Penrose limits in non-Abelian T-dual of Klebanov-Tseytlin Background

In this paper we consider the Klebanov-Tseytlin background and its non-Abelian T-dual geometry along a suitably chosen $SU(2)$ subgroup of isometries. We analyse the Penrose limits along various null geodesics of both the geometries. We observe that, the Klebanov-Tseytlin geometry does not admit any pp-wave solutions. However, the T-dual background gives rise to pp-wave solution upon taking the Penrose limit along some appropriate null geodesic. We comment on the possible gauge theory dual for our pp-wave background.


Introduction
String theory on pp-wave background is being analysed extensively during the past several decades because they are endowed with a number of unique features [1][2][3][4][5]. These pp-wave solutions appear as Penrose limits of various supergravity backgrounds in ten and eleven dimensions [6][7][8]. They provide exact string theory backgrounds to all orders in α ′ as well as g s [9,10]. In the context of AdS/CFT correspondence these backgrounds give rise to the so called BMN sector with large R-charge of the dual N = 4 superconformal theory in four dimensions [7]. The AdS/CFT correspondence is used to construct interacting string states from perturbative gauge theory [7].
Recently pp-wave backgrounds have been constructed from non-Abelian T-dual geometries of various supergravity theories. Non-Abelian T-duality has turned out to be a wonderful tool to construct new supergravity backgrounds from known ones. This is a nontrivial generalization of the conventional T-duality where a non-Abelian isometry group is used for dualization [11]. However, these non-Abelian T-dualities are not symmetries of the full string theory [12]. They are used to relate the low energy supergravity theories among each other. Originally non-Abelian T-duality was formulated for the NS sector of supergravity theories. Subsequently, this formalism has been generalized to include the RR fields [13]. This, in turn played a crucial role in relating different supergravity backgrounds among each other. Several examples of new supergravity backgrounds have also been constructed using the non-Abelian T-duality [14][15][16][17][18][19][20][21].
Of particular interest in the present context is the impact of these developments in understanding several aspects of AdS/CFT correspondence [22][23][24][25][26][27][28][29]. This has opened up the possibility of constructing several new CFT duals corresponding to these non-Abelian T-dual geometries. Relationships between a number of these dual geometries with the Penrose limits [30] of some of the prevailing supergravity backgrounds have also been revealed [21][22][23]. An important development in this context is the non-Abelian T-dual of type-IIB supergravity compactified on certain orbifolds of AdS 5 × S 5 [27]. It has been shown that this geometry indeed admits plane wave solutions upon taking the Penrose limit along appropriate null geodesics [27]. A candidate for the field theory dual of this geometry has also been proposed. These developments have further been generalized for the non-Abelian T-duals of the Klebanov-Witten background, which results in placing a stack of D3 branes near a conifold singularity. The corresponding supergravity background, AdS 5 × T 1,1 , is obtained by blowing up the singularities of AdS 5 × S 5 orbifolds [31,32]. An appropriate SU(2) subrgoup of isometries of T 1,1 can be used to obtain the non-Abelian T-dual geometry [28]. These dual goemetries also give rise to pp-wave solutions upon considering the Penrose limits along appropriate null geodesics [33].
Placing a stack of M fractional along with N regular D3 branes at the conifold singularity gives rise to a N = 1 supersymmetric SU(N + M) × SU(N) gauge theory. The gravity dual is a nontrivial modification of the AdS 5 × T 1,1 background resulting the well-known Klebanov-Tseytlin geometry [34]. The goemetry admits non-Abelian isometries, an SU (2) subgroup of which is used to construct a non-Abelian T-dual background. This gives rise to a new massive type-IIA supergravity background [16,24]. In the present work we will analyse the Penrose limits of this massive type-IIA background in addition to the original Klebanov-Tseytlin background. We will show that, for the type-IIA theory, the resulting background indeed admits a pp-wave solution. The plan of this paper is as follows. In the next section we will review the Klebanov-Tseytlin background and its non-Abelian T-dual. Subsequently, in §3 we will analyse the Penrose limits and obtain pp-wave solution. In §4 we consider the supersymmetry analysis and show that the resulting pp-wave background preserves 16 supercharges. Finally, we will discuss some aspects of the dual quiver theory before summarising the results.

Klebanov-Tseytlin Background
Placing a stack of N regular and M fractional branes at a conifold singularity modifies the spacetime geometry in a non-trivial way. The gravity dual of this non-conformal SU(N +M)×SU(N) gauge theory has been accomplished in a pioneering work by Klebanov and Tseytlin [34]. The geometry of the resulting supergravity background is given by Here we will use the conventions of [35]. The warp factor is given by with η µν denoting the stranded (3 + 1)-dimensional Minkowski metric. The metric of T 1,1 is given by [32] ds 2 In the above, dΩ 2 2 (θ, φ) denotes the round metric on a two-sphere. For T 1,1 the parameters λ, λ 1 , λ 2 take the numerical values λ 2 1 = λ 2 2 = 1 6 , λ 2 = 1 9 . In addition, we need to specify non-vanishing background fields in NS-NS and RR sectors. The background NS-NS two form field B 2 has the expression with the corresponding field strengths The non-vanishing RR fields strengths F 3 and F 5 are given respectively by and Here, * 10 denotes the Hodge dual with respect to the ten dimensional metric (2.1). For convenience, in the above we have used the notation [34] . (2.8) The numbers N of regular D3 branes, and M of fractional D3 branes corresponds respectively to the flux of F 5 and F 3 . It is important to note that, the constant P is proportional to the number M of fractional D3 branes.
We will now consider the non-Abelian T-dual of the Klebanov-Tseytlin background. The non-Abelian T-duality with respect to an SU(2) isometry has been obtained in [16,24]. The corresponding metric of the T-dual geometry is given by where, the one form σ3 is defined as and the functions ∆ and V are given by The expression for the background NS-NS two formB 2 and dilatonΦ of the dual geometry are given bŷ (2.12) The field strengths corresponding to the RR sector arê

The Penrose Limits
In this section we will study Penrose limits for both of the above backgrounds. We will first consider the original type-IIB background. The Penrose limit along a suitable null geodesics for this background has already been studied in [36]. Here we will first outline the main result of this work. Considering the motion of a massless particle in the (r, ψ) plane of the background results the following geometry The background gauge fields behave as It has been noted that [36] this background leads to pp-wave upon setting P = 0. As we have noted earlier, the constant P is proportional to M. Thus, setting P to zero amounts to removing the fractional D3 branes, there by restoring to the undeformed Klebanov-Witten background. This indicates that the deformed background does not support pp-wave upon taking Penrose limits. In appendix A, we consider an extensive study of Penrose limits along the remaining null geodesics. Some of these geometries become singular where as some other are smooth. Nevertheless none of these limits give rise to pp-wave solution. However as we will see in the following, this is not the case upon considering the non-Abelian T-duality. Thus, non-Abelian T-duality gives rise to new exactly solvable backgrounds that are absent in the original type IIB configuration.
In order to carry out the Penrose limit along appropriate null geodesics of the non-Abelian T-dual background, we will first rescale various quantities appropriately. Let us first consider the wrap factor H(r). Introducing the parameterr via we can rewrite it as We will now rescale the Minkowski coordinates x µ → L 2 x µ and the T-dual coordinates v 2,3 → L 2 v 2,3 . In terms of these rescaled coordinates, the T-dual metric (2.9) becomes In the above, for easy reading we have introduced the notation After the rescaling, the NS-NS two formB 2 and the dilatonΦ becomeŝ Similarly, the field strengths in the RR sectors becomeŝ ln r r + 1 9 We will now consider the Penrose limits of the above T-dual background along appropriate null geodesics. Denoting the spacetime coordinates as {x µ }, the geodesic equation is expressed as Here we use u to denote the affine parameter along the geodesic. We are interested to examine the motion along various isometry directions. Denote x λ one such isometry direction. Thus, we need to set the velocity and acceleration along any direction x µ , µ = λ zero: Substituting the above in (3.7), we find that the geodesic equation for motion along an isometry direction takes the simple form In additon to the above condition, we need to impose ds 2 = 0 in order to obtain null geodesics for our purpose.
Let us now focus on various isometry directions of the T-dual geometry. A quick inspection of the rescaled geometry (3.4) indicates that both ψ and φ 1 are isometry directions. Let us first consider the motion along ψ direction. The geodesics equation for this case is (3.10) From (3.4) we note the relevant component of the metric: The metric component g ψψ depends upon r, v 2 and v 3 . For µ = r, the geodesic condition (3.10) leads to v 2 = 0. Similarly, for µ = v 2 , v 3 we obtain {r =r, v 2 = 0}. However, for all the above values, the metric component g ψψ in (3.11) vanishes leading to singular geometries. In the following, we will no longer consider Penrose limits for such singular geometries.
Finally, we will consider Penrose limit around θ 1 = π 2 , v 2 = 0 = v 3 , keeping the rcoordinate constant, i.e., r = c for some constant c =r = 0. Consider the following expansion around this geodesic: In addition, we do the rescaling of the coordinates v 2 and v 3 as v L while keeping the ψ-coordinate unchanged. In the above, a and b are some constant parameters. The null geodesic condition relates the parameters a, b and c as: (3.14) Using the above expansion we consider the leading terms of the T-dual metric in the limit L → ∞. We find This contains a divergent term which can't be remove for any choice of the parameter b. Note that, from the null geodesic condition (3.14), setting b = 0 is not allowed. Hence, motion along the isometric direction φ 1 by keeping r = constant does not lead to any smooth geometry. We can repeat similar analysis for motion along the isometry direction ψ. Recall that the ψψ component of the T-dual metric is given as The null geodesic condition for µ = r leads to v 2 = 0, where as, for µ = v 2 , v 3 we obtain {r =r, v 2 = 0}. However, the metric component g ψψ vanishes for all these values. Thus, we do not have a regular geometry for any of the above geodesics.
To the end we will consider null geodesics for the motion of a particle carrying nonzero angluar momentum in the (r, φ 1 ) plane. We will subject our analysis to a small neighbourhood of θ 1 = π 2 and v 2 = v 3 = 0. Consider the Lagrangian for a massless particle moving along this geodesic: Let u be the affine parameter along the geodesic. The dots in the above equation correspond to derivative with respect to u. Substituting the explicit expression for the background metric (3.4) in the above Lagrangian we find We will now obtain the conserved quantities corresponding to the above system. Note that, the Lagrangian (3.19) does not depend on the generalized coordinates t and φ 1 explicitly. Denoting −EL 2 to be the conserved momentum associated with t, we find Similarly, let −JL 2 be the conserved momentum associated with the generalized coordinate φ 1 . We find In addition, we will require the geodesic to be null. This gives rise to the condition: We will now concentrate on obtaining the Penrose limit for a null geodesic carrying angular momentum J around x i = 0, i = 1, 2, 3 , θ 1 = π 2 and v 2 = v 3 = 0. We redefine the coordinates as We keep the ψ-coordinate unchanged, and redefine the string coupling as g s = L 3g s , in order to keep the dilaton finite at the Penrose limit. Finally, we will consider the following expansion in the limit L → ∞: We need to determine the coefficients c i . Requiring the geodesic to be null determines the values of the coefficients c 1 , c 2 and c 4 as follows: We now substitute the expansion (3.24) in the T-dual metric (3.4) and retain the leading terms. Apriory, this metric will contain divergent terms of order L and L 2 . Imposing the null geodesic condition automatically cancels the O(L 2 ) terms. It can be easily verified that the O(L) term is removed upon setting Using the value of c 2 and c 4 from (3.25) in the above equation we can express the coefficient c 3 in terms c 5 as We will see later that the coefficient c 5 can be determined by requiring the background fields to satisfy the Einstein's equations. Finally, we need to determine the coefficient c 6 . This is easily obtained by upon setting appropriate normalization for the cross term dudv in the metric. We find Substituting the above results in (3.4), and taking the limit L → ∞, we find the pp-wave metric of the form We will subsequently show that this is indeed a pp-wave solution by rewriting it in the standard Brinkmann form. Let us now consider the Penrose limit for the remaining background fields. In this limit, the NS-NS two-form field and dilaton takes the form The field strengths corresponding to the RR sector are given bŷ For later use, we will also compute the field strengthĤ 3 corresponding to the NS-NS two-formB 2 :  Note that, in obtaining the above, we have used dr = c 2 du where the expression for the coefficient c 2 is given by (3.25).
As pointed out earlier, the metric obtained in (3.29) is not in the standard Brinkmann form [5]. A formalism has been developed in [27] in order to transform the line element to the Brinkmann form. Following [27] consider a line element of the form Now, replace the coordinates x i and v as The line element in (3.33) now takes the Brinkmann form with the functions F i being For the case of our pp-wave metric (3.29) we have (3.37) Hence after making the following replacement we find where the functions F i can be read from the expression (3.36).
We will now express the background fields in the Brinkmann form. The dilatonΦ and the NS-NS three form fluxĤ 3 are given as Similarly, the expressions for the RR field strengths are found to be of the form We will now verify that these fields indeed satisfy the Bianchi identities and the gauge field equation of motion. A quick inspection of the background fields in (3.40) -(3.41) shows that the Bianchi identities hold. The field strengthsĤ 3 ,F 2 andF 4 are all closed and bothF 0 as well asĤ 3 ∧F 2 are indeed zero.
Let us now inspect the type-IIA supergravity equations for the gauge fields The Hodge duals for the above background fields are In deriving the above, we have used det(g pp ) = −v 2 2 , g vv = −g uu , g uv = g vu = 1 , g ij = δ ij only g ψψ = It is straightforward to see that, the background fields (3.40)-(3.41) together with (3.44) indeed satisfy the gauge field equations for type-IIA supergravity. Both ⋆Ĥ 3 as well as e −2Φ ⋆Ĥ 3 are closed. Also,F 2 ∧ ⋆F 4 andF 4 ∧F 4 vanish identically. Thus, the first of the equations in (3.43) is satisfied. Similarly, both ⋆F 2 and ⋆F 4 are exact forms. In addition, H 3 ∧F 4 andĤ 3 ∧⋆F 4 vanish as well. Thus, the last two equations in (3.43) are also satisfied.
It is interesting to note that the gauge field equations as well as the Bianchi identities hold irrespective of the value of the coefficient c 5 . However, as we will see in the following, this is not the case with the Einstein's equations. For type-IIA supergravity, the Einstein's equations are given aŝ

.(3.45)
Similarly, the dilation equations arê In the appendix we have analysed these equations in detail. We find that the equation of motion for dilation (3.46) holds automatically. In addition, we observe that the Einstein's equations (3.45) are trivially satisfied for all values of µ, ν except for µ = ν = u. In this case we have the nontrivial condition

Supersymmetry of pp-wave
The supersymmatry analysis of non-Abelian T-dual backgrounds have been studied extensively [14][15][16][37][38][39]. Unlike the AdS 5 × S 5 case, the non-Abelian T-dual of Klebanov-Witten as well as the Klebanov-Tseytlin background preserves all the supersymmetries of the original background. This is because in the later two cases, the Killing spinor of the original background does not carry any SU(2) charge of the isometry group used for non-Abelian T-dualization [14,39]. In this context, it is worth investigating whether the pp-wave we obtained in the above preserves any supersymmetry. In order to analyse this, we will first introduce the Brinkmann coordinates X i such that In these coordinates the pp-wave background (3.39)-(3.41) reads as where we have introduced the notation The functions F ij are defined by Now we introduce the frame {e a } as Hdu , e i = dX i , (4.5) such that the pp-wave metric (4.2) can be written as with η +− = η −+ = 1 and η ij = δ ij . The non-vanishing components of spin-connections are given by In terms of the frame (4.5), the background fields (4.2) take form (4.8) We will now analyse the spinor conditions in detail. Consider the supersymmetric variations of the dilatino and gravitino Here we follow the conventions of [15,27]. In particular, we have the covariant derivative D µǫ = ∂ µǫ + 1 4 ω ab µ Γ abǫ , and in addition we use the notation ✓ ✓ F n ≡F i 1 ...in Γ i 1 ...in . In the above, σ i denote the Pauli matrices. The Killing spinorǫ consists of real Majorana-Weyl spinorsǫ ± , such thatǫ In type-IIA supergravity,ǫ satisfies Γ 11ǫ = −σ 3ǫ . We also introduce We now proceed to solve the spinor conditions. Substituting the background fields in (4.9), and setting the dilatino variation to zero, we obtain after some simplification (4.12) The above condition holds provided Γ −ǫ = 0. This indicates that, subject to the compatibility with the gravitino variation, the pp-wave background (4.2) preserves 16 supercharges. We now proceed to varify the spinor condition arising from the variation of the gravitino. Let us first consider the δψ + variation. The NS-NS three-form does not have any leg along e + . Together with Γ +ǫ = Γ −ǫ = 0, the variation δψ + = 0 leads to ∂ +ǫ = 0. Thus, we find that the Killing spinorǫ is independent of v, i.e.ǫ =ǫ(u, X i ). Now we focus on the variation δψ i , i = 1, ..., 8. The vanishing of δψ i implies that where we have introduced the notation Now, Γ − anticommutes with R and Γ −ǫ = 0. Thus, we have ∂ iǫ = 0 leading toǫ = χ(u) for some χ(u) satisfying Γ − χ(u) = 0. Finally, we consider the variation δψ − = 0. Note that, in this case the covariant derivative D − becomes After some simplification, we find that the condition δψ − = 0 gives rise to This proves that the gravitino condition is compatible with the dilatino variation for the above choice of χ(u), provided Γ − χ 0 = 0. Thus, from the above analysis we find that the pp-wave background (4.9) indeed preserves 16 supercharges.

Gauge theory duals
It is well known that the field theory dual to the Klebanov-Tseytlin geometry consists of a nonconformal N = 1 supersymmetric SU(N + M) × SU(N) gauge theory [34]. It describes the dynamics of N regular and M fractional D3 branes placed near a conifold singularity. The fractional D3 branes arise due to D5 branes wraping the vanishing twocycle at conifold singularity. The nonconformal gauge theory has a nontrivial RG flow. Near UV the supergravity description is valid and the dual geometry is given by the Klebanov-Tseytlin background. As the theory is flown to IR it undergoes to a cascade of Seiberg dualities there by changing the number of D3 branes from N to N − M in each step, resulting a singular geometry at the end. However, for suitably chosen initial condition the conifold geometry gets deformed at IR by strong coupling effects there by leading to the Klebanov-Strassler background [40].
For the non-Abelian T-dual geometry of the Klebanov-Tseytlin background the field theory dual has been considered in [16,24]. The existance of domain wall configurations play a key role in understanding the dual field theories. In the type-IIB theory domain walls can be formed by wraping D5 branes on suitably chosen two cycles of the internal manifold. For the Klebanov-Tseytlin background (2.1)-(2.4) one such two cycle can be constructed upon the identification: with a constant ψ 0 . This gives rise to the following two cycle Σ 2 for the T-dual background (3.4)-(3.6): Furthermore, it is possible to construct a three cycle Σ 3 in the T-dual geometry as Following [24] we will now analyse the construction of domain wall in the Klebanov-Tseytlin background and its T-dual. For the Klebanov-Tseytlin background we will consider the domain wall formed by a D5 brane extended along R 1,2 ∈ R 1,3 of the (1, 3) Minkowski spacetime and wraping the compact directions parametrized by {θ 2 , φ 2 , ψ}. The dynamics of the low energy excitations are captured in terms of the corresponding Born-Infeld action on the world volume of the D5 brane. This gives rise to the corresponding effective tension.
The non-Abelian T-duality is performed along an SU(2) isometry parametrized by the coordinates {θ 2 , φ 2 , ψ}. Thus the D5 brane wraping the SU(2) directions gives rise to a D2 brane extending along the R 1,2 of the T-dual geometry. We will place this domain wall at the origin of the internal manifold: Once again we can consider the corresponding Born-Infeld action and compute the effective tension for it. The effective tension of the domain wall in the Klebanov-Tesytlin background matches with the effective tension of the corresponding configuration in the T-dual geometry upto a constant factor [24]. In addition, it has been shown that [24] the central charge as well as entanglement entropy of both the theories match upto an RG independent coefficient. While T-duality maintains the essential features of the central charge and entanglement entropy, this is not the case for the four dimensional gauge coupling.
The T-dual geometry gives rise to a very unusual behaviour for the gauge coupling. It has been demonstrated that 1 /g 2 ∼ ln r 3 /2 , unlike the case for a conventional field theory where a logarithmic behaviour is observed.
The Maxwell and Page charges of the D-branes in the theory also play a significant role in understanding the field theory dual. For the Klebanov-Tseytlin background we consider D3 and D5 brane charges Q Max, D3 = 1 2k 2 10 T D3 T 1,1 and where Q(r) = K(r) + P T (r) and we choose the normalization factor as described in [18]. For the T-dual background, the Maxwell and Page charges of D6 and D8 are given respectively asQ The above shows that, after dualization we find D8 branes for each of the D5 branes and twice the number of D6 branes for each of the D3 branes in the original background. It has been noticed in [24] that the changes induced in the page charge of D3 brane in the Klebanov-Tseytlin background by a large gauge transformation of the NS-NS two from B 2 is the same as the changes in the Maxwell charge by a suitable change in the radial coordinate. Similar phenomenon is observed in the dual gauge theory, where the page charge of the D6 brane now undergoes a shift under the large gauge transformation. This suggests that the quiver theory corresponding to the T-dual geometry undergoes to a cascade of Seiberg dualities much the same way as the gauge theory corresponding to the original geometry. Since the change in D6 brane charge is twice the change in D3 charge, the T-dual theory undergoes a Seiberg duality by a change of 2M units of D6 brane charge for a change of M units of the D3 brane charge in the Klebanov-Tseytlin background.
We will now consider the Maxwell and Page charges for pp-wave background. Recall that the RR field strengths for this background in Brinkmann coordinates are given aŝ The Maxwell and page charge for various brane in type-IIA theory is given bŷ SinceF 0 is zero for our background, the D8 charges are all zero. Moreover, the Maxwell and Page charges for D6 branes are both equal. The Maxwell and Page charges for D2 brenes also vanish. We havê For fixed v 2 , bothF 4 andF 6 are zero andF 0 as well asB 2 ∧F 2 vanish for the pp-wave background. From (5.9), we find that there is no longer any cascading due to large gauge transformation ofB 2 . This indicates that the quiver theory dual to the pp-wave geometry correspond to the end point of the cascade.

Conclusion
In this paper we have considered Penrose limits for the Klebaov-Tseytlin geometry and its non-Abelian T-dual around a suitable SU(2) isometry. We have scrutinized various null geodesics in these geometries. A direct investigation of the Penrose limits for the Klebanov-Tseytlin geometry gives rise to singular geometries for most of the null geodesics. We found one smooth geometry with a nonvanishing scalar curvature. However upon taking the non-Abelian T-duality results a pp-wave solution around a suitably chosen null geodesic. The holographic dual of the T-dual background exhibits a cascade of Seiberg dualities under large gauge transformation of the NS-NS two form there by reducing the number of D6-branes in each step. However, an analysis of the Maxwell and Page chrages shows the absence of similar phenomenon for the pp-wave background. Thus, the holographic dual in this case appears to be the end point of the cascade of quivers corresponding to the T-dual geometry. The gauge coupling analysis shows that the quiver in the Tdual case is a non-conventional field theory. Further investigation is required to precisely identify the quiver and also the corresponding BMN sector and to establish a map between holographic quantities and field theory observables. It would also be interesting to explore the possibility of obtaining pp-wave geometries for the non-Abelian T-dual of Klebanov-Strassler background as well as backgrounds with AdS 3 factors. Dualization of the Baryonic branch of the Klebanov-Strassler geometry has already been carried out [29]. We hope to address some of these issues in future.

A Penrose limits in Klebanov-Tseytlin background
In this appendix we will examine Penrose limits for various null geodesics in the Klebanov-Tseytlin geometry. Recall the metric of Klebanov-Tseytlin background The T 1,1 metric is given by The parameters λ, λ 1 , λ 2 in the T 1,1 metric have the numerical values λ 2 1 = λ 2 2 = 1 6 and λ 2 = 1 9 . We will now rescale the Minkowski coordinates (x µ ) as x µ → L 2 x µ . The metric then becomes The above metric has U(1) isometries along φ 1 , φ 2 and ψ directions. We will examine the Penrose limits by considering the motion along these isometry directions. Let us first consider motion along ψ-direction. The ψψ-component of the metric is given by Now by imposing the null geodesic condition we get which does not admit any smooth solution. Now we shall consider motion along the φ 1 -direction. The relevant metric component is The geodesic condition for µ = r does not give any solution. For µ = θ 1 the condition leads to the solution r =r and θ 1 = {0, π 2 , π}. However, r =r is a singular point as the metric component g φ 1 φ 1 vanishes for this value of r. Hence we shall consider Penrose limit around θ 1 = {0, π 2 , π}, θ 2 = φ 2 = ψ = 0 while the r-coordinate fixed. Consider the following expansion around the geodesic θ 1 = θ 2 = φ 2 = ψ = 0 and r = constant: where a, b & c are some nonvanishing parameters. The null geodesic condition gives For the above expansion, the leading terms of the T-dual metric in the limit L → ∞ are given by We can see that the divergent term of order O(L) can't be removed for any choice of the parameters a, b, c. Now we consider the following expansion around the geodesic θ 1 = π 2 and θ 2 = φ 2 = ψ = 0 and r = constant: The null geodesic condition gives The leading terms of the T-dual metric in the limit L → ∞ are given by Once again, the metic is divergent in the limit L → ∞ due to presence of O(L) term.
We will now consider a null geodesic which carries angular momentum. To obtain such a geodesic, we consider motion along r and φ 1 directions and concentrate in a small neighbourhood of θ 1 = θ 2 = φ 2 = ψ = 0. The Lagrangian for a massless particle moving along this geodesic is We notice that the above Lagrangian does not depend on t and φ 1 explicitly. Hence the momenta conjugate to the generalized coordinates t and φ 1 are conserved. Denoting these quantities by E and J (upto a factor of −L 2 ), we find 14) The condition that the geodesic becomes null gives rise tȯ To obtain the Penrose limit, we redefine the coordinates as 16) and consider the following expansion in the limit L → ∞: Finally, we will consider the expansion around θ 1 = π 2 and concentrate in a small neighbourhood of θ 2 = φ 2 = ψ = 0 . The Lagrangian for a massless particle then gives rise to there by giving rise to a smooth geometry. However, a straightforward calculation gives rise to a nonvanishing scalar curvature for this geometry. Hence, this does not correspond to a pp-wave.

B Einstein's Equations
In this appendix, we will analyse the Einstein's equations for our pp-wave background. . (B.1) Here we use the conventions of [15]. In particular, we haveĤ 2 µν =Ĥ µαβĤνρσ g αρ g βσ and similar expressions for (F 2 2 ) µν and (F 2 4 ) µν . The equation of motion for the dilation is given byR We will first focus on the dilation equation. For the pp-wave backgroundR = 0. Now, consider evaluating D 2Φ . Note that D 2Φ = g µν D µ D νΦ = g uv D u D vΦ + g vu D v D uΦ + g vv D v D vΦ + g ij D i D jΦ , (B.3) for i, j = {u, v}. Consider the covariant derivatives of the form D µ D νΦ appearing in the above equation. Since, ∂ vΦ = 0 = ∂ iΦ , we find It is straightforward to evaluate the Christoffel symbols. We find Γ u uv = 1 2 g uv ∂ v g vu + ∂ u g vv − ∂ v g uv = 0 ,