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Two-Center Black Holes, Qubits and Elliptic Curves

We relate the U-duality invariants characterizing two-center extremal black hole solutions in the stu, st^2 and t^3 models of N=2, d=4 supergravity to the basic invariants used to characterize entanglement classes of four-qubit systems. For the elementary example of a D0D4-D2D6 composite in the t^3 model we illustrate how these entanglement invariants are related to some of the physical properties of the two-center solution. Next we show that it is possible to associate elliptic curves to charge configurations of two-center composites. The hyperdeterminant of the hypercube, a four-qubit polynomial invariant of order 24 with 2894276 terms, is featuring the j invariant of the elliptic curve. We present some evidence that this quantity and its straightforward generalization should play an important role in the physics of two-center solutions.


I. INTRODUCTION
The aim of the present paper is to show that it is possible to relate the entanglement measures usually used in studies concerning four-qubit systems to the U-duality invariants found recently by Ferrara et. al. 1 characterizing extremal two-center black hole solutions in the stu, st 2 and t 3 models. Interestingly as a byproduct of establishing this correspondence one can also come across an interesting connection between charge configurations of such black holes and a special class of elliptic curves.
Multi center black hole solutions provide an interesting research direction within the rapidly evolving field of black hole solutions in supergravity, string and M-theory. For such solutions the attractor mechanism 2 has been generalized giving rise to split attractor flows 3,4 . For two-center solutions the latter term refers to the situation, when in moduli space after crossing walls of marginal stability the attractor flows are separately evolving to the attractor points of the constituent single center solutions. Recently these developments have triggered activity in a variety of new research fields such as attractor flow trees, entropy enigmas, microstate counting, and bound state recombination [5][6][7][8][9][10] .
For single center solutions it has become obvious that the notion of duality charge orbits and their invariants 11 are useful concepts for classifying black hole solutions together with their supersymmetry properties. Except for a special case 12 for multicenter solutions the corresponding structure of orbits and invariants is still unknown. In order the remedy this situation in a recent paper Ferrara et.al. conducted 1,13 a systematic study on the structure of invariants characterizing the charge configurations and invariants of two center solutions.
In the case of the stu, st 2 and t 3 models looking at the structure of such invariants one immediately notices structural similarities to the well known sets of four-qubit invariants 16,17 discussed in the seemingly unrelated field of quantum information.
Since the advent of the black hole qubit correspondence 18 such coincidences should not come as a surprise. It is well-known by now that few qubit entangled systems are capable of providing interesting new insight into the structure of black hole solutions and their attractor flows 19 . The occurrence of these qubits is related to the presence of tensor products of the spin 1 2 irreducible representations of SL(2) groups. Such products of SL(2)s show up as subgroups of U-duality groups governing the entangled web of dualities of supergravity models giving rise to black hole solutions. Initially mathematical coincidences were recorded merely for two and three qubit systems and the corresponding axion-dilaton and stu black holes 18 , however evidence for n-qubit systems with n > 3 to make their presence in this context started to accumulate.
In this line of development the relevance of four qubit systems to black hole solutions in supergravity was first pointed out in our paper 20 where the isomorphism so(4, 4) ≃ sl(2) 4 ⊕ (2, 2, 2, 2) (1) has been used to describe 7 × 16 of the 133 E 7 (7) generators of N = 8, d = 4 supergravity.
These generators describing seven copies of four-qubit states are not belonging to the sl(2) 7 subalgebra of E 7 (7) . A suitable incidence geometry accounting for the relationship between the 7 groups of 16 generators is that of the dual Fano plane giving rise to a geometry, dual to the one describing the "tripartite entanglement of seven qubits" interpretation 20,21 of the quartic E 7 (7) black hole entropy formula. The isomorphism of Eq.(1) was also discussed in the review paper of Borsten et. al. 22 providing further interesting examples of simple qubit systems.
In our next paper 23 in a four-qubit entanglement based formalism the structure of extremal stationary spherically symmetric black-hole solutions in the STU model of N = 2, d = 4 supergravity was described. The basic idea facilitating this interpretation was the fact that stationary solutions in d = 4 supergravity can be described by dimensional reduction along the time direction 24 . In this d = 3 picture the global symmetry group SL(2, R) 3 of the STU model is extended by the Ehlers SL(2, R) accounting for the fourth qubit. One can then introduce a four-qubit state depending on the charges, the moduli, and the warp factor. Here it was also noticed that in the terminology of four-qubit entanglement extremal black-hole solutions should correspond to nilpotent, and nonextremal ones to semisimple states. The upshot of these considerations was the emerging possibility of relating the entanglement properties of such and similar states to different classes of black-hole solutions in the STU model. The challenge of elaborating on this idea was recently taken up in the papers of Borsten et.al. 25 In these papers the authors applied the black-hole qubit correspondence to the problem of classifying four-qubit entanglement. The key technical ingredient was the Kostant-Sekiguchi theorem which establishes the link between nilpotent orbits of extremal black holes and four qubit entanglement types. The emerging picture is: we have 31 entanglement families which reduce to nine up to permutations of the qubits. These nice papers confirmed once again that the input coming from string theory can be useful in establishing results in a different field, since the literature until now on four qubit entanglement classification was confusing and seemingly contradictory.
In this paper we would like to show that the charge orbit classification of two-center black hole solutions in the stu model is another arena where four-qubit systems naturally make their appearance. As a first possible step in this direction here we establish a correspondence between the U-duality invariants of Ferrara et.al. 1  It is the SL(2) ×4 and permutation invariant hyperdeterminant of type 2 × 2 × 2 × 2. This is a polynomial of order 24 in the 16 amplitudes of the four-qubit state. Mapping the 16 amplitudes to the 16 charges characterizing two-center solutions in the stu model, for a special case we show that the structure of this hyperdeterminant seem to govern issues of consistency in the realm of two-center solutions. These ideas also suggest a natural way for associating an elliptic curve of a special kind to a particular charge configuration.
The coefficients of our elliptic curve are the algebraically independent four-qubit invariants, and its discriminant is just the hyperdeterminant. We also present some evidence for the conjecture that the structure of the j invariant of the elliptic curve should play an important role in the physical properties of the two-center solution. The idea that elliptic functions and the j invariant might play some role in four-qubit systems and the black hole qubit correspondence was first suggested by P. Gibbs 26 some related discussion appeared in the paper of Bellucci et.al. 27 .
The organization of this paper is as follows. In Section II. we summarize the background material on four-qubit invariants, reduced density matrices, and the structure of the hyperdeterminant of the hypercube. We introduce a quartic polynomial featuring the algebraically independent four-qubit SL(2, C) invariants. In Section III. we are discussing extremal two center black hole charge states in a four-qubit based picture. Here we work out a dictionary between the invariants found by Ferrara et. al. 1 in the so called Calabi-Visentini basis and the algebraically independent four-qubit ones in the "special coordinates" basis. We give some of the invariants used in Ref. 1  In Section V. we examine the status of our rather ad hoc assignment: two-center charge configuration-elliptic curve, more thoroughly. By switching to the most general Tate form of an elliptic curve we show that our association of elliptic curves to two-center black hole charge configurations in the stu model is a natural one. This means that the nonzero coefficients a j with j = 1, 2, 3, 4 appearing in the Tate form are algebraically independent four-qubit invariant homogeneous polynomials of order 2j. These coefficients have important physical meaning: a 1 is just the canonical symplectic pairing between the charge vectors of the two centers. Vanishing of a 4 gives rise to the st 2 , and a further vanishing of a 2 results in the t 3 truncation. One can also see that in the st 2 and t 3 models sending a 3 to zero corresponds to the limit when our elliptic curve degenerates. One of the nontrivial coefficients a 6 in the Tate form is always zero for the stu model. This is related to the vanishing of a nontrivial polynomial constraint of homogeneous degree 12 valid in the stu model, already observed by Ferrara et.al. 1 Based on the latest results of Andrianopoli et. al. 13 we conjecture that we should be able to generalize our correspondence between charge orbits and elliptic curves also for the case of maximal N = 8, d = 4 supergravity. In this case the stu model should arise as an a 6 = 0 truncation implemented by the vanishing of a polynomial of order 12.
The aim of our last speculative Section VI. is to draw the readers attention to some interesting structural similarities showing up in a variety of physical contexts where our four-qubit invariants parametrizing elliptic curves might play a crucial role. Here we give a new look and interpretation to a triality invariant curve originally introduced by Seiberg and Witten 14 . Now this curve is paramaterized by four-qubit invariants also displaying permutation invariance. In this new setting we also invoke the F-theory interpretation of this curve as was given by Sen 15 . Finally our conclusions and some comments are left for Section VI.

II. FOUR QUBIT SYSTEMS
In order to facilitate a four-qubit description of the two center charge configurations in the stu model our aim in this subsection is to review the background material on four-qubit states and their entanglement measures. A four qubit state can be written in the form where V 0,1,2,3 ≡ C 2 . Let the subgroup of stochastic local operations and classical communication 28 representing admissible fourpartite manipulations on the qubits be just Our aim is to give a unified description of four-qubit states taken together with their SLOCC transformations and their associated invariants. As we will see states and transformations taken together can be described in a unified manner using the group SO(8, C).
Let us discuss the structure of four-qubit SL(2, C) ×4 invariants 16,17,31,32 . The number of algebraically independent four-qubit invariants is four. We have one quadratic, two quartic, and one sextic invariant. In our recent paper 17 we investigated the structure of these invariants in the special frame where two of our qubits played a distinguished role. As we will see this scenario is just the one needed in the two-center STU black hole context since in this setting one of the special qubits (the one labelled by the number 0) will be associated to the horizontal SL h (2, R) of Ferrara et.al. 1 and the other (the one labelled by the number 1) is arising as the first factor from the structure SL(2, R)×SO(2, 2) known from the STU model.
Indeed such structure is the one arising as a special case of the infinite Jordan symmetric sequence of N = 2 d = 4 supergravity theories 29 .
To an arbitrary state |Λ we can also associate a 4 × 4 matrix or four four-vectors. The splitting of the amplitudes of |Λ into such four-vectors reflects our special choice for the distinguished qubits compatible with our conventions. We will also need the matrices Notice that in four-qubit notation the matrices M and N are arising from L by the permutations (012)(3) and (0)(123) meaning that the index structure of these matrices is Notice that these matrices are entering in the reduced density matrices as where overline denotes complex conjugation.
Now we introduce on the vector space C 4 ≃ C 2 × C 2 corresponding to the third and fourth qubit a symmetric bilinear form g : This means that we have an SL(2, C) ×2 invariant quantity with the explicit form We can also introduce a dual four-qubit state with the associated matrix λ 0000 λ 0001 λ 0010 λ 0011 λ 0100 λ 0101 λ 0110 λ 0111 where Here ǫ 1234 = +1, and indices are lowered by the matrix of g. Notice that the amplitudes of the dual four-qubit state are cubic in the original ones.
Using these definitions we define the quadratic and sextic invariants as The explicit form of the sextic invariant in terms of the dot product of Eq. (10) is We also recall that the explicit form of I 1 is hiding its permutation invariance. Moreover, though the expression of I 3 of Eq. (14) is similar to the one of I 1 the invariant I 3 is not invariant under the permutation of the qubits.
Now we turn to the structure of quartic invariants. We have two independent of such invariants 16 and the simplest of them is the obvious expression i.e. the determinant of the 4 × 4 matrix of Eq. (4). In order to present the definition of the second one we define separable bivectors of the form Here our labelling convention Λ µα indicates that µ = 1, 2, 3, 4 identifies the four-vector in question (i.e. A, B, C or D of Eq.(4)), and the label α = 1, 2, 3, 4 refers to the component of the particular vector. Now our last invariant is the quartic combination Obviously the symmetric nondegenerate bilinear form g of Eq.(10) acting on four vectors like A, B, C, D ∈ C 4 is inducing a corresponding symmetric nondegenerate bilinear form on the space of bivectors 2 C 4 . By an abuse of notation we use again the symbol · for this new bilinear form with the definition 17 Now I 2 can also be written in the equivalent form An important comment here is in order. Let us have a look at I 4 and also at the determinants of the matrices of Eq.(5) Then one can prove 16,30 L + M + N = 0.
One also has the constraint that we will need later.
It is known 16 that the minimal set of algebraically independent SL(2) ×4 invariants is consisting of a quadratic, two quartic and one sextic invariant. Our choice for this set will be 17 : I 1 , I 2 , I 4 and I 3 . Let us now present the reason for this choice. Let us consider the matrix Ω ≡ LgL T g.
Then its characteristic polynomial is Clearly by Newton's identities we have TrΩ, This form of writing our invariants is related to the fact that there is a 1 − 1 correspondence between the SL(2, C) ⊗4 orbits of four-qubit states and the SO(4, C) × SO(4, C) ones of 4 × 4 matrices.
The polynomial of Eq. (25) in the four-qubit context appeared in our recent paper 17 its role as a characteristic polynomial has been emphasized in Ref. 31 . The discriminant of this fourth order polynomial is the hyperdeterminant 37 It is a polynomial of degree 24 in the 16 amplitudes and has 2894276 terms 33 . D 4 can be expressed 17 in terms of our fundamental invariants as where S = (I 2 4 − I 2 2 ) + 4(I 2 For an alternative form of D 4 see the papers of Refs. 16,32 . In closing this section we briefly discuss some results on the classification of entanglement classes for four qubits 31,34 . By entanglement classes we mean orbits under SL(2, C) ×4 · Sym 4 where Sym 4 is the symmetric group on four symbols. The basic result states that four qubits can be entangled in nine different ways 31,34 . It is to be contrasted with the two entanglement classes 28 obtained for three qubits. For a refined classification of four qubit entanglement motivated by the black hole qubit correspondence see the papers of Borsten et.al. 25 Let us consider the matrix which now can be regarded as an element of the Lie algebra of SO(8, C). If the matrix R Λ is diagonalizable under the action we say that the corresponding four-qubit state |Λ is semisimple. If R Λ is nilpotent then we call the corresponding state |Λ nilpotent too. It is known that a nilpotent orbit is conical i.e. if |Λ is an element of the orbit then λ|Λ is also an element for all nonzero complex numbers λ. Hence a nilpotent orbit is also a GL(2, C) ×4 orbit. It is clear that for nilpotent states all of our algebraically independent invariants are zero.
A semisimple state of four qubits can always be transformed to the form 34 where a, b, c, d are complex numbers. This class corresponds to the so called GHZ class found in the three-qubit case 28 . For this state the reduced density matrices obtained by tracing out all but one of the qubits are proportional to the identity. This is the state with maximal four-partite entanglement. Another interesting property of this state is that it does not contain true three-partite entanglement. A straightforward calculation shows that the values of our invariants (I 1 , I 2 , I 3 , I 4 ) occurring for the state |G abcd representing the generic class are hence the values of the invariants (4I 1 , 6I 2 , 4I 3 , I 2 4 ) are given in terms of the elementary symmetric polynomials in the variables (t 1 , t 2 , t 3 , t 4 ) = (a 2 , b 2 , c 2 , d 2 ). For the semisimple states |G abcd the value of D 4 can be expressed as 16,17 Notice that for the states |G abcd with D 4 nonvanishing (t i = t j ) the corresponding matrix of Eq. (30) belongs to a Cartan subalgebra of SO(8, C). The stabilizer of such states corresponds to the Weyl group of SO(8, C). This stabilizer is the Klein group generated by the four elements I ⊗ I ⊗ I ⊗ I and σ a ⊗ σ a ⊗ σ a ⊗ σ a for a = 1, 2, 3.

III. TWO CENTER EXTREMAL BLACK HOLES AS FOUR QUBITS SYSTEMS.
In the paper 1 of Ferrara et.al. in order to describe the structure of the U-duality invariant polynomials associated to the two-center extremal black hole solutions the Calabi-Visentini (CV) basis has been used. Here by U-duality we mean the continuous limit 1 valid for large values of the charges of the usual-nonperturbative string theory symmetries. First we describe the connection of the CV basis to the one making the four-qubit structures explicit.
Next we turn to an entanglement based understanding of the structure of the two-center U-duality invariants.
In the CV basis the two-center black hole solutions are characterized by a pair of real As we see these charge vectors are containing two four-vectors each namely P I , Q I ≡ η IJ Q J , and p I , q I ≡ η IJ q J I, J = 0, 1, 2, 3 where the raising and lowering of the indices I and J are effected by the metric η IJ and η IJ of SO(2, 2) answering the symmetric bilinear form h acting on the charge four-vectors P I and Q J in the CV basis as Let us now relate the 16 component charge vector in the CV basis characterizing a particular two center extremal black hole solution in the STU model to a real unnormalized four-qubit pure state |Λ by relating the four-vectors P I , Q I , p I , q I ∈ R 4 of Eq.(36) and A α , B α , C α , D α ∈ R 4 of Eq.(4) as follows.
Then using the definitions above we clearly have for example with the bilinear form g defined as in Eq.(10).
Let us also give the connection between the Calabi-Visentini basis and the one usually used in special geometry, i.e. the special coordinates (SC) symplectic frame. This frame yields the usual set of electric and magnetic charges i.e. (P I , Q I ) SC and (p I , q I ) SC . In the following we will use these charges so it is important to clarify their relationship to the components of our four qubit state |Λ (see Eq. (4)).
We note here however, that our conventions are slightly different from the ones used in Ref. Notice also that in our four-qubit state |Λ with amplitudes Λ i 0 i 1 i 2 i 3 sit two three-qubit states with amplitudes Λ 0i 1 i 2 i 3 , and Λ 1i 1 i 2 i 3 . The first set of amplitudes labelled by i 0 = 0 corresponds to the charge configuration of the first black hole and the second labelled by Let us now define three bivectors In component notation we have for example Switching to the Calabi-Visentini basis these objects are the T -tensors used in the paper of Ferrara et.al. 1 As we can see the bivectors X and Y are separable i.e. they are precisely the ones satisfying the Plücker relations on the other hand the bivector Z is entangled i.e. in the nomenclature of fermionic entanglement 40 it has Slater rank two.
Now we introduce the shorthand notation for the product of two separable bivectors as Notice that in the notation of Eq.(41)-(42) are just two times the quartic invariants of the charges characterizing the two black holes Here With these definitions we can define the quantities (49) and the ones When reinterpreted in the CV basis these are precisely the SL(2, R) 1 ×SL(2, R) 2 ×SL(2, R) 3 invariants of Ferrara et.al. 1 The important property of the invariants of Eq.(49) is that they are covariants with respect to SL(2, R) 0 acting on the distinguished (horizontal) qubit.
Indeed, they are sitting in the 5 (spin 2) irreducible representation of this group.

Now we elucidate another aspect of this important property of the set of invariants of
Eq. (49). In order to do this we first recall that for two-qubits the canonical measure of pure state entanglement is the concurrence 35 which is related to the determinant of an ordinary 2 × 2 matrix. For three-qubits the basic quantity characterizing genuine three-qubit entanglement 28 is the three-tangle where now D 3 is Cayley's hyperdeterminant 35,36 . According to the method of Schläfli 37 D 3 is related to the discriminant ∆ 2 of the quadratic polynomial where Λ 0 and Λ 1 are four-vectors with components (Λ 000 , Λ 001 , Λ 010 , Λ 011 ) and (Λ 100 , Λ 101 , Λ 110 , Λ 111 ) and the · product is the usual one of Eq.(10). Moreover, due to permutation invariance of D 3 we obtain the same expression whenever the first or the second qubit plays a distinguished role. Obviously the quantities invariants, however they are covariants under the "horizontal" SL(2) 0 . Indeed, the triple J +1 , J 0 , J −1 transforms according to the irreducible representation 3 of spin 1 of this "horizontal" group which can be regarded as some sort of generalized exchange symmetry working between the two two-qubit systems. It is also clear that searching singlets with respect to this "horizontal" symmetry group can reveal some new properties of our pair of two qubit systems, namely that they are secretly comprising a system having a higher degree of symmetry.
Now proceeding by analogy we define the polynomial Notice that Σ 4 is a polynomial with its coefficients I 1 , I 2 , I 3 , I 4 also being SL(2) 0 singlets however, Π 4 is a polynomial with coefficients I ±2 , I ±1 , I 0 being merely SL(2) 0 covariants.
Using this observation we can construct new SL(2) 0 singlets from the quantities of Eq. (49) by relating them to the known algebraically independent four-qubit ones namely I 1 , I 2 , I 3 and I 4 .
In order to relate the SL(2) 0 singlets found by Ferrara et.al. to our four algebraically independent four-qubit invariants one just has to compare the relevant expressions. In fact many of their invariants and the constraints satisfied by them take in this four-qubit setting a much simpler and instructive form. In particular their complete set of invariants with corresponding degrees 2, 4, 6, 8 denoted by W, χ, I 6 , Tr(J 2 ) is related to ours as By virtue of Eqs. (23) and (50) we also have the relations We Using the dictionary of Eq. (55) we can cast this constraint in the nice form There is one more invariant 1 of order 8 which is directly related to Tr(J 2 ) Recall that the constraint P 8 = 0 implements the reduction of the stu model to the st 2 model 1 in a manifestly SL(2) 0 invariant manner. Using again Eq.(55) the new form of P 8 is Putting this into Eq.(58) and recalling Eq.(29) one obtains the simple expressions provided The first result of these considerations is that − 1 12 P 8 is really the square of the basic fourth order invariant L = I 4 . According to Eqs. (4) and (21) L is just the determinant of the matrix L we have started our four-qubit considerations with. Moreover, according to Eqs. (7)(8) we also see that for unnormalized four-qubit states we have hence this invariant is related to the determinant of the reduced density matrices of our four-qubit state |Λ corresponding to the bipartite split of the form: (01) (23). We can also conclude that in the four-qubit picture the reduction of the stu model to the st 2 is effected by sending one of the eigenvalues of the reduced density matrices corresponding to the (01)(23) split to zero. Notice however, that the remaining density matrices corresponding to the remaining two splits (02)(13) and (03)(12) are generally not sharing this property. This means that for the st 2 model we have Recall now that a suitable further reduction to the t 3 model is obtained by employing the following two SL(2) 0 invariant constraints 1 Moreover, since the discriminant of a quartic is the same as the discriminant of its resolvent cubic one can also show that the two quartic equations Σ 4 = 0 and Π 4 = 0 of Eqs. (25) and (54) are having the same resolvent cubics. Indeed, a straightforward calculation shows that the corresponding resolvent qubics in both cases are of the form In the first case we get back to the known expressions for S and T of Eq. (29), and in the second one we get 1 In order to uncover the role of our four-qubit invariants playing in the theory of two center black hole solutions let us consider a special class of two center black hole solutions in the t 3 model featuring a D0D4 − D2D6 split in the type IIA duality frame. In this case the vectors of Eq. (41)- (42) in the special coordinate (SC) basis are corresponding to the first black hole with charge configuration Q 1 of a BPS D0D4 system and  corresponding to the second black hole with charge configuration Q 2 of a BPS D2D6 system.
For BPS configurations in the first case we should have and in the second the corresponding constraint is Notice that our charge split for the special values of U = 4, P = q = 1 and v = −4 incorporates the illustrative example of Bates and Denef 4 (in that paper v is related to ours via a sign flip). The four-vectors A α , B β , C γ and D δ are comprising the 16 amplitudes of a real four-qubit state as displayed in Eqs. (2) and (4). Now using Eqs. (14)- (15), (16) and (20) the algebraically independent four-qubit invariants I 1 , I 2 , I 3 and I 4 can be calculated.
The explicit forms of these invariants are Notice that by virtue of Eqs.
The discriminant of this cubic equation is According to Eqs. (28)-(29) the hyperdeterminant D 4 is also related to this discriminant and is of the form

B. Consistency condition
Let us now consider the necessary condition 3,4 for our two center charge configuration supporting a corresponding two center stationary extremal BPS black hole solution. As it is well-known this condition is of the form where x 1,2 are the locations of the centers, r = |x|, Z 1,2 are the central charges corresponding to the charges Q 1,2 , and the symplectic product of the charge vectors Q 1 , Q 2 is related to our quadratic four-qubit invariant as The explicit forms of the central charges for our centers are Here τ is as usual the complex scalar field of the t 3 model with negative imaginary part 42,45 , and K = − log(8y 3 ) is the Kähler potential. Now the constraint dictated by Eq. (77) is Explicitly we have We are interested in the structure of BPS D0D4 − D2D6 composites hence according to For all possible sign combinations satisfying these constraints we have Let us now write our consistency condition as Now a calculation shows that by virtue of Eq.(84) consistency demands that Let us now look at the expression for our hyperdeterminant of format 2 × 2 × 2 × 2 as Unfortunately the second of our conditions is featuring −P v which is not coming from any of our four-qubit invariants.
In order to eliminate this shortcoming let us now take another look at the form of our polynomial of Eq. (25). By employing the substitution this polynomial can be transformed to the reduced form and S is our well-known quantity defined by Eq. (29) yielding the st 2 model and the further specialization M = 0 results in and the definitions this cubic equation and its discriminant takes the form where for the explicit form of ∆ see Eq.(75). According to Eq.(76) ∆ < 0, this yields for Cardano's formula the case of "casus irreducibilis" 47 with explicit solutions s j+1 , j = 0, 1, 2.
Transforming back these solutions by using Eq.(93) to the variables t j+1 we obtain the final where Note that for BPS solutions we have ef < 0. Notice also that for our special case 2I 1 = Q 1 , Q 2 = 3(e − f ) cannot be zero for BPS splits so this charge configuration is mutually nonlocal.
As our first example let us consider the nontrivial cases f = 0, and e = 0 when D 4 = 0.
The first case is characterized by the constraint e + 3f = 0. In this case I 3 = 0 and from Eq.(99) we get Φ = π hence Using now Eq.(35) up to permutations one can associate a canonical form to this configuration as shown in Eq. (32). This situation arises for example when U = −v = 3 and P = q = 1.
Notice that this is a highly degenerate case since now D 3 (Q) = 0, and W = 3(1 − τ )(1 + τ 2 ), hence Z = 0 for τ = ±1 which is on the boundary of the lower half plane. One can also see that for e + 3f = the superpotential W can be written in a factorized form.
Our second example is associated with the case when f + 3e = 0 producing the other nontrivial zero for D 4 . In this case I 3 = 0, and Φ = 0 hence we have From Eq.(32) one can see that the canonical state is proportional to the one |0101 + |1010 .
Our last example is a one with Φ = π 2 . In this case e(e + 3f ) 2 + f (f + 3e) 2 = 0 which is of the form It is easy to check that λ = −1/3 i.e. e+f = 0 is a solution, hence (λ+1/3)(λ 2 +14λ/3+1/9) is a factorized form of our polynomial, yielding the solutions Let us consider the rational solution λ = − 1 3 . In this case we get Now the real roots are all different hence D 4 = 0. However, now Uv = −3P q hence (Uv + P q)(Uv + 9P q) < 0. A consequence of this is that the consistency condition of Eq.(86) cannot be satisfied so no charge configuration of this kind supports a two-center solution. The canonical form can again be read off from Eqs.(32)- (35). Notice also that for all three cases t 1 + t 2 + t 3 + t 4 = 4I 1 as it has to be, moreover the corresponding values for λ can be written in the form λ k = −1/3 k for k = 0, 1, 2.
It is interesting to realize that all these special charge configurations giving rise to special four-qubit canonical forms are outside the domain of legitimite two-center solutions. Later when we connect the special values of e and f to properties of the j function we have something more to say about this phenomenon. In order to get charge configurations supporting BPS two center solutions we have to chose the asymptotic moduli from the eligible region boundend by the usual wall of marginal stability 4 . In our case this wall is given by the locus

D. Splitting of invariants
Notice that the quartic invariant for the t 3 model has the explicit form Now I +2 = 4UP 3 and I −1 = −4vq 3 hence according to Eq.(54) we can write From this expression we see that the relationship between the quantities −D 3 (Q 1 + Q 2 ), −D 3 (Q 1,2 ) is governed by the combination 4I +1 + 6I 0 + 4I −1 of SL(2) 0 covariants. Such relationships are needed for studying situations when the two-center solution is BPS, but the corresponding single center one is not.
For D0D4 − D2D6 splits the quantity above can also be written in two equivalent forms featuring quantities related to factors of SL(2) 0 invariants Hence if both of the constituents are BPS i.e. −D 3 (Q 1 ) > 0 and −D 3 (Q 2 ) > 0, the conditions for the single centered system to be BPS too i.e. −D 3 (Q 1 + Q 2 ) > 0 are again governed by a four qubit invariant I 1 or by the factors of another four qubit one namely D 4 .
It is amusing to realize that all of the factors of D 4 are appearing in these two equations.
As an example one can see that if I 1 < 0 and D > 0 and moreover Uv + P q > 0 (which is a factor of D 4 ), then two BPS configurations again yield a BPS one. In the example of As a next step to this resolvent cubic we associate an elliptic curve of Weierstrass canonical form as The j invariant of this curve is defined as In particular for the special case of the t 3 model we get After recalling the definitions of Eq.(99) a further specification to our case of the BPS D0D4 − D2D6 split one obtains where Interestingly the number ξ 1 can also be written in the form moreover, one can check that ξ 1 + ξ 2 + ξ 3 = −15, ξ 1 ξ 2 + ξ 2 ξ 3 + ξ 1 ξ 3 = 75 and ξ 1 ξ 2 ξ 3 = 3.
Finally one obtains for the j invariant the expression (We will have something more to say about this embedding later.) Hence we can use the algebraically closed field of complex numbers to put our considerations in a more general setting.
Let us now notice that according to Luque and Thibon 16 we can write our key quantity 256D 4 = S 3 − 27T 2 in yet another form by rewriting the polynomial invariants S and T as where The algebraically independent invariants 16 It is known that a nonsingular projective curve of genus 1 is isomorphic to a plane cubic curve of the Tate form 51 Here the coefficients a 1 , a 2 , a 3 , a 4 , a 6 can be taken from an algebraically closed field F of arbitrary characteristic and (x 0 , x 1 , x 2 ) are homogeneous coordinates of the projective plane FP 2 . (Of course for qubits we are merely interested in the complex case i.e. our concern will be curves in CP 2 .) Using inhomogeneous coordinates (x, y) ≡ (x 1 /x 0 , x 2 /x 0 ) this can be rewritten as Now the j invariant and discriminant ∆ of a plane cubic curve of this form is defined 51 as where A necessary and sufficient condition for our curve to be nonsingular is ∆ = 0. Now looking at Eqs.(119)-(120) one can easily check that four-qubit states provide a particularly nice parametrization for a special class of plane cubic curves with a 6 ≡ 0.
Indeed setting yields Moreover, one can also see that For all of these curves the hyperdeterminant is given by the usual formula 256D 4 = S 3 − 27T 2 with the j invariant having the form of Eq.(111). As an example let us take a look at the hyperdeterminant D 4 for the st 2 model which is given by the expression One can see that in accordance with our association of cubic curves to the stu model and In the following as usual by an elliptic curve we will mean a nonsingular plane cubic curve as given by Eq.(122) and the "point at infinity" (x 0 , x 1 , x 2 ) = (0, 0, 1). Since for these curves we have only (0, 0, 1) as a point at infinity by an abuse of notation one can context. In the stu case it is easy to establish a correspondence between this formalism and the one where complex three and four-qubit states appear 53 . We conjecture that such moduli dependent complex states and their associated invariants featuring elliptic curves could be the relevant mathematical objects hiding behind the considerations of this paper based merely on charge states. Note that moduli dependent curves of that kind would also display explicit dependence on extra complex parameters giving rise to elliptic fibrations.
We will have something more to say about this possibility in Section V. Indeed, let us use the parametrization as introduced by Sen 8 for such two-center compos- Here P and Q are four-vectors related to the ones of Eq.(38)-(42) as follows  where we introduced the notation Now the elliptic curve of Eq.(130) is of the form Completing the square on the left hand side (and by an abuse of notation using y for the new variable again) yields Notice that the quantitiesP andQ hence the Tate form of the cubic curve is featuring the parameters a, b, c, d characterizing the two-center split, however the latter form is not.
This is due to the fact that D 3 (Q) = (P · Q) 2 − P 2 Q 2 = (P ·Q) 2 −P 2Q2 . This leads us to the important observation that at least for this singular example two-center charge configurations labelled by elements of SL(2, Z) correspond to the same singular curve with the (142) canonical form . Hence apart from the four copies of SL(2)s related to four-qubit systems there seems to be a fifth SL(2) parametrizing a hidden torus. We will return to this important point later. Notice also that Eq.(144) can also be parametrized as Using Eqs.(137)-(138) we can write this as with j = ∞ just like in the previous example. However, as far as physics is concerned in a special case of the t 3 model we have also found some connection between the structure of SL(2, R) 0 × G 4 (four-qubit) invariants and the structure of the consistency condition of Eq.(77). This connection at first sight is not surprising since the consistency condition contains a four-qubit invariant 2I 1 = Q 1 , Q 2 , however more importantly it is also featuring the quantity Im(Z 2 Z 1 ) which is not invariant with respect to the full group of SL(2, R) 0 × G 4 transformations.
In order to show this let us observe however, that this quantity is invariant under transformations belonging to the horizontal subgroup SL(2, R) 0 . For let us recall our definition of Eq.(36) and then write the central charges as then for a, b, c, d ∈ R we have Z ′ 1 = aZ 1 + bZ 2 , and Z ′ 2 = cZ 1 + dZ 2 , hence Im(Z ′ 2 Z ′ 1 ) = Im(Z 2 Z 1 ) due to ad − bc = 1. However, this quantity is invariant merely under a special subgroup 54 of G 4 . Indeed, we have where K is the subgroup of G 4 transformations of the form   1 0 Notice that the group K with k a ∈ Z is precisely the stabilizer of the cusps in the three copies of the fundamental domain of the modular group, i.e. the stabilizer of τ a = −i∞ for a = 1, 2, 3.
Since the walls of marginal stability are determined by the central charges and for the four qubit invariant 2I 1 we have KQ 1 , KQ 2 = Q 1 , Q 2 clearly wall-crossing does not obstruct the K subgroup. Therefore we see that unlike our elliptic curves the consistency condition is left invariant (in the above sense) merely with respect to the subgroup SL(2, R) 0 × K.
Hence the correspondence between the physics of two center solutions and our special class of elliptic curves should be refined (see in this respect the comments at the end of Section V.A).
Now we turn to another important issue we have not discussed yet. Based on our experience with the degenerate case studied in the previous subsection we expect that for the nondegenerate case our mapping of two center charge configurations characterized by four-qubit invariants to elliptic curves should be many to one. Moreover, since in the Tate form we should have a 6 = 0 our mapping in the stu case cannot be onto either. We also know from the theory of elliptic functions that if we work in the algebraic closure of our field F i.e. Now over C every elliptic curve E is isomorphic to an elliptic curve E(Λ) where Λ is a lattice in the complex plane and the mapping between C/Λ (a torus) and E(Λ) is an isomorphism provided by the usual map defined by the Weierstrass P function. Explicitly where g 2 (τ ) = 60G 4 (τ ) and g 3 (τ ) = 140G 6 (τ ) with G 2k (τ ) are the Eisenstein series and if x ≡ x 1 /x 0 and y = x 2 /x 0 then x = P(z) and y = P ′ (z). Here the lattice vectors of Λ in C are ω 1 = 1 and ω 2 =τ withτ ∈ H being the modular parameter of the torus and the Weierstrass P function is defined as Here we have used the notationτ for the modular parameter in order not to confuse it with the moduli τ 1 , τ 2 , τ 3 of the stu and with the moduli τ of the t 3 model. Now the j invariant gives rise to the j function Comparing this with the expression of the j invariant as given in terms of the four-qubit invariants S and T of Eq.(111) to a two center charge configuration we can associate a torus with modular parameterτ . Now j : H → C is an automorphic function i.e.
It can be regarded 55 as a map providing a holomorphic universal covering of the orbifold H/P SL(2, Z) (i.e. the fundamental domain for P SL(2, Z)). Its Fourier series atτ = i∞ is provided by the variable u ≡ exp 2πiτ as Since the stu model can be regarded as a consistent truncation of N = 8, d = 4 maximal supergravity with continuous 13 U-duality group G 4 ≡ E 7 (7) one might conjecture that it should be possible to substantially generalize our considerations concerning two-center charge configurations by studying elliptic curves of the form where P 2j are polynomial invariants of the group SL(2) 0 × G 4 where SL(2) 0 refers to the horizontal symmetry group 1 acting as a generalized exchange symmetry group on the two centers. Now we have two sets of 56 component charge vectors where each of these vectors is transforming according to the symplectic irreducible representation of G 4 which is now the fundamental of E 7 (7) . According to an analysis of two centered magical charge orbits 13 working in the complexification of G 4 one discovers that the two centered charge orbits correspond to different real forms of the quotient of the complex groups E 7 /SO (8). In particular we have two 1 8 -BPS two-centered charge orbits with one of them being E 7(7) /SO (4,4). Recall now Eq.(1) displaying in its structure the group theoretical reason for our occurrence of four-qubit states. According to this equation one should be able to obtain our four-qubit invariants as special cases of SL(2) 0 × G 4 ones. Hence we conjecture that an elliptic curve of the (160) form should display this reduction procedure via implementing the constraint P 12 = 0 via a suitable reduction of some unknown SL(2) 0 × G 4 polynomial invariant P 12 .
In fact many pieces of such invariants are already at our disposal. For example the Indeed precisely these polynomials of order 8 and 12 has been suggested as obvious candidates for members of a complete basis for SL(2) 0 × G 4 invariants. Now one can define an elliptic curve as the one corresponding to Eq.(147). As was commented there this form can always be obtained from the Tate form if the characteristic of the field is neither 2 nor 3. Hence we conclude that our Tate form Eq.(160) featuring the unknown invariants P 2j should reduce to Eq.(162) after completing the square on the left and the cube on the right hand side.
Explicitly this process amounts to the transition from using the coefficients a j = P 2j to using the ones c 2 = − 1 4 S and c 4 = 1 4 T of Eqs.(125)-(126). As a solid piece of evidence it is also obvious that P 2 = Q 1 , Q 2 with , being the usual symplectic product of charge vectors which is a singlet with respect to SL(2) 0 × G 4 . Hence for mutually local charge configurations our cubic curve of Eq.(160) falls short of the term proportional to xy just like in the stu truncation. Moreover, as shown by Eqs.(12)- (14) in the four-qubit case our quadratic and sextic invariants I 1 and I 3 are duals of each other . Hence we expect that the invariant P 6 should be related to a quadratic combination of the Freudenthal duals of the corresponding charge vectors which are cubic in terms of the original charges 57 . Moreover, these dual quantities should be antisymmetric with respect to the exchange of the centers. Luckily a quantity J 6 , satisfying these criteria is also available (see Eq. (3.23) of Andrianopoli et.al. 13 ).
Unfortunately since according to Eq.(121) D = I 3 − I 1 I 4 in the stu case this quantity is probably not directly related to our invariant P 6 . Hence one is still left with the problem of finding the invariants P 4 , P 8 and P 12 .
In order to gain some insight into the physical meaning of the invariant P 4 let us look again at the stu (four-qubit) case. In this case we know that the corresponding invariant is M − L. As a first step we would like to somehow relate this quantity to some invariant known within the context of N = 8, d = 4 supergravity.
It is also known that by an SU(8) transformation it is possible to remove three phases, so for instance z 1 , z 2 , z 3 can be chosen to be real (or else having the same phase). Hence the canonical form can be characterized by four real numbers r n = |z n | and a phase φ. The form of J 4 reflecting these considerations is It is also interesting to notice that for z n real i.e. φ = 0 we have where H is the Hadamard matrix used for implementing discrete Fourier transformation in quantum information theory. Notice that with the definition Q n = 2r n we have J 4 = Q 0 Q 1 Q 2 Q 3 the standard expression giving rise to the Bekenstein-Hawking entropy S = The quantities displaying explicit dependence onτ are defined as with g 2 and g 3 given by Eqs.
with q = e 2πiτ . One also has 14,55 Notice that e 1 + e 2 + e 3 = 0 similarly to the property of four qubit quartic invariants L + M + N = 0. In the limitτ → i∞ we have αβ → 1 hence from Eq. (171) we get back to our usual polynomial of Eq. (25). Clearly introducing the factor αβ amounts to rescaling the amplitudes of our four-qubit state of Eq.(2) from Λ to theτ dependent ones √ αβΛ.
Let us now introduce an extra complex parameter u to be specified later and consider the following family of elliptic curves Now the right hand side is a quadratic polynomial in u. Its discriminant turns out to be just (α − β) 2 Σ 4 (Λ,τ, −x). This gives the desired clue for yet another way of associating an elliptic curve to two-center charge configurations.
Notice that in the limitτ → i∞ our curve boils down to y 2 = x(x − u) 2 a curve similar to the one of Eq. (142) with j invariant ∞. Moreover, via the transformation v = u+(α−β) 1 2 H one can also obtain a new form Here E and F are defined as 16 Let us now consider the triality symmetric curve 14 Notice that after using Eq.(185) and the canonical form for our four qubit states of Eq. (32) and (35) with (t 1 , t 2 , t 3 , t 4 ) = (a 2 , b 2 , c 2 , d 2 ) for the polynomial Γ of sixth order we obtain the and performing the (weak coupling 14 ) limitτ → i∞ it boils down to the form Moreover, in order to make contact with our original curve of Eq.(181) one just has to perform the change of variables in Eq.(186) x → x − e 1ũ − e 2 1 H, u =ũ + 1 2 It is also interesting to realize that our identification of the quark masses squared with the canonical four-qubit parameters automatically incorporates the special cases when some of the masses are zero with the singular cases in the four-qubit classification scheme of Finally it is amusing to recall yet another context where elliptic curves parametrized by four-qubit invariants reveal some intriguing structural similarities with interesting physics.
First of all it is well-known that in F-theory an extra SL(2, Z) and a hidden torus also makes its presence in a spectacular way 58,59 . In this case one considers an elliptic fibration M with some basis manifold B and fiber a two dimensional torus with modular parameter τ . Now F -theory is defined on M as type IIB string theory on B with the axion dilaton modulus of type IIB string theory identified with the modular parameter of the two-torus.
In the original setting 59 an elliptically fibered K3 surface was considered of the form y 2 = Notice also that the study of F -theory compactifications 60 is effected by looking at elliptic fibrations described by elliptic curves of the Tate form similar to the one of Eq.(160) where now the coefficients are polynomials in the coordinates of the base manifold B. An important limit studied in these investigations is the Sen limit 61 or orientifold limit. This is achieved by demanding that the axio-dilaton to be constant almost everywhere in type IIB space-time.
Explicitly this limit is realized by setting a 3 → εa 3 , a 4 → εa 4 , a 6 → ε 2 a 6 , We have also introduced a quartic polynomial featuring the algebraically independent four-qubit invariants. For our simple example we have shown that the property that this polynomial has real roots provides a necessary condition for the consistency condition to hold. The resolvent cubic of this fundamental polynomial can be cast into a cubic of Weierstrass canonical form which in turn can be used to define an elliptic curve associated to the two-center charge configuration. After switching to the more convenient Tate form of this curve we have shown that this association is natural, meaning that the Tate form is displaying combinations of the algebraically independent invariants of physical meaning.
Indeed, the Tate form falls short of terms step by step as we perform truncations to the st 2 and t 3 models, restrict attention to mutually local charge configurations, or perform the degenerate limit resulting in splits into small black holes. The discriminant of this elliptic curve is just the hyperdeterminant, a quantity also featuring the j invariant of the curve. Finally we presented other physical contexts where our four-qubit invariants parametrizing elliptic curves also play an important role. Here we have given a new look to the triality invariant curve originally introduced by Seiberg and Witten 14 . In its new form this curve is featuring four-qubit invariants also displaying permutation invariance. In this new setting we have also invoked the F-theory interpretation of this curve as was given by Sen 15 .
Taken together with the two-center black hole context studied in this paper the main unifying theme in these scenarios seems to be the presence of a hidden torus and its extra set of P SL(2, Z) modular transformations. Note that in our investigations elliptic curves appeared merely as natural mathematical objects nicely encapsulating the information on the structure of two-center U-duality invariants. However, apart from this we have also