Smeared versus localized sources in flux compactifications Smeared - PowerPoint PPT Presentation

Smeared versus localized sources in flux compactifications Smeared vs. localized sources Timm Wrase Timm Wrase Flux compact. BPS case non-BPS case Localiz. effects Conclusion Based on: TW, Zagermann 1003.0029 Bl˚ ab¨ ack, Danielsson, Junghans, Van Riet, TW, Zagermann 1009.1877 String Vacuum Project meeting Fall 2010 1 / 16

Classical type II flux compactifications Most constructions of dS vacua use non-perturbative effects for moduli stabilization Smeared vs. localized sources dS after uplift which breaks explicitly SUSY Timm Wrase KKLT, Large Volume Flux compact. BPS case non-BPS case Localiz. effects Conclusion 2 / 16

Classical type II flux compactifications Most constructions of dS vacua use non-perturbative effects for moduli stabilization Smeared vs. localized sources dS after uplift which breaks explicitly SUSY Timm Wrase KKLT, Large Volume Flux compact. It is in principle possible to stabilize all moduli classically BPS case Villadoro, Zwirner hep-th/0503169 non-BPS case DeWolfe, Giryavets, Kachru, Taylor hep-th/0505160 Localiz. effects C´ amara, Font, Ib´ a˜ nez hep-th/0506066 Conclusion 2 / 16

Classical type II flux compactifications Most constructions of dS vacua use non-perturbative effects for moduli stabilization Smeared vs. localized sources dS after uplift which breaks explicitly SUSY Timm Wrase KKLT, Large Volume Flux compact. It is in principle possible to stabilize all moduli classically BPS case Villadoro, Zwirner hep-th/0503169 non-BPS case DeWolfe, Giryavets, Kachru, Taylor hep-th/0505160 Localiz. effects C´ amara, Font, Ib´ a˜ nez hep-th/0506066 Conclusion 2 / 16

Classical type II flux compactifications Most constructions of dS vacua use non-perturbative effects for moduli stabilization Smeared vs. localized sources dS after uplift which breaks explicitly SUSY Timm Wrase KKLT, Large Volume Flux compact. It is in principle possible to stabilize all moduli classically BPS case Villadoro, Zwirner hep-th/0503169 non-BPS case DeWolfe, Giryavets, Kachru, Taylor hep-th/0505160 Localiz. effects C´ amara, Font, Ib´ a˜ nez hep-th/0506066 Conclusion Can we find classical dS vacua? Hertzberg, Tegmark, Kachru, Shelton, Ozcan 0709.0002 [astro-ph] 2 / 16

Type II supergravity The classical ingredients for type II supergravity theories are RR-fluxes F p , NSNS H -flux, R 6 , Oq -planes, ... Smeared vs. localized sources Timm Wrase Flux compact. BPS case non-BPS case Localiz. effects Conclusion 3 / 16

Type II supergravity The classical ingredients for type II supergravity theories are RR-fluxes F p , NSNS H -flux, R 6 , Oq -planes, ... Smeared vs. localized sources Timm Wrase For smeared Oq -planes we find a 4D scalar potential Flux compact. BPS case � V ( ρ, φ, . . . ) = V F p + V H + V R 6 − V Oq , non-BPS case Localiz. effects p Conclusion where ρ = ( vol 6 ) 1 / 3 and φ is the dilaton. 3 / 16

Type II supergravity The classical ingredients for type II supergravity theories are RR-fluxes F p , NSNS H -flux, R 6 , Oq -planes, ... Smeared vs. localized sources Timm Wrase For smeared Oq -planes we find a 4D scalar potential Flux compact. BPS case � V ( ρ, φ, . . . ) = V F p + V H + V R 6 − V Oq , non-BPS case Localiz. effects p Conclusion where ρ = ( vol 6 ) 1 / 3 and φ is the dilaton. When is ∂ ρ V = ∂ φ V = 0 and V > 0 possible? Hertzberg, Kachru, Taylor, Tegmark 0711.2512 [hep-th] 3 / 16

We can evade a no-go theorem involving ρ and φ with the following minimal ingredients Curvature IIA IIB V R 6 ∼ − R 6 ≤ 0 O4, H , F 0 O3, H , F 1 Smeared vs. localized sources O3, F 1 O4, F 0 Timm Wrase O3, F 3 V R 6 ∼ − R 6 > 0 O4, F 2 O3, F 5 Flux compact. O6, F 0 BPS case O5, F 1 non-BPS case Localiz. effects Conclusion Hertzberg, Kachru, Taylor, Tegmark 0711.2512 [hep-th] Silverstein 0712.1196 [hep-th] Haque, Shiu, Underwood, Van Riet 0810.5328 [hep-th] Caviezel, Koerber, K¨ ors, L¨ ust, TW, M. Zagermann 0812.3551 [hep-th] Flauger, Robbins, Paban, TW 0812.3886 [hep-th] Danielsson, Haque, Shiu, Van Riet 0907.2041 [hep-th] de Carlos, Guarino, Moreno 0907.5580, 0911.2876 [hep-th] Caviezel, TW, Zagermann 0912.3287 [hep-th] TW, Zagermann 1003.0029 [hep-th] Danielsson, Koerber, Van Riet 1003.3590 [hep-th] Danielsson, Haque, Koerber, Shiu, Van Riet, TW 1011.xxxx [hep-th] 4 / 16

Smeared versus localized sources O-planes are localized objects Smeared vs. Smearing was necessary to solve equations of motion localized sources Timm Wrase Flux compact. BPS case non-BPS case Localiz. effects Conclusion 5 / 16

Smeared versus localized sources O-planes are localized objects Smeared vs. Smearing was necessary to solve equations of motion localized sources Timm Wrase Flux compact. BPS case non-BPS case When is smearing δ ( Oq ) ≈ 1 a valid approximation? Localiz. effects Conclusion 5 / 16

Smeared versus localized sources O-planes are localized objects Smeared vs. Smearing was necessary to solve equations of motion localized sources Timm Wrase Flux compact. BPS case non-BPS case When is smearing δ ( Oq ) ≈ 1 a valid approximation? Localiz. effects Conclusion Negative curvature R 6 < 0 requires (in the localized case) large warping or large stringy corrections Douglas, Kallosh 1001.4008 [hep-th] 5 / 16

An example with BPS sources Giddings, Kachru and Polchinski found localized no-scale Minkowski solutions with O3-planes Smeared vs. localized sources Giddings, Kachru, Polchinski hep-th/0105097 Timm Wrase Flux compact. BPS case non-BPS case Localiz. effects Conclusion 6 / 16

An example with BPS sources Giddings, Kachru and Polchinski found localized no-scale Minkowski solutions with O3-planes Smeared vs. localized sources Giddings, Kachru, Polchinski hep-th/0105097 Timm Wrase Flux compact. BPS case non-BPS case smeared case Localiz. effects H, F 3 , O3 Conclusion ds 2 = ds 2 4 + ds 2 6 0 = d F 5 = H ∧ F 3 − ˜ µ 3 6 / 16

An example with BPS sources Giddings, Kachru and Polchinski found localized no-scale Minkowski solutions with O3-planes Smeared vs. localized sources Giddings, Kachru, Polchinski hep-th/0105097 Timm Wrase Flux compact. BPS case non-BPS case smeared case localized case Localiz. effects H, F 3 , O3 H, F 3 , O3, F 5 , A Conclusion ds 2 = ds 2 ds 2 = e 2 A ds 2 4 + ds 2 4 + e − 2 A ds 2 6 6 0 = d F 5 = H ∧ F 3 − ˜ d F 5 = H ∧ F 3 − ˜ µ 3 δ ( O 3) µ 3 6 / 16

An example with BPS sources Can solve the 10D equations of motions in both cases Smeared vs. localized sources Find no-scale Minkowski vacua Timm Wrase Flux compact. Internal space is (conformally) Ricci-flat BPS case non-BPS case Localiz. effects Conclusion 7 / 16

An example with BPS sources Can solve the 10D equations of motions in both cases Smeared vs. localized sources Find no-scale Minkowski vacua Timm Wrase Flux compact. Internal space is (conformally) Ricci-flat BPS case non-BPS case Localiz. effects Conclusion But localization effects are large ∇ 2 e − 4 A = − e − φ | H | 2 + ˜ µ 3 δ ( O 3) 7 / 16

An example with BPS sources Can solve the 10D equations of motions in both cases Smeared vs. localized sources Find no-scale Minkowski vacua Timm Wrase Flux compact. Internal space is (conformally) Ricci-flat BPS case non-BPS case Complex structure moduli and φ are stabilized Localiz. effects Conclusion smeared case localized case F 3 = − e − φ ⋆ 6 H F 3 = − e − φ ⋆ 6 H 7 / 16

An example with BPS sources BUT F 3 = − e − φ ⋆ 6 H = − e − φ ⋆ 6 H Smeared vs. localized sources � 3 H e − 2 A g − 1 � � since warp factor cancels: ⋆ 6 H ≈ det (e 2 A g 6 ) Timm Wrase 6 Flux compact. BPS case non-BPS case Localiz. effects Conclusion 8 / 16

An example with BPS sources BUT F 3 = − e − φ ⋆ 6 H = − e − φ ⋆ 6 H Smeared vs. localized sources � 3 H e − 2 A g − 1 � � since warp factor cancels: ⋆ 6 H ≈ det (e 2 A g 6 ) Timm Wrase 6 Flux compact. BPS case Moduli values at minimum unchanged! non-BPS case Localiz. effects Approximation δ ( O 3) ≈ 1 is “ok” Conclusion 8 / 16

An example with BPS sources BUT F 3 = − e − φ ⋆ 6 H = − e − φ ⋆ 6 H Smeared vs. localized sources � 3 H e − 2 A g − 1 � � since warp factor cancels: ⋆ 6 H ≈ det (e 2 A g 6 ) Timm Wrase 6 Flux compact. BPS case Moduli values at minimum unchanged! non-BPS case Localiz. effects Approximation δ ( O 3) ≈ 1 is “ok” Conclusion smeared: H and F 3 stabilize moduli localized: ˜ µ 3 δ ( O 3) , F 5 , A give corrections of equal size ⇒ corrections from ˜ µ 3 δ ( O 3) , F 5 , A cancel each other 8 / 16

A T-dual example with BPS sources T-duality along one H -flux direction ↔ Douglas, Kallosh H → R 6 < 0 Smeared vs. localized sources F 3 → F 4 Timm Wrase O 3 → O 4 Flux compact. F 5 → F 4 BPS case A → A non-BPS case Localiz. effects Conclusion 9 / 16