Marginally deformed AdS5/CFT4 Backgrounds in Type IIB

Paul<EMAIL_ADDRESS>

Department of Physics, Swansea University, Swansea SA2 8PP, United Kingdom

Abstract

Multi-parameter families of $\mathcal{N}=0$ Type IIA and Type IIB AdS5
solutions are presented, promoting to $\mathcal{N}=1$ in some special cases.
The G-Structure description of each $\mathcal{N}=1$ solution is given,
requiring an Abelian T-Duality of the G-Structure conditions and Pure Spinors.
Investigations at the boundaries are performed for a two-parameter family of
Type IIA and a three-parameter family of type IIB solutions, finding the
presence of orbifold singularities in some backgrounds. All parameters drop
out of the Holographic Central Charge calculation, pointing to marginal
deformations in the dual CFT description.

###### Contents

  1. 1 Introduction
  2. 2 Review of Type IIA reductions of Gaiotto-Maldacena
    1. 2.1 Gaiotto-Maldacena Background
    2. 2.2 G-Structure description and review
    3. 2.3 Gaiotto-Maldacena G-Structures
    4. 2.4 $SL(3,\mathds{R})$ Transformations
  3. 3 More Type IIA solutions
    1. 3.1 $\chi$ Reduction
      1. 3.1.1 Unique $\mathcal{N}=1$ IIA Solution
    2. 3.2 $\phi$ Reduction
  4. 4 Type IIB T-Duals
    1. 4.1 Three Parameter Family
    2. 4.1.1 $\mathcal{N}=1$ Type IIB - Background
    3. 4.1.2 $S^{2}$ preserved $\mathcal{N}=0$ Solutions
    4. 4.2 More Solutions
      1. 4.2.1 Unique $\mathcal{N}=1$ IIB Solution
  5. 5 Conclusions and future study
  6. A G-Structure Calculations
    1. A.1 Gaiotto-Maldacena
    2. A.2 G-Structure rules for Abelian T-Duality
    3. A.3 $\mathcal{N}=1$ G-Structures
      1. A.3.1 Unique Solutions
      2. A.3.2 One-parameter Type IIB
  7. B Dimensional Reductions
  8. C Abelian T-Duality
    1. C.1 Deriving the TST of NRSZ
  9. D TST deformations of Gaiotto-Maldacena
  10. E Brane rotation interpretation
  11. F Holographic Central Charge for Type IIB
  12. G Values of $f_{i}$ at the boundaries

##  1 Introduction

Since it’s inception by Maldacena in 1997 [1], the AdS/CFT correspondence has
proved very fruitful in the study of Supergravity backgrounds (with an AdSd+1
factor) and their dual CFTs (of $d$ spatial dimensions). In this paper we
focus our attention on AdS5 backgrounds of both Type IIA and Type IIB.

The work presented in this paper is complementary to the recent work of [2],
where a new two-parameter family of Type IIA solutions were derived via
dimensional reduction of the Gaiotto-Maldacena backgrounds [3], following an
$SL(3,\mathds{R})$ transformation amongst it’s three $U(1)$ directions. In
[2], it was shown in that these solutions give rise to $\mathcal{N}=(2,1,0)$
backgrounds, depending on the choice of the parameters. Central to this
analysis was the method of G-Structures, which we employ again in this work.
In the cases of the $SU(2)\times U(1)$ preserving $\mathcal{N}=0$, and
$U(1)\times U(1)$ preserving $\mathcal{N}=1$ solutions, it was shown that
these additional parameters change the quantization of D brane charge from
integer to rational. Following careful study of the boundaries, this was
nicely interpreted as branes back-reacted onto a Spindle and it’s higher
dimensional analogue, in the two cases respectively. The Holographic central
charge demonstrated that these transformation parameters drop out neatly,
suggesting that such deformations are in fact marginal in the dual CFT. Some
stability analysis of the backgrounds was then presented.

In this work we begin with the boundary analysis given in [2], but for the
full two-parameter family of solutions. We will show that both NS5 and D6
branes are present in all backgrounds, but D4 branes only appear for a
preserved $S^{2}$. In addition, by introducing a new term to the large gauge
transformation of $B_{2}$, we are able to eliminate the effect of the Spindle
on the D4 charge, restoring an integer quantization. This is of course not
possible for the D6 branes. However, as we will show in this paper, the D6
charge does have the same rational form for all of the solutions.

We then move on to investigate Type IIA backgrounds which are derived via
dimensional reductions along the other two $U(1)$ directions with respect to
the case of [2], and see that this gives rise to a few new and unique
solutions. Amongst these is a one-parameter family of $\mathcal{N}=0$
solutions which promotes to a new and unique zero-parameter $\mathcal{N}=1$
solution, when the parameter is fixed to zero. The G-Structure description of
this background is also given.

The main focus of this work then follows, investigating the Abelian T-Duality
(ATD) of the two-parameter family of Type IIA solutions given in [2]. This
derives a three-parameter family of Type IIB solutions, picking up the
additional non-trivial parameter in the T-Duality. This three-parameter
solution contains within it a new one-parameter family of $\mathcal{N}=1$ Type
IIB backgrounds. Throughout this analysis, the Supersymmetry is kept track of
by G-Structures. The transformation generating new backgrounds involved
performing an ATD of the G-Structure forms and conditions. This derived new
expressions for the Pure Spinors in Type IIB, playing a central role in the
analysis of the $\mathcal{N}=1$ solutions. This is presented in detail in an
Appendix.

We go on to investigate the boundaries of the three parameter family, and see
the presence of certain orbifold singularities. However, they do not appear to
be as straightforward as the Spindles found in [2]. It would be interesting to
study this topology further in the future. In analogy with the IIA analysis,
we find the presence of NS5 and D7 branes in all backgrounds. In the case of
the D5 branes, they only exist in the T-Dual solutions of the $S^{2}$
preserved IIA backgrounds. However, if the $S^{2}$ is broken under the ATD
itself, the D5 branes still remain. Once again, by including an additional
term in the gauge transformation of $B_{2}$, one can recover integer
quantization of D5 charge.

Finally, some additional Type IIB backgrounds are presented corresponding to
ATDs of different dimensional reductions. Amongst these we find a new and
unique zero-parameter family of $\mathcal{N}=1$ solutions (corresponding to
the ATD of the unique $\mathcal{N}=1$ Type IIA mentioned above). The
G-Structure forms are again included. Some topics are left for future study,
including the stability analysis conducted in [2]. By deriving the Holographic
central charge, we note that the deformations presented throughout this paper
are again marginal in the dual CFT, but do not recover the $\mathcal{N}=2$
solution in Supergravity.

The content of this paper is as follows:

  * •

In Section 2 we review the $\mathcal{N}=2$ Gaiotto-Maldacena solutions in
M-Theory. We include a brief summary of some relevant G-Structure theory,
setting up a discussion of the G-Structures specific to the Gaiotto-Maldacena
solutions, it’s IIA reductions and subsequent IIB ATD. The new material here
is the 11D G-Structure forms corresponding to a simple dimensional reduction
along the other two $U(1)$ directions to [2] (namely, $\chi$ and $\phi$), and
the IIB G-Structure Pure Spinors and conditions derived in an Appendix. We
conclude the section with a brief review of the $SL(3,\mathds{R})$
transformation being performed in M-Theory, along with the two-parameter
family presented in [2]. We then perform the boundary analysis for the full
two-parameter family, finding the presence of NS5 and D6 branes in all cases,
with the D4 branes only appearing for a preserved $S^{2}$.

  * •

In Section 3 we present some more IIA results, derived via dimensional
reductions along the alternative $U(1)$ directions. Most notable is the zero-
parameter $\mathcal{N}=1$ background. These results are largely just
presented, further analysis is left to future study. However, the G-Structure
forms are included for the $\mathcal{N}=1$ solution. The full calculations are
given in an Appendix.

  * •

In Section 4, we present the three-parameter family of Type IIB solutions. We
then study the boundary of the $(\sigma,\eta)$ space, following the analysis
which gave rise to the Spindle interpretation given in [2]. We again find the
presence of orbfold singularities, the topology of which would be nice to
study further. We find analogous behaviour with the IIA backgrounds, with the
existence of NS5 and D7 branes in all solutions. The D5 branes are only
present for descendants of the $S^{2}$ preserved IIA solutions, but do not
require a preserved $S^{2}$ themselves. We then present the one-parameter
family of $\mathcal{N}=1$ solutions and zone in on a one-parameter family of
$S^{2}$ preserved $\mathcal{N}=0$ backgrounds. In these examples, the charge
quantization is broken without the presence of an orbifold singularity in the
metric, a possible physical interpretation of this is suggested in an Appendix
- corresponding to rotated D-branes. Following this, some additional IIB
solutions are given, including a zero-parameter $\mathcal{N}=1$ background.
The G-structure forms and conditions are derived and presented at length in an
Appendix. Once again, further analysis is needed for these backgrounds, this
is left for future work. The full ATD calculations are again given at length
in an Appendix, and along the way, the TST solution of [4] is re-derived (and
argued to be $\mathcal{N}=0$ in general, promoting to $\mathcal{N}=1$ when
$\gamma=-1$). Finally in an Appendix, a one-parameter M-Theory deformation of
the Gaiotto-Maldacena solution is given, believed to break the Supersymmetry
completely. This is derived by reducing to IIA, performing a TST
transformation and uplifting. It proved highly useful to perform an ATD and
TST once for a general form, allowing one to easily derive the ATD solutions
of many backgrounds at once. These forms are again derived and presented in an
Appendix.

  * •

Finally, Section 5 gives some conclusions and an outline of future analysis
required. This work is complimented by lengthy Appendices which will hopefully
be of use.

##  2 Review of Type IIA reductions of Gaiotto-Maldacena

We begin with a brief review of the $\mathcal{N}=2$ AdS5 class of solutions
found by Gaiotto-Maldacena (GM) in $d=11$ [3], along with it’s G-Structure
description. The method of G-Structures allows one to easily keep track of the
Supersymmetry under reduction, a feature which will be further exploited
throughout this paper. We then proceed to review the $SL(3,\mathds{R})$
transformations and the corresponding two parameter family of dimensional
reductions discussed in [2], finishing the section by investigating the
boundary.

###  2.1 Gaiotto-Maldacena Background

If one begins with the more general AdS5 $\mathcal{N}=2$ solutions of Lin-
Lunin-Maldacena (LLM) [5], with bosonic isometry group $SO(4,2)\times
SU(2)_{R}\times U(1)_{r}$222Where $SU(2)_{R}\times U(1)_{r}$ is the
$\mathcal{N}=2$ R-Symmetry. and magnetic four-form, $G_{4}=dC_{3}$, one can
derive the following Electrostatic form of the Gaiotto-Maldacena Background
via the B$\ddot{\text{a}}$cklund transformation333Noting that the GM U(1)
directions $(\chi,\beta)$ and the LLM U(1) directions
$(\bar{\chi},\bar{\beta})$ are related as follows
$(\chi,\beta)\rightarrow(\bar{\chi}+\bar{\beta},~{}-\bar{\beta}).$ (2.1) This
is discussed in [6, 7, 8] and shown in detail in [2].. See [2] for a more in
depth review. The resulting metric and four form are then defined as follows

$\begin{gathered}ds_{11}^{2}=f_{1}\Bigg{[}4ds^{2}(\text{AdS}_{5})+f_{2}ds^{2}(\text{S}^{2})+f_{3}d\chi^{2}+f_{4}\big{(}d\sigma^{2}+d\eta^{2}\big{)}+f_{5}\Big{(}d\beta+f_{6}d\chi\Big{)}^{2}\Bigg{]},\\\
C_{3}=\Big{(}f_{7}d\chi+f_{8}d\beta\Big{)}\,\wedge\text{Vol}(\text{S}^{2}),\\\
\\\
f_{1}=\kappa^{\frac{2}{3}}\bigg{(}\frac{\dot{V}\tilde{\Delta}}{2V^{\prime\prime}}\bigg{)}^{\frac{1}{3}},~{}~{}~{}~{}f_{2}=\frac{2V^{\prime\prime}\dot{V}}{\tilde{\Delta}},~{}~{}~{}~{}~{}~{}~{}~{}f_{3}=\frac{4\sigma^{2}}{\Lambda},~{}~{}~{}~{}~{}~{}~{}~{}f_{4}=\frac{2V^{\prime\prime}}{\dot{V}},~{}~{}~{}~{}~{}~{}~{}~{}f_{5}=\frac{2\Lambda
V^{\prime\prime}}{\dot{V}\tilde{\Delta}},\\\
f_{6}=\frac{2\dot{V}\dot{V}^{\prime}}{V^{\prime\prime}\Lambda},~{}~{}~{}~{}f_{7}=-\frac{4\kappa\dot{V}^{2}V^{\prime\prime}}{\tilde{\Delta}},~{}~{}~{}~{}~{}~{}~{}f_{8}=2\kappa\bigg{(}\frac{\dot{V}\dot{V}^{\prime}}{\tilde{\Delta}}-\eta\bigg{)},\\\
\tilde{\Delta}=\Lambda(V^{\prime\prime})^{2}+(\dot{V}^{\prime})^{2},~{}~{}~{}~{}~{}~{}~{}~{}~{}\Lambda=\frac{2\dot{V}-\ddot{V}}{V^{\prime\prime}},\end{gathered}$
(2.2)

where the warp factors $f_{i}=f_{i}(\eta,\sigma)$ depend on a constant
$\kappa$ and the function $V(\eta,\sigma)$, with

$\dot{V}\equiv\sigma\partial_{\sigma}V,~{}~{}~{}~{}~{}~{}~{}~{}V^{\prime}\equiv\partial_{\eta}V.$
(2.3)

The Supergravity equations are satisfied by imposing the following
cylindrically symmetric Laplace equation for $V(\eta,\sigma)$

$\frac{1}{\sigma}\partial_{\sigma}(\sigma\partial_{\sigma}V)+\partial_{\eta}^{2}\equiv\ddot{V}+\sigma^{2}V^{\prime\prime}=0,$
(2.4)

along with the Boundary Conditions [4] (where $0<\sigma<\infty$ and
$0<\eta<P$, with $P$ finite)

$\dot{V}\Big{|}_{\eta=0,P}=0,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\dot{V}\Big{|}_{\sigma=0}=\mathcal{R}(\eta).$
(2.5)

One now has an Electrostatic description of the background (2.2), where a line
of charge is extended along $\eta$. The charge density is defined by the “Rank
Function” of the dual quiver, $\mathcal{R}(\eta)$, with Fourier decomposition

$\mathcal{R}(\eta)=\sum_{n=1}^{\infty}\mathcal{R}_{n}\sin\bigg{(}\frac{n\pi}{P}\eta\bigg{)},$
(2.6)

which for quantized fluxes, must be a continuous, linear-by-pieces and convex
function. Hence, in general, it takes the form

$\mathcal{R}(\eta)=\begin{cases}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}N_{1}\eta&\eta\in[0,1]\\\
N_{k}+(N_{k+1}-N_{k})(\eta-k)&\eta\in[k,k+1]\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}N_{P-1}(P-\eta)&\eta\in[P-1,P]\end{cases}$ (2.7)

where $\mathcal{R}(0)=\mathcal{R}(P)=0$. One can find an appropriate solution
to (2.4) using the separation of variables,

$V(\sigma,\eta)=-\sum_{n=1}^{\infty}\mathcal{R}_{n}\sin\bigg{(}\frac{n\pi}{P}\eta\bigg{)}K_{0}\bigg{(}\frac{n\pi}{P}\sigma\bigg{)},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mathcal{R}_{n}=\frac{1}{P}\int_{-P}^{P}\mathcal{R}(\eta)\sin\bigg{(}\frac{n\pi}{P}\eta\bigg{)}d\eta,$
(2.8)

which indeed satisfies the boundary conditions (2.5). Here, $K_{0}$ is a
modified Bessel function of the second kind.

It proves useful when investigating the boundary to introduce an alternative
form for $\dot{V}$, following the arguments of [2], and shown to be equivalent
in [9]

$\displaystyle\dot{V}$ $\displaystyle=\frac{\pi}{P}\sum_{n=1}^{\infty}{\cal
R}_{n}\sigma\sin\bigg{(}\frac{n\pi}{P}\eta\bigg{)}K_{1}\bigg{(}\frac{n\pi}{P}\sigma\bigg{)},$
(2.9)
$\displaystyle=\frac{1}{2}\sum_{m=-\infty}^{\infty}\sum_{k=1}^{P}b_{k}\left(\sqrt{\sigma^{2}+(\eta-2mP+k)^{2}}-\sqrt{\sigma^{2}+(\eta-2mP-k)^{2}}\right),$

where $b_{k}=2N_{k}-N_{k+1}-N_{k-1}$ (see [2] for a more in depth discussion).

We now summarise some key material on the method of G-Structures.

###  2.2 G-Structure description and review

In order to easily keep track of the Supersymmetry when dimensionally reducing
$\mathcal{N}=2$ Gaiotto-Maldacena to Type IIA, it proves highly useful to use
the method of G-Structures. We now provide a brief review, with a focus on the
GM description used throughout this paper. This will set up the next section,
where we discuss the G-Structure formalism for the Gaiotto-Maldacena
background, it’s IIA reduction and subsequent IIB T-Dual solution.

Supersymmetric solutions of M-Theory or Type II Supergravity must satisfy the
necessary Supersymmetry conditions, usually given in terms of a spinor and
metric, as well as the Bianchi identities444Where we follow the convention
that $d_{H}$ is defined with a minus, as follows  $d_{H}\equiv d-H\wedge\,.$

$\begin{gathered}\partial_{\epsilon}\psi_{M}=0,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\partial_{\epsilon}\lambda=0,\\\
d_{H}F=0,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}dH_{3}=0,\end{gathered}$ (2.10)

where $\psi_{m}$ are the Gravitinos, $\lambda$ the Dilatini and $\epsilon^{a}$
are two 2D Majorana-Weyl spinors (with $a=(1,2)$).

The technique of G-structures allows one to recast the above Supersymmetry
equations (involving a spinor and metric) in terms of non-spinorial and
geometrical objects (forms). For our purposes, we will focus on the
Supersymmetry conditions for Gaiotto-Maldacena and it’s daughter Supergravity
solutions, namely Mink${}_{4}\times M^{7}$ and Mink${}_{4}\times M^{6}$
backgrounds. The interested reader is directed to [10, 11, 12, 13, 14, 15, 16,
17] for more detailed and insightful discussions on the topic.

In this discussion, we are interested in the internal (compact) space, writing
the metric as a Warp product. In the Mink${}_{4}\times$M7 M-Theory case, we
have

$ds_{11}^{2}=e^{2\hat{A}}ds^{2}(\text{Mink}_{4})+ds^{2}(M_{7}),$ (2.11)

where $\hat{A}$ is a function of the internal space, $M_{7}$, coordinates. We
decompose the spinors on the non-compact/compact parts of the space, as
$\epsilon=\psi_{+}\otimes\theta_{+}+\psi_{-}\otimes\theta_{-}$ (following the
notation of [13]) where $\theta$ are 7 dimensional spinors, with
$\theta_{+}=\theta_{-}^{*}$.

In the case of Mink${}_{4}\times$M6 Supergravity, we have

$ds_{10}^{2}=e^{2A}ds^{2}(\text{Mink}_{4})+ds^{2}(M_{6}),$ (2.12)

where $A$ is a function of the $M_{6}$ coordinates. In addition, the fields
depend only on the internal space coordinates, with the Poly-form

$\begin{gathered}F=g+e^{4A}\text{Vol}(\text{Mink}_{4})\wedge*_{6}\lambda(g),\\\
\text{Type IIA:
}~{}~{}~{}~{}~{}~{}~{}~{}~{}F_{tot}=\sum_{j=0}^{5}F_{2j},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\lambda(g)=g_{0}-g_{2}+g_{4}-g_{6},\\\
\text{Type IIB:
}~{}~{}~{}~{}~{}~{}~{}~{}~{}F_{tot}=\sum_{j=0}^{4}F_{2j+1},~{}~{}~{}~{}~{}~{}~{}~{}~{}\lambda(g)=g_{1}-g_{3}+g_{5}.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\end{gathered}$
(2.13)

Decomposing the spinor yields555In IIA, $\epsilon^{1}$ has + chirality and
$\epsilon^{2}$ has - chirality. In IIB, they are both + chirality. (adopting
the notation given in [10])

$\begin{gathered}\text{Type IIA:
}~{}~{}~{}~{}~{}~{}~{}~{}~{}\epsilon^{1}=\zeta_{+}\otimes\eta_{+}^{1}+\zeta_{-}\otimes\eta_{-}^{1},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\epsilon^{2}=\zeta_{+}\otimes\eta_{-}^{2}+\zeta_{-}\otimes\eta_{+}^{2},\\\
\text{Type IIB:
}~{}~{}~{}~{}~{}~{}~{}~{}~{}\epsilon^{i}=\zeta_{+}\otimes\eta_{+}^{i}+\zeta_{-}\otimes\eta_{-}^{i},\end{gathered}$
(2.14)

where $\eta$ lives on $M_{6}$. Plugging these decompositions back into the
SUSY conditions given in (2.10), allows one to factor out $\zeta$ \- leading
to six fairly ugly equations in terms of $\eta^{1,2}$. One can however re-
write such conditions in a more elegant manner in the language of generalized
complex geometry and differential forms. The spinor $\eta$ naturally defines a
G-Structure as it’s stabilizer group (the group of rotations which keep $\eta$
invariant). In addition, a G-Structure is typically equivalent to a set of
invariant tensors, allowing one to swap the spinor with non-spinorial objects
(forms) - this is the upshot of the method of G-Structures.
The Supersymmetry conditions can be split into 2 parts (see for example [10]
for a nice discussion)

  * •

Algebraic part: The existence of a non-vanishing, globally well defined
spinor, $\eta$. This is a topological requirement on the manifold666That
$T\otimes T^{*}$ must have an $SU(3)\times SU(3)$ structure.. This implies the
existence of two nowhere vanishing, globally defined Clifford$(6,6)$ pure
spinors, $\Psi_{+},\Psi_{-}$

$\Psi_{+}\equiv\eta_{+}^{1}\otimes\eta_{+}^{2\dagger},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\Psi_{-}\equiv\eta_{+}^{1}\otimes\eta_{-}^{2\dagger}.$
(2.15)

These are sums of even or odd forms, respectively.

  * •

Differential part: The spinor, $\eta$, is covariantly constant, defining the
Killing Spinor equation. The amount of Supersymmetry depends on the number of
spinors satisfying this equation, with the R-Symmetry of the theory (on the
non-compact space) rotating the Killing Spinors into each other. The more
Supersymmetry, the more structure is required on the manifold. This is a
differential condition on the metric, saying that the two spinors should
satisfy

$e^{-2A+\Phi}d_{H}(e^{2A-\Phi}\Psi_{1})=0,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}e^{-2A+\Phi}d_{H}(e^{2A-\Phi}\Psi_{2})=dA\wedge\bar{\Psi}_{2}+F,$
(2.16)

with an additional normalization requirement. These conditions contain the
same information as the Supersymmetry variations from (2.10). Once again, to
find a vacuum, they must be supplemented with the Bianchi identities and
equations of motion for the fluxes.
Writing this more schematically as in [18], we have

$\begin{gathered}d_{H_{3}}\Psi_{1}=0,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}d_{H_{3}}\Psi_{2}=F_{RR},\end{gathered}$
(2.17)

with

$\text{Type
IIA:}~{}~{}~{}~{}~{}~{}\Psi_{1}=\Psi_{+},~{}\Psi_{2}=\Psi_{-},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{Type
IIB:}~{}~{}~{}~{}~{}~{}\Psi_{1}=\Psi_{-},~{}\Psi_{2}=\Psi_{+},$ (2.18)

given that $F_{RR}$ has an even form in IIA and an odd form in IIB. Clearly
the pure spinors must transform in essentially the same manner under T-Duality
as the Ramond fields, this will become important to us later.

#### Example Structures

We now summarise some key (and relevant) examples. See [10][15][16] for
further details.

  * •

$SU(3)$ Structure
An $SU(3)$ structure on an $M_{6}$ can be defined in three ways: via a metric
and spinor $(g,\eta)$; a metric and a complex, decomposable and non-degenerate
3-form $(g,\Omega)$; or a real 2-form and the complex 3-form $(J,\Omega)$.
Hence there is a bijection between the descriptions, notably
$(g,\eta)\leftrightarrow(J,\Omega)$777Using the Fierz identities, one gets the
following mapping
$\Omega_{mnp}=-\eta_{-}^{\dagger}\gamma_{mnp}\eta_{+},~{}~{}~{}~{}~{}~{}~{}\eta_{-}=(\eta_{+})^{c},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}J_{mn}=-i\,\eta_{+}^{\dagger}\gamma_{mn}\eta_{+}.$
. The following conditions are necessary

$J\wedge\Omega=0,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}J\wedge J\wedge
J=\frac{3}{4}i\,\Omega\wedge\bar{\Omega},$ (2.19)

with the positive definite metric defined as $g=-JI_{\Omega}$.888Where
$I_{\Omega}$ is an Almost Complex Structure - a tensor $I^{m}_{n}$ with
$I^{2}=-1$. This allows one to embed the $GL(d/2,\mathds{C})\in
GL(d,\mathds{R})$. The stabilizer group is given as follows
$\text{Stab}_{GL(d,\mathds{R})}(I)=GL(d/2,\mathds{C}),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{Stab}(I)=O(d)\,\cap\,GL(d/2,\mathds{C})\cong
U(d/2)$ so adding a metric on $M_{6}$ means the stabilizer group of $I$ is
$U(3)$. One then defines $J\equiv gI$, or alternatively, $g=-JI$. To move to
an $SU(3)$ one requires a nowhere vanishing holomorphic 3-form $\Omega$, with
$\Omega\rightarrow\text{det}(U)\Omega$ and $det(U)=1$. See [15] for further
details.

One can then build the normalized pure spinors, as follows

$\Psi_{+}=\eta_{+}\otimes\eta_{+}^{\dagger}=\frac{1}{8}e^{-iJ},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\Psi_{-}=\eta_{+}\otimes\eta_{-}^{\dagger}=-\frac{i}{8}\Omega.$
(2.20)

To generalize this to an $SU(3)$ structure on an $M_{7}$, one must add a real
one-form, $K$, where

$ds_{7}^{2}=ds_{6}^{2}+K^{2},$ (2.21)

so an $SU(3)$ structure on an $M_{7}$ is described by $(J,\Omega,K)$. This is
the case for Gaiotto-Maldacena, we will return to this case in the next
section.

  * •

$SU(d)$ in $d=$ even Structure
More generally, this is extended to $SU(d/2)$ in $d=$ even dimensions. Here
the structure is specified by a real 2-form $J$ and a complex $d/2$ form,
$\Omega$

$\Omega=E^{1}\wedge...\wedge
E^{\frac{d}{2}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}J=\frac{i}{2}\sum_{a=1}^{d/2}E^{a}\wedge\bar{E}^{a},$
(2.22)

where $E^{a}$ is a holomorphic vielbein $E^{a}=e^{a}+i\,e^{a+d/2}$, and the
metric is defined by

$g=\sum_{a=1}^{d/2}E^{a}\bar{E}^{a}.$ (2.23)

As a noteworthy example, an $SU(2)$ structure on $M_{4}$ is defined by a real
2-form, $j$, and complex 2-form, $\omega$. The following conditions must then
be met

$j\wedge\omega=\omega\wedge\omega=0,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\omega\wedge\bar{\omega}=2j\wedge
j.$ (2.24)

To generalize to an $SU(2)$ structure on an $M_{6}$, one must now add a
complex one-form, $z$, with

$ds_{6}^{2}=ds_{4}^{2}+z\bar{z},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}z\equiv
u+i\,v.$ (2.25)

The pure spinors are then constructed as follows

$\Psi_{+}=\frac{1}{8}e^{\frac{1}{2}z\wedge\overline{z}}\wedge\omega,~{}~{}~{}~{}~{}\Psi_{-}=\frac{i}{8}e^{-ij}\wedge
z.$ (2.26)

This description will become relevant when dimensionally reducing Gaiotto-
Maldacena, once again, we return to this in the next section.

  * •

$SU(3)\times SU(3)$ Structure
Having two spinors, $\eta^{1,2}$, describes an $SU(3)\times SU(3)$ structure.
When $\eta^{1}$ and $\eta^{2}$ are parallel, they describe an $SU(3)$
structure. When they are nowhere parallel, they define what is called a static
$SU(2)$ structure. In general, we have

$\Psi_{+}=\eta_{+}^{1}\otimes\eta_{+}^{2\dagger}=\frac{1}{8}e^{\frac{1}{2}z\wedge\bar{z}}\wedge(\bar{c}e^{-ij}-i\omega),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\Psi_{-}=\eta_{+}^{1}\otimes\eta_{-}^{2\dagger}=-\frac{1}{8}(e^{-ij}+ic\,\omega)\wedge
z.$ (2.27)

Hence, for static $SU(2)$, one derives (2.26). For $SU(3)$, one gets (2.20)
with

$J=j+\frac{i}{2}z\wedge\bar{z},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\Omega=\omega\wedge
z.$ (2.28)

###  2.3 Gaiotto-Maldacena G-Structures

We will now discuss the G-Structure formalism for the Gaiotto-Maldacena
background, which describes an SU(3) structure on an $M_{7}$. The
$\mathcal{N}=1$ G-Structure conditions for Mink4 vacua of the form given in
(2.11) were derived in [13]. They are defined by a real 2-form, $J$, a
holomorphic 3-form, $\Omega$, (giving an SU(3) structure on $d=6$), with an
additional (orthogonal and unit normed) real one-form, $K$ (to move to $d=7$).
The G-Structure conditions then read

$\displaystyle
d(e^{2\hat{A}}K)=0,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}d(e^{4\hat{A}}J)=e^{4\hat{A}}\star_{7}G_{4},$
(2.29) $\displaystyle
d(e^{3\hat{A}}\Omega)=0,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}d(e^{2\hat{A}}J\wedge
J)=-2e^{2\hat{A}}G_{4}\wedge K,$

where $e^{\hat{A}}$ and $G_{4}$ live on $M_{7}$, with the condition given in
(2.19). One must then impose the Bianchi identities separately. These
conditions allow $(J,\Omega)$ to be written in the manner outlined in (2.22),
written here explicitly for clarity

$\Omega=E^{1}\wedge E^{2}\wedge
E^{3},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}J=\frac{i}{2}(E^{1}\wedge\overline{E}^{1}+E^{2}\wedge\overline{E}^{2}+E^{3}\wedge\overline{E}^{3}),$
(2.30)

where the complex vielbein, $E^{a}$, is orthogonal to $K$ (with $a=1,2,3$).
Hence, for the GM background, using

$ds^{2}(AdS_{5})=e^{2\rho}ds^{2}(\text{Mink}_{4})+d\rho^{2},$ (2.31)

we have (recalling (2.11), (2.21) and (2.23))

$\displaystyle ds^{2}$
$\displaystyle=e^{2\hat{A}}ds^{2}(\text{Mink}_{4})+ds_{7}^{2},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}e^{2\hat{A}}=4f_{1}e^{2\rho},$
(2.32) $\displaystyle ds_{7}^{2}$
$\displaystyle=\sum_{a=1}^{3}E^{a}\bar{E}^{\bar{a}}+K^{2}=f_{1}\bigg{[}4d\rho^{2}+f_{2}ds^{2}(S^{2})+f_{3}d\chi^{2}+f_{4}(d\sigma^{2}+d\eta^{2})+f_{5}(d\beta+f_{6}d\chi)^{2}\bigg{]}.$

For the $\beta$ reduction frame, one derives the following results (given in
[2])999Where the reduction frame can be read off directly from $E_{3}$ using
the reduction formula
$e^{-\frac{2}{3}\Phi_{IIA}}ds_{IIA}^{2}=ds^{2}-e^{\frac{4}{3}\Phi_{IIA}}(d\psi+C_{1})^{2}.$
In the case of (2.3), $\psi=\beta$, $C_{1}=f_{6}d\chi$ and
$e^{\frac{4}{3}\Phi_{IIA}}=f_{1}f_{5}$.

$\beta$ reduction frame $\displaystyle
K=\frac{\kappa\,e^{-2\rho}}{f_{1}}d(\cos\theta
e^{2\rho}\dot{V}),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}E_{1}=-\sqrt{f_{1}f_{3}}\bigg{(}\frac{1}{\sigma}d\sigma+d\rho+id\chi\bigg{)},$
$\displaystyle
E_{2}=e^{i\phi}\bigg{[}\frac{\kappa}{f_{1}}e^{-2\rho}d(\sin\theta
e^{2\rho}\dot{V})+i\,\sqrt{f_{1}f_{2}}\sin\theta d\phi\bigg{]},$ (2.33)
$\displaystyle
E_{3}=-e^{i\chi}\sqrt{f_{1}f_{5}}\,\Bigg{[}-\frac{1}{4}f_{3}\frac{\dot{V}^{\prime}}{\sigma}d\sigma-V^{\prime\prime}d\eta+f_{6}d\rho+i\Big{(}d\beta+f_{6}d\chi\Big{)}\Bigg{]},$

where $K$ and $E_{2}$ are independent of $(\beta,\chi)$, so remain constant
under a rotation to the $\chi$ reduction frame. The $(E_{1},E_{3})$ vielbeins
in this frame then read (now labelled with a tilde for clarity)

$\chi$ reduction frame
$\displaystyle\tilde{E}_{1}=\sqrt{\frac{f_{1}f_{3}}{f_{6}^{2}+\frac{f_{3}}{f_{5}}}}\bigg{(}d(V^{\prime})-i\,d\beta\bigg{)},$
(2.34)
$\displaystyle\tilde{E}_{3}=-e^{i\chi}\sqrt{f_{1}f_{5}\Big{(}f_{6}^{2}+\frac{f_{3}}{f_{5}}\Big{)}}\,\Bigg{[}d\rho-\frac{V^{\prime\prime}}{(\dot{V}^{\prime})^{2}-\ddot{V}V^{\prime\prime}}d(\dot{V})+i\,\bigg{(}d\chi+\frac{f_{6}}{f_{6}^{2}+\frac{f_{3}}{f_{5}}}d\beta\bigg{)}\Bigg{]},$

where, from the conditions in (2.30) and (2.32), one can check that the
following relations indeed hold

$\displaystyle E_{1}\wedge\bar{E_{1}}+E_{3}\wedge\bar{E_{3}}$
$\displaystyle=\tilde{E}_{1}\wedge\bar{\tilde{E}}_{1}+\tilde{E}_{3}\wedge\bar{\tilde{E}}_{3},$
(2.35) $\displaystyle E_{1}\wedge E_{3}$
$\displaystyle=\tilde{E}_{1}\wedge\tilde{E}_{3},$ $\displaystyle
E_{1}\bar{E_{1}}+E_{3}\bar{E_{3}}$
$\displaystyle=\tilde{E}_{1}\bar{\tilde{E}}_{1}+\tilde{E}_{3}\bar{\tilde{E}}_{3}.$

For a final rotation to the $\phi$ reduction frame, $K$ is independent of
$\phi$, so once again remains untouched. The $E_{2}$ given in (2.3) does
however depend on $\phi$, and has the appropriate form for a $\phi$ reduction
$E_{3}$ (with $C_{1}=0$). Hence, performing a trivial relabelling of the
vielbeins, this $E_{2}$ now becomes the $E_{3}$ for the $\phi$-reduction
frame. The $E_{1}$ and $E_{2}$ are then simply either $(E_{1},E_{3})$ from
(2.3) or $(\tilde{E}_{1},\tilde{E}_{3})$ from (2.3). See Appendix A.1 for
further details.

It is worth noting, using (2.30), (2.3) and (2.3), we can read off the
$U(1)_{r}$ component of the $SU(2)_{R}\times U(1)_{r}$ R-Symmetry directly
from the overall phase of $\Omega$, namely $\chi+\phi$. Hence, as it stands,
the only KK reduction which will preserve any Supersymmetry is a $\beta$
reduction. This in fact leads to the Type IIA solution given in [4], and is
the only background which preserves the full $\mathcal{N}=2$ Supersymmetry.
However, as discussed in [2], by performing an $SL(3,\mathds{R})$
transformation prior to reduction, further possibilities emerge (including the
additional $\mathcal{N}=1$ Type IIA and Type IIB solutions which we uncover in
this paper).

Before reviewing this $SL(3,\mathds{R})$ transformation, we must first discuss
the G-Structure analysis of the Type IIA and Type IIB daughter backgrounds.

#### IIA reduction

Now we turn to the G-Structure description of the daughter Type IIA
backgrounds. By performing a dimensional reduction along a $U(1)$ which lies
strictly outside $K$, the resulting theory has an SU(2) structure on an
$M_{6}$. In this case, the G-structure analysis will be defined by a real two-
form, $j$, a holomorphic two-form, $\omega$, (defining an SU(2) structure on
$d=4$), with an additional complex one-form, $z$ (to move to $d=6$). Hence,
the G-structure analysis of the Supersymmetric background outlined here will
be defined by $(j,\omega,z)$101010If the dimensional reduction was performed
along $K$, the background would define an $SU(3)$ on $M_{6}$, described by
$(J,\Omega)$ from (2.20) - namely, a real 2-form and holomorphic 3-form,
respectively. .

These forms can easily be calculated from the results of the previous section
using the standard reduction formula111111 One could of course use the
generalised reduction formula (for reductions along $n\psi$, with
$n\in\mathds{Z}$),
$e^{-\frac{2}{3}\Phi_{IIA}}ds_{IIA}^{2}=ds^{2}-\frac{1}{n^{2}}e^{\frac{4}{3}\Phi_{IIA}}(n\,d\psi+n\,C_{1})^{2},$
where one can easily see  $C_{1}\rightarrow
nC_{1},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}e^{\frac{4}{3}\Phi}\rightarrow\frac{1}{n^{2}}e^{\frac{4}{3}\Phi},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}B_{2}\rightarrow\frac{1}{n}B_{2},$
however, this serves no obvious purpose. We thus fix $n=1$.

$ds^{2}=e^{-\frac{2}{3}\Phi}ds_{IIA}^{2}+e^{\frac{4}{3}\Phi}(d\psi+C_{1})^{2},$
(2.36)

and the following relations given in [19]

$\begin{gathered}\Omega=\omega\wedge\Big{(}e^{-\Phi}u+i(d\psi+C_{1})\Big{)}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}J=j\,e^{-\frac{2}{3}\Phi}+e^{\frac{1}{3}\Phi}u\wedge(d\psi+C_{1}),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}K=v\,e^{-\frac{1}{3}\Phi},\\\
z=u+i\,v.\end{gathered}$ (2.37)

The G-structure conditions for type II Mink4 solutions of the form given in
(2.12) and (2.13) 121212With
$\lambda(\alpha_{n})=(-1)^{\frac{n}{2}(n-1)}\alpha_{n}$ ensuring the condition
$F_{n}=(-1)^{\frac{n}{2}(n-1)}*_{10}F_{10-n}$ still holds. were derived in
[10], with $(A,\Phi,H_{3},g$) living on $M_{6}$. We note the following
relation with the GM Warp factor

$e^{A}=e^{\hat{A}+\frac{1}{3}\Phi},$ (2.38)

with $\Phi$ the dilaton. The differential conditions on the metric, equivalent
to (2.16) from before, are expressed in terms of the SU(2) Pure Spinors
$\Psi_{\pm}$ given in (2.26) and read 131313Where the Pure spinors can be re-
written here in a more convenient form  $\displaystyle\Psi_{+}$
$\displaystyle=\frac{1}{8}\Big{(}1+\frac{1}{2}z\wedge\bar{z}\Big{)}\wedge\omega,$
$\displaystyle\Psi_{-}$
$\displaystyle=\frac{i}{8}z\wedge\Big{(}1-i\,j-\frac{1}{2}j\wedge
j\Big{)}=\frac{1}{8}\bigg{[}u\wedge j-v\wedge\Big{(}1-\frac{1}{2}j\wedge
j\Big{)}\bigg{]}+\frac{i}{8}\,\bigg{[}u\wedge\Big{(}1-\frac{1}{2}j\wedge
j\Big{)}+v\wedge j\bigg{]},$ with the real and imaginary parts of $\Psi_{-}$
made clear.

$\displaystyle d_{H_{3}}(e^{3A-\Phi}\Psi_{+})$ $\displaystyle=0,$ (2.39a)
$\displaystyle d_{H_{3}}(e^{2A-\Phi}\text{Re}\Psi_{-})$ $\displaystyle=0,$
(2.39b) $\displaystyle d_{H_{3}}(e^{4A-\Phi}\text{Im}\Psi_{-})$
$\displaystyle=\frac{e^{4A}}{8}*_{6}\lambda(g),$ (2.39c)

with $g$ the total internal RR flux, and with the conditions given in (2.24),
where $(j,\omega)$ are orthogonal to $z$. These conditions are equivalent to a
set of equations written explicitly in terms of $(j,\omega,z)$, as outlined in
appendix D of [19].

From (2.13) we have
$e^{4A}\text{Vol}_{4}\wedge*_{6}\lambda(g)=F_{6}+F_{8}+F_{10}$, which allows
one to re-write (2.39c) as

$\text{Vol}_{4}\wedge
d_{H_{3}}(e^{4A-\Phi}\text{Im}\Psi_{-})=\frac{1}{8}(F_{6}+F_{8}+F_{10})$
(2.40)

We will utilise this form in the derivation of the Type IIB G-Structure
conditions, which we now summarise. See Appendix A.2 for the full
calculations.

#### IIB Abelian T-Dual

Finally, we investigate the G-Structure description of the Abelian T-Dual
solution. As mentioned at the beginning of this section, and made more
explicit in (2.40), the pure spinors will transform under T-Duality in the
same manner as the Ramond fields. In the G-Structure conditions, the roles of
$\Psi_{\pm}$ switch when moving from Type IIA to Type IIB. This is because
Ramond fields are even in Type IIA and odd in Type IIB.

Using the rules given in [20] (see also [21]), first make the following
decomposition in Type IIA

$ds_{10}^{2}=ds_{9,\mathcal{A}}^{2}+e^{2C}(dy+A_{1})^{2},~{}~{}~{}~{}~{}~{}~{}B=B_{2}+B_{1}\wedge
dy,~{}~{}~{}~{}~{}~{}~{}~{}~{}F=F_{\perp}+F_{||}\wedge E^{y},$ (2.41)

with $E^{y}=e^{C}(dy+A_{1})$. Then the Type IIB T-dual theory is defined as
follows

$\begin{gathered}ds_{9,\mathcal{B}}^{2}=ds_{9,\mathcal{A}}^{2},~{}~{}~{}~{}~{}\Phi^{\mathcal{B}}=\Phi^{\mathcal{A}}-C^{\mathcal{A}},~{}~{}~{}~{}~{}~{}~{}C^{\mathcal{B}}=-C^{\mathcal{A}},\\\
B_{2}^{\mathcal{B}}=B_{2}^{\mathcal{A}}+A_{1}^{\mathcal{A}}\wedge
B_{1}^{\mathcal{A}},~{}~{}~{}~{}~{}~{}~{}~{}A_{1}^{\mathcal{B}}=-B_{1}^{\mathcal{A}},~{}~{}~{}~{}~{}~{}~{}~{}B_{1}^{\mathcal{B}}=-A_{1}^{\mathcal{A}},\\\
F_{\perp}^{\mathcal{B}}=e^{C^{\mathcal{A}}}F_{||}^{\mathcal{A}},~{}~{}~{}~{}~{}~{}~{}F_{||}^{\mathcal{B}}=e^{C^{\mathcal{A}}}F_{\perp}^{\mathcal{A}}.\end{gathered}$
(2.42)

These rules then justify the following decompositions

$\begin{gathered}\Psi_{\pm}^{\mathcal{A}}=\Psi_{\pm_{\perp}}^{\mathcal{A}}+\Psi_{\pm_{||}}^{\mathcal{A}}\wedge
E^{y}_{\mathcal{A}},\\\
\omega^{\mathcal{A}}=\omega^{\mathcal{A}}_{\perp}+\omega^{\mathcal{A}}_{||}\wedge
E^{y}_{\mathcal{A}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}j^{\mathcal{A}}=j^{\mathcal{A}}_{\perp}+j^{\mathcal{A}}_{||}\wedge
E^{y}_{\mathcal{A}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}z^{\mathcal{A}}=z^{\mathcal{A}}_{\perp}+z^{\mathcal{A}}_{||}\wedge
E^{y}_{\mathcal{A}},\\\
z^{\mathcal{A}}_{\perp}=u^{\mathcal{A}}_{\perp}+i\,v^{\mathcal{A}}_{\perp},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}z^{\mathcal{A}}_{||}=u^{\mathcal{A}}_{||}+i\,v^{\mathcal{A}}_{||}.\end{gathered}$
(2.43)

One can then use the rules in (2.42) to transform the G-Structure conditions
and Pure Spinors. After switching the roles of $\Psi_{\pm}$, we get

$e^{-\Phi_{\mathcal{B}}}\Psi_{\mp}^{\mathcal{B}}=e^{-\Phi_{\mathcal{A}}}\Big{[}e^{C^{\mathcal{A}}}\Psi_{\pm_{||}}^{\mathcal{A}}+\Psi_{\pm_{\perp}}^{\mathcal{A}}\wedge(dy-
B_{1}^{\mathcal{A}})\Big{]},$ (2.44)

where the factors of $\Phi_{\mathcal{A}}$ and $\Phi_{\mathcal{B}}$ were
introduced to derive the following forms for the IIB G-Structure conditions
(solely in terms of IIB quantities)

$\displaystyle
d_{H^{\mathcal{B}}_{3}}(e^{3A-\Phi_{\mathcal{B}}}\Psi^{\mathcal{B}}_{-})$
$\displaystyle=0,$ (2.45) $\displaystyle
d_{H^{\mathcal{B}}_{3}}(e^{2A-\Phi_{\mathcal{B}}}\text{Re}\Psi^{\mathcal{B}}_{+})$
$\displaystyle=0,$ $\displaystyle
d_{H^{\mathcal{B}}_{3}}(e^{4A-\Phi_{\mathcal{B}}}\text{Im}\Psi^{\mathcal{B}}_{+})$
$\displaystyle=\frac{e^{4A}}{8}*_{6}\lambda(g).$

Working through the calculation, one gets the following forms

$\displaystyle\Psi_{-}^{\mathcal{B}}$
$\displaystyle=\frac{1}{8}e^{\Phi_{\mathcal{B}}-\Phi_{\mathcal{A}}}\bigg{[}e^{\frac{1}{2}z^{\mathcal{A}}_{\perp}\wedge\overline{z}^{\mathcal{A}}_{\perp}}\wedge\Big{(}e^{C^{\mathcal{A}}}\omega^{\mathcal{A}}_{||}+\omega^{\mathcal{A}}_{\perp}\wedge(dy-
B_{1}^{\mathcal{A}})\Big{)}+e^{C^{\mathcal{A}}}\frac{1}{2}(z^{\mathcal{A}}_{\perp}\wedge\overline{z}^{\mathcal{A}}_{||}-z^{\mathcal{A}}_{||}\wedge\overline{z}^{\mathcal{A}}_{\perp})\wedge\omega^{\mathcal{A}}_{\perp}\bigg{]},$
(2.46) $\displaystyle\Psi_{+}^{\mathcal{B}}$
$\displaystyle=\frac{i}{8}e^{\Phi_{\mathcal{B}}-\Phi_{\mathcal{A}}}e^{-ij^{\mathcal{A}}_{\perp}}\wedge\bigg{[}(e^{C^{\mathcal{A}}}z^{\mathcal{A}}_{||}+z^{\mathcal{A}}_{\perp}\wedge(dy-
B_{1}^{\mathcal{A}}))+ie^{C^{\mathcal{A}}}j^{\mathcal{A}}_{||}\wedge
z^{\mathcal{A}}_{\perp}\bigg{]},$

which for our purposes, in which
$z^{\mathcal{A}}_{||}=\bar{z}^{\mathcal{A}}_{||}=0$, reduces to141414Comparing
these results with (2.26) (and (2.27) more generally), from the form of
$\Psi_{+}^{\mathcal{B}}$ this would suggest the following relations for an
$SU(2)$ structure (noting
$\frac{1}{2}z^{\mathcal{B}}\wedge\bar{z}^{\mathcal{B}}=iv^{\mathcal{B}}\wedge
u^{\mathcal{B}}$)  $v^{\mathcal{B}}\wedge
u^{\mathcal{B}}=-j_{\perp}^{\mathcal{A}},~{}~{}~{}~{}~{}~{}~{}\omega^{\mathcal{B}}=i\,e^{\Phi_{\mathcal{B}}-\Phi_{\mathcal{A}}}\,z^{\mathcal{A}}_{\perp}\wedge\Big{(}(dy-
B_{1}^{\mathcal{A}})-ie^{C^{\mathcal{A}}}j^{\mathcal{A}}_{||}\Big{)}.$ (2.47)
The $\Psi_{-}^{\mathcal{B}}$ comparison is less clear but the dimensions of
the Polyform match as they should.

$\displaystyle\Psi_{-}^{\mathcal{B}}$
$\displaystyle=\frac{1}{8}e^{\Phi_{\mathcal{B}}-\Phi_{\mathcal{A}}}e^{\frac{1}{2}z^{\mathcal{A}}_{\perp}\wedge\overline{z}^{\mathcal{A}}_{\perp}}\wedge\Big{(}e^{C^{\mathcal{A}}}\omega^{\mathcal{A}}_{||}+\omega^{\mathcal{A}}_{\perp}\wedge(dy-
B_{1}^{\mathcal{A}})\Big{)},$ (2.48) $\displaystyle\Psi_{+}^{\mathcal{B}}$
$\displaystyle=\frac{i}{8}e^{\Phi_{\mathcal{B}}-\Phi_{\mathcal{A}}}e^{-ij^{\mathcal{A}}_{\perp}}\wedge
z^{\mathcal{A}}_{\perp}\wedge\Big{(}(dy-
B_{1}^{\mathcal{A}})-ie^{C^{\mathcal{A}}}j^{\mathcal{A}}_{||}\Big{)}.$

See Appendix A.2 for more details.

###  2.4 $SL(3,\mathds{R})$ Transformations

Now that we have reviewed the Gaiotto-Maldacena background and the necessary
G-Structure analysis, we should introduce the $SL(3,\mathds{R})$
transformations which generate the various solutions presented in this paper
(along with the results of [2]).

Given that there are three $U(1)$ directions in the GM background
$(\beta,\chi,\phi)$, one can make the following $SL(3,\mathds{R})$
transformation

$\begin{gathered}d\beta=a\,d\chi+b\,d\beta+c\,d\phi,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}d\chi=p\,d\chi+q\,d\beta+m\,d\phi,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}d\phi=s\,d\chi+v\,d\beta+u\,d\phi,\\\
\begin{vmatrix}~{}p&q&m~{}\\\ ~{}a&b&c~{}\\\ ~{}s&v&u~{}\\\
\end{vmatrix}=p(bu-vc)-q(au-sc)+m(av-sb)=1.\end{gathered}$ (2.49)

The utility of this transformation becomes clear under dimensional reductions
to Type IIA151515These nine parameters must reduce to three free parameters
(corresponding to the three $U(1)$ directions being mixed). For our purposes,
we fix $(p,b,u)=1$ by trivially absorbing them into the definitions of
$(\chi,\beta,\phi)$, respectively. This avoids re-defining the three $U(1)$
directions amongst themselves and immediately eliminates three of the nine
parameters. However, as we note where appropriate, there can be some utility
in choosing different values for these parameters., with the $\beta$ reduction
case already considered in [2]. In that case, one can derive
$\mathcal{N}=(2,1,0)$ Type IIA backgrounds. However, as we will investigate in
this paper, the $\chi$ and $\phi$ reduction cases lead only to $\mathcal{N}=1$
and $\mathcal{N}=0$ solutions. Using the forms given in (2.3), one can see
that the $U(1)$ component of the $SU(2)_{R}\times U(1)_{r}$ R-symmetry of the
Gaiotto-Maldacena background now becomes

$\displaystyle U(1)_{r}$ $\displaystyle=\chi+\phi$ (2.50)
$\displaystyle\rightarrow(p+s)\chi+(q+v)\beta+(m+u)\phi.$

This immediately provides more possibilities to preserve Supersymmetry under
dimensional reduction, i.e. in the case of a $\beta$ reduction, one can now
preserve this $U(1)_{r}$ component by fixing $v=-q$. Throughout this paper we
use the reduction formula given in (2.36). We now review the $\beta$ reduction
case.

### The $\beta$ reduction case

As outlined in [2], by performing the following 11D transformations (with
$q=\xi,~{}v=\zeta$)

$d\beta\rightarrow
d\beta,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}d\chi\rightarrow
d\chi+\xi\,d\beta,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}d\phi\rightarrow
d\phi+\zeta\,d\beta,$ (2.51)

before dimensionally reducing along $\beta$, one gets the following two
parameter family of solutions (re-written below in a form which will be useful
for investigating the boundary)

$\displaystyle
ds_{10,st}^{2}=f_{1}^{\frac{3}{2}}f_{5}^{\frac{1}{2}}\sqrt{\Xi}\bigg{[}4ds^{2}(\text{AdS}_{5})+f_{4}(d\sigma^{2}+d\eta^{2})+ds^{2}(M_{3})\bigg{]},$
$\displaystyle
ds^{2}(M_{3})=f_{2}\bigg{(}d\theta^{2}+\frac{\Delta}{\Xi}\sin^{2}\theta
D\phi^{2}\bigg{)}+\frac{f_{3}}{\Delta}d\chi^{2}=f_{2}\bigg{(}d\theta^{2}+\frac{1}{\Pi}\sin^{2}\theta
d\phi^{2}\bigg{)}+\frac{\Pi}{\Xi}f_{3}D\chi^{2},$ $\displaystyle
B_{2}=\sin\theta\,\bigg{[}\zeta f_{7}d\chi-(f_{8}+\xi
f_{7})d\phi\bigg{]}\wedge
d\theta,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}e^{\frac{4}{3}\Phi}=f_{1}f_{5}\Xi,$
$\displaystyle C_{1}=\frac{1}{\Xi}\bigg{(}\Big{(}f_{6}(1+\xi
f_{6})+\,\xi\frac{f_{3}}{f_{5}}\Big{)}d\chi+\zeta\frac{f_{2}}{f_{5}}\sin^{2}\theta
d\phi\bigg{)},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}C_{3}=f_{7}d\chi\wedge\text{Vol}(S^{2}),$
$\displaystyle\Xi=\Delta+\zeta^{2}\frac{f_{2}}{f_{5}}\sin^{2}\theta,~{}~{}~{}\Delta=(1+\xi
f_{6})^{2}+\xi^{2}\frac{f_{3}}{f_{5}},~{}~{}~{}\Pi=1+\zeta^{2}f_{2}\frac{f_{3}+f_{5}f_{6}^{2}}{f_{3}f_{5}}\sin^{2}\theta,$
$\displaystyle D\phi=d\phi-\frac{\zeta}{\Delta}\bigg{(}f_{6}(1+\xi
f_{6})+\xi\frac{f_{3}}{f_{5}}\bigg{)}d\chi,~{}~{}~{}~{}~{}~{}~{}D\chi=d\chi-\frac{\zeta}{\Pi}\frac{f_{2}}{f_{3}}\bigg{(}f_{6}(1+\xi
f_{6})+\xi\frac{f_{3}}{f_{5}}\bigg{)}\sin^{2}\theta\,d\phi,$ (2.52)

leading to the backgrounds given in Table 1 of [2]. The metric has the nice
property that all dependence on the transformation parameters drops out neatly
when calculating the Holographic Central charge. Hence, one derives the
following (parameter independent) result

$c_{hol}=\frac{\kappa^{3}}{\pi^{4}}\sum_{k=1}^{\infty}P\mathcal{R}_{k}^{2}.$
(2.53)

This means that the two parameters correspond to marginal deformations of the
$d=4$ $\mathcal{N}=2$ long linear quiver dual CFTs, see [2] for a detailed
calculation.

In [2] the $SU(2)\times U(1)$ preserved $\mathcal{N}=0$ background (with
$\zeta=0$) and the $\mathcal{N}=1$ case (with $\zeta=-\xi$) were investigated
in further detail. Studying the boundary of the $(\sigma,\eta)$ coordinates,
one can show that these two backgrounds contain branes which are backreacted
on a Spindle and it’s higher dimensional analogue, respectively. In the next
subsection we will review this analysis by following the same procedure for
the general two-parameter family.

#### Investigations at the Boundary

We now follow the procedure of [2] for the more general two-parameter
solution, using the limits of the Warp Factors which are quoted for
convenience in Appendix G. This analysis will follow closely that of the
$\mathcal{N}=1$ case, with the parameter $\zeta$ left general.

First, for general values of $(\eta,\sigma,\theta)$, we see that $(\Xi,\Pi)$
are non-zero and finite. The deformed $S^{2}$ given by $(\theta,\phi)$ has
$\Pi\rightarrow 1$ at the poles, which given the expression for $M_{3}$ in
(2), means it still behaves topologically as an $S^{2}$. This follows the
argument given in [2], where $D\chi\rightarrow d\chi$ up to leading order at
the poles. We now turn to the behaviour at the boundaries. In the following
analysis, to be consistent with [2] we will use the following large gauge
transformation

$B_{2}\rightarrow B_{2}+2\kappa k\text{Vol}(S^{2}),$ (2.54)

but we will note a potential alternative which leads to quantized D4 charges
in the $\zeta=0$ case.

  * •

At the $\sigma\rightarrow\infty$ boundary, all parameters drop out, so the
behaviour remains the same as the cases studied in [2]. We summarise the
presentation for completeness. One finds

$\displaystyle
ds^{2}=\kappa\Bigg{[}4\sigma\Big{(}ds^{2}(\text{AdS}_{5})+d\chi^{2}\Big{)}+\frac{2P}{\pi}\bigg{(}d\Big{(}\frac{\pi}{P}\sigma\Big{)}^{2}+d\Big{(}\frac{\pi}{P}\eta\Big{)}^{2}+\sin^{2}\bigg{(}\frac{\pi\eta}{P}\bigg{)}ds^{2}(S^{2})\bigg{)}\Bigg{]},$
$\displaystyle
e^{-\Phi}=\frac{\mathcal{R}_{1}\pi^{2}}{2P^{\frac{3}{2}}\sqrt{\kappa}}e^{-\frac{\pi}{P}\sigma}\Big{(}\frac{\pi}{P}\sigma\Big{)}^{-\frac{1}{2}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}H_{3}=-\frac{4\kappa
P}{\pi}\sin^{2}\Big{(}\frac{\pi}{P}\eta\Big{)}d\Big{(}\frac{\pi}{P}\eta\Big{)}\wedge\text{Vol}(S^{2}),$
(2.55)

where we see that $(\eta,S^{2})$ span a unit-radius 3-sphere. Hence, at this
boundary the $\mathcal{N}=2$ case is recovered, giving a stack of $P$ NS5
branes (after fixing $2\kappa=\pi$)161616The Page Charges are calculated using
the following relations  $\displaystyle
Q_{Dp/NS5}=\frac{1}{(2\pi)^{7-p}}\int_{\Sigma_{8-p}}\hat{F}_{8-p},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\hat{F}^{\text{Page}}=F^{\text{Max}}\wedge
e^{-B_{2}}=d(C\wedge
e^{-B_{2}}),~{}~{}~{}~{}~{}~{}~{}~{}~{}F_{p+2}^{\text{Max}}=dC_{p+1}-H_{3}\wedge
C_{p-1}.$

$Q_{NS5}=-\frac{1}{(2\pi)^{2}}\int_{S^{3}}H_{3}=P.$ (2.56)

  * •

At $\eta=0$ with $\sigma\neq 0$, using (G), one finds using the second form
for $M_{3}$ that $(\eta,\theta,\phi)$ vanish as $\mathds{R}^{3}$ in polar
coordinates, namely

$f_{4}d\eta^{2}+f_{2}\bigg{(}d\theta^{2}+\frac{1}{\Pi}\sin^{2}\theta
d\phi^{2}\bigg{)}=\frac{2|\dot{f}|}{\sigma\,f}\Big{(}d\eta^{2}+\eta^{2}ds^{2}(S^{2})\Big{)},$
(2.57)

this once again matches the specific cases studied in [2].

  * •

At $\sigma=0,~{}\eta\in(k,k+1)$ we have that along the $\sigma=0$ boundary,
$\ddot{V}=0$ to leading order. Using (G.6) we find

$\displaystyle\Xi=l_{k}^{2}+\frac{1}{2}\zeta^{2}\mathcal{R}V^{\prime\prime}\sin^{2}\theta,~{}~{}~{}\Delta\rightarrow
l_{k}^{2},~{}~{}~{}D\phi=d\phi-\frac{\zeta(N_{k+1}-N_{k})}{l_{k}}d\chi,~{}~{}~{}~{}l_{k}=1+\xi(N_{k+1}-N_{k}),$
(2.58)

giving an $\mathds{R}^{2}/\mathds{Z}_{l_{k}}$ orbifold singularity in
$(\sigma,\chi)$, as follows

$f_{4}d\sigma^{2}+\frac{f_{3}}{\Delta}d\chi^{2}\rightarrow\frac{2V^{\prime\prime}}{\mathcal{R}}\bigg{(}d\sigma^{2}+\frac{\sigma^{2}}{l_{k}^{2}}d\chi^{2}\bigg{)}.$
(2.59)

  * •

At $\sigma=0,~{}\eta=0$, to approach this boundary one makes the following
coordinate change
$(\eta=r\cos\alpha,~{}\sigma=r\sin\alpha)$, expanding about $r=0$. Using (G)
we find

$\displaystyle\Xi\rightarrow\Delta\rightarrow
l_{0}^{2},~{}~{}~{}~{}~{}~{}l_{0}=1+\xi
N_{1},~{}~{}~{}~{}~{}~{}~{}~{}~{}D\phi=d\phi-\frac{\zeta N_{1}}{l_{0}}d\chi,$
(2.60) $\displaystyle
f_{4}(d\sigma^{2}+d\eta^{2})+ds^{2}(M_{3})\rightarrow\frac{2Q}{N_{1}}\bigg{(}dr^{2}+r^{2}d\alpha^{2}+r^{2}\cos^{2}\alpha\bigg{(}d\theta^{2}+\sin^{2}\theta\,D\phi^{2}\bigg{)}+r^{2}\sin^{2}\alpha\frac{d\chi^{2}}{l_{0}^{2}}\bigg{)},$

where we say that the internal space vanishes as
$\mathds{R}^{5}/\mathds{Z}_{l_{0}}$.

  * •

At $\sigma=0,~{}\eta=k$, using the coordinate change
$(\eta=k-r\cos\alpha,~{}\sigma=r\sin\alpha)$ and (G), we see that the term
$f_{2}/f_{5}$ dominates, unless we are at a pole of the deformed $S^{2}$. We
first assume we are away from a pole, finding

$\displaystyle\Xi\rightarrow\frac{b_{k}\zeta^{2}N_{k}\sin^{2}\theta}{4r},~{}~{}~{}~{}~{}~{}~{}~{}~{}\Delta\rightarrow\Delta_{k},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\Delta_{k}\equiv\frac{1}{4}\xi^{2}b_{k}^{2}\sin^{2}\alpha+\Big{(}1+\xi
g(\alpha)\Big{)}^{2},$
$\displaystyle\Delta_{k}(\alpha=0)=l_{k-1}^{2},~{}~{}~{}~{}~{}~{}\Delta_{k}(\alpha=\pi)=l_{k}^{2}.$
(2.61)

Using $r=z^{2}$ we get a very similar result to the $\mathcal{N}=1$ case
studied in [2],

$\displaystyle
ds^{2}=\zeta\kappa\sin\theta\Big{[}N_{k}\Big{(}4ds^{2}(AdS_{5})+d\theta^{2}\Big{)}+4b_{k}\Big{(}dz^{2}+z^{2}ds^{2}(\mathds{B}_{3})\Big{)}\Big{]},~{}~{}~{}~{}~{}~{}e^{4\Phi}=\kappa^{2}\zeta^{6}N_{k}^{2}\sin^{6}\theta,$
$\displaystyle
ds^{2}(\mathds{B}_{3})=\frac{1}{4}\Big{(}d\alpha^{2}+\frac{\sin^{2}\alpha}{\Delta_{k}}d\chi^{2}\Big{)}+\frac{\Delta_{k}}{\zeta^{2}b_{k}^{2}}(d\phi+\mathcal{A}_{k})^{2},~{}~{}~{}~{}~{}\mathcal{A}_{k}=-\frac{\zeta}{\Delta_{k}}\Big{[}g(\alpha)\Big{(}1+\xi
g(\alpha)\Big{)}+\frac{\xi}{4}b_{k}^{2}\sin^{2}\alpha\Big{]}d\chi,$
$\displaystyle B_{2}=-2\kappa N_{k}\,\sin\theta\Big{(}\zeta d\chi-\xi
d\phi\Big{)}\wedge
d\theta,~{}~{}~{}~{}~{}~{}~{}C_{1}=\frac{1}{\zeta}d\phi,~{}~{}~{}~{}~{}C_{3}=-2\kappa
N_{k}\sin\theta d\theta\wedge d\phi\wedge d\chi.$ (2.62)

Now

$\displaystyle ds^{2}(\mathds{B}_{3})\bigg{|}_{\alpha\sim
0}=\frac{1}{4}\Big{(}d\alpha^{2}+\frac{\alpha^{2}}{l_{k-1}^{2}}d\chi^{2}\Big{)}+\frac{l_{k-1}^{2}}{\zeta^{2}b_{k}^{2}}\bigg{(}d\phi-\frac{\zeta(N_{k}-N_{k-1})}{l_{k-1}}d\chi\bigg{)}^{2},$
$\displaystyle
ds^{2}(\mathds{B}_{3})\bigg{|}_{\alpha\sim\pi}=\frac{1}{4}\Big{(}d\alpha^{2}+\frac{(\pi-\alpha)^{2}}{l_{k}^{2}}d\chi^{2}\Big{)}+\frac{l_{k}^{2}}{\zeta^{2}b_{k}^{2}}\bigg{(}d\phi-\frac{\zeta(N_{k+1}-N_{k})}{l_{k}}d\chi\bigg{)}^{2},$
$\displaystyle\frac{1}{2\pi}\int_{\mathds{W}\mathds{C}\mathds{P}_{[l_{k-1},l_{k}]}}d\mathcal{A}=\frac{\zeta
b_{k}}{l_{k-1}l_{k}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}b_{k}=2N_{k}-N_{k+1}-N_{k-1}.$
(2.63)

So we see that the general case doesn’t differ too much from the
$\mathcal{N}=1$ case in [2]. Hence, we interpret $\mathds{B}_{3}$ as $U(1)$
fibration over $\mathds{W}\mathds{C}\mathds{P}_{[l_{k-1},l_{k}]}$.

Notice here that $C_{3}-C_{1}\wedge B_{2}=0$, meaning that there is no D4
branes present (for $\zeta\neq 0$), as $\hat{F}_{4}=d(C_{3}\wedge
e^{-B_{2}})$. However, in the $S^{2}$ preserved case presented in [2] (with
$\zeta=0$), D4 branes are in fact recovered.

    * –

$\zeta=0$: We now make an aside and quote the $\zeta=0$ case given in [2]
(keeping the parameter $r$). We first present the $B_{2}$ with no gauge
transformation

$\displaystyle\frac{ds^{2}}{2\kappa\sqrt{N_{k}}}=\sqrt{\Delta_{k}}\bigg{[}\frac{1}{\sqrt{\frac{b_{k}}{r}}}\bigg{(}4ds^{2}(\text{AdS}_{5})+ds^{2}(\text{S}^{2})\bigg{)}+\frac{\sqrt{\frac{b_{k}}{r}}}{N_{k}}\bigg{(}dr^{2}+r^{2}\left(d\alpha^{2}+\frac{\sin^{2}\alpha}{\Delta_{k}}d\chi^{2}\right)\bigg{)}\bigg{]},$
$\displaystyle
e^{-\Phi}=\left(\frac{N_{k}b_{k}^{3}}{2^{6}\kappa^{2}r^{3}}\right)^{\frac{1}{4}}\Delta_{k}^{-\frac{3}{4}},~{}~{}~{}~{}~{}C_{1}=\frac{\frac{1}{4}\xi
k^{2}b_{k}^{2}\sin^{2}\alpha+g(\alpha)(1+\xi g(\alpha))}{\Delta_{k}}d\chi,$
(2.64) $\displaystyle C_{3}=-2\kappa
N_{k}d\chi\wedge\text{Vol}(\text{S}^{2}),~{}~{}~{}~{}~{}~{}~{}~{}~{}B_{2}=-2\kappa(k+\xi
N_{k})\text{Vol}(S^{2}).$ (2.65)

Now, using the large gauge transformation of [2], we have (with $2\kappa=\pi$)

$\displaystyle B_{2}\rightarrow B_{2}+2\kappa
k\text{Vol}(S^{2}),~{}~{}~{}~{}~{}~{}~{}~{}B_{2}=-2\kappa\xi
N_{k}\text{Vol}(\text{S}^{2}),$ $\displaystyle
Q_{D4}=-\frac{1}{(2\pi)^{2}}\int_{S^{2}\times\mathds{W}\mathds{C}\mathds{P}^{1}_{[l_{k-1},l_{k}]}}\hat{F}_{4}=\frac{N_{k}}{l_{k}}-\frac{N_{k-1}}{l_{k-1}}.$
(2.66)

However, if we add an additional term to the gauge transformation, noting
$(\xi,N_{k})\in\mathds{Z}$,

$\displaystyle B_{2}\rightarrow B_{2}+2\kappa(k+\xi
N_{k})\text{Vol}(S^{2}),~{}~{}~{}~{}~{}~{}~{}~{}B_{2}=0,$ $\displaystyle
Q_{D4}=-\frac{1}{(2\pi)^{2}}\int_{S^{2}\times\mathds{W}\mathds{C}\mathds{P}^{1}_{[l_{k-1},l_{k}]}}\hat{F}_{4}=N_{k}-N_{k-1},$
(2.67)

we recover the $\mathcal{N}=2$ quantization. This is a potential oddity given
that we are integrating over the $(\alpha,\chi)$ Spindle. It would be
interesting to investigate this further in future.

We now return to our general analysis and study this limit when approaching
the pole $(\sigma=0,$ $\eta=k,\sin\theta=0)$. Following [2], we use

$\eta=k-\rho\cos\alpha\sin^{2}\mu,~{}~{}~{}~{}~{}\sigma=\rho\sin\alpha\sin^{2}\mu,~{}~{}~{}~{}~{}~{}\sin\theta=2\sqrt{\frac{b_{k}\rho}{N_{k}}}\cos\mu,$
(2.68)

noting that $r=\rho\sin^{2}\mu$, and expanding about $\rho=0$. Now,

$\sin^{2}\mu\,\Xi\rightarrow\tilde{\Xi}_{k}=\Delta_{k}\sin^{2}\mu+\zeta^{2}b_{k}^{2}\cos^{2}\mu,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}B_{2}\rightarrow
B_{2}+\frac{4\kappa\,k\,b_{k}\cos^{2}\mu}{N_{k}}d\rho\wedge d\phi,$ (2.69)

giving

$\displaystyle\frac{ds^{2}}{2\kappa\sqrt{N_{k}}}=\sqrt{\tilde{\Xi}_{k}}\Bigg{[}\frac{4}{\sqrt{\frac{b_{k}}{\rho}}}ds^{2}(AdS_{5})+\frac{\sqrt{\frac{b_{k}}{\rho}}}{N_{k}}\bigg{(}d\rho^{2}+4\rho^{2}ds^{2}(\mathds{B}_{4})\bigg{)}\Bigg{]},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}e^{-\Phi}=\bigg{(}\frac{b_{k}^{3}N_{k}}{2^{6}\kappa^{2}\tilde{\Xi}_{k}^{3}\rho^{3}}\bigg{)}^{\frac{1}{4}},$
$\displaystyle
ds^{2}(\mathds{B}_{4})=d\mu^{2}+\frac{1}{4}\sin^{2}\mu\bigg{(}d\alpha^{2}+\frac{\sin^{2}\alpha}{\Delta_{k}}d\chi^{2}\bigg{)}+\frac{\sin^{2}\mu\cos^{2}\mu\,\Delta_{k}}{\tilde{\Xi}_{k}}(d\phi+\mathcal{A}_{k})^{2},$
$\displaystyle B_{2}=4\kappa b_{k}\cos^{2}\mu\Big{(}\xi d\phi-\zeta
d\chi\Big{)}\wedge
d\rho,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}C_{3}=-4\kappa\,b_{k}\cos^{2}\mu\,d\rho\wedge
d\phi\wedge d\chi,$ $\displaystyle
C_{1}=\frac{1}{\tilde{\Xi}_{k}}\bigg{[}b_{k}^{2}\zeta\cos^{2}\mu\,d\phi+\bigg{(}g(\alpha)\Big{(}1+\xi
g(\alpha)\Big{)}+\frac{1}{4}\xi
b_{k}^{2}\sin^{2}\alpha\bigg{)}\sin^{2}\mu\,d\chi\bigg{]}.$ (2.70)

We now find that we simply re-derive the $\mathds{B}_{4}$ of [2] with all
$\zeta$ dependence given in $\Delta_{k}$ and $\mathcal{A}_{k}$. Hence,
following the arguments given there, we say that $\mathds{B}_{4}$ is
topologically a $\mathds{C}\mathds{P}^{2}$, but with additional orbifold
singularities.

Now calculating the charge at $\mu=\frac{\pi}{2}$, we notice that all $\zeta$
dependence drops out of the calculation, re-deriving the D6 charge given in
[2]

$Q_{D6}=-\frac{1}{2\pi}\int_{\mathds{W}\mathds{C}\mathds{P}^{1}_{[l_{k-1},l_{k}]}}F_{2}=-\frac{1}{2\pi}\int_{\chi=0}^{\chi=2\pi}C_{1}\Big{|}_{\alpha=0}^{\alpha=\pi}=\frac{2N_{k}-N_{k+1}-N_{k-1}}{l_{k}l_{k-1}}.$
(2.71)

This rational charge is a consequence of the Spindle, see for example [22]. We
now see that for all values of $(\xi,\zeta)$ we have NS5 and D6 branes, with
the respective charges given above. The case of the D4 branes is special, they
are only present when $\zeta=0$ \- that is, when there is a preserved $S^{2}$.
Picturing the $(\xi,\zeta)$ plot given in [2], the D4 branes are only present
along the $\zeta=0$ axis. Strangely, the rational charge quantization of the
D4 branes, which derives from integrating over the spindle, can be eliminated
by an additional term in the gauge transformation of $B_{2}$.

##  3 More Type IIA solutions

In this section we present some noteworthy solutions derived from
dimensionally reducing Gaiotto-Maldacena along the $\chi$ and $\phi$
directions. The reader is referred to Appendix B for a more in-depth
derivation and discussion of the backgrounds presented in this section.

The form of the metrics presented here match (2), with all parameter
dependence in the dilaton and $ds_{2}^{2}$. By the same calculation, the
Holographic Central charge remains the same, given in (2.53). These solutions
then correspond to marginal deformations in the dual CFT. However, in the
following cases, fixing all parameters to zero does not recover the
$\mathcal{N}=2$ solution in Supergravity.

###  3.1 $\chi$ Reduction

One can derive a two-parameter family of solutions by dimensionally reducing
along $\chi$. This solution can be mapped to (2) by the transformations given
in (B.13) and (B.14). Under this mapping, one requires $\xi\rightarrow 1/\xi$.
Hence, by fixing $\xi=0$ we derive a unique background, which reads

$\displaystyle
ds_{10,st}^{2}=e^{\frac{2}{3}\Phi}f_{1}\bigg{[}4ds^{2}(\text{AdS}_{5})+f_{2}d\theta^{2}+f_{4}(d\sigma^{2}+d\eta^{2})\bigg{]}+f_{1}^{2}e^{-\frac{2}{3}\Phi}ds^{2}_{2},$
$\displaystyle ds^{2}_{2}=\big{(}f_{3}f_{5}+\zeta^{2}\sin^{2}\theta
f_{2}f_{5}\big{)}d\beta^{2}+\sin^{2}\theta
f_{2}\big{(}f_{3}+f_{5}f_{6}^{2}\big{)}d\phi^{2}-2\zeta\big{(}\sin^{2}\theta
f_{2}f_{5}f_{6}\big{)}d\beta d\phi,$ $\displaystyle
e^{\frac{4}{3}\Phi}=f_{1}\Big{(}f_{3}+f_{5}f_{6}^{2}+\zeta^{2}\sin^{2}\theta
f_{2}\Big{)},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}B_{2}=\sin\theta\big{(}\zeta
f_{8}\,d\beta-f_{7}d\phi\big{)}\wedge\,d\theta,$ $\displaystyle
C_{1}=f_{1}e^{-\frac{4}{3}\Phi}\bigg{(}f_{5}f_{6}d\beta+\zeta\sin^{2}\theta
f_{2}d\phi\bigg{)},~{}~{}~{}~{}~{}~{}~{}~{}C_{3}=f_{8}\sin\theta\,d\beta\wedge\,d\theta\wedge
d\phi.$ (3.1)

This is a one-parameter family of $\mathcal{N}=0$ solutions, promoting to
$\mathcal{N}=1$ for $\zeta=-1$ (see Appendix B for further details).

####  3.1.1 Unique $\mathcal{N}=1$ IIA Solution

One now derives the following (unique) $\mathcal{N}=1$ solution (corresponding
to the $\xi=0$ case of the one-parameter family of $\mathcal{N}=1$ solutions
given in (– ‣ • ‣ B))171717Here
$(p,b,u)=1,(m,q,c)=0,v\equiv\gamma=0,a\equiv\xi=0,s\equiv\zeta=-1$.

$\displaystyle
ds_{10,st}^{2}=e^{\frac{2}{3}\Phi}f_{1}\bigg{[}4ds^{2}(\text{AdS}_{5})+f_{2}d\theta^{2}+f_{4}(d\sigma^{2}+d\eta^{2})\bigg{]}+f_{1}^{2}e^{-\frac{2}{3}\Phi}ds^{2}_{2},$
$\displaystyle ds^{2}_{2}=\big{(}f_{3}f_{5}+\sin^{2}\theta
f_{2}f_{5}\big{)}d\beta^{2}+\sin^{2}\theta
f_{2}\big{(}f_{3}+f_{5}f_{6}^{2}\big{)}d\phi^{2}+2\sin^{2}\theta
f_{2}f_{5}f_{6}d\beta d\phi,$ $\displaystyle
e^{\frac{4}{3}\Phi}=f_{1}f_{5}\Big{(}f_{6}^{2}+\frac{f_{3}}{f_{5}}+\frac{f_{2}}{f_{5}}\sin^{2}\theta\Big{)},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}B_{2}=-\sin\theta\big{(}f_{8}d\beta+f_{7}d\phi\big{)}\wedge\,d\theta,$
$\displaystyle C_{1}=f_{1}e^{-\frac{4}{3}\Phi}\bigg{(}f_{5}f_{6}d\beta-
f_{2}\sin^{2}\theta
d\phi\bigg{)},~{}~{}~{}~{}~{}~{}~{}~{}C_{3}=f_{8}\sin\theta\,d\beta\wedge\,d\theta\wedge
d\phi.$ (3.2)

See Appendix A.3.1 for the G-Structure description.

###  3.2 $\phi$ Reduction

As one would expect, dimensionally reducing along $\phi$ again leads to a two-
parameter family of solutions (given in (* ‣ – ‣ • ‣ B)). Once again, this
solution can be mapped to (2) using the transformations outlined in (B.23) and
(B.24). In this case, the mapping involves transformations involving both
$1/\xi$ and $1/\zeta$. We therefore derive the following backgrounds.

When $\zeta=0$, one gets the following one-parameter family of $\mathcal{N}=0$
solutions

$\displaystyle
ds_{10,st}^{2}=e^{\frac{2}{3}\Phi}f_{1}\bigg{[}4ds^{2}(\text{AdS}_{5})+f_{2}d\theta^{2}+f_{4}(d\sigma^{2}+d\eta^{2})\bigg{]}+f_{1}^{2}e^{-\frac{2}{3}\Phi}ds^{2}_{2},$
$\displaystyle ds^{2}_{2}=\Big{(}\xi^{2}f_{3}f_{5}+\sin^{2}\theta
f_{2}(f_{3}+f_{5}f_{6}^{2})\Big{)}d\chi^{2}+f_{2}f_{5}\sin^{2}\theta
d\beta^{2}+2f_{2}f_{5}f_{6}\sin^{2}\theta d\chi d\beta,$ $\displaystyle
e^{\frac{4}{3}\Phi}=f_{1}\Big{(}\xi^{2}f_{5}+\sin^{2}\theta
f_{2}\Big{)},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}C_{1}=\xi
f_{1}f_{5}e^{-\frac{4}{3}\Phi}\Big{(}f_{6}d\chi+d\beta\Big{)},$ $\displaystyle
B_{2}=\sin\theta\Big{(}f_{7}d\chi+f_{8}d\beta\Big{)}\wedge
d\theta,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}C_{3}=0.$ (3.3)

When $\xi=0$, one now derives

$\displaystyle
ds_{10,st}^{2}=e^{\frac{2}{3}\Phi}f_{1}\bigg{[}4ds^{2}(\text{AdS}_{5})+f_{2}d\theta^{2}+f_{4}(d\sigma^{2}+d\eta^{2})\bigg{]}+f_{1}^{2}e^{-\frac{2}{3}\Phi}ds^{2}_{2},$
$\displaystyle
ds^{2}_{2}=f_{2}(f_{3}+f_{5}f_{6}^{2})\sin^{2}\theta\,d\chi^{2}+\Big{(}\zeta^{2}f_{3}f_{5}+\sin^{2}\theta
f_{2}f_{5}\Big{)}d\beta^{2}+2f_{2}f_{5}f_{6}\sin^{2}\theta d\chi d\beta,$
$\displaystyle
e^{\frac{4}{3}\Phi}=f_{1}\Big{[}\zeta^{2}(f_{5}f_{6}^{2}+f_{3})+\sin^{2}\theta
f_{2}\Big{]},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}C_{1}=\zeta
f_{1}e^{-\frac{4}{3}\Phi}\Big{[}\Big{(}f_{3}+f_{5}f_{6}^{2}\Big{)}d\chi+f_{5}f_{6}d\beta\Big{]},$
$\displaystyle B_{2}=\sin\theta\Big{(}f_{7}d\chi+f_{8}d\beta\Big{)}\wedge
d\theta,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}C_{3}=0.$ (3.4)

This is a one-parameter family of $\mathcal{N}=0$ solutions which promote to
the $\mathcal{N}=1$ background given in (3.1.1) when $\zeta=-1$ (following
appropriate gauge transformations - see (B.26) with $\xi=0$).

##  4 Type IIB T-Duals

We now present IIB solutions derived from Abelian T-Dualising the IIA
reductions. In all backgrounds presented throughout this paper, the
transformation parameters drop out exactly when calculating the Holographic
Central Charge (see Appendix F), matching the IIA result in (2.53). This is
not particularly surprising given the arguments presented in Section 4.3 of
[23]. Hence, the transformation parameters once again correspond to marginal
deformations in a dual CFT. We now present a three-parameter family of
solutions.

###  4.1 Three Parameter Family

From the $U(1)_{r}$ component given in (2.50), fixing
$(p,b,u)=1,~{}(a,c,m)=0,~{}q\equiv\xi,v\equiv\zeta,s\equiv\gamma$, we have

$\displaystyle U(1)_{r}$ $\displaystyle=(1+\gamma)\chi+(\xi+\zeta)\beta+\phi.$
(4.1)

Performing a dimensional reduction along $\beta$, followed by an Abelian
T-Duality along $\chi$, one gets the following three-parameter family of Type
IIB solutions181818See also the 3-parameter families given in (– ‣ • ‣ C), (–
‣ • ‣ C) and (– ‣ • ‣ C), which can all be mapped to (4.1) as outlined there.

$\displaystyle
ds_{10,B}^{2}=f_{1}^{\frac{3}{2}}f_{5}^{\frac{1}{2}}\sqrt{\Xi}\Bigg{[}4ds^{2}(\text{AdS}_{5})+f_{4}(d\sigma^{2}+d\eta^{2})+ds^{2}(M_{3})\Bigg{]},$
$\displaystyle
ds^{2}(M_{3})=f_{2}\bigg{(}d\theta^{2}+\frac{1}{\Pi}\sin^{2}\theta\,d\phi^{2}\bigg{)}+\frac{1}{f_{1}^{3}f_{3}f_{5}\Pi}\,D\chi^{2},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}e^{2\Phi_{\mathcal{B}}}=\frac{f_{5}}{f_{3}}\frac{\Xi^{2}}{\Pi},$
$\displaystyle C_{0}=\frac{1}{\Xi}\bigg{(}f_{6}(1+\xi
f_{6})+\xi\frac{f_{3}}{f_{5}}+\gamma\zeta\frac{f_{2}}{f_{5}}\sin^{2}\theta\bigg{)},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}D\chi=d\chi-\big{(}(\gamma\xi-\zeta)f_{7}+\gamma
f_{8}\big{)}\sin\theta d\theta,$ (4.2) $\displaystyle
B_{2}=-\frac{1}{\Pi}\bigg{(}f_{8}+\xi
f_{7}-\zeta\frac{f_{2}}{f_{3}}\Big{(}(\gamma\xi-\zeta)\frac{f_{3}}{f_{5}}f_{8}-(f_{7}-f_{6}f_{8})\big{(}\gamma+(\gamma\xi-\zeta)f_{6}\big{)}\Big{)}\sin^{2}\theta\bigg{)}\sin\theta\,d\phi\wedge
d\theta$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}-\frac{1}{\Pi}\frac{f_{2}}{f_{3}}\Big{(}\xi(\gamma\xi-\zeta)\frac{f_{3}}{f_{5}}+(1+\xi
f_{6})\big{(}\gamma+(\gamma\xi-\zeta)f_{6}\big{)}\Big{)}\sin^{2}\theta
d\phi\wedge d\chi,$ $\displaystyle
C_{2}=\frac{1}{\Pi}\bigg{(}f_{7}-\gamma\frac{f_{2}}{f_{3}}\Big{(}(\gamma\xi-\zeta)\frac{f_{3}}{f_{5}}f_{8}+(f_{6}f_{8}-f_{7})\big{(}\gamma+(\gamma\xi-\zeta)f_{6}\big{)}\Big{)}\sin^{2}\theta\bigg{)}\sin\theta
d\theta\wedge d\phi$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}-\frac{1}{\Pi}\frac{f_{2}}{f_{3}}\Big{(}(\gamma\xi-\zeta)\frac{f_{3}}{f_{5}}+f_{6}\big{(}\gamma+(\gamma\xi-\zeta)f_{6}\big{)}\Big{)}\sin^{2}\theta
d\phi\wedge d\chi,$
$\displaystyle\Pi=1+\frac{f_{2}}{f_{3}}\bigg{(}\frac{f_{3}}{f_{5}}(\gamma\xi-\zeta)^{2}+\big{(}\gamma+(\gamma\xi-\zeta)f_{6}\big{)}^{2}\bigg{)}\sin^{2}\theta,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\Xi=(1+\xi
f_{6})^{2}+\xi^{2}\frac{f_{3}}{f_{5}}+\zeta^{2}\frac{f_{2}}{f_{5}}\sin^{2}\theta.$
(4.3)

We can preserve the $U(1)_{r}$ component given in (B.3) by first fixing
$\zeta=-\xi$ (as in the IIA case), followed by fixing $\gamma=-1$ under the
T-Duality. These conditions together mean $\gamma\xi-\zeta=0$. Due to the
transformation given in (2.49), such conditions break the $SU(2)_{R}$
component of the R-Symmetry - leading to an $\mathcal{N}=1$ background. All
other solutions are of course $\mathcal{N}=0$ backgrounds. One can recover the
$S^{2}$ by enforcing $\gamma=\zeta=0$, giving a one-parameter family of
$S^{2}$ preserved solutions (corresponding to the ATD of the $S^{2}$ preserved
$\mathcal{N}=0$ solution studied in [2]). Unlike in the IIA case however, this
is $\mathcal{N}=0$ for all values of $\xi$ (including $\xi=0$). This is still
a noteworthy example, so we present it explicitly in a later subsection. We
also present the $\mathcal{N}=1$ solution explicitly. One could split (4.1)
into two categories of 2-parameter families, each promoting to the
$\mathcal{N}=1$ solution in a different manner191919It may prove useful to
relabel $\gamma\equiv\hat{\gamma}-1$ such that the $\mathcal{N}=1$ solution is
recovered for $\hat{\gamma}=0$, however given that $\gamma=0$ can lead to a
recovered $S^{2}$, we leave as is.

  * •

Case 1: with $\zeta=-\xi$ (or $\zeta=\gamma\xi$) and $\gamma\in\mathds{Z}$
(promoting to $\mathcal{N}=1$ for $\gamma=-1$),

  * •

Case 2: with $\gamma=-1$ and $(\zeta,\xi)\in\mathds{Z}$ (promoting to
$\mathcal{N}=1$ for $\zeta=-\xi$).

We will however focus on a more general discussion of the full three parameter
family, investigating the boundary by the same procedure as the IIA solutions.

### Investigations at the Boundary

We first note, for general values of $(\eta,\sigma,\theta)$, $(\Xi,\Pi)$ are
non-zero and finite. The deformed $S^{2}$ given by $(\theta,\phi)$ has
$\Pi\rightarrow 1$ at the poles, which given the expression for $M_{3}$ in
(4.1), means it still behaves topologically as an $S^{2}$. This follows the
argument given in [2]. We now turn to the behaviour at the boundaries, keeping
all three parameters non-zero. We will now use the large gauge transformation
which includes the additional term introduced in the IIA discussion

$B_{2}\rightarrow B_{2}+2\kappa(k+\xi N_{k})\text{Vol}(S^{2}).$ (4.4)

  * •

At $\sigma\rightarrow\infty$, we use utilise (G.1) and (G.2) to find

$\begin{gathered}ds_{10,B}^{2}=\kappa\Bigg{[}4\sigma
ds^{2}(\text{AdS}_{5})+\frac{2P}{\pi}\bigg{(}d\Big{(}\frac{\pi}{P}\sigma\Big{)}^{2}+d\Big{(}\frac{\pi}{P}\eta\Big{)}^{2}+\sin^{2}\bigg{(}\frac{\pi\eta}{P}\bigg{)}(d\theta^{2}+\sin^{2}\theta\,d\phi^{2})\bigg{)}\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\frac{1}{4\kappa^{2}\sigma}\,\bigg{(}d\chi+\gamma\frac{\kappa
P}{\pi}\Big{(}\frac{2\pi}{P}\eta-\sin\Big{(}\frac{2\pi}{P}\eta\Big{)}\Big{)}\sin\theta
d\theta\bigg{)}^{2}\Bigg{]},\\\
e^{2\Phi_{\mathcal{B}}}=\frac{P^{2}}{\pi^{3}\mathcal{R}_{1}^{2}}e^{\frac{2\pi\sigma}{P}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}H_{3}=-\frac{4\kappa
P}{\pi}\sin^{2}\Big{(}\frac{\pi}{P}\eta\Big{)}\sin\theta\,d\Big{(}\frac{\pi}{P}\eta\Big{)}\wedge
d\theta\wedge d\phi.\end{gathered}$ (4.5)

It is immediately clear that $\gamma$ plays a special role here, the only
parameter of the three which remains in this limit - when $\gamma=0$, the
$S^{2}$ is recovered. So too is the $S^{3}$ spanned by $(\eta,S^{2})$. This
makes intuitive sense given that the $\mathcal{N}=1$ solution (with a broken
$SU(2)$) is derived when $\gamma=-1$. The background tends to a stack of NS5
branes. The $S^{1}$ shrinks in this limit, in contrast to the IIA cases where
it grows alongside the AdS5 \- this makes intuitive sense following the
T-Duality. Again we find $P$ NS5 branes, as follows (with $2\kappa=\pi$)

$Q_{NS5}=-\frac{1}{(2\pi)^{2}}\int_{(\eta,\theta,\phi)}H_{3}=P.$ (4.6)

  * •

At $\eta=0$ with $\sigma\neq 0$, using the Warp Factor limits in (G), gives

$\displaystyle
f_{4}(d\sigma^{2}+d\eta^{2})+ds^{2}(M_{3})=\frac{1}{\sigma^{2}}\bigg{[}\frac{2|\dot{f}|}{f}\bigg{(}d\sigma^{2}+d\eta^{2}+\eta^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})\bigg{)}+\frac{1}{4\kappa^{2}}D\chi^{2}\bigg{]},$ $\displaystyle
D\chi=d\chi+2\kappa\,\eta\Big{[}2(\gamma\xi-\zeta)\frac{|\dot{f}|}{\sigma^{2}}+\gamma\Big{(}1-\frac{1}{f}\Big{)}\Big{]}\sin\theta
d\theta.$ (4.7)

In this limit we notice that the $\mathds{R}^{3}$ of $(\eta,S^{2})$ is
recovered only when all three parameters are zero.

  * •

At $\sigma=0,~{}\eta\in(k,k+1)$, recall that along the $\sigma=0$ boundary,
$\ddot{V}=0$ to leading order. We now see

$\begin{gathered}\Xi\rightarrow
l_{k}^{2}+\frac{1}{2}\zeta^{2}\mathcal{R}V^{\prime\prime}\sin^{2}\theta,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\frac{\Pi}{f_{2}}\rightarrow\frac{\mathcal{R}}{2\sigma^{2}V^{\prime\prime}}\hat{l}_{k}^{2}\sin^{2}\theta,\\\
l_{k}\equiv
1+\xi(N_{k+1}-N_{k}),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\hat{l}_{k}\equiv\gamma+(\gamma\xi-\zeta)(N_{k+1}-N_{k})=\gamma\,l_{k}-\zeta(N_{k+1}-N_{k}),\end{gathered}$
(4.8)

where

$f_{4}(d\sigma^{2}+d\eta^{2})+\frac{f_{2}}{\Pi}\sin^{2}\theta\,d\phi^{2}\rightarrow\frac{2V^{\prime\prime}}{\mathcal{R}}\bigg{(}d\eta^{2}+\Big{(}d\sigma^{2}+\frac{\sigma^{2}}{\hat{l}_{k}^{2}}d\phi^{2}\Big{)}\bigg{)}.$
(4.9)

So there is a $\mathds{R}^{2}/\mathds{Z}_{\hat{l}_{k}}$ orbifold singularity
in $(\sigma,\phi)$, with $\hat{l}_{k}$. Recall, in the IIA case, this orbifold
singularity was over $(\sigma,\chi)$, with $l_{k}$ \- see (2.59). So in moving
from IIA to IIB, $\chi$ has been replaced by $\phi$, and the orbifold
singularity made a little more general ($l_{k}\rightarrow\hat{l}_{k}$). It is
worth noting that the case of $(\gamma,\zeta)=0$ must be treated separately
because $\hat{l}_{k}=0$ (this is the $S^{2}$ preserving condition). Now we
turn to the remaining internal metric component

$\begin{gathered}f_{2}d\theta^{2}+\frac{1}{f_{1}^{3}f_{3}f_{5}\Pi}\,D\chi^{2}\rightarrow\frac{2\mathcal{R}V^{\prime\prime}}{2\mathcal{R}V^{\prime\prime}+(\mathcal{R}^{\prime})^{2}}d\theta^{2}+\frac{2\mathcal{R}V^{\prime\prime}+(\mathcal{R}^{\prime})^{2}}{4\kappa^{2}\mathcal{R}^{2}}\frac{1}{\hat{l}_{k}^{2}\sin^{2}\theta}D\chi^{2},\\\
D\chi\rightarrow
d\chi+2\kappa\bigg{(}\gamma\,\eta+\frac{\mathcal{R}}{2\mathcal{R}V^{\prime\prime}+(\mathcal{R}^{\prime})^{2}}\Big{(}2(\gamma\xi-\zeta)\mathcal{R}V^{\prime\prime}-\gamma\mathcal{R}^{\prime}\Big{)}\bigg{)}\sin\theta
d\theta.\end{gathered}$ (4.10)

Notice that for the $\mathcal{N}=1$ condition, $\gamma\xi-\zeta=0$, we have
$\hat{l}_{k}=\gamma=-1$.

  * •

At $\sigma=0,~{}\eta=0$, we again adopt the coordinate change (G), where

$\Xi\rightarrow l_{0}^{2},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\Pi\rightarrow
1+\hat{l}_{0}^{2}\,\text{cot}^{2}\alpha\sin^{2}\theta,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}l_{0}=1+\xi
N_{1},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\hat{l}_{0}=\gamma\,l_{0}-\zeta N_{1}.$
(4.11)

We note in general

$\displaystyle
f_{4}(d\sigma^{2}+d\eta^{2})+ds^{2}(M_{3})\rightarrow\frac{2Q}{N_{1}}\bigg{(}dr^{2}+r^{2}d\alpha^{2}+r^{2}\cos^{2}\alpha\bigg{(}d\theta^{2}+\frac{\sin^{2}\theta\,d\phi^{2}}{1+\hat{l}_{0}^{2}\,\text{cot}^{2}\alpha\sin^{2}\theta}\bigg{)}\bigg{)}$
$\displaystyle~{}~{}~{}~{}~{}~{}+\frac{D\chi^{2}}{4\kappa^{2}r^{2}\sin^{2}\alpha(1+\hat{l}_{0}^{2}\,\text{cot}^{2}\alpha\sin^{2}\theta)},~{}~{}~{}~{}~{}~{}~{}~{}~{}D\chi=d\chi+4\kappa(\gamma\xi-\zeta)Qr^{3}\cos^{3}\alpha\sin\theta
d\theta.$ (4.12)

So we see, as one would expect following a T-Duality, the internal space no
longer vanishes as $\mathds{R}^{5}/\mathds{Z}_{l_{0}}$. The orbifold
singularity here is also less clean. Note that for the $\mathcal{N}=1$ case,
$D\chi\rightarrow d\chi$ and again, $\hat{l}_{0}=-1$.

  * •

At $\sigma=0,~{}\eta=k$, we use (G) as in the IIA case, recalling that in this
limit, the term $f_{2}/f_{5}$ term dominates away from the pole. We first
assume then that we are indeed away from the pole, finding

$\displaystyle\Xi\rightarrow\frac{b_{k}\zeta^{2}N_{k}\sin^{2}\theta}{4r},~{}~{}~{}~{}~{}~{}~{}~{}~{}\Pi\rightarrow\frac{N_{k}\sin^{2}\theta}{b_{k}r\sin^{2}\alpha}\Pi_{k}(\alpha),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\Pi_{k}\equiv(\gamma\xi-\zeta)^{2}\frac{b_{k}^{2}}{4}\sin^{2}\alpha+\Big{(}\gamma+(\gamma\xi-\zeta)g(\alpha)\Big{)}^{2},$
$\displaystyle\Pi_{k}(\alpha=0)=\hat{l}_{k-1}^{2},~{}~{}~{}~{}~{}~{}~{}\Pi_{k}(\alpha=\pi)=\hat{l}_{k}^{2}.$
(4.13)

We make the same coordinate trasformation as the IIA case, $r=z^{2}$, giving

$\displaystyle ds_{10,B}^{2}=\zeta\kappa
N_{k}\sin\theta\Big{(}4ds^{2}(\text{AdS}_{5})+ds^{2}(\mathds{B}_{5})\Big{)},$
$\displaystyle
ds^{2}(\mathds{B}_{5})=d\theta^{2}+\frac{4b_{k}}{N_{k}}\bigg{(}dz^{2}+\frac{z^{2}}{4}\Big{(}d\alpha^{2}+\frac{\sin^{2}\alpha}{\Pi_{k}}\,d\phi^{2}\Big{)}+\frac{1}{z^{2}\sin^{2}\theta}\frac{(d\chi+\mathcal{A}_{k})^{2}}{16\,\kappa^{2}\Pi_{k}}\bigg{)},$
$\displaystyle
e^{2\Phi_{\mathcal{B}}}=\frac{\zeta^{4}N_{k}b_{k}\sin^{2}\theta}{4z^{2}\Pi_{k}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mathcal{A}_{k}=2\kappa\big{(}(\gamma\xi-\zeta)N_{k}+\gamma
k\big{)}\sin\theta
d\theta,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}C_{0}=\frac{\gamma}{\zeta}$
$\displaystyle B_{2}=\frac{1}{\Pi_{k}}\bigg{(}\Big{(}1+\xi
g(\alpha)\Big{)}\Big{(}\gamma+(\gamma\xi-\zeta)g(\alpha)\Big{)}+\frac{1}{4}\xi
b_{k}^{2}(\gamma\xi-\zeta)\sin^{2}\alpha\bigg{)}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times\bigg{(}2\kappa\Big{(}\gamma\,k+(\gamma\xi-\zeta)N_{k}\Big{)}\sin\theta
d\theta\wedge d\phi-d\phi\wedge d\chi\bigg{)},$ $\displaystyle
C_{2}=\frac{1}{\Pi_{k}}\bigg{(}\Big{(}(\gamma\xi-\zeta)\frac{1}{4}b_{k}^{2}\sin^{2}\alpha+g(\alpha)\big{(}\gamma+(\gamma\xi-\zeta)g(\alpha)\big{)}\Big{)}d\chi\wedge
d\phi~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(4.14)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+2\kappa\gamma\Big{(}(\gamma\xi-\zeta)\,\frac{k}{4}\,b_{k}^{2}\sin^{2}\alpha+(k\,g(\alpha)-N_{k})\big{(}\gamma+(\gamma\xi-\zeta)g(\alpha)\big{)}\Big{)}\sin\theta
d\theta\wedge d\phi\bigg{)}.$

Now, we find

$\displaystyle ds^{2}(\mathds{B}_{5})\Big{|}_{\alpha\sim
0}=d\theta^{2}+\frac{4b_{k}}{N_{k}}\bigg{(}dz^{2}+\frac{z^{2}}{4}\Big{(}d\alpha^{2}+\frac{\alpha^{2}}{\hat{l}_{k-1}^{2}}\,d\phi^{2}\Big{)}+\frac{1}{z^{2}\sin^{2}\theta}\frac{(d\chi+\mathcal{A}_{k})^{2}}{16\,\kappa^{2}\hat{l}_{k-1}^{2}}\bigg{)},$
(4.15) $\displaystyle
ds^{2}(\mathds{B}_{5})\Big{|}_{\alpha\sim\pi}=d\theta^{2}+\frac{4b_{k}}{N_{k}}\bigg{(}dz^{2}+\frac{z^{2}}{4}\Big{(}d\alpha^{2}+\frac{(\pi-\alpha)^{2}}{\hat{l}_{k}^{2}}\,d\phi^{2}\Big{)}+\frac{1}{z^{2}\sin^{2}\theta}\frac{(d\chi+\mathcal{A}_{k})^{2}}{16\,\kappa^{2}\hat{l}_{k}^{2}}\bigg{)}.$

Note the presence of a $\sin\theta$ out the front of the whole metric! We
observe that $(\alpha,\phi)$ form a
$\mathds{W}\mathds{C}\mathds{P}_{[\hat{l}_{k-1},\hat{l}_{k}]}$. Now we see
that $C_{2}-C_{0}B_{2}=\frac{1}{\zeta}d\phi\wedge d\chi$, meaning that there
are no D5 branes present for $\zeta\neq 0$, noting $\hat{F}_{3}=d(C_{2}\wedge
e^{-B_{2}})$. Before we approach the pole, let’s make an aside and look at the
$\zeta=0$ case in this limit.

    * –

$\zeta=0$: When $\gamma\neq 0$ we have
$\Pi_{k}\rightarrow\gamma^{2}\Delta_{k}$, hence

$\displaystyle\Xi\rightarrow\Delta_{k},~{}~{}~{}~{}~{}~{}~{}~{}\Pi\rightarrow\gamma^{2}\frac{N_{k}\sin^{2}\theta}{b_{k}r\sin^{2}\alpha}\Delta_{k},~{}~{}~{}~{}~{}~{}~{}~{}\Delta_{k}\equiv\frac{1}{4}\xi^{2}b_{k}^{2}\sin^{2}\alpha+\Big{(}1+\xi
g(\alpha)\Big{)}^{2},$ (4.16)

we then see (with the same $\mathcal{A}_{k}$ as above and skipping the gauge
transformation in $B_{2}$ for the moment)

$\displaystyle
ds^{2}=2\kappa\sqrt{\frac{N_{k}}{b_{k}}}\sqrt{\Delta_{k}}\,z\Big{[}4ds^{2}(AdS_{5})+ds^{2}(\mathds{B}_{5})\Big{]},~{}~{}~{}~{}~{}~{}~{}e^{2\Phi_{\mathcal{B}}}=\frac{4z^{2}\Delta_{k}}{\gamma^{2}b_{k}N_{k}\sin^{2}\theta},$
$\displaystyle
ds^{2}(\mathds{B}_{5})=d\theta^{2}+\frac{4b_{k}}{N_{k}}\bigg{(}dz^{2}+\frac{1}{4}z^{2}\Big{(}d\alpha^{2}+\frac{\sin^{2}\alpha}{\gamma^{2}\Delta_{k}}d\phi^{2}\Big{)}+\frac{1}{z^{2}}\frac{(d\chi+\mathcal{A}_{k})^{2}}{16\gamma^{2}\kappa^{2}\Delta_{k}\sin^{2}\theta}\bigg{)},$
$\displaystyle B_{2}=-\frac{1}{\gamma}d\phi\wedge
d\chi,~{}~{}~{}~{}~{}~{}~{}C_{0}=\frac{1}{\Delta_{k}}\Big{(}g(\alpha)\Big{(}1+\xi
g(\alpha)\Big{)}+\frac{1}{4}b_{k}^{2}\xi\sin^{2}\alpha\Big{)},$ $\displaystyle
C_{2}=\frac{1}{\Delta_{k}}\bigg{[}2\kappa\bigg{(}\Big{(}1+\xi
g(\alpha)\Big{)}\Big{(}kg(\alpha)-N_{k}\Big{)}+\frac{1}{4}b_{k}^{2}\xi
k\sin^{2}\alpha\bigg{)}\sin\theta d\theta\wedge d\phi$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-\frac{1}{\gamma}\bigg{(}g(\alpha)\Big{(}1+\xi
g(\alpha)\Big{)}+\frac{1}{4}b_{k}^{2}\xi\sin^{2}\alpha\bigg{)}d\phi\wedge
d\chi\bigg{]}.$ (4.17)

Note, similar to the IIA case, the $S^{2}$ has been replaced by a Spindle
here. Now we again impose the two gauge transformations separately. Using
(2.54), we have

$\displaystyle B_{2}\rightarrow B_{2}+2\kappa
k\text{Vol}(S^{2}),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}B_{2}=-\frac{1}{\gamma}d\phi\wedge
d\chi+2\kappa k\sin\theta d\theta\wedge d\phi,$ $\displaystyle
Q_{D5}=-\frac{1}{(2\pi)^{2}}\int_{(\alpha,\theta,\phi)}\hat{F}_{3}=-\frac{1}{(2\pi)^{2}}\int_{\phi=0}^{\phi=2\pi}\int_{\theta=0}^{\theta=\pi}\Big{[}C_{3}-C_{1}\wedge
B_{2}\Big{]}_{\alpha=0}^{\alpha=\pi}=\frac{N_{k}}{l_{k}}-\frac{N_{k-1}}{l_{k-1}}.$
(4.18)

Alternatively, as in the IIA case, we can include an additional term in the
gauge transformation

$\displaystyle B_{2}\rightarrow B_{2}+2\kappa(k+\xi
N_{k})\text{Vol}(S^{2}),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}B_{2}=-\frac{1}{\gamma}d\phi\wedge
d\chi+2\kappa(k+\xi N_{k})\sin\theta d\theta\wedge d\phi,$ $\displaystyle
Q_{D5}=-\frac{1}{(2\pi)^{2}}\int_{(\alpha,\theta,\phi)}\hat{F}_{3}=N_{k}-N_{k-1}.$
(4.19)

So this additional term in the gauge transformation recovers the quantization
of D5 charge, eliminating the effect of the orbifold singularity present. The
same result is true in the $\gamma=0$ case (with $\zeta=0$). In the IIA
background, D4 branes were only present when $\zeta=0$. Here we see the same
behaviour with the D5 branes, but now the $S^{2}$ preserved condition is
$(\gamma,\zeta)=0$. Therefore, in the IIB case, we can have D5 branes without
the requirement of a preserved $S^{2}$ (as long as the $S^{2}$ is broken under
the ATD itself).

We now return to the general case at this boundary. As in the IIA case, we
approach the pole $(\sigma=0,\eta=k,\sin\theta=0)$ using

$\eta=k-\rho\cos\alpha\sin^{2}\mu,~{}~{}~{}~{}~{}\sigma=\rho\sin\alpha\sin^{2}\mu,~{}~{}~{}~{}~{}~{}\sin\theta=2\sqrt{\frac{b_{k}\rho}{N_{k}}}\cos\mu,$
(4.20)

recalling that $r=\rho\sin^{2}\mu$, and expanding about $\rho=0$. Now

$\displaystyle\sin^{2}\alpha\sin^{2}\mu\,\Pi\rightarrow\sin^{2}\alpha\sin^{2}\mu+4\cos^{2}\mu\,\Pi_{k},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\sin^{2}\mu\,\Xi\rightarrow\tilde{\Xi}_{k}=\Delta_{k}\sin^{2}\mu+\zeta^{2}b_{k}^{2}\cos^{2}\mu,$
$\displaystyle\Delta_{k}\equiv\frac{1}{4}\xi^{2}b_{k}^{2}\sin^{2}\alpha+\Big{(}1+\xi
g(\alpha)\Big{)}^{2},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}B_{2}\rightarrow
B_{2}+\frac{4\kappa b_{k}\cos^{2}\mu}{N_{k}}(k+\xi N_{k})d\rho\wedge d\phi,$
(4.21)

giving

$\displaystyle\frac{ds^{2}}{2\kappa\sqrt{N_{k}}}=\sqrt{\tilde{\Xi}_{k}}\Bigg{[}\frac{4}{\sqrt{\frac{b_{k}}{\rho}}}ds^{2}(AdS_{5})+\frac{\sqrt{\frac{b_{k}}{\rho}}}{N_{k}}ds^{2}(\mathds{B}_{5})\Bigg{]},~{}~{}~{}~{}~{}~{}~{}~{}~{}e^{\Phi_{\mathcal{B}}}=\frac{2\,\tilde{\Xi}_{k}}{b_{k}\sin\mu\sqrt{\sin^{2}\alpha\sin^{2}\mu+4\cos^{2}\mu\,\Pi_{k}}},$
$\displaystyle
ds^{2}(\mathds{B}_{5})=d\rho^{2}+4\rho^{2}\bigg{[}d\mu^{2}+\frac{1}{4}\sin^{2}\mu\bigg{(}\,d\alpha^{2}+\frac{\sin^{2}\alpha}{\Pi_{k}+\frac{1}{4}\sin^{2}\alpha\tan^{2}\mu}d\phi^{2}\bigg{)}\bigg{]}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\frac{1}{\rho}\frac{N_{k}}{4\kappa^{2}b_{k}\sin^{2}\mu}\frac{(d\chi+\mathcal{A}_{k})^{2}}{(\sin^{2}\alpha\sin^{2}\mu+4\,\cos^{2}\mu\,\Pi_{k})},$
$\displaystyle\mathcal{A}_{k}=\frac{4\kappa b_{k}\cos\mu}{N_{k}}\Big{(}\gamma
k+(\gamma\xi-\zeta)N_{k}\Big{)}\Big{(}\cos\mu\,d\rho-2\rho\sin\mu\,d\mu\Big{)},$
$\displaystyle
B_{2}=\frac{4\cos^{2}\mu}{\sin^{2}\alpha\sin^{2}\mu+4\cos^{2}\mu\,\Pi_{k}}\bigg{(}\Big{(}1+\xi
g(\alpha)\Big{)}\Big{(}\gamma+(\gamma\xi-\zeta)g(\alpha)\Big{)}+\frac{1}{4}\xi
b_{k}^{2}(\gamma\xi-\zeta)\sin^{2}\alpha\bigg{)}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times\bigg{(}\frac{4\kappa
b_{k}}{N_{k}}\cos^{2}\mu\Big{(}\gamma
k+(\gamma\xi-\zeta)N_{k}\Big{)}d\rho\wedge d\phi-d\phi\wedge d\chi\bigg{)},$
$\displaystyle C_{0}=\frac{1}{\tilde{\Xi}_{k}}\bigg{(}g(\alpha)\Big{(}1+\xi
g(\alpha)\Big{)}\sin^{2}\mu+\frac{1}{4}b_{k}^{2}\Big{(}4\gamma\zeta\,\cos^{2}\mu+\xi\sin^{2}\alpha\sin^{2}\mu\Big{)}\bigg{)}$
$\displaystyle
C_{2}=\frac{4}{\sin^{2}\alpha\sin^{2}\mu+4\cos^{2}\mu\,\Pi_{k}}\bigg{[}\frac{b_{k}\kappa\cos^{2}\mu}{N_{k}}\bigg{(}\gamma
b_{k}^{2}k(\gamma\xi-\zeta)\cos^{2}\mu\sin^{2}\alpha+4\gamma
k(\gamma\xi-\zeta)g(\alpha)^{2}\,\cos^{2}\mu$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}-N_{k}(\sin^{2}\alpha\sin^{2}\mu+4\gamma^{2}\cos^{2}\mu)+4\gamma\,g(\alpha)\,\cos^{2}\mu\Big{(}\gamma
k-(\gamma\xi-\zeta)N_{k}\Big{)}\bigg{)}d\rho\wedge d\phi$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}-\cos^{2}\mu\bigg{(}\frac{1}{4}b_{k}^{2}\sin^{2}\alpha(\gamma\xi-\zeta)+g(\alpha)\Big{(}\gamma+(\gamma\xi-\zeta)g(\alpha)\Big{)}\bigg{)}d\phi\wedge
d\chi\bigg{]}.$ (4.22)

So again, we see that the $S^{2}$ has inherited orbifold singularities. On
first site, these appear more complicated than the Spindle found in IIA (in
the general case at least). It would be nice to understand this further
topologically. Now we calculate the charge of the D7 branes at
$\mu=\frac{\pi}{2}$ on one of the components of the ‘Spindle like’ space,
$\alpha$, as follows

$Q_{D7}=-\int_{\alpha}F_{1}=C_{0}\Big{|}_{\alpha=0}^{\alpha=\pi}=\frac{2N_{k}-N_{k+1}-N_{k-1}}{l_{k}l_{k-1}}.$
(4.23)

This is the result for all values of $(\gamma,\zeta)$ \- including the
$\mathcal{N}=1$ background. It is important to note that this broken
quantization of charge is due to the $\tilde{\Xi}_{k}$ (and hence
$\Delta_{k}$) in the denominator of $C_{0}$, and not $\Pi_{k}$ which is what
gives rise to singularities in the internal manifold. It appears then on first
sight that this breaking of quantization here is simply a remnant of the IIA
orbifold singularity. As a result of this, one can find situations where the
background has the above charge without having orbifold singularities in it’s
metric - the $\mathcal{N}=1$ and $S^{2}$ preserved $\mathcal{N}=0$ cases are
two such example. We now write them explicitly.

####  4.1.1 $\mathcal{N}=1$ Type IIB - Background

Let’s now present the one-parameter $\mathcal{N}=1$ family as promised, where
$\gamma=-1,\zeta=-\xi$

$\displaystyle
ds_{10,st}^{2}=e^{\frac{2}{3}\Phi_{\mathcal{A}}}f_{1}\bigg{[}4ds^{2}(\text{AdS}_{5})+f_{2}d\theta^{2}+f_{4}(d\sigma^{2}+d\eta^{2})+\frac{f_{2}f_{3}}{f_{3}+f_{2}\sin^{2}\theta}\sin^{2}\theta
d\phi^{2}+\frac{(d\chi+f_{8}\sin\theta
d\theta)^{2}}{f_{1}^{3}(f_{3}f_{5}+\sin^{2}\theta f_{2}f_{5})}\bigg{]},$
$\displaystyle e^{\frac{4}{3}\Phi_{\mathcal{A}}}=f_{1}\Big{[}f_{5}(1+\xi
f_{6})^{2}+\xi^{2}f_{3}+\xi^{2}f_{2}\sin^{2}\theta\Big{]},$ $\displaystyle
e^{2\Phi^{\mathcal{B}}}=\frac{\big{(}f_{5}(1+\xi
f_{6})^{2}+\xi^{2}f_{3}+\xi^{2}\sin^{2}\theta
f_{2}\big{)}^{2}}{f_{3}f_{5}+\sin^{2}\theta
f_{2}f_{5}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}C_{0}=\frac{f_{5}f_{6}(1+\xi
f_{6})+\xi f_{3}+\xi f_{2}\sin^{2}\theta}{f_{5}(1+\xi
f_{6})^{2}+\xi^{2}f_{3}+\xi^{2}f_{2}\sin^{2}\theta},$ $\displaystyle
B_{2}=\bigg{(}\frac{f_{2}f_{8}(1+\xi
f_{6})\sin^{2}\theta}{f_{3}+f_{2}\sin^{2}\theta}-(f_{8}+\xi
f_{7})\bigg{)}\sin\theta d\phi\wedge d\theta+\frac{f_{2}(1+\xi
f_{6})\sin^{2}\theta}{f_{3}+f_{2}\sin^{2}\theta}d\phi\wedge d\chi,$
$\displaystyle
C_{2}=\frac{f_{3}f_{7}+f_{2}(f_{7}-f_{6}f_{8})\sin^{2}\theta}{f_{3}+f_{2}\sin^{2}\theta}\sin\theta
d\theta\wedge
d\phi+\frac{f_{2}f_{6}\sin^{2}\theta}{f_{3}+f_{2}\sin^{2}\theta}d\phi\wedge
d\chi.$ (4.24)

See Appendix A.3.2 for the G-Structure description. As already discussed, this
background has NS5 branes, D7 branes (with charge given in (4.23)), but no D5
branes. As previously mentioned, there is no orbifold singularity present in
this metric, but the D7 branes still inherit the broken quantization of charge
from it’s type IIA ancestor.

####  4.1.2 $S^{2}$ preserved $\mathcal{N}=0$ Solutions

In this case, fixing $(\gamma,\zeta)=0$ preserves the $S^{2}$ whilst breaking
the Supersymmetry. Of course to get the non-deformed solution, we fix $\xi=0$
in the following discussion. The background reads

$\displaystyle
ds_{10,B}^{2}=e^{\frac{2}{3}\Phi_{A}}f_{1}\Bigg{[}4ds^{2}(\text{AdS}_{5})+f_{2}ds^{2}(S^{2})+f_{4}(d\sigma^{2}+d\eta^{2})+\frac{1}{f_{1}^{3}f_{3}f_{5}}\,d\chi^{2}\Bigg{]},$
$\displaystyle e^{2\Phi_{B}}=\frac{1}{f_{3}f_{5}}\Big{[}f_{5}(1+\xi
f_{6})^{2}+\xi^{2}f_{3}\Big{]}^{2},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}e^{\frac{4}{3}\Phi_{A}}=f_{1}\Big{[}f_{5}(1+\xi
f_{6})^{2}+\xi^{2}f_{3}\Big{]},$ $\displaystyle C_{0}=\frac{f_{5}f_{6}(1+\xi
f_{6})+\xi f_{3}}{f_{5}(1+\xi
f_{6})^{2}+\xi^{2}f_{3}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}B_{2}=(f_{8}+\xi
f_{7})\text{Vol}(S^{2}),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}C_{2}=f_{7}\text{Vol}(S^{2}).$
(4.25)

We can see from the metric in this case that there are no orbifold
singularities present here. However, from the boundary analysis just
presented, we know that the charges have the same broken quantization as
backgrounds containing spindles - in both the D7 branes and (depending on the
gauge transformation chosen) in the D5 branes. Therefore, perhaps a different
interpretation may be required, one possible proposal is the idea of rotating
the D branes - see Appendix E.

###  4.2 More Solutions

In a similar vein to the IIA discussion, the three-parameter family of
solutions given in (4.1) is not the only one which can be derived. We have in
addition (– ‣ • ‣ C), (– ‣ • ‣ C) and (– ‣ • ‣ C) which all map to (4.1) under
the transformations given in (C.20), (C.27) and (C.37), respectively. As in
the IIA section, one can still derive unique solutions from these backgrounds
because of the nature of the required mappings. We refrain from including ATDs
along $\phi$ here, as typically performing a T-Duality along the $U(1)$ of an
$S^{2}$ can lead to singularities in the dual description. Perhaps the same is
true in the Spindle case (see (– ‣ • ‣ C) and (– ‣ • ‣ C) for the solutions).

#### ATD along $\beta$ of the $\chi$ reduction

Performing an ATD along $\beta$ of the $\chi$ reduction, one derives (– ‣ • ‣
C), which requires $\xi\rightarrow 1/\xi$ to map to (4.1). Hence, by fixing
$\xi=0$, we derive another unique solution

$\displaystyle
ds_{10,st}^{2}=e^{\frac{2}{3}\Phi_{\mathcal{A}}}f_{1}\bigg{[}4ds^{2}(\text{AdS}_{5})+f_{2}d\theta^{2}+f_{4}(d\sigma^{2}+d\eta^{2})+\frac{1}{\hat{\Xi}}\bigg{(}f_{2}f_{3}f_{5}\sin^{2}\theta
d\phi^{2}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\frac{1}{f_{1}^{3}}\Big{(}d\beta+\big{(}\zeta
f_{8}-\gamma f_{7}\big{)}\sin\theta d\theta\Big{)}^{2}\bigg{)}\bigg{]}$
$\displaystyle
e^{2\Phi_{\mathcal{B}}}=\frac{1}{\hat{\Xi}}\big{(}f_{5}f_{6}^{2}+f_{3}+\zeta^{2}f_{2}\sin^{2}\theta\big{)}^{2},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}e^{\frac{4}{3}\Phi_{\mathcal{A}}}=f_{1}\Big{[}f_{5}f_{6}^{2}+f_{3}+\zeta^{2}f_{2}\sin^{2}\theta\Big{]},$
$\displaystyle B_{2}=\sin\theta\frac{1}{\hat{\Xi}}\bigg{[}f_{2}\Big{(}\gamma
f_{3}+f_{5}f_{6}(\gamma f_{6}-\zeta)\Big{)}\sin\theta
d\beta~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\Big{(}f_{3}f_{5}f_{7}+\zeta
f_{2}\Big{(}\gamma f_{3}f_{8}+f_{5}(f_{6}f_{8}-f_{7})(\gamma
f_{6}-\zeta)\Big{)}\sin^{2}\theta\Big{)}d\theta\bigg{]}\wedge d\phi,$
$\displaystyle C_{2}=\sin\theta\frac{1}{\hat{\Xi}}\bigg{[}f_{2}f_{5}(\gamma
f_{6}-\zeta)\sin\theta d\beta+\Big{(}f_{3}f_{5}f_{8}+\gamma f_{2}\Big{(}\gamma
f_{3}f_{8}+f_{5}(f_{6}f_{8}-f_{7})(\gamma
f_{6}-\zeta)\Big{)}\sin^{2}\theta\Big{)}d\theta\bigg{]}\wedge d\phi,$
$\displaystyle
C_{0}=f_{1}e^{-\frac{4}{3}\Phi_{\mathcal{A}}}\Big{(}f_{5}f_{6}+\gamma\zeta
f_{2}\sin^{2}\theta\Big{)},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\hat{\Xi}=f_{3}f_{5}+f_{2}\big{(}\gamma^{2}f_{3}+f_{5}(\gamma
f_{6}-\zeta)^{2}\big{)}\sin^{2}\theta.$ (4.26)

This time, the background is a two-parameter family of $\mathcal{N}=0$
solutions which promotes to $\mathcal{N}=1$ for $\zeta=-1,~{}\gamma=0$. We now
turn to this solution.

####  4.2.1 Unique $\mathcal{N}=1$ IIB Solution

We now derive a new and unique $\mathcal{N}=1$ background (corresponding to
the $\xi=0$ case of (– ‣ • ‣ C)202020Notice that the transformations given in
(C.29) and (C.30) also demonstrate that the $\xi=0$ solution is unique. Here
$(p,b,u)=1,~{}(q,c,m)=0,~{}a\equiv\xi=0,~{}s\equiv\zeta=-1,~{}v\equiv\gamma=0$.).

$\displaystyle
ds_{10,st}^{2}=e^{\frac{2}{3}\Phi_{A}}f_{1}\bigg{[}4ds^{2}(\text{AdS}_{5})+f_{2}d\theta^{2}+f_{4}(d\sigma^{2}+d\eta^{2})+\frac{f_{2}f_{3}}{f_{3}+f_{2}\sin^{2}\theta}\sin^{2}\theta
d\phi^{2}+\frac{(d\beta-f_{8}\sin\theta
d\theta)^{2}}{f_{1}^{3}(f_{3}f_{5}+\sin^{2}\theta f_{2}f_{5})}\bigg{]}$
$\displaystyle
e^{\frac{4}{3}\Phi_{A}}=f_{1}\Big{[}f_{5}f_{6}^{2}+f_{3}+f_{2}\sin^{2}\theta\Big{]},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}e^{2\Phi^{B}}=\frac{\big{(}f_{5}f_{6}^{2}+f_{3}+\sin^{2}\theta
f_{2}\big{)}^{2}}{f_{3}f_{5}+\sin^{2}\theta f_{2}f_{5}},$ $\displaystyle
B_{2}=\frac{\sin\theta}{f_{3}+f_{2}\sin^{2}\theta}\bigg{(}f_{2}f_{6}\sin\theta
d\beta+\Big{(}f_{3}f_{7}+f_{2}(f_{7}-f_{6}f_{8})\sin^{2}\theta\Big{)}d\theta\bigg{)}\wedge
d\phi,$ $\displaystyle
C_{2}=\frac{\sin\theta}{f_{3}+f_{2}\sin^{2}\theta}\Big{(}f_{2}\sin\theta
d\beta+f_{3}f_{8}d\theta\Big{)}\wedge
d\phi~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}C_{0}=\frac{f_{5}f_{6}}{f_{5}f_{6}^{2}+f_{3}+\sin^{2}\theta
f_{2}}.$ (4.27)

This solution corresponds to the SUSY preserving abelian T-Duality along
$\beta$ of the unique IIA $\mathcal{N}=1$ solution given in (3.1.1).

#### ATD along $\beta$ of the $\phi$ reduction

Performing an ATD along $\beta$ of the $\phi$ reduction, one derives (– ‣ • ‣
C). To map this solution to (4.1) we require transformations involving
$1/\xi,~{}1/\zeta,1/\gamma$ and $1/(\zeta-\gamma\xi)$ (see (C.37)).

Fixing $\xi=0$, we get

$\displaystyle
ds_{10,st}^{2}=e^{\frac{2}{3}\Phi_{\mathcal{A}}}f_{1}\bigg{[}4ds^{2}(\text{AdS}_{5})+f_{2}d\theta^{2}+f_{4}(d\sigma^{2}+d\eta^{2})~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle~{}~{}~{}~{}~{}~{}+\frac{1}{\hat{\Xi}}\bigg{(}f_{2}f_{3}f_{5}\sin^{2}\theta
d\chi^{2}+\frac{1}{f_{1}^{3}}\Big{(}d\beta+(f_{8}+\gamma f_{7})\sin\theta
d\theta\Big{)}^{2}\bigg{)}\bigg{]},~{}~{}~{}~{}~{}~{}~{}e^{\frac{4}{3}\Phi_{\mathcal{A}}}=f_{1}\Big{(}\zeta^{2}(f_{5}f_{6}^{2}+f_{3})+f_{2}\sin^{2}\theta\Big{)},$
$\displaystyle
e^{2\Phi_{\mathcal{B}}}=\frac{1}{\hat{\Xi}}\big{(}\zeta^{2}(f_{5}f_{6}^{2}+f_{3})+f_{2}\sin^{2}\theta\big{)}^{2},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\hat{\Xi}=\zeta^{2}f_{3}f_{5}+f_{2}\Big{(}\gamma^{2}f_{3}+f_{5}(1+\gamma
f_{6})^{2}\Big{)}\sin^{2}\theta,$ $\displaystyle
B_{2}=\frac{1}{\hat{\Xi}}\bigg{[}f_{2}(\gamma f_{3}+f_{5}f_{6}(1+\gamma
f_{6}))\sin^{2}\theta
d\beta~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-\sin\theta\Big{(}\zeta^{2}f_{3}f_{5}f_{7}+f_{2}\Big{(}-\gamma
f_{3}f_{8}+f_{5}(f_{7}-f_{6}f_{8})(1+\gamma
f_{6})\Big{)}\sin^{2}\theta\Big{)}d\theta\bigg{]}\wedge d\chi,$ $\displaystyle
C_{0}=\zeta f_{1}e^{-\frac{4}{3}\Phi_{\mathcal{A}}}\Big{(}\gamma
f_{3}+f_{5}f_{6}(1+\gamma
f_{6})\Big{)},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}C_{2}=-\frac{\zeta}{\hat{\Xi}}f_{3}f_{5}\Big{(}d\beta+(f_{8}+\gamma
f_{7})\sin\theta d\theta\Big{)}\wedge d\chi.$ (4.28)

Fixing $\zeta=0$, we find

$\displaystyle
ds_{10,st}^{2}=e^{\frac{2}{3}\Phi_{\mathcal{A}}}f_{1}\bigg{[}4ds^{2}(\text{AdS}_{5})+f_{2}d\theta^{2}+f_{4}(d\sigma^{2}+d\eta^{2})~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle~{}~{}~{}~{}~{}+\frac{1}{\hat{\Xi}}\bigg{(}f_{2}f_{3}f_{5}\sin^{2}\theta
d\chi^{2}+\frac{1}{f_{1}^{3}}\Big{(}d\beta+(f_{8}+\gamma f_{7})\sin\theta
d\theta\Big{)}^{2}\bigg{)}\bigg{]},~{}~{}~{}~{}~{}~{}~{}~{}e^{\frac{4}{3}\Phi_{\mathcal{A}}}=f_{1}(\xi^{2}f_{5}+f_{2}\sin^{2}\theta),$
$\displaystyle
e^{2\Phi_{\mathcal{B}}}=\frac{1}{\hat{\Xi}}\big{(}\xi^{2}f_{5}+f_{2}\sin^{2}\theta\big{)}^{2},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\hat{\Xi}=\gamma^{2}\xi^{2}f_{3}f_{5}+f_{2}\Big{(}\gamma^{2}f_{3}+f_{5}(1+\gamma
f_{6})^{2}\Big{)}\sin^{2}\theta$ $\displaystyle
B_{2}=\frac{1}{\hat{\Xi}}\bigg{[}\Big{(}\gamma\xi^{2}f_{3}f_{5}+f_{2}(\gamma
f_{3}+f_{5}f_{6}(1+\gamma
f_{6}))\sin^{2}\theta\Big{)}d\beta~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle~{}~{}~{}~{}~{}~{}+\sin\theta\Big{(}\gamma\xi^{2}f_{3}f_{5}f_{8}-f_{2}\Big{(}-\gamma
f_{3}f_{8}+f_{5}(f_{7}-f_{6}f_{8})(1+\gamma
f_{6})\Big{)}\sin^{2}\theta\Big{)}d\theta\bigg{]}\wedge d\chi,$ $\displaystyle
C_{0}=\xi\,f_{1}f_{5}(1+\gamma
f_{6})\,e^{-\frac{4}{3}\Phi_{\mathcal{A}}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}C_{2}=\frac{\gamma\xi}{\hat{\Xi}}f_{3}f_{5}\Big{(}d\beta+(f_{8}+\gamma
f_{7})\sin\theta d\theta\Big{)}\wedge d\chi,$ (4.29)

Fixing $\gamma=0$, we get

$\displaystyle
ds_{10,st}^{2}=e^{\frac{2}{3}\Phi_{\mathcal{A}}}f_{1}\bigg{[}4ds^{2}(\text{AdS}_{5})+f_{2}d\theta^{2}+f_{4}(d\sigma^{2}+d\eta^{2})+\frac{1}{\hat{\Xi}}\bigg{(}f_{2}f_{3}f_{5}\sin^{2}\theta
d\chi^{2}+\frac{1}{f_{1}^{3}}\Big{(}d\beta+f_{8}\sin\theta
d\theta\Big{)}^{2}\bigg{)}\bigg{]}$ $\displaystyle
e^{2\Phi_{\mathcal{B}}}=\frac{1}{\hat{\Xi}}\big{(}f_{5}(\zeta
f_{6}+\xi)^{2}+\zeta^{2}f_{3}+f_{2}\sin^{2}\theta\big{)}^{2},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}e^{\frac{4}{3}\Phi_{\mathcal{A}}}=f_{1}\Big{[}f_{5}(\zeta
f_{6}+\xi)^{2}+\zeta^{2}f_{3}+f_{2}\sin^{2}\theta\Big{]},$ $\displaystyle
B_{2}=\frac{1}{\hat{\Xi}}\bigg{[}f_{5}(-\xi\zeta
f_{3}+f_{2}f_{6}\sin^{2}\theta)d\beta+\sin\theta\Big{(}-\zeta f_{3}f_{5}(\zeta
f_{7}+\xi
f_{8})-f_{2}f_{5}(f_{7}-f_{6}f_{8})\sin^{2}\theta\Big{)}d\theta\bigg{]}\wedge
d\chi,$ $\displaystyle C_{0}=f_{1}f_{5}(\zeta
f_{6}+\xi)\,e^{-\frac{4}{3}\Phi_{\mathcal{A}}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}C_{2}=-\frac{\zeta}{\hat{\Xi}}f_{3}f_{5}\Big{(}d\beta+f_{8}\sin\theta
d\theta\Big{)}\wedge
d\chi,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\hat{\Xi}=\zeta^{2}f_{3}f_{5}+f_{2}f_{5}\sin^{2}\theta.$
(4.30)

Then, by setting $\zeta=-1$ in this background, one re-derives the
$\mathcal{N}=1$ solution in (– ‣ • ‣ C) (following the transformations given
in footnote 30).

Finally, fixing $\zeta=\gamma\,\xi$, we find

$\displaystyle
ds_{10,st}^{2}=e^{\frac{2}{3}\Phi_{\mathcal{A}}}f_{1}\bigg{[}4ds^{2}(\text{AdS}_{5})+f_{2}d\theta^{2}+f_{4}(d\sigma^{2}+d\eta^{2})~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\frac{1}{\hat{\Xi}}\bigg{(}f_{2}f_{3}f_{5}\sin^{2}\theta
d\chi^{2}+\frac{1}{f_{1}^{3}}\Big{(}d\beta+(f_{8}+\gamma f_{7})\sin\theta
d\theta\Big{)}^{2}\bigg{)}\bigg{]}$ $\displaystyle
e^{2\Phi_{\mathcal{B}}}=\frac{1}{\hat{\Xi}}\big{(}\xi^{2}f_{5}(\gamma
f_{6}+1)^{2}+\xi^{2}\gamma^{2}f_{3}+f_{2}\sin^{2}\theta\big{)}^{2},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}e^{\frac{4}{3}\Phi_{\mathcal{A}}}=f_{1}\Big{[}\xi^{2}f_{5}(\gamma
f_{6}+1)^{2}+\xi^{2}\gamma^{2}f_{3}+f_{2}\sin^{2}\theta\Big{]},$
$\displaystyle B_{2}=\frac{1}{\hat{\Xi}}\bigg{[}\Big{(}f_{2}(\gamma
f_{3}+f_{5}f_{6}(1+\gamma
f_{6}))\sin^{2}\theta\Big{)}d\beta+\sin\theta\Big{(}f_{2}\Big{(}\gamma
f_{3}f_{8}-f_{5}(f_{7}-f_{6}f_{8})(1+\gamma
f_{6})\Big{)}\sin^{2}\theta\Big{)}d\theta\bigg{]}\wedge d\chi,$ $\displaystyle
C_{0}=\xi
f_{1}e^{-\frac{4}{3}\Phi_{\mathcal{A}}}\Big{(}\gamma^{2}f_{3}+f_{5}(1+\gamma
f_{6})^{2}\Big{)},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}C_{2}=0,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\hat{\Xi}=f_{2}\Big{(}\gamma^{2}f_{3}+f_{5}(1+\gamma
f_{6})^{2}\Big{)}\sin^{2}\theta.$ (4.31)

#### ATD along $\chi$ of the $\phi$ reduction

Performing an ATD along $\chi$ of the $\phi$ reduction leads to the following
$\mathcal{N}=0$ family of solutions

$\displaystyle
ds_{10,st}^{2}=e^{\frac{2}{3}\Phi_{A}}f_{1}\bigg{[}4ds^{2}(\text{AdS}_{5})+f_{2}d\theta^{2}+f_{4}(d\sigma^{2}+d\eta^{2})~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\frac{1}{\hat{\Xi}}\bigg{(}f_{2}f_{3}f_{5}\sin^{2}\theta
d\beta^{2}+\frac{1}{f_{1}^{3}}(d\chi+f_{7}\sin\theta
d\theta)^{2}\bigg{)}\bigg{]}$ $\displaystyle
e^{2\Phi^{B}}=\frac{1}{\hat{\Xi}}\big{(}f_{5}(\zeta
f_{6}+\xi)^{2}+\zeta^{2}f_{3}+f_{2}\sin^{2}\theta\big{)}^{2},~{}~{}~{}~{}~{}~{}~{}~{}~{}e^{\frac{4}{3}\Phi_{A}}=f_{1}\Big{[}f_{5}(\zeta
f_{6}+\xi)^{2}+\zeta^{2}f_{3}+f_{2}\sin^{2}\theta\Big{]},$ $\displaystyle
C_{0}=f_{1}e^{-\frac{4}{3}\Phi_{A}}\Big{(}f_{5}f_{6}(\zeta f_{6}+\xi)+\zeta
f_{3}\Big{)},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\hat{\Xi}=\xi^{2}f_{3}f_{5}+f_{2}(f_{3}+f_{5}f_{6}^{2})\sin^{2}\theta,$
$\displaystyle B_{2}=-\frac{1}{\hat{\Xi}}\bigg{[}f_{5}(\zeta\xi
f_{3}-f_{2}f_{6}\sin^{2}\theta)d\chi~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle~{}~{}+\Big{[}\xi f_{3}f_{5}(\zeta f_{7}+\xi
f_{8})+f_{2}\Big{(}f_{3}f_{8}+f_{5}f_{6}(f_{6}f_{8}-f_{7})\Big{)}\sin^{2}\theta\Big{]}\sin\theta
d\theta\bigg{]}\wedge d\beta+\gamma d\chi\wedge d\beta,$ $\displaystyle
C_{2}=-\frac{\xi}{\hat{\Xi}}f_{3}f_{5}\Big{(}d\chi+f_{7}\sin\theta
d\theta\Big{)}\wedge d\beta.$ (4.32)

##  5 Conclusions and future study

We now briefly summarize the new results presented throughout this paper.

  * •

After reviewing the Gaiotto-Maldacena solutions in M-Theory and some relevant
material on the method of G-Structures, we included the M-Theory G-Structure
forms corresponding to a reduction along $\chi$ and $\phi$. We then presented
the new IIB G-Structure conditions in terms of the Pure Spinors $\Psi_{\pm}$,
derived in detail in Appendix A.

  * •

Investigations at the boundary were then conducted for the general two-
parameter family of Type IIA solutions presented in [2], following the
analysis of the $\mathcal{N}=1$ solution. We find NS5 and D6 branes in all
backgrounds, with the D4 branes only present for $S^{2}$ preserved solutions.
Introducing an additional parameter into the gauge transformation of $B_{2}$,
we find the integer quantization of D4 charge is recovered, negating the
effect of the Spindle. This is not possible for the D6 branes, but the
rational charge has the same form for all backgrounds.

  * •

Following an $SL(3,\mathds{R})$ transformation in M-Theory, by performing a
dimensional reduction along $\chi$ and $\phi$ in turn, some new IIA solutions
are presented (including a new $\mathcal{N}=1$ solution). An in-depth
derivation is given in Appendix B.

  * •

We then present a three-parameter family of type IIB backgrounds, derived by
performing an ATD of the two-parameter IIA background. This solution contains
within it a one-parameter family of $\mathcal{N}=1$ solutions, as well as a
one-parameter family of $S^{2}$ preserved $\mathcal{N}=0$ solutions.
Investigations at the boundary were conducted in the same manner as the IIA
case, finding the presence of orbifold singularities. In analogy with the IIA
solutions, NS5 and D7 branes are present in all cases. However, D5 branes only
appear in backgrounds which descend from the $S^{2}$ preserved IIA solutions.
In studying the ATD for multiple cases, with full calculations given in
Appendix C, the TST solution of [4] is re-derived - now implying this is an
$\mathcal{N}=0$ solution in general, promoting to $\mathcal{N}=1$ for
$\gamma=-1$.

  * •

This then motivated the uplifting of a TST solution, deriving a deformation of
the M-Theory Gaiotto-Maldacena solution (believed to be Supersymmetry
breaking, and given in Appendix D). It proved useful to perform an ATD and TST
for a general form, deriving the ATD solutions for many backgrounds at once.
These forms are presented in Appendix C. The breaking of quantization seems to
occur without the presence of an orbifold singularity in some solutions, a
proposal for the possible interpretation of this is given in Appendix E\-
involving rotated D-branes.

  * •

Some additional IIB backgrounds are presented, including a zero-parameter
$\mathcal{N}=1$ solution.

  * •

The Holographic Central charge was derived for the general metric, showing as
in [2], the deformations are marginal in the dual CFT. See Appendix F. The
G-Structure forms for all $\mathcal{N}=1$ solutions are derived, and presented
in Appendix A.3.

As outlined throughout this paper, there is still analysis which could and
should be conducted in future work. In addition to the list presented in [2],

  * •

Investigate more thoroughly the backgrounds presented in this paper, including
the charge quantization, behaviour at the boundaries and stability analysis,
where required.

  * •

Use the TST formula presented in Appendix C to check whether performing a TST
transformation on these multi-parameter backgrounds leads to an additional
(non-trivial) parameter.

  * •

It would be interesting to take a closer look at the deformed Gaiotto-
Maldacena solution in M-Theory.

  * •

It would be nice to investigate specific Rank Function examples for these
backgrounds, including the Triangular, Trapezium and Sfetsos-Thompson cases.

  * •

It would be interesting to use the multi-parameter solutions presented here
within the context of Black Holes. Notably, the presence of Spindles in these
solutions.

  * •

It would be nice to investigate whether something can be done along the lines
of [24][25], in the context of 5D Minimal Gauged Supergravity.

Hopefully further work related to these topics will follow in the near future.

## Acknowledgements

I thank Carlos Nunez for his continued help and guidance. I thank Niall T.
Macpherson for many very informative discussions, especially regarding
G-Structures. I thank Christopher Couzens for an insightful discussion.

##  Appendix A G-Structure Calculations

In order to validate the Supersymmetry preservation of the Type IIA and Type
IIB theories presented throughout the paper, our goal is to study the
G-Structure analysis in each case. We must first calculate the Gaiotto-
Maldacena G-Structures for each $\mathcal{N}=1$ reduction frame in turn,
allowing one to derive the corresponding IIA and IIB G-Structures as required.

###  A.1 Gaiotto-Maldacena

One can use the B$\ddot{\text{a}}$cklund transformation to calculate the GM
G-Structure forms from the LLM G-Structure analysis presented in Appendix D of
[26]. This calculation derives the G-Structures corresponding to the $\chi$
reduction frame, given in (2.3) and (2.3). We now wish to rotate these forms
in a general fashion to a reduction frame which accounts for the
$SL(3,\mathds{R})$ transformations performed in this paper. One can then
easily restrict to the $\mathcal{N}=1$ reduction frames of interest.

Following the $SL(3,\mathds{R})$ transformation given in (2.49), one needs to
re-write the metric in the following form (for a dimensional reduction along
$\phi_{3}$)

$\displaystyle ds_{11}^{2}$
$\displaystyle=f_{1}\bigg{[}4ds^{2}(\text{AdS}_{5})+f_{2}d\theta^{2}+f_{4}(d\sigma^{2}+d\eta^{2})\bigg{]}+f_{1}^{2}e^{-\frac{4}{3}\Phi_{IIA}}ds^{2}_{2}+e^{\frac{4}{3}\Phi_{IIA}}(d\phi_{3}+C_{1,\phi_{1}}d\phi_{1}+C_{1,\phi_{2}}d\phi_{2})^{2}$
$\displaystyle ds^{2}_{2}$
$\displaystyle=h_{\phi_{1}}(\eta,\sigma,\theta)d\phi_{1}^{2}+h_{\phi_{2}}(\eta,\sigma,\theta)d\phi_{2}^{2}+h_{\phi_{1}\phi_{2}}(\eta,\sigma,\theta)d\phi_{1}d\phi_{2}\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$
$\displaystyle=h_{\phi_{1}}(\eta,\sigma,\theta)\bigg{(}d\phi_{1}+\frac{1}{2}\frac{h_{\phi_{1}\phi_{2}}(\eta,\sigma,\theta)}{h_{\phi_{1}}(\eta,\sigma,\theta)}d\phi_{2}\bigg{)}^{2}+\bigg{(}h_{\phi_{2}}(\eta,\sigma,\theta)-\frac{1}{4}\frac{h_{\phi_{1}\phi_{2}}(\eta,\sigma,\theta)^{2}}{h_{\phi_{1}}(\eta,\sigma,\theta)}\bigg{)}d\phi_{2}^{2},$
(A.1)

with $(\phi_{1},\phi_{2},\phi_{3})$ representing any arrangement of
$(\beta,\chi,\phi)$ as required. The general G-Structure forms for the GM
background then read

$\displaystyle K$ $\displaystyle=\frac{\kappa\,e^{-2\rho}}{f_{1}}d(\cos\theta
e^{2\rho}\dot{V}),$ $\displaystyle E_{1}$
$\displaystyle=f_{1}e^{-\frac{2}{3}\Phi}\sqrt{h_{1}(\eta,\sigma,\theta)}\bigg{[}b_{1}(\eta,\sigma,\theta)d\sigma+b_{2}(\eta,\sigma,\theta)d\eta+b_{3}(\eta,\sigma,\theta)d\rho+b_{4}(\eta,\sigma,\theta)d\theta$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-i\bigg{(}d\phi_{1}+\frac{1}{2}\frac{h_{\phi_{1}\phi_{2}}(\eta,\sigma,\theta)}{h_{\phi_{1}}(\eta,\sigma,\theta)}d\phi_{2}\bigg{)}\bigg{]},$
$\displaystyle E_{2}$
$\displaystyle=e^{i\bar{\phi}}\bigg{[}b_{5}(\eta,\sigma,\theta)d\sigma+b_{6}(\eta,\sigma,\theta)d\eta+b_{7}(\eta,\sigma,\theta)d\rho+b_{8}(\eta,\sigma,\theta)d\theta$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+i\,f_{1}e^{-\frac{2}{3}\Phi}\sqrt{h_{\phi_{2}}(\eta,\sigma,\theta)-\frac{1}{4}\frac{h_{\phi_{1}\phi_{2}}(\eta,\sigma,\theta)^{2}}{h_{\phi_{1}}(\eta,\sigma,\theta)}}d\phi_{2}\bigg{]},$
$\displaystyle E_{3}$
$\displaystyle=-\sqrt{f_{1}}e^{i\bar{\chi}}\,\Bigg{[}b_{9}(\eta,\sigma,\theta)d\sigma+b_{10}(\eta,\sigma,\theta)d\eta+b_{11}(\eta,\sigma,\theta)d\rho+b_{12}(\eta,\sigma,\theta)d\theta$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+i\frac{e^{\frac{2}{3}\Phi}}{\sqrt{f_{1}}}\big{(}d\phi_{3}+C_{1,\phi_{1}}d\phi_{1}+C_{1,\phi_{2}}d\phi_{2}\big{)}\Bigg{]},$
$\displaystyle\bar{\phi}=s\,\chi+v\,\beta+u\,\phi,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\bar{\chi}=p\,\chi+q\,\beta+m\,\phi,$

where $(b_{1},..,~{}b_{12})$ are functions which must be derived in each case.

Notice here that $K$ remains intact under the rotation of frames because it is
independent of $(\beta,\chi,\phi)$. Hence, in order to still satisfy all four
G-Structure equations in (2.29), $J$ must also remain intact under a frame
rotation. After enforcing this requirement, expressions for
$(b_{1},..,~{}b_{12})$ are easily derived by performing the $SL(3,\mathds{R})$
transformation (2.49) on the original forms for the $\chi$ reduction frame,
given in (2.3) and (2.3). This process must be repeated for each arrangement
of $(\phi_{1},\phi_{2},\phi_{3})$ in (A.1), giving a separate set of functions
for each. These functions, in their most general form, depend on the nine
$SL(3,\mathds{R})$ parameters and the derivatives of $V(\eta,\sigma)$. These
results are too cumbersome to justify including here, but one can easily
derive them using Mathematica and the method just outlined.

To derive these functions for the $\beta$ reduction frame, we could choose
$(\phi_{1},\phi_{2},\phi_{3})=(\chi,\phi,\beta)$. One must then specify the
relevant values of the nine $SL(3,\mathds{R})$ parameters. For the
$\mathcal{N}=2$ $\beta$ reduction frame presented in (2.3) (and given in [2]),
we fix $(p,b,u)=1$ with all other parameters set to zero. Analogously, for the
$\chi$ reduction forms, we pick
$(\phi_{1},\phi_{2},\phi_{3})=(\beta,\phi,\chi)$, re-deriving (2.3) with
$(p,b,u)=1$ and all other parameters set to zero212121Alternatively, one can
derive the $\chi$ reduction forms (up to $E_{2}\rightarrow-E_{2}$) from the
$\beta$ reduction case just discussed by utilising the $SL(3,\mathds{R})$
transformation, with $a=1,q=1,u=-1$ and all others set to zero (noting that
from the determinant in (2.49), we now require $qau=-1$). Hence, in this case,
we have made the transformations
$\beta\rightarrow\chi,~{}\chi\rightarrow\beta,~{}\phi\rightarrow-\phi$.. In
the $\phi$ reduction case, we can simply cycle the roles of $E_{i}$ such that
$(E_{1},E_{2},E_{3})\rightarrow(E_{2},E_{3},E_{1})$, meaning we get the
following two alternative forms (from (2.3) and (2.3), respectively)

$\phi$ reduction frame $\displaystyle
K=\frac{\kappa\,e^{-2\rho}}{f_{1}}d(\cos\theta
e^{2\rho}\dot{V}),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}E_{3}=e^{i\phi}\bigg{[}\frac{\kappa}{f_{1}}e^{-2\rho}d(\sin\theta
e^{2\rho}\dot{V})+i\,\sqrt{f_{1}f_{2}}\,\sin\theta d\phi\bigg{]},$ (A.2a) with
$\displaystyle
E_{1}=-e^{i\chi}\sqrt{f_{1}f_{5}\Big{(}f_{6}^{2}+\frac{f_{3}}{f_{5}}\Big{)}}\,\Bigg{[}d\rho-\frac{V^{\prime\prime}}{(\dot{V}^{\prime})^{2}-\ddot{V}V^{\prime\prime}}d(\dot{V})+i\,\bigg{(}d\chi+\frac{f_{6}}{f_{6}^{2}+\frac{f_{3}}{f_{5}}}d\beta\bigg{)}\Bigg{]},$
(A.2b) $\displaystyle
E_{2}=\sqrt{\frac{f_{1}f_{3}}{f_{6}^{2}+\frac{f_{3}}{f_{5}}}}\bigg{(}d(V^{\prime})-i\,d\beta\bigg{)},$
or $\displaystyle
E_{1}=-e^{i\chi}\sqrt{f_{1}f_{5}}\,\Bigg{[}-\frac{1}{4}f_{3}\frac{\dot{V}^{\prime}}{\sigma}d\sigma-V^{\prime\prime}d\eta+f_{6}d\rho+i\Big{(}d\beta+f_{6}d\chi\Big{)}\Bigg{]},$
(A.2c) $\displaystyle
E_{2}=-\sqrt{f_{1}f_{3}}\bigg{(}\frac{1}{\sigma}d\sigma+d\rho+id\chi\bigg{)},$

where $(E_{1}\bar{E_{1}}+E_{2}\bar{E_{2}}$),
$(E_{1}\wedge\bar{E_{1}}+E_{2}\wedge\bar{E_{2}})$ and $E_{1}\wedge E_{2}$ for
(A.2b) and (A.2c) are equivalent (as required). Alternatively, (A.2b) and
(A.2c) are derived directly from (A.1) with
$(\phi_{1},\phi_{2},\phi_{3})=(\chi,\beta,\phi)$ and
$(\phi_{1},\phi_{2},\phi_{3})=(\beta,\chi,\phi)$, respectively (with
$(p,b,u)=1$ and all other parameters set to zero).

The forms presented in (2.3), (2.3) and (A.2) correspond to naive dimensional
reductions along $\beta,\chi$ and $\phi$ respectively, with no
$SL(3,\mathds{R})$ transformations taking place. Of course, as we’ve discussed
throughout this paper, in order to derive $\mathcal{N}=1$ Type IIA and Type
IIB backgrounds, one must include an appropriate transformation prior to
reduction. The framework just outlined is now general enough to easily derive
the G-Structure forms for all such solutions, allowing for the verification of
the preservation of SUSY in each case. One could use the G-Structure
reductions given in (2.37) to find analogous general IIA forms in terms of
these 12 functions. Supersymmetry will only be preserved in a special set of
cases, so in general, such general forms would be largely redundant.

To thenderive the IIA forms, we simply use (2.36) and (2.37).

###  A.2 G-Structure rules for Abelian T-Duality

Here we present the Abelian T-Duality of the G-Structure conditions. Note that
in this calculation the convention $d_{H_{3}}=d+H_{3}\wedge$ is required - in
order to use the minus sign convention (used throughout the rest of the paper)
we would need to appropriately flip the sign of the $B$ field in the T-Dual
rules given in (2.42) (such that
$E_{\mathcal{B}}^{y}=e^{-C^{\mathcal{A}}}(dy+B_{1}^{\mathcal{A}})$).

Motivated by the Type IIA G-Structure condition given in (2.40), quoted here
for convenience

$\text{Vol}_{4}\wedge
d_{H^{\mathcal{A}}_{3}}(e^{4A-\Phi_{\mathcal{A}}}\text{Im}\Psi^{\mathcal{A}}_{-})=\frac{1}{8}(F_{6}+F_{8}+F_{10}),$

one can use the T-Dual rules of (2.42) to make the decomposition

$\Psi_{\pm}^{\mathcal{A}}=\Psi_{\pm_{\perp}}^{\mathcal{A}}+\Psi_{\pm_{||}}^{\mathcal{A}}\wedge
E^{y}_{\mathcal{A}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\Psi_{\mp}^{\mathcal{B}}=e^{C^{\mathcal{A}}}\Psi_{\pm_{||}}^{\mathcal{A}}+\Psi_{\pm_{\perp}}^{\mathcal{A}}\wedge(dy-
B_{1}^{\mathcal{A}}),$

with $E^{y}_{\mathcal{A}}=e^{C^{\mathcal{A}}}(dy+A_{1}^{\mathcal{A}})$ and
$E^{y}_{\mathcal{B}}=e^{C^{\mathcal{B}}}(dy+A_{1}^{\mathcal{B}})=e^{-C^{\mathcal{A}}}(dy-
B_{1}^{\mathcal{A}})$. The above form of $\Psi_{\mp}^{\mathcal{B}}$ is an
initial ansatz - the final form will be derived at the end of the following
subsection.

#### G-Structure conditions

We begin by transforming the G-Structure condition under Abelian T-Duality
from IIA to IIB. For convenience, we will summarise the left hand side of the
following IIA G-Structure conditions simply as
$d_{H^{\mathcal{A}}_{3}}(e^{\alpha
A-\Phi_{\mathcal{A}}}\Psi_{\pm}^{\mathcal{A}})$,

$\displaystyle
d_{H^{\mathcal{A}}_{3}}(e^{3A-\Phi_{\mathcal{A}}}\Psi^{\mathcal{A}}_{+})$
$\displaystyle=0,$ $\displaystyle
d_{H^{\mathcal{A}}_{3}}(e^{2A-\Phi_{\mathcal{A}}}\text{Re}\Psi^{\mathcal{A}}_{-})$
$\displaystyle=0,$ $\displaystyle
d_{H^{\mathcal{A}}_{3}}(e^{4A-\Phi_{\mathcal{A}}}\text{Im}\Psi^{\mathcal{A}}_{-})$
$\displaystyle=\frac{e^{4A}}{8}*_{6}\lambda(g).$

with the choice of $\alpha\in(2,3,4)$ depending on the specific condition.
Now, the condition which will transform in the same manner as the Ramond
fields is

$\text{Vol}_{4}\wedge d_{H^{\mathcal{A}}_{3}}(e^{\alpha
A-\Phi_{\mathcal{A}}}\Psi_{\pm}^{\mathcal{A}})=\text{Vol}_{4}\wedge\Big{[}d(e^{\alpha
A-\Phi_{\mathcal{A}}}\Psi_{\pm}^{\mathcal{A}})+dB^{\mathcal{A}}\wedge
e^{\alpha A-\Phi_{\mathcal{A}}}\Psi_{\pm}^{\mathcal{A}}\Big{]}.$ (A.3)

From the T-Dual rules (2.42), we have

$\displaystyle dB^{\mathcal{A}}$
$\displaystyle=dB_{2}^{\mathcal{A}}+dB_{1}^{\mathcal{A}}\wedge dy$ (A.4)
$\displaystyle=dB_{2}^{\mathcal{A}}+e^{-C^{\mathcal{A}}}dB_{1}^{\mathcal{A}}\wedge
e^{C^{\mathcal{A}}}(dy+A_{1}^{\mathcal{A}})-dB_{1}^{\mathcal{A}}\wedge
A_{1}^{\mathcal{A}}$
$\displaystyle=(dB_{2}^{\mathcal{A}}-dB_{1}^{\mathcal{A}}\wedge
A_{1}^{\mathcal{A}})+e^{-C^{\mathcal{A}}}dB_{1}^{\mathcal{A}}\wedge
E^{y}_{\mathcal{A}},$

and using the decomposition for $\Psi_{\pm}^{\mathcal{A}}$, we get

$\displaystyle dB^{\mathcal{A}}\wedge e^{\alpha
A-\Phi_{\mathcal{A}}}\Psi_{\pm}^{\mathcal{A}}$
$\displaystyle=[(dB_{2}^{\mathcal{A}}-dB_{1}^{\mathcal{A}}\wedge
A_{1}^{\mathcal{A}})+e^{-C^{\mathcal{A}}}dB_{1}^{\mathcal{A}}\wedge
E^{y}_{\mathcal{A}}]\wedge[e^{\alpha
A-\Phi_{\mathcal{A}}}\Psi_{\pm_{\perp}}^{\mathcal{A}}+e^{\alpha
A-\Phi_{\mathcal{A}}}\Psi_{\pm_{||}}^{\mathcal{A}}\wedge E^{y}_{\mathcal{A}}]$
(A.5) $\displaystyle=(dB_{2}^{\mathcal{A}}-dB_{1}^{\mathcal{A}}\wedge
A_{1}^{\mathcal{A}})\wedge e^{\alpha
A-\Phi_{\mathcal{A}}}\Psi_{\pm_{\perp}}^{\mathcal{A}}$
$\displaystyle~{}~{}~{}~{}+\Big{[}(dB_{2}^{\mathcal{A}}-dB_{1}^{\mathcal{A}}\wedge
A_{1}^{\mathcal{A}})\wedge e^{\alpha
A-\Phi_{\mathcal{A}}}\Psi_{\pm_{||}}^{\mathcal{A}}-e^{\alpha
A-\Phi_{\mathcal{A}}}\Psi_{\pm_{\perp}}^{\mathcal{A}}\wedge
e^{-C^{\mathcal{A}}}dB_{1}^{\mathcal{A}}\Big{]}\wedge E^{y}_{\mathcal{A}}.$

In addition,

$\displaystyle d(e^{\alpha A-\Phi_{\mathcal{A}}}\Psi_{\pm}^{\mathcal{A}})$
$\displaystyle=d(e^{\alpha
A-\Phi_{\mathcal{A}}}\Psi_{\pm_{\perp}}^{\mathcal{A}}+e^{\alpha
A-\Phi_{\mathcal{A}}}\Psi_{\pm_{||}}^{\mathcal{A}}\wedge E^{y}_{\mathcal{A}})$
(A.6) $\displaystyle=d(e^{\alpha
A-\Phi_{\mathcal{A}}}\Psi_{\pm_{\perp}}^{\mathcal{A}})+d(e^{\alpha
A-\Phi_{\mathcal{A}}}\Psi_{\pm_{||}}^{\mathcal{A}})\wedge
E^{y}_{\mathcal{A}}+e^{\alpha
A-\Phi_{\mathcal{A}}}\Psi_{\pm_{||}}^{\mathcal{A}}\wedge\Big{(}d(e^{C^{\mathcal{A}}})\wedge
dy+d(e^{C^{\mathcal{A}}}A_{1}^{\mathcal{A}})\Big{)}$
$\displaystyle=d(e^{\alpha
A-\Phi_{\mathcal{A}}}\Psi_{\pm_{\perp}}^{\mathcal{A}})+e^{\alpha
A-\Phi_{\mathcal{A}}}\Psi_{\pm_{||}}^{\mathcal{A}}\wedge
e^{C^{\mathcal{A}}}d(A_{1}^{\mathcal{A}})$
$\displaystyle~{}~{}~{}~{}~{}+\Big{[}d(e^{\alpha
A-\Phi_{\mathcal{A}}}\Psi_{\pm_{||}}^{\mathcal{A}})+e^{\alpha
A-\Phi_{\mathcal{A}}}\Psi_{\pm_{||}}^{\mathcal{A}}\wedge
e^{-C^{\mathcal{A}}}d(e^{C^{\mathcal{A}}})\Big{]}\wedge E^{y}_{\mathcal{A}}.$

Hence, we have

$\begin{gathered}\text{Vol}_{4}\wedge d_{H^{\mathcal{A}}_{3}}(e^{\alpha
A-\Phi_{\mathcal{A}}}\Psi_{\pm}^{\mathcal{A}})=\text{Vol}_{4}\wedge(\Gamma_{\perp}^{\mathcal{A}}+\Gamma_{||}^{\mathcal{A}}\wedge
E^{y}_{\mathcal{A}}),\\\ \\\ \Gamma_{\perp}^{\mathcal{A}}=d(e^{\alpha
A-\Phi_{\mathcal{A}}}\Psi_{\pm_{\perp}}^{\mathcal{A}})+e^{\alpha
A-\Phi_{\mathcal{A}}}\Psi_{\pm_{||}}^{\mathcal{A}}\wedge
e^{C^{\mathcal{A}}}d(A_{1}^{\mathcal{A}})+(dB_{2}^{\mathcal{A}}-dB_{1}^{\mathcal{A}}\wedge
A_{1}^{\mathcal{A}})\wedge e^{\alpha
A-\Phi_{\mathcal{A}}}\Psi_{\pm_{\perp}}^{\mathcal{A}},\\\
\Gamma_{||}^{\mathcal{A}}=d(e^{\alpha
A-\Phi_{\mathcal{A}}}\Psi_{\pm_{||}}^{\mathcal{A}})+e^{\alpha
A-\Phi_{\mathcal{A}}}\Psi_{\pm_{||}}^{\mathcal{A}}\wedge
e^{-C^{\mathcal{A}}}d(e^{C^{\mathcal{A}}})~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\\\
+(dB_{2}^{\mathcal{A}}-dB_{1}^{\mathcal{A}}\wedge A_{1}^{\mathcal{A}})\wedge
e^{\alpha A-\Phi_{\mathcal{A}}}\Psi_{\pm_{||}}^{\mathcal{A}}-e^{\alpha
A-\Phi_{\mathcal{A}}}\Psi_{\pm_{\perp}}^{\mathcal{A}}\wedge
e^{-C^{\mathcal{A}}}dB_{1}^{\mathcal{A}}.\end{gathered}$ (A.7)

Now, applying the T-Dual rules, one gets for the IIB equations

$\text{Vol}_{4}\wedge\Big{(}e^{C^{\mathcal{A}}}\Gamma_{||}^{\mathcal{A}}+\Gamma_{\perp}^{\mathcal{A}}\wedge(dy-
B_{1}^{\mathcal{A}})\Big{)}.$ (A.8)

Before proceeding, from the transformation rules

$B^{\mathcal{B}}=B_{2}^{\mathcal{B}}+B_{1}^{\mathcal{B}}\wedge
dy=B_{2}^{\mathcal{A}}-A_{1}^{\mathcal{A}}\wedge(dy-B_{1}^{\mathcal{A}}),$
(A.9)

the following result will be required

$dB_{2}^{\mathcal{A}}-dB_{1}^{\mathcal{A}}\wedge
A_{1}^{\mathcal{A}}=dB^{\mathcal{B}}+dA_{1}^{\mathcal{A}}\wedge(dy-
B_{1}^{\mathcal{A}}),$ (A.10)

which we substitute directly into (A.7), giving

$\begin{gathered}\Gamma_{\perp}^{\mathcal{A}}=d(e^{\alpha
A-\Phi_{\mathcal{A}}}\Psi_{\pm_{\perp}}^{\mathcal{A}})+dB^{\mathcal{B}}\wedge
e^{\alpha A-\Phi_{\mathcal{A}}}\Psi_{\pm_{\perp}}^{\mathcal{A}}+e^{\alpha
A-\Phi_{\mathcal{A}}}\Psi_{\pm_{||}}^{\mathcal{A}}\wedge
e^{C^{\mathcal{A}}}d(A_{1}^{\mathcal{A}})-dA_{1}^{\mathcal{A}}\wedge e^{\alpha
A-\Phi_{\mathcal{A}}}\Psi_{\pm_{\perp}}^{\mathcal{A}}\wedge(dy-
B_{1}^{\mathcal{A}}),\\\ \Gamma_{||}^{\mathcal{A}}=d(e^{\alpha
A-\Phi_{\mathcal{A}}}\Psi_{\pm_{||}}^{\mathcal{A}})+e^{\alpha
A-\Phi_{\mathcal{A}}}\Psi_{\pm_{||}}^{\mathcal{A}}\wedge
e^{-C^{\mathcal{A}}}d(e^{C^{\mathcal{A}}})~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\\\
+\Big{(}dB^{\mathcal{B}}+dA_{1}^{\mathcal{A}}\wedge(dy-
B_{1}^{\mathcal{A}})\Big{)}\wedge e^{\alpha
A-\Phi_{\mathcal{A}}}\Psi_{\pm_{||}}^{\mathcal{A}}-e^{\alpha
A-\Phi_{\mathcal{A}}}\Psi_{\pm_{\perp}}^{\mathcal{A}}\wedge
e^{-C^{\mathcal{A}}}dB_{1}^{\mathcal{A}},\end{gathered}$ (A.11)

hence

$\displaystyle
e^{C^{\mathcal{A}}}\Gamma_{||}^{\mathcal{A}}+\Gamma_{\perp}^{\mathcal{A}}\wedge(dy-
B_{1}^{\mathcal{A}})$ $\displaystyle=e^{C^{\mathcal{A}}}d(e^{\alpha
A-\Phi_{\mathcal{A}}}\Psi_{\pm_{||}}^{\mathcal{A}})+e^{\alpha
A-\Phi_{\mathcal{A}}}\Psi_{\pm_{||}}^{\mathcal{A}}\wedge
d(e^{C^{\mathcal{A}}})+e^{C^{\mathcal{A}}}dB^{\mathcal{B}}\wedge e^{\alpha
A-\Phi_{\mathcal{A}}}\Psi_{\pm_{||}}^{\mathcal{A}}$ (A.12)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}-e^{C^{\mathcal{A}}}dA_{1}^{\mathcal{A}}\wedge
e^{\alpha A-\Phi_{\mathcal{A}}}\Psi_{\pm_{||}}^{\mathcal{A}}\wedge(dy-
B_{1}^{\mathcal{A}})-e^{\alpha
A-\Phi_{\mathcal{A}}}\Psi_{\pm_{\perp}}^{\mathcal{A}}\wedge
dB_{1}^{\mathcal{A}}$ $\displaystyle+\bigg{[}d(e^{\alpha
A-\Phi_{\mathcal{A}}}\Psi_{\pm_{\perp}}^{\mathcal{A}})+dB^{\mathcal{B}}\wedge
e^{\alpha A-\Phi_{\mathcal{A}}}\Psi_{\pm_{\perp}}^{\mathcal{A}}+e^{\alpha
A-\Phi_{\mathcal{A}}}\Psi_{\pm_{||}}^{\mathcal{A}}\wedge
e^{C^{\mathcal{A}}}d(A_{1}^{\mathcal{A}})\bigg{]}\wedge(dy-
B_{1}^{\mathcal{A}})$ $\displaystyle=d(e^{\alpha
A-\Phi_{\mathcal{A}}}e^{C^{\mathcal{A}}}\Psi_{\pm_{||}}^{\mathcal{A}})+dB^{\mathcal{B}}\wedge
e^{\alpha
A-\Phi_{\mathcal{A}}}e^{C^{\mathcal{A}}}\Psi_{\pm_{||}}^{\mathcal{A}}$
$\displaystyle~{}~{}~{}~{}~{}~{}+d\Big{(}e^{\alpha
A-\Phi_{\mathcal{A}}}\Psi_{\pm_{\perp}}^{\mathcal{A}}\wedge(dy-
B_{1}^{\mathcal{A}})\Big{)}+dB^{\mathcal{B}}\wedge e^{\alpha
A-\Phi_{\mathcal{A}}}\Psi_{\pm_{\perp}}^{\mathcal{A}}\wedge(dy-
B_{1}^{\mathcal{A}})$ $\displaystyle=d_{H^{\mathcal{B}}_{3}}\Big{[}e^{\alpha
A-\Phi_{\mathcal{A}}}\Big{(}e^{C^{\mathcal{A}}}\Psi_{\pm_{||}}^{\mathcal{A}}+\Psi_{\pm_{\perp}}^{\mathcal{A}}\wedge(dy-
B_{1}^{\mathcal{A}})\Big{)}\Big{]}$
$\displaystyle=d_{H^{\mathcal{B}}_{3}}(e^{\alpha
A-\Phi_{\mathcal{A}}}\Psi_{\mp}^{\mathcal{B}}).$

The IIB G-Structure equations now read

$\displaystyle
d_{H^{\mathcal{B}}_{3}}(e^{3A-\Phi_{\mathcal{A}}}\Psi^{\mathcal{B}}_{-})$
$\displaystyle=0,$ $\displaystyle
d_{H^{\mathcal{B}}_{3}}(e^{2A-\Phi_{\mathcal{A}}}\text{Re}\Psi^{\mathcal{B}}_{+})$
$\displaystyle=0,$ $\displaystyle
d_{H^{\mathcal{B}}_{3}}(e^{4A-\Phi_{\mathcal{A}}}\text{Im}\Psi^{\mathcal{B}}_{+})$
$\displaystyle=\frac{e^{4A}}{8}*_{6}\lambda(g).$

Of course, by adjusting the transformation as follows

$e^{-\Phi_{\mathcal{B}}}\Psi_{\mp}^{\mathcal{B}}=e^{-\Phi_{\mathcal{A}}}\Big{[}e^{C^{\mathcal{A}}}\Psi_{\pm_{||}}^{\mathcal{A}}+\Psi_{\pm_{\perp}}^{\mathcal{A}}\wedge(dy-
B_{1}^{\mathcal{A}})\Big{]},$

one can re-write the above G-Structure conditions in terms of
$\Phi_{\mathcal{B}}$,

$\displaystyle
d_{H^{\mathcal{B}}_{3}}(e^{3A-\Phi_{\mathcal{B}}}\Psi^{\mathcal{B}}_{-})$
$\displaystyle=0,$ $\displaystyle
d_{H^{\mathcal{B}}_{3}}(e^{2A-\Phi_{\mathcal{B}}}\text{Re}\Psi^{\mathcal{B}}_{+})$
$\displaystyle=0,$ $\displaystyle
d_{H^{\mathcal{B}}_{3}}(e^{4A-\Phi_{\mathcal{B}}}\text{Im}\Psi^{\mathcal{B}}_{+})$
$\displaystyle=\frac{e^{4A}}{8}*_{6}\lambda(g).$

#### G-Structure forms

Now we wish to calculate the IIB pure spinors which we need for the
G-Structure conditions just derived. The $SU(2)$ pure spinors for IIA are
given in (2.26), and re-written here for clarity

$\Psi^{\mathcal{A}}_{+}=\frac{1}{8}e^{\frac{1}{2}z^{\mathcal{A}}\wedge\overline{z}^{\mathcal{A}}}\wedge\omega^{\mathcal{A}},~{}~{}~{}~{}~{}\Psi^{\mathcal{A}}_{-}=\frac{i}{8}e^{-ij^{\mathcal{A}}}\wedge
z^{\mathcal{A}}.$

In what follows, it will prove useful to make the following decompositions

$\begin{gathered}\omega^{\mathcal{A}}=\omega^{\mathcal{A}}_{\perp}+\omega^{\mathcal{A}}_{||}\wedge
E^{y}_{\mathcal{A}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}j^{\mathcal{A}}=j^{\mathcal{A}}_{\perp}+j^{\mathcal{A}}_{||}\wedge
E^{y}_{\mathcal{A}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}z^{\mathcal{A}}=z^{\mathcal{A}}_{\perp}+z^{\mathcal{A}}_{||}\wedge
E^{y}_{\mathcal{A}},\\\
z^{\mathcal{A}}_{\perp}=u^{\mathcal{A}}_{\perp}+i\,v^{\mathcal{A}}_{\perp},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}z^{\mathcal{A}}_{||}=u^{\mathcal{A}}_{||}+i\,v^{\mathcal{A}}_{||}.\end{gathered}$
(A.13)

We then note

$\displaystyle e^{\frac{1}{2}z^{\mathcal{A}}\wedge\overline{z}^{\mathcal{A}}}$
$\displaystyle=1+\frac{1}{2}z^{\mathcal{A}}\wedge\overline{z}^{\mathcal{A}}$
(A.14)
$\displaystyle=\Big{(}1+\frac{1}{2}z^{\mathcal{A}}_{\perp}\wedge\overline{z}^{\mathcal{A}}_{\perp}\Big{)}+\frac{1}{2}(z^{\mathcal{A}}_{\perp}\wedge\overline{z}^{\mathcal{A}}_{||}-z^{\mathcal{A}}_{||}\wedge\overline{z}^{\mathcal{A}}_{\perp})\wedge
E^{y}_{\mathcal{A}},$

as
$(z^{\mathcal{A}}\wedge\overline{z}^{\mathcal{A}})\wedge(z^{\mathcal{A}}\wedge\overline{z}^{\mathcal{A}})=0$,
and

$\displaystyle e^{-ij^{\mathcal{A}}}$
$\displaystyle=1+(-ij^{\mathcal{A}})+\frac{1}{2}(-ij^{\mathcal{A}})\wedge(-ij^{\mathcal{A}})+\frac{1}{3!}(-ij^{\mathcal{A}})\wedge(-ij^{\mathcal{A}})\wedge(-ij^{\mathcal{A}})+....$
(A.15) $\displaystyle=1-ij^{\mathcal{A}}-\frac{1}{2}j^{\mathcal{A}}\wedge
j^{\mathcal{A}}+\frac{i}{3!}j^{\mathcal{A}}\wedge j^{\mathcal{A}}\wedge
j^{\mathcal{A}}+....$
$\displaystyle=1-i(j^{\mathcal{A}}_{\perp}+j^{\mathcal{A}}_{||}\wedge
E^{y}_{\mathcal{A}})-\frac{1}{2}(j^{\mathcal{A}}_{\perp}\wedge
j^{\mathcal{A}}_{\perp}+2j^{\mathcal{A}}_{\perp}\wedge
j^{\mathcal{A}}_{||}\wedge
E^{y}_{\mathcal{A}})+\frac{i}{3!}(j^{\mathcal{A}}_{\perp}\wedge
j^{\mathcal{A}}_{\perp}\wedge
j^{\mathcal{A}}_{\perp}+3j^{\mathcal{A}}_{\perp}\wedge
j^{\mathcal{A}}_{\perp}\wedge j^{\mathcal{A}}_{||}\wedge
E^{y}_{\mathcal{A}})+..$
$\displaystyle=\Big{(}1-ij^{\mathcal{A}}_{\perp}-\frac{1}{2}j^{\mathcal{A}}_{\perp}\wedge
j^{\mathcal{A}}_{\perp}+\frac{i}{3!}j^{\mathcal{A}}_{\perp}\wedge
j^{\mathcal{A}}_{\perp}\wedge
j^{\mathcal{A}}_{\perp}+...\Big{)}+\Big{(}-ij^{\mathcal{A}}_{||}-j^{\mathcal{A}}_{\perp}\wedge
j^{\mathcal{A}}_{||}+\frac{i}{2}j^{\mathcal{A}}_{\perp}\wedge
j^{\mathcal{A}}_{\perp}\wedge j^{\mathcal{A}}_{||}+...\Big{)}\wedge
E^{y}_{\mathcal{A}}$
$\displaystyle=e^{-ij^{\mathcal{A}}_{\perp}}\wedge(1-ij^{\mathcal{A}}_{||}\wedge
E^{y}_{\mathcal{A}}).$

Hence, we get

$\hskip
14.22636pt\begin{gathered}\Psi_{\pm}^{\mathcal{A}}=\Psi_{\pm_{\perp}}^{\mathcal{A}}+\Psi_{\pm_{||}}^{\mathcal{A}}\wedge
E^{y}_{\mathcal{A}}\\\
\Psi_{+_{\perp}}^{\mathcal{A}}=\frac{1}{8}\Big{(}1+\frac{1}{2}z^{\mathcal{A}}_{\perp}\wedge\overline{z}^{\mathcal{A}}_{\perp}\Big{)}\wedge\omega^{\mathcal{A}}_{\perp},~{}~{}~{}~{}~{}~{}~{}\Psi_{+_{||}}^{\mathcal{A}}=\frac{1}{8}\bigg{[}\Big{(}1+\frac{1}{2}z^{\mathcal{A}}_{\perp}\wedge\overline{z}^{\mathcal{A}}_{\perp}\Big{)}\wedge\omega^{\mathcal{A}}_{||}+\frac{1}{2}(z^{\mathcal{A}}_{\perp}\wedge\overline{z}^{\mathcal{A}}_{||}-z^{\mathcal{A}}_{||}\wedge\overline{z}^{\mathcal{A}}_{\perp})\wedge\omega^{\mathcal{A}}_{\perp}\bigg{]}\\\
\Psi_{-_{\perp}}^{\mathcal{A}}=\frac{i}{8}e^{-ij^{\mathcal{A}}_{\perp}}\wedge