Weaving the Exotic Web Yuho Sakatani (Kyoto Prefectural Univ. of - - PowerPoint PPT Presentation

Yuho Sakatani

(Kyoto Prefectural Univ. of Medicine)

Strings and Fields 2018 @ YITP

Weaving the Exotic Web

arxiv:1805.12117, a collaboration with Jose J. Fernandez-Melgarejo (Univ. of Murcia) and Tetsuji Kimura (Nihon U). Next speaker

Weaving the Exotic Web

Duality web of branes

Duality web

Standard branes Exotic branes Type II string /

𝟖

Exotic branes

[Elitzur, Giveon, Kutasov, Rabinovici ‘97; Blau, O’Loughlin ’97;

Obers, Pioline ’99; Eyras, Lozano ’00;

Lozano-Tellechea, Ortin ’00]

If a brane (wrapped on a torus) has the mass we call the brane a b𝒐

𝒅𝒕,…,𝒅𝟑-brane. 𝒐

discovered in

Exotic branes

Mass of a b𝒐

𝒅𝒕,…,𝒅𝟑-brane

Mass of a Dp-brane ( p𝟐 -brane ) Mass of a KKM ( 5𝟑

𝟐-brane )

Mass of a 5𝟑

𝟑-brane

T-dual [Obers, Pioline ’99]

Other than the standard branes, all branes in the web are called exotic branes. Standard branes

Here, we haven’t considered co-dimension 1 or 0–branes, such as D8 or D9.

Standard branes

If we consider domain-wall/space-filling branes as well, we find a huge number of exotic branes! (co-dim. 1) (co-dim. 0)

Exotic branes

Type IIA /

𝟖 [T.Kimura, J.J. Fernandez-Melgarejo, YS] defect branes, domain-wall branes, space-filling branes

Exotic branes

Type IIB /

𝟖 [T.Kimura, J.J. Fernandez-Melgarejo, YS] defect branes, domain-wall branes, space-filling branes

Exotic Web

IIA IIA IIA IIB IIB IIB

…

S-dual [T.Kimura, J.J. Fernandez-Melgarejo, YS] T-dual

Our motivation

SUGRA solutions for the standard branes/defect-branes are well-known. How about domain-wall branes? space-filling branes?

[Meessen, Ortin, hep-th/9806120]

(co-dim.2)

Previous works

In type IIA theory, only two domain-wall solutions are known:

[Meessen, Ortin hep-th/9806120]

For other exotic branes, SUGRA solutions are not obtained. KK8A

Our result

In the following, I will show that we can construct SUGRA solutions for all of the exotic branes, by employing duality-covariant formulations of SUGRA, Double Field Theory / Exceptional Field Theory.

Standard Dp-brane solution

Dp-brane … …

Dp Dp Dp Dp Dp

• 1. Smear the Dp-brane
• 2. Perform a T-duality

along the smeared direction becomes isometric

transverse coordinates

D(p+1)-brane

Standard Dp-brane solution

co-dimension 3 D6-brane co-dimension 2 D7-brane

T-duality along Smear along

• direction becomes isometric

depends only on 2 coordinates

D8-brane?

Smeared-D7 We want to obtain a domain-wall solution.

Smear the D7 along

• direction is not isometric

We cannot perform T-duality along -direction. We cannot straightforwardly get D8-brane solution.

To obtain domain-wall solutions

We employ generalized formulations

• f supergravity

Double field theory Exceptional field theory

[Siegel ’93; Hull, Zwiebach ’09; Hohm, Hull, Zwiebach ’10; Jeon, Lee, Park ’10; ...] [West ’00; Berman, Perry ’11; Berman, Godazgar, Perry, West ’12; Hohm, Samtleben ’13; ...]

(T-duality) (U-duality)

Double Field Theory

In DFT, we introduce the doubled spacetime. F1 P All of the SUGRA fields are defined on the doubled spacetime. We organize them into

generalized metric T-duality-inv. dilaton

(20 dim.)

DFT

E.O.M. of DFT is covariant under the T-duality transformation:

Buscher’s rule coordinate exchange This maps a solution of DFT to another DFT solution.

Example 1

A solution of the usual SUGRA.

T𝟗-dual A solution of DFT. Smeared-D7 “D8-brane” solution

[Hohm, Kwak, arXiv:1108.4937]

Example 2

(smeared) 5𝟑

𝟑-brane solution:

A solution of the usual SUGRA.

5𝟑

𝟒-brane solution:

A solution of DFT.

[Meessen, Ortin, hep-th/9806120; de Boer-Shigemori, arXiv:1004.2521]

T𝟗-dual

Dual parameterization

To describe backgrounds of exotic branes it’s more convenient to introduce the dual fields.

[Duff ’89; Andriot, Larfors, Lust, Patalong, arXiv:1106.4015]

In the following, we describe backgrounds with . Just a redefinition of .

Dual parameterization

Dual coordinate appears only in the β-field. Moreover, the dependence is just linear. 5𝟑

𝟑-brane solution:

5𝟑

𝟒-brane solution:

In this way, we can obtain SUGRA solutions for all of the exotic branes:

In order to get all of the exotic branes, T-duality is not enough. We also need to perform S-duality.

IIA IIA IIA IIB IIB IIB

…

S-dual!

DFT → EFT

To perform S-duality, we need to extend DFT to U-duality covariant EFT.

Exceptional Field Theory

In EFT, we introduce the generalized coordinates

F1 P generalized metric D1 D3 NS5 dilaton, R-R fields e.t.c. Natural extension of DFT type IIB branes DFT

Dual parameterization

Usual type IIB fields

Similar to DFT, we can introduce the dual fields:

field redefinitions dual type IIB fields

T-duality in EFT

The T-duality rule in terms of the dual fields is dual version of Buscher’s rule Coordinate exchange: [YS, Uehara arXiv:1701.07819]

S-duality in EFT

The S-duality rule is Coordinate permutations: These map a solution of EFT to another EFT solution. [YS, Uehara arXiv:1701.07819]

Exotic brane solutions in EFT

Using these duality rules, we obtained exotic brane solutions in type II / M-theory.

Examples of EFT solution

dual coordinate of D3-brane

2𝟔

𝟒,𝟒-brane solution (type IIB):

1𝟑𝟓

𝟐,𝟓,𝟒-brane solution (M-theory):

dual coordinate of M5-brane

Relation to massive/deformed SUGRAs

In the literature, D8-brane background is known to be a solution of massive IIA SUGRA

[Bergshoeff, de Roo, Green, Papadopoulos, Townsend, ‘96]

massive IIA SUGRA

In the D8 solution of DFT, R-R 1-form has a linear dual-coordinate dependence: E.O.M. of DFT Ansatz E.O.M. of type IIA SUGRA with a mass deformation E.O.M. of massive type IIA SUGRA

[Hohm, Kwak, arXiv:1108.4937]

massive IIA SUGRA

“D8-brane” solution in DFT

We can convert the linear-dual coordinate dependence into a deformation parameter of SUGRA.

D8-brane solution in massive type IIA theory

[Bergshoeff, de Roo, Green, Papadopoulos, Townsend, ‘96]

5 -brane

Solution of DFT deformation This is a solution of the deformed SUGRA. Again, we can convert the linear-dual coordinate dependence into a deformation parameter of SUGRA.

7

• brane (IIA)

Again, we can in principle calculate the deformed SUGRA action.

[Meessen, Ortin, hep-th/9806120]

KK8A solution of

EFT solution

Our domain-wall solution in EFT can reproduce the known domain-wall solution.

Summary

We constructed exotic-brane solutions in DFT/EFT. All domain-wall solutions have a certain linear-dual-coordinate dependence. The linear-dual-coordinate dependence can be always converted into deformation parameter of SUGRA. Each domain-wall in DFT/EFT can be regarded as a domain-wall solution of a deformed SUGRA.

Comments

DFT/EFT is a useful framework to reproduce various deformed supergravities. Recently, some deformed supergravity, known as the generalized type II supergravity was proposed. This also can be reproduced from DFT/EFT by introducing a linear-dual-coordinate dependence into the dilaton:

[YS, S.Uehara, K.Yoshida, arXiv:1611.05856; J.Sakamoto, YS, K.Yoshida, arXiv:1703.09213] [Arutyunov-Frolov-Hoare-Roiban-Tseytlin, 1511.05795; Tseytlin-Wulff, 1605.04884]

Comments

Usually, we believe that string theory can be consistently defined

• nly in the usual SUGRA background.

However, recently it was discussed that the kappa-invariance / Weyl invariance

• f string theory can be realized even in

the generalized supergravity backgrounds. It must be important to study string theory in backgrounds of deformed SUGRAs or DFT/EFT.

[Tseytlin-Wulff, arXiv:1605.04884; J.Sakamoto, YS, K.Yoshida, arXiv:1703.09213]