Supersymmetric DBI Equations in Diverse Dimensions from BRS Invariance of Pure Spinor Superstring

We examine the BRS invariance of the open pure spinor superstring in the presence of background superfields on a Dp-brane. It is shown that the BRS invariance leads not only to boundary conditions on the spacetime spinors, but also to supersymmetric DBI equations of motion for the background superfields on the Dp-brane. These DBI equations are consistent with the supersymmetric DBI equations for a D9-brane.


Introduction
Dirac-Born-Infeld (DBI) theory is known as a non-linear generalization of Maxwell theory and may describe, along with the Wess-Zumino action, the low-energy effective dynamics on a single D-brane in string theory. The bosonic DBI action is derived from the world-sheet analysis of the bosonic open string [1]. A supersymmetric DBI action should be a part of the effective action on a D-brane in type II superstring theory. In the Ramond-Neveu-Schwarz (RNS) formulation, however, it is difficult to read off the target space geometry coupling to Ramond-Ramond fields, because space-time supersymmetry becomes manifest only after the GSO projection. So the RNS superstring has led to the only bosonic sector of the supersymmetric DBI action [2].
The Green-Schwarz (GS) formulation has an advantage in this direction. The Wess-Zumino term which ensures the κ-invariance of the world-volume action of a D-brane is constructed in [3]. In [4], the κ-symmetric approach, so called the superembedding formalism [5], is shown to lead to linearised supersymmetric DBI equations of motion for a D9-brane, which have the ten-dimensional N = 2 supersymmetry. Furthermore, in [6], the classical κ-invariance of an open GS superstring in an abelian background is shown to imply that the invariance of this action. In section 4.1, background superfield coupling is found by considering the modification of a vertex operator in the open pure spinor superstring. In section 4.2, we confirm that these background superfields satisfy supersymmetric DBI equations of motion. The last section is devoted to summary and discussions. In addition, we give a brief review of the covariant approach for the ten-dimensional N = 1 super Yang-Mills theory in Appendix A. We will formulate a vertex operator in the open pure spinor superstring in Appendix B. We show that our result can be derived also from improving the method used in [21] to include Dirichlet components in Appendix C.
A dot on a field denotes the τ -derivative of the field, while a prime does the σ-derivative. If there are no background fields, the surface term (2.3) can be eliminated by imposing usual boundary conditions for Dp-branes † .
Instead of imposing boundary conditions, we will consider coupling to the background superfields preserving the N = 1 supersymmetry specified by ǫ = γ 1···p ǫ. To preserve N = 1 supersymmetry, we must introduce a boundary term which eliminates (2.3).

N = 1 supersymmetry and boundary term
Here we will introduce a boundary term S b which leaves S 0 + S b invariant under the N = 1 supersymmetry up to equations of motion. For this purpose, it is convenient to introduce the following objects under the ǫ-and ǫ-supersymmetry in (2.2), respectively. By using these variables, the N = 1 supersymmetry transformations specified by ǫ = γ 1···p ǫ are represented as 3) † We must impose the same boundary condition on θ and λ since BRS transformations relate them each other. These boundary conditions also eliminate the surface term which comes from the BRS transformation of the world-sheet action S 0 . See [26,27] for related topics.
where we have used the Fierz identity.
The boundary term S b we found is where c 1 and c 2 are constants. We have introduced the followings where Π m = ∂x m + 1 2 θγ m ∂θ and Π m =∂x m + 1 2 θγ m∂ θ are ǫ-and ǫ-supersymmetry invariants, respectively. Objects in (3.5) are invariant under the N = 1 supersymmetry (up to equations of motion except for ∆ + α and y i ). The equations of motion∂θ α = ∂ θ α = 0 implies Throughout this paper, we will use them only after transformations for supersymmetry and BRST symmetry. We note that the last three terms in (3.4) are invariant under the N = 1 supersymmetry separately. This implies that they are not determined from the N = 1 supersymmetry. It is worth noting that (3.4) cannot be extracted as a dimensional reduction of the one for the D9-brane.

BRS symmetry
We shall show that the last term y i Π i + in (3.4) is required by the BRS invariance of S 0 + S b , when there is no background superfield coupling.
The action (2.1) is invariant under a pair of BRS variations, say δ 1 and δ 2 . In the presence of the boundary, these BRS variation must satisfy δ 1 = δ 2 at the boundary. This implies that the BRS transformations δ Q = δ 1 + δ 2 remain unbroken in the presence of the boundary where equations of motion (3.6) are used after the BRS transformations. Again, we find the world-sheet action S 0 is BRS invariant δ Q S 0 = 0 up to a surface term, and satisfies Let us assume that there are no background fields. In this case, the (3.8) must be eliminated by the usual boundary conditions θ α − = λ α − = 0. It is obvious to see that these boundary conditions eliminate (3.8) as expected. It should be noted that this happens only when we include the term y i Π i + in (3.4). Finally we comment on y i . Remarkably, we confirmed that S 0 + S b is independent of y i . This strongly suggests that y i should represent the position of the Dp-brane.

Supersymmetric DBI equations of motion
In this section, we will give the background coupling V in terms of superfields on a Dp-brane.
Examining the BRS variation of S 0 + S b + V , we obtain supersymmetric DBI equations of motion on the Dp-brane.

Background superfield coupling for Dp-branes
In Appendix A, we define the ten-dimensional N = 1 superfield A M = (A m , A α ). We introduce background superfields on a Dp-brane as a dimensional reduction of A M : A m = (A µ (x µ , θ + ), A i (x µ , θ + )) and A α = A α (x µ , θ + ). Obviously they are invariant under the N = 1 supersymmetry. Similarly we introduce W α = W α (x µ , θ + ) and F mn = F mn (x µ , θ + ) . We use the ten-dimensional Majorana-Weyl spinor notation throughout this paper. This means that we are considering the DBI equations with 16 supersymmetries, for example N = 4 supersymmetric DBI equations on a D3-brane.
The background coupling V used in [21] is regarded as an extension of the vertex operator of the open pure spinor superstring. We give a brief review of the vertex operator in Appendix B.
The background coupling V we introduce is mn and F (4) mnpq are some possible products of any number of vector field strengths F mn ‡ , which is consistent with analysis for D-brane boundary states [23] from the viewpoint of the pure spinor closed superstring. Needless to say, the V is invariant under the N = 1 supersymmetry. Since we have made the factor 1/(2πα ′ ) manifest in V , dimensions of these superfields differ from conventional ones. In this sense, we assign dimensions to and [F mn ] as − 3 2 , −1, − 1 2 and 0, respectively.

DBI equations from BRS symmetry
In this subsection, we will add the background superfield coupling V in (4.1) to the action S 0 + S b and then require that the BRS variation δ Q (S 0 + S b + V ) vanishes. This requirement leads to boundary conditions on spacetime spinors and conditions on background superfields.
The latter is found to be supersymmetric DBI equations of motion for them. ‡ There are no more higher forms because of the property where the sign in the right hand side depends on the chirality of λ.
We find that the BRS variation δ Q V may be expressed as Note that the supercovariant derivative on the Dp-brane is defined by Gathering (3.8) and (4.4) together, we obtain the BRS variation of S 0 + S b + V as where X m , Y m , Λ β , Z β α and Θ ± α are given as follows In the following, we will examine conditions that each term in (4.6) vanishes.
First of all, it is noted that we may fix degrees of freedom for the D-brane position by a static-like gauge We will comment on this issue later.
To achieve our purpose, first, we focus on the term Π i + X i in (4.6) which takes the form (4.14) (4.13) suggests δ Q (y i + A i ) = 0. In addition, we obtain the boundary condition on θ − as This eliminates θ α − from (4.6) completely. Hereafter we understand θ α − as (4.15). Note that (4.15) also leads toθ and imply the boundary condition on λ − This eliminates λ α − from (4.6) completely. Hereafter we understand λ α − as (4.18). Here, it is better to comment on two consequences of the boundary conditions (4.15) and (4.18). First, consider the limit α ′ → 0. The limit α ′ → 0, after rescaling A α → (2πα ′ )A α , This is one of the DBI equations.
Let us return to the subject. Thirdly, the term ∆ + β Λ β in (4.6) is examined. We see that Λ β = 0 reduces to This is one of the DBI equations on a Dp-brane. This equation ensures that conditions (4.15) and (4.18) are consistent with BRS transformations δ Q θ α − = λ α − and δ Q λ α − = 0. As was done in [21], it is convenient to introduce a covariant derivative D α by Applying it to 1
Fifthly, we consider terms including Π µ + in (4.6), Π µ where the second term comes from (4.16). It is straightforward to see that it is eliminated by Finally, we consider terms includingθ α + in (4.6),θ α where the second term comes from (4.16). These terms are eliminated by By eliminating ∂ m A β + D β A m by (4.28), it reduces to Finally substituting (4.20) into the expression in the curly braces in (4.30), we obtain As a result, we have obtained not only boundary conditions (4.15) and (4.18), but also independent equations for background superfields (4.28), (4.31) and (4.20) which eliminates (4.6). We note that c 1 and c 2 can be absorbed into redefinitions of W α and F β α as 1 c 1 W α → W α and 1 c 2 F β α → F β α . So we will set c 1 = c 2 = 1 without loss of generality § . Summarizing, we have obtained supersymmetric DBI equations of motion on a Dp-brane In the last equation, the index µ may be replaced with m because ∂ i W β = 0. Now it is manifest that our DBI equations on a Dp-brane can be expressed in a ten-dimensional covariant fashion. In other words, our result coincides with the dimensional reduction of those for a D9-brane, though the ten-dimensional covariance was absent in the beginning of our analysis.

Summary and discussions
We have examined the BRS invariance of the open pure spinor superstring in the presence of background superfields on a Dp-brane. It was shown that the BRS invariance leads not only to boundary conditions on the spacetime spinors, but also to supersymmetric DBI equations of motion for the background superfields on a Dp-brane. These DBI equations precisely coincide with those obtained by a dimensional reduction of the supersymmetric DBI equations for the abelian D9-brane given in [21,22].
We have introduced the boundary term S b and the background coupling V . Both are determined by the BRS symmetry. In fact, S b was shown to satisfy δ Q (S 0 + S b ) = 0, when we take the limit α ′ → 0 and turn off the background couplings. As for V , we have shown that the conditions for δ Q (S 0 + S b + V ) = 0 reduce to the dimensional reduction of the super-Yang-Mills equations when α ′ → 0. In fact, taking the limit α ′ → 0, after with an appropriate dimensional reduction.
We note that the ten-dimensional Lorentz covariance is manifestly broken by the boundary term S b as well as the background coupling V . However the obtained DBI equations can be expressed in a covariant form. This implies that our result is consistent with that for a D9-brane.
We expect that we can extend our result so that the BRS invariance should lead to supersymmetric non-Abelian DBI equations of motion on a Dp-brane. We would like to report this issue in the near future [30].
As an alternative to our study, non-abelian deformations of the maximally supersymmetric Yang-Mills theory can be specified based on spinorial cohomology [31], which may be closely related to the pure spinor fields in ten-and eleven-dimensional spacetime [32][33][34].
The structure of higher-derivative invariants in the maximally supersymmetric Yang-Mills theories are studied in [35]. Moreover, in [36,37] the pure spinor superspace formalism is developed, which contains not only (minimal) pure spinor variables but also non-minimal pure spinor variables [38]. This enables us to construct the BRS invariant action for the tendimensional supersymmetric DBI theory. Recently, this off-shell action is studied further in [39,40]. It is interesting to pursue these issues from the open string point of view.
On the other hand, the classical BRS invariance of a closed pure spinor superstring in a curved background is shown to imply that the background fields satisfy full non-linear equations of motion for the type II supergravity [41]. This is similar to the result for the classical κ-invariance of a closed Green-Schwarz superstring [42]. Moreover, recently in [43] the classical κ-invariance also leads to the generalized type II supergravity equations of motion ¶ whose solutions originally have found out in the context of integrable deformations of AdS 5 ×S 5 sigma models [44]. It is also interesting to consider whether the generalization of DBI equations can be derived analogously from the κ-or BRS-invariance of an open superstring.
An immediate task is to clarify contribution of the dilaton superfield to Bianchi identities.
In that case we need to investigate closely the DBI equation corresponding to I mnα = 0 in the super Yang-Mills theory as we see in Appendix A. This equation is also useful to confirm that our result agrees with the one which comes from the bosonic part of the DBI action.
Finally, it is interesting for us to calculate quantum higher-derivative corrections to our result by analyzing the quantum BRS invariance of the open pure spinor superstring. ¶ See [45] for further investigations based on double field theory.

A Ten-dimensional N = 1 super Yang-Mills space
We will review the ten-dimensional N = 1 super-Yang-Mills theory [46].
where D α is the supercovariant derivative defined by where T M N R are flat torsion tensors whose components are fixed to zero except for T m αβ = γ m αβ . According to this definition, these field strengths are invariant under the gauge transformations with a superfield parameter Ω For the on-shell super Yang-Mills theory, we might adopt a constraint [32] (see also [33]) which implies If we consider a dimensional reduction to four-dimensions, we see that this constraint reduces to the one in the four-dimensional N = 4 super Yang-Mills theory [47].
In the following, let us solve the Bianchi identities represented as The first identity I αβγ = 0 implies Thanks to the Fierz identity, we find that the field strength F mα must take the form of In other words, Next the second identity I mαβ = 0 together (A.9) implies Multiplying this by γ αβ p , we find that which is equivalent to The third identity I mnα = 0 implies Taking (A.13) into account, (A.14) yields the result Furthermore, multiplying (A.15) by γ nγα ∇ γ we find The (A.16) and (A.15) imply the Maxwell equation for the gauge field ∇ m f mn = 0 and the Dirac equation for the gaugino γ m αβ ∇ m ξ β = 0, respectively. Finally, the remaining identity I mnp = 0 implies and it suggests that F mn is just the curl of a gauge field A m ; The θ-expansion of these superfields are studied in [48].

B Massless vertex operator for pure spinor open superstring
We present a review of the vertex operators in the open pure spinor superstring [20] (see also [29]). For simplicity, we focus on the left-moving sector only.
We consider a ghost number 1 massless vertex operator given by where A α (x, θ) is a spinor superfield. The BRS transformation law is represented as where Q denotes δ 1 in section 3.1. Note that Q 2 ω α = −Π m (γ m λ) α turns out the gauge transformation for ω α . Then cohomology condition, QU = 0 up to the gauge transformation where Ω(x, θ) is a gauge parameter and the derivative D α is given in (A.2).
To derive (B.3), we use the pure spinor constraint for the commutative bispinor λ λ α λ β = 1 2 5 5! γ αβ mnpqr λ γ γ mnpqr Next, we derive an integrated vertex operator such as V = dz V. Recalling the RNS formulation, V is given as the anticommutator of the unintegrated vertex operator U and the b-ghost. However, in the pure spinor formulation, the reparametrization b-ghost is unclear without introducing the non-minimal part [38] . Fortunately, the above facts can be rephrased in terms of the BRS charge Q as * * We find the vertex operator V takes the form of where N mn = 1 2 λγ mn ω is the ghost Lorentz current. Indeed, since where (B.11) is used. If (B.9) is contracted with (γ mnpqr ) αβ , we obtain the equation of motion for A α in (B.3). Contraction of (B.9) with γ αβ n also leads to 14) The non-minimal pure spinor formalism extended to the Maxwell background is investigated in [49].

C BRS charge conservation
We will derive the supersymmetric DBI equations by modifying the method used in [21] to include the Dirichlet components.
We require that the general variation δ(S 0 + S b + V ) vanishes. This leads to boundary conditions in the presence of background superfields. Under these conditions, it is shown that the BRS charge conservation implies superfield equations for DBI fields.
Let us begin to examine a general variation of the world-sheet action S 0 in (2.1), its tendimensional N = 1 supersymmetry counter-term S b in (3.4) and the background coupling V in (4.1). We find that variations δ(S 0 + S b ) and δV may be expressed as where δy µ defined by is invariant under the N = 1 supersymmetry. To lead the representation (C.2), we have used equations of motion (3.6). We also see that δ(S 0 + S b )/δy i = 0 as mentioned in section 3.
To obtain boundary conditions from δ(S 0 + S b + V ) = 0, first we focus on the terms with δ∆ + α and δω + α , and derive They also lead tȯ Next, examining the terms with δλ α + in δ(S 0 + S b + V ) we find Boundary conditions for λ α − in (C.4) and ω − α in (C.6) are consistent with the ghost number charge conservation λ α ω α = λ α ω α , where " " means "evaluated at the boundary". On the other hand, we can eliminate the terms with δ Π i + in δ(S 0 + S b + V ) by fixing the static-like gauge (4.13). After substituting above conditions into δ(S 0 + S b + V ) = 0, we examine the terms with δy µ + and δθ α + . They lead to complicated boundary conditions The (C.7) is regarded as a modified Neumann boundary condition. Boundary conditions for ω − α in (C.6) and d − α in (C.8) must be consistent with the BRS transformation δ Q ω − α = d − α up to the Λ-gauge transformation in section 2. In the following discussion, we will absorb c 1 and c 2 by rescaling W α → c 1 W α and F β α → c 2 F β α . To extract DBI equations, we impose the following relation for BRS currents which implies BRS charge conservation 0 = ∂ τ Q total = dσ ∂ τ (j τ BRS ) = dσ ∂ σ (j σ BRS ) = dσ ∂ σ (j z BRS − jz BRS ) = λ α d α − λ α d α . (C.10) Then we assume the Dirichlet boundary condition This is parallel with the Neumann boundary condition in (C.7) and just the derivation of the static-like gauge (4.13) respect to the time-coordinate τ .