$\lambda$-Deformation Of The $AdS_{5}\times S^{5}$ Pure Spinor Superstring

The lambda deformation of the pure spinor formalism of the superstring in the $AdS_{5}\times S^{5}$ background is introduced. It is shown that the deformation preserves the integrability as well as the one-loop conformal invariance of its parent theory. It is also shown that the effective action takes the standard form of the Berkovits-Howe action functional, allowing to calculate the deformed background supergeometry in a straightforward way. The background fields coincide with those of the lambda model of the Green-Schwarz formalism, hence satisfying the same set of supergravity equations of motion.


Introduction
A fundamental characteristic of the gravity sector of the AdS/CFT correspondence is that superstrings propagating in the AdS 5 ×S 5 background can be described by a (classsical) integrable model [1]. This geometry rises as a solution of the type IIB supergravity equations of motion when supported by a self-dual Ramond-Ramond (RR) five-form flux. It is well known that a superstring moving on a curved background including a RR flux can be correctly formulated by either the Green-Schwarz (GS) formalism [2] or the pure spinor (PS) formalism [3] and that both formulations present manifest target-space supersymmetry. In both formulations, classical integrability is ensured since the equations of motion can be cast into a zero curvature equation satisfied by a Lax connection, see [1,4]. Although the complete quantization of the superstring in the AdS 5 × S 5 geometry has never been fulfilled and still remains as an open problem, the use of integrability techniques yield significant progress in understanding the excitation spectrum (see [5] for a review). It is then reasonable to believe that integrability could play a prominent role on an eventual first principle quantization approach of the theory.
Concerning the GS formalism, in the last years, two different but complementary types of integrable deformations of the GS AdS 5 ×S 5 superstring, have attracted a considerable deal of attention. On one hand we have the Yang-Baxter (YB) deformations, so named because they are characterized by a linear operator acting on the Lie superalgebra psu(2, 2|4), which solves the modified classical Yang-Baxter equation mCYBE. This deformation was first introduced for the principal chiral model by Klimčík in [6] and subsequenty developed in [7][8][9] for (super)-strings on (semi)-symmetric spaces (see [10,11] as well). In order to provide a target space which solves the equations of motion of type IIB supergravity, these R-matrices must satisfy the unimodular condition [12]. On the other hand we have the lambda deformations, which are based on a G/G gauged WZW model and are better understood as deformations of the non-Abelian T-dual of the original theory, they were first introduced for the principal chiral model by Sfetsos in [13] and subsequently developed in [14][15][16][17] for (super)-strings on (semi)-symmetric spaces. An important fact of the lambda deformation is that it produces string theory backgrounds solving the equations of motion of type IIB supergravity, see [12,[18][19][20]. Although each family of deformations produce different target space supergeometries, it has been proposed that, through an analytic continuation, the lambda deformations are Poisson-Lie T-dual to the YB deformations with R-matrices satisfying the nonsplit mCYBE [21,22]. In relation to the classical stringy configurations on η and λ backgrounds, integrability conditions are discussed in [23].
In respect to the PS formalism, integrable deformations of the AdS 5 × S 5 superstring have received much less attention and this is an unsatisfactory scenario for many reasons. For instance, in the PS formalism the world-sheet metric is already in the conformal gauge and the problematic κ-symmetry, signature of the GS superstring, is replaced by a global and much better behaved BRST symmetry, therefore avoiding delicate issues involving the light-cone gauge not to mention the lack of a satisfactory covariant quantization scheme due to the fact that the first and second class fermionic constraints cannot be disentangled covariantly in the GS formalism (a problem not present in the PS formalism by construction). Recently, the homogeneous YB deformations of the PS superstring were introduced in [24], by following the homological perturbation theory developed in [25]. It was shown that its target space background turns out to be the same found for the YB deformation of the GS superstring [12]. From the PS point of view, the deformed space solving the type IIB supergravity equations of motion is produced by a particular set of primary vertex operators belonging to the BRST cohomology in AdS 5 × S 5 . In this context, the mCYBE condition on the R-matrices, needed for integrability of the deformed action, arises by imposing the nilpotency of the deformed BRST charge, revealing a profound connection between the integrability of the deformed theory and its BRST symmetry.
With this vision, in this paper we introduce the lambda deformation of the AdS 5 × S 5 PS superstring. The deformation preserves the main characteristics of the undeformed theory: its BRST symmetry, its classical integrability, their local symmetries and its conformal symmetry at one-loop. In addition, it describes exactly the same supergeometry associated to its lambda deformed GS counterpart, much in the same way as the YB deformations of the GS and PS formulations describe the same background and are equivalent as string theories, at least, from the classical theory point of view.
The paper is organized as follows. In section (2), we introduce the action functional of the lambda deformed AdS 5 × S 5 pure spinor superstring and consider the equations of motion, the classical integrability and the BRST symmetry of the theory. In section (3), we run the Dirac procedure and study the integrability of the deformed theory from the Hamiltonian theory point of view. The analysis is simpler than in the GS formalism because of the absence of kappa symmetry and is essentially the same of the lambda deformed hybrid superstring. It is shown that the classical exchange algebra for the spatial component of the Lax connection takes Maillet r/s form, as expected. In section (4), we consider the quantum conformal symmetry and compute the one-loop beta function of the deformed theory and show that it vanishes. Finally, we show that the effective action for the deformed theory can be cast into the standard form of the Berkovits-Howe action functional. Once in this form, the deformed target space fields can be easily identified. They are exactly the same as the ones entering the geometry of the lambda deformed GS superstring, meaning that both formulations describe the same classical system.

Lambda deformed pure spinor superstring
In this section we recall the lambda deformed action of the PS superstring in the AdS 5 × S 5 background originally constructed in [26]. It is shown that the deformation preserves the integrability of its parent theory, a discussion not covered previously in [26]. The main characteristic of the deformed theory integrable structure is that its associated Lax connection depends explicitly on the deformation parameter λ, a feature not observed before in any of the known lambda models. BRST symmetry is briefly considered as well from the symplectic theory point of view.

Action functional
Consider the Lie superalgebra f = psu(2, 2|4) and its Z 4 decomposition induced by the automorphism Φ (2.1) From this decomposition we associate a twisted loop superagebra given bŷ required later on for describing the integrable structure of the field theory, where z plays the role of the spectral parameter.
where S W ZW (F) k is the level k WZW model action (2.7) The κ 2 is the coupling constant of the undeformed theory and the P (i) are projectors along the subspaces f (i) ⊂ f induced by the Z 4 decomposition.
The second contribution is given by the ghost sector where N (0) are the pure spinor Lorentz currents, (w − ) are the conjugate fields of the bosonic pure spinor ghosts (l (1) ,l (3) ) and D where [ * , * ] + denotes the anti-commutator.

Equations of motion
Before considering the equations of motion in detail, let us first prove an useful identity. Consider the gauge field equations of motion derived from the action 2 (2.3), i.e.
(2.11) Then, the Maurer-Cartan identity for the flat current F −1 ∂ ± F, in the presence of the equations (2.11), takes the form where (2.13) 2 We have that D( * ) ≡ Ad F ( * ) = F( * )F −1 and D T ( * ) ≡ Ad F −1 ( * ) = F −1 ( * )F. Now, the F equations of motion when combined with (2.11) are equivalent to having ξ 1 = 0 and ξ 2 = 0 separately, while the ghosts equations of motion imply that the PS Lorentz currents (2.9) obey (2.14) In terms of the dual currents introduced in [26] for the lambda deformed hybrid formulation of the superstring, the F equations of motion, i.e. ξ 1 = ξ 2 = 0 are (for generic values of λ) equivalent to the following set of equations (2.17) The full set of equations of motion (2.14) and (2.16) follow from the zero curvature condition of the Lax connection satisfying the condition Φ(L ± (z)) = L ± (iz), (2.19) under the action of Φ in (2.1).
However, in contrast to all known lambda models, see for instance [13,14,16,26], this theory has an explicit λ-dependent Lax connection, i.e. the parameter λ can not be absorbed by the ghost currents. Indeed, the un-deformed theory has a Lax pair given by [4] L + (z) = J (0) where J ± = f −1 ∂ ± f is a flat current defined in terms of the Lagrangian field f . The integrability of the action (2.3) was not considered in [26] because of the discrepancy of the equations of motion (2.16) with the equations of motion of the un-deformed theory. Notice it explicit λ-dependence. However, this apparent anomalous behavior is quite natural once we realize it is just a consequence of the pole structure of the deformed theory, materialized in the twisting function, see (3.29) below.
Using the Kac-Moody currents expressions defined below in (3.1), we can write (2.11) in the equivalent forms where z ± = λ ±1/2 . This important result will be invoked later on.
The lambda deformation preserves the integrability of the original theory albeit with a slight modification of the Lax connection when compared to its un-deformed counterpart.

BRST symmetry
The action (2.3) is invariant under the following BRST variations [26] where a, b ∈ R are arbitrary real numbers 3 and We can verify the consistency of the BRST variations by showing that its square is formally a gauge transformation. We find that where we have used the constraints (2.10) in order to show that α 2 = β 2 . Classical nilpotency of the BRST action must be accomplished up to classical equations of motion and local gauge transformations. In relation to the action functional (2.3), we find that such an action is indeed nilpotent In showing this last result, derivatives of the constraints (2.10) are to be used. This is consistent with the bosonic gauge symmetry of (2.3) generated by the grade zero subalgebra f (0) . In particular, (2.25) shows that for the lambda deformation of the PS superstring the Lorentz transformation must be modified as as expected for a lambda deformation. Recall that in the un-deformed case the Lorentz transformation is of the form In order to find the associated BRST charge in an elegant manner, we consider the symplectic form of the action (2.3). Namely, where (1) , (2.30) Here, the δ is to be understood as the exterior derivative in phase space. Using the following contractions This result is found only after using the gauge field equations of motion (2.11), which allow to write the right hand side as a total differential in phase space. Furthermore, it follows from the ghost equations of motion, (2.16) and (2.10) that the currents defined by are chiral, just as their counterparts in the un-deformed theory.
In summary, all the properties of the original action functional are preserved under the deformation, i.e. its BRST symmetry, its integrability and its local symmetries. Below, we will show that its 1-loop conformal symmetry is also maintained.

Hamiltonian structure and integrability
In this section, we run the Dirac procedure and study the integrable structure from the Hamiltonian theory point of view. We will follow the strategy of [27][28][29][30] and show that the Poisson bracket of the spatial component of the extended Lax connection takes the Maillet algebra form [31]. This is possible after constructing a suitable extension of the Lax connection outside the constraint surface. As expected, the extended monodromy matrix is conserved and their charges preserve the constraint surface where the classical motion of the deformed string theory takes place.

Dirac procedure
The phase space associated to the action functional (2.3) is described by the following phase space coordinates: two currents J ± given by obeying the relations of two commuting Kac-Moody algebras 4 2) two conjugated pairs of fields (A ± , P ∓ ) with Poisson brackets and two pairs of conjugated ghosts (l (1) , w − ), satisfying The time flow is determined by the canonical Hamiltonian density where

.7) and
through the relation Above, f is an arbitrary functional of the phase space variables. Now, we run the Dirac algorithm. There are two primary constraints By adding them to the canonical Hamiltonian (3.6) we construct the total Hamiltonian where u ± are arbitrary Lagrange multipliers.
Stability of the primary constraints (3.10) under the flow of H T leads to two secondary constraints given by which are nothing but the gauge field equations of motion (2.11). In this formulation of the superstring we must add by hand the pure spinor constraints (2.10) to the set of constraints found so far, i.e.
From this, we construct the extended Hamiltonian where µ ± and v,v are arbitrary Lagrange multipliers.
Stability of the constraints under the time flow of H E produce no new constraints but rather determine some of the Lagrange multipliers. However, their explicit form will not be required in what follows and algorithm stops.
We now consider the constraints and split them between first and second class. Along the coset directions, the constraints P (i) form three second class pairs of constraints and we impose them strongly by means of a Dirac bracket. The brackets (3.2) and (3.3) are not modified in this process, so we continue using their usual definitions. Then, we have the strong relations I Along the grade zero part of the algebra, we notice that the combination is a first class constraint, while form a pair of second class constraints. The first class constraint (3.17) can be gauge fixed by means of the condition A (0) This is a good gauge fixing condition and now we impose (3.17), (3.18) and (3.19) strongly by means of a Dirac bracket. Fortunately, the brackets (3.2) and (3.3) are not modified at this step. Then, we get the strong relation At this level of analysis, the remaining constraints are (ϕ (0) , Φ,Φ), where They weakly commute among themselves and are first class.
When compared to the Hamiltonian analysis of the lambda deformed GS superstring [30], we realize that for the PS formalism the same analysis is simpler and less involved because of the absence of the fermionic constraints associated to the kappa symmetry.

Maillet algebra
Due to the presence of Hamiltonian constraints, the Poisson bracket of the spatial component of the Lax connection does not take the standard Maillet form [31] and an extension of the Lax connection (2.18) outside the constraint surface must be considered. Only after a proper extension have been chosen, the Maillet algebra is recovered. See [27,28] for string theories with this characteristic, see also [30] for a more direct approach devoted to lambda models. Here, we will apply the same strategy to the PS superstring lambda model.
We start by writing the spatial component of the Lax connection (2.18) in terms of the Kac-Moody currents and in the partial gauge considered so far, where (3.16), (3.19) and (3.20) are valid in the strong sense. We have where f ± (z) = α(z 4 − z 4 ± ). (3.23) The extension of the Lax connection we will consider is defined by imposing that the relations (2.22) are valid outside the constraint surface and hence on the whole phase space. We consider the following obvious extension 6 L σ (z) = L σ (z) + f + (z)ϕ (0) (3.24) and obtain The Poisson bracket of (3.25) with itself takes the r/s Maillet algebra form [31] (3.28) are the anti-symmetric and symmetric parts of an R 12 (z, w) matrix and ϕ λ (z) is the associated deformed twisting function given by (3.29) The algebra (3.27) is the same found for the Green-Schwarz [30] and the hybrid formulations [26]. At the points z = z ± , the Maillet algebra (3.27) reduce to the Kac-Moody algebras we wrote above in (3.2).
The extended Lax connection is given by and from it we get an extension of the equations of motion found from (2.18) by terms involving the bosonic constraint ϕ (0) . Furthermore, from its flatness, it follows that the time derivative of the (super)-trace of the monodromy matrix The Poisson brackets of (3.25) with (3.13) weakly vanish, while the Poisson bracket of (3.25) with ϕ (0) is a gauge transformation, in the sense that

Conformal invariance
In this section we consider the one loop conformal symmetry of the deformed theory by following the method of [32] used to compute the 1-loop beta function of the lambda deformed GS superstring. The same method was used in [26] to deal with the lambda deformation of the hybrid superstring. As the PS superstring is essentially based on the hybrid formulation plus the addition of the pure spinor ghosts, the calculation is quite straightforward. We also consider the supergeometry underlying the action functional (2.3) and show that it is the same of the GS superstring in the AdS 5 × S 5 lambda background [12]. Thus, both theories describe the same classical theory.

1-loop beta function: un-deformed case
For the sake of completeness and in order to understand the method of [32] in a known situation, we will compute here the 1-loop beta function of the PS superstring in the AdS 5 × S 5 background.
The equations of motion of the un-deformed theory are obtained from the flatness condition of the Lax connection (2.20). They are given by (2.16) after the substitutions The ghost equations of motion are still given by (2.14) without any modification. The approach of [32] is based on taking the variations of the equations of motion in order to obtain the operators governing the fluctuations of the currents in an straightforward way.
By choosing a purely bosonic background, the bosonic and fermionic sectors completely decouple at the 1-loop order. We choose satisfying the conditions Other current components being zero, meaning that at the group level this choice corresponds to f = exp x µ θ µ , θ µ ∈ f (2) .
The equations of motion are to be supplemented with a gauge-fixing condition associated to the f (0) gauge symmetry. We choose the following one After variation, we obtain the operators governing the fluctuations. For the bosonic sector we get where the fourth line from top to bottom corresponds to the variation of the gauge fixing condition. For the fermionic sector, we obtain In (4.6) and (4.7) by acting on, we mean the adjoint action, e.g. θ + ( * ) means [θ + , * ] and so on.

1-loop beta function: deformed case
In order to compute the 1-loop beta function of the deformed theory, we consider the following classical background fields where Λ µ ∈ f (2) . From this choice and (2.11), we get the dual currents (4.14) The advantage of this choice lies in the fact that the matter and ghost sectors decouple. Now, from the equations of motion (2.14) and (2.16), we obtain the operators governing the fluctuations of the bosonic and fermionic sectors. For the bosonic sector we get The 1-loop contribution to the effective Lagrangian, in Euclidean signature, is where and The contributions associated to the logarithmic divergences (denoted by adj + T r (2) adj ] θ + θ − + 2λ −4 n +n− , (4.20) Then, L E ≈ c 2 (f) = 0, because of the dual Coxeter number c 2 (f) of f = psu(2, 2|4) vanishes. As a consequence, the deformation preserves the 1-loop conformal invariance of its parent theory. The same fate is found in the Green-Schwarz formalism in the AdS 5 × S 5 lambda background [32] and the hybrid formalism in the AdS 2 × S 2 lambda background [26].

Relating the lambda deformed PS/GS background fields
To compute the target space supergeometry, the gauge fields A ± must be integrated out completely. Then, after using the gauge fields equations of motion we obtain the following effective action S eff = S matter + S ghost , (4.21) where  From the expressions of the projectors in the PS and the GS lambda models, i.e. , we get an important relation between both formalisms where O gs = Ω gs − D. (4.26) The strategy for obtaining a clear and direct relation between the target space geometry of the PS and the GS lambda models start by introducing the following GS super-vielbeins defined by This strategy was used successfully in [24] for understanding the relation between the Yang-Baxter deformations of the PS and the GS formulations of the AdS 5 × S 5 superstring. Our purpose here is to apply the same approach to the case at hand.
Consider now the quantity (4.28) The metric and the antisymmetric fields are extracted, respectively, from the symmetric and the anti-symmetric parts of X, namely and (4.30) Then, we have that where Form these results, the matter contribution to the effective action take the form where Above, we have written the WZ term, locally, in the form Before focusing on the ghost contribution, consider first the following two identities It follows from these expressions that (4.39) Using these results, we have the ghost contribution to the effective action 40) where S lw is given by the usual ghost term (4.41) 7 Notice that P : f (1) → f (1) . 8 Notice thatP : f (3) → f (3) and C,Ĉ = C T : f → f (0) . and Remarkably, the effective action of the lambda deformed PS superstring (4.21) takes the standard form of the Berkovits-Howe (BH) action functional [34] where (4.44) The BH action is the most general action functional which possesses BRST symmetry, classical world-sheet conformal symmetry and zero ghost number. Once the action is in this canonical form, the background fields are easily identified and are encoded in the objects of the following list:  The vielbeins, the metric, the B-field, the RR-bispinor P (actually its inverse in index notation) and the spin connections Θ + ,Θ − are consistent with the background fields of the AdS 5 × S 5 Green-Schwarz lambda model as found in [12]. The C,Ĉ and S are auxiliary fields, i.e. can be defined in terms of the vielbeins [34,35].
Thus, we conclude that both models (the lambda deformations of the GS and PS superstring) describe the same type IIB supergravity background and this means that our action (4.21) corresponds to the pure spinor formulation of the AdS 5 × S 5 superstring in the lambda background. In order to accomplish the correct equations of motion for type IIB supergravity, the dilaton must be the same as that obtained for the lambda model of the GS superstring [12]. However, the contribution from integrating out the gauge fields A ± , gives rise to the following dilaton field e −2φ = sdet O, should be interpreted as a Jacobian determinant arising from a change of group field variables in the path integral Haar measure 9 DF → DF .
Finally, we write the BRST currents (2.34) in the standard curved space form [35] j + = l which is the correct form for the BRST chiral currents in the BH approach [34].

Conclusions
We have shown how to lambda deform the pure spinor formalism of the superstring in the AdS 5 × S 5 background in a consistent way. The deformation preserves the BRST symmetry, the classical integrability, the local symmetries and the 1-loop conformal symmetry of its parent theory. Furthermore, the target space supergeometry is exactly the same as that of the lambda model of the Green-Schwarz formulation of the superstring and satisfy the same set of supergravity equations of motion. This result complements an analogous equivalence, found recently, between the Yang-Baxter deformations of the PS and GS formulations of the superstring in the AdS 5 ×S 5 background.
As a future work, it would be interesting to study the relation of the PS formalism of the AdS 5 × S 5 superstring in terms of a Chern-Simons theory, as it is the case for the GS formalism [29,30]. In case of a positive answer, the non-ultralocality term present in the algebra (3.27) of the PS formalism could be eliminated (for any value of the deformation parameter λ) with the added advantages of not having to deal neither with the kappa symmetry nor the light-cone gauge. This will be considered in a companion paper.