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 ′ , g s 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 = e K ( K A ¯ B ¯ B D A WD ¯ W − 3 | W | 2 ) , D A = ∂ A + K A Φ A - moduli K (Φ A ) = K cl + K pert + K non − pert , W (Φ A ) = W cl + W non − pert , W pert ≡ 0 Standard CY(3) compactifications ⇒ V ≡ 0 → Need to consider additional effects ! Classical: background fluxes → W cl Perturbative: α ′ , g s corrections → K pert Non-perturbative: brane instantons → K np , W np

– IIB Large Volume Comp.: CY(3): complex structure moduli and K¨ ahler structure moduli N = 1 SUGRA in 4d: V = e K ( K A ¯ B ¯ B D A WD ¯ W − 3 | W | 2 ) , D A = ∂ A + K A potential for complex str. moduli z i Background fluxes ⇒ � [ due to W flux = ( F 3 + iτH 3 ) ∧ Ω ] → { z i } fixed by D z i W flux = 0 , i = 1 , ..., h 2 , 1 ahler moduli t α : need quantum effects ! K¨ LVC (Quevedo et al.): essential to include α ′ corrections

Balancing of α ′ corrections and np effects ⇒ Minimum at V ∼ e a s τ s > > 1 , V - CY volume , τ s - cycle wrapped by D3 brane Advantages of LVC: - No fine-tuning of � W flux � (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: d 10 x √− g e − 2Φ � 1 � R + 4( ∂ Φ) 2 − 1 2 H 2 S = 2 κ 2 10 − α ′ � 4 (tr F 2 − tr R 2 + ) + O ( α ′ 3 ) , tr R 2 + = 1 2 R MNAB (Ω + ) R MNAB (Ω + ) , Ω ± = Ω ± H , Ω - spin connection , H = dB + α ′ 4 [ CS (Ω + ) − CS ( A )] , F - YM field strength, A - its connection

– Susy transformations: � ∂ M + 1 � AB + α ′ P M AB ) + O ( α ′ 3 ) δ Ψ M = 4Γ AB (Ω − M ǫ δλ = − 1 � / Φ − 1 / + 3 � / + O ( α ′ 3 ) 2 α ′ P √ ∂ 2 H ǫ 2 2 δχ = − 1 /ǫ + O ( α ′ 3 ) , 2 F 6 H MNP Γ MNP , P MAB = 6 e 2Φ ∇ ( − ) N ( e − 2Φ d H ) MNAB / = 1 H 4d Effective Action: Reduce on a 6d solution → 4d K¨ ahler potential Want α ′ -corrected K¨ ahler pot. ⇒ need α ′ -corrected solution Note: δ Ψ m = 0 and δλ = 0 ⇒ 6d manifold - complex

– Perturbative 6d solution: j + α ′ h (1) j + α ′ 2 h (2) G i ¯ j = g i ¯ j + ... i ¯ i ¯ Φ = φ 0 + α ′ φ (1) + α ′ 2 φ (2) + ... H = α ′ H (1) + α ′ 2 H (2) + ... H (0) = 0 : - 0 th order solution: CY (3) ( H (0) � = 0 ⇒ ∃ O ( α ′ ) cycles) - reliable SUGRA approx. Can show: Φ = φ 0 + O ( α ′ 3 ) with φ 0 = const

– Effective action for the moduli: Reduction ansatz: ds 2 = ˆ g µν ( x ) dx µ dx ν + G mn ( y, M ( x )) dy m dy n , B = B µν ( x ) dx µ ∧ dx ν + B mn ( y, M ( x )) dy m ∧ dy n , Φ = ϕ ( x ) + φ ( y, M ( x )) , T α = b α + it α M I ( x ) = { T α , Z I } , ahler potential for T α : K¨ K = − log( V ) − α ′ 2 � J (0) ∧ ˜ h (1) ∧ ˜ h (1) + O ( α ′ 3 ) , 2 V J = J (0) + α ′ ˜ h (1) + α ′ 2 ˜ h (2) + ...

No order α ′ correction: Varying the moduli: h (1) → h (1) + δh (1) ⇒ g → g + δg , J (0) → J (0) + δJ (0) , h (1) → ˜ h (1) + δ ˜ ˜ h (1) ⇒ Known: ∆ L δg mn = 0 , ∆ L - Lichnerowicz operator Can show: h (1) mn , δh (1) - orthogonal to zero modes of ∆ L mn d 6 y √ g h (1) m � i . e . : m = 0 d 6 y √ g J (0) i ¯ j = 1 � � J (0) ∧ J (0) ∧ ˜ h (1) = 0 j h (1) ⇔ i ¯ 2

– Aside on mirror symmetry: EFT of IIA on CY(3) ⇔ EFT IIB on mirror CY(3) ( h 1 , 1 = p , h 2 , 1 = q ) ( h 1 , 1 = q , h 2 , 1 = p ) EFT: N = 2 susy, Moduli space: M = M V ⊗ M H in IIB: M V - classically exact M V - α ′ corrections in IIA: → compute via IIB on mirror [COGP, Nucl. Phys. B359 (1991) 21] pert. correction: K = − log V + α ′ 3 const ⇒ V agrees with α ′ 3 R 4 computation! → World-sheet description: (2 , 2) SCFT How about (0 , 2) compactifications? [Het. with nonstand. emb.]

Heterotic Moduli Stabilization – Classical stabilization: � W tree = ( H + i dJ ) ∧ Ω , H -flux: complex structure moduli non-K¨ ahler manifolds ( SU (3) structure): K¨ ahler moduli SU (3) structure: 3 � � dJ = 4 i W 1 Ω − W 1 Ω + W 3 + J ∧ W 4 d Ω = W 1 J ∧ J + J ∧ W 2 + Ω ∧ W 5 - 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 A α e ia α T α � W inst = α Generically W inst � = 0 for non-standard embedding - gaugino condensation W GC = Ae ia ( S + β α T α ) , β α < < 1 - NS5-brane instantons V = 1 W NS 5 = ˆ Ae − ˆ a V 6 κ αβγ t α t β t γ ,

– Moduli stabilization set-up: Could take: W = W flux + W np , K = K tree (as in KKLT) Instead: ∃ large volume minima ? (as in Quevedo et al.) ⇒ take W = W flux + W np , K = K tree + K ( α ′ ) Only W np = W inst suitable! W GC = Ae iaS Keep to stabilize axion-dilaton S [ D S W = 0 ⇒ � W flux + W GC � ≡ W 0 � = 0 ] Note on consistency: In our backgrounds: dJ ∧ Ω ≡ 0 ⇒ W tree = W flux

Large Volume Minima T α = b α + it α , J = t α ω α , α = 1 , ..., h 1 , 1 K¨ ahler moduli: α A α e ia α T α ] [ W = W 0 + � – Scalar potential: V = V np 1 + V np 2 + V α ′ , β � A β e i ( a α T α − a β ¯ T β ) � e K G α ¯ a α A α a β ¯ V np 1 = β a α A α e ia α T α W 0 K ¯ ie K � G α ¯ � V np 2 = β − c.c. e K � � G α ¯ β K α K ¯ | W 0 | 2 V α ′ = β − 3 Basic assumption: ∃ 2-cycle t ℓ , t ℓ > > t α for ∀ α � = ℓ e − a ℓ t ℓ < < e − a α t α ⇒ for ∀ α � = ℓ

Consider two moduli: t ℓ > > t s K = − ln V + α ′ 2 f ( t ) → K¨ ahler potential: V 2 / 3 Scalar potential: λ e − 2 a s t s � e − 2 a s t s � V np 1 = V 1 / 3 + O V − µ t s e − a s t s � e − a s t s � V np 2 = + O V 2 V � 1 − ν f ( t ) � ν ∼ α ′ 2 V α ′ = V 5 / 3 + O , V 8 / 3 3 2 a s t s : V np 1 , V np 2 , V α ′ - same order of magnitude! For V ∼ e V = λ e − 2 a s t s V 1 / 3 − µ t s e − a s t s − ν f ( t ) ⇒ V 5 / 3 + ... V

– Type IIB (for comparison): [hep-th/0502058 by BBCQ] • leading correction: O ( α ′ 3 ) • V np 1 , V np 2 , V α ′ comparable for V ∼ e a s τ s , τ s - 4-cycle vol. √ τ s e − 2 a s τ s µ τ s e − a s τ s + ˆ ν V = ˆ → λ − ˆ V 2 V 3 V � 2 √ τ s e a s τ s , � ν ˆ 3 µ ˆ 4ˆ λ At min.: V = τ s = [ a s τ s > > 1 ] 2ˆ µ 2 ˆ λ – Heterotic two-moduli case: V = λ e − 2 a s t s V 1 / 3 − µ t s e − a s t s − ν f ( t ) V 5 / 3 V 3 ∂V 2 a s t s Minimum: ∂ V = 0 → V ∼ e ∂V BUT: ∂t s = 0 has no solution for t s !

– Heterotic LV minima: Need at least three moduli: t ℓ > > t 1 , t 2 • f ( t ) - homogeneous of degree 0 in t 1 , 2 • f ( t ) - nontrivial function (i.e. f � = const ) e − 2 a 1 t 1 e − 2 a 2 t 2 + σ e − a 1 t 1 e − a 2 t 2 V = λ 1 + λ 2 V 1 / 3 V 1 / 3 V 1 / 3 � � t 1 f t 1 e − a 1 t 1 t 2 e − a 2 t 2 t 2 − µ 1 − µ 2 − ν V 5 / 3 V V 3 3 2 a 1 t 1 ∼ e 2 a 2 t 2 Minimum: V ∼ e Now can solve for t 1 , t 2 ; easy to obtain t 1 , 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] 3 2 a s t s , t s - small 2-cycle] • Large volume: ∃ minimum [ V ∼ e – Open issues: • Gauge bundle moduli • Realistic examples • Cosmological applications