Penrose limits of Abelian and non-Abelian T-duals of AdS5 × S5 and their field theory duals

We consider the backgrounds obtained by Abelian and non-Abelian T-duality applied on AdS5 × S5. We study geodesics, calculate Penrose limits and find the associated plane-wave geometries. We quantise the weakly coupled type-IIA string theory on these backgrounds. We study the BMN sector, finding operators that wrap the original quiver CFT. For the non-Abelian plane wave, we find a ‘flow’ in the frequencies. We report some progress to understand this, in terms of deconstruction of a higher dimensional field theory. We explore a relation with the plane-wave limit of the Janus solution, which we also provide.


pp-waves and non-Abelian T-duality
In the mid-seventies 't Hooft suggested that there was a correspondence (a 'Duality') between string and gauge theories. Indeed, analyzing the diagrammatic expansion of an SU(N ) gauge theory with arbitrary matter content, 't Hooft proposed that the relevant expansion parameter was λ = g 2 Y M N . The perturbation theory was written as a doubleexpansion in λ and 1 N 2 [1]. The correspondence was proposed to be with an unknown string theory with expansion parameter g s = 1 N and world-sheet expansion parameter α = λ −1/2 . The discovery of a precise duality between gauge theory and string theory -the Maldacena Conjecture [2], has allowed important progress in the stringy description of gauge theories. The string theoretic descriptions of several (confining or conformal) gauge theories with a large number of colors have since been found. In fact, backgrounds on which the dual type-II/M-theory should be formulated have been written for various gauge theories. Unfortunately, the string theory on these backgrounds is not soluble in general. Even when the masses of low-lying low-spin hadrons (or dimensions of special chiral operators in the conformal case) can be computed from supergravity, the full hadron spectrum (or spectrum of dimensions for non-chiral operators) requires calculating with the full string theory.
It was the remarkable work of BMN [3], that opened avenues to truly stringy calculations in gauge theories. Let us summarise the most salient features of the topic.

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and R-charge J such that Hence, the full plane wave light-cone string theory Hamiltonian (denoted above by p − ) is equated with the difference between the field theory dilatation and R-charge operators. Similarly, the spectrum of strings is related to the spectrum of the dilatation operator, when restricted to the BMN sector. Also, the BMN proposal in (1.4), states that the string theory Hilbert space (consisting on the direct sum of zero, one, two, etc, string states) equals the Hilbert space of N = 4 SYM generated by the BMN operators acting on the vacuum. In the same line, single string states map to single trace operators in the CFT. In particular the string theory vacuum state is in correspondence with a (suitably normalized) BMN field theory operator acting on the CFT vacuum state |0, p + >→ N Tr(Z J )|0 > , (1.5) where Z is a chiral multiplet with R-charge J[Z] = 1.
For the string theory excitations, BMN proposed that one should consider field theory operators that are 'nearly BPS', that is, those with ∆ − J ∼ O(1). The chiral multiplets mix with the nearly BPS operators and the precise expression of the excited string states in terms of field theory states is more elaborated. See the review [6] for its most updated version.
As we see from (1.4), the t' Hooft coupling λ diverges. Perturbative calculations in the CFT are generically not trustable. However, for the case of the BMN operators, being 'nearly BPS', the nice properties of the chiral primaries are inherited. The BMN proposal provided a first example of a full-fledged interacting string theory that calculates perturbative aspects of a gauge theory. Indeed, since the pp-waves are exact solutions of the string theory to all orders in α [7], this makes them the most suitable scenario to perform such computations. For these reasons, the BMN proposal stands, sixteen years after it appeared, as one of the important developments in the area of gauge-strings duality. This idea generated various interesting spin-offs, for example, the relation between AdS/CFT and integrability. Now, we summarise recent progress in a different area of supergravity and string theory, namely non-Abelian T-duality. The bulk of this paper will study the synergy between this duality and the BMN-Penrose limits of the corresponding geometries in our continual effort to elucidate the nature of their field theoretical duals.

A quick review of non-Abelian T-duality
Non-Abelian T-duality [8], the generalization of the Abelian T-duality symmetry of string theory to non-Abelian isometry groups, is a transformation between world-sheet field theories. Its extension to all orders in g s and α remains however a technically hard open JHEP01(2018)071 problem [9][10][11][12][13][14][15]. As a result, non-Abelian T-duality does not stand as a String Theory duality symmetry, as its Abelian counterpart does. In addition, the transformation of the RR-flux fields under non-Abelian T-duality remained unknown for many years which made non-Abelian T-duality of limited use in relation with the AdS/CFT developments.
The authors of [16] reignited the interest in this transformation by extending it to include RR-fluxes and by highlighting its potential powerful applications as a solution generating technique in supergravity. The interesting synergy between the Maldacena conjecture and non-Abelian T-duality was first pointed out in [17][18][19] and further exploited and developed in . These works have widely applied non-Abelian T-duality to generate new AdS backgrounds of relevance in different contexts.
In the papers [44][45][46][47], the field theoretical interpretation of non-Abelian T-duality (in the context of Holography) was first addressed in detail. One outcome of these works is that non-Abelian T-duality changes the dual field theory. In fact, the new backgrounds generated through non-Abelian T-duality have dual CFTs different from those dual to the original backgrounds. This is in accordance with that fact that, contrary to its Abelian counterpart, non-Abelian T-duality has not been proven to be a string theory symmetry. The results in [44][45][46][47], open up an exciting new way to generate new quantum field theories in the context of Holography.
The general idea of this paper is to consider both the Abelian and non-Abelian T-dual of AdS 5 × S 5 , and study the Penrose limit for each of these geometries. The associated field theory interpretation for these two T-dual backgrounds [44], will be used in the study of the resulting pp-waves and the corresponding BMN operators.
Before moving into the discussion of these topics, we summarize the various results and achievements of the present work.

Summary and plan of this paper
As anticipated above, in this work we focus on the Penrose limits of both the Abelian and non-Abelian T-dual backgrounds of AdS 5 × S 5 . We will study the BMN operators and the spectrum of the strings in each background. In more detail, the outline and salient achievements of this paper are: • In section 2, we study the Abelian T-dual of the solution in (1.1). We consider various geodesics and their Penrose limit. One of the pp-waves generated is especially interesting, so that we study the quantization of the string theory in that background and compute the different eigen-frequencies. We restrict the range of the conserved angular momentum quantity J to avoid tachyons. We present an argument explaining why geodesics like the ones discussed in relation to the Abelian T-dual background have the possibility of tachyonic states.
• In section 3, we perform a very similar treatment to that in section 2, but now for non-Abelian T-duality. We study geodesics, the corresponding Penrose limits and also present the Brinkmann form of the resulting pp-wave. In addition, we perform the quantization of the string on this geometry. The eigen-frequencies are in this case depending on the light-cone time to be denoted by u in what follows.

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This is an unexpected feature. Indeed, in spite of having an AdS 5 factor in our original geometry, we encounter a 'flow' in the string frequencies. In a precise sense, our system is explicitly realising the 'flows in pp-waves' that [48,49] attempted to construct in the past.
For small values of the light-cone time u, the eigen-frequencies (masses) become constants. This regime of the pp-wave has a more canonical BMN interpretation.
On the other hand, the large-u behaviour is similar to the pp-wave of the Abelian T-dual, studied in section 2.
• In section 4, we discuss the supersymmetry preserved by the geometries in sections 2 and 3, by carefully discussing the fermionic variations. Noticeably, there does not seem to be any enhancement of the half-supersymmetry preserved by all pp-wave solutions, even at the 'ends of the flow'.
• In section 5, following the BMN proposal, we study the field theoretical aspects of our new pp-wave backgrounds. We focus our attention mostly on the Abelian pp-wave.
In terms of the parameters of the system N and k (the modding of the R-symmetry direction by Z k ), we will show that our strings interact very weakly, with either g s ∼ 1/k or g s ∼ 1/N for the Abelian and non-Abelian T-dual pp waves respectively.
We study the global symmetries and charges labelling the BMN operators and we write precise expressions for the vacuum and near BPS operators. Our BMN-operators 'wrap' around the quiver field theory dual to the original field theory and have non-zero winding. We present an intuitive argument explaining this fact. Also, we argue that (in our scaling of the parameters) there is a mixing between various BMN operators, that corrects the eigen-frequencies from the naively expected values. Finally, we discuss the BMN-sector associated with the non-Abelian pp-wave. In that case, we present arguments showing that the non-Abelian T-dual of AdS 5 × S 5 is dual to a higher dimensional field theory. Both for small and large values of a coordinate this reduces to a 4d CFT. This is reflected by the pp-wave and its eigenfrequencies.
Some conclusions and future directions of research are written in section 6. Various appendices with interesting and important technical details complement the presentation.
2 Penrose limit of an Abelian T-dual of AdS 5 × S 5 In this section we perform the Penrose limit on the Abelian T-dual of AdS 5 × S 5 , and analyze closed string quantization in the resulting pp-wave background.

Abelian T-dual solution
We calculate the Abelian T-dual of AdS 5 × S 5 , in a direction on S 5 . Here we adopt the conventions of [44]. More precisely, we consider the T-dual of AdS 5 × S 5 /Z k .

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After the Abelian T-duality the coordinate acted upon by Z k is renamedψ and takes values in [0, 2πk]. In order to restore the common factor of L 2 (the "radius" of the solution), we rescaleψ by L 2 /α ,ψ The new coordinate has generically small range, where g s is the string coupling in the type-IIB theory. In summary, after these scalings, the Abelian T-dual of the AdS 5 × S 5 solution along the ψ-direction is In the following, we analyze geodesics and Penrose limits in this background.

Penrose limits
A Penrose limit is a focusing on the region near a null geodesic. Denoting the affine parameter along the null geodesic by λ, the condition for a geodesic to exist is For motion (velocity) in the direction x κ , we need to have no acceleration in the other directions, hence where we have assumed that x κ corresponds to an isometric coordinate and that g µν is finite. Aside from this, we impose the condition for the geodesic to be null, ds 2 = 0. For the metric (2.3) it is natural to consider motion in the isometric directions ξ, β and ψ. For motion along ξ, the conditions (2.5)-(2.6) give The above two conditions lead either to χ = (0, π/2, π) and α = (0, π/2, π) or χ = (0, π). However, the values χ = (0, π) are out of consideration, since otherwise the coordinate ξ, JHEP01(2018)071 on which we take the limit, has a vanishing pre-factor. The same is true for α = π/2. That means that we need to have two geodesics, one for α = 0 and χ = π/2 and one for α = π and χ = π/2. It turns out that both geodesics lead to the same pp-wave solution, thus in the following we are going to consider the first one of them. Around the first geodesic, we make the expansions while keeping β unchanged (we do not scale it). In order to keep finite e −2Φ and F 4 we need to redefine the string coupling as The resulting pp wave metric, is already in the Brinkmann form and has to be complemented with the fields For motion in β, the condition (2.5)-(2.6) gives g αα ∂ α g ββ = 2 sin α cos α = 0 , (2.12) which means that either α = (0, π/2, π), with arbitrary values for ψ = ψ 0 , χ = χ 0 , ξ = ξ 0 . For the geodesic with α = π/2 and ψ = ψ 0 , χ = χ 0 , ξ = ξ 0 , we expand the coordinates as and we keep (χ, ξ) unchanged. We also keep the original coordinateψ unchanged, i.e. we undo the rescaling (2.1). Then we obtain the metric complemented by the fields As we see, in this limit the original coordinateψ (before the rescaling) and g s were the appropriate quantities to use. We have checked that the metric (2.14) has non-vanishing JHEP01(2018)071 scalar curvature and therefore it is not of the plane wave form. Since the null geodesic sits at a singular position (α = π/2) in the background, in the exact Penrose limit we don't obtain a pp wave form, though by a coordinate transformation of (2.3) (that would become singular in the exact Penrose limit), it might be possible to do so and thus respect Penrose's construction. Whilst it would be interesting to consider the field theory interpretation of these two Penrose limits, we will be more interested in understanding the effect of T-duality through the Penrose limit. With this purpose, it is useful to consider the Penrose limit for motion along the ψ-direction. In this case, the geodesic condition in (2.5)-(2.6) is just leading to α = 0. However, we will see that moving just along the ψ direction involves some pathologies for the string propagation. We consider instead the combined motion on both the ψ and ξ directions.

Penrose limit in ψ and ξ and pp wave
For motion along the ψ and ξ directions, we must satisfy the geodesic conditions. This implies α = 0 and χ = π/2. We can then construct a Lagrangian for a (massless) particle moving on a geodesic as above, where the dot refers to differentiation with respect to the affine parameter u along the geodesic. We will obtain the geodesic needed and using an expansion around it, the corresponding pp-wave. The Lagrangian is given by which is independent of (t, ψ, ξ), since the coordinates are cyclic. This leads to the conservation of their conjugate momenta where we have appropriately fixed the energy L 2 p t which in turn fixes the relation between time and the affine parameter u. The conserved quantity associated with the cyclic variable ξ is parametrized by J and that for ψ is determined from the fact that the geodesic must be null, i.e. L = 0. This giveṡ where in the solution we have ignored an additive integration constant. We note that for the reality of the solution we need J 2 1/4, or (ignoring the trivial sign of J that just defines the direction of motion in ξ)

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We also note that ψ ∼ O(1) meansψ ∼ O(L 2 /α ), but we also needψ ≤ 2πk. That means that in order to have ψ of order one (as suggested by the above formulas), we need to have where g B s is the string coupling in the type IIB theory (before the T-duality). In (2.20) above, u is the affine parameter along the null geodesic, i.e., the "lightcone time" (usually denoted by x + ). To define the Penrose limit around r = 0, α = 0, χ = π/2, we expand In principle there are several ways to take the Penrose limit (see appendix A for details), but the more physical one is where we do not have a compact time [50]. Hence, the coordinate t must be proportional to the lightcone time u as we found above. We propose an expansion near the non-trivial geodesic in the ψ direction, with a component in the ξ direction. That restricts us to the ansatz where c i , i = 1, . . . , 6 are constants. Moreover, the leading terms c 1 , c 2 , c 4 must be the ones from the null geodesic above This ensures cancellation of the O(L 2 ) terms in the would be plane wave metric, when L → ∞ as needed for the Penrose limit. For the cancellation of the O(L) terms, we obtain c 2 c 3 + c 4 c 5 = 0. Finally, demanding that the coefficient of dw 2 term is normalized to unity we obtain c 2 3 + c 2 5 = 1 and from the condition to have a 2 du dv term we get c 4 c 6 = 1. These conditions are solved by We then obtain a plane wave solution in Brinkmann coordinates, with ds 2 = 2 du dv+dr 2 +r 2 dΩ 2 3 +dz 2 +dx 2 +x 2 dβ 2 +dw 2 − and (2.28)

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Note that the coefficient of the x 2 (du) 2 term may be negative for low enough values for J which implies the appearance of tachyonic modes. Combining with (2.21) we obtain the condition for physical string propagation on this background which excludes J = 0, as advertised. We will come back to this in the next subsection. Technically, the origin of the potential tachyonic mode leading to (2.29) is the difference of the AdS 5 × S 5 in (1.1) and the Abelian T-dual metric in (2.3) in the ψ direction. Using the scaling in (2.23), we find that It is the sign difference in the above expansions that introduces the sign flip responsible for the potential tachyonic mode. Subsequently we study string quantization in the background given by (2.27)-(2.28).

Closed string quantization on the pp wave
The action for a string moving in a space with metric G µν , in the presence of the background fields B µν and Φ is (2.31) where the antisymmetric tensor density, the combination αβ / √ g, transforms as a tensor and αβ has non-vanishing components τ σ = − στ = 1. We fix the Weyl symmetry such that √ −g g αβ = η αβ , −η τ τ = η σσ = 1 .
(2.32) We will use the notation In order to fix the residual diffeomorphism invariance we take Then, for the plane wave in (2.27)-(2.28), we obtain that

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where the inner product is defined using the metric η αβ and we have considered the following gauge choice for the NS two-form, The equations of motion for the scalars in the above action read (2.37) The corresponding boundary conditions are (2.38) Using the above equations of motion and an ansatz of the form X i ∼ e −iωt+inσ , we obtain the frequencies for the oscillators  As we anticipated, the frequencies for i = 5, 6 are not tachyonic if the restriction (2.29) is obeyed.
3 Penrose limit of the non-Abelian T-dual of AdS 5 × S 5 In this section we study the non-Abelian case by repeating the same steps as for the Abelian one of the previous section. Namely, we write the T-dual solution, take Penrose limits and quantize closed strings in the resulting background.

Non-Abelian T-dual solution
We consider the non-Abelian T-dual along an SU(2) isometry of the same AdS 5 × S 5 /Z k background. After T-duality the coordinates are denoted byρ ∈ 0, 2πk and the two angles (χ, ξ). 1

Penrose limits
We start by discussing various possible geodesics. Considering the metric above we impose the condition in (2.6), for a geodesic moving in β and we find the same constraint as in the Abelian case, namely (2.12). As in that case only the case α = π/2 is viable with arbitrary values for ρ, χ and ξ. Then, we propose the expansion (2.13) keeping χ, ξ andρ unchanged. This leads to the metric where x 2 4 is the radial distance squared in the four-dimensional space spanned by the spherical coordinatesr, Ω 3 , so x 2 4 =r 2 . The NS-NS fields become, in the limit of large L, 4) and the RR fields are We have checked that the metric (3.3), similar to (2.14), has non-vanishing scalar curvature and therefore it is not in a plane wave form. We will not discuss this background in the following.
On the other hand, if we consider a geodesic along ξ, the conditions in (2.6) give which leads to the restrictions

Penrose limit in ρ and ξ
We may consider a combined motion inρ and ξ, in which case we can find a geodesic. In this case it is convenient to first rescale the coordinateρ, analogously to (2.1) in order to restore the common L 2 factor to the metric Hence, we obtain a solution that takes parametrically small values for ρ unless k is comparable to L 2 /α . In addition we rescale the string coupling as Consider the null geodesic for motion in ρ and ξ, at χ = π/2, α = 0, which would solve the last two conditions in (3.6), and r = 0. The non-Abelian T-dual solution in the rescaled coordinate ρ, and in terms of the rescaled coupling reads with the R-R fields (3.10) The Lagrangian for a particle moving on a null geodesic reads where as before, the dot indicates a derivative with respect to the affine parameter u. The Lagrangian is independent of t and ξ (these coordinates are cyclic), which means that their conjugate momenta are conserved,

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As in the Abelian case, we fix the energy and define the angular momentum in the ξ direction by the conditions Thenρ is determined from the condition that the geodesic be null, L = 0, givinġ This has solution As in the Abelian case, the affine parameter u is now the lightcone time (usually denoted by x + ). Note that, for the right hand side of the equation to be positive, we need that J 2 1/4 and, since J → −J is an irrelevant choice of direction of motion on ξ, we have For the same reason, ρ is restricted to be which in terms of the original coordinateρ reads This must fit inside the interval 0, 2πk , which means that we need k ∼ L 2 /α . To define the Penrose limit, we expand around the null geodesic considered above, and we set dt = c 1 du , with

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Substituting everything in the metric, we obtain the pp wave At this point, we can already observe that, unless J 2 ≥ 1/8, the x 2 du 2 term will have the wrong sign, which will lead to tachyonic modes for closed string propagation in this background. We will explore this in more detail in the next subsection, but for now we just point out that the consistent range of J is which is the same as (2.29). In particular, we see that J = 0, i.e. motion solely along ρ, is not consistent, as anticipated above.
The field strength of the Neveu-Schwarz B-field in the Penrose limit becomes which tells us that the B-field is The dilaton in the limit becomes simply (3.26) The Ramond fields in the L → ∞ limit are The pp wave obtained above is not yet in Brinkmann coordinates, which is what we would need for a simple string quantization. Below, we discuss the coordinate change that brings the background to the desired form.

Brinkmann form
In order to bring the metric into the Brinkmann form, we notice that for a line element of the form we can replace

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Implementing these, the line element takes the form In the particular case of (3.22) we have Hence after replacing we get The NS-NS and R-R fields become The dilaton is given by (3.26). The NS-NS field strength has a quite simple expression, In general, one can consider different gauge choices for the NS two-form. In the analysis of the quantization of the string we make the choice with κ 1 + κ 2 = 1.
Next, we study the string quantization in the pp-wave of (3.34)-(3.38).

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3.4 Closed string quantization on the pp wave Using the same notation for coordinates as in (2.33), the closed string action in the background of (3.34)-(3.38) is (3.39) Again we fixed the Weyl symmetry like in (2.32) and the residual diffeomorphism invariance by considering (2.34). The equations of motion for the scalars X i , i = 1, . . . , 8 now are The corresponding boundary conditions are Notice that generically, the masses (or frequencies) of the system depend on ρ. Hence, masses do depend on the light-cone time. We will discuss this phenomenon in more generality in section 5 and in appendix B. From the formulas above it is obvious that the equations of motion for the first six scalars can be solved easily. We will not present the solutions for the scalars here, just the oscillator frequencies for each scalar, which we will need for the study of dual field theory. They are There is no unstable (tachyonic) mode in the (5 th , 6 th ) directions only if J 2 1/8, as advertised.

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3.4.1 The limit u 1 We can solve explicitly the equations in the limit u 1, which however does not imply that ρ 1. It is simply the starting region of the trajectory in the affine parameter of the wave. As an aside, notice that the opposite limit, u 1 (that corresponds to ρ → ∞), the solution becomes the Abelian T-dual solution treated previously (except for the dilaton, which is now non-constant, whereas the dilaton in the Abelian case is constant). This correspondence was observed in [41], [44].
In the u 1 limit, ρ becomes a constant, This can be further simplified by an appropriate choice of the integration constant c ρ , for example (3.44) The pp-wave solution in the small u limit becomes Notice that background in (3.45) is an exact solution (on its own) of the type-IIA supergravity equations of motion. In a similar way, the equations of motion for the scalars simplify to The oscillator frequencies of the scalars are (3.47)

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This completes our analysis of the quantization of the string in the pp-wave corresponding to the non-Abelian T-dual of AdS 5 ×S 5 . We will now analyze the amount of supersymmetry preserved by the pp-wave background discussed in this section.

Supersymmetry of the pp waves
In this section we count the number of possible supersymmetries preserved by the ppwave corresponding to the non-Abelian T-dual solution. For convenience we work with the pp-wave in the Brinkmann form and we define the coordinates y i , i = 1, . . . , 8 as In these coordinates the non-Abelian T-dual pp-wave solution reads where now Using the frame proposed in (C.1) the formulas above take the form (4.4) Notice that this solution preserves a global SO(4)×SO(2) symmetry, where the SO(4) factor corresponds to rotations in the y 1 , . . . , y 4 directions and the SO(2) factor to rotations in the y 5 , y 6 directions.
For the susy analysis below we follow the conventions in appendix D of [18,19]. Thus, the susy variations of the dilatino and gravitino in our case are and the Killing spinor consists of two Majorana-Weyl spinors ± , such that In the type IIA supergravity, the Killing spinor satisfies the chirality condition Γ 11 = −σ 3 . We denote the Pauli matrices by σ i , i = 1, 2, 3.
Due to the form of the spin connection, the covariant derivatives can be written as where we assume the summation for repeated indices. Also, the equation (C.7) suggests that we should define the Γ ± matrices as 2 From this formula we easily verify that Γ ± 2 = 0.

Susy of the pp-wave solution for motion along ρ and ξ
For convenience we consider a more general form of the pp-wave where now the function H is given by the expression (instead of the one in (4.3)) Moreover, we turn on a NS three-form and a RR four-form with expressions and we also consider a non-trivial dilaton which is a function of u only, i.e. Φ = Φ(u).
In the analysis that follows, we take the functions F ij , f, g and Φ to be arbitrary. If we want to make contact with the non-Abelian T-dual pp-wave of the previous section we just restrict to the following expressions for those functions: (4.12) We now start our analysis with the dilatino equation.
2 With this definition it is guaranteed that

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The dilatino equation. If we plug the above ansatz into the dilatino variation (4.5) and require that it vanishes, we get For the functions in (4.12), which correspond to the pp-wave of our interest, we realize that the only projection that guarantees the vanishing of the dilatino equation is Γ − = 0. This tells us that the pp-wave that we studied above preserves 16 supercharges. A supersymmetry enhancement would imply the existence of more projections in the dilatino equation which should be also compatible with the gravitino variations. For completeness, let us now turn to the variation of gravitino and examine it component by component (the analysis is done using the frame components and not the natural ones).
The δψ i , i = 1, . . . , 8 components. The vanishing of δψ i implies Notice that since ∂ i is proportional to Γ − , if we act with one more derivative ∂ j , j = 1, . . . , 8 we obtain This tells us that the Killing spinor is linear in y i and thus its solution is where we assume the summation of the index i.
The δψ − component. We now turn to the variation δψ − . Notice that in this case the covariant derivative D − becomes If we expand δψ − then we find

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where we took into account that 4.20) and the fact that does not depend on v. If we now require the vanishing of δψ − and substitute the solution for we find From the last equation we obtain nine total conditions. In order to find those conditions we first isolate the piece that is independent of the y i 's. This piece gives The other eight conditions come from the requirement that the coefficient of each of the y i 's in (4.21) must vanish. Hence, for each y i , after eliminating ∂ u χ using (4.22) we get This completes our analysis of the preserved supersymmetry. Both Abelian and non-Abelian plane waves preserve sixteen supercharges. In the following section, we will discuss different aspects of the field theory dual to the backgrounds in sections 2 and 3.

Quantum field theory interpretation
Here we present a field theoretical interpretation for the results of the strings on the pp waves studied in previous sections. We will mostly refer to the Abelian pp-waves in section 2, but we will also discuss the non-Abelian case of section 3 as well.

The AdS/CFT map
We start by discussing the field theories dual to the backgrounds in (2.3) and (3.9)-(3.10).
The field theory dual to the Abelian and non-Abelian backgrounds was considered in [44]. The field theory dual to the background in (2.3)-the T-dual of AdS 5 × S 5 /Z k -is a circular quiver with k nodes and gauge group SU(N ) k . The theory has N = 2 supersymmetry in 3+1 dimensions. At each node we have a N = 2 vector multiplet and a N = 2 bifundamental hypermultiplet between each two adjacent nodes.

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In the non-Abelian case, the field theory dual proposed in [44] is an infinitely-long quiver, with increasing gauge group, i.e. SU(N )×SU(2N )×SU(3N )×. . .×SU(kN )×. . .. The quiver terminates only when a 'completion' of the background is considered, at an arbitrary positionρ = P π by the addition of a suitable flavour group. After this completion each node has an N = 2 vector multiplet, and each link between two adjacent nodes contains an N = 2 bifundamental hypermultiplet (hence satisfying the condition for conformality N f = 2N c for each node). Similar completions have been found in different backgrounds obtained using non-Abelian T-duality, see for example the papers in [45][46][47].
In the context of Penrose limits and pp-waves, the quiver field theory dual to AdS 5 × S 5 /Z k has been considered in the papers [51] and [52]. As we will see discuss, our Penrose limits in sections 2 and 3 are different. Yet, they still have common ground with the analyses in [51] and [52]. The T-duality will play an important role. A T-dual picture was also considered in [52], nevertheless the order of 'operations' is important. Indeed, the paper [52] considered first a Penrose limit, and then T-duality, whereas in this work we consider T-duality first, and then a Penrose limit. Another important difference is the scaling of the parameters, which in our work is different to that in the papers [51] and [52].
Field theory limit. We have seen that in both the Abelian and non-Abelian cases, in order for the rescaled coordinates ψ and ρ in (2.1) and in (3.7) to be of order one, we must have that Here g B s is the string coupling of the type IIB theory (before the T-duality), related to the string coupling in the type IIA theory by Note that in the Abelian case, the effective string coupling isg s . Using (2.9), we have If we keep fixed the YM coupling, it means g B s is fixed, leading to N ∼ k 2 , and theñ In the non-Abelian case, the effective string couplingg s was defined in (3.8). In this case we have Both results imply that strings on the pp waves are classical, i.e. they cannot break or join. The only relevant computation is the calculation of eigenergies of the strings, to all orders in α .
As discussed in section 4, the Abelian and non-Abelian pp waves have only the standard 1/2 supersymmetry, there is no susy enhancement. We should expect the same from the field theory analysis.
Various Penrose limits of AdS 5 × S 5 /Z k and their field theory dual. Below, we will compare our pp wave with the cases previously considered in the bibliography [51,52] and [53].
In the paper [51] it was pointed out that we can have two possible Penrose limits for AdS 5 × S 5 /Z k : • Along a direction different from that of the Z k identification, leading to an N = 2 supersymmetric result. Basically, this is the Z k orbifold of the pp wave of maximal supersymmetry (the Penrose limit of the AdS 5 × S 5 background). The field theory was analyzed in [51]. In this case, one has the field theory scaling and one considers also a large R-charge limit for operators, but the scaling does not affect the amount of supersymmetry of the sector, which is still N = 2.
• Along the direction of the Z k identification, leading to a maximally supersymmetric background. The result of the Penrose limit has a periodic x − ("lightcone space") direction. This was analyzed by [52]. In this case one considers the scaling together with a large R-charge limit on operators. However, now the large R-charge limit, coupled with the fixed ratio above, leads to a supersymmetry enhancement to maximal for the field theory sector.
Our study of sections 2 and 3 corresponds to a Penrose limit along the direction of the Z k identification, yet with the scaling limit and amount of supersymmetry of the first item above. This means that we will have a different field theory sector from the one analyzed in [51,52] and [53].
The field theory and its symmetries. In order to understand the field theory sector dual to our pp-waves, we first find useful to summarize the properties of the full quiver field theory. We focus on the scalars, since they are the easiest to identify with the gravitational side.

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The scalar fields are: an adjoint complex scalar X i at each node i (belonging to the vector multiplet) and the two bifundamental complex scalars V i and W i , between nodes i and i + 1, one of them in the (i, i + 1), the other in the complex conjugate (i, i + 1) representation of the i-th gauge group. The superpotential, in N = 1 superfields notation, is where the trace is necessary, because the term is in the gauge group of the i + 1 node (the i node indices are all contracted). The kinetic terms are The field theory symmetries are: • An SU(2) R that rotates V i and W i (the two complex scalars in the hypermultiplet that transform in the same representation of the gauge groups). It also rotates the fermionic d 2 θ-terms. Under this symmetry, the fields (V i ,W i ) and (V i , W i ) are doublets in conjugate representations. Inside this SU(2) R , there is a U(1)-symmetry with Cartan generator J 1 (denoted by J in [52]).
• An U(1) R that rotates only the chiral field X i inside the N = 2 vector multiplet and d 2 θ-terms, leaving the bifundamentals invariant, • An extra (non-R) U(1) that multiplies the bifundamental scalars, V i , W i , by the opposite phases e iα , the generator of which is denoted by J 2 (this was called J in [52]). Having sumarized the field content and symmetries of the quiver field theory, let us discuss now how these are realized in our backgrounds and the corresponding pp-waves.
The gravity dual and identification of coordinates and symmetries. Our pp waves correspond to taking first a T-duality along a direction made periodic by a Z k identification, and then taking the Penrose limit along the same direction. In the non-Abelian case we dualize on an SU(2)-isometry containing the Z k -identified coordinate.
The procedure that [52] followed is in reverse order. Indeed, they took first the Penrose limit in the Z k direction, and then considered the Abelian T-dual interpretation (for T-duality on the same Z k -identified direction). Let us analyze the Abelian case in what follows.
From the form of the metric in our case, the T-dual of AdS 5 × S 5 /Z k -see (2.3), we see that the background has explicit symmetries: SU(2) × U(1), where the SU(2) is JHEP01(2018)071 Table 1. Quantum numbers (charges) of the bosonic operators and their conjugates the symmetry of the S 2 parametrized by the angles (χ,ξ), and the U(1) corresponds to translations in the angle β. There is also a shift symmetry corresponding to translations along the compact coordinate ψ, which takes values in [0, 2πk], and has a periodicity 2πk. This corresponds to another U(1) symmetry. 3 Therefore, in total, we have the isometries SU(2) χ,ξ × U(1) β × U(1) ψ , which are identified with the symmetries of the field theory. Indeed, the identification of the dual gravity coordinates with field theory scalars can be achieved by embedding the five-sphere of (1.1) into Euclidean space. In order to do this, we parametrize the six dimensional space in terms of three complex scalars Z i , i = 1, 2, 3, given explicitly by Here i |Z i | 2 = L 2 , L corresponds to the size of the space, and the coordinate ψ has periodicity 2π/k. Indeed, Z 1 is the scalar inside the vector multiplet, that we denote by X, and Z 2 , Z 3 are the bifundamentals, denoted by V, W above.
We will now identify symmetries of our quiver field theory: • The SU(2) R acts on the scalars Z 2 , Z 3 . Indeed, it acts on the S 2 angles described by (χ, ξ) in our parametrization of the gravity dual (after T-duality). This means that in particular, our gravity dual symmetry U(1) ξ ⊂ SU(2) R gives a charge J 1 .
• The U(1) R acts on the scalar Z 1 by multiplication by a phase e iα , and Z 1 is identified with our X-superfield, which means the field theory symmetry U(1) R corresponds to the U(1) β in (5.15).
• The extra U(1) acts on the scalars Z 2 , Z 3 (corresponding to our V, W ) by Z 2 → e iα Z 2 , Z 3 → e −iα Z 3 , which means that ψ → ψ + α, in (5.15). This is just a shift in the direction of orbifolding, ψ. Therefore this extra symmetry, U(1) ψ , corresponds to the non-R U(1) the action of which was defined in (5.14).
The charge assignments are summarized in table 5.1. We see that the resulting value of the energy defined by [52], H = 2p − = ∆ − kJ − J (which plays the role of "∆ − J" in the BMN limit) is H = 0 for V i and H = 1 for W i and X i . This allows us to consider V i as the ground state and W i , X i as the oscillator modes. This finishes our analysis of symmetries. Let us discuss the states and operators in the field theory dual to the type IIA string on the Abelian pp-wave.

Abelian case: operators and field theory sector
As described in the previous subsection, we expect the field theory operators corresponding to our pp wave to be the same ones considered by [52,53], since the Penrose limit is taken in the same direction. However, the scaling of parameters and amount of supersymmetry is different, so we expect the eigenenergies and eigenstates to be different.
The operators we want to describe have a large charge in the direction corresponding to the gravity dual symmetry U(1) ψ , which is U(1) extra in the field theory. The scalars charged under it are V i andW i (its charge is denoted by J in the work [52]). We also have a large charge in the direction corresponding to the gravity dual symmetry U(1) ξ ⊂ SU(2) R (called J in [52]). The ratio of the two charges, J /J in [52], is identified with -see (2.19)-(2.20), We have used the values for the null geodesic around which we expand. The minimum value for the quantity is reached when our parameter J takes its minimum value, J = 1/(2 √ 2), when we get (J 1 /J 2 ) = 1, and the maximum is reached when ou J takes its maximum value, J = 1/2, when we get (J 1 /J 2 ) = ∞.
From the list of charges in table 5.1, we see that there is no way to vary J 1 /J 2 , since we need the vacuum to have zero energy H = 2p − = 0, and the only field with that property is V i , which has J 1 = J 2 = 1/2, corresponding for us to J = 1/(2 √ 2). It is not clear how to see this restriction of the value of J from our pp wave analysis.
As explained also in [52], we construct the vacuum using only V i -fields, that must wrap completely around the circular quiver. The reason is that the V i 's are bifundamental between the adjacent gauge groups SU(N ) i × SU(N ) i+1 . Such a vacuum has a "winding" around the quiver. The authors of [52] associated the vacuum with a state of momentum p = 1 and winding m = 0. However, because of the T-duality between our picture and theirs, their state a † n |p, m (5.17) where p is momentum and m is winding along the Z k (lightcone) direction, and n is a transverse direction, becomes a † n |m, p (5.18) in our T-dual case, thus exchanging momentum with winding. That means that the state of zero-winding and one-lightcone unit of momentum in [52], becomes a state of winding-one and zero-lightcone momentum for us, corresponding to the operator winding once around the quiver. More formally, we write 19) with conformal dimension ∆ = k = J 1 + J 2 (∆ = J + J in the Type IIB analysis of [52]), so with zero energy, H = 0. A state of winding p (momentum p in Type IIB) just winds p times around the quiver.
These are the usual 4+4 bosonic oscillators also present in the standard BMN case. Note however that D a , X i andX i can be inserted inside the operators (at the site i), whereas an insertion of W i needs also an extra insertion of V i , in order to return to the same gauge group SU(N ) i (both W i and V i are bifundamental, but with opposite 'orientations'). Similarly, an insertion ofW i necessitates the removal of the corresponding V i inside the operator, sinceW i acts in the same way as V i . Then the insertions of D a , X i andX i at zero transverse momentum give whereas the insertions of W i andW i give To insert a field of mode number n, we put phases as usual, for instance a † X,n |m, p = 1 us = The important difference is that now putting several n i , the sum of all gives our momentum (the winding in the Type IIB analysis), There are no states of vanishing momentum in the Type IIB analysis. This is because there is no vanishing momentum state in Discrete Light Cone Quantization, DLCQ. As a consequence, there are no operators of vanishing winding in the Type IIA description.

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Indeed, we cannot construct a state of zero energy with charges J 1 = J 2 = 0 other than from V i 's. We could imagine using (V iVi ) pairs, but those have vanishing total charge and dimension 2, so H = 2p − = 2, changing the energy of the state.
Let us understand better the absence of zero-winding operators in the Type IIA pp-wave.
Windings on the pp wave. Here, we provide an interpretation, from the point of view of the string theory on the pp wave, of the fact that we need winding in our states. Specially those states associated with the (T-duality) ψ-direction, and no winding in the other directions. Below we present a plausability argument explaining this fact.
As is well known, strings in flat space with some compact directions and with a constant Neveu-Scharz B-field along those directions change their spectrum (and its states have nonzero winding). Indeed, in the book [54,55] (more precisely in Volume I, around page 250), the effect is studied. We suggest that a similar mechanism may be at work here, even when we are not working in flat space and our B-field is not constant. For this, we need to show that some of the directions of our pp-wave in (2.27)-(2.28) can be considered periodic.
Then we obtain the boundary conditions If we work in the gauge where the coefficient κ 2 = 0, we can take the solutions to these equations such that X 7 (σ = 2π) = X 7 (σ = 0) + C, (5.25) which implies that the first of (5.24) is satisfied. The second equation would indicate the non-periodic character of X 8 . One can see the winding of the lightcone direction by deriving the periodicities on the pp wave. From the first relation in (2.24), t ∝ u, we see that lightcone time u is not periodic. Note that in our case, we have (choosing 8J 2 = 1), From this periodicity and the periodicity of ψ by 2πα k/L 2 , we obtain That means that the lightcone space direction v is compactified with radius α k.

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In summary, we have made plausible the argument that the presence of the B-field on a compact direction adds winding to the states. As we mentioned, it also changes the energy levels. To this we move now.
Eigenenergies. In (2.39), we discussed that at n = 0, the frequencies of the closed string moving on the pp-wave of the Abelian background are We should obtain the same results from the energies H of the operator states described above, at n = 0.
The insertion of D a on the state of zero-momentum and winding-one gives an energy H = 1, since D a has no charges, but has mass dimension ∆ = 1. This corresponds to the ω 0,a . Naively the insertions of X,X, W,W would also give energies H = 1, and this does not seem to match.
An important difference with the analysis of [52], where g 2 Y M N/k was kept finite, is that for us this quantity is divergent.
This leads to an absence of the cancellations that allow for a simple result. In our case, there will be interactions that cannot be neglected, that will correct the eigenenergies. We did not study the calculations of the Feynman diagrams, since they are very involved. We will just present below a sketchy argument, showing the type of interactions we expect to be relevant to change the spectrum.
Because of the W ∼ g Y M Tr i+1 [V i X i W i ] term in the superpotential, we have the interaction term in the scalar potential 29) and this term will mix O W,0 and OW ,0 , the operators with zero momentum insertions of W andW defined in (5.21). Indeed, introduce in OW ,0 the "particle-antiparticle" pair V iVi beforeW i , which would annihilate, though they have a perturbative "energy" of ∆ − J 1 − J 2 = 2, so this would seem to change the energy of the state. Since the effective coupling g 2 Y M N/k is large, interaction effects will be dominant, and they can plausibly reduce the energy of the state with "particle-antiparticle" pair insertion. Then the insertions of (W i V i ) and (V iWi ) could mix via the interaction term from the potential described above.
That means that the energy levels, which start off equal at g 2 Y M = 0, could split off into a higher one and a lower one, as observed in the pp wave -see (5.28).
Similarly, the same superpotential term g Y M Tr i+1 [V i X i W i ] leads also to the potential term which will mix O X,0 and OX ,0 in (5.21), the last with the introduction of a particleantiparticle pairV i V i . That means that the eigenenergies of these states could also be corrected.

Non-Abelian case: RG flow
In the case of the pp wave obtained from the non-Abelian T-dual solution, as we have already observed, the pp wave at large ρ, obtained from the u 1 limit (u being the lightcone time on the pp wave), reduces to the Abelian pp wave, except for the varying dilaton -compare (2.27)-(2.28) with (3.34) and (3.26). However, as was also pointed out in [57], the matching becomes exact if we instead consider ρ = ρ 0 + ψ and take the scaling limit On the other hand, at small u (u 1), leading to a finite ρ = ρ c , the pp wave solution is given in (3.45), and the frequencies of the string modes were calculated in (3.47).
This means that the pp wave solution, and the corresponding eigenenergies for closed string modes on the pp wave, "flow" in the lightcone time u, from u = 0 to u = ∞. This flow not only appears in the Penrose limit of the Sfetsos-Thompson solution. It is indeed a more generic phenomenon for certain solutions of the Gaiotto-Maldacena type [56], as we explain in appendix B. Also, in appendix D, we see a similar flow in the Penrose limit of the Janus solution, dual to a defect CFT.
The idea of an RG flow related to a nontrivial pp wave was considered before, in [48] and [49]. However, those authors considered simply an RG flow in the radial coordinate of the gravity dual. They discussed Penrose limits near geodesics moving on different coordinates, both in the IR and in the UV of the holographic RG flow.
Here, we have a different situation: we have an RG flow for the masses (or frequencies) of string modes, ω = ω(u), between the UV and the IR. This is obtained from the way in which we took the Penrose limit. We are actually realizing the picture advocated (but not obtained) in [48] and [49].
The flow is between a theory in the UV, which corresponds to ρ → ∞ or u → ∞, and the IR theory, at u → 0. Indeed, we see that at large ρ, the equivalent Abelian coupling g s → 0 (corresponding to the UV), by the identification in (5.31) above with the non-Abelian case. This UV fixed point theory is conformal, namely it is the Abelian quiver field theory (which is N = 2 superconformal). The IR fixed point theory is defined by the u 1 limit. We can naturally ask: how is it possible to have an RG flow, when the gravity dual has an AdS factor, dual to conformal symmetry? We can provide two possible non-exclusive answers: • One answer is that we must now consider the full string theory, including winding modes, since we consider the T-dual to AdS 5 ×S 5 /Z k , and winding modes correspond to momentum modes before T-duality. In the full string theory, the conformal invariance is broken by these winding modes, which make possible the flow. This is not fully satisfactory, as there are winding modes in the Abelian T-dual plane wave, and no flow in frequencies. Nevertheless being both backgrounds and dynamics different, the winding modes on the non-Abelian plane wave might have the effect argued above.

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• Another possibility is that the non-Abelian T-dual background in (3.9)-(3.10) is actually dual to a non-conformal field theory in higher dimensions, reducing itself to a four dimensional CFT only for large and small values of the ρ-coordinate. We explore this deconstruction possibility in the next sub-section. This picture is also supported by the material in appendix D.
Note that, since in our Penrose limit we identify t = u, and AdS space is in global coordinates, meaning that on the boundary we have S 3 × R t . Then t = u is the radial time in the field theory, and e ±t = e ±u is the radius in the Euclidean field theory, i.e. distance scale (or energy scale). Thus the flow in u is really an RG flow, that is flow in energy (or distance) scale.
Unfortunately, there isn't much we can calculate in the field theory corresponding to the non-Abelian case; we must consider the string theory calculations as simply predictions for the field theory.
Notice that the non-Abelian T-dual background does not have the U(1) extra . This symmetry only appears in the metric for large values of the ρ. We expect the operators corresponding to the string states to be extensions to the new (infinite) quiver of the operators in the Abelian case. Then the excitations are extensions of those in the Abelian case. Correspondingly we find the same string eigenenergies in the UV (at large u). But in the non-Abelian case, we have a more interesting dynamics, leading to the RG flow in eigenenergies.
We now discuss the second alternative bullet-point explanation of the flow in the non-Abelian plane wave.

Deconstruction
The large quiver describing the field theory dual to the non-Abelian T-dual background is reminiscent of the "dimensional deconstruction" ideas of [58] and [59].
In "deconstruction", at intermediate energies a fifth dimension appears. More precisely, after a particular choice of Higgsed vacuum, a KK tower of states associated with a compact dimension appears. As usual in KK theories, at even lower energies (smaller than the inverse radius of the compact dimension), the fifth dimension becomes invisible again. The UV theory is the full 4-dimensional quiver theory as the effects of the VEV become negligible.
In the previous subsection we discussed that there is an RG flow in u between the UV theory (the quiver) and the IR theory. The deconstruction picture suggests that for intermediate u, u ∼ 1, we should see a higher dimensional field theory.
We have seen that the quiver implied long spin chain operators describing a discretized dimension. The normal interpretation for the spin chain is as a discretized string worldsheet in the gravity dual, but in the field theory itself it can now be also described as an extra dimension, since as we saw we need to consider only operators with "winding" around it.
We now make more precise in which sense the background in (3.9)-(3.10) represents a higher dimensional field theory.

Wilson loops and signs of extra dimensions
To give support to the proposal that the non-Abelian of AdS 5 × S 5 , the Sfetsos-Thompson solution, is actually representing a field theory in dimension higher than four, we will calculate the potential between two static charges. We shall observe that, in a certain regime, there are deviations from the usual V QQ ∼ 1/L QQ characteristic of four dimensional CFTs.
We first present the intuitive argument. The solution in (3.9)-(3.10) was interpreted in [44] as an strictly infinite length quiver SU(N ) × SU(2N ) × SU(3N ) × . . .. This CFT has infinite central charge, hence it is not a four dimensional field theory. The treatment of [44] was to complete the quiver to make sense of it as a four dimensional CFT. This leads us to consider that the solution in (3.9)-(3.10) might be dual to a field theory in higher dimension, realising ideas similar to those in deconstruction.
To make this more precise, we will consider the Sfetsos-Thompson solution with the AdS 5 space written in Poincare patch, with coordinates [x 1,3 , R]. We shall calculate the Wilson loop for a heavy quark-antiquark pair placed on the space directions [x, ρ], separated by distances L x , L ρ , respectively. This is inspired by the fact that ρ plays the role of the field space in the interpretation of [44]. We shall then consider a configuration t = τ, R = σ, (5.32) x = x(σ), ρ = ρ(σ).
In this configuration, the distance between the pair of quarks is L = L 2 x + L 2 ρ . In the following, for computational purposes, we restrict ourselves to the regime in which the distance L ρ is much smaller than L x , hence L ∼ L x . The action of the Nambu-Goto string is (we set L = α = 1 and choose to place the string at the value of the coordinate α = 0) (5.33) We use the conserved quantities We will now produce two 'string effective actions' . One will be useful for small values of separations L QQ ∼ 0, the second will be trustable for strings that explore large parts of the bulk.
Short strings, large R effective action. Using (5.34), we replace ρ (J, R, x ) in the action to obtain the action for an 'effective' string propagating on the non-Abelian T-dual background of (3.9)-(3.10), (5.35)

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This effective action represents a string that moves in a background dual to a relevant deformation of N = 4 SYM. The effective background is singular at R = J, hence long strings probing that position are not faithfully described by this effective action. For small J and/or for large R, the action approximates that of the string on AdS 5 . This is indeed intuitively correct, since J measures how much the string explores the ρ-direction.
We write now the expressions for the separation L QQ ∼ L x (a good approximation when Lρ Lx is small) and the Energy E QQ in terms of R 0 , the point where the Nambu-Goto string turns back in the bulk, Since we want to consider a string that extends very little in the ρ-direction, we expand the expressions in (5.37), to first order in J 2 . The integrals can be exactly calculated. We obtain In this equation above, the numbers l AdS , AdS are those appearing in the original calculation [60]. On the other hand, l 1 , 1 are just numerical constants that appear when evaluating the integral. The expansion in (5.38) can be continued to higher orders. The same structure is preserved. Inverting R 0 (L QQ ) we can write Notice that this expression is trustable for small values of L QQ , effective strings that barely explore the bulk of the space. The point to be kept in mind is that there is a deviation from the CFT behaviour.
We will now analyze long strings that penetrate deeply into the bulk.
Long strings, small R effective action. We consider (5.34) and define 39) and replace this in the action of (5.33). We obtain

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This effective action corresponds to a string moving in a background dual to N = 4 SYM deformed by an irrelevant operator (this is reflected by the change in the large R asymptotics of the metric). We should only trust these string configurations for large values of L QQ . In this case we write integrals for the separation and energy of the quarks pair, that are the analog expressions to those in (5.37). We expand these integrals for small values of x = J/M and obtain Here l 1 , 1 are numerical constants. It is important to notice that the sub-leading terms in this expansion are not convergent (unless we cutoff the space), the expansions cease to make sense for smaller values of L QQ or larger values of the coordinate R. From (5.41) we obtain Here C 1 , C 2 are numerical constants. This equation is again showing deviations from the characteristic conformal law.
In conclusion, we have made the point that the Wilson loop calculation shows indications that the Sfetsos-Thompson solution in (3.9)-(3.10), is actually dual to a field theory in dimension higher than four. For either very short or very long strings, we recover a four dimensional conformal behaviour, but for strings that have a size that explore the ρ-coordinate) we deviate from it. This suggest a form of deconstruction and provides some understanding to our flow within the pp-wave. In fact, this is the reason why we have constant masses/frequencies for the plane wave in (3.34) both for large and small values of the light-cone time u. In some sense, the result of the non-Abelian T-dual plane wave suggests a defect-CFT behaviour. To compare, we calculate in the appendix D the Penrose limit of the Janus solution, dual to a defect-CFT. We will find similarities with the non-Abelian pp-wave.

Conclusions
To close this paper, we present some final remarks.
We have found the plane wave geometries associated with the Abelian and non-Abelian T-duals of AdS 5 ×S 5 . These plane waves are consistent Type IIA backgrounds for a window of parameters and preserve sixteen supercharges.
We quantised the weakly coupled type IIA string on the Abelian T-dual plane wave and wrote the equations describing the frequencies of the non-Abelian T-dual case. The unconventional feature is that the non-Abelian plane wave presents a 'flow' in the frequencies (betwen two constant values).
After a careful analysis of the isometries of the backgrounds and global symmetries of the dual field theories we proposed a precise form for the BMN sector of operators described JHEP01(2018)071 by the plane waves. Scaling limits were analyzed. We gave intuitive arguments to explain the reason why our strings generate states with non-zero winding. Using deconstruction ideas, we presented arguments to explain the above-mentioned flow in eigenfrequencies for the non-Abelian T-dual plane wave.
It would be interesting to explore our BMN sector in more detail. In particular, to have more control over the necessity of non-zero winding and the discrepancy in eigenenergies for the string on the plane wave. Indeed, it is important to understand precisely the corrections discussed around (5.29). On the other hand, the proposal for deconstruction and our calculation supporting it should be further explored and extended.
It would be interesting to investigate the plane waves associated with other backgrounds obtained via non-Abelian T-duality. In order to study similarities and differences we suggest to study the backgrounds found in [17,25,31,34,[37][38][39]. These seem to be the natural candidates given their isometries.
This paper exploits a combination between non-Abelian T-duality in the context of supergravity and the study of the full string theory sigma model in (a limit of) that background. It should be interesting to push forward ideas that exploit the synergy between non-Abelian T-duality and an all order sigma model calculation. This kind of studies might illuminate the nature of non-Abelian T-duality.

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• Identify the scale with the boost, L = e β , and take this quantity to infinity. We obtain µ + t = x + and µ − t = x − /L 2 , or (with an extra factor of 2) which implies the expansion for the original coordinates Then for the derivatives we obtain ∂ + = ∂ t + ∂ µ and ∂ − = (∂ µ − ∂ t )/L 2 . In the context of the pp wave correspondence, one obtains the field theory map (here p − is conjugate to x + and p + to Method 2. But it was noted, originally by Tseytlin, that if µ is periodic (like it is often the case), the expansion for t and µ above implies that the lightcone time x + = (t + µ)/2 is periodic, which is not good. Instead, one can write the expansion In this form, lightcone time is identified with the usual time, which is not periodic, so we have a consistent story. Then also ∂ + = ∂ t + ∂ µ and ∂ − = ∂ µ /L 2 , which leads to the field theory map Both ways of taking the Penrose limit lead to the same pp wave metric in cases of interest. We can verify explicitly, that both in the case of AdS 5 × S 5 , and in the Janus case (next subsection), the resulting limits coincide. More generally, this is always true for the nontrivial terms, with (du) 2 times coordinates squared, since they come from the leading terms in the expansion of t, µ (which are unchanged), times the subleading terms in the transverse coordinates (which are also unchanged).
In a more general case, we can always fix the transformation with x + = t to have the same Penrose limit as the usual one, since we have the same number of conditions as extra constants to be fixed.
The only difference in this correct method amounts to the periodicities of the various coordinates, corresponding to periodicities in field theory. But these are important in our analysis, so in the text we will always use this method for taking the Penrose limit.

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Method 3. However, for completeness, we note that we could in principle also take the "maximally wrong" limit, and put all periodicity in x + , and none in the lightlike space coordinate. This would mean so also In this case ∂ + = ∂ t + ∂ µ and ∂ − = −∂ t /L 2 , so we find for the field theory map but the metric obtained from the limit is the same as for the first two methods, or in general can be put to the same form.
Generalization to motion in 2 coordinates. Consider a Penrose limit in coordinates (t, ξ, ρ) defined by (analogous to method 3 above) where u is lightcone time, and v is lightcone space.
The new (correct) Penrose limit, with no periodic time, is defined by (A.11) Since the leading terms (L-independent) in the Penrose expansion of (t, ρ, ξ) are not corrected, being fixed by the need to cancel the O(L 2 ) terms in the metric, it means that the nontrivial Penrose terms, proportional to du 2 and with coordinates squared as factors, are the same. Therefore all we need is to have the same result for the terms First, the leading order, O(L 2 ), terms cancel in the same way as before, since as we said, that fixed the leading terms in dt, dξ, dρ: − α 2 + γ 2 + δ 2 = 0. (A.13)

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Next, the condition for the first subleading, yet still infinite, O(L) terms to cancel gives a condition on the coefficients. In the original transformation, this was L du dw − 4 α 2 β + 2 δ = 0. (A.14) Now, it is We can also impose the condition that the constant term proportional to dw 2 to have the same coefficient. Before, we had whereas now we have But with the condition η = 2 α 2 β λ/γ from before, we get The condition that the constant term proportional to dudv is the same is as follows. Before, the term was As we see, in this case we had 3 equations for 3 unknowns, and the same matching will be true for motion in more general direction, involving several coordinates.

B More general CFTs: Gaiotto-Maldacena backgrounds
One puzzling result of (3.34) is that the masses for the pp-wave obtained from the Sfetsos-Thompson background depend on the affine parameter u. This motivates a question about the genericity of such phenomenon. In other words: is this a particularity of the Sfetsos-Thompson solution or is it more general? To investigate this, we study here the same type of Penrose limit discussed above, but applied to backgrounds classified by Gaiotto and Maldacena [56]. These solutions are dual to linear quiver field theories with N = 2 SUSY. The correspondence between CFT and geometry was very clearly spelled in [61,62]. Below, we start with a brief summary of the formalism developed in studying these backgrounds and then discuss the Penrose limit and pp waves obtained from them.

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The Gaiotto-Maldacena solutions can be written in terms of a potential function V (σ, η). In particular, denoting V = ∂ η V andV = σ∂ σ V , one can write the Type IIA generic solution as The radius of the space is µ 2 α = L 2 . The parameter µ is related to the number of branes present in the solution. In the following we set µ = 1, with no important effect on the results. The two-sphere dΩ 2 2 (χ, ξ) is parametrised by the angles χ and ξ with corresponding volume form dΩ 2 = sin χdχ ∧ dξ. The usual definition F 4 = dC 3 + A 1 ∧ H 3 is also used.
The problem of writing IIA/M-theory solutions in this class, reduces to finding the function V (σ, η) that solves a Laplace equation with a given charge density λ(η) and boundary conditions Interestingly, the background and fluxes depend onV ,V ,V and V = −σ −2V . Hence, givenV , we have all that is needed to write the Type IIA background.
The backgrounds discussed in sections 2 and 3 turn out to be particular cases of this general class of solutions. Indeed, for the Abelian T-dual background in (2.3), after redefining For the non-Abelian T-dual background in (2.3), redefining variables again as in (B.3), the potential function is In general, given a solution to the Laplace-like equation (B.2) that satisfies the boundary conditions, we have a background dual to a well defined CFT. We will now study the pp-wave obtained from these backgrounds.

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Notice that the function f 6 (σ = 0, η) =f 6 (η) = a, is assumed to be a constant. We will discuss the generality of the proposal and assumptions below. Now, we scale the coordinates as in previous sections, 4 This produces a pp-wave metric While the rest of the Neveu-Schwarz and Ramond fields are, in the large-L expansion,

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If we specify for the particular potential function in (B.5), we find the same masses we obtained for the non-Abelian pp-wave in (3.34). Same goes for the potential in (B.4) and the pp-wave of section 2, that in the Abelian T-dual case was already in Brinkmann coordinates. The general lesson this exercise teaches is that the masses of the pp-wave associated with Gaiotto-Maldacena backgrounds are dependent on the affine parameter. This, we believe, is an effect of the particular geodesic and expansion we are choosing.
As promised above, we now discuss the genericity of the expansion in (B.12). The general solution to a Gaiotto-Maldacena problem, as defined in (B.2) was discussed in the papers [61,62]. Here, we will consider a particular situation, where the solution is expanded for values small values σ ∼ 0. We propose a potential function, For this potential to satisfy the Laplace equation (B.2) we impose Every function in the expansion is given in terms of derivatives of the function F (η). For example, the Sfetsos-Thompson solution corresponds to F (η) = η 3 3 . Also, notice that the charge density -see (B.2)-is λ = aη for any solution in this family. Hence the second boundary condition is not satisfied (the quiver field theory is infinitely long).
It would be interesting to solve for the function F (η), that makes the coefficients for the masses in (B.16) the simplest possible.

C Geometric aspects of the pp-wave
In this appendix we collect useful formulas for various geometric quantities of the ppwave, such as the spin-connection, the Riemann tensor and the Ricci tensor. For a tendimensional pp-wave in Brinkmann coordinates with line element ds 2 = 2 du dv + where the non-vanishing components of η ab are η +− = η −+ = 1 and η ij = δ ij .

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From the previous frame, one can easily compute the components of the spinconnection, From the spin connection we find that the only non-vanishing components of the Riemann tensor are Also, the Ricci tensor has only one non-trivial component, namely

D Penrose limit of the Janus solution
In this appendix we consider the Penrose limit of the Janus solution, in order to explore the origin of the flow in the radial coordinate, and of the possible tachyon, for strings moving in the corresponding pp wave.

D.1 The solution
The Janusian solution [63] is a solution of the type IIB supergravity whose metric consists of a five-dimensional part that is asymptotically AdS 5 , and an internal part which is a five-dimensional sphere, S 5 . This solution can be seen as a deformation of the well-known AdS 5 × S 5 solution and in string frame it takes the form f (µ) dµ 2 + ds 2 AdS 4 + ds 2 S 5 , Φ = Φ(µ) , where Vol AdS 4 and Vol S 5 are the volume forms of AdS 4 and S 5 respectively and they are oriented in such a way that F 5 is self-dual. For later convenience we choose the following parametrization for AdS 4 and S 5 : AdS 4 = − cosh 2 r dt 2 + sinh 2 r dΩ 2 2 + dr 2 , ds 2 S 5 = dθ 2 + cos 2 θ dφ 2 + sin 2 θ dΩ 2 3 . (D. 2) The above ansatz is a solution of the type IIB supergravity provided that the dilaton Φ and the function f satisfy the first order differential equations In order to compare with the Penrose limits for T-dual solutions considered in the main text, we need to consider a Penrose limit giving a pp wave for motion in µ and φ, situated at r = 0, θ = 0. The condition for a null geodesic for motion on µ is g il ∂ l g µµ = 0, ∀l = µ, (D.5) which is indeed satisfied. For motion in φ, we obtain g θθ ∂ θ g φφ = 0 ⇒ sin θ cos θ = 0 , (D. 6) which implies θ = 0 or θ = π/2. The latter gives zero coefficient for dφ 2 , so we must put θ = 0 for this geodesic.
Considering a geodesic moving only in µ, we expand around it as Expanding and using cosh 2 r 1 + ρ 2 /L 2 , we get the metric the dilaton is now φ = φ(x + ), and the 5-form is We see that in this case, the overall function f (x + ) flows in the same way as in the solution before the Penrose limit. Note however that the pp wave solution is not in Brinkmann form.
Consider next the more general case of motion in both µ and φ. The Lagrangian for a null particle moving along the directions t, µ, φ is given by f μ 2 −ṫ 2 +φ 2 , (D.10) where dots stand for differentiation with respect to the affine parameter u. The Lagrangian is cyclic in (independent of) t and φ, meaning that we have the conservation laws ∂L ∂ṫ = p t = const , ∂L ∂φ = p φ = const . (D.11)

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We fix L −2 p t 2 = −e φ/2 fṫ = −1 , L −2 p φ 2 =φe φ//2 = J. (D.12) Then the null condition, L = 0, giveṡ Consider the expansion, near the null geodesic, r =r L , α a = y a L , a = 1, 2, 3, θ i = x i L , i = 1, . . . , 4 , (D.14) where we have not rescaled φ. For the coordinates along which the geodesic moves, we write the expansion The resulting geometry is 1 − J 2 f du ∧ dw ∧ dy 1 ∧ dy 2 ∧ dy 3 (D.20) From the above expression for the line element we can read the coefficients of dx i 2 , dy i 2 and dw 2 , where A x , A y and A w are functions of u. Then one can apply the coordinate transformations (see also (3.29)) (D. 22) Under these transformations the line element (D.19) becomes Similarly the RR form in (D.20) transforms as (D.26)

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To summarize, the pp-wave of the Janusian solution in Brinkmann form is (D. 28) Or, after implementing (D. 18), 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.