aainstitutetext: Instituut voor Theoretische Fysica, KU Leuven.
Celestijnenlaan 200D, B-3001 Leuven, Belgium bbinstitutetext: Departament de
Física Quàntica i Astrofísica, Institut de Ciències del Cosmos Universitat de
Barcelona, Martí i Franquès 1, E-08028 Barcelona, Spain

# Near-horizon geometries and black hole thermodynamics in higher-derivative
AdS5 supergravity

Pablo A. Cano a Marina David<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Higher-derivative corrections in the AdS/CFT correspondence allow us to
capture finer details of the dual CFT and to explore the holographic
dictionary beyond the infinite $N$ and strong coupling limits. Following an
effective field theory approach, we investigate extremal AdS black hole
solutions in five-dimensional supergravity with higher-derivative corrections.
We provide a general analysis of near-horizon geometries of rotating extremal
black holes and show how to obtain their corresponding charges and chemical
potentials. We discuss the near-horizon solutions of the two-derivative
theory, which we write using a novel parametrization that eases our
computation of the higher-derivative corrections. The charges and
thermodynamic properties of the black hole are computed while clarifying the
ambiguities in their definitions. The charges and potentials turn out to
satisfy a near-horizon version of the first law of thermodynamics whose
interpretation we make clear. In the supersymmetric case, the results are
shown to match the field theory prediction as well as previous results
obtained from the on-shell action.

##  1 Introduction

The study of black holes in AdS spacetimes has driven a great deal of
investigation in fundamental physics, as the AdS/CFT correspondence
Maldacena:1997re ; Witten:1998qj ; Gubser:1998bc  has led to significant
progress in our understanding of both gravity and gauge field theories. In
particular, precision holography, which aims to investigate the duality
between gravity and conformal field theories beyond leading order in the large
$N$ and strong coupling regime, is currently the focus of intensive research.
In the gravity side, this involves studying corrections to AdS spacetimes,
such as higher derivative corrections Bobev:2020zov ; Bobev:2020egg ;
Bobev:2021qxx ; Bobev:2021oku ; Bobev:2022bjm ; Cassani:2022lrk ;
Cassani:2023vsa ; Cano:2023dyg  or 1-loop effects, like those of logarithmic
type Bhattacharyya:2012ye ; David:2019lhr ; GonzalezLezcano:2020yeb ;
David:2021eoq ; David:2021qaa ; Karan:2022dfy ; Bobev:2023dwx . By their own
nature, the study of these corrections is a challenging endeavour. In this
work, we focus on higher derivative corrections to black hole solutions of
minimal gauged five-dimensional supergravity, with a special focus (but not
only) on supersymmetric solutions.

The black hole solutions in this theory are in general characterized by the
mass, electric charge and two different angular momenta. As usual in the case
of AdS, the supersymmetric solutions must have rotation, which increases the
complexity of their analysis. The simplest of those solutions was first
written by Gutowski and Reall Gutowski:2004ez , describing a black hole with
electric charge and two equal angular momenta. More general solutions of this
theory — with arbitrary mass, charge and distinct angular momenta, and
containing the most general supersymmetric solution as a particular case —
were found in Chong:2005hr . Our focus will be on studying corrections to this
solution, which we call the CCLP solution.

There has been recent work in understanding higher derivative corrections in
five-dimensional supergravity. The starting point involves applying the off-
shell formalism of $\mathcal{N}=2$ gauged supergravity Lauria:2020rhc  with
all possible four-derivative invariants Zucker:1999ej ; deWit:2006gn ;
Hanaki:2006pj ; Cremonini:2008tw ; Bergshoeff:2011xn ; Coomans:2012cf ;
Ozkan:2013nwa ; Baggio:2014hua ; Butter:2014xxa ; Bobev:2021qxx ;
Cassani:2022lrk ; Liu:2022sew ; Gold:2023dfe ; Gold:2023ykx ; Gold:2023ymc .
Integrating out the auxiliary degrees of freedom leads to an explicit action
for the metric and gauge field Bobev:2021qxx ; Liu:2022sew ; Cassani:2022lrk ,
which can be further simplified by implementing field redefinitions
Cassani:2022lrk ; Gold:2023ymc . From the point of view of holography, such
theory is now dual to a superconformal field theory with unequal central
charges $\mathfrak{a}\neq\mathfrak{c}$, the difference
$(\mathfrak{a}-\mathfrak{c})$ being proportional to the higher-derivative
couplings Hofman:2008ar .

Refs. Bobev:2022bjm ; Cassani:2022lrk  have recently been able to compute the
corrections to the thermodynamic quantities of the CCLP black hole111See also
Gold:2023ymc  for a similar analysis., showing that, in the supersymmetric
case, the result precisely agrees with the corresponding superconformal index
of the dual CFT, which contains corrections controlled by difference of
central charges $(\mathfrak{a}-\mathfrak{c})$. These references made use of
the on-shell action in order to compute the free energy in the grand canonical
ensemble, from which one derive the subsequent thermodynamic quantities, such
as energy, charges and entropy. A key aspect of their computation is that, in
order to obtain the free energy at first order in the higher-derivative
corrections, it suffices to evaluate the on-shell action on the two-derivative
solution, following the Reall and Santos method Reall:2019sah  — see also
Melo:2020amq ; Ma:2023qqj ; Hu:2023gru . Thus, their approach avoids the
complicated task of solving the equations of motion.

Here we take a different approach in order to study the corrections to the
CCLP solution. We consider extremal (but not necessarily supersymmetric) black
holes and we study their near-horizon geometries. Due to the enhanced symmetry
of the near-horizon region, solving the equations of motion is much easier
than for the full solution. Thus, our goal is to compute the corrections to
these solutions and to their thermodynamic quantities by studying their near-
horizon geometries.

There are several reasons that motivate us to pursue this goal. First, even
though the results of Bobev:2022bjm ; Cassani:2022lrk  perfectly agree with
field theory expectations, the Reall-Santos approach depends on carefully
dealing with boundary terms in the on-shell action, which is even more subtle
in the case of AdS spacetimes Hu:2023gru . Thus, we consider it wise to derive
those results by a different method. In addition, the on-shell action approach
does not tell us much about the interpretation of the thermodynamic charges
derived from it. In fact, Cassani:2023vsa , which studied supersymmetric near-
horizon geometries with equal angular momenta, already observed a discrepancy
between the thermodynamic electric charge and the standard definition of
charge understood as a surface integral. Here we further analyze this issue
for general near-horizon geometries. Another reason for following the near-
horizon approach is that it presumably allows for an easier generalization to
subleading corrections with respect to the on-shell action method, as we argue
in Section 7. Finally, obtaining a solution of the equations of motion (in our
case, the near-horizon geometry) gives us much more information than just the
thermodynamic properties and can be useful for other purposes, like e.g. the
study of perturbations Castro:2021wzn .

The study of near-horizon extremal geometries does present some challenges.
Unlike the case of spherically symmetric black holes, where one can elegantly
reduce the problem to an algebraic system of equations by using the entropy
function formalism Sen:2005wa ; Sen:2005iz ; Dabholkar:2006tb ; Sen:2007qy ,
for rotating black holes the story is more involved Astefanesei:2006dd ;
Morales:2006gm ; LopesCardoso:2007hen ; Kunduri:2007vf ; Kunduri:2007qy ;
Cano:2019ozf ; Cano:2023dyg . In the case of five-dimensional solutions, black
holes with two equal angular momenta are of cohomogeneity 1 and have an
enhanced symmetry group that leads to a drastic simplification of the near-
horizon solutions Gutowski:2004ez ; Cassani:2023vsa . However, when the two
angular momenta are different, the solutions take a more complicated form and
we are required to solve a very intricate system of differential equations. On
top of this, there is the question of identifying the various charges and
potentials of the black hole from the near-horizon region. By following the
approaches of Bazanski:1990qd ; Kastor:2008xb ; Kastor:2009wy ; Ortin:2021ade
; Cano:2023dyg ; Cassani:2023vsa , we will compute the charges (electric
charge and angular momenta) by using Komar integrals Komar:1958wp  that are
independent of the surface of integration and hence can be evaluated at the
horizon. The potentials (electrostatic potential and angular velocities of the
horizon) are on the other hand defined as differences of certain quantities
between the horizon and infinity, and hence cannot be determined from the
near-horizon region alone. Nevertheless, one can identify unambiguously
certain quantities that we will refer to as near-horizon potentials. It turns
out that the entropy, charges and these potentials satisfy a near-horizon
version of the first law of thermodynamics, which was already observed (in the
case of equal angular momenta supersymmetric solutions) in Cassani:2023vsa .
Here we show that this law holds for general near-horizon extremal geometries
(including higher-derivative corrections) and we provide an explanation for it
that allows us to interpret the near-horizon potentials.222See Hajian:2013lna
; Hajian:2014twa  for a more general investigation of the laws of black hole
thermodynamics at the near-horizon region.

In the supersymmetric case, our results reproduce those of Bobev:2022bjm ;
Cassani:2022lrk  up to an ambiguity in the definition of the electric charge.
On the other hand, for most of our analysis we do not restrict ourselves to
supersymmetric solutions and we consider general extremal black holes with
arbitrary charge and different angular momenta.

This work is organized as follows. In Section 2, we review the details of the
five-dimensional supergravity action with higher derivative corrections. We
discuss how to compute the conserved charges and we review the CCLP solution
of the two-derivative theory. Section 3 is devoted to a general analysis of
near-horizon geometries. We introduce an appropriate ansatz and show how to
identify the angular coordinates and near-horizon potentials in full
generality. We then move on to Section 4 where we obtain the near-horizon
geometry of the extremal CCLP black holes. We introduce a novel
parametrization that allows us to write the solution in a fully explicit and
compact form, and we derive and study the various thermodynamic quantities and
relations. The analysis of the four-derivative corrections to the solution
comprises Section 5 while we focus on the corrected thermodynamics in Section
6. We conclude with some remarks and open problems in Section 7. In Appendix A
we provide some explicit formulas for the near-horizon solutions that are too
long to be displayed in the main text. We include as well a Mathematica
notebook with our (even longer) complete results.

##  2 Review of higher-derivative AdS5 supergravity

###  2.1 Action

In this section we briefly review the five-dimensional gauged supergravity
theory of interest. We focus on preparing ourselves to study the four-
derivative corrections of the theory and the corresponding Noether surface
charges. After a careful series of field redefinitions starting from the off-
shell formalism Bobev:2021qxx ; Cassani:2022lrk , the result of the
supergravity action is relatively compact

$\displaystyle S\,=$ $\displaystyle\,\frac{1}{16\pi
G_{5}}\int\mathrm{d}^{5}x\,\sqrt{|g|}\left\\{c_{0}R+12c_{1}g^{2}-\frac{c_{2}}{4}F^{2}-\frac{c_{3}}{12\sqrt{3}}\epsilon^{\mu\nu\rho\sigma\lambda}F_{\mu\nu}F_{\rho\sigma}A_{\lambda}\right.$
(1) $\displaystyle\left.\,+\,\lambda_{1}\alpha g^{2}\left[{\cal
X}_{\text{GB}}-\frac{1}{2}C_{\mu\nu\rho\sigma}F^{\mu\nu}F^{\rho\sigma}+\frac{1}{8}F^{4}-\frac{1}{2\sqrt{3}}\epsilon^{\mu\nu\rho\sigma\lambda}R_{\mu\nu\alpha\beta}R_{\rho\sigma}{}^{\alpha\beta}A_{\lambda}\right]\right\\}\,,$

where $g$ is equal to the inverse length of AdS and the Gauss-Bonnet invariant
$\mathcal{X}_{\text{GB}}$, Weyl tensor $C_{\mu\nu\rho\sigma}$ and the Maxwell
field invariants $F^{2}$ and $F^{4}$ are given by

$\displaystyle{\cal X}_{\text{GB}}$
$\displaystyle=R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}-4R_{\mu\nu}R^{\mu\nu}+R^{2}\,,$
(2a) $\displaystyle C_{\mu\nu\rho\sigma}$
$\displaystyle=R_{\mu\nu\rho\sigma}-\frac{2}{3}\left(R_{\mu[\rho}g_{\sigma]\nu}+R_{\nu[\sigma}g_{\rho]\mu}\right)+\frac{1}{6}Rg_{\mu[\rho}g_{\sigma]\nu}\,,$
(2b) $\displaystyle F^{2}$ $\displaystyle=F_{\mu\nu}F^{\mu\nu}\,,$ (2c)
$\displaystyle F^{4}$
$\displaystyle=F_{\mu\nu}F^{\nu\rho}F_{\rho\sigma}F^{\sigma\mu}\,.$ (2d)

The various constants $c_{i}$ are defined as ${c}_{i}=1+\alpha g^{2}{\delta
c}_{i}$ with

${\delta c}_{0}=4\lambda_{2}\,,\hskip 14.22636pt{\delta
c}_{1}=-10\lambda_{1}+4\lambda_{2}\,,\hskip 14.22636pt{\delta
c}_{2}=4\lambda_{1}+4\lambda_{2}\,,\hskip 14.22636pt{\delta
c}_{3}=-12\lambda_{1}+4\lambda_{2}\,,$ (3)

where $\lambda_{1}$ and $\lambda_{2}$ are two free coupling constants that
control the corrections. The action is written in a way such that the first
line in (1) is the usual two-derivative contribution when $\alpha$ is set to
zero. We note that the effect of $\lambda_{2}$ is just to renormalize the
coupling constants and fields of the two-derivative action. In particular,
since it changes the normalization of the Einstein-Hilbert term, it is natural
to introduce an effective Newton’s constant,

$G_{\rm eff}=\frac{G_{5}}{1+4\alpha g^{2}\lambda_{2}}\,,$ (4)

which will naturally appear in some of our results. From the holographic
perspective, the $\lambda_{i}$ couplings modify the CFT central charges
$\mathfrak{a}$ and $\mathfrak{c}$, and in particular they break the degeneracy
between them — this is a generic effect of higher-derivative terms
Hofman:2008ar ; Buchel:2009sk ; Myers:2010jv ; Bueno:2018xqc ; Cano:2022ord .
In fact, we have Bobev:2021qxx ; Cassani:2022lrk

$\mathfrak{a}=\frac{\pi}{8g^{3}G_{\rm
eff}}\,,\qquad\mathfrak{a}-\mathfrak{c}=-\frac{\pi\alpha\lambda_{1}}{gG_{\rm
eff}}\,.$ (5)

###  2.2 Equations of motion and conserved quantities

From the action (1), we can derive the equations of motion
$\mathcal{E}_{\mu\nu}$ and $\mathcal{E}_{\mu}$ by varying the metric as well
as the gauge potential, respectively. This gives us

$\displaystyle\mathcal{E}_{\mu\nu}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}g_{\mu\nu}{\cal
L}^{\prime}+P^{\alpha\beta\gamma}{}_{(\mu|}R_{\alpha\beta\gamma|\nu)}-2\nabla^{\alpha}\nabla^{\beta}(P_{\beta(\mu\nu)\alpha}+\Pi_{\beta(\mu\nu)\alpha})-\frac{1}{2}\mathcal{F}_{(\mu|\rho}F_{|\nu)}{}^{\rho}\,,$
(6) $\displaystyle\mathcal{E}_{\mu}$ $\displaystyle=$
$\displaystyle-\nabla_{\mu}\mathcal{F}^{\mu\nu}+\frac{c_{3}}{4\sqrt{3}}\epsilon^{\nu\alpha\beta\gamma\delta}F_{\alpha\beta}F_{\gamma\delta}+\frac{\lambda_{1}\,\alpha}{2\sqrt{3}}\epsilon^{,nu\alpha\beta\gamma\delta}R_{\alpha\beta\rho\sigma}R_{\gamma\delta}{}^{\rho\sigma}\,.$
(7)

where

$\displaystyle P_{\mu\nu\rho\sigma}=$ $\displaystyle\frac{\partial{\cal
L}^{\prime}}{\partial
R_{\mu\nu\rho\sigma}}=\,{c}_{0}g_{\mu[\rho}g_{\sigma]\nu}+\lambda_{1}\,\alpha\left[2R_{\mu\nu\rho\sigma}-4\left(R_{\mu[\rho}g_{\sigma]\nu}-R_{\nu[\rho}g_{\sigma]\mu}\right)+2g_{\mu[\rho}g_{\sigma]\nu}R\right.$
$\displaystyle\left.-\frac{1}{2}F_{\mu\nu}F_{\rho\sigma}-\frac{1}{12}g_{\mu[\rho}g_{\sigma]\nu}F^{2}+\frac{1}{3}\left(F_{\mu\alpha}F_{[\rho}{}^{\alpha}g_{\sigma]\nu}-F_{\nu\alpha}F_{[\rho}{}^{\alpha}g_{\sigma]\mu}\right)\right]\,,$
(8a) $\displaystyle\Pi^{\mu\nu\rho\sigma}=$
$\displaystyle-\frac{\lambda_{1}\,\alpha}{\sqrt{3}}\epsilon^{\mu\nu\alpha\beta\gamma}R_{\alpha\beta}{}^{\rho\sigma}A_{\gamma}\,,$
(8b) $\displaystyle\mathcal{F}^{\mu\nu}=$ $\displaystyle-2\frac{\partial{\cal
L}^{\prime}}{\partial
F_{\mu\nu}}=c_{2}F^{\mu\nu}-2\lambda_{1}\,\alpha\left(-C^{\mu\nu\rho\sigma}F_{\rho\sigma}+\frac{1}{2}F^{\mu\rho}F_{\rho\sigma}F^{\nu\sigma}\right)\,,$
(8c)

and $\mathcal{L}^{\prime}=\mathcal{L}-\mathcal{L}_{\rm CS}$ is the Lagrangian
without the Chern-Simons terms.333We also point out that we define the
Lagrangian without the $(16\pi G_{5})^{-1}$ factor, so
$\mathcal{L}=c_{0}R+\ldots$ .  It is manifest that these nonlinear coupled
differential equations are highly complicated to solve. Nevertheless, our goal
is to solve them for near-horizon geometries and to evaluate the charges from
a first principles approach. Here we review how obtain the conserved charges
via the Noether current associated to spacetime symmetries and gauge
transformations. For a thorough analysis of the conserved charges of the
theory (1) we refer to Cassani:2023vsa . Here we quote their results and
remark on some of the main subtleties.

Due to the Chern-Simons terms, there are several notions of electric charge
that one may consider Marolf:2000cb . Here we will use the “Page” charge
PhysRevD.28.2976 , which is given by

$\displaystyle Q$ $\displaystyle=-\frac{1}{16\pi
G_{5}}\int_{\Sigma}\left(\star{\cal F}-\frac{c_{3}}{\sqrt{3}}F\wedge
A-\frac{2\lambda_{1}\alpha}{\sqrt{3}}\Omega_{\rm{CS}}\right)\,,$ (9)

where $\Omega_{\rm{CS}}$ is the Lorentz-Chern-Simons three-form defined by

$\Omega_{\rm{CS}}=\mathrm{d}\omega^{ab}\wedge\omega_{ab}-\frac{2}{3}\,\omega^{a}{}_{b}\wedge\omega^{b}{}_{c}\wedge\omega^{c}{}_{a}\,,$
(10)

and $\omega^{ab}=\,\omega^{ab}{}_{\mu}\,\mathrm{d}x^{\mu}$ is the spin
connection and the Latin indices are flat. This charge satisfies a Gauss law,
so the result is independent of the choice of integration surface $\Sigma$, as
long as this is a co-dimension 2 spacelike hypersurface homeomorphic to the
sphere at infinity. Thus, it can be evaluated at the black hole horizon,
$\Sigma=\mathcal{H}$. On the other hand, $Q$ is gauge and frame dependent, due
to the gauge and Lorentz-Chern-Simons three-forms, respectively. The gauge
ambiguity can be fixed by demanding regularity of the vector field at the
horizon, but the frame ambiguity is more worrisome as there is no canonical
choice of frame. We revisit this point in detail in Section 6.1.

If the solution has a Killing vector $\xi_{\mu}$, one also finds the conserved
current

$\boldsymbol{J}_{\xi}=\mathrm{d}\boldsymbol{Q}_{\xi}\,,$ (11)

where $\boldsymbol{Q}_{\xi}$ is the Noether charge three-form, given by

$\boldsymbol{Q}_{\xi}=-\frac{1}{32\pi
G_{5}}\boldsymbol{\epsilon}_{\mu\nu}\left[4\nabla_{\sigma}\mathcal{P}^{\mu\nu\sigma\rho}\xi_{\rho}-2\mathcal{P}^{\mu\nu\sigma\rho}\nabla_{\sigma}\xi_{\rho}-\left(\xi_{\gamma}A^{\gamma}\right)\left(\mathcal{F}^{\mu\nu}+\frac{c_{3}}{3\sqrt{3}}\epsilon^{\mu\nu\rho\sigma\lambda}A_{\rho}F_{\sigma\lambda}\right)\right]\,,$
(12)

and where

${\cal
P}^{\mu\nu\rho\sigma}=\,P^{\mu\nu\rho\sigma}+\Pi^{\mu\nu\rho\sigma}-(P^{\mu[\nu\rho\sigma]}+\Pi^{\mu[\nu\rho\sigma]})\,,$
(13)

where the antisymmetrized terms are there to guarantee that ${\cal
P}^{\mu[\nu\rho\sigma]}=0$.444Those terms do not affect the equations of
motion, but they are crucial in the computation of the Noether charge. The
integral of $\boldsymbol{J}_{\xi}$ over any Cauchy slice $\mathcal{C}$ then
yields the conserved charge, and through the use of Stokes’ theorem one can
reduce this to an integral over the boundary, $\Sigma=\partial\mathcal{C}$,

$J[\xi]=\int_{\Sigma}\boldsymbol{Q}_{\xi}\,.$ (14)

In this case, it is important that $\Sigma$ is the sphere at infinity, since
$\mathrm{d}\boldsymbol{Q}_{\xi}\neq 0$ and therefore we do not have a Gauss
law. Hence, the conserved charges such as the total energy and angular momenta
must in principle be computed at infinity. A way around this consists in
defining a “Noether-Komar” charge three-form

$\widetilde{\boldsymbol{Q}}_{\xi}=\boldsymbol{Q}_{\xi}-\boldsymbol{\Omega}\,,$
(15)

where $\mathrm{d}\boldsymbol{\Omega}=\boldsymbol{J}_{\xi}\big{|}_{\rm on-
shell}$, so that we have $\mathrm{d}\widetilde{\boldsymbol{Q}}_{\xi}=0$ on-
shell. Finding such a three-form $\boldsymbol{\Omega}$ is always possible by
noting that, on-shell, one has
$\boldsymbol{J}_{\xi}=-\frac{1}{2}\star\boldsymbol{\xi}\mathcal{L}$. For more
details about the construction of these Komar charges, we refer to
Bazanski:1990qd ; Kastor:2008xb ; Kastor:2009wy ; Ortin:2021ade ; Cano:2023dyg
; Cassani:2023vsa . Now, the integral of $\widetilde{\boldsymbol{Q}}_{\xi}$ is
independent of the surface of integration, and it computes the Noether charge
as long as the integral of $\boldsymbol{\Omega}$ at infinity vanishes. Thus,
one can use $\tilde{\boldsymbol{Q}}_{\xi}$ to evaluate the charges as an
integral over the horizon. However, there is a final twist in this story. If
one only has access to the near-horizon region, then one usually cannot fix
the ambiguity in $\boldsymbol{\Omega}$ by demanding that its integral at
infinity vanishes. This is one of the reasons why it is not possible to
compute the mass from the near-horizon region. Fortunately, in the case of the
angular momenta, it is possible to fix this ambiguity. It turns out that, in a
stationary spacetime, one can choose the $\boldsymbol{\Omega}$ for the
rotational Killing vectors in a way that its integral on any constant time
surface555The time coordinate is taken as the coordinate associated to the
time-like Killing vector. vanishes Cassani:2023vsa . For a very explicit
example of this we refer to Cano:2023dyg . Thus, to make the long story short,
it turns out one can actually compute the angular momenta by integrating (14)
on the near-horizon region as long as we integrate on a constant time surface.

Finally, we are also interested in the entropy of the black hole, which is
given by the Iyer-Wald formula Wald:1993nt ; Iyer:1994ys

$S=-\frac{1}{8G_{5}}\int_{\cal H}\mathrm{d}^{3}x\,\sqrt{\gamma}\,{\cal
P}^{\mu\nu\rho\sigma}\,n_{\mu\nu}\,n_{\rho\sigma}\,,$ (16)

where $\gamma$ is the determinant of the induced three-dimensional horizon
metric and $n_{\mu\nu}$ is the binormal to the horizon normalized via
$n_{\mu\nu}n^{\mu\nu}=-2$. We note that this formula, just like those for the
charge and the angular momenta, is gauge-dependent. It is possible to refine
Wald’s formalism in order to derive a gauge-invariant entropy formula
Elgood:2020svt ; Elgood:2020mdx ; Elgood:2020nls , but here we will fix this
ambiguity by demanding regularity of the gauge field at the horizon. The
validity of this approach is ultimately verified by checking that our results
are compatible with the first law of thermodynamics and that they agree with
the results of Bobev:2022bjm ; Cassani:2022lrk , obtained by different
methods.

###  2.3 Black holes: review of the CCLP solution

As we are interested in finding corrections to the five-dimensional AdS black
hole with two distinct angular momenta and electric charge as was first
constructed in Chong:2005hr , we briefly review the solution, along with its
thermodynamic quantities. The metric and gauge field are given by

$\displaystyle ds^{2}=$
$\displaystyle-\frac{\Delta_{\theta}\left[\left(1+g^{2}r^{2}\right)\rho^{2}dt+2q\nu\right]dt}{\Xi_{a}\Xi_{b}\rho^{2}}+\frac{2q\nu\omega}{\rho^{2}}+\frac{\gamma}{\rho^{4}}\left(\frac{\Delta_{\theta}dt}{\Xi_{a}\Xi_{b}}-\omega\right)^{2}+\frac{\rho^{2}dr^{2}}{\Delta_{r}}+\frac{\rho^{2}d\theta^{2}}{\Delta_{\theta}}$
(17) $\displaystyle+\frac{r^{2}+a^{2}}{\Xi_{a}}\sin^{2}\theta
d\phi^{2}+\frac{r^{2}+b^{2}}{\Xi_{b}}\cos^{2}\theta d\psi^{2},$ $\displaystyle
A=$
$\displaystyle\frac{\sqrt{3}q}{2\rho^{2}}\left(\frac{\Delta_{\theta}dt}{\Xi_{a}\Xi_{b}}-\omega\right),$

where

$\displaystyle\nu$ $\displaystyle=b\sin^{2}\theta d\phi+a\cos^{2}\theta
d\psi,\quad$ $\displaystyle\Delta_{r}$
$\displaystyle=g^{2}\left(r^{2}+a^{2}\right)\left(r^{2}+b^{2}\right)\left(1+\frac{1}{g^{2}r^{2}}\right)+\frac{q^{2}+2abq}{r^{2}}-2m,$
(18) $\displaystyle\omega$
$\displaystyle=a\sin^{2}\theta\frac{d\phi}{\Xi_{a}}+b\cos^{2}\theta\frac{d\psi}{\Xi_{b}},\quad$
$\displaystyle\Delta_{\theta}$
$\displaystyle=1-a^{2}g^{2}\cos^{2}\theta-b^{2}g^{2}\sin^{2}\theta,$
$\displaystyle\rho^{2}$
$\displaystyle=r^{2}+a^{2}\cos^{2}\theta+b^{2}\sin^{2}\theta,\quad$
$\displaystyle\Xi_{a}$ $\displaystyle=1-a^{2}g^{2},$ $\displaystyle\gamma$
$\displaystyle=2m\rho^{2}-q^{2}+2abqg^{2}\rho^{2},\quad$
$\displaystyle\Xi_{b}$ $\displaystyle=1-b^{2}g^{2}.$

The general non-extremal black hole is paramatrized by the four parameters
$\\{a,b,m,q\\}$ associated to the angular momenta $J_{1},J_{2}$, energy $E$
and electric charge $Q$. The conserved quantities $J_{1},J_{2}$ and $Q$ may be
found via the Komar integrals we introduced above (but specified for the two-
derivative theory)

$\displaystyle J_{1}$ $\displaystyle=\frac{1}{16\pi G_{5}}\int_{S^{3}}\star
d\xi_{\phi}=\frac{\pi\left[2am+qb\left(1+a^{2}g^{2}\right)\right]}{4G_{5}\Xi_{a}^{2}\Xi_{b}},$
(19) $\displaystyle J_{2}$ $\displaystyle=\frac{1}{16\pi
G_{5}}\int_{S^{3}}\star
d\xi_{\psi}=\frac{\pi\left[2bm+qa\left(1+b^{2}g^{2}\right)\right]}{4G_{5}\Xi_{b}^{2}\Xi_{a}},$
(20) $\displaystyle Q$ $\displaystyle=\frac{1}{16\pi
G_{5}}\int_{S^{3}}\left(\star F-\frac{1}{\sqrt{3}}F\wedge
A\right)=\frac{\sqrt{3}\pi q}{4G_{5}\Xi_{a}\Xi_{b}},$ (21)

where $\xi_{\phi}=-g_{\mu\phi}dx^{\mu}$ and $\xi_{\psi}=-g_{\mu\psi}dx^{\mu}$
are the Killing vectors $-\partial_{\phi}$ and $-\partial_{\psi}$ expressed as
1-forms.

The Hawking temperature is derived by requiring appropriate periodic
identifications in Euclidean time, which leads to

$\displaystyle T=\beta^{-1}$
$\displaystyle=\frac{r_{+}^{4}\left[\left(1+g^{2}\left(2r_{+}^{2}+a^{2}+b^{2}\right)\right]-(ab+q)^{2}\right.}{2\pi
r_{+}\left[\left(r_{+}^{2}+a^{2}\right)\left(r_{+}^{2}+b^{2}\right)+abq\right]}.$
(22)

It is important to also find the chemical potentials associated to the angular
momenta and the electric charge. The angular velocities $\Omega_{1}$ and
$\Omega_{2}$ are given by

$\displaystyle\Omega_{1}$
$\displaystyle=\frac{a\left(r_{+}^{2}+b^{2}\right)\left(1+g^{2}r_{+}^{2}\right)+bq}{\left(r_{+}^{2}+a^{2}\right)\left(r_{+}^{2}+b^{2}\right)+abq},\quad\Omega_{2}=\frac{b\left(r_{+}^{2}+a^{2}\right)\left(1+g^{2}r_{+}^{2}\right)+aq}{\left(r_{+}^{2}+a^{2}\right)\left(r_{+}^{2}+b^{2}\right)+abq}.$
(23)

Via the Killing field that generates the horizon
$\chi^{\mu}=\partial_{t}+\Omega_{1}\partial_{\phi}+\Omega_{2}\partial_{\psi}$,
the electrostatic potential is

$\displaystyle\Phi$
$\displaystyle=\left.\chi^{\mu}A_{\mu}\right|_{r\to\infty}-\left.\chi^{\mu}A_{\mu}\right|_{r\to
r_{+}}=\frac{\sqrt{3}gqr_{+}^{2}}{\left(r_{+}^{2}+a^{2}\right)\left(r_{+}^{2}+b^{2}\right)+abq}.$
(24)

Finally, the entropy can be computed via the area of the horizon

$S=\frac{\pi^{2}\left[\left(r_{+}^{2}+a^{2}\right)\left(r_{+}^{2}+b^{2}\right)+abq\right]}{2G_{5}\Xi_{a}\Xi_{b}r_{+}}.$
(25)

We are particularly interested in extremal solutions, i.e., those where the
inner and outer horizons coincide and the Hawking temperature vanishes. This
condition is satisfied for

$\displaystyle\Delta_{r}(r_{+})=0,\quad\partial_{r}\Delta_{r}(r_{+})=0.$ (26)

For extremal solutions, the parametrization chosen in Chong:2005hr  with
$\\{a,b,m,q\\}$ is not optimal as $\Delta_{r}$ is an order 6 polynomial and in
order to write the thermodynamic quantities in a compact way, one would also
require to use the parameter $r_{+}$ which is dependent on the others. We
remedy this difficulty by finding a novel parametrization that automatically
solves (26). The result is that the metric and thermodynamic quantities can be
written compactly in terms of three parameters. We discuss this in detail in
Section 4.

Finally, the CCLP solution becomes supersymmetric when the constraint

$q=\frac{m}{1+(a+b)g}$ (27)

is satisfied. In general, the Lorentzian supersymmetric solutions are
pathological unless one imposes an additional condition, corresponding
precisely to the extremality condition $\partial_{r}\Delta_{r}(r_{+})=0$. For
this supersymmetric and extremal solution the entropy can be expressed
explicitly as a function of the charges as

$S=\pi\sqrt{4g^{-2}Q^{2}-\frac{\pi}{G_{5}g^{3}}(J_{1}+J_{2})}\,.$ (28)

In addition, since the extremal supersymmetric solution only depends on two
parameters, the charges satisfy a nonlinear constraint

$\left[2\sqrt{3}g^{-1}Q+\frac{\pi}{2G_{5}g^{3}}\right]\left[4g^{-2}Q^{2}-\frac{\pi}{G_{5}g^{3}}(J_{1}+J_{2})\right]-\left(\frac{2Q}{\sqrt{3}g}\right)^{3}-\frac{2\pi}{G_{5}g^{3}}J_{1}J_{2}=0\,.$
(29)

These results can be matched with the field theory prediction for the entropy
associated to the superconformal index as first shown in Cabo-Bizet:2018ehj ;
Benini:2018ywd ; Choi:2018hmj . References Bobev:2022bjm ; Cassani:2022lrk
found the higher-derivative corrections to these expressions and matched them
again with the field theory results at subleading order in
$(\mathfrak{a}-\mathfrak{c})$. Here we intend to re-derive those results by
analyzing the higher-derivative corrections to near-horizon geometries.

##  3 Near-horizon geometries with
$\mathrm{SL}(2,\mathbb{R})\times\mathrm{U}(1)\times\mathrm{U}(1)$ symmetry

###  3.1 Ansatz

The near-horizon geometries in which we are interested consist of an AdS2
geometry fibered over two circles, with a total symmetry group
$\mathrm{SL}(2,\mathbb{R})\times\mathrm{U}(1)\times\mathrm{U}(1)$. Near-
horizon geometries of the same type have been studied in previous literature,
e.g. Kunduri:2007qy ; Kunduri:2008rs . However, here we perform an independent
analysis that we deem to be more suitable for the applications we pursue. In
particular, it is crucial to us not only analyzing the local form of the
solution but also its global structure.

Without loss of generality, a metric and vector field with
$\mathrm{SL}(2,\mathbb{R})\times\mathrm{U}(1)\times\mathrm{U}(1)$ symmetry can
always be written as

$\displaystyle ds^{2}$
$\displaystyle=W(x)\left(-r^{2}dt^{2}+\frac{dr^{2}}{r^{2}}\right)+\frac{W(x)}{G(x)}dx^{2}+\sum_{a,b=1}^{2}H_{ab}(x)\left(d\phi_{a}-\omega_{a}rdt\right)\left(d\phi_{b}-\omega_{b}rdt\right)\,,$
(30) $\displaystyle A$ $\displaystyle=\psi
rdt+\sum_{a=1}^{2}A_{a}(x)\left(d\phi_{a}-\omega_{a}rdt\right)\,,$ (31)

where $a,b=1,2$, the coordinate $x$ is compact and $\phi_{1}$, $\phi_{2}$ are
angular coordinates with periodicity $2\pi$.666As we will see, the metric
functions $G(x)$ and $H_{ab}(x)$ have to satisfy certain properties in order
to ensure absence of conical defects. The quantities $\omega_{a}$ are constant
— required by the $\mathrm{SL}(2,\mathbb{R})$ symmetry — and they are
interpreted as chemical potentials conjugate to the angular momenta in each of
the directions $\phi_{1}$ and $\phi_{2}$. They are related to the angular
velocity of the horizon, but in the near-horizon geometry they lose that
meaning as we lack an asymptotic region from where we can measure those
velocities. Analogously, the constant $\psi$ is a variable conjugate to the
electric charge and hence it is related to the electrostatic potential,
although again the lack of an asymptotic region forbids us to interpret it as
the usual electrostatic potential at infinity. Importantly, the three
potentials $(\psi,\omega_{1},\omega_{2})$ are univocally determined from the
near-horizon geometry once we impose regularity of the metric (absence of
conical defects) and of the gauge field (absence of divergences). We detail
this below.

The metric (30) still has some gauge freedom associated to reparametrizations
of the $x$ coordinate. This allows us to set an extra condition in order to
fix the gauge, and we find it useful to set

$\frac{W(x)}{G(x)}\det\left[H_{ab}(x)\right]=\text{const.}$ (32)

Now, even though (30) is the most intuitive way of writing a metric with the
given symmetries, it is not the most convenient form when solving the
equations of motion. In fact, by performing a change of coordinates

$\displaystyle y_{1}$ $\displaystyle=s_{1}\phi_{1}+s_{2}\phi_{2}\,,$ (33)
$\displaystyle y_{2}$ $\displaystyle=s_{3}\phi_{1}+s_{4}\phi_{2}\,,$

where the $s_{i}$ are constants, we can rearrange the solution as

$\displaystyle ds^{2}$
$\displaystyle=W(x)\left(-r^{2}dt^{2}+\frac{dr^{2}}{r^{2}}\right)+\frac{W(x)}{G(x)}dx^{2}+\frac{G(x)}{W(x)B(x)}dy_{1}^{2}+B(x)\left(dy_{2}+J(x)dy_{1}-rdt\right)^{2}\,,$
(34) $\displaystyle A$ $\displaystyle=\psi rdt+\left(\hat{A}_{1}(x)-\psi
J(x)+k\right)dy_{1}+\left(\hat{A}_{2}(x)+\psi\right)\left(dy_{2}+J(x)dy_{1}-rdt\right)\,,$

where we have set $g_{xx}g_{y_{1}y_{1}}g_{y_{2}y_{2}}=1$ following the gauge
condition (32). Furthermore, we have decomposed the $A_{y_{1}}$ and
$A_{y_{2}}$ components of the gauge field in a way in which $\psi$ as well as
the new constant $k$ are gauge parameters. Thus, they do not appear in the
equations of motion, but nevertheless they are fixed by regularity, as we show
next.

###  3.2 Angular coordinates and chemical potentials

In practice, once a solution of the form (34) has been found, one must then
undo the change of variables (33) in order to identify the angular coordinates
$\phi_{a}$ and the chemical potentials $\omega_{a}$. However, one does not
know a priori this change of variables and one must derive it entirely from
the properties of the near-horizon metric (34). Here we show how to do this in
full generality upon the assumption of $\mathbb{S}^{3}$ topology, which is the
topology of the CCLP black hole horizon Chong:2005hr .

As was mentioned, the coordinate $x$ is compact and thus it has a range
$x\in[x_{1},x_{2}]$, which is determined by the vanishing of $G(x)$ at those
points. However, by doing a linear change of variable
$x\rightarrow\alpha+\beta x$ we can always set $x_{1}=-1$, $x_{2}=1$. Observe
that the effect of this change of coordinates can be reabsorbed by performing
$y_{1}\rightarrow y_{1}/\beta^{2}$ and redefining
$\left\\{G(x),J(x),\hat{A}_{1}(x),k\right\\}\rightarrow\left\\{\beta^{2}G(x),\beta
J(x),\beta\hat{A}_{1}(x),\beta k\right\\}$. Hence, without loss of generality
we assume $x\in[-1,1]$, which allows us to interpret $x=\cos\tilde{\theta}$
with $\tilde{\theta}\in[0,\pi]$, although we will work in terms of $x$ for
convenience.

The points $x=\pm 1$ are then identified as the poles of the horizon. Near
those points the angular coordinates are identified by demanding regularity of
the metric, i.e., absence of conical defects. Let us examine the behavior of
the metric near those points for a constant $t$ and $r$ slice

$\displaystyle ds^{2}_{3}$
$\displaystyle=\frac{W(x)}{G(x)}dx^{2}+\frac{G(x)}{W(x)B(x)}dy_{1}^{2}+B(x)\left(dy_{2}+J(x)dy_{1}\right)^{2}\,.$
(35)

First, we introduce the coordinate $z$ such that

$dz=dx\sqrt{\frac{W(x)}{G(x)}}\,.$ (36)

Integrating this relation near $x=\pm 1$ yields

$x=1+\frac{z_{+}^{2}G^{\prime}(1)}{4W(1)}+\mathcal{O}(z_{+}^{4})\,,\quad
x=-1+\frac{z_{-}^{2}G^{\prime}(-1)}{4W(-1)}+\mathcal{O}(z_{-}^{4})\,.$ (37)

We then rewrite the $y_{a}$ coordinates in terms of the angular coordinates
$\phi_{a}$ as in (33). We demand the periodicity of $\phi_{a}$ to be $2\pi$
and $s_{i}$ are undetermined coefficients. We series expand the metric around
$x=-1$ and $x=+1$ up to order $z^{2}$. Then, in order for the geometry to be
regular, the metric around those points must take the form

$\displaystyle ds^{2}_{3}(x\rightarrow+1)$
$\displaystyle=dz_{+}^{2}+z_{+}^{2}\left(d\phi_{2}+C_{+}d\phi_{1}\right)^{2}+(D_{+}+E_{+}z_{+}^{2})d\phi_{1}^{2}+\mathcal{O}(z_{+}^{4})\,,$
(38) $\displaystyle ds^{2}_{3}(x\rightarrow-1)$
$\displaystyle=dz_{-}^{2}+z_{-}^{2}\left(d\phi_{1}+C_{-}d\phi_{2}\right)^{2}+(D_{-}+E_{-}z_{-}^{2})d\phi_{2}^{2}+\mathcal{O}(z_{-}^{4})\,,$
(39)

where $C_{\pm}$, $D_{\pm}$, $E_{\pm}$ are constants. In particular, this means
that

$g_{\phi_{2}\phi_{2}}=z_{+}^{2}+\mathcal{O}(z_{+}^{4})\,,\quad
g_{\phi_{1}\phi_{2}}=0+\mathcal{O}(z_{+}^{2})$ (40)

near $x=1$ while

$g_{\phi_{1}\phi_{1}}=z_{-}^{2}+\mathcal{O}(z_{-}^{2})\,,\quad
g_{\phi_{1}\phi_{2}}=0+\mathcal{O}(z_{-}^{2})$ (41)

near $x=-1$. These are four conditions that fix the four coefficients $s_{i}$.
We find the unique solution up to orientation changes to be

$\displaystyle y_{1}$
$\displaystyle=2\mathcal{H}(1)\phi_{2}+2\mathcal{H}(-1)\phi_{1}\,,$ (42)
$\displaystyle y_{2}$
$\displaystyle=-2J(1)\mathcal{H}(1)\phi_{2}-2J(-1)\mathcal{H}(-1)\phi_{1}\,,$

where we have defined

$\displaystyle\mathcal{H}(x)=\frac{\sqrt{B(x)}W(x)}{G^{\prime}(x)}\,.$ (43)

From the relations (42) we find the generators of rotations

$\displaystyle\partial_{\phi_{1}}$
$\displaystyle=2\mathcal{H}(-1)\left(\partial_{y_{1}}-J(-1)\partial_{y_{2}}\right)\,,$
(44) $\displaystyle\partial_{\phi_{2}}$
$\displaystyle=2\mathcal{H}(1)\left(\partial_{y_{1}}-J(1)\partial_{y_{2}}\right)\,,$

which we need in order to compute the angular momenta. We can also obtain the
area element of the horizon in terms of the angular coordinates,

$d\Sigma_{3}=dx\wedge dy_{1}\wedge dy_{2}=4dx\wedge d\phi_{1}\wedge
d\phi_{2}\mathcal{H}(1)\mathcal{H}(-1)(J(-1)-J(1))\,,$ (45)

which allows us to compute the area right away

$\mathcal{A}=32\pi^{2}|\mathcal{H}(1)\mathcal{H}(-1)(J(1)-J(-1))|\,.$ (46)

After identifying the angular coordinates, we can apply the change of
coordinates (42) in (34) and rewrite the metric in the canonical form (30).
This allows us to read off the chemical potentials $\omega_{a}$

$\displaystyle\omega_{1}=\frac{1}{2\mathcal{P}(-1)(J(1)-J(-1))}\,,\quad\omega_{2}=-\frac{1}{2\mathcal{P}(1)(J(1)-J(-1))}\,.$
(47)

On the other hand, the electrostatic potential is determined by the regularity
of the gauge field at the poles of the horizon. This can be analyzed by using
(34). Since the metric component $g_{y_{1}y_{1}}$ vanishes at $x=\pm 1$, the
component of the gauge field $A_{y_{1}}$ must also vanish at those points —
otherwise the gauge field would be singular, as one can check by computing its
norm. Thus, we impose

$\hat{A}_{1}(x)-\psi J(x)+k\Big{|}_{x=\pm 1}=0$ (48)

and we find

$\displaystyle\psi=$
$\displaystyle\frac{\hat{A}_{1}(1)-\hat{A}_{1}(-1)}{J(1)-J(-1)}\,,$ (49)
$\displaystyle k=$
$\displaystyle\frac{J(-1)\hat{A}_{1}(1)-J(1)\hat{A}_{1}(-1)}{J(1)-J(-1)}\,.$
(50)

In this way, all the chemical potentials are intrinsically determined by the
regularity of the solution.

##  4 Near-horizon geometry of the two-derivative theory

###  4.1 Solving the equations of motion

The near-horizon extremal geometry in minimal gauged supergravity can be
obtained by taking the appropriate near-horizon limit of the CCLP solution.
However, the resulting metric and gauge potential are considerably involved
and not in the form of (34). Another obstacle is that in the case of higher-
derivative gravity, we do not know the complete solution and we must work
directly in the near-horizon geometry. Thus, here we solve directly the
equations of motion of the two-derivative theory for the ansatz (34). This
allows us to extract some insight on how the equations may be solved and then
to implement it in the case when higher derivatives are turned on. Along the
way, we use inspiration from the CCLP metric in some steps of the derivation.

First, it is useful to express the equations in the frame

$\displaystyle e^{0}$ $\displaystyle=\sqrt{W}rdt\,,\quad$ $\displaystyle
e^{1}$ $\displaystyle=\sqrt{W}\frac{dr}{r}\,,\quad
e^{2}=\sqrt{\frac{W}{G}}dx\,,$ (51) $\displaystyle\quad e^{3}$
$\displaystyle=\sqrt{\frac{G}{WB}}dy_{1}\,,\quad$ $\displaystyle e^{4}$
$\displaystyle=\sqrt{B}\left(dy_{2}+Jdy_{1}-rdt\right)\,.$

We note that the functions $J(x)$ and $\hat{A}_{1}(x)$ always appear with at
least one derivative acting upon them, and we can instead consider our
variables to be $J^{\prime}(x)$ and $\hat{A}_{1}^{\prime}(x)$, reducing the
order of the equations. In fact, it proves interesting to perform the change
of variables

$J^{\prime}=\frac{j}{WB}\,,\quad\hat{A}_{1}^{\prime}=\frac{\zeta
B-j\hat{A}_{2}}{WB^{2}}\,,$ (52)

where we have introduced the functions $j(x)$ and $\zeta(x)$. In terms of
these variables, the $\mathcal{E}_{3}$ component of the Maxwell equation and
the $\mathcal{E}_{34}$ component of the Einstein equation yield, respectively

$\displaystyle
3\zeta^{\prime}-2\sqrt{3}\hat{A}_{2}\hat{A}_{2}^{\prime}=0\,,\quad
j^{\prime}+\zeta\hat{A}_{2}^{\prime}=0\,.$ (53)

These can be integrated immediately to find

$\zeta=\kappa_{1}+\frac{1}{\sqrt{3}}\hat{A}_{2}^{2}\,,\quad
j=\kappa_{2}-\kappa_{1}\hat{A}_{2}-\frac{1}{3\sqrt{3}}\hat{A}_{2}^{2}\,,$ (54)

where $\kappa_{1}$ and $\kappa_{2}$ are integration constants. On the other
hand, we have the following combination of components of Einstein equations

$\mathcal{E}_{00}-\mathcal{E}_{22}=-\frac{G^{\prime\prime}-24g^{2}W+2}{2W}=0\,,$
(55)

so that we can express $W$ in terms of $G$ as

$W=\frac{G^{\prime\prime}+2}{24g^{2}}\,.$ (56)

Another interesting combination of Einstein equations is the following

$\displaystyle\mathcal{E}_{00}+\mathcal{E}_{22}+\mathcal{E}_{33}+\mathcal{E}_{44}=-\frac{3}{2BW^{2}}\Bigg{[}B^{2}+\frac{B}{3W}\left(W\left(2\hat{A}_{2}^{2}-3\left(G^{\prime}W^{\prime}+GW^{\prime\prime}\right)\right)\right.$
(57)
$\displaystyle\left.+3GW^{\prime}{}^{2}+24g^{2}W^{3}-6W^{2}\right)+\frac{1}{9}\left(3GW\hat{A}_{2}^{\prime}{}^{2}+\hat{A}_{2}^{4}+2\sqrt{3}\hat{A}_{2}^{2}\kappa_{1}+3\kappa_{1}{}^{2}\right)\Bigg{]}=0\,.$

This equation determines algebraically $B$ in terms of $G$ and $\hat{A}_{2}$,
since $W$ is also determined by $G$. The remaining Einstein equations and the
$\mathcal{E}_{4}$ component of Maxwell equations form a highly nonlinear
system of equations of $G$ and $\hat{A}_{2}$. Furthermore, these contain up to
six derivatives of $G$, making finding the solution highly challenging. To
remedy the difficulty, we take inspiration from the near-horizon limit of the
CCLP metric and consider the following ansatz for these functions

$G=(1-x^{2})(\kappa_{3}+\kappa_{4}x)\,,\quad\hat{A}_{2}=\frac{\kappa_{5}}{W}\,,$
(58)

with undetermined constants $\kappa_{i}$. With this, we can obtain an explicit
expression for $B$ from (57). For general values of $\kappa_{i}$, the solution
for $B$ contains square roots — being the solution of the quadratic equation
(57). However, we constraint the $\kappa_{i}$ constants in order to form a
perfect square inside the square root, so that $B$ becomes a rational function
of $x$. It turns out that doing so already solves most of the components of
the equations of motion. The remaining components simply set an additional
constraint on the remaining integration constants $\kappa_{i}$.

Finally, $J$ and $\hat{A}_{1}$ can be obtained by integrating (52). Thus, the
solution is determined up to solving the constraint equations between the
$\kappa_{i}$ coefficients. We do not explicitly show the constraint here due
to its complicated nature. In fact, one of the most challenging aspects about
the solution is finding an appropriate parametrization in which these
coefficients take a simple form. We have been able to identify a set of three
parameters $(X,Y,Z)$ that allow us to express the solution in an explicit and
fairly compact form. The result reads

$\displaystyle W=$ $\displaystyle\frac{\Delta}{4g^{2}\left(3-R^{2}\right)}\,,$
(59) $\displaystyle G=$
$\displaystyle\left(1-x^{2}\right)\frac{(x-1)X^{2}-(x+1)Y^{2}+2Z^{2}}{3-R^{2}}\,,$
$\displaystyle B=$
$\displaystyle\frac{\Delta}{4g^{2}\left(3-R^{2}\right)}-\frac{\left(1-R^{2}\right)}{2g^{2}\left(3-R^{2}\right)^{2}\Delta^{2}}\bigg{[}\left(1-R^{2}-2XYZ\right)^{2}$
$\displaystyle+\frac{\Delta}{2}(-1+X-Y-Z)(1+X+Y-Z)(1+X-Y+Z)(-1+X+Y+Z)\bigg{]}\,,$
$\displaystyle J=$
$\displaystyle-\frac{4\sqrt{2}g\sqrt{1-R^{2}}}{\left(3-R^{2}\right)\left(X^{2}-Y^{2}\right)B\Delta^{2}}\bigg{[}\left(1-R^{2}-2XYZ\right)^{2}$
$\displaystyle-\Delta\left(1-R^{2}+2Z^{2}(X^{2}+Y^{2})+2X^{2}Y^{2}+XYZ\left(1+R^{2}\right)\right)\bigg{]}\,,$
$\displaystyle\hat{A}_{1}=$
$\displaystyle\frac{2\sqrt{3}\left(1-R^{2}-2XYZ\right)\left(\left(1-R^{2}\right)\left(1-R^{2}-2XYZ\right)-\Delta^{2}\right)}{\left(3-R^{2}\right)(X^{2}-Y^{2})B\Delta^{2}}\,,$
$\displaystyle\hat{A}_{2}=$
$\displaystyle-\frac{\sqrt{\frac{3}{2}}\left(1-R^{2}-2XYZ\right)\sqrt{1-R^{2}}}{g\left(3-R^{2}\right)\Delta}\,,$

where we have introduced the notation777This $R$ should not be confused with
the Ricci scalar.

$\displaystyle\Delta$ $\displaystyle=1-x\left(X^{2}-Y^{2}\right)-Z^{2}\,,$
(60) $\displaystyle R$ $\displaystyle=\sqrt{X^{2}+Y^{2}+Z^{2}}\,.$ (61)

We note that the expressions for $J$ and $\hat{A}_{1}$ are divergent for
$Y\rightarrow X$, but the divergent piece is a constant term that can be
reabsorbed in the choice of integration constant from (52), so the solution is
actually regular at $X=Y$.888We discuss the case of $X=Y$ in Subsection 5.2
and the solution of the various functions in the ansatz is written in (83).

From these expressions it is quite clear that a real Lorentzian solution only
exists if $R\leq 1$ and therefore in terms of the $(X,Y,Z)$ parameters the
space of solutions lies within the unit ball in $\mathbb{R}^{3}$. Note that at
$R=1$ the horizon area is zero yielding a naked singularity. For this reason,
we choose to consider only the regime $R<1$. These parameters turn out to be
related to the original parameters $(a,b,q,m,r_{+})$ of the CCLP solution
Chong:2005hr  via

$\displaystyle a=$ $\displaystyle\frac{X}{gZ}\,,$ $\displaystyle\quad m=$
$\displaystyle\frac{3-2R^{2}-R^{4}+4X^{2}Y^{2}+4X^{2}Z^{2}+4Y^{2}Z^{2}}{8g^{2}Z^{4}}\,,\quad$
$\displaystyle r_{+}^{2}=$ $\displaystyle\frac{1-R^{2}}{2g^{2}Z^{2}}\,,$ (62)
$\displaystyle b=$ $\displaystyle\frac{Y}{gZ}\,,$ $\displaystyle q=$
$\displaystyle\frac{1-R^{2}-2XYZ}{2g^{2}Z^{3}}\,.$

This can be checked by comparing some of the metric components at the horizon
taking into account that $x=\cos(2\theta)$999In particular, the $g_{xx}$ and
$g_{rr}$ components are the easiest ones to check. and by comparing the
expressions for the charges and entropy that we provide below. In addition, we
verify that the relations (62) automatically solve the equation that
determines the position of the horizon as well as extremality condition for
CCLP black holes (26). Therefore, (62) provides an interesting and useful
parametrization of the space of extremal CCLP solutions.

Besides the constraint $R<1$, these relations also inform us that we should
restrict our space of solutions to $X^{2}<Z^{2}$ and $Y^{2}<Z^{2}$, which are
equivalent to the usual constraints $(ag)^{2}<1$, $(bg)^{2}<1$. In this way,
the space of solutions is restricted to two pyramidal spherical caps as shown
in Figure 1.

Figure 1: The allowed region in the parameter space $(X,Y,Z)$ of extremal
black holes is given by two sections of a unit sphere as shaded in blue. The
unit sphere is denoted in gray. Supersymmetry is allowed on the pink plane
defined by $X+Y+Z=1$. Extremal and supersymmetric black holes are allowed for
regions where the supersymmetric plane intersects with the extremal solutions.
Note that the supersymmetric plane only intersects in one of the two blue
regions.

Finally, in terms of the $(X,Y,Z)$ parameters the supersymmetric condition
(27) takes the simple form

$\quad X+Y+Z=1\,.$ (63)

In this case, we can write these variables explicitly in terms of $a$ and $b$
as

$X^{\star}=\frac{ag}{1+(a+b)g}\,,\quad Y^{\star}=\frac{bg}{1+(a+b)g}\,,\quad
Z^{\star}=\frac{1}{1+(a+b)g}\,,$ (64)

where the star $\star$ is meant to represent supersymmetric extremal values.

###  4.2 Thermodynamics

With the solution (59), we evaluate the various thermodynamic quantities of
the black hole. Using the general formulas (47) and (49), it is
straightforward to obtain the near-horizon chemical potentials

$\displaystyle\omega_{1}=$
$\displaystyle-\frac{\sqrt{2}\left(Z^{2}-X^{2}\right)\left[XY^{2}Z+\left(2Y+XZ\right)\left(X^{2}+Z^{2}-1\right)\right]}{\sqrt{1-R^{2}}\left(3-R^{2}\right)\left[2XY+Z\left(1+X^{2}+Y^{2}-Z^{2}\right)\right]}\,,$
(65) $\displaystyle\omega_{2}=$
$\displaystyle-\frac{\sqrt{2}\left(Z^{2}-Y^{2}\right)\left[YX^{2}Z+\left(2X+YZ\right)\left(Y^{2}+Z^{2}-1\right)\right]}{\sqrt{1-R^{2}}\left(3-R^{2}\right)\left[2XY+Z\left(1+X^{2}+Y^{2}-Z^{2}\right)\right]}\,,$
$\displaystyle\psi=$
$\displaystyle\frac{\sqrt{\frac{3}{2}}\left(1-R^{2}-2XYZ\right)\left[Z\left(1-R^{2}\right)-2XY\right]}{g\sqrt{1-R^{2}}\left(3-R^{2}\right)\left[2XY+Z\left(1+X^{2}+Y^{2}-Z^{2}\right)\right]}\,.$

We can also obtain the black hole entropy $S=\mathcal{A}/(4G_{5})$ by direct
application of (46), yielding

$S=\frac{\pi^{2}\sqrt{1-R^{2}}\left(Z\left(X^{2}+Y^{2}+1\right)+2XY-Z^{3}\right)}{4\sqrt{2}G_{5}g^{3}(Z^{2}-X^{2})(Z^{2}-Y^{2})}\,.$
(66)

Then, the charge and angular momenta can be obtained by evaluating the
integrals (21), (19) and (20), respectively, where in the latter we have to
use the Killing vectors defined in (44). This yields

$\displaystyle Q=$ $\displaystyle\frac{\sqrt{3}\pi
Z\left(1-R^{2}-2XYZ\right)}{8G_{5}g^{2}\left(Z^{2}-X^{2}\right)\left(Z^{2}-Y^{2}\right)}\,,$
(67) $\displaystyle J_{1}=$
$\displaystyle-\frac{\pi}{16G_{5}g^{3}\left(Z^{2}-X^{2}\right)^{2}(Z^{2}-Y^{2})}\Big{[}2X^{3}Z\left(Y^{2}-Z^{2}+1\right)+2X^{2}Y\left(Y^{2}+2Z^{2}-1\right)$
$\displaystyle+2X^{4}Y+X^{5}Z+XZ\left(Y^{2}+Z^{2}-1\right)\left(Y^{2}+Z^{2}+3\right)+2YZ^{2}\left(Y^{2}+Z^{2}-1\right)\Big{]}\,,$
$\displaystyle J_{2}=$ $\displaystyle J_{1}(X\leftrightarrow Y)\,.$

These results deserve several comments. First, the expressions for the charges
and entropy coincide with those of Chong:2005hr , specified for extremal
solutions, upon using the relations (62). Obviously this includes the
supersymmetric case given by (64). Second, the near-horizon potentials and the
charges satisfy a near-horizon version of the first law of thermodynamics
Hajian:2013lna  that reads

$\frac{1}{2\pi}dS=\omega_{1}dJ_{1}+\omega_{2}dJ_{2}+\psi dQ\,.$ (68)

This allows us to understand the meaning of the potentials in (65). In
general, the complete first law of thermodynamics for the free energy $F$ in
the canonical ensemble reads

$dF=-SdT+\sum_{i=1}^{2}\Omega_{i}dJ_{i}+\Psi dQ\,,$ (69)

where $\Omega_{i}$ and $\Psi$ are the angular velocities and the electrostatic
potential, respectively, which are given by the difference between values at
the near-horizon and infinity. From this, taking the second derivatives of the
free energy and using that they commute, we can derive the relations

$\left(\frac{\partial S}{\partial
J_{i}}\right)_{T,\,Q}=-\left(\frac{\partial\Omega_{i}}{\partial
T}\right)_{J_{i}\,,Q}\,,\quad\left(\frac{\partial S}{\partial
Q}\right)_{T,\,J_{i}}=-\left(\frac{\partial\Psi}{\partial
T}\right)_{J_{i}\,,Q}\,,$ (70)

where the subindices denote which variables are kept fixed when taking the
derivative. On the other hand, (68) is equivalent to

$\left(\frac{\partial S}{\partial
J_{i}}\right)_{T=0,\,Q}=2\pi\omega_{i}\,,\quad\left(\frac{\partial S}{\partial
Q}\right)_{T=0,\,J_{i}}=2\pi\psi\,,$ (71)

Comparing both expressions allows us to identify the near-horizon potentials
as “fugacities”

$\omega_{i}=-\frac{1}{2\pi}\frac{\partial\Omega_{i}}{\partial
T}\bigg{|}_{T=0}\,,\quad\psi=-\frac{1}{2\pi}\frac{\partial\Psi}{\partial
T}\bigg{|}_{T=0}\,,$ (72)

where $\Omega_{i}$ and $\Psi$ have to be understood as variables of $T$,
$J_{i}$ and $Q$. We can rewrite these relations in a suggestive way as

$\omega_{i}=-\lim_{T\rightarrow 0}\frac{\Omega_{i}-\Omega_{i}^{\rm ext}}{2\pi
T}\,,\quad\psi=-\lim_{T\rightarrow 0}\frac{\Psi-\Psi^{\rm ext}}{2\pi T}\,,$
(73)

where $\Omega_{i}^{\rm ext}$ and $\Psi^{\rm ext}$ represent the values at
extremality. This is very similar to the definition of the supersymmetric
potentials which are often used in the literature as in Silva:2006xv ;
Dias:2007dj ; Sen:2008yk ; Cabo-Bizet:2018ehj ; Choi:2018hmj , with the
difference that, in those cases, the whole expression is evaluated on the
(complex) supersymmetric solution. In fact, in the supersymmetric case, our
potentials (65) satisfy the constraint

$\omega_{1}+\omega_{2}-\sqrt{3}g\psi\Big{|}_{\rm susy}=0\,.$ (74)

This is entirely analogous to the complex constraint satisfied by the
supersymmetric potentials Cabo-Bizet:2018ehj ; Benini:2018ywd ; Choi:2018hmj ,
although in that case the right-hand side of the expression would be $+i$ (in
our conventions). Such difference appears due to the different order in which
the supersymmetric and extremal limits are taken. In our case, we consider the
extremal limit first and then impose supersymmetry, so that the solution is
Lorentzian and the potentials are real. In the next sections, we will use (74)
as the defining property of a supersymmetric solution.

##  5 Higher-derivative corrections

###  5.1 The setup

We now study the higher-derivative corrections in the action (1) to the
solution (59). Let us first explain the logic behind the resolution of the
equations of motion. We work perturbatively at first order in $\alpha$, so
that we assume that the metric and gauge field are expanded as

$g_{\mu\nu}=g_{\mu\nu}^{(0)}+\alpha
g_{\mu\nu}^{(1)}+\mathcal{O}(\alpha^{2})\,,\quad A_{\mu}=A_{\mu}^{(0)}+\alpha
A_{\mu}^{(1)}+\mathcal{O}(\alpha^{2})\,,$ (75)

where $g_{\mu\nu}^{(0)}$ and $A_{\mu}^{(0)}$ satisfy the two-derivative
equations of motion. This allows us to write the Einstein and Maxwell
equations in (6) as

$\displaystyle
G_{\mu\nu}+6g^{2}g_{\mu\nu}-\frac{1}{2}F_{(\mu|\rho}F_{|\nu)}{}^{\rho}+\frac{1}{8}g_{\mu\nu}F^{2}$
$\displaystyle=\alpha T_{\mu\nu}^{\rm eff}(g^{(0)},A^{(0)})\,,$ (76)
$\displaystyle-\nabla_{\mu}\mathcal{F}^{\mu\nu}+\frac{1}{4\sqrt{3}}\epsilon^{\nu\alpha\beta\gamma\delta}F_{\alpha\beta}F_{\gamma\delta}$
$\displaystyle=\alpha J^{\nu}_{\rm eff}(g^{(0)},A^{(0)})\,,$ (77)

where in the right-hand side we collect all the terms proportional to $\alpha$
and we evaluate them on the zeroth-order solution. Thus, the equations that we
have to solve are equivalent to the original two-derivative equations with a
fixed source. The process of solving the equations of motion can in principle
be carried out along the same lines as in Section 4. We consider the ansatz
(34) and we introduce the functions $j(x)$ and $\zeta(x)$ as in (52). Then, we
expand all the functions linearly in $\alpha$ as

$\displaystyle W$ $\displaystyle=W^{(0)}+\alpha g^{2}W^{(1)}\,,\quad$
$\displaystyle G$ $\displaystyle=G^{(0)}+\alpha g^{2}G^{(1)}\,,\quad$
$\displaystyle B$ $\displaystyle=B^{(0)}+\alpha g^{2}B^{(1)}\,,$ (78)
$\displaystyle j$ $\displaystyle=j^{(0)}+\alpha g^{2}j^{(1)}\,,\quad$
$\displaystyle\zeta$ $\displaystyle=\zeta^{(0)}+\alpha
g^{2}\zeta^{(1)}\,,\quad$ $\displaystyle\hat{A}_{2}$
$\displaystyle=\hat{A}_{2}^{(0)}+\alpha g^{2}\hat{A}_{2}^{(1)}\,,$

where the functions with the $(0)$ superscript are those in (59). We arrive
once more to the equations (53), (55), (57), but in this case the right-hand
side to these equations depends only on the zeroth-order solution. These four
equations (52), (53), (55), (57) can be integrated completely to find
$j^{(1)}$, $\zeta^{(1)}$, $W^{(1)}$ and $B^{(1)}$ in terms of $G^{(1)}$ and
$\hat{A}_{2}^{(1)}$. The two remaining variables $G^{(1)}$ and
$\hat{A}_{2}^{(1)}$ satisfy a system of two coupled linear differential
equations with up to sixth-order derivatives. Solving these equations is
highly challenging, but it becomes much simpler in the case of equal angular
momenta or when the difference between the angular momenta is small, as we
consider in the next subsections. Finally, once the zeroth- and first-order
solutions have been found for the variables in (78), it is straight forward to
obtain

$J=J^{(0)}+\alpha g^{2}J^{(1)}\,,\quad\hat{A}_{1}=\hat{A}_{1}^{(0)}+\alpha
g^{2}\hat{A}_{1}^{(1)}\,,$ (79)

from (52).

In order to specify a solution, we also need to fix boundary conditions.
Indeed, in our analysis, we find some integration constants that correspond to
a relabeling of the parameters of the two-derivative solution as
$(X,Y,Z)\rightarrow(X+\alpha\delta X,Y+\alpha\delta Y,Z+\alpha\delta Z)$.
Thus, in order to fix the solution we work in the grand canonical ensemble, so
that we fix $\omega_{1}$, $\omega_{2}$ and $\psi$ to be given by the same
expressions (65) of the two-derivative solution. We recall that these
potentials are given by the general formulas (47) and (49), so when we expand
the functions as in (78) and (79), the potentials take the form

$\omega_{i}=\omega_{i}^{(0)}\left(1+\alpha
g^{2}\delta\omega_{i}\right)\,,\quad\psi=\psi^{(0)}\left(1+\alpha
g^{2}\delta\psi\right)\,,$ (80)

where

$\displaystyle\delta\omega_{1}=$
$\displaystyle\frac{G^{(1)}{}^{\prime}(-1)}{G^{(0)}{}^{\prime}(-1)}-\frac{B^{(1)}(-1)}{2B^{(0)}(-1)}-\frac{W^{(1)}(-1)}{W^{(0)}(-1)}-\frac{J^{(1)}(1)-J^{(1)}(-1)}{J^{(0)}(1)-J^{(0)}(-1)}\,,$
(81) $\displaystyle\delta\omega_{2}=$
$\displaystyle\frac{G^{(1)}{}^{\prime}(1)}{G^{(0)}{}^{\prime}(1)}-\frac{B^{(1)}(1)}{2B^{(0)}(1)}-\frac{W^{(1)}(1)}{W^{(0)}(1)}-\frac{J^{(1)}(1)-J^{(1)}(-1)}{J^{(0)}(1)-J^{(0)}(-1)}\,,$
$\displaystyle\delta\psi=$
$\displaystyle\frac{\hat{A}_{1}^{(1)}(1)-\hat{A}_{1}^{(1)}(-1)}{\hat{A}_{1}^{(0)}(1)-\hat{A}_{1}^{(0)}(-1)}-\frac{J^{(1)}(1)-J^{(1)}(-1)}{J^{(0)}(1)-J^{(0)}(-1)}\,.$

Thus, we impose the conditions

$\delta\omega_{1}=\delta\omega_{2}=\delta\psi=0\,,$ (82)

which in turn imply boundary conditions on the functions of the solution at
$x=\pm 1$.

###  5.2 Equal angular momenta

When the black hole has equal angular momenta, corresponding to $X=Y$ in the
parametrization introduced in (59), the $\mathrm{U}(1)\times\mathrm{U}(1)$
symmetry of the solution is enhanced to $\mathrm{SU}(2)\times\mathrm{U}(1)$.
As a result, all the functions in the ansatz have a fixed functional
dependence. We have $G(x)\propto(1-x^{2})$, while the rest of functions in
(78) must be constant. In turn, the functions $J(x)$ and $\hat{A}_{1}(x)$
become linear in $x$ up to an irrelevant additive constant, as only
derivatives of $J(x)$ and $\hat{A}_{1}(x)$ appear in the equations. This
allows us to express the solution as

$\displaystyle W=$
$\displaystyle\frac{1-Z^{2}}{4g^{2}\left(3-2X^{2}-Z^{2}\right)}\left(1+\alpha
g^{2}\delta W\right)\,,$ (83) $\displaystyle G=$
$\displaystyle\frac{2\left(1-x^{2}\right)\left(Z^{2}-X^{2}\right)}{3-2X^{2}-Z^{2}}\left(1+\alpha
g^{2}\delta G\right)\,,$ $\displaystyle B=$
$\displaystyle\frac{X^{2}\left(2-2X^{2}-Z^{2}-Z\right)^{2}}{g^{2}(1-Z)^{2}\left(3-2X^{2}-Z^{2}\right)^{2}}\left(1+\alpha
g^{2}\delta B\right)\,,$ $\displaystyle J=$ $\displaystyle
x\frac{2\sqrt{2}g^{3}\sqrt{1-2X^{2}-Z^{2}}\left(2X^{2}+Z-Z^{2}\right)\left(3-2X^{2}-Z^{2}\right)^{2}}{X^{2}(1+Z)\left(2-2X^{2}-Z^{2}-Z\right)^{2}}\left(1+\alpha
g^{2}\delta J\right)\,,$ $\displaystyle\hat{A}_{1}=$
$\displaystyle-x\frac{2\sqrt{3}g^{2}\left(1-2X^{2}-Z\right)\left(2X^{2}+(-1+Z)Z\right)\left(3-2X^{2}-Z^{2}\right)}{X^{2}\left(2-2X^{2}-Z^{2}-Z\right)^{2}}\left(1+\alpha
g^{2}\delta\hat{A}_{1}\right),$ $\displaystyle\hat{A}_{2}=$
$\displaystyle-\frac{\sqrt{\frac{3}{2}}\left(1-2X^{2}-Z\right)\sqrt{1-2X^{2}-Z^{2}}}{g(1-Z)\left(3-2X^{2}-Z^{2}\right)}\left(1+\alpha
g^{2}\delta\hat{A}_{2}\right)\,,$

where the leading terms correspond to the two-derivative solution (59) with
$X=Y$ (in the case of $J$ and $\hat{A}_{1}$ we took the $Y\rightarrow X$ limit
and removed the constant term) and the higher-derivative corrections are
captured by the constant coefficients $\delta W$, $\delta G$, $\delta B$,
$\delta J$, $\delta\hat{A}_{i}$. Thus, the higher-derivative Einstein and
Maxwell equations become a linear system of algebraic equations for these
coefficients whose solution is straightforward to obtain. Moreover, we also
use the equations in (82), which impose additional constraints on the
coefficients. When we do so, the solution is completely fixed. The full
expressions for these coefficients are somewhat lengthy and thus we provide
them in Appendix A.

#### Zero chemical potential

There are two interesting limits of this solution. One is the supersymmetric
limit, which is contained in the analysis of the next section. The other is
the case of zero chemical potential. By using (65) with $Y=X$, we can see that
imposing $\psi=0$ amounts to setting

$Z=1-2X^{2}\,.$ (84)

Naturally, in the two-derivative solution, this also implies the vanishing of
the gauge field, $A_{\mu}=0$, so that the solution is purely gravitational.
Interestingly, this is no longer the case when the higher-derivative
corrections are included. Indeed, at first order in $\alpha$, the solution
with $\psi=0$ and equal angular momenta reads

$\displaystyle W=$
$\displaystyle\frac{X^{2}}{2g^{2}(2X^{2}+1)}\left(1+\frac{2\alpha
g^{2}\lambda_{1}\left(8X^{8}-26X^{6}+41X^{4}-12X^{2}+1\right)}{X^{4}\left(8X^{4}-10X^{2}+5\right)}\right)\,,$
(85) $\displaystyle G=$
$\displaystyle\frac{\left(1-x^{2}\right)\left(1-4X^{2}\right)}{2X^{2}+1}\left(1-\frac{32\alpha
g^{2}\lambda_{1}\left(2X^{6}-3X^{2}+1\right)}{X^{2}\left(8X^{4}-10X^{2}+5\right)}\right)\,,$
$\displaystyle B=$
$\displaystyle\left(\frac{X^{2}}{g(2X^{2}+1)}\right)^{2}\left(1+\frac{2\alpha
g^{2}\lambda_{1}\left(X^{2}-1\right)^{2}\left(40X^{4}-2X^{2}+1\right)}{X^{4}\left(8X^{4}-10X^{2}+5\right)}\right)\,,$
$\displaystyle J=$
$\displaystyle\frac{2g^{3}x}{X^{3}}\sqrt{1-2X^{2}}\left(2X^{2}+1\right)^{2}\left(1-\frac{\alpha
g^{2}\lambda_{1}x\left(120X^{8}-134X^{6}+31X^{4}+4X^{2}+3\right)}{X^{4}\left(8X^{4}-10X^{2}+5\right)}\right)\,,$
$\displaystyle\hat{A}_{1}=$ $\displaystyle 0\,,$ $\displaystyle\hat{A}_{2}=$
$\displaystyle-\frac{4\alpha g\lambda_{1}\sqrt{3-6X^{2}}}{X(2X^{2}+1)}\,,$

so a non-trivial $\hat{A}_{2}$ is generated. The reason for this is the
gravitational Chern-Simons term, which acts as a source in Maxwell equation,
as $d\star F\sim\alpha R\wedge R$. Therefore, gravity — and more precisely,
angular momentum — can generate an electric charge. We obtain the precise
value of this charge in the next section.

###  5.3 Supersymmetric solutions with $J_{1}\neq J_{2}$

When $J_{1}\neq J_{2}$, the equations of motion are much more involved and we
have not been able to find an exact solution. However, we can find an
approximate solution when the difference between angular momenta is small by
performing an expansion in powers of $(J_{1}-J_{2})$ around the solution with
equal angular momenta. To simplify things further, we also restrict ourselves
to the more holographically interesting supersymmetric case.

Let us note that, in the two-derivative theory, we identified supersymmetry
with the constraint (63), which in turn implied that the chemical potentials
satisfy the relation (74). In general, a solution is supersymmetric when it
allows for a non-zero Killing spinor. Now, the Killing spinor equations are in
general modified by the higher-derivative corrections, and hence one should
compute what the effect is on the supersymmetry conditions. Following
Cassani:2022lrk , we assume that the constraint (74) is not modified by the
higher-derivative corrections and we take this to be the defining property of
a supersymmetric solution at subleading order in $\alpha$. Since we are
working in the grand-canonical ensemble, we make sure that the correction to
the two-derivative supersymmetric solution still satisfies (74) and hence is
still supersymmetric by setting

$Z=1-X-Y\,.$ (86)

In order to find a solution, our goal is to expand the higher-derivative
corrections in (78) in a power series of $(Y-X)$, since this measures the
difference between the two angular momenta.

There is another key property of the solution that makes the expansion less
burdensome. We observe that, whenever the variable $x$ appears
nontrivially101010With this we mean that there is an additional $x$-dependence
with respect to the $X=Y$ case. in the two-derivative solution (59), it is
always multiplied by $(Y-X)$. Thus, when we expand this solution in powers of
$(Y-X)$, the $n$-th power contains, at most, $n$ powers of $x$. It is very
natural to assume that the higher-derivative corrections will have the same
structure. Taking this into account, we expand the correction to the functions
in (78) as

$\displaystyle W^{(1)}(x)=$
$\displaystyle\sum_{n=0}^{\infty}(Y-X)^{n}\sum_{k=0}^{n}W_{n,k}x^{k}\,,\quad$
$\displaystyle G^{(1)}(x)=$
$\displaystyle(1-x^{2})\sum_{n=0}^{\infty}(Y-X)^{n}\sum_{k=0}^{n}G_{n,k}x^{k}\,,$
(87) $\displaystyle B^{(1)}(x)=$
$\displaystyle\sum_{n=0}^{\infty}(Y-X)^{n}\sum_{k=0}^{n}B_{n,k}x^{k}\,,\quad$
$\displaystyle j^{(1)}(x)=$
$\displaystyle\sum_{n=0}^{\infty}(Y-X)^{n}\sum_{k=0}^{n}j_{n,k}x^{k}\,,$
$\displaystyle\zeta^{(1)}(x)=$
$\displaystyle\sum_{n=0}^{\infty}(Y-X)^{n}\sum_{k=0}^{n}\zeta_{n,k}x^{k}\,,\quad$
$\displaystyle\hat{A}_{2}^{(1)}(x)=$
$\displaystyle\sum_{n=0}^{\infty}(Y-X)^{n}\sum_{k=0}^{n}\hat{A}_{2,n,k}x^{k}\,,$

where $W_{n,k}$, $G_{n,k}$, $B_{n,k}$, $j_{n,k}$, $\zeta_{n,k}$ and
$\hat{A}_{2,n,k}$ are constant coefficients. By plugging this ansatz into the
equations of motion, expanding again in powers of $(Y-X)$, and collecting the
terms with different powers of $x$,111111Observe that we do not perform a
series expansion in $x$, we keep all the $x$-dependence at each order of
$(Y-X)$. This dependence on $x$ is automatically polynomial.  we obtain a
system of algebraic equations for these coefficients. We check that this
system indeed solves the equations of motion order by order, hence validating
the form of the ansatz (87).

The solution in fact contains free coefficients that must be fixed by imposing
the conditions (82). To this end, we first obtain the functions $J(x)$ and
$\hat{A}_{1}(x)$ by plugging (78) with (87) into (52), and expanding again in
$(Y-X)$. In this case, the expansion of these functions takes the form

$J^{(1)}(x)=\sum_{n=0}^{\infty}(Y-X)^{n}\sum_{k=0}^{n}J_{n,k}x^{k+1}\,,\quad\hat{A}_{1}^{(1)}(x)=\sum_{n=0}^{\infty}(Y-X)^{n}\sum_{k=0}^{n}\hat{A}_{1,n,k}x^{k+1}\,,$
(88)

and the coefficients $J_{n,k}$, $\hat{A}_{1,n,k}$ are determined by the ones
in (87). This allows us to evaluate the correction to the potentials in (81),
which also take the form of a series expansion, and hence (82) provides three
constraints at every order of $(Y-X)$. When these constraints are imposed
together with the equations of motion, we find that the solution of the form
(87) is unique. We have been able to obtain the solution to order $(Y-X)^{6}$
and we provide the corresponding coefficients in the ancillary Mathematica
notebook. For the sake of illustration, we show the solution to first order in
$(Y-X)$ in Appendix A.

We have attempted to find a pattern that would allow us to sum the whole
series expansion and obtain the exact solution. In particular, we tried to fit
the solution to a rational function of $x$, in analogy with the two-derivative
solution (59). However, the expressions are too complicated and we could not
find a simple pattern. Nevertheless, we will see in Section 6.4 that there is
a way in which this can be done in the case of the thermodynamic quantities.

With the solution at hand we can compute right away the area of the horizon by
applying (46). We get

$\displaystyle\mathcal{A}=$
$\displaystyle\frac{2\pi^{2}(X+Y)\sqrt{X+Y-X^{2}-XY-Y^{2}}}{g^{3}(1-2X-Y)(1-2Y-X)}\Big{[}1+\alpha
g^{2}\lambda_{1}\delta\mathcal{A}\Big{]}\,,$ (89)

where, to second order in $(Y-X)$, $\delta\mathcal{A}$ reads

$\displaystyle\delta\mathcal{A}=$
$\displaystyle-\frac{-180X^{3}+177X^{2}-32X+9}{X(9X^{2}-12X+1)}-\frac{9(Y-X)\left(63X^{4}+24X^{3}-50X^{2}+24X-1\right)}{2X^{2}\left(9X^{2}-12X+1\right)^{2}}$
(90)
$\displaystyle+\frac{(Y-X)^{2}}{12(X-1)^{2}X^{3}\left(3X^{2}-1\right)\left(9X^{2}-12X+1\right)^{3}}\Big{(}229635X^{11}-946971X^{10}+1665765X^{9}$
$\displaystyle-1705617X^{8}+1153926X^{7}-495990X^{6}+67410X^{5}+76662X^{4}-56937X^{3}$
$\displaystyle+14737X^{2}-1047X+27\Big{)}+\mathcal{O}\left((Y-X)^{3}\right)\,.$

Although this is again terribly complicated to guess a pattern, we will find a
way to sum the whole series as we elaborate on in Section 6.4.

##  6 Thermodynamics

###  6.1 Electric charge: ambiguity due to the Lorentz-Chern-Simons terms

The formulas for the electric charge (9), angular momenta (14) and entropy
(16) present gauge ambiguities due to the presence of Chern-Simons terms.
However, these ambiguities are fixed by working in the regular gauge that we
are utilizing, i.e., imposing regularity at the horizon. However, in the
formula for the electric charge there is also a Lorentz-Chern-Simons three-
form that introduces frame ambiguities. These are harder to fix, since there
is not a priori a canonical choice of frame. Instead, there are infinitely
many regular frames, which are related by “large” Lorentz transformations. In
our case, since the spacetime is a twisted product
AdS${}_{2}\times\mathbb{S}^{3}$, it is natural to restrict to the rotations of
frame generated by the sphere $\mathbb{S}^{3}$, leading to a discrete set of
different notions of charge — one for each homotopy class of $\mathbb{S}^{3}$.
Each of these is, in principle, a valid definition of electric charge.
However, only one particular charge (modulo an additive constant independent
of the parameters of the solution) enters in the first law of thermodynamics.
Then, the question is whether we can find a frame in which the electric charge
is the thermodynamically relevant charge.

With this goal, we consider a family of frames $\tilde{e}^{a}$ related to (51)
according to

$\displaystyle\tilde{e}^{0}$ $\displaystyle=e^{0}\,,$ (91)
$\displaystyle\tilde{e}^{1}$ $\displaystyle=e^{1}\,,$
$\displaystyle\tilde{e}^{2}$
$\displaystyle=\cos(n\phi_{1}+m\phi_{2})e^{2}+\sin(n\phi_{1}+m\phi_{2})e^{3}\,,$
$\displaystyle\tilde{e}^{3}$
$\displaystyle=-\sin(n\phi_{1}+m\phi_{2})e^{2}+\cos(n\phi_{1}+m\phi_{2})e^{3}\,,$
$\displaystyle\tilde{e}^{4}$ $\displaystyle=e^{4}\,,$

where $\phi_{1}$ and $\phi_{2}$ are the angular coordinates introduced in (42)
and $n,m$ must be integers in order to ensure the regularity of the
transformation. Although this is clearly not the most general choice of frame,
we show below that this is indeed enough for our purposes. Due to the frame
transformation, the Lorentz-Chern-Simons three-form introduces a non-trivial
contribution to the charge, which reads

$\tilde{\Omega}_{\rm LCS}=\Omega_{\rm LCS}+\mathcal{W}\,,$ (92)

where

$\mathcal{W}=dy_{1}\wedge dy_{2}\wedge
d\left[-\sigma_{1}B^{3/2}J^{\prime}+\frac{\sigma_{2}}{B^{3/2}W^{2}}\left(BW\left(G^{\prime}+B^{2}JWJ^{\prime}\right)-G\left(WB^{\prime}+BW^{\prime}\right)\right)\right]\,,$
(93)

and

$\displaystyle\sigma_{1}$
$\displaystyle=\frac{1}{2(J(-1)-J(1))}\left[\frac{mJ(-1)G^{\prime}(1)}{\sqrt{B(1)}W(1)}-\frac{nJ(1)G^{\prime}(-1)}{\sqrt{B(-1)}W(-1)}\right]\,,$
(94) $\displaystyle\sigma_{2}$
$\displaystyle=\frac{1}{2(J(-1)-J(1))}\left[\frac{mG^{\prime}(1)}{\sqrt{B(1)}W(1)}-\frac{nG^{\prime}(-1)}{\sqrt{B(-1)}W(-1)}\right]\,.$

These coefficients come from the relation (42) between the angles $\phi_{i}$
and the $y_{i}$ coordinates. When we integrate the charge, we get the
following difference due to the frame change

$\Delta Q=\frac{3\alpha\lambda_{1}}{16\sqrt{3}\pi G_{5}}\int\mathcal{W}\,.$
(95)

At first order in $\alpha$, we can just evaluate this term at the two-
derivative solution, and we get the following explicit result for the
ambiguous contribution to the charge as a function of $n$ and $m$,

$\Delta Q_{n,m}=\frac{\alpha\lambda_{1}\pi}{2\sqrt{3}\pi
G_{5}}\left[nf(X,Y,Z)+mf(Y,X,Z)\right]\,,$ (96)

where

$\displaystyle f(X,Y,Z)=$
$\displaystyle\frac{1}{\left(Y^{2}-Z^{2}\right)\left(-1-X^{2}+Y^{2}+Z^{2}\right)^{3}}\bigg{[}X^{7}Y+X^{5}Y\left(5-3Y^{2}+Z^{2}\right)$
(97) $\displaystyle-
XY\left(-1+Y^{2}+Z^{2}\right)^{2}\left(1+Y^{2}+Z^{2}\right)+X^{6}\left(-Y^{2}+Z(2+Z)\right)$
$\displaystyle+\left(-1+Y^{2}+Z^{2}\right)^{2}\left(Y^{4}+Z^{2}-Z^{4}-Y^{2}(1+2Z)\right)+X^{4}\left(3Y^{4}+Y^{2}(-3+10Z)\right.$
$\displaystyle\left.-3Z^{2}\left(-1+Z^{2}\right)\right)+X^{3}Y\left(-5+3Y^{4}+6Z^{2}-Z^{4}+2Y^{2}\left(1+5Z^{2}\right)\right)$
$\displaystyle+X^{2}\left(-3Y^{6}+Y^{4}\left(6+6Z-3Z^{2}\right)+Z(-2+3Z)\left(-1+Z^{2}\right)^{2}\right.$
$\displaystyle\left.+Y^{2}\left(-3-4Z+4Z^{3}+3Z^{4}\right)\right)\bigg{]}\,.$

Since this is not a constant value, it affects the variations of the charge
$dQ_{n,m}$ and therefore it affects the first law. Our hope is that there is a
choice of $n$ and $m$ for which $Q_{n,m}$ is the thermodynamic charge. We
study this in careful detail below.

###  6.2 Equal angular momenta

By direct evaluation of the integrals (9), (14) and (16) in the solution (83),
we obtain the electric charge, angular momenta and the entropy of the black
holes. The result takes the form

$\displaystyle S$
$\displaystyle=\frac{\pi^{2}(Z+1)\left(2X^{2}-Z^{2}+Z\right)\sqrt{-2X^{2}-Z^{2}+1}}{4\sqrt{2}G_{5}g^{3}\left(X^{2}-Z^{2}\right)^{2}}\left[1+\alpha
g^{2}(4\lambda_{2}+\lambda_{1}\Delta S)\right]\,,$ (98) $\displaystyle J_{i}$
$\displaystyle=\frac{\pi
X(Z+1)\left(X^{2}(6Z-2)+4X^{4}+Z^{4}+Z^{3}+Z^{2}-3Z\right)}{16G_{5}g^{3}\left(X^{2}-Z^{2}\right)^{3}}\left[1+\alpha
g^{2}(4\lambda_{2}+\lambda_{1}\Delta J)\right]\,,$ (99) $\displaystyle Q$
$\displaystyle=\frac{\sqrt{3}\pi
Z(Z+1)\left(1-Z-2X^{2}\right)}{8G_{5}g^{2}\left(X^{2}-Z^{2}\right)^{2}}\left[1+\alpha
g^{2}(4\lambda_{2}+\lambda_{1}\Delta Q)\right]$ (100)
$\displaystyle+\frac{\pi(m+n)\alpha\lambda_{1}\left[(Z-1)^{2}\left(X^{2}(4Z+2)+4X^{4}+Z^{2}\left(Z^{2}-1\right)\right)-8X^{6}\right]}{2\sqrt{3}G_{5}(Z-1)^{3}(Z+1)(Z^{2}-X^{2})}\,,$

where the expressions of $\Delta S$, $\Delta J$, $\Delta Q$ are somewhat long
and hence we provide them in the ancillary Mathematica notebook. Observe that
the correction due to $\lambda_{2}$ is universal and it is equivalent to
replacing $G_{5}$ by the effective Newton’s constant $G_{\rm eff}$ in (4).

In the case of $Q$, we have made explicit the ambiguous Chern-Simons
contribution due to the frame choice, that in this case only depends on the
sum of the two integers $n+m$. In order to fix this ambiguity, we evaluate the
near-horizon version of the first law of black hole mechanics (68), which
should hold as well in the presence of higher-derivative corrections. We get

$\frac{1}{2\pi}dS-\omega_{1}\,dJ_{1}-\omega_{2}\,dJ_{2}-\psi\,dQ=-(m+n+4)\frac{\pi\alpha\lambda_{1}}{2\sqrt{3}G}\psi
d[f(X,X,Z)]\,,$ (101)

where $\omega_{i}$ and $\psi$ are the potentials given by (65) evaluated at
$Y=X$ — these receive no corrections since we are working in the grand-
canonical ensemble — and $f$ is the function introduced in (97). Thus, for

$n+m=-4,$ (102)

the first law is satisfied and hence $Q$ becomes the thermodynamic electric
charge.

#### Supersymmetric limit

In the supersymmetric case, (64) with $a=b$, i.e., $X=ag/(1+2ag)$,
$Z=1/(1+2ag)$, the expressions above simplify dramatically and we get

$\displaystyle S=$ $\displaystyle\frac{\pi^{2}a\sqrt{ag(ag+2)}}{g^{2}G_{\rm
eff}(1-ag)^{2}}\left[1+\alpha
g^{2}\lambda_{1}\frac{48(ag+2)(2ag+1)}{11a^{2}g^{2}+8ag-1}\right]\,,$ (103)
$\displaystyle J_{i}=$ $\displaystyle\frac{\pi a^{2}(ag+3)}{2gG_{\rm
eff}(1-ag)^{3}}\left[1+\alpha
g^{2}\lambda_{1}\frac{24\left(8a^{4}g^{4}+25a^{3}g^{3}+29a^{2}g^{2}+9ag+1\right)}{ag(ag+3)\left(11a^{2}g^{2}+8ag-1\right)}\right]\,,$
$\displaystyle Q=$ $\displaystyle\frac{\sqrt{3}\pi a}{2gG_{\rm
eff}(1-ag)^{2}}\left[1+\alpha
g^{2}\lambda_{1}\frac{8\left(23a^{4}g^{4}+73a^{3}g^{3}+57a^{2}g^{2}+7ag+2\right)}{3ag\left(11a^{2}g^{2}+8ag-1\right)}\right]\,.$

These results match exactly those of Cassani:2023vsa , also obtained from the
near-horizon geometry. However, as also noticed by Cassani:2023vsa , the
charge obtained from this procedure differs by a pure constant — independent
of the parameters of the black hole — from the charge obtained from the on-
shell action Cassani:2022lrk  (the rest of the results are identical). In the
next subsection we analyze this discordance in the case of different angular
momenta.

#### Zero chemical potential

As we noted in Section 5.2, the solutions with $\psi=0$ actually acquire a
nontrivial gauge field on account of the electro-gravitational Chern-Simons
term. These solutions even have a nonzero charge, and in fact we get

$Q=-\frac{4\pi\alpha\lambda_{1}\left(3-4X^{2}-8X^{4}\right)}{\sqrt{3}G_{5}\left(1-4X^{2}\right)^{2}}\,,$
(104)

where we have set $n+m=-4$, as before. Observe that, since $Q$ depends on $X$,
it cannot be taken to zero by adding a pure constant to it. In fact, it is not
zero in any of the frames we have analyzed, so the generation of a non-zero
charge is a genuine effect of the higher-derivative corrections.
Interestingly, $Q$ is a function of the angular momentum through its
dependence on $X$, so it is not a free parameter of the solution. On the other
hand, one may also consider solutions with zero charge, but these will
necessarily have a nonzero potential.

###  6.3 Supersymmetric solutions with $J_{1}\neq J_{2}$

Let us now consider the supersymmetric solutions that we have found in Section
5.3 as an expansion in $(Y-X)$. We can again evaluate the charges and the
entropy through the integrals (9), (14) and (16). In the case of the electric
charge we also add the frame-dependent term (96). In order to make contact
with previous literature, we express the result in terms of the parameters $a$
and $b$ by using the relationships (64). Then, we also translate the series
expansion in $(Y-X)$ into a series expansion in $(b-a)$. The result, to order
$(b-a)^{2}$ and first order in $\alpha$, reads

$\displaystyle S$ $\displaystyle=\frac{\pi^{2}(a+b)\sqrt{ab+a+b}}{2G_{\rm
eff}(1-a)(1-b)}\bigg{[}1+\alpha\lambda_{1}\delta S\bigg{]}\,,$ (105)
$\displaystyle J_{1}$ $\displaystyle=\frac{\pi(a+b)(ab+2a+b)}{4G_{\rm
eff}(1-a)^{2}(1-b)}\bigg{[}1+\alpha\lambda_{1}\delta J_{1}\bigg{]}\,,$
$\displaystyle J_{2}$ $\displaystyle=\frac{\pi(a+b)(ab+2b+a)}{4G_{\rm
eff}(1-b)^{2}(1-a)}\bigg{[}1+\alpha\lambda_{1}\delta J_{2}\bigg{]}\,,$
$\displaystyle Q$ $\displaystyle=\frac{\sqrt{3}\pi(a+b)}{4G_{\rm
eff}(1-a)(1-b)}\bigg{[}1+\alpha\lambda_{1}\delta Q_{n,m}\bigg{]}\,,$

where we have set $g=1$ and

$\displaystyle\delta S$
$\displaystyle=\frac{48(a+2)(2a+1)}{11a^{2}+8a-1}-\frac{72(b-a)\left(13a^{2}+16a+7\right)}{\left(11a^{2}+8a-1\right)^{2}}$
$\displaystyle-\frac{2(b-a)^{2}}{a^{2}(a+1)^{2}\left(a^{2}+4a+1\right)\left(11a^{2}+8a-1\right)^{3}}\big{(}33a^{10}-5520a^{9}-33975a^{8}-81460a^{7}$
$\displaystyle-108482a^{6}-88572a^{5}-43066a^{4}-11036a^{3}-1167a^{2}-4a+1\big{)}+\mathcal{O}\left((b-a)^{3}\right)\,,$
(106) $\displaystyle\delta J_{1}$
$\displaystyle=\frac{24\left(8a^{4}+25a^{3}+29a^{2}+9a+1\right)}{a(a+3)\left(11a^{2}+8a-1\right)}+\frac{8(b-a)}{a^{2}(a+3)^{2}\left(a^{2}+4a+1\right)\left(11a^{2}+8a-1\right)^{2}}\big{(}88a^{9}$
$\displaystyle+1189a^{8}+5762a^{7}+13249a^{6}+14422a^{5}+7889a^{4}+3094a^{3}+895a^{2}+74a-6\big{)}$
$\displaystyle+\frac{2(b-a)^{2}}{a^{3}(a+1)^{2}(a+3)^{3}\left(a^{2}+4a+1\right)^{2}\left(11a^{2}+8a-1\right)^{3}}\big{(}2475a^{16}+73157a^{15}$
$\displaystyle+778745a^{14}+4365729a^{13}+14805057a^{12}+32337717a^{11}+47141825a^{10}$
$\displaystyle+47441601a^{9}+34726215a^{8}+19894695a^{7}+9289651a^{6}+3306235a^{5}+754083a^{4}$
$\displaystyle+76383a^{3}-2157a^{2}-605a+42\big{)}+\mathcal{O}\left((b-a)^{3}\right)\,,$
(107) $\displaystyle\delta Q_{n,m}$
$\displaystyle=\frac{2\left(2+17a+63a^{2}+85a^{3}+49a^{4}\right)}{a\left(-1+8a+11a^{2}\right)}+\frac{\left(2-7a+5a^{2}\right)(m+n)}{6a}$
$\displaystyle+(b-a)\Bigg{[}\frac{2-32a-265a^{2}-544a^{3}-160a^{4}+784a^{5}+539a^{6}}{a^{2}\left(-1+8a+11a^{2}\right)^{2}}$
$\displaystyle-\frac{\left(1+a-5a^{2}-3a^{3}\right)m+\left(1+a-2a^{3}\right)n}{6a^{2}(1+a)}\Bigg{]}$
$\displaystyle+(b-a)^{2}\Bigg{[}\frac{1}{6a^{3}(1+a)^{2}\left(1+4a+a^{2}\right)\left(-1+8a+11a^{2}\right)^{3}}\big{(}6-111a+93a^{2}+14383a^{3}$
$\displaystyle+95277a^{4}+282930a^{5}+497994a^{6}+614982a^{7}+522456a^{8}+208221a^{9}-50727a^{10}$
$\displaystyle-70485a^{11}-15499a^{12}\big{)}+\frac{\left(2+3a+3a^{2}-5a^{4}\right)(n+m)-a^{4}(11m+3n)}{24a^{3}(1+a)^{2}}\Bigg{]}$
$\displaystyle+\mathcal{O}\left((b-a)^{3}\right)\,.$ (108)

and, as usual, $\delta J_{2}=\delta J_{1}(a\leftrightarrow b)$. We observe
that the results for $S$ and $J_{i}$ coincide with those of Cassani:2022lrk
when expanded in $(b-a)$, which represents a highly nontrivial check of our
computations. On the other hand, we have to determine for which value of
$(n,m)$ the electric charge is the thermodynamic one. For this charge, the
near-horizon version of the first law of thermodynamics should hold. By
plugging (105) and (65) in the supersymmetric case into (68), we
obtain121212Naturally, we obtain the result as a series expansion in $(b-a)$,
but we verify that it corresponds to the expansion of this expression.

$\frac{1}{2\pi}dS-\omega_{1}\,dJ_{1}-\omega_{2}\,dJ_{2}-\psi\,dQ=-\frac{\pi\alpha\lambda_{1}}{2\sqrt{3}G_{5}}\psi\,d\left[(n+2)f^{*}(a,b)+(m+2)f^{*}(b,a)\right]\,,$
(109)

where

$f^{*}(a,b)\equiv
f(X^{*},Y^{*},Z^{*})=\frac{3a^{2}b+2ab^{2}+ab+2b^{2}-a-b}{(a+1)(b-1)(a+b)}\,.$
(110)

Therefore, the first law is satisfied for

$n=m=-2\,,$ (111)

and thus the frame (91) is fully fixed by this criterion. For these values of
$n$ and $m$, the charge can be rewritten as

$Q=\frac{\sqrt{3}\pi(a+b)}{4G_{5}(1-a)(1-b)}\bigg{[}1+4\alpha\lambda_{2}+\alpha\lambda_{1}\delta
Q\bigg{]}-\frac{2\pi\alpha\lambda_{1}}{\sqrt{3}G_{5}}\,,$ (112)

with

$\displaystyle\delta Q$
$\displaystyle=\frac{4\left(19a^{4}+44a^{3}+36a^{2}+8a+1\right)}{a\left(11a^{2}+8a-1\right)}$
(113)
$\displaystyle+\frac{(b-a)\left(418a^{6}+608a^{5}-202a^{4}-528a^{3}-266a^{2}-32a+2\right)}{a^{2}\left(11a^{2}+8a-1\right)^{2}}$
$\displaystyle+\frac{(b-a)^{2}}{3a^{3}(a+1)^{2}\left(a^{2}+4a+1\right)\left(11a^{2}+8a-1\right)^{3}}\big{(}-5753a^{12}-20904a^{11}+7026a^{10}$
$\displaystyle+129304a^{9}+260413a^{8}+301360a^{7}+250612a^{6}+143104a^{5}+47449a^{4}+7128a^{3}$
$\displaystyle+58a^{2}-56a+3\big{)}+\mathcal{O}\left((b-a)^{3}\right)\,.$

In this case, the electric charge coincides with the one in Cassani:2022lrk
except for the last term in (112). Indeed, we have

$Q-Q_{\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Cassani:2022lrk}{\@@citephrase{(}}{\@@citephrase{)}}}}}=-\frac{2\pi\alpha\lambda_{1}}{\sqrt{3}G_{5}}\,.$
(114)

This constant shift obviously does not affect the first law and it coincides
with the shift obtained in Cassani:2023vsa  from the near-horizon geometry of
the solution with $a=b$. It would be interesting to look for a different frame
— outside of the family of frames (91) — in which one can further remove this
constant. However, given that this constant is a pure number — does not depend
on the solution — this mismatch is not too worrisome.

###  6.4 Exact expressions

With the equations (106), (6.3), (113) at hand it is quite challenging to
study the relation between the entropy and the charges, due the complexity of
these expressions. However, most of this complexity is due to our requirement
of working in the grand-canonical ensemble. Instead, we can do a relabeling of
the $a$ and $b$ parameters as

$(a,b)\rightarrow(a+\alpha\lambda_{1}\delta a,b+\alpha\lambda_{1}\delta b)\,,$
(115)

which has the effect of changing the corrections $\delta S$, $\delta J_{i}$
and $\delta Q$ in (105) and (113). We can use this freedom to cancel for
instance the $\lambda_{1}$ corrections to the expressions of the angular
momenta $J_{i}$. Thus, in this “fixed angular momenta ensemble” our results
read

$\displaystyle S=$ $\displaystyle\frac{\pi^{2}(a+b)\sqrt{ab+a+b}}{2G_{\rm
eff}(1-a)(1-b)}\bigg{[}1+\alpha\lambda_{1}\delta\tilde{S}\bigg{]}\,,$ (116)
$\displaystyle J_{1}=$ $\displaystyle\frac{\pi(a+b)(ab+2a+b)}{4G_{\rm
eff}(1-a)^{2}(1-b)}\,,$ $\displaystyle J_{2}=$
$\displaystyle\frac{\pi(a+b)(ab+2b+a)}{4G_{\rm eff}(1-b)^{2}(1-a)}\,,$
$\displaystyle Q=$ $\displaystyle\frac{\sqrt{3}\pi(a+b)}{4G_{\rm
eff}(1-a)(1-b)}\bigg{[}1+\alpha\lambda_{1}\delta\tilde{Q}\bigg{]}\,,$

where $\delta\tilde{S}$ and $\delta\tilde{Q}$ are now given by

$\displaystyle\delta\tilde{S}$
$\displaystyle=\frac{12(a+1)}{a(a+2)}-\frac{6\left(a^{2}+2a+2\right)(b-a)}{a^{2}(a+2)^{2}}+\frac{\left(a^{5}+9a^{4}+21a^{3}+26a^{2}+24a+12\right)(b-a)^{2}}{a^{3}(a+1)(a+2)^{3}}$
(117)
$\displaystyle-\frac{\left(2a^{7}+20a^{6}+62a^{5}+91a^{4}+88a^{3}+88a^{2}+72a+24\right)(b-a)^{3}}{2a^{4}(a+1)^{2}(a+2)^{4}}+\mathcal{O}\left((b-a)^{4}\right)\,,$
$\displaystyle\delta\tilde{Q}$
$\displaystyle=\frac{4\left(2a^{2}+5a-1\right)}{3a}+\frac{2\left(2a^{2}+1\right)(b-a)}{3a^{2}}-\frac{\left(3a^{3}+5a^{2}+2a+1\right)(b-a)^{2}}{3a^{3}(a+1)}$
$\displaystyle+\frac{\left(3a^{4}+10a^{3}+9a^{2}+4a+1\right)(b-a)^{3}}{6a^{4}(a+1)^{2}}+\mathcal{O}\left((b-a)^{4}\right)\,.$

These expressions are now simple enough that we can guess the exact result. We
find that the simplest rational functions of $a$ and $b$ — with the lowest
order numerator and denominator — leading to the expansions above are

$\displaystyle\delta\tilde{S}=$
$\displaystyle\frac{4\left[ab\left((a-b)^{2}+5(a+b)+18\right)+a^{3}+b^{3}+3(a^{2}+b^{2})+6(a+b)\right]}{(a+b)(a+b+2)(ab+a+b)}\,,$
(118) $\displaystyle\delta\tilde{Q}=$
$\displaystyle\frac{8\left[ab\left(a^{2}+6ab+b^{2}+13(a+b)+10\right)+a^{3}+b^{3}+3(a^{2}+b^{2})-2(a+b)\right]}{3(a+b)^{2}(a+b+2)}\,.$

We then check that these expressions also capture higher-order terms in the
$(b-a)$ expansion — we checked up to order $(b-a)^{6}$ — and therefore this
result must be exact. With these expressions we can now study the dependence
of the entropy on the charges as well as the non-linear constraint satisfied
by these. One can directly check that the following expression

$S=\pi\sqrt{4Q^{2}-\frac{\pi}{G_{\rm
eff}}(J_{1}+J_{2})+\frac{8\pi\alpha\lambda_{1}}{G_{\rm
eff}}\left[\frac{2Q}{\sqrt{3}}+\frac{\pi(J_{1}-J_{2})^{2}}{16G_{\rm
eff}Q^{2}/3-\pi(J_{1}+J_{2})}\right]}\,,$ (119)

holds up to $\mathcal{O}(\alpha^{2})$ terms. This coincides with the result of
Bobev:2022bjm ; Cassani:2022lrk  except for the linear term in the charge,
which comes precisely from the difference in the definition of the charges
(114). Now, in order to derive the non-linear constraint between the charges,
the easiest way consists in evaluating the zeroth-order constraint (29) to
obtain how much it fails to be satisfied. Since in this parametrization only
the electric charge is modified, we get

$\left[2\sqrt{3}Q+\frac{\pi}{2G_{\rm
eff}}\right]\left[4Q^{2}-\frac{\pi}{G_{\rm
eff}}(J_{1}+J_{2})\right]-\left(\frac{2Q}{\sqrt{3}}\right)^{3}-\frac{2\pi}{G_{\rm
eff}}J_{1}J_{2}=\frac{\alpha\lambda_{1}}{G_{\rm
eff}^{2}}\Sigma+\mathcal{O}(\alpha^{2})\,,$ (120)

with

$\Sigma=\frac{2Q\delta\tilde{Q}}{3G_{\rm eff}}\left(32\sqrt{3}G_{\rm
eff}Q^{2}-3\pi\sqrt{3}(J_{1}+J_{2})+6\pi Q\right)\,.$ (121)

Thus, in order to obtain an explicit constraint we only need to express
$\delta\tilde{Q}$ as function of the charges. For this it suffices to use the
two-derivative relations between $(a,b)$ and $(J_{i},Q)$, since the right-hand
side of (120) is already proportional to $\alpha$. After some massaging, we
find that $\Sigma$ can be written as

$\displaystyle\Sigma=$ $\displaystyle 48\pi G_{\rm
eff}J_{1}J_{2}+4\pi^{2}\left(J_{1}+J_{2}\right)-\frac{8\pi
Q}{\sqrt{3}}\left(\pi-6G_{\rm
eff}\left(J_{1}+J_{2}\right)\right)-\frac{80}{3}\pi G_{\rm eff}Q^{2}$ (122)
$\displaystyle+\frac{4\pi^{2}\left(J_{1}-J_{2}\right){}^{2}\left(16\sqrt{3}G_{\rm
eff}Q+3\pi\right)}{3\pi\left(J_{1}+J_{2}\right)-16G_{\rm eff}Q^{2}}\,.$

Finally, we can rewrite the entropy and the constraint in terms of the central
charges of the dual CFT (5). We have

$S=\pi\sqrt{4Q^{2}-8\mathfrak{a}(J_{1}+J_{2})-8(\mathfrak{a}-\mathfrak{c})\left[\frac{2Q}{\sqrt{3}}+\frac{\mathfrak{a}(J_{1}-J_{2})^{2}}{2Q^{2}/3-\mathfrak{a}(J_{1}+J_{2})}\right]}\,,$
(123)

which again matches the results of Bobev:2022bjm ; Cassani:2022lrk  — and
hence the field theory prediction — once we take into account (114). On the
other hand, the non-linear constraint can be expressed as

$\displaystyle\left[\sqrt{3}Q+2\mathfrak{a}\right]\left[Q^{2}-2\mathfrak{c}\left(J_{1}+J_{2}\right)\right]=$
$\displaystyle\frac{Q^{3}}{3\sqrt{3}}+2(3\mathfrak{c}-2\mathfrak{a})J_{1}J_{2}+\frac{2}{3}(\mathfrak{a}-\mathfrak{c})\Bigg{[}4\sqrt{3}\mathfrak{a}Q+5Q^{2}$
(124)
$\displaystyle+\frac{6\mathfrak{a}\left(3\mathfrak{a}+2\sqrt{3}Q\right)\left(J_{1}-J_{2}\right){}^{2}}{2Q^{2}-3\mathfrak{a}\left(J_{1}+J_{2}\right)}\Bigg{]}\,.$

This again coincides with the results of Cassani:2022lrk  taking into account
(114) once more.

##  7 Conclusions

We have studied extremal black holes in AdS5 supergravity with four-derivative
corrections by analyzing their near-horizon geometries. We introduced a novel
parametrization of these solutions that allowed us to write them explicitly
and to solve the corrected Maxwell and Einstein equations. Then, we were able
to compute the charges of the black holes as integrals over the horizon, thus
allowing us to study the different thermodynamic relations even for different
angular momenta. We observed that the entropy, charges and near-horizon
potentials satisfy a near-horizon version of the first-law of thermodynamics
(68) which holds as well in the presence of corrections — see Hajian:2013lna
for a general analysis of the laws of near-horizon black hole mechanics. This
allowed us to identify the near-horizon potentials as fugacities: the
derivatives of the actual potentials with respect to the temperature at $T=0$
— see (73).

Our results for the thermodynamic quantities in the supersymmetric case match
those of Bobev:2022bjm ; Cassani:2022lrk  obtained from evaluation of the on-
shell action following the Reall-Santos method Reall:2019sah , which does not
require solving any equations of motion. This agreement serves as a nontrivial
check of both approaches, but nevertheless our computation reveals a number of
nuances.

First, the need to solve the equations of motion makes the computation
considerably involved in the case of $J_{1}\neq J_{2}$ and in fact we have not
been able to obtain an exact solution. Rather, we had to restrict ourselves to
a solution in the form of a series expansion in $(J_{1}-J_{2})$. It would be
interesting to obtain a closed form solution but our results indicate that
this would be rather complicated as we were not able to find a simple pattern
in the series. On the other hand, we did manage to sum up the series expansion
for the thermodynamic quantities by working in a fixed angular momentum
ensemble — see (118) — which makes the expressions much simpler.

Second, the computation of the electric charge at the near-horizon region is
ambiguous due to the presence of the Lorentz-Chern-Simons three-form in the
charge integral, whose contribution depends on the frame. We have been able go
around this issue by looking for a frame in which the electric charge
satisfies the first law of black hole mechanics. We stress that in general,
and unlike what is stated in Cassani:2023vsa , a change of frame modifies the
charge by a term that depends on the parameters of the solution — see (97) —
and not just by a pure constant.131313Note that the near-horizon geometry is
not a direct product AdS${}_{2}\times\mathbb{S}^{3}$ but a fibered product, so
one cannot directly apply arguments of 3d gravity to analyze the ambiguities
in the Lorentz-Chern-Simons three-form Witten:2007kt . Thus, only in one frame
(or restricted family of frames) the charge defined in this way satisfies the
first law. However, our analysis was not exhaustive and it leaves open the
question of adding a pure constant — independent of the parameters of the
solution — to the charge. In fact, our charge disagrees with the one of
Bobev:2022bjm ; Cassani:2022lrk  by a pure constant. The same constant was
found in Cassani:2023vsa  for near-horizon geometries with equal angular
momenta in the frame given by the left-invariant Maurer-Cartan forms of
$\mathrm{SU}(2)$, and here we showed that it persists unchanged for $J_{1}\neq
J_{2}$. It would be interesting to characterize all the possible frame
transformations and their effect on the charge and to check if one can remove
this constant shift in another frame. More importantly, we would like to
understand what makes special the frame (91) with $n=m=-2$, in which the
electric charge is the correct one for thermodynamics. In any case, this shows
the difficulty in finding a first-principles interpretation of the charge
derived from the on shell action in Cassani:2022lrk ; Bobev:2022bjm .

Despite these struggles, the analysis of near-horizon geometries does have an
advantage over the on-shell action computation: it allows for a direct
generalization to subleading higher-derivative corrections. Indeed, the
methods and techniques presented here can be extended quite straightforwardly
to compute even higher-order corrections. The only real complication would be
the increasing length of the equations, but there would not be new
obstructions. On the other hand, in order to compute second-order corrections
from the on-shell action, one has to obtain the first-order corrections of the
full solution Ma:2023qqj  — not only the near-horizon geometry. In view of our
results here, obtaining the full solution would be very challenging, hence
making this approach probably less accessible than the near-horizon analysis.

On a different note, our analysis was not restricted to supersymmetric
solutions, since the space of extremal solutions is larger, although the
expressions get substantially longer, even with equal angular momenta. A
particularly interesting phenomenon we observed is that, due to the electro-
gravitational Chern-Simons term, gravity sources electromagnetism. In
particular, angular momentum sources electric charge. For instance, the
extremal Myers-Perry-AdS solution, which is a member of the family of
solutions we are analyzing, necessarily acquires a nontrivial gauge field and
it gets a non-zero electric charge given by (104) (in the case of equal
angular momenta). In fact, one cannot have rotating solutions with both zero
charge and zero electrostatic potential. It would be interesting to understand
the meaning of this effect for the dual field theory, which must certainly be
related to the mixed $\langle TT\mathcal{R}\rangle$ anomaly Hanaki:2006pj .

There are several interesting avenues that are natural to pursue regarding
this work. Our method of studying the near-horizon extremal geometry can be
extended to study six-derivative corrections. For that, one must first derive
the form of the Lagrangian, which involves starting from the off-shell action
formalism, integrating out all auxiliary fields and implementing field
redefinitions to find a hopefully relatively compact six-derivative action,
see for example Zucker:1999ej ; deWit:2006gn ; Hanaki:2006pj ;
Cremonini:2008tw ; Bergshoeff:2011xn ; Coomans:2012cf ; Ozkan:2013nwa ;
Baggio:2014hua ; Bobev:2021qxx ; Cassani:2022lrk ; Liu:2022sew . The
difficulty of the analysis of near-horizon geometries in that theory would
arise in the length of the computations but it should be in fact feasible. It
would then be interesting to match the results with the subleading corrections
in the field theory dual.

On the other hand, we can also consider additional matter fields. To the best
of our knowledge, higher derivative corrections have not been studied in the
context of five-dimensional gauged supergravity with several vector
multiplets. This is a rather challenging endeavor as the solutions now involve
the metric, gauge potentials and scalars. For example, this would correspond
to corrections of the supersymmetric Kunduri-Lucietti-Reall black hole
Kunduri:2006ek .

Finally, our novel $(X,Y,Z)$ parametrization of extremal AdS5 black holes
seems to be a very natural form of expressing these solutions, and it has an
interesting geometric interpretation as depicted in Figure 1. This
parametrization leads to a compact and explicit way of writing the solution
and to a very appealing form of the supersymmetric condition (63). It would be
interesting to see if this kind of parametrization can be generalized to other
extremal AdS black holes Chong:2004na ; Chow:2008ip ; Wu:2011gq ;
Bobev:2023bxl . There are numerous approaches in literature that require the
study of extremal solutions and finding such a parametrization for diverse
dimensions would be quite advantageous.

## Acknowledgments

We thank Nikolay Bobev, Davide Cassani, Alejandro Ruipérez, Enrico Turetta and
Annelien Vekemans for useful discussions. The work of PAC received the support
of a fellowship from “la Caixa” Foundation (ID 100010434) with code
LCF/BQ/PI23/11970032. MD is supported in part by the Odysseus grant (G0F9516N
Odysseus) as well as from the Postdoctoral Fellows of the Research Foundation
- Flanders grant (1235324N).

##  Appendix A Higher-derivative corrections

In this appendix, we present the explicit expressions for the higher
derivative corrections found in the limit where the angular momenta are equal
with no supersymmetry imposed in Section A.1 and in the supersymmetric case
with unequal angular momenta in Section A.2. We note that the results are
written in terms of the $(X,Y,Z)$ parametrization.

###  A.1 Equal $J$ solution

We write the various corrections for the $J_{1}=J_{2}$ solution. We find that
$\delta J=\delta\hat{A}_{1}$ while the remaining corrections are

$\displaystyle\delta W$
$\displaystyle=\frac{\lambda_{1}}{\mathcal{D}}\Big{[}57344X^{18}-2048X^{16}\left(92-79Z+5Z^{2}\right)-512X^{14}\left(-507+878Z-514Z^{2}\right.$
(125)
$\displaystyle\left.+22Z^{3}+65Z^{4}\right)+2(-1+Z)^{7}Z^{5}\left(26+39Z+Z^{2}-18Z^{3}-4Z^{4}+3Z^{5}+Z^{6}\right)$
$\displaystyle-64X^{12}\left(2997-7610Z+9159Z^{2}-4436Z^{3}-1717Z^{4}+1550Z^{5}+57Z^{6}\right)$
$\displaystyle-32X^{10}(-1+Z)^{2}\left(-2423+3006Z-7319Z^{2}-1012Z^{3}+3803Z^{4}+502Z^{5}+115Z^{6}\right)$
$\displaystyle-32X^{8}(-1+Z)^{3}\left(-461+329Z-3408Z^{2}-596Z^{3}+1995Z^{4}+45Z^{5}+58Z^{6}+70Z^{7}\right)$
$\displaystyle+16X^{6}(-1+Z)^{4}\left(21+122Z+1338Z^{2}-375Z^{3}-1157Z^{4}+656Z^{5}+268Z^{6}-11Z^{7}\right.$
$\displaystyle\left.+50Z^{8}\right)+4X^{4}(-1+Z)^{5}\left(-60+118Z-200Z^{2}-1488Z^{3}-569Z^{4}+1225Z^{5}+94Z^{6}\right.$
$\displaystyle\left.-190Z^{7}+207Z^{8}+63Z^{9}\right)-2X^{2}(-1+Z)^{6}Z^{2}\left(360+518Z-179Z^{2}-490Z^{3}+199Z^{4}\right.$
$\displaystyle\left.+322Z^{5}+27Z^{6}+2Z^{7}+9Z^{8}\right)\Big{]}\,,$

$\displaystyle\delta G$
$\displaystyle=\frac{2\lambda_{1}(1-Z^{2})}{\mathcal{D}}\Big{[}39936X^{16}+3072X^{14}\left(-43+34Z+9Z^{2}\right)+256X^{12}\left(721-1144Z\right.$
(126)
$\displaystyle\left.+156Z^{2}+284Z^{3}+7Z^{4}\right)+128X^{10}\left(-1087+2545Z-1441Z^{2}-574Z^{3}+587Z^{4}\right.$
$\displaystyle\left.-35Z^{5}+5Z^{6}\right)-16X^{8}(-1+Z)^{2}\left(-3655+3845Z-393Z^{2}-2606Z^{3}+1007Z^{4}\right.$
$\displaystyle\left.+385Z^{5}+25Z^{6}\right)+(-1+Z)^{6}Z^{3}\left(54+27Z-52Z^{2}-26Z^{3}+22Z^{4}+11Z^{5}+8Z^{6}\right.$
$\displaystyle\left.+4Z^{7}\right)-8X^{6}(-1+Z)^{3}\left(-1578+1219Z-1531Z^{2}-2356Z^{3}+1320Z^{4}+803Z^{5}\right.$
$\displaystyle\left.+253Z^{6}+142Z^{7}\right)+2X^{2}(-1+Z)^{5}Z\left(27+225Z+68Z^{2}-256Z^{3}+119Z^{4}+197Z^{5}\right.$
$\displaystyle\left.+2Z^{6}-6Z^{7}+8Z^{8}\right)-4X^{4}(-1+Z)^{4}\left(-267+139Z-1071Z^{2}-1008Z^{3}+704Z^{4}\right.$
$\displaystyle\left.+99Z^{5}-35Z^{6}+226Z^{7}+61Z^{8}\right)\Big{]}\,.$

$\displaystyle\delta B$
$\displaystyle=-\frac{2\lambda_{1}}{\mathcal{D}}\Big{[}112640X^{18}+1536X^{16}\left(-267+212Z+63Z^{2}\right)$
(127)
$\displaystyle+256X^{14}\left(2589-3962Z+194Z^{2}+1126Z^{3}+141Z^{4}\right)$
$\displaystyle+128X^{12}\left(-4863+10460Z-3828Z^{2}-4210Z^{3}+1835Z^{4}+534Z^{5}+72Z^{6}\right)$
$\displaystyle-32X^{10}(-1+Z)^{2}\left(-11313+7582Z+6767Z^{2}-5740Z^{3}-1615Z^{4}+454Z^{5}+297Z^{6}\right)$
$\displaystyle-8X^{8}(-1+Z)^{3}\left(-15819+2635Z+8769Z^{2}-11265Z^{3}-3925Z^{4}+4181Z^{5}+3535Z^{6}\right.$
$\displaystyle\left.+849Z^{7}\right)-(-1+Z)^{7}Z^{3}\left(54+81Z+Z^{2}-39Z^{3}-3Z^{4}+15Z^{5}+15Z^{6}+15Z^{7}+5Z^{8}\right)$
$\displaystyle-4X^{6}(-1+Z)^{4}\left(-5973-1626Z+1744Z^{2}-8694Z^{3}-5370Z^{4}+3426Z^{5}+4696Z^{6}\right.$
$\displaystyle\left.+1822Z^{7}+87Z^{8}\right)+X^{2}(-1+Z)^{6}Z\left(54+261Z+208Z^{2}-144Z^{3}-324Z^{4}-42Z^{5}\right.$
$\displaystyle\left.+336Z^{6}+480Z^{7}+206Z^{8}+21Z^{9}\right)+2X^{4}(-1+Z)^{5}\left(930+610Z+333Z^{2}+3179Z^{3}\right.$
$\displaystyle\left.+2641Z^{4}-889Z^{5}-1749Z^{6}-451Z^{7}+453Z^{8}+159Z^{9}\right)\Big{]}\,,$

$\displaystyle\delta\hat{A}_{1}$
$\displaystyle=\frac{\lambda_{1}}{\mathcal{D}}\Big{[}55296X^{18}+512X^{16}\left(-277+320Z+53Z^{2}\right)$
(128)
$\displaystyle+256X^{14}\left(543-1390Z+414Z^{2}+354Z^{3}+55Z^{4}\right)$
$\displaystyle+64X^{12}\left(-961+4158Z-3017Z^{2}-1432Z^{3}+761Z^{4}+346Z^{5}+145Z^{6}\right)$
$\displaystyle-64X^{10}(-1+Z)^{2}\left(-97+804Z+1143Z^{2}-204Z^{3}-601Z^{4}-124Z^{5}+111Z^{6}\right)$
$\displaystyle-8X^{8}(-1+Z)^{3}\left(645+559Z+8593Z^{2}+3115Z^{3}-5509Z^{4}-1567Z^{5}+1663Z^{6}\right.$
$\displaystyle\left.+469Z^{7}\right)-3(-1+Z)^{7}Z^{3}\left(54+81Z+Z^{2}-39Z^{3}-3Z^{4}+15Z^{5}+15Z^{6}+15Z^{7}\right.$
$\displaystyle\left.+5Z^{8}\right)+4X^{6}(-1+Z)^{4}\left(-423-298Z-8344Z^{2}-5354Z^{3}+5854Z^{4}+2442Z^{5}\right.$
$\displaystyle\left.-1544Z^{6}-198Z^{7}+281Z^{8}\right)+2X^{4}(-1+Z)^{5}\left(-18-138Z-2995Z^{2}-2161Z^{3}\right.$
$\displaystyle\left.+2563Z^{4}-127Z^{5}-1681Z^{6}+693Z^{7}+1187Z^{8}+277Z^{9}\right)+X^{2}(-1+Z)^{6}Z\left(-54\right.$
$\displaystyle\left.-27Z+72Z^{2}+250Z^{3}-756Z^{4}-908Z^{5}+184Z^{6}+550Z^{7}+202Z^{8}+7Z^{9}\right)\Big{]}\,,$
$\displaystyle\delta\hat{A}_{2}$
$\displaystyle=\frac{4\lambda_{1}X^{2}}{(1-2X^{2}-Z)\mathcal{D}}\Big{[}56320X^{18}+512X^{16}\left(-483+373Z+122Z^{2}\right)$
(129)
$\displaystyle+256X^{14}\left(1834-2831Z+168Z^{2}+789Z^{3}+84Z^{4}\right)$
$\displaystyle+128X^{12}\left(-3940+8968Z-3991Z^{2}-3099Z^{3}+1631Z^{4}+403Z^{5}+28Z^{6}\right)$
$\displaystyle-16X^{10}(-1+Z)^{2}\left(-20713+19650Z+9361Z^{2}-12280Z^{3}-2591Z^{4}+590Z^{5}\right.$
$\displaystyle\left.+175Z^{6}\right)-8X^{8}(-1+Z)^{3}\left(-16471+9646Z+5647Z^{2}-14740Z^{3}-2013Z^{4}+3910Z^{5}\right.$
$\displaystyle\left.+2077Z^{6}+504Z^{7}\right)-4X^{6}(-1+Z)^{4}\left(-7201+1949Z-1834Z^{2}-15013Z^{3}-3262Z^{4}\right.$
$\displaystyle\left.+5335Z^{5}+4494Z^{6}+1993Z^{7}+275Z^{8}\right)+2X^{4}(-1+Z)^{5}\left(1353-252Z+3201Z^{2}\right.$
$\displaystyle\left.+9152Z^{3}+3133Z^{4}-2454Z^{5}-2829Z^{6}-1788Z^{7}-274Z^{8}+54Z^{9}\right)$
$\displaystyle+X^{2}(-1+Z)^{6}\left(54-27Z+1461Z^{2}+2629Z^{3}+417Z^{4}-817Z^{5}-329Z^{6}-149Z^{7}\right.$
$\displaystyle\left.+281Z^{8}+236Z^{9}+36Z^{10}\right)-(-1+Z)^{7}Z^{2}\left(-162-189Z+108Z^{2}+161Z^{3}-38Z^{4}\right.$
$\displaystyle\left.-99Z^{5}-104Z^{6}-45Z^{7}+4Z^{8}+4Z^{9}\right)\Big{]}\,,$

where the denominator reads

$\displaystyle\mathcal{D}=$
$\displaystyle(Z-1)^{4}(Z+1)^{2}\Big{[}96X^{12}+48X^{10}\left(Z^{2}+8Z-5\right)$
(130) $\displaystyle-16X^{8}\left(2Z^{4}-11Z^{3}-33Z^{2}+49Z-13\right)$
$\displaystyle-8X^{6}\left(5Z^{6}+13Z^{5}-42Z^{4}-38Z^{3}+113Z^{2}-63Z+12\right)$
$\displaystyle-2X^{4}(Z-1)^{2}\left(9Z^{6}+44Z^{5}+75Z^{4}-70Z^{3}-150Z^{2}+32Z-12\right)$
$\displaystyle-X^{2}(Z-1)^{3}Z^{2}\left(Z^{5}+9Z^{4}+25Z^{3}+35Z^{2}-46Z-72\right)$
$\displaystyle+(Z-1)^{4}Z^{5}\left(Z^{3}+2Z^{2}+Z+2\right)\Big{]}\,.$

###  A.2 Supersymmetric solution with $J_{1}\neq J_{2}$

When the angular momenta are distinct, the corrections to the supersymmetric
solution are given by

$\displaystyle\begin{split}W_{0,0}=&-\frac{1+5X-15X^{2}+45X^{3}}{1-9X-27X^{2}+27X^{3}}\,,\\\
W_{1,0}=&\frac{-7-12X-162X^{2}+540X^{3}+405X^{4}}{(1+3X)^{2}\left(1-12X+9X^{2}\right)^{2}}\,,\\\
W_{1,1}=&\frac{-1-16X+198X^{2}-600X^{3}+639X^{4}-216X^{5}}{(-1+X)(1+3X)\left(-1+3X^{2}\right)\left(1-12X+9X^{2}\right)}\,,\end{split}$
(131)
$\displaystyle\begin{split}G_{0,0}=&\frac{4(-1+3X)\left(-1+11X-69X^{2}+99X^{3}\right)}{X(1+3X)\left(1-12X+9X^{2}\right)}\,,\\\
G_{1,0}=&\frac{2\left(-1+18X+57X^{2}-1476X^{3}+7533X^{4}-10854X^{5}+243X^{6}\right)}{X^{2}(1+3X)^{2}\left(1-12X+9X^{2}\right)^{2}}\,,\\\
G_{1,1}=&\frac{-11+188X-551X^{2}-1248X^{3}+6603X^{4}-7740X^{5}+2727X^{6}}{2(-1+X)X(1+3X)\left(-1+3X^{2}\right)\left(1-12X+9X^{2}\right)}\,,\end{split}$
(132)

$\displaystyle\begin{split}B_{0,0}=&-\frac{9X\left(3+2X-69X^{2}+126X^{3}\right)}{2(1+3X)^{2}\left(1-12X+9X^{2}\right)}\,,\\\
B_{1,0}=&\frac{9\left(-3+5X+42X^{2}-1782X^{3}+4509X^{4}+405X^{5}\right)}{4(1+3X)^{3}\left(1-12X+9X^{2}\right)^{2}}\,,\\\
B_{1,1}=&-\frac{3\left(6+31X-807X^{2}+2958X^{3}-3096X^{4}-513X^{5}+1701X^{6}\right)}{4(-1+X)(1+3X)^{2}\left(-1+3X^{2}\right)\left(1-12X+9X^{2}\right)}\,,\end{split}$
(133)
$\displaystyle\begin{split}J_{0,0}=&-\frac{4\eta(1+3X)^{2}\left(-9+32X-177X^{2}+180X^{3}\right)}{9X^{4}\left(1-12X+9X^{2}\right)}\,,\\\
J_{1,0}=&-\frac{2(1+3X)\left(63-1132X+3801X^{2}-12024X^{3}+20709X^{4}-19116X^{5}+9963X^{6}\right)}{9X^{4}\eta\left(1-12X+9X^{2}\right)^{2}}\,,\\\
J_{1,1}=&-\frac{\eta(1+3X)^{2}\left(36-133X+1605X^{2}-6234X^{3}+13446X^{4}-14661X^{5}+2997X^{6}\right)}{27(-1+X)X^{5}\left(-1+3X^{2}\right)\left(1-12X+9X^{2}\right)}\,,\end{split}$
(134)
$\displaystyle\begin{split}\hat{A}_{1,0,0}=&\frac{4(1+3X)\left(9-59X+273X^{2}-711X^{3}+540X^{4}\right)}{3\sqrt{3}X^{3}\left(1-12X+9X^{2}\right)}\,,\\\
\hat{A}_{1,1,0}=&-\frac{2\left(27-496X+1653X^{2}-3456X^{3}+2889X^{4}-1296X^{5}+5103X^{6}\right)}{3\sqrt{3}X^{4}\left(1-12X+9X^{2}\right)^{2}}\,,\\\
\hat{A}_{1,1,1}=&-\frac{(1+3X)}{9\sqrt{3}(-1+X)X^{4}\left(-1+3X^{2}\right)\left(1-12X+9X^{2}\right)}\\\
&\times\left(36-217X+1968X^{2}-5559X^{3}-3708X^{4}+31293X^{5}-36936X^{6}+13851X^{7}\right)\,,\end{split}$
(135)
$\displaystyle\begin{split}\hat{A}_{2,0,0}=&\frac{3\sqrt{3}\eta\left(5-36X+117X^{2}\right)}{2(1+3X)\left(1-12X+9X^{2}\right)}\,,\\\
\hat{A}_{2,1,0}=&\frac{3\sqrt{3}\left(5-78X+1530X^{2}-7236X^{3}+6885X^{4}+3402X^{5}\right)}{4\eta(1+3X)^{2}\left(1-12X+9X^{2}\right)^{2}}\,,\\\
\hat{A}_{2,1,1}=&\frac{\sqrt{3}\eta\left(-4-51X+924X^{2}-2754X^{3}+2160X^{4}+81X^{5}\right)}{4(-1+X)X(1+3X)\left(-1+3X^{2}\right)\left(1-12X+9X^{2}\right)}\,,\end{split}$
(136)

where

$\displaystyle\eta=\sqrt{X(3X-2)}\,.$ (137)

## References

  * (1) J. M. Maldacena, _The Large N limit of superconformal field theories and supergravity_ , _Adv. Theor. Math. Phys._ 2 (1998) 231–252, [hep-th/9711200].
  * (2) E. Witten, _Anti-de Sitter space and holography_ , _Adv. Theor. Math. Phys._ 2 (1998) 253–291, [hep-th/9802150].
  * (3) S. S. Gubser, I. R. Klebanov and A. M. Polyakov, _Gauge theory correlators from noncritical string theory_ , _Phys. Lett. B_ 428 (1998) 105–114, [hep-th/9802109].
  * (4) N. Bobev, A. M. Charles, D. Gang, K. Hristov and V. Reys, _Higher-derivative supergravity, wrapped M5-branes, and theories of class $\mathrm{\mathcal{R}}$_, _JHEP_ 04 (2021) 058, [2011.05971].
  * (5) N. Bobev, A. M. Charles, K. Hristov and V. Reys, _The Unreasonable Effectiveness of Higher-Derivative Supergravity in AdS 4 Holography_, _Phys. Rev. Lett._ 125 (2020) 131601, [2006.09390].
  * (6) N. Bobev, K. Hristov and V. Reys, _AdS 5 holography and higher-derivative supergravity_, _JHEP_ 04 (2022) 088, [2112.06961].
  * (7) N. Bobev, A. M. Charles, K. Hristov and V. Reys, _Higher-derivative supergravity, AdS 4 holography, and black holes_, _JHEP_ 08 (2021) 173, [2106.04581].
  * (8) N. Bobev, V. Dimitrov, V. Reys and A. Vekemans, _Higher derivative corrections and AdS5 black holes_ , _Phys. Rev. D_ 106 (2022) L121903, [2207.10671].
  * (9) D. Cassani, A. Ruipérez and E. Turetta, _Corrections to AdS 5 black hole thermodynamics from higher-derivative supergravity_, _JHEP_ 11 (2022) 059, [2208.01007].
  * (10) D. Cassani, A. Ruipérez and E. Turetta, _Boundary terms and conserved charges in higher-derivative gauged supergravity_ , _JHEP_ 06 (2023) 203, [2304.06101].
  * (11) P. A. Cano and M. David, _The extremal Kerr entropy in higher-derivative gravities_ , _JHEP_ 05 (2023) 219, [2303.13286].
  * (12) S. Bhattacharyya, A. Grassi, M. Marino and A. Sen, _A One-Loop Test of Quantum Supergravity_ , _Class. Quant. Grav._ 31 (2014) 015012, [1210.6057].
  * (13) M. David, R. De León Ardón, A. Faraggi, L. A. Pando Zayas and G. A. Silva, _One-loop holography with strings in $AdS_{4}\times\mathbb{CP}^{3}$_, _JHEP_ 10 (2019) 070, [1907.08590].
  * (14) A. González Lezcano, J. Hong, J. T. Liu and L. A. Pando Zayas, _Sub-leading Structures in Superconformal Indices: Subdominant Saddles and Logarithmic Contributions_ , _JHEP_ 01 (2021) 001, [2007.12604].
  * (15) M. David, V. Godet, Z. Liu and L. A. Pando Zayas, _Non-topological logarithmic corrections in minimal gauged supergravity_ , _JHEP_ 08 (2022) 043, [2112.09444].
  * (16) M. David, A. Lezcano González, J. Nian and L. A. Pando Zayas, _Logarithmic corrections to the entropy of rotating black holes and black strings in AdS 5_, _JHEP_ 04 (2022) 160, [2106.09730].
  * (17) S. Karan and G. S. Punia, _Logarithmic correction to black hole entropy in universal low-energy string theory models_ , _JHEP_ 03 (2023) 028, [2210.16230].
  * (18) N. Bobev, M. David, J. Hong, V. Reys and X. Zhang, _A compendium of logarithmic corrections in AdS/CFT_ , 2312.08909.
  * (19) J. B. Gutowski and H. S. Reall, _Supersymmetric AdS(5) black holes_ , _JHEP_ 02 (2004) 006, [hep-th/0401042].
  * (20) Z. W. Chong, M. Cvetic, H. Lu and C. N. Pope, _General non-extremal rotating black holes in minimal five-dimensional gauged supergravity_ , _Phys. Rev. Lett._ 95 (2005) 161301, [hep-th/0506029].
  * (21) E. Lauria and A. Van Proeyen, _${\cal N}=2$ Supergravity in $D=4,5,6$ Dimensions_, vol. 966.  3, 2020, 10.1007/978-3-030-33757-5.
  * (22) M. Zucker, _Minimal off-shell supergravity in five-dimensions_ , _Nucl. Phys. B_ 570 (2000) 267–283, [hep-th/9907082].
  * (23) B. de Wit and F. Saueressig, _Off-shell N=2 tensor supermultiplets_ , _JHEP_ 09 (2006) 062, [hep-th/0606148].
  * (24) K. Hanaki, K. Ohashi and Y. Tachikawa, _Supersymmetric Completion of an R**2 term in Five-dimensional Supergravity_ , _Prog. Theor. Phys._ 117 (2007) 533, [hep-th/0611329].
  * (25) S. Cremonini, K. Hanaki, J. T. Liu and P. Szepietowski, _Black holes in five-dimensional gauged supergravity with higher derivatives_ , _JHEP_ 12 (2009) 045, [0812.3572].
  * (26) E. A. Bergshoeff, J. Rosseel and E. Sezgin, _Off-shell D=5, N=2 Riemann Squared Supergravity_ , _Class. Quant. Grav._ 28 (2011) 225016, [1107.2825].
  * (27) F. Coomans and M. Ozkan, _An off-shell formulation for internally gauged D=5, N=2 supergravity from superconformal methods_ , _JHEP_ 01 (2013) 099, [1210.4704].
  * (28) M. Ozkan and Y. Pang, _All off-shell $R^{2}$ invariants in five dimensional $\mathcal{N}=$ 2 supergravity_, _JHEP_ 08 (2013) 042, [1306.1540].
  * (29) M. Baggio, N. Halmagyi, D. R. Mayerson, D. Robbins and B. Wecht, _Higher Derivative Corrections and Central Charges from Wrapped M5-branes_ , _JHEP_ 12 (2014) 042, [1408.2538].
  * (30) D. Butter, S. M. Kuzenko, J. Novak and G. Tartaglino-Mazzucchelli, _Conformal supergravity in five dimensions: New approach and applications_ , _JHEP_ 02 (2015) 111, [1410.8682].
  * (31) J. T. Liu and R. J. Saskowski, _Four-derivative corrections to minimal gauged supergravity in five dimensions_ , _JHEP_ 05 (2022) 171, [2201.04690].
  * (32) G. Gold, J. Hutomo, S. Khandelwal and G. Tartaglino-Mazzucchelli, _Curvature-squared invariants of minimal five-dimensional supergravity from superspace_ , _Phys. Rev. D_ 107 (2023) 106013, [2302.14295].
  * (33) G. Gold, J. Hutomo, S. Khandelwal and G. Tartaglino-Mazzucchelli, _Components of curvature-squared invariants of minimal supergravity in five dimensions_ , 2311.00679.
  * (34) G. Gold, J. Hutomo, S. Khandelwal, M. Ozkan, Y. Pang and G. Tartaglino-Mazzucchelli, _All Gauged Curvature-Squared Supergravities in Five Dimensions_ , _Phys. Rev. Lett._ 131 (2023) 251603, [2309.07637].
  * (35) D. M. Hofman and J. Maldacena, _Conformal collider physics: Energy and charge correlations_ , _JHEP_ 05 (2008) 012, [0803.1467].
  * (36) H. S. Reall and J. E. Santos, _Higher derivative corrections to Kerr black hole thermodynamics_ , _JHEP_ 04 (2019) 021, [1901.11535].
  * (37) J. a. F. Melo and J. E. Santos, _Stringy corrections to the entropy of electrically charged supersymmetric black holes with $\mathrm{AdS}_{5}\times S^{5}$ asymptotics_, _Phys. Rev. D_ 103 (2021) 066008, [2007.06582].
  * (38) L. Ma, Y. Pang and H. Lu, _Higher derivative contributions to black hole thermodynamics at NNLO_ , _JHEP_ 06 (2023) 087, [2304.08527].
  * (39) P.-J. Hu, L. Ma, H. Lu and Y. Pang, _Improved Reall-Santos method for AdS black holes in general higher derivative gravities_ , 2312.11610.
  * (40) A. Castro and E. Verheijden, _Near-AdS2 Spectroscopy: Classifying the Spectrum of Operators and Interactions in N=2 4D Supergravity_ , _Universe_ 7 (2021) 475, [2110.04208].
  * (41) A. Sen, _Black hole entropy function and the attractor mechanism in higher derivative gravity_ , _JHEP_ 09 (2005) 038, [hep-th/0506177].
  * (42) A. Sen, _Entropy function for heterotic black holes_ , _JHEP_ 03 (2006) 008, [hep-th/0508042].
  * (43) A. Dabholkar, A. Sen and S. P. Trivedi, _Black hole microstates and attractor without supersymmetry_ , _JHEP_ 01 (2007) 096, [hep-th/0611143].
  * (44) A. Sen, _Black Hole Entropy Function, Attractors and Precision Counting of Microstates_ , _Gen. Rel. Grav._ 40 (2008) 2249–2431, [0708.1270].
  * (45) D. Astefanesei, K. Goldstein, R. P. Jena, A. Sen and S. P. Trivedi, _Rotating attractors_ , _JHEP_ 10 (2006) 058, [hep-th/0606244].
  * (46) J. F. Morales and H. Samtleben, _Entropy function and attractors for AdS black holes_ , _JHEP_ 10 (2006) 074, [hep-th/0608044].
  * (47) G. Lopes Cardoso, J. M. Oberreuter and J. Perz, _Entropy function for rotating extremal black holes in very special geometry_ , _JHEP_ 05 (2007) 025, [hep-th/0701176].
  * (48) H. K. Kunduri, J. Lucietti and H. S. Reall, _Near-horizon symmetries of extremal black holes_ , _Class. Quant. Grav._ 24 (2007) 4169–4190, [0705.4214].
  * (49) H. K. Kunduri and J. Lucietti, _Near-horizon geometries of supersymmetric AdS(5) black holes_ , _JHEP_ 12 (2007) 015, [0708.3695].
  * (50) P. A. Cano and D. Pereñiguez, _Extremal Rotating Black Holes in Einsteinian Cubic Gravity_ , _Phys. Rev. D_ 101 (2020) 044016, [1910.10721].
  * (51) S. L. Bazanski and P. Zyla, _A Gauss type law for gravity with a cosmological constant_ , _Gen. Rel. Grav._ 22 (1990) 379–387.
  * (52) D. Kastor, _Komar Integrals in Higher (and Lower) Derivative Gravity_ , _Class. Quant. Grav._ 25 (2008) 175007, [0804.1832].
  * (53) D. Kastor, S. Ray and J. Traschen, _Enthalpy and the Mechanics of AdS Black Holes_ , _Class. Quant. Grav._ 26 (2009) 195011, [0904.2765].
  * (54) T. Ortín, _Komar integrals for theories of higher order in the Riemann curvature and black-hole chemistry_ , _JHEP_ 08 (2021) 023, [2104.10717].
  * (55) A. Komar, _Covariant conservation laws in general relativity_ , _Phys. Rev._ 113 (1959) 934–936.
  * (56) K. Hajian, A. Seraj and M. M. Sheikh-Jabbari, _NHEG Mechanics: Laws of Near Horizon Extremal Geometry (Thermo)Dynamics_ , _JHEP_ 03 (2014) 014, [1310.3727].
  * (57) K. Hajian, A. Seraj and M. M. Sheikh-Jabbari, _Near Horizon Extremal Geometry Perturbations: Dynamical Field Perturbations vs. Parametric Variations_ , _JHEP_ 10 (2014) 111, [1407.1992].
  * (58) A. Buchel, J. Escobedo, R. C. Myers, M. F. Paulos, A. Sinha and M. Smolkin, _Holographic GB gravity in arbitrary dimensions_ , _JHEP_ 03 (2010) 111, [0911.4257].
  * (59) R. C. Myers, M. F. Paulos and A. Sinha, _Holographic studies of quasi-topological gravity_ , _JHEP_ 08 (2010) 035, [1004.2055].
  * (60) P. Bueno, P. A. Cano and A. Ruipérez, _Holographic studies of Einsteinian cubic gravity_ , _JHEP_ 03 (2018) 150, [1802.00018].
  * (61) P. A. Cano, A. J. Murcia, A. Rivadulla Sánchez and X. Zhang, _Higher-derivative holography with a chemical potential_ , _JHEP_ 07 (2022) 010, [2202.10473].
  * (62) D. Marolf, _Chern-Simons terms and the three notions of charge_ , in _International Conference on Quantization, Gauge Theory, and Strings: Conference Dedicated to the Memory of Professor Efim Fradkin_ , pp. 312–320, 6, 2000.  hep-th/0006117.
  * (63) D. N. Page, _Classical stability of round and squashed seven-spheres in eleven-dimensional supergravity_ , _Phys. Rev. D_ 28 (Dec, 1983) 2976–2982.
  * (64) R. M. Wald, _Black hole entropy is the Noether charge_ , _Phys. Rev. D_ 48 (1993) R3427–R3431, [gr-qc/9307038].
  * (65) V. Iyer and R. M. Wald, _Some properties of Noether charge and a proposal for dynamical black hole entropy_ , _Phys. Rev. D_ 50 (1994) 846–864, [gr-qc/9403028].
  * (66) Z. Elgood, P. Meessen and T. Ortín, _The first law of black hole mechanics in the Einstein-Maxwell theory revisited_ , _JHEP_ 09 (2020) 026, [2006.02792].
  * (67) Z. Elgood, D. Mitsios, T. Ortín and D. Pereñíguez, _The first law of heterotic stringy black hole mechanics at zeroth order in $\alpha$’_, _JHEP_ 07 (2021) 007, [2012.13323].
  * (68) Z. Elgood, T. Ortín and D. Pereñíguez, _The first law and Wald entropy formula of heterotic stringy black holes at first order in $\alpha^{\prime}$_, _JHEP_ 05 (2021) 110, [2012.14892].
  * (69) A. Cabo-Bizet, D. Cassani, D. Martelli and S. Murthy, _Microscopic origin of the Bekenstein-Hawking entropy of supersymmetric AdS 5 black holes_, _JHEP_ 10 (2019) 062, [1810.11442].
  * (70) F. Benini and E. Milan, _Black Holes in 4D $\mathcal{N}$=4 Super-Yang-Mills Field Theory_, _Phys. Rev. X_ 10 (2020) 021037, [1812.09613].
  * (71) S. Choi, J. Kim, S. Kim and J. Nahmgoong, _Large AdS black holes from QFT_ , 1810.12067.
  * (72) H. K. Kunduri and J. Lucietti, _A Classification of near-horizon geometries of extremal vacuum black holes_ , _J. Math. Phys._ 50 (2009) 082502, [0806.2051].
  * (73) P. J. Silva, _Thermodynamics at the BPS bound for Black Holes in AdS_ , _JHEP_ 10 (2006) 022, [hep-th/0607056].
  * (74) O. J. C. Dias and P. J. Silva, _Euclidean analysis of the entropy functional formalism_ , _Phys. Rev. D_ 77 (2008) 084011, [0704.1405].
  * (75) A. Sen, _Entropy Function and AdS(2) / CFT(1) Correspondence_ , _JHEP_ 11 (2008) 075, [0805.0095].
  * (76) E. Witten, _Three-Dimensional Gravity Revisited_ , 0706.3359.
  * (77) H. K. Kunduri, J. Lucietti and H. S. Reall, _Supersymmetric multi-charge AdS(5) black holes_ , _JHEP_ 04 (2006) 036, [hep-th/0601156].
  * (78) Z. W. Chong, M. Cvetic, H. Lu and C. N. Pope, _Charged rotating black holes in four-dimensional gauged and ungauged supergravities_ , _Nucl. Phys. B_ 717 (2005) 246–271, [hep-th/0411045].
  * (79) D. D. K. Chow, _Charged rotating black holes in six-dimensional gauged supergravity_ , _Class. Quant. Grav._ 27 (2010) 065004, [0808.2728].
  * (80) S.-Q. Wu, _General Nonextremal Rotating Charged AdS Black Holes in Five-dimensional $U(1)^{3}$ Gauged Supergravity: A Simple Construction Method_, _Phys. Lett. B_ 707 (2012) 286–291, [1108.4159].
  * (81) N. Bobev, M. David, J. Hong and R. Mouland, _AdS 7 black holes from rotating M5-branes_, _JHEP_ 09 (2023) 143, [2307.06364].