n+\frac{1}{2}$, given the D1 and D3 brane quantised charges found in (4.31)
and (4.29). This quiver is depicted in Figure 8 in subsection 4.3. Therefore,
our notation for both quivers is that circles denote (4,4) vector multiplets
(coming from the reduction of 5d $\mathcal{N}=1$ vector multiplets for the
D5-D7-NS5 subsystem) and black lines (4,4) bifundamental hypermultiplets
(coming from the reduction of 5d $\mathcal{N}=1$ bifundamentals for the
D5-D7-NS5 subsystem). These massless modes arise from the quantisation of the
open strings stretched between the D1-branes (or wrapped D5-branes) and
between the D1 and the D3 branes (or wrapped D5 and D7 branes), as follows:

  * •

D1-D1 strings: There are two cases to consider. Open strings with both ends on
the same stack of D1-branes between NS5-branes give rise to a (4,4) vector
multiplet, composed of a (0,4) vector multiplet plus a (0,4) adjoint twisted
hypermultiplet coming from the motion of the D1-branes along the
$(x^{6},x^{7},x^{8},x^{9})$ directions. Since these scalars are charged under
the R-symmetry they combine into a twisted hypermultiplet. In turn, the
strings with end points on adjacent stacks of D1-branes separated by an
NS5-brane contribute with a (4,4) bifundamental hypermultiplet, since the
intersection with the NS5-branes fixes the degrees of freedom along the
$(x^{6},x^{7},x^{8},x^{9})$ directions, leaving behind the scalars associated
to the directions along the torus plus the scalar arising from the KK
reduction of the 2d gauge field along the field theory direction, all of them
uncharged under the R-symmetry.

  * •

D1-D3 strings: Strings with one end on D1-branes and the other on D3-branes in