1/2-BPS D-branes from covariant open superstring in AdS4 × CP3 background

Journal of High Energy Physics, May 2018

Abstract We consider the open superstring action in the AdS4 × CP3 background and investigate the suitable boundary conditions for the open superstring describing the 1/2-BPS D-branes by imposing the κ-symmetry of the action. This results in the classification of 1/2-BPS D-branes from covariant open superstring. It is shown that the 1/2-BPS D-brane configurations are restricted considerably by the Kähler structure on CP3. We just consider D-branes without worldvolume fluxes.

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1/2-BPS D-branes from covariant open superstring in AdS4 × CP3 background

Accepted: May D-branes from covariant open superstring in Jaemo Park 0 1 3 Hyeonjoon Shin 0 1 2 3 0 Pohang , Gyeongbuk 37673 , South Korea 1 POSTECH , Pohang, Gyeongbuk 37673 , South Korea 2 Asia Paci c Center for Theoretical Physics 3 Department of Physics & Center for Theoretical Physics We consider the open superstring action in the AdS4 investigate the suitable boundary conditions for the open superstring describing the 1/2BPS D-branes by imposing the -symmetry of the action. This results in the classi cation of 1/2-BPS D-branes from covariant open superstring. It is shown that the 1/2-BPS Dbrane con gurations are restricted considerably by the Kahler structure on CP3. We just consider D-branes without worldvolume uxes. AdS-CFT Correspondence; D-branes; Extended Supersymmetry - AdS4 CP 3 background 1 Introduction 2 It has been proposed that the Type IIA string theory on the AdS4 CP3 background is dual to the three-dimensional superconformal N = 6 Chern-Simons theory with gauge group U(N )k U(N ) k known as the Aharony-Bergman-Ja eris-Maldacena (ABJM) theory [ 1 ]. To be more precise, since the ABJM theory is motivated by the description of multiple M2-branes, it is dual to the M-theory on AdS4 S7=Zk geometry with N units of four-form ux turned on AdS4, where N and k correspond to the rank of the gauge group and the integer Chern-Simons level respectively. When 1 N 1=5 k N , the M-theory can be dimensionally reduced to the Type IIA string theory on the AdS4 After the proposal of this new type of duality, various supersymmetric embeddings of D-branes have been considered. Embeddings for the giant graviton [2{10], adding avor [11{14], and some other purposes [15, 16] are the examples studied extensively. With some other motivations, we may also consider other types of supersymmetric D-brane embeddings or con gurations. Since each of them would correspond to a speci c object in the dual gauge theory, the exploration of supersymmetric D-branes may be regarded as an important subject to enhance our understanding of duality. However, unlike the case of at spacetime, the sturucture of AdS4 CP3 background is not so trivial and the solution of the associated Killing spinor equation is rather complicated. This makes the case by case study of supersymmetric D-branes laborious, and thus it seems to be desirable to have some guideline. In this paper, we focus especially on the most supersymmetric cases and are trying to classify the 1/2-BPS D-branes in the AdS4 CP3 background. In doing so, we are aiming at obtaining the classi cation data as a guideline for further exploration of which is especially useful in classifying the 1/2-BPS D-branes. It has been developed in [17] for the at spacetime background, and successfully applied to some important backgrounds in superstring theory [18{23]. To carry out such classi cation, we need the Type IIA superstring action in the AdS4 CP3 background, which has been constructed by using the super coset structure [24{27]. However, the action is the one where the -symmetry is { 1 { partially xed, and might be inadequate in describing all possible motions of the string as already pointed out in [24]. The fully -symmetric complete action has been constructed in [28], which we take in this paper. In the next section, we consider the Wess-Zumino (WZ) term of the complete superstring action in the AdS4 CP3 background, which is the ingredient for the covariant open string description of 1/2-BPS D-branes, and set our notation and convention. In section 3, we investigate the suitable boundary conditions for open string in a way to keep the -symmetry and classify the 1/2-BPS D-branes. The discussion with some comments follows in section 4. 2 trary variation of the open superstring action and looks for suitable open string boundary conditions to make the action invariant. However, it has been pointed out in [18] that the -symmetry is enough at least for the description of supersymmetric D-branes. The basic reason is that the -symmetry is crucial for matching the dynamical degrees of freedom for bosons and fermions on the string worldsheet and hence ensuring the object described by the open string supersymmetric. The -symmetry transformation rules in superspace are1 ZM EMA = 0 ; ZM EM = (1 + ) ; 1 2 (2.1) where ZM = (X ; ) is the supercoordinate, EMA (EM ) is the vector (spinor) super eld,2 is the 32 component -symmetry transformation parameter, and is basically the pullback of the antisymmetric product of two Dirac gamma matrices onto the string worldsheet with the properties, 2 = 1 and Tr = 0, whose detailed expression is not needed here. By construction, the bulk part of the superstring action is invariant under this -symmetry transformation. In the case of open superstring, however, we have non-vanishing contributions from the worldsheet boundary, the boundary contributions, under the -symmetry variation. Interestingly, as noted in [18], the kinetic part of the superstring action does not give any boundary contribution due to the rst equation of (2.1). Thus, only the WZ term rather than the full superstring action is of our concern in considering the boundary contributions. 1The notation and convention for indices are as follows. The spinor index for the fermionic object is that of Majorana spinor having 32 real components and suppressed as long as there is no confusion. is explicit expressions have been derived in [28]. 2In the present case, EMA and EM are of course the super elds for the AdS4 CP3 background whose { 2 { HJEP05(218) The WZ term has an expansion in terms of the fermionic coordinate up to the order of 32. Here, we will consider the expansion up to quartic order. From the complete Type IIA superstring action in the AdS4 the WZ term has the following form. CP3 background [28], we see that the expansion of SWZ = S(2) + S(4) + O( 6) ; where S(2) and S(4) represent the quadratic and quartic part respectively. The quadratic part is read o as3 S(2) = R Z k ieA ^ A 11D origin of the appearance of k in the action. R is the radius of S7 in the eleven dimensional Planck unit and has the relation with the CP3 radius, RCP3 in string unit, as R2 R3=k = 4 p2N=k. The radius of AdS4 is half of RCP3 . The ten dimensional gamma CP3 = matrices A are represented through the tensor product of four and six dimensional gamma matrices as where 11 measures the ten dimensional chirality and a = a 1 ; a0 = 5 a0 ; 11 = 5 7 ; 5 = i 0 1 2 3 ; 7 = i 10 20 : : : 60 : The ten dimensional Weyl spinor with 32 real components can be split into two parts in a way to respect the supersymmetry structure of the AdS4 CP3 background as where P6 and P2 are the projectors de ned by 1 8 1 8 P6 = (6 J ) ; P2 = (2 + J ) ; P6 + P2 = 1 ; and J is a quantity depending on the Kahler form 12 Ja0b0 ea0 ^ eb0 on CP3, = P6 ; = P2 ; J = iJa0b0 a0b0 7 : Because J satis es J 2 = 4J + 12 and hence has six eigenvalues 2 and two eigenvalues 6, ( ) has 24 (8) independent components after taking into account the spinorial structure 3In the practical calculation, we utilize the expressions of super elds given in [29], a subsequent paper after [28]. We mostly follow the notation and convention of [28, 29]. As an exception, we use rather than to represent spinor components corresponding to the eight broken supersymmetries of the AdS4 background. { 3 { (2.2) (2.3) (2.4) (2.5) (2.6) (2.7) (2.8) CP3 corresponds to the eight supersymmetries broken by where where D From this, we see that the Ramond-Ramond one-form gauge potential A in (2.10) has the following expression A = 1 8 Ja0b0 !a0b0 through an identity P2 a0b0 P2 = 6i Ja0b0 P2J 7P2 = iJa0b0 P2 7P2. 4 If we now move on to the quartic part S(4) in the expansion of the WZ term (2.2), it is read o as S(4) = ( a0 5 )(D ^ a0 7D ) R Z 2k ( + + + + 2 R(M ) 0 The dots in the last line denote the terms which lead to the boundary contributions of higher order in ( 5 order) under the symmetry transformation and hence should be considered together with the transformation of sextic oder part of the WZ term. 3 Covariant description of 1/2-BPS D-branes In this section, we take the -symmetry variation of the WZ term considered in the previous section and obtain the boundary contributions. We then investigate the suitable open string boundary conditions which make the boundary contributions vanish and hence guarantee the -symmetry, the boundary -symmetry. The resulting open string boundary conditions give the covariant description of 1/2-BPS D-branes. In taking the -symmetry variation, it is convenient to express the variation of X in terms of by using the rst equation of (2.1) as X = i + O( 3) ; where we retain the variations up to the quadratic order in because we are interested in the -symmetry variation of the WZ term up to the quartic order in . By exploiting this, we rst consider the boundary contributions from the -symmetry variation of quadratic part independent of the spin connection, which are as follows: S(2) ! i R Z A A )( A 11d ) + + 2 a e ( 2 a e ( R R b a0 5 )( ab represents the boundary of open string worldsheet . For the boundary symmetry, each term should vanish under a suitable set of open string boundary conditions. Let us look at the rst term. Because dX e A = 0 (A 2 D) ; where A 2 D (N ) implies that A is a Dirichlet (Neumann) direction, the fermion bilinear should vanish for A 2 N . In order to check this at the worldsheet boundary, we rstly split the ten dimensional Majorana spinor into two Majorana-Weyl spinors and 2 with opposite ten diemensional chiralities as = 1 + 2 ; { 5 { (3.1) (3.3) 1 (3.4) HJEP05(218) where we take 11 1 = 1 and 11 2 = 2. Secondly, we impose the following boundary condition breaking the background supersymmetry by half with 2 = P 1 P = s A1:::Ap+1 ; ( 1 for X0 i for X0 In order to see when this condition is satis ed, it is convenient to introduce two integers n and n0 to denote the number of Neumann directions in AdS4 and CP3 respectively. Then we have the relation, (3.5) (3.6) (3.7) (3.8) (3.9) (3.10) (3.11) (3.12) A 11d = 0 (A 2 N ). The rst identity of this equation clearly shows that the rst term of (3.2) vanishes under the boundary condition of eq. (3.5). Another consequence of eq. (3.8) is that the second term of (3.2) becomes zero automatically since = 0 (A 2 N ) also implies Now we consider the fourth term of (3.2) prior to the third one which requires us some care. From eqs. (3.3) and (3.8), the vanishing condition for the term is where all the indices A1; : : : ; Ap+1 are those for Neumann directions, and HJEP05(218) must be even because to be depending on the boundary condition for the time direction X0. It should be noted that p 1 and 2 have opposite chiralities. Then 1 for A 2 D, which means that 1 1 + { 6 { and the matrix P of (3.6) for the boundary condition (3.5) is expressed as P = s a1:::ana01:::a0n0 = s a1:::an ( 5)n0 a01:::a0n0 : A bit of calculation by using this P shows that the condition (3.9) is satis ed for the following cases: according to which the possible candidates of 1/2-BPS Dp-brane are listed as The rst two terms and the fourth term on the right hand side of eq. (3.2) that we have considered are written in terms of the Weyl spinor alone. On the other hand, the third and the last two terms have explicit dependence on ( ), the speci c part of corresponding to the (un-)broken supersymmetry of AdS4 As for the third term, the condition making it vanish is due to eqs. (3.3) and (3.8). It is not di cult to check that these conditions are satis ed for the cases of (3.12) if we split and as (3.4) and if we can apply the boundary conditions similar to (3.5). However, the condition (3.15) is incompatible with (3.5). If we recall the de nitions of and given in eq. (2.6), we see that these boundary conditions (3.15) assume implicitly the commutativity of P with P6 and P2, or more basically [P; J ] = 0 from eq. (2.7). This assumption is too naive because [P; J ] 6= 0 generically. In fact, if P6 (P2) acts on the boundary condition (3.5) and the de nition of the correct boundary condition for ( ) turns out to be ( ) of eq. (2.6) is used, 2 = P 1 + [P; J ] 1 2 = P 1 [P; J ] 1 ; : 1 8 1 8 As one may guess, the conditions of (3.14) are not satis ed under these boundary conditions due to 1 dependent terms which do not vanish by themselves. We may introduce additional suitable boundary condition for 1 to get desired situation. However, this leads to lower supersymmetry. Since we are focusing on the 1/2-BPS D-branes, we are not trying to consider such additional boundary condition. Instead we explore the cases in which P commutes with J . The matrix J depends on the Kahler form 12 Ja0b0 ea0 ^ eb0 on CP3 as one can see from eq. (2.8). It is convenient to choose a local frame such that the tangent space components Ja0b0 take the canonical form [30] 0 " 0 0 0 0 " 1 A Ja0b0 = B0 " 0C ; " = { 7 { Since three two dimensional subspaces are equivalent in this form, it is enough to consider one subspace when investigating the commutativity between P with J . For a given two dimensional subspace, we can now check that [P; a0b0 7] = 0 when a0; b0 2 N or D (n0 = even) ; a0 2 N (D) ; b0 2 D(N ) (n0 = odd) : This implies that [P; J ] = 0 under the following conditions: 1. for even n0, both of two directions in each two dimensional subspace are Neumann or Dirichlet one. HJEP05(218) 2. for odd n0, one of two directions in each two dimensional subspace is Neumann one and another is Dirichlet one. This restricts the value of odd n0 to 3. These two conditions make the boundary condition for ( ) of eq. (3.16) have the same form with (3.5), and in turn eq. (3.14) is satis ed. They also constrain the con gurations of 1/2-BPS D-branes. Especially, the condition (ii) that speci es n0 = 3 for odd n0 informs us that the two D-branes in (3.13) are not 1/2-BPS and thus should be excluded from the list of 1/2-BPS D-branes. As a result, we see that the possible con gurations of 1/2-BPS D-branes are restricted considerably by the Kahler structure on CP3. From eqs. (3.3) and (3.8), we see that the last two terms of (3.2) vanish if a a0 5 7 = 0 (a; a0 2 N ) : It is not di cult to check that this is indeed satis ed for the cases of (3.12) and under the conditions (i) and (ii) of the previous paragraph. Having investigated the vanishing conditions for the boundary contributions from the quadratic part independent of the spin connection, we now move on to the boundary contributions from the -symmetry variation of the spin connection dependent terms. They are obtained as R Z (3.18) (3.19) (3.20) dX ( c0 5 !ab D + eD( + dX ( 1 2 1 2 1 2 dX ( dX ( 1 2 dX ( )( ) hec( c ab )( c0 7 c0 7 c c0 ab c ab c )( 7 ) c ab 5 7 ) 5 7 ) + ec0( c0 ab 7 ) i ) dX ( c0 7 ab )( c0 5 ) ab )( c0 5 ) + (a ! a0; b ! b0) ; (3.21) { 8 { which are cubic order in the fermionic coordinate. After imposing the boundary condition of (3.5) as we did in previous paragraphs, we see that the constraints of (3.12) and the conditions (i) and (ii) su ce for showing that majority of terms vanish. However, the contributions involving !ab with a 2 N (D), b 2 D(N ) and !a0b0 with a0 2 N (D), b0 2 D(N ) do not vanish. At this point, we would like to note that the spin connection for AdS4 (CP3) has the schematic structure of !ab X[adXb] (!a0b0 X[a0 dXb0]). This implies that the non-vanishing contributions vanish if the Dirichlet directions are set to zero. In other words, a given D-brane in the list of (3.13) except for (2; 1) and (2; 5) is 1/2-BPS if it is placed at the coordinate origin in its transverse directions. Finally, we consider the terms in S(4) independent of the spin connection. In this case, it is enough to take the -symmetry variation only for , since as seen from (3.1) X leads to the contributions of higher order in which should be treated with the boundary contributions from the -symmetry variation are read o as S(6). Then S(4) R Z )( a 5 7D ) + ( aD )( We see that there are lots of boundary contributions. One may wonder if all of them vanish without any extra condition after imposing the boundary condition (3.5) with the constraints (3.12) and the conditions (i) and (ii) below (3.18). However, lengthy but straightforward calculation indeed shows that the above boundary contibutions vanish without introducing any additional condition. We have completed the investigation of the open string boundary condition for the -symmetry of the action expanded up to quartic order in . The resulting classi cation of 1/2-BPS D-branes is summarized in table 1. { 9 { a0 7D ) i (n,n0) (1,0) (0,3) (1,4) (3,2) (4,3) (3,6) (i) and (ii) below eq. (3.18). Each D-brane is supposed to have no worldvolume ux. 4 Discussion We have given the covariant open string description of 1/2-BPS D-branes by investigating the suitable boundary condition which makes the boundary contributions from the symmetry variation of the WZ term vanish up to the quartic order in . As the main CP3 background have been classi ed as listed result, the 1/2-BPS D-branes in the AdS4 in table 1. Although we do not have a rigorous proof, we expect that the classi cation is valid even at higher orders in . In other words, any extra condition is expected to be unnecessary in showing the boundary -symmetry of the full WZ term. The reasoning behind this is due to the observation that the constraints of (3.12) for the possible 1/2-BPS D-bane con gurations originate solely from the covariant derivative for (2.10) incorporating the e ects of background elds.5 Note that the third term and the fourth term of (3.2) essentially comes from the variation of the rst term of (2.3) involving the covariant derivative. This means that all the constraints are obtained just from the consideration of quadratic part S(2) (2.3). Of course, S(2) has the terms independent of the covariant derivative. However, if we trace the process of checking = 0, we see that they lead to the vanishing boundary contributions consistently without requiring any additional constraint and have the boundary -symmetry. As we have checked in the previous section, for the quartic part S(4), the rst non-trivial higher order part, again nontrivial contributions come from the quartic terms containing the covariant derivative. We expect that this situation continues to hold even for the higher order of in the expansion of WZ term. Actually, the above reasoning can be explicitly checked for the analogous open string descriptions of 1/2-BPS D-branes in some important supersymmetric backgrounds including Type IIA/IIB plane waves [18, 19] and AdS5 S5 [20{23] backgrounds. In all these cases, the quadratic part including the covariant derivative in the WZ term also determines the full classi cation of the 1/2-BPS D-branes. In particular, the result for the AdS5 S5 background has been shown to be valid at full orders in the fermionic coordinate. That is, except from the quadratic part, we do not have any extra condition from higher order parts which might give further restriction on the 1/2-BPS D-brane con gurations. For the AdS5 S5 background, the string action can be obtained from the supercoset structure. Since AdS4 S7 has the similar supercoset structure and the AdS4 CP3 is obtained as an orbifold of AdS4 S7, we expect to prove the above reasoning explicitly, which will be an interesting topic to pursue. 5In order to describe 1/2-BPS D-branes, open string end points are placed at the coordinate origin of the Dirichlet directions. This eliminates the boundary contributions from the spin connection dependent terms. One interesting fact about the AdS4 CP3 background is that it is related to the Type IIA plane wave background through the Penrose limit [31]. The superstring action in the Type IIA plane wave background has been constructed in [32{34], and the open string description has been used to classify the 1/2-BPS D-branes in the background [19]. From the relation between two coordinate systems for the AdS4 CP3 and the Type IIA plane wave backgrounds, we may compare the class cation data of table 1 with that obtained in [19]. Then we realize an agreement between them except for D0-brane. We note that, since non-trivial Kahler structure does not exist in the Type IIA plane wave background, the conditions below (3.18) due to the Kahler structure on CP3 disappear after taking the Penrose limit and hence two D-branes in (3.19) excluded from the 1/2-BPS D-branes turn supersymmetric in the Type IIA plane wave background. The basic reason is simply the impossibility of taking a suitable open string boundary condition for D0-brane in a way of preserving supersymmetry. Given this discrepancy, one might wonder the fate of the supersymmetric D0-brane in the plane-wave limit. Starting from the usual AdS4 metric ds2 = cosh2 dt2 + d 2 + sinh 2d 22 we consider the boosted limit along an angle direction ~ in CP3. Thus we de ne (4.1) (4.2) (4.3) (4.4) x+ = t + ~ 2 ; x = R~2 t 2 ~ : ds2 = 4dx+dx + : t ~ = o 1 R~2 : Thus the possible D0-brane con guration carried over to the plane-wave limit should satisfy eq. (4.4), which is necessarily nonsupersymmetric in AdS4 CP3. In other words, the plane wave limit is the geometry seen by the particle moving fast along the angle direction in CP3, D0-brane also should be comoving with that particle in order to have a sensible limit in the plane-wave geometry. We also would like to note that there is similar discrepancy between D1-branes in the AdS5 S5 and the type IIB plane-wave backgrounds also related through the Penrose limit [35]. As shown in [20{23], a Lorentzian D1-brane can be 1/2-BPS only when it is placed in the AdS5 space. However, such D1-brane is not supersymmetric in the plane wave background and completely di erent type of con guration [18] appears to be supersymmetric which is furthermore not half but quarter BPS. The classi cation of 1/2-BPS D-branes given in table 1 is `primitive' in a sense that it gives no more information about 1/2-BPS D-branes. For example, it does not tell us about which con guration of a given D-brane is really 1/2-BPS and which part of the Taking R~ ! IIA plane-wave metric of x , The explicit construction was given at [31]. Note that in order to have the nite values 1 limit with some additional scaling of other coordinates, we obtain the Type background supersymmetry is preserved on the D-brane worldvolume. We should consider these questions by using other methods. One possible way would be to take the process adopted in [36, 37] for studying worldvolume theories on 1/2-BPS D-branes in the AdS5 S background. An important point we would like to note here is that it is enough to consider 5 D-brane con gurations based on the classi cation shown in table 1 . We do not need to investigate all possible con gurations for the study of 1/2-BPS D-branes. Therefore, the classi cation provides us a good guideline or starting point for further exploration of the 1/2-BPS D-branes. 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Jaemo Park, Hyeonjoon Shin. 1/2-BPS D-branes from covariant open superstring in AdS4 × CP3 background, Journal of High Energy Physics, 2018, 158, DOI: 10.1007/JHEP05(2018)158