N=2 Born-Infeld Attractors

We derive new types of $U(1)^n$ Born-Infeld actions based on N=2 special geometry in four dimensions. As in the single vector multiplet (n=1) case, the non--linear actions originate, in a particular limit, from quadratic expressions in the Maxwell fields. The dynamics is encoded in a set of coefficients $d_{ABC}$ related to the third derivative of the holomorphic prepotential and in an SU(2) triplet of N=2 Fayet-Iliopoulos charges, which must be suitably chosen to preserve a residual N=1 supersymmetry.


Introduction
The supersymmetric Born-Infeld (BI) Lagrangian [1], obtained in [2] and in a closed superspace form in [3], was shown in [4] to encode the dominant low-energy couplings of the goldstino sector in the presence of a 2 → 1 partial breaking of supersymmetry. The original Volkov-Akulov [5] action plays a similar role in the 1 → 0 case, so that this result can be also summarized by saying that in the 2 → 1 case the goldstino is accompanied by an N = 1 partner, the Abelian vector field strength (1.1) The supersymmetric BI action possesses a number of special features. Clearly, setting to zero the gaugino it reduces to the standard BI action for the vector field, while setting to zero the vector field it reduces to the standard Volkov-Akulov action. Moreover, it is invariant under a second non-linearly realized supersymmetry, whose transformations can be conveniently expressed in terms of W α and of the chiral superfield X, related to W α by the non-linear constraint [4] W 2 + X m − 1 4D Here m is a parameter with dimension of [mass] 2 and the additional supersymmetry transformations read Eq. (1.2) can be regarded as a non-linear nilpotency constraint for an N = 2 chiral superfield X [9], which can be built by combining a pair of N = 1 chiral superfields X and W α according to X (θ 1 , θ 2 ) = X(θ 1 ) − 2 θ α 2 W α (θ 1 ) − θ α 2 θ 2 α m −
Alternatively, in terms of X , the Lagrangian becomes the half-integral of an N = 2 Fayet-Iliopoulos term, projected on the SU(2) triplet half-chiral measure (see [7][8][9]). The authors of [9] also showed that the non-linear action (1.9) can be obtained starting from the quadratic N = 2 action considered by Antoniadis, Partouche and Taylor in [7]. In that paper, the superpotential and the N = 1 Fayet-Iliopoulos terms were chosen to give an N = 1 vacuum with broken N = 2 supersymmetry. A convenient way to obtain this result is via an electric charge (e 1 , e 2 , 0), aligned with the first two components of a triplet and a magnetic charge (m, 0, 0), aligned with the first component. In this fashion, the first supersymmetry is unbroken and the N = 1 Fayet-Iliopoulos terms vanish [9]. On the other hand, the partial breaking N = 2 → N = 1 is only possible if the N = 2 Fayet-Iliopoulos magnetic charge m does not vanish [13] [7,8].
Our goal here is to extend the construction to an arbitrary N = 2 special geometry with n vector multiplets, thus identifying the U(1) n generalization of eqs. (1.2) and (1.9).
The model is defined by magnetic and electric charges, m A and e A , which will be defined in the next sections, and by the superpotential 2 where C AB and d ABC are totally symmetric and real and M sets the scale of the problem.
For brevity, in the following we shall set M = 1, keeping in mind that the dimensionless charge triplets Q x = (m x A , e xA ) are meant to be accompanied by a factor M 2 in the final result.
We shall find it convenient to introduce shifted superfields Y A , (A = 1, . . . , n), defined by The vacuum expectation values (VEV)s x A = X A are determined by the N = 1 vacuum with As we shall see, the Y A satisfy the generalized BI constraints which involve the totally symmetric sets of coefficients d ABC and reduce to eq. (1.2) for n = 1, up to a slight change of conventions. As a result, the U(1) n generalized BI actions will depend on the choice of such symmetric tensors. We shall also examine in detail the available choices for the d ABC in the n = 2 case. Moreover, we shall see that the n-extended Lagrangians can be cast in the form 15) or alternatively, making use of the vacuum condition (1.12) and of the non-linear constraint (1.14), in the form Note that in the n = 1 case the second term in eq. (1.15) vanishes identically on account of the constraint (1.14). This reflects the fact the single C AB that is present in that case can be eliminated by a field redefinition. However, for n > 1 the C AB are needed, in general, to guarantee positivity, as is manifest from the alternative form of the Lagrangian in eq. (1.16).

Special Geometry, Fayet-Iliopoulos Terms and N = 1 Attractors
In this section we generalize the models of refs. [7] and [9] to the multi-field case. To this end, let us first observe that the data of the problem are the N = 2 Fayet-Iliopoulos terms, which build up an Sp(2n) symplectic triplet of electric and magnetic charges Q x = (m x A , e xA ), with x = 1, 2, 3, A = 1, .., n, and the prepotential of eq. (1.10).
Eq. (1.10) clearly identifies the d ABC as third derivatives of the prepotential U. Moreover, the N = 2 Lagrangian with an N = 2 Fayet-Iliopoulos term, written in N = 1 language, acquires a symplectic structure due to the underlying special geometry, which is encoded in the symplectic vector [14][15][16][17] The scalar-field dependent n × n symmetric matrices g AB and θ AB determine the quadratic terms in the vector fields as Moreover, in N = 2 special geometry and it is convenient to define the symplectic metric The 2n × 2n matrix M, with entries then satisfies the two conditions of being symplectic and positive definite: for a positive definite g, as required by the Lagrangian terms in eq. (2.2).
The contributions to the potential involve the triplets Q x = (m A x , e xA ) of electric and magnetic charges. The first two combine into the complex sets and determine the superpotential The last, is real and determines, in N = 1 language, magnetic and electric Fayet-Iliopoulos D-terms.
The potential of the theory can thus be expressed, in N = 1 language, as where Vacua preserving an N = 1 supersymmetry aligned with the N = 1 superspace [9] are determined by critical points of the potential 3 , and thus by the attractor equations which are in this case This equation can admit a solution for nonzero Q only if while the condition V D = 0 implies Q 3 = 0, since at the critical point M is positive definite.
In solving the attractor equations we shall take m A real and e A complex, so that eq. (2.15) will translate into the condition These are indeed attractor equations for the N = 2 theory quadratic in vector field strengths. It is interesting to stress the analogy with the attractor equations for N = 2 extremal black holes with symplectic vector Q = (m A , e A ). In terms of the M matrix the black hole potential [19][20][21][22][23], is also determined by the last expression in eq. (2.11), but for a real Q, so that the Ω term vanishes identically. However, in this case the value attained by V BH at the attractor point is positive and gives the Bekenstein-Hawking entropy On the other hand, when expressed in terms of the central charge Z, which is the counterpart of W, the black-hole potential contains an additional term [19][20][21][22][23], and reads Instead, in our case V crit = 0, which implies ∂ W ∂X A = 0 in order to leave N = 1 supersymmetry unbroken.

Born-Infeld Attractors
We can now exhibit a limit where the original theory quadratic in the field strengths gives rise to a generalized supersymmetric BI system, characterized by eqs. (1.14) and (1.15). In N = 1 language, the initial action reads where Therefore, the Euler-Lagrange equations for X A are The closure of the supersymmetry algebra fixes the parameter c = 4 b, and in the following we shall choose b = 1 4 . Note that only the magnetic charges, and not the electric ones, enter the supersymmetry transformations. The reason is that the contribution to the superpotential W containing the electric charge is linear in X A , and therefore is also invariant under the second supersymmetry [4]. Note also that the action (3.1) contains no other parameters.
The explicit form of the vacuum equations (2.13) and (2.14) is given in (3.10) and implies that the goldstino is therefore, the corresponding superfield takes the form Its non-linear variation under the second supersymmetry, making use of eqs. (2.15) and (3.4), reads so that, in units of M, the supersymmetry breaking scale is which is a symplectic invariant, as expected.
Because of the nilpotency constraints on X, some care will be needed to obtain the nonlinear actions of eqs. (1.15) and (1.16) from the spontaneously broken theory of n linear vector multiplets of Section 2. In particular, in order to satisfy the vacuum conditions (2.13) it is necessary to introduce VEVs X A = x A = 0. In fact, eq. (2.14) is (3.10) and implies the two real equations (3.11) A non-vanishing C AB is needed to restore positivity of the kinetic term when the matrix d ABC m C is not positive definite.
If we now define chiral superfields Y A with vanishing VEV, letting Only the last term depends on x A (and also on e A via the vacuum equations (3.10)).
The BI Lagrangians emerge in the limit in which U AB (x) is negligible with respect to the d ABC , where the equations of motion reduce to The last contribution contains only overall derivatives, and therefore can be neglected in the IR limit where our effective actions will be well defined. One can then insert the ansatz in (3.13), solve the resulting equation and check the self-consistency of the solution. This leads to the multi-field generalization of the BI constraint of [4,9], which was already presented in eq. (1.14) in the Introduction. Taking into account that the equations of motion are solved by D A = 0, the θ 2 component of (3.15) reads where G + is the self-dual vector field strength and, here and in the next section, "dots" indicate full Lorentz contractions 5 5 The superfield expansion corresponding to our definition of These complex algebraic equations determine the auxiliary fields F A as non-linear functions of G + · G + and G − · G − , and are the seed of the generalized BI non-linear Lagrangians.
For n = 1 the Lagrangian corresponding to eq. (1.15) reduce to the form (3.17) A simple way to verify that a field solving eq. (3.15) does indeed satisfy the ansatz (3.14) is to notice that the lowest component of (3.15) is by λ A and using the Fierz identity λ (A λ B λ C) = 0 implies that with y A = Y A | θ=0 . Since the factor within parentheses is arbitrary, this condition requires and multiplying eq. (3.18) by y A and using (3.20) one then finds Therefore, eq. (3.14) holds at θ = 0, and N = 1 supersymmetry then implies that the entire multiplets vanishes. We have thus shown that the Y A obey the nilpotency equations which is the multi-field generalization of eq. (1.7) of the Introduction.
The Lagrangian corresponding to eqs. (3.12) is Before concluding this section, we would like to comment on two aspects of multi-field BI actions. First of all, let us emphasize some analogies and some differences with the multifield case considered in [9,[24][25][26]. In those papers, the chiral superfield X is a matrix, while in our case it is a vector. Moreover, their constraints are stronger. In fact, a U(n) e.m.
duality is imposed, while in our case we generally expect only a U(1) n duality even if the vectors are coupled.
Finally, we notice that our U(1) n construction is not a mere complexification of the construction in [4], since for one matter it also applies for odd values of n. Moreover the terms containing C AB are crucial, in general, to grant positivity. This will be manifest in the simple examples that we are about to discuss, one of which could be related to the complexified nilpotency constraints (X ± iY ) 2 = 0 in superspace. However, the corresponding action would contain ghosts unless a quadratic term involving C AB were added to the prepotential, and this term necessarily breaks the complex structure. Therefore, even in that particular case the model is different from the U(1) 2n generalizations proposed in [24].

Explicit Examples: the n = 2 Case
The generalized BI Lagrangians are determined by the superfield constraints To find them explicitly one needs only the F-term equations (3.16), since the D A -terms vanish. Since eq. (4.1) is clearly solved by F A = 0 when G A + = 0, it is useful to perform the change of variables thus turning imaginary and real parts of eq. (4.1) into Any specific class of models solving these constraints is defined by the U polynomial modulo field redefinitions by Sl(n, R) transformations 6 . Inequivalent theories are thus classified by the Sl(n, R) orbits of the cubic polynomials As a first nontrivial example, let us consider the n = 2 case, where the d ABC , with A, B, C = 1, 2, take values in the spin-3 2 representation of Sl(2, R). This possesses a unique quartic invariant, which also corresponds to the discriminant of the cubic. The quartic invariant is and is a truncation of Cayley's hyperdeterminant, an object that also emerges from studies of black-hole entropies [27,28] and of q-bit entanglement in Quantum Information Theory [28,29]. Different types of roots are associated to different properties of its four orbits: For The four inequivalent theories can be associated to the four representative polynomials determined by the conditions which read Hence, aside from the O s case a C AB term is needed in the generalized BI Lagrangians.
We can now consider the solutions of the constraints given in eqs. where (4.14) On the other hand, eqs. (4.4) become where and (4.17) In terms of these quantities, the explicit solutions for H X and H Y read (4.18)