An approach to BPS black hole microstate counting in an N=2 STU model

We consider four-dimensional dyonic single-center BPS black holes in the $N=2$ STU model of Sen and Vafa. By working in a region of moduli space where the real part of two of the three complex scalars $S, T, U$ are taken to be large, we evaluate the quantum entropy function for these BPS black holes. In this regime, the subleading corrections point to a microstate counting formula partly based on a Siegel modular form of weight two. This is supplemented by another modular object that takes into account the dependence on $Y^0$, a complex scalar field belonging to one of the four off-shell vector multiplets of the underlying supergravity theory. We also observe interesting connections to the rational Calogero model and to formal deformation of a Poisson algebra, and suggest a string web picture of our counting proposal.


Introduction
A highly active area of focused research in string theory aims at a formulation of the microscopic degeneracy of states of black holes to arrive at a statistical description of black hole thermodynamics. This is a critical first step in gleaning insights into the organization of the degrees of freedom in any purported theory of quantum gravity. A series of steady definitive advancements has been accomplished in the last decades in this field, specifically w.r.t the exact counting of microstates of 1 4 BPS and 1 8 BPS black holes in four-dimensional N = 4 and N = 8 string theories, respectively [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. In this paper, we address the problem of black hole microstate counting for 1 2 BPS asymptotically flat black hole backgrounds. We do this in a specific four-dimensional N = 2 string theory model, namely the STU model of Sen and Vafa (example D in [20]), for which the holomorphic function F , which encodes (part of) the Wilsonian effective action, has been determined recently [21]. The function F encodes an infinite set of gravitational coupling functions ω (n) (n ≥ 1). A proposal for the counting of microstates in this model has been made in [22], by only taking into account the gravitational coupling function ω (1) . In this work, by using the detailed knowledge about F given in [21], we arrive at a different proposal. While doing so, we will work in a certain region of moduli space, as we will explain below.
In four-dimensional N = 2 string models, extremal black hole backgrounds are characterized by electric and magnetic charges (q I , p I ) w.r.t the various U (1) gauge fields in these models. These backgrounds, which at weak string coupling correspond to BPS states, possess a near-horizon AdS 2 geometry [23]. This geometry is an exact solution of the equations of motions that is decoupled from the asymptotics of the background. The decoupling ensures that the values of scalar fields {φ} in the near-horizon black hole background as well as all relevant length scales, such as the AdS 2 radius, are solely fixed in terms of the charges and are independent of the asymptotic data. The near-horizon geometry, in effect, acts as an attractor [24][25][26] for the scalar fields {φ} in the black hole background: the attractor values of these scalars determine the near-horizon length scale, and hence the black hole entropy S BH . This, in turn, implies that the quantum gravity partition function Z AdS 2 (φ, p) in this attractor background, evaluated in a canonical ensemble defined at fixed magnetic charges, is the generator of the dimension of the black hole microstate Hilbert space, i.e. of the black hole microstate degeneracy d BH (q, p), graded by electric and magnetic charges. Hence, formally, one has Z AdS 2 (φ, p) = q d BH (q, p)e πq·φ . (1.1) The SL(2, R)-isometries of AdS 2 indicate that the generator of degeneracies has modular properties, an expectation that has been extensively borne out in all cases where the exact microscopic degeneracy is known, such as in four-dimensional N = 4 and N = 8 string theories. Here the microstate degeneracy d BH (q, p) is related to the macroscopic entropy of the black hole by the Boltzmann relation S BH = ln d BH (q, p) . (1. 2) The macroscopic entropy S BH has an expansion in terms of the dimensionless black hole area A BH as 1 For extremal black holes with a near-horizon AdS 2 geometry, a proposal for their exact macroscopic entropy was put forward in [28,29]. This proposal is called the quantum entropy function, and is defined in terms of a regulated path integral over AdS 2 spacetimes with an insertion of a Wilson line corresponding to electric charges. This in turn provides an operational definition for (1.1). For BPS black holes in four-dimensional N = 2 string theories, the quantum entropy function W (q, p) can be expressed as [30] (when strictly restricted to smooth configurations) (1. 4) The various ingredients that go into this formula will be reviewed in section 2. Here we note that the holomorphic function F appearing in the exponent is the holomorphic function mentioned above that encodes (part of) the Wilsonian effective action. The function F can be decomposed as F = F (0) + 2iΩ, where F (0) denotes the prepotential of the model, whereas Ω encodes all the gravitational coupling functions ω (n) , as a series expansion in powers of Υ/(Y 0 ) 2 . Here, (Υ, Y 0 ) denote rescaled supergravity fields (A, X 0 ), where A denotes the lowest component of the square of a chiral superfield that describes the Weyl multiplet, and where X 0 denotes a complex scalar field belonging to one of the off-shell vector multiplets in the associated supergravity theory. The STU model of Sen and Vafa (example D in [20]) is based on four off-shell vector multiplets, and Ω will therefore depend on four complex scalar fields Y I that reside in these multiplets. Introducing projective coordinates (1. 5) This STU model possesses duality symmetries, namely [Γ 0 (2)] 3 symmetry and triality symmetry. The former refers to the duality symmetry Γ 0 (2) S × Γ 0 (2) T × Γ 0 (2) U , associated with each of the moduli S, T, U . Here, Γ 0 (2) denotes a certain congruence subgroup of SL(2, Z). In addition, the model possesses triality symmetry, i.e. invariance under the exchange of S, T and U . Using all these symmetries, it was demonstrated in [21] that the gravitational coupling functions ω (n+1) (S, T, U ) are expressed in terms of one single function ω, and derivatives thereof with respect to S, T, U . The duality symmetries were implemented by adding to Ω in (1.5) a term proportional to Υ ln Y 0 . However, such a term is, strictly speaking, not part of the Wilsonian effective action. Since we will take Ω as an input in (1.4), we will refrain from adding this term to Ω. This term will be effectively generated from Z 1−loop in (1.4), as already noted in [31]. The explicit expressions for the ω (n+1) (S, T, U ) in the STU model are complicated. However, in [21] it was observed that they dramatically simplify in the limit where the real part of two of the three moduli S, T, U are taken to be large. This is the limit in which we will work in this paper. We will thus fix two of the moduli, say T and U , to large values, which we will denote by T 0 , U 0 . In this limit, Ω becomes effectively replaced by Ω(Y 0 , S, Υ) = Υ ω(S) + Υ (Y 0 ) 2 α γ ∂ω(S) ∂S + Ξ(Y 0 , S, Υ) , (1. 6) where α, γ denotes constants, and where Ξ encodes the higher gravitational couplings ω (n+1) with n ≥ 2. The latter are expressed in terms of modular forms I n (S) and products thereof. The I n are themselves products of Eisenstein series for Γ 0 (2), see (B.38). We are thus led to an expansion of Ξ in terms of I n as Ξ(Y 0 , S, Υ) = Here, the right hand side is organised in powers of I n . We call the first sector (which is linear in I n ) the monomial sector, the second sector (which is quadratic in I n ) the binomial sector, and so on. In [21], explicit expressions were only given for the coefficients of the first two sectors, i.e. for α n and α m,n . Notwithstanding this limited knowledge about the expansion coefficients, there are several lessons that one can draw from the power series expansion (1.6), (1.7). Firstly, under Γ 0 (2)-transformations, Ω transforms in the same manner as the logarithm of ϑ 8 2 (τ, z), where ϑ 8 2 (τ, z) denotes one of the Jacobi theta functions, and where τ = iS, z = 1/Y 0 . Moreover, the series expansion (1.6), (1.7) is reminiscent of the Taylor series expansion around z = 0 of ln ϑ 8 2 (τ, z), see (D. 21). In this expansion, the coefficients α n in the monomial sector behave as 1/(2n)!, whereas in (1.7) they behave as 1/n!. Secondly, each of the I n -sectors in the expansion of ln ϑ 8 2 (τ, z) defines a function that is given as a series expansion in z, and the sum over all these functions gives rise to ln ϑ 8 2 (τ, z), which is naturally expressed in terms of variables q = exp(2πiτ ) and y = exp(2πiz). It is tempting to conclude that a similar story may apply to Ω(Y 0 , S, Υ). Thirdly, ln ϑ 8 2 (τ, z) is a solution to a non-linear PDE derived from the heat equation. So we may similarly ask whether Ξ given in (1.7) arises as solution to a non-linear PDE. The answer is affirmative, and we give a PDE whose solution yields the monomial and binomial sectors displayed in (1.7). The higher sectors in (1.7), whose expansion coefficients have not been determined in [21], will lead to modifications of this PDE, and we give a candidate for the modified PDE. Note that Ξ has weight 0 under Γ 0 (2)-transformations. Finally, we note that the monomial sector exhibits a connection with the two-particle rational Calogero model [32][33][34], as well as a relation with Serre-Rankin-Cohen brackets which is suggestive of formal deformation [35].
In this paper we will compute the quantum entropy function (1.4) for large singlecenter BPS black holes in the STU model. We follow [36,37] and add a boundary term to the quantum entropy function, so as to bring it into a manifestly duality invariant form. This modified form is what we thenceforth call the quantum entropy function. It requires specifying an integration contour C, which we take to be the one constructed in [38] in the context of microstate counting for N = 4 BPS black holes. The quantum entropy function will depend on the three ingredients displayed in (1.6), namely ω(S), I 1 (S) ≡ ∂ω/∂S and Ξ(Y 0 , S). Since Ω(Y 0 , S, Υ) depends on Y 0 , so will the quantum entropy function. The dependence on Y 0 brings in an explicit dependence on the magnetic charges (p 0 , p 1 ), as we will see. A microstate counting formula that reproduces the quantum entropy function will have to take this dependence on Y 0 into account. Microstate counting formulae for BPS black holes in four-dimensional N = 4, 8 string theories [1, 3,5,9] are based on modular objects. In the N = 4 context, these are Siegel modular forms, and [22,39] indicates that they should also play a role in N = 2 theories. However, in the latter case, microstate counting formulae cannot be solely based on Siegel modular forms due to the dependence of the quantum entropy function on the additional modulus Y 0 .
Let us briefly describe our microstate proposal. We find that ω(S), which is proportional to ln ϑ 8 2 (S), together with subleading corrections in the quantum entropy function, point to a dependence of the microstate counting formula on a Siegel modular form Φ 2 of weight 2. This Siegel modular form can be constructed by applying a Hecke lift [5] to a specific Jacobi form constructed from the seed ϑ 8 2 , and it differs from the Siegel modular form proposed in [22], which did not take into account the subleading corrections just mentioned. Due to the dependence on Y 0 , this needs to be supplemented by a modular object that depends on Y 0 . By focusing attention onto the leading divisor of the Siegel modular form Φ 2 , we give an expression for this modular object: one of the ingredients that goes into it is Ξ, which we view as a solution to the aforementioned non-linear PDE. We then verify that on the leading divisor, the proposed microstate counting formula is invariant under the subgroup H ⊂ Sp(4, Z) that acts on the Siegel upper half plane and implements the Γ 0 (2) S -symmetry of the STU model. We stress that the various approximations that we have implemented (such as working in a certain region of moduli space, or focussing attention on the leading divisor of Φ 2 ), render the microstate proposal to be only an approximate one.
The paper is structured as follows. In section 2, we give a brief review of the quantum entropy function for BPS black holes in N = 2 supergravity theories in four dimensions. In section 3 we evaluate the quantum entropy function for large single-center BPS black holes in the STU model. While doing so, we work in a certain region of moduli space in which the function F = F (0) + 2iΩ that encodes the Wilsonian effective action simplifies: Ω becomes effectively replaced by (1.6). The modular invariant function Ξ in (1.6) arises as a solution to a non-linear PDE, and exhibits a relation with the two-particle rational Calogero model, as well as with Serre-Rankin-Cohen brackets and formal deformation. We discuss our integration contour C, which we take to be the one constructed in [38]. The result for the quantum entropy function is based on several assumptions and approximations which we explain. In section 4, we propose a microstate counting formula that reproduces this result for the quantum entropy function. It is based on the Siegel modular form Φ 2 as well as on another modular object that captures the dependence on the complex scalar Y 0 . Given the various approximations that went into computing the quantum entropy function, the proposed counting function will be an approximation to the (as yet unkown) exact counting function. In section 5, we conclude with a brief summary and a few observations, and we suggest a string web picture of our counting proposal. In appendices A -G, we collect results about modular forms for SL(2, Z) and Γ 0 (2), Jacobi forms, Siegel modular forms, Rankin-Cohen brackets and Hecke lifts.
2 Quantum entropy function for BPS black holes in N = 2 supergravity theories

Generic structure
The equations of motion of N = 2 supergravity coupled to n V Abelian vector multiplets in four dimensions admit dyonic single-center BPS black hole solutions [24]. These are static, extremal black hole solutions that carry electric/magnetic charges (q I , p I ), where the index I = 0, . . . , n V labels the Maxwell fields in the theory. The near-horizon geometry of these solutions contains an AdS 2 factor. These BPS black holes are supported by complex scalar fields Y I that reside in the vector multiplets. At the horizon, these scalar fields are fixed in terms of specific values that are entirely specified by the charges carried by the black hole.
In the presence of higher-derivative terms proportional to the square of the Weyl tensor, the associated N = 2 Wilsonian effective action is encoded in a holomorphic function F (Y, Υ) [40], where the complex field Υ is related to the lowest component of the square of the Weyl superfield. At the horizon of a BPS black hole, the field Υ takes the real value Υ = −64, and Wald's entropy of the BPS black hole [41] can be written as [42] S BH (q, p) = π 4 ImF (Y, Υ) − q I (Y I +Ȳ I ) , are evaluated at the horizon. The horizon value of φ I is determined by extremizing the right hand side of (2.1) with respect to φ I , Wald's entropy (2.1) constitutes the semi-classical approximation to the exact macroscopic entropy of a BPS black hole. For extremal black holes whose near-horizon geometry contains an AdS 2 factor, a proposal for the exact macroscopic entropy of extremal black holes has been put forward in [28,29]. This proposal is called the quantum entropy function, and it is a regulated partition function for quantum gravity in a near-horizon AdS 2 space-time.
The quantum entropy function is a functional integral, which for the subclass of BPS black holes can be defined by means of equivariant localization techniques [30,36,37,[43][44][45][46] that reduce the infinite dimensional functional integral to a finite dimensional integral over a bosonic localization manifold. When restricting to smooth field configurations in near-horizon AdS 2 backgrounds, the resulting expression for the quantum entropy function W (q, p) for BPS black holes takes the form where q · φ = q I φ I , with I = 0, . . . , n V . The localization manifold is labelled by n V + 1 parameters {φ I }. Note that the integral in (2.4) requires specifying a contour C. The measure µ(φ) arises as a result of the localization procedure implemented on the field configuration space. The term Z 1−loop (φ) describes the semi-classical correction, giving rise to a super determinant, that arises when performing the Gaussian integration over terms quadratic in quantum fluctuations around the localization manifold. The function F entering in (2.4) is the holomorphic function that defines the N = 2 Wilsonian effective action (with Υ = −64). Let us describe the quantities that enter in (2.4) in more detail. The Wilsonian function F can be decomposed into where the dependence on the field Υ is solely contained in Ω. The first term F (0) (Y ) is the so-called prepotential of the N = 2 model. Local supersymmetry requires both F (0) and Ω to be homogeneous of degree 2 under complex rescalings [47], , and hence they satisfy the homogeneity relations Here, we have introduced the notation F I = ∂F/∂Y I , Ω Υ = ∂Ω/∂Υ. The Wilsonian effective action is based on a function F where Ω(Y, Υ) is given in terms of a power series expansion in Υ. Choosing λ = 1/Y 0 in the homogeneity relation (2.7) implies that Ω(Y, Υ) takes the form (2.8) Here, the gravitational coupling functions ω (n+1) only depend on the projective coordinates z A = Y A /Y 0 (A = 1, . . . , n V ). In a generic N = 2 model, this perturbative series in Υ/(Y 0 ) 2 may not be convergent. Next, let us discuss the measure factor µ. While this measure factor has not yet been worked out from first principles 2 , an approximate expression for it can be obtained by demanding consistency under electric-magnetic duality transformations as well as consistency with semi-classical results for BPS entropy, as follows.
Electric-magnetic duality is a characteristic feature of systems with N = 2 Abelian vector multiplets in four dimensions, also in the presence of a chiral background field [40]. Duality transformations act as symplectic transformations on the Abelian field strengths, and hence, on the associated charge vector (p I , q I ), as well as on the vector (Y I , F I ) [47]. 2 A formalism for carrying out localization calculations in the presence of a supersymmetric background has recently been given in [45,46].
The semi-classical entropy (2.1) of a BPS black hole transforms as a function under symplectic transformations, that is, under changes of the duality frame, and this should also be the case for the exact macroscopic entropy W (q, p) of the BPS black hole. To ensure that (2.4) transforms as a function under symplectic transformations, the measure factor has to be proportional to [43,48] det ImF KL , (2.9) as we will review in the next subsection. Here, F KL denotes the second derivative of the Wilsonian function F (2.5) with respect to the Y I . In addition, the measure factor µ should also include non-holomorphic terms, which we will denote by Σ, that have their origin in the holomorphic anomaly equations for the free energies of perturbative topological string theory, and are needed for consistency with semi-classical results [49]. Keeping only non-holomorphic terms associated with the topological free energy F (1) , Σ takes the form where χ = 2(n V − n H + 1), which is determined in terms of the number of vector and hyper multiplets of the N = 2 model (n V and n H , respectively), denotes the Euler number of the Calabi-Yau threefold underlying the model, and We are thus led to consider a measure factor of the form which, when approximating F by F (0) takes the form (c.f. eq. (4.21) in [49]) The expressions (2.12) and (2.13) are evaluated at (2.2). We stress that the measure factor given in (2.13) is an approximate measure factor that will receive further corrections stemming from Ω in (2.5). We will, in due course, make use of this observation. Note that in this approximation, µ only depends on the projective coordinates z A ,z A . Now let us turn to the 1-loop determinant Z 1−loop , which reads [43,44,46] Z 1−loop = e −(2−χ/24) K . (2.14) Here, the quantity is computed from the Wilsonian function F (Y, Υ), and not just from the prepotential F (0) (Y ). The factor 2 in the exponent of (2.14) denotes the contribution from fluctuations of the Weyl multiplet. Note that the expression for Z 1−loop depends on Y 0 and z A (and their complex conjugates), and that it is a symplectic function. Z 1−loop is again evaluated at (2.2). The quantum entropy function (2.4) requires a choice of integration contour C, which we will specify in subsection 3.5 for the specific N = 2 model under consideration.
Let us close this subsection by mentioning three checks that one can perform on the proposed approximate measure factor µ and on Z 1−loop . Firstly, when replacing K by K (0) in Z 1−loop , one infers [43,44] that the macroscopic entropy S BH (q, p) = ln W (q, p) receives a logarithmic correction given by (2.16) Here, * indicates that the expression is evaluated at the attractor values (2.3). For a large supersymmetric black hole, π e −K (0) | * equals the area A BH of the event horizon at the two-derivative level [50], and hence which reproduces the logarithmic area correction to the BPS entropy of a large black hole computed in [31]. Secondly, approximating K by K (0) in (2.14) results in [31] µ which is the combination that plays the role of a measure factor in the study [51] of the OSV conjecture [42]. Thirdly, as mentioned above, the measure factor µ is required to ensure that the quantum entropy function is a symplectic function. To verify this, we will now compute W (q, p) in saddle-point approximation.

Saddle-point approximation
Let us denote the exponent in (2.4) by  and by integrating over the fluctuations δφ I ∈ C in Gaussian approximation, We note that the factor √ det ImF KL has cancelled out against the corresponding factor in (2.12).
In the absence of non-holomorphic corrections (Σ = 0), H is a symplectic function [48], and hence, the result (2.24) is a symplectic function. This justifies the presence of the factor √ det ImF KL in (2.12). In the presence of non-holomorphic corrections (Σ = 0), the discussion of symplectic covariance is more involved [48,49,52], but it can be shown that the combination Σ + πH in (2.24) is a symplectic function.
In this paper, we will focus on a specific N = 2 model, namely the STU model of Sen and Vafa (example D in [20]). This is a model for which the duality symmetries are known exactly. We will analyze the expression (2.24) for this model in subsection 3.4.1, and verify that it is consistent with the duality invariance of the model. with Y I = 1 2 (φ I + ip I ). The attractor value H(φ * , p, q) can be expressed as [48] H(φ * , p, q) by means of homogeneity relations (2.7).
3 Quantum entropy function for the N = 2 STU model of Sen and Vafa Next, we specialize to the N = 2 STU model of Sen and Vafa (example D in [20]). This is a model with duality symmetries which are so restrictive that they have recently led to the determination [21] of Ω in (2.5). We will consider large BPS black holes in this model, and we will evaluate the quantum entropy function (2.4) for these black holes. In doing so, we will work in a certain region of moduli space of the STU model in which the function F simplifies. We begin by reviewing the form of F for this model. The N = 2 STU model of [20] (example D), obtained by a Z 2 ×Z 2 orbifold compactification of type II string theory, is a model with n V = 3 vector multiplets and n H = 4 hyper multiplets, and hence vanishing Euler number χ = 2(n V − n H + 1). The complex scalar fields residing in the three vector multiplets are denoted by S, T and U , The prepotential F (0) (Y ) is exact and given by The model possesses symmetries, in particular a duality symmetry Γ 0 (2) S ×Γ 0 (2) T ×Γ 0 (2) U , where Γ 0 (2) denotes a congruence subgroup of the group SL(2, Z), where * can take any value in Z. Duality transformations act as symplectic transformations on the vector (Y I , F I ). Under Γ 0 (2) S transformations, the complex scalars S and Y 0 transform as while the scalars T, U transform as [49] T → T + 2ic and hence are not inert in the presence of Ω. Similar transformation rules apply under Γ 0 (2) T,U transformations.
The symmetries of the model, namely Γ 0 (2) S × Γ 0 (2) T × Γ 0 (2) U and triality symmetry under exchange of S, T and U , are very restrictive, and have recently been used [21] to determine the coupling functions ω (n+1) (z A ) in Ω, c.f. (2.8). This was achieved by adding to Ω the term 2Υ γ ln Y 0 , with the constant γ given by (3.7). This term, which is not of Wilsonian type, since it is not of the power law type, was crucial to implement the duality symmetries of the STU model while maintaining holomorphy. It was found that the higher gravitational coupling functions ω (n+1) (z A ) (with n ≥ 1) are determined in terms of the first gravitational coupling function ω (1) (z A ). The latter takes the form [53] where with ϑ 2 (S) = 2η 2 (2S)/η(S), and likewise for ω(T ) and ω(U ). Here, ϑ 8 2 (S) denotes a modular form of weight 4 under Γ 0 (2) S (with trivial multiplier system [54]), We refer to appendices A and B for a brief review of modular forms. We pick a single-valued analytic branch of ln ϑ 8 2 (S) such that under Γ 0 (2) S , ω(S) transforms into Now let us describe the region in moduli space in which we will work. It was observed in [21] that the coupling functions ω (n+1) (z A ) in Ω simplify dramatically when working in a regime where two of the three moduli S, T, U (say T and U ) are taken to be large, i.e. Re T, Re U 1. In this limit, We introduce the combination which is constant in this limit. In this region of moduli space, the coupling functions ω (n+1) with n ≥ 1 in (2.8) are functions of S only, and are moreover expressed in terms of derivatives of ω(S), as we now describe. In this limit, the coupling function ω (2) is given by (3.14) and the Wilsonian function Ω(Y, Υ) in (2.8) takes the form 3 , where we used (3.6). Here, Ξ contains the coupling functions ω (n+1) (S) with n ≥ 2, and it denotes a function that is invariant under Γ 0 (2) S transformations. This can be established as follows.
Instead of working with variables S, Y 0 , we find it convenient to work with variables S, z, Under Γ 0 (2) S transformations, these variables transform according to (3.4), and hence, z → z/∆. The derivatives of Ω(Y 0 , S, Υ) are required [21] to transform according to where denotes the transformed quantity. Inserting the expression for Ω given in (3.15) into (3.17) results in where we used (3.10). Note that (3.18) is linear in Ξ. Expanding Ξ in powers of Υ z 2 [21], (3.20) I 2 (S) transforms as a modular form under Γ 0 (2) S transformations, whereas I 1 (S) does not in view of (3.10). Therefore, and for latter use, we introduce the combination which transforms covariantly (with weight 2) under Γ 0 (2) S transformations by virtue of The variable z ≡ 1/Y 0 should not be confused with the projective coordinates z A defined in (3.1). 5 Here we assume that we may differentiate (3.19) term by term, with respect to both z and S.
Next, define higher I n (S) by Here, D S denotes a holomorphic covariant derivative [55], an analogue of Serre derivative, which acts as follows on Γ 0 (2) S modular forms f n (S) of weight 2n, Under Γ 0 (2) S transformations, the I n (with n ≥ 2) transform as modular forms of weight 2n, i.e. I n (S) → ∆ 2n I n (S). We refer to appendix B for further properties of the I n , in particular their relation with Eisenstein series of Γ 0 (2) [55]. The explicit form of Ξ(z, S, Υ), which was determined in [21], is in terms of a power series in I n (S) with n ≥ 2, In this expansion, each summand is invariant under Γ 0 (2) S transformations. This expansion contains an infinite number of different sectors, characterized by different powers of I n . The expansion coefficients in (3.25) can, in principle, be determined by following the rather laborious procedure described in [21], which consists in working at a generic point in moduli space, solving the associated conditions on Ω imposed by duality, and only then taking T and U to be large. In [21], only the coefficients α n and α m,n of the two first sectors were explicitly determined. The expressions for these coefficients are (with m, n ≥ 2) The coefficients β m,n satisfy the following recursion relation (with m, n ≥ 2), with β 1,n = β m,1 ≡ 0. Note that the terms in the bracket of β m,n are related to triangular numbers. In this paper we will focus on the two first sectors displayed in (3.25), since [21] only computed explicit expressions for the coefficients in these two sectors. Both series are convergent, as we will show in appendix D. The quantity Ξ(z, S, Υ) will play a role in the microstate counting proposal for large BPS black holes in section 4.
The expansion (3.25) in powers of I n may look unfamiliar. In appendix D, we discuss another example of such an expansion, namely the expansion of ln ϑ 8 2 (S, z).

Properties of Ξ(z, S, Υ)
The expansion (3.25) with coefficients (3.26) satisfies various interesting relations, which we now describe. First, we present a non-linear PDE that is satisfied by Ξ in (3.25). We also display a candidate for a non-linear PDE governing the all-order completion of (3.25). Next, we relate the monomial sector in (3.25) to (Serre-) Rankin-Cohen brackets at level 1 and comment on its relation with formal deformation. And finally, we deduce a Hamilton-Jacobi equation for Ω in (3.15), whose Hamiltonian describes a time-dependent deformation of a rational Calogero model.

Non-linear PDE
Let S = −iτ , with τ taking values in the complex upper half plane H, and z ∈ C. Consider the following non-linear PDE for a complex function Ξ depending on S and z 2 , where we set u ≡ z 2 for ease of notation in this subsection, and where ν = αΥ/γ. Here we defined I 1 and I 2 denote the combinations given in (3.20).
Proposition: The non-linear PDE (3.28), subject to Ξ| u=0 = 0 and (∂ u Ξ)| u=0 = 0, admits a Γ 0 (2) S invariant solution that is analytic in u in an open neighbourhood of u = 0. This solution is given by the series (3.25) with coefficients (3.26), up to terms that involve products of three I n or higher.

Proof.
We begin by showing that if a solution Ξ to (3.28) exists that satisfies Ξ| u=0 = 0 and (∂ u Ξ)| u=0 = 0, and that is analytic in u in an open neighbourhood of u = 0, then it is unique. Such a solution will be given by the series Ξ(u, S) = ∞ n=2 u n h n (S), with h 2 (S) = ν 2 2 I 2 (S). The latter follows immediately by inserting this series into (3.28) and assuming that one can differentiate it on a term by term basis. Now suppose that there are two such solutions, namely Ξ 1 (u, S) = ∞ n=2 u n h n (S) and Ξ 2 (u, S) = ∞ n=2 u n g n (S), such that h 2 (S) = g 2 (S). Then, if the n-th coefficient functions are the same, i.e. h n (S) = g n (S), also the (n + 1)-st coefficient functions will agree, i.e. h n+1 (S) = g n+1 (S). This follows by direct inspection of the differential equation (3.28), which shows that the (n + 1)-st coefficient function is determined in terms of the lower coefficient functions. This also shows that the coefficient functions h n (S) are modular forms of weight 2n under Γ 0 (2) S transformations, and hence Ξ(u, S) is a Γ 0 (2) S invariant solution.
Next, let us construct this Γ 0 (2) S invariant solution Ξ(u, S) = ∞ n=2 u n h n (S). This solution will be given in terms of an expansion in powers of I n (S) (with n ≥ 2). To distinguish between these various powers, we rescale Ξ and I 2 in (3.28) by a real parameter λ ∈ R\{0}, to obtain We then construct a solution to (3.30) order by order in λ. Since λ is a dummy variable, we set λ = 1 at the end, thereby arriving at a Γ 0 (2) S invariant solution Ξ(u, S) that is organized in powers of I n (S), as in (3.25).
To lowest order in λ, (3.30) reduces to We seek a solution to this differential equation of the form Inserting (3.32) into (3.31), and assuming that we can differentiate (3.32) term by term 6 , one immediately infers that f n (S) = ν n n! I n (S) for n ≥ 2 by virtue of (3.24). This reproduces the first (monomial) sector in the expansion (3.25).
At the next order in λ, we return to (3.30) and substitute (3.32) into the term (∂Ξ/∂u) 2 . At this order, (3.30) is solved by the second (binomial) sector in the series (3.25), with the coefficients satisfying the recursion relation (3.27), thus reproducing the result (3.26).
Setting λ = 1 we conclude that (3.25) solves the non-linear PDE (3.28), up to terms that involve products of three I n or higher.
The expansion Ξ in (3.25) receives corrections that are of higher order (higher than two) in the I n . These will in turn lead to a modification of the PDE (3.28). The coefficients of these higher order terms were not determined in [21]. Inspection of (3.28) suggests the following all-order completion of (3.25).

Conjecture:
The non-linear PDE governing the all-order completion of (3.25) is given by However, at this stage, we cannot verify this, since we do not have at our disposal the coefficients of the higher-order terms.

I n and Rankin-Cohen brackets
Second, let us focus on the terms linear in I n in the expansion (3.25), Note that the coefficients have a 1/n!-suppression. This series is convergent, at least so long as Re S and |Y 0 | are taken to be large, as we show in appendix D.
Using the property I n+1 (S) = D S I n (S) for n ≥ 2, together with I 2 (S) = 1 2 πγ D SẼ2 (S), whereẼ 2 (S) denotes the basis vector of the vector space M 2 (Γ 0 (2)), see (B.37), we obtain (3.37) where the operator on the right hand side is defined by the power series.
Next, let us add to Ξ 1 (z, S, Υ) the term proportional to I 1 (S) = ∂ω(S)/∂S that appears in Ω given in (3.15), so that now we consider As we review in appendix C, the I n , with n ≥ 3, can be expressed in terms of 1st Rankin-Cohen brackets for modular forms, while I 2 can be expressed in terms of the quasi-modular form I 1 by making use of 1st Rankin-Cohen brackets for quasi-modular forms of depth 1 [56], where g(S) = ϑ 8 2 (S) has weight 4. We thus have the following proposition.
Proposition: Let g(S) = ϑ 8 2 (S). Then, Ξ 1 (z, S, Υ) can be expressed as with the understanding that this equals I 1 when n = 0. In the last step, the bracket [·, ·] 1 denotes the 1st Rankin-Cohen bracket for quasi-modular forms of depth 1.
Proof. This follows immediately, using the results of appendix C.
The expression (3.40) exhibits a formal similarity with the differential (C.14) of the exponential map exp : g → G from the Lie algebra g into the linear group G. Now consider the nth Serre-Rankin-Cohen bracket (C.16) for Γ 0 (2), where we used (B.34). Thus, we can write Ξ 1 (z, S, Υ) as We relate this to the formal deformation Eholzer product (C.17) by means of the generalized hypergeometric function where (a) n = a(a + 1) . . . (a + n − 1). Using we obtain the formal expression where I 2 #g is given as in (C.17), with ∈ C a deformation parameter, and the integration contour encloses the origin. It would be interesting to further study the relation of Ξ 1 (z, S, Υ) with formal deformation.

Deformed Calogero model
We return to (3.17) and deduce a Hamilton-Jacobi equation from it, as follows.
Taking the square of the second equation in (3.17), and suitably combining it with the first equation in (3.17), we obtain This implies that the combination 8 Υ 2 α ∂ Ω/∂S + (∂ Ω/∂z − 2Υ γ/z) 2 transforms as a modular form of weight 2 (with a trivial multiplier system) under Γ 0 (2) S transformations. We denote this combination by V (S, z)/z 2 , where V (S, z) denotes a modular invariant function, Then, inserting the expression (3.15) for Ω into (3.48) gives where ν = αΥ/γ, as before. By comparing this equation with (3.28), we infer Proposition: The partial differential equation (3.48) is a Hamilton-Jacobi equation with Hamilton's principal function S(t, z) given by Using (3.51), we obtain Next, using (3.10) and (3.15), we infer the transformation behaviour (3.52). We can compensate for the term proportional to ∂∆/∂t on the right hand side of (3.52) by considering the Γ 0 (2) invariant combination S − S 1 , where (3.57) When t and z are real, S 1 describes Hamilton's principal function for a free particle.
Using (3.15), we note that the combination S − S 1 , when expressed in terms of S and z, takes the form withÎ 1 given in (3.21).
On the other hand, the full V (S, z) in (3.50) results in a time-dependent Hamiltonian (3.55), which can be viewed as a time-dependent deformation of the rational Calogero Hamiltonian by an infinite set of terms involving powers of I n (S), starting with terms quadratic in I n (S).

Large single-center BPS black holes
Now we turn to the computation of the quantum entropy function (2.4) for large singlecenter BPS black holes in the STU model.
Definition: A large single-center BPS black hole is a dyonic spherically symmetric BPS black hole carrying electric/magnetic charges (q I , p I ) such that the charge bilinears m, n and the charge combination 4nm − l 2 are positive, which ensures that the black hole has a non-vanishing horizon area, proportional to √ 4nm − l 2 , at the two-derivative level.
At the two-derivative level, the horizon area A BH equals A BH = 4π(S +S) m [50], where S denotes the value at the horizon. Hence S +S > 0 for a large BPS black hole, which implies that p 0 and p 1 cannot be simultaneously zero. 7 The semi-classical macroscopic entropy S BH of a BPS black hole equals S BH = πH(φ * , p, q), with H(φ * , p, q) given by (2.26). The quantum entropy function computes corrections to the semi-classical entropy.
To compute the quantum entropy function, we will work in a regime where T and U are large, so as to be able to use (3.15). We will expand H(φ, p, q) given in (2.19) around large values Re T 0 , Re U 0 defined below in (3.67). These values, which depend on Y 0 and on S, are invariant under Γ 0 (2) S transformations. When evaluated at the horizon of the BPS black hole, T 0 and U 0 become entirely expressed in terms of the charges carried by the black hole, and this implies that we will have to choose the charges of the black hole in such a way as to ensure that the horizon values of Re T 0 and Re U 0 are large.
We will now evaluate H(φ, p, q) on the attractor values φ 2 * , φ 3 * that satisfy We will denote the resulting expression by H(τ 1 , τ 2 , p, q), which we subsequently expand around large values Re T 0 , Re U 0 . In doing so, we will keep all the charges, including p 0 .
We set Y 2 = 1 2 (φ 2 + ip 2 ) and Y 3 = 1 2 (φ 3 + ip 3 ), and we solve the attractor equations (3.62). We obtain where and we note that ∆ T and ∆ U depend on Y 0 , S (and on their complex conjugates) as well as on φ 2 * , φ 3 * . This yields We infer Note that T 0 and U 0 are inert under Γ 0 (2) S transformations, as can be checked by using the transformation rules (3.59) and (3.4); t and u, on the other hand, are not inert under Γ 0 (2) S transformations. Next, we evaluate H(φ, p, q) given in (2.19) at the values T, U in (3.65). To this end, we use the parametrization [48] We will also set in the following. Then, H(φ, p, q) becomes where we made use of the charge bilinears (3.60). This yields The terms on the right hand side are evaluated at φ 2 * , φ 3 * . The first term in this expression is invariant under Γ 0 (2) S . When extremized with respect to τ 1 and τ 2 (c.f. (3.92)), it yields the entropy of a large BPS black hole at the two-derivative level. The terms proportional to 1/|Y 0 | 2 are also invariant under Γ 0 (2) S , since both ∂Ω/∂T and ∂Ω/∂U are invariant [49].
and drop terms that involve higher order derivatives of ω with respect to T and U in view of (3.11). Inserting this in the combination that appears in (3.71), we obtain, using (3.15), Inserting the above into (3.71) gives with Y 0 expressed in terms of τ = iS and charges (p 0 , p 1 ) as in (3.68), and where in the last line we replaced ∂ω ∂T ∂ω ∂Ū + ∂ω ∂T ∂ω ∂U | T 0 ,U 0 by 2α in view of (3.13). Summarizing, (3.75) gives the value of H(φ, p, q) evaluated at φ 2 * and φ 3 * , in the approximation where the real part of T 0 , U 0 is taken to be large, so that terms involving higher derivatives of ω with respect to T and U can be dropped.
In this approximation, all the terms in (3.75) are invariant under Γ 0 (2) S transformations, except for the term in the second line, whose duality invariance can be repaired by adding a term proportional to ln Y 0 and its complex conjugate to H(τ 1 , τ 2 , p, q), We recall that for BPS black holes the horizon value of Υ is real and given by Υ = −64, and that 4π Υ γ = 1 . (3.77) The combination on the left hand side of (3.76) is thus H(τ 1 , τ 2 , p, q) + (4/π) ln |Y 0 |. This combination will play a role in the quantum entropy function below, c.f. (3.100).

Evaluating W (q, p) beyond saddle point approximation
In (2.24) we displayed the value of the quantum entropy function W (q, p) in a saddle point approximation. Now, we proceed with the evaluation of W (q, p) beyond the saddle point approximation. In doing so, we will impose approximations that we will clearly delineate in what follows. We decompose Using (3.68) and (3.69) we infer to obtain Then, the quantum entropy function reads (using the approximate measure factor (2.18) with χ = 0 ) Next, we integrate out φ 2 and φ 3 , by expanding the exponent 4 ImF (φ+ip)−q·φ around the attractor values φ 2 * and φ 3 * computed in (3.63), and retaining only quadratic fluctuations in φ 2 and φ 3 . The associated quadratic form takes the form given in (2.21), with the indices I, J restricted to I, J = 2, 3. We approximate the resulting fluctuation determinant by replacing F by F (0) , in which case it takes the value τ 2 2 . Thus, in this approximation, the Gaussian integration 8 over fluctuations in φ 2 , φ 3 yields a factor 1/τ 2 , (3.82) and we obtain the following approximate expression for the quantum entropy function, with H(τ 1 , τ 2 , p, q) given by (3.71). Here, e −K (0) is evaluated at the values T, U given in (3.65).
In what follows, we will use the approximate result (3.94) as a starting point for various considerations. One should keep in mind that there are subleading corrections to (3.94) that will not be considered in this paper.
Proposition: For large black holes, and for large attractor values T 0 , U 0 , the quantum entropy function W (q, p) of the STU model is approximately given by where F (Y, Υ) denotes the Wilsonian function The functionF is the one that was obtained recently in [21] using the duality symmetries of the model, in the limit of large T, U , see above (3.6).
Proof. It is straightforward to verify that for large values of T 0 , U 0 , the quantum entropy function (3.101) takes the form where φ 2 * , φ 3 * , given in (3.63), are evaluated at large attractor values of T 0 , U 0 (c.f. the discussion below (3.93)), and where Y 0 is given by (3.78) and (3.79).

Microstate proposal
The quantum entropy function computes the macroscopic entropy of a BPS black hole. Ideally, we would like to reproduce it by state counting. Microstate counting formulae for BPS black holes in N = 4, 8 superstring theories [1, 3,5,9] suggest that the state counting will be based on modular objects. In the N = 4 context, these are Siegel modular forms, whose Fourier expansion yields integer coefficients that count microstates of 1 4 BPS black holes. Encoding microstate degeneracies of BPS black holes in terms of modular forms is a powerful principle that we will also use in the context of the N = 2 STU model. This gives a microstate proposal that can be tested by a state counting process as and when it is divised.
As already mentioned in the previous subsection, the approximate result for the quantum entropy function W (q, p) in (3.108) exhibits a dependence on the parameter k = 2 that is characteristic of a microstate counting formula based on a Siegel modular form Φ 2 of weight k = 2 [8]. Thus, we expect the microstate counting formula for large BPS black holes to involve Φ 2 . On the other hand, the integrand in (3.108) also depends on Y 0 , onỸ 0 as well as on ln(p 1 − ρ p 0 ) and ln(p 1 + σ p 0 ), which means that the counting formula cannot solely be given in terms of Siegel modular form Φ 2 . The microstate counting formula will, in particular, have to depend on the charges (p 0 , p 1 ), which transform as a doublet under Γ 0 (2) S , c.f. (3.59).
Let us first focus on the Siegel modular form Φ 2 . For a given weight k with respect to a subgroup of the full modular group Sp(4, Z), there may exist one or more Siegel modular forms. We propose that the Siegel modular form Φ 2 (ρ, σ, v) relevant for the microstate counting formula of large BPS black holes in the STU model is the Siegel modular form of weight k = 2 briefly discussed in [9] in the context of N = 4 BPS black holes, which in the limit v → 0 behaves as This Siegel modular form can be constructed as follows [5].
Proposition: There exists a Siegel modular form Φ 2 (ρ, σ, v) of weight k = 2, symmetric in ρ and σ, with the property as v → 0, that can be constructed by applying a Hecke lift to the Jacobi form of weight k = 2 and index m = 1.
Proof. We refer to appendix G for the proof, which is based on the construction given in [5]. Our proof of the property (4.2) relies on the relation (G.22), which we prove by considering Hecke eigenforms.
As mentioned above, the microstate counting formula for large BPS black holes cannot be solely based on the Siegel modular form Φ 2 . It should be encoded in various modular objects, one of them being Φ 2 . In the following, we make a proposal for reproducing W (q, p) e ln |ϑ 2 (T 0 )| 8 +ln |ϑ 2 (U 0 )| 8 in terms of modular objects.
Proposition: W (q, p) e ln |ϑ 2 (T 0 )| 8 +ln |ϑ 2 (U 0 )| 8 is approximately captured by the following integral in Siegel's upper half plane H 2 , Here, Y 0 ,Ỹ 0 andÎ 1 are given by (3.110), and Ξ is a solution to the non-linear PDE (3.28). The contour C denotes a contour that encircles v = 0, and that in the (τ 1 , τ 2 )-plane is identified with the contour C discussed in subsection 3.5. Note that G is given in terms of three distinct building blocks.
Proof. Let us begin by considering the integral .

(4.6)
As shown in [5], the exponent in (4.6) as well as are invariant under symplectic transformations acting on Siegel's upper half plane that belong to the subgroup H ⊂ Sp(4, Z), which consists of elements h ∈ H given by with g 1 and g 2 given in (E.6) and (E.10), respectively. Indeed, using (E.4), one finds that under H-transformations, The charge bilinears (m, n, l) transform as in (3.61). Using these transformation rules, one establishes that the exponent in (4.6) as well as (4.7) are invariant under H-transformations.
The integral (4.6) depends on a contour C , which transforms as follows under H. Under H, v gets mapped to v/∆, see (4.9). Thus, a small contour around v = 0 gets mapped to a small contour around v = 0. On the other hand, ρ and σ get mapped to ρ and σ according to (4.9). As mentioned in subsection 3.5, this means that in the limit v = 0, the contour C passing through the attractor point (τ * 1 , τ * 2 ) gets transformed into a new contour that passes through the transformed attractor point.
Thus, our proposal for a microstate counting formula for large BPS black holes which reproduces the corresponding approximate quantum entropy function (3.108) is . (4.15) Being approximate, this formula, which is based on modular objects, gives a non-integer value of d (p 0 ,p 1 ) (m, n, l). Note that we have attached the label (p 0 , p 1 ) to the degeneracy d (p 0 ,p 1 ) (m, n, l), to indicate that it also depends on the Γ 0 (2) S doublet (p 0 , p 1 ).

Conclusions
We conclude with a brief summary and a few observations. We computed the quantum entropy function (3.81) for large BPS black holes in the N = 2 model of Sen and Vafa, by integrating out the moduli φ 2 and φ 3 in Gaussian approximation, to arrive at the intermediate result (3.94), which we then converted into the integral (3.101) by adding a total derivative term (3.97). In doing so, we resorted to various approximations. The integrand in (3.101) is invariant under Γ 0 (2) transformations of (τ 1 , τ 2 ). However, it also depends on T 0 , U 0 , and is not invariant under Γ 0 (2) transformations of T 0 , U 0 . This is due to the fact that when evaluating (3.81), we expanded around background values T 0 and U 0 which we took to be large. There will then be subleading corrections in T 0 , U 0 that will restore the invariance under Γ 0 (2) transformations of T 0 , U 0 . Proceeding in the manner described above, namely integrating out the moduli {φ a } a=2,3 and retaining the dependence on φ 0 , φ 1 , one obtains an integral which resembles, in part, the expression obtained by integrating the inverse of a Siegel modular form Φ 2 (ρ, σ, v) of weight 2 along a closed contour surrounding v = 0, as in (4.6). However, the result of the quantum entropy function calculation also depends on the Γ 0 (2) S doublet (p 0 , p 1 ) through the dependence on Y 0 , and hence, a microstate counting formula cannot solely be based on the inverse of a Siegel modular forms, since the evaluation of the latter gives a result that depends only on the three charge bilinears m, n, l, but not on the individual charges (p 0 , p 1 ). Thus, the proposal for a microstate counting formula has to depend on an additional modular object G(ρ, Y 0 (σ, ρ)), c.f. (4.5).
We can generalize the above discussion to a certain class of N = 2 models with n V vector multiplets, as follows. We take these models to have a holomorphic Wilsonian function F (Y, Υ) = F (0) (Y ) + 2iΩ(Y, Υ) with a heterotic type prepotential of the form Moreover, introducing τ = Y 1 /Y 0 , we assume that the τ -dependence of the first gravitational coupling function ω (1) in Ω is encoded in a modular form of a certain weight under (a subgroup of) SL(2, Z). Suppressing the dependence on the other moduli, we set 4πΥω (1) (τ ) = g(τ ). This modular form will then be related to the seed of an associated Siegel modular form, as we will discuss momentarily. Setting Y I = 1 2 (φ I + ip I ), and using the approximate measure factor (2.18), the quantum entropy function for large BPS black holes in these models becomes with e −K (0) given by (2.11), and with χ = 2(n V − n H + 1) determined in terms of the number of vector and hyper multiplets of the N = 2 model (n V and n H , respectively). Following the steps in subsection 3.4.2 and integrating out the (n V − 1) moduli φ a in Gaussian approximation gives the approximate result up to an overall constant. Here, H(τ 1 , τ 2 , p, q) takes a form similar to (3.75), πH(τ 1 , τ 2 , p, q) = π n + lτ 1 + mτ 2 1 + mτ 2 where h encodes the dependence on Y 0 (τ 1 , τ 2 ) due to presence of the higher gravitational coupling functions, and where we have suppressed the dependence on the other moduli. By adding an appropriate total derivative term to (5.6), the latter can be brought into a form analogous to (3.98), with an appropriately chosen contour C. Note the dependence of the measure on k + 3, which suggests a microstate counting formula based on a Siegel modular form Φ(ρ, σ, v) of weight k, with the property that as v → 0, where f k+2 is a modular form of weight k + 2 under (a subgroup of) SL(2, Z). Requiring k to be integer valued imposes a restriction on the allowed values of (n V , n H ). We assume that this Siegel modular form can be constructed by applying a Hecke lift to a Jacobi form φ k,1 (τ, z) of weight k and index 1 (c.f. (G.2)), f k+2 (τ ) should be related to the modular form g(τ ) that appears in the first gravitational coupling function ω (1) , as mentioned below (5.1), but need not coincide with it. Evaluating the analogue of (4.6), by performing a contour integration over v around v = 0 gives [8] Using (3.105) this reproduces part of (5.9). This then needs to be further supplemented by modular objects that depend on (p 0 , p 1 ) through Y 0 (ρ, σ), as in (4.15).
We conclude by suggesting a string web picture of our counting proposal. We have proposed an approximate degeneracy formula for 1 2 BPS states that gravitate to form a large black hole at strong 't Hooft coupling in terms of a contour integral of the inverse of a Siegel modular form and of additional subleading contributions expressed as a series in 1/(Y 0 ) 2 . A physical picture of this mathematical structure is suggested by the N = 4 picture discussed in [4,11,13], whereby a dyonic BPS black hole is viewed as a string web wrapping a two-torus in type IIB string theory. In this picture, a dyonic black hole in type IIB is described in terms of a web of effective strings. Viewing the six-dimensional compact manifold as a fibration over a two-torus, these effective strings wrap either one of the two cycles of the two-torus. Electric states correspond to wrappings on one of these cycles, while magnetic states correspond to wrappings on the other cycle. This effective description is valid at points in moduli space where the volume of the fibre is small compared to the volume of the two-torus. The electric-magnetic duality group Γ 0 (2) is viewed as a subgroup of the large diffeomorphism group of the two-torus. A computation of the Euclidean partition function of this string network requires time to be compactified, so that the arms of the torus are tube-like and the resultant partition function of the string web is a genus two partition function of an effective string theory that encodes BPS states of the original type II theory. In the present N = 2 case, topological string theory is a candidate for this effective string theory, since amplitude calculations in topological string theory result in corrections to the free energy of the type II BPS partition function. In this picture, the world sheet path integral will encode contributions from worldsheet instantons corresponding to higher genera string webs. These contributions should correspond to the series in 1/(Y 0 ) 2 which appears in the proposed degeneracy formula, and which is absent in the N = 4 case as a result of the larger supersymmetry in the system.

A Modular forms for SL(2, Z)
We review basic properties of modular forms for SL(2, Z), following [59].
Remark: SL(2, Z) is generated by two elements,

Remark:
Consider an open subset F ⊂ H such that no two distinct points of F are equivalent under the action of SL(2, Z), and every τ ∈ H is equivalent to a point in the closure D ≡ F. Then, D is called a fundamental domain for SL(2, Z), Definition: Modular forms are weakly modular functions that are also holomorphic on H and at ∞, i.e. on H * .
Remark: To show that a weakly modular function f is holomorphic at ∞, it suffices to show that f (τ ) is bounded at Im τ → ∞, i.e. there exists C ∈ R such that |f (τ )| ≤ C ∀τ with Im τ 1. Definition: The set of modular forms of weight k ∈ Z is denoted by M k (SL(2, Z)).
Remark: M k (SL(2, Z)) forms a finite-dimensional vector space over C, and the direct sum forms a graded ring.
Definition: A cusp form of weight k ∈ Z is a modular form of weight k whose Fourier expansion has a coefficient a 0 = 0, i.e.
Definition: The set of cusp forms of weight k ∈ Z is denoted by S k (SL(2, Z)).
It has a simple zero at q = 0.
Example: Let k ∈ N with k ≥ 2. The Eisenstein series, defined by is a modular form of weight 2k. It has the following Fourier expansion, where the sum σ p (n) = d|n d p is over positive divisors of n, and ζ(z) is Riemann's zeta function.
Setting k = 1 in (A.12) yields G 2 , a quasi-modular form of weight 2 and depth s = 1, i.e. a holomorphic function G 2 : H * → C that, compared to (A.1), transforms with an additional shift proportional to c/(cτ + d) under (A.2), More generally, a quasi-modular function of weight k and depth s is defined as follows [56]: Definition: Let k ∈ Z and s ∈ N. A holomorphic function f : H * → C is a quasi-modular form of weight k and depth s if there exist holomorphic functions Q 1 (f ), . . . , Q s (f ) on H such that for all a, b, c, d ∈ Z with ad − bc = 1, and such that Q s (f ) is not identically zero.
The normalized Eisenstein series are E 2k = G 2k /(2ζ(2k)). For k ≥ 2, the normalized Eisenstein functions can also be defined by [60]  For large n, the Fourier coefficients a n grow as a n ∼ e 4π Thus, they exhibit exponential growth, as required for the microstate degeneracy of small BPS black holes in N = 4 string theories [62].

B Congruence subgroups of SL(2, Z)
We review basic properties of modular forms for congruence subgroups of SL(2, Z), following [59].
Definition: Let N be a positive integer. The principal congruence subgroup of SL(2, Z) of level N is In particular, Γ(1) = SL(2, Z).
Definition: A subgroup Γ of SL(2, Z) is a congruence subgroup if Γ(N ) ⊂ Γ for some N ∈ Z + . The least such N is called the level of Γ.
Remark: Γ 0 (2) is generated by two elements, Remark: Each congruence subgroup Γ of SL(2, Z) contains a translation matrix of the form for some minimal h ∈ Z + . Hence, every function f : H → C that is weakly modular with respect to Γ is h Z-periodic, and has a corresponding function g : D → C with f (τ ) = g(q h ), where q h = e 2πiτ /h . Then, f is defined to be holomorphic at ∞ if g extends holomorphically to q = 0, in which case Proposition (fundamental domain): Let Γ be a congruence subgroup of SL(2, Z), and let R be a set of coset representatives for the quotient Γ\SL(2, Z). Then, the set D Γ = ∪γ ∈Rγ D is a fundamental domain for Γ [63]. Here, D denotes a fundamental domain for SL(2, Z).
Definition: Let Γ be a congruence subgroup of SL(2, Z), and let k ∈ Z. A modular form of weight k for the subgroup Γ is a holomorphic function f : H → C that is weakly modular of weight k for Γ and holomorphic at all cusps of Γ.
Definition: Let Γ be a congruence subgroup of SL(2, Z), and let k ∈ Z. Writing any element s ∈ {∞} ∪ Q as s = γ(∞) for some γ ∈ SL(2, Z), holomorphy of f at s is defined in terms of holomorphy of f [γ] k at ∞ for all γ ∈ SL (2, Z). (γ(τ )). If a 0 = 0 in the Fourier expansion of f [γ] k for all γ ∈ SL(2, Z), then f is called a cusp form of weight k with respect to Γ.
Remark: For Γ, the finite-dimensional vector space over C of modular forms of weight k is denoted by M k (Γ), and the vector subspace of cusp forms of weight k is denoted by S k (Γ). The direct sums form a graded ring.
Proof. Using absolute convergence, we write where G 2k denote Eisenstein functions for SL(2, Z). Using (A.12) in the form we obtain, using absolute convergence, Expanding in a geometric series, and interchanging the order of summations yields (B.17).

(B.23)
Proposition: E 2 can be expressed as [55] where the theta function ϑ 2 has the product representation valid in the open complex unit disc D = {q ∈ C : |q| < 1}. Thus, E 2 is a quasi-modular form for Γ 0 (2), Proof. Using (B.25) and (B.23) as well as absolute convergence, we compute Using the relation it follows that the right hand side of (B.27) vanishes. Since I 1 is quasi-modular, so is E 2 .
Proof. The proof is by induction. The claim holds for k = 2. Assume that it holds for a k 0 with k 0 > 2. Then, consider operating with (B.33) on a summand (Ẽ 2 ) m (E 4 ) n of weight 2k 0 = 2m + 4n. Using the relations (B.31), one infers Note that the terms on the right hand side have weight 2k 0 + 2, and that they have the same structure as in (B.38). Thus, when operating with (B.33) on I k 0 , the resulting sum is of the form (B.38) with m, n satisfying the relations m ≥ 0, n ≥ 1 and m + 2n = k 0 + 1. Multiplying (B.39) with (−2π), so as to obtain D S on the left hand side, we infer that the coefficients a m,n of I k 0 +1 are real and positive.
Next, we discuss the growth properties of the Fourier coefficients of modular forms for Γ 0 (2).
Proposition: Consider the modular formẼ 2 of weight 2 given in (B.30), for some C ∈ R with C > 0.
Proof. We considerẼ 2 2 and use (B.29), The right hand side is a linear combination of modular forms of weight 4. Each of them has q-expansion coefficients a n that exhibit the property |a n | ≤ C n 3 , c.f. (B.42). Hence, if we denote the q-expansion ofẼ 2 2 byẼ 2 2 = N ≥0 c N q N , the coefficients c N will also satisfy |c N | ≤ D N 3 , for some constant D > 0. Now consider (B.45), and let us assume that its coefficients a n satisfy the bound |a n | ≤ A n p , with p ∈ N, A > 0. Then |Ẽ 2 | 2 ≤ 1 + A m≥1 m p |q| m 1 + A n≥1 n p |q| n . Using absolute convergence, we obtain, for N > 0, |c N | ≤ 2A N p + A 2 m+n=N, m,n≥1 m p n p . We now place an upper bound on the sum, for a given N , as follows. Extremizing m p n p = (N − n) p n p with respect to n gives n * = N/2, in which case m p * n p * = N 2p /2 2p . Using that the number of partitions of N into two positive integers m and n with m + n = N is N − 1, we obtain the bound m+n=N, m,n≥1 m p n p ≤ m p * n p * (N − 1) ≤ B N 2p+1 , for some constant B > 0. Hence, |c N | ≤ D N 2p+1 , for some constant D > 0. On the other hand, we had already concluded that |c N | ≤ D N 3 , so that p = 1.
Proof. The claim holds for k = 2, since I 2 ∝ E 4 . Let us consider the case k > 2.
We begin by considering a summand E 2 4 of weight 8 in (B.38). Then, using (B.42), we obtain |E 2 4 | ≤ 1 + a m≥1 m 3 |q| m 1 + a n≥1 n 3 |q| n for some constant a > 0. Denoting the q-expansion of E 2 4 by E 2 4 = N ≥0 c N q N , and following the same steps as in the proof given above, we obtain the bound |c N | ≤ D N 8−1 , for some constant D > 0.
Proceeding in a similar manner, one finds that the expansion coefficients ofẼ 2 E p 4 exhibit the growth property |c N | ≤ D N 4p+1 . By induction, one then shows that the expansion coefficients ofẼ l 2 E p 4 exhibit the growth property |c N | ≤ D N 4p+2l−1 . This growth goes as N 2k−1 with 2k = 4p + 2l.
Hence we conclude that expansion coefficients of I k in (B.38), with k ≥ 2, exhibit the growth property (B.49).

C Rankin-Cohen brackets
Definition: Let n ∈ N 0 . The nth Rankin-Cohen bracket is a bilinear, differential operator that acts on modular forms f, g of SL(2, Z), of weight k ∈ N and l ∈ N, respectively, by [68] [f, g] n (τ ) = 1 (2πi) n n r=0 (−) r k + n − 1 n − r where f (r) (τ ) denotes the rth derivative of f with respect to τ , and similarly for g (n−r) (τ ).
In the following, we drop the normalization factor 1 (2πi) n .
Example: Let f (S) be a modular form for SL(2, Z) of weight k, and let g(S) = ϑ 8 2 (S), which has weight l = 4. Then, where (c.f. (B.32)) In particular, consider the case f (S) = I 2 (S), which we recall is proportional to the Eisenstein series E 4 of weight 4, c.f. (B.35). Then, For n > 2 we obtain , g] 1 , . . . , g] 1 , g] 1 . (C.5) The above definition can be extended to include quasi-modular forms as well [56]. Here we focus on quasi-modular forms of weight k and depth s = 1 of SL(2, Z). An example thereof is provided by (A.13).
Definition: Let f be a quasi-modular form of weight k and depth s = 1, and let g be a modular form of weight l, respectively, for SL(2, Z). Then, their nth Rankin-Cohen bracket is given by where f (r) (τ ) denotes the rth derivative of f with respect to τ , and similarly for g (n−r) (τ ). In the following, we drop the normalization factor 1 (2πi) n .
Example: As an application, set with weights k = 2 and l = 4, respectively. The depth of f is s = 1. Then which yields a modular form of weight 4 and depth s = 0.

D Jacobi forms
In this section, we follow [61,70]. Let H denote the complex upper half plane. Let τ ∈ H and z ∈ C.
Definition: A Jacobi form of SL(2, Z) is a holomorphic function φ k,m : H × C → C that transforms as follows under the modular group SL(2, Z), and under translations of z by Zτ + Z as Here, k ∈ Z is called the weight, and m ∈ N is called the index of the Jacobi form.
Remark: τ is called modular parameter, while z is called elliptic parameter.
Remark: ϑ 2 is even with respect to z, ϑ 2 (τ, −z) = ϑ 2 (τ, z) , (D. 15) and it solves the heat equation 16) with periodic boundary conditions (z → z + 1) imposed in the z direction. Here τ = iS. Also observe that when taking an odd number of derivatives with respect to z, we have Imposing Γ 0 (2) invariance shows that p n (S) are modular forms for Γ 0 (2) of weight 2n. Inserting (D.30) into (D.29) and differentiating term by term, which will be justified below, one infers p 2 (S) = I 2 (S) and p n+1 (S) = D S p n (S) for n ≥ 2. Hence, p n = I n .
In the main part of the paper, we also encountered the series (c.f.

E Siegel modular forms
In this section, we follow [60,70] as well as appendix A of [71].
The Siegel modular group Sp(4, Z) acts on H 2 as follows. An element M ∈ Sp(4, Z), A standard fundamental domain for the action of Sp(4, Z) on H 2 is the set defined by [72] − The condition | det(CΩ + D)| ≥ 1 applies to all Sp(4, Z) transformations of Ω.
Theorem: Let Φ k be a Siegel modular form of weight k ∈ N. It has the Fourier development [70] Φ k (ρ, σ, v) = m≥0 ψ k,m (ρ, v) e 2πimσ . (E.13) For m > 0, the function ψ k,m is a holomorphic Jacobi form of weight k ∈ N and index m. For m = 0, the function ψ k,0 transforms as a Jacobi form with m = 0. If the first coefficient ψ k,0 is identically zero, the Siegel form is called Siegel cusp form.
Theorem: Let ψ k,1 be a holomorphic Jacobi form of weight k and index 1. Then the functions T m ψ k,1 , with m ≥ 1, defined in terms of the Hecke lift below, are the Fourier coefficients of a Siegel modular form of weight k [70].
Remark: These theorems also hold if Sp(4, Z) is replaced by a congruence subgroup Γ ⊂ Sp(4, Z). We shall be interested in the congruence subgroup Γ = Γ 2,0 (2) of Sp(4, Z), where one restricts to C = 0 mod 2. In particular, we shall be interested in the subgroup H given by (E.11). Then, the functions ψ k,m in the Fourier development are Jacobi forms for Γ 0 (2) ⊂ SL(2, Z).

F Hecke operators for SL(2, Z)
We follow [60]. Let f ∈ M k (SL(2, Z)) with Fourier expansion f (τ ) = n≥0 a n q n . Let m be an integer with m ≥ 1.
Definition: The m-th Hecke operator is a linear operator T m that acts on modular forms of weight k by with α, β, δ ∈ Z.