The plane-wave limit of AdS 3 × S 3 × S 3 × S 1

: The plane-wave limit of AdS 3 × S 3 × S 3 × S 1 is analysed for generic null-geodesics that are not necessarily BPS. For the case of pure NS-NS ﬂux it is shown how the resulting spectrum can be reproduced as a suitable limit of the world-sheet description in terms of WZW models. Since supersymmetry is broken, most of the degeneracies are lifted, and thus the identiﬁcation of states is quite unambiguous.


JHEP08(2018)097
1 Introduction and conclusions Strings on AdS 3 have a rich dynamics. Even when restricting to the most supersymmetric backgrounds, which preserve 16 Killing spinors, the geometry is more varied than in higher dimensions. We have two geometries: AdS 3 × S 3 × T 4 and AdS 3 × S 3 × S 3 × S 1 . The latter is actually a one-parameter family of inequivalent geometries, which can be labelled by the ratio of the curvature radii of the two spheres. In the limit when one of the two spheres becomes flat, we find AdS 3 × S 3 × R 3 × S 1 which is closely related to AdS 3 × S 3 × T 4 in the sector without winding and momentum on the flat coordinates. All these geometries can be supported by a mixture of Ramond-Ramond (R-R) and Neveu-Schwarz-Neveu-Schwarz (NS-NS) three-form fluxes, see, e.g. [1] for a review. Some of these backgrounds are directly accessible from a world-sheet perspective. In particular, for pure NS-NS backgrounds, the AdS 3 factor can be described by a Wess-Zumino-Witten (WZW) model based on sl(2, R) [2][3][4][5][6][7][8][9] that is explicitly solvable. The situation for mixed (or pure R-R) flux is more complicated, but there exists the hybrid formalism of [10] that gives an explicit description away from the pure WZW point, see also [11][12][13][14][15][16][17][18] for further developments. On the other hand, the pure R-R and mixed-flux backgrounds are known to be classically integrable fields theories [19][20][21]. This raises the hope of describing their quantum spectrum by the techniques that proved remarkably successful for strings on AdS 5 × S 5 and AdS 4 × CP 3 , see refs. [22][23][24] for reviews. Indeed an integrable world-sheet S-matrix was found for pure R-R backgrounds on AdS 3 × S 3 × S 3 × S 1 [25] and AdS 3 × S 4 × T 4 [26,27], and later was generalized to mixed-flux backgrounds [28][29][30]. However, not all the dressing factors have been computed and a closed set of equations for the finite-size spectrum is lacking. 1 Some quantitative progress can be achieved by perturbatively quantizing the theory around a free point, i.e. by considering the string non-linear sigma model at large tension. It is well known that in the Penrose limit [33][34][35][36] we recover a free light-cone Hamiltonian which for these backgrounds was already discussed in refs. [19,36], see also [37][38][39]. We can also compute perturbatively the corrections to these free theories; this was done for the purpose of computing the world-sheet S matrix up to one [40,41] and two loops [42,43].
In this paper we study a family of inequivalent Penrose limits of AdS 3 × S 3 × S 3 × S 1 in detail. As we will explain below (and as was observed already in refs. [19,[37][38][39]), geodesics on AdS 3 ×S 3 ×S 3 are described by three parameters, one of which can be eliminated by the JHEP08(2018)097 null condition. Of the resulting two-parameter family, only those having the same angular momenta along the two three-sphere are supersymmetric [19]; 2 however, the whole two parameter family yields a consistent Penrose limit -though the spectrum of excitations around the "plane-wave" geometry does not need to be supersymmetric for generic choices of (non-BPS) geodesic.
Our main aim is to obtain this "plane-wave" spectrum for generic fluxes and geodesics. For pure NS-NS backgrounds we shall also compare our results with the WZW result, extending the analysis of [44] to these more general cases. In the Green-Schwarz description of the string spectrum, states are obtained by acting with free oscillators (subject only to the level-matching constraint). This is to be compared to the WZW approach in which we need different "spectrally flowed" sectors in order to produce the whole spectrum. While of course a matching is expected, it is nonetheless quite instructive to see how this comes about, and how the different "flowed sectors" are identified with states of the Green-Schwarz (GS) description.
Since our Penrose limits are generically taken around non-BPS geodesics, the resulting theories are typically non-supersymmetric. In particular, most of the degeneracies of the plane-wave spectrum are lifted, and the resulting spectrum is not just fixed by supersymmetry. In the most general case, our light-cone Hamiltonian depends on four parameters: two describe the original background (i.e. the relative curvature of the two three-spheres and the mixture of R-R/NS-NS fluxes) while two describe the geodesic chosen for the Penrose limit (i.e. how fast it is spinning and the relative speed on the two-three spheres). This wealth of parameters will allow for a precision-spectroscopy comparison of new proposals for computing the spectrum of strings on AdS 3 backgrounds, see e.g. [45,46]. This paper is organised as follows. In section 2 we describe the different Penrose limits of AdS 3 × S 3 × S 3 × S 1 with mixed background fluxes. In section 3 we derive the lightcone Hamiltonian for bosons and fermions in the Green-Schwarz formalism. After briefly reviewing the N = 1 WZW construction for this background in section 4, we expand it in the near-plane wave limit in section 5. Finally we compare these findings with the GS results in section 6. There are four appendices to which we have relegated a number of technical considerations.

Penrose limits of AdS
In this section we shall review how to obtain the plane-wave background from a Penrose limit [33][34][35][36] of AdS 3 geometries. We shall consider both the cases with Ramond-Ramond (R-R) and Neveu-Schwarz-Neveu-Schwarz (NS-NS) three-form fluxes. The resulting metric will take the form where x ± are light-cone coordinates, while A ij will be a constant matrix (that will depend on our choice of background fluxes and of the geodesic). JHEP08(2018)097 Let us begin by writing the metric of AdS 3 × S 3 × S 3 × S 1 as In order for string theory to be critical, 3 the radius R of AdS 3 and the radii of the two three-spheres R 1 , R 2 must satisfy It is convenient to parameterize this relation as We shall use the conventions of [30], in which the metric on AdS 3 and S 3 is given by where t, φ and ψ are isometric coordinates. This background solves the supergravity equations and in fact has 16 Killing spinors when it is supported by a mixture of R-R and NS-NS 3-form fluxes 4 and Here 0 ≤ q ≤ 1 describes the ratio between R-R and NS-NS flux, with q = 0 corresponding to pure R-R and q = 1 to pure NS-NS flux, respectively. The 16 Killing spinors close on the superisometry algebra d(2, 1; α) L ⊕ d(2, 1; α) R [47], where the labels L and R stand for left and right, and refer to the chirality in the dual CFT 2 . Each copy of d(2, 1; α) has as a bosonic subalgebra su(1, 1) ⊕ su(2) 1 ⊕ su(2) 2 , so that the global isometry is so(2, 2) ⊕ so(4) 1 ⊕ so(4) 2 . The d(2, 1; α) L algebra is characterized by the BPS bound [48,49] j ≥ cos 2 ϕ j 1 + sin 2 ϕ j 2 , where j is the lowest weight of su(1, 1) L , while j 1 and j 2 are the spins of su(2) 1L ⊕ su(2) 2L . A similar formula holds for d(2, 1; α) R , where the corresponding quantum numbers are (, 1 , 2 ). JHEP08(2018)097 As an aside we mention that in the limit ϕ → 0, equation (2.2) reduces to the metric of AdS 3 × S 3 × T 4 . Geometrically, this corresponds to one of the radii of the threespheres going to infinity so that we are left with four flat coordinates. From the point of view of representation theory, this is captured by a suitable contraction of the algebra d(2, 1; α) 2 into psu(1, 1|2) 2 for α → 0. Most of the formulae that we will derive below will admit a smooth α → 0 limit, and thus our results also apply to the plane-wave limit of AdS 3 × S 3 × T 4 .

Null geodesics
We are interested in studying null geodesics of AdS 3 × S 3 × S 3 × S 1 . The geodesic equation is solved by a constant motion along the isometric directions t, φ, ψ. Let where τ is the affine parameter. 5 We have also chosen the normalization so that ∆, J 1 and J 2 are the energy and angular momenta around the three-spheres, respectively; this follows from Noether's theorem, using the form of the Lagrangian The condition for this class of geodesics to be null is which is solved by where ω is a free parameter. 6 The BPS bound of d(2, 1; α) requires which holds true for any ω, In this equation we make explicit the string tension 1/(2πα ′ ); the dimensionful parameter α ′ should not be confused with the dimensionless geometric parameter α of d(2, 1; α). We will often suppress factors of 2πα ′ . 6 Since J1, J2, ∆ ≥ 0 we should always take the positive roots of cos 2 ω, cos 2 ϕ, etc.; to streamline our notation we drop the resulting absolute values and assume, without loss of generality, that cos ω cos ϕ ≥ 0 and sin ω sin ϕ ≥ 0.

JHEP08(2018)097
We note that among the one-parameter family of geodesics (2.13), there is only one which saturates the BPS bound. This happens when ω = ϕ [19], and the corresponding solutions preserve one quarter of the supersymmetry and have (2.16) Indeed all protected states of AdS 3 ×S 3 ×S 3 ×S 1 satisfy eq. (2.16) [50,51]. Because of this, the geodesic in eq. (2.13) at generic ω is a supergravity state that is typically not BPS.

The Penrose limit
We will now see how to obtain a plane-wave background (2.1) from the Penrose limit around the geodesics of section 2.2. We introduce light-cone coordinates adapted to the geodesic given by (2.10) and (2.13) by defining X ± and X 7 via We also rescale the transverse coordinates (except for the X 8 coordinate on S 1 ) by the radii of the respective spheres, i.e. we define and set Expanding for large R, the metric in (2.2) reduces to the form (2.1), i.e.
where A ij is the mass-squared matrix for the eight transverse coordinates, 7 Notice that we have two directions with A ii = 0, which will result in massless modes in the Green-Schwarz string formulation [19]. One is the flat direction corresponding to the S 1 factor, while the other is the linear combination of (φ, ψ) on S 3 × S 3 that is orthogonal to the light-cone geodesic. The 3-form fluxes (2.6) and (2.7) reduce to (2.24) 7 The superscript b in the masses µ b i indicates that these will be the masses of the bosonic string excitations. Later we will also introduce Green-Schwarz fermions with mass µ f i .

JHEP08(2018)097
In the following, x + and x − will play the role of light-cone coordinates. The conserved charges associated to the light-cone coordinates follow from the change of coordinates and eq. (2.13) 8 (2.26) The momentum P + plays a special role: it defines the light-cone Hamiltonian H, whose density H is given by On the other hand, using (2.13) and keeping track of the factors of 2πα ′ , the momentum P − reduces to which parameterizes how fast the string is spinning along the chosen geodesic, cf. eq. (2.10), where small (large) values of P − correspond to slowly (rapidly) spinning geodesics. Since we are interested in configurations with finite momenta, we will from now on assume that ∆, J 1 and J 2 scale as R 2 in the Penrose limit, but in such a way that H remains finite and P − is a free parameter. One can easily verify that this is consistent with (2.13); in fact, if J 1 , J 2 are as in (2.13), H = p + = 0, as p + is the momentum transverse to the geodesic. Note that the ground-state energy in light-cone gauge will receive quantum corrections, unless the ground state is BPS.

Penrose limit of AdS
Again, as an aside, we note that for AdS 3 × S 3 × T 4 , i.e. for ϕ → 0, the null condition (2.12) for the geodesics is simply that J 1 = ∆. The null geodesic then runs along AdS 3 × S 3 , and it is automatically BPS. In fact, it preserves one half, rather than one quarter, of the Killing spinors. This leads to the same metric as in eq. (2.1), where the masses in the matrix A ij now take the values

The Green-Schwarz action
In this section we will outline the computation of the Green-Schwarz action for strings moving in the plane-wave background described in section 2. We shall study the case of generic (mixed) flux, parametrised by 0 ≤ q ≤ 1, see the discussion below eq. (2.7).

The bosonic action
The bosonic part of the action is 9 where the determinant of the world-sheet metric γ αβ has been set to −1. Notice that for point-like strings, this action reduces to (2.11). To obtain the Hamiltonian it is convenient to work directly in first order formalism, introducing the conjugate momenta where we denote with a prime and a dot the derivative with respect to the world-sheet coordinates σ and τ , respectively. The action can be rewritten in the form [22,24,29,30] The Virasoro constraints amount to the equations of motion for the auxiliary fields γ αβ . For the bosonic action they are equivalent to setting C 1 = 0 and C 2 = 0.

The fermionic action
The Green-Schwarz action for AdS 3 backgrounds can be written explicitly up to quartic order in the fermions [52][53][54], and it reads where S B = L B has been defined in subsection 3.1, and Here we have only kept the terms that are quadratic in the fermions as the higher order terms will not be important in the plane-wave limit. The indices α, β ∈ {τ, σ} are worldsheet indices, while I, J, K ∈ {1, 2}. For the vielbeins E A µ we have while the fluxes are defined by More technical details about (3.6) and (3.7) can be found in appendix B. 9 Here we suppressed a factor of 1/(2πα ′ ) in front of the action.

Gauge fixing and kappa-symmetry fixing
The gamma matrices associated to the light-cone directions x + and x − are where our gamma matrix conventions are spelled out in appendix A. We note that, despite the involved form of the light-cone directions, Γ ± still satisfy the familiar identities We fix light-cone gauge and kappa symmetry by setting where κ and κ ′ are two positive constants. This is called uniform light-cone gauge because the momentum density p − along the string is constant [55][56][57]. We should mention that the choice of p − determines the size r of the world-sheet after gauge-fixing. In particular we have, in the Penrose limit, where we have reinstated the factor of 2πα ′ for future convenience. In what follows we find it convenient to set so that the world-sheet length equals As described in appendix C, eq. (3.5) simplifies drastically for the plane-wave background described in section 2. The resulting world-sheet Hamiltonian is quadratic in the fields, where and The parameter q labels again the ratio of background fluxes. We should mention that the fermionic masses µ f 1 , . . . , µ f 4 differ in general from the bosonic masses µ b 1 , . . . , µ b 4 defined in (2.22), and coincide only for BPS geodesics, i.e. for ω = φ.

Mode expansion
The eight real coordinates x 1 , . . . , x 8 enjoy an so(2) ⊕4 symmetry. The Hamiltonian (3.17) can be diagonalised by introducing complex coordinates for each so(2) doublet, yielding four complex bosons. We introduce for each complex boson a pair of creation and annihilation operators a i,± † n and a i,± n with i = 1, . . . , 4 and 10 Then the Hamiltonian takes the form where the terms in the second line arise from normal ordering. Here, µ b i are the bosonic masses defined in eq. (2.22), and the dispersion relation is We can think of the quantity p = 2πn r as the world-sheet momentum p, subject to the free-particle quantization condition exp(ipr) = 1; thus we can also write the dispersion relation as The fermionic Hamiltonian (3.18) can be diagonalised introducing creation and annihilation operators b i,± † n and b i,± n with i = 1, . . . , 4 and where the explicit mode expansion is again given in appendix C. Then eq. (3.18) reduces to where the dispersion relation is the same as the one defined in eq. (3.22), but the fermionic masses are now given by (3.19). 10 The explicit mode-expansion of the complex bosons can be found in appendix C.

JHEP08(2018)097
3.5 Dispersion relation and q → 1 limit The plane-wave dispersion relation (3.22) has the same form for bosons and fermions, even if for ω = ϕ the mass-spectrum of the excitations is not supersymmetric, µ b i = µ f i . For 0 ≤ q < 1 the dispersion we find is consistent with the one required by supersymmetry [25,29,30,58], which takes the form 11 where h is the string tension. In the near-plane-wave limit the momentum is small relative to the tension, and we can approximate 4h 2 sin 2 (p/2h) ∼ = p 2 , where p is the momentum from above, see eq. (3.23).
The case q = 1 is quite special since then the exact dispersion relation (3.26) simplifies to the linear expression, which becomes in the plane-wave limit The modulus signals that the world-sheet theory is chiral, i.e. it distinguishes between leftand right-movers on the world-sheet. A left-moving mode is one which has ∂ω ± /∂p > 0, which happens when n > ∓ µr 2π . To illustrate this point, let us consider the contribution of a complex bosonic mode of mass µ to the Hamiltonian. Suppressing the explicit µ dependence and setting ν ≡ µr 2π , we get for the Hamiltonian where in the last line we have changed variables toñ = −n and setã ± n ≡ a ∓ −n . Thus the complex boson of the GS description has split into two "left" (i.e. holomorphic) real bosons a ± , and two "right" (i.e. anti-holomorphic) real bosonsã ± . Furthermore, their creation operators are shifted by ±ν. This shift was already observed by Maldacena and Ooguri [3], and, as we will see below, is closely related to the spectral flow. Indeed, loosely speaking the spectral flow parameter w is the integer part of ν. Here we have tacitly assumed that ν = µr 2π / ∈ N; otherwise there are zero modes that require more care.

Ground-state energy
It is worth noticing that the normal ordering constant does not vanish; this is due to the fact that the bosonic and fermionic masses, see eqs. (2.22) and (3.19), are different. The resulting shift of the ground state energy is finite, as can be seen by observing that For pure NS-NS flux, i.e. q = 1, the summand of the first sum in eq. (3.29) is independent of n, and E 0 can be readily computed. In particular, if the masses µ i satisfy µ i < 2π r , we find the simple expression This will play an important role below for the comparison with the WZW spectrum.

The WZW description
String theory on AdS 3 × S 3 × S 3 × S 1 with pure NS-NS flux can be described by a WZW model based on the N = 1 affine algebras sl(2) k ⊕ sl(2) k 1 ⊕ su(2) k 2 ⊕ u(1) [49,51,61]; our conventions for the relevant algebras are described in appendix D. The levels k, k 1 , k 2 are all positive integers, and are proportional to the radius squared of the corresponding space, e.g. k α ′ = R 2 , and similarly for the 3-spheres. In terms of our previous parametrization, see eq. (2.4), we therefore have The fermions can be decoupled from the Kač-Moody currents as explained e.g. in ref. [7]. The levels of the resulting sl(2) and su(2) affine algebras are then shifted by +2 and −2, respectively. We will denote the (decoupled) holomorphic currents for sl(2) k by K a −n , and for the two su(2) k I algebras by J I,a −n , where I = 1, 2 and a ∈ {+, −, 3}. The corresponding fermions will be denoted by ψ a −r and χ I,a −r , respectively. We finally have an uncharged boson α n , together with an uncharged fermion η r , coming from the u(1) factor. The corresponding anti-holomorphic fields will be denoted by a tilde.
The fermions are either half-integer moded (NS sector) or integer moded (R sector). In the unflowed sector -there are also spectrally flowed sectors that will be explained in more detail below -the representations of the affine algebras are conventional (Virasoro) highest weight representations generated from a ground state |j 0 ; j 0,1 ; j 0,2 . Here j 0 labels JHEP08(2018)097 a lowest weight state of sl(2), while j 01 and j 02 are the spins with respect to the two su(2) algebras, i.e. K a n |j 0 ; j 0,1 ; j 0,2 = 0 , J I,a n |j 0 ; j 0,1 ; j 0,2 = 0 , n > 0 , ψ a r |j 0 ; j 0,1 ; j 0,2 = 0 , χ I,a r |j 0 ; j 0,1 ; j 0,2 = 0 , r > 0 , (4.2) The ground state representation is the same for left-and right-movers, and j 0 satisfies the Maldacena-Ooguri bound [3] while unitarity requires for the su(2) spins The level matching condition is N eff =Ñ eff , where 5) and N is the total number operator. The spacetime energy and angular momenta are given by ∆ = j + , where j is the sl(2) lowest weight of the relevant state, and similarly for the su(2) spins. These quantum numbers differ in general from those of the ground states, and we define Thus δj counts the number of sl(2) modes with a = + minus those with a = −, while δj I counts the number of su(2) I currents with a = − minus those with a = +. In the R sector, s, s 1 , s 2 ∈ {0, 1} label the different ground states under the action of the fermionic zero-modes, see e.g. [7]. Physical states are annihilated by the positive super Virasoro modes, and obey the mass-shell condition Finally, the GSO projection requires that we have an odd number of fermionic excitations modes in the NS sector. (In the R sector, the GSO projection depends on whether we consider type IIA or type IIB string theory, but as we will see below, this does not affect the spacetime charges of the states.)

Low-lying states
In order to illustrate the construction of the spectrum, let us review briefly the structure of some of the low-lying states in the unflowed sector (including the BPS states).

NS sector
The GSO projection requires an odd number of fermions. Since they are half-integer moded, this is equivalent to N eff ∈ N. The lowest lying states occur at level N eff = 0. We can solve the mass-shell condition (4.8) for j 0 as a function of j 0,1 and j 0,2 , obeying (4.4). For each solution in the interval (4.3), we then have a physical state. The physical states also have to be annihilated by the positive super Virasoro modes, and this leads to the following eight excitations

R sector
In the R sector the different fermionic ground states are parameterized by the quantum numbers s, s 1 , s 2 , s 3 ∈ {0, 1}, see [7] for a careful discussion. Here s arises from AdS 3 , while s 1,2 are associated with S 3 1,2 , and s 3 is the quantum number associated to the fermionic zero modes η 0 andη 0 . The value of s 3 does not have an effect on the spacetime charges (4.6), and hence we can always adjust s 3 ∈ {0, 1} so as to satisfy the GSO projection. At level N eff = 0 we then find eight states, corresponding to the eight possible choices of (s, s 1 , s 2 ). We can label them as |j 0 , j 0,1 , j 0,2 (s,s 1 ,s 2 ) . (4.10) The state |j 0 ; j 0 − 1; j 0 − 1 (0,0,0) has j = j 1 = j 2 and is the only BPS state arising in the unflowed R sector [51].

Spectral flow
Because of the Maldacena-Ooguri bound (4.3), the spacetime spectrum arising from the unflowed sector leads to states with a finite maximal energy. In order to overcome this unphysical feature, it was argued in [3] that the spectrum must also contain the spectrally flowed images of these representations. 12 Spectral flow is an outer automorphism of the affine algebra sl(2) k , see appendix D for the explicit description, and hence relates in general inequivalent representations to one another. For su(2) k one can also define a spectral flow automorphism, but this does not lead to new representations. However, it is sometimes convenient to flow also in the su(2) k I factors, as we shall do below. Since spectral flow shifts the eigenvalues of the Cartan generators, see eq. (D.3), we have now instead of (4.7) j = kw 2 + j 0 + δj , j I = k I w I 2 + j 0,I − δj I NS sector j = kw 2 + j 0 + δj + s − 1 2 , j I = k I w I 2 + j 0,I − δj I − s I + 1 2 R sector. (4.11) 12 The significance of spectral flow was first noticed in [62] based on modular invariance arguments.

JHEP08(2018)097
The spectral flow also affects the eigenvalue of L 0 , see eq. (D. 3), and hence the mass-shell condition, eq. (4.8), becomes in the spectrally flowed NS sector and in the spectrally flowed R sector Furthermore, the level matching condition now becomes in the NS sector where N eff andÑ eff are the number operators before spectral flow. Similarly in the R sector we have In the R sector the GSO projection can still be satisfied using (if necessary) the action of the fermionic zero modes η 0 andη 0 , while in the NS sector it reads [7,63] N eff + w + w 1 + w 2 2 ∈ N . The construction of the BPS states in the spectrally flowed sectors is quite involved for AdS 3 × S 3 × S 3 × S 1 , and we refer the reader to [51,64] for a detailed explanation.

Plane-wave limit of the WZW spectrum
In this subsection we study the plane-wave limit of the WZW model for AdS 3 ×S 3 ×S 3 ×S 1 , generalising the analysis of [44] for AdS 3 ×S 3 ×T 4 . We will consider the limit in which both the levels k, k 1 , k 2 are large (so that we are in the supergravity regime), as well as the charges ∆, J 1 , J 2 (as dictated by the Penrose limit, see section 2.3). Note that it follows from (2.13) that all these charges scale then as k ∼ R 2 .
Since we are interested in states that are obtained from the ground states by a finite number of creation operators, the actual eigenvalues of the states differ only by a finite (and hence small) amount from those of the ground states. Since the ground states have the same spin for left-and right-movers -this is true both for the sl(2, R) lowest weight, as well as for the two su(2) spins -all spins scale the same way for left-and right-movers, i.e. j ∼ and j I ∼ I ; this is also required by the form of the geodesic (2.10).

JHEP08(2018)097
In the Penrose limit, i.e. up to subleading corrections in 1/k, it follows from eq. (2.28) that Note that the Maldacena-Ooguri bound, eq. (4.3), implies that P − ≤ 2π, but as mentioned above, the spectrum also contains spectrally flowed sectors that effectively shift up the spin. Indeed, in the sector with spectral flow w we have 2πw r 2π(w + 1) .
Note that the different 'sectors' are also visible from the viewpoint of the plane-wave dispersion relation (3.22): at q = 1, i.e. for pure NS-NS flux, it can vanish when r = P − = 0 mod 2π. As P − crosses such a value, it is necessary to adjust the definition of positiveand negative-energy modes in the light-cone Hamiltonian. As a consequence, the spectrum is composed of different sectors depending on "how fast" the geodesic is spinning, cf. section 3.5. It is therefore convenient to discuss the WZW spectrum separately for the case where the geodesic is slowly spinning, so that w = 0 and we can use the formulae for the "unflowed" spectrum; and the case where the geodesic is rapidly spinning, where we will need the more general formulae of section 4.2.

Slowly spinning geodesics (unflowed sector)
In the following we shall analyse the spacetime spectrum of string theory for large level k, as well as for large spins j, j I . 13 Since we are interested in the Penrose limit of a specific null geodesic, we have in addition, see eq. (2.13), so that j 2 1 cos 2 ϕ + j 2 2 sin 2 ϕ ∼ j 2 , j 1 cos ϕ cos ω + j 2 sin ϕ sin ω ∼ j . (5.4) This will further simplify our expressions.
For large k and large charges j, j 1 , j 2 , the leading term of the square root in (5.6) is Taking this combination out of the square root and expanding the remaining terms to leading order, we then obtain, where we have written k N eff 2j = 2πN eff r , see eq. (5.1). Essentially the same formula also holds for the anti-holomorphic contribution H R . Note that since δj and δj I can take both positive and negative values, one may worry that the Hamiltonian (5.9) is not bounded from below. However this is not the case, as can be checked directly; in fact, even before taking the Penrose limit, the non-linear WZW energy is also bounded from below [65], and this therefore also remains true in the Penrose limit.

Rapidly-spinning geodesics (flowed sector)
Fast spinning solutions are characterised by the condition that j > k 2 , so that they come from sectors with spectral flow w > 0, see eq. (5.2). It is convenient to spectrally flow also in the two su(2) sectors, where the relevant spectral flows are determined by 2πw 1 cos ϕ cos ω r 2π(w 1 + 1) , 2πw 2 sin ϕ sin ω r 2π(w 2 + 1) . (5.12) In particular, this guarantees that δj I is of order one, and hence we can proceed very similarly as in the unflowed sector. We shall again first deal with the R sector. JHEP08(2018)097

R sector
Using eq. (4.11), the mass-shell condition (4.13) in the spectrally flowed R sector can be rewritten in terms of the "true spins" j, j I as which can be solved for j to give j = δj + s + (j 1 + δj 1 + s 1 ) 2 cos 2 ϕ + (j 2 + δj 2 + s 2 ) 2 sin 2 ϕ 14) The leading terms in the large k and large charge limit are the quadratic charge terms in the first line of the square root. Proceeding as in the unflowed sector we therefore find in the Penrose limit Again one can convince oneself that the Hamiltonian is bounded from below, as must be the case.

NS sector
In the NS flowed sector the analysis works completely similarly. Using now (4.11) we find from the spectrally flowed mass-shell condition in the NS sector j = δj + 1 2 + j 1 + δj 1 + 1 2 2 cos 2 ϕ + j 2 + δj 2 + 1 2 2 sin 2 ϕ The holomorphic contribution to the Hamiltonian in the Penrose limit then becomes 6 Comparison of the spectra at q = 1 Now we have everything at our disposal to compare the Hamiltonian in the Penrose limit of the WZW spectrum that we have just derived, see eqs. (5.9), (5.11), (5.15), and (5.17), with the one that was obtained from the GS formulation at q = 1 in section 3, see eqs. (3.21) and (3.25). As we shall see, this is quite straightforward for the bosonic excitations, but requires a little more work for states that involve fermions. We shall first consider the states at the lowest level, N eff = 0, before discussing systematically the general case. The GS Hamiltonian H of eqs. (3.21) and (3.25), splits naturally into a holomorphic and an anti-holomorphic part and it is hence sufficient to compare (and match) the spectrum for the holomorphic (and the anti-holomorphic) part separately. For r < 2π (corresponding to the unflowed sector in the WZW model) the holomorphic part of the GS Hamiltonian at q = 1 can be written as The anti-holomorphic part has essentially the same form, except that the sums run instead over n < 0, and that the zero mode part only involves the a i,− 0 and b i,− 0 generators (instead of a i,+ 0 and b i,+ 0 ). The bosonic oscillators can now be identified straightforwardly, and for the holomorphic sector we have written out the details in table 1; a similar identification also applies to the anti-holomorphic generators.

The identification of the fermions
The identification for the fermions is a bit more involved, reflecting the usual difficulty in relating the NS-R formulation (as in the WZW model) to the GS formulation. In order to understand how things work out, it is instructive to describe first the lowest lying states systematically. They arise from the unflowed sector of the WZW model at level N eff = 0.  Table 2. Low-lying states in the R sector. In the left column we describe the states in terms of the fermionic zero-modes in the WZW language. In the second column, we describe them in term of GS fermionic oscillators; note that R sector states correspond to an odd number of GS fermions.

JHEP08(2018)097
In the remaining two columns we write the contribution to the (holomorphic) Hamiltonian and to the partition function that is described below.
6.1.1 Level N eff = 0 WZW states at level N eff = 0 correspond to GS states built acting only with fermionic zeromodes. More precisely the NS sector states will be identified with the GS states involving an even number of GS zero modes, while the R sector states will come from an odd number of GS zero-modes. More specifically, in the R sector at level N eff = 0 the WZW Hamiltonian in eq. (5.9) simplifies to H L = s + s 1 cos ϕ cos ω + s 2 sin ϕ sin ω . Here s, s 1 and s 2 take the values {0, 1}, and the resulting eight states have been tabulated in table 2, together with the GS states to which they correspond.

The general case
At higher level, an explicit identification as above becomes cumbersome, but we can demonstrate that the spectra match by considering the generating function (with suitable chemical potentials). More specifically, let us define in the WZW model  Table 3. Low-lying states in the NS sector. In the left column we describe the states in terms of the fermionic modes in the WZW language. In the second column, we describe them in terms of the GS fermionic zero-modes; note that NS sector states correspond to an even number of GS fermions.
In the remaining two columns we write the contribution to the (holomorphic) Hamiltonian and to the partition function described below.
where d(N, j, j 1 , j 2 ) is the multiplicity of states with N = N eff and spins j, j 1 and j 2 ; as we are interested in the plane-wave limit, it makes sense to subtract out the spins of the ground states, i.e. j 0 and j 0,I . In the following we shall only keep track of the states that are generated by the fermions; the contribution of the bosons matches directly, see table 1.
In the NS sector we then find where the difference of the two terms implements the GSO projection, while the R sector contribution is It follows from the abstruse identity, see e.g. ( [66], Chapter 21), that

JHEP08(2018)097
where (6.10) In order to compare the WZW spectrum with that obtained from the GS formalism we now expand the WZW spectrum according to its H L eigenvalues, i.e. we consider where d(H) is now the multiplicity of states with H L = H; here we have used the form of the light-cone Hamiltonian in the unflowed sector, see eq. (5.9) and (5.11). It now follows from the specific form of the bosonic and fermionic mass eigenvalues, see eqs. (2.22) and (3.19), that with these chemical potentials, the z i parameters defined in (6.11) equal Thus the right-hand-side of the abstruse identity (6.8) gives directly the GS spectrum. This proves the matching of the spectrum in the unflowed sector. Since the light-cone Hamiltonian of the flowed sectors have essentially the same form as in the unflowed sector, cf. eqs. (5.15) and (5.9) or eqs. (5.17) and (5.11), the argument goes through also for the flowed sectors.

A Conventions
For the indices we adopt the following notation. The world-sheet coordinates σ and τ are denoted by α, β; we reserve µ, ν, ρ, . . . for spacetime coordinate and capital letters A, B, . . . for tangent space coordinates. Finally, I, J, K ∈ {1, 2} label rows and columns of Pauli matrices.
Essentially following the conventions of [30], we define the three-dimensional gamma matrices for AdS 3 and S 3 as
For light-cone tangent space coordinates we have

B Details of the Green-Schwarz action
The Kalb-Ramond field is given by The fermionsθ I are defined asθ where θ I are 32-components spinors andq = 1 − q 2 . Following once more the conventions of [30], the conjugation of spinors is defined as Moreover, Finally, the covariant derivative is defined as where ω is the spin connection.

C.1 Bosonic Hamiltonian
Exploiting (2.27) the explicit form of the light-cone bosonic Hamiltonian can be found by solving the quadratic equation C 2 = 0, where C 2 has been defined in (3.4). In the chosen gauge-fixing (3.14), for the plane-wave background we find where we have used the short-hand notation ω ± n = ω ± n (µ b i ). The fields x i obey periodic boundary conditions x i (τ, σ) = x i (τ, σ + r) (C.11) and canonical commutation relations that are in turn equivalent to (3.20).

JHEP08(2018)097
In each case one can define decoupled bosonic currents commuting with the fermions. These decoupled currents then have levels k and k ′ that are shifted by +2 and −2, respectively, see [7] for a detailed description of this construction. The N = 1 supersymmetric affine algebra is in fact N = 1 superconformal since the generators of the N = 1 superconformal algebra can be constructed out of the currents and fermions. (For the stress-energy tensor this is just the Sugawara construction applied to the decoupled bosonic currents together with the usual free field formula for the fermions, see e.g. [7] for a detailed exposition.) For any w, w ′ ∈ Z, the spectral flow automorphism of sl (2) (1) k and su (2) (1) k ′ is defined by replacing the original unhatted generators by the hatted expressions defined bŷ K ± n = K ± n∓w ,Ĵ ± n = J ± n±w ′ , K 3 n = K 3 n + k 2 w δ n,0 ,Ĵ 3 n = J 3 n + k ′ 2 w ′ δ n,0 , L sl n = L sl n − w K 3 n − k 4 w 2 δ n,0 ,L su n = L su n + w ′ J 3 n + k ′ 4 w ′ 2 δ n,0 , ψ 3 r = ψ 3 r ,χ 3 r = χ 3 r , ψ ± r = ψ ± r∓w ,χ ± r = χ ± r±w ′ .

(D.3)
It is straightforward to verify that this map defines indeed an automorphism of the two superconformal affine algebras, i.e. that the hatted generators satisfy the same commutation and anti-commutation relations as the original generators.
Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.