Cardy-like asymptotics of the 4d $\mathcal{N}=4$ index and AdS$_5$ blackholes

Choi, Kim, Kim, and Nahmgoong have recently pioneered analyzing a Cardy-like limit of the superconformal index of the 4d $\mathcal{N}=4$ theory with complexified fugacities which encodes the entropy of the dual supersymmetric AdS$_5$ blackholes. Here we study the Cardy-like asymptotics of the index within the rigorous framework of elliptic hypergeometric integrals, thereby filling a gap in their derivation of the blackhole entropy function, finding a new blackhole saddle-point, and demonstrating novel bifurcation phenomena in the asymptotics of the index as a function of fugacity phases. We also comment on the relevance of the supersymmetric Casimir energy to the blackhole entropy function in the present context.


Introduction
It has been a long-standing challenge in AdS 5 /CFT 4 to reproduce the entropy of the charged, rotating, BPS, asymptotically AdS 5 blackholes of [1][2][3][4][5] from a microscopic counting of BPS states in the 4d N = 4 CFT. Several attempts in this direction were made in the past fifteen years or so (e.g. [6][7][8][9][10]), leading to various new lessons for holography and superconformal field theory (SCFT), but not to the desired microscopic count.
In particular, an index was devised in [6,11] for counting the BPS states of general 4d SCFTs. The index counts all of the states-in the radial quantization of the SCFT-that are annihilated by a chosen supercharge. We adopt conventions in which such states satisfy the "BPS condition" ∆ − J 1 − J 2 − 3 2 r = 0, where ∆, J 1 , J 2 , r are the quantum numbers of the 4d N = 1 superconformal group SU(2, 2|1). The index I(p, q, u k ) := Tr (−1) F eβ (∆−J 1 −J 2 − 3 2 r) p J 1 + r 2 q J 2 + r 2 k u q k k , (1.1) is thus independent ofβ, but it does depend on the spacetime fugacities p, q, as well as the flavor fugacities u k associated with flavor quantum numbers q k commuting with the supercharge. In the case of the N = 4 theory, the SU(4) R-symmetry of the N = 4 superconformal algebra decomposes into SU(3)×U(1) r , so there is an SU(3) "flavor" symmetry group commuting with the chosen supercharge; hence there are three q k with 3 k=1 q k = 0, and three u k satisfying 3 k=1 u k = 1. It is customary to define y k := (pq) 1/3 u k and Q k := q k + r/2. Then, dismissingβ, we can rewrite the index of the N = 4 theory as I(p, q, y 1,2,3 ) = Tr (−1) F p J 1 q J 2 y Q 1 1 y Q 2 2 y Q 3 3 .

(1.2)
This index was computed at finite rank for the U(N ) N = 4 theory in the original paper [6]. Then, in an initial attempt to make contact with holography, the large-N limit of the index was evaluated for real-valued fugacities and was seen to be O(N 0 ); the result perfectly matched the index of the KK supergravity multi-particle states in the dual AdS 5 theory, but clearly could not account for the O(N 2 ) entropy of the bulk supersymmetric AdS 5 blackholes [6]. For some time this negative result was interpreted as an indication that the index does not encode the bulk blackhole microstates. Very recently it has been discovered by Choi, Kim, Kim, and Nahmgoong (CKKN) [12], and independently by Benini and Milan [13], that allowing the five fugacities in the index to take complex values one can achieve the desired O(e N 2 ) behavior in the large-N limit of the index. Benini and Milan have succeeded in directly obtaining the AdS 5 blackhole entropy function in the large-N limit of the index [13], while CKKN took a different route and derived the entropy function in a double-scaling-Cardy-like as well as large-N -limit [12,14]. In the present paper we derive the entropy function in a Cardy-like limit of the index at finite rank; although our analysis is closely related to that of CKKN [14], ours is more analogous to the Cardy-formula [15] derivations of blackhole entropy in AdS 3 /CFT 2 (e.g. [16][17][18]) where the central charge is kept fixed.
The study of the Cardy-like asymptotics of 4d superconformal indices had some history prior to [14], but was again mostly limited to real-valued fugacities (e.g. [19][20][21][22][23][24]). The idea that blackhole microstate counting requires complex-valued fugacities in the N = 4 index was not properly appreciated until the recent work of Hosseini, Hristov, and Zaffaroni (HHZ) [25]. This work provided the impetus for the later investigations of CKKN [12,14] and Benini-Milan [13]. HHZ started from the supergravity side and bridged half-way towards the CFT by presenting a "grand-canonical" functional-henceforth the HHZ functional-from which a Legendre transform gives the micro-canonical entropy of the AdS 5 blackholes; the remaining challenge was to extract the HHZ functional in an appropriate asymptotic regime from the index. In particular, it was understood by HHZ [25] (based on recent lessons from AdS 4 /CFT 3 [26,27]) that complexified fugacities are needed in the index in order to make contact with the grand-canonical functional of the AdS 5 blackholes. As alluded to above, CKKN [14] and independently Benini and Milan [13] have recently completed the bridge between the CFT and the bulk by deriving the HHZ functional through asymptotic analysis of the N = 4 theory index, the first group in a double-scaling limit and the second group in a large-N limit.
In the present paper we analyze the Cardy-like asymptotics of the N = 4 theory index with complexified fugacities using the rigorous machinery of elliptic hypergeometric integrals [28][29][30][31] in various Cardy-like regimes of parameters where the flavor fugacities approach the unit circle and the spacetime fugacities approach 1. In particular, we fill a gap in the CKKN derivation of the HHZ functional in this limit by showing that the eigenvalue configuration they chose in their asymptotic analysis of the matrix-integral expression for the index is indeed the dominant configuration in the regime of parameters pertaining to the blackhole saddle-point they considered. Moreover, we discover a new blackhole saddle-point in a differ-ent regime of parameters, corresponding to fugacities that are complex conjugate to those at the CKKN saddle-point. We present intuitive arguments suggesting that no other blackhole saddle-points exist in the Cardy-like limit. We also demonstrate interesting dependence of the qualitative behavior of the Cardy-like asymptotics of the index on the complex phases of the fugacities.
In the rest of this introduction we give a sketchy account of the asymptotic analysis extracting the blackhole entropy function from the appropriate Cardy-like limit of the superconformal index of the N = 4 theory. The main body of the paper starts in Section 2 where we elaborate on the sketchy derivation of the present section; we study the Cardy-like asymptotics of the N = 4 theory index with all its fugacities complexified, clarifying-and addressing a gap in-the CKKN derivation of the HHZ functional. A thorough enough understanding of the asymptotics of the index in different Cardy-like regimes of parameters results in that section which reveals a second blackhole saddle-point in a regime complementary to that of CKKN, and moreover allows us to argue intuitively that no other relevant saddlepoints exist. In Section 3 we keep the spacetime fugacities real-valued, and demonstrate novel bifurcation phenomena in the asymptotics of the index as a function of the flavor-fugacity phases. Section 4 discusses the relation between the Hamiltonian superconformal index and the Lagrangian index computed through path-integration; the two differ by a Casimir-energy factor which is argued to be irrelevant to the blackhole entropy function in the present context. Finally, Section 5 discusses the important open ends of the present work.

Outline of the CKKN derivation in the elliptic hypergeometric language
We now present an outline of the CKKN derivation [14] of the HHZ functional [25], translated to the language of elliptic hypergeometric integrals. More precisely, the problem we consider differs from that of [14] in two respects: • while [14] considered the N = 4 theory with U(N ) gauge group, we consider the SU(N ) theory-the details are rather similar and the end results are related via N 2 → N 2 − 1 shifts; • while in [14] a double-scaling-Cardy-like as well as large-N -limit is taken to simplify the analysis, here in analogy with the Cardy-formula derivations of blackhole entropy in AdS 3 /CFT 2 we keep N finite and only take a Cardy-like limit.
The special function as the starting point The superconformal index of the SU(N ) N = 4 theory is given by the following elliptic hypergeometric integral (see e.g. [32]): with the unit-circle contour for the z j = e 2πix j while N j=1 z j = 1, and with p, q, y k strictly inside the unit circle such that 3 k=1 y k = pq. The two special functions (·; ·) and Γ(·) are respectively the Pochhammer symbol and the elliptic gamma function [33]: The integral expression gives the index as a meromorphic function of p, q, y k in the domain 0 < |p|, |q|, |y k | < 1. A contour deformation can presumably allow meromorphic continuation of the index to 0 < |p|, |q| < 1, y k ∈ C * (c.f. [34]).

Asymptotic analysis in the limit encoding blackholes
The Cardy-type limit analyzed prior to the work of CKKN [14] was of the form p, q, y k → 1; more precisely, it was what in the mathematics literature is referred to as the hyperbolic limit of the elliptic hypergeometric integral [31,35]. CKKN considered instead limits of the type p, q → 1, y i → e iθ i , with θ i / ∈ 2πZ: they correctly recognized that giving finite (non-vanishing) phases to the flavor fugacities can obstruct the bose-fermi cancelations 1 occurring in the hyperbolic limit. For future reference we define σ, τ, T k through p = e 2πiσ , q = e 2πiτ , y k = e 2πiT k , and write the appropriate limit explicitly as the CKKN limit: |σ|, |τ |, ImT k → 0, with τ σ ∈ R >0 , ReT k fixed, and Imτ, Imσ > 0.
Note that the "balancing condition" 3 k=1 y k = pq implies 3 k=1 T k − σ − τ ∈ Z, and that the restriction Imτ, Imσ > 0 keeps us in the domain of meromorphy of the index. 1 A similar obstruction mechanism is at work in the AdS3/CFT2 context, where the entropy of the AdS3 blackholes is derived from a Cardy-like limit of the CFT2 elliptic genus χ(q, y): the limit q, y → 1 does not encode the bulk blackholes, but the limit q → 1, y → e iθ with θ / ∈ 2πZ does. However, note that while in the AdS3/CFT2 context q can be kept real, in AdS5/CFT4 the spacetime fugacities p, q should take off the real line to meet the blackhole saddle-points. See [13,26,27] for related discussions of "I-extremization" in the large-N analysis.
The asymptotic analysis of the integral (1.3) now proceeds as follows. As will be explained in Section 2, the leading asymptotics comes from the elliptic gamma functions Γ(·), so the Pochhammer symbols (·; ·) and the N ! in the pre-factor can be neglected. The required estimate, reviewed in Section 2, follows from Proposition 2.11 of Rains [31]: for |τ |, |σ| → 0, with Imτ, Imσ > 0, and τ σ ∈ R >0 , x ∈ R. Here κ(·) is the continuous, odd, piecewise cubic 2 , periodic function In order to apply the estimate (1.6) to the gamma functions in (1.3) we have to identify the phase of the arguments with 2πx; then, for instance, we can apply (1.6) to the gamma function in the numerator of the integrand of (1.3) by identifying x with ReT k ± (x i − x j ). This way we can simplify (1.3) to where κ(A ± B) stands for κ(A + B) + κ(A − B). It only remains to evaluate the asymptotics of the integral (1.8).
Note that we are assuming Im(τ σ) = 0; this corresponds to complexifying the "temperature" as explained below. When Im(τ σ) = 0 the integrand of (1.8)-or already the RHS of (1.6)-would be a pure phase, and not sufficient to describe the exponential growth of the blackhole microstates. The Im(τ σ) = 0 case is therefore not directly relevant to the AdS 5 blackhole physics, but it exhibits some interesting asymptotic bifurcation phenomena that are discussed in Section 3.
The last step of the asymptotic analysis of the index involves arguing that in the appropriate range of parameters the dominant small-|τ |, |σ| configuration in (1.8) is x = 0. CKKN presented [14] numerical evidence for this in the N = 2 case, and left it as a conjecture that the same is true for N > 2. In Section 2 we will prove that for the range of parameters relevant to the AdS 5 blackholes (e.g. for Im(τ σ) > 0 and −1 < ReT 1,2 , −1 − ReT 1 − ReT 2 < 0) their conjecture is correct. Hence the asymptotics of the index becomes 4 log I(p, q, (1.10) The right-hand-side is a nonanalytic function of the ReT k , manifestly invariant under ReT k → ReT k + 1 as it should be.
To match the grand-canonical functional of HHZ [25] we now pick a particular chamber in the parameter-space so that an analytic expression can be written down. Specifically, assuming Im(τ σ) > 0, going into the chamber −1 < ReT 1,2,3 < 0 with ReT 3 = −1 − ReT 1 − ReT 2 , we can simplify 3 k=1 κ(ReT k ) to 6ReT 1 ReT 2 ReT 3 , and arrive at Analytic continuation of (1.11) to complex T k (i.e. replacing every ReT k with T k ) allows recovering the subleading terms in the CKKN limit and connecting with the complex HHZ functional [25]: So far in this subsection we have been essentially rephrasing the developments due to CKKN [14]. One of the novel contributions of the present paper is to demonstrate in Section 2 that when Im(τ σ) < 0 another chamber with 0 < ReT 1,2,3 < 1 and ReT 3 = 1 − ReT 1 − ReT 2 yields the asymptotics (1.11), this time with

Legendre transform and blackhole entropy
Thinking of the index (1.2) as the generating function of the degeneracies d(J 1,2 , Q 1,2,3 ) of the BPS states 5 in the N = 4 theory, methods of elementary analytic combinatorics can be used to extract the large-J 1,2 , Q 1,2,3 asymptotics of d(J 1,2 , Q 1,2,3 ) from the Cardy-like asymptotics of the index. The CKKN limit of the index encodes the degeneracy of the BPS states as of the bulk AdS 5 blackholes is satisfied [14]. 4 Compare with Eq. (2.34) of CKKN [14]; note that 2πiT Although at first glance it appears that because of the (−1) F factor in it the index (1.2) counts the number of bosonic states minus the number of fermionic states, as argued in [13], on the blackhole saddlepoints essentially all the states are bosonic, so the index counts a degeneracy.
The degeneracies can be obtained from the generating function through with all the contours slightly inside the unit circle; note that y 3 is not independent, so is not integrated over on the RHS (c.f. Section 5 of [13]). The asymptotic degeneracy can be obtained using a saddle-point evaluation of the integral on the right-hand side. Using the Cardy-like asymptotics in (1.12), the result for the asymptotic entropy S( The subscript "ext" on the RHS means picking its extremized value on the saddle-point. The extremization problem was addressed for the Imτ σ > 0 case by HHZ [25], but was made completely explicit and analytic by CKKN [14] (and independently in Appendix B of [36] by Cabo-Bizet, Cassani, Martelli, and Murthy), who found the blackhole saddle-point at and giving the entropy which thanks to the charge relation (1.13) can be written in the alternative form Both of the relations (1.17), (1.18) correctly reproduce the Bekenstein-Hawking entropy of the BPS AdS 5 blackholes of [1][2][3][4][5] in the scaling limit of CKKN, upon using the AdS/CFT dictionary , with AdS 5 , G AdS 5 respectively the radius and the Newton constant of the bulk AdS 5 .
In Section 2 we show that (1.11) is valid also when 0 < ReT 1,2,3 < 1, Imτ σ < 0, though this time with T 1 + T 2 + T 3 − τ − σ = +1, and find a new blackhole saddle-point at with the same entropy S as that of the CKKN saddle-point 6 . We moreover argue that besides the two just described-having complex conjugate fugacities p, q, y 1,2,3 -no other blackhole saddle-points exist in the Cardy-like asymptotics of the N = 4 theory index.

Final remarks
A remaining gap for unequal Q k . A rather serious gap in the above derivation is revealed upon closer inspection of the critical T k in (1.16) and (1.19): while our asymptotic analysis is valid only in the limit ImT k → 0, the blackhole saddle-points have nonzero ImT k unless Q 1 = Q 2 = Q 3 . It is therefore only in the special case with equal-or approximately equal-charges that the above derivation (augmented with the refinements of Section 2) is satisfactory. CKKN assumed in a leap of faith [14] that the asymptotics (1.12) remains valid away from the limit ImT k → 0, and thus the blackhole entropy derivation can be extended to the general case with unequal charges. In Section 2 we present a partial justification for this extrapolation; the rigorous justification is beyond the scope of the present paper, and its absence constitutes the most important open end of this work.
Cardy-like versus large-N. The above derivation extracts the AdS 5 blackhole entropy from a "high-temperature" (Cardy-like) limit of the 4d superconformal index at finite N . This is analogous to how the classic papers of Strominger-Vafa [16], BMPV [17], and Strominger [18] derived the Bekenstein-Hawking entropy of certain blackholes in what nowadays might be called an AdS 3 /CFT 2 context. From the holographic perspective, a more conceptually satisfying derivation would involve the large-N limit of the index. In AdS 3 /CFT 2 such conceptually satisfactory derivations can be found in [37,38]. In the AdS 5 /CFT 4 context this was achieved very recently by Benini and Milan [13], leveraging the Bethe Ansatz formula of Closset, Kim, and Willett [39]. Curiously, although the derivation in [13] is not limited to the equal-charge blackholes, because of certain technical obstacles it so far applies only to the case with equal angular momenta J 1 = J 2 and the general case with J 1 = J 2 is still open. The more general Bethe Ansatz formula of [40] seems promising in that direction.
The elliptic gamma function estimate (1.6) Let us define the parameters b, β through τ = iβb −1 2π , σ = iβb 2π . For p, q ∈ R, the parameter β defined as such was referred to as the inverse-temperature in [21,22]; here we similarly refer to β as the complexified inverse-temperature. Throughout the present work we assume b ∈ R >0 (i.e. τ /σ ∈ R >0 ); this simplifies the analysis and suffices for making contact with blackhole physics in the Cardy-like limit. We also take Reβ > 0 (i.e. |argβ| < π 2 ) to stay within the domain of meromorphy of the index (1.3). In terms of b, β we have the CKKN limit: |β|, ImT k → 0, with b ∈ R >0 , ReT k fixed, and Reβ > 0.
The starting point for deriving the estimate (1.6) is the following identity, essentially due to Narukawa [41]: and ψ b (x) a function [see Appendix A of [22] for its definition in terms of the hyperbolic gamma function] with the important property that for argx ∈ (−π, 0) and fixed b > 0 with an exponentially small error, of the type e −|x| -see Corollary 2.3 of Rains [31] for the precise statement and see Appendix B of [42] for an earlier analysis in a different notation. This property guarantees that the infinite product in (2.1) is convergent when Reβ > 0. For x strictly inside the strip as |β| → 0 with Reβ > 0 and with b > 0 fixed, all the ψ b functions on the RHS of (2.1) approach unity exponentially fast. Moreover, the dominant piece of Q + in the limit is of order 1 τ σ and gives Since the LHS of the above relation is periodic in x → x + 1, we can extend it beyond x ∈ S + by replacing every x on the RHS with its horizontal shift {x} := x − Rex + Imx · tan(argβ) to inside S + . For x ∈ R we have {x} = x − x ; this yields our desired estimate (1.6).
A somewhat subtle point is that the estimate (1.6) is not uniform with respect to x when applied to the ("vector multiplet") gamma functions in the denominator of the RHS of (1.3)-or more generally (2.5) is not uniform when x approaches the boundaries of the strip S + . We need a uniform estimate because we want to apply the estimate in the integrand of the index. We expect though that an argument similar to that at the top of page 23 of [22] can be given implying that the non-uniform estimate introduces a negligible error on the leading asymptotics of the index.
Cardy-like asymptotics of the index (1.10) It follows from the relation between the Pochhammer symbol and the Dedekind eta function η(τ ) = e 2πiτ /24 (e 2πiτ ; e 2πiτ ), (2.6) and the modular property η(−1/τ ) = √ −iτ η(τ ) of the eta function that in the Cardy-like limit the Pochhammer symbols on the RHS of (1.3) contribute an exponential growth of They can hence be neglected, along with the N ! in the denominator of (1.3), in the Cardy-like limit when 0 < |argβ| < π/2. We thus end up with (1.8) as promised.
We remind the reader that if β ∈ R >0 the integrand of (1.8) becomes a pure phase, and the more precise asymptotic analysis of Section 3 has to be performed.
For 0 < |argβ| < π 2 , in the small-|β| limit the integral (1.8) is localized around the minima of − sin(2argβ) · Q h (x; ReT k ), whose x-dependent part can be read from (1.9) to be − sin(2argβ) 12 is thus roughly a pair-wise potential for the "holonomies" x i . We take argβ and ReT 1,2 to be our control-parameters; ReT 3 is determined (mod Z to be precise, which is enough) by the balancing condition. We take the fundamental region of ReT 1,2 to be [−1/2, 1/2]. The two qualitatively different behaviors that the function V Q can exhibit in various regions of the space of the control-parameters ReT 1,2 are shown in Figure 1 for −π/2 < argβ < 0. This figure can be deduced either by numerically scanning  (using Mathematica for instance) the fundamental region ReT 1,2 ∈ [−1/2, 1/2], for some fixed argβ ∈ (−π/2, 0), or by analytically investigating the function 3 k=1 κ(ReT k ± x ij ) in its various regions of analyticity. Note that an M -type potential means x ij = 0 is preferred in the small-|β| limit, while a W -type potential means some x ij = 0 (always a neighborhood of x ij = ±1/2 it turns out) is preferred. Since Figure 1 is a bit too featureful, we use the equivalence ReT 1,2 → ReT 1,2 ± 1 to shift its triangular regions so that the equivalent Figure 2 is obtained, which is one of the main results of the present paper. It should be clear from the sin(2argβ) factor in (2.8) that the M and W wings in Figure 2 switch places if argβ is taken to be inside (0, π/2) instead.
To be specific, let us continue with the argβ ∈ (−π/2, 0) case for the moment. Then on the M wing of Figure 2 the minimum value of V Q occurs at x = 0. Moreover, since V Q is stationary at x = 0, the phase of the integral (1.8) is stationary there. We conclude that for − π 2 < argβ < 0 and −1 < ReT 1,2 , −1 − ReT 1 − ReT 2 < 0, the leading small-β asymptotics of the index is dominated by the x ij = 0 configuration-which in our SU(N ) case implies x i = 0. This proves CKKN's conjecture in [14] and fills the gap in their derivation of the HHZ functional in the appropriate region of the parameter-space.
On the bifurcation set, indicated by the dashed lines in Figure 2, the functions V Q and Q h vanish; a more precise analysis using the techniques of Section 3 is then required, but in any case it is clear that the asymptotic growth of the index is much slower (with Re log I = O( 1 |β| )) there, so we do not discuss this set any further.
For N = 2 the reason is that the x-independent piece of Q h in (1.9) moves κ(−1/3 ± 0) = −4/9 further down by −2/9, while it moves κ(+1/3 ± 1/2) = −5/9 further up by +2/9. Thus in the CKKN limit with −π/2 < argβ < 0 we have while I N =2 (p, q, y 1,2,3 ) In short, for N = 2 the fastest asymptotic growth in the CKKN limit with −π/2 < argβ < 0 occurs on the M wing of Figure 2. For higher ranks there is a more important reason why points on the W wing do not exhibit a faster asymptotic growth. That is because for N > 2 it is impossible to distribute N holonomies x i on the fundamental region [−1/2, 1/2] (with −1/2 and 1/2 identified, and with x N determined from the rest via N i=1 x i ∈ Z) and have all of them at equal distance |x ij | = 1/2 from each other. Colloquially speaking, it is not possible to capitalize on the minima of V Q on the W wing at |x ij | = 1/2 with all the holonomies, whereas it is possible to do so on the minima at |x ij | = 0 on the M wing; hence as we increase the rank it becomes more and more intuitively likely that the asymptotic growth of the index should be faster on the M wing, and so we expect that only this region potentially bears blackhole entropy functions. Based on a numerical study of the contribution of the equal-distanced configuration for the holonomies we conjecture that on the W wing, as N increases, the index exhibits an asymptotic growth with an O(N 0 ) exponent in the CKKN limit, and so asymptotic growth with an O(N 2 ) exponent is viable only on the M wing.
Let us recapitulate our findings so far. We have demonstrated that the |x ij | = 0 point is preferred in the CKKN limit on the M wing of the space of the control-parameters ReT 1,2 , and thus the asymptotic result (1.10) is valid there. We have also argued intuitively that the W wing yields slower asymptotic growth and is not expected to bear blackhole entropy functions.
It is straightforward to deduce the analogous statements for 0 < argβ < π 2 . In that case the M and W wings of Figure 2 are swapped. Hence this time it is on the upper-right wing that the |x ij | = 0 configuration is preferred in the CKKN limit, and the asymptotic result (1.10) is valid, though this time with ReT 3 = 1 − ReT 1 − ReT 2 . We also know that for N = 2 the asymptotic growth of the index is slower on the lower-left wing, and as N increases we conjecture that the asymptotic growth has an O(N 0 ) exponent there. Now we ask: in the case −π/2 < argβ < 0 does the lower-left wing, and in the case 0 < argβ < π 2 does the upper-right wing contain blackhole saddle-points? To make contact with the AdS 5 blackholes we have to find the critical points of the Legendre transform of log I in the CKKN limit. In both cases it turns out that one blackhole saddle-point exists. The latter blackhole saddle-point seems to have been overlooked in [14], but can be obtained with minor modification of the computations in their Section 2.3 as we now outline. Recall that when 0 < argβ < π 2 we impose k T k = τ + σ + 1 rather than k T k = τ + σ − 1; while CKKN [14] (following HHZ [25]) impose the latter relation via (2.11) the former relation can be simply imposed by putting (2.12) We now would like to argue that the z * 1,2,3,4 which solve the extremization problem for 0 < argβ < π 2 are indeed the complex conjugates of the z 1,2,3,4 that CKKN found solving the extremization problem for −π/2 < argβ < 0. To demonstrate this, we present some of the details of the extremization problem, in parallel with Section 2.3 of CKKN [14]. Setting the derivatives of (1.15) with respect to z * 1,2,3,4 to zero, we get with a different sign on the RHS compared to the CKKN case-c.f. their Eq. (2.79). As a result, the equations for z * 1,2,3,4 following from the above relations read To obtain S, we can follow CKKN and write things in terms of f * : and then use the definition of f * to obtain which is a cubic relation for f * . The cubic equation that follows for S = 2πi(f * + J 2 ) will then have the entropy functions (1.17) and (1.18) as its solutions in the CKKN scaling limit.
To demonstrate the self-consistency of our computations we need to show that the CKKN/HHZ saddle-point (1.16) is indeed on the lower-left wing of Figure 2 and has −π/2 < argβ < 0, while the new saddle-point lies on the upper-right wing and has 0 < argβ < π 2 . We show only the second statement, as the first follows using the fact that the two saddle-points have their T k , τ, σ negative complex conjugate of each other.
A quick way to the desired result is to note that S + 2πiQ k are on a straight line in the complex plane, so that their reciprocals are on a circle. This observation motivates the change of variables 1 S+2πiQ k = 1 2S (1 + e −iφ k ), with φ k ∈ (0, π). Then the desired ranges of ReT k and argβ follow easily from the vector representation of the complex numbers 1 + e −iφ k .
In summary, we have shown that when 0 < argβ < π 2 a blackhole saddle-point exists on the upper-right wing of Figure 2; as comparison of Eqs. (2.11) and (2.12) shows, the new saddle-point has fugacities p * , q * , y * k that are complex conjugates of the fugacities at the CKKN/HHZ saddle-point. Moreover, we have argued that besides this and the CKKN/HHZ saddle-point no other (inequivalent) blackhole saddle-points exist in the Cardy-like limit.

Moving the flavor fugacities away from the unit circle
As we noted at the end of Subsection 1.1, unless Q 1 = Q 2 = Q 3 , the critical T k have nonzero imaginary parts, and thus the critical fugacities u k (and also y k ) lie away from the unit circle. Hence to complete the blackhole entropy derivation for the general case with unequal Q k , we need to be able to justify the Cardy-like asymptotics (1.12) when ImT k are not sent to zero.
A partial justification is as follows. Let us assume that ImT k are small enough so that the integral (1.3) still represents the index, albeit with a slightly deformed contour of integration 7 . We can then use (2.5) to arrive at the following variant of (1.8): where κ(x) is still defined as in (1.7), but with {x} := x− Rex+Imx·tan(argβ) as discussed around (2.5). We expect that for fixed argβ (either in (−π/2, 0) or in (0, π/2)), and for small enough ImT k , the catastrophic behavior of the pair-wise potential for the holonomies to remain similar to that discussed above, with the two complementary "wings" T 1,2 , 1 − T 1 − T 2 ∈ S + and T 1,2 , −1 − T 1 − T 2 ∈ S + − 1 being associated to M -or W -type behaviors, with one or the other having x = 0 as its preferred configuration depending on the sign of argβ. Then for argβ ∈ (0, π/2) one can use (2.5) on the wing T 1,2 , 1 − T 1 − T 2 ∈ S + to arrive at (1.12) with k T k = τ + σ + 1, while for argβ ∈ (−π/2, 0) one can use (2.5) with x → x + 1 on the wing Beyond a small neighborhood of ImT k = 0 the methods of the present paper do not seem powerful enough to demonstrate (1.12). Whether the fascinating formalism of [39,40] can help addressing the general case with nonzero ImT k is currently being investigated.

Real-valued temperature
In this section we keep the spacetime fugacities p, q real-valued and define b, β ∈ R >0 through p = e −βb , q = e −βb −1 . We also keep the flavor fugacities u k = e 2πiT k on the unit circle (hence T k ∈ R), and study the effect of finite nonzero T k on the small-β asymptotics of the index.
In order to provide some conceptual context for the somewhat technical analysis in the rest of this section we now briefly discuss the path-integral interpretation of the index with real-valued p, q. We will still be analyzing the Hamiltonian index I, and only importing intuition from the path-integral picture-until the next section where the path-integral partition function is analyzed.
The superconformal index with real p, q can be obtained via the path-integral SUSY partition function of the theory on S 3 b × S 1 β , where S 3 b is the squashed three-sphere with unit radius and squashing parameter b, while S 1 β is the circle with circumference β [43]. The integration variables z i in the index (1.3) correspond to the eigenvalues of the holonomy 7 It appears like we might only need the contour-deformation to be small near x = 0, which is the dominant eigenvalue configuration in the regime of parameters pertaining to the blackhole saddle-points. matrix P exp(i S 1 β A 0 ), with A 0 the component along S 1 β of the SU(N ) gauge field. The u k correspond to the eigenvalues of the background holonomy matrix P exp(i S 1 β A u 0 ), with A u 0 the component along S 1 β of the background gauge field A u associated to the "flavor" SU(3) of the N = 4 theory. The path-integral partition function is actually a Casimir-energy factor different from the index; this factor is irrelevant for the present analysis and we postpone its discussion to the next section. Interpreting the S 3 b as the spatial manifold and the S 1 β as the Euclidean time circle, we refer to β as the inverse-temperature in analogy with thermal quantum physics-even though our fermions have supersymmetric (i.e. periodic) boundary conditions around S 1 β . Next, we note that while large-N QFTs (N → ∞) on compact spatial manifolds can have finite-temperature phases associated to large-N saddle-points, in the present work we are considering a finite-N QFT on a compact spatial manifold (namely S 3 b ), which can not be assigned a phase at any finite temperature. In the high-temperature limit (β → 0), however, infinite-temperature phases can be associated to the small-β saddle-points. In particular, we will say that the infinite-temperature phase of the index is Higgsed if the dominant small-β saddle-point(s) of its matrix-integral lie away from the "origin" x = 0. For example, the infinite-temperature phase of the index of the SU(2) ISS model is Higgsed, but that of the N = 1 SU(N ) SQCD (say in the conformal window) is not [22].
Moreover, we will say that the infinite-temperature phase of the index is deconfined if for the leading small-β asymptotics we have Re log I ≈ A/β with A > 0; in other words if the index exhibits exponential growth in the high-temperature limit.
Below we will see that for generic non-zero T k ∈ R the infinite-temperature phase of the index of the SU(N ) N = 4 theory is Higgsed, and in the N = 2 case for a specific range of T k also deconfined. We suspect, but could not demonstrate, that for large enough N no values of T k can make the infinite-temperature phase of the index deconfined.
Taking p, q to be real means taking τ, σ to be pure imaginary. Then we have Im(τ σ) = 0, so that the estimate (1.6) gives only a pure phase; hence we have to consider the subleading terms in the exponent of its RHS to get information about the modulus of the index. The improved estimate is [22] log where the continuous, positive, even, periodic function is defined after Rains [31].
In order to apply the estimate (3.1) to the gamma functions in (1.3) we have to interpret the modulus of the arguments of the gamma functions as (pq) r/2 , and interpret the phase of the arguments as 2πx; then, for instance, we can apply (3.1) to the gamma function in the numerator of the integrand of (1.3) by identifying r, x as r = 2/3, x = T k ± (x i − x j ); note that the balancing condition 3 k=1 y k = pq implies 3 k=1 T k ∈ Z. Since the Pochhammer symbols in (1.3) yield asymptotics that cancel the contribution of the gamma functions from the third term on the RHS of (3.1) [20,22], applying (3.1) to (1.3) we get where we have used τ = iβb −1 /2π and σ = iβb/2π. The functions L h and Q h are the natural generalizations of those defined in [22] for T k = 0, and are explicitly given by 8 (3.5) Note that for T k = 0 both functions identically vanish, as in [22]. Since the 1/β 2 term in the exponent of the RHS of (3.3) gives a pure phase, the dominant contribution to the integral presumably comes from the locus of minima of L h (x, r k = 2/3; T k ). One has to make sure that Q h (x; T k ) is stationary at that locus though, otherwise a more careful analysis is required.

The SU(2) case
Take for example the N = 2 case. Figure 3 shows the L h function of the SU(2) N = 4 theory for sample values of T k . As the picture clearly shows, at the point x 1 = 0 the integrand is maximally suppressed.
It is easy to check that the correct "saddle-point" for N = 2 lies at |x 1 | = 1/4 ( Figure 3 is suggestive of this also); not only L h is minimized there, but also Q h is stationary as desired. Moreover, we see from Figure 3 that depending on T k the minimum of L h can be positive, negative, or zero. Only when the minimum is negative the infinite-temperature phase is  deconfined. The contours of L h (x 1 = ±1/4, r k = 2/3; T k ) are shown in Figure 4: outside the blue contour we have L h (x 1 = ±1/4, r k = 2/3; T k ) < 0, so the index is deconfined, except on the blue dots at Let us review what we have observed. While for T k = 0 both functions L h and Q h are zero and the index has a power-law asymptotics (more precisely an I ≈ 1/β behavior as β → 0 [22]), finite nonzero T k can induce Mexican-hat potentials for the holonomies in the high-temperature limit, triggering an infinite-temperature deconfinement in the index.

Higher ranks
We now show that the integrand of the index is maximally suppressed at x = 0 in fact for arbitrary N ≥ 2 and T k ∈ R/Z.
Let us study the behavior of the L h function in (3.4) with respect to x i . For this purpose, we use the following equality derived in [22] (c.f. Eq. (3.51) there), valid for −1/2 ≤ u i ≤ 1/2: min(|u l |, |u m |). (3.6) We will use the above identity with M = 4 and u 1,2,3 = T 1,2,3 ; we would moreover like to take u 4 = x i − x j , but this is not allowed since the range −1 < x i − x j < 1 is incompatible with −1/2 ≤ u 4 ≤ 1/2; to fix that we put instead u 4 = {x i − x j + 1/2} − 1/2. Using (3.6) we can now rewrite the L h function in (3.4) such that its only x-dependent piece is The above expression is obviously negative-semi-definite as a function of x i , and it is maximized when x i − x j = 0. So the index is Higgsed for any T k ∈ R/Z at infinite temperature. Just as we argued for the W wings of the previous section, here we expect that with increasing rank it becomes increasingly difficult to have the holonomies distributed such that they can pair-wise yield the negative minima of the L h function. Here one might speculate that for large enough N , since in the dominant configuration we would likely have a significant portion of the holonomies close to each other (giving x ij near zero and thus yielding nearmaxima of L h ) the total L h function would likely have a positive minimum. In other words, we find it tempting to speculate that for large enough N the infinite-temperature phase of the index is not deconfined, no matter how T k ∈ R are tuned. This is of course not incompatible with the asymptotic exponential growth arising for complexified temperature discussed in the previous section (and the related discussions of "deconfinement" in [12][13][14]).

Supersymmetric Casimir energy with complex chemical potentials
When all the fugacities p, q, u k are real-valued, the index I(p, q, u k ) is related to the pathintegral SUSY partition function Z(β, b, m k ) of the theory on where E SUSY (b, m k ) is known as the supersymmetric Casimir energy, β, b, m k are defined through p = e −βb , q = e −βb −1 , u k = e −βm k , (4.2) and S 3 b is the squashed three-sphere with unit radius and squashing parameter b, while S 1 β is the circle with circumference β. (The special case of (4.1) with m k = 0 was understood already in [21,45], based on earlier slightly contrasting computations of [43].) As made clear by HHZ [25] (and further elucidated in [12][13][14]36]) making contact with the AdS 5 BPS blackholes requires considering complex fugacities p, q, u k in the index. With the goal of understanding the role of the supersymmetric Casimir energy in the blackhole entropy discussion, in this section we study the relation between Z and I for complex fugacities such that b ∈ R >0 and β ∈ C with Reβ > 0 as in Section 2, while u k are on the unit circle as in Section 3. Rather than modifying the background geometry to achieve such complexified β (c.f. [43]), we simply analytically continue the results obtained for real p, q.
Let us consider a free chiral multiplet to begin with; as in [21,43], we expect that solving this case leads to the solution of the interacting non-abelian case as well.
Following Appendix A of [21], we start with the one-loop determinant of the nth KK mode on S 3 × S 1 . Eq. (A.15) in [21] now generalizes to where b is the special function discussed in [21], the R-charge of the multiplet is denoted by R, and T k := iβm k 2π ∈ R, with m k the only chemical potential the chiral multiplet couples to. Define X := (R − 1) b+b −1 2 for notational convenience. Following [21] step by step, we now rewrite log Z (n) in terms of ψ b which has a simple asymptotic behavior. Eq. (A.2) of [21] implies that in terms of ψ b : )]sgn(n + T k ). (4.5) One way to check the above equation is to check it separately for sgn(n + T k ) = +1 and sgn(n + T k ) = −1, using b (−x) = − b (x) and Eq. (A.2) in [21]. The reason for this rewriting is to divide ψ b s into the numerator and denominator of Z, so we can eventually relate Z to I using expressions such as (2.1). Finally, we sum (4.5) over n ∈ Z. In doing so, we use the relations Di Pietro and Honda used [24] for analyzing the high-temperature asymptotics of the index: n∈Z sgn(n + T k ) n + T k 2 = − 1 3 κ(T k ). With this regularization-combining techniques from [21] and [24]-we obtain (4.9) Putting T k = 0 we can compare with Eq. (A.16) in [21], noting that κ(0) = ϑ(0) = 0, so that the only surviving term on the second line of the RHS of the above relation gives the Di Pietro-Komargodski asymptotics [20] as β → 0; the first and the third lines combine to give the first and the third terms on the RHS of Eq. (A.16) in [21].
We are done with our regularization. We believe our method of regularization is correct because we have been careful with the convergence of the infinite product appearing in Z-or equivalently the convergence of the infinite sum appearing in log Z-after regularization, and because we have used well-established tools of analytic continuation 9 for evaluating the sums (4.6)-(4.8). As a byproduct, from the second line on the RHS of (4.9) we can read off the high-temperature asymptotics of the partition function of a chiral multiplet with a flavor fugacity on the unit circle.
We now would like to relate Z as obtained in (4.9) to the index I. We use (2.1) and the fact that the index of the chiral multiplet is Γ((pq) R/2 u k ). For simplicity we assume 0 < T k < 1, and replace all {T k } in (4.9) with T k . Then we set (4.9) equal to The end result is that E SUSY comes out just as in [21,45]: there is no dependence on T k ! In other words, for u k = e 2πiT k on the unit circle (which is relevant to the equal-charge AdS 5 blackholes) we have E SUSY (b, m k ) = E SUSY (b, 0). Since in the small-|β| limit with b > 0 fixed we have βE SUSY (b, 0) → 0, we conclude that on the saddle-point associated to the equal-charge blackholes the supersymmetric Casimir energy has no significance in the leading Cardy-like asymptotics of the partition function Z. In particular, the Casimir-energy factor relating Z and I is irrelevant to the blackhole entropy function arising in the Cardy-like limit of either.
The relation between the above discussion and the interesting proposal of [36] which seems to involve analytic continuation of Z with respect to τ and σ is currently under study.

Open problems
We have presented a careful analysis of the asymptotics of the SU(N ) N = 4 theory index in the CKKN limit where the flavor fugacities approach the unit circle and the spacetime 9 See Chapter VII of [46] for some context. fugacities approach 1. For 0 < |argβ| < π/2, we have demonstrated that in the CKKN limit, depending on the sign of argβ, there are complementary M or W wings in the space of the control-parameters ReT 1,2 . On the M wings we have given the leading asymptotics of the index, and from it extracted the two blackhole saddle-points discussed above. On the W wings, except for the N = 2 case, the analysis seems difficult; we have only presented intuitive arguments suggesting that for N > 2 the index has a slower asymptotic growth there (with an exponent that based on our numerical investigation we conjecture to be O(N 0 ) as N → ∞) and therefore no blackhole saddle-points are expected in those regions.
Problem 1) In the CKKN limit |σ|, |τ |, ImT k → 0, with τ σ ∈ R >0 , ReT k fixed, and Imτ, Imσ > 0, (5.1) find the asymptotics of the SU(N ) N = 4 theory index for N > 2 when τ, σ are inside the 2nd quadrant and ReT 1,2 are on the lower-left wing of Figure 2, or when τ, σ are inside the 1st quadrant and ReT 1,2 are on the upper-right wing of Figure 2. In particular, prove (or disprove) that in those regions the growth exponent in the CKKN limit is O(N 0 ) as N → ∞.
Even without addressing the above problem, we have successfully derived two blackhole saddle-points in the CKKN limit. However, the saddle-points have flavor fugacities that are away from the unit circle unless the three charges Q k are equal (or approximately equal). Therefore our derivation of the blackhole entropy function is incomplete for the general blackholes with unequal Q k . To complete the analysis for the general case we have to derive the asymptotic relation (1.12) when ImT k are not sent to zero.
We would like to emphasize that although we have not given a complete derivation of the entropy function for the general case with unequal charges, our analysis in the equalcharge case already allows addressing various conceptual issues in the derivation. One such conceptual issue has been the significance of the rather special relation k T k = τ + σ − 1 in the HHZ functional [25]. In the present paper we have shown that a similar asymptotics arises with k T k = τ +σ +1 in a separate region of parameters, leading to a second blackhole saddle-point with fugacities that are complex conjugate to those of the CKKN saddle-point. (See [13] for related statements in the large-N analysis.) Another conceptual point that we were able to clarify in the special case with equal charges was the insignificance of the supersymmetric Casimir energy to the blackhole entropy function in the Cardy-like limit. Generalizing that discussion to the case with the flavor fugacities away from the unit circle constitutes another important open problem related to the present work.
Problem 3) Study the supersymmetric Casimir energy of the N = 4 theory with flavor fugacities away from the unit circle. In particular, investigate its relevance to the blackhole entropy function in the Cardy-like limit.
Note added: While this work was nearing completion the preprint [47] appeared on arXiv which has some overlap with our Section 2 and moreover suggests that extra hairy-blackhole [48,49] saddlepoints might reside in the regions of Problem 1 above. As discussed in Section 2, we find it more likely that no such extra blackhole saddle-points (with O(N 2 ) entropy) exist in the Cardy-like limit of the index. The existence/interpretation of extra saddle-points in the large-N analysis [13] is of course a separate issue.