VII Latin American Symposium on High Energy Physics and IX Argentine Symposium of Particles and Fields. Bariloche, January 17, 2009.

String compactifications

with fluxes

Anamarıa Font

Universidad Central de Venezuela

based on Algebras and non-geometric flux vacua (JHEP12 (2008) 050)

in collaboration with A. Guarino and J. Moreno (IFT UAM/CSIC)

– p.1/27

Scenario

• N=1 type II compactifications on toroidal orientifoldsinternal manifold Y = T6

Z2×Z2

orientifold actionΩP (−1)FL σA , σA(xi, yi) = (xi,−yi) IIA/O6-planes

ΩP (−1)FL σB , σB(xi, yi) = −(xi, yi) IIB/O3-planes

MAYM potential

gMN BMN

Ceven

Codd

branes and open stringsclosed strings

gravity sector

gauge sector

charged multiplets

, φ ,

RR p−form potentials

IIB

IIA

NSNS 2−form

– p.2/27

Fluxes

non-trivial backgrounds for YM, NSNS and RR field strengths −→ fluxes

e.g. Maxwell flux

∫

Π2

〈F2〉 = g 6= 0

e.g. NSNS flux

∫

Π3

〈H3〉 = h 6= 0 (H3 = dB2)

Πn are non-trivial cycles

– p.3/27

Fluxes thread non-trivial cycles in extra dimensions

Fluxes are quantized (Dirac quantization condition)

– p.4/27

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

• to trigger symmetry breaking and induce chirality

• . . .– p.5/27

We look at new vacua and moduli fixing

Outline

• Typical closed string moduli

• Review of NSNS, RR and geometric flux effects

• T-duality and non-geometric (NG) fluxes

• NG Q-fluxes in IIB orientifolds on T6/Z2 × Z2

• Q-subalgebras and superpotentials

• Families of AdS4 vacua

• Final comments

– p.6/27

References

• 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; Camara, Font, Ibanez

• IIA with geometric fluxesGurrieri, Louis, Micu, Waldram; Kachru, Schulz, Tripathy, Trivedi; Derendinger, Kounnas, Petropoulos, Zwirner;

Villadoro, Zwirner; Camara, Font, Ibanez, Grana, Minasian, Petrini, Tomasiello

• Non-geometric fluxesDabholkar, Hull; Shelton, Taylor, Wecht; Aldazabal, Camara, Font, Ibanez ; Grana, Louis, Waldram; Marchesano, Schulgin

Reviews: Grana; Douglas, Kachru; Blumenhagen, Kors, Lust, Stieberger; Wecht

For a more complete list of references see: JHEP12 (2008) 050

– p.7/27

Typical modulisize shape

dilaton Kahler complex structure

φ A = RxRy sin θ τ = −iRy

Rxeiθ

R in units of ℓs =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-planesfrom RR C4

S = ie−φ + C0 ; Ti = ie−φAjAk + ηi ; Ui = τi

Isotropic Ansatz

T1 = T2 = T3 = T ; U1 = U2 = U3 = U

– p.8/27

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.9/27

Flux induced moduli potentials in 4d

S =1

ℓ8s

∫

d10x√−G

e−2φ[

R− H2]

−∑

n

F 2n + · · ·

∫

Π3

〈H3〉 = h ⇒ V =h2e2φ

R12⇒ W =

h heterotic

hS orientifolds

In generic type II orientifoldsZ

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,T,U

ln(−i(Φ − Φ∗)) ; V = eK˘

X

Φ=S,T,U

− (Φ − Φ∗)2|DΦW |2 − 3|W |2¯

(gs → 0, α′ → 0) (DΦW = ∂ΦW + W∂ΦK)

– p.10/27

NSNS and RR fluxes in IIB/O3

W (S, U) =

∫

Y

(F 3 − SH3) ∧ Ω depends on U

〈F3〉 ≡ F 3 , 〈H3〉 ≡ H3 expanded in basis of 3-forms in Y

Flux quantization, i.e.

Z

Π3

F 3 ∈ Z ;

Z

Π3

H3 ∈ Z ⇒ coefficients aA, bA ∈ Z

W =(

a0−3a1U+3a2U2−a3U

3)

+ S(

−b0+3b1U−3b2U2+b3U

3)

Cancellation of C4 tadpole

Z

M4×Y

C4 ∧H3 ∧ F 3 +X

D3

Z

M4

C4 − 32

Z

M4

C4

a0b3 − 3a1b2 + 3a2b1 − a3b0 = N3 ; N3 = 32 − ND3

N3 > 0 ⇒ O3-planes ; N3 < 0 ⇒ D3-branes

– p.11/27

NSNS, RR and geometric fluxes in IIA/O6

Algebra of internal isometries : [ZM , ZN ] = ωPMNZP ; M, N = 1, · · · , 6

structure constants ωPMN = geometric fluxes ⇒ dJ = ωJ 6= 0 (ω forbidden in IIB/O3)

W (S, U, T ) =

∫

Y

eJc∧FRR + Ωc∧(

H3+ωJc

)

,Jc = iB + J ; Ωc = iC3 + Re (e−φΩ)

F RR = F0+F2+F4+F6

W =(

a0−3a1T+3a2T2−a3T

3)

+ S(

−b0+3b1U)

+ 3U(

c0+(2c1−c1)T)

b0 = Hyyy ; c0 = Hxxy ; b1 = ωxyy ; c1 = ωy

xy ; c1 = ωxxx

Flux induced C7 tadpole

Z

M4×Y

C7 ∧`

− F 0 ∧H3 + ωF 2

´

– p.12/27

T-duality

IIB/O3Txxx−−−→ IIA/O6

Txxx = T-duality in x1, x2, x3 (horizontal directions in T6)

IIB → IIA moduli : UTxxx−−−→ T ; T

Txxx−−−→ U ; STxxx−−−→ S

WIIB =`

a0−3a1U+3a2U2−a3U3´

+ S`

−b0+3b1U−3b2U2+b3U3´

WIIA =`

a0−3a1T+3a2T 2−a3T 3´

+ S`

−b0+3b1T´

+ 3U`

c0+(2c1−c1)T´

∗ RR → RR : F 3 → F 0 + F 2 + F 4 + F 6

∗ NSNS → NSNS : b0 = Hyyy → b0 = Hyyy

∗ NSNS → geometric : b1 = Hxyy → b1 = ωxyy (applying Buscher rules)

∗ missing fluxes ?

– p.13/27

Non-geometric fluxes

Shelton, Taylor, Wecht

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

NPTN←→ QMN

PTP←→ RMNP

Ex. H3 = b2 dx1 ∧ dx2 ∧ dy (B = b2ydx1 ∧ dx2)

ds2 Txxx−−−→1

1 + b22y2[(dx1)2 +(dx2)2]+ (dy)2 + · · ·

BTxxx−−−→

b2y

1 + b22y2dx1 ∧ 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 =b2)

HxxyTxxx−−−→ Qxx

y = b2

Ex. H3 = b3 dx1 ∧ dx2 ∧ dx3 Txxx formal

HxxxTxxx−−−→ Rxxx = b3

– p.14/27

T-duality chain : HMNPTM←→ ωM

NPTN←→ QMN

PTP←→ RMNP

STW

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

IIB/O3 H, Q

IIA/O6 H, ω, Q, 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.15/27

Non-geometric fluxes in IIB/O3 - General results

W =

Z

Y

`

F 3 − SH3 + QJc

´

∧ Ω

Jc∼ T, in T6, Jc = C4 + i2 e−φJ ∧ J =

3X

i=1

Tieωi eωi : basis of 4-forms

Flux tadpoles: Q’s induce new C8 tadpoles (to be cancelled by O7/D7-branes)

Z

M4×Y

C4 ∧H3 ∧ F 3 −

Z

M4×Y

C8 ∧QF 3

Aldazabal, Camara, Font, Ibanez

– p.16/27

IIB/O3 on T6/Z2 × Z2 with isotropic Ansatz

Fxxx Fyxx Fxyy Fyyy Hxxx Hyxx Hxyy Hyyy

a3 a2 a1 a0 b3 b2 b1 b0

Qxxx Qyx

y Qxxy Qyy

x Qyxx Qyy

y

c1 c1 c0 c3 c2 c2

W = P1(U) + SP2(U) + TP3(U)

RR P1(U) = a0 − 3a1U + 3a2U2 − a3U3

NSNS P2(U) = −b0 + 3b1U − 3b2U2 + b3U3

non-geo P3(U) = 3ˆ

c0 + (2c1 − c1) U − (2c2 − c2) U2 − c3 U3˜

a0 b3 − 3 a1 b2 + 3 a2 b1 − a3 b0 = N3 N3 = 32 − ND3

a0 c3 + a1 (2 c2 − c2) − a2 (2 c1 − c1) − a3 c0 = N7 N7 = −32 + ND7

– p.17/27

Constraints on non-geometric and NSNS fluxes

[XM , XN ] = QMNP XP ; [ZM , ZN ] = HMNP XP ; [ZM , XP ] = QPR

M ZR

Jacobi identities

Q[MNP Q

L]PR = 0 QRP

[LHMN ]P = 0

c0 (c2 − c2) + c1 (c1 − c1) = 0 b2c0 − b0c2 + b1(c1 − c1) = 0

c2 (c2 − c2) + c3 (c1 − c1) = 0 b3c0 − b1c2 + b2(c1 − c1) = 0

c0c3 − c1c2 = 0 b2c1 − b0c3 − b1(c2 − c2) = 0

b3c1 − b1c3 − b2(c2 − c2) = 0

Strategy: solve QQ = 0 by classifying Q-subalgebras consistent with

symmetries. Then solve QH = 0.

– p.18/27

Q-subalgebras

ˆX2i, X2j

˜= ǫijk

`c3 X2k−1 + c2 X2k

´,ˆX2i−1, X2j

˜= ǫijk

`c2 X2k−1 + c1 X2k

´

ˆX2i−1, X2j−1

˜= ǫijk

`c1 X2k−1 + c0 X2k

´; i, j, k = 1, 2, 3

change of basis

Ei

eEi

!= 1

|Γ|2

−α β

−γ δ

! X2i−1

X2i

!; |Γ| = αδ − βγ 6= 0

5 allowed subalgebras

semisimple: so(4)ˆEi, Ej

˜= ǫijkEk ;

ˆ eEi, eEj˜

= ǫijkeEk ;

ˆEi, eEj

˜= 0

so(3, 1)

nonsemisimple: nilˆEi, Ej

˜= ǫijkEk ;

ˆ eEi, eEj˜

= 0 ;ˆEi, eEj

˜= 0

su(2) ⊕ u(1)3

iso(3)

– p.19/27

Flux parametrization. Example so(4) ∼ su(2)2

Q NSNS

c0 = β δ (β + δ) ; c3 = −α γ (α + γ) ; b0 = − (ǫ1 β3 + ǫ2 δ3)

c1 = β δ (α + γ) ; c2 = −α γ (β + δ) ; b1 = ǫ1 α β2 + ǫ2 γ δ2

c2 = γ2 β + α2 δ ; c1 = − (γ β2 + α δ2) ; b2 = − (ǫ1 α2 β + ǫ2 γ2 δ)

b3 = ǫ1 α3 + ǫ2 γ3

Fluxes transform under U modular invariance (eK |W |2 is invariant)

SL(2, Z)U : U ′ =k U + ℓ

m U + n; k, ℓ, m, n ∈ Z ; kn − ℓm = 1

0

@

α′ β′

γ′ δ′

1

A =

0

@

α β

γ δ

1

A

0

@

n −ℓ

−m k

1

A ; ǫ′1 = ǫ1 ; ǫ′2 = ǫ2

New modular invariant variable Z =αU + β

γU + δ

– p.20/27

Q-subalgebras and superpotentials

new variables : Z =αU + β

γU + δ; S = S + ξs ; T = T + ξt

ξs and ξt can be reabsorbed in RR fluxes

W = |Γ|3/2 [T P3(Z) + S P2(Z) + P1(Z)] and tadpoles depend on Q-subalgebra

Q-subalgebra P3(Z)/3 P2(Z) P1(Z)

so(4) Z(Z + 1) ǫ1Z3 + ǫ2 ξ3(ǫ1 − ǫ2Z3) + 3ξ7Z(1 − Z)

ξ3(ǫ1 + 3ǫ2Z − 3ǫ1Z2 − ǫ2Z3)so(3, 1) −Z(Z2 + 1) ǫ1Z3 − 3ǫ2Z2 − 3ǫ1Z + ǫ2

+3ξ7(Z2 + 1)

su(2) + u(1)3 Z ǫ1Z3 + ǫ2 ξ3(ǫ1 − ǫ2Z3) − 3ξ7Z2

iso(3) 1 − Z ǫ1Z + ǫ2 3λ1Z + 3λ2Z2 + λ3Z3

nil 1 ǫ1Z + ǫ2 3λ1Z + 3λ2Z2 + λ3Z3

Q-subalgebra N3/|Γ|3 N7/|Γ|3

so(4) (ǫ21 + ǫ22) ξ3 2 ξ7

so(3, 1) 4(ǫ21 + ǫ22) ξ3 4 ξ7

su(2) + u(1)3 (ǫ21 + ǫ22) ξ3 ξ7

iso(3) λ2 ǫ1 − λ3 ǫ2 λ2 + λ3

nil λ2 ǫ1 − λ3 ǫ2 λ3

|Γ| = αδ − βγ

N3 = 32 − ND3

N7 = −32 + ND7

– p.21/27

Supersymmetric vacua

V = eK

8<:

X

Φ=Z,S,T

KΦΦ|DΦW|2 − 3|W|29=; has susy minima when F-terms=0

DTW = ∂W∂T + 3iW

2Im T = 0 ; DSW = ∂W∂S + iW

2Im S = 0 ; DZW = ∂W∂Z + 3iW

2Im Z = 0

at minimum V0 = −3eK0 |W0|2 ⇒ Minkowski (W0 = 0) or AdS (W0 6= 0)

Further physical requirements

∗ ImS0 > 0 (gs = 1/ImS0) ; Im T0 > 0 (Im T ∼ e−φA2) ; ImZ0 6= 0

There exist no Minkowski vacua for any of the Q-subalgebras

∗ Effective supergravity is a reliable approximation to string theory

Small string coupling, gs < 1

Conventionally large internal volume, Vint = (Im T0/ImS0)3/2 < 1

With Q-fluxes there could be light winding modes.

∗ Small cosmological constant, |V0| < 1

– p.22/27

Vacuum degeneracy

new variables : Z =αU + β

γU + δ; S = S + ξs ; T = T + ξt

superpotential : W = |Γ|3/2 [T P3(Z) + S P2(Z) + P1(Z)] ; Γ =`

α βγ δ

´

∗ Z, S, T invariant under SL(2, Z)U

inequivalent vacua can be labelled by vevs Z0, S0, T0 and choice of Γ

∗ W invariant under S → S − ξs , T → T − ξt

(because ξs and ξt can be reabsorbed in RR fluxes in P1)

– p.23/27

Supersymmetric AdS vacua Example su(2)2

W = |Γ|3/23T Z(Z + 1) + S(ǫ1Z3+ ǫ2) + ξ3(ǫ1 − ǫ2Z3

) + 3ξ7Z(1− Z)

N3 = |Γ|3(ǫ21 + ǫ22)ξ3 ; N7 = 2|Γ|3ξ7

ǫ1 = 0

Re T0 = ξ7 − ǫ2ξ32 ; ǫ2ReS0 = 3ξ7 − ǫ2ξ3

2 ; x0 = ReZ0 = − 12

Im T0 = − 4ǫ2Im S0(1+4y2

0); ǫ2ImS0 = −y0

ˆ3ξ7 +

ǫ2ξ38 (4y2

0 − 3)˜

; y0 = ImZ0

ǫ2ξ3(4y20 − 1)(4y2

0 + 5)− 8ξ7(4y20 − 5) = 0 ; ǫ2 < 0

∗ ξ7 = 0 , N3 > 0 O3-planes ; ∗ ξ3 = 0 , N7 > 0 D7-branes

∗ ξ3ξ7 6= 0 , 0 < ξ7ǫ2ξ3

< 18

, N3 > 0 , N7 > 0 O3-planes, D7-branes

ξ7ǫ2ξ3

≤ 0 , N3 > 0 , N7 ≤ 0 O3-planes, O7-planes

metastable vacuaξ7

ǫ2ξ3> 7+2

√10

4, N3 < 0 , N7 > 0 D3-branes, D7-branes

ǫ1ǫ2 6= 0 (x0, y0) determined by two coupled polynomials

there can exist multiple vacua even for ξ3ξ7 = 0– p.24/27

Families of vacua Example su(2)2 , ǫ1 = 0 , ξ7 = 0

By modular transformations can go to canonical gauge β = γ = 0

α and δ determined by surviving non-geometric fluxes: c1 = −2m ; c2 = 2n

NSNS fluxes b1, b2 and b3 vanish identically

RR fluxes a0, a1 and a2 can be set to zero by shifts in S and T

b0 = −2m2

nǫ2 ; a3 =

2n2

mǫ2ξ3 ; N3 = −a3b0 = |Γ|3ǫ22ξ3

V0 = −48 n6b3

0

m3N23

; gs =8 n3b2

0

m3N3

V0 and gs can be made small by keeping n and b0 fixed while m is taken large

– p.25/27

Final Comments

• T-duality requires extra non-geometric fluxes. In IIB the Q-fluxes form a

subalgebra

• Constraints on Q-fluxes can be solved by classifyng the Q-subalgebras.

There are 5 subalgebras compatible with the symmetries. Each leads to a

characteristic superpotential. Parametrization of fluxes leads to a modular

invariant variable Z = (αU + β)/(γU + δ).

• In IIB toroidal orientifolds with R-R, NS-NS plus non-geometric Q-fluxes,

and (S, T, U) moduli, there are no susy Minkowski vacua at all. However,

there are susy AdS vacua with moduli stabilized in perturbative region.

• Vevs Im S0 > 0 and Im T0 > 0 are correlated with net RR charges N3 and

N7. The type of sources needed to cancel tadpoles depends on the

Q-subalgebra. E.g. N3 > 0 and N7 = 0 occurs only for su(2)2 ,

N3 = N7 = 0 only for iso(3). Also, sources for compact so(4) and

non-compact so(3, 1) have opposite charges.

– p.26/27

• There can be multiple vacua but gs < 1 requires large N3 and/or N7.

• To cancel RR tadpoles it might be necessary to add D3 and/or D7-branes.

Allowed branes must be free of Freed-Witten anomalies, i.e.

W must be invariant under S → S + qsν and T → T + qtν, where (qs, qt)

depend on U(1) gauged by the D-brane.

D3-branes and unmagnetized D7-branes, which do not give rise to charged

chiral matter are anomaly-free. Magnetized branes can be added only for

the nilpotent and iso(3) cases in which only a linear combination of Re S

and Re T is determined. Different with non-isotropic Ansatz.

• 10d origin of non-geometric fluxes ?

Compactification on manifolds of SU(3)× SU(3) structure.

Benmachiche, Grimm; Grana, Louis, Waldram; Grana, Minasian, Petrini, Waldram.

Supergravity with electric and magnetic gaugings.

D’Auria, Ferrara, Trigiante; Hull, Reid-Edwards, Dall’Agata, Prezas, Samtleben, Trigiante; Aldazabal, Camara, Rosabal

• World-sheet description of non-geometric fluxes ?

T-folds. Hull

Asymmetric orbifolds. Hellerman, McGreevy, Williams; Dabholkar, Hull; Flournoy, Wecht, Williams

– p.27/27