A superfield realization of the integrated vertex operator in an $AdS_5\times S^5$ background

The integrated massless vertex operator in an $AdS_5\times S^5$ background in the pure spinor formalism is constructed in terms of superfields.

This paper is organized as follows. Section 2 contains a short review of the pure spinor formalism and a detailed description of its massless vertex operators. In section 3 we begin with the description of the pure spinor version of type IIB superstring in an AdS 5 × S 5 background. After it we discuss the unintegrated vertex operator, pointing out some similarities with the flat space case. In section 4 we find the chain of operators (1.1) for AdS 5 × S 5 background with all superfields containing the physical fluctuations. We conclude the work in section 5, discussing future problems and possible applications.
It is useful to work with supersymmetric combinations of the world-sheet variables. They are the d,d defined above, world-sheet derivatives of θ,θ and Π m = ∂X m + 1 2 (θγ m ∂θ) + 1 2 (θγ m ∂θ),Π m =∂X m + 1 2 (θγ m∂ θ) + 1 2 (θγ m∂θ ). (2.6) The BRST transformations of these supersymmetric invariants are given by Note that on-shell λ only appears in the holomorphic sector andλ only appears in the antiholomorphic sector. This is due to the fact that Q is only part of a much larger symmetry generated by the holomorphic and anti-holomorphic currents j = λ α d α andj =λᾱdᾱ.
The BRST transformation of any superfield is QΨ(X, θ,θ) = λ α D α Ψ +λᾱDᾱΨ, where D α = ∂ α + 1 2 (γ m θ) α ∂ m and Dᾱ = ∂ᾱ + 1 2 (γ mθ )ᾱ∂ m . The algebra of theses superspace covariant derivatives is given by Physical states are defined to be in the cohomology of Q. The massless states are described by the unintegrated vertex operator with vanishing classical dimension 1 U = λ αλβ A αβ (X, θ,θ). (2.10) Since U is in the cohomology of Q, its gauge transformation is given by In order to preserve the original form of U the parameters (Λ α ,Λᾱ) are constrained to satisfy These conditions resembles the equations that define two on-shell vector multiplet in ten dimensions.
The main difference is that here (Λ α ,Λᾱ) are functions of N = 2 superspace variables. From them we can also obtain
The condition that U is BRST closed implies which is solved by 1 A more general form of the vertex operator is given by Aᾱβ(X, θ,θ). Although this for is useful for some applications [18,19], all physical states can be described by the gauge fixed version (2.10) Using the covariant derivatives algebra and gamma matrix identities these equations imply a chain of equations that define the supergravity fields as higher components of A ββ (X, θ,θ) and put them on-shell. The first few are given by The integrated vertex operator can be seen as a deformation of the flat space action This deformation induces a change of order µ in the flat space BRST transformations (2.7) and (2.8). We will call the generator of these new transformations Q 1 . Invariance of S def under BRST transformations up to order µ implies that The integrated vertex operator is not uniquely defined; we can make a non-linear field redefinition of the fundamental fields at order µ that will change V by terms proportional to the world-sheet equations of motion. Since Q 1 S 0 is proportional to the equations of motion, we can first solve (2.23) on-shell and require that QV = ∂W −∂W (2.24) up to terms proportional to the flat space equations of motion. The off-shell solution of (2.23) can then be constructed.
Since (2.7) and (2.8) are on-shell nilpotent on all matter fields and combinations of the ghost variables that are invariant under (2.5), (2.24) implies that QW = ∂U, QW =∂U, (2.25) and finally we have that QU = 0, where U is the unintegrated vertex operator defined before.
The idea is to find V from U up to world-sheet equations of motion. We first start with (2.25) to find W andW . Consider the equation for W first. Its form can be guessed knowing that we can use ∂λᾱ = 0 and that it should have classical dimension (1, 0). W turns out to be (2.29) Using this, W transforms as Similarly,W can be found using, which are consequences of (2.16)-(2.21). We obtain that The last step is to find V such that QV = ∂W −∂W . Knowing that V has to have classical dimension (1, 1), vanishing ghost number and we can ignore term proportional to the world-sheet equations of motion, we can guess the following form Using the BRST transformations and QV = ∂W −∂W is possible to show that A mn is defined by Ω andΩ are defined by

37)
C mn β and C mnβ are defined by and finally S is defined by It can be shown that these definitions are equivalent among themselves after writing all the superfields in terms of A αβ and superspace derivatives. Note that the gauge transformation of V is given by and vanishes after integration. There are no BRST exact terms above because the only way to have well defined covariant operators with ghost number −1 invariant under (2.5) is using the nonminimal pure spinor formalism [20]. In particular, we can see from (2.34) and (2.35) that the gauge transformations of A mn , E m α and E mᾱ are given by which are the expected gauge transformations for the graviton plus the B-field and local supersymmetries for the gravitini. Note that the γ-trace of E m α and E mᾱ are invariant by a consequence of (2.13) and (2.14) and are identified as the dilatini.
3 Unintegrated vertex operator in type IIB superstring in AdS 5 ×S 5 We will now focus on the Type IIB superstring in an AdS 5 ×S 5 background using the pure spinor description. It is well known that this background is described by the coset P SU (2, 2|4)/Sp(1, 1) × Sp(2). We will denote the generators of the psu(2, 2|4) algebra by The non-vanishing structure constants f BC A can read off from these (anti-)commutators. The geometric quantities are defined using the Maurer-Cartan currents. They are defined in terms of the coset element g as where ∂ M are derivatives with respect to local coordinates Z M .
Additionally, we need the background values of the RR field-strength and the NSNS two-form, which are not defined by the geometry, where η αβ = γ 01234 αβ , η αβ = γ αβ 01234 . We can also calculate the non-zero values of the torsion and the curvature Here a = (a, a ′ ) with a(a ′ ) refers to the tangent vectorial index of AdS 5 (S 5 ). With all these ingredients, the world-sheet action in the pure spinor formalism for this background is where we are using It is important to note that the action does not depend on all components of ghost currents, however, fluctuations of these background can and will depend on them. We have integrated out the d α and dᾱ variables of the pure spinor formalism. After this, the BRST charge (2.2) becomes The BRST transformation of the coset element is Qg = g(λ +λ), where we used the shorthand λ = λ α Q α andλ =λᾱQᾱ. The Maurer-Cartan currents J = g −1 ∂g transform as and similarly forJ = g −1∂ g. The pure spinor ghosts λ,λ are BRST invariant and their conjugate momenta transform as The structure of the unintegrated vertex operator will be the same as in the flat space case where the superfield A αᾱ (g) is a function of the coset element g. The equations coming from QU = 0 are [2] ᾱ} are the covariant derivatives. The residual gauge symmetry from δU = QΛ implies that where the gauge parameters Λ andΛ satisfy which are the generalization of (2.12) and (2.28) for the AdS 5 × S 5 case. As in the flat space case, there are consequences of these constraint equations. They are These equations are similar to the equation satisfied by the super-Maxell fields in ten-dimensions, that is (2.16) and (2.17). Our goal is to find the remaining fields with equations similar to (2.18), (2.19) and (2.20), (2.21). Consider the first equation in (3.12) . We note that the combination The proof is the same as in the flat space case [21], using the general (anti-)commutator of covariant derivatives acting on a superfield A CD the relevant commutators are the same as in flat space. Therefore we have that The next equation in the chain is for ∇ α W βγ . From (3.19), we obtain after some covariant derivative algebra and the background values for the curvature where This is again very similar to the flat ten-dimensional superspace [21], the difference is in the terms involving commutators. Using (3.18), we obtain, This is solved by Similarly from the second equation in (3.12), we obtain Let us summarize our results up to now. The unintegrated vertex operator is U = λ αλβ A αβ and its BRST invariance imply a chain of superfields defined by The gauge transformations for F abβ and F αab require a separate note. The combination (γ [a ηγ b] ) vanishes when one indices is in AdS 5 and the other in S 5 . Similarly, R ab cd is zero unless all its four indices are in AdS 5 or all four are in S 5 . In these cases, the transformations simplify to On the other hand, when both indices are in, for example, AdS 5 , the transformation for F abβ is Combinations of the types (Λ δ − 2η δᾱΛᾱ ) and (Λᾱ + 2η αᾱ Λ α ) will appear again when we discuss the local supersymmetry transformation for the gravitino, they are the local supersymmetry parameters for the AdS 5 × S 5 superspace.
We will use the transformations the to verify that the analogous of (2.26) and (2.32) in AdS 5 ×S 5 transform adequately.

Integrated vertex operator in type IIB superstring in AdS 5 × S 5
After the initial discussion on the superstring in AdS 5 × S 5 background, the unintegrated vertex operator and some of its consequences, we want to explicitly solve the equations (2.24), (2.25) that define the integrated vertex operator (2.22). Unlike the flat space case, the b ghosts of the AdS 5 ×S 5 superstring can be constructed without the introduction of non-minimal fields [22,23]. Therefore it is guaranteed that ∂U is cohomologically trivial.
As before we will work on-shell. The first equation of motion we will need is the one forλ obtained from (3.5) From the same argument as before, working on-shell allows us to chose W to have the following form Acting with Q and using the equations (3.12), (3.19) and (3.25) we obtain, QW =∂U − λ α ∇λβA αβ +λβ(λγ a )γJγA aβ +λβλγJ a ∇γA aβ − 1 2λβ (λγ a η) α J a A αβ − 2η αᾱλβ ∇λᾱW αβ + 2η αᾱλβ Jᾱλγ∇γW αβ Let us consider first the fourth term above. Using (3.13), (3.18), (3.4) and we obtain which cancels with the last two terms in the first line of (4.3). Consider now the second term in the second line of (4.3), it contains Using (3.18) and (4.5), Finally, after using (3.4) we obtain which helps to cancel the second term in the second line with the third term in the first line of (4.3).
Up to now we have The last term can be written as after using the definition of F abγ (3.21) and commuting derivatives.
Note that the last term vanishes on-shell. It remains to compute ∇ γ F abβ . Using (3.19) and commuting derivatives, From here and using λ γ N ab (γ a ) γδ = − 1 2 (λω)(λγ b ) δ we obtain The first term in this expression contains we finally obtain A similar calculation shows that the operatorW that satisfies QW =∂U is given bȳ Performing the same steps one can show that Note that the second term is proportional to the equation of motion for λ.
In summary, we have found the conformal weights (1, 0) and (0, 1) superfields as in flat space. Using the gauge transformations of the superfields inside W , we obtain that which fixes Ψ to be An analogous result holds forW It is interesting to note that this looks like the vertex operator for N = 1 vector multiplet in ten dimensions for an open superstring, but now in an AdS 5 × S 5 background. Perhaps this identification can be made more precise by studying boundary conditions for the pure spinor string in this space. Appropriate boundary conditions will reduce the number of independent components of J α and Jᾱ.
Now that W andW have been constructed, we can find the conformal weight (1, 1) operator V that satisfies QV = ∂W −∂W . We will need to use the equations of motion of the world-sheet action and some Maurer-Cartan identities. Also, the equations for the pure spinor ghosts imply that the ghost Lorenz currents satisfȳ The equations of morion for the currents J A for A = (a, α,ᾱ) arē They are calculated varying the action with respect to δg = g(δX a T a + δX α T α + δXᾱTᾱ) and using the Maurer-Cartan identities The vertex operator V couples all gauge covariant currents with conformal weight (1, 0) with the ones with weight (0, 1) and it has the form From the structure of W ,W and the BRST transformations of the currents J α andJᾱ we can immediately identify the first seven superfields in V with the ones that already appeared in W and W . The remaining superfields, collectively called V, are constrained by QV = ∂W −∂W .
For example,V aα is constrained by the equation (4.34) Using the definition (3.31) for W γγ and (anti-)commuting covariant derivatives, one can obtain that where O aα is a combination of superfields and derivatives. This allows us to findV aα . The resulting expression contains the superfield A αβ and its derivatives. The full expression forV aα is not very illuminating and it turns out to bē All other superfields V in (4.33) will appear in equations that can be inverted. We will write all their defining equations. Their explicit expressions can be obtained in the same way as the one for V aα in (4.35).
The superfields V aᾱ satisfies the equation The superfield V ab which, from its coupling in (4.33), can be identified with fluctuations of the metric plus the NSNS two-form, satisfies the equations which are equivalent when they are expressed in terms of A αβ . Note that V ab contains the components V ab ′ that twist the AdS 5 and S 5 spaces.
The superfieldV αβ satisfies the equations which are also equivalent when written in terms of A αβ .
The next four superfields are related to fluctuations of the connections. First, V abβ is given by andV abα is determined by and V a bc satisfies the equation Finally, the background fieldV a bc satisfies the equation The last superfield to be defined is related to fluctuations of the curvature V ab cd and it satisfies the equations As a simple application one can calculate the integrated vertex operator (4.33) for the radius operator found in [22]. The unintegrated vertex operator is U = λ αλβ η αβ and the integrated vertex operator should be proportional to the lagrangian in (3.5). Using A αᾱ = η αᾱ the solution for (4.34)-(4.43) gives Which is proportional to the lagrangian up to the equations of motion for the ghosts.

Local symmetries and field content
Due to the extra mixing coming from the RR background, non vanishing torsions and curvatures, it is quite involved to get the equations of motion for the superfields defined by V in a way that we can identify what is their lowest component. However, we can use the gauge invariance and look for fields that transform in the expected way. As in flat space, V inherits a gauge transformation from the unintegrated vertex operator δU = Q(λ α Λ α +λᾱΛᾱ), after finding the gauge transformations of W andW , we have that V transforms as The transformation of each field in V (4.33) can be found using the definition above and the equations for the Maurer-Cartan currents and N ab ,N ab . Alternatively, we can use the definition of each V in terms of the superfields in W andW and use (3.13), (3.33) and (3.34).
We can find a very simple expression for the superfield whose lowest component gives the metric plus the NSNS two-form Its gauge transformation can be computed and it is given by where we can see that Λ a −Λ a is related to the diffeomorphism parameter and Λ a +Λ a is the parameter of the gauge transformation for the NSNS two-form.
The superfield whose lowest component is the gravitino can be identified by looking for the correct gauge transformation. It turns out that the combinationV aα − 1 4 (γ a b ) α β A βb transforms as From this we will identify one of the gravitini as the lowest component of the superfield which will transform as δE aᾱ = 1 2 (ηγ a η)ᾱβ(Λβ + 2η δβ Λ δ ) + ∇ a (η αᾱ Λ α − 2Λᾱ). (4.50) Note that (ηγ a η) is γ a and −γ a ′ , which is the expected behavior for the term depending on the cosmological constant in the local transformation of the gravitino. Furthermore, it reduces to the usual local supersymmetry transformation in the limit where the RR background goes to zero.
Now we want to find the bi-spinor superfield whose lowest component is bi-spinor RR fieldstrength. Comparing with the flat superstring we start with V αᾱ and find that it transforms as (4.51) After using (3.15) and (3.16) we have that Since the RR field-strengths should be gauge invariant, the following combination has the desired property. The usual definition using upper indices can be achieved using the η tensors. The zero mode of the gauge invariant scalar φ = η αᾱ F αᾱ is the operator that changes the radius of the AdS 5 × S 5 space. The superfields whose lowest components are the dilatini can be found in similar ways.

Conclusion and prospects
In this paper we did the explicit construction of the integrated vertex operator in an AdS 5 × S 5 background using the pure spinor formalism. We have found how all superfields present in the integrated vertex are related to the initial superfield A αᾱ (g) in the unintegrated vertex operator. The analysis done here complements the formal construction previously done in [13]. However, the integrated vertex operator found in that work does not depend on the Lorentz ghost currents with mixed indices. It would be interesting to find the origin of this discrepancy.
The final answer resembles the construction of the integrated vertex operator in a flat background, replacing the flat space supersymmetric currents (Π m , ∂θ α , d α , N mn ) with the the isometry invariant currents (J a , J α , Jᾱ, N ab ). The superspace description of linearized supergravity found here should be related to the supergravity pre-potential found in [24]. A possible future application is to find explicit expressions for vertex operators for some specific supergravity state defined by its psu(2, 2|4) labels, in the same way single trace operators of N = 4 SYM are defined. Progress in this problem was made in [25] where explicits states were found expanding the vertex operator close to the boundary of AdS 5 . These results were later used in [26] to compute amplitudes with one closed string state and open string states. Another known explicit vertex operator is the one corresponding to the beta deformation [19]. A possible way to describe more general explicit states is to use the formalism described in [27].
Another interesting direction is to study vertex operators for open strings in AdS 5 × S 5 . As we noticed before, the gauge parameters for the local symmetries of the supergravity states satisfy equations that resembles the equations of motion of an N = 1 vector multiplet in ten dimensions. Of course this identification has to be made more precise since it is known that there are no D9-branes in AdS 5 × S 5 . The boundary interaction of a D-brane should be an operator of the form I boundary = A a (g)J a + A α (g)J α + Aᾱ(g)Jᾱ + F ab (g)N ab , (5 .1) where (A a , A α , Aᾱ, F ab ) describe the massless states of the D-brane and (J a , J α , Jᾱ, N ab ) are appropriate isometry invariant world-sheet one-forms evaluated at the boundary. A BRST analysis of the boundary conditions with the interaction I boundary along the lines of [28] should give the correct physical state conditions for (A a , A α , Aᾱ, F ab ) and the allowed D-branes.
Perhaps the most important problem is to understand the quantum correction to the physical states conditions. For the case of superstrings in flat space, since it is a free theory, the primary state condition receives only a one loop quantum correction, the anomalous dimension for the exponential operator e ik·X . From the BRST cohomology point of view there is no correction at all. This will not be true for fluctuations around a general background since the physical state condition will receive corrections from vertices of the action for both computations. However, for the case of AdS 5 × S 5 the situation for the massless spectrum should be similar to the flat space case. These states are dual to the protected BPS operators in the N = 4 SYM theory. The only quantum correction to the primary state condition should be a one loop correction, proportional to the quadratic Casimir of psu(2, 2|4) algebra [29]. The BRST cohomology condition should receive no correction. It would be very interesting to understand how all possible quantum corrections to the classical calculations in this work cancel. We plan to address this in a future work.