Blackhole/blackring transition

We consider BPS states in M theory compactified on a Calabi-Yau threefold with one K\"ahler parameter, and investigate their relation to blackholes and blackrings. On the microscopic side, a characterization of the BPS degeneracy can be obtained from the non-perturbative part of the topological string free energy according to the Gopakumar-Vafa conjecture. On the macroscopic side, the Wald entropy of relevant black objects in the four-dimensional IIA description is computed from the perturbative part of the same topological string free energy following the work of Cardoso-de Wit-Mohaupt and then lifted to five-dimensional M theory through the Gaiotto-Strominger-Yin correspondence. For a given value of the M2 brane charge, we find that for small angular momenta, the microscopic BPS degeneracy matches the entropy of a rotating blackhole, whereas an apparent transition occurs at a critical value of the angular momentum. Beyond this value, we find encouraging evidence that the microscopic counting is well approximated by the entropy of a blackring. We conclude by formulating a new puzzle involving the corrections due to degenerate worldsheet instantons.


Introduction
A microscopic explanation of the Bekenstein-Hawking entropy [1,2] of blackholes is an important and fascinating question that any consistent quantum theory of gravity must address. A landmark achievement was made by Strominger and Vafa [3] using the degrees of freedom on the constituent D-branes. 1 The precise version of the argument relies highly on supersymmetry 2 , and at the time the only known supersymmetric black solutions to supergravity were single-centered black holes. The Cardy growth of the supersymmetric index of the theory on the brane reproduced to leading order (in the large charge limit) the Bekenstein-Hawking entropy of a five-dimensional single-centered blackhole. Soon after, supersymmetric multi-centered blackhole solutions were discovered [15,16]. A supergravity analysis shows these multi-centered blackholes can sometimes entropically dominate over the single-centered ones with the same conserved charges [17]. It has been an open question since then if there is a precise microscopic counting in which we get the dominant contribution to entropy from both single-and multi-centered blackholes with macroscopic horizons in different ranges of charges. Consider M theory compactified on a Calabi-Yau threefold (CY 3 ). The microscopic entropy of the BPS solutions in 5d is captured by the non-perturbative part of the free energy of topological strings on the Calabi-Yau, following the Gopakumar-Vafa conjecture [18,19]. 3 We will investigate the existence of a transition between blackholes and blackrings in this context. In fact, under the Gaiotto-Strominger-Yin (GSY) lift [22,23], this transition is closely related to the aforementioned transition between 4d single-and multi-centered macroscopic blackholes in IIA superstrings with N = 2 supersymmetry (in compactifications with higher supersymmetry, say compactification on K3 × T 2 [24,25], it is well known that the only blackhole with macroscopic horizon that contributes to the suitable supersymmetric index is a singlecentered one [26,27]). However, to compare the microscopic and macroscopic formulae, we need to be careful about the quantum corrections. The advantage of the GSY lift is the expectation that the BPS entropies of black solutions in 4d and 5d are the same, including perturbative quantum corrections. This allows us to systematically take into account corrections to the Bekenstein-Hawking entropy of black solutions in 5d in terms of the well-developed 4d formalism that we discuss below.
Corrections to the Bekenstein-Hawking entropy of a blackhole in 4d consist of various contributions. The first set of corrections come from the Wald entropy [28] of higher derivative terms in the Wilsonian effective action of massless fields-present due to integrating out other massive degrees of freedom. In the context of 4d N = 2 compactifications, such contributions were systematically analyzed by Cardoso, de Wit, and Mohaupt in [29][30][31]. Among these higher derivative terms, there is the R∧R coupling that comes from the classical dimensional reduction of the R 4 term in the Wilsonian effective action of M theory [32][33][34]. All other higher derivative terms come from a degenerate limit of the contribution of Euclidean M2 brane instantons (wrapping a two-cycle in the Calabi-Yau and the M theory circle). In the language of topological string theory, these contributions are captured by the perturbative part of the free energy. According to the Ooguri-Strominger-Vafa conjecture [35], 4 apart from these higher derivative terms there also exist the contributions of nondegenerate M2 brane instantons and the one-loop fluctuations of massless fields in 4d that can be calculated systematically in Sen's formalism [36][37][38]. 5 In this work, by carefully analyzing the attractor equations [42,43], we will show that for the D6-D2-D0 blackhole with one unit of D6 charge, the volume of the Calabi-Yau at the horizon in string units is fixed to a large value (scales with the D2 charge) when the D0 charge is small. The radius of the horizon is the same as the M theory circle, and the blackhole is prone to all-order corrections from the degenerate worldsheet instantons. As a result of this phenomenon, the microscopic origin of this blackhole has qualitatively new features compared to its higher-supersymmetric analogs (for N = 4 blackholes these contributions vanish [44,45]). Since the volume of the Calabi-Yau is large we can ignore the contribution of nontrivial M2 brane instantons. As a result, the entropy from the 4d point of view is captured by the perturbative part of the free energy of the topological strings, in sharp contrast to the microscopic calculation in 5d that gets contributions from the non-perturbative part only. This gives a nontrivial consistency check for the topological strings itself.
As a concrete example, we consider the compactification on the quintic threefold. The calculation of Gopakumar-Vafa (GV) invariants 6 on a compact manifold such as the quintic is a notoriously difficult task (more so because the quintic does not have any elliptic or K3 fibration-at least not before making a nontrivial geometric transition). A systematic procedure for the calculation of the GV invariants up to all genera is not known currently. The difficulty is related to the fact that the holomorphic anomaly equation [50,51] satisfied by the free energy of topological strings requires suitable boundary conditions to give a unique solution. A procedure for calculating GV invariants up to genus 51 for quintic was formulated in [52], taking into account a set of physical boundary conditions at the orbifold point, the conifold point, and the large volume limit. However, the explicit information available in [52] for the GV invariants was up to much lower genera. Therefore, we initially carried out the explicit calculation of GV invariants up to genus 49 (available at [53]) microscopic description [57] of blackrings given by the Maldacena-Strominger-Witten (MSW) CFT [58]. The comparison of the entropy coming from GV invariants with that coming from the MSW index with p = 1 shows a spectacular agreement. This gives unequivocal evidence that blackring microstates indeed contribute to the 5d BPS index captured by the GV invariants and that p = 1 is a dominant contribution up to the M2 brane charge d for which the GV data is available. It is unclear whether p = 1 should remain a dominant contribution in the large d limit, or whether p > 1 blackrings even contribute. If all values of p contribute, then for very large d, a better characterization of the blackring entropy would be an interpolation of the semi-classical entropy optimized over continuous (positive) values of p for a range of j L , and the p = 1 MSW entropy for j L beyond the point where the optimal p becomes O(1).
The remainder of this paper is organized as follows. In section 2, we study the blackhole/blackring transition within supergravity, using their Bekenstein-Hawking entropies. In section 3, we introduce the Gopakumar-Vafa invariants counting 5d BPS states, review how they are computed, and compare the corresponding microscopic entropy with supergravity. In section 4, we systematically study the Wald entropy as well as further corrections, first in the 4d language by way of the Ooguri-Strominger-Vafa conjecture, and then uplifted to 5d through the Gaiotto-Strominger-Yin correspondence. In section 5, we perform a microscopic/microscopic comparison between the GV invariants and the microstate counting of blackrings with unit M5 brane charge. Finally, section 6 ends with a summary and discussion of the open problem concerning degenerate instantons.

Blackhole/blackring transition in supergravity
In this section, we investigate the phase diagram of 5d minimal supergravity coming from M theory compactified on a Calabi-Yau (CY 3 ). In the regime of large angular momentum, we will show that there is a transition (in the sense of entropic dominance) from a single-centered blackhole with a spherical horizon (i.e. S 3 ) to a blackring with a different horizon topology (more precisely S 2 × S 1 ). 10 While this is certainly expected when the angular momentum is above the blackhole extremality bound, the actual transition occurs slightly below it. 11 10 See [59][60][61] for similar discussions involving other horizon topologies. We thank Jorge E. Santos for pointing this out to us. 11 It is not completely unreasonable to wonder if there is a manifestation of this phenomena at low temperatures as a dynamical instability of the blackhole. For a recent account of instability in the background of fast rotating blackholes (at high enough temperature) in AdS see [62] and references therein.

Field content of 5d supergravity
We work in units where the tension of an M2 brane is unity unless otherwise mentioned. There are eight real supercharges unbroken in this situation. We will describe the bosonic field content in the language of d = 5 N = 1 supergravity at a generic point on the moduli space (at special points one might need to add additional matter fields that we will not discuss here). In 11d the only degrees of freedom are the metric gμν and the three form gauge field Aμνρ (μ = 0, 1, . . . , 10). We decompose the coordinates into the 5d part µ = 0, 1, . . . , 4 and the internal CY 3 part i,ī = 1, 2, 3.
Under the reduction to 5d, we obtain the 5d metric g µν , h 2,1 complex scalars g ij , h 1,1 real scalars g ij , a real scalar Φ related by Hodge star to A µνρ , one complex scalar A ijk = ϵ ijk c, h 2,1 complex scalars A ijk , and h 1,1 real vectors A µjk . Before discussing the multiplet structure of these fields it is convenient to introduce an integer basis α A , A = 1, 2, .., h 1,1 for H 1,1 (CY 3 , Z). The Kähler class is harmonic and can be expanded as where v A are real scalar fields in 5d. The scalar field associated with the volume of the CY 3 is where the intersection numbers are defined by In total we get n H = h 2,1 + 1 hypermultiplets whose scalars are related to g ij , A ijk , c, Φ, V. The remaining n V = h 1,1 − 1 scalars φ a related to the g ij above are in the vector multiplet. Those can be parameterized as The abelian real vector fields in 5d coming from A µjk can be also discussed by expanding the three form field strength as One of the field strengths is in the graviton multiplet along with g µν (enters in the SUSY variation of the gravitino) and given by and is called the graviphoton field strength (in general it is not gauge invariant), and the remaining n V = h 1,1 − 1 of them are in the vector multiplet mentioned above along with the φ a s. All the fields in the vector and hypermultiplets are uncharged under the h 1,1 vector fields mentioned here, and as a result, we get ungauged d = 5 N = 1 supergravity.

Entropy of the half-BPS blackhole
The Breckenridge-Myers-Peet-Vafa (BMPV) blackhole [56] is a solution to 5d N = 1 supergravity that preserves half the supersymmetries. In this solution, only the fields in the graviton and vector multiplets are turned on [63]. The entropy of the blackhole is given by where Here y A are related to the horizon values of the scalars in the vector multiplet φ a , and fixed by the attractor mechanism [42,43]. The blackhole solution preserves half of the supersymmetries. Here d A is the charge with respect to V A , normalized such that at the quantum level, it is integer-valued. The flat 5d has a spatial rotation isometry of SU(2) L × SU(2) R that is broken down to U(1) L × SU(2) R by the blackhole solution. And j L is the half-integer quantized (in the full quantum theory) angular momentum for U(1) L .

Entropy of the half-BPS blackring
Another half-BPS solution to 5d N = 1 supergravity is the Elvang-Emparan-Mateos-Reall/Bena-Warner/Gauntlett-Gutowski (EEMR/BW/GG) blackring [64][65][66][67], which has Bekenstein-Hawking entropy given by [57] (2.10) Here p A are the charges associated with M5 brane, and q 0 , q A are related to charges mentioned above as follows  Table 1: One-parameter Calabi-Yau threefolds studied in [68]. The first column gives their definitions as complete intersections in weighted projective space; we refer to them by X • without the weights. The next three columns provide their basic topological data, including the Euler characteristic χ, the self-intersection number k, and the second Chern class c 2 . The last two columns specify the maximal genus g avail at which the topological string data is currently available from [54,55], and the corresponding maximal M2 brane charge d avail . Although not needed in the supergravity analysis of this section, we list in table 1 the 13 one-parameter CY 3 studied in e.g. [52,54,68] and their relevant topological data, including the Euler character χ and second Chern class c 2 . For such CY 3 , the entropy of the blackhole in (2.7) simplifies to From (2.13), (2.14), it is clear that the rescaled entropy

Blackhole/blackring transition on CY
is independent of the geometric data (k) of the Calabi-Yau for both the blackhole and blackring solutions. For the blackhole, s 0 depends only on y, whereas for the blackring, s 0 depends on both x, y. For a given value of y > 0, the scaled entropy of the blackring s 0 (x, y) has a local maximum as x > 0 is varied, by which we get a function of y alone. The resulting curves are plotted in figure 1. From the plot, one can see that for angular momentum the blackring is entropically dominant over the blackhole. We refer to this phenomenon as the blackhole/blackring transition. In the rest of the paper, we will study this from the point of view of M theory.

5d BPS indices from topological string theory
Having established the existence of a blackhole/blackring transition in supergravity, we can ask if this transition manifests in a quantum setting. Had the entropic transition been for supergravity solutions in anti-de-Sitter space, the transition would have had ramifications in the dual conformal field theory. In flat space, while the microstates describing a single black object admit a quantum description in terms of the worldvolume theory of the underlying brane, it is less clear whether the microstates of all objects (with the appropriate boundary conditions) can be captured by a coherent quantum theory.
Fortunately, a well-defined counting of microstates in flat space can be done by studying supersymmetric indices, where supersymmetry reduces the problem to geometric questions about the internal Calabi-Yau. However, not all supersymmetric objects contribute to indices, 12 and thus it is unclear whether the blackhole/blackring transition should manifest in the index, which is a central question we investigate.
The first two subsections are reviews, where we introduce 5d indices known as Gopakumar-Vafa (GV) invariants, and explain how they are computed using the machinery of topological string theory. We then numerically compare the microscopic index with the macroscopic entropy, the results of which will motivate us to study in later sections the corrections to the semiclassical entropy from a macroscopic point of view, as well as the comparison of GV invariants with blackring indices for small charges.

Gopakumar-Vafa conjecture
Consider the worldvolume theory of an M2 brane wrapping two-cycle of class [d] ( see section 2.2 for the appropriate definition). One obtains an effective supersymmetric quantum mechanics of a BPS particle of charge d propagating in 5d (see [21] for a careful discussion). These states can be further labeled by the representation [(j L , j R )] under the spatial rotation group SU(2) L × SU(2) R . Here both j L,R are taken to be half-integers. Abstractly, one can write the representation content as (3.1) Following [18,19], we can define a supersymmetric index by tracing over SU(2) R with signs, and further organize the results into SU(2) L representations to define the Gopakumar-Vafa (GV) invariants n g d , i.e.,

(3.2)
12 This phenomenon is well studied in 4d N = 4 string theory (II on K3 × T 2 ), where none of the multi-centered black hole solutions (with at least one macroscopic horizon) contributes to the index (inverse Igusa cusp form) [26,27].
The 5d supersymmetric index-a proxy for the BPS degeneracy at fixed d, j L -can be obtained from Gopakumar and Vafa argued that the GV invariants n g d can be computed by studying topological strings on the CY 3 (see [20] for more details and [21,69] for the subtleties involved).
For concreteness of discussion, let us focus on the quintic X 5 with χ = −200, k = 5, and c 2 = 50. The GV conjecture can be expressed more explicitly as 13 (3.4) Here F (g) is the free energy of topological strings at genus g > 0 and F (0) is the prepotential (in a specific physical holomorphic gauge that we will discuss in detail later), (t,t) keeps track of the Kähler moduli of the CY 3 . Each C n is an nth order polynomial with the constant term set to C n (0) = 0 by absorbing it into n g 0 . All the terms with d > 0 are coming from worldsheet instantons and would be referred to as the non-perturbative part and the rest of it would be called the perturbative part.
In IIA, F (g) for g > 1 is expected to capture the coefficient in quantum effective action involving two anti-self dual gravitons and 2g − 2 anti-self dual graviphotons [70,71], so the right side of (3.4) can be thought of as encoding the coefficients of the F terms [50] in the Wilsonian effective action coming from integrating out a Euclidean M2 brane wrapping a holomorphic two-cycle in the CY 3 and the M theory circle. However, the conjecture relates them at the large volume limit.
Physically, we expect the number of states to go to zero as we increase the angular momentum j L for fixed mass d, and this expectation is met by the following bound on the GV invariants known as the Castelnuovo bound [20] n g d = 0, g > 10 + 5d + d 2 10 . (3.5)

Computation of topological string free energy
In this section, we review the method in [52] for computing the topological string free energy, which is based on the Bershadsky-Cecotti-Ooguri-Vafa (BCOV) holomorphic anomaly equation [50,51] as well as [72,73], and focus on the quintic for concreteness. The reader who is familiar with the methodology, or is content with knowing the existence of an algorithm, can safely skip to the next subsection.
At genus zero, F (0) is determined in terms of the geometry of the moduli space. However, the Kähler moduli space parameterized by t that we are after gets nontrivial α ′ corrections. On the other hand Mirror symmetry maps the Kähler structure moduli space to the complex structure moduli space parameterized by z which is α ′ exact. Therefore we will study the topological strings on the mirror quintic and then we will use mirror map t = t(z) to get the answer for the quintic. The coordinate z is chosen such that at z = 0 we have an orbifold description of the worldsheet CFT, at z = 1 we encounter the conifold transition, and z → ∞ is the large volume limit.
We rewrite the topological string free energy as Here ω 0 reflects the choice of the very specific holomorphic Kähler gauge (from the point of view of the worldsheet CFT of topological strings, there are nontrivial contact terms in the OPE at a generic point on the moduli space. Because of these contact terms, topological string free energy depends sensitively on the choice of the U(1) line bundle gauge field A and the Zamolodchikov metric G on the moduli space. Once a holomorphic gauge is chosen for A, the answer still depends on the choice of the holomorphic gauge which can be further fixed at a given value oft =t 0 by demanding that the covariant derivative with respect to A simplifies. It has been argued that this special choice is what is necessary to compare the results with the algebraic results on holomorphic curves). The mirror map can be calculated from the Picard-Fuchs equations at a point on the moduli space [74] ( The boundary condition on ω is subtle and contains the choice of gauge. We will specify it at specific points below. Near the large volume pointt = −i∞, we have following solutions The geometry of the moduli space in this holomorphic limit is determined by (the constant of proportionality would be unimportant for the calculation and we will not bother to fix it) Near the orbifold pointt = 0, we have the following solutions (these are related to the solutions in the large volume limit but not exactly the same) where (k) n is the Pochhammer symbol. Again the geometry of the moduli space is given by (3.9) with ω 0,1 defined as in (3.10). A closed-form expression is not available near the conifold pointt = 1. We fix it by the following conditions (these are related to the solutions in the large volume limit and at the orbifold point presented earlier but not exactly the same) (3.11) Here (3.9) remains valid. Now we turn to state the results for the GV invariants. It will be convenient to define the following decomposition (at a generic point on the moduli space) (3.12) The coefficients a g,i will be called the holomorphic ambiguity, the reason for which will be clear in a moment. The upper and lower limit on the sum has to do with the boundary conditions on the moduli space that we will discuss later. We define a covariant derivative by the following formula (3.13) At genus zero the data of GV invariants can be conveniently encoded into For higher genus non-holomorphic dependence of the topological string free energy is completely fixed by the holomorphic "anomaly" equations [51]. Conceptually this can be understood as follows. F (g) (t,t) for the topological string B model can be written as an integration over the moduli space of genus h Riemann surface just like the usual IIB string theory. An anti-holomorphic derivative (∂t) of F (g) (t,t) inserts an operator in the path integral representation that is BRST exact and therefore gets non-zero contribution only from the boundary of the moduli space where one can factorize the answer through lower genus data. The equation is most conveniently expressed through Yamaguchi-Yau variables [72] All other A n , B n can be expressed as a polynomial of A 1 , B 1,2,3 , X. For example Also, note that Genus one GV invariants can be obtained from The holomorphic anomaly equations for g > 1 are given by To use these equations one has to treat u, v 1,2,3 , X as independent variables and P (g) n (u, v 1 , v 2 , v 3 , X) as a function of these variables. Now start from eq. (3.14) and eq. (3.18) and use (3.19) recursively. It is clear from this structure that the holomorphic anomaly terms parameterized by a i are not fixed in this procedure.
There is no systematic procedure to fix the holomorphic ambiguity for all genera. However, from the additional input of boundary conditions at the orbifold point, the conifold point, and the large volume limit, we can fix them up to genus g = 51 as follows. The lower limit in holomorphic ambiguity sum (3.12) is contained by the contribution of degenerate worldsheet instantons (these instantons are constant maps in the target space. These contributions can be calculated on the worldsheet for their relative simplicity) in the large volume limit [75] . (3.20) The upper limit is constrained by the knowledge of the gapless modes at the conifold point [76,77] In the IIA language, they come from a D2 brane wrapping the S 2 that shrinks as we approach the conifold point. At this point and until the new massless degrees of freedom are taken into account, the string background is expected to be singular. When the new massless degrees of freedom are taken into account, we go through a geometric transition to a different Calabi-Yau.
This gives 2h − 2 conditions on a g,i . More specifically this formula can be obtained by using the information of massless modes in the RHS of (3.4). On the other hand, we do not expect anything singular at the Gepner point The number of unknown a g,i reduces to The boundary conditions (3.20), (3.21), (3.22) together with the knowledge of Castelnuovo bound (3.5) and explicit knowledge of n 51 20 = 165 14 in principle determines the GV invariants up to genus g = 51. 15 However, in practice, the evaluation is rather complex.
Note As we computed the GV invariants for the quintic up to g = 49, we came to learn of the beautiful recent work [54] that established new constraints on the GV invariants of various one-parameter CY 3 , and pushed the determinable GV invariants beyond past limitations (in principle up to g = 68 for quintic). Our results for quintic are in complete agreement with [54]. The details of our calculation can be found in appendix B and on the website [53]. For the subsequent numerical analysis, we use the data publicly available at [55].

Microscopic/macroscopic comparison
We now perform a numerical study of the 5d BPS degeneracies (3.3) for M theory on quintic, using the GV invariants calculated in the previous section. For fixed M2 brane charge d, the entropy defined as is plotted against the angular momentum j L in figure 2. We pause for a moment to notice a few basic features of the graph. For a given value of d, there is a kink at a 14 Near the Castelnuovo bound some GV invariants are determined in the mathematics literature. 15 Note that the maximal genus for a non-zero GV invariant at d = 20 is g max (20) = 51. critical angular momentum j L,c (d), which can be defined by maximizing over j L the discrete second derivative Thus we expect that the contribution to the entropy for angular momenta j L < j L,c and j L > j L,c are coming from different black objects from the macroscopic point of view. Is this the blackhole/blackring transition? A direct comparison of the BPS index entropy at d = 22 with supergravity exhibits apparent discrepancy. However, as we see on the right of figure 2, the BPS index entropy continues to decrease as d is increased and is far from stabilizing at d = 22.
Blackhole For angular momenta below the transition point, an agreement in the d → ∞ limit seems plausible. In fact, near j L = 0, this agreement has already been established by careful numerical extrapolation using the Richardson transform [68]. In figure 3, we perform linear extrapolations in 1/d to d → ∞ for a range of j L /Q 3 2 below the transition point, and find good agreement with supergravity. To be precise, since the GV invariants define Ω(j L , d) for half-integral j L , in order to fix j L /Q non-half-integral values of j L . 16 The strength of the linear relationship between S(d) and 1/d as shown in figure 3 suggests that the leading order correction to supergravity takes the form This is the kind of correction that would arise from the dimensional reduction of the higher derivative terms (TrR 2 ) 2 and TrR 4 in M theory. Although such corrections have been studied in the past [29][30][31], the structure of these corrections has not been carefully analyzed to the full extent possible. We devote the entirety of section 4 to expounding the details of these corrections.
Blackring Now we focus our attention on the transition point. The critical angular momentum is plotted in figure 4 as a function of 1/d. Again we use a linear fit to give the extrapolated value for j c (d → ∞) = 0.9Q(d) 3/2 . By comparing this value with the discussion of blackhole/blackring transition in section 2.4, we conclude that it is possible that the black object dominating the entropy in the microscopic plot for angular momentum larger than the transition point is a blackring. However, there is no direct evidence for the index entropy to agree with the blackring entropy in the d → ∞ limit. We work under the hypothesis that it counts blackrings. In the next few sections, we will analyze in detail the following two viable scenarios: 1. The leading contribution to entropy is coming from a blackring whose M5 brane parameter p is macroscopic (scales as d 1 2 ). In this situation, we will carefully discuss the corrections to Bekenstein-Hawking entropy similar to the blackhole above.
2. The leading contribution to entropy is coming from a blackring whose p is microscopic, invalidating the large d analysis. In section 5.2, we will compare the tail with the exact blackring microstate counting for p = 1, and find surprisingly good agreement for small values of d.

Wald entropy and quantum corrections in 4d
In 5d supergravity there is no systematic understanding of the superspace formulation for the higher derivative terms that are relevant for the corrections to the Bekenstein-Hawking entropy. However, the BPS black solutions in 5d can be considered as the Gaiotto-Strominger-Yin uplift [22,23] of the solutions in 4d where we have a much better understanding of the superspace formulation. We turn to describe the uplift very briefly. Consider a configuration in IIA with p 0 D6 branes wrapping the Calabi-Yau. This configuration in M theory language corresponds to a factorized geometry involving the Calabi-Yau and a Taub-NUT space. The Taub-NUT radius R T N in this example is determined by the 10d IIA string coupling. At strong coupling, R T N becomes large in string units, and then for radial distance in Taub-NUT much smaller compared to R T N we have an effective flat Minkowski space for p 0 = 1. On the other hand, for weak string coupling in the region of Taub-NUT at distances much larger compared to R T N , we have an effective 4d IIA description. The radius of the Taub-NUT is related to an asymptotic value of the moduli in the IIA language, and due to the attractor mechanism, the leading order entropy of the BPS blackhole is independent of the asymptotic moduli. It is expected that this fact survives beyond leading order. This allows us to calculate the quantum corrected entropy of the blackholes in 5d in terms of that in 4d. Keeping this in mind, we first discuss the quantum correction in 4d following the Ooguri-Strominger-Vafa conjecture [35] and delay the detailed mapping of solutions to section 4.4.

Ooguri-Strominger-Vafa conjecture
In this section, we consider extremal blackholes in 4d N = 2 supergravity obtained by compactifying M theory on S 1 ×CY 3 . This can be thought of as a compactification of the 5d N = 1 supergravity we discussed in the previous section. The compactification gives rise to an additional Kaluza-Klein gauge field a. The gauge fields in 4d can be related to the 5d ones as follows where α I = V I S 1 is the component of 5d gauge fields in the M theory circle direction. Among h 1,1 + 1 gauge fields A Λ , one combination becomes the 4d graviphoton field (we will describe it in detail in the later part of this section) and each of the remaining h 1,1 is in a vector multiplet. Along with the gauge fields, a scalar field is generated e σ = e γ v where e γ keeps track of the radius of the S 1 circle in M theory units. 17 The complex scalar fields in the h 1,1 vector multiplets can be parameterized as follows Now we consider extremal blackholes carrying magnetic and electric charges given by (p Λ , q Λ ). We associate the magnetic charges p 0 , p A with the D6 and D4 branes, respectively, and the electric charges q 0 , q A with the D0 and D2 branes. In our convention, all charges are integer quantized. The leading order Bekenstein-Hawking entropy of the blackhole may be written as [42,43] where the Kähler potential is determined from the holomorphic prepotential F 0 by 18 The moduli fields in the vector multiplet at the horizon are determined entirely in terms of the charges by the attractor mechanism (the BPS entropy does not depend on the fields in the hypermultiplet) Clearly, this determines X Λ , C only up to Kähler gauge transformations 19 The entropy can be written in a very suggestive thermodynamic form following Ooguri-Strominger-Vafa (OSV) [35] where we used the following definition for the electric potential ϕ Λ (4.8) 17 Note that the 4d string coupling is given by g s,4 = v − 3 2 and it is in the hypermultiplet in 4d. Just as in 5d, in 4d we still have h 2,1 + 1 hypermultiplets. 18 Here sum is over {Λ} = 0, {A}. We are using the notationf (X) = (f (X)) * . 19 The Kähler potential transforms as K → K − f (X) −f (X).
Note that here ϕ Λ is a real number and we can determine the electric charge in terms of it as follows, 20 . This thermodynamic form suggests that we consider the partition function given by (4.9) The prepotential can be calculated from the topological string amplitude in holomorphic gauge The OSV conjecture [35] (here we present a refined formulation following Denef-Moore [17]) is the statement that the exact degeneracy of the blackhole can be calculated from a formula analogous to (4.9), (4.10) (near large volume limit of the Calabi-Yau threefold) with Here the partition function of the topological strings is given by (4.14) The perturbative contribution F local (g top , t) is the same as C 3,1 in (3.4). These contributions can be understood from the M theory point of view as follows: in 11d, lowest derivative order supergravity action [78] gives the cubic term in F local as expected. Apart from that, there is a specific R 4 term [32][33][34] that is required by the consistency 20 Compared with [17], the parameter ϕ(here) = −2πϕ(there).
of supersymmetry and anomaly cancellation [79,80]. Upon dimensional reduction of M theory to 4d IIA, this contains the R ∧ R coupling (the two other factors of R are in the Calabi-Yau directions and captured by c 2 above) that contributes to the linear term in F local [29].
Other contributions from degenerate worldsheet instantons are given by (see [45] for more details) 21 .
On the other hand, Z ′ (g top , e 2πit ) comes from nontrivial worldsheet instanton contributions in topological strings. For this paper, we will always stay in the domain where the nontrivial worldsheet instanton corrections are negligible. The precise form of the measure is not known, but the following form was suggested in [17] and found to be consistent with the analysis of [37]: ). (4.16) The simplest way of calculating the degeneracy as in (4.12) would be to perform a saddle point approximation taking the contribution only from the F local (g top , t) term (we will come to the validity of such an approximation in details later in this section). The saddle point value of the charges of the blackhole is given by The entropy is given by a thermodynamic formula with In this subsection, we focus on the discussion of the entropy of a single-centered large blackhole carrying one unit of D6 charge (p 0 = 1) and D2-D0 charges, but no D4 charge (p A = 0). For notational convenience that will be closer to the 5d setting which we aim to discuss later, we introduce This is motivated by the fact under the 4d/5d uplift the Taub-NUT circle plays the role of the M theory circle and the momentum on the circle naturally keeps track of the D0 brane charge in the IIA description. In terms of these parameters, we have As mentioned earlier, for the saddle point evaluation of the entropy, we approximate the topological string free energy by Here we have treated ω, ϕ as real numbers while taking complex conjugate. Since we will be dealing with only Calabi-Yau compactification with one Kähler parameter in this paper, for simplicity we will denote ϕ := ϕ 1 , q := q 1 . The blackhole partition function is given by and the charges evaluate to have the following simple expressions (4.24) We pause for a moment to note that when both the charges q, j L are large, |ϕ/ω| is a large number. This justifies the assumption that we can ignore nontrivial worldsheet instanton effects in this domain. In addition, in this limit, g top is a small number and therefore we can ignore most of the degenerate worldsheet instantons from higher genera. 22 These discussions are much more subtle when we are looking at the limit 22 Strictly speaking, we cannot ignore the effect of the genus-zero degenerate instantons, or the effect of the measure factor in the OSV formula. The first effect is important only when we are too close to the extremality. Both these effects will be numerically sub-dominant in the parameter range where we will use the formula below.
of small angular momenta. This issue will be discussed in detail later in this section. By inverting the expression of the charges, we can write down the entropy as a series expansion in where the first few coefficients are explicitly computed to be (4.26) The corrections associated with d (0) m do not vanish at zero angular momentum. Whereas the corrections associated with d (1) m,n remain small for slowly rotating blackhole and in the extremal limit. The value of d (0) 1 in the formula above is the same as in [82]. Other corrections have their origin in the fact that the topological string Kähler parameter gets various Q −1 corrections.

D6, D4-D2-D0 multi-centered blackhole
First, we consider a single-centered blackhole carrying D4-D2-D0 charge, but no D6 charge (p 0 = 0). We will again use (4.20) and work with one parameter CY 3 , but now we have (as before p := p 1 ) which gives It immediately follows that the entropy is given by [45] S = 2π (kp 3 + c 2 p) 6 (−q 0 + 1 2kp (4.29) In the large p limit, the expression for the entropy above reduces to the more general formula in [23] when p 0 is set to zero. The whole effect of the Wald entropy correction 23 At this stage, we do not have a clear logic for this particular choice of the expansion parameters.
The explicit comparison of BPS entropy with the GV data will serve as an indirect justification. We thank Cumrun Vafa for an insightful discussion on this topic. Further details of the formula will appear in [81].
was to shift kp 3 → kp 3 +c 2 p. This is very different from the case considered previously with p 0 ̸ = 0.
In this paper, we will be interested in the D6, D4-D2-D0 multi-centered blackhole with one unit of D6 charge. Now we turn to the discussion of the wall of marginal stability for this bound state following [17,83]. We demand the distance between the centers to be positive and finite Here Γ = (p Λ , q Λ ) is the charge vector and the central charge of single-center is given by (in large volume limit, we set the B field to zero) This suggests that the bound state exists for the following choice of signs In this work, we will talk about the bound state only in this domain. For the charges in this domain, the 4d solution when uplifted to 5d is certainly healthy. We propose this is the physical domain of the uplifted blackring solution (as far as its contribution to the GV formula is concerned). Keeping in mind the identification q 0 = 2j L , we need to take into account the zero point shift in entropy as follows [57] S = 2π (kp 3 + c 2 p) 6 −q 0 + 1 2kp q 2 + (kp 3 + c 2 p) 24 1 2 . (4.33)

Loop corrections
We pause for a moment to specialize to the background that is suitable for the 4d/5d lift of the BMPV blackhole This gives the following mapping of parameters The chemical potential ω is conjugate to j L (and ϕ A is conjugate to q A ). Note that the topological string coupling is not small in this setting for small ω g top (ω = 0) = −4πi, =⇒ |g top (ω = 0)|> 1. Therefore to discuss slowly rotating BMPV blackhole we need to sum over all order contributions of the topological string amplitudes. In particular, we need to resum (4.15). Now we explain this from a more physical point of view as follows. We focus on the case ω = 0 and make some order of magnitude estimate using the attractor equations (4.24), (4.2). Before we do that we want to restore the units for this section from M theory units to string units (see appendix A for more details). Note that so far we are working in M theory units in which we have set  The blackhole has a very tiny horizon-it is of the same order as the M theory circle! We can summarize this as follows. The blackhole has a large horizon in 4d Planck units but since the volume of the Calabi-Yau in string units is fixed by the attractor mechanism to scale with the entropy, from 10d units the radius of the horizon is too small. Therefore it is highly quantum. The contributions in (4.15) are precisely the Wald entropy coming from higher derivative F-terms in the 4d Wilsonian effective action whose bosonic part contains Here R − is the anti-self dual curvature, and W − is the anti-self dual graviphoton field strength. Now we want to estimate the order of magnitude of this term entirely from the 10d IIA perspective when ω = 0, i.e. for the D6-D2 system. The attractor 24 We thank Xi Yin for pointing this out, and Andrew Strominger for an insightful discussion.
value of the graviphoton field strength is given by [30] 25 (4.42) Here we used the result in (4.40). Since from 10d IIA perspective, the term originates exactly in genus g, as a result, we note that the effective dilaton dependence of the term is independent of the genus g.
We conclude that it is necessary to resum Z 0 GW and re-expand around (4.36). A resummation is proposed in appendix A of [45], assuming a certain choice of the contour (at this point without further physical inputs we do not have any justification for any particular choice) in terms of the MacMahon function: Note that the value (4.36) is on the boundary of the domain of convergence and therefore there might be new non-analytic terms in g top in this limit. We will leave a careful analysis of such questions to future work. Similar conclusions apply to the slowly rotating EEMR/BW/GG blackring.

Gaiotto-Strominger-Yin uplift to 5d
In this subsection, we will compare the result for the entropy obtained on the macroscopic side around the large d limit with c 2 corrections, and then compare it to the microscopic answer obtained through the GV invariants. This process has two limitations. The first is that it is valid for large d, but here we are in a range of d that is comparable to geometric data of the Calabi-Yau. Secondly, as we have discussed extensively before, at subleading orders in 1/d, there are an infinite number of quantum corrections that are naively large term by term (we do not know their resummed contribution at this stage). Despite these issues, we will show that in four different Calabi-Yau compactifications, our large d results are very close to the microscopic index. The D6-D2-D0 blackhole in 4d uplifts to BMPV blackhole [56] solution in 5dthe D0 brane charge becomes the angular momentum and D2 brane charge becomes the M2 brane charge in 5d.
(4.44) 25 We used the following mapping of fields:  The multi-centered bound state of D6 and D4-D2-D0 blackhole in 4d uplifts to the EEMR/BW/GG blackring in 5d for which the mapping of the charges is a little different The entropy of the EEMR/BW/GG blackring solution [64][65][66][67] for one parameter Calabi-Yau in terms of 5d parameters takes the following form Our microscopic calculation does not keep track of the M5 brane parameter p. For d as large as the currently available GV data, extremizing the blackring entropy (4.47) over (positive and continuous) p for individual values of j L results in small values of p especially when j L is large. Therefore, this procedure, while reasonable for much larger d, does not provide a meaningful comparison to the available GV data. We need some additional rules for fixing this parameter.
A natural proposal might be the following: For a given d from the plot of the Gopakumar-Vafa curve one can see that there is a critical angular momentum j L,c (d) at which there is a sharp transition (studied in detail for quintic in the previous section). We will maximize the leading order (in large d, i.e. c 2 → 0) entropy of the blackring at j L = j L,c (d) subjected to (4.32) to obtain p = p c (d). While comparing to the microscopic calculation, we will add c 2 corrections using (4.47) with p = p c (d) fixed. For X 5 , X 4,2 , X 4,4 , X 6,2 we compare resulting plots in figure 5 below for the maximal available value of d, and find that the numerical curves are close to each other. However, for the small range of d that we can discuss, the extremized value of p is pretty small-near or smaller than one. This suggests that the discussion here in terms of supergravity is at best an extrapolation. To take this fact into 26 At this point, it is natural to question how reasonable it is to keep the 1 d 2 correction while ignoring other corrections of the form (at j L = 0) − log 2 9k 1 3 d as expected from the 5d analysis of [38]. One way to justify this for the quintic is to note that we are interested in the range where d ≈ 20, c 2 = 50, k = 5, and the numerical value of the log term is smaller. As we mentioned previously, other O(d 0 ) contributions from degenerate instantons are also not understood for the BMPV blackhole. account, in the next section, we discuss the comparison of the GV calculation with the MSW index which captures the microstates of the blackring, and discuss why the contribution to the tail of the GV curve might arise solely from bound states based on a mathematical conjecture.

Microscopic observations and a quantum transition
Whereas 5d BPS states in Calabi-Yau compactification of M theory are counted by the Gopakumar-Vafa invariants, 4d BPS states in that of IIA (in the large B field chamber) are counted by the rank-one Donaldson-Thomas (DT) invariants [84,85]. The latter in the case of unit D6 charge is related to the former by a mathematical conjecture [86,87] that was motivated by [88] and physically justified by [89] using the GSY 4d/5d lift [22], as will be reviewed in section 5.1. It was argued in [57] that a microscopic description of 5d blackring microstates is provided by the Maldacena-Strominger-Witten (MSW) CFT [58], which arises in the infrared limit of the worldvolume theory of M5 branes wrapping a 4-cycle (ample divisor) of the Calabi-Yau. Therefore, the comparison of the tail in the 5d entropy with the blackring entropy becomes, at the microscopic level, the comparison of the GV invariants with the MSW indices. In section 5.2, by invoking the 4d/5d lift and wall-crossing, we propose an approximate relation (5.17) between the two with the latter having unit D4 charge, and find strong numerical evidence for the proposal. In section 5.3, we observe that the mathematical Bayer-Macrì-Toda (BMT) inequality [90,91], a necessary condition for the stability (with respect to a reduced central charge function) of 4d BPS states everywhere in the Kähler moduli space, coincides under the 4d/5d lift with the extremality bound of 5d BMPV blackholes. 27 As before, we consider one-parameter (h 1,1 = 1) Calabi-Yau threefolds X whose divisor has self-intersection number k ∈ Z and second Chern class c 2 .

Free gas of M2 branes and the DT/GV relation
We first review an exact relation between the 4d and 5d BPS counting as well as the physical interpretation, setting the stage for later subsections.
DT/GV relation Given the GSY 4d/5d lift [22], one may expect that the 4d counting of DT invariants is related to the 5d counting of GV invariants in a deterministic fashion. Indeed, such a relation was conjectured [86,87] even before the GSY discovery, and can be stated as follows. 28 The generating function of DT invariants Z DT (y, q) := Q,m DT(Q, m)y Q q m (5.1) 27 Given the GSY correspondence [23] between 5d and 4d solutions [92], the 5d blackhole extremality bound is equivalent to a 4d bound on the D2 and D0 charges for the existence of single-centered macroscopic blackhole solutions (with unit D6 charge) to the attractor equations [42,43]. 28 Mathematicians formulated a conjectural relation between Donaldson-Thomas and Gromov-Witten invariants, the latter of which can be transformed into the Gopakumar-Vafa invariants.
admits an expression in terms of the GV invariants n g d as where M (q) := n>0 (1 − q n ) −n is the MacMahon function, and χ is the Euler characteristic of X. The formula (5.2) admits a rewriting as 29 which has the symmetry N m d = N −m d . The MacMahon factor can also be absorbed into the triple-product formula with n g 0 = −χδ g 0 (and hence N m 0 = −χδ m 0 ).
Free M2 gas It was suggested in [89] that the DT/GV relation (5.2) or (5.7) can be physically justified by a free M2 gas picture, the basic idea of which is the following. Suppose one already knows the full collection C of 5d single-particle M2brane bound states on flat spatial R 4 , where 'single-particle' ('irreducible' in the words of [89]) means a state with a normalizable wave function as opposed to a multi-particle scattering state which is δ-function normalizable. Now, instead of R 4 let us consider spatial Taub-NUT, which is a circle fibration over R 3 such that the circle shrinks at the origin and asymptotes to a fixed radius R T N at infinity. Near the origin, the spatial section of Taub-NUT looks like R 4 , so we can imagine putting inside it particles belonging to C. Since Taub-NUT with large flux acts like a box 29 The second line of (5.2) can be rewritten through combinatoric identities as where preventing the particles from forming scattering states (the importance of large flux was later pointed out in [17,93]), if we put multiple particles inside, we expect them to form normalizable 'multi-particle' bound states which become scattering states in the R T N → ∞ limit as the Taub-NUT circle opens up. While the actual (unsigned) degeneracies have a complicated dependence on the Taub-NUT circle size R T N , supersymmetric indices are protected and have no R T N dependence. In the small R T N limit, we get the index of 4d D6-D2-D0 bound states with unit D6 charge, whereas in the large R T N limit, we get the index of a free gas of M2-branes, counting all subsets of C. Since the indices of 4d BPS states in the chamber of large B field give the DT invariants, and the indices of C give the GV invariants, it is intuitively clear that DT is a plethystic exponential of GV. The precise relation is (5.7), where m + 1 and n − 1 are interpreted as the intrinsic and orbital angular momenta, respectively, of each 5d BPS particle [89].
Quantum GSY lift According to the free gas picture [89], when series expanding the DT/GV relation (5.7), the terms with coefficients linear in N m d give the indices of single-particle states, i.e.
Moreover, the terms with n = 1 count the states with no orbital angular momenta, We will also be interested in the 5d index defined in (3.3), which can be related to the above by 5d blackhole extremality bound A key role in the following subsections is played by the extremality bound of a 5d BMPV blackhole [56]. Let us recall this bound, and translate it to 4d single-particle language. The extremality bound on Ω(j L , d) can be read off from (2.7), giving In terms of the counting of 5d supersymmetry multiplets by N m d , this translates via (5.11) to And in terms of the 4d counting by DT(Q, m), this translates via (5.10) to (5.14)

Blackring microstates with unit M5 charge
Blackrings enter the story as their microscopic index is related to DT via wallcrossing, which can then be lifted to a relation with GV. Of course, underlying this somewhat roundabout argument is the physical expectation that the GV tail counts blackring microstates.
Blackring microstates and the MSW CFT In [57], the authors constructed 5d BPS rotating cosmic closed strings by wrapping M5 branes on four-cycles (ample divisor) of Calabi-Yau threefolds. On the one hand, the macroscopic limit of this cosmic string becomes the EEMR/BW/GG blackring [64][65][66][67]; on the other hand, the low energy limit of the M5 brane worldvolume theory is the Maldacena-Strominger-Witten (MSW) CFT [58]. Hence, the MSW CFT describes the microscopic constituents of a blackring. Originally, the MSW CFT was formulated to describe D4-D2-D0 bound states in Calabi-Yau compactifications of IIA string theory. In particular, the generating function of indices is (encoded in) the modified index of the MSW CFT. The relation between the D4-D2-D0 context and the blackring context was later elucidated by the GSY 4d/5d lift [22,23]: by adding a single D6 brane and considering two-centered bound states, with the D6 at one center and the D4-D2-D0 at the other, the resulting 4d configuration lifts to a 5d blackring.
Wall-crossing and a GV/MSW proposal Suppose there is a wall of marginal stability for a D6-D2-D0 bound state (counted by a rank-one DT invariant) to decay into a single D6 brane and a D4-D2-D0 bound state, where the extra D4 dissolves into background flux that shifts the asymptotic D2 and D0 charges. Let MSW p,Q (q 0 ) denote the MSW index with charge vector γ = (0, p, Q, q 0 ). For unit D4 charge p = 1, the wall-crossing formula [94,95] gives [54] 30 (5.15) 30 Instead of DT, the wall-crossing analysis of [54] considers stable-pair invariants [96], also known as Pandharipande-Thomas (PT) invariants, whose generating function is related to that of DT by stripping of a MacMahon factor M (−q) χ . As explained in [17], the PT invariants count 4d BPS bound states in the chamber of large B + iJ with arctan(J/B) = (2π/3) + , such that the D0-halos become unbound. One could thus regard (5.15) as first crossing the wall at arctan(J/B) = 2π/3 and then proceeding as in [54]. We now make two assumptions, whose regime of validity will be empirically justified: 1. The contributions from multi-particle states, SU(2) L descendants, and 5d supersymmetry descendants are subleading in (5.10) and (5.11).
Then combined with (5.10) and (5.11), we would have an approximate relation in the context of 5d counting 31 According to [54], the above is an exact equality at the maximal value of j L (saturating the Castelnuovo bound) when k | d.
Numerical evidence In figure 6, we compare the 5d index entropy log|Ω(j L , d)|, related to the GV invariants by (3.3), with the Maldacena-Strominger-Witten entropy log|MSW ′ (d, 2j L − 2)| for several one-parameter Calabi-Yau threefolds. We find surprisingly good agreement for large enough angular momentum j L even for small d. 32 Notice that for X 42 , even the kink at the far right matches. 33 31 Here, A ∼ B means A/B − 1 ≪ 1. 32 The 'missing' black dot in each plot is due to the vanishing of χ(Q, m). 33 The comparison of log|N m d | with log|MSW ′ (d, m − 1)| yields a similar degree of agreement, while the comparison of log|PT(Q, m)| with log|MSW ′ (Q, −m)| seems to be even better. It would be interesting to understand how to account for the difference between Ω(j L , d) and MSW ′ (d, 2j L − 2) in the tail region. The 4d interpretation of this difference comes from crossing other walls [54], the analysis of which is notoriously complicated, hence we hope that 5d microstate counting might offer a useful complementary perspective. For instance, there can be contributions from blackrings with non-unit D4 charges, but the MSW index in this case is not known (see [54,97] for the current status).

4d instability of 5d superextremal states
In this subsection, we observe that the GV invariants counting 5d BPS states with angular momenta beyond the 5d blackhole extremality bound give rise to 4d BPS states that are stable in the large B + iJ limit but become unbound in a chamber of the Kähler moduli space for a 'reduced' central charge function corresponding to what is known as a (very) weak stability condition. Although this stability condition is not the physical one away from large J, we argue in the end the conclusion should be valid in the large charge limit.
In the context of the blackhole/blackring transition, we view this observation as a consistency check for the proposal that superextremal GV invariants count 5d blackring microstates. Under the GSY 4d/5d correspondence [22,23], blackrings descend to 4d two-centered bound states that can disassociate in certain chambers, whereas 5d blackholes descend to 4d single-centered objects that should be stable everywhere.
Stability and wall-crossing The index counting of stable BPS states in Calabi-Yau compactifications of IIA string theory is a rich and intricate subject. Inside the Kähler moduli space, the index remains constant in domains (chambers) but jumps across walls of marginal stability [94,95,98,99]. For unit D6 charge p 0 = 1, an important chamber is that of large J and B with B ≫ J, where BPS states are counted by the rank-one Donaldson-Thomas (DT) invariants [84,85]. Given knowledge of the DT invariants, one can apply the celebrated wall-crossing formulae [94,95] to compute the indices in various other chambers. Conversely, starting from an empty chamber, one can cross walls to compute the DT invariants.
The stability of a BPS state is defined by a central charge function that maps charge vectors to complex numbers. The central charge function of IIA on Calabi-Yau is extremely complicated at a generic point on the Kähler moduli space (as it depends on the exact prepotential), albeit simplifying in the large J regime. To circumvent this difficulty, mathematicians study not the physical central charge, but a space of central charge functions whose corresponding stability conditions satisfy a set of axioms formulated by Bridgeland [100]. The physical stability condition, known as Π-stability, is a particular map (sometimes an embedding) from the Kähler moduli space to the space of Bridgeland stability conditions. Here is a key point: given a simpler 'unphysical' map/embedding, and as long as it agrees with the physical one in the large J limit, we can still use wall-crossing to compute the DT invariants starting from an empty chamber, as was done in [54] to formulate new constraints on GV invariants.
In the following, we closely follow the discussions and conventions of [54].
Central charge function For convenience, we first translate the charge (Mukai) vector γ := (p 0 , p 1 , q 1 , q 0 ) into the Chern characters (of line bundles describing Dbrane bound states in the large J limit) as follows: and The Bridgeland stability conditions are defined by a 4-parameter family of central charge functions with the following important special cases: • The large J limit of the physical central charge is given by (with J 0 and e −J corrections) • A weak stability condition (which is located at the boundary of the space Bridgeland stability conditions) is defined with respect to a reduced central charge function independent of the D0 charge q 0 . A crucial point is the following: With the identification a = J/ √ 3 and b = B, the weak stability condition agrees with the physical one in the large J limit. In the following it is convenient to define Bayer-Macrì-Toda conjecture The Bayer-Macrì-Toda (BMT) conjecture [90,91], which has been proven by [101] for the quintic threefold, states that is a necessary condition for there to exist a weakly-semistable object of class γ. In other words, for a fixed charge vector γ, the region on the side of the line L b,w that violates (5.23) is in an empty chamber. It turns out that C 2 1 − 2C 0 C 2 > 0, and hence the BMT inequality (5.23) is always satisfied for large enough Kähler modulus. If L b,w intersects w = 1 2 b 2 at two points, which happens for a positive discriminant then an empty chamber exists. We now investigate the existence of empty chambers for BPS states counted by the DT invariants.

Donaldson-Thomas
The rank-one DT invariants DT(Q, m) are defined with the charge assignments The BMT line (5.23) and discriminant (5.24) read 26) and the BMT conjecture states that an empty chamber exists when The largest value of a = J/ √ 3 in the empty chamber is In the large Q limit with m/Q 3 2 fixed and (5.28) satisfied, the empty chamber contains a subregion with large J ∼ Q 1 2 , such that the weak stability condition becomes a good approximation of the physical stability condition. It is therefore reasonable to expect that at least for large charges, the existence of an empty chamber under weak stability implies the existence of an empty chamber under physical stability.
We now observe that the condition (5.28) for the existence of an empty chamber is, up to a constant shift, the complement of the 5d blackhole extremality bound (5.14). Hence, the BMT conjecture can be regarded as a microscopic version of the 5d blackhole extremality bound, which lends credence to the conjecture's validity. Conversely, the BMT conjecture provides a consistency check of our proposal that superextremal GV invariants count blackring microstates.

On the contributions of degenerate instantons
In this work, we noticed that the one-loop corrected Wald entropy for the blackhole approximated the microscopic entropy coming from the topological strings well for smaller values of the angular momentum, whereas the individual contributions from higher-genus degenerate instantons were all of the same order, demanding a summation to all orders (see section 4.3 for more details). In this section, we look into this puzzle from the viewpoint of a gravitational path integration of supergravity.
For supersymmetric extremal blackholes with a near horizon AdS 2 factor, the entropy can be calculated by performing path integration over the fields localized in the near horizon region using Sen's formalism [36] 34 . In this process, Z local comes from the Wald entropy associated with the classical reduction of the 5d (M theory on CY 3 ) Wilsonian effective action to 4d. The sub-leading contributions come from quantum fluctuations. It is tempting to identify the measure factor µ in the OSV formula as coming from the one-loop fluctuations of the massless fields in 4d [37]. This fact has been established by the explicit one-loop localization of supergravity in [104,105] following the developments of [39]. The origin of Z 0 (g top ) is much more subtle; note that for higher supersymmetric compactifications (N = 4, 8 in the 4d language), these contributions are not present [44,45]. It has been observed in [21] that one-loop contribution from the Kaluza-Klein (KK) modes of a massless charged hypermultiplet in 5d reproduces part of the structure in Z 0 (g top ). It is natural to wonder if in Taub-NUT space with one unit of D6 charge, a one-loop calculation of massless fields around BMPV blackhole in 5d (similar to the ones done for the flat space in [38]) 35 can be recast as a calculation of massless 4d modes along with the contribution of the KK modes on the Taub-NUT circle, effectively capturing the contribution of degenerate instantons. 36 In the context of the 4d/5d lift considered here, the KK modes on the Taub-NUT circle would be the KK modes on the M theory circle (for toroidal compactifications, such one-loop effects in M theory have been analyzed in [106,107]

A Length scales and units
In this Appendix we will explicitly write down all the relevant length scales under compactification (we will not set any length scale to one) without being careful about order one factors. The length of the M theory in asymptotic infinity is Here g s is the 10d IIA string coupling (defined to be constant). 10d dilaton ϕ d is defined with the convention that in asymptotic infinity ϕ d (∞) = 0. The length of the M theory circle at a generic point would be given by where l 11 is the Planck scale in 11d. The M2 brane tension will be given by B GV invariants for quintic up to genus 4 In this appendix, we will present some details (up to genus 4-we necessarily need to rely on Castelnuovo bound to determine the holomorphic ambiguity for g > 3) of the calculation of GV invariants for the quintic. For higher genera, the details are extremely complicated and we won't mention that here. The Mathematica file used for calculation up to genus 49 can be found at [53]. We will present results only in holomorphic limit at respective points on the moduli space.

625z
The mirror map q = e 2πit(z) near the large volume limit is given by (up to higher order in q corrections) (B.14) z = 3211453397717989716q 6  The GV invariants n g d are given by