The AdS 5 non-Abelian T-dual of Klebanov-Witten as a \( \mathcal{N}=1 \) linear quiver from M5-branes

Journal of High Energy Physics, Sep 2017

In this paper we study an AdS5 solution constructed using non-Abelian T-duality, acting on the Klebanov-Witten background. We show that this is dual to a linear quiver with two tails of gauge groups of increasing rank. The field theory dynamics arises from a D4-NS5-NS5’ brane set-up, generalizing the constructions discussed by Bah and Bobev. These realize \( \mathcal{N}=1 \) quiver gauge theories built out of \( \mathcal{N}=1 \) and \( \mathcal{N}=2 \) vector multiplets flowing to interacting fixed points in the infrared. We compute the central charge using a-maximization, and show its precise agreement with the holographic calculation. Our result exhibits n 3 scaling with the number of five-branes. This suggests an eleven-dimensional interpretation in terms of M5-branes, a generic feature of various AdS backgrounds obtained via non-Abelian T-duality.

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The AdS 5 non-Abelian T-dual of Klebanov-Witten as a \( \mathcal{N}=1 \) linear quiver from M5-branes

Received: June The AdS5 non-Abelian T-dual of Klebanov-Witten as Georgios Itsios 0 1 2 5 6 Yolanda Lozano 0 1 2 3 6 Jesus Montero 0 1 2 3 6 Carlos Nun~ez 0 1 2 4 6 0 Swansea SA2 8PP , United Kingdom 1 Avda. Calvo Sotelo 18 , 33007 Oviedo , Spain 2 R. Dr. Bento T. Ferraz 271, Bl. II , Sao Paulo 01140-070, SP , Brazil 3 Department of Physics, University of Oviedo 4 Department of Physics, Swansea University 5 Instituto de F sica Teorica, UNESP-Universidade Estadual Paulista 6 Bobev. These realize In this paper we study an AdS5 solution constructed using non-Abelian Tduality, acting on the Klebanov-Witten background. We show that this is dual to a linear quiver with two tails of gauge groups of increasing rank. The eld theory dynamics arises from a D4-NS5-NS5' brane set-up, generalizing the constructions discussed by Bah and = 1 quiver gauge theories built out of N String Duality; AdS-CFT Correspondence; Conformal Field Models in String - M5-branes = 1 and N = 2 vector multiplets owing to interacting xed points in the infrared. We compute the central charge using a-maximization, and show its precise agreement with the holographic calculation. Our result exhibits n3 scaling with the number of ve-branes. This suggests an eleven-dimensional interpretation in terms of M5-branes, a generic feature of various AdS backgrounds obtained via non-Abelian T-duality. Theory 1 Introduction 1.1 General framework and organization of this paper 2 The non-Abelian T-dual of the Klebanov-Witten solution 2.1 2.2 The non-Abelian T-dual solution 2.2.1 2.2.2 2.2.3 Asymptotics Quantized charges Central charge 3 Basics of Bah-Bobev 4d N = 1 theories 3.1 N = 1 linear quivers 3.2 IIA brane realization and M-theory uplift 4.1 4.2 Proposed N = 1 linear quiver -functions and R-symmetry anomalies 4.3 Field-theoretical central charge 5 Solving the INST-BBBW puzzle 6 Conclusions and future directions 4 The non-Abelian T-dual of Klebanov-Witten as a N = 1 linear quiver A Connection with the GMSW classi cation A.1 Uplift of the non-Abelian T-dual solution A.2 Review of GMSW A.3 Recovering the non-Abelian T-dual from GMSW B The Abelian T-dual of the Klebanov-Witten solution B.1 Background B.2 Quantized charges and brane set-up C Some eld theory elaborations C.1 A summary of the Klebanov-Witten CFT C.2 Central Charge of the N = 2 UV CFT C.3 Central charge of the Klebanov-Witten theory modded by Zk Introduction Non-Abelian T-duality [1], the generalization of the Abelian T-duality symmetry of String Theory to non-Abelian isometry groups, is a transformation between world-sheet eld theories known since the nineties. Its extension to all orders in gs and 0 remains however a technically-hard open problem [2{8]. As a result, non-Abelian T-duality does not stand as a String Theory duality symmetry, as its Abelian counterpart does. In the paper [9], Sfetsos and Thompson reignited the interest in this transformation by highlighting its potential powerful applications as a solution generating technique in supergravity. An interesting synergy between Holography (the Maldacena conjecture) [ 10 ] and non-Abelian T-duality was also pointed out. This connection was further exploited These works have widely applied non-Abelian T-duality to generate new AdS backgrounds of relevance in di erent contexts. While some of the new solutions avoid previously existing classi cations [11, 28, 31, 32], which has led to generalizations of existing families [38{41], some others provide the only known explicit solutions belonging to a given family [32, 33], which can be used to test certain conjectures, such as 3d-3d duality [42, 43]. Some of these works also put forward some ideas to de ne the associated holographic duals. Nevertheless, these initial attempts always encountered some technical or conceptual puzzle, rendering these proposals only partially satisfactory. It was in the papers [44{46], where the eld theoretical interpretation of non-Abelian T-duality (in the context of Holography) was rst addressed in detail. One outcome of these works is that non-Abelian T-duality changes the dual eld theory. In other words, that new AdS backgrounds generated through non-Abelian T-duality have dual CFTs di erent from those dual to the original backgrounds. This is possible because, contrary to its Abelian counterpart, non-Abelian T-duality has not been proven to be a String Theory symmetry. The results in [44{46] open up an exciting new way to generate new quantum eld theories in the context of Holography. In these examples the dual CFT arises in the low energy limit of a given Dp-NS5 brane intersection. This points to an interesting relation between AdS non-Abelian T-duals and M5-branes, that is con rmed by the n3 scaling of the central charges. Reversing the logic, the understanding of the eld theoretical realization of non-Abelian T-duality brings in a surprising new way (using Holography!) to extract global information about the new backgrounds. Indeed, as discussed in the various papers [2{8], one of the long-standing open problems of non-Abelian T-duality is that it fails in determining global aspects of the dual background. The idea proposed in [44] and further elaborated in [45, 46], relies on using the dual eld theory to globally de ne (or complete) the background obtained by non-Abelian T-duality. In this way the Sfetsos-Thompson solution [9], constructed acting with non-Abelian Tduality on the AdS5 S5 background, was completed and understood as a superposition of Maldacena-Nun~ez solutions [47], dual to a four dimensional CFT. This provides a global de nition of the background and also smoothes out its singularity. This idea was also put to work explicitly in [45] in the context of N = 4 AdS4 solutions. In this case the nonAbelian T-dual solution was shown to arise as a patch of a geometry discussed in [48{51], { 2 { In this paper we follow the methods in [44] to propose a CFT interpretation for the dual to the renormalization xed point of a T ^(SU(N )) quiver eld theory, belonging to the general class introduced by Gaiotto and Witten in [52]. In the two examples discussed in [44, 45] the non-Abelian T-dual solution arose as the result of zooming-in on a particular region of a completed and well-de ned background. Remarkably, this process of zooming-in has recently been identi ed more precisely as a Penrose limit of a well-known solution. The particular example studied in the paper [46], a background with isometries R SO(3) SO(6), was shown to be the Penrose limit of a given Superstar solution [53]. This provides an explicit realization of the ideas in [44] that is clearly applicable in more generality. N = 1 AdS5 background obtained in [12, 13, 28], by acting with non-Abelian T-duality on a subspace of the Klebanov-Witten solution [54]. We show that, similarly to the examples in [44, 45], the dual CFT is given by a linear quiver with gauge groups of increasing rank. The dynamics of this quiver is shown to emerge from a D4-NS5-NS5' brane construction that generalizes the Type IIA brane set-ups discussed by Bah and Bobev in [55], realizing N = 1 linear quivers built out of N = 1 and N = 2 vector multiplets that ow to interacting xed points in the infrared. These quivers can be thought of as N = 1 twisted compacti cations of the six-dimensional (2; 0) theory on a punctured sphere, thus providing a generalization to N = 1 of the N = 2 CFTs discussed in [56]. The results in this paper suggest that the non-Abelian T-dual solution under consideration could provide the rst explicit gravity dual to an ordinary N = 1 linear quiver associated to a D4-NS5 brane intersection [55]. In this construction, the N = 2 SUSY D4-NS5 brane set-up associated to the Sfetsos-Thompson solution (see [44]) is reduced to N = 1 SUSY through the addition of extra orthogonal NS5-branes, as in [55]. The quiver that we propose does not involve the TN theories introduced by Gaiotto [57], and is in contrast with the classes of N = 1 CFTs constructed in [58{61]. We support our proposal with the computation of the central charge associated to the quiver, which is shown to match exactly the holographic result. We also clarify a puzzle posed in [12, 13], where the non-Abelian T-dual background was treated as a solution in the general class constructed in [58, 59], involving the TN theories, whose corresponding central charge was however in disagreement with the holographic result. Before describing the plan of this paper, let us put the present work in a wider framework, discussing in some more detail the general ideas behind it. 1.1 General framework and organization of this paper In the papers [12, 13], the non-Abelian T-dual of the Klebanov-Witten background was constructed. There, it was loosely suggested that the dual eld theory could have some relation to the N = 1 version of Gaiotto's CFTs. Indeed, following the ideas in [60], the nonAbelian T-dual of the Klebanov-Witten solution could be thought of as a mass deformation of the non-Abelian T-dual of AdS5 S5=Z2, as indicated in the following diagram, AdS5 AdS5 S5=Z2 mass T 1;1 / NATD of AdS5 Nevertheless, there were many unknowns and not-understood subtle issues when the papers [12, 13] were written. To begin with, the dual CFT to the non-Abelian T-dual of S5 was not known, the holographic central charge of such background was not expressed in a way facilitating the comparison with the CFT result, the important role played by large gauge transformations [19, 25] had not been identi ed, etc. In hindsight, the papers [12, 13] did open an interesting line of research, but left various uncertainties and loose ends. This line of investigations evolved to culminate in the works [44{46], that gave a precise dual eld theoretical description of di erent backgrounds obtained by non-Abelian T-duality. This led to a eld-theory-inspired completion or regularization of the non notably, the central charge is a quantity that nicely matches the eld theory calculation with the holographic computation in the completed (regulated) background. In this paper we will apply the ideas of [44{46] and the eld theory methods of [55] to the non-Abelian T-dual of the Klebanov-Witten background. A summary of our results is: We perform a study of the background and its quantized charges, and deduce the Hanany-Witten [62] brane set-up, in terms of D4 branes and two types of ve-branes NS5 and NS5'. We calculate the holographic central charge. This requires a regularization of the background, particularly in one of its coordinates. The regularization we adopt here is a hard-cuto . Whilst geometrically unsatisfactory, previous experience in [44] shows that this leads to sensible results, easy to compare with a eld theoretical calculation. Based on the brane set-up, we propose a precise linear quiver eld theory. This, we conjecture, is dual to the regulated non-Abelian T-dual background. We check that the quiver is at a strongly coupled xed point by calculating the beta functions and R-symmetry anomalies. The quiver that we propose is a generalization of those studied in [55]. It can be thought of as a mass deformation of the N = 2 quiver dual to the non-Abelian Tdual of AdS5 S5=Z2, that is constructed following the ideas in [44]. It is the presence of a avor group in the CFT that regulates the space generated by non-Abelian Tduality. We calculate the eld theoretical central charge applying the methods in [55]. We nd precise agreement with the central charge computed holographically for the regulated non-Abelian T-dual solution. In more detail, the present paper is organized as follows. In section 2, we summarize the main properties of the solution constructed in [12, 13]. We perform a detailed study of the quantized charges, with special attention to the role played by large gauge transformations. Our analysis suggests a D4, NS5, NS5' brane set-up associated to the solution, similar to { 4 { that associated to the Abelian T-dual of Klebanov-Witten, studied in [63, 64]. In section 3 we summarize the brane set-up and N = 1 linear quivers of [55], which we use in section 4 for the proposal of a linear quiver that, we conjecture, is dual to the regulated version of the non-Abelian T-dual solution of AdS5 T 1;1. We provide support for our proposal with the detailed computation of the ( eld theoretical) central charge which we show to be in full agreement with the (regulated) holographic result. We give an interpretation for the eld theory dual to our background in terms of a mass deformation of the N = 2 CFT associated to the non-Abelian T-dual of AdS5 S5=Z2. This suggests the geometrically sensible way of completing our background. Section 5 contains a discussion where we further elaborate on the relation between our proposal and previous results in [12, 13]. We also resolve a puzzle raised there regarding the relation between the non-Abelian T-dual solution and the solutions in [59]. Concluding remarks and future research directions are presented in section 6. Detailed appendices complement our presentation. In appendix A, we explicitly calculate the di erential forms showing that the non-Abelian T-dual solution ts in the classi cation of [65], for N = 1 SUSY spaces with an AdS5-factor. Appendix B studies in detail the relation between the non-Abelian T-dual solution and its (Abelian) T-dual counterpart. Finally in appendix C we present some eld theory results relevant for the analysis in section 4. 2 The non-Abelian T-dual of the Klebanov-Witten solution In this section we summarize the Type IIA supergravity solution obtained after a nonAbelian T-duality transformation acts on the T 1;1 of the Klebanov-Witten background [54]. This solution was rst derived in [12, 13]. It was later studied in [28] where a more suitable set of coordinates was used. More general solutions in Type IIA were constructed in [26] as non-Abelian T-duals of AdS5 Y p;q Sasaki-Einstein geometries. Following our paper, the study of their dual CFTs appears to be a natural next step to investigate. We start by introducing our conventions for the background and by summarizing the calculation of the holographic central charge of the AdS5 T 1;1 solution. 2.1 The metric is given by, ds2AdS5 = ds2T 1;1 = ds2 = ds2AdS5 + L2 ds2T 1;1 ; r 2 L2 dx12;3 + L2 r2 dr2; 12( ^12 + ^22) + 22( 12 + 22) + 2 ( 3 + cos 1 d 1)2; (2.1) where 2 = 19 ; 21 = 22 = 16 and ^1 = sin 1 d 1; 1 = cos sin 2 d 2 sin d 2; 3 = d + cos 2 d 2: { 5 { Using that 2 210 TDp = (2 )7 p gs 0 7 2 p this leads to a quantization of the size of the space, To calculate the holographic central charge of this background, we use the formalism developed in [28, 66]. Indeed, for a generic background and dilaton of the form, ds2 = a(r; i) hdx12;d + b(r) dr2i + gij (r; i) d i d j ; (r; i); we de ne the quantities V^int; H^ as, (2.5) The associated charge is given by { 6 { The holographic central charge for the (d + 1)-dimensional QFT is calculated as, V^int = Z d iq det[gij ] e 4 ad ; H^ = V^in2t : c = d d bd=2H^ 2d2+1 GN;10 H^ 0 d ; GN;10 = 8 6gs2 04: Using these expressions for the background in eq. (2.1), we have r 2 L2 L4 r4 a = ; b = ; d = 3; e 4 det[gij ] a3 = gs 2L2r3 41 sin 1 sin 2 : After some algebra, we obtain the well-known result [ 67 ], q cKW = L8 108 3gs4 04 = 27 64 N32 : We now study the action of non-Abelian T-duality on one of the SU(2) isometries displayed by the background in eq. (2.1). We use the notation and conventions in [28]. 2.2 The non-Abelian T-dual solution The NS-NS sector of the non-Abelian T-dual solution constructed in [12, 13, 28] is composed of a metric, a NS-NS two-form and a dilaton. Using the variables in [28], the metric reads,1 ds^2 = r 2 L2 dx12;3 + L2 r2 dr2 + L 2 21 d 12 + sin2 1d 12 + + 2 21 sin d + cos d also substitute 2 = 1 for convenience. ! L20 so that all factors in the internal metric scale with L2. We The NS two-form is, and the dilaton is given by,2 B2 = For convenience we have de ned the following functions, d ^ d i VolAdS5 ^ d ; 1 2 sin VolAdS5 ^ d ^ d ^ d + cos 1 d 1 : The higher rank RR elds which are related to the previous ones through Fp = F10 p read, The associated RR potentials C1 and C3, de ned through the formulas F2 = dC1 and F4 = dC3 H3 ^ C1, are given by, 1 2 cos 1 sin h 2 1 2 sin 2 d ^ d (2.17) In the papers [12, 13] this solution of the Type IIA equations of motion was shown to preserve N = 1 supersymmetry. In the coordinates used in this paper the Killing vector In appendix A we promote the background in eqs. (2.11){(2.17) to a solution of elevendimensional supergravity. We show that this background ts in the classi cation of N = 2As in the original paper [1], the dilaton needs to transform as well in order to ful l the equations of motion. { 7 { di erential relations and de ne the SU(2)-structure. The eleven dimensional lift suggests that this solution is associated to M5-branes wrapped on a spherical 2d manifold. We discuss this picture further in section 5. As indicated, one goal of this paper is to propose a conformal eld theory dual to the Type IIA non-Abelian T-dual solution. We will do this by combining di erent insights coming from the large -asymptotics, the quantized charges and the calculation of eld theoretical observables using the background. 2.2.1 Asymptotics In complicated systems, like those corresponding to intersections of branes, it is often illuminating to consider the asymptotic behavior of the background. In the case at hand, for the background in eqs. (2.11){(2.17), we consider the leading-order behavior of the solution, when ! 1. This allows us to read the brane intersection that in the decoupling limit and for a very large number of branes generates the solution. Indeed, for ! 1, the leading behavior of the NS- elds is HJEP09(217)38 ds2 B2 e 2 + L2 sin 2 P ( ) L6 gs2 03 P ( ) 2 ; 2 cos P ( ) 2 21 sin 2 d ^ d ; ds2AdS5 + L cos P ( ) d where we have performed a gauge transformation in B2, of the form B2 + d 1, with 1 = L 2 2 cos 1 cos P ( ) ! d 1 : Intuitively, this result suggests that we have two di erent types of NS- ve branes. One type of ve-branes (which we refer to as NS ) extend along R1;3 S2( 1; 1). The second type of ve branes (referred to as NS' ) extend along R1;3 S~2( ; ) . To preserve SUSY, the spaces S2( 1; 1) and S~2( ; ) are bered by the monopole gauge eld A1 = cos 1d 1. This bration is also re ected in the B2- eld, that contains a term that mixes the spheres. The asymptotics of the RR- elds can be easily read from eq. (2.16). Indeed, the expression F6 = dC5, generates asymptotically C5 of D4 branes extended along the directions R1;3 r4dx1;3 ^ d . This suggests an array . D6 branes appear due to the presence of the B2- eld, that blows up the D4 branes due to the Myers e ect [68]. In summary, the asymptotic analysis suggests that the background in eqs. (2.11){ (2.17), is generated in the decoupling limit of an intersection of NS5-NS5'-D4 branes. This will be con rmed by the calculation of the quantized charges associated to this solution. { 8 { In the papers [44, 45], the brane set-ups encoding the dynamics of the CFTs dual to the corresponding non-Abelian T-dual backgrounds were proposed after a careful analysis of the quantized charges. The charges that are relevant for the study of the non-Abelian T-dual of the Klebanov-Witten background are those related to D4, D6 and N S5 branes. Based on this analysis we will propose an array of branes, from which the dynamics of a linear quiver with gauge groups of increasing rank will be obtained. For D6 branes the Page charge reads, QD6 = 1 Z 2 210 TD6 ( 1; 1) F2 = where we have absorbed an overall minus sign by choosing an orientation for the integrals. Imposing the quantization of the D6 charge, the AdS radius L is quantized in terms of N6, (2.20) (2.21) (2.22) (2.23) (2.24) (2.25) L4 = 2 27 gs2 02N6 : B2 ! B2 + B2 ; This relation di ers from that for the original background, see eq. (2.5), which is a common feature already observed in the bibliography [19]. In turn, the Page charge associated to D4-branes vanishes, QD4 = 1 Z under which the Page charges transform as, QD4 = 1 Z Indeed, consider a four-manifold M4 = [ 1; 1] 2, with the two-cycle given by 2 = [ ; ].3 Under a large gauge transformation of the form, the Page charges transform as The rst relation shows that n units of D4-brane charge are induced in each D6-brane. Conversely, nN6 D4-branes can expand in the presence of the B2 eld given by eq. (2.24) into N6 D6-branes wrapped on 2, through Myers dielectric e ect. Consider now the (conveniently normalized) integral of the B2 eld, given by eq. (2.12), along the non-trivial 3Note that this 2-cycle vanishes at ! 0, while at ! 1 it is almost a two sphere of nite size. { 9 { 2 = [ ; ]. Following the paper [25], this must take values in the interval [0; 1].4 Imposing this condition implies that jb0j b0 = 1 The asymptotic behavior of b0 for small and large values of is given by, The expression given by eq. (2.26) is monotonically increasing for all the value jb0j = 1 only once. In order to bring the function jb0( )j back to the interval [0; 1] we need to perform a large gauge transformation of the type de ned in eq. (2.24), whenever jb0( n)j = n; n 2 N. The number of D4-branes in the con guration then increases by a multiple of N6, as implied by eq. (2.25), each time we cross the position = n. The form of the B2 potential in eq. (2.12) suggests that it is also possible to take a 2 [0; 1), and takes di erent 2-cycle, 02 = [ 1; 1] =0 ; which is a rounded S2( 1; 1) at = 0. As in the case analyzed above, large gauge transformations are needed as we move in in order to render b0 in the fundamental region, b0 2 [0; 1]. This shift does not modify however the number of D4 or D6-branes, while it induces NS5-brane charge (we call these NS5' for later convenience) in the con guration. Indeed, let us discuss the NS5-brane charges associated to the solution. Let us rst consider the three-cycle, 3 = [ ; ; ] ; built out of the rst 2-cycle 2 = [ ; ] and the -coordinate. Taking into account the expression for the B2 eld given by eq. (2.12) one nds, sin sin 2 Q 3 Q P sin (2.30) The rst term does not contribute to the charge, which reads, we take the cycle de ned by sin d = b0( n) = n : (2.31) 2 [0; n]. If, on the other hand, 03 = [ ; 1; 1] =0; 4A physical interpretation of this condition in terms of a fundamental string action was presented in [35]. (2.26) (2.27) The conclusion of this analysis is that one can de ne two types of NS5-branes in the non-Abelian T-dual background: NS5-branes located at n and transverse to S~2( ; ), and NS5'-branes located at 0n = n 0=L2 and transverse to S2( 1; 1). These branes are localized in the direction, such that a NS5'-brane lies in between each pair of NS5-branes, as illustrated in gure 1. Further, as implied by eq. (2.25), N6 D4-branes are created each time a NS5-brane is crossed. This brane set-up will be the basis of our proposed quiver in section 4, and will be instrumental in de ning the dual CFT of the non-Abelian Tdual solution. As we will see, it will allow us to identify the global symmetries and the parameters characterizing the associated eld theory. Let us study now an important eld theoretical quantity, calculated from the Type IIA solution, the central charge. 2.2.3 Central charge In this section, we compute the holographic central charge associated to the non-Abelian T-dual solution in eqs. (2.11){(2.17). This will be the main observable to check the validity of the N = 1 quiver proposed in section 4. We must be careful about the following subtle point. The calculation of the quantity V^int in eq. (2.7), will involve an integral in the -direction of the metric in eq. (2.11). The range of this coordinate is not determined by the process of non-Abelian T-duality (the global issues we referred to in the Introduction). If we take 0 that the central charge will be strictly in nite. A process of regularization or completion of the background of eqs. (2.11){(2.17) is needed. In this paper we choose to end the space with a hard cut-o , namely 0 n. We do know that this is geometrically unsatisfactory. Nevertheless, the eld theoretical analysis of section 4 will teach us that a avor group, represented by D6 branes added to the background of eqs. (2.11){(2.17), should end the space in the correct fashion. Previous experience [44] tells us that the hardcuto used here does capture the result for the holographic central charge that is suitable to compare with the eld theoretical one found in section 4. < 1, we face the problem We then proceed, by considering the metric in eq. (2.11), the dilaton in eq. (2.13) and eqs. (2.6){(2.8). We obtain, where we have integrated between two arbitrary values [ a; b]. We have also used the quantization condition of eq. (2.20). For coincides with the central charge of the Abelian T-dual of the Klebanov-Witten background, that we discuss in detail in appendix B. This is that of the original background | see eq. (2.10), with N3 replaced by N4, cKWATD = 27 64 N42: For completeness, we also reproduce in appendix C.3 this value of the central charge from the eld theory, using a-maximization. This matching between the central charges of nonAbelian and Abelian T-duals was found in previous examples [44, 45]. Next, we review aspects of the N = 1 quivers discussed in [55]. These will be the basis of the quiver proposed to describe the eld theory associated to the non-Abelian T-dual solution. In section 4, the holographic result in eq. (2.34) will be found by purely eld theoretical means. 3 Basics of Bah-Bobev 4d N = 1 theories In this section, we provide a summary of the results in [55], which will be instrumental for our proposal of a eld theory dual to the background in eqs. (2.11){(2.17). 3.1 N = 1 linear quivers In [55], Bah and Bobev introduced N = 1 linear quiver gauge theories built out of N = 2 and N = 1 vector multiplets and ordinary matter multiplets. These theories were argued to ow to interacting 4d N = 1 SCFTs in the infrared. They consist of products of ` 1 copies of SU(N ) gauge groups, with either N = 1 (shaded) or N = 2 (unshaded) vector multiplets | see gure 2. Let n1 be the number of N = 1 vector multiplets and n2 the number of N = 2 vector multiplets. There are also ` 2 bifundamental hypermultiplets of SU(N ) SU(N ), depicted in gure 2 as lines between the nodes, and two sets of N (2.33) (2.34) (2.35) (2.36) (N = 2) vector multiplets. Lines between them represent bifundamentals of SU(N ) SU(N ). The boxes at the two ends represent SU(N ) fundamentals. hypermultiplets transforming in the fundamental of the two end SU(N ) gauge groups. 1 = n1 + n2 gauge groups and ` matter multiplets. The total Thus, there are in total ` global symmetry is, SU(N ) SU(N ) U(1)`+n2 U(1)R; corresponding to the SU(N ) avor symmetries acting on the end hypermultiplets, the U(1) avor symmetry acting on each of the ` hypermultiplets, the U(1) avor acting on the chiral adjoint super elds (there are as many as N = 2 vector multiplets) and the R-symmetry. Out of these U(1)0s only a certain non-anomalous linear combination will survive in the IR SCFT. Similarly, the xed point R-charge is computed through a-maximization [ 69 ] as a non-anomalous linear combination of the U(1)'s and U(1)R. As shown in [55], it is convenient to assign a charge i = 1 to each matter hypermultiplet, with the rule that N = 1 vector multiplets connect hypermultiplets with opposite sign, while N = 2 vector multiplets connect hypermultiplets with the same sign. Let p be the number of hypermultiplets with i = +1 and q = ` p those with i = 1, and let us introduce the twist parameter z, z = p ` q : Thus, z = 1 corresponds to a quiver with only N = 2 nodes, involving hypermultiplets of the same charge. z = 0 corresponds in turn to a quiver with the same number of hypermultiplets of each type, so it includes the quiver with only N = 1 nodes. We will focus on 0 z 1 (q p) without loss of generality. We also introduce = ( 0 + l)=2, which can take values = 1; 0; +1. This will later be associated to the type of punctures on the Riemann surface on which M5-branes are wrapped. In a superconformal xed point the a and c central charges can be computed from the 't Hooft anomalies associated to the R-symmetry [ 70 ], where the R-symmetry is given by 3 32 a = and R0 is the anomaly free R-symmetry, F is the non-anomalous global U(1) symmetry and is a number that is determined by a-maximization [ 69 ]. This was used in [55] to compute the a and c central charges associated to the general quiver represented in gure 2. Their values were shown to depend only on the set of parameters f ; z; `; N g. It was then conjectured that all quivers with the same f ; z; `; N g should be dual to each (3.1) (3.2) (3.3) represent NS5-branes extended along fx4; x5g, denoted in [55] as v-branes, while diagonal lines represent the NS5'-branes extended along fx7; x8g, denoted as w-branes. The same number of D4-branes extended along the x6 direction stretch between adjacent 5-branes. other and ow to the same SCFT in the infrared. Moreover, for ` ! 1 the two central charges were shown to agree. Therefore, in this limit the quivers can admit holographic AdS duals. In section 4 we will provide a variation of these N = 1 quivers for which this condition is satis ed, and argue that it is associated to the AdS5 non-Abelian T-dual solution presented in section 2. 3.2 IIA brane realization and M-theory uplift Interestingly, it was shown in [55] that the linear quivers discussed above have a natural description in terms of D4, NS5, NS5' brane set-ups that generalize the N = 2 brane constructions in [56], and allow for an M-theory interpretation. The two types of NS5branes in this construction are taken to be orthogonal to each other, explicitly breaking N = 2 supersymmetry to N = 1. The speci c locations of the branes involved are N coincident D4-branes extend along R1;3 and the x6 direction. p non-coincident NS5-branes extend along R1;3 fx4; x5g, and sit at x6 = x6 for q non-coincident NS5'-branes extend along R1;3 fx7; x8g, and sit at x6 = x6 for = 1; : : : ; p. = 1; : : : ; q. The corresponding brane set-up is depicted in gure 3, see also [55]. In this con guration, open strings connecting D4-branes stretched between two parallel NS5-branes are described at long distances and weak coupling by an N = 2 SU(N ) vector multiplet, while those connecting D4-branes stretched between perpendicular NS5 and NS5' branes are described by an N = 1 SU(N ) vector multiplet. In turn, open strings connecting adjacent D4-branes separated by a NS5-brane (NS5'-brane) are described at low energies by bifundamental hypermultiplets with charge i = 1 ( i = 1). Finally, semi-in nite N D4-branes (or D6 branes) at both ends of the con guration yield two sets { 14 { of hypermultiplets in the fundamental representation of SU(N ). The resulting eld theory is e ectively four dimensional at low energies compared to the inverse size of the D4 along x6. The e ective gauge coupling behaves as g12 4 x6;n+1 x6;n . Given that the 5-branes can be freely moved along the x6 direction, the gauge couplings are marginal parameters. Rotations in the v = x4 + ix5 and w = x7 + ix8 planes of the NS5 and NS5' branes give a U(1)v and a U(1)w global symmetry, so that the IR xed point R-symmetry and avor U(1) are realized geometrically as linear combinations of them: R0 = U(1)v + U(1)w ; F = U(1)v U(1)w : Relying on similar N = 2 constructions in [56], it is possible to describe the previous system of intersecting branes at strong coupling in M-theory. The x6 direction is combined with the M-theory circle x11 to form a complex coordinate s = (x6 + ix11)=R11 describing a Riemann surface 2, which is a punctured sphere or, equivalently, a punctured cylinder. The uplift of this system yields, (3.4) HJEP09(217)38 N M5-branes wrapping the cylinder, from the N D4-branes extended on x6. p simple punctures (in the language of [57]) on the cylinder, coming from the p transversal M5-branes with avor charge i = 1. q simple punctures on the cylinder, coming from the q transversal M5-branes with avor charge i = 1. Two maximal punctures, coming from the stacks of N transversal M5-branes at both ends of the cylinder. They are also assigned 0 ; ` = 1, from which the additional parameter = ( 0 + `)=2 is de ned, taking values = 1; 0; +1. The cylinder or sphere the M5-branes wrap can be viewed as a Riemann surface Cg;n of genus g = 0 and n = p + q + 2 punctures, so that 2 = C0;n. This Riemann surface can be deformed by bringing some of the punctures close to each other (which corresponds to certain weak and strong coupling limits of the dual 6d N = (0; 2) AN 1 eld theory living on the M5-branes) to a collection of higher-genus and less-punctured surfaces. The parameter is associated to the type of punctures on the Cg;n Riemann surface. This closes our summary of the ndings of the paper [55], that we will use in the next section. Let us now propose a dual CFT to our background in eqs. (2.11){(2.17). 4 The non-Abelian T-dual of Klebanov-Witten as a N = 1 linear quiver As we showed in section 2.2.2, the analysis of the quantized charges of the non-Abelian T-dual solution is consistent with a D4, NS5, NS5' brane set-up in which the number of D4-branes stretched between the NS5 and NS5' branes increases by N6 units every time a NS5-brane is crossed. This con guration thus generalizes the brane set-ups discussed in the previous section and in [55]. In this section, inspired by the previous analysis, we will use the brane set-up depicted in gure 1 to propose a linear quiver dual to the background in eqs. (2.11){(2.17). As AdS5 AdS5 a consistency check we will compute its central charge using a-maximization and show that it is in perfect agreement with the holographic study in section 2.2.3 and the result of eq. (2.34), in particular. We will show that the central charge also satis es the wellknown 27/32 ratio [72] with the central charge associated to the non-Abelian T-dual of S5=Z2. This suggests de ning our N = 1 conformal eld theory as the result of deforming by mass terms the N = 2 CFT associated to the non-Abelian T-dual of Proposed N = 1 linear quiver HJEP09(217)38 The quantized charges associated to the non-Abelian T-dual solution are consistent with a brane set-up, depicted in gure 1, in which D4-branes extend on IR1;3 branes on IR1;3 0=L2], n ! 1 and upon compacti cation, the brane set-up, depicted f g , NS52 in Figure 7 in appendix B, associated to the Abelian limit of the solution. We conjecture that, in a similar fashion, the non-Abelian T-dual background in eqs. (2.11){(2.17), arises as the decoupling limit of a D4, NS5, NS5' brane intersection. As opposed to its Abelian counterpart, the precise way in which D-branes transform under non-Abelian T-duality has not been worked out in the literature. This would require analysing the transformation of the boundary conditions at the level of the sigma model (see [71] for some preliminary steps in this direction). Still, as stressed in the previous works [44, 45], similar assumptions based on the analysis of the quantized charges of the supergravity background have produced consistent successful outcomes. Given that the precise D4, NS5, NS5' brane intersection is not known prior to the near horizon limit, it is unclear, on the other hand, how the original D3-brane con guration associated to the Klebanov-Witten solution would be recovered. In fact, even after taking the near horizon limit it is unclear how the Klebanov-Witten background would be recovered from the background de ned by eqs. (2.11){(2.17), given that the original SU(2) symmetry used to construct it is no longer present.5 These issues make non-Abelian T-duality substantially di erent from its Abelian counterpart, and underlie the fact that it can non-trivially change the dual CFT. Coming back to our proposal, we would have an in nite-length quiver with (in the notation of section 3) p = n, q = n, ` = p + q = 2n and z = (p associated eld theory would consist on (2n q)=` = 0 with n ! 1. The 1) N = 1 vector multiplets and matter elds connecting them. However, this in nitely-long quiver does not describe a four dimensional eld theory (its central charge is strictly in nite, among other problematic aspects). This is the same issue that we discussed when calculating the holographic central charge in section 2.2.3. Some regularization is needed and, as we will see, the eld theory precisely provides the way to do this. Elaborating on the ideas in [44], we propose to study this quiver for nite n, completing it as shown in gure 4. The proposed eld theory has the following characteristics: 5This is related to the well-known non-invertibility of non-Abelian T-duality, noticed in the early works [2{8]. matter elds Qj; Q~j in the bifundamental and anti-bifundamental of each pair of nodes, associated to a 5-brane connecting adjacent D4-stacks, with a total number of j = 1; : : : ; n 1 hypermultiplets Hj = (Qj; Q~j) at each side of the quiver. We label r = 1; : : : ; [n=2] the j = +1 hypermultiplets corresponding to NS5-branes and s = 1; 2; : : : ; [n=2] the j = 1 hypermultiplets from NS5'-branes, assuming an alternating distribution of both types of 5-branes. This con guration comes from a re-ordering of the branes in gure 1 that is consistent with Seiberg self-duality and the vanishing of the beta functions and R-symmetry anomalies. The squares in the middle of the quiver denote avor groups corresponding either to semi-in nite D4-branes ending on the NS5 and NS5' branes or to D6-branes transversal to the D4-branes. They complete the quiver at nite n. We choose f1 = f2 for the corresponding fundamental hypermultiplets. It is strongly coupled. This is in correspondence with the fact that it should be dual to an AdS solution whose internal space is smooth in a large region and reduces to our non-Abelian T-dual background in eqs. (2.11){(2.17) in some limit. The eld theory is self-dual under Seiberg duality. This can be quickly seen, by observing that each node is at the self-dual point (with Nf = 2Nc). The beta function and the R-symmetry anomalies vanish, in correspondence with the SO(2; 4) isometry of the background and the number of preserved SUSYs. The central charge calculated by eld theoretical means coincides (for long enough quivers) with the holographic result of eq. (2.34). non-Abelian T-dual of AdS5 S5=Z2. The quiver can be thought of as a mass deformation of the N = 2 quiver dual to the Below, we show that the eld theory represented in gure 4 has all these characteristics. As it happens in the paper [44], the completion we propose with the avor groups has the e ect of ending the space at a given nite value in the direction. 4.2 -functions and R-symmetry anomalies In this section we study the -functions and the anomalies associated to the linear quiver proposed in gure 4. This analysis clari es that the quantum eld theory ows to a conformal xed point in the infrared. In a supersymmetric gauge theory, the -function for a coupling constant g is given by the well-known Novikov-Shifman-Vainshtein-Zakharov (NSVZ) formula [73], which can be HJEP09(217)38 U(1) written in terms of the number of colors, Nc, the number of avors, Nfq , and the anomalous dimensions for the matter elds, q, as Here, we considered the Wilsonian beta function. The denominator in the NSVZ formula is not relevant for us (see [74] for a nice explanation of this). Another important quantity is the R-symmetry anomaly, given by the correlation function of three currents and represented by the Feynman diagram in gure 5. The anomaly is given by the relation, F F~ ; X Rf T (Rf ) ; 1 2 1 2 (4.1) (4.2) (4.3) (4.4) (4.5) where Rf is the R-charge of the fermions in the multiplet. In the case of an SU(N ) gauge group 8 : 1; T (Rf ) = < 2N; for fermions in the adjoint representation for fermions in the fundamental representation Moreover, at the conformal point, one should take into account the relation between the physical dimension of a gauge invariant operator O (with engineering dimension O ) and its R-charge R , O dim O = O + 2 O = 3 2 R O : In the appendix C, we present details of these calculations for the well-known example of the Klebanov-Witten CFT. Readers unfamiliar with that example can study the details in appendix C and then come back to the more demanding calculation presented below. Let us now analyze the quiver depicted in gure 4. We propose for the anomalous dimensions and R-charges of the matter elds and gauginos the same values as in the Klebanov-Witten CFT, 1 2 Q = Q~ = ; RQ = RQ~ = ; R( ) = 1: Notice that in our proposal only one bifundamental eld runs in each arrow. We call them Q or Q~ depending on the direction of the arrow. We nd, substituting in eq. (4.1) for the nodes with rank kN6, k 3 k N6 (k + 1) N6 + (k 1) N6) 1 + = 0 ; The rst term re ects the contribution of the gauge multiplets and the second that of the matter elds. For the anomaly we nd, k = 2 k N6 + 2 (k + 1) N6 + (k 1) N6 = 0 ; (4.7) The rst term indicates the contribution of the gauginos and the second one the contribution of the fermions in the Q; Q~ multiplets. These calculations indicate that both R-symmetry anomalies and beta functions are vanishing. Indeed, they belong to the same anomaly multiplet. Also, notice that the large anomalous dimensions indicate that the CFT is strongly coupled. With this numerology, This allows for the presence of superpotential terms involving four matter multiplets, like the ones proposed in [55]. Let us move now to the calculation of the central charge. Field-theoretical central charge In this section we compute the central charge of the quiver depicted in gure 4 at the xed point, using the a-maximization procedure [ 69 ]. As recalled in section 3, the a and c central charges can be computed from the N = 1 R-symmetry t'Hooft anomalies of the fermionic degrees of freedom of the theory, 3 32 to the chiral multiplet scalars, we have that and, for the fermions R (Qj ) = R (Q~j ) = 1 + j ; R ( j ) = R ( ~j ) = 1 + j : 1 2 1 2 Tr R (Vt) = Tr R3(Vt) = Na2 1 ; where we have used that R ( ) = R0( ) = 1 for the gaugino. Now, we can compute the linear contribution to the anomaly coming from the hypermultiplet Hj = (Qj ; Q~j ), whose chiral elds transform in the fundamental of a gauge group with rank Na and in the anti-fundamental of another gauge group with rank Nb, and vice-versa: Tr R (Hj ) = Na Nb R ( j ) + R ( ~j ) = Na Nb( j 1) : The cubic contribution is Vt are given by, Tr R3(Hj ) = Na Nb R ( j ) + R3( ~j ) = 2Na Nb 2 3 ( j 1 3 1) : In turn, the linear and cubic anomaly contributions from an N = 1 vector multiplet (4.10) (4.11) (4.12) We now consider the completed quiver in gure 4. Hypermultiplets with j = +1 and j = 1 (transforming in the bifundamental of gauge groups of ranks Nj, Nj+1) alternate along the quiver, and f1 = f2. In this way all nodes are equipped with N = 1 vector multiplets. Moreover, we have z = 0 exactly, as well as = 0. The total linear contribution of the hypermultiplets is then: Tr R (H) = Tr R (Hj) + Tr R (Hj) + X Tr R (Hfi) n 1 X j=1;right : 0 ( j;left + j;right) 1 + n X fi 1 ) ( 2 i=1 : 0 In the last line the approximation of a long quiver (large n) has been used. Similarly, the total cubic contribution of the hypermultiplets can be readily computed to be, where long quivers have been considered in the last expression. In turn, recalling that each node appears twice in the quiver depicted in gure 4, with the exception of the central one, the trace anomaly coming from the N = 1 vector multiplets becomes, Tr R (V ) = Tr R3(V ) = 2 X Tr R (Vt) + Tr R (Vn) = 2 X t2N62 1 + n2N62 1 n 1 t=1 3 = N62 2n3 + n 2(n 1) 3 2 n3 N62 + O(n) : (4.15) From this result we see that Tr R (V ) Tr R (H) in the large n limit, so that the overall linear trace anomaly is of order n N62 at most. Putting all these expressions together we S5. Each line represents a hypermultiplet of N = 2 SUSY. nd, for the exact charges in eq. (4.9), From these expressions we see that a( ) is clearly maximized for = 0, as expected for the N = 1 xed point [55]. The superconformal central charges are thus found to be They give, in the large n limit, aN =1 cN =1 a( = 0) = c( = 0) = 3 64 1 64 (3n3 + 2n)N62 (9n3 + 10n)N62 4(2n 1) ; 8(2n 1) : cN =1 aN =1 n3 N62 + O(n) : 9 64 This nal result matches the holographic calculation given by eq. (2.34). This provides a non-trivial check of the validity of the linear quiver in gure 4 as dual to the background in eqs. (2.11){(2.17). It is noteworthy that the agreement with the holographic result occurs in the large number of nodes limit, n ! 1. A further non-trivial check of the validity of our proposed quiver is that the central charge given by (4.18) and that associated with the non-Abelian T-dual of AdS5 satisfy the same 27/32 relation [72], that is, as the central charges of the corresponding theories prior to dualization. Indeed, the quiver associated to the non-Abelian T-dual of AdS5 by Z2 the quiver describing the non-Abelian T-dual of AdS5 S5=Z2 can be obtained by modding out S5, constructed in [44] and depicted in gure 6. This quiver was completed at nite n by a avor group with gauge group SU(nN6). It thus satis es the condition to be conformal (preserving N = 2 SUSY), i.e. that the number of avors is twice the number of colors at each node. Modding out by Z2 results in the same quiver in gure 4, but built out of 2n N = 2 vector and matter multiplets. Taking the central charge, computed in [44], for the non-Abelian T-dual of S5 and doubling it, we obtain the central charge of the non-Abelian T-dual of cNAT D AdS5 S5=Z2 2 1 12 n3N62 + O n ; (4.16) (4.17) (4.18) S5=Z2 (4.19) (4.20) nd that eq. (4.19) indeed holds with cN =1 as in eq. (4.18) and cN =2 as in eq. (4.20). We have checked in appendix C.2 that the same result (4.20) is reproduced using a-maximization. The a-charge is maximized for in [55]. = 13 , as previously encountered Further, one can check that also at nite n, aN =1 and cN =1 satisfy the relation [72],6 9 32 1 32 aN =1 = 4 aN =2 cN =2 ; with the aN =2, cN =2 exact central charges of the N = 2 quiver. The explicit expressions for aN =2 and cN =2 are given in eq. (C.11) in appendix C.2. This precisely de nes our dual CFT as the result of deforming by mass terms the CFT dual to the Sfetsos-Thompson solution modded by Z2. The material presented in this section makes very precise the somewhat loose ideas proposed in the works [12, 13]. In particular, we have identi ed the concrete relation via a RG- ow between the non-Abelian T-dual of AdS5 S2=Z2 and the non-Abelian T-dual of the Klebanov-Witten solution. Notice that here, we are providing precisions about the CFT dual to the non-Abelian T-dual backgrounds. This more precise information is matched by the regularized form of the non-Abelian T-dual solution. The diagram in the Introduction section summarizes the connections between the UV and IR eld theories discussed in this section. We repeat it here for the perusal of the reader. AdS5 AdS5 S5=Z2 mass T 1;1 / NATD of AdS5 S5=Z2 / NATD of AdS5 T 1;1 mass As a closing remark, an explicit ow (triggered by a VEV) between the N = 1 and the N = 2 non-Abelian T-dual backgrounds was constructed in [35]. It should be interesting to use the detailed eld theoretical picture developed above and in [44], to be more precise about various aspects of this RG- ow. 5 Solving the INST-BBBW puzzle The non-Abelian T-dual of the Klebanov-Witten background was rst written in [12, 13] (INST). Further, in that paper an attempt was made to match the non-Abelian T-dual background with a Bah, Beem, Bobev and Wecht (BBBW) solution [59]. This matching was feasible assuming a particular split of the metric into a seven-dimensional and a fourdimensional internal space (see below). The formula in [59] for the central charge of BBBW solutions led however to a c 0 + O(N ) for the non-Abelian T-dual solution, in blatant disagreement with the holographic result. This was the puzzle that the authors of [12, 13] pointed out. In this section we present its resolution. We start by summarizing the most relevant aspects of the work [59]. 6We would like to thank Nikolay Bobev for suggesting this to us. In the work of Bah, Beem, Bobev and Wecht new N = 1 AdS5 solutions in M-theory were constructed, describing the xed points of new N = 1 eld theories associated to M5branes wrapped on complex curves. The central charges of these SCFTs were computed using the six dimensional anomaly polynomial and a-maximization, and were shown to match, in the large number of M5-branes limit, the holographic results. The solutions constructed in [59] were obtained by considering M-theory compacti ed on a deformed four-sphere. In principle, this compacti cation leads to an SO(5)-gauged supergravity in seven dimensions. Following the ideas in [47], BBBW searched for their solutions in the seven dimensional gravity theory (a U(1)2 truncation of the full SO(5) theory) discussed in [75]. They proposed a background consisting of a metric, two gauge elds A(i) and two scalars (i), of the form ds72 = e2f(r)[dx12;3 + dr2] + e2g(r)d k(x1; x2); F (1) = p 8g 8 vol k; F (2) = vol k; (i)(r): q 8g 8 (5.1) (5.2) They then searched for ` xed point' solutions, namely, those where ddr (i) = ddr g = 0 and f log r, leading to backgrounds of the form AdS5 k. They found general solutions depending on four parameters (N; ; z; g). For excitations with wavelength longer than the size of k, these are dual to four dimensional CFTs. In the dual CFT N is the number of M5-branes, g is its genus and z is the so-called `twisting parameter', de ned as z = 2((pg q1)) from the integer numbers p; q that indicate the twisting applied to the M5-branes. The holographic central charge computed in [59] depends on these parameters, and reads c = a = N 3(1 1 9z2 + (1 + 3z2)3=2 48z2 : BBBW completed their analysis deriving various of their formulas, in particular the holographic central charge, using purely 4d CFT arguments. Their CFTs are combinations of Gaiotto's TN -theories, conveniently gauged and connected with other TN factors, with either N = 1 or N = 2 vector multiplets (shaded and unshaded TN 's in the same line as what we explained in section 3). The key-point to be kept in mind after this discussion is that these results were obtained in the context of a compacti cation of eleven-dimensional supergravity to seven dimensions. Let us now come back to the paper [12, 13]. The matching of the non-Abelian Tdual solution with a BBBW geometry assumed that the seven dimensional part of the metric in (5.1) was AdS5 S2( 1; 1) and that the internal space contained the coordinates [ ; ; ; x11]. Also, the authors of [12, 13] chose the parameters = z = 1 for such matching. Using the formula (5.2) in BBBW for the central charge they then found that at leading order the central charge vanished. What was not-correct in the analysis of [12, 13] was the assumption that the nonAbelian T-dual solution could be obtained from a compacti cation of M-theory on a deformed four-sphere (and hence be in the BBBW class of solutions). In fact, inspecting the BPS equations of BBBW | eq. (3.10) of [59] | one nds that a xed point solution does not exist for the set of values = jzj = 1. Even more, the generic solution that BBBW wrote in their eq. (3.8) is troublesome for those same values. A parallel argument can be made by comparing the BBBW and non-Abelian T-dual solutions in the language of the paper [76]. Indeed, the comparison in the appendix C of [76], shows that these solutions t in their formalism in section 4.2 for values of parameters that are incompatible. Either BBBW is t or the non-Abelian T-dual solution is, for a chosen set of parameters. The resolution to this problem is that the non-Abelian T-dual background should instead be thought of as providing a non-compacti cation of eleven dimensional supergravity. Strictly speaking, our coordinate runs in [0; 1], the four manifold is non-compact. In our calculation of the central charge, we assumed that the -coordinate was bounded in [0; n L20 ], but this hard cut-o , as we emphasized, is not a geometrically satisfactory way of bounding a coordinate. There should be another, more general solution, that contains our non-Abelian T-dual metric in a small patch of the space (for small values of ), and 0 closes the -coordinate at some large value n = n L2 . But this putative new metric, especially its behaviour near n, will di er considerably from the one obtained via non-Abelian T-duality. Below, we will comment more about this putative solution. Let us close with some eld theoretical remarks. The class of CFTs studied by BBBW [59] are quite di erent from those studied by Bah and Bobev in [55]. Their central charges are di erent, and the rst involve Gaiotto's TN theories while the second do not. In the same line, our CFT discussed in section 4 is a generalization, but strictly di erent, of the theories in [55], and is certainly di erent from those in [59]. The quiver we presented in section 4 encodes the dynamics of a solution in Type IIA/M-theory where the -coordinate is bounded in a geometrically sounding fashion. The addition of the avor groups in our quiver encode the way in which the -coordinate should be ended. Indeed, in analogy with what was observed in [44, 45], we expect the metric behaving like that of D6 branes close to the end of the space. In M-theory language, we expect to nd a puncture on the Riemann surface, representing the presence of avor groups in the dual CFT. We will be slightly more precise about this in the Conclusions section. 6 Conclusions and future directions Let us brie y summarize the main achievements of this paper. After discussing details of the Type IIA solution obtained by non-Abelian T-duality applied on the Klebanov-Witten background, we carefully studied its quantized charges and holographic central charge (section 2). We lifted the solution to M-theory and showed by explicit calculation of the relevant di erential forms that the background has SU(2)structure and ts the classi cation of [65]. Based on the quantized charges, we proposed a brane set-up (section 4) and a precise quiver gauge theory, generalizing the class of theories discussed by Bah and Bobev in [55] (and summarized in our section 3). This quiver was used to calculate the central charge, one of the important observables of a conformal eld theory at strong coupling. Indeed, in section 4, we showed the precise agreement of this observable, computed by eld theoretical means, with the holographic central charge. We also showed that the quiver has a strongly coupled IR- xed point. Finally, section 5, solves a puzzle raised in previous bibliography. Various appendices discuss technical points in detail. In particular, relations of the nonAbelian T-dual of the Klebanov-Witten background and the more conventional T-dual, details about the dual eld theory, etc, are carefully explained there. To close this paper let us state the most obvious and natural continuation of our work. As we discussed, the holographic central charge calculation in section 2 was done for a regulated version of the Type IIA background. Indeed, the integral over the internal space was taken to range in a nite interval for the -coordinate. We introduced a hard-cuto , but emphasized that this form of regularization is not rigorous from a geometric viewpoint. Fortunately, the dual CFT provides a rationale to regulate the space. The avor groups SU(N6) that end our quiver eld theory (see gure 4), will be re ected in the Type IIA background by the presence of avor branes that will backreact and end the geometry, solving the Einstein's equations. In eleven dimensions, the same e ect will be captured by punctures on the S2 that the M5 branes are wrapping. A phenomenon like this was at work in the papers [44, 45]. The formalism to backreact these avor D6 branes is far-less straightforward in the present case, as the number of isometries and SUSY is less than in the cases of [44, 45]. Qualitatively one may think of de ning the completed solution by deforming with mass terms the superposition of N = 2 Maldacena-Nunez solutions [47] used in [44] to complete the Sfetsos-Thompson background. This would give rise to a superposition of N = 1 MN solutions de ning the completed non-Abelian T-dual solution. It is unclear however in which precise way this superposition would solve the (very non-linear) PDEs associated to N = 1 solutions [65, 77]. We see two possible paths to follow: In the paper [77], Bah rewrote the general M-theory background of [65] in terms of a new set of coordinates that are more useful to discuss the addition of punctures on the Riemann surface. In the type IIA language the new solutions found using Bah's non-linear and coupled PDEs should represent the addition of the avor D6 branes argued above. The equations need to be solved close to the singularity (the puncture or the avor D6 brane) and then numerically matched with the rest of the non-Abelian T-dual background. In [76] generic backgrounds in massive Type IIA were found with an AdS5 factor in the metric and preserving eight SUSYs. For the particular case in which the internal space contains a Riemann surface of constant curvature, the involved set of non-linear and coupled PDEs simpli es considerably. One of the solutions, for the case in which the massive parameter vanishes, is the one studied in this paper | named INST in [76]. Since the paper [76] and some follow-up works have discussed ways of ending these spaces by the addition of D6 and D8 branes, we could consider these technical developments together with the ideas discussed above. Finding a completed or regularized solution would provide the rst example for a background dual to a CFT like that discussed in section 4. The natural following steps would be to extend the formalism to discuss the situations for a cascading QFT. In fact, the precise knowledge of the CFT we have achieved in this paper can be used to improve the understanding and cure the singularity structure of the backgrounds written in the rst paper in [12, 13], in [20], etc. We reserve these problems to be discussed in forthcoming publications. Acknowledgments We would like to thank Fabio Apruzzi, Antoine Bourget, Noppadol Mekareeya, Achilleas Passias, Alessandro Tomasiello, Daniel Thompson, Salomon Zacar as and especially Nikolay Bobev and Niall Macpherson for very useful discussions. G.I. is supported by the FAPESP grants 2016/08972-0 and 2014/18634-9. Y.L. and J.M. are partially supported by the Spanish and Regional Government Research Grants FPA2015-63667-P and FC-15GRUPIN-14-108. J.M. is supported by the FPI grant BES-2013-064815 of the Spanish MINECO, and the travel grant EEBB-I-17-12390 of the same institution. Y.L. would like to thank the Physics Department of Swansea U. and the Mainz Institute for Theoretical Physics (MITP) for the warm hospitality and support. J.M. is grateful to the Physics Department of Milano-Bicocca U. for the warm hospitality and exceptional working atmosphere. C.N. is Wolfson Fellow of the Royal Society. A Connection with the GMSW classi cation In this appendix we prove that the uplift of the non-Abelian T-dual of the Klebanov-Witten solution ts in the classi cation of N = 1 AdS5 backgrounds in M-theory of GMSW [65]. A.1 Uplift of the non-Abelian T-dual solution The eleven dimensional uplift of the non-Abelian T-dual solution consists of metric and 4-form ux. The metric is given by ds121 = e 23 dsI2IA + e 43 dx11 + C1 2 ; where x11 stands for the 11th coordinate, dsI2IA is the ten dimensional metric, given by is the dilaton, given by eq. (2.13), and C1 is the RR potential given in eq. (2.17). The eleven dimensional four-form eld, F4M , is derived from F4M = dC3M , where C3M = C3 + B2 ^ dx11 ; and C3 and B2 are given by (2.17) and (2.12), respectively. The nal expression for F4M is given by (A.1) (A.2) F4M = + 0 2 L4 41 + 02 2 Q 0 L4 4 2 1 Q cos d 2( 1; 1) ^ d ^ dx11 sin d 2( 1; 1) ^ d ^ dx11 gs Q 2 cos sin2 3 d 2( 1; 1) ^ d 2( ; ) S 2 cos 1 d 2( ; ) ^ d ^ d 1 d + cos 1 d 1 ^ d ^ d ^ dx11 ; where for the sake of clarity we have de ned, HJEP09(217)38 Q In the large n limit this expression takes the simpler form ( 2 = 1=9, 21 = 1=6, 0 = gs = 1), which tells us that the M5-branes sourcing this ux are transversal to both squashed two-spheres S2( 1; 1) and S~2( ; ). These are associated to the global isometries U(1)w and U(1)v, whose product lies in the Cartan of both the local R-symmetry and the nonanomalous avor symmetry. A.2 Review of GMSW Before matching the previous solution within the classi cation in [65], let us brie y review the most general N = 1 eleven dimensional solutions with an AdS5 factor found in that paper. These solutions are described by a metric of the form, ds121 = e2 ds2AdS5 + ds2M4 + e 6 cos2 dy2 + cos2 9 m2 d ~ + ~ Here ~ is the R-symmetry direction, ~ is a one-form de ned on M4, whose components depend on both the M4 coordinates and y, and and are functions also depending on the M4 coordinates and y. The coordinate y is related to the warping factor and the function through, 2 m y = e3 sin ; with m being the inverse radius of AdS5. The four-dimensional manifold M4 admits an SU(2) structure which is characterized by a (1; 1)-form J and a complex (2; 0)-form . The SU(2) structure forms, together with the frame components K1 and K2, de ned as, K1 dy ; K2 cos 3 m d ~ + ~ ; e 3 cos = 3 m = e 3 (A.3) (A.4) (A.5) (A.6) (A.7) (A.8) (A.9) (A.10) (A.11) e 3 d e3 sin = 2 m cos K1 ; e 6 d e6 cos sin K1 + i K2 ; e 6 d e6 cos K2 = e 3 ? G + 4 m J sin K1 ^ K2 ; must satisfy the following set of di erential conditions dictated by supersymmetry, e 6 d e6 cos J ^ K2 sin G + m J ^ J 2 sin J ^ K1 ^ K2 : (A.12) e 6 = 4 y2 + ; cos = q 9 r q 36 y2 + q ; ~ = 1 6 q dw 1 12 q 12 q cos 1 d 1 : where q is a function of the coordinates on M4 and y, determined below. We also take Then the one-forms K1 and K2 read, 3 K1 = pq dy ; K2 = 1 r 3 m q 36 y2 + q d ~ 1 6 q dw 1 12 q 12 q cos 1 d 1 : (A.16) Moreover, we de ne an orthogonal frame for the four-dimensional space M4, 1 6 1 r 18 e1 = p sin 1 d 1 ; e3 = 36 q 36 y2 q + q2 dz ; e2 = p d 1 e4 = 1 6 1 r 18 36 q 1 36 y2 q + q2 dw + 1 2 cos 1 d 1 : 1 z (A.13) (A.14) (A.15) (A.17) (A.18) (A.19) In the above formulas, ? stands for Hodge duality in the six-dimensional space spanned by M4 and the one-forms K1 and K2. G is an eleven-dimensional four-form whose components lie along the six-dimensional space that is transverse to AdS5,7 e 9 cos cos3 3 m G = ^ K1 e 3 3 m cos2 ^?4d4 ~ + 4 m e 3 J^ ^ K1 ^ K2 : In this expression the hatted quantities are referred to the four-dimensional metric g^(4) = e 6 g(4). Finally, d4 is the exterior derivative on the four-dimensional space that is transverse to AdS5 and K1; K2. A.3 Recovering the non-Abelian T-dual from GMSW Let us now nd the explicit map between the GMSW geometry and the lifted non-Abelian T-dual geometry. In order to do this we rst identify the functions and according to, In the above expressions, q can be thought of as a function of z and y through the relation, 162 y2 36 q 1 12 1 12 ln 36 q 1 = 0 : Solving this equation for q one nds,8 q = 1 h 36 1 + ProductLog e12 (z 162 y2) i : 7There is a sign di erence between the rst term in the second line of (A.13) and the corresponding term in eq. (2.50) of [65], that is due to our di erent conventions for Hodge duality. 8With ProductLog(Z) we mean the solution of the equation Z = W eW in terms of W. M4 as, identifying,9 and up. There are N4 D4-branes stretched between the NS5 and NS5' branes. NS5 and NS5'-branes are represented by transversal black and red dashed lines, respectively. From the above frame one can construct the forms J and of the SU(2) structure on J = e1 ^ e2 + e3 ^ e4 ; = ei ~ e1 + i e2 ^ e3 + i e4 : Both the metric and the 4-form ux associated to our solution are then obtained after y = cos 6 q = w = 9 x11 + 1 36 2 6 ; : ~ = (A.20) (A.21) (A.22) One can also check that with the above de nitions the constraints (A.9){(A.12), proving that the solution of appendix A.1 ts into the class of solutions found in [65], are satis ed. B The Abelian T-dual of the Klebanov-Witten solution The Abelian T-dual, Type IIA description, of the Klebanov-Witten theory is particularly useful for the study of certain properties of this theory [63, 64]. One interesting aspect is that the eld theory can directly be read from the D4, NS5, NS5' brane set-up associated to this solution. We have depicted both the brane set-up and the associated quiver in gure 7. objective of this work. B.1 Background In this appendix we discuss some aspects of this description that are relevant for the understanding of the CFT interpretation of the non-Abelian T-dual solution, the main The paper [63] considered an Abelian T-duality transformation along the Hopf- ber direction of the T 1;1. This dualization gives rise to a well-de ned string theory background. 9We take L = m = 0 = gs = 1 for convenience. There is a minus overall sign between G, from (A.13), and F4, from (A.3), due to our di erent conventions. It is however a typical example of Supersymmetry without supersymmetry [78], being the low energy supergravity background non-supersymmetric. Since our ultimate goal in this section will be to compare with the non-Abelian T-dual solution, which is only guaranteed to be a well-de ned string theory background at low energies, we will instead dualize along the 2 azimuthal direction of the T 1;1. This preserves the N = 1 supersymmetry of the Klebanov-Witten solution, and can be matched directly with the non-Abelian T-dual solution in the large limit. We start by rewriting the Klebanov-Witten metric in terms of the T-duality preferred frame, in which 2 does only appear in the form d 2 and just in one vielbein, e1 = L 1 d 1 ; e2 = L 1 sin 1 d 1 ; HJEP09(217)38 sin 2 2 pP ( 2) d + cos 1 d 1 ; e x = dx ; r L e^1 = L 2 d 2 ; e3 = eC d 2 + A~1 ; connection The Klebanov-Witten metric thus reads er = dr ; L r e^2 = L A~1 = 2 cos 2 d 22 sin2 2, and we have introduced the ds2 = ds2AdS5 + L 2 21 d 22( 1; 1) + 22 d 22 + + cos 1d 1 + cos 1 d 1 : 2 sin2 2 d : + P ( 2) d 2 + + cos 1d 1 (B.2) ; (B.1) (B.3) (B.4) is then given by:10 A U(1) T-duality performed on the 2 direction trades the vielbein e3 for e^ = 0e C d 2, and generates a NS-NS 2-form B2 = 0A~1 ^ d 2. The NS-NS sector for the dual solution ds2ATD = ds2AdS5 + L 2 21 d 22( 1; 1)+ d 22 + B2ATD = e 2 ATD = We can see in the metric the geometrical realization of the U(1) R-symmetry in the direction. We can also see that it agrees with the asymptotic form of the metric of the non-Abelian T-dual solution, given by the rst equation in (2.18), under the replacements ! 2 ; ! ; ! 2 : 10We rescale 2 ! L2 0 2, so that the metric of the internal space scales with L2. We also use that 2 = 1 for later comparison with the NATD solution. The B2 elds do also agree, once a gauge transformation of parameter L2 cos 2 2 d 2 cos 1 is performed, giving rise to B2 = L 2 2 d 2( 2; ) + L2 sin 2 2 P ( 2) 2 cos 2 (B.5) (B.6) HJEP09(217)38 (B.7) (B.8) ! 1 We will use this expression for the B2- eld in the remaining of this section. As in [44, 45], the two dilatons satisfy e 2 NAT D 2 e 2 ATD for large (after re-absorbing the scaling factors in ! L20 ). As explained in [44, 45], this relation has its origin in the di erent measures in the partition functions of the non-Abelian and Abelian T-dual sigma models. Finally, the RR elds are: F4 = F6 = gs 01=21 sin 1 sin 2 d 1 ^ d 1 ^ d 2 ^ d ; gs 01=2 VolAdS5 ^ d 2 : One can check that, as in [45], for large the uxes polyforms satisfy e NAT D FNAT D e ATD FATD : The previous relations show that the non-Abelian T-dual solution reduces in the limit to the Abelian T-dual one. This connection between non-Abelian and Abelian Tduals was discussed previously in examples where the dualization took place on a round S3 [44, 45]. Our results show that it extends more generally. It is worth stressing however that in this case the relation is more subtle globally. Indeed, the relations in eq. (B.4) identify setting 2 [0; 4 ]. The reason for this apparent mismatch is that the dualization on 2 generates a bolt singularity in the metric, and this must be cured by 2 [0; 2 ], such that the bolt singularity reduces to the coordinate singularity of R2 written in polar coordinates. Once this is taken into account the ranges of both coordinates also agree. As encountered in [12, 13], the dualization has enforced a Z2 quotient on . Our Abelian T-dual is thus describing the Klebanov-Witten theory modded by Z2. This is consistent with the brane set-up that is implied by the quantized charges of the background, as we now show. B.2 Quantized charges and brane set-up The background uxes of the Abelian T-dual solution support D4 and N S5-brane charges. The Page charge for the D4 branes is given by: QD4 = 1 Z F4 = 2 L4 27 gs2 02 = N4 : (B.9) cos 2 ! d 2 ^ d 1 Imposing the quantization of this charge we nd that the radius L is related to the number of D4 branes through the formula: We nd a factor of 2 of di erence with respect to the original background. This is due to the change in the periodicity of the direction from [0; 4 ] to [0; 2 ]. In turn, the charge of N S5 branes is calculated from: L4 = 27 2 g s2 02 N4 : H3 : (B.10) (B.11) Taking M3 to be any of these cycles we nd that there are two units of NS5, or NS5', charge. This is consistent with a brane picture of two alternating NS5, NS5' branes, transverse to either of the two 2-cycles S~2( 2; ), S2( 1; 1), located along the compact 2-direction. This is the brane set-up discussed in [64], describing the Klebanov-Witten theory modded by Z2 in Type IIA. The general Zk case is depicted in gure 9 of appendix C.3. Note that, as discussed in [64], the positions of the branes in the 2-circle are not speci ed by the geometry, so generically we can only think that they de ne four intervals in the 2-circle.11 The same number of D4-branes are stretched between each pair of NS5, NS5' branes since even if large gauge transformations are required as we pass the value 2 = D4-brane charge does not change in the absence of F2- ux. Coming back to section 2.2.3, the relation found there between the central charges of the non-Abelian and Abelian T-dual solutions helps us understand now the connection between agree when and 2 globally. The computation in that section showed that the central charges 2 [n L20 ; (n + 1) L20 ] and n is sent to in nity. This is consistent with the globally the ! 1 limit that must be taken at the level of the solutions. Furthermore, it clari es why direction is identi ed, through the replacements in (B.4), with 2 2 [0; 2 L20 ]. This is just implied by the Z2 quotient enforced by the Abelian T-duality transformation. L2= 0, the C Some eld theory elaborations In this appendix we discuss some aspects of the eld theory analysis presented in section 4. We start with the calculation of the beta functions and anomalies for the Klebanov Witten CFT. C.1 A summary of the Klebanov-Witten CFT The eld content of the Klebanov-Witten theory consists on a SU(N ) SU(N ) gauge group with bifundamental matter elds A1; A2 and B1; B2, transforming in the (N; N ) 11In [64] it was argued that the di erent orderings correspond to di erent phases in the Kahler moduli space of the orbifold singularity. This is interpreted in the eld theory side in terms of Seiberg duality [79, 80], so the corresponding theories should ow to the same CFT in the infrared. B1, B2 1 N N 2 3 4 Ai = Bi = 1 2 2 dim(Ai) = dim(Bi) = 1 R[A] = R[B] = R A = R B = 1 2 : and (N ; N ) representations of SU(N ), respectively. This theory is represented by the quiver depicted in gure 8. The anomalous dimensions of the matter elds are, and thus the physical dimensions and the R-charges are given by, Substituting in eq. (4.1) we see that the -functions for the couplings g1 and g2 vanish: i i = 1; 2 : We can also check the vanishing of the anomaly, i = 2 N + 2 (2 N ) 1 2 = 0 ; where we took into account that the R-charge of the gaugino is 1 while that of the two Weyl fermions is 1=2. We hope that this has prepared the reader unfamiliar with these formalities to understand the material in our section 4. C.2 Central Charge of the N = 2 UV CFT In this appendix we compute the central charge of the N = 2 quiver associated to the nonAbelian T-dual of AdS5 is maximized for S5=Z2, using a-maximization. We obtain that the central charge = 13 , as for the equal rank quivers considered in [55]. Furthermore, we show that the result of this calculation leads, consistently, to the holographic central charge given by eq. (4.20). We consider the Z2-re ection of the quiver of gure 6 and take i = +1 for all hypermultiplets, including the ones associated with the avor groups. We then nd for the trace anomalies (N N6): Tr R (H) = 2 X Tr R (Hj ) + X Tr R (Hfi ) n 1 j=1 n 1 j=1 2 i=1 j 1 + n N 2 X fi 1 2 i=1 (C.1) (C.2) (C.3) (C.4) Tr R3(H) = 2 X Tr R3(Hj) + X Tr R3(Hfi) 1 + 2n( 1 + O(n) ; (C.5) 1 3 HJEP09(217)38 1 3 + 2n 6 1 n3 N 2 1 3 + O(n) : (C.6) = N 2 2 n3 + 4 n 3 n 1 j=1 3 n 1 j=1 = 2 N 2 X j j + 1 N 2 2 4 3 = N 2 1 n3 + 1 n 6 3 Tr R (Vj) = Nj2 Tr R3(Vj) = Nj2 1 1 1 1 n 1 j=1 1 3 1 3 ( ( 3 2 n3 N 2 + n N 2 X 2 i=1 1 3 j 1 + j j 1 + j 3 : 3 2 n3 N 2 1 3 2 n3 N 2 1 2 i=1 1 4 1 3 1 3 1 2 1 3 For the N = 2 vector multiplets (N = 1 vector + chiral adjoint) the non-anomalous Rcharge R = R0 + F =2 is obtained from the R-charge for the gaugino, R0( ) = 1, plus the non-anomalous avor charge of the fermion in the chiral adjoint F ( j) = ( 1) j 1 + j , being R0( j) = 0. We thus have: Tr R (V ) = 2 X Tr R (Vj) + Tr R (Vn) = 2 X j2N 2 1 + n2N 2 These are summed up easily for all j = +1: 2n3 + n N 2 1) 1 The cubic term follows most readily: Tr R3(V ) = 2n3 + n N 2 1) 1 3 We thus see that both linear contributions (C.5) and (C.8) from the hypermultiplets and vector multiplets cancel at leading order, so that Tr R Tr R (H) + Tr R (V ) O(n) : Now both a( ) and c( ) charges can be computed exactly, ) h3n3(1 + )2 + 2(1 + 3 )niN62 ) h9n3(1 + )2 + 2(5 + 9 )niN62 2(2n 1) 2 + 3 (1 + ) 2(2n 1) 4 + 9 (1 + ) ; (C.10) 1 1 + O(n) : 3 + O(n) : ) ) ; (C.7) (C.8) (C.9) k k k set-up. There are N4 D4-branes stretched between p = k NS5-branes, labeled by r = 1; : : : ; k (as for the corresponding hypermultiplets) and q = k NS5'-branes labeled by s = 1; : : : ; k. NS5 and NS5'-branes are represented by transversal black and red dashed lines, respectively. and a( ) is maximized for = 1=3, yielding the superconformal charges: In the long quiver approximation, we recover the holographic result aN =2 cN =2 a( = 1=3) = c( = 1=3) = 1 1 6 (4n3 + 3n)N62 10n + 5 ; (n3 + n)N62 2n + 1 : cN =2 aN =2 6 1 n3 N62 + O n ; (C.11) (C.12) as expected. It is noteworthy that = 1=3 is the value of quivers with nodes of the same rank. predicted in [55] for N = 2 C.3 Central charge of the Klebanov-Witten theory modded by Zk In this appendix we include, for completeness, the eld theory calculation of the central charge of the Klebanov-Witten theory, using a-maximization. We will center in the more general case in which the theory is modded by Zk. The computation of the eld theoretical central charge in this example is very illustrative of the a-maximization technique used throughout the paper. In this case we have, in the Type IIA description, p = k NS5-branes and q = k NS5'branes, and ` = p + q = 2k hypermultiplets connecting ` N = 1 vector multiplets [64]. The rst and the last nodes are made to coincide, as depicted in Figure 9. We closely follow the eld-theoretical computation of the central charge for the linear quiver proposed in section 4.3. We just need to take Na = Nb = N4 for the bifundamentals in (4.10) for all the ` = 2k nodes. This yields the linear contribution for the hypermultiplets: Tr R (H) X Tr R (Hj ) = X Tr R (Hr) + X Tr R (Hs) ` j=1 = ` N42 z 1 z=0 = 2 k N42 ; s=1 r=1 where we have used z = (p q)=`. Similarly, the cubic contribution is given by Tr R3(H) = N42 z 3 + 3 3 2 1 Contributions from N = 1 vector multiplets are computed straightforwardly to be: Tr R (V ) = Tr R3(V ) = ` N42 1 = 2 k N42 1 : We can now use (4.9) to get which, upon a-maximization for = 0, yields the xed point central charge (for large N4): 27 N42 1 16 ; c k N42 ; (C.13) (C.14) which coincides, as expected, with the holographic value (given by eq. (2.36) for k = 1). This expression is valid for any k 1, i.e. no large ` = 2k limit has been assumed. Note that in the absence of avor groups it is not possible to de ne 0 ; `, and neither = 0 + `, as we have done for the linear quivers discussed in section 3. Still, the result in (C.14) agrees with the central charge of a Bah-Bobev type of linear quiver (see eq. (3.20) in [55]) for = 0 and large `. 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Georgios Itsios, Yolanda Lozano, Jesús Montero, Carlos Núñez. The AdS 5 non-Abelian T-dual of Klebanov-Witten as a \( \mathcal{N}=1 \) linear quiver from M5-branes, Journal of High Energy Physics, 2017, 38, DOI: 10.1007/JHEP09(2017)038