Yang-Baxter deformations of the AdS5 × S5 pure spinor superstring

We present integrable Yang-Baxter deformations of the AdS5 × S5 pure spinor superstring theory which were obtained by using homological perturbation theory. Its equations of motion and BRST symmetry are discussed and its Lax connection is derived. We also show that its target space background is the same generalized supergravity background found for Yang-Baxter deformations of the Green-Schwarz superstring in AdS5 × S5.


Introduction
There are many instances in the context of the AdS/CFT correspondence where both sides of the duality present an integrable structure [1,2]. On the string theory side it is well known that the AdS 5 × S 5 superstring equations of motion either in the Green-Schwarz (GS) or in the pure spinor (PS) formulation can be cast into a zero curvature equation satisfied by a Lax pair [3,4]. More recently, significant progress has been made in deforming the AdS 5 × S 5 structure while preserving its integrability and the fermionic κsymmetry. On one hand, the GS λ-deformed models [5] are based on a G/G gauged WZW model and yields a target superspace corresponding to a supergravity background. On the other hand, the GS Yang-Baxter (YB) deformed models make use of a linear operator, called R-matrix, which solves the modified classical Yang-Baxter equation (mCYBE) (also known as η-deformations) [6,7], or solves the homogeneous CYBE [8,9]. Even though κ-symmetry is preserved for these deformations, its target superspace does not solve the equations of motion of type IIB supergravity [10][11][12] but rather a generalization of them. It was also argued that, even without a dilaton to preserve Weyl invariance at one-loop level, the generalized supergravity backgrounds still define a two-dimensional scale invariant theory [13]. This apparent conflict was solved by Tseytlin and Wulff [14] who showed that contrary to the standard assumption that κ-symmetry implies the equations of motion of type IIB supergravity, it in fact leads to generalized supergravity equations of motion. In this framework, the supergravity equations of motion can be recovered when the R-matrix is unimodular [15].

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These generalized backgrounds are also related to the standard supergravity equations by T -dualizing a supergravity target space in a isometric direction which is a symmetry of all the fields except for the dilaton transforming linearly in this direction [16,17]. Also, the relation between homogeneous YB deformations and T-duality have been extensively studied [18][19][20][21][22]. It has also been shown that these deformations and the T-duality transformations of the original model can be formulated in a unified description [23][24][25]. The emergence of generalized supergravity backgrounds has also been further explored in the context of open-closed string map and in the double field theory formalism [19,[26][27][28][29][30][31][32][33].
In the PS formalism the world-sheet metric is in the conformal gauge and the κsymmetry of the GS superstring is replaced by a global BRST symmetry avoiding well known issues with the light-cone gauge. This provides a powerful tool to gain new insights like properties of vertex operators and correlation functions [34][35][36]. It also allowed to show that the integrability of the PS string in AdS 5 × S 5 [4] persists to all loops in the quantum theory [37]. Regarding deformations of PS superstrings just a few cases are known, like the β-deformation, obtained by a TsT transformation on the supergravity background [38], or the λ-deformation for the matter sector of the pure spinor model [39].
A more systematic way to deform the PS superstring in AdS 5 × S 5 was proposed in [40] and makes use of homological perturbation theory. The main idea is to find a deformed action and a deformed nilpotent BRST operator which can be constructed as a series expansion in the deformation parameter. At first order it was found that the action is proportional to an integrated vertex operator parametrized by an antisymmetric R-matrix. When acting on the matter sector, the deformed BRST operator has the same structure as the η-deformed κ-symmetry of the GS superstring [15], thus suggesting that the PS deformed model could give rise to an η-deformed background. Moreover, nilpotency of the BRST charge at first order implies the CYBE or the mCYBE for the R-matrix [40]. It was also found that in the flat space limit BRST invariance is not enough to characterize the linearised equations of motion for type IIB supergravity [41]. This is due to a conflict with the conformal invariance of the deformed PS action. In order to preserve the conformal symmetry for the deformed world-sheet theory the corresponding vertex operator must be given by a primary operator of AdS 5 × S 5 . This means that the double pole of the OPE for the vertex operator and the energy-momentum tensor, which is proportional to the action of the Laplacian on psu(2, 2|4), must vanish [42]. As shown in [40] this requirement implies the unimodular condition for the R-matrix. It has also been pointed out that the cohomology of the deformed BRST charge in the flat space limit does not give rise to a supergravity background but rather to a generalized supergravity one [43,44]. Since these results were found at first order in the deformation parameter it would be very interesting to see which of these properties hold for the fully deformed theory. Besides that, since the YB deformations of the GS superstring in AdS 5 × S 5 do not require a small deformation parameter we would expect the same to be true for the PS case.
The aim of this paper is to extend the approach of [40] to all orders in the deformation parameter to obtain full YB deformations of the PS superstring in AdS 5 ×S 5 . For simplicity we will consider only the case in which the R-matrix satisfies the CYBE. The case where the R-matrices satisfy the mCYBE should follow the same lines as in [40]. A troublesome JHEP02(2019)056 issue of this construction is that the BRST charge acts through a non-local operator on the anti-field sector turning the full deformed action non-polynomial. We then find that it is possible to remove the anti-field sector so that the complete deformed action is a polynomial expansion in the R-matrix. As a consequence, the BRST operator has now an infinite series expansion in the ghost sector and its nilpotency holds only on-shell. Even so, this local action allows us to read the background superfields using the general action of Berkovits and Howe [45]. By rewriting the PS deformed action in terms of the GS deformed variables we show that both, the GS and the PS YB deformed models have the same geometry and target space fields.
The YB deformations of the GS superstring can be constructed in a way which manifestly preserves the integrability of the undeformed theory [7]. As expected [6,7,46], it is possible to recast the deformed equations of motion of the PS superstring in such a way that it presents the same algebraic structure as the undeformed ones. This allow us to find a Lax representation for them establishing integrability for the PS case.
The contents of this paper is the following. In section 2 we review some important properties about the PS superstring in AdS 5 × S 5 like its BRST symmetry and the Lax representation for the equations of motion. In section 3 we present the full deformation of the AdS 5 × S 5 PS superstring extending the linearised results of [40] to all orders. Then, in section 4, we show how to get rid of the awkward non-polynomial terms appearing in the deformed action. In section 5 we show the integrability of the deformed action by constructing a suitable Lax connection. Finally, in section 6, we make contact with the GS deformed superstring and show that the target space superfields of our model correspond to those of generalized supergravity.
2 Review of the pure spinor superstring in AdS 5 × S 5 It is well known that the superstring theory in AdS 5 ×S 5 can be formulated as a supercoset sigma model on PSU(2, 2|4)/(SO(4, 1) × SO(5)) [47]. In this construction, a key role is played by the Z 4 grading of the superalgebra psu(2, 2|4) which allows us to decompose it as g = g 0 + g 1 + g 2 + g 3 . (2.1) If g is an element of the supergroup PSU(2, 2|4) we define the Maurer-Cartan form as Since it takes values in psu(2, 2|4) we can decompose it as The pure spinor action in AdS 5 × S 5 is given by [42] 1 3)

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with d PS = P 1 + 2P 2 + 3P 3 , where P i projects an element of the superalgebra g on its g i -component. The Lie algebra valued ghost fields are defined as where η αα = −ηα α is numerically equal to the identity matrix. The bosonic ghosts λ α and λα are constrained to satisfy the pure spinor condition λγ a λ =λγ aλ = 0 , (2.5) and the pure spinor Lorentz currents are given by The action is invariant under a BRST symmetry whose classical charge Q = Q L + Q R is given by and acts on a group element as a derivative (2.14) For the matter sector the equations of motion are obtained from variations δg = gξ i , i = 1, 2, 3, where ξ i is an element of g i . Defining the covariant derivatives as we find that 20)

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Similarly, the equations of motion for the ghost sector are obtained by varying the action with respect of λ and ω and expressing the result in terms of the Lorentz currents Classical integrability can be proven by constructing the Lax pair [4] L where z is the spectral parameter, in such a way that the equations of motion (2.16)-(2.21) and (2.22) are equivalent to the zero curvature condition Defining z = e l it is possible to express the density of the conserved currents as It is also useful to find the BRST transformation of the currents We recall that the BRST charge is nilpotent up to equations of motion [37]. In order to make the BRST charge nilpotent off-shell we introduce a pair of fermionic anti-fields (ω * 1+ , ω * 3− ) which must satisfy the following condition [48] The BRST transformations (2.9) and (2.10) are then be modified to and the BRST transformation of the anti-fields is given by Notice that the BRST transformations of the conserved currents j ± are proportional to the equations of motion for λ, therefore the conserved charge is BRST invariant only when

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the classical equations of motion hold. It is possible to avoid this situation by including the anti-fields in the currents (2.26) We then find that the action 3 Deformation of the AdS 5 × S 5 pure spinor superstring In this section we will follow the general deformation procedure for a BRST invariant action presented in [40]. The deformed action and BSRT charge are constructed as a series expansion in η where the coefficients of the expansion are determined by imposing BRST invariance We find, up to order η 3 , that The first of these equations is satisfied since the undeformed model is assumed to be BSRT invariant. In [40] equations (3.5) and (3.6) were solved and consequently V 1 , V 2 , Q 1 and Q 2 were obtained. Here we review this procedure and solve the equations to all orders. In particular, we will show that the expansion of the BRST charge stops at first order. The first step is to make an ansatz to solve (3.5) as [40] To cancel this contribution at first level, Q 1 is required to satisfy . This is achieved by taking Q 1 as [40] where the projectors P 13 and P 31 are defined as where S 2 is a vector of grading 2, such that [49] [λ 1 , Some important identities involving the projectors P can be found in appendix A. The next step is to discuss the nilpotency of the BSRT charge. Noticing that Q 0 g = gΛ( ) forΛ( ) = g −1 ( λ 1 + λ 3 )g and Q 1 g = gRΛ( ) then [Λ( ),Λ( )] = 0 is a consequence of the pure spinor condition. We also find that so that {Q 0 , Q 1 } = 0. We also need to find the action of Q 2 1 over g. We start with which is also proportional to the CYBE. Nilpotency on ω 1+ is satisfied because is proportional to the gauge symmetry generated by the pure spinor constraint (2.5). An analogous result can be obtained for ω 3− . Now, for ω * 1+ we find which is proportional to the CYBE. An analogous result is obtained for ω * 3− . Thus, the BRST charge Q = Q 0 + ηQ 1 is nilpotent if R is a solution of the CYBE [49].
Notice that the R-matrix is not constrained to lie in the kernel of the CYBE. We can clearly see this when considering that the left-hand side of the (3.17) is required to vanish up to a local transformation of g [37]. The term g[Λ( ), Λ( )] is proportional to a

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Lorentz rotation on g, hence, the nilpotency of the BRST charge allows the introduction of R-matrices that solve the mCYBE. This has been extensively discussed in [40].
Having found Q 1 the next step is to find V 2 using (3.6). First of all, we compute the action of Q 1 on the local currentsj ± where we introducedd = 3P 1 + 2P 2 + P 3 , the transpose operator of d. Then, we obtain where we have made use of the CYBE for R. It is important to take into account the following identity valid for any X and Y invariant fields under BRST transformations. Then, it follows that which cancels the contribution from (3.21). Then, Since there is no other local fermionic symmetry of the undeformed pure spinor action then it follows that Q 2 = 0.
Having understood this procedure it is not too difficult to find an expression for V 3 . First, we note that so that Q 3 = 0. At this stage it is clear the pattern for the higher order terms. Then, the complete deformed action, invariant under the BRST charge Q = Q 0 + ηQ 1 , takes the form (3.25)

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We can check this by splitting the total BRST variation of the action in three parts. First of all we note that Similarly, taking into account (3.23), the action of Q 1 over the same expression is given by When considering the CYBE the above equation is rewritten as Taking into account (3.26), (3.27) and (3.28) it is easy to verify that the BRST variation of the complete deformed action vanishes.

The polynomial deformed action
Recall that the anti-fields were introduced in order to get an off-shell nilpotent BRST charge. The deformed action obtained in the last section (3.25) is non polynomial due to the projectors P which emerged as a consequence of the BRST transformation on the anti-fields (3.11) and (3.12). In this section we abandon the anti-fields formulation from the outset and, as a consequence, the advantages of working with an off-shell nilpotent BRST charge. What we get is a local action which is a polynomial expansion of local fields. First of all we solve (3.5) for Taking into account (2.27) we have that

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We look for a BRST operator Q 1 consistent with the PS formulation [45] so this requires Q 1 (λ) = 0. Our ansatz then takes the form where α and β are to be fixed by imposing BRST invariance at first order in η. Its action over S 0 is given by Then, to cancel (4.2) the action of Q 1 on the ghost sector must be Having obtained Q 1 we look for V 2 by solving (3.6). Let us now write the Q 1 transformations of j ± . (4.7) This means that (4.8) The natural ansatz for V 2 is then 1 4 Str(j + , R • Ad −1 g • d PS • Ad g • Rj + ). In order to show this we need to find a Q 2 such that (3.6) holds. First, we observe that These results altogether give and we find

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where R g = Ad g • R • Ad −1 g . The last two terms of (4.11) are proportional to the equations of motion of ω 1 and ω 3 and can be removed by Q 2 S 0 if Q 2 is defined as The remaining terms in (4.11) cancel with the contribution coming from (4.8). This result allows us to infer the pattern of deformation for the higher order terms. Taking into account the above results the complete deformed action is given by the local expansion (4.14) We can rearrange this action into a more familiar form by defining the operators Also, it is useful to consider the deformed pure spinor currents Hence, the deformed action can be rewritten as 2 This action must be invariant under the BRST transformations This can be shown by splitting the action into four sectors. Taking into account that the deformed currents vary under δg = gξ i as the first term of (4.17), the matter sector, contributes with

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In particular, for δgg −1 = Q(g)g −1 = (1−ηR g ) λ 1 +(1+ηR g ) λ 3 , and taking into account the following identities For the matter-ghost sector, the second term in (4.17), we note that the BRST transformations of J − andJ + are After a lengthy calculation we obtain that On the other hand, considering (4.19) and (4.20), we have so that In the ghost sector we have to consider (4.19) and (4.20) to set For the third term of (4.17) we have

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Considering (4.32), the first term of the above equation can be expressed as while the second one is given by After a lengthly computation the last term in (4.36) can be expressed as

Integrability
To derive the equations of motion from (4.17) it is convenient to split the action in three sectors. For the matter sector we implement the variation in the form δg = gξ i to get a contribution as in (4.23). For the matter-ghost sector we have to consider (Str(δJ 0− , N 0+ ) + Str(δJ 0+ , N 0− )) = Str(δgg −1 , E 1 ) , In the N 0− N 0+ sector we have where

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Collecting all contributions the equations of motion take the form For the ghost currents the equations of motion can be expressed as We notice that the equations of motion (5.4)-(5.6) present the same structure as the undeformed ones as expected [6,7,46]. We can write them in a more suggestive manner by defining the currents J + and J − as so that the equation of motion (5.4) takes the form and for the ghost currents (5.5) and (5.6), At this stage it should be clear that the ansatz for the Lax pair can be found by exchanging J − and J + for J andJ + , respectively, in the undeformed Lax pair (2.23). This is expected since the pair of currents (J − , J + ) satisfy the zero curvature condition when the classical equations of motion are imposed. This can be shown by inverting (5.7) as (5.10) Each term in the Maurer-Cartan equation ∂ − A + −∂ + A − +[A − , A + ] = 0 can be expressed as so that, after a lengthy calculation, the Maurer-Cartan equation takes the form which shows that the pair (J − , J + ) satisfies the zero curvature condition when the equations of motion hold.

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Defining the covariant derivatives as the equations of motion can be written as 14) Similarly, the equations of motion for the ghost sector in terms of the Lorentz currents are We have then shown that the equations of motion for the deformed action admits a zero-curvature representation given by the Lax pair: where z is the spectral parameter. An interesting property of the matter sector is that the equations of motion accept a Lax representation as can be seen by switching off the ghost contributions.
Considering (4.19) and (4.20) the BRST charges are given by The (anti) holomorphicity of the BRST currents can be easily proven by using the above equations of motion.

Relation to the homogeneous YB deformations of the GS superstring
In this section we will look for the background fields of the deformed pure spinor action. This is achieved by comparing the deformed model (4.17) with the standard Berkovits-Howe action [45] S BH = 1 2πα dz 2 1 2 E aĒb η ab + 1 2 E AĒB B AB +d αĒ α +dαEα +d α dαP αα + +Ω β α λ α ω β +Ωβ αλαωβ +λ α ω βdγ C βγ α +λαωβd γCβ γ α +λ α ω βλαωβ S ββ αα +S gh . and the left and right-moving spin connection, and A = (a, α,α) is a tangent space index. The action also includes the ghosts (λ α , ω β ,λα,ωβ) and the world-sheet auxiliary fields (d α , dα). These world-sheet fields are coupled through target space fields. The superfield B AB is a superspace two-form. The leading component of P αα is the Ramond-Ramond bispinor. The (C βγ α ,Cβ γ α ) are related to the gravitini and dilatini, and S ββ αα is related to the Riemann curvature. As stated above, the pair (d α , dα) are auxiliary fields and can be integrated out when P αα is invertible. Defining its inverse as P αα P βα = δ β α , the equations of motion for d α and dα give us Substituting these values in (6.1) the action takes the form In this way we have split the action into four sectors depending of their ghost content.
We are almost ready to read the target space superfields by comparing the action described above with the deformed action (4.17) and to show that the deformation of the PS AdS 5 × S 5 superstring yields the same target space supergeometry as the homogeneous YB deformation of the GS AdS 5 × S 5 superstring [15]. The YB deformations of the GS superstring [6,8] is implemented through the Lie algebra operator (6.5) Their components are linear combinations of the projectors d GS = P 1 + 2η 2 P 2 − P 3 ,d GS = −P 1 + 2η 2 P 2 + P 3 , (6.6) whereη = (1 − cη 2 ) 1/2 . Since we are interested in the case when R satisfies the CYBE this means thatη = 1. It is convenient to define the GS deformed currents so that the action of the GS η-model is written as A nice approach was introduced in [15] in order to read the target space supergeometry.
In particular, the supervielbiens E A of the deformed geometry are given by where h is an element of the isotropy group SO(5, 1) × SO(6).

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To illustrate the correspondence between the two superstrings it is worthwhile to rewrite (4.17) in GS language. For this purpose we define the operators which relates the PS deformed currents J ± defined in (4.16) to J GS as It is useful to keep in mind some useful identities involving (6.10): We start by examining the matter sector of (4.17). This contribution should be compared with the matter sector coming from (6.4) which will allow us to read the background fields B and P αα . It reads Str(P 3 J GS− , P 1 • ϑ + P 1JGS+ ) . (6.14) The first two terms of (6.14) can be compared with the first two terms of (6.4). so that the metric and the B-field are They match the metric and B field of the GS YB deformations [15]. Now we look for the Ramond-Ramond bispinor. Substituting (6.9) in the last term in (6.14), we have which can be compared with (6.4) to get

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where Kα α = Str(t 3 α , t 1 α ). Taking into account (6.12) we can write P αα as which is the same RR bispinor found for the homogeneous YB deformations [15]. Now we move to the matter-ghost sector. First of all we notice that Using these equations in the matter-ghost sector of (4.17) we have The above equation can be compared with the second line of (6.4) to read the spin connection (Ω,Ω) and the pair (C,C) Finally, the same analysis can be done for the N 0− N 0+ sector and we find S αα ββ = (γ ab )α β (γ cd ) α β (1 − ηO −1 PS− R g ) ab cd . (6.23) This shows that the homogeneous YB deformations of the GS superstring and our deformation of pure spinor in AdS 5 × S 5 have the same geometry and target space contents, that is, the same generalized supergravity background.

Concluding remarks
We have shown how to build homogeneous YB deformations for the PS superstring in AdS 5 × S 5 by exploiting its BRST symmetry and using homological perturbation theory. Even though we restricted our analysis to the case where the R-matrix satisfies the homogeneous CYBE, the extension to the non-homogeneous case should proceed along the same lines as in [40] and we expect a simple relation between them as in the case of deformed GS superstrings [6,8].
We have found a one to one correspondence between deformations of the action and the cohomology of the BRST charge for the PS superstring in AdS 5 ×S 5 . Having found the deformed BRST operator it would be interesting to study the elements in its cohomology as, for instance, the deformation of the dilaton vertex operator considered in [50].

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It is important to remark that our analysis is completely classical and it is plausible to expect that extra quantum requirements may enforce on-shell supergravity. As it was shown in [13] the YB deformed GS model preserves the original scale invariance and defines a UV finite theory. In the PS case we expect that the Weyl symmetry can only be recovered when the deformed target space allows a type IIB supergravity solution suggesting that the central charge of the deformed model must be proportional to the unimodular condition for the R-matrix.
As remarked in the introduction, the GS superstring propagates in a background which is restricted by κ-symmetry to be a solution of generalized supergravity [14]. Our results strongly suggest that, at least classically, the constraints imposed on the target superspace by the nilpotency and holomorphicity of the BRST charge [45] should also imply the equations of motion for generalized supergravity. As in the case of the GS superstring [13], we expect that this condition would be, in principle, sufficient to get a vanishing one-loop beta function.

Proof
Analogously, it can be proved that Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.