On supersymmetric $Dp - \bar D p$ brane solutions

We analyze in the spirit of hep-th/0110039 the possible existence of supersymmetric $D p - \bar D p$ brane systems in flat ten dimensional Minkowski space. For $p=3,4$ we show that besides the solutions related by T-duality to the $D2 - \bar D 2$ systems found by Bak and Karch there exist other ansatz whose compatibility is shown from general arguments and that preserve also eight supercharges, in particular a $D4 - \bar D4$ system with D2-branes dissolved on it and Taub-NUT charge. We carry out the explicit construction in Weyl basis of the corresponding Killing spinors and conjecture the existence of new solutions for higher dimensional branes with some compact directions analogous to the supertube recently found.


Introduction
The discovery in type II string theories of cylinder-like branes preserving a quarter of the supersymmetries of the flat Minkowski space-time, the so-called "supertubes" [1], [2], [3] has attracted much attention recently. The stabilizing factor at the origin of their BPS character that prevent them from collapse is the angular momentum generated by the non-zero gauge field living on the brane. The solution in [1] presenting circular section was extended to arbitrary section in [4]; supertubes in the matrix model context can be found in [5], [6].
A feature of the supertube is that it has D0 and F 1 charges, but not D2 charge. An interesting observation related to this fact was made by Bak and Karch (BK); if we take the elliptical supertube with semi-axis a and b in the limit when for example a goes to infinity that is equivalent to see the geometry near the tube where it looks flat, the system should become like two flat branes separated by a distance b. But because of the absence of D2 charge it is natural to suspect that indeed the system could be a D2-D2 one. The existence of this system as well as systems with arbitrary numbers of D2 and D2 branes was proved in the context of the Born-Infeld action in reference [7] where the conditions to be satisfied by the Killing spinors were identified, while the absence of tachyonic instabilities was shown in [2], [8]. The aim of this letter is to extend the results of [7] to higher dimensional brane-antibrane systems. In the course of the investigation we will find, other than the T-dual solutions to that of BK, also new solutions for p = 3, 4.
We are now ready to start with the analysis of various cases. We will restrict in this paper to work on the flat ten-dimensional Minkowski vacuum of type II string theories, the spinors being constants (in cartesian coordinates) of the type mentioned above.

The Bak-Karch ansatz
Let us consider a flat D2-brane extended in directions (X 0 , X 1 , X 2 ), with field strength F 20 = E, F 12 = B. The general case with F 10 = 0 can be reached from this using the Lorentz invariance SO(1, 2) of the setting by means of a rotation. We can also take E < 0 or E > 0, it will not be relevant to fix a particular sign. The Γ-matrix is It is well-known that equation (1.1) has solutions preserving 1 2 of SUSY, i.e. 16 supercharges, for any constant F µν [10]. The ansatz introduced in [7] consists in subdividing condition (1.1) in two (or maybe more, if possible) parts in such a way they result compatible; in other words they show we can get novel non-trivial solutions at expenses of SUSY. In particular their solutions, as well as all the solutions presented in this paper, preserve 1 4 SUSY. We will refer to the two conditions to solve (1.1) with the labels a and b ; thus we write where "sg" stands for the sign-function. Equation (2.2) is equivalent to ask for the annihilation of the first two terms in (2.1) when acting on the spinor while (2.3) enforces (1.1) (with a minus sign on the rhs when theD-brane case is considered). They sign out the presence of dissolved F 1 ině 2 -direction and D0 branes respectively [7]. Consistency conditions for a and b say that But Γ a 2 = E 2 1, so we must take E 2 = 1 for the electric field, constraint used in writing (2.3). Assumed it, we get Γ b 2 = 1 and so nothing new is added. Finally the compatibility between the two conditions is assured from the fact that [Γ a ; Γ b ] = 0. The analysis of these consistency conditions will determine the compatibility conditions and will be the route to be followed to assure the existence of such solutions. From the further properties trΓ a = trΓ b = trΓ a Γ b = 0 is straightforward to conclude that each condition preserves

Explicit solution
We can obtain the Killing spinors explicitly working in the Weyl basis, we refer the reader to the appendix for a brief review. The relevant operators are The spinorial space is divided in two 16-dimensional (complex) subspaces with Γ a = ±1; each one is expanded by the vectors The last property is a result of the strict commutation of both operators that allows for the common diagonalisation in each subspace separately. We are of course interested in the subspace with Γ a = +1; on it we can introduce the following set of basis vectors From here we conclude that the spinors {η The observation made by BK is that aD2 with different fields (E 2 = E, −B), |E| = 1, must have the same Killing spinors. This fact follows immediately from the equality η Therefore we should be able to put arbitrary (parallel) number of D2-branes andD2-branes of the characteristics defined above and such configurations must preserve 1 4 SUSY with the corresponding Killing spinors given by (2.8). Finally it is worth to spend some words about the Majorana condition to be imposed. In a Majorana basis where all the Γ-matrices are real (or purely imaginary) the constraint is straightforward because we can take D = 1 in such a basis. But it is not so in a Weyl basis; this is a price to be paid for working in a setting where computations are relatively easy to handle in any dimension. We get from imposing (A.5) on the spinor (2.8) This completes the characterization of the Killing spinors of the BK solution.
After sketched with this known example the route to follow we move to higher dimensional cases.
3 The D3-D3 system Let us consider a flat D3-brane extended in directions (X 0 , X 1 , X 2 , X 3 ) with field strength where we have taken with no lost of generality the electromagnetic fields in the plane (12). The Γ-matrix is then given by The equation to solve (1.1) is written as where (ǫ 1 , ǫ 2 ) are Majorana-Weyl spinors of the same chirality. A minus sign on the rhs applies for the antibrane. We have found two possible solutions to (3.3).

Solution I
Consistency of a will require the constraint E 1 2 + E 2 2 = 1 which implies d = B 2 E 2 2 and then B and E 2 non zero, fact that we will assume; the ansatz is with a minus sign in (3.5) for theD3-brane. The compatibility for both constraints requires that A short computation shows that says that in the subspace with Γ a = 1 to which ǫ 1 belongs (3.6) is obeyed. The explicit solution can be worked out; first we introduce ( The last line shows the compatibility of both constraints; the imposition of (3.5) gives for both spinors the following general form The complex parameters ǫ (s 1 ...s 4 ) expand a 32-dimensional space; however the spinors must be Weyl and Majorana. Because Γ 11 (ǫ (+) (s 1 ...s 4 ) ) = (−) 1+ 4 k=1 s k we must constraint this value to +1 (−1) if we decide to take both of them left-(right) handed, so one index, e.g. s 4 , will be fixed by this condition. Finally the Majorana condition (3.10) shows that the solution has indeed 8 supercharges. Conditions (3.4), (3.5) can be interpreted as dissolved fundamental strings in the plane (12) at angle arctan E 2 E 1 and D1branes in theě 1 -direction respectively. This solution for a D3-brane can be extended to aD3-brane with the same Killing spinors by reverting the direction of the magnetic field, B → −B, as in the BK solution. In fact it is easy to see that we can reach it by T-dualizing ině 1 and boosting with β = −E 1 in that direction, remaining with the dual D2-brane in the (023)-hyperplane, electric field E = sg(E 2 ) ině 2 and magnetic field B.

Solution II
Here we will work out a new ansatz. The idea is to ask for the cancellation of the fielddependent part in (3.2). Compatibility of this ansatz imposes the constraints E 1 = 0 and B 2 = E 2 2 = 0, i.e. it is just a solution for orthogonal electric and magnetic fields with the same module. The corresponding conditions are The further properties Γ a 2 = −Γ b 2 = 1 , [Γ a ; Γ b ] = 0 assure the existence of solutions to these equations.
To get the Killing spinors associated to these systems we first solve a through  it is shown that the solution preserves 1 4 SUSY. Furthermore conditions (3.11) and (3.12) correspond to the SUSY's preserved by a p-p wave moving oně 3 direction and a D3brane in (0123), the existence of such configuration of intersecting branes being known since time ago (see for example reference [11]); the SUSY of the D3-brane configuration results then only sensitive to that of the constituents induced by the background fields.
On the other hand for theD3-branes (3.12) has a minus sign on the rhs; what is the same it is replaced by Γ b ǫ 2 = ǫ 1 , i.e. both spinors are interchanged, operation that obviously leaves invariant (3.11). In view of the existence of the O(2) automorphism group in type IIB string theory which rotates the supersymmetry generators we are tempted to conjecture the existence of systems of branes and antibranes in arbitrary number provided that on each one live fields (E 2 , B) with |E 2 | = |B| but otherwise arbitrary, the only condition common to all of them being to have the same sg(B E 2 ).

The D4-Dsystem
Now we take a flat D4-brane extended in directions (X 0 , X 1 , X 2 , X 3 , X 4 ) We again do not loose generality doing so because the anti-symmetric matrix (f i j ) = (F ij ), i, j = 1, . . . , 4 , can be put in the standard form iσ 2 ⊗ B 1 0 0 B 2 by means of a SO(4)-rotation; a SO(2) × SO (2) is then left over that we fix by putting the electric field E i ≡ F i0 to the form in (4.1) 3 . The Γ-matrix is We have found two solutions to equation (1.1).

Solution I
Consistency of a will require the constraint E 1 2 + E 2 2 = 1 which we will assume; the ansatz is as in (2.2), (2.3) where the relevant operators are 3 More generically, any anti-symmetric matrix in d-dimensions can be written as ) (a 0 is added if d is odd). It can be shown that in the general case when the magnetic field matrix (B ij ) ≡ (F ij ) has inverse there is no boost compatible with (4.1). However it is possible in this case to go to a frame where there is no electric field, being the boost velocity β = −B −1 E, if the condition β 2 = E t (−B 2 ) −1 E < 1 is fulfilled.
From standard arguments we should get again a solution preserving 1 4 SUSY. The explicit solution can be obtained as it was made in the precedent cases; first we introduce a basis for the subspace of spinors obeying a ,  In terms of them we can express a basis which also diagonalice Γ b , (s 1 s 2 s 3 ) } it is for aD4-brane (with the same fields as the D4brane). However to get brane-antibrane BPS systems the preserved supersymmetries must coincide. Condition a implies that both D4 andD4 branes must have the same electric field, but b implies that the magnetic fields must have opposite signs and not only this, direct inspection of (4.4) (or (4.7)) shows that B 1 B 2 = 0 necessary must hold. This gives two possible solutions, related by permutations of the planes (12) and (34).
Then E 2 and B 2 are non zero; theD4-brane will have E 2 and −B 2 fields as said. From   Again we can identify (4.3) and (4.4) with F 1 charge in (01) and D2-brane charge in (034) respectively; these solutions are T-duals to the Solutions I just considered (for example in the last case, by means of a T-duality ině 3 , a boost in that direction with β = −E 2 and further T-duality ině 4 we go back to BK solution). The Majorana condition for a generic Killing spinor (2.8) reads ǫ (s 1 s 2 s 3 ) * = (−) 1+s 1 +s 3 ǫ (s 1s2s3 ) (4.11)

Solution II
Here we present a new solution giving a D4-D4 system. The ansatz a this time consists in imposing the cancellation of the part of (4.2) even in the fields by itself. Consistency imposes the constraint which replaces the E 2 = 1 constraint of the BK-type solutions and that we will assume henceforth. Let us note that |B 1 B 2 | ≥ 1, in particular B ij must be non singular; also from (4.12 Consistency of the ansatz follows from Γ b where . From here it is clear that a basis for the space obeying Γ a ǫ = ǫ consists of the spinors With the definitions It is easy to see that theD4-brane solution (corresponding to have a minus sign in the b -condition) will have the same Killing spinors provided that it has both electric and magnetic fields with opposite signs wrt that of the D4-brane. Therefore (2.8) with η (+) (s 1 s 2 s 3 ) given in (4.18) is the general Killing spinor of such brane-antibrane systems.
It is worth to note however that (4.12) implies 0 < ||B −1 E|| 2 = 1 + E 1 , and therefore from the footnote at the beginning of this Section we know that there exists a boost with β = −B −1 E that eliminates the electric field; a further rotation will lead us to the case E = 0 (with different B 1 , B 2 ). The Majorana condition in this case looks like in (4.11).

Conclusions
We have studied in the context of the Born-Infeld effective action the existence of supersymmetric, presumably stable solutions of D2, D3 and D4-branes preserving a quarter of the supersymmetries of the flat background in which they are embedded. These results allow to conjecture at the light of the compatible ansatz in solutions II, the existence of configurations other than the T-dual to those presented here, for D5 and D6 branes , and from here for D7 −D7 and D8 −D8 and so on. An explicit prove of theses facts along the lines followed here should be straightforward.
In the case of the D4 brane we note that equations (4.13), (4.14) imply the existence of Taub-NUT charge and a sort of D2-brane charge for this solution, no D4-brane charge is present; the D4-D4 systems should represent genuine bound states of these components. What is more, it is plausible that a five dimensional supertube-like solution exists, leading in a certain limit to the brane-antibrane system much as it happens with the supertube.
Another interesting open problem is certainly to find the explicit form of the corresponding supergravity solutions since as it is stressed in reference [2] the low energy analysis presented here does not assure by itself the complete absence of instabilities. To this goal the knowledge of the world-volume fields as well as the explicit form of the Killing spinors (although they do not take into account the back-reaction) could be of great help. All these items are under current investigation [12].
A Majorana condition is a reality condition; to be able to define it in some fixed basis we need a matrix D satisfying D −1 S M N D = S * M N in such a way that if Ψ is a spinor, e.g. a 32-dimensional vector (in d = 10) transforming linearly in the spinorial representation of the Lorentz group, then Ψ c ≡ D Ψ * also it is; Ψ c is called the conjugate spinor [14]. Then it has sense to impose the Majorana condition Ψ c ≡ D Ψ * = Ψ (A. 5) In the Weyl basis a matrix D verifying D −1 Γ M D = Γ * M , M, N, = 0, 1, . . . , 9 , can be taken as D ≡ −Γ 2 Γ 4 Γ 6 Γ 8 = σ 1 σ 2 σ 1 σ 2 1 (A.6) It is worth to note that if D is such a matrix in a basis {|α >, α = 1, . . . , 32}, under a change of basis |α >= P β α |β > ′ it transforms as D ′ = P −1 D P * .