Yang-Baxter Deformations of the $AdS_{5}\times S^{5}$ Pure Spinor Superstring

We present an integrable deformation of the $AdS_{5}\times S^{5}$ pure spinor action based on homological perturbation theory. Its equations of motion, Lax connection and BRST symmetry are discussed. The resulting model describes a pure spinor superstring in a generalized supergravity background.


Introduction
There are many instances in the context of the AdS/CFT correspondence where both sides of the duality present an integrable structure [1]. 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 [2,3]. More recently significant progress has been made in deforming the AdS 5 × S 5 structure while preserving the integrability and the fermionic κ-symmetry. The GS λ-deformed model [4] is based on a G/G gauged WZW model and yields a target superspace corresponding to a supergravity background. On the other hand, in the GS η-deformation [5,6] the main ingredient is a linear operator R which solves the modified classical Yang-Baxter equation (mCYBE) The supergravity equations can be recovered when then killing vector I a vanishes and Z a = ∂ a φ. 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, which transforms linearly in this direction [11,12]. Consequently, even without a dilaton to preserve Weyl invariance, the generalized supergravity backgrounds still define a two-dimensional scale invariant theory. The emergence of generalized supergravity backgrounds has been further explored in the context of open-closed string map and doubled formalism [13][14][15][16]. Then, contrary to the standard assumption that κ-symmetry implies the equations of motion of the type IIB supergravity it leads, in fact, to generalized supergravity equations of motion [17]. In the context of ηdeformation, the condition to have a supergravity background translates into an algebraic relation on the R-matrices, the so-called unimodular condition [18] R AB f C AB = 0 . For pure spinors on the other side, just a few cases of integrable deformations of AdS 5 × S 5 are known. The β-deformation, obtained by TsT transformations on the supergravity background, is a well known case [19]. An integrable λ-deformation was also found for the matter sector of the pure spinor model [20]. Another approach is to deform the pure spinor superstring in AdS 5 ×S 5 in a BRST invariant way based on homological perturbation theory [21]. This work opened the possibility to formulate pure spinor superstrings in backgrounds which do not correspond to any supergravity solution [22,23], but this endeavour is still incomplete and its relation to GS deformations not well understood.
In the pure spinor model the BRST symmetry is expected to play a very important role in determining the structure of any consistent deformation so the path followed in [21] starts with an infinitesimal deformation corresponding to the massless vertex operator parametrized by a constant antisymmetric tensor B AB 1 . At the linearized level, the deformation of the action is parametrized by its integrated vertex operator which is obtained from (1.4) by applying the standard descent procedure. Here j A are the global symmetry currents of the undeformed action and η is a small parameter controlling the deformation. Once (1.5) is known, the deformed action S def and the deformed BRST operator Q def can be constructed as a series expansion in the deformation parameter η In this paper we will follow the approach developed in [21] to find the full η deformed pure spinor action. We find that the BRST charge expansion (1.7) stops at first order and its nilpotency holds only for B-matrices which solve the mCYBE. For the sake of clarity and simplicity we will work with R-matrices which solve the CYBE. The case c = 0 was also considered in [21] and could be constructed in a straightforward way.
It is important to notice that not all vertices of the type (1.4) belong to the cohomology of the pure spinor operators in AdS 5 × S 5 . It includes a family of trivial states parametrized by a matrix of the form These states, which transform in the adjoint representation, do not appear in the type IIB supergravity spectrum, even though they give rise to a well defined deformation. It was found that in the flat space limit of this deformation BRST invariance is not enough to characterize the linearised equations of motion for type IIB supergravity [22]. The solution for this conflict lies in conformal invariance since the vertex V [R] is a primary field only when the unimodular condition (1.3) holds. Furthermore, the cohomology of the deformed BRST charge in the flat limit does not reproduce a supergravity background but rather a generalization involving a pair of vectors which satisfy the conditions [23] In the light of η deformations, these states have been interpreted as the pair of vectors arising in generalized supergravity [24]. Moreover, the deformed BRST action in the matter sector possess the same structure as the η-deformed GS superstring, which allows the preservation of kappa symmetry [18] δg = gǫ , ǫ = (1 − ηR g )λ 1 + (1 + ηR g )λ 3 (1. 10) thus suggesting that the in the pure spinor case we have an η-deformed background. We then find that Q 1 acts through a non-local operator on the anti-field sector turning the full deformed action non-polynomial. Then, we show that it is possible to remove the anti-field sector so that the complete deformed action is a polynomial expansion in the R-matrices but the BRST charge 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 [25]. By rewriting the PS deformed action in terms of the GS deformed variables we show that both, the GS and the PS, η-deformed models have the same geometry and target space fields. This means that classically the nilpotency and holomorphicity of the pure spinor BRST operator leads to a generalized supergravity background.
The η deformation of the GS superstring can be constructed in a way which manifestly preserves their integrability [6]. On the pure spinor side the main criterion to find the deformation is BRST invariance. Surprisingly, it is possible to recast the deformed equations of motion into the same algebraic structure of the undeformed equations of motion, allowing us to find a Lax representation for them.
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 of the equations of motion. In section 3 we present the full deformation of the AdS 5 ×S 5 PS superstring extending the results of [21] to all orders in η. 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 η-deformation and show that the target space superfields of our model correspond to those of generalized supergravity [18].
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 P SU (2, 2|4)/(SO(4, 1) × SO(5)) [26]. 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 P SU (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 2 3) with d P S = 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 (2.4) The bosonic ghosts λ α andλα are constrained to satisfy the pure spinor condition λγ a λ =λγ aλ = 0 , (2.5) and the pure spinor Lorentz generators are given by The action is invariant under a BRST symmetry whose classical charge is and acts on a group element as a derivative ǫ Q(g) = (ǫλ 1 + ǫλ 3 )g , (2.14) For the matter sector the equations of motion are obtained from small 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 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 [3] L where z is the spectral parameter, in such a way that the equations of motion (2.17)-(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 local conserved charges as such that ∂ + j − + ∂ − j + = 0. Explicitly, the j ± currents are whered P S is the transpose of the projector d P S . It is also useful to find the BRST transformation of the global currents where, The main motivation to introduce fermionic anti-fields (ω * 1+ , ω * 3− ) in the pure spinor formalism is to make the BRST charge Q nilpotent off-shell [27]. The fermionic anti-fields must satisfy the following constraints The BRST transformations (2.9) and (2.10) must then be modified Notice that the BRST transformation of the local conserved currents j ± is proportional to the equations of motion for λ (2.27), therefore the local conserved charge is BRST invariant only when the classical equations of motion hold. It is possible to avoid this situation by modifying the local currents as so that, 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 theory for a BRST invariant action developed in [21]. Once V 1 is known, the full deformation can be constructed as a series expansion in η for the action and the BRST charge where the coefficients are determined by imposing BRST invariance We then find up to order η 3 that The first of these equations is satisfied by construction. In reference [21] equations (3.5) and (3.6) were solved and consequently V 2 1 , V 2 2 , Q 1 and Q 2 were obtained. Here we review this procedure and solve the equations to all orders. We will also find that the expansion of the BRST charge stops at first order, that is, Q n = 0, for n ≥ 2.
The first step is to solve (3.5) as To show this we take the BRST transformation ofj ± (2.33) and compute Therefore, to cancel this contribution at first level, Q 1 is required to satisfy . This is achieved by taking Q 1 as where the projectors P 13 and P 31 are defined as so that, Some important identities involving the projectors P can be found in Appendix A. Nilpotency of the BRST charge requires that the R-matrices must satisfy the mCYBE. This item was extensively analyzed in reference [21]. As remarked in the Introduction we will restrict ourselves to the CYBE case and show that the nilpotency of Q 0 + ηQ 1 is proportional to the CYBE equation up to local transformations.
Noticing that ǫQ 0 g = gΛ(ǫ) forΛ(ǫ) = g −1 (ǫλ 1 + ǫλ 3 )g and ǫQ 1 g = gRΛ(ǫ) then [Λ(ǫ),Λ(ǫ ′ )] = 0 is a consequence of the pure spinor condition. We then find that We also need to find the action of Q 2 1 over g which is 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 it is obtained for ω * 3− . Thus, we have shown the BRST charge Q = Q 0 + ηQ 1 is nilpotent if R is a solution of the CYBE.
Having found Q 1 the next step is to find V 2 2 using (3.6). First of all, we compute the action of Q 1 on the local currentsj ± Then, we obtain where we have made use of the CYBE for R. It is important to take into account the following identity which cancels the contribution from (3.21). This solution is consistent if Q 2 S 0 = 0, but 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 2 3 . First, we note that In the light of the above discussion we conclude 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 given by Q = Q 0 + ηQ 1 takes the following form We can show this by splitting the total BRST variation of the action in three parts. First of all we note that Considering (3.22) we find Similarly, taking into account (3.24), 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 (3.27), (3.28), and (3.29) it is easy to verify that the BRST variation of the complete deformed action vanishes.

The polynomial deformed action
The deformed action obtained in the last section (3.26) depends on the non-local projectors P which emerged as a consequence of the BRST transformation on the anti-fields. Antifields were introduced in order to define a nilpotent BRST charge off-shell. In this section we abandon the anti-fields formulation and 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 terms. First of all we solve (3.5) for Taking into account (2.27) we have that We look for a BRST operator such that Q 1 (λ) = 0, then, our ansatz takes the form and its action over S 0 is given by Then we conclude that to cancel (4.2) the action of Q 1 on the ghost sector must be Having obtained Q 1 we look for V 2 2 by solving (3.6). Let us now write the Q 1transformations of j ± This means that In order to show that we need to find a Q 2 such that (3.6) holds. First, we observe that These results all together give and we find The last two terms of (4.12) are proportional to the equations of motion of ω 1,3 and can be removed by Q 2 S 0 if Q 2 is defined as (4.14) The remaining terms in (4.12) cancel with the contribution coming from (4.9). 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 We can rearrange this action into a more familiar form by defining the operators where R g = Adg •R•Adg −1 . Also, it is useful to consider the deformed pure spinor currents (4.17) Hence, the deformed action can be rewritten as 3 This action must be invariant under the BRST transformations This can be shown by splitting the action into four sectors. The first term of (4.18), the matter sector, contributes with where δgg −1 = ǫQ(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 (4.18), we note that the BRST transformations of J − andJ + are After a lengthy calculation we obtain that On the other hand, considering (4.20) and (4.21), we have In the ghost sector we have to consider (4.20) and (4.21) to set For the third term of (4.18) we have 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 (4.18) it is convenient to split the action in three sectors and implement the variation in the form δg = gξ i . An interesting property of the matter sector is that the equations on motions accept a Lax representation. For a proof we refer the reader to appendix B. Now, we look for Taking into account that it is easy to express (5.1) as where, For the matter-ghost sector we have to consider then, it is easy to see that where In the N 0− N 0+ sector we have Collecting all contributions the equations of motion take the form For the ghost currents the equations of motion can be expressed as It is remarkable that the equations of motion, (5.10)-(5.12) present the same structure as the undeformed ones. We can write them in a more suggestive manner by defining the currents J − and J + as so that the equations of motion take the form (5.14) and for the ghost currents, At this stage it should be clear that the ansatz for the Lax pair should 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 (5.13) as

17)
Taking into account the Maurer-Cartan equation we can express its components as as well as, After a lengthly calculation it is possible to express the Maurer-Cartan equation as 20) which shows that the pair (J − , J + ) satisfies the zero curvature condition when the equations of motion hold.
Defining the covariant derivatives as the equations of motion can be written as 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. Moreover, taking into account (4.20) and (4.21), the BRST density charges can be written as The (anti) holomorphicity of (j B + ) j B − can be easily proven by using the above equations of motion.
6 Relation to the η-deformation 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 (4.18) with the standard Berkovits-Howe action [25] This is the most general action which possesses BRST symmetry, classical world-sheet conformal invariance and zero ghost number. The world-sheet matter fields are the vielbiens E A , where 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 (6.2) 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 (4.18) and to show that the deformation of the pure spinor AdS 5 × S 5 superstring yields the same target space supergeometry as the η-deformation of the GS AdS 5 × S 5 superstring [18]. The η-model of the GS superstring is an example of the so-called Yang-Baxter deformations [5]. This deformation is implemented through the Lie algebra operator 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, when c = 0, it 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 [18] 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.
To illustrate the correspondence between the two superstrings it is worthwhile to rewrite (4.18) in GS language. For this purpose we define the operators which relates the 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 (4.18). This sector should be compared with the matter sector coming from (6.4) so the background fields B and P αα have to be found in this sector giving 1 4 Str(Ā, d P S J − ) (6.13) After a convenient rearrangement of the last term, , the matter part of the action takes the following form and can be compared with (6.4). The first two terms reproduce the deformed GS superstring (6.14) and the last term gives the contribution coupled to the Ramond-Ramond bispinor in (6.4). The metric and the B-field can be read from the GS sector which match the the B field of the GS η-deformation [18]. Now we look for the Ramond-Ramond bispinor. Substituting (6.9) in the last term in (6.14), we have that and comparing with (6.4) it follows that where Kα α = Str(t 3 α , t 1 α ). Taking into account (6.12) we write P αα as which is the same RR bispinor found for the GS η-deformation when the R-matrices are of the CYBE type [18]. Now we move to the matter-ghost sector and find that Using these equations in the matter-ghost sector coming from the deformed action (4.18) we have The above equation can be compared with the second line of (6.4) to read the spin connection and the pair (C,C) For completeness, the same analysis can be done for the N 0− N 0+ sector and we find This shows that the GS η-model and the 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 presented the η-deformation of the PS superstring in AdS 5 × S 5 . It leads to a consistent deformation of AdS 5 × S 5 which is not a type IIB supergravity solution but rather a generalized supergravity solution. 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 [28] the η-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 R-matrices (1.3). As remarked in the introduction the GS superstring propagates in a background which is restricted by kappa-symmetry to be a solution of generalized supergravity [17]. Our results strongly suggest that at least classically the constraints imposed on the target superspace by the requirement of nilpotency and holomorphicity of the BRST charge [25] should also imply the equations of motion for generalized supergravity. This condition would be sufficient to get a vanishing one-loop beta function.
Analogously, it can be proved that This result presents the same pattern of the undeformed model when we put the ghost sector to zero and replace A + →J + and A − →J − . Therefore, the Lax pair for this system is constructed by replacing A + →J + and A − →J − in the Lax pair of the undeformed model.