� � Very difficult to get de Sitter solutions of 10D SUGRA. Several reasons: David ANDRIOT technical: SUSY is broken ⇒ resolution is difficult. inherent to 10D SUGRA: generally R 4 ≤ 0 ... Introduction not talking of stability. De Sitter sol. Field redefinition 4D perspective: no-go theorems: V | 0 ≤ 0 Conclusion (and ways of circumventing them) hep-th/0007018 by J. M. Maldacena, C. Núñez arXiv:0711.2512 by M. P. Hertzberg, S. Kachru, W. Taylor, M. Tegmark arXiv:0810.5328 by S. S. Haque, G. Shiu, B. Underwood, T. Van Riet In particular, non-geometric terms: specific terms in the potential V ( ϕ ) of 4D SUGRA. + they generically help to get dS solutions of 4D SUGRA ! arXiv:0907.5580, arXiv:0911.2876 by B. de Carlos, A. Guarino, J. M. Moreno ֒ → use them to get dS solutions of 10D SUGRA? Major issue: 10D origin of non-geometric terms? gauging � SUGRA 4D V non − geo . ( ϕ ) SUGRA 4D V geo . ( ϕ )
� � Very difficult to get de Sitter solutions of 10D SUGRA. Several reasons: David ANDRIOT technical: SUSY is broken ⇒ resolution is difficult. inherent to 10D SUGRA: generally R 4 ≤ 0 ... Introduction not talking of stability. De Sitter sol. Field redefinition 4D perspective: no-go theorems: V | 0 ≤ 0 Conclusion (and ways of circumventing them) hep-th/0007018 by J. M. Maldacena, C. Núñez arXiv:0711.2512 by M. P. Hertzberg, S. Kachru, W. Taylor, M. Tegmark arXiv:0810.5328 by S. S. Haque, G. Shiu, B. Underwood, T. Van Riet In particular, non-geometric terms: specific terms in the potential V ( ϕ ) of 4D SUGRA. + they generically help to get dS solutions of 4D SUGRA ! arXiv:0907.5580, arXiv:0911.2876 by B. de Carlos, A. Guarino, J. M. Moreno ֒ → use them to get dS solutions of 10D SUGRA? Major issue: 10D origin of non-geometric terms? SUGRA 10D : 4D × 6D M compactif . on M gauging � SUGRA 4D V non − geo . ( ϕ ) SUGRA 4D V geo . ( ϕ )
� � Very difficult to get de Sitter solutions of 10D SUGRA. Several reasons: David ANDRIOT technical: SUSY is broken ⇒ resolution is difficult. inherent to 10D SUGRA: generally R 4 ≤ 0 ... Introduction not talking of stability. De Sitter sol. Field redefinition 4D perspective: no-go theorems: V | 0 ≤ 0 Conclusion (and ways of circumventing them) hep-th/0007018 by J. M. Maldacena, C. Núñez arXiv:0711.2512 by M. P. Hertzberg, S. Kachru, W. Taylor, M. Tegmark arXiv:0810.5328 by S. S. Haque, G. Shiu, B. Underwood, T. Van Riet In particular, non-geometric terms: specific terms in the potential V ( ϕ ) of 4D SUGRA. + they generically help to get dS solutions of 4D SUGRA ! arXiv:0907.5580, arXiv:0911.2876 by B. de Carlos, A. Guarino, J. M. Moreno ֒ → use them to get dS solutions of 10D SUGRA? Major issue: 10D origin of non-geometric terms? SUGRA 10D : 4D × 6D M ?? compactif . on M gauging � SUGRA 4D V non − geo . ( ϕ ) SUGRA 4D V geo . ( ϕ )
10D origin of 4D non-geometric terms: two problems: David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion
10D origin of 4D non-geometric terms: two problems: David ANDRIOT g mn , ˆ φ, ˆ B mn , ˆ H kmn = 3 ∂ [ k ˆ Focus on NSNS sector: ˆ B mn ] . Introduction De Sitter sol. Field redefinition Conclusion
10D origin of 4D non-geometric terms: two problems: David ANDRIOT g mn , ˆ φ, ˆ B mn , ˆ H kmn = 3 ∂ [ k ˆ Focus on NSNS sector: ˆ B mn ] . Compactification over M Introduction → in 4D V ( ϕ ), terms generated by ˆ mn | 0 , ˆ Γ k De Sitter sol. H kmn | 0 . ֒ Field redefinition Conclusion
10D origin of 4D non-geometric terms: two problems: David ANDRIOT g mn , ˆ φ, ˆ B mn , ˆ H kmn = 3 ∂ [ k ˆ Focus on NSNS sector: ˆ B mn ] . Compactification over M Introduction → in 4D V ( ϕ ), terms generated by ˆ mn | 0 , ˆ Γ k De Sitter sol. H kmn | 0 . ֒ Field redefinition But 4D non-geo. terms generated by “fluxes”: Q kmn , R kmn Conclusion + different dependence on scalars.
10D origin of 4D non-geometric terms: two problems: David ANDRIOT g mn , ˆ φ, ˆ B mn , ˆ H kmn = 3 ∂ [ k ˆ Focus on NSNS sector: ˆ B mn ] . Compactification over M Introduction → in 4D V ( ϕ ), terms generated by ˆ mn | 0 , ˆ Γ k De Sitter sol. H kmn | 0 . ֒ Field redefinition But 4D non-geo. terms generated by “fluxes”: Q kmn , R kmn Conclusion + different dependence on scalars. → 10D origin of Q and R -fluxes in NSNS sector? ֒
10D origin of 4D non-geometric terms: two problems: David ANDRIOT g mn , ˆ φ, ˆ B mn , ˆ H kmn = 3 ∂ [ k ˆ Focus on NSNS sector: ˆ B mn ] . Compactification over M Introduction → in 4D V ( ϕ ), terms generated by ˆ mn | 0 , ˆ Γ k De Sitter sol. H kmn | 0 . ֒ Field redefinition But 4D non-geo. terms generated by “fluxes”: Q kmn , R kmn Conclusion + different dependence on scalars. → 10D origin of Q and R -fluxes in NSNS sector? ֒ 10D non-geometry: different. Relation to 4D unclear.
10D origin of 4D non-geometric terms: two problems: David ANDRIOT g mn , ˆ φ, ˆ B mn , ˆ H kmn = 3 ∂ [ k ˆ Focus on NSNS sector: ˆ B mn ] . Compactification over M Introduction → in 4D V ( ϕ ), terms generated by ˆ mn | 0 , ˆ Γ k De Sitter sol. H kmn | 0 . ֒ Field redefinition But 4D non-geo. terms generated by “fluxes”: Q kmn , R kmn Conclusion + different dependence on scalars. → 10D origin of Q and R -fluxes in NSNS sector? ֒ 10D non-geometry: different. Relation to 4D unclear. A 10D non-geometric config. of fields: not single-valued, global issues, M not a standard compact manifold (see talk of Dieter Lüst) → compactification, integration over M ? ֒
10D origin of 4D non-geometric terms: two problems: David ANDRIOT g mn , ˆ φ, ˆ B mn , ˆ H kmn = 3 ∂ [ k ˆ Focus on NSNS sector: ˆ B mn ] . Compactification over M Introduction → in 4D V ( ϕ ), terms generated by ˆ mn | 0 , ˆ Γ k De Sitter sol. H kmn | 0 . ֒ Field redefinition But 4D non-geo. terms generated by “fluxes”: Q kmn , R kmn Conclusion + different dependence on scalars. → 10D origin of Q and R -fluxes in NSNS sector? ֒ 10D non-geometry: different. Relation to 4D unclear. A 10D non-geometric config. of fields: not single-valued, global issues, M not a standard compact manifold (see talk of Dieter Lüst) → compactification, integration over M ? ֒ Here: progress in relating 10D/4D non-geometry A way to overcome these two issues:
10D origin of 4D non-geometric terms: two problems: David ANDRIOT g mn , ˆ φ, ˆ B mn , ˆ H kmn = 3 ∂ [ k ˆ Focus on NSNS sector: ˆ B mn ] . Compactification over M Introduction → in 4D V ( ϕ ), terms generated by ˆ mn | 0 , ˆ Γ k De Sitter sol. H kmn | 0 . ֒ Field redefinition But 4D non-geo. terms generated by “fluxes”: Q kmn , R kmn Conclusion + different dependence on scalars. → 10D origin of Q and R -fluxes in NSNS sector? ֒ 10D non-geometry: different. Relation to 4D unclear. A 10D non-geometric config. of fields: not single-valued, global issues, M not a standard compact manifold (see talk of Dieter Lüst) → compactification, integration over M ? ֒ Here: progress in relating 10D/4D non-geometry A way to overcome these two issues: - field redefinition on NSNS fields ⇒ Q , R in 10D Lag.
10D origin of 4D non-geometric terms: two problems: David ANDRIOT g mn , ˆ φ, ˆ B mn , ˆ H kmn = 3 ∂ [ k ˆ Focus on NSNS sector: ˆ B mn ] . Compactification over M Introduction → in 4D V ( ϕ ), terms generated by ˆ mn | 0 , ˆ Γ k De Sitter sol. H kmn | 0 . ֒ Field redefinition But 4D non-geo. terms generated by “fluxes”: Q kmn , R kmn Conclusion + different dependence on scalars. → 10D origin of Q and R -fluxes in NSNS sector? ֒ 10D non-geometry: different. Relation to 4D unclear. A 10D non-geometric config. of fields: not single-valued, global issues, M not a standard compact manifold (see talk of Dieter Lüst) → compactification, integration over M ? ֒ Here: progress in relating 10D/4D non-geometry A way to overcome these two issues: - field redefinition on NSNS fields ⇒ Q , R in 10D Lag. - new fields globally defined ⇒ compactify, get V ( ϕ ) �
10D origin of 4D non-geometric terms: two problems: David ANDRIOT g mn , ˆ φ, ˆ B mn , ˆ H kmn = 3 ∂ [ k ˆ Focus on NSNS sector: ˆ B mn ] . Compactification over M Introduction → in 4D V ( ϕ ), terms generated by ˆ mn | 0 , ˆ Γ k De Sitter sol. H kmn | 0 . ֒ Field redefinition But 4D non-geo. terms generated by “fluxes”: Q kmn , R kmn Conclusion + different dependence on scalars. → 10D origin of Q and R -fluxes in NSNS sector? ֒ 10D non-geometry: different. Relation to 4D unclear. A 10D non-geometric config. of fields: not single-valued, global issues, M not a standard compact manifold (see talk of Dieter Lüst) → compactification, integration over M ? ֒ Here: progress in relating 10D/4D non-geometry A way to overcome these two issues: - field redefinition on NSNS fields ⇒ Q , R in 10D Lag. - new fields globally defined ⇒ compactify, get V ( ϕ ) � For dS solutions: extend field redef. to full 10D SUGRA.
David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion Plan: de Sitter solutions of IIA SUGRA From 10D and 4D: generally Λ ≤ 0 4D non-geometric terms ⇒ help for de Sitter. Field redefinition ⇒ Q , R -fluxes in 10D. Compactification ⇒ V ( ϕ ) � .
De Sitter solutions of 10D and 4D SUGRA David ANDRIOT 10D discussion Introduction 10D IIA SUGRA, with O6-planes/D6-branes De Sitter sol. 10D (common set-up to look for dS solutions). 4D geometric Bosonic content: NSNS sector, RR sector: F 0 , F 2 , (no F 4 , F 6 ). 4D non-geometric Field redefinition O6/D6 source of F 2 . Conclusion
De Sitter solutions of 10D and 4D SUGRA David ANDRIOT 10D discussion Introduction 10D IIA SUGRA, with O6-planes/D6-branes De Sitter sol. 10D (common set-up to look for dS solutions). 4D geometric Bosonic content: NSNS sector, RR sector: F 0 , F 2 , (no F 4 , F 6 ). 4D non-geometric Field redefinition O6/D6 source of F 2 . � Conclusion � 1 d 10 x S IIA = | ˆ g 10 | [ L NSNS + L RR + L sources ] , 2 κ 2 φ | 2 − 1 L NSNS = e − 2 ˆ φ ( � R 10 + 4 |∇ ˆ 2 | ˆ H | 2 ) , 2( | F 0 | 2 + | F 2 | 2 ) , F m 1 ... m p F m 1 ... m p L RR = − 1 = | F p | 2 . p !
De Sitter solutions of 10D and 4D SUGRA David ANDRIOT 10D discussion Introduction 10D IIA SUGRA, with O6-planes/D6-branes De Sitter sol. 10D (common set-up to look for dS solutions). 4D geometric Bosonic content: NSNS sector, RR sector: F 0 , F 2 , (no F 4 , F 6 ). 4D non-geometric Field redefinition O6/D6 source of F 2 . � Conclusion � 1 d 10 x S IIA = | ˆ g 10 | [ L NSNS + L RR + L sources ] , 2 κ 2 φ | 2 − 1 L NSNS = e − 2 ˆ φ ( � R 10 + 4 |∇ ˆ 2 | ˆ H | 2 ) , 2( | F 0 | 2 + | F 2 | 2 ) , F m 1 ... m p F m 1 ... m p L RR = − 1 = | F p | 2 . p ! Derive 10D e.o.m.
De Sitter solutions of 10D and 4D SUGRA David ANDRIOT 10D discussion Introduction 10D IIA SUGRA, with O6-planes/D6-branes De Sitter sol. 10D (common set-up to look for dS solutions). 4D geometric Bosonic content: NSNS sector, RR sector: F 0 , F 2 , (no F 4 , F 6 ). 4D non-geometric Field redefinition O6/D6 source of F 2 . � Conclusion � 1 d 10 x S IIA = | ˆ g 10 | [ L NSNS + L RR + L sources ] , 2 κ 2 φ | 2 − 1 L NSNS = e − 2 ˆ φ ( � R 10 + 4 |∇ ˆ 2 | ˆ H | 2 ) , 2( | F 0 | 2 + | F 2 | 2 ) , F m 1 ... m p F m 1 ... m p L RR = − 1 = | F p | 2 . p ! Derive 10D e.o.m. Compactification ansatz: d s 2 10 = d s 2 4 + d s 2 6 (no warp factor), φ = g s . non-trivial fluxes only along M , constant dilaton e ˆ
De Sitter solutions of 10D and 4D SUGRA David ANDRIOT 10D discussion Introduction 10D IIA SUGRA, with O6-planes/D6-branes De Sitter sol. 10D (common set-up to look for dS solutions). 4D geometric Bosonic content: NSNS sector, RR sector: F 0 , F 2 , (no F 4 , F 6 ). 4D non-geometric Field redefinition O6/D6 source of F 2 . � Conclusion � 1 d 10 x S IIA = | ˆ g 10 | [ L NSNS + L RR + L sources ] , 2 κ 2 φ | 2 − 1 L NSNS = e − 2 ˆ φ ( � R 10 + 4 |∇ ˆ 2 | ˆ H | 2 ) , 2( | F 0 | 2 + | F 2 | 2 ) , F m 1 ... m p F m 1 ... m p L RR = − 1 = | F p | 2 . p ! Derive 10D e.o.m. Compactification ansatz: d s 2 10 = d s 2 4 + d s 2 6 (no warp factor), φ = g s . non-trivial fluxes only along M , constant dilaton e ˆ Combine (4D, 6D) trace of Einstein and dilaton e.o.m.
De Sitter solutions of 10D and 4D SUGRA David ANDRIOT 10D discussion Introduction 10D IIA SUGRA, with O6-planes/D6-branes De Sitter sol. 10D (common set-up to look for dS solutions). 4D geometric Bosonic content: NSNS sector, RR sector: F 0 , F 2 , (no F 4 , F 6 ). 4D non-geometric Field redefinition O6/D6 source of F 2 . � Conclusion � 1 d 10 x S IIA = | ˆ g 10 | [ L NSNS + L RR + L sources ] , 2 κ 2 φ | 2 − 1 L NSNS = e − 2 ˆ φ ( � R 10 + 4 |∇ ˆ 2 | ˆ H | 2 ) , 2( | F 0 | 2 + | F 2 | 2 ) , F m 1 ... m p F m 1 ... m p L RR = − 1 = | F p | 2 . p ! Derive 10D e.o.m. Compactification ansatz: d s 2 10 = d s 2 4 + d s 2 6 (no warp factor), φ = g s . non-trivial fluxes only along M , constant dilaton e ˆ Combine (4D, 6D) trace of Einstein and dilaton e.o.m.: � � � H | 2 � 3Λ = 3 R 4 = 1 = 1 R 6 − 1 s | F 0 | 2 − | ˆ � − � g 2 2 g 2 s | F 2 | 2 4 2 3
David ANDRIOT Introduction De Sitter sol. Obtain de Sitter solution: 10D � � � H | 2 � 4D geometric 3Λ = 3 R 4 = 1 = 1 R 6 − 1 s | F 0 | 2 − | ˆ 4D non-geometric � g 2 − � 2 g 2 s | F 2 | 2 > 0 Field redefinition 4 2 3 Conclusion
David ANDRIOT Introduction De Sitter sol. Obtain de Sitter solution: 10D � � � H | 2 � 4D geometric 3Λ = 3 R 4 = 1 = 1 R 6 − 1 s | F 0 | 2 − | ˆ 4D non-geometric � g 2 − � 2 g 2 s | F 2 | 2 > 0 Field redefinition 4 2 3 Conclusion → need at least F 0 � = 0, � R 6 < 0 ! ֒ hep-th/0007018 by J. M. Maldacena, C. Núñez arXiv:0810.5328 by S. S. Haque, G. Shiu, B. Underwood, T. Van Riet
David ANDRIOT Introduction De Sitter sol. Obtain de Sitter solution: 10D � � � H | 2 � 4D geometric 3Λ = 3 R 4 = 1 = 1 R 6 − 1 s | F 0 | 2 − | ˆ 4D non-geometric � g 2 − � 2 g 2 s | F 2 | 2 > 0 Field redefinition 4 2 3 Conclusion → need at least F 0 � = 0, � R 6 < 0 ! ֒ hep-th/0007018 by J. M. Maldacena, C. Núñez arXiv:0810.5328 by S. S. Haque, G. Shiu, B. Underwood, T. Van Riet But F 0 and ˆ H not independent ( B -field e.o.m., F 2 B.I.)...
David ANDRIOT Introduction De Sitter sol. Obtain de Sitter solution: 10D � � � H | 2 � 4D geometric 3Λ = 3 R 4 = 1 = 1 R 6 − 1 s | F 0 | 2 − | ˆ 4D non-geometric � g 2 − � 2 g 2 s | F 2 | 2 > 0 Field redefinition 4 2 3 Conclusion → need at least F 0 � = 0, � R 6 < 0 ! ֒ hep-th/0007018 by J. M. Maldacena, C. Núñez arXiv:0810.5328 by S. S. Haque, G. Shiu, B. Underwood, T. Van Riet But F 0 and ˆ H not independent ( B -field e.o.m., F 2 B.I.)... → In most of the examples: AdS, Minkowski. ֒
David ANDRIOT Introduction De Sitter sol. Obtain de Sitter solution: 10D � � � H | 2 � 4D geometric 3Λ = 3 R 4 = 1 = 1 R 6 − 1 s | F 0 | 2 − | ˆ 4D non-geometric � g 2 − � 2 g 2 s | F 2 | 2 > 0 Field redefinition 4 2 3 Conclusion → need at least F 0 � = 0, � R 6 < 0 ! ֒ hep-th/0007018 by J. M. Maldacena, C. Núñez arXiv:0810.5328 by S. S. Haque, G. Shiu, B. Underwood, T. Van Riet But F 0 and ˆ H not independent ( B -field e.o.m., F 2 B.I.)... → In most of the examples: AdS, Minkowski. ֒ F 4 , F 6 do not help: enter with a minus... de Sitter not favored.
4D discussion: the scalar potential David V ( ρ, σ ): only consider volume ρ and dilaton σ , always appear. ANDRIOT Introduction De Sitter sol. 10D 4D geometric 4D non-geometric Field redefinition Conclusion
4D discussion: the scalar potential David V ( ρ, σ ): only consider volume ρ and dilaton σ , always appear. ANDRIOT φ → e ˆ φ (0) e ˆ ˆ ϕ , σ = ρ 3 ϕ , ˆ Introduction g (0) H , F p → ˆ 2 e − ˆ H (0) , F (0) g 6 ij → ˆ ˆ 6 ij ρ , e p De Sitter sol. 10D 4D geometric 4D non-geometric Field redefinition Conclusion
4D discussion: the scalar potential David V ( ρ, σ ): only consider volume ρ and dilaton σ , always appear. ANDRIOT φ → e ˆ φ (0) e ˆ ˆ ϕ , σ = ρ 3 ϕ , ˆ Introduction g (0) H , F p → ˆ 2 e − ˆ H (0) , F (0) g 6 ij → ˆ ˆ 6 ij ρ , e p De Sitter sol. 10D Compactification procedure: from 10D S IIA , get 4D geometric � � � 4D non-geometric � 4 + kin − V Field redefinition � S = M 2 d 4 x R E g E | ˆ 4 | 4 M 2 Conclusion 4 V ( ρ, σ ) = σ − 2 ( ρ − 1 V R + ρ − 3 V H ) + ρ 3 σ − 4 ( V F 0 + ρ − 2 V F 2 ) − σ − 3 V sources M 2 4 R 6 , V H = 1 2 | H | 2 , V F p = 1 s | F p | 2 . where V R = − � 2 g 2 arXiv:0712.1196 by E. Silverstein
4D discussion: the scalar potential David V ( ρ, σ ): only consider volume ρ and dilaton σ , always appear. ANDRIOT φ → e ˆ φ (0) e ˆ ˆ ϕ , σ = ρ 3 ϕ , ˆ Introduction g (0) H , F p → ˆ 2 e − ˆ H (0) , F (0) g 6 ij → ˆ ˆ 6 ij ρ , e p De Sitter sol. 10D Compactification procedure: from 10D S IIA , get 4D geometric � � � 4D non-geometric � 4 + kin − V 1 Field redefinition � S = M 2 d 4 x R E g E | ˆ 4 | ⇒ Λ = V | 0 4 M 2 2 M 2 Conclusion 4 4 V ( ρ, σ ) = σ − 2 ( ρ − 1 V R + ρ − 3 V H ) + ρ 3 σ − 4 ( V F 0 + ρ − 2 V F 2 ) − σ − 3 V sources M 2 4 R 6 , V H = 1 2 | H | 2 , V F p = 1 s | F p | 2 . where V R = − � 2 g 2 arXiv:0712.1196 by E. Silverstein
4D discussion: the scalar potential David V ( ρ, σ ): only consider volume ρ and dilaton σ , always appear. ANDRIOT φ → e ˆ φ (0) e ˆ ˆ ϕ , σ = ρ 3 ϕ , ˆ Introduction g (0) H , F p → ˆ 2 e − ˆ H (0) , F (0) ˆ g 6 ij → ˆ 6 ij ρ , e p De Sitter sol. 10D Compactification procedure: from 10D S IIA , get 4D geometric � � � 4D non-geometric � 4 + kin − V 1 Field redefinition � S = M 2 d 4 x R E g E | ˆ 4 | ⇒ Λ = V | 0 4 M 2 2 M 2 Conclusion 4 4 V ( ρ, σ ) = σ − 2 ( ρ − 1 V R + ρ − 3 V H ) + ρ 3 σ − 4 ( V F 0 + ρ − 2 V F 2 ) − σ − 3 V sources M 2 4 R 6 , V H = 1 2 | H | 2 , V F p = 1 s | F p | 2 . where V R = − � 2 g 2 arXiv:0712.1196 by E. Silverstein ∂ V ∂ρ | 0 = ∂ V Get V | 0 ? Extremize the potential: ∂σ | 0 = 0.
4D discussion: the scalar potential David V ( ρ, σ ): only consider volume ρ and dilaton σ , always appear. ANDRIOT φ → e ˆ φ (0) e ˆ ˆ ϕ , σ = ρ 3 ϕ , ˆ Introduction g (0) H , F p → ˆ 2 e − ˆ H (0) , F (0) ˆ g 6 ij → ˆ 6 ij ρ , e p De Sitter sol. 10D Compactification procedure: from 10D S IIA , get 4D geometric � � � 4D non-geometric � 4 + kin − V 1 Field redefinition � S = M 2 d 4 x R E g E | ˆ 4 | ⇒ Λ = V | 0 4 M 2 2 M 2 Conclusion 4 4 V ( ρ, σ ) = σ − 2 ( ρ − 1 V R + ρ − 3 V H ) + ρ 3 σ − 4 ( V F 0 + ρ − 2 V F 2 ) − σ − 3 V sources M 2 4 R 6 , V H = 1 2 | H | 2 , V F p = 1 s | F p | 2 . where V R = − � 2 g 2 arXiv:0712.1196 by E. Silverstein ∂ V ∂ρ | 0 = ∂ V Get V | 0 ? Extremize the potential: ∂σ | 0 = 0. ֒ → Combine and get 3Λ = 3 V | 0 = V F 0 − V H = 1 3 ( V R − V F 2 ) M 2 2 4
4D discussion: the scalar potential David V ( ρ, σ ): only consider volume ρ and dilaton σ , always appear. ANDRIOT φ → e ˆ φ (0) e ˆ ˆ ϕ , σ = ρ 3 ϕ , ˆ Introduction g (0) H , F p → ˆ 2 e − ˆ H (0) , F (0) ˆ g 6 ij → ˆ 6 ij ρ , e p De Sitter sol. 10D Compactification procedure: from 10D S IIA , get 4D geometric � � � 4D non-geometric � 4 + kin − V 1 Field redefinition � S = M 2 d 4 x R E g E | ˆ 4 | ⇒ Λ = V | 0 4 M 2 2 M 2 Conclusion 4 4 V ( ρ, σ ) = σ − 2 ( ρ − 1 V R + ρ − 3 V H ) + ρ 3 σ − 4 ( V F 0 + ρ − 2 V F 2 ) − σ − 3 V sources M 2 4 R 6 , V H = 1 2 | H | 2 , V F p = 1 s | F p | 2 . where V R = − � 2 g 2 arXiv:0712.1196 by E. Silverstein ∂ V ∂ρ | 0 = ∂ V Get V | 0 ? Extremize the potential: ∂σ | 0 = 0. ֒ → Combine and get 3Λ = 3 V | 0 = V F 0 − V H = 1 3 ( V R − V F 2 ) M 2 2 4 Same relations as in 10D... Same difficulty to get de Sitter.
4D discussion: the scalar potential David V ( ρ, σ ): only consider volume ρ and dilaton σ , always appear. ANDRIOT φ → e ˆ φ (0) e ˆ ˆ ϕ , σ = ρ 3 ϕ , ˆ Introduction g (0) H , F p → ˆ 2 e − ˆ H (0) , F (0) g 6 ij → ˆ ˆ 6 ij ρ , e p De Sitter sol. 10D Compactification procedure: from 10D S IIA , get 4D geometric � � � 4D non-geometric � 4 + kin − V 1 Field redefinition � S = M 2 d 4 x R E g E | ˆ 4 | ⇒ Λ = V | 0 4 M 2 2 M 2 Conclusion 4 4 V ( ρ, σ ) = σ − 2 ( ρ − 1 V R + ρ − 3 V H ) + ρ 3 σ − 4 ( V F 0 + ρ − 2 V F 2 ) − σ − 3 V sources M 2 4 R 6 , V H = 1 2 | H | 2 , V F p = 1 s | F p | 2 . where V R = − � 2 g 2 arXiv:0712.1196 by E. Silverstein ∂ V ∂ρ | 0 = ∂ V Get V | 0 ? Extremize the potential: ∂σ | 0 = 0. ֒ → Combine and get 3Λ = 3 V | 0 = V F 0 − V H = 1 3 ( V R − V F 2 ) M 2 2 4 Same relations as in 10D... Same difficulty to get de Sitter. → Non-geometric terms... ֒
4D non-geometric terms David ANDRIOT Long history, motivation for these terms... Here, pratical point of view: allowed in 4D (super)potential. Introduction De Sitter sol. 10D 4D geometric 4D non-geometric Field redefinition Conclusion
4D non-geometric terms David ANDRIOT Long history, motivation for these terms... Here, pratical point of view: allowed in 4D (super)potential. Introduction Scalar potential: De Sitter sol. 10D 4D geometric V NSNS ∼ ρ − 3 V H + ρ − 1 V R + ρ V Q + ρ 3 V R 4D non-geometric Field redefinition Conclusion arXiv:0711.2512 by M. P. Hertzberg, S. Kachru, W. Taylor, M. Tegmark
4D non-geometric terms David ANDRIOT Long history, motivation for these terms... Here, pratical point of view: allowed in 4D (super)potential. Introduction Scalar potential: De Sitter sol. 10D 4D geometric V NSNS ∼ ρ − 3 V H + ρ − 1 V R + ρ V Q + ρ 3 V R 4D non-geometric Field redefinition Conclusion arXiv:0711.2512 by M. P. Hertzberg, S. Kachru, W. Taylor, M. Tegmark With Q , R , and F 4 , F 6 , one gets de Sitter for: ( V F 0 − V H ) + ( V Q − V F 4 ) + 2( V R − V F 6 ) > 0 ( V R − V F 2 ) + 2( V Q − V F 4 ) + 3( V R − V F 6 ) > 0 arXiv:0907.5580, arXiv:0911.2876 by B. de Carlos, A. Guarino, J. M. Moreno
4D non-geometric terms David ANDRIOT Long history, motivation for these terms... Here, pratical point of view: allowed in 4D (super)potential. Introduction Scalar potential: De Sitter sol. 10D 4D geometric V NSNS ∼ ρ − 3 V H + ρ − 1 V R + ρ V Q + ρ 3 V R 4D non-geometric Field redefinition Conclusion arXiv:0711.2512 by M. P. Hertzberg, S. Kachru, W. Taylor, M. Tegmark With Q , R , and F 4 , F 6 , one gets de Sitter for: ( V F 0 − V H ) > 0 ( V R − V F 2 ) > 0 arXiv:0907.5580, arXiv:0911.2876 by B. de Carlos, A. Guarino, J. M. Moreno
4D non-geometric terms David ANDRIOT Long history, motivation for these terms... Here, pratical point of view: allowed in 4D (super)potential. Introduction Scalar potential: De Sitter sol. 10D 4D geometric V NSNS ∼ ρ − 3 V H + ρ − 1 V R + ρ V Q + ρ 3 V R 4D non-geometric Field redefinition Conclusion arXiv:0711.2512 by M. P. Hertzberg, S. Kachru, W. Taylor, M. Tegmark With Q , R , and F 4 , F 6 , one gets de Sitter for: ( V F 0 − V H ) + ( V Q − V F 4 ) + 2( V R − V F 6 ) > 0 ( V R − V F 2 ) + 2( V Q − V F 4 ) + 3( V R − V F 6 ) > 0 arXiv:0907.5580, arXiv:0911.2876 by B. de Carlos, A. Guarino, J. M. Moreno
4D non-geometric terms David ANDRIOT Long history, motivation for these terms... Here, pratical point of view: allowed in 4D (super)potential. Introduction Scalar potential: De Sitter sol. 10D 4D geometric V NSNS ∼ ρ − 3 V H + ρ − 1 V R + ρ V Q + ρ 3 V R 4D non-geometric Field redefinition Conclusion arXiv:0711.2512 by M. P. Hertzberg, S. Kachru, W. Taylor, M. Tegmark With Q , R , and F 4 , F 6 , one gets de Sitter for: ( V F 0 − V H ) + ( V Q − V F 4 ) + 2( V R − V F 6 ) > 0 ( V R − V F 2 ) + 2( V Q − V F 4 ) + 3( V R − V F 6 ) > 0 arXiv:0907.5580, arXiv:0911.2876 by B. de Carlos, A. Guarino, J. M. Moreno
4D non-geometric terms David ANDRIOT Long history, motivation for these terms... Here, pratical point of view: allowed in 4D (super)potential. Introduction Scalar potential: De Sitter sol. 10D 4D geometric V NSNS ∼ ρ − 3 V H + ρ − 1 V R + ρ V Q + ρ 3 V R 4D non-geometric Field redefinition Conclusion arXiv:0711.2512 by M. P. Hertzberg, S. Kachru, W. Taylor, M. Tegmark With Q , R , and F 4 , F 6 , one gets de Sitter for: ( V F 0 − V H ) + ( V Q − V F 4 ) + 2( V R − V F 6 ) > 0 ( V R − V F 2 ) + 2( V Q − V F 4 ) + 3( V R − V F 6 ) > 0 arXiv:0907.5580, arXiv:0911.2876 by B. de Carlos, A. Guarino, J. M. Moreno Q , R , help to get de Sitter solutions ! Some examples in 4D...
4D non-geometric terms David ANDRIOT Long history, motivation for these terms... Here, pratical point of view: allowed in 4D (super)potential. Introduction Scalar potential: De Sitter sol. 10D 4D geometric V NSNS ∼ ρ − 3 V H + ρ − 1 V R + ρ V Q + ρ 3 V R 4D non-geometric Field redefinition Conclusion arXiv:0711.2512 by M. P. Hertzberg, S. Kachru, W. Taylor, M. Tegmark With Q , R , and F 4 , F 6 , one gets de Sitter for: ( V F 0 − V H ) + ( V Q − V F 4 ) + 2( V R − V F 6 ) > 0 ( V R − V F 2 ) + 2( V Q − V F 4 ) + 3( V R − V F 6 ) > 0 arXiv:0907.5580, arXiv:0911.2876 by B. de Carlos, A. Guarino, J. M. Moreno Q , R , help to get de Sitter solutions ! Some examples in 4D... Obtain this from a compactification of 10D SUGRA? 10D interpretation of such solutions?
Field redefinition David Presentation ANDRIOT Key object: ˜ β : antisymmetric bivector ˜ β mn . Introduction De Sitter sol. Field redefinition Presentation Lagrangians Back to 4D Conclusion
Field redefinition David Presentation ANDRIOT Key object: ˜ β : antisymmetric bivector ˜ β mn . Introduction De Sitter sol. Motivations from Generalized Complex Geometry Field redefinition math.DG/0209099 by N. Hitchin, math.DG/0401221 by M. Gualtieri Presentation Lagrangians Back to 4D Conclusion
Field redefinition David Presentation ANDRIOT Key object: ˜ β : antisymmetric bivector ˜ β mn . Introduction De Sitter sol. Motivations from Generalized Complex Geometry Field redefinition math.DG/0209099 by N. Hitchin, math.DG/0401221 by M. Gualtieri Presentation Arguments in GCG: ˜ ab , R abc Lagrangians β related to non-geometry / to Q c Back to 4D hep-th/0609084, arXiv:0708.2392 by P. Grange, S. Schäfer-Nameki Conclusion arXiv:0807.4527 by M. Graña, R. Minasian, M. Petrini, D. Waldram
Field redefinition David Presentation ANDRIOT Key object: ˜ β : antisymmetric bivector ˜ β mn . Introduction De Sitter sol. Motivations from Generalized Complex Geometry Field redefinition math.DG/0209099 by N. Hitchin, math.DG/0401221 by M. Gualtieri Presentation Arguments in GCG: ˜ ab , R abc Lagrangians β related to non-geometry / to Q c Back to 4D hep-th/0609084, arXiv:0708.2392 by P. Grange, S. Schäfer-Nameki Conclusion arXiv:0807.4527 by M. Graña, R. Minasian, M. Petrini, D. Waldram ˜ β appears via a reparametrization of the gen. metric H : � � � � g − 1 ˆ g − ˆ ˆ g ˜ g − 1 ˆ B ˆ B B ˆ ˜ g ˜ β H = = , ˜ g : new metric g − 1 ˆ g − 1 − ˜ − ˜ g ˜ g − 1 β ˜ ˜ β ˜ − ˆ B ˆ g β
Field redefinition David Presentation ANDRIOT Key object: ˜ β : antisymmetric bivector ˜ β mn . Introduction De Sitter sol. Motivations from Generalized Complex Geometry Field redefinition math.DG/0209099 by N. Hitchin, math.DG/0401221 by M. Gualtieri Presentation Arguments in GCG: ˜ ab , R abc Lagrangians β related to non-geometry / to Q c Back to 4D hep-th/0609084, arXiv:0708.2392 by P. Grange, S. Schäfer-Nameki Conclusion arXiv:0807.4527 by M. Graña, R. Minasian, M. Petrini, D. Waldram ˜ β appears via a reparametrization of the gen. metric H : � � � � g − 1 ˆ g − ˆ ˆ g ˜ g − 1 ˆ B ˆ B B ˆ g ˜ ˜ β H = = , ˜ g : new metric g − 1 ˆ g − 1 − ˜ − ˜ g ˜ g − 1 β ˜ ˜ β ˜ − ˆ B ˆ g β g − 1 + ˜ g − 1 − ˜ g − 1 − ˜ β ) − 1 ˜ g − 1 (˜ β ) − 1 g + ˆ β ) − 1 ⇔ ˆ g = (˜ ⇔ (ˆ B ) = (˜ g − 1 + ˜ β ) − 1 ˜ g − 1 − ˜ ˆ β ) − 1 B = (˜ β (˜
Field redefinition David Presentation ANDRIOT Key object: ˜ β : antisymmetric bivector ˜ β mn . Introduction De Sitter sol. Motivations from Generalized Complex Geometry Field redefinition math.DG/0209099 by N. Hitchin, math.DG/0401221 by M. Gualtieri Presentation Arguments in GCG: ˜ ab , R abc Lagrangians β related to non-geometry / to Q c Back to 4D hep-th/0609084, arXiv:0708.2392 by P. Grange, S. Schäfer-Nameki Conclusion arXiv:0807.4527 by M. Graña, R. Minasian, M. Petrini, D. Waldram ˜ β appears via a reparametrization of the gen. metric H : � � � � g − 1 ˆ g − ˆ ˆ g ˜ g − 1 ˆ B ˆ B B ˆ ˜ g ˜ β H = = , ˜ g : new metric g − 1 ˆ g − 1 − ˜ − ˜ g ˜ g − 1 β ˜ ˜ β ˜ − ˆ B ˆ g β g − 1 + ˜ g − 1 − ˜ g − 1 − ˜ β ) − 1 ˜ g − 1 (˜ β ) − 1 g + ˆ β ) − 1 ⇔ g = (˜ ˆ ⇔ (ˆ B ) = (˜ g − 1 + ˜ β ) − 1 ˜ g − 1 − ˜ ˆ β ) − 1 B = (˜ β (˜ φ � φ � e − 2 ˜ g | = e − 2 ˆ | ˜ | ˆ g |
Field redefinition David Presentation ANDRIOT Key object: ˜ β : antisymmetric bivector ˜ β mn . Introduction De Sitter sol. Motivations from Generalized Complex Geometry Field redefinition math.DG/0209099 by N. Hitchin, math.DG/0401221 by M. Gualtieri Presentation Arguments in GCG: ˜ ab , R abc Lagrangians β related to non-geometry / to Q c Back to 4D hep-th/0609084, arXiv:0708.2392 by P. Grange, S. Schäfer-Nameki Conclusion arXiv:0807.4527 by M. Graña, R. Minasian, M. Petrini, D. Waldram ˜ β appears via a reparametrization of the gen. metric H : � � � � g − 1 ˆ g − ˆ ˆ g ˜ g − 1 ˆ B ˆ B B ˆ g ˜ ˜ β H = = , ˜ g : new metric g − 1 ˆ g − 1 − ˜ − ˜ g ˜ g − 1 β ˜ ˜ β ˜ − ˆ B ˆ g β g − 1 + ˜ g − 1 − ˜ g − 1 − ˜ β ) − 1 ˜ g − 1 (˜ β ) − 1 g + ˆ β ) − 1 ⇔ g = (˜ ˆ ⇔ (ˆ B ) = (˜ g − 1 + ˜ β ) − 1 ˜ g − 1 − ˜ ˆ β ) − 1 B = (˜ β (˜ φ � φ � e − 2 ˜ g | = e − 2 ˆ | ˜ | ˆ g | g , ˆ B , ˆ g , ˜ β, ˜ φ ), ˜ Field redefinition: (ˆ φ ) ↔ (˜ β favored for non-geom.
Field redefinition David Presentation ANDRIOT Key object: ˜ β : antisymmetric bivector ˜ β mn . Introduction De Sitter sol. Motivations from Generalized Complex Geometry Field redefinition math.DG/0209099 by N. Hitchin, math.DG/0401221 by M. Gualtieri Presentation Arguments in GCG: ˜ ab , R abc Lagrangians β related to non-geometry / to Q c Back to 4D hep-th/0609084, arXiv:0708.2392 by P. Grange, S. Schäfer-Nameki Conclusion arXiv:0807.4527 by M. Graña, R. Minasian, M. Petrini, D. Waldram ˜ β appears via a reparametrization of the gen. metric H : � � � � g − 1 ˆ g − ˆ ˆ g ˜ g − 1 ˆ B ˆ B B ˆ g ˜ ˜ β H = = , ˜ g : new metric g − 1 ˆ g − 1 − ˜ − ˜ g ˜ g − 1 β ˜ ˜ β ˜ − ˆ B ˆ g β g − 1 + ˜ g − 1 − ˜ g − 1 − ˜ β ) − 1 ˜ g − 1 (˜ β ) − 1 g + ˆ β ) − 1 ⇔ g = (˜ ˆ ⇔ (ˆ B ) = (˜ g − 1 + ˜ β ) − 1 ˜ g − 1 − ˜ ˆ β ) − 1 B = (˜ β (˜ φ � φ � e − 2 ˜ g | = e − 2 ˆ | ˜ | ˆ g | g , ˆ B , ˆ g , ˜ β, ˜ φ ), ˜ Field redefinition: (ˆ φ ) ↔ (˜ β favored for non-geom. Apply it on NSNS Lagrangian? ˜ β could be related to non-geo. fluxes ⇒ would they appear?
David Rewriting of the NSNS Lagrangian ANDRIOT Introduction � H | 2 � φ � φ | 2 − 1 L = e − 2 ˆ ˆ R + 4 | dˆ � 2 | ˆ De Sitter sol. | ˆ g | Field redefinition Presentation Lagrangians Back to 4D Conclusion
David Rewriting of the NSNS Lagrangian ANDRIOT Introduction � H | 2 � φ � φ | 2 − 1 L = e − 2 ˆ ˆ R + 4 | dˆ � 2 | ˆ De Sitter sol. | ˆ g | = ? Field redefinition Presentation Lagrangians Back to 4D Conclusion (assumption: ˜ β km ∂ m · = 0 ) gpq � gkp � 1 R = � � gkm ˜ guq ˜ gps + 2˜ gpq ˜ gks ˜ gmu + guq ˜ gsm ˜ R − ∂ k ˜ gsu ∂ m ˜ 2˜ ˜ 2 1 β pk ∂ m ˜ β qm − β qm ∂ m ˜ β pk gpq ∂ k ˜ gpq ∂ k ˜ − ˜ ˜ 2 gkm ˜ gpq ∂ k ∂ m ˜ gkm ( G − 1) pq ∂ k ∂ mGqp + 2˜ gpq + 2˜ + ∂ mGvl � gks ( G − 1) lv ∂ k ˜ gkm ( G − 1) lv ∂ k ˜ gmr ˜ grs ˜ − 2˜ grs − ˜ grs gvr � gru ( G − 1) lu ∂ v ˜ grs ( G − 1) ls ∂ k ˜ gms ˜ gkm ˜ + ˜ grs − ˜ + ∂ mGvl � glq ∂ vGmq � 1 ( G − 1) lq ∂ vGqm + ˆ 2 gkm � gpv � − ∂ mGvl ∂ k Gps 1 2( G − 1) lv ( G − 1) sp + 5( G − 1) sv ( G − 1) lp + ˆ ˜ gsl ˜ 2 g − 1 + ˜ where G = ˜ β .
David Rewriting of the NSNS Lagrangian ANDRIOT Introduction � H | 2 � φ � φ | 2 − 1 L = e − 2 ˆ ˆ R + 4 | dˆ � 2 | ˆ De Sitter sol. | ˆ g | Field redefinition � 2 | Q | 2 � φ � φ | 2 − 1 Presentation = e − 2 ˜ R + 4 | d˜ � + ∂ ( . . . ) = ˜ | ˜ g | L + ∂ ( . . . ) Lagrangians Back to 4D Conclusion (assumption: ˜ β km ∂ m · = 0 ) gpq � gkp � 1 R = � � gkm ˜ guq ˜ gps + 2˜ gpq ˜ gks ˜ gmu + guq ˜ gsm ˜ R − ∂ k ˜ gsu ∂ m ˜ 2˜ ˜ 2 1 β pk ∂ m ˜ β qm − β qm ∂ m ˜ β pk gpq ∂ k ˜ gpq ∂ k ˜ − ˜ ˜ 2 gkm ˜ gpq ∂ k ∂ m ˜ gkm ( G − 1) pq ∂ k ∂ mGqp + 2˜ gpq + 2˜ + ∂ mGvl � gks ( G − 1) lv ∂ k ˜ gkm ( G − 1) lv ∂ k ˜ gmr ˜ grs ˜ − 2˜ grs − ˜ grs gvr � gru ( G − 1) lu ∂ v ˜ grs ( G − 1) ls ∂ k ˜ gms ˜ gkm ˜ + ˜ grs − ˜ + ∂ mGvl � glq ∂ vGmq � 1 ( G − 1) lq ∂ vGqm + ˆ 2 gkm � gpv � − ∂ mGvl ∂ k Gps 1 2( G − 1) lv ( G − 1) sp + 5( G − 1) sv ( G − 1) lp + ˆ ˜ gsl ˜ 2 g − 1 + ˜ where G = ˜ β .
David Rewriting of the NSNS Lagrangian ANDRIOT Introduction � H | 2 � φ � φ | 2 − 1 L = e − 2 ˆ ˆ R + 4 | dˆ � 2 | ˆ De Sitter sol. | ˆ g | Field redefinition � 2 | Q | 2 � φ � φ | 2 − 1 Presentation = e − 2 ˜ R + 4 | d˜ � + ∂ ( . . . ) = ˜ | ˜ g | L + ∂ ( . . . ) Lagrangians Back to 4D mn = ∂ k ˜ β mn , | Q | 2 = qr ˜ Conclusion 1 mn Q p g kp ˜ where Q k 2! Q k g mq ˜ g nr (assumption: ˜ β km ∂ m · = 0 )
David Rewriting of the NSNS Lagrangian ANDRIOT Introduction � H | 2 � φ � φ | 2 − 1 L = e − 2 ˆ ˆ R + 4 | dˆ � 2 | ˆ De Sitter sol. | ˆ g | Field redefinition � 2 | R | 2 � φ � φ | 2 − 1 2 | Q | 2 + · · · − 1 Presentation = e − 2 ˜ R + 4 | d˜ � | ˜ g | + ∂ ( . . . ) Lagrangians Back to 4D mn = ∂ k ˜ β mn , | Q | 2 = qr ˜ Conclusion 1 mn Q p g kp ˜ where Q k 2! Q k g mq ˜ g nr (assumption: ˜ β km ∂ m · = 0 ) Without the assumption ⇒ also get R mnp = 3 ˜ β np ] , | R | 2 = 3! R kmn R pqr ˜ β k [ m ∂ k ˜ 1 g kp ˜ g mq ˜ g nr
David Rewriting of the NSNS Lagrangian ANDRIOT Introduction � H | 2 � φ � φ | 2 − 1 L = e − 2 ˆ ˆ R + 4 | dˆ � 2 | ˆ De Sitter sol. | ˆ g | Field redefinition � 2 | R | 2 � φ � φ | 2 − 1 2 | Q | 2 + · · · − 1 Presentation = e − 2 ˜ R + 4 | d˜ � | ˜ g | + ∂ ( . . . ) Lagrangians Back to 4D mn = ∂ k ˜ β mn , | Q | 2 = qr ˜ Conclusion 1 mn Q p g kp ˜ where Q k 2! Q k g mq ˜ g nr (assumption: ˜ β km ∂ m · = 0 ) Without the assumption ⇒ also get R mnp = 3 ˜ β np ] , | R | 2 = 3! R kmn R pqr ˜ β k [ m ∂ k ˜ 1 g kp ˜ g mq ˜ g nr Q -, R -fluxes appear in 10D NSNS via field redefinition Relation to 4D Q -, R -fluxes/non-geo. terms? ⇒ compactification
David Relation to the 4D potential ANDRIOT Introduction We have shown ˆ L = ˜ L + ∂ ( . . . ). De Sitter sol. Field redefinition Presentation Lagrangians Back to 4D Conclusion
David Relation to the 4D potential ANDRIOT Introduction We have shown ˆ L = ˜ L + ∂ ( . . . ). De Sitter sol. → compactify ˜ Field redefinition L ... ֒ Presentation Lagrangians Back to 4D Conclusion
David Relation to the 4D potential ANDRIOT Introduction We have shown ˆ L = ˜ L + ∂ ( . . . ). De Sitter sol. g (0) → compactify ˜ Field redefinition L ... If ˜ g 6 ij → ˜ 6 ij ρ , ֒ Presentation Lagrangians 1 2 | Q | 2 = 1 → ρ V Q , V Q = 1 Back to 4D qr ˜ 2 | Q (0) | 2 , mn Q p g kp ˜ g mq ˜ g nr − 4 Q k Conclusion 1 2 | R | 2 = 1 → ρ 3 V R , V R = 1 12 R kmn R pqr ˜ 2 | R (0) | 2 . g kp ˜ g mq ˜ g nr −
David Relation to the 4D potential ANDRIOT Introduction We have shown ˆ L = ˜ L + ∂ ( . . . ). De Sitter sol. g (0) → compactify ˜ Field redefinition L ... If ˜ g 6 ij → ˜ 6 ij ρ , ֒ Presentation Lagrangians 1 2 | Q | 2 = 1 → ρ V Q , V Q = 1 Back to 4D qr ˜ 2 | Q (0) | 2 , mn Q p g kp ˜ g mq ˜ g nr − 4 Q k Conclusion 1 2 | R | 2 = 1 → ρ 3 V R , V R = 1 12 R kmn R pqr ˜ 2 | R (0) | 2 . g kp ˜ g mq ˜ g nr − ˜ L can give the � potential in 4D (gives a 10D origin) Q -, R -fluxes in 10D ⇒ Q -, R -fluxes in 4D �
David Relation to the 4D potential ANDRIOT Introduction We have shown ˆ L = ˜ L + ∂ ( . . . ). De Sitter sol. g (0) → compactify ˜ Field redefinition L ... If ˜ g 6 ij → ˜ 6 ij ρ , ֒ Presentation Lagrangians 1 2 | Q | 2 = 1 → ρ V Q , V Q = 1 Back to 4D qr ˜ 2 | Q (0) | 2 , mn Q p g kp ˜ g mq ˜ g nr − 4 Q k Conclusion 1 2 | R | 2 = 1 → ρ 3 V R , V R = 1 12 R kmn R pqr ˜ 2 | R (0) | 2 . g kp ˜ g mq ˜ g nr − ˜ L can give the � potential in 4D (gives a 10D origin) Q -, R -fluxes in 10D ⇒ Q -, R -fluxes in 4D � Subtleties on global aspects... (see talk of Dieter Lüst) Discussion on ˜ L rather than ˆ L , discarding ∂ ( . . . ).
David Relation to the 4D potential ANDRIOT Introduction We have shown ˆ L = ˜ L + ∂ ( . . . ). De Sitter sol. g (0) → compactify ˜ Field redefinition L ... If ˜ g 6 ij → ˜ 6 ij ρ , ֒ Presentation Lagrangians 2 | Q | 2 = 1 1 → ρ V Q , V Q = 1 Back to 4D qr ˜ 2 | Q (0) | 2 , mn Q p g kp ˜ g mq ˜ g nr − 4 Q k Conclusion 1 2 | R | 2 = 1 → ρ 3 V R , V R = 1 12 R kmn R pqr ˜ 2 | R (0) | 2 . g kp ˜ g mq ˜ g nr − ˜ L can give the � potential in 4D (gives a 10D origin) Q -, R -fluxes in 10D ⇒ Q -, R -fluxes in 4D � Subtleties on global aspects... (see talk of Dieter Lüst) Discussion on ˜ L rather than ˆ L , discarding ∂ ( . . . ). Good low-en. effective description ( L , fields) of string theory depends on the background. Prescription: use ˜ L for a non-geometric background.
Conclusion David ANDRIOT Introduction De Sitter sol. de Sitter sol. of 10D / 4D SUGRA are difficult to obtain. Field redefinition Unless additional ingredients ⇒ non-geometric terms � Conclusion 10D origin of these terms (of Q , R -fluxes?) g , ˆ B , ˆ g , ˜ β, ˜ GCG ⇒ Field redefinition (ˆ φ ) ↔ (˜ φ ) Rewriting NSNS Lag.: ˆ L = ˜ L + ∂ ( . . . ) , β mn (for ˜ mn = ∂ k ˜ β km ∂ m · = 0 ), R mnp = 3 ˜ β k [ m ∂ k ˜ β np ] 10D Q k Compactification of ˜ L ⇒ 4D potential � Extend to RR sector (S-duality) and D-brane/O-plane sources (new objects?) ֒ → de Sitter solutions in 10D More...
David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion
Non-geometry in 10D and 4D SUGRA David ANDRIOT Non-geometry in 10D Introduction Original idea of non-geometry De Sitter sol. hep-th/0208174 by S. Hellerman, J. McGreevy, B. Williams Field redefinition hep-th/0210209 by A. Dabholkar, C. Hull Conclusion hep-th/0404217 by A. Flournoy, B. Wecht, B. Williams
Non-geometry in 10D and 4D SUGRA David ANDRIOT Non-geometry in 10D Introduction Original idea of non-geometry De Sitter sol. hep-th/0208174 by S. Hellerman, J. McGreevy, B. Williams Field redefinition hep-th/0210209 by A. Dabholkar, C. Hull Conclusion hep-th/0404217 by A. Flournoy, B. Wecht, B. Williams A (target) space, divided in patches Fields glue with transition functions: diffeomorphisms, gauge transformation (point-like symmetries)
Non-geometry in 10D and 4D SUGRA David ANDRIOT Non-geometry in 10D Introduction Original idea of non-geometry De Sitter sol. hep-th/0208174 by S. Hellerman, J. McGreevy, B. Williams Field redefinition hep-th/0210209 by A. Dabholkar, C. Hull Conclusion hep-th/0404217 by A. Flournoy, B. Wecht, B. Williams A (target) space, divided in patches Fields glue with transition functions: diffeomorphisms, gauge transformation (point-like symmetries) String theory has more... ֒ → use stringy symmetries for gluing
Non-geometry in 10D and 4D SUGRA David ANDRIOT Non-geometry in 10D Introduction Original idea of non-geometry De Sitter sol. hep-th/0208174 by S. Hellerman, J. McGreevy, B. Williams Field redefinition hep-th/0210209 by A. Dabholkar, C. Hull Conclusion hep-th/0404217 by A. Flournoy, B. Wecht, B. Williams A (target) space, divided in patches Fields glue with transition functions: diffeomorphisms, gauge transformation (point-like symmetries) String theory has more... ֒ → use stringy symmetries for gluing Away from standard geometry ֒ → non-geometry
Non-geometry in 10D and 4D SUGRA David ANDRIOT Non-geometry in 10D Introduction Original idea of non-geometry De Sitter sol. hep-th/0208174 by S. Hellerman, J. McGreevy, B. Williams Field redefinition hep-th/0210209 by A. Dabholkar, C. Hull Conclusion hep-th/0404217 by A. Flournoy, B. Wecht, B. Williams A (target) space, divided in patches Fields glue with transition functions: diffeomorphisms, gauge transformation (point-like symmetries) String theory has more... ֒ → use stringy symmetries for gluing Away from standard geometry ֒ → non-geometry Fields look ill-defined: not single-valued, global issues Simple example: T-duality: circle of radius R → 1 R
Non-geometry in 10D and 4D SUGRA David ANDRIOT Non-geometry in 10D Introduction Original idea of non-geometry De Sitter sol. hep-th/0208174 by S. Hellerman, J. McGreevy, B. Williams Field redefinition hep-th/0210209 by A. Dabholkar, C. Hull Conclusion hep-th/0404217 by A. Flournoy, B. Wecht, B. Williams A (target) space, divided in patches Fields glue with transition functions: diffeomorphisms, gauge transformation (point-like symmetries) R String theory has more... 1 /R ֒ → use stringy symmetries for gluing Away from standard geometry ֒ → non-geometry 0 ≡ 2 πR y Fields look ill-defined: not single-valued, global issues G. Moutsopoulos PhD 2008 Simple example: T-duality: circle of radius R → 1 R
Non-geometry in 10D and 4D SUGRA David ANDRIOT Non-geometry in 10D Introduction Original idea of non-geometry De Sitter sol. hep-th/0208174 by S. Hellerman, J. McGreevy, B. Williams Field redefinition hep-th/0210209 by A. Dabholkar, C. Hull Conclusion hep-th/0404217 by A. Flournoy, B. Wecht, B. Williams A (target) space, divided in patches Fields glue with transition functions: diffeomorphisms, gauge transformation (point-like symmetries) R String theory has more... 1 /R ֒ → use stringy symmetries for gluing Away from standard geometry ֒ → non-geometry 0 ≡ 2 πR y Fields look ill-defined: not single-valued, global issues G. Moutsopoulos PhD 2008 Simple example: T-duality: circle of radius R → 1 R Famous toroidal example torus T 3 + ˆ T a T b → twisted torus ( f a − − bc ) − → non − geometric config . H abc hep-th/0211182 by S. Kachru, M. B. Schulz, P. K. Tripathy, S. P. Trivedi hep-th/0303173 by D. A. Lowe, H. Nastase, S. Ramgoolam
In 4D: non-geometric terms in the potential David Terms introduced in the superpot., for T-duality covariance. ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion
In 4D: non-geometric terms in the potential David Terms introduced in the superpot., for T-duality covariance. ANDRIOT T-duality: symmetry of string theory on a background with Introduction isometries. De Sitter sol. Field redefinition Conclusion
In 4D: non-geometric terms in the potential David Terms introduced in the superpot., for T-duality covariance. ANDRIOT T-duality: symmetry of string theory on a background with Introduction isometries. De Sitter sol. Compactification on such a background Field redefinition ⇒ 4D effective theory inherits this symmetry. Conclusion
In 4D: non-geometric terms in the potential David Terms introduced in the superpot., for T-duality covariance. ANDRIOT T-duality: symmetry of string theory on a background with Introduction isometries. De Sitter sol. Compactification on such a background Field redefinition ⇒ 4D effective theory inherits this symmetry. Conclusion ֒ → require T-duality covariance on the 4D superpotential.
In 4D: non-geometric terms in the potential David Terms introduced in the superpot., for T-duality covariance. ANDRIOT T-duality: symmetry of string theory on a background with Introduction isometries. De Sitter sol. Compactification on such a background Field redefinition ⇒ 4D effective theory inherits this symmetry. Conclusion ֒ → require T-duality covariance on the 4D superpotential. ˆ H abc on M generates a term in the 4D potential. ֒ → if T-duality along a , b , c , how does the term transform?
In 4D: non-geometric terms in the potential David Terms introduced in the superpot., for T-duality covariance. ANDRIOT T-duality: symmetry of string theory on a background with Introduction isometries. De Sitter sol. Compactification on such a background Field redefinition ⇒ 4D effective theory inherits this symmetry. Conclusion ֒ → require T-duality covariance on the 4D superpotential. ˆ H abc on M generates a term in the 4D potential. ֒ → if T-duality along a , b , c , how does the term transform? T a Term generated by ˆ → term gen. by f a − − bc (curvature) H abc
In 4D: non-geometric terms in the potential David Terms introduced in the superpot., for T-duality covariance. ANDRIOT T-duality: symmetry of string theory on a background with Introduction isometries. De Sitter sol. Compactification on such a background Field redefinition ⇒ 4D effective theory inherits this symmetry. Conclusion ֒ → require T-duality covariance on the 4D superpotential. ˆ H abc on M generates a term in the 4D potential. ֒ → if T-duality along a , b , c , how does the term transform? T a Term generated by ˆ → term gen. by f a − − bc (curvature) H abc T b , T c ab , R abc − − − − → new terms needed ! Generated by Q c
In 4D: non-geometric terms in the potential David Terms introduced in the superpot., for T-duality covariance. ANDRIOT T-duality: symmetry of string theory on a background with Introduction isometries. De Sitter sol. Compactification on such a background Field redefinition ⇒ 4D effective theory inherits this symmetry. Conclusion ֒ → require T-duality covariance on the 4D superpotential. ˆ H abc on M generates a term in the 4D potential. ֒ → if T-duality along a , b , c , how does the term transform? T a Term generated by ˆ → term gen. by f a − − bc (curvature) H abc T b , T c ab , R abc − − − − → new terms needed ! Generated by Q c 4D T-duality chain: T a T b T c ˆ ab → f a → R abc − − − → Q c − − H abc bc hep-th/0508133, hep-th/0607015 by J. Shelton, W. Taylor, B. Wecht
In 4D: non-geometric terms in the potential David Terms introduced in the superpot., for T-duality covariance. ANDRIOT T-duality: symmetry of string theory on a background with Introduction isometries. De Sitter sol. Compactification on such a background Field redefinition ⇒ 4D effective theory inherits this symmetry. Conclusion ֒ → require T-duality covariance on the 4D superpotential. ˆ H abc on M generates a term in the 4D potential. ֒ → if T-duality along a , b , c , how does the term transform? T a Term generated by ˆ → term gen. by f a − − bc (curvature) H abc T b , T c ab , R abc − − − − → new terms needed ! Generated by Q c 4D T-duality chain: T a T b T c ˆ ab → f a → R abc − − − → Q c − − H abc bc hep-th/0508133, hep-th/0607015 by J. Shelton, W. Taylor, B. Wecht Toroidal example: Q , R , correspond to non-geometric config... On the contrary to ˆ H , f , no 10D interpretation of Q , R .
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