# Weak Gravity Conjecture from Conformal Field Theory: A Challenge from
Hyperscaling Violating and Kerr-Newman-AdS Black Holes

Jafar Sadeghi<EMAIL_ADDRESS>Department of Physics, University of Mazandaran,
P. O. Box 47416-95447, Babolsar, Iran  Mehdi Shokri
<EMAIL_ADDRESS>School of Physics, Damghan University, P. O. Box
3671641167, Damghan, Iran  Department of Physics, University of Tehran, North
Karegar Ave., Tehran 14395-547, Iran  Canadian Quantum Research Center
204-3002 32 Avenue Vernon, British Columbia V1T 2L7 Canada  Mohammad Reza
Alipour<EMAIL_ADDRESS>Department of Physics, University of
Mazandaran, P. O. Box 47416-95447, Babolsar, Iran  Saeed Noori Gashti
<EMAIL_ADDRESS>Department of Physics, University of
Mazandaran, P. O. Box 47416-95447, Babolsar, Iran

###### Abstract

We search for a possible relation between weak gravity conjecture and
conformal field theory in the hyperscaling violating and Kerr-Newman-AdS black
holes. We deal with the critical points of the black hole systems through the
correlation function introduced in conformal field theory. We discuss the weak
gravity conjecture conditions by the imaginary part of the energy obtained
from the critical points and their poles. Under the assumptions $z=1$, $d=1$
and $\theta\rightarrow 0^{-}$, we link weak gravity conjecture to the
hyperscaling violating black holes due to the existence of $r_{H}$ larger and
smaller than one. As the second black hole system, we study the conditions of
weak gravity conjecture for the Kerr-Newman-AdS black hole by rotation and
radius parameters. Then, we show that the conditions of weak gravity
conjecture are satisfied when the charged particle near the hyperscaling
violating and Kerr-Newman black holes is $\frac{1}{a}$ with a ratio
$\frac{a}{\ell}\ll 1$.

##  I Introduction

The AdS/CFT correspondence has been known as one of the essential dualities
between quantum field theory (QFT) and gravity that corresponds to classical
dynamics of gravity in a higher dimension. The duality was first introduced to
connect four dimensions conformal field theory to the Anti-de Sitter space
(AdS) in five dimensions Maldacena:1997re ; Gubser:1998bc ; Witten:1998qj .
The AdS/CFT correspondence has been applied to a wide range of physical
science e.g. strong-coupling dynamics (QCD), the physics of black holes and
quantum gravity, electroweak theories, relativistic hydrodynamics
Klebanov:1999tb ; Kioumarsipour:2021zyg ; Bu:2021jlp ; Fujiwara:2021xgu ;
Evans:2021zzm ; MartinContreras:2021bis ; Gherghetta:2009ac ; Brodsky:2007hb ;
Nakano:2006js ; Katz:2005ir ; Meltzer:2019nbs ; Karch:2006pv ; Andreev:2006ct
; Cavaglia:2021mft ; Harmark:2020vll ; BitaghsirFadafan:2020lkh ;
DeLeeuw:2020ahx ; DeWolfe:2020uzb ; Ishigaki:2020vtr ; Terashima:2020uqu ;
Yin:2021zhs ; Mes:2020vgy ; Berenstein:2020cll ; Berman:2022idl ;
Aharony:1999ti ; DHoker:2002nbb ; Hartnoll:2009sz ; McGreevy:2009xe ;
Kim:2012ey ; Adams:2012th .

One of the most significant challenges in theoretical particle physics is
finding an effective field theory (EFT) compatible with quantum gravity. Such
theories are situated in the landscape while other incompatible EFTs live in
the swampland. A universal test to distinguish between these two classes of
theories is weak gravity conjecture (WGC). This means that gravity is always
the weakest force and shows an extremality state of the black hole Vafa:2005ui
; Ooguri:2006in ; Arkani-Hamed:2006emk ; Sadeghi:2021plz ; Cordova:2022rer ;
Henriksson:2022oeu ; Kaya:2022edp ; Collazuol:2022jiy ; Capozziello:2011nr ;
McInnes:2022tut ; Cheung:2014vva ; Rudelius:2022gyu ; Cribiori:2022trc ;
Klaewer:2020lfg ; Heidenreich:2015nta ; Heidenreich:2016aqi ; Andriolo:2018lvp
; Ooguri:2006in ; Polchinski:2003bq ; Ooguri:2016pdq ; Ibanez:2017kvh ;
Shiu:2016weq ; Fisher:2017dbc ; Cheung:2018cwt ; Crisford:2017gsb ;
Harlow:2015lma ; Hamada:2018dde ; Kinney:2018nny ; Harlow:2022gzl ;
Saraswat:2016eaz ; Nakayama:2015hga ; Bachlechner:2015qja ; Bellazzini:2019xts
; Aharony:2021mpc ; Heidenreich:2019zkl ; Benakli:2020vng ; Lee:2018spm ;
deAlwis:2019aud ; Andriolo:2020lul ; Cremonini:2020smy ; Ibanez:2015fcv ;
Banerjee:2020xcn .

In the present manuscript, we attempt to prove WGC from CFT by studying two
critical classes of black holes e.g. hyperscaling violating (HSV) and Kerr-
Newman-AdS (KNA) black holes. Since WGC is more consistent at critical points,
we use CFT calculations in order to obtain the critical points associated with
the black holes under consideration. The above discussion motivates us to
arrange the paper as follows. In section II, we briefly explain how the WGC
could be related to the CFT. In section III, we investigate the relationship
between the WGC and the CFT in the HSV black holes. We perform the similar
analysis for the KNA black holes in section IV. Conclusions and remarks are
drawn in section V.

##  II A bridge between WGC and CFT

The swampland suffers from the lack of global symmetries and completeness of
the charge spectrum. Consequently, it cannot present an acceptable explanation
for phenomena unless we restrict the global symmetry and whether there exists
an upper bound on the mass of some charged conditions Heidenreich:2021yda ;
Fichet:2019ugl ; Daus:2020vtf ; Kats:2006xp ; vanBeest:2021lhn . The swampland
can restrain a complete theory, not EFTs with low energy. Phenomenologically,
it is important to answer whether all charged particles are heavy and actual
enough concerning the black holes or if there exist several concepts of the
spectrum completeness that prevails at low energies. The swampland conjectures
discuss some important issues, such as they aspire to point out how near we
can obtain to the status of recovering the global symmetries. One can refresh
a global symmetry of $U(1)$ by transmitting the gauge coupling to 0, which
should not be permitted in quantum gravity.

Attempting to comprehend how string theory prohibits this issue and what
drives incorrect if one tries to accomplish this can supply facts about the
restrictions that EFT must meet to be compatible with quantum gravity. WGC
prohibits this methodology by the existence of unique light-charged forms that
exclude the definition of EFT. This supplies an upper bound on the mass of
charged states. In general, WGC includes the magnetic and electric versions
providing a gauge theory coupled to gravity. So we deal with electrically
charged Planck units as $Q/m\geq\mathcal{Q}/M|_{ext}=\mathcal{O}(1)$
Heidenreich:2021yda ; Fichet:2019ugl ; Daus:2020vtf ; Kats:2006xp ;
vanBeest:2021lhn . Here, $Q$ is charge, and $M$ determines mass in the
extremal black hole. Also, $Q=qg$, where $q$ and $g$ are the quantized charges
of the state and the gauge coupling, respectively.

To have WGC with the existence of charge and mass need to apply the
charge/mass greater than the one in the extremal black hole. The most specific
topic coordinates with the theory of Maxwell, which is coupled to gravity; no
massless scalar fields exist. So, we can construct solutions of R-N black
hole, so provided a $p$-form gauge field in d dimensions. WGC indicates the
presence of a ($p-1$)-brane can meet the $p(d-p-2)T^{2}/d-2\leq
Q^{2}M_{p}^{d-2}$ Harlow:2022gzl ; Saraswat:2016eaz ; Nakayama:2015hga ;
Bachlechner:2015qja ; Bellazzini:2019xts ; Aharony:2021mpc ;
Heidenreich:2019zkl ; Benakli:2020vng ; Lee:2018spm ; Daus:2020vtf ;
Katz:2005ir ; vanBeest:2021lhn . Here we note that the motivation for
understanding WGC is well seen in the physics of black holes. This conjecture
expresses that the entire lattice of authorized gauge charges must be settled
by physical conditions in an approach with a gauge coupled to gravity. This is
not required in quantum field theory since a charged particle can be decoupled
from this theory by transmitting the mass to $\infty$. The second issue is
breaking global symmetries. There are fascinating relationships between the
lack of global symmetries in quantum gravity and spectrum completeness. A
typical method of breaking higher form global symmetries includes charged
conditions. While exclusively if the charged conditions spectrum exists
entire, can one break the whole groupArkani-Hamed:2006emk ; Sadeghi:2021plz ;
Cordova:2022rer ; Henriksson:2022oeu ; Kaya:2022edp ; Collazuol:2022jiy ;
Capozziello:2011nr ; McInnes:2022tut ; Cheung:2014vva ; Harlow:2022gzl ;
Saraswat:2016eaz ; Nakayama:2015hga ; Bachlechner:2015qja ; Bellazzini:2019xts
; Aharony:2021mpc ; Heidenreich:2019zkl ; Benakli:2020vng ; Lee:2018spm .

In this article, we want to prove or emerge WGC from CFT equations. Hence, we
discuss the mixed Klein-Gordon equation against the background of the black
hole as a general perturbation, and we focus on a charged scalar $\Phi$ with
charge $q$ and mass $m$ Konoplya:2013rxa

$\frac{1}{\sqrt{-g}}\partial_{\mu}(g^{\mu\nu}\sqrt{-g}\partial_{\nu}\Phi)-2iqg^{\mu\nu}A_{\mu}\partial_{\nu}\Phi-q^{2}g^{\mu\nu}A_{\mu}A_{\nu}\Phi-m^{2}\Phi=0.$
(1)

By inserting $\Phi=e^{-i\omega t}\phi(\bm{x})$, we get its scalar function
also $\omega$ is related to the energy part. Then, we use CFT to find the two-
point correlation function of the scalar operator $J_{k}$. We also can find
the correlation function based on the ratio of its sub-coefficients as
follows. Urbano:2018kax ; Alishahiha:2012ad

$\Upsilon_{R}^{(k)}(\omega)=<J_{k}(-\omega)J_{k}(\omega)>=\frac{B_{k}(\omega)}{A_{k}(\omega)}.$
(2)

Setting $A_{k}(\omega)=0$, one can find the location of the poles in the Green
function. In such a case, we have
$\omega=\operatorname{Re}(\omega)+\operatorname{Im}(\omega)$ in which it has
real part (normal mode) and imaginary part (quasi-normal mode). The imaginary
part, which is negative, introduces inverse relaxation time $\tau_{d}$ of the
desired mode as
$\omega=\operatorname{Re}[\omega]+i\operatorname{Im}[\omega]\equiv\operatorname{Re}[\omega]-i2\pi/\tau_{d}$.
Moreover, it describes exponential damping with a characteristic time scale
set by $\tau_{d}$.
On the other hand, the quasi-normal modes play the role of controller for the
black hole ringdown in order to decay or collapse perturbed black holes
towards their hairless cases or a relaxation towards thermal equilibrium after
a perturbation period. Also, the quasi-normal modes with the smallest value of
$\operatorname{Im}[\omega]$ have a less damped state and supervise to overlook
the thermalization time- scale. Since the black hole ringdown is dual to the
thermalization process of the CFT Horowitz:1999jd , the quasi-normal modes
appear as the poles of the Fourier transform of the retarded Green’s function
in the context of the Ruelle resonances Birmingham:2001pj .
In the context of the WGC, we encounter $\tau_{d}>0$. Because the negative
values of $\tau_{d}$ correspond to an exponentially-growing unstable
fundamental state, which threatens the stability of the geometry of black
holes. Besides the positive value, it should have a specific bound because the
thermalization process can not be happened arbitrarily fast. Hence, we engage
a conjecture Urbano:2018kax

$\tau_{d}\geq\frac{1}{T},$ (3)

see Refs.Horowitz:1999jd ; Hod:2006jw ; Hod:2017uqc ; Sachdev:1992fk ;
Maldacena:2016hyu ; Eberlein:2017wah ; Gruzinov:2007ai  for more discussion.
Due to the perturbations with different thermalization time scales, the key
point is that the perturbations decay very fast in order to satisfy the
condition $\tau_{d}>c/T$. From exiting the perturbations from the metric
background, this could be realized that for a near-extremal charged black
hole, there must be at least one particle to fulfill the thermalization rate
$\tau_{d}>c/T$.
In fact, considering a particle near a black hole leads to producing some
perturbations that could be described by the Klein-Gordon equation (1) as the
Equation of Motion of the particle. By solving this equation, we find an
imaginary value of energy for the particle which is connected to eq.(3). On
the other hand, the existence of such perturbations makes the black holes
unstable so that they approach the extremal limit and consequently connect to
the WGC condition. By using the mentioned relationship between eq.(3) and the
WGC and also the polarity quantity $w$ that we obtain during our calculations,
we can study the WGC condition from the universal equation (3) (see
Refs.Urbano:2018kax ; Hod:2017uqc ; Hod:2010hw ).

In the next section, we use the above description for two black holes, e.g.,
HSV and KNA.

##  III Hyperscaling violating black holes

The metric of the HSV black hole with mass $M$ and charge $Q$ is given by
Urbano:2018kax ; Pedraza:2018eey ; Hod:2017uqc ; Horowitz:1999jd

$ds^{2}=r^{\frac{-2\theta}{\textrm{d}}}\left[-r^{2\textrm{z}}f(r)dt^{2}+\frac{dr^{2}}{r^{2}f(r)}+r^{2}d\Omega^{2}_{k,\textrm{d}}\right].$
(4)

Where $z$ and $\theta$ are dynamical and Hyperscaling violation parameters,
respectively. Also, $f(r)$ takes the form

$f(r)=1+\frac{k}{r^{2}}\frac{(\textrm{d}-1)^{2}}{(\textrm{z}+\textrm{d}-\theta-2)^{2}}-\frac{M}{r^{\textrm{z}+\textrm{d}-\theta}}+\frac{Q^{2}}{r^{2(\textrm{z}+\textrm{d}-\theta)}}.$
(5)

Here, $k=-1,0,1$ determines the hyperboloid, planar and spherical topology for
the black hole horizon Pedraza:2018eey . In this work, we consider the $k=0$,
so we have

$\displaystyle
ds^{2}=r^{\frac{-2\theta}{\textrm{d}}}\left[-r^{2\textrm{z}}f(r)dt^{2}+\frac{dr^{2}}{r^{2}f(r)}+r^{2}d\bm{x}^{2}\right],\hskip
28.45274ptf(r)=1-\frac{M}{r^{\textrm{z}+\textrm{d}-\theta}}+\frac{Q^{2}}{r^{2(\textrm{z}+\textrm{d}-\theta)}},$
(6)

where $d\bm{x}^{2}=\sum_{i=1}^{d}dx_{i}^{2}$ and $x_{i}$ are spatial
coordinates of a $d$ dimensional space. By setting $f(r_{H})=0$, we obtain the
event horizon radius, $r_{H}$, for a charged black hole solution

$r_{H}^{2(\textrm{d}+\textrm{z}-\theta-1)}-Mr_{H}^{\textrm{d}+\textrm{z}-\theta-2}+Q^{2}=0.$
(7)

Using the formula $T=\frac{r_{H}^{z+1}}{4\pi}|\acute{f}(r_{H})|$ and (6), one
can obtain the Hawking temperature Alishahiha:2012qu ; Caldarelli:1999xj

$T=\frac{(\textrm{z}+\textrm{d}-\theta)r_{H}^{\textrm{z}}}{4\pi}\left(1-\frac{(\textrm{z}+\textrm{d}-\theta-2)Q^{2}}{\textrm{z}+\textrm{d}-\theta}r_{H}^{2(-\textrm{z}-\textrm{d}+\theta+1)}\right).$
(8)

Therefore, by putting $T=0$, the extremality bound is obtained by

$r_{H}^{2(\textrm{z}+\textrm{d}-\theta-1)}=\frac{(\textrm{z}+\textrm{d}-\theta-2)}{\textrm{z}+\textrm{d}-\theta}Q^{2}.$
(9)

In this case, we use relations (7) and (9) and rewrite $f(r)$ in terms of
$r_{H}$

$f(r)=1-\frac{2(\textrm{z}+\textrm{d}-\theta-1)}{\textrm{z}+\textrm{d}-\theta-2}\left(\frac{r_{H}}{r}\right)^{\textrm{z}+\textrm{d}-\theta}+\frac{\textrm{z}+\textrm{d}-\theta}{\textrm{z}+\textrm{d}-\theta-2}\left(\frac{r_{H}}{r}\right)^{2(\textrm{z}+\textrm{d}-\theta-1)}.$
(10)

Also the potential is given by Pedraza:2018eey

$A_{t}=\frac{\sqrt{2(\textrm{z}+\textrm{d}-\theta)(d-\theta)}}{\textrm{z}+\textrm{d}-\theta-2}r_{H}^{\textrm{z}+\textrm{d}-\theta-1}\left(\frac{1}{r^{\textrm{z}+\textrm{d}-\theta-2}}-\frac{1}{r_{H}^{\textrm{z}+\textrm{d}-\theta-2}}\right).$
(11)

By changing the corresponding coordinates and examining $r$ near the event
horizon, we can write the geometry of $AdS_{2}\times R^{d-1}$
Alishahiha:2012qu

$r=r_{H}+\frac{\epsilon
r_{H}^{2}}{(\textrm{z}+\textrm{d}-\theta)(\textrm{z}+\textrm{d}-\theta-1)\zeta},\hskip
14.22636ptt=\frac{\tau}{\epsilon r_{H}^{\textrm{z}}}.$ (12)

Here we note that equations (12),(10) and (6) are obtained by the limit
$\epsilon\rightarrow 0$ as follows

$ds^{2}=r_{H}^{2-\frac{2\theta}{\textrm{d}}}\left[\frac{-d\tau^{2}+d\zeta^{2}}{(\textrm{z}+\textrm{d}-\theta)(\textrm{z}+\textrm{d}-\theta-1)\zeta^{2}}+d\bm{x}^{2}\right].$
(13)

Also, we will have

$\begin{split}A_{\tau}=\frac{\sqrt{2(\textrm{z}+\textrm{d}-\theta)(\textrm{d}-\theta)}}{(\textrm{z}+\textrm{d}-\theta)(\textrm{z}+\textrm{d}-\theta-1)\zeta}\times
r_{H}^{-\textrm{z}+2}\end{split}.$ (14)

Now, using (1), (13) and (14) also with respect to that fact that metric
allows for the separation of variables,
$\Phi(\tau,\zeta,\bm{x})=e^{-i\omega\tau}e^{i\bm{k}.\bm{x}}\phi(\zeta)$, one
can calculate

$\begin{split}\partial_{\zeta}^{2}\phi(\zeta)+\left(\omega+\frac{qr_{H}^{-z+\textrm{2}}\sqrt{2(\textrm{z}+\textrm{d}-\theta)(\textrm{d}-\theta)}}{(\textrm{z}+\textrm{d}-\theta)(\textrm{z}+\textrm{d}-\theta-1)\zeta}\right)^{2}\phi(\zeta)-\frac{k^{2}+m^{2}r_{H}^{2(1-\frac{\theta}{d})}}{(\textrm{z}+\textrm{d}-\theta)(\textrm{z}+\textrm{d}-\theta-1)\zeta^{2}}\phi(\zeta)=0\end{split}.$
(15)

According to the above equation, $-k^{2}$ is the eigenvalue of the Laplacian
in the flat base sub-manifold. Concerning the above equations, $\phi(\zeta)$
is obtained in terms of the Whittaker functions, Which are given by

$\begin{split}\phi(\zeta)=&c_{1}WhittakerM\left[\frac{-iqr_{H}^{-\textrm{z}+2}\sqrt{2(\textrm{z}+\textrm{d}-\theta)(d-\theta)}}{(\textrm{z}+\textrm{d}-\theta)(\textrm{z}+\textrm{d}-\theta-1)},\nu_{k},2ir\omega\right]\\\
&+c_{2}WhittakerW\left[\frac{-iqr_{H}^{-\textrm{z}+2}\sqrt{2(\textrm{z}+\textrm{d}-\theta)(\textrm{d}-\theta)}}{(\textrm{z}+\textrm{d}-\theta)(\textrm{z}+\textrm{d}-\theta-1)},\nu_{k},2ir\omega\right].\end{split}$
(16)

Also, with respect to the above equation, the $\nu_{k}$ is defined by

$\displaystyle\nu_{k}=\sqrt{\frac{1}{4}+\frac{k^{2}}{(\textrm{z}+\textrm{d}-\theta)(\textrm{z}+\textrm{d}-\theta-1)}-\frac{q^{2}r_{H}^{-\textrm{2z}+4}2(\textrm{z}+\textrm{d}-\theta)(\textrm{d}-\theta)}{(\textrm{z}+\textrm{d}-\theta)^{2}(\textrm{z}+\textrm{d}-\theta-1)^{2}}+m^{2}r_{H}^{2(1-\frac{\theta}{\textrm{d}})}}.$
(17)

We want to examine the obtained function $\phi(\zeta)$ near the event horizon.
To do this, we use the limit $\zeta\longrightarrow 0$. Given the properties of
the Whittaker function and the definition of $\Delta_{k}=\frac{1}{2}-\nu_{k}$,
we will have

$\begin{split}\phi(\zeta)_{\zeta\rightarrow
0}\approx&C\bigg{[}\frac{\Gamma(2\nu_{k})(-2i\omega)^{\frac{1}{2}-\nu_{k}}}{\Gamma\bigg{(}\frac{1}{2}+\nu_{k}-\frac{iqr_{H}^{-\textrm{z}+2}\sqrt{2(\textrm{z}+\textrm{d}-\theta)(\textrm{d}-\theta)}}{(\textrm{z}+\textrm{d}-\theta)(\textrm{z}+\textrm{d}-\theta-1)}\bigg{)}}\zeta^{{\frac{1}{2}-\nu_{k}}}+\frac{\Gamma(-2\nu_{k})(-2i\omega)^{\frac{1}{2}+\nu_{k}}}{\Gamma\bigg{(}\frac{1}{2}-\nu_{k}-\frac{iqr_{H}^{-\textrm{z}+2}\sqrt{2(\textrm{z}+\textrm{d}-\theta)(\textrm{d}-\theta)}}{(\textrm{z}+\textrm{d}-\theta)(\textrm{z}+\textrm{d}-\theta-1)}\bigg{)}}\zeta^{{\frac{1}{2}+\nu_{k}}}\bigg{]}\\\
&\equiv
B_{k}(\omega)\zeta^{\Delta_{k}}+A_{k}(\omega)\zeta^{1-\Delta_{k}}.\end{split}$
(18)

Using equations (2) and (18), the correlation function is obtained by
following equation

$\begin{split}&\mathcal{X}=\Gamma\big{(}2\nu_{k}\big{)}\Gamma\bigg{(}\frac{1}{2}-\nu_{k}-\frac{iqr_{H}^{-\textrm{z}+2}\sqrt{2(\textrm{z}+\textrm{d}-\theta)(\textrm{d}-\theta)}}{(\textrm{z}+\textrm{d}-\theta)(\textrm{z}+\textrm{d}-\theta-1)}\bigg{)}\\\
&\mathcal{Y}=\Gamma\big{(}-2\nu_{k}\big{)}\Gamma\bigg{(}\frac{1}{2}+\nu_{k}-\frac{iqr_{H}^{-\textrm{z}+2}\sqrt{2(\textrm{z}+\textrm{d}-\theta)(\textrm{d}-\theta)}}{(\textrm{z}+\textrm{d}-\theta)(\textrm{z}+\textrm{d}-\theta-1)}\bigg{)}\\\
&\Upsilon_{R}^{(k)}(\omega)=(2\omega)^{-2\nu_{k}}e^{i\pi\nu_{k}}\times\frac{\mathcal{X}}{\mathcal{Y}}\end{split}.$
(19)

According to equation (19), do not have $\omega$ with an imaginary part; as a
result, WGC can not be discussed. Now we consider condition $r_{H}\rightarrow
r_{H}+\frac{\epsilon
r_{H}^{2}}{(\textrm{z}+\textrm{d}-\theta)(\textrm{z}+\textrm{d}-\theta-1)\zeta_{0}}$
with respect to (13) and (14), so we can obtain

$ds^{2}=\frac{r_{H}^{2-\frac{2\theta}{\textrm{d}}}}{(\textrm{z}+\textrm{d}-\theta)(\textrm{z}+\textrm{d}-\theta-1)\zeta^{2}}\left[-(1-\frac{\zeta^{2}}{\zeta_{0}^{2}})d\tau^{2}+(1-\frac{\zeta^{2}}{\zeta_{0}^{2}})^{-1}d\zeta^{2}\right]+r_{H}^{2-\frac{2\theta}{\textrm{d}}}d\bm{x}^{2},$
(20)

and

$A_{\tau}=\frac{\sqrt{2(\textrm{z}+\textrm{d}-\theta)(\textrm{d}-\theta)}}{(\textrm{z}+\textrm{d}-\theta)(\textrm{z}+\textrm{d}-\theta-1)\zeta}r_{H}^{-\textrm{z}+2}\left(1-\frac{\zeta}{\zeta_{0}}\right).$
(21)

The temperature associated to the metric in equation (20) is
$T=1/2\pi\zeta_{0}$. Using (1), (20) and (21) also with respect to this issue
that the metric allows for the separation of variables,
$\Phi(\tau,\zeta,\bm{x})=e^{-i\omega\tau}e^{i\bm{k}.\bm{x}}\phi(\zeta)$, one
can calculate

$\begin{split}&\partial_{\zeta}^{2}\phi(\zeta)+\frac{2\zeta}{\zeta^{2}-\zeta_{0}^{2}}\partial_{\zeta}\phi(\zeta)+\frac{\left[\omega+\frac{qr_{H}^{-\textrm{z}+2}\sqrt{2(\textrm{z}+\textrm{d}-\theta)(\textrm{d}-\theta)}(1-\frac{\zeta}{\zeta_{0}})}{(\textrm{z}+\textrm{d}-\theta)(\textrm{z}+\textrm{d}-\theta-1)\zeta}\right]^{2}}{(1-\frac{\zeta^{2}}{\zeta_{0}^{2}})^{2}}\phi(\zeta)\\\
&-\frac{k^{2}+m^{2}r_{H}^{2(1-\frac{\theta}{\textrm{d}})}}{(\textrm{z}+\textrm{d}-\theta)(\textrm{z}+\textrm{d}-\theta-1)\zeta^{2}(1-\frac{\zeta^{2}}{\zeta_{0}^{2}})}\phi(\zeta)=0.\end{split}$
(22)

By solving the above equation, $\phi(\zeta,\omega)$ is obtained by

$\begin{split}&\phi_{k}(\zeta,\omega)\sim\left(\frac{1}{\zeta}-\frac{1}{\zeta_{0}}\right)^{-\frac{1}{2}\mp\nu_{k}}\left(\frac{\zeta_{0}+\zeta}{\zeta_{0}-\zeta}\right)^{i\omega\zeta_{0}-i\frac{qr_{H}^{-\textrm{z}+2}\sqrt{2(\textrm{z}+\textrm{d}-\theta)(\textrm{d}-\theta)}}{(\textrm{z+d}-\theta)(\textrm{z+d}-\theta-1)}}\\\
&\times
F_{1}\bigg{\\{}\frac{1}{2}\pm\nu_{k}+i\omega\zeta_{0}-i\frac{q\sqrt{2(\textrm{z}+\textrm{d}-\theta)(\textrm{d}-\theta)}r_{H}^{-\textrm{z}+2}}{(\textrm{z}+\textrm{d}-\theta)(\textrm{z}+\textrm{d}-\theta-1)},\frac{1}{2}\\\
&\pm\nu_{k}+-i\frac{q\sqrt{2(\textrm{z}+\textrm{d}-\theta)(\textrm{d}-\theta)}r_{H}^{-\textrm{z}+2}}{(\textrm{z}+\textrm{d}-\theta)(\textrm{z}+\textrm{d}-\theta-1)},\frac{1}{2}\pm\nu_{k},\frac{2\zeta}{\zeta-\zeta_{0}}\bigg{\\}}.\end{split}$
(23)

We examine the above solution on the AdS boundary. In that case, using the
properties of the hypergeometric functions and the equation (2), we obtain the
retarded Green’s function as follows

$\begin{split}&\mathcal{A}=\Gamma\big{(}1+2\nu_{k}\big{)}\Gamma\bigg{(}\frac{1}{2}-\nu_{k}-\frac{i\omega}{2\pi
T}+\frac{iqr_{H}^{-\textrm{z}+2}\sqrt{2(\textrm{z}+\textrm{d}-\theta)(\textrm{d}-\theta)}}{(\textrm{z}+\textrm{d}-\theta)(\textrm{z}+\textrm{d}-\theta-1)}\bigg{)}\\\
&\times\Gamma\bigg{(}\frac{1}{2}-\nu_{k}+\frac{iqr_{H}^{-\textrm{z}+2}\sqrt{2(\textrm{z}+\textrm{d}-\theta)(\textrm{d}-\theta)}}{(\textrm{z}+\textrm{d}-\theta)(\textrm{z}+\textrm{d}-\theta-1)}\bigg{)}\\\
,&\mathcal{B}=\Gamma\big{(}1-2\nu_{k}\big{)}\Gamma\bigg{(}\frac{1}{2}+\nu_{k}-i\frac{\omega}{2\pi
T}+\frac{iqr_{H}^{-\textrm{z}+2}\sqrt{2(\textrm{z}+\textrm{d}-\theta)(\textrm{d}-\theta)}}{(\textrm{z}+\textrm{d}-\theta)(\textrm{z}+\textrm{d}-\theta-1)}\bigg{)}\\\
&\times\Gamma\bigg{(}\frac{1}{2}+\nu_{k}+\frac{iqr_{H}^{-\textrm{z}+2}\sqrt{2(\textrm{z}+\textrm{d}-\theta)(\textrm{d}-\theta)}}{(\textrm{z}+\textrm{d}-\theta)(\textrm{z}+\textrm{d}-\theta-1)}\bigg{)},\\\
&\Upsilon_{R}^{(k)}(\omega)=(4\pi
T)^{-2\nu_{k}}\times\frac{\mathcal{A}}{\mathcal{B}}\end{split}.$ (24)

We used the explicit definition of $T$ instead of $\zeta_{0}$. When we get the
polarity of Equation (2), $\omega$ has two parts, real and imaginary as
$\omega=Re(\omega)+Im(\omega)$.

$\omega=\frac{qr_{H}^{-\textrm{z}+2}\sqrt{2(\textrm{z}+\textrm{d}-\theta)(\textrm{d}-\theta)}}{(\textrm{z}+\textrm{d}-\theta)(\textrm{z}+\textrm{d}-\theta-1)}-i2\pi
T\left(\frac{1}{2}+n+\nu_{k}\right).$ (25)

Here, we can talk about WGC because we have a quasi-normal mode. So we note
here that we are interested in the system’s response at low frequencies
Hod:2010hw ; Hod:2006jw ; Horowitz:1999jd .

$\tau_{d}\equiv\tau_{d}^{(0,0)}=\frac{1}{T(\frac{1}{2}+\nu_{0})},\hskip
14.22636pt\nu_{0}=\sqrt{\frac{1}{4}-\frac{2q^{2}r_{H}^{-2\textrm{z}+4}(\textrm{z}+\textrm{d}-\theta)(\textrm{d}-\theta)}{(\textrm{z}+\textrm{d}-\theta)^{2}(\textrm{z}+\textrm{d}-\theta-1)^{2}}+m^{2}r_{H}^{2(1-\frac{\theta}{\textrm{d}})}}.$
(26)

Now using equations (3) and (26), we get the following condition

$\left(mr_{H}^{1-\frac{\theta}{\textrm{d}}}-\frac{qr_{H}^{-\textrm{z}+2}\sqrt{2(\textrm{z}+\textrm{d}-\theta)(\textrm{d}-\theta)}}{(\textrm{z}+\textrm{d}-\theta)(\textrm{z}+\textrm{d}-\theta-1)}\right)\left(mr_{H}^{1-\frac{\theta}{\textrm{d}}}+\frac{qr_{H}^{-\textrm{z}+2}\sqrt{2(\textrm{z}+\textrm{d}-\theta)(\textrm{d}-\theta)}}{(\textrm{z}+\textrm{d}-\theta)(\textrm{z}+\textrm{d}-\theta-1)}\right)<0.$
(27)

The WGC condition is satisfied when we have the following expression

$r_{H}^{1-\frac{\theta}{\textrm{d}}}=\frac{r_{H}^{-\textrm{z}+2}\sqrt{2(\textrm{z}+\textrm{d}-\theta)(\textrm{d}-\theta)}}{(\textrm{z}+\textrm{d}-\theta)(\textrm{z}+\textrm{d}-\theta-1)}.$
(28)

According to eq.(27), we can rewrite the radius of the event horizon in terms
of dynamic parameters as follows,

$r_{H}^{z-\frac{\theta}{d}-1}=\frac{\sqrt{2(z+d-\theta)(d-\theta)}}{(z+d-\theta)(z+d-\theta-1)}.$
(29)

This relation shows the compatibility of the WGC and the CFT only on the
horizon of a specific event obtained from eq.(29) According to the relation
(9), $r$ is positive when we have relation as
$\textrm{z}+\textrm{d}-\theta>2$. The answer to the above relationship is when
we have $\textrm{z}=1,\textrm{d}=1$ and $\theta\rightarrow 0^{-}$ or
$\textrm{z}\rightarrow 1^{+},\textrm{d}=1$ and $\theta=0$.

To drive the WGC from the CFT viewpoint, we need to specify a series of
specific points and discuss the compatibility of these two structures in those
points. Since $r_{H}$ represents the position of the whole horizon, we can not
discuss the relation between the WGC of the CFT for the entire horizon
parameters. Therefore, we must inevitably determine a specific value for
$r_{H}$ by setting some important parameters such as z, d, and $\theta$ in
order to find specific points of $r_{H}$. Consequently, we can say that by
setting $r_{H}<1$ and for a series of specific points, we can say the WGC and
the CFT are aligned with each other.

Figure 1: Left plot: $z=1$ , $d=1$, and $\theta=-0.01$ Right plot: $z=1$ ,
$d=1$, and $\theta=-0.1$

As we can see in fig. 1, when $\theta$ is closer to zero, WGC holds for bigger
values of $r_{H}$. In that case, it is set for both $r_{H}$ greater and less
than one.

Figure 2: Left plot: $z=1.01$ , $d=1$, and $\theta=0$ Right plot: $z=1.1$ ,
$d=1$, and $\theta=0$

In fig 2, also we see that when for z that is closer to one, $r_{H}$ can be
established for different values, but z is slightly larger than one, WGC for
$r_{H}<1$ will be established. When $\theta<\textrm{d}$ we have an solution
for a value of $r_{H}<1$.

Figure 3: Left plot: $z=1.5$ , $d=2$, and $\theta=-1$ Right plot: $z=2$ ,
$d=3$, and $\theta=-2$

From fig 3, when we consider $\theta$ to be a number less than zero and z and
d to be greater than one, then WGC will hold for a value of $r_{H}<1$.

##  IV Kerr-Newman-AdS black holes

Analogous to the previous black hole, we attempt to obtain the wave function
and the energy spectrum of the KNA black hole. Here, we find that our black
hole is unstable, which is proof of WGC. To see such condition for the
corresponding black hole, we consider the Boyer-Lindquist-type coordinates
Caldarelli:1999xj ,

$ds^{2}=-\frac{f(r)}{\rho^{2}}\left(dt-\frac{a\sin^{2}\theta}{\Xi}d\phi\right)^{2}+\frac{\rho^{2}}{f(r)}dr^{2}+\frac{\rho^{2}}{f(\theta)}d\theta^{2}+\frac{f(\theta)}{\rho^{2}}\sin^{2}\theta\left(adt-\frac{(r^{2}+a^{2})}{\Xi}d\phi\right)^{2},$
(30)

where

$\displaystyle f(r)={(r^{2}+a^{2})(1+\frac{r^{2}}{\ell^{2}})-2Mr+Q^{2}},\hskip
21.33955ptf(\theta)={1-\frac{a^{2}}{\ell^{2}}\cos^{2}\theta},\hskip
21.33955pt\rho^{2}={r^{2}+a^{2}\cos^{2}\theta},\hskip
21.33955pt\Xi={1-\frac{a^{2}}{\ell^{2}}}.$ (31)

Here, $a$ is the rotational parameter, and $Q$ is the electric charge.
$\ell^{2}$ is related to the cosmological constant. If it is positive, we have
$dS$; when it is negative, we have $AdS$. Its vector potential is considered
as follows

$A=-\frac{Qr}{\rho^{2}}\left(dt-\frac{a\sin^{2}\theta}{\Xi}d\phi\right).$ (32)

The Hawking temperature, entropy, and angular velocity of the horizon are
obtained by

$\displaystyle{T_{H}}={\frac{r_{+}(1+\frac{a^{2}}{\ell^{2}}+\frac{3r_{+}^{2}}{\ell^{2}}-\frac{a^{2}+Q^{2}}{r_{+}^{2}})}{4\pi(r_{+}^{2}+a^{2})}},\hskip
21.33955ptS={\frac{\pi(r_{+}^{2}+a^{2})}{\Xi}},\hskip 21.33955pt{\Omega_{H}}$
$\displaystyle=$ $\displaystyle{\frac{a\Xi}{r_{+}^{2}+a^{2}}}.$ (33)

We can rewrite $f(r)$ near the event horizon in quadratic order as follows
Chen:2010bh

$f(r)\simeq k(r-r_{+})(r-r_{*}),\hskip
28.45274ptk=1+\frac{a^{2}}{\ell^{2}}+\frac{6r_{+}^{2}}{\ell^{2}},$ (34)

Where $r_{+}$ is the radius of the external event horizon and $r_{*}$ is the
radius of another horizon. In the extremality limit, the following conditions
also apply. Chen:2010bh

$r_{+}^{2}\geq\frac{\ell^{2}}{6}\left(\sqrt{(1+\frac{a^{2}}{\ell^{2}})^{2}+12\frac{a^{2}}{\ell^{2}}}-(1+\frac{a^{2}}{\ell^{2}})\right).$
(35)

In the limit $\frac{a}{\ell}\ll 1$ we can consider the minimum value of
$r_{+}$ equal to $a$. By putting $f(r)=0$ and $T_{H}=0$ we can get the extreme
state of the black hole with the condition $r_{+}=a$ as follows

$k=\frac{M_{exe}}{r_{+}}+\frac{4r_{+}^{2}}{\ell^{2}}.$ (36)

We now use (1) to solve Klein Gordon’s equation in the background of the KNA
black hole and obtain the correlation function, which is calculated by the
following equation Chen:2010bh

$\Upsilon_{R}=\frac{\Gamma(1-2h_{Q})}{\Gamma(2h_{Q}-1)}\times\frac{\Gamma(h_{Q}+i\frac{\omega_{L}-Q_{L}\mu_{L}}{2\pi
T_{L}})\Gamma(h_{Q}+i\frac{\omega_{R}-Q_{R}\mu_{R}}{2\pi
T_{R}})}{\Gamma(1-h_{Q}+i\frac{\omega_{L}-Q_{L}\mu_{L}}{2\pi
T_{L}})\Gamma(1-h_{Q}+i\frac{\omega_{R}-Q_{R}\mu_{R}}{2\pi T_{R}})},$ (37)

where

$\displaystyle{\omega_{L}}$ $\displaystyle=$
$\displaystyle{\frac{r_{+}^{2}+r_{*}^{2}+2a^{2}}{2a\Xi}\omega},\hskip
56.9055pt{\omega_{R}}={\frac{r_{+}^{2}+r_{*}^{2}+2a^{2}}{2a\Xi}\omega-M},$
$\displaystyle{Q_{L}}$ $\displaystyle=$ $\displaystyle Q_{R}={e}$
$\displaystyle{\mu_{L}}$ $\displaystyle=$
$\displaystyle{\frac{Q(r_{+}^{2}+r_{*}^{2}+2a^{2})}{2a\Xi(r_{+}+r_{*})}},\hskip
56.9055pt{\mu_{R}}={\frac{Q(r_{+}+r_{*})}{2a\Xi}},$ $\displaystyle{T_{L}}$
$\displaystyle=$ $\displaystyle{\frac{k(r_{+}^{2}+r_{*}^{2}+2a^{2})}{4\pi
a\Xi(r_{+}+r_{*})}},\hskip 56.9055pt{T_{R}}={\frac{k(r_{+}-r_{*})}{4\pi
a\Xi}},$ (38)

also we have

$h_{Q}=\frac{1}{2}+\frac{1}{2}\sqrt{1-4\left(\frac{e^{2}Q^{2}}{k^{2}}-\frac{K_{Q}}{k}\right)},$
(39)

where $K_{Q}$ is the separation constant Chen:2010bh . The pole of equation
(37) is given as follows

$1-h_{Q}+i\frac{\omega_{L}-Q_{L}\mu_{L}}{2\pi T_{L}}=-n,\hskip
42.67912pt1-h_{Q}+i\frac{\omega_{R}-Q_{R}\mu_{R}}{2\pi T_{R}}=-n.$ (40)

Since the above two relations are similar to each other, we consider one of
them and place it in relation (3) with the condition of $K_{Q}=-3k$ and obtain
the following equation

$\frac{eQ}{k}>1.$ (41)

Using equations (35), (36) and (41) also with condition $\frac{a}{\ell}\ll 1$,
we achieve the following equation

$\frac{eaQ}{M_{exe}+\frac{4a^{3}}{\ell^{2}}}>1.$ (42)

The above statement satisfies WGC condition for $\frac{a}{\ell}\ll 1$ and
$e=\frac{1}{a}$.

The initial arguments about the WGC were presented about the physics of black
holes and then were followed by some expansions in different parts.
The primary characteristics of black holes are addressed through the laws of
thermodynamics and its relation with gravity based on the AdS/CFT
correspondence. In this work, we show the WGC is related to the thermalization
dynamics governing the relaxation process after a perturbation process. The
validity of such result is guaranteed by having a lower bound on the
thermalization time scale. In fact, the CFT gives a lower thermalization bound
on the thermalization time-scale to charged-near-extremal black holes, which
can be used as a necessary condition to reach the definition of the WGC. By
using the model’s free parameters, we determine a series of parametric points
and regions and then check the WGC in such points. Note that imposing more
restrictions might determine the exact range of compatibility between the WGC
and the CFT. By considering the common points coming from gravity,
thermodynamics and the relationship between gravity and the WGC, the
thermodynamics of black holes can be the desired issue in order to enter the
WGC and challenge different theories for a better understanding between gauge
theories and quantum gravity.

##  V Conclusions and outlook

The idea of WGC has been applied in many cosmological structures, such as
inflation, dark energy, and black holes physics. On the other hand, CFT is
engaged a lot in the literature of theoretical particle physics, particularly
in the AdS/CFT correspondence. But the connection between the two theories
navigates us to some interesting results. The main aim of this paper is to
study the relationship between WGC and CFT using HSV and KNA black holes. To
fulfill this, we used the correlation function of CFT. Hence, we obtained
proof for predicting WGC on the CFT side. This means the emergence of WGC from
correlation function calculations in the considered black holes. Since WGC at
critical points has compatibility results, it also appears in extremality
bound of black holes. Assuming this and the CFT correlation function, we
calculated the black holes’ critical points. Hence, WGC appeared for both
black holes. In addition to proving WGC, we have also shown the exciting close
relation between these two ideas, i.e., WGC and CFT. In that case, we used the
correlation function in CFT and its poles. Also, we obtained the energy
spectrum of the black holes, including two parts, i.e., real (normal mode) and
imaginary (quasi-normal mode). We found that when $z=1$, $d=1$ and
$\theta\rightarrow 0^{-}$, WGC emerges in the HSV black holes since it
contains $r_{H}$ larger and smaller than one. In other cases, WGC is only
valid for $r_{H}$ less than one. The condition of WGC for the KNA black hole
is related to the rotation and radius parameters if the charged particle near
the black hole is $\frac{1}{a}$ and has a ratio such as $\frac{a}{\ell}\ll 1$.
Since a relationship between the two ideas has somehow emerged, we can do more
calculations to look at the results and even reach a correspondence between
the two theories, which we shall examine in future work.

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# City-GAN: Learning architectural styles using a custom Conditional GAN
architecture

Maximilian Bachl<EMAIL_ADDRESS>and  Daniel C. Ferreira
<EMAIL_ADDRESS>
###### Abstract.

Generative Adversarial Networks (GANs) are a well-known technique that is
trained on samples (e.g. pictures of fruits) and which after training is able
to generate realistic new samples. Conditional GANs (CGANs) additionally
provide label information for subclasses (e.g. apple, orange, pear) which
enables the GAN to learn more easily and increase the quality of its output
samples. We use GANs to learn architectural features of major cities and to
generate images of buildings which do not exist. We show that currently
available GAN and CGAN architectures are unsuited for this task and propose a
custom architecture and demonstrate that our architecture has superior
performance for this task and verify its capabilities with extensive
experiments.

##  1\. Introduction

Generative Adversarial Networks (GANs) are a neural network architecture whose
main purpose is generating previously unseen instances of data following a
previously learned distribution. For example, they can be used to generate new
handwritten digits or faces of people who do not exist (Goodfellow et al.,
2014).

This works by having two separate neural networks, the generator and the
discriminator. The generator’s task is to create realistic samples and the
discriminator’s task is to determine whether a given sample is real or a fake
from the generator. The input to the generator is a vector of noise, and its
output a data sample. The discriminator’s input is a data sample and its
output is a decision whether the sample if real or fake. Over the course of
training a GAN, the generator learns to create more realistic samples while
the discriminator gets better at distinguishing real and fake instances.

Conditional Generative Adversarial Networks (CGANs) (Mirza and Osindero, 2014)
were introduced recently after GANs. Their modification is an additional label
input for both the generator and the discriminator. For example, for learning
to generate digits, the labels would be digits from 0 to 9. For example, if
one gave the label “2” to the generator it would try to generate a digit that
resembles “2”. The discriminator then would also get the label “2” so that it
knows that the sample it is looking at should be “2” no matter if it is real
or fake.

CGANs can be used for tagging images by utilizing the discriminator (Mirza and
Osindero, 2014). It is also possible to use the labels in the generator to
transition between different cases (Gauthier, 2015), like for example
transitioning from “1” to “7”.

Commonly, CGANs are combined with Convolutional Neural Networks (CNNs) to be
able to process image data (Radford et al., 2015). For the generator this is
done by concatenating the label with the noise vector that is fed into the
generator. For the discriminator the image is processed by a regular CNN.
After the CNN there is one (or more than one) fully connected layer, which
takes as its input the output of the CNN as well as the label. Finally the
discriminator outputs its decision regarding the realness of the sample.

We present one new dataset for GANs: The dataset consists of thousands of
images of facades of buildings in various cities. Each image also has a label,
which indicates the city of the building depicted.

We show that both regular Convolutional GANs as well as Convolutional CGANs
have significant shortcomings for our use case. We then propose a new custom
architecture, which delivers higher-quality results as well as more stable
convergence for our datasets.

In summary, we make the following contributions:

  * •

a new neural network architecture for Convolutional CGANs

  * •

two datasets of facades

  * •

insight about the architectural features of the cities under study

  * •

program code that allows practitioners to reproduce our experiments and
generate similar datasets

All our code and the dataset we used are available at
https://github.com/muxamilian/city-gan.

##  2\. Overview

We implemented City-GAN, a Generative Adversarial Network (Goodfellow et al.,
2014) that can generate facades of buildings in cities it was trained on.

To achieve this, we followed these steps:

  1. (1)

We chose four cities for our analysis: Vienna, Paris, Amsterdam and Manhattan.

  2. (2)

We created a list of valid addresses in these cities.

  3. (3)

We downloaded images from Google Street View from a randomly sampled subset of
these addresses in each city. We chose the altitude angle of the camera to be
20 degrees.

  4. (4)

We filtered images which fulfilled the following criteria:

     * •

It is possible to find a way from the left to the right edge of the image that
only goes over facades and has no gaps in between.

     * •

The building does not show anomalous structures. For example, we consider
construction sites as anomalous because they do not represent the typical
cityscape.

Figure 1. Two examples of pictures that we did not include in our dataset: on
the left image there are gaps in the facades, the right one is anomalous (not
a facade).

  5. (5)

After filtering we had 1000 images per city (4000 in total).

  6. (6)

We experimented with different types of GANs and different architectures.

  7. (7)

We used another, larger dataset (Zamir, 2019) and checked whether a
significantly higher number of samples improves the learning performance. This
dataset consists of 500,000 facades each for Amsterdam, Washington DC,
Florence, Las Vegas and Manhattan. These images are, however, not curated. The
only restriction we made is that the altitude angle must be between 0 and 40
degrees. Otherwise, these images are randomly sampled from the cities with no
further restrictions.

We tried different GAN architectures during the course of our experimentation.
For all our experiments we used the following data augmentation methods:
horizontal flipping with probability 0.5 as well as random cropping of
approximately 85% of the original width and height of the images. We
experimented with different cropped image sizes: 64, 128 and 256 pixels in
height and width. For a final image of 256 pixels, we would input an image of
300 pixels and randomly crop it to 256.

##  3\. Regular GAN

First, we used a regular deep convolutional GAN (Radford et al., 2015) and
didn’t distinguish between the different cities. The code we used we got from
the official PyTorch examples (Pytorch, 2019) which implements the procedure
outlined by Radford et al. (2015). However, from visually inspecting the
results, we concluded that the GAN has a hard time learning the data
distribution.

The generator starts with a vector of multivariate gaussian noise (length 100)
and continuously deconvolutes it (kernel size 4, stride 2) until it outputs
the image of the final size.

The discriminator gets the image and continuously convolutes it (kernel size
4, stride 2) until it is of dimension $1\times 1\times n_{\text{channels}}$.
Then there is one linear layer followed by a sigmoid function that outputs
whether the sample is real or fake.

Figure 2. Eight random samples from the generator of the regular GAN.

Figure 2 shows the results of our generator after sufficient training
iterations. We see that while its output looks like facades, the output is
clearly faulty and can be easily characterized by a human as being fake.

##  4\. Conditional GAN 1

After realizing that a regular GAN has problems learning the data distribution
we used a conditional GAN (CGAN) (Mirza and Osindero, 2014). The rationale
behind this was that if we tell the GAN which city a picture is from, it can
focus on learning facades and doesn’t get confused by different cities’
styles. The difference between a GAN and a CGAN is that the latter receives
label information both in the generator and discriminator. As such, the
discriminator has more information to distinguish between real and fake
pictures, and the generator has an incentive to develop features specifically
for each label.

We implemented it ourselves based on the code of the regular GAN. The
generator of the CGAN gets a one-hot-encoded vector of the cities alongside
the multivariate Gaussian distribution and can thus consider the additional
information about the city that the discriminator expects in the image
generation process. For instance, when the generator is supposed to output an
image of Amsterdam, it gets the vector
$\begin{pmatrix}1,0,0,0\end{pmatrix}^{\top}$ attached to its input. Thus the
input to the generator network is a vector of length 104 with our four cities
(100+4).

The discriminator of the GAN proceeds as in the regular GAN and uses a
convolutional neural network to process the input image. After the
convolutional layers we concatenate their output with the one-hot-encoded
vector of the city and pass it through a dense layer followed by a ReLU
function. After that we have another dense layer followed by a sigmoid
function that outputs whether the sample is real or fake.

Figure 3. Our first attempt at a CGAN. Each column represents one random input
to the generator. The rows represent: the average of all cities, Amsterdam,
Manhattan, Paris and Vienna.

Figure 3 shows that the results are even worse than for the regular GAN.

##  5\. Conditional GAN with custom architecture

After achieving suboptimal results with this CGAN architecture we tried a
different approach and modified the discriminator: instead of adding the one-
hot-encoded vector with the city information after the convolution, we decided
to add it as an input to each pixel in the beginning of the convolution. Thus,
additional to the RGB values, each pixel also has the one-hot-encoded
information of the city. For example, with four cities, each pixel has 7 color
channels: 3 for the colors and 4 for the one-hot-encoded cities (Figure 4).

Figure 4. The architecture of our neural network is inspired by (Pytorch,
2019). The difference in our custom architecture is that labels are fed into
the generator (top) as well as into the discriminator (bottom) alongside the
data.

While this approach sounds wasteful because of the duplicate information that
is fed into the convolutional layer, it achieved significantly better results.
One possible explanation for this improvement is that by adding the city
information to the input picture, we help the kernels of the convolution to
work differently for each city.

Figure 5. Our custom CGAN architecture. Each column represents one random
input to the generator. The rows represent: the average of all cities,
Amsterdam, Manhattan, Paris and Vienna.

Figure 5 shows that the results of our custom CGAN are significantly better
than with our previous attempt and are also better than the normal GAN.

##  6\. Influence of image size on the quality of generated images

We also tested the effect of image size on the quality of the generated
images. For this, we used images with a length and width of 64, 128 and 256
pixels. Our general conclusion is that smaller image sizes improve quality:
with 64 pixels Figure 6, the generated facades looked good and had no or only
very minor flaws. With 128 we got a larger number of artifacts and faulty
images but we deemed quality to be still acceptable. With 256, we couldn’t get
satisfying results any longer: the generator never converged and changed its
output over time when fed the same random input. Furthermore, the generator
did not generalize over different cities and, when inputting an average of all
cities using the input vector
$\begin{pmatrix}\frac{1}{4},\frac{1}{4},\frac{1}{4},\frac{1}{4}\end{pmatrix}^{\top}$—which
gives equal importance to all cities—the obtained image was only noise. Thus,
the generator only learned to output facades for each city but did not learn
which features all cities have in common. During the course of our further
experiments we only used images of resolution $128\times 128$.

Figure 6. Our custom CGAN trained on images of size $64\times 64$. Each column
represents one random input to the generator. The rows represent: the average
of all cities, Amsterdam, Manhattan, Paris and Vienna.

##  7\. Using a larger, uncurated dataset

Finally, we wanted to investigate whether a larger dataset could significantly
improve the quality of generated images. For this we used a dataset of
Stanford University (Zamir, 2019) and extracted 500,000 images for the cities
of Amsterdam, Washington DC, Florence, Las Vegas and Manhattan. These images
are not curated, thus it is not guaranteed that they actually show facades.

We noticed that Washington DC and Las Vegas are especially problematic for our
neural network. In Washington DC, the majority of images depict trees, and in
Las Vegas there are few multi-story buildings.

Thus we also tried training the CGAN with only these three cities, hoping this
would increase the quality of learned images.

Figure 7. Our custom CGAN trained on the Stanford dataset. Each column
represents one random input to the generator. The rows represent: the average
of all cities, Amsterdam, Washington D.C., Florence, Las Vegas and Manhattan.

Generally speaking we think that the quality of the generated images is the
same as the one of our own dataset Figure 7. However, while with our dataset
we only learn facades, with the Stanford dataset we also learn entire streets
with building on the left and right side of the street.

##  8\. Traveling Between Cities

To understand if our network learns characteristics of each city, we can
compare the output of the generator for the same vector with different labels.
We show a 5-step interpolation for 3 distinct random vectors in Figure 8. We
show results interpolating from Amsterdam to each one of Florence, Washington
DC and Manhattan. Additionally, we show also results of interpolating from
“Europe” (half Amsterdam/half Florence) to the “US” (a third of each American
city). These results are from the CGAN with the uncurated dataset described in
the last section.

(a) Amsterdam$\to$Florence

(b) Amsterdam$\to$Washington DC

(c) Amsterdam$\to$Manhattan

(d) Europe$\to$US

Figure 8. Outputs of our CGAN when interpolating between locations. Each row
contains one cityscape, interpolated across the locations specified in the
subtitle.

The first thing we notice is that the color palette of the cities is quite
different. For example, Amsterdam is quite redish, while Florence has tones
closer to beige (8a). Another obvious characteristic is that Washington DC has
a lot of trees (8b). In all the examples with it, the amount of trees greatly
increases when compared to Amsterdam.

There are also some more nuanced changes that characterize the stylistic
architectural differences between cities. A good example of this is show
between Amsterdam and Florence: distinct roofs, with breaks between them can
be seen. This is most noticeable in the second row of 8a. Between Amsterdam
and Manhattan there are also two strong distinctions: Manhattan has taller
buildings, with more smaller windows. In the second row of Figure 8c, the
buildings grow as the label approaches Manhattan, and in the third row the
windows split into smaller windows.

We also show the transition between Europe and the US. The US images look
quite unrealistic, as they are an average between 3 cities with quite
different styles. However, one can identify an increase of trees (Washington
DC has really a lot of trees) and different angles of photography (buildings
are farther from the camera in Washington DC and Las Vegas). Both of these
differences are evident in the second row of 8d. Another good example of the
Europe/US relation is in Figure 9, which shows an European style church being
transformed into an American style skyscraper.

Figure 9. Transition from a tall European church to an American skyscraper.

Another thing we can try is to experiment with subtracting stylistic features.
We can for example subtract Manhattan’s label from Amsterdam’s, and
intuitively that should remove typical Manhattan features from the Amsterdam
image. Indeed, we do just that in Figure 10, and observe that the modern
looking building in the background in Amsterdam completely disappears, after
subtracting Manhattan. Also, if you subtract Manhattan, more trees appear.
Apparently, in Manhattan suppresses the output of trees because there are not
many there.

Figure 10. Transition from Amsterdam to Amsterdam-Manhattan (This means that
we set the one-hot vector with 1 for Amsterdam and -1 for Manhattan).

Not all images obtained are realistic, and some are even quite bad. In Figure
11, we show some of these bad examples. The transitions are in the same order
as in Figure 8, but with only one fixed random vector instead of 3.

Figure 11. Transition from Amsterdam to each of Florence, Washington DC, and
Manhattan (first 3 rows), and from Europe to the US (last row).

##  9\. Conclusion

In this paper we sought to generate images of facades containing the stylistic
features of specific cities. We experimented with multiple GAN architectures,
and propose a custom architecture. Our custom architecture produced more
realistic images than the alternative architectures, and was capable of
generating architectural features specific to each city. By using a continuous
label space, we were also able to transition one facade between cities, or do
averages over them.

The results suggest that our architecture is also better suited for other
tasks than the existing architectures. Investigating its effects on other
tasks is a feasible future research path.

We have made all our code and used data available, so that the community can
both reproduce and improve on our results.

## Acknowledgement

The Titan Xp used for this research was donated by the NVIDIA Corporation.

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