Large black hole entropy from the giant brane expansion

We show that the Bekenstein-Hawking entropy of large supersymmetric black holes in AdS$_5\times S^5$ emerges from remarkable cancellations in the giant graviton expansions recently proposed by Imamura, and Gaiotto and Lee, independently. A similar cancellation mechanism is shown to happen in the exact expansion in terms of free fermions recently put-forward by Murthy. These two representations can be understood as sums over independent systems of giant D3-branes and free fermions, respectively. At large charges, the free energy of each independent system localizes to its asymptotic expansion near the leading singularity. The sum over the independent systems maps their localized free energy to the localized free energy of the superconformal index of $U(N)$ $\mathcal{N}=4$ SYM. This result constitutes a non-perturbative test of the giant graviton expansion valid at any value of $N$. Moreover, in the holographic scaling limit $N\to\infty$ at fixed ratio $\frac{\text{Entropy}}{N^2}\,$, it recovers the 1/16 BPS black hole entropy by a saddle-point approximation of the giant graviton expansion.


Introduction
Recently, the counting of small 1 16 -BPS states in 4d U (N ) N = 4 SYM on R × S 3 [1,2] has been nicely related to the problem of counting graviton and giant graviton BPS excitations in AdS 5 ×S 5 [3,4] [5,6]. It is known that upon truncation to charges of order N or smaller, the 1 16 -BPS index I(q) matches the truncation of the index I KK (q) of a gas of gravitons in AdS 5 × S 5 . This match happens at any value of the gauge rank N .
The index counting multigravitons I KK (q) is independent of the gauge rank N [1]. The N -dependence of the complete index I(q) appears when it is truncated at powers of order N or larger [7,8]. These N -dependent contributions can be always reorganized in linear combinations of subsums with overall weights q nN 1 , where n is a non-negative integer.
As recognized in [3,4] [5,6], there is at least one such reorganization for which the q nNweighted subsums correspond to truncated indices of n D3 brane excitations wrapping supersymmetric and contractible S 3 -cycles in S 5 . However, such reorganization is obviously non-unique. 2 Whether this correspondence holds for the complete q-series I(q) or it is just a property of its truncation at certain order of charges remains an open question. 3 For example, it is plausible that new stringy excitations in AdS 5 × S 5 are required at large enough N and charges of order N 2 in order to keep the correspondence going. For such charges the number of 1 16 -BPS gauge-invariant states in the gauge theory grows as the exponential of the Bekenstein Hawking entropy of the dual BPS black holes [9][10][11]. Thus, in a sense, it is a priori unclear whether such an entropy growth can be understood by working solely within the D3 brane systems prescribed by the proposal of [3].
Another giant graviton-like reorganization of the index, an exact one by construction, has been recently put forward in [12]. 4 This reorganization is not quite the same as the proposal of [3] -as explained in [13] -but it seems to be closely related to it as argued in [12] and [14] . Being an exact expansion, it would be useful to understand the physics behind it and how close it is to the physics of the proposals of [3] and [5]. 5 The main goal of this paper is to study the giant graviton representations of [3,5,6] and [12] at large charges and to compare the results with the ones obtained with the canonical matrix integral representation [10,[18][19][20][21].
Using the representation of [3] and working in the macrocanonical ensemble, we will show that an exponentially large number of cancellations occurs when summing over individual giant graviton contributions at large charges and for all N . Such cancellations can be explained in terms of an extremization mechanism for the giant graviton number n . At N ≫ 1 this mechanism explains how the dual black-hole entropy is recovered within the giant graviton expansion, and its derivation provides, in particular, a first-principle explanation of the large-N extremization mechanism proposed in [22]. More generally, the mechanism here identified implies that the latter cancellations continue to happen at large charges for any value of N , not just in the large-N expansion. It will be also shown that a similar extremization mechanism holds for the exact giant graviton-like representation of [12] and checked -against numerics -how such mechanism exactly accounts for the exponentially large cancellations happening after summing over individual giant graviton-like 2 For instance, assume N = 8 then a monomial q 16 in the total index can be divided in many ways into contributions coming from the subsums labelled by n = 1 and n = 2 . 3 For the Schur limit of the 1 16 -BPS index the correspondence applies to the complete q-series [5,6]. 4 This study covers a family a matrix integrals that include the superconformal index as a particular case. 5 It would very interesting to understand whether there is a systematic way to identify holographic dualities of this kind starting from the partition function of free gauge theories. The approach put forward in [5,6] seems natural to start thinking about this problem. The approach of [12] gives a first step in such a direction as well. The next step though, which would be to understand how to translate the averages over free-fermion systems to partition functions of brane systems in AdS5 × S 5 , seems more involved. Perhaps some of the ideas in [15,16] may be useful, at least to study 1 4 [17] and 1 8 -BPS indices, and to understand what stringy/brane excitations the individual free-fermion contributions are counting.
In the representation of [3,5,6], this extremization mechanism will tell us that the black hole entropy [23][24][25] comes from the superposition of a pair of complex conjugated saddle points whose semiclassical contributions evaluate the sum over giant graviton brane number n. The canonical matrix integral representation of the index [1] is known to be dominated by a pair of complex conjugated eigenvalue configurations too [11,[26][27][28][29] [8]. The latter and the former pairs are related: they provide two different interpretations of the very same contributions to the index at large charges of order N 2 . 6 It remains for the future to understand the physics of the excitations accounting for subleading corrections in both, the canonical matrix model and giant graviton(-like) expansions, and for both small and large black holes. 7 8 The paper is organized as follows. After a summary of results, in section 2 we explain how the large-charge approximation simplifies the counting of states, and introduce tools that will be useful later on. In section 3 we introduce conventions, and the two representations of the superconformal index that we will study. In section 3.3, and as warm-up for the analysis of the giant graviton indices, we compute the large charge asymptotics of the superconformal index using a novel approach that turns out to be convenient for our scope. In section 4 we apply the previously mentioned asymptotic tools to understand how the large-charge growth of the index is matched by the large-charge counting of giant gravitons for all N not just at N ≫ 1 . In appendix C we explain the role played by the choice of contour of integration [6] [17] in the large charge expansion. In appendix D we move on to answer the very same question from the exact representation of the superconformal index put-forward in [12] and conclude explaining how exponentially large cancellations among individual giant graviton-like contributions are understood in the macrocanonical ensemble.

Summary of main results
Let us briefly summarize our main results. Detailed expositions will be presented in the main body of the paper.
As mentioned in the introduction, the authors of [3,5,6] proposed that the superconformal index I of four-dimensional U (N ) N = 4 SYM on S 3 can be expanded in a sum over indices I n := I KK I n 1 ,n 2 ,n 3 of stacks of n 1 , n 2 , n 3 giant graviton D3-branes wrapping three contractible S 3 -cycles in S 5 . 9 The details of this proposal will be given in subsection 3.1. Schematically, it looks as follows I(τ ) = ? n 1 ,n 2 ,n 3 where a(Q)e 2πiτ Q , I n (τ ) = Qgg ∈ Sgg a n (Q gg )e 2πiτ Qgg . (1. 2) The letter τ denotes the set of chemical potentials {φ 1 , φ 2 , φ 3 , . . .} dual to the conserved global charges -including spin J and R-charges -whose individual eigenvalues in both N = 4 and the infinitely many stacks of giant graviton branes 10 will be denoted by the letters Q and Q gg , respectively. S N =4 and S gg , are the respective charge-lattices. These two lattices are very different and in consequence the domains of convergence of the Hamiltonian traces that define I(τ ) and I n 1 ,n 2 ,n 3 (τ ) are different. Thus, to check (1.1), analytic continuation in τ is necessary, either in the left or right-hand side. The microcanonical version of the proposal (1.1) is n a n (Q gg ) = For later convenience it should be said that the sum over n in (1.3) is not really a series because a n (Q gg ) vanishes for large enough values n , at a fixed Q gg . 11 (1.3) says that the sum over giant graviton numbers n must project the BPS giant graviton spectrum S gg to the much smaller gauge-theory spectrum S N =4 . As said before, the proposal (1.3) has been checked for small enough values of Q ∼ N [3] [6].
Our goal is to show that at large charges Q → ∞ (and for all N ) a precise version of the following asymptotic relation holds 12 n a n (Q) ∼ a(Q) ∼ e ( √ 3)3 1/3 π c J 2/3 N 2/3 , (1.4) meaning that at large charges and for all N the sum over the giant graviton microcanonical indices matches the exponential growth of 1 16 -BPS states. In this relation the quantity c is an order 1 real contribution that depends on how fast the spin J grows in relation to the R-charges, we will come back to comment on it below (e.g. a particularly simple case where c is simply a c-number will be reported in (3.74)).
Let us briefly explain how (1.4) will be obtained. In subsection 2.3 we will introduce a large-charge localization Lemma that will help us to compute localized contributions a loc ±,n to the giant graviton index. If we define the microcanonical index of giant gravitons to be a n (Q) := with Γ being a period of the integrand, the a loc ±,n are two equally-dominating contributions to (1.5) in its asymptotic expansion at large R-charges, fixed J , and fixed n a n (Q) ∼ a loc +,n (Q) + a loc −,n (Q) . (1.6) These two contributions ± are complex conjugated to each other The large-charge localization Lemma of 2.3 will tell us that the contours Γ ± can be understood as small subpieces of the contour Γ , centered at the leading (exponential) divergencies of I n (τ ) . The latter divergencies of I n (τ ) are in the subset of chemical potentials φ = {φ 1 , φ 2 , φ 3 } dual to R-charges. In the cases of interest to us, there are two types of such divergences that we label by the two choices of signs ± . The localized integrands I (±) n (τ ) are the leading asymptotic expansions of I n (τ ) around the divergencies ± and are exact in the chemical potentials τ that are not φ's.
After commuting the sum over n with the integrals over τ in (1.5) one obtains n a loc ±,n (Q) = As it will be explained in the main body of the paper, the sum over n can be replaced by an integral over a compact domain whose asymptotic behaviour around the singularities ± (and at large R-charges) can be obtained by the saddle point method The saddle point condition ends up taking a simple linear form that fixes n = n ⋆ := n ⋆ (τ ) as a function of τ . The function n ⋆ (τ ) is defined by a linear relation of the schematic form where f ± (τ ) are cubic polynomials in τ such that |f ± (0)| is finite and non zero. The explicit form of this equation will be specified in the main body of the paper. 13 To compute the asymptotic behaviour of Γ ± dτ I (±) n ⋆ ± (τ )e −2πiτ Q at large Q, not just at large R-charges as before, but also at large spin J , we use again a saddle point evaluation (1.11) This time the saddle-point condition fixes the τ 's as a function of Q, for instance schematically one obtains with c τ ± being order 1 contributions that depend on how fast the spin J grows in relation to the R-charges. At this point we simply collect results and obtain n a n (Q) ∼ a loc which after trivial algebraic manipulations leads to the announced asymptotic relations (1.4). By composing (1.10) with (1.12) we obtain the scaling properties of the complex saddle point configuration that dominates the sum over giant gravitons In this equation c 1,± and c 2,± , again, represent order 1 14 contributions that depend on how fast the Lorentz spin J grows in comparison with the R-charges. In particular, we note that c 2,± are complex quantities. 15 In summary, the asymptotic relations (1.4) show that the giant graviton proposals of [3] [5,6] capture the large charge (for all N ) asymptotic growth of the microcanonical superconformal index. In particular, in the limit 15) they tell us that the latter proposals match the exponential growth of 1 16 -BPS black holes in AdS 5 . 16 2 State-counting at large charges The large charge approximation has been a useful tool in varied contexts as, for example, the computation of anomalous dimensions, correlation functions, partition functions, the conformal bootstrap, cf. [32][33][34][35][36] [37]. Let us explain briefly how this tool applies to the counting of operators in quantum statistical system. 17 14 If one fixes the angular momentum J to be small and instead considers large R-charges then the conlcusions are different (See the discussion in the last paragraph of subsection 3.3). In this paper we will not study in detail this other domain of the spectrum of charges. 15 They are related to the constant c in (1.4). 16 Note that for the black hole scaling (1.15) the absolute value of the complex saddle points (1.14) becomes of order N as expected. 17 In the context of superconformal and topologically twisted indices a particular case of one such largecharge approximation known as the Cardy-like approximation has been thoroughly studied in the last few years [10,[18][19][20] [21,[38][39][40][41][42][43][44][45][46][47][48][49-52] [53]. Perturbative corrections to the leading asymptotic behaviour of four-dimensional N = 1 superconformal indices in the large charge expansion have been exactly matched against higher-derivative corrections to the leading semiclassical onshell action of AdS5 black holes in the relevant dual supergravities [54][55][56]. It would be very interesting to study the large charge expansion of the partition function at non-vanishing coupling , of say N = 4 SYM , at least in near-BPS sectors, in [57][58][59][60] [61]. The goal being to try to extract universal lessons that could be compared against recent holographic expectations e.g. [62,63].
Consider a 2π-periodic complex function f = f (x) = f (x+2π) with a set of singularities at x = x a,sing ∈ R , a = 1, 2, . . ., such that where the definition of the symbol ∼

Λ→0
, which denotes an asymptotic relation, is given in appendix A. Let us consider the average over a cycle Γ that can be decomposed in an integral combination of Lefschetz thimbles Γ x ⋆ ending at saddle points x = x ⋆ of the exponent f (x) + ixQ . Under these assumptions, the leading asymptotic behaviour of d(Q) in the large charge approximation is determined by the asymptotic form of the saddle points x * , which in the large charge regime become infinitelly close to the singularities x a,sing , Then, under the previous assumptions and in the large charge approximation, we have where a ⋆ , δx ⋆ a ⋆ label the singularity a = a * and the solution δx ⋆ = δx ⋆ a ⋆ of (2.5), respectively, that maximize the real part of the exponent Λ n f a (δx ⋆ )−i(x a,sing Λ+δx ⋆ )q . The definition of the symbol ∼ exp Λ→∞ , which denotes an asymptotic relation, is given in appendix A.

An illustrative example
As an example, we briefly discuss a simple toy model. Let us assume Q = qΛ 3 ∈ Z + , In this case, we have n = 2 , and Let us fix the integration cycle as follows Obviously d(Q) is convergent, because Γ is compact and it does not intersect the set of singularities x a,sing = 0 + 2πi(a − 1) , a = 1 , 2 , . . . . (2.10) There are three saddle points around each singularity x a,sing . At large charge, they take the form Notice that we have engineered the integration cycle Γ to intersect the last saddle. This guarranties |d(Q)| ↗ ∞ for Q ↗ ∞ . Indeed, one can check numerically for Q ∼ 100 and larger, that the integral d(Q) localizes to the integrals over the infinitesimal vicinity of the contour Γ that becomes infinitely close to the singularities. More precisely, at large charges, d(Q) localizes to its saddle-point approximation which is, at leading order, 18 The prediction coming from the saddle point intersected by Γ for Q ≪ 0 is d(Q) = 0 , which happens to be the correct answer as well, i.e., the answer we computed from the direct numerical evaluation of the integral d(Q) at Q ≪ 0 . This happens because the cycle Γ has zero intersection number with the Lefschetz thimble ending at the saddle point that produces exponential growth of the quantity e f (x ⋆ )−ix ⋆ Q at Q ≪ 0 , which is the first one in (2.11).

Application to the superconformal index
In the case of the superconformal index, we are interested in computing integrals over multidimensional cycles of the form at large charges. Here, x denotes the set of four chemical potentials dual to four global charges Q ′ . Γ and Γ gauge are integration cycles that we assume can be decomposed in integral combinations of Lefschetz thimbles of S eff (x; u) . The effective action −S eff (x; u) is the logarithm of the integrand of the superconformal index I(x) := Γgauge du e −S eff (x;u) . As it will be shown below, S eff (x; u) has leading singularities located at x 4,sing , x 5,sing = 2πi(a 4,5 − 1) .
(2.15) 18 Having into consideration the contributions from the two saddles whose thimbles are intersected by the contour of integration [0, 2π], labelled by a = 1, 2, and one-loop logarithmic corrections about each one of them, one obtains an improvement of (2.13). Comparing absolute values for simplicity, as we will eventually do, one obtains Now the quotient between the left and right-hand sides is 1 at Λ → ∞.
The free energy takes the form and most importantly, these two functions are asymptotically-equal Notice that the singularities (2.15) are not points but a 2-cycle Γ 1,2 spanned by the variables x 1,2 (times the integration cycle Γ gauge over the gauge potentials). Then, following analogous reasoning as before and using (2.18), it follows that at large charges where the ⋆ denotes one of the saddle points that maximize the real part of the exponent in (2.20) among those intersected by the Lefschetz thimbles that compose the original cycle Γ⊗Γ c . As in the simplest toy example before, such saddle points will be asymptotically close to the singular locus Γ 1,2 × Γ gauge of S eff (x; u) in the scaling limit (2.19). Note that the first perturbative corrections in the 1 Λ -expansion are also captured by (2.20). They are encoded in the Laurent expansion of s(x) around x 4 = x 5 = 0.

Large-charge limit as a localization mechanism
Let us come back to a generic function f = f (x) with a regular singularity x sing ix sing Q = 2πin , n ∈ Z , (2.21) such that with a single dominating saddle x ⋆ = x sing + δx ⋆ Λ . Let us further assume that given the equality Then, as we explained before, in the large-charge scaling limit where Γ δx ⋆ is the Lefschetz thimble of f Λ (δx) − iδxq intersecting the dominating saddle point δx ⋆ . After scaling the variable δx → yΛ , equations (2.26) and (2.23) imply where Γ y ⋆ is a Lefschetz thimble of f (y) − iyQ that ends up at the dominating saddle point y ⋆ . 19 Thus, to compute the asymptotic behaviour of d(Q) at large values of charges we only need to plug the asymptotic expansion of f (x) around x = 0 This integral will be called the large-charge-localization or large-charge coarse grain of the original integral Γ e f (x)−ixQ , and it is much simpler to study. Roughly speaking, this localization mechanism tells us that at large charges the function f (x), which could be rather complicated, can be substituted by its asymptotic expansion f (y) around the singularity y = 0 , i.e., the singularity that attracts the leading saddle point y = y ⋆ at large charges. It should be also noted that the integration cycle needs also to be modified as indicated before. The subleading and perturbative terms in the asymptotic expansion of f (y) give exact perturbative corrections to the leading prediction for the asymptotic growth of d(Q). The generalization of this localization mechanism to the case where f (x) depends on more than one variable (when the singularities can be not only points, but also cycles), is straightforward. For example, the microcanonical index (2.14), is such that the localized action s (essentially the series expansion of the complete effective action about the leading singularity) Then as a consequence of (2.20) it follows the large-charge localization formula or lemma: In this equation Γ y ⋆ ,u ⋆ is a 4 + N -dimensional integration contour. It is also a combination of Lefschetz thimbles of − s(y; u) − iy · Q and it intersects the leading saddle point(s) y ⋆ , u ⋆ ∂ u s(y; u) y=y ⋆ ,u=u ⋆ = 0 , ∂ y s(y; u) with intersection numbers defined by the decomposition of the original integration contour Γ×Γ gauge in terms of the Lefschetz thimbles associated to the original exponent S(y; u) − iy · Q.
In conclusion, to compute the asymptotic behaviour of d(Q) at large values of charges we need, first, to compute the asymptotic expansion of s(y 1 , y 2 , y 4 , y 5 ) around y 4 = y 5 = 0 s(y; u) = s (1,1) (y 1 , y 2 ; u) y 4 y 5 + s (1,0) (y 1 , y 2 , y 4 , y 5 ; u) y 4 + s (0,1) (y 1 , y 2 , y 4 , y 5 ; u) y 5 + subleading (2.35) which, by construction, is the same as the asymptotic expansion of the complete effective action S(y, u) around y 4 = y 5 = 0. Second, we must compute the leading saddle point values y ⋆ and u ⋆ of the desired truncation of (2.35). Then, at last, we obtain the following asymptotic formula In the following sections we will use this recipe, and particularly its integral version, the large-charge localization formula (2.33), to compute asymptotic behaviours.

The 1 16 -BPS index at large charges
The superconformal index of 4d N = 4 SYM on R × S 3 is defined as [1] Substituting it in (3.1) fixes the four-dimensional lattice of charges within the five-dimensional lattice spanned by J, J, Q 1 , Q 2 , Q 3 that commutes with the two super (conformal) charges that define the index I . The commuting charges in (3.1) are defined as follows in terms of the dilation operator E, the left and right angular momenta J L,R in the Cartan of the SO(4) = SU (2) × SU (2) isometries of S 3 , and q 1 , q 2 and q 3 are the Cartan elements of the SO(6) R-symmetry. 20 The following definitions of rapidities and chemical potentials will be useful later on For gauge group U (N ) the index can be written in the form [1] where U a is the a-th diagonal component of a diagonal unitary matrix, and U ab = U a /U b . The measure in (3.5) is defined as and The plethystic exponential is defined as usual for any rational function R of d rapidities x 1 , . . . , x d . In particular, Summarizing different representation for the index that can be found in various references [1][2] [65] (see also, for instance [66]) we recall that where the normalization (or zero modes) factor is defined as and Pexp (i(w; p 1 , p 2 )) : (3.12) 20 We use the conventions and values of charges of fundamental letters of e.g. [64].

Two ways of implementing the constraint among rapidities
The constraint can be implemented in various ways.
Expansion A) The implementation (A) (and analogously for the case obtained by the permutation of the indices 1, 2, 3 of w's) defines the following series expansion in terms of the four charges For (3.15) to be a well-defined expansion, i.e. for it to follow from the original representation (3.5), requires imposing the following condition (A) guarantees absolute convergence of the series in the exponent of the plethystic exponential defining the index (3.10).

Scaling limit A)
For later purposes, we note that the condition (3.17) implies that in a scaling limit to the boundary of the convergence region of representation A) Thus, we are free to assume that in such a scaling limit where Im(∆ J,K ) is a generic real number (which eventually we will require to be different from 2πn, with n integer).

Expansion B) The implementation (B)
(and analogously for the case obtained by the permutation of the indices 1, 2 of p's) defines the following series expansion that counts degeneracies as a function of the four charges These charges relate to (3.16) as follows Obviously, the two degeneracies d and d are related by the composition conditions (3.25). For (3.23) to be a well-defined expansion of the index I, i.e. for it to follow from the original representation (3.5), requires imposing the following condition (B) (3.26) Scaling limit B) For later purposes, we note that the condition (3.26) implies that in a scaling limit to the boundary of the convergence region of representation B) Hence, we are free to assume where Im(ω 1 ) is generic real number (which eventually we will require to be different from 2πn, where n is an arbitrary integer number). We will use the expansion B) for the study of the giant graviton representation. As mentioned before, the domain of convergence of the giant graviton Hamiltonian traces is different from the one of the 1 16 -BPS index of N = 4 SYM. In such an analysis, extensive use of analytic continuation will be required.

The giant graviton proposal
The giant graviton expansion proposed in [3] is where I KK is the generating function of 1 16 -BPS multi-graviton excitations at N = ∞ (closed strings contributions) and I GG is the giant graviton index Here, I n 1 ,n 2 ,n 3 is the index of n 1 , n 2 and n 3 stacks of D3 branes wrapping three different S 3 cycles within the internal space S 5 (i = 1, 2, 3), times the index of open strings ending on pairs of stacks [3]. Concretely, with measure The closed contour Γ gauge , which is not the trivial unit-circle, has been proposed and tested at small values of N and charges in [6] [3]. Another seemingly valid definition has been given in [6]. 21 For reasons that will be explained in Appendix C the explicit form of the closed contour Γ gauge plays (almost) no role in the large-charge expansion. To understand this one must rely on results that will be derived in subsection 4.2. So, from now on we postpone any discussion on Γ gauge until appendix C. The objects: The 4d adjoint contributions are for I ̸ = J ̸ = K = 1, 2, 3 , and the zero-mode contributions are defined as .
The contributions to the index coming from a 2d U (n 1 ) × U (n 2 ) bi-fundamental field are In this expression J ̸ = I, I + 1 mod 3 . We define the quotient of diagonal components of different unitary matrices as U (I,I+1) ab

The free fermion representation of the index
An exact expansion of the index as an average over an ensemble of free fermion systems was put-forward in [12]. As we explained in the introduction, it takes again the form of a giant-graviton expansion, different from the physically motivated D-brane expansion. Still, it is a mathematical exact rearrangement of the index and it will be interesting to consider 22 Following the conventions of the original proposal of [3] here we have assumed a loop = a12a23a31 = 1 .
In that case, without loss of generality we can assume a (I,I+1) = 1 (See equation (11) in [3]). More generally, the analysis in section (4.2) can be straightforwardly reproduced for any other choice of a (I,I+1) , however, the only for a loop = 1 we obtain consitent results.
its properties. In particular, we will discuss in Appendix D the detailed way it reproduces the large black hole entropy. In this representation, the index reads and The object J n (N ) is a Hubbard-Stratonovich transformation of a determinant of two-point functions in an auxiliary theory of free fermions [12].
Using the identity (and ignoring the ′ in the z ′ i 's from now on) one reaches the form that we will work with where . (3.47)

The index at large charges
Let us fix the constraint (3.14) and study the large charge asymptotic behaviour of the microcanonical index The 4πi is because the charges Q ′ 4,5 are quantized in units of 1/2. The two saddle point positions ω ⋆ 1,2 (which are not pure imaginary) will be determined below. The effective action has singularities located at Around these singularities: Using the formal Taylor expansion [67] 1 on the denominator in the right-hand side of (3.50) one computes the small-1/Λ expansion of the effective action S eff s(y; u) = s (1,1) (y 1 , y 2 ; u) y 4 y 5 + s (1,0) (y 1 , y 2 , y 4 , y 5 ; u) y 4 + s (0,1) (y 1 , y 2 , y 4 , y 5 ; u) y 5 + c 4 N log y 4 + c 5 N log y 5 + c 6 N log(y 4 + y 5 ) + . . . , where s (1,0) (y 1 , y 2 , y 4 , y 5 ; u) , s (1,0) (y 1 , y 2 , y 4 , y 5 ; u) , (3.55) are linear functions of y 4 and y 5 , and dots denote contributions that vanish in the infinitely large scale transformation y 4,5 → For simplicity we choose to focus on the leading contribution Below we will show how to compute the subleading contributions. Recalling the expansion (3.56) can be rewritten as: is the periodic Bernoulli polynomial of order n . For example, for n = 3 one gets The contributions s (1,0) and s (0,1) , can be computed analogously. At leading order, the large-charge prediction for the degeneracy of states is: i.e. the saddle points of (3.62) with respect to (ω 1,2 , ∆ 1,2 ) -those that maximize the real part of (3.62).
The contribution of zero modes: computing c 4 , c 5 , and c 6 The contribution of zero modes in the second line of (3.50) determines the coefficients of the logarithmic divergencies log y 4,5 and log(y 4 + y 5 ). The easiest way to compute these contributions is to write and Taylor-expand the denominator, keeping as many terms as necessary. Then, we sum (over l) the coefficients of each monomial in the Taylor expansion. The result is a linear combination of polylogarithms. Many of such polylogarithms contribute to the terms (3.55). The remaining ones take the form where for 0 < ω 1,2 = −iy 4,5 < 1 c 4 = c 5 = 1 12 the dots in (3.64) denote terms that vanish after rescaling y 4,5 → y 4,5 /Λ and taking Λ → ∞ .
Logarithmic contributions with similar origins as (3.64) will appear in the study of the giant graviton expansions. They are subleading contributions (of type-F ) that will not affect the leading asymptotics we are looking for, but for future developments it may be useful to explain how to compute them.

Evaluating the saddle points
The saddle-point condition has a leading solution (independent of other chemical potentials) [20,27], The remaining saddle point conditions are piecewise polynomial conditions and can be solved straightforwardly. In this subsection we focus on counting operators with charges The solvability of conditions (3.68) requires Then the leading solutions of (3.68) are to the asymptotic growth of the microcanonical index along the region of charges (3.69) We note that this result is valid at any finite values of the rank N . Note also that in order to have order N 2 growth it is necessary to require Λ 3 ∼ N 2 .
Comments on the more general cases Let us assume Working with the analytic continuation to complex χ := ∆ 1,2 /(2πi) of the function (D.32), which was originally defined for χ ∈ R, the extremization conditions take the form (3.76) where the complex saddle value α = α ⋆ is defined by the cubic equation The asymptotic growth of degeneracies comes from the root α ⋆ with positive and maximal imaginary part of and one recovers the asymptotic growth computed in the previous case (3.74). On the contrary if r ≈ 0 (i.e. for small enough J ′ at fixed Q ′ 1,2 ) none of the saddle points of − s(x; u ⋆ ) − ix · Q carries exponential growth: the leading saddle value becomes a highly oscillating phase times a bounded function. This feature is not surprising because we expect many more operators at large spin and fixed R-charge, than the other way around.

Large charge entropy from giant gravitons
We move on to compute the asymptotic growth of the giant graviton index (3.30) or more precisely, in a large-charge expansion (Λ ≫ 1) defined by the scaling properties at any J ′ . Let us define the following particularization of chemical potentials x and charges Q ′ where the functions I 2d,4d n 1 ,n 2 ,n 3 , defined in (3.35), depend on the 3N gauge potentials := e 2πiu (I) a . Note that we have truncated the sums over n 1,2,3 . That is because the truncated terms do not contribute to the counting of degeneracies at charges smaller or equal than Q ′ 1 , Q ′ 2 , and Q ′ 3 (the explanation was given in footnote 11).
The procedure to follow is summarized in the following steps: 3. Use (4.7) in (4.6) and substitute the result in (4.1). Then commute the integral over x with the sums over n to obtain 24 4. Evaluate the asymptotic behaviour of the sum over n (in the large charge regime (4.2) we can safely drop the floor's) e − s (n 1 ,n 2 ,n 3 ) (x; u ⋆ ) . (4.9) 23 The integral over x can be commuted with the truncated sum over n , which is finite. 24 These integrals can be commuted because the localized integrand does not have poles: the logarithmic divergencies in the exponential are either suppressed or can be absorbed in a redefinition of gauge variables u .

5.
Substitute the entropy function of the gas of giant gravitons s GG (x) into (4.8), and localize the remaining integral over x to the leading saddle point x ⋆ which is the one attracted by the leading singularity of s GG (x) . At last one obtains and in their vicinity (the details behind the derivation of this formula are postponed to the following subsection) (4.14) Assuming (for the moment) we obtain for all n and for all N with where, again, these equations will be derived from scratch in the following section. The B 2 (x) in equation (4.17) is the periodic Bernoulli polynomial of order 2 In this equation, r comes from a subleading contribution to s (n 1 ,n 2 ,n 3 ) (x; u ⋆ ) which is a scale-invariant combination of ∆ 1 , ∆ 2 , and ∆ 3 . Naively, one would say that discarding this contribution would not change the leading asymptotic behaviour of the giant graviton index (in microcanonical ensemble) at large charges and spin. However, as we will show below such an assumption turns out to be incorrect. In particular, at large N , discarding r does not give a chance to recover the counting of microstates of large BPS black holes. Instead, it allows, at most, to recover the entropy of small black holes i.e. those with large values of charges Q, such that N 2 ≫ |Q| ≫ 1 [22].
The contribution r(x) turns out to be such that Step 4. further clarifies the relevance of r . The entropy functional − s GG (x) of the gas of giant gravitons is defined from To compute (4.20) at large Q ′ 1,2,3 (as detailed in (4.2)) it is convenient to change variables: In the new variables the sums over n 1,2,3 become integrals Precisely, From (4.16) it follows that this integral is Gaussian. Assuming for the time being that the x a are real and positive (the general result can be obtained by analytic continuation) then in the variables the integral measure (which acts upon an integrand that depends only on X) becomes is the area of a two-dimensional region Σ(X) spanned by pairs (Y, Z) ∈ R 2 such that As A Σ(X) is the area of a polygonal surface whose perimeter has length growing linearly with X and/or q ′ a x a , then A Σ(X) is always bounded from above by a polynomial function of X and q ′ a x a . This is all we need to know about A Σ(X) . Implementing the change of variables (4.24) and evaluating the one-loop saddle point approximation at large Λ one obtains x 1 x 2 x 3 at the saddle point locus Collecting results one obtains . This is because at leading order in such an expansion the integrand of (4.23) depends on a single direction in the three-dimensional space of (n 1 , n 2 , n 3 ) 's: the other two directions become flat, and thus, summing over giant gravitons configurations along such directions produces an overall factor proportional to Λ 4 N 2 . If and only if x is close enough to the zeroes of T (x), i.e. at distances of order 1 Λ of them, then grows exponentially fast with Λ . Indeed, our large-charge localization lemma implies that the zeroes of T (x) which are the leading singularities of s GG , determine the leading largecharge asymptotic behaviour of the integral

Control over subleading corrections in
Step 2 is essential to recover large spin growth Step 6: Is the asymptotic growth in the index of giant gravitons d GG ( Q ′ ) equal to the asymptotic growth of the superconformal index? i.e.
In the chambers If one naively substitutes (4.36) into the saddle point formula (4.33) assuming r(x) → 0, then one does not obtain the exponential growth at large spin of the superconformal index (3.74) (i.e. the degree of the singularity ω 1 → 0 or ∓ 2πi would be 1 < 2). Indeed, at large N and assuming r(x) → 0, the localized action (4.32) can lead, at best, to the asymptotic growth of microstates of small black holes [22] [1]. For example, if we assume r = 0 and focus on the particular locus of charges [22] J ′ =: j = 0 , then extremizing the entropy function with respect to the chemical potentials x one obtains at the saddle point values and for q > 0 the following prediction for the entropy (4.41) 25 In the asymptotic regime N 2 ≫ q ≫ 1 this is the leading term of the Bekenstein-Hawking entropy of small and supersymmetric black holes in AdS 5 with equal left and right angular momenta j = 0 [22]. 26 In order for (4.34) to hold, namely in order to obtain the asymptotic growth of the most generic index at large charges which are not too small in comparison with N 2 , it is necessary that which means that if the underlined contribution does not match the microscopic prediction of r(x) then the growth of the series of giant graviton indices can not account for the large charge growth of the complete superconformal index. In the following subsection we proceed to check whether r(x) equals (4.43)

Refined calculation and large black hole entropy
In this subsection the localized form s (n 1 ,n 2 ,n 3 ) of the giant graviton effective action S (n 1 ,n 2 ,n 3 ) eff is computed. We follow the steps summarized below the equation (4.6). The first step is to compute the asymptotic expansion S (n 1 ,n 2 ,n 3 ) eff near its leading singularity(ies).
Let us divide the effective action in three pieces (and omit the supra indices n 1,2,3 for a moment)  We proceed to compute the expansion Λ → ∞ of S (n 1 ,n 2 ,n 3 ) eff 25 Note that for finite N and q → 1 the singularity of the localized action N 2 4T (x) that attracts the saddle (4.40) is not ω1 = 0 but ∆ = ∞ . 26 Compare with the leading contribution in the first line of equation (2.26) of [22].
or equivalently the expansion of each of the four contributions in (4.44) and extract its localized form s (n 1 ,n 2 ,n 3 ) .
To compute this expansion we proceed as follows 2. Perform the sums ∞ l=1 in the result obtained after step 1.

Substitute
in the result obtained after steps 1. and 2. and expand the answer around ϵ = 0 up to order O(ϵ 0 ) being careful about logarithmic singularities.
4. Lastly, truncate the series at order O(ϵ 0 ), and re-scale back the variables to obtain an ϵ-independent effective action. Such an answer is the contribution of S ZM, Vect, Hypers , respectively, to the localized action s n 1 ,n 2 ,n 3 (x; u) .
Using these steps allows us to keep control over logarithmic corrections that appear in the expansion Λ → ∞ (coming from the action of vector zero-modes). Proceeding otherwise these non-analyticities would evidence themselves as infinite coefficients in the would-be-Laurent expansion around Λ = ∞ .
The large charge effective action of zero modes: Let us start computing the large charge effective action of zero modes following steps 1-4. To illustrate the procedure let us focus on a single zero mode contribution of the vector multiplet 1: .

(4.53)
After steps 1. and 2. we obtain for all n 1 , at order and at order O(ϵ −1 ) and at order O(ϵ 0 ) Then, after adding (4.56), (4.55) and (4.56) and implementing step 3., we obtain at order ϵ −1 and at order ϵ 0 where (assuming for the moment ϵ > 0, ∆ I > 0) (4.59) 27 At last, implementing step 4 we obtain the contribution of the zero mode 1 to the large charge action s (n 1 ,n 2 ,n 3 ) (x; u) (4.60) The contribution coming from the zero modes 2 and 3 are computed analogously. The general result is where the definition of log Ξ I can be recovered from (4.59) by the obvious permutation of subscripts. Assuming (4.35), equation (4.61) can be rewritten as Note that the contributions log Ξ I (x)n I coming from zero-modes are of the type F defined around (A.3) and thus we can ignore them in the following. However, to gain insight into their meaning, we will keep track of them from now on. The large charge effective action of vector non-zero modes To start let us focus on the contribution of the vector multiplet 1: and at order ϵ 0 (4.65) 27 We note that the term in the first line can be absorbed in a redefinition of the argument of the second out of the three logarithms in the second line and the second and third out of the six logarithms in the fourth line. This implies that the log Ξ1(x)n1 contribution is of type-F and thus it will not contribute at the degree of accuracy we are looking for. We will keep track of these contributions though, as we may learn something for the future.
At last, adding (4.64) and (4.67) and implementing step 4 we obtain the contribution of the vector modes 1 to the coarse grained action s (n 1 ,n 2 ,n 3 ) (x; u) (4.68) The contribution coming from the vector modes 2 and 3 are computed analogously. The general result is (4.69) The large charge effective action of hypermultiplets To start let us focus on the contribution of the hypermultiplet 3: After steps 1.-3. we obtain for all n 1 , at order ϵ −1 (4.71) and at order ϵ 0 (4.72) ( At last, adding (4.71) and (4.75) and implementing step 4 we obtain the contribution of the hypermultiplet 3 to the coarse grained action s (n 1 ,n 2 ,n 3 ) (x; u) (4.76) The contribution coming from the hypermultiplets 1 and 2 are computed analogously. The general result is (4.77) The coarse grained action: Collecting (4.62), (4.69), and (4.77) we obtain, at last, We note that contributions coming from zero modes have been obtained in a certain choice of branch that has simplified computations for us. Other choices of branch would give us a different answer. However, as we will show next, these contributions can be absorbed in a redefinition of the gauge variables which does not affect the leading saddle point evaluation. This is, at least at large charges the ambiguities coming from zero modes are indistinguishable from gauge-choice ambiguities and thus they do not affect the indices of giant graviton branes.
The gauge saddle point u ⋆ The next step is to find the leading saddle points u ⋆ of i ·e − s (n 1 ,n 2 ,n 3 ) (x;u) (4.79) in the small ϵ = 1 Λ expansion at fixed n 1,2,3 , this is, assuming After changing integration variables and plug the ansatz

equation (4.79) transforms into an integral over a new contour Γ
a,0 (x) ϵ 2 log ϵ + . . . , (4.85) 28 into the saddle point equations following from the action Then, we expand about Λ = ∞ and extract recurrence relations among the u's and the h's. One obvious saddle point solution to this recurrence relations is 29 where u 0 is a zero mode that is integrated out trivially and we can set it to u 0 = 0 without loss of generality (the integrand does not depend on this mode and thus, the corresponding integral gives 1). (4.87) is a saddle point of the coarse grained action (4.86) 30 , it is, on the other hand, a logarithmic singularity of the original effective action S (n 1 ,n 2 ,n 3 ) eff . The saddlepoint of the original effective action must have a non vanishing ϵ−subleading contribution which must be non-coincident, i.e., such that For our purposes knowing the explicit form of the small-ϵ correction to (4.87) is not necessary. All that we need to know is of its existence, which as it was just explained, it has to be the case. The existence of one such non-coincident solution implies the existence of other ∼ n 1 !n 2 !n 3 ! − 1 identical copies obtained by permutations of the gauge indices. Summing over these solutions cancels the 1 n 1 !n 2 !n 3 ! prefactor in (4.79). Thus, in the large R-charge expansion (4.2) (4.90) At last, in Step 6. we conclude that Namely, that the large charge asymptotic growth of the superconformal index of U (N ) N = 4 SYM on S 3 , at any N , equals the large charge asymptotic growth of the giant graviton index. The large-charge analysis for the representation of [12] is summarized in appendix D. 29 There are other saddle points u ⋆ of s (n 1 ,n 2 ,n 3 ) (x; u) corresponding to n1,2,3 -th roots of unity. Here we will focus on the leading ones (4.87) (See analogous discussions in [20,27,28]). 30 First, because s Ξ(n 1 ,n 2 ,n 3 ) is even in u; second, because s Ξ(n 1 ,n 2 ,n 3 ) depends only on differences of u's, and third, because s Ξ(n 1 ,n 2 ,n 3 ) has continuous first derivatives on u .

A Conventions
Let us assume two functions X Λ = X Λ (µ) and Y = Y (µ) of a set of variables µ. Let us assume the explicit dependence of X Λ on Λ to be such that its limit function in the Λ → ∞ is well-defined. Let us select a subset of variables α ⊂ µ and denote its complement as γ.
Let us assume that α = α 0 is a singularity of X and Y . Thus, if we define it follows that X Λ , Y → ∞ in the limit Λ → ∞ defined by keeping δα ren fixed. Based on the previous definitions we will say that in such Λ → ∞ limit The A, and F are functions of the δα (they can also depend on the γ) such that i.e. is invariant under homogeneous scaling of the δα and C is a c-number (independent of the µ). Obviously, if δα is a single variable then A is a c-number as well. Also, if the explicit dependence of X Λ on Λ is trivial, we can safely assume A = 0. The function F does not need to be scale invariant, but it needs to have only power-like zeroes and singularities in such a way that log F has only logarithmic divergences. The function c branch is fixed in terms of log F it is defined as a generic choice of branch cut of log F and thus choosing the appropriate branch we can always assume, and we will do so from now on, that c branch = 0 . Contributions to F can have perturbative 32 or non-perturbative 33 origin. Perturbatively, they could originate from subleading one-loop determinant contributions or subleading corrections to the effective action. Non-perturbatively, they could originate from the superposition of complex conjugated saddle-points (e.g., see subsection 3.3, second line of 31 In this paper → x→x 0 means that the limit x → x0 of the quotient among the left and right-hand sides of the symbol is 1 . 32 Explcit examples of this kind of contributions are given in equations (3.64) and (4.59). 33 An explicit example of this kind of contributions is given in equation (3.74).

equation (3.73)
). 34 In this paper we will not try to fix all these contributions (which are subleading with respect to the leading asymptotics we are looking after). Analogously, we will say that in the expansion Λ → ∞, as defined before,

B Elliptic functions
The q-Pochammer symbol (ζ; q) ≡ (ζ; q) ∞ has the following product representation The quasi-elliptic function has the following product representation The elliptic Gamma functions has the following product representation

C On the contour of integration Γ gauge
In this appendix we explain how the details of the contour of integration Γ gauge , cf. (3.33), are relevant to compute the asymptotic growth of I n 1 ,n 2 ,n 3 in the expansion ∆ a → 0 at fixed ratios among ∆'s.

Resolving the physical poles
Let us comeback to the definition of giant-graviton indices (3.33) I n 1 ,n 2 ,n 3 = Γgauge dµ 1 dµ 2 dµ 3 I 4d n 1 ,n 2 ,n 3 I 2d n 1 ,n 2 ,n 3 . The contour prescription of [6] indicates that all physical poles selected by Γ gauge should be located at . This means that in the new variables the contour of integration can be divided in two components, a co-dimension 3 loop that we denote below as Γ gauge and a 3-dimensional infinitesimal loop picking up the residue at U (Vandermonde Det's).
(C.7) At this point we can proceed to evaluate the 3-dimensional integral over the diagonal modes U = 0 . This technical complication makes ill-defined the naive residue evaluation, due to the contribution coming from the fundamental strings stretching among different stacks of branes I 2d .
To simplify this residue computation it is convenient to deform the integration measure by substituting where µ should be thought of as a parameter that will be taken to zero after evaluating the non-degenerate residues. After this modification the degenerate poles transform into non-degenerate ones U 2πi U (I) a,a+1 I 4d n 1 ,n 2 ,n 3 I 2d n 1 ,n 2 ,n 3 , (C.12) where we have not written down the limit µ → 0 in the right-hand side because the integrand I 4d n 1 ,n 2 ,n 3 I 2d n 1 ,n 2 ,n 3 does not depend on µ . In what follows we assume either that there is no other remaining degenerate residue in the affine integration variables U (I) a,a+1 , or that, if there is any one such, then it has been resolved [6,17]. Anyways, the poles that dominate the expansion ∆ a → 0 at fixed ratios among ∆'s, which are the ones we will be concerned with, are non-degenerate in the affine variables U (I) a,a+1 and thus they do not require any further resolution. This will be explained below.
The residues at ∆ a → 0 The integral (C.12) can be written as I n 1 ,n 2 ,n 3 = α Res . . . I 4d n 1 ,n 2 ,n 3 I 2d n 1 ,n 2 ,n 3 ; U = U α . (C. 13) where α runs over whichever are the poles selected by the choice of contour Γ gauge . In these expressions we have removed the indices I and a, a + 1, and the products over I = 1, 2, 3 and a = 1, . . . , n I − 1, to ease presentation. For generic values of n 1 , n 2 and n 3 α Res . . . I 4d n 1 ,n 2 ,n 3 I 2d n 1 ,n 2 ,n 3 ; U = U α ̸ = 0 . (C.14) The results in subsection 4.2 imply the following asymptotic condition for residues 37 38 Res . . . I 4d n 1 ,n 2 ,n 3 I 2d n 1 ,n 2 ,n 3 ; U = U α → ∆a → 0 with ratios fixed α;a,a+1 ) that needs to be multiplied to the integrand in order to extract its residue at U = U α , does not affect the leading exponential growth of the integrand in the limit ∆ a → 0 at fixed ratios of ∆ a 's.
The function Res α [x, U ] is a subleading contribution defined as where the 0 + is an auxiliary regulator whose only function is to keep finite the two terms in the exponent of (C.18) at U = U α (for the combination of the two quantities, this regulator plays no role because the logarithmic term is cancelled by the first term). 37 To derive this relation below it is important not to truncate the infinite products in the residues and to work with their plethystic exponential representations. 38 We recall that the symbol → ∆a → 0 with ratios fixed means that the quotient between the left and right-hand side expressions tends to 1 in the corresponding limit.
As it was explained in the main body of the paper in a certain region of chemical potentials x the magnitude of the exponential growth of the factor |e − s (n 1 ,n 2 ,n 3 ) (...) | is maximized by the configuration U if and only if: • Γ gauge encloses some of them and the sum over their residues is non-vanishing.
For the contour prescription proposed in [3] there are infinitely many such poles. In the integrand . . . I 4d n 1 ,n 2 ,n 3 I 2d n 1 ,n 2 ,n 3 these poles always come in pairs 39 (denoted as positive and negative poles). For example, assuming n I > 1 there are simple poles defined by selecting n I − 1 pairs (a, b) for each I = 1, 2, 3 such that (for I ̸ = J ̸ = K and generic w I,J,K 40 ) for any two choices of integers c 1 (a, b) ≥ 0 and c 2 (a, b) > 0 . These poles come from the elliptic gamma functions [68] in the vector contributions (3.36). The first family comes from the poles of the first factor in the numerator of (3.36). The second family comes from the zeroes of the denominator of (3.36). They can be organized in two groups that map into each other under a Z 2 operation. One could denote such two subsets as positive and negative. This separation in two, which is non unique, comes from the fact that for every pole U = U α there is a pole located at the inverse position U = U −1 α . This bijection implies the existence of many Z 2 operations, out of which one can pick up one, and declare that it maps half of the number of poles coming from vector multiplets (positive) into the other 39 At least at large charges, these pairs mutually cancel each other, as it will be shown below. 40 It is sufficient, not necessary, to assume wI to be different from any product of rational powers of wJ . half (negative). For the indices studied here there are ∞ many such poles. 41 As it will be shown below, in order to have a non-trivial answer at large charges, Γ gauge must necessarily pick up an unbalanced number of positive and negative poles in order for the corresponding integral not to vanish trivially at large charges.
In the concrete example of I 0,0,2 it is easy to identify poles in the first family in (C.22) for the choices c 1 = 0 and 1 as: Using both, the identifications and the constraint below the three positive poles (C.23) map into the three positive poles corresponding to the tachyonic and zero mode terms f 1 , f 2 and f 3 depicted in figure 2 of [3]. As recalled in the latter example, the pole-selection prescriptions of [3] and [6], pick up an unbalanced number of positive and negative poles of type β which happen to come solely from vector multiplets (the positions of the poles coming from the chiral multiplets reduce to some power of p 1 in the scaling ∆ a → 0. Please refer to (3.38)) 42 .
Let us proceed to explain why the sum over residues of type β selected by contours Γ gauge breaking the Z 2 symmetries among the latter, does not vanish for generic values of chemical potentials. For the poles of type β, we can always use the Taylor expan- where the coefficients c i,β := c i 1 ,i 2 ,i 3 ,β are c-numbers. In particular for every β, 0,β (x) , whose dependence on β can be constrained in a simple way. We will do so in the following subsection. After substituting (C.25) in the leading contribution to (C.20) coming from the residues of type β, and 41 If the giant graviton expansion is complete and not asymptotic, then one must expect, and we will assume so, that the corresponding infinite sum over poles will be convergent in some continuous domain of rapidities. 42 The poles for this bi-fundamental contribution come from the zeroes of the Jacobi theta functions [68] in the denominator. 43 We identify poles β and β ′ that are identical after a permutation of their gauge indices aI = 1, . . . nI −1 .
keeping in the exponent the terms that do not vanish trivially as ∆ a → 0 , one obtains where -after reinstating the indices I and a -it follows that vanishes trivially because U = 1 is a saddle point of the action s . Thus, we conclude that in the limit ∆ a → 0 with ratios fixed, the total -and leading -residue contribution to I n 1 ,n 2 ,n 3 takes the asymptotic form In such a case, in virtue of (C.31), one concludes that, provided the sum over β's is either finite or a convergent series, where the ambiguity in the choice of β 0 is shielded in the ambiguity in the relations ∼ .

Constraining the relative contribution of poles
Let us come back to the function (reduced residue) The goal is to constrain the dependence on β of the quantity in the asymptotic expansion near its singularity U β → 1 . It is convenient to compute such asymptotic expansion in two-steps, starting from the function (C.34) For example, in a two-step expansion defined by the quadratic differential variations where δ U β ≪ δ U the exponent of (C.34) takes the form Using the expansion (C.38) for two different poles β = β 1 and β = β 2 we conclude, after exponentiation, that in a limit δ U β 1,2 → 0 the quotient among the reduced residues of roots of type β approaches a universal expression, This expression and (C.25) imply the following relation .

(C.40)
This relation is telling us that the relative contribution of poles of type β is defined, unambiguously, by their corresponding coefficients c 1,β . This is very useful, because the latter coefficients can be computed easily, and consequently using (C.40) one can straightforwardly predict what poles in the integrand would cancel among each other should Γ gauge pick them all. This implies that, should Γ gauge not break the Z 2 symmetries for poles of type β, then the contributions of the latter would vanish at large charges. On the contrary for a Γ gauge that breaks the Z 2 symmetries for poles of type β the analytic analysis above presented predicts that the answer will not vanish. For choices of Γ gauge that pick up an infinite number of unpaired positive and negative poles of type β it may be possible that the sum where the integer number deg(n) receives contributions from every β selected by Γ gauge with n(β) = n : precisely, +1 contributions from positive poles and −1 contributions from negative poles. Obviously, only if deg(n) = 0 for n > L where L is a positive integer, then the sum in the right hand side of (C.45) becomes finite. Assuming Γ gauge does select an infinite number of unpaired positive and negative poles of typeβ , we interpret the infinity above as signature that the infinite sum over residues can not be blindly commuted with the expansion ∆ a → 0 at fixed ratios. At the level of computing asymptotic expansions though, it is enough to truncate the convergent sum over poles β to a large sum, say with only L ≫ 1 elements, those with the minimum values of n(β) out of the infinitely many selected by the contour. In the presence of this intermediate cut-off L the asymptotic relation (C.33) follows from the fact that the dependence on L is shielded in the subleading ambiguity of the relations ∼ . 44

D Large charge entropy from averages over free Fermi systems
Brief summary of results in this appendix In [12] the author proposed an exact giant graviton-like expansion for a large family of matrix integrals that include the 1 16 -BPS index as a particular example. Schematically, this expansion looks like ζ = e −t is an auxiliary integration variable, whose string theory interpretation is unclear to us, and which we find evidence that -at least at large charges -it may be related to the 44 In other words, in the regions of chemical potentials ∆a's where an infinite sum over poles of type β converges, U (I) ab = 0 happens to be an accumulation point for such type of poles, i.e., in those regions of ∆ ′ a s the larger |n(β)| the closer β is to U (I) ab = 0 . This is the reason why a series over poles of type β can not be commuted with the limit ∆a → 0 with ratios fixed: for a a fast enough limit of poles towards U (I) ab = 0 it is not always true that the posterior limit ∆a → 0 implies the condition (C.21). That said, for any finite sum over poles of type β the latter issue is not present. linear combination of giant graviton numbers c 1,± · n in the representation of [3]. On the other hand, n is a non-negative natural number that reminisces, as well, one of the three numbers of giant gravitons in the expansions of [3] and [5]. From now on, when referring to the representation of [12], n will be called the giant graviton-like number.
At large enough charges, the microcanonical index grows slower than the giant gravitonlike contributions dt a n,t (Q) [13] |a(Q)| | dζ a n,ζ (Q)| ∼ Indeed, we will check that these cancellations can be understood as a transition in between two pairs of complex conjugated saddle-point configurations of The large-charge localization Lemma of subsection (2.3) implies that the integral over t must localize -at large-R charges and fixed J and n -around exponentially fast singularities of the integrand I n,ζ (τ ) . Indeed, we find that the two relevant exponential singularities are located around ζ = ±1, respectively. If we denote the asymptotic expansion of I n,ζ (τ ) around them as I n,ζ (τ ) → I n,±1 + t (τ ) , (D.6) then the saddle points obtained after extremizing the substitution of the choice of sign − in (D.6) on (D.5) are the ones determining a n (Q) at large Q and fixed n . On the other hand the saddle points obtained after extremizing the substitution of the choice of sign + in (D.6) on (D.5) dominate the counting after the sum over n is evaluated and exponentially large cancellations happen. The details of this analysis will be summarized in section D. In summary, we will check that at large charges Q → ∞ (for all N ) the following asymptotic formulae hold n dt a n,t (Q) ∼ a loc n,t ⋆ where the complex conjugated contributions a loc n,t ⋆ ± (Q) , (D.8) come from the saddle points of the localized effective action − log I n,ζ (τ ) around the singular region ζ → 1. In this case the two complex conjugated saddle point values are where again, the c ± are order 1 complex contributions that happen to match the abovequoted c 2± in equation (1.14) -up to a normalization factor-. The asymptotic relations (D.7) tell us how the exponential growth of 1 16 -BPS states in the boundary gauge theory is recovered from the giant graviton-like representation of [12]. The relevant computations are summarized below. Curiously, the similarity among (1.14) and (D.9) suggests that there may be a relation between the sum of the n auxiliary integration variables of type t = − log ζ (in representation (3.40)) and a single linear combination of giant graviton numbers n in the proposal of [3] (in representation (3.30)).

The cancellation mechanism
Let us explain how the cancellation mechanism among giant-graviton-like contributions happens in the exact expansion (3.40). In this expansion the microcanonical index of giant graviton-like contribution, n J n , can be written as: To apply our large-charge localization Lemma, we must compute the asymptotic expansion of the effective action around the relevant power-like singularity(ies). There are many singularities, but we will show that the two relevant ones (±) are located at ∆ a → 0 and ζ i → ±1 . 45 We find and check (in the following section), that the singularity locus at ζ i = −1 is the one relevant to compute asymptotic growth of states at fixed giant graviton-like number n. As in the cases before, these previous singularities serve as attractors to saddle-points. The localization of the effective action around ζ i = ±1 determines different saddle-point contributions to the total integral (D.10). In this subsection, we will focus on the vicinity of the singularity locus (or equivalently, on the saddle-points obtained after localization) at ζ i = 1, which is the one making explicit contact with the index at large charges.
If we substitute ∆ a → ϵ∆ a , (D.15) and ζ i → e −ϵt i , z i → e −2πi u i , (D. 16) in the effective action (D.11) and expand it 46 around ϵ = 1 Λ = 0 then the first term in the right-hand side of (D.11) reduces to n i̸ =j=1 (D.17) Evaluating this at the asymptotic form of the n! inequivalent saddle points for gaugerapidities u ij → u ⋆ ij = 0, and expanding the T (n) we obtain −S  45 In particular, from now on we will only pay attention to the leading asymptotic behaviour, thus will not pay attention to the F -type contributions (See the definitions given around (A.3)) coming from the log Det[z, ζ] term. 46 Really, we first make the substitution in the denominator, then expand the result and keep leading contributions. Then, finally, we re-sum over the variable l and obtain a sum over polylogarithms at diverse level. Then we substitute (D.26) and (D. 27) and expand the answer around ϵ = 0 up to the desired order. In this way we are able to avoid finding undesired infinities due to mistreatment of logarithmic divergencies (See the discussion in pargraph 4.2).
where we have changed to variables (with unit Jacobian) This result matches the exponential of the entropy function accounting for the asympotic growth of states at large charges and spin, and thus one concludes that This had to be the case because the representation (3.40) is an exact representation of the index.
The large charge growth at fixed n We finalize by showing that the contribution coming from the (pair of complex conjugated) saddle points of the localized action at ζ i = −1 , dominate the counting of states relative to a single giant graviton-like block n , for any finite n . And second, by understanding from a macrocanonical perspective how the latter contributions cancel at large charges after summing over n , letting the complex conjugated pairs of solutions of the localized action at ζ i = 1 to dominate the counting of 1 16 -BPS states at large charges. This time we substitute ∆ a → ϵ∆ a , (D. 26) and 27) in the effective action −S (n) M (x; z, ζ) .
(D.28) 47 We assume the change of variables is implemented before taking any expansion that breaks the periodicity ti → ti + 2πi ϵ . In this way we are safe to consider that the n new variables are such that ϵ 2πi t and ϵ 2πi t1,2,3,...,n−1 range over the segment [0, 1) .
Then we expand the answer around ϵ = 1 Λ = 0 . Being careful with contributions coming from zero-modes (as detailed in previous analysis) and picking up the leading gauge saddle point u * ij = 0 we obtain the following leading contribution Notice first that this does not depend on N and second, that at finite N it grows faster than the 1 16 -BPS microcanonical index | d M | which grows exponentially fast in O(N 2/3 Q ′2/3 ) . Let us define q = e − ∆ 1 2 = e − ∆ 2 2 = e − ∆ 3 2 = e − ω 1 3 , (D. 35) and proceed to compute the q-series J n=1 = J n=1 (q) by computing residues of the inte- (D.36) We note that Q ̸ = Q ′ (see the discussion in section 3), however they are related in the asymptotic limit (D.31) as follows Q ∼ The result is presented in figure 1. Notice that the giant graviton index at fixed n grows faster than the total giant graviton index in the limit (D.31). How are these cancellations explained in the present approach? Let us define the variable δn δn := N Λ 4 n , (D. 39) which ranges over a continuum domain in the limit as Λ → ∞. Then we can trade the sum over n by an integral over a finite segment of length q ′ = O(1) as Λ → ∞ that can be This mechanism explains how these contributions do not compete with the ones coming from the singularity locus at ζ i = 1 (encoded in (D.24)) in determinining the total microcanonical giant graviton index (D.10) at large charges.