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

Weaving the Exotic Web Yuho Sakatani (Kyoto Prefectural Univ. of Medicine) arxiv:1805.12117, a collaboration with Jose J. Fernandez-Melgarejo (Univ. of Murcia) and Tetsuji Kimura (Nihon U). Next speaker Strings and Fields 2018 @ YITP

Weaving the Exotic Web Duality web of branes

Duality web Type II string / 𝟖 Exotic branes Standard branes

Exotic branes discovered in [Elitzur, Giveon, Kutasov, Rabinovici ‘97; 𝒐 Blau, O’Loughlin ’97; Obers, Pioline ’99 ; Eyras, Lozano ’00; Lozano-Tellechea, Ortin ’00] If a brane (wrapped on a torus) has the mass �𝒅 𝒕 ,…,𝒅 𝟑 � -brane. we call the brane a b 𝒐

Exotic branes �𝒅 𝒕 ,…,𝒅 𝟑 � -brane Mass of a b 𝒐 [Obers, Pioline ’99] Mass of a D p -brane ( p 𝟐 -brane ) Mass of a KKM 𝟐 -brane ) ( 5 𝟑 T-dual 𝟑 -brane Mass of a 5 𝟑

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

Here, we haven’t considered co-dimension 1 or 0–branes, such as D8 or D9. Standard branes (co-dim. 1) (co-dim. 0) If we consider domain-wall/space-filling branes as well, we find a huge number of exotic branes !

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

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

Exotic Web IIA T-dual IIB S-dual IIA IIB IIA IIB … [T.Kimura, J.J. Fernandez-Melgarejo, YS]

Our motivation SUGRA solutions for (co-dim.2) the standard branes/defect-branes are well-known. [Meessen, Ortin, hep-th/9806120] domain-wall branes ? How about space-filling branes ?

Previous works In type IIA theory, only two [Meessen, Ortin domain-wall solutions are known: hep-th/9806120] KK8A For other exotic branes, SUGRA solutions are not obtained.

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 D p -brane solution transverse coordinates D p -brane D( p+1 )-brane 1. Smear the D p -brane D p D p D p D p D p smeared direction … … becomes isometric 2. Perform a T -duality along the

Standard D p -brane solution co-dimension 3 D6-brane Smear along -direction becomes isometric T -duality along co-dimension 2 D7-brane depends only on 2 coordinates

D8-brane ? We want to obtain a domain-wall solution. Smear the D7 along Smeared-D7 -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 of supergravity ( T -duality) Double field theory [Siegel ’93; Hull, Zwiebach ’09; Hohm, Hull, Zwiebach ’10; Jeon, Lee, Park ’10; ...] Exceptional field theory ( U -duality) [West ’00; Berman, Perry ’11; Berman, Godazgar, Perry, West ’12; Hohm, Samtleben ’13; ...]

Double Field Theory In DFT, we introduce the doubled spacetime. (20 dim.) P F1 All of the SUGRA fields are defined on the doubled spacetime. We organize them into T-duality-inv. generalized metric dilaton

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 Smeared-D7 A solution of the usual SUGRA. T 𝟗 -dual “D8-brane” solution [Hohm, Kwak, A solution of DFT. arXiv:1108.4937]

Example 2 (smeared) [Meessen, Ortin, hep-th/9806120; 𝟑 -brane solution: 5 𝟑 de Boer-Shigemori, arXiv:1004.2521] A solution of the usual SUGRA. T 𝟗 -dual 𝟒 -brane solution: 5 𝟑 A solution of DFT.

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

Dual parameterization 𝟑 -brane solution: 5 𝟑 𝟒 -brane solution: 5 𝟑 Dual coordinate appears only in the β -field. Moreover, the dependence is just linear.

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 IIB S-dual ! IIA IIB IIA IIB …

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

Dual parameterization Similar to DFT, we can introduce the dual fields: Usual type IIB fields dual type IIB fields field redefinitions

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

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

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

Examples of EFT solution �𝟒,𝟒� -brane solution (type IIB): 2 𝟔 dual coordinate of D3-brane �𝟐,𝟓,𝟒� -brane solution (M-theory): 1 𝟑𝟓 dual coordinate of M5-brane

In the literature, D8-brane background is known to be a solution of massive IIA SUGRA [Bergshoeff, de Roo, Green, Papadopoulos, Townsend, ‘96] Relation to massive/deformed SUGRAs

massive IIA SUGRA In the D8 solution of DFT, R-R 1-form has a linear dual-coordinate dependence: Ansatz E.O.M. of DFT E.O.M. of type IIA SUGRA with a mass deformation [Hohm, Kwak, arXiv:1108.4937] E.O.M. of massive type IIA SUGRA

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

7 -brane (IIA) EFT solution Again, we can in principle calculate the deformed SUGRA action. KK8A solution of [Meessen, Ortin, hep-th/9806120] 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. [ A rutyunov- F rolov- H oare- R oiban- T seytlin , 1511.05795 ; Tseytlin-Wulff, 1605.04884] 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]

Comments Usually, we believe that string theory can be consistently defined only in the usual SUGRA background. However, recently it was discussed that the kappa-invariance / Weyl invariance of string theory can be realized even in the generalized supergravity backgrounds. [Tseytlin-Wulff, arXiv:1605.04884; J.Sakamoto, YS, K.Yoshida, arXiv:1703.09213] It must be important to study string theory in backgrounds of deformed SUGRAs or DFT/EFT.