††institutetext: Leinweber Center for Theoretical Physics,
Department of Physics, University of Michigan,
450 Church St, Ann Arbor, MI 48109, USA

# The Phase Diagram of BPS Black Holes in AdS5

Nizar Ezroura  Finn Larsen  Zhihan Liu  and Yangwenxiao Zeng
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Motivated by recent studies of supersymmetric black holes, we revisit the
phase diagram of AdS black holes, whether BPS or not, with particular emphasis
on the role of rotation. We develop BPS thermodynamics systematically and, in
many explicit examples, we study its striking similarities with more familiar
AdS black holes, as well as some differences. We highlight an important
fugacity that preserves BPS saturation but is not captured by the
supersymmetric index.

###### Keywords:

AdS Black Holes, BPS Limit

††preprint: LCTP-21-22

##  1 Introduction and Summary

The thermodynamics of AdS black holes has been studied by numerous researchers
Hawking:1982dh ; Chamblin:1999tk ; Caldarelli:1999xj ; Silva:2006xv ;
Benini:2015noa ; Almheiri:2016fws ; Hosseini:2016tor ; Benini:2015eyy ;
Hosseini:2017mds ; Benini:2016rke ; Choi:2018hmj ; Choi:2018vbz ; Choi:2019miv
; Cabo-Bizet:2018ehj ; Larsen:2019oll ; Copetti:2020dil ; Larsen:2020lhg .
However, in the last few years there has been a resurgence of interest in BPS
black holes, primarily focussed on the supersymmetric index of their dual
CFT’s Larsen:2019oll ; Larsen:2020lhg ; Cabo-Bizet:2018ehj ; Choi:2018hmj ;
Benini:2018ywd , and we are unaware of studies that interpolate between the
very well known thermodynamics of generic AdS black holes and their BPS limit.
The basic raison d’être of this article is to address this gap in the
literature, with emphasis on the complications due to angular momentum. It is
a welcome bonus that this poorly explored corner of the black hole phase
diagram turns out to illuminate questions that are central to ongoing
microscopic studies of BPS black holes.

BPS black holes are characterized, first of all, by their mass satisfying the
BPS equality111In this article we set the AdS scale $\ell=1$ unless
specifically noted. With this convention the black hole quantum numbers $M$,
$Q_{I}$, $J_{a}$ are all integers for bosons while, for fermions, they are
half-integral.

$M=\sum_{I}Q_{I}+\sum_{i}J_{i}~{},$ (1)

relating their mass and charges. For AdS5 the ranges $I=1,2,3$ and $i=a,b$ but
the formula applies in other dimensions when ranges are adapted appropriately.
In the context of a microscopic theory (or the Euclidean path integral) we can
isolate such states by taking the extremal ($T\rightarrow 0$) limit of the
partition function

$Z={\rm
Tr}~{}e^{-\beta(M-\Phi_{I}Q_{I}-\Omega_{i}J_{i})}~{}~{}\underset{T\rightarrow
0}{\rightarrow}~{}~{}Z_{\rm BPS}={\rm
Tr}~{}e^{\Phi^{\prime}_{I}Q_{I}+\Omega^{\prime}_{i}J_{i}}~{},$ (2)

where primes denote derivatives with respect to $T=\beta^{-1}$ and we have
absorbed the supersymmetric Casimir Energy in the black hole mass. The limit
$T\rightarrow 0$ is sufficient to project onto states that satisfy (1). That
in turn is possible only for BPS states, due to unitarity, because the
difference between the two sides of (1) appears on the right hand side of the
anticommutator between a supercharge and its conjugate.

This setup is entirely standard, perhaps even boring, but it is also just the
background for our story Kinney:2005ej ; Choi:2018hmj . Contemporary research
on BPS black holes in AdS, whether in gravity or in CFT, nearly always imposes
the additional condition on potentials that

$\varphi^{\prime}=\sum_{I}\Phi^{\prime}_{I}-\sum_{i}\Omega^{\prime}_{i}\overset{?}{=}0~{},$
(3)

vanishes. There are excellent reasons to do so. The supersymmetric index is
defined via the BPS partition function (2), with the grading $(-)^{F}$ with
respect to fermion number inserted. The index is independent of temperature
$T$, so the limit $T\rightarrow 0$ is optional, and the limit
$T\rightarrow\infty$ may well be more convenient for computations. The reason
that the index does not depend on $T$ is, as usual, that states violating the
BPS bound are “long”, in that the relevant supercharge acts nontrivially, and
so contributions from bosons and fermions in the supermultiplet cancel. The
brute force limit $T\rightarrow 0$ accomplishes this task as well, albeit with
less elegance.

However, the index is more powerful, it also ensures that the states that do
satisfy the BPS bound cannot combine into long multiplets as continuous
parameters, such as the coupling constant, vary. This only works if the
fugacities assigned to BPS states are consistent with with the preserved
supercharge, a condition that amounts precisely to $\varphi^{\prime}=0$. Thus,
if $\varphi^{\prime}\neq 0$, the partition function generally depends on
continuous parameters whether or not $(-)^{F}$ is inserted. Such loss of
control discourages analysis in supersymmetric quantum field theory at weak
coupling but, from the vantage point of the gravitational dual, there is no
reason not to proceed. Quite the contrary, the AdS/CFT correspondence is
routinely exploited to compute partition functions in strong coupling regimes
that are inaccessible to other methods Larsen:2019oll ; Larsen:2020lhg ;
Drukker:2010nc ; Aharony:2003sx .

The processes whereby BPS states combine or disassociate into fewer or more
BPS states preserve the linear mass relation $\eqref{eqn:MBPS}$ by rearranging
the mass between $R$-charge $\sum_{I}Q_{I}$ and angular momentum
$\sum_{i}J_{i}$. The index is, by construction, insensitive to such
reorganizations but, for macroscopic black holes, it is fundamental that all
conserved charges are specified. The dependence of black hole thermodynamics
on $\varphi^{\prime}$ amounts to a probe on how equilibrium forms along the
chain of competing partitions of $(\sum_{I}Q_{I})+(\sum_{i}J_{i})$
Larsen:2021wnu .

To appreciate the central significance of this question, recall that the hope
of describing BPS black holes in AdS in terms of weakly coupled CFT’s has a
history of disappointment that was only partially addressed by recent
progress. It was long thought that the supersymmetric index was order ${\cal
O}(1)$ in the rank $N$ of the underlying theory, unlike the exponential growth
needed to account for the black hole entropy Kinney:2005ej ; Berkooz:2006wc ;
Grant:2008sk ; Chang:2013fba . That would be unfortunate but there is no
general argument against the cancellations brought about by insertion of
$(-)^{F}$ being very efficient, effectively removing the leading asymptotics
completely. Over the last few years evidence has emerged that such large
cancellations can be prevented, after all, while preserving the protective
abilities of the index, by introducing complex fugacities that compensate for
the sensitivity to the statistics factor $(-)^{F}$ Cabo-Bizet:2018ehj ;
Choi:2018hmj ; Benini:2018ywd ; Choi:2018vbz ; Choi:2019miv , at least in
part. Our study offers a complementary perspective on the dynamics along this
direction in configuration space.

As we noted already, early literature on the thermodynamics of AdS black holes
did not highlight the BPS limit. The renewed interest in recent years was
stimulated by the Hosseini-Hristov-Zaffaroni (HHZ) free energies
Hosseini:2017mds , and rightfully so. The AdS5 version can be written as

$\ln
Z=\frac{1}{2}N^{2}\frac{\Delta_{1}\Delta_{2}\Delta_{3}}{\omega_{a}\omega_{b}}~{},$
(4)

where complex potentials $\Delta_{I}$, $\omega_{i}$ have real parts that can
be identified with $\Phi^{\prime}_{I}$, $\Omega^{\prime}_{i}$ but, in
addition, they are endowed with complex parts. The HHZ free energy is at a
central nexus because it can be derived, with some caveats, either from the
gravitational action or from the supersymmetric index, and it may be related
to the supersymmetric Casimir energy. As a point of comparison between such
varied aspects of BPS black holes, it has a strict regime of applicability,
required by the principles of supersymmetric localization. In particular, it
is based on the index, the grading $(-)^{F}$ must be inserted, and it only
applies when $\varphi^{\prime}=0$. Therefore, by its very nature, it is not
sensitive to the thermodynamic questions we study in this paper. We discuss
the precise relation between the more conservative notion of gravitational
free energy that we study and the index motivated HHZ expression in subsection
3.5.

The phase diagram for BPS black holes is surprisingly elaborate. It has been
known for a few years that, with $\varphi^{\prime}=0$, it is qualitatively
similar to that of the AdS-Schwarzschild black hole Hawking:1982dh ;
Kubiznak:2016qmn . There are small and large black hole branches joined by a
cusp, and a Hawking-Page transition point to a thermal gas phase, all within a
setting that does not break supersymmetry. Our work augments the BPS phase
diagram and show that, in various ranges of the “primed” BPS potentials, the
small black holes may exhibit a maximal temperature, their free energy at high
temperature may asymptote to a nonzero value, or the small black hole branch
may disappear altogether. The large black holes, on the other hand, exhibit
simple scaling laws at high temperature. These structures are reminiscent of
the phase diagram for non-supersymmetric AdS black holes but here they appear
in the context of BPS thermodynamics, justifying hope for a precision
understanding of these interesting features starting from microscopic
principles.

The intricate phase diagram for BPS black holes, and its striking resemblance
to analogous plots for generic AdS black holes, motivate a closer study of the
entire phase diagram. In section 2 we explore the general thermodynamics of
$\text{AdS}_{5}$ black holes in the grand canonical ensemble, including the
effects of electric potential $\Phi$ and angular velocities $\Omega_{a,b}$.
The critical values $\Phi^{*}=3$ and $\Omega^{*}_{a,b}=1$ correspond to BPS
saturation. They give rise to phase boundaries where physical properties
change qualitatively. We detail the behavior as one or more critical values is
approached, while paying special attention to the orders of limits.

All BPS black holes have the same potentials, they take their critical values,
and vanishing free energy. In section 3 we introduce the BPS free energy $W$
and the “primed” BPS potentials $\Phi^{\prime}_{I},\Omega^{\prime}_{i}$ that
it depends on. The thermodynamics of BPS black holes in AdS5 is developed with
emphasis on a “benchmark” case, the simplest example. We also connect the
gravitational thermodynamics to the dual CFT via the complex HHZ potential,
establishing an exact match whether $(-)^{F}$ is inserted
($\varphi^{\prime}=2\pi i$) or not ($\varphi^{\prime}=0$).

Finally, in section 4 we study the thermodynamics of BPS black holes for
general BPS potentials. We determine the physical ranges of the BPS potentials
and show that generic values of $\varphi^{\prime}$ can modify the phase
diagram qualitatively. Throughout this article our main example is the Kerr-
Newman AdS5 black hole and its BPS limit. By this we mean “diagonal” charges
$Q_{I}=Q$ and two angular momenta $J_{a},J_{b}$ that may or may not be equal.
In particular, we study $J_{a}\neq J_{b}$ for BPS black holes in subsection
4.4.

##  2 General Thermodynamics of $\text{AdS}_{5}$ Black Holes

In this section we study the thermodynamics of $\text{AdS}_{5}$ black holes
for the case with equal charges ($Q_{I}=Q$) but general non-equal angular
momenta $J_{a,b}$.

Taking Schwarzschild-AdS as a benchmark, the phase diagram deforms smoothly as
electric potential and angular velocities are added. Angular momentum proves
more destabilizing than charge.

###  2.1 Black Hole Thermodynamics

The Kerr-Newman AdS5 family of black holes is characterized by four conserved
charges $(M,Q,J_{a},J_{b})$. These physical charges are parametrized by $4$
variables $(m,q,a,b)$ through:

$\displaystyle M$ $\displaystyle=$
$\displaystyle\frac{\pi}{4G_{5}}\frac{m(3-a^{2}g^{2}-b^{2}g^{2}-a^{2}b^{2}g^{4})+2qabg^{2}(2-a^{2}g^{2}-b^{2}g^{2})}{(1-a^{2}g^{2})^{2}(1-b^{2}g^{2})^{2}}~{},$
(5) $\displaystyle Q$ $\displaystyle=$
$\displaystyle\frac{\pi}{4G_{5}}\frac{q}{(1-a^{2}g^{2})(1-b^{2}g^{2})}~{},$
(6) $\displaystyle J_{a}$ $\displaystyle=$
$\displaystyle\frac{\pi}{4G_{5}}\frac{2ma+qb(1+a^{2}g^{2})}{(1-a^{2}g^{2})^{2}(1-b^{2}g^{2})}~{},$
(7) $\displaystyle J_{b}$ $\displaystyle=$
$\displaystyle\frac{\pi}{4G_{5}}\frac{2mb+qa(1+b^{2}g^{2})}{(1-a^{2}g^{2})(1-b^{2}g^{2})^{2}}~{}.$
(8)

We will assume that all the conserved charges are nonnegative. Their scale is
set by the dimensionless parameter

$\frac{\pi\ell^{3}_{5}}{4G_{5}}=\frac{1}{2}N^{2}~{},$

where $N$ refers to the dual $SU(N)$ gauge group of ${\cal N}=4$ SYM. The only
other dimensionful parameter entering the thermodynamic formulae is the AdS5
scale $\ell_{5}$ that is equivalent to the coupling of gauged supergravity
$g=\ell^{-1}_{5}$. Henceforth we set $\ell_{5}=1$ (and so $g=1$), to avoid
clutter, but the AdS5 scale is easily restored. For example, $M,Q$ are inverse
lengths, $a,b$ are lengths, and $q,m$ are lengths squared.

The event horizon is at the coordinate location $r_{+}$, a combination of
parameters that is ubiquitous in thermodynamic formulae for AdS black holes.
It is determined as the largest root of the horizon equation

$\displaystyle\Delta_{r}(r)=\frac{(r^{2}+a^{2})(r^{2}+b^{2})(1+r^{2})+q^{2}+2abq}{r^{2}}-2m=0~{}.$
(9)

With $r_{+}$ given implicitly through this equation, the black hole
temperature is

$\displaystyle T=\frac{r_{+}^{4}[1+(2r_{+}^{2}+a^{2}+b^{2})]-(ab+q)^{2}}{2\pi
r_{+}[(r_{+}^{2}+a^{2})(r_{+}^{2}+b^{2})+abq]}~{},$ (10)

and the electric potential and angular velocities are

$\displaystyle\Phi$ $\displaystyle=$
$\displaystyle\frac{3qr_{+}^{2}}{(r_{+}^{2}+a^{2})(r_{+}^{2}+b^{2})+abq}~{},$
(11) $\displaystyle\Omega_{a}$ $\displaystyle=$
$\displaystyle\frac{a(r_{+}^{2}+b^{2})(1+r_{+}^{2})+bq}{(r_{+}^{2}+a^{2})(r_{+}^{2}+b^{2})+abq}~{},$
(12) $\displaystyle\Omega_{b}$ $\displaystyle=$
$\displaystyle\frac{b(r_{+}^{2}+a^{2})(1+r_{+}^{2})+aq}{(r_{+}^{2}+a^{2})(r_{+}^{2}+b^{2})+abq}~{}.$
(13)

The linchpin for thermodynamics is, of course, the black hole entropy computed
from the area law:

$\displaystyle
S=2\pi\cdot\frac{1}{2}N^{2}\cdot\frac{(r_{+}^{2}+a^{2})(r_{+}^{2}+b^{2})+abq}{(1-a^{2})(1-b^{2})r_{+}}~{},$
(14)

The conserved charges (5-8) and the thermodynamic potentials (10-14), with the
subsidiary condition (9) that determines $r_{+}$ implicitly, completely
determine the thermodynamics of Kerr-Newman AdS5 black holes. The
nonlinearities evident in these parametric equations contain a great deal of
physics.

The black hole mass $M$ is bounded from below by the BPS mass

$M_{\rm BPS}=\Phi^{*}Q+\Omega^{*}_{a}J_{a}+\Omega^{*}_{b}J_{b}~{},$

where $\Phi_{*}=3$ and $\Omega^{*}_{a}=\Omega^{*}_{b}=1$. In this section we
consider generic variables, a detailed study of BPS thermodynamics follows
later. However, we will find that the critical values of the potentials,
denoted by stars, play a central role also away from the BPS limit. In
particular, we consider only $\Omega_{a,b}\leq\Omega^{*}_{a,b}=1$, because the
“overspinning” black holes with larger values are classically unstable due to
superradiance.

###  2.2 The Grand Canonical Ensemble

In this subsection we study the phase diagram in the grand canonical ensemble,
i.e. as a function of temperature $T$, electric potential $\Phi$, and angular
velocities $\Omega_{a,b}$. These potentials correspond to periodicities of the
(Euclidean) time, the gauge function and two azimuthal angles, respectively.
Therefore, the grand canonical ensemble is the natural description in the path
integral formalism where asymptotic boundary conditions are specified as
geometrical data.

The Gibbs free energy is given by

$\displaystyle G$ $\displaystyle=$ $\displaystyle M-TS-\Phi
Q-\Omega_{a}J_{a}-\Omega_{b}J_{b}~{}.$ (15)

The extensive variables $M$, $Q$, $J_{a,b}$, $S$ are presented in (5-8, 14).
After trading $m$ for $r_{+}$ through the horizon equation (9) they become
functions of the parameters $(r_{+},q,a,b)$. We can explicitly eliminate $q$
in favor of the potential $\Phi$

$\displaystyle q$ $\displaystyle=$
$\displaystyle\frac{\Phi(r_{+}^{2}+a^{2})(r_{+}^{2}+b^{2})}{3r_{+}^{2}-ab\Phi}~{},$
(16)

by inverting (11). However, the parameters $(r_{+},a,b)$ must remain as
implicit functions of the potentials $T$, $\Phi$ and $\Omega_{a,b}$ because it
is impractical to solve (10, 12-13). Roughly, but not precisely, the horizon
location $r_{+}$ is a proxy for the temperature $T$, and the rotation
parameters $a,b$ represent rotation velocities $\Omega_{a,b}$.

After the elimination of the parameter $q$, the angular velocities become

$\displaystyle 1-\Omega_{a}$ $\displaystyle=$
$\displaystyle\frac{1-a}{r_{+}^{2}+a^{2}}\left[r_{+}^{2}-r_{\ast}^{2}+b(1+a)(1-\frac{1}{3}\Phi)\right]~{},$
(17) $\displaystyle 1-\Omega_{b}$ $\displaystyle=$
$\displaystyle\frac{1-b}{r_{+}^{2}+b^{2}}\left[r_{+}^{2}-r_{\ast}^{2}+a(1+b)(1-\frac{1}{3}\Phi)\right]~{}.$
(18)

These formulae are completely general, albeit presented in a manner that
anticipate a special role for the BPS horizon position
$r^{2}_{\ast}=\sqrt{a+b+ab}$ and the critical values of the potentials
$\Omega^{*}_{a}=\Omega^{*}_{b}=1$, $\Phi^{*}=3$. In a similar spirit, we can
present the general temperature as

$\begin{split}T=&\frac{r_{+}(r_{+}^{2}+a+b-ab)}{2\pi\left(r_{+}^{2}-ab\right)}\left(\frac{1-\Omega_{a}}{1-a}+\frac{1-\Omega_{b}}{1-b}\right)\\\
&-\frac{r_{+}}{6\pi}\frac{\left(3+\Phi-3a-3b\right)r_{+}^{2}+(a^{2}+b^{2}+a^{2}b+ab^{2})\Phi-
ab(\Phi-3)}{(r_{+}^{2}-ab)(3r_{+}^{2}-ab\Phi)}\left(\Phi-3\right)~{}.\end{split}$
(19)

In this form it is manifest that the temperature vanishes when the potentials
take their critical values. We are not able to explicitly invert the three
equations (17-19) any further but, taken together, they relate the physical
potentials $(T,\Omega_{a},\Omega_{b})$ to the parameters $(r_{+},a,b)$ for a
given $\Phi$.

After elimination of the parameter $q$ in favor of the electric potential
$\Phi$, Gibbs’ free energy $G$ becomes:

$\begin{split}G=&-\frac{N^{2}}{4}\frac{(r_{+}^{2}+a^{2})(r_{+}^{2}+b^{2})(r_{+}^{2}+a+b-ab)(r_{+}^{2}-r_{\ast}^{2})}{(1-a^{2})(1-b^{2})(r_{+}^{2}-ab)^{2}}\\\
&-\frac{N^{2}\left(r_{+}^{2}+a^{2}\right)\left(r_{+}^{2}+b^{2}\right)\left(\Phi-3\right)}{12(1-a^{2})(1-b^{2})\left(r_{+}^{2}-ab\right)^{2}\left(3r_{+}^{2}-ab\Phi\right)^{2}}\left[(r^{2}_{+}-ab)^{2}(9r^{2}_{+}+3r^{2}_{+}\Phi-2ab\Phi^{2})\right.\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\left.-3(a+b)^{2}r^{2}_{+}(3r^{2}_{+}+r^{2}_{+}\Phi-2ab\Phi)\right]~{}.\end{split}$
(20)

Again, we have pulled out factors that make it manifest that the expression
vanishes when the potentials take their critical values.

In the following subsections we study the phase diagram for various ranges or
special values of $\Phi$, with any $\Omega$, and for two exceptional values of
$\Omega$, for any $\Phi$. When combined, these special cases map out the
entire phase diagram.

####  2.2.1 Angular Velocity $\Omega$ with Electric Potential $\Phi=0$

If, in addition to taking the electric potential $\Phi=0$, we also consider
vanishing angular velocities $\Omega_{a,b}=0$, we are left with the
Schwarzschild-AdS black hole. In this case the phase diagram, reproduced in
Figure 1, is extremely well-known Witten:1998qj . Some of its features are:

  * •

There is a strictly positive lower bound on the black hole temperature $T\geq
T_{\text{min}}$.

  * •

For each temperature $T\geq T_{\text{min}}$ there are two branches. Black
holes on the upper one are “small” in units of the AdS5 radius. They are
qualitatively similar to asymptotically flat black holes. The “large” black
holes on the lower branch are influenced significantly by the AdS5 background.

  * •

A thermal AdS phase has the same boundary conditions at infinity as the black
holes but vanishing free energy. It dominates when the temperature is below
the Hawking-Page temperature $T_{\text{HP}}$ Hawking:1982dh , where the large
black hole also has zero free energy. The large black holes are
thermodynamically preferred only when they have temperature $T\geq
T_{\text{HP}}$.

The Hawking-Page transition is particularly interesting because it can also be
interpreted as the confinement/deconfinement transition in QCD-like theories
tHooft:1977nqb ; Polyakov:1978vu ; Susskind:1979up ; Aharony:2003sx  living on
the boundary via the AdS/CFT correspondenceWitten:1998zw ; Witten:1998qj ;
Maldacena:1999 .

Figure 1: AdS-Schwarzschild phase diagram ($\Phi=0,~{}\Omega_{a,b}=0$). The
cusp is at the minimal temperature $T_{\text{min}}=\frac{\sqrt{2}}{\pi}$, and
the maximal free energy $G_{\text{max}}=\frac{N^{2}}{16}$. The large black
hole branch meets the thermal AdS phase at the Hawking-Page temperature
$T_{\text{HP}}=\frac{3}{2\pi}$.

As the angular velocities $\Omega_{a,b}$ are turned on and increased, with the
electric potential kept at $\Phi=0$, the qualitative features of the AdS-
Schwarzschild phase diagram are preserved. Naturally, there are quantitative
changes:

  * •

For any given $T$, on either branch, the angular velocities lower the free
energy. This follows from the first law that gives
$\partial_{\Omega_{a,b}}G=-J_{a,b}<0$.

  * •

The transition temperature $T_{\text{HP}}$ decreases as either angular
velocity increases, as it must because the large black hole branch is lowered.

  * •

The minimal black hole temperature $T_{\text{min}}$ also decreases as either
angular velocity increases. The maximal free energy, attained at the cusp,
increases. Thus the cusp travels “towards North-West” when angular velocity
increases.

  * •

In the limit where the angular velocities $\Omega_{a,b}\rightarrow 1$ the
height of the cusp diverges $G_{\text{max}}\rightarrow\infty$ but its
temperature approaches a finite (nonzero) limit
$T_{\text{min}}\rightarrow\frac{1}{2\pi}$ . In the strict limit
$\Omega_{a}=\Omega_{b}=1$ the “large” branch disappears altogether. This limit
is detailed in subsection 2.2.6.

The existence of an absolute minimum for the temperature for any angular
velocities and the increase in maximal free energy both suggest that,
everything else being equal, rotation tends to destabilize the black hole.

When $\Phi=0$, the two angular velocities are “decoupled” in that

$\Omega_{a}=\frac{a(1+r_{+}^{2})}{r_{+}^{2}+a^{2}}~{},$ (21)

is a function only of $r_{+}$ and $a$, but not $b$. The formula for
$\Omega_{b}$ is analogous.

The free energy (20) also simplifies greatly when $\Phi=0$:

$\displaystyle
G=-\frac{N^{2}}{4}\frac{\left(r_{+}^{2}-1\right)\left(r_{+}^{2}+a^{2}\right)\left(r_{+}^{2}+b^{2}\right)}{r_{+}^{2}\left(1-a^{2}\right)\left(1-b^{2}\right)}~{}.$
(22)

The Hawking-Page transition, where $G=0$, corresponds to the horizon parameter
$r_{+}=1$. With this value we can invert (21) (and its analogue for
$\Omega_{b}$) and find $a$ ($b$) as a function of $\Omega_{a}$ ($\Omega_{b}$).
These expressions, along with $r_{+}=1$, give a simple formula for the
Hawking-Page transition temperature

$T~{}\underset{\Phi=0}{=}~{}\frac{r_{+}^{4}\left(2r_{+}^{2}+a^{2}+b^{2}+1\right)-a^{2}b^{2}}{2\pi
r_{+}\left(r_{+}^{2}+a^{2}\right)\left(r_{+}^{2}+b^{2}\right)}~{}\underset{\rm
HP}{=}~{}\frac{1+\sqrt{1-\Omega_{a}^{2}}+\sqrt{1-\Omega_{b}^{2}}}{2\pi}~{}.$
(23)

It is a decreasing function of the two angular velocities independently, as
expected. It interpolates between the AdS-Schwarzschild value $T_{\rm
HP}=\frac{3}{2\pi}$ when there is no rotation, and approaches the finite value
$T_{\rm HP}=\frac{1}{2\pi}$ towards $\Omega_{a}=\Omega_{b}=1$. where G
diverges. We discuss the subtle limiting case in subsection 2.2.6.

The phase diagram for vanishing potential $\Phi=0$ and a sample of angular
velocities is presented in Figure 2.

Figure 2: Gibbs’ free energy $G$ as function of the temperature $T$. The
electric potential vanishes $\Phi=0$ and the angular velocities increase from
right to left as $\Omega_{a}=\Omega_{b}=0,~{}0.5,~{}0.75,~{}0.95$. The cusp
moves “North-West” as angular velocities increase, staying above the minimum
temperature $T_{\rm min}=\frac{1}{2\pi}\approx 0.16$ even in the limit
$\Omega_{a}=\Omega_{b}\rightarrow 1$.

There has been suggestions that the “thermal gas in AdS” that competes with
the black hole phase may not exist, except for AdS-Schwarzschild. The free
energy is evaluated by a gravitational path integral with asymptotic boundary
conditions that are satisfied by a black hole, but also compatible with a bulk
spacetime that has no black hole. The latter has $G=0$ (up to a Casimir term)
and, whatever its precise physical nature, it must be taken into account. It
is on this basis that we will consider black holes with $G>0$ unstable, also
for spacetimes with angular velocity and/or electric potential.

####  2.2.2 Subcritical Electric Potential: $0<\Phi<3$

Starting from vanishing electric potential $\Phi$, but arbitrary angular
velocities (below their critical values $\Omega_{a}^{*}=\Omega_{b}^{*}=1$), we
now consider increasing the electric potential toward its critical value
$\Phi^{*}=3$. The phase diagram remains qualitatively similar to the AdS-
Schwarzschild case as the electric potential increases, but quantitative
features are modified. Some of these changes are similar to an increase in
angular velocity at fixed electric potential:

  * •

The free energy at a given temperature decreases, on both the small and the
large black hole branches. This follows from the thermodynamic relation
$\partial_{\Phi}G=-Q$.

  * •

The transition temperature $T_{\text{HP}}$ decreases, as it must because the
entire large black hole branch is lowered. The minimal temperature, attained
at the cusp where the two branches meet, also decreases.

However, despite these similarities, increasing electric potential differs
significantly from increasing angular velocity:

  * •

The free energy at the cusp $G_{\rm max}$ decreases with increasing electric
potential.

  * •

The transition temperature $T_{\text{HP}}$ (and so the minimal temperature
$T_{\rm min}<T_{\text{HP}}$) decreases with no minimum nonzero value as the
electric potential approaches the critical value $\Phi=\Phi^{*}$.

Both of these features indicate that increased potential $\Phi$ stabilizes the
black hole. The phase diagram with electric potential in the range $0<\Phi<3$
is presented in Figure 3.

Figure 3: Gibbs’ free energy $G$ as function of the temperature $T$. The
electric potential $\Phi=1.5$ is in the range $0<\Phi<3$ and the angular
velocities increase from right to left as
$\Omega_{a}=\Omega_{b}=0,~{}0.5,~{}0.75,~{}0.95$. For a given $\Omega_{a,b}$,
the free energy and the temperature are both smaller than for $\Phi=0$
(compare with Figure 2.)

####  2.2.3 Critical Electric Potential $\Phi=\Phi^{*}=3$

When $\Phi$ increases to the critical value $\Phi^{*}=3$, the phase diagram
changes qualitatively.

The expressions for potentials

$\displaystyle
T|_{\Phi=3}=\frac{r_{+}(2r_{+}^{2}+a^{2}+b^{2})\left(r_{+}^{2}+a+b-ab\right)}{2\pi(r_{+}^{2}+a^{2})(r_{+}^{2}+b^{2})(r_{+}^{2}-ab)}(r_{+}^{2}-r_{*}^{2})~{},$
(24) $\displaystyle
1-\Omega_{a}|_{\Phi=3}=\frac{1-a}{r^{2}_{+}+a^{2}}(r^{2}_{+}-r_{*}^{2})~{},$
(25) $\displaystyle
1-\Omega_{b}|_{\Phi=3}=\frac{1-b}{r^{2}_{+}+b^{2}}(r^{2}_{+}-r_{*}^{2})~{},$
(26)

and for Gibbs free energy

$\displaystyle
G|_{\Phi=3}=-\frac{N^{2}(r_{+}^{2}+a^{2})(r_{+}^{2}+b^{2})\left(r_{+}^{2}+a+b-ab\right)}{4(1-a^{2})(1-b^{2})(r_{+}^{2}-ab)^{2}}(r_{+}^{2}-r_{*}^{2})~{},$
(27)

simplify somewhat. As in earlier formulae, $r_{*}=\sqrt{a+b+ab}$ is the
horizon location for BPS black holes with parameters $(a,b)$.

The equations show that, when $\Phi=\Phi^{*}=3$, the requirement of positive
temperature $T\geq 0$ is equivalent to $r_{+}\geq r_{*}$. Therefore, taking
$\Phi=\Phi^{*}=3$ automatically prevents overspinning (it keeps
$\Omega_{a,b}\leq 1$) and it also ensures non-positive free energy $G\leq 0$.
Because of these inequalities, the phase diagram simplifies greatly when
$\Phi=\Phi^{*}$. There is no small black hole branch, no matter what the
temperature and angular velocities are. In particular, there is no “cusp”
where two branches meet. Additionally, since black hole states have non-
positive free energy, the thermal AdS gas with $G=0$ is never preferred.

When any of the bounds noted in the previous paragraph are saturated we have
$r^{2}_{+}=r^{2}_{*}$ and so they all become equalities. Moreover, in this
case the black hole is BPS. In other words, the following are equivalent:

  * •

$\Phi=\Phi^{*}$ and any of $T=0$, $\Omega_{a}=1$, $\Omega_{b}=1$, $G=0$.

  * •

$\Phi=\Phi^{*}$ and all of $T=0$, $\Omega_{a}=1$, $\Omega_{b}=1$, $G=0$.

  * •

The black hole is supersymmetric, i.e. the mass is exactly the BPS mass:
$M=M_{\rm BPS}=\Phi^{*}Q+\Omega^{*}_{a}J_{a}+\Omega^{*}_{b}J_{b}$.

The formulae (24-26) suggest a specific approach to the BPS limit: simply fix
the parameters $a,b$ and tune black hole parameters so that
$r^{2}_{+}-r^{2}_{*}=\epsilon\rightarrow 0$. This corresponds to physical
potentials approaching the BPS limit linearly as

$\displaystyle
T|_{\Phi=3}=\frac{(2+a+b)\sqrt{a+b+ab}}{\pi(1+a)(1+b)(a+b)}\epsilon~{},$ (28)
$\displaystyle 1-\Omega_{a}|_{\Phi=3}=\frac{1-a}{(1+a)(a+b)}\epsilon~{},$ (29)
$\displaystyle 1-\Omega_{b}|_{\Phi=3}=\frac{1-b}{(1+b)(a+b)}\epsilon~{},$ (30)

with Gibbs free energy (20) also approaching zero linearly

$\displaystyle
G|_{\Phi=3}=-\frac{1}{2}N^{2}\frac{a+b}{(1-a)(1-b)}\epsilon~{}.$ (31)

When implementing the BPS limit in this way it is manifest that
$G,T,1-\Omega_{a,b}$ all reach $0$ simultaneously, with specified relative
rate. In section 3 we will identify this approach with the BPS limit itself.

The somewhat formal statements on the BPS limit in the previous paragraphs
fail to reflect all physical aspects. Therefore, we now consider the physical
realization of thermodynamics already discussed for $\Phi<3$: start with high
temperature and then cool the system all the way to $T=0$, while keeping the
physical angular velocities $\Omega_{a,b}$ fixed. Figure 4 shows the resulting
free energy $G$ as function of temperature $T$, for various values of
$\Omega_{a}=\Omega_{b}$.

Figure 4: The free energy $G$ as function of temperature $T$ for
$\Phi=\Phi^{*}=3$. The angular velocity
$\Omega_{a}=\Omega_{b}=0,~{}0.5,~{}0.75,~{}0.95$ from right to left. The
curves approach the origin $G=T=0$ without reaching it. The special case
$\Omega_{a}=\Omega_{b}=1$ is not a curve, it is the single point $G=T=0$.

To implement the physical thermodynamics underlying Figure 4 we would lower
the parameter $r_{+}$, a proxy for temperature, and simultaneously lower the
parameters $a,b$, in order that the left hand sides of (25-26) remain
constant. The approach to the BPS limit, described in (28-30), is different
because it keeps the parameters $a,b$ fixed. The distinction is dramatic
because Figure 4 depicts a smooth approach to $T=G=0$ for various
$\Omega_{a,b}$, in apparent contradiction with the statement that vanishing
temperature $T=0$ is possible only when $\Omega_{a}=\Omega_{b}=1$ identically.

To resolve the tension, it is sufficient to consider the regime $r^{2}_{+}\gg
a^{2},b^{2},ab$ where formulae simplify so that:

$\displaystyle T$ $\displaystyle=\frac{r^{4}_{+}-(a+b)^{2}}{\pi
r^{3}_{+}}~{},$ (32) $\displaystyle\frac{1-\Omega_{a}}{1-a}$
$\displaystyle=\frac{1-\Omega_{b}}{1-b}=1-\frac{a+b}{r^{2}_{+}}~{},$ (33)
$\displaystyle G$
$\displaystyle=-\frac{1}{2}N^{2}\frac{r^{4}_{+}-(a+b)^{2}}{(1-a^{2})(1-b^{2})}~{}.$
(34)

We further take $a,b\ll 1$ so $r^{2}_{+}\sim a+b$ (which easily satisfies
$r^{2}_{+}\gg a^{2},b^{2},ab$) in a manner where the rotational velocities can
take any value $\Omega_{a,b}=\frac{a+b}{r^{2}_{+}}$, as long as
$\Omega_{a}=\Omega_{b}$. Then

$T=\frac{1}{\pi}r_{+}(1-\Omega^{2})~{},$ (35)

and the free energy becomes:

$G=-\frac{1}{2}N^{2}\frac{\pi^{4}}{(1-\Omega^{2})^{3}}T^{4}~{}.$ (36)

These formulae are not the most general, but they exhibit important features
clearly.

The BPS limit (28-30) lowers $r_{+}$ so
$r^{2}_{+}-r^{2}_{*}=\epsilon\rightarrow 0$. In this case (35-36) are realized
by $G$, $T$, $1-\Omega^{2}$ all approaching zero at the same rate, while
$r_{+}$ is near the constant $r_{*}$.

The low temperature limit with the angular velocity $\Omega$ fixed is also
described by the simplified free energy (36) but, due to (35), the strict
limit $T\rightarrow 0$ requires $r^{2}_{+}\rightarrow 0$ (with $a,b\rightarrow
0$ such that $\Omega_{a,b}=\frac{a+b}{r^{2}_{+}}$ is fixed). The limiting
geometry with $r_{+}=0$ is singular, so it is not actually a solution. It is a
“small” black hole, not just in the AdS/CFT vernacular, where the term refers
to black hole size that is much smaller than the AdS-scale $r_{+}\ll\ell_{5}$,
but in the sense that unknown higher derivative curvature corrections dominate
the classical “solution”. Therefore, the BPS limit is well-controlled only if
it is taken as specified in (28-30).

The free energy (36) is reminiscent of the high temperature regime $r_{+}\gg
1$. In this limit the rotational velocities $\Omega_{a}=a$, $\Omega_{b}=b$ ,
the temperature (24) simplifies to $T=\frac{r_{+}}{\pi}$ and the free energy
(34) becomes

$G=-\frac{1}{2}N^{2}\frac{\pi^{4}}{(1-\Omega_{a}^{2})(1-\Omega_{b}^{2})}T^{4}~{}.$
(37)

This formula shares with the low temperature expression (36) the power $T^{4}$
that is characteristic of CFT’s in four dimensions. However, factors of
$1-\Omega^{2}$ differ. The high temperature regime is dual to the conformal
fluid Bhattacharyya:2007vjd ; Rangamani:2009xk .

####  2.2.4 Super-critical Electric Potential $\Phi>3$

When $\Phi$ increases above the critical value $\Phi_{*}=3$ all black holes
acquire strictly negative free energy. Therefore, they are thermodynamically
preferred to thermal AdS5. The free energy decreases monotonically as $\Phi$
increases further. Moreover, it decreases further as the angular velocities
increase with fixed $\Phi$ and $T$. All these trends follow straightforwardly
from the thermodynamic relations $\partial_{\Phi}G=-Q$ and
$\partial_{\Omega_{a,b}}G=-J_{a,b}$. They are illustrated in Figure 5.

Figure 5: The free energy $G$ when $\Phi=3.5$. The angular velocities
$\Omega_{a}=\Omega_{b}=0,~{}0.5,~{}0.75,~{}0.9$ increase from right to left.

To be more specific, we consider the case $\Omega_{a}=\Omega_{b}\equiv\Omega$,
corresponding to $a=b$. Then we can invert the angular velocity (17) and solve
for $r_{+}^{2}$

$r_{+}^{2}=\frac{a(3(1-a\Omega)+\Phi(1-a^{2}))}{3(\Omega-a)}~{},$ (38)

and the temperature (19) becomes

$T=\frac{a(3+\Phi-6\Omega^{2})-(\Phi-3)\Omega}{6\pi a(1-a\Omega)}r_{+}~{}.$
(39)

The nontrivial solution for extremality ($T=0$)

$a=\frac{(\Phi-3)\Omega}{3+\Phi-6\Omega^{2}}~{},$ (40)

yields the free energy (20) in terms of $\Phi$ and
$\Omega_{a}=\Omega_{b}=\Omega$ at extremality:

$G_{\text{ext}}=-\frac{N^{2}\left(\frac{1}{3}\Phi-1\right)^{2}}{16(1-\Omega^{2})^{2}}\left[3\left(\frac{1}{3}\Phi+1\right)^{2}-4\Omega^{2}\left(1+\frac{2}{3}\Phi\right)\right].$
(41)

This analytical formula gives the strictly negative free energy at $T=0$ for
any $\Phi>3$. In the nonrotating limit $\Omega\rightarrow 0$ (further
discussed in the following subsection) the formula is regular, so non-rotating
black holes in AdS5 are stable when the electric potential is large. This
result applies also in the flat space limit where it is significant for
studies of the weak gravity conjecture (WGC) Harlow:2021trr ; Harlow:2022gzl .

We are particularly interested in the BPS limit which requires $\Omega=1$.
When $\Phi>3$ the free energy is singular for this value: the physical range
of the angular velocity $\Omega$ is $[0,1)$, so $\Omega$ can be arbitrarily
close to the critical value $\Omega^{*}=1$, but it cannot reach it. However,
the square bracket in (41) vanishes when both $\Omega=1$ and $\Phi=3$.
Therefore, in the limit $\Phi\rightarrow 3$, the singularity at $\Omega=1$ is
suppressed by three powers of $\frac{1}{3}\Phi-1$. This shows that the free
energy vanishes $G\rightarrow 0$ when $\Omega\rightarrow 1$ and
$\Phi\rightarrow 3$ at identical “speed”.

The reason that the angular velocities $\Omega_{a,b}$ can be arbitrarily close
to their critical values $\Omega^{*}_{a,b}=1$, but cannot reach it, is that
the conserved charges $M,Q,J_{a,b}$ must be finite: they can be arbitrarily
large, but they cannot diverge. This in turn requires $a,b\in[0,1)$, neither
of these parameters can be exactly equal to one.

To see this, first consider the temperature (19) near $a=1$ and/or $b=1$:

$\displaystyle
T\underset{a=1-\epsilon_{a},b=1-\epsilon_{b}}{=}\frac{r_{+}\Big{(}r_{+}^{2}-(\frac{2}{3}\Phi+1)-\frac{1}{18}(\Phi^{2}-9)\Big{)}}{\pi\left(r_{+}^{2}-\frac{\Phi}{3}\right)}+\mathcal{O}(\epsilon_{a},\epsilon_{b})~{}.$
(42)

Nonnegativity of the temperature in this regime shows
$r_{+}^{2}\geq(\frac{2}{3}\Phi+1)>0$. Therefore, since $\epsilon_{a,b}$ can be
small but not strictly zero, the angular velocities (17) and (18)

$\displaystyle
1-\Omega_{a}\underset{a=1-\epsilon_{a},b=1-\epsilon_{b}}{=}\frac{\left(r_{+}^{2}-(\frac{2}{3}\Phi+1)\right)}{(1+r_{+}^{2})}\epsilon_{a}+\mathcal{O}(\epsilon^{2})~{},$
(43) $\displaystyle
1-\Omega_{b}\underset{a=1-\epsilon_{a},b=1-\epsilon_{b}}{=}\frac{\left(r_{+}^{2}-(\frac{2}{3}\Phi+1)\right)}{(1+r_{+}^{2})}\epsilon_{b}+\mathcal{O}(\epsilon^{2})~{}.$
(44)

can be arbitrarily close, but not equal to, the critical values
$\Omega^{*}_{a,b}=1$. In other words, when $\Phi>3$ either of the limits
$\Omega_{a}\rightarrow 1$ or $\Omega_{b}\rightarrow 1$ takes all the quantum
numbers $M,Q,J_{a,b}\rightarrow\infty$

####  2.2.5 Nonrotating Black Holes: $\Omega=0$

In this subsection we turn off rotation by setting $a=b=0$ and focus on the
effect of the electric potential $\Phi$. This case was well developed early on
Chamblin:1999tk ; Caldarelli:1999xj  so it serves as an important benchmark
for the effects of rotation.

For zero angular velocity $\Omega_{a}=\Omega_{b}=0$, the temperature (19) and
the free energy (20) reduce to

$\displaystyle T=$
$\displaystyle\frac{2g^{2}r_{+}^{2}+1-\left(\frac{1}{3}\Phi\right)^{2}}{2\pi
r_{+}}~{},$ (45) $\displaystyle G=$
$\displaystyle-\frac{N^{2}}{4}r_{+}^{2}\left(g^{2}r_{+}^{2}+\left(\frac{1}{3}\Phi\right)^{2}-1\right)~{}.$
(46)

For given $(T,\Phi)$, the first equation yields $2$, $1$, or $0$ solutions for
$r_{+}$; and then the second equation gives the applicable values of the free
energy $G(T,\Phi)$. It is presented in Figure 6.

The AdS-Schwarzschild black hole, reviewed in the beginning of subsection
2.2.1 and plotted in Figure 1, is the curve to the right. Increasing electric
potential $\Phi$ lowers both the minimal temperature $T_{\rm min}$ and the
maximal free energy $G_{\rm max}$. These effects make the black hole more
stable.

Figure 6: The free energy for non-rotating black holes $\Omega=0$. The
electric potential $\Phi$ increases from right to left. There is a qualitative
transition from AdS-Schwarzschild type at $\Phi<\Phi_{*}=3$ to the high
potential regime $\Phi\geq\Phi_{*}=3$.

There is a qualitative change as the potential increases from the regime
$\Phi<3$ to $\Phi\geq 3$. The lower range of potential $\Phi<3$ is
“Schwarzchild-type”, it has two black hole branches that are joined at a cusp
where the temperature reaches its minimum $T_{\rm min}$ and the free energy
its maximum $G_{\rm max}$. The higher range of potential $\Phi\geq 3$ has just
a ”large” black hole branch. Therefore, this range corresponds to particularly
stable black holes, as identified by the weak gravity conjecture.

The earlier parts of this subsection considered the effect of rotation on
various ranges of $\Phi$. In particular, each curve in Figure 6 is the same as
first curve shown in Figures 2-5. The results of those earlier subsections
showed that angular velocities strictly below maximal $0\leq\Omega_{a,b}<1$ do
not change the phase diagrams qualitatively. For $\Phi<3$, angular velocity
lowers the temperature $T_{\rm min}$ but the free energy $G_{\rm max}$
increases. For $\Phi>3$ rotational velocity lowers the free energy of the
large black hole branch, the only one there is. The case where the electric
potential is exactly critical $\Phi=\Phi^{*}=3$ is subtle, as discussed in
subsection 2.2.3.

Black hole thermodynamics is much simpler in the absence of rotation so more
formulae can be made explicit. In this case, we can invert (45) and solve for
the coordinate location of the horizon $r_{+}$:

$r_{+}=\frac{\sqrt{2g^{2}(\Phi^{2}-9)+(3\pi T)^{2}}+3\pi T}{6g^{2}}~{}.$ (47)

We picked the positive root. Then (46) gives the free energy:

$G=-\frac{N^{2}}{4}\frac{\left(\sqrt{2g^{2}\left(\Phi^{2}-9\right)+(3\pi
T)^{2}}+3\pi T\right)^{2}\left(\pi
T\left(\sqrt{2g^{2}\left(\Phi^{2}-9\right)+(3\pi T)^{2}}+3\pi
T\right)+g^{2}\left(\Phi^{2}-9\right)\right)}{216g^{6}}~{}.$ (48)

We restored the AdS5 length scale $g=\ell_{5}^{-1}$ in order to facilitate
comparison to the flat space limit $g\rightarrow 0$. When
$\Phi<\Phi_{\ast}=3$, the argument of the square roots may become negative so,
as we have elaborated on extensively, there is a minimal temperature. For
$\Phi>\Phi_{\ast}=3$, the case discussed in the previous subsection, there is
a lower bound $r_{+}>\frac{1}{6}\ell_{5}\sqrt{2(\Phi^{2}-9)}$ and the free
energy is strictly negative in the limit.

It is instructive to take $\Phi=\Phi_{\ast}=3$ from the outset and then take
$T\rightarrow 0$ with the AdS scale $\ell_{5}\rightarrow\infty$ such that the
product $T\ell_{5}=\Delta$ is fixed. In this limit the coordinate horizon

$r_{+}=\pi T\ell^{2}_{5}=\pi\Delta\ell_{5}~{},$ (49)

diverges. In this limit $\ell_{5}\rightarrow\infty$ the underlying black hole
geometry is not ”small”, the position of the coordinate horizon $r_{+}$
approaches a finite limit, and here this corresponds to finite spatial extent.
Indeed, the underlying ”large” black hole geometry reduces to the
asymptotically flat Reissner-Nordström black hole.

The low temperature regime with no rotation also makes contact with a large
body of recent literature that studies universal features shared with the SYK
model (some entry points to the literature Maldacena:2016upp ; Sachdev:2019bjn
). In this context general thermodynamic arguments show that the approach to
extremality is quadratic in temperature Preskill:1991tb ; Almheiri:2016fws
and it is interesting that the coefficient of the quadratic temperature
dependence can be computed directly in the extremal ($T=0$) geometry
Larsen:2018iou ; Hong:2019tsx . Here the low temperature of the free energy
(48) is linear in the temperature

$G=-\frac{N^{2}\left(\Phi^{2}-9\right)^{2}}{432g^{2}}-\frac{N^{2}(\Phi^{2}-9)^{\frac{3}{2}}}{54\sqrt{2}g^{3}}\pi
T-\frac{N^{2}\left(\Phi^{2}-9\right)}{24g^{4}}\left(\pi T\right)^{2}+\ldots$
(50)

for a given $\Phi>\Phi_{*}=3$, There is an apparent tension, but it is
resolved because our study is in the grand canonical ensemble (fixed
potentials), while the quadratic behavior pertains to the canonical ensemble
(fixed charges).

####  2.2.6 Maximal Rotational Velocity: $\Omega=1$

It is instructive to consider black holes with maximal horizon velocities
$\Omega_{a}=\Omega_{b}=1$ for any temperature $T$ and electric potential
$\Phi$. This example shows that generally such spins are destabilizing, but
the BPS case is an exception.

The general expressions for $\Omega_{a,b}$ (17-18) give the horizon position

$r_{+}^{2}=r_{\ast}^{2}-b(1+a)(1-\frac{1}{3}\Phi)~{}.$ (51)

It also gives the analogous expression with $a\leftrightarrow b$ so, for
consistency, the rotational parameters must be identical $a=b$. The general
formula for temperature (10) becomes

$\displaystyle T=$
$\displaystyle\frac{1}{2\pi}(1-\frac{1}{3}\Phi)\sqrt{(1+\frac{1}{3}\Phi)a^{-1}+\frac{1}{3}\Phi}~{}.$
(52)

Since the temperature is nonnegative $T\geq 0$, we must demand $\Phi\leq 3$.
This agrees with the analysis in subsection 2.2.4: the rotational velocities
can not reach their maximal values $\Omega_{a,b}=1$ when $\Phi>3$.

In most of the examples we consider, the horizon location $r_{+}$ is larger
than the corresponding BPS radius $r_{*}$, at the same values of $a,b$. One
might wonder if this is a physical condition. For example, it could be that
black holes that are excited above the BPS limit must be bigger because they
have more entropy. On the other hand, the parameter $r_{+}$ is not a physical
variable, nor are $a,b$, so this comparison cannot be taken too seriously.
Indeed, according to (51), black holes with $\Omega=1$ give a counterexample
because they have $r_{+}<r_{*}$.

The temperature $T$ given in (52) is a monotonically decreasing function of
the parameter $a$. Since black holes with finite charges must have $a<1$, the
temperature is bounded from below by the value at $a=1$:

$T_{\rm min}=\frac{1}{2\pi}(1-\frac{1}{3}\Phi)\sqrt{1+\frac{2}{3}\Phi}~{}.$
(53)

The inequality $a<1$ is strict, $a=1$ is not possible, so the lower bound
$T\geq T_{\rm min}$ can be saturated only when $\Phi=3$. In fact, when
$\Phi=3$, the minimal temperature $T_{\rm min}=0$ is not just a bound, (52)
shows it is the only possible temperature. That the temperature must vanish in
this case is precisely as expected, because when $\Phi=3$, $\Omega_{a,b}=1$
all potentials are critical and so the black hole is BPS $M=M_{\rm BPS}$. Here
we focus on the interesting discontinuity between $\Phi\rightarrow 3^{-}$ and
$\Phi=3$.

For maximal rotational velocities $\Omega_{a,b}=1$ and any generic potential
$0\leq\Phi<3$, all temperatures $T>T_{\rm min}$ are possible. Gibbs’ free
energy (15) becomes:

$\begin{split}G&=\frac{1}{2}N^{2}\frac{a(3-\Phi)(\Phi+3)^{2}}{54(1-a)}\cr&=\frac{1}{4}N^{2}\frac{\left(1-\frac{1}{9}\Phi^{2}\right)^{3}}{(2\pi
T)^{2}-(1+\frac{2}{3}\Phi)(1-\frac{1}{3}\Phi)^{2}}~{}.\end{split}$ (54)

In the second expression the rotational parameter $a$ was eliminated by taking
advantage of (52). This free energy, plotted in Figure 7, is positive
definite, so the corresponding black holes are always “small”. The minimal
temperature $T_{\rm min}$ given in (53) is such that the denominator is
positive for all temperatures $T>T_{\rm min}$ and vanishes in the limit
$T\rightarrow T_{\rm min}$. Therefore, the free energy diverges in this limit.

The challenges faced by the approach to the BPS limit $T=0$, $\Phi=3$ are
stark when $\Omega_{a,b}=1$ is set from the outset:

  * •

There is a minimal temperature $T_{\rm min}$. However, it decreases
monotonically as a function of the potential $\Phi$, from $T_{\rm
min}=\frac{1}{2\pi}$ when $\Phi=0$ to $T_{\rm min}\rightarrow 0$ as
$\Phi\rightarrow 3_{-}$.

  * •

The free energy diverges if, for a given potential, the temperature $T$ is
lowered to $T_{\rm min}$. However, the “height” of the divergence is set by
$(1-\frac{1}{3}\Phi)^{3}$.

  * •

By increasing $\Phi\rightarrow 3_{-}$ and lowering $T$ simultaneously, the
free energy can vanish in the limit, after all. However, the free energy
necessarily approaches zero from above $G\rightarrow 0_{+}$.

The “safest” approach to the BPS limit are from the large black hole branch
$G\rightarrow 0_{-}$. It requires taking electric potential
$\Phi=\Phi_{\ast}=3$ first, or at least approaching it from above
$\Phi\rightarrow 3_{+}$, and only then tuning other parameters.

Figure 7: Free energy as function of temperature when $\Omega_{a,b}=1$ and
$\Phi\neq 3$. There are only “small” black holes. They all have $G>0$ and
negative specific heat ($G$ decreases with temperature). The curves represent
$\Phi=0,1,2$ (from right to left). The vertical asymptote indicating the
minimal temperature is shown only for $\Phi=2$, to reduce clutter. As $\Phi$
increases to $\Phi\rightarrow 3_{-}$ the minimal temperature decreases to
$T_{\rm min}=0$. The BPS limit where $\Phi=3$ exactly is not a curve, it is
just the origin where $T=0$ and $G=0$.

##  3 Thermodynamics of Supersymmetric Black Holes

In this section we study the thermodynamics of black holes that are strictly
supersymmetric. We consider general aspects, as well as their realization by
black holes in AdS5. As an instructive example, we develop new details on a
“benchmark” case that was discussed previously. The following section 4
studies more general cases.

###  3.1 General Structure of BPS Thermodynamics

Suppose we have a general free energy222In this subsection we refer to a
single electric potential $\Phi$ and a single angular velocity $\Omega$, to
keep terminology simple. However, one or both kinds of potentials can have
multiple components, corresponding to several independent charges and/or
angular momenta.

$G=G(T,\Phi,\Omega)=M-TS-\Phi Q-\Omega J~{},$ (55)

and want to study the states that satisfy the BPS mass relation

$M=\Phi^{*}Q+\Omega^{*}J~{},$ (56)

where $\Phi^{*}$ and $\Omega^{*}$ are given numbers that depend on the system
we consider. The general free energy (55) satisfies the first law

$dG=-SdT-Qd\Phi-Jd\Omega~{},$

so it yields the mass

$M=G+TS+\Phi Q+\Omega
J=(1-T\partial_{T}-\Phi\partial_{\Phi}-\Omega\partial_{\Omega})G~{}.$

The BPS mass relation (56) is satisfied if and only if

$\Big{[}1-T\partial_{T}G-(\Phi-\Phi^{*})\partial_{\Phi}-(\Omega-\Omega^{*})\partial_{\Omega}\Phi\Big{]}G=0~{}.$

This is the condition that the free energy is homogeneous of degree $1$ in the
variables $T,\Phi-\Phi^{*},\Omega-\Omega^{*}$.

Considering any $G$ that is homogeneous of degree $1$, we define the BPS free
energy

$W(\Phi^{\prime},\Omega^{\prime})=\frac{G}{T}=-S-\Phi^{\prime}Q-\Omega^{\prime}J~{}.$
(57)

The second form used the BPS mass relation (56) and introduced primed
potentials

$\displaystyle\Phi^{\prime}$ $\displaystyle=\frac{\Phi-\Phi^{*}}{T}~{},$ (58)
$\displaystyle\Omega^{\prime}$ $\displaystyle=\frac{\Omega-\Omega^{*}}{T}~{}.$
(59)

The function W inherits a homogeneity property from $G$ and, because we have
divided by $T$, it is in fact precisely homogeneous in the variables
$T,\Phi-\Phi^{*},\Omega-\Omega^{*}$, it has “weight” zero. Therefore, it is
possible to interpret it as a function of the ratios
$\Phi^{\prime},\Omega^{\prime}$, as we have indicated in our notation for $W$.
In the BPS limit we necessarily have temperature $T\rightarrow 0$ so these
“primed” variables can be identified as derivatives with respect to the
temperature, evaluated at $T=0$. An alternative but equivalent interpretation
is that the BPS surface is a projective manifold defined by ratios of
thermodynamic variables.

The BPS free energy $W$ satisfies the first law

$dW=-Qd\Phi^{\prime}-Jd\Omega^{\prime}~{},$ (60)

so

$\displaystyle Q=-\frac{\partial W}{\partial\Phi^{\prime}}~{},$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}J=-\frac{\partial
W}{\partial\Omega^{\prime}}~{}.$ (61)

It yields the entropy

$S=(-1+\Phi^{\prime}\partial_{\Phi^{\prime}}+\Omega^{\prime}\partial_{\Omega^{\prime}})W~{}.$
(62)

However, the BPS mass has been “integrated out” of the BPS thermodynamics, it
is not encoded in $W$ and only follows from $Q$ and $J$ via the BPS mass
relation (56).

In statistical mechanics, the BPS free energy $W$ arises when introducing a
BPS partition function as a trace over all BPS states

$Z_{\rm BPS}={\rm Tr}_{\rm
BPS}\left[e^{\Phi^{\prime}Q+\Omega^{\prime}J}\right]~{},$ (63)

with no explicit reference to mass. Then the BPS free energy $W=-\ln Z_{\rm
BPS}$.

###  3.2 Parameters for BPS Black Holes in AdS5

In this paper we primarily study $\text{AdS}_{5}$ black holes with a single
charge and two independent angular momenta. We want to make the general
considerations in the preceding subsection explicit in this specific case.

The BPS condition is

$M=3Q+J_{a}+J_{b}~{}.$ (64)

Comparing with the generic formula (56) we have

$\Phi^{*}=3~{},$ (65)

and, for $a=b$, $\Omega^{*}=2$. In the general case where $a\neq b$ there are
two independent rotational velocities $\Omega_{a,b}$ with BPS values

$\Omega^{*}_{a}=\Omega^{*}_{b}=1~{}.$ (66)

The general parametric expressions for the black hole quantum numbers
$(M,Q,J_{a,b})$ in (5 \- 8) give the mass excess above the BPS bound

$M-3Q-(J_{a}+J_{b})=\frac{1}{2}N^{2}\frac{3+(a+b)-ab}{(1-a)(1+a)^{2}(1-b)(1+b)^{2}}\Big{(}m-q(1+a+b)\Big{)}~{}.$
(67)

The prefactor on the right hand side is manifestly positive for all
$a,b\in[0,1[$, so the BPS formula (64) yields the condition on the parameters
$(m,q,a,b)$:

$m=q(1+a+b)~{}.$ (68)

We can therefore consider $m$ a dependent variable and parametrize the BPS
black holes by $(q,a,b)$.

Even when the parameter $m$ satisfies (68), the equation (9) for the
coordinate location of the event horizon is a cubic equation in $r^{2}$.
Instead of proceeding by brute force, it is useful to recall that BPS black
holes are extremal and so we expect a double root. With this clue, and the BPS
condition (68), it is manageable to find the solution

$r_{*}^{2}=a+b+ab~{},$ (69)

for a specific value of the parameter $q$:

$q^{*}=(1+a)(1+b)(a+b)~{}.$ (70)

Whichever way this solution for particular values of $m,q$ was arrived at, we
can, for any $m,q$, recast the cubic equation (9) satisfied by the horizon
location $r^{2}$ as

$(r^{2}-r^{2}_{*})^{2}r^{2}+\Big{(}(q-q^{*})-(r^{2}-r^{2}_{*})(1+a+b)\Big{)}^{2}+2r^{2}(m-q(1+a+b))=0~{}.$
(71)

When the BPS condition (68) is satisfied this expression is the sum of two
squares so, for BPS black holes, it is manifest that a real solution exists
only for $q=q^{*}$ and for this value of $q$ there is indeed a double root
when $r^{2}=r^{2}_{*}$. We will refer to the starred variables as the BPS
values.

When $m$ satisfies the BPS condition (68) and $q$ takes its BPS value (70),
the parametric formulae (5 \- 8) for the physical variables $(M,Q,J_{a,b})$
simplify to:

$\displaystyle M^{*}$
$\displaystyle=\frac{1}{2}N^{2}\frac{(a+b)(3-a^{2}+ab-b^{2}-ab(a+b))}{(1-a)^{2}(1-b)^{2}}~{},$
(72) $\displaystyle Q^{*}$
$\displaystyle=\frac{1}{2}N^{2}\frac{a+b}{(1-a)(1-b)}~{},$ (73) $\displaystyle
J_{a}^{*}$
$\displaystyle=\frac{1}{2}N^{2}\frac{(a+b)(2a+b+ab)}{(1-a)^{2}(1-b)}~{},$ (74)
$\displaystyle J_{b}^{*}$
$\displaystyle=\frac{1}{2}N^{2}\frac{(a+b)(a+2b+ab)}{(1-a)(1-b)^{2}}~{}.$ (75)

The potential $\Phi$ and the angular velocities $\Omega_{a,b}$ given in (11 \-
13) reduce to the BPS values (65-66), as they should. The entropy $S$ (14)
simplifies to:

$S^{*}=2\pi\frac{1}{2}N^{2}\frac{(a+b)\sqrt{a+b+ab}}{(1-a)(1-b)}~{}.$ (76)

As we have stressed, once we impose the BPS mass formula (64), black holes
exist only when the parameters $q,a,b$ satisfy the relation (70). Thus the BPS
mass cannot be reached for all black hole variables $(Q,J_{a},J_{b})$. Rather,
according to (73), (74) and (75), the conserved charges must satisfy the
constraint

$\left(Q^{*3}+\frac{1}{2}N^{2}J_{a}^{*}J_{b}^{*}\right)-\left(3Q^{*}+\frac{1}{2}N^{2}\right)\left(3Q^{*2}-\frac{1}{2}N^{2}(J^{*}_{a}+J^{*}_{b})\right)=0~{}.$
(77)

It is surprising that there is this condition on the charges.

###  3.3 The BPS Free Energy

Usually, a good starting point for the study of thermodynamics is the free
energy $G$, an extensive function of the intensive thermodynamic potentials
$T,\Phi,\Omega_{a,b}$. However, for BPS black holes the free energy vanishes
identically $G=0$ and the “variables” it would ordinarily be a “function” of
are fixed at $T=0,\Phi^{*}=3$, and $\Omega^{*}_{a,b}=1$.

The general structure of BPS thermodynamics discussed in subsection 3.1
addresses this obstacle. A free energy $G$ that is homogeneous of degree one
in $T$, $\Phi-\Phi^{*}$, $\Omega_{a,b}-\Omega_{a,b}^{*}$ depends on variables
that nominally depart from their BPS values, but the homogeneity property
shows that the mass is exactly the BPS mass. Therefore, a free energy of this
form describes thermodynamics intrinsic to the BPS surface.

####  3.3.1 Derivation of the BPS Free Energy

In the AdS5 example we focus on, the explicit free energy $G$ (20) of a
general AdS5 black hole depends on the electric potential $\Phi$, as it
should, but it is also a function of the parameters $r_{+},a,b$, rather than
$T$, $\Omega_{a,b}$. Conveniently, the expression for $G$ given in (20)
vanishes when, in addition to $\Phi=3$, the auxiliary horizon position
$r^{2}_{+}$ take the BPS value $r^{2}_{*}$ given in (69). At linear order in
both $r^{2}_{+}-r^{2}_{*}$ and $\Phi-3$ we have

$G=-\frac{N^{2}}{2}\frac{a+b}{(1-a)(1-b)}(r^{2}_{+}-r^{2}_{*})-\frac{N^{2}}{6}\frac{(a+b)(1-a-b-
ab)}{(1-a)(1-b)}(\Phi-3)~{}.$

The combination $r^{2}_{+}-r^{2}_{*}$ has a geometrical interpretation in
terms of horizon position, but it is not a traditional thermodynamic
potential. It is related to the angular velocities through the two equations
(17-18) which are straightforward to expand at linear order in
$\Phi-\Phi^{*}$, $\Omega_{a,b}-\Omega_{a,b}^{*}$. The result

$r^{2}_{+}-r^{2}_{*}=-\frac{(a+b)(1+a)}{2(1-a)}(\Omega_{a}-1)-\frac{(a+b)(1+b)}{2(1-b)}(\Omega_{b}-1)+\frac{a+b+2ab}{6}(\Phi-3)~{},$
(78)

leads to the free energy

$G=-\frac{N^{2}}{12}\frac{(a+b)(2-a-b)}{(1-a)(1-b)}(\Phi-3)+\frac{N^{2}}{4}\frac{(a+b)^{2}(1+a)}{(1-a)^{2}(1-b)}(\Omega_{a}-1)+\frac{N^{2}}{4}\frac{(a+b)^{2}(1+b)}{(1-a)(1-b)^{2}}(\Omega_{b}-1)~{}.$
(79)

Moreover, the condition that the two equations (17-18) give identical values
for $r^{2}_{+}-r^{2}_{*}$ yields the constraint on the potentials

$\frac{(a+b)(1+a)}{1-a}(\Omega_{a}-1)-\frac{(a+b)(1+b)}{1-b}(\Omega_{b}-1)+\frac{1}{3}(a-b)(\Phi-3)=0~{}.$
(80)

The formula for temperature (19), also expanded at linear order in
$\Phi-\Phi^{*}$, $\Omega_{a,b}-\Omega_{a,b}^{*}$, imposes a second constraint
on the potentials

$T=-\frac{\sqrt{a+b+ab}}{\pi(1-a)}(\Omega_{a}-1)-\frac{\sqrt{a+b+ab}}{\pi(1-b)}(\Omega_{b}-1)-\frac{\sqrt{a+b+ab}}{3\pi(a+b)}(\Phi-3)~{}.$
(81)

We can interpret the two constraints (80-81) together, as definitions of the
auxiliary parameters $a,b$. Since both constraints are invariant under
simultaneous linear scaling of $T,\Phi-\Phi^{*}$,
$\Omega_{a,b}-\Omega_{a,b}^{*}$, the $a,b$ defined implicitly this way are
homogeneous functions of these potentials.

The free energy (79) transforms linearly under simultaneous linear scaling of
$T,\Phi-\Phi^{*}$, $\Omega_{a,b}-\Omega_{a,b}^{*}$ so, as a function of these
variables, it is homogeneous of degree one. According to the discussion in
subsection 3.1, it follows that it describes BPS black holes. To reach this
conclusion it is important that the parameters $a,b$ are invariant under such
rescalings.

To avoid possible confusion, we reiterate the reasoning. The free energy (79)
and the constraints (80-81) were all derived by expanding to linear order in
$\Phi-\Phi^{*}$, $\Omega_{a,b}-\Omega_{a,b}^{*}$. That could leave the
impression that they are approximations that are valid near the BPS limit but
in fact they describe the BPS black holes themselves, these equations contain
no information beyond BPS. Once a free energy is presented as a homogeneous
function, the additional step of taking the limit $T\rightarrow 0$ is
possible, but not required, doing so corresponds to a choice of coordinates in
projective geometry.

To understand how this is possible, consider the mass excess over the BPS mass
(67), rewritten using (71, 78):

$M-M_{*}=\frac{N^{2}}{2}\frac{(3+a+b-ab)(a+b)^{2}}{4(1-a)(1-b)(1+3a+3b+a^{2}+3ab+b^{2})}\Big{(}(2\pi
T)^{2}+\varphi^{2}\Big{)}~{},$ (82)

where

$\varphi=(\Phi-\Phi^{*})-(\Omega_{a}-\Omega_{a}^{*})-(\Omega_{b}-\Omega_{b}^{*})=\Phi-\Omega_{a}-\Omega_{b}-1~{}.$
(83)

This mass excess $M-M_{*}$ is quadratic in the small variables and so the
black hole is BPS because the equality $M=M_{*}$ holds at linear order.333The
quadratic correction in (82) is not important in this article but it is
interesting in its own right Preskill:1991tb ; Almheiri:2016fws ;
Moitra:2018jqs ; Nayak:2018qej ; Larsen:2019oll ; Iliesiu:2020qvm ;
Heydeman:2020hhw ; Larsen:2020lhg . It is described in effective quantum field
theory by the Schwarzian theory Maldacena:2016upp ; kitaev2015simple ;
Maldacena:2016hyu ; Charles:2019tiu ; Choi:2021nnq .

The BPS energy $W$ (57) that is intrinsic to the BPS surface is computed from
(79) by dividing with $T$:

$\displaystyle W$
$\displaystyle=\frac{G}{T}=\frac{1}{2}N^{2}\left(-\frac{(a+b)(2-a-b)}{6(1-a)(1-b)}\Phi^{\prime}+\frac{(a+b)^{2}(1+a)}{2(1-a)^{2}(1-b)}\Omega_{a}^{\prime}+\frac{(a+b)^{2}(1+b)}{2(1-a)(1-b)^{2}}\Omega_{b}^{\prime}\right)~{}.$
(84)

The primed potentials

$\displaystyle\Phi^{\prime}$ $\displaystyle=\frac{\Phi-3}{T}$
$\displaystyle\Omega^{\prime}_{a,b}$
$\displaystyle=\frac{\Omega_{a,b}-1}{T}~{},$ (85)

were previously introduced in the general BPS formalism through (58).
Depending on the point of view, they are either coordinates in the projective
geometry defining the BPS surface or, via the limiting procedure, thermal
derivatives evaluated at $T=0$.

The parameters $a$ and $b$ are complicated functions of $\Phi^{\prime}$,
$\Omega_{a}^{\prime}$ and $\Omega_{b}^{\prime}$ defined by the constraints
(80-81) recast in the form:

$\displaystyle-\frac{\sqrt{a+b+ab}}{1-a}\Omega_{a}^{\prime}-\frac{\sqrt{a+b+ab}}{1-b}\Omega_{b}^{\prime}-\frac{\sqrt{a+b+ab}}{3(a+b)}\Phi^{\prime}$
$\displaystyle=\pi~{},$ (86)
$\displaystyle\frac{(a+b)(1+a)}{1-a}\Omega_{a}^{\prime}-\frac{(a+b)(1+b)}{1-b}\Omega_{b}^{\prime}+\frac{a-b}{3}\Phi^{\prime}$
$\displaystyle=0~{}.$ (87)

Differentiation of the BPS free energy $W$ (84) with respect to the primed
potentials must give the conserved charges through the thermodynamic relations
(61). When computing the derivatives, note that, in addition to the explicit
dependence of $W$ on the primed potentials, there is implicit dependence
through $a,b$ that is given by (86-87). It is a consistency check on the
various formulae that the black hole charges computed this way do in fact
agree with (73-75).

####  3.3.2 Simplifications of the Constraints

In general, it is unpractical to solve the constraints (86-87) and give $a,b$
as explicit functions of the potentials $\Omega^{\prime}_{a,b}$ and
$\Phi^{\prime}$. However, we can do so in the regime where $1-a\ \sim\ 1-b\
\sim\ \epsilon$ is small and positive. This is a version of the Cardy-like
limit Choi:2018hmj ; David:2020ems .

In this limit, the constraint (87) shows that the $\Phi^{\prime}$ term is
dominated by the large $\Omega^{\prime}_{a,b}$ terms. We can partially address
this by tuning $\Omega^{\prime}_{a}\sim\Omega^{\prime}_{b}\sim\epsilon$ but,
because $a,b$ are both near unity we have $(a-b)\sim\epsilon$, and so it is
insufficient that the $\Omega^{\prime}_{a,b}$ terms are order unity, they must
nearly cancel. Thus:

$\frac{\Omega^{\prime}_{a}}{1-a}=\frac{\Omega^{\prime}_{b}}{1-b}~{},$ (88)

up to terms of order $\epsilon$. With this equality, and using $a\sim b\sim 1$
extensively, the other constraint (86) then yields

$\frac{\Omega^{\prime}_{a}}{1-a}=\frac{\Omega^{\prime}_{b}}{1-b}=\frac{\Phi^{\prime}+2\pi\sqrt{3}}{12}~{}.$
(89)

This equation is equivalent to presenting $a,b$ as explicit functions of the
BPS potentials $\Phi^{\prime},\Omega^{\prime}_{a,b}$, as we wanted to do. With
this result it is straightforward to also express the BPS free energy $W$ in
(84) as function of these variables:

$W_{\rm
Cardy}=-\frac{N^{2}}{432}\frac{(\Phi^{\prime}+2\pi\sqrt{3})^{3}}{\Omega_{a}^{\prime}\Omega_{b}^{\prime}}~{}.$
(90)

This is expression is completely explicit, but it applies only in the Cardy-
like limit.

A complementary approach to the awkward constraints (86-87) that define $a,b$
as functions of the primed potentials is to implement them using Lagrange
multipliers. Introducing two multipliers $\Lambda_{1,2}$, we have:

$\begin{split}W&=\frac{1}{2}N^{2}\left[\frac{-(a+b)(2-(a+b))}{6(1-a)(1-b)}\Phi^{\prime}+\frac{(a+b)^{2}(1+a)}{2(1-a)^{2}(1-b)}\Omega_{a}^{\prime}+\frac{(a+b)^{2}(1+b)}{2(1-a)(1-b)^{2}}\Omega_{b}^{\prime}\right]\\\
&+\Lambda_{1}\left(-\frac{\sqrt{a+b+ab}}{1-a}\Omega_{a}^{\prime}-\frac{\sqrt{a+b+ab}}{1-b}\Omega_{b}^{\prime}-\frac{\sqrt{a+b+ab}}{3(a+b)}\Phi^{\prime}-\pi\right)\\\
&+\Lambda_{2}\left(\frac{(a+b)(1+a)}{1-a}\Omega_{a}^{\prime}-\frac{(a+b)(1+b)}{1-b}\Omega_{b}^{\prime}+\frac{a-b}{3}\Phi^{\prime}\right)~{}.\end{split}$
(91)

Extremization with respect to $a,b$ give conditions that are solved when:

$\displaystyle\Lambda_{1}$
$\displaystyle=\frac{1}{2}N^{2}\frac{2(a+b)\sqrt{a+b+ab}}{(1-a)(1-b)}~{},$
(92) $\displaystyle\Lambda_{2}$
$\displaystyle=\frac{1}{2}N^{2}\frac{-(a-b)}{2(1-a)(1-b)}~{},$ (93)

without imposing any additional relation on $\Phi^{\prime}$ and
$\Omega_{a,b}^{\prime}$. Substituting these equations into (91), we find:

$\begin{split}W_{\text{ext}}=-\frac{1}{2}N^{2}&\Big{[}\frac{2\pi(a+b)\sqrt{a+b+ab}}{(1-a)(1-b)}-\frac{(a+b)(2a+b+ab)}{(1-a)^{2}(1-b)}\Omega^{\prime}_{a}\\\
&-\frac{(a+b)(a+2b+ab)}{(1-a)(1-b)^{2}}\Omega^{\prime}_{b}-\frac{a+b}{(1-a)(1-b)}\Phi^{\prime}\Big{]}~{}.\end{split}$
(94)

Importantly, in this equation $a,b$ are auxiliary variables: $W_{\rm ext}$ is
defined only after extremizing $a,b$ with primed potentials kept fixed.

At this point differentiation with respect to the primed potentials is
trivial, there is no need to take implicit dependence via $a,b$ into account,
because these auxiliary variables are anyway evaluated at their extremum.
Thus, in (94) the coefficient of each BPS potential $\Phi^{\prime}$,
$\Omega^{\prime}_{a,b}$ must reproduce its conjugate conserved charge (73,
74,75), as indeed it does. Additionally, the BPS black hole entropy can be
computed from the BPS potential through the formula (62), which amounts to
extracting the independent-of-primed potentials constant in the BPS free
energy $W_{\rm ext}$ given in (94). It agrees with the BPS entropy (14), as it
should.

The free energy (94) is thus completely general, unlike the expression (90) in
the Cardy-limit no assumptions were made on the parameters. The drawback is
that, in this formalism, $a,b$ must be extremized over, and the resulting
conditions are precisely the constraints (86-87).

####  3.3.3 Thermodynamics of BPS Black Holes

In our study of BPS thermodynamics, we will facilitate comparisons with the
literature by trading the BPS electric potential $\Phi^{\prime}$ for

$\varphi^{\prime}=\Phi^{\prime}-\Omega_{a}^{\prime}-\Omega_{b}^{\prime}~{},$
(95)

which is the obvious BPS analogue of (83). The constraints (80-81) for the
angular BPS potentials then become:

$\displaystyle\Omega_{a}^{\prime}=-\frac{1-a}{1+3a+3b+a^{2}+3ab+b^{2}}\left[\frac{\pi(a+2b+2ab+b^{2})}{\sqrt{a+b+ab}}+\frac{1+a}{2}\varphi^{\prime}\right]~{},$
(96)
$\displaystyle\Omega_{b}^{\prime}=-\frac{1-b}{1+3a+3b+a^{2}+3ab+b^{2}}\left[\frac{\pi(2a+b+2ab+a^{2})}{\sqrt{a+b+ab}}+\frac{1+b}{2}\varphi^{\prime}\right]~{}.$
(97)

For reference, we also record the inverse formula that converts from
$\varphi^{\prime}$ back to the electric BPS potential $\Phi^{\prime}$:

$\displaystyle\Phi^{\prime}=-\frac{3(a+b)}{1+3a+3b+a^{2}+3ab+b^{2}}\left[\frac{\pi(1-ab)}{\sqrt{a+b+ab}}-\frac{(2+a+b)}{2}\varphi^{\prime}\right]~{}.$
(98)

In the new variables that eliminate $\Phi^{\prime}$, the free BPS energy (94)
becomes:

$\begin{split}\frac{W}{N^{2}}=&\frac{\pi(a+b)^{2}\left[1-2a-2b-a^{2}-5ab-b^{2}-a^{2}b-ab^{2}\right]}{2(1-a)(1-b)\left(1+3a+3b+a^{2}+3ab+b^{2}\right)\sqrt{a+b+ab}}\\\
&-\frac{(a+b)^{2}(3+a+b-ab)}{4(1-a)(1-b)\left(1+3a+3b+a^{2}+3ab+b^{2}\right)}\varphi^{\prime}~{}.\end{split}$
(99)

Even though BPS thermodynamics applies only at strictly vanishing temperature
$T=0$, it is meaningful to construct phase diagrams that depend on the BPS
potentials. In order to interpret them, it is useful to introduce the “BPS
temperature”

$\displaystyle\tau\equiv-\frac{2}{\Omega_{a}^{\prime}+\Omega_{b}^{\prime}}=\frac{4\left(1+3a+3b+a^{2}+3ab+b^{2}\right)}{\frac{6\pi(a+b)(1-ab)}{\sqrt{a+b+ab}}+\left(2-a^{2}-b^{2}\right)\varphi^{\prime}}~{},$
(100)

that shares some properties with the usual physical temperature. For example,
it is positive because physical black holes have $1-\Omega_{i}\geq 0$ and so
the primed potentials $\Omega^{\prime}_{i}$ introduced in (85) are negative.
Also, for small and fixed $1-\Omega_{i}$, the effective temperature $\tau$ is
proportional to the physical temperature $T$, again by (85). Finally, it
generalizes the analogous definition in Choi:2018vbz ; Choi:2018hmj .

In the preceding, and in our entire study of the phase diagram for BPS black
holes, we have opted to use $(a,b,\varphi^{\prime})$ as the variables for all
thermodynamic quantities. The $a$, $b$ arise as conventional parameters for
presenting the underlying BPS black hole geometry, they do not have a meaning
directly in thermodynamics. When we write them, we refer to them as the
functions of $\Omega^{\prime}_{i}$ and $\varphi^{\prime}$ that are defined
implicitly by the constraints (86-87) which cannot be tractably inverted,
except in specific cases like the Cardy-like limit where we have the explicit
relation (89).

###  3.4 The BPS Phase Diagram: a Benchmark Case

In this subsection we review the special case where angular momenta are equal
($a=b$) and $\varphi^{\prime}=0$ Choi:2018vbz ; Choi:2018hmj . It will serve
as an important benchmark for our study of the general black holes in section
4.

The BPS temperature (100) simplifies when $a=b$ and $\varphi^{\prime}=0$:

$\displaystyle\tau=\frac{1+5a}{3\pi(1-a)}\sqrt{1+\frac{2}{a}}~{}.$ (101)

This expression diverges at $a=0$ and $a=1$. It has a local minimum when the
parameter $a=a_{\text{cusp}}=\frac{3\sqrt{3}-4}{11}\approx 0.109$ and
$\tau=\tau_{\text{cusp}}\approx 0.809$. It is plotted in Figure 8.

Figure 8: The BPS temperature $\tau$ as function of the rotation parameter $a$
when $a=b$ and $\varphi^{\prime}=0$. The local minimum determines the position
of the cusp in the phase diagram. The divergences at $a=0$ and $a=1$
correspond to the large-$\tau$ extremes of the “small” and “large” black hole
branches, respectively.

The free energy (99) when $a=b$ and $\varphi^{\prime}=0$ is:

$\displaystyle\frac{W}{N^{2}}=$ $\displaystyle\frac{2\pi
a^{\frac{3}{2}}(1-5a-2a^{2})}{(1-a)^{2}\left(1+5a\right)\sqrt{2+a}}~{}.$ (102)

The expression yields $W=0$ for $a=0$ and increases for small and positive
$a$. The BPS temperature $\tau\rightarrow\infty$ as $a\rightarrow 0^{+}$ so in
the limit of small parameter $a$ the BPS free energy $W\rightarrow 0^{+}$ for
large $\tau$. As $a$ increases to the value $a_{\rm cusp}$ where the
temperature is at its minimum $\tau_{\rm cusp}$, the free energy increases
monotonically to $W_{\rm
cusp}/N^{2}=\frac{11\pi}{27\sqrt{273+158\sqrt{3}}}\approx 0.055$. The range
$0<a<a_{\rm cusp}$ maps out the “small” black hole branch of the BPS phase
diagram.

As $a$ increases above $a_{\rm cusp}$, the free energy decreases from $W_{\rm
cusp}$. It crosses $W=0$ at $a_{\text{HP}}=\frac{\sqrt{33}-5}{4}\approx 0.186$
which corresponds to the BPS temperature $\tau_{\text{HP}}=0.863$. Finally, in
the limit $a\rightarrow 1^{-}$, the free energy diverges $W\rightarrow-\infty$
along with the BPS temperature $\tau\rightarrow\infty$. The range where
$a_{\rm cusp}<a<1$ is the “large” BPS black hole branch. We plot the BPS phase
diagram in Figure 9

The phase diagram of the BPS black hole is remarkably similar to that of AdS-
Schwarzschild, reviewed in subsection 2.2.1 Choi:2018vbz . In other words,
Figure 9 is reminiscent of Figure 1. In line with this analogy, we denoted the
point where the BPS free energy crosses $W\equiv 0$ by the subscript ”HP” that
refers to the Hawking-Page transition. However, the notion of thermal AdS is
not trivial in the BPS context. It presupposes the existence of a non-
interacting gas of BPS particles for any $\tau$, corresponding to a general
value of the BPS potential for angular momentum $\Omega^{\prime}$. This
hypothetical phase would always have $W=0$ and be preferred for the $\tau$’s
where the BPS black hole has $W>0$. The equilibrium conditions between such
states of matter have not yet been established so we consider the “BPS gas”
conjectural.

Figure 9: BPS free energy $W/N^{2}$ vs. BPS temperature $\tau$. This phase
diagram displays an upper (small black hole) branch and a lower (large black
hole) branch. The two branches coincide at the cusp
$(\tau_{\text{cusp}},W_{\text{cusp}}/N^{2})\approx(0.809,0.055)$. The large
black hole branch dominates over the BPS gas with $W=0$ (red line) for
$\tau>\tau_{\text{HP}}\approx 0.863$.

The striking qualitative agreement between the AdS-BPS and AdS-Schwarzschild
phase diagrams hides a more detailed distinction between their asymptotic
behaviours.

Starting with the Schwarzschild-AdS spacetime, the Gibbs free energy $G$ and
the temperature $T$ computed from the parameters (5 \- 10) with $a=b=0$

$\displaystyle G$ $\displaystyle=\frac{N^{2}r_{+}^{2}(1-r_{+}^{2})}{4}~{},$
(103) $\displaystyle T$ $\displaystyle=\frac{1+2r_{+}^{2}}{2\pi r_{+}}~{},$
(104)

give the rescaled free energy $W=G/T$:

$\displaystyle\frac{W}{N^{2}}$
$\displaystyle=\frac{\pi}{2}\frac{r_{+}^{3}(1-r_{+}^{2})}{1+2r_{+}^{2}}~{}.$
(105)

This allows us to extract the asymptotic behaviors of the two branches by
focussing on the extreme regimes $r_{+}\ll 1$ and $r_{+}\gg 1$.

For very small black holes $r_{+}\ll 1$, $T$ diverges as $\sim r_{+}^{-1}$ and
$W$ vanishes as $\sim r_{+}^{3}$ so:

$W\sim T^{-3}\text{ as }T\rightarrow\infty~{}.$ (106)

This is equivalent to $M\sim T^{-2}$ for asymptotically flat 5D Schwarzschild
black holes. In contrast, for very large black holes $r_{+}\rightarrow\infty$,
$T$ diverges as $r_{+}$ and $W$ diverges as $-r_{+}^{3}$ so then:

$W\sim-T^{3}\text{ ~{}~{}as ~{}~{}}T\rightarrow\infty~{}.$ (107)

as expected for a conformal fluid in the dual CFT4.

Turning now to the BPS black holes, we use equations (101-102). For very small
BPS black holes $a\rightarrow 0^{+}$, $\tau$ diverges $\sim a^{-1/2}$ and $W$
vanishes as $\sim a^{3/2}$ so:

$W\sim\tau^{-3}\text{ ~{}~{}as~{}~{} }\tau\rightarrow\infty~{}.$ (108)

When $a\rightarrow 1^{-}$ instead, $\tau$ diverges as $\sim(1-a)^{-1}$ and $W$
diverges as $\sim-(1-a)^{-2}$ so:

$W\sim-\tau^{2}\text{ ~{}~{}as ~{}~{}}\tau\rightarrow\infty~{}.$ (109)

Our results show that the asymptotic behaviors (106, 108) for very small black
holes agree but the analogues (107, 109) for large black holes do not. Thus
the qualitative identification between $\tau$ and $T$ is not precise.

###  3.5 The HHZ Free Energy

The HHZ free energy has been central to recent progress on the microscopic
understanding of BPS black holes in AdS Benini:2015noa ; Benini:2015eyy ;
Benini:2016rke ; Hosseini:2017mds ; Cabo-Bizet:2018ehj ; Choi:2018hmj ;
Benini:2018ywd ; Choi:2019miv ; Zaffaroni:2019dhb . It was originally
motivated by analysis of the near-horizon geometry of supersymmetric black
holes rather than conventional black hole thermodynamics. In this subsection
we address this gap in the literature.

####  3.5.1 Extremization of the HHZ Free Energy

The HHZ free energy is a functional of the HHZ potentials $\Delta$ and
$\omega_{a,b}$ that are conjugate to the conserved charges $Q$ and $J_{a,b}$:
444Our reference to the HHZ free energy generalizes the original research
Hosseini:2017mds . In particular, we allow general, non-equal angular momenta
$J_{a,b}$.

$H(\Delta,\omega_{a,b})=-\frac{N^{2}}{2}\frac{\Delta^{3}}{\omega_{a}\omega_{b}}~{}.$
(110)

Unlike the free energy that is familiar from undergraduate studies, the HHZ
free energy is complex and the potentials it depends on are complex as well.
The latter satisfy the constraint

$3\Delta-\omega_{a}-\omega_{b}=2\pi i~{},$ (111)

so they are genuinely complex numbers Hosseini:2017mds ; Zaffaroni:2019dhb .

In order to extremize the free energy (110) over its arguments $\Delta$,
$\omega_{a,b}$, while taking into account the constraint (111), it is
convenient to Legendre transform to an ensemble with fixed charges
$Q,J_{a,b}$. To do so, we introduce the entropy function:

$S(\Delta,\omega_{a,b},Q,J_{a,b})=-\frac{N^{2}}{2}\frac{\Delta^{3}}{\omega_{a}\omega_{b}}-3\Delta
Q-\omega_{a}J_{a}-\omega_{b}J_{b}-\Lambda(3\Delta-\omega_{a}-\omega_{b}-2\pi
i)~{},$ (112)

where $\Lambda$ is a Lagrange multiplier. Extremization over the potentials
then gives:

$\displaystyle\frac{\partial S}{\partial\Delta}=0$
$\displaystyle~{}~{}~{}\implies~{}~{}~{}-\frac{N^{2}}{2}\frac{3\Delta^{2}}{\omega_{a}\omega_{b}}-3(Q+\Lambda)=0~{},$
(113) $\displaystyle\frac{\partial S}{\partial\omega_{i}}=0$
$\displaystyle~{}~{}~{}\implies~{}~{}~{}\frac{N^{2}}{2\omega_{i}}\frac{\Delta^{3}}{\omega_{a}\omega_{b}}-(J_{i}-\Lambda)=0~{}.$
(114)

When these conditions are satisfied, the entropy at the extremum $S^{*}$ is
related to the Lagrange multiplier $\Lambda$ through

$S^{*}=2\pi i\Lambda~{},$ (115)

and $\Lambda$ is determined by the black hole charges $Q$, $J_{a,b}$ via the
cubic equation:

$\Lambda^{3}+\underbrace{(3Q+\frac{N^{2}}{2})}_{A}\Lambda^{2}+\underbrace{\left(3Q^{2}-\frac{N^{2}}{2}(J_{a}+J_{b})\right)}_{B}\Lambda+\underbrace{(Q^{3}+\frac{N^{2}}{2}J_{a}J_{b})}_{C}=0~{}.$
(116)

The HHZ prescription invokes reality and positivity of the entropy, so that
$\Lambda$ must be a purely imaginary number with negative imaginary part. This
is possible only if the cubic polynomial factorizes as
$(\Lambda^{2}+B)(\Lambda+A)$. It follows that the coefficients of the cubic
polynomial satisfy the relation:

$AB-C=(3Q+\frac{N^{2}}{2})\left(3Q^{2}-\frac{N^{2}}{2}(J_{a}+J_{b})\right)-(Q^{3}+\frac{N^{2}}{2}J_{a}J_{b})=0~{},$
(117)

and also that the root with negative purely imaginary part is:

$\Lambda=-i\sqrt{B}=-i\sqrt{3Q^{2}-\frac{N^{2}}{2}(J_{a}+J_{b})}~{}.$ (118)

Inserting this value for $\Lambda$ in (115), and expressing $Q$ and $J_{a,b}$
in terms of $a$ and $b$ using their values from (73-75), we match the BPS
black hole entropy $S^{*}$ (76). Moreover, the condition (117) is precisely
the constraint on black hole charges (77).

The HHZ procedure thus gives the correct functions of charges, but we stress
that we introduced the HHZ free energy $H$ (121) and the potentials
$(\Delta,\omega_{a,b})$ it depends on without committing to an associated
spacetime interpretation: we derived the black hole entropy via a formal
extremization procedure. Some authors simply identify the HHZ formalism with
“gravitational thermodynamics”, and consider it the target for microscopic CFT
considerations. However, the HHZ variables are genuinely complex so their
relation to spacetime physics is not clear a priori. We therefore distinguish
“boundary CFT” (the HHZ free energy) and “bulk interpretation” (black hole
thermodynamics) carefully.

The values of the potentials $\Delta$ and $\omega_{i}$ (with $i=a,b$) that
extremize the HHZ free energy follow from (113-114) as:

$\displaystyle\Delta$ $\displaystyle=2\pi
i(Q+\Lambda)^{-1}\left(\frac{1}{J_{a}-\Lambda}+\frac{1}{J_{b}-\Lambda}+\frac{3}{Q+\Lambda}\right)^{-1}~{},$
(119) $\displaystyle\omega_{i}$ $\displaystyle=-2\pi
i(J_{i}-\Lambda)^{-1}\left(\frac{1}{J_{a}-\Lambda}+\frac{1}{J_{b}-\Lambda}+\frac{3}{Q+\Lambda}\right)^{-1}~{}.$
(120)

with the understanding that the Lagrange multiplier
$\Lambda=\Lambda(Q,J_{a,b})$ is given by (118). Because of the constraint
(111), they yield the on-shell value of the HHZ free energy (110)

$H=-\frac{N^{2}}{2}\frac{1}{\omega_{a}\omega_{b}}\left(\frac{2\pi
i+\omega_{a}+\omega_{b}}{3}\right)^{3}~{}.$ (121)

The values of the potentials $\Delta,\omega_{i}$ (119-120) and the entropy $S$
(115) at the extremum are all given in terms of the charges $Q,J_{i}$. This is
a natural form to present the result of the extremization, and the one that is
most appropriate for comparison with microscopic considerations. However, it
is awkward that all these expressions are defined only modulo the constraint
on the charges (117).

The gravitational thermodynamics of BPS black holes expresses the
corresponding physical quantities in terms of the parameters $(a,b)$, in a
form that automatically solves the constraint on the charges. In particular,
the BPS values of the charge $Q^{*}$ (73) and angular momenta $J_{i}^{*}$
(74-75) satisfy the constraint and give equations for $\omega_{i}$:

$\displaystyle\omega_{a}$
$\displaystyle=\frac{-\pi(1-a)}{a^{2}+b^{2}+3(a+b+ab)+1}\left(\frac{a+2b+b^{2}+2ab}{\sqrt{a+b+ab}}+i(1+a)\right)~{},$
(122) $\displaystyle\omega_{b}$
$\displaystyle=\frac{-\pi(1-b)}{a^{2}+b^{2}+3(a+b+ab)+1}\left(\frac{b+2a+a^{2}+2ab}{\sqrt{a+b+ab}}+i(1+b)\right)~{}.$
(123)

We do not explicitly write the corresponding expression for the electric
potential $\Delta=\Delta(a,b)$ because it follows from the simple linear
relation (111). The on-shell value of the HHZ potential $H$ (121) can be
similarly expressed in terms of $a$ and $b$:

$\begin{split}H(a,b)&=\frac{N^{2}}{2}\frac{-\pi(a+b)^{2}(1-2a-2b-a^{2}-5ab-b^{2}-a^{2}b-ab^{2})}{(1-a)(1-b)\sqrt{a+b+ab}(a^{2}+b^{2}+3(a+b+ab)+1)}\\\
&+\frac{N^{2}}{2}\frac{\pi(a+b)^{2}(3-ab+a+b)}{2(1-a)(1-b)(a^{2}+b^{2}+3(a+b+ab)+1)}i~{}.\end{split}$
(124)

Even though the potentials $\omega_{a,b}$ in (122-123) and the HHZ potential
(124) are functions of the parameters $a,b$ that were introduced in the
context of gravitational solutions, we stress that these formulae are entirely
equivalent to (119-121) which we consider function of charges. The only
difference is that the constraint on charges was solved. As a bonus, the
potentials that extremize the HHZ free energy are now written in a form that
is convenient for comparison with gravitational thermodynamics.

####  3.5.2 Comparing the HHZ Results with Black Hole Thermodynamics

We now reconsider the conventional thermodynamics of BPS black holes that
depends on manifestly real spacetime potentials
$\Phi^{\prime},\Omega^{\prime}_{a,b}$ and the BPS free energy $W$ (99) that is
also real. Our gravitational considerations presented results in terms of
auxiliary variables $(a,b)$ that are related to black hole charges and an
additional physical potential
$\varphi^{\prime}=\Phi^{\prime}-\Omega^{\prime}_{a}-\Omega^{\prime}_{b}$
introduced in (95). In these variables, the BPS free energy $W$ defined in
(84) becomes (99):

$\begin{split}\frac{W}{N^{2}}=&\frac{\pi(a+b)^{2}\left(1-2a-2b-a^{2}-5ab-b^{2}-a^{2}b-ab^{2}\right)}{2(1-a)(1-b)\left(1+3a+3b+a^{2}+3ab+b^{2}\right)\sqrt{a+b+ab}}\\\
&-\frac{(a+b)^{2}(3+a+b-ab)}{4(1-a)(1-b)\left(1+3a+3b+a^{2}+3ab+b^{2}\right)}\varphi^{\prime}~{}~{},\end{split}$
(125)

and the BPS angular velocities defined in (85) are:

$\displaystyle\Omega_{a}^{\prime}$
$\displaystyle=-\frac{1-a}{2(1+3a+3b+a^{2}+3ab+b^{2})}\left[\frac{2\pi(a+2b+2ab+b^{2})}{\sqrt{a+b+ab}}+(1+a)\varphi^{\prime}\right]~{},$
(126) $\displaystyle\Omega_{b}^{\prime}$
$\displaystyle=-\frac{1-b}{2(1+3a+3b+a^{2}+3ab+b^{2})}\left[\frac{2\pi(2a+b+2ab+a^{2})}{\sqrt{a+b+ab}}+(1+b)\varphi^{\prime}\right]~{}.$
(127)

The BPS electric potential $\Phi^{\prime}$ defined in (85) is not an
independent variable because it is given through
$\Phi^{\prime}=\Omega^{\prime}_{a}+\Omega^{\prime}_{b}+\varphi^{\prime}$.

We expect that the spacetime free energy $W$ (99) must be related to the HHZ
free energy $H$ (124). Similarly, the spacetime angular velocities
$\Omega^{\prime}_{a,b}$ (96-97) should be related to the HHZ variables
$\omega_{a,b}$ (122-123), since they are both conjugate to angular momenta
$J_{a,b}$. There are indeed striking similarities between the expressions but
there are also key differences. As we have stressed already, the HHZ variables
are complex while black hole thermodynamics is manifestly real. Moreover, the
thermodynamic formulae depend on one more variable $\varphi^{\prime}$.

There are several options that address these differences, at least formally.
The most conservative is to assign $\varphi^{\prime}$ a specific value that is
imaginary. Indeed, for $\varphi^{\prime}=2\pi i$ we have:

$H(a,b)=-W(a,b,\varphi^{\prime}=2\pi i)~{},$ (128)

for the free energy and

$\omega_{i}(a,b)=\Omega_{i}^{\prime}(a,b,\varphi^{\prime}=2\pi
i)~{}~{}~{},~{}3\Delta(a,b)=\Phi^{\prime}(a,b,\varphi^{\prime}=2\pi i)~{},$
(129)

for the potentials. Moreover, the definition of $\varphi^{\prime}$ reduces
precisely to the HHZ constraint (111) when $\varphi^{\prime}=2\pi i$.

This procedure is well motivated. The general BPS partition function reduces
to the superconformal index precisely for $\varphi^{\prime}=2\pi i$ because
$\varphi^{\prime}$ is integral (half-integral) for bosons (fermions), and so
this value of the potential is equivalent to inserting $(-)^{F}$. Therefore,
for this value of $\varphi^{\prime}$, comparisons between weak and strong
coupling in the CFT is justified, with the latter dual to the semiclassical
gravity description. According to this line of reasoning, black hole
thermodynamics for any real value of $\varphi^{\prime}$ corresponds to
strongly coupled field theory that is unlikely to be accounted for by
computations in weakly coupled CFT.

A distinct procedure, also common in the literature Hosseini:2016tor ;
Liu:2017vbl ; Choi:2018vbz ; Choi:2018hmj ; Larsen:2019oll ; Copetti:2020dil ,
is to identify only the real parts of the HHZ free energy and potentials as
physical, and compare them with black hole thermodynamics only for the special
value $\varphi^{\prime}=0$. Indeed, for the free energy, we find the identity:

$\text{Re}\
H(a,b)=-W(a,b,\varphi^{\prime}=0)=\frac{N^{2}}{2}\frac{-\pi(a+b)^{2}(1-2a-2b-a^{2}-5ab-b^{2}-a^{2}b-ab^{2})}{(1-a)(1-b)\sqrt{a+b+ab}(a^{2}+b^{2}+3(a+b+ab)+1)}~{}.$
(130)

Sometimes authors even refer to the quantity $\text{Re}\ H(a,b)$ (expressed in
terms of the HHZ variables $\Delta$ and $\omega_{a,b}$) as the free energy
$F$, rather than the full complex expression Choi:2018vbz ; Choi:2018hmj ;
Larsen:2019oll ; Copetti:2020dil . The analogous comparison between the real
part of the complex chemical potentials is also successful:

$\begin{split}\text{Re
}\omega_{a}&=\frac{-\pi(1-a)(a+2b+b^{2}+2ab)}{(a^{2}+b^{2}+3(a+b+ab)+1)\sqrt{a+b+ab}}=\Omega_{a}^{\prime}(a,b,\varphi^{\prime}=0)~{},\\\
\text{Re
}\omega_{b}&=\frac{-\pi(1-b)(2a+b+a^{2}+2ab)}{(a^{2}+b^{2}+3(a+b+ab)+1)\sqrt{a+b+ab}}=\Omega_{b}^{\prime}(a,b,\varphi^{\prime}=0)~{}.\end{split}$
(131)

The focus on the real part of the potential is heuristic, but it is striking
that it “works”, in that it yields identities between somewhat elaborate
functions. Within the rigorous framework of the superconformal index, these
agreements are coincidental.

The feature that allows both procedures to work is that the free energy is
linear in $\varphi^{\prime}$. The analogous feature in the AdS3/CFT2
correspondence has an appealing physical interpretation: the BPS black hole
cannot be identified with a specific BPS state, it is an average over all
chiral primaries Larsen:2021wnu . It would be interesting to develop an
analogus interpreation in the AdS5/CFT4 correspondence.

####  3.5.3 HHZ free Energy and Black Hole Thermodynamics: the Phase Diagram

Choi et. al. showed that a phase diagram determined from the HHZ free energy
(110) shares important aspects of the AdS-Schwarzschild black thermodynamics,
including the Hawking-Page phase transition and its characteristic “cusp”
Chamblin:1999tk ; Witten:1998qj ; Natsuume:2014sfa ; Kubiznak:2016qmn . This
is interesting, because it suggests that the confinement/deconfinement
transition in QCD-like theories tHooft:1977nqb ; Polyakov:1978vu ;
Witten:1998zw ; Susskind:1979up ; Aharony:2003sx  can be analyzed while
preserving supersymmetry, possibly with a great deal of precision. However, it
is not obvious that results derived from the HHZ free energy agree with
traditional thermodynamics, derived from black hole geometry. In this
subsection we verify that they do in fact agree.

The extremization conditions on the entropy function (113,114) give:

$J+Q=-\frac{2N^{2}}{27}\frac{(\omega+\pi i)^{2}(\omega-2\pi
i)}{\omega^{3}}~{},$ (132)

after taking $\omega_{a}=\omega_{b}$ and eliminating $\Lambda$. Choi et.al.
follow the “heuristic” procedure yielding (130-131) so they demand that $J,Q$
are positive and real. This requirement relates the real and imaginary parts
of $\omega=\omega_{R}+i\omega_{I}$ as:

$\omega_{R}^{2}=\frac{3\omega_{I}^{2}(\pi+\omega_{I})}{\pi-3\omega_{I}}~{}.$
(133)

The heuristic procedure interprets $\omega_{R}=\text{Re }\omega$ as the
physical potential $\Omega^{\prime}$ through (131) and the condition that
$\Omega^{\prime}<0$ determines phases so:

$\tau=-\omega_{R}^{-1}=-\omega_{I}^{-1}\sqrt{\frac{\pi-3\omega_{I}}{3(\pi+\omega_{I})}}~{},$
(134)

and constrains the imaginary part so $-\pi<\omega_{I}<0$. Further, the
heuristic procedure posits that it is the real part of the HHZ energy (110)
that is physical. Since $\Delta$ is related to $\omega=\omega_{R}+i\omega_{I}$
through the constraint (111) we then find

$\displaystyle F$ $\displaystyle=-\frac{N^{2}}{2}\text{Re
}\frac{\Delta^{3}}{\omega^{2}}=-\frac{N^{2}}{2}\text{Re }\left(\frac{(2\pi
i+2\omega)^{3}}{27\omega^{2}}\right)$ (135)
$\displaystyle=-\frac{N^{2}}{18}\frac{\pi^{3}-9\pi\omega_{I}^{2}-8\omega_{I}^{3}}{\omega_{I}^{2}}\sqrt{\frac{\pi+\omega_{I}}{3(\pi-3\omega_{I})}}~{}.$
(136)

The variable $\omega_{I}$ is related by the BPS temperature $\tau$ through
(134) so this equation gives the free energy $F$ as function of the BPS
temperature $\tau$ which we plot in Figure 10.

Figure 10: The free energy $F$ vs. the BPS temperature $\tau$. This phase
diagram was derived from the HHZ potential using the “heuristic” method.

From visual inspection it seems clear that Figure 10 is precisely the same as
its analogue in Figure 9 that applies to black holes with parameters $a=b$ and
$\varphi^{\prime}=0$. However, they were arrived at very differently: the
former by extremizing the HHZ potential (110) (and imposing various reality
conditions), the latter from the standard thermodynamic interpretation of BPS
black hole geometry.

The analytical comparison is also quite nontrivial. The BPS free energy $W$
derived from gravity is a function of the BPS temperature $\tau$, with both
$W$ and $\tau$ given in terms of the auxiliary parameter $a$ in (101,102), and
reproduced here for convenience:

$\displaystyle W$ $\displaystyle=\frac{N^{2}}{2}\frac{4\pi
a^{2}(1-5a-2a^{2})}{(1-a)^{2}(1+5a)\sqrt{a(a+2)}}~{},$ (137)
$\displaystyle\tau$ $\displaystyle=-\omega^{-1}_{R}=-\frac{3\pi
a(1-a)}{(1+5a)\sqrt{a(a+2)}}~{}.$ (138)

It must be compared with the real part of the HHZ free energy $F$ (135) that
is a function of $\omega_{I}$, the imaginary part of the complexified
rotational velocity $\omega=\omega_{R}+i\omega_{I}$, which serves as an
auxiliary parameter that is related to $\tau=-\omega_{R}^{-1}$ through (134).

As suspected, there is in fact an identification of auxiliary parameters that
transforms all the analytical formulae into one another:

$\omega_{I}=-\pi\frac{1-a}{1+5a}~{}.$ (139)

We stress that, given the nonlinear nature of the formulae involved, this
agreement was far from preordained. Its success is a sensitive test of the HHZ
potential (110).

##  4 Thermodynamics of BPS Black Holes: Detailed Study

In this section we study the phase diagram of general BPS black holes. The
benchmark BPS black hole introduced in subsection 3.4 has equal angular
velocities $\Omega^{\prime}_{a}=\Omega^{\prime}_{b}$ and electric potential
$\Phi^{\prime}$ tuned so $\varphi^{\prime}=0$. We now explore generic values
of the three independent potentials and explain their significance.

###  4.1 Primed Potentials and Their Conjugate Charges

The general BPS black holes we consider in this section are parametrized by
the three primed potentials $\Omega^{\prime}_{a}$, $\Omega^{\prime}_{b}$, and
$\Phi^{\prime}$. The primes remind us that their definitions (85) relate them
to the corresponding non-BPS potentials $\Omega_{a,b}$ and $\Phi$ via a
thermal derivative. The primed potentials are the variables that the BPS free
energy $W$

$W=-S-\Omega^{\prime}_{a}J_{a}-\Omega^{\prime}_{b}J_{b}-\Phi^{\prime}Q~{},$
(140)

defined in (57) depends on. In particular, its derivatives read off from the
first law of BPS black hole thermodynamics

$dW=-J_{a}d\Omega^{\prime}_{a}-J_{b}d\Omega^{\prime}_{b}-Qd\Phi^{\prime}~{},$
(141)

yield the black hole charges. Alternative ensembles that are functions of some
or all of the charges, rather than their conjugate potentials, can be obtained
as usual, by appropriate Legendre transforms.

As we discussed in subsection 3.3.3, the interpretation of the phase diagram
is simplified by introducing a different basis for the potentials. The BPS
temperature $\tau$ (100) is an analogue of the physical temperature. The
modified potential $\varphi^{\prime}$ (95) is a proxy for the electric field
that is defined so the benchmark case, studied in the literature and reviewed
in subsection 3.4, corresponds to $\varphi^{\prime}=0$. To complete a well-
defined transformation from the three potentials
$\Omega^{\prime}_{a,b},\Phi^{\prime}$ we introduce a third potential:

$\mu=\Omega^{\prime}_{a}-\Omega^{\prime}_{b}~{},$ (142)

that is sensitive to departures from two equal angular momenta. The
combinations of charges that are conjugate to the potentials
$(\tau,\varphi^{\prime},\mu)$ follow from the first law of BPS black hole
thermodynamics in the form

$dW=-\frac{1}{\tau^{2}}(J_{a}+J_{b}+2Q)d\tau-
Qd\varphi^{\prime}-\frac{1}{2}(J_{a}-J_{b})d\mu~{}.$ (143)

For completeness, we also record the inverse transform from the variables we
employ to discuss the phase diagram to the original physical BPS potentials:

$\displaystyle\Phi^{\prime}$
$\displaystyle=-\frac{2}{\tau}+\varphi^{\prime}~{},$ (144)
$\displaystyle\Omega^{\prime}_{a}$
$\displaystyle=-\frac{1}{\tau}+\frac{\mu}{2}~{},$ (145)
$\displaystyle\Omega^{\prime}_{b}$
$\displaystyle=-\frac{1}{\tau}-\frac{\mu}{2}~{},$ (146)

that are conjugate the physical charges $Q,J_{a},J_{b}$.

The changes of variables above are routine, at the face of it: BPS black holes
are characterized by three charges charges $\\{Q,J_{a},J_{b}\\}$ which, in the
grand canonical ensemble, correspond to three BPS potentials
$\\{\Phi^{\prime},\Omega^{\prime}_{a},\Omega^{\prime}_{b}\\}$. When presenting
explicit phase diagrams we further change the basis among the potentials to
$\\{\tau,\varphi^{\prime},\mu\\}$, in order to clarify physical
interpretations and connections to the literature. However, straightforward as
these transformations appear, they do not incorporate the fact that all BPS
black holes satisfy the constraint between charges (77). Mathematically, this
means the Legendre transform, from the microcanonical to the canonical
ensemble, is singular.

Our physical interpretation of this peculiar feature, discussed at length in
the introduction, is that all classical BPS black holes correspond to thermal
equilibrium along the direction in parameter space that corresponds to the BPS
potential $\varphi^{\prime}$. The formula for the mass excess $M-M_{*}$ (82)
offers a perspective. BPS black holes $M=M_{*}$ must have zero temperature
$T=0$ as well as the potential $\varphi=0$ but the BPS condition does not
specify the “slope” $\varphi^{\prime}=\varphi/T$ when the approach is realized
as a physical limit where $\varphi\rightarrow 0$ and $T\rightarrow 0$
simultaneously. This ratio is physical: any supersymmetry implies $M-M^{*}=0$
but, once a given supercharge has been committed to, others that differ by a
phase $\varphi^{\prime}$ are inconsistent with the “preferred” supersymmetry.
555Slowly varying fluctuations of $\varphi^{\prime}$ can be identified with
the scalar that carries R-symmetry charge in the ${\cal N}=2$ Schwarzian
theory describing excitations in the near geometry of the BPS black
holeFu:2016vas ; Iliesiu:2020qvm ; Heydeman:2020hhw ; Choi:2021nnq  (and
references therein).

The equilibrium condition along the $\varphi^{\prime}$ direction was
understood in AdS3, where it corresponds to black holes necessarily taking on
the R-charge that is the average of all chiral primaries, in contrast to the
index that is independent of the charges of these states Dijkgraaf:2000fq ;
Larsen:2021wnu . It is important to elucidate the analogous mechanism in AdS5
so, in the following subsections, we treat $\varphi^{\prime}$ as an
independent thermodynamic variable. In particular, we do not take
$\varphi^{\prime}=0$ a priori because, even if some argument were to establish
$\varphi^{\prime}=0$ as the equilibrium value, we would still need to show
that the constraint among charges emerges as a derivative with respect to this
variable.

###  4.2 The Physical Range of $\varphi^{\prime}$ and $\mu$

The BPS free energy $W$ was defined in subsection 3.1 as a function of the BPS
potentials $\Phi^{\prime},\Omega^{\prime}_{a,b}$ but, as have discussed above
(and in subsection 3.3.3), the variables $(\tau,\mu,\varphi^{\prime})$ are
preferable. Unfortunately, in practice the free energy and the various
potentials are known only in parametric forms, as functions of
$(a,b,\varphi^{\prime})$. For example, the BPS temperature $\tau$ was
presented in (100) and (126-127) similarly give

$\mu=\frac{a-b}{2(1+3a+3b+a^{2}+3ab+b^{2})}\left[\frac{2\pi(1+2a+2b+ab)}{\sqrt{a+b+ab}}+(a+b)\varphi^{\prime}\right]~{}.$
(147)

We need to determine the physical regime of the various parameters. The
rotational velocities are bounded by the speed of light $\Omega_{i}\leq 1$
which, via the definition of primed potentials (85), corresponds to
$\Omega^{\prime}_{i}<0$. This in turn implies that $\tau$ introduced in (100)
is strictly positive $\tau>0$ so

$\displaystyle-\varphi^{\prime}\leq\frac{6\pi(a+b)(1-ab)}{(2-a^{2}-b^{2})\sqrt{a+b+ab}}~{}.$
(148)

Additionally, we require $0\leq a,b<1$, in order that the underlying black
hole solutions exist as regular geometries.

The positive temperature condition in the form (148) is satisfied for all
$\varphi^{\prime}\geq 0$ but it is nontrivial for $\varphi^{\prime}<0$. For
$a=b$, it reduces to

$-\varphi^{\prime}\underset{a=b}{\leq}6\pi\sqrt{\frac{a}{2+a}}<2\sqrt{3}\pi~{}.$

The middle formula is a monotonically increasing function of $a$ and the
second inequality would be saturated for $a=1$. Therefore, for equal angular
momenta $a=b$, the BPS potential $\varphi^{\prime}$ is bounded from below by
$\varphi_{-}^{\prime}=-2\sqrt{3}\pi<\varphi^{\prime}$

For given $a+b$, the function on the right hand side of (148) increases
monotonically as function of $(a-b)^{2}$. Therefore, the lower bound on
$\varphi^{\prime}$ is relaxed when the two angular momenta are unequal. Its
global minimum $\varphi^{\prime}_{\text{min}}=-6\pi$ corresponds to
$(a,b)=(1,0)$ or $(a,b)=(0,1)$.

We want to similarly analyze the physical range of $\mu$ (147), the BPS
potential that measures the departure from angular momenta of $a$ and $b$ type
being equal. Because of the antisymmetry between $a$ and $b$ we have
$\mu\equiv 0$ along the line $a=b$ and it is sufficient to study $a\geq b$.
When $\varphi^{\prime}=0$ we find that $\mu$ increases monotonically with
$a-b$ when $a+b$ is fixed. The maximal value $\mu_{\rm
max}=\frac{3\pi}{5}\approx 1.88$ is reached when $(a,b)=(1,0)$. We plot the
BPS potential $\mu$ in Figure 11.

Figure 11: The chemical potential $\mu$ as a function of the black hole
parameters $a$ and $b$, for $\varphi^{\prime}=0$. The maximum is achieved at
$(a,b)=(1,0)$. We plot only $a\geq b$, the mirror for $a\leq b$ (the grey
area) follows from antisymmetry of $\mu(a,b)$ under $a\leftrightarrow b$.

###  4.3 BPS Black Holes with General $\varphi^{\prime}$ and Equal Angular
Momenta

In this subsection we study the significance of the potential
$\varphi^{\prime}$. We take equal angular momenta, i.e. $a=b$ (and so
$\Omega_{a}^{\prime}=\Omega_{b}^{\prime}=\Omega^{\prime}$) which corresponds
to $\mu\equiv 0$.

The BPS free energy (99) reduces to

$\frac{W}{N^{2}}=\frac{2a}{(1-a)^{2}(1+5a)}\left(\frac{\pi(1-5a-2a^{2})}{\sqrt{1+\frac{2}{a}}}-\frac{a(3-a)\varphi^{\prime}}{2}\right)~{},$
(149)

and the BPS “temperature” $\tau$ (100) becomes

$\tau\equiv-\Omega^{\prime-1}=\frac{1+5a}{1-a}\frac{1}{\frac{3\pi}{\sqrt{1+\frac{2}{a}}}+\frac{\varphi^{\prime}}{2}}~{}.$
(150)

We will discuss the dependence of $W$ on $\tau$ for each sign sign of
$\varphi^{\prime}$ in turn.

####  4.3.1 $\varphi^{\prime}\leq 0$

We first consider the BPS temperature (150). For large black holes $\tau$
diverges at $a=1$ and, as $a$ get smaller, it decreases to a minimal
temperature $\tau_{\rm cusp}$ that is attained at $a_{\rm cusp}$. This is
qualitatively similar to the benchmark case $\varphi^{\prime}=0$ that was
discussed in subsection 3.4 (with $\tau(a)$ plotted in figure 8).

The temperature increases again when $a$ decreases below $a_{\rm cusp}$, as
expected for small black holes. However, when $\varphi^{\prime}<0$, the
denominator in the second factor of $\tau$ given in (150) reaches zero at some
positive value

$a_{\min}=\frac{2\varphi^{\prime 2}}{36\pi^{2}-\varphi^{\prime 2}}~{},$ (151)

and then the temperature diverges. Thus, when $\varphi^{\prime}\leq 0$, the
parameter $a$ is limited to the range $1>a>a_{\min}$.

The parameter value $a_{min}$ increases monotonically when the absolute value
$|\varphi^{\prime}|$ increases. The physical range $1>a>a_{\rm min}$ shrinks
to zero if it reaches $a_{\text{min}}=1$ and then no underlying black hole
geometry would exist. Therefore, the corresponding value of the potential
$\varphi^{\prime}$

$\varphi^{\prime}_{-}=-2\sqrt{3}\pi~{},$ (152)

constitutes a lower limit $\varphi^{\prime}\geq\varphi^{\prime}_{-}$.

The finite range of $a$ changes the BPS free energy $W$ (149) qualitatively:
at the high temperature end of the small black hole branch (as $a\rightarrow
a^{+}_{\min}$), it does not approach $0$ but a finite positive value

$W_{\text{asym}}/N^{2}=\frac{4\varphi^{\prime 3}}{9(12\pi^{2}-\varphi^{\prime
2})}~{}.$ (153)

Also, for the ”big” black hole branch, we have an asymptotic relation

$W/N^{2}\sim-\frac{(\varphi^{\prime}+2\sqrt{3}\pi)^{3}}{432}\tau^{2}.$ (154)

When $\varphi^{\prime}=0$, it is in agreement with our benchmark case in
subsection 3.4.

The phase diagram for various values of $\varphi^{\prime}\leq 0$ is presented
in Figure 12.

Figure 12: The BPS free energy $W$ as function of the BPS temperature $\tau$.
The two angular momenta are equal ($\mu=0$) and
$\varphi^{\prime}=0,-\pi,-\frac{3\pi}{2},-2\pi$, from left to right. There are
two branches in the phase diagram that meet in a cusp. Small black holes (the
upper branch) asymptote to a positive BPS free energy $W_{\text{asym}}$ (153)
at large temperature when $\varphi^{\prime}<0$.

The minimal temperature for a given $\varphi^{\prime}$ is attained at
$a_{\text{cusp}}$ determined by $\partial_{a}\tau=0$ which leads to:

$\varphi^{\prime}=\frac{\pi(1-8a_{\text{cusp}}-11a_{\text{cusp}}^{2})}{(2+a_{\text{cusp}})\sqrt{a_{\text{cusp}}(2+a_{\text{cusp}})}}~{}.$
(155)

This equation is equivalent to a sextic in $a$ that cannot be solved
analytically in general. For $\varphi^{\prime}=0$, the condition (155) reduces
to a quadratic equation with the solution
$a_{\text{cusp}}=\frac{3\sqrt{3}-4}{11}$, in agreement with the result in
subsection 3.4. We also note that $a_{\text{cusp}}=1$ is a solution at the
lower bound $\varphi^{\prime}=\varphi^{\prime}_{-}$ given in (152). The BPS
temperature $\tau$ and BPS free energy $W$ both diverge as $a\rightarrow
1^{-}$ so this shows that, as the lower bound is approached
$\varphi^{\prime}\rightarrow\varphi^{\prime}_{-}$, the cusp moves to the far
upper right corner in Figure 12 and no physical black hole remain in the
strict limit.

Generally, the right hand side of (155) is a monotonic function so, as
$|\varphi^{\prime}|$ increases from zero to its maximum
$|\varphi^{\prime}_{-}|$, $a_{\text{cusp}}$ increases monotonically through
$[\frac{3\sqrt{3}-4}{11},1[$. The BPS free energy (149) at the cusp moves as

$\frac{dW_{\text{cusp}}}{da}=\left(\partial_{a}W\right)_{\varphi^{\prime}}+\left(\partial_{\varphi^{\prime}}W\right)_{a}\partial_{a}\varphi^{\prime}=\left.\frac{\pi
a\left(3+14a-5a^{2}\right)}{\sqrt{a(2+a)}(2+a)^{2}(1-a)^{2}}\right|_{a_{\text{cusp}}}>0~{},$
(156)

with $\partial_{a}\varphi^{\prime}$ evaluated from (155), so it increases
monotonically as well. The BPS temperature (150) at the cusp increases
monotonically entirely through its dependence on $\varphi^{\prime}$ because it
is a minimum $\partial_{a}\tau=0$. Taken together, these arguments establish
analytically that the cusp moves monotonically up and to the right when
$|\varphi^{\prime}|$ increases. These trends are also visible in Figure 12.

We can analyze the Hawking-Page temperature similarly. The condition that the
free energy $W=0$ yields the algebraic relation for $a_{\text{HP}}$:

$\varphi^{\prime}=\frac{2\pi(1-5a_{\text{HP}}-2a^{2}_{\text{HP}})}{(3-a_{\text{HP}})\sqrt{a_{\text{HP}}(2+a_{\text{HP}})}}~{}.$
(157)

The derivative of this expression

$\partial_{a_{\text{HP}}}\varphi^{\prime}=-\frac{6\pi(1+a_{\text{HP}})^{2}(1+3a_{\text{HP}})}{(3-a_{\text{HP}})^{2}(a_{\text{HP}}(2+a_{\text{HP}}))^{3/2}}~{},$
(158)

is negative in the entire range $0\leq a\leq 1$. Therefore, the parameter
$a_{\text{HP}}$ at the Hawking-Page transition depends monotonically on
$\varphi^{\prime}$. The corresponding temperature (150) moves as

$\frac{d\tau_{\text{HP}}}{da}=\left(\partial_{a}\tau\right)_{\varphi^{\prime}}+\left(\partial_{\varphi^{\prime}}\tau\right)_{a}\partial_{a}\varphi^{\prime}=\left.\frac{3(1+3a)}{\pi(1-a)^{3}\sqrt{a(2+a)}}\right|_{a_{\text{HP}}}>0~{},$
(159)

so the Hawking-Page temperature $\tau_{\text{HP}}$ is a monotonically
increasing function of $|\varphi^{\prime}|$. This dependence is also apparent
in Figure 12.

It is instructive to compare the phase diagram for $\varphi^{\prime}$ in the
range $(-2\pi\sqrt{3},0)$ with the benchmark case $\varphi^{\prime}=0$
discussed in subsection 3.4:

  * •

At a given BPS temperature $\tau$ the potential $|\varphi^{\prime}|$ increases
the BPS free energy $W$ on both branches.

  * •

The characteristic temperatures $\tau_{\text{HP}}$ and $\tau_{\text{cusp}}$
both increase with $|\varphi^{\prime}|$.

Both effects suggest an instability. Conversely, increased $\varphi^{\prime}$
(decreased $|\varphi^{\prime}|$) stabilises the BPS black hole. This is
reminiscent of how an electric potential modifies the AdS-Reissner-Nordström
black hole, discussed in subsection 2.2.2.

####  4.3.2 $\varphi^{\prime}\geq 0$

When $\varphi^{\prime}>0$, the denominator in the second factor of $\tau$
(150) is strictly positive for all $a\in[0,1)$. Therefore, unlike when the
potential $\varphi^{\prime}<0$, the entire range $a\in(0,1)$ of the parameter
$a$ is physical. At the lower end of the range $a=0$ the free energy $W=0$ and
the temperature is

$\tau_{\rm max}=\frac{2}{\varphi^{\prime}}~{}.$ (160)

This is the maximal temperature on the small black hole branch, a novel
feature of the $\varphi^{\prime}>0$ regime. In the strict limit $a=0$ the
geometry underlying the thermodynamic formulae reverts to pure AdS5, it is not
a black hole, so $\tau_{\rm max}$ is a bound, it cannot be reached. The bound
on the temperature (160) is lowered when the potential $\varphi^{\prime}$
increases. The range of allowed temperatures on the small black hole branch
shrinks and, in the limit $\varphi^{\prime}\rightarrow\infty$, it disappears
altogether. In this limit the large black hole branch starts at
$(W,\tau)=(0,0)$ and there is no small black hole branch. This limit is
similar to the special case $\Phi=\Phi^{*}=3$ of ordinary (non-BPS) AdS-
Reissner-Nordström black holes that was discussed in subsection 2.2.3.

The large black hole branch is not modified qualitatively from the benchmark
case $\varphi^{\prime}=0$ discussed in subsection 3.4, or more generally the
case $\varphi^{\prime}\leq 0$ developed earlier in this subsection. The phase
diagram for various values of $\varphi^{\prime}\geq 0$ is plotted in Figure
13.

Figure 13: The free energy $\frac{W}{N^{2}}$ as function of BPS temperature
$\tau$ for various $\varphi^{\prime}\geq 0$. The values of
$\varphi^{\prime}=0,1,2,4$ increase from right to left. The small black holes
(the upper branch) have $W=0$ at their maximal temperature $\tau_{\rm max}$
(160) which is $\tau_{\rm max}=\frac{1}{2},1$ for $\varphi^{\prime}=4,2$. The
large black hole branch is qualitatively similar to its analogue for
$\varphi^{\prime}<0$ in Figure 12.

The figure indicates that, as we increase $\varphi^{\prime}$ starting from
zero, the coordinates of the cusp $\tau_{\text{cusp}}$, $W_{\text{cusp}}$ both
decrease, as does the Hawking-Page temperature $\tau_{\text{HP}}$ where the
large black hole branch crosses $W=0$. To verify these features analytically,
we note that the parameter $a_{\rm cusp}$, corresponding to the minimal
temperature, is given by (155) for either sign of $\varphi^{\prime}$. It takes
the value $a_{\text{cusp}}=\frac{3\sqrt{3}-4}{11}$ when $\varphi^{\prime}=0$
and decreases monotonically when $\varphi^{\prime}$ increases to positive
values. The result follows because, for and $\varphi^{\prime}$, the
temperature $\tau$ (150) has positive derivative
$\partial_{\varphi^{\prime}}\tau>0$ and the motion (156) of the free energy
$W$. The estimate (159) similarly establishes that $\tau_{\text{HP}}$
decreases as $\varphi^{\prime}$ become larger.

In summary, the evolution of the cusp, the Hawking-Page temperature, and the
asymptotic behavior of the large black hole branch at high temperature, are
all smooth for $\varphi^{\prime}$ in the entire range
$(\varphi^{\prime}_{-},\infty)$. For these features the value
$\varphi^{\prime}=0$ plays no special role. In contrast, the asymptotic
behavior on the small black hole branch depends sensitively on the sign of
$\varphi^{\prime}$: for $\varphi^{\prime}\leq 0$, the temperature
$\tau\rightarrow\infty$ is reached with an asymptotic value $W_{\rm asym}$
(153) of the free energy that is strictly positive except that it vanishes
when $\varphi^{\prime}=0$. For $\varphi^{\prime}\geq 0$, the temperature is
bounded by $\tau_{\rm max}$ (160) that is finite, except that it diverges when
$\varphi^{\prime}=0$.

###  4.4 BPS Black Holes with Unequal Angular Momenta

In this paper we have presented nearly all formulae for general parameters
$a,b$ but most examples focus on the “equal angular momenta” case $a=b$. This
is also the case that has been most studied in the literature, by far. In this
subsection we elucidate the significance of “unequal angular momenta” $a\neq
b$ by taking the chemical potential $\mu\neq 0$.

We will keep $\varphi^{\prime}=0$ so the Gibbs energy $W$ (99) simplifies to

$\frac{W}{N^{2}}=\frac{\pi(a+b)^{2}\left[1-2a-2b-a^{2}-5ab-b^{2}-a^{2}b-ab^{2}\right]}{2(1-a)(1-b)\left(1+3a+3b+a^{2}+3ab+b^{2}\right)\sqrt{a+b+ab}}~{},$
(161)

and the BPS potentials (100) and (147) read

$\displaystyle\tau=\frac{2\left(1+3a+3b+a^{2}+3ab+b^{2}\right)}{3\pi(a+b)(1-ab)}\sqrt{a+b+ab}~{},$
(162)
$\displaystyle\mu=\pi\frac{a-b}{1+3a+3b+a^{2}+3ab+b^{2}}\frac{1+2a+2b+ab}{\sqrt{a+b+ab}}~{}.$
(163)

Because of the antisymmetry of $\mu$ under the exchange $a\leftrightarrow b$,
it is sufficient to analyze $\mu\geq 0$. The phase diagram is shown in Figure
14. From the plot, for $\mu$ that is positive and small, the phase diagram
evolves perturbatively from the benchmark case $\varphi^{\prime}=\mu=0$
discussed in subsection 3.4 with some changes that appear smoothly:

  * •

A maximal temperature of the small black hole branch develops that decreases
when $\mu$ increases. The upper branch shrinks.

  * •

For any given $\tau$, on either branch, the BPS potential $\mu>0$ lowers the
free energy. This is expected because, when $a>b$ (74-75) give $J_{a}>J_{b}$,
so the first law of BPS thermodynamics (143) yields:
$\partial_{\mu}W=-\frac{1}{2}(J_{a}-J_{b})<0$.

  * •

The Hawking-Page transition temperature $\tau_{\text{HP}}$ also decreases as
$\mu$ get larger, because both black hole branches are lowered.

  * •

The maximal free energy $W_{\text{cusp}}$ increases with $\mu$, but the
minimal BPS temperature $\tau_{\text{cusp}}$ decreases. Thus the cusp travels
”towards North-West” as $\mu$ increases.

Our interpretation of these features is that increasing $\mu$ is
destabilising: it takes a lower temperature to achieve $W>0$ and so render the
black holes unstable, and positive $\mu$ allows a higher maximal free energy
$W_{\text{cusp}}$, indicating stronger instability. This is a BPS analogue of
our finding in subsection 2.2.1 that, for non-BPS black holes, angular
velocities destabilise the AdS Reissner-Nordström black holes.

Figure 14: The BPS free energy $W$ vs. the BPS temperature $\tau$ for select
values of the potential $\mu\geq 0$ that parametrizes the asymmetry between
angular momenta. $\mu=0,0.8,1.26,1.6$ from right to left. For small $\mu$, the
phase diagram exhibits the familiar lower (large) and upper (small) black hole
branches that meet in a cusp. For the small black hole branch, there is a
maximal BPS temperature that decreases as function of $\mu$ and reaches the
cusp at some $\mu=\mu_{\rm crit}$. The maximal value of $W$ also increases
with $\mu\leq\mu_{\rm crit}$ but then decreases for $\mu>\mu_{\rm crit}$.

The preceding comments only apply when the chemical potential $\mu$ is
sufficiently small. From Figure 14 we see that when $\mu$ exceeds a certain
critical value $\mu_{\text{crit}}\approx 1.26$ the small black hole branch
disappears altogether, for such potentials only the “large” black hole
solution exists. The evolution when $\mu$ is larger than this critical value
has the following features:

  * •

For any given $\tau$, increased $\mu$ lowers the BPS free energy.

  * •

As $\mu$ increases, the minimal temperature $\tau_{\text{cusp}}$ increases as
well but the maximal free energy $W_{\text{cusp}}$ decreases.

  * •

The motion of the ”cusp” shows that the values $\tau_{\text{min}}$ and
$W_{\text{max}}$ evaluated at the cusp with $\mu=\mu_{\text{crit}}$ are the
global minimum for the temperature and maximum for the BPS free energy,
respectively.

  * •

The Hawking-Page transition temperature $\tau_{\text{HP}}$ (where $W=0$)
generally decreases as the entire large black hole branch is lowered. However,
when the chemical potential is sufficiently large, above $\mu\approx 1.5$, the
entire large black hole branch is below $W=0$ and the transition disappears
altogether.

  * •

As discussed in subsection 4.2, the parameter $\mu$ satisfies an absolute
upper bound $\mu_{\max}=\frac{3\pi}{5}\approx 1.885$ when
$\varphi^{\prime}=0$. As $\mu\rightarrow\mu_{\max}$, (147) shows that
$a\rightarrow 1$ and $b=0$ so the BPS free energy (99) $W\rightarrow-\infty$.
In the strict limit these is no underlying black hole solution.

Here, we observe another effect of perturbing $\mu$ when $\mu$ is sufficiently
large. Our interpretation is that for large $\mu$, it is stabilising: the
entire black hole would lie below $W=0$ and thus render the black holes
stable, and increasing $\mu$ lowers the maximal free energy $W_{\text{cusp}}$
indicating the stability. Since our discussion for non-BPS black holes is
concentrated in the equal angular momenta case, we do not have an analogous
case of it.

###  4.5 Extreme Rotational Asymmetry: $\Omega_{b}^{\prime}=0$

As an extreme example of asymmetry between the two angular momenta, we
consider the special case $\Omega_{b}^{\prime}\equiv 0$. In this case (97)
gives

$\varphi^{\prime}=-\frac{2\pi\left(2a+b+a^{2}+2ab\right)}{(1+b)\sqrt{a+b+ab}}~{}.$
(164)

It follows that that $\varphi^{\prime}$ must be negative. Formally, we can
also achieve $\Omega_{b}^{\prime}=0$ by taking $b=1$, but then the free energy
$W$ (99) diverges, so we dismiss this possibility as an unphysical limit.

When $\Omega_{b}^{\prime}=0$ the BPS temperature (100) and the BPS free energy
(99) simplify

$\begin{split}\tau&=-\frac{2}{\Omega_{a}^{\prime}}=-\frac{2}{\mu}=\frac{2(1+b)\sqrt{a+b+ab}}{\pi(1-a)(b-a)}~{},\\\
W&=\frac{\pi(1+a)(a+b)^{2}}{2(1-a)(1+b)\sqrt{a+b+ab}}~{}.\end{split}$ (165)

We see that $W$ is positive definite. The phase diagram of $\tau$ and $W$ for
various fixed $\varphi^{\prime}$ is presented in Figure 15. There are several
features:

  * •

For any given $-2\sqrt{3}\pi<\varphi^{\prime}<0$, the BPS free energy $W$
decreases monotonically with $\tau$.

  * •

The lower bound $\tau_{\min}$ decreases in the range
$-2\sqrt{3}\pi<\varphi^{\prime}<-\pi$ and increases when
$-\pi<\varphi^{\prime}<0$. It reaches its global minimum
$\tau_{\min}=\frac{4}{\pi}$ at $\varphi^{\prime}=-\pi$.

  * •

$W$ displays vertical asymptotes at $\tau_{\min}$ as long as $b$ can approach
each $1$ (allowed for $-2\pi\sqrt{3}<\varphi^{\prime}<-\pi$), whereas for
$\varphi^{\prime}>-\pi$, $W$ reaches an upper bound but with no asymptote
(such as the $\varphi^{\prime}=-\frac{3}{4}\pi$ curve in Figure 15).

  * •

$\tau$ diverges when $a$ approaches $b$ from below, leading to an asymptotic
value for $W$ at large $\tau$: $\frac{2\varphi^{\prime 2}}{9(\varphi^{\prime
2}-12\pi^{2})}$.

Figure 15: $W$ vs. $\tau$ when $\Omega_{b}^{\prime}=0$ with
$\varphi^{\prime}=-\frac{3}{2}\pi$, $-\pi$ and $-\frac{3}{4}\pi$ (upper to
lower respectively). There are only ”small” black holes: they all have $W>0$
and negative specific heat. For any given $\varphi^{\prime}$, there is a
minimal BPS temperature which corresponds to the maximal BPS free energy.

In summary, the large black hole branch disappears entirely when
$\Omega_{b}^{\prime}=0$, there are only small black holes. This is reminiscent
of the general (not BPS) black holes with $\Omega=1$, discussed in subsection
2.2.6. One aspect of the special case $\Omega_{b}^{\prime}=0$ is that it
corresponds to “maximal asymmetry”, there is a BPS potential for rotation
along the “a” direction, but none along “b”. However, a more illuminating
perspective is that the definition of the “primed” potentials (85) shows that
when the BPS potential $\Omega_{b}^{\prime}=0$, the physical rotational
velocity $\Omega_{b}=1$ reaches the speed of light. From this point of view
the close analogy with the $\Omega=1$ non-BPS black holes is expected.

##  5 Discussion

In this article, we studied the thermodynamics of AdS black holes via
analysing their phase diagrams. Compared with previous studies Chamblin:1999tk
; Caldarelli:1999xj , this work has a particular emphasis on the role of
rotation. We pointed out that rotation tends to destabilise a black hole in
the sense that rotation increases the maximal free energy.

We also developed BPS thermodynamics systematically and, in many explicit
examples, we pointed out that the phase diagram of the BPS black hole exhibits
the similar ”cusp” structure as the well-known AdS Schwarzschild black hole
phase diagram. We emphasised the role of an important fugacity,
$\varphi^{\prime}$, that preserves BPS saturation. This fugacity
$\varphi^{\prime}$ has been set to zero in the BPS limit in previous
discussion Choi:2019miv ; Kinney:2005ej , and paid little attention. We
illustrated how $\varphi^{\prime}$ brings qualitative change of the phase
diagram. Besides, we studied the case where the BPS black holes carry two
unequal angular momenta, and discovered a qualitative change of the phase
diagram.

However, there are many open questions. For example, we introduced a notion of
temperature for BPS black holes in (100) as a generalisation to the work by
Choi et al. in Choi:2019miv . It is for sure that one should be able to obtain
a deeper understanding on this BPS temperature. Another important question is
the physical meaning of $W=0$. Is this phase transition true or, instead, an
illusion. We look forward to pursue these and related questions in future
research work.

## Acknowledgements

This research was supported in part by the U.S. Department of Energy under
grant DE-SC0007859. NE is supported in part by a Leinweber Graduate
Fellowship.

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