Beta deformed sigma model and strong deformation coupling limit

We study the beta deformation of the superstring in $AdS_5\times S^5$ at all orders in the deformation parameter, using the pure spinor formalism. This is necessary to study the regime of strong deformation parameter, which in the field side is related to fishnet theories. We compare the pure spinor sigma model approach to the previously known supergravity description. We find a complete agreement. Moreover, the BRST structure of the worldsheet model provides a natural explanation of the peculiar features of the worldsheet model in the fishnet limit. In particular, we study the degeneracy of the sigma model Lagrangian. We show that the BRST structure is responsible for a particularly"tame"degeneration of the fishnet sigma-model.

The N = 4 Supersymmetric Yang-Mills theory has been one of the most studied theories in high energy physics over the last few decades due to the prominent role it plays in the AdS/CFT correspondence [1], [2].One important advance in its understanding was the introduction of a marginal deformation by Leight and Strassler, called β deformation [3], which preserves N = 1 supersymmetry.A generalization to 3-parameters deformation is also known [4] and does not preserves any supersymmetry in general.This deformation has been recently applied to obtain fishnet theories in the field side [5], which arise in the limit of strong deformation.The gravity dual of such deformed theories was found by Maldacena, Lunin and Frolov [6], [7].Worldsheet description in the pure spinor formalism was proposed in [8].However, the latter only describes the deformation at the linearized level.A generalization to all orders in the deformation was proposed in [9] in the case when the deformation satisfies the Yang-Baxter equation.
In this paper we derive the supergravity solution of Maldacena and Lunin from the worldsheet pure spinor sigma model of the beta deformation.We then proceed to examine the beta deformed theory in the fishnet limit, where we identify a degeneration in the background fields.We explain how the BRST structure of the theory imposes restriction on the structure of degeneration.The space of all possible degenerate limits of a sigmamodel is very rich.But, BRST invariance implies that the degeneration is of a very special, "regular" kind.

Supergravity description
The β−deformed N = 4 SYM theory is related via gauge-gravity duality to a supergravity solution which is a deformation of the standard AdS 5 × S 5 one.This deformation only acts on the S 5 geometry and leaves the AdS 5 part intact.This solution was obtained by Maldacena, Lunin and Frolov by compactifying M theory on a torus of modulus τ [6], [7].The parameter τ is acted upon by the SL(2; R) symmetry of the supergravity equations of motion.Then, the beta deformation of N = 4 SYM with parameter γ corresponds to the transformation which acts as a solution generating transformation.This action is obtained from a T-duality on one of the torus circles, then a shift of coordinates, and then another T-duality.Such procedure is called T sT .More precisely, the solution found by Maldacena-Lunin-Frolov is given by the following metric and B-field: As we can see, this corresponds to a deformation in the sphere metric, which is described by ds 2 = R 2 3  i=1 (dµ 2 i + µ 2 i dφ 2 i ), with 3  i=1 µ 2 i = 1 and R the radius of the sphere.In the limit of γ i = 0 we recover the sphere metric and then the AdS 5 × S 5 undeformed solution.

Deriving supergravity from worldsheet pure spinor description
One of the results of this paper is the explicit demonstration that the sigma model proposed in [9] reproduces the supergravity solution of Maldacena-Lunin-Frolov.The first step to obtain this model is the insertion of a vertex operator in the AdS 5 ×S 5 superstring action [8].However, this procedure only provides a description of the linearized beta deformation.In order to include the non-linear behaviour, one must impose BRST invariance of the action at all orders in the deformation parameter.The reason why this computation is possible is that the expansion of the BRST operator in powers of deformation parameter stops at the first order.
We can introduce embedding coordinates by the embedding AdS 5 × S 5 ⊂ R 6 × R 6 .Doing this, we can parametrize the space-time in an easier way and then write the deformed sigma model action.Using the coordinate system derived from this embedding, the resulting sigma model is where the background fields g mn and B mn are precisely the metric and the B-field in the supergravity solution (1.2), and x m denotes the bosonic space-time coordinates of AdS 5 × S 5 .
We therefore are able to rederive the supergravity solution obtained from TsT in [6, 7] using string theory techniques.

Large γ
Another aim of the paper is to work towards a string dual description of the fishnet field theories [5].In order to do this, we must work with the fully deformed theory since it allows us to study the sigma model in the strong deformation regime.These theories are obtained from the beta deformed N = 4 SYM in a particular Double Scaling limit.This limit is given by taking γ → ∞, λ → 0 with λe −i γ 2 fixed, where γ is the deformation parameter and λ the t'Hooft coupling constant.To study this double scaling limit in the gravity side, we must take into account the relation between t'Hooft constant and the AdS radios given by λ = R 4 /α ′2 , and then impose the fishnet limits In principle, we must impose the condition γ << 1 in the supergravity solution (1.2).The reason is that the transformation (1.1) in the torus τ → τ 1+γτ cannot make the torus smaller than the string scale.Also, we must have R >> 1.Then, it is clear that in order to study the fishnet limit we must use the fully deformed theory, since it is defined at all orders in γ.This is the precise advantage of our proposal.We must emphasize that, in this paper, we study the large γ limit but with constant radius R.This would be the first step to understanding the string dual for the fishnet theories.The small radius limit has not been investigated in this context yet.
The BRST operator of the theory in the strong deformation regime is given by (1.5) The indices a are bosonic indices of the algebra of rotations su(4) ⊂ psu(2, 2|4), Λ a are functions of the sigma model fields, t a are generators of su(4) and B ab an antisymmetric constant matrix.The BRST invariance of the model will imply a degeneracy of the background fields in the t a directions, where G f µν and B f µν are the metric and the B-field of the theory in the fishnet limit.

Plan of the paper
In section 2 we review the Pure Spinor sigma model in AdS 5 × S 5 .In section 3 we construct the beta deformed theory in all orders in the deformation parameter.There, we write the action for the full beta deformation, as well as the BRST operator.We then compute explicitly the bosonic part of our action and find a complete agreement with the Maldacena-Lunin-Frolov supergravity solution.In section 4 we study the fishnet limit applied to our sigma model and show that the background fields degenerate in this regime.
2 Review of AdS 5 × S 5 Superstring: pure spinor formulation In the superstring in the AdS 5 × S 5 background, the fields are maps from the world-sheet Σ to the following coset: Considering (τ + , τ − ) the parametrization of the worldsheet, we will write the action in terms of the invariant current J = J + dτ + + J − dτ − , where J ± = −∂ ± gg −1 and ∂ ± = ∂ ∂τ ± .One decomposes the lie algebra in the following way: psu(2, 2|4) = g 0 ⊕ g 1 ⊕ g 2 ⊕ g 3 , with g 0 = so(4, 1)⊕ so (5).The massive part of the action can be written in terms of the currents J 1 , J 2 , J 3 .In the pure spinor formalism [11], the whole action is It is remarkable here that we introduced two ghost fields λ 1 , λ 3 , which are respectively right and left pure spinors.These are spinor fields with values in the lie algebra, satisfying the pure spinor constraints characterized by 3) The fields w 1+ and w 3− are respectively the conjugate momenta associated to λ 3 and λ 1 .N 0 are the Lorentz currents, defined as , −] is the covariant derivative.In terms of the algebraic structure underlying the ghost fields, λ 1 and w 1+ are elements of g 1 , while λ 3 and w 3− are elements of g 3 .The introduction of the ghost sector is crucial for achieving covariant quantization of the superstring, as discussed in [11].This accomplishment is realized through the definition of a nilpotent BRST-like operator, parameterized by the pure spinor ghosts, as: ǫQ 0 g = (ǫλ 3 + ǫλ 1 )g (2.5) The constraints (2.3) that the ghost fields satisfy imply a non-trivial cohomology for this BRST-like operator.Consequently, it is natural to define the physical states of the system as elements within this cohomology, called vertex operators.A particularly notable example is the construction of the zero-mode dilaton vertex in [12,13], which is defined by (2.7) This vertex can be seem as an state resulting from the rescaling of the overall string coupling constant.Thus, in essence, the string theory is deformed by an expression given by the integral of the Lagrangian itself.The action is invariant under BRST transformation and the Lagrangian cannot be BRST-exact, therefore it is in the cohomology.Then, in order to find the corresponding unintegrated vertex operator, writen in (2.7), one uses that the BRST variation of the Lagrangian is Q 0 L 0 = d Str(λ 3 J 1 − λ 1 J 3 ), which results in the following descent equations: (2.9) In the next section, after a modification of this vertex operator, it will play a crucial rule in the study of the beta deformation.
We should now define an important quantity for the superstring, the density of the global conserved charges, given by j = j + dz + + j − dz − , where (2.10) such that dj = 0.The density current transforms under BRST action as where Λ(ǫ) = g −1 ǫ(λ 3 − λ 1 )g.For any element of the Lie algebra x ∈ g, we can write the components of x by x a = Str(xt a ), (2.12) where {t a } are the generators of the lie algebra g.

β-deformation of the superstring sigma model
In this section, we will first review the beta deformation of the pure spinor superstring, as introduced in [8] and [9].We will review how to obtain the worldsheet action and the BRST operator in the presence of this deformation.After the review, we will establish a new connection between this worldsheet action and the Maldacena-Lunin construction in [6].
To start with, one considers a modification of the dilaton vertex (2.7).This modification was proposed in [8] in order to describe the beta deformation in string theory and is given by the following expression: This vertex will result in an integrated vertex operator given by the product of two conserved currents, V As we will see in the end of this section, this vertex operator gives the supergravity solutions corresponding to the beta deformation, described in [6].
The expression for the integrated vertex in (3.2) is obtained from (3.1) via the following descent equations, computed by using the relation in (2.11): The AdS 5 × S 5 superstring is deformed by adding the integrated vertex (3.2) to its Lagrangian.The deformed action is then where B ab is an antisymmetric matrix with two adjoint indices.This is a linearized deformation of the action in the parameter γ.The next step now is to think about the second order deformation in γ.To do that, a deformation is added to the BRST operator, in such a way that the action remains BRST invariant, which is guaranteed by the equation By acting with B ab j a ∧ dΛ b .Then, equation (3.7) implies that we must find a Q 1 such that In order to find Q 1 , one notice that the deformation of the BRST operator acting on g is a psu(2, 2|4) transformation with a space-time dependent parameter α(z + , z − ), Q 1 g = gα.
By using this in equation (3.8) one finds the following expression for the deformation of the BRST operator at first order: The actions on the w ghost were also computed in [9] in a similar manner.We then write the first order correction for the BRST operator as

Simplification of the BRST operator
The deformation of the BRST operator we are studying is hugely simplified when the matrix B satisfy the following Yang-Baxter equation: Namely, under this condition the BRST operator acting on g is deformed only up to the first order in γ.From this condition it also follows that Q 2 1 g = 0 automatically.Indeed, Now, since Q 0 and Q 1 commute, the operator given by Q 0 + γQ 1 is nillpotent by itself.Then, in conclusion, the correction for the BRST operator acting on the group element stops at first order, and the full operator acting on g is then just

Deformation to all orders
In order to find the fully deformed Lagrangian, one imposes the BRST invariance of the action to all orders in γ.From the BRST invariance condition follow an infinite set of equations, one for each order in γ, n−i = 0, ∀n.
The equation corresponding to the first order in γ is 0 and it was explained in the beginning of this section.The equation for the second order in γ is , from which one obtains expressions for V 2 and Q 2 .This was systematically computed in [9].Finally, the full deformation for the action is where j is the conserved current (2.10) and M = Str (gt a g −1 )d P S (gt b g −1 ) , with d P S = And the full correction to the BRST operator is We then write the full deformation as and then we obtain our full sigma model, with the BRST operator given by In the next sections, we will describe the connection between the worldsheet action in (3.18), obtained in [9], with the Maldacena-Lunin construction in [6].

Maldacena-Lunin background
The Maldacena-Lunin background [6] is a supergravity dual of the beta deformed field theory.In order to obtain it from our sigma model we will restrict to the bosonic part and only deform the 5-sphere, letting AdS 5 undeformed.The sphere can be parametrized by the embedding coordinates Different choices of the matrix B in (3.17) will give us different backgrounds.Let us chose a B such that its only non-vanishing components are where ij are indices for the su(4) subgroup of psu(2, 2|4) (see Appendix A).This B satisfy the Yang-Baxter equation (3.12) In this section, we will compute the bosonic part of the matrix κ in (3.17) and then write the action in therms of the embedding coordinates (3.20).Consider the vector space su(4) given by the basis t ij with i, j anti-symmetric going from 1 to 6. Ordering the basis as {t 12 , t 34 , t 56 , • • • }, we have From now on, we will restrict our vector space to a 3-dimensional space generated by these three elements {t 12 , t 34 , t 56 }.Then, we will write B as a 3 × 3 matrix as well as the matrix The matrix M is rather complicated, It may be decomposed into a symmetric and antisymmetric part, (3.29) The bosonic action will be given by where If we substitute the expression for M (2) given by equation (3.29) into the matrix κ γ in (3.31) and use the matrix B in (3.24), we find where The currents j, also computed in Appendix A, equation (A.11), can be written as j 12 = µ 2 1 dφ 1 , j 34 = µ 2 2 dφ 2 and j 56 = µ 2 3 dφ 3 .We use this expression for the currents and the matrix κ γ above to compute the bosonic action (3.30).The result is (3.33)This gives us the metric and the B-field of the sigma model, which is the Maldacena-Lunin solution [6].

Generalization to γ i deformation
In [7], Frolov studied the supergravity dual to a generalization of the beta deformation by considering 3 parameters γ i .This can be captured in our sigma model by labeling the deformation parameter as with constants α i , for i = 1, 2, 3.Then, we define our theory based on the following matrix B: obtained by setting B [12][34] = α 3 , B [34][56] = α 1 and B [56] [12] = α 2 .The matrix κ becomes Now, we use the expression for M (2) we computed in (3.29) and the matrix B defined above in (3.37) to compute κ (2) γ i in (3.38).As a result we find . The corresponding sigma model can be computed using this result and the expressions for the currents j 12 , j 34 , j 56 applied to equation (3.30).The result is the bosonic action (3.40)This gives us the metric and the B-field of the sigma model already described by Frolov [7], (3.42)

Symmetries
The superconformal transformation of the current j and the matrix M is: The currents j 12 , j 34 , j 56 are all invariant under the plane rotations t 12 , t 34 , t 56 .This happens because all these generators commutes among themselves, since the intersection of the plane defined by then is just a point.Therefore the structure constants f I ab are all zero for a, b ∈ {12, 34, 56}.For example, The same thing happens for the variation of the components M ab with indices a, b ∈ {12, 34, 56}, in such a way that they are also invariant under these three rotations, The action for the beta deformation was constructed using only the components for the currents j 12 , j 34 , j 56 and M ab with indices a, b ∈ {12, 34, 56}.Therefore, the action is invariant under U (1) 3 .This is a residual symmetry from the isometry group of the sphere SU (4).This also happens in the field side [5].This symmetry will important later in the next section.

The large γ limit
In this section, we will study the worldsheet action under strong deformations.This is motivated by the work of [5], where they studied a field theory in the limit of strong deformations.Specifically, the N = 4 SYM theory is γ i -deformed and a double scaling limit is taken, giving rise to a field theory named fishnet theory.This limit is characterized by the imaginary part of the deformation parameter γ j being infinitely large (γ j → i∞), the coupling constant g 2 = N g 2 Y M being zero (g 2 → 0), and ξ 2 j = g 2 e −iγ j being constant, where j = 1, 2, 3.This section makes a similar analysis for the worldsheet, in order to establish a possible candidate for the string dual of this fishnet field theory.
For simplicity, one can start studying the strong deformed sigma model by setting two deformation parameters to zero, say γ 2 and γ 3 , and keep the remaining γ 1 nonzero.In this case, the bosonic part of the sigma model action is Then, in the limit of strong deformation, γ 1 → i∞, we have which makes manifest a degeneracy in the action.This may reflect the decoupling of some fields already present in the fishnet field theory [5].In the general case where three deformation parameters are considered, the bosonic action assumes the form

The pp-wave limit
We shall now investigate the behavior of the theory in the limit of large γ within the pp-wave limit.For this, we focus on the particular case where string concentrates at the equator of the sphere characterized by µ 1 = 1.In this case, the metric of the sphere takes the form which means we can take the following curve as one geodesic of the strongly deformed AdS 5 × S 5 space: u 5 + iu 6 = e iJτ = cos(Jτ ) + i sin(Jτ ).(4.5) We then proceed to examine perturbations around this geodesic.These perturbations can be described using the parameterization ) The corresponding conserved currents are j 56 = J, j 12 = ν 2 dθ and j 34 = ρ 2 dψ.Around this geodesic, the undeformed action in AdS produces 4 massive modes, as shown in Appendix B: In other words, the action for the perturbation is described by a kinetic term for the complex fields u 12 , u 34 , together with a mass term J 2 .In terms of the coordinates ν, ρ, θ, ψ, the action S b is: We now add the deformation.For the model (4.1), the correction is: (4.10) The whole action is then (4.11) Therefore, when we take the limit of large γ 1 , we have: This means that when we add our correction, deform the action, and take the large γ correction, part of the action in equation (4.8) disappears.Namely, the angular part given by ν 2 ∂θ ∂θ + ρ 2 ∂ψ ∂ψ vanishes in the action.This indicates some degeneracy of the metric in the large γ limit.Now, we turn on all deformation parameters, as in (3.33) and take the Penrose limit, obtaining where φ ± = (θ ± ψ).The limit of large γ provides: It is remarkable that taking the Penrose limit in AdS 5 × S 5 , we discover that there are eight massive string modes.The large γ limit of the beta deformed string still has eight such massive modes, but now the action is degenerate at a given angle of propagation.

Studying the matrix κ γ
Let us analyze closely the deformed theory defined by the action (3.18).Initially, the action is described by (2.2), and then the deformation is defined in (3.17).This deformation involves the matrix κ γ = B γ 1−γM B , which plays a central role in the deformation.In this section, our focus will be on a careful examination of this matrix, aiming to understand the behavior of the deformed action as γ becomes large.If the matrices B and M were invertible, the matrix κ γ at large γ would simply reduce to κ = − 1 M .However, this is not the case.The matrix B employed in (3.24) exhibits a non-trivial kernel characterized by v 0 = (1, 1, 1).We will then introduce an appropriate basis that facilitates a simpler representation of the matrix κ γ .
The matrix κ γ involves two other ones, B and M .First, we set the matrix B to the form (3.24) and matrix M to (3.25).We then define Given K, we can find a basis U = {v 0 , v + , v − } consisting of eigenvectors of K, The eigenvectors are v 0 = (1, 1, 1) and v ± are intricate expressions which are not necessary for the scope of this paper.The eigenvalues are given by 2Γ ± = α ± β, with For instance, when we restrict to the bosonic part, we have that M 12,12 = µ 2 1 , M 34,34 = µ 2 2 , M 56,56 = µ 2 3 and the remaining entries are zero, as computed in Appendix A. In this case we find that α = 0 ,

.18)
We now proceed to see how the matrix κ γ is written in terms of our new basis U. First, using (4.16), we act with the matrix 1  1−γK in this basis, obtaining Then, the matrix κ γ = γB 1 1−γK will act in this basis as Finally, we conclude that κ γ in the basis U takes the form Now, we write the deformation of the action (3.17) using this language.Consider the current in terms of this new basis, j = j 0 v 0 + j + v + + j − v − , where j 0 , j − , j + are coefficients.We then use the expression in (4.21) for the matrix κ γ to compute the product j, κ γ • j .At the end, we obtain the deformation of the Lagrangian in the form 22) The advantage of expressing the Lagrangian in this manner is that one can demonstrate its convergence for large γ.To see that, we note that the possible divergence arises when α 2 = β 2 , or equivalently, when α = ∓β.In such cases, Γ ± = 0, making the matrix K degenerate along the v ± directions.However, since the matrix M is non-degenerate, the sole possibility is for v ± to lie within the kernel of B. Yet, this cannot be the case, since Ker(B) possesses dimension one and already contains v 0 , given by v 0 = (1, 1, 1).
Let us contrast our answer (4.21) to the initial guess at the beginning of this section, given by − 1 M .In terms of the basis U, the latter is where h 11 , h 12 , h 13 are nonzero expressions, whose form is not useful to write here.As we commented, this is the answer when B is invertible.The presence of a kernel in B makes the h's disappear.

The BRST operator
In section 3, we studied how the BRST operator is modified under the γ i −deformation.The goal of this section is to investigate the behavior of this operator under the regime of strong deformations.To accomplish this, we decompose the BRST operator into two distinct parts, as follows: where and The BRST operator is split into these two parts to accommodate different orders of the deformation parameter γ.Namely, Q 1 is linear in γ, while Q 0 is the zeroth order in γ, This limit becomes apparent through the examination of the matrix κ γ for large values of γ, as illustrated in equation ( 4.21).We can now redefine the BRST charge as q = 1 γ Q 0 + Q 1 .Consequently, in the limit of large γ, the charge simplifies to q f = Q 1 .
We started with an action and a BRST operator defined in the AdS 5 × S 5 space, and subsequently deformed both.Through our analysis in this section, we derived an explicit formula for the BRST operator at the large γ limit.
The upcoming sections are devoted to investigating the action in the limit of large γ.In order to do that, we will use the fact that our theory remains BRST invariant in this limit, with the BRST charge given by q f = Q 1 .

Action for a deformation with strong coupling
To study the full sigma model (3.18) in the large γ limit, we will use the symmetries of the action.First of all, we have a residual R-symmetry corresponding to rotations under the planes 12, 34 and 56, defined in (3.20).Restricted to the bosonic part, these are shifts of the angles, ∂ ∂φ 1 , ∂ ∂φ 2 , ∂ ∂φ 3 .Therefore, since the bosonic action is invariant under the rotation of these planes, the angle variables φ i only enter through their derivatives.In other words, the only terms in the bosonic part of the action describing the angular part are kinetic terms.Moreover, we also see that some of these kinetic terms disappear (see equations (4.2) and (4.3)).We will explain why this happens based on the BRST symmetry of the action we just described in the last section.Furthermore, based on the BRST structure we will prove a statement concerning the degeneracy of the sigma model background field.
First, we will formulate a general statement for pure spinor theories only based on its BRST operator.

BRST structure and the action for more general backgrounds
We are studying a theory with a BRST operator given by Q 1 = B ab Λ a t b , as shown above in section (4.3).However, we don't have an explicit expression for the action.In this section, we will study some properties of this action only using its invariance under the BRST operator Q 1 .
We will start with a pure spinor sigma model in a particular background that allows us to integrate out the conjugate momenta.Separating the terms with ghosts and without ghosts in the action, this model can be writen as: where A µν (Z) = G µν (Z) + B µν (Z) and Z µ = (Z m , θ α , θ α) are local coordinates of the superspace.
The AdS sigma model and its deformations fall under this category.The case of interest in this paper arises from a deformation of AdS in a specific limit of strong deformation.Therefore, our conclusions in this section apply to the case of interest.
Back to the model (4.26), we will use the BRST symmetry inherent in the pure spinor action [14] to extract information regarding the constraints that the background fields must satisfy.The BRST operator can be expressed as Since this operator has ghost number 1, the matter part can be writen as: where E α is a function of space-time coordinates Z.Let us also split the BRST operator into left and right sectors, The BRST variation of w and λ in S g will possibly cancel with the BRST variation of the matter part.However, it can be argued that S g is BRST invariant when the pure spinors λ and their conjugate momentum w are on-shell.To demonstrate this, we use the equations of motion for the pure spinors λ L and λ R , which are determined by the variation of w: δ δw α S g = 0. Also, we use the equations of motion for the conjugate momentum w + and w − , given by the variation of λ, δ δλ α S g = 0. Therefore, we have for λ and w on-shell.(4.30) The BRST variation of the matter fields Z µ in S g will result in an expression containing w-ghost fields.This variation cannot cancel out with the BRST variation of the matter part of the action because the latter does not involve w-ghosts.Therefore it vanishes on its own: for λ and w on-shell.(4.31) We conclude then that the complete BRST variation of the ghost action S g is zero on-shell, QS g = 0, for λ and w on-shell.(4.32)Therefore, from QS = 0, we also have for λ and w on-shell.( Expanding this equation, we obtain for λ and w on-shell This equality is true for any field configuration of Z and any on-shell field configuration for λ and w.First we consider a field Z µ which does not depend on τ − , with ∂ − Z = 0.In this case, equation (4.34) reduces to and Z such that ∂ − Z = 0.
We further specify the field configuration of Z µ by setting it to: where V µ is an arbitrary constant vector in the superspace, η is a small parameter and H(x) ≡ 1 x≥0 is the Heaviside step function.The purpose of using the Heaviside function is that its derivative is a delta function, ∂ + H(τ + ) = δ(τ + ).This will be used to eliminate the integral in the equation (4.35).
Let us now deal with the pure spinor fields, which are on-shell, so we have to be careful if we want to specify a field configuration for them.First, we choose λ 3 = 0, since this is one possible solution to the equation of motions of λ 3 .Then, we notice that w + has conformal weight (1, 0) and w − has conformal weight (0, 1).Therefore, roughly speaking, the action of conformal weight (1, 1) and ghost number zero contains only terms of the form (w where E α βµ are complicated functions of Z, obtained from varying the action with respect to w 3− and setting λ 3 = 0.This is a partial differential equation and we will now proceed to formulate a Cauchy problem with it.Consider the differential equation (4.37) and a Cauchy surface S defined as a one dimensional surface in the worldsheet Σ by Worldsheet Σ parametrized by τ + and τ − .Cauchy surface S defined on the worldsheet and the initial boundary condition for the fields λ α 1 (τ + , τ − ).Green lines represent those transverse to the Cauchy surface S along Σ.
Then, we define the boundary condition of the problem by defining the functions λ α 1 on the Cauchy surface S, where u α is a constant pure spinor.This is schematically represented in figure 1.Now, we have fully stated our initial value problem.In summary: (4.40) Outside the Cauchy surface, the functions λ α 1 will be determined by the differential equation.To do that, we solve the differential equation along each green line in figure 1, defined by First, we remember that we have specified the field configuration for Z in equation (4.36).
We rewrite the differential equation with this field configuration, where we denote E β α = V µ E β αµ .Now, for each line L y we have an ordinary differential equation, with initial value at the point L y ∩ S. Along the L y lines, we define the functions λ α 1y (τ + ) = λ α (τ + , y), and ordinary differential equations for them: The solution for a fixed y is Indeed, we can check that it is a solution, since and so it satisfies the boundary conditions λ α 1y (0) = u α H(y).Now, we can vary the parameter y and write the full solution as Now that we have specified the field configurations for Z and λ, we go back to the expression of the BRST variation (4.35).Explicitly, we obtain for the derivatives where we consider η 2 = 0.After integration, (4.35) reduces to If we instead specify the field configurations to be Z µ = Z µ 0 + ηV µ H(τ − ), λ α 1 = 0 and a non-vanishing on-shell spinor λ 3 , corresponding to (4.46), we get a new corresponding condition, that is, equation (4.34) reduces to The vector V µ is arbitrary and we can take it out of the equation.In summary, we have A µν E ν R u = 0, (4.52) where angle brackets u means a linear dependence of the operator on u.Although it is written in terms of a set of coordinates, this result is independent of the coordinate system we use, owing to the invariance of equations (4.51) and (4.52) under coordinate transformations.To see this, we notice that under a change of coordinates Z → Z ′ , A µν transforms as a 2-tensor and E µ transforms as a vector.Now, applying a change of coordinates to the equation see how this general picture explains the structure of the Maldacena-Lunin-Frolov solution in the large γ limit, as written in equation (4.3).We begin by applying equations (4.51) and (4.52) to the sigma model with three deformation parameters with a strong coupling, namely where T µ is a space-time vector field which depends linearly to the parameter u, defined by T µ u = t µ a B ab (g −1 t α g) b u α .As argued in the last section, the spaces Z L and Z R are spanned by B ab t b .In turn, when u runs over the pure spinor cone, T µ spans the subspace of T (AdS where ℑ(B ) means the Image of the matrix B ab .In our case, W is any vector of the form W a = B ab C b .Z is the space of degeneracy of the background fields.Next, we write the three parameters as γ 1 = α 1 γ, γ 2 = α 2 γ and γ 3 = α 3 γ, where α i are non-zero constants and γ is the parameter we will tend to infinity.The BRST operator is then If we redefine the generators t a by T 1 := t 12 , T 2 = t 34 − α 2 α 1 t 12 and T 3 := t 56 − α 3 α 2 t 34 , we will have ) and therefore the BRST operator can be written as In the action (3.40), the remaining angular term in the large γ limit is dω 2 , where ω = 3 i=1 α i φ i , as explicitly written in (4.3), Then, the degeneracy space Z is generated by linear combinations of t 12 , t 34 , t 56 on which the 1-form dω = 3 i=1 α i dφ i is zero.These combinations is precisely given by the vectors appearing in the BRST operator (4.74), dω(T 2 ) = dω(T 3 ) = 0. (4.76) An interesting observation is that the 1-form dω is in the kernel of B. Indeed, the fact that B has a kernel of dimension 1 implies that the space of degeneracy defined in (4.71) has only 2 dimensions.As a consequence of this fact, some kinetic terms of the angles still survive, for instance in the bosonic case, where precisely dω 2 survives.In summary, in the general case the kinetic term for the angle of the 12 plane will not vanish in the bosonic action because its generator is no longer present in the BRST operator, as evidenced in (4.74).On the other hand, the planes T 2 and T 3 define angles that have vanishing kinetic terms.The angle of the 12 plane is not a special direction, but only a consequence of the coordinates redefinition (4.73).In fact, we could have chosen any new coordinate redefinition such that some other angle remains in the action.This can be seen in the Penrose limit of this case, (4.14).The general condition for the field Then, in the fishnet model In this case, Z L and Z R collapses to the same space Z, which by itself has the interpretation of a degeneration space of the background fields.

Relation of degeneracy with BRST variation of w
In the last subsection, we supposed that we don't know the structure of the BRST operator Q 1 acting on the w-ghost.In fact, the degeneracy of the background fields will imply that BRST acting on w is zero.Let us prove it.
Without assuming the structure of BRST transformation of w, the background fields satisfy the relations (4.77) and (4.78), for a = 2, 3. Now, this equation implies that the matter part of the action is BRST invariant off-shell.This happens because the degeneracy of the backgrounds is sufficient to guarantee the BRST invariance of the matter action separately, as we can see in the following demonstration.First, we act with the BRST operator in the matter part of the action, Next, we use the invariance of the action under the rotations of the 3 planes explained in section 3.5.This can be implemented by saying that the action is invariant under the action of the t a generators for a = 12, 34, 56, .82) Using this symmetry, equation (4.81) reduces to which is zero from equation (4.80).Therefore, applying Q 1 to equation (4.26), we find that Q 1 S g = 0 off-shell.Therefore, since the BRST variation of S g with respect to w is an expression only involving λ-ghosts, it cannot cancel with the other terms in Q 1 S g , which are expressions containing w-ghosts.Therefore, we conclude that

.84)
The only possible way this to be true for every field configuration is if BRST variation of w is zero, that is, Q α w = 0.This provides a clarification for the BRST variation of the w-ghost being zero.Indeed, another way to see this is by looking for the structure of the matrix κ in the large gamma limit and then seeing what happens to Q w in (3.19) in this limit.We can do this by using the coordinates {v 0 , v + , v − } described in section 4.2.In this case, according to Equation (4.21), the matrix κ in the large γ limit becomes: where Γ ± = α ± β, with α and β defined in (4.17).Therefore, since Q w in (3.19) depends only on κ acting on j a and Ad(t a ), and from (4.85) we have that κ γ − → 0, we conclude that Qw γ − → 0. Which implies that the BRST variation of w in the limit we are considering is zero.

Conclusion
In this paper we established a clearer a connection between the string description of the beta deformation and the supergravity solution originally proposed by Maldacena and Lunin in [6].This was done by first writing the string action with all order deformations and then extracting the supergravity solution.
Formulating the beta deformation in terms of a string action enabled us to investigate the particular problem of strong deformations.Studying the theory in the large γ limit was achievable through the presence of a BRST operator defined by Q 1 = B ab Λ a t b .The final conclusion, represented by the equations in (4.79), is that the background fields present a degeneracy of dimension 2, given by a particular combinations of the t a vectors, which rotate the sphere planes in R 6 .In terms of the ghost sector, we see that this degeneracy implies that the BRST operator acts trivially in the w-ghost.Furthermore, we concluded that as a result of the strong deformation, the geometry of the deformed sphere S 5 def becomes "saturated" as γ approaches infinity, indicating the emergence of a region where the metric degenerates, as indicated in the action (4.3).and the structure constants are given by: We may also translate it to: the generators of su(4) will be: Elements in G 0 are characterized by H = EH T E. Therefore, we can write a generic even element M as a sum M = M 0 + M 2 with M 0 = EM T E and M 2 = M − EM T E. For example, the current of the sigma model is given by R T dR and we may project it: It is remarkable that the projection of the current in 2 can be related to an expression with U: U T dU = −(RER T dRER T + dRR T ) = −R(R T dR) 2 R T (A.9) The bosonic part of the Lagrangian for the sigma-model is Str[J 2 ∧ ⋆J 2 ], where J 2 is the 2-component of the current J = −g −1 dg.The subsector of the sphere S 5 will have the current given by J = −R T dR.Therefore we set an important relation:   6) rotations using embedding coordinates of R 6 .In red we have the so(6) 0 part while in blue the so(6) 2 part.
In order to go from the infinitesimal case to what we really have, we use the relation: where W = 1−e ad X ad X , and we used that the structure constants we are using are invertible matrices to rewrite W in terms of a new w and the coordinates Z.Then, using what we already found for the infinitesimal case, we find the space of u's such that u α W 12 α = 0 and inside this space we choose a specific one such that ūα W 34 α = 0, and vice-versa.

B Penrose Limit
Consider the following null geodesic:

2 5 ConclusionA
Deriving supergravity from worldsheet pure spinor description 1.3 Large γ 1.4 Plan of the paper 2 Review of AdS 5 × S 5 Superstring: pure spinor formulation 3 β-deformation of the superstring sigma model 3.1 Simplification of the BRST operator 3.2 Deformation to all orders 3.3 Maldacena-Lunin background 3.3.1The structure of M ab 3.4 Generalization to γ i deformation 3.5 Symmetries 4 The large γ limit 4.1 The pp-wave limit 4.2 Studying the matrix κ γ 4.3 The BRST operator 4.4 Action for a deformation with strong coupling 4.5 BRST structure and the action for more general backgrounds 4.6 Sigma model under strong coupling deformation: large γ limit 4.6.1 General picture of degeneracy and relation to Maldacena-Lunin-Frolov 4.6.2Relation of degeneracy with BRST variation of w AdS 5 × S 5 embedding coordinates A.1 Grading of the algebra B Penrose Limit 1 Introduction

Figure 2 .
Figure 2. Representation of SO(6) rotations using embedding coordinates of R6 .In red we have the so(6) 0 part while in blue the so(6) 2 part.