Ads Vacua, Attractor Mechanism and Generalized Geometries

Institute for the Physics and Mathematics of the Universe (IPMU), University of Tokyo: Mar 05, 2009

AdS Vacua, Attractor Mechanism and Generalized Geometries

based on arXiv:0810.0937 [hep-th] Tetsuji KIMURA Yukawa Institute for Theoretical Physics, Kyoto University Introduction We are looking for the origin of 4D physics

Physical information Particle contents and spectra (Broken) symmetries Potential, vacuum and cosmological constant

TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES What kind of 4D models come from String Theories? ↓

What kind of Compactiﬁcations?

4 = 10 − 6 = 11 − 7

TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES Q D 32 16 8 4

"12" F

11 M

IIBIIA HE8 HSO I torus 10 S ——

–– 9 U M K3

· · · · · · 8 U F Calabi-Yau

7 U M

F S-duality 6 U M – (2,2) (2,0) (1,1) (1,0)

5 U IIB M

FF 4 U S N=8N=4 N=2 N=1

B. de Wit and J. Louis, in the Proceedings “NATO Advanced Study Institute on Strings, Branes and Dualities (1997)” hep-th/9801132

TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES Many Abelian Supergravities (SUGRA) in lower dimensions

Compactiﬁcations on Tori, Calabi-Yaus, etc.

Minkowski ground state, massless ﬁelds

Global E7 symmetry (4D N = 8 case)

Many Gauged SUGRA in lower dimensions

Compactiﬁcations on group manifolds, torsionful manifolds, etc.

Scalar potential generating masses [Moduli Stabilization]

Non-trivial cosmological constant

TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES There are various Gauged SUGRA which cannot be derived from String Theories compactiﬁed on conventional geometric backgrounds L99 We want to derive all Gauged SUGRA from String Theories

Nongeometric String Backgrounds

TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES What is a Nongeometric String Background?

Structure group = Diﬀeo. (GL(d, R)) + |Duality transf.{z groups} ↑ coming from String dualities

GL(d, R) + duality transf.

d-dim. internal space Md ' monodrofold

TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES SUGRA on Nongeometric String Backgrounds ex.) Lower-dim. Gauged SUGRA compactiﬁed by Scherk-Schwarz mechanism

c c [Za,Zb] = fab Zc + Habc X a b ab c abc “Kaloper-Myers” algebra: [X ,X ] = Q c X + R Zc a a c ac [X ,Zb] = f bc X − Q b Zc

Various “ﬂuxes” are involved

N. Kaloper, R.C. Myers hep-th/9901045

J. Shelton, W. Taylor, B. Wecht hep-th/0508133, A. Dabholkar, C.M. Hull hep-th/0512005

M. Gra˜na,R. Minasian, M. Petrini, D. Waldram arXiv:0807.4527

TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES ' $ String Theories compactiﬁed on Nongeometric Backgrounds ↓

All(?) Gauged SUGRA & %

Hitchin’s Generalized Geometries to study vacua

Hull’s Doubled Formalism to ﬁnd gauge symmetries

IPMU Workshop “Supersymmetry in Complex Geometry” (January 2009)

TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES 4D N = 1 supergravity

4D N = 1 action: ∫ ( ) 1 1 S = R ∗ 1 − F a ∧ ∗F a − K ∇φM ∧ ∗∇φN − V ∗ 1 2 2 MN ( ) K MN 2 1 a 2 V = e K DMW DN W − 3|W| + |D | 2

K : K¨ahlerpotential

K c W : superpotential L99 δψµ = ∇µ ε − e 2 W γµ ε

a L99 a a µν a D : D-term δχ = ImFµνγ ε + iD ε

10D string theory provides K, W, Da

via compactiﬁcations: 10 = 4 + 6

TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES Motivation and Results

Search 4D SUSY vacua in type IIA theory compactiﬁed on generalized geometries

Moduli stabilization We ﬁnd SUSY AdS (or Minkowski) vacua

Mathematical feature We obtain a powerful rule to evaluate vacua: Discriminants of superpotentials governing the cosmological constant

Stringy eﬀects We see α0 corrections in certain conﬁgurations

TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES Contents

Introduction

Killing Spinors and Fluxes

Generalized (Complex) Geometries

Exterior Derivatives and Flux Charges

Setup in N = 1 Theory

My Work: Search of SUSY AdS Vacua

Summary and Discussions Contents

Introduction

Killing Spinors and Fluxes

Generalized (Complex) Geometries

Exterior Derivatives and Flux Charges

Setup in N = 1 Theory

My Work: Search of SUSY AdS Vacua

Summary and Discussions Compactiﬁcations in 10D type IIA string

Decompose 10D type IIA SUSY parameters:

1 ⊗ 1 c ⊗ 1 2 ⊗ 2 c ⊗ 2 = ε1 (a η−) + ε1 (a η+) , = ε2 (b η+) + ε2 (b η−)

TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES Compactiﬁcations in 10D type IIA string

Decompose 10D type IIA SUSY parameters:

1 ⊗ 1 c ⊗ 1 2 ⊗ 2 c ⊗ 2 = ε1 (a η−) + ε1 (a η+) , = ε2 (b η+) + ε2 (b η−)

δ(fermions) = 0 provide Killing spinor equations on the 6D internal space M: ( ) 1 ( ) ( ) δψA = ∂ + ω γab ηA + 3-form ﬂuxes · η A + other ﬂuxes · η A = 0 m m 4 mab

1 2 with a pair of SU(3) invariant Weyl spinors η+, η+: 2 1 0 m 1 − 0 m ≡ 1† m 2 η+ = ck(y)η+ + c⊥(y)(v + iv ) γm η− , (v iv ) η+ γ η−

TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES Compactiﬁcations in 10D type IIA string

Decompose 10D type IIA SUSY parameters:

1 ⊗ 1 c ⊗ 1 2 ⊗ 2 c ⊗ 2 = ε1 (a η−) + ε1 (a η+) , = ε2 (b η+) + ε2 (b η−)

δ(fermions) = 0 provide Killing spinor equations on the 6D internal space M: ( ) 1 ( ) ( ) δψA = ∂ + ω γab ηA + 3-form ﬂuxes · η A + other ﬂuxes · η A = 0 m m 4 mab

1 2 with a pair of SU(3) invariant Weyl spinors η+, η+: 2 1 0 m 1 − 0 m ≡ 1† m 2 η+ = ck(y)η+ + c⊥(y)(v + iv ) γm η− , (v iv ) η+ γ η− ' $ Calabi-Yau three-fold ↓ Information of SU(3)-structure manifold with torsion 1 2 6D SU(3) Killing spinors η+, η+: ↓ Generalized Geometry & %

TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES I Calabi-Yau three-folds 99K Fluxes are highly restricted    type IIA : No ﬂuxes −  type IIB : F3 τH (warped Calabi-Yau)  heterotic : No ﬂuxes

I SU(3)-structure manifolds 99K Some components of ﬂuxes can be interpreted as torsion

Piljin Yi, TK “Comments on heterotic ﬂux compactiﬁcations” JHEP 0607 (2006) 030, hep-th/0605247

TK “Index theorems on torsional geometries” JHEP 0708 (2007) 048, arXiv:0704.2111

I Generalized geometries 99K All (non)geometric ﬂuxes can be introduced “Complete” classiﬁcation of N = 1 SUSY solutions

TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES Compactiﬁcations in 10D type IIA

−| |2 2 2A(y) µ ν 2 10D = 4D (Λcosmo. = µ ) + 6D: ds10 = e gµν dx dx + ds6

0 Consider polyforms Φ± on the internal space M which satisfy

−2A+φ − ∧ 2A−φ 0 − 0 e (d H )(e Φ+) = 2µ ReΦ− e−2A+φ(d − H∧)(e2A−φΦ0 ) = −3i Im(µΦ0 ) + dA ∧ Φ0 − [ + − ] 1 + eφ (|a|2 − |b|2)F + i(|a|2 + |b|2) ∗ λ(F ) 16

M. Gra˜na,R. Minasian, M. Petrini, A. Tomasiello hep-th/0505212

TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES Compactiﬁcations in 10D type IIA

−| |2 2 2A(y) µ ν 2 10D = 4D (Λcosmo. = µ ) + 6D: ds10 = e gµν dx dx + ds6

0 Consider polyforms Φ± on the internal space M which satisfy

TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES Compactiﬁcations in 10D type IIA

−| |2 2 2A(y) µ ν 2 10D = 4D (Λcosmo. = µ ) + 6D: ds10 = e gµν dx dx + ds6

0 Consider polyforms Φ± on the internal space M which satisfy

M. Gra˜na,R. Minasian, M. Petrini, A. Tomasiello hep-th/0505212 I 0 0 6 1 2 On Calabi-Yau (dΦ± = 0) or SU(3)-structure manifolds (dΦ± = 0) (η+ = η+): ' $ 0 −iJ 0 − Φ+ = e , Φ− = Ω − † − † Jmn = 2i η+ γmn η+ , Ωmnp = 2i η− γmnp η+ & % I × 1 6 2 On SU(3) SU(3) generalized geometries (η+ = η+ at some points y): ' $ 0 0 −ij − ∧ −iv∧v 0 −ij ∧ 0 Φ+ = (cke ic⊥w) e , Φ− = (cke + ic⊥w) (v + iv ) J A = j ± v ∧ v0 , ΩA = w ∧ (v ± iv0) & %

TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES Contents

Introduction

Killing Spinors and Fluxes

Generalized (Complex) Geometries

Exterior Derivatives and Flux Charges

Setup in N = 1 Theory

My Work: Search of SUSY AdS Vacua

Summary and Discussions Generalized almost complex structures

Introduce a generalized almost complex structure J on F ⊕ F ∗ s.t.

J : F ⊕ F ∗ −→ F ⊕ F ∗

2 J = −12d ∃ O(d, d) invariant metric L, s.t. J T LJ = L

Structure group on F ⊕ F ∗

∃L GL(2d) 99K O(d, d)

2 J = −12d O(d, d) 99K U(d/2, d/2)

J1, J2 U1(d/2, d/2) ∩ U2(d/2, d/2) 99K U(d/2) × U(d/2)

integrable J1,2 U(d/2) × U(d/2) 99K SU(d/2) × SU(d/2)

TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES I Integrability is discussed by “(0, 1)” part of the complexiﬁed F ⊕ F ∗:

1 Π ≡ (1 − iJ ) 2 2d ΠA = A where A = v + ζ is a section of F ⊕ F ∗

We call this A i-eigenbundle LJ whose dimension is dim LJ = d.

TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES I Integrability is discussed by “(0, 1)” part of the complexiﬁed F ⊕ F ∗:

1 Π ≡ (1 − iJ ) 2 2d ΠA = A where A = v + ζ is a section of F ⊕ F ∗

We call this A i-eigenbundle LJ whose dimension is dim LJ = d.

Integrability condition of J is [ ] ∈ ∈ ∗ Π Π(v + ζ), Π(w + η) Courant = 0 ; v, w section of F ; ζ, η section of F [ ] 1 v + ζ, w + η = [v, w] + L η − L ζ − d(ι η − ι ζ) Courant bracket Courant Lie v w 2 v w

TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES I Two examples of generalized almost complex structures: ( )

I 0 2 J− = w/ I = −1 : almost complex structure 0 −IT d ( ) 0 −J −1 J = w/ J: almost symplectic form + J 0

integrable J− ↔ integrable I

integrable J+ ↔ integrable J

p On a usual geometry, Jmn = gmp I n is given by an SU(3) invariant (pure) spinor η+ as − † i ι 6 Jmn = 2i η+γmnη+ γ η+ = 0 γ η+ = 0 In a similar analogy, we want to ﬁnd pure spinor(s) Φ on gengeralized geometry.

TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES Cliﬀ(6, 6) pure spinors

On F ⊕ F ∗, we can deﬁne Cliﬀ(6, 6) algebra and Spin(6, 6) spinor Φ: { m n} { m e } m {e e } Γ , Γ = 0 Γ , Γn = δn Γm, Γn = 0 Irreducible repr. of Spin(6, 6) spinor is a Majorana-Weyl → a generic Spin(6, 6) spinor bundle S splits to S± (Weyl)

TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES Cliﬀ(6, 6) pure spinors

Weyl spinor bundles S± are isomorphic to bundles of forms F ∗: + even ∗ Φ+ ∈ S ∼ section of ∧ F − odd ∗ Φ− ∈ S ∼ section of ∧ F

A form-valued representation of the algebra m m∧ e Γ = dx , Γn = ι∂n

TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES Cliﬀ(6, 6) pure spinors

Weyl spinor bundles S± are isomorphic to bundles of forms F ∗: + even ∗ Φ+ ∈ S ∼ section of ∧ F − odd ∗ Φ− ∈ S ∼ section of ∧ F

A form-valued representation of the algebra m m∧ e Γ = dx , Γn = ι∂n

IF Φ is annihilated by half numbers of the Cliﬀ(6, 6) generators: → Φ is called a pure spinor

i cf.) SU(3) invariant spinor η+ is a pure spinor: γ η+ = 0

TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES Correspondence

Correspondence between generalized almost complex structures and pure spinors: J ↔ Φ

TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES Correspondence

Correspondence between generalized almost complex structures and pure spinors: J ↔ Φ

Then, we can rewrite the generalized almost complex structure as

J±ΠΣ = ReΦ±, ΓΠΣ ReΦ±

even forms: Ψ+, Φ+ = Ψ6 ∧ Φ0 − Ψ4 ∧ Φ2 + Ψ2 ∧ Φ4 − Ψ0 ∧ Φ6 w/ Mukai pairing: odd forms: Ψ−, Φ− = Ψ5 ∧ Φ1 − Ψ3 ∧ Φ3 + Ψ1 ∧ Φ5

TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES Correspondence

Correspondence between generalized almost complex structures and pure spinors: J ↔ Φ

Then, we can rewrite the generalized almost complex structure as

J±ΠΣ = ReΦ±, ΓΠΣ ReΦ±

even forms: Ψ+, Φ+ = Ψ6 ∧ Φ0 − Ψ4 ∧ Φ2 + Ψ2 ∧ Φ4 − Ψ0 ∧ Φ6 w/ Mukai pairing: odd forms: Ψ−, Φ− = Ψ5 ∧ Φ1 − Ψ3 ∧ Φ3 + Ψ1 ∧ Φ5

J is integrable ←→ ∃ vector v and one-form ζ s.t. dΦ = (vx+ζ∧)Φ generalized CY ←→ ∃Φ is pure s.t. dΦ = 0 “twisted” GCY ←→ ∃Φ is pure, and H is closed s.t. (d − H∧)Φ = 0

TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES Cliﬀord map between generalized geometry and SU(3)-structure manifold

A spinor Φ can also be mapped to a bispinor by using ∑ ∑ 1 1 ··· (k) m1 m (k) m1 mk C ≡ C ··· dx ∧ · · · ∧ dx k ←→ C/ ≡ C ··· γ k! m1 mk k! m1 mk αβ k k

TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES Cliﬀord map between generalized geometry and SU(3)-structure manifold

A spinor Φ can also be mapped to a bispinor by using ∑ ∑ 1 1 ··· (k) m1 m (k) m1 mk C ≡ C ··· dx ∧ · · · ∧ dx k ←→ C/ ≡ C ··· γ k! m1 mk k! m1 mk αβ k k On a geometry of a single SU(3)-structure, the following two SU(3, 3) spinors:

∑6 † 1 1 † m ···m 1 −iJ Φ = η ⊗ η = η γ ··· η γ 1 k = e 0+ + + 4 k! + mk m1 + 8 k=0 ∑6 † 1 1 † m ···m i Φ − = η ⊗ η = η γ ··· η γ 1 k = − Ω 0 + − 4 k! − mk m1 + 8 k=0 n ⊗ † ⊗ † ∓ n Check purity: (δ + iJ)m γn η+ η± = 0 = η+ η± γn(δ iJ) m

One-to-one correspondence: Φ0− ↔ J1, Φ0+ ↔ J2

TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES Cliﬀord map between generalized geometry and SU(3)-structure manifold

One-to-one correspondence: Φ0− ↔ J1, Φ0+ ↔ J2

1 2 On a generic geometry of a pair of SU(3)-structures deﬁned by (η+, η+)

( ) 0 1 2† 1 −ij −iv∧v Φ0+ = η ⊗ η = cke − ic⊥w ∧ e + + 8 | |2 | |2 ( ) ck + c⊥ = 1 1 2† 1 −ij 0 Φ − = η ⊗ η = − c⊥e + ickw ∧ (v + iv ) 0 + − 8

TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES K¨ahlerpotentials

Spaces of Φ± are special K¨ahlergeometries of local type Moduli space of M has K¨ahlerpotentials, prepotentials, projective coordinates ∫ ( ) A A K+ = − log i Φ+, Φ+ = − log i X FA − X FA ∫M ( ) I I K− = − log i Φ−, Φ− = − log i Z GI − Z GI M

Expand the even/odd-forms Φ± by the basis forms:

A A A a Φ+ = X ωA − FAω , ωA = (1, ωa) , ω = (ω , vol(M)) : 0, 2, 4, 6-forms e e e I I I i 0 Φ− = Z αI − GIβ , αI = (α0, αi) , β = (β , β ) : 1, 3, 5-forms

∫ ∫ ∫ ∫ B B J J hωA, ωBi = 0 , hωA, ωe i = δA , hαI, αJi = 0 , hαI, β i = δI M M M M

TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES Contents

Introduction

Killing Spinors and Fluxes

Generalized (Complex) Geometries

Exterior Derivatives and Flux Charges

Setup in N = 1 Theory

My Work: Search of SUSY AdS Vacua

Summary and Discussions Geometric ﬂux charges

1 2 On generalized geometries with a single SU(3)-structure (η+ = η+):

I I A dHωA = mA αI − eIA β dHωe = 0 A I I A dHαI = eIA ωe dHβ = mA ωe where NS three-form H deforms the diﬀerential operator:

ﬂ ﬂ I I dH = 0 ,H = H + dB,H = m0 αI − eI0 β

ﬂ dH ≡ d − H ∧

background charges I NS three-form ﬂux eI0 m0 I torsion eIa ma

TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES Nongeometric ﬂux charges

× 1 6 2 On generalized geometries with SU(3) SU(3) structures (η+ = η+ at some points): Extend to the generalized diﬀerential operator D:

ﬂ ﬂ dH = d − H ∧ → D ≡ d − H ∧ −f · −Q · −R x

I I A IA A I DωA ∼ mA αI − eIA β Dωe ∼ −q αI + pI β

A A I IA I A DαI ∼ pI ωA + eIA ωe Dβ ∼ q ωA + mA ωe

TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES Nongeometric ﬂux charges

× 1 6 2 On generalized geometries with SU(3) SU(3) structures (η+ = η+ at some points): Extend to the generalized diﬀerential operator D:

ﬂ ﬂ dH = d − H ∧ → D ≡ d − H ∧ −f · −Q · −R x

I I A IA A I DωA ∼ mA αI − eIA β Dωe ∼ −q αI + pI β

A A I IA I A DαI ∼ pI ωA + eIA ωe Dβ ∼ q ωA + mA ωe

The internal space becomes nongeometric:

a · ··· ≡ (f C)m1 mk+1 f [m1m2C|a|m3···mk+1] (part of) structure const. in Gauged SUGRA ab · ··· ≡ (Q C)m1 mk−1 Q [m1C|ab|m2···mk−1] T-fold x ≡ abc (R C)m1···mk−3 R Cabcm1···mk−3 locally nongeometric background

Hull’s Doubled formalism Structure group = Diﬀeo. + duality trsf. 99K to study gauge symmetries

TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES Ramond-Ramond Flux charges

RR-fluxes on SU(3) × SU(3) generalized geometries: G = Gfl + DA, DG = 0 fl A − eA I − e I G = mRR ωA eRRA ω ,A = ξ αI ξI β ↓

' $

A e A G ∼ G ωA − GA ωe A ∼ A I A − e IA e ∼ − I e I G mRR + ξ pI ξI q , GA eRRA ξ eIA + ξI mA & %

TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES Flux charges on generalized geometry: summary

fluxes charges I NS three-form H eI0 m0 I torsion eIa ma A IA nongeometric fluxes pI q A RR-fluxes eRRA mRR

backgrounds ﬂux charges Calabi-Yau — I Calabi-Yau with H eI0 m0 I SU(3) geometry eIA mA I A IA SU(3) × SU(3) geometry eIA mA pI q

Note: SU(3) generalized geometry without RR-ﬂuxes ∼ SU(3)-structure manifold

TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES All the information of the internal space is translated into the (non)geometric ﬂux charges and the RR-ﬂux charges.

NEXT STEP Introduce the ﬂux charges into 4D N = 1 physics via various functionals: K, W, Da

( ) K MN 2 1 a 2 V = e K DMW DN W − 3|W| + |D | 2 Contents

Introduction

Killing Spinors and Fluxes

Generalized (Complex) Geometries

Exterior Derivatives and Flux Charges

Setup in N = 1 Theory

My Work: Search of SUSY AdS Vacua

Summary and Discussions N = 1 K¨ahlerpotential

Functionals are given by two K¨ahlerpotentials on two Hodge-K¨ahlergeometries of Φ±:

K = K+ + 4ϕ ∫ ( ) A A K+ = − log i Φ+, Φ+ = − log i X FA − X F A ∫M ( ) I I K− = − log i Φ−, Φ− = − log i Z GI − Z GI ∫ M (10) 1 −K± −2ϕ+2φ vol6 = e = e M 8 √ − (10) K− − Introduce C = 2ab e φ = 4ab e 2 ϕ

|C|2 ∴ e−2ϕ = e−K− 16|a|2|b|2 [ ] 1 = Im(CZI)Re(CG ) − Re(CZI)Im(CG ) 8|a|2|b|2 I I

TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES SUSY truncation: N = 2 → N = 1

4D SUSY variations yield the superpotential and the D-term:

K c δψµ = ∇µε − e 2 W γµ ε

A A µν A δχ = ImFµν γ ε + i D ε

Information of W and DA comes from 10D SUSY variations ↑

Spinors Φ± on 6D internal geometry

TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES Superpotential and D-term

[ ∫ ∫ ] K− i −ϕ 1 W = 4i e 2 Φ+, DIm(abΦ−) + √ Φ+,G 4ab M 2 M ≡ WRR I WQ e WfI + U I + UI Q [ ] i WRR = − XA e − F mA 4ab RRA A RR [ ] [ ] Q i i W = XA e + F p A , WfI = − XA m I + F qIA I 4ab IA A I Q 4ab A A

I I I e e U = ξ + i Im(CZ ) , UI = ξI + i Im(CGI)

[ ]( ) A K+ cd A B C B Px − N PexC D = 2 e (K+) DcX DdX n (σx)C nB B BC

TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES Field contents in N = 1 theory

0 gravity multiplet gµν,Aµ N a a a a + = 2 multiplets: vector multiplets Aµ, t = b + iv a = 1, . . . , b

a a 0 i i 0 i i e − (t = X /X , z = Z /Z ) hypermultiplets z , ξ , ξi i = 1, . . . , b

0 e tensor multiplet Bµν, ϕ, ξ , ξ0

O ≡ − FL ↓ O6 orientifold projection: ΩWS ( 1) σ

gravity multiplet gµν aˆ + − vector multiplets Aµ aˆ = 1,..., nˆv = b nch chiral multiplets taˇ = baˇ + ivaˇ aˇ = 1, . . . , n N = 1 multiplets: ch U Iˇ = ξIˇ + i Im(CZIˇ) chiral/linear multiplets I = (I,ˇ Iˆ) = 0, 1, . . . , b− e e CG UIˆ = ξIˆ + i Im( Iˆ) 0 aˇ aˆ Iˆ e (projected out) Bµν,Aµ,Aµ, t ,U , UIˇ

iθ | |2 | |2 1 Parameters are restricted as a = b e and a = b = 2

T.W. Grimm hep-th/0507153

TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES We are ready to search SUSY vacua in 4D N = 1 theory given by K, W, Da

NEXT: Consider two situations

' $ generalized geometry with RR-ﬂux charges I A IA A eIA, mA , pI , q , eRRA, mRR

SU(3)-structure manifold I eIA, mA

& %

TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES Contents

Introduction

Killing Spinors and Fluxes

Generalized (Complex) Geometries

Exterior Derivatives and Flux Charges

Setup in N = 1 Theory

My Work: Search of SUSY AdS Vacua

Summary and Discussions 4D N = 1 scalar potential

( ) K MN 2 1 a 2 V = e K DMW DN W − 3|W| + |D | 2

≡ VW + VD

V∗ > 0 : de Sitter space (non-SUSY)

Search of vacua ∂PV ∗ = 0 V∗ = 0 : Minkowski space

V∗ < 0 : Anti-de Sitter space

TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES 4D N = 1 scalar potential

( ) K MN 2 1 a 2 V = e K DMW DN W − 3|W| + |D | 2

≡ VW + VD

V∗ > 0 : de Sitter space (non-SUSY)

Search of vacua ∂PV ∗ = 0 V∗ = 0 : Minkowski space

V∗ < 0 : Anti-de Sitter space

{ } K MN MN 0 = ∂PVW = e K DPDMWDN W + ∂PK DMWDN W − 2WDPW

a 0 = ∂PVD 99K D = 0 ( ) Consider the SUSY condition DPW ≡ ∂P + ∂PK W = 0 in various cases.

TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES Example 1: SU(3) × SU(3) generalized geometry with RR-ﬂux charges

XaXbXc 1. Set a simple prepotential: F = D abc X0

2. Consider the simplest model: single modulus t of Φ+ (and U of Φ−)

TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES Example 1: SU(3) × SU(3) generalized geometry with RR-ﬂux charges

XaXbXc 1. Set a simple prepotential: F = D abc X0

2. Consider the simplest model: single modulus t of Φ+ (and U of Φ−)

The superpotential is reduced to

W = WRR + U WQ

WRR 0 3 − 2 = mRR t 3 mRR t + eRR t + eRR0

Q 0 3 2 W = p0 t − 3 p0 t − e0 t − e00

RR Q DtW = 0 99K 0 = DtW + UDtW Consider the SUSY condition: ( ) i D W = 0 99K 0 = WRR + ReU WQ U ImU The discriminant of the superpotential WRR (and WQ) governs the SUSY solutions.

TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES I Discriminant of cubic equation 8 W(t) = a t3 + b t2 + c t + d Consider a cubic function and its derivative: < ∂ W(t) = 3a t2 + 2b t + c : t Discriminants ∆(W) and ∆(∂tW) are

∆(W) ≡ ∆ = −4b3d + b2c2 − 4ac3 + 18abcd − 27a2d2

2 ∆(∂tW) ≡ λ = 4(b − 3ac)

W(t) λ > 0 λ = 0 λ < 0

∆ > 0

∆ = 0

∆ < 0

TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES RR RR RR ∆ > 0 case: always λ > 0, and exists a zero point: DtW = 0

RR D W |∗ = 0 t √ 6 (3 m0 e + m e ) 3 ∆RR tRR = RR RR0 RR RR − 2i ∗ λRR λRR ( ) 24 ∆RR √ WRR = − 36 (m )3 + 36 (m0 )2e − 3 m λRR − 4i m0 3 ∆RR ∗ (λRR)3 RR RR RR0 RR RR

TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES RR RR RR ∆ > 0 case: always λ > 0, and exists a zero point: DtW = 0

RR D W |∗ = 0 t √ 6 (3 m0 e + m e ) 3 ∆RR tRR = RR RR0 RR RR − 2i ∗ λRR λRR ( ) 24 ∆RR √ WRR = − 36 (m )3 + 36 (m0 )2e − 3 m λRR − 4i m0 3 ∆RR ∗ (λRR)3 RR RR RR0 RR RR

∆RR < 0 case: only λRR < 0 is physically allowed, and exists a zero point: WRR = 0

WRR 0 − − − RR ∗ = mRR(t∗ e)(t∗ α)(t∗ α) = 0 , t∗ = α = α1 + i α2 λRR + F 2/3 + 12 m F 1/3 α = RR 1 12 m0 F 1/3 ( RR ) 1 (α )2 = e − 6 m α + 3 m0 (α )2 2 m0 RR RR 1 RR 1 RR ( ) 1 e = − − 3 m + 2 m0 α m0 RR RR 1 RR √ 0 2 0 − RR 3 − RR F = 108 (mRR) eRR0 + 12 mRR 3∆ + 108 (mRR) 9 mRR λ

WRR| 0 − RR Dt ∗ = 2i mRR(e α )α2

... Analysis of WQ is also discussed.

TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES RR Q RR Q Three types of solutions to satisfy 0 = DtW + UDtW and 0 = W + ReU W :

TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES RR Q RR Q Three types of solutions to satisfy 0 = DtW + UDtW and 0 = W + ReU W : SUSY AdS vacuum: moduli are (almost) stabilized ' $

RR Q RR Q ∆ > 0 , ∆ > 0 ; DtW |∗ = 0 = DtW |∗ WRR RR Q − ∗ t∗ = t∗ , Re U∗ = Q W∗ √ Q K 2 4 ∆ ∗ − |W∗| − O V = 3 e = 2 (1) [Re(CG0)] 3 & %

RR Q RR Q ∆ > 0 , ∆ > 0 ; DtW |∗ = 0 = DtW |∗ WRR RR Q − ∗ t∗ = t∗ , Re U∗ = Q W∗ √ Q K 2 4 ∆ ∗ − |W∗| − O V = 3 e = 2 (1) [Re(CG0)] 3 & % SUSY Minkowski vacuum: moduli are stabilized ' $

RR Q RR Q ∆ < 0 , ∆ < 0 ; W∗ = 0 = W∗ WRR| RR Q −Dt ∗ 6 α = α ,U∗ = Q = 0 DtW |∗

V∗ = 0

& %

RR Q RR Q ∆ < 0 , ∆ < 0 ; W∗ = 0 = W∗ WRR| RR Q −Dt ∗ 6 α = α ,U∗ = Q = 0 DtW |∗

V∗ = 0

& % SUSY AdS vacua, but moduli t and U are not ﬁxed: non-stabilized point WRR WRR −Dt (t) − (t) U = Q , Re U = Q DtW (t) W (t)

TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES Example 2: SU(3)-structure manifold

A A IA 1. Set eRRA = 0 = mRR, pI = 0 = q , and single modulus t of Φ+ (and U of Φ−) (Xt)3 2. Set a deformed prepotential: F = X0

TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES Example 2: SU(3)-structure manifold

A A IA 1. Set eRRA = 0 = mRR, pI = 0 = q , and single modulus t of Φ+ (and U of Φ−) (Xt)3 ∑ (Xt)n+3 2. Set a deformed prepotential: F = + N X0 n(X0)n+1 n

TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES Example 2: SU(3)-structure manifold

A A IA 1. Set eRRA = 0 = mRR, pI = 0 = q , and single modulus t of Φ+ (and U of Φ−) (Xt)3 ∑ (Xt)n+3 2. Set a deformed prepotential: F = + N X0 n(X0)n+1 n

Q Superpotential W = UW with a simple setting N1 =6 0, Nn = 0: ( ) 3(t − t)2 − ∂ P D WQ = −e + t e + e t t 00 (t − t)3 − P 00 0 ( ) 4 4 3 3 P ≡ −2 N1 t − N1 t − 2N1 t t + 2N1 tt

' $

Q 2 e00 t∗ = − , Re U∗ = 0 SUSY condition e0 W W Q Dt = DU = 0 W∗ = e00 , ImN1 < 0 4 has a solution K 2 1 3 (e0) ∗ − |W∗| V = 3 e = 2 2 [Re(CG0)] 16 (e00) ImN1 & %

Also heterotic string on SU(3)-structure manifolds with torsion which carries α0 corrections

TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES Contents

Introduction

Killing Spinors and Fluxes

Generalized (Complex) Geometries

Exterior Derivatives and Flux Charges

Setup in N = 1 Theory

My Work: Search of SUSY AdS Vacua

Summary and Discussions Summary

Studied generalized geometries and their applications to string compactiﬁcations

Obtained a powerful rule to discuss SUSY vacua: Discriminants

Exhibited that α0 corrections are included in certain conﬁgurations

Discussions More generic conﬁgurations

Gauge symmetries

Understanding the physical interpretation of nongeometric fluxes de Sitter vacua? In order to build (stable) de Sitter vacua perturbatively in type IIA, in addition to the usual R-R and NS-NS fluxes and O6/D6 sources, one must minimally have geometric fluxes and non-zero Romans’ mass parameter.

S.S. Haque, G. Shiu, B. Underwood, T. Van Riet arXiv:0810.5328

Romans’ mass parameter ∼ G0

Search (meta)stable de Sitter vacua in this formulation Thank You Appendices Contents

Diﬀerential Forms: Geometric Objects

Hitchin Functional

Killing Prepotentials Decompositions of spinors in 10D type IIA string

Decomposition of vector bundle on 10D spacetime: M ⊕ { T 9,1 = T3,1 F T3,1 : a real SO(3, 1) vector bundle F : an SO(6) vector bundle which admits a pair of SU(3) structures

TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES Decompositions of spinors in 10D type IIA string

Decomposition of vector bundle on 10D spacetime: M ⊕ { T 9,1 = T3,1 F T3,1 : a real SO(3, 1) vector bundle F : an SO(6) vector bundle which admits a pair of SU(3) structures

Decomposition of Lorentz symmetry: Spin(9, 1) → Spin(3, 1) × Spin(6) = SL(2, C) × SU(4) 16 = (2, 4) ⊕ (2, 4) 16 = (2, 4) ⊕ (2, 4) Decomposition of supersymmetry parameters (with a, b ∈ C):

1 ⊗ 1 c ⊗ 1 2 ⊗ 2 c ⊗ 2 IIA = ε1 (a η−) + ε1 (a η+) , IIA = ε2 (b η+) + ε2 (b η−)

TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES Decompositions of spinors in 10D type IIA string

Decomposition of vector bundle on 10D spacetime: M ⊕ { T 9,1 = T3,1 F T3,1 : a real SO(3, 1) vector bundle F : an SO(6) vector bundle which admits a pair of SU(3) structures

Decomposition of Lorentz symmetry: Spin(9, 1) → Spin(3, 1) × Spin(6) = SL(2, C) × SU(4) 16 = (2, 4) ⊕ (2, 4) 16 = (2, 4) ⊕ (2, 4) Decomposition of supersymmetry parameters (with a, b ∈ C):

1 ⊗ 1 c ⊗ 1 2 ⊗ 2 c ⊗ 2 IIA = ε1 (a η−) + ε1 (a η+) , IIA = ε2 (b η+) + ε2 (b η−)

A ∇(T ) A A Set SU(3) invariant spinor η+ s.t. η+ = 0 ( = 1, 2) 1 2 ←→ 1 2 a pair of SU(3) on F (η+, η+) a single SU(3) on F (η+ = η+ = η+)

TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES Requirement that we have a pair of SU(3) structures means there is a sub-supermanifold

N9,1|4+4 ⊂ M9,1|16+16 ( ) (9, 1) : bosonic degrees 4 + 4 : eight Grassmann variables as spinors of Spin(3, 1) and singlet of SU(3)s

Equivalence such as

eight SUSY theory reformulation of type II strings m a pair of SU(3) structures on vector bundle F m SU(3) × SU(3) structures on extended F ⊕ F ∗

TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES Geometric objects

† a real two-form Jmn = ∓2i η± γmn η± I with a single SU(3): − † a complex three-form Ωmnp = 2i η− γmnp η+

TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES Geometric objects

† a real two-form Jmn = ∓2i η± γmn η± I with a single SU(3): − † a complex three-form Ωmnp = 2i η− γmnp η+

− 0 m 1† m 2 two real vectors (v iv ) = η+ γ η− J 1 = j + v ∧ v0 , Ω1 = w ∧ (v + iv0) I with a pair of SU(3): (J A, ΩA) J 2 = j − v ∧ v0 , Ω2 = w ∧ (v − iv0)

(j, w): locally SU(2)-invariant two-forms

TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES Geometric objects

† a real two-form Jmn = ∓2i η± γmn η± I with a single SU(3): − † a complex three-form Ωmnp = 2i η− γmnp η+

− 0 m 1† m 2 two real vectors (v iv ) = η+ γ η− J 1 = j + v ∧ v0 , Ω1 = w ∧ (v + iv0) I with a pair of SU(3): (J A, ΩA) J 2 = j − v ∧ v0 , Ω2 = w ∧ (v − iv0)

(j, w): locally SU(2)-invariant two-forms

2 1 0 m 1 | |2 | |2 η+ = ckη+ + c⊥(v + iv ) γm η− , ck + c⊥ = 1

1 6 2 0 If η+ = η+ globally: a single SU(2) w/ (j, w, v, v ) 1 2 If η+ = η+ globally: a single SU(3) w/ (J, Ω)

a pair of SU(3) on F ∼ SU(3) × SU(3) on F ⊕ F ∗

TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES ex.) SU(3)-structure manifolds

(T ) ∃ i Information from Killing spinor eqs. with torsion ∇ η± = 0 ( complex Weyl η±)

I Invariant p-forms on SU(3)-structure manifold:

† a real two-form Jmn = ∓2i η± γmn η±

− † a holomorphic three-form Ωmnp = 2i η− γmnp η+ 3 dJ = Im(W Ω) + W ∧ J + W dΩ = W J ∧ J + W ∧ J + W ∧ Ω 2 1 4 3 1 2 5 I Five classes of (intrinsic) torsion are given as components interpretations SU(3)-representations

W1 J ∧ dΩ or Ω ∧ dJ 1 ⊕ 1 W 2,2 ⊕ 2 (dΩ)0 8 8 W 2,1 1,2 ⊕ 3 (dJ)0 + (dJ)0 6 6

W4 J ∧ dJ 3 ⊕ 3 3,1 W5 (dΩ) 3 ⊕ 3

TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES I Classiﬁcation of SU(3)-structure manifolds:

hermitian W1 = W2 = 0

balanced W1 = W2 = W4 = 0

special hermitian W1 = W2 = W4 = W5 = 0 complex K¨ahler W1 = W2 = W3 = W4 = 0

Calabi-Yau W1 = W2 = W3 = W4 = W5 = 0

conformally Calabi-Yau W1 = W2 = W3 = 3W4 + 2W5 = 0

symplectic W1 = W3 = W4 = 0

nearly K¨ahler W2 = W3 = W4 = W5 = 0

almost K¨ahler W1 = W3 = W4 = W5 = 0 almost complex quasi K¨ahler W3 = W4 = W5 = 0

semi K¨ahler W4 = W5 = 0

half-ﬂat ImW1 = ImW2 = W4 = W5 = 0

TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES Contents

Diﬀerential Forms: Geometric Objects

Hitchin Functional

Killing Prepotentials Hitchin functional

even/odd ∗ Start with a real form χf ∈ ∧ F (associated with a real Spin(6, 6) spinor χs)

Regard χf as a stable form satisfying

1 ΠΣ 6 ∗ 6 ∗ q(χf ) = − χf , ΓΠΣχf χf , Γ χf ∈ ∧ F ⊗ ∧ F 4 { } even/odd ∗ U = χf ∈ ∧ F q(χf ) < 0

TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES Hitchin functional

even/odd ∗ Start with a real form χf ∈ ∧ F (associated with a real Spin(6, 6) spinor χs)

Regard χf as a stable form satisfying

1 ΠΣ 6 ∗ 6 ∗ q(χf ) = − χf , ΓΠΣχf χf , Γ χf ∈ ∧ F ⊗ ∧ F 4 { } even/odd ∗ U = χf ∈ ∧ F q(χf ) < 0 Deﬁne a Hitchin function √ 1 H(χ ) ≡ − q(χ ) ∈ ∧6F ∗ f 3 f which gives an integrable complex structure on U

TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES Hitchin functional

even/odd ∗ Start with a real form χf ∈ ∧ F (associated with a real Spin(6, 6) spinor χs)

Regard χf as a stable form satisfying

Then we can get another real form χˆf and a complex form Φf by Mukai pairing

∂H(χf ) χˆf , χf = −dH(χf ) i.e., χˆf = − ∂χf 1 99K Φ ≡ (χ + iˆχ ) H(Φ ) = i Φ , Φ f 2 f f f f f

Hitchin showed: Φf is a (form corresponding to) pure spinor!

N.J. Hitchin math/0010054, math/0107101, math/0209099

TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES Consider the space of pure spinors Φ ...

Mukai pairing ∗, ∗ −→ symplectic structure Hitchin function H(∗) −→ complex structure ↓

The space of pure spinor is K¨ahler

TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES Consider the space of pure spinors Φ ...

Mukai pairing ∗, ∗ −→ symplectic structure Hitchin function H(∗) −→ complex structure ↓

The space of pure spinor is K¨ahler

Compatible with Φ → λΦ w/ λ ∈ C∗ 99K The space becomes a local special K¨ahlergeometry with K¨ahlerpotential K:

( ) A A 6 ∗ exp(−K) = H(Φ) = i Φ, Φ = i X FA − X F A ∈ ∧ F XA : holomorphic projective coordinates A FA : derivative of prepotential F (FA = ∂F/∂X )

TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES Contents

Diﬀerential Forms: Geometric Objects

Hitchin Functional

Killing Prepotentials 10D gravitino variations

10D spinors in type IIA in Einstein frame

( ) 1 δΨA = ∇ A − e−φ Γ P QRH − 9ΓPQH Γ A M M 96 M P QR MPQ (11) [ ] ∑ − 1 5 nφ N ···N N N ···N n/2 1 A − e 4 (n − 1)Γ 1 n − n(9 − n)δ 1Γ 2 n F ··· (Γ ) (σ ) 64n! M M N1 Nn (11) n=0,2,4,6,8

* 99K W A δψAµ = 0 superpotential δΨM = 0 A 99K δψm = 0 K¨ahlerpotential K

TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES Killing prepotential

See the SUSY variation of 4D N = 2 gravitinos:

B δψAµ = ∇µεA − SAB γµ ε + ... ( ) 1 2 3 i K+ P − iP −P SAB = e 2 2 −P3 −P1 − P2 i AB

The components are also written by Φ±: ∫ ∫ K− K− 1 2 +ϕ 1 2 +ϕ P − iP = 2 e 2 Φ+, DΦ− , P + iP = 2 e 2 Φ+, DΦ− M ∫ M

3 1 2ϕ P = −√ e Φ+,G 2 M

ˆ 1 m A ⊗ ⊗ Note: ΨAµ = ΨAµ + 2Γµ Ψm = ψAµ± η+ + ψAµ∓ η− + ...

TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES SUSY truncation: N = 2 → N = 1

4D N = 1 fermions given by the SUSY truncation from 4D N = 2 system:

A SUSY parameter : ε ≡ n εA = a ε1 + b ε2 ( ) A e gravitino : ψµ ≡ n ψAµ = a ψ1µ + b ψ2µ , ψµ ≡ b ψ1µ − a ψ2µ

K ( ) A + A C Eb gauginos : χ ≡ −2 e 2 D X n CE χ b   ( ) A  0 1  where n = a , b , AB = −1 0

TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES Superpotential and D-term

SUSY variations yield the superpotential and the D-term:

A ∗B c K c δψµ = ∇µε − n SAB n γµ ε ≡ ∇µε − e 2 W γµ ε e δψµ = 0

A A µν A δχ = ImFµν γ ε + i D ε

[ ]( ) A K+ cd A B C B Px − N PexC D = 2 e (K+) DcX DdX n (σx)C nB B BC

TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES References References

(Lower dimensional) supergravity related to this topic J. Maharana, J.H. Schwarz hep-th/9207016 L. Andrianopoli, M. Bertolini, A. Ceresole, R. D’Auria, S. Ferrara, P. Fré,T. Magri hep-th/9605032 P. Fré hep-th/9512043 N. Kaloper, R.C. Myers hep-th/9901045 E. Bergshoeff, R. Kallosh, T. Ortin, D. Roest, A. Van Proeyen hep-th/0103233 M.B. Schulz hep-th/0406001 S. Gurrieri hep-th/0408044 T.W. Grimm hep-th/0507153 B. de Wit, H. Samtleben, M. Trigiante hep-th/0507289

EOM, SUSY, and Bianchi identities on generalized geometry M. Graña,R. Minasian, M. Petrini, A. Tomasiello hep-th/0407249 hep-th/0505212 M. Graña,J. Louis, D. Waldram hep-th/0505264 hep-th/0612237 D. Cassani, A. Bilal arXiv:0707.3125 D. Cassani arXiv:0804.0595 P. Koerber, D. Tsimpis arXiv:0706.1244 A.K. Kashani-Poor, R. Minasian hep-th/0611106 A. Tomasiello arXiv:0704.2613 B.y. Hou, S. Hu, Y.h. Yang arXiv:0806.3393 M. Graña,R. Minasian, M. Petrini, D. Waldram arXiv:0807.4527

SUSY AdS4 vacua D. Lüst,D. Tsimpis hep-th/0412250 C. Kounnas, D. Lüst,P.M. Petropoulos, D. Tsimpis arXiv:0707.4270 P. Koerber, D. Lüst,D. Tsimpis arXiv:0804.0614 C. Caviezel, P. Koerber, S. Kors, D. Lüst,D. Tsimpis, M. Zagermann arXiv:0806.3458

TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES References

D-branes, orientifold projection, calibration, and smeared sources B.S. Acharya, F. Benini, R. Valandro hep-th/0607223 M. Gra˜na,R. Minasian, M. Petrini, A. Tomasiello hep-th/0609124 L. Martucci, P. Smyth hep-th/0507099 P. Koerber, D. Tsimpis arXiv:0706.1244 P. Koerber, L. Martucci arXiv:0707.1038 M. Cederwall, A. von Gussich, B.E.W. Nilsson, P. Sundell, A. Westerberg hep-th/9611159 E. Bergshoeﬀ, P.K. Townsend hep-th/9611173

Mathematics

N.J. Hitchin math/0209099 M. Gualtieri math/0401221

Doubled formalism E. Cremmer, B. Julia, H. L¨u,C.N. Pope hep-th/9710119 hep-th/9806106 C.M. Hull hep-th/0406102 hep-th/0605149 hep-th/0701203 C.M. Hull, R.A. Reid-Edwards hep-th/0503114 arXiv:0711.4818 J. Shelton, W. Taylor, B. Wecht hep-th/0508133 A. Dabholkar, C.M. Hull hep-th/0512005 A. Lawrence, M.B. Schulz, B. Wecht hep-th/0602025 G. Dall’Agata, S. Ferrara hep-th/0502066 G. Dall’Agata, M. Prezas, H. Samtleben, M. Trigiante arXiv:0712.1026 G. Dall’Agata, N. Prezas arXiv:0806.2003 C.M. Hull, R.A. Reid-Edwards arXiv:0902.4032 C. Albertsson, R.A. Reid-Edwards, TK arXiv:0806.1783

and more...

TETSUJI KIMURA: ADS VACUA, ATTRACTOR MECHANISM AND GENERALIZED GEOMETRIES