A. Micu: String Dualities and Manifolds with SU(3) Structure

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Lecture form Spring School on Strings, Cosmology and Particles (SSSCP2009), March 31-4 2009, Belgrade/Nis, Serbia

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A. Micu: String Dualities and Manifolds with SU(3) Structure

  1. 1. Moduli Stabilisation in Heterotic String Compactifications Andrei Micu Department of Theoretical Physics IFIN-HH Bucharest Based on arXiv:0812.2172 [hep-th] Nis, 4 April, 2009
  2. 2. Introduction The Standard Model of Particle Physics → gauge group SU (3) × SU (2) × U (1) – works well at energies of oorder 100 GeV. It is just an effective theory – at higher energies needs to be modified Possibilities: Supersymmetry → Minimal Supersymmetric Standard Model (MSSM) GUT/susy GUT Supersymmetry = fermionic symmetry: ↔ fermion (Super)Multiplets → combinations of fields with different spin Matter → chiral supermultiplets Φ = (φ, ψ) Lagrangean for these fields is given by three functions 2
  3. 3. ¯ • K¨hler potentialK(Φ, Φ) a • superpotential W (Φ) • gauge coupling function fab(Φ) ¯¯ 1 a˜ L ∼ −gi¯∂µφi∂ µφ − 4 ImfabFµν F b µν + 4 RefabFµν F b µν − V , a i  ¯ gi¯ = ∂Φi ∂Φ K(Φ, Φ) , ¯¯  = eK DiW Dj W g i¯ − 3|W |2 + 1 Imfab D aD b  V −1 2 Di W = ∂Φi W + (∂Φi K)W . Supersymmetric solutions: DiW = 0. 3
  4. 4. String theory String theory is supposed to be valid at energies of order MP l = 1019GeV . In the low energy limit → supergravity in 10 space-time dimensions Compactifications on 6-dimensional manifolds → supergravity in 4d: K, W and f can be computed in string theory There exist 5 consistent superstring theories: type IIA/B, type I, heterotic SO(32)/E8 × E8. 4
  5. 5. 4d requirements N=1 supersymmetry Standard Model/GUT • gauge grup G ⊃ SU (3) × SU (2) × U (1) • chiral matter Type II → need additional constructions: intersecting branes, singularities etc. SO(32) gauge group: does not have the right representations for matter fields in 4d We are left with E8 → works pretty well 5
  6. 6. Heterotic models Bosonic spectrum in 10d: graviton gM N ; antisymmetric tensor field BM N ; dilaton (scalar) φ; gauge fields E8 × E8. Constraints: Bianchi identity dH = trF ∧ F − trR ∧ R , H = dB field strength ofB 6
  7. 7. 4d theory N = 1 supersymmetry → compactifications on Calabi–Yau manifolds (SU(3) holonomy). [trR ∧ R] = 0 → need trF ∧ F = 0 → breaks E8 gauge symmetry We can always set F ≡ R - SU (3) - structure E6 × SU (3) = maximal subgrup of E8 → surviving gauge symmetry in 4d is E6. Charged fields: 248 = (78, 1) ⊕ (1, 8) ⊕ (27, 3) ⊕ (27, ¯ 3) 7
  8. 8. Decompositionn of Dirac operator /10 = /4 + /6 → /6 − mass operator in 4d ; Massless fields in 4d ⇔ /6 ψ = 0 For Calabi–Yau manifolds with F = R H 0,1(T 1,0X) ≡ H 2,1(X) ; /6ψ3 = 0 ↔ H 0,1(T 0,1X) ≡ H 1,1(X) ; /6ψ3 = 0 ↔ ¯ Number of generations = |h1,1 − h2,1| = |χ|/2 8
  9. 9. Neutral fields (moduli) • δgab = complex structure deformations – h2,1 - complex • δga¯ K¨hler class deformations– h1,1 - real ba • Ba¯ – h1,1 real b • dilaton φ and axion Bµν : S = a + ieφ In total h1,1 + h2,1 + 1 neutral chiral fields. 9
  10. 10. Results • superpotential: cubic in the charged fields; does not depend on the moduli (for CY manifolds). • can obtain a dependence on the moduli from fluxes and/or manifolds with SU (3) structure. • K¨hler potential for moduli fields: specific to string compactifications a ¯ • K¨hler potential for matter fields: gC C a • fab = Sδab 10
  11. 11. Specific model Heterotic string compactifications on manifolds with SU(3) structure. Effective theory: Supergravity + super Yang-Mills theory E6 gauge group + one chiral superfield in 27 C A + one chiral singlet superfield T (h1,1 = 1, h2,1 = 0) 3 ¯ ¯ C ACA K = −3 ln(T + T ) + ¯ T +T 1 W = ieT + jABC C AC B C C . 3 11
  12. 12. Supersymmetric solutions ¯ I. C = 0: DT W = ie − 3/(T + T )(ieT ) = 0 ⇒ e = 0 not good. II. C = 0 what changes? 1 27 = 10−2 ⊕ 16 ⊕ 14 a. E6 ⊃ SO(10) × U (1) 27 = (3, ¯ 1) ⊕ (¯ 1, ¯ ⊕ (1, 3, 3) b. E6 ⊃ SU (3) × SU (3) × SU (3) 3, 3, 3) 12
  13. 13. a < 1 >= 0, < 10 >=< 16 >= 0 −→ E6 → SO(10) No 13 coupling in W ¯ 3C1 D1 W = 0+ ¯W = 0 =⇒ W = 0 , T +T DT W = e = 0 not good 13
  14. 14. b < (1, 3, 3) >= 0 −→ E6 → SU (3) × SU (2) × SU (2) There exist (1, 3, 3)3 ≡ B 3 coupling in W ¯ DB W = B · B + B · W = 0 B – small fluctuations ⇒ B 1 ⇒ W = eT ∼ B 1 ¯ but e is cuantised and T + T 1 for the supergravity approximation 14
  15. 15. Conclusions • The system under scrutiny (h1,1 = 1, h2,1 = 0) does not have satisfying supersimetric solutions • have to consider more complicated models (h2,1 = 0) • more complicated superpotential and there may exist viable solutions 15

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