Compactification of 6d ${\cal N}=(1,0)$ quivers, 4d SCFTs and their
holographic dual Massive IIA backgrounds

Paul Merrikin<EMAIL_ADDRESS><EMAIL_ADDRESS>Carlos<EMAIL_ADDRESS>and
Ricardo<EMAIL_ADDRESS>

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

Abstract

In this paper we study an infinite family of Massive Type IIA backgrounds that
holographically describe the twisted compactification of ${\cal N}=(1,0)$ six-
dimensional SCFTs to four dimensions. The analysis of the branes involved
motivates an heuristic proposal for a four dimensional linear quiver QFT, that
deconstructs the theory in six dimensions. For the case in which the system
reaches a strongly coupled fixed point, we calculate some observables that we
compare with holographic results. Two quantities measuring the number of
degrees of freedom for the flow across dimensions are studied.

###### Contents

  1. 1 Introduction
  2. 2 Supergravity backgrounds
    1. 2.1 Page fluxes and charges
      1. 2.1.1 Page charges
    2. 2.2 The cases of $H_{2}$ and $S^{2}$ compactifications
  3. 3 Study of the dual field theory
    1. 3.1 Free Energy and Holographic Central Charge
      1. 3.1.1  Example 1
      2. 3.1.2 Example 2
      3. 3.1.3  Energy dependence of the holographic central charge
    2. 3.2 Flow central charge
    3. 3.3 A phenomenological proposal for the QFT
  4. 4 Conclusions
  5. A Appendix 1
    1. A.1 7D $SU(2)$ Topologically Massive Gauged Supergravity
      1. A.1.1 Background Ansatz
    2. A.2 From SUSY Variations to BPS Equations
      1. A.2.1 Dilatino Variation
      2. A.2.2 Gravitino Variation
      3. A.2.3 Rearranging the SUSY variations
    3. A.3 Uplift to 10D Massive Type IIA
      1. A.3.1 Einstein Frame and Normalizations
      2. A.3.2 Uplift of 7D Topologically Massive to Massive IIA
      3. A.3.3 Explicit Uplift of the Interpolating Background
      4. A.3.4 Page Fluxes
      5. A.3.5 Rewriting in terms of $\alpha(z)$
  6. B Numerical solution of the BPS system
    1. B.1 Infrared Fixed Point
    2. B.2 Linear Perturbations
  7. C Analysis of the 4d QFT
    1. C.0.1  Beta functions
    2. C.0.2  R-symmetry anomaly
    3. C.1 Central charges
      1. C.1.1 Example 1
      2. C.1.2  Example 2

##  1 Introduction

Maldacena’s AdS/CFT conjecture [2] motivates the study of both gravity and
field theory topics. In particular, the study of supersymmetric and conformal
field theories in diverse dimensions. In the past few years we witnessed the
definition of new, characteristically non-Lagrangian CFTs, by the existence of
a trustable background of Type II or M-theory, containing an AdS-factor.

In fact, this procedure has been applied to the possible space-time dimensions
for which super conformal field theories exist ($d+1=1,....,6$). With eight
Poincare supercharges, there exists a classification and an algorithmic way of
associating a particular SCFTd+1 with a Type II background containing an
AdSd+2 factor. At present there seems to be exceptions to this statement for
the cases of supergravity solutions containing AdS3 and AdS2 spaces. See
[3]-[15], for references working details of the cases $(d+1)=1,2,3,4,5,6$. A
comprehensive summary of various aspects of SCFTs in diverse dimensions can be
found in [16].

A reasonable extension is the study of RG-flows away from these SCFTsd+1.
These flows can be between two conformal points or between a CFT and a gapped
theory. Less conventional are the flows across dimensions, between a SCFTD+1
and a SCFTd+1 (there is also with the possibility of ending in gapped
systems). There are numerous case-studies of this in the bibliography, see for
example [17], for early examples working with twisted compactifications from
the holographic point of view. The topic progressed considerably after the
paper [18]. This was followed by many works studying compactifications
(twisted or with fluxes) from a purely QFT point of view. In the particular
case of compactifications of 6d to 4d systems (preserving minimal SUSY in both
dimensions), we find the works [19]-[20]. For a very nice summary of these
developments from a field theoretical perspective, see [21].

In this paper, we present an interesting example of flow across dimensions
involving a twisted compactification. In particular, we start from an infinite
family of six-dimensional ${\cal N}=(1,0)$ SCFTs and compactify it on a two
manifold of constant curvature. The end-point of the flow is an infinite
family of strongly coupled four dimensional ${\cal N}=1$ SCFTs (and possibly
gapped QFTs, that we leave for future studies). The holographic study of the
family of 4d SCFTs occupies an important part of this work, calculating
observables that characterise it.

In more detail, the contents of the paper are distributed as follows.
In Section 2, we construct a new infinite family of Massive Type IIA
backgrounds that represent the flow between a family of six-dimensional ${\cal
N}=(1,0)$ SCFTs and four dimensional ${\cal N}=1$ SCFTs. These flows are new
backgrounds, not present in the bibliography. The case of gapped four
dimensional systems leads to singular backgrounds, hence we leave it to future
study. The charges of the brane system are discussed, with emphasis on the
effects of the twisted-compactification.

In Section 3, we present calculations of the holographic central charge in
these supergravity backgrounds (the free energy of the dual CFT). These are
calculations at the AdS5 fixed point and along the flow. We also present a
monotonic quantity interpolating between the conformal points at low and high
energies. After this, based on the branes charges discussed in Section 2, we
give a phenomenological proposal for a suitable quiver capturing the low
energy dynamics. These 4d quiver QFTs are proposed to reach a conformal point
at low energies, their strongly coupled dynamics being described by the
infinite family of Massive Type IIA backgrounds with an AdS5 factor (discussed
in Section 2). We emphasise on the heuristic character of this proposal.
Indeed, whilst the beta functions and R-symmetry anomalies of the proposed QFT
are cancelled and the scaling of the free energy with the quiver parameters
(rank of gauge groups and number of nodes) matches the holographic result, the
precise coefficient of the free energy does not exactly match the one computed
in the holographic dual. Hence the proposed quiver is only a first step
towards the correct field theory dual to our infinite family of geometries. We
discuss possible improvements in the conclusions and Appendices.

In Section 4, we summarise, present conclusions and propose some ideas for
further research. Three very intensive appendices complement the presentation.
The reader wishing to work on these topics should benefit from reading them in
detail.

##  2 Supergravity backgrounds

We start this section by describing an infinite family of supergravity
solutions, the analysis of which, is the main subject of the rest of this
paper. This is a family of Massive Type IIA backgrounds, preserving four
supersymmetries (${\cal N}$=1 in four dimensional notation). The construction
of these backgrounds is described in great detail in Appendix A. From a
quantum field theoretical perspective, these backgrounds are dual to twisted
compactifications of six dimensional ${\cal N}=(1,0)$ SCFTs at the origin of
their tensor branch. We discuss this in more detail in Section 3.

Let us present the family of backgrounds in Massive Type IIA. These are
written in terms of coordinates, parameters and functions,

$\displaystyle\text{Coordinates:}\leavevmode\nobreak\
(t,x_{1},x_{2},x_{3},r,\theta_{1},\phi_{1},z,\theta_{2},\phi_{2}).\leavevmode\nobreak\
\leavevmode\nobreak\ \text{Parameters:}\leavevmode\nobreak\ (\Psi_{0},k).$
(2.1) $\displaystyle\text{Functions:}\leavevmode\nobreak\
\alpha(z),f(r),h(r),X(r)=e^{\frac{2}{5}\Phi(r)},\omega(r,z)=\left(\frac{\alpha^{\prime}(z)^{2}-2\alpha(z)\alpha^{\prime\prime}(z)X(r)^{5}}{\alpha^{\prime}(z)^{2}-2\alpha(z)\alpha^{\prime\prime}(z)}\right).$

The equations constraining these functions are written below. In terms of
these coordinates and functions, the string-frame spacetime metric reads

$\displaystyle ds^{2}_{st}$
$\displaystyle=2\pi\sqrt{2}\sqrt{-\frac{\alpha(z)}{\alpha^{\prime\prime}(z)}}\,X(r)^{-\frac{1}{2}}e^{-\frac{4\Phi(r)}{5}}\left[e^{2f(r)}dx^{2}_{3,1}+dr^{2}+e^{2h(r)}\left(d\theta^{2}_{1}+\frac{1}{k}\sin^{2}(\sqrt{k}\theta_{1})d\phi^{2}_{1}\right)\right]$
(2.2)
$\displaystyle+X(r)^{5/2}\left[\pi\sqrt{2}\sqrt{-\frac{\alpha^{\prime\prime}(z)}{\alpha(z)}}dz^{2}+\frac{\sqrt{2}\,\pi}{\omega(r,z)}\frac{\sqrt{-\alpha^{3}(z)\alpha^{\prime\prime}(z)}}{2\alpha(z)\alpha^{\prime\prime}(z)-\alpha^{\prime
2}}\left(d\theta^{2}_{2}+\sin^{2}(\theta_{2})\left(d\phi_{2}-\frac{1}{k}\cos(\sqrt{k}\theta_{1})d\phi_{1}\right)^{2}\right)\right].$

The Neveu-Schwarz ($B_{2},\Psi$) and Ramond ($F_{0},F_{2},F_{4}$) background
fields are,

$\displaystyle
B_{2}=\left(\frac{\pi}{\omega(r,z)}\frac{\alpha(z)\alpha^{\prime}(z)}{\alpha^{\prime}(z)^{2}-2\alpha(z)\alpha^{\prime\prime}(z)}\sin(\theta_{2})d\theta_{2}-\pi\cos(\theta_{2})dz\right)\wedge\left(d\phi_{2}-\frac{1}{k}\cos(\sqrt{k}\theta_{1})d\phi_{1}\right),$
$\displaystyle
e^{4\Psi(r,z)}=\frac{X^{5}(r)}{\omega^{2}(r,z)}\left(\frac{-\alpha(z)}{\alpha^{\prime\prime}(z)}\right)^{3}\left(\frac{e^{2\Psi_{0}}}{\alpha^{\prime}(z)^{2}-2\alpha(z)\alpha^{\prime\prime}(z)}\right)^{2},$
$\displaystyle
F_{0}=2^{\frac{1}{4}}\frac{e^{-\Psi_{0}}}{\sqrt{\pi}}\alpha^{\prime\prime\prime}(z),$
(2.3) $\displaystyle
F_{2}=2^{\frac{1}{4}}\sqrt{\pi}e^{-\Psi_{0}}\alpha^{\prime\prime}(z)\left[\cos(\theta_{2})\operatorname{Vol}(\Sigma_{k})-\operatorname{Vol}(S^{2}_{c})\right]+F_{0}\frac{\pi}{\omega(r,z)}\frac{\alpha(z)\alpha^{\prime}(z)}{\alpha^{\prime}(z)^{2}-2\alpha(z)\alpha^{\prime\prime}(z)}\,\operatorname{Vol}(S^{2}_{c}),$
$\displaystyle
F_{4}=\left(\frac{2^{\frac{1}{4}}\pi^{\frac{3}{2}}e^{-\Psi_{0}}}{\omega(r,z)}\right)\left(\frac{\alpha(z)\alpha^{\prime}(z)\alpha^{\prime\prime}(z)}{\alpha^{\prime}(z)^{2}-2\alpha(z)\alpha^{\prime\prime}(z)}\right)\cos(\theta_{2})\,\operatorname{Vol}(\Sigma_{k})\wedge\operatorname{Vol}(S^{2})$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
+2^{\frac{1}{4}}\pi^{\frac{3}{2}}e^{-\Psi_{0}}\alpha^{\prime\prime}(z)\sin^{2}(\theta_{2})\,dz\wedge
d\phi_{2}\wedge\operatorname{Vol}(\Sigma_{k}).$

We have defined the volume elements,

$\displaystyle\operatorname{Vol}(S^{2})=\sin(\theta_{2})d\theta_{2}\wedge
d\phi_{2},\leavevmode\nobreak\ \leavevmode\nobreak\
\operatorname{Vol}(S^{2}_{c})=\sin(\theta_{2})d\theta_{2}\wedge\left(d\phi_{2}-\frac{1}{k}\cos(\sqrt{k}\theta_{1})d\phi_{1}\right),$
$\displaystyle\operatorname{Vol}(\Sigma_{k})=\frac{\sin\left(\sqrt{k}\theta_{1}\right)}{\sqrt{k}}d\theta_{1}\wedge
d\phi_{1}.$ (2.4)

The functions $f(r),h(r),\Phi(r)$ must satisfy first order (BPS) ordinary
differential equations. Denoting the derivative respect to the coordinate $r$
with a dot, they read,

$\displaystyle\dot{f}=\pm\frac{m}{2}\,e^{-2\Phi},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\dot{h}=\pm\frac{1}{2}\left(\frac{1}{k}e^{-2h}+m\,e^{-2\Phi}\right),$
$\displaystyle\dot{\Phi}=\pm\left(-1+\frac{1}{4k}e^{-2h}+m\,e^{-2\Phi}\right).$
(2.5)

We choose the positive sign from now on. The derivation of eqs.(2.5) and the
origin of the parameter $m$ are explained in Appendix A. The remaining BPS
equation for $\alpha(z)$ is already written in eq.(2.3). In fact, the mass-
parameter of massive Type IIA $F_{0}$, that should be constant by pieces for
an interpretation in terms of localised D8 branes dictates that
$\alpha^{\prime\prime\prime}(z)$ must be piece-wise constant. It is in the
many possible choices for a piece-wise constant $F_{0}$ that the family of
backgrounds is originated. More on this in Section 2.1 below.

The configurations in eqs.(2.2)-(2.5), are new solutions to the equations of
motion of Massive IIA. In string frame these read,

$\displaystyle\frac{1}{4}R+\nabla^{2}\Psi-(\nabla\Psi)^{2}-\frac{1}{8}H_{3}^{2}=0,$
$\displaystyle dF_{p}+H_{3}\wedge*F_{p-2}=0,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
d(e^{-2\Psi}*H_{3})-\left(F_{0}*F_{2}+F_{2}\wedge*F_{4}+F_{4}\wedge
F_{4}\right)=0,$ $\displaystyle
R_{MN}+2\nabla_{M}\nabla_{N}\Psi-\frac{1}{2}(H^{2}_{3})_{MN}-\frac{1}{4}e^{2\Psi}\sum_{p}(F^{2}_{p})_{MN}=0.$
(2.6)

In eq. (2.6) $p=2,4,6,8,10$, and

$(F^{2}_{p})_{MN}=\frac{1}{(p-1)!}F_{M}^{\phantom{M}N_{1}...N_{p-1}}F_{NN_{1}...N_{p-1}},\;\;\;(H^{2}_{3})_{MN}=\frac{1}{2}H_{M}^{\phantom{M}N_{1}N_{2}}H_{NN_{1}N_{2}}.$

Regarding the volume form ${\hbox{\rm Vol}}(\Sigma_{k})$ in eq.(2.4), for the
allowed values for the parameter $k$, namely $k=(1,-1)$, $\Sigma_{k}$ is
describing a two-sphere or a hyperbolic plane. The case $k=0$, corresponding
to a torus, is slightly more subtle and will be briefly addressed below.

This concludes the presentation of the backgrounds. To interpret these in
terms of branes, we calculate the Page charges (quantised and gauge-variant)
associated with this family of SUSY solutions.

###  2.1 Page fluxes and charges

The Page fluxes $\widehat{F}_{p}$, defined as a polyform
$\widehat{F}=e^{-B_{2}}\wedge F$ are quantised. This implies a certain form
for some of the functions in the background, as we discuss below.

The Page fluxes are gauge-variant. They do change under a gauge transformation
of $B_{2}$. We use this in our favour, performing a particular transformation
that makes explicit the quantised charges present and the role of the function
$\alpha(z)$. To write the Page fluxes in a concise fashion, it proves useful
to define a one form $\Theta_{1}$ and its exterior derivative,

$\Theta_{1}=-\cos(\theta_{2})\left(d\phi_{2}-\frac{1}{k}\cos(\sqrt{k}\theta_{1})d\phi_{1}\right),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
d\Theta_{1}=\operatorname{Vol}(S^{2}_{c})-\cos(\theta_{2})\operatorname{Vol}(\Sigma_{k}).$
(2.7)

It is also convenient to change the $B_{2}$-field by a large gauge
transformation (this has no effect on $H_{3}=dB_{2}$). Below, we explain the
purpose of such transformation,

$\displaystyle B_{2,new}=B_{2,old}-\pi
d\Big{[}(z-\Delta)\Theta_{1}\Big{]},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \Delta\leavevmode\nobreak\ \text{is a constant},$ (2.8)
$\displaystyle
B_{2,new}=\left(\frac{\pi}{\omega(r,z)}\frac{\alpha(z)\alpha^{\prime}(z)}{\alpha^{\prime}(z)^{2}-2\alpha(z)\alpha^{\prime\prime}(z)}\operatorname{Vol}(S^{2}_{c})-\pi(z-\Delta)d\Theta_{1}\right),$
$\displaystyle
H_{3}=\pi\left[-\frac{1}{\omega^{2}}\frac{\alpha(z)\alpha^{\prime}(z)}{\alpha^{\prime}(z)^{2}-2\alpha(z)\alpha^{\prime\prime}(z)}\frac{\partial\omega}{\partial
X}X^{\prime}(r)dr+\frac{d}{dz}\left(\frac{1}{\omega}\frac{\alpha(z)\alpha^{\prime}(z)}{\alpha^{\prime}(z)^{2}-2\alpha(z)\alpha^{\prime\prime}(z)}\right)dz\right]\wedge\operatorname{Vol}(S^{2}_{c})$
$\displaystyle\phantom{=}-\pi\left(\frac{1}{\omega}\frac{\alpha(z)\alpha^{\prime}(z)}{\alpha^{\prime}(z)^{2}-2\alpha(z)\alpha^{\prime\prime}(z)}\right)\sin(\theta_{2})d\theta_{2}\wedge\operatorname{Vol}(\Sigma_{k})-\pi\,dz\wedge\operatorname{Vol}(S^{2}_{c})+\pi\cos(\theta_{2})dz\wedge\operatorname{Vol}(\Sigma_{k}).$

With this new-$B_{2}$ we compute the Page flux $\widehat{F}_{2}$ and obtain,

$\displaystyle\widehat{F}_{2}=F_{2}-B_{2}F_{0}=-2^{\frac{1}{4}}\sqrt{\pi}e^{-\psi_{0}}(\alpha^{\prime\prime}-(z-\Delta)\alpha^{\prime\prime\prime})d\Theta_{1}.$
(2.9)

Similarly we calculate the $\widehat{F}_{4}$ Page flux,

$\displaystyle\widehat{F}_{4}=F_{4}-B_{2}\wedge F_{2}+\frac{1}{2}B_{2}\wedge
B_{2}F_{0},$ (2.10)
$\displaystyle\widehat{F}_{4}={2^{\frac{1}{4}}\pi^{\frac{3}{2}}e^{-\Psi_{0}}\alpha^{\prime\prime}(z)\sin^{2}\theta_{2}\,dz\wedge
d\phi_{2}\wedge\operatorname{Vol}(\Sigma_{k})}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
{+2^{1/4}\pi^{3/2}e^{-\Psi_{0}}(z-\Delta)\left(2\alpha^{\prime\prime}(z)-\alpha^{\prime\prime\prime}(z)(z-\Delta)\right)\cos\theta_{2}\operatorname{Vol}(S^{2}_{c})\wedge\operatorname{Vol}(\Sigma_{k}).}$

Finally, for $F_{0}$ we have the same as in eq.(2.3),
$\widehat{F}_{0}=2^{\frac{1}{4}}\frac{e^{-\Psi_{0}}}{\sqrt{\pi}}\alpha^{\prime\prime\prime}(z)$.
We now calculate the charges associated with these Page fluxes, and impose
their quantisation.

####  2.1.1 Page charges

To calculate the charges, we need to compute the integrals (as in the rest of
the paper we set $g_{s}=\alpha^{\prime}=1$),

$Q_{NS5}=\frac{1}{4\pi^{2}}\int_{M_{3}}H_{3},\leavevmode\nobreak\
\leavevmode\nobreak\
Q_{D_{p}}=\frac{1}{(2\pi)^{7-p}}\int_{M_{8-p}}\widehat{F}_{8-p},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ p=4,6,8.$ (2.11)

These integrals need to be defined over suitable cycles, some of which contain
the sub-manifold $\Sigma_{k}$. Then, it is useful to first calculate the
volume of the two-manifold $\Sigma_{k}$ defined in eq.(2.4). As we stated
above, in the cases $k=(1,0,-1)$ the space described is a two-sphere, a torus
or a hyperbolic plane. Its volume is calculated using Gauss-Bonnet’s theorem
and the fact that $\Sigma_{k}$ has curvature $R=2k$. For a genus $g$ Riemann
surface, we have

$\int_{\Sigma_{k}}d^{2}x\sqrt{g}R=8\pi(1-g)\leavevmode\nobreak\
\leavevmode\nobreak\ \longrightarrow\leavevmode\nobreak\ \leavevmode\nobreak\
\int_{\Sigma_{k}}d^{2}x\sqrt{g}=\frac{4\pi}{k}(1-g).$

For the sphere, $k=1$ and $g=0$, while for the hyperbolic plane, $k=-1$ with
$g>1$. The case of the torus needs some care, as we have $k=0,g=1$ (and the
volume is $4\pi$). We will not discuss the case of $T^{2}$ in what follows
(except for the purpose of making an intuitive argument below). This allows us
to write (for $S^{2},H_{2}$)

$\int_{\Sigma_{k}}d^{2}x\sqrt{g}=4\pi|g-1|.$ (2.12)

After these preliminaries, we calculate the Page charges. Inspecting the
$H_{3}$-field in eq.(2.8) we find two possible three-cycles on which the
integral can be performed. These three-cycles are

${\cal
M}_{1}=(\theta_{1},\phi_{1},z)\big{|}_{\theta_{2}=0},\;\;\;\text{and}\leavevmode\nobreak\
\leavevmode\nobreak\ {\cal
M}_{2}=(\theta_{2},\phi_{2},z)\big{|}_{r\rightarrow+\infty}.$ (2.13)

We find that there are two sets of NS-five branes. Their total numbers being,

$\displaystyle N^{(1)}_{NS5}=\frac{1}{4\pi^{2}}\int_{{\cal
M}_{1}}H_{3}=\frac{1}{4\pi}\operatorname{Vol}(\Sigma_{k})P=|g-1|P,$
$\displaystyle N^{(2)}_{NS5}=\frac{1}{4\pi^{2}}\int_{{\cal M}_{2}}H_{3}=P.$
(2.14)

To calculate $N^{(2)}_{NS5}$ a boundary condition $\alpha(z=0)=\alpha(z=P)=0$
has been imposed, more on this below. We have also taken the orientation of
the manifolds such that all the charges are positive. Importantly, we have set
the range of the $z$-coordinate to be $z\in[0,P]$, with $P$ an integer.

For the charges of D6-branes, we have a pair of two-cycles on which to
integrate the Page flux of eq.(2.9),

${\cal
M}_{3}=(\theta_{1},\phi_{1})\big{|}_{\theta_{2}=0},\;\;\;\text{and}\;\;\;{\cal
M}_{4}=(\theta_{2},\phi_{2}).$

Performing the integrals, we find,

$\displaystyle N^{(1)}_{D6}=\frac{1}{2\pi}\int_{{\cal
M}_{3}}\widehat{F}_{2}=2^{\frac{5}{4}}\sqrt{\pi}e^{-\Psi_{0}}|g-1|(\alpha^{\prime\prime}-(z-\Delta)\alpha^{\prime\prime\prime}),$
$\displaystyle N^{(2)}_{D6}=\frac{1}{2\pi}\int_{{\cal
M}_{4}}\widehat{F}_{2}=2^{\frac{5}{4}}\sqrt{\pi}e^{-\Psi_{0}}(\alpha^{\prime\prime}-(z-\Delta)\alpha^{\prime\prime\prime}).$
(2.15)

For the D8 branes we use eq.(2.3) and find,

$N_{D8}=2\pi\int dzF_{0}^{\prime}=2^{\frac{5}{4}}\sqrt{\pi}e^{-\Psi_{0}}\int
dz\,\alpha^{(4)}.$ (2.16)

Finally, for the D4 branes, the four-cycle is ${\cal
M}_{5}=(\theta_{2},\phi_{2},\theta_{1},\phi_{1})$, at constant values of
$r,z$. Calculating

$Q_{D4}=\frac{1}{8\pi^{3}}\int_{{\cal M}_{5}}\widehat{F}_{4}=0.$

In other words, there is no charge of D4 branes in the system111One might
wonder about computing $Q_{D4}$ integrating $\widehat{F}_{4}$ over the
manifold ${\cal
M}_{6}=[z,\theta_{1},\phi_{1},\phi_{2}]_{\theta_{2}=\frac{\pi}{2}}$. This is
not a well defined four-cycle, as it has a boundary. .

Inspecting the charges in eqs.(2.14), (2.15) and (2.16), suggests to set
$e^{\Psi_{0}}=2^{\frac{5}{4}}\sqrt{\pi}$. Imposing quantisation, it is clear
from eqs. (2.15) that the function $\alpha^{\prime\prime}(z)$ must be a linear
function with integer coefficients $N_{l}$. In fact, we can divide the range
of the $z$-coordinate in intervals of unit size. In each interval,
$\alpha^{\prime\prime}(z)$ should be a linear function. The charge of D8
branes in eq.(2.16) suggests that $\alpha^{\prime\prime}(z)$ should be
piecewise linear and continuous (with integer coefficients), the third
derivative piecewise constant (integer and generically discontinuous), whilst
the fourth-derivative a sum of delta functions with integer coefficients. In
other words, if we choose for $\alpha^{\prime\prime}(z)$,

$\alpha^{\prime\prime}(z)=\begin{cases}N_{1}z&,0\leq z<1\\\
N_{1}+(N_{2}-N_{1})(z-1)&,1\leq z<2\\\ \vdots\\\
N_{l}+(N_{l+1}-N_{l})(z-l)&,l\leq z<l+1\\\ \vdots\\\ N_{P-1}(P-z)&,(P-1)\leq
z<P.\\\ \end{cases}$ (2.17)

This implies,

$\alpha^{\prime\prime\prime}(z)=\begin{cases}N_{1}&,0\leq z<1\\\
N_{2}-N_{1}&,1\leq z<2\\\ \vdots\\\ N_{l+1}-N_{l}&,l\leq z<l+1\\\ \vdots\\\
-N_{P-1}&,(P-1)\leq z<P,\\\ \end{cases}$ (2.18)

and

$\alpha^{(4)}(z)=\sum^{P-1}_{l=1}\left(2N_{l}-N_{l+1}-N_{l-1}\right)\delta(z-l).$
(2.19)

At this point we choose a convenient coefficient $\Delta$ in the large gauge
transformation of $B_{2}$– see eq.(2.8) . In fact, choosing $\Delta=l$ for
$z\in[l,l+1]$ we have in each interval

$(\alpha^{\prime\prime}-(z-l)\alpha^{\prime\prime\prime})=N_{l}.$

In summary, we have two kinds of NS-five branes, their total number is given
in eqs.(2.14). In the interval $z\in[l,l+1]$ we have two types of D6 brane
charges and one kind of D8 charge given by (in each interval),

$\displaystyle
N^{(1)}_{D6}[l,l+1]\,=|g-1|N_{l},\;\;\;N^{(2)}_{D6}[l,l+1]\,=N_{l},$
$\displaystyle N_{D8}[l,l+1]\,=2N_{l}-N_{l+1}-N_{l-1}.$ (2.20)

Though it does not feature in the calculation of Page charges, the function
$\alpha(z)$ is obtained after two integrations of eq.(LABEL:alphasecond). The
integration constants must be chosen such that $\alpha(z)$ is a piecewise
continuous cubic function with continuous derivative $\alpha^{\prime}(z)$. To
avoid singular behaviours (not associated with the presence of localised D8
branes), it must satisfy $\alpha(0)=\alpha(P)=0$.

This is a good point to discuss the physical effect of the large gauge
transformation on the $B_{2}$-field– see eq.(2.8). Being a gauge
transformation, it does not affect the Physics of our system, but it makes the
counting of charges more transparent. Had we not performed it and calculated
Page charges with the $B_{2,old}$, we would have obtained a combination
between charges of D6 branes induced on the D8 branes and those of ’actual’ D6
branes. The large gauge transformation separates these, making the counting
clearer.

To better understand these systems, it is a good (and intuitive) guide to go
back to the case of the torus. In this case, we are compactifying a six
dimensional ${\cal N}=(1,0)$ SCFT on $T^{2}$ without any flux that breaks
SUSY. In other words, we would end with an ${\cal N}=2$ four dimensional SCFT.
For more on this perspective see, for example [22]. What follows in the next
paragraph is an intuitive argument.

The compactification on $T^{2}$ (which sets $k=0$) should be handled with some
care. For example, the ’twisting’ in the one-form $\Theta_{1}$ is absent,
leading to $\Theta_{1}=-\cos\theta_{2}d\phi_{2}$. Also, in the BPS eqs.(2.5),
the terms $\frac{e^{-2h}}{k}$ are absent. This gives $2\Phi(r)=0$ with
$X(r)=1$ (for the parameter $m=1$) and $2f(r)=2h(r)=r$. This leads to a
background metric—see eq.(2.2)– of the form AdS${}_{7}\times S^{2}\times
R_{z}$. For this case the genus is $g=1$ and the charges in eqs.(2.14) and
(2.20) indicate the presence of only one type of NS-five branes and D6 branes,
together with D8 branes. The reader will recognise that this is the
compactification on $T^{2}$ of the backgrounds in [14]-[15].

Intuitively, this is what our background in eqs.(2.2)-(2.5) is describing for
large values of the $r$-coordinate. Of course, since we compactify on a curved
two manifold (either $S^{2}$ or $H_{2}$), the twisting needs to be performed
even at high energies (large values of the $r$-coordinate). The system
preserves four supercharges all along the flow. The twisted compactification
of the six dimensional system of NS-D6-D8 induces new sets of NS and D6
branes. We identify these new branes as those whose charge comes with the
$(g-1)$-factor in front– see eqs.(2.14) and (2.15).

Let us be more precise about the compactifications on $S^{2}$ and $H_{2}$.

###  2.2 The cases of $H_{2}$ and $S^{2}$ compactifications

In both cases ($k=1$ or $k=-1$), we can solve two of the equations in (2.5),
finding

$e^{-2\Phi}=\frac{2k\dot{h}-e^{-2h}}{km},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ f=\frac{m}{2}\int e^{-2\Phi}dr,$
(2.21)

whilst $e^{2h(r)}={\cal G}(r)$ must satisfy a nonlinear second order
differential equation,

$\ddot{{\cal G}}+\frac{(\dot{{\cal G}})^{2}}{{\cal G}}-\frac{5\dot{{\cal
G}}}{2k{\cal G}}-2\dot{{\cal G}}+\frac{3}{2{\cal G}}+\frac{2}{k}=0.$ (2.22)

The numerical resolution of this system is studied in Appendix B. Let us gain
some understanding by discussing asymptotic solutions. For large values of the
coordinate $r$ we find an asymptotic solution,

$e^{2h(r)}\sim e^{r},\;\;\;e^{2f(r)}\sim
e^{2f_{0}+r},\;\;\;e^{2\Phi}\sim\frac{km}{k-e^{-r}}\sim m.$ (2.23)

In the case of the compactification on the hyperbolic plane ($k=-1$), we find
an exact, fixed-point solution. This is the same solution found in [23], after
conventions are matched.

$e^{2f(r)}=e^{\frac{2}{3}r},\;\;\;e^{2h(r)}=\frac{3}{4},\;\;\;e^{2\Phi(r)}=\frac{3m}{4}.$
(2.24)

As is usual, the $r$-coordinate plays the role of energy-coordinate. The
solution in eq.(2.23) asymptotes to $\widehat{\text{AdS}}_{7}\times
S^{2}_{c}\times R_{z}$ when replaced in eqs.(2.2)-(2.3), describing a six
dimensional SCFT formulated on $R^{1,3}\times H_{2}$. The
$\widehat{\text{AdS}}_{7}$ is written as a foliation over this six-space. On
the other end, when the fixed point solution of eq.(2.24) is replaced in the
family of backgrounds of Massive Type IIA of eqs.(2.2)-(2.3), the space time
takes the form AdS${}_{5}\times{\cal M}_{5}$. These describe the dual to a
family of four dimensional ${\cal N}=1$ SCFTs. The numerical solution
connecting the large-$r$ asymptotics in eq.(2.23) with the fixed-point exact
solution (2.24) is described in Appendix B. We present the plot for the
functions $f(r),h(r),g(r)$ in Figure 1.

Figure 1: The plot for the case of the $H_{2}$ compactification ($k=-1$) of
the functions $f(r),h(r),\Phi(r)$. The UV asymptotics at $r\rightarrow+\infty$
and the IR at $r\rightarrow-\infty$. We clearly see the attainment of both IR
and UV fixed points.

The numerical study of the case of the two-sphere ($k=1$) leads to badly
singular behaviour as we decrease the $r$-coordinate. We believe that a
background more elaborated than that in eqs.(2.2)-(2.3) could resolve this
singular behaviour, but leave this for a future investigation. In what follows
we focus our attention on the backgrounds describing the twisted
compactification on $H_{2}$ (preserving 4 SUSYs) of a six dimensional ${\cal
N}=(1,0)$ 6d SCFT.

We close this section emphasising that we have presented a new infinite family
of backgrounds in eqs.(2.2)-(2.5). Each solution in this family is labelled by
each possible $\alpha(z)$. At the IR fix point, this is a Massive IIA
background with an AdS5 factor preserving four supercharges.

##  3 Study of the dual field theory

We start this section with the holographic calculation of the free energy of
the field theory. In particular we start with the free energy of the 4d SCFT
that results in the low energy limit of our compactification. After that, we
study the same quantity along the flow described by our full solution. We then
define a monotonic quantity that captures the flow between the low energy four
dimensional SCFT and the high energy six dimensional conformal point (as
suggested by the family of supergravity backgrounds).

After these holographic calculations we use the charges calculated in Section
2 to give an heuristic proposal for a 4d quiver that at low energies becomes
conformal and captures some aspects of the fixed point AdS5 backgrounds. This
proposal is provisional and certainly incomplete. In fact, whilst various
observables and scaling behaviours do match between supergravity and QFT
calculations, the precise coefficient of the free energy does not, suggesting
that some extra fields should supplement our QFT proposal.

###  3.1 Free Energy and Holographic Central Charge

One meaningful observable of SCFTs is the free energy. For the case of SCFTs
in diverse dimensions this was calculated holographically. See for example the
papers [5], [7], [8], [10], [12], [13], [14]. These calculations were checked
against field theoretical computations (typically a global anomaly coefficient
or a localisation calculation in matrix models). These checks of the AdS/CFT
correspondence are valid in the regime of parameters for which the
supergravity background is a reliable representation of the QFT dynamics. This
is typically for long linear quivers $P\rightarrow\infty$ and for large rank
of the gauge nodes $N_{i}\rightarrow\infty$.

The calculations done with the supergravity backgrounds are either computing a
regularised on-shell action for a putative reduced supergravity or
alternatively the calculation of the Newton constant associated with a reduced
theory of gravity in lower dimensions.

In particular, it was shown in [26], [27] that for any generic holographic
background dual to a QFT in $d+1$ spacetime dimensions, with metric and
dilaton given by,

$ds^{2}=a(r,y^{i})\left(dx^{2}_{d,1}+b(r)dr\right)+g_{ij}dy^{i}dy^{j},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \Psi(r,y^{i}),$
(3.1)

we can define quantities $V_{int},\widehat{H}$ according to

$V_{int}=\int
dy^{i}\sqrt{e^{-4\Psi}a(r,y^{i})^{d}\det[g_{int}]},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\widehat{H}=V_{int}^{2}.$ (3.2)

From these we define the holographic central charge (or free energy),

$c_{hol}=d^{d}\frac{b(r)^{\frac{d}{2}}H^{\frac{2d+1}{2}}}{G^{(10)}_{N}(H^{\prime})^{d}},$
(3.3)

where $G^{(10)}_{N}=8\pi^{6}$ is in our conventions, the ten-dimensional
Newton constant.

For the case of our backgrounds in eq.(2.2), we compare with eq.(3.1) and
read,

$a(r,z)=2\beta\pi\sqrt{2}\sqrt{-\frac{\alpha(z)}{\alpha^{\prime\prime}(z)}}\,X(r)^{-\frac{1}{2}}e^{-\frac{4\Phi(r)}{5}+2f(r)},\leavevmode\nobreak\
\leavevmode\nobreak\ b(r)=e^{-2f(r)},\leavevmode\nobreak\ \leavevmode\nobreak\
d=3.$ (3.4)

The metric of the internal space has determinant,

$\text{det}(g_{\text{int}})=\frac{16\sqrt{2}\pi^{5}}{\omega^{2}}e^{4h(r)+\Phi(r)}\frac{\sqrt{-\alpha(z)^{7}\alpha^{\prime\prime}(z)}}{(\alpha^{\prime}(z)^{2}-2\alpha(z)\alpha^{\prime\prime}(z))^{2}}\frac{\sin^{2}(\sqrt{k}\theta_{1})}{k}\sin^{2}(\theta_{2}).$
(3.5)

This leads, after using eqs.(3.2)-(3.3), to

$\displaystyle
H=\widehat{\mathcal{N}}^{2}e^{6f(r)+4h(r)-4\Phi(r)},\leavevmode\nobreak\
\leavevmode\nobreak\
\widehat{\mathcal{N}}=16\sqrt{2}\pi^{4}e^{-2\Psi_{0}}\int\left[-\alpha(z)\alpha^{\prime\prime}(z)\right]dz\,\operatorname{Vol}(\Sigma_{k})\operatorname{Vol}(S^{2}).$
$\displaystyle
c_{hol}=\frac{27\widehat{\mathcal{N}}}{8G^{(10)_{N}}}\frac{e^{2h(r)-2\Phi(r)}}{(3\dot{f}+2\dot{h}-2\dot{\Phi})^{3}}.$
(3.6)

After using the BPS equations in (2.5), the holographic central charge reads

$c=\frac{27\widehat{\mathcal{N}}}{G^{(10)}_{N}}\frac{e^{2h(r)-2\Phi(r)}}{\left(4+\frac{1}{k}e^{-2h(r)}+m\,e^{-2\Phi(r)}\right)^{3}}.$
(3.7)

Since this computes the central charge of the four dimensional SCFTs– we have
set $d=3$ in eq.(3.4)— we evaluate this quantity for $k=-1$, the hyperbolic
plane twisted compactification of the 6d SCFTs (setting $m=1$), and find

$c_{hol}=\frac{27\widehat{\mathcal{N}}}{64\,G^{(10)}_{N}}.$ (3.8)

Let us analyse the results in eqs.(3.7)-(3.8).

First, notice that the factor $\widehat{{\cal N}}$ in eq.(3.6), has a part
proportional to $\Big{(}-\operatorname{Vol}(S^{2})\int
dz\alpha^{\prime\prime}\alpha\Big{)}$. We recognise this as information coming
from the UV, six dimensional SCFT. Indeed, this factor appears when
calculating the free energy of a six dimensional $(1,0)$ SCFT, see equation
(2.14) in the paper [15] or equation (4.10) in the first paper referred in
[14]. This is coming from the UV-part of the flow. Note that $\widehat{{\cal
N}}$ in eq.(3.6), also contains a factor of ${\hbox{\rm
Vol}}\Sigma_{k}=4\pi|(g-1)|$. This suggests that the number of fields gets
multiplied by $(g-1)$. Both these factors inform the phenomenological proposal
for the QFT in Section 3.3.

Let us evaluate the holographic central charge in eq.(3.8) for two examples.
Whilst these examples are not generic, they capture many aspects of the
dynamics of the QFTs. We also discuss these examples (purely from a QFT
perspective) in Appendix C.

####  3.1.1  Example 1

We study the case corresponding to a function $\alpha(z)$ given by,

$\alpha(z)=\begin{cases}\frac{N}{6}(1-P^{2})z+\frac{N}{6}z^{3}&,0\leq
z\leq(P-1)\\\ \\\
-\frac{1}{6}(2P^{2}-3P+1)(P-z)+\frac{N}{6}(P-1)(P-z)^{3}&,(P-1)\leq z\leq
P.\end{cases}$ (3.9)

This function is associated with a 6d SCFT consisting of a liner quiver with
gauge groups of rank $N_{j}=jN$, ending with a flavour group $SU(PN)$. The
function $\alpha(z)$ vanishes at $z=0$ and $z=P$, is continuous at $z=(P-1)$
and the derivative $\alpha^{\prime}(z)$, is continuous at the same point. The
second derivative is

$\alpha^{\prime\prime}(z)=\begin{cases}Nz&,0\leq z\leq(P-1)\\\ \\\
N(P-1)(P-z)&,(P-1)\leq z\leq P.\end{cases}$ (3.10)

We now use the expressions in eqs. (3.6)-(3.8), the fact that
$e^{2\Psi_{0}}=4\sqrt{2}\pi$, $G_{N}=8\pi^{6}$ and find,

$\displaystyle\widehat{{\cal
N}}=\frac{64}{45}\pi^{5}N^{2}P^{5}(g-1),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
c_{hol,ex1}=\frac{3}{40\pi}N^{2}P^{5}(g-1)\left(1+O(\frac{1}{P^{2}})\right).$
(3.11)

We find a scaling with the length of the quiver and the number of nodes, that
is reminiscent of what occurs for this kind of quiver for the case of six
dimensional SCFTs. Let us see a second example.

####  3.1.2 Example 2

Consider the function $\alpha(z)$,

$\alpha(z)=\begin{cases}\frac{N}{2}(1-P)z+\frac{N}{6}z^{3}&,0\leq z\leq 1\\\
\frac{N}{6}-\frac{PN}{2}z+\frac{N}{2}z^{2}&,1\leq z\leq(P-1)\\\
-\frac{N}{2}(P-1)(P-z)+\frac{N}{6}(P-z)^{3}&,(P-1)\leq z\leq P.\end{cases}$
(3.12)

From the perspective of the 6d SCFT, this function describes a linear quiver
with $(P-1)$ gauge nodes of rank $N_{j}=N$ and flavour nodes
$F_{J}=N(\delta_{J,1}+\delta_{J,P-1})$. The one in eq.(3.12) is a function
that vanishes at $z=0$ and $z=P$, is continuos at $z=1$ and at $z=(P-1)$ and
the derivative $\alpha^{\prime}(z)$ is continuous at the same two points. The
second derivative is

$\alpha^{\prime\prime}(z)=\begin{cases}Nz&,0\leq z\leq 1\\\ N&,1\leq
z\leq(P-1)\\\ N(P-z)&,(P-1)\leq z\leq P.\end{cases}$ (3.13)

Using eqs. (3.6)-(3.8), we calculate,

$\displaystyle
c_{hol,ex2}=\frac{9}{32\pi}N^{2}P^{3}(g-1)\left(1+O(\frac{1}{P^{2}})\right).$
(3.14)

This 4d SCFT has a scaling characteristic of a six dimensional linear quiver
of the same type described above.

This result, together with the one in the first example and their respective
comparison with the field theoretical calculation in eqs.(C.9)-(C.11), inform
our phenomenological proposal of Section 3.3. Let us now discuss the
$r$-dependence of the holographic central charge in eq.(3.6).

####  3.1.3  Energy dependence of the holographic central charge

Let us now interpret the $r$-dependence of our free energy/holographic central
charge in eq.(3.7). Inspecting this quantity, we find at the IR fixed point
the result in eq.(3.8). We associate with this the free energy of a family of
four dimensional SCFTs preserving four Poincare supercharges. The family of
SCFTs in 4d is labelled by the different functions $\alpha(z)$ that we choose
as input, as in the examples above. On the other end, at high energies, we
find using eq.(2.23), that the quantity in eq.(3.7) diverges. The plot of
Figure 2 with the numerical solution found in Appendix B shows this.

Figure 2: Central Charge calculated for the numerical solutions of Figure 1.
Here we see that the central charge is divergent in the UV.

The divergence of this monotonically increasing (towards the UV) free energy
is understandable. Massive fields that originated in the wrapping of both D6’s
and NS-five branes on the surface $\Sigma_{k}=H_{2}$ generically have a mass
inversely proportional to the size of $\Sigma_{k}$ and are frozen at low
energies (of course, there are also massless fields present in the 4d SCFT).
When flowing towards the UV these massive fields become active. The number of
these Kaluza-Klein modes grows with energy and leads to the divergence of
$c_{hol}$. Somewhat, the quantity defined in eqs.(3.4)-(3.8) for a four
dimensional QFT (we have set $d=3$) is not able to recognise that the system
is approximately approaching a six dimensional SCFT.

Similar arguments were expressed by the authors of [28]. In that paper the
authors define a quantity $c_{monotonic}$ using the Entanglement Entropy. The
quantity $c_{monotonic}$ in [28] and our eq.(3.7), are related by a
proportionality factor when studied in our family of backgrounds.

This calls for a quantity that is actually able to detect both fixed points
(this was also one of the objectives in [28]). A quantity called $c_{flow}$ in
[27], [29] can play this role. This quantity is monotonic, but decreases
towards the UV fixed point. We study this below.

###  3.2 Flow central charge

Let us now define a quantity that is capable of detecting both the UV and IR
fixed points of our flow across dimensions. With this in mind, we write a
generic holographic metric dual to a QFT defined on anisotropic spacetime–see
[27], [29] for details,

$ds^{2}=-a_{0}dt^{2}_{0}+\sum^{d-p}_{i=1}a_{i}dx^{2}_{i}+a_{d-p+1}ds^{2}_{p}+\prod^{d-p}_{i=1}(a_{i}\,a^{p}_{d-p+1})^{\frac{1}{d}}\,\tilde{b}\,dr^{2}+h_{ab}dy^{a}dy^{b}.$
(3.15)

where $t$ and $x_{i}$ are coordinates of the macroscopic space, $p$ is the
dimension of the compactification manifold and $ds^{2}_{p}$ its line element.
The metric and the coordinates of the internal manifold are denoted by
$h_{ab}$ and $y^{a}$ respectively. The dilaton is $\Psi(r,y^{a})$. We define
the quantities,

$\displaystyle
G_{ij}dX^{i}dX^{j}=\sum^{d-p}_{i=1}a_{i}dx^{2}_{i}+a_{d-p+1}ds^{2}_{p}+h_{ab}dy^{a}dy^{b},$
(3.16)
$\displaystyle\tilde{H}=\left(\int\prod^{8-d}_{a=1}dX^{a}\sqrt{e^{-4\Psi}\text{det}(G_{ij})}\right)^{2},\;\;\;c_{flow}=d^{d}\frac{\tilde{b}^{\frac{d}{2}}\widehat{H}^{\frac{2d+1}{2}}}{G^{(10)}_{N}(\widehat{H}^{\prime})^{d}}.$

Specialising these quantities in our metric of eq.(2.2), we have

$\displaystyle
d=5,\;\;\;\;p=2,\;\;\;\;a_{0}=a_{1}=a_{2}=a_{3}=2\pi\sqrt{2}\sqrt{-\frac{\alpha(z)}{\alpha^{\prime\prime}(z)}}\,X(r)^{-\frac{1}{2}}e^{-\frac{4\Phi(r)}{5}+2f(r)},$
$\displaystyle
a_{4}=a_{0}\,e^{2h(r)-2f(r)},\;\;\;\tilde{b}=e^{-\frac{6}{5}f(r)-\frac{4}{5}h(r)},$
$\displaystyle\tilde{H}=\widehat{\mathcal{N}}^{2}e^{6f(r)+4h(r)-4\Phi(r)},\;\;\;\widehat{\mathcal{N}}=16\sqrt{2}\pi^{4}e^{-2\Psi_{0}}\int(-\alpha(z)\alpha^{\prime\prime}(z))dz\,\operatorname{Vol}(\Sigma_{k})\operatorname{Vol}(S^{2}).$

Using eq.(3.16) we compute,

$\displaystyle
c_{flow}=\frac{27\widehat{\mathcal{N}}}{32G^{(10)}_{N}}\frac{e^{-2\Phi(r)}}{(3\dot{f}+2\dot{h}-2\dot{\Phi})^{5}},\leavevmode\nobreak\
\leavevmode\nobreak\ \text{use BPS in eq.(\ref{BPSp}) to find},$
$\displaystyle
c_{flow}=\frac{27\widehat{\mathcal{N}}}{G^{(10)}_{N}}\frac{e^{-2\Phi(r)}}{\left(4+\frac{1}{k}e^{-2h(r)}+m\,e^{-2\Phi(r)}\right)^{5}}.$
(3.17)

Evaluating in the IR fixed point of eq.(2.24) and in the asymptotic UV-
expansion of eq.(2.23), leads to

$c_{flow,IR}=\frac{9\widehat{\mathcal{N}}}{256\,G^{(10)}_{N}},\;\;\;c_{flow,UV}=\frac{27\widehat{\mathcal{N}}}{5^{5}\,G^{(10)}_{N}}.$
(3.18)

We can evaluate this quantity numerically on the flow solutions discussed in
Appendix B. The plot in Figure 3 shows this.

Figure 3: Flow central charge calculated for the numerical solutions. We
clearly see the attainment of both IR and UV fixed points.

This plot is very similar to the analogous quantity calculated from the
Entanglement Entropy in reference [28]. The discussion about the factors in
$\widehat{\cal N}$ is similar to that in the previous subsection. The
qualitative lessons for the QFT along the flow are similar. In summary, we
have defined a quantity for an anisotropic QFT realising a flow across
dimensions. It should be nice to tackle this same problem from a purely field
theoretical perspective.

###  3.3 A phenomenological proposal for the QFT

We wish to put forwards a proposal for the brane system and the field theory
dual to the background in eqs.(2.2)-(2.3). Our proposal is motivated by the
scaling of the holographic central charge, calculated in
eqs.(3.8),(3.11),(3.14). We find that our proposal is at best provisional. In
fact, whilst the scaling with free energy of the quiver parameters matches
with the holographic computations, the precise coefficients do not match. We
suggest possible modifications of our proposal to be studied in the future.

We will be particularly interested on the fixed point solution of eq.(2.24),
that gives a family of backgrounds with an AdS5 factor preserving four
Poincare supercharges.

As we advanced in the previous section, for large values of the
$r$-coordinate, the system is dual to a QFT that is to a good approximation
six-dimensional. In fact, using the solution of eq.(2.23) in the background of
eqs.(2.2)-(2.3), we find that the space asymptotes to AdS${}_{7}\times
S^{2}\times R_{z}$. The six dimensional SCFT is defined on a spacetime of the
form $R^{1,3}\times H_{2}$. This forces the twisting of the SCFT (to partially
preserve SUSY). The different fibrations in the Massive IIA background
(encoded by the one form $\Theta_{1}$) are the effects of the twist in the
holographic perspective. Following the formalism developed in the references
[14]-[15], we can roughly think that the field theory is encoded in a long
linear quiver and Hanany-Witten set up of D6-D8-NS5 branes depicted in Figure
4

Figure 4: We show a rough approximation to the UV field theory. This would be
just the torus reduction of a family of ${\cal N}=(1,0)$ six dimensional
SCFTs. As we explain below, this picture is deformed by other brane stacks
appearing.

The numbers $N_{1},N_{2},....,N_{P}$ and $F_{1},....,F_{P}$ must satisfy the
relation $F_{i}=2N_{i}-N_{i-1}-N_{i+1}$ in order for the six dimensional field
theory to be free of gauge anomalies. This condition is ensured by the
function $\alpha^{\prime\prime}(z)$ chosen in eq.(LABEL:alphasecond). Indeed,
note that from the fourth-derivative $\alpha^{(4)}(z)$ we derive the relation
between flavours and colours required, see eq.(2.19). The function
$\alpha(z)$, continuous and cubic by pieces is chosen such that the
$z$-coordinate begins and ends smoothly. This is imposed by the conditions
$\alpha(0)=\alpha(P)=0$. There is one UV conformal point for each choice of
$\alpha(z)$. Of course, this is just a very rough picture.

Indeed, these UV conformal points are deformed (either by VEVs or by relevant
operators). The dimension of these operators can be read from the near-AdS7
expansion of the metric. These deformations (analogously, the presence of the
fibrations) topologically twist the 6d CFT and trigger a RG flow, that ends in
a CFT4. As we discussed around eqs. (2.14), (2.20), a new set of NS and
D6-branes appears due to the twisted compactification. These new branes lead
to the presence of new gauge groups, not present in the original linear quiver
without twisting.

Following this line of arguments, our system is represented by the Hanany-
Witten set up of Figure 5.

Figure 5: We draw the new set of branes–denoted with a tilde–which are induced
by the twisting. These new branes give place to new gauge groups and new
matter fields represented by strings extending between different stacks of
branes.

As expressed in eq.(2.14), a new set of NS-five branes appears due to the
twisting. These new $(g-1)\times P$ NS5 branes share the $R^{1,3}$ directions
with the other branes and extend along the coordinates
$(\theta_{2},\phi_{2})$. They are orthogonal to the original NS branes that
extend along $(R^{1,3},\theta_{1},\phi_{1})$. This leads to the generation of
new gauge groups. In fact, the D6 branes originally extended along
$(R^{1,3},\theta_{1},\phi_{1},z)$ can now extend between the two stacks of NS-
five branes. This generates a new stack of D6 branes extended along
$(R^{1,3},\theta_{2},\phi_{2},z)$. We associate with these the new gauge
groups. As calculated in eq.(2.20), there are $(g-1)\times N_{l}$ of them, for
each interval. There are also D8 sources, playing the role of flavour groups.
The system must cancel all gauge anomalies. We emphasise the $(g-1)\times
P\times P$-replication of the original quiver. This reflects in the free
energy calculated in Section 3.1. It should be interesting to carefully
quantise the open strings between these different stacks of branes, to have a
more concrete handle on the low energy QFT.

A natural question that arises is the following: can we find a Lagrangian
description for the 4d QFT? If so, we could propose a (low energy) duality
between a 4d Lagrangian QFT and a 6d SCFT on a geometry of the form
$R^{1,3}\times\Sigma_{\kappa}$. This is perhaps too much to ask. We should be
content with a 4d quiver (even when strongly coupled) that captures aspects of
the compactified 6d theory. In fact, the above analysis leads us to propose a
quiver, that consist on $P^{2}(g-1)$ copies of the six-dimensional ’mother’
theory. Of course, the system is now four dimensional and preserves ${\cal
N}=1$ SUSY. In the usual notation (lines with arrows indicating chiral
multiplets, circles denoting vector multiplets, boxes indicating flavour
symmetries), we propose the quiver in Figure 6.

Figure 6: The proposed quiver field theory. The adjoint fields $\Phi_{i}$ get
a mass. The bifundamentals fields are denoted by $(Q,\bar{Q})$, the
fundamentals by $(q,\bar{q})$ and the vector multiplets represented by the
circles. All the fields are in four dimensional ${\cal N}=1$ notation.

In Appendix C, we assign values to the anomalous dimensions and R-charges of
each of these four dimensional fields. Using those values we calculate beta
functions and R-symmetry anomalies for each gauge group (finding vanishing
values). Also a suitable superpotential is proposed, this indicates an
interaction between different rows of the quiver in Figure 6. With the
R-charges assignation above, we also calculate $a_{CFT}$ and $c_{CFT}$, the
$a,c$-central charges of the quiver in Figure 6. We compare some of these
results with the analogous holographic ones in Section 3.1. Whilst the scaling
with the quiver parameters (number and rank of gauge nodes) is matched, the
precise numerical coefficient does not coincide.

Let us close this section with three important comments.

  * •

It may be the case that other fields that do not contribute to the beta
functions and the R-symmetry anomaly, but still contribute to the $a,c$
central charges, are present in our quiver of Figure 6. To decide if they
should be there, we should compute the integral of the anomaly polynomial
$I_{8}$ on the compactification manifold $\Sigma_{-}$. We get a hint at those
fields, by comparing the material of Section 3.1 with that in Appendix C.

  * •

The scaling with the number of nodes $P$ of the a-central charge (the free
energy) is two powers higher than usual (see Appendix C for a field
theoretical derivation of this result, and Section 3.1 for a holographic
viewpoint on this). In fact, it actually scales as a six dimensional CFT. The
quiver we write is deconstructing two dimensions, in the sense of [24]. What
is interesting about this case is that this is occurring due to the twisting,
whilst in [24] it is an effect of going to a particular point in the moduli
space, expanding, etc.

  * •

Finally, as explained in Appendix C, a mass term for $\Phi_{i}$ in eq.(C.3)
appears. This generates a new set of $(P-1)-U(1)$’s. These are all anomalous,
except for one. The R-symmetry we have taken is the one after this mixing has
taken place. Importantly, this makes the beautiful result of Bobev and
Crichigno [25], not applicable here. In fact, the authors of [25] assume the
absence of mixings. In this sense, the coefficient in their eq.(3.6) for the
$a$ central charge, is not enforced on us222Special thanks to Nikolay Bobev
for a discussion on this..

Let us close this work by presenting some conclusions.

##  4 Conclusions

Let us start with a brief overview of the contents of this paper.

In Section 2, we construct an infinite family of Massive Type IIA backgrounds.
These describe holographically the flow between a family of six-dimensional
${\cal N}=(1,0)$ SCFTs and four dimensional ${\cal N}=1$ SCFTs. The charges of
the brane system are discussed. New sets of branes appear induced by the
twisted-compactification and are carefully discussed.

In Section 3, we present and calculate two quantities that measure the number
of degrees of freedom along the flow. One of these quantities coincides with
the holographic central charge for the 4d SCFTs, and diverges in the UV. The
second quantity detects both fixed points, it is monotonous, its value being
bigger in the IR than in the UV. We also present a phenomenological proposal
for a family of dual four dimensional SCFTs of the linear quiver type. The
form and the parameters characterising the quiver are inherited from the six
dimensional description. The 4d QFTs are proposed to reach a conformal point
at low energies compared to the inverse size of the compactification manifold.
These 4d SCFTs are holographically described by an infinite family of Massive
Type IIA backgrounds with an AdS5 factor. We use the quiver field theory
description to calculate field theory observables–see Appendix C for details.
Interestingly our field theoretical calculation of the $a$ and $c$ central
charges in comparison with the holographic one, suggests that other fields
should intervene in the dynamics, aside from those in the proposed quiver.

Detailed appendices complement the presentation. We trust that the reader
wishing to work on these topics should enjoy and profit from them.

This paper suggest various interesting topics to work on, as follow up of the
material presented here. We make a small list below.

  * •

It should be interesting to further study the two-sphere compactifications.
These do not lead to conformal 4d fixed points. It is important to de-
singularise those solutions.

  * •

It is important to study the string quantisation in the Hanany-Witten set up
that arises as a result of the twisted compactification. The massless modes
would give firm clues about the four dimensional quiver field theory, for
which we proposed a particular type of quiver.

  * •

The presence of extra fields contributing to the central charges (as
emphasised in Appendix C) is one of the most urgent topics suggested by this
work. May be a treatment along the lines of [20] proves effective. In fact,
involving the anomaly polynomials $I_{8}$ and $I_{6}$ for the SCFT6 and SCFT4
may show the need to add flip fields.

  * •

More generally, bringing together the field theoretical techniques of
[19]-[20] with the holographic results expressed in this paper or in [29],
seems like a problem to study.

  * •

It should be interesting to attempt this kind of compactifications with
systems in different dimensions. The systematic study of examples might
indicate dimension-dependent characteristics.

  * •

It should be interesting to exploit the recently discovered plethora of
embeddings of 5d supergravity into Massive IIA [30]. Finding interesting
compactifications of our family of backgrounds, black holes, and other
solutions may illuminate various aspects of the ${\cal N}=1$, four dimensional
family of SCFTs.

We hope to return to these problems soon.

## Acknowledgments:

The contents and presentation of this work much benefitted from extensive
discussion with various colleagues. We would like to make a special mention to
Nikolay Bobev, Stefano Cremonesi and Alessandro Tomasiello. We would also like
to thank very useful conversations with Mohammad Akhond, Andrea Legramandi,
Yolanda Lozano, Daniel Thompson.

For the purpose of open access, the authors have applied a Creative Commons
Attribution (CC BY) licence to any Author Accepted Manuscript version arising.

We are supported by STFC grant ST/T000813/1. The work of R.S. is supported by
STFC grant ST/W507878/1.

##  Appendix A Appendix 1

In this appendix we outline in detail the construction of the infinite family
of solutions of Massive Type IIA used in this paper. We begin by considering
7D $SU(2)$ Topologically Massive gauged Supergravity. We propose an ansatz and
derive the BPS equations from the SUSY variations. Finally, we outline the
uplift of the 7D background to 10D Massive Type IIA, following the procedure
used in [31].

Starting in Appendix A.1.1 we change slightly the notation respect to the main
body of the paper. We denote with a prime $()^{\prime}$ the derivative with
respect to the $r$-coordinate.

###  A.1 7D $SU(2)$ Topologically Massive Gauged Supergravity

The Lagrangian, in string frame, of the bosonic fields of the 7D $SU(2)$
Topologically Massive Supergravity is given by,

$\displaystyle\mathcal{L}$
$\displaystyle=\sqrt{-g}e^{-2\Phi}\left[R-\frac{1}{8}F^{i}_{\mu\nu}F^{\mu\nu\,i}+4\partial^{\mu}\Phi\partial_{\mu}\Phi-\left(\frac{m^{2}}{2}e^{-4\Phi}-4m\,e^{-2\Phi}-4\right)\right]$
(A.1) $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\phantom{=}-\frac{1}{2}e^{2\Phi}\ast G_{4}\wedge G_{4}+\frac{1}{4}F^{i}\wedge
F^{i}\wedge B_{3}-\frac{m}{2}G_{4}\wedge B_{3},$

where $G_{4}=dB_{3}$, $F=dA+iA\wedge A$, and $A$ is a $su(2)$ valued gauge
1-form, i.e. $A=A^{i}T^{i}$. The SUSY variations of the fermionic fields of
the theory are

$\displaystyle\delta\lambda$
$\displaystyle=\left[\Gamma^{\mu}\partial_{\mu}\Phi-\frac{i}{4}\Gamma^{\mu\nu}F_{\mu\nu}+\frac{1}{48}e^{2\Phi}\Gamma^{\mu\nu\lambda\rho}G_{\mu\nu\lambda\rho}-m\,e^{-2\Phi}+1\right]\epsilon,$
(A.2) $\displaystyle\delta\Psi_{\mu}$
$\displaystyle=\left[D_{\mu}+iA_{\mu}-\frac{i}{2}\Gamma^{\nu}F_{\mu\nu}+\frac{1}{96}e^{2\Phi}\Gamma^{\phantom{\mu}\nu\lambda\rho\sigma}_{\mu}G_{\nu\lambda\rho\sigma}-\frac{m}{4}e^{-2\Phi}\Gamma_{\mu}\right]\epsilon,$
(A.3)

where the covariant derivative is given by

$D_{\mu}\epsilon=\left(\partial_{\mu}+\frac{1}{4}\omega_{\mu}^{\phantom{\mu}ab}\Gamma_{ab}\right)\epsilon.$
(A.4)

Here $\mu,\nu$ are spacetime indices while $a,b$ are tangent space ones.

We want to derive the BPS equations using a method similar to the one showed
in [32], where the author studied the same Supergravity theory but for
vanishing topological mass ($m=0$). For this, we need to redefine our 1-form
gauge field as follows: first we express the $su(2)$ generators, $T^{i}$, in
terms of the Pauli matrices, $\sigma^{i}$, as $T^{i}=\frac{\sigma^{i}}{2}$.
Then we derive the expression for the components of the field strength along
the algebra generators. Using $A\wedge A=\frac{1}{2}[A,A]$, we have

$F=dA+\frac{i}{2}[A,A]=dA^{i}\frac{\sigma^{i}}{2}+\frac{i}{8}\left[\sigma^{j},\sigma^{k}\right]^{i}A^{j}\wedge
A^{k}.$ (A.5)

Using the algebra of the Pauli matrices,
$\left[\sigma^{i},\sigma^{j}\right]=2i\,\varepsilon^{ijk}\sigma^{k}$, and
expanding $F=F^{i}\frac{\sigma^{i}}{2}$, we get

$F^{i}=dA^{i}-\frac{1}{2}\varepsilon^{ijk}A^{j}\wedge A^{k}.$ (A.6)

In order to get both the same field strength and SUSY variations as in [32] in
the limit $m=0$, we redefine $A^{i}\rightarrow-A^{i}$, which leads to

$F^{i}=-\left(dA^{i}+\frac{1}{2}\varepsilon^{ijk}A^{j}\wedge
A^{k}\right)=-F^{\prime\,i}.$ (A.7)

Note that this change doesn’t affect the Lagrangian since it is quadratic in
F, while the SUSY variations now change, since they are linear in $A$ and $F$.
Using these redefinitions and expanding in the algebra generators, one finds

$\displaystyle\delta\lambda$
$\displaystyle=\left[\Gamma^{\mu}\partial_{\mu}\Phi+\frac{i}{8}\Gamma^{\mu\nu}F^{i}_{\mu\nu}\sigma^{i}+\frac{1}{48}e^{2\Phi}\Gamma^{\mu\nu\lambda\rho}G_{\mu\nu\lambda\rho}-m\,e^{-2\Phi}+1\right]\epsilon,$
(A.8) $\displaystyle\delta\Psi_{\mu}$
$\displaystyle=\left[D_{\mu}-\frac{i}{2}A^{i}_{\mu}\sigma^{i}+\frac{i}{4}\Gamma^{\nu}F^{i}_{\mu\nu}\sigma^{i}+\frac{1}{96}e^{2\Phi}\Gamma^{\phantom{\mu}\nu\lambda\rho\sigma}_{\mu}G_{\nu\lambda\rho\sigma}-\frac{m}{4}e^{-2\Phi}\Gamma_{\mu}\right]\epsilon,$
(A.9)

note that we dropped the ′ of $F^{\prime}$.

####  A.1.1 Background Ansatz

For the background metric we consider the following fibred version of
$\mathbb{R}^{3,1}\times\Sigma_{k}\times\mathbb{R}$

$ds^{2}=e^{2f(r)}dx^{2}_{3,1}+e^{2h(r)}\left(d\theta^{2}+\frac{1}{k}\sin^{2}(\sqrt{k}\theta)d\phi^{2}\right)+dr^{2},$
(A.10)

where $k$ is proportional to the curvature of $\Sigma_{k}$. Taking $k=-1,0,1$,
gives $\Sigma_{k}=H^{2},\mathbb{R}^{2}$ or $S^{2}$, respectively. Because we
are interested in studying twisted compactifications we will only consider the
cases $k=\pm 1$.

The rest of the background fields are $B_{3}=0$, $\Phi=\Phi(r)$ and

$A^{1}=-k\,a(r)d\theta,\quad
A^{2}=\frac{1}{\sqrt{k}}a(r)\sin(\sqrt{k}\theta)d\phi,\quad
A^{3}=-\frac{1}{k}\cos(\sqrt{k}\theta)d\phi,$ (A.11)

which leads to the field strengths333In the appendixes, we have denoted the
derivative with respect to $r$ with a prime. In the main body of the paper we
have reserved the primes for derivatives with respect to $z$, whilst denoting
the $r$-derivatives with a dot.

$F^{1}=-k\,a^{\prime}(r)\,dr\wedge d\theta,\quad
F^{2}=\frac{1}{\sqrt{k}}a^{\prime}(r)\sin(\sqrt{k}\theta)\,dr\wedge
d\phi,\quad
F^{3}=\Big{(}1-k\,a^{2}(r)\Big{)}\frac{\sin(\sqrt{k}\theta)}{\sqrt{k}}\,d\theta\wedge
d\phi.$ (A.12)

In what follows we will need the vielbeins

$e^{\widehat{m}}=e^{f(r)}dx^{m},\quad
e^{\widehat{\theta}}=e^{h(r)}d\theta,\quad
e^{\widehat{\phi}}=\frac{1}{\sqrt{k}}e^{h(r)}\sin(\theta)d\phi,\quad
e^{\widehat{r}}=dr,$ (A.13)

and the spin connection

$\omega^{\widehat{m}\widehat{r}}=f^{\prime}(r)\,e^{f(r)}dx^{m},\quad\omega^{\widehat{\theta}\widehat{\phi}}=-\cos(\sqrt{k}\theta)d\phi,\quad\omega^{\widehat{\theta}\widehat{r}}=h^{\prime}(r)\,e^{h(r)}d\theta,\quad\omega^{\widehat{\phi}\widehat{r}}=\frac{1}{\sqrt{k}}h^{\prime}(r)\,e^{h(r)}\sin(\sqrt{k}\theta)d\phi.$
(A.14)

In the vielbein basis, we have

$A^{1}=-k\,a(r)\,e^{-h(r)}e^{\widehat{\theta}},\quad
A^{2}=a(r)\,e^{-h(r)}e^{\widehat{\phi}},\quad
A^{3}=\frac{1}{\sqrt{k}}e^{-h(r)}\text{cotg}(\sqrt{k}\theta)e^{\widehat{\phi}},$
(A.15) $F^{1}=k\,a^{\prime}(r)e^{-h(r)}e^{\widehat{\theta}}\wedge
e^{\widehat{r}},\quad F^{2}=-a^{\prime}(r)e^{-h(r)}e^{\widehat{\phi}}\wedge
e^{\widehat{r}},\quad
F^{3}=\Big{(}1-k\,a^{2}(r)\Big{)}e^{-2h(r)}e^{\widehat{\theta}}\wedge
e^{\widehat{\phi}},$ (A.16)
$\omega^{\widehat{m}\widehat{r}}=f^{\prime}(r)e^{\widehat{m}},\quad\omega^{\widehat{\theta}\widehat{\phi}}=-\sqrt{k}\,e^{-h(r)}\text{cotg}(\sqrt{k}\theta)e^{\widehat{\phi}},\quad\omega^{\widehat{\theta}\widehat{r}}=h^{\prime}(r)e^{\widehat{\theta}},\quad\omega^{\widehat{\phi}\widehat{r}}=h^{\prime}(r)e^{\widehat{\phi}}.$
(A.17)

###  A.2 From SUSY Variations to BPS Equations

Substituting the above ansatz into the expressions for the SUSY variations,
one derives BPS equations as outlined below. We begin with the Dilatino
variation and move onto the Gravitino variations.

####  A.2.1 Dilatino Variation

Substituting the ansatz into the Dilatino variation leads to

$\delta\lambda=\left[\Gamma_{\widehat{r}}\Phi^{\prime}(r)+\frac{i}{4}\left(k\,a^{\prime}(r)e^{-h(r)}\Gamma_{\widehat{\theta}\widehat{r}}\sigma^{1}-a^{\prime}(r)e^{-h(r)}\Gamma_{\widehat{\phi}\widehat{r}}\sigma^{2}+\Big{(}1-k\,a^{2}(r)\Big{)}e^{-2h(r)}\Gamma_{\widehat{\theta}\widehat{\phi}}\sigma^{3}\right)-m\,e^{-2\Phi(r)}+1\right]\epsilon.$
(A.18)

We now impose the projection

$\Gamma_{\widehat{\theta}\widehat{\phi}}\epsilon=\frac{1}{k}\sigma^{1}\sigma^{2}\epsilon.$
(A.19)

Recalling that the Pauli matrices satisfy $\sigma^{1}\sigma^{2}\sigma^{3}=i$,
we have

$\delta\lambda=\left[\Gamma_{\widehat{r}}\Phi^{\prime}(r)+\frac{i}{4}a^{\prime}(r)e^{-h(r)}\left(k\,\Gamma_{\widehat{\theta}\widehat{r}}\sigma^{1}-\Gamma_{\widehat{\phi}\widehat{r}}\sigma^{2}\right)-\frac{1}{4k}\Big{(}1-k\,a^{2}(r)\Big{)}e^{-2h(r)}-m\,e^{-2\Phi(r)}+1\right]\epsilon,$
(A.20)

then, by multiplying by $\Gamma_{\widehat{r}}$, we get

$\Gamma_{\widehat{r}}\delta\lambda=\left[\Phi^{\prime}(r)-\frac{i}{4}a^{\prime}(r)e^{-h(r)}\left(k\,\Gamma_{\widehat{\theta}}\sigma^{1}-\Gamma_{\widehat{\phi}}\sigma^{2}\right)+\left(1-m\,e^{-2\Phi(r)}-\frac{1}{4k}\Big{(}1-k\,a^{2}(r)\Big{)}e^{-2h(r)}\right)\Gamma_{\widehat{r}}\right]\epsilon.$
(A.21)

Using the projection (A.19) and the fact that $k=\pm 1$ one can show that

$\left(k\,\Gamma_{\widehat{\theta}}\sigma^{1}-\Gamma_{\widehat{\phi}}\sigma^{2}\right)\epsilon=2k\Gamma_{\widehat{\theta}}\sigma^{1}\epsilon.$
(A.22)

Replacing this in the variation, yields

$\Gamma_{\widehat{r}}\delta\lambda=\left[\Phi^{\prime}(r)-\frac{i}{2}k\,a^{\prime}(r)e^{-h(r)}\Gamma_{\widehat{\theta}}\sigma^{1}+\left(1-m\,e^{-2\Phi(r)}-\frac{1}{4k}\Big{(}1-k\,a^{2}(r)\Big{)}e^{-2h(r)}\right)\Gamma_{\widehat{r}}\right]\epsilon.$
(A.23)

####  A.2.2 Gravitino Variation

We now switch our focus to the Gravitino variations, focusing on the
$\widehat{m}$, $\widehat{\theta}$, $\widehat{\phi}$ and $\widehat{r}$
components in turn. From the $m=0,1,2,3$ components of the variation of the
gravitino in tangent space indices, we have

$\delta\psi_{\widehat{m}}=\left(\frac{1}{2}f^{\prime}(r)\Gamma_{\widehat{m}\widehat{r}}-\frac{m}{4}e^{-2\Phi(r)}\Gamma_{\widehat{m}}\right)\epsilon,$
(A.24)

and multiplying by $2\Gamma_{\widehat{r}\widehat{m}}$, gives

$2\Gamma_{\widehat{r}\widehat{m}}\delta\psi_{\widehat{m}}=\left(f^{\prime}(r)-\frac{m}{2}e^{-2\Phi(r)}\Gamma_{\widehat{r}}\right)\epsilon.$
(A.25)

From the $\widehat{\theta}$ component, we have

$\delta\psi_{\widehat{\theta}}=\left[\frac{1}{2}h^{\prime}(r)\Gamma_{\widehat{\theta}\widehat{r}}+\frac{i}{2}k\,a(r)\,e^{-h(r)}\sigma^{1}+\frac{i}{4}\left(k\,a^{\prime}(r)e^{-h(r)}\Gamma_{\widehat{r}}\sigma^{1}+\Big{(}1-k\,a^{2}(r)\Big{)}e^{-2h(r)}\Gamma_{\widehat{\phi}}\sigma^{3}\right)-\frac{m}{4}e^{-2\Phi(r)}\Gamma_{\widehat{\theta}}\right]\epsilon.$
(A.26)

Multiplying by $2\Gamma_{\widehat{r}\widehat{\theta}}$, we get

$2\Gamma_{\widehat{r}\widehat{\theta}}\delta\psi_{\widehat{\theta}}=\left[h^{\prime}(r)+i\,k\,a(r)\,e^{-h(r)}\Gamma_{\widehat{r}\widehat{\theta}}\sigma^{1}+\frac{i}{4}\left(-k\,a^{\prime}(r)e^{-h(r)}\Gamma_{\widehat{\theta}}\sigma^{1}+\Big{(}1-k\,a^{2}(r)\Big{)}e^{-2h(r)}\Gamma_{\widehat{r}}\Gamma_{\widehat{\theta}\widehat{\phi}}\sigma^{3}\right)-\frac{m}{4}e^{-2\Phi(r)}\Gamma_{\widehat{r}}\right]\epsilon.$
(A.27)

Using the projection (A.19) leads to

$2\Gamma_{\widehat{r}\widehat{\theta}}\delta\psi_{\widehat{\theta}}=\left[h^{\prime}(r)+i\,k\,a(r)\,e^{-h(r)}\Gamma_{\widehat{r}\widehat{\theta}}\sigma^{1}-\frac{i}{2}k\,a^{\prime}(r)e^{-h(r)}\Gamma_{\widehat{\theta}}\sigma^{1}-\frac{1}{2k}\Big{(}1-k\,a^{2}(r)\Big{)}e^{-2h(r)}\Gamma_{\widehat{r}}-\frac{m}{2}e^{-2\Phi(r)}\Gamma_{\widehat{r}}\right]\epsilon.$
(A.28)

From the $\widehat{\phi}$ component we have

$\displaystyle\delta\psi_{\widehat{\phi}}=$
$\displaystyle\left[\frac{1}{2}\left(h^{\prime}(r)\Gamma_{\widehat{\phi}\widehat{r}}-\sqrt{k}e^{-h(r)}\text{cotg}(\sqrt{k}\theta)\Gamma_{\widehat{\theta}\widehat{\phi}}\right)-\frac{i}{2}\left(a(r)\,e^{-h(r)}\sigma^{2}+\frac{1}{\sqrt{k}}e^{-h(r)}\text{cotg}(\sqrt{k}\theta)\sigma^{3}\right)\right.$
(A.29) $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\left.+\frac{i}{4}\left(-a^{\prime}(r)e^{-h(r)}\Gamma_{\widehat{r}}\sigma^{2}-\Big{(}1-k\,a^{2}(r)\Big{)}e^{-2h(r)}\Gamma_{\widehat{\theta}}\sigma^{3}\right)-\frac{m}{4}e^{-2\Phi(r)}\Gamma_{\widehat{\phi}}\right]\epsilon.$

Using (A.19), the second and the fourth term cancel. This is the effect of the
topological twist. We are left with

$\delta\psi_{\widehat{\phi}}=\left[\frac{1}{2}h^{\prime}(r)\Gamma_{\widehat{\phi}\widehat{r}}-\frac{i}{2}a(r)\,e^{-h(r)}\sigma^{2}+\frac{i}{4}\left(-a^{\prime}(r)e^{-h(r)}\Gamma_{\widehat{r}}\sigma^{2}-\Big{(}1-k\,a^{2}(r)\Big{)}e^{-2h(r)}\Gamma_{\widehat{\theta}}\sigma^{3}\right)-\frac{m}{4}e^{-2\Phi(r)}\Gamma_{\widehat{\phi}}\right]\epsilon,$
(A.30)

then multiplying by $2\Gamma_{\widehat{r}\widehat{\phi}}$, gives

$2\Gamma_{\widehat{r}\widehat{\phi}}\delta\psi_{\widehat{\phi}}=\left[h^{\prime}(r)-i\,a(r)\,e^{-h(r)}\Gamma_{\widehat{r}\widehat{\phi}}\sigma^{2}+\frac{i}{2}\left(a^{\prime}(r)e^{-h(r)}\Gamma_{\widehat{\phi}}\sigma^{2}+\Big{(}1-k\,a^{2}(r)\Big{)}e^{-2h(r)}\Gamma_{\widehat{r}}\Gamma_{\widehat{\theta}\widehat{\phi}}\sigma^{3}\right)-\frac{m}{2}e^{-2\Phi(r)}\Gamma_{\widehat{r}}\right]\epsilon.$
(A.31)

Using the projection again, leads to

$2\Gamma_{\widehat{r}\widehat{\phi}}\delta\psi_{\widehat{\phi}}=\left[h^{\prime}(r)-i\,a(r)\,e^{-h(r)}\Gamma_{\widehat{r}\widehat{\phi}}\sigma^{2}+\frac{i}{2}a^{\prime}(r)e^{-h(r)}\Gamma_{\widehat{\phi}}\sigma^{2}-\frac{1}{2k}\Big{(}1-k\,a^{2}(r)\Big{)}e^{-2h(r)}-\frac{m}{2}e^{-2\Phi(r)}\Gamma_{\widehat{r}}\right]\epsilon,$
(A.32)

and due to (A.22), we have
$k\Gamma_{\widehat{\theta}}\sigma^{1}\epsilon=-\Gamma_{\widehat{\phi}}\sigma^{2}\epsilon$,
giving

$2\Gamma_{\widehat{r}\widehat{\phi}}\delta\psi_{\widehat{\phi}}=\left[h^{\prime}(r)+i\,k\,a(r)\,e^{-h(r)}\Gamma_{\widehat{r}\widehat{\theta}}\sigma^{1}-\frac{i}{2}k\,a^{\prime}(r)e^{-h(r)}\Gamma_{\widehat{\theta}}\sigma^{1}-\frac{1}{2}\Big{(}1-k\,a^{2}(r)\Big{)}e^{-2h(r)}-\frac{m}{2}e^{-2\Phi(r)}\Gamma_{\widehat{r}}\right]\epsilon,$
(A.33)

so we see that
$\Gamma_{\widehat{r}\widehat{\theta}}\delta\psi_{\widehat{\theta}}=\Gamma_{\widehat{r}\widehat{\phi}}\delta\psi_{\widehat{\phi}}$.

Finally, from the $\widehat{r}$ component of the gravitino variation, we have

$\delta\psi_{\widehat{r}}=\left[\partial_{r}+\frac{i}{4}\left(-k\,a^{\prime}(r)e^{-h(r)}\Gamma_{\widehat{\theta}}\sigma^{1}+a^{\prime}(r)e^{-h(r)}\Gamma_{\widehat{\phi}}\sigma^{2}\right)-\frac{m}{4}e^{-2\Phi(r)}\Gamma_{\widehat{r}}\right]\epsilon,$
(A.34)

which can be rewritten as

$\delta\psi_{\widehat{r}}=\left[\partial_{r}+\frac{i}{4}a^{\prime}(r)e^{-h(r)}\left(-k\,\Gamma_{\widehat{\theta}}\sigma^{1}+\Gamma_{\widehat{\phi}}\sigma^{2}\right)-\frac{m}{4}e^{-2\Phi(r)}\Gamma_{\widehat{r}}\right]\epsilon,$
(A.35)

and using the projection, we find

$\delta\psi_{\widehat{r}}=\left[\partial_{r}-\frac{i}{2}k\,a^{\prime}(r)e^{-h(r)}\Gamma_{\widehat{\theta}}\sigma^{1}-\frac{m}{4}e^{-2\Phi(r)}\Gamma_{\widehat{r}}\right]\epsilon.$
(A.36)

After imposing $\delta\lambda=0$ and $\delta\psi_{\mu}=0$, the results
(A.23),(A.25) and (A.28) lead to the following BPS equations

$\displaystyle\Phi^{\prime}(r)\epsilon-\frac{i}{2}k\,a^{\prime}(r)e^{-h(r)}\Gamma_{\widehat{\theta}}\sigma^{1}\epsilon+\left(1-m\,e^{-2\Phi(r)}-\frac{1}{4k}\Big{(}1-k\,a^{2}(r)\Big{)}e^{-2h(r)}\right)\Gamma_{\widehat{r}}\epsilon=0,$
(A.37) $\displaystyle
f^{\prime}(r)\epsilon-\frac{m}{2}e^{-2\Phi(r)}\Gamma_{\widehat{r}}\epsilon=0,$
(A.38) $\displaystyle
h^{\prime}(r)\epsilon+i\,k\,a(r)\,e^{-h(r)}\Gamma_{\widehat{r}\widehat{\theta}}\sigma^{1}\epsilon-\frac{i}{2}k\,a^{\prime}(r)e^{-h(r)}\Gamma_{\widehat{\theta}}\sigma^{1}\epsilon-\frac{1}{2k}\Big{(}1-k\,a^{2}(r)\Big{)}e^{-2h(r)}\Gamma_{\widehat{r}}\epsilon-\frac{m}{2}e^{-2\Phi(r)}\Gamma_{\widehat{r}}\epsilon=0,$
(A.39)
$\displaystyle\partial_{r}\epsilon=\frac{i}{2}k\,a^{\prime}(r)e^{-h(r)}\Gamma_{\widehat{\theta}}\sigma^{1}\epsilon+\frac{m}{4}e^{-2\Phi(r)}\Gamma_{\widehat{r}}\epsilon.$
(A.40)

####  A.2.3 Rearranging the SUSY variations

Following the procedure given in [32], we now rearrange the SUSY variations to
obtain the BPS equations. For this, we use the properties of the $\Gamma$
matrices to simplify the equations obtained in the previous subsection.

Let us start by rewriting the dilatino variation (A.23) as

$\Gamma_{\widehat{r}}\epsilon=\beta\epsilon+\tilde{\beta}i\Gamma_{\widehat{\theta}}\sigma^{1}\epsilon,$
(A.41)

with

$\displaystyle\beta$
$\displaystyle=\frac{-\Phi^{\prime}(r)}{1-m\,e^{-2\Phi(r)}-\frac{1}{4k}(1-k\,a^{2}(r))e^{-2h(r)}},$
(A.42) $\displaystyle\tilde{\beta}$
$\displaystyle=\frac{\frac{1}{2}k\,a^{\prime}(r)e^{-h(r)}}{1-m\,e^{-2\Phi(r)}-\frac{1}{4k}(1-k\,a^{2}(r))e^{-2h(r)}}.$
(A.43)

By applying $\Gamma_{\widehat{r}}$ to (A.41), we get

$\epsilon=\beta\,\Gamma_{\widehat{r}}\epsilon-i\tilde{\beta}\,\Gamma_{\widehat{\theta}}\sigma^{1}\Gamma_{hr}\epsilon.$
(A.44)

Using (A.41), gives

$\epsilon=\beta\left(\beta\,\epsilon+\tilde{\beta}\,i\Gamma_{\widehat{\theta}}\sigma^{1}\epsilon\right)-i\tilde{\beta}\,\Gamma_{\widehat{\theta}}\sigma^{1}\left(\beta\,\epsilon+\tilde{\beta}\,i\Gamma_{\widehat{\theta}}\sigma^{1}\epsilon\right),$
(A.45)

and by simplifying, we obtain

$\epsilon=(\beta^{2}+\tilde{\beta}^{2})\epsilon,$ (A.46)

from which we see that

$\beta^{2}+\tilde{\beta}^{2}=1.$ (A.47)

Multiplying (A.38) by $\Gamma_{\widehat{r}}$ we get

$f^{\prime}(r)\Gamma_{\widehat{r}}\epsilon=\frac{m}{2}\,e^{-2\Phi(r)}\epsilon.$
(A.48)

Combining this with (A.38) leads to

$\left(f^{\prime\,2}(r)-\frac{m^{2}}{4}e^{-4\Phi(r)}\right)\epsilon=0,$ (A.49)

which means that

$f^{\prime}(r)=\pm\frac{m}{2}\,e^{-2\Phi(r)}.$ (A.50)

Now we go back to (A.48) and substitute in (A.41) as follows

$f^{\prime}(r)\left(\beta+\tilde{\beta}\,i\Gamma_{\widehat{\theta}}\sigma^{1}\right)\epsilon-\frac{m}{2}\,e^{-2\Phi(r)}=0.$
(A.51)

Collecting terms with and without $i\Gamma_{\widehat{\theta}}\sigma^{1}$, we
get

$\left(f^{\prime}(r)\,\beta-\frac{m}{2}\,e^{-2\Phi(r)}\right)\epsilon+f^{\prime}(r)\,\tilde{\beta}\,i\Gamma_{\widehat{\theta}}\sigma^{1}=0,$
(A.52)

from where we get two equations

$\displaystyle f^{\prime}(r)\beta$ $\displaystyle=\frac{m}{2}\,e^{-2\Phi(r)},$
(A.53) $\displaystyle f^{\prime}(r)\tilde{\beta}$ $\displaystyle=0.$ (A.54)

From (A.50) and (A.53) we get $\beta=\pm 1$, and from (A.54) we get
$\tilde{\beta}=0$, which in (A.43) gives

$a^{\prime}(r)=0,$ (A.55)

and hence, (A.41) becomes

$\Gamma_{\widehat{r}}\epsilon=\beta\epsilon.$ (A.56)

Now we turn to (A.39). By replacing (A.56) and $a^{\prime}(r)=0$, we get

$h^{\prime}(r)\epsilon-i\beta\,k\,a(r)\,e^{-h(r)}\Gamma_{\widehat{\theta}}\sigma^{1}\epsilon-\beta\frac{1}{2k}\Big{(}1-k\,a^{2}(r)\Big{)}e^{-2h(r)}\epsilon-\beta\frac{m}{2}e^{-2\Phi(r)}\epsilon=0.$
(A.57)

Again, by collecting terms with and without
$i\Gamma_{\widehat{\theta}}\sigma^{1}$, we get

$\left(h^{\prime}(r)-\beta\frac{1}{2k}\Big{(}1-k\,a^{2}(r)\Big{)}e^{-2h(r)}-\beta\frac{m}{2}e^{-2\Phi(r)}\right)\epsilon-i\beta\,k\,a(r)\,e^{-h(r)}\Gamma_{\widehat{\theta}}\sigma^{1}\epsilon=0.$
(A.58)

From which we get

$\displaystyle
h^{\prime}(r)-\beta\frac{1}{2k}\Big{(}1-k\,a^{2}(r)\Big{)}e^{-2h(r)}-\beta\frac{m}{2}e^{-2\Phi(r)}=0,$
(A.59) $\displaystyle a(r)\,e^{-h(r)}=0.$ (A.60)

From the second result, we necessarily get $a(r)=0$. After implementing this
condition into (A.42) and (A.59), we finally derive the following BPS
equations for $f(r)$ (which we got from (A.50)), $h(r)$ and $\Phi(r)$

$\displaystyle f^{\prime}(r)$ $\displaystyle=\pm\frac{m}{2}\,e^{-2\Phi(r)},$
(A.61) $\displaystyle h^{\prime}(r)$
$\displaystyle=\pm\frac{1}{2}\left(\frac{1}{k}e^{-2h(r)}+m\,e^{-2\Phi(r)}\right),$
(A.62) $\displaystyle\Phi^{\prime}(r)$
$\displaystyle=\pm\left(-1+\frac{1}{4k}e^{-2h(r)}+m\,e^{-2\Phi(r)}\right).$
(A.63)

The minus sign can be absorbed by $r\rightarrow-r$.

###  A.3 Uplift to 10D Massive Type IIA

####  A.3.1 Einstein Frame and Normalizations

The procedure which we will follow to uplift our 7D topologically massive
solution is given in [31], however, the action we used for the 7D
$\mathcal{N}=2$ $SU(2)$ Gauged SUGRA is written in string frame, while in [31]
is written in Einstein frame, hence we move our action to Einstein frame and
then we compare normalizations and relevant constants. To move to Einstein
frame we use the transformation

$g^{(E)}_{\mu\nu}=e^{-\alpha\Phi(r)}g^{(S)}_{\mu\nu},$ (A.64)

in the action (A.1). Then, by comparing with [31] we see that
$\alpha=\frac{4}{5}$, $m=1$, $g=\sqrt{2}$, $h=\frac{1}{2}$ and that our
Dilaton $\Phi(r)$ is related to the one in [31], which we call $\varphi(r)$,
by

$\varphi(r)=\frac{2\sqrt{10}}{5}\Phi(r).$ (A.65)

Also, from the kinetic term for the gauge field, we see that in [31] the gauge
field is normalised in a way in which the coupling constant appears in the
definition of $F$ rather than as a coefficient of the kinetic term $F^{2}$,
while in (A.1) we consider the opposite. To match conventions we re-scale our
gauge field and field strength as

$A\rightarrow-\frac{1}{g}A,\quad F\rightarrow-\frac{1}{g}F.$ (A.66)

With this, our 7D ansatz now reads

$ds^{2}=e^{-\frac{4}{5}\Phi(r)}\left(e^{2f(r)}ds^{2}_{3,1}+e^{2h(r)}\left(d\theta^{2}+\frac{1}{k}\sin^{2}(\sqrt{k}\theta)d\phi^{2}\right)+dr^{2}\right),$
(A.67)

together with the gauge field

$A_{1}=0,\quad A_{2}=0,\quad
A_{3}=\frac{1}{g}\frac{1}{k}\cos(\sqrt{k}\theta)d\phi,$ (A.68)

which leads to

$F_{1}=0,\quad F_{2}=0,\quad
F_{3}=-\frac{1}{g}\operatorname{Vol}(\Sigma^{2}_{k}).$ (A.69)

####  A.3.2 Uplift of 7D Topologically Massive to Massive IIA

The massive Type IIA ansatz is given by

$ds^{2}=\frac{1}{\ell}X(r)^{-\frac{1}{2}}e^{2A(\rho)}ds^{2}_{7}+X(r)^{\frac{5}{2}}ds^{2}_{M_{3}},$
(A.70)

with $\ell=4$, $ds^{2}_{7}$ a solution of the 7D $\mathcal{N}=2$ Gauged
Supergravity presented in the previous section, and

$ds^{2}_{M_{3}}=d\rho^{2}+\frac{1-x(\rho)^{2}}{16\omega}e^{2A(\rho)}Ds^{2}_{S^{2}},$
(A.71)

where $X(r)$ is given by the 7D dilaton

$X(r)=e^{\frac{2}{5}\Phi(r)},$ (A.72)

and

$\omega=X(r)^{5}(1-x(\rho)^{2})+x(\rho)^{2}.$ (A.73)

The covariantized metric $Ds^{2}_{S^{2}}$ on the sphere is constructed as
follows. First we consider the normal unitary vector to the sphere

$y^{i}=\left(\cos(\phi_{2})\sin(\theta_{2}),\sin(\phi_{2})\sin(\theta_{2}),\cos(\theta_{2})\right),$
(A.74)

then, the covariant line element is given by

$Ds^{2}_{S^{2}}=Dy^{i}Dy^{i},\quad Dy^{i}=dy^{i}+g\epsilon^{ijk}y^{k}A^{k},$
(A.75)

where $A^{i}$ is the component along the $i$th Pauli matrix of the 7D gauge
field. This spacetime is supported by the 10D Dilaton $\Psi$

$e^{2\Psi}=\frac{X(r)^{\frac{5}{2}}}{\omega}e^{2\psi(\rho)},$ (A.76)

and the background forms

$\displaystyle B_{2}$
$\displaystyle=\left[\frac{1}{16}e^{2A(\rho)}\frac{x(\rho)\sqrt{1-x(\rho)^{2}}}{\omega}\operatorname{Vol}_{2}-\frac{1}{2}e^{A(\rho)}d\rho\wedge\left(a-\frac{1}{g}y^{i}A^{i}\right)\right],$
(A.77) $\displaystyle F_{2}$
$\displaystyle=-q\left(\operatorname{Vol}_{2}+g\,y^{i}F^{i}\right)+\frac{1}{16\omega}\,F_{0}e^{2A(\rho)}x(\rho)\sqrt{1-x(\rho)^{2}}\operatorname{Vol}_{2},$
(A.78) $\displaystyle F_{4}$
$\displaystyle=\frac{\ell}{4}\left[-g\frac{q}{16\omega}e^{2A(\rho)}x(\rho)\sqrt{1-x(\rho)^{2}}y^{i}F^{i}\wedge\operatorname{Vol}_{2}-g\frac{q}{4}e^{A(\rho)}d\rho\wedge\epsilon^{ijk}F^{i}y^{j}Dy^{k}\right]+G_{4}\,\text{terms},$
(A.79)

where Vol2 is the volume element of the covariantised sphere,
$a=\frac{1}{2}\cos(\theta_{2})d\phi_{2}$ and $F_{0}$ is the Ramond mass (which
is constant). Note that we have not written explicitly the terms proportional
to the 7D 4-form, since we will not use them.

This configuration is a solution to the equations of motion

$\displaystyle\frac{1}{4}R+\nabla^{2}\Psi-(\nabla\Psi)^{2}-\frac{1}{8}H_{3}^{2}=0,$
(A.80) $\displaystyle dF_{p}+H_{3}\wedge*F_{p-2}=0,$ (A.81) $\displaystyle
d(e^{-2\Psi}*H_{3})-\left(F_{0}*F_{2}+F_{2}\wedge*F_{4}+F_{4}\wedge
F_{4}\right)=0,$ (A.82) $\displaystyle
R_{MN}+2\nabla_{M}\nabla_{N}\Psi-\frac{1}{2}(H^{2}_{3})_{MN}-\frac{1}{4}e^{2\Psi}\sum_{p}(F^{2}_{p})_{MN},$
(A.83)

with $p=2,4,6,8,10$, and

$(F^{2}_{p})_{MN}=\frac{1}{(p-1)!}F_{M}^{\phantom{M}N_{1}...N_{p-1}}F_{MN_{1}...N_{p-1}},$
(A.84)

and similarly for $(H^{2}_{3})_{MN}$, provided $\psi(\rho)$, $x(\rho)$ and
$A(\rho)$ satisfy

$\displaystyle\frac{d}{d\rho}\psi(\rho)$
$\displaystyle=\frac{1}{4}\frac{e^{-A(\rho)}}{\sqrt{1-x(\rho)^{2}}}\left(12x(\rho)+\Big{(}2x(\rho)^{2}-5\Big{)}F_{0}\,e^{A(\rho)+\psi(\rho)}\right),$
(A.85) $\displaystyle\frac{d}{d\rho}x(\rho)$
$\displaystyle=-\frac{1}{2}e^{-A(\rho)}\sqrt{1-x(\rho)^{2}}\Big{(}4+x(\rho)\,F_{0}\,e^{A(\rho)+\psi(\rho)}\Big{)},$
(A.86) $\displaystyle\frac{d}{d\rho}A(\rho)$
$\displaystyle=\frac{1}{4}\frac{e^{-A(\rho)}}{\sqrt{1-x(\rho)^{2}}}\left(4x(\rho)-F_{0}\,e^{A(\rho)+\psi(\rho)}\right).$
(A.87)

####  A.3.3 Explicit Uplift of the Interpolating Background

We now write explicitly the field configuration for the uplift of
(A.67)-(A.69). The spacetime metric reads (here we rename
$(\theta,\phi)\rightarrow(\theta_{1},\phi_{1})$ with respect to the 7D
solution)

$\displaystyle ds^{2}$
$\displaystyle=\frac{1}{\ell}X(r)^{-\frac{1}{2}}e^{2A(\rho)}e^{-\frac{4\Phi(r)}{5}}\left[e^{2f(r)}dx^{2}_{3,1}+dr^{2}+e^{2h(r)}\left(d\theta^{2}_{1}+\frac{1}{k}\sin^{2}(\sqrt{k}\theta_{1})d\phi^{2}_{1}\right)\right]$
(A.88) $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
+X(r)^{5/2}\left[d\rho^{2}+\frac{1-x(\rho)^{2}}{16\omega}e^{2A(\rho)}\left(d\theta^{2}_{2}+\sin^{2}(\theta_{2})\left(d\phi_{2}-\frac{1}{k}\cos(\sqrt{k}\theta_{1})d\phi_{1}\right)^{2}\right)\right],$

with $\ell=4$, while the background forms read

$\displaystyle
B_{2}=\left(\frac{1}{16\omega}e^{2A(\rho)}x(\rho)\sqrt{1-x(\rho)^{2}}\sin(\theta_{2})d\theta_{2}-\frac{1}{4}e^{A(\rho)}\cos(\theta_{2})d\rho\right)\wedge\left(d\phi_{2}-\frac{1}{k}\cos(\sqrt{k}\theta_{1})d\phi_{1}\right),$
(A.89) $\displaystyle
F_{2}=\frac{1}{4}e^{A(\rho)-\psi(\rho)}\sqrt{1-x(\rho)^{2}}\left[\cos(\theta_{2})\operatorname{Vol}(\Sigma_{k})-\operatorname{Vol}(S^{2}_{c})\right]+\frac{1}{16\omega}\,F_{0}e^{2A(\rho)}x(\rho)\sqrt{1-x(\rho)^{2}}\,\operatorname{Vol}(S^{2}_{c}),$
$\displaystyle
F_{4}=\frac{e^{3A(\rho)-\psi(\rho)}}{64\omega}\cos(\theta_{2})\,x(\rho)\Big{(}1-x(\rho)^{2}\Big{)}\operatorname{Vol}(\Sigma_{k})\wedge\operatorname{Vol}(S^{2})+\frac{e^{2A(\rho)-\psi(\rho)}}{16}\sin^{2}(\theta_{2})\sqrt{1-x^{2}(\rho)}\,d\rho\wedge
d\phi_{2}\wedge\operatorname{Vol}(\Sigma_{k}),$

where

$\operatorname{Vol}(S^{2})=\sin(\theta_{2})d\theta_{2}\wedge d\phi_{2},$
(A.90)

is the volume of the 2-sphere of the internal manifold, while

$\operatorname{Vol}(S^{2}_{c})=\sin(\theta_{2})d\theta_{2}\wedge\left(d\phi_{2}-\frac{1}{k}\cos(k\theta_{1})d\phi_{1}\right),$
(A.91)

is the volume of the covariantised 2-sphere. Note that

$\operatorname{Vol}(\Sigma_{k})\wedge\operatorname{Vol}(S^{2}_{c})=\operatorname{Vol}(\Sigma_{k})\wedge\operatorname{Vol}(S^{2}).$
(A.92)

####  A.3.4 Page Fluxes

In order to get a quantised number of charges, we need to consider the Page
fluxes, given by

$\widehat{F}_{p}=F_{p}\wedge e^{-B_{2}},$ (A.93)

for the RR fields. Explicitly we have

$\displaystyle\widehat{F}_{2}$ $\displaystyle=F_{2}-B_{2}F_{0}$
$\displaystyle=\frac{1}{4}e^{A(\rho)-\psi(\rho)}\sqrt{1-x(\rho)^{2}}\left[\cos(\theta_{2})\operatorname{Vol}(\Sigma_{k})-\operatorname{Vol}(S^{2}_{c})\right]+F_{0}\frac{e^{A(\rho)}}{4}\cos(\theta_{2})d\rho\wedge\left(d\phi_{2}-\frac{1}{k}\cos(\sqrt{k}\theta_{1})d\phi_{1}\right),$
(A.94) $\displaystyle\widehat{F}_{4}$ $\displaystyle=F_{4}-F_{2}\wedge
B_{2}+\frac{1}{2}B_{2}\wedge B_{2}F_{0}$
$\displaystyle=\frac{1}{16}e^{2A(\rho)-\psi(\rho)}\sqrt{1-x(\rho)^{2}}d\rho\wedge
d\phi_{2}\wedge\operatorname{Vol}(\Sigma_{k}).$ (A.95)

####  A.3.5 Rewriting in terms of $\alpha(z)$

Finally, it can be shown (as in [33]) that the 10D BPS equations can be solved
in terms of just one function, $\alpha(z)$, provided $(A,x,\psi)$ are of the
following form

$\displaystyle A(\rho)$
$\displaystyle=\frac{1}{2}\ln\,\left(8\pi\sqrt{2}\sqrt{-\frac{\alpha(z)}{\alpha^{\prime\prime}(z)}}\right),$
(A.96) $\displaystyle\psi(\rho)$
$\displaystyle=\frac{1}{4}\ln\left(e^{4\psi_{0}}\frac{\left(-\frac{\alpha(z)}{\alpha^{\prime\prime}(z)}\right)^{3}}{\Big{(}\alpha^{\prime}(z)^{2}-2\alpha(z)\alpha^{\prime\prime}(z)\Big{)}^{2}}\right),$
(A.97) $\displaystyle x(\rho)$
$\displaystyle=\sqrt{1+\frac{2\alpha(z)\alpha^{\prime\prime}(z)}{\alpha^{\prime}(z)^{2}-2\alpha(z)\alpha^{\prime\prime}(z)}},$
(A.98)

where the coordinate $z$ is related to $\rho$ via the following change of
coordinates

$d\rho=\sqrt{\pi\sqrt{2}}\left(-\frac{\alpha^{\prime\prime}(z)}{\alpha(z)}\right)^{\frac{1}{4}}dz.$
(A.99)

This is a solution of the 10D BPS equations provided $\alpha(z)$ satisfies

$\alpha^{\prime\prime\prime}(z)=2^{-\frac{1}{4}}\sqrt{\pi}e^{\psi_{0}}F_{0}.$
(A.100)

In terms of this new variable, the metric and background fields are the ones
in (2.2)-(2).

##  Appendix B Numerical solution of the BPS system

Here we give a detailed derivation of the numerical solutions that describe
the flow from AdS7 to AdS${}_{5}\times H^{2}$. The starting point is to
consider a linear perturbation around the IR fixed point and use that as
initial conditions for the numerical solution.

###  B.1 Infrared Fixed Point

We are looking for solutions to the BPS equations for $f(r)$, $h(r)$ and
$\Phi(r)$, which we quote here for convenience,

$\displaystyle f^{\prime}$ $\displaystyle=\frac{m}{2}\,e^{-2\Phi},$ (B.1)
$\displaystyle h^{\prime}$
$\displaystyle=\frac{1}{2}\left(\frac{1}{k}e^{-2h}+m\,e^{-2\Phi}\right),$
(B.2) $\displaystyle\Phi^{\prime}$
$\displaystyle=\left(-1+\frac{1}{4k}e^{-2h}+m\,e^{-2\Phi}\right).$ (B.3)

that have constant $h(r)$ and $\Phi(r)$. This will correspond to the IR fixed
point, and we note that it exists only for $k=-1$. The solutions

$f(r)=\frac{2}{3}r,\quad
h(r)=\frac{1}{2}\ln\left(\frac{3}{4}\right),\quad\Phi(r)=\frac{1}{2}\ln\left(\frac{3m}{4}\right),$
(B.4)

are exact solutions to the BPS equations, and the spacetime obtained from this
corresponds to AdS${}_{5}\times H^{2}$.

###  B.2 Linear Perturbations

Here we consider linear deviations from the IR fixed point

$\displaystyle f(r)$ $\displaystyle=\frac{2}{3}r+\epsilon\leavevmode\nobreak\
a(r),$ (B.5) $\displaystyle h(r)$
$\displaystyle=\frac{1}{2}\ln\left(\frac{3}{4}\right)+\epsilon\leavevmode\nobreak\
b(r),$ (B.6) $\displaystyle\Phi(r)$
$\displaystyle=\frac{1}{2}\ln\left(\frac{3m}{4}\right)+\epsilon\leavevmode\nobreak\
c(r).$ (B.7)

Replacing in the BPS equations and keeping only first order terms in
$\epsilon$, we obtain linear equations for the linear perturbations

$\displaystyle a^{\prime}$ $\displaystyle=-\frac{4}{3}c,$ (B.8) $\displaystyle
b^{\prime}$ $\displaystyle=\frac{4}{3}(b-c),$ (B.9) $\displaystyle c^{\prime}$
$\displaystyle=\frac{1}{3}(2b-8c).$ (B.10)

This systems has as solution

$\displaystyle a(r)$
$\displaystyle=\frac{C_{1}}{3}(\sqrt{7}+2)e^{-\frac{2}{3}(\sqrt{7}+1)r}-\frac{C_{2}}{3}(\sqrt{7}-2)e^{\frac{2}{3}(\sqrt{7}-1)r},$
(B.11) $\displaystyle b(r)$
$\displaystyle=C_{1}e^{-\frac{2}{3}(\sqrt{7}+1)r}+C_{2}e^{\frac{2}{3}(\sqrt{7}-1)r},$
(B.12) $\displaystyle c(r)$
$\displaystyle=\frac{C_{1}}{2}(\sqrt{7}+3)e^{-\frac{2}{3}(\sqrt{7}+1)r}-\frac{C_{2}}{2}(\sqrt{7}-3)e^{\frac{2}{3}(\sqrt{7}-1)r}.$
(B.13)

We are interested in solutions to vanish for $r\rightarrow-\infty$, which is
the location of the fixed point, hence we set $C_{1}=0$ and also without loss
of generality we can set $C_{2}=1$.

Finally, to obtain the numerical solutions we use the perturbations around the
fixed point (B.5)-(B.7) as initial conditions at $r=-8$. Figure 1 shows these
numerical solutions.

##  Appendix C Analysis of the 4d QFT

Referring to the four dimensional ${\cal N}=1$ quiver in Figure 6, we assign
R-charges and anomalous dimensions for each of the fields. For the adjoint
scalars $\Phi_{i}$, the vector multiplets $W_{i}$, the bifundamentals between
gauge nodes $(Q,\tilde{Q})$ and the fundamentals $(q,\bar{q})$ we propose

$\displaystyle\text{The R-charges:}\leavevmode\nobreak\ \leavevmode\nobreak\
R[\Phi_{i}]=1,\;\;\;R[W_{i}]=1,\;\;\;R[Q]=R[\tilde{Q}]=R[q]=R[\tilde{q}]=\frac{1}{2}.$
(C.1) $\displaystyle\text{The anomalous dimensions:}\leavevmode\nobreak\
\leavevmode\nobreak\
\gamma_{\Phi_{i}}=1,\;\;\;\gamma_{Q}=\gamma_{\tilde{Q}}=\gamma_{q}=\gamma_{\tilde{q}}=-\frac{1}{2}.$
(C.2)

Notice that the dimension of any combination of fields $\cal{O}$ satisfy

$\text{dim}{\cal O}=\frac{3}{2}R_{{\cal O}}.$

As should occur at conformal points. In particular the following
superpotential terms are present,

${\cal
W}\sim\mu_{i}\Phi_{i}\Phi_{i}+q_{i}\bar{q}_{j}q_{j}\bar{q}_{i}+\text{all other
combinations}.$ (C.3)

The R-charge of each of these possible terms is $R[{\cal W}]=2$ and their
dimension $\text{dim}[{\cal W}]=3$. The mass term for all the adjoint scalars
decouples them from the IR dynamics. They do not participate in the quantities
computed below. The quartic term, on the other hand, generates interactions
between all the different $P^{2}(g-1)$ horizontal lines of the quiver, by
closing loops using the flavour groups. As one can start seeing, the behaviour
of these quivers is somewhat reminiscent of those in [34]. Let us calculate
the beta functions and the R-symmetry anomaly for each gauge group of the
quiver in Figure 6.

####  C.0.1  Beta functions

For any particular node we use the NSVZ beta function $\beta\sim
3N_{c}-N_{fi}(1-\gamma_{i})$, we find

$\beta_{i}\sim
3N_{i}-(N_{i+1}+N_{i-1}+F_{i})(1-(-\frac{1}{2}))=\frac{3}{2}\Big{[}2N_{i}-F_{i}-N_{i+1}-N_{i-1}\Big{]}=0.$
(C.4)

As we start with a balanced quiver in the UV (the six dimensional quiver must
cancel gauge anomalies), the beta function in four dimensions vanishes for
each gauge group. It is nice to see how the six dimensional anomaly transmuted
into the four dimensional beta function condition for conformality.

####  C.0.2  R-symmetry anomaly

We calculate using the expression $\Delta\Theta=T(R_{i})R(f_{i})$. We use that
$T(adj_{i})=2N_{i}$ and $T(fund)=1$. We find

$\Delta\Theta_{i}=2N_{i}\times 1+(F_{i}+N_{i+1}+N_{i-1})\times
2\times(-\frac{1}{2})=2N_{i}-F_{i}-N_{i+1}-N_{i-1}=0.$ (C.5)

The comments written below eq.(C.4) also apply here. Both these calculations
add to the proposal that the quiver in Figure 6 describes the dynamics of our
AdS5 fixed point.
Other interesting quantities are the central charges $a$ and $c$.

###  C.1 Central charges

They are defined as,

$a=\frac{3}{32\pi}\Big{[}3\mbox{Tr}R^{3}-\mbox{Tr}R\Big{]},\;\;\;\;c=\frac{1}{32\pi}\Big{[}9\mbox{Tr}R^{3}-5\mbox{Tr}R\Big{]}.$
(C.6)

Calculating explicitly for the quiver in Figure 6, we find

$\displaystyle\mbox{Tr}R=P^{2}(g-1)\Big{[}\sum_{j=1}^{P-1}(N_{j}^{2}-1)\times
1+N_{j}F_{j}\times(-\frac{1}{2})\times
2+\sum_{j=1}^{P-2}N_{j}N_{j+1}\times(-\frac{1}{2})\times 2\Big{]}=$
$\displaystyle
P^{2}(g-1)\Big{[}\sum_{j=1}^{P-1}(N_{j}^{2}-1-N_{j}F_{j})-\sum_{j=1}^{P-2}N_{j}N_{j+1}\Big{]}.$
$\displaystyle\mbox{Tr}R^{3}=P^{2}(g-1)\Big{[}\sum_{j=1}^{P-1}(N_{j}^{2}-1)\times
1^{3}+N_{j}F_{j}\times(-\frac{1}{2})^{3}\times
2+\sum_{j=1}^{P-2}N_{j}N_{j+1}\times(-\frac{1}{2})^{3}\times 2\Big{]}=$
$\displaystyle=P^{2}(g-1)\Big{[}\sum_{j=1}^{P-1}(N_{j}^{2}-1-\frac{1}{4}N_{j}F_{j})-\frac{1}{4}\sum_{j=1}^{P-2}N_{j}N_{j+1}\Big{]}.$

Using eqs.(C.6) we have,

$\displaystyle
a=\frac{3}{16\pi}P^{2}(g-1)\Big{[}\sum_{j=1}^{P-1}(N_{j}^{2}-1)+\frac{1}{8}N_{j}F_{j}+\frac{1}{8}\sum_{j=1}^{P-2}N_{j}N_{j+1}\Big{]},$
(C.7) $\displaystyle
c=\frac{1}{8\pi}P^{2}(g-1)\Big{[}\sum_{j=1}^{P-1}(N_{j}^{2}-1)+\frac{11}{16}N_{j}F_{j}+\frac{11}{16}\sum_{j=1}^{P-2}N_{j}N_{j+1}\Big{]}.$
(C.8)

To gain some intuition, we discuss explicitly two illustrative examples.

####  C.1.1 Example 1

In this case we take

$N_{j}=jN,\;\;\;\;F_{j}=NP\delta_{j,P-1}.$

We find, for large values of $N$ and $P$ (the holographic limit!)

$a=\frac{9}{128\pi}P^{5}N^{2}(g-1)\left(1+O(\frac{1}{P})\right),\;\;\;\;c\sim
a.$ (C.9)

The comparison between eqs.(C.9) and (3.11) suggests that in the QFT side
there must be other fields that whilst not contributing to the beta function
and R-anomaly, it makes a contribution to the $a$ and $c$ central charges. The
contribution of these fields, that we denote as ${\cal X}$ should modify the
result in eq.(C.9) (at leading order in $P$) according to,

$a=\frac{9}{128\pi}N^{2}P^{5}(g-1)(1+\delta_{{\cal X}}),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ 15\delta_{{\cal X}}=1.$ (C.10)

####  C.1.2  Example 2

In this case we take

$N_{j}=N,\;\;\;\;F_{j}=N(\delta_{j,1}+\delta_{j,P-1}).$

We find, in the holographic limit,

$a=\frac{27}{128\pi}P^{3}N^{2}(g-1)\left(1+O(\frac{1}{P})\right),\;\;\;\;c\sim
a.$ (C.11)

The comparison between eqs.(C.11) and (3.14) suggests that in the QFT side
there must be other fields (not contributing to the beta function and
R-anomaly), but adding a contribution to the $a$ and $c$ central charges. The
contribution of these fields, that we denote as ${\cal X}$ should be of the
form (at leading order in $P$)

$a=\frac{27}{128\pi}N^{2}P^{3}(g-1)(1+\delta_{{\cal X}}),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ 3\delta_{{\cal X}}=1.$ (C.12)

This suggest that in our definition of the QFT quiver, new fields should enter
adding these small contributions to the $a$ central charge. Indeed, the
presence of ’flip fields’ that couple to irrelevant operators of the baryonic
type is common in field theories of the class $S_{k}$. These fields are gauge
singlets and decouple at low energies. They do not contribute to the beta
functions or R-symmetry anomaly, but do change the result of $a,c$. It would
be nice to see that their contribution can be $\delta_{{\cal X}}$ given in
eqs.(C.10)-(C.12).

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