B-RNS-GSS Type II superstring in Ramond-Ramond backgrounds

The B-RNS-GSS formalism containing both RNS and GS worldsheet variables is extended for the description of the Type II superstring in curved backgrounds. The BRST charge of the theory is expressed in terms of $N=(1,1)$ superconformal generators $(T,G; {\widehat{T}}, {\widehat{G}})$, and the classical anticommutation relations $\{G,G\}=-2T$ and $\{{\widehat{G}},{\widehat{G}}\}=-2{\widehat{T}}$ are shown to imply the Type II supergravity equations of motion. The particular case of the Type IIB superstring in an $AdS_5\times S^5$ background is discussed and shown to be classically integrable.


Introduction
The Ramond-Neveu-Schwarz (RNS) formalism of the superstring has an elegant structure based on N = (1, 1) worldsheet supersymmetry. Despite its elegance, vertex operators for spacetime fermions and the description of Ramond-Ramond backgrounds are complicated in this formalism [1] because of the lack of manifest spacetime supersymmetry.
A new formalism for the superstring with both worldsheet and spacetime supersymmetry was recently proposed in [2], where a set of extra worldsheet variables was added to the worldsheet action in such a way that simple vertex operators for spacetime fermions could 1 be constructed. It was named the B-RNS-GSS formalism since it involves features of the pure spinor [3], RNS and Green-Schwarz-Siegel [4] formalisms of the superstring. Consequently, it is expected to play a useful role in resolving the open problem of proving the equivalence of the pure spinor and RNS formalisms [5].
The quantization of the B-RNS-GSS model is performed with a BRST charge that resembles the BRST charge of the RNS formalism. It has the standard form for an N = 1 worldsheet superconformal field theory Q = (cT + γG + γ 2 b − bc∂c), where T is the stress-energy tensor, G is the generator of worldsheet supersymmetry and (b, c, β, γ) are the usual superconformal ghosts. The fact that G and T generate an N = 1 superconformal algebra implies nilpotence and holomorphicity of Q.
This new formalism is also suitable to describe curved backgrounds. Indeed, it was recently shown [6] that, at the classical level, nilpotence and holomorphicity of Q in the context of the B-RNS-GSS string in a curved heterotic background imply the N = 1 D = 10 supergravity and super-Yang-Mills constraints for the background superfields as formulated in [7], using an analysis similar to the case of the pure spinor formalism.
In this paper, the B-RNS-GSS formalism will be defined in curved Type II supergravity backgrounds, and nilpotence and holomorphicity of the BRST charge at the classical level will imply the N = 2 D = 10 Type II supergravity constraints of [7]. It should be stressed that this includes backgrounds with finite Ramond-Ramond flux, and the sigma-model for the B-RNS-GSS in an AdS 5 × S 5 background will be constructed at the end of the paper to illustrate this fact.
The paper is organized as follows: in Section 2 we review the construction of the Type II version of [2] in a flat background. In Section 3 we construct the Type II superstring in a curved background. In Section 4 we show that the equations {G, G} = −2T and { G, G} = −2 T at the classical level impose the constraints of Type II supergravity. In Section 5 we prove that the constraints from Section 4 imply {G, G} = 0. In Section 6 we prove the holomorphicity of G and the antiholomorphicity of G, consequently showing the BRST invariance of the action for Type II supergravity. In Section 7 we specialize our results for the AdS 5 × S 5 background of the Type IIB superstring and establish the classical integrability of the corresponding sigma-model. We conclude in Section 8 by commenting on possible applications of these results. The appendices are organized as follows: Appendix A gathers some gauge transformations of the action and BRST charge, Appendix B contains the results for the necessary Poisson brackets between different worldsheet fields, Appendix C gathers the equations of motion derived from the worldsheet action, and Appendix D gives the conventions for the AdS 5 × S 5 section.
(2. 6) In order to do so, one must perform the similarity transformation O → e R Oe −R with on all operators O. Then, the theory is manifestly invariant under the transformations generated by the usual Green-Schwarz-Siegel supersymmetry generator q α = p α + 1 2 ∂x m (γθ) α + 1 24 (θγ m ∂θ)(γ m θ) (2.8) and similarly for q α . Note that the similarity transformation takes the superconformal current G to G = 1 2 Λγ m Λ ψ m + Λ α d α + ψ m Π m + w α ∂θ α (2.9) and similarly for G. The variables 10) are defined as in the Green-Schwarz-Siegel formalism.
The BRST operator of the B-RNS-GSS formalism has the standard N = (1, 1) worldsheet superconformal form, which for the left-moving charge means Q B−RN S−GSS = cT + γG − bc∂c + γ 2 b , (2.11) where T is the holomorphic stress-energy tensor obtained from the action (2.5) and the spinors (Λ α , w α ) carry holomorphic conformal weight ( 1 2 , 1 2 ). Analogous statements follow for the right-moving BRST charge.
The presence of extra fields implies that the spectrum of the B-RNS-GSS string is larger than that of the RNS string. Physical vertex operators for the usual GSO (+) projected RNS open superstring spectrum have been constructed recently [2,8]. These operators are in the cohomology of Q B−RN S−GSS and satisfy the extra condition that they carry non-positive charge with respect to the U(1) generators (2.12) To understand the non-positive charge requirement on the vertex operators, consider twisting 1 the (Λ α , w α , Λ α , w α ) variables by After performing the twist, the bosonic spinor variables carry integer worldsheet spin as in the pure spinor formalism. But due to the inclusion of inverse powers of γ, vertex operators with positive charge with respect to (2.12) would no longer be in the small Hilbert space with respect to the twisted worldsheet variables. So the additional requirement of nonpositive charge is needed to guarantee that the vertex operators remain in the small Hilbert space after performing the twist.

The integrated left-moving massless open string vertex is given by
In order for the conformal anomaly to cancel such that the "untwisted" theory presented here is consistent at the quantum level, one must also add non-minimal bosonic Λ α , w α and fermionic R α , S α left-moving variables of conformal weight (0, 1), as well as their right-moving counterparts. These variables are not needed for the construction of massless vertex operators, and will therefore be ignored in the remainder of this paper. For more details on their role in the B-RNS-GSS formalism, see [5,6].
with N mn = −ψ m ψ n + 1 2 Λγ mn w, and a similar expression holds for the right-moving vertex. The closed string B-RNS-GSS integrated massless vertex operator can then be obtained by the left-right product of the left and right-moving open vertices, which yields (2.15)

Curved background
The closed vertex (2.15) does not include all possible terms with non-positive U (1) charge, as terms with −2 U(1) charge (i.e. with factors of w α w β and/or w α w β ) can clearly be written. The lessons learned in the analysis of the heterotic string [6] show that such terms must be considered and that they describe non-linear deformations around flat-space. After including these terms and covariantizing the flat action (2.5), we arrive at the most general action satisfying the U(1)-charge/small Hilbert space condition where Π A and Π A are given by 3) The Ω and Ω superfields and the remaining background superfields C, . . . , M will be constrained by BRST invariance.
The BRST symmetry is generated by the left and right-moving BRST charges which retain their standard N = (1, 1) worldsheet supersymmetric form where T and T are the stress-energy tensors given by and (3.6) The supercurrents G and G are generalized to We define ∂ 0 = 1 2 (∂ + ∂) and ∂ 1 = 1 2 (∂ − ∂), where ∂ 0 denotes derivation in the τ -direction and ∂ 1 denotes derivation in the σ-direction of the worldsheet.
Note that these expressions for the supercurrents also satisfy the U(1)-charge/small Hilbert space condition on the deformations.
The background superfields that have been introduced above parametrize the deformations away from the flat background. As in [6], the physical deformations are the ones such that the BRST charge is nilpotent. The BRST charges are nilpotent provided that the supercurrents G, G satisfy where {, } denotes the Poisson bracket. These requirements will constrain the background superfields. As we will see, the constraints are such that the supercurrent G, G are respectively holomorphic and anti-holomorphic, and will imply that the background superfields describe Type II supergravity backgrounds.
Before ending this section, the torsion and curvature components for the background geometry will be defined because the constraints of Type II supergravity are expressed in terms of them. The vielbein E A one-form and connections Ω α β , Ω α β one-forms allow the definition of torsion and curvature two-forms. Note that up to this point, the Ω, Ω connections can be decomposed as Our calculation will show that Ω abcd = Ω abcd = 0 and the torsion and curvatures will satisfy the constraints of [7]. The torsions one-forms T α and T α are defined as Since we have Ω ab and Ω ab , we can define the torsions where the indices a, b, . . . are raised or lowered with the Minkowski metric η ab or η ab . The torsion components will be constrained by imposing (3.10). Similarly, the curvatures are defined as The product between forms is the wedge product. We are going to need the Bianchi Finally, we define two types of covariant derivatives depending on which of the Ω ab or Ω ab connections one uses 3 . For example, the covariant derivatives on a zero-form Ψ aαβ are Type II supergravity also contains the components of the three-form superfield H = dB in its spectrum, where B is the two-form of the action.
It is important to note that the action (3.1) and BRST charges (3.4) are invariant under several gauge symmetries associated to relabelling background fields, which are gathered in Appendix A. This behavior has already been encountered during the analysis of the heterotic string case. Before moving on to the Poisson brackets, we will already make use of the gauge symmetries parameterized by ω αβ γ and ω αβ γ to gauge  and analogously for G αβ γ .
3 It will be shown in this paper that the constraints imposed by BRST invariance relate the connections Ω ab and Ω ab such that ( Ω abc − Ω abc ) ∝ T abc , ( Ω αbc − Ω αbc ) ∝ T αbc and (Ω αbc − Ω αbc ) ∝ T αbc . In this section, we prove the claim that imposing the conditions G, G = −2T and G, G = −2 T implies the N = 2 D = 10 Type II supergravity constraints of [7], as well as determines the background fields in the action and in the supercurrents in terms of supergravity superfields. The canonical Poisson brackets for the fundamental variables in the action (3.1) are where the momentum P M ≡ δS δ(∂ 0 Z M ) is given by The last equation allows us to express d a , d α and d α in terms of canonical variables. Namely, (4.5) From these definitions, the Poisson brackets necessary for the calculation of (3.10) can be computed. They have been listed in Appendix B. The computation of (3.10) will also make use of the equations of motion derived from (3.1). The variation of the action with respect to d α and d α gives rise to 6) and the remaining equations of motion are listed in Appendix C.
Explicitly, we compute that is, these are same-time commutators and the spatial coordinates which are not written explicitly are understood to be integrated over. The computation is long and there are many contributions coming from the different necessary Poisson brackets. However, a pattern appears which is helpful when it comes to analysing the final result. The resulting terms are of two types: where ϕ ∈ Λ α , ψ a , w α , Λ α , ψ a , w α and ∆ ∈ d α , d α , Π α , Π α , Π a , Π a . The notation C AB C and I ABCD denotes functions of the background superfields in the action and in the supercurrents (G, G), as well as of supergravity superfields. The first type of term will be analysed in the next subsection. These terms are the easiest, and the functions C AB C are either N = 2 D = 10 Type II supergravity constraints, or determine the superfields in the supercurrent G and C, Y, D, U, V, Z which appear in the action. The second type of term will be analysed later, as they are more complicated. The functions I ABCD will be equivalent to Bianchi identities, which vanish identically, or will determine the remaining background fields S, E, E, F, F , G, G, H, I, I, J, J, K, L, L, M which appear in the action.

Three worldsheet fields
We will now focus on the contributions with three worldsheet fields coming from the G, G = −2T Poisson bracket computation. We start by considering the contributions involving a factor of Π a . The terms with Λ α Λ β Π a imply the constraint The terms with ψ a Λ α Π b lead to The terms with ψ a ψ b Π c lead to the equation Symmetrizing bc in this equation gives the constraint T a(bc) = 0, whereas antisimmetrizing bc implies the constraint T cba + H cba = 0. From now on, the torsion component T abc is totally antisymmetric. The terms with Λ α w β Π a imply the constraint The terms with ψ a w α Π b lead to defining equations for the C ab γ superfield 14) The terms with w α w β Π a lead to the equation Consider now the terms involving Π a in {G, G}. The terms with Λ α Λ β Π a give the constraint 1 2 (T αβa + H αβa ) + (γ a ) αβ = 0, (4. 16) which together with (4.10) determines the Type II supergravity constraint The terms with Λ α w β Π a determine G aα β of the supercurrent G (3.7) as The terms with ψ a ψ b Π c determine G abc from G as The terms with ψ a Λ α Π b determine G αab of G according to where (4.11) was used. The terms with ψ a w α Π b determine G ab α of G as Finally, the terms with w α w β Π a lead to where (4.15) was used.
Take now the terms with d α in {G, G}. The terms with Λ α Λ β d γ lead to the constraint The terms with ψ a Λ α d β give the constraint where (4.11) was used. The terms with Λ α w β d γ lead to The terms with ψ a w α d β give The terms with ψ a Λ α d γ lead to (4.18). Similarly, the terms with ψ a ψ b d α lead to (4.21).
The terms with Λ α w β d γ imply The terms with ψ a w α d β lead to equation (4.22). Finally, the terms with w α w β d γ give Focus now on the terms with factor of Π α in {G, G}. The term with Λ α Λ β Π γ gives the constraint The contributions with ψ a Λ α Π β give the equation The terms with Λ α w β Π γ lead to the constraint The terms with ψ a ψ b Π α imply The terms with ψ a w α Π β lead to The last contributions we must consider are the ones with a factor of Π α in {G, G}.
The analysis of the terms with three worldsheet fields coming from the G, G = −2 T Poisson bracket is analogous to the one we just covered and gives the "hatted" version of the constraints and definitions above (note that some of the signs are different, though). After gathering all the contributions and simplifying all the expressions with the constraints 13 coming from both Poisson brackets, the analysis of the terms with three worldsheet fields in the commutators G, G = −2T and G, G = −2 T leads to the constraints: which are indeed the N = 2 D = 10 Type II supergravity constraints of [7]. Note that the constraints T αbc = T αbc = 0 and T abc = −T abc = H abc imply as mentioned previously. Furthermore, note that in writing these expressions we have made use of the fact that the constraints imply that T αβ γ = T αβ γ = 0. Indeed, following the analysis done for the heterotic string, imply that F 1440 abcdδ and F 1440 abcdδ must be zero. To see why, note that these components are not in the kernel of the linear maps T (αβ ρ γ d γ)ρ = 0 and T (αβ ρ γ d γ)ρ = 0. Furthermore, since we have the constraints T αβ γ = T αβ γ = T αβ c = 0, there are no other 1440 irrep contributions in the Bianchi identities above. Consequently, we do conclude that F 1440 abcdδ = F 1440 abcdδ = 0 and therefore Finally, we stress the fact that imposing the U(1)-charge/small Hilbert space condition in addition to the anti-commutators was essential to achieve the constraints of (4.39).
This has already been observed in the heterotic string case. Indeed, were we allowed to deform the supercurrents with terms with positive U(1) charge (or equivalently, in the large Hilbert space), we would lose, for example, the essential constraints

Four worldsheet fields
Now we move on to the contributions with four worldsheet fields. Again, we start by focusing on the contributions coming from the G, G = −2T Poisson bracket. As mentioned above, this analysis will give the remaining background superfields of the action (3.1) in terms of the supergravity superfields. It will also determine constraints on the curvature.
Consider first the terms with Λ α Λ β w γ w δ . These are This contribution vanishes because of the Bianchi identity involving R (σαβ) ρ . The terms Using the Bianchi identity involving R [αβa]b in the second line, we obtain the equation which implies Ω a = 0 and Ω a cdef = 0. The vanishing of the term with ψ a ψ b Λ γ Λ δ implies the constraint for the curvature The terms with Λ α Λ β Λ γ w δ are proportional to leads to the constraint for the curvature The terms with ψ a Λ β Λ γ w δ give The first line vanishes because of the Bianchi involving R [aαβ] γ and the vanishing of the second line implies Ω ρ = 0 and Ω ρ bcde = 0. The terms with ψ a Λ β Λ γ w δ are The first line vanishes because of the Bianchi identity involving R [abα] β and the second line vanishes due to the Bianchi identity involving R [αγa]b . The terms with ψ a ψ b Λ γ w δ determine the background superfield E abα β of the action as The terms with ψ a ψ b w γ w δ are The first line is zero because of the Bianchi identity involving R [γab] β and the second line vanishes due to the "hatted" version of the curvature constraint (4.51) that is obtained The terms with ψ a ψ b ψ c w δ are The terms with w α w β w γ w δ are proportional to which is zero due to the D = 10 Fierz identity (γ a ) (στ (γ a ) κ)λ = 0. The terms with w α w β w γ w δ determine the background superfield M αβγρ of the action to be The terms with ψ a ψ b w γ w δ determine the background superfied J ab αβ as The terms with Λ α w β w γ w δ amount to which vanishes because of the "hatted" version of the curvature constraint (4.53). The terms with w α w β Λ γ w δ determine the background superfield G α βγρ to be The terms with ψ a w β w γ w δ are proportional to which is zero because of the Bianchi identity involving R [aρσ] γ . The terms with ψ a w β w γ w δ give the background superfield L a γαβ of the action as The terms with ψ a Λ β w γ w δ are proportional to which vanishes because of the Bianchi identity involving R [aρα] γ . The terms with ψ a Λ β w γ w δ are proportional to After using the equations for Y a βγ and Z ρβγ obtained from the G, G = −2 T bracket, as well as Ω αbc − Ω αbc = T αbc and the Bianchi identities involving Since the connections Ω ab and Ω ab differ by torsion components as in Equation (4.40), the two-forms R ab and R ab also differ by torsion components. In particular which, after using the definition of D β γρ , gives the constraint for the curvature The terms with ψ a ψ b Λ γ w δ are proportional to which vanishes after using the expression for V b γβ in terms of supergravity superfields, together with Ω abc − Ω abc = T abc and the Bianchi identity involving ∇ [α T ab] β . The terms The terms with ψ a ψ b Λ γ w δ determine the background superfied E abα β as The terms with Λ α w β ψ c w δ give the background superfield F aα βγ according to The terms with Λ α w β w γ w δ define the background superfield G α βγρ as The terms with ψ a w β Λ γ w δ determine the background superfield F aβ γα to be The terms with ψ a w β ψ c ψ d give the background superfield I bca α as The terms with ψ a w β ψ c w δ determine the background superfield K ab αβ as The terms with ψ a w β w γ w δ give the background superfield L a αβγ as The terms with ψ a Λ β Λ γ Λ δ are proportional to which is zero because of the Bianchi identity involving R (αβγ)a . Finally, the terms with which vanishes due to the Fierz identity (γ a ) (αβ (γ a ) γ)δ = 0.
Unsurprisingly, the analysis of the terms with four worldsheet fields coming from G, G = −2 T is analogous and gives the "hatted" version of the constraints and definitions above. Besides the constraints for the torsion and the H components given in (4.39), we have also obtained the constraints for the curvature Using the definitions of U ab β and D α βγ found previously, this expression becomes Then, using the Bianchi identity involving R [αab] β and the Bianchi identity involving which corresponds to the alternative definition (4.59) of E abα β given by G, G = −2T .
A similar analysis shows that the other background superfields of the action are uniquely defined.

Simplifying constraints
Before we end this section, we note that the expressions which determine the background superfields in the action and in the supercurrents found above can be simplified, since we have not fixed all of the gauge symmetries in (A.1) to (A.14). Indeed, we may use the transformations parameterized by ρ αb γ , ρ αb γ , ρ α βγ and ρ α βγ to gauge away the C αb γ , C αb γ , Y α βγ and Y α βγ background superfields respectively. As a consequence, they drop out of all the constraints in which they appear, simplifying several expressions.
We will now present the final expression for the background superfields. The background fields U, D, V, Z, C, Y are determined by the equations The supercurrents G, G are given by Note that this result is the same as the one for the heterotic superstring, even though in this case the superfields Y a βγ and Y a βγ are not zero. Finally, the remaining background superfields in the action are determined by the following equations: J ab αβ = 1 2 This concludes our task of determining all the background superfields in the action and in the supercurrents G, G in terms of supergravity superfields. We find a similar behavior to that found in the analysis of the heterotic string [6] and in the pure spinor formalism [7].
More precisely, the superfields S αβ γδ , D α βγ and D α γβ are determined in terms of the same supergravity fields as in the pure spinor formalism, where they also appear in the action. with the commutators from Appendix B. Again, the resulting terms can be organized into a set with three worldsheet fields, and another with four worldsheet fields. Table 1 below shows what constraints or definitions coming from the {G, G} = −2T and { G, G} = −2T Poisson brackets imply the vanishing of the terms with a given factor of three-worldsheet fields. The constraints related to the vanishing of the terms with factors of Λ α w β , ψ a w β and Λ α ψ b are just the "hatted" version of the constraints associated to Λ α w β , ψ a w β and Λ α ψ b from the table. Table 1 Worldsheet field factor Associated constraint or definition It remains to prove that the terms with four worldsheet fields in (5.1) also vanish. We use all the constraints found so far. The terms with Λ α Λ β w γ w δ and with Λ α Λ β w γ w δ are zero trivially. The terms with ψ a ψ b Λ γ Λ δ are proportional to which is zero due to the Bianchi identity involving which vanishes as implied by the Bianchi identity involving R (αβγ) ρ . The terms with Using the Bianchi identity involving R [γab] β in the term with a factor of P αγ and using the definition for Y a αβ , this expression vanishes. The terms with ψ a ψ b ψ c ψ d amount to which vanishes because of the Bianchi identity involving The first line vanishes because of the Bianchi identity for R [abc] α . The second line also vanishes after using the expression for Y a αβ , the constraint T αb γ = −P γγ T γαb , as well as the Bianchi identity involving R [βca]b , the Bianchi identity involving ∇ [β H abc] and the constraint H abc + T abc = 0. After using the definitions for D α βγ , Z αβγ , V a αβ and Y a αβ , the contributions with w α w β w γ w δ amount to which vanishes because of the Bianchi identity involving R (στ κ) ρ . Consider the terms with ψ a ψ b w γ w δ . After trivial cancellations they become Commuting the derivatives in the first term leads to The first line is zero due to the Bianchi identity involving R [abγ] α . The second line also vanishes after using the Bianchi identities involving R [abγ] δ and R [αβc]d . The terms with Λ α w β w γ w δ , after some trivial cancellations, amount to Λ α w β w γ w ρ P βσ {∇ α , ∇ σ }P γρ + P βσ P τ ρ R στ α γ + 1 2 (γ a ) ατ T aσ γ .

(5.12)
Computing the anticommutator one obtains Λ α w β w γ w ρ P βσ P γτ R α(στ ) ρ + P βσ P τ ρ R σ(ατ ) γ + 1 2 (γ a ) ατ T aσ γ . (5.13) The first and second terms vanish due to the Bianchi identities for R (αστ ) ρ and R (σατ ) γ respectively. The terms with ψ a w β w γ w δ amount to After computing the commutator we obtain Using the definition for Y a βγ , as well as the Bianchi identity involving R [aρα] γ , this expression vanishes. The terms with ψ a Λ β w γ w δ are which amounts zero as implied by the Bianchi identity involving R [aαρ] γ . Consider the terms with ψ a ψ b ψ c Λ δ . They amount to The Bianchi identity for R [αab]c , the Bianchi identity for ∇ [α H abc] and the constraint H abc − T abc = 0 imply that this contribution vanishes. Let us consider the terms with ψ a Λ β Λ γ w δ in (5.1). They amount to which is zero as implied by the Bianchi identity involving R [aαβ] γ . The terms with This contribution vanishes because of the Bianchi identity involving R [aαb] β . The terms with Λ α w β w γ w δ sum up to The definition for Y a αβ implies that this expression is equal to which vanishes because of the Fierz identity (γ a ) α(τ (γ a ) σκ) = 0. The terms with the factor of ψ a ψ b Λ γ w δ are proportional to Computing the anti-commutator and imposing the curvature constraint for R aαβ γ implies that this expression becomes which is zero because of the Bianchi identity involving R (σαβ) γ . The terms with ψ a Λ β w γ w δ , after some trivial cancellations, lead to After computing the commutator, this expression becomes The first line vanishes because of the Bianchi identity involving R [aαρ] β and the second line vanishes because of the curvature constraint for R αaρ γ . The terms with a factor of The terms with factor of ψ a Λ β w γ w δ in (5.1) are which is zero because of the Bianchi identity involving R [aαβ] γ . The terms with the factor ψ a w β w γ w δ in (5.1) are Using the Y a αβ definition we obtain The first line vanishes because of the Bianchi identity for ∇ (ρ T σ)a γ and the second line is also zero after noticing that the supergravity constraints imply T αab = 2(γ ab Ω) α [7]. The terms with factor of ψ a Λ β Λ γ Λ δ in (5.1) are proportional to which is zero because of the Bianchi identity for R [αβγ]a . The terms with factor of Λ α Λ β ψ c ψ d and the terms with Λ α Λ β ψ c ψ d (5.1) are trivially zero. Finally, the terms with factor of ψ a Λ β Λ γ w δ in (5.1) are Using T αab = 2(γ ab Ω) α as implied by the supergravity constraints [7] this expression becomes proportional to Above, we have covered only half of the contributions with four worldsheet fields one obtains when computing G, G . The missing half is the "hatted" version of the contributions we have written, that is, we detailed the vanishing of the contribution with a factor of ψ a Λ β Λ γ w δ , but not the vanishing of ψ a Λ β Λ γ w δ . Unsurprisingly, the reason why these omitted contributions vanish is analogous to the reason why their "unhatted" counterparts shown here vanish. This concludes the proof that the constraints from {G, G} = −2T and { G, G} = −2 T imply {G, G} = 0.

Holomorphicity and anti-holomorphicity of G, G
In the previous sections we have shown that requiring the conditions G, G = −2T and G, G = −2 T imply G, G = 0 and, more importantly, constraints for the Type II supergravity fields. In the present section we will show that these three Poisson Brackets imply ∂G = 0, ∂ G = 0. (6.1) Here we should recall the notation that Poisson brackets with omitted σ, σ ′ imply integration over these variables. More precisely: and so on.
The first step in the proof is to use the Jacobi identity to write where f (σ) is some function of σ which does not depend on any of the fields in our action.
Then, using the aforementioned anti-commutators, we find that this equation reduces to Here, note that G, f G = 0 is true even with the inclusion of f (σ). The analogous computation switching G → G, G → G and T → T yields the result It is clear that once we show we are done with the proof, as the computation holds for any f (σ). Indeed, here we see the importance of being able to have non-constant f (σ): were we not, we would conclude only that ∂ 0 G(σ) = 0 because the piece ∂ 1 G(σ) would appear as a total derivative (remember that we are always dealing with an integral over σ).
In order to prove (6.6), we need to show that where ϕ ∈ ψ a , Λ α , w α , ψ a , Λ α , w α , Φ(Z) is some function of the supercoordinates Z M and d A ∈ d a , d α , d α . We will focus on the proof for the brackets that involve T (the proof for the ones with T is analogous).
Consider then using the equations of motion for ψ a , Λ α and w α to rewrite T as Let us start with the case for T , w α . Using the commutators listed in the appendix, one can show that the bracket reduces to T , w α (σ) = 2∂ 1 w α + ∂w α (σ) = ∂w α (σ), (6.10) where the contribution 2∂ 1 w α comes from the commutator (6.11) and the remaining commutators amount to ∂w α using the equations of motion for w α again. The same logic applies to the cases for ψ a and Λ α , where we also find T , ψ a (σ) = 2∂ 1 ψ a + ∂ψ a (σ) = ∂ψ a (σ), (6.12) For the case of ψ a , Λ α and w α , the computation is slightly simpler. The brackets with the are always zero, whereas the remaining brackets can be easily computed to yield ∂ψ a , ∂Λ α and ∂w α after we use the equations of motion for these fields. This concludes the proof that T , ϕ(σ) = ∂ϕ(σ). The proof for T , Φ(Z)(σ) = ∂Φ(Z)(σ) is easy. The only brackets that contribute are They are easily computed and we conclude that Note that none of the results so far are changed by the inclusion of non-constant f (σ).
Finally, the proof that T , f d A = f ∂d A is the one with the longer computations. Still, the only thing one has to do is to compute the relevant commutators and recombine the contributions to get the result. After some long computations using the commutators in the appendix, we conclude that where we have used the equations of motion for the d A derived from Equation (C.12).
We have proven all the commutators in (6.7). Therefore, the Jacobi identity (6.3) does indeed imply

Action and equations of motion
In order to write the action for the B-RNS-GSS string on AdS 5 × S 5 , we must know the values for the RR flux, the B-field, the torsions, the H-fluxes and the curvatures on this background. Using the conventions of [9], these background fields take the values Furthermore, we will favor the supercoset description of AdS 5 × S 5 AdS 5 × S 5 = P SU(2, 2|4) SO(4, 1) × SO(5) (7.4) and we use the conventions shown in appendix D for the psu(2, 2|4) algebra. In this description, we introduce the left-invariant currents T ab + J α T α + J a T a + J α T α , g ∈ P SU(2, 2|4) SO(4, 1) × SO (5) , (7.5) where T [ab] , T α , T a , T α denote the psu(2, 2|4) generators. These currents are related to our familiar background fields Ω [ab] and E A as: Finally, note that since T abc = H abc = 0 we only have one spin-connection Ω ab = Ω ab = J [ab] . Plugging in these values and the constraints derived earlier in the general action for the Type II string, we find where recall the notation Note that the auxiliary fields d α , d α have the same equations of motion as they do in the pure spinor formalism, namely This means that by integrating them out we find (7.10) Consider now the Lie-algebra valued worldsheet fields given by (7.12) and define (7.14) Using these new notations and the conventions for the traces of psu(2, 2|4) generators from Appendix D, the stress-energy tensors are written as whereas the superconformal currents are written as Furthermore, the action can also be rewritten as Note that the part of this action that involves only J's, Λ's and w's has the same form as the pure spinor AdS 5 × S 5 action [10]. Furthermore, the new terms are the simplest possible generalization which respects the Z 4 grading of the psu(2, 2|4) algebra while including all of the N i 's for i = 0 to 3. The equations of motion for the Λ, w, ψ, Λ, w, ψ 20) The graded Jacobi identity then allows us to obtain the equations of motion for the N i 's and N i 's. We find: 25) N 2 ], (7.26) Finally, by varying the action with respect to δg = gX and using the Maurer-Cartan equations, we can derive the equations of motion for the J i 's and J i 's. These are: (7.32)

Integrability
We will now establish classical integrability of the B-RNS-GSS string on AdS 5 × S 5 .
In order to do so, we will find a Lax connection that is flat and that transforms under the action of the BRST charge by a dressing transformation. As a further check, we will also show that the Lax connection generates the usual Noether charges associated to the isometries of AdS 5 × S 5 .
The most standard way to tackle the problem of finding the Lax connection is to write a general expression for it using the J i 's, J i 's, N i 's and N i 's and then fix the coefficients associated to each of these currents by requiring the flatness condition to be satisfied.
This was the method used when integrability was first established for the Green-Schwarz string in AdS 5 × S 5 [11], and was also used to find the Lax connection for the pure spinor formalism [12]. We will, however, take a more direct route which takes advantage of the previous knowledge about the pure spinor Lax connection. Notice that the equations of motion above reduce to the ones from the pure spinor formalism when we take N 1 , N 2 , N 2 , N 3 → 0. Furthermore, it is well known that the flatness condition on the Lax connection implies the equations of motion. Bearing these two facts in mind is enough to easily achieve our goal, as we may start off with the pure spinor Lax connection and add corrections to compensate for the new terms in the equations of motion. It turns out that the needed corrections are really simple.
The pure spinor Lax connection is given by If we compute the flatness condition with these expressions at orders µ −4 and µ 4 using the B-RNS-GSS equations of motion, we find that we are missing two contributions: (7.36) These missing commutators are accounted for if we correct the Lax connection to is equivalent to the equations of motion listed above for all orders from µ 6 to µ −6 . As claimed above, it is also amusing to check that upon taking the coefficient of the first order term in the Taylor expansion of A, A around µ = 1 we find the Noether current associated to the AdS 5 × S 5 isometries which is the expected behavior of a Lax connection.
The last thing to do is to show that the eigenvalues of the monodromy matrix (7.42) associated to this Lax connection are BRST invariant. In order to do so, we must first compute the BRST variation of the fields Λ, w, ψ, Λ, w, ψ and of the currents J (for the discussion on the BRST variation of the Lax connection in the context of the pure spinor formalism see [13]). We start by noting that the action of the stress energy tensor is simply We conclude that the (cT + c T ) piece of the BRST charge transforms the Lax connection by a dressing transformation with dressing parameter cA(µ) + cA(µ).
Next, we note that, up to SO(4, 1) × SO(5) gauge transformations (see [14] for their inclusion in the pure spinor case), and therefore The equation for the J i 's is analogous. Furthermore, we find (γG + γ G)(Λ) = 1 2 γN 1 − γJ 1 + γ ψ, Λ + 1 2 γ ψ, w , (7.49) It then follows that the variation of the N i 's is with the dressing factor ϕ(µ) being given by Note that this is the natural generalization of the dressing factor for the pure spinor case due to the presence of the terms Str ψJ 2 − 1 2 wJ 1 in G and Str ψJ 2 + 1 2 wJ 3 in G. We thus conclude that the complete variation of the Lax connection under the BRST charge is given by a dressing transformation Consequently, the monodromy matrix (7.42) generates an infinite set of BRST-invariant conserved charges and classical integrability has been established for the B-RNS-GSS string in AdS 5 × S 5 .

Conclusion
In this paper, the B-RNS-GSS formalism has been extended for the description of the Type II superstring in curved supergravity backgrounds, including backgrounds with finite Ramond-Ramond flux. More precisely, nilpotence of the N = (1, 1) worldsheet superconformal BRST charge at the classical level has been shown to imply the Type II supergravity equations of motion for the background superfields. As a direct application of this result, an action for the B-RNS-GSS string on an AdS 5 × S 5 background has been proposed. Classical integrability of this sigma-model has also been established.
There are several lines of investigation that can be pursued now that these results have been achieved. One possible direction is to relate the AdS 5 × S 5 action (7.19) to the twistor superstring action proposed in [15]. Note that this twistor superstring formalism is also manifestly invariant under N = (1, 1) worldsheet supersymmetry and N = 2 D = 10 spacetime supersymmetry. Furthermore, physical states in this formalism must satisfy a U(1) charge requirement that is similar to the U(1)-charge/small Hilbert space condition discussed in Section 2. It should also be possible to find B-RNS-GSS vertex operators in AdS 5 × S 5 starting from the pure spinor ones [16] by exploring the similarities between the two formalisms in this background.
Attempts at using the RNS formalism to describe Ramond-Ramond backgrounds have either struggled with ill-defined half-integer picture raising operators [17], or have been limited to a perturbative analysis of Ramond-Ramond deformations using String Field Theory methods [18]. Since the starting point of the B-RNS-GSS formalism can be seen as the variables, action and BRST charge from the RNS formalism, perhaps a more daring application would be to use the results of this paper to better understand how to describe Ramond-Ramond backgrounds with the RNS formalism.
Acknowledgments: The authors are especially grateful to Nathan Berkovits for useful discussions, for suggesting to investigate the integrability of the AdS 5 × S 5 sigmamodel and for carefully reading and commenting on the manuscript. OC would like to thank fondecyt grants 1200342 and 1201550 for partial financial support. JG would also like to thank Lucas N.S. Martins and Rodrigo S. Pitombo for useful discussions, and FAPESP grants 2021/14335-0 (ICTP-SAIFR) and 2022/04105-0 for partial financial support.