Workshop on Generalized Geometries and Flux Compactifications, Hamburg, February 23, 2007

Fluxes and moduli fixing

in toroidal orientifolds

Anamarıa Font

IFT UAM/CSIC and UCV

in collaboration with Gerardo Aldazabal, Pablo G. Camara and Luis E. Ibanez

– p.1/26

Why fluxes ?

• to find 10d susy string vacua beyond the standard model

NSNS 〈H3〉 = 0 ; RR 〈Fn〉 = 0 ; Mink4 × Calabi-Yau

• to fix moduli, i.e. 〈Φ〉, in the 4d effective theory

Φ

V

Φ : massless scalar with flat potential

∃ in generic standard compactifications

– p.2/26

Typical modulisize shape

dilaton Kahler complex structure

φ A = RxRy sin θ τ = −iRy

Rxeiθ

R in units of `s =2π√

α′

Ti=1

3

i=1

3

Rx

Ry

θ==6

In N=1, φ, A, τ → S, T, U ∈ chiral multiplets

Ex. IIB T6 orientifold with O3-planes

from RR C4

S = e−φ + iC0 ; Ti = e−φAjAk + iηi ; Ui = τi

In generic CY orientifolds with O3-planes Grimm, Louis

Tα, α = 1, · · · , h+11 ; Uk, k = 1, · · · , h−12

– p.3/26

Moduli problems

∗ mass

m = 0 or m ∼ msusy ruled out by observations

∗ undetermined vevs

1

g2YM

∼

〈Re S〉 heterotic, D9, D3

〈Re T 〉 D5, D7

〈Re U〉 D6

known for > 20 yrs

First proposal to fix S (heterotic) Dine, Rohm, Seiberg, Witten

W (S) = h + c e−γS

flux 〈H3〉 gaugino condensation

– p.4/26

Flux induced moduli potentials in 4d

S =1

`8s

∫

d10x√−G

e−2φ[R− H2

]−

∑

n

F 2n + · · ·

∫

Π3

〈H3〉 = h ⇒ V =h2e2φ

R12⇒ W =

h heterotic

hS orientifolds

In generic type II orientifolds (3-cycles ↔ h12 , 2,4-cycles ↔ h11)

Z

Π3

〈H3〉

Z

Π3

〈H3〉

IIB W (S, U) ; IIA W (S, U,T )Z

Π3

〈F3〉

Z

Π2m

〈F2m〉

∗ N=1 potential determined from K and W

K = −X

Φ=S,Ti,Ui

ln(Φ + Φ∗) ; V = eK˘

X

Φ=S,Ti,Ui

(Φ + Φ∗)2|DΦW |2 − 3|W |2¯

(gs → 0, α′ → 0) (DΦW = ∂ΦW + W∂ΦK)

– p.5/26

Outline

• NSNS and RR fluxes in IIB and IIA orientifolds

IIB : Gukov, Vafa, Witten; Taylor, Vafa; Giddings, Kachru, Polchinski

IIA : Derendinger, Kounnas, Petropoulos, Zwirner; Grimm, Louis; Villadoro, Zwirner; DeWolfe, Giryavets, Kachru, Taylor; CFI

• IIA with metric fluxes (twisted tori)

Gurrieri, Louis, Micu, Waldram; Kachru, Schulz, Tripathy, Trivedi; Derendinger, Kounnas, Petropoulos, Zwirner; Villadoro, Zwirner

• vacua, MSSM-like models

Camara, Font, Ibanez

• Non-geometric fluxes

Dabholkar, Hull; Shelton, Taylor, Wecht; Aldazabal, Camara, Font, Ibanez

• S-dual fluxes

ACFI

• Final comments

Recent work: Benmachiche, Grimm; Hull; Louis, Micu; Grange, Schafer-Nameki, Grana, Minasian, Petrini, Tomasiello;

Grana, Louis, Waldram; Micu, Palti, Tasinato; D’Auria, Ferrara, Trigiante

– p.6/26

Outline

• NSNS and RR fluxes in IIB and IIA orientifoldsIIB : Gukov, Vafa, Witten; Taylor, Vafa; Giddings, Kachru, Polchinski

IIA : Derendinger, Kounnas, Petropoulos, Zwirner; Grimm, Louis; Villadoro, Zwirner; DeWolfe, Giryavets, Kachru, Taylor; CFI

• IIA with metric fluxes (twisted tori)Gurrieri, Louis, Micu, Waldram; Kachru, Schulz, Tripathy, Trivedi; Derendinger, Kounnas, Petropoulos, Zwirner; Villadoro, Zwirner

• vacua, MSSM-like modelsCamara, Font, Ibanez

• Non-geometric fluxesDabholkar, Hull; Shelton, Taylor, Wecht; Aldazabal, Camara, Font, Ibanez

• S-dual fluxesACFI

• Final comments

Recent work: Benmachiche, Grimm; Hull; Louis, Micu; Grange, Schafer-Nameki, Grana, Minasian, Petrini, Tomasiello;

Grana, Louis, Waldram; Micu, Palti, Tasinato; D’Auria, Ferrara, Trigiante

– p.6/26

Notation - Type II orientifolds T6/O

O =

8

>

>

<

>

>

:

ΩP IIB/O9-planes

ΩP (−1)FLσA , σA(zi) = z∗i IIA/O6-planes

ΩP (−1)FLσB , σB(zi) = −zi IIB/O3-planes

Kahler form, holomorphic 3-form and basis of 3-forms of T6 =

3O

i=1

T2i

J =3X

i=1

Ai(dyi ∧ dx

i) , y

i ≡ xi+3

, Ai, τi : area and complex structure of T2i

Ω = (dx1 + iτ1 dy1) ∧ (dx2 + iτ2 dy2) ∧ (dx3 + iτ3 dy3)

α0 = dx1 ∧ dx2 ∧ dx3 , β0 = dy1 ∧ dy2 ∧ dy3αI , βJ odd under σB

α1 = dx1 ∧ dy2 ∧ dy3 , β1 = dy1 ∧ dx2 ∧ dx3αI even, βJ odd under σA

α2 = dy1 ∧ dx2 ∧ dy3 , β2 = dx1 ∧ dy2 ∧ dx3

α3 = dy1 ∧ dy2 ∧ dx3 , β3 = dx1 ∧ dx2 ∧ dy3

– p.7/26

NSNS and RR fluxes in IIB/O3

W(S, U) =

∫

Y

(F3 − iSH3)∧Ω depends on U, Ω = α0 + · · · − iU1U2U3β0 in Y =T6

O-invariance ⇒ 〈F3〉 ≡ F3, 〈H3〉 ≡ H3 odd under σB

F3 = −mα0 − e0β0 +

3X

i=1

(eiαi − qiβi) ; H3 = h0β0 −

3X

i=1

aiαi + h0α0 −

3X

i=1

aiβi

Flux quantization, e.g.

Z

Π3

H3 ∈ Z, ⇒ m, e0, · · · , ai ∈ Z NSNS, RR forms in units of `-1s

W = e0 + i

3X

i=1

eiUi − q1U2U3 − q2U3U1 − q3U1U2 + imU1U2U3 +

Sˆ

ih0 −

3X

i=1

aiUi + ia1U2U3 + ia2U3U1 + ia3U1U2 − h0U1U2U3

˜

Cancellation of C4 tadpole

Z

M4×Y

C4 ∧H3 ∧ F3 +X

D3

Z

M4

C4 − 32

Z

M4

C4

ND3 +1

2[mh0 − e0h0 +

X

i

(qiai + eiai)] = 16

– p.8/26

NSNS and RR fluxes in IIA/O6

W (S, U, T ) =

∫

Y

eJc ∧ FRR

︸ ︷︷ ︸

WK(T )

+

∫

Y

Ωc ∧ H3

︸ ︷︷ ︸

WQ(S, U)

,

Jc = B + iJ

F RR = F 0+F 2+F 4+F 6

Ωc = C3 + iRe (e−φΩ)

WK Txxx in fiber T3x

Z

Y

F3 ∧ Ω→WK under mirror symmetry = T-duality in x1, x2, x3 ≡ T123

F3T123−−−→ F RR, e.g. F123 → F 0 , F456 → F 142536

WK = e0 + i3

X

i=1

eiTi − q1T2T3 − q2T3T1 − q3T1T2 + imT1T2T3

∗ IIB→IIA moduli : UiT123−−−→ Ti ; Ti

T123−−−→ Ui ; ST123−−−→ S

IIA : S = e−φR1xR2

xR3x + iχ0 ; Ui = e−φRi

xRjyRk

y + iχi , i 6= j 6= k

axions χI arise from C3 , Ωc = iSα0 − i

3X

i=1

Uiαi

– p.9/26

WQ

WQ(S, U) =

∫

Y

Ωc ∧ H3 , Ωc = C3 + iRe (e−φΩ)

O-invariance ⇒ H3 odd under σA ⇒ H3 = h0β0 +

3X

i=1

hiβi

WQ = ih0S − i

3X

i=1

hiUi

only H456T123−−−→ H456 = h0

∗ IIB→IIA moduli : UiT123−−−→ Ti ; Ti

T123−−−→ Ui ; ST123−−−→ S

IIA : S = e−φR1xR2

xR3x + iχ0 ; Ui = e−φRi

xRjyRk

y + iχi , i 6= j 6= k

axions χI arise from C3 , Ωc = iSα0 − i

3X

i=1

Uiαi

– p.9/26

Set-up

W = WK + WQ ; WQ = ih0S − i

3X

i=1

hiUi

WK = e0 + i3

X

i=1

eiTi − q1T2T3 − q2T3T1 − q3T1T2 + imT1T2T3

Integer fluxes constrained by cancellation of C7 tadpoles

−

Z

M4×T6

C7 ∧ F 0 ∧H3 +X

D6a

Na

Z

M4×Πa

C7 − 32

Z

M4×ΠO

C7

Na D6a-branes wrap 3-cycle Πa = (n1a, m1

a) ⊗ (n2a, m2

a) ⊗ (n3a, m3

a) ∈ T6

O6-plane, ΠO = (1, 0) ⊗ (1, 0) ⊗ (1, 0)

susy D6, e.g. (1, 0) ⊗ (1, 0) ⊗ (1, 0), (0, 1) ⊗ (1, 0) ⊗ (0,−1) Im Ω|Πa= 0

X

a

Nan1an

2an

3a +

1

2mh0 = 16 ;

X

a

Nan1am

2am

3a +

1

2mh1 = 0

X

a

Nam1an

2am

3a +

1

2mh2 = 0 ;

X

a

Nam1am

2an

3a +

1

2mh3 = 0

– p.10/26

Results

W = WK + WQ ; WQ = ih0S − i

3X

i=1

hiUi

WK = e0 + i3

X

i=1

eiTi − q1T2T3 − q2T3T1 − q3T1T2 + imT1T2T3

X

a

Nan1an

2an

3a +

1

2mh0 = 16 ;

X

a

Nan1am

2am

3a +

1

2mh1 = 0

X

a

Nam1an

2am

3a +

1

2mh2 = 0 ;

X

a

Nam1am

2an

3a +

1

2mh3 = 0

∃ susy AdS vacua DTiW = DSW = DUi

W = 0 iff m 6= 0, (mei + qjqk) < 0

∗ Ti, Re S ∼ Re Ui and (h0Im S −X

i

hiIm Ui) fixed

∗ moduli lie in region of large radius and small gs, fluxes are diluted

∗ ∃ family of vacua parametrized by RR fluxes F 2 (ei) and F 4 (qi)

∗ fluxes contribute to tadpoles like D6-branes ⇒ O6-planes necessary

– p.10/26

D6-branes with fluxes

Na D6a-branes wrap 3-cycle Πa = (n1a, m1

a) ⊗ (n2a, m2

a) ⊗ (n3a, m3

a) ∈ T6

∗ With H3 6= 0, cancellation of Freed-Witten anomaly requiresZ

Πa

H3 = 0

(dF = H3, F = B + 2πα′F )

H3 = h0β0 +

3X

i=1

hiβiFW=⇒ h0ca

0 +

3X

i=1

hicai = 0 (ca

0 = m1am2

am3a, ca

i = mianj

anka)

∗ The U(1) in U(Na) has a coupling Aµa∂µUa, Ua =

`

ca0Im S −

X

i

cai Im Ui

´

Ua must remain massless (to be swallowed by Aaµ), it cannot be fixed by fluxes

so must be orthogonal to`

h0Im S −X

i

hiIm Ui

´

guaranteed by FW !

∗ In susy AdS vacua, hiRe Ui = −h0Re S ⇒ H3 ∼ ImΩ

FW ⇔ ImΩ|Πa= 0 susy D6a

∗ With metric fluxes FW becomesZ

Πa

H3 + ωJc = 0

– p.11/26

IIA with metric fluxes (twisted tori)

T6 isometries : [ZM , ZN ] = 0 ; M, N = 1, · · · , 6

in twisted T6 : [ZM , ZN ] = ωPMNZP Lie algebra of G

dηP = − 12ωP

MN ηM ∧ ηN

structure constants ωPMN = metric fluxes

Conditions on metric fluxes

∗ ωP[MNωS

R]P = 0 Jacobi/Bianchi

∗ ωPPN = 0 volume form is not exact

∗ ∃ discrete Γ ⊂ G such that G/Γ is compact Hull, Reid-Edwards; Grana et al

e.g. only 34 TT6 nilmanifolds

– p.12/26

Example

(a 56, 0, 0, 0, 0, 0) i.e. only ω156 = a 6= 0 , [Z5, Z6] = aZ1

dη1 = −aη5 ∧ η6 ; dη` = 0 , ` = 2, · · · , 6

η1 = dx1 + ax6dx5 ; η` = dx`

ds2 = (dx1 + ax6dx5)2 + (dx2)2 + · · · + (dx6)2

a = 0 −→ T6 : xi ' xi + mi , mi ∈ Z , i = 1, · · · , 6

a 6= 0 −→ TT6 : x1 ' x1 + m1 − am6x5 so that η1 be globally defined

a ∈ Z

– p.13/26

Compactification on TT = generalized Scherk-Schwarz reduction Kaloper, Myers

O-invariance allows ω in IIA/O6 (and in IIB/O9 but not in IIB/O3)

σA(xi) = xi , σA(yi) = −yi ⇒ ωxyy , ωy

xy , ωxxx

12 metric fluxes with one leg in each T2, i..e

0

B

B

@

a1

a2

a3

1

C

C

A

=

0

B

B

@

ω156

ω264

ω345

1

C

C

A

;

0

B

B

@

b11 b12 b13

b21 b22 b23

b31 b32 b33

1

C

C

A

=

0

B

B

@

−ω123 ω4

53 ω426

ω534 −ω2

31 ω561

ω642 ω6

15 −ω312

1

C

C

A

Txxx-duality of electric Hxyy in IIB/O3 → ωxyy in IIA/O6

Ex. apply Txxx to H3 = −a1 dx1 ∧ dy2 ∧ dy3 = −a1 dx1 ∧ dx5 ∧ dx6

Buscher rules ⇒ H′3 = 0 , ds′2 = (dx1 + a1x6dx5)2 + (dx2)2 + · · · + (dx6)2

a1 =ω156

– p.14/26

ω effects

• WQ(S, U)→WQ(S, U,T )

WQ =

Z

Y

Ωc ∧ (H3 + ωJc) , Jc = B + iJ

ωJc = dJc , (ωX (2))LMN = ωA[LMX

(2)

N ]A

WQ = ih0S −

3X

i=1

[ihiUi + (aiS + biiUi +X

j 6=i

bijUj)Ti]

• New induced C7 tadpoles

Z

M4×Y

C7 ∧`− F 0 ∧H3 + ωF 2

´+ · · ·

X

a

Nan1an

2an

3a +

1

2(mh0 + a1q1 + a2q2 + a3q3) = 16

X

a

Naniam

jam

ka +

1

2(mhi − q1b1i − q2b2i − q3b3i) = 0 , i 6= j 6= k

– p.15/26

Susy Minkowski vacua

∗ H3 = 0 , at most F2 6= 0

∗ can fix only ratios of moduli

∗ solutions for W (S, T1, T2, T3) duals of IIB

∗ W (S, U1, T2, T3) = −T2(a2S + b21U1)− T3(a3S + b3U1)

F n = 0 ⇒ no flux tadpoles, ω’s of flat solvmanifold Grana, Minasian, Petrini, Tomasiello

∗ W (U1, U2, T1, T2, T3) = −T1(q3T2 + q2T3)− U2(b22T2 + b32T3)− U3(b23T2 + b33T3)

flux tadpoles like D6, ω’s of non-compact solvmanifold

– p.16/26

Susy AdS vacua

isotropic fluxes: ei = e, qi = q, ai = a, bji = −bii = bi −→ Ti = T

W = e0 + 3ieT − 3qT 2 + imT 3 + ih0S − 3aST −

3X

k=0

(ihk + bkT )Uk

∗ 3ahk + h0bk = 0 FW

∗ bkuk = 3as , as = −2(q + mIm T )t , (3aIm S +X

k

bkIm Uk) = 24m(Im T )2 + · · ·

F0m = 0⇒ Im T =

h03a ; t2 =

h0e9qa −

h20

27a2 −e09q (s, uk, t) ≡ Re (S, Uk, T )

m 6= 0⇒ Im T =h03a (λ + λ0) ; t2 =

5h20

3a2 λ(λ + λ0 − 1) ; λ0 = − 3aqmh0

160λ3 + 186(λ0 − 1)λ2 + 27(λ0 − 1)2λ + λ20(λ0 − 3) + 27a2

mh30(e0a− eh0) = 0

∗ moduli fixed in region of large radius, small gs and diluted fluxes

∗ flux tadpoles can have opposite sign as D6-branes, or even cancel

X

a

Nan1an

2an

3a +

1

2(mh0 + 3aq) = 16 ;

X

a

Naniam

jam

ka +

1

2(mhi − qbi) = 0

– p.17/26

MSSM-like model with stabilized moduli

Metric, RR and NSNS fluxes:

ωxyy = a = 16 , ωy

xy = ωxxx = bk = 4 , m = 4 , e0 = e = 0 , q = hi − 2

hk = −h0bk/3a , h0 arbitrary

Moduli:

Re T ' |h0| , gs ' |h0|−1/2 large h0 ⇒ large radius, weak coupling

Tadpoles:

X

a

Nan1an

2an

3a = 64 ;

X

a

Naniam

jam

ka = −4

a Na (n1a, m1

a) (n2a, m2

a) (n3a, m3

a)

A 4 (1, 0) (3, 1) (3, −1)

B 1 (0, 1) (1, 0) (0, −1)

C 1 (0, 1) (0, −1) (1, 0)

H1 3 (2, 1) (1, 0) (2, −1)

H2 3 (2, 1) (2, −1) (1, 0)

O 4 (1, 0) (1, 0) (1, 0)

U(4) × SU(2) × U(1)

↓

SU(3) × SU(2) × U(1)Y × U(1)B−L

Cremades, Ibanez, Marchesano

O6

D6 A D6 C

D6 B

– p.18/26

Non-geometric fluxes

IIBTxxx

−−−−−→ IIA

Hyyy Hyyy

S S

Hxyy ωxyy

SU ST

Hxxy ?

SU2 ST2

Hxxx ?

SU3 ST3

? Hxxy

T U

? ωyxy , ωx

xx

UT TU

?? ??

Use T-duality to find missing fluxes

−HMNPTM←→ ωM

NP

TN←→ QMNP

TP←→ −RMNP

Ex. H3 = −a dx1 ∧dx2 ∧dy (B = −aydx1 ∧ dx2)

ds2 Txxx−−−→1

1 + a2y2[(dx1)2 +(dx2)2]+ (dy)2 + · · ·

BTxxx−−−→

ay

1 + a2y2dx1 ∧ dx2

not periodic as y→y + 1

Under y→y + 1, E = G + B of T212 has O(2, 2, Z)

monodromy E → E/(ΘE + 1) (ΘT =−Θ, Θ12 =−a)

−HxxyTxxx−−−→ Qxx

y = a

Ex. H3 = h0 dx1 ∧ dx2 ∧ dx3 Txxx formal

HxxxTxxx−−−→ Rxxx = h0

– p.19/26

T-duality chain : −HMNPTM←→ ωM

NPTN←→ QMN

PTP←→ −RMNP

STW

H, Q odd, ω, R even under the orientifold involution σ

IIB/O3 H,Q

IIA/O6 H, ω, Q, R

IIB/O9 ω, R

Isometry algebra enlarged by generators XM from B-field

[ZM , ZN ] = ωPMNZP − HMNP XP

[ZM , XP ] = −ωPMNXN + QPR

M ZR invariant under ZMTM←→XM

[XM , XN ] = QMNP XP − RMNP ZP

Jacobi identities ⇒ constraints on fluxes

In IIB/O3 : Q[MNP Q

L]PR = 0 , QRP

[L HMN ]P = 0

– p.20/26

IIA/O6 superpotential

−HxxyTxxx−−−→ Qxx

y ⇒ ST 2 ; HxxxTxxx−−−→ Rxxx ⇒ ST 3

Altogether W includes S ·(cubic polynomial in T )

Expect also U ·(cubic polynomial in T ) Qxyx , Qyy

y ⇒ UT 2 ; Rxyy ⇒ UT 3

W =

Z

Y

eJc ∧ F RR +

Z

Y

Ωc ∧`H3 + ωJc + QJ (4)

c + RJ (6)c

´

J(4)c = 1

2Jc ∧ Jc ; (QX (4))LMN = QAB

[L X(4)

MN]AB

J(6)c = 1

6Jc ∧ Jc ∧ Jc ; (RX (6))LMN = RABCX (6)

[LMN]ABC

IIA/O6 flux tadpolesZ

M4×Y

C7 ∧`− F 0 ∧H3 + ωF 2 −QF 4 + RF 6

´

depends on all RR fluxes– p.21/26

IIB/O3 superpotential

Need T ·(cubic polynomial in U) to match IIA/O6

W =

Z

Y

`F3 − iSH3 +QJ (4)

c

´∧ Ω

J (4)c ∼ T, in T6, J (4)

c = C4 + i2 e−φJ ∧ J = i

3X

i=1

Ti eωi eωi : basis of 4-forms

W = E0(U) + SE1(U) + TE2(U) Ea : cubic polynomials

IIB/O3 flux tadpoles

Besides C4, expect C8 tadpoles (to be cancelled by D7-branes)

Z

M4×Y

C4 ∧H3 ∧ F3 −

Z

M4×Y

C8 ∧QF3

mirror to IIA/O6 flux tadpoles

– p.22/26

S-dual fluxes

IIB/O3 with RR and NSNS fluxes is invariant under SL(2, Z) S-duality

S →kS − i`

imS + n;

„F3

H3

«→

„k `

m n

«„F3

H3

«; kn− `m = 1 ; k, `, m, n ∈ Z

True because W =

Z

Y(F3 − iSH3) ∧ Ω transforms as W → W

imS+nso that

K + log |W|2 is invariant

Invariance with non-geometric QMNL requires S-dual PMN

L (non-geometric RR)

W =

Z

Y

ˆ(F3 − iSH3) + (Q− iSP)J (4)

c

˜∧ Ω

„QP

«→

„k `

m n

«„QP

«

In T6, P ⇒ extra STE(U) terms in W

cubic polynomial– p.23/26

Constraints

With SL(2, Z) doublets (F3,H3) and (Q,P) can form

Q[MNP Q

L]PR = 0 ; P

[MNP P

L]PR = 0 ; Q

[MNP P

L]PR + P

[MNP Q

L]PR = 0 triplet

QRP[L HMN ]P − P

RP[L FMN ]P = 0 singlet

Flux tadpoles

C4 is a SL(2, Z) singlet, but C8 belongs to a triplet

(C8, eC8, C′8)

S→1/S−−−−−→ (− eC8,−C8,−C′

8)

sourced by (D7, NS7, I7)-branes Bergshoeff et al

S-duality invariance implies

Z

M4×Y

C4 ∧H3 ∧ F3 − C8 ∧QF3 + eC8 ∧ PH3 + C′8 ∧ (QH3 + PF3)

– p.24/26

Final comments

• Analysis of simple T6/O allows to find general(ized ?) structure of superpotential

and tadpoles induced by metric and non-geometric fluxes.

• Can also study IIB/O9 and the S-dual heterotic. In both several non-geometric

fluxes are in geometric ω.

• There are susy AdS IIA vacua with moduli stabilized in perturbative region.

• Only one linear combination of Im S, Im Ui is fixed (massive). Orthogonality to

combination which couples to potentially anomalous U(1)a is guaranteed byZ

Πa

H3 + ωJc = 0 (cancellation of Freed-Witten anomaly).

• Susy D6a-branes satisfy FW automatically.

• If ω = 0, flux tadpoles with same sign as D6.

• If ω 6= 0, flux tadpoles can have opposite sign as D6, or even cancel.

⇒ can build mssm-like vacua without additional O6-planes.

• Lift to dS ? Adding D6 does not work. Presumably including gs and/or α′

corrections, plus D-terms could work. Saueressig, Theis, Vandoren; Parameswaran, Westphal

– p.25/26

• Switching on non-geometric fluxes imposes additional constraints and finding vacua

is more difficult. E.g. no susy Minkowski in simplest cases.

• 10d origin of non-geometric fluxes ? Compactification on manifolds of

SU(3)× SU(3) structure. Benmachiche, Grimm; Grana, Louis, Waldram

• What about 11d ? Compactification of M-theory on G2 manifolds with fluxes

(Dall’Agata, Prezas) suggests even more fluxes in 10d, including some in our proposed

S-dual P.

• To cancel S-dual tadpoles still need a realization of the S-dual NS7 and I7 branes.

10d origin of the non-geometric RR fluxes P ?

– p.26/26