Generating new $N = 2$ small black holes

We use the exact symmetries of the $N = 2$ STU model of Sen and Vafa to classify BPS orbits in this theory. Subsequently, we construct examples of small BPS black holes in this model by solving the associated attractor equations, and use duality symmetries as a solution generating technique to construct a two-center bound state of small BPS black holes in asymptotically flat space-time.


Introduction
The search for a statistical understanding of black hole entropy as well as attempts to extend the solution space of black hole backgrounds in various theories of gravity has yielded rich insights into the mathematical structures that govern the non-perturbative physics in these theories. In particular, the former endeavour has uncovered modular forms that underlie the microstate organisation of certain classes of supersymmetric dyonic black holes in specific four-dimensional N = 4 and N = 8 string theories [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. Especially in the N = 4 cases, where the counting formulae for a class of dyonic black holes can be written down exactly in terms of Siegel modular forms, studying the solution space of decadent black hole backgrounds restricted to 2-centered small black hole pairs and the lines of marginal stability across which they decay allows one to extract non-perturbative information encoded in the zeros of these modular forms. In both the quest to write down exact counting formulae as well as study the solution space of small black holes, the underlying duality symmetries of the theory not only provide useful checks on allowed solutions and modular forms, but also act as solution generating techniques in the low energy gravity theories obtained from the string theories under consideration. So far, both types of aforementioned analyses have proved elusive in the N = 2 case until very recently, when an approximate counting formula, in terms of Siegel modular forms, Jacobi and quasi-modular forms was written down [20] for supersymmetric dyonic black holes in the N = 2 STU model of Sen and Vafa (example D of [21]). This is a model that possesses exact duality symmetries. These have been used recently [22] to obtain exact results for the function F that encodes the Wilsonian action describing the coupling of vector multiplets to supergravity in the presence of an infinite set of gravitational coupling functions ω (n+1) (S, T, U ) (n ≥ 0). The function F is one of the ingredients that enter in the definition of the quantum entropy function [23][24][25] for BPS black holes in N = 2 theories. In the N = 2 Sen-Vafa STU model, it is this quantum entropy function for large dyonic BPS black holes that was rewritten in [20] as an approximate counting formula, in terms of modular forms.
In this note, we will attempt to harness the exact symmetries of the model to investigate the solution space of 2-centered BPS small black holes. We will begin with a classification of BPS orbits using the Γ 0 (2) duality symmetry of this N = 2 model, following the analysis based on SL(2) given in [26]. We will then turn to the construction of small BPS black holes in this N = 2 model. This is done by solving the associated attractor equations. To this end, we begin by working in a regime where the gravitational coupling functions ω (n+1) (S, T, U ) simplify, namely, we take two of the moduli S, T, U of the model to be small, while the third modulus is taken to be large. In this regime, the small BPS black holes [27][28][29] that we obtain carry more than two charges, in agreement with [30]. Subsequently, with the help of duality symmetries, we construct a two-center bound state of small BPS black holes in asymptotically flat space-time, by taking the asymptotic moduli to equal the attractor values of a dyonic BPS black hole carrying the same amount of charge as the combined two-center configuration.

Classification of orbits
In N = 2 supergravity theories in four dimensions, the couplings of Abelian vector multiplets to supergravity are encoded in a holomorphic function F (Y, Υ) that characterizes the Wilsonian effective action [31]. The function F (Y, Υ) can be decomposed as F (Y, Υ) = F (0) (Y ) + 2i Ω(Y, Υ), where Υ denotes the (rescaled) lowest component of the square of the Weyl superfield, and where Y I ∈ C denote complex scalar fields that reside in the off-shell vector multiplets of the underlying supergravity theory (I = 0, 1, . . . , n). These theories admit BPS black holes, which are extremal dyonic black holes carrying electric/magnetic charges (q I , p I ).
In this note, we will consider small BPS black holes in the N = 2 STU model of Sen and Vafa (example D of [21]). For this model, I = 0, 1, 2, 3. We introduce projective coordinates This model possesses duality symmetries, namely Γ 0 (2) S × Γ 0 (2) T × Γ 0 (2) U symmetry as well as triality symmetry under exchange of S, T, U . Here, Γ 0 (2) denotes the subgroup where * can take any value in Z. We may assemble the eight electric/magnetic charges (q I , p I ) into two vectors, Then, under Γ 0 (2) S -transformations, these two vectors transform as [30], Each of the four-component vectors Q e , Q m corresponds to a point in a four-dimensional lattice made out of two two-dimensional hyperbolic lattices, each with hyperbolic metric h, Introducing the four-dimensional metric η = h ⊕ h, we define the charge bilinears These bilinears transform as a triplet under Γ 0 (2) S [30], and the Γ 0 (2) invariant norm of this vector is ∆ ≡ 4nm − l 2 . Given a generic charge vector Q m , we now show that Q m can be brought into one of the following inequivalent forms by means of sequences of Γ 0 (2)-transformations: be a charge vector with gcd(q 1 , p 0 , p 3 , p 2 ) = 1. Then, it can be brought into one of the following inequivalent forms by Γ 0 (2)-transformations, where m is the magnetic charge invariant defined in (2.7) Proof. Given an integral column vector, V = a b , with non-vanishing entries a and b, we , where (α, β) are coprime, and where g V = gcd(|a|, |b|) denotes the greatest common divisor of |a| and |b|. We then focus on the vector α β . First, consider the case when β is even, in which case α is odd. Then, by a sequence of T , T −1 , S and S −1 transformations, this vector can be brought to the form 1 0 . Here, T and S denote the two generators of Γ 0 (2), given by The Γ 0 (2)-transformation that brings the vector into the form 1 0 must be of the form where the integers (γ, δ) are coprime and satisfy αδ − βγ = 1, with δ odd and β even. The fact that (γ, δ) are coprime is a consequence of the fact that the determinant of any SL(2, Z)-matrix is one. The entries of the second row of the Γ 0 (2)-matrix have to be equal to (−β, α) (up to an overall minus sign). The reason is as follows. Suppose that the entries of the second row are given by coprime integers (−β,α), withβ even (due to the Γ 0 (2) nature of the matrix), such thatβα =αβ. Decompose α into prime factors, α = p 1 . . . p n . Then, none of these prime numbers can divide β, since (α, β) are coprime. Hence, they must divideα, and henceα = mα, with m ∈ Z. It follows thatβ = mβ. Since (α,β) are coprime, it follows that m = ±1. Next, consider the case when β is odd. Then, by a sequence of T , T −1 , S and S −1 transformations, this vector can be brought to the form 0 1 . The Γ 0 (2)-transformation that brings the vector into this form must be of the form where the integers (γ, δ) are coprime and satisfy αδ − βγ = 1, with δ = 0 mod 2 and γ odd. This follows by using a similar reasoning as above. Hence, the vector V can be brought into the form by means of a Γ 0 (2)-transformation. This result will be used in what follows next. Now, associate to the vector Q T m = (q 1 , p 0 , p 3 , p 2 ) the matrix whose determinant equals the charge bilinear m, det Q = m. Let us denote by gcd(Q) the greatest common divisor of the four charges (q 1 , p 3 , p 2 , p 0 ). We now factor out gcd(Q) from the matrix Q, so that in the following, Q will denote the matrix (2.14) with gcd(Q) = 1. Now we operate with matrices A ∈ Γ 0 (2) U , B ∈ Γ 0 (2) T on Q as follows, Q → AQB T , to bring Q either into diagonal form or into anti-diagonal form. Note that gcd(Q) = 1 is preserved by this operation. Here, the matrix A takes the form (2.5), with the entries (b, c) replaced by (−b, −c), while the matrix B is of the form (2.5). We proceed to explain this construction. In the first step, we use the result given above to bring the first column of the matrix Q into the form whenp 2 = 0 mod 2, and into the form Let us first consider the case (2.15). We operate on it from the left with the Γ 0 (2)matrix to obtain where we note that the entries of the first row are coprime, for the following reason. Let p denote a prime factor of g V . Then, it either dividesp 3 , or it does not. If it dividesp 3 , it cannot divide k, and it also cannot dividep 0 , because gcd(Q) = 1, by assumption. If it does not dividep 3 , it divides k. Hence, the two entries of the first row are coprime. Next, we operate on (2.18) from the right to bring the first row into canonical form, i.e. either into the form (1 0) or into the form (0 1), depending on whether −p 3 + kp 0 is even or odd. This is achieved by using the transpose of the matrix given in either (2.11) or (2.12). Therefore, if −p 3 + kp 0 is even, we obtain 1 0 r s , (2.19) with some integer entries r and s, whereas if −p 3 + kp 0 is odd, we obtain If s is even, we operate on this matrix from the right with the matrix Finally, operating from the left with the matrix Now we operate from the right with and subsequently from the left with , which corresponds to a charge vector Q T m = (m, 1, 0, 0). Finally, let us consider the case whenp 2 is odd. Then, we act on the left of (2.16) with the matrix (2.17), to obtain (2.38) The elements in the first line of (2.38) are coprime, for the reason given below (2.18), and therefore, following the same steps as before, we conclude that the case of oddp 2 reduces to the case of evenp 2 .
Hence, we have shown that when gcd(q 1 , p 0 , p 3 , p 2 ) = 1, we can bring Q m into one of the following four inequivalent forms, 3 Small BPS black holes in the STU-model BPS black holes in N = 2 supergravity theories carry electric/magnetic charges (q I , p I ) 1 . These charges give rise to three charge bilinears (n, m, l), which for the STU-model are given by (2.7). Then, one introduces the quartic charge combination ∆ ≡ 4nm − l 2 , to define small BPS black holes, as follows.
Definition: A small BPS black hole is a BPS black hole carrying electric/magnetic charges (q I , p I ) such that the charge combination ∆ = 4nm − l 2 vanishes, and such that its macroscopic (Wald) entropy S is, for large charges, given by S ∝ Q 2 . Here Q 2 denotes a linear combination of products of two charges.
Remark: At the two-derivative level, a BPS black hole has a non-vanishing horizon area proportional to √ ∆, with ∆ > 0. Therefore, a small BPS black hole does not exist as a solution to the equations of motion of the underlying N = 2 supergravity theory at the two-derivative level. For a small BPS black hole to exist, higher-curvature corrections need to be taken into account 2 . These are encoded in Ω(Y, Υ).
For the N = 2 STU-model of Sen and Vafa [21], the Wilsonian function F (Y, Υ) = F (0) (Y ) + 2i Ω(Y, Υ) takes the form [22], 3 The gravitational coupling functions ω (n+1) (S, T, U ) were determined in [22], using the duality symmetries of the model, namely invariance under Γ 0 (2) S × Γ 0 (2) T × Γ 0 (2) U transformations and under triality (i.e. exchanges of S, T and U ). The coupling functions ω (n+1) (S, T, U ) are all non-vanishing when evaluated at a generic point in the S, T, Umoduli space, and are determined in terms of the first gravitational coupling function ω (1) (S, T, U ), where [32] in view of the fact that ϑ 2 (τ ) is defined on the complex upper half-plane {τ ∈ C : Im τ > 0}. The expressions for the coupling functions ω (n+1) (S, T, U ) obtained in [22] are complicated. However, as noted in [22], they do simplify when some of the moduli S, T, U are taken to large or to small values. In the following, we will choose a regime where these couplings simplify, namely the regime where Re S 1 and Re T 1, Re U 1. We proceed with a discussion of the ω (n+1) (S, T, U ) in this regime.
We begin by discussing the behaviour of ω(τ ) in the limit of large and of small Im τ . Using the product representation for ϑ 2 (τ ), valid in the punctured disc {q ∈ C\{0} : |q| < 1}, we obtain for large Im τ 1, and hence, setting S = −iτ , For small Im τ 1, we proceed as follows. Using the relations as well as we obtain, setting T = −iτ , provided Re(1/T ) 1. The latter will be enforced in the next subsection by constructing a BPS configuration that satisfies Im T = 0.
In the regime mentioned above, the higher gravitational coupling functions ω (n+1) (S, T, U ), with n ≥ 1, simplify, and become proportional to This is a consequence of the behaviour ∂ S ω ∼ 1, ∂ T ω ∼ 1/T, ∂ U ω ∼ 1/U in this regime, as well as of the explicit form of the ω (n+1) (S, T, U ), with n ≥ 1, given in [22]. Then, the function F (Y, Υ) takes the approximate form, 12) with constants α n ∈ C. Here we assume that the constant coefficients α n are such that the infinite sum in (3.12), viewed as a power series in z ≡ 1/[(Y 0 ) 2 T U ], has a non-vanishing radius of convergence, and that z = −2 is in the region of convergence, c.f. (3.19). However, we cannot verify this at the present stage, because of lack of knowledge about the precise form of the constant coefficients α n .
In the near-horizon region, a BPS black hole solution is determined in terms of the charges carried by the black hole. The field Υ takes a real constant value at the horizon of the BPS black hole, and the horizon values of the scalar fields Y I supporting the BPS black hole are determined in terms of the charges of the black hole by solving the attractor equations [33] Y I −Ȳ I = ip I , where F I = ∂F/∂Y I . The semi-classical (Wald) entropy of the BPS black hole takes the form [33] S = π p I F I − q I Y I + 4ΥIm F Υ | attractor , (3.14) where F Υ = ∂F/∂Υ. We note the useful relation We now proceed to solve the attractor equations (3.13) in the regime specified above, with F (Y, Υ) given in (3.12). We first construct a small BPS solution with p 0 = 0 (and magnetic bilinear m = 0). Subsequently, we use the Γ 0 (2) S duality symmetry of the model to construct a small BPS black hole with p 0 = 0 (and m = 0).

Small BPS black holes with p 0 = 0
We construct small BPS black holes with non-vanishing charges (p 0 , p 1 , q 2 , q 3 ). The magnetic bilinear m vanishes.
Small BPS black holes with charges (p 0 , q 2 , q 3 ) were considered in [30]. However, they were constructed using a non-holomorphic function F = F (0) + 2iΩ, with a real Ω. This came about by implementing the duality symmetries of the model through the inclusion of non-holomorphic terms. In the present setup, however, the duality symmetries are implemented by adding a term proportional to ln Y 0 , following [22]. We will use this setup to analyze small black holes with four charges (p 0 , p 1 , q 2 , q 3 ), while also paying attention to the behaviour of the higher gravitational coupling functions ω (n+1) (S, T, U ). The threecharge black hole of [30] will arise as a special case of this four-charge system.
The magnetic attractor equations (3.13) imply that as well as Using this, the electric attractor equation Inserting this into the electric attractor equation F 0 −F 0 = 0, we get 20) and therefore Y 0 is purely imaginary, The electric attractor equation where we used (3.19). Similarly, the electric attractor equation (3.23) Comparing (3.22) with (3.23), we infer Multiplying Y 2 Y 3 = Υc 1 with q 2 q 3 and using (3.24) yields Hence we infer q 2 q 3 > 0 (3.26) as well as Inserting this into (3.22) or (3.23) determines n β n (Υc 1 ) n ± q 2 q 3 Υc 1 . (3.28) Next, we display the expressions for S, T, U , Note that Im T = Im U = 0, in agreement with what was stated below (3.10). Demanding Re S 1, and recalling that Υc 1 = 1/2 > 0, we infer that we have to pick the negative sign in the expression for Re S 1 given in (3.29), so that n β n (Υc 1 ) n + q 2 q 3 Υc 1 , (3.30) Demanding Re T > 0, Re U > 0 requires taking and hence, using (3.26), we infer that the charges p 0 , q 2 , q 3 have to be either all positive or all negative. To ensure Re S 1, we require |q 2 | 1, |q 3 | 1. To ensure that Re T and Re U are small, we take |q 2 | ∼ |q 3 |. We therefore take |q 2 | 1, |q 3 | 1, |q 2 | ∼ |q 3 |, |p 0 | 1 to achieve Re T 1, Re U 1. Note that the solution depends crucially on the combination Υc 1 = 0, and hence only exists due to the gravitational coupling functions encoded in Ω, as expected for a small BPS black hole.
Next, we compute the semi-classical Wald entropy (3.14) of this small BPS black hole, retaining only contributions proportional to charges. First, we note that on the solution, as well as which we drop, since it is independent of charges. Next, using (3.24) as well as (3.27) (where we have to take the negative sign), we obtain And finally, using which is valid in the approximation that we drop terms independent of charges, we obtain again up to a charge independent constant. Hence, for large charges, we obtain, with Υc 1 = 1/2, In the limit where the charges p 0 , q 2 , q 3 are taken to be uniformly large, the first term is the leading term, and it equals A/2, where A denotes the area of the event horizon of the small BPS black hole. Note that the result for the BPS entropy (3.37) does not depend on the charge p 1 . The term ln |p 0 | represents a correction term, whose coefficient will however change when including corrections to the semi-classical Wald entropy computed by the quantum entropy function [20,23,24] (c.f. footnote 3), and hence we drop this term in (3.37). Then, the result (3.37) agrees with the one obtained previously in [30].
In the above, we have assumed that the coefficients α n and β n are such that the infinite sums in (3.16) and in (3.28) are convergent. As already mentioned below (3.12), we cannot verify this at the present stage. At any rate, these infinite sums give rise to charge independent contributions to the semi-classical entropy, which we discarded.
When dropping all charge independent terms, the solution given in (3.21) and (3.29) takes the form We will use this solution to generate a small BPS black hole solution with p 0 = 0 in the next subsection.
3.2 Small BPS black holes with p 0 = 0 Now we construct small BPS black holes with p 0 = 0 and non-vanishing magnetic bilinear m. These black holes will have to carry non-vanishing charges p 1 , p 2 , p 3 , to ensure that the real part of the moduli S, T, U is non-vanishing.
In order to obtain a solution with p 0 = 0, we start from the solution (3.38) and apply a duality transformation to it, as follows. We act with a Γ 0 (2) S transformation on the charges, rewriting explicitly (2.4): p 0 → dp 0 + cp 1 , q 0 → aq 0 − bq 1 , (3.39) The moduli Y I transform in a similar manner, by performing the replacement p I → Y I , q I → F I in (3.39). We apply the following Γ 0 (2) S transformation to (3.38), To ensure that the resulting charge configuration has vanishing charge p 0 , we choose the charge p 1 in (3.38) to have the value p 1 = p 0 /2. Then, using (3.39), we obtain a new charge configuration (q I ,p I ) withp 0 = 0 and with non-vanishing charges (q 2 = q 2 ,q 3 = q 3 ,p 1 = p 0 /2,p 2 = 2q 3 ,p 3 = 2q 2 ). The magnetic charge bilinear m is non-vanishing and given by m =p 2p3 = 4q 2 q 3 . The associated moduliỸ I are, to leading order in the charges, given bỹ (3.41) In obtaining these leading order expressions, we used F = F (0) = −Y 1 Y 2 Y 3 /Y 0 , rather than (3.16). These expressions receive subleading corrections in the charges which we have dropped. Since Γ 0 (2) S constitutes a symmetry of the model, the configuration (3.41) describes a small BPS black hole solution, to leading order in the charges. The constraints (3.26), (3.31) on the charges ensure that the transformed fields satisfy ReS, ReT , ReŨ > 0. In addition, using |q 2 |, |q 3 |, |p 0 | 1, |q 2 | ∼ |q 3 | (c.f. below (3.31)), we infer from (3.41) that Re S, Re T, Re U 1.