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JHEP09(2017)038

Published for SISSA by Springer

Received: June 23, 2017

Revised: August 14, 2017

Accepted: August 19, 2017

Published: September 11, 2017

The AdS

non-Abelian T-dual of Klebanov-Witten as

a N = 1 linear quiver from M5-branes

Georgios Itsios,

Yolanda Lozano,

Jes´us Montero

and Carlos N´u˜nez

Instituto de F´ısica Te´orica, UNESP-Universidade Estadual Paulista,

R. Dr. Bento T. Ferraz 271, Bl. II, Sao Paulo 01140-070, SP, Brazil

Department of Physics, University of Oviedo,

Avda. Calvo Sotelo 18, 33007 Oviedo, Spain

Department of Physics, Swansea University,

Swansea SA2 8PP, United Kingdom

E-mail: gitsios@gmail.com, ylozano@uniovi.es,

jesus.montero.a@gmail.com, c.nunez@swansea.ac.uk

Abstract: In this paper we study an AdS

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 N = 1 quiver gauge theories built out of N = 1 and 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

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.

Keywords: String Duality, AdS-CFT Correspondence, Conformal Field Models in String

Theory

ArXiv ePrint: 1705.09661

Open Access,

 The Authors.

Article funded by SCOAP

https://doi.org/10.1007/JHEP09(2017)038

JHEP09(2017)038

1 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 ﬁeld

theories known since the nineties. Its extension to all orders in g

and α

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

in [11]–[37]. These works have widely applied non-Abelian T-duality to generate new

AdS backgrounds of relevance in diﬀerent contexts. While some of the new solutions

avoid previously existing classiﬁcations [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 deﬁne 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 ﬁeld theoretical interpretation of non-Abelian

T-duality (in the context of Holography) was ﬁrst addressed in detail. One outcome of these

works is that non-Abelian T-duality changes the dual ﬁeld theory. In other words, that new

AdS backgrounds generated through non-Abelian T-duality have dual CFTs diﬀerent 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 ﬁeld

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 conﬁrmed by the n

scaling of

the central charges.

Reversing the logic, the understanding of the ﬁeld 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 ﬁeld

theory to globally deﬁne (or complete) the background obtained by non-Abelian T-duality.

In this way the Sfetsos-Thompson solution [9], constructed acting with non-Abelian T-

duality on the AdS

×S

background, was completed and understood as a superposition of

Maldacena-N´u˜nez solutions [47], dual to a four dimensional CFT. This provides a global

deﬁnition 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 AdS

solutions. In this case the non-

Abelian T-dual solution was shown to arise as a patch of a geometry discussed in [48–51],

– 2 –

JHEP09(2017)038

dual to the renormalization ﬁxed point of a T

ˆρ

(SU(N)) quiver ﬁeld 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-deﬁned background.

Remarkably, this process of zooming-in has recently been identiﬁed 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.

In this paper we follow the methods in [44] to propose a CFT interpretation for the

N = 1 AdS

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 ﬂow to

interacting ﬁxed points in the infrared. These quivers can be thought of as N = 1 twisted

compactiﬁcations 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 consid-

eration could provide the ﬁrst 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 T

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 T

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 frame-

work, 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 ﬁeld theory could have some

relation to the N = 1 version of Gaiotto’s CFTs. Indeed, following the ideas in [60], the non-

Abelian T-dual of the Klebanov-Witten solution could be thought of as a mass deformation

of the non-Abelian T-dual of AdS

× S

, as indicated in the following diagram,

AdS

× S

mass



NATD of AdS

× S

mass



AdS

× T

1,1

NATD of AdS

× T

1,1

– 3 –

JHEP09(2017)038

Nevertheless, there were many unknowns and not-understood subtle issues when the pa-

pers [12, 13] were written. To begin with, the dual CFT to the non-Abelian T-dual of

AdS

× S

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 identiﬁed, 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 ﬁeld theoretical description of diﬀerent backgrounds obtained by non-Abelian

T-duality. This led to a ﬁeld-theory-inspired completion or regularization of the non-

Abelian T-dual backgrounds. Diﬀerent checks of this proposal have been performed. Most

notably, the central charge is a quantity that nicely matches the ﬁeld theory calculation

with the holographic computation in the completed (regulated) background.

In this paper we will apply the ideas of [44–46] and the ﬁeld 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 ﬁve-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-cutoﬀ. Whilst geometrically unsatisfactory, previous experience in [44]

shows that this leads to sensible results, easy to compare with a ﬁeld theoretical

calculation.

• Based on the brane set-up, we propose a precise linear quiver ﬁeld theory. This, we

conjecture, is dual to the regulated non-Abelian T-dual background. We check that

the quiver is at a strongly coupled ﬁxed 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 T-

dual of AdS

×S

, that is constructed following the ideas in [44]. It is the presence

of a ﬂavor group in the CFT that regulates the space generated by non-Abelian T-

duality.

• We calculate the ﬁeld theoretical central charge applying the methods in [55]. We ﬁnd

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 –

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