From giant gravitons to black holes

We study AdS5 black holes from a recently suggested giant graviton expansion formula for the index of U(N) maximal super-Yang-Mills theory. We compute the large N entropy at fixed charges and giant graviton numbers nI by a saddle point analysis, and further maximize it in nI. This agrees with the dual black hole entropy in the small black hole limit. To get black holes at general sizes, one should note that various giant graviton indices cancel because gauge theory does not suffer from a Hagedorn-like pathology by an infinite baryonic tower. With one assumption on the mechanism of this cancellation, we account for the dual black hole entropy at general sizes. We interpret our results as analytic continuations of the large N free energies of SCFTs, and based on it compute the entropies of AdS4,7 black holes from M5, M2 giant gravitons.


Introduction
Understanding the Hilbert space is the most basic task in quantum physics.It has also been a key problem of large N gauge theories and AdS/CFT, often with a focus on the emergent bulk descriptions.For instance, consider 4d maximal super-Yang-Mills theory on S 3 × R, dual to the type IIB theory on global AdS 5 × S 5 .We normalize the energy E to be dimensionless by multiplying the AdS radius.At E ∼ N 0 , the spectrum is described by the gas of gravitons.At E ∼ Ng 2 YM , where g YM is the Yang-Mills coupling, stringy excitations enter.At E ∼ N 1 , a novel finite N effect enters.On the QFT side, this comes from the finite size of the N × N matrices, imposing trace relations on gauge-invariant operators.In the gravity dual, this is realized as the gravitons polarizing to D3-branes.The branes can stretch either in S 5 , called giant gravitons [1], or in AdS 5 , called dual giant gravitons [2,3].The trace relations are realized either by giant gravitons having a maximal size, or the dual giant gravitons having a maximal number [4,5].These descriptions use probe D-brane approaches, whose validity requires that the energy is not too large, say E ≪ N 2 .This is merely a technical limitation, and the concept of giant gravitons may exist at higher energies and provide useful insights.This turned out to be the case in the half-BPS sector [6].
At E ∼ N 2 , semiclassical black hole solutions represent ensembles of states.In this paper, we wish to clarify the giant graviton picture at E ∼ N 2 , studying how black holes emerge from a giant graviton description of the spectral problem.We consider the BPS sector of the maximal super-Yang-Mills theory through the index of [7,8].This index has been studied to better understand the dual BPS black holes [9]: see [10,11,12] and references thereof.We shall study the recently suggested reformulation of this index [13,14,15,16,17], called 'giant graviton expansion.' We shall mainly consider the formula by Yosuke Imamura [13]. 1he giant graviton expansion of the index is given by I=1 n I ∆ I Z n 1 ,n 2 ,n 3 (∆ I , ω i ) , I = 1, 2, 3 , i = 1, 2 .(1.1) See section 2 for detailed explanations.n I 's are winding numbers of maximal giant gravitons along three different S 3 cycles in S 5 .Z n 1 ,n 2 ,n 3 is 'formally' an index of a U(n 1 ) × U(n 2 ) × U(n 3 ) quiver gauge theory, consisting of 4d/2d fields on the D-branes or at their intersections.When E ∼ N2 , we expect all n I 's are typically at the order of N 1 .Our strategy of studying this index is roughly as follows.We first find certain large n I (∼ N 1 ) saddle points of the integral representation of Z n 1 ,n 2 ,n 3 .The contour of this integral is complicated and empirically determined only for low n I 's.The contour information is in principle important to decide whether a saddle is relevant for approximating the integral, through the Picard-Lefschetz theory.As often done in practical studies of challenging integrals, we ignore this issue and assume that our saddles are relevant.After Legendre transforming the free energy at large fixed charges q I and n I 's, one obtains a macroscopic entropy S(q I , n I ).Further maximizing it in n I 's to find the dominant term Z n 1 ,n 2 ,n 3 , one would naively find the entropy at fixed q I .For a reason to be explained, this strategy is correct only in the 'small black hole limit' q I N 2 ≪ 1.
To understand why (1.1) is a subtle formula, one should note that n I 's are the numbers of determinant operators in gauge theory, which are morally baryons.Baryons and mesons provide towers of confining spectrum, responsible for fast growth of the high energy density of states.However, since the basic degrees of freedom are gluons, the growth for gauge theories should be much slower.Therefore, for an expansion like (1.1) to correctly capture the gauge theory entropy at high energies, one expects substantial cancellations of different Z n 1 ,n 2 ,n 3 's.For instance, if such cancellations do not happen, we shall see that the series (1.1) exhibits very fast growth at large n I 's and cause a Hagedorn-like pathology [18,19]: due to string and brane states, the canonical partition function becomes ill-defined.This implies that individual S(q I , n I ) and Z n 1 ,n 2 ,n 3 lose physical meanings at high energies, while the series (1.1) itself may be physical after cancellations.To address the cancellations rigorously, one should be able to compute the subleading terms in the large N limit.This is beyond the scope of this paper (and the subleading terms depend on the contour choice for Z n 1 ,n 2 ,n 3 ).We shall rather assume a particular mechanism of how the apparently leading contributions e S(q I ,n I ) cancel, and then proceed to compute the true entropy that exactly accounts for the dual black holes.As we explain in section 2.2, the mechanism we suggest assumes that the summations over discrete n I 's can be approximated by integrations of these variables, which is valid only if the subleading terms in the large N limit are arranged suitably.
As a byproduct, we find an emergent 2d QFT-like structure in our large N calculation when the three chemical potentials for the U(1) 3 ⊂ SO(6) electric charges are equal.This has to do with the integrand of Z n 1 ,n 2 ,n 3 reducing to the ratios of theta functions.This concretely justifies a study of [8], which assumed the existence of a hypothetical 2d CFT on the worldvolume of giant gravitons and then proceeded to count small black holes.
Our large N results can be interpreted as an analytic continuation of the maximal super-Yang-Mills index, extending the idea of [14,17].After establishing this interpretation on AdS 5 × S 5 , we apply it to the large N index on AdS 4 × S 7 based on the expansion of M5-brane giant gravitons [20].Namely, just assuming the existence of such an expansion and very basic structures, we explain the entropy of the AdS 4 black holes from the large N free energy of 6d SCFTs on M5-branes.Similarly, we find a relation between the entropies of black holes on AdS 7 × S 4 and the large N free energies of 3d SCFTs on M2-branes.
The rest of this paper is organized as follows.In section 2.1, we present a saddle point analysis of Z n 1 ,n 2 ,n 3 , and show that it accounts for the small black hole entropy.In section 2.2 we explain a possible way in which different Z n 1 ,n 2 ,n 3 can cancel at general charges.Then assuming this, we compute the true asymptotic large N entropy accounting for the dual black holes.Section 2.3 comments on the similar analysis with three unequal electric charges.In section 3, we make an interpretation of our results from analytic continuations and generalize it to account for the entropies of BPS black holes in AdS 4,7 .Section 4 concludes with discussions.

Giant graviton index and black holes
The index for the N = 4 Yang-Mills theory is defined by subject to the condition 3 , where Q I are U(1) 3 ⊂ SO(6) R-charges and J i are U(1) 2 ⊂ SO(4) angular momenta.See [7,8] for a unitary matrix integral representation of this index for the U(N) gauge group.Recently, an alternative expression for this index was proposed.It takes the form of (1.1), where Z KK is the index of low energy gravitons [8].
while Z 0,0,0 ≡ 1.The functions Z 4d I and Z 2d I,I+1 appearing in the integrand are given as follows.From now on, let us define ∆ I ≡ −2πiτ I = −2πi(τ + z I ) with 3 I=1 z I = 0, and Then Z 4d 3 from the 4d U(n 3 ) adjoint fields is given by . (2.3) Here and below, whenever the argument contains ±, corresponding two functions are multiplied.u b , and Γ(z; σ, τ ) is the elliptic Gamma function defined by 1 − e −2πiz e 2πi((m+1)σ+(n+1)τ ) 1 − e 2πiz e 2πi(mσ+nτ ) . (2.4) Other Z 4d I are given similarly by permuting the I = 1, 2, 3 indices.The integrand from the 2d U(n 1 ) × U(n 2 ) bifundamental fields is given by ab + a 12 ) + τ 3 2 + y, τ 3 ) θ(±(u where u . θ(z, τ ) is the q-theta function (with 'q' given by e 2πiτ ) defined by Other Z 2d I,I+1 are given similarly.The integration contour is complicated, and is related to how the auxiliary parameters a I,I+1 are chosen.Only the value of a loop ≡ a 12 + a 23 + a 31 is important.In [13] and [16], two different choices of a loop were made, also with different choices of the integration contour.Both prescriptions are tested till certain low orders.One of a loop = − 3τ 2 ± y was chosen in [13], while a loop = 0 was chosen in [16].The situation might be that both prescriptions work to all orders in n I 's, or one of the two is correct for higher n I 's.Although we have little to say about this issue, we simply note that our saddle point ansatz below works with the choice a loop = 0 of [16].Perhaps with the choice of [13], residue contributions may be more important when the contour crosses poles during its deformation towards the saddle point.(Such an issue may also arise in the original Yang-Mills matrix integral for the index, as commented on in [21].)So we set a I,I+1 = 0 from now on.
We would like to study the large N behaviors of Z n 1 ,n 2 ,n 3 .Since n I 's contribute Nn I to the electric charges Q I , which we want to scale as N 2 , we let n I 's to scale linearly in N.

Large N saddle points and small black holes
In this subsection we shall consider the index at ∆ 1 = ∆ 2 = ∆ 3 .(We shall comment on the generalization to unequal ∆ I 's in section 2.3.)This corresponds to taking the z I → 0 limit.It was shown [13] that individual Z n 1 ,n 2 ,n 3 diverges in this limit, while the full index after summing them over remains finite.We are interested in the leading order free energy log Z n 1 ,n 2 ,n 3 ∼ N 2 in this limit.To understand this limit more precisely, we first decompose the contributions to Z 4d I from the N Cartans at a = b and the off-diagonals at a = b.The former part can be written in the limit as where z I,J ≡ z ).So the Cartan parts diverge in the limit z I → 0. This accounts for many of the divergences encountered in [13] in this limit.The divergence is linear in N, ∼ N log ε at small z I ∼ ε.So we ignore this part since we are interested in the leading free energy proportional to N 2 .(However, see the later part of this subsection and section 2.2 for important roles of the subleading parts.)The off-diagonal part with a = b contains extra divergences in the limit z I → 0, by the z I dependent poles pinching the integration contour [13].Our precise setting of taking the limit is as follows.We are interested in the behaviors of the integrand near the saddle point of our interest, to be presented below.The saddle point is away from the contour, and will not suffer in any sense from the pinching of the z I dependent poles.So as for this part, we naively take the z I → 0 limit and simplify the integrand.Using Γ(z + σ; σ, τ ) = θ(z, τ )Γ(z; σ, τ ), Γ(z + τ ; σ, τ ) = θ(z, σ)Γ(z; σ, τ ) and Γ(z; σ, τ ) = 1 Γ(σ+τ −z;σ,τ ) , one obtains We have ignored the Cartan part which only makes a subleading N 1 contribution.At the saddle point, all u (I) a 's will be different, so that this function remains finite in the z I → 0 limit.We realize that the contributions from the 4d fields are given in terms of the theta functions after substantial cancellations.Similarly, Z 2d I,I+1 are given in the z I → 0 limit by Z 4d I and Z 2d I,I+1 in this limit are invariant under shifting u As our large N (and large n I ∼ N) saddle point ansatz, we take each set of U(n I ) eigenvalues u (I) a to be uniformly distributed along the τ -circle, (2.10)This is a coarse-grained continuum description of the eigenvalues, which are separated from their nearest neighbor by a distance at order 1 N .We can typically assume that none of these eigenvalues are at precisely the same values.Therefore, (2.8) and (2.9) do not diverge due to u b .If such a divergence apparently seems to happen in the continuum description, it should be avoided by integrating over x I 's with a principal-value prescription.It is easy to see that this distribution solves the large n I saddle point equation.To check this, it is convenient to first S-dualize the integrand using the identity (2.11) We shall set y to be in the range 0 < y 1 < 1, where y ≡ y 1 + y 2 τ with real y 1 , y 2 : this convention can be chosen by a suitable period shift of y.Then, applying the S-dual identities, one obtains (2.12) .
Collecting all, the integrand is given by a constant factor (2.14) In order to show that (2.10) is a saddle point, one should show that the force ∂ ∂u (I) a log Z vanishes in the large N limit.More precisely, one should show that the leading N 1 order term of the force vanishes.This force is given by (2.15) The expression inside the square bracket of the right hand side is given by a linear combination of the function of the form log(1 − e ±2πi(x−x ′ ) e − 2πi(n+α) τ ) with n ∈ Z ≥ 0, α ≥ 0 and x, x ′ ∈ [0, 1].(α may be either 0 or y.)So all these log functions are periodic in x ′ → x ′ + 1 shift without crossing the branch cut.Therefore, we integrate the derivative of a periodic function over a circle, which vanishes.The terms with α = 0 and n = 0 have the branch points on the circle x ′ ∈ [0, 1], but employing the principal-valued integrals as explained, they also vanish.
One can compute log Z n 1 ,n 2 ,n 3 at this saddle point.By evaluating log Z in the continuum limit, similar to the evaluation of (2.15), one finds log Z = 0.This is because the integral of log(1 − e ±2πi(x−x ′ ) e − 2πi(n+α)

τ
) is zero at n ≥ 0, α ≥ 0. So the large N free energy is given by where n ≡ n 1 + n 2 + n 3 .Note that the leading free energy depends only on one combination of n I .So there apparently is a large number of degenerate terms if we only consider the leading free energy.One can Legendre transform this free energy to obtain the macroscopic entropy at fixed charges q, j conjugate to τ , y, also at fixed n.This amounts to the extremization of (2.17) The charges correspond to q = Q 1 +Q2 +Q 3 3 The solution is given by where P ≡ 3q − nN − 9n 2 8 , and the extremized entropy is given by The constant imaginary term −πij can be ignored in the discussions below.
Before proceeding, let us comment on the structure of the asymptotic Z n 1 ,n 2 ,n 3 that we obtained in (2.16).We first investigate the structure of the expansion (1.1) in the grand canonical ensemble with fixed τ , y.Since these parameters are complex, it is helpful to focus on a region which contains the saddle point of the Legendre transformation (2.18). 2 For instance, as for τ , let us take it to be purely imaginary with Im(τ ) > 0. For y, let us freeze y = 1 2 for simplicity of the discussion.This corresponds to setting j = 0 in the microcanonical ensemble, or unrefining the chemical potential y for j in the grand canonical ensemble.Then one finds that the giant graviton expansion (1.1) takes the form of (2.20) where β ≡ −2πiτ is real and positive.Recall that n ≡ n 1 + n 2 + n 3 , and Ω(n I ) come from the subleading contributions to log Z n 1 ,n 2 ,n 3 in the large n ∼ N expansion.At large n, each |Z n 1 ,n 2 ,n 3 | grows very fast like e an 2 with certain a > 0 when β < β c ≡ 2π 3 .So at high temperatures, unless the subleading factors Ω(n I ) are given in a manner that various terms substantially cancel, the sum will diverge very badly at large n.One can be more realistic and insert the complex values of τ (q) as a function of real charge q, at which we know that BPS black hole saddle points exist [11,22].Then one finds that Im 1 τ (q) − 9τ (q) < 0 is always met, again making |Z n 1 ,n 2 ,n 3 | to grow fast.However, if the expansion (1.1) provides an exact expression for the gauge theory partition function, we expect the series (2.20) to better behave at high temperatures where the system deconfines [22,25,24].
Let us elaborate more on why we expect the series (2.20) to behave well for β < β c .For instance, consider a series of the form n Ω(n)e βcn e −nβ (2.21) with β c > 0, and Ω(n) does not affect the exponential growth of e βcn .The series ceases to converge at β < β c outside its radius of convergence.If this series is for a thermal partition function, the divergence is the Hagedorn pathology [18] caused by exponential growth of the density of states at high energy.It happens due to an infinite tower of mesonic states [18], or an infinite tower of string oscillations [19].The apparent divergence ∼ e an 2 of (2.20) would make the series worse-behaved than (2.21).Namely, unless cancellations happen, the radius of convergence is zero.One may interpret this divergence (if present) as coming from a much faster asymptotic growth of baryonic states.However, the notion of baryonic states should become ambiguous at large charge q for which τ (q) enters the deconfining regime.For expressions like (1.1) or (2.20) to remain relevant, Ω(n I )'s should be arranged so that the apparent asymptotic growth e an 2 cancels.The cancellation effects should be more crucial for τ (q) with larger q, as the system is deeper inside the deconfining regime.If such cancellations are not taken into account, each term in the series (2.19) may over-estimate the microcanonical entropy.
With this caution in mind, let us try to extract the microcanonical entropy from the formula (2.19).We first study the case with j = 0. Since n is not a physical charge, we should try to maximize Re[S(q, n)] as a function of non-negative integer n.Re[S(q, n)] is positive when So to study the macroscopic entropy from this index, one only needs to sum over n till n * (q).Thus, we consider e S(q) = dτ e −2πiτ •3q (2.23) We study this quantity at large N and large q ∝ N 2 , naively expecting at this moment that the leading contribution comes from certain n at order N 1 .We first note that the overall phase factor e − 3πin 2 2 oscillates between i and 1, depending on whether n is odd or even.So dividing the sum into even/odd n's and naturally expecting that the maximization will not be sensitive to the even/odd nature of n's, this phase does not matter and the dominant contribution to this entropy is given by the maximum of Re[S(q, n)].This happens at which is in the range 0 < n 0 (q) < n * (q).The maximal entropy is given by We first study its asymptotic behaviors at q ≪ N 2 and q ≫ N 2 , which would respectively correspond to the small and large black hole limits.One finds Re[S(q)] = π(2q) Re[S(q)] = 2πq − 4πNq One can show that this Re[S(q)] is asymptotically equal to the Bekenstein-Hawking entropy S BH (q) of the dual black hole when q ≪ N 2 , in which case Re[S(q)] ≈ π(2q)

N
≈ S BH (q).Away from the asymptotic limit q ≪ N 2 , Re[S(q)] > S BH (q) always holds.In particular, in the two asymptotic limits, S BH is expanded as One finds a small over-estimating deviation Re[S(q)] − S BH (q) ≈ 12πq

√
2N 3 > 0 in the small charge expansion, and Re[S(q)] ≫ S BH (q) in the large charge expansion.As already explained, our interpretation of this over-estimate is that we have been ignoring the possible cancellations of the apparently leading order terms due to nontrivial Ω(n I )'s in (2.20).The large charge behavior Re[S(q)] ∼ 2πq = 3β c q of (2.26) is a Hagedorn growth.

N
with the Bekenstein-Hawking entropy of small black holes might still look a bit miraculous.To better appreciate this, it is first worthwhile to note that small black holes are never dominant saddles in the grand canonical ensemble.Also, they always stay in the confining region in the complex τ space [24].So it makes sense that they admit a description in terms of D-branes, which are baryonic objects in the confining phase.On the other hand, as q gradually grows, giant gravitons will eventually lose their meaning at high energy.This is because the fundamental high energy degrees of freedom are gluons rather than their bound states.Interestingly, the D3-brane giant graviton approach has been already employed in [8] to account for the entropy of small black holes.The calculation we did with S(q, n) was discussed in [8], in precisely the same computational procedure.The rough idea of [8] is to regard the maximal giant gravitons to be similar to the wrapped D-branes which account for 5d asymptotically flat black holes [26].Since most of the microscopic accounts for asymptotically flat black holes use branes, and since small AdS 5 black holes are (at least mathematically) identical to the 5d asymptotically flat black holes embedded in large AdS, it is natural that both objects admit similar D-brane-based descriptions.We find that our studies provide precise logical grounds for the calculations of [8].
From (2.18), the saddle point value of τ for the Legendre transformation at j = 0 is . (2.28) In the small black hole limit q ≪ N 2 , n is ranged in 0 < n < n * (q) ≈ 3q N and the maximum n 0 (q) of S(q, n) is approximately n 0 (q) ≈ 2q N .Around the maximum n 0 (q), τ scales like (2.29)So the small black hole limit q ≪ N 2 corresponds to the 'Cardy limit' τ → i0 in the 2d-like integrand (2.8), (2.9).We find this to be a concrete realization of the studies made in [8], which assumed the existence of a hypothetical 2d CFT living on the worldvolume of maximal giant gravitons at fixed n = n 1 + n 2 + n 3 and used its Cardy formula to account for the small black hole entropy.The 2d CFT was supposed to live on the Hopf fiber circle of the S 5 , which is wrapped by the D3-branes.However, there was no logical justification for the existence of such a 2d CFT, since the 4d worldvolume has no scale separation which justifies the 2d reduction.We found from our index in the limit z I → 0 that the 4d part of the integrand Z 4d I partly canceled to yield Jacobi theta functions, which are 2 dimensional objects.So what justifies the 2d reduction here is the boson-fermion cancellations in the index.This is much more specific than a reduction based on the scale separation.This 2d description may break down if one studies unprotected quantities beyond the index.For instance, [8] studied the charge relation satisfied by small black holes, J 1 + J 2 ∼ q 2 N 2 .The idea of [8] is as follows.If the 2d CFT exists, small AdS 5 black holes are dual to its NS sector.It is related to the CFT in the Ramond sector by a spectral flow.The Ramond sector CFT describes 5 dimensional asymptotically flat black holes [26,27] satisfying a charge relation J 1 + J 2 = 0.So if the 2d description exists universally beyond the index, the spectral flow will connect the charge relation J 1 + J 2 = 0 to that of the AdS black holes.The relation obtained from this route does not agree with the charge relation of AdS black holes [8].We interpret this as the absence of the 2d description beyond the index.
We can generalize the studies to the case with j ≡ J 1 − J 2 = 0 by keeping y = 1  2 .Again we are only able to successfully count small black holes by keeping the leading term of (2.16) at small τ .Making a Legendre transformation of this free energy by extremizing and then maximizing Re[S(q, j; n)] with n, one obtains the entropy given by (2.32) τ , y at these saddles always satisfy |τ | ≪ 1, Re(y) = 1 2 as long as the charges are away from the closed timelike curve (CTC) bound j 2 = 8q 3 N 2 , staying within the regime that we assumed.This is precisely the entropy of small spinning BPS black holes in AdS 5 × S 5 .This can also be regarded as the BMPV black holes [27] embedded in large AdS 5 .We emphasize that this is the first microscopic counting of small spinning AdS 5 black holes at all allowed values of J 1 = J 2 .In fact accounting for black holes at J 1 = J 2 has been technically tricky from the QFT dual.For instance, in the saddle point approach to the Yang-Mills matrix model, general J 1 = J 2 was discussed only in the 4d Cardy limit [11].At general finite charges, [21] found saddle points which cover substantial charge regions for j, but failed to cover the whole parameter space of CTC-free black holes.More precisely, [21] found the saddles when certain inequalities were met, like eqn.(2.41) or (2.45) there.In the parametrization of BPS black holes given by the third reference of [9], these inequalities cover the region 0 < ag, bg < 1.On the other hand, the CTC-free black holes exist in a bigger region ag, bg < 1, a + b + abg > 0. This should be due to our limited understanding of the large N matrix model saddles.At least in the small black hole limit, it is amusing that the giant graviton calculation of this paragraph was able to cover the whole CTC-free black holes satisfying ag, bg ≪ 1, a + b > 0.

Comments on finite size black holes
From the absence of the Hagedorn behavior in gauge theories, we think it is obvious that cancellations of different Z n 1 ,n 2 ,n 3 's happen in general.However, computing such cancellations at large N is technically very challenging.This is because the cancellations happen due to relative minus signs, whose precise determination goes beyond the leading order calculation.For instance, we tried to compute such subleading terms in the small τ regime, but found that the precise integral contours for Z n 1 ,n 2 ,n 3 are needed to compute them.Also, with such a contour dependence, taking z I → 0 limit is trickier than in the previous subsection.
In this subsection, leaving the full microscopic analysis to the future, we shall make a simple assumption on how these subleading corrections should be arranged.This assumption will allow us to compute the true entropy of this index after the cancellations, which precisely reproduces the dual black hole entropy.The claim is that, once we include the 1 N effects to each Z n 1 ,n 2 ,n 3 's contribution, the degeneracy at given n will be lifted by small deviations from (2.16) in a way that the sum over n can be replaced by an integral.The microcanonical sum (2.23) over discrete n can be replaced by e S(q) = dτ e −2πiτ •3q (2.33) The claim asserts that we use the same function S(q, n) but sum over a dense set of n's.Before explaining anything about this claim, we emphasize that we have no derivation of (2.33) except that this formula will give the exact black hole entropy at an arbitrary size.
Let us first explain why this is a nontrivial claim, and in particular why it is related to including 1 n I subleading terms.Our claim is essentially that, once we include the subleading corrections at fixed n, Z n 1 ,n 2 ,n 3 will behave like where ρ(n) is a suitable distribution determined by δ n (n I )'s.Since the total number of summands satisfying n ≤ n * (q) is proportional to n 3 * , one finds . Thus, ignoring it, one obtains (2.33).
If the sum over n is replaced by an integral over n, it is no longer valid to find the dominant contribution by maximizing Re[S(q, n)].Rather, one should find a saddle point of S(q, n) in the complex n plane.Regarding S(q, n) as a complex function of n, the maxima n 0 (q) of Re[S(q, n)] on the real axis is not a saddle point, due to nontrivial Im[S(q, n)] = − 3πn 2 2 for continuous n.In summary, part of our claim is about the 1 n I subleading corrections of Im[S(q, n)], which lift the degeneracy and render substantial cancellations of different Z n 1 ,n 2 ,n 3 's.
Given (2.33), one can identify the saddle point on the complex n plane.In fact it is inconvenient to work directly with the last expression of (2.33).Rather, we keep the variables τ, y unintegrated, and consider the multiple integral formula for e S(q,j) given by e S(q,j) ∼ dτ dy n * (q) 0 dne log Zn 1 ,n 2 ,n 3 −2πiτ (3q−nN )−2πiy•j (2.36) Note that at this stage we reintroduced the refinement with y or j, since the analysis is no more difficult.To find the possible saddle points in the complex n plane, we can simply extremize the 3-dimensional integral (2.36).(Later in this subsection, when numerically discussing the contour deformation, it will be more convenient to use the original 1 dimensional integral (2.33).)It is easy to first extremize in n, since the integrand is Gaussian in n.One finds the saddle point . Inserting this, the remaining τ, y integral is given by S S(q,j) ∼ dτ dy exp − (2.37) Reintroducing where q ≡ Q+ J 1 +J 2 2 , j ≡ J 1 −J 2 .This is precisely the entropy function (at equal electric charges Q I ) for the Bekenstein-Hawking entropy of BPS black holes in AdS 5 × S 5 [28].Although we discussed the saddle points of the 3-dimensional integral (2.36), the same entropy is obtained with S(q, n) from the last expression of (2.33).
We finally show that the contour can be deformed to pass through this saddle, by extending the standard Picard-Lefschetz theory.Consider the following integral where x = n N and x * ( q N 2 ) = n * (q) N .If the integrand vanishes at the two ends x = 0, x * , one can deform the integration contour to the steepest descent contour.The steepest descent contour has maximal Re[f (x)] at the saddle point, and satisfies the stationary phase condition Im[f (x)] = constant.The integration on this contour can be approximated at large N by a Gaussian approximation around the saddle point.In our case, the integrand does not vanish at the two ends.Then the steepest descent contour passing through our complex saddle point x s ≡ ns N does not end on x = 0, x * , so we have to slightly extend this standard method.We combine an interval C 2 of the steepest descent contour (solid red line of Fig. 1) with two more intervals C 1 , C 2 of contours satisfying Re[f ] = constant and ending on x = 0, x * , respectively (solid black).The original contour C 0 = [0, x * ] (green) can be deformed to C 1 ∪ C 2 ∪ C 3 .As q decreases, one can see that the complex saddle x s approaches the real maximum x 0 ≡ n 0 (q) N of Re[S(q, n)].So the complex saddle approach naturally converges the naive analysis with real n of section 2.1 in the small black hole limit.
The dominant term of the integral on C 2 can be computed by the Gaussian approximation around x = x s , yielding a term of the form ∼ e N 2 f (xs) .Then, denoting the two ends of the interval C 2 by x 1 and x 3 , respectively, the integrands on C 1 and C 3 take the form of respectively.Since C 2 is the steepest descent contour, one finds e N 2 Re[f (x 1,3 )] ≪ e N 2 Re[f (xs)] and these integrals are bounded as Therefore, the contribution from C 1 ∪ C 3 is subdominant, justifying the approximation using the Gaussian approximation near x s .As illustrated in Fig. 1, we checked for a wide range of q N 2 that the contour can always be deformed in this way.

Comments on unequal electric charges
So far, we studied the index with the chemical potentials ∆ I for the three electric charges In this subsection we comment on the generalizations with unequal ∆ I 's.Note that in the original Yang-Mills matrix model of [7,8], taking independent ∆ I was rather straightforward, while introducing the refinement y = 1 2 for two independent angular momenta J 1 , J 2 was much trickier [24,21].This was basically because independent ω 1 , ω 2 for the spacetime charges in QFT could introduce branch points in the matrix model potentials which yield nonzero eigenvalue forces.In fact, as explained in section 2.1, [21] found saddle points for the black holes at J 1 = J 2 only when certain inequalities are met: see eqn.
(2.41) or (2.45) of [21].In the giant graviton index Z n 1 ,n 2 ,n 3 , since the role of internal and spacetime symmetries are partly exchanged, such as ∆ 1 , ∆ 2 ↔ ω 1 , ω 2 for the giant gravitons with n 3 = 0, the situation is the other way round.We have seen in section 2.1 that the giant graviton approach sees the black holes at J 1 = J 2 (i.e. at y = 1 2 ) rather easily at least when the black hole size is small.This is because ω 1,2 are the chemical potentials for internal symmetries in this approach.On the other hand, we find it very difficult to construct saddle points of Z n 1 ,n 2 ,n 3 when ∆ I 's are different.
More concretely, we have tried to construct the large n I saddle points by noting that the integrand Z 4d I resembles the integrand of a Yang-Mills index with U(n I ) gauge group, except for the tachyon and contour issues [13,16].So we tried to use the ansatz of [21] to find the saddle point at unequal ∆ I 's.This almost solves the saddle point equations but not quite, due to several branch points in the potential.During the course, however, we could write down a free energy for log Z n 1 ,n 2 ,n 3 whose Legendre transformation and maximization in n I 's yield the dual black hole entropy.Although we have a gap in our derivation, we strongly believe that we found the correct answer.So we simply report our findings without any microscopic derivation.
We find that the refined free energy should be given by and the corresponding entropy function to extremize is Like the analysis we did for equal τ I , we should first extremize this in τ I 's, and then maximize the real part with n I .If we do this calculation, again one generally obtains an entropy Re[S(q I , j)] which overestimates the degeneracy unless cancellations of various Z n 1 ,n 2 ,n 3 are taken into account.
Like the case with equal q I , the entropy estimated with single Z n 1 ,n 2 ,n 3 reproduces the black hole entropy in the small black hole limit q I , j ≪ N 2 .For simplicity, we show this only at j = 0 (y = 1  2 ).The small black hole limit corresponds to |τ I | ≪ 1, in which case one obtains This is a real function for purely imaginary τ I 's.We shall extremize S with τ I 's on this subspace.Since S is real, the next maximization of Re[S] with n I is just maximizing S. We can exchange the order of the two extremizations.S(q I ; n I , τ I ) is quadratic in n I , and depends only on Therefore, only this linear combination of three n I is fixed after extremizing with n I , leaving two parameters unfixed.This is a generalization of sections 2.1 and 2.2 where only n 1 + n 2 + n 3 ≡ n was fixed.After this extremization, one obtains This is precisely the entropy function for the small black holes, whose further extremization yields the Bekenstein-Hawking entropy S(q I ) ≈ π N √ 8q 1 q 2 q 3 of small AdS black holes.
Beyond small black holes, we also expect that subleading order terms should render substantial cancellations of different Z n 1 ,n 2 ,n 3 's, in order for the index not to exhibit a Hagedorn-like pathology at large n I 's.At complex τ I , one can only fix I n I τ I so that one of the three parameters n I remains unfixed in the leading free energy.Summing over them could render cancellations.We also suggest that the concrete mechanism of such cancellation is replacing the sum over discrete n I 's by an integral, as we explained in section 2.2.This allows one to seek a complex saddle point for n I 's in (2.43).Since (2.42) is quadratic in n I 's, one first finds a Gaussian saddle point for This is precisely the entropy function of BPS black holes in AdS 5 × S 5 [28], whose further extremization yields the Bekenstein-Hawking entropies of the dual black holes.
3 Analytic continuation and AdS 4,7 black holes In this section, we interpret our results in section 2 as the analytic continuation of the maximal super-Yang-Mills index.[14] established such an interpretation for Z 0,0,n 3 .See also [17].For general Z n 1 ,n 2 ,n 3 , we find a similar interpretation of its large N free energy.
Let us first review [14] in the language of [13].The integrand for Z 0,0,n 3 is simply Z 4d 3 of our section 2.1, as the quiver consists only of one adjoint node.This is related to the integrand Z YM int of 4d maximal super-Yang-Mills index Z YM U (n 3 ) in a very simple manner.Let us first note that, when we write the arguments of Z YM int as Z YM int (∆ 1 , ∆ 2 , ∆ 3 ; ω 1 , ω 2 ; u a ), the first three denote the U(1) 3 ⊂ SO(6) internal rotations while the next two denote the U(1) 2 ⊂ SO(4) rotations on the spacetime of the QFT.Then Z 4d 3 is given by [13,14] This formula can be understood as follows.Since n 3 D3-branes wrap S 3 ⊂ S 5 , the two worldvolume rotation parameters are ∆ 1 , ∆ 2 .On the other hand, SO(4) rotations on AdS 5 are internal symmetries on D3-branes.So ω 1 , ω 2 are their internal rotation parameters.Finally, since the maximal giant gravitons can shrink rather than grow, losing energies, the corresponding transverse scalar is tachyonic.This demands replacing ∆ 3 by −∆ 3 .In fact, as emphasized in [13,16], (3.1) has to be defined by analytic continuation since Re(−∆ 3 ) < 0. Suitably choosing the integration contour, one finds that Z 0,0,n 3 is obtained from Z YM U (n 3 ) by exchanging ω 1,2 ↔ ∆ 1,2 and replacing ∆ 3 → −∆ 3 with analytic continuation [14].
One can also get its large n 3 ∼ N free energy from analytic continuation.When ω 1,2 , −∆ 3 , ∆ 1,2 on the right hand side of (3.1) have positive real parts, its large N free energy is given by where the imaginary parts of the chemical potentials are suitably shifted by their periods to satisfy either . This result can be understood in two different ways.Firstly, it can be understood as derived from various calculations of the Yang-Mills index [11,12,21].Secondly, one can interpret it as the free energy of dual AdS 5 black holes [28,10].For 4d maximal super-Yang-Mills, both viewpoints are available.Having in mind less explored SCFT's, to be explored later in this section, we emphasize the virtue of understanding log Z SCFT D as the free energy of dual black holes in AdS D+1 .Once (3.2) is known in the region Re(−∆ 3 ) > 0, it is quite immediate to continue it to the physical region Re(∆ 3 ) > 0. Namely, we just keep the expression on the right hand side of (3.2).This continuation assumes the absence of the Stokes' phenomena.Let us assume this and proceed.In the parametrization of section 2, we take which is the giant graviton free energy (2.42) or (2.16) at n 1 = n 2 = 0. Therefore, our formulae of section 2 can be naturally understood as the analytic continuation of the Yang-Mills index.
In fact one can similarly interpret our general formula (2.42) for Z n 1 ,n 2 ,n 3 .Expanding the complete square in the numerator, one obtains where I ∼ I + 3 is understood.The three terms on the first line are log Z n 1 ,0,0 , log Z 0,n 2 ,0 and log Z 0,0,n 3 .One can more concretely identify each of them as the saddle point value of the integrand log Z 4d I .This was derived in section 2.1 at equal τ I 's.At equal τ I 's, the three terms on the second line can also be separately identified as the saddle point values of the integrands log Z 2d I,I+1 .So we naturally find a picture of the large N giant graviton free energy, as the sum of three maximal super-Yang-Mills free energies and three 2d free energies at the intersections.
We comment on two features of (2.42) and (3.4).Firstly, the six 4d and 2d contributions factorize in (3.4).This does not always have to be the case, as there is no reason for these degrees of freedom to decouple.We find that it is a rather exceptional property, perhaps for even dimensional QFT's whose free energies can be read off from anomalies.A more fundamental aspect is the n I dependence through 3 I=1 n I ∆ I .This may be heuristically understood as follows. 1  8 -BPS giant graviton solutions of [29] are given by holomorphic surfaces in where z I are the coordinates of C 3 .∆ I can be regarded as U(1) 3 rotation parameters on the moduli space, transforming 3 .The formula of [13] can be interpreted as an 'equivariant localization' of an integration over the moduli space given by C n 1 ,n 2 ,n 3 's (modded out by an overall multiplication of a complex number) [14,30].When all ∆ I 's assume general values, (3.5) is invariant under the rotation only if a single term is kept on the right hand side.So the moduli are completely lifted.In this case, n I 's label a discrete set of points in the moduli space which are invariant under U(1) 3 .When ∆ I 's assume rational ratios, one may keep multiple terms on the right hand side if the value of I n I ∆ I is the same.In this case, the moduli are partly unlifted.If we can interpret the integration over the unlifted moduli space as the 1-loop zero-mode integral in the large N calculation, only I n I ∆ I would appear in the leading large N free energy since this is the only invariant quantity on the unlifted moduli space.Of course this line of thinking assumes many things, such as the equivariant localization picture of the index, etc.However, we think it is a somewhat natural explanation of the appearance of I n I ∆ I .[20,14,17] explored the index of the 6d SCFTs on N M5-branes from M2-brane giant gravitons, and also the index of the 3d SCFTs on N M2-branes from M5-brane giant gravitons.We shall now study the 6d/3d indices accepting the existence of such giant graviton expansions, only assuming the analytic continuation picture and the n I dependence through I n I ∆ I .In the former case, the M2-brane giant gravitons wrap an internal S 2 ⊂ S 4 .So analytic continuations of the 3d maximal SCFT index will give the giant graviton index in AdS 7 × S 4 .In the latter case, the M5-brane giant gravitons wrap an internal S 5 ⊂ S 7 .So the 6d maximal SCFT index will provide the giant graviton index in AdS 4 × S 7 .Integrating out the giant graviton numbers, we indeed recover the free energies and entropies of AdS 7,4 black holes.
AdS 4 black holes from M5-branes: We assume the following giant graviton expansion [20] of the 3d index of maximal SCFT living on N M2-branes [31]: When only one n I is nonzero, say when n 1 = 0, log Z n 1 ,0,0,0 is obtain by analytically continuing the free energy log Z 6d n 1 of 6d (2, 0) SCFT of A n 1 −1 type at large n 1 .The three spacetime parameters are ∆ 2,3,4 , and the two U(1) 2 internal parameters are ω, −∆ 1 .Either from QFT [11,32] or gravity [33] considerations, one obtains log Z n 1 ,0,0,0 = log Z 6d at I ∆ I − ω = ±2πi.When all n I 's are nonzero, there would be four different 6d QFT's, and also extra modes supported on the intersection S3 of two giants and on the intersection S 1 of three giants.We suggest that the net large n I free energy is given by This formula reduces to the expected ones like (3.7) when only one n I is nonzero.This is the unique expression with the correct limits and the I n I ∆ I dependence.
Expanding the numerator of (3.8), one obtains contributions from the 6d/4d/2d modes: log Z 6d n I are given by (3.7) or its permuted versions, and the other terms are given by log We can independently justify them.log Z 4d n 1 ,n 2 is the free energy of an SCFT at the intersection of n 2 M5-branes on 012345, and n 1 M5-branes on 01236, 10.Its free energy takes the form of , where P is the cubic anomaly polynomial for SO(2) 45 , SO(2) 6,10 , SO(3) 789 . 3The SO(2) 45 -SO(3)-SO(3) anomaly can be computed by separating the n 2 M5-branes along the 6'th direction, and compactifying the 10'th direction to obtain Witten's SU(n 1 ) n 2 −1 MQCD [34].(SO(2) 6,10 is explicitly broken by the deformations.)This is a linear quiver with n 1 fundamentals attached to each SU(n 1 ) node at the end.The large n I anomaly is given by where n V is the number of vector multiplets.This yields the following contribution to the anomaly polynomial P (e.g.see eqn.(2.34) of [35]): This explains the first term of log Z 4d n 1 ,n 2 in (3.10).Similarly, its second term is explained from the SO(2) 6,10 -SO(3) 2 anomaly of the SU(n 2 ) n 1 −1 MQCD.log Z 2d n 1 ,n 2 ,n 3 can also be computed from anomalies, but here we just explain a quick check of its coefficient from the entropy of 4d black holes obtained by triply intersecting M5-strings with momentum p.The entropy is given by 2π √ n 1 n 2 n 3 p, which is obtained by extremizing π 2 n 1 n 2 n 3 β + pβ.This is the Cardy limit of log Z 2d n 1 ,n 2 ,n 3 in (3.10), upon taking ∆ 4 = β ≪ 1 and ω = I ∆ I ∓ 2πi ≈ ∓2πi.
Now we extremize the entropy function given by with (3.8).As in section 2, we may either understand it as maximizing Re[S] with real n I 's for small black holes, or extremizing S with complex n I 's for generic black holes.We present the calculation in the latter viewpoint.Extremizing S in n I , I n I ∆ I is given by We should pick one saddle point solution.To explain this, we employ the convention of [36,37] on the square-root, which sets √ ∆ 4 = −∆ 2 when all ∆ I 's are equal.Then, among the two solutions of (3.14), one should choose the upper/lower sign at 4 I=1 ∆ I −ω = ±2πi, respectively.We showed this by a Picard-Lefschetz analysis like that of section 2.2.With this choice, let us first discuss the small black hole limit, in which case ∆ I ≪ 1 are small positive numbers and ω ≈ ∓2πi.Then (3.14) at equal ∆ I 's reduces to yielding real positive I n I .This ensures that extremizing S with complex n I is equivalent to maximizing Re[S] with real n I in the small black hole limit.
Inserting the solution picked in the previous paragraph into (3.13),one obtains which perfectly reproduces the entropy function of BPS black holes in AdS 4 × S 7 [36].
Assuming the analytic continuation picture, log Z n 1 ,0 at large n 1 ∼ N will be given by the 3d free energies on n 1 M2-branes.Taking ω 1,2,3 , −∆ 1 as the internal parameters and ∆ 2 as the worldvolume parameter, the analytically continued free energy is given by [36,37,38,39] Similar expression is obtained for log Z 0,n 2 .Now we suggest that the general log Z n 1 ,n 2 at large n I ∼ N is given by log with (3.19), one obtains Inserting this back to the entropy function, one obtains which is precisely the entropy function of BPS black holes in AdS 7 × S 4 [33].
Before finishing this exercise on AdS 7 × S 4 , we comment on a puzzle that we resolved only partly.In section 2, it was important to have a leading order degeneracy after fixing I n I ∆ I by extremization, for the continuum conjecture and cancellations of Z n I 's.Here, fixing n 1 ∆ 1 + n 2 ∆ 2 at general complex ∆ I 's does not leave any degeneracy.So one may wonder whether the picture of section 2 is valid here.Curiously, we can see that the apparent real maximum n 0 I of Re[S(Q I , J i , n I )] is generally subject to severe cancellations already with the leading free energy.To simplify the discussions, let us unrefine ∆ I , ω i and keep n The maximum n 0 satisfies Re[f ′ ( n 0 N 2 )] = 0. Now considering a neighborhood of n 0 , one finds Since the real part of the second term vanishes at n 0 , the change of Re[S] is very slow in a wide range of |∆n| (≪ N 2 ).However, Im[S] generally varies fast in this neighborhood due to the second term NIm[f ′ ( n 0 N 2 )]∆n.This causes cancellations of nearby terms around n = n 0 , reducing the apparently over-estimated entropy at n 0 .This cancellation happens because of the large N scalings S ∼ N 3 and n ∼ N 2 : similar cancellations do not happen in AdS 5 /CFT 4 with S ∼ N 2 and n ∼ N.This provides the expected cancellations of the giant graviton index.We would like to further understand what makes the full continuum conjecture for n I 's possible.

Conclusion
In this paper, we studied the microstate counting of BPS black holes in AdS 4,5,7 from the viewpoint of giant graviton expansions.
We employed a saddle point approach to compute the large N giant graviton index Z n 1 ,n 2 ,n 3 in AdS 5 × S 5 , when the three electric chemical potentials are unrefined.Further maximizing the corresponding entropy in n I , we successfully accounted for the microstates of small AdS 5 black holes.To understand the black holes at general sizes, one had to take into account the cancellations of Z n 1 ,n 2 ,n 3 's.We conjectured a particular cancellation mechanism which successfully reproduces the entropies of the black holes at arbitrary sizes.We further proposed the large N free energy for Z n 1 ,n 2 ,n 3 with all chemical potentials refined.
We interpreted the large N giant graviton free energies as the analytically continued indices of SCFT's.Extending this interpretation to other AdS/CFT examples, we found intriguing connections between the SCFT free energies in various dimensions.We obtained the large N free energies of 6d/3d SCFTs on M5/M2-branes from the analytic continuations of vice versa, after suitably dressing them by certain defect free energies.This suggests that the (BPS) spectrum of quantum gravity admits dual formulations, either in terms of electric or magnetic variables.Presumably, there will be more duality relations of this sort that one can find from giant gravitons and analytic continuation.As a crude but nontrivial exercise, we also tried to account for the large N free energy of the 5d SCFTs dual to the massive IIA theory on AdS 6 × S 4 /Z 2 [36,40,41], from analytic continuations of the free energy of D2-brane SCFTs on an orbifold.Although the details need to be clarified, we find that the N Perhaps we should also comment on our current understanding of the giant graviton expansion.The formula was constructed from maximal giant gravitons, wrapping the largest (non-topological) S 3 ⊂ S 5 .The worldvolume degrees of freedom include tachyons which shrink the giant gravitons.The tachyonic quiver theories on these branes were used in a subtle way [13] to write down the giant graviton index formula.Technically, the tachyonic part of the partition function was first analytically continued.Then one chose the path integral contour for the holonomy zero modes empirically.After these ad hoc steps, the resulting Z n 1 ,n 2 ,n 3 appears to correctly describe the spectral problem of the Yang-Mills index.However, its physics looks different from that of the original quiver system.Namely, the 'ground states' of Z n 1 ,n 2 ,n 3 describe the trace relation constraints, subtracting the over-estimated states in the graviton index.We feel that a well-defined QFT cannot show this behavior.So it seems desirable to better understand the relation between the quiver and the formula.
Giant gravitons were originally conceived in [1] as a gravity mechanism of imposing trace relation constraints at E ∼ N 1 .However, in this paper, we found that their worldvolume degrees of freedom are also responsible for enhanced entropies at E ∼ N 2 .Perhaps new BPS states exist thanks to the trace relations.Since crucial roles were played by the open strings connecting various D-branes in our studies, one can construct an ansatz for the BPS cohomologies in terms of open spin chains ending on determinant operators [42].We expect that such an ansatz will be relevant at least in the small black hole regime, q ≪ N 2 .
In section 2.2, we conjectured a possible mechanism in which various Z n 1 ,n 2 ,n 3 's are partly canceled to account for the black hole entropies correctly.A key feature was the discrete sums of n I 's being effectively replaced by integrals.We have no a priori justification for this conjecture, except that the final entropy derived with this assumption is correct.Here we simply want to note that the discrete points labeled by (n 1 , n 2 , n 3 ) are special points on the moduli space of 1  8 -BPS giant gravitons [29].As alluded to in [14,30], they could be the fixed points of the equivariant localization calculation of an integral over this moduli space.Perhaps it may be helpful to seek such an integral reformulation of this sum to justify our continuum conjecture.
34) with a nontrivial function δ n (n I ) ∼ O(1) ≪ n of n I .This makes the distribution of n + δ n (n I ) dense over a range of O(1) width around n. (2.34) is a claim about the 1 N subleading corrections of Z n 1 ,n 2 ,n 3 , since all δ n (n I ) dependent terms are subleading.If this happens to all values of n, one would obtain

Figure 1 :
Figure 1: Contour deformations at various q.Green interval is the original integration contour [0, x * ].Blue arrows denote the gradient flow which determines the contour deformation.The deformed contour is the union of two solid black lines C 1 , C 3 , and the solid red line C 2 .The solid blue line is the steepest ascent contour.

5 2 5 3
scaling of the former free energy is obtained from the n scaling for D2-branes in massive IIA theory.
19))This gives the desired limits when either of n 1 , n 2 vanishes, and depends only on I n I ∆ I .It defies factorization between 3d-1d degrees of freedom.