aainstitutetext: Instituut voor Theoretische Fysica, KU Leuven Celestijnenlaan
200D, B-3001 Leuven, Belgiumbbinstitutetext: Leinweber Center for Theoretical
Physics, University of Michigan, Ann Arbor, MI 48109, U.S.A.ccinstitutetext:
Department of Physics and Stanford Institute for Theoretical Physics,
Stanford University, Palo Alto, CA 94305, USA.

# The Attractor Flow for AdS5 Black Holes in $\mathcal{N}=2$ Gauged
Supergravity

Marina David b , Nizar Ezroura b,c , and Finn Larsen<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study the flow equations for BPS black holes in $\mathcal{N}=2$ five-
dimensional gauged supergravity coupled to any number of vector multiplets via
FI couplings. We develop the Noether-Wald procedure in this context and
exhibit the conserved charges as explicit integrals of motion, in the sense
that they can be computed at any radius on the rotating spacetime. The
boundary conditions needed to solve the first order differential equations are
discussed in great detail. We extremize the entropy function that controls the
near horizon geometry and give explicit formulae for all geometric variables
at their supersymmetric extrema. We have also considered a complexification of
the near-horizon variables that elucidates some features of the theory from
the near-horizon perspective.

††preprint: LCTP-23-09

##  1 Introduction

The radial dependence of physical fields in a black hole background relates
the field configuration far from the black hole to the region near the black
hole horizon. It is important in holography because the radial evolution is
identified with the renormalization group flow in the dual quantum field
theory, so it determines the low energy QFT observables in terms of UV data.
For supersymmetric black holes in asymptotically flat spacetimes, these ideas
are realized beautifully by the so-called attractor flow. Unfortunately, the
analogous construction for BPS black holes in asymptotically AdS spacetimes,
where holography is more precise, is less developed. The goal of this article
is to give an explicit and detailed account of the attractor flow for BPS
black holes in AdS5.

All extremal black holes, whether supersymmetric or not, enjoy an attractor
mechanism, in that the end point of the radial flow is a horizon region with
enhanced $SO(2,1)$ symmetry. The field configuration in this AdS2 region is
determined by the entropy function formalism, an extremization principle that
was studied in many contexts Sen:2005wa ; Sen:2005iz ; David:2006yn ;
Sen:2007qy ; Sen:2008yk ; Sen:2008vm ; Sen:2009vz ; Astefanesei:2006dd ;
Castro:2007sd , including its applications to subleading corrections of the
black hole entropy Kraus:2005vz ; Banerjee:2009af ; Banerjee:2010qc . The
attractor flow refers to the entire evolution from the asymptotic space to the
event horizon. The long throat characterizing the final approach to the
horizon gives a geometric reason to intuit that only a restricted set of
endpoints are possible. In this case the flow is attracted to specific points
in configuration space.

The study of supersymmetric attractor flows was initiated in ${\cal N}=2$
ungauged supergravity Ferrara:1995ih ; Ferrara:1996dd . In this context an
attractor mechanism was realized. The horizon values of scalars in ${\cal
N}=2$ vector multiplets is independent of their freely chosen asymptotic
values. The attractor flow for asymptotically flat black holes was later
generalized to different amounts of supersymmetry in various dimensions
Ferrara:1996um ; Ferrara:2006em ; Larsen:2006xm ; Ceresole:2007wx ;
LopesCardoso:2007qid ; Perz:2008kh ; Galli:2010mg ; Ceresole:2009iy ;
Andrianopoli:2007gt ; Andrianopoli:2009je . In this work we inquire about the
analogous radial flow in gauged supergravity, the setting for asymptotically
AdS black holes. This research direction is very well motivated by holography,
but it is technically more involved and much less developed. Moreover, the
reference to an attractor is a misnomer in gauged supergravity: in general no
initial data is introduced at the asymptotic AdS boundary, so nothing is lost
when the horizon is reached. However, the attractor terminology is so
ingrained by now that we keep it, even though it can be misleading. Indeed,
versions of attractor flows and attractor mechanisms in gauged supergravity
previously appeared in Ceresole:2001wi ; Hosseini:2017mds ; DallAgata:2010ejj
; Cacciatori:2009iz ; Halmagyi:2014qza ; Hristov:2018spe .

In the canonical set-up Ferrara:1995ih ; Ferrara:1996dd , ungauged ${\cal
N}=2$ supergravity coupled to vector multiplets, scalar fields are the only
variables needed to characterize the attractor flow. Classical black hole
solutions involve vector fields as well but, in stationary black hole
backgrounds, their radial dependence is entirely determined by the conserved
electric charges, augmented by magnetic charges in $D=4$. The scalars can take
any values in the asymptotically flat space as they are moduli that
parametrize the vacuum. However, for given charges, their radial flow is
governed by an effective black hole potential that guides them to an attractor
value that depends only on the charges.

BPS black holes in AdS are fundamentally different because the scalar fields
in gauged supergravity are subject to a potential that depends on the
couplings of the theory, so generally the scalars are not moduli that
parametrize vacua. When scalars do not depend on free parameters at infinity,
there is no scope for an attractor mechanism that imposes horizon values
independent of such parameters. There is an exception when FI-couplings are
fine-tuned to create flat directions in the scalar potential. In this case the
scalar fields do obey an attractor mechanism.

This work focuses on black holes with nonzero electrical charge and rotation
in 5D asymptotically AdS spacetimes Gutowski:2004ez ; Gutowski:2004yv ;
Kunduri:2006ek ; Chong:2005hr ; Cvetic:2005zi  whose solution has been well
studied in various contexts. Several of the technical challenges we encounter
in our setup arise because all known BPS black holes in AdS5 have non-
vanishing angular momentum. In our implementation of the attractor flow, the
angular momentum $J$ is a conserved quantity like any other: it can be
computed as a flux integral over any topological sphere surrounding the black
hole. The attractor flow from asymptotically AdS to the horizon corresponds to
decreasing radial coordinate. Because of the rotation, the spatial sphere at
any given radius is squashed and co-rotating.

There has been much previous work on the attractor mechanism in gauged
supergravity, including Ceresole:2001wi ; Hosseini:2017mds ; Halmagyi:2014qza
; Huebscher:2008wob ; Bellucci:2008cb ; Cacciatori:2009iz ; Cassani:2009na ;
Hristov:2010eu ; DallAgata:2010ejj ; Hristov:2010ri ; Hristov:2014eba ;
Bobev:2020jlb ; Hristov:2018spe ; Hosseini:2018usu ; Cabo-Bizet:2017xdr ;
Hristov:2016vbm ; Hristov:2014eza ; Kachru:2011ps ; Chimento:2015rra ;
Astesiano:2021hro  and references therein. As a guide, we highlight some of
the features we focus on:

  * •

We construct the entire attractor flow of gauged supergravity, ie. the radial
solution that interpolates between the AdS5 vacuum far from the black hole and
the horizon vacuum of AdS2-type. We do this directly in 5D, without the
dimensional reduction to 4D pursued by some researchers Hosseini:2017mds ;
Hristov:2018spe .

  * •

We construct conserved charges in terms of flux integrals. They are not
specifically defined asymptotically far from the black hole, nor in the
horizon region. Rather, they can be evaluated on any member of a nested family
of surfaces that interpolate between the horizon and a homologous asymptotic
surface. This is essential for the attractor flow.

Technically, we construct these conserved charges using the covariant Noether-
Wald procedure. This is similar to what is done in Suryanarayana:2007rk ,
although their focus is on the near-horizon region. One can also take the
approach of a generalized Komar integral Rodriguez:2021hks ; Kastor:2008xb ;
Kastor:2009wy ; Magnon:1985sc ; Ortin:2021ade ; Cano:2023dyg , where a
specific gauge must be chosen to deal with diverging asymptotics in AdS
spacetimes.

  * •

Our work incorporates general special geometry, including symplectic
invariance. As such, we incorporate general cubic prepotentials. We provide a
reader friendly guide to the widely studied STU and minimal SUGRA models that
correspond to special cases. For a recent study on the attractor mechanism in
the STU model, see Ntokos:2021duk .

  * •

We carry out the entropy extremization procedure in complete detail,
reconstructing all details of the horizon structure. This elucidates the
relation between real and complex fields.

It may be useful to also note some aspects of the attractor mechanism for
gauged supergravity that are interesting but not developed in this article:

  * •

We focus on electric black holes in only five spacetime dimensions.

  * •

We study supersymmetric attractor flows but other extremal flows are
interesting as well.

  * •

We specialize to black holes with equal angular momenta, so our ansatz for the
geometry preserves an $S^{2}$ throughout the flow. More general AdS5 black
holes with unequal angular momenta, studied in Chong:2005hr ; Kunduri:2006ek ,
depend on a complex structure on $S^{1}\times S^{3}$ that evolves radially and
is defined only in Euclidean signature Closset:2013vra ; Closset:2014uda .

  * •

We study AdS5 supergravity with FI-gauging, so only $U(1)$ gauge groups
appear. This class of theories is technically simpler, because it is entirely
specified by a linear superpotential, and there is no need for the moment map.
Moreover, in this theory all scalar fields are neutral Markeviciute:2018yal .

  * •

We do not connect to standard aspects of the AdS5/CFT4 correspondence, such as
holographic renormalization, time-dependent sources, and so on. This direction
was recently studied in Ntokos:2021duk .

We hope to expand on some of these omissions in future work.

This article is organized as follows. In section 2, we introduce the
$\mathcal{N}=2$ gauged supergravity action, the 5D form of the black hole
solution, and its dimensional reduction to both 2D and 1D. section 3 is
devoted to an analysis of Noether-Wald surface charges, and the subtleties of
gauge invariant conserved charges for actions with Chern-Simons terms. section
4 is the longest and most detailed. We derive the first order differential
equations imposed by preserved supersymmetry, in the context of our ansatz. We
study the boundary conditions needed to solve the equations perturbatively,
from both the horizon and the asymptotic AdS point of view. With both these
perspectives, we recover the known black hole solutions by establishing
truncation of the pertubative expansion. In section 5 we develop the entropy
extremization formalism and compute all near horizon aspects of the black
hole, including its entropy. We also construct a complexification of the near-
horizon variables that elucidates some aspects of the solution. We conclude in
section 6 with a discussion of open problems concerning the attractor flow in
gauged supergravity and related topics. A series of appendices are devoted to
technical details, conventions and notations regarding differential forms in
Appendix A, real special geometry in B, and the supersymmetry conditions in
Appendix C.

##  2 The Effective 2D Lagrangian

In this section we introduce the action of $\mathcal{N}=2$ 5D supergravity
with coupling to $n_{V}$ vector multiplets and gauging by Fayet–Iliopoulos
couplings, as well as its dimensional reduction to a 2D theory. This also
serves to define conventions and notation. For additional details on real
special geometry and supersymmetry we refer to Appendices B and C,
respectively.

###  2.1 The 5D theory

We study five dimensional $\mathcal{N}=2$ gauged supergravity with bosonic
action

$\displaystyle\begin{split}S=\frac{1}{16\pi
G_{5}}&\int_{\mathcal{M}}\mathcal{L}_{5}+\frac{1}{8\pi
G_{5}}\int_{\partial\mathcal{M}}d^{4}x\sqrt{|h|}\Tr K\leavevmode\nobreak\
,\end{split}$ (2.1)

where the 5D Lagrangian density is given by

$\displaystyle\mathcal{L}_{5}$
$\displaystyle=(-\mathcal{R}_{5}-2V)\star_{5}1-G_{IJ}F^{I}_{5}\wedge\star_{5}F^{J}_{5}+G_{IJ}dX^{I}\wedge\star_{5}dX^{J}-\frac{1}{6}c_{IJK}F^{I}_{5}\wedge
F^{J}_{5}\wedge A^{K}_{5}\,.$ (2.2)

We have included the subscript $5$ to emphasize that we are in five dimensions
and the five dimensional Hodge dual is given by $\star_{5}$. The Gibbons-
Hawking-York boundary term must be included to have a well-defined variation
of the action (2.1) and is given by the trace of the second fundamental form
$K$ which is integrated over the induced metric $h$ on the boundary. Other
conventions and notations regarding differential forms and the Hodge dual are
in Appendix A.

The field content includes the field strengths $F^{I}_{5}=dA^{I}_{5}$ where
$I=1,\dots,n$ and the scalars $X^{I}$, correspond to $n-1$ physical scalars,
constrained via the following relation

$\displaystyle\frac{1}{6}c_{IJK}X^{I}X^{J}X^{K}=1\leavevmode\nobreak\ .$ (2.3)

The scalar potential is given by

$\displaystyle V$
$\displaystyle=-c^{IJK}\xi_{I}\xi_{J}X_{K}=-\xi_{I}\xi_{J}\left(X^{I}X^{J}-\frac{1}{2}G^{IJ}\right)\leavevmode\nobreak\
,$ (2.4)

where $\xi_{I}$ are the real Fayet–Iliopoulos parameters. The scalars with
lowered index

$\displaystyle X_{I}$ $\displaystyle=2G_{IJ}X^{J}\leavevmode\nobreak\ ,$ (2.5)

obey the analogous constraint

$\displaystyle\frac{1}{6}c^{IJK}X_{I}X_{J}X_{K}=1\leavevmode\nobreak\ ,$ (2.6)

when closure relation (B.3) is satisfied. For further details on definitions,
conventions and identities, we refer the reader to Appendix B.

Alternatively, the scalar potential can be expressed as

$\displaystyle
V=-\left(\frac{2}{3}W^{2}-\frac{1}{2}G^{IJ}D_{I}WD_{J}W\right)\leavevmode\nobreak\
,$ (2.7)

where the superpotential $W$ is

$\displaystyle W=\xi_{I}X^{I}\leavevmode\nobreak\ ,$ (2.8)

and the Kähler covariant derivative $D_{I}$ takes the constraint (2.6) into
acount. Using this form of the potential $V$, the condition for a
supersymmetric minimum becomes

$D_{I}W=\xi_{I}-\frac{1}{3}X_{I}(\xi\cdot X)\underset{\rm
min}{=}0\leavevmode\nobreak\ .$ (2.9)

According to this equation, the asymptotic values of the scalars
$X_{I,\infty}$ must be parallel to $\xi_{I}$, in the sense of real special
geometry vectors, and the constraint (2.6) determines the proportionality
constant between the two:

$X_{I,\infty}=\left(\frac{1}{6}c^{JKL}\xi_{J}\xi_{K}\xi_{L}\right)^{-1/3}\xi_{I}\leavevmode\nobreak\
.$ (2.10)

The value of the potential $V$ at the minimum must be related to the
$\text{AdS}_{5}$ length scale $\ell$ and the cosmological constant in the
usual manner

$\displaystyle V_{\infty}$
$\displaystyle=-c^{IJK}\xi_{I}\xi_{J}X_{K,\infty}\equiv-6\ell^{-2}\leavevmode\nobreak\
.$ (2.11)

This gives the constraint

$\displaystyle\frac{1}{6}c^{IJK}\xi_{I}\xi_{J}\xi_{K}=\ell^{-3}\leavevmode\nobreak\
,$ (2.12)

on the FI-parameters $\xi_{I}$ and the simple relation for the asymptotic
values of the scalars

$X_{I,\infty}=\ell\xi_{I}\leavevmode\nobreak\ .$ (2.13)

Thus the $n_{V}+1$ independent FI-parameters $\xi_{I}$ determine the
asymptotic values $X^{I}_{\infty}$ of the $n_{V}$ scalars, as well as the AdS5
scale $\ell$.

For contrast, recall that in ungauged supergravity, the scalar fields are
moduli as they experience no potential. Then their asymptotic values
$X_{I,\infty}$ far from the black hole are set arbitrarily by boundary
conditions, which is related to the fact that the spacetime is asymptotically
flat. The fact that the value of the scalars $X_{I}$ at the horizon is
independent of the asymptotic values $X_{I,\infty}$ is the attractor mechanism
for BPS black holes in ungauged supergravity.

As we have seen, the present context is very different in that the asymptotic
values of the scalars are set by the theory through the FI-parameters
$\xi_{I}$, rather than by boundary conditions. This is a generic feature of
gauged supergravity, theories with asymptotically AdS vacuum. It precludes an
attractor mechanism that is analogous to the one in asymptotically flat space.
We will discuss this point in more depth in section 4 when we study the linear
flow equations derived from supersymmetry.

The equations of motion $\mathcal{E}_{\Phi}$, where $\Phi$ is any field in the
theory corresponding to the Lagrangian density (2.2), are the Einstein
equation

$\displaystyle\begin{split}\mathcal{E}_{g}&=R_{AB}-\tfrac{1}{2}g_{AB}R+G_{IJ}\left(F^{I}_{5,AC}F^{J,C}_{5,B}-\tfrac{1}{4}g_{AB}F^{I}_{5,CD}F^{J,CD}_{5}\right)\\\
&\quad-
G_{IJ}\left(\nabla_{A}X^{I}\nabla_{B}X^{J}-\tfrac{1}{2}g_{AB}\nabla_{C}X^{I}\nabla^{C}X^{J}\right)-g_{AB}V=0\,,\end{split}$
(2.14)

and the matter equations for the Maxwell field $A^{I}_{5}$ and the constrained
scalars $X^{I}$

$\displaystyle\begin{split}\mathcal{E}_{A}&=d\left(G_{IJ}\star
F^{J}_{5}\right)+\tfrac{1}{4}c_{IJK}F^{J}_{5}\wedge F^{K}_{5}=0,\end{split}$
(2.15) $\displaystyle\begin{split}\mathcal{E}_{X^{I}}&=-d\star
dX_{I}+\tfrac{1}{3}X_{I}X^{J}d\star
dX_{J}+2c^{JKL}\xi_{K}\xi_{L}\left(\tfrac{2}{3}X_{I}X_{J}-c_{IJM}X^{M}\right)\star
1+\left(X_{J}X^{L}c_{IKL}\right.\\\
&\quad\left.-\tfrac{1}{2}c_{IJK}-\tfrac{2}{3}X_{I}X_{J}X_{K}+\tfrac{1}{6}X_{I}c_{JKN}X^{N}\right)(F^{J}_{5}\wedge\star
F^{K}_{5}-dX^{J}\wedge\star dX^{K})=0\leavevmode\nobreak\ .\end{split}$ (2.16)

###  2.2 The effective 2D theory

We do not study all solutions to the 5D theory (2.1), just stationary black
holes. Then it is sufficient to consider a reduction to 2D — and eventually to
1D. We impose the metric ansatz111In our conventions the metric has a mostly
negative signature.

$\displaystyle
ds^{2}_{5}=ds^{2}_{2}-e^{-U_{1}}d\Omega^{2}_{2}-e^{-U_{2}}(\sigma_{3}+a^{0})^{2}\leavevmode\nobreak\
,$ (2.17)

with $ds_{2}^{2}$ a general 2D metric and the 1-form ansatz for the gauge
potential

$\displaystyle A^{I}_{5}$
$\displaystyle=a^{I}+b^{I}(\sigma_{3}+a^{0})\leavevmode\nobreak\ \,.$ (2.18)

In our conventions, the left invariant 1-forms

$\displaystyle\sigma_{1}$
$\displaystyle=\sin\phi\,d\theta-\cos\phi\sin\theta\,d\psi\leavevmode\nobreak\
,$ (2.19) $\displaystyle\sigma_{2}$
$\displaystyle=\cos\phi\,d\theta+\sin\phi\sin\theta\,d\psi\leavevmode\nobreak\
,$ $\displaystyle\sigma_{3}$
$\displaystyle=d\phi+\cos\theta\,d\psi\leavevmode\nobreak\ ,$

parametrize $SU(2)$ with

$\displaystyle 0\leq\theta\leq\pi\leavevmode\nobreak\ ,\qquad 0\leq\phi\leq
4\pi\leavevmode\nobreak\ ,\qquad 0\leq\psi\leq 2\pi\,.$ (2.20)

The ansatz (2.17) suggests the vielbein

$\displaystyle e^{0}$ $\displaystyle=e^{0}_{\mu}dx^{\mu}\leavevmode\nobreak\
,$ $\displaystyle\quad e^{1}$
$\displaystyle=e^{1}_{\mu}dx^{\mu}\leavevmode\nobreak\ ,$ $\displaystyle\quad
e^{2}$ $\displaystyle=e^{-\frac{1}{2}U_{1}}\sigma_{1}\leavevmode\nobreak\ ,$
(2.21) $\displaystyle e^{3}$
$\displaystyle=e^{-\frac{1}{2}U_{1}}\sigma_{2}\leavevmode\nobreak\ ,$
$\displaystyle\quad e^{4}$
$\displaystyle=e^{-\frac{1}{2}U_{2}}(\sigma_{3}+a^{0})\leavevmode\nobreak\ .$

We use Greek indices to denote the curved coordinates $t$ and $R$ in 2D. For
extremal near-horizon geometries, the 2D coordinates describe the AdS2 throat
of the solution. The dimensional reduction via (2.17) and (2.18) of the 5D
Lagrangian (2.2) introduces the scalar fields $U_{1},U_{2}$ and $b^{I}$, along
with the 1-forms $a^{0},a^{I}$. All these fields depend only on the 2D
coordinates.

The effective 2D Lagrangian density that follows from (2.2) is given by

$\displaystyle\begin{split}\mathcal{L}_{2}&=\frac{\pi}{G_{5}}e^{-U_{1}-\frac{1}{2}U_{2}}\Big{\\{}(-{\cal
R}_{2}+2e^{U_{1}}-\tfrac{1}{2}e^{2U_{1}-U_{2}})\star
1-\tfrac{1}{2}dU_{1}\wedge\star d\left(U_{1}+2U_{2}\right)\\\
&\left.\quad\quad-\tfrac{1}{2}e^{-U_{2}}da^{0}\wedge\star
da^{0}-2V-G_{IJ}\Big{(}(da^{I}+b^{I}da^{0})\wedge\star(da^{J}+b^{J}da^{0})+e^{2U_{1}}b^{I}b^{J}\star
1\right.\\\ &\quad\quad+e^{U_{2}}db^{I}\wedge\star db^{J}-dX^{I}\wedge\star
dX^{J}\Big{)}+\tfrac{1}{3}e^{U_{1}+\frac{1}{2}U_{2}}c_{IJK}\left(\tfrac{3}{2}b^{I}b^{J}da^{K}+b^{I}b^{J}b^{K}da^{0}\right)\Big{\\}}\\\
&\quad\quad+\frac{\pi}{G_{5}}d\left((e^{-U_{1}-\frac{1}{2}U_{2}}\star
d(2U_{1}+U_{2}))-\tfrac{1}{6}b^{I}b^{J}a^{K}\right)\,.\end{split}$ (2.22)

We denote the Ricci scalar of the reduced 2D metric $\mathcal{R}_{2}$ and the
Hodge dual is now in 2D. The overall exponential factor
$e^{-U_{1}-\frac{1}{2}U_{2}}$ comes from the 5D metric on a deformed $S^{3}$.
The first line in (2.22) is due to the reduction of the 5D Ricci scalar, which
introduces additional kinetic and potential terms associated to the scalars
$U_{1}$ and $U_{2}$, as well as for the 1-form $a^{0}$ in the beginning of the
second line. The terms preceded by $G_{IJ}$ are the reduction of the Maxwell
field which yield kinetic terms for the 1-forms $a^{0},a^{I}$ and the scalars
$b^{I}$, and the reduction of the kinetic term of $X^{I}$. The remainder of
the third line of (2.22) is the Chern-Simons term. Finally, in the last line,
there is a total derivative that is inconsequential for the equations of
motion but is required in order that $\mathcal{L}_{2}$ (2.22) is the
dimensional reduction of the 5D Lagrangian (2.2). The latter does not include
the Gibbons-Hawking-York boundary term, the extrinsic curvature that appears
separately in (2.1).

Boundary terms present an important subtlety that we will return to repeatedly
in our study. The Chern-Simons term in the 5D Lagrangian (2.2) is not
manifestly gauge invariant, but it transforms to a total derivative under a
gauge variation. Gauge invariance could be restored by introducing a total
derivative in the action. Such a term does not change the equations of motion
but the resulting theory is not covariant in 5D, so there is a tension between
important principles. The bulk part of the 2D Lagrangian (2.22) is not only
covariant, it is also manifestly gauge invariant: $a^{I}$ appears only as the
field strength $da^{I}$. Manifest gauge invariance also applies to $a^{0}$
which encodes 5D rotational invariance. These are benefits of reducing to 2D.

From the dimensionally reduced Lagrangian density (2.22), we can derive the
equations of motion for the fields $U_{1},U_{2},a^{0},a^{I}$ and $b^{I}$. The
solutions to these 2D equations of motion are solutions of the 5D theory. The
field equations for the 2D scalar fields are given by

$\displaystyle\begin{split}\mathcal{E}_{U_{1}}&=d(e^{-U_{1}-\frac{1}{2}U_{2}}\star(dU_{1}+dU_{2}))+e^{-U_{1}-\frac{1}{2}U_{2}}\\{\left(\mathcal{R}_{2}+2V-\tfrac{1}{2}e^{2U_{1}-U_{2}}\right)\star
1\\\
&\quad+\tfrac{1}{2}dU_{1}\wedge\star(dU_{1}+2dU_{2})+\tfrac{1}{2}e^{-U_{2}}da^{0}\wedge\star
da^{0}+G_{IJ}\left((b^{I}da^{0}+da^{I})\wedge\star(b^{J}da^{0}+da^{J})\right.\\\
&\left.\quad-e^{2U_{1}}b^{I}b^{J}\star 1+e^{U_{2}}db^{I}\wedge\star
db^{J}-dX^{I}\wedge\star dX^{J}\right)\\}=0\,,\end{split}$ (2.23)
$\displaystyle\begin{split}\mathcal{E}_{U_{2}}&=d(e^{-U_{1}-\frac{1}{2}U_{2}}\star
dU_{1})+\tfrac{1}{2}e^{-U_{1}-\frac{1}{2}U_{2}}\\{\left(\mathcal{R}_{2}+2V-2e^{U_{1}}+\tfrac{3}{2}e^{2U_{1}-U_{2}}\right)\star
1\\\
&\quad+\tfrac{1}{2}dU_{1}\wedge\star(dU_{1}+2dU_{2})+\tfrac{3}{2}e^{-U_{2}}da^{0}\wedge\star
da^{0}+G_{IJ}\left((b^{I}da^{0}+da^{I})\wedge\star(b^{J}da^{0}+da^{J})\right.\\\
&\left.\quad+e^{2U_{1}}b^{I}b^{J}\star 1-e^{U_{2}}db^{I}\wedge\star
db^{J}-dX^{I}\wedge\star dX^{J}\right)\\}=0\,,\end{split}$ (2.24)
$\displaystyle\begin{split}\mathcal{E}_{b^{I}}&=2d(e^{-U_{1}+\frac{1}{2}U_{2}}G_{IJ}\star
db^{J})-2e^{-U_{1}-\frac{1}{2}U_{2}}G_{IJ}da^{0}\wedge\star(b^{J}da^{0}+da^{J})-2G_{IJ}e^{U_{1}-\frac{1}{2}U_{2}}b^{J}\star
1\\\ &\quad+c_{IJK}b^{J}da^{K}+c_{IJK}b^{J}b^{K}da^{0}=0\,,\end{split}$ (2.25)

and the 1-forms satisfy

$\displaystyle\begin{split}\mathcal{E}_{a^{0}}&=-d(e^{-U_{1}-\frac{3}{2}U_{2}}\star
da^{0})-2d(G_{IJ}b^{I}e^{-U_{1}-\frac{1}{2}U_{2}}\star(b^{J}da^{0}+da^{J}))+\tfrac{1}{3}c_{IJK}d(b^{I}b^{J}b^{K})=0\leavevmode\nobreak\
,\\\
\mathcal{E}_{a^{I}}&=-2d\left(e^{-U_{1}-\frac{1}{2}U_{2}}G_{IJ}\star\left(b^{J}da^{0}+da^{J}\right)\right)+\tfrac{1}{2}c_{IJK}d\left(b^{J}b^{K}\right)=0\leavevmode\nobreak\
.\end{split}$ (2.26)

###  2.3 An effective 1D theory

We conclude the section by reducing the 2D reduced Lagrangian (2.22) to a one-
dimensional radial effective theory where all of the functions that appear in
the effective Lagrangian (2.22) are set to be exclusively radial functions,
with respect to the radial coordinate $R$. In this additional reduction we
pick a diagonal gauge for the 2d line element of (2.17):

$ds_{2}^{2}=e^{2\rho}dt^{2}-e^{2\sigma}dR^{2}\leavevmode\nobreak\ .$ (2.27)

The operators $d$ and $\star$ acting on the fields in the Lagrangian (2.22)
simplify with this ansatz. For example, the 2D Ricci scalar becomes

$\displaystyle\mathcal{R}_{2}$
$\displaystyle=2e^{-\rho-\sigma}\partial_{R}(e^{-\sigma}\partial_{R}e^{\rho})\,.$
(2.28)

Second derivatives are awkward so it is advantageous to rewrite this term as

$\displaystyle\begin{split}e^{\rho+\sigma-
U_{1}-\frac{1}{2}U_{2}}\mathcal{R}_{2}&=2\partial_{R}\left(e^{\rho-\sigma-
U_{1}-\frac{1}{2}U_{2}}\partial_{R}\rho\right)+e^{\rho+\sigma-
U_{1}-\frac{1}{2}U_{2}}(e^{-2\sigma}\partial_{R}\rho)\partial_{R}\left(2U_{1}+U_{2}\right)\,\leavevmode\nobreak\
.\end{split}$ (2.29)

The first term is a total derivative, an additional boundary term. To examine
the total boundary contribution, we consider a constant radial slice at
infinity. As we are now reducing to 1D, the boundary terms must be evaluated
at the bounds for the time coordinate. This is trivial since there is no
explicit time dependence. After dimensional reduction, the Gibbons-Hawking-
York term in (2.1) corresponds to the total derivative

$\displaystyle\begin{split}\mathcal{L}_{\text{GHY}}&=\frac{2\pi}{G_{5}}\partial_{R}\left(e^{\rho-\sigma-
U_{1}-\frac{1}{2}U_{2}}\partial_{R}(\rho-
U_{1}-\tfrac{1}{2}U_{2})\right)\,.\end{split}$ (2.30)

The total derivative term in (2.29), the Gibbons-Hawking-York term (2.30), and
the boundary terms in the last line of (2.22) after dimensional reduction to
1D, precisely cancel. This leaves only the contribution arising from the
Chern-Simons term

$\displaystyle\begin{split}\mathcal{L}_{\text{bdry}}&=-\frac{1}{6}\frac{\pi}{G_{5}}d\left(c_{IJK}b^{I}b^{J}a^{K}_{t}\right)\,.\end{split}$
(2.31)

This remaining boundary term in (2.31) is crucial as it will affect the
conserved charges we seek to compute. We will comment on this in depth in the
subsequent section 3. In summary, the 1D Lagrangian density takes the form

$\begin{split}\mathcal{L}_{1}&=\frac{\pi}{G_{5}}e^{\rho+\sigma-
U_{1}-\frac{1}{2}U_{2}}\left[-e^{-2\sigma}(\partial_{R}\rho)\partial_{R}\left(2U_{1}+U_{2}\right)-\tfrac{1}{2}e^{-2\sigma}(\partial_{R}U_{1})(\partial_{R}U_{1}+2\partial_{R}U_{2})\right.\\\
&\quad\left.-G_{IJ}e^{-2\sigma}\left(\partial_{R}X^{I}\partial_{R}X^{J}-e^{U_{2}}\partial_{R}b^{I}\partial_{R}b^{J}\right)+\tfrac{1}{2}e^{-U_{2}-2\rho-2\sigma}(\partial_{R}a^{0}_{t})^{2}\right.\\\
&\quad\left.+G_{IJ}e^{-2\rho-2\sigma}(\partial_{R}a^{I}_{t}+b^{I}\partial_{R}a^{0}_{t})(\partial_{R}a^{J}_{t}+b^{J}\partial_{R}a^{0}_{t})-2V+2e^{U_{1}}-\tfrac{1}{2}e^{2U_{1}-U_{2}}\right.\\\
&\quad\left.-G_{IJ}e^{2U_{1}}b^{I}b^{J}\right]+\frac{\pi}{G_{5}}\frac{1}{3}c_{IJK}\left[-\frac{3}{2}b^{I}b^{J}\partial_{R}a^{K}_{t}-b^{I}b^{J}b^{K}\partial_{R}a^{0}_{t}\right]\,.\end{split}$
(2.32)

Having established the effective Lagrangian in 2D (2.22) and 1D (2.32), we
proceed in the next subsection with construction of the Noether-Wald surface
charges in our theory.

##  3 Noether-Wald surface charges

In this section, we review the Noether-Wald procedure for computing the
conserved charge due to a general symmetry Wald:1993nt ; Iyer:1994ys . We
specifically consider an isometry generated by a Killing vector and a gauge
symmetry in the presence of Chern-Simon terms. In each case, we express the
conserved charge as a flux integral that is the same for any surface
surrounding the black hole.

###  3.1 The Noether-Wald surface charge: General formulae

We consider a theory in $D$ dimensions described by a Lagrangian $\mathcal{L}$
that is presented as a $D$-form. The Lagrangian depends on fields $\Phi_{i}$
that include both the metric $g_{\mu\nu}$ and matter fields, as well as the
derivatives of these fields.

A symmetry $\zeta$ is such that the variation of $\mathcal{L}$ with respect to
$\zeta$ is a closed form (locally), i.e. $d$ acting on a $D-1$ form
$\mathcal{J}_{\zeta}$:

$\displaystyle\mathcal{L}\leavevmode\nobreak\ \leavevmode\nobreak\
\underset{\zeta}{\to}\leavevmode\nobreak\ \leavevmode\nobreak\
\mathcal{L}+\delta\mathcal{L}=\mathcal{L}+d\mathcal{J}_{\zeta}\leavevmode\nobreak\
.$ (3.1)

The variation of the Lagrangian due to any change in the fields is given
by222In practice, when we solve for the Einstein equations, we will consider a
variation of the metric and will not directly use (3.2).

$\displaystyle\begin{split}\delta\mathcal{L}&=\delta\Phi_{i}\frac{\partial\mathcal{L}}{\partial\Phi_{i}}+(\partial_{\mu}\delta\Phi_{i})\frac{\delta\mathcal{L}}{\delta\partial_{\mu}\Phi_{i}}\\\
&=\delta\Phi_{i}\left[\frac{\partial\mathcal{L}}{\partial\Phi_{i}}-\partial_{\mu}\left(\frac{\delta\mathcal{L}}{\delta\partial_{\mu}\Phi_{i}}\right)\right]+\partial_{\mu}\left(\delta\Phi_{i}\frac{\delta\mathcal{L}}{\delta\partial_{\mu}\Phi_{i}}\right)\leavevmode\nobreak\
.\end{split}$ (3.2)

The usual variational principle determines the equations of motion
$\mathcal{E}_{\Phi}$ as the vanishing of the expression in the square bracket.
The remaining term, by definition, is the total derivative of the
presymplectic potential

$\displaystyle\Theta^{\mu}$
$\displaystyle\equiv\delta\Phi_{i}\frac{\delta\mathcal{L}}{\delta\partial_{\mu}\Phi_{i}}\leavevmode\nobreak\
.$ (3.3)

In our informal notation, the left hand side of this equation is
indistinguishable from a vector. However, the Lagrangian is a $D$-form and the
$\delta$-type “derivative” removes an entire 1-form. Therefore, the
presymplectic potential $\Theta$ becomes a $D-1$ form, with indices obtained
by contracting the volume form with the vector that is normal to the boundary.
A more precise version of (3.2) reads

$\displaystyle\delta\mathcal{L}=\delta\Phi_{i}\left[\frac{\partial\mathcal{L}}{\partial\Phi_{i}}-\partial_{\mu}\left(\frac{\delta\mathcal{L}}{\delta\partial_{\mu}\Phi_{i}}\right)\right]+d\Theta[\Phi_{i},\delta\Phi_{i}]\leavevmode\nobreak\
.$ (3.4)

Comparing this formula for a general variation with its analogue (3.1) for a
symmetry establishes $d\mathcal{J}_{\zeta}=d\Theta$ and so the $D-1$ form

$\displaystyle J_{\zeta}$
$\displaystyle=\mathcal{J}_{\zeta}-\Theta[\Phi_{i},\delta\Phi_{i}]\leavevmode\nobreak\
$ (3.5)

is closed when the equations of motion $\mathcal{E}_{\Phi}$ are imposed. This
identifies the familiar conserved Noether current associated to the symmetry
$\zeta$. The corresponding Noether charge is

$\displaystyle
Q_{\zeta,\text{Noether}}=\int_{\Sigma}J_{\zeta}\leavevmode\nobreak\ ,$ (3.6)

where $\Sigma$ is a Cauchy surface on the background manifold. Conservation
amounts to this charge being the same on all Cauchy surfaces. Conceptually,
the total charge is the same at all times. That is the point of conservation
in a truly dynamical setting, but it is not terribly interesting in a
stationary black hole spacetime which is, by definition, independent of time.

For black holes it is important that, given the closed $(D-1)$ form
$J_{\zeta}$, there exists a $(D-2)$-form $Q_{\zeta}$ such that

$\displaystyle J_{\zeta}\cong dQ_{\zeta}\leavevmode\nobreak\ .$ (3.7)

The $Q_{\zeta}$ is the Noether-Wald surface charge. It amounts to a conserved
flux in the sense of Gauss’ law: integration of the flux over any surface
enclosing the source gives the same result.

The surface charge $Q_{\zeta}$ is more subtle than the conserved charge
integrated over an entire Cauchy surface. The semi-equality $\cong$ reminds us
that generally the closed form $J_{\zeta}$ is only $d$ of something locally
so, in general, the charge $Q_{\zeta}$, is only defined up to $d$ of some
$D-3$ form. Therefore, it does not necessarily satisfy Gauss’ law.

One way around this is to evaluate the surface charge at infinity. For
example, the presence of a Chern-Simons term can be interpreted physically as
a charge density that obstructs flux conservation but this contribution is
subleading at infinity and will not contribute to $Q_{\zeta,\text{Noether}}$.

Alternatively, following Kastor:2008xb ; Kastor:2009wy ; Ortin:2021ade ;
Cano:2023dyg ; Cassani:2023vsa , we can modify our definition of the surface
charge by adding a $D-2$ form to $Q_{\zeta}$. This new surface charge
satisfies a Gauss law and can be integrated at any given surface $\Sigma$.

A third approach Sen:2007qy , is the one taken in this paper. It is to compute
the surface charges in a dimensionally reduced 2D theory.

Moreover, we integrate by parts such that in the process of dimensional
reduction to 2D, we ensure gauge invariance. We will carry this procedure out
in section 3.4. Therefore, in this case, $Q_{\zeta}$ will satisfy a Gauss law.

The procedure for computing the conserved charges is extremely general. In the
following, we make the abstract procedure explicit for two particular
symmetries: isometries generated by a spacetime Killing vector $\xi$ and gauge
symmetries $\lambda$ in the presence of Chern-Simons terms.

###  3.2 Killing vector fields

A Killing vector $\xi$ generates a spacetime isometry. It transforms the
Lagrangian as

$\displaystyle\delta_{\xi}\mathcal{L}$
$\displaystyle=L_{\xi}\mathcal{L}\leavevmode\nobreak\ .$ (3.8)

Here $L_{\xi}$ is the Lie derivative along the Killing vector $\xi$.
The Lie derivative acting on a general form $\omega$ is given by Cartan’s
magic formula

$\displaystyle
L_{\xi}\omega=d(i_{\xi}\omega)+i_{\xi}d\omega\leavevmode\nobreak\ .$ (3.9)

Since $\mathcal{L}$ is a $D$-form it must be closed $d\mathcal{L}=0$ and then
the Lie derivative becomes

$\displaystyle\delta_{\xi}\mathcal{L}$
$\displaystyle=L_{\xi}\mathcal{L}=d(i_{\xi}\mathcal{L})+i_{\xi}(d\mathcal{L})=d(\xi\cdot\mathcal{L})\leavevmode\nobreak\
,$ (3.10)

where $\cdot$ denotes the contraction of $\xi$ with the first index of
$\mathcal{L}$. Comparing (3.10) with (3.1), we identify

$\displaystyle{\cal J}_{\xi}$
$\displaystyle=\xi\cdot\mathcal{L}\leavevmode\nobreak\ ,$ (3.11)

up to a closed form that is unimportant in our application. Thus, for a
Killing vector $\xi$, the Noether current (3.5) becomes

$\displaystyle
J_{\xi}=\xi\cdot\mathcal{L}-\Theta[\Phi,\mathcal{L}_{\xi}\Phi]\leavevmode\nobreak\
.$ (3.12)

The computations show that this current $(D-1)$ form is closed on-shell. In
other words, it is conserved when the equations of motion are satisfied.

###  3.3 Incorporating gauge invariance

We now consider a gauge invariant Lagrangian and compute the conserved current
as defined in (3.5) for the conserved charges of the theory, whether derived
from spacetime isometries or gauge invariance.

The relevant gauge invariant Lagrangian is the one defined in (2.1) without
the Chern-Simons term. In other words, we consider the Lagrangian density

$\displaystyle\begin{split}\mathcal{L}_{5,\text{pot}}&=-\frac{1}{16\pi
G_{5}}\sqrt{g_{5}}\left(\mathcal{R}_{5}+2V\right)\,,\\\
\mathcal{L}_{5,\text{kin}}&=\frac{1}{16\pi
G_{5}}\sqrt{g_{5}}\left(-\frac{1}{2}G_{IJ}F^{I}_{5,AB}F^{J,AB}_{5}+G_{IJ}\nabla^{A}X^{I}\nabla_{A}X^{J}\right)\leavevmode\nobreak\
.\end{split}$ (3.13)

We use early capital Latin indices $A,B,\ldots$ to denote 5D coordinates. The
Lagrangian $\mathcal{L}_{5,\text{kin}}+\mathcal{L}_{5,\text{pot}}$ is
manifestly gauge invariant

$\displaystyle\delta_{\alpha}(\mathcal{L}_{5,\text{pot}}+\mathcal{L}_{5,\text{kin}})$
$\displaystyle=0\leavevmode\nobreak\ ,$ (3.14)
$\displaystyle\delta_{\xi}(\mathcal{L}_{5,\text{pot}}+\mathcal{L}_{5,\text{kin}})$
$\displaystyle=\nabla_{A}\left(\xi^{A}(\mathcal{L}_{5,\text{pot}}+\mathcal{L}_{5,\text{kin}})\right)\leavevmode\nobreak\
.$ (3.15)

As detailed in the previous subsection, there is a conserved charge for any
Killing vector that generates a spacetime isometry. According to (3.10), the
Lagrangian $\mathcal{L}_{5,\text{kin}}+\mathcal{L}_{5,\text{pot}}$ transforms
as

$\displaystyle\delta_{\xi}(\mathcal{L}_{5,\text{pot}}+\mathcal{L}_{5,\text{kin}})$
$\displaystyle=\nabla_{A}\left(\xi^{A}(\mathcal{L}_{5,\text{pot}}+\mathcal{L}_{5,\text{kin}})\right)\leavevmode\nobreak\
.$ (3.16)

The presymplectic potential (3.3) for $\mathcal{L}_{5,\text{pot}}$ is

$\displaystyle\begin{split}\Theta^{A,5}_{\xi,\text{pot}}&=\frac{1}{16\pi
G_{5}}\sqrt{g_{5}}\left(\nabla_{B}\nabla^{A}\xi^{B}+\nabla_{B}\nabla^{B}\xi^{A}-2\nabla^{A}\nabla_{B}\xi^{B}\right)\\\
&=\frac{1}{16\pi
G_{5}}\sqrt{g_{5}}\left(\nabla_{B}\left(\nabla^{B}\xi^{A}-\nabla^{A}\xi^{B}\right)+2R^{AB}\xi_{B}\right)\,,\end{split}$
(3.17)

where in the second line, we have used the commutator relation for two
covariant derivatives. In addition, the presymplectic potential for the
kinetic terms $\mathcal{L}_{5,\text{kin}}$ is

$\displaystyle\Theta^{A,5}_{\alpha,\xi,\text{kin}}$
$\displaystyle=\frac{1}{16\pi
G_{5}}\sqrt{g_{5}}G_{IJ}\left(2F^{I,AB}_{5}\left(\xi^{C}F^{J}_{5,CB}+\nabla_{B}\left(\xi^{C}A^{J}_{5,C}+\alpha^{J}_{5}\right)\right)\right)\,,$
(3.18)

where we have used the variation

$\displaystyle\delta A^{I}_{A,5}$
$\displaystyle=\delta_{\xi}A^{I}_{5,A}+\delta_{\alpha}A^{I}_{5,A}=\xi^{B}F^{I}_{5,BA}+\nabla_{A}\left(\xi^{B}A^{I}_{5,B}\right)+\nabla_{A}\alpha^{I}\,.$
(3.19)

Inserting the variations (3.16) and the presympletic potentials given in
(3.17) and (3.18) into the current density (3.5), we find

$\displaystyle\begin{split}J^{A}_{\alpha,\xi}&=\frac{1}{16\pi
G_{5}}\sqrt{g_{5}}\Big{[}\left.-\nabla_{B}\left(\nabla^{B}\xi^{A}-\nabla^{A}\xi^{B}\right)-2\nabla_{B}\left(G_{IJ}F^{I,AB}(\xi^{C}A^{J}_{C}+\alpha^{J})\right)\right.\\\
&\left.\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad-2\xi_{B}\mathcal{E}_{g}^{B}-2\mathcal{E}_{J,A_{5}}^{A}(\xi^{C}A^{J}_{C}+\alpha^{J})\right.\Big{]}\leavevmode\nobreak\
,\end{split}$ (3.20)

where the second line is proportional to the equations of motion
$\mathcal{E}_{g}^{B}$ and $\mathcal{E}_{J,A_{5}}^{A}$ and vanish on-shell
giving

$\displaystyle J^{A}_{\alpha,\xi}$ $\displaystyle=-\frac{1}{16\pi
G_{5}}\sqrt{g_{5}}\nabla_{B}\Big{[}\left(\nabla^{B}\xi^{A}-\nabla^{A}\xi^{B}\right)+2\left(G_{IJ}F^{I,AB}(\xi^{C}A^{J}_{C}+\alpha^{J})\right)\Big{]}.$
(3.21)

The Noether-Wald surface charges of the theory can now be read off from the
current (3.21). To find the conserved charges, we integrate over a surface
$\Sigma$ enclosing the source and we find

$\displaystyle\begin{split}Q_{\alpha}&=-\frac{1}{8\pi
G_{5}}\int_{\Sigma}d\Sigma_{AB}\sqrt{g_{5}}G_{IJ}F^{I,AB}\alpha^{J}\,,\\\
Q_{\xi}&=-\frac{1}{16\pi
G_{5}}\int_{\Sigma}d\Sigma_{AB}\sqrt{g_{5}}\left[\left(\nabla^{B}\xi^{A}-\nabla^{A}\xi^{B}\right)+2G_{IJ}F^{I,AB}\xi^{C}A^{J}_{C}\right]\,.\end{split}$
(3.22)

###  3.4 Chern-Simons Terms

The charge $Q_{\xi}$ that corresponds to angular momentum depends explicitly
on the gauge field $A^{J}$ whereas the electric charges $Q_{\alpha}$ depend on
the field strength. When Chern-Simons terms are taken into account,
$Q_{\alpha}$ also depends on the gauge field $A^{J}$. This gauge dependence
renders the value of the charges ambiguous.

To address the situation, we dimensionally reduce the theory (2.1) to 2D, as
was done in subsection 2.3 and express the resulting action as a covariant
theory in 2D Sen:2007qy . As part of the process, we must ensure that the
field strength does not have a nonzero flux through the squashed sphere. This
can be achieved by adding total derivatives before the dimensional reduction
to remove the derivatives acting on the gauge potentials and gauge fields that
act nontrivally through the squashed sphere.

We now show how this can be done. Let us consider the Lagrangian (3.13) along
with the five-dimensional Chern-Simons term of the form

$\displaystyle\mathcal{L}_{5,\text{CS}}=-\frac{1}{16\pi
G_{5}}\frac{1}{6}c_{IJK}F^{I}_{5}\wedge F^{J}_{5}\wedge
A^{K}_{5}\leavevmode\nobreak\ .$ (3.23)

We are interested in transforming (3.23) by the inclusion of total derivatives
such that the potential term associated to the electric charge is manifestly
gauge invariant. Note this procedure is not covariant in 5D and therefore we
explicitly break covariance along the way. However, because of the dimensional
reduction, the 2D Lagrangian still remains covariant.

We consider the ansatz in (2.17) such that the potential and gauge fields are
of the form

$\displaystyle\begin{split}A^{I}_{5}&=A^{I}_{5,A}dx^{A}=A^{I}_{5,\mu}dx^{\mu}+A^{I}_{5,a}dx^{a}\leavevmode\nobreak\
,\\\ F^{I}_{5}&=\frac{1}{2}F^{I}_{5,AB}\,dx^{A}\wedge
dx^{B}=\frac{1}{2}F^{I}_{5,\mu\nu}dx^{\mu}\wedge dx^{\nu}+F^{I}_{5,\mu
a}dx^{\mu}\wedge dx^{a}+\frac{1}{2}F^{I}_{5,ab}dx^{a}\wedge
dx^{b}\leavevmode\nobreak\ ,\end{split}$ (3.24)

where lowercase Latin indices denote the indices on the compact space and as
before, the Greek indices correspond to the 2D space. Expanding out the Chern-
Simons term in component form using (3.24), there are two types of terms,
having the following structure of indices: $F^{I}_{\mu\nu}F^{J}_{bc}A^{K}_{a}$
and $F^{I}_{\mu a}F^{J}_{bc}A^{K}_{\nu}$. Only for the second expression we
must transfer the derivative such that in the process of dimensional
reduction, we find it to be gauge invariant in the 2D theory. This means the
integration by parts of this term takes the form

$\displaystyle c_{IJK}\epsilon^{\mu abc\nu}F^{I}_{\mu
a}F^{J}_{bc}A^{K}_{\nu}=2c_{IJK}\epsilon^{\mu
abc\nu}\left(\partial_{\mu}(A^{K}_{\nu}A^{I}_{a}F^{J}_{bc})-(\partial_{\mu}A^{K}_{\nu})(A^{I}_{a}F^{J}_{bc})\right)\leavevmode\nobreak\
,$ (3.25)

and the presymplectic potential is found to be

$\displaystyle\begin{split}\Theta^{A,5}_{\alpha,\xi,\text{CS}}&=\frac{1}{16\pi
G_{5}}\sqrt{g_{5}}\left[\frac{1}{6}c_{IJK}\left(\epsilon^{ABCDE}F^{I}_{BC}A^{J}_{D}\left(\xi^{F}F^{K}_{FE}+\nabla_{E}(\xi^{F}A^{K}_{F})+\nabla_{E}\alpha^{K}\right)\right)\right]\\\
&\quad-\frac{1}{8\pi
G_{5}}\sqrt{g_{5}}\left[c_{IJK}\epsilon^{Aabc\nu}A^{I}_{a}F^{J}_{bc}\nabla_{\nu}\alpha^{K}\right]\,,\end{split}$
(3.26)

where the last term is the contribution of (3.25) coming from adding a total
derivative. To investigate the current and the Noether-Wald surface charges,
we proceed to dimensionally reduce over the squashed $S^{3}$ where covariance
over the 2D spacetime is still maintained. The 5D rotational isometries in
$\varphi$ and $\psi$ take on a different role in the 2D perspective. Moreover,
we find that they become 2D gauge transformations of $a^{0}$ and $a^{I}$
coming from the dimensionally reduced potential $A^{I}$ (2.18).

###  3.5 The 2D conserved charges

The 2D Lagrangian (2.22) inherits some symmetries from the 5D theory (2.2),
including gauge symmetry associated with the 5D gauge potential $A^{I}$ and
rotational isometries associated to the Killing vectors $\partial_{\phi}$ and
$\partial_{\psi}$. In the 2D theory, all symmetries become gauge symmetries
and have associated charges. We denote the 2D charge originally coming from
the 5D rotational isometries $J$ and the 2D charges originally coming from the
5D gauge transformations $Q_{I}$. These 2D gauge transformations are
associated to $a^{0}$ and $a^{I}$ as they come from the dimensionally reduced
potential $A^{I}$ (2.18). Therefore, we consider the following symmetries

$\displaystyle\delta_{\lambda}a^{0}=d\lambda,\qquad\delta_{\chi}a^{I}=d\chi^{I}\leavevmode\nobreak\
,$ (3.27)

with total corresponding conserved current

$\displaystyle J_{\lambda,\chi}$
$\displaystyle=J_{\lambda}+J_{\chi}=\sum_{i=\lambda,\chi}\left(\mathcal{J}_{i}-\Theta_{i}\right),$
(3.28)

where $J_{\lambda}$ and $J_{\chi}$ are the currents corresponding to $\lambda$
and $\chi$, respectively, and the second equality is given by (3.5). The
effective 2D Lagrangian (2.22) is manifestly gauge invariant and the
variations with respect to each symmetry (3.27) yield

$\displaystyle\delta_{\lambda}\mathcal{L}_{2}$
$\displaystyle=d\mathcal{J}_{\lambda}=0\leavevmode\nobreak\
,\quad\delta_{\chi}\mathcal{L}_{2}=d\mathcal{J}_{\chi}=-\frac{\pi}{G_{5}}\frac{1}{6}c_{IJK}d\left(b^{I}b^{J}d\chi^{K}\right)\leavevmode\nobreak\
.$ (3.29)

The presymplectic potentials given in (3.4) become

$\displaystyle\Theta_{\lambda}$
$\displaystyle=-\frac{\pi}{G_{5}}e^{-U_{1}-\frac{1}{2}U_{2}}\left[e^{-U_{2}}\star
da^{0}+2G_{IJ}b^{I}\star\left(b^{J}da^{0}+da^{J}\right)\right]d\lambda+\frac{\pi}{G_{5}}\frac{1}{3}c_{IJK}b^{I}b^{J}b^{K}d\lambda\leavevmode\nobreak\
,$ (3.30) $\displaystyle\Theta_{\chi}$
$\displaystyle=-\frac{\pi}{G_{5}}e^{-U_{1}-\frac{1}{2}U_{2}}\left[2G_{IJ}d\chi^{I}\wedge\star\left(b^{J}da^{0}+da^{J}\right)\right]+\frac{\pi}{G_{5}}\frac{1}{3}c_{IJK}b^{I}b^{J}d\chi^{K}\leavevmode\nobreak\
.$ (3.31)

We used the symmetries (3.27) and included the additional total derivative
term (3.25). Using the equations of motion (2.26), the on-shell current (3.28)
can be recast in the form of (3.7):

$\displaystyle\begin{split}J_{\lambda,\chi}&\cong\frac{\pi}{G_{5}}d\left[\lambda\left(e^{-U_{1}-\frac{1}{2}U_{2}}\left[e^{-U_{2}}\star
da^{0}+2G_{IJ}b^{I}\star\left(b^{J}da^{0}+da^{J}\right)\right]-\frac{1}{3}c_{IJK}b^{I}b^{J}b^{K}\right)\right.\\\
&\left.\qquad\qquad+\chi^{I}\left(2e^{-U_{1}-\frac{1}{2}U_{2}}G_{IJ}\star\left(b^{J}da^{0}+da^{J}\right)-\frac{1}{2}c_{IJK}b^{J}b^{K}\right)\right].\end{split}$
(3.32)

The conserved charges $J$ and $Q_{I}$ can be directly read off from (3.32) and
we find

$\displaystyle\begin{split}J&=\frac{\pi}{G_{5}}\left[e^{-U_{1}-\frac{1}{2}U_{2}}\left[e^{-U_{2}}\star
da^{0}+2G_{IJ}b^{I}\star\left(b^{J}da^{0}+da^{J}\right)\right]-\frac{1}{3}c_{IJK}b^{I}b^{J}b^{K}\right]\leavevmode\nobreak\
,\\\
Q_{I}&=\frac{\pi}{G_{5}}\left[2e^{-U_{1}-\frac{1}{2}U_{2}}G_{IJ}\star\left(b^{J}da^{0}+da^{J}\right)-\frac{1}{2}c_{IJK}b^{J}b^{K}\right]\leavevmode\nobreak\
.\end{split}$ (3.33)

From now on, we use the rescaled charges

$\displaystyle\widetilde{J}$
$\displaystyle\equiv\frac{4G_{5}}{\pi}J\leavevmode\nobreak\
,\qquad\widetilde{Q}\equiv\frac{4G_{5}}{\pi}Q_{I}\leavevmode\nobreak\ .$
(3.34)

The charges and the current are indeed conserved including the charge
associated to $a^{I}$ since we demanded gauge invariance at the level of the
Lagrangian in (2.22). This added a total derivative that shifted the charge
but did not affect the equations of motion. Moreover, the charges computed in
2D are proportional to those computed in 5D. In the 1D reduction (2.32), the
charges take the following form

$\displaystyle\begin{split}\widetilde{J}&=4\left(e^{-U_{1}-\frac{3}{2}U_{2}-\rho-\sigma}\left(\partial_{R}a^{0}_{t}+2G_{IJ}e^{U_{2}}b^{I}\left(\partial_{R}a^{J}_{t}+b^{J}\partial_{R}a^{0}_{t}\right)\right)-\frac{1}{3}c_{IJK}b^{I}b^{J}b^{K}\right)\leavevmode\nobreak\
,\\\
\widetilde{Q}_{I}&=4\left(2e^{-U_{1}-\frac{1}{2}U_{2}-\rho-\sigma}G_{IJ}\left(\partial_{R}a^{J}_{t}+b^{J}\partial_{R}a^{0}_{t}\right)-\frac{1}{2}c_{IJK}b^{J}b^{K}\right)\leavevmode\nobreak\
.\end{split}$ (3.35)

These formulae are essential for the radial flow in the black hole background.
A very rough reading is that each of the conserved charges $\widetilde{J}$ and
$\widetilde{Q}_{I}$ are radial derivatives of their conjugate potentials
$a^{0}_{t}$, $a^{I}_{t}$, as in elementary electrodynamics. With this naïve
starting point, the overall factors depending on $U_{1},U_{2},\rho$ and
$\sigma$ serve to take on the non-flat spacetime into account and $G_{IJ}$
incorporates special geometry as required by symmetry. All remaining terms
depend on $b^{I}$ and take rotation into account in a manner that combines
kinematics (rotation “looks” like a force) and electrodynamics (electric and
magnetic fields mix in a moving frame). These effects defy simple physical
interpretations.

From our point of view, the formulae (3.35) for the charges $\widetilde{J}$
and $\widetilde{Q}_{I}$ are complicated functions of various fields, each of
which are themselves nontrivial functions of the radial coordinate. Our
construction shows that symmetry guarantees that these combinations must be
independent of the radial position, within the framework of our ansatz.

In the following section we study the conditions that supersymmetric AdS5
black holes must satisfy. The vanishing of the supersymmetry variations of the
theory for a subset of the supersymmetries always imposes first-order radial
differential equations on the joint geometry/matter configuration. We refer to
these first order equations as flow equations. They are very constraining but,
as usual for first order equations imposed by supersymmetry, they are not
sufficient to determine the solution. The raison d’être of this entire section
is that the additional data needed, sometimes referred to as the integrability
conditions, is furnished by the conserved charges.

##  4 The flow equations

In this section we derive the first order flow equations for AdS5 black holes.
They follow from preservation of supersymmetry, complemented by conservation
of the charges. We study the flow equations using two perturbative expansions:
one starting at the near-horizon and one starting at the asymptotic boundary.
Enforcing the conservation of charges at both the horizon and at the
asymptotic boundary allows us to make contact between the two expansions.

###  4.1 Supersymmetry conditions

We study bosonic backgrounds that preserve some supersymmetry
gunaydin1984geometry ; Gauntlett:2002nw ; gauntlett2003all ; Gutowski:2004yv .
Thus there exists a supersymmetric spinor $\epsilon^{\alpha}$ for which the
gravitino and the dilatino variations vanish. This condition amounts to

$\displaystyle 0$
$\displaystyle=\left[G_{IJ}\left(\frac{1}{2}\gamma^{AB}F^{J}_{\leavevmode\nobreak\
AB}-\gamma^{A}\nabla_{A}X^{J}\right)\epsilon^{\alpha}-\xi_{I}\epsilon^{\alpha\beta}\epsilon^{\beta}\right]\partial_{i}X^{I}\leavevmode\nobreak\
,$ (4.1) $\displaystyle 0$
$\displaystyle=\left[(\partial_{A}-\frac{1}{4}\omega_{A}^{BC}\gamma_{BC})+\frac{1}{24}(\gamma_{A}^{\leavevmode\nobreak\
BC}-4\delta^{\leavevmode\nobreak\
B}_{A}\gamma^{C})X_{I}F^{I}_{BC}\right]\epsilon^{\alpha}+\frac{1}{6}\xi_{I}(3A^{I}_{A}-X^{I}\gamma_{A})\epsilon^{\alpha\beta}\epsilon^{\beta}\leavevmode\nobreak\
,$ (4.2)

where $\epsilon^{\alpha}\ (\alpha=1,2)$ are symplectic Majorana spinors. In
the following, we recast these variations as radial flow equations.

For the analysis of supersymmetry, it is convenient to split the 5D spacetime
geometry into $(1+4)$ dimensions as

$\displaystyle ds_{5}^{2}$
$\displaystyle=f^{2}(dt+w\sigma_{3})^{2}-f^{-1}ds_{4}^{2}\leavevmode\nobreak\
,$ (4.3) $\displaystyle ds_{4}^{2}$
$\displaystyle=g_{m}^{-1}dR^{2}+\frac{1}{4}R^{2}(\sigma_{1}^{2}+\sigma_{2}^{2}+g_{m}\sigma_{3}^{2})\,.$
(4.4)

This form highlights the 4D base space $ds_{4}^{2}$ which is automatically
Kähler. This can be shown by picking the flat vielbein

$\begin{split}e^{1}&=g_{m}^{-1/2}dR\leavevmode\nobreak\ ,\quad
e^{2}=\frac{1}{2}R\sigma_{1}\leavevmode\nobreak\ ,\quad
e^{3}=\frac{1}{2}R\sigma_{2}\leavevmode\nobreak\ ,\quad
e^{4}=\frac{1}{2}Rg_{m}^{1/2}\sigma_{3}\leavevmode\nobreak\ ,\end{split}$
(4.5)

which gives the manifestly closed Kähler 2-form

$J^{(1)}=\epsilon(e^{1}\wedge e^{4}-e^{2}\wedge e^{3})\leavevmode\nobreak\ .$
(4.6)

The symbol $\epsilon=\pm 1$ denotes the orientation of the base manifold. It
should not be confused with the supersymmetry parameter $\epsilon^{\alpha}$.

The $(1+4)$ split of the 5D gauge potential $A^{I}$ defined in (2.18) can be
expressed as

$\begin{split}A^{I}&=fY^{I}(dt+w\sigma_{3})+u^{I}\sigma_{3}\leavevmode\nobreak\
.\end{split}$ (4.7)

In the rest of the paper, as well as in Appendix C, we use lowercase Latin
letters for the four spatial indices.

###  4.2 Dictionary between the $(1+4)$ and the $(2+3)$ splits

We can relate the $(1+4)$ split introduced in the previous subsection to
simplify the supersymmetric variations (4.1) and (4.2), to the $(2+3)$ split
(2.17) and (2.18) that was used earlier to perform the reduction from 5D to
2D. The 5D geometry (2.17) with the diagonal gauge (2.27) for the 2D line
element is

$ds_{5}^{2}=e^{2\rho}dt^{2}-e^{2\sigma}dR^{2}-e^{-U_{1}}(\sigma_{1}^{2}+\sigma_{2}^{2})-e^{-U_{2}}(\sigma_{3}+a^{0})^{2}\leavevmode\nobreak\
.$ (4.8)

By identifying the metric components of (4.3) and (4.8), we find the
dictionary of variables in the $(2+3)$ split of the 5D line element
$ds_{5}^{2}$, expressed in terms of the variables in the $(1+4)$ split

$\displaystyle e^{-U_{1}}$
$\displaystyle=\frac{1}{4}R^{2}f^{-1}\leavevmode\nobreak\ ,$
$\displaystyle\quad e^{-U_{2}}$
$\displaystyle=\frac{1}{4}R^{2}g_{m}f^{-1}-f^{2}w^{2}\leavevmode\nobreak\ ,$
$\displaystyle\quad b^{I}$ $\displaystyle=fY^{I}w+u^{I}\leavevmode\nobreak\ ,$
(4.9) $\displaystyle e^{2\sigma}$
$\displaystyle=f^{-1}g_{m}^{-1}\leavevmode\nobreak\ ,$ $\displaystyle\quad
a^{0}_{t}$
$\displaystyle=\frac{-f^{2}w}{\tfrac{1}{4}R^{2}g_{m}f^{-1}-f^{2}w^{2}}\leavevmode\nobreak\
,$ $\displaystyle\quad e^{2\rho}$
$\displaystyle=f^{2}-\frac{f^{4}w^{2}}{(\frac{1}{4}R^{2}g_{m}f^{-1}-f^{2}w^{2})^{2}}\leavevmode\nobreak\
.$

In this section we primarily use the $(1+4)$ variables $fX^{I}$, $u^{I}$, $w$
and $g_{m}$, along with the conserved charges $Q_{I}$ and $J$.

As noted in the previous subsection, the 4D base of the $(1+4)$ split (4.3) is
automatically Kähler. In the variables of the $(2+3)$ split in (4.9), the
Kähler condition amounts to the relation

$e^{\sigma+\rho-U_{2}/2}=\frac{1}{2}R\leavevmode\nobreak\ ,$ (4.10)

between $\rho$, $\sigma$, and $U_{2}$. This is explained further in Appendix
C.1.

###  4.3 The attractor flow equations

The preserved supersymmetries are defined by the projections on the spinors
$\epsilon^{\alpha}$

$\displaystyle\gamma^{0}\epsilon^{\alpha}$
$\displaystyle=\epsilon^{\alpha}\leavevmode\nobreak\ ,$ (4.11)
$\displaystyle\frac{1}{4}J^{(1)}_{mn}\gamma^{mn}\epsilon^{\alpha}$
$\displaystyle=-\epsilon^{\alpha\beta}\epsilon^{\beta}\leavevmode\nobreak\ .$
(4.12)

The $J^{(1)}_{mn}$ are components of the Kähler form $J^{(1)}$ (4.6), and the
spatial gamma matrices $\gamma^{m}$ satisfy the usual Clifford algebra. The
details on the simplification of the equations (4.1) and (4.2) are presented
in Appendix C. The result is the following differential conditions on the
variables $fX^{I}$, $u^{I}$, $w$ and $g_{m}$

$\displaystyle 0$
$\displaystyle=G_{IJ}\left(\partial_{R}(fY^{I})-\partial_{R}(fX^{I})\right)\partial_{i}X^{J}\leavevmode\nobreak\
,$ (4.13) $\displaystyle 0$
$\displaystyle=\left(\partial_{R^{2}}+\frac{1}{R^{2}}\right)u^{I}-\frac{1}{2}\epsilon
f^{-1}c^{IJK}X_{J}\xi_{K}\leavevmode\nobreak\ ,$ (4.14) $\displaystyle 0$
$\displaystyle=\left(\partial_{R^{2}}-\frac{1}{R^{2}}\right)w+\frac{1}{2}f^{-1}X_{I}\left(\partial_{R^{2}}-\frac{1}{R^{2}}\right)u^{I}\leavevmode\nobreak\
,$ (4.15) $\displaystyle 0$ $\displaystyle=-\epsilon
R^{2}(\partial_{R^{2}}g_{m})+2\epsilon(1-g_{m})+2\xi_{I}u^{I}\leavevmode\nobreak\
.$ (4.16)

The variation (4.13) allows for the electric potential $fY^{I}$ in (4.7) to be
identified with the scalar field $fX^{I}$, and thus

$A^{I}=fX^{I}(dt+w\sigma_{3})+u^{I}\sigma_{3}\leavevmode\nobreak\ .$ (4.17)

This sets the variables $a^{I}$, $a^{0}$ and $b^{I}$ in the decomposition
(2.18) to

$a^{I}+b^{I}a^{0}=fX^{I}dt\ ,\ b^{I}=fX^{I}w+u^{I}\leavevmode\nobreak\ ,$
(4.18)

with $a^{0}=a^{0}_{t}dt$ given in (4.9).

In the limit of ungauged supergravity, $fX^{I}$ is a harmonic function. Then,
the identification of $X^{I}=Y^{I}$ means that the corresponding electric
potential is also a harmonic function. We may expect the same functional
dependence in the case of gauged supergravity.333There are $n_{V}+1$
potentials $Y^{I}$ and $n_{V}$ scalars $X^{I}$ so there is freedom to adjust a
single integration constant that we do not exploit. It is unclear to us if
this freedom is physically significant.

In the context of a black hole, solutions to the supersymmetry conditions
(4.13-4.16) are specified in part by the conserved charges of the theory. The
charges in $\widetilde{J}$ and $\widetilde{Q}_{I}$ in (3.35) are expressed in
terms of the $(2+3)$ variables $U_{1},U_{2},b^{I},a^{0}_{t},a^{I}_{t}$ but we
can recast them in terms of $(1+4)$ variables $f,u^{I},w,g_{m}$ using the
dictionary for the geometry (4.9) and the potential (4.18). We can also remove
most of the derivatives in the equations (3.35) for the charges
$\widetilde{J}$ and $\widetilde{Q}_{I}$ using the radial equations for the
variables $u^{I}$, $w$ and $g_{m}$ (4.14-4.16). Our final expressions of the
charges, which we use for the remainder of the section are given by

$\displaystyle\begin{split}\widetilde{J}&=\widetilde{Q}_{I}u^{I}+\frac{2}{3}c_{IJK}u^{I}u^{J}u^{K}-R^{2}g_{m}\left(f^{-1}X\cdot
u+2w-\frac{1}{2}\epsilon R^{2}f^{-3}\xi\cdot fX\right)\\\
&\quad+2R^{2}w(1+\epsilon\xi\cdot u)\leavevmode\nobreak\ ,\\\
\widetilde{Q}_{I}&=-2c_{IJK}u^{J}u^{K}-2\epsilon
wR^{2}\xi_{I}-g_{m}R^{4}\partial_{R^{2}}(f^{-1}X_{I})\leavevmode\nobreak\
.\end{split}$ (4.19)

The second of these equations is a first order differential equation for
$f^{-1}X_{I}$. Together with the three radial differential equations
(4.14-4.16) for the variables $u^{I}$, $g_{m}$ and $w$, we find the four
equations

$\displaystyle\left(\partial_{R^{2}}+\frac{1}{R^{2}}\right)u^{I}$
$\displaystyle=\frac{1}{2}\epsilon
c^{IJK}(f^{-1}X_{J})\xi_{K}\leavevmode\nobreak\ ,$ (4.20a)
$\displaystyle\left(\partial_{R^{2}}+\frac{2}{R^{2}}\right)g_{m}$
$\displaystyle=\frac{2}{R^{2}}+\frac{2}{R^{2}}\epsilon\xi_{I}u^{I}\leavevmode\nobreak\
\,,$ (4.20b) $\displaystyle\left(\partial_{R^{2}}-\frac{1}{R^{2}}\right)w$
$\displaystyle=-\frac{1}{2}f^{-1}X_{I}\left(\partial_{R^{2}}-\frac{1}{R^{2}}\right)u^{I}\leavevmode\nobreak\
,$ (4.20c) $\displaystyle R^{4}\partial_{R^{2}}(f^{-1}X_{I})$
$\displaystyle=-\frac{1}{g_{m}}\left(\widetilde{Q}_{I}+2c_{IJK}u^{J}u^{K}+2\epsilon
wR^{2}\xi_{I}\right)\leavevmode\nobreak\ .$ (4.20d)

We refer to the set of first order differential equations (4.20a-4.20d) as the
attractor flow equations for black hole solutions to the theory (2.1).

###  4.4 Solution of the attractor flow equations

The attractor flow equations (4.20a-4.20d) are first order differential
equations. In this subsection we discuss the boundary conditions needed to
specify their solutions completely. This turns out to be surprisingly subtle.
We then solve the equations using perturbative expansions.

####  4.4.1 Boundary conditions

The attractor flow equations (4.20a-4.20d) are first order differential
equations with $\xi_{I}$ and $\tilde{Q}_{I}$ as given parameters. As such the
superficial expectation is that the specification of all the unknown functions
$u^{I},g_{m},w,f^{-1}X_{I}$ at any coordinate $R^{2}$ yields the corresponding
derivatives at that position. Further iterations should then be sufficient to
reconstruct the entire radial dependence, at least in principle. We seek to
implement this strategy starting from either asymptotically AdS5, or from the
horizon. We consider each in turn.

For a solution to be asymptotically AdS5, the metric ansatz (4.3) requires the
leading order behavior $f\to R^{0}$, $g_{m}\to R^{2}$, and $w\to R^{2}$ as
$R\to\infty$. With these boundary conditions for $f$, $g_{m}$, and $w$,
(4.20a) and (4.20d) yield $u^{I}\to c^{IJK}\xi_{J}\xi_{K}R^{2}$ and
$X_{I}\to\xi_{I}\cdot R^{0}$ for the matter fields as $R\to\infty$.

Alternatively, we can impose boundary conditions at the horizon of the black
hole. There, the near-horizon geometry has a manifest AdS2 factor of the form

$\displaystyle ds^{2}_{2}=R^{4}dt^{2}-\frac{dR^{2}}{R^{2}}\leavevmode\nobreak\
,$ (4.21)

and so $g_{m}\to R^{0}$, $f\to R^{2}$, and $w\to R^{-2}$. With these leading
asymptotics, the attractor flow equations (4.20a) and (4.20d) determine
$u^{I}\to R^{0}$ and $X_{I}\to R^{0}$ near the horizon.

Staying with our superficial expectation, we would start from either
asymptotically AdS5 or from the AdS2 horizon. Mathematically, it could be a
concern that the differential equations are coupled and non-linear, because
then the expansions might fail to converge. This situation is most likely
incompatible with a black hole solution that is regular throughout the entire
flow from asymptotically AdS5 to the AdS2 horizon, or vice versa. Nonlinearity
does not appear to pose a conceptual challenge.

In order to study the unknown functions $u^{I},g_{m},w$ and $f^{-1}X_{I}$
around a regular point, we multiply by an appropriate factor of $R^{2}$. Near
the horizon we consider $R^{2}u^{I},R^{2}g_{m},R^{2}w$, and
$R^{2}f^{-1}X_{I}$. At infinity, we expand the functions
$R^{-2}u^{I},R^{-2}g_{m},R^{-2}w$ and $R^{-2}f^{-1}X_{I}$. After such
rescalings the left hand sides of each of the attractor flow equations
(4.20a-4.20d) will take the form

$\displaystyle\begin{split}\left(\partial_{R^{2}}+\frac{\alpha+\beta}{R^{2}}\right)P=R^{-2\beta}\left(\partial_{R^{2}}+\frac{\alpha}{R^{2}}\right)(R^{2\beta}P),\end{split}$
(4.22)

for some field $P$ and some integers $\alpha$ and $\beta$ that can either be
positive or negative. The challenge we will encounter repeatedly is that, when
$P\sim R^{-2(\alpha+\beta)}$, this expression vanishes. We refer to this
situation as a zero-mode of the perturbative expansion. What it means is that
an attractor flow equation does not reveal a derivative, contrary to
expectation. Instead, it yields a constraint between the unknown functions on
the right hand side of the equation in question. This constraint will be
nonlinear and, in general, difficult to implement. In other words, the initial
value problem, at both the horizon and at asymptotic infinity, turns out to be
unexpectedly complicated.

| $R\to\infty$ | $R\to 0$
---|---|---
$g_{m}$ | $R^{2}$ | $R^{0}$
$f$ | $R^{0}$ | $R^{2}$
$w$ | $R^{2}$ | $R^{-2}$
$u^{I}$ | $R^{2}$ | $R^{0}$
$X_{I}$ | $R^{0}$ | $R^{0}$
Table 1: Asymptotics of the various functions

In the following subsections we develop this general theme explicitly, first
starting from the horizon, and then from asymptotic infinity. We subsequently
merge the two perturbative expansions to seek a global understanding.

####  4.4.2 Perturbative solution starting from the horizon

To satisfy the regularity conditions at the horizon, we take $\beta=1$ in
(4.22) and rewrite the attractor flow equations (4.20a-4.20d) as

$\displaystyle\partial_{R^{2}}(R^{2}u^{I})$ $\displaystyle=\frac{1}{2}\epsilon
c^{IJK}(R^{2}f^{-1}X_{J})\xi_{K}\leavevmode\nobreak\ ,$ (4.23a)
$\displaystyle\left(\partial_{R^{2}}+\frac{1}{R^{2}}\right)(R^{2}g_{m})$
$\displaystyle=2+\frac{2}{R^{2}}\epsilon\xi_{I}(R^{2}u^{I})\leavevmode\nobreak\
,$ (4.23b)
$\displaystyle\left(\partial_{R^{2}}-\frac{2}{R^{2}}\right)(R^{2}w)$
$\displaystyle=-\frac{1}{2}R^{-2}(R^{2}f^{-1}X_{I})\left(\partial_{R^{2}}-\frac{2}{R^{2}}\right)(R^{2}u^{I})\leavevmode\nobreak\
,$ (4.23c)
$\displaystyle\left(\partial_{R^{2}}-\frac{1}{R^{2}}\right)(R^{2}f^{-1}X_{I})$
$\displaystyle=-R^{-2}g_{m}^{-1}\left(\widetilde{Q}_{I}+2R^{-4}c_{IJK}(R^{2}u^{J})(R^{2}u^{K})+2\epsilon(R^{2}w)\xi_{I}\right)\leavevmode\nobreak\
.$ (4.23d)

We then expand the unknown functions near the horizon. Since the radial
dependence is of the form $R^{2n}$ where $n$ is some integer, the expansion
can be written as

$\displaystyle R^{2}u^{I}$
$\displaystyle=\sum_{n=1}^{\infty}u_{(n)}^{I}R^{2n}\leavevmode\nobreak\ ,$
(4.24a) $\displaystyle R^{2}g_{m}$
$\displaystyle=\sum_{n=1}^{\infty}g_{m,(n)}R^{2n}\leavevmode\nobreak\ ,$
(4.24b) $\displaystyle R^{2}w$
$\displaystyle=\sum_{n=0}^{\infty}w_{(n)}R^{2n}\leavevmode\nobreak\ ,$ (4.24c)
$\displaystyle R^{2}f^{-1}X_{I}$
$\displaystyle=\sum_{n=0}^{\infty}x_{I,(n)}R^{2n}\leavevmode\nobreak\ .$
(4.24d)

With the asymptotic structure of Table 1, the expansions for $R^{2}u^{I}$ and
$R^{2}g_{m}$ do not start with a constant term. Moreover, with the horizon
expansions above, the differential operators on the left hand sides of (4.23c)
and (4.23d) are such that the coefficients $w_{(2)}$ and $x_{(1)}$ drop out.
These coefficients are the zero-modes that make the initial value problem more
complicated. There are no analogous zero-modes for $u^{I}$ and $g_{m}$.

To study the structure of the attractor equations (4.23a-4.23d), we
temporarily treat the unknown scalar field $f^{-1}X_{I}$ as a given function
of the radial coordinate $R$. Then the linear flow equation (4.23a), which is
sourced by $f^{-1}X_{I}$, yields all the coefficients $u^{I}_{(n)}$ in terms
of $x_{I,(n)}$. At this point we know both of the functions $f^{-1}X_{I}$ and
$u^{I}$ and then the attractor flow equation (4.23b) similarly yields the
series coefficients $g_{m,(n)}$ in terms of $x_{I,(n)}$. Given all of
$f^{-1}X_{I}$, $u^{I}$, and $g_{m}$, it would seem straightforward to exploit
(4.23c) and find all the coefficients $w_{(n)}$ in terms of $x_{I,(n)}$. This
mostly works, but the zero-mode $w_{(2)}$ can not be determined this way. That
is the obstacle where, as advertized, the derivatives are such that an
expansion coefficient simply drops out.

The final flow equation (4.23d), due to the conserved charge $Q_{I}$ (4.19),
is crucial for the complete story. Assuming for a moment that the zero mode
$w_{(2)}$ is given as an initial condition, along with the entire function
$f^{-1}X_{I}$, this equation determines the expansion parameters $x_{I,(n)}$
in terms of the $x_{I,(n)}$ themselves, and so the entire system would appear
to be solved. However, this last equation also has a zero mode $x_{I,(1)}$
which cannot be determined by the iterative procedure. In short, a careful
analysis must be considered and accordingly, this is what we proceed to do
now: we solve the expansion coefficients order by order, following the
procedure we have outlined.

First, inserting the expansions (4.24a) and (4.24d) into the flow equations
(4.23a), we find

$\displaystyle\begin{split}\sum_{n=1}^{\infty}nu_{(n)}^{I}R^{2n-2}&=\frac{\epsilon}{2}\sum_{n=0}^{\infty}c^{IJK}\xi_{K}x_{J,(n)}R^{2n}\leavevmode\nobreak\
.\end{split}$ (4.25)

Comparing each order in $R^{2}$ leads to a relation between the coefficients
$u^{I}_{(n)}$ and $x_{I,(n)}$:

$\displaystyle\begin{split}u_{(n)}^{I}&=\frac{\epsilon}{2n}c^{IJK}\xi_{K}x_{J,(n-1)}\leavevmode\nobreak\
,\quad n\geq 1\leavevmode\nobreak\ .\end{split}$ (4.26)

We insert this result in the flow equation (4.23b) for the expansion of
$g_{m}$ (4.24b) and find

$\displaystyle\begin{split}\sum_{n=1}^{\infty}(n+1)g_{m,(n)}R^{2n-2}&=2+\sum_{n=1}^{\infty}\frac{1}{n}c^{IJK}\xi_{I}\xi_{J}x_{K,(n-1)}R^{2n-2}\leavevmode\nobreak\
\,.\end{split}$ (4.27)

Thus all expansion coeficients of $g_{m}$ can be expressed in terms of the
$x_{I,(n)}$:

$\displaystyle\begin{split}g_{m,(n)}=\frac{1}{n+1}\left(2\delta_{n,1}+\frac{1}{n}c^{IJK}\xi_{I}\xi_{J}x_{K,(n-1)}\right),\quad
n\geq 1\,.\end{split}$ (4.28)

The steps we have taken so far leads us to express the functions $u^{I}$ and
$g_{m}$ solely in terms of $f^{-1}X_{I}$. This was expected from the general
discussion.

We then start with (4.23c) and use the expansions (4.24a-4.24d) to obtain

$\begin{split}\sum_{n=0}^{\infty}(n-2)w_{(n)}R^{2n-2}=-\frac{1}{2}\sum_{n=0}^{\infty}\left(\sum_{k=0}^{n}(n-1-k)x_{I,(k)}u^{I}_{(n+1-k)}\right)R^{2n-2}\leavevmode\nobreak\
.\end{split}$ (4.29)

Comparing powers of $R^{2}$ we find

$\displaystyle(n-2)w_{(n)}=-\frac{\epsilon}{4}\sum_{k=0}^{n}\frac{n-1-k}{n+1-k}c^{IJK}\xi_{I}x_{J,(k)}x_{K,(n-k)}\leavevmode\nobreak\
.$ (4.30)

We see explicitly that the differential equation (4.23c) fails to express the
zero mode $w_{(2)}$ in the expansion (4.24c) in terms of other data. However,
the right hand side of (4.30) still reveals important information at $n=2$
since it imposes a constraint on the $x_{I,(n)}$ expansion coefficients

$\begin{split}0&=\frac{\epsilon}{3}c^{IJK}x_{I,(0)}x_{J,(2)}\xi_{K}\leavevmode\nobreak\
.\end{split}$ (4.31)

Thus determination of $w_{(2)}$ is replaced by a constraint on the functions
$f^{-1}X_{I}$ which we have considered given so far.

To make further progress it remains to study the constants of motion due to
the conservation (4.19) of electric charge and angular momentum. The electric
charge (4.23d) yields

$\begin{split}&-\sum_{n=1}^{\infty}\sum_{k=1}^{n}(n-1-k)g_{m,(k)}x_{I,(n-k)}R^{2n-2}\\\
&\quad\quad\quad=\widetilde{Q}_{I}+2c_{IJK}\sum_{n=1}^{\infty}\sum_{k=1}^{n}u^{J}_{(k)}u^{K}_{(n+1-k)}R^{2n-2}+2\epsilon\xi_{I}\sum_{n=0}^{\infty}w_{(n)}R^{2n}\leavevmode\nobreak\
.\end{split}$ (4.32)

Comparing each power of $R^{2}$ we find the sequence of relations

$-\sum_{k=1}^{n+1}(n-k)g_{m,(k)}x_{I,(n-k+1)}=\widetilde{Q}_{I}\delta_{n,0}+2c_{IJK}\sum_{k=1}^{n+1}u^{J}_{(k)}u^{K}_{(n+2-k)}+2\epsilon\xi_{I}w_{(n)}\leavevmode\nobreak\
,$ (4.33)

where it is understood in this relation that the coefficients $g_{m,(k)}$,
$u^{I}_{(k)}$, and $w_{(k\neq 2)}$ depend on the $x_{I,(k)}$ through (4.26),
(4.28), and (4.30). Thus the relations (4.33) constrain the given functions
$f^{-1}X_{I}$ significantly. Unfortunately, these constraints are nonlinear
and difficult to solve.

For the constant order in the $R^{2}$ expansion, we take $n=0$ in (4.33) and
find the electric charge

$\displaystyle\begin{split}\widetilde{Q}_{I}&=x_{I,(0)}\left(1+\tfrac{1}{2}c^{JKL}x_{J,(0)}\xi_{K}\xi_{L}\right)-\tfrac{1}{2}c_{IJK}c^{JML}c^{KNP}\xi_{M}\xi_{N}x_{L,(0)}x_{P,(0)}\\\
&\quad+\tfrac{1}{4}\xi_{I}c^{LMN}x_{L,(0)}x_{M,(0)}\xi_{N}\leavevmode\nobreak\
.\end{split}$ (4.34)

The charges $\widetilde{Q}_{I}$ depend only on the scalar fields at the
horizon $x_{I,(0)}$ and the FI-parameters $\xi_{I}$. In fact, if positivity
conditions are imposed on the $x_{I,(0)}$, the $x_{I,(0)}$ in (4.34) can be
inverted in terms of the $\widetilde{Q}_{I}$, allowing to replace the pair of
inputs $(\xi_{I},x_{I,(0)})$ by the pair $(\xi_{I},\widetilde{Q}_{I})$. This
fits nicely with the understanding of the physical inputs and charges that go
in defining the radial flow at every radial hypersurface.

To rewrite (4.34) in a more canonical form, we simplify the second term
involving a triple product of $c_{IJK}$ by contracting (B.3) with
$\xi_{M}\xi_{Q}x_{L,(0)}x_{P,(0)}$. This gives

$\widetilde{Q}_{I}=x_{I,(0)}-\frac{1}{2}\xi_{I}\left(\frac{1}{2}c^{JKL}x_{J,(0)}x_{K,(0)}\xi_{L}\right)+\frac{1}{2}c_{IJK}\left(\frac{1}{2}c^{JNO}\xi_{N}\xi_{O}\right)c^{KLM}x_{L,(0)}x_{M,(0)}\leavevmode\nobreak\
.$ (4.35)

This expression makes contact with the form of the charge given in
Gutowski:2004yv .444The equation agrees with $Q_{I}$ given in (3.53) of
Gutowski:2004yv  with the following map between notations:  $\displaystyle
q_{I}$ $\displaystyle=\frac{1}{3}x_{I,(0)},\
\bar{X}_{I}=\frac{1}{3}\ell\xi_{I}\leavevmode\nobreak\ ,\
\bar{X}^{I}=\frac{1}{2}\ell^{2}c^{IJK}\xi_{J}\xi_{K}\leavevmode\nobreak\ ,\
Q_{\text{there}}=\frac{\pi}{4G}\widetilde{Q}_{\text{here}}\leavevmode\nobreak\
.$ (4.36) The improvement in our work is that we introduce the charge
independently of the radial coordinate so it can be computed at any
hypersurface we choose, which — in this case — is the black hole horizon.

Before analyzing the consequences of electric charge conservation (4.33) for
$n\geq 1$, we consider the analogous equations due to conservation of the
black hole angular momentum $J$ (4.19). As the first line of (4.19) involves
the scalar field $X^{I}$, we must recast it in terms of the scalar field with
a lowered index, as our expansion (4.24d) dictates. Utilizing (B.6), we have

$\begin{split}f^{-3}fX^{I}&=\frac{1}{2}c^{IJK}(f^{-1}X_{J})(f^{-1}X_{K})\leavevmode\nobreak\
.\end{split}$ (4.37)

Introducing the near-horizon expansions (4.24a-4.24d) we find

$\begin{split}\widetilde{J}\delta_{n,0}&=\widetilde{Q}_{I}u^{I}_{(n+1)}+\frac{2}{3}c_{IJK}\sum^{n+1}_{k=1}\sum_{\ell=1}^{k}u^{I}_{(\ell)}u^{J}_{(k+1-\ell)}u^{K}_{(n+2-k)}-\sum_{k=0}^{n}\sum_{\ell=0}^{n-k}g_{m,(k+1)}x_{I,(\ell)}u^{I}_{(n+1-\ell-k)}\\\
&\quad-2\sum_{k=0}^{n}g_{m,(k+1)}w_{(n-k)}+\frac{\epsilon}{4}c^{IJK}\xi_{I}\sum_{k=0}^{n}\sum_{\ell=0}^{n-k}g_{m,(k+1)}x_{J,(\ell)}x_{K,(n-k-\ell)}\\\
&\quad+2w_{(n)}+2\epsilon\xi_{I}\sum_{k=0}^{n}w_{(k)}u_{(n-k+1)}^{I}\leavevmode\nobreak\
.\end{split}$ (4.38)

As before, it is understood that $g_{m,(k)}$, $u^{I}_{(k)}$, and $w_{(k)}$
depends on the $x_{I,(k)}$ according to (4.26), (4.28) and (4.30), and here we
also need the explicit form of $\widetilde{Q}_{I}$ (4.35). Thus angular
momentum conservation gives another infinite set of relations between the
$x_{I,(k)}$. Unfortunately, they are even more nonlinear than their analogues
for conservation of electric charge.

For $n=0$ (4.38) gives the angular momentum expressed in terms of $x_{I(0)}$
and $\xi_{I}$

$\begin{split}\widetilde{J}&=\frac{\epsilon}{4}c^{IJK}x_{I,(0)}x_{J,(0)}\xi_{K}-\frac{\epsilon}{4}c^{IJK}c^{LMN}\xi_{I}\xi_{N}\xi_{K}x_{J,(0)}x_{L,(0)}x_{M,(0)}\\\
&\quad+\epsilon
c_{IJK}c^{ILM}c^{JNO}c^{KPQ}\left(\tfrac{1}{8}x_{L,(0)}x_{P,(0)}x_{Q,(0)}\xi_{N}\xi_{O}\xi_{M}+\tfrac{1}{12}x_{N,(0)}x_{P,(0)}\xi_{M}\xi_{O}\xi_{Q}\right)\leavevmode\nobreak\
,\end{split}$ (4.39)

where we have used the value of $\widetilde{Q}_{I}$ given in (4.35). To make
contact with the form of the angular momentum in Gutowski:2004yv , we rewrite
the formula as555This agrees with the angular momentum reported in (3.50) of
Gutowski:2004yv  with the following map between conventions:  $\displaystyle
q_{I}$ $\displaystyle=\frac{1}{3}x_{I,(0)}\leavevmode\nobreak\ ,\
\bar{X}_{I}=\frac{1}{3}\ell\xi_{I}\leavevmode\nobreak\ ,\
J_{\text{there}}=\frac{\pi}{4G}\widetilde{J}_{\text{here}}\leavevmode\nobreak\
.$ (4.40)

$\widetilde{J}=\frac{\epsilon}{4}c^{IJK}x_{I,(0)}x_{J,(0)}\xi_{K}+\frac{1}{36}\left(c^{IJK}\xi_{I}\xi_{J}\xi_{K}\right)\left(c^{LMN}x_{L,(0)}x_{M,(0)}x_{N,(0)}\right)\leavevmode\nobreak\
.$ (4.41)

Again, we are able to express the final result for the conserved charge
entirely in terms of near horizon data. Moreover, since $\widetilde{Q}_{I}$
and $\widetilde{J}$ depend on the same integration constants $x_{I,(0)}$, the
charges are indeed not independent of each other.

We now turn to the $n=1$ component of (4.33), i.e., electric charge
conservation at order $R^{2}$ away from the horizon. It amounts to

$g_{m,(2)}x_{I,(0)}=4c_{IJK}u^{J}_{(1)}u^{K}_{(2)}+2\epsilon\xi_{I}w_{(1)}\leavevmode\nobreak\
.$ (4.42)

The absence of $x_{I,(1)}$ in this equation is due to the zero-mode in
(4.24d). However, a constraint on the $x_{I,(1)}$ will follow, in analogy with
the zero-mode $w_{(2)}$ giving the condition (4.31). The values of
$g_{m,(2)}$, $u^{I}_{(1)}$, $u^{I}_{(2)}$, and $w_{(1)}$ from (4.26), (4.28)
and (4.30) give the vector relation

$\left[\frac{1}{6}c^{JKL}\xi_{K}\xi_{L}x_{I,(0)}-\frac{1}{2}c_{IKM}c^{JLM}c^{KNP}\xi_{L}\xi_{N}x_{P,(0)}+\frac{1}{2}c^{JKL}\xi_{K}x_{L,(0)}\xi_{I}\right]x_{J,(1)}=0\leavevmode\nobreak\
.$ (4.43)

We see that $x_{I,(1)}$ is constrained even though it is a zero-mode of the
differential operator. Using the cubic condition on the $c_{IJK}$ (B.4), we
can show that the matrix in square brackets has the null vector

$x_{I,(1)}=\ell\xi_{I}\leavevmode\nobreak\ .$ (4.44)

It is unique, at least for generic structure constants $c_{IJK}$ and generic
charges, which are parametrized by $x_{I,(0)}$. The constraint (4.43) does not
determine the overall normalization. However, the scale of the radial
coordinate $R^{2}$ is arbitrary from the near horizon point of view, so the
choice (4.44) involves no loss of generality.

The $n=2$ component of (4.33) gives another vector-valued relation

$T_{I}^{J}x_{J,(2)}=2\epsilon\left(w_{(2)}+\frac{\epsilon}{2\ell}\right)\xi_{I}\leavevmode\nobreak\
,$ (4.45)

where we have simplified using (4.44) and

$T_{I}^{J}=-\left(1+\frac{1}{2}c^{KLM}\xi_{K}\xi_{L}x_{M,(0)}\right)\delta_{I}^{J}+\frac{1}{12}c^{JKL}\xi_{K}\xi_{L}x_{I,(0)}-\frac{1}{3}c_{IKL}c^{KMJ}c^{LNP}\xi_{M}\xi_{N}x_{P,(0)}\leavevmode\nobreak\
.$ (4.46)

The matrix $T_{I}^{J}$ is invertible so, given the inputs
$\xi_{I},x_{I,(0)},w_{(2)}$, the coefficient $x_{J,(2)}$ is completely
determined by (4.45). However, the value of $x_{J,(2)}$ computed this way
fails to satisfy the previously established constraint (4.31). This apparent
contradiction can be avoided only if

$x_{I,(2)}=0\leavevmode\nobreak\ .$ (4.47)

Because the left hand side of (4.45) vanishes, the right side requires

$w_{(2)}=-\frac{\epsilon}{2\ell}\leavevmode\nobreak\ .$ (4.48)

At this point, we can finally consider generic components of the electric
charge conservation (4.33), i.e. the infinite set of equations $n\geq 3$. The
coefficients $w_{(n\geq 3)}$ can be eliminated using (4.30) for $w_{(n)}$. The
$u^{I}_{(n)}$ and $g_{m,(n)}$ are similarly traded for $x_{I,(n)}$, this time
using (4.26) and (4.28). For all $n\geq 3$ this gives

$\begin{split}&-\sum_{k=0}^{n}\frac{n-1-k}{k+2}\left(2\delta_{0,k}+\frac{1}{k+1}c^{JKL}\xi_{J}\xi_{K}x_{L,(k)}\right)x_{I,(n-k)}\\\
&=\frac{1}{2}c_{IJK}c^{JLM}c^{KNP}\xi_{L}\xi_{N}\sum_{k=0}^{n}\frac{1}{(k+1)(n-k+1)}x_{M,(k)}x_{P,(n-k)}\\\
&\quad-\frac{1}{2(n-2)}\xi_{I}\sum_{k=0}^{n}\frac{n-1-k}{n+1-k}c^{JKL}\xi_{J}x_{K,(k)}x_{L,(n-k)}\leavevmode\nobreak\
.\end{split}$ (4.49)

This messy expression can be reorganized as a recurrence relation giving
$x_{I,(n)}$ in terms of the preceding $x_{I,(0\leq k\leq n-1)}$:

$\begin{split}&\left[-\frac{n-1}{2}\left(2+c^{KLM}\xi_{K}\xi_{L}x_{M,(0)}\right)\delta^{J}_{I}+\frac{1}{(n+1)(n+2)}c^{JKL}\xi_{K}\xi_{L}x_{I,(0)}\right.\\\
&\left.-\frac{1}{n+1}c_{IKL}c^{KMJ}c^{LPQ}\xi_{M}\xi_{P}x_{Q,(0)}-\frac{1}{(n-2)(n+1)}c^{JKL}\xi_{K}x_{L,(0)}\xi_{I}\right]x_{J,(n)}\\\
&=\sum_{k=1}^{n-1}\left[\frac{(n-1-k)}{(k+1)(k+2)}c^{JKL}\xi_{J}\xi_{K}x_{L,(k)}x_{I,(n-k)}+\frac{1}{2}c_{IJK}c^{JLM}c^{KNP}\xi_{L}\xi_{N}x_{M,(k)}x_{P,(n-k)}\right.\\\
&\left.\quad-\frac{1}{2(n-2)}\xi_{I}\frac{n-1-k}{n+1-k}c^{JKL}\xi_{J}x_{K,(k)}x_{L,(n-k)}\right]\leavevmode\nobreak\
.\end{split}$ (4.50)

The left hand side can be inverted, at least for some specific models of
$c_{IJK}$, such as the STU model ($c_{IJK}=|\epsilon_{IJK}|$ for $I,J,K$
running from 1 to 3). In such cases the recurrence relation
(LABEL:eq:_nh_x_recursion) determines all higher-order $x_{I,(n\geq 3)}$ in
terms of the coefficients $x_{I,(0)}$, $x_{I,(1)}$ and $x_{I,(2)}$ as well as
the FI-parameters $\xi_{I}$. In fact, the constraints (4.44) and (4.47) from
low $n$ will be sufficient to show that the series truncates at $n=2$. This is
discussed in subsection 4.4.4.

At this point we have exhausted the information that comes from the
conservation of electric charge $\widetilde{Q}_{I}$ charge in (4.33). We did
not yet study the $\widetilde{J}$ conservation relations (4.38). As noted
already, the constant order $n=0$ determines the angular momentum from a near
horizon perspective. We have worked out the first few orders $n\geq 1$ and
found either redundant relations, involving already known coefficients such as
$x_{I,(0)}$, $\xi_{I}$ and $x_{I,(1)}$, or relations that tie together higher
order $x_{I,(k\geq 2)}$ with lower-order ones. We do not foresee any further
constraints due to the $\widetilde{J}$ relations.

In summary, starting from the near horizon region, we have exploited
supersymmetry and found the entire black hole solution. The fields
$u^{I},g_{m},w$ and $f^{-1}X_{I}$ are reported in (4.26), (4.28), (4.30) and
(4.45). Additionally, we computed the electric charges $\widetilde{Q}_{I}$ and
the angular momenta in terms of the horizon values of the scalars $x_{I,(0)}$,
and the subleading coefficients $x_{I,(1)}$ which, according to (4.44),
coincide with the FI-parameters $\xi_{I}$.

####  4.4.3 Perturbative solution starting from asymptotic AdS

We now adapt the approach from the previous subsection and expand the unknown
functions $u^{I}$, $g_{m}$, $w$ and $f^{-1}X_{I}$ at large $R$, near the
asymptotic AdS5 boundary.

Given the asymptotic behaviors listed in Table 1, regularity requires taking
$\beta=-1$ in (4.22). We then recast the flow equations (4.20a–4.20d) as

$\displaystyle\left(\partial_{R^{2}}+\frac{2}{R^{2}}\right)(R^{-2}u^{I})$
$\displaystyle=\frac{1}{2}\epsilon
c^{IJK}(R^{-2}f^{-1}X_{J})\xi_{K}\leavevmode\nobreak\ ,$ (4.51a)
$\displaystyle\left(\partial_{R^{2}}+\frac{3}{R^{2}}\right)(R^{-2}g_{m})$
$\displaystyle=\frac{2}{R^{4}}+\frac{2}{R^{2}}\epsilon\xi_{I}(R^{-2}u^{I})\leavevmode\nobreak\
.$ (4.51b) $\displaystyle\partial_{R^{2}}(R^{-2}w)$
$\displaystyle=-\frac{1}{2}R^{2}(R^{-2}f^{-1}X_{I})\partial_{R^{2}}(R^{-2}u^{I})\leavevmode\nobreak\
,$ (4.51c)
$\displaystyle\begin{split}\left(\partial_{R^{2}}+\frac{1}{R^{2}}\right)(R^{-2}f^{-1}X_{I})&=-\frac{R^{-8}}{(R^{-2}g_{m})}\left(\widetilde{Q}_{I}+2c_{IJK}R^{4}(R^{-2}u^{J})(R^{-2}u^{K})\right.\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\left.+2\epsilon
R^{4}(R^{-2}w)\xi_{I}\right)\leavevmode\nobreak\ \,.\end{split}$ (4.51d)

We define the perturbative expansions at infinity as

$\displaystyle R^{-2}u^{I}$
$\displaystyle=\sum_{n=0}^{\infty}\bar{u}_{(n)}^{I}R^{-2n}\leavevmode\nobreak\
,$ (4.52a) $\displaystyle R^{-2}g_{m}$
$\displaystyle=\sum_{n=0}^{\infty}\bar{g}_{m,(n)}R^{-2n}\leavevmode\nobreak\
\,,$ (4.52b) $\displaystyle R^{-2}w$
$\displaystyle=\sum_{n=0}^{\infty}\bar{w}_{(n)}R^{-2n}\leavevmode\nobreak\ ,$
(4.52c) $\displaystyle R^{-2}f^{-1}X_{I}$
$\displaystyle=\sum_{n=1}^{\infty}\bar{x}_{I,(n)}R^{-2n}\leavevmode\nobreak\
.$ (4.52d)

The bar distinguishes the expansion coefficients at the asymptotically AdS5
boundary from their analogues at the horizon.

As before, we initially specify the entire series $\bar{x}_{I,(n)}$.
Additionally, examination of (4.51a–4.51d) shows that $\bar{u}_{(2)}^{I}$,
$\bar{g}_{m,(3)}$, $\bar{w}_{(0)}$, and $\bar{x}_{I,(1)}$ do not appear on the
left hand sides of the equations. These are the zero modes that we also regard
as inputs, at least provisionally. Among the zero-modes, we can determine
$\bar{x}_{I,(1)}$ from the outset because they give the asymptotic values of
the scalars

$\displaystyle\bar{x}_{I,(1)}=\ell\xi_{I}\,,$ (4.53)

as we found in (2.13), by extremizing the potential of gauged supergravity.

We now proceed to solve for the expansion coefficients of each variable, order
by order. Starting with the $u^{I}$ flow equation (4.20a), and using the
expansions (4.52a) and (4.52d), we find

$\displaystyle\begin{split}\sum_{n=0}^{\infty}(2-n)\bar{u}_{(n)}^{I}R^{-2n-2}&=\frac{1}{2}\epsilon
c^{IJK}\xi_{J}\sum_{n=1}^{\infty}\bar{x}_{K,(n+1)}R^{-2n-2}\leavevmode\nobreak\
\,.\end{split}$ (4.54)

Comparing inverse powers of $R^{2}$, we find $\bar{u}_{(n)}^{I}$ for $n\neq
2$:

$\displaystyle\begin{split}(2-n)\bar{u}_{(n)}^{I}&=\frac{1}{2}\epsilon
c^{IJK}\xi_{J}\bar{x}_{K,(n+1)}\leavevmode\nobreak\ ,\quad n\geq
0\,.\end{split}$ (4.55)

The zero mode $\bar{u}_{(2)}^{I}$ drops out of the equation. Instead, we find
a vectorial constraint on $x_{I,(3)}$

$c^{IJK}\xi_{J}\bar{x}_{K,(3)}=0\leavevmode\nobreak\ \,.$ (4.56)

It has the obvious solution

$\displaystyle\bar{x}_{I,(3)}=0\leavevmode\nobreak\ ,$ (4.57)

for all values of $I$. This solution is unique if the matrix $c^{IJK}\xi_{J}$
is non-singular. In (B.21), we show that it is indeed invertible

$(c^{IJK}\xi_{J})^{-1}=\frac{1}{2}\ell^{3}\left(c_{IJK}c^{JLM}\xi_{L}\xi_{M}-\xi_{I}\xi_{K}\right)\leavevmode\nobreak\
\,.$ (4.58)

Next, we consider the $g_{m}$ flow equation (4.20b). The expansions (4.52a)
and (4.52b) give

$\displaystyle\sum_{n=0}^{\infty}(3-n)\bar{g}_{m,(n)}R^{-2n-2}$
$\displaystyle=2R^{-4}+2\epsilon\xi_{I}\sum_{n=0}^{\infty}\bar{u}_{(n)}^{I}R^{-2n-2}\,.$
(4.59)

The expansion coefficients $\bar{g}_{m,(n)}$ — with the exception of the zero
mode $g_{m,(3)}$ — can be expressed in terms of $x_{I,(n)}$ and the zero mode
$u^{I}_{(2)}$ as

$(3-n)\bar{g}_{m,(n)}=\left\\{\begin{array}[]{cc}2\delta_{1,n}+\frac{1}{2-n}c^{IJK}\xi_{I}\xi_{J}\bar{x}_{K,(n+1)}&n\neq
2,\\\ 2\epsilon\xi_{I}\bar{u}_{(2)}^{I}&\leavevmode\nobreak\
n=2\,.\end{array}\right.$ (4.60)

In compensation for not determining $\bar{g}_{m,(3)}$, we find the constraint
$\xi_{I}\bar{u}_{(3)}^{I}=0$. Rewriting this constraint using (4.55) gives a
projection on $x_{I,(4)}$

$c^{IJK}\xi_{I}\xi_{J}\bar{x}_{K,(4)}=0\leavevmode\nobreak\ .$ (4.61)

This constraint is a real special geometry scalar, unlike the vector-valued
condition (4.56). It will nevertheless prove useful when simplifying results
at large $R$.

We now turn to the nonlinear flow equation for $w$ (4.20c). After using the
expansions (4.52a), (4.52c) and (4.52d), we find

$\displaystyle\sum_{n=1}^{\infty}(n-1)\bar{w}_{(n-1)}R^{-2n}$
$\displaystyle=-\frac{1}{2}\sum_{n=0}^{\infty}\sum_{k=1}^{n}(n-k)\bar{x}_{I,(k)}\bar{u}^{I}_{(n-k)}R^{-2n}\leavevmode\nobreak\
,$ (4.62)

and so

$(n-1)\bar{w}_{(n-1)}=-\frac{1}{2}\sum_{k=1}^{n}(n-k)\bar{x}_{I,(k)}\bar{u}^{I}_{(n-k)}\leavevmode\nobreak\
,\qquad n\geq 1\,.$ (4.63)

For $n=1$, the left hand side vanishes, so the zero-mode $\bar{w}_{(0)}$ is
undetermined. The right hand side also vanishes for $n=1$ so in this case the
equation with a zero-mode offers no additional information. We omit the $n=1$
case and rewrite (4.63) to

$\displaystyle\bar{w}_{(n)}$
$\displaystyle=-\frac{1}{2n}\sum_{k=1}^{n}(n+1-k)\bar{x}_{I,(k)}\bar{u}^{I}_{(n+1-k)}\leavevmode\nobreak\
,\qquad n\geq 1\leavevmode\nobreak\ .$ (4.64)

We have refrained from eliminating $\bar{u}^{I}_{(k)}$ in favor of
$\bar{x}_{I,(n)}$ via (4.55) because, generally, the equation involves the
zero mode $\bar{u}^{I}_{(2)}$ which cannot be removed this way.

The final flow equation (4.51d) was derived by combining supersymmetry with
conservation of electric charge. Using the expansions (4.52a–4.52d), we find

$\begin{split}&R^{-4}\sum_{n=0}^{\infty}\sum_{k=0}^{n}\bar{g}_{m,(k)}(n-k)\bar{x}_{I,(n+1-k)}R^{-2n}\\\
&=R^{-4}\widetilde{Q}_{I}\delta_{n,2}+2c_{IJK}R^{-4}\sum_{n=0}^{\infty}\sum_{k=0}^{n}\bar{u}^{J}_{(k)}\bar{u}^{K}_{(n-k)}R^{-2n}+2\epsilon\xi_{I}R^{-4}\sum_{n=0}^{\infty}\bar{w}_{(n)}R^{-2n}\leavevmode\nobreak\
.\end{split}$ (4.65)

For all $n\geq 0$, this gives

$\sum_{k=0}^{n}\bar{g}_{m,(k)}(n-k)\bar{x}_{I,(n+1-k)}=\widetilde{Q}_{I}\delta_{n,2}+2c_{IJK}\sum_{k=0}^{n}\bar{u}^{J}_{(k)}\bar{u}^{K}_{(n-k)}+2\epsilon\xi_{I}\bar{w}_{(n)}\leavevmode\nobreak\
\,.$ (4.66)

Again, we have chosen to maintain (4.66) as implicit functions of
$\bar{x}_{I,(n)}$ due to the presence of the zero modes.

It is worth examining the first few orders of (4.66) in detail. The $n=0$
component of (4.66) gives

$\begin{split}\xi_{I}\bar{w}_{(0)}=-\epsilon
c_{IJK}\bar{u}^{J}_{(0)}\bar{u}^{K}_{(0)}=-\frac{\epsilon}{16}c_{IJK}c^{JLM}c^{KNP}\bar{x}_{L,(1)}\xi_{M}\bar{x}_{N,(1)}\xi_{P}=-\frac{\epsilon}{2\ell}\xi_{I}\leavevmode\nobreak\
,\end{split}$ (4.67)

where we have used (4.53). Thus, it provides the value of the zero mode
$w_{(0)}$

$\displaystyle\bar{w}_{(0)}=-\frac{\epsilon}{2\ell}\,.$ (4.68)

At the next order, the $n=1$ component of (4.66) is redundant as it just
confirms the value for $\bar{w}_{(1)}$ already obtained from (4.64).

The $n=2$ component of (4.66) is particularly important, because it relates
the electric charge to the expansion parameters at infinity

$\begin{split}\widetilde{Q}_{I}&=\bar{x}_{I,(2)}-\frac{1}{2}\xi_{I}\left(\frac{1}{2}c^{JKL}\bar{x}_{J,(2)}\bar{x}_{K,(2)}\xi_{L}\right)+\frac{1}{2}c_{IJK}\left(\frac{1}{2}c^{JNO}\xi_{N}\xi_{O}\right)c^{KLM}\bar{x}_{L,(2)}\bar{x}_{M,(2)}\\\
&\quad+\epsilon\ell\left(\xi_{I}\xi_{J}-c_{IJK}c^{KLM}\xi_{L}\xi_{M}\right)\bar{u}^{J}_{(2)}\,,\end{split}$
(4.69)

where we imposed (4.53) and (4.57) and recast the charge in a form similar to
(4.35). Since the electric charge is conserved, the expression (4.69) for
$\widetilde{Q}_{I}$, written in terms of the expansion parameters at infinity,
must be equal to its analogue (4.35) obtained from expansion near the horizon.

We have established that $\widetilde{Q}_{I}$ at infinity has been determined
with the only inputs necessary being the coefficients $\bar{x}_{I,(2)}$,
$\xi_{I}$ and $\bar{u}^{I}_{(2)}$. We now move on to the components of (4.66)
for $n\geq 3$, to establish the recursion relation for the coefficients at
infinity.

For $n=3$, (4.66) simplifies after eliminating the $g_{m}$, $u^{I}$ and $w$
coefficients using (4.55), (4.60) and (4.64) to

$4\ell^{-2}\bar{x}_{I,(4)}+2\epsilon\left(\bar{x}_{I,(2)}\xi_{J}+\frac{1}{3}\xi_{I}\bar{x}_{J,(2)}-c_{IJK}c^{KLM}\xi_{L}\bar{x}_{m,(2)}\right)\bar{u}^{J}_{(2)}=0\leavevmode\nobreak\
.$ (4.70)

This relation indicates that $\bar{x}_{I,(4)}$ is described only with the help
of $\bar{x}_{I,(2)}$, $\xi_{I}$ and $\bar{u}^{I}_{(2)}$.

Furthermore, we note that for $n\geq 4$, the simplification of (4.66) yields a
generalization of (4.70)

$\displaystyle\begin{split}&\frac{(n-1)^{2}}{n-2}\ell^{-2}\bar{x}_{I,(n+1)}\\\
&=-(n-1)\left(1+\tfrac{1}{2}(c\cdot\xi\xi\bar{x}_{(2)})\right)\bar{x}_{I,(n)}-2\epsilon(n-2)\xi_{J}\bar{u}^{J}_{(2)}\bar{x}_{I,(n-1)}-(n-3)\bar{g}_{m,(3)}\bar{x}_{I,(n-2)}\\\
&\quad-\sum_{k=4}^{n}\frac{n-k}{(k-2)(k-3)}(c\cdot\xi\xi\bar{x}_{(k+1)})\bar{x}_{I,(n+1-k)}+4c_{IJK}\bar{u}^{J}_{(2)}\bar{u}^{K}_{(n-2)}\\\
&\quad+2c_{IJK}c^{JLM}c^{KNP}\xi_{L}\xi_{N}\sideset{}{{}^{\prime}}{\sum}_{\begin{subarray}{c}k=1\end{subarray}}^{n-1}\frac{\bar{x}_{M,(k+1)}\bar{x}_{P,(n-k+1)}}{4(k-2)(n-k-2)}+\xi_{I}\sideset{}{{}^{\prime}}{\sum}_{\begin{subarray}{c}k=2\end{subarray}}^{n}\frac{n+1-k}{n-1-k}\frac{(c\cdot\xi\bar{x}_{(k)}\bar{x}_{(n-k+2)})}{2n}\,,\end{split}$
(4.71)

where the apostrophes on the summation symbols indicate that we exclude the
terms in the sum with vanishing denominators. We have imposed the value of
$\bar{x}_{I,(1)}$ as given in (4.53) and products of the form $c\cdot xyz$
indicate special geometry contractions under $c^{JKL}$ of the form
$c^{JKL}x_{J}y_{K}z_{L}$. The expression (4.71) has been expanded to
distinguish contributions coming from $\bar{x}_{I,(n+1)}$, given by the left
hand side of (4.71), and $\bar{x}_{I,(2\leq k\leq n)}$, given by the right
hand side of the equality in (4.71). It becomes clear that a given
$\bar{x}_{I,(n+1)}$ depends only on the expansion parameters
$\bar{x}_{I,(1)},\bar{x}_{I,(2)},\dots,\bar{x}_{I,(n)}$. By recursion, i.e.
applying (4.71) repeatedly, we can now determine all $\bar{x}_{I,(4\leq k\leq
n)}$ in terms of of $\bar{x}_{I,(2)}$, $\xi_{I}$, $\bar{u}^{I}_{(2)}$ and
$\bar{g}_{m,(3)}$.

Lastly, we analyse the conservation of angular momentum $\widetilde{J}$ by
expanding the first equation in (4.19) at infinity, making sure to rescale the
functions $u^{I}$, $g_{m}$, $w$ and $f^{-1}X_{I}$ appropriately

$\begin{split}\widetilde{J}&=R^{2}Q_{I}(R^{-2}u^{I})+\tfrac{2}{3}R^{6}c_{IJK}(R^{-2}u^{I})(R^{-2}u^{J})(R^{-2}u^{K})-R^{8}(R^{-2}g_{m})\left(2R^{-2}(R^{-2}w)\right.\\\
&\quad\left.+(R^{-2}f^{-1}X)\cdot(R^{-2}u)-\tfrac{1}{2}\epsilon
R^{-2}f^{-3}\xi\cdot fX)\right)+2R^{4}(R^{-2}w)(1+\epsilon
R^{2}\xi\cdot(R^{-2}u))\leavevmode\nobreak\ .\end{split}$ (4.72)

This can be expanded like (4.38) in terms of the expansions at infinity
(4.52a-4.52b), leading to a relation for the component of $R^{-2n}$. At
constant order ($n=0$), we obtain

$\begin{split}\widetilde{J}&=\frac{\epsilon}{4}c^{IJK}\bar{x}_{I,(2)}\bar{x}_{J,(2)}\xi_{K}+\frac{1}{36}\left(c^{IJK}\xi_{I}\xi_{J}\xi_{K}\right)\left(c^{LMN}\bar{x}_{L,(2)}\bar{x}_{M,(2)}\bar{x}_{N,(2)}\right)+\frac{\epsilon}{\ell}\bar{g}_{m,(3)}\\\
&\quad+\ell\left(\frac{1}{2}\xi_{I}c^{JKL}\xi_{J}\xi_{K}\bar{x}_{L,(2)}+\xi_{I}+\frac{5}{3}\ell^{-3}\bar{x}_{I,(2)}\right)\bar{u}^{I}_{(2)}\leavevmode\nobreak\
,\end{split}$ (4.73)

where we have imposed (4.53) and (4.57). This expression at most depends on
the inputs $\bar{x}_{I,(2)}$, $\xi_{I}$, $\bar{g}_{m,(3)}$ and
$\bar{u}^{I}_{(2)}$. All higher order powers of the $\widetilde{J}$ relation
at infinity in (4.72) are redundant because they yield relations between the
coefficients that have already been established.

In summary, we have studied the first order equations (4.51a-4.51d) due to
supersymmetry and conservation of electric charge, by expanding perturbatively
near infinity. Given the asymptotic values of the scalars fields (4.53), as
well as the conserved charges $\widetilde{Q}_{I}$, $\widetilde{J}$ defined by
fall-off conditions at infinity, the simplest outcome would have been for
supersymmetry to determine the entire black hole geometry. Our finding is much
more complicated: all physical fields can be expressed as a perturbative
series with expansion parameters that depend not only on $\xi_{I}$ and
$\bar{x}_{I,(2)}$ but also the zero-modes $\bar{u}^{I}_{(2)}$ and
$\bar{g}_{m,(3)}$.

####  4.4.4 Summary and discussion of perturbative solutions

The study in the subsection so far focused on technical details. This was
needed because the interplay between supersymmetry, boundary conditions, and
conserved charges proved to be rather intricate. We now conclude the
subsection with a summary of the final results and discussion of their
interpretation.

The black hole solution is parametrized primarily by the matter fields: scalar
fields $f^{-1}X_{I}$, with the prefactor $f$ such that the combination
$f^{-1}X_{I}$ is unconstrained by real special geometry, and the magnetic
potentials $u^{I}$. Because of supersymmetry, the electric potentials $fY^{I}$
can be identified with the scalar fields $fX^{I}$, see (C.49). Given the
matter fields, $f^{-1}X_{I}$ and $u^{I}$, as well as supersymmetry, the
geometry is specified by a Kähler base that depends on the function $g_{m}$,
and a fibre encoding rotation through the potential $w$. All unknown functions
$f^{-1}X_{I}$, $u_{I}$, $g_{m}$ and $w$ can depend only on a single radial
coordinate $R^{2}$ and they must satisfy specified first order differential
equations (4.20a-4.20d).

Supersymmetry is never sufficient to specify an entire solution, because it is
first order, and there is always an integrability condition that is of second
order. Taking into account the Noether-Wald procedure, we find a second order
constraint that satisfies a Gauss’ law that was subject to detailed discussion
in section 3. With this augmentation, the first order differential equations
form a complete system. Angular momentum, with its conservation law also
discussed in section 3, yields nothing new, except for a formula giving the
angular momentum in terms of the same parameters that define the electric
charge in the near-horizon expansion. In the case of the asymptotic infinity
expansion, the electric charge depends on one of the zero modes
$\bar{u}^{I}_{(2)}$ in addition to $\bar{x}_{I,(2)}$, $\xi_{I}$ whereas the
angular momentum depends on both the zero modes $\bar{g}_{m,(3)}$ and
$\bar{u}^{I}_{(2)}$ as well as $\bar{x}_{I,(2)}$ and $\xi_{I}$.

Consistent boundary conditions for the differential equations can be specified
at any radius, in principle. They must depend, at the very least, on the FI-
coupling constants $\xi_{I}$ and the electric charges $Q_{I}$. We find that,
when starting from the black hole horizon, this data is sufficient. Because of
supersymmetry, these parameters specify the entire near horizon geometry,
including the squashing of the horizon due to angular momentum. This explains
why the electric charge and the angular momentum are only described with the
use of the leading $x_{I,(0)}$ term in $f^{-1}X_{I}$, but any further
subleading information about any of the fields $u^{I}$, $g_{m}$ or $w$
requires subleading contributions away from the horizon, with derivative
information of the $f^{-1}X_{I}$ expansion (4.24d) supplied by the $x_{I,(1)}$
coefficients.

The linchpin for establishing this claim about the near horizon expansion is
the scalar field. In the series expansion for $R^{2}f^{-1}X_{I}$, we have the
constant at the horizon $x_{I,(0)}$ and then at ${\cal O}(R^{2})$, we have
$x_{I,(1)}$. For the third expansion coefficent, we find $x_{I,(2)}=0$ (4.47).
With this starting point, the recursion relation (LABEL:eq:_nh_x_recursion)
shows that all $x_{I,(k\geq 2)}$ actually vanish. The fact that the scalar
field $f^{-1}X_{I}$ truncates after the first two terms is the near horizon
version of the fact that $f^{-1}X_{I}$ is a harmonic function, as is familiar
from ungauged supergratity.

When analyzing the supersymmetry conditions we provisionally considered
$f^{-1}X_{I}$ an input that all other variables were expressed in terms of.
The truncation $x_{I,(k\geq 2)}=0$ has the immediate effect of truncating
$u^{I}_{(n\geq 3)}=g_{m,(n\geq 3)}=w_{(n\geq 3)}=0$. The expansions at the
horizon (4.24a-4.24d) simplify and we find

$\displaystyle R^{2}u^{I}$
$\displaystyle=\frac{\epsilon}{2}c^{IJK}\xi_{J}x_{K,(0)}R^{2}+\frac{\epsilon\ell}{4}c^{IJK}\xi_{J}\xi_{K}R^{4}\,,$
(4.74a) $\displaystyle R^{2}g_{m}$
$\displaystyle=\left(1+\frac{1}{2}c^{IJK}\xi_{I}\xi_{J}x_{K,(0)}\right)R^{2}+\frac{1}{\ell^{2}}R^{4}\,,$
(4.74b) $\displaystyle R^{2}w$
$\displaystyle=-\frac{\epsilon}{8}c^{IJK}\xi_{I}x_{J,(0)}x_{K,(0)}-\frac{\epsilon\ell}{4}c^{IJK}\xi_{I}\xi_{J}x_{K,(0)}R^{2}-\frac{\epsilon}{2\ell}R^{4}\,,$
(4.74c) $\displaystyle R^{2}f^{-1}X_{I}$
$\displaystyle=x_{I,(0)}+\ell\xi_{I}R^{2}.$ (4.74d)

These expressions exactly match the well-known Gutowski-Reall solution
Gutowski:2004yv , with the appropriate identifications of notation666$u^{I}$,
$g_{m}$, $w$ and $f^{-1}X_{I}$ match $U^{I}$, $g$, $w$ and $f^{-1}X_{I}$
respectively in Gutowski:2004yv  via:  $q_{I}=\frac{1}{3}\bar{x}_{I,(2)}\ ,\
\bar{X}_{I}=\frac{1}{3}\ell\xi_{I}\leavevmode\nobreak\ .$ (4.75) . The
electric charge $\widetilde{Q}_{I}$ (4.35) and the angular momentum
$\widetilde{J}$ (4.39) computed in the near horizon expansion similarly agree
with the familiar results.

The analogous analysis starting from asymptotic AdS5 turned out to be less
straightforward. Recalling that coefficients starting from infinity are
denoted by barred expansion coefficients, we find that, when shooting in
(going from infinity towards the horizon), we must not only specify $\xi_{I}$
and $\bar{x}_{I,(2)}$, but also $\bar{u}_{I,(2)}$ and $\bar{g}_{m,(3)}$. The
harmonic function we established at the horizon reproduces the Gutowski-Reall
solution and has features in common with their very familiar analogues in
ungauged supergravity. At infinity, it corresponds to

$x_{I,(0)}=\bar{x}_{I,(2)}\leavevmode\nobreak\ ,\leavevmode\nobreak\
\leavevmode\nobreak\ \bar{u}^{I}_{(2)}=\bar{g}_{m,(3)}=0\leavevmode\nobreak\
.$ (4.76)

With these special values, the recursion relation (4.71) simplifies greatly

$\displaystyle\begin{split}&\frac{(n-1)^{2}}{n-2}\ell^{-2}\bar{x}_{I,(n+1)}\\\
&=-(n-1)\left(1+\tfrac{1}{2}(c\cdot\xi\xi\bar{x}_{(2)})\right)\bar{x}_{I,(n)}-\sum_{k=4}^{n}\frac{n-k}{(k-2)(k-3)}(c\cdot\xi\xi\bar{x}_{(k+1)})\bar{x}_{I,(n+1-k)}\\\
&\quad+2c_{IJK}c^{JLM}c^{KNP}\xi_{L}\xi_{N}\sideset{}{{}^{\prime}}{\sum}_{\begin{subarray}{c}k=1\end{subarray}}^{n-1}\frac{\bar{x}_{M,(k+1)}\bar{x}_{P,(n-k+1)}}{4(k-2)(n-k-2)}+\xi_{I}\sideset{}{{}^{\prime}}{\sum}_{\begin{subarray}{c}k=2\end{subarray}}^{n}\frac{n+1-k}{n-1-k}\frac{(c\cdot\xi\bar{x}_{(k)}\bar{x}_{(n-k+2)})}{2n}\leavevmode\nobreak\
.\end{split}$ (4.77)

Since we already know $\bar{x}_{I,(3)}=0$ from (4.57), and vanishing
$\bar{u}^{I}_{(2)}$ leads to vanishing $\bar{x}_{I,(4)}$ as well via (4.61),
it is not difficult to show that the expansion coefficients $\bar{x}_{I,(k\geq
3)}$ all vanish. Thus the perturbative series for $\bar{x}_{I,(n)}$ truncates
after two terms, as expected for a harmonic function. The identification
(4.76) identifies the subleading coefficient in the harmonic function at
infinity with the leading one at the horizon, and vice versa.

While (4.76) are the default, it is interesting that asymptotic boundary
conditions with nonzero $\bar{u}_{I,(2)}$, $\bar{g}_{m,(3)}$ are consistent
with supersymmetry. It has been argued that there may be missing solutions in
certain supergravity theories that may not satisfy the canonical nonlinear
charge constraint, see for example Bhattacharyya:2010yg ; Markeviciute:2016ivy
; Markeviciute:2018yal ; Dias:2022eyq . Since the value of the conserved
charges do not take the canonical form, one way wonder if those parameters are
somehow related to these missing solutions.

From this point of view, the possibility of $\bar{u}_{I,(2)}$,
$\bar{g}_{m,(3)}$ perturbing asymptotic AdS5 might be desirable. In the
following, we discuss this possibility.

First, recall that the electric charge $\widetilde{Q}_{I}$ and the angular
momentum $\widetilde{J}$ are conserved charges, which means that they are the
same whether evaluated at infinity or the horizon. Identifying (4.35) with
(4.69) we find

$\begin{split}&\quad
x_{I,(0)}-\frac{1}{2}\xi_{I}\left(\frac{1}{2}c^{JKL}x_{J,(0)}x_{K,(0)}\xi_{L}\right)+\frac{1}{2}c_{IJK}\left(\frac{1}{2}c^{JNO}\xi_{N}\xi_{O}\right)c^{KLM}x_{L,(0)}x_{M,(0)}\\\
&=\bar{x}_{I,(2)}-\frac{1}{2}\xi_{I}\left(\frac{1}{2}c^{JKL}\bar{x}_{J,(2)}\bar{x}_{K,(2)}\xi_{L}\right)+\frac{1}{2}c_{IJK}\left(\frac{1}{2}c^{JNO}\xi_{N}\xi_{O}\right)c^{KLM}\bar{x}_{L,(2)}\bar{x}_{M,(2)}\\\
&\quad+\epsilon\ell\left(\xi_{I}\xi_{J}-c_{IJK}c^{KLM}\xi_{L}\xi_{M}\right)\bar{u}^{J}_{(2)}\leavevmode\nobreak\
,\end{split}$ (4.78)

from matching $\widetilde{Q}_{I}$, and similarly (4.41) with (4.73) give

$\displaystyle\begin{split}&\quad\frac{\epsilon}{4}c^{IJK}x_{I,(0)}x_{J,(0)}\xi_{K}+\frac{1}{36}\left(c^{IJK}\xi_{I}\xi_{J}\xi_{K}\right)\left(c^{LMN}x_{L,(0)}x_{M,(0)}x_{N,(0)}\right)\\\
&=\frac{\epsilon}{4}c^{IJK}\bar{x}_{I,(2)}\bar{x}_{J,(2)}\xi_{K}+\frac{1}{36}\left(c^{IJK}\xi_{I}\xi_{J}\xi_{K}\right)\left(c^{LMN}\bar{x}_{L,(2)}\bar{x}_{M,(2)}\bar{x}_{N,(2)}\right)\\\
&\quad+\frac{\epsilon}{\ell}\bar{g}_{m,(3)}+\ell\left(\frac{1}{2}\xi_{I}c^{JKL}\xi_{J}\xi_{K}\bar{x}_{L,(2)}+\xi_{I}+\frac{5}{3}\ell^{-3}\bar{x}_{I,(2)}\right)\bar{u}^{I}_{(2)}\leavevmode\nobreak\
,\end{split}$ (4.79)

from matching $\widetilde{J}$. These conservation laws are consistent with a
UV solution specified by $x_{I,(0)}$ (and the FI-couplings $\xi_{I}$) that
flows to an IR configuration with $\bar{x}_{I,(2)}$ that may not even remotely
agree with (4.76). This consideration suggests that supersymmetry and charge
conservation do little to constrain the IR limit of the flow.

However, there is a different source of intuition. If the perturbative series
of $f^{-1}X_{I}$ from infinity did not truncate after exactly two terms, the
third term would diverge at the horizon $R^{2}\to 0$, rather than approaching
a constant. Other fields excited at the same order would similarly suggest a
singularity. It could happen that, taking into account successive powers
$R^{-2k}$ to all orders, there would be a finite limit $R^{2}\to 0$, after
all, but determining by explicit computation whether this possibility is
realized for any $\bar{u}_{I,(2)}$, $\bar{g}_{m,(3)}$ is technically
challenging.

From a different perspective, since the conserved charges from the near-
horizon expansion do satisfy the typical charge constraint, the possibly new
black hole solutions that do not seem to satisfy the typical charge constraint
at infinity, would not flow to the expected near-horizon extremal AdS2
geometry, which implies that these solutions may not be black holes after all.

Moreover, a change in the electric potential $fY^{I}\to fY^{I}+\beta^{I}$ with
$\beta^{I}$ constant is trivial as it does not change the electromagnetic
field strength. However, with the vielbein we have picked, such a shift must
be accompanied by $u^{I}\to u^{I}-w\beta^{I}$. Because $w$ includes a term
$w\sim R^{-2}$ at large $R$, such a gauge transformation has the ability to
remove $\bar{u}^{I}_{(2)}$. This mechanism shows the $\bar{u}^{I}_{(2)}$ are
allowed, in principle, but also that they are not physical deformations.
Indeed, these coefficients diverge at the horizon, so they correspond to a
singular gauge which is ill-advised.

##  5 Entropy Extremization

In this section, we consider the near-horizon limit of the Legendre transform
of the radial Lagrangian (2.32), leading to a near-horizon entropy function.
Extremizing this entropy function with respect to the near-horizon variables
leads to an expression for the entropy in terms of the aforementioned charges.

###  5.1 Near-horizon setup

First, we consider the near-horizon of the line element $ds_{2}^{2}$ (2.27),
where we recall that $e^{2\rho}$ and $e^{2\sigma}$ can be expressed in terms
of the variables $f$, $g_{m}$, and $w$ as in (4.9). At the horizon $R\to 0$,
these variables have known near-horizon behaviors according to Table 1. Thus,
the near-horizon limit of (2.27) becomes

$ds_{2,\text{nh}}^{2}=v\left(\frac{R^{4}}{\ell_{2}^{2}}dt^{2}-\frac{dR^{2}}{R^{2}}\right)\,,$
(5.1)

with $v$ and $\ell_{2}$ defined based on the $R\to 0$ behavior of $e^{2\rho}$
and $e^{2\sigma}$:

$\displaystyle\left.e^{2\sigma}\right|_{\text{nh}}\equiv\frac{v}{R^{2}}\leavevmode\nobreak\
,\qquad\left.e^{2\rho}\right|_{\text{nh}}\equiv\frac{v}{\ell_{2}^{2}}R^{4}\leavevmode\nobreak\
.$ (5.2)

Furthermore, $v^{\frac{1}{2}}$ and $\ell_{2}^{\frac{1}{3}}$ are near-horizon
length scales defining the 2D $(t,R)$ part of the line element (4.8)

$ds_{5,\text{nh}}^{2}=v\left(\frac{R^{4}}{\ell_{2}^{2}}dt^{2}-\frac{dR^{2}}{R^{2}}\right)-e^{-U_{1}}(\sigma_{1}^{2}+\sigma_{2}^{2})-e^{-U_{2}}(\sigma_{3}+a^{0})^{2}\leavevmode\nobreak\
.$ (5.3)

The role of the variable $\ell_{2}$ is elucidated by noting the near-horizon
limit of the Kähler condition (4.10):

$\ell_{2}=2ve^{-\frac{1}{2}U_{2}}\leavevmode\nobreak\ .$ (5.4)

This relation will be used to eliminate $\ell_{2}$ in the rest of the near-
horizon analysis.

Having reviewed the near-horizon 2D line element, and in anticipation of
applying the entropy function formalism Sen:2005wa ; Sen:2005iz ; David:2006yn
; Sen:2007qy ; Sen:2008yk ; Sen:2008vm ; Sen:2009vz ; Astefanesei:2006dd ;
Castro:2007sd  to the Lagrangian density in (2.32), we use the following
coordinate transformation

$\begin{split}dt&\to\frac{1}{2}\ell_{2}dt\leavevmode\nobreak\ ,\\\
dR&\to\frac{1}{2R}dR\,,\end{split}$ (5.5)

to bring the coordinates $(t,R)$ in (5.1) to the canonical $\text{AdS}_{2}$
form

$ds_{2,\text{nh}}^{2}=\frac{v}{4}\left(R^{2}dt^{2}-\frac{dR^{2}}{R^{2}}\right)\leavevmode\nobreak\
,$ (5.6)

where now it is clear that $v^{\frac{1}{2}}$ is related to the
$\text{AdS}_{2}$ length scale.

The coordinate transformation (5.5) will have the effect of rescaling the
Lagrangian 2-form $\mathcal{L}_{2}=\mathcal{L}_{1}dt\wedge dR$ (2.22) by a
factor

$\mathcal{L}_{2}\to\frac{\ell_{2}}{4R}\mathcal{L}_{2}\leavevmode\nobreak\ .$
(5.7)

With the prescription of defining the entropy function through omitting the
$dt\wedge dR$ volume form from the dimensionally-reduced action, we anticipate
dividing the density $\mathcal{L}_{1}$ by a factor of $\frac{4R}{\ell_{2}}$.

Combining the near-horizon behaviors of $e^{2\rho}$ and $e^{2\rho}$ studied
above with the dictionary definitions (4.9), $e^{-U_{1}}$, $e^{-U_{2}}$ and
$b^{I}$ can be shown to be constants to leading order in the near-horizon
limit based on the leading-order behaviors of $f$, $g_{m}$ and $w$ consistent
with the small $R$ asymptotics in Table 1.

Concerning the matter fields, the electric fields $a^{I}$ and $a^{0}$ in
(2.18) become in the near-horizon limit

$\left.a^{I}\right|_{\text{nh}}\equiv\frac{e^{I}R^{2}}{2v}e^{\frac{1}{2}U_{2}}dt,\quad\left.a^{0}\right|_{\text{nh}}\equiv-\frac{e^{0}R^{2}}{2v}e^{\frac{1}{2}U_{2}}dt\leavevmode\nobreak\
.$ (5.8)

The total 1D Lagrangian density (2.32) then becomes

$\begin{split}\mathcal{L}_{1,\text{nh}}&=\frac{\pi}{2G_{5}}e^{-U_{1}-\frac{1}{2}U_{2}}\frac{4R}{\ell_{2}}v\left[\frac{1}{v^{2}}e^{-U_{2}}(e^{0})^{2}-\frac{4}{v}+e^{U_{1}}-\frac{1}{4}e^{2U_{1}-U_{2}}-\frac{1}{2}G_{IJ}e^{2U_{1}}b^{I}b^{J}\right.\\\
&\quad+\left.\frac{2}{v^{2}}G_{IJ}(e^{I}-b^{I}e^{0})(e^{J}-b^{J}e^{0})-V\right]-\frac{\pi}{2G_{5}}\frac{4R}{\ell_{2}}\frac{1}{2}c_{IJK}b^{I}b^{J}(e^{K}-\frac{2}{3}e^{0}b^{K})\,.\end{split}$
(5.9)

Apart from an overall prefactor in the integration measure, every other
appearance of $\ell_{2}$ has been re-expressed in terms of $v$ and $U_{2}$ by
using (5.4). The $\frac{4R}{\ell_{2}}$ factor has been factored out of the
volume element, and we follow the prescription made earlier to exactly cancel
it out with the $\frac{\ell_{2}}{4R}$ factor from (5.7) in order to obtain the
Lagrangian density suitable for the entropy function. We also note the
presence of the Chern-Simons boundary terms in (5.9) that are crucial for
calculating the near-horizon charges.

We now obtain a near-horizon Lagrangian (5.9) that is a function of the
variables $v$, $U_{1}$, $U_{2}$, $e^{I}$, $e^{0}$, and $b^{I}$. We will next
derive the charges $Q_{I}$ and $J$ from $\mathcal{L}_{1,\text{nh}}$, with the
goal of Legendre transforming the Lagrangian into an entropy function
$\mathcal{S}$ that can be ultimately extremized towards a function purely of
the charges $\mathcal{S}=\mathcal{S}(Q_{I},J)$.

We note from the earlier Noether procedure (3.33) that $Q_{I}$ and $J$ were
obtained in terms of radially dependent variables. In terms of the near-
horizon limit of the electric fields (5.8), this becomes

$\displaystyle Q_{I}$
$\displaystyle=\frac{\pi}{G_{5}}\left[e^{-U_{1}-\frac{1}{2}U_{2}}\frac{4}{v}G_{IJ}(e^{J}-b^{J}e^{0})-\frac{1}{2}c_{IJK}b^{J}b^{K}\right]\leavevmode\nobreak\
,$ (5.10) $\displaystyle J$
$\displaystyle=\frac{\pi}{G_{5}}\left[-\frac{2}{v}e^{-U_{1}-\frac{3}{2}U_{2}}e^{0}+\frac{4}{v}e^{-U_{1}-\frac{1}{2}U_{2}}G_{IJ}b^{I}(e^{J}-b^{J}e^{0})-\frac{1}{3}c_{IJK}b^{I}b^{J}b^{K}\right]\leavevmode\nobreak\
.$ (5.11)

This introduces the charges $Q_{I}$ and $J$ as conjugates to the electric
fields $e^{I}$ and $e^{0}$, allowing for the inversion

$\displaystyle e^{I}-b^{I}e^{0}$
$\displaystyle=\frac{v}{16}e^{U_{1}+\frac{1}{2}U_{2}}G^{IJ}\left(\widetilde{Q}_{J}+2c_{JKL}b^{K}b^{L}\right)\leavevmode\nobreak\
,$ (5.12) $\displaystyle e^{0}$
$\displaystyle=-\frac{v}{8}e^{U_{1}+\frac{3}{2}U_{2}}\left(\widetilde{J}-\widetilde{Q}_{I}b^{I}-\frac{2}{3}c_{IJK}b^{I}b^{J}b^{K}\right)\leavevmode\nobreak\
\,,$ (5.13)

where now the rescaled $\widetilde{Q}_{I}$ and $\widetilde{J}$ (3.34) have
been used. The near-horizon entropy function can now be defined as a Legendre
transform of the Lagrangian density (5.9) with fixed charges

$\mathcal{S}=2\pi\left(e^{I}\frac{\partial\mathcal{L}_{1,\text{nh}}}{\partial
e^{I}}+e^{0}\frac{\partial\mathcal{L}_{1,\text{nh}}}{\partial
e^{0}}-\mathcal{L}_{1,\text{nh}}\right)\leavevmode\nobreak\ ,$ (5.14)

which, after eliminating the electric fields through (5.12) and (5.13), yields

$\begin{split}{\cal
S}&=\frac{4\pi^{2}}{G_{5}}e^{-U_{1}-\frac{1}{2}U_{2}}\left[1+\frac{v}{4}\left(\frac{1}{4}e^{2U_{1}-U_{2}}-e^{U_{1}}+\frac{1}{64}e^{2U_{1}+2U_{2}}\left(\widetilde{J}-\widetilde{Q}_{I}b^{I}-\frac{2}{3}c_{IJK}b^{I}b^{J}b^{K}\right)^{2}\right.\right.\cr&\left.\left.+V+\frac{1}{128}e^{2U_{1}+U_{2}}G^{IJ}\left(\widetilde{Q}_{I}+2c_{IKL}b^{K}b^{L}\right)\left(\widetilde{Q}_{J}+2c_{JMN}b^{M}b^{N}\right)+\frac{1}{2}e^{2U_{1}}G_{IJ}b^{I}b^{J}\right)\right]\leavevmode\nobreak\
.\end{split}$ (5.15)

This entropy function depends on the physical variables $v$, $U_{1}$, $U_{2}$,
$b^{I}$, $X^{I}$ describing the near horizon geometry and matter fields, with
the conserved charges $\widetilde{Q}_{I}$, $\widetilde{J}$ appearing as fixed
parameters. At its extremum, it yields the physical variables and the black
hole entropy as a function of the charges.

###  5.2 Extremization of the Entropy Function

It is exceedingly simple to extremize with respect to $v$ which appears only
as a Lagrange multiplier in front of the large round bracket that comprises
nearly all of (5.15). This leaves the extremized value of $\mathcal{S}$:

$\mathcal{S}=\frac{4\pi^{2}}{G_{5}}e^{-U_{1}-\frac{1}{2}U_{2}}\leavevmode\nobreak\
.$ (5.16)

This is exactly the black hole entropy computed via the area law for a horizon
defined by the volume 3-form
$e^{-U_{1}-\frac{1}{2}U_{2}}\sigma_{1}\wedge\sigma_{2}\wedge\sigma_{3}$ with
the angular ranges specified in (2.20). However, the explicit dependence of
$U_{1}$ and $U_{2}$ on the charges remains to be determined. For this we must
extremize with respect to the remaining variables

$\displaystyle\begin{split}\partial_{U_{1}}\mathcal{S}=\partial_{U_{2}}\mathcal{S}=\partial_{b^{I}}\mathcal{S}=D_{I}\mathcal{S}=0\leavevmode\nobreak\
.\end{split}$ (5.17)

Here $D_{I}$ is the Kähler-covariantized derivative with respect to the
scalars $X^{I}$. It is defined such that $X^{I}D_{I}=0$, which is the correct
way to vary the scalars while also implementing the constraint (B.1). The
conditions (5.17) give

$\displaystyle 0=$ $\displaystyle
V-e^{U_{1}}+\frac{1}{4}e^{2U_{1}-U_{2}}+\frac{1}{64}e^{2U_{1}+2U_{2}}\mathcal{M}^{2}+G^{IJ}\frac{1}{128}e^{2U_{1}+U_{2}}\mathcal{K}_{I}\mathcal{K}_{J}+\frac{1}{2}G_{IJ}e^{2U_{1}}b^{I}b^{J}\leavevmode\nobreak\
,$ (5.18) $\displaystyle 0=$
$\displaystyle-4-vV+\frac{v}{4}e^{2U_{1}-U_{2}}+\frac{v}{64}e^{2U_{1}+2U_{2}}\mathcal{M}^{2}+G^{IJ}\frac{v}{128}e^{2U_{1}+U_{2}}\mathcal{K}_{I}\mathcal{K}_{J}+\frac{v}{2}G_{IJ}e^{2U_{1}}b^{I}b^{J}\leavevmode\nobreak\
,$ (5.19) $\displaystyle 0=$
$\displaystyle-2-\frac{v}{2}V+\frac{v}{2}e^{U_{1}}-\frac{3v}{8}e^{2U_{1}-U_{2}}+\frac{3v}{128}e^{2U_{1}+2U_{2}}\mathcal{M}^{2}+G^{IJ}\frac{v}{256}e^{2U_{1}+U_{2}}\mathcal{K}_{I}\mathcal{K}_{J}$
$\displaystyle\qquad-\frac{v}{4}G_{IJ}e^{2U_{1}}b^{I}b^{J}\leavevmode\nobreak\
,$ (5.20)
$\displaystyle\begin{split}0=&vD_{I}V+\frac{v}{128}e^{2U_{1}+U_{2}}(D_{I}G^{JK})\mathcal{K}_{J}\mathcal{K}_{K}+\frac{v}{2}e^{2U_{1}}(D_{I}G_{JK})b^{J}b^{K}\leavevmode\nobreak\
,\end{split}$ (5.21) $\displaystyle 0=$
$\displaystyle\frac{1}{32}e^{U_{2}}\left(-e^{U_{2}}\mathcal{K}_{I}\mathcal{M}\right.\left.+2G^{JK}c_{IJN}b^{N}\mathcal{K}_{K}\right)+G_{IJ}b^{J}\leavevmode\nobreak\
,$ (5.22)

where $\mathcal{M}$ and $\mathcal{K}_{I}$ are shorthand for

$\mathcal{M}\equiv\widetilde{J}-\widetilde{Q}_{I}b^{I}-\frac{2}{3}c_{IJK}b^{I}b^{J}b^{K}\
,\
\mathcal{K}_{I}\equiv\widetilde{Q}_{I}+2c_{IJK}b^{J}b^{K}\leavevmode\nobreak\
.$ (5.23)

Additionally, $D_{I}$ acts on the scalars $X^{J}$ following777This can be
generalized to other quantities via the product rule on $D_{I}$, for instance,
$\begin{split}D_{I}X_{J}&=\frac{1}{2}c_{JKL}D_{I}(X^{K}X^{L})=c_{IJK}X^{K}-\frac{2}{3}X_{I}X_{J}\leavevmode\nobreak\
.\end{split}$ (5.24)

$D_{I}X^{J}=\delta_{I}^{J}-\frac{1}{3}X_{I}X^{J}\leavevmode\nobreak\ .$ (5.25)

Ideally we seek the most general extremum that solves (5.18-5.22) that is
consistent with our ansatz (5.3). However, the extremization equations are
highly nonlinear in the variables of interest $(v,U_{1},U_{2},X^{I},b^{I})$.
Therefore, in the following we specialize and find all supersymmetric
solutions.

#### Near-horizon supersymmetric conditions

It is straightforward to take the near-horizon limit of the supersymmetry
conditions (4.14-4.16), along with the identification $X^{I}=Y^{I}$ (C.49).
After inverting $e^{I}$ and $e^{0}$ in terms of $\widetilde{Q}_{I}$ and
$\widetilde{J}$, following (5.12) and (5.13)), we obtain the following near-
horizon supersymmetric relations

$\displaystyle 0$
$\displaystyle=\widetilde{Q}_{I}+2c_{IJK}b^{J}b^{K}-4e^{-U_{2}}X_{I}\leavevmode\nobreak\
,$ (5.26) $\displaystyle 0$
$\displaystyle=\widetilde{J}-\widetilde{Q}_{I}b^{I}-\frac{2}{3}c_{IJK}b^{I}b^{J}b^{K}-4e^{-U_{1}-U_{2}}(\xi\cdot
X)\leavevmode\nobreak\ ,$ (5.27) $\displaystyle 0$
$\displaystyle=b^{I}-e^{-U_{1}}\left(X^{I}(\xi\cdot
X)-G^{IJ}\xi_{J}\right)\leavevmode\nobreak\ ,$ (5.28) $\displaystyle 0$
$\displaystyle=e^{U_{1}}-\frac{4}{v}-2V\leavevmode\nobreak\ .$ (5.29)

We also need the near-horizon version of the Kähler condition (4.10). We can
trade the variables $\rho$ and $\sigma$ describing the 2D geometry for $f$ and
$g_{m}$ following (4.9) and eliminate $a^{0}_{t}$ that results in favor of the
charges through (5.13). These steps lead to

$\frac{4}{v}-(\xi\cdot X)^{2}=e^{2U_{1}-U_{2}}\leavevmode\nobreak\ .$ (5.30)

We have verified that when the five supersymmetric relations (5.26-5.30) are
satisfied, then the five $\mathcal{S}$ extremization equations (5.18-5.22) are
satisfied as well. The details of this computation are not instructive so we
omit them. The reverse logic would be that all extremal solutions within the
scope of our ansatz are supersymmetric. This we have not shown, and it is
indeed not true, i.e. there are no known nonextremal supersymmetric Lorentzian
black holes. Thus the specialization to supersymmetric solutions addresses a
genuine subset of the extremal black holes.

In the remainder of this subsection we solve the supersymmetry relations
(5.26-5.30) explicitly and find all variables as functions of the charges
$Q_{I}$ and $J$.

####  5.2.1 Solving for the entropy and the charge constraint

The supersymmetry conditions (5.26-5.30) are all algebraic, but they are far
from trivial. Straightforward contractions, followed by taking simple linear
combinations, give scalar identities

$\displaystyle X\cdot b$ $\displaystyle=e^{-U_{1}}\xi\cdot
X\leavevmode\nobreak\ ,$ (5.31) $\displaystyle\widetilde{Q}\cdot b$
$\displaystyle=\frac{3}{2}\widetilde{J}-8e^{-U_{1}-U_{2}}\xi\cdot
X\leavevmode\nobreak\ ,$ (5.32) $\displaystyle\xi\cdot b$
$\displaystyle=e^{U_{1}-U_{2}}-1\leavevmode\nobreak\ ,$ (5.33)

which will prove useful later. Our strategy will be to exploit identities like
these to find simple combinations of variables that can be expressed entirely
in terms of the charges $Q_{I}$, $J$ and the couplings $\xi_{I}$. Combinations
of those will in turn give explicit formulae for physical variables.

In this spirit, we expand $\widetilde{Q}_{J}\widetilde{Q}_{K}$ using the
square of (5.26), and then simplify the terms that are products of $b$’s using
(5.27). We obtain

$\frac{1}{2}c^{IJK}\widetilde{Q}_{J}\widetilde{Q}_{K}=32e^{-2U_{1}-U_{2}}c^{IJK}\xi_{J}\xi_{K}+16e^{-U_{1}-U_{2}}X^{I}-2\widetilde{J}b^{I}\leavevmode\nobreak\
.$ (5.34)

Contracting (5.34) with $\xi_{I}$, the first term on the right becomes
proportional to $e^{-2U_{1}-U_{2}}$, which is related to the black hole
entropy through $\mathcal{S}$ (5.16). There will also be a term $(\xi\cdot X)$
that we can eliminate with the help of

$\displaystyle\widetilde{J}=8e^{-2U_{1}}(\xi\cdot
X)+\frac{32}{3}e^{-3U_{1}}c^{IJK}\xi_{I}\xi_{J}\xi_{K}\leavevmode\nobreak\ ,$
(5.35)

which is a simplification of (5.27) with $\widetilde{Q}_{I}$ and $b^{I}$
eliminated using (5.26) and (5.28), respectively. These steps give

$\frac{1}{6}c^{IJK}\xi_{I}\xi_{J}\xi_{K}\left(\frac{\mathcal{S}}{2\pi}\right)^{2}=\frac{1}{2}c^{IJK}\xi_{I}Q_{J}Q_{K}-\frac{\pi}{4G_{5}}2J\leavevmode\nobreak\
,$ (5.36)

which amounts to an explicit formula for the black hole entropy as function of
the conserved charges

$\mathcal{S}=2\pi\sqrt{\frac{1}{2}c^{IJK}\ell^{3}\xi_{I}Q_{J}Q_{K}-N^{2}J}\leavevmode\nobreak\
.$ (5.37)

This is in full agreement with the entropy of supersymmetric extremal AdS5
black holes Cabo-Bizet:2018ehj ; Benini:2018mlo ; Choi:2018hmj . We have
expressed the entropy using the untilded charges $Q_{I}$ and $J$ (3.34) and
traded $G_{5}$ for $N^{2}$ using $\frac{\pi\ell^{3}}{4G_{5}}=\frac{1}{2}N^{2}$
and applied $\frac{1}{6}c^{IJK}\xi_{I}\xi_{J}\xi_{K}=\ell^{-3}$ from (2.12) to
explicitly show all the dimensionful quantities.

Continuing with the strategy of evaluating natural combinations of the
conserved charges, we evaluate the cubic invariant of the charges
$c^{IJK}\widetilde{Q}_{I}\widetilde{Q}_{J}\widetilde{Q}_{K}$ by taking the
cube of $\widetilde{Q}_{I}$ from (5.26), with the resulting contractions of
$X_{I}$ and $b^{I}$ such as $c_{IJK}X^{I}b^{J}b^{K}$ and $X_{I}b^{I}$
simplified using the $b^{I}$ relation (5.28) as well as (5.29) and (5.30) for
the terms quadratic in $\xi$. This yields

$c^{IJK}\widetilde{Q}_{I}\widetilde{Q}_{J}\widetilde{Q}_{K}=6e^{-U_{1}-2U_{2}}+\frac{1}{8}\widetilde{J}c_{IJK}b^{I}b^{J}b^{K}\leavevmode\nobreak\
.$ (5.38)

Alternatively, we can arrive at the cubic product of electric charges by
contracting (5.34) with $Q_{I}$ from (5.26), giving

$\displaystyle\begin{split}&\tfrac{1}{64}c^{IJK}\widetilde{Q}_{I}\widetilde{Q}_{J}\widetilde{Q}_{K}\\\
&=4e^{-U_{1}-2U_{2}}+2e^{-2U_{1}-U_{2}}+e^{-2U_{1}-U_{2}}c^{IJK}\widetilde{Q}_{I}\xi_{J}\xi_{K}-\tfrac{1}{4}e^{-U_{1}-U_{2}}\widetilde{J}(\xi\cdot
X)+\tfrac{1}{8}\widetilde{J}c_{IJK}b^{I}b^{J}b^{K}\leavevmode\nobreak\
.\end{split}$ (5.39)

Comparing (5.38) and (5.39), we find the identity

$\frac{1}{2}c^{IJK}\widetilde{Q}_{I}\xi_{J}\xi_{K}+1=e^{U_{1}}\left(e^{-U_{2}}+\frac{1}{8}\widetilde{J}\leavevmode\nobreak\
\xi\cdot X\right)\leavevmode\nobreak\ .$ (5.40)

It is useful because it gives access to a useful combination of $U_{1}$,
$U_{2}$, and $\xi\cdot X$. Indeed, we can simplify the cube of the electric
charge in the form (5.38) using the $b$ identity (5.28), and then (5.35) to
eliminate $(\xi\cdot X)$, to give

$\frac{1}{6}c^{IJK}\widetilde{Q}_{I}\widetilde{Q}_{J}\widetilde{Q}_{K}+\widetilde{J}^{2}=64e^{-U_{1}-U_{2}}\left(e^{-U_{2}}+\frac{1}{8}\widetilde{J}\leavevmode\nobreak\
\xi\cdot X\leavevmode\nobreak\ \right)\leavevmode\nobreak\ .$ (5.41)

The right-hand side of this equation differs from that of (5.40) only by a
factor proportional $e^{-2U_{1}-U_{2}}$ which is the square of the geometric
measure on the black hole horizon. As such, it is related to the black hole
entropy $\mathcal{S}$ both through the area law (5.16) and as a function of
charges (5.37). Collecting these relations, and reintroducing $Q_{I}$ and $J$
(3.34) in order to align with the conventional units for this result, we find

$\left(\frac{1}{6}c^{IJK}Q_{I}Q_{J}Q_{K}+\frac{\pi}{4G_{5}}J^{2}\right)=\ell^{3}\left(\frac{1}{2}c^{IJK}\xi_{I}\xi_{J}Q_{K}+\frac{\pi}{4G_{5}}\right)\left(\frac{1}{2}c^{IJK}\xi_{I}Q_{J}Q_{K}-\frac{\pi}{2G_{5}}J\right)\leavevmode\nobreak\
.$ (5.42)

This is the prototypical 5D nonlinear charge constraint Cabo-Bizet:2018ehj ;
Benini:2018mlo ; Choi:2018hmj . However, the charge constraint (5.42) does not
make progress towards solving the supersymmetry equations nor determining the
near horizon solution in terms of conserved charges. Rather, it is a relation
between the conserved charges that, if taken at face value, all supersymmetric
black holes dual to ${\cal N}=4$ SYM must satisfy. This is extremely important
and the continuing questions regarding this constraint is one of the
motivations for the detailed study reported in this article.

Even at this point of our discussion where we are deep into solving certain
nonlinear equations, it is worth noting that that the black hole entropy
(5.36) and the charge constraint (5.42) can be combined into one complex-
valued equation

$\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\frac{1}{6}c^{IJK}\left(Q_{I}+i\frac{\cal
S}{2\pi}\xi_{I}\right)\left(Q_{J}+i\frac{\cal
S}{2\pi}\xi_{J}\right)\left(Q_{K}+i\frac{\cal
S}{2\pi}\xi_{K}\right)+\frac{\pi}{4G_{5}}\left(-J+i\frac{\cal
S}{2\pi}\right)^{2}=0\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ .$ (5.43)

The real part gives the constraint (5.42) and the imaginary part gives the
formula for the entropy (5.36). Complexified equations are natural in problems
involving supersymmetry. It is possible that (5.43) is related to the
condition for a complex saddle point that provides an accounting in ${\cal
N}=4$ SYM for the entropy of black hole preserving $1/16$ of the maximal
supersymmetry cassani2021eft ; grant2008comments .

####  5.2.2 Near-horizon variables as function of conserved charges

Having discussed the black hole entropy and the constraint on charges, we move
on to expressing all other aspects of the near-horizon geometry and the matter
content in terms of the fixed charges $Q_{I}$ and $J$.

For the following computations, we make use of the expressions (5.26) and
(5.27) for $J$ and $Q_{I}$ simplified using the relation for $b^{I}$ in
(5.28). We find

$\displaystyle Q_{I}$
$\displaystyle=\frac{\pi}{G_{5}}e^{-2U_{1}}\left[X_{I}e^{2U_{1}-U_{2}}+X_{I}(\xi\cdot
X)^{2}-2\xi_{I}(\xi\cdot
X)-4G_{IJ}c^{JKL}\xi_{K}\xi_{L}\right]\leavevmode\nobreak\ ,$ (5.44)
$\displaystyle J$
$\displaystyle=\frac{16\pi}{G_{5}}e^{-3U_{1}}\left[\frac{1}{8}e^{U_{1}}\xi\cdot
X+\ell^{-3}\right]\leavevmode\nobreak\ .$ (5.45)

Multiplying both sides of (5.40) by $J$, using the relation (5.45), we can
solve for $e^{U_{1}}\leavevmode\nobreak\ \xi\cdot X$ in terms of the charges

$\frac{1}{8}\xi\cdot
Xe^{U_{1}}=\frac{J\left(\frac{1}{2}c^{IJK}Q_{I}\xi_{J}\xi_{K}+\frac{\pi}{4G_{5}}\right)-\left(\frac{\mathcal{S}}{2\pi}\right)^{2}\ell^{-3}}{J^{2}+\left(\frac{\mathcal{S}}{2\pi}\right)^{2}}\leavevmode\nobreak\
.$ (5.46)

This result ties a geometrical quantity — as it appears in the near-horizon
Kähler relation (5.30) — to the charges. It also has an additional immediate
value as (5.45) relates it to $e^{-3U_{1}}$, giving

$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
e^{-3U_{1}}=\frac{G_{5}}{16\pi}\frac{J^{2}+\left(\frac{\mathcal{S}}{2\pi}\right)^{2}}{\frac{1}{2}c^{IJK}Q_{I}\xi_{J}\xi_{K}+\frac{\pi}{4G_{5}}+J\ell^{-3}}\leavevmode\nobreak\
.$ (5.47)

This expression sets the scale of the non-deformed $S^{3}$ which has line
element $e^{-U_{1}}(\sigma_{1}^{2}+\sigma_{2}^{2}+\sigma_{3}^{2})$.

Due to the rotation of the black hole, the horizon geometry (4.8) is deformed
away from $S^{3}$. We can quantify the deformation by computing
$e^{U_{1}-U_{2}}$ via $e^{3U_{1}}$ from (5.47) and $e^{2U_{1}+U_{2}}$ from
(5.16):

$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
e^{U_{1}-U_{2}}=\frac{\frac{4G_{5}}{\pi}\left(\frac{1}{2}c^{IJK}Q_{I}\xi_{J}\xi_{K}+\frac{\pi}{4G_{5}}+J\ell^{-3}\right)\left(\frac{\cal
S}{2\pi}\right)^{2}}{J^{2}+\left(\frac{\cal
S}{2\pi}\right)^{2}}\leavevmode\nobreak\ .$ (5.48)

The only scalar near-horizon parameter that was not yet computed is the
AdS2-volume $v$. Due to the alternate near-horizon Kähler condition (5.30),
the combination $ve^{U_{1}}$ can be expressed in terms of $\mathcal{S}$,
$(\xi\cdot X)e^{U_{1}}$ and $e^{-3U_{1}}$. These three quantities were given
as functions of the conserved charges in (5.16), (5.46) and (5.47). After
simplifications, we find

$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\frac{v}{4}e^{U_{1}}=\frac{\pi\ell^{3}}{4G_{5}}\frac{\ell^{3}\left(\frac{1}{2}c^{IJK}Q_{I}\xi_{J}\xi_{K}+\frac{\pi}{4G_{5}}\right)+J}{\ell^{6}\left(\frac{1}{2}c^{IJK}Q_{I}\xi_{J}\xi_{K}+\frac{\pi}{4G_{5}}\right)^{2}+\left(\frac{\cal
S}{2\pi}\right)^{2}}.$ (5.49)

This completes the explicit extremization of the entropy function for the
scalar variables which at this point have all been expressed in terms of
conserved charges $Q_{I},J$ and FI-couplings $\xi_{I}$.

We must similarly determine the vectors $b^{I}$ and $X^{I}$ at the extremum
which may be determined, in principle, by the input vectors $\xi_{I}$ and
$\widetilde{Q}_{I}$. However, the position of the vector indices $I$ do not
match so the full real special geometry enters. We exploit only the vectorial
symmetry, and then, $X^{I}$ and $b^{I}$ must be linear combinations of three
vectors: $c^{IJK}\widetilde{Q}_{I}Q_{J}$, $c^{IJK}\widetilde{Q}_{I}\xi_{J}$,
and $c^{IJK}\xi_{I}\xi_{J}$. One linear relation of this kind was given in
(5.34). To find another, we contract (5.34) with $\widetilde{Q}_{I}$ and
simplify using (5.39). This gives

$\widetilde{Q}\cdot X=8e^{-U_{2}}+4e^{-U_{1}}\leavevmode\nobreak\ .$ (5.50)

Combining this with (5.33) and (5.46), we have all four inner products of
$b^{I}$, $X^{I}$, $Q_{I}$ and $\xi_{I}$. We already determined the scalar
combinations $c^{IJK}\xi_{I}\xi_{J}\widetilde{Q}_{K}$ and
$c^{IJK}\xi_{I}\widetilde{Q}_{J}\widetilde{Q}_{K}$ from (5.36) and (5.40), so
we can establish the vectorial equation

$\frac{1}{2}c^{IJK}\widetilde{Q}_{J}\xi_{K}=b^{I}+\frac{1}{8}\widetilde{J}e^{U_{1}}X^{I}\leavevmode\nobreak\
.$ (5.51)

Inversion of (5.34) and (5.51) give

$\displaystyle
b^{I}=\frac{4G_{5}}{\pi}\cdot\frac{1}{2}\frac{c^{IJK}\xi_{J}Q_{K}\left(\frac{\cal