Non-Abelian T-duality and the
AdS/CFT correspondence (I)Yolanda Lozano (U. Oviedo)

Tohoku U, Sendai, March 4, 2019

1. Introduction & motivation:

NATD in AdS/CFT

Non-Abelian T-duality (NATD) has proved to be very usefulas a solution generating technique in AdS/CFT
Many interesting AdS backgrounds (some of which evading
existing classifications) have been constructed
What NATD does to the CFT remains less understood
Examples so far show that NATD may change the CFT dual to
the AdS backgrounds in which it is applied
This may happen because, contrary to its Abelian counterpart,
NATD has not been proved as a String Theory symmetry

In this talk we will discuss different examples for which the
CFT realization of NATD has been studied
In all of them the NATD background will be associated to
QFT living on (Dp,NS5) Hanany-Witten brane set-ups→ Novel way of describing (Dp,NS5) CFTs holographically

These examples have been studied in:- Y.L., Carlos Núñez, 1603.04440- Y.L., Niall Macpherson, Jesús Montero, Carlos Núñez, 1609.09061- Georgios Itsios,Y.L., Jesús Montero, Carlos Núñez, 1705.09661- Y.L., Niall Macpherson, Jesús Montero, 1810.08093

Outline:
1. Introduction and motivation: NATD in AdS/CFT
2. Basics of NATD: i) NATD vs Abelian T-dualityii) NATD as a solution generating technique
3. The NATD of AdS5 × S 5
4. The NATD of AdS5 × S 5 as a Gaiotto-Maldacena geometry
5. The NATD of Klebanov-Witten
6. Conclusions

2. Basics of NATD: i) NATD vs Abelian T-duality
Using the string sigma-model Rocek and Verlinde proved that Abelian
T-duality is a symmetry to all orders in gs and α
(Buscher’88; Rocek,Verlinde’92)

The extension to arbitrary wordsheets determines the global
properties of the dual variable:θ ∈ [0, 2π]T−→θ̃ ∈ [0, 2π]

In the non-Abelian case neither proof works
Variables living in a group manifold are substituted by variables
living in its Lie algebrag ∈ SU (2)NAT−→

χ ∈ R3In the absence of global information the new variables remain noncompact

In more detail:Rocek and Verlinde’s formulation of Abelian T-duality for ST
in a curved background (Buscher’88) :
1
S=
4πα

 

gμν dX ∧ ∗dX + Bμν dX ∧ dX
μ

ν

μ

ν



1
+
4π



i) Identify an Abelian isometry: X μ = {θ, X α } such that
θ → θ +  and ∂θ (backgrounds) = 0

ii) Gauge the isometry:dθ → Dθ = dθ + A

A non-dynamical gauge field / δA = −dR(2) φiii) Add a Lagrange multiplier term: θ̃ dA , such thatDθ̃ → dA = 0 ⇒ A exact(in a topologically trivial worldsheet)+ fix the gauge: A = 0 → Original theoryiv) Integrate the gauge field+ fix the gauge:

θ=0→ Dual sigma model:{θ, X α } → {θ̃, X α } and(g̃, B̃2 , φ̃) given by Buscher’s formulae- Conformal invariance? Original and dual theories can beobtained from the gauged Lagrangian either gauging avectorial or an axial combination of chiral currents

- Arbitrary worldsheets? (symmetry of string perturbation
theory):Large gauge transformations:

(a)



(b)

d = 2πn ; n ∈ ZγTo have invariance:
Multivalued Lagrange multiplier:such that[exact] → dA = 0+dθ̃ = 2πm ; m ∈ Zγ[harmonic] ⇒ A exact

This fixes the periodicity of the dual variableNon-Abelian T-duality(De la Ossa, Quevedo’93)

Non-Abelian isometry:i) Gauge it:n
X m → gmX
,g ∈ GnndX m → DX m = dX m + Am
Xn

A ∈ Lie algebra of G

A → g(A + d)g −1

ii) Add a Lagrange multiplier term: Tr(χF )F = dA − A ∧ A
χ ∈ Lie Algebra of G , χ → gχg −1, such thatDχ → F = 0 ⇒ A pure gauge(in a topologically trivial worldsheet)+ fix the gauge: A = 0 ⇒ Original theoryiii) Integrate the gauge field + fix the gauge → Dual theory
However:• Non-involutive• Higher genus generalization? Set to zero Wγ = P e A• Global properties?
γ

3
χ
∈
R
: Global completion of R3 ?
For SU(2):• Conformal invariance not proved in generaliii) Integrate the gauge field + fix the gauge → Dual theory
However:• Non-involutive• Higher genus generalization? Set to zero Wγ = P e A• Global properties?
γ

3
χ
∈
R
: Global completion of R3 ?
For SU(2):• Conformal invariance not proved in general
True symmetry in String Theory?

Still, NATD has been proved to be a very useful solution
generating technique (Sfetsos and Thompson (2010))

Still, NATD has been proved to be a very useful solution
generating technique (Sfetsos and Thompson (2010))
Some examples:• New AdS background in IIB
6(Y.L., O Colgain, Rodriguez-Gomez, Sfetsos, PRL (2013))

Candidate for the holographical description of a 5d fixed point
theoryMotivated classifications of IIB AdS6 solutions that culminatedwith the work of D’Hoker, Gutperle, (Karch,) Uhlemann’16,17• New AdS M-theory solution beyond existing classifications
3

Only explicit example in KKK with SU(2) structure(Y.L., Macpherson, Montero, O Colgain’15)

ii) NATD as a solution generating technique
Need to know how the RR fields transform
Sfetsos and Thompson (2010) extended Hassan’s derivation in the
Abelian case:
Implement the relative twist between left and right movers in thebispinor formed by the RR fields

But how do we interpret the non-compact directions in thecontext of AdS/CFT?
Can we learn something about their global properties from
Holography?

3. The NATD of AdS5 × S 5
AdS5 × S 5

N=2 AdS5 × S 2 ∈ GM
NATD

N=4 SYM(Sfetsos, Thompson’10)

N=2 SCFT(Y.L., Nunez, ’16)

- Gaiotto-Maldacena geometries encode the information
about the dual CFT- Useful example to study the CFT realization of NATD

5
AdS
×
S
Take
the
background•5

ds2 = ds2AdS5 + L2 dα2 + sin2 αdβ 2 + cos2 αds2 (S 3 )F5 = 8L4 sin α cos3 αdα ∧ dβ ∧ Vol(S 3 ) + Hodge dual

•Dualize it w.r.t. one of the SU (2) symmetries
In spherical coordinates adapted to the remaining SU (2) :ds2= ds2AdS5 + L2



2

2

dα + sin αdβ

ρ3
2
Vol(S
B2 = 2),4
4
ρ + L cos α2L2 cos2 αρ2
dρ2
2
2
+
ds
(S
)
+ 2
2
2
4
4
L cos α ρ + L cos αe−2φ = L2 cos2 α(L4 cos4 α + ρ2 )F2 = L4 sin α cos3 αdα ∧ dβ,

F 4 = B 2 ∧ F2

• New Gaiotto-Maldacena geometry• What about r ?
•Background perfectly smooth for all r ∈ R+
•No global properties inferred from the NATD
•How do we interpret the running of r to infinity in the CFT?

•Singular at α = π/2 where the original S 3 shrinks (due tothe presence of NS5-branes)

This is the tip of a cone with S 2 boundary →Large gauge transformations B2 → B2 − nπVol(S 2 )for r ∈ [nπ, (n + 1)π]This modifies the Page charges such that N4 = nN6 in each[nπ, (n + 1)π] interval
N4←→ D4-branes on R1,3 , r

N6←→ D6-branes on R1,3 , r, S 2

We have also N5 charge

1
N5 =
(2π)2



1
H3 =
π

 r∗
dr
0

such that every time we cross a π interval one unit of NS5
charge is created. For r ∈ [0, nπ], N5 = n.
The picture that arises is that D3 → (D4, NS5)
which seems to be a general pattern

This is consistent with a D4/NS5 brane set-up:
NS5

NS5

D4: R1,3 , r
NS5: R1,3 , α, β

NS5

NS5
n D4

2 D4
D4

........
π

2π

NS5

3π

nπ

..
.

...
(n + 1)πr(in units of N6 )These D4/NS5 brane set-ups realize 4d N = 2 field theories
with gauge groups connected by bifundamentals (Witten’97)

Having the D4 finite extension in the r direction, the field
theory living in them is 4d at low energies, with effective gauge
coupling:
1∼ rn+1 − rn2g4Open strings connecting D4’s stretched between different
NS5 represent bifundamental matter
Strings connecting the D4 to D6 or semi-infinite D4 represent
fundamental matter
For ln D4-branes in [rn , rn+1 ] the gauge group is SU (ln )and there are (ln−1 , ln ) and (ln , ln+1 ) hypermultiplets.

The field theory is then described by a quiverSU (l1 )SU (l2 )SU (ln )The bifundamentals contribute to the SU (ln ) beta function as
ln−1 + ln+1 flavors.
The beta function thus vanishes at each interval if
2ln = ln+1 + ln−1

This condition is satisfied by our brane configuration, which has
ln = nN6

It corresponds to an infinite linear quiver:
SU (N6 )SU (2N6 )SU (3N6 )It corresponds to an infinite linear quiver:
SU (N6 )SU (2N6 )SU (3N6 )Gaiotto-Maldacena geometries were built to study the
CFTs associated to these brane set-ups (Gaiotto, Maldacena’09)
We will now see that, as a GM geometry, the dual quiverassociated to the NATD solution is the same above
This brane intersection has also been recently confirmed by
the analysis in Terrise, Tsimpis, Whiting’18

4. The NATD as a Gaiotto-Maldacena geometryi) Gaiotto-Maldacena geometries:
Generic backgrounds dual to 4d N=2 SCFTs.
Described in terms of a function V (σ, η) solving a Laplace eq.
with a given charge density λ(η) at σ = 0∂σ [σ∂σ V ] + σ∂η2 V = 0,λ(η) = σ∂σ V (σ, η)|σ=0

Regularity and quantization of charges impose strong constraints
on the allowed form of λ(η) , which encodes the information of
the dual CFT:

• It must vanish at η = 0• Its slope must be an integer, always decrease, and it can only change
at integer values of η

• Each time the slope changes by k units there is an Ak−1 singularityii) N = 2 CFTs from Gaiotto-Maldacena geometries:- A SU (ni ) gauge group is associated to each integer value ofη = ηi , with ni given by λ(ηi ) = ni
- A kink in the line profile corresponds to extra ki fundamentals
attached to the gauge group at the node ni

For example:
λ
7
6
5
4
3
2
1

4

3

1 2 3 4

η

5

6

7

1

7

iii) The NATD as a Gaiotto-Maldacena geometry:
The NATD background is a Gaiotto-Maldacena geometrywithλ(η) = η , η ∼ r , σ = sin αλ(η) = η ⇒ Infinite linear quiver, consistent with the brane

set-upiii) The NATD as a Gaiotto-Maldacena geometry:
The NATD background is a Gaiotto-Maldacena geometrywithλ(η) = η , η ∼ r , σ = sin αλ(η) = η ⇒ Infinite linear quiver, consistent with the brane

set-upThis brings strong evidence to the brane set-up predicted fromthe solutioniii) The NATD as a Gaiotto-Maldacena geometry:
The NATD background is a Gaiotto-Maldacena geometrywithλ(η) = η , η ∼ r , σ = sin αλ(η) = η ⇒ Infinite linear quiver, consistent with the brane

set-upThis brings strong evidence to the brane set-up predicted fromthe solution
However, the associated CFT is infinite.
We will see next that we can complete the quiver to produce
a well-defined 4d CFT, and, using holography, complete thegeometry

A natural way to complete the quiver is by adding fundamentals:
SU (N6 )SU (2N6 )SU ((p − 1)N6 )SU (pN6 )This completion reproduces correctly the value of the holographic
central charge:
From the geometry: η∗(Klebanov, Kutasov,
N62 N53f (η)dη =cN AT D ∼Murugan’08)
120
1
In the field theory we can use: c = (2nv + nh ) (Shapere,
Tachikawa’08)
12This gives

1
2
N62 p3 
2  N62 p3
1− − 2 2 + 2 3 ≈c=12p p N6N6 p
12

In the geometry, the completed quiver corresponds toλflavor group

λ(η)
=
N6
p−1 p



η
(1 − p)η + (p2 − p)0≤η ≤p−1(p − 1) ≤ η ≤ p

ηThe solution to the Laplace equation that gives rise to thischarge density completes the non-Abelian T-dual geometry, andresolves its singularity
Close to the kink the behaviour of the completed geometry isthat of D6-branes
The NATD solution arises when zooming-in away from thekinkThis idea also works in other examples

5. The NATD of Klebanov-Witten
N=1 AdS5 ∈ GMSW

AdS5 × T 1,1

NATD
N=1 KW

N=1 SCFT(Itsios, Nunez, Sfetsos, Thompson’13) (Itsios,Y.L., Montero, Nunez, ’17)

- GMSW geometries do not encode the information
about the dual CFT- Still, we can extract useful information using the relation
with GM

The Klebanov-Witten theory:
N = 1 SU (N ) × SU (N ) gauge theory with bifundamental
matter fields transforming in the (N, N̄ ) and (N̄ , N )representations of SU (N )Dual to AdS5 × T 1,1 :
1 2 2
1
2
2
2
dsT 1,1 = (ds (S1 ) + ds (S2 )) + (dψ + cos θ1 dφ1 + cos θ2 dφ2 )2
6
9

4
F5 =
Vol(AdS5 ) − L5 Vol(T 1,1 )LAbelian T-dual realization: (Dasgupta, Mukhi’98; Uranga’98)
IIA background with B2 , F4 fluxes ↔ NS5, D4 branes
Two types of, orthogonal, NS5-branes
Mutual rotation equivalent to a mass deformation in the
CFT breaking N = 2 to N = 1

The non-Abelian T-dual solution
IIA background with B2 , F2 , F4 = B2 ∧ F2 fluxes
Two types of, orthogonal, NS5 and NS5’ branes
Brane set-up:

4

It generalizes the brane set-ups describing the linear quivergauge theories of Bah-Bobev (Bah, Bobev’13)

Bah-Bobev linear quivers: (Bah, Bobev’13)

N = 1 and N = 2 SU (N ) vector multiplets plus bifundamental
matter fields

Realized in brane set-ups with orthogonal NS5 and NS5’ branes:
4

Flow to interacting 4d N = 1 CFT in the IR

In the CFT fixed point the central charges can be determinedfrom the ’t Hooft anomalies associated to the R-symmetry(Anselmi, Freedman, Grisaru, Johansen’97) :3(3Tr R3 − Tr R) ,a=
32

1
(9Tr R3 − 5Tr R)c=32

1
where R is the R-symmetry, R = R0 + F , determined by
2
a -maximization (Intriligator, Wecht’03)

CFT dual to the NATD solution
Infinite linear quiver → Complete it, such that:
Vanishing beta functions and R-symmetry anomalies
Self-dual under Nc → Nf − Nc
Mass deformation of the N = 2 quiver dual to the NATD
of AdS5 × S 5 /Z2 :AdS5 × S 5 /Z2

mass
AdS5 × T 1,1

NATD of AdS5 × S 5 /Z2
mass

NATD of AdS5 × T 1,1

Our proposed completed quiver is:

It is obtained by modding out by Z2 the N = 2 completed
quiver dual to the NATD of AdS5 × S 5 and mass deforming it
A non-trivial check is that the central charges satisfy the
Tachikawa-Wecht UV/IR relations:9
aN =1 =
(4 aN =2 − cN =2 ) ,32

1(−12 aN =2 + 39 cN =2 )cN =1 =
32

(Tachikawa, Wecht’09)

number of nodes limit: c• In the large
27
cN =1 =

•

32

cN =2

N =1 ∼ aN =1

and(Tachikawa, Wecht’09)

9 3 2
Moreover, cN =1 ∼ aN =1 ∼ N5 N6 matches the holographic
64
9 3 2
result: cN AT D ∼ Vint ∼ N5 N664

number of nodes limit: c• In the large
27
cN =1 =

•

32

cN =2

N =1 ∼ aN =1

and(Tachikawa, Wecht’09)

9 3 2
Moreover, cN =1 ∼ aN =1 ∼ N5 N6 matches the holographic
64
9 3 2
result: cN AT D ∼ Vint ∼ N5 N664Open problem: Completion of the NATD solution
Need to introduce flavor D6-branes in the geometry(Bah’15; Apruzzi, Fazzi, Passias, Tomasiello’15)

The NATD would arise as a result of zooming-in in a particular
patch of the completed geometry

6. Conclusions (so far)
- NATD useful as a solution generating technique in AdS/CFT:New Gaiotto-Maldacena geometry
New solution in the class of GMSW
- Significant progress in the CFT interpretation:
Gaiotto-Maldacena geometry dual to an infinite linear quiver
GMSW geometry dual to a mass deformation of the field
theory dual to AdS5 × S 5 /Z2
Non-trivial check: Tachikawa-Wecht UV/IR relations→ Different CFTs after NATD

D3 → (D4, NS5)
D3 → (D4, NS5, NS5’)

AdS solutions constructed through non-Abelian T-duality need
to be completed in order to describe well-defined CFTs
In the example discussed the NATD arises away from the kink

- General pattern? :AdS6 × S 4 : (D4,D8) system → (D5,NS5,D7)
AdS4 × S 3 × S 2 : (D2,D6) system → (D3,NS5,D5)

Connection with (Dp,NS5) brane set-ups

7. The NATD of AdS6 × S 4
The AdS6 /CF T5 correspondence remains much lessunderstood than its cousins in other dimensions
This is largely due to the fact that in 5d there are no maximallysupersymmetric CFTs
Indeed, in 5d the superconformal algebra is unique: F (4)It contains 8 supersymmetries (enhanced to a total of 16)
Accordingly, AdS6 backgrounds are quite unique:
There is a unique AdS6 solution to massive Type IIA (theBrandhuber-Oz solution (Brandhuber, Oz’99)), there are no solutionsin M-theory, and a kind of no-go theorem existed in Type IIB(Passias’12)

AdS6 /CF T5 duals

5d gauge theories are non-renormalizable:[g 2 ] = M −1

→

g2 E

→

UV completion

Indeed, they can flow to strongly coupled CFTs in the UV
for specific gauge groups and matter content(Seiberg’96; Intriligator, Morrison, Seiberg’97)

String theory realizations are known in some cases that canbe used to construct their AdS6 duals
For example, Sp(N ) (with specific matter content) can berealized in Type I’ in a D4/D8/O8 system (Seiberg’96)
The Brandhuber-Oz solution arises as its near horizon
geometryMore general string theory realizations can be given in termsof (p,q) 5-brane webs in Type IIB (Aharony, Hanany’97; Aharony, Hanany,
Kol’97)

The 5d field theory lives on D5-branes which are of finite
extent on one dimension and are embedded in a web ofsemi-infinite (p,q) 5-branes as well as optional 7-branes (DeWolfe,
Hanany, Iqbal, Katz’99)

These webs are 5d realizations of Hanany-Witten brane setups
D5
D7

NS5

Their AdS6 duals remained unknown till the recent work of
D’Hoker, Gutperle, Karch, Uhlemann’16; D’Hoker, Gutperle, Uhlemann’17,18

These works provide a complete classification of AdS6
solutions to Type IIB SUGRA, which are firm candidates forholographic duals of CFTs living in (p,q) 5-brane webs.

T-duality played a very important role in these developments:
The first AdS6 solutions in Type IIB were constructed byacting with T-duality (Abelian and non-Abelian) on the
Brandhuber-Oz solution(Y.L., O Colgain, Rodriguez-Gomez, Sfetsos’12; Y.L., O Colgain, RodriguezGomez’13)

The existence of these solutions raised the interest in the studyof classifications of AdS6 solutions in Type IIB:
Apruzzi, Fazzi, Passias, Rosa, Tomasiello’14; Kim, Kim, Suh’15; Kim, Kim’16;
Gutowski, Papadopoulos’17,

which culminated with the work of
D’Hoker, Gutperle, Karch, Uhlemann’16; D’Hoker, Gutperle, Uhlemann’17,18

The Abelian T-dual solution describes the same 5d Sp(N )fixed point theory as the Brandhuber-Oz solution but now
in terms of a D5, NS5, D7/O7 brane system in Type IIB
In Y.L., Macpherson, Montero: 1810.08093 we showed that both the
Abelian and non-Abelian T-dual solutions fit in the formalismof DGU and gave a field theory interpretation to the NATD

8. The Brandhuber-Oz solution of massive IIA
The near horizon geometry of the D4, D8/O8 system is a
fibration of AdS6 over half- S 4 with an S 3 boundary at theposition of the O8-plane, preserving 16 SUSYs

2 2 
W
L
ds2 =
9 ds2 (AdS6 ) + 4 ds2 (S 4 )
4F4 = 5 L4 W −2 sin3 θ dθ ∧ Vol(S 3 )
3L
−φ
− 16
e =
,
W = (m cos θ)
52Wθ ∈ [0,π]
28 − Nfm=
2πls• SO(5) symmetry broken to SO(4) ∼ SU (2) × SU (2) :SU (2) ↔ SU (2)R R-symmetry of the field theory
SU (2) ↔ global symmetry massless antisym. hyper

9. The Abelian T-dual of Brandhuber-Oz
Take the AdS6 × S 4 background


2 2
L
W
9ds2 (AdS6 ) + 4 dθ2 + sin2 θds2 (S 3 )ds2 =4

F4 = 5L4 W −2 sin3 θ dθ ∧ Vol(S 3 )and dualize w.r.t. the U(1) fiber
The resulting background preserves all SUSY
It realizes the Sp(N ) fixed point theory in a D5,NS5,D7/O7
brane system:
O7
D7
N D5

O7

10. The non-Abelian T-dual of Brandhuber-Oz
Take the AdS6 × S 4 background


2 2
L
W
9ds2 (AdS6 ) + 4 dθ2 + sin2 θds2 (S 3 )ds2 =4

F4 = 5L4 W −2 sin3 θ dθ ∧ Vol(S 3 )Dualize it w.r.t. one of the SU (2) symmetries
In spherical coordinates adapted to the remaining SU (2) :

2 2 
L
W
r 2 e2 A
2
2
2
−2 A
2
2
2
ds =
dr + 2ds(S)
9 ds (AdS6 ) + 4 dθ + e
4
A
4
r +e
r3
3L A
2
−φ
2 + e4 A
B2 = 2
Vol(S
)
e
=
e
r
r + e4 A
2W5
r2
4A
2
F1 = −G1 − m r dr
F3 = 2
[−r
G
+
m
e
dr]
∧
Vol(S
)
1
r + e4 A

• It solves the IIB equations of motion, preserving all SUSY
•What about r ?
+r∈R
Backgroundperfectly
smoothforall••No global properties inferred from the NATD
•How do we interpret the running of r to infinity in the CFT?
•Boundary at θ = π/2 inherited•Singular at θ = 0 where the original S 3 shrinks (due to thepresence of NS5-branes)
This is the tip of a cone with S 2 boundary → We have tocare about large gauge transformations•Large gauge transformations modify the quantised charges
such that ND5 = nND7 in each [nπ, (n + 1)π] intervalWe have also NS5 charge, such that every time we cross a πinterval one unit of NS5 charge is createdThis is compatible with a D5/NS5 brane set-up:
The NS5 are bent due to the different
number of D5 ending on each sidex9r

There is a symmetry under x9 → −x9
due to the orientifold action

Add the orientifold:
A worldsheet analysis shows that under NATD: Ω → Iχ Ω
⇒ The O8 plane is mapped onto a O5 plane, located at r = 0
This O5 is in fact a O7 wrapped on the S 2

Then:

n(1, −1)n(0, ND7 )n(ND7 + 2, 1)n(ND7 , 0)n(2, 1)n(1, 1)Here:n(ND7 − 2, −1)

We have completed the infinite brane set-up with semi-infinite(p,q) 5-branes
Sp(ND7 ) × Sp(2ND7 ) × · · · × Sp(nND7 )fixed point theory

At each interval the condition NF ≤ 2N + 4 for the existenceof a Sp(N ) UV fixed point theory is satisfied, according to theclassification in Intriligator, Morrison, Seiberg’97

This is supported by the central charge and EE, which scale with
N 5/2

11. The AdS4 × S 2 × S 2 example(Y.L., Macpherson, Montero, Nunez, ’16)

Non-Abelian T-duality on a reduction to IIA of AdS4 × S 7 /Zk→ IIB AdS4 × S 2 × S 2 background, N=4 SUSY, in theclassification of D’Hoker, Estes and Gutperle’07
Analysis of charges:
NS5(D3,NS5,D5) brane set-up:
NS5

D5

NS5
D5

......

D3

Gaiotto and Witten’08:

π

NS5
D5

D3

0

NS5

2π

..
.

..

D3

(n − 1)π3d N=4 Tρρ̂ (N ) theoriesnπTρρ̂ (N )field theories flow to CFTs in the infrared if the partitions
satisfy certain conditions, that are satisfied by our brane set-up
ND5

2ND5

3ND5

nND5

k0

2k0

3k0

nk0

The holographic duals of these CFTs are known (Assel, Bachas, Estes andGomis’11)

They belong to the general class of AdS4 × S 2 × S 2 geometries in
D’Hoker, Estes and Gutperle’07

These are fibrations of AdS4 × S 2 × S 2 over a Riemann surface
that can be completely determined from two harmonic functions
h1 (z, z̄), h2 (z, z̄)Assel, Bachas, Estes and Gomis’11 showed how to determine thesefunctions from the (D3, NS5, D5) brane set-ups associated toTρρ̂ (N ) theories:
p

π
i
1
a
2 + δa − z) + cch1 = −
N5 log tanh (
4 a=1
2

1
h2 = −4p̂

z − δ̂bbN̂5 log tanh () + cc

b=1

2





The positions of the D5 and NS5 branes are determined, in turn,from the linking numbers of the configuration:
π la ˆlbδ̂b − δa = log tan ()
2 N

The h1 , h2 functions computed from our completed brane set-up
agree with those associated to the non-Abelian T-dual geometry
far from the location of the added branes
The non-Abelian T-dual arises as a result of zooming-in in aparticular region of the completed solution
This completion smoothes out the singularities and defines thegeometry globally
The ideas about the completion of AdS non-Abelian T-duals arevery explicit in this example
The matching between the field theory and holographic central
charges is however quite non-trivial in this example

The free energy of specific Tρρ̂ (N ) theories was computed byAssel, Estes and Yamazaki’12 both in the field theory and in the geometry:- It was shown to exhibit a F ∼ N 2 log N behaviour- In the geometry the log N comes from the size of theconfiguration- It was argued that this provides an upper bound to the freeenergy of more general Tρρ̂ (N ) theories√

3/2
The NATD has a free energy F ∼ kND5
(n + 1)3

Quite non-trivially, the completion reproduces a N 2 log N behaviourand satisfies the expected bound12. Conclusions
The Abelian and non-Abelian T-duals of Brandhuber-Oz were thefirst AdS6 solutions known in Type IIB SUGRA.
We have shown that they fit within an extension of the AdS6 × S 2
global solutions with 7-branes recently classified by D’Hoker,
Gutperle and Uhlemann
Showing this required extending the formalism of DGU to include
D7-branes in the annulus
Non-trivial examples involving smeared NS5 and D7 branes, andO7
The Abelian T-dual solution provides the first known example forthe annulus

We have given a candidate CFT dual to the non-Abelian T-dual
solution, along the same lines as in previous examples
This 5d fixed point theory belongs to the general classification by
Intriligator, Morrison and Seiberg.

We have given a candidate CFT dual to the non-Abelian T-dual
solution, along the same lines as in previous examples
This 5d fixed point theory belongs to the general classification by
Intriligator, Morrison and Seiberg.
Open problems:
Work out the properties of this CFT:
Computation of observables
Study of global symmetries, enhancement..
Compute the backreaction in the geometry of the external
5-branes added to complete the infinite quiverComplete the non-Abelian T-dual geometry

More generally,
Non-Abelian T-duality has interesting applications to Holography:
Generation of new solutions → Extension of existingclassifications, explicit examples in known classes,
construction of new classes of solutions,..
AdS solutions constructed through NATD are dual to CFTs
living on (Dp,NS5) brane intersections → First step
towards their holographic description
We still need a general prescription for completing the solution

Holography has interesting applications to duality in ST:
Using holography it is possible to provide the sought for
global completion of NATD geometries
The problem remains of course open for backgrounds withouta holographic description

Future directions:
NATD and defect CFTs
NATD and BH solutions
NATD and deconstruction…..

THANKS!