[PPT] - Towards de Sitter solutions Introduction De Sitter sol. of string PowerPoint Presentation

SLIDE 1

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

Towards de Sitter solutions

f string theory via non-geometry

David ANDRIOT

ASC, LMU, Munich, Germany

arXiv:1003.3774 by D. A., E. Goi, R. Minasian, M. Petrini arXiv:1106.4015 by D. A., M. Larfors, D. Lüst, P. Patalong arXiv:1202.3060 by D. A., O. Hohm, M. Larfors, D. Lüst, P. Patalong Work in progress by D. A., O. Hohm, M. Larfors, D. Lüst, P. Patalong

17/03/2012, UPenn, Philadelphia, USA

SLIDE 2

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

Introduction

Cosmological observations ֒ → de Sitter solutions of string theory?

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David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

Introduction

Cosmological observations ֒ → de Sitter solutions of string theory? Set-up: compactification 10D string theory, low en. eff. theory: 10D SUGRA

SLIDE 4

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

Introduction

Cosmological observations ֒ → de Sitter solutions of string theory? Set-up: compactification 10D string theory, low en. eff. theory: 10D SUGRA Split 10D ⇒ 4D max. sym. space-time × 6D compact M

SLIDE 5

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

Introduction

Cosmological observations ֒ → de Sitter solutions of string theory? Set-up: compactification 10D string theory, low en. eff. theory: 10D SUGRA Split 10D ⇒ 4D max. sym. space-time × 6D compact M Compactification procedure: 10D theory ⇒ 4D theory (based on a solution to 10D e.o.m.: vacuum)

SLIDE 6

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

Introduction

Cosmological observations ֒ → de Sitter solutions of string theory? Set-up: compactification 10D string theory, low en. eff. theory: 10D SUGRA Split 10D ⇒ 4D max. sym. space-time × 6D compact M Compactification procedure: 10D theory ⇒ 4D theory (based on a solution to 10D e.o.m.: vacuum) SUGRA 10D : 4D × 6D M

compactif. on M
SUGRA 4D

SLIDE 7

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

Introduction

Cosmological observations ֒ → de Sitter solutions of string theory? Set-up: compactification 10D string theory, low en. eff. theory: 10D SUGRA Split 10D ⇒ 4D max. sym. space-time × 6D compact M Compactification procedure: 10D theory ⇒ 4D theory (based on a solution to 10D e.o.m.: vacuum) SUGRA 10D : 4D × 6D M

compactif. on M
SUGRA 4D (potential V (ϕ))

SLIDE 8

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

Introduction

Cosmological observations ֒ → de Sitter solutions of string theory? Set-up: compactification 10D string theory, low en. eff. theory: 10D SUGRA Split 10D ⇒ 4D max. sym. space-time × 6D compact M Compactification procedure: 10D theory ⇒ 4D theory (based on a solution to 10D e.o.m.: vacuum) SUGRA 10D : 4D × 6D M

compactif. on M
SUGRA 4D (potential V (ϕ))

Vacuum of 4D theory:

∂V ∂ϕ = 0

⇔ solution to 10D e.o.m.

SLIDE 9

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

Introduction

Cosmological observations ֒ → de Sitter solutions of string theory? Set-up: compactification 10D string theory, low en. eff. theory: 10D SUGRA Split 10D ⇒ 4D max. sym. space-time × 6D compact M Compactification procedure: 10D theory ⇒ 4D theory (based on a solution to 10D e.o.m.: vacuum) SUGRA 10D : 4D × 6D M

compactif. on M
SUGRA 4D (potential V (ϕ))

Vacuum of 4D theory:

∂V ∂ϕ = 0

⇔ solution to 10D e.o.m. de Sitter (dS) solution: 4D space-time = dS

SLIDE 10

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

Introduction

Cosmological observations ֒ → de Sitter solutions of string theory? Set-up: compactification 10D string theory, low en. eff. theory: 10D SUGRA Split 10D ⇒ 4D max. sym. space-time × 6D compact M Compactification procedure: 10D theory ⇒ 4D theory (based on a solution to 10D e.o.m.: vacuum) SUGRA 10D : 4D × 6D M

compactif. on M
SUGRA 4D (potential V (ϕ))

Vacuum of 4D theory:

∂V ∂ϕ = 0

⇔ solution to 10D e.o.m. de Sitter (dS) solution: 4D space-time = dS 10D solution: ds2

10 = ds2 4 + ds2 6, R4 > 0 (depends on 6D...)

SLIDE 11

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

Introduction

Cosmological observations ֒ → de Sitter solutions of string theory? Set-up: compactification 10D string theory, low en. eff. theory: 10D SUGRA Split 10D ⇒ 4D max. sym. space-time × 6D compact M Compactification procedure: 10D theory ⇒ 4D theory (based on a solution to 10D e.o.m.: vacuum) SUGRA 10D : 4D × 6D M

compactif. on M
SUGRA 4D (potential V (ϕ))

Vacuum of 4D theory:

∂V ∂ϕ = 0

⇔ solution to 10D e.o.m. de Sitter (dS) solution: 4D space-time = dS 10D solution: ds2

10 = ds2 4 + ds2 6, R4 > 0 (depends on 6D...),

4D perspective: R4 ∼ Λ ∼ V |0 > 0.

SLIDE 12

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

Very difficult to get de Sitter solutions of 10D SUGRA. Several reasons:

SLIDE 13

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

Very difficult to get de Sitter solutions of 10D SUGRA. Several reasons:

technical: SUSY is broken ⇒ resolution is difficult.

SLIDE 14

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

Very difficult to get de Sitter solutions of 10D SUGRA. Several reasons:

technical: SUSY is broken ⇒ resolution is difficult. inherent to 10D SUGRA: generally R4 ≤ 0 ...

SLIDE 15

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

Very difficult to get de Sitter solutions of 10D SUGRA. Several reasons:

technical: SUSY is broken ⇒ resolution is difficult. inherent to 10D SUGRA: generally R4 ≤ 0 ... not talking of stability.

SLIDE 16

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

Very difficult to get de Sitter solutions of 10D SUGRA. Several reasons:

technical: SUSY is broken ⇒ resolution is difficult. inherent to 10D SUGRA: generally R4 ≤ 0 ... not talking of stability.

4D perspective: no-go theorems: V |0 ≤ 0 (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

SLIDE 17

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

Very difficult to get de Sitter solutions of 10D SUGRA. Several reasons:

technical: SUSY is broken ⇒ resolution is difficult. inherent to 10D SUGRA: generally R4 ≤ 0 ... not talking of stability.

4D perspective: no-go theorems: V |0 ≤ 0 (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.

SLIDE 18

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

Very difficult to get de Sitter solutions of 10D SUGRA. Several reasons:

technical: SUSY is broken ⇒ resolution is difficult. inherent to 10D SUGRA: generally R4 ≤ 0 ... not talking of stability.

4D perspective: no-go theorems: V |0 ≤ 0 (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?

SLIDE 19

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

Very difficult to get de Sitter solutions of 10D SUGRA. Several reasons:

technical: SUSY is broken ⇒ resolution is difficult. inherent to 10D SUGRA: generally R4 ≤ 0 ... not talking of stability.

4D perspective: no-go theorems: V |0 ≤ 0 (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?

SLIDE 20

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

Very difficult to get de Sitter solutions of 10D SUGRA. Several reasons:

technical: SUSY is broken ⇒ resolution is difficult. inherent to 10D SUGRA: generally R4 ≤ 0 ... not talking of stability.

4D perspective: no-go theorems: V |0 ≤ 0 (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 4D Vgeo.(ϕ)

SUGRA 4D Vnon−geo.(ϕ)

SLIDE 21

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

Very difficult to get de Sitter solutions of 10D SUGRA. Several reasons:

technical: SUSY is broken ⇒ resolution is difficult. inherent to 10D SUGRA: generally R4 ≤ 0 ... not talking of stability.

4D perspective: no-go theorems: V |0 ≤ 0 (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 4D Vgeo.(ϕ)

gauging

SUGRA 4D Vnon−geo.(ϕ)

SLIDE 22

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

Very difficult to get de Sitter solutions of 10D SUGRA. Several reasons:

technical: SUSY is broken ⇒ resolution is difficult. inherent to 10D SUGRA: generally R4 ≤ 0 ... not talking of stability.

4D perspective: no-go theorems: V |0 ≤ 0 (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
SUGRA 4D Vgeo.(ϕ)

gauging

SUGRA 4D Vnon−geo.(ϕ)

SLIDE 23

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

Very difficult to get de Sitter solutions of 10D SUGRA. Several reasons:

technical: SUSY is broken ⇒ resolution is difficult. inherent to 10D SUGRA: generally R4 ≤ 0 ... not talking of stability.

4D perspective: no-go theorems: V |0 ≤ 0 (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
??
SUGRA 4D Vgeo.(ϕ)

gauging

SUGRA 4D Vnon−geo.(ϕ)

SLIDE 24

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

10D origin of 4D non-geometric terms: two problems:

SLIDE 25

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

10D origin of 4D non-geometric terms: two problems: Focus on NSNS sector: ˆ gmn, ˆ φ, ˆ Bmn, ˆ Hkmn = 3 ∂[k ˆ Bmn].

SLIDE 26

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

10D origin of 4D non-geometric terms: two problems: Focus on NSNS sector: ˆ gmn, ˆ φ, ˆ Bmn, ˆ Hkmn = 3 ∂[k ˆ Bmn]. Compactification over M ֒ → in 4D V (ϕ), terms generated by ˆ Γk

mn|0, ˆ

Hkmn|0.

SLIDE 27

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

10D origin of 4D non-geometric terms: two problems: Focus on NSNS sector: ˆ gmn, ˆ φ, ˆ Bmn, ˆ Hkmn = 3 ∂[k ˆ Bmn]. Compactification over M ֒ → in 4D V (ϕ), terms generated by ˆ Γk

mn|0, ˆ

Hkmn|0. But 4D non-geo. terms generated by “fluxes”: Qkmn, Rkmn + different dependence on scalars.

SLIDE 28

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

10D origin of 4D non-geometric terms: two problems: Focus on NSNS sector: ˆ gmn, ˆ φ, ˆ Bmn, ˆ Hkmn = 3 ∂[k ˆ Bmn]. Compactification over M ֒ → in 4D V (ϕ), terms generated by ˆ Γk

mn|0, ˆ

Hkmn|0. But 4D non-geo. terms generated by “fluxes”: Qkmn, Rkmn + different dependence on scalars. ֒ → 10D origin of Q and R-fluxes in NSNS sector?

SLIDE 29

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

10D origin of 4D non-geometric terms: two problems: Focus on NSNS sector: ˆ gmn, ˆ φ, ˆ Bmn, ˆ Hkmn = 3 ∂[k ˆ Bmn]. Compactification over M ֒ → in 4D V (ϕ), terms generated by ˆ Γk

mn|0, ˆ

Hkmn|0. But 4D non-geo. terms generated by “fluxes”: Qkmn, Rkmn + different dependence on scalars. ֒ → 10D origin of Q and R-fluxes in NSNS sector? 10D non-geometry: different. Relation to 4D unclear.

SLIDE 30

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

10D origin of 4D non-geometric terms: two problems: Focus on NSNS sector: ˆ gmn, ˆ φ, ˆ Bmn, ˆ Hkmn = 3 ∂[k ˆ Bmn]. Compactification over M ֒ → in 4D V (ϕ), terms generated by ˆ Γk

mn|0, ˆ

Hkmn|0. But 4D non-geo. terms generated by “fluxes”: Qkmn, Rkmn + 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?

SLIDE 31

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

10D origin of 4D non-geometric terms: two problems: Focus on NSNS sector: ˆ gmn, ˆ φ, ˆ Bmn, ˆ Hkmn = 3 ∂[k ˆ Bmn]. Compactification over M ֒ → in 4D V (ϕ), terms generated by ˆ Γk

mn|0, ˆ

Hkmn|0. But 4D non-geo. terms generated by “fluxes”: Qkmn, Rkmn + 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:

SLIDE 32

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

10D origin of 4D non-geometric terms: two problems: Focus on NSNS sector: ˆ gmn, ˆ φ, ˆ Bmn, ˆ Hkmn = 3 ∂[k ˆ Bmn]. Compactification over M ֒ → in 4D V (ϕ), terms generated by ˆ Γk

mn|0, ˆ

Hkmn|0. But 4D non-geo. terms generated by “fluxes”: Qkmn, Rkmn + 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.

SLIDE 33

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

10D origin of 4D non-geometric terms: two problems: Focus on NSNS sector: ˆ gmn, ˆ φ, ˆ Bmn, ˆ Hkmn = 3 ∂[k ˆ Bmn]. Compactification over M ֒ → in 4D V (ϕ), terms generated by ˆ Γk

mn|0, ˆ

Hkmn|0. But 4D non-geo. terms generated by “fluxes”: Qkmn, Rkmn + 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)

SLIDE 34

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

10D origin of 4D non-geometric terms: two problems: Focus on NSNS sector: ˆ gmn, ˆ φ, ˆ Bmn, ˆ Hkmn = 3 ∂[k ˆ Bmn]. Compactification over M ֒ → in 4D V (ϕ), terms generated by ˆ Γk

mn|0, ˆ

Hkmn|0. But 4D non-geo. terms generated by “fluxes”: Qkmn, Rkmn + 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.

SLIDE 35

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) .

SLIDE 36

David ANDRIOT Introduction De Sitter sol.

10D 4D geometric 4D non-geometric

Field redefinition Conclusion

De Sitter solutions of 10D and 4D SUGRA

10D discussion

10D IIA SUGRA, with O6-planes/D6-branes (common set-up to look for dS solutions). Bosonic content: NSNS sector, RR sector: F0, F2, (no F4, F6). O6/D6 source of F2.

SLIDE 37

David ANDRIOT Introduction De Sitter sol.

10D 4D geometric 4D non-geometric

Field redefinition Conclusion

De Sitter solutions of 10D and 4D SUGRA

10D discussion

10D IIA SUGRA, with O6-planes/D6-branes (common set-up to look for dS solutions). Bosonic content: NSNS sector, RR sector: F0, F2, (no F4, F6). O6/D6 source of F2. SIIA = 1 2κ2

d10x
|ˆ

g10| [LNSNS + LRR + Lsources] , LNSNS = e−2 ˆ

φ(

R10 + 4|∇ˆ φ|2 − 1 2| ˆ H|2) , LRR = −1 2(|F0|2 + |F2|2) , Fm1...mpF m1...mp p! = |Fp|2 .

SLIDE 38

David ANDRIOT Introduction De Sitter sol.

10D 4D geometric 4D non-geometric

Field redefinition Conclusion

De Sitter solutions of 10D and 4D SUGRA

10D discussion

10D IIA SUGRA, with O6-planes/D6-branes (common set-up to look for dS solutions). Bosonic content: NSNS sector, RR sector: F0, F2, (no F4, F6). O6/D6 source of F2. SIIA = 1 2κ2

d10x
|ˆ

g10| [LNSNS + LRR + Lsources] , LNSNS = e−2 ˆ

φ(

R10 + 4|∇ˆ φ|2 − 1 2| ˆ H|2) , LRR = −1 2(|F0|2 + |F2|2) , Fm1...mpF m1...mp p! = |Fp|2 . Derive 10D e.o.m.

SLIDE 39

David ANDRIOT Introduction De Sitter sol.

10D 4D geometric 4D non-geometric

Field redefinition Conclusion

De Sitter solutions of 10D and 4D SUGRA

10D discussion

10D IIA SUGRA, with O6-planes/D6-branes (common set-up to look for dS solutions). Bosonic content: NSNS sector, RR sector: F0, F2, (no F4, F6). O6/D6 source of F2. SIIA = 1 2κ2

d10x
|ˆ

g10| [LNSNS + LRR + Lsources] , LNSNS = e−2 ˆ

φ(

R10 + 4|∇ˆ φ|2 − 1 2| ˆ H|2) , LRR = −1 2(|F0|2 + |F2|2) , Fm1...mpF m1...mp p! = |Fp|2 . Derive 10D e.o.m. Compactification ansatz: ds2

10 = ds2 4 + ds2 6 (no warp factor),

non-trivial fluxes only along M, constant dilaton e ˆ

φ = gs.

SLIDE 40

David ANDRIOT Introduction De Sitter sol.

10D 4D geometric 4D non-geometric

Field redefinition Conclusion

De Sitter solutions of 10D and 4D SUGRA

10D discussion

10D IIA SUGRA, with O6-planes/D6-branes (common set-up to look for dS solutions). Bosonic content: NSNS sector, RR sector: F0, F2, (no F4, F6). O6/D6 source of F2. SIIA = 1 2κ2

d10x
|ˆ

g10| [LNSNS + LRR + Lsources] , LNSNS = e−2 ˆ

φ(

R10 + 4|∇ˆ φ|2 − 1 2| ˆ H|2) , LRR = −1 2(|F0|2 + |F2|2) , Fm1...mpF m1...mp p! = |Fp|2 . Derive 10D e.o.m. Compactification ansatz: ds2

10 = ds2 4 + ds2 6 (no warp factor),

non-trivial fluxes only along M, constant dilaton e ˆ

φ = gs.

Combine (4D, 6D) trace of Einstein and dilaton e.o.m.

SLIDE 41

David ANDRIOT Introduction De Sitter sol.

10D 4D geometric 4D non-geometric

Field redefinition Conclusion

De Sitter solutions of 10D and 4D SUGRA

10D discussion

10D IIA SUGRA, with O6-planes/D6-branes (common set-up to look for dS solutions). Bosonic content: NSNS sector, RR sector: F0, F2, (no F4, F6). O6/D6 source of F2. SIIA = 1 2κ2

d10x
|ˆ

g10| [LNSNS + LRR + Lsources] , LNSNS = e−2 ˆ

φ(

R10 + 4|∇ˆ φ|2 − 1 2| ˆ H|2) , LRR = −1 2(|F0|2 + |F2|2) , Fm1...mpF m1...mp p! = |Fp|2 . Derive 10D e.o.m. Compactification ansatz: ds2

10 = ds2 4 + ds2 6 (no warp factor),

non-trivial fluxes only along M, constant dilaton e ˆ

φ = gs.

Combine (4D, 6D) trace of Einstein and dilaton e.o.m.: 3Λ = 3 4

R4 = 1

2

g2

s |F0|2 − | ˆ

H|2 = 1 3

−

R6 − 1 2g2

s |F2|2

SLIDE 42

David ANDRIOT Introduction De Sitter sol.

10D 4D geometric 4D non-geometric

Field redefinition Conclusion

Obtain de Sitter solution: 3Λ = 3 4

R4 = 1

2

g2

s |F0|2 − | ˆ

H|2 = 1 3

−

R6 − 1 2g2

s |F2|2

> 0

SLIDE 43

David ANDRIOT Introduction De Sitter sol.

10D 4D geometric 4D non-geometric

Field redefinition Conclusion

Obtain de Sitter solution: 3Λ = 3 4

R4 = 1

2

g2

s |F0|2 − | ˆ

H|2 = 1 3

−

R6 − 1 2g2

s |F2|2

> 0

֒ → need at least F0 = 0, R6 < 0 !

hep-th/0007018 by J. M. Maldacena, C. Núñez arXiv:0810.5328 by S. S. Haque, G. Shiu, B. Underwood, T. Van Riet

SLIDE 44

David ANDRIOT Introduction De Sitter sol.

10D 4D geometric 4D non-geometric

Field redefinition Conclusion

Obtain de Sitter solution: 3Λ = 3 4

R4 = 1

2

g2

s |F0|2 − | ˆ

H|2 = 1 3

−

R6 − 1 2g2

s |F2|2

> 0

֒ → need at least F0 = 0, R6 < 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 F0 and ˆ H not independent (B-field e.o.m., F2 B.I.)...

SLIDE 45

David ANDRIOT Introduction De Sitter sol.

10D 4D geometric 4D non-geometric

Field redefinition Conclusion

Obtain de Sitter solution: 3Λ = 3 4

R4 = 1

2

g2

s |F0|2 − | ˆ

H|2 = 1 3

−

R6 − 1 2g2

s |F2|2

> 0

֒ → need at least F0 = 0, R6 < 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 F0 and ˆ H not independent (B-field e.o.m., F2 B.I.)... ֒ → In most of the examples: AdS, Minkowski.

SLIDE 46

David ANDRIOT Introduction De Sitter sol.

10D 4D geometric 4D non-geometric

Field redefinition Conclusion

Obtain de Sitter solution: 3Λ = 3 4

R4 = 1

2

g2

s |F0|2 − | ˆ

H|2 = 1 3

−

R6 − 1 2g2

s |F2|2

> 0

֒ → need at least F0 = 0, R6 < 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 F0 and ˆ H not independent (B-field e.o.m., F2 B.I.)... ֒ → In most of the examples: AdS, Minkowski. F4, F6 do not help: enter with a minus... de Sitter not favored.

SLIDE 47

David ANDRIOT Introduction De Sitter sol.

10D 4D geometric 4D non-geometric

Field redefinition Conclusion

4D discussion: the scalar potential

V (ρ, σ): only consider volume ρ and dilaton σ, always appear.

SLIDE 48

David ANDRIOT Introduction De Sitter sol.

10D 4D geometric 4D non-geometric

Field redefinition Conclusion

4D discussion: the scalar potential

V (ρ, σ): only consider volume ρ and dilaton σ, always appear. ˆ g6ij → ˆ g(0)

6ij ρ , e ˆ φ → e ˆ φ(0)e ˆ ϕ , σ = ρ

3 2 e− ˆ

ϕ , ˆ

H, Fp → ˆ H (0), F (0)

p

SLIDE 49

David ANDRIOT Introduction De Sitter sol.

10D 4D geometric 4D non-geometric

Field redefinition Conclusion

4D discussion: the scalar potential

V (ρ, σ): only consider volume ρ and dilaton σ, always appear. ˆ g6ij → ˆ g(0)

6ij ρ , e ˆ φ → e ˆ φ(0)e ˆ ϕ , σ = ρ

3 2 e− ˆ

ϕ , ˆ

H, Fp → ˆ H (0), F (0)

p

Compactification procedure: from 10D SIIA, get

S = M 2

4

d4x
|ˆ

gE

4 |

RE

4 + kin − V

M 2

4

V (ρ, σ)

M 2

4

= σ−2(ρ−1VR + ρ−3VH) + ρ3σ−4(VF0 + ρ−2VF2) − σ−3Vsources where VR = − R6 , VH = 1 2|H|2 , VFp = 1 2g2

s |Fp|2 .

arXiv:0712.1196 by E. Silverstein

SLIDE 50

David ANDRIOT Introduction De Sitter sol.

10D 4D geometric 4D non-geometric

Field redefinition Conclusion

4D discussion: the scalar potential

V (ρ, σ): only consider volume ρ and dilaton σ, always appear. ˆ g6ij → ˆ g(0)

6ij ρ , e ˆ φ → e ˆ φ(0)e ˆ ϕ , σ = ρ

3 2 e− ˆ

ϕ , ˆ

H, Fp → ˆ H (0), F (0)

p

Compactification procedure: from 10D SIIA, get

S = M 2

4

d4x
|ˆ

gE

4 |

RE

4 + kin − V

M 2

4

⇒

Λ = 1 2M 2

4

V |0 V (ρ, σ) M 2

4

= σ−2(ρ−1VR + ρ−3VH) + ρ3σ−4(VF0 + ρ−2VF2) − σ−3Vsources where VR = − R6 , VH = 1 2|H|2 , VFp = 1 2g2

s |Fp|2 .

arXiv:0712.1196 by E. Silverstein

SLIDE 51

David ANDRIOT Introduction De Sitter sol.

10D 4D geometric 4D non-geometric

Field redefinition Conclusion

4D discussion: the scalar potential

V (ρ, σ): only consider volume ρ and dilaton σ, always appear. ˆ g6ij → ˆ g(0)

6ij ρ , e ˆ φ → e ˆ φ(0)e ˆ ϕ , σ = ρ

3 2 e− ˆ

ϕ , ˆ

H, Fp → ˆ H (0), F (0)

p

Compactification procedure: from 10D SIIA, get

S = M 2

4

d4x
|ˆ

gE

4 |

RE

4 + kin − V

M 2

4

⇒

Λ = 1 2M 2

4

V |0 V (ρ, σ) M 2

4

= σ−2(ρ−1VR + ρ−3VH) + ρ3σ−4(VF0 + ρ−2VF2) − σ−3Vsources where VR = − R6 , VH = 1 2|H|2 , VFp = 1 2g2

s |Fp|2 .

arXiv:0712.1196 by E. Silverstein

Get V |0? Extremize the potential:

∂V ∂ρ |0 = ∂V ∂σ |0 = 0.

SLIDE 52

David ANDRIOT Introduction De Sitter sol.

10D 4D geometric 4D non-geometric

Field redefinition Conclusion

4D discussion: the scalar potential

V (ρ, σ): only consider volume ρ and dilaton σ, always appear. ˆ g6ij → ˆ g(0)

6ij ρ , e ˆ φ → e ˆ φ(0)e ˆ ϕ , σ = ρ

3 2 e− ˆ

ϕ , ˆ

H, Fp → ˆ H (0), F (0)

p

Compactification procedure: from 10D SIIA, get

S = M 2

4

d4x
|ˆ

gE

4 |

RE

4 + kin − V

M 2

4

⇒

Λ = 1 2M 2

4

V |0 V (ρ, σ) M 2

4

= σ−2(ρ−1VR + ρ−3VH) + ρ3σ−4(VF0 + ρ−2VF2) − σ−3Vsources where VR = − R6 , VH = 1 2|H|2 , VFp = 1 2g2

s |Fp|2 .

arXiv:0712.1196 by E. Silverstein

Get V |0? Extremize the potential:

∂V ∂ρ |0 = ∂V ∂σ |0 = 0.

֒ → Combine and get 3Λ = 3 2 V |0 M 2

4

= VF0 − VH = 1 3 (VR − VF2)

SLIDE 53

David ANDRIOT Introduction De Sitter sol.

10D 4D geometric 4D non-geometric

Field redefinition Conclusion

4D discussion: the scalar potential

V (ρ, σ): only consider volume ρ and dilaton σ, always appear. ˆ g6ij → ˆ g(0)

6ij ρ , e ˆ φ → e ˆ φ(0)e ˆ ϕ , σ = ρ

3 2 e− ˆ

ϕ , ˆ

H, Fp → ˆ H (0), F (0)

p

Compactification procedure: from 10D SIIA, get

S = M 2

4

d4x
|ˆ

gE

4 |

RE

4 + kin − V

M 2

4

⇒

Λ = 1 2M 2

4

V |0 V (ρ, σ) M 2

4

= σ−2(ρ−1VR + ρ−3VH) + ρ3σ−4(VF0 + ρ−2VF2) − σ−3Vsources where VR = − R6 , VH = 1 2|H|2 , VFp = 1 2g2

s |Fp|2 .

arXiv:0712.1196 by E. Silverstein

Get V |0? Extremize the potential:

∂V ∂ρ |0 = ∂V ∂σ |0 = 0.

֒ → Combine and get 3Λ = 3 2 V |0 M 2

4

= VF0 − VH = 1 3 (VR − VF2) Same relations as in 10D... Same difficulty to get de Sitter.

SLIDE 54

David ANDRIOT Introduction De Sitter sol.

10D 4D geometric 4D non-geometric

Field redefinition Conclusion

4D discussion: the scalar potential

V (ρ, σ): only consider volume ρ and dilaton σ, always appear. ˆ g6ij → ˆ g(0)

6ij ρ , e ˆ φ → e ˆ φ(0)e ˆ ϕ , σ = ρ

3 2 e− ˆ

ϕ , ˆ

H, Fp → ˆ H (0), F (0)

p

Compactification procedure: from 10D SIIA, get

S = M 2

4

d4x
|ˆ

gE

4 |

RE

4 + kin − V

M 2

4

⇒

Λ = 1 2M 2

4

V |0 V (ρ, σ) M 2

4

= σ−2(ρ−1VR + ρ−3VH) + ρ3σ−4(VF0 + ρ−2VF2) − σ−3Vsources where VR = − R6 , VH = 1 2|H|2 , VFp = 1 2g2

s |Fp|2 .

arXiv:0712.1196 by E. Silverstein

Get V |0? Extremize the potential:

∂V ∂ρ |0 = ∂V ∂σ |0 = 0.

֒ → Combine and get 3Λ = 3 2 V |0 M 2

4

= VF0 − VH = 1 3 (VR − VF2) Same relations as in 10D... Same difficulty to get de Sitter. ֒ → Non-geometric terms...

SLIDE 55

David ANDRIOT Introduction De Sitter sol.

10D 4D geometric 4D non-geometric

Field redefinition Conclusion

4D non-geometric terms

Long history, motivation for these terms... Here, pratical point of view: allowed in 4D (super)potential.

SLIDE 56

David ANDRIOT Introduction De Sitter sol.

10D 4D geometric 4D non-geometric

Field redefinition Conclusion

4D non-geometric terms

Long history, motivation for these terms... Here, pratical point of view: allowed in 4D (super)potential. Scalar potential: VNSNS ∼ ρ−3VH + ρ−1VR + ρVQ + ρ3VR

arXiv:0711.2512 by M. P. Hertzberg, S. Kachru, W. Taylor, M. Tegmark

SLIDE 57

David ANDRIOT Introduction De Sitter sol.

10D 4D geometric 4D non-geometric

Field redefinition Conclusion

4D non-geometric terms

Long history, motivation for these terms... Here, pratical point of view: allowed in 4D (super)potential. Scalar potential: VNSNS ∼ ρ−3VH + ρ−1VR + ρVQ + ρ3VR

arXiv:0711.2512 by M. P. Hertzberg, S. Kachru, W. Taylor, M. Tegmark

With Q, R, and F4, F6, one gets de Sitter for: (VF0 − VH) + (VQ − VF4) + 2(VR − VF6) > 0 (VR − VF2) + 2(VQ − VF4) + 3(VR − VF6) > 0

arXiv:0907.5580, arXiv:0911.2876 by B. de Carlos, A. Guarino, J. M. Moreno

SLIDE 58

David ANDRIOT Introduction De Sitter sol.

10D 4D geometric 4D non-geometric

Field redefinition Conclusion

4D non-geometric terms

Long history, motivation for these terms... Here, pratical point of view: allowed in 4D (super)potential. Scalar potential: VNSNS ∼ ρ−3VH + ρ−1VR + ρVQ + ρ3VR

arXiv:0711.2512 by M. P. Hertzberg, S. Kachru, W. Taylor, M. Tegmark

With Q, R, and F4, F6, one gets de Sitter for: (VF0 − VH) > 0 (VR − VF2) > 0

arXiv:0907.5580, arXiv:0911.2876 by B. de Carlos, A. Guarino, J. M. Moreno

SLIDE 59

David ANDRIOT Introduction De Sitter sol.

10D 4D geometric 4D non-geometric

Field redefinition Conclusion

4D non-geometric terms

Long history, motivation for these terms... Here, pratical point of view: allowed in 4D (super)potential. Scalar potential: VNSNS ∼ ρ−3VH + ρ−1VR + ρVQ + ρ3VR

arXiv:0711.2512 by M. P. Hertzberg, S. Kachru, W. Taylor, M. Tegmark

With Q, R, and F4, F6, one gets de Sitter for: (VF0 − VH) + (VQ − VF4) + 2(VR − VF6) > 0 (VR − VF2) + 2(VQ − VF4) + 3(VR − VF6) > 0

arXiv:0907.5580, arXiv:0911.2876 by B. de Carlos, A. Guarino, J. M. Moreno

SLIDE 60

David ANDRIOT Introduction De Sitter sol.

10D 4D geometric 4D non-geometric

Field redefinition Conclusion

4D non-geometric terms

Long history, motivation for these terms... Here, pratical point of view: allowed in 4D (super)potential. Scalar potential: VNSNS ∼ ρ−3VH + ρ−1VR + ρVQ + ρ3VR

arXiv:0711.2512 by M. P. Hertzberg, S. Kachru, W. Taylor, M. Tegmark

With Q, R, and F4, F6, one gets de Sitter for: (VF0 − VH) + (VQ − VF4) + 2(VR − VF6) > 0 (VR − VF2) + 2(VQ − VF4) + 3(VR − VF6) > 0

arXiv:0907.5580, arXiv:0911.2876 by B. de Carlos, A. Guarino, J. M. Moreno

SLIDE 61

David ANDRIOT Introduction De Sitter sol.

10D 4D geometric 4D non-geometric

Field redefinition Conclusion

4D non-geometric terms

Long history, motivation for these terms... Here, pratical point of view: allowed in 4D (super)potential. Scalar potential: VNSNS ∼ ρ−3VH + ρ−1VR + ρVQ + ρ3VR

arXiv:0711.2512 by M. P. Hertzberg, S. Kachru, W. Taylor, M. Tegmark

With Q, R, and F4, F6, one gets de Sitter for: (VF0 − VH) + (VQ − VF4) + 2(VR − VF6) > 0 (VR − VF2) + 2(VQ − VF4) + 3(VR − VF6) > 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...

SLIDE 62

David ANDRIOT Introduction De Sitter sol.

10D 4D geometric 4D non-geometric

Field redefinition Conclusion

4D non-geometric terms

Long history, motivation for these terms... Here, pratical point of view: allowed in 4D (super)potential. Scalar potential: VNSNS ∼ ρ−3VH + ρ−1VR + ρVQ + ρ3VR

arXiv:0711.2512 by M. P. Hertzberg, S. Kachru, W. Taylor, M. Tegmark

With Q, R, and F4, F6, one gets de Sitter for: (VF0 − VH) + (VQ − VF4) + 2(VR − VF6) > 0 (VR − VF2) + 2(VQ − VF4) + 3(VR − VF6) > 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?

SLIDE 63

David ANDRIOT Introduction De Sitter sol. Field redefinition

Presentation Lagrangians Back to 4D

Conclusion

Field redefinition

Presentation

Key object: ˜ β: antisymmetric bivector ˜ βmn.

SLIDE 64

David ANDRIOT Introduction De Sitter sol. Field redefinition

Presentation Lagrangians Back to 4D

Conclusion

Field redefinition

Presentation

Key object: ˜ β: antisymmetric bivector ˜ βmn. Motivations from Generalized Complex Geometry

math.DG/0209099 by N. Hitchin, math.DG/0401221 by M. Gualtieri

SLIDE 65

David ANDRIOT Introduction De Sitter sol. Field redefinition

Presentation Lagrangians Back to 4D

Conclusion

Field redefinition

Presentation

Key object: ˜ β: antisymmetric bivector ˜ βmn. Motivations from Generalized Complex Geometry

math.DG/0209099 by N. Hitchin, math.DG/0401221 by M. Gualtieri

Arguments in GCG: ˜ β related to non-geometry / to Qc

ab, Rabc

hep-th/0609084, arXiv:0708.2392 by P. Grange, S. Schäfer-Nameki arXiv:0807.4527 by M. Graña, R. Minasian, M. Petrini, D. Waldram

SLIDE 66

David ANDRIOT Introduction De Sitter sol. Field redefinition

Presentation Lagrangians Back to 4D

Conclusion

Field redefinition

Presentation

Key object: ˜ β: antisymmetric bivector ˜ βmn. Motivations from Generalized Complex Geometry

math.DG/0209099 by N. Hitchin, math.DG/0401221 by M. Gualtieri

Arguments in GCG: ˜ β related to non-geometry / to Qc

ab, Rabc

hep-th/0609084, arXiv:0708.2392 by P. Grange, S. Schäfer-Nameki arXiv:0807.4527 by M. Graña, R. Minasian, M. Petrini, D. Waldram

˜ β appears via a reparametrization of the gen. metric H: H =

ˆ

g − ˆ Bˆ g−1 ˆ B ˆ Bˆ g−1 −ˆ g−1 ˆ B ˆ g−1

=
˜

g ˜ g ˜ β −˜ β˜ g ˜ g−1 − ˜ β˜ g ˜ β

, ˜

g : new metric

SLIDE 67

David ANDRIOT Introduction De Sitter sol. Field redefinition

Presentation Lagrangians Back to 4D

Conclusion

Field redefinition

Presentation

Key object: ˜ β: antisymmetric bivector ˜ βmn. Motivations from Generalized Complex Geometry

math.DG/0209099 by N. Hitchin, math.DG/0401221 by M. Gualtieri

Arguments in GCG: ˜ β related to non-geometry / to Qc

ab, Rabc

hep-th/0609084, arXiv:0708.2392 by P. Grange, S. Schäfer-Nameki arXiv:0807.4527 by M. Graña, R. Minasian, M. Petrini, D. Waldram

˜ β appears via a reparametrization of the gen. metric H: H =

ˆ

g − ˆ Bˆ g−1 ˆ B ˆ Bˆ g−1 −ˆ g−1 ˆ B ˆ g−1

=
˜

g ˜ g ˜ β −˜ β˜ g ˜ g−1 − ˜ β˜ g ˜ β

, ˜

g : new metric

⇔ ˆ g = (˜ g−1 + ˜ β)−1˜ g−1(˜ g−1 − ˜ β)−1 ⇔ (ˆ g + ˆ B) = (˜ g−1 − ˜ β)−1 ˆ B = (˜ g−1 + ˜ β)−1 ˜ β(˜ g−1 − ˜ β)−1

SLIDE 68

David ANDRIOT Introduction De Sitter sol. Field redefinition

Presentation Lagrangians Back to 4D

Conclusion

Field redefinition

Presentation

Key object: ˜ β: antisymmetric bivector ˜ βmn. Motivations from Generalized Complex Geometry

math.DG/0209099 by N. Hitchin, math.DG/0401221 by M. Gualtieri

Arguments in GCG: ˜ β related to non-geometry / to Qc

ab, Rabc

hep-th/0609084, arXiv:0708.2392 by P. Grange, S. Schäfer-Nameki arXiv:0807.4527 by M. Graña, R. Minasian, M. Petrini, D. Waldram

˜ β appears via a reparametrization of the gen. metric H: H =

ˆ

g − ˆ Bˆ g−1 ˆ B ˆ Bˆ g−1 −ˆ g−1 ˆ B ˆ g−1

=
˜

g ˜ g ˜ β −˜ β˜ g ˜ g−1 − ˜ β˜ g ˜ β

, ˜

g : new metric

⇔ ˆ g = (˜ g−1 + ˜ β)−1˜ g−1(˜ g−1 − ˜ β)−1 ⇔ (ˆ g + ˆ B) = (˜ g−1 − ˜ β)−1 ˆ B = (˜ g−1 + ˜ β)−1 ˜ β(˜ g−1 − ˜ β)−1 e−2 ˜

φ

|˜ g| = e−2 ˆ

φ

|ˆ g|

SLIDE 69

David ANDRIOT Introduction De Sitter sol. Field redefinition

Presentation Lagrangians Back to 4D

Conclusion

Field redefinition

Presentation

Key object: ˜ β: antisymmetric bivector ˜ βmn. Motivations from Generalized Complex Geometry

math.DG/0209099 by N. Hitchin, math.DG/0401221 by M. Gualtieri

Arguments in GCG: ˜ β related to non-geometry / to Qc

ab, Rabc

hep-th/0609084, arXiv:0708.2392 by P. Grange, S. Schäfer-Nameki arXiv:0807.4527 by M. Graña, R. Minasian, M. Petrini, D. Waldram

˜ β appears via a reparametrization of the gen. metric H: H =

ˆ

g − ˆ Bˆ g−1 ˆ B ˆ Bˆ g−1 −ˆ g−1 ˆ B ˆ g−1

=
˜

g ˜ g ˜ β −˜ β˜ g ˜ g−1 − ˜ β˜ g ˜ β

, ˜

g : new metric

⇔ ˆ g = (˜ g−1 + ˜ β)−1˜ g−1(˜ g−1 − ˜ β)−1 ⇔ (ˆ g + ˆ B) = (˜ g−1 − ˜ β)−1 ˆ B = (˜ g−1 + ˜ β)−1 ˜ β(˜ g−1 − ˜ β)−1 e−2 ˜

φ

|˜ g| = e−2 ˆ

φ

|ˆ g|

Field redefinition: (ˆ g, ˆ B, ˆ φ) ↔ (˜ g, ˜ β, ˜ φ), ˜ β favored for non-geom.

SLIDE 70

David ANDRIOT Introduction De Sitter sol. Field redefinition

Presentation Lagrangians Back to 4D

Conclusion

Field redefinition

Presentation

Key object: ˜ β: antisymmetric bivector ˜ βmn. Motivations from Generalized Complex Geometry

math.DG/0209099 by N. Hitchin, math.DG/0401221 by M. Gualtieri

Arguments in GCG: ˜ β related to non-geometry / to Qc

ab, Rabc

hep-th/0609084, arXiv:0708.2392 by P. Grange, S. Schäfer-Nameki arXiv:0807.4527 by M. Graña, R. Minasian, M. Petrini, D. Waldram

˜ β appears via a reparametrization of the gen. metric H: H =

ˆ

g − ˆ Bˆ g−1 ˆ B ˆ Bˆ g−1 −ˆ g−1 ˆ B ˆ g−1

=
˜

g ˜ g ˜ β −˜ β˜ g ˜ g−1 − ˜ β˜ g ˜ β

, ˜

g : new metric

⇔ ˆ g = (˜ g−1 + ˜ β)−1˜ g−1(˜ g−1 − ˜ β)−1 ⇔ (ˆ g + ˆ B) = (˜ g−1 − ˜ β)−1 ˆ B = (˜ g−1 + ˜ β)−1 ˜ β(˜ g−1 − ˜ β)−1 e−2 ˜

φ

|˜ g| = e−2 ˆ

φ

|ˆ g|

Field redefinition: (ˆ g, ˆ B, ˆ φ) ↔ (˜ g, ˜ β, ˜ φ), ˜ β favored for non-geom. Apply it on NSNS Lagrangian? ˜ β could be related to non-geo. fluxes ⇒ would they appear?

SLIDE 71

David ANDRIOT Introduction De Sitter sol. Field redefinition

Presentation Lagrangians Back to 4D

Conclusion

Rewriting of the NSNS Lagrangian

ˆ L = e−2 ˆ

φ

|ˆ g|

R + 4|dˆ

φ|2 − 1 2| ˆ H|2

SLIDE 72

David ANDRIOT Introduction De Sitter sol. Field redefinition

Presentation Lagrangians Back to 4D

Conclusion

Rewriting of the NSNS Lagrangian

ˆ L = e−2 ˆ

φ

|ˆ g|

R + 4|dˆ

φ|2 − 1 2| ˆ H|2 = ?

(assumption: ˜

βkm∂m· = 0)

R =

R − ∂k˜ gsu∂m˜ gpq 2˜ gkm˜ guq˜ gps + 2˜ gpq˜ gks˜ gmu + 1 2 ˜ guq˜ gsm˜ gkp − ˜ gpq∂k ˜ βpk ∂m ˜ βqm − 1 2 ˜ gpq∂k ˜ βqm∂m ˜ βpk + 2˜ gkm˜ gpq∂k ∂m˜ gpq + 2˜ gkm(G−1)pq∂k ∂mGqp + ∂mGvl − 2˜ gmr˜ gks(G−1)lv∂k˜ grs − ˜ grs˜ gkm(G−1)lv∂k˜ grs + ˜ gms˜ gru(G−1)lu∂v˜ grs − ˜ gkm˜ grs(G−1)ls∂k˜ gvr + ∂mGvl (G−1)lq∂vGqm + 1 2 ˆ glq∂vGmq − ∂mGvl ∂k Gps 1 2 ˜ gkm 2(G−1)lv(G−1)sp + 5(G−1)sv(G−1)lp + ˆ gsl˜ gpv where G = ˜ g−1 + ˜ β .

SLIDE 73

David ANDRIOT Introduction De Sitter sol. Field redefinition

Presentation Lagrangians Back to 4D

Conclusion

Rewriting of the NSNS Lagrangian

ˆ L = e−2 ˆ

φ

|ˆ g|

R + 4|dˆ

φ|2 − 1 2| ˆ H|2 = e−2 ˜

φ

|˜ g|

R + 4|d˜

φ|2 − 1 2|Q|2 + ∂(. . . ) = ˜ L + ∂(. . . )

(assumption: ˜

βkm∂m· = 0)

R =

R − ∂k˜ gsu∂m˜ gpq 2˜ gkm˜ guq˜ gps + 2˜ gpq˜ gks˜ gmu + 1 2 ˜ guq˜ gsm˜ gkp − ˜ gpq∂k ˜ βpk ∂m ˜ βqm − 1 2 ˜ gpq∂k ˜ βqm∂m ˜ βpk + 2˜ gkm˜ gpq∂k ∂m˜ gpq + 2˜ gkm(G−1)pq∂k ∂mGqp + ∂mGvl − 2˜ gmr˜ gks(G−1)lv∂k˜ grs − ˜ grs˜ gkm(G−1)lv∂k˜ grs + ˜ gms˜ gru(G−1)lu∂v˜ grs − ˜ gkm˜ grs(G−1)ls∂k˜ gvr + ∂mGvl (G−1)lq∂vGqm + 1 2 ˆ glq∂vGmq − ∂mGvl ∂k Gps 1 2 ˜ gkm 2(G−1)lv(G−1)sp + 5(G−1)sv(G−1)lp + ˆ gsl˜ gpv where G = ˜ g−1 + ˜ β .

SLIDE 74

David ANDRIOT Introduction De Sitter sol. Field redefinition

Presentation Lagrangians Back to 4D

Conclusion

Rewriting of the NSNS Lagrangian

ˆ L = e−2 ˆ

φ

|ˆ g|

R + 4|dˆ

φ|2 − 1 2| ˆ H|2 = e−2 ˜

φ

|˜ g|

R + 4|d˜

φ|2 − 1 2|Q|2 + ∂(. . . ) = ˜ L + ∂(. . . )

where Qk

mn = ∂k ˜

βmn, |Q|2 =

1 2!Qk mnQp qr ˜

gkp˜ gmq˜ gnr

(assumption: ˜

βkm∂m· = 0)

SLIDE 75

David ANDRIOT Introduction De Sitter sol. Field redefinition

Presentation Lagrangians Back to 4D

Conclusion

Rewriting of the NSNS Lagrangian

ˆ L = e−2 ˆ

φ

|ˆ g|

R + 4|dˆ

φ|2 − 1 2| ˆ H|2 = e−2 ˜

φ

|˜ g|

R + 4|d˜

φ|2 − 1 2|Q|2 + · · · − 1 2|R|2 + ∂(. . . )

where Qk

mn = ∂k ˜

βmn, |Q|2 =

1 2!Qk mnQp qr ˜

gkp˜ gmq˜ gnr

(assumption: ˜

βkm∂m· = 0)

Without the assumption ⇒ also get Rmnp = 3 ˜

βk[m∂k ˜ βnp], |R|2 =

1 3!RkmnRpqr ˜

gkp˜ gmq˜ gnr

SLIDE 76

David ANDRIOT Introduction De Sitter sol. Field redefinition

Presentation Lagrangians Back to 4D

Conclusion

Rewriting of the NSNS Lagrangian

ˆ L = e−2 ˆ

φ

|ˆ g|

R + 4|dˆ

φ|2 − 1 2| ˆ H|2 = e−2 ˜

φ

|˜ g|

R + 4|d˜

φ|2 − 1 2|Q|2 + · · · − 1 2|R|2 + ∂(. . . )

where Qk

mn = ∂k ˜

βmn, |Q|2 =

1 2!Qk mnQp qr ˜

gkp˜ gmq˜ gnr

(assumption: ˜

βkm∂m· = 0)

Without the assumption ⇒ also get Rmnp = 3 ˜

βk[m∂k ˜ βnp], |R|2 =

1 3!RkmnRpqr ˜

gkp˜ gmq˜ gnr

Q-, R-fluxes appear in 10D NSNS via field redefinition Relation to 4D Q-, R-fluxes/non-geo. terms? ⇒ compactification

SLIDE 77

David ANDRIOT Introduction De Sitter sol. Field redefinition

Presentation Lagrangians Back to 4D

Conclusion

Relation to the 4D potential

We have shown ˆ L = ˜ L + ∂(. . . ).

SLIDE 78

David ANDRIOT Introduction De Sitter sol. Field redefinition

Presentation Lagrangians Back to 4D

Conclusion

Relation to the 4D potential

We have shown ˆ L = ˜ L + ∂(. . . ). ֒ → compactify ˜ L...

SLIDE 79

David ANDRIOT Introduction De Sitter sol. Field redefinition

Presentation Lagrangians Back to 4D

Conclusion

Relation to the 4D potential

We have shown ˆ L = ˜ L + ∂(. . . ). ֒ → compactify ˜ L... If ˜ g6ij → ˜ g(0)

6ij ρ,

1 2|Q|2 = 1 4Qk

mnQp qr ˜

gkp˜ gmq˜ gnr − → ρVQ , VQ = 1 2|Q(0)|2 , 1 2|R|2 = 1 12RkmnRpqr ˜ gkp˜ gmq˜ gnr − → ρ3VR , VR = 1 2|R(0)|2 .

SLIDE 80

David ANDRIOT Introduction De Sitter sol. Field redefinition

Presentation Lagrangians Back to 4D

Conclusion

Relation to the 4D potential

We have shown ˆ L = ˜ L + ∂(. . . ). ֒ → compactify ˜ L... If ˜ g6ij → ˜ g(0)

6ij ρ,

1 2|Q|2 = 1 4Qk

mnQp qr ˜

gkp˜ gmq˜ gnr − → ρVQ , VQ = 1 2|Q(0)|2 , 1 2|R|2 = 1 12RkmnRpqr ˜ gkp˜ gmq˜ gnr − → ρ3VR , VR = 1 2|R(0)|2 . ˜ L can give the potential in 4D (gives a 10D origin) Q-, R-fluxes in 10D ⇒ Q-, R-fluxes in 4D

SLIDE 81

David ANDRIOT Introduction De Sitter sol. Field redefinition

Presentation Lagrangians Back to 4D

Conclusion

Relation to the 4D potential

We have shown ˆ L = ˜ L + ∂(. . . ). ֒ → compactify ˜ L... If ˜ g6ij → ˜ g(0)

6ij ρ,

1 2|Q|2 = 1 4Qk

mnQp qr ˜

gkp˜ gmq˜ gnr − → ρVQ , VQ = 1 2|Q(0)|2 , 1 2|R|2 = 1 12RkmnRpqr ˜ gkp˜ gmq˜ gnr − → ρ3VR , VR = 1 2|R(0)|2 . ˜ 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 ∂(. . . ).

SLIDE 82

David ANDRIOT Introduction De Sitter sol. Field redefinition

Presentation Lagrangians Back to 4D

Conclusion

Relation to the 4D potential

We have shown ˆ L = ˜ L + ∂(. . . ). ֒ → compactify ˜ L... If ˜ g6ij → ˜ g(0)

6ij ρ,

1 2|Q|2 = 1 4Qk

mnQp qr ˜

gkp˜ gmq˜ gnr − → ρVQ , VQ = 1 2|Q(0)|2 , 1 2|R|2 = 1 12RkmnRpqr ˜ gkp˜ gmq˜ gnr − → ρ3VR , VR = 1 2|R(0)|2 . ˜ 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.

SLIDE 83

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

Conclusion

de Sitter sol. of 10D / 4D SUGRA are difficult to obtain. Unless additional ingredients ⇒ non-geometric terms 10D origin of these terms (of Q, R-fluxes?) GCG ⇒ Field redefinition (ˆ

g, ˆ B, ˆ φ) ↔ (˜ g, ˜ β, ˜ φ)

Rewriting NSNS Lag.: ˆ

L = ˜ L + ∂(. . . ),

10D Qk

mn = ∂k ˜

βmn (for ˜ βkm∂m· = 0), Rmnp = 3 ˜ βk[m∂k ˜ βnp]

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...

SLIDE 84

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

SLIDE 85

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

Non-geometry in 10D and 4D SUGRA

Non-geometry in 10D

Original idea of non-geometry

hep-th/0208174 by S. Hellerman, J. McGreevy, B. Williams hep-th/0210209 by A. Dabholkar, C. Hull hep-th/0404217 by A. Flournoy, B. Wecht, B. Williams

SLIDE 86

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

Non-geometry in 10D and 4D SUGRA

Non-geometry in 10D

Original idea of non-geometry

hep-th/0208174 by S. Hellerman, J. McGreevy, B. Williams hep-th/0210209 by A. Dabholkar, C. Hull 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)

SLIDE 87

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

Non-geometry in 10D and 4D SUGRA

Non-geometry in 10D

Original idea of non-geometry

hep-th/0208174 by S. Hellerman, J. McGreevy, B. Williams hep-th/0210209 by A. Dabholkar, C. Hull 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

SLIDE 88

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

Non-geometry in 10D and 4D SUGRA

Non-geometry in 10D

Original idea of non-geometry

hep-th/0208174 by S. Hellerman, J. McGreevy, B. Williams hep-th/0210209 by A. Dabholkar, C. Hull 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

SLIDE 89

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

Non-geometry in 10D and 4D SUGRA

Non-geometry in 10D

Original idea of non-geometry

hep-th/0208174 by S. Hellerman, J. McGreevy, B. Williams hep-th/0210209 by A. Dabholkar, C. Hull 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

SLIDE 90

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

Non-geometry in 10D and 4D SUGRA

Non-geometry in 10D

Original idea of non-geometry

hep-th/0208174 by S. Hellerman, J. McGreevy, B. Williams hep-th/0210209 by A. Dabholkar, C. Hull 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

0 ≡ 2πRy R 1/R

G. Moutsopoulos PhD 2008

Simple example: T-duality: circle of radius R → 1

R

SLIDE 91

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

Non-geometry in 10D and 4D SUGRA

Non-geometry in 10D

Original idea of non-geometry

hep-th/0208174 by S. Hellerman, J. McGreevy, B. Williams hep-th/0210209 by A. Dabholkar, C. Hull 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

0 ≡ 2πRy R 1/R

G. Moutsopoulos PhD 2008

Simple example: T-duality: circle of radius R → 1

R

Famous toroidal example

torus T 3 + ˆ Habc

Ta

− − → twisted torus (f a

bc) Tb

− → non − geometric config.

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

SLIDE 92

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

In 4D: non-geometric terms in the potential

Terms introduced in the superpot., for T-duality covariance.

SLIDE 93

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

In 4D: non-geometric terms in the potential

Terms introduced in the superpot., for T-duality covariance. T-duality: symmetry of string theory on a background with isometries.

SLIDE 94

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

In 4D: non-geometric terms in the potential

Terms introduced in the superpot., for T-duality covariance. T-duality: symmetry of string theory on a background with isometries. Compactification on such a background ⇒ 4D effective theory inherits this symmetry.

SLIDE 95

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

In 4D: non-geometric terms in the potential

Terms introduced in the superpot., for T-duality covariance. T-duality: symmetry of string theory on a background with isometries. Compactification on such a background ⇒ 4D effective theory inherits this symmetry. ֒ → require T-duality covariance on the 4D superpotential.

SLIDE 96

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

In 4D: non-geometric terms in the potential

Terms introduced in the superpot., for T-duality covariance. T-duality: symmetry of string theory on a background with isometries. Compactification on such a background ⇒ 4D effective theory inherits this symmetry. ֒ → require T-duality covariance on the 4D superpotential. ˆ Habc on M generates a term in the 4D potential. ֒ → if T-duality along a, b, c, how does the term transform?

SLIDE 97

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

In 4D: non-geometric terms in the potential

Terms introduced in the superpot., for T-duality covariance. T-duality: symmetry of string theory on a background with isometries. Compactification on such a background ⇒ 4D effective theory inherits this symmetry. ֒ → require T-duality covariance on the 4D superpotential. ˆ Habc on M generates a term in the 4D potential. ֒ → if T-duality along a, b, c, how does the term transform? Term generated by ˆ Habc

Ta

− − → term gen. by f a

bc (curvature)

SLIDE 98

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

In 4D: non-geometric terms in the potential

Terms introduced in the superpot., for T-duality covariance. T-duality: symmetry of string theory on a background with isometries. Compactification on such a background ⇒ 4D effective theory inherits this symmetry. ֒ → require T-duality covariance on the 4D superpotential. ˆ Habc on M generates a term in the 4D potential. ֒ → if T-duality along a, b, c, how does the term transform? Term generated by ˆ Habc

Ta

− − → term gen. by f a

bc (curvature) Tb,Tc

− − − − → new terms needed ! Generated by Qc

ab, Rabc

SLIDE 99

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

In 4D: non-geometric terms in the potential

Terms introduced in the superpot., for T-duality covariance. T-duality: symmetry of string theory on a background with isometries. Compactification on such a background ⇒ 4D effective theory inherits this symmetry. ֒ → require T-duality covariance on the 4D superpotential. ˆ Habc on M generates a term in the 4D potential. ֒ → if T-duality along a, b, c, how does the term transform? Term generated by ˆ Habc

Ta

− − → term gen. by f a

bc (curvature) Tb,Tc

− − − − → new terms needed ! Generated by Qc

ab, Rabc

4D T-duality chain: ˆ Habc

Ta

− − → f a

bc Tb

− → Qc

ab Tc

− − → Rabc

hep-th/0508133, hep-th/0607015 by J. Shelton, W. Taylor, B. Wecht

SLIDE 100

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

In 4D: non-geometric terms in the potential

Terms introduced in the superpot., for T-duality covariance. T-duality: symmetry of string theory on a background with isometries. Compactification on such a background ⇒ 4D effective theory inherits this symmetry. ֒ → require T-duality covariance on the 4D superpotential. ˆ Habc on M generates a term in the 4D potential. ֒ → if T-duality along a, b, c, how does the term transform? Term generated by ˆ Habc

Ta

− − → term gen. by f a

bc (curvature) Tb,Tc

− − − − → new terms needed ! Generated by Qc

ab, Rabc

4D T-duality chain: ˆ Habc

Ta

− − → f a

bc Tb

− → Qc

ab Tc

− − → Rabc

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.

SLIDE 101

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

In 4D: non-geometric terms in the potential

Terms introduced in the superpot., for T-duality covariance. T-duality: symmetry of string theory on a background with isometries. Compactification on such a background ⇒ 4D effective theory inherits this symmetry. ֒ → require T-duality covariance on the 4D superpotential. ˆ Habc on M generates a term in the 4D potential. ֒ → if T-duality along a, b, c, how does the term transform? Term generated by ˆ Habc

Ta

− − → term gen. by f a

bc (curvature) Tb,Tc

− − − − → new terms needed ! Generated by Qc

ab, Rabc

4D T-duality chain: ˆ Habc

Ta

− − → f a

bc Tb

− → Qc

ab Tc

− − → Rabc

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. T-d. cov. of gauge algebra of gauged SUGRA ⇒ Qc

ab, Rabc appear as (new) structure constants

hep-th/0210209, hep-th/0512005 by A. Dabholkar, C. Hull

SLIDE 102

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

Non-geometry in various dimensions

Ten dimensions: the toroidal example

SLIDE 103

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

Non-geometry in various dimensions

Ten dimensions: the toroidal example

Toroidal example: geometric config.

T−dualities

− − − − − − − → non-geometric

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

SLIDE 104

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

Non-geometry in various dimensions

Ten dimensions: the toroidal example

Toroidal example: geometric config.

T−dualities

− − − − − − − → non-geometric

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

NSNS sector: ˆ gmn, ˆ Bmn, ˆ φ

SLIDE 105

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

Non-geometry in various dimensions

Ten dimensions: the toroidal example

Toroidal example: geometric config.

T−dualities

− − − − − − − → non-geometric

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

NSNS sector: ˆ gmn, ˆ Bmn, ˆ φ Geometry Torus T 3(x, y, z) Fields (x)

ˆ gA, ˆ BA

Gluing

ˆ BA:

along S1

x

gauge transfo. Flux

ˆ HA = dˆ BA

quantized

Habc

SLIDE 106

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

Non-geometry in various dimensions

Ten dimensions: the toroidal example

Toroidal example: geometric config.

T−dualities

− − − − − − − → non-geometric

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

NSNS sector: ˆ gmn, ˆ Bmn, ˆ φ T-d frame Frame A Geometry Torus T 3(x, y, z) Fields (x)

ˆ gA, ˆ BA

Gluing

ˆ BA:

along S1

x

gauge transfo. Flux

ˆ HA = dˆ BA

quantized

Habc

SLIDE 107

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

Non-geometry in various dimensions

Ten dimensions: the toroidal example

Toroidal example: geometric config.

T−dualities

− − − − − − − → non-geometric

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

NSNS sector: ˆ gmn, ˆ Bmn, ˆ φ T-d frame Frame A

Ty

− − → Frame B Geometry Torus T 3(x, y, z) Fields (x)

ˆ gA, ˆ BA

Gluing

ˆ BA:

along S1

x

gauge transfo. Flux

ˆ HA = dˆ BA

quantized

Habc

SLIDE 108

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

Non-geometry in various dimensions

Ten dimensions: the toroidal example

Toroidal example: geometric config.

T−dualities

− − − − − − − → non-geometric

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

NSNS sector: ˆ gmn, ˆ Bmn, ˆ φ T-d frame Frame A

Ty

− − → Frame B Geometry Torus Twisted torus T 3(x, y, z) S1

y ֒

→ M → T 2

xz

Fields (x)

ˆ gA, ˆ BA ˆ gB, ˆ BB = 0

Gluing

ˆ BA:

connection in ˆ

gB:

along S1

x

gauge transfo. gauge transfo. Flux

ˆ HA = dˆ BA dea = − 1

2f a bc eb ∧ ec

quantized

Habc

geom. flux: f a

bc

SLIDE 109

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

Non-geometry in various dimensions

Ten dimensions: the toroidal example

Toroidal example: geometric config.

T−dualities

− − − − − − − → non-geometric

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

NSNS sector: ˆ gmn, ˆ Bmn, ˆ φ T-d frame Frame A

Ty

− − → Frame B

Tz

− → Frame C Geometry Torus Twisted torus T 3(x, y, z) S1

y ֒

→ M → T 2

xz

Fields (x)

ˆ gA, ˆ BA ˆ gB, ˆ BB = 0

Gluing

ˆ BA:

connection in ˆ

gB:

along S1

x

gauge transfo. gauge transfo. Flux

ˆ HA = dˆ BA dea = − 1

2f a bc eb ∧ ec

quantized

Habc

geom. flux: f a

bc

SLIDE 110

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

Non-geometry in various dimensions

Ten dimensions: the toroidal example

Toroidal example: geometric config.

T−dualities

− − − − − − − → non-geometric

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

NSNS sector: ˆ gmn, ˆ Bmn, ˆ φ T-d frame Frame A

Ty

− − → Frame B

Tz

− → Frame C Geometry Torus Twisted torus T 3(x, y, z) S1

y ֒

→ M → T 2

xz

Fields (x)

ˆ gA, ˆ BA ˆ gB, ˆ BB = 0 ˆ gC, ˆ BC, ill-def.

Gluing

ˆ BA:

connection in ˆ

gB:

need T-d. along S1

x

gauge transfo. gauge transfo. Non-geom. ! Flux

ˆ HA = dˆ BA dea = − 1

2f a bc eb ∧ ec

quantized

Habc

geom. flux: f a

bc

SLIDE 111

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

Non-geometry in various dimensions

Ten dimensions: the toroidal example

Toroidal example: geometric config.

T−dualities

− − − − − − − → non-geometric

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

NSNS sector: ˆ gmn, ˆ Bmn, ˆ φ T-d frame Frame A

Ty

− − → Frame B

Tz

− → Frame C Geometry Torus Twisted torus Locally: T 3(x, y, z) S1

y ֒

→ M → T 2

xz

S1

x , S1 y, S1 z

Fields (x)

ˆ gA, ˆ BA ˆ gB, ˆ BB = 0 ˆ gC, ˆ BC, ill-def.

Gluing

ˆ BA:

connection in ˆ

gB:

need T-d. along S1

x

gauge transfo. gauge transfo. Non-geom. ! Flux

ˆ HA = dˆ BA dea = − 1

2f a bc eb ∧ ec

? quantized

Habc

geom. flux: f a

bc

SLIDE 112

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

Heterotic string effective description at α′ and non-geometry

arXiv:1102.1434 by D. A. S = 1 2κ2

d10x

|g|e−2φ R + 4|dφ|2 − 1 2 |H|2 + α′ 4 (tr(R2 +) − tr(F2)) + O(α′ 2)

SLIDE 113

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

Heterotic string effective description at α′ and non-geometry

arXiv:1102.1434 by D. A. S = 1 2κ2

d10x

|g|e−2φ R + 4|dφ|2 − 1 2 |H|2 + α′ 4 (tr(R2 +) − tr(F2)) + O(α′ 2)

=

1 2κ2 26

d26x

|g′|e−2φ′ R′ + 4|dφ′|2 − 1 2 |H′|2 + α′ 4 tr(R′2 +)

SLIDE 114

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

Heterotic string effective description at α′ and non-geometry

arXiv:1102.1434 by D. A. S = 1 2κ2

d10x

|g|e−2φ R + 4|dφ|2 − 1 2 |H|2 + α′ 4 (tr(R2 +) − tr(F2)) + O(α′ 2)

=

1 2κ2 26

d26x

|g′|e−2φ′ R′ + 4|dφ′|2 − 1 2 |H′|2 + α′ 4 tr(R′2 +)

for abelian heterotic string, Aa =

√ α′Aa , gab = 1

4tr(tatb),

g′ ∼

gmn + gabAa

mAb n

gbaAa

m

gabAb

n

gab

, B′ ∼
Bmn

gbaAa

m

−gabAb

n

Bab

SLIDE 115

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

Heterotic string effective description at α′ and non-geometry

arXiv:1102.1434 by D. A. S = 1 2κ2

d10x

|g|e−2φ R + 4|dφ|2 − 1 2 |H|2 + α′ 4 (tr(R2 +) − tr(F2)) + O(α′ 2)

=

1 2κ2 26

d26x

|g′|e−2φ′ R′ + 4|dφ′|2 − 1 2 |H′|2 + α′ 4 tr(R′2 +)

for abelian heterotic string, Aa =

√ α′Aa , gab = 1

4tr(tatb),

g′ ∼

gmn + gabAa

mAb n

gbaAa

m

gabAb

n

gab

, B′ ∼
Bmn

gbaAa

m

−gabAb

n

Bab

Equivalence of eq. of motion, Bianchi identity, SUSY cond.

SLIDE 116

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

Heterotic string effective description at α′ and non-geometry

arXiv:1102.1434 by D. A. S = 1 2κ2

d10x

|g|e−2φ R + 4|dφ|2 − 1 2 |H|2 + α′ 4 (tr(R2 +) − tr(F2)) + O(α′ 2)

=

1 2κ2 26

d26x

|g′|e−2φ′ R′ + 4|dφ′|2 − 1 2 |H′|2 + α′ 4 tr(R′2 +)

for abelian heterotic string, Aa =

√ α′Aa , gab = 1

4tr(tatb),

g′ ∼

gmn + gabAa

mAb n

gbaAa

m

gabAb

n

gab

, B′ ∼
Bmn

gbaAa

m

−gabAb

n

Bab

Equivalence of eq. of motion, Bianchi identity, SUSY cond.

Embedding in bosonic string at order α′ from target space

SLIDE 117

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

Heterotic string effective description at α′ and non-geometry

arXiv:1102.1434 by D. A. S = 1 2κ2

d10x

|g|e−2φ R + 4|dφ|2 − 1 2 |H|2 + α′ 4 (tr(R2 +) − tr(F2)) + O(α′ 2)

=

1 2κ2 26

d26x

|g′|e−2φ′ R′ + 4|dφ′|2 − 1 2 |H′|2 + α′ 4 tr(R′2 +)

for abelian heterotic string, Aa =

√ α′Aa , gab = 1

4tr(tatb),

g′ ∼

gmn + gabAa

mAb n

gbaAa

m

gabAb

n

gab

, B′ ∼
Bmn

gbaAa

m

−gabAb

n

Bab

Equivalence of eq. of motion, Bianchi identity, SUSY cond.

Embedding in bosonic string at order α′ from target space ֒ → (abelian) heterotic Double Field Theory

SLIDE 118

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

Heterotic string effective description at α′ and non-geometry

arXiv:1102.1434 by D. A. S = 1 2κ2

d10x

|g|e−2φ R + 4|dφ|2 − 1 2 |H|2 + α′ 4 (tr(R2 +) − tr(F2)) + O(α′ 2)

=

1 2κ2 26

d26x

|g′|e−2φ′ R′ + 4|dφ′|2 − 1 2 |H′|2 + α′ 4 tr(R′2 +)

for abelian heterotic string, Aa =

√ α′Aa , gab = 1

4tr(tatb),

g′ ∼

gmn + gabAa

mAb n

gbaAa

m

gabAb

n

gab

, B′ ∼
Bmn

gbaAa

m

−gabAb

n

Bab

Equivalence of eq. of motion, Bianchi identity, SUSY cond.

Embedding in bosonic string at order α′ from target space ֒ → (abelian) heterotic Double Field Theory Here:

e−2φ′ |g′| R′ + 4|dφ′|2 − 1 2 |H′|2 = e−2 ˜ φ′ |˜ g′| R′ + 4|d ˜ φ′|2 − 1 2 |Q′|2 + ∂(. . . )

SLIDE 119

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

Heterotic string effective description at α′ and non-geometry

arXiv:1102.1434 by D. A. S = 1 2κ2

d10x

|g|e−2φ R + 4|dφ|2 − 1 2 |H|2 + α′ 4 (tr(R2 +) − tr(F2)) + O(α′ 2)

=

1 2κ2 26

d26x

|g′|e−2φ′ R′ + 4|dφ′|2 − 1 2 |H′|2 + α′ 4 tr(R′2 +)

for abelian heterotic string, Aa =

√ α′Aa , gab = 1

4tr(tatb),

g′ ∼

gmn + gabAa

mAb n

gbaAa

m

gabAb

n

gab

, B′ ∼
Bmn

gbaAa

m

−gabAb

n

Bab

Equivalence of eq. of motion, Bianchi identity, SUSY cond.

Embedding in bosonic string at order α′ from target space ֒ → (abelian) heterotic Double Field Theory Here:

e−2φ′ |g′| R′ + 4|dφ′|2 − 1 2 |H′|2 = e−2 ˜ φ′ |˜ g′| R′ + 4|d ˜ φ′|2 − 1 2 |Q′|2 + ∂(. . . )

֒ → higher (26) dimensional ˜ g′, ˜ β′ for heterotic which contain Aa Characterize heterotic non-geometry ...

SLIDE 120

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

Global aspects and 4D potential

Toroidal non-geometric example (assumption automatically )

SLIDE 121

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

Global aspects and 4D potential

Toroidal non-geometric example (assumption automatically ) ˆ g, ˆ φ, ˆ H ×

SLIDE 122

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

Global aspects and 4D potential

Toroidal non-geometric example (assumption automatically ) ˆ g, ˆ φ, ˆ H × ˆ L ⇒ ?

SLIDE 123

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

Global aspects and 4D potential

Toroidal non-geometric example (assumption automatically ) ˆ g, ˆ φ, ˆ H ×

Field redef.

− − − − − − − → ˜ g, ˜ φ, Q ˆ L ⇒ ? (˜ g redefines the geometry: flat torus)

SLIDE 124

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

Global aspects and 4D potential

Toroidal non-geometric example (assumption automatically ) ˆ g, ˆ φ, ˆ H ×

Field redef.

− − − − − − − → ˜ g, ˜ φ, Q ˆ L ⇒ ? ˜ L ⇒ ˜ S (˜ g redefines the geometry: flat torus)

SLIDE 125

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

Global aspects and 4D potential

Toroidal non-geometric example (assumption automatically ) ˆ g, ˆ φ, ˆ H ×

Field redef.

− − − − − − − → ˜ g, ˜ φ, Q ˆ L ⇒ ? ˜ L ⇒ ˜ S (˜ g redefines the geometry: flat torus) General result (always valid): ˆ L = ˜ L + ∂(. . . )

SLIDE 126

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

Global aspects and 4D potential

Toroidal non-geometric example (assumption automatically ) ˆ g, ˆ φ, ˆ H ×

Field redef.

− − − − − − − → ˜ g, ˜ φ, Q ˆ L ⇒ ? ˜ L ⇒ ˜ S (˜ g redefines the geometry: flat torus) General result (always valid): ˆ L = ˜ L + ∂(. . . )

SLIDE 127

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

Global aspects and 4D potential

Toroidal non-geometric example (assumption automatically ) ˆ g, ˆ φ, ˆ H ×

Field redef.

− − − − − − − → ˜ g, ˜ φ, Q ˆ L ⇒ ? ˜ L ⇒ ˜ S (˜ g redefines the geometry: flat torus) General result (always valid): ˆ L = ˜ L + ∂(. . . )

∂(. . . ) = 0 ⇒ Propose: discard total derivative

Prescription: use ˜ L as the good low-energy effective description for non-geometry

SLIDE 128

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

Global aspects and 4D potential

Toroidal non-geometric example (assumption automatically ) ˆ g, ˆ φ, ˆ H ×

Field redef.

− − − − − − − → ˜ g, ˜ φ, Q ˆ L ⇒ ? ˜ L ⇒ ˜ S (˜ g redefines the geometry: flat torus) General result (always valid): ˆ L = ˜ L + ∂(. . . )

∂(. . . ) = 0 ⇒ Propose: discard total derivative

Prescription: use ˜ L as the good low-energy effective description for non-geometry If we do so, 1 2|Q|2 = 1 4Qk

mnQp qr˜

gkp˜ gmq˜ gnr − → ρVQ , VQ = 1 2|Q(0)|2 .

SLIDE 129

David ANDRIOT Introduction De Sitter sol. Field redefinition Conclusion

Global aspects and 4D potential

Toroidal non-geometric example (assumption automatically ) ˆ g, ˆ φ, ˆ H ×

Field redef.

− − − − − − − → ˜ g, ˜ φ, Q ˆ L ⇒ ? ˜ L ⇒ ˜ S (˜ g redefines the geometry: flat torus) General result (always valid): ˆ L = ˜ L + ∂(. . . )

∂(. . . ) = 0 ⇒ Propose: discard total derivative

Prescription: use ˜ L as the good low-energy effective description for non-geometry If we do so, 1 2|Q|2 = 1 4Qk

mnQp qr˜