New gravity duals for higher - dimensional superconformal theories - - PowerPoint PPT Presentation

Alessandro Tomasiello

based on 1309.2949 with F.Apruzzi, M. Fazzi, D. Rosa 1404.0711 with D. Gaiotto 1406.0852 with F.Apruzzi, M. Fazzi, A. Passias, D. Rosa

New gravity duals for higher-dimensional superconformal theories

Introduction

Introduction

d > 4

Introduction

(Fµν)2 d > 4 √−gR d > 2

• Harder to define.

d > 4

Introduction

(Fµν)2 d > 4 √−gR d > 2

• Harder to define.
• N = (2, 0)
• theories arising at singularities
• intersecting branes
• String theory:

[Witten ’96; Seiberg, Witten ’96; (Blum,) Intriligator ’97; Hanany, Zaffaroni ’97; Brunner, Karch ’97…]

d > 4

Introduction

(Fµν)2 d > 4 √−gR d > 2

• Harder to define.
• N = (2, 0)
• theories arising at singularities
• intersecting branes
• String theory:
• d ≤ 4

[Witten ’96; Seiberg, Witten ’96; (Blum,) Intriligator ’97; Hanany, Zaffaroni ’97; Brunner, Karch ’97…]

d > 4

Introduction

(Fµν)2 d > 4 √−gR d > 2

• Harder to define.
• N = (2, 0)
• theories arising at singularities
• intersecting branes
• String theory:

In this talk, we will see some holographic classification results d = 5, 6

• d ≤ 4

[Witten ’96; Seiberg, Witten ’96; (Blum,) Intriligator ’97; Hanany, Zaffaroni ’97; Brunner, Karch ’97…]

d > 4

• 11 7 × S4/Γ
• 7

• 11 7 × S4/Γ

S3

=

7 × M3

D8–D6 bound state [stabilized by flux]

for example

F0 ̸= 0

• 7

• 11 7 × S4/Γ

S3

=

7 × M3

D8–D6 bound state [stabilized by flux]

for example

F0 ̸= 0

• 7

NS5 D6 D8

• near-horizon limits of brane systems
• quiver descriptions on tensor branch
• via T-duality: ‘Hitchin pole’ extension of F-theory

classification in [Heckman, Morrison, V

afa ’13]

• N = (1, 0) 6

• 11 7 × S4/Γ

S3

=

7 × M3

D8–D6 bound state [stabilized by flux]

for example

F0 ̸= 0

• 7

NS5 D6 D8

• near-horizon limits of brane systems
• quiver descriptions on tensor branch
• via T-duality: ‘Hitchin pole’ extension of F-theory

classification in [Heckman, Morrison, V

afa ’13]

• N = (1, 0) 6
• 6

• 1. Methods: Pure spinors

Plan

7 6 6

• I. Methods: Pure spinors

• I. Methods: Pure spinors

1,2

• G(1) ∩ G(2)
• G

• I. Methods: Pure spinors

1,2

• G(1) ∩ G(2)
• G
• r
• G T ⊕ T ∗

⊕

• I. Methods: Pure spinors

1,2

• G(1) ∩ G(2)
• G

nicer equations; easier classifications

forms obeying algebraic constraints:

• ften ‘pure spinors’
• r
• G T ⊕ T ∗

⊕

• riginal example

[Graña, Minasian, Petrini, AT ’05]

4 4} ×M6

• riginal example

[Graña, Minasian, Petrini, AT ’05]

4 4} ×M6

[Hitchin’s “generalized complex geometry”]

(3)×(3)

• riginal example

[Graña, Minasian, Petrini, AT ’05]

4 4} ×M6 (d + H∧)Φ = (ιK + ˜ K∧)F

[Hitchin’s “generalized complex geometry”]

(3)×(3)

[AT ’11]

M10

• riginal example

[Graña, Minasian, Petrini, AT ’05]

4 4} ×M6 (d + H∧)Φ = (ιK + ˜ K∧)F

*simplifying the story a bit…

((7) R8)2

total RR flux NS flux [Hitchin’s “generalized complex geometry”]

(3)×(3)

[AT ’11]

M10

• riginal example

[Graña, Minasian, Petrini, AT ’05]

4 4} ×M6 (d + H∧)Φ = (ιK + ˜ K∧)F

*simplifying the story a bit…

((7) R8)2

total RR flux NS flux [Apruzzi, Fazzi, Rosa, AT ’13]

6 × M4

[Apruzzi, Fazzi, Passias, Rosa, AT, ’14]

7 × M3

[Hitchin’s “generalized complex geometry”]

(3)×(3)

[AT ’11]

M10

• riginal example

[Graña, Minasian, Petrini, AT ’05]

4 4} ×M6 (d + H∧)Φ = (ιK + ˜ K∧)F

*simplifying the story a bit…

((7) R8)2

total RR flux NS flux [Apruzzi, Fazzi, Rosa, AT ’13]

6 × M4

[Apruzzi, Fazzi, Passias, Rosa, AT, ’14]

7 × M3

[Hitchin’s “generalized complex geometry”]

(3)×(3)

[AT ’11]

M10

× ea

• =2 vielbeine:

ea

• riginal example

[Graña, Minasian, Petrini, AT ’05]

4 4} ×M6 (d + H∧)Φ = (ιK + ˜ K∧)F

*simplifying the story a bit…

((7) R8)2

total RR flux NS flux [Apruzzi, Fazzi, Rosa, AT ’13]

6 × M4

[Apruzzi, Fazzi, Passias, Rosa, AT, ’14]

7 × M3

[Hitchin’s “generalized complex geometry”]

(3)×(3)

[AT ’11]

M10

× ea

• =2 vielbeine:

ea

• ea

R(θi) we prefer working with one ‘average’ of the two θi M3

[Apruzzi, Fazzi, Rosa, AT ’13]

IIB:

• θi

[Apruzzi, Fazzi, Rosa, AT ’13]

IIB:

• θi

no solutions!

but: see later about F-theory

[Apruzzi, Fazzi, Rosa, AT ’13]

IIB:

• θi

IIA:

the system contains

• ne-form equations

ea ∼ (θi) d(θi)

no solutions!

but: see later about F-theory

[Apruzzi, Fazzi, Rosa, AT ’13]

IIB:

• θi

IIA:

the system contains

• ne-form equations

ea ∼ (θi) d(θi)

no solutions!

but: see later about F-theory

[Apruzzi, Fazzi, Rosa, AT ’13]

local form

• f the metric:

ds2 ∼ dr2 + v2(r)ds2

S2

S2 (2) (1, 0)

IIB:

• θi

IIA:

the system contains

• ne-form equations

ea ∼ (θi) d(θi)

no solutions!

but: see later about F-theory

[Apruzzi, Fazzi, Rosa, AT ’13]

local form

• f the metric:

ds2 ∼ dr2 + v2(r)ds2

S2

S2 (2) (1, 0)

no Ansatz necessary

IIB:

• θi

IIA:

the system contains

• ne-form equations

ea ∼ (θi) d(θi)

The rest of the system also determines the fluxes

Bianchi id’s automatically satisfied

no solutions!

but: see later about F-theory

[Apruzzi, Fazzi, Rosa, AT ’13]

local form

• f the metric:

ds2 ∼ dr2 + v2(r)ds2

S2

S2 (2) (1, 0)

no Ansatz necessary

IIB:

• θi

IIA:

the system contains

• ne-form equations

ea ∼ (θi) d(θi)

The rest of the system also determines the fluxes

Bianchi id’s automatically satisfied

When the dust settles: we have a local solution provided we solve a system of 3 ODEs

}

∂rA = . . . ∂rv = . . . ∂rφ = . . .

dilaton warping

no solutions!

but: see later about F-theory

[Apruzzi, Fazzi, Rosa, AT ’13]

local form

• f the metric:

ds2 ∼ dr2 + v2(r)ds2

S2

S2 (2) (1, 0)

no Ansatz necessary

F0 = 0

M4

• 7 × M4

11 M4 = S4/Γ

F0 = 0

• 7 × S4

M4

• 7 × M4

11 M4 = S4/Γ

F0 = 0

S3

S4 =

• 7 × S4

M4

• 7 × M4

11 M4 = S4/Γ

F0 = 0

S3

S4 =

• S

3

• S

2

• D6
• S2

S3 ∼ =

• 7 × S4

M4

• 7 × M4

11 M4 = S4/Γ

F0 = 0

S3

S4 =

• S

3

• S

2

• D6
• S2

S3 ∼ =

• 7 × S4

M4

• 7 × M4

11 M4 = S4/Γ agrees with results in previous slide

• F0 = 0

local solutions also in [Blåbäck, Danielsson, Junghans, V an Riet, Wrase, Zagermann ’11] susy-breaking? in [Junghans, Schmidt, Zagermann ’14]

• F0 = 0

we can make

• ne of the poles regular:

D6 stack

local solutions also in [Blåbäck, Danielsson, Junghans, V an Riet, Wrase, Zagermann ’11] susy-breaking? in [Junghans, Schmidt, Zagermann ’14]

• F0 = 0

we can make

• ne of the poles regular:

D6 stack

• r both,

if we include D8’s:

D8–D6 stack actually, ‘magnetized’ D8’s D8–D6 bound states

=

local solutions also in [Blåbäck, Danielsson, Junghans, V an Riet, Wrase, Zagermann ’11] susy-breaking? in [Junghans, Schmidt, Zagermann ’14]

• F0 = 0

we can make

• ne of the poles regular:

D6 stack

2 4 6 8

r

1 2 3

S2 dilaton warping

• r both,

if we include D8’s:

D8–D6 stack actually, ‘magnetized’ D8’s D8–D6 bound states

=

local solutions also in [Blåbäck, Danielsson, Junghans, V an Riet, Wrase, Zagermann ’11] susy-breaking? in [Junghans, Schmidt, Zagermann ’14]

• F0 = 0

we can make

• ne of the poles regular:

D6 stack

2 4 6 8

r

1 2 3

S2 dilaton warping

• r both,

if we include D8’s:

D8–D6 stack actually, ‘magnetized’ D8’s D8–D6 bound states

=

• r include several

D8/D6 stacks:

stacks with opposite D6 charge

intuitively: D8’s don’t slip off because of electric attraction

local solutions also in [Blåbäck, Danielsson, Junghans, V an Riet, Wrase, Zagermann ’11] susy-breaking? in [Junghans, Schmidt, Zagermann ’14]

Most general solution is characterized by

[Apruzzi, Fazzi, Rosa, AT ’13; Gaiotto, AT ’14]

Most general solution is characterized by

[D8’s with same D6 charge stay together.]

• Ni µi

N

3 µ 3

N

1 µ 1

N

2 µ 2

N

1 µ 1

N

2 µ 2

[Apruzzi, Fazzi, Rosa, AT ’13; Gaiotto, AT ’14]

Most general solution is characterized by

[D8’s with same D6 charge stay together.]

• Ni µi
• N ≡

1 4π2

• H

N

3 µ 3

N

1 µ 1

N

2 µ 2

N

1 µ 1

N

2 µ 2

[Apruzzi, Fazzi, Rosa, AT ’13; Gaiotto, AT ’14]

Most general solution is characterized by

[D8’s with same D6 charge stay together.]

• Ni µi
• N ≡

1 4π2

• H

subject to constraints:

N

3 µ 3

N

1 µ 1

N

2 µ 2

N

1 µ 1

N

2 µ 2

[Apruzzi, Fazzi, Rosa, AT ’13; Gaiotto, AT ’14]

Most general solution is characterized by

[D8’s with same D6 charge stay together.]

• Ni µi
• N ≡

1 4π2

• H

subject to constraints:

µi B

F0 = 0

F0 = 0

N

3 µ 3

N

1 µ 1

N

2 µ 2

N

1 µ 1

N

2 µ 2

[Apruzzi, Fazzi, Rosa, AT ’13; Gaiotto, AT ’14]

Most general solution is characterized by

[D8’s with same D6 charge stay together.]

• Ni µi
• N ≡

1 4π2

• H

subject to constraints:

µi B

F0 = 0

F0 = 0

N

3 µ 3

N

1 µ 1

N

2 µ 2

N

1 µ 1

N

2 µ 2

ρ

µ

1

{

{

−µ

1

ρ

• µi

F0 > 0 F0 < 0

• ρ ρ

[Apruzzi, Fazzi, Rosa, AT ’13; Gaiotto, AT ’14]

Most general solution is characterized by

[D8’s with same D6 charge stay together.]

• Ni µi
• N ≡

1 4π2

• H

subject to constraints:

µi B

F0 = 0

F0 = 0

N

3 µ 3

N

1 µ 1

N

2 µ 2

N

1 µ 1

N

2 µ 2

N ρ

µ

1

{

{

−µ

1

ρ

• µi

F0 > 0 F0 < 0

• ρ ρ
• N ≥ |µ

1| + |µ 1 |

• F0 = 0

[Apruzzi, Fazzi, Rosa, AT ’13; Gaiotto, AT ’14]

[Gaiotto, AT ’14]

Often one finds a CFT dual using a brane configuration.

[Gaiotto, AT ’14]

Often one finds a CFT dual using a brane configuration.

F0 = 0

[Gaiotto, AT ’14]

Often one finds a CFT dual using a brane configuration.

7 × S4/Zk

reduction to IIA

7 × S3

k

F0 = 0

[Gaiotto, AT ’14]

Often one finds a CFT dual using a brane configuration. near-horizon N R × R4/Zk

7 × S4/Zk

reduction to IIA

7 × S3

k

F0 = 0

[Gaiotto, AT ’14]

Often one finds a CFT dual using a brane configuration. near-horizon N R × R4/Zk N reduction to IIA near-horizon k {

7 × S4/Zk

reduction to IIA

7 × S3

k

F0 = 0

[Gaiotto, AT ’14]

Often one finds a CFT dual using a brane configuration.

6

• (1, 0)
• N 3k2
• (k) × (k)

near-horizon N R × R4/Zk N reduction to IIA near-horizon k {

7 × S4/Zk

reduction to IIA

7 × S3

k

F0 = 0

[Gaiotto, AT ’14]

F0 = 0

Adapting methods developed in [Gaiotto, Witten ’08] for 3d theories

F0 = 0

Each D6 on a separate D8: Dirichlet b.c. for fields on the D6.

D8 stacks D6’s N

Adapting methods developed in [Gaiotto, Witten ’08] for 3d theories

F0 = 0

brane configurations studied long ago in [Hanany, Zaffaroni ’97; Brunner, Karch ’97] Each D6 on a separate D8: Dirichlet b.c. for fields on the D6. (triggered by Higgs vev)

RG flow D8 stacks D6’s N

Adapting methods developed in [Gaiotto, Witten ’08] for 3d theories

N

F0 = 0

brane configurations studied long ago in [Hanany, Zaffaroni ’97; Brunner, Karch ’97] Each D6 on a separate D8: Dirichlet b.c. for fields on the D6. (triggered by Higgs vev)

RG flow D8 stacks D6’s N

W

• uld the D8 survive a near-horizon limit?

Adapting methods developed in [Gaiotto, Witten ’08] for 3d theories

N

F0 = 0 D8’s can be thought of as Nahm poles for the D6’s.

brane configurations studied long ago in [Hanany, Zaffaroni ’97; Brunner, Karch ’97] Each D6 on a separate D8: Dirichlet b.c. for fields on the D6. (triggered by Higgs vev)

RG flow D8 stacks D6’s N

Xi ∼ ti

z

BPS equations on D6: Nahm equations ∂zX1 = [X2, X3] Xi z

W

• uld the D8 survive a near-horizon limit?

Adapting methods developed in [Gaiotto, Witten ’08] for 3d theories

N

F0 = 0 D8’s can be thought of as Nahm poles for the D6’s.

brane configurations studied long ago in [Hanany, Zaffaroni ’97; Brunner, Karch ’97] Each D6 on a separate D8: Dirichlet b.c. for fields on the D6. (triggered by Higgs vev)

RG flow D8 stacks D6’s N

Xi ∼ ti

z

BPS equations on D6: Nahm equations ∂zX1 = [X2, X3] Xi z

∼ = W

• uld the D8 survive a near-horizon limit?

∼ =

• (2) (k)

Adapting methods developed in [Gaiotto, Witten ’08] for 3d theories

N

∼ = So a better picture is

N

∼ =

near-horizon

• n the NS5 stack...

Conjecture: So a better picture is

N

∼ = …and the funnels become

• ur D8–D6 bound states!

near-horizon

• n the NS5 stack...

Conjecture: So a better picture is

N

∼ = …and the funnels become

• ur D8–D6 bound states!

near-horizon

• n the NS5 stack...

Conjecture: So a better picture is

N

More precisely:

• M3 H

µ ≡ N =

∼ = …and the funnels become

• ur D8–D6 bound states!

near-horizon

• n the NS5 stack...

Conjecture: So a better picture is Indeed the brane pictures would generically lead to predict a CFT:

• coinciding NS5’s

tensionless strings

[Hanany, Zaffaroni ’97; Brunner, Karch ’97]

• nontrivial moduli spaces

[singularities in the Higgs moduli space of massless theory]

N

More precisely:

• M3 H

µ ≡ N =

However, finer analysis shows constraints:

However, finer analysis shows constraints:

• ordering constraint in [Gaiotto, Witten ’08]

3 > 1

1 < 3

However, finer analysis shows constraints:

• ordering constraint in [Gaiotto, Witten ’08]

3 > 1

1 < 3

• separating the NS5’s

not a CFT: presence of free hypers

2 tensor branch, effective quiver description:

• 5

F0 = 0

• N ≥ |µ

1| + |µ 1 |

However, finer analysis shows constraints:

• µi

F0 = 0

• ordering constraint in [Gaiotto, Witten ’08]

3 > 1

1 < 3 same constraints as for AdS7:

• separating the NS5’s

not a CFT: presence of free hypers

2 tensor branch, effective quiver description:

• 5

F0 = 0

• N ≥ |µ

1| + |µ 1 |

However, finer analysis shows constraints:

• µi

F0 = 0

• ordering constraint in [Gaiotto, Witten ’08]

3 > 1

1 < 3

One-to-one correspondence with AdS7 solutions!

same constraints as for AdS7:

• separating the NS5’s

not a CFT: presence of free hypers

2 tensor branch, effective quiver description:

• 5

[Heckman, Morrison, V afa ’13]

Coda: Comparison to F-theory

[wip. with del Zotto, Heckman, V afa]; see also V afa’s talk

(1, 0)

[Heckman, Morrison, V afa ’13]

Coda: Comparison to F-theory

• one T-duality:

D8’s D6’s

. . .

NS5’s

. . .

D7’s D7’s

[wip. with del Zotto, Heckman, V afa]; see also V afa’s talk

(1, 0)

[Heckman, Morrison, V afa ’13]

Coda: Comparison to F-theory

• one T-duality:

D8’s D6’s

. . .

NS5’s

. . .

D7’s D7’s Nahm’s equations Hitchin’s equations

. . .

∼ =

but actually the D7’s fuse together: One of their chains of intersecting curves, decorated by Hitchin poles

[wip. with del Zotto, Heckman, V afa]; see also V afa’s talk

(1, 0)

[Heckman, Morrison, V afa ’13]

Coda: Comparison to F-theory

• one T-duality:

D8’s D6’s

. . .

NS5’s

. . .

D7’s D7’s Nahm’s equations Hitchin’s equations

. . .

∼ =

but actually the D7’s fuse together: One of their chains of intersecting curves, decorated by Hitchin poles

[wip. with del Zotto, Heckman, V afa]; see also V afa’s talk

(1, 0) This suggests that one should add Hitchin poles to the chains of non-perturbative F-theory 7-branes as well.

• some chains with exceptional gauge groups

also have gravity duals in M-theory

• some chains with exceptional gauge groups

also have gravity duals in M-theory

e.g. a chain of curves with self-intersection

12; 1, 2, 2, 3, 1, 5, 1, 3, 2, 2, 1, 12; 1, 2, 2, . . .

{

N

and gauge groups

(2) × G2 × F4 × G2 × (2) × E8 × (2) × . . .

{

“fractional M5’s” 7 × S4/ΓE8

• [Apruzzi, Fazzi, Passias, Rosa, AT ’13]

• IIA:
• F0 = 0
• F0 = 0

near-horizon of D4’s near D8–O8 wall [Brandhuber, Oz’99] unique: [Passias ’12]

[Apruzzi, Fazzi, Passias, Rosa, AT ’13]

• IIA:
• F0 = 0
• F0 = 0

near-horizon of D4’s near D8–O8 wall [Brandhuber, Oz’99] unique: [Passias ’12]

IIB: again we determined the local form of the metric:

S2 Σ2

we also determined all fluxes; they obey Bianchi automatically

S2 (2)

• [Apruzzi, Fazzi, Passias, Rosa, AT ’13]

for warping and dilaton

• 2 Σ2

two known solutions:

abelian and non-abelian [non-compact!] T-dual of D4–D8–O8

[Cvetic, Lu, Pope, V azquez-Poritz ’00; Lozano, Colgain, Sfetsos, Rodriguez-Gomez ’12] for warping and dilaton

• 2 Σ2

two known solutions:

abelian and non-abelian [non-compact!] T-dual of D4–D8–O8

[Cvetic, Lu, Pope, V azquez-Poritz ’00; Lozano, Colgain, Sfetsos, Rodriguez-Gomez ’12] (p, q) (p, q)

• (p, q)

e.g. [Aharony, Hanany ’97; deW

• lfe, Hanany Iqbal, Katz ’99;

Benini, Benvenuti, Tachikawa ’08; Bergman, Rodriguez-Gomez ’12] for warping and dilaton

• 2 Σ2

Conclusions

N T N ρ,ρ ρ ρ

NS5-branes pattern of D6’s ending on D8’s

• Possible generalizations: include O6’s, O8’s
• Hints of more general story from F-theory
• Gravity duals for 5d CFTs?

with effective quiver description on ‘tensor branch’

• (1, 0)