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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