Open-closed superstring amplitudes using vertex operators in $\mathrm{AdS}_5 \times \mathrm{S}^5$

Using the pure spinor formalism, a particular superstring scattering amplitude involving one closed string and $N$ open string vertex operators in $\mathrm{AdS}{}_5 \times \mathrm{S}^5$ is studied. It is shown that the tree-level amplitude containing one supergravity state and $N$ super-Yang-Mills states located on D3-branes near the AdS${}_5$ boundary can be expressed as a $d=4$ ${\cal N}=4$ harmonic superspace integral in terms of the supergravity and super-Yang-Mills superfields.


Introduction
In order to study superstrings in an AdS 5 × S 5 background [1], it is possible to use both the Green-Schwarz [2,3] and the pure-spinor [4,5] formalisms. Although the superstring action is known in both formalisms, the explicit superfield construction of the vertex operators of the theory is still an open problem. Vertex operators correspond to physical states in string theory, and knowing their expressions is necessary to compute scattering amplitudes. Even though the RNS formalism cannot be used to describe AdS 5 × S 5 , since it is a Ramond-Ramond background, flat-space RNS vertex operators have been used to compute scattering amplitudes in that background in certain limits [6][7][8]. In the pure-spinor formalism, the first work on vertex operators in AdS 5 × S 5 was [9]. There the authors have constructed massless vertex operators corresponding to on-shell fluctuations around that background, but the expansion in powers of θ was not computed and the connection between the vertex operators and the duals of the half-BPS operators was not found.
Recently [10], a step in that direction was taken. Using the pure-spinor formalism, vertex operator expressions have been found for a particular case, namely the massless states (supergravity) in the limit z → 0, i.e. close to the AdS boundary (z is the radial coordinate of AdS 5 ). The vertex operators are in the ghost-number +2 cohomology of a BRST operator, and come in a family {V (N ) } (N = 1, 2, . . .) such that V (N ) is dual to a half-BPS operator involving N super-Yang-Mills fields.
Half-BPS operators can be described in an elegant manner in harmonic superspace as [11][12][13] where W ij is the N = 4, d = 4 Sohnius superfield strength [14]. In addition, the supergravity vertex operators V (N ) in [10] were written in terms of harmonic superfields T (4−N ) which where shown by Heslop and Howe [12] to couple naturally to W (N ) via This led to the following conjecture: the tree-level (disk) scattering amplitude with one closed string supergravity vertex operator V (N ) ∝ T (4−N ) in the bulk and N open string SYM vertex operators located on D3-branes near the AdS 5 boundary would be proportional to the coupling (1.2).
In this paper, we will prove this conjecture is indeed true. Using (super)symmetry and BRST arguments, we will show that can be written as (1.2). Here V SYM is the unintegrated vertex operator of SYM, U SYM is the integrated one and the "D3-brane" subscript indicates that these vertex operators are located on D3-branes parallel and close to the AdS 5 boundary. Each of the vertex operators depends on the four transverse D3-brane directions x a in the plane-wave form e ik (r) a x a (ξr) . However, when scattered with the closed string state represented by T (4−N ) in the limit where the D-branes approach the AdS 5 boundary, there are no poles in k (r) · k (s) and the amplitude only depends on k (r) a through the usual conservation term δ 4 ( r k (r) a ). So it can be expressed as a local integral over d 4 x as in (1.2).
In principle, the complete expression for the supergravity vertex operators is needed to compute scattering amplitudes, while, as mentioned above, only their leading-order behavior near the boundary is known. However, in the limit where the D-branes approach the AdS 5 boundary, the leading-order behavior is sufficient.
To understand why the structure of (1.3) is so simple, note that W (N ) of (1.1) is the half-BPS super-Yang-Mills operator which is dual to the supergravity state represented by T (4−N ) . For half-BPS states, the duality relation between super-Yang-Mills operators and closed string states is protected against corrections and is given in harmonic superspace by the relation of (1.2). This paper is organized as follows. In section 2, we give a brief review of the computation of the zero-mode cohomology of the AdS 5 × S 5 BRST operator, obtaining explicit superfield expressions for the behavior of supergravity vertex operators near the boundary of AdS 5 . In section 3, we use the pure-spinor prescription and supersymmetry to compute the above-mentioned tree-level amplitude, first in the case N = 1 and then extending the result to N > 1. Finally, we summarize our results in section 4. The appendix contains notations, conventions and further information useful for the reader.
2 Review of zero-mode cohomology in AdS 5 × S 5 In this section, we briefly review the computation of the AdS 5 × S 5 supergravity vertex operators done in [10].
In the pure-spinor formalism, physical vertex operators must be in the cohomology of the BRST operator Q. In [10], the authors computed Q for the AdS 5 × S 5 background using the coset instead of the more usual coset PSU(2,2|4) SO(4,1)×SO (5) . The AdS 5 superspace is parameterized by five bosonic variables denoted z and x a for a = 0 to 3 and thirty-two fermionic variables denoted θ αi ,θα i , ψ β j ,ψβ j for α,α = 1 to 2 and i, j = 1 to 4. We use the standard d=4 two-component spinor notation as described in Appendix A.1. These variables appear in the PSU(2,2|4) SO(4,1)×SO(6) coset representative as where P a , q αi ,qα i are generators of the N = 4, d = 4 supersymmetry algebra, s j β ,sβ j are the N = 4, d = 4 superconformal generators and ∆ is the generator of dilatations. With this choice of coset representative, the boundary of AdS 5 is located at z = 0 and, at the boundary, the variables x a , θ αi ,θα i transform in the usual N = 4, d = 4 superconformal manner under the action of global PSU(2, 2|4) transformations.
The S 5 is parameterized by an SO(6) vector y I for I = 1 to 6 satisfying y I y I = 1. The SO(6) Pauli matrices (ρ I ) ij described in Appendix A.2 can be used to define y ij := (ρ I ) ij y I , which satisfies y ij = 1 2 ε ijk y k and y ij y jk = δ k i . The final ingredients needed to study the pure-spinor string in AdS 5 × S 5 are the leftand right-moving ghost variables λ, λ and their conjugate momenta w, w. The λ's are pure spinors, i.e. they satisfy λγ µ λ = 0 and λγ µ λ = 0 for µ = 0 to 9. Note these expressions have been written in ten-dimensional notation where λα and λα are chiral spinors (α = 1 to 16) which can be decomposed into SO(3, 1) × SO(6) spinors (λ αi ,λα j ) and ( λ αi ,¯ λα j ) in the usual manner. The gauge invariance under δw = (γ µ λ) Λ µ for any Λ µ implies that w can only appear in combinations of the gauge-invariant quantities which are respectively the SO(9, 1) Lorentz currents in the ghost sector and the ghostnumber current. Of course, similar expressions hold for the hatted quantities.

Worldsheet action
To construct the BRST-invariant worldsheet action using the coset (2.1), we need to define the left-invariant current J = g −1 dg, taking values in the PSU(2, 2|4) Lie algebra. Here d = dζ ∂ ∂ζ +dζ ∂ ∂ζ and the variables ζ,ζ parameterize the string worldsheet. The components of this current are defined via where K a are the generators of special conformal transformations in four dimensions, R j k are the SU(4) R-symmetry generators and the dots stand for terms proportional to generators in the isotropy group of AdS 5 , i.e. in SO(4, 1). Analogously, one can defineJ a ,J ∆ , . . . through the calculation of g −1 ∂ ∂ζ g. In terms of these currents, the matter part of the worldsheet action is given by where (∇y) jk = ∂y jk + 2J [j y k] , ∂ := ∂/∂ζ. One can show that this action is in agreement with the one written in terms of the usual coset (??) by using the SO(6) gauge invariance to gauge-fix y I = δ I 6 in S matter and then comparing with the well-known result in [5], for example.

The BRST operator and its cohomology
In the last subsection we introduced the worldsheet action for the pure-spinor superstring in an AdS 5 × S 5 background, which is given by S = S matter + S ghost . This action is invariant under the BRST transformations generated by To simplify the analysis of the cohomology, it is convenient to express the BRST charge in terms of the worldsheet variables and their canonical momenta, defined as P x = δS δ(∂τ x) , P z = δS δ(∂τ z) and so on. Here τ is the variable associated with the time direction of the worldsheet, whereas σ is associated with the space direction. In our conventions, After substituting the currents, the operator Q can be organized in the form where Q n ∝ z n . Near the boundary of AdS 5 , it is also possible to expand a physical vertex operator V as where V n ∝ z n and d 0 is the minimum degree of V , i.e. V d 0 is the leading term in the expansion of V near the boundary. Equations (2.11) and (2.12) imply that, in order to compute the cohomology of Q, one can first compute the cohomology of Q − 1 2 , then compute the cohomology of Q 1 2 restricted to that of Q − 1 2 , and so on. The reason is that, collecting the terms with the same power of z, one has , and so on. where We do not consider terms in Q containing σ-derivatives, since we are only interested in its zero-mode cohomology. In other words, we take the limit in which the string length goes to zero (or the tension goes to infinity), which corresponds to supergravity (massless states). Because of the usual quartet argument, we assume the zero-mode cohomology of Q − 1 2 is independent of λ + . 1 Moreover, the states in the cohomology depend on ψ only through λ − γ µψ , whereψ := y J (γ J ψ) and It is easy to see that λ − γ µψ is annihilated by which vanishes because of the pure-spinor conditions for λ and λ. One can also show that This identity implies that, when considering states in the cohomology of The next step is to compute the zero-mode cohomology of Q 1 2 + Q 3 2 + · · · restricted to states in the cohomology of Q − 1 2 . This means we can neglect terms containing λ + and we can consider λ − a pure spinor. It turns out that Q . This is because the terms depending on ψ in their expansions cannot be expressed in terms of the λ − γ µψ . Thus, the zero-mode cohomology of Q near the boundary of AdS 5 is determined by Q − 1 2 and Q 1 2 only. Writing the canonical momenta as derivatives, the part of Q 1 2 that is relevant for us is, then, where Dα = ∂ ∂θα + (θγ a )α∂ a is the dimensional reduction of the d=10 supersymmetric derivative and we have dropped the minus superscript from λ − . The second term in (2.18) is understood not to act on λγ µψ , even thoughψ depends on y.
In order to express the vertex operators in a convenient way using harmonic superspace, we need to introduce non-minimal pure-spinor variables [15]. They consist of a bosonic spinor λα and a fermionic spinor rα, as well as their conjugates wα and sα. These variables satisfy the constraints λγ µ λ = 0 , λγ µ r = 0 . The first of these equations implies λα is a pure spinor.
After introducing the non-minimal variables, we need to modify the BRST operator Q 1 2 as: (2.20) The addition of this term implies, using the quartet argument, that the cohomology is independent of the non-minimal variables.
Then, for arbitrary N , it was shown in [10] that Q 1 2 + wαrα annihilates the following vertex operator: is a G-analytic superfield of harmonic U(1) charge 4 − N , and du denotes an integral over the compact space SU(4)/S(U(2)×U(2)) parameterized by the harmonic variables u I i . See appendix B for more information on harmonic superspace. The operators Ω (n) in (3.2.2) are defined by ) ) ) It is easy to check that the 2z ∂ ∂z + y k ∂ Moreover, the Ω (n) 's have been designed so that the following equations are satisfied:

Connection to N = 4 SYM
If the operators (3.2.2) correspond to supergravity states in AdS 5 × S 5 , then AdS/CFT predicts they should be related to half-BPS states in N = 4 SYM. Indeed, we can make this relation explicit by making use of the harmonic superspace. Consider the following family of gauge-invariant operators introduced in [11]: [14] and the trace is taken over the gauge group. In components, we have iσ abθ j f ab (x) + · · · , (2.25) where φ ij , ξ,ξ and f ab are, respectively, the N = 4 SYM scalars, chiral and anti-chiral gluinos and gluon field-strength. It is easy to see W (N ) describes a gauge-invariant half-BPS operator constructed from N SYM fields.
For each value of N , one can show that W (N ) is an analytic superfield. Thus, it is possible to define a superfield dual to W (N ) through the coupling In [12], these superfields U (4−N ) were shown to be in one-to-one correspondence with the chiral superfields describing type IIB supergravity states in AdS 5 × S 5 . It is natural to identify U (4−N ) with the T (4−N ) of (3.2.2), since they have exactly the same properties. One way to see it is consistent is by first noting that the coupling (2.26) is invariant under the transformation Then one can show that, when T (4−N ) changes according to (2.27), V (N ) changes by a BRST-trivial amount.
Thus, the superfield T (4−N ) appearing in the expression for the vertex operator V (N ) is dual to the half-BPS operator W (N ) in the sense of (2.26). This in turn implies a correspondence between the supergravity state itself and W (N ) . For example, when . This PSU(2, 2|4) scalar is the zero-momentum dilaton vertex operator, which is dual to the linearized SYM action d 4 x duD 4 W (2) , as can be seen from (2.26).

Open-closed amplitudes
Because of the duality presented at the end of the last section, and also because of symmetry arguments, the disk scattering amplitude with the supergravity vertex operator V (N ) and N massless open superstring (SYM) vertex operators was conjectured in [10] to be proportional to the coupling (2.26). The SYM vertex operators would be located on D3-branes parallel and close to the AdS 5 boundary, at some fixed value of y ij and z near 0. Since the disk has an SL(2, R) symmetry which allows us to fix the positions of one open and one closed superstring vertex operator, the disk amplitude has the form where V SYM is the unintegrated vertex operator of SYM and U SYM is the integrated one. The angle brackets in the above equation contain integrations over the x, λ and θ zero modes, but they do not contain integrations over the z and y ij zero modes, since the position of the D3-brane is fixed. Schematically, one has λαλβλγ f (x, z, y, θ) where the powers of z ensure the measure is PSU(2, 2|4)-invariant, since d 4 x and d 5 θ have dimension −4 and 5 2 , respectively. More details on the integration of the λ and θ zero modes will be given shortly.
Proving that (3.1) is indeed proportional to (2.26) is the main purpose of this paper, and what we begin to do in the following.

The case N = 1
We now proceed to the computation of the amplitude (3.1) for the case N = 1. This is the simplest case, not only because the amplitude does not involve integrated vertex operators, but also because V (1) is simpler than any other supergravity vertex operator in AdS 5 × S 5 . Indeed, from (3.2.2) we have which has no y-or ψ-dependence. These operators are the duals to SYM "singleton" operators, i.e. the duals to abelian SYM fields.
In fact, the expression for V (1) can be further simplified. By adding the BRST-trivial quantity to V (1) , we get an equivalent expression which does not depend on the non-minimal pure spinor variables. It looks the same as (3.3), but with Ω (0) replaced by Based on the conjecture referred to at the beginning of this section, we expect the following relation to hold: and Aα is the dimensional reduction of the N = 1, d = 10 SYM superfield, whose properties are reviewed in Appendix C. Substituting the expressions for the vertex operators, one can write the amplitude as where we used (3.3) and replaced Ω (0) by the Ω (0) of (3.5). Now the angle brackets denote the integrations of the λ and θ zero modes only. In order to perform these integrations, we need to find a λαDα-invariant, SO(3, 1) × SU(4) scalar measure factor, since this is the symmetry of the boundary of AdS 5 × S 5 . At first, it might seem to be just a matter of dimensionally reducing the ten-dimensional measure (λγ µ θ)(λγ ν θ)(λγ ρ θ)(θγ µνρ θ) . (3.8) However, although this particular combination of λ's and θ's is special in ten flat dimensions, since it is the unique (up to an overall factor) SO(9, 1) scalar which can be built out of three λ's and five θ's, there is no reason why its dimensional reduction should be preferred over any other BRST-invariant, SO(3, 1) × SU(4) scalar in four dimensions. Hence, there could be some ambiguity in the definition of the N = 4, d = 4 measure factor. Fortunately, though, it was shown in [16] that it is unique up to BRST-trivial terms and an overall factor, so we can use (3.8) consistently. Using the measure factor (3.8), the pure-spinor prescription for the computation of tree-level scattering amplitudes states that [4] This definition ensures that (T D 5 )α 1α2α3 is totally symmetric and γ-traceless, i.e.
as is the product of three pure spinors.
In the case at hand, with (3.14) Thus, we arrive at the final form of the amplitude:

Possible contributions
Computing (3.15) explicitly would be very complicated, but fortunately we need not do that. Instead, we can use symmetry arguments and equations of motion to determine the form of the terms that might appear in the computation, and then use supersymmetry to obtain the relative coefficients.
Let us see what kind of terms one can expect to find when computing M 1 . Recalling that 16) we see that the amplitude has the schematic form where the index contractions need to be worked out. Depending on the number of Dα's that act on Aβ, there can be several possible contributions, as we show in the following. In our search for the possible contributions to (3.15), we are guided by dimensional analysis, SO(3, 1)×SU(4)-invariance, the SYM equations of motion and gauge invariance. One can see M 1 is gauge-invariant because, under a gauge transformation δAα = DαΛ, one has since BRST-exact terms decouple and since Q 1 We also take into account that T (3) is a G-analytic superfield, such that independent contributions only contain (uDα) derivatives acting on it.
When the five Dα's in (3.17) act on Aβ, we get a dimension-3 superfield. So this term could in principle give a contribution proportional to ∂ a ∂ b W ij or ∂ a F bc . We do not consider ∂ a ∂ b A c because this term is not gauge-invariant. It is easy to convince oneself that one cannot construct a non-zero term out of ∂ a F bc , and hence the only possible contribution is where we used the definition ∂ αα := (σ a ) αα ∂ a . When four Dα's hit Aβ, we get a dimension- 5 2 superfield. This could be either ∂ a W βj or its conjugate, ∂ aWβ j . Keeping in mind the SYM equation of motion ∂ αα W αi = 0, we are left with two possibilities: (3.20) Going on with this analysis, we find all the possible gauge-invariant contributions coming from different numbers of derivatives acting on Aβ, until the last case: (3.21) Note that, since M 1 is gauge-invariant, any contributions coming partly from a term in which no Dα acts on Aβ can be expressed as a linear combination of gauge-invariant terms in which at least one Dα acts on Aβ .
In summary, we get the following list of possible terms: To conclude this subsubsection, let us argue that is also given by a linear combination of the terms listed above. This expression contains eight derivatives of the type (uDα). When all these derivatives hit T (3) , one obviously gets the first term in the list. If only one derivative acts on W (1) , the resulting expression is proportional to either the second or the third term in the list, depending on the chirality of the derivative (D αi orD iα ). And so on until the case in which four (uDα)'s hit W (1) to give either zero or something proportional to the last term. Acting with five or more derivatives on W (1) gives zero, as can be seen from the fact that (uDα) 5 W (1) has U(1) charge − 3 2 , while the SYM fields {φ ij , ξ αi ,ξα i , f ab } have U(1) charges ranging from −1 to +1.

Proof using supersymmetry
Defining t n (n = 1, . . . , 9) to be the n-th possible term as listed at the end of the previous subsubsection, the amplitude we are computing must have the form for some constants c 1 , . . . , c 9 . In this subsection, we will prove that these constants are uniquely determined (up to an overall factor) by supersymmetry. Because both the left and right-hand sides of (3.6) are supersymmetric, this proves our conjecture since both the left and right-hand sides of (3.6) must be proportional to (3.22). Note that the left-hand side of (3.6) is supersymmetric (up to total derivatives), because λαAα Ω (0) T (3) is annihilated by λαDα so BRST invariance of the pure spinor measure factor implies supersymmetry as usual [4]. To see that the right-hand side of (3.6) is also supersymmetric (up to total derivatives), it suffices to write the supersymmetry generators as qα ∼ Dα + ∂/∂x and note that D αiD 4 D 4 F andD jαD 4 D 4 F vanish for any G-analytic superfield F (since there are only four independent (uD) αA and they are fermionic).
In order to uniquely determine the constants c 1 , . . . , c 9 , we have to impose supersymmetry which implies that Dα 9 n=1 c n t n = 0. We begin by acting on the possible terms with D αi . We have: 1.
We see that acting with D αi on t n produces various terms. In order to impose that the amplitude is supersymmetric, we need to collect the terms which should cancel independently. In the following, we organize the results according to the superfields appearing in each term. The imposition that they cancel gives rise to a system of many equations for the constants c n , which have to be solved at the same time.
• Terms proportional to W βj (without x-derivatives): Hence we get our first equation: • Terms proportional to F γ β : . Therefore, • Terms proportional to ∂F γ β : where we used ∂ αα F β γ = ∂ βα F α γ . So we get • Terms proportional to ∂Wα j : where we used (uu) m (uu) mi + (uu) m (uu) mi = δ i . Thus we obtain two equations: • Terms proportional to ∂W βj : • Terms proportional to ∂ 2 W βj : whence 3c 7 + 2c 9 = 0 . (3.28) • Terms proportional to ∂W ij : so that we must have    8c 1 + c 3 = 0 , • Terms proportional to ∂ 2 W ij : then we find the last pair of equations: Putting together (3.23), . . . , (3.30), we get the following system of equations:  We see there are a few more equations than unknowns, but they turn out to be not all independent. Setting c 1 = 1 as our normalization, 2 this system can be solved to give Note that M 1 is real, which implies D αi M 1 = 0 ⇐⇒D iα M 1 = 0.
Hence we have proved that, up to an overall factor, there is only one combination of the possible terms which is supersymmetric. This in turn shows that equation (3.6) is indeed true.

The cases N > 1
In the last subsection, we have shown that Now we want to generalize this result to any N , i.e. we want to show where we recall U SYM is the integrated version of V SYM = √ z λαAα and V (N ) was defined in (3.2.2).

The N = 2 case
As a natural first step, let us analyze the case N = 2. In this case, we want to compute the following scattering amplitude: where we have fixed the worldsheet positions of the unintegrated vertex operators (here is a positive infinitesimal). 2 Here we are implicitly neglecting the case in which all the constants cn vanish, which of course is also a solution to the system of equations. It is straightforward to show the right-hand side of (3.6) is nonvanishing, and the left-hand side can also be shown to be non-zero by direct computation of one of the possible terms, for example by choosing the gauge Aα = (θγ [ij] )αWij with Wij constant.
The easiest way to do this computation is by taking the OPE of V (2) (i , −i ) with U SYM (ξ) and looking for terms which can contribute to the dual to V SYM . Here we use the word "dual" meaning an object O such that V SYM O is nonzero. This object must be in the ghost-number +2 cohomology of the BRST operator Q and its product with V SYM should include terms proportional to the measure factor (λγ µ θ)(λγ ν θ)(λγ ρ θ)(θγ µνρ θ) .
Moreover, the amplitude must of course be PSU(2, 2|4)-invariant, and in particular SU(4)-invariant. A superfield in the [0, p, 0] representation of SU(4) should couple to another superfield in the same representation, so that a scalar ([0, 0, 0]) is present in their tensor product decomposition. The Dynkin label p is related to the number of y's (n y ) in a vertex operator by p = n y + 1, as can be seen from the coupling (2.26). The argument goes as follows: T (4−N ) couples to (W ij ) N and hence V (N ) corresponds to (W ij ) N . Now, As seen in subsection 2.2, an object in the ghost-number +2 cohomology of Q has the form where G (4−N ) (x, θ, u, u) is some G-analytic superfield of harmonic U(1) charge 4 − N and the operators Ω (n) were defined in (2.22). Since, by the argument given in the previous paragraph, the dual to V SYM must be independent of y, it follows that it can be written as where we have explicitly included a possible BRST-trivial term. Thus our problem is equivalent to finding the superfield G (3) . To find terms in the OPE of V (2) (i , −i ) with U SYM (ξ) which can contribute to O, one first considers the OPE's coming from the conformal-weight +1 operators of the integrated vertex operator. Since we are only interested in the z → 0 limit, we can consider the flatspace expression for the integrated vertex operator corresponding to states propagating in a D3-brane world-volume, i.e.
where Pψα is the momentum conjugate toψα. The term 1 2 zPψαWα is just the dαWα term written in AdS 5 × S 5 notation. The easiest way to see Pψα corresponds to dα in the z → 0 limit is by recalling the expression for Q − 1 2 , the lowest term in the Q expansion in powers of z, which is reproduced below: Since this expression can be written in ten-dimensional notation as z − 1 2 λ +α Pψα and the expression in flat space would be λαdα, it follows that Pψα corresponds to dα. The factor of z in 1 2 zPψαWα enters by dimensional analysis, and the numerical factor is needed for BRST-invariance.
The terms of order z 2 in (3.37) will not contribute. For example, one of these terms is z 2 N ab F ab . Because the OPE of N ab and λ is independent of z, this term cannot give a contribution of order z, and therefore cannot contribute to the dual of V SYM (cf. (3.36)) in the z → 0 limit.
Another term in U SYM which will not contribute is ∂x a A a . Since the kinetic term for x a in the Lagrangian is 1 2 it turns out the OPE of ∂x a and a superfield depending on x is also of order z 2 .
In fact, since the dual to V SYM does not depend on y orψ, only the terms z ∂y ij W ij and 1 2 zPψαWα in U SYM may contribute to M 2 . The reason is that these are the only terms which can remove the y-andψ-dependence of V (2) via the OPE's and where the dots include terms depending on y.
The closed superstring vertex operator for N = 2 is given by (cf. Hence, there are two contributions to the amplitude. The Ω (0) -term in V (2) is contracted with the z ∂y ij W ij in U SYM to give where Ω (n) is equal to Ω (n) with the substitutionψα −→ Wα. So, performing the integral over dξ in the complex plane, choosing a contour that encloses the pole ξ = i , we obtain (3.41) Since √ zλαDα(λγ µ W ) = 0, the Ω (n) 's also satisfy the equations (2.23), with the substitutionψα −→ Wα. Hence, it is not difficult to show that while the modified version of (2.23b) gives Thus the BRST-variations of the two terms in (3.42) cancel each other, implying that expression is indeed BRST-invariant. Previously we argued that any object in the ghost-number +2 cohomology of Q which does not depend on y can be expressed in the form (3.36) for some G (3) which is Ganalytic. Therefore, (3.42) can be expressed in the form (3.36). Moreover, when W ij is constant, it is easy to see that (3.42) can be expressed in this form with (1) is not G-analytic and there are no G-analytic terms of the appropriate dimension that can be constructed out of derivatives of W ij . So G (3) must be equal to W (1) T (2) even when W ij is not constant, i.e. (3.42) must be equal to where the BRST-trivial term Qχ 2 vanishes when W ij is constant. This implies Note this is consistent with the gauge transformation (2.27), since δT (2) and the analyticity of W (1) imply which is BRST-trivial. Finally, using (3.33), we obtain thus proving (3.34) in the case N = 2.

Generalization to any N > 1
We are now in position to prove (3.34) for any N . Let us copy it here for the sake of readability: Again, we are looking for the dual to V SYM in the form (3.36), i.e. we are looking for the expression of the G-analytic superfield G (3) in the case of the amplitude M N . As argued in the previous subsubsection, only the terms z ∂y ij W ij and 1 2 zPψαWα in the integrated vertex operators can remove the y-andψ-dependence from the supergravity vertex operator through their OPE's and thus contribute to M N . This also implies that there can be no contribution coming from contractions between two or more integrated vertex operators. Recalling that the OPE's give, after performing the (N − 1) integrations over the dξ's, is BRST-invariant. The calculation is similar to the one performed below (3.42), but with more terms. Then, by arguments completely analogous to the ones given at the end of the previous subsubsection, we conclude that (3.49) must be equal to where the BRST-trivial term Qχ N vanishes when W ij is constant, i.e.
for arbitrary N . This implies and thus, using (3.33), (3.52) thus proving (3.34) in the general case.

Summary
In this paper, after reviewing the work done in [10], we have computed the first scattering amplitude involving pure spinor vertex operators in AdS 5 ×S 5 . We have verified a conjecture according to which the tree-level scattering amplitude containing a supergravity state and N massless open superstring states close to the boundary of AdS 5 can be written as a harmonic superspace integral involving the supergravity and super-Yang-Mills (SYM) fields. More precisely, we have shown that where V (N ) is the supergravity vertex operator defined in (3.2.2) and the "D3-brane" subscript indicates that the open superstring (SYM) vertex operators are located on D3-branes parallel and close to the AdS 5 boundary, at some fixed value of y ij and z ∼ 0.
The harmonic superspace coupling above has been known for some time [12]. Here we have shown that it can be obtained as a superstring scattering amplitude computation involving open and closed superstring vertex operators. This can be seen as a consistency check for the vertex operator found in [10], as well as one more test of the AdS/CFT conjecture, in that the expected relation between supergravity and SYM was found. Future and perhaps more interesting applications would involve the computation of scattering amplitudes with closed superstring vertex operators only, which could be compared with correlation functions in the SYM side.
The other fundamental representation, called 0, 1 2 or right-handed chiral, is obtained by complex conjugation:ψ The dot over the indices indicates the representation to which we refer.
The indices with and without dot are raised and lowered in the following way: where ε is antisymmetric and has the properties In SL(2, C) notation, a four-component Dirac spinor is represented by a pair of chiral spinors: For a Majorana spinor,χα = (ψ α ). The Dirac matrices are with I 2 the 2 × 2 identity matrix and σ the Pauli matrices and have the following properties: with η ab = diag(−1, 1, 1, 1). These properties imply {Σ a , Σ b } = −2η ab I 4 .
for (γ µ )αβ. It is straightforward to show that the above matrices satisfy the usual relation with η [ij][k ] := 1 2 ε ijk . As an example, we show how to obtain the dimensional reduction of the pure spinor constraints λγ µ λ = 0 using (A.17). For λγ a λ = 0, we have

B Harmonic Superspace
In this work we make use of a harmonic superspace composed by an N = 4, d = 4 Minkowski superspace and the coset space SU(4)/S(U(2)×U(2)) [12,13,20]. In addition to the usual coordinates x a , θ αi andθα i , this superspace is parameterized by new variables u ∈ SU(4), called harmonic coordinates. In terms of indices, we write u as u I i = (u A i , u A i ), and denote its inverse by u i I = (u i A , u i A ). The index I is transformed by the isotropy group S(U(2)×U(2)) and thus splits naturally into A = 1, 2 and A = 3, 4. The u's have the following properties: The bars on some of the u's reflect their U(1) charge, which is opposite to that of the unbarred ones. More precisely, the U(1) charge of an object is defined as the eigenvalue of the operator i.e. u (resp. u) has U(1) charge 1 2 (resp. − 1 2 ). The introduction of harmonic variables allows the definition of superfields which satisfy generalized chirality constraints. A superfield F which satisfies is said to be G-analytic, whereas a superfield which satisfies is said to be H-analytic. A superfield that is both G-and H-analytic is called an analytic superfield, for short. In this work, the following conventions are used: where ε AB , ε A B are completely analogous to ε αβ , εαβ. Using (B.1), the following identities (among others) can be derived:

C SYM equations
The N = 1, d = 10 super-Yang-Mills theory admits a formulation in superspace in terms of the on-shell superfields A µ and Aα. Defining the supercovariant derivatives as one can show that, in the linearized theory, which, using (A. 19), can be written as Note that this equation is equivalent to (γ µ 1 ···µ 5 )αβDαAβ = 0. Forα = (αi) andβ = (βj), we have (recall (γ a ) (αi)(βj) = 0) Because their θ = 0 components are the same (the scalars φ ij ), we claim that where W ij is the Sohnius superfield of N = 4 SYM [14] and we made use of (C.4). In four-dimensional notation, we get where we made use of (C.7). Note that (C.6) then implies Indeed, we can show that this superfield satisfies the same constraints as the Sohnius one. First note that (C.7) implies (W ij ) † = 1 2 ε ijk W k . Then, writing W jk = − 1 8 ε jk mD αĀα m , we have where we made use of (C.8). Therefore, Moreover, it can be shown that the Hermitian conjugate of the above equation implies Note that these constraints imply (uu) ij W ij is an analytic superfield. Studying the Bianchi identities for the field-strength superfields leads to the following (linearized) equations of motion [21]: where Wα is the superfield whose θ = 0 component is the gluino ξα. These in turn imply, by dimensional reduction, as well as their Hermitian conjugates, where ∂ αβ := (σ a ) αβ ∂ a , F β γ := (σ ab ) β γ F ab and Fβα := (σ ab )βαF ab .

D Dimensionally reduced expressions
Although we do not explicitly use them in the text, we derive here, for completeness, the dimensionally reduced forms of Ω (0)αβ and (T D 5 )αβγ, as they might be useful for the reader.

D.2 Reduced form of (T D 5 )αβγ
In principle, the dimensional reduction of T D 5 (cf. (3.10)) could also be obtained directly by using the formulas (A.17) and (A. 18), but that would be a very long and tedious task. Fortunately, there is an easier way of obtaining this result, as we describe in the following. We begin by noting that, if Λα is the derivative of λα, such that Λγ µ Λ = 0, then The left-hand side of the above equation is not so difficult to compute. Up to an overall factor, we find (Λγ µ D)(Λγ ν D)(Λγ ρ D)(Dγ µνρ D) = ε mnj (Λ iΛk )(Λ m D n )(D i D j )(D k D ) where "H.c." stands for the "Hermitian conjugate". One can check that the expression obtained from the above by substituting Λ for λ and D for θ is annihilated by λαDα.