Notes on worldvolume supersymmetries for D-branes on AdS5×S5 background

We revisit the 1/2-BPS D-branes on the AdS$_5\times$S$^5$ background. Based only on the classification of 1/2-BPS D-branes obtained by the covariant open string description, we consider various purely static configurations of D-branes without any worldvolume flux on the AdS$_5\times$S$^5$ background. Under the covariant $\kappa$ symmetry fixing condition, we investigate which part the spacetime supersymmetries is preserved on the D-brane worldvolume and obtain the associated worldvolume supersymmetry transformation rules to leading order in the worldvolume fluctuating fields. It is shown that, for purely static configurations without any worldvolume flux, only the AdS type D-branes, in which the AdS radial direction is one of worldvolume coordinates, are 1/2-BPS.


Introduction
D-brane in a given background containing AdS spacetime is an interesting object to explore.
It carries the information about the open string sector in the background. In the AdS/CFT correspondence [1,2], a certain D-brane configuration on the background involving the AdS spacetime is related to the defect conformal field theory (dCFT) [3,4]. For example, a certain D5-brane on the AdS 5 ×S 5 background is dual to the three dimensional dCFT of the N=4 SYM theory [4]. In a suitable approximation, D branes on curved spacetime can be described by Born-Infeld type action. The Born-Infeld type action of supersymmetric D-brane has an important symmetry, κ symmetry. By a suitable gauge fixing, one can obtain the supersymmetric worldvolume theory on the Born-Infeld type action of D-brane. In Ref. [5], by choosing a static gauge combined with a suitable κ gauge fixing, the supersymmetric worldvolume theory of D-brane in flat space is obtained, with the explicit supersymmetric transformation worked out.
One can certainly adopt the same strategy to D-branes on the background involving the AdS spacetime. In fact, κ symmetric action of D-branes on the AdS 5 ×S 5 background was obtained using supercoset approach. One can guess that again by taking the static gauge with suitable κ gauge fixing condition one can obtain the worldvolume theories of supersymmetric D-branes on the AdS 5 ×S 5 background. 1 However there is a subtlety in this program. In [6] , the author considered the D3-brane whose world volume spans the four directions in AdS 5 other than the radial direction but could not find the suitable gauge fixing condition leading to a supersymmetric worldvolume theory on the D-brane. 2 Note that the configuration is supersymmetric since the D3 brane of interest is parallel to the D3 branes, whose near horizon geometry turns into AdS 5 ×S 5 . Indeed in [7], the D3 brane is shown to satisfy the generalized calibration, hence is supersymmetric. 3 Analogous result is worked out at [9], where they consider M2 brane in AdS 4 ×S 7 where M2 brane is parallel to M2 branes which make AdS geometry. They show that Killing spinor gauge is incompatible with the static gauge of the M2 brane worldvolume action. Given these results, one might wonder if there are some restrictions on the possible supersymmetric worldvolume thoery on the AdS spacetime. It turns out that this problem is intimately related to the classification of supersymmetric D-brane embeddings into the AdS spacetime.
In this paper we are looking for the supersymmetric worldvolume theories of 1/2-BPS D-branes on the AdS 5 ×S 5 background . Since the worldvolume supersymmetry is of our concern, it is natural to take the probe brane analysis. In order to study the worldvolume theory of 1/2-BPS D-branes, we start from the data obtained from the covariant open string description of supersymmetric D-branes developed in [10,11]. The covariant description can be applied to any background if the superstring action on it is given and, as the first nontrivial application, has led to the classification of 1/2-BPS D-branes in some plane wave backgrounds [11,12]. 4 As for the AdS 5 ×S 5 background, the classification has been worked out in [15,16] and Table 1   same data listed in Table 1 have been also obtained in [17] in the context of pure spinor formalism [18].
The covariant open string description provides us a definite guideline for further study of supersymmetric D-branes, although it gives no more information about D-branes other than the classification data. Starting from the Table 1, we consider all possible types of corresponding D-brane configurations and use a suitable static gauge for the D-brane worldvolume diffeomorphism. For the fermionic κ symmetry of the D-brane action, the covariant κ symmetry gauge is adopted. For each of the configurations, we identify the worldvolume supersymmetry realized on the D-brane worldvolume and obtain the associated worldvolume supersymmetry transformation rules for the worldvolume fields.
We restrict ourselves to the purely static Lorentzian D-brane configurations without any worldvolume flux. Here the Lorentzian means that the worldvolume time is identified with that of the background spacetime. Thus, the configurations for (0,0) and (0,2) D-branes of Table 1 are not considered. As one may realize, there are twelve types of Lorentzian configurations. Six of them correspond to the AdS type D-branes in which the AdS radial direction is one of worldvolume coordinates, and the remaining six are of non-AdS type in which the AdS radial direction is transverse to the D-brane worldvolume. We will treat the AdS and non-AdS type branes separately. It turns out that, for purely static configurations without any worldvolume flux, only AdS type branes admit the supersymmetric worldvolume theories. Note that the corresponding D-brane configurations are obtained from the supersymmmetric intersecting D3⊥Dp brane configurations in flat spacetime after turning D3 branes into AdS geometry [4]. For non-AdS type branes, the analysis suggests that world volume fluxes should be turned on or some motions in transverse directions should be considered to have the supersymmetric world volume theory. Our work suggests that only D-branes tabulated at Table 1 admit the supersymmetric worldvolume theory. In order to confirm it, the worldvolume theories of non-AdS type branes should be worked out, which is beyond the scope of this paper.
The organization of this paper is as follows. In Sec. 2, we describe the way of realizing the worldvolume supersymmetry for a given D-brane configuration after reviewing some necessary elements. In Sec. 3, we investigate the worldvolume supersymmetry for the AdS type D-branes. Then the non-AdS branes are considered in Sec. 4. The discussion with our conclusion follows in Sec. 5. Finally, appendix A contains our notation and convention with the expressions of superfields.

Generalities
In this section, we briefly review the AdS 5 ×S 5 background with its associated Killing spinor and the symmetries of Dp-brane action. We then describe how to identify the supersymmetry realized on the brane worldvolume for a given brane configuration.

AdS 5 ×S 5 background
In the Poincaré patch coordinates, the metric for the AdS 5 ×S 5 geometry is written as is the metric of S 5 parametrized by five angular coordinates φ α (α = 1, . . . , 5), with ranges of 0 ≤ φ 1 , φ 2 , φ 3 , φ 4 ≤ π and 0 ≤ φ 5 ≤ 2π. Here, we have taken the common radius R of the AdS 5 and S 5 to be one, R = 1. The ten dimensional coordinates are aligned as and from the metric (2.1) the zehnbein is chosen to be In addition to the metric (2.1), another constituent of the AdS 5 ×S 5 background is the Ramond-Ramond five form field strength given by (2.5) The AdS 5 ×S 5 background composed of (2.1) and (2.5) is maximally supersymmetric. Its supersymmetry structure is encoded in the spacetime Killing spinor η I , which is the solution of the spacetime Killing spinor equation D µ η I (X) = 0 for the AdS 5 ×S 5 background. 5 The Killing spinor equation has been solved in Refs. [19,20], and its solution is expressed in a simpler form if we split η I as where η I ± are defined by with the projection operator In this splitting, we see that η 1 ± and η 2 ± are not independent from each other because Thus, to avoid this redundancy, it is convenient to define to which η 1 and η 2 are related by where ǫ ± are constant spinors, is a spinorial function of five angles of S 5 given by We note that, since η I is taken to have positive chirality in this paper, ǫ + (ǫ − ) is a positive (negative) chirality spinor, and has sixteen independent free components. 7 5 The explicit form of the covariant derivative D µ can be found in Eq. (A.5). 6 While the complex spinor notation is adopted in [19], we use the real or Majorana-Weyl spinor notation throughout the paper. 7 The definition of Γ 11 is given in (A.6).

Symmetries of Dp-brane action
The Dp-brane action S p is composed of the Dirac-Born-Infeld (DBI) and the Wess-Zumino (WZ) parts: Here, M p+1 represents the Dp-brane worldvolume and M p+2 is a (p + 2)-dimensional manifold whose boundary is identified with M p+1 , that is, ∂M p+2 = M p+1 .
In the DBI part, G ij is the pullback of the AdS 5 ×S 5 supergeometry described by the Cartan one-form vector superfield Lâ onto the worldvolume, where i, j are the worldvolume indices (i, j = 0, 1, . . . , p). F ij is a combination of the field strength F ij of the worldvolume gauge field A i (F ij = ∂ i A j − ∂ j A i ) and the pulled-back background NS-NS two-form superfield B. In the form notation, F is given by where L I is the Cartan one-form spinorial superfield and the subscript s in the superfields means that the fermionic coordinate Θ inside the superfields is replaced by Θ → sΘ. In the WZ part, H p+2 is the supersymmetric closed (p + 2)-form consisting of various combinations of the Cartan one-form superfields and F . 8 The Dp-brane action has three manifest symmetries. Firstly, it is invariant under the worldvolume reparametrization where λ i (σ) is the local reparametrization parameter. Under this, the worldvolume fields transform as follows.
We note that both of Θ I and X µ are scalars from the worldvolume viewpoint. Secondly, the action is spacetime supersymmetric under the transformations where η I is the Killing spinor of Eq. (2.11) with Eq. (2.12). More precisely, the DBI and the WZ parts of the action are supersymmetric separately. Actually, supersymmetry is natural because the super coset method respects the background supersymmetry by construction.
If we expand the spacetime supersymmetry transformation of Eq. (2.21) in terms of Θ, we get where e μ a is the inverse of the zehnbein eâ µ given in Eq. (2.4) and The transformation for the worldvolume gauge field A i is determined from the invariance of The last one is the local fermionic κ symmetry, which is in some sense the most important one since it guarantees the worldvolume supersymmetry after gauge fixing. The κ symmetry transformation rules are given by where the transformation parameter κ satisfies, for the κ symmetric projection Γ (p) , The κ symmetry projection is basically the pullback of various gamma matrix products onto the Dp-brane worldvolume and, for the type IIB case, its explicit expression [22] is where γ i 1 ···in = γ [i 1 · · · γ in] and γ i is the pullback of Γâ, γ i = Lâ i Γâ. The important properties of Γ (p) are and, as is verified with the τ matrices (A.3), Γ (p) can always be put into the form (2.28) The two blocks β (p) , which is Γ (p)2 = 1 of (2.27), and their expressions for each p will be given in the next section. If we now write down the κ symmetry transformation rules for the worldvolume fields by expanding Eq. (2.24) in terms of Θ, then they are

Worldvolume supersymmetry
If a given configuration or embedding of Dp-brane in a specific supersymmetric background preserves some fraction of the background supersymmetries, then the preserved supersymmetries should be respected on the D-brane worldvolume theory. How are they realized and described?
One way to answer this practical question is to follow the procedure developed in Ref. [5]. In this subsection, following Ref. [5], we describe how to identify the supersymmetry on the worldvolume and give the associated supersymmetry transformation rules for the worldvolume fields.
For a given Dp brane, we first consider its configuration based on the data of Table 1 and align the worldvolume coordinates with those of spacetime as which is equivalent to specify indices (ℓ 0 , ℓ 1 , . . . ℓ p ) among ten spacetime coordinates (2.3).
This is nothing but the static gauge which fixes the worldvolume reparametrization symmetry (2.19). Since the Lorentzian branes are of our concern, X ℓ 0 will be always X 0 (= x 0 ), that is, ℓ 0 = 0 or x 0 (σ) = σ 0 . The remaining spacetime coordinates transverse to X ℓ i will be denoted by X f , which describe the transverse fluctuations of Dp-brane. Since the brane may be placed in some transverse position, it is convenient to split X f as 31) 9 The expression for δ κ F itself has bee derived in Ref. [23].
where X f 0 denote the constant transverse position of brane andX f are the fluctuations around them. We note that we could consider more general configurations where X f depend on X ℓ i as X f = X f (X ℓ i ). One typical example would be the constant motion along certain transverse directions: ∂ 0 X f = constant. In this paper, however, we will restrict ourselves to purely static configurations and turn off any worldvolume fluxes As alluded to in the last subsection, the Dp brane has a local worldvolume symmetry, the κ symmetry. As we do in a theory with local gauge symmetries, we should fix it properly before doing any actual calculation. Here, we take the covariant κ symmetry fixing condition given by One possible way of resolving this situation is to introduce the compensating κ and the worldvolume reparametrization transformations and define a new transformation δ as The parameters κ and λ of the compensating transformations are determined in terms of η I such that the new transformation δ keeps the gauge-fixing conditions, that is, δΘ 1 = 0 and δX ℓ i = 0. They can be found order by oder in θ and are, at the leading order, − appears due to Eq. (2.28). In this way, we have the transformation δ consistent with the gauge-fixing conditions and interpret it as the worldvolume supersymmetry.
Let us now turn to the theory on the Dp-brane worldvolume. From the viewpoint of the worldvolume theory, the static gauge describing the embedding of the brane can be regarded as the 'vacuum' configuration. Then, as usual, the supersymmetry preserved by the 'vacuum' is specified by the free components of the supersymmetry parameter satisfying the equation δ(fermion) = 0, that is, δθ = 0, which we call the worldvolume Killing spinor equation. If we rewrite δθ = 0 by plugging (2.20), (2.22), (2.29) into (2.33) with (2.34), it becomes a fairly simple equation, 0 depends on the 'vacuum' configuration and is defined by We would like to note that the worldvolume Killing spinor equation (2.35) is an exact one because θ = 0 in the 'vacuum' configuration.
Further evaluation of (2.35) using (2.11) and (2.12) leads us to have to the right of S 0 (φ), which is done by evaluating Generically, the resulting expression is not of the form of projection operator but a sum of many gamma matrix products with coefficients composed of trigonometric functions.
However, as we will see in the next section, it becomes a projection operator for some special values of the transverse angular coordinates and can be used to pick out the free components among ǫ ± . This is interesting in a sense that the transverse position is determined by insisting on the supersymmetry in the D-brane worldvolume theory without resort to the equations of motion. 11 After identifying the supersymmetry preserved on the D-brane worldvolume, we can read off how the transformation δ acts on the remaining worldvolume fields. Let us denote the worldvolume fields collectively as Its transformation δΦ can be written as an expansion in terms of the power of Φ. If we consider the terms up to linear order in Φ, then the transformation rules for the worldvolume fields are − is also expanded as 1 is the collection of terms linear order in Φ. In the process of calculation leading to (2.40), we will encounter many trigonometric functions. Thus, for notational simplicity, we would like to define the following quantities before moving on to the next section.

AdS branes
When the radial direction of the AdS space u is one of the worldvolume directions for a given brane configuration, the brane is usually called the AdS brane since the induced metric on the worldvolume contains the AdS space. If we take a look at the Table 1 and 11 Strictly speaking, in some cases, we should also impose the non-degeneracy condition for the induced worldvolume metric.
consider the Lorentzian branes in which the time x 0 is always a worldvolume direction, we see that there are six types of AdS brane configurations. In this section, for each of them, following the procedure outlined in Sec. 2.3, we investigate the supersymmetry realized on the worldvolume and give the worldvolume supersymmetry transformation rules for the worldvolume fields . We note that the following subsections and subsubsections are selfcontained and completely independent from each other.

D1
The D1-brane configuration (2,0) of Table 1 leads us to take the static gauge as or (ℓ 0 , ℓ 1 ) = (0, 4) in Eq. (2.30), which corresponds to the AdS 2 brane. The coordinates transverse to this configuration are then In order to identify which part of the spacetime supersymmetry is preserved on the worldvolume of AdS 2 brane, we should solve the worldvolume Killing spinor equation (2.35).
What is necessary to do this is β ± , which is read off from Eqs. (2.26) and (2.28) as Having the projection operators, the next step described in Sec. 2.3 is to send Γ 0123 β 0 to the right of S 0 (φ) in (2.37). If we denote the resulting expression asΓ, we get the relation Γ 0123 β The solution of this equation is readily found to be Since other components except for those of (3.9) are undetermined, we conclude that the supersymmetry preserved on the AdS 2 brane is characterized by where α = 1, . . . , 5 andη (3.12) As for the worldvolume gauge field, the transformation rule is obtained as Finally, we get the transformation rule for the fermionic field as where α = 1, . . . , 5 and

D3
The D3-brane configuration (3,1) of Table 1 leads us to take the static gauge as 16) or (ℓ 0 , ℓ 1 , ℓ 2 , ℓ 3 ) = (0, 1,4,9) in Eq. (2.30), which corresponds to the AdS 3 ×S 1 brane. The coordinates transverse to this configuration are then In order to identify which part of the spacetime supersymmetry is preserved on the worldvolume of AdS 3 ×S 1 brane, we should solve the worldvolume Killing spinor equation (2.35). What is necessary to do this is β ± , which is read off from Eqs. (2.26) and (2.28) as  Having the projection operators, the next step described in Sec. 2.3 is to send Γ 0123 β 0 to the right of S 0 (φ) in (2.37). If we denote the resulting expression asΓ, we get the relation With the fact that Γ 0123 β (3) 0 = Γ 2349 ,Γ is evaluated by repeated use of the identity (2.38) as follows: Except for the last one, the first three choices lead to the singular or degenerate induced worldvolume metric. Thus, if one wishes to have a regular theory on the worldvolume, the last choice is quite natural and henceΓ = Γ 2389 . If we now split ǫ ± according to the eigenvalues of Γ 2389 as then the worldvolume Killing spinor equation (2.37) becomes The solution of this equation is readily found to be Since other components except for those of (3.24) are undetermined, we conclude that the supersymmetry preserved on the AdS 3 ×S 1 brane is characterized by As for the worldvolume gauge field, the transformation rule is obtained as Finally, we get the transformation rule for the fermionic field as 1 (η + +η − ) + . . . , (3.29) where α = 1, 2, 3, 4 and β One may wonder if the D-brane configuration considered above is stable since S 1 in S 5 is not a topological cycle. The similar problem was worked out by [25]. The scalar mode corresponding to slipping off the S 1 in S 5 satisfies Breitenlohner-Friedmann bound, hence it does not lead to the instability. Also the above D-brane configuration satisfies the so called generalized calibration, which is the condition for the supersymmetric cycle of D-branes to satisfy on the general supergravity background with various fluxes. [7]. These remarks hold as well for other D-brane configurations in subsequent subsections.

D5
In this subsection, we are led to consider two kinds of D5-brane configurations. The common content for them is β (5) ± appearing in the κ symmetry projection Γ (5) , which is read off from Eqs. (2.26) and (2.28) as (3.31)

(2, 4)-brane
The D5-brane configuration (2,4) of Table 1 leads us to take the static gauge as brane. The coordinates transverse to this configuration are then In the static gauge (3.46), β Having the projection operators, the next step described in Sec. 2.3 is to send Γ 0123 β (5) 0 to the right of S 0 (φ) in (2.37). If we denote the resulting expression asΓ, we get the relation Γ 0123 β (5) 0 S 0 (φ) = S 0 (φ)Γ. With the fact that Γ 0123 β (5) 0 = −Γ 12346789 = −Γ 05 Γ 11 where Γ 11 of (A.6) has been used,Γ is evaluated by repeated use of the identity (2.38) as follows: where we have used the definitions of Eq. (2.42). This shows clearly that 1 ±Γ do not have the form of projection operators. One can make them have the desired form by fixing the transverse angular position, and realize that there is a unique choice of φ 1 0 = π 2 which leads toΓ = −Γ 04 Γ 11 . If we now use the chirality property of ǫ ± in Eq. (2.14) and split ǫ ± according to the eigenvalues of Γ 04 as The solution of this equation is readily found to be Since other components except for those of (3.53) are undetermined, we conclude that the supersymmetry preserved on the AdS 2 ×S 4 brane is characterized by whereη As for the worldvolume gauge field, we obtain where α = 2, 3, 4, 5 and φ 1 = π 2 should be imposed on eâ φ α . Finally, we get the transformation rule for the fermionic field as (3.59)

D7
In this subsection, we are led to consider two kinds of D7-brane configurations. The common content for them is β ± appearing in the κ symmetry projection Γ (7) , which is read off from Eqs. (2.26) and (2.28) as (3.60)

Invariance of quadratic action
The transformation rules obtained in the previous subsections are explicit to the leading linear order in the worldvolume fluctuating fields. This means that they can be used to confirm the invariance of the quadratic action coming from the expansion of the gauge fixed D-brane action in terms of the worldvolume fields. In this last subsection, we would like to verify the transformation rules by showing the invariance of the quadratic action. However, we will not consider all the six kinds of AdS branes but take one representative, since the transformation rules have the same pattern.
We consider the (3,1) configuration of D3-brane of Sec. 3.2, the AdS 3 ×S 1 brane, as the representative. The D3-brane action is given in Eq. (2.15) for p = 3 and the closed five-form H 5 in the WZ term [23] is where m = 2, 3, α = 1, 2, 3, 4,ā = 0, 1, 4, 9, and the terms linear in the derivative ∂ φ 5 and the fermionic mass term originate from the WZ term. The metric g ij is the induced one on the worldvolume given by This induced metric is also expressed as eā i eb j ηāb, where eā i is defined by the worldvolume field independent part of the pullback of zehnbein (2.23), that is, eā i ≡ ∂ i X µ eā µ |Xf =0 . In the covariant derivative for the spinor given by ∇ i = ∂ i + 1 4 ωāb i Γāb, the spin connection is determined from eā i by using the Cartan structure equation or can be defined, like the definition of eā i , as ωāb i ≡ ∂ i X µ ωāb µ |Xf =0 from the spacetime spin connection ωāb µ . In the present case, the nonvanishing components are ω 04 = udx 0 and ω 14 = udx 1 .
We note that the Lagrangian density forx m in the quadratic action (3.88) is not of the canonical form because of the overall u 2 factor. In order to make it canonical, we take the under which the Lagrangian density for the fieldx m becomes This invariance of the quadratic action clearly shows that the transformation rules realize the supersymmetry of the worldvolume theory.
If the non-AdS brane takes a certain constant motion along a transverse direction or has non-vanishing fluxes on its worldvolume, the situation may change completely. A typical example is the giant graviton [26], one type of which is a (1,3) configuration of D3 brane and takes a constant motion along a transverse angular direction. It is known to be 1/2-BPS for some particular angular speed.

Discussion
Starting from the data in Table 1 One interesting point in the study of supersymmetric configuration is that the transverse angular position has been determined without resort to the equations of motion. The position is fixed only by requiring the worldvolume supersymmetry and sometimes the nondegeneracy of induced worldvolume metric. For example, let us consider the AdS 4 ×S 2 embedding of D5-brane of Sec. 3.3.1, which was also explored in Ref. [4] related to the holographic description of the defect conformal field theory [3]. As shown in [4], there are two solutions of the equations of motion for the transverse angular position and it turns out that only one of them leads to the supersymmetric configuration which is the same as the angular position determined in this paper as it should be. In fact, this kind of situation seems to be natural. Usually, the solution of the Killing spinor equation satisfies also the equations of motion. Thus the fact that the transverse angular position is determined in the process of solving the Killing spinor equation may not be surprising.
As shown in Sec. 4, all the static non-AdS branes without any worldvolume fluxes are not supersymmetric. We emphasize again that this may change completely when there are motions in transverse directions or the worldvolume fluxes are turned on, since we already know at least one definite example, the giant graviton. Beyond the static case, there will be lots of possibilities. Having said that, we expect that there will be a suitable classification facilitating the study of them similar to the static D-brane configurations classified as the AdS and non-AdS types.