Tetsuji KIMURA Yukawa Institute for Theoretical Physics, Kyoto - PowerPoint PPT Presentation

Global COE “Quest for Fundamental Principles in the Universe,” Nagoya University: Nov 27, 2008 Realization of AdS Vacua in Attractor Mechanism on Generalized Geometries arXiv:0810.0937 [hep-th] Tetsuji KIMURA Yukawa Institute for Theoretical Physics, Kyoto University

Introduction We are looking for the origin of 4D physics ✓ ✏ Physical information • particle contents and spectra • (broken) symmetries and interactions • potential, vacuum and cosmological constant ✒ ✑ Realization of AdS vacua in attractor mechanism on generalized geometries - 1 -

Introduction 4D N = 1 supergravity: � � 1 � 2 R ∗ 1 − 1 2 F a ∧ ∗ F a − K MN ∇ φ M ∧ ∗∇ φ N − V ∗ 1 S = = e K � K MN D M W D N W − 3 |W| 2 � + 1 2 | D a | 2 V K : K¨ ahler potential K 2 W γ µ ε c W : superpotential δψ µ = ∇ µ ε − e ��� δχ a = Im F a D a : D-term µν γ µν ε + i D a ε ��� V ∗ > 0 : de Sitter space � � Search of vacua ∂ P V ∗ = 0 V ∗ = 0 : Minkowski space V ∗ < 0 : Anti-de Sitter space Realization of AdS vacua in attractor mechanism on generalized geometries - 2 -

Introduction 10D string theories could provide information via compactifications 10 = 4 + 6 A typical success: ✓ ✏ E 8 × E 8 heterotic string compactified on Calabi-Yau three-fold • number of generations = | χ ( CY 3 ) | / 2 • E 6 gauge symmetry • zero cosmological constant ✒ ✑ P. Candelas, G.T. Horowitz, A. Strominger, E. Witten “ Vacuum configurations for superstrings ,” Nucl. Phys. B 258 (1985) 46 Realization of AdS vacua in attractor mechanism on generalized geometries - 3 -

Introduction However, Calabi-Yau manifold is not sufficient ��� Fluxes are strongly restricted H 3 , ∇ φ , torsion common type IIA F 0 , F 2 , F 4 , F 6 type IIB F 1 , F 3 , F 5   type IIA : No fluxes   On Calabi-Yau three-fold: type IIB : F 3 − τH (warped Calabi-Yau)    heterotic : No fluxes Realization of AdS vacua in attractor mechanism on generalized geometries - 4 -

Compactifications in 10D type IIA Decompose 10D type IIA SUSY parameters: ǫ 1 = ε 1 ⊗ ( a η 1 ǫ 2 = ε 2 ⊗ ( b η 2 + ) + ε c 1 ⊗ ( a η 1 − ) + ε c 2 ⊗ ( b η 2 − ) , + ) δψ A m = 0 gives the Killing spinor equation on the 6D compactified space M : � 4 ω mab γ ab � � � A + � � A = 0 ∂ m + 1 δψ A η A m = + + 3-form fluxes · η other fluxes · η ✬ ✩ Calabi-Yau three-fold ↓ Information of SU (3) -structure manifold with torsion 6D SU (3) Killing spinors η 1 + , η 2 + : ↓ generalized geometry ✫ ✪ Realization of AdS vacua in attractor mechanism on generalized geometries - 5 -

Beyond Calabi-Yau ◮ Calabi-Yau three-fold ��� Fluxes are strongly restricted   type IIA : No fluxes   type IIB : F 3 − τH (warped Calabi-Yau)    heterotic : No fluxes Realization of AdS vacua in attractor mechanism on generalized geometries - 6 -

Beyond Calabi-Yau ◮ Calabi-Yau three-fold ��� Fluxes are strongly restricted   type IIA : No fluxes   type IIB : F 3 − τH (warped Calabi-Yau)    heterotic : No fluxes ◮ SU (3) -structure manifold ��� Some components of fluxes can be interpreted as torsion   type IIA   restricted fluxes are turned on 2 type IIB    heterotic 1 1 : Piljin Yi, TK “ Comments on heterotic flux compactifications ,” JHEP 0607 (2006) 030, hep-th/0605247 2 : TK “ Index theorems on torsional geometries ,” JHEP 0708 (2007) 048, arXiv:0704.2111 Realization of AdS vacua in attractor mechanism on generalized geometries - 6 -

Beyond Calabi-Yau ◮ Calabi-Yau three-fold ��� Fluxes are strongly restricted   type IIA : No fluxes   type IIB : F 3 − τH (warped Calabi-Yau)    heterotic : No fluxes ◮ SU (3) -structure manifold ��� Some components of fluxes can be interpreted as torsion   type IIA   restricted fluxes are turned on 2 type IIB    heterotic 1 1 : Piljin Yi, TK “ Comments on heterotic flux compactifications ,” JHEP 0607 (2006) 030, hep-th/0605247 2 : TK “ Index theorems on torsional geometries ,” JHEP 0708 (2007) 048, arXiv:0704.2111 ◮ Generalized geometry ��� Any types of fluxes can be included All the N = 1 SUSY solutions can be classified Realization of AdS vacua in attractor mechanism on generalized geometries - 6 -

What is a generalized geometry? Consider the compactified space M 6 • Ordinary complex structure J mn lives in T M : J 2 = − 1 6 , n = − 2i η † J m + γ m n η + η + : SU (3) invariant spinor • Generalized complex structures J ΛΣ in T M ⊕ T ∗ M with basis { d x m ∧ , ι ∂ n } and (6 , 6) -signature � � J 2 = − 1 12 , J Λ Re Φ ± , Γ Λ ± Σ = Σ Re Φ ± Φ ± : SU (3 , 3) invariant spinors Φ ± can be described by means of η 1 ± and η 2 ± in SUSY parameters Realization of AdS vacua in attractor mechanism on generalized geometries - 7 -

Strategy 10D type IIA supergravity as a low energy theory of IIA string ↓ compactifications on a certain compact space in the presence of fluxes 4D N = 2 supergravity ↓ SUSY truncation 4D N = 1 supergravity Realization of AdS vacua in attractor mechanism on generalized geometries - 8 -

4D N = 1 Minkowski vacua in type IIA Classification of SUSY solutions on the SU (3) generalized geometries ( η 1 + = η 2 + ): M. Gra˜ na, R. Minasian, M. Petrini, A. Tomasiello hep-th/0407249 a = b e i β (type BC) IIA a = 0 or b = 0 (type A) W 1 = H (1) = 0 3 1 F (1) = ∓ F (1) = F (1) = ∓ F (1) F (1) 2 n = 0 0 2 4 6 generic β β = 0 W 2 = F (8) = F (8) Re W 2 = e φ F (8) Re W 2 = e φ F (8) + e φ F (8) 8 = 0 2 2 4 2 4 Im W 2 = 0 Im W 2 = 0 W 3 = ∓ ∗ 6 H (6) W 3 = H (6) = 0 6 3 3 W 5 = 2 W 4 = ∓ 2i H (3) F (3) = 2i W 5 = − 2i ∂A = 2i = ∂φ 3 ∂φ 3 2 3 ∂A = ∂a = 0 W 4 = 0 NS-flux only (common to IIA , IIB , heterotic) type A W 1 = W 2 = 0 , W 3 � = 0 : complex RR-flux only type BC W 1 = Im W 2 = W 3 = W 4 = 0 , Re W 2 � = 0 , W 5 � = 0 : symplectic For N = 1 AdS 4 vacua: hep-th/0403049, hep-th/0407263, hep-th/0412250, hep-th/0502154, hep-th/0609124, etc. Realization of AdS vacua in attractor mechanism on generalized geometries - 9 -

4D N = 1 Minkowski vacua in type IIA Classification of SUSY solutions on the SU (3) generalized geometries ( η 1 + = η 2 + ): M. Gra˜ na, R. Minasian, M. Petrini, A. Tomasiello hep-th/0407249 a = b e i β (type BC) IIA a = 0 or b = 0 (type A) W 1 = H (1) = 0 3 1 F (1) = ∓ F (1) = F (1) = ∓ F (1) F (1) 2 n = 0 0 2 4 6 generic β β = 0 W 2 = F (8) = F (8) Re W 2 = e φ F (8) Re W 2 = e φ F (8) + e φ F (8) 8 = 0 2 2 4 2 4 Im W 2 = 0 Im W 2 = 0 W 3 = ∓ ∗ 6 H (6) W 3 = H (6) = 0 6 3 3 W 5 = 2 W 4 = ∓ 2i H (3) F (3) = 2i W 5 = − 2i ∂A = 2i = ∂φ 3 ∂φ 3 2 3 ∂A = ∂a = 0 W 4 = 0 NS-flux only (common to IIA , IIB , heterotic) type A W 1 = W 2 = 0 , W 3 � = 0 : complex RR-flux only type BC W 1 = Im W 2 = W 3 = W 4 = 0 , Re W 2 � = 0 , W 5 � = 0 : symplectic For N = 1 AdS 4 vacua: hep-th/0403049, hep-th/0407263, hep-th/0412250, hep-th/0502154, hep-th/0609124, etc. SU (3) × SU (3) generalized geometries ( η 1 + � = η 2 + at some points) would complete the classification. (But, it’s quite hard to find all solutions.) Realization of AdS vacua in attractor mechanism on generalized geometries - 9 -

Motivation and Results Search 4D SUSY vacua in type IIA theory compactified on generalized geometries Moduli stabilization We find SUSY AdS (or Minkowski) vacua Mathematical feature We obtain a powerful rule to evaluate vacua: Discriminant of the superpotential governs the cosmological constant Stringy effects We see that α ′ corrections are included in certain configurations Realization of AdS vacua in attractor mechanism on generalized geometries - 10 -

Contents N = 1 scalar potential from generalized geometry Search of SUSY vacua Summary and discussions

N = 1 scalar potential from generalized geometry

What is a generalized geometry? Consider a compact space M 6 • Ordinary complex structure J mn in T M is given by SU (3) invariant Weyl spinor η + : J 2 = − 1 6 , n = − 2i η † J m + γ m n η + • Generalized complex structures J ΛΣ in T M ⊕ T ∗ M : basis { d x m ∧ , ι ∂ n } , (6 , 6) -signature � � J 2 J Λ Re Φ ± , Γ Λ ± = − 1 12 , ± Σ = Σ Re Φ ± SU (3 , 3) invariant Weyl spinors isomorphic to even/odd-forms on T ∗ M Φ ± : Γ Λ : (repr. = (d x m ∧ , ι ∂ n ) ) Cliff (6 , 6) gamma matrix � � even forms: Ψ + , Φ + = Ψ 6 ∧ Φ 0 − Ψ 4 ∧ Φ 2 + Ψ 2 ∧ Φ 4 − Ψ 0 ∧ Φ 6 Mukai pairing: � � = Ψ 5 ∧ Φ 1 − Ψ 3 ∧ Φ 3 + Ψ 1 ∧ Φ 5 odd forms: Ψ − , Φ − Realization of AdS vacua in attractor mechanism on generalized geometries - 12 -