Microscopic Entropy of AdS_3 Black Holes Revisited
LLCTP-21-03
Microscopic Entropy of AdS Black Holes Revisited
Finn Larsen a and Siyul Lee a a Leinweber Center for Theoretical Physics, University of Michigan, Ann Arbor, MI 48109, U.S.A.
E-mail: [email protected] , [email protected] Abstract:
We revisit the microscopic description of AdS black holes in light of recentprogress on their higher dimensional analogues. The grand canonical partition functionthat follows from the AdS /CFT correspondence describes BPS and nearBPS black holethermodynamics. We formulate an entropy extremization principle that accounts for boththe black hole entropy and a constraint on its charges, in close analogy with asymptoticallyAdS black holes in higher dimensions. We are led to interpret supersymmetric black holesas ensembles of BPS microstates satisfying a charge constraint that is not respected byindividual states. This interpretation provides a microscopic understanding of the hithertomysterious charge constraints satisfied by all BPS black holes in AdS. We also developthermodynamics and a nAttractor mechanism of AdS black holes in the nearBPS regime. a r X i v : . [ h e p - t h ] J a n ontents The microscopic origin of the Bekenstein-Hawking entropy [1] has been one of the mostprominent topics in all of theoretical physics for several decades. It is largely what triggeredthe celebrated AdS/CFT correspondence [2] and it continues to serve as an indispensabletheoretical laboratory for many aspects of quantum gravity. However, despite very sig-nificant early investigations [3–7], only in the last few years was progress made towardsunderstanding the entropy of asymptotically AdS d> black holes microscopically [8–16].Moreover, the physical picture behind these recent developments remains blurred by vari-ous technicalities even now. The purpose of this paper is to exploit well-established insightsinto black holes in AdS to illuminate these conceptual challenges.– 1 –upersymmetric black holes in AdS × S , dual to 4D N = 4 super-Yang-Mills with SU ( N ) gauge group, have entropy that scales as S ∼ N [17–23]. The entropy cannotbe accounted for by the conventional superconformal index of SYM which has asymptoticbehavior O ( e N ) [4, 5]. However, it is now understood that the superconformal index growsas O ( e N ) [12–14] (see also [24–30]) when studied as a function of complex chemical po-tentials, rather than real ones. Moreover, the resulting density of states accounts preciselyfor the Bekenstein-Hawking entropy of the dual BPS black hole: S = 2 π (cid:114) Q Q + Q Q + Q Q − N ( J + J ) , (1.1)where Q I (with I = 1 , ,
3) denote the R-charges (rotations on S ) and J i (with i = 1 , .The Legendre transform from the canonical (potentials specified) to the microcanonical(charges specified) ensemble can be formulated as an extremization principle for an entropyfunction [11, 13] that is necessarily complex. Its extremum successfully yields the correctentropy (1.1) but the requirement that it be real imposes an extra constraint on the blackhole charges: Q Q Q + 12 N J J = (cid:18) Q + Q + Q + 12 N (cid:19) (cid:18) Q Q + Q Q + Q Q − N ( J + J ) (cid:19) . (1.2)The physical origin of this constraint is somewhat mysterious, and the way it arises tech-nically is unfamiliar from previous studies of the microscopic black hole entropy in othersettings. On the other hand, the extra constraint (1.2) is very much anticipated from thegravity side where it is satisfied by all BPS black holes in AdS [17–22], in addition to themore conventional BPS mass condition M = (cid:88) I =1 Q I + (cid:88) i =1 J i . (1.3)In other words, all black holes that satisfy the mass formula (1.3) also obey the constraint(1.2) [31].The necessity of angular momentum, the complexification of potentials, and the extraconstraint are features of all BPS black holes in AdS d> . They may give the impressionthat BPS black holes in higher dimensional AdS are fundamentally different from theirasymptotically flat relatives which are closely related to the BTZ black holes in AdS . Inthis article we show that, on the contrary, BPS black holes in AdS are very similar to thosein AdS d> and vice versa . Indeed, most of the material in the paper is not genuinely new,but it has been reworked so the analysis of AdS black holes closely follows contemporarydiscussions of the higher dimensional case, in an effort to demystify some of the newerdevelopments.The AdS /CFT correspondence is simpler, and therefore more transparent, than itshigher dimensional counterparts because: • There are fewer charges. – 2 –
The charge constraint analogous to (1.2) is linear. • The superconformal algebra in two dimensions factorizes into two independent fac-tors. • There is a powerful tool in CFT : modular invariance. It is for these reasons that the AdS problem has already been “solved”, to a large extent.We consider general CFT ’s with (4 ,
4) supersymmetry that are not necessarily chiral,we allow distinct levels k R,L in the two sectors. In this theory we study the high temperaturegrand canonical partition function, computed via modular invariance from the vacuumstate, and their dual BTZ black holes. From this simple starting point we derive BPSproperties of black holes in several ways.The most direct approach is to take an appropriate limit of the thermodynamic ex-pressions. This isolates the zero temperature sector. However, supersymmetry demandsthat, in addition, we engage a gauge field for an SU (2) R symmetry that is interpretedin spacetime as rotation on an S fibered over AdS . Thus the BPS limit involves twoconditions on the thermodynamic potentials.In terms of charges, one of the conditions satisfied by BPS black holes in AdS is alinear mass condition that we present as: E − E SUSY = P + J L , (1.4)where P and J L = J + J are conserved charges of the black hole. The left hand side,including the supersymmetric Casimir energy E SUSY = − k L , corresponds to the blackhole mass in the higher dimensional examples. We see that the form of the mass formulain AdS is completely analogous to (1.3).The second condition satisfied by BPS black holes in AdS is a constraint on the blackhole charges, namely J L = k L . (1.5)We interpret this relation as the AdS analogue of the constraint (1.2). Despite its simplic-ity, it is far from trivial. The BPS states identified by the superconformal algebra are, inour conventions, the chiral primaries. They all satisfy the mass formula (1.4) and unitarityfurther demands that 0 ≤ J L ≤ k L [34, 35]. The charge constraint (1.5) is much stronger,it shows that black holes are possible only for a single value of J L . As we explain furtherbelow, we interpret this fact as a result of ensemble average.Following the cue from recent work on BPS black holes in higher dimensional AdS,we also study the supersymmetric index I , i.e. the elliptic genus in CFT . It is simple tocompute via an analytical condition from the partition function and, in the case k R = k L ,we find ln I = k ˜ ω ˜ ω ˜ µ . (1.6)The variables are potentials that are subject to the constraint˜ µ − ˜ ω − ˜ ω = 2 πi . (1.7) Interesting modular-like properties of 4D CFT are being studied as well, see [32, 33] for examples. – 3 –hese formulae give an AdS version of the HHZ free energy that plays a central role indiscussions of AdS black holes in higher dimensions [11]. We analyze it by defining theentropy function as a Legendre transform of (1.6), or more precisely its generalization (3.6)to k R (cid:54) = k L . After extremization over all potentials, our entropy function becomes S = 2 π (cid:114) k R ( P + 12 J L − k L ) − J R + πi ( J L − k L ) . (1.8)Upon requiring this to be real, we recover the charge constraint J L = k L given in (1.5) andwe further find the correct BPS entropy S BPS = 2 π (cid:114) k R ( P + 14 k L ) − J R . (1.9)The fact that these manipulations are much simpler than their higher dimensional ana-logues facilitates a critical evaluation of the procedure. Alas, we find the reasoning un-satisfying: the imaginary part of (1.8) is immaterial to the reality of physical quantitiesbecause J L − k L ∈ Z and so the degeneracy e S is manifestly real, even before imposing anycondition.In the AdS context we can examine why the manipulations “work”. The real part ofthe index condition (1.7) indicates that the index does not distinguish the two charges P and J L , it only depends on the combination P + J L . It is extremization over the potentialsindependently, rather than their combination, that gives the correct charge constraint froma principled point of view. That the reality condition gives the same result appears to bean artifact of special mathematical properties of the BPS partition function.Instead, we provide a physical interpretation of the AdS charge constraint (1.5) thatis purely microscopic: the ensemble average . While it is not a novel claim that blackholes are described by thermal ensembles in the dual field theory, we show that the veryconcept of thermal ensemble, that macroscopic charges are obtained by taking averagesover the ensemble, leads to their constraint. We expect this observation to be central tounderstanding more intricate problems in higher dimensions, despite its simplicity.The rest of this paper is organized as follows. In section 2 we develop the thermo-dynamics of asymptotically AdS BPS black holes with all chemical potentials treated asreal. In section 3 we define the supersymmetric index, as opposed to partition function, andpotentials become complex. We formulate an entropy extremization principle and examinewhy this procedure works. We also introduce a nAttractor mechanism for the BTZ blackholes, to give a clear spacetime interpretation of the potentials. In section 4 we generalizethe thermodynamics of the black holes to the nearBPS regime. Finally, in section 5, wediscuss how the charge constraint (1.5) arises from an ensemble average, by consideringthe representation theory of (4 ,
4) SCFT ’s. In this section we study the thermodynamics of BPS black holes in AdS . The startingpoint is the high temperature partition function which we motivate from both sides of the– 4 –dS /CFT correspondence. We show that the BPS limit imposes two conditions on theblack hole parameters. We consider the standard set-up that describes BPS black holes in 5 asymptotically flatdimensions. Such black holes lift to the 6D geometry AdS × S and are dual to CFT ’swith (4 ,
4) supersymmetry. The SU (2) × SU (2) isometry of S corresponds to rotationof the original black hole in five dimensions and is identified with the R-symmetry of theCFT .We define the grand canonical partition function as Z = Tr e − β ( (cid:15) − µp − ω R j R − ω L j L ) , (2.1)where the quantum numbers (cid:15), p, j R , j L characterize individual states. The correspondingmacroscopic charges, evaluated as averages over many states, are denoted E, P, J R , J L . Theconjugate potentials of both microscopic and macroscopic quantities are β, βµ, βω R,L withsigns specified by the definition (2.1). Alternatively, the first law of thermodynamics
T dS = dE − µdP − ω R dJ R − ω L dJ L , summarizes conventions conveniently in a form that is well adapted to black holes.In CFT the eigenvalues of Virasoro generators are introduced through L − k R (cid:15) + p , ˜ L − k L (cid:15) − p . The constants k L,R are levels of the SU (2) R-currents. They are related to central chargesas c L,R = k L,R by N = 4 supersymmetry. The unique SL (2) × SL (2) invariant groundstate annihilated by L , ˜ L has strictly negative energy E vac = − ( k R + k L ) and correspondsto the AdS vacuum. It is separated by a gap from the black holes which have nonnegativeenergy in the CFT terminology. The momentum P corresponds to angular momentumof the AdS black hole but for the 5D black hole it is momentum along a compact 6thdimension. The high temperature partition function plays a central role in our considerations. In fact,we will regularly refer to it as the “general” partition function despite the restriction to hightemperature, in order to stress that it depends on all the continuous variables appearingin the definition (2.1). We write it in either of the two formsln Z = k R β (1 − µ ) (cid:0) π + β ω R (cid:1) + k L β (1 + µ ) (cid:0) π + β ω L (cid:1) = πik R τ (cid:0) − z (cid:1) − πik L τ (cid:0) − z (cid:1) . (2.2)The second line is a rewriting of the first that introduces standard CFT notation for thefugacities: 2 πiτ = − β (1 − µ ) , – 5 – πi ¯ τ = β (1 + µ ) , πiz = βω R , πi ¯ z = − βω L . (2.3)Note that, in either notation, the partition function is a function of four independent realvariables. In contrast, the index corresponds to a boundary condition that sets ¯ z = andis automatically independent of ¯ τ . Thus the index depends on only two real variables andthe dependence on the anti-holomorphic ( L ) sector disappears entirely. We study the indexin section 3.The simplest derivation of the partition function (2.2) applies a modular transformationto the ground state contribution. However, the result is very robust and can be reachedin many ways. For example, a more refined derivation was given in [36], from both bulk(AdS ) and boundary (CFT ) points of view. It showed that, when starting from bulkprinciples, all (local) higher derivative corrections are incorporated.From the general partition function (2.2), thermodynamic properties such as macro-scopic variables of the ensemble are readily obtained. Differentiation of the partition func-tion (2.2) by β gives E − µP − ω R J R − ω L J L = − ∂ ln Z∂β = k R β (1 − µ ) (cid:0) π − β ω R (cid:1) + k L β (1 + µ ) (cid:0) π − β ω L (cid:1) , (2.4)and we similarly find the conserved charges P = 1 β ∂ ln Z∂µ = k R β (1 − µ ) (cid:0) π + β ω R (cid:1) − k L β (1 + µ ) (cid:0) π + β ω L (cid:1) , (2.5) J L,R = 1 β ∂ ln Z∂ω
L,R = 2 k L,R ± µ ω L,R . (2.6)A combination of these expressions gives the energy E = k R β (1 − µ ) (cid:0) π + β ω R (cid:1) + k L β (1 + µ ) (cid:0) π + β ω L (cid:1) , (2.7)and the macroscopic entropy S = β ( E − µP − ω R J R − ω L J L ) + ln Z = 2 k R π β (1 − µ ) + 2 k L π β (1 + µ )= 2 π (cid:114) k R ( E + P ) − J R + 2 π (cid:114) k L ( E − P ) − J L . (2.8)Equations (2.5-2.8) are starting points for various limits we study in the rest of this section. Up to this point we did not impose any conditions on the black hole parameters. We nowimpose supersymmetry and show that the resulting BPS black holes satisfy two conditions.– 6 –n the 2D superconformal theory with (4 ,
4) supersymmetry, there are four -BPSsectors. Each sector preserves two real supersymmetries that are either holomorphic ( R )or anti-holomorphic ( L ), and that either raise or lower the R-charge. We focus without lossof generality throughout the article to the -BPS sector which preserves supersymmetriesthat are anti-holomorphic ( L ) and raise the R-charge. Then the unitarity bound from theanticommutator of the supercharges on individual CFT states in the NS sector is: (cid:15) − p + 12 k L ≥ j L , (2.9)from which a bound for black hole energy and charges follows: E − P + 12 k L ≥ J L . (2.10)Microscopic states whose quantum numbers saturate the inequality (2.9) are called chiralprimaries. Unitarity further requires that chiral primaries have 0 ≤ j L ≤ k L [34, 35].Saturation of the inequality (2.10) is a necessary condition for a supersymmetric blackhole but it is not sufficient. Indeed, the black hole entropy formula (2.8) does not makesense unless [37]: 12 ( E − P ) ≥ k L J L . (2.11)A hypothetical black hole solution that violates this inequality would have event horizonwith imaginary area. Such geometries are not regular so black holes with these quantumnumbers simply do not exist. This regularity condition is variously referred to as the cosmiccensorship bound or the condition for absence of closed time-like curves.The BPS condition demands that the inequality (2.10) be saturated but then compat-ibility with regularity (2.11) gives J L = k L . (2.12)This is the charge constraint on BPS black holes in AdS advertised in the introduction(1.5). Thus BPS black holes have the same quantum numbers as the particular chiralprimaries situated in the middle of the interval 0 ≤ j L ≤ k L allowed by unitarity. In the previous subsection we established that BPS black holes in AdS are co-dimension2 in parameter space: saturation of two inequalities (2.10-2.11) introduces two relationsbetween the four parameters E , P , and J R,L . In this and the next subsection we elaborateon this property from a thermodynamic point of view.In discussions of black holes two notions of “ground state” appear: • Extremality : the temperature T = 0. • Supersymmetry : the BPS inequality for the energy is saturated.These conditions are similar in that both determine the black hole energy in terms ofits charges. However, they are not at all equivalent. On the contrary, it may be useful to– 7 –nterpret them as two complementary requirements that each imposes one relation betweenthe black hole parameters. The supersymmetric black holes are co-dimension 2 in parameterspace because of these two conditions. The two concepts of ground state can be applied in either order. In the previoussubsection our starting point was the supersymmetry algebra:1. Supersymmetry gives the BPS condition E = P + J L − k L that determines theenergy E in terms of conserved charges. In CFT terminology the eigenvalue of L is J L .2. Among configurations with charges that satisfy the BPS formula for the energy, aregular black hole exists only if, in addition, the extremality condition T − = β = (cid:18) ∂S∂E (cid:19) P,J
L,R = k L π (cid:113) k L ( E − P ) − J L + k R π (cid:113) k R ( E + P ) − J R → ∞ , is met. This is only possible when the charges are further restricted to J L = k L .From this point of view the second condition on charges is “additional” and perhaps sur-prising. However, thermodynamic reasoning suggests that we impose extremality first:1. The lowest possible energy allowing a regular black hole geometry for given conservedcharges ( P, J
L,R ) is the extremal energy E ext .2. Considering only extremal black holes, we further require that the geometry permitssupersymmetry: a spacetime Killing spinor must exist. This imposes an independentconstraint on the charges.From the thermodynamic point of view it is supersymmetry that imposes an additionalcondition on the charges that may appear surprising. In the next subsection we willimplement the BPS limit with extremality imposed first. In particular, we will derive thetwo inequalities (2.10-2.11) defining the BPS limit from the general partition function (2.2). Recall the formulae (2.5-2.7) that relate the quantum numbers to potentials, reproducedhere for convenience: E = k R β (1 − µ ) (cid:0) π + β ω R (cid:1) + k L β (1 + µ ) (cid:0) π + β ω L (cid:1) , (2.13a) P = k R β (1 − µ ) (cid:0) π + β ω R (cid:1) − k L β (1 + µ ) (cid:0) π + β ω L (cid:1) , (2.13b) J L,R = 2 k L,R ± µ ω L,R . (2.13c)In the canonical ensemble the extremal limit amounts to vanishing temperature β → ∞ .However, we must be careful with what remains finite in this limit. In this paper we just consider conditions on continuous black hole parameters. There are also importantdiscrete distinctions that must be made, such as the ones defining the nonBPS branch [38]. – 8 –onsider a pair of particular combinations of these charges: E + P − J R k R = 2 k R π β (1 − µ ) ≥ , (2.14a) E − P − J L k L = 2 k L π β (1 + µ ) ≥ . (2.14b)If one na¨ıvely takes β → ∞ with the chemical potential µ finite and generic, both of theseinequalities will be saturated. However, when the expressions on the left hand sides of bothequations in (2.14) vanish, the black hole entropy (2.8) will be zero as well. Therefore,the limit taken this way yields an extremal “black hole” with an event horizon that hasvanishing area. Such a geometry is singular, it is not a black hole solution.In order to circumvent this obstacle, we need to saturate only one of the inequalities(2.14). We pick the latter without loss of generality, because this choice is analogous to theone leading to (2.11). Accordingly, we take β → ∞ while rescaling µ so that ˜ µ ≡ β ( µ − µ ≤ µ ≤
1. It further follows from (2.13c) that, inorder to describe black holes with generic values of J R , we must further take ω R → ω R ≡ βω R is also kept finite. In contrast, ω L does not require any rescaling, it can bekept finite by itself.In summary, the extremal limit of a general AdS black hole is:Extremal limit: β → ∞ ,µ → µ ≡ β ( µ −
1) finite, ω R → ω R ≡ βω R finite, ω L finite. (2.15)This limit was designed so that (2.13) gives expressions that are finite: E = k R ˜ µ (cid:0) π + ˜ ω R (cid:1) + k L ω L , (2.16a) P = k R ˜ µ (cid:0) π + ˜ ω R (cid:1) − k L ω L , (2.16b) J R = − k R ˜ µ ˜ ω R , (2.16c) J L = k L ω L . (2.16d)The explicit sign in the formula for J R compensates ˜ µ < J R has the same sign as the rescaled angular velocity ˜ ω R , as expected. These formulae forthe conserved charges give the energy as a function of the charges E ext = P + 12 k L J L . (2.17)This is the ground state energy for these conserved charges. It saturates (2.11) and is– 9 –dentified with the extremal black hole mass. The extremal entropy becomes S ext = − k R π ˜ µ = 2 π (cid:114) k R ( E ext + P ) − J R = 2 π (cid:114) k R P + k R k L J L − J R . (2.18)The last equation eliminated the energy using the extremality condition (2.17).As we have stressed, the extremal black holes are not necessarily supersymmetric. Asthe second and last step of implementing the BPS limit, we now examine supersymmetry.Recall from (2.10) that charges of supersymmetric black holes must saturate the inequality E − P − J L + 12 k L ≥ . The left hand side can be recast as a sum of two squares E − P − J L + 12 k L = 2 k L π β (1 + µ ) + k L − ω L µ ) , (2.19)using (2.13). The first square is precisely (2.14b) so it vanishes in the extremal limit.In order to saturate the BPS bound (2.10) the second square must vanish as well so wedemand that the potentials satisfy ϕ ≡ µ − ω L = 0 , (2.20)in addition to conditions for extremality. We defined the parameter ϕ for future use. Since µ = 1 at extremality we must have ω L = 1 in the BPS limit. However, just as the extremallimit is taken with ˜ µ ≡ β ( µ −
1) kept finite there is no obstacle to taking the BPS limit ω L → ω L ≡ β ( ω L −
1) remains finite. The value of ˜ ω L is, like ˜ µ and ˜ ω R , not constrained.To summarize, the BPS AdS black holes are limits of generic AdS black holes as T = β − → , (2.21)while the potentials ˜ µ = β ( µ − , ˜ ω R = βω R , ˜ ω L = β ( ω L − , (2.22)are kept finite. In this limit two inequalities (2.10) and (2.11) are saturated.The BPS limit of the extremal expressions (2.16) gives E = k R ˜ µ (cid:0) π + ˜ ω R (cid:1) + k L , (2.23a) P = k R ˜ µ (cid:0) π + ˜ ω R (cid:1) − k L , (2.23b) J R = − k R ˜ µ ˜ ω R , (2.23c)and notably, J L = k L . (2.24)– 10 –he extremal black hole entropy (2.18) also simplifies further in the BPS limit S BPS = 2 π (cid:114) k R ( P + 14 k L ) − J R . (2.25)The four macroscopic quantities E, P, J
L,R are parametrized by only two potentials ˜ µ and ˜ ω R , they are independent of the third potential ˜ ω L . This confirms the expectationthat the parameters of a BPS black hole form a co-dimension 2 surface in the space of allpossible charges. On the other hand, there really are three independent rescaled potentials˜ µ, ˜ ω L,R . This is possible because ˜ ω L parametrizes a flat direction along which the BPSblack hole does not change. We now implement the BPS limit discussed in the previous subsection on the partitionfunction rather than the macroscopic variables.As before, we first take the extremal (zero temperature) limit β → ∞ in the mannerspecified in (2.15). The trace (2.1) that defines the partition function becomes Z = Tr e − β ( (cid:15) − p )+˜ µp +˜ ω R j R + βω L j L . (2.26)This expression is schematic because β appears explicitly even though we take β → ∞ .However, it captures an important qualitative feature of the physics. Disregarding tem-porarily the term βω L j L (which will be addressed below), as β → ∞ the first term in theexponent assures that only states with (cid:15) = p contribute insofar as such states exist andthey are separated from the states with (cid:15) > p by a gap. The states singled out this waywill be the BPS states, except for the proviso that we have yet to account for the term βω L j L .To do so we proceed and implement the second part of the BPS prescription (2.21-2.22) which specifies the BPS energy. It is taken into account by rewriting the extremalpartition function (2.26) as Z = e βk L Tr e − β ( (cid:15) − p − j L + k L )+˜ µp +˜ ω R j R +˜ ω L j L = e βk L Tr e − β (˜ L − j L ) e ˜ µp +˜ ω R j R +˜ ω L j L . (2.27)In the second expression we introduced ˜ L − k L = ( (cid:15) − p ) and reorganized in order toisolate the term β ( ˜ L − j L ) in the exponent which, because the limit β → ∞ is implied,singles out the chiral primary states annihilated by ˜ L − j L . We assume that such statesare separated by a gap from the states where ˜ L − j L is positive and unitarity ensuresthat this operator cannot be negative. Thus the partition function receives contributionsonly from the chiral primaries, precisely the states that preserve supersymmetry.The overall factor e βk L in (2.27) diverges as β → ∞ but, because no other potentialenters, it does not depend on the state. This term incorporates the supersymmetric Casimirenergy [39] E SUSY = − k L , (2.28)– 11 –hat is common to all states. Note that it is not the regular Casimir energy E C = − ( k L + k R ) that enters here and the two notions of Casimir energy agree only when the levels k L = k R . The Casimir energy appears explicitly because we study the partition function defined as a path integral rather than as a trace over a Hilbert space normalized such thatthe vacuum contributes unity.It is the convention in CFT that the Virasoro generators L , ˜ L annihilate the SL (2) invariant (NS-NS)-vacuum which, therefore, is assigned a negative Casimir energy E C = − ( c R + c L ) = − ( k R + k L ). This usage has been adopted in discussions of AdS /CFT correspondence. The supersymmetric Casimir energy (2.28) is a variant that is betterprotected by supersymmetry, but it follows the same conventions. In contrast, in thecontext of black holes in higher dimensional AdS spaces, it is customary to assign mass M = 0 to the AdS vacuum. Adaptation of our AdS treatment to this practice amountsto defining the BPS black hole mass as M = E − E SUSY = P + J L . (2.29)This simple linear formula, with numerical value “1” in front of each quantum number P and J L , is the AdS version of the standard supersymmetric mass formulae for supersym-metric black holes in AdS , , , .Taking the extremal limit (2.15) explicitly on the general partition function (2.2) wefind ln Z ext = − k R ˜ µ (cid:0) π + ˜ ω R (cid:1) + 12 k L ( β −
12 ˜ µ ) ω L . (2.30)We retained the divergent linear-in- β term which encodes the supersymmetric Casimirenergy but does not contribute to the entropy. Other terms were computed by expandingfor small temperature and retaining the terms that are finite in the extremal limit. Theextremal partition function (2.30) simplifies further in the BPS limitln Z BPS = 12 k L β − k R ˜ µ (cid:0) π + ˜ ω R (cid:1) + k L (cid:18) ˜ ω L −
14 ˜ µ (cid:19) . (2.31)This BPS partition function reproduces the formulae for BPS limits of macroscopic charges(2.23-2.24). For example, the potential ˜ ω L now appears entirely as a linear term that givesthe correct value J L = ∂∂ ˜ ω L ln Z BPS = k L . (2.32) In the previous section we discussed black hole thermodynamics with the partition functionas starting point, as in conventional thermodynamics. However, recent progress on BPSblack holes in AdS with dimensions larger than three is based on the superconformal index.Therefore, in this section, we study the thermodynamics of BTZ black holes on the basis ofthe supersymmetric index. In particular, we develop an entropy extremization prescriptionfor BTZ black holes that mimics its analogues in the literature on higher dimensional cases[11]. – 12 – .1 The Partition Function and the Index
The grand canonical partition function was defined in (2.1), as a trace over all states: Z = Tr e − β ( (cid:15) − µp − ω R j R − ω L j L ) . (3.1)In subsection 2.6 we isolated the BPS states by taking β → ∞ with certain rescaledpotentials (identified by their tilde) kept finite. This gave the BPS partition function(2.27): Z BPS = e βk L Tr e − β (˜ L − j L ) e ˜ µp +˜ ω R j R +˜ ω L j L (cid:12)(cid:12)(cid:12) β →∞ = e βk L (cid:12)(cid:12)(cid:12) β →∞ Tr BPS e ˜ µp +˜ ω R j R +˜ ω L j L . (3.2)The limit β → ∞ ensures that only the chiral primaries contribute to the trace since theoperator ˜ L − j L vanishes exactly on those and is positive on others. Equivalently, thetrace is taken only over the chiral primaries (BPS states) in the second line.In this section we study the supersymmetric index, also known as the elliptic genusin CFT , rather than the partition function. As usual, the index is the general partitionfunction (3.1), except for insertion into the trace of a sign ( − F that depends on thefermion number F . The goal is that when the supercharge Q that defines the BPS sectordoes not annihilate some state | ψ (cid:105) , it creates a nontrivial partner Q| ψ (cid:105) that cancels theoriginal state | ψ (cid:105) in the trace, because the two members of the pair are counted withopposite signs ( − F . The general partition function (3.1) with ( − F inserted shouldtherefore receive contributions only from states that are annihilated by Q and so reduceto the BPS partition function (3.2), also with ( − F inserted.However, for the two members of each pair to cancel properly, they must have the samefugacities, their weight depending on the potentials with tilde must be the same. This canbe arranged by considering only fugacities that satisfy the constraint β (1 + µ − ω L ) = ˜ µ − ω L = 0 , (3.3)which commutes with the supercharge Q in the anti-holomorphic ( L ) sector. More con-cisely, the insertion of ( − F and the requirement ˜ µ − ω L = 0 can be elegantly combinedas the complex constraint ˜ µ − ω L = 2 πi , (3.4)on the potentials. With this constraint the general partition function (3.1) automaticallyreduces to the BPS partition function (3.2). In particular, the dependence on β disappears,except for the factor e βk L that accounts for the supersymmetric Casimir energy. It isconventional to omit this overall factor from definitions of supersymmetric indices, or ofelliptic genus.To summarize, I ≡ e βE SUSY Z (cid:12)(cid:12)(cid:12) ˜ ω L = ˜ µ − iπ = Tr BPS e ˜ µp +˜ ω R j R +˜ ω L j L (cid:12)(cid:12) ˜ ω L = ˜ µ − iπ = e βE SUSY Z BPS (cid:12)(cid:12)(cid:12) ˜ ω L = ˜ µ − iπ , (3.5)– 13 –here E SUSY = − k L was given in (2.28). Going from the first to the third line is non-trivial, it is valid because the aforementioned cancellations within pairs allow one to restrictthe trace to BPS states. In other words, the index is independent of β , as expressed by thesecond line of (3.5), so β → ∞ is not needed in its definition.The BPS partition function Z BPS depends on three independent potentials: ˜ µ and˜ ω L,R , apart from the formal e − βE SUSY (cid:12)(cid:12) β →∞ factor. Since the dependence on ˜ ω L can beeliminated by the complex constraint (3.4), the index depends on only two independentparameters which we take as ˜ µ and ˜ ω R .We can compute the index for supersymmetric black holes in AdS explicitly by startingfrom the general partition function (2.2), introducing tilde potentials through (2.22), andthen imposing the index constraint (3.4):ln I = − k L β + k R β (1 − µ ) (cid:0) π + β ω R (cid:1) + k L β (1 + µ ) (cid:0) π + β ω L (cid:1) = − k L β − k R ˜ µ (cid:0) π + ˜ ω R (cid:1) + k L ˜ µ + 2 β (cid:0) π + (˜ ω L + β ) (cid:1) = − k R ˜ µ (cid:0) π + ˜ ω R (cid:1) + k L µ − πi )= − k R ˜ µ (cid:0) π + ˜ ω R (cid:1) + k L ˜ µ ( π + ˜ ω L ) . (3.6)We present the manipulations in detail to highlight that they are exact and that thedependence on β disappears without any limit taken, as anticipated. The final expressionwith the constraint (3.4) implied agrees with the BPS partition function (2.31), again asanticipated. A simpler but less illuminating route to the formula for the index given in thelast line of (3.6) is to evaluate the partition function and take the high temperature limit β → β = 0.The computation illustrates how the index (3.5) and the BPS partition function (2.31)are closely related, yet they are different in significant ways such that they complementone another: • The BPS partition function restricts the trace to the chiral primary states by anexplicit limit β → ∞ . In contrast, the index is independent of β , the limit β → ∞ ispossible but not mandatory. This is one aspect of the index being protected undercontinuous deformations of the theory, while the BPS partition function is not. • The supersymmetric index is defined not only by an insertion of ( − ) F , its fugacitiesmust be constrained by (3.4) or else it is not protected under continuous deforma-tions. In contrast, the BPS partition function keeps all three potentials ˜ µ and ˜ ω R,L independent. It is possible to focus on variables that satisfy the constraint, but thegeneral case incorporates more information about the theory. • The supersymmetric index is defined with the supersymmetric Casimir energy strippedoff, while the partition function retains it.– 14 –hese distinctions between the supersymmetric index and the BPS partition function arecentral to this paper.In the non-chiral case k L = k R = k we can recast our result for the index (3.6) asln I = k ˜ ω ˜ ω ˜ µ , (3.7)by choosing the basis ˜ ω L,R = (˜ ω ± ˜ ω ) for the potentials. This result is reminiscent ofthe HHZ free energy that plays a central role in discussions of black hole entropy in higherdimensional AdS spaces. For example, in AdS /CFT [11],ln Z = 12 N ˜∆ ˜∆ ˜∆ ˜ ω a ˜ ω b . (3.8)The three potentials ˜∆ I ( I = 1 , ,
3) for R-charges in the higher dimensional setting (rota-tion on S ) are analogous to ˜ ω , for R-charges in CFT (rotation on S ). The rotationalvelocities ˜ ω a,b (not to be confused with ˜ ω , in (3.7)) in AdS correspond to the potentialfor angular momentum ˜ µ in AdS . The overall coefficient k is two times the Casimir energyin AdS while N is two times the Casimir energy in AdS .We interpret our result for the supersymmetric index (3.6) as the HHZ free energyin AdS . It is more general than the version (3.7) that is more directly analogous tothe HHZ formulae in higher dimensions, because it includes the non-chiral case k R (cid:54) = k L .In each dimension, the index nature of the HHZ free energy requires imposing a linearconstraint between the complexified potentials: the 3D free energy (3.6) satisfies (3.4) andthe constraint ˜∆ + ˜∆ + ˜∆ − ˜ ω a − ˜ ω b = 2 πi , is imposed on the 5D free energy (3.8). In the AdS example we can make completelyexplicit the distinction between the HHZ free energy (3.6) and the BPS partition function(2.31) that depends on unconstrained potentials. This comparison also highlights the roleof the supersymmetric Casimir energy. Whereas we have derived the supersymmetric index (3.6) for AdS black holes by imposinga complex condition (3.4) on the more general BPS partition function, in higher dimensionalAdS spaces it is only the index that can be reliably computed. In that context a procedureto extract the entropy and the charge constraint of supersymmetric black holes directlyfrom the index has been developed [11]. In this subsection we apply this procedure to theAdS case and show that it reproduces the results derived from the BPS partition functionin section 2.The claim that is now standard in higher dimensional AdS spaces is that we can processthe index as if it was an ordinary free energy. It is with this procedure in mind that wehave referred to the (logarithm of the) index as the HHZ free energy. According to thisprescription, the black hole entropy is given by the Legendre transform of the index (3.6),– 15 –ubject to the complex constraint (3.4). Following [11], it can be computed efficiently byextremizing the entropy function S [˜ µ, ˜ ω R , ˜ ω L ] = k L (cid:0) ˜ ω L + π (cid:1) − k R (cid:0) ˜ ω R + π (cid:1) ˜ µ − ˜ ω L J L − ˜ ω R J R − ˜ µP − Λ(˜ µ − ω L − πi ) , (3.9)with respect to the potentials ˜ µ , ˜ ω R,L and the Lagrange multiplier Λ that enforces thecondition (3.4).The entropy function is homogeneous of degree one in the potentials ˜ µ , ˜ ω R,L , exceptfor 2 πi Λ which is constant, and for the terms proportional to π which are homogeneous ofdegree minus one. Keeping track of the inhomogeneous terms, the extremization conditionsgive 0 = (˜ ω L ∂ ˜ ω L + ˜ ω R ∂ ˜ ω R + ˜ µ∂ ˜ µ ) S = S − πi Λ + 2 π ( k R − k L )˜ µ , so that S = 2 πi Λ − π ( k R − k L )˜ µ . (3.10)The second term vanishes when k R = k L but otherwise not. It represents a novel refinementwhen compared to analogous computations in higher dimensional AdS spaces.The individual entropy extremization conditions are ∂ ˜ ω L S = k L ω L ˜ µ + (2Λ − J L ) = 0 , (3.11a) ∂ ˜ ω R S = − k R ω R ˜ µ − J R = 0 , (3.11b) ∂ ˜ µ S = − k L (cid:0) ˜ ω L + π (cid:1) − k R (cid:0) ˜ ω R + π (cid:1) ˜ µ − (Λ + P ) = 0 . (3.11c)Using the constraint (3.4), the first equation gives k L ˜ µ − πi ˜ µ = J L − ⇒ πik L ˜ µ = Λ −
12 ( J L − k L ) . (3.12)The entropy function therefore becomes S = 2 πi (cid:20) Λ + iπ ˜ µ ( k R − k L ) (cid:21) = 2 πi (cid:20) k R k L Λ − k L ( k R − k L )( J L − k L ) (cid:21) ≡ πi Λ eff , (3.13)where we defined Λ eff = k R k L Λ − k L ( k R − k L )( J L − k L ) . (3.14)Rewriting the last extremization condition (3.11c) using the others (3.11a-3.11b) and theexpression for ˜ µ (3.12) we find − k L (Λ − J L ) + 14 k R J R − (Λ + P ) − k L ( k R − k L )(Λ −
12 ( J L − k L )) = 0 , (3.15)which we reorganize into a quadratic equation for Λ eff :Λ − ( J L − k L )Λ eff + 14 ( J L − k L ) + k R ( P + J L − k L − J R = 0 . (3.16)– 16 –electing the root with negative imaginary part we find the extremized entropy functionin terms of charges: S = 2 πi Λ eff = 2 π (cid:114) k R ( P + J L − k L − J R πi ( J L − k L ) . (3.17)For BPS black holes in higher dimensional AdS the standard prescription posits thatcharges must be constrained such that the extremized entropy function is real [11, 13].Applying this rule in AdS as well we find J L = k L , in agreement with the charge constraint (2.24) that we inferred from gravitational consid-erations. After fixing the charges this way, the entropy function (3.17) is real with thevalue S BPS = 2 π (cid:114) k R ( P + 14 k L ) − J R , (3.18)in agreement with the entropy (2.25) of a BPS black holes in AdS .In summary, in this subsection we applied the entropy extremization procedure torecover thermodynamic properties from the supersymmetric index (3.6). The computationis novel in that the index (3.6) used here is more refined than the version (3.7) that isdirectly analogous to higher dimensional cases, as explained at the end of subsection 3.1. The result of entropy extremization agrees with the gravitational side for the BPS blackholes in AdS discussed here, as it does for their analogues in AdS , , , . However, in allthese cases it is not entirely clear why the procedure works. In particular, it is somewhatmysterious how the reality condition on the entropy function gives the charge constraintobeyed by BPS black holes. In this subsection we address this question in the AdS context.In order to understand the reality condition on the entropy function, recall how complexnumbers enter in the first place. We compute the supersymmetric index from the BPSpartition function in (3.5), by imposing the complex constraint (3.4) on the potentials: I = Tr BPS e ˜ µp +˜ ω R j R +˜ ω L j L (cid:12)(cid:12) ˜ ω L = ˜ µ − iπ = Tr BPS e − iπj L e ˜ µ ( p + j L )+˜ ω R j R . (3.19)However, despite the appearance of a complex constraint, the index remains real as long asall potentials other than ˜ ω L remain real, because the R-symmetry quantum number j L isquantized as an integer. This, of course, is unsurprising since the complex number simplyencodes the real grading ( − F .Entropy extremization computes degeneracies (with negative signs for fermions) d = e S for states with specified quantum numbers from the index through a Legendre transform.Schematically for a system with one quantum number j and chemical potential ˜ ω we have I = (cid:88) j ( − F ( j ) d ( j ) (cid:0) e ˜ ω (cid:1) j ⇔ ( − F ( j ) d ( j ) = (cid:73) d ˜ ω πi e log I− ˜ ω ( j +1) , (3.20)– 17 –nd entropy extremization amounts to computing the contour integral from a saddle point.However, this procedure does not introduce any genuinely complex numbers. We alreadynoted that the index is real and the resulting degeneracies d ( j ) must also be real, bydefinition. Indeed, that is what our explicit result for the entropy function (3.17) shows:although πi ( J L − k L ) is complex, this term simply accounts for fermion statistics e πi ( J L − k L ) because J L − k L ∈ Z . Thus the imaginary part of the entropy function has a perfectlyacceptable physical interpretation and so there is no good reason a priori to demand thatit vanish. It is puzzling, then, that the charge constraint required for regularity of the blackhole geometry is precisely equivalent to reality of the entropy function.Our resolution of the puzzle is that the charge constraint originates from the BPSpartition function which, as we stressed in subsection 3.1, contains more information thanthe index. However, due to a particular property of the BPS partition function (2.31), theindex inherits the data needed to infer the charge constraint.To see this, consider the entropy function (3.9) that we extremized in subsection 3.2,written in terms of the BPS partition function: S [˜ µ, ˜ ω R ] = (ln Z BPS − ˜ µP − ˜ ω R J R − ˜ ω L J L ) | ˜ ω L = ˜ µ − iπ . Here we explicitly substitute ˜ ω L = ˜ µ − iπ , rather than employing a Lagrange multiplier.Also, we omitted the supersymmetric Casimir energy for clarity, as it is immaterial to ourargument. Extremization of the entropy function over ˜ µ gives dd ˜ µ (ln Z BPS ) | ˜ ω L = ˜ µ − iπ = ∂∂ ˜ µ ln Z BPS + 12 ∂∂ ˜ ω L ln Z BPS = P + 12 J L . (3.21)This reproduces the standard formulae for macroscopic charges P and J L in the canonicalensemble, but only for the combination P + J L . The outcome that only one combinationof P and J L appears is expected because, as seen in (3.19), the index does not distinguishthe two charges P , J L , it only depends on their combination P + J L . However, we foundin (2.32) that the charge constraint J L = k L originates from averaging over the j L quantumnumber alone. In other words, the charge constraint follows from separating (3.21) intotwo independent equations, one for P and another for J , a step that is usually not justified.However, the situation at hand is special, because the BPS partition function (2.31),and so the entropy function S = ln Z BPS − ˜ µP − ˜ ω R J R − ˜ ω L J L , are linear functions of ˜ ω L ,and also real functions of all other potentials. Therefore, provided that ˜ µ and ˜ ω R are realand ˜ ω L = ˜ µ − iπ is the only source of complex numbers,Im(ln Z BPS ) = (Im ˜ ω L ) · ∂∂ ˜ ω L ln Z BPS , ⇒ Im S = (Im ˜ ω L ) · (cid:18) ∂∂ ˜ ω L ln Z BPS − J L (cid:19) . (3.22)The requirement that S be real givesIm S = 0 ⇔ ∂∂ ˜ ω L ln Z BPS − J L = 0 , (3.23)– 18 –hich becomes the charge constraint J L = k L . This is how, upon introduction of complexnumbers via ˜ ω L = ˜ µ − iπ , reality of the entropy function mimics extremization with respectto a potential that is an independent variable only in the BPS partition function and notin the index.To summarize, the BPS charge constraint J L = k L is a piece of information that iscontained in the partition function (2.31) but not in the index (3.6), because dependenceon two potentials ˜ µ and ˜ ω L are lumped together in the index. It is only because theBPS partition function is i) a real function of all potentials and ii) linear in ˜ ω L , that thedependence on ˜ ω L alone can be extracted from the index, as it is encoded in the imaginarypart. Were it not for these features, a principled derivation of the charge constraint wouldfollow only from the BPS partition function and not from the index, which depends on onefugacity less.It is unclear if the analogous mechanism applies to asymptotically AdS BPS blackholes where, in fact, the correct charge constraint can be derived by demanding reality ofthe entropy function. Recent progress on the superconformal index of the dual N = 4Super-Yang-Mills theory relies heavily on the modified index [13, 24, 29] where the role of( − F is played by e iπr with r the U (1) R-charge of 4D N = 1 theory. The modified indexis a Witten index that counts only -BPS states and exhibits deconfined behavior for somecomplex phases of the fugacities. However, it is no longer a manifestly real function evenwhen the fugacities are real, because r is not integer-quantized. In this situation realityof the extremized entropy function is not an a priori principled way to extract additionalinformation from the index. The entropy function is constructed from the index, yet it encodes data characterizing blackholes that are not even BPS. In this subsection we illustrate this claim and, in the process,develop a spacetime interpretation of the potentials that extremize the entropy function,following analogous computations for black holes in higher dimensional AdS spaces [31, 40].The value of the potential for 3D angular momentum at the extremum of the entropyfunction was determined in (3.12):˜ µ = 2 πik L − ( J L − k L ) = − πk R (cid:113) k R ( P + k L ) − J R . (3.24a)In the second equation we first take J L = k L satisfied by BPS black holes and then thedenominator Λ becomes purely imaginary with value given implicitly in (3.15). We chooseits sign consistently with (3.17) and with ˜ µ <
0. The constraint (3.4) and the extremizationcondition on ˜ ω R (3.11b) then easily give˜ ω L = 12 (˜ µ − πi ) = − πk R (cid:113) k R ( P + k L ) − J R − iπ , (3.24b)˜ ω R = − J R k R ˜ µ = πJ R (cid:113) k R ( P + k L ) − J R . (3.24c)– 19 –hese potentials are real, except for the imaginary part of ˜ ω L which implements the bound-ary condition needed for the index. They are derived from the index, an object protected bysupersymmetry, yet their real parts can be identified with physical potentials in spacetime[11]. More precisely, they correspond to features of the potentials that break supersymme-try. In order to establish this we adapt the nearAdS attractor mechanism known in higherdimensions [41, 42] to BTZ black holes [43, 44]. Accordingly, consider a general asymptot-ically AdS geometry of the form ds = − r − r (cid:96) R ( r ) dt + (cid:96) r r − r dr + R ( r ) (cid:18) dφ + µ ( r ) (cid:96) dt (cid:19) , (3.25)where the function R ( r ) ∼ r for large r to ensure the correct asymptotics. The BTZblack hole at hand is the special case where r = ( r − r − ) and the functions specifyingthe geometry are R ( r ) = r + 12 ( r + r − ) ,µ ( r ) = r + r − R ( r ) , (3.26)in terms of the parameters r ± that are related to physical black hole variables as M = r + r − G (cid:96) ,P = r + r − G (cid:96) . (3.27)We denote 3D angular momentum by P to conform with notation elsewhere in this article.Regularity of the Euclidean geometry at the horizon r = r determines the tempera-ture of any black hole of the form (3.25) as T = r π(cid:96) R ( r ) . In the extremal case r = 0 and the inner and outer horizons coincide at r = 0, but atnon-zero temperature they move to ± r , respectively. The associated entropy change isentirely captured by the increase in “area” due to the event horizon moving outwards by∆ r = r : ∆ S = 14 G · π∂ r R (cid:12)(cid:12)(cid:12)(cid:12) r = r ∆ r = π (cid:96) G ∂ r R (cid:12)(cid:12)(cid:12)(cid:12) r = r T .
The BTZ black hole (3.26) has ∂ r R = 1 so the near extremal heat capacity is linear intemperature C T ∼ T with constant of proportionality C T T = π (cid:96) G = π k L (cid:96) , (3.28) The standard radial coordinate for the BTZ black hole is r = R . The shifted radial coordinatehere is a close analogue of the radial coordinate that is appropriate in higher dimensional cases. – 20 –here we used the Brown-Henneaux formula (cid:96) G = k L [45] for excitations of a BPS blackhole with its L -sector in the ground state. Similarly, the dimensionless 3D rotational velocity (3.26) is µ (0) = 1 for the BPS blackhole where R = r = r − . For a nearBPS black hole it is changed by∆ µ = − r + r − R ∂ r R (cid:12)(cid:12)(cid:12) r = r ∆ r = − π(cid:96) R T . (3.29)This contribution is negative because the nearBPS rotational velocity is below the speedof light. The “area” of the event horizon is 2 πR so we can rewrite the rescaled potential(2.22) in terms of the BPS entropy and find˜ µ = ∆ µT = − π (cid:96)k R S BPS . (3.30)We used the Brown-Henneaux formula (cid:96) G = k R for BPS states preserving the L -sectorground state. The result agrees in the unit (cid:96) = 1 with (3.24a) from entropy extremization,given (2.25), as expected.We defined both the specific heat (3.28) and the nearBPS rotational velocity (3.30)as response coefficients for the black hole becoming near-extremal, by adding a small tem-perature. However, the computation in this subsection shows that we can equally inter-pret these parameters as characterizing the BPS black hole, albeit slightly away from itsevent horizon. This is the situation described in low energy effective field theory by thenAdS /nCFT correspondence and seems like the most appropriate for discussions of theindex. The thermodynamics of black holes in AdS spacetimes sheds light on the phase diagram ofgauge theories (and their relatives) at strong coupling [3, 47]. This relation is interestingeven for BPS black holes described by an index, despite the protection against phase tran-sitions due to supersymmetry. For example, interpreting the index as a conventional freeenergy gives, for BPS black holes in AdS , a phase diagram that is surprisingly similar tothat of the Schwarzschild-AdS black hole [29, 48]. In this subsection we give a perspectiveon such higher dimensional BPS phase diagrams by discussing their analogue in AdS .The BPS partition function (2.31) gives the free energy in the BPS limit as W = − ln Z BPS = − k L β − k L ˜ ω L + k R ˜ µ (cid:0) π + ˜ ω R (cid:1) + 14 k L ˜ µ . (3.31)We define the BPS free energy without the factor 1 /β appearing in standard thermody-namics. Local thermodynamic stability can be probed by the compressibility matrix K ij = − (cid:18) ∂ W ∂ Φ i ∂ Φ j (cid:19) T , (3.32) The refinements needed to distinguish between k L and k R in AdS were discussed in [46]. – 21 –here { Φ i } = { ˜ µ, ˜ ω R , ˜ ω L } collectively refer to the potentials. The potential ˜ ω L parametrizesa direction that decouples and is entirely flat. The remaining two directions are spannedby ˜ ω R and ˜ µ , and the free energy has response coefficients − ∂ W ∂ ˜ µ ∂ W ∂ ˜ µ∂ ˜ ω R ∂ W ∂ ˜ ω R ∂ ˜ µ ∂ W ∂ ˜ ω R = − k R ˜ µ ( π + ˜ ω R ) 2 k R ˜ ω R ˜ µ k R ˜ ω R ˜ µ − k R ˜ µ . (3.33)Recalling that ˜ µ <
0, both eigenvalues of the matrix are positive. Therefore, the compress-ibility matrix is positive definite and the system is locally stable.The formula (3.31) expresses the standard Cardy asymptotics of CFT but in a no-tation that is adapted for comparison with BPS black holes in higher dimensional AdS.The linear-in- β term encodes the supersymmetric Casimir energy (2.28). Similarly, thelinear-in-˜ ω L term encodes the charge constraint J L = k L (2.32). Both of these linear con-tributions depend only on k L so they are properties of the theory rather than the state.They can be removed without losing any physical information, by Legendre transform to amicrocanonical ensemble that fixes the charges E and J L rather than the potentials β and˜ ω L . This feature shows that a linear shift in the potentials β , ˜ ω L is inconsequential so wecan remove the first two terms in (3.31) entirely, not even a constant is left behind.The remaining two terms in (3.31) are negative because ˜ µ is required to be negative,as discussed above (2.15). Apart from the sign, the potential ˜ µ can be interpreted as aninverse “temperature” T eff = − ˜ µ − . (3.34)The physical temperature vanishes, as always for BPS states, but this effective BPS tem-perature expresses the usual physical intuition that a large value corresponds to largeoccupancy numbers. The sum of the two “thermal” contributions to the free energy (3.31)are bounded from above W ≤ W max = − π (cid:112) k R k L , with equality when ˜ ω R = 0 and ˜ µ = ˜ µ HP = − π (cid:114) k R k L . (3.35)The index “HP” anticipates that we shortly interpret the special value (3.35) as theHawking-Page transition temperature.The standard modular S-transformation takes ˜ µ → π ˜ µ and, at least at a first glance,the free energy (3.31) suggests that such a high/low temperature duality could persist inthe effective description, perhaps inherited from an underlying SL (2 , Z ) symmetry andsubject to the interesting refinement that the self-dual point would have to be rescaledfrom 2 π to 2 π (cid:113) k R k L . Unfortunately, as we explain next, this suggestion does not hold upto closer scrutiny.In bulk AdS quantum gravity, modular transformation interchanges the high tem-perature black hole phase where (Euclidean) temperature is contractible with the low– 22 – igure 1 . Index-inspired free energy (3.36) as a function of effective temperature T eff = − ˜ µ − ,not drawn to scale. The red dashed line represents the thermal gas phase for lower temperature,and the large black hole phase for higher temperature. The red dot represents the Hawking-Pagetransition point between the two. temperature AdS gas phase where it is the spatial circle that is contractible. Indeed, inthe complete CFT there are infinitely many saddle points related by SL (2 , Z ) symmetry,corresponding to the thermal gas and a family of black hole images [49]. However, the freeenergy (2.2) that we study throughout this paper does not represent a complete CFT , itis just the classical contribution from a single saddle point, that of the simplest black hole.It is related to the thermal gas saddle point by the SL (2 , Z ) symmetry in the full theory,but the map is nontrivial. Duality takes ˜ ω R → πi ˜ ω R ˜ µ , flipping the sign of the term in (3.31)that is proportional to ˜ ω R . Moreover, the free energy is not invariant, its transformationadds a term proportional to ˜ ω R such that no term of this form remains, and it adds yetanother term proportional to ˜ ω L . In this way, the underlying high/low temperature dualityrelates the black hole and the thermal gas while also exchanging the L and R sectors ofthe CFT . We expect that similar mechanisms are possible in higher dimensions.The procedure followed when analyzing BPS black holes in higher dimensional AdSspaces suggests yet another perspective on the free energy (3.31). Motivated by the su-persymmetric index (3.5), we cancel the linear-in- β term that gives the supersymmetricCasimir energy but we then evaluate the linear-in-˜ ω L term by imposing the constraint(3.4). This gives the index-inspired free energy W I = k R ˜ µ (cid:0) π + ˜ ω R (cid:1) − k L ˜ µ = − k R (cid:0) π + ˜ ω R (cid:1) T eff + k L T eff , (3.36)that is an AdS analogue of the free energy taken as a basis for discussions of the confine-ment/deconfinement transition for black holes in higher dimensional AdS [29, 48]. Notethat the second term k L T eff now gives a positive contribution to the free energy.The index-inspired free energy (3.36) is plotted, in units of k , as a function of effectivetemperature T eff in Figure 1. We interpret the phase diagram in analogy with the AdS-Schwarzschild case and discussions of BPS black holes in higher dimensional AdS. The hightemperature phase where W I < W I > W I = 0 at all temperatures. The Hawking-Page transitionpoint is where the line crosses W I = 0, at the temperature corresponding to (3.35).The index-inspired free energy assigns the entire expression (3.36) to the black holewhile the BPS free energy (3.31) interprets the two last terms in (3.31) as the black holeand thermal contributions, respectively. The two approaches therefore differ physically,but they give the same transition temperature, because acting with − ˜ µ∂ ˜ µ | ˜ ω L on the freeenergy (3.31) is exactly equivalent to imposing the real part of the constraint (3.4). As insubsection 3.3 this is possible because the free energy depends linearly on ˜ ω L . In this section we generalize the description of AdS BPS black holes discussed in theprevious sections and study the thermodynamics of small deviations away from the BPSlimit. This adapts to AdS the nearBPS black hole thermodynamics in AdS , , that wasstudied in [31, 40]. The simplifications in AdS clarify their higher dimensional analogues. We first evaluate the macroscopic quantum numbers for AdS black holes slightly awayfrom the BPS limit. The organizing principle, stressed in subsection 2.4, is that BPSblack holes are co-dimension two in parameter space. The two conditions satisfied byBPS black holes were presented, by the thermodynamic interpretation in subsection 2.5,as extremality T = 0 and, in addition, the vanishing of the potential ϕ = 1 + µ − ω L introduced in (2.20). Therefore, the nearBPS regime is characterized by T and ϕ that aresmall but not necessarily zero. We take the two parameters T and ϕ to be of the sameorder in smallness: T ∼ ϕ ∼ (cid:15) (cid:28) . In the canonical ensemble, the four macroscopic charges of a generic nonBPS blackhole (2.5-2.7) are functions of four independent conjugate potentials. We can pick a basiswhere the potentials are ˜ µ = β ( µ − ω R = βω R , T , and ϕ . For given values of ˜ µ and ˜ ω R ,we now expand (2.5-2.6) to linear order in T , ϕ and find P = k R ˜ µ ( π + ˜ ω R ) − k L k L ϕ + . . . = P ∗ + k L ϕ + . . . , (4.1a) J L = k L − k L ϕ + . . . = J L ∗ − k L ϕ + . . . , (4.1b) J R = − k R ˜ µ ˜ ω R = J R ∗ . (4.1c)The dots denote terms of order O ( (cid:15) ) that we neglect. The quantities with an asterisk referto the values of the charges (2.23-2.24) in the strict BPS limit where T = 0 and ϕ = 0. The The deviations T and ϕ need not be small, the general partition function (2.2) is valid for any non-BPSblack hole. Taking them small illuminates the relation between the BPS and nearBPS regimes. Additionally,considerations for small T and ϕ are direct analogues of discussions of black holes in higher dimensionalAdS spaces. – 24 –ormulae show that, in our basis of potentials, none of the charges depend on temperature T to linear order, and J R depends on neither T nor ϕ to any order. The potential ϕ is asource for the charges but leaves fixed the combination P + J L that the index is sensitiveto. We also want to expand the energy (2.7) in T and ϕ . However, recall that, for givencharges P and J L , the energy is bounded from below by E BPS = P + J L − k L . Therefore,rather than computing the energy by itself, it is instructive to expand the excitation energy E − E BPS above the BPS bound. It vanishes at linear order but at quadratic order wefind E − E BPS = 2 k L β (1 + µ ) ( π + β ω L ) − k L ω L µ + 12 k L = 18 k L (cid:16) (2 πT ) + ϕ (cid:17) . (4.2)The formulae (4.1-4.2) characterize the low lying excitations of a BPS black hole which,by definition, is both extremal and supersymmetric. This ground state has the smallestpossible mass for its charges and, to preserve supersymmetry, the charges are constrainedby J L = k L . The formulae make explicit that these two conditions correspond to twoorthogonal directions that violate BPS-ness of the black hole: • One direction raises the temperature, so that the mass increases by k L (2 πT ) whilecharges remain unchanged. Conversely, as noted after (4.1), all charges are indepen-dent of T . • Another direction turns on the potential ϕ while maintaining zero temperature. Asa result, the charges (4.1) are shifted by terms that are linear in ϕ . The energy ofthe resulting extremal but non-supersymmetric black hole is given by (2.17), whichis higher than E BPS by E ext − E BPS = (cid:18) P + J L k L (cid:19) − (cid:18) P + J L − k L (cid:19) = 12 k L ( J L − k L ) = 18 k L ϕ , (4.3)in agreement with (4.2).Expanding the entropy (2.8) at linear order in T and ϕ gives S = − k R π ˜ µ + π k L T + . . . = S ∗ + π k L T + . . . . (4.4) Because we are also considering J L away from its BPS limit k L , the “BPS” energy E BPS is not necessarilythe energy of a BPS black hole. It is the energy of a hypothetical “black hole” that is supersymmetric butnot necessarily regular, for given charges. – 25 –he entropy has no term that is linear in ϕ , but only a term that is linear in T . This termindicates a heat capacity C T that is linear in temperature with a value C T T = π k L . (4.5)This coefficient, computed from black hole thermodynamics, agrees with the result of thenAttractor mechanism (3.28) in the unit (cid:96) = 1, which is derived directly from the geometryof the supersymmetric black hole.In the expression for the excitation energy (4.2), the heat capacity enters as a term thatis quadratic in the temperature T . Furthermore, drawing analogy between the potential ϕ π and an electric potential, we interpret the coefficient of the term quadratic in ϕ asthe capacitance. The energy formula shows that these two linear response coefficients areidentical, up to possible differences in notation and terminology. We introduce a parameter C ϕ in lieu of capacitance, in order to stress this fact: C T T = C ϕ T = π k L . (4.6)This agreement is a nontrivial consequence of N = 2 supersymmetry. For example, it isbuilt into the N = 2 superschwarzian description of the low energy excitations, i.e. thenAdS /nCFT correspondence. As a check on our computations and our understanding, we can now explicitly verify thefirst law of thermodynamics
T dS = dE − µdP − ω R dJ R − ω L dJ L = d ( E − P − J L ) − ( µ − dP − ω R dJ R − ( ω L − dJ L , (4.7)in the nearBPS regime. For variations within the BPS surface, J L ∗ = k L is constant so dJ L = 0, and d ( E − P − J L ) = 0 follows from E ∗ = P ∗ + J L ∗ − k L because k L is constant.Therefore, the first law within the BPS surface reduces to: dS ∗ = − ˜ µdP ∗ − ˜ ω R dJ R ∗ . (4.8)This is indeed satisfied by the BPS expressions (2.23-2.25): the variables S ∗ , P ∗ , and J R ∗ depend on the potentials ˜ ω R and ˜ µ only, and in such a way that the linear relation (4.8) issatisfied. Thus (4.8) parametrizes the 2D surface of BPS black holes.Taking into account the BPS surface (4.8), we can rewrite the more general first law(4.7) as an equation for excitations above the BPS surface: T d ( S − S ∗ ) = d ( E − P − J L ) − ( µ − d ( P − P ∗ ) − ( ω L − dJ L . (4.9)There is no differential dJ R because J R ∗ = J R . Variations of J R do not influence theexcitations, they correspond to motion entirely within the BPS surface. We now use (4.1)to evaluate two of the terms on the right hand side:( µ − d ( P − P ∗ ) + ( ω L − dJ L = 14 k L [( µ − − ω L − dϕ = 14 k L ϕdϕ . (4.10)– 26 –t this point we can verify that (4.2) and (4.4) satisfy the first law for excitations abovethe BPS surface (4.9): T d ( S − S ∗ ) (cid:124) (cid:123)(cid:122) (cid:125) π k L T = d ( E − P − J L ) (cid:124) (cid:123)(cid:122) (cid:125) π k L T + k L ϕ − [( µ − d ( P − P ∗ ) + ( ω L − dJ L ] (cid:124) (cid:123)(cid:122) (cid:125) k L ϕdϕ . On the gravitational side of the AdS/CFT correspondence it is natural to study thermo-dynamics in the canonical ensemble, with potentials specified and the conjugate chargesincorporated as subsidiary variables. In this subsection we first discuss the nearBPS po-tentials and then the nearBPS free energy.Inverting the relations (4.1a) and (4.1c) between (
P, J R ) and (˜ µ, ˜ ω R ) we find˜ µ = µ − µ ∗ T = − πk R (cid:113) k R ( P + k L ) − J R + O ( (cid:15) ) , (4.11a)˜ ω R = ω R − ω R ∗ T = πJ R (cid:113) k R ( P + k L ) − J R + O ( (cid:15) ) , (4.11b)where the BPS values of the potentials are µ ∗ = 1, ω R ∗ = 0 and the sign for the squareroot was chosen so ˜ µ <
0. The nearBPS corrections of order (cid:15) ∼ T ∼ ϕ are not needed.Therefore, at this order, the equations are essentially the same as the BPS relations (2.23),and they also agree with the real part of the potentials (3.24a, 3.24c) determined byextremization of the BPS entropy function, and with the value (3.30) from the spacetimesolution. However, in the nearBPS thermodynamics, terms of O ( (cid:15) ) are merely small, thestrict limit (cid:15) → ω L , using the definition of ϕ (2.20)˜ ω L = ω L − ω L ∗ T = − πk R (cid:113) k R ( P + ) k L − J R − ϕ πT · π + O ( (cid:15) ) . (4.11c)Here ω L ∗ = 1 and we used ω L < ϕ and T both vanish in the strict BPS limit (cid:15) → a priori the ratio ϕ πT can take anyvalue without obstructing BPS saturation.The index corresponds to an analytical continuation of the black hole that takes ϕ πT → i , as one can see from (3.4). In this sense the real and imaginary parts of the result (4.11c)for the potential ˜ ω L both coincide with the complex value (3.24b) that was derived byextremization of the entropy function. The agreement between imaginary parts is notvery impressive in AdS because it is very simple, ˜ µ and ˜ ω R are both independent of ϕ πT .However, analogous agreements persist in higher dimensional AdS where they are moreelaborate, with multiple potentials involved [31, 40].In the canonical ensemble all thermodynamic data — charges, energy, entropy — iscontained in Gibbs’ free energy G ≡ − β ln Z . (4.12)– 27 –n the nearBPS regime where we expand in small T , ϕ for given ˜ µ , ˜ ω R , G = − β (cid:18) − k R ˜ µ (cid:16) π + ˜ ω R (cid:17) + k L ˜ µ + 2 β (cid:16) π + (˜ ω L + β ) (cid:17)(cid:19) = G BPS − k L (cid:16) ϕ + (2 πT ) (cid:17) + . . . , (4.13)up to quadratic order in T and ϕ , and we have G BPS = − k L − k L T (cid:16) ˜ ω L −
14 ˜ µ (cid:17) + k R T ˜ µ (cid:16) π + ˜ ω R (cid:17) , (4.14)as in (2.31).Gibbs’ free energy generates extensive variables through the first law of thermodynam-ics in the form dG = − SdT − P dµ − J L dω L − J R dω R . (4.15)Note that these are potentials without tilde, before rescaling by T . For example, theentropy is given by a thermal derivative taken with fixed µ , ω L , ω R : S = − ∂ T G = − π k R µ − T + 14 k L (2 π ) T = S ∗ + π k L T , (4.16)up to linear order in T , in agreement with (4.4). In the nearBPS regime we can alsoquantify the magnitude of thermal fluctuations in the standard manner. For example, (cid:68) j L − (cid:104) j L (cid:105) (cid:69) = − ∂ ω L G = k L , with the average value of J L = (cid:104) j L (cid:105) = k L . The levels k L,R are both huge for semiclassicalblack holes, but they are finite. The relative fluctuations in the value of J L are of order ∼ k − L . In the previous sections, we reached the BPS limit of AdS black holes from a thermo-dynamic point of view and stressed that the supersymmetric limit is reached by tuning two potentials. In this section, we revisit this property of BPS black holes from a micro-scopic point of view, noting that chiral primaries are co-dimension one in parameter space.We argue that black holes are ensemble averages, effectively restricting their macroscopiccharges to that of a particular chiral primary, thus yielding a second condition on theparameters. This gives a complementary and fully microscopic understanding of the BPScharge constraint J L = k L , which was derived from the thermodynamic partition functionin section 2. We consider black holes in AdS × S described by supersymmetric CFT ’s with (4 , N = 4 superconformal algebra [50] in2D [34, 35].It is sufficient to analyze one chiral sector of the (4 ,
4) algebra, either left or right,and we denote by c = 6 k the central charge of this sector. Each unitary representation ofthe algebra is labeled by the L - and J -eigenvalues ( h, j ) of its superconformal primary,and the whole multiplet consists of the primary and its descendants. We can focus on theNS sector because representations in the Ramond sector are isomorphic through spectralflow by half-integral unit. Then there are just two types of representations: the massless (a.k.a. short) with a superconformal primary that saturates the unitarity bound h ≥ j ,and the massive (a.k.a. long) with a primary that does not. The massless multipletsare enumerated by the representation of the SU (2) R-charge in the range j = 0 , , · · · , k that fixes the conformal weight h = j . The massive multiplets only permit the range j = 0 , , · · · k − h can take any real value strictly larger than the bound h > j .Massive representations with identical j and distinct h all have the same structure so it isnot of our interest to distinguish them, they are not essentially distinct.The representations are conveniently described by their characters Tr q h y j . Note thatthe Casimir term − c in the exponent is absent by convention, it must be restored inphysical partition functions. The character formulae for the two classes of multiplets are[35]: Massive : ch h,j ( q, y ) = q h F NS ∞ (cid:88) m = −∞ (cid:16) y k +1) m + j +1 − y − k +1) m − j − (cid:17) q ( k +1) m +( j +1) m y − y − , ( j = 0 , , · · · k − χ j ( q, y ) = q j F NS ∞ (cid:88) m = −∞ (cid:32) y k +1) m + j +1 (1 + yq m + ) − y − k +1) m − j − (1 + y − q m + ) (cid:33) q ( k +1) m +( j +1) m y − y − , ( j = 0 , , · · · k ) (5.1)where F NS = (cid:89) n ≥ (cid:16) yq n − (cid:17) (cid:16) y − q n − (cid:17) (1 − y q n )(1 − q n ) (1 − y − q n ) , accounts for the action of creation operators, i.e. the negative frequency modes { G r< } and { L n< , J in< } of the four fermionic and four bosonic fields. Since the massive characterch h,j ( q, y ) depends on the conformal weight h only via q h , it is convenient to define an h -independent massive character by shifting out the h in excess of the unitarity bound j : (cid:101) ch j ( q, y ) ≡ ch h,j ( q, y ) q − h + j . (5.2)The transformation under spectral flow follows from these formulae. In particular, thesum over m in (5.1) guarantees invariance of each character under spectral flow by integral We use the Dynkin convention where the label j is always an integer and the j ’th representation hasdimension j + 1. The half-integral spin familiar from quantum mechanics is j QM = j here . We turn off a U (1) fugacity called y in [35] and the SU (2) fugacity is renamed ( e iθ ) there = y here . – 29 – : (cid:40) h → h η = h − ηj + kη j → j η = j − kη ⇔ q h y j → q h η y j η = q h ( yq − η ) j q kη y − kη . (5.3)Although massless multiplets have no continuous parameter, it is possible that a com-bination of them continuously deform into a massive multiplet, at least group theoretically.Such recombination rules are fairly simple. Notice that the massless character formula in(5.1) differ from the massive one only by the factors (1 + y ± q m + ) in the denominator.Inspecting how χ j ( q, y ) depends on j , one can see that these factors are precisely cancelledby adding four characters with different j ’s, thus yielding the mathematical identity: (cid:101) ch j ( q, y ) = χ j ( q, y ) + 2 χ j +1 ( q, y ) + χ j +2 ( q, y ) . (5.4)The identity holds literally for j = 0 , . . . , k −
2; for j = k − j + 2 isundefined but the identity is valid with this term omitted.The supersymmetric index is protected against recombinations because contributionsfrom the four massless representations on the right hand side of (5.4) cancel one another inthe index, in agreement with the vanishing result for the index of the massive representa-tions with any value of h . The BPS partition function includes all massless representationsand is not protected in this way. Given the unitary representations described by their characters, we are now ready to extractan extra constraint on macroscopic charges imposed by supersymmetry.In the N = 4 superconformal algebra, the R-symmetry is SU (2), rather than SO (2)as in N = 2, so any chiral primary with J -eigenvalue j is part of an SU (2) representationthat, in particular, contains the anti-chiral primary with J -eigenvalue − j and the same L -eigenvalue h = j as the initial chiral primary, which saturates the anti-chiral unitaritybound h ≥ − j . The anti-chiral primary is related to a state with eigenvalues ( h, j ) =( k − j, k − j ) via spectral flow and this state is itself a chiral primary. Thus a chiralprimary with R-charge j always comes in pair with another chiral primary that has R-charge 2 k − j .This pairing is easily observed in explicit expansion of the characters (5.1). For exam-ple, for k = 5, χ j =0 ( q, y ) = 1 + y q + y q + y q + y q + y q + · · · ,χ j =1 ( q, y ) = yq / + y q / + y q / + y q / + y q / + · · · , where ellipses represent terms with strictly h > j .In fact the argument is not restricted to chiral primaries, it shows that any statewith eigenvalues ( h, j ) is paired with another state with ( h + k − j, k − j ). The pairis characterized by having R-charge mirrored about k and the same conformal weight inexcess of the unitarity bound: h − j = ( h + k − j ) −
12 (2 k − j ) . – 30 –he claim can be explicitly proved using the characters (5.1). The Z exchange operationwithin the pairs corresponds to a substitution y → q − y − followed by multiplication by q k y k , because q h y j → q h ( q − y − ) j q k y k = q h + k − j y k − j . (5.5)Then one can verify that all characters (5.1) are invariant (or, even ) under this Z trans-formation: χ j ( q, y ) = χ j ( q, q − y − ) · q k y k , (cid:101) ch j ( q, y ) = (cid:101) ch j ( q, q − y − ) · q k y k , (5.6)proving that all states appear in pairs, as claimed.Provided an ensemble of microscopic states that come packaged in multiplets, macro-scopic charges are obtained by taking ensemble averages. We have seen that every statewithin any multiplet comes in a pair with another state with respective R-charges j and2 k − j . It is obvious that the ensemble average of the angular momentum turns out to be k , regardless of which and how many multiplets of each type appear in the ensemble.An important caveat in this argument is that both microscopic states within a pair mustbe weighed with equal probability within the canonical ensemble. Given the eigenvalues( h, j ) and ( h + k − j, k − j ) of the two states, this assumption translates into a relationbetween chemical potentials: τ + 2 z = 0 , (5.7)where τ and z define the canonical partition function by Z = Tr e πiτL +2 πizJ . (5.8)To see how this argument applies to microscopic accounting of BPS black holes, westart again from the definition of the partition function (2.1), as rewritten in (2.27): Z = e βk L Tr e − β ( (cid:15) − p − j L + k L )+˜ µp +˜ ω R j R +˜ ω L j L , (5.9)where we recall the definitions ˜ µ = β ( µ − ω R = βω R , and ˜ ω L = β ( ω L − L -sector where for a state with quantum numbers ( p, j L , j R )that saturates the BPS bound there is another BPS state that has quantum numbers( p − k L + j L , k L − j L , j R ). In the supersymmetric partition function we choose potentialsso − ˜ µ + 2˜ ω L = 0 , (5.10)which guarantees that the two members of the pair have the same weight. It follows thatthe contribution from the two states in the pair to the expectation value (cid:104) j L (cid:105) is k L .The discussion in this subsection is based on the partition function and we do notappeal to cancellations, unlike in the reasoning based on the index. Rather, we interpretthe splitting and joining of the BPS states in the chiral ring as a thermodynamic processwhere there are many possible values of the quantum number j L but, in the ensemblerealized by a black hole, thermodynamic equilibrium forces the macroscopic value J L = k L ,even though this is not the value in most microstates by themselves.– 31 – cknowledgements We would like to thank James Liu, Jun Nian, Leopoldo Pando Zayas, Shruti Paranjape andYangwenxiao Zeng for helpful discussions and communications. This work was supportedin part by the U.S. Department of Energy under grant DE-SC0007859.
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