The effective theory of type IIA AdS 4 compactifications on nilmanifolds and cosets Based on: 0804.0614 (PK, Tsimpis, Lüst), 0806.3458 (Caviezel, PK, Körs, Lüst, Tsimpis, Zagermann) Paul Koerber Max-Planck-Institut für Physik, Munich Zürich, 11 August 2008 1 / 24
Motivation Type IIB orientifold with D3/D7-branes on conformal CY: well-studied 2 / 24
Motivation Type IIB orientifold with D3/D7-branes on conformal CY: well-studied stabilization Kähler moduli through non-perturbative effects uplift susy vacuum to ds, models of inflation 2 / 24
Motivation Type IIB orientifold with D3/D7-branes on conformal CY: well-studied stabilization Kähler moduli through non-perturbative effects uplift susy vacuum to ds, models of inflation Type IIA compactifications with AdS 4 space-time 2 / 24
Motivation Type IIB orientifold with D3/D7-branes on conformal CY: well-studied stabilization Kähler moduli through non-perturbative effects uplift susy vacuum to ds, models of inflation Type IIA compactifications with AdS 4 space-time Early type IIA models on torus orientifolds: Derendinger et al., DeWolfe et al., Cámara et al. All moduli can be stabilized at tree level 2 / 24
Motivation Type IIB orientifold with D3/D7-branes on conformal CY: well-studied stabilization Kähler moduli through non-perturbative effects uplift susy vacuum to ds, models of inflation Type IIA compactifications with AdS 4 space-time Early type IIA models on torus orientifolds: Derendinger et al., DeWolfe et al., Cámara et al. All moduli can be stabilized at tree level Uplift susy vacuum to ds, models of inflation: problems 2 / 24
Motivation Type IIB orientifold with D3/D7-branes on conformal CY: well-studied stabilization Kähler moduli through non-perturbative effects uplift susy vacuum to ds, models of inflation Type IIA compactifications with AdS 4 space-time Early type IIA models on torus orientifolds: Derendinger et al., DeWolfe et al., Cámara et al. All moduli can be stabilized at tree level Uplift susy vacuum to ds, models of inflation: problems No-go theorem modular inflation: fluxes, D6/O6 Hertzberg, Kachru, Taylor, Tegmark 2 / 24
Motivation Type IIB orientifold with D3/D7-branes on conformal CY: well-studied stabilization Kähler moduli through non-perturbative effects uplift susy vacuum to ds, models of inflation Type IIA compactifications with AdS 4 space-time Early type IIA models on torus orientifolds: Derendinger et al., DeWolfe et al., Cámara et al. All moduli can be stabilized at tree level Uplift susy vacuum to ds, models of inflation: problems No-go theorem modular inflation: fluxes, D6/O6 Hertzberg, Kachru, Taylor, Tegmark Way-out: geometric fluxes, NS5-branes, non-geometric fluxes 2 / 24
Motivation Type IIB orientifold with D3/D7-branes on conformal CY: well-studied stabilization Kähler moduli through non-perturbative effects uplift susy vacuum to ds, models of inflation Type IIA compactifications with AdS 4 space-time Early type IIA models on torus orientifolds: Derendinger et al., DeWolfe et al., Cámara et al. All moduli can be stabilized at tree level Uplift susy vacuum to ds, models of inflation: problems No-go theorem modular inflation: fluxes, D6/O6 Hertzberg, Kachru, Taylor, Tegmark Way-out: geometric fluxes, NS5-branes, non-geometric fluxes 2 / 24
Motivation Type IIB orientifold with D3/D7-branes on conformal CY: well-studied stabilization Kähler moduli through non-perturbative effects uplift susy vacuum to ds, models of inflation Type IIA compactifications with AdS 4 space-time Early type IIA models on torus orientifolds: Derendinger et al., DeWolfe et al., Cámara et al. All moduli can be stabilized at tree level Uplift susy vacuum to ds, models of inflation: problems No-go theorem modular inflation: fluxes, D6/O6 Hertzberg, Kachru, Taylor, Tegmark Way-out: geometric fluxes, NS5-branes, non-geometric fluxes Geometric fluxes: deviation from Calabi-Yau 2 / 24
Motivation Type IIB orientifold with D3/D7-branes on conformal CY: well-studied stabilization Kähler moduli through non-perturbative effects uplift susy vacuum to ds, models of inflation Type IIA compactifications with AdS 4 space-time Early type IIA models on torus orientifolds: Derendinger et al., DeWolfe et al., Cámara et al. All moduli can be stabilized at tree level Uplift susy vacuum to ds, models of inflation: problems No-go theorem modular inflation: fluxes, D6/O6 Hertzberg, Kachru, Taylor, Tegmark Way-out: geometric fluxes, NS5-branes, non-geometric fluxes Geometric fluxes: deviation from Calabi-Yau Other application: type IIA on AdS 4 CP 3 AdS/CFT Aharony, Bergman, Jafferis, Maldacena M2-branes 2 / 24
Type IIA susy vacua with AdS 4 space-time Lüst, Tsimpis RR-forms: Romans mass F 0 = m, F 2, F 4, NSNS 3-form: H, dilaton: Φ SU(3)-structure: J, Ω 3 / 24
Type IIA susy vacua with AdS 4 space-time Lüst, Tsimpis Solution susy equations: 3 / 24
Type IIA susy vacua with AdS 4 space-time Lüst, Tsimpis Solution susy equations: Constant warp factor 3 / 24
Type IIA susy vacua with AdS 4 space-time Lüst, Tsimpis Solution susy equations: Constant warp factor Geometric flux i.e. non-zero torsion classes: dj = 3 2 Im(W1Ω ) + W 4 J + W 3 dω = W 1J J + W 2 J + W 5 Ω 3 / 24
Type IIA susy vacua with AdS 4 space-time Lüst, Tsimpis Solution susy equations: Constant warp factor Geometric flux i.e. non-zero torsion classes: dj = 3 2 Im(W1Ω ) dω = W 1J J + W 2 J with W 1 = 4i 9 eφ f W 2 = ie Φ F 2 3 / 24
Type IIA susy vacua with AdS 4 space-time Lüst, Tsimpis Solution susy equations: Constant warp factor Geometric flux i.e. non-zero torsion classes: dj = 3 2 Im(W1Ω ) dω = W 1J J + W 2 J with W 1 = 4i 9 eφ f W 2 = ie Φ F 2 Form-fluxes: AdS 4 superpotential W: H = 2m 5 eφ ReΩ F 2 = f 9 J + F 2 F 4 = fvol 4 + 3m 10 J J µζ = 1 2 Wγµζ+ definition We iθ = 1 5 eφ m + i 3 eφ f 3 / 24
Bianchi identities Automatically satisfied except for df 2 + HF 0 = j 6 4 / 24
Bianchi identities Automatically satisfied except for df 2 + HF 0 = j 6 Source j 6 (O6/D6) must be calibrated (here SLAG): j 6 J = 0 j 6 ReΩ = 0 4 / 24
Bianchi identities Automatically satisfied except for df 2 + HF 0 = j 6 Source j 6 (O6/D6) must be calibrated (here SLAG): j 6 J = 0 j 6 ReΩ = 0 j 6 = 2 5 e Φ µreω + w 3 w 3 simple (1,2)+(2,1) 4 / 24
Bianchi identities Automatically satisfied except for df 2 + HF 0 = j 6 Source j 6 (O6/D6) must be calibrated (here SLAG): j 6 J = 0 j 6 ReΩ = 0 j 6 = 2 5 e Φ µreω + w 3 w 3 simple (1,2)+(2,1) µ > 0: net orientifold charge, µ < 0: net D-brane charge 4 / 24
Bianchi identities Automatically satisfied except for df 2 + HF 0 = j 6 Source j 6 (O6/D6) must be calibrated (here SLAG): j 6 J = 0 j 6 ReΩ = 0 j 6 = 2 5 e Φ µreω + w 3 w 3 simple (1,2)+(2,1) µ > 0: net orientifold charge, µ < 0: net D-brane charge Bianchi: e 2Φ m 2 = µ + 5 ( 3 W1 2 W 2 2) 0 16 w 3 = ie Φ (2,1)+(1,2) dw 2 4 / 24
Smeared orientifolds Warp factor constant only smeared orientifolds 5 / 24
Smeared orientifolds Warp factor constant only smeared orientifolds Solutions without orientifolds: 5 / 24
Smeared orientifolds Warp factor constant only smeared orientifolds Solutions without orientifolds: possible! No-go theorem Maldacena,Núñez only for Minkowski, not AdS 4 5 / 24
Smeared orientifolds Warp factor constant only smeared orientifolds Solutions without orientifolds: possible! No-go theorem Maldacena,Núñez only for Minkowski, not AdS 4 We will introduce orientifolds to obtain N = 1 5 / 24
Smeared orientifolds Warp factor constant only smeared orientifolds Solutions without orientifolds: possible! No-go theorem Maldacena,Núñez only for Minkowski, not AdS 4 We will introduce orientifolds to obtain N = 1 Smearing: j 6 = T O6 δ(x 4, x 5, x 6 )dx 4 dx 5 dx 6 T Op c dx 4 dx 5 dx 6 5 / 24
Smeared orientifolds Warp factor constant only smeared orientifolds Solutions without orientifolds: possible! No-go theorem Maldacena,Núñez only for Minkowski, not AdS 4 We will introduce orientifolds to obtain N = 1 Smearing: j 6 = T O6 δ(x 4, x 5, x 6 )dx 4 dx 5 dx 6 T Op c dx 4 dx 5 dx 6 We will still associate orientifold involution: O6 : dx 4 dx 4, dx 5 dx 5, dx 6 dx 6 5 / 24
Smeared orientifolds Warp factor constant only smeared orientifolds Solutions without orientifolds: possible! No-go theorem Maldacena,Núñez only for Minkowski, not AdS 4 We will introduce orientifolds to obtain N = 1 Smearing: j 6 = T O6 δ(x 4, x 5, x 6 )dx 4 dx 5 dx 6 T Op c dx 4 dx 5 dx 6 We will still associate orientifold involution: O6 : dx 4 dx 4, dx 5 dx 5, dx 6 dx 6 Write j 6 as sum of decomposable forms 5 / 24
Smeared orientifolds Warp factor constant only smeared orientifolds Solutions without orientifolds: possible! No-go theorem Maldacena,Núñez only for Minkowski, not AdS 4 We will introduce orientifolds to obtain N = 1 Smearing: j 6 = T O6 δ(x 4, x 5, x 6 )dx 4 dx 5 dx 6 T Op c dx 4 dx 5 dx 6 We will still associate orientifold involution: O6 : dx 4 dx 4, dx 5 dx 5, dx 6 dx 6 Write j 6 as sum of decomposable forms B, H,F 2,ReΩ odd, F 0, F 4,ImΩ even 5 / 24
Calabi-Yau solution Calabi-Yau solution Acharya, Benini, Valandro j 6 = 2 5 e Φ µreω + w 3 e 2Φ m 2 = µ + 5 ( 3 W1 2 W 2 2) 16 6 / 24
Calabi-Yau solution Calabi-Yau solution Acharya, Benini, Valandro Put W 1 = 0, W 2 = 0 w 3 = 0 j 6 = 2 5 e Φ µreω + w 3 e 2Φ m 2 = µ + 5 ( 3 W1 2 W 2 2) 16 6 / 24
Calabi-Yau solution Calabi-Yau solution Acharya, Benini, Valandro Put W 1 = 0, W 2 = 0 w 3 = 0 j 6 = 2 5 e Φ µreω e 2Φ m 2 = µ 6 / 24
Calabi-Yau solution Calabi-Yau solution Acharya, Benini, Valandro Put W 1 = 0, W 2 = 0 w 3 = 0 j 6 = 2 5 e Φ µreω e 2Φ m 2 = µ Torus orientifolds 6 / 24
What about equations of motion? IIA: Lüst,Tsimpis, IIB: Gauntlett, Martelli, Sparks, Waldram With sources: PK, Tsimpis 0706.1244 Under mild conditions (subtleties time direction): 7 / 24
What about equations of motion? IIA: Lüst,Tsimpis, IIB: Gauntlett, Martelli, Sparks, Waldram With sources: PK, Tsimpis 0706.1244 Under mild conditions (subtleties time direction): Bulk supersymmetry conditions Bianchi identities form-fields with source Supersymmetry conditions source = generalized calibration conditions PK; Martucci, Smyth 7 / 24
What about equations of motion? IIA: Lüst,Tsimpis, IIB: Gauntlett, Martelli, Sparks, Waldram With sources: PK, Tsimpis 0706.1244 Under mild conditions (subtleties time direction): imply Bulk supersymmetry conditions Bianchi identities form-fields with source Supersymmetry conditions source = generalized calibration conditions PK; Martucci, Smyth Einstein equations with source Dilaton equation of motion with source Form field equations of motion 7 / 24
Some older solutions Nearly-Kähler: only W 1 0 Behrndt, Cveti c SU(2) SU(2) and the coset spaces G 2 SU(3), Sp(2) S(U(2) U(1)), SU(3) U(1) U(1) 8 / 24
Some older solutions Nearly-Kähler: only W 1 0 Behrndt, Cveti c SU(2) SU(2) and the coset spaces Iwasawa manifold Lüst, Tsimpis: G 2 SU(3), Sp(2) S(U(2) U(1)), A certain nilmanifold or twisted torus = a group manifold associated to a nilpotent algebra Left-invariant forms e i obeying Maurer-Cartan relation: de i = 1 2 fi jke j e k Singular limit of T 2 bundle over K3 SU(3) U(1) U(1) 8 / 24
Group manifolds SU(3)-structures with J,Ω constant in terms of left-invariant forms on a group manifold: fertile source of examples algebraic relations de i = 1 2 fi jke j e k 9 / 24
Group manifolds SU(3)-structures with J,Ω constant in terms of left-invariant forms on a group manifold: fertile source of examples algebraic relations SU(2) SU(2) de i = 1 2 fi jke j e k Nilmanifolds (divide discrete group: compact) Only Iwasawa & torus Needs smeared orientifolds Solvmanifolds (compact?): no solution Extend to cosets 9 / 24
Coset manifolds Tomasiello: W 2 0 on two examples PK, Lüst, Tsimpis: classification Sp(2) S(U(2) U(1)), SU(3) U(1) U(1) 10 / 24
Coset manifolds Tomasiello: W 2 0 on two examples PK, Lüst, Tsimpis: classification Maurer-Cartan more complicated: Sp(2) S(U(2) U(1)), de i = 1 2 fi jke j e k f i ajω a e j SU(3) U(1) U(1) 10 / 24
Coset manifolds Tomasiello: W 2 0 on two examples PK, Lüst, Tsimpis: classification Maurer-Cartan more complicated: Sp(2) S(U(2) U(1)), de i = 1 2 fi jke j e k f i ajω a e j SU(3) U(1) U(1) Condition left-invariance p-form φ: constant and components f j a[i 1 φ i2...i p]j = 0 10 / 24
Type IIA AdS 4 susy vacua on coset manifolds PK, Lüst, Tsimpis 0804.0614 SU(2) SU(2) SU(3) U(1) U(1) Sp(2) S(U(2) U(1)) G 2 SU(3) SU(3) U(1) SU(2) # of parameters 2 4 4 3 2 4 W 2 0 No Yes Yes Yes No Yes j 6 ReΩ Yes No Yes Yes Yes No Note: geometric flux: W 1 0, W 2 0 11 / 24
Type IIA AdS 4 susy vacua on coset manifolds PK, Lüst, Tsimpis 0804.0614 SU(2) SU(2) SU(3) U(1) U(1) Sp(2) S(U(2) U(1)) G 2 SU(3) SU(3) U(1) SU(2) # of parameters 2 4 4 3 2 4 W 2 0 No Yes Yes Yes No Yes j 6 ReΩ Yes No Yes Yes Yes No Note: geometric flux: W 1 0, W 2 0 Parameters: Geometric: scale and shape J = ae 12 + be 34 + ce 56 Scale a and shape ρ = b/a, σ = c/a: Orientifold charge µ 11 / 24
Generalization SU(3) SU(3) susy ansatz: 10d 4d for IIA/IIB ǫ 1 =ζ + η (1) + + (c.c.) ǫ 2 =ζ + η (2) + (c.c.) 12 / 24
Generalization SU(3) SU(3) susy ansatz: 10d 4d for IIA/IIB ǫ 1 =ζ + η (1) + + (c.c.) ǫ 2 =ζ + η (2) + (c.c.) Strict SU(3)-structure: η (2) + = e iθ η (1) + Type: (0,3) 12 / 24
Generalization SU(3) SU(3) susy ansatz: 10d 4d for IIA/IIB ǫ 1 =ζ + η (1) + + (c.c.) ǫ 2 =ζ + η (2) + (c.c.) Strict SU(3)-structure: η (2) + = e iθ η (1) + Type: (0,3) Static SU(2)-structure: η (1), η (2) orthogonal everywhere Type: (2,1) 12 / 24
Generalization SU(3) SU(3) susy ansatz: 10d 4d for IIA/IIB ǫ 1 =ζ + η (1) + + (c.c.) ǫ 2 =ζ + η (2) + (c.c.) Strict SU(3)-structure: η (2) + = e iθ η (1) + Type: (0,3) Static SU(2)-structure: η (1), η (2) orthogonal everywhere Type: (2,1) intermediate SU(2)-structure: η (1), η (2) fixed angle, but neither a zero angle nor a right angle Type: (0,1) 12 / 24
Generalization SU(3) SU(3) susy ansatz: 10d 4d for IIA/IIB ǫ 1 =ζ + η (1) + + (c.c.) ǫ 2 =ζ + η (2) + (c.c.) Strict SU(3)-structure: η (2) + = e iθ η (1) + Type: (0,3) Static SU(2)-structure: η (1), η (2) orthogonal everywhere Type: (2,1) intermediate SU(2)-structure: η (1), η (2) fixed angle, but neither a zero angle nor a right angle Type: (0,1) dynamic SU(3) SU(3)-structure: angle changes, type may change 12 / 24
SU(3) SU(3)-structure susy equations Graña, Minasian, Petrini, Tomasiello with d H ( e 4A Φ ImΨ 1 ) = 3e 3A Φ Im(W Ψ 2 ) + e 4A F d H [ e 3A Φ Re(W Ψ 2 ) ] = 2 W 2 e 2A Φ ReΨ 1 d H [ e 3A Φ Im(W Ψ 2 ) ] = 0 d H ( e 2A Φ ReΨ 1 ) = 0. d H = d + H Ψ 1 = Ψ, Ψ 2 = Ψ ± /Ψ + = 8 η (1) η (2) η(1) + η(2) +, /Ψ = 8 η (1) η (2) η(1) + η(2). 13 / 24
SU(3) SU(3)-structure susy equations Graña, Minasian, Petrini, Tomasiello ( d H e 4A Φ ) ImΨ 1 = 3e 3A Φ Im(W Ψ 2 ) + e 4A F [ d H e 3A Φ Re(W Ψ 2 ) ] = 2 W 2 e 2A Φ ReΨ 1 [ d H e 3A Φ Im(W Ψ 2 ) ] = 0 ( d H e 2A Φ ) ReΨ 1 = 0. Example: strict SU(3)-structure in IIA Ψ 1 = Ψ = Ω, Ψ 2 = Ψ + = e iθ e ij leads to susy equations Lüst, Tsimpis 13 / 24
SU(3) SU(3)-structure susy equations Graña, Minasian, Petrini, Tomasiello ( d H e 4A Φ ) ImΨ 1 = 3e 3A Φ Im(W Ψ 2 ) + e 4A F [ d H e 3A Φ Re(W Ψ 2 ) ] = 2 W 2 e 2A Φ ReΨ 1 [ d H e 3A Φ Im(W Ψ 2 ) ] = 0 ( d H e 2A Φ ) ReΨ 1 = 0. 13 / 24
Generalizations: no-go theorems d H [ e 3A Φ Re(W Ψ 2 ) ] = 2 W 2 e 2A Φ ReΨ 1 14 / 24
Generalizations: no-go theorems d H [ e 3A Φ Re(W Ψ 2 ) ] = 2 W 2 e 2A Φ ReΨ 1 Strict SU(3)-structure in IIB: (Ψ 1, Ψ 2 ) = (Ψ +, Ψ ) 14 / 24
Generalizations: no-go theorems d H [ e 3A Φ Re(W Ψ 2 ) ] = 2 W 2 e 2A Φ ReΨ 1 Strict SU(3)-structure in IIB: (Ψ 1, Ψ 2 ) = (Ψ +, Ψ ) Type: (0,3) Ψ 2 3-form ReΨ 1 0 = ReΨ 1 2 = 0 14 / 24
Generalizations: no-go theorems d H [ e 3A Φ Re(W Ψ 2 ) ] = 2 W 2 e 2A Φ ReΨ 1 Strict SU(3)-structure in IIB: (Ψ 1, Ψ 2 ) = (Ψ +, Ψ ) Type: (0,3) Ψ 2 3-form ReΨ 1 0 = ReΨ 1 2 = 0 But we also need Ψ 1, Ψ 1 = 8ivol 0 with the Mukai pairing φ 1, φ 2 = φ 1 α(φ 2 ) top and α reverses the indices of a form 14 / 24
Generalizations: no-go theorems d H [ e 3A Φ Re(W Ψ 2 ) ] = 2 W 2 e 2A Φ ReΨ 1 Strict SU(3)-structure in IIB: (Ψ 1, Ψ 2 ) = (Ψ +, Ψ ) Type: (0,3) Ψ 2 3-form ReΨ 1 0 = ReΨ 1 2 = 0 But we also need Ψ 1, Ψ 1 = 8ivol 0 impossible 14 / 24
Generalizations: no-go theorems d H [ e 3A Φ Re(W Ψ 2 ) ] = 2 W 2 e 2A Φ ReΨ 1 Strict SU(3)-structure in IIB: (Ψ 1, Ψ 2 ) = (Ψ +, Ψ ) Type: (0,3) Ψ 2 3-form ReΨ 1 0 = ReΨ 1 2 = 0 But we also need Ψ 1, Ψ 1 = 8ivol 0 impossible Intermediate SU(2)-structure: 14 / 24
Generalizations: no-go theorems d H [ e 3A Φ Re(W Ψ 2 ) ] = 2 W 2 e 2A Φ ReΨ 1 Strict SU(3)-structure in IIB: (Ψ 1, Ψ 2 ) = (Ψ +, Ψ ) Type: (0,3) Ψ 2 3-form ReΨ 1 0 = ReΨ 1 2 = 0 But we also need Ψ 1, Ψ 1 = 8ivol 0 impossible Intermediate SU(2)-structure: Type: (0,1) general form e 2A Φ Ψ 1 = ce iω+b 14 / 24
Generalizations: no-go theorems d H [ e 3A Φ Re(W Ψ 2 ) ] = 2 W 2 e 2A Φ ReΨ 1 Strict SU(3)-structure in IIB: (Ψ 1, Ψ 2 ) = (Ψ +, Ψ ) Type: (0,3) Ψ 2 3-form ReΨ 1 0 = ReΨ 1 2 = 0 But we also need Ψ 1, Ψ 1 = 8ivol 0 impossible Intermediate SU(2)-structure: Type: (0,1) general form e 2A Φ Ψ 1 = ce iω+b Rec = 0, cω exact 14 / 24
Generalizations: no-go theorems d H [ e 3A Φ Re(W Ψ 2 ) ] = 2 W 2 e 2A Φ ReΨ 1 Strict SU(3)-structure in IIB: (Ψ 1, Ψ 2 ) = (Ψ +, Ψ ) Type: (0,3) Ψ 2 3-form ReΨ 1 0 = ReΨ 1 2 = 0 But we also need Ψ 1, Ψ 1 = 8ivol 0 impossible Intermediate SU(2)-structure: Type: (0,1) general form e 2A Φ Ψ 1 = ce iω+b Rec = 0, cω exact Imc e 2A Φ Ψ 1, e 2A Φ, Ψ 1 = 8 3! (cω)3 14 / 24
Generalizations: no-go theorems d H [ e 3A Φ Re(W Ψ 2 ) ] = 2 W 2 e 2A Φ ReΨ 1 Strict SU(3)-structure in IIB: (Ψ 1, Ψ 2 ) = (Ψ +, Ψ ) Type: (0,3) Ψ 2 3-form ReΨ 1 0 = ReΨ 1 2 = 0 But we also need Ψ 1, Ψ 1 = 8ivol 0 impossible Intermediate SU(2)-structure: Type: (0,1) general form e 2A Φ Ψ 1 = ce iω+b Rec = 0, cω exact Imc e 2A Φ Ψ 1, e 2A Φ, Ψ 1 = 8 3! (cω)3 c not everywhere non-vanishing 14 / 24
Generalizations: conclusion of no-go theorems Type IIB: only static SU(2) or dynamic SU(3) SU(3) that changes type to static SU(2) at least somewhere is possible 15 / 24
Generalizations: conclusion of no-go theorems Type IIB: only static SU(2) or dynamic SU(3) SU(3) that changes type to static SU(2) at least somewhere is possible Example: type IIB static SU(2) on nilmanifold 5.1 (table of Graña, Minasian, Petrini, Tomasiello) 15 / 24
Generalizations: conclusion of no-go theorems Type IIB: only static SU(2) or dynamic SU(3) SU(3) that changes type to static SU(2) at least somewhere is possible Example: type IIB static SU(2) on nilmanifold 5.1 (table of Graña, Minasian, Petrini, Tomasiello) Type IIA: only strict SU(3) or dynamic SU(3) SU(3) for which d ( e 2A Φ η (2) η (1)) 0 15 / 24
Low energy effective theory: nilmanifolds Torus (IIA), nilmanifold 5.1 (IIB), Iwasawa (IIA): T-dual to each other Calculation mass spectrum Using effective 4D sugra techniques For Iwasawa & torus: direct KK reduction of equations of motion Perfect agreement 16 / 24
Low energy effective theory: nilmanifolds Torus (IIA), nilmanifold 5.1 (IIB), Iwasawa (IIA): T-dual to each other Calculation mass spectrum Using effective 4D sugra techniques For Iwasawa & torus: direct KK reduction of equations of motion Perfect agreement Result for M 2 / W 2 Complex structure 2, 2, 2 Kähler & dilaton 70, 18, 18, 18 Three axions of δc 3 0, 0, 0 δb & one more axion 88, 10, 10, 10 16 / 24
Issue: decoupling KK tower Consistency approximation: g s 1, l s /L int 1 17 / 24
Issue: decoupling KK tower Consistency approximation: g s 1, l s /L int 1 17 / 24
Issue: decoupling KK tower Consistency approximation: g s 1, l s /L int 1 Decoupling KK modes: W 2 L 2 int = e2φ m 2 L 2 int 25 + e2φ f 2 L 2 int 9 1 17 / 24
Issue: decoupling KK tower Consistency approximation: g s 1, l s /L int 1 Decoupling KK modes: W 2 L 2 int = e2φ m 2 L 2 int 25 + e2φ f 2 L 2 int 9 1 Can be tuned with orientifold charge µ: e 2Φ m 2 = µ + 5 ( 3 W1 2 W 2 2) 0 16 17 / 24
Issue: decoupling KK tower Consistency approximation: g s 1, l s /L int 1 Decoupling KK modes: e Φ fl int W 1L int 1: W 2 L 2 int = e2φ m 2 L 2 int 25 + e2φ f 2 L 2 int 9 1 17 / 24
Issue: decoupling KK tower Consistency approximation: g s 1, l s /L int 1 Decoupling KK modes: e Φ fl int W 1L int 1: nearly Calabi-Yau W 2 L 2 int = e2φ m 2 L 2 int 25 + e2φ f 2 L 2 int 9 1 17 / 24
Issue: decoupling KK tower Consistency approximation: g s 1, l s /L int 1 Decoupling KK modes: W 2 L 2 int = e2φ m 2 L 2 int 25 + e2φ f 2 L 2 int 9 1 e Φ fl int W 1L int 1: nearly Calabi-Yau Can be arranged for torus, Iwasawa, nilmanifold 5.1 17 / 24
Issue: decoupling KK tower Consistency approximation: g s 1, l s /L int 1 Decoupling KK modes: W 2 L 2 int = e2φ m 2 L 2 int 25 + e2φ f 2 L 2 int 9 1 e Φ fl int W 1L int 1: nearly Calabi-Yau Can be arranged for torus, Iwasawa, nilmanifold 5.1 Harder for coset examples 17 / 24
Low energy theory: superpotential and Kähler potential We use the superpotential of Graña, Louis, Waldram; Benmachiche, Grimm; PK, Martucci Generalizes Gukov, Vafa, Witten 18 / 24
Low energy theory: superpotential and Kähler potential We use the superpotential of Graña, Louis, Waldram; Benmachiche, Grimm; PK, Martucci Generalizes Gukov, Vafa, Witten W E = i 4κ 2 10 M Z, T holomorphic coordinates, Ψ 2 e } {{ δb, } ˆF + idĥ(e δb e Φ ImΨ 1 iδc) } {{ } Z T 18 / 24
Low energy theory: superpotential and Kähler potential We use the superpotential of Graña, Louis, Waldram; Benmachiche, Grimm; PK, Martucci Generalizes Gukov, Vafa, Witten W E = i 4κ 2 10 M Ψ 2 e } {{ δb, } ˆF + idĥ(e δb e Φ ImΨ 1 iδc) } {{ } Z T Z, T holomorphic coordinates, and the Kähler potential K = lni Z, Z 2ln i e Φ Ψ 1,e Φ Ψ1 + 3ln(8κ 2 10MP) 2 M where e Φ e δb Ψ 1 : function of ReT = e Φ e δb ImΨ 1 Hitchin. M 18 / 24
Superpotential and Kähler potential: SU(3) Specialized to SU(3): Ψ 1 = Ω, Ψ 2 = e iθ e ij 19 / 24
Superpotential and Kähler potential: SU(3) Specialized to SU(3): Ψ 1 = Ω, Superpotential: W E = ie iθ 4κ 2 10 M Ψ 2 = e iθ e ij e i(j iδb), ˆF ( idĥ e Φ ) ImΩ + iδc 3 19 / 24
Superpotential and Kähler potential: SU(3) Specialized to SU(3): Ψ 1 = Ω, Superpotential: W E = ie iθ 4κ 2 10 M Ψ 2 = e iθ e ij e i(j iδb), ˆF ( idĥ e Φ ) ImΩ + iδc 3 Kähler potential: 4 K = ln 3 J3 2ln M M 2e Φ ImΩ e Φ ReΩ+3ln(8κ 2 10M 2 P), 19 / 24
Superpotential and Kähler potential: SU(3) Specialized to SU(3): Ψ 1 = Ω, Superpotential: W E = ie iθ 4κ 2 10 M Ψ 2 = e iθ e ij e i(j iδb), ˆF ( idĥ e Φ ) ImΩ + iδc 3 Kähler potential: 4 K = ln 3 J3 2ln Expansion: M M 2e Φ ImΩ e Φ ReΩ+3ln(8κ 2 10M 2 P), J c = J iδb = (k i ib i )Y (2 ) i e Φ ImΩ + iδc 3 = (u i + ic i )e = t i Y (2 ) i ˆΦY (3+) i = z i e ˆΦY (3+) i 19 / 24
Comments We used the same superpotential and Kähler potential for static SU(2) 20 / 24
Comments We used the same superpotential and Kähler potential for static SU(2) F-flatness conditions: D i W E = i W E + i K W E = 0 20 / 24
Comments We used the same superpotential and Kähler potential for static SU(2) F-flatness conditions: PK, Martucci: D i W E = i W E + i K W E = 0 If W = M 1 P W E 0 i.e. AdS 4 compactification F-flatness conditions 10D susy conditions vacuum 20 / 24
Comments We used the same superpotential and Kähler potential for static SU(2) F-flatness conditions: PK, Martucci: D i W E = i W E + i K W E = 0 If W = M 1 P W E 0 i.e. AdS 4 compactification F-flatness conditions 10D susy conditions vacuum If W = 0 i.e. Minkowski compactification F-flatness & D-flatness conditions 10D susy conditions vacuum 20 / 24
Comments We used the same superpotential and Kähler potential for static SU(2) F-flatness conditions: PK, Martucci: D i W E = i W E + i K W E = 0 If W = M 1 P W E 0 i.e. AdS 4 compactification F-flatness conditions 10D susy conditions vacuum If W = 0 i.e. Minkowski compactification F-flatness & D-flatness conditions 10D susy conditions vacuum The scalar potential is V = M 2 P ek ( K i j D i W E D j W E 3 W E 2) 20 / 24
Comments We used the same superpotential and Kähler potential for static SU(2) F-flatness conditions: PK, Martucci: D i W E = i W E + i K W E = 0 If W = M 1 P W E 0 i.e. AdS 4 compactification F-flatness conditions 10D susy conditions vacuum If W = 0 i.e. Minkowski compactification F-flatness & D-flatness conditions 10D susy conditions vacuum The scalar potential is V = M 2 P ek ( K i j D i W E D j W E 3 W E 2) Mass spectrum from quadratic terms 20 / 24
Low energy theory: cosets For all cosets (but not for SU(2) SU(2)): all moduli stabilized at tree level 21 / 24
Low energy theory: cosets For all cosets (but not for SU(2) SU(2)): all moduli stabilized at tree level Example Sp(2) S(U(2) U(1)) M 2 / W 2 25 M 2 / W 2 25 20 20 15 15 10 10 5 2 4 6 8 10 µ 5 2 4 6 8 10 12 µ (a) σ = 1 (b) σ = 2 5 Movie 21 / 24
Low energy theory: cosets For all cosets (but not for SU(2) SU(2)): all moduli stabilized at tree level Example Sp(2) S(U(2) U(1)) M 2 / W 2 25 M 2 / W 2 25 20 20 15 15 10 10 5 2 4 6 8 10 µ 5 2 4 6 8 10 12 µ (a) σ = 1 (b) σ = 2 5 Movie µ big enough: all mass-squared positive 21 / 24
Nearly-Calabi Yau limit Important decoupling KK-modes 22 / 24
Nearly-Calabi Yau limit Important decoupling KK-modes Sp(2) Example S(U(2) U(1)) analytic continuation to negative σ = 2 Twistor bundle description more appropriate Xu 22 / 24
Nearly-Calabi Yau limit Important decoupling KK-modes Sp(2) Example S(U(2) U(1)) analytic continuation to negative σ = 2 Twistor bundle description more appropriate Xu 120 M 2 / W 2 100 80 60 40 20 10 20 30 40 50 µ 22 / 24
Nearly-Calabi Yau limit Important decoupling KK-modes Sp(2) Example S(U(2) U(1)) analytic continuation to negative σ = 2 Twistor bundle description more appropriate Xu 120 M 2 / W 2 100 80 60 40 20 10 20 30 40 50 µ Light modes: M2 / W 2 = ( 38/49, 130/49) 22 / 24
Inflation No-go theorem modular inflation IIA Hertzberg, Kachru, Taylor, Tegmark Dependence on volume-modulus ρ and dilaton-modulus τ Ingredients: form-fluxes, D6-branes & O6-planes ǫ > 27/13 23 / 24
Inflation No-go theorem modular inflation IIA Hertzberg, Kachru, Taylor, Tegmark Dependence on volume-modulus ρ and dilaton-modulus τ Ingredients: form-fluxes, D6-branes & O6-planes ǫ > 27/13 Way out: geometric flux, NS5-branes, other branes, non-geometric flux 23 / 24
Inflation No-go theorem modular inflation IIA Hertzberg, Kachru, Taylor, Tegmark Dependence on volume-modulus ρ and dilaton-modulus τ Ingredients: form-fluxes, D6-branes & O6-planes ǫ > 27/13 Way out: geometric flux, NS5-branes, other branes, non-geometric flux potential geometric flux V f > 0 R < 0 23 / 24
Inflation No-go theorem modular inflation IIA Hertzberg, Kachru, Taylor, Tegmark Dependence on volume-modulus ρ and dilaton-modulus τ Ingredients: form-fluxes, D6-branes & O6-planes ǫ > 27/13 Way out: geometric flux, NS5-branes, other branes, non-geometric flux potential geometric flux V f > 0 R < 0 not possible for, possible for all other cosets and SU(2) SU(2) G 2 SU(3) 23 / 24
Conclusions Tractable models with geometric flux: nilmanifold & coset models Other models: e.g. twistor bundles with negative σ Uplift: uplifting term or look for ds minimum e.g. Silverstein Study inflation 24 / 24
Conclusions Tractable models with geometric flux: nilmanifold & coset models Other models: e.g. twistor bundles with negative σ Uplift: uplifting term or look for ds minimum e.g. Silverstein Study inflation The end...the end...the end... 24 / 24