Strings in Ramond-Ramond Backgrounds from the Neveu-Schwarz-Ramond Formalism

We treat RR flux backgrounds of type II string theory in the framework of closed superstring field theory based on the NSR formalism, focusing on two examples: (1) the pp-wave background supported by 5-form flux, and (2) $AdS_3\times S^3\times M_4$ supported by mixed 3-form fluxes. In both cases, we analyze the classical string field solution perturbatively, and compute the correction to the dispersion relation of string states to quadratic order in the RR flux. In the first example, our result is in a delicate way consistent with that obtained from lightcone quantization of the Green-Schwarz string. In the second example, we will obtain numerically the mass corrections to pulsating type IIB strings in $AdS_3\times S^3\times M_4$. Our results, valid at finite $AdS$ radius, agree with previously known answers in the semiclassical limit and in the BMN limit respectively.


Introduction
The Neveu-Schwarz-Ramond (NSR) formulation [1][2][3][4][5][6] of the perturbation theory of superstrings is based on a worldsheet (1, 1) superconformal field theory coupled to the b, c, β, γ ghost system, with total central charge zero, and a BRST symmetry such that the superconformal currents are the BRST transformation of b and β ghosts. This framework can be used to formulate the perturbation theory of type II superstrings propagating in a spacetime background that corresponds to a solution of the supergravity equations of motion in which the NSNS fields are turned on. A deformation of the NSNS background corresponds to deforming the worldsheet SCFT by a marginal primary V 0,0 NS , known as the NSNS vertex operator in (0, 0)-picture. Generic supergravity solutions also involve nonzero RR field strengths (fluxes), and the latter play key roles in string compactifications and holographic dualities [7][8][9]. As the RR vertex operators come with half-integer picture numbers, it does not seem sensible to deform the worldsheet SCFT by the RR vertex operators. This is often thought of as a fundamental limitation of the NSR formalism, and alternative formalisms were deemed necessary [10][11][12][13][14][15].
It was suggested in the seminal work of Friedan, Martinec, and Shenker [1] that the fermionic contributions to supergravity equations can be recovered by the inclusion of an even number of Ramond-sector vertex operator insertions. This idea was investigated in detail two decades ago by Berenstein and Leigh [16,17]. It was proposed that the NSR formalism can be extended to RR backgrounds by deforming the worldsheet action by ∆S = d 2 z V 0,0 NS (z,z) + d 2 z P 1 where V 0,0 NS stands for an NSNS vertex operator in the (0, 0) picture, and V − 1 2 ,− 1 2 R stands for an RR vertex operator in the (− 1 2 , − 1 2 ) picture. The formal object P1 2 was introduced in [16,17] such that a pair of P1 2 insertions is to be thought of as a single picture changing operator (PCO) X (and similarly for the anti-holomorphic object P1 2 ). The obvious difficulty with this prescription is that P1 2 is not well defined, as the locations of PCOs are unspecified. While in a BRST invariant correlator the locations of PCOs are unimportant [1], in superstring perturbation theory the moduli space integrand is typically only BRST invariant up to total derivatives.
One may attempt to make (1.1) precise by making a specific choice of the location of PCOs, and replace the insertion of exp(−∆S) with exp − d 2 z V 0,0 NS (z,z) by picture raising. From the worldsheet CFT point of view, such a deformation is nonlocal, but nonetheless Weyl invariance ought to be preserved. Indeed, it was shown in [16,17] that the supergravity equations involving RR fields can be recovered from the cancellation of the Weyl anomaly on the worldsheet. It is not a priori clear, however, how to define (1.2) precisely beyond leading order in the deformation parameter, as the vertex operators in question are expected to be off-shell.
The closed superstring field theory (SFT) based on the NSR formalism [18][19][20][21][22][23][24] is a framework that in principle provides a systematic formulation of superstring perturbation theory in any closed string background, at the classical as well as the quantum level. While the NSNS and RR deformation operators in (1.2) can loosely be identified with string fields to leading orders, in the SFT approach, the background deformation is entirely captured by a solution to the string field equation which lives in the Hilbert space of the original, undeformed, worldsheet CFT.
In this paper we will adopt the classical SFT framework to analyze RR background deformations and the string spectrum. Our general strategy is laid out in section 2, where we consider perturbative solutions to the string field equation that represent RR background deformations, and outline the steps of extracting the string spectrum from the linearized string field fluctuations around the background.
In section 3 we will consider type IIB strings in the pp-wave background supported by the self-dual RR 5-form flux as a basic example to illustrate our framework. We explicitly solve the string fields to leading nontrivial orders in the RR flux, and investigate linearized fluctuations that represent a family of maximally spinning string states. In particular, we compute corrections to their dispersion relations to quadratic order in the RR flux. In a rather intricate manner, our result is consistent with that of [25][26][27], obtained by quantizing the Green-Schwarz action in the lightcone gauge.
In section 4, we consider type IIB strings in AdS 3 × S 3 × M 4 (where M 4 is K3 or T 4 ) in mixed three-form flux backgrounds, viewed as a deformation of the pure NSNS background by turning on RR flux. We will focus on pulsating string states due to the simple form of their vertex operators, and compute their mass corrections to quadratic order in the RR flux. The crux of the computation is a straightforward but tedious exercise of evaluating (and integrating) correlation functions of SL(2, R) current algebra descendants, the details of which are given in the Appendices.
Our explicit result for the mass corrections at finite AdS radius is numerical. In the large radius and large oscillator number limit, our result agrees with that obtained from semiclassical quantization of a pulsating string in AdS 3 [28]. In the Berenstein-Maldacena-Nastase (BMN) limit [27], our result also agrees with the previously known string spectrum in the mixed flux pp-wave background obtained by lightcone quantization of the Green-Schwarz action. We conclude and comment on future perspectives in section 5.
2 Background deformation in NSR superstring field theory 2.1 Closed NSR superstring field theory A systematic treatment of background deformation in superstring perturbation theory is provided by the closed superstring field theory based on the NSR formalism, as formulated in [19][20][21][22]24]. In this paper we will only be concerned with classical string field theory. More precisely, we will work with the string field equation derived from the tree level 1PI off-shell amplitudes.
One defines the closed string field Ψ as a state in the worldsheet CFT Hilbert space of picture number −1 in the NS sector and − 1 2 in the R sector, that is subject to the constraints One further defines the operator G as the identity when acting on an NS sector state, or the zero mode of the PCO, X 0 or X 0 , when acting on an R sector state on the left or on the right, respectively. Our convention for the PCO X (z) is ∂(e 2φ ηb), (2.2) where G m is the supercurrent of the matter CFT. ξ, η, φ are the standard re-bosonization of the β, γ ghost system, β = e −φ ∂ξ, γ = e φ η. (2.3) The string field equation takes the form whereas gauge transformations are given by Here Q B is the BRST operator in the worldsheet CFT. The bracket [Ψ ⊗(n−1) ] is a multi-linear map on the string field defined through where {Ψ 1 ⊗· · ·⊗Ψ n } is the 1PI part of the tree-level off-shell amplitude A[Ψ 1 , · · · , Ψ n ]. That is, we subtract from A[Ψ 1 , · · · , Ψ n ] all 1-particle reducible contributions that correspond to off-shell amplitudes of sub-diagrams connected by free string field propagators. The off-shell amplitude A, as defined in [19,20,24], takes the form where the integrand Ω and the integration domain S n are defined as follows. In the holomorphic sector, suppose n NS of the string fields among Ψ i are in the NS sector (of picture number −1), and n R of them are in the R sector (of picture number − 1 2 ). We then need n NS + 1 2 n R −2 holomorphic PCO insertions. Similarly we need n NS + 1 2 n R −2 antiholomorphic PCO insertions. Let M 0,n be the moduli space of the n-punctured Riemann sphere. P 0,n is the space of the n-punctured Riemann sphere together with the choice of a holomorphic coordinate system w i on a disc D i = {|w i | ≤ 1} containing each puncture, i = 1, · · · , n. P 0,n is the space of the data that define P 0,n together with the choice of the locations of the PCOs, all of which are inserted outside of n i=1 D i . S n is a suitable subspace of P 0,n that projects onto the entire M 0,n if we forget the data of w i and PCOs.
Ω is a degree (2n − 6) differential form on P 0,n defined by a suitable correlation function of the string fields inserted at the punctures on the sphere, together with insertions of b ghosts associated with the moduli, and insertions of PCOs. For instance, if we choose the coordinate w i on the disc D i to be related to the coordinate z i on the Riemann sphere by the P SL(2, C) map the form Ω can be written as 1 , (2.9) 1 Compared to the convention of [19,24], our string field is rescaled by a factor of 2πi.
where C z i is a counterclockwise contour around z i , and · · · | (p,q) means that we are keeping only the (p, q)-form component. The integration slice S n is defined by a choice of z i , y m , q i , s i as functions of the moduli (t k ,t k ) that parameterize M 0,n . Note that the b ghost insertion in (2.9) is multiplied by ∂z where the o(z − z i ) terms depend on the choice of q i and s i as functions of t k . The result of the b ghost contour integral is of the form dz i b −1 + dt k n≥0 h i k,n b n , for some coefficients h i k,n , acting on the string field inserted at z i . S n should be arranged such that the amplitude is completely symmetric in the n string fields. To achieve this in practice it is convenient to average over different choices of S n . For instance, we can define the "ordered off-shell amplitude" A ord [Ψ 1 , · · · , Ψ n ] by performing the integration (2.7) over a slice S ord n defined through fixed z 1 = 0, z 2 = 1, z 3 = ∞, with the moduli of M 0,n parameterized by z 4 , · · · , z n , and an appropriate assignment of y m , q i , s i that respects permutation symmetry on Ψ 4 , · · · , Ψ n . We then obtain the off-shell amplitude A by symmetrizing all string fields in A ord .
Note that deforming S n amounts to a field redefinition of the string field. In order for the amplitude to factorize consistently when an intermediate closed string state goes on-shell, one further demands that the choices of S n for different n's are compatible with respect to gluing together punctured spheres via plumbing fixture in the degeneration limits where the punctures cluster. Such choices are possible because the plumbing can be performed using P SL(2, C) maps.
As an example, let us consider the 3-point ordered amplitude with the choice of S ord 3 , in this case a point in P 0,3 , labeled by z i , y m , q i , s i . For simplicity let us ignore the PCO for the moment, and choose z 1 = 0, z 2 = 1, z 3 = ∞, with the transition maps (2.8) given by This corresponds to the choice q 3 = −r, s 3 = 1. Thus, the insertion of the third string field takes the form e −L 1 − L 1 r 2L 0 Ψ 3 (∞). Note that, if all three string fields Ψ i are SL(2)×SL(2) primaries, the ordered amplitude defined through the transition map (2.11) is the same as the symmetrized off-shell amplitude.

RR deformation and Weyl invariance
An RR deformation of a closed string background corresponds to a solution Ψ 0 of the string field equation (2.4) of the form where the superscripts indicate the picture numbers, and µ is the RR deformation parameter. The string field equation expanded to first order in µ implies that V is BRST closed, and thus takes the form of the usual on-shell RR vertex operator. It can be put in the form where O R,1 is a matter CFT RR operator of weight ( 5 8 , 5 8 ). The NSNS component of the string field equation at order µ 2 is is not BRST-exact, it would present an obstruction to turning on the RR background. We will proceed by assuming that the string field solution does exist. Let P be the projection operator onto states with L 0 + L 0 = 0. A solution to (2.16) can be written in the form where W NS is a state that obeys In practice, we will be able to solve W (2) NS relatively easily, up to a BRST-closed term that must in principle be adjusted to allow for a solution at higher orders in µ.
where O NS,2 is a matter CFT operator, and · · · represents higher weight operators that generally involve other combinations of ghosts. To illustrate the point, let us contract (2.16) for an arbitrary NSNS string field Ψ that obeys the Siegel gauge constraint Here {· · · } is the 1PI part of the off-shell amplitude introduced in section 2.1, which in the three-point case coincides with the off-shell amplitude itself. The latter is particularly simple, as neither moduli nor PCO are involved. If we take Ψ = c ce −φ− φ Φ for some matter CFT operator Φ that is an SL(2) × SL(2) primary, then (2.20) reduces to the following condition on O NS,2 at the level of matter CFT correlators, where we have adopted the transition maps (2.11) in writing the 3-point off-shell amplitude.
From the point of view of deforming the worldsheet CFT along the lines of (1.2), one may identify the RR deformation operator with µe − φ 2 − φ 2 O R,1 , and identify the NSNS deformation operator with the picture-raised version of µ 2 e −φ− φ O NS,2 (note however that O NS,2 is offshell). (2.21) amounts to the cancellation of the Weyl anomaly due to a pair of colliding RR deformation operators by that of the NSNS deformation operator, where the "radius" parameter r now plays the role of a UV cutoff on the worldsheet. In contrast, in the SFT approach the worldsheet CFT is never deformed, and changing the parameter r amounts to a redefinition of the string field.

String spectrum from linearized string field equation
Expanding the string field Ψ around a background solution Ψ 0 , Ψ = Ψ 0 + Ψ, we can extract the string spectrum from the linearized equation in Ψ, Let us consider the RR background of the form (2.14), (2.15), and restrict our attention to bosonic fluctuations described by Ψ. We will split Ψ into its NSNS component Ψ NS and the RR component Ψ R , and expand each of them as a power series in µ, (2.23) At zeroth order in µ, (2.23) amounts to a linear combination of the BRST-closed NSNS state Ψ One should be cautious that, while it is typically possible to arrange Ψ to be a deltafunction normalizable state with respect to spacetime momenta, the series expansion of Ψ to a given order in µ are not quite delta-function normalizable. 2 For instance, if Ψ (0) NS/R contains e ik·X for some spacetime momentum k µ , Ψ may involve e ik ·X for a shifted spacetime momentum k µ = k µ + δk µ that obeys a corrected dispersion relation. Ψ (1) NS/R would involve states like δk ·Xe ik·X whose inner products are not delta-functions in k but rather derivatives of delta-functions. Acting on states whose inner products are defined in this distributional sense, L 0 is not diagonalizable (as is standard in logarithmic CFTs). Instead, we can work with a basis of states on which L 0 acts in its Jordan normal form. The projector P onto L 0 = 0 states defined in section 2.2 will now be generalized to the projection operator onto the invariant subspace on which L 0 acts nilpotently.

First order in µ and mixing between NSNS and RR states
Expanding the equation (2.22) to first order in µ gives (2.24) We can write the solution for the first order string field in the form NS are P-invariant states that obey

(2.26)
A priori, the solution for Y NS/R exists only if the string field brackets appearing on the RHS (2.26) are BRST-exact. However, as already remarked, Y NS/R are not necessarily deltafunction normalizable states, which allows for the possibility of trivializing a larger class of BRST-closed states.
We will focus on solutions that represent string states of well-defined dispersion relation in the deformed background. Suppose the µ 0 order string field takes the form where Φ m NS (k) is a matter superconformal primary, and Φ m R is an RR state in the matter CFT that is annihilated by G m r and G m r for r ≥ 0. The parameter k represents the spacetime momentum, or the analogous quantum numbers if our starting point is a nontrivial NSNS background. We will assume a continuation of Φ m NS/R (k) to off-shell values of k such that Φ m NS/R (k) are still annihilated by L m n>0 and G m r>0 in the matter CFT.
Up to order µ, we will demand that Ψ where · · · represent orthogonal states that either involve other ghost structures or matter SCFT descendants. Comparing with (2.26), we need (2.29) (2.29) will determine the RR deformed dispersion relation as well as the mixing of NSNS and RR string states to first order in µ.

Second order in µ
Let us proceed to analyze the order µ 2 string field equation for the linearized fluctuations, whose solution takes the form NS/R obeys the equation restricted to the P-invariant subspace, (2.32) Analogously to (2.28), in order to have a well-defined dispersion relation, we need (2.33) Assuming this is the case, we can determine the order µ 2 correction to the dispersion relation from L 0 Y NS/R . To compute the latter, we can contract We will refer to A as a modified amplitude of string fields, in the following sense. If we omit the projector P on the RHS of (2.35), it would give 1 2 times the off-shell tree level 4-point amplitude A of the string fields Ψ , V R,1 , V R,1 , and Ψ (0) NS . A differs from A in that, when the latter diverges as an intermediate string state going on-shell, a pole contribution is subtracted off so that A is finite. From the point of view of the spacetime S-matrix, this would be a contrived definition. The amplitude considered here involves the insertion of background string fields whose momenta may not be adjustable, and the subtraction is useful in organizing practical computations.
We can compute the 4-point off-shell amplitude A by averaging the ordered amplitude A ord over permutations on the four string fields. For instance, one of the ordered amplitudes is 3 (2.36) where s and r = |q| are parameters of the transition map for the coordinate system on the disc containing Ψ (∞), which are generally functions of (z,z) to ensure the correct factorization limits. In writing (2.36) we have used the fact that V R,1 and Ψ (0) NS are on-shell. A priori, the location of the PCOs y = y(z,z) as specified by the integration slice S ⊂ P 0,4 should be arranged such that when z approaches either 0, 1, or ∞, y is away from z. If Ψ is on-shell, then the form Ω (2.9) is closed, and one can deform S by moving the PCOs to z, to a string field in the ( 1 2 , 1 2 ) picture, up to a possible boundary term of the moduli integral that can typically be shown to vanish. When Ψ is off-shell, deforming S to S "vertically" by moving the PCOs changes the amplitude A by the integral of dΩ along a chain in P 0,4 bounded by S − S. The latter in particular involves the "vertical" integration of dΩ, with the latter expressed as a correlator involving Q B Ψ and explicit insertions of ∂ξ or∂ ξ. In simple situations where Q B Ψ does not involve η or η, such vertical integration gives a vanishing result.
The corresponding ordered modified amplitude A differs from (2.36) by subtracting off possible log divergences in the integration near z = 0, 1, ∞. Note that the finite part of the subtraction is generally not invariant under field redefinition of the string field.

Type IIB strings in pp-wave with RR 5-form flux
As the first example, we consider the simplest nontrivial solution of type IIB supergravity with RR flux, namely the pp-wave solution supported by the self-dual 5-form flux [25][26][27], (3.1) We will first analyze the string field solution corresponding to the background (3.1) to quadratic order in µ, and will then study the spectrum of a family of maximally spinning strings from the linearized fluctuations of the string field.

The background string field
Viewed as a deformation of Minkowskian spacetime, the order µ string field takes the form of the vertex operator corresponding to the RR 5-form field strength. In the notation of (2.14) and (2.15), we have Here S α and S α are the left and right spin fields, where we follow the conventions of [6] and have included both chiral and anti-chiral components. The overall normalization constant N 1 will be determined shortly.
At order µ 2 , the NSNS string field takes the form (2.17), with the equation for W This can be solved with W Note that Z 2 is not an eigenstate of the dilatation operator, as L 0 Z 2 = −4α . Furthermore, we are free to shift Z 2 in (3.4) by a constant. In the supergravity description this amounts to shifting the metric by a constant times µ 2 (dx + ) 2 , which can be absorbed into a redefinition of the coordinate x − .
It is useful to consider the picture-raised string field components, where O ↑ R,1 ad O ↑ NS,2 are matter CFT operators, given by whereas · · · in (3.5) stands for higher weight operators, which also involve more general combinations of ghost fields. From the point of view of deforming the worldsheet CFT along the lines of (1.2), O ↑ NS,2 can be identified with the worldsheet Lagrangian deformation. This also suggests that no further BRST-closed terms are needed in (3.4). Comparison with the NSNS part of the supergravity solution (3.1) then fixes the normalization constant Since O ↑ NS,2 is not marginal, a deformation of the worldsheet CFT by V 0,0 NS = µ 2 O ↑ NS,2 would lead to a Weyl anomaly of the form T a a = −16µ 2 ∂X +∂ X + , consistent with the Ricci cuvature of (3.1) having the only nonzero component R ++ = 8µ 2 . This is cancelled against the Weyl anomaly due to the logarithmic divergence that arises when a pair of RR deformation operators V where δ is a short-distance cutoff.

The maximally spinning string states
Let us consider a family of NSNS string states on the leading Regge trajectory in Minkowskian spacetime, described by the vertex operator or string field where the operator product is understood to be normal ordered. For simplicity, we have taken the wave function F to be independent of Z 1 , Z 2 . This is such that (3.9) is BRST-closed, provided that In the pp-wave background, the order µ string field equation (2.26) for Y (1) R is very simple, as one can verify that NS of the form (3.9). We will proceed by setting Y R to zero. At order µ 2 , we will compute Y (2) NS from (2.34). First, we need to compute the projected string field bracket P W where it is understood that the result should be averaged over a set of locations (y,ȳ) of the PCO to ensure symmetry with respect to the three string field insertions.
If Ψ takes the form c ce −φ− φ Φ for some matter superconformal primary Φ (not necessarily marginal), so that Q B Ψ does not involve η, then we would be free to move the PCOs in (3.12). From the 3-point off-shell amplitude with such Ψ , we can determine the projected string field bracket to be of the form 13) where · · · stands for states that involve either other ghost structures or matter superconformal descendants.
Next, we will examine the modified amplitude where the subscript "prim" stands for projection onto the matter superconformal primary component. Note that PΨ = Ψ , even though such Ψ is not necessarily BRST-closed. In this case, the corresponding 4-point off-shell amplitude of the string fields evaluates to In deriving the first equality above, we have moved the PCOs to convert one of the RR background string fields to ( 1 2 , 1 2 ) picture, and reduced the correlator in the moduli integrand to one in the free fermion CFT. The latter is then evaluated using (1,9) spinors, and indices on gamma matrices are raised/lowered with C.
In the last step of (3.14), the z-integral is regularized by restricting to |z − 1| > δ for a short-distance cutoff δ, and λ is a finite constant. Note the absence of log divergence near z = 0 or ∞, which is consistent with (3.11). The log divergence near z = 1 is due to the , and is projected out in the modified amplitude A (2.35).
The latter then evaluates to for some finite constant λ that depends on the choice of integration slice S 4 ⊂ P 0,4 .
Substituting Y where · · · represents terms of other ghost structures or matter superconformal descendants.
Recall that we are free to shift Z 2 by a constant via a spacetime coordinate transformation, which we can use to remove the n-independent constant −8α ln r + 8α λ in (3.16). From (3.16) and moreover the first equation of (2.32), we deduce that Y NS is of the form where the function F (2) obeys To summarize, we started with the µ 0 maximally spinning NSNS string state of momentum (k + , k − ) in (X + , X − ) directions, and a wave function F (Z 3 , · · · , Z 8 ) that is independent of Z 1 , Z 2 , and found that the NSNS string field receives a µ 2 order correction to the Z-wave function of the form µ 2 F (2) (Z 1 , · · · , Z 8 ) subject to (3.18).
To understand the meaning of (3.18), let us compare it with the result obtained from quantizing the Green-Schwarz effective action [25][26][27]. In the pp-wave background (3.1), the Green-Schwarz action reduces in a lightcone gauge to that of eight free massive bosons and eight free massive fermions, whose masses are equal and related to the RR flux. The spinning string states of our consideration correspond in the Green-Schwarz lightcone description to a closed string state with n left-moving and n right-moving oscillators excited in the first Fourier mode, as well as 4 fermionic zero mode excitations (out of the total 8 fermion zero modes). If we further excite zero modes of the Z i 's, the dispersion relation of the string state is Let f (Z) be the wave function in the Z i zero modes of the string. We can re-express in terms of a harmonic oscillator Hamiltonian in the Z i directions acting on f (Z), by substituting Expanding (3.19) to order µ 2 then gives where M 4 is either the K3 manifold or the four-torus T 4 . These backgrounds arise as the near-horizon limit of the D1-D5-F1-NS5 system and play a central role in AdS 3 /CFT 2 dualities [8,9,29,30]. Furthermore, they provide a nontrivial but tractable laboratory for the application of our formalism for perturbative strings in mixed-flux backgrounds. This is because we can start with the AdS 3 ×S 3 ×M 4 background supported purely by NSNS fluxes, where the worldsheet theory in the NSR formalism admits a solvable CFT description that involves the N = 1 supersymmetric SL(2, R) × SU (2) WZW model [31][32][33][34]. We can then consider the deformed background by turning on RR 3-form fluxes. Note that while in the full non-perturbative string theory, both the NSNS and RR fluxes are quantized, at the level of string perturbation theory only the quantization of the NSNS flux is visible while the RR flux can be treated as a continuous parameter.
The type IIB supergravity solution for AdS 3 × S 3 × M 4 takes the form where ω AdS 3 and ω S 3 are volume forms on AdS 3 and S 3 of unit radius, and q is a parameter between 0 and 1. The quantization of NSNS flux is such that R 2 = α k for a positive integer k. We begin with the case q = 1, and turn on the RR flux by deforming to so that the NSNS flux is held fixed. As already mentioned, since we will be working at the level of string perturbation theory, the quantization of RR flux is irrelevant, and we can treat q as a continuous parameter.

Worldsheet theory and the string spectrum in the purely NSNS background
The type IIB string in the purely NSNS AdS 3 × S 3 × M 4 background is described in the NSR formalism by a worldsheet "matter" SCFT that is the tensor product of the N = 1 SL(2, R) k × SU (2) k WZW model and a c = 6 superconformal nonlinear sigma model on M 4 , along with the superconformal ghosts and a suitable GSO projection. The supersymmetric SL(2, R) k WZW model, whose currents we denote J a , a = 0, ±, can be constructed as the bosonic SL(2, R) k+2 WZW model whose currents are denoted J a , together with three free fermions ψ a and their anti-holomorphic counterparts. The supersymmetric SU (2) k WZW model is constructed from the bosonic SU (2) k−2 WZW model whose currents are denoted K a , a = 0, ±, together with three free fermions χ a and their anti-holomorphic counterparts.
The detailed conventions are summarized in Appendix A.
As building blocks of vertex operators, the current algebra primaries of the bosonic SL(2, R) k+2 are denoted V sl j,m,m , whose conformal weights are h =h = − j(j−1) k , whereas the current algebra primaries of the bosonic SU (2) k−2 are denoted V su j ,m ,m with weight h =h = j (j +1) k . Physical string states will be described by (delta-function) normalizable vertex operators that involve unitary representations of the SL(2, R) as well as states related by spectral flow [33]. It is important here that spectral flow is that of the N = 1 SL(2, R) k . In this paper, we will restrict to principal discrete representations D ± j of SL(2, R) with no spectral flow. The RR deformation of the background will be described using nonnormalizable vertex operators that involve finite dimensional non-unitary representations of SL(2, R), such as the fundamental (j = − 1 2 ) or the adjoint (j = −1) representation. For simplicity we will analyze massive pulsating string modes, described by (−1, −1) picture NSNS vertex operators of the form c ce −φ−φ V j,j ,n , with or equivalently (j + j )(j − j − 1) = (n + h int )k. (4.5) Note that the spacetime energy is ∆ = 2(j −n−1) whereas the angular momentum quantum number on the S 3 is j . The positivity of spacetime energy requires j − n − 1 > 0. This also leads, by spectral flow symmetry, to an upper bound on j so that which is stronger than the no-ghost bound [35]. We also have 0 ≤ j ≤ k−2 2 for half-integer j of SU (2) k−2 . To facilitate a comparison with the BMN limit [27], we take h int = 0 and define where H l.c. = 2p − can be identified with the light-cone Hamiltonian. The mass-shell condition (4.4) can be written as (kp + + n + 1) p − + n = nk. (4.8) The BMN limit corresponds to taking j, j , k 1, holding j − j and n fixed, so that p + and p − are both of order 1. In this limit (4.8) reduces to p + p − = n(1 − p + ), (4.9) which agrees with the string spectrum in the purely NSNS pp-wave background, in the case where only oscillators of the first Fourier mode on the string are excited [27,36].

Turning on RR fluxes
We now turn on RR flux by taking µ to be nonzero in (4.1), (4.2), while fixing the NSNS flux number k. Consequently the overall radius R of AdS 3 × S 3 is deformed according to (4.2). The deformed supergravity solution, a priori valid in the large k limit, corresponds to a solution to the string field equation of the form (2.14), (2.15), with Here we denote by Θ ± and Θ ± the holomorphic and anti-holomorphic spin fields in the superconformal nonlinear sigma model on M 4 , where the subscript ± labels their chirality. Note that only the combination Θ + Θ + appears in the first order RR string field. 4 S and S are holomorphic and anti-holomorphic spin fields constructed from the bosonization of the 6 free fermions as part of the N = 1 SL(2, R) × SU (2) WZW model. The components of S are denoted S αα ± where α and α are doublet indices of the sl(2, R) so(1, 2) and su(2) so(3) respectively, and the subscript ± indicates the chirality of the spin field with respect to the SO(1, 5) acting on the 6 fermions.
The order µ 2 NSNS string field can be solved from (2.17) and (2.18). In this case, the RHS of (2.18) vanishes, and we can simply set W where · · · represents higher weight operators. The coefficients A sl and A su can be computed from (2.21), with the result where r is the disc radius parameter in the definition of the string field bracket as in (2.21).
Here we denote by C sl j 1 ,j 2 ,j 3 and C su j 1 ,j 2 ,j 3 the structure constants of primaries of the corresponding spins in the bosonic SL(2, R) k+2 and SU (2) k−2 WZW models, respectively. In the case of interest, the fundamental-fundamental-adjoint structure constants are given by [37] C su (4.13) Both C sl − 1 2 ,− 1 2 ,−1 and C su 1 2 , 1 2 ,1 approach 2 √ 3 in the large radius limit k → ∞. In this limit, the picture-raised version of the NSNS string field e −φ− φ O NS,2 is proportional to the kinetic term in the WZW Lagrangian density, as expected from the radius deformation of AdS 3 × S 3 . From the point of view of worldsheet CFT deformation (1.2), the parameter r in (4.11), (4.12) again plays the role of a short distance cutoff, as the non-marginality of O NS,2 serves to cancel the Weyl anomaly due to a pair of colliding RR deformation operators.
For later convenience, we also write down the picture-raised RR string field at order µ, (4.14) The terms involving Θ − or Θ − will not be important for the computations below, as they will not contribute to the mass correction to the string states of our consideration at order µ 2 .

Corrections to the pulsating string states
We now study the linearized string field Ψ describing the pulsating string state in the RR deformed background, whose order µ 0 term is Ψ where we have moved the two PCOs to z,z to convert the string field insertion V Note that there is no log divergence from the region near z = 0 or ∞, indicating that A is the same as the modified amplitude A , and that we can set Y where δh or the corresponding mass squared correction δm 2 are given by δh = α 4 δm 2 = − π 2 µ 2 d 2 zf j,j ,n (z,z). (4.17) From the viewpoint of deforming the worldsheet CFT by (1.2), we may view δh as an anomalous weight of the vertex operator V j,j ,n at order µ 2 , hence the notation.
The explicit evaluation of f j,j ,n involves two steps. First, due to the appearance of SL(2, R) k+2 current algebra descendants in V 1 2 , 1 2 R,1 as well as in V j,j ,n , we need to apply Ward identities to reduce the computation to that of correlators of current algebra primaries, as described in Appendix B. Next, correlators of current algebra primaries are computed using the Knizhnik-Zamolodchikov (KZ) equation [38], which is performed in Appendices C, D.
Using these results, the four point function is written as

BPS states
An important consistency check is the case in which the pulsating string vertex operators V j,j ,n are chiral primaries, namely n = 0, j = j + 1, h int = 0 [39], describing BPS supergraviton states. Such states should receive no mass corrections under the RR deformation. Instead of demonstrating the preservation of spacetime supersymmetry under the deformation by solving for the gauge symmetries perserving the background, we can verify the absence of mass corrections to the BPS states at order µ 2 , by evaluating the integral on the RHS of (4.17) including a counter term that subtracts off the power divergence at z = 1, − zg sl,j +1 +−;−+ (z,z)g su,j −+;+− (z,z) −zg sl,j +1 −+;+− (z,z)g su,j +−;−+ (z,z) where the last equality is confirmed numerically. Here, the counter term coefficient B j is determined by inspecting the z → 1 limit of the 4-point function. g sl,j and g su,j are four point functions of SL(2, R) k+2 and SU (2) k−2 current algebra primaries given by (D. 15) and (C.20).
There are two cases of (4.19), where j = j + 1 is either 1 or k 2 , that require special treatment. Writing g sl,j as a linear combination of SL(2) current algebra conformal blocks (D.15), the coefficient h 24 − diverges at j = 1, whereas h 24 + diverges at j = k 2 . However, j = 1 is excluded by the bound (4.6). On the other hand, j = k 2 is allowed, and while the 4-point function in the integrand of (4.15) diverges in this case, the divergent coefficient is multiplied by an integral in the cross ratio that vanishes, thus a more careful regularization is required. While j a priori takes discrete values, as constrained by the mass-shell condition, we can consider AdS 3 × S 3 × T 4 where the T 4 is large and a near-BPS supergraviton mode that carries small momentum on the T 4 , corresponding to a string vertex operator of the form (4.3) with j = k 2 − 1, n = 0, and small h int > 0. We can then take the limit h int → 0, in which j approaches k 2 from above. One can indeed verify numerically that δh converges to zero in this limit.  Table 1: The leading anomalous weight in units of µ 2 for some small values of the level k (or equivalently, the AdS radius) at n = 1 (with h int = 0). For each value of k, the range of j considered eventually truncates once the SL(2, R) k+2 spin j fixed by the mass-shell condition (4.4) saturates or exceeds the upper bound k 2 (cf (4.6)).

Non-BPS states
Now we turn to the mass correction due to RR flux at order µ 2 for the non-BPS pulsating string states, described by the vertex operators V j,j ,n , where n ≥ 1, 0 ≤ j ≤ k 2 − 1, with j determined by the mass-shell condition (4.4) and subject to the bound (4.6). Here we compute δh by numerically integrating the 4-point function in (4.15). Some explicit numerical values of δh/µ 2 for small values of k are shown in Table 1.
In Figure 1, we plot our numerical results of δh/µ 2 for larger fixed level k, and for fixed SU (2) spin j , as a function of the oscillator number n. The results are compared to those obtained from semiclassical quantization, as we analyze in the next subsection.  The spectrum of j is of course discrete once h int is specified. However, since at this order h int enters the computation in a trivial way, the SL(2, R) k+2 spin j can essentially be regarded as a continuous variable. As j approaches k 2 from below, we observe a divergence in the anomalous weight, which has the interpretation of a short pulsating string becoming long [33]. Indeed, we will see this feature explicitly in the semiclssical regime (4.27) in the next subsection.

The semiclassical limit
The classical pulsating string solutions in AdS 3 with mixed flux were studied in [28]. At the classical level, the RR flux only affects the relation between the radius R and the Hflux number k, namely kα R 2 = q = 1 − µ 2 2 . In the conformal gauge, the relevant worldsheet Lagrangian is 20) where (t, r, φ) are the worldsheet fields corresponding to AdS 3 global coordinates (r ≡ sinh ρ), · stands for derivative with respect to the worldsheet time τ , and the derivative with respect to worldsheet spatial coordinate σ. We will set α = 1 in the rest of this section. A (nonspinning) pulsating string is represented by It is easy to see that the δφ component of the equation of motion (EOM) and the Virasoro constraint T στ = 0 are trivially satisfied by (4.21). Nontrivial components of the EOM read 5 The δt-EOM givesṫ = 1 1+r 2 The Virasoro constraint can then be written aṡ The canonical momentum conjugate to r is given by p r = R 2ṙ 1+r 2 . We can now proceed with semiclassical quantization, by the Bohr-Sommerfeld quantization condition The integral on the LHS is over the orbit in the (r, p r ) phase space. Note that as the pulsating string goes from shrinking to expanding at r = 0, the sign of p r flips. r 0 is the maximal value of r, whereṙ = 0.
The r-integral in (4.25) can be evaluated by changing the variable r ≡ r 0 sn(uτ, v), where , and sn(u, v) is the Jacobi elliptic sn function.
The resulting quantization condition is , v) , (4.26) where K, E, Π, am are complete elliptic integral of the first kind, elliptic integral of the second kind, incomplete elliptic integral, and Jacobi amplitude respectively. Expanding this relation in µ to order µ 2 leads to the following energy spectrum ∆ = ∆ 0 + δ∆, Note the divergence of δ∆ at n = k 4 , which corresponds to j ∼ k 2 in the semiclassical regime (cf (4.4)), where the short strings become long. This can be compared to the order µ 2 quantum results computed in the previous section. In the semiclassical limit, the anomalous weight δh is related to the correction to spacetime energy δ∆ in the following way (in the case j = 0 with no internal excitation) (4.28) With this identification, a numerical extrapolation of our quantum results to the k → ∞ limit is in accurate agreement with the semiclassical quantization, as shown in Figure 2. 6

The BMN limit
We can compare our results with the previously known spectrum of strings in pp-wave background, obtained from the Penrose/BMN limit of AdS 3 ×S 3 [27]. Writing the AdS 3 ×S 3 metric as R 2 (ds 2 AdS 3 + ds 2 S 3 ) =R 2 (− cosh 2 ρ dt 2 + dρ 2 + sinh 2 ρ dφ 2 1 + cos 2 θ dψ 2 + dθ 2 + sin 2 θ dφ 2 2 ), the Penrose limit is defined by zooming in near a null geodesic corresponding to a massless particle moving in ψ direction on the S 3 while staying at the center of AdS 3 . To this end we introduce coordinatesx ± = 1 √ 2 (t ± ψ), define the rescaled coordinates and take the limit R → ∞. The resulting background is that of the pp-wave geometry supported by three-form fluxes where we have redefined re iφ 1 = x 1 + ix 2 and ye iφ 2 = x 3 + ix 4 . The spectrum of strings in the pp-wave background is naturally described in terms of the lightcone momenta where ∆ and J ψ are the charges associated with the isometries generated by ∂ t and ∂ ψ . (4.33) is the same as (4.7) in the purely NSNS case. In the mixed flux case, we must keep in mind that the radius R is given by (4.2).
The full spectrum of type IIB strings in the above pp-wave background has been determined from the lightcone quantization of the Green-Schwarz effective string action [27,36], and recently reproduced in the hybrid formalism [10,40]. Here we briefly recap the results following the notation of [36]. In the lightcone gauge, the Green-Schwarz action of strings in the pp-wave background is that of free massive bosons and fermions on the worldsheet. Focusing on bosonic oscillator excitations transverse to the M 4 directions, the lightcone Hamiltonian can be written, up to a ground state energy, as the following where a i,± n and (a i,± n ) † are the annihilation and creation operators for a pair of complex bosons, and satisfy the commutation relations The oscillator energies ω ± n are given by the dispersion relation (4.36) In addition, the total momentum along the string is constrained to be zero (level matching condition).
It was shown in [36] that at the purely NSNS point, the SL(2, R) oscillators J ± −n ,J ± −n in the NSR formalism of the worldsheet theory correspond in the BMN limit to (a 1,± n ) † , (a 1,∓ −n ) † among the above described oscillators of the lightcone Green-Schwarz string. Therefore, the pulsating string states (4.3) correspond to, in the BMN limit, the Green-Schwarz string with n (a 1,− 1 ) † and n (a 1,+ −1 ) † excitations. This obeys the level matching condition. Note that, importantly, (a 1,− 1 ) † and (a 1,+ −1 ) † create oscillator modes of the same lightcone energy, ω − 1 = ω + −1 . After turning on RR flux, the spectrum of the pulsating strings in the pp-wave is determined by In the previous section, we have computed numerically the order µ 2 correction to the spectrum of the pulsating strings in AdS 3 × S 3 of finite radius, through the anomalous weight δh. The lightcone momentum (4.33) is expressed in terms of the quantum numbers j, j , n as The anomalous weight is related to the lightcone momenta p + , p − as follows kp + 1 − µ 2 2 + n + 1 (p − + n) = (n + δh)k. (4.39) The BMN limit amounts to sending k → ∞ while keeping γ ≡ j /k finite and n fixed. The mass-shell condition then sets j = j + 1 + n+δh 2γ + O(k −1 ). We perform the computation described in section 4.3.2 for δh at order µ 2 as a function of k and γ, for fixed n, and take the BMN limit by numerically extrapolating to k = ∞. The result as a function of γ fits accurately to as demonstrated in Figure 3. 7 This is precisely in agreement with the Green-Schwarz string spectrum (4.37).

Discussion
In a rough sense, Ramond-Ramond flux backgrounds can be treated perturbatively as a nonlocal but nonetheless conformally invariant deformation of the worldsheet CFT, within the Neveu-Schwarz-Ramond formalism [16,17]. We redefined such a prescription using the framework of closed superstring field theory. As a basic check of our formalism, we considered type IIB strings in pp-wave background supported by 5-form RR flux, viewed as a deformation of Minkowskian spacetime, and analyzed corrections to the dispersion relation of a family of spinning string states to second order in the RR flux. This is a rather delicate example as the deformation changes the asymptotics and thereby normalizability of the string wave functions. While our result agrees with a special case of the previously known string spectrum in pp-wave obtained from the Green-Schwarz formalism, various features of the string field solution illustrated through our derivation may provide useful guidance for analyzing, say, the spectrum of type IIB string theory in AdS 5 × S 5 viewed as a deformation of Minkowskian spacetime.
We then considered a more nontrivial example, namely type IIB strings in AdS 3 ×S 3 ×M 4 supported by mixed 3-form fluxes, viewed as an RR deformation of the purely NSNS background. The general structure of the perturbative string field solution in this case is in fact much simpler than the pp-wave case, as the RR deformation does not change the spacetime geometry qualitatively, although the detailed computation requires some SL(2, R) × SU (2) current algebra gymnastics. Our main result is the second order RR correction to the mass spectrum of the pulsating strings at finite AdS radius. To the best of our knowledge, this result has not been attainable in alternative formulations of superstring perturbation theory in RR backgrounds. Our result passes two consistency checks. In the limit of large AdS radius and large oscillation quantum number n, the mass corrections agree with the Bohr-Sommerfeld quantization of a pulsating string. In the BMN limit, our result is in agreement with the previously known spectrum of strings in the pp-wave background with mixed fluxes.
Let us briefly comment on other attempts of formulating superstring perturbation theory in RR backgrounds. The Green-Schwarz formalism [41][42][43] is based on a worldsheet effective action constrained by spacetime local super-Poincaré symmetry. Such an effective string action is subject to the ambiguity of deformation by higher dimensional operators [44,45], and the rule of quantization is a priori unclear except in simple backgrounds, such as in Minkowskian or pp-wave spacetimes in which the string action happens to be free in a suitable gauge. Nonetheless, substantial progress has been made in quantizing the GS action for strings in AdS 5 × S 5 [46][47][48][49][50][51][52]. Under the assumption that an appropriate quantization of the GS action for strings in AdS 3 × S 3 will respect the integrability observed of the classical string in this background, the spectrum of quantum strings (in particular in the case of mixed 3-form flux) has been analyzed in [53][54][55][56][57][58][59][60]. It would be very interesting to compare our results to those obtained from integrability.
The hybrid [10] and pure spinor [12] formalisms are based on the BRST framework as in the NSR formalism, while replacing the worldsheet CFT by ones defined through local actions that appear to accommodate RR flux backgrounds and respect spacetime symmetries. The main difficulty with these formalisms, in our opinion, is that the worldsheet ghost-matter coupling is not fully defined at the quantum level, which appears to be the main obstacle in its application to, for instance, strings in AdS at finite radius. 8 Note that in the type IIB pp-wave background supported by mixed 3-form fluxes, the ghost-matter coupling is absent, and indeed the string spectrum in this case has been recovered from the hybrid formalism in [40].
The investigation of the string spectrum in AdS 3 ×S 3 ×M 4 with mixed flux is particularly interesting due to the role of the latter in the AdS 3 /CF T 2 correspondence [8,62,63]. So far, for simplicity, we have restricted our attention to a special class of short string states, namely the pulsating strings (that may also be orbiting on the S 3 ). Another class of short string states of interest are those on the leading Regge trajectory. In the presence of NSNS H-flux, as was pointed out in [64], the folded spinning string solutions of [65] are unstable, and the leading Regge trajectory string states have more intricate classical limits. At the purely NSNS point, such states have been studied in [66]. It should be a straightforward extension of the computation performed in the present paper to analyze the leading RR corrections to all short string states starting from the purely NSNS background.
The purely NSNS background has the distinguishing feature that it admits long strings [33,67] that can be viewed as scattering states in global AdS 3 , whose energy spectrum forms a continuum above a gap. When RR fluxes are turned on, the long string spectrum is expected to become discrete. The simple perturbative treatment of the RR flux in this paper is not directly applicable in computing the mass spectrum of the now-discrete long string states.
In this paper we merely made use of the classical version of SFT, for the sake of a consistent treatment of off-shell deformation operators in the worldsheet theory. It is conceivable that, at the level of string spectrum and tree level string perturbation theory, the SFT framework is ultimately not necessary and one may be able to formulate the RR background deformation in terms of a nonlocal conformal worldsheet theory. The nonlocality under consideration here is in a mild sense; that is, they should be "local up to BRST exact terms" [16,17]. Other types of nonlocal deformations on the worldsheet have been considered previously in [68][69][70]. Nonetheless, from the point of view of effective string theory [44,45,[71][72][73], there is no a priori reason to expect the worldsheet theory of perturbative strings to be described by a local quantum field theory. It is perhaps no coincidence that holographic confining backgrounds for QCD-like strings generally involve RR fluxes.
Here t a β α = (t a ) β α and T a β α = T a β α are SL(2, R) and SU (2) matrices in fundamental representations. In the standard basis, they are given by t 0 ±∓ = 1 2 , t ± ±± = ∓i, and T 0 ±∓ = 1 2 , T ± ±± = ∓1. The fermionic OPEs lead to the following four point function where we have also used the relation We furthermore define (A.6)

B Current algebra Ward identities
In this section we derive some useful Ward identities that reduce correlation functions of SL(2, R) current algebra descendants to those of the current algebra primaries. We start by stating the commutation relations among SL(2, R) k+2 generators, A current algebra primary V sl obeys commutation relation of the form where t a i are the SL(2, R) generator matrices; the subscript i is meant to denote that they are acting on the current algebra primary at z i . Focusing on the holomorphic part, the basic correlators that will be needed for the computation of the 4-point function f j,j ,n in (4.15) are of the form Here the operator V sl j,j corresponds to the lowest weight state |j, j of D + j , which in particular obeys J − 0 |j, j = 0. We begin by evaluating the following chain of current algebra generators that will be useful in the computations that follow: where (a) n = Γ(a+n) Γ(a) is the Pochhammer symbol. In particular, f n,n = n!(k − 2j + 2) n will enter the normalization of the pulsating string vertex operators.
Next, we compute the correlator W (in intermediate steps of computations we will omit the fundamental indices on V sl −1/2 , with the understanding that the matrices t i act where we have used (J ± 0 ) 2 V sl −1/2,m = 0, and W(z 1 , z 2 ) is defined as a matrix acting on the holomorphic SL(2, R) spinor indices. Explicitly writing out the fundamental indices, we have The matrix acting on the anti-holomorphic spinor indices is analogously defined.
To compute X a , we use the contour representation of J a −1 at z 2 , and deform the contour around z 1 , giving (B.7) Using the lowest weight property of |j, j one finds that the first sum on the RHS reduces to the = 0 term, whereas the second sum reduces to = 0, 1 terms, giving (B.8) The first term, for instance, can be evaluated as where we have used j, j|V sl −1/2,m 1 (z 1 )V sl −1/2,m 2 (z 2 )(J − −1 ) n |j, j = 0 for n > 2. The remaining terms are handled similarly. In the end, we find where we again defined a matrix X a (z 1 , z 2 ) acting on the holomorphic spinor indices, and the anti-holomorphic analogue is similarly defined.
The relevant SU (2) k−2 correlator is much simpler to compute. The correlator of interest is One easily finds where now |j , j = V su j ,j (0)|0 and T a i are the SU (2) generator matrices acting on the current algebra primaries at z i .