D-branes from Pure Spinor Superstring in AdS_5 x S^5 Background

We examine the surface term for the BRST transformation of the open pure spinor superstring in an AdS_5 x S^5 background. We find that the boundary condition to eliminate the surface term leads to a classification of possible configurations of 1/2 supersymmetric D-branes.

is composed of psu(2, 2|4) currents J, and the left-and right-moving ghosts, (λ α , w α ), and (λα,ŵα), respectively. The ghosts satisfy the pure spinor constraints λγ A λ =λγ Aλ = 0 (A = 0, 1, · · · , 9). The pure spinor superstring in the AdS 5 ×S 5 background, as well as the GS superstring in the AdS 5 ×S 5 background given in [7], is integrable in the sense that infinitely many conserved charges are constructed [8,9] (see also [10]). Nevertheless, for the detailed study of the AdS/CFT correspondence [11], covariant quantization of the superstring should be useful. Though the action of the pure spinor superstring in the AdS 5 ×S 5 background is bilinear in the current J, its quantization is still difficult because the J is not (anti-)holomorphic unlike the principal chiral model. We need more effort to quantize the pure spinor superstring covariantly.
The purpose of this paper is to study D-branes in the AdS 5 ×S 5 background. A D-brane is a solitonic object in string theory, and is characterized by the Dirichlet boundary condition of an open string. The classical BRST invariance of the open pure spinor superstring in a background implies that the background fields satisfy full non-linear equations of motion for a supersymmetric Born-Infeld action [12]. This is the open string version of [13] in which the classical BRST invariance of the closed pure spinor superstring in a curved background was shown to imply that the background fields satisfy full non-linear equations of motion for the type-II supergravity. For D-branes in the AdS 5 ×S 5 background, supersymmetric D-brane configurations are derived in [14] by examining equations of motion for a Dirac-Born-Infeld action for each D-brane embedding ansatz.
In the present paper, we will examine D-branes in the AdS 5 ×S 5 background by using the open pure spinor superstring. Especially, we concentrate ourselves on the BRST invariance in the presence of the boundary. Namely, we examine the surface term for the BRST transformation of the open pure spinor superstring in the AdS 5 ×S 5 background. We will find that the boundary condition to eliminate the surface term leads to a classification of possible configurations of 1/2 supersymmetric D-branes. This approach is the pure spinor superstring version of [15,18] § . In [15,18], the boundary condition for the κ-symmetry surface term of the GS superstring in the AdS 5 ×S 5 background was shown to lead to a classification of possible configurations of 1/2 supersymmetric D-branes. We find that our result is consistent with those obtained in [14,15,18]. One of the main advantages in our approach is that the derivation is much simpler than the one by using the Dirac-Born-Infeld § A covariant approach to study D-branes in flat spacetime was proposed by Lambert and West in [19]. action and the GS superstring action. This is because the pure spinor superstring action is bilinear in the currents, and because we don't need to deal with the κ-symmetry variation which is highly non-linear. This paper is organized as follows. In section 2, after introducing the pure spinor superstring in the AdS 5 ×S 5 background, we examine the BRST invariance of the open superstring action and extract the surface term. For the BRST invariance to be preserved even in the presence of the boundary, the surface term must be eliminated by a certain boundary condition. In section 3, we fix the boundary conditions by examining a few terms contained in the surface term. The boundary conditions lead us to a classification of possible 1/2 supersymmetric D-branes in the AdS 5 ×S 5 background. In section 4, the boundary conditions fixed above are shown to eliminate all terms contained in the surface term. The last section is devoted to a summary and discussions. Our notation and convention are summarized in Appendix.

BRST invariance
The BRST transformation of the action is examined below. We will not drop any surface term here. In the next section we will consider the boundary condition for the surface term to be eliminated, and show that the condition leads us to a classification of possible configurations of 1/2 supersymmetric D-branes in the AdS 5 ×S 5 background.
First we examine S σ . The BRST transformation law of currents with a Grassmann odd parameter ε is given as [5] εQ Next we consider S WZ . Using (2.7), one derives By using Maurer-Cartan equations the second line of the right-hand side of (2.9) may be rewritten as Finally we examine S gh . The BRST transformation law of ghosts Further noting that which follow from the pure spinor conditions {λ, λ} = {λ,λ} = 0, we can derive Gathering all results obtained above together, we find that the BRST transformation of S is In the second equality, we have used the fact that Str([J 0 , ελJ 3 ]) = 0 and the similar relations.
We can conclude that S is BRST invariant as long as this surface term vanishes. For a closed string, the surface term always vanishes. For an open string, however, appropriate boundary conditions are required. In the next section we will examine these boundary conditions.

Boundary BRST invariance to D-brane configurations
In this section we will examine boundary conditions for the surface term to be eliminated, and show that they lead us to a classification of possible 1/2 supersymmetric D-brane configurations in the AdS 5 ×S 5 background.
The surface term (2.16) turns to ¶ where we have assumed that the surface term at τ = ±∞ vanishes as usual. The open string boundaries are at σ = σ * with σ * = 0, π. As seen in (A.21), J 1τ and J 3τ correspond to q α L 1α τ andqαL 2α τ , respectively, where the corresponding Cartan one-form L I is given in (A.18). It follows that J 1τ and J 3τ are polynomials in θ I . We should note that the surface terms do not cancel out each other. So we may examine each surface term separately without loss of generality. Our strategy is as follows. First we examine a few terms contained in J 1 and J 3 , and fix the boundary condition. Next we will show that the boundary condition we have fixed would eliminate all terms in (3.1).
We shall examine each term contained in (3.6) below so that we will fix p and ρ. Let us begin with examining the second term in the right hand side of (3.6). Because e A τ = 0, must be satisfied. It follows from (3.11) that in order to delete the third term in the right hand side of (3.6),θ Γ A ∂ τ θ = 0 (3.12) must be satisfied. We examine (3.12) first. Noting that CM = ∓αM T C with ρ = αρ T for p = { 1 3 mod 4, respectively, we derivē so that α is fixed as α = ±1 for (3.12). It means that ρ = ±ρ T for p = { 1 3 mod 4. Now, we return to (3.11). We derive, defining β by σ 3 ρ = βρσ 3 , It implies that β is fixed as β = −1 for (3.11). This means that ρ = σ 1 or iσ 2 . Combining this with the result obtained from (3.11), we can conclude that ρ = σ 1 for p = 1 mod 4, and that ρ = iσ 2 for p = 3 mod 4. For consistency we require that M 2 = 1. This implies that the time direction 0 is a Neumann direction since s 2 = 1. The results so far coincide with the boundary condition for 1/2 supersymmetric D-branes in flat spacetime.
Finally, we examine the first term in the right hand side of (3.6) which leads to the additional condition specific to the AdS 5 ×S 5 background. One may show that In the second equality we have used σ 1 ρ = ±ρσ 1 for p = { 1 3 mod 4. The third equality follows from IM = (−1) n MI where n is the number of Neumann directions contained in AdS 5 spanned by {0,1,2,3,4}. As a result, forλIσ 1 ∂ τ θ = 0 we must impose n = even for p = 1 mod 4 and n = odd for p = 3 mod 4. We summarize the result in the Table 1 where (n, n ′ ) means a D-brane of which world-volume is extended along AdS n ×S n ′ . This gives a classification of 1/2 supersymmetric D-brane configurations in the AdS 5 ×S 5 background.
This result is consistent with the ones obtained by using the κ-symmetry variation of the Green-Schwarz superstring [15,18] and by examining Dp-brane field equations [14].  For this purpose, it is convenient to rewrite the surface term (3.1) in the 32-component notation as where the corresponding Cartan one-form L is given in (A.6). We will show that the validity of the boundary conditions for p = 1 mod 4 and for p = 3 mod 4, in turn.

p = 1 mod 4
We shall show that the surface term (4.1) is eliminated by the boundary conditions for p = 1 mod 4: θ = P + θ and λ = P + λ where P + = 1 2 (1 + M) with M = sΓĀ 0 ···Āp ⊗ σ 1 and n = even. First we examine D τ θ defined in (A.7). It follows from θ = P + θ that In order to derive this relation, we have used Here we have assumed that wĀ B = 0. This is because a D-brane breaks rotational invariance in the plane spanned by one of Neumann directions and one of Dirichlet directions.
Next we will examine M 2 P + where M 2 is defined in (A.8). Noting that CP ± = P T ∓ C one derives It follows that Gathering the results (4.2) and (4.8) together we obtain L τ = P + L τ . Using this we may deriveλ This shows that the boundary conditions for p = 1 mod 4 eliminate the surface term (4.1).
Summarizing we have shown that the boundary condition (3.16) with (3.17) eliminates the surface term (3.1) of the BRST transformation εQ(S).

Summary and discussions
We examined the BRST invariance of the open pure spinor superstring action in the AdS 5 ×S 5 background. In order for the BRST symmetry to be preserved even in the presence of the boundary, the surface term of the BRST transformation must be eliminated by appropriate boundary conditions. We determined such boundary conditions and found that the boundary conditions lead to a classification of possible configurations of 1/2 supersymmetric D-branes in the AdS 5 ×S 5 background. Our result is summarized in the Table 1. This is consistent with the results obtained by the other approaches [14,15,18].
We have used an exponential parametrization of the coset representative g throughout this paper. In [21] a GS superstring action in the AdS 5 ×S 5 background was derived based on an alternate version of the coset superspace construction in terms of GL(4|4). The pure spinor superstring action in this coset superspace construction was given in [22]. This action is expected to make it more transparent to examine the surface term for the BRST transformation of the action and to derive possible D-brane configurations in the AdS 5 ×S 5 background.
The method used in this paper can be applied easily to a superstring in the other background, for example the superstring in the type IIB pp-wave background [23] [24]. The result will be consistent with the one obtained by the boundary κ-invariance of the open GS superstring [15,20] and by examining equations of motion for a D-brane [14].
It is also known that in the presence of a constant flux, the boundary condition to ensure the κ-invariance of the GS superstring action leads to possible (non-commutative) D-branes [25]. Furthermore the boundary condition to ensure the κ-invariance of the supermembrane action leads to the self-duality condition for the three-form flux on the M5-brane world-volume [26]. The same result is expected to be obtained by using the pure spinor supermembrane action [27]. We hope to report this issue in another place [28].
Finally let us comment on a characterization of the Wess-Zumino (WZ) action. The WZ action is necessary for the κ-symmetry of the action, and then halves fermionic degrees of freedom on the world-volume so as to match bosonic and fermionic degrees of freedom.
It is shown that the WZ term of a (D)p-brane in flat spacetime is characterized as a nontrivial element of the Chevalley-Eilenberg (CE) cohomology in [29] for p-branes, and in [30,31] for Dp-branes. In [32], Dp-brane actions in the extended pure spinor formalism [2] are characterized as a non-trivial element of the BRST cohomology of the extended BRST symmetry. It is interesting for us to extend this analysis to the Dp-brane action in the AdS 5 ×S 5 background. and where Γ A (A = 0, 1, · · · , 9) are 32 × 32 gamma matrices, and Q I = Q I h + (I = 1, 2) are a pair of Majorana-Weyl spinors. We introduced ǫ IJ = We have defined I = Γ 01234 and h ± = 1 2 (1 ± Γ 11 ) with Γ 11 ≡ Γ 012···9 . The charge conjugation matrix C satisfies CΓ A = −Γ T A C. The AdS 5 isometry is generated by P a an M ab (a, b = 0, 1, · · · , 4), while the S 5 isometry is by P a ′ and M a ′ b ′ (a ′ , b ′ = 5, 6, · · · , 9).
In this notation, the left-invariant Cartan one-forms are given as The currents used in (2.5) are related to the above objects by where we have replaced (q 1 , q 2 ) with (q α ,qα) and correspondingly (θ 1 , θ 2 ) with (θ α ,θα).