aa r X i v : . [ h e p - t h ] N ov The M2/M5 BPS Partition Functions from Supergravity
Pedro J. Silva ∗ Institut de Ci`encies de l’Espai (IEEC-CSIC) and Institut de F´ısica d’Altes Energies (IFAE)UAB, E-08193 Bellaterra (Barcelona) Spain (Dated: October 2008)In the framework of the AdS/CFT duality, we calculate the supersymmetric partition function ofthe superconformal field theories living in the world volume of either
N M
N M
AdS d +1 co-ordinates, thedual SCFTs are in R x S d for d = 2 ,
5. The resulting partition function shows phase transitions,constraints on the phase space and allowed us to identify unstable BPS Black holes in the
AdS phase. These configurations should correspond to unstable configurations in the dual theory. Wealso report an intriguing relation between the most general Witten Index, computed in the abovetheories, and our BPS partition functions.
PACS numbers:
1. Introduction
Partition functions over supersymmetric states Z bps , are fascinating objects from which we can ex-tract key information on the corresponding supersym-metric theory, like for example the real number of in-dependent degrees of freedom and the structure of thevacua moduli space. They provide a handle to studyextensive properties of the theory including the de-scription of the possible different phases, the order oftheir transitions, etc.In general, the calculation of Z bps is not an easytask. If the theory happens to be strongly cou-pled, things get worse, since not even perturbativeapproaches can be applied. In the above cases, wemay try to calculate other type of objects that hope-fully are somehow similar to the unknown partitionfunction. Among these objects we have the so-calledIndices of the theory. Their construction is based onrepresentation theory such that, they are by defini-tion invariants of the couplings. Basically, the indicescount (modulo some weights) the number of short rep-resentations (or BPS rep.), that do not contribute tolong representations (general rep.) as the couplingchanges. These objects are more easy to calculate,and in certain cases, they are a good approximationto Z bps provided there are no strong cancelations onthe characteristic sum within the index.With the new models of Bager and Lambert [1],Gustavsom [2] and more recently with the ABJM pro-posal [3], there has been a revival on the research ofthe M /M ∗ E-mail address: [email protected] theories (SCFT). Unfortunately, up to date, we arenot able to calculate the corresponding supersymmet-ric Z bps , although there are some partial results andvigourous programs currently under development (see[4] and references there in). On the other hand, thereis a proposal for the most general super-conformal In-dex in four, three, five and six dimensions [5, 6]. ThisIndices have been applied to the specific superconfor-mal theories of N = 4 SU ( N ) SYM in 4D and tothe N = 6 U ( N ) × U ( N ) k-level Chern-Simon theoryof the ABJM proposal, corresponding to D M AdS / matches exactly the Index calculated on thedual D /M Z bps . It isa fact that these indices show no phase transitions asfunctions of the different fugacities (or chemical poten-tials) and therefore do not capture (at least one of themore important features), the associated supersym-metric partition function Z bps . Another perspectiveon this same result comes from the point of view ofthe dual gravity theory, where these indices are blindto Black Hole (BH) physics.The main idea of this work, is to obtain the super-symmetric partition function Z bps in the large N limitof the M /M AdS/CF T duality. To be more precise,we calculate the supergravity partition function usinga saddle point approximation on supersymmetric BHsolutions such that Z bps = e − I bps . Then, based on theabove duality, this object reproduce the correspondingsuperconformal partition function of the dual theoryin the large N limit. We work with supersymmetricM-theory BHs that are asymptotically AdS n xS − n , n = 4 ,
7, leaving other types of asymptotic behaviorfor future research.To define the supergravity partition function onBPS BH solutions, we need to calculate the super-symmetric Euclidean action I bps , in any of the follow-ing ensembles; Micro canonical, Canonical or Grandcanonical. We define I bps as the supersymmetric limitof the Euclidean action calculated on non-extremalBHs, in the Grand canonical ensemble. This approachwas defined in [8, 9], and not only provides a naturalconnection between SCFT BPS statistical mechanicsand BPS Euclidean methods in supergravity, but alsomakes connection with the attractor mechanism andthe entropy function of Sen [10]. In field theory, to define the supersymmetric limitof given partition function, at some point, we haveto use that all supersymmetric states saturate a BPSinequality. This equality translates into constraintsbetween the different labeling charges of the associ-ated Hilbert space. To illustrate a general procedureto calculate Z bps , let us consider a simple example aHilbert space characterized by only two labels, say en-ergy E and charge Q . We take the BPS bound E = Q [22]. The Grand canonical partition function Z is afunction of two potentials ( β, Ω) conjugated to (
E, Q )respectively. Define then, the left and right variables E ± = ( E ± Q ), β ± = β (1 ± Ω) such that, Z ( β, Ω) = X e − βE + β Ω Q = X e − β − E + − β + E − . (1)The supersymmetric partition function is obtainedtaking the limit β + → ∞ while β − → ξ (constant).The above limiting procedure takes T = 1 /β to zero, while Ω goes to Ω = 1 − ξT + O ( T ). Where the newsupersymmetric conjugated variable ξ corresponds tothe next to leading order in T . Note that among allavailable states, only those that satisfy the BPS boundare not suppress in the sum, giving our resulting su-persymmetric partition function Z bps = X bps e − ξQ = X Q d Q e − ξQ = X E e − ξE e S ( E ) , (2)where the first sum is over all supersymmetric states( bps ) with E = J , in the second sum we have isolatedthe multiplicity at each Q as d Q and in the third wesolved for E with S equal to the usual entropy.This limiting procedure can be implemented on su-pergravity BH solutions, by considering a careful near- to-BPS expansion of the usual Euclidean Action, po-tentials and charges (the detail explanation and exam-ples can be found in [8, 9]). From the above limitingprocedure we are able to define the Euclidean actionfor BPS BH as a function of the different fugacities ω i conjugated to the conserved charges p i . Therefore wecan write Z bps = e − I bps , I bps = X ω i p i − S ( p i ) , i = 1 , , ... (3)where Z bps stands for the saddle point approximationof the supergravity partition function. There is An-other method to calculate the same quantities, usingthe entropy function of Sen calculated in the near-horizon of the BPS BH. In this case, the differentelectric charges correspond to the fugacities, while thefunction f (Legendre transformed of the entropy func-tion on-shell) is the Euclidean action (see [10] for de-tails).With the above techniques we have been able tocompute the partition function over supersymmetricstates corresponding to the known BPS BH in AdS / .Our Z bps , shows phase transitions as function of thedifferent fugacities, contrary to the behavior of knownIndices. We found small/big BH associated to thesephase transitions, where small BH are unstable whilebig are stable. We also report on a peculiar relationbetween the partition function Z bps and the associatedIndex. In short, both objects come equipped with the same constraint among its fugacities (that also can beread as a constraint among its charges). Presently, wedo not understand this issue, but we believe it shouldplay an important role with deep implications to theunderstanding of superconformal partition function ofthe world-volume theory of the M /M
2. M-theory Black holes
The low energy effective theory of M-theory is con-jecture to be N = 1 11D supergravity. The rele-vant set up to consider the AdS/CFT duality is tofix boundary conditions such that the asymptotic be-havior on each solution is AdS d xS − d with d = 3 , N M M S − d , defining d -dimensional gauge supergravitywith R-symmetry group SO (12 − d ). This theory canbe further truncated to the maximal abelian subalge-bra of the R-symmetry, corresponding to U (1) m gaugesupergravity, where m = [13 − d/ U (1) R-charges to get the socalled minimal models. Therefore, M-theory BH so-lutions can appear at all the above different levels oftruncations. Presently, BPS BHs are known only atthe last two levels and it is not clear how generic thesesolutions are (see [15] for a study on almost all knownsolutions).To calculate the supergravity Euclidean action I bps on BPS Bh, as a function of the different fugacities( ω, φ, .. ) in the Grand canonical ensemble, we followthe procedure introduce in our previous discussions ofsupersymmetric limits in statistical mechanics. Wetake the relevant thermodynamic quantities of ourfamily of non-extremal BHs, and study the leadingand next-to-leading behavior in a near BPS expan-sion i.e. as β → ∞ . It is a non-trivial fact that we getthe following relations, E → E bps + O ( β − ) , J → J bps + O ( β − ) ,Q → Q bps + O ( β − ) , S → S bps + O ( β − ) (4)while Ω → Ω bps − wβ + O ( β − ) , Φ → Φ bps − φβ + O ( β − ) . (5)Where the ” bps ” subscript defines the correspondingsupersymmetric values of generic angular momentum J and electric charge Q , while the next-to-leadingterms in the conjugated chemical potentials, definethe supersymmetric fugacities. M case: BPS Black holes in AdS xS BPS BHs in asymptotic
AdS xS space-time, areelectrically charged, rotating extremal solutions of N = 1 11D supergravity. They are conjecture to bedual to BPS ensembles of the three dimensional N = 8SCFT on the world-volume N M N .BPS BHs are label by the maximal compact subgroupsof the asymptotic isometry group, and therefore arelabel by it energy E , 4 d -angular momentum J andfour U (1) R-charges Q i i = 1 , ...,
4. Notice that thisis precisely the set of labels used to characterize thedual states in the SCFT.Non-extremal BH solutions of N = 8 SO (8) gaugesupergravity truncated to its maximal abelian subal-gebra, where found in [16, 17], while its supersym-metric limits and thermodynamics are review in [15].On these BHs there is an extra parameter labeling themagnetic charge, that is zero in the BPS case. TheBPS case only conserves two real supercharges. Forsimplicity, we only work explicitly the minimal casewhere all R-charges set equal. Nevertheless, our re-sult are easily generalized to the U (1) case. In the minimal case, the non-extremal electricallycharge rotating BH solution comes as a function ofthree parameter ( m, a, q ) [23]. The thermodynamicpotentials are, β = 4 π ( r − a ) r + [1 + a + 3 r + − ( a + q ) /r ] , Ω = a (1 + r )( r + a ) , Φ = qr + ( r + a ) , (6)where r + is a function of ( m, a, q ) corresponding tothe radial position of the outer horizon. The threedifferent charges and entropy S are E = m (1 − a ) , J = am (1 − a ) ,Q = q − a ) , S = π ( r + a )1 − a (7)while the corresponding Euclidean can be written as I = β E − β Ω J + 4 β Φ Q − S ( E, J, Q ) . (8)The extremal BH in the BPS regime is obtainedimposing the BPS constrain E = J + 4 Q , (9)together with the requirement that no closed time-like(CTC) curves are found outside the horizon (see [15]for details). This last two conditions, reduce the totalnumber of independent degree of freedom to only one.If we oxidate the BPS BH to 11D, using co-ordinatesthat are asymptotically static to
AdS xS , the solu-tion rotates in both factors, AdS and S , in all possi-ble directions, with velocities equal to the velocity oflight, i.e. Ω bps = 1 , Φ bps = 1.To obtain the correct expressions for the BPS parti-tion function and to define the different chemical po-tentials, we have to calculate the leading and next toleading term in a near BPS expansion on the abovefamily of non-extremal BHs. To do this, we chose aoff-BPS parameter µ such that, m = a (1 + a ) + µ , q = √ a (1 + a ) (10)where µ = 0 reproduces the BPS BH. The correspond-ing expansion of the charges and entropy in terms of µ gives E = √ a (1 − a ) + O ( µ ) , J = √ aa (1 − a ) + O ( µ ) ,Q = √ a − a ) + O ( µ ) , S = πa (1 − a ) + O ( √ µ ) , (11)while the expansion of its conjugated potentials gives; β = √ πa / (1 − a ) √ a + 6 a + 1 1 √ µ + O (0) , Ω = 1 − √ − a ) a / √ a + 6 a + 1 √ µ + O ( µ ) , Φ = 1 − √ − a ) a / √ a + 6 a + 1 √ µ + O ( µ ) . (12)From the above expansion, using eqn. (5), we can readoff the parametric form of the BPS charges, entropyand fugacities, J bps = √ aa (1 − a ) , ω = 4 π √ a (1 − a ) (1 + a ) √ a + 6 a + 1 ,Q bps = √ a − a ) , φ = 2 π √ a (1 − a ) (1 + a ) √ a + 6 a + 1 ,E bps = √ a (1 − a ) , S bps = πa (1 − a ) , (13)that allows us to write the Euclidean action as a func-tion of ( ω, φ ) as follows, I bps = ωJ bps + 4 φ Q bps − S bps . (14)Finally we can use the BPS equation (9), to rewrite I bps in terms of the fugacities related to ( E, Q ) asfollows, I bps = ξ E bps − µ Q bps − S bps , , (15)with ξ = ω , µ = ω − φ . M Phase transitions, stable/unstable Bh andconstraints
The Euclidean action for the M a is given by the expression I bps = πa ( a + 8 a − a − a + 6 a + 1) , (16)where a runs on the interval (0 ,
1) and the BH radiusis r = √ a . The above is our working expression sum-marizing our results for the Euclidean action in theGrand canonical ensemble of BPS BHs.The corresponding partition function shows a clearphase transition between two phases (see fig 1). Thesenon-coexisting phases should correspond to a sea ofsupergravitons in AdS and a BH in AdS . In termsof the SCFT degrees of freedom, both phases are verydifferent, since the action scales like N / in the BHphase in contrast to the scale N characteristic of the AdS phase. The phase transition is of first order as r - - - - Figure 1: Plot of the Euclidean action of the BPS BH asa function of the BH radius. r Figure 2: Plot of the fugacity ξ as a function of the BHradius. It can be seen that given one value of ξ therecorrespond two solutions for the BH radius r , a small andbig BH respectively. can be seen from the calculation of the different sus-ceptibilities.The role of the unstable BPS BHs in the AdS phaseis similar to the that of small BHs at finite tempera-ture in
AdS . They are unstable saddle points of theEuclidean action, and should therefore be dual to un-stable configurations in the SCFT (see [18] for a simi-lar phenomenology on the D ξ (potential conjugated to the Energy) as a function of r . We can indeed see, that ξ is double value showingtwo branches. The first/second branch correspondsto small/big BHs. Small BHs are found between theorigin and the maximum value of ξ , while big BHs arefound from this maximum until the end.As a last comment, it is important to stress thatthese BPS BHs are constraint systems. There is onlyone free degree of freedom (that we parameterizedwith a ), while in principle, from naive expatiations onthe dual SCFT there should be two. Recalled that ourBH solutions are labeled by three charges ( E, J, Q ),that we have the BPS constraint of eqn. (9) and acausality constraint to eliminate CTC. This last extraconstraint adopts a complicated form in terms of theBH charges (we give it implicitly in eqn.(10)), but isparticularly simple when written in terms of the con-jugated fugacities, giving ξ = 2 µ equivalent to ω = 2 φ . (17)We will come back to this issue when discussing theSCFT partition function and its Index in section 3. M case: BPS Black holes in AdS xS BPS BHs in asymptotic
AdS xS space-time, areelectrically charged, rotating extremal solutions of N = 1 11D supergravity. They are conjecture tobe dual to BPS ensembles of the six dimensional N = (2 ,
0) SCFT on the world-volume
N M N . BPS BHs are label by the maximal com-pact subgroups of the asymptotic isometry group, andtherefore are label by it energy E , three 7 d -angularmomenta J I I = 1 , , U (1) R-charges Q i i = 1 ,
2. Again, this is precisely the set of labels usedto characterize the dual states in the SCFT.Non-extremal BH solutions of N = 4 SO (5) gaugesupergravity truncated to its maximal abelian subal-gebra, where found in [19, 20] and its supersymmet-ric limits and thermodynamics are review in [15, 20].The general non-extremal BH solution depending onall six charges in not known, where the known solu-tion have either same angular momenta and differentelectric charges [19], or same electric charges and un-equal angular momenta [20]. For simplicity, we onlywork out the minimal case, where all R-charges are setequal to Q and all the angular momenta are set equal J . Nevertheless, our result are easily generalized tothe other available cases. All known BPS cases onlyconserve two real supercharges.In this minimal case, the non-extremal electricallycharge rotating BH solution comes as a function ofthree parameter ( m, a, d ) [24]. The thermodynamicpotentials are, β = πr + [( r + a ) + q ( r − a )] − q +3 r (1+ r )( a + r ) − ( a + r ) +2 q ( a + r ) , Ω = a [( r + a ) (1+ r )+ q ( r − a )]( r + a ) + q ( r − a ) , Φ = π cosh( d ) sinh( d )(1 − a ) , (18)where q = 2 m sinh( d ) and r + is a function of ( m, a, d )corresponding to the position of the outer horizon. The three different charges and entropy S are E = mπ [ − − a +( − a +12 a +3 a ) sinh( d ) ]8(1 − a ) ,J = amπ ( − cosh( d ) + a (1+ a ) sinh( d ) )(1 − a ) ,Q = πm sinh( d ) cosh( d )2(1 − a ) ,S = π [( a + r ) + q ( ro − a ))4(1 − a ) r + . (19)Then, in the Grand canonical ensemble the Euclideanaction I can be written as I = β E − β Ω J − β Φ Q − S ( E, J, Q ) . (20)In the above conventions, supersymmetric BHs areobtained imposing the BPS constrain E + 3 J − Q = 0 , (21)together with the requirement that no closed time-likecurves (CTC) are found outside the horizon (see [15]for details). This last two conditions, reduce the to-tal number of independent degree of freedom to onlyone. As in the M AdS xS ), is rotating in both factors, AdS and S , with velocities equal to the velocity of light, i.e. | Ω bps | = 1, Φ bps = 1.To obtain the corresponding BPS partition func-tion, we calculate the leading and next to leading termin a near BPS expansion on the family of non-extremalBHs. The off-BPS parameter is µ such that, m = 3 qa (3 a − µ , q = 8 a ( a − (1 − a ) (22)where µ = 0 reproduces the BPS BH. The correspond-ing expansion of the charges and entropy in terms of µ gives E = π a ( − a − a + 3 a )(1 + a ) (1 − a ) + O ( µ ) ,J = − π a (1 − a + a )(1 + a ) (1 − a ) + O ( µ ) ,Q = 2 π a (1 + a ) (3 a −
1) + O ( µ ) ,S = 2 π a √ − a (1 + a ) p (1 − a ) + O ( √ µ ) , (23)while the expansion of its conjugated potentials gives; β = π [( r + a ) + q ( r − a )] Br + [2 q +3( a + r )(1+ a +2 r )] 1 √ µ + O (0) , Φ = 1 − Ba [( r + a )( a − r )+ qa ] r [( r + a ) − q ( r + − a )] √ µ + O ( µ ) , Ω = − B (1+ a )( a +2 a +4 a r +2 ar +3 r + q )[( r + a ) − q ( r + − a )] √ µ ++ O ( µ ) . (24)Where r = a (3 − a ) / (1 − a ) is the BPS BH radiusand B a polynomial in a . From the above expansion,using eqn. (5), we can read-off the parametric form ofthe BPS charges, entropy and fugacities E bps = π a ( − a − a +3 a )(1+ a ) (1 − a ) , S bps = π a √ − a (1+ a ) √ (1 − a ) ,J bps = − π a (1 − a + a )(1+ a ) (1 − a ) , ω = − πa (1+ a ) √ − a √ (3 − a ) a [3+ a (19 a − ,Q bps = π a (1+ a ) (3 a − , φ = πa (1+ a ) √ − a √ (3 − a ) a [3+ a (19 a − . (25)These expressions allow us to write the Grand canon-ical Euclidean action, I bps = 3 ωJ bps + 2 φ Q bps − S bps , (26)as a function of ( ω, φ ) only. As we proceeded in the M I bps in terms of the fugacities related to ( E, Q ) as follows, I bps = ξ E bps − µ Q bps − S bps , (27)with ξ = − ω , µ = − ( ω + φ ). M Phase transitions, stable/unstable BH andconstraints
The parametric form of the Euclidean action for the M I bps = π a {− a [3+(31 − a ) a ] }√ − a (1 − a ) (1+ a ) (3+ a ( − a )) √ ((3 − a ) a , (28)where a runs on ( − , M AdS and a BHin AdS . In terms of the SCFT degrees of freedom,the action scales like N in the BH phase in contrastto the scale N characteristic of the AdS phase. Thephase transition is of first order as can be seen fromthe calculation of the different susceptibilities.We have the same situation as in the M AdS phasecorresponds to unstable saddle points of the Euclideanaction, and should therefore be dual to unstable con-figurations in the SCFT. In fig. 4, the potential con-jugated to the Energy ξ is ploted as a function of a.Here it can be seen that it is double valued function of a , showing two branches of small/big BHs. Small BHsare found between the origin and the maximum value - a - - - Figure 3: Plot of the Euclidean action of the BPS BH asa function of the parameter − a . - a Figure 4: Plot of the fugacity ξ as a function of - a . Givenone value of ξ there correspond two different BHs, withdifferent radius; small and big BHs. of ξ , while big BHs are found from this maximum untilthe end.To closed this section, this system also shows an ex-tra constraint over the BPS relation of (21). There isonly one free degree of freedom (that we parameter-ized with the letter a ), while in principle, from naiveexpatiations on the dual SCFT there should be two.This extra constraint adopts a complicated form interms of the BH charges (we give it implicitly in eqn.(22)), but is particularly simple when written in termsof the conjugated fugacities, giving ξ = 4 µ equivalent to ω = 43 φ . (29)In the next section, we will elaborate on this and the M
3. Indices and Partition functions
The field theory calculation of the M /M N limit are unknown.Our incomplete understanding is due to the fact thatwe do not have a good description of the world-volumetheory of multiple M /M
5. Even in the case of theABJM proposal, where there is a candidate for theSCFT at level k the calculation is presently out ofour possibilities (see [4] for results on this direction.).Nevertheless at least formally, it should be a functionof all the potentials conjugated to the charges thatlabel our supersymmetric states in the ensemble.The situation is better when considering supersym-metric Witten indices. In [5, 6], it was possible todefine and to give a prescription of how to calcu-late the most general Witten Index I w for SCFT in D = 3 , , ,
6. This Index has already been computedin the AdS/CFT framework, showing perfect agre-ment on both sides of the duality [5, 7] for the caseswhere we have a realization the SCFT.To define the most general I w in SCFT, we firstchoose an arbitrary conjugated pair of supersymmet-ric generators ( Q, S ), that define the unbroken super-symmetries of our BPS ensembles of states. Then, theindex is written as a weighted sum over the Hilbertsubspace of states that are annihilated by (
Q, S ).Since all our BPS states transform in a irreducible rep-resentation of the subalgebra of the superconformalalgebra that commutes with our chosen pair (
Q, S ),it is clear that this maximal commuting subalgebra(MCS) plays a key role in the definition of I w . Thecorresponding trace formula for I w is I w = T r H [( − F exp( − β { Q, S } + G )] , (30)where we traced over the full Hilbert space H , F is theFermion number operator, G is an element of MCS.The resulting index is independent of β and is there-fore label by elements of the MCS only (see [6] for adetail explanation on all the above). Due to this lastpoint, I w is in general a function of less parametersthan the one needed to label a BPS state and there-fore is defined as a constrained function on the generalphase space of the theory.In the following, we will extract these constraints interms of the natural fugacities conjugated to the la-beling charges of our BPS states. We work this out forthe particular cases of the 3D SCFT with R-symmetry SO (8) and for the 6D SCFT with R-symmetry Sp (4).The resulting constraints should corresponds to the M /M I w .For the large N limit case of N M I w can be calculated over multi-gravitons states in AdS xS .In this situation, we first calculate the index over eachgraviton representation I sp , to then sum over all singlegravitons and multi-gravitons [25]. The index on eachgraviton rep. goes as I Rn = T r bps " ( − F exp( − ρE + X i =1 γ i H i ) , (31)where ( E, H i ) are the Cartan charges of the bosonicsubgroup SO (2 , xSO (6) of the MCS. Its relation tothe full set of superconformal Cartan charges ( e, j, h i )is E = e + j , H i = h i +1 (32)where i runs over 2 , , e, j, h i ) stands for energy,angular momentum and R-charge respectively. TheBPS constraint in these conventions turns out to be e − j − h = 0 . (33)To find the form of the constraint among the differentfugacities, we write ηj + X i =1 λ i h i = − ρE + X i =1 γ i H i (34)from which we get, after using the above BPS relation; ρ = − λ , γ i = λ i +1 for i = 1 , , η − λ = − ρ .That allow us to find the corresponding constraint ap-pearing in I w , 2 η − γ = 0 . (35)Rewriting the above constrain in terms of the U (1) gauge supergravity in 4D of previous sections, gives2 ω − X i =1 φ i = 0 , (36)that in the case of equally R-charged ensemble (i.e. φ i = φ for all i ), gives2 ω = φ ←→ ξ = 2 µ . (37)Where in the last expression, we have used the fugac-ities of eqn. (15). Therefore, we have found that: The constraint appearing in the Witten index ofour M -SCFT at large N is exactly the same con-straint that appears in known M -BHs .For the M I Rn in AdS xS , I Rn = T r bps h ( − F e ( − ρE + P i =1 , γ i H i + ζK ) i , (38)where ( E, H , H , K ) are the Cartan charges of thebosonic subgroup SO (5 , xSp (2) of the MCS. Its re-lation to the full set of superconformal Cartan charges( e, h I , k , k ) with I = 1 , , E = 3 e + h + h − h , H = h − h ,H = h + h , K = k , (39)where ( e, h I , k , k ) are respectively energy, angularmomenta and R-charges. The BPS constraint in theseconventions turns out to be e − h − h + h − k = 0 . (40)To find the form of the constraint among the differentfugacities, we write X I =1 η I h I + X i =0 , λ i k i = − ρE + X l =1 , γ l H l + ζK . (41)From the above equation and the BPS relation, we getthat the constraint in this case is η + η − η − γ = 0 . (42)Rewriting the above constrain in terms of the U (1) gauge supergravity in 7D of previous sections, gives X I =1 ω I + 12 X i =1 φ i = 0 . (43)In the case of equal angular momenta and equal R-charge (i.e. ω I = ω and φ i = φ for all I, i ), the con-straint reduces to3 ω − φ = 0 ←→ ξ = 4 µ , (44)where in the last expression we have used the fugaci-ties of eqn. (27). Therefore we have found that: The constraint appearing in the definition ofthe Witten index of the M -SCFT at large N , is thesame constraints that appears in known M -BHs.
5. Discussion
In this work, we have used the AdS/CFT dualityto define the supersymmetric partition function of the M /M N , Z bps . We have calculatedthe M-theory supergravity partition function as a sad-dle point approximation on supersymmetric BH. Inturns, to calculate this supergravity partition func-tion, we need the Euclidean action on these BPS BHs,as a function of all the fugacities conjugated to all thedifferent charges that label our BPS states. The Euclidean action on the BPS BHs is defined asthe supersymmetric limit of the Euclidean action ofnon-extremal BHs in M-theory [8, 9]. There is anothermethod, base on the attractor mechanism and Sen’sentropy function, that is equivalent as shown in [10].Our result are summarized by eqns. (13,15) for the M M AdS phasewe found the existence of small BHs that are unstable,representing local maximums of the Euclidean actionand therefore should be dual to unstable configura-tions in the SCFT (see [18] for similar but differentbehavior in the finite temperature for the case D I w ofthe corresponding SCFT. These constraints are givenin eqn. (36,43) for the M M D AdS isthe same constraint of the Index on the dual N = 44 D SYM theory [9]. One interesting application wasstudied in [21], where CFT information regarding theextra-constraint, was used to treat the problem of”BH uniqueness in 5D gauge supergravity” .
We thank M. Panareda for her infinite patience andsupport. This work was partially funded by the Min-isterio de Educacion y Ciencia under grant CICYT-FEDER-FPA2005-02211 and CSIC under the I3P pro-gram. [1] J. Bagger and N. Lambert, “Modeling multi-ple M2’s,” Phys. Rev. D (2007) 045020[arXiv:hep-th/0611108].J. Bagger and N. Lambert, “Gauge Symmetry andSupersymmetry of Multiple M2-Branes,” Phys. Rev.D (2008) 065008 [arXiv:0711.0955 [hep-th]].[2] A. Gustavsson, “Algebraic structures on parallel M2-branes,” arXiv:0709.1260 [hep-th].A. Gustavsson, “Selfdual strings and loop space Nahmequations,” JHEP (2008) 083 [arXiv:0802.3456[hep-th]]. [3] O. Aharony, O. Bergman, D. L. Jafferis andJ. Maldacena, “N=6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals,”arXiv:0806.1218 [hep-th].[4] A. Hanany, N. Mekareeya and A. Zaffaroni, “Parti-tion Functions for Membrane Theories,” JHEP (2008) 090 [arXiv:0806.4212 [hep-th]].[5] J. Kinney, J. M. Maldacena, S. Minwalla andS. Raju, “An index for 4 dimensional super confor-mal theories,” Commun. Math. Phys. (2007) 209[arXiv:hep-th/0510251].[6] J. Bhattacharya, S. Bhattacharyya, S. Minwalla andS. Raju, “Indices for Superconformal Field Theoriesin 3,5 and 6 Dimensions,” JHEP (2008) 064[arXiv:0801.1435 [hep-th]].[7] J. Bhattacharya and S. Minwalla, “Superconfor-mal Indices for N = 6 Chern Simons Theories,”arXiv:0806.3251 [hep-th].[8] P. J. Silva, “Thermodynamics at the BPS boundfor black holes in AdS,” JHEP (2006) 022[arXiv:hep-th/0607056].[9] P. J. Silva, “Phase transitions and statistical mechan-ics for BPS black holes in JHEP (2007) 015[arXiv:hep-th/0610163].[10] O. J. C. Dias and P. J. Silva, “Euclidean analysis ofthe entropy functional formalism,” Phys. Rev. D (2008) 084011 [arXiv:0704.1405 [hep-th]].[11] S. Corley, A. Jevicki and S. Ramgoolam, “Exact cor-relators of giant gravitons from dual N = 4 SYMtheory,” Adv. Theor. Math. Phys. (2002) 809[arXiv:hep-th/0111222].[12] D. Berenstein, “A toy model for the AdS/CFTcorrespondence,” JHEP , 018 (2004)[arXiv:hep-th/0403110].[13] M. M. Caldarelli and P. J. Silva, “Giant gravitons inAdS/CFT. I: Matrix model and back reaction,” JHEP (2004) 029 [arXiv:hep-th/0406096]. [14] H. Lin, O. Lunin and J. M. Maldacena, “BubblingAdS space and 1/2 BPS geometries,” JHEP (2004) 025 [arXiv:hep-th/0409174].[15] M. Cvetic, G. W. Gibbons, H. Lu and C. N. Pope,“Rotating black holes in gauged supergravities: Ther-modynamics, supersymmetric limits, topological soli-tons and time machines,” arXiv:hep-th/0504080.[16] Z. W. Chong, M. Cvetic, H. Lu and C. N. Pope,“Charged rotating black holes in four-dimensionalgauged and ungauged supergravities,” Nucl. Phys. B (2005) 246 [arXiv:hep-th/0411045].[17] M. M. Caldarelli and D. Klemm, “Supersymmetry ofanti-de Sitter black holes,” Nucl. Phys. B , 434(1999) [arXiv:hep-th/9808097].[18] T. Hollowood, S. P. Kumar and A. Naqvi, “Instabili-ties of the small black hole: A view from N = 4 SYM,”JHEP (2007) 001 [arXiv:hep-th/0607111].[19] Z. W. Chong, M. Cvetic, H. Lu and C. N. Pope,“Non-extremal charged rotating black holes in seven-dimensional gauged supergravity,” Phys. Lett. B (2005) 215 [arXiv:hep-th/0412094].[20] D. D. K. Chow, “Equal charge black holes and sevendimensional gauged supergravity,” Class. Quant.Grav. (2008) 175010 [arXiv:0711.1975 [hep-th]].[21] P. J. Silva, “On Uniqueness of supersymmetric Blackholes in AdS(5),” Class. Quant. Grav. (2008)195016 [arXiv:0712.0132 [hep-th]].[22] This type of BPS bound appears in two dimensionalsupersymmetric models like, e.g. , the effective theoryof 1 / N = 4 SYM in R ⊗ S (see [11, 12, 13, 14]).[23] here we follow the conventions of [17] with AdS ra-dius set equal to 1.[24] here we follow the conventions of [20] with AdS7