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On the Uniqueness of Supersymmetric Attractors

In this paper we discuss the uniqueness of supersymmetric attractors in four dimensional N=2 supergravity theories coupled to n vector multiplets. We prove that for a given charge configuration the supersymmetry preserving axion free attractors are unique. We generalise the analysis to axionic attractors and state the conditions for uniqueness explicitly. We consider the example of a two-parameter model and find all solutions to the supersymmetric attractor equations and discuss their uniqueness.


Introduction
Understanding the origin of black hole entropy has remained to be an important topic of research in gravity and string theory since the seminal work by Bekenstein [1] on this issue. One of the important developments in this area is the so called attractor mechanism, which states that, in a theory of gravity coupled to several scalar fields admitting a single centred extremal black hole, the scalar fields run into a fixed point at the horizon whose value depends only on the black hole charges [2][3][4][5]. There are several aspects of attractor mechanism which have been studied thoroughly [6,7]. Multiplicity of the attractors is one of the puzzling issues which remains to be understood better. Because of the presence of multiple basin of attractors, the near horizon geometry of the black hole is no longer uniquely determined by its charges and one needs to specify the area code in addition to the black hole charges.
The existence of multiple basin of attractors for a given set of charges has been first discussed in [8,9]. Area codes in the context of flux vacua and black hole attractors has been studied [10,11]. Subsequently, multiple supersymmetric attractors in five dimensional N = 2 supergravity theory has been discussed and explicit constructions in the simple case of a two parameter model has been carried out [12]. The analysis has been extended to four dimensional N = 2 supergravity [13] by using the know 4D − 5D correspondence of the attractor points [14]. Further, new multiple non-supersymmetric attractors which does not have obvious five dimensional embedding has been constructed [13]. Multiple attractors in a one parameter model in the presence of quantum corrections has already been studied [15].
The existence of multiple single centred supersymmetric attractors might at first sight appear to be in contradiction with the uniqueness results [16]. (For homogeneous moduli spaces, the solution is always unique up to a duality transformation [17]). However, as explained by Kallosh [18], this is not always the case, because the moduli space might in general possess several disconnected branches. The attractor solution in each of these branches remains unique. One might expect similar results in four dimensional N = 2 supergravity. However, though there exists multiple non-supersymmetric attractors and also multiple attractors with one of the attractor points being supersymmetric in these four dimensional supergravity theories there is no known example where both the attractor points are supersymmetric for these N = 2 supergravity theories in four dimensions [13]. This suggests that, unlike the five dimensional case, the supersymmetric attractors might be unique in these four dimensional supergravity theories. The present work aims to investigate this issue in detail.
The plan of this paper is as follows. In the following section, we will briefly overview the N = 2 supergravity theory. In §3 we will prove that the axion free attractors in four dimensions are unique. Subsequently, we will generalise this result for axionic attractors. This will be followed by an explicit construction of all supersymmetric attractors in a simple two-parameter model in §4. Finally, we will be summarise our results in §5.

Overview
The Lagrangian density for the bosonic part of the four dimensional N = 2 supergravity theory coupled t o n vector multiplet, is given by Here h µν is the space-time metric, R is the corresponding Ricci scalar, g ab is the metric on the vector multiplet moduli space parameterized by the corresponding n complex scalar fields x a and A Λ µ are the (n + 1) gauge fields with corresponding field strength F Λ µν . The gauge couplings µ ΛΣ , ν ΛΣ and the moduli space metric g ab are uniquely determined by the N = 2 prepotential F .
We are interested in static, spherically symmetric configurations. The line element corresponding to the space time metric h µν in this case is given by (2. 2) The wrap factor U depends only on the radial coordinate r. For extremal black holes, the metric of the spacial section γ mn must be identity. The equations of motion for these configurations simplifies and the system can now be described in terms of an effective one dimensional theory with a potential which is extremized at the horizon.
For the N = 2 Lagrangian (2.1), the effective black hole potential takes the form [4]: Here W and K are respectively the superpotential and the Kähler potential. The superpotential W is related to the central charge by Z = e K/2 W . In terms of the dyonic charges (q Λ , p Λ ) and the prepotential F , the expression for W is given by The symplectic sections X Λ are related to the physical scalar fields by x a = X a /X 0 . The Kähler potential is given in terms of F by the relation: The covariant derivative is defined as ∇ a W = ∂ a W +∂ a KW . For supersymmetric attractors ∇ a W = 0. In general, the attractor points are determined by ∂ a V = 0.
Throughout this paper, we will focus on the N = 2 prepotential which is of the form The above prepotential appears as the leading term in the compactification of type IIA string theory on a Calabi-Yau manifold M in the large volume limit. In this case, D abc are the triple intersection numbers D abc = M α a ∧α b ∧α c , where the two forms α a form a basis of H 2 (M, Z). In this paper, we will use string theory terminologies to describe various charge configurations irrespective of whether the coefficients D abc are actually associated with a Calabi-Yau compactification or not.
In the following we will describe some of the well known supersymmetric attractor solutions. For this purpose we need explicit expressions for the Kähler and the superpotentials. The Kähler potential K corresponding to the N = 2 prepotential F has the following simple form (Now on we set the gauge X 0 = 1 without any loss of generality and express our formulae in terms of the physical scalar fields x a .) The superpotential depends on the specific charge configurations. In this paper we will mainly focus on D0 − D4 and D0 − D4 − D6 configurations. For the D0 − D4 configuration, the superpotential is given by whereas for the D0 − D4 − D6 configuration, we have These configurations possess well known supersymmetric attractor solutions [19]. For the D0 − D4 configuration, we have From here onwards we use the standard notations [20] real where as M ab and M are pure imaginary.) Setting the ansatz, x a = p a t, we find and hence, for the supersymmetric D0 − D4 configuration. The entropy of the corresponding super- The solution can be generalised in a straightforward manner upon adding D6 branes.
We find Setting the ansatz x a = p a t, we find the supersymmetric configuration corresponds to [19] (2.10) The entropy for this configuration is

The general solution
In this section, we will focus on the supersymmetric conditions more carefully and obtain the general solution without assuming any specific ansatz. We will first focus on the D0 − D4 configuration. Note that, in this case the superpotential contains only even powers of x a . Thus we can set the axionic parts of the scalar fields to zero: x a = ix a 2 . The supersymmetry condition now becomes Note that, for any configuration x a 2 satisfying the above equation, we have q 0 = − 1 4 M a p a . We can see this by multiplying by (x a −x a ) and simplifying the above equation. Thus, we can further simplify Eq.(3.1) by substituting 1 4 M a p a = −q 0 in it. We find Assuming the matrix M ab to be invertible, we can rewrite the above equation as This is a cubic equation in We will now generalise this result in the presence of D6 branes. Note that in the presence of D6 branes it is no longer possible to set the axionic parts of the scalar fields to zero. We denote x a = x a 1 +ix a 2 and express the real and imaginary parts the supersymmetric condition ∇ a W = 0 as For convenience we introduce ω a = p a − p 0 x a 1 . Expressing the above equations in terms of ω a and x a 2 , we find We would like to find the most general solution of the above equations for the variables ω a , x a 2 . We first rewrite these equations in a simpler form so that it will be easier for us to solve them. Consider first (3. (3.8) Using the above relation in (3.7) we obtain We can similarly simplify (3.6). Multiplication of (x a −x a ) on both sides of (3.6) provides Putting back (3.10) in (3.6) we find Introducing µ = ( 2iMaω a /M) the above equation can be rewritten as w a = µx a 2 . Substituting ω a = µx a 2 in (3.9) we get which implies Defining we can rewrite Eqs.(3.12) along with ω a = µx a 2 as (3.14) It can be shown that the matrix I a b is involutory: I a b I b c = δ a c and it satisfies the relation Using the explicit expressions for µ and after some simplifications, we can rewrite Eqs. (3.13) and (3.14) in terms of the variables x a 1 , x a 2 as This is the most general solution for the supersymmetry conditions (3.6) and (3.7). Any involution I a b satisfying the relation (3.15) will give us a new supersymmetric attractor. The standard solution (2.10) can be recovered by setting I a b = δ a b . We will have multiple attractors if there exists nontrivial involutions satisfying (3.15) and if the moduli space metric as well as the gauge kinetic terms remain positive definite at more than one attractor points for the same charge configuration.
For supersymmetric black holes the entropy is given by S = πe K 0 |W 0 | 2 , where K 0 and W 0 are the values of the Kähler and superpotential at the attractor point respectively.
Substituting the value of K 0 and W 0 in the expression for entropy, we find (3.18)

An explicit example
In the previous section we have derived the most general expression for the supersymmetric D0 − D4 − D6 attractors. They are given in terms of the involution I a b satisfying the constraint (3.15). In general it is not possible to solve (3.15) for arbitrary number of vector multiplets. Here we will consider the simplest case of a two-parameter model where this condition can be solved exactly to obtain new supersymmetric attractors.
It is straightforward to solve the above set of equations. For L 2 − 4MN > 0 they admit a solution of the form: Thus we obtain a new D0 − D4 − D6 supersymmetric attractor in the two parameter case in addition to the standard solution (2.10). Using the above solution for the involutory matrix I a b we can obtain explicit expressions for the vector multiplet moduli x 1 = x 1 1 + ix 1 2 and x 2 = x 2 1 + ix 2 2 (for easy reading we denote χ = D a I a b p b in the following): We will first consider the moduli space metric g ab = ∂ a ∂bK. From the expression for it Kähler potential (2.7) it is straightforward to find At the attractor point (2.10) it takes the form For the two parameter model it is straightforward to diagonalise both the metrics. The explicit expressions for the eigenvalues are lengthy and we will not reproduce them here.
For our purpose it will be sufficient to consider the determinant of the metric. From (4.5) we find There are n such terms and adding them all we get M det(M ab ). Thus, the determinant of the moduli space metric is found to be −3 n 2 (n−1) det M ab M . Substituting the explicit solutions, we find, for (2.10),  We can explicitly verify that the eigenvalues become positive in these respective regions of the charge lattice. We have numerically verified that the gauge kinetic terms can also simultaneously be made positive definite by suitable choice of charges.

Summary
In this paper we have studied the uniqueness of supersymmetric attractors in N = 2 supergravity theories in four dimensions arising from type IIA compactification on a Calabi-Yau manifold. We have proved the uniqueness for D0 − D4 attractors. We found that the supersymmetry conditions admit more general solutions if we include D6 charges in addition.
These solutions are determined by involutions which satisfies certain constraints. For the two parameter model we can explicitly solve the constraint to find two independent solutions for the attractor equation. However, they exist in mutually exclusive domains of the charge lattice. Hence, the attractors are unique in the respective domains.