The AdS 5 nonAbelian Tdual of KlebanovWitten as a \( \mathcal{N}=1 \) linear quiver from M5branes
Received: June
The AdS5 nonAbelian Tdual of KlebanovWitten 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 01140070, SP , Brazil
3 Department of Physics, University of Oviedo
4 Department of Physics, Swansea University
5 Instituto de F sica Teorica, UNESPUniversidade Estadual Paulista
6 Bobev. These realize
In this paper we study an AdS5 solution constructed using nonAbelian Tduality, acting on the KlebanovWitten 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 D4NS5NS5' brane setup, generalizing the constructions discussed by Bah and = 1 quiver gauge theories built out of N
String Duality; AdSCFT Correspondence; Conformal Field Models in String

M5branes
= 1 and N = 2
vector multiplets
owing to interacting
xed points in the infrared. We compute the
central charge using amaximization, and show its precise agreement with the holographic
calculation. Our result exhibits n3 scaling with the number of vebranes. This suggests
an elevendimensional interpretation in terms of M5branes, a generic feature of various
AdS backgrounds obtained via nonAbelian Tduality.
Theory
1 Introduction
1.1
General framework and organization of this paper
2 The nonAbelian Tdual of the KlebanovWitten solution
2.1
2.2 The nonAbelian Tdual solution
2.2.1
2.2.2
2.2.3
Asymptotics
Quantized charges
Central charge
3 Basics of BahBobev 4d N = 1 theories
3.1
N = 1 linear quivers
3.2 IIA brane realization and Mtheory uplift
4.1
4.2
Proposed N = 1 linear quiver
functions and Rsymmetry anomalies
4.3 Fieldtheoretical central charge
5 Solving the INSTBBBW puzzle
6 Conclusions and future directions
4 The nonAbelian Tdual of KlebanovWitten as a N = 1 linear quiver
A Connection with the GMSW classi cation
A.1 Uplift of the nonAbelian Tdual solution
A.2 Review of GMSW A.3 Recovering the nonAbelian Tdual from GMSW
B The Abelian Tdual of the KlebanovWitten solution
B.1 Background B.2 Quantized charges and brane setup
C Some eld theory elaborations
C.1 A summary of the KlebanovWitten CFT
C.2 Central Charge of the N = 2 UV CFT
C.3 Central charge of the KlebanovWitten theory modded by Zk
Introduction
NonAbelian Tduality [1], the generalization of the Abelian Tduality symmetry of String
Theory to nonAbelian isometry groups, is a transformation between worldsheet
eld
theories known since the nineties. Its extension to all orders in gs and 0 remains however
a technicallyhard open problem [2{8]. As a result, nonAbelian Tduality 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 nonAbelian Tduality was also pointed out. This connection was further exploited
These works have widely applied nonAbelian Tduality 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 3d3d
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 nonAbelian
Tduality (in the context of Holography) was rst addressed in detail. One outcome of these
works is that nonAbelian Tduality changes the dual eld theory. In other words, that new
AdS backgrounds generated through nonAbelian Tduality have dual CFTs di erent from
those dual to the original backgrounds. This is possible because, contrary to its Abelian
counterpart, nonAbelian Tduality 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 DpNS5 brane intersection. This points to an interesting relation
between AdS nonAbelian Tduals and M5branes, that is con rmed by the n3 scaling of
the central charges.
Reversing the logic, the understanding of the eld theoretical realization of nonAbelian
Tduality 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
longstanding open problems of nonAbelian Tduality 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 nonAbelian Tduality.
In this way the SfetsosThompson solution [9], constructed acting with nonAbelian
Tduality on the AdS5
S5 background, was completed and understood as a superposition of
MaldacenaNun~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 Tdual 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 nonAbelian Tdual solution arose as the
result of zoomingin on a particular region of a completed and wellde ned background.
Remarkably, this process of zoomingin has recently been identi ed more precisely as a
Penrose limit of a wellknown 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 nonAbelian Tduality on
a subspace of the KlebanovWitten 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 D4NS5NS5' brane construction
that generalizes the Type IIA brane setups 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 sixdimensional (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 nonAbelian Tdual solution under
consideration could provide the rst explicit gravity dual to an ordinary N = 1 linear quiver
associated to a D4NS5 brane intersection [55]. In this construction, the N = 2 SUSY
D4NS5 brane setup associated to the SfetsosThompson solution (see [44]) is reduced to
N = 1 SUSY through the addition of extra orthogonal NS5branes, 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
nonAbelian Tdual 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 nonAbelian Tdual of the KlebanovWitten 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 Tdual of the KlebanovWitten solution could be thought of as a mass deformation
of the nonAbelian Tdual 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 notunderstood subtle issues when the
papers [12, 13] were written. To begin with, the dual CFT to the nonAbelian Tdual 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 nonAbelian
Tduality. This led to a eldtheoryinspired 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 nonAbelian Tdual of the KlebanovWitten background. A summary of our results is:
We perform a study of the background and its quantized charges, and deduce the
HananyWitten [62] brane setup, in terms of D4 branes and two types of vebranes
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 hardcuto .
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 setup, we propose a precise linear quiver eld theory. This, we
conjecture, is dual to the regulated nonAbelian Tdual background. We check that
the quiver is at a strongly coupled xed point by calculating the beta functions and
Rsymmetry 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 nonAbelian
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 nonAbelian
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
nonAbelian Tdual 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 setup associated to the solution, similar to
{ 4 {
that associated to the Abelian Tdual of KlebanovWitten, studied in [63, 64]. In section 3
we summarize the brane setup 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 nonAbelian Tdual 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 nonAbelian Tdual 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 nonAbelian Tdual
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 nonAbelian Tdual solution
ts in the classi cation of [65], for N = 1 SUSY spaces with an AdS5factor. Appendix B
studies in detail the relation between the nonAbelian Tdual solution and its (Abelian)
Tdual counterpart. Finally in appendix C we present some eld theory results relevant
for the analysis in section 4.
2
The nonAbelian Tdual of the KlebanovWitten solution
In this section we summarize the Type IIA supergravity solution obtained after a
nonAbelian Tduality transformation acts on the T 1;1 of the KlebanovWitten 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 nonAbelian Tduals of AdS5
Y p;q SasakiEinstein 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 wellknown result [
67
],
q
cKW =
L8
108 3gs4 04 =
27
64 N32 :
We now study the action of nonAbelian Tduality 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 nonAbelian Tdual solution
The NSNS sector of the nonAbelian Tdual solution constructed in [12, 13, 28] is composed
of a metric, a NSNS twoform 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 twoform 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 M5branes 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 nonAbelian Tdual 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 leadingorder 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 vebranes (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 NS5NS5'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 setups encoding the dynamics of the CFTs dual to the
corresponding nonAbelian Tdual backgrounds were proposed after a careful analysis of
the quantized charges. The charges that are relevant for the study of the nonAbelian
Tdual of the KlebanovWitten 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 D4branes vanishes,
QD4 =
1
Z
under which the Page charges transform as,
QD4 =
1
Z
Indeed, consider a fourmanifold M4 = [ 1; 1]
2, with the twocycle 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 D4brane charge are induced in each D6brane.
Conversely, nN6 D4branes can expand in the presence of the B2 eld given by eq. (2.24)
into N6 D6branes 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 nontrivial
3Note that this 2cycle 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 D4branes 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 2cycle,
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 D6branes, while it
induces NS5brane charge (we call these NS5' for later convenience) in the con guration.
Indeed, let us discuss the NS5brane charges associated to the solution. Let us rst
consider the threecycle,
3 = [ ; ; ] ;
built out of the rst 2cycle
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 NS5branes in the
nonAbelian Tdual background: NS5branes 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 NS5branes,
as illustrated in gure 1. Further, as implied by eq. (2.25), N6 D4branes are created each
time a NS5brane is crossed. This brane setup will be the basis of our proposed quiver
in section 4, and will be instrumental in de ning the dual CFT of the nonAbelian
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 nonAbelian
Tdual 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 nonAbelian Tduality (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 cuto , 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 Tdual of the KlebanovWitten 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 amaximization. This matching between the central charges of
nonAbelian and Abelian Tduals 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 nonAbelian Tdual
solution. In section 4, the holographic result in eq. (2.34) will be found by purely eld
theoretical means.
3
Basics of BahBobev 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 Rsymmetry.
Out of these U(1)0s only a certain nonanomalous linear combination will survive in the IR
SCFT. Similarly, the xed point Rcharge is computed through amaximization [
69
] as a
nonanomalous 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 M5branes are wrapped.
In a superconformal xed point the a and c central charges can be computed from the
't Hooft anomalies associated to the Rsymmetry [
70
],
where the Rsymmetry is given by
3
32
a =
and R0 is the anomaly free Rsymmetry, F is the nonanomalous global U(1) symmetry
and
is a number that is determined by amaximization [
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 NS5branes extended along fx4; x5g, denoted in [55] as vbranes, while diagonal lines
represent the NS5'branes extended along fx7; x8g, denoted as wbranes. The same number of
D4branes extended along the x6 direction stretch between adjacent 5branes.
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 nonAbelian Tdual
solution presented in section 2.
3.2
IIA brane realization and Mtheory uplift
Interestingly, it was shown in [55] that the linear quivers discussed above have a natural
description in terms of D4, NS5, NS5' brane setups that generalize the N = 2 brane
constructions in [56], and allow for an Mtheory 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 D4branes extend along R1;3 and the x6 direction.
p noncoincident NS5branes extend along R1;3
fx4; x5g, and sit at x6 = x6 for
q noncoincident NS5'branes extend along R1;3
fx7; x8g, and sit at x6 = x6 for
= 1; : : : ; p.
= 1; : : : ; q.
The corresponding brane setup is depicted in gure 3, see also [55].
In this con guration, open strings connecting D4branes stretched between two parallel
NS5branes are described at long distances and weak coupling by an N = 2 SU(N ) vector
multiplet, while those connecting D4branes stretched between perpendicular NS5 and
NS5' branes are described by an N
= 1 SU(N ) vector multiplet. In turn, open strings
connecting adjacent D4branes separated by a NS5brane (NS5'brane) are described at
low energies by bifundamental hypermultiplets with charge i = 1 ( i =
1). Finally, semiin nite N D4branes (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 5branes
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 Rsymmetry 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 Mtheory. The x6 direction is combined
with the Mtheory 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 M5branes wrapping the cylinder, from the N D4branes extended on x6.
p simple punctures (in the language of [57]) on the cylinder, coming from the p
transversal M5branes with avor charge i = 1.
q simple punctures on the cylinder, coming from the q transversal M5branes with
avor charge i =
1.
Two maximal punctures, coming from the stacks of N transversal M5branes 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 M5branes 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 M5branes) to a collection of highergenus and lesspunctured 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 nonAbelian Tdual of KlebanovWitten as a N
= 1 linear quiver
As we showed in section 2.2.2, the analysis of the quantized charges of the nonAbelian
Tdual solution is consistent with a D4, NS5, NS5' brane setup in which the number of
D4branes stretched between the NS5 and NS5' branes increases by N6 units every time
a NS5brane is crossed. This con guration thus generalizes the brane setups discussed in
the previous section and in [55].
In this section, inspired by the previous analysis, we will use the brane setup 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 amaximization 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 nonAbelian Tdual 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 nonAbelian Tdual of
Proposed N = 1 linear quiver
HJEP09(217)38
The quantized charges associated to the nonAbelian Tdual solution are consistent with
a brane setup, depicted in
gure 1, in which D4branes extend on IR1;3
branes on IR1;3
0=L2], n ! 1 and upon compacti cation, the brane setup, 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 nonAbelian Tdual 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 Dbranes transform
under nonAbelian Tduality 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 D3brane con guration associated to the
KlebanovWitten solution would be recovered. In fact, even after taking the near
horizon limit it is unclear how the KlebanovWitten 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 nonAbelian Tduality substantially
di erent from its Abelian counterpart, and underlie the fact that it can nontrivially change
the dual CFT.
Coming back to our proposal, we would have an in nitelength 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 nitelylong 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 wellknown noninvertibility of nonAbelian Tduality, noticed in the early
works [2{8].
matter elds Qj; Q~j in the bifundamental and antibifundamental of each pair of nodes, associated
to a 5brane connecting adjacent D4stacks, 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 NS5branes and s = 1; 2; : : : ; [n=2] the j =
1 hypermultiplets from NS5'branes,
assuming an alternating distribution of both types of 5branes. This con guration comes from a
reordering of the branes in
gure 1 that is consistent with Seiberg selfduality and the vanishing
of the beta functions and Rsymmetry anomalies. The squares in the middle of the quiver denote
avor groups corresponding either to semiin nite D4branes ending on the NS5 and NS5' branes
or to D6branes transversal to the D4branes. 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 nonAbelian Tdual background in eqs. (2.11){(2.17) in some limit.
The eld theory is selfdual under Seiberg duality. This can be quickly seen, by
observing that each node is at the selfdual point (with Nf = 2Nc).
The beta function and the Rsymmetry 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).
nonAbelian Tdual 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 Rsymmetry 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 wellknown NovikovShifmanVainshteinZakharov (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 Rsymmetry 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 Rcharge 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 Rcharge R ,
O
dim O =
O +
2
O =
3
2
R
O :
In the appendix C, we present details of these calculations for the wellknown example of
the KlebanovWitten 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 Rcharges of the matter elds and gauginos the same values as in the
KlebanovWitten 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 Rsymmetry 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.
Fieldtheoretical central charge
In this section we compute the central charge of the quiver depicted in gure 4 at the xed
point, using the amaximization procedure [
69
].
As recalled in section 3, the a and c central charges can be computed from the N = 1
Rsymmetry 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 antifundamental of another gauge group with rank Nb, and viceversa:
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
nontrivial 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 nontrivial check of the validity of our proposed quiver is that the central
charge given by (4.18) and that associated with the nonAbelian Tdual 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 nonAbelian Tdual of AdS5
by Z2 the quiver describing the nonAbelian Tdual 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 nonAbelian Tdual of
S5 and doubling it, we obtain the central charge of the nonAbelian Tdual 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 amaximization. The acharge 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 SfetsosThompson
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 nonAbelian Tdual of AdS5
S2=Z2 and the nonAbelian Tdual of
the KlebanovWitten solution. Notice that here, we are providing precisions about the CFT
dual to the nonAbelian Tdual backgrounds. This more precise information is matched
by the regularized form of the nonAbelian Tdual 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 nonAbelian Tdual 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 INSTBBBW puzzle
The nonAbelian Tdual of the KlebanovWitten background was rst written in [12, 13]
(INST). Further, in that paper an attempt was made to match the nonAbelian Tdual
background with a Bah, Beem, Bobev and Wecht (BBBW) solution [59]. This matching
was feasible assuming a particular split of the metric into a sevendimensional 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 nonAbelian Tdual 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 Mtheory
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 amaximization, and were shown to
match, in the large number of M5branes limit, the holographic results.
The solutions constructed in [59] were obtained by considering Mtheory compacti ed
on a deformed foursphere. 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
M5branes,
g is its genus and z is the socalled `twisting parameter', de ned as z = 2((pg q1)) from the
integer numbers p; q that indicate the twisting applied to the M5branes. 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 keypoint to be kept in mind after this discussion is that these results were obtained
in the context of a compacti cation of elevendimensional supergravity to seven dimensions.
Let us now come back to the paper [12, 13]. The matching of the nonAbelian
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 notcorrect in the analysis of [12, 13] was the assumption that the
nonAbelian Tdual solution could be obtained from a compacti cation of Mtheory on a
deformed foursphere (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 nonAbelian Tdual
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 nonAbelian Tdual solution is, for a chosen
set of parameters.
The resolution to this problem is that the nonAbelian Tdual background should
instead be thought of as providing a noncompacti cation of eleven dimensional supergravity.
Strictly speaking, our coordinate
runs in [0; 1], the four manifold is noncompact. In
our calculation of the central charge, we assumed that the coordinate was bounded in
[0; n L20 ], but this hard cuto , as we emphasized, is not a geometrically satisfactory way
of bounding a coordinate. There should be another, more general solution, that contains
our nonAbelian Tdual 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 nonAbelian
Tduality. 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/Mtheory 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 Mtheory 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 nonAbelian Tduality
applied on the KlebanovWitten background, we carefully studied its quantized charges
and holographic central charge (section 2). We lifted the solution to Mtheory 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 setup (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 Tdual of the KlebanovWitten background and the more conventional Tdual,
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 hardcuto ,
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 farless 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 MaldacenaNunez solutions [47] used in [44] to complete
the SfetsosThompson background. This would give rise to a superposition of N = 1 MN
solutions de ning the completed nonAbelian Tdual solution. It is unclear however in
which precise way this superposition would solve the (very nonlinear) PDEs associated to
N = 1 solutions [65, 77]. We see two possible paths to follow:
In the paper [77], Bah rewrote the general Mtheory 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 nonlinear 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
nonAbelian Tdual 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 nonlinear
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 followup 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 2016/089720 and 2014/186349. Y.L. and J.M. are partially supported
by the Spanish and Regional Government Research Grants FPA201563667P and
FC15GRUPIN14108. J.M. is supported by the FPI grant BES2013064815 of the Spanish
MINECO, and the travel grant EEBBI1712390 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 MilanoBicocca 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 nonAbelian Tdual of the KlebanovWitten
solution ts in the classi cation of N = 1 AdS5 backgrounds in Mtheory of GMSW [65].
A.1
Uplift of the nonAbelian Tdual solution
The eleven dimensional uplift of the nonAbelian Tdual solution consists of metric and
4form
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 fourform
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 M5branes sourcing this ux are transversal to both squashed
twospheres 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 Rsymmetry 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 Rsymmetry direction, ~ is a oneform 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 fourdimensional 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 oneforms 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 fourdimensional 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 sixdimensional space spanned by
M4 and the oneforms K1 and K2. G is an elevendimensional fourform whose components
lie along the sixdimensional 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 fourdimensional metric
g^(4) = e
6 g(4). Finally, d4 is the exterior derivative on the fourdimensional space that is
transverse to AdS5 and K1; K2.
A.3
Recovering the nonAbelian Tdual from GMSW
Let us now nd the explicit map between the GMSW geometry and the lifted nonAbelian
Tdual 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 D4branes 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 4form
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 Tdual of the KlebanovWitten solution
The Abelian Tdual, Type IIA description, of the KlebanovWitten 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 setup associated
to this solution. We have depicted both the brane setup 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 nonAbelian Tdual solution, the main
The paper [63] considered an Abelian Tduality transformation along the Hopf ber
direction of the T 1;1. This dualization gives rise to a wellde 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 nonsupersymmetric. Since our ultimate goal in this
section will be to compare with the nonAbelian Tdual solution, which is only
guaranteed to be a wellde 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 KlebanovWitten solution, and can be matched directly with the nonAbelian Tdual
solution in the large
limit.
We start by rewriting the KlebanovWitten metric in terms of the Tduality 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 KlebanovWitten 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) Tduality performed on the 2 direction trades the vielbein e3 for e^ = 0e C d 2,
and generates a NSNS 2form B2 = 0A~1 ^ d 2. The NSNS 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) Rsymmetry in the
direction. We can also see that it agrees with the asymptotic form of the metric of the
nonAbelian Tdual 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 reabsorbing 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 nonAbelian and Abelian Tdual 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 nonAbelian Tdual solution reduces in the
limit to the Abelian Tdual one. This connection between nonAbelian 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 Tdual is thus describing the KlebanovWitten theory modded by Z2. This is
consistent with the brane setup that is implied by the quantized charges of the background,
as we now show.
B.2
Quantized charges and brane setup
The background uxes of the Abelian Tdual solution support D4 and N S5brane 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 2cycles S~2( 2; ), S2( 1; 1), located along the compact
2direction.
This is the brane setup discussed in [64], describing the KlebanovWitten 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 2circle are not speci ed by the
geometry, so generically we can only think that they de ne four intervals in the 2circle.11
The same number of D4branes are stretched between each pair of NS5, NS5' branes since
even if large gauge transformations are required as we pass the value 2 =
D4brane 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 nonAbelian and Abelian Tdual 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 Tduality 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 KlebanovWitten CFT
The eld content of the KlebanovWitten 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 Rcharges 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 Rcharge 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 Tdual of AdS5
is maximized for
S5=Z2, using amaximization. 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 Z2re 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 nonanomalous
Rcharge R = R0 +
F =2 is obtained from the Rcharge for the gaugino, R0( ) = 1, plus the
nonanomalous 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
setup. There are N4 D4branes stretched between p = k NS5branes, 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 KlebanovWitten theory modded by Zk
In this appendix we include, for completeness, the eld theory calculation of the central
charge of the KlebanovWitten theory, using amaximization. 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 amaximization technique used
throughout the paper.
In this case we have, in the Type IIA description, p = k NS5branes 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
eldtheoretical 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 amaximization 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 BahBobev type of linear quiver (see eq.
(3.20) in [55]) for
= 0 and large `. Indeed, even if there is no clear de nition for
in
this case, the uplift of the circular brane setup is interpreted as M5branes wrapping a
torus (
= 0) with minimal punctures, as the gauging of the end avor groups of the linear
quiver corresponds in Mtheory to gluing the two leftover maximal punctures, closing up
the Riemann surface.
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