Thermodynamics of Near BPS Black Holes in AdS₄ and AdS₇

Given the central importance of supersymmetry in nearly all current work in the area we now address, before getting to further details, how any progress can be made at all. To do so, recall the microstate counting of asymptotically flat black holes which has a long history and is understood incomparably better. Many precision agreements were established, not just at the leading order but also for higher derivative corrections, quantum corrections, and far beyond Castro:2008ne ; Dabholkar:2010rm ; Sen:2014aja . Moreover, in most cases it has been understood why these agreements hold with the precision they do. The reasons vary according to the setting and, although they often involve supersymmetry, that is not always the case. In particular, it has proven fruitful to study black holes that are solutions to theories with supersymmetry without themselves preserving any supersymmetry.

Specifically, experience with asymptotically flat black holes suggests that small deformations away from the BPS limit are under good control. One avenue is revealed geometrically by the near horizon AdS₂ being enlarged to an AdS₃. In this situation a combination of anomaly arguments and modular invariance ensures agreements at leading and subleading order, even when supersymmetry is broken Maldacena:1997de ; Kraus:2005vz . This success is not a feature of AdS₃ alone, the entropy of extremal but non-BPS black holes can be accounted for correctly even at four-derivative level, as shown by application of the entropy extremization formalism in AdS₂ Banerjee:2016qvj . Such successes for asymptotically flat black holes motivate studying nearly supersymmetric black holes in AdS.

As we mentioned tangentially already, a somewhat complementary motivation for studying near BPS black holes is presented by recent progress on their holographic description through the nearAdS₂/nearCFT₁ correspondence MSZ ; Almheiri:2016fws . A central aspect of this duality is a nontrivial symmetry breaking pattern which coincides between the two sides. It is realized in melonic quantum theories such as the SYK model and its avatars Sachdev:1992fk ; Kitaev ; Maldacena:2016hyu , novel settings that have justifiably generated much interest. Importantly, the symmetry breaking pattern is also realized in gravitational theories such as the Jackiw-Teitelboim model and its relatives Almheiri:2014cka ; MSZ ; Kitaev:2017awl . This sets the scene for a holographic duality.

However, nearAdS₂/nearCFT₁ correspondence is not a straightforward equivalence, it is an IR duality where the effective theory in the bulk and on the boundary are dual to one another only at large distances. The significance of the near BPS AdS black holes we study is that they offer a UV completion of the description on both sides of the duality. Thus, for each of the black holes, there is a specific dual theory that is well-defined in the UV.

The most precise studies of black holes focus on supersymmetric ground states and their ability to describe the entropy of BPS black holes microscopically. This is very interesting, of course, but BPS states are relatively inert ground states, unsuitable as proxies for physical black holes. It is therefore important to explore the low lying excited states as well Larsen:2019oll . That is one of our motivations.

The low energy effective theory is usefully summarized as a theory of Schwarzian type, characterized by one or more dimensionful coupling constants. These coefficients are response parameters such as the specific heat of the black hole. They are arbitrary inputs from the effective field theory point of view. However, in the context of a UV complete theory the response parameters can be determined from microscopic principles Moitra:2018jqs ; Castro:2018ffi ; Moitra:2019bub ; Castro:2019crn . One goal of this article is to do so explicitly.

The bosonic symmetry groups of AdS${}_{4}\times S^{7}$, AdS${}_{5}\times S^{5}$, and AdS${}_{7}\times S^{4}$ all have rank $6$. Therefore, in each case the BPS condition derived from supersymmetry becomes a linear relation between $6$ conserved charges that arise as eigenvalues of their respective Cartan generators. In other words, it expresses the mass $M$ as a sum of $5$ conserved charges as:

in appropriate conventions. Here the range of the indices $I$ and $i$ depend on the spacetime dimension but in all cases they enumerate a total of $5$ values.

However, it turns out that regular black holes in AdS only exist if, in addition, a certain constraint is imposed on the $5$ charges $(Q_{I},J_{i})$. For example, BPS black holes in gauged ${\cal N}=8$ supergravity in five dimensions are characterized by $5$ charges that satisfy the constraint

where $N^{2}$ refers to the dual ${\cal N}=4$ SYM theory with $SU(N)$ gauge group. This type of constraint may at first appear novel and special to AdS but actually it is not, it is just more complicated in AdS than in the more familiar settings with asymptotically flat spacetime. For example, within the family of Kerr-Newman black holes in four-dimensional Einstein-Maxwell theory (supersymmetrized as minimal ungauged ${\cal N}=2$ supergravity), the BPS black holes have mass given in terms of charge as $M=Q$ and they have angular momentum $J=0$. There are certainly regular extremal Kerr-Newman black holes that rotate, including the extremal Kerr black hole that is neutral under the gauge field. Alas, supersymmetry demands more than vanishing temperature, it imposes the constraint that the angular momentum $J=0$.

The recognition of the constraint plays an important role in this article. We study thermodynamics of near BPS black holes, i.e. parameter values that differ slightly from those of a reference BPS black hole. Importantly, there is a two-dimensional space of deformations. The most obvious is to increase the black hole mass beyond its BPS value while keeping charges fixed. This is equivalent to increasing temperature to $T>0$. The alternative is to maintain extremality ($T=0$) but modify charges so they violate the constraint.

The two directions away from the BPS surface both have a preferred sign: stability imposes not only $T\geq 0$ but also $h\geq 0$. Thus the parameter space is a quadrant of the plane with the BPS point at the origin. Moreover, the interplay between the two directions is quite nontrivial. It is characterized by $3$ response functions, each of which depend on all the conserved charges (subject to the constraint). A successful microscopic theory of the near BPS black holes must account for all these nontrivial functions.

In this article we do not attempt to derive a microscopic theory of the AdS black holes we study ab initio but we explain how recent progress on the strict BPS limit can be leveraged towards that goal.

The understanding of the AdS black hole entropy from microscopic principles has admittedly proven quite subtle even in the BPS limit. However, even though significant technical questions remain, the outline is now generally agreed upon. Accordingly, we take it as a given that the entropy of BPS black holes in AdS can be interpreted in terms of a dual field theory. We think of the required estimate of the asymptotic density of states as a two step-process: enumeration of all states in a “large” Hilbert space, followed by identifying physical states as those satisfying the constraint (2), or one of its analogues in other dimensions,

Our strategy for addressing near BPS black holes is to take the “large” Hilbert space established in the course of investigating BPS black holes as a given starting point. We then identify physical states by imposing a constraint that has been relaxed to accommodate a departure from the BPS limit. The less demanding constraint permits more physical states and this allows computation of the excess entropy enjoyed by near BPS black holes. We find that the entropy computed from this microscopic reasoning agrees with the one found from gravitational thermodynamics.

Our prescription describing near BPS thermodynamics is intuitive and physically reasonable but it goes against conventional wisdom on what quantities can be reliably computed. It suggests that agreements are justified not by preserved supersymmetry alone but also by broken supersymmetry and/or anomalies. We stress again that we do not claim fully principled comparisons between nonsupersymmetric black holes in AdS and their holographically dual boundary theory. It would in fact be premature to expect such results since the BPS agreements themselves remain beset by questions. However, the agreements we report are quite impressive as they involve many parameters and apply in each of the dimensions we develop. This persuades us that they are accurate and we expect they will ultimately acquire a solid justification.

This paper is divided into two parts: section 2 on AdS₄ black holes and section 3 on AdS₇ black holes. We have purposely written these sections on AdS₄ and AdS₇ so they are largely independent and can be studied in any order.

Within each section, we first discuss black hole thermodynamics generally and then consider the nature of the BPS limit. This sets up the development of several distinct near BPS regions and their interplay, all from the gravitational point of view. Our discussion of the microscopic description is collected in subsection 2.7 for AdS₄ and subsections 3.6-3.8 for AdS₇.

The $6$ quantum numbers of the maximally supersymmetric black holes in AdS₄ are the mass $M$, the angular momentum $J$, and four $R$-charges $Q_{I}$ ($I=1,2,3,4$) that correspond to the Cartan generators of the $S^{7}$ isometry group $SO(8)$. We specialize to the Kerr-Newman AdS₄ black holes where the four R-charges are identical so the solution depends on just three quantum numbers: $M$, $J$, and $Q$. The only other parameters that enter are the asymptotically AdS₄ radius $\ell_{4}$ (related to the coupling $g=\ell_{4}^{-1}$ of gauged supergravity) and the gravitational coupling $G_{4}$.

The explicit solution (first presented in Chong:2004dy ) is fairly elaborate, as expected for a rotating black hole, so we will not present the geometry and its associated matter here. The only feature that is needed in our study is the radial function

that appears prominently in the metric. The event horizon of the black hole is located at the coordinate $r=r_{+}$ that is the largest real root of the quartic equation $\Delta_{r}(r)=0$. The parameters $(m,a,q)$ in $\Delta_{r}$ are related to the physical variables $(M,J,Q)$ of the black hole as Caldarelli:1999xj ; Gibbons:2004ai ; Papadimitriou:2005ii

where $\Xi=1-a^{2}g^{2}$. The parameters $m,q$ are positive while $0\leq ag<1$. The charges are normalized so $2Q\ell_{4}$ and $J$ are integral (for bosons) or half-integral (for fermions).

In black hole thermodynamics a central role is played by the potentials that are conjugate to the three quantum numbers $(M,J,Q)$. They are:

As usual, the size of the quantum configuration space is encoded in the black hole entropy

We also record the on-shell Euclidean action of the black hole

It satisfies the quantum statistical relation

as it should.

The supersymmetry algebra realized by the theory demands that the black hole mass satisfies the BPS bound¹¹1In comparison with the schematic formula (1) each of the four Cartan generators are taken as $\frac{1}{2}Q_{I}=Q_{\rm here}$. Also, here mass $M$ and charge $Q$ have dimension of inverse length while $J$ is a pure number.

with the inequality saturated (i.e. satisfied as an equality) precisely when the black hole preserves a fraction of the supersymmetry. The * designates that variables refer to the BPS black holes. In this subsection we take the view that the BPS bound is a hypothesis that we seek to validate through explicit computation, by showing that it is satisfied by all the aforementioned black hole solutions.

As a first step we rewrite the BPS bound (12) using our parametric formulae (4)

Thus the inequality on physical variables is equivalent to the bound on parameters

with BPS saturation corresponding to equality in both cases.

To make further progress we differentiate the BPS equality $M^{*}=2Q^{*}+gJ^{*}$ and find the potentials $\Phi^{*}=2$, $\Omega^{*}=g$ for BPS configurations. We can invert the parametric formulae (7) for these potentials to find the corresponding BPS relation between parameters

and also find the coordinate position of the horizon

It is a consistency check that the temperature $T=0$ for these values of parameters, as expected in the BPS limit.

It follows from the facts established so far that the radial function $\Delta_{r}(r)$ (3) must vanish at $r=r^{*}$ for black hole parameters such that $q=q^{*}$ and $m=m^{*}=(1+ag)q^{*}$. We can make this feature manifest by rewriting the general formula for the radial function as:

The location of the horizon $r=r_{+}$ is the largest solution to $\Delta_{r}(r)=0$ so, for any value of black hole parameters, it satisfies the exact equation

The terms on the right hand side are manifestly positive so we conclude that the black hole parameters must satisfy $m-q(1+ag)\geq 0$. This agrees with the parametric bound (14) so we have established by explicit computation that the physical BPS bound $M\geq M^{*}$ is satisfied for all the black holes solutions, as we wanted to show.

This result was expected from supersymmetry of the theory. However, there is a less obvious corollary of the computation. The identity (19) shows that the BPS bound is saturated if and only if both of the square brackets on its right hand side vanish. This is clearly the case for “the” BPS black holes with $q=q^{*}$ and $r_{+}=r^{*}$ that we have already identified but it is not difficult to check that this is the unique solution. In other words, we have shown by explicit computation that the BPS bound on the black hole mass (12) is saturated if and only if $q=q^{*}$ and $r_{+}=r^{*}$. We will see below that these conditions on parameters correspond to vanishing temperature $T=0$ and an additional constraint on the physical potentials $\Phi-\Omega\ell_{4}=1$ or on the charges $Q$, $J$.

We will discuss general black holes with frequent reference to the BPS limit. Therefore, we collect formulae for this special case in this short subsection. We label the one-parameter family of BPS black holes by $ag$ and express the other dimensionless parameters as

Inserting these values into into (4) we find the electric charge

This formula can be inverted as

We can use this equation to eliminate the parameter $ag$ in favor of the charge when considering any physical variable of a BPS black hole. As an important example, after inserting the BPS parameters (22) into (4) for the angular momentum, we can eliminate $ag$ and find

This relation between physical conserved charges is the constraint that must be satisfied for all BPS black holes.

We can similarly find the BPS black hole entropy by substituting the parametric formulae (22) in the general equation for the entropy (10) and then eliminate $ag$ using (24). However, because of the constraint (25) the dependence of the BPS entropy on the conserved charges $Q^{*},J^{*}$ is not unique, it can take many different forms. Our “preferred” formula will be to eliminate the angular momentum entirely and express the black hole entropy it in terms of charges alone

where the omnipresent dimensionless ratio

is a large pure number that sets the scale for the conserved charges. It quantifies that the black hole is much bigger than the Planck scale with a precise value that is characteristic of the microscopic theory. The second equality applies when the AdS₄ background arises from $N$ $M2$-branes, or from their dual description by ABJM theory.

In this subsection we initiate our study of thermodynamics in the nearBPS regime. The parametric representation of the BPS limit is $q=q^{*}$ and $r_{+}=r^{*}$ so we define nearBPS black holes as those where

are small. The identity (19) shows that this is possible only if $m$ is such that $m-q(1+ag)\sim\epsilon^{2}$.

The black hole temperature is

at linear order in the small parameter $\epsilon$ introduced in (28). The nearBPS potentials are

at the same order. It will prove advantageous to introduce a “nearBPS potential” $\varphi$ defined by the linear combination

Then the conditions (28) on the parameters of nearBPS black holes are equivalent to physical potentials of order

When black holes depart from the supersymmetric limit, their mass exceeds the BPS mass $M^{*}$. We showed earlier that the excitation energy $M-M^{*}$ is proportional to $m-q(1+ag)$ and the identity (19) established that this quantity is positive definite, by casting it as a sum of two squares. We now observe that the linear combination of parameters that appear in those two squares coincide with the temperature $T$ and the nearBPS potential $\varphi$ at linear order. Therefore, the nearBPS mass is given by the quadratic mass formula

where, after collecting various proportionality factors, we find the dimensionless coefficient

In the second form of the expression we used (24) and (27) to convert the gravitational formula into a remarkably economical form that can later be compared with microscopic results. The notation $C_{T}$ in (33) refers to the specific heat of the nearBPS black hole. The specific heat is proportional to the temperature $T$ in the nearBPS regime so the coefficient of interest is the ratio $\frac{C_{T}}{T}$.

Recall that general perturbations away from the BPS locus are characterized by two variables: the temperature $T$ adds energy with charges kept fixed while the nearBPS potential $\varphi$ parametrizes deformations along the extremal surface $T=0$ that violate the constraint (25) on conserved charges. The specific heat refers to the first of these, the addition of energy through the nearBPS potential $\varphi$ is physically quite distinct. To the extent $\varphi$ can be identified with an electric potential the corresponding coefficient in the mass excess $M-M^{*}$ is the black hole capacitance. Interestingly, our mass formula (33) indicates that, for the black hole studied here, the capacitance is identical to the specific heat. ²²2They arguably differ by a factor of temperature. We highlight the parallel between the two response functions by introducing capacitance as $M-M^{*}=\frac{C_{\varphi}}{2T}\varphi^{2}$. In this notation $C_{\varphi}=C_{T}$.

An analogous equality between two physically distinct linear response coefficients was previously noticed for nearBPS black holes in AdS₅ Larsen:2019oll and in Section 3 we will establish it also in AdS₇. The common feature of these settings is the supersymmetry breaking pattern. The gravitational theory has (at least) ${\cal N}=2$ supersymmetry which is mildly broken by an excess energy (conjugate to temperature) or R-charge (conjugate to the nearBPS potential). The reasonable expectation that the corresponding symmetry breaking scales are themselves related by supersymmetry is born out in the ${\cal N}=2$ version of the SYK model which realizes the analogous symmetry breaking pattern in a nongravitational setting Fu:2016vas . The nearBPS black holes developed in this paper offer an appropriate setting for this physical mechanism on the bulk side of the nAdS₂/CFT₁ correspondence. It would be interesting to further study the supersymmetry breaking pattern in supergravity.

The constraint on conserved charges (25) can be presented as the vanishing of the “height” function

This form of the constraint makes it manifest that nonrotating ($J=0$) and uncharged ($Q=0$) black holes are both inconsistent with supersymmetry in AdS₄. In the nearBPS region we can relax the condition $h=0$. Any surface with constant height function $h$ is characterized by the vanishing of the differential

The constraint $h=0$ defines a line in the two-dimensional space of conserved charges $(Q\ell_{4},J)$ and we can interpret nonzero values of $h$ as a coordinate along the normal to this line, quantifying the departure from the BPS line. However, we have already introduced the nearBPS potential $\varphi$ such that $\varphi=0$ on the constraint surface and $\varphi$ is an equally good measure of the distance from the BPS surface. Indeed, the geometry of embedded surfaces guarantees that for small values these coordinates must be proportional. In the following we calculate their constant of proportionality.

In the nearBPS regime, the parameters $m$ and $q$ are proportional up to second order in $\epsilon$, as noted after (28). Therefore, at linear order the general physical charges $Q$, $J$ given in (4) are both proportional to $q$, with distinct proportionality factors depending $a$. Rescaling of $q$ with $a$ fixed therefore changes $Q$ and $J$ by a common factor. It modifies the height function as

where we simplified using the constraint $h=0$. However, the nearBPS potential $\varphi$ (32) essentially measures the scale of $q$ via $d\varphi=2q^{-1}_{*}dq$ so this calculation determines the constant of proportionality that we seek:

We can further illuminate the nearBPS regime by explicitly verifying the first law of thermodynamics. We write it as

and consider the left and right hand sides in turn.

The quadratic formula for the nearBPS mass (33) gives

with the coefficient $\frac{C_{T}}{T}$ given in (34). The entropy $S-S^{*}$ in excess of its BPS value will turn out to involve a subtlety, as we discuss below. For now we compute the difference between the general area law (10) evaluated at the respective horizon positions $r_{+},r^{*}$, each at the same value of $a$. This procedure gives

in the nearBPS regime. The linear-in-$T$ term has the correct coefficient to cancel the analogous term in the mass formula (42) so the left hand side of (41) yields an expression proportional to $d\varphi$:

The right hand side of the first law (41) involves the BPS entropy $S^{*}$. In our “preferred” expression (26), it is a function of electric charge $Q$ that gives

We also need the potentials (30-31) recast in terms of the temperature $T$ (29) and the nearBPS potential $\varphi$ (32) as

and

These expressions quantify the linear changes as we move off the BPS line so the terms on the right hand sides are equivalent to derivatives with respect to temperature $T$ and potential $\varphi$. In this subsection the formulae in terms of the intrinsic coordinate $a$ are sufficient but we record these results also in microscopic units for later reference.

Returning to our ongoing verification of the first law (41), we combine the potentials (47-49) with the differential of the BPS entropy (46) and find

In the second line we took advantage of the fact that the particular linear combination of $dJ$ and $dQ$ that appears is proportional to $dh$ given in (37). Thus the relative change in the conserved charges preserves the “height” function (36) $h={\rm constant}$, for example by remaining within the constraint surface $h=0$. Because of this property, we can invoke (40) and rewrite the differential in terms of the nearBPS potential $\varphi$

The first law of thermodynamics (41) demands that this expression agrees with (45). The fact that both are proportional to $d\varphi$ shows that temperature changes $dT$ match, as they should. The coefficients of $\varphi d\varphi$ also coincide but the terms proportional to $Td\varphi$ do not agree. The reason, previously uncovered in the analogous 5D setting Larsen:2019oll (and alluded to as a subtlety earlier in this subsection), is a dependence on the reference point on the BPS surface.

The BPS entropy $S^{*}$ is defined only modulo the constraint $h=0$ on the charges. Therefore expressions that are equivalent when $h=0$ is imposed may have differentials that differ by $dh$. Indeed, the BPS entropy $S^{*}$ employed as reference when computing $S-S^{*}$ in (44) is the general area law (10), evaluated at the BPS point. In contrast, the differential of $S^{*}$ was derived in (46) from the “preferred” form of the entropy (26), expressed in terms of the physical charge $Q$. The former amounts to a formula depending entirely on the coordinate $ag$ along the BPS line, but the latter also takes into account that $Q$ is proportional to $q$. This amounts to an additional contribution:

Adding this expression to (52) we recover (45), as required by the first law.

The quantitative output of this subsection is the evaluation of the entropy due to the violation of the constraint. Its value read off from (44) at $T=0$

defines a third response coefficient, above and beyond the two implied by the quadratic mass formula (33). In view of the ambiguity discussed above we must specify that the differentiation in its definition is taken at fixed value of the intensive parameter $a$ which, as for Kerr-Newman black holes in asymptotically flat space, equals the ratio of physical variables $J/M$ .

The potential $\varphi$ was introduced in (32) as a linear function of $\Phi$ and $\Omega$ that vanishes for BPS black holes and measures the departures from the BPS line that preserve extremality $T=0$. We see from (44) that it was defined such that the entropy increases for $\varphi\geq 0$. This inequality suggests that the physical configuration space is restricted to $\varphi\geq 0$. An equivalent statement is that the constraint relating physical charges $Q,J$ can be violated, but only such that the height function introduced in (36) is positive $h\geq 0$. Yet another version of the inequality is that the charge parameter $q\geq q^{*}$.

For a perspective on these conditions we analyze Gibbs’ free energy. Starting from the on-shell action (11) we can write it as

where the horizon position $r_{+}$ and the angular momentum to mass ratio $a$ are interpreted as functions of the potentials $T,\Omega$. These functions are determined implicitly through

and

We want to examine the range of parameters that corresponds to physical black holes. We first demand that the extensive variables mass $M$, angular momentum $J$, charge $Q$, and entropy $S$ are finite and non-negative. This restricts the parameters so $0\leq ag<1$. The inequality $a\geq 0$ (and so $\Omega\geq 0$) does not limit generality because $a\rightarrow-a$ leaves all thermodynamic formulae (and the entire geometry) invariant, except for a flip of parity. Our second physical requirement is that $\Omega\ell_{4}\leq 1$, in order that the speed of light is not exceeded in the dual boundary theory. Since we already took $|ag|<1$ this condition amounts to

Interestingly, the first inequality in (55) can be recast as $r_{+}\geq r_{*}$ so the nonBPS black holes are all larger than their BPS relatives when measured in the conventional $r$ coordinate.

Our goal is ultimately to describe nearBPS black holes as excitations of BPS black holes. In the grand canonical ensemble considered here such states can be reached by deforming the potentials $(\Phi,\Omega)$ away from their BPS values $(\Phi_{*},\Omega_{*})$ while staying at extremality $T=0$, followed by raising the temperature while keeping $(\Phi,\Omega)$ fixed. This motivates our third physical condition: the potentials $(\Phi,\Omega)$ must be consistent with extremality $T=0$. Given the general restrictions (55) already imposed, vanishing temperature is possible only for $\Phi^{2}\geq\Phi^{2}_{*}=4$. This leaves a physical domain defined by

The Gibbs’ free energy (54) is automatically negative semidefinite in this entire region. It vanishes only for BPS black holes where $\Omega=\Omega_{*}=1$, $\Phi=\Phi_{*}=2$. Therefore the nonBPS black holes in the entire region (56) are stable with the proviso that the BPS black holes are only marginally stable, they can be in equilibrium with a gas of BPS particles.

The conditions we impose may be overly strict for some purposes and there can be good reasons to relax them. On the other hand, the region (56) is natural for microscopic studies. For fixed $(\Phi,\Omega)$ the temperature $T$ can be increased from zero (extremality) all the way to the high temperature conformal regime without any phase transitions being encountered. Specifically, the deconfined phase reigns in the entire domain, the entropy is of ${\cal O}(k)$ throughout.

As an example that is widely studied in the literature consider the Gibbs’ free energy for nonrotating black holes ($\Omega=0$). In this case it is elementary to solve the equations above explicitly. This yields the formula:

This function is manifestly smooth everywhere in the interior of the domain (56). For large temperature (at fixed $\Phi$) it takes the conformal form $G\sim-\frac{4k}{27}(2\pi T\ell_{4})^{3}$ with the numerical coefficient familiar from studies of large AdS-Schwarzchild black holes. In particular, it appears in the hydrodynamic description that applies at large temperature Bhattacharyya:2007vs ; Rangamani:2009xk ; Hubeny:2011hd . The opposite regime of small temperature is more delicate. For example, the dependence $G\sim-\frac{k}{4\sqrt{27}}(\Phi^{2}-\Phi^{2}_{*})^{\frac{3}{2}}$ on the potential along the extremal surface $T=0$ describes small nonrotating black holes.

The BPS region can be reached by tuning the potentials so $\frac{r^{2}_{+}}{a\ell_{4}}\rightarrow 1^{+}$ while simultaneously taking $\Phi\rightarrow\Phi_{*}^{+}$ with the ratio $(\Phi-\Phi_{*})/(r^{2}_{+}-a\ell_{4})$ kept fixed. In this limit Gibbs’ free energy becomes

where the dimensionless parameter $ag$ is defined implicitly by the equation

This equation determines $a$ as a homogeneous function of $\Phi-\Phi^{*}$, $1-\Omega\ell_{4}$, and $T$. It is a quartic in the variable $\sqrt{a}$ and its general solution in terms of radicals is not illuminating.

We can “solve” (58) for $1-\Omega\ell_{4}$, $\Phi-\Phi_{*}$ by expressing each of these quantities as a linear combination of $T$ and $\varphi$ with coefficients that depend on $a$. Such equations were already found in (47-49) using other thermodynamic arguments and we can verify that they satisfy (58). This is a useful consistency check.

In the Cardy limit $1-ag\ll 1$ we can solve the constraint (58) explicitly:

and find the free energy

Its derivatives with respect to $T$, $\Phi$, $\Omega$ yield BPS values $S^{*}$, $Q^{*}$and $J^{*}$ that agree with our previous results (26, 23, 25) in the Cardy limit. It is interesting that the free energy approaches the BPS limit linearly in the temperature because this reflects a BPS remnant of the familiar deconfinement transition Witten:1998zw ; Aharony:2003sx ; ArabiArdehali:2019orz ; Cabo-Bizet:2019osg ; Cabo-Bizet:2020nkr ; Copetti:2020dil .

The Gibbs’ free energy (57) with $a$ given implicitly by (58) actually describes the entire BPS surface, not just the Cardy limit. For example, we can determine the mass as

This is the exact BPS equation without assumptions on the black hole parameters. This computation is possible without knowing $a$ in detail because $G$ is homogeneous of degree one in the variables $(T,\Phi-\Phi^{*},\Omega-\Omega^{*})$ that the function $a$ depends homogeneously on. Similarly, the general derivatives of $G$ with respect to $T$, $\Phi$ and $\Omega$ depend on the unknown derivatives $\partial_{T}a,\partial_{\Phi}a,\partial_{\Omega}a$ but only in combinations that follow from parametric differentiation of (58). This procedure recovers the general expressions for $S^{*}$, $Q^{*}$and $J^{*}$ without imposing the Cardy limit.

The comparison between microscopic and macroscopic thermodynamics of BPS black holes in AdS₄ can be implemented conveniently by considering the free energy ³³3The dimensionless numerical constant $k$ was introduced in (27). In this subsection we absorb the length scale $\ell_{4}$ into the charges $Q$ which are, therefore, quantized as integers. Hosseini:2018dob ; Zaffaroni:2019dhb

On the microscopic side the partition function is identified with an index that can be found by methods such as supersymmetric localization. These computations are comparatively rigorous but the extraction of the black hole entropy from the free energy has some heuristic aspects, as we review below. Additionally, it unclear why the index yields the black hole entropy rather than just a lower bound.

The focus of our study is the macroscopic thermodynamics. In order to facilitate comparisons with microscopic ideas, we will repackage our gravitational results into the free energy (59). Importantly, we will do so not just for the BPS limit but for the entire nearBPS regime. Therefore, any microscopic computation that yields (59) accounts for the nearBPS entropy as well.