A Penrose limit for type IIB $AdS_6$ solutions

In this paper a Penrose limit is constructed for type IIB $AdS_6\times S^2\times \Sigma$ supergravity solutions. These solutions are dual to five dimensional SCFTs related to (p,q) five brane webs, which can often be described in terms of long quiver gauge theories. The null geodesic from which the Penrose limit is constructed is localized at a unique point on the two dimensional Riemann surface $\Sigma$, where the $AdS_6$ and $S^2$ metric factors are extremal. The resulting pp-wave spacetime takes a universal form. The world sheet action of the Green-Schwarz string is quadratic in the light cone gauge and the spectrum of string excitations is obtained.


Introduction
The quantization of superstrings in the presence of Ramond-Ramond fields is a technically challenging problem and not solved in general. This constitutes a challenge to go beyond the semi-classical supergravity approximation in many examples of AdS/CFT. Instead of solving the general problem one approach is to consider limits or deformations of the supergravity solution which may lead to a simpler quantization problem. One such limit is given by taking the Penrose limit of an AdS p × S q background. The Penrose limit [1] (see also [2,3,4]) corresponds to zooming in on the close vicinity of a null geodesic in the spacetime and produces a plane wave geometry. In the case of the Penrose limit of the type IIB AdS 5 × S 5 solution, where the null geodesic sits at the center of AdS 5 and on a great circle of S 5 , one obtains a maximally supersymmetric plane wave [5,6,7]. It was proposed in [8] that on the field theory side this limit corresponds to singling out a special subset of CFT operators where both the conformal dimension ∆ and a U (1) R-charge J are taken to be of order √ N , with the difference ∆ − J finite, as N → ∞. One important feature of this type IIB plane wave background is that the Green-Schwarz string can be quantized exactly [9,10,11]. The world sheet theory in the light cone gauge corresponds to the action of eight massive bosons and fermions. Furthermore the machinery of light-cone string field theory can be used to calculate string interactions and compare the results to field theory [12,13,14,15]. For an incomplete list discussing Penrose limits of other supergravity backgrounds see [16,17,18,19,20].
The aim of this paper is to study the Penrose limit for a different AdS solution, namely the type IIB solutions found in [21,22], which are realized as warped products of AdS 6 × S 2 over a Riemann surface Σ with boundary. These solutions are supported by R-R and NS-NS three form fluxes. Indeed the data characterizing the solution can be identified with the (p, q) five brane charges of semi-infinite five branes forming a five brane web.
Unlike the type IIB AdS 5 × S 5 solution or the AdS 4|7 × S 7|4 solution of M-theory these backgrounds are warped product geometries and preserve only sixteen of the thirty-two supersymmetries of ten dimensional type IIB supergravity. This fact makes finding a suitable Penrose limit more challenging, since the radii of the AdS 6 and S 2 vary with the location on the Riemann surface Σ and therefore a general null geodesic will also have a nontrivial dependence on the Riemann surface. There is a special point on the Riemann surface, namely the critical point of a function G. At this point the metric factors of the AdS 6 and S 2 are extremized with respect to the coordinates on Σ. We choose the null-geodesic on which the Penrose limit is based to be localized at this critical point on Σ.
The structure of the paper is as follows: In section 2 we review the type IIB supergravity solutions first found in [21,22] which we use in the rest of the paper. In section 3 we define the null geodesic on which the Penrose limit is based and present the resulting type IIB plane wave background, including all the other bosonic supergravity fields. In section 4 we discuss the quantization of the Green-Schwarz superstring action in the light cone gauge for this background. We discuss our results and possible directions for further research in section 5.
Some detailed calculations and supplementary materials are relegated to appendices.

Type IIB AdS 6 solutions
Solutions of superstring theories with AdS 6 factors are candidates for holographic duals of five dimensional superconformal field theories (SCFTs). The AdS/CFT correspondence is an important tool in studying these theories. Since the unique superconformal algebra F (4) with an SO(2, 5) factor has sixteen fermionic generators [23,24,25], the supergravity background preserves only 16 of the 32 supersymmetries of maximally symmetric type II or M-theory vacua. Furthermore the solutions are all realized as warped products of the AdS 6 over a base space. The first such solution was obtained in massive type IIA [26]. In this paper we will focus on type IIB solutions first constructed in [21,22], which is realized as a warped product of AdS 6 × S 2 over a two dimensional Riemann surface Σ with boundary.
where w is a complex coordinate on Σ and ds 2 AdS 6 and ds 2 S 2 are the line elements for unitradius AdS 6 and S 2 , respectively 1 . The AdS 6 solutions are defined in terms of locally holomorphic functions A ± on a Riemann surface Σ. The metric functions read and they are expressed in terms of the holomorphic functions A ± as follows In order to obtain regular and geodesically complete solutions one has to impose that G vanishes along the boundary of the Riemann surface, which implies that the S 2 shrinks to zero size and the spacetime closes off. In [22] a large class of regular solutions were constructed by choosing Σ to be the upper half plane and the holomorphic functions such that ∂ w A ± have L simple poles localized on the boundary which is the real line.
The regularity condition translates into L conditions [22] A 0 The other non vanishing type IIB supergravity fields of the solution are the complex scalar B which is related to the axion dilaton τ = χ + ie −φ via B = (1 + iτ )/(1 − iτ ) and the complex two form antisymmetric tensor potential C (2) (2.6) As discussed in [22] the residues of A ± can be identified with the charges of the (p, q) five brane web which realizes the dual SCFT via

Explicit examples
For definiteness we consider two explicit examples of the regular supergravity solution for which the dual SCFT are well studied, namely the T N theory and the + M,N theory where the holomorphic functions A ± are given by The T N solution has three poles located at w = ±1, 0 whereas the + M,N has four poles located at w = 0, 1 2 , 2 3 , 1. The relevant brane webs for the two theories are given by the junction of N D5, N NS5 and N (1,1) 5-branes for the T N theory and the intersection of N D5-branes and M NS5-branes for the + M,N theory. The quiver theories which at their conformal fixed points realize the SCFTs are given by long linear quivers with SU (n) gauge nodes and bi-fundamental matter connecting them and a fundamental matter node at the end. For the T N theories we have a linear quiver with SU (k) nodes with increasing k along the quiver and fundamental representations attached to the end of the quiver (2.9) For the + M N theory we have M − 1 SU (N ) nodes with matter in the fundamental representation of the end nodes attached at the end of the quiver.
For the two example cases the function G defined in (2.3) takes the following compact form where D is the Bloch-Wigner function given by where Li 2 is the dilogarithm function.

Penrose limit and the plane wave background
The idea behind taking a Penrose limit of AdS 5 × S 5 solution of type IIB is to consider the trajectory of a particle sitting at the center of AdS 5 and moving very fast along a great circle on S 5 , such that in the limit the trajectory becomes a null geodesic. Zooming in on the vicinity of the null geodesic produces a pp-wave solution which preserves 32 supercharges. For the AdS 6 × S 2 solutions presented in section 2, the geometry is more complicated since the AdS 6 and S 2 are warped over the two dimensional surface Σ and in general a nice Penrose limit does not exist for geodesics through a generic point on Σ. Furthermore as discussed in appendix B considering a geodesic with a nontrivial dependence on Σ looks daunting due to the complicated form of the geodesic equation.
Fortunately, we can find a special point on the Riemann surface Σ by considering critical points of the function G defined in (2.3), i.e w c ∈ Σ for which It was observed in [27] that in all known examples, G has a unique critical point on the Riemann surface Σ. For our two examples the critical point on w c on Σ is located at In [27] it was shown that the critical point w c correspond to the location of BPS probe D3-branes that realize surface defects in the dual SCFT. As we shall see in the following, the existence of the critical point is essential for our construction of Penrose limit in the warped spacetime.
We can examine our solution near the critical point by expanding w = w c + εζ. The most important expression, G, can be expanded in a power series in ε Using (2.3) we find the following expansions for the other functions which appear in the metric factors Using the expressions for the metric factors (2.2) it is straightforward to show that the metric factors of the AdS 6 and S 2 are extremized at w = w c . For the T N and + N,M examples we find the following expressions for the expansion coefficients G 0 and G 2 where C is Catalan's constant.

Penrose limit
Using the expansions (3.3) and (3.4) the Einstein frame metric (2.1) can be expanded to second order which corresponds to the metric close to the critical point w = w c .
We can drop the expansion parameter ε in the following since it can be absorbed into a redefinition of the coordinate ζ. The null geodesic which gives the proper Penrose limit is defined by Note that ζ = 0 implies that the null geodesic is localized at the critical point w = w c on the Riemann surface. We express the metric in terms of rescaled and relabelled coordinates The Penrose limit is obtained by taking G 0 → ∞ , which corresponds to taking N → ∞ (for the T N case) and M N → ∞ (for the + N,M example) and keeping the finite terms in the metric (3.6). All expressions involving the ratio G 2 /G 0 are finite in this limit. Note that the curvature radius of the AdS 6 and S 2 factors are controlled by G 0 and hence the limit of large G 0 is precisely the one taken for a Penrose limit in AdS p × S q spacetimes, i.e. zooming in on a region close to the null geodesic, which is much smaller than the curvature radius of the spacetime.
The resulting metric is given by The only difference between this metric and the one of the Penrose limit of AdS 5 × S 5 that the coordinate x 6 is singled out and has a different normalization in the (dx + ) 2 term in the metric, which in the world sheet action leads to a different mass term for the field associated with x 6 .
The energy in global AdS is associated with i∂ t whereas −i∂ φ is related to a U (1) generator of the SU (2) R symmetry, which is realized as the group of isometries of the S 2 . Hence using the argument of [8] for the Penrose limit (3.8) we can identify the light cone momentum with the conformal dimension and the R-charge of a dual operator via where we have identified the AdS radius with R 2 = √ 6G 0 . Since we take R to be very large in the Penrose limit excitations with finite p ± correspond to states with large energy and angular momentum which on the CFT side correspond to operators with large conformal dimension ∆ and R-charge J, which are close to the BPS bound ∆ = 3J [28,29].

Other supergravity fields
In the Penrose limit, the complex scalar field B of type IIB supergravity becomes a constant Note that the standard RR axion χ and the dilaton φ is related to the complex scalar B via Consequently, the axion dilaton has the following expression in terms of the holomorphic functions at the critical point w c For the two examples we consider in this paper one finds 2 14) The complex rank two antisymmetric tensor potential C (2) is related to the NS-NS and R-R two form potentials by Note that the constant term c 0 is not important since in the bosonic part of the worldsheet action, a constant NS-NS anti-symmetric tensor potential will be a total derivative. In the fermionic worldsheet action only the field strength, which does not depend on c 0 , appears. In the Penrose limit the finite part which survives (after using the rescaled variables) Note that in the scaling limit which produces the Penrose limit discussed below (3.8) the ∂A ± and |G 2 | scale in the same way and hence the term above is the expression which survives the Penrose limit. The NS-NS 3 form field strength H 3 N S and the RR field strength F 3 RR are given by For the examples we get 3.18) and It is important to mention that the supergravity solutions of section 2 are given in the Einstein frame. In order to quantize the Green-Schwarz string in this background we have to transform to the string frame Note that the particular combinations of the anti-symmetric tensor field strength which appear in the string frame supersymmetry transformations as well as the Green-Schwarz action are H In the Penrose limit, the transformation to the Einstein frame is just a constant rescaling of the metric. However, it is convenient for the string frame metric in to be the canonical plane wave form (3.9). This can be accomplished by a rescaling of all coordinates (except x + ) by a factor of e −φ/4 . This has the effect of introducing a factor of e −φ/2 in each of the antisymmetric tensor fields. The end result is the following combination of fields which appears in the string worldsheet: We will drop "string" from the subscripts moving forward since we will always be working in the string frame and there will be no confusion.
N S ) tensors are orthogonal to each other and square to four. This means that we could rotate x 7,8 so that the fields which appear in the worldsheet action take exactly the same form in both T N and + M N theories and therefore give the same string spectrum. In fact, this statement also holds for all other solutions [30,31] that we have checked. We conjecture that all global solutions (2.1) -(2.6) share this property and in the quantization the follows, take our fields to have the form of the T N example (3.21).

Light cone Green-Schwarz string spectrum
In this section we quantize the Green-Schwarz string the pp-wave spacetime obtained in the previous section. As shown in [10,9,11,32] the fermionic part of the Green-Schwarz string becomes quadratic in fermionic fields for the pp-wave spacetimes in the light cone gauge, which makes the free string spectrum exactly solvable.

Bosonic Spectrum
We will start by examining the bosonic spectrum for the Green-Schwarz string in the plane wave background with NS-NS flux. The general action is given by As in the AdS 5 × S 5 case, this action simplifies considerably after light cone gauge-fixing, leaving us with a free theory which can be easily quantized following [33]. We set the target space coordinate x + = τ and worldsheet metric to h στ = 0, h σσ = −h τ τ = 1 and after plugging in the background (3.9), (3.21), we are left with the following action for the eight remaining transverse scalar fields where m 2 I = 1 for all I = 6 in which case m 2 6 = 9. The equations of motion deduced from this action are given by Together with the periodicity condition X I (σ + 2πα p + , τ ) = X I (σ, τ ), the first set of equations where the frequencies will determine the energy of the string excitations ω I n = n 2 + (α p + ) 2 (4.7) To solve the remaining two equations of motion (4.4) and (4.5) for I = 6, 7 we use an ansatz of the following form where A I n andÃ I n for I = 6, 7 can be expressed in terms of eigenmodes For the zero mode A 0 there only one eigenfunction which can be obtained from A I n in (4.9) by setting n = 0. The eigenfrequencies for J = 6, 7 and (V n ) I J , J = 6, 7 are the orthonormal set of vectors which satisfy With this mode equation the the canonical quantization conditions where p I = 1 2πα ∂ τ x I , yield the commutation relation for the creation and annihilation operators where now we allow the indices I, J to run through all eight transverse oscillators 1, 2, · · · , 8. The light-cone Hamiltonian can be expressed in terms of the creation and annihilation operators (4.6) and (4.8) ω I n α I † n α I n +α I † nα I n + ν bos (4.14) Here ν bos denotes a normal ordering constant for the bosonic creation and annihilation operators. The zero mode modes x I 0 , p I 0 , I = 1, 2, · · · , 5, 8 have been expressed in terms of creation and annihilation operatorsα to maintain consistent normalization. It will be convenient to write ν bos in terms of the oscillator frequencies:

Fermionic Spectrum
We can now turn our attention to the fermionic part of the Green-Schwarz action for type IIB strings, the term quadratic in the world sheet fermions is given by (see appendix A for details on the notation).

S
(2) This Green-Schwarz action possesses a κ symmetry which is required to obtain spacetime supersymmetry for the on-shell string modes [9]. To obtain these physical fermionic modes, we can gauge fix (as in flat space) by choosing . After enforcing the light cone gauge condition x + = τ and worldsheet metric to h στ = 0, h σσ = −h τ τ = 1, the fermionic part of the Green-Schwarz action reduces to the quadratic one in the plane wave background [9,10,11] and takes the following form after inserting the antisymmetric tensor fields (3.21) for the T N theory, the action becomes where we have used the fact that Γ + Γ + = 1 + Γ 09 and hence (Γ + Γ + ) 2 = 2Γ + Γ + . The chirality condition Γ 11 θ a = θ a together with the light cone gauge condition (4.18) projects a general thirty-two component spinor onto an eight dimensional subspace. It is possible to find a representation of the Gamma matrices in this subspace and combine them with the σ matrices encoding the two chiral spinors such that the action can be written as Due to the common 1 4 factor in all the terms it follows that there is a fourfold degeneracy and equation of motion for each degenerate spinor component is given by where α = 1, ..., 4 labels four degenerate spinor components acting on the first 1 4 factor in the tensor product. We can proceed just as in the bosonic case with an ansatz of the form the positive eigenfrequencies are given by The vectors V ± n satisfy the following equation ensuring that the mode expansion of θ (4.24) satisfies the equation of motion (4.22). For each n > 0 the four vectors V ± n , V ± −n can be chosen to form a orthonormal basis. The canonical anti-commutation relations following from the action (4.21) imply the following algebra for the fermionic creation and annihilation operators with all other anti commutators vanishing. Interestingly, we can see from (4.25) that our spectrum contains eight fermionic zero modes with ω − n = 0 when n = 0. The zero modes are associated with θ in the original fermionic action (4.20) for which where we have dropped the index a for notational simplicity. Notice that (1 − iΓ 78 σ 2 ) is a projector and hence has an eight dimensional eigenspace with eigenvalue zero, matching precisely with the eight zero modes found above. Lastly, we can write the fermionic light cone Hamiltonian Where ν f is the constant obtained by normal ordering the fermionic creation and anilhilation operators.

Light cone spectrum
The Hamiltonian of the light cone string is the sum of the bosonic (4.14) and fermionic (4.30) parts The normal ordering constant is the sum of the bosonic (4.16) and fermionic (4.31) contribution and the bosonic and fermionic normal ordering contributions cancel up to a finite number of terms It is interesting to note that in the limit of vanishing α p + the finite normal ordering constant goes to zero, in agreement with the zero normal ordering constant for the light cone string in flat space. On the other hand for large α p + , a large but finite number of modes contribute to the normal ordering constant 3 . In the light cone gauge the level matching constraint is obtained by considering the variation of the action with respect to h τ σ in terms of the bosonic and fermionic modes one finds the constraint The spectrum of the light cone string has several features which distinguish it from the Penrose limit of AdS 5 × S 5 discussed in [8]. First due to the absence of world sheet supersymmetry the bosonic and fermionic worldsheet energies ω I n and ω α,± n are not the same. As discussed in appendix A.1 this is a consequence of the fact that there are no "supernumerary" [34] supersymmetries in the pp-wave background we consider. A second difference is the presence of fermionic zero modes associated with the β α,− 0 , (β α,− 0 ) † modes. These modes imply that even though the bosonic and fermionic oscillators have different energies that there are an equal number of bosonic and fermionic states at each energy level due to the 8 bosonic and 8 fermionic states created by (β α,− 0 ) † , α = 1, 2, 3, 4. The third feature is the presence of a normal ordering constant and the dependence on α p + . We discuss some possible reasons for the different properties of the lightcone string action in the next section. Presently, we have not been able to find an identification of the string spectrum with BMN like operators on the field theory side.

Discussion
In this paper we have constructed a Penrose limit for type IIB warped AdS 6 × S 2 solutions. We have constructed the limit by zooming in on a null geodesic at the center of the AdS 6 and along a great circle of the S 2 . This construction only works if the geodesic is localized at the critical point w = w c of the function G on the two dimensional Riemann surface Σ. We constructed the pp-wave supergravity solution one obtains from taking the Penrose limit. After appropriate coordinate transformation the resulting pp-wave background is universal for all the cases we have considered. It would be interesting to consider a more general class of examples, such as the solutions including seven branes [35] or O7-branes [36] and confirm that the behavior of the pp-wave at the critical point is of the same form. Another observation is that all known regular solutions have a unique critical point w c on the Riemann surface. This statement has not been proven but checked in all cases constructed in [37]. It is interesting that this critical point w c is also relevant for supersymmetric embedding of D3-branes in the AdS 6 × S 2 solutions which realizes BPS co-dimension two defects in the dual SCFT [27].
We presented the quadratic bosonic and fermionic world sheet actions of the Green-Schwarz string in this background and calculated the spectrum of bosonic and fermionic excitations of the string in the light cone gauge. The spectrum has some interesting features which are different from other cases such as the Penrose limit of AdS 5 × S 5 . Namely that the frequencies associated with the bosonic and fermionic creation and annihilation operators do not coincide, which is a consequence of the absence of any "extra" supersymmetries which would be associated with linearly realized supersymmetries. An additional new feature is the presence of fermionic zero modes, which makes sure that there are equal number of fermions and bosons at each level of the light cone hamiltonian.
One of the exciting features of the BMN correspondence was the identification of the light cone vacuum with a special class of BPS operators in the N=4 SYM theory, which can be viewed as a long spin chain. Furthermore, the action of creation operators on the string vacuum can be related to the insertion of impurities into the spin chain and diagonalizing the Hamiltonian. It is an interesting question whether a similar identification can be found for the light cone vacuum and excited states of the Green-Schwarz string in this plane wave background. This identification is more challenging for two reasons. While the Penrose limit identifies the light cone string vacuum with BPS states which satisfy ∆ = 3J, where J is a U (1) inside the SU (2) R symmetry of the SCFT and therefore related to protected BPS multiplet, these theories are realized as conformal fixed points of long quiver theories and hence strongly coupled. For d = 4, N = 4 SYM the identification of these operators was possible in terms of the scalar fields of the SYM theory. For long quiver theories it is not clear how to construct these operators from the fundamental fields of the nodes of the quiver and the bi-fundamental matter. A class of such operators corresponding to "stringy" long meson and baryon operators was given in [30] which are constructed by taking products of the bi-fundamental hyper multiplets, from one end of the quiver to the other 4 . Since our null geodesic is at a fixed location on Σ is is natural to speculate that the BMN like operator would be associated with a single node in the quiver unlike the string like operators in [30] which stretch across the Riemann surface Σ. It would be interesting to investigate what mechanism on the field theory side singles out a specific node in the quiver theory. One could also investigate more general null geodesics which are not at a fixed point on Σ and consider the their Penrose limits. A very preliminary discussion of this issue can be found in appendix B.
In [39,40,41] sphere partition functions and expectation values of BPS Wilson line operators for five dimensional SCFTs were calculated using localization techniques. For long quivers in the limit of large number of nodes and large gauge groups, the eigenvalues for the nodes where replaced by a continuous distribution and the saddle point of the path integral was reduced to an analogue electrostatic problem. It would be interesting to see whether these methods could be adapted to identify the BMN operators or whether the two dimensional electrostatic problem can be related to the light cone world sheet in some way and explain some of the curious properties of the light cone spectrum described in section 4.3.
We leave these interesting questions for future work.

A Supersymmetry of the plane wave background
The fermionic supersymmetry transformations in the string frame can be expressed in terms of the following two operators [42], that are also used to construct the part of the Green-Schwarz string action which is quadratic in space time fermions.
Here σ a , a = 1, 3 are the Pauli matrices which act on two component Majorna-Weyl spinors 5 . The supersymmetry transformations in the two component formalism are given by The susy transformations for the gravitino in the string frame are is different from the one in the Einstein frame given in [43], this is because the transformation g M N → e φ/2 g M N induces a mixing of the dilatino supersymmetry transformation with the gravitino supersymmetry transformation [44]. The supersymmetry transformations (A.2) are given in the string frame. The string world sheet action which is quadratic in fermions is given by where θ a , a = 1, 2 are the two ten dimensional Weyl-Majorana spinor world sheet fields with the same chirality as the supersymmetry transformation parameters ε a in (A.2) . The operator (D j ) b c is the pullback of the covariant derivative to the world sheet is related to (A.1) by

A.1 Integrability of supersymmetry transformations
To examine the supersymmetries of of the background (3.9), (3.21), we will first need to work out the spin connection.
The frame forms fields are given by where m k = 1 for all x k except for k = 6 for which m 6 = 3. We can calculate the spin connection using the Cartan structure equations The the frame forms given above the only nontrivial one is which implies that the connection 1-form is given by and we find that the only non vanishing components of the spin connection are given by Note that because (A.10) has two legs along the x + directions, the spin connection will not contribute to the fermion action (A.3), since the light-cone gauge Γ + θ a = 0 condition will make it vanish. In the following we will present the explicit form of the supersymmetry transformations in the string frame two component formalism following from (A.2). For the antisymmetric tensor fields of the T N example (3.21) the dilatino variation takes the form 6 . .11) and the gravitino variations are given by It was pointed out in [34] that pp waves have 16 + N sup Killing spinors; 16 of which must occur in any background while the remaining 0 ≤ N sup ≤ 16 so-called "supernumerary" Killing spinors occur only in special backgrounds. After light-cone gauge fixing, only these extra spinors give rise to linearly-realized supersymmetries. Such linearly realized supersymmetries also act as two dimensional world sheet supersymmetries implying a degeneracy of the world sheet energies of bosonic and fermionic excitations. In our case, the sixteen ε which satisfy Γ + ε = 0 are the "automatic" supersymmetries. It's easy to check that for those the conditions (A.12) are integrable as only δψ + is not automatically vanishing and can easily be integrated. For the "supernumerary" Killing spinors the vanishing of the dilatino variation (A.11) imposes a projection condition on the supersymmetry transformation parameters ε. The integrability condition in the i, + directions of δψ M = 0 becomes Where we used the identities Γ + Γ + = 1 + Γ 09 , Γ + Γ + = 1 − Γ 09 (A.14) Using the dilatino projection condition the +, 7 and +, 8 integrability conditions are also satisfied. However the +, 6 condition takes the following form (∂ + ∂ 6 − ∂ 6 ∂ + )ε a = 1 2 Γ 6+ 3 + 1 2 Γ + Γ + δ a b − 2iΓ 78 1 + Where the factor in parenthesis is not a projector and consequently there are no "supernumerary" Killing spinors in our background. This is to be contrasted with the Penrose limit of AdS 5 × S 5 where there are 16 additional supersymmetries [6].

B Null geodesics
In this appendix we consider null geodesics in the warped AdS 6 × S 2 metric ds 2 = f 2 6 (− cosh ρ 2 dt + dρ 2 + sinh ρ 2 ds 2 S 4 ) + f 2 2 (dθ 2 + sin 2 θdφ 2 ) + 4ρ 2 |dw| 2 , (B.1) We consider geodesics which are located at the center of AdS 6 at ρ = 0 and at the equator of the two sphere at θ = π 2 . These choices preserve the U (1) of the two-sphere which we identify with a U (1) inside the SU (2) R symmetry of the dual SCFT as well as the symmetries of the four-sphere of AdS 6 , which means that the dual state has no angular momentum.
We consider however the possibility that the geodesic moves along the Riemann surface Σ, i.e. the coordinates t, φ, w andw depend on an affine parameter λ and the geodesic equation and null condition. With these choices the geodesic equation C Plane wave limit for AdS 6 solution of massive type IIA In [26] a solution of massive type IIA was found which is a warped product of AdS 6 over part of a four sphere S 4 . This background was constructed by taking a near horizon limit of a D4-D8 semi-localized brane solution. In this appendix we briefly compare the Penrose limit for this supergravity background [45] to the one for the type IIB AdS 6 solutions described in the body of this paper. The metric and four form antisymmetric tensor field strength of the solution are given by ds 2 = 9 2 W (ξ) − cosh 2 ρdt 2 + dρ 2 + sinh 2 ρds 2 S 4 + 4 9 dξ 2 + sin 2 ξds 2 (cos ξ) Here the warp factor is given by W (ξ) = cos ξ 1 6 in the string frame. The range of the warping coordinate ξ ∈ [0, π/2] means that the three sphere warped over ξ produces only half of a four-sphere. The critical point of the warp factor of the three sphere is at ξ = π/2 and this corresponds to a Penrose limit considered in [45], which is analogous to the one discussed in our paper. There is however one important difference since ξ = π/2 corresponds to the S 3 along the equator of the S 4 and therefore to the strong coupling region where the supergravity approximation breaks down. In contrast, the critical point in the type IIB AdS 6 is a regular point where the supergravity approximation is valid for large N . The same conclusion can be drawn from considering the T-dual of the Brandhuber-Oz solution [46] for which the holomorphic functions A ± and G are [21] A ± = a 2 w 2 ∓ bw, G = ab 3 1 + (w +w) 3 (C.2) Here the condition ∂ w G = 0 is solved by Im(w) = 0 which under T-duality is mapped to ξ = π/2 on the type IIA side.