Heterotic Large Volume Compactifications Lilia Anguelova - - PowerPoint PPT Presentation

Heterotic Large Volume Compactifications

Lilia Anguelova

(University of Cincinnati) arXiv:1007.4793 [hep-th]; arXiv:1007.5047 [hep-th] (with S. Sethi, C. Quigley)

Motivation

– String compactifications: 6 internal dimensions: ∃ many deformations with no energy cost In 4d effective theory: Deformation parameters → massless fields without a potential (moduli) BUT: Moduli vevs determine many 4d properties ... ⇒ huge vacuum degeneracy, lack of predictability of 4d coupling constants → Need to stabilize the moduli !

– Moduli stabilization:

• background fluxes

generically ⇒ generalized compact. (SU(3)×SU(3) structure) IIB : can have warped CY(3) compactifications

• perturbative corrections

α′ , gs corrections (affect 4d K¨ ahler potential only)

• non-perturbative effects

brane instantons, gaugino condensation For phenomenology: need large volume and weak coupling

– Effective 4d action: N = 1 Supergravity V = eK(KA ¯

BDAWD ¯ B ¯

W − 3|W|2) , DA = ∂A + KA K(ΦA) = Kcl + Kpert + Knon−pert , ΦA - moduli W(ΦA) = Wcl + Wnon−pert , Wpert ≡ 0 Standard CY(3) compactifications ⇒ V ≡ 0 → Need to consider additional effects ! Classical: background fluxes → Wcl Perturbative: α′, gs corrections → Kpert Non-perturbative: brane instantons → Knp, Wnp

– IIB Large Volume Comp.: CY(3): complex structure moduli and K¨ ahler structure moduli N = 1 SUGRA in 4d: V = eK(KA ¯

BDAWD ¯ B ¯

W − 3|W|2) , DA = ∂A + KA Background fluxes ⇒ potential for complex str. moduli zi [ due to Wflux =

• (F3 + iτH3) ∧ Ω ]

→ {zi} fixed by DziWflux = 0, i = 1, ..., h2,1 K¨ ahler moduli tα: need quantum effects ! LVC (Quevedo et al.): essential to include α′ corrections

Balancing of α′ corrections and np effects ⇒ Minimum at V ∼ easτs > > 1 , V - CY volume , τs - cycle wrapped by D3 brane Advantages of LVC:

• No fine-tuning of Wflux

(unlike in KKLT)

• Reliable low-energy EFT
• Tool for generation of hierarchies

Question: Are there heterotic LVC ? (Leading α′ correction in heterotic compactifications?)

Plan

• Motivation
• α′-Corrections in the Heterotic String

– Perturbative Solution – K¨ ahler Potential for K¨ ahler Moduli

• Heterotic Moduli Stabilization

– Non-perturbative Superpotential

• Large Volume Minima
• Summary

α′-Corrections in the Heterotic String

– 10d Action: S = 1 2κ2

10

• d10x √−g e−2Φ
• R + 4(∂Φ)2 − 1

2H2 − α′ 4 (trF2 − trR2

+) + O(α′3)

• ,

trR2

+ = 1 2RMNAB(Ω+)RMNAB(Ω+) ,

Ω± = Ω ± H , Ω - spin connection , H = dB + α′

4 [CS(Ω+) − CS(A)] ,

F - YM field strength, A - its connection

– Susy transformations: δΨM =

• ∂M + 1

4ΓAB(Ω−M

AB + α′PM AB) + O(α′3)

• ǫ

δλ = − 1 2 √ 2

• ∂

/Φ − 1 2H / + 3 2α′P / + O(α′3)

• ǫ

δχ = −1 2F /ǫ + O(α′3) , H / = 1

6HMNPΓMNP , PMAB = 6e2Φ∇(−)N(e−2ΦdH)MNAB

4d Effective Action: Reduce on a 6d solution → 4d K¨ ahler potential Want α′-corrected K¨ ahler pot. ⇒ need α′-corrected solution Note: δΨm = 0 and δλ = 0 ⇒ 6d manifold - complex

– Perturbative 6d solution: Gi¯

j = gi¯ j + α′h(1) i¯ j + α′2h(2) i¯ j + ...

Φ = φ0 + α′φ(1) + α′2φ(2) + ... H = α′ H(1) + α′2 H(2) + ... H(0) = 0:

• 0th order solution: CY (3)
• reliable SUGRA approx.

(H(0)= 0 ⇒ ∃ O(α′) cycles) Can show: Φ = φ0 + O(α′3) with φ0 = const

– Effective action for the moduli: Reduction ansatz: ds2 = ˆ gµν(x) dxµdxν + Gmn (y, M(x)) dymdyn , B = Bµν(x) dxµ∧dxν + Bmn (y, M(x)) dym∧dyn , Φ = ϕ(x) + φ (y, M(x)) , M I(x) = {T α, ZI} , T α = bα + itα K¨ ahler potential for T α: K = − log(V) − α′2 2V

• J(0) ∧ ˜

h(1) ∧ ˜ h(1) + O(α′3) , J = J(0) + α′˜ h(1) + α′2˜ h(2) + ...

No order α′ correction: Varying the moduli: ⇒ g → g + δg , h(1) → h(1) + δh(1) ⇒ J(0) → J(0) + δJ(0) , ˜ h(1) → ˜ h(1) + δ˜ h(1) Known: ∆Lδgmn = 0 , ∆L - Lichnerowicz operator Can show: h(1)

mn , δh(1) mn

• rthogonal to zero modes of ∆L

i.e. :

• d6y √g h(1)m

m = 0

⇔

• d6y √g J(0)i¯

jh(1) i¯ j = 1

2

• J(0) ∧ J(0) ∧ ˜

h(1) = 0

– Aside on mirror symmetry: EFT of IIA on CY(3) ⇔ EFT IIB on mirror CY(3)

(h1,1 = p, h2,1 = q) (h1,1 = q, h2,1 = p)

EFT: N = 2 susy, Moduli space: M = MV ⊗ MH in IIB: MV - classically exact in IIA: MV - α′ corrections → compute via IIB on mirror

[COGP, Nucl. Phys. B359 (1991) 21]

⇒

• pert. correction: K = − log V + α′3 const

V

→ agrees with α′3R4 computation! World-sheet description: (2, 2) SCFT How about (0, 2) compactifications? [Het. with nonstand. emb.]

Heterotic Moduli Stabilization

– Classical stabilization: Wtree =

• (H + i dJ) ∧ Ω ,

H-flux: complex structure moduli non-K¨ ahler manifolds (SU(3) structure): K¨ ahler moduli SU(3) structure: dJ = 3 4i

• W1Ω − W 1Ω
• + W3 + J ∧ W4

dΩ = W1J ∧ J + J ∧ W2 + Ω ∧ W5

• not well-understood moduli space
• ∃ O(α′) internal cycles

(hep-th/0304001 by BBDP)

– Quantum effects: (No perturbative corrections to W) Non-perturbative corrections:

• world-sheet instantons

Winst =

• α

AαeiaαT α Generically Winst = 0 for non-standard embedding

• gaugino condensation

WGC = Aeia(S+βαT α) , βα < < 1

• NS5-brane instantons

WNS5 = ˆ Ae−ˆ

aV

, V = 1 6καβγtαtβtγ

– Moduli stabilization set-up: Could take: W = Wflux+Wnp , K = Ktree (as in KKLT) Instead: ∃ large volume minima ? (as in Quevedo et al.) ⇒ take W = Wflux + Wnp , K = Ktree + K(α′) Only Wnp = Winst suitable! Keep WGC = AeiaS to stabilize axion-dilaton S [ DSW = 0 ⇒ Wflux + WGC ≡ W0 = 0 ] Note on consistency: In our backgrounds: dJ ∧ Ω ≡ 0 ⇒ Wtree = Wflux

Large Volume Minima

K¨ ahler moduli: T α = bα + itα, J = tαωα, α = 1, ..., h1,1 – Scalar potential: [ W = W0 +

α AαeiaαT α ]

V = Vnp1 + Vnp2 + Vα′ , Vnp1 = eKGα ¯

β

aαAα aβ ¯ Aβ ei(aαT α−aβ ¯

T β)

Vnp2 = ieK Gα ¯

β aαAα eiaαT α W 0 K ¯ β − c.c.

• Vα′

= eK Gα ¯

βKαK ¯ β − 3

• |W0|2

Basic assumption: ∃ 2-cycle tℓ , tℓ > > tα for ∀α = ℓ ⇒ e−aℓtℓ < < e−aαtα for ∀α = ℓ

Consider two moduli: tℓ > > ts → K¨ ahler potential: K = −lnV + α′2 f(t) V2/3 Scalar potential: Vnp1 = λ e−2asts V1/3 + O e−2asts V

• Vnp2

= −µ ts e−asts V + O e−asts V2

• Vα′

= −ν f(t) V5/3 + O 1 V8/3

• ,

ν ∼ α′2 For V ∼ e

3 2asts:

Vnp1, Vnp2, Vα′ - same order of magnitude! ⇒ V = λ e−2asts V1/3 − µ ts e−asts V − ν f(t) V5/3 + ...

– Type IIB (for comparison):

[hep-th/0502058 by BBCQ]

• leading correction: O(α′3)
• Vnp1, Vnp2, Vα′ comparable for V ∼ easτs , τs - 4-cycle vol.

→ V = ˆ λ √τse−2asτs V − ˆ µ τse−asτs V2 + ˆ ν V3 At min.: V =

ˆ µ 2ˆ λ

√τs easτs , τs =

• 4ˆ

νˆ λ ˆ µ2

2

3

[asτs > > 1] – Heterotic two-moduli case: V = λ e−2asts V1/3 − µ ts e−asts V − ν f(t) V5/3 Minimum:

∂V ∂V = 0

→ V ∼ e

3 2asts

BUT:

∂V ∂ts = 0

has no solution for ts !

– Heterotic LV minima: Need at least three moduli: tℓ > > t1, t2

• f(t) - homogeneous of degree 0 in t1,2
• f(t) - nontrivial function (i.e. f = const)

V = λ1 e−2a1t1 V1/3 + λ2 e−2a2t2 V1/3 + σ e−a1t1e−a2t2 V1/3 − µ1 t1 e−a1t1 V − µ2 t2 e−a2t2 V − ν f

• t1

t2

• V5/3

Minimum: V ∼ e

3 2a1t1 ∼ e 3 2a2t2

Now can solve for t1, t2 ; easy to obtain t1,2 ∼ O(100)

Summary

– Heterotic Large Volume Comp.:

• Leading α′ corrections in effective action

[α′-pert. solution]

• K¨

ahler potential for K¨ ahler moduli − No order α′ corrections! [leading effect: O(α′2)]

• Heterotic scalar potential

− World-sheet instantons [nonstandard embeddings]

• Large volume: ∃ minimum

[V ∼ e

3 2asts , ts - small 2-cycle]

– Open issues:

• Gauge bundle moduli
• Realistic examples
• Cosmological applications